# Properties

 Label 507.4.a.r.1.1 Level $507$ Weight $4$ Character 507.1 Self dual yes Analytic conductor $29.914$ Analytic rank $0$ Dimension $10$ Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, 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("507.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 507.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: $$29.9139683729$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} - 70x^{8} + 1645x^{6} - 14700x^{4} + 44100x^{2} - 27648$$ x^10 - 70*x^8 + 1645*x^6 - 14700*x^4 + 44100*x^2 - 27648 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{3}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-5.36472$$ of defining polynomial Character $$\chi$$ $$=$$ 507.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-5.36472 q^{2} +3.00000 q^{3} +20.7803 q^{4} +2.69631 q^{5} -16.0942 q^{6} +15.2025 q^{7} -68.5626 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-5.36472 q^{2} +3.00000 q^{3} +20.7803 q^{4} +2.69631 q^{5} -16.0942 q^{6} +15.2025 q^{7} -68.5626 q^{8} +9.00000 q^{9} -14.4650 q^{10} -66.8848 q^{11} +62.3408 q^{12} -81.5570 q^{14} +8.08894 q^{15} +201.577 q^{16} +4.16354 q^{17} -48.2825 q^{18} +26.0850 q^{19} +56.0301 q^{20} +45.6074 q^{21} +358.819 q^{22} +47.3242 q^{23} -205.688 q^{24} -117.730 q^{25} +27.0000 q^{27} +315.911 q^{28} +257.007 q^{29} -43.3949 q^{30} +206.242 q^{31} -532.906 q^{32} -200.655 q^{33} -22.3362 q^{34} +40.9906 q^{35} +187.022 q^{36} +175.686 q^{37} -139.939 q^{38} -184.866 q^{40} +156.463 q^{41} -244.671 q^{42} +51.9845 q^{43} -1389.88 q^{44} +24.2668 q^{45} -253.881 q^{46} -354.222 q^{47} +604.732 q^{48} -111.885 q^{49} +631.588 q^{50} +12.4906 q^{51} -10.4723 q^{53} -144.848 q^{54} -180.342 q^{55} -1042.32 q^{56} +78.2550 q^{57} -1378.77 q^{58} +445.114 q^{59} +168.090 q^{60} +119.696 q^{61} -1106.43 q^{62} +136.822 q^{63} +1246.28 q^{64} +1076.46 q^{66} -22.4078 q^{67} +86.5195 q^{68} +141.973 q^{69} -219.903 q^{70} +285.207 q^{71} -617.064 q^{72} +740.989 q^{73} -942.507 q^{74} -353.190 q^{75} +542.053 q^{76} -1016.81 q^{77} -547.679 q^{79} +543.516 q^{80} +81.0000 q^{81} -839.378 q^{82} +603.056 q^{83} +947.734 q^{84} +11.2262 q^{85} -278.882 q^{86} +771.021 q^{87} +4585.80 q^{88} +215.668 q^{89} -130.185 q^{90} +983.409 q^{92} +618.726 q^{93} +1900.31 q^{94} +70.3333 q^{95} -1598.72 q^{96} +1447.50 q^{97} +600.233 q^{98} -601.964 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + 30 q^{3} + 60 q^{4} + 90 q^{9}+O(q^{10})$$ 10 * q + 30 * q^3 + 60 * q^4 + 90 * q^9 $$10 q + 30 q^{3} + 60 q^{4} + 90 q^{9} + 80 q^{10} + 180 q^{12} - 60 q^{14} + 500 q^{16} + 210 q^{17} + 580 q^{22} - 120 q^{23} + 960 q^{25} + 270 q^{27} + 990 q^{29} + 240 q^{30} - 120 q^{35} + 540 q^{36} + 1380 q^{38} + 2000 q^{40} - 180 q^{42} - 740 q^{43} + 1500 q^{48} + 1550 q^{49} + 630 q^{51} + 330 q^{53} + 520 q^{55} - 5340 q^{56} + 2750 q^{61} - 1560 q^{62} + 3140 q^{64} + 1740 q^{66} + 1200 q^{68} - 360 q^{69} - 4380 q^{74} + 2880 q^{75} + 4320 q^{77} + 1100 q^{79} + 810 q^{81} - 4780 q^{82} + 2970 q^{87} + 6340 q^{88} + 720 q^{90} - 1740 q^{92} + 6460 q^{94} - 2760 q^{95}+O(q^{100})$$ 10 * q + 30 * q^3 + 60 * q^4 + 90 * q^9 + 80 * q^10 + 180 * q^12 - 60 * q^14 + 500 * q^16 + 210 * q^17 + 580 * q^22 - 120 * q^23 + 960 * q^25 + 270 * q^27 + 990 * q^29 + 240 * q^30 - 120 * q^35 + 540 * q^36 + 1380 * q^38 + 2000 * q^40 - 180 * q^42 - 740 * q^43 + 1500 * q^48 + 1550 * q^49 + 630 * q^51 + 330 * q^53 + 520 * q^55 - 5340 * q^56 + 2750 * q^61 - 1560 * q^62 + 3140 * q^64 + 1740 * q^66 + 1200 * q^68 - 360 * q^69 - 4380 * q^74 + 2880 * q^75 + 4320 * q^77 + 1100 * q^79 + 810 * q^81 - 4780 * q^82 + 2970 * q^87 + 6340 * q^88 + 720 * q^90 - 1740 * q^92 + 6460 * q^94 - 2760 * q^95

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −5.36472 −1.89672 −0.948358 0.317201i $$-0.897257\pi$$
−0.948358 + 0.317201i $$0.897257\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 20.7803 2.59753
$$5$$ 2.69631 0.241165 0.120583 0.992703i $$-0.461524\pi$$
0.120583 + 0.992703i $$0.461524\pi$$
$$6$$ −16.0942 −1.09507
$$7$$ 15.2025 0.820856 0.410428 0.911893i $$-0.365379\pi$$
0.410428 + 0.911893i $$0.365379\pi$$
$$8$$ −68.5626 −3.03007
$$9$$ 9.00000 0.333333
$$10$$ −14.4650 −0.457423
$$11$$ −66.8848 −1.83332 −0.916661 0.399666i $$-0.869126\pi$$
−0.916661 + 0.399666i $$0.869126\pi$$
$$12$$ 62.3408 1.49969
$$13$$ 0 0
$$14$$ −81.5570 −1.55693
$$15$$ 8.08894 0.139237
$$16$$ 201.577 3.14965
$$17$$ 4.16354 0.0594004 0.0297002 0.999559i $$-0.490545\pi$$
0.0297002 + 0.999559i $$0.490545\pi$$
$$18$$ −48.2825 −0.632239
$$19$$ 26.0850 0.314963 0.157482 0.987522i $$-0.449662\pi$$
0.157482 + 0.987522i $$0.449662\pi$$
$$20$$ 56.0301 0.626436
$$21$$ 45.6074 0.473921
$$22$$ 358.819 3.47729
$$23$$ 47.3242 0.429034 0.214517 0.976720i $$-0.431182\pi$$
0.214517 + 0.976720i $$0.431182\pi$$
$$24$$ −205.688 −1.74941
$$25$$ −117.730 −0.941839
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 315.911 2.13220
$$29$$ 257.007 1.64569 0.822845 0.568266i $$-0.192386\pi$$
0.822845 + 0.568266i $$0.192386\pi$$
$$30$$ −43.3949 −0.264093
$$31$$ 206.242 1.19491 0.597455 0.801903i $$-0.296179\pi$$
0.597455 + 0.801903i $$0.296179\pi$$
$$32$$ −532.906 −2.94392
$$33$$ −200.655 −1.05847
