Properties

 Label 9702.2.a.u.1.1 Level $9702$ Weight $2$ Character 9702.1 Self dual yes Analytic conductor $77.471$ 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] = [9702,2,Mod(1,9702)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9702, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9702.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9702.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: $$77.4708600410$$ 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 $$\chi$$ $$=$$ 9702.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{8} -2.00000 q^{10} -1.00000 q^{11} +2.00000 q^{13} +1.00000 q^{16} +6.00000 q^{17} +4.00000 q^{19} +2.00000 q^{20} +1.00000 q^{22} -6.00000 q^{23} -1.00000 q^{25} -2.00000 q^{26} -6.00000 q^{29} -2.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} +2.00000 q^{37} -4.00000 q^{38} -2.00000 q^{40} +10.0000 q^{41} -6.00000 q^{43} -1.00000 q^{44} +6.00000 q^{46} +1.00000 q^{50} +2.00000 q^{52} +2.00000 q^{53} -2.00000 q^{55} +6.00000 q^{58} -4.00000 q^{59} +14.0000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} +12.0000 q^{67} +6.00000 q^{68} +2.00000 q^{71} +6.00000 q^{73} -2.00000 q^{74} +4.00000 q^{76} +2.00000 q^{80} -10.0000 q^{82} +6.00000 q^{83} +12.0000 q^{85} +6.00000 q^{86} +1.00000 q^{88} +6.00000 q^{89} -6.00000 q^{92} +8.00000 q^{95} +O(q^{100})$$

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
$$4$$ 1.00000 0.500000
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ −2.00000 −0.632456
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 2.00000 0.447214
$$21$$ 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
$$25$$ −1.00000 −0.200000
$$26$$ −2.00000 −0.392232
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 0 0
$$40$$ −2.00000 −0.316228
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ −6.00000 −0.914991 −0.457496 0.889212i $$-0.651253\pi$$
−0.457496 + 0.889212i $$0.651253\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ 6.00000 0.884652
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ 2.00000 0.277350
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 0 0
$$55$$ −2.00000 −0.269680
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 6.00000 0.787839
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ 2.00000 0.254000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 4.00000 0.496139
$$66$$ 0 0
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 2.00000 0.237356 0.118678 0.992933i $$-0.462134\pi$$
0.118678 + 0.992933i $$0.462134\pi$$
$$72$$ 0 0
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 2.00000 0.223607
$$81$$ 0 0
$$82$$ −10.0000 −1.10432
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 0 0
$$85$$ 12.0000 1.30158
$$86$$ 6.00000 0.646997
$$87$$ 0 0
$$88$$ 1.00000 0.106600
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −6.00000 −0.625543
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 8.00000 0.820783
$$96$$ 0 0
$$97$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −1.00000 −0.100000
$$101$$ −12.0000 −1.19404 −0.597022 0.802225i $$-0.703650\pi$$
−0.597022 + 0.802225i $$0.703650\pi$$
$$102$$ 0 0
$$103$$ −6.00000 −0.591198 −0.295599 0.955312i $$-0.595519\pi$$
−0.295599 + 0.955312i $$0.595519\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ 8.00000 0.766261 0.383131 0.923694i $$-0.374846\pi$$
0.383131 + 0.923694i $$0.374846\pi$$
$$110$$ 2.00000 0.190693
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 4.00000 0.376288 0.188144 0.982141i $$-0.439753\pi$$
0.188144 + 0.982141i $$0.439753\pi$$
$$114$$ 0 0
$$115$$ −12.0000 −1.11901
$$116$$ −6.00000 −0.557086
$$117$$ 0 0
$$118$$ 4.00000 0.368230
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −14.0000 −1.26750
$$123$$ 0 0
