# Properties

 Label 7650.2.a.o.1.1 Level $7650$ Weight $2$ Character 7650.1 Self dual yes Analytic conductor $61.086$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7650,2,Mod(1,7650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7650, 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("7650.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7650 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7650.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: $$61.0855575463$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2550) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 7650.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} +3.00000 q^{11} +4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} -5.00000 q^{19} -3.00000 q^{22} +8.00000 q^{23} -4.00000 q^{26} -1.00000 q^{28} +4.00000 q^{29} -3.00000 q^{31} -1.00000 q^{32} -1.00000 q^{34} +7.00000 q^{37} +5.00000 q^{38} -2.00000 q^{41} +1.00000 q^{43} +3.00000 q^{44} -8.00000 q^{46} -7.00000 q^{47} -6.00000 q^{49} +4.00000 q^{52} +7.00000 q^{53} +1.00000 q^{56} -4.00000 q^{58} +8.00000 q^{59} -2.00000 q^{61} +3.00000 q^{62} +1.00000 q^{64} +11.0000 q^{67} +1.00000 q^{68} +6.00000 q^{71} +2.00000 q^{73} -7.00000 q^{74} -5.00000 q^{76} -3.00000 q^{77} -15.0000 q^{79} +2.00000 q^{82} +16.0000 q^{83} -1.00000 q^{86} -3.00000 q^{88} -2.00000 q^{89} -4.00000 q^{91} +8.00000 q^{92} +7.00000 q^{94} -10.0000 q^{97} +6.00000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −1.00000 −0.377964 −0.188982 0.981981i $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 0 0
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 1.00000 0.242536
$$18$$ 0 0
$$19$$ −5.00000 −1.14708 −0.573539 0.819178i $$-0.694430\pi$$
−0.573539 + 0.819178i $$0.694430\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −3.00000 −0.639602
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −4.00000 −0.784465
$$27$$ 0 0
$$28$$ −1.00000 −0.188982
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ 0 0
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −1.00000 −0.171499
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 7.00000 1.15079 0.575396 0.817875i $$-0.304848\pi$$
0.575396 + 0.817875i $$0.304848\pi$$
$$38$$ 5.00000 0.811107
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ 1.00000 0.152499 0.0762493 0.997089i $$-0.475706\pi$$
0.0762493 + 0.997089i $$0.475706\pi$$
$$44$$ 3.00000 0.452267
$$45$$ 0 0
$$46$$ −8.00000 −1.17954
$$47$$ −7.00000 −1.02105 −0.510527 0.859861i $$-0.670550\pi$$
−0.510527 + 0.859861i $$0.670550\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 4.00000 0.554700
$$53$$ 7.00000 0.961524 0.480762 0.876851i $$-0.340360\pi$$
0.480762 + 0.876851i $$0.340360\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ −4.00000 −0.525226
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 3.00000 0.381000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 11.0000 1.34386 0.671932 0.740613i $$-0.265465\pi$$
0.671932 + 0.740613i $$0.265465\pi$$
$$68$$ 1.00000 0.121268
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 0 0
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ −7.00000 −0.813733
$$75$$ 0 0
$$76$$ −5.00000 −0.573539
$$77$$ −3.00000 −0.341882
$$78$$ 0 0
$$79$$ −15.0000 −1.68763 −0.843816 0.536633i $$-0.819696\pi$$
−0.843816 + 0.536633i $$0.819696\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 2.00000 0.220863
$$83$$ 16.0000 1.75623 0.878114 0.478451i $$-0.158802\pi$$
0.878114 + 0.478451i $$0.158802\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −1.00000 −0.107833
$$87$$ 0 0
$$88$$ −3.00000 −0.319801
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ 8.00000 0.834058
$$93$$ 0 0
$$94$$ 7.00000 0.721995
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ 6.00000 0.606092
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −5.00000 −0.497519 −0.248759 0.968565i $$-0.580023\pi$$
−0.248759 + 0.968565i $$0.580023\pi$$
$$102$$ 0 0
$$103$$ −10.0000 −0.985329 −0.492665 0.870219i $$-0.663977\pi$$
−0.492665 + 0.870219i $$0.663977\pi$$
$$104$$ −4.00000 −0.392232
$$105$$ 0 0
$$106$$ −7.00000 −0.679900