$$34$$ −22.3362 −0.112666
$$35$$ 40.9906 0.197962
$$36$$ 187.022 0.865845
$$37$$ 175.686 0.780611 0.390305 0.920685i $$-0.372369\pi$$
0.390305 + 0.920685i $$0.372369\pi$$
$$38$$ −139.939 −0.597396
$$39$$ 0 0
$$40$$ −184.866 −0.730748
$$41$$ 156.463 0.595984 0.297992 0.954568i $$-0.403683\pi$$
0.297992 + 0.954568i $$0.403683\pi$$
$$42$$ −244.671 −0.898895
$$43$$ 51.9845 0.184362 0.0921809 0.995742i $$-0.470616\pi$$
0.0921809 + 0.995742i $$0.470616\pi$$
$$44$$ −1389.88 −4.76211
$$45$$ 24.2668 0.0803885
$$46$$ −253.881 −0.813755
$$47$$ −354.222 −1.09933 −0.549666 0.835384i $$-0.685245\pi$$
−0.549666 + 0.835384i $$0.685245\pi$$
$$48$$ 604.732 1.81845
$$49$$ −111.885 −0.326196
$$50$$ 631.588 1.78640
$$51$$ 12.4906 0.0342948
$$52$$ 0 0
$$53$$ −10.4723 −0.0271412 −0.0135706 0.999908i $$-0.504320\pi$$
−0.0135706 + 0.999908i $$0.504320\pi$$
$$54$$ −144.848 −0.365023
$$55$$ −180.342 −0.442134
$$56$$ −1042.32 −2.48725
$$57$$ 78.2550 0.181844
$$58$$ −1378.77 −3.12141
$$59$$ 445.114 0.982185 0.491092 0.871107i $$-0.336598\pi$$
0.491092 + 0.871107i $$0.336598\pi$$
$$60$$ 168.090 0.361673
$$61$$ 119.696 0.251238 0.125619 0.992079i $$-0.459908\pi$$
0.125619 + 0.992079i $$0.459908\pi$$
$$62$$ −1106.43 −2.26640
$$63$$ 136.822 0.273619
$$64$$ 1246.28 2.43413
$$65$$ 0 0
$$66$$ 1076.46 2.00761
$$67$$ −22.4078 −0.0408589 −0.0204294 0.999791i $$-0.506503\pi$$
−0.0204294 + 0.999791i $$0.506503\pi$$
$$68$$ 86.5195 0.154295
$$69$$ 141.973 0.247703
$$70$$ −219.903 −0.375478
$$71$$ 285.207 0.476731 0.238365 0.971176i $$-0.423388\pi$$
0.238365 + 0.971176i $$0.423388\pi$$
$$72$$ −617.064 −1.01002
$$73$$ 740.989 1.18803 0.594015 0.804454i $$-0.297542\pi$$
0.594015 + 0.804454i $$0.297542\pi$$
$$74$$ −942.507 −1.48060
$$75$$ −353.190 −0.543771
$$76$$ 542.053 0.818128
$$77$$ −1016.81 −1.50489
$$78$$ 0 0
$$79$$ −547.679 −0.779983 −0.389992 0.920818i $$-0.627522\pi$$
−0.389992 + 0.920818i $$0.627522\pi$$
$$80$$ 543.516 0.759586
$$81$$ 81.0000 0.111111
$$82$$ −839.378 −1.13041
$$83$$ 603.056 0.797518 0.398759 0.917056i $$-0.369441\pi$$
0.398759 + 0.917056i $$0.369441\pi$$
$$84$$ 947.734 1.23103
$$85$$ 11.2262 0.0143253
$$86$$ −278.882 −0.349682
$$87$$ 771.021 0.950139
$$88$$ 4585.80 5.55509
$$89$$ 215.668 0.256863 0.128431 0.991718i $$-0.459006\pi$$
0.128431 + 0.991718i $$0.459006\pi$$
$$90$$ −130.185 −0.152474
$$91$$ 0 0
$$92$$ 983.409 1.11443
$$93$$ 618.726 0.689881
$$94$$ 1900.31 2.08512
$$95$$ 70.3333 0.0759583
$$96$$ −1598.72 −1.69967
$$97$$ 1447.50 1.51517 0.757586 0.652735i $$-0.226378\pi$$
0.757586 + 0.652735i $$0.226378\pi$$
$$98$$ 600.233 0.618700
$$99$$ −601.964 −0.611107
$$100$$ −2446.46 −2.44646
$$101$$ −883.450 −0.870362 −0.435181 0.900343i $$-0.643316\pi$$
−0.435181 + 0.900343i $$0.643316\pi$$
$$102$$ −67.0087 −0.0650476
$$103$$ 1251.74 1.19745 0.598726 0.800954i $$-0.295674\pi$$
0.598726 + 0.800954i $$0.295674\pi$$
$$104$$ 0 0
$$105$$ 122.972 0.114293
$$106$$ 56.1812 0.0514792
$$107$$ −341.614 −0.308645 −0.154323 0.988021i $$-0.549319\pi$$
−0.154323 + 0.988021i $$0.549319\pi$$
$$108$$ 561.067 0.499896
$$109$$ 775.177 0.681179 0.340589 0.940212i $$-0.389373\pi$$
0.340589 + 0.940212i $$0.389373\pi$$
$$110$$ 967.487 0.838603
$$111$$ 527.058 0.450686
$$112$$ 3064.47 2.58541
$$113$$ 1279.05 1.06480 0.532402 0.846492i $$-0.321290\pi$$
0.532402 + 0.846492i $$0.321290\pi$$
$$114$$ −419.816 −0.344907
$$115$$ 127.601 0.103468
$$116$$ 5340.67 4.27473
$$117$$ 0 0
$$118$$ −2387.91 −1.86293
$$119$$ 63.2961 0.0487592
$$120$$ −554.599 −0.421898
$$121$$ 3142.58 2.36107
$$122$$ −642.137 −0.476528
$$123$$ 469.388 0.344091
$$124$$ 4285.77 3.10382
$$125$$ −654.476 −0.468305
$$126$$ −734.013 −0.518977
$$127$$ −1113.82 −0.778233 −0.389117 0.921188i $$-0.627220\pi$$
−0.389117 + 0.921188i $$0.627220\pi$$
$$128$$ −2422.68 −1.67294
$$129$$ 155.953 0.106441
$$130$$ 0 0
$$131$$ 2100.12 1.40068 0.700339 0.713811i $$-0.253032\pi$$
0.700339 + 0.713811i $$0.253032\pi$$
$$132$$ −4169.65 −2.74941
$$133$$ 396.556 0.258540
$$134$$ 120.211 0.0774977
$$135$$ 72.8004 0.0464123
$$136$$ −285.463 −0.179987
$$137$$ −1205.02 −0.751471 −0.375736 0.926727i $$-0.622610\pi$$
−0.375736 + 0.926727i $$0.622610\pi$$
$$138$$ −761.643 −0.469822
$$139$$ 322.890 0.197030 0.0985149 0.995136i $$-0.468591\pi$$
0.0985149 + 0.995136i $$0.468591\pi$$
$$140$$ 851.796 0.514213
$$141$$ −1062.67 −0.634700
$$142$$ −1530.06 −0.904223
$$143$$ 0 0
$$144$$ 1814.20 1.04988
$$145$$ 692.971 0.396884
$$146$$ −3975.20 −2.25336
$$147$$ −335.655 −0.188329
$$148$$ 3650.80 2.02766
$$149$$ 1128.86 0.620669 0.310335 0.950627i $$-0.399559\pi$$
0.310335 + 0.950627i $$0.399559\pi$$
$$150$$ 1894.77 1.03138
$$151$$ −2940.44 −1.58470 −0.792350 0.610066i $$-0.791143\pi$$
−0.792350 + 0.610066i $$0.791143\pi$$
$$152$$ −1788.46 −0.954361
$$153$$ 37.4719 0.0198001
$$154$$ 5454.93 2.85436
$$155$$ 556.093 0.288171
$$156$$ 0 0
$$157$$ 629.388 0.319940 0.159970 0.987122i $$-0.448860\pi$$
0.159970 + 0.987122i $$0.448860\pi$$
$$158$$ 2938.15 1.47941
$$159$$ −31.4170 −0.0156700
$$160$$ −1436.88 −0.709972
$$161$$ 719.444 0.352175
$$162$$ −434.543 −0.210746
$$163$$ −394.912 −0.189766 −0.0948832 0.995488i $$-0.530248\pi$$
−0.0948832 + 0.995488i $$0.530248\pi$$
$$164$$ 3251.33 1.54809
$$165$$ −541.027 −0.255266
