# Properties

 Label 363.4.a.l.1.1 Level $363$ Weight $4$ Character 363.1 Self dual yes Analytic conductor $21.418$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,4,Mod(1,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, 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("363.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 363.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: $$21.4176933321$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 363.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-5.19615 q^{2} -3.00000 q^{3} +19.0000 q^{4} +9.00000 q^{5} +15.5885 q^{6} -24.2487 q^{7} -57.1577 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-5.19615 q^{2} -3.00000 q^{3} +19.0000 q^{4} +9.00000 q^{5} +15.5885 q^{6} -24.2487 q^{7} -57.1577 q^{8} +9.00000 q^{9} -46.7654 q^{10} -57.0000 q^{12} +71.0141 q^{13} +126.000 q^{14} -27.0000 q^{15} +145.000 q^{16} +88.3346 q^{17} -46.7654 q^{18} -145.492 q^{19} +171.000 q^{20} +72.7461 q^{21} +90.0000 q^{23} +171.473 q^{24} -44.0000 q^{25} -369.000 q^{26} -27.0000 q^{27} -460.726 q^{28} -88.3346 q^{29} +140.296 q^{30} -188.000 q^{31} -296.181 q^{32} -459.000 q^{34} -218.238 q^{35} +171.000 q^{36} +133.000 q^{37} +756.000 q^{38} -213.042 q^{39} -514.419 q^{40} +36.3731 q^{41} -378.000 q^{42} -72.7461 q^{43} +81.0000 q^{45} -467.654 q^{46} +72.0000 q^{47} -435.000 q^{48} +245.000 q^{49} +228.631 q^{50} -265.004 q^{51} +1349.27 q^{52} -45.0000 q^{53} +140.296 q^{54} +1386.00 q^{56} +436.477 q^{57} +459.000 q^{58} +378.000 q^{59} -513.000 q^{60} +623.538 q^{61} +976.877 q^{62} -218.238 q^{63} +379.000 q^{64} +639.127 q^{65} -386.000 q^{67} +1678.36 q^{68} -270.000 q^{69} +1134.00 q^{70} -198.000 q^{71} -514.419 q^{72} +76.2102 q^{73} -691.088 q^{74} +132.000 q^{75} -2764.35 q^{76} +1107.00 q^{78} -152.420 q^{79} +1305.00 q^{80} +81.0000 q^{81} -189.000 q^{82} +1247.08 q^{83} +1382.18 q^{84} +795.011 q^{85} +378.000 q^{86} +265.004 q^{87} +45.0000 q^{89} -420.888 q^{90} -1722.00 q^{91} +1710.00 q^{92} +564.000 q^{93} -374.123 q^{94} -1309.43 q^{95} +888.542 q^{96} +89.0000 q^{97} -1273.06 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} + 38 q^{4} + 18 q^{5} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 + 38 * q^4 + 18 * q^5 + 18 * q^9 $$2 q - 6 q^{3} + 38 q^{4} + 18 q^{5} + 18 q^{9} - 114 q^{12} + 252 q^{14} - 54 q^{15} + 290 q^{16} + 342 q^{20} + 180 q^{23} - 88 q^{25} - 738 q^{26} - 54 q^{27} - 376 q^{31} - 918 q^{34} + 342 q^{36} + 266 q^{37} + 1512 q^{38} - 756 q^{42} + 162 q^{45} + 144 q^{47} - 870 q^{48} + 490 q^{49} - 90 q^{53} + 2772 q^{56} + 918 q^{58} + 756 q^{59} - 1026 q^{60} + 758 q^{64} - 772 q^{67} - 540 q^{69} + 2268 q^{70} - 396 q^{71} + 264 q^{75} + 2214 q^{78} + 2610 q^{80} + 162 q^{81} - 378 q^{82} + 756 q^{86} + 90 q^{89} - 3444 q^{91} + 3420 q^{92} + 1128 q^{93} + 178 q^{97}+O(q^{100})$$ 2 * q - 6 * q^3 + 38 * q^4 + 18 * q^5 + 18 * q^9 - 114 * q^12 + 252 * q^14 - 54 * q^15 + 290 * q^16 + 342 * q^20 + 180 * q^23 - 88 * q^25 - 738 * q^26 - 54 * q^27 - 376 * q^31 - 918 * q^34 + 342 * q^36 + 266 * q^37 + 1512 * q^38 - 756 * q^42 + 162 * q^45 + 144 * q^47 - 870 * q^48 + 490 * q^49 - 90 * q^53 + 2772 * q^56 + 918 * q^58 + 756 * q^59 - 1026 * q^60 + 758 * q^64 - 772 * q^67 - 540 * q^69 + 2268 * q^70 - 396 * q^71 + 264 * q^75 + 2214 * q^78 + 2610 * q^80 + 162 * q^81 - 378 * q^82 + 756 * q^86 + 90 * q^89 - 3444 * q^91 + 3420 * q^92 + 1128 * q^93 + 178 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −5.19615 −1.83712 −0.918559 0.395285i $$-0.870646\pi$$
−0.918559 + 0.395285i $$0.870646\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 19.0000 2.37500
$$5$$ 9.00000 0.804984 0.402492 0.915423i $$-0.368144\pi$$
0.402492 + 0.915423i $$0.368144\pi$$
$$6$$ 15.5885 1.06066
$$7$$ −24.2487 −1.30931 −0.654654 0.755929i $$-0.727186\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ −57.1577 −2.52604
$$9$$ 9.00000 0.333333
$$10$$ −46.7654 −1.47885
$$11$$ 0 0
$$12$$ −57.0000 −1.37121
$$13$$ 71.0141 1.51506 0.757529 0.652801i $$-0.226406\pi$$
0.757529 + 0.652801i $$0.226406\pi$$
$$14$$ 126.000 2.40535
$$15$$ −27.0000 −0.464758
$$16$$ 145.000 2.26562
$$17$$ 88.3346 1.26025 0.630126 0.776493i $$-0.283003\pi$$
0.630126 + 0.776493i $$0.283003\pi$$
$$18$$ −46.7654 −0.612372
$$19$$ −145.492 −1.75675 −0.878374 0.477974i $$-0.841371\pi$$
−0.878374 + 0.477974i $$0.841371\pi$$
$$20$$ 171.000 1.91184
$$21$$ 72.7461 0.755929
$$22$$ 0 0
$$23$$ 90.0000 0.815926 0.407963 0.912998i $$-0.366239\pi$$
0.407963 + 0.912998i $$0.366239\pi$$
$$24$$ 171.473 1.45841
$$25$$ −44.0000 −0.352000
$$26$$ −369.000 −2.78334
$$27$$ −27.0000 −0.192450
$$28$$ −460.726 −3.10960
$$29$$ −88.3346 −0.565632 −0.282816 0.959174i $$-0.591269\pi$$
−0.282816 + 0.959174i $$0.591269\pi$$
$$30$$ 140.296 0.853815
$$31$$ −188.000 −1.08922 −0.544610 0.838690i $$-0.683322\pi$$
−0.544610 + 0.838690i $$0.683322\pi$$
$$32$$ −296.181 −1.63618
$$33$$ 0 0
$$34$$ −459.000 −2.31523
$$35$$ −218.238 −1.05397
$$36$$ 171.000 0.791667
$$37$$ 133.000 0.590948 0.295474 0.955351i $$-0.404522\pi$$
0.295474 + 0.955351i $$0.404522\pi$$