$$107$$ 15.0000 1.45010 0.725052 0.688694i $$-0.241816\pi$$
0.725052 + 0.688694i $$0.241816\pi$$
$$108$$ 0 0
$$109$$ −9.00000 −0.862044 −0.431022 0.902342i $$-0.641847\pi$$
−0.431022 + 0.902342i $$0.641847\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −1.00000 −0.0944911
$$113$$ −9.00000 −0.846649 −0.423324 0.905978i $$-0.639137\pi$$
−0.423324 + 0.905978i $$0.639137\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 4.00000 0.371391
$$117$$ 0 0
$$118$$ −8.00000 −0.736460
$$119$$ −1.00000 −0.0916698
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 2.00000 0.181071
$$123$$ 0 0
$$124$$ −3.00000 −0.269408
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −2.00000 −0.177471 −0.0887357 0.996055i $$-0.528283\pi$$
−0.0887357 + 0.996055i $$0.528283\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ 5.00000 0.433555
$$134$$ −11.0000 −0.950255
$$135$$ 0 0
$$136$$ −1.00000 −0.0857493
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −6.00000 −0.503509
$$143$$ 12.0000 1.00349
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ 0 0
$$148$$ 7.00000 0.575396
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 0 0
$$151$$ 20.0000 1.62758 0.813788 0.581161i $$-0.197401\pi$$
0.813788 + 0.581161i $$0.197401\pi$$
$$152$$ 5.00000 0.405554
$$153$$ 0 0
$$154$$ 3.00000 0.241747
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 4.00000 0.319235 0.159617 0.987179i $$-0.448974\pi$$
0.159617 + 0.987179i $$0.448974\pi$$
$$158$$ 15.0000 1.19334
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ 0 0
$$163$$ 18.0000 1.40987 0.704934 0.709273i $$-0.250976\pi$$
0.704934 + 0.709273i $$0.250976\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ −16.0000 −1.24184
$$167$$ −10.0000 −0.773823 −0.386912 0.922117i $$-0.626458\pi$$
−0.386912 + 0.922117i $$0.626458\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 1.00000 0.0762493
$$173$$ 2.00000 0.152057 0.0760286 0.997106i $$-0.475776\pi$$
0.0760286 + 0.997106i $$0.475776\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 3.00000 0.226134
$$177$$ 0 0
$$178$$ 2.00000 0.149906
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ −19.0000 −1.41226 −0.706129 0.708083i $$-0.749560\pi$$
−0.706129 + 0.708083i $$0.749560\pi$$
$$182$$ 4.00000 0.296500
$$183$$ 0 0
$$184$$ −8.00000 −0.589768
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 3.00000 0.219382
$$188$$ −7.00000 −0.510527
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −15.0000 −1.08536 −0.542681 0.839939i $$-0.682591\pi$$
−0.542681 + 0.839939i $$0.682591\pi$$
$$192$$ 0 0
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ −6.00000 −0.428571
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ 0 0
$$199$$ 13.0000 0.921546 0.460773 0.887518i $$-0.347572\pi$$
0.460773 + 0.887518i $$0.347572\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 5.00000 0.351799
$$203$$ −4.00000 −0.280745
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 10.0000 0.696733
$$207$$ 0 0
$$208$$ 4.00000 0.277350
$$209$$ −15.0000 −1.03757
$$210$$ 0 0
$$211$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$212$$ 7.00000 0.480762
$$213$$ 0 0
$$214$$ −15.0000 −1.02538
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 3.00000 0.203653
$$218$$ 9.00000 0.609557
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 4.00000 0.269069
$$222$$ 0 0
$$223$$ 2.00000 0.133930 0.0669650 0.997755i $$-0.478668\pi$$
0.0669650 + 0.997755i $$0.478668\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ 9.00000 0.598671
$$227$$ −27.0000 −1.79205 −0.896026 0.444001i $$-0.853559\pi$$
−0.896026 + 0.444001i $$0.853559\pi$$
$$228$$ 0 0
$$229$$ 28.0000 1.85029 0.925146 0.379611i $$-0.123942\pi$$
0.925146 + 0.379611i $$0.123942\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −4.00000 −0.262613
$$233$$ 22.0000 1.44127 0.720634 0.693316i $$-0.243851\pi$$
0.720634 + 0.693316i $$0.243851\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 8.00000 0.520756
$$237$$ 0 0
$$238$$ 1.00000 0.0648204
$$239$$ 3.00000 0.194054 0.0970269 0.995282i $$-0.469067\pi$$
0.0970269 + 0.995282i $$0.469067\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ 2.00000 0.128565
