# Properties

 Label 1323.4.a.bh.1.2 Level $1323$ Weight $4$ Character 1323.1 Self dual yes Analytic conductor $78.060$ Analytic rank $1$ Dimension $7$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,4,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1323.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: $$78.0595269376$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - x^{6} - 43x^{5} + 10x^{4} + 513x^{3} + 258x^{2} - 936x - 504$$ x^7 - x^6 - 43*x^5 + 10*x^4 + 513*x^3 + 258*x^2 - 936*x - 504 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{3}$$ Twist minimal: no (minimal twist has level 189) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$4.76215$$ of defining polynomial Character $$\chi$$ $$=$$ 1323.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.76215 q^{2} +14.6781 q^{4} +18.7013 q^{5} -31.8020 q^{8} +O(q^{10})$$ $$q-4.76215 q^{2} +14.6781 q^{4} +18.7013 q^{5} -31.8020 q^{8} -89.0586 q^{10} -59.4578 q^{11} +12.2112 q^{13} +34.0213 q^{16} +11.5215 q^{17} -49.8054 q^{19} +274.500 q^{20} +283.147 q^{22} +125.792 q^{23} +224.740 q^{25} -58.1516 q^{26} -228.511 q^{29} +189.567 q^{31} +92.4014 q^{32} -54.8669 q^{34} +33.1458 q^{37} +237.181 q^{38} -594.740 q^{40} -524.149 q^{41} +234.349 q^{43} -872.727 q^{44} -599.039 q^{46} -273.742 q^{47} -1070.25 q^{50} +179.237 q^{52} -255.936 q^{53} -1111.94 q^{55} +1088.20 q^{58} +168.833 q^{59} +195.141 q^{61} -902.749 q^{62} -712.200 q^{64} +228.366 q^{65} +515.148 q^{67} +169.113 q^{68} -319.048 q^{71} -635.518 q^{73} -157.845 q^{74} -731.048 q^{76} -852.002 q^{79} +636.244 q^{80} +2496.08 q^{82} -264.419 q^{83} +215.467 q^{85} -1116.01 q^{86} +1890.88 q^{88} -914.197 q^{89} +1846.38 q^{92} +1303.60 q^{94} -931.428 q^{95} +455.582 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q - q^{2} + 31 q^{4} - q^{5} - 84 q^{8}+O(q^{10})$$ 7 * q - q^2 + 31 * q^4 - q^5 - 84 * q^8 $$7 q - q^{2} + 31 q^{4} - q^{5} - 84 q^{8} - 12 q^{10} - 98 q^{11} + 124 q^{13} + 139 q^{16} + 30 q^{17} - 182 q^{19} - 110 q^{20} + 276 q^{22} + 6 q^{23} + 388 q^{25} - 245 q^{26} - 323 q^{29} - 26 q^{31} - 398 q^{32} - 114 q^{34} - 112 q^{37} + 1015 q^{38} + 147 q^{40} - 524 q^{41} + 8 q^{43} - 937 q^{44} - 339 q^{46} + 288 q^{47} - 2576 q^{50} + 1075 q^{52} - 1353 q^{53} - 156 q^{55} - 81 q^{58} + 165 q^{59} - 56 q^{61} - 1215 q^{62} - 1706 q^{64} - 1694 q^{65} - 988 q^{67} + 2625 q^{68} - 792 q^{71} - 1487 q^{73} - 2736 q^{74} - 1952 q^{76} - 1273 q^{79} - 2501 q^{80} + 2049 q^{82} - 1170 q^{83} + 216 q^{85} + 160 q^{86} + 9 q^{88} + 1058 q^{89} - 3834 q^{92} - 1653 q^{94} - 3260 q^{95} + 3730 q^{97}+O(q^{100})$$ 7 * q - q^2 + 31 * q^4 - q^5 - 84 * q^8 - 12 * q^10 - 98 * q^11 + 124 * q^13 + 139 * q^16 + 30 * q^17 - 182 * q^19 - 110 * q^20 + 276 * q^22 + 6 * q^23 + 388 * q^25 - 245 * q^26 - 323 * q^29 - 26 * q^31 - 398 * q^32 - 114 * q^34 - 112 * q^37 + 1015 * q^38 + 147 * q^40 - 524 * q^41 + 8 * q^43 - 937 * q^44 - 339 * q^46 + 288 * q^47 - 2576 * q^50 + 1075 * q^52 - 1353 * q^53 - 156 * q^55 - 81 * q^58 + 165 * q^59 - 56 * q^61 - 1215 * q^62 - 1706 * q^64 - 1694 * q^65 - 988 * q^67 + 2625 * q^68 - 792 * q^71 - 1487 * q^73 - 2736 * q^74 - 1952 * q^76 - 1273 * q^79 - 2501 * q^80 + 2049 * q^82 - 1170 * q^83 + 216 * q^85 + 160 * q^86 + 9 * q^88 + 1058 * q^89 - 3834 * q^92 - 1653 * q^94 - 3260 * q^95 + 3730 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.76215 −1.68367 −0.841837 0.539732i $$-0.818526\pi$$
−0.841837 + 0.539732i $$0.818526\pi$$
$$3$$ 0 0
$$4$$ 14.6781 1.83476
$$5$$ 18.7013 1.67270 0.836350 0.548196i $$-0.184685\pi$$
0.836350 + 0.548196i $$0.184685\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −31.8020 −1.40546
$$9$$ 0 0
$$10$$ −89.0586 −2.81628
$$11$$ −59.4578 −1.62975 −0.814873 0.579639i $$-0.803193\pi$$
−0.814873 + 0.579639i $$0.803193\pi$$
$$12$$ 0 0
$$13$$ 12.2112 0.260521 0.130261 0.991480i $$-0.458419\pi$$
0.130261 + 0.991480i $$0.458419\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 34.0213 0.531583
$$17$$ 11.5215 0.164374 0.0821872 0.996617i $$-0.473809\pi$$
0.0821872 + 0.996617i $$0.473809\pi$$
$$18$$ 0 0
$$19$$ −49.8054 −0.601376 −0.300688 0.953723i $$-0.597216\pi$$
−0.300688 + 0.953723i $$0.597216\pi$$
$$20$$ 274.500 3.06900
$$21$$ 0 0
$$22$$ 283.147 2.74396
$$23$$ 125.792 1.14041 0.570204 0.821503i $$-0.306864\pi$$
0.570204 + 0.821503i $$0.306864\pi$$
$$24$$ 0 0
$$25$$ 224.740 1.79792
$$26$$ −58.1516 −0.438633
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −228.511 −1.46322 −0.731610 0.681723i $$-0.761231\pi$$
−0.731610 + 0.681723i $$0.761231\pi$$
$$30$$ 0 0
$$31$$ 189.567 1.09830 0.549150 0.835724i $$-0.314951\pi$$
0.549150 + 0.835724i $$0.314951\pi$$
$$32$$ 92.4014 0.510451
$$33$$ 0 0
$$34$$ −54.8669 −0.276753
