# Properties

 Label 7605.2.a.a.1.1 Level $7605$ Weight $2$ Character 7605.1 Self dual yes Analytic conductor $60.726$ 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] = [7605,2,Mod(1,7605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7605, 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("7605.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7605 = 3^{2} \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7605.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: $$60.7262307372$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 195) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 7605.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} +5.00000 q^{7} +O(q^{10})$$ $$q-2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} +5.00000 q^{7} -2.00000 q^{10} -2.00000 q^{11} -10.0000 q^{14} -4.00000 q^{16} -2.00000 q^{17} +2.00000 q^{20} +4.00000 q^{22} -6.00000 q^{23} +1.00000 q^{25} +10.0000 q^{28} +4.00000 q^{29} -7.00000 q^{31} +8.00000 q^{32} +4.00000 q^{34} +5.00000 q^{35} -2.00000 q^{37} -6.00000 q^{41} +1.00000 q^{43} -4.00000 q^{44} +12.0000 q^{46} +8.00000 q^{47} +18.0000 q^{49} -2.00000 q^{50} +4.00000 q^{53} -2.00000 q^{55} -8.00000 q^{58} -12.0000 q^{59} -13.0000 q^{61} +14.0000 q^{62} -8.00000 q^{64} -7.00000 q^{67} -4.00000 q^{68} -10.0000 q^{70} -12.0000 q^{71} +15.0000 q^{73} +4.00000 q^{74} -10.0000 q^{77} +3.00000 q^{79} -4.00000 q^{80} +12.0000 q^{82} -8.00000 q^{83} -2.00000 q^{85} -2.00000 q^{86} -14.0000 q^{89} -12.0000 q^{92} -16.0000 q^{94} -5.00000 q^{97} -36.0000 q^{98} +O(q^{100})$$

## 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 $$0.750000\pi$$
$$3$$ 0 0
$$4$$ 2.00000 1.00000
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 5.00000 1.88982 0.944911 0.327327i $$-0.106148\pi$$
0.944911 + 0.327327i $$0.106148\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ −2.00000 −0.632456
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ −10.0000 −2.67261
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 2.00000 0.447214
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ −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
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 10.0000 1.88982
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ 0 0
$$31$$ −7.00000 −1.25724 −0.628619 0.777714i $$-0.716379\pi$$
−0.628619 + 0.777714i $$0.716379\pi$$
$$32$$ 8.00000 1.41421
$$33$$ 0 0
$$34$$ 4.00000 0.685994
$$35$$ 5.00000 0.845154
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 1.00000 0.152499 0.0762493 0.997089i $$-0.475706\pi$$
0.0762493 + 0.997089i $$0.475706\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ 12.0000 1.76930
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ 18.0000 2.57143
$$50$$ −2.00000 −0.282843
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 4.00000 0.549442 0.274721 0.961524i $$-0.411414\pi$$
0.274721 + 0.961524i $$0.411414\pi$$
$$54$$ 0 0
$$55$$ −2.00000 −0.269680
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −8.00000 −1.05045
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ −13.0000 −1.66448 −0.832240 0.554416i $$-0.812942\pi$$
−0.832240 + 0.554416i $$0.812942\pi$$
$$62$$ 14.0000 1.77800
$$63$$ 0 0
$$64$$ −8.00000 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −7.00000 −0.855186 −0.427593 0.903971i $$-0.640638\pi$$
−0.427593 + 0.903971i $$0.640638\pi$$
$$68$$ −4.00000 −0.485071
$$69$$ 0 0
$$70$$ −10.0000 −1.19523
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ 15.0000 1.75562 0.877809 0.479012i $$-0.159005\pi$$
0.877809 + 0.479012i $$0.159005\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −10.0000 −1.13961
$$78$$ 0 0
$$79$$ 3.00000 0.337526 0.168763 0.985657i $$-0.446023\pi$$
0.168763 + 0.985657i $$0.446023\pi$$
$$80$$ −4.00000 −0.447214
$$81$$ 0 0
$$82$$ 12.0000 1.32518
$$83$$ −8.00000 −0.878114 −0.439057 0.898459i $$-0.644687\pi$$
−0.439057 + 0.898459i $$0.644687\pi$$
$$84$$ 0 0
$$85$$ −2.00000 −0.216930
$$86$$ −2.00000 −0.215666
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −14.0000 −1.48400 −0.741999 0.670402i $$-0.766122\pi$$
