# Properties

 Label 525.4.a.n.1.2 Level $525$ Weight $4$ Character 525.1 Self dual yes Analytic conductor $30.976$ 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] = [525,4,Mod(1,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, 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("525.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 525.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: $$30.9760027530$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 14$$ x^2 - x - 14 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$4.27492$$ of defining polynomial Character $$\chi$$ $$=$$ 525.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.27492 q^{2} -3.00000 q^{3} +19.8248 q^{4} -15.8248 q^{6} -7.00000 q^{7} +62.3746 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+5.27492 q^{2} -3.00000 q^{3} +19.8248 q^{4} -15.8248 q^{6} -7.00000 q^{7} +62.3746 q^{8} +9.00000 q^{9} +34.7492 q^{11} -59.4743 q^{12} +37.2990 q^{13} -36.9244 q^{14} +170.423 q^{16} +10.5498 q^{17} +47.4743 q^{18} -58.5980 q^{19} +21.0000 q^{21} +183.299 q^{22} +125.347 q^{23} -187.124 q^{24} +196.749 q^{26} -27.0000 q^{27} -138.773 q^{28} -35.4020 q^{29} +291.794 q^{31} +399.969 q^{32} -104.248 q^{33} +55.6495 q^{34} +178.423 q^{36} +259.897 q^{37} -309.100 q^{38} -111.897 q^{39} -338.248 q^{41} +110.773 q^{42} -6.80397 q^{43} +688.894 q^{44} +661.196 q^{46} -250.694 q^{47} -511.268 q^{48} +49.0000 q^{49} -31.6495 q^{51} +739.444 q^{52} +536.900 q^{53} -142.423 q^{54} -436.622 q^{56} +175.794 q^{57} -186.743 q^{58} -35.8904 q^{59} +57.7940 q^{61} +1539.19 q^{62} -63.0000 q^{63} +746.423 q^{64} -549.897 q^{66} -481.691 q^{67} +209.148 q^{68} -376.042 q^{69} +363.752 q^{71} +561.371 q^{72} -581.299 q^{73} +1370.94 q^{74} -1161.69 q^{76} -243.244 q^{77} -590.248 q^{78} -693.691 q^{79} +81.0000 q^{81} -1784.23 q^{82} -1334.39 q^{83} +416.320 q^{84} -35.8904 q^{86} +106.206 q^{87} +2167.47 q^{88} -353.038 q^{89} -261.093 q^{91} +2484.98 q^{92} -875.382 q^{93} -1322.39 q^{94} -1199.91 q^{96} -1445.88 q^{97} +258.471 q^{98} +312.743 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} - 6 q^{3} + 17 q^{4} - 9 q^{6} - 14 q^{7} + 87 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q + 3 * q^2 - 6 * q^3 + 17 * q^4 - 9 * q^6 - 14 * q^7 + 87 * q^8 + 18 * q^9 $$2 q + 3 q^{2} - 6 q^{3} + 17 q^{4} - 9 q^{6} - 14 q^{7} + 87 q^{8} + 18 q^{9} - 6 q^{11} - 51 q^{12} - 16 q^{13} - 21 q^{14} + 137 q^{16} + 6 q^{17} + 27 q^{18} + 64 q^{19} + 42 q^{21} + 276 q^{22} - 6 q^{23} - 261 q^{24} + 318 q^{26} - 54 q^{27} - 119 q^{28} - 252 q^{29} + 40 q^{31} + 279 q^{32} + 18 q^{33} + 66 q^{34} + 153 q^{36} + 248 q^{37} - 588 q^{38} + 48 q^{39} - 450 q^{41} + 63 q^{42} - 376 q^{43} + 804 q^{44} + 960 q^{46} + 12 q^{47} - 411 q^{48} + 98 q^{49} - 18 q^{51} + 890 q^{52} + 1104 q^{53} - 81 q^{54} - 609 q^{56} - 192 q^{57} + 306 q^{58} + 804 q^{59} - 428 q^{61} + 2112 q^{62} - 126 q^{63} + 1289 q^{64} - 828 q^{66} - 148 q^{67} + 222 q^{68} + 18 q^{69} + 954 q^{71} + 783 q^{72} - 1072 q^{73} + 1398 q^{74} - 1508 q^{76} + 42 q^{77} - 954 q^{78} - 572 q^{79} + 162 q^{81} - 1530 q^{82} - 1944 q^{83} + 357 q^{84} + 804 q^{86} + 756 q^{87} + 1164 q^{88} + 366 q^{89} + 112 q^{91} + 2856 q^{92} - 120 q^{93} - 1920 q^{94} - 837 q^{96} - 808 q^{97} + 147 q^{98} - 54 q^{99}+O(q^{100})$$ 2 * q + 3 * q^2 - 6 * q^3 + 17 * q^4 - 9 * q^6 - 14 * q^7 + 87 * q^8 + 18 * q^9 - 6 * q^11 - 51 * q^12 - 16 * q^13 - 21 * q^14 + 137 * q^16 + 6 * q^17 + 27 * q^18 + 64 * q^19 + 42 * q^21 + 276 * q^22 - 6 * q^23 - 261 * q^24 + 318 * q^26 - 54 * q^27 - 119 * q^28 - 252 * q^29 + 40 * q^31 + 279 * q^32 + 18 * q^33 + 66 * q^34 + 153 * q^36 + 248 * q^37 - 588 * q^38 + 48 * q^39 - 450 * q^41 + 63 * q^42 - 376 * q^43 + 804 * q^44 + 960 * q^46 + 12 * q^47 - 411 * q^48 + 98 * q^49 - 18 * q^51 + 890 * q^52 + 1104 * q^53 - 81 * q^54 - 609 * q^56 - 192 * q^57 + 306 * q^58 + 804 * q^59 - 428 * q^61 + 2112 * q^62 - 126 * q^63 + 1289 * q^64 - 828 * q^66 - 148 * q^67 + 222 * q^68 + 18 * q^69 + 954 * q^71 + 783 * q^72 - 1072 * q^73 + 1398 * q^74 - 1508 * q^76 + 42 * q^77 - 954 * q^78 - 572 * q^79 + 162 * q^81 - 1530 * q^82 - 1944 * q^83 + 357 * q^84 + 804 * q^86 + 756 * q^87 + 1164 * q^88 + 366 * q^89 + 112 * q^91 + 2856 * q^92 - 120 * q^93 - 1920 * q^94 - 837 * q^96 - 808 * q^97 + 147 * q^98 - 54 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 5.27492 1.86496 0.932482 0.361215i $$-0.117638\pi$$
0.932482 + 0.361215i $$0.117638\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 19.8248 2.47809
$$5$$ 0 0
$$6$$ −15.8248 −1.07674
$$7$$ −7.00000 −0.377964
$$8$$ 62.3746 2.75659
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 34.7492 0.952479 0.476240 0.879316i $$-0.342000\pi$$
0.476240 + 0.879316i $$0.342000\pi$$
$$12$$ −59.4743 −1.43073
$$13$$ 37.2990 0.795760 0.397880 0.917437i $$-0.369746\pi$$
0.397880 + 0.917437i $$0.369746\pi$$
$$14$$ −36.9244 −0.704890
$$15$$ 0 0
$$16$$ 170.423 2.66286
$$17$$ 10.5498 0.150512 0.0752562 0.997164i $$-0.476023\pi$$
0.0752562 + 0.997164i $$0.476023\pi$$
$$18$$ 47.4743 0.621655
$$19$$ −58.5980 −0.707542 −0.353771 0.935332i $$-0.615101\pi$$
