Properties

 Label 637.2.a.c.1.1 Level $637$ Weight $2$ Character 637.1 Self dual yes Analytic conductor $5.086$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [637,2,Mod(1,637)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(637, 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("637.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.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: $$5.08647060876$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 637.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{4} -3.00000 q^{8} -3.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{4} -3.00000 q^{8} -3.00000 q^{9} -3.00000 q^{11} -1.00000 q^{13} -1.00000 q^{16} +7.00000 q^{17} -3.00000 q^{18} -7.00000 q^{19} -3.00000 q^{22} -6.00000 q^{23} -5.00000 q^{25} -1.00000 q^{26} -5.00000 q^{29} +5.00000 q^{32} +7.00000 q^{34} +3.00000 q^{36} +8.00000 q^{37} -7.00000 q^{38} +2.00000 q^{43} +3.00000 q^{44} -6.00000 q^{46} +7.00000 q^{47} -5.00000 q^{50} +1.00000 q^{52} -3.00000 q^{53} -5.00000 q^{58} -7.00000 q^{59} -7.00000 q^{61} +7.00000 q^{64} -3.00000 q^{67} -7.00000 q^{68} -5.00000 q^{71} +9.00000 q^{72} +14.0000 q^{73} +8.00000 q^{74} +7.00000 q^{76} -6.00000 q^{79} +9.00000 q^{81} +2.00000 q^{86} +9.00000 q^{88} +6.00000 q^{92} +7.00000 q^{94} -14.0000 q^{97} +9.00000 q^{99} +O(q^{100})$$

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 0.353553 0.935414i $$-0.384973\pi$$
0.353553 + 0.935414i $$0.384973\pi$$
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −3.00000 −1.06066
$$9$$ −3.00000 −1.00000
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 7.00000 1.69775 0.848875 0.528594i $$-0.177281\pi$$
0.848875 + 0.528594i $$0.177281\pi$$
$$18$$ −3.00000 −0.707107
$$19$$ −7.00000 −1.60591 −0.802955 0.596040i $$-0.796740\pi$$
−0.802955 + 0.596040i $$0.796740\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −3.00000 −0.639602
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 0 0
$$25$$ −5.00000 −1.00000
$$26$$ −1.00000 −0.196116
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −5.00000 −0.928477 −0.464238 0.885710i $$-0.653672\pi$$
−0.464238 + 0.885710i $$0.653672\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 5.00000 0.883883
$$33$$ 0 0
$$34$$ 7.00000 1.20049
$$35$$ 0 0
$$36$$ 3.00000 0.500000
$$37$$ 8.00000 1.31519 0.657596 0.753371i $$-0.271573\pi$$
0.657596 + 0.753371i $$0.271573\pi$$
$$38$$ −7.00000 −1.13555
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 2.00000 0.304997 0.152499 0.988304i $$-0.451268\pi$$
0.152499 + 0.988304i $$0.451268\pi$$
$$44$$ 3.00000 0.452267
$$45$$ 0 0
$$46$$ −6.00000 −0.884652
$$47$$ 7.00000 1.02105 0.510527 0.859861i $$-0.329450\pi$$
0.510527 + 0.859861i $$0.329450\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −5.00000 −0.707107
$$51$$ 0 0
$$52$$ 1.00000 0.138675
$$53$$ −3.00000 −0.412082 −0.206041 0.978543i $$-0.566058\pi$$
−0.206041 + 0.978543i $$0.566058\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −5.00000 −0.656532
$$59$$ −7.00000 −0.911322 −0.455661 0.890153i $$-0.650597\pi$$
−0.455661 + 0.890153i $$0.650597\pi$$
$$60$$ 0 0
$$61$$ −7.00000 −0.896258 −0.448129 0.893969i $$-0.647910\pi$$
−0.448129 + 0.893969i $$0.647910\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −3.00000 −0.366508 −0.183254 0.983066i $$-0.558663\pi$$
−0.183254 + 0.983066i $$0.558663\pi$$
$$68$$ −7.00000 −0.848875
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −5.00000 −0.593391 −0.296695 0.954972i $$-0.595885\pi$$
−0.296695 + 0.954972i $$0.595885\pi$$
$$72$$ 9.00000 1.06066
$$73$$ 14.0000 1.63858 0.819288 0.573382i $$-0.194369\pi$$
0.819288 + 0.573382i $$0.194369\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 0 0
$$76$$ 7.00000 0.802955
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −6.00000 −0.675053 −0.337526 0.941316i $$-0.609590\pi$$
−0.337526 + 0.941316i $$0.609590\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 2.00000 0.215666
$$87$$ 0 0
$$88$$ 9.00000 0.959403
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 6.00000 0.625543
