# Properties

 Label 507.2.a.e.1.1 Level $507$ Weight $2$ Character 507.1 Self dual yes Analytic conductor $4.048$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,2,Mod(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("507.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.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: $$4.04841538248$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 507.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -2.00000 q^{4} -3.46410 q^{5} -1.73205 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -2.00000 q^{4} -3.46410 q^{5} -1.73205 q^{7} +1.00000 q^{9} +3.46410 q^{11} +2.00000 q^{12} +3.46410 q^{15} +4.00000 q^{16} -3.46410 q^{19} +6.92820 q^{20} +1.73205 q^{21} +6.00000 q^{23} +7.00000 q^{25} -1.00000 q^{27} +3.46410 q^{28} +6.00000 q^{29} -1.73205 q^{31} -3.46410 q^{33} +6.00000 q^{35} -2.00000 q^{36} -6.92820 q^{41} +1.00000 q^{43} -6.92820 q^{44} -3.46410 q^{45} +3.46410 q^{47} -4.00000 q^{48} -4.00000 q^{49} +12.0000 q^{53} -12.0000 q^{55} +3.46410 q^{57} -3.46410 q^{59} -6.92820 q^{60} +1.00000 q^{61} -1.73205 q^{63} -8.00000 q^{64} +8.66025 q^{67} -6.00000 q^{69} +10.3923 q^{71} +1.73205 q^{73} -7.00000 q^{75} +6.92820 q^{76} -6.00000 q^{77} -11.0000 q^{79} -13.8564 q^{80} +1.00000 q^{81} +13.8564 q^{83} -3.46410 q^{84} -6.00000 q^{87} -6.92820 q^{89} -12.0000 q^{92} +1.73205 q^{93} +12.0000 q^{95} +5.19615 q^{97} +3.46410 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 4 q^{4} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 4 * q^4 + 2 * q^9 $$2 q - 2 q^{3} - 4 q^{4} + 2 q^{9} + 4 q^{12} + 8 q^{16} + 12 q^{23} + 14 q^{25} - 2 q^{27} + 12 q^{29} + 12 q^{35} - 4 q^{36} + 2 q^{43} - 8 q^{48} - 8 q^{49} + 24 q^{53} - 24 q^{55} + 2 q^{61} - 16 q^{64} - 12 q^{69} - 14 q^{75} - 12 q^{77} - 22 q^{79} + 2 q^{81} - 12 q^{87} - 24 q^{92} + 24 q^{95}+O(q^{100})$$ 2 * q - 2 * q^3 - 4 * q^4 + 2 * q^9 + 4 * q^12 + 8 * q^16 + 12 * q^23 + 14 * q^25 - 2 * q^27 + 12 * q^29 + 12 * q^35 - 4 * q^36 + 2 * q^43 - 8 * q^48 - 8 * q^49 + 24 * q^53 - 24 * q^55 + 2 * q^61 - 16 * q^64 - 12 * q^69 - 14 * q^75 - 12 * q^77 - 22 * q^79 + 2 * q^81 - 12 * q^87 - 24 * q^92 + 24 * q^95

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ −2.00000 −1.00000
$$5$$ −3.46410 −1.54919 −0.774597 0.632456i $$-0.782047\pi$$
−0.774597 + 0.632456i $$0.782047\pi$$
$$6$$ 0 0
$$7$$ −1.73205 −0.654654 −0.327327 0.944911i $$-0.606148\pi$$
−0.327327 + 0.944911i $$0.606148\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 3.46410 1.04447 0.522233 0.852803i $$-0.325099\pi$$
0.522233 + 0.852803i $$0.325099\pi$$
$$12$$ 2.00000 0.577350
$$13$$ 0 0
$$14$$ 0 0
$$15$$ 3.46410 0.894427
$$16$$ 4.00000 1.00000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ −3.46410 −0.794719 −0.397360 0.917663i $$-0.630073\pi$$
−0.397360 + 0.917663i $$0.630073\pi$$
$$20$$ 6.92820 1.54919
$$21$$ 1.73205 0.377964
$$22$$ 0 0
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 0 0
$$25$$ 7.00000 1.40000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 3.46410 0.654654
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ −1.73205 −0.311086 −0.155543 0.987829i $$-0.549713\pi$$
−0.155543 + 0.987829i $$0.549713\pi$$
$$32$$ 0 0
$$33$$ −3.46410 −0.603023
$$34$$ 0 0
$$35$$ 6.00000 1.01419
$$36$$ −2.00000 −0.333333
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −6.92820 −1.08200 −0.541002 0.841021i $$-0.681955\pi$$
−0.541002 + 0.841021i $$0.681955\pi$$
$$42$$ 0 0
$$43$$ 1.00000 0.152499 0.0762493 0.997089i $$-0.475706\pi$$
0.0762493 + 0.997089i $$0.475706\pi$$
$$44$$ −6.92820 −1.04447
$$45$$ −3.46410 −0.516398
$$46$$ 0 0
$$47$$ 3.46410 0.505291 0.252646 0.967559i $$-0.418699\pi$$
0.252646 + 0.967559i $$0.418699\pi$$
$$48$$ −4.00000 −0.577350
$$49$$ −4.00000 −0.571429
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 12.0000 1.64833 0.824163 0.566352i $$-0.191646\pi$$
0.824163 + 0.566352i $$0.191646\pi$$
$$54$$ 0 0
$$55$$ −12.0000 −1.61808
$$56$$ 0 0
$$57$$ 3.46410 0.458831
$$58$$ 0 0
$$59$$ −3.46410 −0.450988 −0.225494 0.974245i $$-0.572400\pi$$
−0.225494 + 0.974245i $$0.572400\pi$$
$$60$$ −6.92820 −0.894427
$$61$$ 1.00000 0.128037 0.0640184 0.997949i $$-0.479608\pi$$
0.0640184 + 0.997949i $$0.479608\pi$$
$$62$$ 0 0
$$63$$ −1.73205 −0.218218
$$64$$ −8.00000 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 8.66025 1.05802 0.529009 0.848616i $$-0.322564\pi$$