−0.353771 + 0.935332i $$0.615101\pi$$
$$20$$ 0 0
$$21$$ 21.0000 0.218218
$$22$$ 183.299 1.77634
$$23$$ 125.347 1.13638 0.568189 0.822898i $$-0.307644\pi$$
0.568189 + 0.822898i $$0.307644\pi$$
$$24$$ −187.124 −1.59152
$$25$$ 0 0
$$26$$ 196.749 1.48406
$$27$$ −27.0000 −0.192450
$$28$$ −138.773 −0.936631
$$29$$ −35.4020 −0.226689 −0.113345 0.993556i $$-0.536156\pi$$
−0.113345 + 0.993556i $$0.536156\pi$$
$$30$$ 0 0
$$31$$ 291.794 1.69057 0.845286 0.534313i $$-0.179430\pi$$
0.845286 + 0.534313i $$0.179430\pi$$
$$32$$ 399.969 2.20954
$$33$$ −104.248 −0.549914
$$34$$ 55.6495 0.280700
$$35$$ 0 0
$$36$$ 178.423 0.826031
$$37$$ 259.897 1.15478 0.577389 0.816469i $$-0.304072\pi$$
0.577389 + 0.816469i $$0.304072\pi$$
$$38$$ −309.100 −1.31954
$$39$$ −111.897 −0.459432
$$40$$ 0 0
$$41$$ −338.248 −1.28842 −0.644212 0.764847i $$-0.722815\pi$$
−0.644212 + 0.764847i $$0.722815\pi$$
$$42$$ 110.773 0.406969
$$43$$ −6.80397 −0.0241301 −0.0120651 0.999927i $$-0.503841\pi$$
−0.0120651 + 0.999927i $$0.503841\pi$$
$$44$$ 688.894 2.36033
$$45$$ 0 0
$$46$$ 661.196 2.11931
$$47$$ −250.694 −0.778033 −0.389016 0.921231i $$-0.627185\pi$$
−0.389016 + 0.921231i $$0.627185\pi$$
$$48$$ −511.268 −1.53740
$$49$$ 49.0000 0.142857
$$50$$ 0 0
$$51$$ −31.6495 −0.0868984
$$52$$ 739.444 1.97197
$$53$$ 536.900 1.39149 0.695745 0.718289i $$-0.255075\pi$$
0.695745 + 0.718289i $$0.255075\pi$$
$$54$$ −142.423 −0.358913
$$55$$ 0 0
$$56$$ −436.622 −1.04189
$$57$$ 175.794 0.408500
$$58$$ −186.743 −0.422767
$$59$$ −35.8904 −0.0791955 −0.0395977 0.999216i $$-0.512608\pi$$
−0.0395977 + 0.999216i $$0.512608\pi$$
$$60$$ 0 0
$$61$$ 57.7940 0.121308 0.0606538 0.998159i $$-0.480681\pi$$
0.0606538 + 0.998159i $$0.480681\pi$$
$$62$$ 1539.19 3.15286
$$63$$ −63.0000 −0.125988
$$64$$ 746.423 1.45786
$$65$$ 0 0
$$66$$ −549.897 −1.02557
$$67$$ −481.691 −0.878327 −0.439164 0.898407i $$-0.644725\pi$$
−0.439164 + 0.898407i $$0.644725\pi$$
$$68$$ 209.148 0.372984
$$69$$ −376.042 −0.656088
$$70$$ 0 0
$$71$$ 363.752 0.608021 0.304010 0.952669i $$-0.401674\pi$$
0.304010 + 0.952669i $$0.401674\pi$$
$$72$$ 561.371 0.918864
$$73$$ −581.299 −0.931999 −0.465999 0.884785i $$-0.654305\pi$$
−0.465999 + 0.884785i $$0.654305\pi$$
$$74$$ 1370.94 2.15362
$$75$$ 0 0
$$76$$ −1161.69 −1.75336
$$77$$ −243.244 −0.360003
$$78$$ −590.248 −0.856825
$$79$$ −693.691 −0.987928 −0.493964 0.869482i $$-0.664453\pi$$
−0.493964 + 0.869482i $$0.664453\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ −1784.23 −2.40287
$$83$$ −1334.39 −1.76468 −0.882341 0.470611i $$-0.844033\pi$$
−0.882341 + 0.470611i $$0.844033\pi$$
$$84$$ 416.320 0.540764
$$85$$ 0 0
$$86$$ −35.8904 −0.0450019
$$87$$ 106.206 0.130879
$$88$$ 2167.47 2.62560
$$89$$ −353.038 −0.420472 −0.210236 0.977651i $$-0.567423\pi$$
−0.210236 + 0.977651i $$0.567423\pi$$
$$90$$ 0 0
$$91$$ −261.093 −0.300769
$$92$$ 2484.98 2.81605
$$93$$ −875.382 −0.976053
$$94$$ −1322.39 −1.45100
$$95$$ 0 0
$$96$$ −1199.91 −1.27568
$$97$$ −1445.88 −1.51347 −0.756735 0.653722i $$-0.773207\pi$$
−0.756735 + 0.653722i $$0.773207\pi$$
$$98$$ 258.471 0.266424
$$99$$ 312.743 0.317493
$$100$$ 0 0
$$101$$ 474.852 0.467817 0.233909 0.972259i $$-0.424848\pi$$
0.233909 + 0.972259i $$0.424848\pi$$
$$102$$ −166.949 −0.162062
$$103$$ 1999.59 1.91287 0.956433 0.291951i $$-0.0943044\pi$$
0.956433 + 0.291951i $$0.0943044\pi$$
$$104$$ 2326.51 2.19359
$$105$$ 0 0
$$106$$ 2832.10 2.59508
$$107$$ −1166.74 −1.05414 −0.527068 0.849823i $$-0.676709\pi$$
−0.527068 + 0.849823i $$0.676709\pi$$
$$108$$ −535.268 −0.476909
$$109$$ −1337.18 −1.17503 −0.587515 0.809213i $$-0.699894\pi$$
−0.587515 + 0.809213i $$0.699894\pi$$
$$110$$ 0 0
$$111$$ −779.691 −0.666712
$$112$$ −1192.96 −1.00646
$$113$$ −906.578 −0.754723 −0.377361 0.926066i $$-0.623169\pi$$
−0.377361 + 0.926066i $$0.623169\pi$$
$$114$$ 927.299 0.761838
$$115$$ 0 0
$$116$$ −701.836 −0.561757
$$117$$ 335.691 0.265253
$$118$$ −189.319 −0.147697
$$119$$ −73.8488 −0.0568883
$$120$$ 0 0
$$121$$ −123.495 −0.0927836
$$122$$ 304.859 0.226235
$$123$$ 1014.74 0.743872
$$124$$ 5784.74 4.18940
$$125$$ 0 0
$$126$$ −332.320 −0.234963
$$127$$ 1714.89 1.19820 0.599101 0.800674i $$-0.295525\pi$$
0.599101 + 0.800674i $$0.295525\pi$$
$$128$$ 737.564 0.509313
$$129$$ 20.4119 0.0139315
$$130$$ 0 0
$$131$$ 470.611 0.313874 0.156937 0.987609i $$-0.449838\pi$$
0.156937 + 0.987609i $$0.449838\pi$$
$$132$$ −2066.68 −1.36274
$$133$$ 410.186 0.267426
$$134$$ −2540.88 −1.63805
$$135$$ 0 0
$$136$$ 658.042 0.414901
$$137$$ 443.910 0.276831 0.138415 0.990374i $$-0.455799\pi$$
0.138415 + 0.990374i $$0.455799\pi$$
$$138$$ −1983.59 −1.22358
$$139$$ 1669.98 1.01904 0.509518 0.860460i $$-0.329824\pi$$
0.509518 + 0.860460i $$0.329824\pi$$
$$140$$ 0 0
$$141$$ 752.083 0.449197
$$142$$ 1918.76 1.13394
$$143$$ 1296.11 0.757945
$$144$$ 1533.80 0.887619
$$145$$ 0 0
$$146$$ −3066.30 −1.73814
$$147$$ −147.000 −0.0824786
$$148$$ 5152.39 2.86165
$$149$$ 743.871 0.408995 0.204497 0.978867i $$-0.434444\pi$$
0.204497 + 0.978867i $$0.434444\pi$$
$$150$$ 0 0
$$151$$ 606.764 0.327005 0.163503 0.986543i $$-0.447721\pi$$
0.163503 + 0.986543i $$0.447721\pi$$
$$152$$ −3655.03 −1.95041
$$153$$ 94.9485 0.0501708