−0.741999 + 0.670402i $$0.766122\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −12.0000 −1.25109
$$93$$ 0 0
$$94$$ −16.0000 −1.65027
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −5.00000 −0.507673 −0.253837 0.967247i $$-0.581693\pi$$
−0.253837 + 0.967247i $$0.581693\pi$$
$$98$$ −36.0000 −3.63655
$$99$$ 0 0
$$100$$ 2.00000 0.200000
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ 0 0
$$103$$ −7.00000 −0.689730 −0.344865 0.938652i $$-0.612075\pi$$
−0.344865 + 0.938652i $$0.612075\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −8.00000 −0.777029
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 0 0
$$109$$ −11.0000 −1.05361 −0.526804 0.849987i $$-0.676610\pi$$
−0.526804 + 0.849987i $$0.676610\pi$$
$$110$$ 4.00000 0.381385
$$111$$ 0 0
$$112$$ −20.0000 −1.88982
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 0 0
$$115$$ −6.00000 −0.559503
$$116$$ 8.00000 0.742781
$$117$$ 0 0
$$118$$ 24.0000 2.20938
$$119$$ −10.0000 −0.916698
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 26.0000 2.35393
$$123$$ 0 0
$$124$$ −14.0000 −1.25724
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −11.0000 −0.976092 −0.488046 0.872818i $$-0.662290\pi$$
−0.488046 + 0.872818i $$0.662290\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 4.00000 0.349482 0.174741 0.984614i $$-0.444091\pi$$
0.174741 + 0.984614i $$0.444091\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 14.0000 1.20942
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ 0 0
$$139$$ −3.00000 −0.254457 −0.127228 0.991873i $$-0.540608\pi$$
−0.127228 + 0.991873i $$0.540608\pi$$
$$140$$ 10.0000 0.845154
$$141$$ 0 0
$$142$$ 24.0000 2.01404
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 4.00000 0.332182
$$146$$ −30.0000 −2.48282
$$147$$ 0 0
$$148$$ −4.00000 −0.328798
$$149$$ 12.0000 0.983078 0.491539 0.870855i $$-0.336434\pi$$
0.491539 + 0.870855i $$0.336434\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 20.0000 1.61165
$$155$$ −7.00000 −0.562254
$$156$$ 0 0
$$157$$ −15.0000 −1.19713 −0.598565 0.801074i $$-0.704262\pi$$
−0.598565 + 0.801074i $$0.704262\pi$$
$$158$$ −6.00000 −0.477334
$$159$$ 0 0
$$160$$ 8.00000 0.632456
$$161$$ −30.0000 −2.36433
$$162$$ 0 0
$$163$$ −15.0000 −1.17489 −0.587445 0.809264i $$-0.699866\pi$$
−0.587445 + 0.809264i $$0.699866\pi$$
$$164$$ −12.0000 −0.937043
$$165$$ 0 0
$$166$$ 16.0000 1.24184
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 4.00000 0.306786
$$171$$ 0 0
$$172$$ 2.00000 0.152499
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ 5.00000 0.377964
$$176$$ 8.00000 0.603023
$$177$$ 0 0
$$178$$ 28.0000 2.09869
$$179$$ 6.00000 0.448461 0.224231 0.974536i $$-0.428013\pi$$
0.224231 + 0.974536i $$0.428013\pi$$
$$180$$ 0 0
$$181$$ −22.0000 −1.63525 −0.817624 0.575753i $$-0.804709\pi$$
−0.817624 + 0.575753i $$0.804709\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −2.00000 −0.147043
$$186$$ 0 0
$$187$$ 4.00000 0.292509
$$188$$ 16.0000 1.16692
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 0 0
$$193$$ −11.0000 −0.791797 −0.395899 0.918294i $$-0.629567\pi$$
−0.395899 + 0.918294i $$0.629567\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ 36.0000 2.57143
$$197$$ 12.0000 0.854965 0.427482 0.904024i $$-0.359401\pi$$
0.427482 + 0.904024i $$0.359401\pi$$
$$198$$ 0 0
$$199$$ 17.0000 1.20510 0.602549 0.798082i $$-0.294152\pi$$
0.602549 + 0.798082i $$0.294152\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −36.0000 −2.53295
$$203$$ 20.0000 1.40372
$$204$$ 0 0
$$205$$ −6.00000 −0.419058
$$206$$ 14.0000 0.975426
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 15.0000 1.03264 0.516321 0.856395i $$-0.327301\pi$$
0.516321 + 0.856395i $$0.327301\pi$$
$$212$$ 8.00000 0.549442
$$213$$ 0 0
$$214$$ 8.00000 0.546869
$$215$$ 1.00000 0.0681994
$$216$$ 0 0
$$217$$ −35.0000 −2.37595
$$218$$ 22.0000 1.49003
$$219$$ 0 0
$$220$$ −4.00000 −0.269680
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −8.00000 −0.535720 −0.267860 0.963458i $$-0.586316\pi$$
−0.267860 + 0.963458i $$0.586316\pi$$
$$224$$ 40.0000 2.67261
$$225$$ 0 0
$$226$$ −4.00000 −0.266076
$$227$$ −10.0000 −0.663723 −0.331862 0.943328i $$-0.607677\pi$$