0.529009 + 0.848616i $$0.322564\pi$$
$$68$$ 0 0
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ 10.3923 1.23334 0.616670 0.787222i $$-0.288481\pi$$
0.616670 + 0.787222i $$0.288481\pi$$
$$72$$ 0 0
$$73$$ 1.73205 0.202721 0.101361 0.994850i $$-0.467680\pi$$
0.101361 + 0.994850i $$0.467680\pi$$
$$74$$ 0 0
$$75$$ −7.00000 −0.808290
$$76$$ 6.92820 0.794719
$$77$$ −6.00000 −0.683763
$$78$$ 0 0
$$79$$ −11.0000 −1.23760 −0.618798 0.785550i $$-0.712380\pi$$
−0.618798 + 0.785550i $$0.712380\pi$$
$$80$$ −13.8564 −1.54919
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 13.8564 1.52094 0.760469 0.649374i $$-0.224969\pi$$
0.760469 + 0.649374i $$0.224969\pi$$
$$84$$ −3.46410 −0.377964
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −6.00000 −0.643268
$$88$$ 0 0
$$89$$ −6.92820 −0.734388 −0.367194 0.930144i $$-0.619682\pi$$
−0.367194 + 0.930144i $$0.619682\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −12.0000 −1.25109
$$93$$ 1.73205 0.179605
$$94$$ 0 0
$$95$$ 12.0000 1.23117
$$96$$ 0 0
$$97$$ 5.19615 0.527589 0.263795 0.964579i $$-0.415026\pi$$
0.263795 + 0.964579i $$0.415026\pi$$
$$98$$ 0 0
$$99$$ 3.46410 0.348155
$$100$$ −14.0000 −1.40000
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ 0 0
$$103$$ 1.00000 0.0985329 0.0492665 0.998786i $$-0.484312\pi$$
0.0492665 + 0.998786i $$0.484312\pi$$
$$104$$ 0 0
$$105$$ −6.00000 −0.585540
$$106$$ 0 0
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ 2.00000 0.192450
$$109$$ 15.5885 1.49310 0.746552 0.665327i $$-0.231708\pi$$
0.746552 + 0.665327i $$0.231708\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −6.92820 −0.654654
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ −20.7846 −1.93817
$$116$$ −12.0000 −1.11417
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 6.92820 0.624695
$$124$$ 3.46410 0.311086
$$125$$ −6.92820 −0.619677
$$126$$ 0 0
$$127$$ −13.0000 −1.15356 −0.576782 0.816898i $$-0.695692\pi$$
−0.576782 + 0.816898i $$0.695692\pi$$
$$128$$ 0 0
$$129$$ −1.00000 −0.0880451
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 6.92820 0.603023
$$133$$ 6.00000 0.520266
$$134$$ 0 0
$$135$$ 3.46410 0.298142
$$136$$ 0 0
$$137$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$138$$ 0 0
$$139$$ 5.00000 0.424094 0.212047 0.977259i $$-0.431987\pi$$
0.212047 + 0.977259i $$0.431987\pi$$
$$140$$ −12.0000 −1.01419
$$141$$ −3.46410 −0.291730
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 4.00000 0.333333
$$145$$ −20.7846 −1.72607
$$146$$ 0 0
$$147$$ 4.00000 0.329914
$$148$$ 0 0
$$149$$ 6.92820 0.567581 0.283790 0.958886i $$-0.408408\pi$$
0.283790 + 0.958886i $$0.408408\pi$$
$$150$$ 0 0
$$151$$ −3.46410 −0.281905 −0.140952 0.990016i $$-0.545016\pi$$
−0.140952 + 0.990016i $$0.545016\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 6.00000 0.481932
$$156$$ 0 0
$$157$$ 11.0000 0.877896 0.438948 0.898513i $$-0.355351\pi$$
0.438948 + 0.898513i $$0.355351\pi$$
$$158$$ 0 0
$$159$$ −12.0000 −0.951662
$$160$$ 0 0
$$161$$ −10.3923 −0.819028
$$162$$ 0 0
$$163$$ −19.0526 −1.49231 −0.746156 0.665771i $$-0.768103\pi$$
−0.746156 + 0.665771i $$0.768103\pi$$
$$164$$ 13.8564 1.08200
$$165$$ 12.0000 0.934199
$$166$$ 0 0
$$167$$ 6.92820 0.536120 0.268060 0.963402i $$-0.413617\pi$$
0.268060 + 0.963402i $$0.413617\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ −3.46410 −0.264906
$$172$$ −2.00000 −0.152499
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ −12.1244 −0.916515
$$176$$ 13.8564 1.04447
$$177$$ 3.46410 0.260378
$$178$$ 0 0
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 6.92820 0.516398
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 0 0
$$183$$ −1.00000 −0.0739221
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −6.92820 −0.505291
$$189$$ 1.73205 0.125988
$$190$$ 0 0
$$191$$ −18.0000 −1.30243 −0.651217 0.758891i $$-0.725741\pi$$
−0.651217 + 0.758891i $$0.725741\pi$$
$$192$$ 8.00000 0.577350
$$193$$ 15.5885 1.12208 0.561041 0.827788i $$-0.310401\pi$$
0.561041 + 0.827788i $$0.310401\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 8.00000 0.571429
$$197$$ 13.8564 0.987228 0.493614 0.869681i $$-0.335676\pi$$
0.493614 + 0.869681i $$0.335676\pi$$
$$198$$ 0 0
$$199$$ −7.00000 −0.496217 −0.248108 0.968732i $$-0.579809\pi$$
−0.248108 + 0.968732i $$0.579809\pi$$
$$200$$ 0 0
$$201$$ −8.66025 −0.610847
$$202$$ 0 0
$$203$$ −10.3923 −0.729397
$$204$$ 0 0
$$205$$ 24.0000 1.67623
$$206$$ 0 0
$$207$$ 6.00000 0.417029
