# Properties

 Label 9702.2.a.q.1.1 Level $9702$ Weight $2$ Character 9702.1 Self dual yes Analytic conductor $77.471$ 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] = [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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3234) 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)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{8} -2.00000 q^{10} -1.00000 q^{11} -4.00000 q^{13} +1.00000 q^{16} +6.00000 q^{17} -2.00000 q^{19} +2.00000 q^{20} +1.00000 q^{22} -1.00000 q^{25} +4.00000 q^{26} +6.00000 q^{29} -2.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} +2.00000 q^{37} +2.00000 q^{38} -2.00000 q^{40} -2.00000 q^{41} -12.0000 q^{43} -1.00000 q^{44} -6.00000 q^{47} +1.00000 q^{50} -4.00000 q^{52} -10.0000 q^{53} -2.00000 q^{55} -6.00000 q^{58} +8.00000 q^{59} -4.00000 q^{61} +2.00000 q^{62} +1.00000 q^{64} -8.00000 q^{65} +6.00000 q^{68} +8.00000 q^{71} -6.00000 q^{73} -2.00000 q^{74} -2.00000 q^{76} +12.0000 q^{79} +2.00000 q^{80} +2.00000 q^{82} +6.00000 q^{83} +12.0000 q^{85} +12.0000 q^{86} +1.00000 q^{88} -12.0000 q^{89} +6.00000 q^{94} -4.00000 q^{95} +O(q^{100})$$

## 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
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 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
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 2.00000 0.447214
$$21$$ 0 0
$$22$$ 1.00000 0.213201
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 4.00000 0.784465
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 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 $$0.447432\pi$$
$$38$$ 2.00000 0.324443
$$39$$ 0 0
$$40$$ −2.00000 −0.316228
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ −12.0000 −1.82998 −0.914991 0.403473i $$-0.867803\pi$$
−0.914991 + 0.403473i $$0.867803\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −6.00000 −0.875190 −0.437595 0.899172i $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ −4.00000 −0.554700
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ 0 0
$$55$$ −2.00000 −0.269680
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −6.00000 −0.787839
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ −4.00000 −0.512148 −0.256074 0.966657i $$-0.582429\pi$$
−0.256074 + 0.966657i $$0.582429\pi$$
$$62$$ 2.00000 0.254000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −8.00000 −0.992278
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 0 0
$$73$$ −6.00000 −0.702247 −0.351123 0.936329i $$-0.614200\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ −2.00000 −0.229416
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 12.0000 1.35011 0.675053 0.737769i $$-0.264121\pi$$
0.675053 + 0.737769i $$0.264121\pi$$
$$80$$ 2.00000 0.223607
$$81$$ 0 0
$$82$$ 2.00000 0.220863
$$83$$ 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
$$86$$ 12.0000 1.29399
$$87$$ 0 0
$$88$$ 1.00000 0.106600
$$89$$ −12.0000 −1.27200 −0.635999 0.771690i $$-0.719412\pi$$
−0.635999 + 0.771690i $$0.719412\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 6.00000 0.618853
$$95$$ −4.00000 −0.410391
$$96$$ 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 $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ 0 0
$$103$$ −6.00000 −0.591198 −0.295599 0.955312i $$-0.595519\pi$$
−0.295599 + 0.955312i $$0.595519\pi$$
$$104$$ 4.00000 0.392232
$$105$$ 0 0
$$106$$ 10.0000 0.971286
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 2.00000 0.190693
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 0 0
$$118$$ −8.00000 −0.736460
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 4.00000 0.362143
