# Properties

 Label 525.4.d.h.274.4 Level $525$ Weight $4$ Character 525.274 Analytic conductor $30.976$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,4,Mod(274,525)]

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, 1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("525.274");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 525.d (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$30.9760027530$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{65})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 33x^{2} + 256$$ x^4 + 33*x^2 + 256 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 274.4 Root $$4.53113i$$ of defining polynomial Character $$\chi$$ $$=$$ 525.274 Dual form 525.4.d.h.274.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+4.53113i q^{2} -3.00000i q^{3} -12.5311 q^{4} +13.5934 q^{6} -7.00000i q^{7} -20.5311i q^{8} -9.00000 q^{9} +O(q^{10})$$ $$q+4.53113i q^{2} -3.00000i q^{3} -12.5311 q^{4} +13.5934 q^{6} -7.00000i q^{7} -20.5311i q^{8} -9.00000 q^{9} -19.0623 q^{11} +37.5934i q^{12} +2.93774i q^{13} +31.7179 q^{14} -7.21984 q^{16} -6.49806i q^{17} -40.7802i q^{18} +5.43580 q^{19} -21.0000 q^{21} -86.3735i q^{22} -49.3774i q^{23} -61.5934 q^{24} -13.3113 q^{26} +27.0000i q^{27} +87.7179i q^{28} +291.494 q^{29} +244.307 q^{31} -196.963i q^{32} +57.1868i q^{33} +29.4436 q^{34} +112.780 q^{36} -193.121i q^{37} +24.6303i q^{38} +8.81323 q^{39} +315.113 q^{41} -95.1537i q^{42} +300.996i q^{43} +238.872 q^{44} +223.735 q^{46} +86.5058i q^{47} +21.6595i q^{48} -49.0000 q^{49} -19.4942 q^{51} -36.8132i q^{52} -509.677i q^{53} -122.340 q^{54} -143.718 q^{56} -16.3074i q^{57} +1320.80i q^{58} +83.3852 q^{59} -5.25291 q^{61} +1106.99i q^{62} +63.0000i q^{63} +834.706 q^{64} -259.121 q^{66} +205.992i q^{67} +81.4281i q^{68} -148.132 q^{69} +1004.31 q^{71} +184.780i q^{72} +1007.29i q^{73} +875.055 q^{74} -68.1168 q^{76} +133.436i q^{77} +39.9339i q^{78} +863.237 q^{79} +81.0000 q^{81} +1427.82i q^{82} -1334.72i q^{83} +263.154 q^{84} -1363.85 q^{86} -874.483i q^{87} +391.370i q^{88} -326.249 q^{89} +20.5642 q^{91} +618.755i q^{92} -732.922i q^{93} -391.969 q^{94} -590.889 q^{96} +1526.77i q^{97} -222.025i q^{98} +171.560 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 34 q^{4} + 6 q^{6} - 36 q^{9}+O(q^{10})$$ 4 * q - 34 * q^4 + 6 * q^6 - 36 * q^9 $$4 q - 34 q^{4} + 6 q^{6} - 36 q^{9} - 44 q^{11} + 14 q^{14} - 174 q^{16} - 204 q^{19} - 84 q^{21} - 198 q^{24} + 108 q^{26} + 392 q^{29} + 300 q^{31} + 924 q^{34} + 306 q^{36} + 132 q^{39} - 352 q^{41} + 504 q^{44} - 1040 q^{46} - 196 q^{49} + 696 q^{51} - 54 q^{54} - 462 q^{56} + 1688 q^{59} - 408 q^{61} + 1678 q^{64} - 456 q^{66} - 1560 q^{69} + 3340 q^{71} + 2436 q^{74} + 824 q^{76} + 1776 q^{79} + 324 q^{81} + 714 q^{84} - 2424 q^{86} - 1176 q^{89} + 308 q^{91} + 2560 q^{94} - 90 q^{96} + 396 q^{99}+O(q^{100})$$ 4 * q - 34 * q^4 + 6 * q^6 - 36 * q^9 - 44 * q^11 + 14 * q^14 - 174 * q^16 - 204 * q^19 - 84 * q^21 - 198 * q^24 + 108 * q^26 + 392 * q^29 + 300 * q^31 + 924 * q^34 + 306 * q^36 + 132 * q^39 - 352 * q^41 + 504 * q^44 - 1040 * q^46 - 196 * q^49 + 696 * q^51 - 54 * q^54 - 462 * q^56 + 1688 * q^59 - 408 * q^61 + 1678 * q^64 - 456 * q^66 - 1560 * q^69 + 3340 * q^71 + 2436 * q^74 + 824 * q^76 + 1776 * q^79 + 324 * q^81 + 714 * q^84 - 2424 * q^86 - 1176 * q^89 + 308 * q^91 + 2560 * q^94 - 90 * q^96 + 396 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.53113i 1.60200i 0.598667 + 0.800998i $$0.295697\pi$$
−0.598667 + 0.800998i $$0.704303\pi$$
$$3$$ − 3.00000i − 0.577350i
$$4$$ −12.5311 −1.56639
$$5$$ 0 0
$$6$$ 13.5934 0.924913
$$7$$ − 7.00000i − 0.377964i
$$8$$ − 20.5311i − 0.907356i
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ −19.0623 −0.522499 −0.261249 0.965271i $$-0.584135\pi$$
−0.261249 + 0.965271i $$0.584135\pi$$
$$12$$ 37.5934i 0.904356i
$$13$$ 2.93774i 0.0626756i 0.999509 + 0.0313378i $$0.00997677\pi$$
−0.999509 + 0.0313378i $$0.990023\pi$$
$$14$$ 31.7179 0.605498
$$15$$ 0 0
$$16$$ −7.21984 −0.112810
$$17$$ − 6.49806i − 0.0927066i −0.998925 0.0463533i $$-0.985240\pi$$
0.998925 0.0463533i $$-0.0147600\pi$$
$$18$$ − 40.7802i − 0.533999i
$$19$$ 5.43580 0.0656347 0.0328173 0.999461i $$-0.489552\pi$$
0.0328173 + 0.999461i $$0.489552\pi$$
$$20$$ 0 0
$$21$$ −21.0000 −0.218218
$$22$$ − 86.3735i − 0.837041i
$$23$$ − 49.3774i − 0.447648i −0.974630 0.223824i $$-0.928146\pi$$
0.974630 0.223824i $$-0.0718541\pi$$
$$24$$ −61.5934 −0.523862
$$25$$ 0 0
$$26$$ −13.3113 −0.100406
$$27$$ 27.0000i 0.192450i
$$28$$ 87.7179i 0.592040i
$$29$$ 291.494 1.86652 0.933261 0.359200i $$-0.116950\pi$$
0.933261 + 0.359200i $$0.116950\pi$$
$$30$$ 0 0
$$31$$ 244.307 1.41545 0.707724 0.706489i $$-0.249722\pi$$
0.707724 + 0.706489i $$0.249722\pi$$
$$32$$ − 196.963i − 1.08808i
$$33$$ 57.1868i 0.301665i
$$34$$ 29.4436 0.148516
$$35$$ 0 0
$$36$$ 112.780 0.522130
$$37$$ − 193.121i − 0.858077i −0.903286 0.429038i $$-0.858853\pi$$
0.903286 0.429038i $$-0.141147\pi$$
$$38$$ 24.6303i 0.105147i
$$39$$ 8.81323 0.0361858
$$40$$ 0 0
$$41$$ 315.113 1.20030 0.600151 0.799887i $$-0.295107\pi$$