$$93$$ 0 0
$$94$$ 7.00000 0.721995
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ 0 0
$$99$$ 9.00000 0.904534
$$100$$ 5.00000 0.500000
$$101$$ −14.0000 −1.39305 −0.696526 0.717532i $$-0.745272\pi$$
−0.696526 + 0.717532i $$0.745272\pi$$
$$102$$ 0 0
$$103$$ 14.0000 1.37946 0.689730 0.724066i $$-0.257729\pi$$
0.689730 + 0.724066i $$0.257729\pi$$
$$104$$ 3.00000 0.294174
$$105$$ 0 0
$$106$$ −3.00000 −0.291386
$$107$$ 8.00000 0.773389 0.386695 0.922208i $$-0.373617\pi$$
0.386695 + 0.922208i $$0.373617\pi$$
$$108$$ 0 0
$$109$$ 4.00000 0.383131 0.191565 0.981480i $$-0.438644\pi$$
0.191565 + 0.981480i $$0.438644\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 9.00000 0.846649 0.423324 0.905978i $$-0.360863\pi$$
0.423324 + 0.905978i $$0.360863\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 5.00000 0.464238
$$117$$ 3.00000 0.277350
$$118$$ −7.00000 −0.644402
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ −7.00000 −0.633750
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 14.0000 1.22319 0.611593 0.791173i $$-0.290529\pi$$
0.611593 + 0.791173i $$0.290529\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −3.00000 −0.259161
$$135$$ 0 0
$$136$$ −21.0000 −1.80074
$$137$$ 4.00000 0.341743 0.170872 0.985293i $$-0.445342\pi$$
0.170872 + 0.985293i $$0.445342\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −5.00000 −0.419591
$$143$$ 3.00000 0.250873
$$144$$ 3.00000 0.250000
$$145$$ 0 0
$$146$$ 14.0000 1.15865
$$147$$ 0 0
$$148$$ −8.00000 −0.657596
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ −3.00000 −0.244137 −0.122068 0.992522i $$-0.538953\pi$$
−0.122068 + 0.992522i $$0.538953\pi$$
$$152$$ 21.0000 1.70332
$$153$$ −21.0000 −1.69775
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 7.00000 0.558661 0.279330 0.960195i $$-0.409888\pi$$
0.279330 + 0.960195i $$0.409888\pi$$
$$158$$ −6.00000 −0.477334
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 9.00000 0.707107
$$163$$ −13.0000 −1.01824 −0.509119 0.860696i $$-0.670029\pi$$
−0.509119 + 0.860696i $$0.670029\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −7.00000 −0.541676 −0.270838 0.962625i $$-0.587301\pi$$
−0.270838 + 0.962625i $$0.587301\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 21.0000 1.60591
$$172$$ −2.00000 −0.152499
$$173$$ 7.00000 0.532200 0.266100 0.963945i $$-0.414265\pi$$
0.266100 + 0.963945i $$0.414265\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 3.00000 0.226134
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −10.0000 −0.747435 −0.373718 0.927543i $$-0.621917\pi$$
−0.373718 + 0.927543i $$0.621917\pi$$
$$180$$ 0 0
$$181$$ −7.00000 −0.520306 −0.260153 0.965567i $$-0.583773\pi$$
−0.260153 + 0.965567i $$0.583773\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 18.0000 1.32698
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −21.0000 −1.53567
$$188$$ −7.00000 −0.510527
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −20.0000 −1.44715 −0.723575 0.690246i $$-0.757502\pi$$
−0.723575 + 0.690246i $$0.757502\pi$$
$$192$$ 0 0
$$193$$ 4.00000 0.287926 0.143963 0.989583i $$-0.454015\pi$$
0.143963 + 0.989583i $$0.454015\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 9.00000 0.639602
$$199$$ −14.0000 −0.992434 −0.496217 0.868199i $$-0.665278\pi$$
−0.496217 + 0.868199i $$0.665278\pi$$
$$200$$ 15.0000 1.06066
$$201$$ 0 0
$$202$$ −14.0000 −0.985037
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 14.0000 0.975426
$$207$$ 18.0000 1.25109
$$208$$ 1.00000 0.0693375
$$209$$ 21.0000 1.45260
$$210$$ 0 0
$$211$$ −26.0000 −1.78991 −0.894957 0.446153i $$-0.852794\pi$$
−0.894957 + 0.446153i $$0.852794\pi$$
$$212$$ 3.00000 0.206041
$$213$$ 0 0
$$214$$ 8.00000 0.546869
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 4.00000 0.270914
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −7.00000 −0.470871
$$222$$ 0 0
$$223$$ 21.0000 1.40626 0.703132 0.711059i $$-0.251784\pi$$
0.703132 + 0.711059i $$0.251784\pi$$
$$224$$ 0 0
$$225$$ 15.0000 1.00000
$$226$$ 9.00000 0.598671