−0.331862 + 0.943328i $$0.607677\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 12.0000 0.791257
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 14.0000 0.917170 0.458585 0.888650i $$-0.348356\pi$$
0.458585 + 0.888650i $$0.348356\pi$$
$$234$$ 0 0
$$235$$ 8.00000 0.521862
$$236$$ −24.0000 −1.56227
$$237$$ 0 0
$$238$$ 20.0000 1.29641
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 14.0000 0.899954
$$243$$ 0 0
$$244$$ −26.0000 −1.66448
$$245$$ 18.0000 1.14998
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ −2.00000 −0.126491
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 12.0000 0.754434
$$254$$ 22.0000 1.38040
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 22.0000 1.37232 0.686161 0.727450i $$-0.259294\pi$$
0.686161 + 0.727450i $$0.259294\pi$$
$$258$$ 0 0
$$259$$ −10.0000 −0.621370
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −8.00000 −0.494242
$$263$$ 10.0000 0.616626 0.308313 0.951285i $$-0.400236\pi$$
0.308313 + 0.951285i $$0.400236\pi$$
$$264$$ 0 0
$$265$$ 4.00000 0.245718
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −14.0000 −0.855186
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ 29.0000 1.76162 0.880812 0.473466i $$-0.156997\pi$$
0.880812 + 0.473466i $$0.156997\pi$$
$$272$$ 8.00000 0.485071
$$273$$ 0 0
$$274$$ −4.00000 −0.241649
$$275$$ −2.00000 −0.120605
$$276$$ 0 0
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ 6.00000 0.359856
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 12.0000 0.715860 0.357930 0.933748i $$-0.383483\pi$$
0.357930 + 0.933748i $$0.383483\pi$$
$$282$$ 0 0
$$283$$ 5.00000 0.297219 0.148610 0.988896i $$-0.452520\pi$$
0.148610 + 0.988896i $$0.452520\pi$$
$$284$$ −24.0000 −1.42414
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −30.0000 −1.77084
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ −8.00000 −0.469776
$$291$$ 0 0
$$292$$ 30.0000 1.75562
$$293$$ −16.0000 −0.934730 −0.467365 0.884064i $$-0.654797\pi$$
−0.467365 + 0.884064i $$0.654797\pi$$
$$294$$ 0 0
$$295$$ −12.0000 −0.698667
$$296$$ 0 0
$$297$$ 0 0
$$298$$ −24.0000 −1.39028
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 5.00000 0.288195
$$302$$ 16.0000 0.920697
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −13.0000 −0.744378
$$306$$ 0 0
$$307$$ −31.0000 −1.76926 −0.884632 0.466290i $$-0.845590\pi$$
−0.884632 + 0.466290i $$0.845590\pi$$
$$308$$ −20.0000 −1.13961
$$309$$ 0 0
$$310$$ 14.0000 0.795147
$$311$$ 22.0000 1.24751 0.623753 0.781622i $$-0.285607\pi$$
0.623753 + 0.781622i $$0.285607\pi$$
$$312$$ 0 0
$$313$$ −31.0000 −1.75222 −0.876112 0.482108i $$-0.839871\pi$$
−0.876112 + 0.482108i $$0.839871\pi$$
$$314$$ 30.0000 1.69300
$$315$$ 0 0
$$316$$ 6.00000 0.337526
$$317$$ −12.0000 −0.673987 −0.336994 0.941507i $$-0.609410\pi$$
−0.336994 + 0.941507i $$0.609410\pi$$
$$318$$ 0 0
$$319$$ −8.00000 −0.447914
$$320$$ −8.00000 −0.447214
$$321$$ 0 0
$$322$$ 60.0000 3.34367
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 30.0000 1.66155
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 40.0000 2.20527
$$330$$ 0 0
$$331$$ 9.00000 0.494685 0.247342 0.968928i $$-0.420443\pi$$
0.247342 + 0.968928i $$0.420443\pi$$
$$332$$ −16.0000 −0.878114
$$333$$ 0 0
$$334$$ 24.0000 1.31322
$$335$$ −7.00000 −0.382451
$$336$$ 0 0
$$337$$ −1.00000 −0.0544735 −0.0272367 0.999629i $$-0.508671\pi$$
−0.0272367 + 0.999629i $$0.508671\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ −4.00000 −0.216930
$$341$$ 14.0000 0.758143
$$342$$ 0 0
$$343$$ 55.0000 2.96972
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 16.0000 0.858925 0.429463 0.903085i $$-0.358703\pi$$
0.429463 + 0.903085i $$0.358703\pi$$
$$348$$ 0 0
$$349$$ 3.00000 0.160586 0.0802932 0.996771i $$-0.474414\pi$$
0.0802932 + 0.996771i $$0.474414\pi$$
$$350$$ −10.0000 −0.534522
$$351$$ 0 0
$$352$$ −16.0000 −0.852803
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ −12.0000 −0.636894
$$356$$ −28.0000 −1.48400
$$357$$ 0 0
$$358$$ −12.0000 −0.634220
$$359$$ −2.00000 −0.105556 −0.0527780 0.998606i $$-0.516808\pi$$
−0.0527780 + 0.998606i $$0.516808\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 44.0000 2.31259
