# Properties

 Label 525.4.d.h.274.3 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 274.3 Root $$3.53113i$$ of defining polynomial Character $$\chi$$ $$=$$ 525.274 Dual form 525.4.d.h.274.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.53113i q^{2} +3.00000i q^{3} -4.46887 q^{4} -10.5934 q^{6} +7.00000i q^{7} +12.4689i q^{8} -9.00000 q^{9} +O(q^{10})$$ $$q+3.53113i q^{2} +3.00000i q^{3} -4.46887 q^{4} -10.5934 q^{6} +7.00000i q^{7} +12.4689i q^{8} -9.00000 q^{9} -2.93774 q^{11} -13.4066i q^{12} -19.0623i q^{13} -24.7179 q^{14} -79.7802 q^{16} -122.498i q^{17} -31.7802i q^{18} -107.436 q^{19} -21.0000 q^{21} -10.3735i q^{22} +210.623i q^{23} -37.4066 q^{24} +67.3113 q^{26} -27.0000i q^{27} -31.2821i q^{28} -95.4942 q^{29} -94.3074 q^{31} -181.963i q^{32} -8.81323i q^{33} +432.556 q^{34} +40.2198 q^{36} -97.1206i q^{37} -379.370i q^{38} +57.1868 q^{39} -491.113 q^{41} -74.1537i q^{42} -43.0039i q^{43} +13.1284 q^{44} -743.735 q^{46} -473.494i q^{47} -239.340i q^{48} -49.0000 q^{49} +367.494 q^{51} +85.1868i q^{52} -183.677i q^{53} +95.3405 q^{54} -87.2821 q^{56} -322.307i q^{57} -337.202i q^{58} +760.615 q^{59} -198.747 q^{61} -333.012i q^{62} -63.0000i q^{63} +4.29373 q^{64} +31.1206 q^{66} +309.992i q^{67} +547.428i q^{68} -631.868 q^{69} +665.693 q^{71} -112.220i q^{72} +621.288i q^{73} +342.945 q^{74} +480.117 q^{76} -20.5642i q^{77} +201.934i q^{78} +24.7626 q^{79} +81.0000 q^{81} -1734.18i q^{82} -406.724i q^{83} +93.8463 q^{84} +151.852 q^{86} -286.483i q^{87} -36.6303i q^{88} -261.751 q^{89} +133.436 q^{91} -941.245i q^{92} -282.922i q^{93} +1671.97 q^{94} +545.889 q^{96} +1004.77i q^{97} -173.025i q^{98} +26.4397 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 34 q^{4} + 6 q^{6} - 36 q^{9}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.53113i 1.24844i 0.781248 + 0.624221i $$0.214584\pi$$
−0.781248 + 0.624221i $$0.785416\pi$$
$$3$$ 3.00000i 0.577350i
$$4$$ −4.46887 −0.558609
$$5$$ 0 0
$$6$$ −10.5934 −0.720789
$$7$$ 7.00000i 0.377964i
$$8$$ 12.4689i 0.551051i
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ −2.93774 −0.0805239 −0.0402619 0.999189i $$-0.512819\pi$$
−0.0402619 + 0.999189i $$0.512819\pi$$
$$12$$ − 13.4066i − 0.322513i
$$13$$ − 19.0623i − 0.406686i −0.979108 0.203343i $$-0.934819\pi$$
0.979108 0.203343i $$-0.0651807\pi$$
$$14$$ −24.7179 −0.471867
$$15$$ 0 0
$$16$$ −79.7802 −1.24656
$$17$$ − 122.498i − 1.74766i −0.486236 0.873828i $$-0.661630\pi$$
0.486236 0.873828i $$-0.338370\pi$$
$$18$$ − 31.7802i − 0.416148i