$$124$$ −2.00000 −0.179605
$$125$$ −12.0000 −1.07331
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ −4.00000 −0.350823
$$131$$ 14.0000 1.22319 0.611593 0.791173i $$-0.290529\pi$$
0.611593 + 0.791173i $$0.290529\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ 8.00000 0.683486 0.341743 0.939793i $$-0.388983\pi$$
0.341743 + 0.939793i $$0.388983\pi$$
$$138$$ 0 0
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −2.00000 −0.167836
$$143$$ −2.00000 −0.167248
$$144$$ 0 0
$$145$$ −12.0000 −0.996546
$$146$$ −6.00000 −0.496564
$$147$$ 0 0
$$148$$ 2.00000 0.164399
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ 20.0000 1.62758 0.813788 0.581161i $$-0.197401\pi$$
0.813788 + 0.581161i $$0.197401\pi$$
$$152$$ −4.00000 −0.324443
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −4.00000 −0.321288
$$156$$ 0 0
$$157$$ 22.0000 1.75579 0.877896 0.478852i $$-0.158947\pi$$
0.877896 + 0.478852i $$0.158947\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ −2.00000 −0.158114
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 24.0000 1.87983 0.939913 0.341415i $$-0.110906\pi$$
0.939913 + 0.341415i $$0.110906\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 0 0
$$166$$ −6.00000 −0.465690
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ −12.0000 −0.920358
$$171$$ 0 0
$$172$$ −6.00000 −0.457496
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ 0 0
$$178$$ −6.00000 −0.449719
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 18.0000 1.33793 0.668965 0.743294i $$-0.266738\pi$$
0.668965 + 0.743294i $$0.266738\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 6.00000 0.442326
$$185$$ 4.00000 0.294086
$$186$$ 0 0
$$187$$ −6.00000 −0.438763
$$188$$ 0 0
$$189$$ 0 0
$$190$$ −8.00000 −0.580381
$$191$$ −6.00000 −0.434145 −0.217072 0.976156i $$-0.569651\pi$$
−0.217072 + 0.976156i $$0.569651\pi$$
$$192$$ 0 0
$$193$$ 22.0000 1.58359 0.791797 0.610784i $$-0.209146\pi$$
0.791797 + 0.610784i $$0.209146\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ −2.00000 −0.141776 −0.0708881 0.997484i $$-0.522583\pi$$
−0.0708881 + 0.997484i $$0.522583\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 0 0
$$202$$ 12.0000 0.844317
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 20.0000 1.39686
$$206$$ 6.00000 0.418040
$$207$$ 0 0
$$208$$ 2.00000 0.138675
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ −10.0000 −0.688428 −0.344214 0.938891i $$-0.611855\pi$$
−0.344214 + 0.938891i $$0.611855\pi$$
$$212$$ 2.00000 0.137361
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ −12.0000 −0.818393
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −8.00000 −0.541828
$$219$$ 0 0
$$220$$ −2.00000 −0.134840
$$221$$ 12.0000 0.807207
$$222$$ 0 0
$$223$$ 2.00000 0.133930 0.0669650 0.997755i $$-0.478668\pi$$
0.0669650 + 0.997755i $$0.478668\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −4.00000 −0.266076
$$227$$ −18.0000 −1.19470 −0.597351 0.801980i $$-0.703780\pi$$
−0.597351 + 0.801980i $$0.703780\pi$$
$$228$$ 0 0
$$229$$ 18.0000 1.18947 0.594737 0.803921i $$-0.297256\pi$$
0.594737 + 0.803921i $$0.297256\pi$$
$$230$$ 12.0000 0.791257
$$231$$ 0 0
$$232$$ 6.00000 0.393919
$$233$$ −14.0000 −0.917170 −0.458585 0.888650i $$-0.651644\pi$$
−0.458585 + 0.888650i $$0.651644\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −4.00000 −0.260378
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −20.0000 −1.29369 −0.646846 0.762620i $$-0.723912\pi$$
−0.646846 + 0.762620i $$0.723912\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ 0 0
$$244$$ 14.0000 0.896258
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 8.00000 0.509028
$$248$$ 2.00000 0.127000
$$249$$ 0 0
$$250$$ 12.0000 0.758947
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 6.00000 0.377217