$$154$$ −1283.09 −0.671393
$$155$$ 0 0
$$156$$ −2218.33 −1.13852
$$157$$ −3114.78 −1.58336 −0.791678 0.610939i $$-0.790792\pi$$
−0.791678 + 0.610939i $$0.790792\pi$$
$$158$$ −3659.16 −1.84245
$$159$$ −1610.70 −0.803377
$$160$$ 0 0
$$161$$ −877.430 −0.429511
$$162$$ 427.268 0.207218
$$163$$ −2413.07 −1.15955 −0.579774 0.814777i $$-0.696859\pi$$
−0.579774 + 0.814777i $$0.696859\pi$$
$$164$$ −6705.67 −3.19284
$$165$$ 0 0
$$166$$ −7038.81 −3.29107
$$167$$ 610.475 0.282874 0.141437 0.989947i $$-0.454828\pi$$
0.141437 + 0.989947i $$0.454828\pi$$
$$168$$ 1309.87 0.601538
$$169$$ −805.784 −0.366766
$$170$$ 0 0
$$171$$ −527.382 −0.235847
$$172$$ −134.887 −0.0597968
$$173$$ −3793.81 −1.66727 −0.833636 0.552315i $$-0.813745\pi$$
−0.833636 + 0.552315i $$0.813745\pi$$
$$174$$ 560.228 0.244085
$$175$$ 0 0
$$176$$ 5922.05 2.53631
$$177$$ 107.671 0.0457235
$$178$$ −1862.25 −0.784165
$$179$$ −2804.68 −1.17112 −0.585562 0.810627i $$-0.699126\pi$$
−0.585562 + 0.810627i $$0.699126\pi$$
$$180$$ 0 0
$$181$$ 3106.04 1.27553 0.637763 0.770232i $$-0.279860\pi$$
0.637763 + 0.770232i $$0.279860\pi$$
$$182$$ −1377.24 −0.560924
$$183$$ −173.382 −0.0700370
$$184$$ 7818.48 3.13253
$$185$$ 0 0
$$186$$ −4617.57 −1.82030
$$187$$ 366.598 0.143360
$$188$$ −4969.95 −1.92804
$$189$$ 189.000 0.0727393
$$190$$ 0 0
$$191$$ 261.952 0.0992365 0.0496182 0.998768i $$-0.484200\pi$$
0.0496182 + 0.998768i $$0.484200\pi$$
$$192$$ −2239.27 −0.841694
$$193$$ −4051.07 −1.51089 −0.755447 0.655210i $$-0.772580\pi$$
−0.755447 + 0.655210i $$0.772580\pi$$
$$194$$ −7626.88 −2.82257
$$195$$ 0 0
$$196$$ 971.413 0.354013
$$197$$ 2874.83 1.03971 0.519855 0.854254i $$-0.325986\pi$$
0.519855 + 0.854254i $$0.325986\pi$$
$$198$$ 1649.69 0.592113
$$199$$ −3066.97 −1.09252 −0.546261 0.837615i $$-0.683949\pi$$
−0.546261 + 0.837615i $$0.683949\pi$$
$$200$$ 0 0
$$201$$ 1445.07 0.507103
$$202$$ 2504.81 0.872463
$$203$$ 247.814 0.0856804
$$204$$ −627.444 −0.215342
$$205$$ 0 0
$$206$$ 10547.7 3.56743
$$207$$ 1128.12 0.378793
$$208$$ 6356.60 2.11899
$$209$$ −2036.23 −0.673919
$$210$$ 0 0
$$211$$ 595.422 0.194268 0.0971340 0.995271i $$-0.469032\pi$$
0.0971340 + 0.995271i $$0.469032\pi$$
$$212$$ 10643.9 3.44824
$$213$$ −1091.26 −0.351041
$$214$$ −6154.44 −1.96593
$$215$$ 0 0
$$216$$ −1684.11 −0.530507
$$217$$ −2042.56 −0.638976
$$218$$ −7053.49 −2.19139
$$219$$ 1743.90 0.538090
$$220$$ 0 0
$$221$$ 393.498 0.119772
$$222$$ −4112.81 −1.24339
$$223$$ 3779.79 1.13504 0.567520 0.823360i $$-0.307903\pi$$
0.567520 + 0.823360i $$0.307903\pi$$
$$224$$ −2799.79 −0.835127
$$225$$ 0 0
$$226$$ −4782.12 −1.40753
$$227$$ −1827.62 −0.534376 −0.267188 0.963644i $$-0.586094\pi$$
−0.267188 + 0.963644i $$0.586094\pi$$
$$228$$ 3485.07 1.01230
$$229$$ −850.249 −0.245354 −0.122677 0.992447i $$-0.539148\pi$$
−0.122677 + 0.992447i $$0.539148\pi$$
$$230$$ 0 0
$$231$$ 729.733 0.207848
$$232$$ −2208.18 −0.624890
$$233$$ 6591.10 1.85321 0.926604 0.376039i $$-0.122714\pi$$
0.926604 + 0.376039i $$0.122714\pi$$
$$234$$ 1770.74 0.494688
$$235$$ 0 0
$$236$$ −711.518 −0.196254
$$237$$ 2081.07 0.570381
$$238$$ −389.547 −0.106095
$$239$$ −182.556 −0.0494083 −0.0247042 0.999695i $$-0.507864\pi$$
−0.0247042 + 0.999695i $$0.507864\pi$$
$$240$$ 0 0
$$241$$ 1523.90 0.407315 0.203657 0.979042i $$-0.434717\pi$$
0.203657 + 0.979042i $$0.434717\pi$$
$$242$$ −651.426 −0.173038
$$243$$ −243.000 −0.0641500
$$244$$ 1145.75 0.300612
$$245$$ 0 0
$$246$$ 5352.68 1.38730
$$247$$ −2185.65 −0.563034
$$248$$ 18200.5 4.66022
$$249$$ 4003.18 1.01884
$$250$$ 0 0
$$251$$ 2357.73 0.592903 0.296451 0.955048i $$-0.404197\pi$$
0.296451 + 0.955048i $$0.404197\pi$$
$$252$$ −1248.96 −0.312210
$$253$$ 4355.71 1.08238
$$254$$ 9045.89 2.23460
$$255$$ 0 0
$$256$$ −2080.79 −0.508006
$$257$$ −2782.55 −0.675372 −0.337686 0.941259i $$-0.609644\pi$$
−0.337686 + 0.941259i $$0.609644\pi$$
$$258$$ 107.671 0.0259818
$$259$$ −1819.28 −0.436465
$$260$$ 0 0
$$261$$ −318.618 −0.0755630
$$262$$ 2482.44 0.585364
$$263$$ −2043.78 −0.479183 −0.239591 0.970874i $$-0.577013\pi$$
−0.239591 + 0.970874i $$0.577013\pi$$
$$264$$ −6502.40 −1.51589
$$265$$ 0 0
$$266$$ 2163.70 0.498740
$$267$$ 1059.11 0.242759
$$268$$ −9549.41 −2.17658
$$269$$ 3452.84 0.782614 0.391307 0.920260i $$-0.372023\pi$$
0.391307 + 0.920260i $$0.372023\pi$$
$$270$$ 0 0
$$271$$ 2644.29 0.592728 0.296364 0.955075i $$-0.404226\pi$$
0.296364 + 0.955075i $$0.404226\pi$$
$$272$$ 1797.93 0.400793
$$273$$ 783.279 0.173649
$$274$$ 2341.59 0.516280
$$275$$ 0 0
$$276$$ −7454.93 −1.62585
$$277$$ −2679.49 −0.581208 −0.290604 0.956843i $$-0.593856\pi$$
−0.290604 + 0.956843i $$0.593856\pi$$
$$278$$ 8809.01 1.90046
$$279$$ 2626.15 0.563524
$$280$$ 0 0
$$281$$ −1019.69 −0.216476 −0.108238 0.994125i $$-0.534521\pi$$
−0.108238 + 0.994125i $$0.534521\pi$$
$$282$$ 3967.18 0.837737
$$283$$ −432.206 −0.0907844 −0.0453922 0.998969i $$-0.514454\pi$$
−0.0453922 + 0.998969i $$0.514454\pi$$
$$284$$ 7211.30 1.50673
$$285$$ 0 0
$$286$$ 6836.87 1.41354
$$287$$ 2367.73 0.486979
$$288$$ 3599.72 0.736513
$$289$$ −4801.70 −0.977346
$$290$$ 0 0
$$291$$ 4337.63 0.873802
$$292$$ −11524.1 −2.30958
$$293$$ 2245.92 0.447809 0.223904 0.974611i $$-0.428120\pi$$