$$243$$ 0 0
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −20.0000 −1.27257
$$248$$ 3.00000 0.190500
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −10.0000 −0.631194 −0.315597 0.948893i $$-0.602205\pi$$
−0.315597 + 0.948893i $$0.602205\pi$$
$$252$$ 0 0
$$253$$ 24.0000 1.50887
$$254$$ 2.00000 0.125491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 20.0000 1.24757 0.623783 0.781598i $$-0.285595\pi$$
0.623783 + 0.781598i $$0.285595\pi$$
$$258$$ 0 0
$$259$$ −7.00000 −0.434959
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −12.0000 −0.741362
$$263$$ 9.00000 0.554964 0.277482 0.960731i $$-0.410500\pi$$
0.277482 + 0.960731i $$0.410500\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −5.00000 −0.306570
$$267$$ 0 0
$$268$$ 11.0000 0.671932
$$269$$ 8.00000 0.487769 0.243884 0.969804i $$-0.421578\pi$$
0.243884 + 0.969804i $$0.421578\pi$$
$$270$$ 0 0
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ 1.00000 0.0606339
$$273$$ 0 0
$$274$$ 18.0000 1.08742
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −5.00000 −0.300421 −0.150210 0.988654i $$-0.547995\pi$$
−0.150210 + 0.988654i $$0.547995\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 14.0000 0.835170 0.417585 0.908638i $$-0.362877\pi$$
0.417585 + 0.908638i $$0.362877\pi$$
$$282$$ 0 0
$$283$$ 26.0000 1.54554 0.772770 0.634686i $$-0.218871\pi$$
0.772770 + 0.634686i $$0.218871\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ −12.0000 −0.709575
$$287$$ 2.00000 0.118056
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 2.00000 0.117041
$$293$$ −10.0000 −0.584206 −0.292103 0.956387i $$-0.594355\pi$$
−0.292103 + 0.956387i $$0.594355\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −7.00000 −0.406867
$$297$$ 0 0
$$298$$ −18.0000 −1.04271
$$299$$ 32.0000 1.85061
$$300$$ 0 0
$$301$$ −1.00000 −0.0576390
$$302$$ −20.0000 −1.15087
$$303$$ 0 0
$$304$$ −5.00000 −0.286770
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 16.0000 0.913168 0.456584 0.889680i $$-0.349073\pi$$
0.456584 + 0.889680i $$0.349073\pi$$
$$308$$ −3.00000 −0.170941
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −2.00000 −0.113410 −0.0567048 0.998391i $$-0.518059\pi$$
−0.0567048 + 0.998391i $$0.518059\pi$$
$$312$$ 0 0
$$313$$ 8.00000 0.452187 0.226093 0.974106i $$-0.427405\pi$$
0.226093 + 0.974106i $$0.427405\pi$$
$$314$$ −4.00000 −0.225733
$$315$$ 0 0
$$316$$ −15.0000 −0.843816
$$317$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$318$$ 0 0
$$319$$ 12.0000 0.671871
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 8.00000 0.445823
$$323$$ −5.00000 −0.278207
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −18.0000 −0.996928
$$327$$ 0 0
$$328$$ 2.00000 0.110432
$$329$$ 7.00000 0.385922
$$330$$ 0 0
$$331$$ 17.0000 0.934405 0.467202 0.884150i $$-0.345262\pi$$
0.467202 + 0.884150i $$0.345262\pi$$
$$332$$ 16.0000 0.878114
$$333$$ 0 0
$$334$$ 10.0000 0.547176
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 24.0000 1.30736 0.653682 0.756770i $$-0.273224\pi$$
0.653682 + 0.756770i $$0.273224\pi$$
$$338$$ −3.00000 −0.163178
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −9.00000 −0.487377
$$342$$ 0 0
$$343$$ 13.0000 0.701934
$$344$$ −1.00000 −0.0539164
$$345$$ 0 0
$$346$$ −2.00000 −0.107521
$$347$$ 5.00000 0.268414 0.134207 0.990953i $$-0.457151\pi$$
0.134207 + 0.990953i $$0.457151\pi$$
$$348$$ 0 0
$$349$$ −22.0000 −1.17763 −0.588817 0.808267i $$-0.700406\pi$$
−0.588817 + 0.808267i $$0.700406\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −3.00000 −0.159901
$$353$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −2.00000 −0.106000
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 25.0000 1.31945 0.659725 0.751507i $$-0.270673\pi$$
0.659725 + 0.751507i $$0.270673\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ 19.0000 0.998618
$$363$$ 0 0
$$364$$ −4.00000 −0.209657
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 13.0000 0.678594 0.339297 0.940679i $$-0.389811\pi$$
0.339297 + 0.940679i $$0.389811\pi$$
$$368$$ 8.00000 0.417029
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −7.00000 −0.363422
$$372$$ 0 0