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −22.0000 −1.37232 −0.686161 0.727450i $$-0.740706\pi$$
−0.686161 + 0.727450i $$0.740706\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 4.00000 0.248069
$$261$$ 0 0
$$262$$ −14.0000 −0.864923
$$263$$ 20.0000 1.23325 0.616626 0.787256i $$-0.288499\pi$$
0.616626 + 0.787256i $$0.288499\pi$$
$$264$$ 0 0
$$265$$ 4.00000 0.245718
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 12.0000 0.733017
$$269$$ 30.0000 1.82913 0.914566 0.404436i $$-0.132532\pi$$
0.914566 + 0.404436i $$0.132532\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 0 0
$$274$$ −8.00000 −0.483298
$$275$$ 1.00000 0.0603023
$$276$$ 0 0
$$277$$ 4.00000 0.240337 0.120168 0.992754i $$-0.461657\pi$$
0.120168 + 0.992754i $$0.461657\pi$$
$$278$$ 20.0000 1.19952
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −14.0000 −0.835170 −0.417585 0.908638i $$-0.637123\pi$$
−0.417585 + 0.908638i $$0.637123\pi$$
$$282$$ 0 0
$$283$$ 28.0000 1.66443 0.832214 0.554455i $$-0.187073\pi$$
0.832214 + 0.554455i $$0.187073\pi$$
$$284$$ 2.00000 0.118678
$$285$$ 0 0
$$286$$ 2.00000 0.118262
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 12.0000 0.704664
$$291$$ 0 0
$$292$$ 6.00000 0.351123
$$293$$ −16.0000 −0.934730 −0.467365 0.884064i $$-0.654797\pi$$
−0.467365 + 0.884064i $$0.654797\pi$$
$$294$$ 0 0
$$295$$ −8.00000 −0.465778
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ 10.0000 0.579284
$$299$$ −12.0000 −0.693978
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −20.0000 −1.15087
$$303$$ 0 0
$$304$$ 4.00000 0.229416
$$305$$ 28.0000 1.60328
$$306$$ 0 0
$$307$$ −4.00000 −0.228292 −0.114146 0.993464i $$-0.536413\pi$$
−0.114146 + 0.993464i $$0.536413\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 4.00000 0.227185
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 0 0
$$313$$ −28.0000 −1.58265 −0.791327 0.611393i $$-0.790609\pi$$
−0.791327 + 0.611393i $$0.790609\pi$$
$$314$$ −22.0000 −1.24153
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −6.00000 −0.336994 −0.168497 0.985702i $$-0.553891\pi$$
−0.168497 + 0.985702i $$0.553891\pi$$
$$318$$ 0 0
$$319$$ 6.00000 0.335936
$$320$$ 2.00000 0.111803
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 24.0000 1.33540
$$324$$ 0 0
$$325$$ −2.00000 −0.110940
$$326$$ −24.0000 −1.32924
$$327$$ 0 0
$$328$$ −10.0000 −0.552158
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ 6.00000 0.329293
$$333$$ 0 0
$$334$$ −12.0000 −0.656611
$$335$$ 24.0000 1.31126
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 9.00000 0.489535
$$339$$ 0 0
$$340$$ 12.0000 0.650791
$$341$$ 2.00000 0.108306
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 6.00000 0.323498
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 4.00000 0.214731 0.107366 0.994220i $$-0.465758\pi$$
0.107366 + 0.994220i $$0.465758\pi$$
$$348$$ 0 0
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 1.00000 0.0533002
$$353$$ 6.00000 0.319348 0.159674 0.987170i $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ 0 0
$$355$$ 4.00000 0.212298
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 4.00000 0.211112 0.105556 0.994413i $$-0.466338\pi$$
0.105556 + 0.994413i $$0.466338\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ −18.0000 −0.946059
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 12.0000 0.628109
$$366$$ 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
$$369$$ 0 0
$$370$$ −4.00000 −0.207950
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 12.0000 0.621336 0.310668 0.950518i $$-0.399447\pi$$
0.310668 + 0.950518i $$0.399447\pi$$
$$374$$ 6.00000 0.310253
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −12.0000 −0.618031
$$378$$ 0 0
