# Properties

 Label 1100.4.b.c.749.2 Level $1100$ Weight $4$ Character 1100.749 Analytic conductor $64.902$ 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] = [1100,4,Mod(749,1100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1100.749");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1100 = 2^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1100.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$64.9021010063$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 44) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 749.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1100.749 Dual form 1100.4.b.c.749.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.00000i q^{3} -26.0000i q^{7} +2.00000 q^{9} +O(q^{10})$$ $$q+5.00000i q^{3} -26.0000i q^{7} +2.00000 q^{9} -11.0000 q^{11} -52.0000i q^{13} +46.0000i q^{17} +96.0000 q^{19} +130.000 q^{21} -27.0000i q^{23} +145.000i q^{27} -16.0000 q^{29} -293.000 q^{31} -55.0000i q^{33} -29.0000i q^{37} +260.000 q^{39} -472.000 q^{41} +110.000i q^{43} -224.000i q^{47} -333.000 q^{49} -230.000 q^{51} -754.000i q^{53} +480.000i q^{57} -825.000 q^{59} -548.000 q^{61} -52.0000i q^{63} -123.000i q^{67} +135.000 q^{69} +1001.00 q^{71} +1020.00i q^{73} +286.000i q^{77} -526.000 q^{79} -671.000 q^{81} +158.000i q^{83} -80.0000i q^{87} +1217.00 q^{89} -1352.00 q^{91} -1465.00i q^{93} -263.000i q^{97} -22.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^9 $$2 q + 4 q^{9} - 22 q^{11} + 192 q^{19} + 260 q^{21} - 32 q^{29} - 586 q^{31} + 520 q^{39} - 944 q^{41} - 666 q^{49} - 460 q^{51} - 1650 q^{59} - 1096 q^{61} + 270 q^{69} + 2002 q^{71} - 1052 q^{79} - 1342 q^{81} + 2434 q^{89} - 2704 q^{91} - 44 q^{99}+O(q^{100})$$ 2 * q + 4 * q^9 - 22 * q^11 + 192 * q^19 + 260 * q^21 - 32 * q^29 - 586 * q^31 + 520 * q^39 - 944 * q^41 - 666 * q^49 - 460 * q^51 - 1650 * q^59 - 1096 * q^61 + 270 * q^69 + 2002 * q^71 - 1052 * q^79 - 1342 * q^81 + 2434 * q^89 - 2704 * q^91 - 44 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$551$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 5.00000i 0.962250i 0.876652 + 0.481125i $$0.159772\pi$$
−0.876652 + 0.481125i $$0.840228\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 26.0000i − 1.40387i −0.712242 0.701934i $$-0.752320\pi$$
0.712242 0.701934i $$-0.247680\pi$$
$$8$$ 0 0
$$9$$ 2.00000 0.0740741
$$10$$ 0 0
$$11$$ −11.0000 −0.301511
$$12$$ 0 0
$$13$$ − 52.0000i − 1.10940i −0.832050 0.554700i $$-0.812833\pi$$
0.832050 0.554700i $$-0.187167\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 46.0000i 0.656273i 0.944630 + 0.328136i $$0.106421\pi$$
−0.944630 + 0.328136i $$0.893579\pi$$
$$18$$ 0 0
$$19$$ 96.0000 1.15915 0.579577 0.814918i $$-0.303218\pi$$
0.579577 + 0.814918i $$0.303218\pi$$
$$20$$ 0 0
$$21$$ 130.000 1.35087
$$22$$ 0 0
$$23$$ − 27.0000i − 0.244778i −0.992482 0.122389i $$-0.960944\pi$$
0.992482 0.122389i $$-0.0390555\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 145.000i 1.03353i
$$28$$ 0 0
$$29$$ −16.0000 −0.102453 −0.0512263 0.998687i $$-0.516313\pi$$
−0.0512263 + 0.998687i $$0.516313\pi$$
$$30$$ 0 0
$$31$$ −293.000 −1.69756 −0.848780 0.528746i $$-0.822662\pi$$
−0.848780 + 0.528746i $$0.822662\pi$$
$$32$$ 0 0
$$33$$ − 55.0000i − 0.290129i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 29.0000i − 0.128853i −0.997922 0.0644266i $$-0.979478\pi$$
0.997922 0.0644266i $$-0.0205218\pi$$
$$38$$ 0 0
$$39$$ 260.000 1.06752
$$40$$ 0 0
$$41$$ −472.000 −1.79790 −0.898951 0.438048i $$-0.855670\pi$$
−0.898951 + 0.438048i $$0.855670\pi$$
$$42$$ 0 0
$$43$$ 110.000i 0.390113i 0.980792 + 0.195056i $$0.0624890\pi$$
−0.980792 + 0.195056i $$0.937511\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 224.000i − 0.695186i −0.937645 0.347593i $$-0.886999\pi$$
0.937645 0.347593i $$-0.113001\pi$$
$$48$$ 0 0
$$49$$ −333.000 −0.970845
$$50$$ 0 0
$$51$$ −230.000 −0.631499
$$52$$ 0 0
$$53$$ − 754.000i − 1.95415i −0.212899 0.977074i $$-0.568291\pi$$
0.212899 0.977074i $$-0.431709\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 480.000i 1.11540i
$$58$$ 0 0
$$59$$ −825.000 −1.82044 −0.910219 0.414127i $$-0.864087\pi$$
−0.910219 + 0.414127i $$0.864087\pi$$
$$60$$ 0 0
$$61$$ −548.000 −1.15023 −0.575116 0.818072i $$-0.695043\pi$$
−0.575116 + 0.818072i $$0.695043\pi$$