$$227$$ −28.0000 −1.85843 −0.929213 0.369546i $$-0.879513\pi$$
−0.929213 + 0.369546i $$0.879513\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 15.0000 0.984798
$$233$$ −27.0000 −1.76883 −0.884414 0.466702i $$-0.845442\pi$$
−0.884414 + 0.466702i $$0.845442\pi$$
$$234$$ 3.00000 0.196116
$$235$$ 0 0
$$236$$ 7.00000 0.455661
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −19.0000 −1.22901 −0.614504 0.788914i $$-0.710644\pi$$
−0.614504 + 0.788914i $$0.710644\pi$$
$$240$$ 0 0
$$241$$ 28.0000 1.80364 0.901819 0.432113i $$-0.142232\pi$$
0.901819 + 0.432113i $$0.142232\pi$$
$$242$$ −2.00000 −0.128565
$$243$$ 0 0
$$244$$ 7.00000 0.448129
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 7.00000 0.445399
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −14.0000 −0.883672 −0.441836 0.897096i $$-0.645673\pi$$
−0.441836 + 0.897096i $$0.645673\pi$$
$$252$$ 0 0
$$253$$ 18.0000 1.13165
$$254$$ 2.00000 0.125491
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ −14.0000 −0.873296 −0.436648 0.899632i $$-0.643834\pi$$
−0.436648 + 0.899632i $$0.643834\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 15.0000 0.928477
$$262$$ 14.0000 0.864923
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 3.00000 0.183254
$$269$$ −21.0000 −1.28039 −0.640196 0.768211i $$-0.721147\pi$$
−0.640196 + 0.768211i $$0.721147\pi$$
$$270$$ 0 0
$$271$$ −7.00000 −0.425220 −0.212610 0.977137i $$-0.568196\pi$$
−0.212610 + 0.977137i $$0.568196\pi$$
$$272$$ −7.00000 −0.424437
$$273$$ 0 0
$$274$$ 4.00000 0.241649
$$275$$ 15.0000 0.904534
$$276$$ 0 0
$$277$$ −17.0000 −1.02143 −0.510716 0.859750i $$-0.670619\pi$$
−0.510716 + 0.859750i $$0.670619\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −12.0000 −0.715860 −0.357930 0.933748i $$-0.616517\pi$$
−0.357930 + 0.933748i $$0.616517\pi$$
$$282$$ 0 0
$$283$$ −14.0000 −0.832214 −0.416107 0.909316i $$-0.636606\pi$$
−0.416107 + 0.909316i $$0.636606\pi$$
$$284$$ 5.00000 0.296695
$$285$$ 0 0
$$286$$ 3.00000 0.177394
$$287$$ 0 0
$$288$$ −15.0000 −0.883883
$$289$$ 32.0000 1.88235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −14.0000 −0.819288
$$293$$ 14.0000 0.817889 0.408944 0.912559i $$-0.365897\pi$$
0.408944 + 0.912559i $$0.365897\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −24.0000 −1.39497
$$297$$ 0 0
$$298$$ −6.00000 −0.347571
$$299$$ 6.00000 0.346989
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −3.00000 −0.172631
$$303$$ 0 0
$$304$$ 7.00000 0.401478
$$305$$ 0 0
$$306$$ −21.0000 −1.20049
$$307$$ 21.0000 1.19853 0.599267 0.800549i $$-0.295459\pi$$
0.599267 + 0.800549i $$0.295459\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ −14.0000 −0.791327 −0.395663 0.918396i $$-0.629485\pi$$
−0.395663 + 0.918396i $$0.629485\pi$$
$$314$$ 7.00000 0.395033
$$315$$ 0 0
$$316$$ 6.00000 0.337526
$$317$$ −6.00000 −0.336994 −0.168497 0.985702i $$-0.553891\pi$$
−0.168497 + 0.985702i $$0.553891\pi$$
$$318$$ 0 0
$$319$$ 15.0000 0.839839
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −49.0000 −2.72643
$$324$$ −9.00000 −0.500000
$$325$$ 5.00000 0.277350
$$326$$ −13.0000 −0.720003
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 0 0
$$333$$ −24.0000 −1.31519
$$334$$ −7.00000 −0.383023
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 23.0000 1.25289 0.626445 0.779466i $$-0.284509\pi$$
0.626445 + 0.779466i $$0.284509\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 21.0000 1.13555
$$343$$ 0 0
$$344$$ −6.00000 −0.323498
$$345$$ 0 0
$$346$$ 7.00000 0.376322
$$347$$ 4.00000 0.214731 0.107366 0.994220i $$-0.465758\pi$$
0.107366 + 0.994220i $$0.465758\pi$$
$$348$$ 0 0
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −15.0000 −0.799503
$$353$$ 14.0000 0.745145 0.372572 0.928003i $$-0.378476\pi$$
0.372572 + 0.928003i $$0.378476\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ −10.0000 −0.528516
$$359$$ 8.00000 0.422224 0.211112 0.977462i $$-0.432292\pi$$
0.211112 + 0.977462i $$0.432292\pi$$
$$360$$ 0 0
$$361$$ 30.0000 1.57895
$$362$$ −7.00000 −0.367912