0.600151 + 0.799887i $$0.295107\pi$$
$$42$$ − 95.1537i − 0.349584i
$$43$$ 300.996i 1.06748i 0.845650 + 0.533738i $$0.179213\pi$$
−0.845650 + 0.533738i $$0.820787\pi$$
$$44$$ 238.872 0.818437
$$45$$ 0 0
$$46$$ 223.735 0.717130
$$47$$ 86.5058i 0.268472i 0.990949 + 0.134236i $$0.0428580\pi$$
−0.990949 + 0.134236i $$0.957142\pi$$
$$48$$ 21.6595i 0.0651309i
$$49$$ −49.0000 −0.142857
$$50$$ 0 0
$$51$$ −19.4942 −0.0535242
$$52$$ − 36.8132i − 0.0981745i
$$53$$ − 509.677i − 1.32093i −0.750855 0.660467i $$-0.770358\pi$$
0.750855 0.660467i $$-0.229642\pi$$
$$54$$ −122.340 −0.308304
$$55$$ 0 0
$$56$$ −143.718 −0.342948
$$57$$ − 16.3074i − 0.0378942i
$$58$$ 1320.80i 2.99016i
$$59$$ 83.3852 0.183997 0.0919985 0.995759i $$-0.470674\pi$$
0.0919985 + 0.995759i $$0.470674\pi$$
$$60$$ 0 0
$$61$$ −5.25291 −0.0110257 −0.00551283 0.999985i $$-0.501755\pi$$
−0.00551283 + 0.999985i $$0.501755\pi$$
$$62$$ 1106.99i 2.26754i
$$63$$ 63.0000i 0.125988i
$$64$$ 834.706 1.63029
$$65$$ 0 0
$$66$$ −259.121 −0.483266
$$67$$ 205.992i 0.375611i 0.982206 + 0.187806i $$0.0601375\pi$$
−0.982206 + 0.187806i $$0.939862\pi$$
$$68$$ 81.4281i 0.145215i
$$69$$ −148.132 −0.258450
$$70$$ 0 0
$$71$$ 1004.31 1.67872 0.839362 0.543573i $$-0.182929\pi$$
0.839362 + 0.543573i $$0.182929\pi$$
$$72$$ 184.780i 0.302452i
$$73$$ 1007.29i 1.61499i 0.589876 + 0.807494i $$0.299177\pi$$
−0.589876 + 0.807494i $$0.700823\pi$$
$$74$$ 875.055 1.37464
$$75$$ 0 0
$$76$$ −68.1168 −0.102810
$$77$$ 133.436i 0.197486i
$$78$$ 39.9339i 0.0579695i
$$79$$ 863.237 1.22939 0.614695 0.788765i $$-0.289279\pi$$
0.614695 + 0.788765i $$0.289279\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 1427.82i 1.92288i
$$83$$ − 1334.72i − 1.76512i −0.470200 0.882560i $$-0.655818\pi$$
0.470200 0.882560i $$-0.344182\pi$$
$$84$$ 263.154 0.341815
$$85$$ 0 0
$$86$$ −1363.85 −1.71009
$$87$$ − 874.483i − 1.07764i
$$88$$ 391.370i 0.474093i
$$89$$ −326.249 −0.388565 −0.194283 0.980946i $$-0.562238\pi$$
−0.194283 + 0.980946i $$0.562238\pi$$
$$90$$ 0 0
$$91$$ 20.5642 0.0236892
$$92$$ 618.755i 0.701192i
$$93$$ − 732.922i − 0.817210i
$$94$$ −391.969 −0.430091
$$95$$ 0 0
$$96$$ −590.889 −0.628202
$$97$$ 1526.77i 1.59815i 0.601232 + 0.799075i $$0.294677\pi$$
−0.601232 + 0.799075i $$0.705323\pi$$
$$98$$ − 222.025i − 0.228857i
$$99$$ 171.560 0.174166
$$100$$ 0 0
$$101$$ 96.8716 0.0954365 0.0477182 0.998861i $$-0.484805\pi$$
0.0477182 + 0.998861i $$0.484805\pi$$
$$102$$ − 88.3307i − 0.0857455i
$$103$$ − 1321.99i − 1.26466i −0.774700 0.632329i $$-0.782099\pi$$
0.774700 0.632329i $$-0.217901\pi$$
$$104$$ 60.3152 0.0568691
$$105$$ 0 0
$$106$$ 2309.41 2.11613
$$107$$ 1745.71i 1.57724i 0.614883 + 0.788619i $$0.289203\pi$$
−0.614883 + 0.788619i $$0.710797\pi$$
$$108$$ − 338.340i − 0.301452i
$$109$$ −476.856 −0.419032 −0.209516 0.977805i $$-0.567189\pi$$
−0.209516 + 0.977805i $$0.567189\pi$$
$$110$$ 0 0
$$111$$ −579.362 −0.495411
$$112$$ 50.5389i 0.0426382i
$$113$$ − 1641.65i − 1.36666i −0.730108 0.683332i $$-0.760530\pi$$
0.730108 0.683332i $$-0.239470\pi$$
$$114$$ 73.8910 0.0607064
$$115$$ 0 0
$$116$$ −3652.75 −2.92370
$$117$$ − 26.4397i − 0.0208919i
$$118$$ 377.829i 0.294763i
$$119$$ −45.4864 −0.0350398
$$120$$ 0 0
$$121$$ −967.630 −0.726995
$$122$$ − 23.8016i − 0.0176631i
$$123$$ − 945.339i − 0.692994i
$$124$$ −3061.45 −2.21715
$$125$$ 0 0
$$126$$ −285.461 −0.201833
$$127$$ − 844.016i − 0.589719i −0.955541 0.294859i $$-0.904727\pi$$
0.955541 0.294859i $$-0.0952728\pi$$
$$128$$ 2206.46i 1.52363i
$$129$$ 902.988 0.616308
$$130$$ 0 0
$$131$$ −2796.20 −1.86493 −0.932463 0.361265i $$-0.882345\pi$$
−0.932463 + 0.361265i $$0.882345\pi$$
$$132$$ − 716.615i − 0.472525i
$$133$$ − 38.0506i − 0.0248076i
$$134$$ −933.377 −0.601728
$$135$$ 0 0
$$136$$ −133.413 −0.0841179
$$137$$ − 2057.13i − 1.28287i −0.767179 0.641433i $$-0.778340\pi$$
0.767179 0.641433i $$-0.221660\pi$$
$$138$$ − 671.206i − 0.414035i
$$139$$ 1745.12 1.06489 0.532444 0.846465i $$-0.321274\pi$$
0.532444 + 0.846465i $$0.321274\pi$$
$$140$$ 0 0
$$141$$ 259.517 0.155002
$$142$$ 4550.65i 2.68931i
$$143$$ − 56.0000i − 0.0327479i
$$144$$ 64.9786 0.0376033
$$145$$ 0 0
$$146$$ −4564.15 −2.58720
$$147$$ 147.000i 0.0824786i
$$148$$ 2420.02i 1.34408i
$$149$$ 1173.57 0.645254 0.322627 0.946526i $$-0.395434\pi$$
0.322627 + 0.946526i $$0.395434\pi$$
$$150$$ 0 0
$$151$$ 1540.07 0.829994 0.414997 0.909823i $$-0.363783\pi$$
0.414997 + 0.909823i $$0.363783\pi$$
$$152$$ − 111.603i − 0.0595540i
$$153$$ 58.4826i 0.0309022i
$$154$$ −604.615 −0.316372
$$155$$ 0 0
$$156$$ −110.440 −0.0566811
$$157$$ − 2544.53i − 1.29348i −0.762712 0.646738i $$-0.776133\pi$$
0.762712 0.646738i $$-0.223867\pi$$
$$158$$ 3911.44i 1.96948i
$$159$$ −1529.03 −0.762642
$$160$$ 0 0
$$161$$ −345.642 −0.169195
$$162$$ 367.021i 0.178000i
$$163$$ 594.708i 0.285774i 0.989739 + 0.142887i $$0.0456385\pi$$
−0.989739 + 0.142887i $$0.954361\pi$$
$$164$$ −3948.72 −1.88014
$$165$$ 0 0
$$166$$ 6047.81 2.82772
$$167$$ 928.498i 0.430236i 0.976588 + 0.215118i $$0.0690135\pi$$
−0.976588 + 0.215118i $$0.930986\pi$$
$$168$$ 431.154i 0.198001i
$$169$$ 2188.37 0.996072
$$170$$ 0 0
$$171$$ −48.9222 −0.0218782