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 33.1458 0.147274 0.0736370 0.997285i $$-0.476539\pi$$
0.0736370 + 0.997285i $$0.476539\pi$$
$$38$$ 237.181 1.01252
$$39$$ 0 0
$$40$$ −594.740 −2.35092
$$41$$ −524.149 −1.99654 −0.998272 0.0587584i $$-0.981286\pi$$
−0.998272 + 0.0587584i $$0.981286\pi$$
$$42$$ 0 0
$$43$$ 234.349 0.831115 0.415558 0.909567i $$-0.363586\pi$$
0.415558 + 0.909567i $$0.363586\pi$$
$$44$$ −872.727 −2.99019
$$45$$ 0 0
$$46$$ −599.039 −1.92007
$$47$$ −273.742 −0.849560 −0.424780 0.905297i $$-0.639649\pi$$
−0.424780 + 0.905297i $$0.639649\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −1070.25 −3.02712
$$51$$ 0 0
$$52$$ 179.237 0.477994
$$53$$ −255.936 −0.663312 −0.331656 0.943400i $$-0.607607\pi$$
−0.331656 + 0.943400i $$0.607607\pi$$
$$54$$ 0 0
$$55$$ −1111.94 −2.72608
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 1088.20 2.46359
$$59$$ 168.833 0.372545 0.186272 0.982498i $$-0.440359\pi$$
0.186272 + 0.982498i $$0.440359\pi$$
$$60$$ 0 0
$$61$$ 195.141 0.409595 0.204797 0.978804i $$-0.434346\pi$$
0.204797 + 0.978804i $$0.434346\pi$$
$$62$$ −902.749 −1.84918
$$63$$ 0 0
$$64$$ −712.200 −1.39102
$$65$$ 228.366 0.435774
$$66$$ 0 0
$$67$$ 515.148 0.939334 0.469667 0.882844i $$-0.344374\pi$$
0.469667 + 0.882844i $$0.344374\pi$$
$$68$$ 169.113 0.301587
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −319.048 −0.533295 −0.266648 0.963794i $$-0.585916\pi$$
−0.266648 + 0.963794i $$0.585916\pi$$
$$72$$ 0 0
$$73$$ −635.518 −1.01893 −0.509464 0.860492i $$-0.670156\pi$$
−0.509464 + 0.860492i $$0.670156\pi$$
$$74$$ −157.845 −0.247961
$$75$$ 0 0
$$76$$ −731.048 −1.10338
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −852.002 −1.21339 −0.606694 0.794935i $$-0.707505\pi$$
−0.606694 + 0.794935i $$0.707505\pi$$
$$80$$ 636.244 0.889178
$$81$$ 0 0
$$82$$ 2496.08 3.36153
$$83$$ −264.419 −0.349684 −0.174842 0.984597i $$-0.555941\pi$$
−0.174842 + 0.984597i $$0.555941\pi$$
$$84$$ 0 0
$$85$$ 215.467 0.274949
$$86$$ −1116.01 −1.39933
$$87$$ 0 0
$$88$$ 1890.88 2.29055
$$89$$ −914.197 −1.08882 −0.544409 0.838820i $$-0.683246\pi$$
−0.544409 + 0.838820i $$0.683246\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 1846.38 2.09237
$$93$$ 0 0
$$94$$ 1303.60 1.43038
$$95$$ −931.428 −1.00592
$$96$$ 0 0
$$97$$ 455.582 0.476880 0.238440 0.971157i $$-0.423364\pi$$
0.238440 + 0.971157i $$0.423364\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 3298.76 3.29876
$$101$$ −900.185 −0.886849 −0.443424 0.896312i $$-0.646237\pi$$
−0.443424 + 0.896312i $$0.646237\pi$$
$$102$$ 0 0
$$103$$ 114.380 0.109419 0.0547095 0.998502i $$-0.482577\pi$$
0.0547095 + 0.998502i $$0.482577\pi$$
$$104$$ −388.341 −0.366153
$$105$$ 0 0
$$106$$ 1218.81 1.11680
$$107$$ 904.066 0.816816 0.408408 0.912799i $$-0.366084\pi$$
0.408408 + 0.912799i $$0.366084\pi$$
$$108$$ 0 0
$$109$$ 1446.61 1.27119 0.635596 0.772022i $$-0.280754\pi$$
0.635596 + 0.772022i $$0.280754\pi$$
$$110$$ 5295.23 4.58982
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −10.2655 −0.00854596 −0.00427298 0.999991i $$-0.501360\pi$$
−0.00427298 + 0.999991i $$0.501360\pi$$
$$114$$ 0 0
$$115$$ 2352.47 1.90756
$$116$$ −3354.10 −2.68466
$$117$$ 0 0
$$118$$ −804.006 −0.627244
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 2204.23 1.65607
$$122$$ −929.292 −0.689624
$$123$$ 0 0
$$124$$ 2782.49 2.01512
$$125$$ 1865.28 1.33468
$$126$$ 0 0
$$127$$ −1590.09 −1.11101 −0.555503 0.831514i $$-0.687474\pi$$
−0.555503 + 0.831514i $$0.687474\pi$$
$$128$$ 2652.39 1.83157
$$129$$ 0 0
$$130$$ −1087.51 −0.733701
$$131$$ 224.882 0.149985 0.0749924 0.997184i $$-0.476107\pi$$
0.0749924 + 0.997184i $$0.476107\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −2453.21 −1.58153
$$135$$ 0 0
$$136$$ −366.406 −0.231022
$$137$$ −2251.72 −1.40421 −0.702106 0.712072i $$-0.747757\pi$$
−0.702106 + 0.712072i $$0.747757\pi$$
$$138$$ 0 0
$$139$$ −3015.34 −1.83998 −0.919992 0.391937i $$-0.871805\pi$$
−0.919992 + 0.391937i $$0.871805\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 1519.35 0.897896
$$143$$ −726.051 −0.424584
$$144$$ 0 0
$$145$$ −4273.46 −2.44753
$$146$$ 3026.43 1.71554
$$147$$ 0 0
$$148$$ 486.517 0.270212
$$149$$ 387.308 0.212950 0.106475 0.994315i $$-0.466044\pi$$
0.106475 + 0.994315i $$0.466044\pi$$
$$150$$ 0 0
$$151$$ 1643.03 0.885480 0.442740 0.896650i $$-0.354006\pi$$
0.442740 + 0.896650i $$0.354006\pi$$
$$152$$ 1583.91 0.845212
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 3545.17 1.83713
$$156$$ 0 0
$$157$$ 693.668 0.352616 0.176308 0.984335i $$-0.443585\pi$$
0.176308 + 0.984335i $$0.443585\pi$$
$$158$$ 4057.36 2.04295
$$159$$ 0 0
$$160$$ 1728.03 0.853830
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 1745.54 0.838781 0.419390 0.907806i $$-0.362244\pi$$
0.419390 + 0.907806i $$0.362244\pi$$
$$164$$ −7693.50 −3.66318
