# Properties

 Label 2205.4.a.v.1.1 Level $2205$ Weight $4$ Character 2205.1 Self dual yes Analytic conductor $130.099$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,4,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2205 = 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2205.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: $$130.099211563$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.70156$$ of defining polynomial Character $$\chi$$ $$=$$ 2205.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.70156 q^{2} +14.1047 q^{4} -5.00000 q^{5} -28.7016 q^{8} +O(q^{10})$$ $$q-4.70156 q^{2} +14.1047 q^{4} -5.00000 q^{5} -28.7016 q^{8} +23.5078 q^{10} -24.5969 q^{11} +35.0156 q^{13} +22.1047 q^{16} -18.4187 q^{17} +67.4031 q^{19} -70.5234 q^{20} +115.644 q^{22} +145.675 q^{23} +25.0000 q^{25} -164.628 q^{26} -214.419 q^{29} +88.6594 q^{31} +125.686 q^{32} +86.5969 q^{34} +162.125 q^{37} -316.900 q^{38} +143.508 q^{40} -337.769 q^{41} +122.156 q^{43} -346.931 q^{44} -684.900 q^{46} +354.219 q^{47} -117.539 q^{50} +493.884 q^{52} -676.691 q^{53} +122.984 q^{55} +1008.10 q^{58} +501.319 q^{59} +708.931 q^{61} -416.837 q^{62} -767.758 q^{64} -175.078 q^{65} -907.956 q^{67} -259.791 q^{68} -430.334 q^{71} -41.3406 q^{73} -762.241 q^{74} +950.700 q^{76} +890.388 q^{79} -110.523 q^{80} +1588.04 q^{82} -1057.15 q^{83} +92.0937 q^{85} -574.325 q^{86} +705.969 q^{88} +1473.72 q^{89} +2054.70 q^{92} -1665.38 q^{94} -337.016 q^{95} -555.034 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + 9 q^{4} - 10 q^{5} - 51 q^{8}+O(q^{10})$$ 2 * q - 3 * q^2 + 9 * q^4 - 10 * q^5 - 51 * q^8 $$2 q - 3 q^{2} + 9 q^{4} - 10 q^{5} - 51 q^{8} + 15 q^{10} - 62 q^{11} + 6 q^{13} + 25 q^{16} + 40 q^{17} + 122 q^{19} - 45 q^{20} + 52 q^{22} - 16 q^{23} + 50 q^{25} - 214 q^{26} - 352 q^{29} - 66 q^{31} + 309 q^{32} + 186 q^{34} - 188 q^{37} - 224 q^{38} + 255 q^{40} + 16 q^{41} - 396 q^{43} - 156 q^{44} - 960 q^{46} - 188 q^{47} - 75 q^{50} + 642 q^{52} - 982 q^{53} + 310 q^{55} + 774 q^{58} + 516 q^{59} + 880 q^{61} - 680 q^{62} - 479 q^{64} - 30 q^{65} - 356 q^{67} - 558 q^{68} - 310 q^{71} - 326 q^{73} - 1358 q^{74} + 672 q^{76} + 1832 q^{79} - 125 q^{80} + 2190 q^{82} - 680 q^{83} - 200 q^{85} - 1456 q^{86} + 1540 q^{88} + 796 q^{89} + 2880 q^{92} - 2588 q^{94} - 610 q^{95} + 670 q^{97}+O(q^{100})$$ 2 * q - 3 * q^2 + 9 * q^4 - 10 * q^5 - 51 * q^8 + 15 * q^10 - 62 * q^11 + 6 * q^13 + 25 * q^16 + 40 * q^17 + 122 * q^19 - 45 * q^20 + 52 * q^22 - 16 * q^23 + 50 * q^25 - 214 * q^26 - 352 * q^29 - 66 * q^31 + 309 * q^32 + 186 * q^34 - 188 * q^37 - 224 * q^38 + 255 * q^40 + 16 * q^41 - 396 * q^43 - 156 * q^44 - 960 * q^46 - 188 * q^47 - 75 * q^50 + 642 * q^52 - 982 * q^53 + 310 * q^55 + 774 * q^58 + 516 * q^59 + 880 * q^61 - 680 * q^62 - 479 * q^64 - 30 * q^65 - 356 * q^67 - 558 * q^68 - 310 * q^71 - 326 * q^73 - 1358 * q^74 + 672 * q^76 + 1832 * q^79 - 125 * q^80 + 2190 * q^82 - 680 * q^83 - 200 * q^85 - 1456 * q^86 + 1540 * q^88 + 796 * q^89 + 2880 * q^92 - 2588 * q^94 - 610 * q^95 + 670 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.70156 −1.66225 −0.831127 0.556083i $$-0.812304\pi$$
−0.831127 + 0.556083i $$0.812304\pi$$
$$3$$ 0 0
$$4$$ 14.1047 1.76309
$$5$$ −5.00000 −0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −28.7016 −1.26844
$$9$$ 0 0
$$10$$ 23.5078 0.743382
$$11$$ −24.5969 −0.674203 −0.337102 0.941468i $$-0.609447\pi$$
−0.337102 + 0.941468i $$0.609447\pi$$
$$12$$ 0 0
$$13$$ 35.0156 0.747045 0.373523 0.927621i $$-0.378150\pi$$
0.373523 + 0.927621i $$0.378150\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 22.1047 0.345386
$$17$$ −18.4187 −0.262777 −0.131388 0.991331i $$-0.541943\pi$$
−0.131388 + 0.991331i $$0.541943\pi$$
$$18$$ 0 0
$$19$$ 67.4031 0.813860 0.406930 0.913459i $$-0.366599\pi$$
0.406930 + 0.913459i $$0.366599\pi$$
$$20$$ −70.5234 −0.788476
$$21$$ 0 0
$$22$$ 115.644 1.12070
$$23$$ 145.675 1.32067 0.660333 0.750973i $$-0.270415\pi$$
0.660333 + 0.750973i $$0.270415\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ −164.628 −1.24178
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −214.419 −1.37298 −0.686492 0.727137i $$-0.740851\pi$$
−0.686492 + 0.727137i $$0.740851\pi$$
$$30$$ 0 0
$$31$$ 88.6594 0.513667 0.256834 0.966456i $$-0.417321\pi$$
0.256834 + 0.966456i $$0.417321\pi$$
$$32$$ 125.686 0.694323
$$33$$ 0 0
$$34$$ 86.5969 0.436801
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 162.125 0.720356 0.360178 0.932884i $$-0.382716\pi$$
0.360178 + 0.932884i $$0.382716\pi$$
$$38$$ −316.900 −1.35284
$$39$$ 0 0
$$40$$ 143.508 0.567264
$$41$$ −337.769 −1.28660 −0.643300 0.765614i $$-0.722435\pi$$
−0.643300 + 0.765614i $$0.722435\pi$$
$$42$$ 0 0