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 15.0000 0.785136
$$366$$ 0 0
$$367$$ 7.00000 0.365397 0.182699 0.983169i $$-0.441517\pi$$
0.182699 + 0.983169i $$0.441517\pi$$
$$368$$ 24.0000 1.25109
$$369$$ 0 0
$$370$$ 4.00000 0.207950
$$371$$ 20.0000 1.03835
$$372$$ 0 0
$$373$$ 13.0000 0.673114 0.336557 0.941663i $$-0.390737\pi$$
0.336557 + 0.941663i $$0.390737\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −5.00000 −0.256833 −0.128416 0.991720i $$-0.540989\pi$$
−0.128416 + 0.991720i $$0.540989\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 24.0000 1.22795
$$383$$ 18.0000 0.919757 0.459879 0.887982i $$-0.347893\pi$$
0.459879 + 0.887982i $$0.347893\pi$$
$$384$$ 0 0
$$385$$ −10.0000 −0.509647
$$386$$ 22.0000 1.11977
$$387$$ 0 0
$$388$$ −10.0000 −0.507673
$$389$$ −8.00000 −0.405616 −0.202808 0.979219i $$-0.565007\pi$$
−0.202808 + 0.979219i $$0.565007\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −24.0000 −1.20910
$$395$$ 3.00000 0.150946
$$396$$ 0 0
$$397$$ −15.0000 −0.752828 −0.376414 0.926451i $$-0.622843\pi$$
−0.376414 + 0.926451i $$0.622843\pi$$
$$398$$ −34.0000 −1.70427
$$399$$ 0 0
$$400$$ −4.00000 −0.200000
$$401$$ −16.0000 −0.799002 −0.399501 0.916733i $$-0.630817\pi$$
−0.399501 + 0.916733i $$0.630817\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 36.0000 1.79107
$$405$$ 0 0
$$406$$ −40.0000 −1.98517
$$407$$ 4.00000 0.198273
$$408$$ 0 0
$$409$$ −15.0000 −0.741702 −0.370851 0.928692i $$-0.620934\pi$$
−0.370851 + 0.928692i $$0.620934\pi$$
$$410$$ 12.0000 0.592638
$$411$$ 0 0
$$412$$ −14.0000 −0.689730
$$413$$ −60.0000 −2.95241
$$414$$ 0 0
$$415$$ −8.00000 −0.392705
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −38.0000 −1.85642 −0.928211 0.372055i $$-0.878653\pi$$
−0.928211 + 0.372055i $$0.878653\pi$$
$$420$$ 0 0
$$421$$ −23.0000 −1.12095 −0.560476 0.828171i $$-0.689382\pi$$
−0.560476 + 0.828171i $$0.689382\pi$$
$$422$$ −30.0000 −1.46038
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −2.00000 −0.0970143
$$426$$ 0 0
$$427$$ −65.0000 −3.14557
$$428$$ −8.00000 −0.386695
$$429$$ 0 0
$$430$$ −2.00000 −0.0964486
$$431$$ 28.0000 1.34871 0.674356 0.738406i $$-0.264421\pi$$
0.674356 + 0.738406i $$0.264421\pi$$
$$432$$ 0 0
$$433$$ 1.00000 0.0480569 0.0240285 0.999711i $$-0.492351\pi$$
0.0240285 + 0.999711i $$0.492351\pi$$
$$434$$ 70.0000 3.36011
$$435$$ 0 0
$$436$$ −22.0000 −1.05361
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 15.0000 0.715911 0.357955 0.933739i $$-0.383474\pi$$
0.357955 + 0.933739i $$0.383474\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −26.0000 −1.23530 −0.617649 0.786454i $$-0.711915\pi$$
−0.617649 + 0.786454i $$0.711915\pi$$
$$444$$ 0 0
$$445$$ −14.0000 −0.663664
$$446$$ 16.0000 0.757622
$$447$$ 0 0
$$448$$ −40.0000 −1.88982
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 0 0
$$451$$ 12.0000 0.565058
$$452$$ 4.00000 0.188144
$$453$$ 0 0
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 35.0000 1.63723 0.818615 0.574342i $$-0.194742\pi$$
0.818615 + 0.574342i $$0.194742\pi$$
$$458$$ −28.0000 −1.30835
$$459$$ 0 0
$$460$$ −12.0000 −0.559503
$$461$$ 2.00000 0.0931493 0.0465746 0.998915i $$-0.485169\pi$$
0.0465746 + 0.998915i $$0.485169\pi$$
$$462$$ 0 0
$$463$$ 3.00000 0.139422 0.0697109 0.997567i $$-0.477792\pi$$
0.0697109 + 0.997567i $$0.477792\pi$$
$$464$$ −16.0000 −0.742781
$$465$$ 0 0
$$466$$ −28.0000 −1.29707
$$467$$ 4.00000 0.185098 0.0925490 0.995708i $$-0.470499\pi$$
0.0925490 + 0.995708i $$0.470499\pi$$
$$468$$ 0 0
$$469$$ −35.0000 −1.61615
$$470$$ −16.0000 −0.738025
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −2.00000 −0.0919601
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −20.0000 −0.916698
$$477$$ 0 0
$$478$$ 24.0000 1.09773
$$479$$ −42.0000 −1.91903 −0.959514 0.281659i $$-0.909115\pi$$
−0.959514 + 0.281659i $$0.909115\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −20.0000 −0.910975
$$483$$ 0 0
$$484$$ −14.0000 −0.636364
$$485$$ −5.00000 −0.227038
$$486$$ 0 0
$$487$$ −28.0000 −1.26880 −0.634401 0.773004i $$-0.718753\pi$$