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ 8.00000 0.410391
$$381$$ 0 0
$$382$$ 6.00000 0.306987
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −22.0000 −1.11977
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 2.00000 0.101404 0.0507020 0.998714i $$-0.483854\pi$$
0.0507020 + 0.998714i $$0.483854\pi$$
$$390$$ 0 0
$$391$$ −36.0000 −1.82060
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 22.0000 1.10415 0.552074 0.833795i $$-0.313837\pi$$
0.552074 + 0.833795i $$0.313837\pi$$
$$398$$ 2.00000 0.100251
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ 24.0000 1.19850 0.599251 0.800561i $$-0.295465\pi$$
0.599251 + 0.800561i $$0.295465\pi$$
$$402$$ 0 0
$$403$$ −4.00000 −0.199254
$$404$$ −12.0000 −0.597022
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −2.00000 −0.0991363
$$408$$ 0 0
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ −20.0000 −0.987730
$$411$$ 0 0
$$412$$ −6.00000 −0.295599
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ −2.00000 −0.0980581
$$417$$ 0 0
$$418$$ 4.00000 0.195646
$$419$$ −36.0000 −1.75872 −0.879358 0.476162i $$-0.842028\pi$$
−0.879358 + 0.476162i $$0.842028\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 10.0000 0.486792
$$423$$ 0 0
$$424$$ −2.00000 −0.0971286
$$425$$ −6.00000 −0.291043
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ 12.0000 0.578691
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ 0 0
$$433$$ 16.0000 0.768911 0.384455 0.923144i $$-0.374389\pi$$
0.384455 + 0.923144i $$0.374389\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 8.00000 0.383131
$$437$$ −24.0000 −1.14808
$$438$$ 0 0
$$439$$ −4.00000 −0.190910 −0.0954548 0.995434i $$-0.530431\pi$$
−0.0954548 + 0.995434i $$0.530431\pi$$
$$440$$ 2.00000 0.0953463
$$441$$ 0 0
$$442$$ −12.0000 −0.570782
$$443$$ −36.0000 −1.71041 −0.855206 0.518289i $$-0.826569\pi$$
−0.855206 + 0.518289i $$0.826569\pi$$
$$444$$ 0 0
$$445$$ 12.0000 0.568855
$$446$$ −2.00000 −0.0947027
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 12.0000 0.566315 0.283158 0.959073i $$-0.408618\pi$$
0.283158 + 0.959073i $$0.408618\pi$$
$$450$$ 0 0
$$451$$ −10.0000 −0.470882
$$452$$ 4.00000 0.188144
$$453$$ 0 0
$$454$$ 18.0000 0.844782
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 18.0000 0.842004 0.421002 0.907060i $$-0.361678\pi$$
0.421002 + 0.907060i $$0.361678\pi$$
$$458$$ −18.0000 −0.841085
$$459$$ 0 0
$$460$$ −12.0000 −0.559503
$$461$$ −12.0000 −0.558896 −0.279448 0.960161i $$-0.590151\pi$$
−0.279448 + 0.960161i $$0.590151\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ 14.0000 0.648537
$$467$$ −20.0000 −0.925490 −0.462745 0.886492i $$-0.653135\pi$$
−0.462745 + 0.886492i $$0.653135\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 4.00000 0.184115
$$473$$ 6.00000 0.275880
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 20.0000 0.914779
$$479$$ 12.0000 0.548294 0.274147 0.961688i $$-0.411605\pi$$
0.274147 + 0.961688i $$0.411605\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ −2.00000 −0.0910975
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ −14.0000 −0.633750
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ 0 0
$$493$$ −36.0000 −1.62136
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ −2.00000 −0.0898027
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 8.00000 0.358129 0.179065 0.983837i $$-0.442693\pi$$
0.179065 + 0.983837i $$0.442693\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −28.0000 −1.24846 −0.624229 0.781241i $$-0.714587\pi$$
−0.624229 + 0.781241i $$0.714587\pi$$
$$504$$ 0 0
$$505$$ −24.0000 −1.06799
$$506$$ −6.00000 −0.266733
$$507$$ 0 0
$$508$$ −8.00000 −0.354943
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 22.0000 0.970378
$$515$$ −12.0000 −0.528783