$$43$$ 122.156 0.433224 0.216612 0.976258i $$-0.430499\pi$$
0.216612 + 0.976258i $$0.430499\pi$$
$$44$$ −346.931 −1.18868
$$45$$ 0 0
$$46$$ −684.900 −2.19528
$$47$$ 354.219 1.09932 0.549661 0.835388i $$-0.314757\pi$$
0.549661 + 0.835388i $$0.314757\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −117.539 −0.332451
$$51$$ 0 0
$$52$$ 493.884 1.31710
$$53$$ −676.691 −1.75378 −0.876892 0.480687i $$-0.840387\pi$$
−0.876892 + 0.480687i $$0.840387\pi$$
$$54$$ 0 0
$$55$$ 122.984 0.301513
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 1008.10 2.28225
$$59$$ 501.319 1.10621 0.553103 0.833113i $$-0.313444\pi$$
0.553103 + 0.833113i $$0.313444\pi$$
$$60$$ 0 0
$$61$$ 708.931 1.48802 0.744011 0.668167i $$-0.232921\pi$$
0.744011 + 0.668167i $$0.232921\pi$$
$$62$$ −416.837 −0.853845
$$63$$ 0 0
$$64$$ −767.758 −1.49953
$$65$$ −175.078 −0.334089
$$66$$ 0 0
$$67$$ −907.956 −1.65559 −0.827795 0.561031i $$-0.810405\pi$$
−0.827795 + 0.561031i $$0.810405\pi$$
$$68$$ −259.791 −0.463298
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −430.334 −0.719314 −0.359657 0.933085i $$-0.617106\pi$$
−0.359657 + 0.933085i $$0.617106\pi$$
$$72$$ 0 0
$$73$$ −41.3406 −0.0662816 −0.0331408 0.999451i $$-0.510551\pi$$
−0.0331408 + 0.999451i $$0.510551\pi$$
$$74$$ −762.241 −1.19741
$$75$$ 0 0
$$76$$ 950.700 1.43490
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 890.388 1.26806 0.634028 0.773310i $$-0.281400\pi$$
0.634028 + 0.773310i $$0.281400\pi$$
$$80$$ −110.523 −0.154461
$$81$$ 0 0
$$82$$ 1588.04 2.13866
$$83$$ −1057.15 −1.39804 −0.699020 0.715102i $$-0.746380\pi$$
−0.699020 + 0.715102i $$0.746380\pi$$
$$84$$ 0 0
$$85$$ 92.0937 0.117517
$$86$$ −574.325 −0.720129
$$87$$ 0 0
$$88$$ 705.969 0.855188
$$89$$ 1473.72 1.75522 0.877610 0.479376i $$-0.159137\pi$$
0.877610 + 0.479376i $$0.159137\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 2054.70 2.32845
$$93$$ 0 0
$$94$$ −1665.38 −1.82735
$$95$$ −337.016 −0.363969
$$96$$ 0 0
$$97$$ −555.034 −0.580981 −0.290491 0.956878i $$-0.593819\pi$$
−0.290491 + 0.956878i $$0.593819\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 352.617 0.352617
$$101$$ 1890.14 1.86214 0.931071 0.364838i $$-0.118876\pi$$
0.931071 + 0.364838i $$0.118876\pi$$
$$102$$ 0 0
$$103$$ −662.700 −0.633959 −0.316979 0.948432i $$-0.602669\pi$$
−0.316979 + 0.948432i $$0.602669\pi$$
$$104$$ −1005.00 −0.947583
$$105$$ 0 0
$$106$$ 3181.50 2.91523
$$107$$ −1614.53 −1.45872 −0.729358 0.684132i $$-0.760181\pi$$
−0.729358 + 0.684132i $$0.760181\pi$$
$$108$$ 0 0
$$109$$ 217.206 0.190868 0.0954339 0.995436i $$-0.469576\pi$$
0.0954339 + 0.995436i $$0.469576\pi$$
$$110$$ −578.219 −0.501191
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1658.20 1.38044 0.690221 0.723598i $$-0.257513\pi$$
0.690221 + 0.723598i $$0.257513\pi$$
$$114$$ 0 0
$$115$$ −728.375 −0.590620
$$116$$ −3024.31 −2.42069
$$117$$ 0 0
$$118$$ −2356.98 −1.83879
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −725.994 −0.545450
$$122$$ −3333.08 −2.47347
$$123$$ 0 0
$$124$$ 1250.51 0.905640
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ −1108.81 −0.774734 −0.387367 0.921926i $$-0.626615\pi$$
−0.387367 + 0.921926i $$0.626615\pi$$
$$128$$ 2604.17 1.79827
$$129$$ 0 0
$$130$$ 823.141 0.555340
$$131$$ 185.488 0.123711 0.0618554 0.998085i $$-0.480298\pi$$
0.0618554 + 0.998085i $$0.480298\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 4268.81 2.75201
$$135$$ 0 0
$$136$$ 528.647 0.333317
$$137$$ 37.9907 0.0236917 0.0118458 0.999930i $$-0.496229\pi$$
0.0118458 + 0.999930i $$0.496229\pi$$
$$138$$ 0 0
$$139$$ −183.609 −0.112040 −0.0560199 0.998430i $$-0.517841\pi$$
−0.0560199 + 0.998430i $$0.517841\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 2023.24 1.19568
$$143$$ −861.275 −0.503660
$$144$$ 0 0
$$145$$ 1072.09 0.614018
$$146$$ 194.366 0.110177
$$147$$ 0 0
$$148$$ 2286.72 1.27005
$$149$$ 1383.34 0.760587 0.380293 0.924866i $$-0.375823\pi$$
0.380293 + 0.924866i $$0.375823\pi$$
$$150$$ 0 0
$$151$$ 765.256 0.412422 0.206211 0.978508i $$-0.433887\pi$$
0.206211 + 0.978508i $$0.433887\pi$$
$$152$$ −1934.57 −1.03233
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −443.297 −0.229719
$$156$$ 0 0
$$157$$ 2366.76 1.20311 0.601554 0.798832i $$-0.294548\pi$$
0.601554 + 0.798832i $$0.294548\pi$$
$$158$$ −4186.21 −2.10783
$$159$$ 0 0
$$160$$ −628.430 −0.310511
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −3137.69 −1.50775 −0.753875 0.657018i $$-0.771817\pi$$
−0.753875 + 0.657018i $$0.771817\pi$$
$$164$$ −4764.12 −2.26839
$$165$$ 0 0
$$166$$ 4970.26 2.32390
$$167$$ 146.469 0.0678688 0.0339344 0.999424i $$-0.489196\pi$$
0.0339344 + 0.999424i $$0.489196\pi$$
$$168$$ 0 0
$$169$$ −970.906 −0.441924
$$170$$ −432.984 −0.195343
$$171$$ 0 0
$$172$$ 1722.98 0.763812