$$166$$ −3235.23 −1.51267
$$167$$ −151.860 −0.0703669 −0.0351834 0.999381i $$-0.511202\pi$$
−0.0351834 + 0.999381i $$0.511202\pi$$
$$168$$ −3126.96 −1.43601
$$169$$ 0 0
$$170$$ −60.2255 −0.0271711
$$171$$ 234.765 0.104988
$$172$$ 1080.25 0.478886
$$173$$ 538.813 0.236793 0.118397 0.992966i $$-0.462225\pi$$
0.118397 + 0.992966i $$0.462225\pi$$
$$174$$ −4136.31 −1.80214
$$175$$ −1789.78 −0.773114
$$176$$ −13482.5 −5.77432
$$177$$ 1335.34 0.567065
$$178$$ −1157.00 −0.487196
$$179$$ −2220.80 −0.927319 −0.463659 0.886014i $$-0.653464\pi$$
−0.463659 + 0.886014i $$0.653464\pi$$
$$180$$ 504.271 0.208812
$$181$$ −3822.78 −1.56986 −0.784932 0.619582i $$-0.787302\pi$$
−0.784932 + 0.619582i $$0.787302\pi$$
$$182$$ 0 0
$$183$$ 359.089 0.145052
$$184$$ −3244.67 −1.30000
$$185$$ 473.704 0.188256
$$186$$ −3319.30 −1.30851
$$187$$ −278.478 −0.108900
$$188$$ −7360.84 −2.85555
$$189$$ 410.467 0.157974
$$190$$ −377.319 −0.144071
$$191$$ 3464.19 1.31236 0.656178 0.754606i $$-0.272172\pi$$
0.656178 + 0.754606i $$0.272172\pi$$
$$192$$ 3738.83 1.40535
$$193$$ 4697.40 1.75195 0.875975 0.482357i $$-0.160219\pi$$
0.875975 + 0.482357i $$0.160219\pi$$
$$194$$ −7765.46 −2.87385
$$195$$ 0 0
$$196$$ −2325.00 −0.847304
$$197$$ −2887.89 −1.04443 −0.522217 0.852813i $$-0.674895\pi$$
−0.522217 + 0.852813i $$0.674895\pi$$
$$198$$ 3229.37 1.15910
$$199$$ 63.0092 0.0224453 0.0112226 0.999937i $$-0.496428\pi$$
0.0112226 + 0.999937i $$0.496428\pi$$
$$200$$ 8071.87 2.85384
$$201$$ −67.2233 −0.0235899
$$202$$ 4739.47 1.65083
$$203$$ 3907.14 1.35087
$$204$$ 259.558 0.0890820
$$205$$ 421.872 0.143731
$$206$$ −6715.24 −2.27123
$$207$$ 425.918 0.143011
$$208$$ 0 0
$$209$$ −1744.69 −0.577429
$$210$$ −659.710 −0.216782
$$211$$ −1049.70 −0.342484 −0.171242 0.985229i $$-0.554778\pi$$
−0.171242 + 0.985229i $$0.554778\pi$$
$$212$$ −217.618 −0.0705002
$$213$$ 855.622 0.275241
$$214$$ 1832.66 0.585412
$$215$$ 140.166 0.0444617
$$216$$ −1851.19 −0.583137
$$217$$ 3135.39 0.980848
$$218$$ −4158.61 −1.29200
$$219$$ 2222.97 0.685909
$$220$$ −3747.56 −1.14846
$$221$$ 0 0
$$222$$ −2827.52 −0.854823
$$223$$ 2313.49 0.694722 0.347361 0.937732i $$-0.387078\pi$$
0.347361 + 0.937732i $$0.387078\pi$$
$$224$$ −8101.49 −2.41653
$$225$$ −1059.57 −0.313946
$$226$$ −6861.74 −2.01963
$$227$$ 3799.46 1.11092 0.555460 0.831543i $$-0.312542\pi$$
0.555460 + 0.831543i $$0.312542\pi$$
$$228$$ 1626.16 0.472347
$$229$$ −4321.07 −1.24692 −0.623459 0.781856i $$-0.714273\pi$$
−0.623459 + 0.781856i $$0.714273\pi$$
$$230$$ −684.543 −0.196250
$$231$$ −3050.44 −0.868850
$$232$$ −17621.1 −4.98655
$$233$$ 5279.77 1.48450 0.742251 0.670122i $$-0.233758\pi$$
0.742251 + 0.670122i $$0.233758\pi$$
$$234$$ 0 0
$$235$$ −955.094 −0.265121
$$236$$ 9249.59 2.55126
$$237$$ −1643.04 −0.450323
$$238$$ −339.566 −0.0924823
$$239$$ 1547.92 0.418939 0.209469 0.977815i $$-0.432826\pi$$
0.209469 + 0.977815i $$0.432826\pi$$
$$240$$ 1630.55 0.438547
$$241$$ −4918.01 −1.31451 −0.657255 0.753669i $$-0.728282\pi$$
−0.657255 + 0.753669i $$0.728282\pi$$
$$242$$ −16859.1 −4.47828
$$243$$ 243.000 0.0641500
$$244$$ 2487.32 0.652600
$$245$$ −301.677 −0.0786671
$$246$$ −2518.14 −0.652644
$$247$$ 0 0
$$248$$ −14140.5 −3.62066
$$249$$ 1809.17 0.460447
$$250$$ 3511.08 0.888241
$$251$$ 1155.78 0.290646 0.145323 0.989384i $$-0.453578\pi$$
0.145323 + 0.989384i $$0.453578\pi$$
$$252$$ 2843.20 0.710734
$$253$$ −3165.27 −0.786556
$$254$$ 5975.34 1.47609
$$255$$ 33.6786 0.00827073
$$256$$ 3026.80 0.738965
$$257$$ 2351.95 0.570859 0.285429 0.958400i $$-0.407864\pi$$
0.285429 + 0.958400i $$0.407864\pi$$
$$258$$ −836.647 −0.201889
$$259$$ 2670.86 0.640769
$$260$$ 0 0
$$261$$ 2313.06 0.548563
$$262$$ −11266.6 −2.65669
$$263$$ 5521.88 1.29465 0.647326 0.762213i $$-0.275887\pi$$
0.647326 + 0.762213i $$0.275887\pi$$
$$264$$ 13757.4 3.20723
$$265$$ −28.2367 −0.00654553
$$266$$ −2127.41 −0.490376
$$267$$ 647.004 0.148300
$$268$$ −465.639 −0.106132
$$269$$ 3916.95 0.887810 0.443905 0.896074i $$-0.353593\pi$$
0.443905 + 0.896074i $$0.353593\pi$$
$$270$$ −390.554 −0.0880310
$$271$$ −2777.53 −0.622593 −0.311297 0.950313i $$-0.600763\pi$$
−0.311297 + 0.950313i $$0.600763\pi$$
$$272$$ 839.276 0.187090
$$273$$ 0 0
$$274$$ 6464.58 1.42533
$$275$$ 7874.34 1.72669
$$276$$ 2950.23 0.643416
$$277$$ 6583.08 1.42794 0.713969 0.700177i $$-0.246896\pi$$
0.713969 + 0.700177i $$0.246896\pi$$
$$278$$ −1732.21 −0.373710
$$279$$ 1856.18 0.398303
$$280$$ −2810.42 −0.599839
$$281$$ 2871.66 0.609640 0.304820 0.952410i $$-0.401404\pi$$
0.304820 + 0.952410i $$0.401404\pi$$
$$282$$ 5700.92 1.20385
$$283$$ 7518.04 1.57916 0.789578 0.613651i $$-0.210300\pi$$
0.789578 + 0.613651i $$0.210300\pi$$
$$284$$ 5926.69 1.23832
$$285$$ 211.000 0.0438546
$$286$$ 0 0
$$287$$ 2378.62 0.489217
$$288$$ −4796.16 −0.981307
$$289$$ −4895.66 −0.996472
$$290$$ −3717.60 −0.752776
$$291$$ 4342.51 0.874785
$$292$$ 15397.9 3.08595
$$293$$ 4506.57 0.898555 0.449278 0.893392i $$-0.351681\pi$$
0.449278 + 0.893392i $$0.351681\pi$$
$$294$$ 1800.70 0.357207
$$295$$ 1200.17 0.236869
$$296$$ −12045.5 −2.36530
$$297$$ −1805.89 −0.352823
$$298$$ −6056.02 −1.17723
$$299$$ 0 0
$$300$$ −7339.38 −1.41246
$$301$$ 790.292 0.151335
$$302$$ 15774.7 3.00573
$$303$$ −2650.35 −0.502504
$$304$$ 5258.15 0.992024