$$172$$ − 3771.82i − 1.67209i
$$173$$ 315.642i 0.138716i 0.997592 + 0.0693578i $$0.0220950\pi$$
−0.997592 + 0.0693578i $$0.977905\pi$$
$$174$$ 3962.39 1.72637
$$175$$ 0 0
$$176$$ 137.626 0.0589431
$$177$$ − 250.156i − 0.106231i
$$178$$ − 1478.28i − 0.622480i
$$179$$ 1445.49 0.603581 0.301791 0.953374i $$-0.402416\pi$$
0.301791 + 0.953374i $$0.402416\pi$$
$$180$$ 0 0
$$181$$ −1843.81 −0.757180 −0.378590 0.925564i $$-0.623591\pi$$
−0.378590 + 0.925564i $$0.623591\pi$$
$$182$$ 93.1790i 0.0379499i
$$183$$ 15.7587i 0.00636567i
$$184$$ −1013.77 −0.406176
$$185$$ 0 0
$$186$$ 3320.97 1.30917
$$187$$ 123.868i 0.0484391i
$$188$$ − 1084.02i − 0.420532i
$$189$$ 189.000 0.0727393
$$190$$ 0 0
$$191$$ −244.074 −0.0924637 −0.0462318 0.998931i $$-0.514721\pi$$
−0.0462318 + 0.998931i $$0.514721\pi$$
$$192$$ − 2504.12i − 0.941246i
$$193$$ 1733.03i 0.646355i 0.946338 + 0.323178i $$0.104751\pi$$
−0.946338 + 0.323178i $$0.895249\pi$$
$$194$$ −6918.01 −2.56023
$$195$$ 0 0
$$196$$ 614.025 0.223770
$$197$$ 358.230i 0.129557i 0.997900 + 0.0647787i $$0.0206342\pi$$
−0.997900 + 0.0647787i $$0.979366\pi$$
$$198$$ 777.362i 0.279014i
$$199$$ 3203.63 1.14120 0.570601 0.821227i $$-0.306710\pi$$
0.570601 + 0.821227i $$0.306710\pi$$
$$200$$ 0 0
$$201$$ 617.977 0.216859
$$202$$ 438.938i 0.152889i
$$203$$ − 2040.46i − 0.705479i
$$204$$ 244.284 0.0838398
$$205$$ 0 0
$$206$$ 5990.12 2.02598
$$207$$ 444.397i 0.149216i
$$208$$ − 21.2100i − 0.00707044i
$$209$$ −103.619 −0.0342940
$$210$$ 0 0
$$211$$ 4943.16 1.61280 0.806401 0.591369i $$-0.201412\pi$$
0.806401 + 0.591369i $$0.201412\pi$$
$$212$$ 6386.83i 2.06910i
$$213$$ − 3012.92i − 0.969211i
$$214$$ −7910.05 −2.52673
$$215$$ 0 0
$$216$$ 554.340 0.174621
$$217$$ − 1710.15i − 0.534989i
$$218$$ − 2160.70i − 0.671288i
$$219$$ 3021.86 0.932414
$$220$$ 0 0
$$221$$ 19.0896 0.00581044
$$222$$ − 2625.16i − 0.793646i
$$223$$ − 3160.15i − 0.948965i −0.880265 0.474482i $$-0.842635\pi$$
0.880265 0.474482i $$-0.157365\pi$$
$$224$$ −1378.74 −0.411255
$$225$$ 0 0
$$226$$ 7438.51 2.18939
$$227$$ − 3651.11i − 1.06755i −0.845628 0.533773i $$-0.820774\pi$$
0.845628 0.533773i $$-0.179226\pi$$
$$228$$ 204.350i 0.0593571i
$$229$$ 4083.70 1.17842 0.589210 0.807980i $$-0.299439\pi$$
0.589210 + 0.807980i $$0.299439\pi$$
$$230$$ 0 0
$$231$$ 400.307 0.114019
$$232$$ − 5984.70i − 1.69360i
$$233$$ 3682.51i 1.03540i 0.855561 + 0.517702i $$0.173212\pi$$
−0.855561 + 0.517702i $$0.826788\pi$$
$$234$$ 119.802 0.0334687
$$235$$ 0 0
$$236$$ −1044.91 −0.288211
$$237$$ − 2589.71i − 0.709789i
$$238$$ − 206.105i − 0.0561336i
$$239$$ −2658.78 −0.719591 −0.359796 0.933031i $$-0.617154\pi$$
−0.359796 + 0.933031i $$0.617154\pi$$
$$240$$ 0 0
$$241$$ −4820.39 −1.28842 −0.644209 0.764850i $$-0.722813\pi$$
−0.644209 + 0.764850i $$0.722813\pi$$
$$242$$ − 4384.46i − 1.16464i
$$243$$ − 243.000i − 0.0641500i
$$244$$ 65.8249 0.0172705
$$245$$ 0 0
$$246$$ 4283.45 1.11017
$$247$$ 15.9690i 0.00411369i
$$248$$ − 5015.91i − 1.28432i
$$249$$ −4004.17 −1.01909
$$250$$ 0 0
$$251$$ 1672.27 0.420530 0.210265 0.977644i $$-0.432567\pi$$
0.210265 + 0.977644i $$0.432567\pi$$
$$252$$ − 789.461i − 0.197347i
$$253$$ 941.245i 0.233896i
$$254$$ 3824.34 0.944727
$$255$$ 0 0
$$256$$ −3320.09 −0.810569
$$257$$ − 3697.74i − 0.897506i −0.893656 0.448753i $$-0.851868\pi$$
0.893656 0.448753i $$-0.148132\pi$$
$$258$$ 4091.56i 0.987322i
$$259$$ −1351.84 −0.324323
$$260$$ 0 0
$$261$$ −2623.45 −0.622174
$$262$$ − 12670.0i − 2.98760i
$$263$$ − 7319.00i − 1.71600i −0.513646 0.858002i $$-0.671706\pi$$
0.513646 0.858002i $$-0.328294\pi$$
$$264$$ 1174.11 0.273717
$$265$$ 0 0
$$266$$ 172.412 0.0397416
$$267$$ 978.747i 0.224338i
$$268$$ − 2581.32i − 0.588354i
$$269$$ 815.097 0.184749 0.0923743 0.995724i $$-0.470554\pi$$
0.0923743 + 0.995724i $$0.470554\pi$$
$$270$$ 0 0
$$271$$ −5106.02 −1.14453 −0.572267 0.820068i $$-0.693936\pi$$
−0.572267 + 0.820068i $$0.693936\pi$$
$$272$$ 46.9150i 0.0104582i
$$273$$ − 61.6926i − 0.0136769i
$$274$$ 9321.13 2.05515
$$275$$ 0 0
$$276$$ 1856.26 0.404833
$$277$$ 1398.72i 0.303398i 0.988427 + 0.151699i $$0.0484744\pi$$
−0.988427 + 0.151699i $$0.951526\pi$$
$$278$$ 7907.39i 1.70595i
$$279$$ −2198.77 −0.471816
$$280$$ 0 0
$$281$$ −7102.38 −1.50780 −0.753901 0.656988i $$-0.771830\pi$$
−0.753901 + 0.656988i $$0.771830\pi$$
$$282$$ 1175.91i 0.248313i
$$283$$ − 4465.18i − 0.937907i −0.883223 0.468953i $$-0.844631\pi$$
0.883223 0.468953i $$-0.155369\pi$$
$$284$$ −12585.1 −2.62954
$$285$$ 0 0
$$286$$ 253.743 0.0524621
$$287$$ − 2205.79i − 0.453671i
$$288$$ 1772.67i 0.362692i
$$289$$ 4870.78 0.991405
$$290$$ 0 0
$$291$$ 4580.32 0.922692
$$292$$ − 12622.5i − 2.52970i
$$293$$ − 7590.61i − 1.51348i −0.653718 0.756738i $$-0.726792\pi$$
0.653718 0.756738i $$-0.273208\pi$$
$$294$$ −666.076 −0.132130
$$295$$ 0 0
$$296$$ −3964.98 −0.778581
$$297$$ − 514.681i − 0.100555i
$$298$$ 5317.61i 1.03369i
$$299$$ 145.058 0.0280566
$$300$$ 0 0
$$301$$ 2106.97 0.403468
$$302$$ 6978.26i 1.32965i
$$303$$ − 290.615i − 0.0551003i
$$304$$ −39.2456 −0.00740425
$$305$$ 0 0
$$306$$ −264.992 −0.0495052
$$307$$ 9480.12i 1.76241i 0.472737 + 0.881203i $$0.343266\pi$$