$$62$$ 0 0
$$63$$ − 52.0000i − 0.103990i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 123.000i − 0.224281i −0.993692 0.112141i $$-0.964229\pi$$
0.993692 0.112141i $$-0.0357707\pi$$
$$68$$ 0 0
$$69$$ 135.000 0.235538
$$70$$ 0 0
$$71$$ 1001.00 1.67319 0.836597 0.547818i $$-0.184541\pi$$
0.836597 + 0.547818i $$0.184541\pi$$
$$72$$ 0 0
$$73$$ 1020.00i 1.63537i 0.575666 + 0.817685i $$0.304743\pi$$
−0.575666 + 0.817685i $$0.695257\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 286.000i 0.423282i
$$78$$ 0 0
$$79$$ −526.000 −0.749109 −0.374555 0.927205i $$-0.622204\pi$$
−0.374555 + 0.927205i $$0.622204\pi$$
$$80$$ 0 0
$$81$$ −671.000 −0.920439
$$82$$ 0 0
$$83$$ 158.000i 0.208949i 0.994528 + 0.104474i $$0.0333160\pi$$
−0.994528 + 0.104474i $$0.966684\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 80.0000i − 0.0985851i
$$88$$ 0 0
$$89$$ 1217.00 1.44946 0.724729 0.689034i $$-0.241965\pi$$
0.724729 + 0.689034i $$0.241965\pi$$
$$90$$ 0 0
$$91$$ −1352.00 −1.55745
$$92$$ 0 0
$$93$$ − 1465.00i − 1.63348i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 263.000i − 0.275295i −0.990481 0.137647i $$-0.956046\pi$$
0.990481 0.137647i $$-0.0439541\pi$$
$$98$$ 0 0
$$99$$ −22.0000 −0.0223342
$$100$$ 0 0
$$101$$ −814.000 −0.801941 −0.400970 0.916091i $$-0.631327\pi$$
−0.400970 + 0.916091i $$0.631327\pi$$
$$102$$ 0 0
$$103$$ − 376.000i − 0.359693i −0.983695 0.179847i $$-0.942440\pi$$
0.983695 0.179847i $$-0.0575601\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 1494.00i − 1.34982i −0.737901 0.674909i $$-0.764183\pi$$
0.737901 0.674909i $$-0.235817\pi$$
$$108$$ 0 0
$$109$$ −842.000 −0.739899 −0.369949 0.929052i $$-0.620625\pi$$
−0.369949 + 0.929052i $$0.620625\pi$$
$$110$$ 0 0
$$111$$ 145.000 0.123989
$$112$$ 0 0
$$113$$ − 1281.00i − 1.06643i −0.845980 0.533214i $$-0.820984\pi$$
0.845980 0.533214i $$-0.179016\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 104.000i − 0.0821778i
$$118$$ 0 0
$$119$$ 1196.00 0.921321
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 0 0
$$123$$ − 2360.00i − 1.73003i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 1736.00i 1.21295i 0.795101 + 0.606477i $$0.207418\pi$$
−0.795101 + 0.606477i $$0.792582\pi$$
$$128$$ 0 0
$$129$$ −550.000 −0.375386
$$130$$ 0 0
$$131$$ 6.00000 0.00400170 0.00200085 0.999998i $$-0.499363\pi$$
0.00200085 + 0.999998i $$0.499363\pi$$
$$132$$ 0 0
$$133$$ − 2496.00i − 1.62730i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 1439.00i − 0.897387i −0.893686 0.448694i $$-0.851889\pi$$
0.893686 0.448694i $$-0.148111\pi$$
$$138$$ 0 0
$$139$$ 318.000 0.194046 0.0970231 0.995282i $$-0.469068\pi$$
0.0970231 + 0.995282i $$0.469068\pi$$
$$140$$ 0 0
$$141$$ 1120.00 0.668943
$$142$$ 0 0
$$143$$ 572.000i 0.334497i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 1665.00i − 0.934196i
$$148$$ 0 0
$$149$$ 922.000 0.506934 0.253467 0.967344i $$-0.418429\pi$$
0.253467 + 0.967344i $$0.418429\pi$$
$$150$$ 0 0
$$151$$ −1030.00 −0.555101 −0.277550 0.960711i $$-0.589523\pi$$
−0.277550 + 0.960711i $$0.589523\pi$$
$$152$$ 0 0
$$153$$ 92.0000i 0.0486128i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 1017.00i 0.516977i 0.966014 + 0.258489i $$0.0832245\pi$$
−0.966014 + 0.258489i $$0.916776\pi$$
$$158$$ 0 0
$$159$$ 3770.00 1.88038
$$160$$ 0 0
$$161$$ −702.000 −0.343636
$$162$$ 0 0
$$163$$ 2444.00i 1.17441i 0.809438 + 0.587205i $$0.199772\pi$$
−0.809438 + 0.587205i $$0.800228\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 1452.00i − 0.672809i −0.941718 0.336405i $$-0.890789\pi$$
0.941718 0.336405i $$-0.109211\pi$$
$$168$$ 0 0
$$169$$ −507.000 −0.230769
$$170$$ 0 0
$$171$$ 192.000 0.0858632
$$172$$ 0 0
$$173$$ − 1914.00i − 0.841149i −0.907258 0.420574i $$-0.861829\pi$$
0.907258 0.420574i $$-0.138171\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 4125.00i − 1.75172i
$$178$$ 0 0
$$179$$ −1293.00 −0.539907 −0.269954 0.962873i $$-0.587008\pi$$
−0.269954 + 0.962873i $$0.587008\pi$$
$$180$$ 0 0
$$181$$ 455.000 0.186850 0.0934251 0.995626i $$-0.470218\pi$$
0.0934251 + 0.995626i $$0.470218\pi$$
$$182$$ 0 0
$$183$$ − 2740.00i − 1.10681i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 506.000i − 0.197874i
$$188$$ 0 0
$$189$$ 3770.00 1.45094
$$190$$ 0 0