$$208$$ 0 0
$$209$$ −12.0000 −0.830057
$$210$$ 0 0
$$211$$ 13.0000 0.894957 0.447478 0.894295i $$-0.352322\pi$$
0.447478 + 0.894295i $$0.352322\pi$$
$$212$$ −24.0000 −1.64833
$$213$$ −10.3923 −0.712069
$$214$$ 0 0
$$215$$ −3.46410 −0.236250
$$216$$ 0 0
$$217$$ 3.00000 0.203653
$$218$$ 0 0
$$219$$ −1.73205 −0.117041
$$220$$ 24.0000 1.61808
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 17.3205 1.15987 0.579934 0.814664i $$-0.303079\pi$$
0.579934 + 0.814664i $$0.303079\pi$$
$$224$$ 0 0
$$225$$ 7.00000 0.466667
$$226$$ 0 0
$$227$$ −20.7846 −1.37952 −0.689761 0.724037i $$-0.742285\pi$$
−0.689761 + 0.724037i $$0.742285\pi$$
$$228$$ −6.92820 −0.458831
$$229$$ 27.7128 1.83131 0.915657 0.401960i $$-0.131671\pi$$
0.915657 + 0.401960i $$0.131671\pi$$
$$230$$ 0 0
$$231$$ 6.00000 0.394771
$$232$$ 0 0
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ 0 0
$$235$$ −12.0000 −0.782794
$$236$$ 6.92820 0.450988
$$237$$ 11.0000 0.714527
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 13.8564 0.894427
$$241$$ 20.7846 1.33885 0.669427 0.742878i $$-0.266540\pi$$
0.669427 + 0.742878i $$0.266540\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ −2.00000 −0.128037
$$245$$ 13.8564 0.885253
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −13.8564 −0.878114
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 3.46410 0.218218
$$253$$ 20.7846 1.30672
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 0 0
$$263$$ 12.0000 0.739952 0.369976 0.929041i $$-0.379366\pi$$
0.369976 + 0.929041i $$0.379366\pi$$
$$264$$ 0 0
$$265$$ −41.5692 −2.55358
$$266$$ 0 0
$$267$$ 6.92820 0.423999
$$268$$ −17.3205 −1.05802
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 5.19615 0.315644 0.157822 0.987468i $$-0.449553\pi$$
0.157822 + 0.987468i $$0.449553\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 24.2487 1.46225
$$276$$ 12.0000 0.722315
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ 0 0
$$279$$ −1.73205 −0.103695
$$280$$ 0 0
$$281$$ −24.2487 −1.44656 −0.723278 0.690557i $$-0.757366\pi$$
−0.723278 + 0.690557i $$0.757366\pi$$
$$282$$ 0 0
$$283$$ 11.0000 0.653882 0.326941 0.945045i $$-0.393982\pi$$
0.326941 + 0.945045i $$0.393982\pi$$
$$284$$ −20.7846 −1.23334
$$285$$ −12.0000 −0.710819
$$286$$ 0 0
$$287$$ 12.0000 0.708338
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ −5.19615 −0.304604
$$292$$ −3.46410 −0.202721
$$293$$ −17.3205 −1.01187 −0.505937 0.862570i $$-0.668853\pi$$
−0.505937 + 0.862570i $$0.668853\pi$$
$$294$$ 0 0
$$295$$ 12.0000 0.698667
$$296$$ 0 0
$$297$$ −3.46410 −0.201008
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 14.0000 0.808290
$$301$$ −1.73205 −0.0998337
$$302$$ 0 0
$$303$$ −18.0000 −1.03407
$$304$$ −13.8564 −0.794719
$$305$$ −3.46410 −0.198354
$$306$$ 0 0
$$307$$ −1.73205 −0.0988534 −0.0494267 0.998778i $$-0.515739\pi$$
−0.0494267 + 0.998778i $$0.515739\pi$$
$$308$$ 12.0000 0.683763
$$309$$ −1.00000 −0.0568880
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ 13.0000 0.734803 0.367402 0.930062i $$-0.380247\pi$$
0.367402 + 0.930062i $$0.380247\pi$$
$$314$$ 0 0
$$315$$ 6.00000 0.338062
$$316$$ 22.0000 1.23760
$$317$$ −6.92820 −0.389127 −0.194563 0.980890i $$-0.562329\pi$$
−0.194563 + 0.980890i $$0.562329\pi$$
$$318$$ 0 0
$$319$$ 20.7846 1.16371
$$320$$ 27.7128 1.54919
$$321$$ 6.00000 0.334887
$$322$$ 0 0
$$323$$ 0 0
$$324$$ −2.00000 −0.111111
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −15.5885 −0.862044
$$328$$ 0 0
$$329$$ −6.00000 −0.330791
$$330$$ 0 0
$$331$$ −5.19615 −0.285606 −0.142803 0.989751i $$-0.545612\pi$$
−0.142803 + 0.989751i $$0.545612\pi$$
$$332$$ −27.7128 −1.52094
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −30.0000 −1.63908
$$336$$ 6.92820 0.377964
$$337$$ 5.00000 0.272367 0.136184 0.990684i $$-0.456516\pi$$
0.136184 + 0.990684i $$0.456516\pi$$
$$338$$ 0 0
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ −6.00000 −0.324918
$$342$$ 0 0
$$343$$ 19.0526 1.02874
$$344$$ 0 0
$$345$$ 20.7846 1.11901
$$346$$ 0 0
$$347$$ −24.0000 −1.28839 −0.644194 0.764862i $$-0.722807\pi$$
−0.644194 + 0.764862i $$0.722807\pi$$
$$348$$ 12.0000 0.643268
$$349$$ 19.0526 1.01986 0.509930 0.860216i $$-0.329671\pi$$
0.509930 + 0.860216i $$0.329671\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 10.3923 0.553127 0.276563 0.960996i $$-0.410804\pi$$
0.276563 + 0.960996i $$0.410804\pi$$
$$354$$ 0 0
$$355$$ −36.0000 −1.91068
