# Properties

 Label 440.4.a.b.1.1 Level $440$ Weight $4$ Character 440.1 Self dual yes Analytic conductor $25.961$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [440,4,Mod(1,440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(440, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("440.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$440 = 2^{3} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 440.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: $$25.9608404025$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 440.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.00000 q^{3} +5.00000 q^{5} +8.00000 q^{7} -11.0000 q^{9} +O(q^{10})$$ $$q-4.00000 q^{3} +5.00000 q^{5} +8.00000 q^{7} -11.0000 q^{9} +11.0000 q^{11} -58.0000 q^{13} -20.0000 q^{15} +114.000 q^{17} -4.00000 q^{19} -32.0000 q^{21} -152.000 q^{23} +25.0000 q^{25} +152.000 q^{27} -138.000 q^{29} +208.000 q^{31} -44.0000 q^{33} +40.0000 q^{35} -226.000 q^{37} +232.000 q^{39} -294.000 q^{41} +276.000 q^{43} -55.0000 q^{45} -240.000 q^{47} -279.000 q^{49} -456.000 q^{51} -370.000 q^{53} +55.0000 q^{55} +16.0000 q^{57} -716.000 q^{59} -650.000 q^{61} -88.0000 q^{63} -290.000 q^{65} +124.000 q^{67} +608.000 q^{69} +232.000 q^{71} -454.000 q^{73} -100.000 q^{75} +88.0000 q^{77} -144.000 q^{79} -311.000 q^{81} -692.000 q^{83} +570.000 q^{85} +552.000 q^{87} -1206.00 q^{89} -464.000 q^{91} -832.000 q^{93} -20.0000 q^{95} -1438.00 q^{97} -121.000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −4.00000 −0.769800 −0.384900 0.922958i $$-0.625764\pi$$
−0.384900 + 0.922958i $$0.625764\pi$$
$$4$$ 0 0
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ 8.00000 0.431959 0.215980 0.976398i $$-0.430705\pi$$
0.215980 + 0.976398i $$0.430705\pi$$
$$8$$ 0 0
$$9$$ −11.0000 −0.407407
$$10$$ 0 0
$$11$$ 11.0000 0.301511
$$12$$ 0 0
$$13$$ −58.0000 −1.23741 −0.618704 0.785624i $$-0.712342\pi$$
−0.618704 + 0.785624i $$0.712342\pi$$
$$14$$ 0 0
$$15$$ −20.0000 −0.344265
$$16$$ 0 0
$$17$$ 114.000 1.62642 0.813208 0.581974i $$-0.197719\pi$$
0.813208 + 0.581974i $$0.197719\pi$$
$$18$$ 0 0
$$19$$ −4.00000 −0.0482980 −0.0241490 0.999708i $$-0.507688\pi$$
−0.0241490 + 0.999708i $$0.507688\pi$$
$$20$$ 0 0
$$21$$ −32.0000 −0.332522
$$22$$ 0 0
$$23$$ −152.000 −1.37801 −0.689004 0.724757i $$-0.741952\pi$$
−0.689004 + 0.724757i $$0.741952\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 152.000 1.08342
$$28$$ 0 0
$$29$$ −138.000 −0.883654 −0.441827 0.897100i $$-0.645669\pi$$
−0.441827 + 0.897100i $$0.645669\pi$$
$$30$$ 0 0
$$31$$ 208.000 1.20509 0.602547 0.798084i $$-0.294153\pi$$
0.602547 + 0.798084i $$0.294153\pi$$
$$32$$ 0 0
$$33$$ −44.0000 −0.232104
$$34$$ 0 0
$$35$$ 40.0000 0.193178
$$36$$ 0 0
$$37$$ −226.000 −1.00417 −0.502083 0.864819i $$-0.667433\pi$$
−0.502083 + 0.864819i $$0.667433\pi$$
$$38$$ 0 0
$$39$$ 232.000 0.952557
$$40$$ 0 0
$$41$$ −294.000 −1.11988 −0.559940 0.828533i $$-0.689176\pi$$
−0.559940 + 0.828533i $$0.689176\pi$$
$$42$$ 0 0
$$43$$ 276.000 0.978828 0.489414 0.872052i $$-0.337211\pi$$
0.489414 + 0.872052i $$0.337211\pi$$
$$44$$ 0 0
$$45$$ −55.0000 −0.182198
$$46$$ 0 0
$$47$$ −240.000 −0.744843 −0.372421 0.928064i $$-0.621472\pi$$
−0.372421 + 0.928064i $$0.621472\pi$$
$$48$$ 0 0
$$49$$ −279.000 −0.813411
$$50$$ 0 0
$$51$$ −456.000 −1.25202
$$52$$ 0 0
$$53$$ −370.000 −0.958932 −0.479466 0.877560i $$-0.659170\pi$$
−0.479466 + 0.877560i $$0.659170\pi$$
$$54$$ 0 0
$$55$$ 55.0000 0.134840
$$56$$ 0 0
$$57$$ 16.0000 0.0371799
$$58$$ 0 0
$$59$$ −716.000 −1.57992 −0.789960 0.613159i $$-0.789899\pi$$
−0.789960 + 0.613159i $$0.789899\pi$$
$$60$$ 0 0
$$61$$ −650.000 −1.36433 −0.682164 0.731199i $$-0.738961\pi$$
−0.682164 + 0.731199i $$0.738961\pi$$
$$62$$ 0 0
$$63$$ −88.0000 −0.175983
$$64$$ 0 0
$$65$$ −290.000 −0.553386
$$66$$ 0 0
$$67$$ 124.000 0.226105 0.113052 0.993589i $$-0.463937\pi$$
0.113052 + 0.993589i $$0.463937\pi$$
$$68$$ 0 0
$$69$$ 608.000 1.06079
$$70$$ 0 0
$$71$$ 232.000 0.387793 0.193897 0.981022i $$-0.437887\pi$$
0.193897 + 0.981022i $$0.437887\pi$$
$$72$$ 0 0
$$73$$ −454.000 −0.727900 −0.363950 0.931419i $$-0.618572\pi$$