−0.634401 + 0.773004i $$0.718753\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ −36.0000 −1.62631
$$491$$ 24.0000 1.08310 0.541552 0.840667i $$-0.317837\pi$$
0.541552 + 0.840667i $$0.317837\pi$$
$$492$$ 0 0
$$493$$ −8.00000 −0.360302
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 28.0000 1.25724
$$497$$ −60.0000 −2.69137
$$498$$ 0 0
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 2.00000 0.0894427
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 6.00000 0.267527 0.133763 0.991013i $$-0.457294\pi$$
0.133763 + 0.991013i $$0.457294\pi$$
$$504$$ 0 0
$$505$$ 18.0000 0.800989
$$506$$ −24.0000 −1.06693
$$507$$ 0 0
$$508$$ −22.0000 −0.976092
$$509$$ 30.0000 1.32973 0.664863 0.746965i $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ 0 0
$$511$$ 75.0000 3.31780
$$512$$ −32.0000 −1.41421
$$513$$ 0 0
$$514$$ −44.0000 −1.94076
$$515$$ −7.00000 −0.308457
$$516$$ 0 0
$$517$$ −16.0000 −0.703679
$$518$$ 20.0000 0.878750
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$524$$ 8.00000 0.349482
$$525$$ 0 0
$$526$$ −20.0000 −0.872041
$$527$$ 14.0000 0.609850
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ −8.00000 −0.347498
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −4.00000 −0.172935
$$536$$ 0 0
$$537$$ 0 0
$$538$$ −12.0000 −0.517357
$$539$$ −36.0000 −1.55063
$$540$$ 0 0
$$541$$ 29.0000 1.24681 0.623404 0.781900i $$-0.285749\pi$$
0.623404 + 0.781900i $$0.285749\pi$$
$$542$$ −58.0000 −2.49131
$$543$$ 0 0
$$544$$ −16.0000 −0.685994
$$545$$ −11.0000 −0.471188
$$546$$ 0 0
$$547$$ −9.00000 −0.384812 −0.192406 0.981315i $$-0.561629\pi$$
−0.192406 + 0.981315i $$0.561629\pi$$
$$548$$ 4.00000 0.170872
$$549$$ 0 0
$$550$$ 4.00000 0.170561
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 15.0000 0.637865
$$554$$ −20.0000 −0.849719
$$555$$ 0 0
$$556$$ −6.00000 −0.254457
$$557$$ 20.0000 0.847427 0.423714 0.905796i $$-0.360726\pi$$
0.423714 + 0.905796i $$0.360726\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −20.0000 −0.845154
$$561$$ 0 0
$$562$$ −24.0000 −1.01238
$$563$$ −26.0000 −1.09577 −0.547885 0.836554i $$-0.684567\pi$$
−0.547885 + 0.836554i $$0.684567\pi$$
$$564$$ 0 0
$$565$$ 2.00000 0.0841406
$$566$$ −10.0000 −0.420331
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 20.0000 0.838444 0.419222 0.907884i $$-0.362303\pi$$
0.419222 + 0.907884i $$0.362303\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 60.0000 2.50435
$$575$$ −6.00000 −0.250217
$$576$$ 0 0
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 26.0000 1.08146
$$579$$ 0 0
$$580$$ 8.00000 0.332182
$$581$$ −40.0000 −1.65948
$$582$$ 0 0
$$583$$ −8.00000 −0.331326
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 32.0000 1.32191
$$587$$ −28.0000 −1.15568 −0.577842 0.816149i $$-0.696105\pi$$
−0.577842 + 0.816149i $$0.696105\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 24.0000 0.988064
$$591$$ 0 0
$$592$$ 8.00000 0.328798
$$593$$ 10.0000 0.410651 0.205325 0.978694i $$-0.434175\pi$$
0.205325 + 0.978694i $$0.434175\pi$$
$$594$$ 0 0
$$595$$ −10.0000 −0.409960
$$596$$ 24.0000 0.983078
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 16.0000 0.653742 0.326871 0.945069i $$-0.394006\pi$$
0.326871 + 0.945069i $$0.394006\pi$$
$$600$$ 0 0
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ −10.0000 −0.407570
$$603$$ 0 0
$$604$$ −16.0000 −0.651031
$$605$$ −7.00000 −0.284590
$$606$$ 0 0
$$607$$ 16.0000 0.649420 0.324710 0.945814i $$-0.394733\pi$$
0.324710 + 0.945814i $$0.394733\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 26.0000 1.05271
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −15.0000 −0.605844 −0.302922 0.953015i $$-0.597962\pi$$
−0.302922 + 0.953015i $$0.597962\pi$$
$$614$$ 62.0000 2.50212
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 0 0
$$619$$ 37.0000 1.48716 0.743578 0.668649i $$-0.233127\pi$$
0.743578 + 0.668649i $$0.233127\pi$$
$$620$$ −14.0000 −0.562254
$$621$$ 0 0
$$622$$ −44.0000 −1.76424
$$623$$ −70.0000 −2.80449
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 62.0000 2.47802
$$627$$ 0 0
$$628$$ −30.0000 −1.19713
$$629$$ 4.00000 0.159490
$$630$$ 0 0
$$631$$ 7.00000 0.278666 0.139333 0.990246i $$-0.455504\pi$$