$$373$$ 38.0000 1.96757 0.983783 0.179364i $$-0.0574041\pi$$
0.983783 + 0.179364i $$0.0574041\pi$$
$$374$$ −3.00000 −0.155126
$$375$$ 0 0
$$376$$ 7.00000 0.360997
$$377$$ 16.0000 0.824042
$$378$$ 0 0
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 15.0000 0.767467
$$383$$ 20.0000 1.02195 0.510976 0.859595i $$-0.329284\pi$$
0.510976 + 0.859595i $$0.329284\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 14.0000 0.712581
$$387$$ 0 0
$$388$$ −10.0000 −0.507673
$$389$$ −9.00000 −0.456318 −0.228159 0.973624i $$-0.573271\pi$$
−0.228159 + 0.973624i $$0.573271\pi$$
$$390$$ 0 0
$$391$$ 8.00000 0.404577
$$392$$ 6.00000 0.303046
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −25.0000 −1.25471 −0.627357 0.778732i $$-0.715863\pi$$
−0.627357 + 0.778732i $$0.715863\pi$$
$$398$$ −13.0000 −0.651631
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ −12.0000 −0.597763
$$404$$ −5.00000 −0.248759
$$405$$ 0 0
$$406$$ 4.00000 0.198517
$$407$$ 21.0000 1.04093
$$408$$ 0 0
$$409$$ −2.00000 −0.0988936 −0.0494468 0.998777i $$-0.515746\pi$$
−0.0494468 + 0.998777i $$0.515746\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −10.0000 −0.492665
$$413$$ −8.00000 −0.393654
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −4.00000 −0.196116
$$417$$ 0 0
$$418$$ 15.0000 0.733674
$$419$$ −20.0000 −0.977064 −0.488532 0.872546i $$-0.662467\pi$$
−0.488532 + 0.872546i $$0.662467\pi$$
$$420$$ 0 0
$$421$$ 20.0000 0.974740 0.487370 0.873195i $$-0.337956\pi$$
0.487370 + 0.873195i $$0.337956\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ −7.00000 −0.339950
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 2.00000 0.0967868
$$428$$ 15.0000 0.725052
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −26.0000 −1.25238 −0.626188 0.779672i $$-0.715386\pi$$
−0.626188 + 0.779672i $$0.715386\pi$$
$$432$$ 0 0
$$433$$ 5.00000 0.240285 0.120142 0.992757i $$-0.461665\pi$$
0.120142 + 0.992757i $$0.461665\pi$$
$$434$$ −3.00000 −0.144005
$$435$$ 0 0
$$436$$ −9.00000 −0.431022
$$437$$ −40.0000 −1.91346
$$438$$ 0 0
$$439$$ −12.0000 −0.572729 −0.286364 0.958121i $$-0.592447\pi$$
−0.286364 + 0.958121i $$0.592447\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −4.00000 −0.190261
$$443$$ 26.0000 1.23530 0.617649 0.786454i $$-0.288085\pi$$
0.617649 + 0.786454i $$0.288085\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −2.00000 −0.0947027
$$447$$ 0 0
$$448$$ −1.00000 −0.0472456
$$449$$ −13.0000 −0.613508 −0.306754 0.951789i $$-0.599243\pi$$
−0.306754 + 0.951789i $$0.599243\pi$$
$$450$$ 0 0
$$451$$ −6.00000 −0.282529
$$452$$ −9.00000 −0.423324
$$453$$ 0 0
$$454$$ 27.0000 1.26717
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 11.0000 0.514558 0.257279 0.966337i $$-0.417174\pi$$
0.257279 + 0.966337i $$0.417174\pi$$
$$458$$ −28.0000 −1.30835
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 9.00000 0.419172 0.209586 0.977790i $$-0.432788\pi$$
0.209586 + 0.977790i $$0.432788\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ 4.00000 0.185695
$$465$$ 0 0
$$466$$ −22.0000 −1.01913
$$467$$ −18.0000 −0.832941 −0.416470 0.909149i $$-0.636733\pi$$
−0.416470 + 0.909149i $$0.636733\pi$$
$$468$$ 0 0
$$469$$ −11.0000 −0.507933
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −8.00000 −0.368230
$$473$$ 3.00000 0.137940
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −1.00000 −0.0458349
$$477$$ 0 0
$$478$$ −3.00000 −0.137217
$$479$$ 10.0000 0.456912 0.228456 0.973554i $$-0.426632\pi$$
0.228456 + 0.973554i $$0.426632\pi$$
$$480$$ 0 0
$$481$$ 28.0000 1.27669
$$482$$ 0 0
$$483$$ 0 0
$$484$$ −2.00000 −0.0909091
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −40.0000 −1.81257 −0.906287 0.422664i $$-0.861095\pi$$
−0.906287 + 0.422664i $$0.861095\pi$$
$$488$$ 2.00000 0.0905357
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −22.0000 −0.992846 −0.496423 0.868081i $$-0.665354\pi$$
−0.496423 + 0.868081i $$0.665354\pi$$
$$492$$ 0 0
$$493$$ 4.00000 0.180151
$$494$$ 20.0000 0.899843
$$495$$ 0 0
$$496$$ −3.00000 −0.134704
$$497$$ −6.00000 −0.269137
$$498$$ 0 0
$$499$$ −6.00000 −0.268597 −0.134298 0.990941i $$-0.542878\pi$$