$$38$$ 756.000 3.22735
$$39$$ −213.042 −0.874720
$$40$$ −514.419 −2.03342
$$41$$ 36.3731 0.138549 0.0692746 0.997598i $$-0.477932\pi$$
0.0692746 + 0.997598i $$0.477932\pi$$
$$42$$ −378.000 −1.38873
$$43$$ −72.7461 −0.257993 −0.128996 0.991645i $$-0.541176\pi$$
−0.128996 + 0.991645i $$0.541176\pi$$
$$44$$ 0 0
$$45$$ 81.0000 0.268328
$$46$$ −467.654 −1.49895
$$47$$ 72.0000 0.223453 0.111726 0.993739i $$-0.464362\pi$$
0.111726 + 0.993739i $$0.464362\pi$$
$$48$$ −435.000 −1.30806
$$49$$ 245.000 0.714286
$$50$$ 228.631 0.646665
$$51$$ −265.004 −0.727607
$$52$$ 1349.27 3.59826
$$53$$ −45.0000 −0.116627 −0.0583134 0.998298i $$-0.518572\pi$$
−0.0583134 + 0.998298i $$0.518572\pi$$
$$54$$ 140.296 0.353553
$$55$$ 0 0
$$56$$ 1386.00 3.30736
$$57$$ 436.477 1.01426
$$58$$ 459.000 1.03913
$$59$$ 378.000 0.834092 0.417046 0.908885i $$-0.363065\pi$$
0.417046 + 0.908885i $$0.363065\pi$$
$$60$$ −513.000 −1.10380
$$61$$ 623.538 1.30879 0.654393 0.756155i $$-0.272924\pi$$
0.654393 + 0.756155i $$0.272924\pi$$
$$62$$ 976.877 2.00102
$$63$$ −218.238 −0.436436
$$64$$ 379.000 0.740234
$$65$$ 639.127 1.21960
$$66$$ 0 0
$$67$$ −386.000 −0.703842 −0.351921 0.936030i $$-0.614471\pi$$
−0.351921 + 0.936030i $$0.614471\pi$$
$$68$$ 1678.36 2.99310
$$69$$ −270.000 −0.471075
$$70$$ 1134.00 1.93627
$$71$$ −198.000 −0.330962 −0.165481 0.986213i $$-0.552918\pi$$
−0.165481 + 0.986213i $$0.552918\pi$$
$$72$$ −514.419 −0.842012
$$73$$ 76.2102 0.122188 0.0610941 0.998132i $$-0.480541\pi$$
0.0610941 + 0.998132i $$0.480541\pi$$
$$74$$ −691.088 −1.08564
$$75$$ 132.000 0.203227
$$76$$ −2764.35 −4.17228
$$77$$ 0 0
$$78$$ 1107.00 1.60696
$$79$$ −152.420 −0.217071 −0.108536 0.994093i $$-0.534616\pi$$
−0.108536 + 0.994093i $$0.534616\pi$$
$$80$$ 1305.00 1.82379
$$81$$ 81.0000 0.111111
$$82$$ −189.000 −0.254531
$$83$$ 1247.08 1.64921 0.824605 0.565709i $$-0.191397\pi$$
0.824605 + 0.565709i $$0.191397\pi$$
$$84$$ 1382.18 1.79533
$$85$$ 795.011 1.01448
$$86$$ 378.000 0.473963
$$87$$ 265.004 0.326568
$$88$$ 0 0
$$89$$ 45.0000 0.0535954 0.0267977 0.999641i $$-0.491469\pi$$
0.0267977 + 0.999641i $$0.491469\pi$$
$$90$$ −420.888 −0.492950
$$91$$ −1722.00 −1.98368
$$92$$ 1710.00 1.93782
$$93$$ 564.000 0.628861
$$94$$ −374.123 −0.410509
$$95$$ −1309.43 −1.41416
$$96$$ 888.542 0.944650
$$97$$ 89.0000 0.0931606 0.0465803 0.998915i $$-0.485168\pi$$
0.0465803 + 0.998915i $$0.485168\pi$$
$$98$$ −1273.06 −1.31223
$$99$$ 0 0
$$100$$ −836.000 −0.836000
$$101$$ 1288.65 1.26955 0.634777 0.772695i $$-0.281092\pi$$
0.634777 + 0.772695i $$0.281092\pi$$
$$102$$ 1377.00 1.33670
$$103$$ 1358.00 1.29910 0.649552 0.760317i $$-0.274956\pi$$
0.649552 + 0.760317i $$0.274956\pi$$
$$104$$ −4059.00 −3.82709
$$105$$ 654.715 0.608511
$$106$$ 233.827 0.214257
$$107$$ 1299.04 1.17367 0.586835 0.809706i $$-0.300374\pi$$
0.586835 + 0.809706i $$0.300374\pi$$
$$108$$ −513.000 −0.457069
$$109$$ 1203.78 1.05781 0.528903 0.848683i $$-0.322604\pi$$
0.528903 + 0.848683i $$0.322604\pi$$
$$110$$ 0 0
$$111$$ −399.000 −0.341184
$$112$$ −3516.06 −2.96640
$$113$$ 711.000 0.591905 0.295952 0.955203i $$-0.404363\pi$$
0.295952 + 0.955203i $$0.404363\pi$$
$$114$$ −2268.00 −1.86331
$$115$$ 810.000 0.656808
$$116$$ −1678.36 −1.34338
$$117$$ 639.127 0.505020
$$118$$ −1964.15 −1.53232
$$119$$ −2142.00 −1.65006
$$120$$ 1543.26 1.17400
$$121$$ 0 0
$$122$$ −3240.00 −2.40439
$$123$$ −109.119 −0.0799914
$$124$$ −3572.00 −2.58690
$$125$$ −1521.00 −1.08834
$$126$$ 1134.00 0.801784
$$127$$ 651.251 0.455033 0.227516 0.973774i $$-0.426939\pi$$
0.227516 + 0.973774i $$0.426939\pi$$
$$128$$ 400.104 0.276285
$$129$$ 218.238 0.148952
$$130$$ −3321.00 −2.24055
$$131$$ 2608.47 1.73972 0.869859 0.493301i $$-0.164210\pi$$
0.869859 + 0.493301i $$0.164210\pi$$
$$132$$ 0 0
$$133$$ 3528.00 2.30012
$$134$$ 2005.71 1.29304
$$135$$ −243.000 −0.154919
$$136$$ −5049.00 −3.18344
$$137$$ 2394.00 1.49294 0.746472 0.665417i $$-0.231746\pi$$
0.746472 + 0.665417i $$0.231746\pi$$
$$138$$ 1402.96 0.865420
$$139$$ 183.597 0.112033 0.0560163 0.998430i $$-0.482160\pi$$
0.0560163 + 0.998430i $$0.482160\pi$$
$$140$$ −4146.53 −2.50318
$$141$$ −216.000 −0.129011
$$142$$ 1028.84 0.608015
$$143$$ 0 0
$$144$$ 1305.00 0.755208
$$145$$ −795.011 −0.455325
$$146$$ −396.000 −0.224474
$$147$$ −735.000 −0.412393
$$148$$ 2527.00 1.40350
$$149$$ −3611.33 −1.98558 −0.992790 0.119869i $$-0.961753\pi$$
−0.992790 + 0.119869i $$0.961753\pi$$
$$150$$ −685.892 −0.373352
$$151$$ 1357.93 0.731832 0.365916 0.930648i $$-0.380756\pi$$
0.365916 + 0.930648i $$0.380756\pi$$
$$152$$ 8316.00 4.43761
$$153$$ 795.011 0.420084
$$154$$ 0 0
$$155$$ −1692.00 −0.876805
$$156$$ −4047.80 −2.07746
$$157$$ 1306.00 0.663886 0.331943 0.943299i $$-0.392296\pi$$
0.331943 + 0.943299i $$0.392296\pi$$
$$158$$ 792.000 0.398786
$$159$$ 135.000 0.0673346
$$160$$ −2665.63 −1.31710
$$161$$ −2182.38 −1.06830
$$162$$ −420.888 −0.204124
$$163$$ 2546.00 1.22342 0.611712 0.791081i $$-0.290481\pi$$
0.611712 + 0.791081i $$0.290481\pi$$
$$164$$ 691.088 0.329054
$$165$$ 0 0
$$166$$ −6480.00 −3.02979
$$167$$ −675.500 −0.313004 −0.156502 0.987678i $$-0.550022\pi$$