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 14.0000 0.730794 0.365397 0.930852i $$-0.380933\pi$$
0.365397 + 0.930852i $$0.380933\pi$$
$$368$$ 6.00000 0.312772
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 15.0000 0.776671 0.388335 0.921518i $$-0.373050\pi$$
0.388335 + 0.921518i $$0.373050\pi$$
$$374$$ −21.0000 −1.08588
$$375$$ 0 0
$$376$$ −21.0000 −1.08299
$$377$$ 5.00000 0.257513
$$378$$ 0 0
$$379$$ −12.0000 −0.616399 −0.308199 0.951322i $$-0.599726\pi$$
−0.308199 + 0.951322i $$0.599726\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −20.0000 −1.02329
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 4.00000 0.203595
$$387$$ −6.00000 −0.304997
$$388$$ 14.0000 0.710742
$$389$$ −3.00000 −0.152106 −0.0760530 0.997104i $$-0.524232\pi$$
−0.0760530 + 0.997104i $$0.524232\pi$$
$$390$$ 0 0
$$391$$ −42.0000 −2.12403
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 2.00000 0.100759
$$395$$ 0 0
$$396$$ −9.00000 −0.452267
$$397$$ 14.0000 0.702640 0.351320 0.936255i $$-0.385733\pi$$
0.351320 + 0.936255i $$0.385733\pi$$
$$398$$ −14.0000 −0.701757
$$399$$ 0 0
$$400$$ 5.00000 0.250000
$$401$$ 22.0000 1.09863 0.549314 0.835616i $$-0.314889\pi$$
0.549314 + 0.835616i $$0.314889\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 14.0000 0.696526
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −24.0000 −1.18964
$$408$$ 0 0
$$409$$ −28.0000 −1.38451 −0.692255 0.721653i $$-0.743383\pi$$
−0.692255 + 0.721653i $$0.743383\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −14.0000 −0.689730
$$413$$ 0 0
$$414$$ 18.0000 0.884652
$$415$$ 0 0
$$416$$ −5.00000 −0.245145
$$417$$ 0 0
$$418$$ 21.0000 1.02714
$$419$$ 14.0000 0.683945 0.341972 0.939710i $$-0.388905\pi$$
0.341972 + 0.939710i $$0.388905\pi$$
$$420$$ 0 0
$$421$$ 30.0000 1.46211 0.731055 0.682318i $$-0.239028\pi$$
0.731055 + 0.682318i $$0.239028\pi$$
$$422$$ −26.0000 −1.26566
$$423$$ −21.0000 −1.02105
$$424$$ 9.00000 0.437079
$$425$$ −35.0000 −1.69775
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −8.00000 −0.386695
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 0 0
$$433$$ 21.0000 1.00920 0.504598 0.863355i $$-0.331641\pi$$
0.504598 + 0.863355i $$0.331641\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −4.00000 −0.191565
$$437$$ 42.0000 2.00913
$$438$$ 0 0
$$439$$ 14.0000 0.668184 0.334092 0.942541i $$-0.391570\pi$$
0.334092 + 0.942541i $$0.391570\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −7.00000 −0.332956
$$443$$ −20.0000 −0.950229 −0.475114 0.879924i $$-0.657593\pi$$
−0.475114 + 0.879924i $$0.657593\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 21.0000 0.994379
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −12.0000 −0.566315 −0.283158 0.959073i $$-0.591382\pi$$
−0.283158 + 0.959073i $$0.591382\pi$$
$$450$$ 15.0000 0.707107
$$451$$ 0 0
$$452$$ −9.00000 −0.423324
$$453$$ 0 0
$$454$$ −28.0000 −1.31411
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −6.00000 −0.280668 −0.140334 0.990104i $$-0.544818\pi$$
−0.140334 + 0.990104i $$0.544818\pi$$
$$458$$ 14.0000 0.654177
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −28.0000 −1.30409 −0.652045 0.758180i $$-0.726089\pi$$
−0.652045 + 0.758180i $$0.726089\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ 5.00000 0.232119
$$465$$ 0 0
$$466$$ −27.0000 −1.25075
$$467$$ 14.0000 0.647843 0.323921 0.946084i $$-0.394999\pi$$
0.323921 + 0.946084i $$0.394999\pi$$
$$468$$ −3.00000 −0.138675
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 21.0000 0.966603
$$473$$ −6.00000 −0.275880
$$474$$ 0 0
$$475$$ 35.0000 1.60591
$$476$$ 0 0
$$477$$ 9.00000 0.412082
$$478$$ −19.0000 −0.869040
$$479$$ 7.00000 0.319838 0.159919 0.987130i $$-0.448877\pi$$
0.159919 + 0.987130i $$0.448877\pi$$
$$480$$ 0 0
$$481$$ −8.00000 −0.364769
$$482$$ 28.0000 1.27537
$$483$$ 0 0
$$484$$ 2.00000 0.0909091
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 25.0000 1.13286 0.566429 0.824110i $$-0.308325\pi$$
0.566429 + 0.824110i $$0.308325\pi$$
$$488$$ 21.0000 0.950625
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 30.0000 1.35388 0.676941 0.736038i $$-0.263305\pi$$