−0.472737 + 0.881203i $$0.656734\pi$$
$$308$$ − 1672.10i − 0.309340i
$$309$$ −3965.98 −0.730151
$$310$$ 0 0
$$311$$ 7078.01 1.29054 0.645268 0.763956i $$-0.276746\pi$$
0.645268 + 0.763956i $$0.276746\pi$$
$$312$$ − 180.945i − 0.0328334i
$$313$$ − 5593.84i − 1.01017i −0.863070 0.505084i $$-0.831461\pi$$
0.863070 0.505084i $$-0.168539\pi$$
$$314$$ 11529.6 2.07214
$$315$$ 0 0
$$316$$ −10817.3 −1.92571
$$317$$ 3567.81i 0.632139i 0.948736 + 0.316070i $$0.102363\pi$$
−0.948736 + 0.316070i $$0.897637\pi$$
$$318$$ − 6928.24i − 1.22175i
$$319$$ −5556.54 −0.975255
$$320$$ 0 0
$$321$$ 5237.14 0.910618
$$322$$ − 1566.15i − 0.271050i
$$323$$ − 35.3222i − 0.00608477i
$$324$$ −1015.02 −0.174043
$$325$$ 0 0
$$326$$ −2694.70 −0.457809
$$327$$ 1430.57i 0.241928i
$$328$$ − 6469.62i − 1.08910i
$$329$$ 605.541 0.101473
$$330$$ 0 0
$$331$$ −4389.67 −0.728936 −0.364468 0.931216i $$-0.618749\pi$$
−0.364468 + 0.931216i $$0.618749\pi$$
$$332$$ 16725.6i 2.76487i
$$333$$ 1738.09i 0.286026i
$$334$$ −4207.14 −0.689236
$$335$$ 0 0
$$336$$ 151.617 0.0246172
$$337$$ 2348.83i 0.379671i 0.981816 + 0.189835i $$0.0607955\pi$$
−0.981816 + 0.189835i $$0.939205\pi$$
$$338$$ 9915.79i 1.59570i
$$339$$ −4924.94 −0.789044
$$340$$ 0 0
$$341$$ −4657.05 −0.739570
$$342$$ − 221.673i − 0.0350488i
$$343$$ 343.000i 0.0539949i
$$344$$ 6179.79 0.968581
$$345$$ 0 0
$$346$$ −1430.21 −0.222222
$$347$$ 558.436i 0.0863931i 0.999067 + 0.0431965i $$0.0137542\pi$$
−0.999067 + 0.0431965i $$0.986246\pi$$
$$348$$ 10958.3i 1.68800i
$$349$$ −3233.89 −0.496006 −0.248003 0.968759i $$-0.579774\pi$$
−0.248003 + 0.968759i $$0.579774\pi$$
$$350$$ 0 0
$$351$$ −79.3190 −0.0120619
$$352$$ 3754.56i 0.568519i
$$353$$ 7516.35i 1.13330i 0.823959 + 0.566650i $$0.191761\pi$$
−0.823959 + 0.566650i $$0.808239\pi$$
$$354$$ 1133.49 0.170181
$$355$$ 0 0
$$356$$ 4088.27 0.608646
$$357$$ 136.459i 0.0202302i
$$358$$ 6549.70i 0.966935i
$$359$$ 6577.76 0.967021 0.483511 0.875338i $$-0.339361\pi$$
0.483511 + 0.875338i $$0.339361\pi$$
$$360$$ 0 0
$$361$$ −6829.45 −0.995692
$$362$$ − 8354.56i − 1.21300i
$$363$$ 2902.89i 0.419731i
$$364$$ −257.693 −0.0371065
$$365$$ 0 0
$$366$$ −71.4048 −0.0101978
$$367$$ 8307.17i 1.18155i 0.806835 + 0.590777i $$0.201179\pi$$
−0.806835 + 0.590777i $$0.798821\pi$$
$$368$$ 356.497i 0.0504992i
$$369$$ −2836.02 −0.400101
$$370$$ 0 0
$$371$$ −3567.74 −0.499266
$$372$$ 9184.34i 1.28007i
$$373$$ 4551.09i 0.631760i 0.948799 + 0.315880i $$0.102300\pi$$
−0.948799 + 0.315880i $$0.897700\pi$$
$$374$$ −561.261 −0.0775992
$$375$$ 0 0
$$376$$ 1776.06 0.243599
$$377$$ 856.335i 0.116985i
$$378$$ 856.383i 0.116528i
$$379$$ 1788.29 0.242370 0.121185 0.992630i $$-0.461331\pi$$
0.121185 + 0.992630i $$0.461331\pi$$
$$380$$ 0 0
$$381$$ −2532.05 −0.340474
$$382$$ − 1105.93i − 0.148126i
$$383$$ 1358.47i 0.181240i 0.995886 + 0.0906199i $$0.0288848\pi$$
−0.995886 + 0.0906199i $$0.971115\pi$$
$$384$$ 6619.37 0.879670
$$385$$ 0 0
$$386$$ −7852.60 −1.03546
$$387$$ − 2708.97i − 0.355825i
$$388$$ − 19132.2i − 2.50333i
$$389$$ −9722.54 −1.26723 −0.633615 0.773649i $$-0.718430\pi$$
−0.633615 + 0.773649i $$0.718430\pi$$
$$390$$ 0 0
$$391$$ −320.858 −0.0414999
$$392$$ 1006.03i 0.129622i
$$393$$ 8388.61i 1.07672i
$$394$$ −1623.19 −0.207551
$$395$$ 0 0
$$396$$ −2149.84 −0.272812
$$397$$ − 4788.04i − 0.605302i −0.953101 0.302651i $$-0.902128\pi$$
0.953101 0.302651i $$-0.0978717\pi$$
$$398$$ 14516.1i 1.82820i
$$399$$ −114.152 −0.0143227
$$400$$ 0 0
$$401$$ 9681.41 1.20565 0.602826 0.797873i $$-0.294041\pi$$
0.602826 + 0.797873i $$0.294041\pi$$
$$402$$ 2800.13i 0.347408i
$$403$$ 717.712i 0.0887141i
$$404$$ −1213.91 −0.149491
$$405$$ 0 0
$$406$$ 9245.58 1.13017
$$407$$ 3681.32i 0.448344i
$$408$$ 400.238i 0.0485655i
$$409$$ −11113.1 −1.34353 −0.671767 0.740763i $$-0.734464\pi$$
−0.671767 + 0.740763i $$0.734464\pi$$
$$410$$ 0 0
$$411$$ −6171.40 −0.740663
$$412$$ 16566.1i 1.98095i
$$413$$ − 583.696i − 0.0695443i
$$414$$ −2013.62 −0.239043
$$415$$ 0 0
$$416$$ 578.627 0.0681959
$$417$$ − 5235.37i − 0.614814i
$$418$$ − 469.510i − 0.0549389i
$$419$$ 1230.09 0.143421 0.0717107 0.997425i $$-0.477154\pi$$
0.0717107 + 0.997425i $$0.477154\pi$$
$$420$$ 0 0
$$421$$ −12356.5 −1.43044 −0.715222 0.698897i $$-0.753674\pi$$
−0.715222 + 0.698897i $$0.753674\pi$$
$$422$$ 22398.1i 2.58370i
$$423$$ − 778.552i − 0.0894906i
$$424$$ −10464.2 −1.19856
$$425$$ 0 0
$$426$$ 13651.9 1.55267
$$427$$ 36.7703i 0.00416731i
$$428$$ − 21875.7i − 2.47057i
$$429$$ −168.000 −0.0189070
$$430$$ 0 0
$$431$$ −7375.27 −0.824256 −0.412128 0.911126i $$-0.635214\pi$$
−0.412128 + 0.911126i $$0.635214\pi$$
$$432$$ − 194.936i − 0.0217103i
$$433$$ 690.067i 0.0765877i 0.999267 + 0.0382939i $$0.0121923\pi$$
−0.999267 + 0.0382939i $$0.987808\pi$$
$$434$$ 7748.92 0.857051
$$435$$ 0 0
$$436$$ 5975.55 0.656369
$$437$$ − 268.406i − 0.0293812i
$$438$$ 13692.5i 1.49372i
$$439$$ −8408.79 −0.914191 −0.457095 0.889418i $$-0.651110\pi$$
−0.457095 + 0.889418i $$0.651110\pi$$
$$440$$ 0 0
$$441$$ 441.000 0.0476190
$$442$$ 86.4976i 0.00930830i
$$443$$ − 6568.55i − 0.704473i −0.935911 0.352236i $$-0.885421\pi$$
0.935911 0.352236i $$-0.114579\pi$$