$$191$$ −1115.00 −0.422401 −0.211200 0.977443i $$-0.567737\pi$$
−0.211200 + 0.977443i $$0.567737\pi$$
$$192$$ 0 0
$$193$$ − 5012.00i − 1.86928i −0.355591 0.934642i $$-0.615720\pi$$
0.355591 0.934642i $$-0.384280\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 4146.00i − 1.49944i −0.661753 0.749721i $$-0.730187\pi$$
0.661753 0.749721i $$-0.269813\pi$$
$$198$$ 0 0
$$199$$ 1240.00 0.441715 0.220857 0.975306i $$-0.429114\pi$$
0.220857 + 0.975306i $$0.429114\pi$$
$$200$$ 0 0
$$201$$ 615.000 0.215815
$$202$$ 0 0
$$203$$ 416.000i 0.143830i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 54.0000i − 0.0181317i
$$208$$ 0 0
$$209$$ −1056.00 −0.349498
$$210$$ 0 0
$$211$$ −2820.00 −0.920080 −0.460040 0.887898i $$-0.652165\pi$$
−0.460040 + 0.887898i $$0.652165\pi$$
$$212$$ 0 0
$$213$$ 5005.00i 1.61003i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 7618.00i 2.38315i
$$218$$ 0 0
$$219$$ −5100.00 −1.57363
$$220$$ 0 0
$$221$$ 2392.00 0.728069
$$222$$ 0 0
$$223$$ − 3695.00i − 1.10958i −0.831992 0.554788i $$-0.812799\pi$$
0.831992 0.554788i $$-0.187201\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 486.000i − 0.142101i −0.997473 0.0710506i $$-0.977365\pi$$
0.997473 0.0710506i $$-0.0226352\pi$$
$$228$$ 0 0
$$229$$ −4231.00 −1.22093 −0.610464 0.792044i $$-0.709017\pi$$
−0.610464 + 0.792044i $$0.709017\pi$$
$$230$$ 0 0
$$231$$ −1430.00 −0.407303
$$232$$ 0 0
$$233$$ 3336.00i 0.937977i 0.883204 + 0.468988i $$0.155381\pi$$
−0.883204 + 0.468988i $$0.844619\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 2630.00i − 0.720831i
$$238$$ 0 0
$$239$$ −3610.00 −0.977036 −0.488518 0.872554i $$-0.662462\pi$$
−0.488518 + 0.872554i $$0.662462\pi$$
$$240$$ 0 0
$$241$$ −6408.00 −1.71276 −0.856381 0.516345i $$-0.827292\pi$$
−0.856381 + 0.516345i $$0.827292\pi$$
$$242$$ 0 0
$$243$$ 560.000i 0.147835i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 4992.00i − 1.28596i
$$248$$ 0 0
$$249$$ −790.000 −0.201061
$$250$$ 0 0
$$251$$ −1787.00 −0.449380 −0.224690 0.974430i $$-0.572137\pi$$
−0.224690 + 0.974430i $$0.572137\pi$$
$$252$$ 0 0
$$253$$ 297.000i 0.0738033i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 354.000i − 0.0859218i −0.999077 0.0429609i $$-0.986321\pi$$
0.999077 0.0429609i $$-0.0136791\pi$$
$$258$$ 0 0
$$259$$ −754.000 −0.180893
$$260$$ 0 0
$$261$$ −32.0000 −0.00758908
$$262$$ 0 0
$$263$$ 2026.00i 0.475013i 0.971386 + 0.237507i $$0.0763302\pi$$
−0.971386 + 0.237507i $$0.923670\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 6085.00i 1.39474i
$$268$$ 0 0
$$269$$ 8750.00 1.98326 0.991630 0.129112i $$-0.0412128\pi$$
0.991630 + 0.129112i $$0.0412128\pi$$
$$270$$ 0 0
$$271$$ −5036.00 −1.12884 −0.564419 0.825488i $$-0.690900\pi$$
−0.564419 + 0.825488i $$0.690900\pi$$
$$272$$ 0 0
$$273$$ − 6760.00i − 1.49866i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 4306.00i − 0.934016i −0.884253 0.467008i $$-0.845332\pi$$
0.884253 0.467008i $$-0.154668\pi$$
$$278$$ 0 0
$$279$$ −586.000 −0.125745
$$280$$ 0 0
$$281$$ −1202.00 −0.255179 −0.127590 0.991827i $$-0.540724\pi$$
−0.127590 + 0.991827i $$0.540724\pi$$
$$282$$ 0 0
$$283$$ 6620.00i 1.39052i 0.718757 + 0.695262i $$0.244712\pi$$
−0.718757 + 0.695262i $$0.755288\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 12272.0i 2.52402i
$$288$$ 0 0
$$289$$ 2797.00 0.569306
$$290$$ 0 0
$$291$$ 1315.00 0.264903
$$292$$ 0 0
$$293$$ − 6968.00i − 1.38933i −0.719331 0.694667i $$-0.755552\pi$$
0.719331 0.694667i $$-0.244448\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 1595.00i − 0.311620i
$$298$$ 0 0
$$299$$ −1404.00 −0.271557
$$300$$ 0 0
$$301$$ 2860.00 0.547667
$$302$$ 0 0
$$303$$ − 4070.00i − 0.771668i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 7640.00i 1.42032i 0.704041 + 0.710159i $$0.251377\pi$$
−0.704041 + 0.710159i $$0.748623\pi$$
$$308$$ 0 0
$$309$$ 1880.00 0.346115
$$310$$ 0 0
$$311$$ −652.000 −0.118880 −0.0594398 0.998232i $$-0.518931\pi$$
−0.0594398 + 0.998232i $$0.518931\pi$$
$$312$$ 0 0
$$313$$ − 8055.00i − 1.45462i −0.686310 0.727309i $$-0.740771\pi$$
0.686310 0.727309i $$-0.259229\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 5675.00i − 1.00549i −0.864435 0.502744i $$-0.832324\pi$$
0.864435 0.502744i $$-0.167676\pi$$
$$318$$ 0 0
$$319$$ 176.000 0.0308906
$$320$$ 0 0
$$321$$ 7470.00 1.29886