−0.156502 + 0.987678i $$0.550022\pi$$
$$168$$ −4158.00 −1.90950
$$169$$ 2846.00 1.29540
$$170$$ −4131.00 −1.86372
$$171$$ −1309.43 −0.585583
$$172$$ −1382.18 −0.612732
$$173$$ −3180.05 −1.39754 −0.698770 0.715347i $$-0.746269\pi$$
−0.698770 + 0.715347i $$0.746269\pi$$
$$174$$ −1377.00 −0.599943
$$175$$ 1066.94 0.460876
$$176$$ 0 0
$$177$$ −1134.00 −0.481563
$$178$$ −233.827 −0.0984610
$$179$$ −2556.00 −1.06729 −0.533644 0.845709i $$-0.679178\pi$$
−0.533644 + 0.845709i $$0.679178\pi$$
$$180$$ 1539.00 0.637279
$$181$$ −775.000 −0.318261 −0.159131 0.987258i $$-0.550869\pi$$
−0.159131 + 0.987258i $$0.550869\pi$$
$$182$$ 8947.77 3.64425
$$183$$ −1870.61 −0.755627
$$184$$ −5144.19 −2.06106
$$185$$ 1197.00 0.475704
$$186$$ −2930.63 −1.15529
$$187$$ 0 0
$$188$$ 1368.00 0.530700
$$189$$ 654.715 0.251976
$$190$$ 6804.00 2.59797
$$191$$ 2628.00 0.995578 0.497789 0.867298i $$-0.334145\pi$$
0.497789 + 0.867298i $$0.334145\pi$$
$$192$$ −1137.00 −0.427375
$$193$$ 2790.33 1.04069 0.520344 0.853957i $$-0.325804\pi$$
0.520344 + 0.853957i $$0.325804\pi$$
$$194$$ −462.458 −0.171147
$$195$$ −1917.38 −0.704136
$$196$$ 4655.00 1.69643
$$197$$ −3590.54 −1.29856 −0.649278 0.760551i $$-0.724929\pi$$
−0.649278 + 0.760551i $$0.724929\pi$$
$$198$$ 0 0
$$199$$ 1618.00 0.576367 0.288183 0.957575i $$-0.406949\pi$$
0.288183 + 0.957575i $$0.406949\pi$$
$$200$$ 2514.94 0.889165
$$201$$ 1158.00 0.406363
$$202$$ −6696.00 −2.33232
$$203$$ 2142.00 0.740586
$$204$$ −5035.07 −1.72807
$$205$$ 327.358 0.111530
$$206$$ −7056.37 −2.38661
$$207$$ 810.000 0.271975
$$208$$ 10297.0 3.43255
$$209$$ 0 0
$$210$$ −3402.00 −1.11791
$$211$$ 1860.22 0.606934 0.303467 0.952842i $$-0.401856\pi$$
0.303467 + 0.952842i $$0.401856\pi$$
$$212$$ −855.000 −0.276989
$$213$$ 594.000 0.191081
$$214$$ −6750.00 −2.15617
$$215$$ −654.715 −0.207680
$$216$$ 1543.26 0.486136
$$217$$ 4558.76 1.42612
$$218$$ −6255.00 −1.94331
$$219$$ −228.631 −0.0705453
$$220$$ 0 0
$$221$$ 6273.00 1.90936
$$222$$ 2073.26 0.626795
$$223$$ −964.000 −0.289481 −0.144740 0.989470i $$-0.546235\pi$$
−0.144740 + 0.989470i $$0.546235\pi$$
$$224$$ 7182.00 2.14227
$$225$$ −396.000 −0.117333
$$226$$ −3694.46 −1.08740
$$227$$ −2857.88 −0.835614 −0.417807 0.908536i $$-0.637201\pi$$
−0.417807 + 0.908536i $$0.637201\pi$$
$$228$$ 8293.06 2.40887
$$229$$ −5519.00 −1.59260 −0.796301 0.604901i $$-0.793213\pi$$
−0.796301 + 0.604901i $$0.793213\pi$$
$$230$$ −4208.88 −1.20663
$$231$$ 0 0
$$232$$ 5049.00 1.42881
$$233$$ −3611.33 −1.01539 −0.507695 0.861537i $$-0.669502\pi$$
−0.507695 + 0.861537i $$0.669502\pi$$
$$234$$ −3321.00 −0.927780
$$235$$ 648.000 0.179876
$$236$$ 7182.00 1.98097
$$237$$ 457.261 0.125326
$$238$$ 11130.2 3.03135
$$239$$ 3263.18 0.883171 0.441585 0.897219i $$-0.354416\pi$$
0.441585 + 0.897219i $$0.354416\pi$$
$$240$$ −3915.00 −1.05297
$$241$$ 3457.17 0.924050 0.462025 0.886867i $$-0.347123\pi$$
0.462025 + 0.886867i $$0.347123\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 11847.2 3.10836
$$245$$ 2205.00 0.574989
$$246$$ 567.000 0.146954
$$247$$ −10332.0 −2.66158
$$248$$ 10745.6 2.75141
$$249$$ −3741.23 −0.952172
$$250$$ 7903.35 1.99941
$$251$$ 6714.00 1.68838 0.844191 0.536042i $$-0.180081\pi$$
0.844191 + 0.536042i $$0.180081\pi$$
$$252$$ −4146.53 −1.03653
$$253$$ 0 0
$$254$$ −3384.00 −0.835949
$$255$$ −2385.03 −0.585712
$$256$$ −5111.00 −1.24780
$$257$$ 1341.00 0.325484 0.162742 0.986669i $$-0.447966\pi$$
0.162742 + 0.986669i $$0.447966\pi$$
$$258$$ −1134.00 −0.273642
$$259$$ −3225.08 −0.773732
$$260$$ 12143.4 2.89655
$$261$$ −795.011 −0.188544
$$262$$ −13554.0 −3.19606
$$263$$ 956.092 0.224164 0.112082 0.993699i $$-0.464248\pi$$
0.112082 + 0.993699i $$0.464248\pi$$
$$264$$ 0 0
$$265$$ −405.000 −0.0938828
$$266$$ −18332.0 −4.22560
$$267$$ −135.000 −0.0309433
$$268$$ −7334.00 −1.67162
$$269$$ 3087.00 0.699694 0.349847 0.936807i $$-0.386234\pi$$
0.349847 + 0.936807i $$0.386234\pi$$
$$270$$ 1262.67 0.284605
$$271$$ 4035.68 0.904613 0.452306 0.891863i $$-0.350601\pi$$
0.452306 + 0.891863i $$0.350601\pi$$
$$272$$ 12808.5 2.85526
$$273$$ 5166.00 1.14528
$$274$$ −12439.6 −2.74271
$$275$$ 0 0
$$276$$ −5130.00 −1.11880
$$277$$ −3791.46 −0.822407 −0.411203 0.911544i $$-0.634891\pi$$
−0.411203 + 0.911544i $$0.634891\pi$$
$$278$$ −954.000 −0.205817
$$279$$ −1692.00 −0.363073
$$280$$ 12474.0 2.66237
$$281$$ −2722.78 −0.578034 −0.289017 0.957324i $$-0.593328\pi$$
−0.289017 + 0.957324i $$0.593328\pi$$
$$282$$ 1122.37 0.237007
$$283$$ −3238.94 −0.680335 −0.340167 0.940365i $$-0.610484\pi$$
−0.340167 + 0.940365i $$0.610484\pi$$
$$284$$ −3762.00 −0.786034
$$285$$ 3928.29 0.816463
$$286$$ 0 0
$$287$$ −882.000 −0.181404
$$288$$ −2665.63 −0.545394
$$289$$ 2890.00 0.588235
$$290$$ 4131.00 0.836485
$$291$$ −267.000 −0.0537863
$$292$$ 1447.99 0.290197
$$293$$ 4338.79 0.865101 0.432551 0.901610i $$-0.357614\pi$$
0.432551 + 0.901610i $$0.357614\pi$$
$$294$$ 3819.17 0.757614
$$295$$ 3402.00 0.671431
$$296$$ −7601.97 −1.49276
$$297$$ 0 0
$$298$$ 18765.0 3.64774
$$299$$ 6391.27 1.23618
$$300$$ 2508.00 0.482665
$$301$$ 1764.00 0.337792
