# Properties

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

## $q$-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 $$-0.704303\pi$$
0.598667 0.800998i $$-0.295697\pi$$
$$3$$ 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 $$-0.990023\pi$$
0.999509 0.0313378i $$-0.00997677\pi$$
$$14$$ 31.7179 0.605498
$$15$$ 0 0
$$16$$ −7.21984 −0.112810
$$17$$ 6.49806i 0.0927066i 0.998925 + 0.0463533i $$0.0147600\pi$$
−0.998925 + 0.0463533i $$0.985240\pi$$
$$18$$ 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 $$0.0718541\pi$$
−0.974630 + 0.223824i $$0.928146\pi$$
$$24$$ −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 $$0.141147\pi$$
−0.903286 + 0.429038i $$0.858853\pi$$
$$38$$ − 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 $$-0.820787\pi$$
0.845650 0.533738i $$-0.179213\pi$$
$$44$$ 238.872 0.818437
$$45$$ 0 0
$$46$$ 223.735 0.717130
$$47$$ − 86.5058i − 0.268472i −0.990949 0.134236i $$-0.957142\pi$$
0.990949 0.134236i $$-0.0428580\pi$$
$$48$$ − 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 $$0.229642\pi$$
−0.750855 + 0.660467i $$0.770358\pi$$
$$54$$ −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 $$-0.939862\pi$$
0.982206 0.187806i $$-0.0601375\pi$$
$$68$$ − 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 $$-0.700823\pi$$
0.589876 0.807494i $$-0.299177\pi$$
$$74$$ 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 $$0.344182\pi$$
−0.470200 + 0.882560i $$0.655818\pi$$
$$84$$ 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 $$-0.705323\pi$$
0.601232 0.799075i $$-0.294677\pi$$
$$98$$ 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 $$0.217901\pi$$
−0.774700 + 0.632329i $$0.782099\pi$$
$$104$$ 60.3152 0.0568691
$$105$$ 0 0
$$106$$ 2309.41 2.11613
$$107$$ − 1745.71i − 1.57724i −0.614883 0.788619i $$-0.710797\pi$$
0.614883 0.788619i $$-0.289203\pi$$
$$108$$ 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 $$0.239470\pi$$
−0.730108 + 0.683332i $$0.760530\pi$$
$$114$$ 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 $$0.0952728\pi$$
−0.955541 + 0.294859i $$0.904727\pi$$
$$128$$ − 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 $$0.221660\pi$$
−0.767179 + 0.641433i $$0.778340\pi$$
$$138$$ 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 $$0.223867\pi$$
−0.762712 + 0.646738i $$0.776133\pi$$
$$158$$ − 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 $$-0.954361\pi$$
0.989739 0.142887i $$-0.0456385\pi$$
$$164$$ −3948.72 −1.88014
$$165$$ 0 0
$$166$$ 6047.81 2.82772
$$167$$ − 928.498i − 0.430236i −0.976588 0.215118i $$-0.930986\pi$$
0.976588 0.215118i $$-0.0690135\pi$$
$$168$$ − 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 $$-0.977905\pi$$
0.997592 0.0693578i $$-0.0220950\pi$$
$$174$$ 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 $$-0.895249\pi$$
0.946338 0.323178i $$-0.104751\pi$$
$$194$$ −6918.01 −2.56023
$$195$$ 0 0
$$196$$ 614.025 0.223770
$$197$$ − 358.230i − 0.129557i −0.997900 0.0647787i $$-0.979366\pi$$
0.997900 0.0647787i $$-0.0206342\pi$$
$$198$$ − 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 $$0.157365\pi$$
−0.880265 + 0.474482i $$0.842635\pi$$
$$224$$ −1378.74 −0.411255
$$225$$ 0 0
$$226$$ 7438.51 2.18939
$$227$$ 3651.11i 1.06755i 0.845628 + 0.533773i $$0.179226\pi$$
−0.845628 + 0.533773i $$0.820774\pi$$
$$228$$ − 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 $$-0.826788\pi$$
0.855561 0.517702i $$-0.173212\pi$$
$$234$$ 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 $$0.148132\pi$$
−0.893656 + 0.448753i $$0.851868\pi$$
$$258$$ − 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 $$0.328294\pi$$
−0.513646 + 0.858002i $$0.671706\pi$$
$$264$$ 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 $$-0.951526\pi$$
0.988427 0.151699i $$-0.0484744\pi$$
$$278$$ − 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 $$0.155369\pi$$