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −4.00000 −0.175412
$$521$$ −14.0000 −0.613351 −0.306676 0.951814i $$-0.599217\pi$$
−0.306676 + 0.951814i $$0.599217\pi$$
$$522$$ 0 0
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ 14.0000 0.611593
$$525$$ 0 0
$$526$$ −20.0000 −0.872041
$$527$$ −12.0000 −0.522728
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ −4.00000 −0.173749
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 20.0000 0.866296
$$534$$ 0 0
$$535$$ −24.0000 −1.03761
$$536$$ −12.0000 −0.518321
$$537$$ 0 0
$$538$$ −30.0000 −1.29339
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 40.0000 1.71973 0.859867 0.510518i $$-0.170546\pi$$
0.859867 + 0.510518i $$0.170546\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ 0 0
$$544$$ −6.00000 −0.257248
$$545$$ 16.0000 0.685365
$$546$$ 0 0
$$547$$ −38.0000 −1.62476 −0.812381 0.583127i $$-0.801829\pi$$
−0.812381 + 0.583127i $$0.801829\pi$$
$$548$$ 8.00000 0.341743
$$549$$ 0 0
$$550$$ −1.00000 −0.0426401
$$551$$ −24.0000 −1.02243
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −4.00000 −0.169944
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ 14.0000 0.593199 0.296600 0.955002i $$-0.404147\pi$$
0.296600 + 0.955002i $$0.404147\pi$$
$$558$$ 0 0
$$559$$ −12.0000 −0.507546
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 14.0000 0.590554
$$563$$ 10.0000 0.421450 0.210725 0.977545i $$-0.432418\pi$$
0.210725 + 0.977545i $$0.432418\pi$$
$$564$$ 0 0
$$565$$ 8.00000 0.336563
$$566$$ −28.0000 −1.17693
$$567$$ 0 0
$$568$$ −2.00000 −0.0839181
$$569$$ −34.0000 −1.42535 −0.712677 0.701492i $$-0.752517\pi$$
−0.712677 + 0.701492i $$0.752517\pi$$
$$570$$ 0 0
$$571$$ −14.0000 −0.585882 −0.292941 0.956131i $$-0.594634\pi$$
−0.292941 + 0.956131i $$0.594634\pi$$
$$572$$ −2.00000 −0.0836242
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 6.00000 0.250217
$$576$$ 0 0
$$577$$ 28.0000 1.16566 0.582828 0.812596i $$-0.301946\pi$$
0.582828 + 0.812596i $$0.301946\pi$$
$$578$$ −19.0000 −0.790296
$$579$$ 0 0
$$580$$ −12.0000 −0.498273
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −2.00000 −0.0828315
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ 16.0000 0.660954
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 0 0
$$589$$ −8.00000 −0.329634
$$590$$ 8.00000 0.329355
$$591$$ 0 0
$$592$$ 2.00000 0.0821995
$$593$$ 34.0000 1.39621 0.698106 0.715994i $$-0.254026\pi$$
0.698106 + 0.715994i $$0.254026\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −10.0000 −0.409616
$$597$$ 0 0
$$598$$ 12.0000 0.490716
$$599$$ 14.0000 0.572024 0.286012 0.958226i $$-0.407670\pi$$
0.286012 + 0.958226i $$0.407670\pi$$
$$600$$ 0 0
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 20.0000 0.813788
$$605$$ 2.00000 0.0813116
$$606$$ 0 0
$$607$$ −28.0000 −1.13648 −0.568242 0.822861i $$-0.692376\pi$$
−0.568242 + 0.822861i $$0.692376\pi$$
$$608$$ −4.00000 −0.162221
$$609$$ 0 0
$$610$$ −28.0000 −1.13369
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$614$$ 4.00000 0.161427
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −36.0000 −1.44931 −0.724653 0.689114i $$-0.758000\pi$$
−0.724653 + 0.689114i $$0.758000\pi$$
$$618$$ 0 0
$$619$$ 40.0000 1.60774 0.803868 0.594808i $$-0.202772\pi$$
0.803868 + 0.594808i $$0.202772\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ 0 0
$$622$$ −8.00000 −0.320771
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 28.0000 1.11911
$$627$$ 0 0
$$628$$ 22.0000 0.877896
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 6.00000 0.238290
$$635$$ −16.0000 −0.634941
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −6.00000 −0.237542
$$639$$ 0 0
$$640$$ −2.00000 −0.0790569
$$641$$ 12.0000 0.473972 0.236986 0.971513i $$-0.423841\pi$$