0.223904 + 0.974611i $$0.428120\pi$$
$$294$$ −775.413 −0.153820
$$295$$ 0 0
$$296$$ 16211.0 3.18325
$$297$$ −938.228 −0.183305
$$298$$ 3923.86 0.762761
$$299$$ 4675.33 0.904284
$$300$$ 0 0
$$301$$ 47.6278 0.00912034
$$302$$ 3200.63 0.609853
$$303$$ −1424.56 −0.270094
$$304$$ −9986.44 −1.88408
$$305$$ 0 0
$$306$$ 500.846 0.0935668
$$307$$ 3197.08 0.594354 0.297177 0.954822i $$-0.403955\pi$$
0.297177 + 0.954822i $$0.403955\pi$$
$$308$$ −4822.26 −0.892122
$$309$$ −5998.76 −1.10439
$$310$$ 0 0
$$311$$ −3355.60 −0.611829 −0.305915 0.952059i $$-0.598962\pi$$
−0.305915 + 0.952059i $$0.598962\pi$$
$$312$$ −6979.53 −1.26647
$$313$$ 2256.39 0.407472 0.203736 0.979026i $$-0.434692\pi$$
0.203736 + 0.979026i $$0.434692\pi$$
$$314$$ −16430.2 −2.95290
$$315$$ 0 0
$$316$$ −13752.3 −2.44818
$$317$$ 6139.19 1.08773 0.543866 0.839172i $$-0.316960\pi$$
0.543866 + 0.839172i $$0.316960\pi$$
$$318$$ −8496.31 −1.49827
$$319$$ −1230.19 −0.215917
$$320$$ 0 0
$$321$$ 3500.21 0.608606
$$322$$ −4628.37 −0.801022
$$323$$ −618.199 −0.106494
$$324$$ 1605.80 0.275344
$$325$$ 0 0
$$326$$ −12728.8 −2.16252
$$327$$ 4011.53 0.678404
$$328$$ −21098.0 −3.55166
$$329$$ 1754.86 0.294069
$$330$$ 0 0
$$331$$ 7029.81 1.16735 0.583676 0.811987i $$-0.301614\pi$$
0.583676 + 0.811987i $$0.301614\pi$$
$$332$$ −26454.0 −4.37305
$$333$$ 2339.07 0.384926
$$334$$ 3220.21 0.527550
$$335$$ 0 0
$$336$$ 3578.88 0.581083
$$337$$ −10328.4 −1.66951 −0.834757 0.550619i $$-0.814392\pi$$
−0.834757 + 0.550619i $$0.814392\pi$$
$$338$$ −4250.44 −0.684005
$$339$$ 2719.73 0.435740
$$340$$ 0 0
$$341$$ 10139.6 1.61024
$$342$$ −2781.90 −0.439847
$$343$$ −343.000 −0.0539949
$$344$$ −424.395 −0.0665170
$$345$$ 0 0
$$346$$ −20012.0 −3.10940
$$347$$ −1967.54 −0.304389 −0.152194 0.988351i $$-0.548634\pi$$
−0.152194 + 0.988351i $$0.548634\pi$$
$$348$$ 2105.51 0.324330
$$349$$ −4365.46 −0.669564 −0.334782 0.942296i $$-0.608663\pi$$
−0.334782 + 0.942296i $$0.608663\pi$$
$$350$$ 0 0
$$351$$ −1007.07 −0.153144
$$352$$ 13898.6 2.10454
$$353$$ 6071.59 0.915462 0.457731 0.889091i $$-0.348662\pi$$
0.457731 + 0.889091i $$0.348662\pi$$
$$354$$ 567.957 0.0852728
$$355$$ 0 0
$$356$$ −6998.90 −1.04197
$$357$$ 221.547 0.0328445
$$358$$ −14794.4 −2.18411
$$359$$ 9638.04 1.41693 0.708463 0.705748i $$-0.249389\pi$$
0.708463 + 0.705748i $$0.249389\pi$$
$$360$$ 0 0
$$361$$ −3425.27 −0.499384
$$362$$ 16384.1 2.37881
$$363$$ 370.485 0.0535687
$$364$$ −5176.10 −0.745334
$$365$$ 0 0
$$366$$ −914.576 −0.130617
$$367$$ −522.725 −0.0743488 −0.0371744 0.999309i $$-0.511836\pi$$
−0.0371744 + 0.999309i $$0.511836\pi$$
$$368$$ 21362.0 3.02601
$$369$$ −3044.23 −0.429475
$$370$$ 0 0
$$371$$ −3758.30 −0.525934
$$372$$ −17354.2 −2.41875
$$373$$ −3229.84 −0.448351 −0.224175 0.974549i $$-0.571969\pi$$
−0.224175 + 0.974549i $$0.571969\pi$$
$$374$$ 1933.77 0.267361
$$375$$ 0 0
$$376$$ −15637.0 −2.14472
$$377$$ −1320.46 −0.180390
$$378$$ 996.959 0.135656
$$379$$ 6639.71 0.899892 0.449946 0.893056i $$-0.351443\pi$$
0.449946 + 0.893056i $$0.351443\pi$$
$$380$$ 0 0
$$381$$ −5144.66 −0.691782
$$382$$ 1381.77 0.185073
$$383$$ 14224.4 1.89774 0.948871 0.315664i $$-0.102227\pi$$
0.948871 + 0.315664i $$0.102227\pi$$
$$384$$ −2212.69 −0.294052
$$385$$ 0 0
$$386$$ −21369.1 −2.81777
$$387$$ −61.2358 −0.00804338
$$388$$ −28664.2 −3.75052
$$389$$ 2921.82 0.380828 0.190414 0.981704i $$-0.439017\pi$$
0.190414 + 0.981704i $$0.439017\pi$$
$$390$$ 0 0
$$391$$ 1322.39 0.171039
$$392$$ 3056.35 0.393799
$$393$$ −1411.83 −0.181215
$$394$$ 15164.5 1.93902
$$395$$ 0 0
$$396$$ 6200.04 0.786778
$$397$$ −811.940 −0.102645 −0.0513226 0.998682i $$-0.516344\pi$$
−0.0513226 + 0.998682i $$0.516344\pi$$
$$398$$ −16178.0 −2.03751
$$399$$ −1230.56 −0.154398
$$400$$ 0 0
$$401$$ 2338.63 0.291237 0.145618 0.989341i $$-0.453483\pi$$
0.145618 + 0.989341i $$0.453483\pi$$
$$402$$ 7622.64 0.945728
$$403$$ 10883.6 1.34529
$$404$$ 9413.83 1.15930
$$405$$ 0 0
$$406$$ 1307.20 0.159791
$$407$$ 9031.21 1.09990
$$408$$ −1974.12 −0.239543
$$409$$ −2727.57 −0.329755 −0.164877 0.986314i $$-0.552723\pi$$
−0.164877 + 0.986314i $$0.552723\pi$$
$$410$$ 0 0
$$411$$ −1331.73 −0.159828
$$412$$ 39641.3 4.74026
$$413$$ 251.233 0.0299331
$$414$$ 5950.76 0.706435
$$415$$ 0 0
$$416$$ 14918.5 1.75826
$$417$$ −5009.94 −0.588340
$$418$$ −10741.0 −1.25684
$$419$$ 13306.3 1.55144 0.775721 0.631076i $$-0.217386\pi$$
0.775721 + 0.631076i $$0.217386\pi$$
$$420$$ 0 0
$$421$$ −11007.5 −1.27428 −0.637138 0.770750i $$-0.719882\pi$$
−0.637138 + 0.770750i $$0.719882\pi$$
$$422$$ 3140.80 0.362303
$$423$$ −2256.25 −0.259344
$$424$$ 33488.9 3.83577
$$425$$ 0 0
$$426$$ −5756.29 −0.654679
$$427$$ −404.558 −0.0458500
$$428$$ −23130.2 −2.61225
$$429$$ −3888.33 −0.437600
$$430$$ 0 0
$$431$$ −6525.62 −0.729300 −0.364650 0.931145i $$-0.618811\pi$$
−0.364650 + 0.931145i $$0.618811\pi$$
$$432$$ −4601.41 −0.512467
$$433$$ 11716.3 1.30034 0.650171 0.759788i $$-0.274697\pi$$
0.650171 + 0.759788i $$0.274697\pi$$
$$434$$ −10774.3 −1.19167
$$435$$ 0 0
$$436$$ −26509.2 −2.91183
$$437$$ −7345.10 −0.804036
$$438$$ 9198.91 1.00352
$$439$$ −14611.4 −1.58853 −0.794264 0.607573i $$-0.792143\pi$$
−0.794264 + 0.607573i $$0.792143\pi$$