−0.883223 + 0.468953i $$0.844631\pi$$
$$284$$ −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 $$0.273208\pi$$
−0.653718 + 0.756738i $$0.726792\pi$$
$$294$$ −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 $$-0.656734\pi$$
0.472737 0.881203i $$-0.343266\pi$$
$$308$$ 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 $$0.168539\pi$$
−0.863070 + 0.505084i $$0.831461\pi$$
$$314$$ 11529.6 2.07214
$$315$$ 0 0
$$316$$ −10817.3 −1.92571
$$317$$ − 3567.81i − 0.632139i −0.948736 0.316070i $$-0.897637\pi$$
0.948736 0.316070i $$-0.102363\pi$$
$$318$$ 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 $$-0.939205\pi$$
0.981816 0.189835i $$-0.0607955\pi$$
$$338$$ − 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 $$-0.986246\pi$$
0.999067 0.0431965i $$-0.0137542\pi$$
$$348$$ − 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 $$-0.808239\pi$$
0.823959 0.566650i $$-0.191761\pi$$
$$354$$ 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 $$-0.798821\pi$$
0.806835 0.590777i $$-0.201179\pi$$
$$368$$ − 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 $$-0.897700\pi$$
0.948799 0.315880i $$-0.102300\pi$$
$$374$$ −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 $$-0.971115\pi$$
0.995886 0.0906199i $$-0.0288848\pi$$
$$384$$ 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 $$0.0978717\pi$$
−0.953101 + 0.302651i $$0.902128\pi$$
$$398$$ − 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 $$-0.987808\pi$$
0.999267 0.0382939i $$-0.0121923\pi$$
$$434$$ 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 $$0.114579\pi$$
−0.935911 + 0.352236i $$0.885421\pi$$
$$444$$ 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 $$0.136865\pi$$
−0.908976 + 0.416849i $$0.863135\pi$$
$$458$$ − 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 $$-0.906722\pi$$
0.957370 0.288864i $$-0.0932776\pi$$
$$464$$ −2104.54 −0.210562
$$465$$ 0 0
$$466$$ −16685.9 −1.65871
$$467$$ 4143.73i 0.410598i 0.978699 + 0.205299i $$0.0658166\pi$$
−0.978699 + 0.205299i $$0.934183\pi$$
$$468$$ − 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 $$-0.904377\pi$$
0.955215 0.295912i $$-0.0956235\pi$$
$$488$$ − 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 $$0.168267\pi$$
−0.863501 + 0.504347i $$0.831733\pi$$
$$504$$ 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 $$-0.942347\pi$$
0.983642 0.180134i $$-0.0576532\pi$$
$$524$$ 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 $$-0.887647\pi$$
0.938351 0.345683i $$-0.112353\pi$$
$$548$$ − 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 $$0.139337\pi$$
−0.905712 + 0.423894i $$0.860663\pi$$
$$558$$ 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 $$-0.695025\pi$$
0.575071 0.818104i $$-0.304975\pi$$
$$564$$ −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 $$-0.735335\pi$$
0.673791 0.738922i $$-0.264665\pi$$
$$578$$ − 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 $$0.308394\pi$$
−0.566248 + 0.824235i $$0.691606\pi$$
$$588$$ 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 $$0.0487362\pi$$
−0.988302 + 0.152512i $$0.951264\pi$$
$$594$$ 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 $$-0.874994\pi$$
0.923872 0.382701i $$-0.125006\pi$$
$$608$$ 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 $$-0.778803\pi$$
0.768111 0.640316i $$-0.221197\pi$$
$$614$$ −42955.6 −2.82337
$$615$$ 0 0
$$616$$ 2739.59 0.179190
$$617$$ 20530.1i 1.33956i 0.742558 + 0.669781i $$0.233612\pi$$
−0.742558 + 0.669781i $$0.766388\pi$$
$$618$$ 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 $$0.0697821\pi$$
−0.976066 + 0.217475i $$0.930218\pi$$
$$644$$ 4331.28 0.265026
$$645$$ 0 0
$$646$$ 160.049 0.00974777
$$647$$ − 27773.0i − 1.68758i −0.536670 0.843792i $$-0.680318\pi$$
0.536670 0.843792i $$-0.319682\pi$$
$$648$$ 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 $$0.221332\pi$$
−0.767839 + 0.640643i $$0.778668\pi$$