$$322$$ 0 0
$$323$$ 4416.00i 0.760721i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 4210.00i − 0.711968i
$$328$$ 0 0
$$329$$ −5824.00 −0.975950
$$330$$ 0 0
$$331$$ −2245.00 −0.372799 −0.186399 0.982474i $$-0.559682\pi$$
−0.186399 + 0.982474i $$0.559682\pi$$
$$332$$ 0 0
$$333$$ − 58.0000i − 0.00954469i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 694.000i − 0.112180i −0.998426 0.0560899i $$-0.982137\pi$$
0.998426 0.0560899i $$-0.0178633\pi$$
$$338$$ 0 0
$$339$$ 6405.00 1.02617
$$340$$ 0 0
$$341$$ 3223.00 0.511834
$$342$$ 0 0
$$343$$ − 260.000i − 0.0409291i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 4092.00i 0.633055i 0.948583 + 0.316527i $$0.102517\pi$$
−0.948583 + 0.316527i $$0.897483\pi$$
$$348$$ 0 0
$$349$$ −334.000 −0.0512281 −0.0256141 0.999672i $$-0.508154\pi$$
−0.0256141 + 0.999672i $$0.508154\pi$$
$$350$$ 0 0
$$351$$ 7540.00 1.14660
$$352$$ 0 0
$$353$$ − 891.000i − 0.134343i −0.997741 0.0671716i $$-0.978603\pi$$
0.997741 0.0671716i $$-0.0213975\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 5980.00i 0.886541i
$$358$$ 0 0
$$359$$ −2476.00 −0.364006 −0.182003 0.983298i $$-0.558258\pi$$
−0.182003 + 0.983298i $$0.558258\pi$$
$$360$$ 0 0
$$361$$ 2357.00 0.343636
$$362$$ 0 0
$$363$$ 605.000i 0.0874773i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 1379.00i 0.196140i 0.995180 + 0.0980698i $$0.0312668\pi$$
−0.995180 + 0.0980698i $$0.968733\pi$$
$$368$$ 0 0
$$369$$ −944.000 −0.133178
$$370$$ 0 0
$$371$$ −19604.0 −2.74337
$$372$$ 0 0
$$373$$ 6266.00i 0.869816i 0.900475 + 0.434908i $$0.143219\pi$$
−0.900475 + 0.434908i $$0.856781\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 832.000i 0.113661i
$$378$$ 0 0
$$379$$ −151.000 −0.0204653 −0.0102327 0.999948i $$-0.503257\pi$$
−0.0102327 + 0.999948i $$0.503257\pi$$
$$380$$ 0 0
$$381$$ −8680.00 −1.16717
$$382$$ 0 0
$$383$$ 1989.00i 0.265361i 0.991159 + 0.132680i $$0.0423584\pi$$
−0.991159 + 0.132680i $$0.957642\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 220.000i 0.0288972i
$$388$$ 0 0
$$389$$ −6817.00 −0.888523 −0.444262 0.895897i $$-0.646534\pi$$
−0.444262 + 0.895897i $$0.646534\pi$$
$$390$$ 0 0
$$391$$ 1242.00 0.160641
$$392$$ 0 0
$$393$$ 30.0000i 0.00385064i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 4170.00i − 0.527170i −0.964636 0.263585i $$-0.915095\pi$$
0.964636 0.263585i $$-0.0849049\pi$$
$$398$$ 0 0
$$399$$ 12480.0 1.56587
$$400$$ 0 0
$$401$$ 10914.0 1.35915 0.679575 0.733606i $$-0.262164\pi$$
0.679575 + 0.733606i $$0.262164\pi$$
$$402$$ 0 0
$$403$$ 15236.0i 1.88327i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 319.000i 0.0388507i
$$408$$ 0 0
$$409$$ 1102.00 0.133228 0.0666142 0.997779i $$-0.478780\pi$$
0.0666142 + 0.997779i $$0.478780\pi$$
$$410$$ 0 0
$$411$$ 7195.00 0.863511
$$412$$ 0 0
$$413$$ 21450.0i 2.55565i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 1590.00i 0.186721i
$$418$$ 0 0
$$419$$ −11028.0 −1.28581 −0.642903 0.765947i $$-0.722270\pi$$
−0.642903 + 0.765947i $$0.722270\pi$$
$$420$$ 0 0
$$421$$ 2622.00 0.303536 0.151768 0.988416i $$-0.451503\pi$$
0.151768 + 0.988416i $$0.451503\pi$$
$$422$$ 0 0
$$423$$ − 448.000i − 0.0514953i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 14248.0i 1.61478i
$$428$$ 0 0
$$429$$ −2860.00 −0.321870
$$430$$ 0 0
$$431$$ 16598.0 1.85498 0.927491 0.373845i $$-0.121961\pi$$
0.927491 + 0.373845i $$0.121961\pi$$
$$432$$ 0 0
$$433$$ 5763.00i 0.639612i 0.947483 + 0.319806i $$0.103618\pi$$
−0.947483 + 0.319806i $$0.896382\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 2592.00i − 0.283735i
$$438$$ 0 0
$$439$$ 3128.00 0.340071 0.170036 0.985438i $$-0.445612\pi$$
0.170036 + 0.985438i $$0.445612\pi$$
$$440$$ 0 0
$$441$$ −666.000 −0.0719145
$$442$$ 0 0
$$443$$ − 6369.00i − 0.683071i −0.939869 0.341535i $$-0.889053\pi$$
0.939869 0.341535i $$-0.110947\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 4610.00i 0.487798i
$$448$$ 0 0
$$449$$ −8691.00 −0.913483 −0.456741 0.889600i $$-0.650983\pi$$
−0.456741 + 0.889600i $$0.650983\pi$$
$$450$$ 0 0
$$451$$ 5192.00 0.542088
$$452$$ 0 0
$$453$$ − 5150.00i − 0.534146i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 2260.00i 0.231331i 0.993288 + 0.115666i $$0.0369001\pi$$
−0.993288 + 0.115666i $$0.963100\pi$$
$$458$$ 0 0