$$123$$ 0 0
$$124$$ −2.00000 −0.179605
$$125$$ −12.0000 −1.07331
$$126$$ 0 0
$$127$$ 4.00000 0.354943 0.177471 0.984126i $$-0.443208\pi$$
0.177471 + 0.984126i $$0.443208\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 8.00000 0.701646
$$131$$ 2.00000 0.174741 0.0873704 0.996176i $$-0.472154\pi$$
0.0873704 + 0.996176i $$0.472154\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ −10.0000 −0.854358 −0.427179 0.904167i $$-0.640493\pi$$
−0.427179 + 0.904167i $$0.640493\pi$$
$$138$$ 0 0
$$139$$ 10.0000 0.848189 0.424094 0.905618i $$-0.360592\pi$$
0.424094 + 0.905618i $$0.360592\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −8.00000 −0.671345
$$143$$ 4.00000 0.334497
$$144$$ 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
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 2.00000 0.162221
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −4.00000 −0.321288
$$156$$ 0 0
$$157$$ −14.0000 −1.11732 −0.558661 0.829396i $$-0.688685\pi$$
−0.558661 + 0.829396i $$0.688685\pi$$
$$158$$ −12.0000 −0.954669
$$159$$ 0 0
$$160$$ −2.00000 −0.158114
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −12.0000 −0.939913 −0.469956 0.882690i $$-0.655730\pi$$
−0.469956 + 0.882690i $$0.655730\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 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
$$169$$ 3.00000 0.230769
$$170$$ −12.0000 −0.920358
$$171$$ 0 0
$$172$$ −12.0000 −0.914991
$$173$$ −24.0000 −1.82469 −0.912343 0.409426i $$-0.865729\pi$$
−0.912343 + 0.409426i $$0.865729\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ 0 0
$$178$$ 12.0000 0.899438
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ −18.0000 −1.33793 −0.668965 0.743294i $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 4.00000 0.294086
$$186$$ 0 0
$$187$$ −6.00000 −0.438763
$$188$$ −6.00000 −0.437595
$$189$$ 0 0
$$190$$ 4.00000 0.290191
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ 0 0
$$199$$ −14.0000 −0.992434 −0.496217 0.868199i $$-0.665278\pi$$
−0.496217 + 0.868199i $$0.665278\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 0 0
$$202$$ −12.0000 −0.844317
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −4.00000 −0.279372
$$206$$ 6.00000 0.418040
$$207$$ 0 0
$$208$$ −4.00000 −0.277350
$$209$$ 2.00000 0.138343
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ −10.0000 −0.686803
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ −24.0000 −1.63679
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 10.0000 0.677285
$$219$$ 0 0
$$220$$ −2.00000 −0.134840
$$221$$ −24.0000 −1.61441
$$222$$ 0 0
$$223$$ 14.0000 0.937509 0.468755 0.883328i $$-0.344703\pi$$
0.468755 + 0.883328i $$0.344703\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 14.0000 0.931266
$$227$$ −6.00000 −0.398234 −0.199117 0.979976i $$-0.563807\pi$$
−0.199117 + 0.979976i $$0.563807\pi$$
$$228$$ 0 0
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ −26.0000 −1.70332 −0.851658 0.524097i $$-0.824403\pi$$
−0.851658 + 0.524097i $$0.824403\pi$$
$$234$$ 0 0
$$235$$ −12.0000 −0.782794
$$236$$ 8.00000 0.520756
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 16.0000 1.03495 0.517477 0.855697i $$-0.326871\pi$$
0.517477 + 0.855697i $$0.326871\pi$$
$$240$$ 0 0
$$241$$ −22.0000 −1.41714 −0.708572 0.705638i $$-0.750660\pi$$
−0.708572 + 0.705638i $$0.750660\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ 0 0
$$244$$ −4.00000 −0.256074
$$245$$ 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
$$253$$ 0 0
$$254$$ −4.00000 −0.250982
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 20.0000 1.24757 0.623783 0.781598i $$-0.285595\pi$$