$$302$$ −7056.00 −1.34446
$$303$$ −3865.94 −0.732978
$$304$$ −21096.4 −3.98013
$$305$$ 5611.84 1.05355
$$306$$ −4131.00 −0.771744
$$307$$ −5781.59 −1.07483 −0.537415 0.843318i $$-0.680599\pi$$
−0.537415 + 0.843318i $$0.680599\pi$$
$$308$$ 0 0
$$309$$ −4074.00 −0.750038
$$310$$ 8791.89 1.61079
$$311$$ 5220.00 0.951765 0.475883 0.879509i $$-0.342129\pi$$
0.475883 + 0.879509i $$0.342129\pi$$
$$312$$ 12177.0 2.20957
$$313$$ −3977.00 −0.718190 −0.359095 0.933301i $$-0.616915\pi$$
−0.359095 + 0.933301i $$0.616915\pi$$
$$314$$ −6786.18 −1.21964
$$315$$ −1964.15 −0.351324
$$316$$ −2895.99 −0.515545
$$317$$ 9918.00 1.75726 0.878628 0.477506i $$-0.158459\pi$$
0.878628 + 0.477506i $$0.158459\pi$$
$$318$$ −701.481 −0.123702
$$319$$ 0 0
$$320$$ 3411.00 0.595877
$$321$$ −3897.11 −0.677619
$$322$$ 11340.0 1.96259
$$323$$ −12852.0 −2.21395
$$324$$ 1539.00 0.263889
$$325$$ −3124.62 −0.533301
$$326$$ −13229.4 −2.24757
$$327$$ −3611.33 −0.610724
$$328$$ −2079.00 −0.349980
$$329$$ −1745.91 −0.292568
$$330$$ 0 0
$$331$$ 8756.00 1.45400 0.726999 0.686639i $$-0.240915\pi$$
0.726999 + 0.686639i $$0.240915\pi$$
$$332$$ 23694.5 3.91687
$$333$$ 1197.00 0.196983
$$334$$ 3510.00 0.575026
$$335$$ −3474.00 −0.566582
$$336$$ 10548.2 1.71265
$$337$$ 9919.45 1.60340 0.801702 0.597724i $$-0.203928\pi$$
0.801702 + 0.597724i $$0.203928\pi$$
$$338$$ −14788.2 −2.37981
$$339$$ −2133.00 −0.341736
$$340$$ 15105.2 2.40940
$$341$$ 0 0
$$342$$ 6804.00 1.07578
$$343$$ 2376.37 0.374088
$$344$$ 4158.00 0.651699
$$345$$ −2430.00 −0.379208
$$346$$ 16524.0 2.56744
$$347$$ −1330.22 −0.205792 −0.102896 0.994692i $$-0.532811\pi$$
−0.102896 + 0.994692i $$0.532811\pi$$
$$348$$ 5035.07 0.775598
$$349$$ 2842.30 0.435944 0.217972 0.975955i $$-0.430056\pi$$
0.217972 + 0.975955i $$0.430056\pi$$
$$350$$ −5544.00 −0.846684
$$351$$ −1917.38 −0.291573
$$352$$ 0 0
$$353$$ −1431.00 −0.215763 −0.107882 0.994164i $$-0.534407\pi$$
−0.107882 + 0.994164i $$0.534407\pi$$
$$354$$ 5892.44 0.884688
$$355$$ −1782.00 −0.266419
$$356$$ 855.000 0.127289
$$357$$ 6426.00 0.952661
$$358$$ 13281.4 1.96073
$$359$$ −5393.61 −0.792935 −0.396467 0.918049i $$-0.629764\pi$$
−0.396467 + 0.918049i $$0.629764\pi$$
$$360$$ −4629.77 −0.677807
$$361$$ 14309.0 2.08616
$$362$$ 4027.02 0.584683
$$363$$ 0 0
$$364$$ −32718.0 −4.71123
$$365$$ 685.892 0.0983595
$$366$$ 9720.00 1.38818
$$367$$ 3554.00 0.505497 0.252748 0.967532i $$-0.418666\pi$$
0.252748 + 0.967532i $$0.418666\pi$$
$$368$$ 13050.0 1.84858
$$369$$ 327.358 0.0461831
$$370$$ −6219.79 −0.873924
$$371$$ 1091.19 0.152700
$$372$$ 10716.0 1.49354
$$373$$ −5577.20 −0.774200 −0.387100 0.922038i $$-0.626523\pi$$
−0.387100 + 0.922038i $$0.626523\pi$$
$$374$$ 0 0
$$375$$ 4563.00 0.628353
$$376$$ −4115.35 −0.564450
$$377$$ −6273.00 −0.856965
$$378$$ −3402.00 −0.462910
$$379$$ −13186.0 −1.78712 −0.893561 0.448942i $$-0.851801\pi$$
−0.893561 + 0.448942i $$0.851801\pi$$
$$380$$ −24879.2 −3.35862
$$381$$ −1953.75 −0.262713
$$382$$ −13655.5 −1.82899
$$383$$ −3330.00 −0.444269 −0.222135 0.975016i $$-0.571302\pi$$
−0.222135 + 0.975016i $$0.571302\pi$$
$$384$$ −1200.31 −0.159513
$$385$$ 0 0
$$386$$ −14499.0 −1.91186
$$387$$ −654.715 −0.0859975
$$388$$ 1691.00 0.221256
$$389$$ 3537.00 0.461010 0.230505 0.973071i $$-0.425962\pi$$
0.230505 + 0.973071i $$0.425962\pi$$
$$390$$ 9963.00 1.29358
$$391$$ 7950.11 1.02827
$$392$$ −14003.6 −1.80431
$$393$$ −7825.41 −1.00443
$$394$$ 18657.0 2.38560
$$395$$ −1371.78 −0.174739
$$396$$ 0 0
$$397$$ −4501.00 −0.569014 −0.284507 0.958674i $$-0.591830\pi$$
−0.284507 + 0.958674i $$0.591830\pi$$
$$398$$ −8407.37 −1.05885
$$399$$ −10584.0 −1.32798
$$400$$ −6380.00 −0.797500
$$401$$ 3303.00 0.411332 0.205666 0.978622i $$-0.434064\pi$$
0.205666 + 0.978622i $$0.434064\pi$$
$$402$$ −6017.14 −0.746537
$$403$$ −13350.6 −1.65023
$$404$$ 24484.3 3.01519
$$405$$ 729.000 0.0894427
$$406$$ −11130.2 −1.36054
$$407$$ 0 0
$$408$$ 15147.0 1.83796
$$409$$ −9576.51 −1.15777 −0.578885 0.815409i $$-0.696512\pi$$
−0.578885 + 0.815409i $$0.696512\pi$$
$$410$$ −1701.00 −0.204894
$$411$$ −7182.00 −0.861951
$$412$$ 25802.0 3.08537
$$413$$ −9166.01 −1.09208
$$414$$ −4208.88 −0.499651
$$415$$ 11223.7 1.32759
$$416$$ −21033.0 −2.47891
$$417$$ −550.792 −0.0646820
$$418$$ 0 0
$$419$$ −6750.00 −0.787015 −0.393507 0.919322i $$-0.628738\pi$$
−0.393507 + 0.919322i $$0.628738\pi$$
$$420$$ 12439.6 1.44521
$$421$$ −13481.0 −1.56063 −0.780313 0.625389i $$-0.784940\pi$$
−0.780313 + 0.625389i $$0.784940\pi$$
$$422$$ −9666.00 −1.11501
$$423$$ 648.000 0.0744843
$$424$$ 2572.10 0.294604
$$425$$ −3886.72 −0.443609
$$426$$ −3086.51 −0.351038
$$427$$ −15120.0 −1.71360
$$428$$ 24681.7 2.78747
$$429$$ 0 0
$$430$$ 3402.00 0.381533
$$431$$ −1382.18 −0.154471 −0.0772356 0.997013i $$-0.524609\pi$$
−0.0772356 + 0.997013i $$0.524609\pi$$
$$432$$ −3915.00 −0.436020
$$433$$ −1531.00 −0.169920 −0.0849598 0.996384i $$-0.527076\pi$$
−0.0849598 + 0.996384i $$0.527076\pi$$
$$434$$ −23688.0 −2.61995
$$435$$ 2385.03 0.262882
$$436$$ 22871.7 2.51229
$$437$$ −13094.3 −1.43338
$$438$$ 1188.00 0.129600
$$439$$ 1919.11 0.208643 0.104321 0.994544i $$-0.466733\pi$$