$$459$$ −6670.00 −0.678277
$$460$$ 0 0
$$461$$ −12756.0 −1.28873 −0.644367 0.764717i $$-0.722879\pi$$
−0.644367 + 0.764717i $$0.722879\pi$$
$$462$$ 0 0
$$463$$ 6887.00i 0.691287i 0.938366 + 0.345644i $$0.112339\pi$$
−0.938366 + 0.345644i $$0.887661\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 1535.00i − 0.152101i −0.997104 0.0760507i $$-0.975769\pi$$
0.997104 0.0760507i $$-0.0242311\pi$$
$$468$$ 0 0
$$469$$ −3198.00 −0.314861
$$470$$ 0 0
$$471$$ −5085.00 −0.497462
$$472$$ 0 0
$$473$$ − 1210.00i − 0.117623i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 1508.00i − 0.144752i
$$478$$ 0 0
$$479$$ 17564.0 1.67541 0.837703 0.546126i $$-0.183898\pi$$
0.837703 + 0.546126i $$0.183898\pi$$
$$480$$ 0 0
$$481$$ −1508.00 −0.142950
$$482$$ 0 0
$$483$$ − 3510.00i − 0.330664i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 7541.00i − 0.701674i −0.936437 0.350837i $$-0.885897\pi$$
0.936437 0.350837i $$-0.114103\pi$$
$$488$$ 0 0
$$489$$ −12220.0 −1.13008
$$490$$ 0 0
$$491$$ 12552.0 1.15369 0.576847 0.816852i $$-0.304283\pi$$
0.576847 + 0.816852i $$0.304283\pi$$
$$492$$ 0 0
$$493$$ − 736.000i − 0.0672369i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 26026.0i − 2.34894i
$$498$$ 0 0
$$499$$ 8396.00 0.753220 0.376610 0.926372i $$-0.377090\pi$$
0.376610 + 0.926372i $$0.377090\pi$$
$$500$$ 0 0
$$501$$ 7260.00 0.647411
$$502$$ 0 0
$$503$$ 12194.0i 1.08092i 0.841369 + 0.540461i $$0.181750\pi$$
−0.841369 + 0.540461i $$0.818250\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 2535.00i − 0.222058i
$$508$$ 0 0
$$509$$ −18295.0 −1.59315 −0.796573 0.604542i $$-0.793356\pi$$
−0.796573 + 0.604542i $$0.793356\pi$$
$$510$$ 0 0
$$511$$ 26520.0 2.29584
$$512$$ 0 0
$$513$$ 13920.0i 1.19802i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 2464.00i 0.209607i
$$518$$ 0 0
$$519$$ 9570.00 0.809396
$$520$$ 0 0
$$521$$ 7101.00 0.597122 0.298561 0.954391i $$-0.403493\pi$$
0.298561 + 0.954391i $$0.403493\pi$$
$$522$$ 0 0
$$523$$ 4912.00i 0.410682i 0.978690 + 0.205341i $$0.0658304\pi$$
−0.978690 + 0.205341i $$0.934170\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 13478.0i − 1.11406i
$$528$$ 0 0
$$529$$ 11438.0 0.940084
$$530$$ 0 0
$$531$$ −1650.00 −0.134847
$$532$$ 0 0
$$533$$ 24544.0i 1.99459i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 6465.00i − 0.519526i
$$538$$ 0 0
$$539$$ 3663.00 0.292721
$$540$$ 0 0
$$541$$ 11496.0 0.913589 0.456794 0.889572i $$-0.348997\pi$$
0.456794 + 0.889572i $$0.348997\pi$$
$$542$$ 0 0
$$543$$ 2275.00i 0.179797i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 19048.0i 1.48891i 0.667673 + 0.744455i $$0.267291\pi$$
−0.667673 + 0.744455i $$0.732709\pi$$
$$548$$ 0 0
$$549$$ −1096.00 −0.0852024
$$550$$ 0 0
$$551$$ −1536.00 −0.118758
$$552$$ 0 0
$$553$$ 13676.0i 1.05165i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 10718.0i 0.815325i 0.913133 + 0.407663i $$0.133656\pi$$
−0.913133 + 0.407663i $$0.866344\pi$$
$$558$$ 0 0
$$559$$ 5720.00 0.432791
$$560$$ 0 0
$$561$$ 2530.00 0.190404
$$562$$ 0 0
$$563$$ − 6660.00i − 0.498553i −0.968432 0.249277i $$-0.919807\pi$$
0.968432 0.249277i $$-0.0801929\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 17446.0i 1.29217i
$$568$$ 0 0
$$569$$ 10960.0 0.807499 0.403750 0.914870i $$-0.367707\pi$$
0.403750 + 0.914870i $$0.367707\pi$$
$$570$$ 0 0
$$571$$ 18596.0 1.36290 0.681452 0.731863i $$-0.261349\pi$$
0.681452 + 0.731863i $$0.261349\pi$$
$$572$$ 0 0
$$573$$ − 5575.00i − 0.406455i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 21119.0i − 1.52374i −0.647733 0.761868i $$-0.724283\pi$$
0.647733 0.761868i $$-0.275717\pi$$
$$578$$ 0 0
$$579$$ 25060.0 1.79872
$$580$$ 0 0
$$581$$ 4108.00 0.293337
$$582$$ 0 0
$$583$$ 8294.00i 0.589198i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 2836.00i − 0.199411i −0.995017 0.0997055i $$-0.968210\pi$$
0.995017 0.0997055i $$-0.0317901\pi$$
$$588$$ 0 0
$$589$$ −28128.0 −1.96773
$$590$$ 0 0
$$591$$ 20730.0 1.44284
$$592$$ 0 0
$$593$$ − 18044.0i − 1.24954i −0.780808 0.624771i $$-0.785192\pi$$
0.780808 0.624771i $$-0.214808\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 6200.00i 0.425040i
$$598$$ 0 0
$$599$$ 9264.00 0.631914 0.315957 0.948773i $$-0.397674\pi$$
0.315957 + 0.948773i $$0.397674\pi$$
$$600$$ 0 0