0.623783 + 0.781598i $$0.285595\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −8.00000 −0.496139
$$261$$ 0 0
$$262$$ −2.00000 −0.123560
$$263$$ −4.00000 −0.246651 −0.123325 0.992366i $$-0.539356\pi$$
−0.123325 + 0.992366i $$0.539356\pi$$
$$264$$ 0 0
$$265$$ −20.0000 −1.22859
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ 28.0000 1.70088 0.850439 0.526073i $$-0.176336\pi$$
0.850439 + 0.526073i $$0.176336\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 0 0
$$274$$ 10.0000 0.604122
$$275$$ 1.00000 0.0603023
$$276$$ 0 0
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ −10.0000 −0.599760
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −2.00000 −0.119310 −0.0596550 0.998219i $$-0.519000\pi$$
−0.0596550 + 0.998219i $$0.519000\pi$$
$$282$$ 0 0
$$283$$ −14.0000 −0.832214 −0.416107 0.909316i $$-0.636606\pi$$
−0.416107 + 0.909316i $$0.636606\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 0 0
$$286$$ −4.00000 −0.236525
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ −12.0000 −0.704664
$$291$$ 0 0
$$292$$ −6.00000 −0.351123
$$293$$ 8.00000 0.467365 0.233682 0.972313i $$-0.424922\pi$$
0.233682 + 0.972313i $$0.424922\pi$$
$$294$$ 0 0
$$295$$ 16.0000 0.931556
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ 10.0000 0.579284
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 16.0000 0.920697
$$303$$ 0 0
$$304$$ −2.00000 −0.114708
$$305$$ −8.00000 −0.458079
$$306$$ 0 0
$$307$$ 26.0000 1.48390 0.741949 0.670456i $$-0.233902\pi$$
0.741949 + 0.670456i $$0.233902\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 4.00000 0.227185
$$311$$ −34.0000 −1.92796 −0.963982 0.265969i $$-0.914308\pi$$
−0.963982 + 0.265969i $$0.914308\pi$$
$$312$$ 0 0
$$313$$ 20.0000 1.13047 0.565233 0.824931i $$-0.308786\pi$$
0.565233 + 0.824931i $$0.308786\pi$$
$$314$$ 14.0000 0.790066
$$315$$ 0 0
$$316$$ 12.0000 0.675053
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ 0 0
$$319$$ −6.00000 −0.335936
$$320$$ 2.00000 0.111803
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −12.0000 −0.667698
$$324$$ 0 0
$$325$$ 4.00000 0.221880
$$326$$ 12.0000 0.664619
$$327$$ 0 0
$$328$$ 2.00000 0.110432
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −24.0000 −1.31916 −0.659580 0.751635i $$-0.729266\pi$$
−0.659580 + 0.751635i $$0.729266\pi$$
$$332$$ 6.00000 0.329293
$$333$$ 0 0
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −26.0000 −1.41631 −0.708155 0.706057i $$-0.750472\pi$$
−0.708155 + 0.706057i $$0.750472\pi$$
$$338$$ −3.00000 −0.163178
$$339$$ 0 0
$$340$$ 12.0000 0.650791
$$341$$ 2.00000 0.108306
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ 24.0000 1.29025
$$347$$ −20.0000 −1.07366 −0.536828 0.843692i $$-0.680378\pi$$
−0.536828 + 0.843692i $$0.680378\pi$$
$$348$$ 0 0
$$349$$ 16.0000 0.856460 0.428230 0.903670i $$-0.359137\pi$$
0.428230 + 0.903670i $$0.359137\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 1.00000 0.0533002
$$353$$ −12.0000 −0.638696 −0.319348 0.947638i $$-0.603464\pi$$
−0.319348 + 0.947638i $$0.603464\pi$$
$$354$$ 0 0
$$355$$ 16.0000 0.849192
$$356$$ −12.0000 −0.635999
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −20.0000 −1.05556 −0.527780 0.849381i $$-0.676975\pi$$
−0.527780 + 0.849381i $$0.676975\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 18.0000 0.946059
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −12.0000 −0.628109
$$366$$ 0 0
$$367$$ 22.0000 1.14839 0.574195 0.818718i $$-0.305315\pi$$
0.574195 + 0.818718i $$0.305315\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ −4.00000 −0.207950
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 6.00000 0.310668 0.155334 0.987862i $$-0.450355\pi$$