0.139333 + 0.990246i $$0.455504\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 24.0000 0.953162
$$635$$ −11.0000 −0.436522
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 16.0000 0.633446
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 0 0
$$643$$ −19.0000 −0.749287 −0.374643 0.927169i $$-0.622235\pi$$
−0.374643 + 0.927169i $$0.622235\pi$$
$$644$$ −60.0000 −2.36433
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −38.0000 −1.49393 −0.746967 0.664861i $$-0.768491\pi$$
−0.746967 + 0.664861i $$0.768491\pi$$
$$648$$ 0 0
$$649$$ 24.0000 0.942082
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −30.0000 −1.17489
$$653$$ −42.0000 −1.64359 −0.821794 0.569785i $$-0.807026\pi$$
−0.821794 + 0.569785i $$0.807026\pi$$
$$654$$ 0 0
$$655$$ 4.00000 0.156293
$$656$$ 24.0000 0.937043
$$657$$ 0 0
$$658$$ −80.0000 −3.11872
$$659$$ 24.0000 0.934907 0.467454 0.884018i $$-0.345171\pi$$
0.467454 + 0.884018i $$0.345171\pi$$
$$660$$ 0 0
$$661$$ 35.0000 1.36134 0.680671 0.732589i $$-0.261688\pi$$
0.680671 + 0.732589i $$0.261688\pi$$
$$662$$ −18.0000 −0.699590
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −24.0000 −0.929284
$$668$$ −24.0000 −0.928588
$$669$$ 0 0
$$670$$ 14.0000 0.540867
$$671$$ 26.0000 1.00372
$$672$$ 0 0
$$673$$ 33.0000 1.27206 0.636028 0.771666i $$-0.280576\pi$$
0.636028 + 0.771666i $$0.280576\pi$$
$$674$$ 2.00000 0.0770371
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 12.0000 0.461197 0.230599 0.973049i $$-0.425932\pi$$
0.230599 + 0.973049i $$0.425932\pi$$
$$678$$ 0 0
$$679$$ −25.0000 −0.959412
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −28.0000 −1.07218
$$683$$ 20.0000 0.765279 0.382639 0.923898i $$-0.375015\pi$$
0.382639 + 0.923898i $$0.375015\pi$$
$$684$$ 0 0
$$685$$ 2.00000 0.0764161
$$686$$ −110.000 −4.19982
$$687$$ 0 0
$$688$$ −4.00000 −0.152499
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 37.0000 1.40755 0.703773 0.710425i $$-0.251497\pi$$
0.703773 + 0.710425i $$0.251497\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ −32.0000 −1.21470
$$695$$ −3.00000 −0.113796
$$696$$ 0 0
$$697$$ 12.0000 0.454532
$$698$$ −6.00000 −0.227103
$$699$$ 0 0
$$700$$ 10.0000 0.377964
$$701$$ −40.0000 −1.51078 −0.755390 0.655276i $$-0.772552\pi$$
−0.755390 + 0.655276i $$0.772552\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 16.0000 0.603023
$$705$$ 0 0
$$706$$ 12.0000 0.451626
$$707$$ 90.0000 3.38480
$$708$$ 0 0
$$709$$ −23.0000 −0.863783 −0.431892 0.901926i $$-0.642154\pi$$
−0.431892 + 0.901926i $$0.642154\pi$$
$$710$$ 24.0000 0.900704
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 42.0000 1.57291
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 0 0
$$718$$ 4.00000 0.149279
$$719$$ −40.0000 −1.49175 −0.745874 0.666087i $$-0.767968\pi$$
−0.745874 + 0.666087i $$0.767968\pi$$
$$720$$ 0 0
$$721$$ −35.0000 −1.30347
$$722$$ 38.0000 1.41421
$$723$$ 0 0
$$724$$ −44.0000 −1.63525
$$725$$ 4.00000 0.148556
$$726$$ 0 0
$$727$$ 9.00000 0.333792 0.166896 0.985975i $$-0.446626\pi$$
0.166896 + 0.985975i $$0.446626\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −30.0000 −1.11035
$$731$$ −2.00000 −0.0739727
$$732$$ 0 0
$$733$$ 7.00000 0.258551 0.129275 0.991609i $$-0.458735\pi$$
0.129275 + 0.991609i $$0.458735\pi$$
$$734$$ −14.0000 −0.516749
$$735$$ 0 0
$$736$$ −48.0000 −1.76930
$$737$$ 14.0000 0.515697
$$738$$ 0 0
$$739$$ −36.0000 −1.32428 −0.662141 0.749380i $$-0.730352\pi$$
−0.662141 + 0.749380i $$0.730352\pi$$
$$740$$ −4.00000 −0.147043
$$741$$ 0 0
$$742$$ −40.0000 −1.46845
$$743$$ −24.0000 −0.880475 −0.440237 0.897881i $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ 0 0
$$745$$ 12.0000 0.439646
$$746$$ −26.0000 −0.951928
$$747$$ 0 0
$$748$$ 8.00000 0.292509
$$749$$ −20.0000 −0.730784
$$750$$ 0 0
$$751$$ −28.0000 −1.02173 −0.510867 0.859660i $$-0.670676\pi$$
−0.510867 + 0.859660i $$0.670676\pi$$
$$752$$ −32.0000 −1.16692
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −8.00000 −0.291150
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 10.0000 0.363216
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 36.0000 1.30500 0.652499 0.757789i $$-0.273720\pi$$