−0.134298 + 0.990941i $$0.542878\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 10.0000 0.446322
$$503$$ 2.00000 0.0891756 0.0445878 0.999005i $$-0.485803\pi$$
0.0445878 + 0.999005i $$0.485803\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −24.0000 −1.06693
$$507$$ 0 0
$$508$$ −2.00000 −0.0887357
$$509$$ 29.0000 1.28540 0.642701 0.766117i $$-0.277814\pi$$
0.642701 + 0.766117i $$0.277814\pi$$
$$510$$ 0 0
$$511$$ −2.00000 −0.0884748
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −20.0000 −0.882162
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −21.0000 −0.923579
$$518$$ 7.00000 0.307562
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 11.0000 0.481919 0.240959 0.970535i $$-0.422538\pi$$
0.240959 + 0.970535i $$0.422538\pi$$
$$522$$ 0 0
$$523$$ −36.0000 −1.57417 −0.787085 0.616844i $$-0.788411\pi$$
−0.787085 + 0.616844i $$0.788411\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ −9.00000 −0.392419
$$527$$ −3.00000 −0.130682
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 5.00000 0.216777
$$533$$ −8.00000 −0.346518
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −11.0000 −0.475128
$$537$$ 0 0
$$538$$ −8.00000 −0.344904
$$539$$ −18.0000 −0.775315
$$540$$ 0 0
$$541$$ 17.0000 0.730887 0.365444 0.930834i $$-0.380917\pi$$
0.365444 + 0.930834i $$0.380917\pi$$
$$542$$ 20.0000 0.859074
$$543$$ 0 0
$$544$$ −1.00000 −0.0428746
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 42.0000 1.79579 0.897895 0.440209i $$-0.145096\pi$$
0.897895 + 0.440209i $$0.145096\pi$$
$$548$$ −18.0000 −0.768922
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −20.0000 −0.852029
$$552$$ 0 0
$$553$$ 15.0000 0.637865
$$554$$ 5.00000 0.212430
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −11.0000 −0.466085 −0.233042 0.972467i $$-0.574868\pi$$
−0.233042 + 0.972467i $$0.574868\pi$$
$$558$$ 0 0
$$559$$ 4.00000 0.169182
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −14.0000 −0.590554
$$563$$ 20.0000 0.842900 0.421450 0.906852i $$-0.361521\pi$$
0.421450 + 0.906852i $$0.361521\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −26.0000 −1.09286
$$567$$ 0 0
$$568$$ −6.00000 −0.251754
$$569$$ −30.0000 −1.25767 −0.628833 0.777541i $$-0.716467\pi$$
−0.628833 + 0.777541i $$0.716467\pi$$
$$570$$ 0 0
$$571$$ 30.0000 1.25546 0.627730 0.778431i $$-0.283984\pi$$
0.627730 + 0.778431i $$0.283984\pi$$
$$572$$ 12.0000 0.501745
$$573$$ 0 0
$$574$$ −2.00000 −0.0834784
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −11.0000 −0.457936 −0.228968 0.973434i $$-0.573535\pi$$
−0.228968 + 0.973434i $$0.573535\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −16.0000 −0.663792
$$582$$ 0 0
$$583$$ 21.0000 0.869731
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ 10.0000 0.413096
$$587$$ 22.0000 0.908037 0.454019 0.890992i $$-0.349990\pi$$
0.454019 + 0.890992i $$0.349990\pi$$
$$588$$ 0 0
$$589$$ 15.0000 0.618064
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 7.00000 0.287698
$$593$$ −12.0000 −0.492781 −0.246390 0.969171i $$-0.579245\pi$$
−0.246390 + 0.969171i $$0.579245\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 18.0000 0.737309
$$597$$ 0 0
$$598$$ −32.0000 −1.30858
$$599$$ −15.0000 −0.612883 −0.306442 0.951889i $$-0.599138\pi$$
−0.306442 + 0.951889i $$0.599138\pi$$
$$600$$ 0 0
$$601$$ −16.0000 −0.652654 −0.326327 0.945257i $$-0.605811\pi$$
−0.326327 + 0.945257i $$0.605811\pi$$
$$602$$ 1.00000 0.0407570
$$603$$ 0 0
$$604$$ 20.0000 0.813788
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −32.0000 −1.29884 −0.649420 0.760430i $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ 5.00000 0.202777
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −28.0000 −1.13276
$$612$$ 0 0
$$613$$ 30.0000 1.21169 0.605844 0.795583i $$-0.292835\pi$$
0.605844 + 0.795583i $$0.292835\pi$$
$$614$$ −16.0000 −0.645707
$$615$$ 0 0
$$616$$ 3.00000 0.120873
$$617$$ 33.0000 1.32853 0.664265 0.747497i $$-0.268745\pi$$
0.664265 + 0.747497i $$0.268745\pi$$
$$618$$ 0 0
$$619$$ 44.0000 1.76851 0.884255 0.467005i $$-0.154667\pi$$
0.884255 + 0.467005i $$0.154667\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 2.00000 0.0801927