$$173$$ −1424.12 −0.625860 −0.312930 0.949776i $$-0.601311\pi$$
−0.312930 + 0.949776i $$0.601311\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −543.706 −0.232860
$$177$$ 0 0
$$178$$ −6928.81 −2.91762
$$179$$ −1244.70 −0.519737 −0.259869 0.965644i $$-0.583679\pi$$
−0.259869 + 0.965644i $$0.583679\pi$$
$$180$$ 0 0
$$181$$ 3879.09 1.59299 0.796493 0.604648i $$-0.206686\pi$$
0.796493 + 0.604648i $$0.206686\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −4181.10 −1.67519
$$185$$ −810.625 −0.322153
$$186$$ 0 0
$$187$$ 453.044 0.177165
$$188$$ 4996.14 1.93820
$$189$$ 0 0
$$190$$ 1584.50 0.605009
$$191$$ −1574.90 −0.596628 −0.298314 0.954468i $$-0.596424\pi$$
−0.298314 + 0.954468i $$0.596424\pi$$
$$192$$ 0 0
$$193$$ −4775.67 −1.78114 −0.890572 0.454843i $$-0.849695\pi$$
−0.890572 + 0.454843i $$0.849695\pi$$
$$194$$ 2609.53 0.965738
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2803.58 1.01394 0.506971 0.861963i $$-0.330765\pi$$
0.506971 + 0.861963i $$0.330765\pi$$
$$198$$ 0 0
$$199$$ −4102.92 −1.46155 −0.730774 0.682620i $$-0.760841\pi$$
−0.730774 + 0.682620i $$0.760841\pi$$
$$200$$ −717.539 −0.253688
$$201$$ 0 0
$$202$$ −8886.63 −3.09535
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 1688.84 0.575385
$$206$$ 3115.72 1.05380
$$207$$ 0 0
$$208$$ 774.009 0.258019
$$209$$ −1657.91 −0.548707
$$210$$ 0 0
$$211$$ −823.512 −0.268687 −0.134343 0.990935i $$-0.542893\pi$$
−0.134343 + 0.990935i $$0.542893\pi$$
$$212$$ −9544.51 −3.09207
$$213$$ 0 0
$$214$$ 7590.82 2.42476
$$215$$ −610.781 −0.193744
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −1021.21 −0.317271
$$219$$ 0 0
$$220$$ 1734.66 0.531593
$$221$$ −644.944 −0.196306
$$222$$ 0 0
$$223$$ −817.194 −0.245396 −0.122698 0.992444i $$-0.539155\pi$$
−0.122698 + 0.992444i $$0.539155\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −7796.12 −2.29465
$$227$$ 3655.85 1.06893 0.534465 0.845190i $$-0.320513\pi$$
0.534465 + 0.845190i $$0.320513\pi$$
$$228$$ 0 0
$$229$$ −939.393 −0.271078 −0.135539 0.990772i $$-0.543277\pi$$
−0.135539 + 0.990772i $$0.543277\pi$$
$$230$$ 3424.50 0.981760
$$231$$ 0 0
$$232$$ 6154.15 1.74155
$$233$$ 7.64701 0.00215010 0.00107505 0.999999i $$-0.499658\pi$$
0.00107505 + 0.999999i $$0.499658\pi$$
$$234$$ 0 0
$$235$$ −1771.09 −0.491631
$$236$$ 7070.94 1.95034
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 889.115 0.240636 0.120318 0.992735i $$-0.461609\pi$$
0.120318 + 0.992735i $$0.461609\pi$$
$$240$$ 0 0
$$241$$ −2140.23 −0.572051 −0.286026 0.958222i $$-0.592334\pi$$
−0.286026 + 0.958222i $$0.592334\pi$$
$$242$$ 3413.30 0.906676
$$243$$ 0 0
$$244$$ 9999.25 2.62351
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2360.16 0.607990
$$248$$ −2544.66 −0.651557
$$249$$ 0 0
$$250$$ 587.695 0.148676
$$251$$ −6749.81 −1.69739 −0.848693 0.528886i $$-0.822610\pi$$
−0.848693 + 0.528886i $$0.822610\pi$$
$$252$$ 0 0
$$253$$ −3583.15 −0.890398
$$254$$ 5213.15 1.28780
$$255$$ 0 0
$$256$$ −6101.62 −1.48965
$$257$$ 3068.64 0.744811 0.372405 0.928070i $$-0.378533\pi$$
0.372405 + 0.928070i $$0.378533\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −2469.42 −0.589027
$$261$$ 0 0
$$262$$ −872.081 −0.205639
$$263$$ 4674.12 1.09589 0.547944 0.836515i $$-0.315411\pi$$
0.547944 + 0.836515i $$0.315411\pi$$
$$264$$ 0 0
$$265$$ 3383.45 0.784316
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −12806.4 −2.91895
$$269$$ 2417.38 0.547919 0.273960 0.961741i $$-0.411667\pi$$
0.273960 + 0.961741i $$0.411667\pi$$
$$270$$ 0 0
$$271$$ −7724.30 −1.73143 −0.865715 0.500537i $$-0.833136\pi$$
−0.865715 + 0.500537i $$0.833136\pi$$
$$272$$ −407.141 −0.0907593
$$273$$ 0 0
$$274$$ −178.616 −0.0393816
$$275$$ −614.922 −0.134841
$$276$$ 0 0
$$277$$ −4576.17 −0.992620 −0.496310 0.868145i $$-0.665312\pi$$
−0.496310 + 0.868145i $$0.665312\pi$$
$$278$$ 863.250 0.186239
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 1358.56 0.288415 0.144208 0.989547i $$-0.453937\pi$$
0.144208 + 0.989547i $$0.453937\pi$$
$$282$$ 0 0
$$283$$ −3885.04 −0.816048 −0.408024 0.912971i $$-0.633782\pi$$
−0.408024 + 0.912971i $$0.633782\pi$$
$$284$$ −6069.73 −1.26821
$$285$$ 0 0
$$286$$ 4049.34 0.837211
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −4573.75 −0.930948
$$290$$ −5040.52 −1.02065
$$291$$ 0 0
$$292$$ −583.097 −0.116860
$$293$$ −4033.91 −0.804312 −0.402156 0.915571i $$-0.631739\pi$$
−0.402156 + 0.915571i $$0.631739\pi$$
$$294$$ 0 0
$$295$$ −2506.59 −0.494710
$$296$$ −4653.24 −0.913730
$$297$$ 0 0
$$298$$ −6503.85 −1.26429
$$299$$ 5100.90 0.986598
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −3597.90 −0.685549
$$303$$ 0 0
$$304$$ 1489.92 0.281096
$$305$$ −3544.66 −0.665464
$$306$$ 0 0
$$307$$ 4620.36 0.858950 0.429475 0.903079i $$-0.358699\pi$$