$$440$$ 0 0
$$441$$ 441.000 0.0476190
$$442$$ 2075.67 0.223370
$$443$$ 15239.8 1.63446 0.817228 0.576314i $$-0.195510\pi$$
0.817228 + 0.576314i $$0.195510\pi$$
$$444$$ −15457.2 −1.65217
$$445$$ 0 0
$$446$$ 19938.1 2.11681
$$447$$ −2231.61 −0.236133
$$448$$ −5224.96 −0.551018
$$449$$ 10678.8 1.12241 0.561206 0.827676i $$-0.310338\pi$$
0.561206 + 0.827676i $$0.310338\pi$$
$$450$$ 0 0
$$451$$ −11753.8 −1.22720
$$452$$ −17972.7 −1.87027
$$453$$ −1820.29 −0.188796
$$454$$ −9640.53 −0.996592
$$455$$ 0 0
$$456$$ 10965.1 1.12607
$$457$$ −4228.23 −0.432797 −0.216399 0.976305i $$-0.569431\pi$$
−0.216399 + 0.976305i $$0.569431\pi$$
$$458$$ −4484.99 −0.457577
$$459$$ −284.846 −0.0289661
$$460$$ 0 0
$$461$$ 910.121 0.0919492 0.0459746 0.998943i $$-0.485361\pi$$
0.0459746 + 0.998943i $$0.485361\pi$$
$$462$$ 3849.28 0.387629
$$463$$ −4456.16 −0.447290 −0.223645 0.974671i $$-0.571796\pi$$
−0.223645 + 0.974671i $$0.571796\pi$$
$$464$$ −6033.30 −0.603640
$$465$$ 0 0
$$466$$ 34767.5 3.45617
$$467$$ 4429.42 0.438907 0.219453 0.975623i $$-0.429573\pi$$
0.219453 + 0.975623i $$0.429573\pi$$
$$468$$ 6654.99 0.657323
$$469$$ 3371.84 0.331977
$$470$$ 0 0
$$471$$ 9344.35 0.914151
$$472$$ −2238.65 −0.218310
$$473$$ −236.432 −0.0229835
$$474$$ 10977.5 1.06374
$$475$$ 0 0
$$476$$ −1464.03 −0.140975
$$477$$ 4832.10 0.463830
$$478$$ −962.970 −0.0921448
$$479$$ 2752.85 0.262591 0.131296 0.991343i $$-0.458086\pi$$
0.131296 + 0.991343i $$0.458086\pi$$
$$480$$ 0 0
$$481$$ 9693.90 0.918927
$$482$$ 8038.43 0.759628
$$483$$ 2632.29 0.247978
$$484$$ −2448.26 −0.229927
$$485$$ 0 0
$$486$$ −1281.80 −0.119638
$$487$$ 670.598 0.0623977 0.0311989 0.999513i $$-0.490067\pi$$
0.0311989 + 0.999513i $$0.490067\pi$$
$$488$$ 3604.88 0.334396
$$489$$ 7239.22 0.669466
$$490$$ 0 0
$$491$$ −8244.70 −0.757797 −0.378898 0.925438i $$-0.623697\pi$$
−0.378898 + 0.925438i $$0.623697\pi$$
$$492$$ 20117.0 1.84338
$$493$$ −373.485 −0.0341195
$$494$$ −11529.1 −1.05004
$$495$$ 0 0
$$496$$ 49728.3 4.50175
$$497$$ −2546.27 −0.229810
$$498$$ 21116.4 1.90010
$$499$$ 8164.91 0.732488 0.366244 0.930519i $$-0.380644\pi$$
0.366244 + 0.930519i $$0.380644\pi$$
$$500$$ 0 0
$$501$$ −1831.43 −0.163317
$$502$$ 12436.8 1.10574
$$503$$ −8175.59 −0.724715 −0.362357 0.932039i $$-0.618028\pi$$
−0.362357 + 0.932039i $$0.618028\pi$$
$$504$$ −3929.60 −0.347298
$$505$$ 0 0
$$506$$ 22976.0 2.01859
$$507$$ 2417.35 0.211752
$$508$$ 33997.2 2.96926
$$509$$ −878.448 −0.0764961 −0.0382480 0.999268i $$-0.512178\pi$$
−0.0382480 + 0.999268i $$0.512178\pi$$
$$510$$ 0 0
$$511$$ 4069.09 0.352262
$$512$$ −16876.5 −1.45673
$$513$$ 1582.15 0.136167
$$514$$ −14677.7 −1.25955
$$515$$ 0 0
$$516$$ 404.661 0.0345237
$$517$$ −8711.42 −0.741060
$$518$$ −9596.55 −0.813992
$$519$$ 11381.4 0.962600
$$520$$ 0 0
$$521$$ 11712.6 0.984910 0.492455 0.870338i $$-0.336100\pi$$
0.492455 + 0.870338i $$0.336100\pi$$
$$522$$ −1680.68 −0.140922
$$523$$ 7341.82 0.613834 0.306917 0.951736i $$-0.400703\pi$$
0.306917 + 0.951736i $$0.400703\pi$$
$$524$$ 9329.75 0.777809
$$525$$ 0 0
$$526$$ −10780.8 −0.893659
$$527$$ 3078.38 0.254452
$$528$$ −17766.1 −1.46434
$$529$$ 3544.92 0.291355
$$530$$ 0 0
$$531$$ −323.014 −0.0263985
$$532$$ 8131.84 0.662707
$$533$$ −12616.3 −1.02528
$$534$$ 5586.74 0.452738
$$535$$ 0 0
$$536$$ −30045.3 −2.42119
$$537$$ 8414.03 0.676149
$$538$$ 18213.4 1.45955
$$539$$ 1702.71 0.136068
$$540$$ 0 0
$$541$$ −15868.7 −1.26109 −0.630545 0.776153i $$-0.717169\pi$$
−0.630545 + 0.776153i $$0.717169\pi$$
$$542$$ 13948.4 1.10542
$$543$$ −9318.13 −0.736426
$$544$$ 4219.61 0.332563
$$545$$ 0 0
$$546$$ 4131.73 0.323850
$$547$$ −2315.26 −0.180975 −0.0904875 0.995898i $$-0.528843\pi$$
−0.0904875 + 0.995898i $$0.528843\pi$$
$$548$$ 8800.41 0.686013
$$549$$ 520.146 0.0404359
$$550$$ 0 0
$$551$$ 2074.49 0.160392
$$552$$ −23455.4 −1.80857
$$553$$ 4855.84 0.373402
$$554$$ −14134.1 −1.08393
$$555$$ 0 0
$$556$$ 33106.9 2.52526
$$557$$ 4819.05 0.366588 0.183294 0.983058i $$-0.441324\pi$$
0.183294 + 0.983058i $$0.441324\pi$$
$$558$$ 13852.7 1.05095
$$559$$ −253.781 −0.0192018
$$560$$ 0 0
$$561$$ −1099.79 −0.0827689
$$562$$ −5378.79 −0.403720
$$563$$ −2540.86 −0.190203 −0.0951017 0.995468i $$-0.530318\pi$$
−0.0951017 + 0.995468i $$0.530318\pi$$
$$564$$ 14909.9 1.11315
$$565$$ 0 0
$$566$$ −2279.85 −0.169310
$$567$$ −567.000 −0.0419961
$$568$$ 22688.9 1.67607
$$569$$ −24220.0 −1.78445 −0.892227 0.451587i $$-0.850858\pi$$
−0.892227 + 0.451587i $$0.850858\pi$$
$$570$$ 0 0
$$571$$ −11772.1 −0.862778 −0.431389 0.902166i $$-0.641976\pi$$
−0.431389 + 0.902166i $$0.641976\pi$$
$$572$$ 25695.1 1.87826
$$573$$ −785.855 −0.0572942
$$574$$ 12489.6 0.908198
$$575$$ 0 0
$$576$$ 6717.80 0.485952
$$577$$ −10584.3 −0.763655 −0.381827 0.924234i $$-0.624705\pi$$
−0.381827 + 0.924234i $$0.624705\pi$$
$$578$$ −25328.6 −1.82272
$$579$$ 12153.2 0.872315
$$580$$ 0 0
$$581$$ 9340.74 0.666987
$$582$$ 22880.6 1.62961
$$583$$ 18656.8 1.32536
$$584$$ −36258.3 −2.56914
$$585$$ 0 0
$$586$$ 11847.0 0.835148
$$587$$ 8712.63 0.612621 0.306311 0.951932i $$-0.400905\pi$$
0.306311 + 0.951932i $$0.400905\pi$$
$$588$$ −2914.24 −0.204390
$$589$$ −17098.6 −1.19615
$$590$$ 0 0