$$654$$ −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 $$-0.637838\pi$$
0.419624 0.907698i $$-0.362162\pi$$
$$674$$ −10642.9 −0.608231
$$675$$ 0 0
$$676$$ −27422.7 −1.56024
$$677$$ − 20440.3i − 1.16039i −0.814477 0.580195i $$-0.802976\pi$$
0.814477 0.580195i $$-0.197024\pi$$
$$678$$ 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 $$-0.778362\pi$$
0.767223 0.641381i $$-0.221638\pi$$
$$684$$ 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 $$-0.705604\pi$$
0.601937 0.798544i $$-0.294396\pi$$
$$728$$ 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 $$-0.869975\pi$$
0.917723 0.397220i $$-0.130025\pi$$
$$734$$ −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 $$0.0650844\pi$$
−0.979169 + 0.203047i $$0.934916\pi$$
$$744$$ −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 $$0.316266\pi$$
−0.545695 + 0.837984i $$0.683734\pi$$
$$758$$ − 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 $$0.151048\pi$$
−0.889507 + 0.456921i $$0.848952\pi$$
$$774$$ 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 $$-0.808990\pi$$
0.825293 0.564705i $$-0.191010\pi$$
$$788$$ 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 $$0.00826506\pi$$
−0.999663 + 0.0259625i $$0.991735\pi$$
$$798$$ 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 $$0.0155940\pi$$
−0.998800 + 0.0489705i $$0.984406\pi$$
$$824$$ −27142.0 −1.14750
$$825$$ 0 0
$$826$$ 2644.80 0.111410
$$827$$ − 10422.4i − 0.438238i −0.975698 0.219119i $$-0.929682\pi$$
0.975698 0.219119i $$-0.0703183\pi$$
$$828$$ 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 $$0.251322\pi$$
−0.704165 + 0.710037i $$0.748678\pi$$
$$854$$ −166.611 −0.00667602
$$855$$ 0 0
$$856$$ 35841.4 1.43112
$$857$$ − 6697.57i − 0.266960i −0.991052 0.133480i $$-0.957385\pi$$
0.991052 0.133480i $$-0.0426152\pi$$
$$858$$ 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 $$0.163707\pi$$
−0.870637 + 0.491926i $$0.836293\pi$$
$$864$$ 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 $$0.101452\pi$$
−0.949637 + 0.313353i $$0.898548\pi$$
$$878$$ 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 $$-0.863713\pi$$
0.909732 0.415196i $$-0.136287\pi$$
$$884$$ −239.215 −0.00910142
$$885$$ 0 0
$$886$$ 29763.0 1.12856
$$887$$ 26813.2i 1.01499i 0.861654 + 0.507496i $$0.169429\pi$$
−0.861654 + 0.507496i $$0.830571\pi$$
$$888$$ − 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 $$-0.908170\pi$$
0.958674 0.284507i $$-0.0918300\pi$$
$$908$$ − 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 $$0.105510\pi$$
−0.945565 + 0.325432i $$0.894490\pi$$
$$938$$ − 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 $$-0.884202\pi$$
0.934554 0.355820i $$-0.115798\pi$$
$$948$$ − 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 $$-0.716240\pi$$
0.628279 0.777988i $$-0.283760\pi$$
$$954$$ −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 $$-0.805778\pi$$
0.819552 0.573005i $$-0.194222\pi$$
$$968$$ − 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 $$-0.974908\pi$$
0.996895 0.0787470i $$-0.0250919\pi$$
$$978$$ − 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 $$-0.852269\pi$$
0.894220 0.447628i $$-0.147731\pi$$
$$984$$ −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 $$-0.839732\pi$$
0.875901 0.482491i $$-0.160268\pi$$
$$998$$ − 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.274.1 4
5.2 odd 4 105.4.a.f.1.2 2
5.3 odd 4 525.4.a.k.1.1 2
5.4 even 2 inner 525.4.d.h.274.4 4
15.2 even 4 315.4.a.i.1.1 2
15.8 even 4 1575.4.a.w.1.2 2
20.7 even 4 1680.4.a.bg.1.2 2
35.27 even 4 735.4.a.p.1.2 2
105.62 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 5.2 odd 4
315.4.a.i.1.1 2 15.2 even 4
525.4.a.k.1.1 2 5.3 odd 4
525.4.d.h.274.1 4 1.1 even 1 trivial
525.4.d.h.274.4 4 5.4 even 2 inner
735.4.a.p.1.2 2 35.27 even 4
1575.4.a.w.1.2 2 15.8 even 4
1680.4.a.bg.1.2 2 20.7 even 4
2205.4.a.z.1.1 2 105.62 odd 4