$$601$$ −19326.0 −1.31169 −0.655844 0.754897i $$-0.727687\pi$$
−0.655844 + 0.754897i $$0.727687\pi$$
$$602$$ 0 0
$$603$$ − 246.000i − 0.0166134i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 10082.0i 0.674161i 0.941476 + 0.337081i $$0.109439\pi$$
−0.941476 + 0.337081i $$0.890561\pi$$
$$608$$ 0 0
$$609$$ −2080.00 −0.138400
$$610$$ 0 0
$$611$$ −11648.0 −0.771240
$$612$$ 0 0
$$613$$ − 13088.0i − 0.862348i −0.902269 0.431174i $$-0.858100\pi$$
0.902269 0.431174i $$-0.141900\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 23426.0i 1.52852i 0.644910 + 0.764259i $$0.276895\pi$$
−0.644910 + 0.764259i $$0.723105\pi$$
$$618$$ 0 0
$$619$$ −23587.0 −1.53157 −0.765785 0.643097i $$-0.777649\pi$$
−0.765785 + 0.643097i $$0.777649\pi$$
$$620$$ 0 0
$$621$$ 3915.00 0.252985
$$622$$ 0 0
$$623$$ − 31642.0i − 2.03485i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 5280.00i − 0.336304i
$$628$$ 0 0
$$629$$ 1334.00 0.0845629
$$630$$ 0 0
$$631$$ 19683.0 1.24179 0.620894 0.783895i $$-0.286770\pi$$
0.620894 + 0.783895i $$0.286770\pi$$
$$632$$ 0 0
$$633$$ − 14100.0i − 0.885347i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 17316.0i 1.07706i
$$638$$ 0 0
$$639$$ 2002.00 0.123940
$$640$$ 0 0
$$641$$ 375.000 0.0231070 0.0115535 0.999933i $$-0.496322\pi$$
0.0115535 + 0.999933i $$0.496322\pi$$
$$642$$ 0 0
$$643$$ 21055.0i 1.29133i 0.763619 + 0.645667i $$0.223421\pi$$
−0.763619 + 0.645667i $$0.776579\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 6427.00i − 0.390528i −0.980751 0.195264i $$-0.937444\pi$$
0.980751 0.195264i $$-0.0625563\pi$$
$$648$$ 0 0
$$649$$ 9075.00 0.548883
$$650$$ 0 0
$$651$$ −38090.0 −2.29319
$$652$$ 0 0
$$653$$ 7617.00i 0.456472i 0.973606 + 0.228236i $$0.0732958\pi$$
−0.973606 + 0.228236i $$0.926704\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 2040.00i 0.121138i
$$658$$ 0 0
$$659$$ 17630.0 1.04214 0.521068 0.853515i $$-0.325534\pi$$
0.521068 + 0.853515i $$0.325534\pi$$
$$660$$ 0 0
$$661$$ 4605.00 0.270974 0.135487 0.990779i $$-0.456740\pi$$
0.135487 + 0.990779i $$0.456740\pi$$
$$662$$ 0 0
$$663$$ 11960.0i 0.700585i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 432.000i 0.0250781i
$$668$$ 0 0
$$669$$ 18475.0 1.06769
$$670$$ 0 0
$$671$$ 6028.00 0.346808
$$672$$ 0 0
$$673$$ 2818.00i 0.161406i 0.996738 + 0.0807028i $$0.0257165\pi$$
−0.996738 + 0.0807028i $$0.974284\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 8438.00i 0.479023i 0.970894 + 0.239512i $$0.0769873\pi$$
−0.970894 + 0.239512i $$0.923013\pi$$
$$678$$ 0 0
$$679$$ −6838.00 −0.386478
$$680$$ 0 0
$$681$$ 2430.00 0.136737
$$682$$ 0 0
$$683$$ 17344.0i 0.971669i 0.874051 + 0.485834i $$0.161484\pi$$
−0.874051 + 0.485834i $$0.838516\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 21155.0i − 1.17484i
$$688$$ 0 0
$$689$$ −39208.0 −2.16793
$$690$$ 0 0
$$691$$ −3947.00 −0.217295 −0.108648 0.994080i $$-0.534652\pi$$
−0.108648 + 0.994080i $$0.534652\pi$$
$$692$$ 0 0
$$693$$ 572.000i 0.0313542i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 21712.0i − 1.17991i
$$698$$ 0 0
$$699$$ −16680.0 −0.902569
$$700$$ 0 0
$$701$$ −7998.00 −0.430928 −0.215464 0.976512i $$-0.569126\pi$$
−0.215464 + 0.976512i $$0.569126\pi$$
$$702$$ 0 0
$$703$$ − 2784.00i − 0.149361i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 21164.0i 1.12582i
$$708$$ 0 0
$$709$$ 881.000 0.0466666 0.0233333 0.999728i $$-0.492572\pi$$
0.0233333 + 0.999728i $$0.492572\pi$$
$$710$$ 0 0
$$711$$ −1052.00 −0.0554896
$$712$$ 0 0
$$713$$ 7911.00i 0.415525i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 18050.0i − 0.940153i
$$718$$ 0 0
$$719$$ 26093.0 1.35341 0.676707 0.736252i $$-0.263406\pi$$
0.676707 + 0.736252i $$0.263406\pi$$
$$720$$ 0 0
$$721$$ −9776.00 −0.504962
$$722$$ 0 0
$$723$$ − 32040.0i − 1.64811i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 7481.00i − 0.381644i −0.981625 0.190822i $$-0.938885\pi$$
0.981625 0.190822i $$-0.0611153\pi$$
$$728$$ 0 0
$$729$$ −20917.0 −1.06269
$$730$$ 0 0
$$731$$ −5060.00 −0.256020
$$732$$ 0 0
$$733$$ − 17788.0i − 0.896337i −0.893949 0.448168i $$-0.852077\pi$$
0.893949 0.448168i $$-0.147923\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1353.00i 0.0676233i
$$738$$ 0 0
$$739$$ 32182.0 1.60194 0.800970 0.598704i $$-0.204317\pi$$
0.800970 + 0.598704i $$0.204317\pi$$
$$740$$ 0 0