0.104321 + 0.994544i $$0.466733\pi$$
$$440$$ 0 0
$$441$$ 2205.00 0.238095
$$442$$ −32595.5 −3.50771
$$443$$ 14598.0 1.56563 0.782813 0.622258i $$-0.213784\pi$$
0.782813 + 0.622258i $$0.213784\pi$$
$$444$$ −7581.00 −0.810312
$$445$$ 405.000 0.0431435
$$446$$ 5009.09 0.531810
$$447$$ 10834.0 1.14637
$$448$$ −9190.26 −0.969194
$$449$$ −12591.0 −1.32340 −0.661699 0.749769i $$-0.730164\pi$$
−0.661699 + 0.749769i $$0.730164\pi$$
$$450$$ 2057.68 0.215555
$$451$$ 0 0
$$452$$ 13509.0 1.40577
$$453$$ −4073.78 −0.422523
$$454$$ 14850.0 1.53512
$$455$$ −15498.0 −1.59683
$$456$$ −24948.0 −2.56206
$$457$$ −16778.4 −1.71742 −0.858708 0.512465i $$-0.828733\pi$$
−0.858708 + 0.512465i $$0.828733\pi$$
$$458$$ 28677.6 2.92580
$$459$$ −2385.03 −0.242536
$$460$$ 15390.0 1.55992
$$461$$ −14138.7 −1.42843 −0.714215 0.699926i $$-0.753216\pi$$
−0.714215 + 0.699926i $$0.753216\pi$$
$$462$$ 0 0
$$463$$ 6902.00 0.692793 0.346396 0.938088i $$-0.387405\pi$$
0.346396 + 0.938088i $$0.387405\pi$$
$$464$$ −12808.5 −1.28151
$$465$$ 5076.00 0.506223
$$466$$ 18765.0 1.86539
$$467$$ 15894.0 1.57492 0.787459 0.616367i $$-0.211396\pi$$
0.787459 + 0.616367i $$0.211396\pi$$
$$468$$ 12143.4 1.19942
$$469$$ 9360.00 0.921545
$$470$$ −3367.11 −0.330453
$$471$$ −3918.00 −0.383295
$$472$$ −21605.6 −2.10695
$$473$$ 0 0
$$474$$ −2376.00 −0.230239
$$475$$ 6401.66 0.618375
$$476$$ −40698.0 −3.91889
$$477$$ −405.000 −0.0388756
$$478$$ −16956.0 −1.62249
$$479$$ 9716.81 0.926873 0.463436 0.886130i $$-0.346616\pi$$
0.463436 + 0.886130i $$0.346616\pi$$
$$480$$ 7996.88 0.760429
$$481$$ 9444.87 0.895320
$$482$$ −17964.0 −1.69759
$$483$$ 6547.15 0.616782
$$484$$ 0 0
$$485$$ 801.000 0.0749929
$$486$$ 1262.67 0.117851
$$487$$ −7622.00 −0.709211 −0.354606 0.935016i $$-0.615385\pi$$
−0.354606 + 0.935016i $$0.615385\pi$$
$$488$$ −35640.0 −3.30604
$$489$$ −7638.00 −0.706344
$$490$$ −11457.5 −1.05632
$$491$$ −14902.6 −1.36974 −0.684871 0.728664i $$-0.740141\pi$$
−0.684871 + 0.728664i $$0.740141\pi$$
$$492$$ −2073.26 −0.189980
$$493$$ −7803.00 −0.712839
$$494$$ 53686.6 4.88963
$$495$$ 0 0
$$496$$ −27260.0 −2.46776
$$497$$ 4801.24 0.433331
$$498$$ 19440.0 1.74925
$$499$$ 13006.0 1.16679 0.583395 0.812188i $$-0.301724\pi$$
0.583395 + 0.812188i $$0.301724\pi$$
$$500$$ −28899.0 −2.58481
$$501$$ 2026.50 0.180713
$$502$$ −34887.0 −3.10176
$$503$$ 14185.5 1.25746 0.628728 0.777626i $$-0.283576\pi$$
0.628728 + 0.777626i $$0.283576\pi$$
$$504$$ 12474.0 1.10245
$$505$$ 11597.8 1.02197
$$506$$ 0 0
$$507$$ −8538.00 −0.747901
$$508$$ 12373.8 1.08070
$$509$$ 18234.0 1.58783 0.793917 0.608026i $$-0.208038\pi$$
0.793917 + 0.608026i $$0.208038\pi$$
$$510$$ 12393.0 1.07602
$$511$$ −1848.00 −0.159982
$$512$$ 23356.7 2.01607
$$513$$ 3928.29 0.338086
$$514$$ −6968.04 −0.597952
$$515$$ 12222.0 1.04576
$$516$$ 4146.53 0.353761
$$517$$ 0 0
$$518$$ 16758.0 1.42144
$$519$$ 9540.14 0.806870
$$520$$ −36531.0 −3.08075
$$521$$ 20502.0 1.72401 0.862005 0.506900i $$-0.169209\pi$$
0.862005 + 0.506900i $$0.169209\pi$$
$$522$$ 4131.00 0.346377
$$523$$ 10600.2 0.886257 0.443128 0.896458i $$-0.353869\pi$$
0.443128 + 0.896458i $$0.353869\pi$$
$$524$$ 49560.9 4.13183
$$525$$ −3200.83 −0.266087
$$526$$ −4968.00 −0.411816
$$527$$ −16606.9 −1.37269
$$528$$ 0 0
$$529$$ −4067.00 −0.334265
$$530$$ 2104.44 0.172474
$$531$$ 3402.00 0.278031
$$532$$ 67032.0 5.46279
$$533$$ 2583.00 0.209910
$$534$$ 701.481 0.0568465
$$535$$ 11691.3 0.944787
$$536$$ 22062.9 1.77793
$$537$$ 7668.00 0.616199
$$538$$ −16040.5 −1.28542
$$539$$ 0 0
$$540$$ −4617.00 −0.367933
$$541$$ −6082.96 −0.483414 −0.241707 0.970349i $$-0.577707\pi$$
−0.241707 + 0.970349i $$0.577707\pi$$
$$542$$ −20970.0 −1.66188
$$543$$ 2325.00 0.183748
$$544$$ −26163.0 −2.06200
$$545$$ 10834.0 0.851517
$$546$$ −26843.3 −2.10401
$$547$$ −19378.2 −1.51472 −0.757360 0.652998i $$-0.773511\pi$$
−0.757360 + 0.652998i $$0.773511\pi$$
$$548$$ 45486.0 3.54574
$$549$$ 5611.84 0.436262
$$550$$ 0 0
$$551$$ 12852.0 0.993673
$$552$$ 15432.6 1.18995
$$553$$ 3696.00 0.284213
$$554$$ 19701.0 1.51086
$$555$$ −3591.00 −0.274648
$$556$$ 3488.35 0.266077
$$557$$ 14216.7 1.08147 0.540736 0.841192i $$-0.318146\pi$$
0.540736 + 0.841192i $$0.318146\pi$$
$$558$$ 8791.89 0.667008
$$559$$ −5166.00 −0.390874
$$560$$ −31644.6 −2.38791
$$561$$ 0 0
$$562$$ 14148.0 1.06192
$$563$$ 2130.42 0.159479 0.0797394 0.996816i $$-0.474591\pi$$
0.0797394 + 0.996816i $$0.474591\pi$$
$$564$$ −4104.00 −0.306400
$$565$$ 6399.00 0.476474
$$566$$ 16830.0 1.24985
$$567$$ −1964.15 −0.145479
$$568$$ 11317.2 0.836021
$$569$$ −17251.2 −1.27102 −0.635509 0.772094i $$-0.719210\pi$$
−0.635509 + 0.772094i $$0.719210\pi$$
$$570$$ −20412.0 −1.49994
$$571$$ −9079.41 −0.665432 −0.332716 0.943027i $$-0.607965\pi$$
−0.332716 + 0.943027i $$0.607965\pi$$
$$572$$ 0 0
$$573$$ −7884.00 −0.574797
$$574$$ 4583.01 0.333260
$$575$$ −3960.00 −0.287206
$$576$$ 3411.00 0.246745
$$577$$ −7427.00 −0.535858 −0.267929 0.963439i $$-0.586339\pi$$
−0.267929 + 0.963439i $$0.586339\pi$$
$$578$$ −15016.9 −1.08066
$$579$$ −8371.00 −0.600841
$$580$$ −15105.2 −1.08140
$$581$$ −30240.0 −2.15932