$$356$$ 13.8564 0.734388
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 6.92820 0.365657 0.182828 0.983145i $$-0.441475\pi$$
0.182828 + 0.983145i $$0.441475\pi$$
$$360$$ 0 0
$$361$$ −7.00000 −0.368421
$$362$$ 0 0
$$363$$ −1.00000 −0.0524864
$$364$$ 0 0
$$365$$ −6.00000 −0.314054
$$366$$ 0 0
$$367$$ 17.0000 0.887393 0.443696 0.896177i $$-0.353667\pi$$
0.443696 + 0.896177i $$0.353667\pi$$
$$368$$ 24.0000 1.25109
$$369$$ −6.92820 −0.360668
$$370$$ 0 0
$$371$$ −20.7846 −1.07908
$$372$$ −3.46410 −0.179605
$$373$$ −11.0000 −0.569558 −0.284779 0.958593i $$-0.591920\pi$$
−0.284779 + 0.958593i $$0.591920\pi$$
$$374$$ 0 0
$$375$$ 6.92820 0.357771
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −22.5167 −1.15660 −0.578302 0.815823i $$-0.696284\pi$$
−0.578302 + 0.815823i $$0.696284\pi$$
$$380$$ −24.0000 −1.23117
$$381$$ 13.0000 0.666010
$$382$$ 0 0
$$383$$ −27.7128 −1.41606 −0.708029 0.706183i $$-0.750416\pi$$
−0.708029 + 0.706183i $$0.750416\pi$$
$$384$$ 0 0
$$385$$ 20.7846 1.05928
$$386$$ 0 0
$$387$$ 1.00000 0.0508329
$$388$$ −10.3923 −0.527589
$$389$$ −12.0000 −0.608424 −0.304212 0.952604i $$-0.598393\pi$$
−0.304212 + 0.952604i $$0.598393\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −6.00000 −0.302660
$$394$$ 0 0
$$395$$ 38.1051 1.91728
$$396$$ −6.92820 −0.348155
$$397$$ 15.5885 0.782362 0.391181 0.920314i $$-0.372067\pi$$
0.391181 + 0.920314i $$0.372067\pi$$
$$398$$ 0 0
$$399$$ −6.00000 −0.300376
$$400$$ 28.0000 1.40000
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −36.0000 −1.79107
$$405$$ −3.46410 −0.172133
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −8.66025 −0.428222 −0.214111 0.976809i $$-0.568685\pi$$
−0.214111 + 0.976809i $$0.568685\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −2.00000 −0.0985329
$$413$$ 6.00000 0.295241
$$414$$ 0 0
$$415$$ −48.0000 −2.35623
$$416$$ 0 0
$$417$$ −5.00000 −0.244851
$$418$$ 0 0
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 12.0000 0.585540
$$421$$ 12.1244 0.590905 0.295452 0.955357i $$-0.404530\pi$$
0.295452 + 0.955357i $$0.404530\pi$$
$$422$$ 0 0
$$423$$ 3.46410 0.168430
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −1.73205 −0.0838198
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 20.7846 1.00116 0.500580 0.865690i $$-0.333120\pi$$
0.500580 + 0.865690i $$0.333120\pi$$
$$432$$ −4.00000 −0.192450
$$433$$ 23.0000 1.10531 0.552655 0.833410i $$-0.313615\pi$$
0.552655 + 0.833410i $$0.313615\pi$$
$$434$$ 0 0
$$435$$ 20.7846 0.996546
$$436$$ −31.1769 −1.49310
$$437$$ −20.7846 −0.994263
$$438$$ 0 0
$$439$$ −35.0000 −1.67046 −0.835229 0.549902i $$-0.814665\pi$$
−0.835229 + 0.549902i $$0.814665\pi$$
$$440$$ 0 0
$$441$$ −4.00000 −0.190476
$$442$$ 0 0
$$443$$ 24.0000 1.14027 0.570137 0.821549i $$-0.306890\pi$$
0.570137 + 0.821549i $$0.306890\pi$$
$$444$$ 0 0
$$445$$ 24.0000 1.13771
$$446$$ 0 0
$$447$$ −6.92820 −0.327693
$$448$$ 13.8564 0.654654
$$449$$ 38.1051 1.79829 0.899146 0.437649i $$-0.144189\pi$$
0.899146 + 0.437649i $$0.144189\pi$$
$$450$$ 0 0
$$451$$ −24.0000 −1.13012
$$452$$ −12.0000 −0.564433
$$453$$ 3.46410 0.162758
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −36.3731 −1.70146 −0.850730 0.525603i $$-0.823840\pi$$
−0.850730 + 0.525603i $$0.823840\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 41.5692 1.93817
$$461$$ 41.5692 1.93607 0.968036 0.250812i $$-0.0806976\pi$$
0.968036 + 0.250812i $$0.0806976\pi$$
$$462$$ 0 0
$$463$$ −36.3731 −1.69040 −0.845200 0.534450i $$-0.820519\pi$$
−0.845200 + 0.534450i $$0.820519\pi$$
$$464$$ 24.0000 1.11417
$$465$$ −6.00000 −0.278243
$$466$$ 0 0
$$467$$ −6.00000 −0.277647 −0.138823 0.990317i $$-0.544332\pi$$
−0.138823 + 0.990317i $$0.544332\pi$$
$$468$$ 0 0
$$469$$ −15.0000 −0.692636
$$470$$ 0 0
$$471$$ −11.0000 −0.506853
$$472$$ 0 0
$$473$$ 3.46410 0.159280
$$474$$ 0 0
$$475$$ −24.2487 −1.11261
$$476$$ 0 0
$$477$$ 12.0000 0.549442
$$478$$ 0 0
$$479$$ −34.6410 −1.58279 −0.791394 0.611306i $$-0.790644\pi$$
−0.791394 + 0.611306i $$0.790644\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 10.3923 0.472866
$$484$$ −2.00000 −0.0909091
$$485$$ −18.0000 −0.817338
$$486$$ 0 0
$$487$$ 24.2487 1.09881 0.549407 0.835555i $$-0.314854\pi$$
0.549407 + 0.835555i $$0.314854\pi$$
$$488$$ 0 0
$$489$$ 19.0526 0.861586
$$490$$ 0 0
$$491$$ −6.00000 −0.270776 −0.135388 0.990793i $$-0.543228\pi$$
−0.135388 + 0.990793i $$0.543228\pi$$
$$492$$ −13.8564 −0.624695