$$19$$ −107.436 −1.29723 −0.648617 0.761115i $$-0.724652\pi$$
−0.648617 + 0.761115i $$0.724652\pi$$
$$20$$ 0 0
$$21$$ −21.0000 −0.218218
$$22$$ − 10.3735i − 0.100529i
$$23$$ 210.623i 1.90947i 0.297455 + 0.954736i $$0.403862\pi$$
−0.297455 + 0.954736i $$0.596138\pi$$
$$24$$ −37.4066 −0.318150
$$25$$ 0 0
$$26$$ 67.3113 0.507724
$$27$$ − 27.0000i − 0.192450i
$$28$$ − 31.2821i − 0.211134i
$$29$$ −95.4942 −0.611477 −0.305738 0.952116i $$-0.598903\pi$$
−0.305738 + 0.952116i $$0.598903\pi$$
$$30$$ 0 0
$$31$$ −94.3074 −0.546391 −0.273195 0.961959i $$-0.588081\pi$$
−0.273195 + 0.961959i $$0.588081\pi$$
$$32$$ − 181.963i − 1.00521i
$$33$$ − 8.81323i − 0.0464905i
$$34$$ 432.556 2.18185
$$35$$ 0 0
$$36$$ 40.2198 0.186203
$$37$$ − 97.1206i − 0.431528i −0.976446 0.215764i $$-0.930776\pi$$
0.976446 0.215764i $$-0.0692242\pi$$
$$38$$ − 379.370i − 1.61952i
$$39$$ 57.1868 0.234800
$$40$$ 0 0
$$41$$ −491.113 −1.87071 −0.935353 0.353716i $$-0.884918\pi$$
−0.935353 + 0.353716i $$0.884918\pi$$
$$42$$ − 74.1537i − 0.272433i
$$43$$ − 43.0039i − 0.152512i −0.997088 0.0762562i $$-0.975703\pi$$
0.997088 0.0762562i $$-0.0242967\pi$$
$$44$$ 13.1284 0.0449814
$$45$$ 0 0
$$46$$ −743.735 −2.38387
$$47$$ − 473.494i − 1.46949i −0.678341 0.734747i $$-0.737301\pi$$
0.678341 0.734747i $$-0.262699\pi$$
$$48$$ − 239.340i − 0.719705i
$$49$$ −49.0000 −0.142857
$$50$$ 0 0
$$51$$ 367.494 1.00901
$$52$$ 85.1868i 0.227178i
$$53$$ − 183.677i − 0.476038i −0.971261 0.238019i $$-0.923502\pi$$
0.971261 0.238019i $$-0.0764980\pi$$
$$54$$ 95.3405 0.240263
$$55$$ 0 0
$$56$$ −87.2821 −0.208278
$$57$$ − 322.307i − 0.748959i
$$58$$ − 337.202i − 0.763394i
$$59$$ 760.615 1.67837 0.839183 0.543849i $$-0.183034\pi$$
0.839183 + 0.543849i $$0.183034\pi$$
$$60$$ 0 0
$$61$$ −198.747 −0.417163 −0.208582 0.978005i $$-0.566885\pi$$
−0.208582 + 0.978005i $$0.566885\pi$$
$$62$$ − 333.012i − 0.682137i
$$63$$ − 63.0000i − 0.125988i
$$64$$ 4.29373 0.00838618
$$65$$ 0 0
$$66$$ 31.1206 0.0580407
$$67$$ 309.992i 0.565247i 0.959231 + 0.282624i $$0.0912048\pi$$
−0.959231 + 0.282624i $$0.908795\pi$$
$$68$$ 547.428i 0.976256i
$$69$$ −631.868 −1.10243
$$70$$ 0 0
$$71$$ 665.693 1.11272 0.556360 0.830941i $$-0.312197\pi$$
0.556360 + 0.830941i $$0.312197\pi$$
$$72$$ − 112.220i − 0.183684i
$$73$$ 621.288i 0.996113i 0.867145 + 0.498057i $$0.165953\pi$$
−0.867145 + 0.498057i $$0.834047\pi$$
$$74$$ 342.945 0.538738
$$75$$ 0 0
$$76$$ 480.117 0.724647
$$77$$ − 20.5642i − 0.0304352i
$$78$$ 201.934i 0.293135i
$$79$$ 24.7626 0.0352659 0.0176330 0.999845i $$-0.494387\pi$$