$$165$$ 0 0
$$166$$ 1259.20 0.588754
$$167$$ −3320.55 −1.53863 −0.769316 0.638868i $$-0.779403\pi$$
−0.769316 + 0.638868i $$0.779403\pi$$
$$168$$ 0 0
$$169$$ −2047.89 −0.932129
$$170$$ −1026.09 −0.462924
$$171$$ 0 0
$$172$$ 3439.80 1.52490
$$173$$ 54.1150 0.0237820 0.0118910 0.999929i $$-0.496215\pi$$
0.0118910 + 0.999929i $$0.496215\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −2022.83 −0.866345
$$177$$ 0 0
$$178$$ 4353.54 1.83321
$$179$$ 3720.54 1.55355 0.776776 0.629776i $$-0.216854\pi$$
0.776776 + 0.629776i $$0.216854\pi$$
$$180$$ 0 0
$$181$$ −1280.85 −0.525992 −0.262996 0.964797i $$-0.584711\pi$$
−0.262996 + 0.964797i $$0.584711\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −4000.43 −1.60280
$$185$$ 619.871 0.246345
$$186$$ 0 0
$$187$$ −685.041 −0.267889
$$188$$ −4018.00 −1.55874
$$189$$ 0 0
$$190$$ 4435.60 1.69364
$$191$$ −940.010 −0.356108 −0.178054 0.984021i $$-0.556980\pi$$
−0.178054 + 0.984021i $$0.556980\pi$$
$$192$$ 0 0
$$193$$ −1412.92 −0.526965 −0.263483 0.964664i $$-0.584871\pi$$
−0.263483 + 0.964664i $$0.584871\pi$$
$$194$$ −2169.55 −0.802910
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −3007.74 −1.08778 −0.543890 0.839156i $$-0.683049\pi$$
−0.543890 + 0.839156i $$0.683049\pi$$
$$198$$ 0 0
$$199$$ 22.0447 0.00785280 0.00392640 0.999992i $$-0.498750\pi$$
0.00392640 + 0.999992i $$0.498750\pi$$
$$200$$ −7147.19 −2.52691
$$201$$ 0 0
$$202$$ 4286.82 1.49316
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −9802.29 −3.33962
$$206$$ −544.693 −0.184226
$$207$$ 0 0
$$208$$ 415.441 0.138489
$$209$$ 2961.32 0.980090
$$210$$ 0 0
$$211$$ −4881.84 −1.59279 −0.796397 0.604774i $$-0.793264\pi$$
−0.796397 + 0.604774i $$0.793264\pi$$
$$212$$ −3756.65 −1.21702
$$213$$ 0 0
$$214$$ −4305.30 −1.37525
$$215$$ 4382.65 1.39021
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −6888.96 −2.14027
$$219$$ 0 0
$$220$$ −16321.2 −5.00169
$$221$$ 140.691 0.0428230
$$222$$ 0 0
$$223$$ −144.053 −0.0432579 −0.0216290 0.999766i $$-0.506885\pi$$
−0.0216290 + 0.999766i $$0.506885\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 48.8857 0.0143886
$$227$$ 4922.14 1.43918 0.719590 0.694400i $$-0.244330\pi$$
0.719590 + 0.694400i $$0.244330\pi$$
$$228$$ 0 0
$$229$$ 15.7889 0.00455616 0.00227808 0.999997i $$-0.499275\pi$$
0.00227808 + 0.999997i $$0.499275\pi$$
$$230$$ −11202.8 −3.21171
$$231$$ 0 0
$$232$$ 7267.10 2.05650
$$233$$ 727.505 0.204551 0.102276 0.994756i $$-0.467388\pi$$
0.102276 + 0.994756i $$0.467388\pi$$
$$234$$ 0 0
$$235$$ −5119.34 −1.42106
$$236$$ 2478.14 0.683530
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −63.6850 −0.0172361 −0.00861807 0.999963i $$-0.502743\pi$$
−0.00861807 + 0.999963i $$0.502743\pi$$
$$240$$ 0 0
$$241$$ −3349.78 −0.895345 −0.447672 0.894198i $$-0.647747\pi$$
−0.447672 + 0.894198i $$0.647747\pi$$
$$242$$ −10496.9 −2.78829
$$243$$ 0 0
$$244$$ 2864.30 0.751508
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −608.184 −0.156671
$$248$$ −6028.62 −1.54362
$$249$$ 0 0
$$250$$ −8882.74 −2.24717
$$251$$ 4922.71 1.23792 0.618961 0.785422i $$-0.287554\pi$$
0.618961 + 0.785422i $$0.287554\pi$$
$$252$$ 0 0
$$253$$ −7479.30 −1.85857
$$254$$ 7572.26 1.87057
$$255$$ 0 0
$$256$$ −6933.49 −1.69275
$$257$$ −2873.40 −0.697422 −0.348711 0.937230i $$-0.613381\pi$$
−0.348711 + 0.937230i $$0.613381\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 3351.97 0.799540
$$261$$ 0 0
$$262$$ −1070.92 −0.252526
$$263$$ −4367.70 −1.02405 −0.512023 0.858972i $$-0.671104\pi$$
−0.512023 + 0.858972i $$0.671104\pi$$
$$264$$ 0 0
$$265$$ −4786.35 −1.10952
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 7561.39 1.72345
$$269$$ 5354.90 1.21373 0.606866 0.794804i $$-0.292426\pi$$
0.606866 + 0.794804i $$0.292426\pi$$
$$270$$ 0 0
$$271$$ 3042.25 0.681931 0.340966 0.940076i $$-0.389246\pi$$
0.340966 + 0.940076i $$0.389246\pi$$
$$272$$ 391.975 0.0873786
$$273$$ 0 0
$$274$$ 10723.0 2.36424
$$275$$ −13362.6 −2.93016
$$276$$ 0 0
$$277$$ −8580.03 −1.86110 −0.930549 0.366167i $$-0.880670\pi$$
−0.930549 + 0.366167i $$0.880670\pi$$
$$278$$ 14359.5 3.09793
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 953.058 0.202330 0.101165 0.994870i $$-0.467743\pi$$
0.101165 + 0.994870i $$0.467743\pi$$
$$282$$ 0 0
$$283$$ 7531.60 1.58200 0.791002 0.611813i $$-0.209560\pi$$
0.791002 + 0.611813i $$0.209560\pi$$
$$284$$ −4683.00 −0.978469
$$285$$ 0 0
$$286$$ 3457.57 0.714861
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −4780.26 −0.972981
$$290$$ 20350.9 4.12084
$$291$$ 0 0
$$292$$ −9328.19 −1.86949
$$293$$ −1325.09 −0.264207 −0.132104 0.991236i $$-0.542173\pi$$
−0.132104 + 0.991236i $$0.542173\pi$$
$$294$$ 0 0
$$295$$ 3157.40 0.623155
$$296$$ −1054.10 −0.206988
$$297$$ 0 0
$$298$$ −1844.42 −0.358538
$$299$$ 1536.07 0.297100