0.155334 + 0.987862i $$0.450355\pi$$
$$374$$ 6.00000 0.310253
$$375$$ 0 0
$$376$$ 6.00000 0.309426
$$377$$ −24.0000 −1.23606
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ −4.00000 −0.205196
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 6.00000 0.306586 0.153293 0.988181i $$-0.451012\pi$$
0.153293 + 0.988181i $$0.451012\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 14.0000 0.712581
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −22.0000 −1.11544 −0.557722 0.830028i $$-0.688325\pi$$
−0.557722 + 0.830028i $$0.688325\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −18.0000 −0.906827
$$395$$ 24.0000 1.20757
$$396$$ 0 0
$$397$$ 34.0000 1.70641 0.853206 0.521575i $$-0.174655\pi$$
0.853206 + 0.521575i $$0.174655\pi$$
$$398$$ 14.0000 0.701757
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 0 0
$$403$$ 8.00000 0.398508
$$404$$ 12.0000 0.597022
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −2.00000 −0.0991363
$$408$$ 0 0
$$409$$ −2.00000 −0.0988936 −0.0494468 0.998777i $$-0.515746\pi$$
−0.0494468 + 0.998777i $$0.515746\pi$$
$$410$$ 4.00000 0.197546
$$411$$ 0 0
$$412$$ −6.00000 −0.295599
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ 4.00000 0.196116
$$417$$ 0 0
$$418$$ −2.00000 −0.0978232
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ 14.0000 0.682318 0.341159 0.940006i $$-0.389181\pi$$
0.341159 + 0.940006i $$0.389181\pi$$
$$422$$ −20.0000 −0.973585
$$423$$ 0 0
$$424$$ 10.0000 0.485643
$$425$$ −6.00000 −0.291043
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ 24.0000 1.15738
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ 0 0
$$433$$ −8.00000 −0.384455 −0.192228 0.981350i $$-0.561571\pi$$
−0.192228 + 0.981350i $$0.561571\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −10.0000 −0.478913
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −28.0000 −1.33637 −0.668184 0.743996i $$-0.732928\pi$$
−0.668184 + 0.743996i $$0.732928\pi$$
$$440$$ 2.00000 0.0953463
$$441$$ 0 0
$$442$$ 24.0000 1.14156
$$443$$ −24.0000 −1.14027 −0.570137 0.821549i $$-0.693110\pi$$
−0.570137 + 0.821549i $$0.693110\pi$$
$$444$$ 0 0
$$445$$ −24.0000 −1.13771
$$446$$ −14.0000 −0.662919
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ 2.00000 0.0941763
$$452$$ −14.0000 −0.658505
$$453$$ 0 0
$$454$$ 6.00000 0.281594
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −6.00000 −0.280668 −0.140334 0.990104i $$-0.544818\pi$$
−0.140334 + 0.990104i $$0.544818\pi$$
$$458$$ −6.00000 −0.280362
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 12.0000 0.558896 0.279448 0.960161i $$-0.409849\pi$$
0.279448 + 0.960161i $$0.409849\pi$$
$$462$$ 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
$$466$$ 26.0000 1.20443
$$467$$ 28.0000 1.29569 0.647843 0.761774i $$-0.275671\pi$$
0.647843 + 0.761774i $$0.275671\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 12.0000 0.553519
$$471$$ 0 0
$$472$$ −8.00000 −0.368230
$$473$$ 12.0000 0.551761
$$474$$ 0 0
$$475$$ 2.00000 0.0917663
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −16.0000 −0.731823
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ −8.00000 −0.364769
$$482$$ 22.0000 1.00207
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ 4.00000 0.181071
$$489$$ 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
$$499$$ −40.0000 −1.79065 −0.895323 0.445418i $$-0.853055\pi$$
−0.895323 + 0.445418i $$0.853055\pi$$
$$500$$ −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
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 4.00000 0.177471
$$509$$ −30.0000 −1.32973 −0.664863 0.746965i $$-0.731510\pi$$