0.429475 + 0.903079i $$0.358699\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 2084.19 0.381851
$$311$$ 6675.89 1.21722 0.608609 0.793470i $$-0.291728\pi$$
0.608609 + 0.793470i $$0.291728\pi$$
$$312$$ 0 0
$$313$$ −2836.78 −0.512283 −0.256141 0.966639i $$-0.582451\pi$$
−0.256141 + 0.966639i $$0.582451\pi$$
$$314$$ −11127.5 −1.99987
$$315$$ 0 0
$$316$$ 12558.6 2.23569
$$317$$ −4010.63 −0.710597 −0.355299 0.934753i $$-0.615621\pi$$
−0.355299 + 0.934753i $$0.615621\pi$$
$$318$$ 0 0
$$319$$ 5274.03 0.925671
$$320$$ 3838.79 0.670609
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −1241.48 −0.213863
$$324$$ 0 0
$$325$$ 875.391 0.149409
$$326$$ 14752.1 2.50626
$$327$$ 0 0
$$328$$ 9694.49 1.63198
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 11087.5 1.84117 0.920583 0.390546i $$-0.127714\pi$$
0.920583 + 0.390546i $$0.127714\pi$$
$$332$$ −14910.8 −2.46486
$$333$$ 0 0
$$334$$ −688.631 −0.112815
$$335$$ 4539.78 0.740402
$$336$$ 0 0
$$337$$ 12118.7 1.95890 0.979450 0.201689i $$-0.0646431\pi$$
0.979450 + 0.201689i $$0.0646431\pi$$
$$338$$ 4564.78 0.734589
$$339$$ 0 0
$$340$$ 1298.95 0.207193
$$341$$ −2180.74 −0.346316
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −3506.07 −0.549520
$$345$$ 0 0
$$346$$ 6695.58 1.04034
$$347$$ 6361.22 0.984116 0.492058 0.870562i $$-0.336245\pi$$
0.492058 + 0.870562i $$0.336245\pi$$
$$348$$ 0 0
$$349$$ 3115.18 0.477799 0.238899 0.971044i $$-0.423213\pi$$
0.238899 + 0.971044i $$0.423213\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −3091.48 −0.468115
$$353$$ −11927.4 −1.79839 −0.899194 0.437550i $$-0.855846\pi$$
−0.899194 + 0.437550i $$0.855846\pi$$
$$354$$ 0 0
$$355$$ 2151.67 0.321687
$$356$$ 20786.4 3.09460
$$357$$ 0 0
$$358$$ 5852.02 0.863935
$$359$$ 6143.95 0.903245 0.451623 0.892209i $$-0.350845\pi$$
0.451623 + 0.892209i $$0.350845\pi$$
$$360$$ 0 0
$$361$$ −2315.82 −0.337632
$$362$$ −18237.8 −2.64794
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 206.703 0.0296420
$$366$$ 0 0
$$367$$ 1927.67 0.274178 0.137089 0.990559i $$-0.456225\pi$$
0.137089 + 0.990559i $$0.456225\pi$$
$$368$$ 3220.10 0.456139
$$369$$ 0 0
$$370$$ 3811.20 0.535500
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 10452.0 1.45090 0.725449 0.688276i $$-0.241632\pi$$
0.725449 + 0.688276i $$0.241632\pi$$
$$374$$ −2130.01 −0.294493
$$375$$ 0 0
$$376$$ −10166.6 −1.39443
$$377$$ −7508.01 −1.02568
$$378$$ 0 0
$$379$$ 7066.43 0.957726 0.478863 0.877890i $$-0.341049\pi$$
0.478863 + 0.877890i $$0.341049\pi$$
$$380$$ −4753.50 −0.641709
$$381$$ 0 0
$$382$$ 7404.51 0.991747
$$383$$ 7168.04 0.956318 0.478159 0.878273i $$-0.341304\pi$$
0.478159 + 0.878273i $$0.341304\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 22453.1 2.96071
$$387$$ 0 0
$$388$$ −7828.58 −1.02432
$$389$$ 7414.06 0.966344 0.483172 0.875525i $$-0.339485\pi$$
0.483172 + 0.875525i $$0.339485\pi$$
$$390$$ 0 0
$$391$$ −2683.15 −0.347040
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −13181.2 −1.68543
$$395$$ −4451.94 −0.567092
$$396$$ 0 0
$$397$$ 8936.01 1.12969 0.564843 0.825198i $$-0.308937\pi$$
0.564843 + 0.825198i $$0.308937\pi$$
$$398$$ 19290.1 2.42946
$$399$$ 0 0
$$400$$ 552.617 0.0690771
$$401$$ −1782.91 −0.222031 −0.111015 0.993819i $$-0.535410\pi$$
−0.111015 + 0.993819i $$0.535410\pi$$
$$402$$ 0 0
$$403$$ 3104.46 0.383733
$$404$$ 26659.9 3.28312
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −3987.77 −0.485667
$$408$$ 0 0
$$409$$ 8759.92 1.05905 0.529524 0.848295i $$-0.322371\pi$$
0.529524 + 0.848295i $$0.322371\pi$$
$$410$$ −7940.20 −0.956436
$$411$$ 0 0
$$412$$ −9347.17 −1.11772
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 5285.75 0.625222
$$416$$ 4400.97 0.518691
$$417$$ 0 0
$$418$$ 7794.75 0.912090
$$419$$ −3212.74 −0.374588 −0.187294 0.982304i $$-0.559972\pi$$
−0.187294 + 0.982304i $$0.559972\pi$$
$$420$$ 0 0
$$421$$ 15757.8 1.82420 0.912101 0.409965i $$-0.134459\pi$$
0.912101 + 0.409965i $$0.134459\pi$$
$$422$$ 3871.79 0.446626
$$423$$ 0 0
$$424$$ 19422.1 2.22457
$$425$$ −460.469 −0.0525553
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −22772.5 −2.57184
$$429$$ 0 0
$$430$$ 2871.63 0.322051
$$431$$ 405.917 0.0453650 0.0226825 0.999743i $$-0.492779\pi$$
0.0226825 + 0.999743i $$0.492779\pi$$
$$432$$ 0 0
$$433$$ 7845.25 0.870713 0.435357 0.900258i $$-0.356622\pi$$
0.435357 + 0.900258i $$0.356622\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 3063.63 0.336516
$$437$$ 9818.95 1.07484
$$438$$ 0 0
$$439$$ −423.029 −0.0459911 −0.0229955 0.999736i $$-0.507320\pi$$
−0.0229955 + 0.999736i $$0.507320\pi$$
$$440$$ −3529.84 −0.382452
$$441$$ 0 0
$$442$$ 3032.24 0.326310
$$443$$ 16058.7 1.72229 0.861143 0.508362i $$-0.169749\pi$$
0.861143 + 0.508362i $$0.169749\pi$$