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −7824.33 −1.49086
$$303$$ 0 0
$$304$$ −1694.45 −0.319681
$$305$$ 3649.41 0.685129
$$306$$ 0 0
$$307$$ 8871.73 1.64930 0.824652 0.565640i $$-0.191371\pi$$
0.824652 + 0.565640i $$0.191371\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −16882.6 −3.09312
$$311$$ −4531.09 −0.826156 −0.413078 0.910696i $$-0.635546\pi$$
−0.413078 + 0.910696i $$0.635546\pi$$
$$312$$ 0 0
$$313$$ −4316.61 −0.779518 −0.389759 0.920917i $$-0.627442\pi$$
−0.389759 + 0.920917i $$0.627442\pi$$
$$314$$ −3303.35 −0.593691
$$315$$ 0 0
$$316$$ −12505.7 −2.22628
$$317$$ −6697.55 −1.18666 −0.593331 0.804959i $$-0.702187\pi$$
−0.593331 + 0.804959i $$0.702187\pi$$
$$318$$ 0 0
$$319$$ 13586.8 2.38468
$$320$$ −13319.1 −2.32675
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −573.831 −0.0988508
$$324$$ 0 0
$$325$$ 2744.35 0.468397
$$326$$ −8312.53 −1.41223
$$327$$ 0 0
$$328$$ 16669.0 2.80607
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −8065.66 −1.33936 −0.669681 0.742649i $$-0.733569\pi$$
−0.669681 + 0.742649i $$0.733569\pi$$
$$332$$ −3881.16 −0.641586
$$333$$ 0 0
$$334$$ 15812.9 2.59056
$$335$$ 9633.97 1.57122
$$336$$ 0 0
$$337$$ −480.216 −0.0776232 −0.0388116 0.999247i $$-0.512357\pi$$
−0.0388116 + 0.999247i $$0.512357\pi$$
$$338$$ 9752.34 1.56940
$$339$$ 0 0
$$340$$ 3162.64 0.504465
$$341$$ −11271.3 −1.78995
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −7452.78 −1.16810
$$345$$ 0 0
$$346$$ −257.704 −0.0400412
$$347$$ 2237.12 0.346094 0.173047 0.984914i $$-0.444639\pi$$
0.173047 + 0.984914i $$0.444639\pi$$
$$348$$ 0 0
$$349$$ −2602.84 −0.399217 −0.199609 0.979876i $$-0.563967\pi$$
−0.199609 + 0.979876i $$0.563967\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −5493.99 −0.831905
$$353$$ −6938.47 −1.04617 −0.523084 0.852281i $$-0.675219\pi$$
−0.523084 + 0.852281i $$0.675219\pi$$
$$354$$ 0 0
$$355$$ −5966.62 −0.892043
$$356$$ −13418.7 −1.99772
$$357$$ 0 0
$$358$$ −17717.8 −2.61568
$$359$$ −2153.11 −0.316538 −0.158269 0.987396i $$-0.550591\pi$$
−0.158269 + 0.987396i $$0.550591\pi$$
$$360$$ 0 0
$$361$$ −4378.42 −0.638347
$$362$$ 6099.59 0.885600
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −11885.0 −1.70436
$$366$$ 0 0
$$367$$ −1848.98 −0.262986 −0.131493 0.991317i $$-0.541977\pi$$
−0.131493 + 0.991317i $$0.541977\pi$$
$$368$$ 4279.60 0.606221
$$369$$ 0 0
$$370$$ −2951.92 −0.414765
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −44.2183 −0.00613816 −0.00306908 0.999995i $$-0.500977\pi$$
−0.00306908 + 0.999995i $$0.500977\pi$$
$$374$$ 3262.27 0.451037
$$375$$ 0 0
$$376$$ 8705.53 1.19403
$$377$$ −2790.39 −0.381200
$$378$$ 0 0
$$379$$ 3338.62 0.452489 0.226245 0.974071i $$-0.427355\pi$$
0.226245 + 0.974071i $$0.427355\pi$$
$$380$$ −13671.6 −1.84562
$$381$$ 0 0
$$382$$ 4476.47 0.599571
$$383$$ 8000.54 1.06739 0.533693 0.845678i $$-0.320804\pi$$
0.533693 + 0.845678i $$0.320804\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 6728.54 0.887238
$$387$$ 0 0
$$388$$ 6687.07 0.874960
$$389$$ 1125.83 0.146740 0.0733701 0.997305i $$-0.476625\pi$$
0.0733701 + 0.997305i $$0.476625\pi$$
$$390$$ 0 0
$$391$$ 1449.30 0.187454
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 14323.3 1.83147
$$395$$ −15933.6 −2.02963
$$396$$ 0 0
$$397$$ −182.743 −0.0231023 −0.0115512 0.999933i $$-0.503677\pi$$
−0.0115512 + 0.999933i $$0.503677\pi$$
$$398$$ −104.980 −0.0132216
$$399$$ 0 0
$$400$$ 7645.96 0.955745
$$401$$ 9397.81 1.17033 0.585167 0.810913i $$-0.301029\pi$$
0.585167 + 0.810913i $$0.301029\pi$$
$$402$$ 0 0
$$403$$ 2314.85 0.286131
$$404$$ −13213.0 −1.62715
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −1970.78 −0.240019
$$408$$ 0 0
$$409$$ −2608.55 −0.315366 −0.157683 0.987490i $$-0.550402\pi$$
−0.157683 + 0.987490i $$0.550402\pi$$
$$410$$ 46680.0 5.62283
$$411$$ 0 0
$$412$$ 1678.87 0.200758
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −4944.99 −0.584916
$$416$$ 1128.33 0.132983
$$417$$ 0 0
$$418$$ −14102.3 −1.65015
$$419$$ −9533.73 −1.11158 −0.555792 0.831322i $$-0.687585\pi$$
−0.555792 + 0.831322i $$0.687585\pi$$
$$420$$ 0 0
$$421$$ −11430.2 −1.32322 −0.661611 0.749848i $$-0.730127\pi$$
−0.661611 + 0.749848i $$0.730127\pi$$
$$422$$ 23248.1 2.68175
$$423$$ 0 0
$$424$$ 8139.28 0.932260
$$425$$ 2589.34 0.295532
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 13269.9 1.49866
$$429$$ 0 0
$$430$$ −20870.8 −2.34065
$$431$$ −6992.79 −0.781510 −0.390755 0.920495i $$-0.627786\pi$$
−0.390755 + 0.920495i $$0.627786\pi$$
$$432$$ 0 0
$$433$$ 4730.76 0.525048 0.262524 0.964925i $$-0.415445\pi$$
0.262524 + 0.964925i $$0.415445\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 21233.4 2.33233
$$437$$ −6265.10 −0.685814
$$438$$ 0 0
$$439$$ 8609.29 0.935988 0.467994 0.883732i $$-0.344977\pi$$
0.467994 + 0.883732i $$0.344977\pi$$