−0.363950 + 0.931419i $$0.618572\pi$$
$$74$$ 0 0
$$75$$ −100.000 −0.153960
$$76$$ 0 0
$$77$$ 88.0000 0.130241
$$78$$ 0 0
$$79$$ −144.000 −0.205079 −0.102540 0.994729i $$-0.532697\pi$$
−0.102540 + 0.994729i $$0.532697\pi$$
$$80$$ 0 0
$$81$$ −311.000 −0.426612
$$82$$ 0 0
$$83$$ −692.000 −0.915143 −0.457571 0.889173i $$-0.651281\pi$$
−0.457571 + 0.889173i $$0.651281\pi$$
$$84$$ 0 0
$$85$$ 570.000 0.727355
$$86$$ 0 0
$$87$$ 552.000 0.680237
$$88$$ 0 0
$$89$$ −1206.00 −1.43636 −0.718178 0.695859i $$-0.755024\pi$$
−0.718178 + 0.695859i $$0.755024\pi$$
$$90$$ 0 0
$$91$$ −464.000 −0.534510
$$92$$ 0 0
$$93$$ −832.000 −0.927682
$$94$$ 0 0
$$95$$ −20.0000 −0.0215995
$$96$$ 0 0
$$97$$ −1438.00 −1.50522 −0.752612 0.658464i $$-0.771206\pi$$
−0.752612 + 0.658464i $$0.771206\pi$$
$$98$$ 0 0
$$99$$ −121.000 −0.122838
$$100$$ 0 0
$$101$$ 590.000 0.581259 0.290630 0.956836i $$-0.406135\pi$$
0.290630 + 0.956836i $$0.406135\pi$$
$$102$$ 0 0
$$103$$ −104.000 −0.0994896 −0.0497448 0.998762i $$-0.515841\pi$$
−0.0497448 + 0.998762i $$0.515841\pi$$
$$104$$ 0 0
$$105$$ −160.000 −0.148709
$$106$$ 0 0
$$107$$ 1364.00 1.23236 0.616182 0.787604i $$-0.288679\pi$$
0.616182 + 0.787604i $$0.288679\pi$$
$$108$$ 0 0
$$109$$ −826.000 −0.725839 −0.362920 0.931820i $$-0.618220\pi$$
−0.362920 + 0.931820i $$0.618220\pi$$
$$110$$ 0 0
$$111$$ 904.000 0.773008
$$112$$ 0 0
$$113$$ 2322.00 1.93306 0.966528 0.256560i $$-0.0825892\pi$$
0.966528 + 0.256560i $$0.0825892\pi$$
$$114$$ 0 0
$$115$$ −760.000 −0.616264
$$116$$ 0 0
$$117$$ 638.000 0.504129
$$118$$ 0 0
$$119$$ 912.000 0.702545
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 0 0
$$123$$ 1176.00 0.862084
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ 1296.00 0.905523 0.452761 0.891632i $$-0.350439\pi$$
0.452761 + 0.891632i $$0.350439\pi$$
$$128$$ 0 0
$$129$$ −1104.00 −0.753502
$$130$$ 0 0
$$131$$ −2100.00 −1.40059 −0.700297 0.713851i $$-0.746949\pi$$
−0.700297 + 0.713851i $$0.746949\pi$$
$$132$$ 0 0
$$133$$ −32.0000 −0.0208628
$$134$$ 0 0
$$135$$ 760.000 0.484521
$$136$$ 0 0
$$137$$ 2394.00 1.49294 0.746472 0.665417i $$-0.231746\pi$$
0.746472 + 0.665417i $$0.231746\pi$$
$$138$$ 0 0
$$139$$ 2340.00 1.42789 0.713943 0.700204i $$-0.246907\pi$$
0.713943 + 0.700204i $$0.246907\pi$$
$$140$$ 0 0
$$141$$ 960.000 0.573380
$$142$$ 0 0
$$143$$ −638.000 −0.373093
$$144$$ 0 0
$$145$$ −690.000 −0.395182
$$146$$ 0 0
$$147$$ 1116.00 0.626164
$$148$$ 0 0
$$149$$ −1826.00 −1.00397 −0.501986 0.864876i $$-0.667397\pi$$
−0.501986 + 0.864876i $$0.667397\pi$$
$$150$$ 0 0
$$151$$ −3128.00 −1.68578 −0.842891 0.538085i $$-0.819148\pi$$
−0.842891 + 0.538085i $$0.819148\pi$$
$$152$$ 0 0
$$153$$ −1254.00 −0.662614
$$154$$ 0 0
$$155$$ 1040.00 0.538934
$$156$$ 0 0
$$157$$ −730.000 −0.371085 −0.185542 0.982636i $$-0.559404\pi$$
−0.185542 + 0.982636i $$0.559404\pi$$
$$158$$ 0 0
$$159$$ 1480.00 0.738186
$$160$$ 0 0
$$161$$ −1216.00 −0.595244
$$162$$ 0 0
$$163$$ −2500.00 −1.20132 −0.600660 0.799505i $$-0.705095\pi$$
−0.600660 + 0.799505i $$0.705095\pi$$
$$164$$ 0 0
$$165$$ −220.000 −0.103800
$$166$$ 0 0
$$167$$ 3560.00 1.64959 0.824794 0.565434i $$-0.191291\pi$$
0.824794 + 0.565434i $$0.191291\pi$$
$$168$$ 0 0
$$169$$ 1167.00 0.531179
$$170$$ 0 0
$$171$$ 44.0000 0.0196770
$$172$$ 0 0
$$173$$ 2406.00 1.05737 0.528684 0.848818i $$-0.322686\pi$$
0.528684 + 0.848818i $$0.322686\pi$$
$$174$$ 0 0
$$175$$ 200.000 0.0863919
$$176$$ 0 0
$$177$$ 2864.00 1.21622
$$178$$ 0 0
$$179$$ 636.000 0.265569 0.132785 0.991145i $$-0.457608\pi$$
0.132785 + 0.991145i $$0.457608\pi$$
$$180$$ 0 0
$$181$$ −1842.00 −0.756435 −0.378218 0.925717i $$-0.623463\pi$$
−0.378218 + 0.925717i $$0.623463\pi$$
$$182$$ 0 0
$$183$$ 2600.00 1.05026
$$184$$ 0 0
$$185$$ −1130.00 −0.449077
$$186$$ 0 0
$$187$$ 1254.00 0.490383
$$188$$ 0 0
$$189$$ 1216.00 0.467995
$$190$$ 0 0
$$191$$ 240.000 0.0909204 0.0454602 0.998966i $$-0.485525\pi$$
0.0454602 + 0.998966i $$0.485525\pi$$
$$192$$ 0 0
$$193$$ 3778.00 1.40905 0.704524 0.709680i $$-0.251160\pi$$
0.704524 + 0.709680i $$0.251160\pi$$
$$194$$ 0 0
$$195$$ 1160.00 0.425997
$$196$$ 0 0
$$197$$ −3378.00 −1.22169 −0.610844 0.791751i $$-0.709170\pi$$