$$444$$ 7260.06 0.776007
$$445$$ 0 0
$$446$$ 14319.0 1.52024
$$447$$ − 3520.72i − 0.372537i
$$448$$ − 5842.94i − 0.616190i
$$449$$ −2954.55 −0.310543 −0.155271 0.987872i $$-0.549625\pi$$
−0.155271 + 0.987872i $$0.549625\pi$$
$$450$$ 0 0
$$451$$ −6006.76 −0.627156
$$452$$ 20571.7i 2.14073i
$$453$$ − 4620.21i − 0.479197i
$$454$$ 16543.7 1.71020
$$455$$ 0 0
$$456$$ −334.810 −0.0343835
$$457$$ − 8144.84i − 0.833697i −0.908976 0.416849i $$-0.863135\pi$$
0.908976 0.416849i $$-0.136865\pi$$
$$458$$ 18503.8i 1.88782i
$$459$$ 175.448 0.0178414
$$460$$ 0 0
$$461$$ 2495.26 0.252095 0.126048 0.992024i $$-0.459771\pi$$
0.126048 + 0.992024i $$0.459771\pi$$
$$462$$ 1813.84i 0.182657i
$$463$$ 5755.66i 0.577728i 0.957370 + 0.288864i $$0.0932776\pi$$
−0.957370 + 0.288864i $$0.906722\pi$$
$$464$$ −2104.54 −0.210562
$$465$$ 0 0
$$466$$ −16685.9 −1.65871
$$467$$ − 4143.73i − 0.410598i −0.978699 0.205299i $$-0.934183\pi$$
0.978699 0.205299i $$-0.0658166\pi$$
$$468$$ 331.319i 0.0327248i
$$469$$ 1441.95 0.141968
$$470$$ 0 0
$$471$$ −7633.60 −0.746789
$$472$$ − 1711.99i − 0.166951i
$$473$$ − 5737.67i − 0.557755i
$$474$$ 11734.3 1.13708
$$475$$ 0 0
$$476$$ 569.996 0.0548860
$$477$$ 4587.09i 0.440312i
$$478$$ − 12047.3i − 1.15278i
$$479$$ 6765.96 0.645396 0.322698 0.946502i $$-0.395410\pi$$
0.322698 + 0.946502i $$0.395410\pi$$
$$480$$ 0 0
$$481$$ 567.339 0.0537805
$$482$$ − 21841.8i − 2.06404i
$$483$$ 1036.93i 0.0976848i
$$484$$ 12125.5 1.13876
$$485$$ 0 0
$$486$$ 1101.06 0.102768
$$487$$ 6360.42i 0.591824i 0.955215 + 0.295912i $$0.0956235\pi$$
−0.955215 + 0.295912i $$0.904377\pi$$
$$488$$ 107.848i 0.0100042i
$$489$$ 1784.12 0.164992
$$490$$ 0 0
$$491$$ −7072.54 −0.650060 −0.325030 0.945704i $$-0.605374\pi$$
−0.325030 + 0.945704i $$0.605374\pi$$
$$492$$ 11846.2i 1.08550i
$$493$$ − 1894.15i − 0.173039i
$$494$$ −72.3576 −0.00659012
$$495$$ 0 0
$$496$$ −1763.86 −0.159677
$$497$$ − 7030.15i − 0.634498i
$$498$$ − 18143.4i − 1.63258i
$$499$$ 18473.9 1.65732 0.828661 0.559751i $$-0.189103\pi$$
0.828661 + 0.559751i $$0.189103\pi$$
$$500$$ 0 0
$$501$$ 2785.49 0.248397
$$502$$ 7577.28i 0.673687i
$$503$$ − 11379.2i − 1.00869i −0.863501 0.504347i $$-0.831733\pi$$
0.863501 0.504347i $$-0.168267\pi$$
$$504$$ 1293.46 0.114316
$$505$$ 0 0
$$506$$ −4264.90 −0.374700
$$507$$ − 6565.11i − 0.575082i
$$508$$ 10576.5i 0.923730i
$$509$$ −6064.48 −0.528101 −0.264051 0.964509i $$-0.585059\pi$$
−0.264051 + 0.964509i $$0.585059\pi$$
$$510$$ 0 0
$$511$$ 7051.02 0.610408
$$512$$ 2607.89i 0.225105i
$$513$$ 146.767i 0.0126314i
$$514$$ 16755.0 1.43780
$$515$$ 0 0
$$516$$ −11315.5 −0.965379
$$517$$ − 1649.00i − 0.140276i
$$518$$ − 6125.38i − 0.519563i
$$519$$ 946.926 0.0800875
$$520$$ 0 0
$$521$$ 2682.88 0.225603 0.112801 0.993618i $$-0.464018\pi$$
0.112801 + 0.993618i $$0.464018\pi$$
$$522$$ − 11887.2i − 0.996720i
$$523$$ 4309.02i 0.360268i 0.983642 + 0.180134i $$0.0576532\pi$$
−0.983642 + 0.180134i $$0.942347\pi$$
$$524$$ 35039.6 2.92120
$$525$$ 0 0
$$526$$ 33163.3 2.74903
$$527$$ − 1587.52i − 0.131221i
$$528$$ − 412.879i − 0.0340308i
$$529$$ 9728.87 0.799611
$$530$$ 0 0
$$531$$ −750.467 −0.0613323
$$532$$ 476.817i 0.0388584i
$$533$$ 925.720i 0.0752296i
$$534$$ −4434.83 −0.359389
$$535$$ 0 0
$$536$$ 4229.25 0.340813
$$537$$ − 4336.47i − 0.348478i
$$538$$ 3693.31i 0.295966i
$$539$$ 934.051 0.0746427
$$540$$ 0 0
$$541$$ 4081.47 0.324355 0.162178 0.986762i $$-0.448148\pi$$
0.162178 + 0.986762i $$0.448148\pi$$
$$542$$ − 23136.0i − 1.83354i
$$543$$ 5531.44i 0.437158i
$$544$$ −1279.88 −0.100872
$$545$$ 0 0
$$546$$ 279.537 0.0219104
$$547$$ 8844.82i 0.691366i 0.938351 + 0.345683i $$0.112353\pi$$
−0.938351 + 0.345683i $$0.887647\pi$$
$$548$$ 25778.2i 2.00947i
$$549$$ 47.2762 0.00367522
$$550$$ 0 0
$$551$$ 1584.51 0.122509
$$552$$ 3041.32i 0.234506i
$$553$$ − 6042.66i − 0.464666i
$$554$$ −6337.80 −0.486042
$$555$$ 0 0
$$556$$ −21868.4 −1.66803
$$557$$ − 11144.7i − 0.847787i −0.905712 0.423894i $$-0.860663\pi$$
0.905712 0.423894i $$-0.139337\pi$$
$$558$$ − 9962.90i − 0.755848i
$$559$$ −884.249 −0.0669047
$$560$$ 0 0
$$561$$ 371.603 0.0279663
$$562$$ − 32181.8i − 2.41549i
$$563$$ 21857.5i 1.63621i 0.575071 + 0.818104i $$0.304975\pi$$
−0.575071 + 0.818104i $$0.695025\pi$$
$$564$$ −3252.05 −0.242794
$$565$$ 0 0
$$566$$ 20232.3 1.50252
$$567$$ − 567.000i − 0.0419961i
$$568$$ − 20619.6i − 1.52320i
$$569$$ −23496.4 −1.73115 −0.865573 0.500783i $$-0.833046\pi$$
−0.865573 + 0.500783i $$0.833046\pi$$
$$570$$ 0 0
$$571$$ 11067.8 0.811164 0.405582 0.914059i $$-0.367069\pi$$
0.405582 + 0.914059i $$0.367069\pi$$
$$572$$ 701.743i 0.0512961i
$$573$$ 732.222i 0.0533839i
$$574$$ 9994.72 0.726780
$$575$$ 0 0
$$576$$ −7512.36 −0.543429
$$577$$ 20482.9i 1.47784i 0.673791 + 0.738922i $$0.264665\pi$$
−0.673791 + 0.738922i $$0.735335\pi$$
$$578$$ 22070.1i 1.58823i
$$579$$ 5199.10 0.373173
$$580$$ 0 0
$$581$$ −9343.07 −0.667153
$$582$$ 20754.0i 1.47815i
$$583$$ 9715.60i 0.690187i
$$584$$ 20680.8 1.46537
$$585$$ 0 0
$$586$$ 34394.1 2.42458
$$587$$ − 23444.3i − 1.64847i −0.566248 0.824235i $$-0.691606\pi$$
0.566248 0.824235i $$-0.308394\pi$$
$$588$$ − 1842.08i − 0.129194i