−0.664863 + 0.746965i $$0.731510\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −20.0000 −0.882162
$$515$$ −12.0000 −0.528783
$$516$$ 0 0
$$517$$ 6.00000 0.263880
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 8.00000 0.350823
$$521$$ 28.0000 1.22670 0.613351 0.789810i $$-0.289821\pi$$
0.613351 + 0.789810i $$0.289821\pi$$
$$522$$ 0 0
$$523$$ 2.00000 0.0874539 0.0437269 0.999044i $$-0.486077\pi$$
0.0437269 + 0.999044i $$0.486077\pi$$
$$524$$ 2.00000 0.0873704
$$525$$ 0 0
$$526$$ 4.00000 0.174408
$$527$$ −12.0000 −0.522728
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 20.0000 0.868744
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 8.00000 0.346518
$$534$$ 0 0
$$535$$ 24.0000 1.03761
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 6.00000 0.258678
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ −28.0000 −1.20270
$$543$$ 0 0
$$544$$ −6.00000 −0.257248
$$545$$ −20.0000 −0.856706
$$546$$ 0 0
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ −10.0000 −0.427179
$$549$$ 0 0
$$550$$ −1.00000 −0.0426401
$$551$$ −12.0000 −0.511217
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −10.0000 −0.424859
$$555$$ 0 0
$$556$$ 10.0000 0.424094
$$557$$ 26.0000 1.10166 0.550828 0.834619i $$-0.314312\pi$$
0.550828 + 0.834619i $$0.314312\pi$$
$$558$$ 0 0
$$559$$ 48.0000 2.03018
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 2.00000 0.0843649
$$563$$ −14.0000 −0.590030 −0.295015 0.955493i $$-0.595325\pi$$
−0.295015 + 0.955493i $$0.595325\pi$$
$$564$$ 0 0
$$565$$ −28.0000 −1.17797
$$566$$ 14.0000 0.588464
$$567$$ 0 0
$$568$$ −8.00000 −0.335673
$$569$$ −34.0000 −1.42535 −0.712677 0.701492i $$-0.752517\pi$$
−0.712677 + 0.701492i $$0.752517\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 4.00000 0.167248
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 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
$$583$$ 10.0000 0.414158
$$584$$ 6.00000 0.248282
$$585$$ 0 0
$$586$$ −8.00000 −0.330477
$$587$$ −24.0000 −0.990586 −0.495293 0.868726i $$-0.664939\pi$$
−0.495293 + 0.868726i $$0.664939\pi$$
$$588$$ 0 0
$$589$$ 4.00000 0.164817
$$590$$ −16.0000 −0.658710
$$591$$ 0 0
$$592$$ 2.00000 0.0821995
$$593$$ 22.0000 0.903432 0.451716 0.892162i $$-0.350812\pi$$
0.451716 + 0.892162i $$0.350812\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −10.0000 −0.409616
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 32.0000 1.30748 0.653742 0.756717i $$-0.273198\pi$$
0.653742 + 0.756717i $$0.273198\pi$$
$$600$$ 0 0
$$601$$ 46.0000 1.87638 0.938190 0.346122i $$-0.112502\pi$$
0.938190 + 0.346122i $$0.112502\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −16.0000 −0.651031
$$605$$ 2.00000 0.0813116
$$606$$ 0 0
$$607$$ 8.00000 0.324710 0.162355 0.986732i $$-0.448091\pi$$
0.162355 + 0.986732i $$0.448091\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ 0 0
$$610$$ 8.00000 0.323911
$$611$$ 24.0000 0.970936
$$612$$ 0 0
$$613$$ −6.00000 −0.242338 −0.121169 0.992632i $$-0.538664\pi$$
−0.121169 + 0.992632i $$0.538664\pi$$
$$614$$ −26.0000 −1.04927
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 18.0000 0.724653 0.362326 0.932051i $$-0.381983\pi$$
0.362326 + 0.932051i $$0.381983\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ 0 0
$$622$$ 34.0000 1.36328
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ −20.0000 −0.799361
$$627$$ 0 0
$$628$$ −14.0000 −0.558661
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ −12.0000 −0.477334
$$633$$ 0 0
$$634$$ 18.0000 0.714871
$$635$$ 8.00000 0.317470
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 6.00000 0.237542
$$639$$ 0 0
$$640$$ −2.00000 −0.0790569