0.676941 + 0.736038i $$0.263305\pi$$
$$492$$ 0 0
$$493$$ −35.0000 −1.57632
$$494$$ 7.00000 0.314945
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 8.00000 0.358129 0.179065 0.983837i $$-0.442693\pi$$
0.179065 + 0.983837i $$0.442693\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −14.0000 −0.624851
$$503$$ −28.0000 −1.24846 −0.624229 0.781241i $$-0.714587\pi$$
−0.624229 + 0.781241i $$0.714587\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 18.0000 0.800198
$$507$$ 0 0
$$508$$ −2.00000 −0.0887357
$$509$$ −28.0000 −1.24108 −0.620539 0.784176i $$-0.713086\pi$$
−0.620539 + 0.784176i $$0.713086\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −11.0000 −0.486136
$$513$$ 0 0
$$514$$ −14.0000 −0.617514
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −21.0000 −0.923579
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −14.0000 −0.613351 −0.306676 0.951814i $$-0.599217\pi$$
−0.306676 + 0.951814i $$0.599217\pi$$
$$522$$ 15.0000 0.656532
$$523$$ −14.0000 −0.612177 −0.306089 0.952003i $$-0.599020\pi$$
−0.306089 + 0.952003i $$0.599020\pi$$
$$524$$ −14.0000 −0.611593
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 21.0000 0.911322
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 9.00000 0.388741
$$537$$ 0 0
$$538$$ −21.0000 −0.905374
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 8.00000 0.343947 0.171973 0.985102i $$-0.444986\pi$$
0.171973 + 0.985102i $$0.444986\pi$$
$$542$$ −7.00000 −0.300676
$$543$$ 0 0
$$544$$ 35.0000 1.50061
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 2.00000 0.0855138 0.0427569 0.999086i $$-0.486386\pi$$
0.0427569 + 0.999086i $$0.486386\pi$$
$$548$$ −4.00000 −0.170872
$$549$$ 21.0000 0.896258
$$550$$ 15.0000 0.639602
$$551$$ 35.0000 1.49105
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −17.0000 −0.722261
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 18.0000 0.762684 0.381342 0.924434i $$-0.375462\pi$$
0.381342 + 0.924434i $$0.375462\pi$$
$$558$$ 0 0
$$559$$ −2.00000 −0.0845910
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −12.0000 −0.506189
$$563$$ 28.0000 1.18006 0.590030 0.807382i $$-0.299116\pi$$
0.590030 + 0.807382i $$0.299116\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −14.0000 −0.588464
$$567$$ 0 0
$$568$$ 15.0000 0.629386
$$569$$ 1.00000 0.0419222 0.0209611 0.999780i $$-0.493327\pi$$
0.0209611 + 0.999780i $$0.493327\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ −3.00000 −0.125436
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 30.0000 1.25109
$$576$$ −21.0000 −0.875000
$$577$$ 14.0000 0.582828 0.291414 0.956597i $$-0.405874\pi$$
0.291414 + 0.956597i $$0.405874\pi$$
$$578$$ 32.0000 1.33102
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 9.00000 0.372742
$$584$$ −42.0000 −1.73797
$$585$$ 0 0
$$586$$ 14.0000 0.578335
$$587$$ 21.0000 0.866763 0.433381 0.901211i $$-0.357320\pi$$
0.433381 + 0.901211i $$0.357320\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −8.00000 −0.328798
$$593$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ 0 0
$$598$$ 6.00000 0.245358
$$599$$ 32.0000 1.30748 0.653742 0.756717i $$-0.273198\pi$$
0.653742 + 0.756717i $$0.273198\pi$$
$$600$$ 0 0
$$601$$ −7.00000 −0.285536 −0.142768 0.989756i $$-0.545600\pi$$
−0.142768 + 0.989756i $$0.545600\pi$$
$$602$$ 0 0
$$603$$ 9.00000 0.366508
$$604$$ 3.00000 0.122068
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 14.0000 0.568242 0.284121 0.958788i $$-0.408298\pi$$
0.284121 + 0.958788i $$0.408298\pi$$
$$608$$ −35.0000 −1.41944
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −7.00000 −0.283190
$$612$$ 21.0000 0.848875
$$613$$ 32.0000 1.29247 0.646234 0.763139i $$-0.276343\pi$$
0.646234 + 0.763139i $$0.276343\pi$$
$$614$$ 21.0000 0.847491
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 30.0000 1.20775 0.603877 0.797077i $$-0.293622\pi$$
0.603877 + 0.797077i $$0.293622\pi$$
$$618$$ 0 0
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ −14.0000 −0.559553
$$627$$ 0 0
$$628$$ −7.00000 −0.279330