0.652499 + 0.757789i $$0.273720\pi$$
$$762$$ 0 0
$$763$$ −55.0000 −1.99113
$$764$$ −24.0000 −0.868290
$$765$$ 0 0
$$766$$ −36.0000 −1.30073
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −34.0000 −1.22607 −0.613036 0.790055i $$-0.710052\pi$$
−0.613036 + 0.790055i $$0.710052\pi$$
$$770$$ 20.0000 0.720750
$$771$$ 0 0
$$772$$ −22.0000 −0.791797
$$773$$ −46.0000 −1.65451 −0.827253 0.561830i $$-0.810097\pi$$
−0.827253 + 0.561830i $$0.810097\pi$$
$$774$$ 0 0
$$775$$ −7.00000 −0.251447
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 16.0000 0.573628
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 24.0000 0.858788
$$782$$ −24.0000 −0.858238
$$783$$ 0 0
$$784$$ −72.0000 −2.57143
$$785$$ −15.0000 −0.535373
$$786$$ 0 0
$$787$$ 17.0000 0.605985 0.302992 0.952993i $$-0.402014\pi$$
0.302992 + 0.952993i $$0.402014\pi$$
$$788$$ 24.0000 0.854965
$$789$$ 0 0
$$790$$ −6.00000 −0.213470
$$791$$ 10.0000 0.355559
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 30.0000 1.06466
$$795$$ 0 0
$$796$$ 34.0000 1.20510
$$797$$ −30.0000 −1.06265 −0.531327 0.847167i $$-0.678307\pi$$
−0.531327 + 0.847167i $$0.678307\pi$$
$$798$$ 0 0
$$799$$ −16.0000 −0.566039
$$800$$ 8.00000 0.282843
$$801$$ 0 0
$$802$$ 32.0000 1.12996
$$803$$ −30.0000 −1.05868
$$804$$ 0 0
$$805$$ −30.0000 −1.05736
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −4.00000 −0.140633 −0.0703163 0.997525i $$-0.522401\pi$$
−0.0703163 + 0.997525i $$0.522401\pi$$
$$810$$ 0 0
$$811$$ 45.0000 1.58016 0.790082 0.613001i $$-0.210038\pi$$
0.790082 + 0.613001i $$0.210038\pi$$
$$812$$ 40.0000 1.40372
$$813$$ 0 0
$$814$$ −8.00000 −0.280400
$$815$$ −15.0000 −0.525427
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 30.0000 1.04893
$$819$$ 0 0
$$820$$ −12.0000 −0.419058
$$821$$ 22.0000 0.767805 0.383903 0.923374i $$-0.374580\pi$$
0.383903 + 0.923374i $$0.374580\pi$$
$$822$$ 0 0
$$823$$ 20.0000 0.697156 0.348578 0.937280i $$-0.386665\pi$$
0.348578 + 0.937280i $$0.386665\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 120.000 4.17533
$$827$$ −46.0000 −1.59958 −0.799788 0.600282i $$-0.795055\pi$$
−0.799788 + 0.600282i $$0.795055\pi$$
$$828$$ 0 0
$$829$$ −11.0000 −0.382046 −0.191023 0.981586i $$-0.561180\pi$$
−0.191023 + 0.981586i $$0.561180\pi$$
$$830$$ 16.0000 0.555368
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −36.0000 −1.24733
$$834$$ 0 0
$$835$$ −12.0000 −0.415277
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 76.0000 2.62538
$$839$$ 34.0000 1.17381 0.586905 0.809656i $$-0.300346\pi$$
0.586905 + 0.809656i $$0.300346\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 46.0000 1.58526
$$843$$ 0 0
$$844$$ 30.0000 1.03264
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −35.0000 −1.20261
$$848$$ −16.0000 −0.549442
$$849$$ 0 0
$$850$$ 4.00000 0.137199
$$851$$ 12.0000 0.411355
$$852$$ 0 0
$$853$$ 9.00000 0.308154 0.154077 0.988059i $$-0.450760\pi$$
0.154077 + 0.988059i $$0.450760\pi$$
$$854$$ 130.000 4.44851
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 12.0000 0.409912 0.204956 0.978771i $$-0.434295\pi$$
0.204956 + 0.978771i $$0.434295\pi$$
$$858$$ 0 0
$$859$$ 43.0000 1.46714 0.733571 0.679613i $$-0.237852\pi$$
0.733571 + 0.679613i $$0.237852\pi$$
$$860$$ 2.00000 0.0681994
$$861$$ 0 0
$$862$$ −56.0000 −1.90737
$$863$$ −6.00000 −0.204242 −0.102121 0.994772i $$-0.532563\pi$$
−0.102121 + 0.994772i $$0.532563\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −2.00000 −0.0679628
$$867$$ 0 0
$$868$$ −70.0000 −2.37595
$$869$$ −6.00000 −0.203536
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 5.00000 0.169031
$$876$$ 0 0
$$877$$ 6.00000 0.202606 0.101303 0.994856i $$-0.467699\pi$$
0.101303 + 0.994856i $$0.467699\pi$$
$$878$$ −30.0000 −1.01245
$$879$$ 0 0
$$880$$ 8.00000 0.269680
$$881$$ −20.0000 −0.673817 −0.336909 0.941537i $$-0.609381\pi$$
−0.336909 + 0.941537i $$0.609381\pi$$
$$882$$ 0 0
$$883$$ −25.0000 −0.841317 −0.420658 0.907219i $$-0.638201\pi$$
−0.420658 + 0.907219i $$0.638201\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 52.0000 1.74697
$$887$$ 44.0000 1.47738 0.738688 0.674048i $$-0.235446\pi$$