0.236986 + 0.971513i $$0.423841\pi$$
$$642$$ 0 0
$$643$$ −4.00000 −0.157745 −0.0788723 0.996885i $$-0.525132\pi$$
−0.0788723 + 0.996885i $$0.525132\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −24.0000 −0.944267
$$647$$ 8.00000 0.314512 0.157256 0.987558i $$-0.449735\pi$$
0.157256 + 0.987558i $$0.449735\pi$$
$$648$$ 0 0
$$649$$ 4.00000 0.157014
$$650$$ 2.00000 0.0784465
$$651$$ 0 0
$$652$$ 24.0000 0.939913
$$653$$ 42.0000 1.64359 0.821794 0.569785i $$-0.192974\pi$$
0.821794 + 0.569785i $$0.192974\pi$$
$$654$$ 0 0
$$655$$ 28.0000 1.09405
$$656$$ 10.0000 0.390434
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 0 0
$$661$$ −30.0000 −1.16686 −0.583432 0.812162i $$-0.698291\pi$$
−0.583432 + 0.812162i $$0.698291\pi$$
$$662$$ −12.0000 −0.466393
$$663$$ 0 0
$$664$$ −6.00000 −0.232845
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 36.0000 1.39393
$$668$$ 12.0000 0.464294
$$669$$ 0 0
$$670$$ −24.0000 −0.927201
$$671$$ −14.0000 −0.540464
$$672$$ 0 0
$$673$$ −6.00000 −0.231283 −0.115642 0.993291i $$-0.536892\pi$$
−0.115642 + 0.993291i $$0.536892\pi$$
$$674$$ 14.0000 0.539260
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −12.0000 −0.460179
$$681$$ 0 0
$$682$$ −2.00000 −0.0765840
$$683$$ 32.0000 1.22445 0.612223 0.790685i $$-0.290275\pi$$
0.612223 + 0.790685i $$0.290275\pi$$
$$684$$ 0 0
$$685$$ 16.0000 0.611329
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −6.00000 −0.228748
$$689$$ 4.00000 0.152388
$$690$$ 0 0
$$691$$ −4.00000 −0.152167 −0.0760836 0.997101i $$-0.524242\pi$$
−0.0760836 + 0.997101i $$0.524242\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ −4.00000 −0.151838
$$695$$ −40.0000 −1.51729
$$696$$ 0 0
$$697$$ 60.0000 2.27266
$$698$$ 2.00000 0.0757011
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −2.00000 −0.0755390 −0.0377695 0.999286i $$-0.512025\pi$$
−0.0377695 + 0.999286i $$0.512025\pi$$
$$702$$ 0 0
$$703$$ 8.00000 0.301726
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ −4.00000 −0.150117
$$711$$ 0 0
$$712$$ −6.00000 −0.224860
$$713$$ 12.0000 0.449404
$$714$$ 0 0
$$715$$ −4.00000 −0.149592
$$716$$ 0 0
$$717$$ 0 0
$$718$$ −4.00000 −0.149279
$$719$$ 40.0000 1.49175 0.745874 0.666087i $$-0.232032\pi$$
0.745874 + 0.666087i $$0.232032\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 3.00000 0.111648
$$723$$ 0 0
$$724$$ 18.0000 0.668965
$$725$$ 6.00000 0.222834
$$726$$ 0 0
$$727$$ 42.0000 1.55769 0.778847 0.627214i $$-0.215805\pi$$
0.778847 + 0.627214i $$0.215805\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −12.0000 −0.444140
$$731$$ −36.0000 −1.33151
$$732$$ 0 0
$$733$$ 18.0000 0.664845 0.332423 0.943131i $$-0.392134\pi$$
0.332423 + 0.943131i $$0.392134\pi$$
$$734$$ −10.0000 −0.369107
$$735$$ 0 0
$$736$$ 6.00000 0.221163
$$737$$ −12.0000 −0.442026
$$738$$ 0 0
$$739$$ −34.0000 −1.25071 −0.625355 0.780340i $$-0.715046\pi$$
−0.625355 + 0.780340i $$0.715046\pi$$
$$740$$ 4.00000 0.147043
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 48.0000 1.76095 0.880475 0.474093i $$-0.157224\pi$$
0.880475 + 0.474093i $$0.157224\pi$$
$$744$$ 0 0
$$745$$ −20.0000 −0.732743
$$746$$ −12.0000 −0.439351
$$747$$ 0 0
$$748$$ −6.00000 −0.219382
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 24.0000 0.875772 0.437886 0.899030i $$-0.355727\pi$$
0.437886 + 0.899030i $$0.355727\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 12.0000 0.437014
$$755$$ 40.0000 1.45575
$$756$$ 0 0
$$757$$ −30.0000 −1.09037 −0.545184 0.838316i $$-0.683540\pi$$
−0.545184 + 0.838316i $$0.683540\pi$$
$$758$$ 8.00000 0.290573
$$759$$ 0 0
$$760$$ −8.00000 −0.290191
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −6.00000 −0.217072
$$765$$ 0 0
$$766$$ −24.0000 −0.867155
$$767$$ −8.00000 −0.288863