$$623$$ 2.00000 0.0801283
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −8.00000 −0.319744
$$627$$ 0 0
$$628$$ 4.00000 0.159617
$$629$$ 7.00000 0.279108
$$630$$ 0 0
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ 15.0000 0.596668
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −24.0000 −0.950915
$$638$$ −12.0000 −0.475085
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 0 0
$$643$$ −32.0000 −1.26196 −0.630978 0.775800i $$-0.717346\pi$$
−0.630978 + 0.775800i $$0.717346\pi$$
$$644$$ −8.00000 −0.315244
$$645$$ 0 0
$$646$$ 5.00000 0.196722
$$647$$ 32.0000 1.25805 0.629025 0.777385i $$-0.283454\pi$$
0.629025 + 0.777385i $$0.283454\pi$$
$$648$$ 0 0
$$649$$ 24.0000 0.942082
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 18.0000 0.704934
$$653$$ 2.00000 0.0782660 0.0391330 0.999234i $$-0.487540\pi$$
0.0391330 + 0.999234i $$0.487540\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −2.00000 −0.0780869
$$657$$ 0 0
$$658$$ −7.00000 −0.272888
$$659$$ 48.0000 1.86981 0.934907 0.354892i $$-0.115482\pi$$
0.934907 + 0.354892i $$0.115482\pi$$
$$660$$ 0 0
$$661$$ 18.0000 0.700119 0.350059 0.936727i $$-0.386161\pi$$
0.350059 + 0.936727i $$0.386161\pi$$
$$662$$ −17.0000 −0.660724
$$663$$ 0 0
$$664$$ −16.0000 −0.620920
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 32.0000 1.23904
$$668$$ −10.0000 −0.386912
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −6.00000 −0.231627
$$672$$ 0 0
$$673$$ −4.00000 −0.154189 −0.0770943 0.997024i $$-0.524564\pi$$
−0.0770943 + 0.997024i $$0.524564\pi$$
$$674$$ −24.0000 −0.924445
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ −26.0000 −0.999261 −0.499631 0.866239i $$-0.666531\pi$$
−0.499631 + 0.866239i $$0.666531\pi$$
$$678$$ 0 0
$$679$$ 10.0000 0.383765
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 9.00000 0.344628
$$683$$ 24.0000 0.918334 0.459167 0.888350i $$-0.348148\pi$$
0.459167 + 0.888350i $$0.348148\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −13.0000 −0.496342
$$687$$ 0 0
$$688$$ 1.00000 0.0381246
$$689$$ 28.0000 1.06672
$$690$$ 0 0
$$691$$ 18.0000 0.684752 0.342376 0.939563i $$-0.388768\pi$$
0.342376 + 0.939563i $$0.388768\pi$$
$$692$$ 2.00000 0.0760286
$$693$$ 0 0
$$694$$ −5.00000 −0.189797
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −2.00000 −0.0757554
$$698$$ 22.0000 0.832712
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −30.0000 −1.13308 −0.566542 0.824033i $$-0.691719\pi$$
−0.566542 + 0.824033i $$0.691719\pi$$
$$702$$ 0 0
$$703$$ −35.0000 −1.32005
$$704$$ 3.00000 0.113067
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 5.00000 0.188044
$$708$$ 0 0
$$709$$ −1.00000 −0.0375558 −0.0187779 0.999824i $$-0.505978\pi$$
−0.0187779 + 0.999824i $$0.505978\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 2.00000 0.0749532
$$713$$ −24.0000 −0.898807
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ −25.0000 −0.932992
$$719$$ 10.0000 0.372937 0.186469 0.982461i $$-0.440296\pi$$
0.186469 + 0.982461i $$0.440296\pi$$
$$720$$ 0 0
$$721$$ 10.0000 0.372419
$$722$$ −6.00000 −0.223297
$$723$$ 0 0
$$724$$ −19.0000 −0.706129
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 12.0000 0.445055 0.222528 0.974926i $$-0.428569\pi$$
0.222528 + 0.974926i $$0.428569\pi$$
$$728$$ 4.00000 0.148250
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 1.00000 0.0369863
$$732$$ 0 0
$$733$$ 14.0000 0.517102 0.258551 0.965998i $$-0.416755\pi$$
0.258551 + 0.965998i $$0.416755\pi$$
$$734$$ −13.0000 −0.479839
$$735$$ 0 0
$$736$$ −8.00000 −0.294884
$$737$$ 33.0000 1.21557
$$738$$ 0 0
$$739$$ 11.0000 0.404642 0.202321 0.979319i $$-0.435152\pi$$
0.202321 + 0.979319i $$0.435152\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 7.00000 0.256978
$$743$$ 6.00000 0.220119 0.110059 0.993925i $$-0.464896\pi$$
0.110059 + 0.993925i $$0.464896\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −38.0000 −1.39128
$$747$$ 0 0
$$748$$ 3.00000 0.109691
$$749$$ −15.0000 −0.548088
$$750$$ 0 0
$$751$$ −4.00000 −0.145962 −0.0729810 0.997333i $$-0.523251\pi$$
−0.0729810 + 0.997333i $$0.523251\pi$$
$$752$$ −7.00000 −0.255264