$$305$$ 322.738 0.0605900
$$306$$ −201.026 −0.0375552
$$307$$ 9538.89 1.77333 0.886667 0.462409i $$-0.153015\pi$$
0.886667 + 0.462409i $$0.153015\pi$$
$$308$$ −21129.7 −3.90901
$$309$$ 3755.22 0.691350
$$310$$ −2983.29 −0.546578
$$311$$ −7466.28 −1.36133 −0.680666 0.732594i $$-0.738309\pi$$
−0.680666 + 0.732594i $$0.738309\pi$$
$$312$$ 0 0
$$313$$ −1821.65 −0.328964 −0.164482 0.986380i $$-0.552595\pi$$
−0.164482 + 0.986380i $$0.552595\pi$$
$$314$$ −3376.49 −0.606836
$$315$$ 368.915 0.0659874
$$316$$ −11380.9 −2.02603
$$317$$ 3125.14 0.553708 0.276854 0.960912i $$-0.410708\pi$$
0.276854 + 0.960912i $$0.410708\pi$$
$$318$$ 168.543 0.0297215
$$319$$ −17189.9 −3.01708
$$320$$ 3360.35 0.587029
$$321$$ −1024.84 −0.178196
$$322$$ −3859.62 −0.667976
$$323$$ 108.606 0.0187090
$$324$$ 1683.20 0.288615
$$325$$ 0 0
$$326$$ 2118.60 0.359933
$$327$$ 2325.53 0.393279
$$328$$ −10727.5 −1.80587
$$329$$ −5385.05 −0.902394
$$330$$ 2902.46 0.484167
$$331$$ −1553.67 −0.257999 −0.128999 0.991645i $$-0.541177\pi$$
−0.128999 + 0.991645i $$0.541177\pi$$
$$332$$ 12531.7 2.07158
$$333$$ 1581.17 0.260204
$$334$$ 814.686 0.133466
$$335$$ −60.4183 −0.00985375
$$336$$ 9193.42 1.49269
$$337$$ −3190.43 −0.515709 −0.257855 0.966184i $$-0.583016\pi$$
−0.257855 + 0.966184i $$0.583016\pi$$
$$338$$ 0 0
$$339$$ 3837.15 0.614764
$$340$$ 233.283 0.0372105
$$341$$ −13794.5 −2.19065
$$342$$ −1259.45 −0.199132
$$343$$ −6915.37 −1.08862
$$344$$ −3564.19 −0.558629
$$345$$ 382.802 0.0597373
$$346$$ −2890.59 −0.449130
$$347$$ −5718.68 −0.884712 −0.442356 0.896840i $$-0.645857\pi$$
−0.442356 + 0.896840i $$0.645857\pi$$
$$348$$ 16022.0 2.46802
$$349$$ −3328.46 −0.510511 −0.255256 0.966874i $$-0.582160\pi$$
−0.255256 + 0.966874i $$0.582160\pi$$
$$350$$ 9601.70 1.46638
$$351$$ 0 0
$$352$$ 35643.4 5.39715
$$353$$ −12306.5 −1.85555 −0.927774 0.373142i $$-0.878281\pi$$
−0.927774 + 0.373142i $$0.878281\pi$$
$$354$$ −7163.74 −1.07556
$$355$$ 769.008 0.114971
$$356$$ 4481.64 0.667210
$$357$$ 189.888 0.0281511
$$358$$ 11914.0 1.75886
$$359$$ 8539.97 1.25549 0.627747 0.778418i $$-0.283977\pi$$
0.627747 + 0.778418i $$0.283977\pi$$
$$360$$ −1663.80 −0.243583
$$361$$ −6178.57 −0.900798
$$362$$ 20508.2 2.97759
$$363$$ 9427.74 1.36316
$$364$$ 0 0
$$365$$ 1997.94 0.286512
$$366$$ −1926.41 −0.275123
$$367$$ 2496.65 0.355107 0.177553 0.984111i $$-0.443182\pi$$
0.177553 + 0.984111i $$0.443182\pi$$
$$368$$ 9539.49 1.35130
$$369$$ 1408.16 0.198661
$$370$$ −2541.29 −0.357069
$$371$$ −159.205 −0.0222790
$$372$$ 12857.3 1.79199
$$373$$ −1142.91 −0.158653 −0.0793264 0.996849i $$-0.525277\pi$$
−0.0793264 + 0.996849i $$0.525277\pi$$
$$374$$ 1493.96 0.206552
$$375$$ −1963.43 −0.270376
$$376$$ 24286.4 3.33105
$$377$$ 0 0
$$378$$ −2202.04 −0.299632
$$379$$ −12181.8 −1.65102 −0.825512 0.564384i $$-0.809114\pi$$
−0.825512 + 0.564384i $$0.809114\pi$$
$$380$$ 1461.54 0.197304
$$381$$ −3341.46 −0.449313
$$382$$ −18584.4 −2.48917
$$383$$ −10180.6 −1.35824 −0.679118 0.734029i $$-0.737638\pi$$
−0.679118 + 0.734029i $$0.737638\pi$$
$$384$$ −7268.04 −0.965874
$$385$$ −2741.65 −0.362928
$$386$$ −25200.3 −3.32295
$$387$$ 467.860 0.0614539
$$388$$ 30079.5 3.93571
$$389$$ 5845.83 0.761941 0.380971 0.924587i $$-0.375590\pi$$
0.380971 + 0.924587i $$0.375590\pi$$
$$390$$ 0 0
$$391$$ 197.036 0.0254848
$$392$$ 7671.13 0.988395
$$393$$ 6300.37 0.808681
$$394$$ 15492.7 1.98100
$$395$$ −1476.71 −0.188105
$$396$$ −12509.0 −1.58737
$$397$$ −2500.92 −0.316166 −0.158083 0.987426i $$-0.550531\pi$$
−0.158083 + 0.987426i $$0.550531\pi$$
$$398$$ −338.027 −0.0425723
$$399$$ 1189.67 0.149268
$$400$$ −23731.7 −2.96646
$$401$$ −9189.25 −1.14436 −0.572181 0.820127i $$-0.693902\pi$$
−0.572181 + 0.820127i $$0.693902\pi$$
$$402$$ 360.634 0.0447433
$$403$$ 0 0
$$404$$ −18358.3 −2.26080
$$405$$ 218.401 0.0267962
$$406$$ −20960.7 −2.56223
$$407$$ −11750.7 −1.43111
$$408$$ −856.390 −0.103916
$$409$$ 9214.38 1.11399 0.556995 0.830516i $$-0.311954\pi$$
0.556995 + 0.830516i $$0.311954\pi$$
$$410$$ −2263.23 −0.272617
$$411$$ −3615.05 −0.433862
$$412$$ 26011.5 3.11042
$$413$$ 6766.83 0.806232
$$414$$ −2284.93 −0.271252
$$415$$ 1626.03 0.192334
$$416$$ 0 0
$$417$$ 968.669 0.113755
$$418$$ 9359.78 1.09522
$$419$$ −6494.41 −0.757214 −0.378607 0.925558i $$-0.623597\pi$$
−0.378607 + 0.925558i $$0.623597\pi$$
$$420$$ 2555.39 0.296881
$$421$$ 3059.56 0.354190 0.177095 0.984194i $$-0.443330\pi$$
0.177095 + 0.984194i $$0.443330\pi$$
$$422$$ 5631.33 0.649595
$$423$$ −3188.00 −0.366444
$$424$$ 718.010 0.0822398
$$425$$ −490.173 −0.0559456
$$426$$ −4590.18 −0.522054
$$427$$ 1819.68 0.206230
$$428$$ −7098.82 −0.801716
$$429$$ 0 0
$$430$$ −751.954 −0.0843313
$$431$$ 7937.05 0.887040 0.443520 0.896264i $$-0.353729\pi$$
0.443520 + 0.896264i $$0.353729\pi$$
$$432$$ 5442.59 0.606150
$$433$$ 7294.37 0.809573 0.404786 0.914411i $$-0.367346\pi$$
0.404786 + 0.914411i $$0.367346\pi$$
$$434$$ −16820.5 −1.86039
$$435$$ 2078.91 0.229141
$$436$$ 16108.4 1.76938
$$437$$ 1234.45 0.135130
$$438$$ −11925.6 −1.30098
$$439$$ 15214.7 1.65412 0.827059 0.562115i $$-0.190012\pi$$
0.827059 + 0.562115i $$0.190012\pi$$
$$440$$ 12364.7 1.33970
$$441$$ −1006.97 −0.108732
$$442$$ 0 0
$$443$$ 1517.05 0.162703 0.0813515 0.996685i $$-0.474076\pi$$