$$768$$ 0 0
$$769$$ 50.0000 1.80305 0.901523 0.432731i $$-0.142450\pi$$
0.901523 + 0.432731i $$0.142450\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 22.0000 0.791797
$$773$$ 34.0000 1.22290 0.611448 0.791285i $$-0.290588\pi$$
0.611448 + 0.791285i $$0.290588\pi$$
$$774$$ 0 0
$$775$$ 2.00000 0.0718421
$$776$$ 0 0
$$777$$ 0 0
$$778$$ −2.00000 −0.0717035
$$779$$ 40.0000 1.43315
$$780$$ 0 0
$$781$$ −2.00000 −0.0715656
$$782$$ 36.0000 1.28736
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 44.0000 1.57043
$$786$$ 0 0
$$787$$ −52.0000 −1.85360 −0.926800 0.375555i $$-0.877452\pi$$
−0.926800 + 0.375555i $$0.877452\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 28.0000 0.994309
$$794$$ −22.0000 −0.780751
$$795$$ 0 0
$$796$$ −2.00000 −0.0708881
$$797$$ 6.00000 0.212531 0.106265 0.994338i $$-0.466111\pi$$
0.106265 + 0.994338i $$0.466111\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 1.00000 0.0353553
$$801$$ 0 0
$$802$$ −24.0000 −0.847469
$$803$$ −6.00000 −0.211735
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 4.00000 0.140894
$$807$$ 0 0
$$808$$ 12.0000 0.422159
$$809$$ 10.0000 0.351581 0.175791 0.984428i $$-0.443752\pi$$
0.175791 + 0.984428i $$0.443752\pi$$
$$810$$ 0 0
$$811$$ 28.0000 0.983213 0.491606 0.870817i $$-0.336410\pi$$
0.491606 + 0.870817i $$0.336410\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 2.00000 0.0701000
$$815$$ 48.0000 1.68137
$$816$$ 0 0
$$817$$ −24.0000 −0.839654
$$818$$ 26.0000 0.909069
$$819$$ 0 0
$$820$$ 20.0000 0.698430
$$821$$ 30.0000 1.04701 0.523504 0.852023i $$-0.324625\pi$$
0.523504 + 0.852023i $$0.324625\pi$$
$$822$$ 0 0
$$823$$ 16.0000 0.557725 0.278862 0.960331i $$-0.410043\pi$$
0.278862 + 0.960331i $$0.410043\pi$$
$$824$$ 6.00000 0.209020
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 36.0000 1.25184 0.625921 0.779886i $$-0.284723\pi$$
0.625921 + 0.779886i $$0.284723\pi$$
$$828$$ 0 0
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ −12.0000 −0.416526
$$831$$ 0 0
$$832$$ 2.00000 0.0693375
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 24.0000 0.830554
$$836$$ −4.00000 −0.138343
$$837$$ 0 0
$$838$$ 36.0000 1.24360
$$839$$ 24.0000 0.828572 0.414286 0.910147i $$-0.364031\pi$$
0.414286 + 0.910147i $$0.364031\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 10.0000 0.344623
$$843$$ 0 0
$$844$$ −10.0000 −0.344214
$$845$$ −18.0000 −0.619219
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 2.00000 0.0686803
$$849$$ 0 0
$$850$$ 6.00000 0.205798
$$851$$ −12.0000 −0.411355
$$852$$ 0 0
$$853$$ −42.0000 −1.43805 −0.719026 0.694983i $$-0.755412\pi$$
−0.719026 + 0.694983i $$0.755412\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ −30.0000 −1.02478 −0.512390 0.858753i $$-0.671240\pi$$
−0.512390 + 0.858753i $$0.671240\pi$$
$$858$$ 0 0
$$859$$ −40.0000 −1.36478 −0.682391 0.730987i $$-0.739060\pi$$
−0.682391 + 0.730987i $$0.739060\pi$$
$$860$$ −12.0000 −0.409197
$$861$$ 0 0
$$862$$ 12.0000 0.408722
$$863$$ −2.00000 −0.0680808 −0.0340404 0.999420i $$-0.510837\pi$$
−0.0340404 + 0.999420i $$0.510837\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −16.0000 −0.543702
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ −8.00000 −0.270914
$$873$$ 0 0
$$874$$ 24.0000 0.811812
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 28.0000 0.945493 0.472746 0.881199i $$-0.343263\pi$$
0.472746 + 0.881199i $$0.343263\pi$$
$$878$$ 4.00000 0.134993
$$879$$ 0 0
$$880$$ −2.00000 −0.0674200
$$881$$ 30.0000 1.01073 0.505363 0.862907i $$-0.331359\pi$$
0.505363 + 0.862907i $$0.331359\pi$$
$$882$$ 0 0
$$883$$ −16.0000 −0.538443 −0.269221 0.963078i $$-0.586766\pi$$
−0.269221 + 0.963078i $$0.586766\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 0 0
$$886$$ 36.0000 1.20944