$$753$$ 0 0
$$754$$ −16.0000 −0.582686
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −50.0000 −1.81728 −0.908640 0.417579i $$-0.862879\pi$$
−0.908640 + 0.417579i $$0.862879\pi$$
$$758$$ −16.0000 −0.581146
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 48.0000 1.74000 0.869999 0.493053i $$-0.164119\pi$$
0.869999 + 0.493053i $$0.164119\pi$$
$$762$$ 0 0
$$763$$ 9.00000 0.325822
$$764$$ −15.0000 −0.542681
$$765$$ 0 0
$$766$$ −20.0000 −0.722629
$$767$$ 32.0000 1.15545
$$768$$ 0 0
$$769$$ 11.0000 0.396670 0.198335 0.980134i $$-0.436447\pi$$
0.198335 + 0.980134i $$0.436447\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −14.0000 −0.503871
$$773$$ −18.0000 −0.647415 −0.323708 0.946157i $$-0.604929\pi$$
−0.323708 + 0.946157i $$0.604929\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 10.0000 0.358979
$$777$$ 0 0
$$778$$ 9.00000 0.322666
$$779$$ 10.0000 0.358287
$$780$$ 0 0
$$781$$ 18.0000 0.644091
$$782$$ −8.00000 −0.286079
$$783$$ 0 0
$$784$$ −6.00000 −0.214286
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −22.0000 −0.784215 −0.392108 0.919919i $$-0.628254\pi$$
−0.392108 + 0.919919i $$0.628254\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 9.00000 0.320003
$$792$$ 0 0
$$793$$ −8.00000 −0.284088
$$794$$ 25.0000 0.887217
$$795$$ 0 0
$$796$$ 13.0000 0.460773
$$797$$ −51.0000 −1.80651 −0.903256 0.429101i $$-0.858830\pi$$
−0.903256 + 0.429101i $$0.858830\pi$$
$$798$$ 0 0
$$799$$ −7.00000 −0.247642
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 30.0000 1.05934
$$803$$ 6.00000 0.211735
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 12.0000 0.422682
$$807$$ 0 0
$$808$$ 5.00000 0.175899
$$809$$ −5.00000 −0.175791 −0.0878953 0.996130i $$-0.528014\pi$$
−0.0878953 + 0.996130i $$0.528014\pi$$
$$810$$ 0 0
$$811$$ −2.00000 −0.0702295 −0.0351147 0.999383i $$-0.511180\pi$$
−0.0351147 + 0.999383i $$0.511180\pi$$
$$812$$ −4.00000 −0.140372
$$813$$ 0 0
$$814$$ −21.0000 −0.736050
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −5.00000 −0.174928
$$818$$ 2.00000 0.0699284
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ 0 0
$$823$$ −32.0000 −1.11545 −0.557725 0.830026i $$-0.688326\pi$$
−0.557725 + 0.830026i $$0.688326\pi$$
$$824$$ 10.0000 0.348367
$$825$$ 0 0
$$826$$ 8.00000 0.278356
$$827$$ 47.0000 1.63435 0.817175 0.576390i $$-0.195539\pi$$
0.817175 + 0.576390i $$0.195539\pi$$
$$828$$ 0 0
$$829$$ −28.0000 −0.972480 −0.486240 0.873825i $$-0.661632\pi$$
−0.486240 + 0.873825i $$0.661632\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 4.00000 0.138675
$$833$$ −6.00000 −0.207888
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −15.0000 −0.518786
$$837$$ 0 0
$$838$$ 20.0000 0.690889
$$839$$ 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
$$842$$ −20.0000 −0.689246
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 2.00000 0.0687208
$$848$$ 7.00000 0.240381
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 56.0000 1.91966
$$852$$ 0 0
$$853$$ 9.00000 0.308154 0.154077 0.988059i $$-0.450760\pi$$
0.154077 + 0.988059i $$0.450760\pi$$
$$854$$ −2.00000 −0.0684386
$$855$$ 0 0
$$856$$ −15.0000 −0.512689
$$857$$ 41.0000 1.40053 0.700267 0.713881i $$-0.253064\pi$$
0.700267 + 0.713881i $$0.253064\pi$$
$$858$$ 0 0
$$859$$ −45.0000 −1.53538 −0.767690 0.640821i $$-0.778594\pi$$
−0.767690 + 0.640821i $$0.778594\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 26.0000 0.885564
$$863$$ 1.00000 0.0340404 0.0170202 0.999855i $$-0.494582\pi$$
0.0170202 + 0.999855i $$0.494582\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −5.00000 −0.169907
$$867$$ 0 0
$$868$$ 3.00000 0.101827
$$869$$ −45.0000 −1.52652
$$870$$ 0 0
$$871$$ 44.0000 1.49088
$$872$$ 9.00000 0.304778
$$873$$ 0 0
$$874$$ 40.0000 1.35302
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 46.0000 1.55331 0.776655 0.629926i $$-0.216915\pi$$
0.776655 + 0.629926i $$0.216915\pi$$
$$878$$ 12.0000 0.404980
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 57.0000 1.92038 0.960189 0.279350i $$-0.0901189\pi$$
0.960189 + 0.279350i $$0.0901189\pi$$
$$882$$ 0 0
$$883$$ −56.0000 −1.88455 −0.942275 0.334840i $$-0.891318\pi$$