$$741$$ 24960.0 1.23742
$$742$$ 0 0
$$743$$ 32044.0i 1.58221i 0.611682 + 0.791104i $$0.290493\pi$$
−0.611682 + 0.791104i $$0.709507\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 316.000i 0.0154777i
$$748$$ 0 0
$$749$$ −38844.0 −1.89497
$$750$$ 0 0
$$751$$ −33779.0 −1.64130 −0.820648 0.571434i $$-0.806387\pi$$
−0.820648 + 0.571434i $$0.806387\pi$$
$$752$$ 0 0
$$753$$ − 8935.00i − 0.432416i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 15630.0i − 0.750439i −0.926936 0.375219i $$-0.877567\pi$$
0.926936 0.375219i $$-0.122433\pi$$
$$758$$ 0 0
$$759$$ −1485.00 −0.0710172
$$760$$ 0 0
$$761$$ 1948.00 0.0927923 0.0463962 0.998923i $$-0.485226\pi$$
0.0463962 + 0.998923i $$0.485226\pi$$
$$762$$ 0 0
$$763$$ 21892.0i 1.03872i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 42900.0i 2.01959i
$$768$$ 0 0
$$769$$ 17420.0 0.816881 0.408440 0.912785i $$-0.366073\pi$$
0.408440 + 0.912785i $$0.366073\pi$$
$$770$$ 0 0
$$771$$ 1770.00 0.0826783
$$772$$ 0 0
$$773$$ − 11122.0i − 0.517504i −0.965944 0.258752i $$-0.916689\pi$$
0.965944 0.258752i $$-0.0833112\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 3770.00i − 0.174064i
$$778$$ 0 0
$$779$$ −45312.0 −2.08404
$$780$$ 0 0
$$781$$ −11011.0 −0.504487
$$782$$ 0 0
$$783$$ − 2320.00i − 0.105888i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 29648.0i 1.34287i 0.741064 + 0.671434i $$0.234321\pi$$
−0.741064 + 0.671434i $$0.765679\pi$$
$$788$$ 0 0
$$789$$ −10130.0 −0.457082
$$790$$ 0 0
$$791$$ −33306.0 −1.49712
$$792$$ 0 0
$$793$$ 28496.0i 1.27607i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 23219.0i − 1.03194i −0.856606 0.515972i $$-0.827431\pi$$
0.856606 0.515972i $$-0.172569\pi$$
$$798$$ 0 0
$$799$$ 10304.0 0.456232
$$800$$ 0 0
$$801$$ 2434.00 0.107367
$$802$$ 0 0
$$803$$ − 11220.0i − 0.493082i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 43750.0i 1.90839i
$$808$$ 0 0
$$809$$ −9232.00 −0.401211 −0.200606 0.979672i $$-0.564291\pi$$
−0.200606 + 0.979672i $$0.564291\pi$$
$$810$$ 0 0
$$811$$ −32286.0 −1.39792 −0.698961 0.715160i $$-0.746354\pi$$
−0.698961 + 0.715160i $$0.746354\pi$$
$$812$$ 0 0
$$813$$ − 25180.0i − 1.08623i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 10560.0i 0.452200i
$$818$$ 0 0
$$819$$ −2704.00 −0.115367
$$820$$ 0 0
$$821$$ −31706.0 −1.34780 −0.673902 0.738821i $$-0.735383\pi$$
−0.673902 + 0.738821i $$0.735383\pi$$
$$822$$ 0 0
$$823$$ − 41139.0i − 1.74242i −0.490906 0.871212i $$-0.663334\pi$$
0.490906 0.871212i $$-0.336666\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 15252.0i − 0.641311i −0.947196 0.320655i $$-0.896097\pi$$
0.947196 0.320655i $$-0.103903\pi$$
$$828$$ 0 0
$$829$$ −369.000 −0.0154595 −0.00772973 0.999970i $$-0.502460\pi$$
−0.00772973 + 0.999970i $$0.502460\pi$$
$$830$$ 0 0
$$831$$ 21530.0 0.898757
$$832$$ 0 0
$$833$$ − 15318.0i − 0.637140i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 42485.0i − 1.75448i
$$838$$ 0 0
$$839$$ 10257.0 0.422063 0.211032 0.977479i $$-0.432318\pi$$
0.211032 + 0.977479i $$0.432318\pi$$
$$840$$ 0 0
$$841$$ −24133.0 −0.989503
$$842$$ 0 0
$$843$$ − 6010.00i − 0.245546i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 3146.00i − 0.127624i
$$848$$ 0 0
$$849$$ −33100.0 −1.33803
$$850$$ 0 0
$$851$$ −783.000 −0.0315404
$$852$$ 0 0
$$853$$ 34386.0i 1.38025i 0.723690 + 0.690126i $$0.242445\pi$$
−0.723690 + 0.690126i $$0.757555\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 23464.0i − 0.935257i −0.883925 0.467628i $$-0.845109\pi$$
0.883925 0.467628i $$-0.154891\pi$$
$$858$$ 0 0
$$859$$ 22475.0 0.892709 0.446355 0.894856i $$-0.352722\pi$$
0.446355 + 0.894856i $$0.352722\pi$$
$$860$$ 0 0
$$861$$ −61360.0 −2.42874
$$862$$ 0 0
$$863$$ − 2880.00i − 0.113599i −0.998386 0.0567997i $$-0.981910\pi$$
0.998386 0.0567997i $$-0.0180897\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 13985.0i 0.547815i
$$868$$ 0 0
$$869$$ 5786.00 0.225865
$$870$$ 0 0
$$871$$ −6396.00 −0.248818
$$872$$ 0 0
$$873$$ − 526.000i − 0.0203922i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 11084.0i − 0.426773i −0.976968 0.213387i $$-0.931551\pi$$
0.976968 0.213387i $$-0.0684494\pi$$
$$878$$ 0 0
$$879$$ 34840.0 1.33689
$$880$$ 0 0
$$881$$ 41797.0 1.59838 0.799192 0.601076i $$-0.205261\pi$$
0.799192 + 0.601076i $$0.205261\pi$$