$$440$$ 35362.0 3.83140
$$441$$ 0 0
$$442$$ −669.991 −0.0721000
$$443$$ −5271.91 −0.565408 −0.282704 0.959207i $$-0.591231\pi$$
−0.282704 + 0.959207i $$0.591231\pi$$
$$444$$ 0 0
$$445$$ −17096.7 −1.82126
$$446$$ 686.003 0.0728323
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 13513.8 1.42039 0.710197 0.704003i $$-0.248606\pi$$
0.710197 + 0.704003i $$0.248606\pi$$
$$450$$ 0 0
$$451$$ 31164.8 3.25386
$$452$$ −150.677 −0.0156798
$$453$$ 0 0
$$454$$ −23440.0 −2.42311
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 7288.39 0.746031 0.373016 0.927825i $$-0.378324\pi$$
0.373016 + 0.927825i $$0.378324\pi$$
$$458$$ −75.1891 −0.00767108
$$459$$ 0 0
$$460$$ 34529.8 3.49991
$$461$$ 5654.21 0.571243 0.285621 0.958343i $$-0.407800\pi$$
0.285621 + 0.958343i $$0.407800\pi$$
$$462$$ 0 0
$$463$$ 4102.17 0.411758 0.205879 0.978577i $$-0.433995\pi$$
0.205879 + 0.978577i $$0.433995\pi$$
$$464$$ −7774.24 −0.777823
$$465$$ 0 0
$$466$$ −3464.49 −0.344397
$$467$$ −5104.11 −0.505761 −0.252880 0.967498i $$-0.581378\pi$$
−0.252880 + 0.967498i $$0.581378\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 24379.1 2.39260
$$471$$ 0 0
$$472$$ −5369.22 −0.523598
$$473$$ −13933.9 −1.35451
$$474$$ 0 0
$$475$$ −11193.3 −1.08123
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 303.277 0.0290200
$$479$$ −4200.44 −0.400674 −0.200337 0.979727i $$-0.564204\pi$$
−0.200337 + 0.979727i $$0.564204\pi$$
$$480$$ 0 0
$$481$$ 404.750 0.0383680
$$482$$ 15952.1 1.50747
$$483$$ 0 0
$$484$$ 32353.9 3.03850
$$485$$ 8520.00 0.797677
$$486$$ 0 0
$$487$$ 4300.99 0.400198 0.200099 0.979776i $$-0.435874\pi$$
0.200099 + 0.979776i $$0.435874\pi$$
$$488$$ −6205.88 −0.575670
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −1774.44 −0.163095 −0.0815474 0.996669i $$-0.525986\pi$$
−0.0815474 + 0.996669i $$0.525986\pi$$
$$492$$ 0 0
$$493$$ −2632.78 −0.240516
$$494$$ 2896.26 0.263783
$$495$$ 0 0
$$496$$ 6449.33 0.583838
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −6854.78 −0.614954 −0.307477 0.951556i $$-0.599485\pi$$
−0.307477 + 0.951556i $$0.599485\pi$$
$$500$$ 27378.7 2.44883
$$501$$ 0 0
$$502$$ −23442.7 −2.08426
$$503$$ −6584.76 −0.583698 −0.291849 0.956464i $$-0.594270\pi$$
−0.291849 + 0.956464i $$0.594270\pi$$
$$504$$ 0 0
$$505$$ −16834.7 −1.48343
$$506$$ 35617.5 3.12923
$$507$$ 0 0
$$508$$ −23339.5 −2.03843
$$509$$ 4220.29 0.367507 0.183754 0.982972i $$-0.441175\pi$$
0.183754 + 0.982972i $$0.441175\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 11799.2 1.01847
$$513$$ 0 0
$$514$$ 13683.5 1.17423
$$515$$ 2139.05 0.183025
$$516$$ 0 0
$$517$$ 16276.1 1.38457
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −7262.49 −0.612464
$$521$$ −5111.00 −0.429783 −0.214891 0.976638i $$-0.568940\pi$$
−0.214891 + 0.976638i $$0.568940\pi$$
$$522$$ 0 0
$$523$$ −14147.7 −1.18286 −0.591429 0.806357i $$-0.701436\pi$$
−0.591429 + 0.806357i $$0.701436\pi$$
$$524$$ 3300.83 0.275186
$$525$$ 0 0
$$526$$ 20799.7 1.72416
$$527$$ 2184.09 0.180532
$$528$$ 0 0
$$529$$ 3656.53 0.300528
$$530$$ 22793.3 1.86807
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −6400.49 −0.520142
$$534$$ 0 0
$$535$$ 16907.2 1.36629
$$536$$ −16382.7 −1.32020
$$537$$ 0 0
$$538$$ −25500.8 −2.04353
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 21627.4 1.71874 0.859368 0.511357i $$-0.170857\pi$$
0.859368 + 0.511357i $$0.170857\pi$$
$$542$$ −14487.6 −1.14815
$$543$$ 0 0
$$544$$ 1064.60 0.0839050
$$545$$ 27053.5 2.12632
$$546$$ 0 0
$$547$$ −11453.9 −0.895305 −0.447652 0.894208i $$-0.647740\pi$$
−0.447652 + 0.894208i $$0.647740\pi$$
$$548$$ −33050.9 −2.57639
$$549$$ 0 0
$$550$$ 63634.6 4.93343
$$551$$ 11381.1 0.879946
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 40859.4 3.13348
$$555$$ 0 0
$$556$$ −44259.4 −3.37593
$$557$$ −1041.42 −0.0792219 −0.0396109 0.999215i $$-0.512612\pi$$
−0.0396109 + 0.999215i $$0.512612\pi$$
$$558$$ 0 0
$$559$$ 2861.69 0.216523
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −4538.61 −0.340658
$$563$$ −19938.9 −1.49259 −0.746293 0.665618i $$-0.768168\pi$$
−0.746293 + 0.665618i $$0.768168\pi$$
$$564$$ 0 0
$$565$$ −191.978 −0.0142948
$$566$$ −35866.6 −2.66358
$$567$$ 0 0
$$568$$ 10146.4 0.749527
$$569$$ 14051.2 1.03525 0.517625 0.855608i $$-0.326816\pi$$
0.517625 + 0.855608i $$0.326816\pi$$
$$570$$ 0 0
$$571$$ −21502.4 −1.57591 −0.787956 0.615732i $$-0.788861\pi$$
−0.787956 + 0.615732i $$0.788861\pi$$
$$572$$ −10657.0 −0.779009
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 28270.4 2.05036
$$576$$ 0 0
$$577$$ −16860.9 −1.21652 −0.608259 0.793739i $$-0.708132\pi$$
−0.608259 + 0.793739i $$0.708132\pi$$
$$578$$ 22764.3 1.63818
$$579$$ 0 0
$$580$$ −62726.2 −4.49063
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 15217.4 1.08103
$$584$$ 20210.8 1.43207