$$641$$ −6.00000 −0.236986 −0.118493 0.992955i $$-0.537806\pi$$
−0.118493 + 0.992955i $$0.537806\pi$$
$$642$$ 0 0
$$643$$ −40.0000 −1.57745 −0.788723 0.614749i $$-0.789257\pi$$
−0.788723 + 0.614749i $$0.789257\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 12.0000 0.472134
$$647$$ −10.0000 −0.393141 −0.196570 0.980490i $$-0.562980\pi$$
−0.196570 + 0.980490i $$0.562980\pi$$
$$648$$ 0 0
$$649$$ −8.00000 −0.314027
$$650$$ −4.00000 −0.156893
$$651$$ 0 0
$$652$$ −12.0000 −0.469956
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 0 0
$$655$$ 4.00000 0.156293
$$656$$ −2.00000 −0.0780869
$$657$$ 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
$$661$$ −18.0000 −0.700119 −0.350059 0.936727i $$-0.613839\pi$$
−0.350059 + 0.936727i $$0.613839\pi$$
$$662$$ 24.0000 0.932786
$$663$$ 0 0
$$664$$ −6.00000 −0.232845
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 12.0000 0.464294
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 4.00000 0.154418
$$672$$ 0 0
$$673$$ −18.0000 −0.693849 −0.346925 0.937893i $$-0.612774\pi$$
−0.346925 + 0.937893i $$0.612774\pi$$
$$674$$ 26.0000 1.00148
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ −48.0000 −1.84479 −0.922395 0.386248i $$-0.873771\pi$$
−0.922395 + 0.386248i $$0.873771\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −12.0000 −0.460179
$$681$$ 0 0
$$682$$ −2.00000 −0.0765840
$$683$$ −16.0000 −0.612223 −0.306111 0.951996i $$-0.599028\pi$$
−0.306111 + 0.951996i $$0.599028\pi$$
$$684$$ 0 0
$$685$$ −20.0000 −0.764161
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −12.0000 −0.457496
$$689$$ 40.0000 1.52388
$$690$$ 0 0
$$691$$ −40.0000 −1.52167 −0.760836 0.648944i $$-0.775211\pi$$
−0.760836 + 0.648944i $$0.775211\pi$$
$$692$$ −24.0000 −0.912343
$$693$$ 0 0
$$694$$ 20.0000 0.759190
$$695$$ 20.0000 0.758643
$$696$$ 0 0
$$697$$ −12.0000 −0.454532
$$698$$ −16.0000 −0.605609
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −14.0000 −0.528773 −0.264386 0.964417i $$-0.585169\pi$$
−0.264386 + 0.964417i $$0.585169\pi$$
$$702$$ 0 0
$$703$$ −4.00000 −0.150863
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ 12.0000 0.451626
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −38.0000 −1.42712 −0.713560 0.700594i $$-0.752918\pi$$
−0.713560 + 0.700594i $$0.752918\pi$$
$$710$$ −16.0000 −0.600469
$$711$$ 0 0
$$712$$ 12.0000 0.449719
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 8.00000 0.299183
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 20.0000 0.746393
$$719$$ 46.0000 1.71551 0.857755 0.514058i $$-0.171858\pi$$
0.857755 + 0.514058i $$0.171858\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 15.0000 0.558242
$$723$$ 0 0
$$724$$ −18.0000 −0.668965
$$725$$ −6.00000 −0.222834
$$726$$ 0 0
$$727$$ 30.0000 1.11264 0.556319 0.830969i $$-0.312213\pi$$
0.556319 + 0.830969i $$0.312213\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 12.0000 0.444140
$$731$$ −72.0000 −2.66302
$$732$$ 0 0
$$733$$ −12.0000 −0.443230 −0.221615 0.975134i $$-0.571133\pi$$
−0.221615 + 0.975134i $$0.571133\pi$$
$$734$$ −22.0000 −0.812035
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −4.00000 −0.147142 −0.0735712 0.997290i $$-0.523440\pi$$
−0.0735712 + 0.997290i $$0.523440\pi$$
$$740$$ 4.00000 0.147043
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ −20.0000 −0.732743
$$746$$ −6.00000 −0.219676
$$747$$ 0 0
$$748$$ −6.00000 −0.219382
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$752$$ −6.00000 −0.218797
$$753$$ 0 0
$$754$$ 24.0000 0.874028
$$755$$ −32.0000 −1.16460
$$756$$ 0 0
$$757$$ −30.0000 −1.09037 −0.545184 0.838316i $$-0.683540\pi$$
−0.545184 + 0.838316i $$0.683540\pi$$