$$582$$ 1387.37 0.0988118
$$583$$ 0 0
$$584$$ −4356.00 −0.308652
$$585$$ 5752.14 0.406533
$$586$$ −22545.0 −1.58929
$$587$$ −9972.00 −0.701173 −0.350586 0.936530i $$-0.614018\pi$$
−0.350586 + 0.936530i $$0.614018\pi$$
$$588$$ −13965.0 −0.979433
$$589$$ 27352.5 1.91348
$$590$$ −17677.3 −1.23350
$$591$$ 10771.6 0.749721
$$592$$ 19285.0 1.33887
$$593$$ 21922.6 1.51813 0.759066 0.651014i $$-0.225656\pi$$
0.759066 + 0.651014i $$0.225656\pi$$
$$594$$ 0 0
$$595$$ −19278.0 −1.32827
$$596$$ −68615.2 −4.71575
$$597$$ −4854.00 −0.332765
$$598$$ −33210.0 −2.27100
$$599$$ 576.000 0.0392900 0.0196450 0.999807i $$-0.493746\pi$$
0.0196450 + 0.999807i $$0.493746\pi$$
$$600$$ −7544.81 −0.513360
$$601$$ 7740.54 0.525363 0.262681 0.964883i $$-0.415393\pi$$
0.262681 + 0.964883i $$0.415393\pi$$
$$602$$ −9166.01 −0.620563
$$603$$ −3474.00 −0.234614
$$604$$ 25800.6 1.73810
$$605$$ 0 0
$$606$$ 20088.0 1.34657
$$607$$ 14500.7 0.969632 0.484816 0.874616i $$-0.338887\pi$$
0.484816 + 0.874616i $$0.338887\pi$$
$$608$$ 43092.0 2.87436
$$609$$ −6426.00 −0.427577
$$610$$ −29160.0 −1.93550
$$611$$ 5113.01 0.338544
$$612$$ 15105.2 0.997700
$$613$$ −5298.34 −0.349100 −0.174550 0.984648i $$-0.555847\pi$$
−0.174550 + 0.984648i $$0.555847\pi$$
$$614$$ 30042.0 1.97459
$$615$$ −982.073 −0.0643919
$$616$$ 0 0
$$617$$ 6939.00 0.452761 0.226381 0.974039i $$-0.427311\pi$$
0.226381 + 0.974039i $$0.427311\pi$$
$$618$$ 21169.1 1.37791
$$619$$ 1286.00 0.0835036 0.0417518 0.999128i $$-0.486706\pi$$
0.0417518 + 0.999128i $$0.486706\pi$$
$$620$$ −32148.0 −2.08241
$$621$$ −2430.00 −0.157025
$$622$$ −27123.9 −1.74850
$$623$$ −1091.19 −0.0701728
$$624$$ −30891.1 −1.98179
$$625$$ −8189.00 −0.524096
$$626$$ 20665.1 1.31940
$$627$$ 0 0
$$628$$ 24814.0 1.57673
$$629$$ 11748.5 0.744743
$$630$$ 10206.0 0.645423
$$631$$ −15110.0 −0.953280 −0.476640 0.879099i $$-0.658145\pi$$
−0.476640 + 0.879099i $$0.658145\pi$$
$$632$$ 8712.00 0.548330
$$633$$ −5580.67 −0.350413
$$634$$ −51535.4 −3.22829
$$635$$ 5861.26 0.366294
$$636$$ 2565.00 0.159920
$$637$$ 17398.5 1.08218
$$638$$ 0 0
$$639$$ −1782.00 −0.110321
$$640$$ 3600.93 0.222405
$$641$$ 13293.0 0.819098 0.409549 0.912288i $$-0.365686\pi$$
0.409549 + 0.912288i $$0.365686\pi$$
$$642$$ 20250.0 1.24487
$$643$$ 20528.0 1.25901 0.629506 0.776995i $$-0.283257\pi$$
0.629506 + 0.776995i $$0.283257\pi$$
$$644$$ −41465.3 −2.53721
$$645$$ 1964.15 0.119904
$$646$$ 66781.0 4.06728
$$647$$ −11124.0 −0.675934 −0.337967 0.941158i $$-0.609739\pi$$
−0.337967 + 0.941158i $$0.609739\pi$$
$$648$$ −4629.77 −0.280671
$$649$$ 0 0
$$650$$ 16236.0 0.979736
$$651$$ −13676.3 −0.823372
$$652$$ 48374.0 2.90563
$$653$$ 5562.00 0.333320 0.166660 0.986014i $$-0.446702\pi$$
0.166660 + 0.986014i $$0.446702\pi$$
$$654$$ 18765.0 1.12197
$$655$$ 23476.2 1.40045
$$656$$ 5274.09 0.313901
$$657$$ 685.892 0.0407294
$$658$$ 9072.00 0.537482
$$659$$ −16378.3 −0.968144 −0.484072 0.875028i $$-0.660843\pi$$
−0.484072 + 0.875028i $$0.660843\pi$$
$$660$$ 0 0
$$661$$ −28385.0 −1.67027 −0.835135 0.550045i $$-0.814611\pi$$
−0.835135 + 0.550045i $$0.814611\pi$$
$$662$$ −45497.5 −2.67116
$$663$$ −18819.0 −1.10237
$$664$$ −71280.0 −4.16596
$$665$$ 31752.0 1.85156
$$666$$ −6219.79 −0.361880
$$667$$ −7950.11 −0.461514
$$668$$ −12834.5 −0.743386
$$669$$ 2892.00 0.167132
$$670$$ 18051.4 1.04088
$$671$$ 0 0
$$672$$ −21546.0 −1.23684
$$673$$ 6948.99 0.398015 0.199007 0.979998i $$-0.436228\pi$$
0.199007 + 0.979998i $$0.436228\pi$$
$$674$$ −51543.0 −2.94564
$$675$$ 1188.00 0.0677424
$$676$$ 54074.0 3.07658
$$677$$ 3923.10 0.222713 0.111357 0.993781i $$-0.464480\pi$$
0.111357 + 0.993781i $$0.464480\pi$$
$$678$$ 11083.4 0.627810
$$679$$ −2158.14 −0.121976
$$680$$ −45441.0 −2.56262
$$681$$ 8573.65 0.482442
$$682$$ 0 0
$$683$$ 2034.00 0.113951 0.0569757 0.998376i $$-0.481854\pi$$
0.0569757 + 0.998376i $$0.481854\pi$$
$$684$$ −24879.2 −1.39076
$$685$$ 21546.0 1.20180
$$686$$ −12348.0 −0.687243
$$687$$ 16557.0 0.919489
$$688$$ −10548.2 −0.584514
$$689$$ −3195.63 −0.176697
$$690$$ 12626.7 0.696650
$$691$$ 4598.00 0.253135 0.126567 0.991958i $$-0.459604\pi$$
0.126567 + 0.991958i $$0.459604\pi$$
$$692$$ −60420.9 −3.31916
$$693$$ 0 0
$$694$$ 6912.00 0.378063
$$695$$ 1652.38 0.0901845
$$696$$ −15147.0 −0.824922
$$697$$ 3213.00 0.174607
$$698$$ −14769.0 −0.800881
$$699$$ 10834.0 0.586236
$$700$$ 20271.9 1.09458
$$701$$ 5949.59 0.320561 0.160280 0.987072i $$-0.448760\pi$$
0.160280 + 0.987072i $$0.448760\pi$$
$$702$$ 9963.00 0.535654
$$703$$ −19350.5 −1.03815
$$704$$ 0 0
$$705$$ −1944.00 −0.103851
$$706$$ 7435.69 0.396382
$$707$$ −31248.0 −1.66224
$$708$$ −21546.0 −1.14371
$$709$$ 24814.0 1.31440 0.657200 0.753716i $$-0.271741\pi$$
0.657200 + 0.753716i $$0.271741\pi$$
$$710$$ 9259.54 0.489443
$$711$$ −1371.78 −0.0723571
$$712$$ −2572.10 −0.135384
$$713$$ −16920.0 −0.888722
$$714$$ −33390.5 −1.75015
$$715$$ 0 0
$$716$$ −48564.0 −2.53481
$$717$$ −9789.55 −0.509899
$$718$$ 28026.0 1.45671
$$719$$ 26442.0 1.37152 0.685758 0.727829i $$-0.259471\pi$$
0.685758 + 0.727829i $$0.259471\pi$$
$$720$$ 11745.0 0.607931
$$721$$ −32929.7 −1.70093