$$585$$ 0 0
$$586$$ 6310.28 0.444839
$$587$$ −3952.45 −0.277913 −0.138957 0.990298i $$-0.544375\pi$$
−0.138957 + 0.990298i $$0.544375\pi$$
$$588$$ 0 0
$$589$$ −9441.48 −0.660492
$$590$$ −15036.0 −1.04919
$$591$$ 0 0
$$592$$ 1127.66 0.0782883
$$593$$ −15161.7 −1.04995 −0.524973 0.851119i $$-0.675924\pi$$
−0.524973 + 0.851119i $$0.675924\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 5684.93 0.390711
$$597$$ 0 0
$$598$$ −7314.98 −0.500220
$$599$$ 7411.38 0.505544 0.252772 0.967526i $$-0.418658\pi$$
0.252772 + 0.967526i $$0.418658\pi$$
$$600$$ 0 0
$$601$$ −10776.0 −0.731388 −0.365694 0.930735i $$-0.619168\pi$$
−0.365694 + 0.930735i $$0.619168\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 24116.5 1.62464
$$605$$ 41222.1 2.77011
$$606$$ 0 0
$$607$$ −11.6383 −0.000778225 0 −0.000389113 1.00000i $$-0.500124\pi$$
−0.000389113 1.00000i $$0.500124\pi$$
$$608$$ −4602.09 −0.306973
$$609$$ 0 0
$$610$$ −17379.0 −1.15353
$$611$$ −3342.71 −0.221329
$$612$$ 0 0
$$613$$ 15621.4 1.02927 0.514635 0.857410i $$-0.327928\pi$$
0.514635 + 0.857410i $$0.327928\pi$$
$$614$$ −42248.5 −2.77689
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −26376.5 −1.72103 −0.860517 0.509422i $$-0.829859\pi$$
−0.860517 + 0.509422i $$0.829859\pi$$
$$618$$ 0 0
$$619$$ 21156.6 1.37375 0.686877 0.726774i $$-0.258981\pi$$
0.686877 + 0.726774i $$0.258981\pi$$
$$620$$ 52036.2 3.37069
$$621$$ 0 0
$$622$$ 21577.7 1.39098
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 6790.67 0.434603
$$626$$ 20556.3 1.31245
$$627$$ 0 0
$$628$$ 10181.7 0.646966
$$629$$ 381.888 0.0242081
$$630$$ 0 0
$$631$$ 18757.1 1.18337 0.591686 0.806169i $$-0.298463\pi$$
0.591686 + 0.806169i $$0.298463\pi$$
$$632$$ 27095.4 1.70537
$$633$$ 0 0
$$634$$ 31894.7 1.99795
$$635$$ −29736.9 −1.85838
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −64702.2 −4.01502
$$639$$ 0 0
$$640$$ 49603.3 3.06366
$$641$$ −3996.83 −0.246280 −0.123140 0.992389i $$-0.539296\pi$$
−0.123140 + 0.992389i $$0.539296\pi$$
$$642$$ 0 0
$$643$$ 22237.8 1.36388 0.681940 0.731408i $$-0.261136\pi$$
0.681940 + 0.731408i $$0.261136\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 2732.67 0.166433
$$647$$ 22992.2 1.39709 0.698545 0.715566i $$-0.253831\pi$$
0.698545 + 0.715566i $$0.253831\pi$$
$$648$$ 0 0
$$649$$ −10038.4 −0.607153
$$650$$ −13069.0 −0.788628
$$651$$ 0 0
$$652$$ 25621.2 1.53896
$$653$$ −29699.3 −1.77982 −0.889912 0.456132i $$-0.849234\pi$$
−0.889912 + 0.456132i $$0.849234\pi$$
$$654$$ 0 0
$$655$$ 4205.59 0.250880
$$656$$ −17832.2 −1.06133
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −388.786 −0.0229817 −0.0114909 0.999934i $$-0.503658\pi$$
−0.0114909 + 0.999934i $$0.503658\pi$$
$$660$$ 0 0
$$661$$ −9988.03 −0.587730 −0.293865 0.955847i $$-0.594942\pi$$
−0.293865 + 0.955847i $$0.594942\pi$$
$$662$$ 38409.9 2.25505
$$663$$ 0 0
$$664$$ 8409.06 0.491468
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −28744.7 −1.66867
$$668$$ −48739.2 −2.82302
$$669$$ 0 0
$$670$$ −45878.4 −2.64543
$$671$$ −11602.7 −0.667536
$$672$$ 0 0
$$673$$ 19996.8 1.14535 0.572674 0.819783i $$-0.305906\pi$$
0.572674 + 0.819783i $$0.305906\pi$$
$$674$$ 2286.86 0.130692
$$675$$ 0 0
$$676$$ −30059.0 −1.71023
$$677$$ −25123.8 −1.42627 −0.713135 0.701027i $$-0.752725\pi$$
−0.713135 + 0.701027i $$0.752725\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −6852.28 −0.386431
$$681$$ 0 0
$$682$$ 53675.5 3.01370
$$683$$ −23089.7 −1.29356 −0.646782 0.762675i $$-0.723886\pi$$
−0.646782 + 0.762675i $$0.723886\pi$$
$$684$$ 0 0
$$685$$ −42110.1 −2.34883
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 7972.88 0.441807
$$689$$ −3125.29 −0.172807
$$690$$ 0 0
$$691$$ 10755.9 0.592147 0.296074 0.955165i $$-0.404323\pi$$
0.296074 + 0.955165i $$0.404323\pi$$
$$692$$ 794.304 0.0436343
$$693$$ 0 0
$$694$$ −10653.5 −0.582710
$$695$$ −56390.9 −3.07774
$$696$$ 0 0
$$697$$ −6038.96 −0.328181
$$698$$ 12395.1 0.672152
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −31197.1 −1.68088 −0.840442 0.541901i $$-0.817705\pi$$
−0.840442 + 0.541901i $$0.817705\pi$$
$$702$$ 0 0
$$703$$ −1650.84 −0.0885670
$$704$$ 42345.9 2.26700
$$705$$ 0 0
$$706$$ 33042.0 1.76141
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 28895.4 1.53059 0.765296 0.643678i $$-0.222592\pi$$
0.765296 + 0.643678i $$0.222592\pi$$
$$710$$ 28413.9 1.50191
$$711$$ 0 0
$$712$$ 29073.3 1.53029
$$713$$ 23846.0 1.25251
$$714$$ 0 0
$$715$$ −13578.1 −0.710201
$$716$$ 54610.3 2.85040
$$717$$ 0 0
$$718$$ 10253.5 0.532946
$$719$$ −190.317 −0.00987151 −0.00493575 0.999988i $$-0.501571\pi$$
−0.00493575 + 0.999988i $$0.501571\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 20850.7 1.07477
$$723$$ 0 0
$$724$$ −18800.4 −0.965069
$$725$$ −51355.6 −2.63076
$$726$$ 0 0
$$727$$ 5281.27 0.269424 0.134712 0.990885i $$-0.456989\pi$$