$$493$$ 0 0
$$494$$ 0 0
$$495$$ −12.0000 −0.539360
$$496$$ −6.92820 −0.311086
$$497$$ −18.0000 −0.807410
$$498$$ 0 0
$$499$$ −31.1769 −1.39567 −0.697835 0.716258i $$-0.745853\pi$$
−0.697835 + 0.716258i $$0.745853\pi$$
$$500$$ 13.8564 0.619677
$$501$$ −6.92820 −0.309529
$$502$$ 0 0
$$503$$ −30.0000 −1.33763 −0.668817 0.743427i $$-0.733199\pi$$
−0.668817 + 0.743427i $$0.733199\pi$$
$$504$$ 0 0
$$505$$ −62.3538 −2.77471
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 26.0000 1.15356
$$509$$ −17.3205 −0.767718 −0.383859 0.923392i $$-0.625405\pi$$
−0.383859 + 0.923392i $$0.625405\pi$$
$$510$$ 0 0
$$511$$ −3.00000 −0.132712
$$512$$ 0 0
$$513$$ 3.46410 0.152944
$$514$$ 0 0
$$515$$ −3.46410 −0.152647
$$516$$ 2.00000 0.0880451
$$517$$ 12.0000 0.527759
$$518$$ 0 0
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ −24.0000 −1.05146 −0.525730 0.850652i $$-0.676208\pi$$
−0.525730 + 0.850652i $$0.676208\pi$$
$$522$$ 0 0
$$523$$ −28.0000 −1.22435 −0.612177 0.790721i $$-0.709706\pi$$
−0.612177 + 0.790721i $$0.709706\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 12.1244 0.529150
$$526$$ 0 0
$$527$$ 0 0
$$528$$ −13.8564 −0.603023
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ −3.46410 −0.150329
$$532$$ −12.0000 −0.520266
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 20.7846 0.898597
$$536$$ 0 0
$$537$$ 12.0000 0.517838
$$538$$ 0 0
$$539$$ −13.8564 −0.596838
$$540$$ −6.92820 −0.298142
$$541$$ −29.4449 −1.26593 −0.632967 0.774179i $$-0.718163\pi$$
−0.632967 + 0.774179i $$0.718163\pi$$
$$542$$ 0 0
$$543$$ 14.0000 0.600798
$$544$$ 0 0
$$545$$ −54.0000 −2.31311
$$546$$ 0 0
$$547$$ −19.0000 −0.812381 −0.406191 0.913788i $$-0.633143\pi$$
−0.406191 + 0.913788i $$0.633143\pi$$
$$548$$ 0 0
$$549$$ 1.00000 0.0426790
$$550$$ 0 0
$$551$$ −20.7846 −0.885454
$$552$$ 0 0
$$553$$ 19.0526 0.810197
$$554$$ 0 0
$$555$$ 0 0
$$556$$ −10.0000 −0.424094
$$557$$ 27.7128 1.17423 0.587115 0.809504i $$-0.300264\pi$$
0.587115 + 0.809504i $$0.300264\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 24.0000 1.01419
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 42.0000 1.77009 0.885044 0.465506i $$-0.154128\pi$$
0.885044 + 0.465506i $$0.154128\pi$$
$$564$$ 6.92820 0.291730
$$565$$ −20.7846 −0.874415
$$566$$ 0 0
$$567$$ −1.73205 −0.0727393
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 44.0000 1.84134 0.920671 0.390339i $$-0.127642\pi$$
0.920671 + 0.390339i $$0.127642\pi$$
$$572$$ 0 0
$$573$$ 18.0000 0.751961
$$574$$ 0 0
$$575$$ 42.0000 1.75152
$$576$$ −8.00000 −0.333333
$$577$$ 34.6410 1.44212 0.721062 0.692870i $$-0.243654\pi$$
0.721062 + 0.692870i $$0.243654\pi$$
$$578$$ 0 0
$$579$$ −15.5885 −0.647834
$$580$$ 41.5692 1.72607
$$581$$ −24.0000 −0.995688
$$582$$ 0 0
$$583$$ 41.5692 1.72162
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −31.1769 −1.28681 −0.643404 0.765526i $$-0.722479\pi$$
−0.643404 + 0.765526i $$0.722479\pi$$
$$588$$ −8.00000 −0.329914
$$589$$ 6.00000 0.247226
$$590$$ 0 0
$$591$$ −13.8564 −0.569976
$$592$$ 0 0
$$593$$ 3.46410 0.142254 0.0711268 0.997467i $$-0.477341\pi$$
0.0711268 + 0.997467i $$0.477341\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −13.8564 −0.567581
$$597$$ 7.00000 0.286491
$$598$$ 0 0
$$599$$ −30.0000 −1.22577 −0.612883 0.790173i $$-0.709990\pi$$
−0.612883 + 0.790173i $$0.709990\pi$$
$$600$$ 0 0
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ 0 0
$$603$$ 8.66025 0.352673
$$604$$ 6.92820 0.281905
$$605$$ −3.46410 −0.140836
$$606$$ 0 0
$$607$$ 8.00000 0.324710 0.162355 0.986732i $$-0.448091\pi$$
0.162355 + 0.986732i $$0.448091\pi$$
$$608$$ 0 0
$$609$$ 10.3923 0.421117
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 8.66025 0.349784 0.174892 0.984588i $$-0.444042\pi$$
0.174892 + 0.984588i $$0.444042\pi$$
$$614$$ 0 0
$$615$$ −24.0000 −0.967773
$$616$$ 0 0
$$617$$ 10.3923 0.418378 0.209189 0.977875i $$-0.432918\pi$$
0.209189 + 0.977875i $$0.432918\pi$$
$$618$$ 0 0
$$619$$ 25.9808 1.04425 0.522127 0.852867i $$-0.325139\pi$$
0.522127 + 0.852867i $$0.325139\pi$$
$$620$$ −12.0000 −0.481932
$$621$$ −6.00000 −0.240772
$$622$$ 0 0
$$623$$ 12.0000 0.480770
$$624$$ 0 0
$$625$$ −11.0000 −0.440000
$$626$$ 0 0
$$627$$ 12.0000 0.479234
$$628$$ −22.0000 −0.877896
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 1.73205 0.0689519 0.0344759 0.999406i $$-0.489024\pi$$
0.0344759 + 0.999406i $$0.489024\pi$$
$$632$$ 0 0
$$633$$ −13.0000 −0.516704