$$722$$ −74351.7 −3.83253
$$723$$ −10371.5 −0.533501
$$724$$ −14725.0 −0.755871
$$725$$ 3886.72 0.199102
$$726$$ 0 0
$$727$$ −28370.0 −1.44730 −0.723649 0.690169i $$-0.757536\pi$$
−0.723649 + 0.690169i $$0.757536\pi$$
$$728$$ 98425.5 5.01084
$$729$$ 729.000 0.0370370
$$730$$ −3564.00 −0.180698
$$731$$ −6426.00 −0.325136
$$732$$ −35541.7 −1.79462
$$733$$ 20332.5 1.02456 0.512278 0.858820i $$-0.328802\pi$$
0.512278 + 0.858820i $$0.328802\pi$$
$$734$$ −18467.1 −0.928657
$$735$$ −6615.00 −0.331970
$$736$$ −26656.3 −1.33500
$$737$$ 0 0
$$738$$ −1701.00 −0.0848437
$$739$$ −9048.23 −0.450399 −0.225199 0.974313i $$-0.572303\pi$$
−0.225199 + 0.974313i $$0.572303\pi$$
$$740$$ 22743.0 1.12980
$$741$$ 30996.0 1.53666
$$742$$ −5670.00 −0.280529
$$743$$ −9758.37 −0.481830 −0.240915 0.970546i $$-0.577448\pi$$
−0.240915 + 0.970546i $$0.577448\pi$$
$$744$$ −32236.9 −1.58853
$$745$$ −32501.9 −1.59836
$$746$$ 28980.0 1.42230
$$747$$ 11223.7 0.549737
$$748$$ 0 0
$$749$$ −31500.0 −1.53670
$$750$$ −23710.0 −1.15436
$$751$$ −34010.0 −1.65252 −0.826260 0.563289i $$-0.809536\pi$$
−0.826260 + 0.563289i $$0.809536\pi$$
$$752$$ 10440.0 0.506260
$$753$$ −20142.0 −0.974788
$$754$$ 32595.5 1.57435
$$755$$ 12221.4 0.589113
$$756$$ 12439.6 0.598444
$$757$$ −5345.00 −0.256628 −0.128314 0.991734i $$-0.540957\pi$$
−0.128314 + 0.991734i $$0.540957\pi$$
$$758$$ 68516.5 3.28315
$$759$$ 0 0
$$760$$ 74844.0 3.57221
$$761$$ 22629.2 1.07794 0.538968 0.842326i $$-0.318814\pi$$
0.538968 + 0.842326i $$0.318814\pi$$
$$762$$ 10152.0 0.482635
$$763$$ −29190.0 −1.38499
$$764$$ 49932.0 2.36450
$$765$$ 7155.10 0.338161
$$766$$ 17303.2 0.816174
$$767$$ 26843.3 1.26370
$$768$$ 15333.0 0.720419
$$769$$ 34961.4 1.63946 0.819728 0.572753i $$-0.194124\pi$$
0.819728 + 0.572753i $$0.194124\pi$$
$$770$$ 0 0
$$771$$ −4023.00 −0.187918
$$772$$ 53016.3 2.47163
$$773$$ 29610.0 1.37775 0.688873 0.724882i $$-0.258106\pi$$
0.688873 + 0.724882i $$0.258106\pi$$
$$774$$ 3402.00 0.157988
$$775$$ 8272.00 0.383405
$$776$$ −5087.03 −0.235327
$$777$$ 9675.24 0.446714
$$778$$ −18378.8 −0.846930
$$779$$ −5292.00 −0.243396
$$780$$ −36430.2 −1.67232
$$781$$ 0 0
$$782$$ −41310.0 −1.88906
$$783$$ 2385.03 0.108856
$$784$$ 35525.0 1.61830
$$785$$ 11754.0 0.534418
$$786$$ 40662.0 1.84525
$$787$$ 5785.05 0.262026 0.131013 0.991381i $$-0.458177\pi$$
0.131013 + 0.991381i $$0.458177\pi$$
$$788$$ −68220.3 −3.08407
$$789$$ −2868.28 −0.129421
$$790$$ 7128.00 0.321016
$$791$$ −17240.8 −0.774985
$$792$$ 0 0
$$793$$ 44280.0 1.98289
$$794$$ 23387.9 1.04535
$$795$$ 1215.00 0.0542033
$$796$$ 30742.0 1.36887
$$797$$ −4626.00 −0.205598 −0.102799 0.994702i $$-0.532780\pi$$
−0.102799 + 0.994702i $$0.532780\pi$$
$$798$$ 54996.1 2.43965
$$799$$ 6360.09 0.281607
$$800$$ 13032.0 0.575936
$$801$$ 405.000 0.0178651
$$802$$ −17162.9 −0.755664
$$803$$ 0 0
$$804$$ 22002.0 0.965113
$$805$$ −19641.5 −0.859963
$$806$$ 69372.0 3.03167
$$807$$ −9261.00 −0.403969
$$808$$ −73656.0 −3.20694
$$809$$ −31239.3 −1.35762 −0.678810 0.734314i $$-0.737504\pi$$
−0.678810 + 0.734314i $$0.737504\pi$$
$$810$$ −3788.00 −0.164317
$$811$$ −44340.5 −1.91986 −0.959929 0.280242i $$-0.909585\pi$$
−0.959929 + 0.280242i $$0.909585\pi$$
$$812$$ 40698.0 1.75889
$$813$$ −12107.0 −0.522278
$$814$$ 0 0
$$815$$ 22914.0 0.984837
$$816$$ −38425.5 −1.64848
$$817$$ 10584.0 0.453228
$$818$$ 49761.0 2.12696
$$819$$ −15498.0 −0.661226
$$820$$ 6219.79 0.264884
$$821$$ 38493.1 1.63632 0.818160 0.574991i $$-0.194994\pi$$
0.818160 + 0.574991i $$0.194994\pi$$
$$822$$ 37318.8 1.58351
$$823$$ 8678.00 0.367553 0.183776 0.982968i $$-0.441168\pi$$
0.183776 + 0.982968i $$0.441168\pi$$
$$824$$ −77620.1 −3.28158
$$825$$ 0 0
$$826$$ 47628.0 2.00628
$$827$$ 27238.2 1.14530 0.572652 0.819799i $$-0.305915\pi$$
0.572652 + 0.819799i $$0.305915\pi$$
$$828$$ 15390.0 0.645941
$$829$$ −19789.0 −0.829072 −0.414536 0.910033i $$-0.636056\pi$$
−0.414536 + 0.910033i $$0.636056\pi$$
$$830$$ −58320.0 −2.43894
$$831$$ 11374.4 0.474817
$$832$$ 26914.3 1.12150
$$833$$ 21642.0 0.900180
$$834$$ 2862.00 0.118828
$$835$$ −6079.50 −0.251964
$$836$$ 0 0
$$837$$ 5076.00 0.209620
$$838$$ 35074.0 1.44584
$$839$$ −1800.00 −0.0740678 −0.0370339 0.999314i $$-0.511791\pi$$
−0.0370339 + 0.999314i $$0.511791\pi$$
$$840$$ −37422.0 −1.53712
$$841$$ −16586.0 −0.680061
$$842$$ 70049.3 2.86705
$$843$$ 8168.35 0.333728
$$844$$ 35344.2 1.44147
$$845$$ 25614.0 1.04278
$$846$$ −3367.11 −0.136836
$$847$$ 0 0
$$848$$ −6525.00 −0.264233
$$849$$ 9716.81 0.392791
$$850$$ 20196.0 0.814961
$$851$$ 11970.0 0.482170
$$852$$ 11286.0 0.453817
$$853$$ −33513.5 −1.34523 −0.672614 0.739994i $$-0.734828\pi$$
−0.672614 + 0.739994i $$0.734828\pi$$
$$854$$ 78565.8 3.14809
$$855$$ −11784.9 −0.471385
$$856$$ −74250.0 −2.96473
$$857$$ −26687.4 −1.06374 −0.531870 0.846826i $$-0.678511\pi$$
−0.531870 + 0.846826i $$0.678511\pi$$
$$858$$ 0 0
$$859$$ 46694.0 1.85469 0.927345 0.374207i $$-0.122085\pi$$
0.927345 + 0.374207i $$0.122085\pi$$
$$860$$ −12439.6 −0.493240
$$861$$ 2646.00 0.104733
$$862$$ 7182.00 0.283782
$$863$$ −36018.0 −1.42070 −0.710352 0.703847i $$-0.751464\pi$$