$$589$$ 1328.01 0.0929025
$$590$$ 0 0
$$591$$ 1074.69 0.0748000
$$592$$ 1394.30i 0.0967996i
$$593$$ − 4404.69i − 0.305024i −0.988302 0.152512i $$-0.951264\pi$$
0.988302 0.152512i $$-0.0487362\pi$$
$$594$$ 2332.09 0.161089
$$595$$ 0 0
$$596$$ −14706.2 −1.01072
$$597$$ − 9610.89i − 0.658874i
$$598$$ 657.277i 0.0449466i
$$599$$ −3327.05 −0.226945 −0.113472 0.993541i $$-0.536197\pi$$
−0.113472 + 0.993541i $$0.536197\pi$$
$$600$$ 0 0
$$601$$ −14244.8 −0.966818 −0.483409 0.875395i $$-0.660602\pi$$
−0.483409 + 0.875395i $$0.660602\pi$$
$$602$$ 9546.97i 0.646354i
$$603$$ − 1853.93i − 0.125204i
$$604$$ −19298.8 −1.30010
$$605$$ 0 0
$$606$$ 1316.81 0.0882704
$$607$$ 11446.5i 0.765402i 0.923872 + 0.382701i $$0.125006\pi$$
−0.923872 + 0.382701i $$0.874994\pi$$
$$608$$ − 1070.65i − 0.0714156i
$$609$$ −6121.38 −0.407308
$$610$$ 0 0
$$611$$ −254.132 −0.0168266
$$612$$ − 732.852i − 0.0484049i
$$613$$ 19436.4i 1.28063i 0.768111 + 0.640316i $$0.221197\pi$$
−0.768111 + 0.640316i $$0.778803\pi$$
$$614$$ −42955.6 −2.82337
$$615$$ 0 0
$$616$$ 2739.59 0.179190
$$617$$ − 20530.1i − 1.33956i −0.742558 0.669781i $$-0.766388\pi$$
0.742558 0.669781i $$-0.233612\pi$$
$$618$$ − 17970.4i − 1.16970i
$$619$$ −5833.35 −0.378776 −0.189388 0.981902i $$-0.560650\pi$$
−0.189388 + 0.981902i $$0.560650\pi$$
$$620$$ 0 0
$$621$$ 1333.19 0.0861499
$$622$$ 32071.4i 2.06744i
$$623$$ 2283.74i 0.146864i
$$624$$ −63.6301 −0.00408212
$$625$$ 0 0
$$626$$ 25346.4 1.61829
$$627$$ 310.856i 0.0197997i
$$628$$ 31885.9i 2.02609i
$$629$$ −1254.91 −0.0795493
$$630$$ 0 0
$$631$$ 24776.6 1.56314 0.781568 0.623820i $$-0.214420\pi$$
0.781568 + 0.623820i $$0.214420\pi$$
$$632$$ − 17723.2i − 1.11549i
$$633$$ − 14829.5i − 0.931152i
$$634$$ −16166.2 −1.01268
$$635$$ 0 0
$$636$$ 19160.5 1.19460
$$637$$ − 143.949i − 0.00895366i
$$638$$ − 25177.4i − 1.56235i
$$639$$ −9038.77 −0.559574
$$640$$ 0 0
$$641$$ 27219.4 1.67723 0.838613 0.544728i $$-0.183367\pi$$
0.838613 + 0.544728i $$0.183367\pi$$
$$642$$ 23730.1i 1.45881i
$$643$$ − 7091.79i − 0.434950i −0.976066 0.217475i $$-0.930218\pi$$
0.976066 0.217475i $$-0.0697821\pi$$
$$644$$ 4331.28 0.265026
$$645$$ 0 0
$$646$$ 160.049 0.00974777
$$647$$ 27773.0i 1.68758i 0.536670 + 0.843792i $$0.319682\pi$$
−0.536670 + 0.843792i $$0.680318\pi$$
$$648$$ − 1663.02i − 0.100817i
$$649$$ −1589.51 −0.0961382
$$650$$ 0 0
$$651$$ −5130.46 −0.308876
$$652$$ − 7452.37i − 0.447634i
$$653$$ − 21380.4i − 1.28129i −0.767839 0.640643i $$-0.778668\pi$$
0.767839 0.640643i $$-0.221332\pi$$
$$654$$ −6482.09 −0.387568
$$655$$ 0 0
$$656$$ −2275.06 −0.135406
$$657$$ − 9065.59i − 0.538329i
$$658$$ 2743.78i 0.162559i
$$659$$ 17232.3 1.01863 0.509315 0.860580i $$-0.329899\pi$$
0.509315 + 0.860580i $$0.329899\pi$$
$$660$$ 0 0
$$661$$ 26577.7 1.56392 0.781962 0.623326i $$-0.214219\pi$$
0.781962 + 0.623326i $$0.214219\pi$$
$$662$$ − 19890.1i − 1.16775i
$$663$$ − 57.2689i − 0.00335466i
$$664$$ −27403.4 −1.60159
$$665$$ 0 0
$$666$$ −7875.49 −0.458212
$$667$$ − 14393.2i − 0.835544i
$$668$$ − 11635.1i − 0.673917i
$$669$$ −9480.44 −0.547885
$$670$$ 0 0
$$671$$ 100.132 0.00576090
$$672$$ 4136.22i 0.237438i
$$673$$ 31695.2i 1.81540i 0.419624 + 0.907698i $$0.362162\pi$$
−0.419624 + 0.907698i $$0.637838\pi$$
$$674$$ −10642.9 −0.608231
$$675$$ 0 0
$$676$$ −27422.7 −1.56024
$$677$$ 20440.3i 1.16039i 0.814477 + 0.580195i $$0.197024\pi$$
−0.814477 + 0.580195i $$0.802976\pi$$
$$678$$ − 22315.5i − 1.26405i
$$679$$ 10687.4 0.604044
$$680$$ 0 0
$$681$$ −10953.3 −0.616348
$$682$$ − 21101.7i − 1.18479i
$$683$$ 22896.9i 1.28276i 0.767223 + 0.641381i $$0.221638\pi$$
−0.767223 + 0.641381i $$0.778362\pi$$
$$684$$ 613.051 0.0342699
$$685$$ 0 0
$$686$$ −1554.18 −0.0864997
$$687$$ − 12251.1i − 0.680361i
$$688$$ − 2173.14i − 0.120422i
$$689$$ 1497.30 0.0827904
$$690$$ 0 0
$$691$$ −23764.0 −1.30829 −0.654143 0.756371i $$-0.726971\pi$$
−0.654143 + 0.756371i $$0.726971\pi$$
$$692$$ − 3955.35i − 0.217283i
$$693$$ − 1200.92i − 0.0658287i
$$694$$ −2530.34 −0.138401
$$695$$ 0 0
$$696$$ −17954.1 −0.977800
$$697$$ − 2047.62i − 0.111276i
$$698$$ − 14653.2i − 0.794600i
$$699$$ 11047.5 0.597791
$$700$$ 0 0
$$701$$ 26259.5 1.41485 0.707423 0.706791i $$-0.249858\pi$$
0.707423 + 0.706791i $$0.249858\pi$$
$$702$$ − 359.405i − 0.0193232i
$$703$$ − 1049.77i − 0.0563196i
$$704$$ −15911.4 −0.851822
$$705$$ 0 0
$$706$$ −34057.6 −1.81554
$$707$$ − 678.101i − 0.0360716i
$$708$$ 3134.73i 0.166399i
$$709$$ −12783.0 −0.677116 −0.338558 0.940945i $$-0.609939\pi$$
−0.338558 + 0.940945i $$0.609939\pi$$
$$710$$ 0 0
$$711$$ −7769.14 −0.409797
$$712$$ 6698.26i 0.352567i
$$713$$ − 12063.3i − 0.633623i
$$714$$ −618.315 −0.0324087
$$715$$ 0 0
$$716$$ −18113.6 −0.945444
$$717$$ 7976.35i 0.415456i
$$718$$ 29804.7i 1.54916i
$$719$$ 27609.0 1.43205 0.716025 0.698075i $$-0.245960\pi$$
0.716025 + 0.698075i $$0.245960\pi$$
$$720$$ 0 0
$$721$$ −9253.95 −0.477996
$$722$$ − 30945.1i − 1.59509i
$$723$$ 14461.2i 0.743868i
$$724$$ 23105.1 1.18604
$$725$$ 0 0
$$726$$ −13153.4 −0.672407
$$727$$ 31306.2i 1.59709i 0.601937 + 0.798544i $$0.294396\pi$$
−0.601937 + 0.798544i $$0.705604\pi$$
$$728$$ − 422.206i − 0.0214945i