$$882$$ 0 0
$$883$$ − 23780.0i − 0.906298i −0.891435 0.453149i $$-0.850301\pi$$
0.891435 0.453149i $$-0.149699\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 14334.0i − 0.542603i −0.962494 0.271301i $$-0.912546\pi$$
0.962494 0.271301i $$-0.0874540\pi$$
$$888$$ 0 0
$$889$$ 45136.0 1.70283
$$890$$ 0 0
$$891$$ 7381.00 0.277523
$$892$$ 0 0
$$893$$ − 21504.0i − 0.805827i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 7020.00i − 0.261305i
$$898$$ 0 0
$$899$$ 4688.00 0.173919
$$900$$ 0 0
$$901$$ 34684.0 1.28245
$$902$$ 0 0
$$903$$ 14300.0i 0.526992i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 25716.0i 0.941440i 0.882283 + 0.470720i $$0.156006\pi$$
−0.882283 + 0.470720i $$0.843994\pi$$
$$908$$ 0 0
$$909$$ −1628.00 −0.0594030
$$910$$ 0 0
$$911$$ −28300.0 −1.02922 −0.514611 0.857424i $$-0.672064\pi$$
−0.514611 + 0.857424i $$0.672064\pi$$
$$912$$ 0 0
$$913$$ − 1738.00i − 0.0630004i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 156.000i − 0.00561786i
$$918$$ 0 0
$$919$$ −21410.0 −0.768499 −0.384250 0.923229i $$-0.625540\pi$$
−0.384250 + 0.923229i $$0.625540\pi$$
$$920$$ 0 0
$$921$$ −38200.0 −1.36670
$$922$$ 0 0
$$923$$ − 52052.0i − 1.85624i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 752.000i − 0.0266439i
$$928$$ 0 0
$$929$$ −19938.0 −0.704138 −0.352069 0.935974i $$-0.614522\pi$$
−0.352069 + 0.935974i $$0.614522\pi$$
$$930$$ 0 0
$$931$$ −31968.0 −1.12536
$$932$$ 0 0
$$933$$ − 3260.00i − 0.114392i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 26728.0i 0.931874i 0.884818 + 0.465937i $$0.154283\pi$$
−0.884818 + 0.465937i $$0.845717\pi$$
$$938$$ 0 0
$$939$$ 40275.0 1.39971
$$940$$ 0 0
$$941$$ 8634.00 0.299108 0.149554 0.988754i $$-0.452216\pi$$
0.149554 + 0.988754i $$0.452216\pi$$
$$942$$ 0 0
$$943$$ 12744.0i 0.440087i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 24841.0i 0.852401i 0.904629 + 0.426201i $$0.140148\pi$$
−0.904629 + 0.426201i $$0.859852\pi$$
$$948$$ 0 0
$$949$$ 53040.0 1.81428
$$950$$ 0 0
$$951$$ 28375.0 0.967531
$$952$$ 0 0
$$953$$ − 51234.0i − 1.74148i −0.491742 0.870741i $$-0.663640\pi$$
0.491742 0.870741i $$-0.336360\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 880.000i 0.0297245i
$$958$$ 0 0
$$959$$ −37414.0 −1.25981
$$960$$ 0 0
$$961$$ 56058.0 1.88171
$$962$$ 0 0
$$963$$ − 2988.00i − 0.0999865i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 26752.0i − 0.889645i −0.895619 0.444822i $$-0.853267\pi$$
0.895619 0.444822i $$-0.146733\pi$$
$$968$$ 0 0
$$969$$ −22080.0 −0.732004
$$970$$ 0 0
$$971$$ 21155.0 0.699172 0.349586 0.936904i $$-0.386322\pi$$
0.349586 + 0.936904i $$0.386322\pi$$
$$972$$ 0 0
$$973$$ − 8268.00i − 0.272415i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 56323.0i − 1.84435i −0.386770 0.922176i $$-0.626409\pi$$
0.386770 0.922176i $$-0.373591\pi$$
$$978$$ 0 0
$$979$$ −13387.0 −0.437028
$$980$$ 0 0
$$981$$ −1684.00 −0.0548073
$$982$$ 0 0
$$983$$ − 24683.0i − 0.800880i −0.916323 0.400440i $$-0.868857\pi$$
0.916323 0.400440i $$-0.131143\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 29120.0i − 0.939108i
$$988$$ 0 0
$$989$$ 2970.00 0.0954909
$$990$$ 0 0
$$991$$ −26816.0 −0.859574 −0.429787 0.902930i $$-0.641411\pi$$
−0.429787 + 0.902930i $$0.641411\pi$$
$$992$$ 0 0
$$993$$ − 11225.0i − 0.358726i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 18614.0i 0.591285i 0.955299 + 0.295643i $$0.0955337\pi$$
−0.955299 + 0.295643i $$0.904466\pi$$
$$998$$ 0 0
$$999$$ 4205.00 0.133173
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 1100.4.b.c.749.2 2
5.2 odd 4 1100.4.a.d.1.1 1
5.3 odd 4 44.4.a.a.1.1 1
5.4 even 2 inner 1100.4.b.c.749.1 2
15.8 even 4 396.4.a.e.1.1 1
20.3 even 4 176.4.a.e.1.1 1
35.13 even 4 2156.4.a.b.1.1 1
40.3 even 4 704.4.a.c.1.1 1
40.13 odd 4 704.4.a.j.1.1 1
55.43 even 4 484.4.a.a.1.1 1
60.23 odd 4 1584.4.a.p.1.1 1
220.43 odd 4 1936.4.a.m.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
44.4.a.a.1.1 1 5.3 odd 4
176.4.a.e.1.1 1 20.3 even 4
396.4.a.e.1.1 1 15.8 even 4
484.4.a.a.1.1 1 55.43 even 4
704.4.a.c.1.1 1 40.3 even 4
704.4.a.j.1.1 1 40.13 odd 4
1100.4.a.d.1.1 1 5.2 odd 4
1100.4.b.c.749.1 2 5.4 even 2 inner
1100.4.b.c.749.2 2 1.1 even 1 trivial
1584.4.a.p.1.1 1 60.23 odd 4
1936.4.a.m.1.1 1 220.43 odd 4
2156.4.a.b.1.1 1 35.13 even 4