$$444$$ 0 0
$$445$$ −7368.62 −0.784958
$$446$$ 3842.09 0.407911
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −2186.75 −0.229842 −0.114921 0.993375i $$-0.536662\pi$$
−0.114921 + 0.993375i $$0.536662\pi$$
$$450$$ 0 0
$$451$$ 8308.05 0.867430
$$452$$ 23388.3 2.43384
$$453$$ 0 0
$$454$$ −17188.2 −1.77683
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −5799.22 −0.593602 −0.296801 0.954939i $$-0.595920\pi$$
−0.296801 + 0.954939i $$0.595920\pi$$
$$458$$ 4416.62 0.450600
$$459$$ 0 0
$$460$$ −10273.5 −1.04131
$$461$$ 9873.35 0.997500 0.498750 0.866746i $$-0.333793\pi$$
0.498750 + 0.866746i $$0.333793\pi$$
$$462$$ 0 0
$$463$$ −6181.84 −0.620506 −0.310253 0.950654i $$-0.600414\pi$$
−0.310253 + 0.950654i $$0.600414\pi$$
$$464$$ −4739.66 −0.474209
$$465$$ 0 0
$$466$$ −35.9529 −0.00357400
$$467$$ 6145.50 0.608950 0.304475 0.952520i $$-0.401519\pi$$
0.304475 + 0.952520i $$0.401519\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 8326.91 0.817216
$$471$$ 0 0
$$472$$ −14388.6 −1.40316
$$473$$ −3004.66 −0.292081
$$474$$ 0 0
$$475$$ 1685.08 0.162772
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −4180.23 −0.399999
$$479$$ 10879.4 1.03777 0.518887 0.854843i $$-0.326347\pi$$
0.518887 + 0.854843i $$0.326347\pi$$
$$480$$ 0 0
$$481$$ 5676.91 0.538139
$$482$$ 10062.4 0.950894
$$483$$ 0 0
$$484$$ −10239.9 −0.961675
$$485$$ 2775.17 0.259823
$$486$$ 0 0
$$487$$ 8087.51 0.752526 0.376263 0.926513i $$-0.377209\pi$$
0.376263 + 0.926513i $$0.377209\pi$$
$$488$$ −20347.4 −1.88747
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 6959.90 0.639707 0.319853 0.947467i $$-0.396366\pi$$
0.319853 + 0.947467i $$0.396366\pi$$
$$492$$ 0 0
$$493$$ 3949.32 0.360788
$$494$$ −11096.4 −1.01063
$$495$$ 0 0
$$496$$ 1959.79 0.177413
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 18632.0 1.67151 0.835756 0.549101i $$-0.185030\pi$$
0.835756 + 0.549101i $$0.185030\pi$$
$$500$$ −1763.09 −0.157695
$$501$$ 0 0
$$502$$ 31734.6 2.82149
$$503$$ 4627.62 0.410209 0.205105 0.978740i $$-0.434247\pi$$
0.205105 + 0.978740i $$0.434247\pi$$
$$504$$ 0 0
$$505$$ −9450.72 −0.832775
$$506$$ 16846.4 1.48007
$$507$$ 0 0
$$508$$ −15639.4 −1.36592
$$509$$ −11351.8 −0.988528 −0.494264 0.869312i $$-0.664562\pi$$
−0.494264 + 0.869312i $$0.664562\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 7853.76 0.677911
$$513$$ 0 0
$$514$$ −14427.4 −1.23806
$$515$$ 3313.50 0.283515
$$516$$ 0 0
$$517$$ −8712.67 −0.741166
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 5025.02 0.423772
$$521$$ 19096.1 1.60579 0.802893 0.596123i $$-0.203293\pi$$
0.802893 + 0.596123i $$0.203293\pi$$
$$522$$ 0 0
$$523$$ 3145.11 0.262956 0.131478 0.991319i $$-0.458028\pi$$
0.131478 + 0.991319i $$0.458028\pi$$
$$524$$ 2616.24 0.218113
$$525$$ 0 0
$$526$$ −21975.7 −1.82164
$$527$$ −1632.99 −0.134980
$$528$$ 0 0
$$529$$ 9054.20 0.744160
$$530$$ −15907.5 −1.30373
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −11827.2 −0.961148
$$534$$ 0 0
$$535$$ 8072.66 0.652358
$$536$$ 26059.8 2.10002
$$537$$ 0 0
$$538$$ −11365.5 −0.910781
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 8776.12 0.697440 0.348720 0.937227i $$-0.386616\pi$$
0.348720 + 0.937227i $$0.386616\pi$$
$$542$$ 36316.3 2.87808
$$543$$ 0 0
$$544$$ −2314.98 −0.182452
$$545$$ −1086.03 −0.0853587
$$546$$ 0 0
$$547$$ −13695.1 −1.07049 −0.535247 0.844696i $$-0.679781\pi$$
−0.535247 + 0.844696i $$0.679781\pi$$
$$548$$ 535.847 0.0417705
$$549$$ 0 0
$$550$$ 2891.09 0.224139
$$551$$ −14452.5 −1.11742
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 21515.2 1.64999
$$555$$ 0 0
$$556$$ −2589.75 −0.197536
$$557$$ −7850.44 −0.597188 −0.298594 0.954380i $$-0.596518\pi$$
−0.298594 + 0.954380i $$0.596518\pi$$
$$558$$ 0 0
$$559$$ 4277.38 0.323638
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −6387.33 −0.479419
$$563$$ −4948.81 −0.370457 −0.185229 0.982695i $$-0.559303\pi$$
−0.185229 + 0.982695i $$0.559303\pi$$
$$564$$ 0 0
$$565$$ −8290.98 −0.617353
$$566$$ 18265.7 1.35648
$$567$$ 0 0
$$568$$ 12351.3 0.912408
$$569$$ 8115.76 0.597945 0.298972 0.954262i $$-0.403356\pi$$
0.298972 + 0.954262i $$0.403356\pi$$
$$570$$ 0 0
$$571$$ 5656.42 0.414560 0.207280 0.978282i $$-0.433539\pi$$
0.207280 + 0.978282i $$0.433539\pi$$
$$572$$ −12148.0 −0.887996
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 3641.87 0.264133
$$576$$ 0 0
$$577$$ 9536.77 0.688078 0.344039 0.938955i $$-0.388205\pi$$
0.344039 + 0.938955i $$0.388205\pi$$
$$578$$ 21503.8 1.54747
$$579$$ 0 0
$$580$$ 15121.5 1.08257
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 16644.5 1.18241
$$584$$ 1186.54 0.0840743
$$585$$ 0 0
$$586$$ 18965.7 1.33697
$$587$$ −13089.6 −0.920383 −0.460191 0.887820i $$-0.652219\pi$$
−0.460191 + 0.887820i $$0.652219\pi$$