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ 1955.89 0.0989621
$$732$$ − 197.475i − 0.00997113i
$$733$$ 15765.8i 0.794441i 0.917723 + 0.397220i $$0.130025\pi$$
−0.917723 + 0.397220i $$0.869975\pi$$
$$734$$ −37640.8 −1.89285
$$735$$ 0 0
$$736$$ −9725.53 −0.487076
$$737$$ − 3926.68i − 0.196256i
$$738$$ − 12850.4i − 0.640959i
$$739$$ −3966.51 −0.197443 −0.0987216 0.995115i $$-0.531475\pi$$
−0.0987216 + 0.995115i $$0.531475\pi$$
$$740$$ 0 0
$$741$$ 47.9070 0.00237504
$$742$$ − 16165.9i − 0.799823i
$$743$$ − 8224.50i − 0.406094i −0.979169 0.203047i $$-0.934916\pi$$
0.979169 0.203047i $$-0.0650844\pi$$
$$744$$ −15047.7 −0.741500
$$745$$ 0 0
$$746$$ −20621.6 −1.01208
$$747$$ 12012.5i 0.588373i
$$748$$ − 1552.20i − 0.0758745i
$$749$$ 12220.0 0.596140
$$750$$ 0 0
$$751$$ 18929.2 0.919754 0.459877 0.887983i $$-0.347894\pi$$
0.459877 + 0.887983i $$0.347894\pi$$
$$752$$ − 624.558i − 0.0302863i
$$753$$ − 5016.82i − 0.242793i
$$754$$ −3880.16 −0.187410
$$755$$ 0 0
$$756$$ −2368.38 −0.113938
$$757$$ − 34906.8i − 1.67597i −0.545695 0.837984i $$-0.683734\pi$$
0.545695 0.837984i $$-0.316266\pi$$
$$758$$ 8102.96i 0.388276i
$$759$$ 2823.74 0.135040
$$760$$ 0 0
$$761$$ −13683.4 −0.651803 −0.325902 0.945404i $$-0.605668\pi$$
−0.325902 + 0.945404i $$0.605668\pi$$
$$762$$ − 11473.0i − 0.545438i
$$763$$ 3337.99i 0.158379i
$$764$$ 3058.52 0.144834
$$765$$ 0 0
$$766$$ −6155.42 −0.290345
$$767$$ 244.964i 0.0115321i
$$768$$ 9960.28i 0.467982i
$$769$$ −41837.3 −1.96189 −0.980943 0.194294i $$-0.937758\pi$$
−0.980943 + 0.194294i $$0.937758\pi$$
$$770$$ 0 0
$$771$$ −11093.2 −0.518175
$$772$$ − 21716.9i − 1.01245i
$$773$$ − 19640.0i − 0.913843i −0.889507 0.456921i $$-0.848952\pi$$
0.889507 0.456921i $$-0.151048\pi$$
$$774$$ 12274.7 0.570031
$$775$$ 0 0
$$776$$ 31346.4 1.45009
$$777$$ 4055.53i 0.187248i
$$778$$ − 44054.1i − 2.03010i
$$779$$ 1712.89 0.0787814
$$780$$ 0 0
$$781$$ −19144.4 −0.877131
$$782$$ − 1453.85i − 0.0664827i
$$783$$ 7870.34i 0.359212i
$$784$$ 353.772 0.0161157
$$785$$ 0 0
$$786$$ −38009.9 −1.72489
$$787$$ 24935.3i 1.12941i 0.825293 + 0.564705i $$0.191010\pi$$
−0.825293 + 0.564705i $$0.808990\pi$$
$$788$$ − 4489.02i − 0.202938i
$$789$$ −21957.0 −0.990736
$$790$$ 0 0
$$791$$ −11491.5 −0.516551
$$792$$ − 3522.33i − 0.158031i
$$793$$ − 15.4317i 0 0.000691041i
$$794$$ 21695.2 0.969692
$$795$$ 0 0
$$796$$ −40145.1 −1.78757
$$797$$ − 1168.33i − 0.0519251i −0.999663 0.0259625i $$-0.991735\pi$$
0.999663 0.0259625i $$-0.00826506\pi$$
$$798$$ − 517.237i − 0.0229448i
$$799$$ 562.120 0.0248891
$$800$$ 0 0
$$801$$ 2936.24 0.129522
$$802$$ 43867.7i 1.93145i
$$803$$ − 19201.2i − 0.843829i
$$804$$ −7743.95 −0.339686
$$805$$ 0 0
$$806$$ −3252.05 −0.142120
$$807$$ − 2445.29i − 0.106665i
$$808$$ − 1988.88i − 0.0865949i
$$809$$ 35175.7 1.52869 0.764345 0.644807i $$-0.223062\pi$$
0.764345 + 0.644807i $$0.223062\pi$$
$$810$$ 0 0
$$811$$ −15256.5 −0.660577 −0.330288 0.943880i $$-0.607146\pi$$
−0.330288 + 0.943880i $$0.607146\pi$$
$$812$$ 25569.3i 1.10506i
$$813$$ 15318.1i 0.660797i
$$814$$ −16680.5 −0.718245
$$815$$ 0 0
$$816$$ 140.745 0.00603806
$$817$$ 1636.16i 0.0700635i
$$818$$ − 50354.7i − 2.15233i
$$819$$ −185.078 −0.00789639
$$820$$ 0 0
$$821$$ −15971.9 −0.678956 −0.339478 0.940614i $$-0.610250\pi$$
−0.339478 + 0.940614i $$0.610250\pi$$
$$822$$ − 27963.4i − 1.18654i
$$823$$ − 2312.41i − 0.0979409i −0.998800 0.0489705i $$-0.984406\pi$$
0.998800 0.0489705i $$-0.0155940\pi$$
$$824$$ −27142.0 −1.14750
$$825$$ 0 0
$$826$$ 2644.80 0.111410
$$827$$ 10422.4i 0.438238i 0.975698 + 0.219119i $$0.0703183\pi$$
−0.975698 + 0.219119i $$0.929682\pi$$
$$828$$ − 5568.79i − 0.233731i
$$829$$ 13213.4 0.553584 0.276792 0.960930i $$-0.410729\pi$$
0.276792 + 0.960930i $$0.410729\pi$$
$$830$$ 0 0
$$831$$ 4196.17 0.175167
$$832$$ 2452.15i 0.102179i
$$833$$ 318.405i 0.0132438i
$$834$$ 23722.2 0.984929
$$835$$ 0 0
$$836$$ 1298.46 0.0537179
$$837$$ 6596.30i 0.272403i
$$838$$ 5573.67i 0.229761i
$$839$$ 10119.6 0.416409 0.208205 0.978085i $$-0.433238\pi$$
0.208205 + 0.978085i $$0.433238\pi$$
$$840$$ 0 0
$$841$$ 60579.9 2.48390
$$842$$ − 55988.7i − 2.29157i
$$843$$ 21307.1i 0.870530i
$$844$$ −61943.4 −2.52628
$$845$$ 0 0
$$846$$ 3527.72 0.143364
$$847$$ 6773.41i 0.274778i
$$848$$ 3679.79i 0.149015i
$$849$$ −13395.5 −0.541501
$$850$$ 0 0
$$851$$ −9535.80 −0.384116
$$852$$ 37755.3i 1.51816i
$$853$$ − 35378.1i − 1.42007i −0.704165 0.710037i $$-0.748678\pi$$
0.704165 0.710037i $$-0.251322\pi$$
$$854$$ −166.611 −0.00667602
$$855$$ 0 0
$$856$$ 35841.4 1.43112
$$857$$ 6697.57i 0.266960i 0.991052 + 0.133480i $$0.0426152\pi$$
−0.991052 + 0.133480i $$0.957385\pi$$
$$858$$ − 761.230i − 0.0302890i
$$859$$ 24298.4 0.965135 0.482568 0.875859i $$-0.339704\pi$$
0.482568 + 0.875859i $$0.339704\pi$$
$$860$$ 0 0
$$861$$ −6617.37 −0.261927
$$862$$ − 33418.3i − 1.32045i
$$863$$ − 24942.9i − 0.983853i −0.870637 0.491926i $$-0.836293\pi$$
0.870637 0.491926i $$-0.163707\pi$$
$$864$$ 5318.00 0.209401
$$865$$ 0 0
$$866$$ −3126.78 −0.122693
$$867$$ − 14612.3i − 0.572388i
$$868$$ 21430.1i 0.838002i
$$869$$ −16455.3 −0.642355
$$870$$ 0 0
$$871$$ −605.152 −0.0235417