0.134712 + 0.990885i $$0.456989\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 56598.4 2.86959
$$731$$ 2700.05 0.136614
$$732$$ 0 0
$$733$$ 27817.2 1.40171 0.700853 0.713305i $$-0.252803\pi$$
0.700853 + 0.713305i $$0.252803\pi$$
$$734$$ 8805.12 0.442783
$$735$$ 0 0
$$736$$ 11623.3 0.582122
$$737$$ −30629.6 −1.53088
$$738$$ 0 0
$$739$$ −8553.43 −0.425768 −0.212884 0.977077i $$-0.568286\pi$$
−0.212884 + 0.977077i $$0.568286\pi$$
$$740$$ 9098.52 0.451984
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 14150.4 0.698693 0.349346 0.936994i $$-0.386404\pi$$
0.349346 + 0.936994i $$0.386404\pi$$
$$744$$ 0 0
$$745$$ 7243.18 0.356201
$$746$$ 210.574 0.0103347
$$747$$ 0 0
$$748$$ −10055.1 −0.491511
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −24243.3 −1.17796 −0.588981 0.808147i $$-0.700471\pi$$
−0.588981 + 0.808147i $$0.700471\pi$$
$$752$$ −9313.05 −0.451612
$$753$$ 0 0
$$754$$ 13288.3 0.641817
$$755$$ 30726.8 1.48114
$$756$$ 0 0
$$757$$ −13428.2 −0.644722 −0.322361 0.946617i $$-0.604476\pi$$
−0.322361 + 0.946617i $$0.604476\pi$$
$$758$$ −15899.0 −0.761844
$$759$$ 0 0
$$760$$ 29621.3 1.41379
$$761$$ 10525.9 0.501398 0.250699 0.968065i $$-0.419340\pi$$
0.250699 + 0.968065i $$0.419340\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −13797.5 −0.653373
$$765$$ 0 0
$$766$$ −38099.8 −1.79713
$$767$$ 2061.65 0.0970558
$$768$$ 0 0
$$769$$ −31855.3 −1.49380 −0.746900 0.664936i $$-0.768459\pi$$
−0.746900 + 0.664936i $$0.768459\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −20739.0 −0.966855
$$773$$ −14812.7 −0.689232 −0.344616 0.938744i $$-0.611991\pi$$
−0.344616 + 0.938744i $$0.611991\pi$$
$$774$$ 0 0
$$775$$ 42603.4 1.97466
$$776$$ −14488.4 −0.670237
$$777$$ 0 0
$$778$$ −5361.38 −0.247063
$$779$$ 26105.5 1.20067
$$780$$ 0 0
$$781$$ 18969.9 0.869136
$$782$$ −6901.80 −0.315611
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 12972.5 0.589821
$$786$$ 0 0
$$787$$ −38359.3 −1.73744 −0.868718 0.495307i $$-0.835056\pi$$
−0.868718 + 0.495307i $$0.835056\pi$$
$$788$$ −44147.9 −1.99582
$$789$$ 0 0
$$790$$ 75878.1 3.41724
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 2382.91 0.106708
$$794$$ 870.251 0.0388968
$$795$$ 0 0
$$796$$ 323.574 0.0144080
$$797$$ 26078.1 1.15901 0.579506 0.814968i $$-0.303246\pi$$
0.579506 + 0.814968i $$0.303246\pi$$
$$798$$ 0 0
$$799$$ −3153.90 −0.139646
$$800$$ 20766.3 0.917751
$$801$$ 0 0
$$802$$ −44753.8 −1.97046
$$803$$ 37786.5 1.66060
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −11023.6 −0.481751
$$807$$ 0 0
$$808$$ 28627.7 1.24643
$$809$$ 33877.7 1.47228 0.736142 0.676827i $$-0.236646\pi$$
0.736142 + 0.676827i $$0.236646\pi$$
$$810$$ 0 0
$$811$$ −15463.4 −0.669537 −0.334769 0.942300i $$-0.608658\pi$$
−0.334769 + 0.942300i $$0.608658\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 9385.14 0.404114
$$815$$ 32644.0 1.40303
$$816$$ 0 0
$$817$$ −11671.9 −0.499813
$$818$$ 12422.3 0.530973
$$819$$ 0 0
$$820$$ −143879. −6.12740
$$821$$ −29146.6 −1.23901 −0.619503 0.784994i $$-0.712666\pi$$
−0.619503 + 0.784994i $$0.712666\pi$$
$$822$$ 0 0
$$823$$ −1831.30 −0.0775640 −0.0387820 0.999248i $$-0.512348\pi$$
−0.0387820 + 0.999248i $$0.512348\pi$$
$$824$$ −3637.50 −0.153784
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −19801.5 −0.832609 −0.416304 0.909225i $$-0.636675\pi$$
−0.416304 + 0.909225i $$0.636675\pi$$
$$828$$ 0 0
$$829$$ −20707.6 −0.867557 −0.433778 0.901020i $$-0.642820\pi$$
−0.433778 + 0.901020i $$0.642820\pi$$
$$830$$ 23548.8 0.984808
$$831$$ 0 0
$$832$$ −8696.82 −0.362389
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −62098.7 −2.57367
$$836$$ 43466.5 1.79823
$$837$$ 0 0
$$838$$ 45401.1 1.87154
$$839$$ 28880.2 1.18838 0.594192 0.804323i $$-0.297472\pi$$
0.594192 + 0.804323i $$0.297472\pi$$
$$840$$ 0 0
$$841$$ 27828.2 1.14101
$$842$$ 54432.5 2.22787
$$843$$ 0 0
$$844$$ −71656.0 −2.92240
$$845$$ −38298.2 −1.55917
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −8707.28 −0.352605
$$849$$ 0 0
$$850$$ −12330.8 −0.497580
$$851$$ 4169.46 0.167952
$$852$$ 0 0
$$853$$ −34006.5 −1.36502 −0.682509 0.730877i $$-0.739111\pi$$
−0.682509 + 0.730877i $$0.739111\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −28751.1 −1.14801
$$857$$ 27810.0 1.10849 0.554243 0.832355i $$-0.313008\pi$$
0.554243 + 0.832355i $$0.313008\pi$$
$$858$$ 0 0
$$859$$ −15538.4 −0.617186 −0.308593 0.951194i $$-0.599858\pi$$
−0.308593 + 0.951194i $$0.599858\pi$$
$$860$$ 64328.9 2.55069
$$861$$ 0 0
$$862$$ 33300.7 1.31581
$$863$$ 34590.4 1.36439 0.682196 0.731169i $$-0.261025\pi$$
0.682196 + 0.731169i $$0.261025\pi$$
$$864$$ 0 0
$$865$$ 1012.02 0.0397802
$$866$$ −22528.6 −0.884010
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 50658.2 1.97752
$$870$$ 0 0
$$871$$ 6290.58 0.244717