$$758$$ 20.0000 0.726433
$$759$$ 0 0
$$760$$ 4.00000 0.145095
$$761$$ 34.0000 1.23250 0.616250 0.787551i $$-0.288651\pi$$
0.616250 + 0.787551i $$0.288651\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −6.00000 −0.216789
$$767$$ −32.0000 −1.15545
$$768$$ 0 0
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −14.0000 −0.503871
$$773$$ −26.0000 −0.935155 −0.467578 0.883952i $$-0.654873\pi$$
−0.467578 + 0.883952i $$0.654873\pi$$
$$774$$ 0 0
$$775$$ 2.00000 0.0718421
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 22.0000 0.788738
$$779$$ 4.00000 0.143315
$$780$$ 0 0
$$781$$ −8.00000 −0.286263
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −28.0000 −0.999363
$$786$$ 0 0
$$787$$ 38.0000 1.35455 0.677277 0.735728i $$-0.263160\pi$$
0.677277 + 0.735728i $$0.263160\pi$$
$$788$$ 18.0000 0.641223
$$789$$ 0 0
$$790$$ −24.0000 −0.853882
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 16.0000 0.568177
$$794$$ −34.0000 −1.20661
$$795$$ 0 0
$$796$$ −14.0000 −0.496217
$$797$$ 6.00000 0.212531 0.106265 0.994338i $$-0.466111\pi$$
0.106265 + 0.994338i $$0.466111\pi$$
$$798$$ 0 0
$$799$$ −36.0000 −1.27359
$$800$$ 1.00000 0.0353553
$$801$$ 0 0
$$802$$ −30.0000 −1.05934
$$803$$ 6.00000 0.211735
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ 0 0
$$808$$ −12.0000 −0.422159
$$809$$ −14.0000 −0.492214 −0.246107 0.969243i $$-0.579151\pi$$
−0.246107 + 0.969243i $$0.579151\pi$$
$$810$$ 0 0
$$811$$ 34.0000 1.19390 0.596951 0.802278i $$-0.296379\pi$$
0.596951 + 0.802278i $$0.296379\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 2.00000 0.0701000
$$815$$ −24.0000 −0.840683
$$816$$ 0 0
$$817$$ 24.0000 0.839654
$$818$$ 2.00000 0.0699284
$$819$$ 0 0
$$820$$ −4.00000 −0.139686
$$821$$ −42.0000 −1.46581 −0.732905 0.680331i $$-0.761836\pi$$
−0.732905 + 0.680331i $$0.761836\pi$$
$$822$$ 0 0
$$823$$ 40.0000 1.39431 0.697156 0.716919i $$-0.254448\pi$$
0.697156 + 0.716919i $$0.254448\pi$$
$$824$$ 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
$$829$$ −10.0000 −0.347314 −0.173657 0.984806i $$-0.555558\pi$$
−0.173657 + 0.984806i $$0.555558\pi$$
$$830$$ −12.0000 −0.416526
$$831$$ 0 0
$$832$$ −4.00000 −0.138675
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 24.0000 0.830554
$$836$$ 2.00000 0.0691714
$$837$$ 0 0
$$838$$ −12.0000 −0.414533
$$839$$ −42.0000 −1.45000 −0.725001 0.688748i $$-0.758161\pi$$
−0.725001 + 0.688748i $$0.758161\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ −14.0000 −0.482472
$$843$$ 0 0
$$844$$ 20.0000 0.688428
$$845$$ 6.00000 0.206406
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −10.0000 −0.343401
$$849$$ 0 0
$$850$$ 6.00000 0.205798
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −36.0000 −1.23262 −0.616308 0.787505i $$-0.711372\pi$$
−0.616308 + 0.787505i $$0.711372\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 42.0000 1.43469 0.717346 0.696717i $$-0.245357\pi$$
0.717346 + 0.696717i $$0.245357\pi$$
$$858$$ 0 0
$$859$$ 8.00000 0.272956 0.136478 0.990643i $$-0.456422\pi$$
0.136478 + 0.990643i $$0.456422\pi$$
$$860$$ −24.0000 −0.818393
$$861$$ 0 0
$$862$$ 12.0000 0.408722
$$863$$ −56.0000 −1.90626 −0.953131 0.302558i $$-0.902160\pi$$
−0.953131 + 0.302558i $$0.902160\pi$$
$$864$$ 0 0
$$865$$ −48.0000 −1.63205
$$866$$ 8.00000 0.271851
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −12.0000 −0.407072
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 10.0000 0.338643
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −2.00000 −0.0675352 −0.0337676 0.999430i $$-0.510751\pi$$
−0.0337676 + 0.999430i $$0.510751\pi$$
$$878$$ 28.0000 0.944954
$$879$$ 0 0
$$880$$ −2.00000 −0.0674200