−0.610844 + 0.791751i $$0.709170\pi$$
$$198$$ 0 0
$$199$$ −1784.00 −0.635500 −0.317750 0.948175i $$-0.602927\pi$$
−0.317750 + 0.948175i $$0.602927\pi$$
$$200$$ 0 0
$$201$$ −496.000 −0.174055
$$202$$ 0 0
$$203$$ −1104.00 −0.381703
$$204$$ 0 0
$$205$$ −1470.00 −0.500826
$$206$$ 0 0
$$207$$ 1672.00 0.561411
$$208$$ 0 0
$$209$$ −44.0000 −0.0145624
$$210$$ 0 0
$$211$$ 3804.00 1.24113 0.620564 0.784156i $$-0.286904\pi$$
0.620564 + 0.784156i $$0.286904\pi$$
$$212$$ 0 0
$$213$$ −928.000 −0.298524
$$214$$ 0 0
$$215$$ 1380.00 0.437745
$$216$$ 0 0
$$217$$ 1664.00 0.520552
$$218$$ 0 0
$$219$$ 1816.00 0.560337
$$220$$ 0 0
$$221$$ −6612.00 −2.01254
$$222$$ 0 0
$$223$$ 3520.00 1.05703 0.528513 0.848925i $$-0.322750\pi$$
0.528513 + 0.848925i $$0.322750\pi$$
$$224$$ 0 0
$$225$$ −275.000 −0.0814815
$$226$$ 0 0
$$227$$ −4996.00 −1.46078 −0.730388 0.683032i $$-0.760661\pi$$
−0.730388 + 0.683032i $$0.760661\pi$$
$$228$$ 0 0
$$229$$ 286.000 0.0825302 0.0412651 0.999148i $$-0.486861\pi$$
0.0412651 + 0.999148i $$0.486861\pi$$
$$230$$ 0 0
$$231$$ −352.000 −0.100259
$$232$$ 0 0
$$233$$ −678.000 −0.190632 −0.0953160 0.995447i $$-0.530386\pi$$
−0.0953160 + 0.995447i $$0.530386\pi$$
$$234$$ 0 0
$$235$$ −1200.00 −0.333104
$$236$$ 0 0
$$237$$ 576.000 0.157870
$$238$$ 0 0
$$239$$ 4016.00 1.08692 0.543459 0.839436i $$-0.317114\pi$$
0.543459 + 0.839436i $$0.317114\pi$$
$$240$$ 0 0
$$241$$ 2866.00 0.766039 0.383019 0.923740i $$-0.374884\pi$$
0.383019 + 0.923740i $$0.374884\pi$$
$$242$$ 0 0
$$243$$ −2860.00 −0.755017
$$244$$ 0 0
$$245$$ −1395.00 −0.363768
$$246$$ 0 0
$$247$$ 232.000 0.0597644
$$248$$ 0 0
$$249$$ 2768.00 0.704477
$$250$$ 0 0
$$251$$ −4428.00 −1.11352 −0.556759 0.830674i $$-0.687955\pi$$
−0.556759 + 0.830674i $$0.687955\pi$$
$$252$$ 0 0
$$253$$ −1672.00 −0.415485
$$254$$ 0 0
$$255$$ −2280.00 −0.559918
$$256$$ 0 0
$$257$$ 7042.00 1.70921 0.854607 0.519276i $$-0.173798\pi$$
0.854607 + 0.519276i $$0.173798\pi$$
$$258$$ 0 0
$$259$$ −1808.00 −0.433759
$$260$$ 0 0
$$261$$ 1518.00 0.360007
$$262$$ 0 0
$$263$$ 1288.00 0.301983 0.150991 0.988535i $$-0.451753\pi$$
0.150991 + 0.988535i $$0.451753\pi$$
$$264$$ 0 0
$$265$$ −1850.00 −0.428848
$$266$$ 0 0
$$267$$ 4824.00 1.10571
$$268$$ 0 0
$$269$$ 7894.00 1.78924 0.894620 0.446827i $$-0.147446\pi$$
0.894620 + 0.446827i $$0.147446\pi$$
$$270$$ 0 0
$$271$$ −5104.00 −1.14408 −0.572040 0.820225i $$-0.693848\pi$$
−0.572040 + 0.820225i $$0.693848\pi$$
$$272$$ 0 0
$$273$$ 1856.00 0.411466
$$274$$ 0 0
$$275$$ 275.000 0.0603023
$$276$$ 0 0
$$277$$ −4418.00 −0.958310 −0.479155 0.877730i $$-0.659057\pi$$
−0.479155 + 0.877730i $$0.659057\pi$$
$$278$$ 0 0
$$279$$ −2288.00 −0.490964
$$280$$ 0 0
$$281$$ −2358.00 −0.500592 −0.250296 0.968169i $$-0.580528\pi$$
−0.250296 + 0.968169i $$0.580528\pi$$
$$282$$ 0 0
$$283$$ 644.000 0.135271 0.0676357 0.997710i $$-0.478454\pi$$
0.0676357 + 0.997710i $$0.478454\pi$$
$$284$$ 0 0
$$285$$ 80.0000 0.0166273
$$286$$ 0 0
$$287$$ −2352.00 −0.483743
$$288$$ 0 0
$$289$$ 8083.00 1.64523
$$290$$ 0 0
$$291$$ 5752.00 1.15872
$$292$$ 0 0
$$293$$ 1678.00 0.334573 0.167286 0.985908i $$-0.446500\pi$$
0.167286 + 0.985908i $$0.446500\pi$$
$$294$$ 0 0
$$295$$ −3580.00 −0.706562
$$296$$ 0 0
$$297$$ 1672.00 0.326664
$$298$$ 0 0
$$299$$ 8816.00 1.70516
$$300$$ 0 0
$$301$$ 2208.00 0.422814
$$302$$ 0 0
$$303$$ −2360.00 −0.447454
$$304$$ 0 0
$$305$$ −3250.00 −0.610146
$$306$$ 0 0
$$307$$ 5676.00 1.05520 0.527600 0.849493i $$-0.323092\pi$$
0.527600 + 0.849493i $$0.323092\pi$$
$$308$$ 0 0
$$309$$ 416.000 0.0765871
$$310$$ 0 0
$$311$$ 4312.00 0.786209 0.393105 0.919494i $$-0.371401\pi$$
0.393105 + 0.919494i $$0.371401\pi$$
$$312$$ 0 0
$$313$$ −7190.00 −1.29841 −0.649206 0.760613i $$-0.724899\pi$$
−0.649206 + 0.760613i $$0.724899\pi$$
$$314$$ 0 0
$$315$$ −440.000 −0.0787022
$$316$$ 0 0
$$317$$ −4474.00 −0.792697 −0.396348 0.918100i $$-0.629723\pi$$
−0.396348 + 0.918100i $$0.629723\pi$$
$$318$$ 0 0
$$319$$ −1518.00 −0.266432
$$320$$ 0 0
$$321$$ −5456.00 −0.948674
$$322$$ 0 0
$$323$$ −456.000 −0.0785527
$$324$$ 0 0
$$325$$ −1450.00 −0.247482
$$326$$ 0 0
$$327$$ 3304.00 0.558751
$$328$$ 0 0
$$329$$ −1920.00 −0.321742
$$330$$ 0 0