$$588$$ 0 0
$$589$$ 5975.92 0.418053
$$590$$ 11784.9 0.822334
$$591$$ 0 0
$$592$$ 3583.72 0.248801
$$593$$ 4281.96 0.296524 0.148262 0.988948i $$-0.452632\pi$$
0.148262 + 0.988948i $$0.452632\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 19511.5 1.34098
$$597$$ 0 0
$$598$$ −23982.2 −1.63997
$$599$$ −3699.92 −0.252378 −0.126189 0.992006i $$-0.540275\pi$$
−0.126189 + 0.992006i $$0.540275\pi$$
$$600$$ 0 0
$$601$$ 17286.1 1.17323 0.586616 0.809865i $$-0.300460\pi$$
0.586616 + 0.809865i $$0.300460\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 10793.7 0.727135
$$605$$ 3629.97 0.243933
$$606$$ 0 0
$$607$$ −14456.7 −0.966689 −0.483344 0.875430i $$-0.660578\pi$$
−0.483344 + 0.875430i $$0.660578\pi$$
$$608$$ 8471.63 0.565082
$$609$$ 0 0
$$610$$ 16665.4 1.10617
$$611$$ 12403.2 0.821243
$$612$$ 0 0
$$613$$ 17981.9 1.18480 0.592400 0.805644i $$-0.298181\pi$$
0.592400 + 0.805644i $$0.298181\pi$$
$$614$$ −21722.9 −1.42779
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −19614.7 −1.27983 −0.639916 0.768445i $$-0.721031\pi$$
−0.639916 + 0.768445i $$0.721031\pi$$
$$618$$ 0 0
$$619$$ 10462.9 0.679385 0.339692 0.940537i $$-0.389677\pi$$
0.339692 + 0.940537i $$0.389677\pi$$
$$620$$ −6252.56 −0.405014
$$621$$ 0 0
$$622$$ −31387.1 −2.02332
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 13337.3 0.851544
$$627$$ 0 0
$$628$$ 33382.4 2.12118
$$629$$ −2986.14 −0.189293
$$630$$ 0 0
$$631$$ 24481.9 1.54454 0.772272 0.635292i $$-0.219120\pi$$
0.772272 + 0.635292i $$0.219120\pi$$
$$632$$ −25555.5 −1.60846
$$633$$ 0 0
$$634$$ 18856.2 1.18119
$$635$$ 5544.06 0.346471
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −24796.2 −1.53870
$$639$$ 0 0
$$640$$ −13020.9 −0.804211
$$641$$ 1109.39 0.0683595 0.0341797 0.999416i $$-0.489118\pi$$
0.0341797 + 0.999416i $$0.489118\pi$$
$$642$$ 0 0
$$643$$ −30112.5 −1.84684 −0.923422 0.383787i $$-0.874620\pi$$
−0.923422 + 0.383787i $$0.874620\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 5836.90 0.355495
$$647$$ −4260.27 −0.258869 −0.129435 0.991588i $$-0.541316\pi$$
−0.129435 + 0.991588i $$0.541316\pi$$
$$648$$ 0 0
$$649$$ −12330.9 −0.745808
$$650$$ −4115.70 −0.248356
$$651$$ 0 0
$$652$$ −44256.2 −2.65829
$$653$$ 10576.8 0.633844 0.316922 0.948452i $$-0.397351\pi$$
0.316922 + 0.948452i $$0.397351\pi$$
$$654$$ 0 0
$$655$$ −927.438 −0.0553252
$$656$$ −7466.27 −0.444373
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −3394.70 −0.200666 −0.100333 0.994954i $$-0.531991\pi$$
−0.100333 + 0.994954i $$0.531991\pi$$
$$660$$ 0 0
$$661$$ 33174.4 1.95210 0.976048 0.217554i $$-0.0698079\pi$$
0.976048 + 0.217554i $$0.0698079\pi$$
$$662$$ −52128.7 −3.06048
$$663$$ 0 0
$$664$$ 30341.9 1.77333
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −31235.4 −1.81326
$$668$$ 2065.89 0.119658
$$669$$ 0 0
$$670$$ −21344.1 −1.23074
$$671$$ −17437.5 −1.00323
$$672$$ 0 0
$$673$$ 753.881 0.0431797 0.0215899 0.999767i $$-0.493127\pi$$
0.0215899 + 0.999767i $$0.493127\pi$$
$$674$$ −56976.9 −3.25619
$$675$$ 0 0
$$676$$ −13694.3 −0.779149
$$677$$ 15668.8 0.889511 0.444756 0.895652i $$-0.353291\pi$$
0.444756 + 0.895652i $$0.353291\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −2643.23 −0.149064
$$681$$ 0 0
$$682$$ 10252.9 0.575665
$$683$$ 11557.4 0.647485 0.323742 0.946145i $$-0.395059\pi$$
0.323742 + 0.946145i $$0.395059\pi$$
$$684$$ 0 0
$$685$$ −189.953 −0.0105952
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 2700.22 0.149630
$$689$$ −23694.7 −1.31016
$$690$$ 0 0
$$691$$ 18503.1 1.01866 0.509328 0.860572i $$-0.329894\pi$$
0.509328 + 0.860572i $$0.329894\pi$$
$$692$$ −20086.8 −1.10344
$$693$$ 0 0
$$694$$ −29907.7 −1.63585
$$695$$ 918.046 0.0501057
$$696$$ 0 0
$$697$$ 6221.28 0.338088
$$698$$ −14646.2 −0.794223
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −22580.4 −1.21662 −0.608311 0.793699i $$-0.708153\pi$$
−0.608311 + 0.793699i $$0.708153\pi$$
$$702$$ 0 0
$$703$$ 10927.7 0.586269
$$704$$ 18884.4 1.01099
$$705$$ 0 0
$$706$$ 56077.4 2.98938
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −27426.6 −1.45279 −0.726394 0.687278i $$-0.758805\pi$$
−0.726394 + 0.687278i $$0.758805\pi$$
$$710$$ −10116.2 −0.534725
$$711$$ 0 0
$$712$$ −42298.2 −2.22639
$$713$$ 12915.5 0.678383
$$714$$ 0 0
$$715$$ 4306.37 0.225244
$$716$$ −17556.1 −0.916342
$$717$$ 0 0
$$718$$ −28886.1 −1.50142
$$719$$ 19383.0 1.00538 0.502688 0.864468i $$-0.332344\pi$$
0.502688 + 0.864468i $$0.332344\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 10888.0 0.561230
$$723$$ 0 0
$$724$$ 54713.3 2.80857
$$725$$ −5360.47 −0.274597
$$726$$ 0 0
$$727$$ 12317.3 0.628368 0.314184 0.949362i $$-0.398269\pi$$
0.314184 + 0.949362i $$0.398269\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −971.828 −0.0492726