$$634$$ 0 0
$$635$$ 45.0333 1.78709
$$636$$ 24.0000 0.951662
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 10.3923 0.411113
$$640$$ 0 0
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 0 0
$$643$$ −19.0526 −0.751360 −0.375680 0.926750i $$-0.622591\pi$$
−0.375680 + 0.926750i $$0.622591\pi$$
$$644$$ 20.7846 0.819028
$$645$$ 3.46410 0.136399
$$646$$ 0 0
$$647$$ 18.0000 0.707653 0.353827 0.935311i $$-0.384880\pi$$
0.353827 + 0.935311i $$0.384880\pi$$
$$648$$ 0 0
$$649$$ −12.0000 −0.471041
$$650$$ 0 0
$$651$$ −3.00000 −0.117579
$$652$$ 38.1051 1.49231
$$653$$ 36.0000 1.40879 0.704394 0.709809i $$-0.251219\pi$$
0.704394 + 0.709809i $$0.251219\pi$$
$$654$$ 0 0
$$655$$ −20.7846 −0.812122
$$656$$ −27.7128 −1.08200
$$657$$ 1.73205 0.0675737
$$658$$ 0 0
$$659$$ 48.0000 1.86981 0.934907 0.354892i $$-0.115482\pi$$
0.934907 + 0.354892i $$0.115482\pi$$
$$660$$ −24.0000 −0.934199
$$661$$ 25.9808 1.01053 0.505267 0.862963i $$-0.331394\pi$$
0.505267 + 0.862963i $$0.331394\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −20.7846 −0.805993
$$666$$ 0 0
$$667$$ 36.0000 1.39393
$$668$$ −13.8564 −0.536120
$$669$$ −17.3205 −0.669650
$$670$$ 0 0
$$671$$ 3.46410 0.133730
$$672$$ 0 0
$$673$$ 1.00000 0.0385472 0.0192736 0.999814i $$-0.493865\pi$$
0.0192736 + 0.999814i $$0.493865\pi$$
$$674$$ 0 0
$$675$$ −7.00000 −0.269430
$$676$$ 0 0
$$677$$ 48.0000 1.84479 0.922395 0.386248i $$-0.126229\pi$$
0.922395 + 0.386248i $$0.126229\pi$$
$$678$$ 0 0
$$679$$ −9.00000 −0.345388
$$680$$ 0 0
$$681$$ 20.7846 0.796468
$$682$$ 0 0
$$683$$ 24.2487 0.927851 0.463926 0.885874i $$-0.346441\pi$$
0.463926 + 0.885874i $$0.346441\pi$$
$$684$$ 6.92820 0.264906
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −27.7128 −1.05731
$$688$$ 4.00000 0.152499
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 43.3013 1.64726 0.823629 0.567129i $$-0.191946\pi$$
0.823629 + 0.567129i $$0.191946\pi$$
$$692$$ −12.0000 −0.456172
$$693$$ −6.00000 −0.227921
$$694$$ 0 0
$$695$$ −17.3205 −0.657004
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ −18.0000 −0.680823
$$700$$ 24.2487 0.916515
$$701$$ 12.0000 0.453234 0.226617 0.973984i $$-0.427233\pi$$
0.226617 + 0.973984i $$0.427233\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ −27.7128 −1.04447
$$705$$ 12.0000 0.451946
$$706$$ 0 0
$$707$$ −31.1769 −1.17253
$$708$$ −6.92820 −0.260378
$$709$$ −19.0526 −0.715534 −0.357767 0.933811i $$-0.616462\pi$$
−0.357767 + 0.933811i $$0.616462\pi$$
$$710$$ 0 0
$$711$$ −11.0000 −0.412532
$$712$$ 0 0
$$713$$ −10.3923 −0.389195
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 24.0000 0.896922
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 6.00000 0.223762 0.111881 0.993722i $$-0.464312\pi$$
0.111881 + 0.993722i $$0.464312\pi$$
$$720$$ −13.8564 −0.516398
$$721$$ −1.73205 −0.0645049
$$722$$ 0 0
$$723$$ −20.7846 −0.772988
$$724$$ 28.0000 1.04061
$$725$$ 42.0000 1.55984
$$726$$ 0 0
$$727$$ 35.0000 1.29808 0.649039 0.760755i $$-0.275171\pi$$
0.649039 + 0.760755i $$0.275171\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 2.00000 0.0739221
$$733$$ −39.8372 −1.47142 −0.735710 0.677297i $$-0.763151\pi$$
−0.735710 + 0.677297i $$0.763151\pi$$
$$734$$ 0 0
$$735$$ −13.8564 −0.511101
$$736$$ 0 0
$$737$$ 30.0000 1.10506
$$738$$ 0 0
$$739$$ −45.0333 −1.65658 −0.828289 0.560301i $$-0.810685\pi$$
−0.828289 + 0.560301i $$0.810685\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −10.3923 −0.381257 −0.190628 0.981662i $$-0.561053\pi$$
−0.190628 + 0.981662i $$0.561053\pi$$
$$744$$ 0 0
$$745$$ −24.0000 −0.879292
$$746$$ 0 0
$$747$$ 13.8564 0.506979
$$748$$ 0 0
$$749$$ 10.3923 0.379727
$$750$$ 0 0
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ 13.8564 0.505291
$$753$$ 12.0000 0.437304
$$754$$ 0 0
$$755$$ 12.0000 0.436725
$$756$$ −3.46410 −0.125988
$$757$$ 34.0000 1.23575 0.617876 0.786276i $$-0.287994\pi$$
0.617876 + 0.786276i $$0.287994\pi$$
$$758$$ 0 0
$$759$$ −20.7846 −0.754434
$$760$$ 0 0
$$761$$ −20.7846 −0.753442 −0.376721 0.926327i $$-0.622948\pi$$
−0.376721 + 0.926327i $$0.622948\pi$$
$$762$$ 0 0
$$763$$ −27.0000 −0.977466
$$764$$ 36.0000 1.30243
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ −16.0000 −0.577350
$$769$$ −6.92820 −0.249837 −0.124919 0.992167i $$-0.539867\pi$$
−0.124919 + 0.992167i $$0.539867\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −31.1769 −1.12208
$$773$$ 51.9615 1.86893 0.934463 0.356060i $$-0.115880\pi$$