$$881$$ 24.0000 0.808581 0.404290 0.914631i $$-0.367519\pi$$
0.404290 + 0.914631i $$0.367519\pi$$
$$882$$ 0 0
$$883$$ −52.0000 −1.74994 −0.874970 0.484178i $$-0.839119\pi$$
−0.874970 + 0.484178i $$0.839119\pi$$
$$884$$ −24.0000 −0.807207
$$885$$ 0 0
$$886$$ 24.0000 0.806296
$$887$$ 40.0000 1.34307 0.671534 0.740973i $$-0.265636\pi$$
0.671534 + 0.740973i $$0.265636\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 24.0000 0.804482
$$891$$ 0 0
$$892$$ 14.0000 0.468755
$$893$$ 12.0000 0.401565
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −18.0000 −0.600668
$$899$$ −12.0000 −0.400222
$$900$$ 0 0
$$901$$ −60.0000 −1.99889
$$902$$ −2.00000 −0.0665927
$$903$$ 0 0
$$904$$ 14.0000 0.465633
$$905$$ −36.0000 −1.19668
$$906$$ 0 0
$$907$$ −8.00000 −0.265636 −0.132818 0.991140i $$-0.542403\pi$$
−0.132818 + 0.991140i $$0.542403\pi$$
$$908$$ −6.00000 −0.199117
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −40.0000 −1.32526 −0.662630 0.748947i $$-0.730560\pi$$
−0.662630 + 0.748947i $$0.730560\pi$$
$$912$$ 0 0
$$913$$ −6.00000 −0.198571
$$914$$ 6.00000 0.198462
$$915$$ 0 0
$$916$$ 6.00000 0.198246
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −12.0000 −0.395199
$$923$$ −32.0000 −1.05329
$$924$$ 0 0
$$925$$ −2.00000 −0.0657596
$$926$$ −16.0000 −0.525793
$$927$$ 0 0
$$928$$ −6.00000 −0.196960
$$929$$ −44.0000 −1.44359 −0.721797 0.692105i $$-0.756683\pi$$
−0.721797 + 0.692105i $$0.756683\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −26.0000 −0.851658
$$933$$ 0 0
$$934$$ −28.0000 −0.916188
$$935$$ −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
$$940$$ −12.0000 −0.391397
$$941$$ 48.0000 1.56476 0.782378 0.622804i $$-0.214007\pi$$
0.782378 + 0.622804i $$0.214007\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 8.00000 0.260378
$$945$$ 0 0
$$946$$ −12.0000 −0.390154
$$947$$ −12.0000 −0.389948 −0.194974 0.980808i $$-0.562462\pi$$
−0.194974 + 0.980808i $$0.562462\pi$$
$$948$$ 0 0
$$949$$ 24.0000 0.779073
$$950$$ −2.00000 −0.0648886
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 2.00000 0.0647864 0.0323932 0.999475i $$-0.489687\pi$$
0.0323932 + 0.999475i $$0.489687\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 16.0000 0.517477
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 8.00000 0.257930
$$963$$ 0 0
$$964$$ −22.0000 −0.708572
$$965$$ −28.0000 −0.901352
$$966$$ 0 0
$$967$$ 8.00000 0.257263 0.128631 0.991692i $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 60.0000 1.92549 0.962746 0.270408i $$-0.0871586\pi$$
0.962746 + 0.270408i $$0.0871586\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 16.0000 0.512673
$$975$$ 0 0
$$976$$ −4.00000 −0.128037
$$977$$ 2.00000 0.0639857 0.0319928 0.999488i $$-0.489815\pi$$
0.0319928 + 0.999488i $$0.489815\pi$$
$$978$$ 0 0
$$979$$ 12.0000 0.383522
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −20.0000 −0.638226
$$983$$ −38.0000 −1.21201 −0.606006 0.795460i $$-0.707229\pi$$
−0.606006 + 0.795460i $$0.707229\pi$$
$$984$$ 0 0
$$985$$ 36.0000 1.14706
$$986$$ −36.0000 −1.14647
$$987$$ 0 0
$$988$$ 8.00000 0.254514
$$989$$ 0 0
$$990$$ 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
$$995$$ −28.0000 −0.887660
$$996$$ 0 0
$$997$$ 8.00000 0.253363 0.126681 0.991943i $$-0.459567\pi$$
0.126681 + 0.991943i $$0.459567\pi$$
$$998$$ 40.0000 1.26618
$$999$$ 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.q.1.1 1
3.2 odd 2 3234.2.a.u.1.1 yes 1
7.6 odd 2 9702.2.a.g.1.1 1
21.20 even 2 3234.2.a.r.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.r.1.1 1 21.20 even 2
3234.2.a.u.1.1 yes 1 3.2 odd 2
9702.2.a.g.1.1 1 7.6 odd 2
9702.2.a.q.1.1 1 1.1 even 1 trivial