$$629$$ 56.0000 2.23287
$$630$$ 0 0
$$631$$ 16.0000 0.636950 0.318475 0.947931i $$-0.396829\pi$$
0.318475 + 0.947931i $$0.396829\pi$$
$$632$$ 18.0000 0.716002
$$633$$ 0 0
$$634$$ −6.00000 −0.238290
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 15.0000 0.593856
$$639$$ 15.0000 0.593391
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 0 0
$$643$$ −7.00000 −0.276053 −0.138027 0.990429i $$-0.544076\pi$$
−0.138027 + 0.990429i $$0.544076\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −49.0000 −1.92788
$$647$$ −42.0000 −1.65119 −0.825595 0.564263i $$-0.809160\pi$$
−0.825595 + 0.564263i $$0.809160\pi$$
$$648$$ −27.0000 −1.06066
$$649$$ 21.0000 0.824322
$$650$$ 5.00000 0.196116
$$651$$ 0 0
$$652$$ 13.0000 0.509119
$$653$$ −6.00000 −0.234798 −0.117399 0.993085i $$-0.537456\pi$$
−0.117399 + 0.993085i $$0.537456\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −42.0000 −1.63858
$$658$$ 0 0
$$659$$ −40.0000 −1.55818 −0.779089 0.626913i $$-0.784318\pi$$
−0.779089 + 0.626913i $$0.784318\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$662$$ −20.0000 −0.777322
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ −24.0000 −0.929981
$$667$$ 30.0000 1.16160
$$668$$ 7.00000 0.270838
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 21.0000 0.810696
$$672$$ 0 0
$$673$$ −26.0000 −1.00223 −0.501113 0.865382i $$-0.667076\pi$$
−0.501113 + 0.865382i $$0.667076\pi$$
$$674$$ 23.0000 0.885927
$$675$$ 0 0
$$676$$ −1.00000 −0.0384615
$$677$$ −35.0000 −1.34516 −0.672580 0.740025i $$-0.734814\pi$$
−0.672580 + 0.740025i $$0.734814\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −24.0000 −0.918334 −0.459167 0.888350i $$-0.651852\pi$$
−0.459167 + 0.888350i $$0.651852\pi$$
$$684$$ −21.0000 −0.802955
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −2.00000 −0.0762493
$$689$$ 3.00000 0.114291
$$690$$ 0 0
$$691$$ 35.0000 1.33146 0.665731 0.746191i $$-0.268120\pi$$
0.665731 + 0.746191i $$0.268120\pi$$
$$692$$ −7.00000 −0.266100
$$693$$ 0 0
$$694$$ 4.00000 0.151838
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 14.0000 0.529908
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ 0 0
$$703$$ −56.0000 −2.11208
$$704$$ −21.0000 −0.791467
$$705$$ 0 0
$$706$$ 14.0000 0.526897
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 50.0000 1.87779 0.938895 0.344204i $$-0.111851\pi$$
0.938895 + 0.344204i $$0.111851\pi$$
$$710$$ 0 0
$$711$$ 18.0000 0.675053
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 10.0000 0.373718
$$717$$ 0 0
$$718$$ 8.00000 0.298557
$$719$$ −42.0000 −1.56634 −0.783168 0.621810i $$-0.786397\pi$$
−0.783168 + 0.621810i $$0.786397\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 30.0000 1.11648
$$723$$ 0 0
$$724$$ 7.00000 0.260153
$$725$$ 25.0000 0.928477
$$726$$ 0 0
$$727$$ 28.0000 1.03846 0.519231 0.854634i $$-0.326218\pi$$
0.519231 + 0.854634i $$0.326218\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ 14.0000 0.517809
$$732$$ 0 0
$$733$$ 42.0000 1.55131 0.775653 0.631160i $$-0.217421\pi$$
0.775653 + 0.631160i $$0.217421\pi$$
$$734$$ 14.0000 0.516749
$$735$$ 0 0
$$736$$ −30.0000 −1.10581
$$737$$ 9.00000 0.331519
$$738$$ 0 0
$$739$$ 4.00000 0.147142 0.0735712 0.997290i $$-0.476560\pi$$
0.0735712 + 0.997290i $$0.476560\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 9.00000 0.330178 0.165089 0.986279i $$-0.447209\pi$$
0.165089 + 0.986279i $$0.447209\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 15.0000 0.549189
$$747$$ 0 0
$$748$$ 21.0000 0.767836
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −20.0000 −0.729810 −0.364905 0.931045i $$-0.618899\pi$$
−0.364905 + 0.931045i $$0.618899\pi$$
$$752$$ −7.00000 −0.255264
$$753$$ 0 0
$$754$$ 5.00000 0.182089
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 9.00000 0.327111 0.163555 0.986534i $$-0.447704\pi$$
0.163555 + 0.986534i $$0.447704\pi$$
$$758$$ −12.0000 −0.435860
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 20.0000 0.723575
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 7.00000 0.252755
$$768$$ 0 0