0.0813515 + 0.996685i $$0.474076\pi$$
$$444$$ 10952.4 1.17067
$$445$$ 581.509 0.0619464
$$446$$ −12411.3 −1.31769
$$447$$ 3386.58 0.358343
$$448$$ 18946.5 1.99807
$$449$$ −705.247 −0.0741262 −0.0370631 0.999313i $$-0.511800\pi$$
−0.0370631 + 0.999313i $$0.511800\pi$$
$$450$$ 5684.30 0.595467
$$451$$ −10465.0 −1.09263
$$452$$ 26579.0 2.76586
$$453$$ −8821.33 −0.914927
$$454$$ −20383.1 −2.10710
$$455$$ 0 0
$$456$$ −5365.37 −0.551001
$$457$$ −7277.73 −0.744940 −0.372470 0.928044i $$-0.621489\pi$$
−0.372470 + 0.928044i $$0.621489\pi$$
$$458$$ 23181.4 2.36505
$$459$$ 112.416 0.0114316
$$460$$ 2651.58 0.268762
$$461$$ 1961.88 0.198208 0.0991041 0.995077i $$-0.468402\pi$$
0.0991041 + 0.995077i $$0.468402\pi$$
$$462$$ 16364.8 1.64796
$$463$$ −10374.1 −1.04131 −0.520653 0.853768i $$-0.674311\pi$$
−0.520653 + 0.853768i $$0.674311\pi$$
$$464$$ 51806.8 5.18334
$$465$$ 1668.28 0.166376
$$466$$ −28324.5 −2.81568
$$467$$ 8788.92 0.870883 0.435442 0.900217i $$-0.356592\pi$$
0.435442 + 0.900217i $$0.356592\pi$$
$$468$$ 0 0
$$469$$ −340.653 −0.0335392
$$470$$ 5123.82 0.502860
$$471$$ 1888.16 0.184718
$$472$$ −30518.2 −2.97609
$$473$$ −3476.97 −0.337995
$$474$$ 8814.44 0.854136
$$475$$ −3070.98 −0.296645
$$476$$ 1315.31 0.126654
$$477$$ −94.2510 −0.00904707
$$478$$ −8304.14 −0.794608
$$479$$ −11141.7 −1.06279 −0.531394 0.847125i $$-0.678332\pi$$
−0.531394 + 0.847125i $$0.678332\pi$$
$$480$$ −4310.65 −0.409903
$$481$$ 0 0
$$482$$ 26383.8 2.49325
$$483$$ 2158.33 0.203328
$$484$$ 65303.7 6.13295
$$485$$ 3902.92 0.365407
$$486$$ −1303.63 −0.121674
$$487$$ −19640.5 −1.82750 −0.913752 0.406273i $$-0.866828\pi$$
−0.913752 + 0.406273i $$0.866828\pi$$
$$488$$ −8206.69 −0.761269
$$489$$ −1184.74 −0.109562
$$490$$ 1618.41 0.149209
$$491$$ −3410.31 −0.313453 −0.156726 0.987642i $$-0.550094\pi$$
−0.156726 + 0.987642i $$0.550094\pi$$
$$492$$ 9754.00 0.893789
$$493$$ 1070.06 0.0977546
$$494$$ 0 0
$$495$$ −1623.08 −0.147378
$$496$$ 41573.8 3.76354
$$497$$ 4335.86 0.391327
$$498$$ −9705.68 −0.873338
$$499$$ −5032.44 −0.451469 −0.225735 0.974189i $$-0.572478\pi$$
−0.225735 + 0.974189i $$0.572478\pi$$
$$500$$ −13600.2 −1.21644
$$501$$ −455.580 −0.0406263
$$502$$ −6200.44 −0.551274
$$503$$ 17189.4 1.52373 0.761866 0.647735i $$-0.224284\pi$$
0.761866 + 0.647735i $$0.224284\pi$$
$$504$$ −9380.89 −0.829083
$$505$$ −2382.06 −0.209901
$$506$$ 16980.8 1.49187
$$507$$ 0 0
$$508$$ −23145.5 −2.02149
$$509$$ −930.560 −0.0810341 −0.0405170 0.999179i $$-0.512901\pi$$
−0.0405170 + 0.999179i $$0.512901\pi$$
$$510$$ −180.676 −0.0156872
$$511$$ 11264.9 0.975201
$$512$$ 3143.50 0.271337
$$513$$ 704.295 0.0606148
$$514$$ −12617.6 −1.08276
$$515$$ 3375.08 0.288784
$$516$$ 3240.75 0.276485
$$517$$ 23692.1 2.01543
$$518$$ −14328.4 −1.21536
$$519$$ 1616.44 0.136713
$$520$$ 0 0
$$521$$ −9869.60 −0.829933 −0.414966 0.909837i $$-0.636207\pi$$
−0.414966 + 0.909837i $$0.636207\pi$$
$$522$$ −12408.9 −1.04047
$$523$$ −21420.6 −1.79093 −0.895466 0.445129i $$-0.853158\pi$$
−0.895466 + 0.445129i $$0.853158\pi$$
$$524$$ 43641.1 3.63831
$$525$$ −5369.35 −0.446358
$$526$$ −29623.4 −2.45559
$$527$$ 858.697 0.0709781
$$528$$ −40447.4 −3.33380
$$529$$ −9927.42 −0.815930
$$530$$ 151.482 0.0124150
$$531$$ 4006.03 0.327395
$$532$$ 8240.54 0.671565
$$533$$ 0 0
$$534$$ −3471.00 −0.281283
$$535$$ −921.097 −0.0744345
$$536$$ 1536.34 0.123805
$$537$$ −6662.39 −0.535388
$$538$$ −21013.4 −1.68392
$$539$$ 7483.41 0.598021
$$540$$ 1512.81 0.120558
$$541$$ 7771.50 0.617602 0.308801 0.951127i $$-0.400072\pi$$
0.308801 + 0.951127i $$0.400072\pi$$
$$542$$ 14900.7 1.18088
$$543$$ −11468.3 −0.906361
$$544$$ −2218.78 −0.174870
$$545$$ 2090.12 0.164277
$$546$$ 0 0
$$547$$ −15577.5 −1.21763 −0.608817 0.793310i $$-0.708356\pi$$
−0.608817 + 0.793310i $$0.708356\pi$$
$$548$$ −25040.6 −1.95197
$$549$$ 1077.27 0.0837461
$$550$$ −42243.7 −3.27505
$$551$$ 6704.02 0.518332
$$552$$ −9734.01 −0.750556
$$553$$ −8326.07 −0.640254
$$554$$ −35316.4 −2.70840
$$555$$ 1421.11 0.108690
$$556$$ 6709.74 0.511792
$$557$$ −22804.5 −1.73475 −0.867377 0.497652i $$-0.834196\pi$$
−0.867377 + 0.497652i $$0.834196\pi$$
$$558$$ −9957.89 −0.755468
$$559$$ 0 0
$$560$$ 8262.78 0.623511
$$561$$ −835.433 −0.0628734
$$562$$ −15405.7 −1.15631
$$563$$ −3517.41 −0.263306 −0.131653 0.991296i $$-0.542028\pi$$
−0.131653 + 0.991296i $$0.542028\pi$$
$$564$$ −22082.5 −1.64865
$$565$$ 3448.71 0.256794
$$566$$ −40332.2 −2.99521
$$567$$ 1231.40 0.0912062
$$568$$ −19554.6 −1.44453
$$569$$ −6093.44 −0.448946 −0.224473 0.974480i $$-0.572066\pi$$
−0.224473 + 0.974480i $$0.572066\pi$$
$$570$$ −1131.96 −0.0831797
$$571$$ 10460.2 0.766630 0.383315 0.923618i $$-0.374782\pi$$
0.383315 + 0.923618i $$0.374782\pi$$
$$572$$ 0 0
$$573$$ 10392.6 0.757689
$$574$$ −12760.6 −0.927906
$$575$$ −5571.47 −0.404081
$$576$$ 11216.5 0.811378
$$577$$ −9648.19 −0.696117 −0.348058 0.937473i $$-0.613159\pi$$
−0.348058 + 0.937473i $$0.613159\pi$$
$$578$$ 26263.9 1.89002
$$579$$ 14092.2 1.01149
$$580$$ 14400.1 1.03092
$$581$$ 9167.93 0.654647
$$582$$ −23296.4 −1.65922
$$583$$ 700.440 0.0497586
$$584$$ −50804.1 −3.59981
$$585$$ 0 0
$$586$$ −24176.5 −1.70430
$$587$$ 2170.73 0.152633 0.0763166 0.997084i $$-0.475684\pi$$
0.0763166 + 0.997084i $$0.475684\pi$$
$$588$$ −6975.01 −0.489191