$$331$$ −7644.00 −1.26934 −0.634671 0.772782i $$-0.718864\pi$$
−0.634671 + 0.772782i $$0.718864\pi$$
$$332$$ 0 0
$$333$$ 2486.00 0.409105
$$334$$ 0 0
$$335$$ 620.000 0.101117
$$336$$ 0 0
$$337$$ −2926.00 −0.472966 −0.236483 0.971636i $$-0.575995\pi$$
−0.236483 + 0.971636i $$0.575995\pi$$
$$338$$ 0 0
$$339$$ −9288.00 −1.48807
$$340$$ 0 0
$$341$$ 2288.00 0.363349
$$342$$ 0 0
$$343$$ −4976.00 −0.783320
$$344$$ 0 0
$$345$$ 3040.00 0.474400
$$346$$ 0 0
$$347$$ −2140.00 −0.331070 −0.165535 0.986204i $$-0.552935\pi$$
−0.165535 + 0.986204i $$0.552935\pi$$
$$348$$ 0 0
$$349$$ −8522.00 −1.30708 −0.653542 0.756890i $$-0.726718\pi$$
−0.653542 + 0.756890i $$0.726718\pi$$
$$350$$ 0 0
$$351$$ −8816.00 −1.34064
$$352$$ 0 0
$$353$$ 12834.0 1.93508 0.967542 0.252709i $$-0.0813215\pi$$
0.967542 + 0.252709i $$0.0813215\pi$$
$$354$$ 0 0
$$355$$ 1160.00 0.173427
$$356$$ 0 0
$$357$$ −3648.00 −0.540820
$$358$$ 0 0
$$359$$ −4264.00 −0.626867 −0.313434 0.949610i $$-0.601479\pi$$
−0.313434 + 0.949610i $$0.601479\pi$$
$$360$$ 0 0
$$361$$ −6843.00 −0.997667
$$362$$ 0 0
$$363$$ −484.000 −0.0699819
$$364$$ 0 0
$$365$$ −2270.00 −0.325527
$$366$$ 0 0
$$367$$ 9840.00 1.39957 0.699787 0.714351i $$-0.253278\pi$$
0.699787 + 0.714351i $$0.253278\pi$$
$$368$$ 0 0
$$369$$ 3234.00 0.456247
$$370$$ 0 0
$$371$$ −2960.00 −0.414220
$$372$$ 0 0
$$373$$ 1502.00 0.208500 0.104250 0.994551i $$-0.466756\pi$$
0.104250 + 0.994551i $$0.466756\pi$$
$$374$$ 0 0
$$375$$ −500.000 −0.0688530
$$376$$ 0 0
$$377$$ 8004.00 1.09344
$$378$$ 0 0
$$379$$ −10700.0 −1.45019 −0.725095 0.688649i $$-0.758204\pi$$
−0.725095 + 0.688649i $$0.758204\pi$$
$$380$$ 0 0
$$381$$ −5184.00 −0.697072
$$382$$ 0 0
$$383$$ 4000.00 0.533657 0.266828 0.963744i $$-0.414024\pi$$
0.266828 + 0.963744i $$0.414024\pi$$
$$384$$ 0 0
$$385$$ 440.000 0.0582454
$$386$$ 0 0
$$387$$ −3036.00 −0.398782
$$388$$ 0 0
$$389$$ 286.000 0.0372771 0.0186385 0.999826i $$-0.494067\pi$$
0.0186385 + 0.999826i $$0.494067\pi$$
$$390$$ 0 0
$$391$$ −17328.0 −2.24121
$$392$$ 0 0
$$393$$ 8400.00 1.07818
$$394$$ 0 0
$$395$$ −720.000 −0.0917143
$$396$$ 0 0
$$397$$ 2230.00 0.281916 0.140958 0.990016i $$-0.454982\pi$$
0.140958 + 0.990016i $$0.454982\pi$$
$$398$$ 0 0
$$399$$ 128.000 0.0160602
$$400$$ 0 0
$$401$$ 3186.00 0.396761 0.198381 0.980125i $$-0.436432\pi$$
0.198381 + 0.980125i $$0.436432\pi$$
$$402$$ 0 0
$$403$$ −12064.0 −1.49119
$$404$$ 0 0
$$405$$ −1555.00 −0.190787
$$406$$ 0 0
$$407$$ −2486.00 −0.302768
$$408$$ 0 0
$$409$$ 5290.00 0.639544 0.319772 0.947494i $$-0.396394\pi$$
0.319772 + 0.947494i $$0.396394\pi$$
$$410$$ 0 0
$$411$$ −9576.00 −1.14927
$$412$$ 0 0
$$413$$ −5728.00 −0.682461
$$414$$ 0 0
$$415$$ −3460.00 −0.409264
$$416$$ 0 0
$$417$$ −9360.00 −1.09919
$$418$$ 0 0
$$419$$ 3372.00 0.393157 0.196579 0.980488i $$-0.437017\pi$$
0.196579 + 0.980488i $$0.437017\pi$$
$$420$$ 0 0
$$421$$ −1410.00 −0.163228 −0.0816142 0.996664i $$-0.526008\pi$$
−0.0816142 + 0.996664i $$0.526008\pi$$
$$422$$ 0 0
$$423$$ 2640.00 0.303454
$$424$$ 0 0
$$425$$ 2850.00 0.325283
$$426$$ 0 0
$$427$$ −5200.00 −0.589334
$$428$$ 0 0
$$429$$ 2552.00 0.287207
$$430$$ 0 0
$$431$$ −7728.00 −0.863677 −0.431838 0.901951i $$-0.642135\pi$$
−0.431838 + 0.901951i $$0.642135\pi$$
$$432$$ 0 0
$$433$$ 466.000 0.0517195 0.0258597 0.999666i $$-0.491768\pi$$
0.0258597 + 0.999666i $$0.491768\pi$$
$$434$$ 0 0
$$435$$ 2760.00 0.304211
$$436$$ 0 0
$$437$$ 608.000 0.0665551
$$438$$ 0 0
$$439$$ 8296.00 0.901928 0.450964 0.892542i $$-0.351080\pi$$
0.450964 + 0.892542i $$0.351080\pi$$
$$440$$ 0 0
$$441$$ 3069.00 0.331390
$$442$$ 0 0
$$443$$ −5692.00 −0.610463 −0.305231 0.952278i $$-0.598734\pi$$
−0.305231 + 0.952278i $$0.598734\pi$$
$$444$$ 0 0
$$445$$ −6030.00 −0.642358
$$446$$ 0 0
$$447$$ 7304.00 0.772858
$$448$$ 0 0
$$449$$ 5698.00 0.598898 0.299449 0.954112i $$-0.403197\pi$$
0.299449 + 0.954112i $$0.403197\pi$$
$$450$$ 0 0
$$451$$ −3234.00 −0.337657
$$452$$ 0 0
$$453$$ 12512.0 1.29772
$$454$$ 0 0
$$455$$ −2320.00 −0.239040
$$456$$ 0 0
$$457$$ −5990.00 −0.613130 −0.306565 0.951850i $$-0.599180\pi$$
−0.306565 + 0.951850i $$0.599180\pi$$
$$458$$ 0 0
$$459$$ 17328.0 1.76210
$$460$$ 0 0