$$731$$ −2249.96 −0.113841
$$732$$ 0 0
$$733$$ −1234.02 −0.0621822 −0.0310911 0.999517i $$-0.509898\pi$$
−0.0310911 + 0.999517i $$0.509898\pi$$
$$734$$ −9063.05 −0.455754
$$735$$ 0 0
$$736$$ 18309.3 0.916970
$$737$$ 22332.9 1.11620
$$738$$ 0 0
$$739$$ −15257.3 −0.759473 −0.379736 0.925095i $$-0.623985\pi$$
−0.379736 + 0.925095i $$0.623985\pi$$
$$740$$ −11433.6 −0.567984
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 35565.1 1.75606 0.878032 0.478602i $$-0.158856\pi$$
0.878032 + 0.478602i $$0.158856\pi$$
$$744$$ 0 0
$$745$$ −6916.69 −0.340145
$$746$$ −49140.8 −2.41176
$$747$$ 0 0
$$748$$ 6390.04 0.312357
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 14266.7 0.693209 0.346605 0.938011i $$-0.387335\pi$$
0.346605 + 0.938011i $$0.387335\pi$$
$$752$$ 7829.89 0.379690
$$753$$ 0 0
$$754$$ 35299.4 1.70494
$$755$$ −3826.28 −0.184441
$$756$$ 0 0
$$757$$ −15927.9 −0.764744 −0.382372 0.924009i $$-0.624893\pi$$
−0.382372 + 0.924009i $$0.624893\pi$$
$$758$$ −33223.3 −1.59198
$$759$$ 0 0
$$760$$ 9672.87 0.461674
$$761$$ −2566.48 −0.122253 −0.0611266 0.998130i $$-0.519469\pi$$
−0.0611266 + 0.998130i $$0.519469\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −22213.5 −1.05191
$$765$$ 0 0
$$766$$ −33701.0 −1.58964
$$767$$ 17554.0 0.826386
$$768$$ 0 0
$$769$$ −14433.1 −0.676816 −0.338408 0.940999i $$-0.609888\pi$$
−0.338408 + 0.940999i $$0.609888\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −67359.4 −3.14031
$$773$$ −29443.2 −1.36999 −0.684993 0.728550i $$-0.740195\pi$$
−0.684993 + 0.728550i $$0.740195\pi$$
$$774$$ 0 0
$$775$$ 2216.48 0.102733
$$776$$ 15930.4 0.736941
$$777$$ 0 0
$$778$$ −34857.7 −1.60631
$$779$$ −22766.7 −1.04711
$$780$$ 0 0
$$781$$ 10584.9 0.484964
$$782$$ 12615.0 0.576869
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −11833.8 −0.538046
$$786$$ 0 0
$$787$$ −26390.6 −1.19533 −0.597664 0.801747i $$-0.703904\pi$$
−0.597664 + 0.801747i $$0.703904\pi$$
$$788$$ 39543.6 1.78767
$$789$$ 0 0
$$790$$ 20931.1 0.942650
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 24823.7 1.11162
$$794$$ −42013.2 −1.87783
$$795$$ 0 0
$$796$$ −57870.3 −2.57683
$$797$$ −3738.33 −0.166146 −0.0830730 0.996543i $$-0.526473\pi$$
−0.0830730 + 0.996543i $$0.526473\pi$$
$$798$$ 0 0
$$799$$ −6524.26 −0.288876
$$800$$ 3142.15 0.138865
$$801$$ 0 0
$$802$$ 8382.48 0.369072
$$803$$ 1016.85 0.0446873
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −14595.8 −0.637861
$$807$$ 0 0
$$808$$ −54250.1 −2.36202
$$809$$ −43204.1 −1.87760 −0.938798 0.344468i $$-0.888059\pi$$
−0.938798 + 0.344468i $$0.888059\pi$$
$$810$$ 0 0
$$811$$ 30192.4 1.30727 0.653637 0.756809i $$-0.273242\pi$$
0.653637 + 0.756809i $$0.273242\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 18748.7 0.807301
$$815$$ 15688.5 0.674286
$$816$$ 0 0
$$817$$ 8233.71 0.352584
$$818$$ −41185.3 −1.76041
$$819$$ 0 0
$$820$$ 23820.6 1.01445
$$821$$ 40274.7 1.71206 0.856028 0.516929i $$-0.172925\pi$$
0.856028 + 0.516929i $$0.172925\pi$$
$$822$$ 0 0
$$823$$ 25184.2 1.06667 0.533334 0.845905i $$-0.320939\pi$$
0.533334 + 0.845905i $$0.320939\pi$$
$$824$$ 19020.5 0.804140
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 38941.7 1.63741 0.818703 0.574218i $$-0.194694\pi$$
0.818703 + 0.574218i $$0.194694\pi$$
$$828$$ 0 0
$$829$$ 8327.05 0.348867 0.174433 0.984669i $$-0.444191\pi$$
0.174433 + 0.984669i $$0.444191\pi$$
$$830$$ −24851.3 −1.03928
$$831$$ 0 0
$$832$$ −26883.5 −1.12021
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −732.343 −0.0303518
$$836$$ −23384.2 −0.967418
$$837$$ 0 0
$$838$$ 15104.9 0.622660
$$839$$ 8784.41 0.361468 0.180734 0.983532i $$-0.442153\pi$$
0.180734 + 0.983532i $$0.442153\pi$$
$$840$$ 0 0
$$841$$ 21586.4 0.885087
$$842$$ −74086.4 −3.03229
$$843$$ 0 0
$$844$$ −11615.4 −0.473718
$$845$$ 4854.53 0.197634
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −14958.0 −0.605732
$$849$$ 0 0
$$850$$ 2164.92 0.0873602
$$851$$ 23617.6 0.951350
$$852$$ 0 0
$$853$$ 9076.15 0.364316 0.182158 0.983269i $$-0.441692\pi$$
0.182158 + 0.983269i $$0.441692\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 46339.6 1.85030
$$857$$ 36396.7 1.45074 0.725372 0.688357i $$-0.241668\pi$$
0.725372 + 0.688357i $$0.241668\pi$$
$$858$$ 0 0
$$859$$ −8915.27 −0.354115 −0.177058 0.984200i $$-0.556658\pi$$
−0.177058 + 0.984200i $$0.556658\pi$$
$$860$$ −8614.88 −0.341587
$$861$$ 0 0
$$862$$ −1908.44 −0.0754081
$$863$$ 6148.26 0.242514 0.121257 0.992621i $$-0.461308\pi$$
0.121257 + 0.992621i $$0.461308\pi$$
$$864$$ 0 0
$$865$$ 7120.59 0.279893
$$866$$ −36884.9 −1.44735
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −21900.8 −0.854928
$$870$$ 0 0
$$871$$ −31792.6 −1.23680
$$872$$ −6234.16 −0.242105
$$873$$ 0 0