0.934463 + 0.356060i $$0.115880\pi$$
$$774$$ 0 0
$$775$$ −12.1244 −0.435520
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 24.0000 0.859889
$$780$$ 0 0
$$781$$ 36.0000 1.28818
$$782$$ 0 0
$$783$$ −6.00000 −0.214423
$$784$$ −16.0000 −0.571429
$$785$$ −38.1051 −1.36003
$$786$$ 0 0
$$787$$ 32.9090 1.17308 0.586539 0.809921i $$-0.300490\pi$$
0.586539 + 0.809921i $$0.300490\pi$$
$$788$$ −27.7128 −0.987228
$$789$$ −12.0000 −0.427211
$$790$$ 0 0
$$791$$ −10.3923 −0.369508
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 41.5692 1.47431
$$796$$ 14.0000 0.496217
$$797$$ −12.0000 −0.425062 −0.212531 0.977154i $$-0.568171\pi$$
−0.212531 + 0.977154i $$0.568171\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −6.92820 −0.244796
$$802$$ 0 0
$$803$$ 6.00000 0.211735
$$804$$ 17.3205 0.610847
$$805$$ 36.0000 1.26883
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 12.0000 0.421898 0.210949 0.977497i $$-0.432345\pi$$
0.210949 + 0.977497i $$0.432345\pi$$
$$810$$ 0 0
$$811$$ 25.9808 0.912308 0.456154 0.889901i $$-0.349227\pi$$
0.456154 + 0.889901i $$0.349227\pi$$
$$812$$ 20.7846 0.729397
$$813$$ −5.19615 −0.182237
$$814$$ 0 0
$$815$$ 66.0000 2.31188
$$816$$ 0 0
$$817$$ −3.46410 −0.121194
$$818$$ 0 0
$$819$$ 0 0
$$820$$ −48.0000 −1.67623
$$821$$ 24.2487 0.846286 0.423143 0.906063i $$-0.360927\pi$$
0.423143 + 0.906063i $$0.360927\pi$$
$$822$$ 0 0
$$823$$ −32.0000 −1.11545 −0.557725 0.830026i $$-0.688326\pi$$
−0.557725 + 0.830026i $$0.688326\pi$$
$$824$$ 0 0
$$825$$ −24.2487 −0.844232
$$826$$ 0 0
$$827$$ 48.4974 1.68642 0.843210 0.537584i $$-0.180663\pi$$
0.843210 + 0.537584i $$0.180663\pi$$
$$828$$ −12.0000 −0.417029
$$829$$ −31.0000 −1.07667 −0.538337 0.842729i $$-0.680947\pi$$
−0.538337 + 0.842729i $$0.680947\pi$$
$$830$$ 0 0
$$831$$ −10.0000 −0.346896
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −24.0000 −0.830554
$$836$$ 24.0000 0.830057
$$837$$ 1.73205 0.0598684
$$838$$ 0 0
$$839$$ 31.1769 1.07635 0.538173 0.842834i $$-0.319115\pi$$
0.538173 + 0.842834i $$0.319115\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ 24.2487 0.835170
$$844$$ −26.0000 −0.894957
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −1.73205 −0.0595140
$$848$$ 48.0000 1.64833
$$849$$ −11.0000 −0.377519
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 20.7846 0.712069
$$853$$ −25.9808 −0.889564 −0.444782 0.895639i $$-0.646719\pi$$
−0.444782 + 0.895639i $$0.646719\pi$$
$$854$$ 0 0
$$855$$ 12.0000 0.410391
$$856$$ 0 0
$$857$$ −18.0000 −0.614868 −0.307434 0.951569i $$-0.599470\pi$$
−0.307434 + 0.951569i $$0.599470\pi$$
$$858$$ 0 0
$$859$$ 13.0000 0.443554 0.221777 0.975097i $$-0.428814\pi$$
0.221777 + 0.975097i $$0.428814\pi$$
$$860$$ 6.92820 0.236250
$$861$$ −12.0000 −0.408959
$$862$$ 0 0
$$863$$ 17.3205 0.589597 0.294798 0.955559i $$-0.404747\pi$$
0.294798 + 0.955559i $$0.404747\pi$$
$$864$$ 0 0
$$865$$ −20.7846 −0.706698
$$866$$ 0 0
$$867$$ 17.0000 0.577350
$$868$$ −6.00000 −0.203653
$$869$$ −38.1051 −1.29263
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 5.19615 0.175863
$$874$$ 0 0
$$875$$ 12.0000 0.405674
$$876$$ 3.46410 0.117041
$$877$$ −41.5692 −1.40369 −0.701846 0.712328i $$-0.747641\pi$$
−0.701846 + 0.712328i $$0.747641\pi$$
$$878$$ 0 0
$$879$$ 17.3205 0.584206
$$880$$ −48.0000 −1.61808
$$881$$ 12.0000 0.404290 0.202145 0.979356i $$-0.435209\pi$$
0.202145 + 0.979356i $$0.435209\pi$$
$$882$$ 0 0
$$883$$ 5.00000 0.168263 0.0841317 0.996455i $$-0.473188\pi$$
0.0841317 + 0.996455i $$0.473188\pi$$
$$884$$ 0 0
$$885$$ −12.0000 −0.403376
$$886$$ 0 0
$$887$$ −42.0000 −1.41022 −0.705111 0.709097i $$-0.749103\pi$$
−0.705111 + 0.709097i $$0.749103\pi$$
$$888$$ 0 0
$$889$$ 22.5167 0.755185
$$890$$ 0 0
$$891$$ 3.46410 0.116052
$$892$$ −34.6410 −1.15987
$$893$$ −12.0000 −0.401565
$$894$$ 0 0
$$895$$ 41.5692 1.38951
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −10.3923 −0.346603
$$900$$ −14.0000 −0.466667
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 1.73205 0.0576390
$$904$$ 0 0
$$905$$ 48.4974 1.61211
$$906$$ 0 0
$$907$$ 44.0000 1.46100 0.730498 0.682915i $$-0.239288\pi$$
0.730498 + 0.682915i $$0.239288\pi$$
$$908$$ 41.5692 1.37952
$$909$$ 18.0000 0.597022
$$910$$ 0 0
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ 13.8564 0.458831
$$913$$ 48.0000 1.58857
$$914$$ 0 0
$$915$$ 3.46410 0.114520
$$916$$ −55.4256 −1.83131
$$917$$ −10.3923 −0.343184
$$918$$ 0 0