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −4.00000 −0.143963
$$773$$ 42.0000 1.51064 0.755318 0.655359i $$-0.227483\pi$$
0.755318 + 0.655359i $$0.227483\pi$$
$$774$$ −6.00000 −0.215666
$$775$$ 0 0
$$776$$ 42.0000 1.50771
$$777$$ 0 0
$$778$$ −3.00000 −0.107555
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 15.0000 0.536742
$$782$$ −42.0000 −1.50192
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 7.00000 0.249523 0.124762 0.992187i $$-0.460183\pi$$
0.124762 + 0.992187i $$0.460183\pi$$
$$788$$ −2.00000 −0.0712470
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −27.0000 −0.959403
$$793$$ 7.00000 0.248577
$$794$$ 14.0000 0.496841
$$795$$ 0 0
$$796$$ 14.0000 0.496217
$$797$$ −42.0000 −1.48772 −0.743858 0.668338i $$-0.767006\pi$$
−0.743858 + 0.668338i $$0.767006\pi$$
$$798$$ 0 0
$$799$$ 49.0000 1.73350
$$800$$ −25.0000 −0.883883
$$801$$ 0 0
$$802$$ 22.0000 0.776847
$$803$$ −42.0000 −1.48215
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 42.0000 1.47755
$$809$$ −31.0000 −1.08990 −0.544951 0.838468i $$-0.683452\pi$$
−0.544951 + 0.838468i $$0.683452\pi$$
$$810$$ 0 0
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −24.0000 −0.841200
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −14.0000 −0.489798
$$818$$ −28.0000 −0.978997
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 36.0000 1.25641 0.628204 0.778048i $$-0.283790\pi$$
0.628204 + 0.778048i $$0.283790\pi$$
$$822$$ 0 0
$$823$$ 4.00000 0.139431 0.0697156 0.997567i $$-0.477791\pi$$
0.0697156 + 0.997567i $$0.477791\pi$$
$$824$$ −42.0000 −1.46314
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −5.00000 −0.173867 −0.0869335 0.996214i $$-0.527707\pi$$
−0.0869335 + 0.996214i $$0.527707\pi$$
$$828$$ −18.0000 −0.625543
$$829$$ 35.0000 1.21560 0.607800 0.794090i $$-0.292052\pi$$
0.607800 + 0.794090i $$0.292052\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −7.00000 −0.242681
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −21.0000 −0.726300
$$837$$ 0 0
$$838$$ 14.0000 0.483622
$$839$$ 7.00000 0.241667 0.120833 0.992673i $$-0.461443\pi$$
0.120833 + 0.992673i $$0.461443\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 30.0000 1.03387
$$843$$ 0 0
$$844$$ 26.0000 0.894957
$$845$$ 0 0
$$846$$ −21.0000 −0.721995
$$847$$ 0 0
$$848$$ 3.00000 0.103020
$$849$$ 0 0
$$850$$ −35.0000 −1.20049
$$851$$ −48.0000 −1.64542
$$852$$ 0 0
$$853$$ −42.0000 −1.43805 −0.719026 0.694983i $$-0.755412\pi$$
−0.719026 + 0.694983i $$0.755412\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −24.0000 −0.820303
$$857$$ 21.0000 0.717346 0.358673 0.933463i $$-0.383229\pi$$
0.358673 + 0.933463i $$0.383229\pi$$
$$858$$ 0 0
$$859$$ −56.0000 −1.91070 −0.955348 0.295484i $$-0.904519\pi$$
−0.955348 + 0.295484i $$0.904519\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −24.0000 −0.817443
$$863$$ −20.0000 −0.680808 −0.340404 0.940279i $$-0.610564\pi$$
−0.340404 + 0.940279i $$0.610564\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 21.0000 0.713609
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 18.0000 0.610608
$$870$$ 0 0
$$871$$ 3.00000 0.101651
$$872$$ −12.0000 −0.406371
$$873$$ 42.0000 1.42148
$$874$$ 42.0000 1.42067
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −48.0000 −1.62084 −0.810422 0.585846i $$-0.800762\pi$$
−0.810422 + 0.585846i $$0.800762\pi$$
$$878$$ 14.0000 0.472477
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 42.0000 1.41502 0.707508 0.706705i $$-0.249819\pi$$
0.707508 + 0.706705i $$0.249819\pi$$
$$882$$ 0 0
$$883$$ −40.0000 −1.34611 −0.673054 0.739594i $$-0.735018\pi$$
−0.673054 + 0.739594i $$0.735018\pi$$
$$884$$ 7.00000 0.235435
$$885$$ 0 0
$$886$$ −20.0000 −0.671913
$$887$$ 42.0000 1.41022 0.705111 0.709097i $$-0.250897\pi$$
0.705111 + 0.709097i $$0.250897\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −27.0000 −0.904534
$$892$$ −21.0000 −0.703132
$$893$$ −49.0000 −1.63972
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −12.0000 −0.400445
$$899$$ 0 0
$$900$$ −15.0000 −0.500000
$$901$$ −21.0000 −0.699611