$$591$$ −8624.48 −0.600277
$$592$$ 44292.4 3.07501
$$593$$ 15362.9 1.06387 0.531937 0.846784i $$-0.321464\pi$$
0.531937 + 0.846784i $$0.321464\pi$$
$$594$$ −4949.07 −0.341857
$$595$$ 0 0
$$596$$ 14747.0 1.01353
$$597$$ 9200.91 0.630768
$$598$$ 24662.0 1.68646
$$599$$ 26003.8 1.77377 0.886883 0.461994i $$-0.152866\pi$$
0.886883 + 0.461994i $$0.152866\pi$$
$$600$$ 0 0
$$601$$ 20567.7 1.39596 0.697982 0.716115i $$-0.254082\pi$$
0.697982 + 0.716115i $$0.254082\pi$$
$$602$$ 251.233 0.0170091
$$603$$ −4335.22 −0.292776
$$604$$ 12029.0 0.810349
$$605$$ 0 0
$$606$$ −7514.42 −0.503717
$$607$$ −19642.1 −1.31342 −0.656711 0.754142i $$-0.728053\pi$$
−0.656711 + 0.754142i $$0.728053\pi$$
$$608$$ −23437.4 −1.56334
$$609$$ −743.442 −0.0494676
$$610$$ 0 0
$$611$$ −9350.65 −0.619127
$$612$$ 1882.33 0.124328
$$613$$ −8454.59 −0.557060 −0.278530 0.960428i $$-0.589847\pi$$
−0.278530 + 0.960428i $$0.589847\pi$$
$$614$$ 16864.3 1.10845
$$615$$ 0 0
$$616$$ −15172.3 −0.992383
$$617$$ 24168.4 1.57696 0.788479 0.615061i $$-0.210869\pi$$
0.788479 + 0.615061i $$0.210869\pi$$
$$618$$ −31643.0 −2.05966
$$619$$ −2037.56 −0.132305 −0.0661523 0.997810i $$-0.521072\pi$$
−0.0661523 + 0.997810i $$0.521072\pi$$
$$620$$ 0 0
$$621$$ −3384.37 −0.218696
$$622$$ −17700.5 −1.14104
$$623$$ 2471.27 0.158923
$$624$$ −19069.8 −1.22340
$$625$$ 0 0
$$626$$ 11902.3 0.759921
$$627$$ 6108.70 0.389088
$$628$$ −61749.8 −3.92370
$$629$$ 2741.87 0.173808
$$630$$ 0 0
$$631$$ 12339.5 0.778489 0.389244 0.921135i $$-0.372736\pi$$
0.389244 + 0.921135i $$0.372736\pi$$
$$632$$ −43268.7 −2.72332
$$633$$ −1786.27 −0.112161
$$634$$ 32383.7 2.02858
$$635$$ 0 0
$$636$$ −31931.7 −1.99084
$$637$$ 1827.65 0.113680
$$638$$ −6489.15 −0.402677
$$639$$ 3273.77 0.202674
$$640$$ 0 0
$$641$$ −10222.6 −0.629906 −0.314953 0.949107i $$-0.601989\pi$$
−0.314953 + 0.949107i $$0.601989\pi$$
$$642$$ 18463.3 1.13503
$$643$$ 1211.75 0.0743187 0.0371594 0.999309i $$-0.488169\pi$$
0.0371594 + 0.999309i $$0.488169\pi$$
$$644$$ −17394.8 −1.06437
$$645$$ 0 0
$$646$$ −3260.95 −0.198607
$$647$$ 2817.22 0.171184 0.0855922 0.996330i $$-0.472722\pi$$
0.0855922 + 0.996330i $$0.472722\pi$$
$$648$$ 5052.34 0.306288
$$649$$ −1247.16 −0.0754320
$$650$$ 0 0
$$651$$ 6127.67 0.368913
$$652$$ −47838.6 −2.87347
$$653$$ −20986.2 −1.25766 −0.628831 0.777542i $$-0.716466\pi$$
−0.628831 + 0.777542i $$0.716466\pi$$
$$654$$ 21160.5 1.26520
$$655$$ 0 0
$$656$$ −57645.1 −3.43089
$$657$$ −5231.69 −0.310666
$$658$$ 9256.74 0.548428
$$659$$ −2384.09 −0.140927 −0.0704635 0.997514i $$-0.522448\pi$$
−0.0704635 + 0.997514i $$0.522448\pi$$
$$660$$ 0 0
$$661$$ −7577.10 −0.445862 −0.222931 0.974834i $$-0.571562\pi$$
−0.222931 + 0.974834i $$0.571562\pi$$
$$662$$ 37081.7 2.17707
$$663$$ −1180.50 −0.0691503
$$664$$ −83232.2 −4.86451
$$665$$ 0 0
$$666$$ 12338.4 0.717874
$$667$$ −4437.54 −0.257605
$$668$$ 12102.5 0.700989
$$669$$ −11339.4 −0.655315
$$670$$ 0 0
$$671$$ 2008.30 0.115543
$$672$$ 8399.36 0.482161
$$673$$ −11724.6 −0.671547 −0.335774 0.941943i $$-0.608998\pi$$
−0.335774 + 0.941943i $$0.608998\pi$$
$$674$$ −54481.7 −3.11358
$$675$$ 0 0
$$676$$ −15974.5 −0.908880
$$677$$ −32304.3 −1.83390 −0.916952 0.398997i $$-0.869358\pi$$
−0.916952 + 0.398997i $$0.869358\pi$$
$$678$$ 14346.4 0.812639
$$679$$ 10121.1 0.572038
$$680$$ 0 0
$$681$$ 5482.85 0.308522
$$682$$ 53485.6 3.00303
$$683$$ −33367.1 −1.86934 −0.934669 0.355519i $$-0.884304\pi$$
−0.934669 + 0.355519i $$0.884304\pi$$
$$684$$ −10455.2 −0.584452
$$685$$ 0 0
$$686$$ −1809.30 −0.100699
$$687$$ 2550.75 0.141655
$$688$$ −1159.55 −0.0642551
$$689$$ 20025.8 1.10729
$$690$$ 0 0
$$691$$ −1043.67 −0.0574577 −0.0287288 0.999587i $$-0.509146\pi$$
−0.0287288 + 0.999587i $$0.509146\pi$$
$$692$$ −75211.3 −4.13166
$$693$$ −2189.20 −0.120001
$$694$$ −10378.6 −0.567674
$$695$$ 0 0
$$696$$ 6624.55 0.360780
$$697$$ −3568.46 −0.193924
$$698$$ −23027.4 −1.24871
$$699$$ −19773.3 −1.06995
$$700$$ 0 0
$$701$$ −11305.7 −0.609143 −0.304572 0.952489i $$-0.598513\pi$$
−0.304572 + 0.952489i $$0.598513\pi$$
$$702$$ −5312.23 −0.285608
$$703$$ −15229.4 −0.817055
$$704$$ 25937.6 1.38858
$$705$$ 0 0
$$706$$ 32027.1 1.70730
$$707$$ −3323.97 −0.176818
$$708$$ 2134.55 0.113307
$$709$$ −13306.8 −0.704860 −0.352430 0.935838i $$-0.614645\pi$$
−0.352430 + 0.935838i $$0.614645\pi$$
$$710$$ 0 0
$$711$$ −6243.22 −0.329309
$$712$$ −22020.6 −1.15907
$$713$$ 36575.6 1.92113
$$714$$ 1168.64 0.0612538
$$715$$ 0 0
$$716$$ −55602.0 −2.90216
$$717$$ 547.669 0.0285259
$$718$$ 50839.9 2.64252
$$719$$ 10701.2 0.555062 0.277531 0.960717i $$-0.410484\pi$$
0.277531 + 0.960717i $$0.410484\pi$$
$$720$$ 0 0
$$721$$ −13997.1 −0.722996
$$722$$ −18068.0 −0.931333
$$723$$ −4571.69 −0.235163
$$724$$ 61576.5 3.16088
$$725$$ 0 0
$$726$$ 1954.28 0.0999037
$$727$$ 2121.14 0.108210 0.0541051 0.998535i $$-0.482769\pi$$
0.0541051 + 0.998535i $$0.482769\pi$$
$$728$$ −16285.6 −0.829098
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −71.7808 −0.00363189
$$732$$ −3437.26 −0.173558
$$733$$ 21584.0 1.08762 0.543809 0.839209i $$-0.316981\pi$$
0.543809 + 0.839209i $$0.316981\pi$$
$$734$$ −2757.33 −0.138658
$$735$$ 0 0
$$736$$ 50135.0 2.51087