$$589$$ 5379.82 0.376353
$$590$$ −6438.56 −0.449274
$$591$$ −8663.66 −0.603004
$$592$$ 35414.3 2.45865
$$593$$ 22885.9 1.58484 0.792421 0.609975i $$-0.208820\pi$$
0.792421 + 0.609975i $$0.208820\pi$$
$$594$$ 9688.11 0.669205
$$595$$ 170.666 0.0117590
$$596$$ 23458.0 1.61221
$$597$$ 189.028 0.0129588
$$598$$ 0 0
$$599$$ −23978.7 −1.63563 −0.817815 0.575482i $$-0.804815\pi$$
−0.817815 + 0.575482i $$0.804815\pi$$
$$600$$ 24215.6 1.64766
$$601$$ 12873.2 0.873728 0.436864 0.899528i $$-0.356089\pi$$
0.436864 + 0.899528i $$0.356089\pi$$
$$602$$ −4239.70 −0.287039
$$603$$ −201.670 −0.0136196
$$604$$ −61103.2 −4.11631
$$605$$ 8473.38 0.569408
$$606$$ 14218.4 0.953107
$$607$$ −7117.15 −0.475908 −0.237954 0.971276i $$-0.576477\pi$$
−0.237954 + 0.971276i $$0.576477\pi$$
$$608$$ −13900.9 −0.927227
$$609$$ 11721.4 0.779927
$$610$$ −1731.40 −0.114922
$$611$$ 0 0
$$612$$ 778.675 0.0514315
$$613$$ 173.297 0.0114183 0.00570913 0.999984i $$-0.498183\pi$$
0.00570913 + 0.999984i $$0.498183\pi$$
$$614$$ −51173.5 −3.36351
$$615$$ 1265.62 0.0829830
$$616$$ 69715.5 4.55993
$$617$$ −6102.75 −0.398197 −0.199099 0.979979i $$-0.563801\pi$$
−0.199099 + 0.979979i $$0.563801\pi$$
$$618$$ −20145.7 −1.31129
$$619$$ −14867.8 −0.965409 −0.482705 0.875783i $$-0.660346\pi$$
−0.482705 + 0.875783i $$0.660346\pi$$
$$620$$ 11555.8 0.748534
$$621$$ 1277.75 0.0825676
$$622$$ 40054.6 2.58206
$$623$$ 3278.69 0.210847
$$624$$ 0 0
$$625$$ 12951.6 0.828900
$$626$$ 9772.64 0.623951
$$627$$ −5234.07 −0.333379
$$628$$ 13078.9 0.831056
$$629$$ 731.475 0.0463686
$$630$$ −1979.13 −0.125159
$$631$$ −17210.6 −1.08580 −0.542902 0.839796i $$-0.682675\pi$$
−0.542902 + 0.839796i $$0.682675\pi$$
$$632$$ 37550.3 2.36340
$$633$$ −3149.09 −0.197733
$$634$$ −16765.5 −1.05023
$$635$$ −3003.21 −0.187683
$$636$$ −652.853 −0.0407033
$$637$$ 0 0
$$638$$ 92218.9 5.72254
$$639$$ 2566.87 0.158910
$$640$$ −6532.30 −0.403456
$$641$$ −12636.0 −0.778616 −0.389308 0.921108i $$-0.627286\pi$$
−0.389308 + 0.921108i $$0.627286\pi$$
$$642$$ 5497.99 0.337988
$$643$$ −9586.37 −0.587947 −0.293973 0.955814i $$-0.594978\pi$$
−0.293973 + 0.955814i $$0.594978\pi$$
$$644$$ 14950.2 0.914786
$$645$$ 420.499 0.0256700
$$646$$ −582.641 −0.0354856
$$647$$ 5244.32 0.318664 0.159332 0.987225i $$-0.449066\pi$$
0.159332 + 0.987225i $$0.449066\pi$$
$$648$$ −5553.57 −0.336674
$$649$$ −29771.4 −1.80066
$$650$$ 0 0
$$651$$ 9406.17 0.566293
$$652$$ −8206.39 −0.492925
$$653$$ 18869.0 1.13078 0.565392 0.824822i $$-0.308725\pi$$
0.565392 + 0.824822i $$0.308725\pi$$
$$654$$ −12475.8 −0.745938
$$655$$ 5662.59 0.337795
$$656$$ 31539.3 1.87714
$$657$$ 6668.90 0.396010
$$658$$ 28889.3 1.71159
$$659$$ −24299.8 −1.43639 −0.718197 0.695839i $$-0.755033\pi$$
−0.718197 + 0.695839i $$0.755033\pi$$
$$660$$ −11242.7 −0.663062
$$661$$ 29915.0 1.76030 0.880151 0.474694i $$-0.157441\pi$$
0.880151 + 0.474694i $$0.157441\pi$$
$$662$$ 8335.03 0.489350
$$663$$ 0 0
$$664$$ −41347.1 −2.41653
$$665$$ 1069.24 0.0623508
$$666$$ −8482.56 −0.493532
$$667$$ 12162.6 0.706056
$$668$$ −3155.69 −0.182780
$$669$$ 6940.48 0.401098
$$670$$ 324.128 0.0186898
$$671$$ −8005.86 −0.460600
$$672$$ −24304.5 −1.39519
$$673$$ 15493.7 0.887426 0.443713 0.896169i $$-0.353661\pi$$
0.443713 + 0.896169i $$0.353661\pi$$
$$674$$ 17115.8 0.978154
$$675$$ −3178.71 −0.181257
$$676$$ 0 0
$$677$$ 11729.7 0.665891 0.332945 0.942946i $$-0.391958\pi$$
0.332945 + 0.942946i $$0.391958\pi$$
$$678$$ −20585.2 −1.16603
$$679$$ 22005.6 1.24374
$$680$$ −769.698 −0.0434067
$$681$$ 11398.4 0.641390
$$682$$ 74003.5 4.15505
$$683$$ −1168.42 −0.0654587 −0.0327294 0.999464i $$-0.510420\pi$$
−0.0327294 + 0.999464i $$0.510420\pi$$
$$684$$ 4878.48 0.272709
$$685$$ −3249.10 −0.181229
$$686$$ 37099.1 2.06480
$$687$$ −12963.2 −0.719909
$$688$$ 10478.9 0.580675
$$689$$ 0 0
$$690$$ −2053.63 −0.113305
$$691$$ 32992.9 1.81637 0.908183 0.418574i $$-0.137470\pi$$
0.908183 + 0.418574i $$0.137470\pi$$
$$692$$ 11196.7 0.615078
$$693$$ −9151.33 −0.501631
$$694$$ 30679.2 1.67805
$$695$$ 870.612 0.0475168
$$696$$ −52863.2 −2.87899
$$697$$ 651.438 0.0354017
$$698$$ 17856.3 0.968295
$$699$$ 15839.3 0.857078
$$700$$ −37192.2 −2.00819
$$701$$ −14785.8 −0.796651 −0.398326 0.917244i $$-0.630409\pi$$
−0.398326 + 0.917244i $$0.630409\pi$$
$$702$$ 0 0
$$703$$ 4582.77 0.245864
$$704$$ −83357.0 −4.46255
$$705$$ −2865.28 −0.153068
$$706$$ 66021.0 3.51945
$$707$$ −13430.6 −0.714442
$$708$$ 27748.8 1.47297
$$709$$ 12634.0 0.669225 0.334612 0.942356i $$-0.391395\pi$$
0.334612 + 0.942356i $$0.391395\pi$$
$$710$$ −4125.52 −0.218067
$$711$$ −4929.11 −0.259994
$$712$$ −14786.8 −0.778312
$$713$$ 9760.24 0.512656
$$714$$ −1018.70 −0.0533947
$$715$$ 0 0
$$716$$ −46148.7 −2.40874
$$717$$ 4643.75 0.241874
$$718$$ −45814.6 −2.38132
$$719$$ −27296.5 −1.41584 −0.707920 0.706292i $$-0.750366\pi$$
−0.707920 + 0.706292i $$0.750366\pi$$
$$720$$ 4891.64 0.253195
$$721$$ 19029.5 0.982936
$$722$$ 33146.3 1.70856
$$723$$ −14754.0 −0.758932
$$724$$ −79438.5 −4.07777
$$725$$ −30257.4 −1.54997
$$726$$ −50577.2 −2.58553
$$727$$ 4658.21 0.237639 0.118819 0.992916i $$-0.462089\pi$$
0.118819 + 0.992916i $$0.462089\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ −10718.4 −0.543432
$$731$$ 216.439 0.0109512
$$732$$ 7461.96 0.376779