0.0176330 + 0.999845i $$0.494387\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ − 1734.18i − 2.33547i
$$83$$ − 406.724i − 0.537876i −0.963157 0.268938i $$-0.913327\pi$$
0.963157 0.268938i $$-0.0866727\pi$$
$$84$$ 93.8463 0.121898
$$85$$ 0 0
$$86$$ 151.852 0.190403
$$87$$ − 286.483i − 0.353036i
$$88$$ − 36.6303i − 0.0443728i
$$89$$ −261.751 −0.311748 −0.155874 0.987777i $$-0.549819\pi$$
−0.155874 + 0.987777i $$0.549819\pi$$
$$90$$ 0 0
$$91$$ 133.436 0.153713
$$92$$ − 941.245i − 1.06665i
$$93$$ − 282.922i − 0.315459i
$$94$$ 1671.97 1.83458
$$95$$ 0 0
$$96$$ 545.889 0.580360
$$97$$ 1004.77i 1.05175i 0.850563 + 0.525873i $$0.176261\pi$$
−0.850563 + 0.525873i $$0.823739\pi$$
$$98$$ − 173.025i − 0.178349i
$$99$$ 26.4397 0.0268413
$$100$$ 0 0
$$101$$ −128.872 −0.126962 −0.0634812 0.997983i $$-0.520220\pi$$
−0.0634812 + 0.997983i $$0.520220\pi$$
$$102$$ 1297.67i 1.25969i
$$103$$ 806.008i 0.771051i 0.922697 + 0.385526i $$0.125980\pi$$
−0.922697 + 0.385526i $$0.874020\pi$$
$$104$$ 237.685 0.224105
$$105$$ 0 0
$$106$$ 648.587 0.594305
$$107$$ 769.712i 0.695429i 0.937600 + 0.347714i $$0.113042\pi$$
−0.937600 + 0.347714i $$0.886958\pi$$
$$108$$ 120.660i 0.107504i
$$109$$ 780.856 0.686169 0.343085 0.939304i $$-0.388528\pi$$
0.343085 + 0.939304i $$0.388528\pi$$
$$110$$ 0 0
$$111$$ 291.362 0.249143
$$112$$ − 558.461i − 0.471157i
$$113$$ − 1115.65i − 0.928771i −0.885633 0.464386i $$-0.846275\pi$$
0.885633 0.464386i $$-0.153725\pi$$
$$114$$ 1138.11 0.935032
$$115$$ 0 0
$$116$$ 426.751 0.341576
$$117$$ 171.560i 0.135562i
$$118$$ 2685.83i 2.09534i
$$119$$ 857.486 0.660552
$$120$$ 0 0
$$121$$ −1322.37 −0.993516
$$122$$ − 701.802i − 0.520804i
$$123$$ − 1473.34i − 1.08005i
$$124$$ 421.448 0.305219
$$125$$ 0 0
$$126$$ 222.461 0.157289
$$127$$ 1875.98i 1.31076i 0.755299 + 0.655381i $$0.227492\pi$$
−0.755299 + 0.655381i $$0.772508\pi$$
$$128$$ − 1440.54i − 0.994744i
$$129$$ 129.012 0.0880530
$$130$$ 0 0
$$131$$ 364.203 0.242905 0.121452 0.992597i $$-0.461245\pi$$
0.121452 + 0.992597i $$0.461245\pi$$
$$132$$ 39.3852i 0.0259700i
$$133$$ − 752.051i − 0.490309i
$$134$$ −1094.62 −0.705679
$$135$$ 0 0
$$136$$ 1527.41 0.963048
$$137$$ − 1603.13i − 0.999743i −0.866099 0.499872i $$-0.833380\pi$$
0.866099 0.499872i $$-0.166620\pi$$
$$138$$ − 2231.21i − 1.37633i
$$139$$ −2431.12 −1.48349 −0.741746 0.670681i $$-0.766002\pi$$
−0.741746 + 0.670681i $$0.766002\pi$$
$$140$$ 0 0
$$141$$ 1420.48 0.848413
$$142$$ 2350.65i 1.38917i
$$143$$ 56.0000i 0.0327479i
$$144$$ 718.021 0.415522
$$145$$ 0 0
$$146$$ −2193.85 −1.24359
$$147$$ − 147.000i − 0.0824786i