$$461$$ −8186.00 −0.827028 −0.413514 0.910498i $$-0.635699\pi$$
−0.413514 + 0.910498i $$0.635699\pi$$
$$462$$ 0 0
$$463$$ −8272.00 −0.830308 −0.415154 0.909751i $$-0.636272\pi$$
−0.415154 + 0.909751i $$0.636272\pi$$
$$464$$ 0 0
$$465$$ −4160.00 −0.414872
$$466$$ 0 0
$$467$$ 44.0000 0.00435991 0.00217995 0.999998i $$-0.499306\pi$$
0.00217995 + 0.999998i $$0.499306\pi$$
$$468$$ 0 0
$$469$$ 992.000 0.0976680
$$470$$ 0 0
$$471$$ 2920.00 0.285661
$$472$$ 0 0
$$473$$ 3036.00 0.295128
$$474$$ 0 0
$$475$$ −100.000 −0.00965961
$$476$$ 0 0
$$477$$ 4070.00 0.390676
$$478$$ 0 0
$$479$$ −12864.0 −1.22708 −0.613540 0.789664i $$-0.710255\pi$$
−0.613540 + 0.789664i $$0.710255\pi$$
$$480$$ 0 0
$$481$$ 13108.0 1.24256
$$482$$ 0 0
$$483$$ 4864.00 0.458219
$$484$$ 0 0
$$485$$ −7190.00 −0.673157
$$486$$ 0 0
$$487$$ −5672.00 −0.527768 −0.263884 0.964554i $$-0.585004\pi$$
−0.263884 + 0.964554i $$0.585004\pi$$
$$488$$ 0 0
$$489$$ 10000.0 0.924776
$$490$$ 0 0
$$491$$ 11268.0 1.03568 0.517839 0.855478i $$-0.326737\pi$$
0.517839 + 0.855478i $$0.326737\pi$$
$$492$$ 0 0
$$493$$ −15732.0 −1.43719
$$494$$ 0 0
$$495$$ −605.000 −0.0549348
$$496$$ 0 0
$$497$$ 1856.00 0.167511
$$498$$ 0 0
$$499$$ −13348.0 −1.19747 −0.598736 0.800946i $$-0.704330\pi$$
−0.598736 + 0.800946i $$0.704330\pi$$
$$500$$ 0 0
$$501$$ −14240.0 −1.26985
$$502$$ 0 0
$$503$$ −14504.0 −1.28569 −0.642844 0.765997i $$-0.722246\pi$$
−0.642844 + 0.765997i $$0.722246\pi$$
$$504$$ 0 0
$$505$$ 2950.00 0.259947
$$506$$ 0 0
$$507$$ −4668.00 −0.408902
$$508$$ 0 0
$$509$$ 4134.00 0.359993 0.179996 0.983667i $$-0.442391\pi$$
0.179996 + 0.983667i $$0.442391\pi$$
$$510$$ 0 0
$$511$$ −3632.00 −0.314423
$$512$$ 0 0
$$513$$ −608.000 −0.0523272
$$514$$ 0 0
$$515$$ −520.000 −0.0444931
$$516$$ 0 0
$$517$$ −2640.00 −0.224578
$$518$$ 0 0
$$519$$ −9624.00 −0.813963
$$520$$ 0 0
$$521$$ 16410.0 1.37991 0.689957 0.723850i $$-0.257629\pi$$
0.689957 + 0.723850i $$0.257629\pi$$
$$522$$ 0 0
$$523$$ −4748.00 −0.396970 −0.198485 0.980104i $$-0.563602\pi$$
−0.198485 + 0.980104i $$0.563602\pi$$
$$524$$ 0 0
$$525$$ −800.000 −0.0665045
$$526$$ 0 0
$$527$$ 23712.0 1.95998
$$528$$ 0 0
$$529$$ 10937.0 0.898907
$$530$$ 0 0
$$531$$ 7876.00 0.643671
$$532$$ 0 0
$$533$$ 17052.0 1.38575
$$534$$ 0 0
$$535$$ 6820.00 0.551130
$$536$$ 0 0
$$537$$ −2544.00 −0.204435
$$538$$ 0 0
$$539$$ −3069.00 −0.245253
$$540$$ 0 0
$$541$$ −5930.00 −0.471258 −0.235629 0.971843i $$-0.575715\pi$$
−0.235629 + 0.971843i $$0.575715\pi$$
$$542$$ 0 0
$$543$$ 7368.00 0.582304
$$544$$ 0 0
$$545$$ −4130.00 −0.324605
$$546$$ 0 0
$$547$$ 15836.0 1.23784 0.618920 0.785454i $$-0.287571\pi$$
0.618920 + 0.785454i $$0.287571\pi$$
$$548$$ 0 0
$$549$$ 7150.00 0.555837
$$550$$ 0 0
$$551$$ 552.000 0.0426787
$$552$$ 0 0
$$553$$ −1152.00 −0.0885859
$$554$$ 0 0
$$555$$ 4520.00 0.345700
$$556$$ 0 0
$$557$$ −10554.0 −0.802850 −0.401425 0.915892i $$-0.631485\pi$$
−0.401425 + 0.915892i $$0.631485\pi$$
$$558$$ 0 0
$$559$$ −16008.0 −1.21121
$$560$$ 0 0
$$561$$ −5016.00 −0.377497
$$562$$ 0 0
$$563$$ 24332.0 1.82144 0.910721 0.413023i $$-0.135527\pi$$
0.910721 + 0.413023i $$0.135527\pi$$
$$564$$ 0 0
$$565$$ 11610.0 0.864489
$$566$$ 0 0
$$567$$ −2488.00 −0.184279
$$568$$ 0 0
$$569$$ 18570.0 1.36818 0.684090 0.729397i $$-0.260199\pi$$
0.684090 + 0.729397i $$0.260199\pi$$
$$570$$ 0 0
$$571$$ −2092.00 −0.153323 −0.0766615 0.997057i $$-0.524426\pi$$
−0.0766615 + 0.997057i $$0.524426\pi$$
$$572$$ 0 0
$$573$$ −960.000 −0.0699905
$$574$$ 0 0
$$575$$ −3800.00 −0.275602
$$576$$ 0 0
$$577$$ 16898.0 1.21919 0.609595 0.792713i $$-0.291332\pi$$
0.609595 + 0.792713i $$0.291332\pi$$
$$578$$ 0 0
$$579$$ −15112.0 −1.08469
$$580$$ 0 0
$$581$$ −5536.00 −0.395305
$$582$$ 0 0
$$583$$ −4070.00 −0.289129
$$584$$ 0 0
$$585$$ 3190.00 0.225453
$$586$$ 0 0
$$587$$ 26996.0 1.89820 0.949101 0.314973i $$-0.101995\pi$$
0.949101 + 0.314973i $$0.101995\pi$$
$$588$$ 0 0
$$589$$ −832.000 −0.0582037
$$590$$ 0 0
$$591$$ 13512.0 0.940456
$$592$$ 0 0
$$593$$ −5934.00 −0.410928 −0.205464 0.978665i $$-0.565870\pi$$
−0.205464 + 0.978665i $$0.565870\pi$$
$$594$$ 0 0
$$595$$ 4560.00 0.314188
$$596$$ 0 0
$$597$$ 7136.00 0.489208
$$598$$ 0 0