$$872$$ −46005.0 −1.78661
$$873$$ 0 0
$$874$$ 29835.4 1.15469
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −13884.4 −0.534597 −0.267298 0.963614i $$-0.586131\pi$$
−0.267298 + 0.963614i $$0.586131\pi$$
$$878$$ −40998.7 −1.57590
$$879$$ 0 0
$$880$$ −37829.7 −1.44914
$$881$$ −50252.5 −1.92173 −0.960867 0.277009i $$-0.910657\pi$$
−0.960867 + 0.277009i $$0.910657\pi$$
$$882$$ 0 0
$$883$$ −22551.2 −0.859466 −0.429733 0.902956i $$-0.641392\pi$$
−0.429733 + 0.902956i $$0.641392\pi$$
$$884$$ 2065.07 0.0785700
$$885$$ 0 0
$$886$$ 25105.6 0.951964
$$887$$ 28015.3 1.06050 0.530248 0.847842i $$-0.322099\pi$$
0.530248 + 0.847842i $$0.322099\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 81417.1 3.06641
$$891$$ 0 0
$$892$$ −2114.43 −0.0793679
$$893$$ 13633.8 0.510905
$$894$$ 0 0
$$895$$ 69579.0 2.59863
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −64354.9 −2.39148
$$899$$ −43318.2 −1.60706
$$900$$ 0 0
$$901$$ −2948.76 −0.109031
$$902$$ −148411. −5.47844
$$903$$ 0 0
$$904$$ 326.462 0.0120110
$$905$$ −23953.6 −0.879827
$$906$$ 0 0
$$907$$ 26825.9 0.982071 0.491035 0.871140i $$-0.336619\pi$$
0.491035 + 0.871140i $$0.336619\pi$$
$$908$$ 72247.5 2.64055
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 9637.55 0.350501 0.175250 0.984524i $$-0.443926\pi$$
0.175250 + 0.984524i $$0.443926\pi$$
$$912$$ 0 0
$$913$$ 15721.8 0.569896
$$914$$ −34708.4 −1.25607
$$915$$ 0 0
$$916$$ 231.751 0.00835945
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 3759.24 0.134936 0.0674678 0.997721i $$-0.478508\pi$$
0.0674678 + 0.997721i $$0.478508\pi$$
$$920$$ −74813.3 −2.68100
$$921$$ 0 0
$$922$$ −26926.2 −0.961787
$$923$$ −3895.95 −0.138935
$$924$$ 0 0
$$925$$ 7449.20 0.264787
$$926$$ −19535.2 −0.693267
$$927$$ 0 0
$$928$$ −21114.7 −0.746902
$$929$$ −4162.56 −0.147007 −0.0735033 0.997295i $$-0.523418\pi$$
−0.0735033 + 0.997295i $$0.523418\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 10678.4 0.375302
$$933$$ 0 0
$$934$$ 24306.6 0.851536
$$935$$ −12811.2 −0.448097
$$936$$ 0 0
$$937$$ 39048.7 1.36144 0.680718 0.732545i $$-0.261668\pi$$
0.680718 + 0.732545i $$0.261668\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −75142.0 −2.60730
$$941$$ −15015.4 −0.520179 −0.260089 0.965585i $$-0.583752\pi$$
−0.260089 + 0.965585i $$0.583752\pi$$
$$942$$ 0 0
$$943$$ −65933.6 −2.27687
$$944$$ 5743.91 0.198038
$$945$$ 0 0
$$946$$ 66355.4 2.28055
$$947$$ −12467.9 −0.427825 −0.213913 0.976853i $$-0.568621\pi$$
−0.213913 + 0.976853i $$0.568621\pi$$
$$948$$ 0 0
$$949$$ −7760.44 −0.265453
$$950$$ 53304.1 1.82044
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −26584.1 −0.903612 −0.451806 0.892116i $$-0.649220\pi$$
−0.451806 + 0.892116i $$0.649220\pi$$
$$954$$ 0 0
$$955$$ −17579.4 −0.595662
$$956$$ −934.773 −0.0316242
$$957$$ 0 0
$$958$$ 20003.1 0.674605
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 6144.81 0.206264
$$962$$ −1927.48 −0.0645992
$$963$$ 0 0
$$964$$ −49168.3 −1.64274
$$965$$ −26423.5 −0.881455
$$966$$ 0 0
$$967$$ 26613.6 0.885041 0.442521 0.896758i $$-0.354084\pi$$
0.442521 + 0.896758i $$0.354084\pi$$
$$968$$ −70099.0 −2.32755
$$969$$ 0 0
$$970$$ −40573.5 −1.34303
$$971$$ −10272.7 −0.339512 −0.169756 0.985486i $$-0.554298\pi$$
−0.169756 + 0.985486i $$0.554298\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −20481.9 −0.673803
$$975$$ 0 0
$$976$$ 6638.96 0.217734
$$977$$ −36897.7 −1.20825 −0.604127 0.796888i $$-0.706478\pi$$
−0.604127 + 0.796888i $$0.706478\pi$$
$$978$$ 0 0
$$979$$ 54356.2 1.77450
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 8450.17 0.274599
$$983$$ −21443.4 −0.695766 −0.347883 0.937538i $$-0.613099\pi$$
−0.347883 + 0.937538i $$0.613099\pi$$
$$984$$ 0 0
$$985$$ −56248.8 −1.81953
$$986$$ 12537.7 0.404951
$$987$$ 0 0
$$988$$ −8926.97 −0.287454
$$989$$ 29479.2 0.947810
$$990$$ 0 0
$$991$$ −16667.9 −0.534281 −0.267141 0.963658i $$-0.586079\pi$$
−0.267141 + 0.963658i $$0.586079\pi$$
$$992$$ 17516.3 0.560628
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 412.266 0.0131354
$$996$$ 0 0
$$997$$ −20248.7 −0.643213 −0.321606 0.946873i $$-0.604223\pi$$
−0.321606 + 0.946873i $$0.604223\pi$$
$$998$$ 32643.5 1.03538
$$999$$ 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 1323.4.a.bh.1.2 7
3.2 odd 2 1323.4.a.bk.1.6 7
7.3 odd 6 189.4.e.g.163.6 yes 14
7.5 odd 6 189.4.e.g.109.6 yes 14
7.6 odd 2 1323.4.a.bi.1.2 7
21.5 even 6 189.4.e.f.109.2 14
21.17 even 6 189.4.e.f.163.2 yes 14
21.20 even 2 1323.4.a.bj.1.6 7

By twisted newform
Twist Min Dim Char Parity Ord Type
189.4.e.f.109.2 14 21.5 even 6
189.4.e.f.163.2 yes 14 21.17 even 6
189.4.e.g.109.6 yes 14 7.5 odd 6
189.4.e.g.163.6 yes 14 7.3 odd 6
1323.4.a.bh.1.2 7 1.1 even 1 trivial
1323.4.a.bi.1.2 7 7.6 odd 2
1323.4.a.bj.1.6 7 21.20 even 2
1323.4.a.bk.1.6 7 3.2 odd 2