−0.710352 + 0.703847i $$0.751464\pi$$
$$864$$ 7996.88 0.314883
$$865$$ −28620.4 −1.12500
$$866$$ 7955.31 0.312162
$$867$$ −8670.00 −0.339618
$$868$$ 86616.4 3.38704
$$869$$ 0 0
$$870$$ −12393.0 −0.482945
$$871$$ −27411.4 −1.06636
$$872$$ −68805.0 −2.67205
$$873$$ 801.000 0.0310535
$$874$$ 68040.0 2.63328
$$875$$ 36882.3 1.42497
$$876$$ −4343.98 −0.167545
$$877$$ 9191.99 0.353924 0.176962 0.984218i $$-0.443373\pi$$
0.176962 + 0.984218i $$0.443373\pi$$
$$878$$ −9972.00 −0.383301
$$879$$ −13016.4 −0.499466
$$880$$ 0 0
$$881$$ −40005.0 −1.52986 −0.764928 0.644116i $$-0.777225\pi$$
−0.764928 + 0.644116i $$0.777225\pi$$
$$882$$ −11457.5 −0.437409
$$883$$ 4492.00 0.171198 0.0855990 0.996330i $$-0.472720\pi$$
0.0855990 + 0.996330i $$0.472720\pi$$
$$884$$ 119187. 4.53472
$$885$$ −10206.0 −0.387651
$$886$$ −75853.4 −2.87624
$$887$$ −43554.1 −1.64871 −0.824355 0.566074i $$-0.808462\pi$$
−0.824355 + 0.566074i $$0.808462\pi$$
$$888$$ 22805.9 0.861843
$$889$$ −15792.0 −0.595778
$$890$$ −2104.44 −0.0792596
$$891$$ 0 0
$$892$$ −18316.0 −0.687517
$$893$$ −10475.4 −0.392550
$$894$$ −56295.0 −2.10603
$$895$$ −23004.0 −0.859150
$$896$$ −9702.00 −0.361742
$$897$$ −19173.8 −0.713706
$$898$$ 65424.8 2.43124
$$899$$ 16606.9 0.616097
$$900$$ −7524.00 −0.278667
$$901$$ −3975.06 −0.146979
$$902$$ 0 0
$$903$$ −5292.00 −0.195024
$$904$$ −40639.1 −1.49517
$$905$$ −6975.00 −0.256195
$$906$$ 21168.0 0.776225
$$907$$ 7634.00 0.279474 0.139737 0.990189i $$-0.455374\pi$$
0.139737 + 0.990189i $$0.455374\pi$$
$$908$$ −54299.8 −1.98458
$$909$$ 11597.8 0.423185
$$910$$ 80530.0 2.93356
$$911$$ −43830.0 −1.59402 −0.797010 0.603966i $$-0.793586\pi$$
−0.797010 + 0.603966i $$0.793586\pi$$
$$912$$ 63289.1 2.29793
$$913$$ 0 0
$$914$$ 87183.0 3.15510
$$915$$ −16835.5 −0.608268
$$916$$ −104861. −3.78243
$$917$$ −63252.0 −2.27782
$$918$$ 12393.0 0.445566
$$919$$ 32531.4 1.16769 0.583847 0.811864i $$-0.301547\pi$$
0.583847 + 0.811864i $$0.301547\pi$$
$$920$$ −46297.7 −1.65912
$$921$$ 17344.8 0.620553
$$922$$ 73467.0 2.62419
$$923$$ −14060.8 −0.501426
$$924$$ 0 0
$$925$$ −5852.00 −0.208014
$$926$$ −35863.8 −1.27274
$$927$$ 12222.0 0.433035
$$928$$ 26163.0 0.925477
$$929$$ 15651.0 0.552737 0.276368 0.961052i $$-0.410869\pi$$
0.276368 + 0.961052i $$0.410869\pi$$
$$930$$ −26375.7 −0.929992
$$931$$ −35645.6 −1.25482
$$932$$ −68615.2 −2.41155
$$933$$ −15660.0 −0.549502
$$934$$ −82587.6 −2.89331
$$935$$ 0 0
$$936$$ −36531.0 −1.27570
$$937$$ 6337.57 0.220960 0.110480 0.993878i $$-0.464761\pi$$
0.110480 + 0.993878i $$0.464761\pi$$
$$938$$ −48636.0 −1.69299
$$939$$ 11931.0 0.414647
$$940$$ 12312.0 0.427205
$$941$$ −9846.71 −0.341120 −0.170560 0.985347i $$-0.554558\pi$$
−0.170560 + 0.985347i $$0.554558\pi$$
$$942$$ 20358.5 0.704158
$$943$$ 3273.58 0.113046
$$944$$ 54810.0 1.88974
$$945$$ 5892.44 0.202837
$$946$$ 0 0
$$947$$ −738.000 −0.0253239 −0.0126620 0.999920i $$-0.504031\pi$$
−0.0126620 + 0.999920i $$0.504031\pi$$
$$948$$ 8687.97 0.297650
$$949$$ 5412.00 0.185122
$$950$$ −33264.0 −1.13603
$$951$$ −29754.0 −1.01455
$$952$$ 122432. 4.16810
$$953$$ 15541.7 0.528274 0.264137 0.964485i $$-0.414913\pi$$
0.264137 + 0.964485i $$0.414913\pi$$
$$954$$ 2104.44 0.0714191
$$955$$ 23652.0 0.801425
$$956$$ 62000.5 2.09753
$$957$$ 0 0
$$958$$ −50490.0 −1.70277
$$959$$ −58051.4 −1.95472
$$960$$ −10233.0 −0.344030
$$961$$ 5553.00 0.186399
$$962$$ −49077.0 −1.64481
$$963$$ 11691.3 0.391224
$$964$$ 65686.3 2.19462
$$965$$ 25113.0 0.837737
$$966$$ −34020.0 −1.13310
$$967$$ 37692.9 1.25349 0.626743 0.779226i $$-0.284387\pi$$
0.626743 + 0.779226i $$0.284387\pi$$
$$968$$ 0 0
$$969$$ 38556.0 1.27822
$$970$$ −4162.12 −0.137771
$$971$$ −12402.0 −0.409886 −0.204943 0.978774i $$-0.565701\pi$$
−0.204943 + 0.978774i $$0.565701\pi$$
$$972$$ −4617.00 −0.152356
$$973$$ −4452.00 −0.146685
$$974$$ 39605.1 1.30290
$$975$$ 9373.86 0.307901
$$976$$ 90413.1 2.96522
$$977$$ 31203.0 1.02177 0.510887 0.859648i $$-0.329317\pi$$
0.510887 + 0.859648i $$0.329317\pi$$
$$978$$ 39688.2 1.29764
$$979$$ 0 0
$$980$$ 41895.0 1.36560
$$981$$ 10834.0 0.352602
$$982$$ 77436.0 2.51638
$$983$$ −36540.0 −1.18560 −0.592800 0.805350i $$-0.701978\pi$$
−0.592800 + 0.805350i $$0.701978\pi$$
$$984$$ 6237.00 0.202061
$$985$$ −32314.9 −1.04532
$$986$$ 40545.6 1.30957
$$987$$ 5237.72 0.168914
$$988$$ −196308. −6.32124
$$989$$ −6547.15 −0.210503
$$990$$ 0 0
$$991$$ −56888.0 −1.82352 −0.911759 0.410725i $$-0.865276\pi$$
−0.911759 + 0.410725i $$0.865276\pi$$
$$992$$ 55682.0 1.78216
$$993$$ −26268.0 −0.839466
$$994$$ −24948.0 −0.796079
$$995$$ 14562.0 0.463966
$$996$$ −71083.4 −2.26141
$$997$$ −15711.4 −0.499083 −0.249542 0.968364i $$-0.580280\pi$$
−0.249542 + 0.968364i $$0.580280\pi$$
$$998$$ −67581.2 −2.14353
$$999$$ −3591.00 −0.113728
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 363.4.a.l.1.1 2
3.2 odd 2 1089.4.a.s.1.2 2
11.10 odd 2 inner 363.4.a.l.1.2 yes 2
33.32 even 2 1089.4.a.s.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
363.4.a.l.1.1 2 1.1 even 1 trivial
363.4.a.l.1.2 yes 2 11.10 odd 2 inner
1089.4.a.s.1.1 2 33.32 even 2
1089.4.a.s.1.2 2 3.2 odd 2