$$599$$ 24120.0 1.64527 0.822635 0.568570i $$-0.192503\pi$$
0.822635 + 0.568570i $$0.192503\pi$$
$$600$$ 0 0
$$601$$ −6070.00 −0.411981 −0.205990 0.978554i $$-0.566042\pi$$
−0.205990 + 0.978554i $$0.566042\pi$$
$$602$$ 0 0
$$603$$ −1364.00 −0.0921167
$$604$$ 0 0
$$605$$ 605.000 0.0406558
$$606$$ 0 0
$$607$$ −1424.00 −0.0952197 −0.0476099 0.998866i $$-0.515160\pi$$
−0.0476099 + 0.998866i $$0.515160\pi$$
$$608$$ 0 0
$$609$$ 4416.00 0.293835
$$610$$ 0 0
$$611$$ 13920.0 0.921674
$$612$$ 0 0
$$613$$ 1742.00 0.114778 0.0573888 0.998352i $$-0.481723\pi$$
0.0573888 + 0.998352i $$0.481723\pi$$
$$614$$ 0 0
$$615$$ 5880.00 0.385536
$$616$$ 0 0
$$617$$ −16518.0 −1.07778 −0.538889 0.842376i $$-0.681156\pi$$
−0.538889 + 0.842376i $$0.681156\pi$$
$$618$$ 0 0
$$619$$ −1084.00 −0.0703871 −0.0351936 0.999381i $$-0.511205\pi$$
−0.0351936 + 0.999381i $$0.511205\pi$$
$$620$$ 0 0
$$621$$ −23104.0 −1.49297
$$622$$ 0 0
$$623$$ −9648.00 −0.620448
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 176.000 0.0112101
$$628$$ 0 0
$$629$$ −25764.0 −1.63319
$$630$$ 0 0
$$631$$ 248.000 0.0156462 0.00782308 0.999969i $$-0.497510\pi$$
0.00782308 + 0.999969i $$0.497510\pi$$
$$632$$ 0 0
$$633$$ −15216.0 −0.955421
$$634$$ 0 0
$$635$$ 6480.00 0.404962
$$636$$ 0 0
$$637$$ 16182.0 1.00652
$$638$$ 0 0
$$639$$ −2552.00 −0.157990
$$640$$ 0 0
$$641$$ 21378.0 1.31729 0.658643 0.752456i $$-0.271131\pi$$
0.658643 + 0.752456i $$0.271131\pi$$
$$642$$ 0 0
$$643$$ 18364.0 1.12629 0.563146 0.826358i $$-0.309591\pi$$
0.563146 + 0.826358i $$0.309591\pi$$
$$644$$ 0 0
$$645$$ −5520.00 −0.336976
$$646$$ 0 0
$$647$$ 31064.0 1.88756 0.943780 0.330573i $$-0.107242\pi$$
0.943780 + 0.330573i $$0.107242\pi$$
$$648$$ 0 0
$$649$$ −7876.00 −0.476364
$$650$$ 0 0
$$651$$ −6656.00 −0.400721
$$652$$ 0 0
$$653$$ 13398.0 0.802916 0.401458 0.915877i $$-0.368504\pi$$
0.401458 + 0.915877i $$0.368504\pi$$
$$654$$ 0 0
$$655$$ −10500.0 −0.626365
$$656$$ 0 0
$$657$$ 4994.00 0.296552
$$658$$ 0 0
$$659$$ −14724.0 −0.870358 −0.435179 0.900344i $$-0.643315\pi$$
−0.435179 + 0.900344i $$0.643315\pi$$
$$660$$ 0 0
$$661$$ 23182.0 1.36411 0.682054 0.731302i $$-0.261087\pi$$
0.682054 + 0.731302i $$0.261087\pi$$
$$662$$ 0 0
$$663$$ 26448.0 1.54925
$$664$$ 0 0
$$665$$ −160.000 −0.00933013
$$666$$ 0 0
$$667$$ 20976.0 1.21768
$$668$$ 0 0
$$669$$ −14080.0 −0.813698
$$670$$ 0 0
$$671$$ −7150.00 −0.411360
$$672$$ 0 0
$$673$$ −7902.00 −0.452600 −0.226300 0.974058i $$-0.572663\pi$$
−0.226300 + 0.974058i $$0.572663\pi$$
$$674$$ 0 0
$$675$$ 3800.00 0.216685
$$676$$ 0 0
$$677$$ −27826.0 −1.57968 −0.789838 0.613316i $$-0.789835\pi$$
−0.789838 + 0.613316i $$0.789835\pi$$
$$678$$ 0 0
$$679$$ −11504.0 −0.650196
$$680$$ 0 0
$$681$$ 19984.0 1.12451
$$682$$ 0 0
$$683$$ 5780.00 0.323815 0.161907 0.986806i $$-0.448235\pi$$
0.161907 + 0.986806i $$0.448235\pi$$
$$684$$ 0 0
$$685$$ 11970.0 0.667665
$$686$$ 0 0
$$687$$ −1144.00 −0.0635318
$$688$$ 0 0
$$689$$ 21460.0 1.18659
$$690$$ 0 0
$$691$$ 10044.0 0.552955 0.276477 0.961020i $$-0.410833\pi$$
0.276477 + 0.961020i $$0.410833\pi$$
$$692$$ 0 0
$$693$$ −968.000 −0.0530610
$$694$$ 0 0
$$695$$ 11700.0 0.638570
$$696$$ 0 0
$$697$$ −33516.0 −1.82139
$$698$$ 0 0
$$699$$ 2712.00 0.146749
$$700$$ 0 0
$$701$$ −5322.00 −0.286746 −0.143373 0.989669i $$-0.545795\pi$$
−0.143373 + 0.989669i $$0.545795\pi$$
$$702$$ 0 0
$$703$$ 904.000 0.0484993
$$704$$ 0 0
$$705$$ 4800.00 0.256423
$$706$$ 0 0
$$707$$ 4720.00 0.251080
$$708$$ 0 0
$$709$$ 19902.0 1.05421 0.527105 0.849800i $$-0.323277\pi$$
0.527105 + 0.849800i $$0.323277\pi$$
$$710$$ 0 0
$$711$$ 1584.00 0.0835508
$$712$$ 0 0
$$713$$ −31616.0 −1.66063
$$714$$ 0 0
$$715$$ −3190.00 −0.166852
$$716$$ 0 0
$$717$$ −16064.0 −0.836710
$$718$$ 0 0
$$719$$ −18688.0 −0.969325 −0.484663 0.874701i $$-0.661058\pi$$
−0.484663 + 0.874701i $$0.661058\pi$$
$$720$$ 0 0
$$721$$ −832.000 −0.0429754
$$722$$ 0 0
$$723$$ −11464.0 −0.589697
$$724$$ 0 0
$$725$$ −3450.00 −0.176731
$$726$$ 0 0
$$727$$ 6760.00 0.344862 0.172431 0.985022i $$-0.444838\pi$$
0.172431 + 0.985022i $$0.444838\pi$$
$$728$$ 0 0
$$729$$ 19837.0 1.00782
$$730$$ 0 0
$$731$$ 31464.0 1.59198
$$732$$ 0 0