$$737$$ −16738.4 −0.836588
$$738$$ −16058.0 −0.800955
$$739$$ −9945.21 −0.495048 −0.247524 0.968882i $$-0.579617\pi$$
−0.247524 + 0.968882i $$0.579617\pi$$
$$740$$ 0 0
$$741$$ 6556.94 0.325068
$$742$$ −19824.7 −0.980848
$$743$$ −2867.01 −0.141562 −0.0707808 0.997492i $$-0.522549\pi$$
−0.0707808 + 0.997492i $$0.522549\pi$$
$$744$$ −54601.6 −2.69058
$$745$$ 0 0
$$746$$ −17037.1 −0.836158
$$747$$ −12009.5 −0.588227
$$748$$ 7267.71 0.355259
$$749$$ 8167.15 0.398426
$$750$$ 0 0
$$751$$ −10824.1 −0.525934 −0.262967 0.964805i $$-0.584701\pi$$
−0.262967 + 0.964805i $$0.584701\pi$$
$$752$$ −42724.0 −2.07179
$$753$$ −7073.19 −0.342313
$$754$$ −6965.31 −0.336421
$$755$$ 0 0
$$756$$ 3746.88 0.180255
$$757$$ 14512.0 0.696761 0.348381 0.937353i $$-0.386732\pi$$
0.348381 + 0.937353i $$0.386732\pi$$
$$758$$ 35023.9 1.67827
$$759$$ −13067.1 −0.624910
$$760$$ 0 0
$$761$$ −33075.8 −1.57556 −0.787778 0.615959i $$-0.788769\pi$$
−0.787778 + 0.615959i $$0.788769\pi$$
$$762$$ −27137.7 −1.29015
$$763$$ 9360.23 0.444120
$$764$$ 5193.13 0.245917
$$765$$ 0 0
$$766$$ 75032.8 3.53922
$$767$$ −1338.68 −0.0630206
$$768$$ 6242.38 0.293297
$$769$$ 6728.44 0.315518 0.157759 0.987478i $$-0.449573\pi$$
0.157759 + 0.987478i $$0.449573\pi$$
$$770$$ 0 0
$$771$$ 8347.65 0.389926
$$772$$ −80311.5 −3.74414
$$773$$ 24233.3 1.12757 0.563784 0.825922i $$-0.309345\pi$$
0.563784 + 0.825922i $$0.309345\pi$$
$$774$$ −323.014 −0.0150006
$$775$$ 0 0
$$776$$ −90186.0 −4.17202
$$777$$ 5457.84 0.251993
$$778$$ 15412.4 0.710231
$$779$$ 19820.6 0.911615
$$780$$ 0 0
$$781$$ 12640.1 0.579127
$$782$$ 6975.51 0.318982
$$783$$ 955.854 0.0436263
$$784$$ 8350.72 0.380408
$$785$$ 0 0
$$786$$ −7447.31 −0.337960
$$787$$ 17200.4 0.779069 0.389535 0.921012i $$-0.372636\pi$$
0.389535 + 0.921012i $$0.372636\pi$$
$$788$$ 56992.7 2.57650
$$789$$ 6131.35 0.276656
$$790$$ 0 0
$$791$$ 6346.05 0.285258
$$792$$ 19507.2 0.875199
$$793$$ 2155.66 0.0965318
$$794$$ −4282.92 −0.191430
$$795$$ 0 0
$$796$$ −60801.9 −2.70737
$$797$$ 4208.87 0.187059 0.0935295 0.995617i $$-0.470185\pi$$
0.0935295 + 0.995617i $$0.470185\pi$$
$$798$$ −6491.09 −0.287948
$$799$$ −2644.78 −0.117104
$$800$$ 0 0
$$801$$ −3177.34 −0.140157
$$802$$ 12336.1 0.543146
$$803$$ −20199.7 −0.887709
$$804$$ 28648.2 1.25665
$$805$$ 0 0
$$806$$ 57410.2 2.50892
$$807$$ −10358.5 −0.451842
$$808$$ 29618.7 1.28958
$$809$$ −23632.1 −1.02702 −0.513511 0.858083i $$-0.671656\pi$$
−0.513511 + 0.858083i $$0.671656\pi$$
$$810$$ 0 0
$$811$$ 28425.1 1.23075 0.615377 0.788233i $$-0.289004\pi$$
0.615377 + 0.788233i $$0.289004\pi$$
$$812$$ 4912.85 0.212324
$$813$$ −7932.88 −0.342212
$$814$$ 47638.9 2.05128
$$815$$ 0 0
$$816$$ −5393.80 −0.231398
$$817$$ 398.699 0.0170731
$$818$$ −14387.7 −0.614981
$$819$$ −2349.84 −0.100256
$$820$$ 0 0
$$821$$ 39409.6 1.67528 0.837640 0.546223i $$-0.183935\pi$$
0.837640 + 0.546223i $$0.183935\pi$$
$$822$$ −7024.77 −0.298074
$$823$$ −16346.6 −0.692352 −0.346176 0.938170i $$-0.612520\pi$$
−0.346176 + 0.938170i $$0.612520\pi$$
$$824$$ 124723. 5.27300
$$825$$ 0 0
$$826$$ 1325.23 0.0558241
$$827$$ 3738.87 0.157211 0.0786054 0.996906i $$-0.474953\pi$$
0.0786054 + 0.996906i $$0.474953\pi$$
$$828$$ 22364.8 0.938684
$$829$$ −45196.2 −1.89352 −0.946761 0.321937i $$-0.895666\pi$$
−0.946761 + 0.321937i $$0.895666\pi$$
$$830$$ 0 0
$$831$$ 8038.46 0.335561
$$832$$ 27840.8 1.16010
$$833$$ 516.942 0.0215018
$$834$$ −26427.0 −1.09723
$$835$$ 0 0
$$836$$ −40367.8 −1.67004
$$837$$ −7878.44 −0.325351
$$838$$ 70189.5 2.89338
$$839$$ 15899.7 0.654254 0.327127 0.944980i $$-0.393920\pi$$
0.327127 + 0.944980i $$0.393920\pi$$
$$840$$ 0 0
$$841$$ −23135.7 −0.948612
$$842$$ −58063.4 −2.37648
$$843$$ 3059.07 0.124982
$$844$$ 11804.1 0.481414
$$845$$ 0 0
$$846$$ −11901.5 −0.483668
$$847$$ 864.465 0.0350689
$$848$$ 91500.0 3.70534
$$849$$ 1296.62 0.0524144
$$850$$ 0 0
$$851$$ 32577.4 1.31227
$$852$$ −21633.9 −0.869913
$$853$$ −33926.7 −1.36182 −0.680908 0.732369i $$-0.738415\pi$$
−0.680908 + 0.732369i $$0.738415\pi$$
$$854$$ −2134.01 −0.0855086
$$855$$ 0 0
$$856$$ −72774.7 −2.90583
$$857$$ 35432.4 1.41231 0.706154 0.708058i $$-0.250428\pi$$
0.706154 + 0.708058i $$0.250428\pi$$
$$858$$ −20510.6 −0.816108
$$859$$ −6780.17 −0.269309 −0.134655 0.990893i $$-0.542992\pi$$
−0.134655 + 0.990893i $$0.542992\pi$$
$$860$$ 0 0
$$861$$ −7103.20 −0.281157
$$862$$ −34422.1 −1.36012
$$863$$ 30675.1 1.20995 0.604977 0.796243i $$-0.293182\pi$$
0.604977 + 0.796243i $$0.293182\pi$$
$$864$$ −10799.2 −0.425226
$$865$$ 0 0
$$866$$ 61802.4 2.42509
$$867$$ 14405.1 0.564271
$$868$$ −40493.2 −1.58344
$$869$$ −24105.2 −0.940981
$$870$$ 0 0
$$871$$ −17966.6 −0.698938
$$872$$ −83405.8 −3.23908
$$873$$ −13012.9 −0.504490
$$874$$ −38744.8 −1.49950
$$875$$ 0 0
$$876$$ 34572.3 1.33344
$$877$$ −40861.3 −1.57330 −0.786652 0.617397i $$-0.788187\pi$$
−0.786652 + 0.617397i $$0.788187\pi$$
$$878$$ −77073.9 −2.96255
$$879$$ −6737.76 −0.258543
$$880$$ 0 0
$$881$$ −43839.0 −1.67647 −0.838236 0.545308i $$-0.816413\pi$$
−0.838236 + 0.545308i $$0.816413\pi$$
$$882$$ 2326.24 0.0888079
$$883$$ −44625.1 −1.70074 −0.850371 0.526183i $$-0.823623\pi$$
−0.850371 + 0.526183i $$0.823623\pi$$
$$884$$ 7801.01 0.296806