0.738688 + 0.674048i $$0.235446\pi$$
$$888$$ 0 0
$$889$$ −55.0000 −1.84464
$$890$$ 28.0000 0.938562
$$891$$ 0 0
$$892$$ −16.0000 −0.535720
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 6.00000 0.200558
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 36.0000 1.20134
$$899$$ −28.0000 −0.933852
$$900$$ 0 0
$$901$$ −8.00000 −0.266519
$$902$$ −24.0000 −0.799113
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −22.0000 −0.731305
$$906$$ 0 0
$$907$$ 20.0000 0.664089 0.332045 0.943264i $$-0.392262\pi$$
0.332045 + 0.943264i $$0.392262\pi$$
$$908$$ −20.0000 −0.663723
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ 16.0000 0.529523
$$914$$ −70.0000 −2.31539
$$915$$ 0 0
$$916$$ 28.0000 0.925146
$$917$$ 20.0000 0.660458
$$918$$ 0 0
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −4.00000 −0.131733
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −2.00000 −0.0657596
$$926$$ −6.00000 −0.197172
$$927$$ 0 0
$$928$$ 32.0000 1.05045
$$929$$ −52.0000 −1.70606 −0.853032 0.521858i $$-0.825239\pi$$
−0.853032 + 0.521858i $$0.825239\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 28.0000 0.917170
$$933$$ 0 0
$$934$$ −8.00000 −0.261768
$$935$$ 4.00000 0.130814
$$936$$ 0 0
$$937$$ −30.0000 −0.980057 −0.490029 0.871706i $$-0.663014\pi$$
−0.490029 + 0.871706i $$0.663014\pi$$
$$938$$ 70.0000 2.28558
$$939$$ 0 0
$$940$$ 16.0000 0.521862
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ 0 0
$$943$$ 36.0000 1.17232
$$944$$ 48.0000 1.56227
$$945$$ 0 0
$$946$$ 4.00000 0.130051
$$947$$ −18.0000 −0.584921 −0.292461 0.956278i $$-0.594474\pi$$
−0.292461 + 0.956278i $$0.594474\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −6.00000 −0.194359 −0.0971795 0.995267i $$-0.530982\pi$$
−0.0971795 + 0.995267i $$0.530982\pi$$
$$954$$ 0 0
$$955$$ −12.0000 −0.388311
$$956$$ −24.0000 −0.776215
$$957$$ 0 0
$$958$$ 84.0000 2.71392
$$959$$ 10.0000 0.322917
$$960$$ 0 0
$$961$$ 18.0000 0.580645
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 20.0000 0.644157
$$965$$ −11.0000 −0.354103
$$966$$ 0 0
$$967$$ −56.0000 −1.80084 −0.900419 0.435023i $$-0.856740\pi$$
−0.900419 + 0.435023i $$0.856740\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 10.0000 0.321081
$$971$$ −20.0000 −0.641831 −0.320915 0.947108i $$-0.603990\pi$$
−0.320915 + 0.947108i $$0.603990\pi$$
$$972$$ 0 0
$$973$$ −15.0000 −0.480878
$$974$$ 56.0000 1.79436
$$975$$ 0 0
$$976$$ 52.0000 1.66448
$$977$$ −60.0000 −1.91957 −0.959785 0.280736i $$-0.909421\pi$$
−0.959785 + 0.280736i $$0.909421\pi$$
$$978$$ 0 0
$$979$$ 28.0000 0.894884
$$980$$ 36.0000 1.14998
$$981$$ 0 0
$$982$$ −48.0000 −1.53174
$$983$$ −38.0000 −1.21201 −0.606006 0.795460i $$-0.707229\pi$$
−0.606006 + 0.795460i $$0.707229\pi$$
$$984$$ 0 0
$$985$$ 12.0000 0.382352
$$986$$ 16.0000 0.509544
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −6.00000 −0.190789
$$990$$ 0 0
$$991$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$992$$ −56.0000 −1.77800
$$993$$ 0 0
$$994$$ 120.000 3.80617
$$995$$ 17.0000 0.538936
$$996$$ 0 0
$$997$$ 29.0000 0.918439 0.459220 0.888323i $$-0.348129\pi$$
0.459220 + 0.888323i $$0.348129\pi$$
$$998$$ −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 7605.2.a.a.1.1 1
3.2 odd 2 2535.2.a.m.1.1 1
13.3 even 3 585.2.j.b.451.1 2
13.9 even 3 585.2.j.b.406.1 2
13.12 even 2 7605.2.a.s.1.1 1
39.29 odd 6 195.2.i.a.61.1 yes 2
39.35 odd 6 195.2.i.a.16.1 2
39.38 odd 2 2535.2.a.c.1.1 1
195.29 odd 6 975.2.i.i.451.1 2
195.68 even 12 975.2.bb.f.724.2 4
195.74 odd 6 975.2.i.i.601.1 2
195.107 even 12 975.2.bb.f.724.1 4
195.113 even 12 975.2.bb.f.874.1 4
195.152 even 12 975.2.bb.f.874.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.i.a.16.1 2 39.35 odd 6
195.2.i.a.61.1 yes 2 39.29 odd 6
585.2.j.b.406.1 2 13.9 even 3
585.2.j.b.451.1 2 13.3 even 3
975.2.i.i.451.1 2 195.29 odd 6
975.2.i.i.601.1 2 195.74 odd 6
975.2.bb.f.724.1 4 195.107 even 12
975.2.bb.f.724.2 4 195.68 even 12
975.2.bb.f.874.1 4 195.113 even 12
975.2.bb.f.874.2 4 195.152 even 12
2535.2.a.c.1.1 1 39.38 odd 2
2535.2.a.m.1.1 1 3.2 odd 2
7605.2.a.a.1.1 1 1.1 even 1 trivial
7605.2.a.s.1.1 1 13.12 even 2