$$733$$ −166.474 −0.00838864 −0.00419432 0.999991i $$-0.501335\pi$$
−0.00419432 + 0.999991i $$0.501335\pi$$
$$734$$ −13393.9 −0.673537
$$735$$ −905.031 −0.0454185
$$736$$ −25219.4 −1.26304
$$737$$ 1498.74 0.0749074
$$738$$ −7554.41 −0.376804
$$739$$ 12738.1 0.634069 0.317035 0.948414i $$-0.397313\pi$$
0.317035 + 0.948414i $$0.397313\pi$$
$$740$$ 9843.70 0.489002
$$741$$ 0 0
$$742$$ 854.092 0.0422570
$$743$$ −30724.5 −1.51705 −0.758527 0.651641i $$-0.774081\pi$$
−0.758527 + 0.651641i $$0.774081\pi$$
$$744$$ −42421.5 −2.09039
$$745$$ 3043.76 0.149684
$$746$$ 6131.38 0.300919
$$747$$ 5427.50 0.265839
$$748$$ −5786.84 −0.282871
$$749$$ −5193.37 −0.253353
$$750$$ 10533.2 0.512826
$$751$$ −39538.6 −1.92115 −0.960575 0.278021i $$-0.910322\pi$$
−0.960575 + 0.278021i $$0.910322\pi$$
$$752$$ −71403.2 −3.46251
$$753$$ 3467.34 0.167805
$$754$$ 0 0
$$755$$ −7928.35 −0.382175
$$756$$ 8529.61 0.410342
$$757$$ 23035.1 1.10598 0.552990 0.833188i $$-0.313487\pi$$
0.552990 + 0.833188i $$0.313487\pi$$
$$758$$ 65352.1 3.13153
$$759$$ −9495.81 −0.454119
$$760$$ −4822.23 −0.230159
$$761$$ 32454.0 1.54594 0.772968 0.634445i $$-0.218771\pi$$
0.772968 + 0.634445i $$0.218771\pi$$
$$762$$ 17926.0 0.852220
$$763$$ 11784.6 0.559150
$$764$$ 71986.8 3.40889
$$765$$ 101.036 0.00477511
$$766$$ 54616.1 2.57619
$$767$$ 0 0
$$768$$ 9080.40 0.426641
$$769$$ −32216.2 −1.51072 −0.755362 0.655307i $$-0.772539\pi$$
−0.755362 + 0.655307i $$0.772539\pi$$
$$770$$ 14708.2 0.688372
$$771$$ 7055.86 0.329586
$$772$$ 97613.2 4.55075
$$773$$ 2924.60 0.136081 0.0680404 0.997683i $$-0.478325\pi$$
0.0680404 + 0.997683i $$0.478325\pi$$
$$774$$ −2509.94 −0.116561
$$775$$ −24280.9 −1.12541
$$776$$ −99244.7 −4.59108
$$777$$ 8012.58 0.369948
$$778$$ −31361.2 −1.44519
$$779$$ 4081.32 0.187713
$$780$$ 0 0
$$781$$ −19076.1 −0.874001
$$782$$ −1057.04 −0.0483374
$$783$$ 6939.19 0.316713
$$784$$ −22553.5 −1.02740
$$785$$ 1697.03 0.0771586
$$786$$ −33799.8 −1.53384
$$787$$ 25507.1 1.15531 0.577656 0.816280i $$-0.303967\pi$$
0.577656 + 0.816280i $$0.303967\pi$$
$$788$$ −60011.1 −2.71295
$$789$$ 16565.6 0.747468
$$790$$ 7922.16 0.356782
$$791$$ 19444.7 0.874050
$$792$$ 41272.2 1.85170
$$793$$ 0 0
$$794$$ 13416.8 0.599677
$$795$$ −84.7100 −0.00377906
$$796$$ 1309.35 0.0583023
$$797$$ 5448.98 0.242174 0.121087 0.992642i $$-0.461362\pi$$
0.121087 + 0.992642i $$0.461362\pi$$
$$798$$ −6382.24 −0.283119
$$799$$ −1474.82 −0.0653008
$$800$$ 62739.0 2.77270
$$801$$ 1941.01 0.0856209
$$802$$ 49297.8 2.17053
$$803$$ −49560.9 −2.17804
$$804$$ −1396.92 −0.0612755
$$805$$ 1939.85 0.0849324
$$806$$ 0 0
$$807$$ 11750.9 0.512577
$$808$$ 60571.7 2.63726
$$809$$ −2453.12 −0.106610 −0.0533048 0.998578i $$-0.516975\pi$$
−0.0533048 + 0.998578i $$0.516975\pi$$
$$810$$ −1171.66 −0.0508247
$$811$$ −5133.85 −0.222286 −0.111143 0.993804i $$-0.535451\pi$$
−0.111143 + 0.993804i $$0.535451\pi$$
$$812$$ 81191.4 3.50894
$$813$$ −8332.58 −0.359454
$$814$$ 63039.4 2.71441
$$815$$ −1064.81 −0.0457651
$$816$$ 2517.83 0.108017
$$817$$ 1356.01 0.0580673
$$818$$ −49432.6 −2.11292
$$819$$ 0 0
$$820$$ 8766.61 0.373345
$$821$$ −23854.0 −1.01402 −0.507009 0.861941i $$-0.669249\pi$$
−0.507009 + 0.861941i $$0.669249\pi$$
$$822$$ 19393.8 0.822913
$$823$$ 757.156 0.0320690 0.0160345 0.999871i $$-0.494896\pi$$
0.0160345 + 0.999871i $$0.494896\pi$$
$$824$$ −85822.6 −3.62836
$$825$$ 23623.0 0.996907
$$826$$ −36302.2 −1.52919
$$827$$ 28621.2 1.20345 0.601726 0.798702i $$-0.294480\pi$$
0.601726 + 0.798702i $$0.294480\pi$$
$$828$$ 8850.68 0.371476
$$829$$ 27429.2 1.14916 0.574582 0.818447i $$-0.305165\pi$$
0.574582 + 0.818447i $$0.305165\pi$$
$$830$$ −8723.19 −0.364803
$$831$$ 19749.2 0.824421
$$832$$ 0 0
$$833$$ −465.838 −0.0193761
$$834$$ −5196.64 −0.215761
$$835$$ −409.462 −0.0169701
$$836$$ −36255.1 −1.49989
$$837$$ 5568.54 0.229960
$$838$$ 34840.7 1.43622
$$839$$ −4633.62 −0.190668 −0.0953339 0.995445i $$-0.530392\pi$$
−0.0953339 + 0.995445i $$0.530392\pi$$
$$840$$ −8431.27 −0.346317
$$841$$ 41663.6 1.70829
$$842$$ −16413.7 −0.671798
$$843$$ 8614.98 0.351976
$$844$$ −21813.0 −0.889614
$$845$$ 0 0
$$846$$ 17102.7 0.695041
$$847$$ 47775.0 1.93810
$$848$$ −2110.99 −0.0854853
$$849$$ 22554.1 0.911726
$$850$$ 2629.64 0.106113
$$851$$ 8314.19 0.334908
$$852$$ 17780.1 0.714947
$$853$$ −14854.6 −0.596261 −0.298131 0.954525i $$-0.596363\pi$$
−0.298131 + 0.954525i $$0.596363\pi$$
$$854$$ −9762.07 −0.391161
$$855$$ 632.999 0.0253194
$$856$$ 23421.9 0.935216
$$857$$ −42799.5 −1.70595 −0.852977 0.521948i $$-0.825206\pi$$
−0.852977 + 0.521948i $$0.825206\pi$$
$$858$$ 0 0
$$859$$ −8246.47 −0.327551 −0.163775 0.986498i $$-0.552367\pi$$
−0.163775 + 0.986498i $$0.552367\pi$$
$$860$$ 2912.70 0.115491
$$861$$ 7135.85 0.282450
$$862$$ −42580.1 −1.68246
$$863$$ 17695.4 0.697983 0.348991 0.937126i $$-0.386524\pi$$
0.348991 + 0.937126i $$0.386524\pi$$
$$864$$ −14388.5 −0.566558
$$865$$ 1452.81 0.0571064
$$866$$ −39132.3 −1.53553
$$867$$ −14687.0 −0.575313
$$868$$ 65154.2 2.54779
$$869$$ 36631.4 1.42996
$$870$$ −11152.8 −0.434615
$$871$$ 0 0
$$872$$ −53148.2 −2.06402
$$873$$ 13027.5 0.505058
$$874$$ −6622.49 −0.256303
$$875$$ −9949.64 −0.384411
$$876$$ 46193.8 1.78167
$$877$$ −14346.8 −0.552401 −0.276200 0.961100i $$-0.589075\pi$$