$$733$$ −25162.0 −1.26791 −0.633956 0.773369i $$-0.718570\pi$$
−0.633956 + 0.773369i $$0.718570\pi$$
$$734$$ 0 0
$$735$$ 5580.00 0.280029
$$736$$ 0 0
$$737$$ 1364.00 0.0681731
$$738$$ 0 0
$$739$$ −14548.0 −0.724164 −0.362082 0.932146i $$-0.617934\pi$$
−0.362082 + 0.932146i $$0.617934\pi$$
$$740$$ 0 0
$$741$$ −928.000 −0.0460067
$$742$$ 0 0
$$743$$ −2168.00 −0.107047 −0.0535237 0.998567i $$-0.517045\pi$$
−0.0535237 + 0.998567i $$0.517045\pi$$
$$744$$ 0 0
$$745$$ −9130.00 −0.448990
$$746$$ 0 0
$$747$$ 7612.00 0.372836
$$748$$ 0 0
$$749$$ 10912.0 0.532331
$$750$$ 0 0
$$751$$ −8768.00 −0.426030 −0.213015 0.977049i $$-0.568328\pi$$
−0.213015 + 0.977049i $$0.568328\pi$$
$$752$$ 0 0
$$753$$ 17712.0 0.857186
$$754$$ 0 0
$$755$$ −15640.0 −0.753904
$$756$$ 0 0
$$757$$ −19794.0 −0.950363 −0.475182 0.879888i $$-0.657618\pi$$
−0.475182 + 0.879888i $$0.657618\pi$$
$$758$$ 0 0
$$759$$ 6688.00 0.319841
$$760$$ 0 0
$$761$$ −6262.00 −0.298288 −0.149144 0.988815i $$-0.547652\pi$$
−0.149144 + 0.988815i $$0.547652\pi$$
$$762$$ 0 0
$$763$$ −6608.00 −0.313533
$$764$$ 0 0
$$765$$ −6270.00 −0.296330
$$766$$ 0 0
$$767$$ 41528.0 1.95501
$$768$$ 0 0
$$769$$ −39454.0 −1.85013 −0.925063 0.379813i $$-0.875988\pi$$
−0.925063 + 0.379813i $$0.875988\pi$$
$$770$$ 0 0
$$771$$ −28168.0 −1.31575
$$772$$ 0 0
$$773$$ −20034.0 −0.932177 −0.466089 0.884738i $$-0.654337\pi$$
−0.466089 + 0.884738i $$0.654337\pi$$
$$774$$ 0 0
$$775$$ 5200.00 0.241019
$$776$$ 0 0
$$777$$ 7232.00 0.333908
$$778$$ 0 0
$$779$$ 1176.00 0.0540880
$$780$$ 0 0
$$781$$ 2552.00 0.116924
$$782$$ 0 0
$$783$$ −20976.0 −0.957370
$$784$$ 0 0
$$785$$ −3650.00 −0.165954
$$786$$ 0 0
$$787$$ −35572.0 −1.61119 −0.805594 0.592468i $$-0.798154\pi$$
−0.805594 + 0.592468i $$0.798154\pi$$
$$788$$ 0 0
$$789$$ −5152.00 −0.232466
$$790$$ 0 0
$$791$$ 18576.0 0.835002
$$792$$ 0 0
$$793$$ 37700.0 1.68823
$$794$$ 0 0
$$795$$ 7400.00 0.330127
$$796$$ 0 0
$$797$$ −35994.0 −1.59972 −0.799858 0.600190i $$-0.795092\pi$$
−0.799858 + 0.600190i $$0.795092\pi$$
$$798$$ 0 0
$$799$$ −27360.0 −1.21142
$$800$$ 0 0
$$801$$ 13266.0 0.585182
$$802$$ 0 0
$$803$$ −4994.00 −0.219470
$$804$$ 0 0
$$805$$ −6080.00 −0.266201
$$806$$ 0 0
$$807$$ −31576.0 −1.37736
$$808$$ 0 0
$$809$$ −19110.0 −0.830497 −0.415248 0.909708i $$-0.636305\pi$$
−0.415248 + 0.909708i $$0.636305\pi$$
$$810$$ 0 0
$$811$$ 3300.00 0.142884 0.0714418 0.997445i $$-0.477240\pi$$
0.0714418 + 0.997445i $$0.477240\pi$$
$$812$$ 0 0
$$813$$ 20416.0 0.880714
$$814$$ 0 0
$$815$$ −12500.0 −0.537247
$$816$$ 0 0
$$817$$ −1104.00 −0.0472755
$$818$$ 0 0
$$819$$ 5104.00 0.217763
$$820$$ 0 0
$$821$$ 4862.00 0.206681 0.103340 0.994646i $$-0.467047\pi$$
0.103340 + 0.994646i $$0.467047\pi$$
$$822$$ 0 0
$$823$$ 7176.00 0.303936 0.151968 0.988385i $$-0.451439\pi$$
0.151968 + 0.988385i $$0.451439\pi$$
$$824$$ 0 0
$$825$$ −1100.00 −0.0464207
$$826$$ 0 0
$$827$$ 10276.0 0.432082 0.216041 0.976384i $$-0.430686\pi$$
0.216041 + 0.976384i $$0.430686\pi$$
$$828$$ 0 0
$$829$$ 40646.0 1.70289 0.851444 0.524446i $$-0.175727\pi$$
0.851444 + 0.524446i $$0.175727\pi$$
$$830$$ 0 0
$$831$$ 17672.0 0.737707
$$832$$ 0 0
$$833$$ −31806.0 −1.32294
$$834$$ 0 0
$$835$$ 17800.0 0.737718
$$836$$ 0 0
$$837$$ 31616.0 1.30563
$$838$$ 0 0
$$839$$ 8808.00 0.362439 0.181219 0.983443i $$-0.441996\pi$$
0.181219 + 0.983443i $$0.441996\pi$$
$$840$$ 0 0
$$841$$ −5345.00 −0.219156
$$842$$ 0 0
$$843$$ 9432.00 0.385356
$$844$$ 0 0
$$845$$ 5835.00 0.237550
$$846$$ 0 0
$$847$$ 968.000 0.0392690
$$848$$ 0 0
$$849$$ −2576.00 −0.104132
$$850$$ 0 0
$$851$$ 34352.0 1.38375
$$852$$ 0 0
$$853$$ −18658.0 −0.748931 −0.374465 0.927241i $$-0.622174\pi$$
−0.374465 + 0.927241i $$0.622174\pi$$
$$854$$ 0 0
$$855$$ 220.000 0.00879981
$$856$$ 0 0
$$857$$ 22410.0 0.893245 0.446623 0.894722i $$-0.352627\pi$$
0.446623 + 0.894722i $$0.352627\pi$$
$$858$$ 0 0
$$859$$ −4780.00 −0.189862 −0.0949310 0.995484i $$-0.530263\pi$$
−0.0949310 + 0.995484i $$0.530263\pi$$
$$860$$ 0 0
$$861$$ 9408.00 0.372385
$$862$$ 0 0
$$863$$ 5088.00 0.200692 0.100346 0.994953i $$-0.468005\pi$$
0.100346 + 0.994953i $$0.468005\pi$$
$$864$$ 0 0
$$865$$ 12030.0 0.472870
$$866$$ 0 0
$$867$$ −32332.0 −1.26650