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −27.0000 −0.898007
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −38.0000 −1.26177 −0.630885 0.775877i $$-0.717308\pi$$
−0.630885 + 0.775877i $$0.717308\pi$$
$$908$$ 28.0000 0.929213
$$909$$ 42.0000 1.39305
$$910$$ 0 0
$$911$$ −54.0000 −1.78910 −0.894550 0.446968i $$-0.852504\pi$$
−0.894550 + 0.446968i $$0.852504\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −6.00000 −0.198462
$$915$$ 0 0
$$916$$ −14.0000 −0.462573
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 50.0000 1.64935 0.824674 0.565608i $$-0.191359\pi$$
0.824674 + 0.565608i $$0.191359\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −28.0000 −0.922131
$$923$$ 5.00000 0.164577
$$924$$ 0 0
$$925$$ −40.0000 −1.31519
$$926$$ 16.0000 0.525793
$$927$$ −42.0000 −1.37946
$$928$$ −25.0000 −0.820665
$$929$$ 14.0000 0.459325 0.229663 0.973270i $$-0.426238\pi$$
0.229663 + 0.973270i $$0.426238\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 27.0000 0.884414
$$933$$ 0 0
$$934$$ 14.0000 0.458094
$$935$$ 0 0
$$936$$ −9.00000 −0.294174
$$937$$ 7.00000 0.228680 0.114340 0.993442i $$-0.463525\pi$$
0.114340 + 0.993442i $$0.463525\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 28.0000 0.912774 0.456387 0.889781i $$-0.349143\pi$$
0.456387 + 0.889781i $$0.349143\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 7.00000 0.227831
$$945$$ 0 0
$$946$$ −6.00000 −0.195077
$$947$$ −27.0000 −0.877382 −0.438691 0.898638i $$-0.644558\pi$$
−0.438691 + 0.898638i $$0.644558\pi$$
$$948$$ 0 0
$$949$$ −14.0000 −0.454459
$$950$$ 35.0000 1.13555
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 9.00000 0.291539 0.145769 0.989319i $$-0.453434\pi$$
0.145769 + 0.989319i $$0.453434\pi$$
$$954$$ 9.00000 0.291386
$$955$$ 0 0
$$956$$ 19.0000 0.614504
$$957$$ 0 0
$$958$$ 7.00000 0.226160
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ −8.00000 −0.257930
$$963$$ −24.0000 −0.773389
$$964$$ −28.0000 −0.901819
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −5.00000 −0.160789 −0.0803946 0.996763i $$-0.525618\pi$$
−0.0803946 + 0.996763i $$0.525618\pi$$
$$968$$ 6.00000 0.192847
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −28.0000 −0.898563 −0.449281 0.893390i $$-0.648320\pi$$
−0.449281 + 0.893390i $$0.648320\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 25.0000 0.801052
$$975$$ 0 0
$$976$$ 7.00000 0.224065
$$977$$ −38.0000 −1.21573 −0.607864 0.794041i $$-0.707973\pi$$
−0.607864 + 0.794041i $$0.707973\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −12.0000 −0.383131
$$982$$ 30.0000 0.957338
$$983$$ −7.00000 −0.223265 −0.111633 0.993750i $$-0.535608\pi$$
−0.111633 + 0.993750i $$0.535608\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −35.0000 −1.11463
$$987$$ 0 0
$$988$$ −7.00000 −0.222700
$$989$$ −12.0000 −0.381578
$$990$$ 0 0
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −7.00000 −0.221692 −0.110846 0.993838i $$-0.535356\pi$$
−0.110846 + 0.993838i $$0.535356\pi$$
$$998$$ 8.00000 0.253236
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 637.2.a.c.1.1 1
3.2 odd 2 5733.2.a.c.1.1 1
7.2 even 3 91.2.e.a.53.1 2
7.3 odd 6 637.2.e.a.79.1 2
7.4 even 3 91.2.e.a.79.1 yes 2
7.5 odd 6 637.2.e.a.508.1 2
7.6 odd 2 637.2.a.d.1.1 1
13.12 even 2 8281.2.a.f.1.1 1
21.2 odd 6 819.2.j.b.235.1 2
21.11 odd 6 819.2.j.b.352.1 2
21.20 even 2 5733.2.a.d.1.1 1
28.11 odd 6 1456.2.r.g.625.1 2
28.23 odd 6 1456.2.r.g.417.1 2
91.25 even 6 1183.2.e.b.170.1 2
91.51 even 6 1183.2.e.b.508.1 2
91.90 odd 2 8281.2.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.a.53.1 2 7.2 even 3
91.2.e.a.79.1 yes 2 7.4 even 3
637.2.a.c.1.1 1 1.1 even 1 trivial
637.2.a.d.1.1 1 7.6 odd 2
637.2.e.a.79.1 2 7.3 odd 6
637.2.e.a.508.1 2 7.5 odd 6
819.2.j.b.235.1 2 21.2 odd 6
819.2.j.b.352.1 2 21.11 odd 6
1183.2.e.b.170.1 2 91.25 even 6
1183.2.e.b.508.1 2 91.51 even 6
1456.2.r.g.417.1 2 28.23 odd 6
1456.2.r.g.625.1 2 28.11 odd 6
5733.2.a.c.1.1 1 3.2 odd 2
5733.2.a.d.1.1 1 21.20 even 2
8281.2.a.e.1.1 1 91.90 odd 2
8281.2.a.f.1.1 1 13.12 even 2