$$887$$ −44.0000 −1.47738 −0.738688 0.674048i $$-0.764554\pi$$
−0.738688 + 0.674048i $$0.764554\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −12.0000 −0.402241
$$891$$ 0 0
$$892$$ 2.00000 0.0669650
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −12.0000 −0.400445
$$899$$ 12.0000 0.400222
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ 10.0000 0.332964
$$903$$ 0 0
$$904$$ −4.00000 −0.133038
$$905$$ 36.0000 1.19668
$$906$$ 0 0
$$907$$ 16.0000 0.531271 0.265636 0.964073i $$-0.414418\pi$$
0.265636 + 0.964073i $$0.414418\pi$$
$$908$$ −18.0000 −0.597351
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 38.0000 1.25900 0.629498 0.777002i $$-0.283261\pi$$
0.629498 + 0.777002i $$0.283261\pi$$
$$912$$ 0 0
$$913$$ −6.00000 −0.198571
$$914$$ −18.0000 −0.595387
$$915$$ 0 0
$$916$$ 18.0000 0.594737
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −20.0000 −0.659739 −0.329870 0.944027i $$-0.607005\pi$$
−0.329870 + 0.944027i $$0.607005\pi$$
$$920$$ 12.0000 0.395628
$$921$$ 0 0
$$922$$ 12.0000 0.395199
$$923$$ 4.00000 0.131662
$$924$$ 0 0
$$925$$ −2.00000 −0.0657596
$$926$$ −16.0000 −0.525793
$$927$$ 0 0
$$928$$ 6.00000 0.196960
$$929$$ 34.0000 1.11550 0.557752 0.830008i $$-0.311664\pi$$
0.557752 + 0.830008i $$0.311664\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −14.0000 −0.458585
$$933$$ 0 0
$$934$$ 20.0000 0.654420
$$935$$ −12.0000 −0.392442
$$936$$ 0 0
$$937$$ −38.0000 −1.24141 −0.620703 0.784046i $$-0.713153\pi$$
−0.620703 + 0.784046i $$0.713153\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ 0 0
$$943$$ −60.0000 −1.95387
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ −6.00000 −0.195077
$$947$$ −24.0000 −0.779895 −0.389948 0.920837i $$-0.627507\pi$$
−0.389948 + 0.920837i $$0.627507\pi$$
$$948$$ 0 0
$$949$$ 12.0000 0.389536
$$950$$ 4.00000 0.129777
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 26.0000 0.842223 0.421111 0.907009i $$-0.361640\pi$$
0.421111 + 0.907009i $$0.361640\pi$$
$$954$$ 0 0
$$955$$ −12.0000 −0.388311
$$956$$ −20.0000 −0.646846
$$957$$ 0 0
$$958$$ −12.0000 −0.387702
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ −4.00000 −0.128965
$$963$$ 0 0
$$964$$ 2.00000 0.0644157
$$965$$ 44.0000 1.41641
$$966$$ 0 0
$$967$$ 20.0000 0.643157 0.321578 0.946883i $$-0.395787\pi$$
0.321578 + 0.946883i $$0.395787\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 24.0000 0.770197 0.385098 0.922876i $$-0.374168\pi$$
0.385098 + 0.922876i $$0.374168\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −8.00000 −0.256337
$$975$$ 0 0
$$976$$ 14.0000 0.448129
$$977$$ 56.0000 1.79160 0.895799 0.444459i $$-0.146604\pi$$
0.895799 + 0.444459i $$0.146604\pi$$
$$978$$ 0 0
$$979$$ −6.00000 −0.191761
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −20.0000 −0.638226
$$983$$ −32.0000 −1.02064 −0.510321 0.859984i $$-0.670473\pi$$
−0.510321 + 0.859984i $$0.670473\pi$$
$$984$$ 0 0
$$985$$ −12.0000 −0.382352
$$986$$ 36.0000 1.14647
$$987$$ 0 0
$$988$$ 8.00000 0.254514
$$989$$ 36.0000 1.14473
$$990$$ 0 0
$$991$$ −40.0000 −1.27064 −0.635321 0.772248i $$-0.719132\pi$$
−0.635321 + 0.772248i $$0.719132\pi$$
$$992$$ 2.00000 0.0635001
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −4.00000 −0.126809
$$996$$ 0 0
$$997$$ 14.0000 0.443384 0.221692 0.975117i $$-0.428842\pi$$
0.221692 + 0.975117i $$0.428842\pi$$
$$998$$ −8.00000 −0.253236
$$999$$ 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 9702.2.a.u.1.1 yes 1
3.2 odd 2 9702.2.a.bh.1.1 yes 1
7.6 odd 2 9702.2.a.d.1.1 1
21.20 even 2 9702.2.a.cb.1.1 yes 1

By twisted newform
Twist Min Dim Char Parity Ord Type
9702.2.a.d.1.1 1 7.6 odd 2
9702.2.a.u.1.1 yes 1 1.1 even 1 trivial
9702.2.a.bh.1.1 yes 1 3.2 odd 2
9702.2.a.cb.1.1 yes 1 21.20 even 2