$$874$$ −46164.4 −1.78665
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −14287.0 −0.550101 −0.275050 0.961430i $$-0.588694\pi$$
−0.275050 + 0.961430i $$0.588694\pi$$
$$878$$ 1988.90 0.0764488
$$879$$ 0 0
$$880$$ 2718.53 0.104138
$$881$$ −13315.9 −0.509221 −0.254610 0.967044i $$-0.581947\pi$$
−0.254610 + 0.967044i $$0.581947\pi$$
$$882$$ 0 0
$$883$$ −5271.78 −0.200917 −0.100458 0.994941i $$-0.532031\pi$$
−0.100458 + 0.994941i $$0.532031\pi$$
$$884$$ −9096.73 −0.346104
$$885$$ 0 0
$$886$$ −75501.1 −2.86288
$$887$$ 2606.07 0.0986507 0.0493253 0.998783i $$-0.484293\pi$$
0.0493253 + 0.998783i $$0.484293\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 34644.0 1.30480
$$891$$ 0 0
$$892$$ −11526.3 −0.432654
$$893$$ 23875.4 0.894694
$$894$$ 0 0
$$895$$ 6223.48 0.232434
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 10281.1 0.382056
$$899$$ −19010.2 −0.705258
$$900$$ 0 0
$$901$$ 12463.8 0.460854
$$902$$ −39060.8 −1.44189
$$903$$ 0 0
$$904$$ −47592.8 −1.75101
$$905$$ −19395.4 −0.712405
$$906$$ 0 0
$$907$$ 18610.6 0.681317 0.340659 0.940187i $$-0.389350\pi$$
0.340659 + 0.940187i $$0.389350\pi$$
$$908$$ 51564.6 1.88462
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −41091.7 −1.49443 −0.747216 0.664581i $$-0.768610\pi$$
−0.747216 + 0.664581i $$0.768610\pi$$
$$912$$ 0 0
$$913$$ 26002.6 0.942563
$$914$$ 27265.4 0.986716
$$915$$ 0 0
$$916$$ −13249.8 −0.477934
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 38891.3 1.39598 0.697990 0.716107i $$-0.254078\pi$$
0.697990 + 0.716107i $$0.254078\pi$$
$$920$$ 20905.5 0.749167
$$921$$ 0 0
$$922$$ −46420.2 −1.65810
$$923$$ −15068.4 −0.537360
$$924$$ 0 0
$$925$$ 4053.12 0.144071
$$926$$ 29064.3 1.03144
$$927$$ 0 0
$$928$$ −26949.4 −0.953295
$$929$$ 18699.4 0.660396 0.330198 0.943912i $$-0.392885\pi$$
0.330198 + 0.943912i $$0.392885\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 107.859 0.00379080
$$933$$ 0 0
$$934$$ −28893.4 −1.01223
$$935$$ −2265.22 −0.0792305
$$936$$ 0 0
$$937$$ −21509.6 −0.749933 −0.374967 0.927038i $$-0.622346\pi$$
−0.374967 + 0.927038i $$0.622346\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −24980.7 −0.866788
$$941$$ −11241.7 −0.389448 −0.194724 0.980858i $$-0.562381\pi$$
−0.194724 + 0.980858i $$0.562381\pi$$
$$942$$ 0 0
$$943$$ −49204.5 −1.69917
$$944$$ 11081.5 0.382068
$$945$$ 0 0
$$946$$ 14126.6 0.485513
$$947$$ 36556.3 1.25441 0.627203 0.778856i $$-0.284200\pi$$
0.627203 + 0.778856i $$0.284200\pi$$
$$948$$ 0 0
$$949$$ −1447.57 −0.0495153
$$950$$ −7922.50 −0.270568
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 36633.4 1.24520 0.622598 0.782542i $$-0.286077\pi$$
0.622598 + 0.782542i $$0.286077\pi$$
$$954$$ 0 0
$$955$$ 7874.52 0.266820
$$956$$ 12540.7 0.424263
$$957$$ 0 0
$$958$$ −51150.3 −1.72504
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −21930.5 −0.736146
$$962$$ −26690.3 −0.894523
$$963$$ 0 0
$$964$$ −30187.3 −1.00858
$$965$$ 23878.4 0.796551
$$966$$ 0 0
$$967$$ −35515.8 −1.18109 −0.590544 0.807006i $$-0.701087\pi$$
−0.590544 + 0.807006i $$0.701087\pi$$
$$968$$ 20837.2 0.691871
$$969$$ 0 0
$$970$$ −13047.6 −0.431891
$$971$$ −39661.0 −1.31080 −0.655398 0.755283i $$-0.727499\pi$$
−0.655398 + 0.755283i $$0.727499\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −38023.9 −1.25089
$$975$$ 0 0
$$976$$ 15670.7 0.513942
$$977$$ −50325.3 −1.64795 −0.823977 0.566624i $$-0.808249\pi$$
−0.823977 + 0.566624i $$0.808249\pi$$
$$978$$ 0 0
$$979$$ −36249.0 −1.18337
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −32722.4 −1.06335
$$983$$ 51189.0 1.66091 0.830456 0.557084i $$-0.188080\pi$$
0.830456 + 0.557084i $$0.188080\pi$$
$$984$$ 0 0
$$985$$ −14017.9 −0.453449
$$986$$ −18568.0 −0.599721
$$987$$ 0 0
$$988$$ 33289.3 1.07194
$$989$$ 17795.1 0.572145
$$990$$ 0 0
$$991$$ −55137.3 −1.76740 −0.883700 0.468054i $$-0.844955\pi$$
−0.883700 + 0.468054i $$0.844955\pi$$
$$992$$ 11143.2 0.356651
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 20514.6 0.653624
$$996$$ 0 0
$$997$$ −41606.5 −1.32166 −0.660828 0.750537i $$-0.729795\pi$$
−0.660828 + 0.750537i $$0.729795\pi$$
$$998$$ −87599.6 −2.77848
$$999$$ 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 2205.4.a.v.1.1 2
3.2 odd 2 735.4.a.q.1.2 2
7.6 odd 2 315.4.a.g.1.1 2
21.20 even 2 105.4.a.g.1.2 2
35.34 odd 2 1575.4.a.y.1.2 2
84.83 odd 2 1680.4.a.y.1.2 2
105.62 odd 4 525.4.d.j.274.4 4
105.83 odd 4 525.4.d.j.274.1 4
105.104 even 2 525.4.a.i.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.g.1.2 2 21.20 even 2
315.4.a.g.1.1 2 7.6 odd 2
525.4.a.i.1.1 2 105.104 even 2
525.4.d.j.274.1 4 105.83 odd 4
525.4.d.j.274.4 4 105.62 odd 4
735.4.a.q.1.2 2 3.2 odd 2
1575.4.a.y.1.2 2 35.34 odd 2
1680.4.a.y.1.2 2 84.83 odd 2
2205.4.a.v.1.1 2 1.1 even 1 trivial