$$885$$ 0 0
$$886$$ 80388.6 3.04820
$$887$$ 43967.5 1.66436 0.832178 0.554509i $$-0.187094\pi$$
0.832178 + 0.554509i $$0.187094\pi$$
$$888$$ −48632.9 −1.83785
$$889$$ −12004.2 −0.452878
$$890$$ 0 0
$$891$$ 2814.68 0.105831
$$892$$ 74933.5 2.81273
$$893$$ 14690.2 0.550491
$$894$$ −11771.6 −0.440380
$$895$$ 0 0
$$896$$ −5162.95 −0.192502
$$897$$ −14026.0 −0.522089
$$898$$ 56329.7 2.09326
$$899$$ −10330.1 −0.383234
$$900$$ 0 0
$$901$$ 5664.21 0.209436
$$902$$ −62000.4 −2.28868
$$903$$ −142.883 −0.00526563
$$904$$ −56547.4 −2.08046
$$905$$ 0 0
$$906$$ −9601.89 −0.352099
$$907$$ 13584.3 0.497309 0.248654 0.968592i $$-0.420012\pi$$
0.248654 + 0.968592i $$0.420012\pi$$
$$908$$ −36232.1 −1.32423
$$909$$ 4273.67 0.155939
$$910$$ 0 0
$$911$$ −16421.6 −0.597226 −0.298613 0.954374i $$-0.596524\pi$$
−0.298613 + 0.954374i $$0.596524\pi$$
$$912$$ 29959.3 1.08778
$$913$$ −46369.0 −1.68082
$$914$$ −22303.6 −0.807152
$$915$$ 0 0
$$916$$ −16856.0 −0.608010
$$917$$ −3294.28 −0.118633
$$918$$ −1502.54 −0.0540208
$$919$$ −29487.3 −1.05843 −0.529214 0.848488i $$-0.677513\pi$$
−0.529214 + 0.848488i $$0.677513\pi$$
$$920$$ 0 0
$$921$$ −9591.23 −0.343151
$$922$$ 4800.81 0.171482
$$923$$ 13567.6 0.483839
$$924$$ 14466.8 0.515067
$$925$$ 0 0
$$926$$ −23505.9 −0.834181
$$927$$ 17996.3 0.637622
$$928$$ −14159.7 −0.500878
$$929$$ 3441.85 0.121554 0.0607769 0.998151i $$-0.480642\pi$$
0.0607769 + 0.998151i $$0.480642\pi$$
$$930$$ 0 0
$$931$$ −2871.30 −0.101077
$$932$$ 130667. 4.59242
$$933$$ 10066.8 0.353240
$$934$$ 23364.8 0.818545
$$935$$ 0 0
$$936$$ 20938.6 0.731196
$$937$$ −5646.60 −0.196869 −0.0984346 0.995144i $$-0.531384\pi$$
−0.0984346 + 0.995144i $$0.531384\pi$$
$$938$$ 17786.2 0.619125
$$939$$ −6769.18 −0.235254
$$940$$ 0 0
$$941$$ −44680.1 −1.54785 −0.773927 0.633275i $$-0.781710\pi$$
−0.773927 + 0.633275i $$0.781710\pi$$
$$942$$ 49290.7 1.70486
$$943$$ −42398.4 −1.46414
$$944$$ −6116.54 −0.210886
$$945$$ 0 0
$$946$$ −1247.16 −0.0428633
$$947$$ 48924.6 1.67881 0.839406 0.543505i $$-0.182903\pi$$
0.839406 + 0.543505i $$0.182903\pi$$
$$948$$ 41256.8 1.41346
$$949$$ −21681.9 −0.741647
$$950$$ 0 0
$$951$$ −18417.6 −0.628002
$$952$$ −4606.29 −0.156818
$$953$$ −52014.3 −1.76801 −0.884003 0.467482i $$-0.845161\pi$$
−0.884003 + 0.467482i $$0.845161\pi$$
$$954$$ 25488.9 0.865026
$$955$$ 0 0
$$956$$ −3619.14 −0.122439
$$957$$ 3690.57 0.124660
$$958$$ 14521.1 0.489723
$$959$$ −3107.37 −0.104632
$$960$$ 0 0
$$961$$ 55352.8 1.85804
$$962$$ 51134.5 1.71377
$$963$$ −10500.6 −0.351379
$$964$$ 30210.9 1.00936
$$965$$ 0 0
$$966$$ 13885.1 0.462470
$$967$$ 47117.7 1.56691 0.783456 0.621448i $$-0.213455\pi$$
0.783456 + 0.621448i $$0.213455\pi$$
$$968$$ −7702.95 −0.255767
$$969$$ 1854.60 0.0614843
$$970$$ 0 0
$$971$$ −8195.04 −0.270846 −0.135423 0.990788i $$-0.543239\pi$$
−0.135423 + 0.990788i $$0.543239\pi$$
$$972$$ −4817.41 −0.158970
$$973$$ −11689.9 −0.385159
$$974$$ 3537.35 0.116370
$$975$$ 0 0
$$976$$ 9849.42 0.323025
$$977$$ −4643.51 −0.152056 −0.0760282 0.997106i $$-0.524224\pi$$
−0.0760282 + 0.997106i $$0.524224\pi$$
$$978$$ 38186.3 1.24853
$$979$$ −12267.8 −0.400490
$$980$$ 0 0
$$981$$ −12034.6 −0.391677
$$982$$ −43490.1 −1.41326
$$983$$ −43986.5 −1.42721 −0.713607 0.700546i $$-0.752940\pi$$
−0.713607 + 0.700546i $$0.752940\pi$$
$$984$$ 63294.1 2.05055
$$985$$ 0 0
$$986$$ −1970.10 −0.0636317
$$987$$ −5264.58 −0.169781
$$988$$ −43329.9 −1.39525
$$989$$ −852.859 −0.0274210
$$990$$ 0 0
$$991$$ 1595.21 0.0511337 0.0255668 0.999673i $$-0.491861\pi$$
0.0255668 + 0.999673i $$0.491861\pi$$
$$992$$ 116709. 3.73539
$$993$$ −21089.4 −0.673971
$$994$$ −13431.3 −0.428588
$$995$$ 0 0
$$996$$ 79362.0 2.52478
$$997$$ −21501.2 −0.682998 −0.341499 0.939882i $$-0.610935\pi$$
−0.341499 + 0.939882i $$0.610935\pi$$
$$998$$ 43069.2 1.36606
$$999$$ −7017.22 −0.222237
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 525.4.a.n.1.2 2
3.2 odd 2 1575.4.a.p.1.1 2
5.2 odd 4 525.4.d.g.274.4 4
5.3 odd 4 525.4.d.g.274.1 4
5.4 even 2 21.4.a.c.1.1 2
15.14 odd 2 63.4.a.e.1.2 2
20.19 odd 2 336.4.a.m.1.2 2
35.4 even 6 147.4.e.l.79.2 4
35.9 even 6 147.4.e.l.67.2 4
35.19 odd 6 147.4.e.m.67.2 4
35.24 odd 6 147.4.e.m.79.2 4
35.34 odd 2 147.4.a.i.1.1 2
40.19 odd 2 1344.4.a.bo.1.1 2
40.29 even 2 1344.4.a.bg.1.1 2
60.59 even 2 1008.4.a.ba.1.1 2
105.44 odd 6 441.4.e.q.361.1 4
105.59 even 6 441.4.e.p.226.1 4
105.74 odd 6 441.4.e.q.226.1 4
105.89 even 6 441.4.e.p.361.1 4
105.104 even 2 441.4.a.r.1.2 2
140.139 even 2 2352.4.a.bz.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.c.1.1 2 5.4 even 2
63.4.a.e.1.2 2 15.14 odd 2
147.4.a.i.1.1 2 35.34 odd 2
147.4.e.l.67.2 4 35.9 even 6
147.4.e.l.79.2 4 35.4 even 6
147.4.e.m.67.2 4 35.19 odd 6
147.4.e.m.79.2 4 35.24 odd 6
336.4.a.m.1.2 2 20.19 odd 2
441.4.a.r.1.2 2 105.104 even 2
441.4.e.p.226.1 4 105.59 even 6
441.4.e.p.361.1 4 105.89 even 6
441.4.e.q.226.1 4 105.74 odd 6
441.4.e.q.361.1 4 105.44 odd 6
525.4.a.n.1.2 2 1.1 even 1 trivial
525.4.d.g.274.1 4 5.3 odd 4
525.4.d.g.274.4 4 5.2 odd 4
1008.4.a.ba.1.1 2 60.59 even 2
1344.4.a.bg.1.1 2 40.29 even 2
1344.4.a.bo.1.1 2 40.19 odd 2
1575.4.a.p.1.1 2 3.2 odd 2
2352.4.a.bz.1.1 2 140.139 even 2