$$868$$ 0 0
$$869$$ −1584.00 −0.0618337
$$870$$ 0 0
$$871$$ −7192.00 −0.279784
$$872$$ 0 0
$$873$$ 15818.0 0.613240
$$874$$ 0 0
$$875$$ 1000.00 0.0386356
$$876$$ 0 0
$$877$$ −22714.0 −0.874569 −0.437285 0.899323i $$-0.644060\pi$$
−0.437285 + 0.899323i $$0.644060\pi$$
$$878$$ 0 0
$$879$$ −6712.00 −0.257554
$$880$$ 0 0
$$881$$ 38354.0 1.46672 0.733359 0.679841i $$-0.237951\pi$$
0.733359 + 0.679841i $$0.237951\pi$$
$$882$$ 0 0
$$883$$ 10892.0 0.415113 0.207557 0.978223i $$-0.433449\pi$$
0.207557 + 0.978223i $$0.433449\pi$$
$$884$$ 0 0
$$885$$ 14320.0 0.543911
$$886$$ 0 0
$$887$$ −32104.0 −1.21527 −0.607636 0.794215i $$-0.707882\pi$$
−0.607636 + 0.794215i $$0.707882\pi$$
$$888$$ 0 0
$$889$$ 10368.0 0.391149
$$890$$ 0 0
$$891$$ −3421.00 −0.128628
$$892$$ 0 0
$$893$$ 960.000 0.0359744
$$894$$ 0 0
$$895$$ 3180.00 0.118766
$$896$$ 0 0
$$897$$ −35264.0 −1.31263
$$898$$ 0 0
$$899$$ −28704.0 −1.06489
$$900$$ 0 0
$$901$$ −42180.0 −1.55962
$$902$$ 0 0
$$903$$ −8832.00 −0.325482
$$904$$ 0 0
$$905$$ −9210.00 −0.338288
$$906$$ 0 0
$$907$$ −20396.0 −0.746679 −0.373340 0.927695i $$-0.621787\pi$$
−0.373340 + 0.927695i $$0.621787\pi$$
$$908$$ 0 0
$$909$$ −6490.00 −0.236809
$$910$$ 0 0
$$911$$ −10432.0 −0.379394 −0.189697 0.981843i $$-0.560751\pi$$
−0.189697 + 0.981843i $$0.560751\pi$$
$$912$$ 0 0
$$913$$ −7612.00 −0.275926
$$914$$ 0 0
$$915$$ 13000.0 0.469690
$$916$$ 0 0
$$917$$ −16800.0 −0.605000
$$918$$ 0 0
$$919$$ 3080.00 0.110555 0.0552774 0.998471i $$-0.482396\pi$$
0.0552774 + 0.998471i $$0.482396\pi$$
$$920$$ 0 0
$$921$$ −22704.0 −0.812293
$$922$$ 0 0
$$923$$ −13456.0 −0.479859
$$924$$ 0 0
$$925$$ −5650.00 −0.200833
$$926$$ 0 0
$$927$$ 1144.00 0.0405328
$$928$$ 0 0
$$929$$ −15582.0 −0.550300 −0.275150 0.961401i $$-0.588728\pi$$
−0.275150 + 0.961401i $$0.588728\pi$$
$$930$$ 0 0
$$931$$ 1116.00 0.0392862
$$932$$ 0 0
$$933$$ −17248.0 −0.605224
$$934$$ 0 0
$$935$$ 6270.00 0.219306
$$936$$ 0 0
$$937$$ −21862.0 −0.762220 −0.381110 0.924530i $$-0.624458\pi$$
−0.381110 + 0.924530i $$0.624458\pi$$
$$938$$ 0 0
$$939$$ 28760.0 0.999518
$$940$$ 0 0
$$941$$ −33850.0 −1.17267 −0.586333 0.810070i $$-0.699429\pi$$
−0.586333 + 0.810070i $$0.699429\pi$$
$$942$$ 0 0
$$943$$ 44688.0 1.54320
$$944$$ 0 0
$$945$$ 6080.00 0.209294
$$946$$ 0 0
$$947$$ −26836.0 −0.920858 −0.460429 0.887696i $$-0.652304\pi$$
−0.460429 + 0.887696i $$0.652304\pi$$
$$948$$ 0 0
$$949$$ 26332.0 0.900709
$$950$$ 0 0
$$951$$ 17896.0 0.610218
$$952$$ 0 0
$$953$$ 25194.0 0.856363 0.428181 0.903693i $$-0.359154\pi$$
0.428181 + 0.903693i $$0.359154\pi$$
$$954$$ 0 0
$$955$$ 1200.00 0.0406608
$$956$$ 0 0
$$957$$ 6072.00 0.205099
$$958$$ 0 0
$$959$$ 19152.0 0.644891
$$960$$ 0 0
$$961$$ 13473.0 0.452251
$$962$$ 0 0
$$963$$ −15004.0 −0.502074
$$964$$ 0 0
$$965$$ 18890.0 0.630146
$$966$$ 0 0
$$967$$ −10296.0 −0.342396 −0.171198 0.985237i $$-0.554764\pi$$
−0.171198 + 0.985237i $$0.554764\pi$$
$$968$$ 0 0
$$969$$ 1824.00 0.0604699
$$970$$ 0 0
$$971$$ 46116.0 1.52413 0.762066 0.647499i $$-0.224185\pi$$
0.762066 + 0.647499i $$0.224185\pi$$
$$972$$ 0 0
$$973$$ 18720.0 0.616789
$$974$$ 0 0
$$975$$ 5800.00 0.190511
$$976$$ 0 0
$$977$$ −37070.0 −1.21389 −0.606947 0.794742i $$-0.707606\pi$$
−0.606947 + 0.794742i $$0.707606\pi$$
$$978$$ 0 0
$$979$$ −13266.0 −0.433078
$$980$$ 0 0
$$981$$ 9086.00 0.295712
$$982$$ 0 0
$$983$$ 13896.0 0.450879 0.225439 0.974257i $$-0.427618\pi$$
0.225439 + 0.974257i $$0.427618\pi$$
$$984$$ 0 0
$$985$$ −16890.0 −0.546355
$$986$$ 0 0
$$987$$ 7680.00 0.247677
$$988$$ 0 0
$$989$$ −41952.0 −1.34883
$$990$$ 0 0
$$991$$ −39856.0 −1.27757 −0.638783 0.769387i $$-0.720562\pi$$
−0.638783 + 0.769387i $$0.720562\pi$$
$$992$$ 0 0
$$993$$ 30576.0 0.977140
$$994$$ 0 0
$$995$$ −8920.00 −0.284204
$$996$$ 0 0
$$997$$ −13138.0 −0.417337 −0.208668 0.977986i $$-0.566913\pi$$
−0.208668 + 0.977986i $$0.566913\pi$$
$$998$$ 0 0
$$999$$ −34352.0 −1.08794
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 440.4.a.b.1.1 1
4.3 odd 2 880.4.a.l.1.1 1
5.4 even 2 2200.4.a.h.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
440.4.a.b.1.1 1 1.1 even 1 trivial
880.4.a.l.1.1 1 4.3 odd 2
2200.4.a.h.1.1 1 5.4 even 2