−0.276200 + 0.961100i $$0.589075\pi$$
$$878$$ −81622.7 −3.13739
$$879$$ 13519.7 0.518781
$$880$$ −36353.0 −1.39257
$$881$$ −2063.51 −0.0789121 −0.0394560 0.999221i $$-0.512563\pi$$
−0.0394560 + 0.999221i $$0.512563\pi$$
$$882$$ 5402.09 0.206233
$$883$$ −34137.1 −1.30103 −0.650513 0.759495i $$-0.725446\pi$$
−0.650513 + 0.759495i $$0.725446\pi$$
$$884$$ 0 0
$$885$$ 3600.50 0.136756
$$886$$ −8138.58 −0.308602
$$887$$ −22238.5 −0.841821 −0.420910 0.907102i $$-0.638289\pi$$
−0.420910 + 0.907102i $$0.638289\pi$$
$$888$$ −36136.5 −1.36561
$$889$$ −16932.8 −0.638817
$$890$$ −3119.63 −0.117495
$$891$$ −5417.67 −0.203702
$$892$$ 48075.0 1.80456
$$893$$ −9239.89 −0.346250
$$894$$ −18168.0 −0.679676
$$895$$ −5987.96 −0.223637
$$896$$ −36830.7 −1.37325
$$897$$ 0 0
$$898$$ 3783.45 0.140596
$$899$$ 53005.7 1.96645
$$900$$ −22018.1 −0.815486
$$901$$ −43.6019 −0.00161220
$$902$$ 56141.7 2.07241
$$903$$ 2370.88 0.0873730
$$904$$ −87694.9 −3.22643
$$905$$ −10307.4 −0.378597
$$906$$ 47324.0 1.73536
$$907$$ −44160.3 −1.61667 −0.808335 0.588723i $$-0.799631\pi$$
−0.808335 + 0.588723i $$0.799631\pi$$
$$908$$ 78953.8 2.88565
$$909$$ −7951.05 −0.290121
$$910$$ 0 0
$$911$$ 11916.3 0.433376 0.216688 0.976241i $$-0.430475\pi$$
0.216688 + 0.976241i $$0.430475\pi$$
$$912$$ 15774.4 0.572745
$$913$$ −40335.3 −1.46211
$$914$$ 39043.0 1.41294
$$915$$ 968.215 0.0349816
$$916$$ −89793.0 −3.23891
$$917$$ 31927.1 1.14975
$$918$$ −603.078 −0.0216825
$$919$$ −20128.9 −0.722516 −0.361258 0.932466i $$-0.617653\pi$$
−0.361258 + 0.932466i $$0.617653\pi$$
$$920$$ −8748.64 −0.313515
$$921$$ 28616.7 1.02383
$$922$$ −10525.0 −0.375945
$$923$$ 0 0
$$924$$ −63389.0 −2.25687
$$925$$ −20683.5 −0.735210
$$926$$ 55654.1 1.97506
$$927$$ 11265.7 0.399151
$$928$$ −136961. −4.84478
$$929$$ −31428.4 −1.10994 −0.554969 0.831871i $$-0.687270\pi$$
−0.554969 + 0.831871i $$0.687270\pi$$
$$930$$ −8949.86 −0.315567
$$931$$ −2918.52 −0.102740
$$932$$ 109715. 3.85605
$$933$$ −22398.9 −0.785965
$$934$$ −47150.1 −1.65182
$$935$$ −750.863 −0.0262629
$$936$$ 0 0
$$937$$ 42473.2 1.48083 0.740416 0.672149i $$-0.234629\pi$$
0.740416 + 0.672149i $$0.234629\pi$$
$$938$$ 1827.51 0.0636144
$$939$$ −5464.94 −0.189927
$$940$$ −19847.1 −0.688661
$$941$$ −42644.0 −1.47732 −0.738659 0.674079i $$-0.764541\pi$$
−0.738659 + 0.674079i $$0.764541\pi$$
$$942$$ −10129.5 −0.350357
$$943$$ 7404.46 0.255697
$$944$$ 89725.0 3.09354
$$945$$ 1106.75 0.0380978
$$946$$ 18653.0 0.641080
$$947$$ 4282.95 0.146966 0.0734831 0.997296i $$-0.476588\pi$$
0.0734831 + 0.997296i $$0.476588\pi$$
$$948$$ −34142.7 −1.16973
$$949$$ 0 0
$$950$$ 16475.0 0.562651
$$951$$ 9375.42 0.319683
$$952$$ −4339.74 −0.147744
$$953$$ 40593.9 1.37982 0.689908 0.723897i $$-0.257651\pi$$
0.689908 + 0.723897i $$0.257651\pi$$
$$954$$ 505.630 0.0171597
$$955$$ 9340.54 0.316495
$$956$$ 32166.1 1.08821
$$957$$ −51569.6 −1.74191
$$958$$ 59771.9 2.01581
$$959$$ −18319.2 −0.616850
$$960$$ 10081.1 0.338922
$$961$$ 12744.8 0.427808
$$962$$ 0 0
$$963$$ −3074.52 −0.102882
$$964$$ −102198. −3.41448
$$965$$ 12665.7 0.422510
$$966$$ −11578.9 −0.385656
$$967$$ 17709.4 0.588931 0.294466 0.955662i $$-0.404858\pi$$
0.294466 + 0.955662i $$0.404858\pi$$
$$968$$ −215464. −7.15420
$$969$$ 325.818 0.0108016
$$970$$ −20938.1 −0.693074
$$971$$ 4038.97 0.133488 0.0667440 0.997770i $$-0.478739\pi$$
0.0667440 + 0.997770i $$0.478739\pi$$
$$972$$ 5049.61 0.166632
$$973$$ 4908.72 0.161733
$$974$$ 105366. 3.46626
$$975$$ 0 0
$$976$$ 24128.1 0.791312
$$977$$ −2764.30 −0.0905197 −0.0452598 0.998975i $$-0.514412\pi$$
−0.0452598 + 0.998975i $$0.514412\pi$$
$$978$$ 6355.79 0.207807
$$979$$ −14424.9 −0.470912
$$980$$ −6268.93 −0.204340
$$981$$ 6976.59 0.227060
$$982$$ 18295.4 0.594531
$$983$$ −17804.5 −0.577695 −0.288848 0.957375i $$-0.593272\pi$$
−0.288848 + 0.957375i $$0.593272\pi$$
$$984$$ −32182.4 −1.04262
$$985$$ −7786.65 −0.251881
$$986$$ −5740.57 −0.185413
$$987$$ −16155.2 −0.520997
$$988$$ 0 0
$$989$$ 2460.12 0.0790974
$$990$$ 8707.39 0.279534
$$991$$ −8129.69 −0.260593 −0.130297 0.991475i $$-0.541593\pi$$
−0.130297 + 0.991475i $$0.541593\pi$$
$$992$$ −109908. −3.51772
$$993$$ −4661.02 −0.148956
$$994$$ −23260.7 −0.742237
$$995$$ 169.893 0.00541302
$$996$$ 37595.0 1.19603
$$997$$ −29452.9 −0.935589 −0.467795 0.883837i $$-0.654951\pi$$
−0.467795 + 0.883837i $$0.654951\pi$$
$$998$$ 26997.7 0.856309
$$999$$ 4743.52 0.150229
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 507.4.a.r.1.1 10
3.2 odd 2 1521.4.a.bk.1.10 10
13.2 odd 12 39.4.j.c.4.1 10
13.5 odd 4 507.4.b.i.337.10 10
13.7 odd 12 39.4.j.c.10.1 yes 10
13.8 odd 4 507.4.b.i.337.1 10
13.12 even 2 inner 507.4.a.r.1.10 10
39.2 even 12 117.4.q.e.82.5 10
39.20 even 12 117.4.q.e.10.5 10
39.38 odd 2 1521.4.a.bk.1.1 10
52.7 even 12 624.4.bv.h.49.3 10
52.15 even 12 624.4.bv.h.433.3 10

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.1 10 13.2 odd 12
39.4.j.c.10.1 yes 10 13.7 odd 12
117.4.q.e.10.5 10 39.20 even 12
117.4.q.e.82.5 10 39.2 even 12
507.4.a.r.1.1 10 1.1 even 1 trivial
507.4.a.r.1.10 10 13.12 even 2 inner
507.4.b.i.337.1 10 13.8 odd 4
507.4.b.i.337.10 10 13.5 odd 4
624.4.bv.h.49.3 10 52.7 even 12
624.4.bv.h.433.3 10 52.15 even 12
1521.4.a.bk.1.1 10 39.38 odd 2
1521.4.a.bk.1.10 10 3.2 odd 2