$$872$$ 9790.39i 0.380212i
$$873$$ − 13741.0i − 0.532716i
$$874$$ 1216.18 0.0470686
$$875$$ 0 0
$$876$$ −37867.4 −1.46052
$$877$$ − 16276.6i − 0.626705i −0.949637 0.313353i $$-0.898548\pi$$
0.949637 0.313353i $$-0.101452\pi$$
$$878$$ − 38101.3i − 1.46453i
$$879$$ −22771.8 −0.873806
$$880$$ 0 0
$$881$$ 26636.5 1.01862 0.509311 0.860582i $$-0.329900\pi$$
0.509311 + 0.860582i $$0.329900\pi$$
$$882$$ 1998.23i 0.0762855i
$$883$$ 21788.3i 0.830392i 0.909732 + 0.415196i $$0.136287\pi$$
−0.909732 + 0.415196i $$0.863713\pi$$
$$884$$ −239.215 −0.00910142
$$885$$ 0 0
$$886$$ 29763.0 1.12856
$$887$$ − 26813.2i − 1.01499i −0.861654 0.507496i $$-0.830571\pi$$
0.861654 0.507496i $$-0.169429\pi$$
$$888$$ 11895.0i 0.449514i
$$889$$ −5908.11 −0.222893
$$890$$ 0 0
$$891$$ −1544.04 −0.0580554
$$892$$ 39600.2i 1.48645i
$$893$$ 470.229i 0.0176211i
$$894$$ 15952.8 0.596803
$$895$$ 0 0
$$896$$ 15445.2 0.575879
$$897$$ − 435.174i − 0.0161985i
$$898$$ − 13387.4i − 0.497488i
$$899$$ 71214.2 2.64196
$$900$$ 0 0
$$901$$ −3311.91 −0.122459
$$902$$ − 27217.4i − 1.00470i
$$903$$ − 6320.92i − 0.232942i
$$904$$ −33704.8 −1.24005
$$905$$ 0 0
$$906$$ 20934.8 0.767672
$$907$$ 15543.0i 0.569014i 0.958674 + 0.284507i $$0.0918300\pi$$
−0.958674 + 0.284507i $$0.908170\pi$$
$$908$$ 45752.6i 1.67219i
$$909$$ −871.844 −0.0318122
$$910$$ 0 0
$$911$$ 48711.1 1.77154 0.885768 0.464128i $$-0.153632\pi$$
0.885768 + 0.464128i $$0.153632\pi$$
$$912$$ 117.737i 0.00427485i
$$913$$ 25442.8i 0.922273i
$$914$$ 36905.3 1.33558
$$915$$ 0 0
$$916$$ −51173.3 −1.84587
$$917$$ 19573.4i 0.704876i
$$918$$ 794.976i 0.0285818i
$$919$$ −1030.47 −0.0369883 −0.0184941 0.999829i $$-0.505887\pi$$
−0.0184941 + 0.999829i $$0.505887\pi$$
$$920$$ 0 0
$$921$$ 28440.4 1.01753
$$922$$ 11306.3i 0.403855i
$$923$$ 2950.40i 0.105215i
$$924$$ −5016.30 −0.178598
$$925$$ 0 0
$$926$$ −26079.6 −0.925518
$$927$$ 11897.9i 0.421553i
$$928$$ − 57413.6i − 2.03092i
$$929$$ −879.756 −0.0310698 −0.0155349 0.999879i $$-0.504945\pi$$
−0.0155349 + 0.999879i $$0.504945\pi$$
$$930$$ 0 0
$$931$$ −266.354 −0.00937638
$$932$$ − 46146.0i − 1.62185i
$$933$$ − 21234.0i − 0.745092i
$$934$$ 18775.8 0.657776
$$935$$ 0 0
$$936$$ −542.836 −0.0189564
$$937$$ − 18668.1i − 0.650864i −0.945565 0.325432i $$-0.894490\pi$$
0.945565 0.325432i $$-0.105510\pi$$
$$938$$ 6533.64i 0.227432i
$$939$$ −16781.5 −0.583221
$$940$$ 0 0
$$941$$ 29613.4 1.02590 0.512948 0.858420i $$-0.328553\pi$$
0.512948 + 0.858420i $$0.328553\pi$$
$$942$$ − 34588.8i − 1.19635i
$$943$$ − 15559.5i − 0.537313i
$$944$$ −602.028 −0.0207567
$$945$$ 0 0
$$946$$ 25998.1 0.893521
$$947$$ 20738.9i 0.711640i 0.934554 + 0.355820i $$0.115798\pi$$
−0.934554 + 0.355820i $$0.884202\pi$$
$$948$$ 32452.0i 1.11181i
$$949$$ −2959.15 −0.101220
$$950$$ 0 0
$$951$$ 10703.4 0.364966
$$952$$ 933.888i 0.0317936i
$$953$$ 45776.5i 1.55598i 0.628279 + 0.777988i $$0.283760\pi$$
−0.628279 + 0.777988i $$0.716240\pi$$
$$954$$ −20784.7 −0.705377
$$955$$ 0 0
$$956$$ 33317.5 1.12716
$$957$$ 16669.6i 0.563064i
$$958$$ 30657.4i 1.03392i
$$959$$ −14399.9 −0.484878
$$960$$ 0 0
$$961$$ 29895.1 1.00349
$$962$$ 2570.68i 0.0861561i
$$963$$ − 15711.4i − 0.525746i
$$964$$ 60404.9 2.01817
$$965$$ 0 0
$$966$$ −4698.44 −0.156491
$$967$$ 34461.0i 1.14601i 0.819552 + 0.573005i $$0.194222\pi$$
−0.819552 + 0.573005i $$0.805778\pi$$
$$968$$ 19866.5i 0.659643i
$$969$$ −105.967 −0.00351304
$$970$$ 0 0
$$971$$ −22762.8 −0.752309 −0.376154 0.926557i $$-0.622754\pi$$
−0.376154 + 0.926557i $$0.622754\pi$$
$$972$$ 3045.06i 0.100484i
$$973$$ − 12215.9i − 0.402490i
$$974$$ −28819.9 −0.948099
$$975$$ 0 0
$$976$$ 37.9251 0.00124381
$$977$$ 4809.57i 0.157494i 0.996895 + 0.0787470i $$0.0250919\pi$$
−0.996895 + 0.0787470i $$0.974908\pi$$
$$978$$ 8084.10i 0.264316i
$$979$$ 6219.04 0.203025
$$980$$ 0 0
$$981$$ 4291.70 0.139677
$$982$$ − 32046.6i − 1.04139i
$$983$$ 27591.6i 0.895256i 0.894220 + 0.447628i $$0.147731\pi$$
−0.894220 + 0.447628i $$0.852269\pi$$
$$984$$ −19408.9 −0.628793
$$985$$ 0 0
$$986$$ 8582.63 0.277207
$$987$$ − 1816.62i − 0.0585853i
$$988$$ − 200.109i − 0.00644365i
$$989$$ 14862.4 0.477854
$$990$$ 0 0
$$991$$ −22263.4 −0.713643 −0.356822 0.934173i $$-0.616140\pi$$
−0.356822 + 0.934173i $$0.616140\pi$$
$$992$$ − 48119.5i − 1.54012i
$$993$$ 13169.0i 0.420851i
$$994$$ 31854.5 1.01646
$$995$$ 0 0
$$996$$ 50176.8 1.59630
$$997$$ 30378.2i 0.964983i 0.875901 + 0.482491i $$0.160268\pi$$
−0.875901 + 0.482491i $$0.839732\pi$$
$$998$$ 83707.4i 2.65502i
$$999$$ 5214.26 0.165137
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.d.h.274.4 4
5.2 odd 4 525.4.a.k.1.1 2
5.3 odd 4 105.4.a.f.1.2 2
5.4 even 2 inner 525.4.d.h.274.1 4
15.2 even 4 1575.4.a.w.1.2 2
15.8 even 4 315.4.a.i.1.1 2
20.3 even 4 1680.4.a.bg.1.2 2
35.13 even 4 735.4.a.p.1.2 2
105.83 odd 4 2205.4.a.z.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.f.1.2 2 5.3 odd 4
315.4.a.i.1.1 2 15.8 even 4
525.4.a.k.1.1 2 5.2 odd 4
525.4.d.h.274.1 4 5.4 even 2 inner
525.4.d.h.274.4 4 1.1 even 1 trivial
735.4.a.p.1.2 2 35.13 even 4
1575.4.a.w.1.2 2 15.2 even 4
1680.4.a.bg.1.2 2 20.3 even 4
2205.4.a.z.1.1 2 105.83 odd 4