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ 0 0
$$921$$ 1.73205 0.0570730
$$922$$ 0 0
$$923$$ 0 0
$$924$$ −12.0000 −0.394771
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 1.00000 0.0328443
$$928$$ 0 0
$$929$$ 20.7846 0.681921 0.340960 0.940078i $$-0.389248\pi$$
0.340960 + 0.940078i $$0.389248\pi$$
$$930$$ 0 0
$$931$$ 13.8564 0.454125
$$932$$ −36.0000 −1.17922
$$933$$ −24.0000 −0.785725
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −14.0000 −0.457360 −0.228680 0.973502i $$-0.573441\pi$$
−0.228680 + 0.973502i $$0.573441\pi$$
$$938$$ 0 0
$$939$$ −13.0000 −0.424239
$$940$$ 24.0000 0.782794
$$941$$ 10.3923 0.338779 0.169390 0.985549i $$-0.445820\pi$$
0.169390 + 0.985549i $$0.445820\pi$$
$$942$$ 0 0
$$943$$ −41.5692 −1.35368
$$944$$ −13.8564 −0.450988
$$945$$ −6.00000 −0.195180
$$946$$ 0 0
$$947$$ 48.4974 1.57595 0.787977 0.615704i $$-0.211128\pi$$
0.787977 + 0.615704i $$0.211128\pi$$
$$948$$ −22.0000 −0.714527
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 6.92820 0.224662
$$952$$ 0 0
$$953$$ −6.00000 −0.194359 −0.0971795 0.995267i $$-0.530982\pi$$
−0.0971795 + 0.995267i $$0.530982\pi$$
$$954$$ 0 0
$$955$$ 62.3538 2.01772
$$956$$ 0 0
$$957$$ −20.7846 −0.671871
$$958$$ 0 0
$$959$$ 0 0
$$960$$ −27.7128 −0.894427
$$961$$ −28.0000 −0.903226
$$962$$ 0 0
$$963$$ −6.00000 −0.193347
$$964$$ −41.5692 −1.33885
$$965$$ −54.0000 −1.73832
$$966$$ 0 0
$$967$$ 24.2487 0.779786 0.389893 0.920860i $$-0.372512\pi$$
0.389893 + 0.920860i $$0.372512\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 36.0000 1.15529 0.577647 0.816286i $$-0.303971\pi$$
0.577647 + 0.816286i $$0.303971\pi$$
$$972$$ 2.00000 0.0641500
$$973$$ −8.66025 −0.277635
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 4.00000 0.128037
$$977$$ −27.7128 −0.886611 −0.443306 0.896370i $$-0.646194\pi$$
−0.443306 + 0.896370i $$0.646194\pi$$
$$978$$ 0 0
$$979$$ −24.0000 −0.767043
$$980$$ −27.7128 −0.885253
$$981$$ 15.5885 0.497701
$$982$$ 0 0
$$983$$ −10.3923 −0.331463 −0.165732 0.986171i $$-0.552999\pi$$
−0.165732 + 0.986171i $$0.552999\pi$$
$$984$$ 0 0
$$985$$ −48.0000 −1.52941
$$986$$ 0 0
$$987$$ 6.00000 0.190982
$$988$$ 0 0
$$989$$ 6.00000 0.190789
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 0 0
$$993$$ 5.19615 0.164895
$$994$$ 0 0
$$995$$ 24.2487 0.768736
$$996$$ 27.7128 0.878114
$$997$$ 35.0000 1.10846 0.554231 0.832363i $$-0.313013\pi$$
0.554231 + 0.832363i $$0.313013\pi$$
$$998$$ 0 0
$$999$$ 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 507.2.a.e.1.1 2
3.2 odd 2 1521.2.a.h.1.2 2
4.3 odd 2 8112.2.a.bu.1.1 2
13.2 odd 12 39.2.j.a.4.1 2
13.3 even 3 507.2.e.f.22.1 4
13.4 even 6 507.2.e.f.484.2 4
13.5 odd 4 507.2.b.c.337.2 2
13.6 odd 12 507.2.j.b.361.1 2
13.7 odd 12 39.2.j.a.10.1 yes 2
13.8 odd 4 507.2.b.c.337.1 2
13.9 even 3 507.2.e.f.484.1 4
13.10 even 6 507.2.e.f.22.2 4
13.11 odd 12 507.2.j.b.316.1 2
13.12 even 2 inner 507.2.a.e.1.2 2
39.2 even 12 117.2.q.a.82.1 2
39.5 even 4 1521.2.b.f.1351.1 2
39.8 even 4 1521.2.b.f.1351.2 2
39.20 even 12 117.2.q.a.10.1 2
39.38 odd 2 1521.2.a.h.1.1 2
52.7 even 12 624.2.bv.b.49.1 2
52.15 even 12 624.2.bv.b.433.1 2
52.51 odd 2 8112.2.a.bu.1.2 2
65.2 even 12 975.2.w.d.199.1 4
65.7 even 12 975.2.w.d.49.2 4
65.28 even 12 975.2.w.d.199.2 4
65.33 even 12 975.2.w.d.49.1 4
65.54 odd 12 975.2.bc.c.901.1 2
65.59 odd 12 975.2.bc.c.751.1 2
156.59 odd 12 1872.2.by.f.1297.1 2
156.119 odd 12 1872.2.by.f.433.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.j.a.4.1 2 13.2 odd 12
39.2.j.a.10.1 yes 2 13.7 odd 12
117.2.q.a.10.1 2 39.20 even 12
117.2.q.a.82.1 2 39.2 even 12
507.2.a.e.1.1 2 1.1 even 1 trivial
507.2.a.e.1.2 2 13.12 even 2 inner
507.2.b.c.337.1 2 13.8 odd 4
507.2.b.c.337.2 2 13.5 odd 4
507.2.e.f.22.1 4 13.3 even 3
507.2.e.f.22.2 4 13.10 even 6
507.2.e.f.484.1 4 13.9 even 3
507.2.e.f.484.2 4 13.4 even 6
507.2.j.b.316.1 2 13.11 odd 12
507.2.j.b.361.1 2 13.6 odd 12
624.2.bv.b.49.1 2 52.7 even 12
624.2.bv.b.433.1 2 52.15 even 12
975.2.w.d.49.1 4 65.33 even 12
975.2.w.d.49.2 4 65.7 even 12
975.2.w.d.199.1 4 65.2 even 12
975.2.w.d.199.2 4 65.28 even 12
975.2.bc.c.751.1 2 65.59 odd 12
975.2.bc.c.901.1 2 65.54 odd 12
1521.2.a.h.1.1 2 39.38 odd 2
1521.2.a.h.1.2 2 3.2 odd 2
1521.2.b.f.1351.1 2 39.5 even 4
1521.2.b.f.1351.2 2 39.8 even 4
1872.2.by.f.433.1 2 156.119 odd 12
1872.2.by.f.1297.1 2 156.59 odd 12
8112.2.a.bu.1.1 2 4.3 odd 2
8112.2.a.bu.1.2 2 52.51 odd 2