−0.942275 + 0.334840i $$0.891318\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 0 0
$$886$$ −26.0000 −0.873487
$$887$$ 28.0000 0.940148 0.470074 0.882627i $$-0.344227\pi$$
0.470074 + 0.882627i $$0.344227\pi$$
$$888$$ 0 0
$$889$$ 2.00000 0.0670778
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 2.00000 0.0669650
$$893$$ 35.0000 1.17123
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ 13.0000 0.433816
$$899$$ −12.0000 −0.400222
$$900$$ 0 0
$$901$$ 7.00000 0.233204
$$902$$ 6.00000 0.199778
$$903$$ 0 0
$$904$$ 9.00000 0.299336
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 40.0000 1.32818 0.664089 0.747653i $$-0.268820\pi$$
0.664089 + 0.747653i $$0.268820\pi$$
$$908$$ −27.0000 −0.896026
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 48.0000 1.59031 0.795155 0.606406i $$-0.207389\pi$$
0.795155 + 0.606406i $$0.207389\pi$$
$$912$$ 0 0
$$913$$ 48.0000 1.58857
$$914$$ −11.0000 −0.363848
$$915$$ 0 0
$$916$$ 28.0000 0.925146
$$917$$ −12.0000 −0.396275
$$918$$ 0 0
$$919$$ 26.0000 0.857661 0.428830 0.903385i $$-0.358926\pi$$
0.428830 + 0.903385i $$0.358926\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −9.00000 −0.296399
$$923$$ 24.0000 0.789970
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −16.0000 −0.525793
$$927$$ 0 0
$$928$$ −4.00000 −0.131306
$$929$$ −19.0000 −0.623370 −0.311685 0.950186i $$-0.600893\pi$$
−0.311685 + 0.950186i $$0.600893\pi$$
$$930$$ 0 0
$$931$$ 30.0000 0.983210
$$932$$ 22.0000 0.720634
$$933$$ 0 0
$$934$$ 18.0000 0.588978
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −42.0000 −1.37208 −0.686040 0.727564i $$-0.740653\pi$$
−0.686040 + 0.727564i $$0.740653\pi$$
$$938$$ 11.0000 0.359163
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 58.0000 1.89075 0.945373 0.325991i $$-0.105698\pi$$
0.945373 + 0.325991i $$0.105698\pi$$
$$942$$ 0 0
$$943$$ −16.0000 −0.521032
$$944$$ 8.00000 0.260378
$$945$$ 0 0
$$946$$ −3.00000 −0.0975384
$$947$$ 13.0000 0.422443 0.211222 0.977438i $$-0.432256\pi$$
0.211222 + 0.977438i $$0.432256\pi$$
$$948$$ 0 0
$$949$$ 8.00000 0.259691
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 1.00000 0.0324102
$$953$$ 4.00000 0.129573 0.0647864 0.997899i $$-0.479363\pi$$
0.0647864 + 0.997899i $$0.479363\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 3.00000 0.0970269
$$957$$ 0 0
$$958$$ −10.0000 −0.323085
$$959$$ 18.0000 0.581250
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ −28.0000 −0.902756
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −28.0000 −0.900419 −0.450210 0.892923i $$-0.648651\pi$$
−0.450210 + 0.892923i $$0.648651\pi$$
$$968$$ 2.00000 0.0642824
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −6.00000 −0.192549 −0.0962746 0.995355i $$-0.530693\pi$$
−0.0962746 + 0.995355i $$0.530693\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 40.0000 1.28168
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ −12.0000 −0.383914 −0.191957 0.981403i $$-0.561483\pi$$
−0.191957 + 0.981403i $$0.561483\pi$$
$$978$$ 0 0
$$979$$ −6.00000 −0.191761
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 22.0000 0.702048
$$983$$ −32.0000 −1.02064 −0.510321 0.859984i $$-0.670473\pi$$
−0.510321 + 0.859984i $$0.670473\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −4.00000 −0.127386
$$987$$ 0 0
$$988$$ −20.0000 −0.636285
$$989$$ 8.00000 0.254385
$$990$$ 0 0
$$991$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$992$$ 3.00000 0.0952501
$$993$$ 0 0
$$994$$ 6.00000 0.190308
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −25.0000 −0.791758 −0.395879 0.918303i $$-0.629560\pi$$
−0.395879 + 0.918303i $$0.629560\pi$$
$$998$$ 6.00000 0.189927
$$999$$ 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 7650.2.a.o.1.1 1
3.2 odd 2 2550.2.a.v.1.1 yes 1
5.4 even 2 7650.2.a.cc.1.1 1
15.2 even 4 2550.2.d.d.2449.2 2
15.8 even 4 2550.2.d.d.2449.1 2
15.14 odd 2 2550.2.a.m.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
2550.2.a.m.1.1 1 15.14 odd 2
2550.2.a.v.1.1 yes 1 3.2 odd 2
2550.2.d.d.2449.1 2 15.8 even 4
2550.2.d.d.2449.2 2 15.2 even 4
7650.2.a.o.1.1 1 1.1 even 1 trivial
7650.2.a.cc.1.1 1 5.4 even 2