$$148$$ 434.020i 0.241055i
$$149$$ −2341.57 −1.28744 −0.643722 0.765260i $$-0.722611\pi$$
−0.643722 + 0.765260i $$0.722611\pi$$
$$150$$ 0 0
$$151$$ −2104.07 −1.13395 −0.566976 0.823734i $$-0.691887\pi$$
−0.566976 + 0.823734i $$0.691887\pi$$
$$152$$ − 1339.60i − 0.714843i
$$153$$ 1102.48i 0.582552i
$$154$$ 72.6148 0.0379966
$$155$$ 0 0
$$156$$ −255.560 −0.131162
$$157$$ 593.467i 0.301680i 0.988558 + 0.150840i $$0.0481979\pi$$
−0.988558 + 0.150840i $$0.951802\pi$$
$$158$$ 87.4399i 0.0440275i
$$159$$ 551.031 0.274840
$$160$$ 0 0
$$161$$ −1474.36 −0.721712
$$162$$ 286.021i 0.138716i
$$163$$ 2178.71i 1.04693i 0.852047 + 0.523465i $$0.175361\pi$$
−0.852047 + 0.523465i $$0.824639\pi$$
$$164$$ 2194.72 1.04499
$$165$$ 0 0
$$166$$ 1436.19 0.671508
$$167$$ − 799.502i − 0.370463i −0.982695 0.185231i $$-0.940697\pi$$
0.982695 0.185231i $$-0.0593035\pi$$
$$168$$ − 261.846i − 0.120249i
$$169$$ 1833.63 0.834606
$$170$$ 0 0
$$171$$ 966.922 0.432412
$$172$$ 192.179i 0.0851947i
$$173$$ − 1444.36i − 0.634754i −0.948299 0.317377i $$-0.897198\pi$$
0.948299 0.317377i $$-0.102802\pi$$
$$174$$ 1011.61 0.440745
$$175$$ 0 0
$$176$$ 234.374 0.100378
$$177$$ 2281.84i 0.969005i
$$178$$ − 924.276i − 0.389199i
$$179$$ −3343.49 −1.39611 −0.698056 0.716043i $$-0.745952\pi$$
−0.698056 + 0.716043i $$0.745952\pi$$
$$180$$ 0 0
$$181$$ 2251.81 0.924729 0.462365 0.886690i $$-0.347001\pi$$
0.462365 + 0.886690i $$0.347001\pi$$
$$182$$ 471.179i 0.191902i
$$183$$ − 596.241i − 0.240849i
$$184$$ −2626.23 −1.05222
$$185$$ 0 0
$$186$$ 999.035 0.393832
$$187$$ 359.868i 0.140728i
$$188$$ 2115.98i 0.820873i
$$189$$ 189.000 0.0727393
$$190$$ 0 0
$$191$$ −1001.93 −0.379565 −0.189782 0.981826i $$-0.560778\pi$$
−0.189782 + 0.981826i $$0.560778\pi$$
$$192$$ 12.8812i 0.00484177i
$$193$$ − 4054.97i − 1.51235i −0.654372 0.756173i $$-0.727067\pi$$
0.654372 0.756173i $$-0.272933\pi$$
$$194$$ −3547.99 −1.31304
$$195$$ 0 0
$$196$$ 218.975 0.0798013
$$197$$ 5140.23i 1.85902i 0.368802 + 0.929508i $$0.379768\pi$$
−0.368802 + 0.929508i $$0.620232\pi$$
$$198$$ 93.3619i 0.0335098i
$$199$$ −585.631 −0.208614 −0.104307 0.994545i $$-0.533263\pi$$
−0.104307 + 0.994545i $$0.533263\pi$$
$$200$$ 0 0
$$201$$ −929.977 −0.326346
$$202$$ − 455.062i − 0.158505i
$$203$$ − 668.459i − 0.231116i
$$204$$ −1642.28 −0.563642
$$205$$ 0 0
$$206$$ −2846.12 −0.962614
$$207$$ − 1895.60i − 0.636490i
$$208$$ 1520.79i 0.506961i
$$209$$ 315.619 0.104458
$$210$$ 0 0
$$211$$ −1055.16 −0.344266 −0.172133 0.985074i $$-0.555066\pi$$
−0.172133 + 0.985074i $$0.555066\pi$$
$$212$$ 820.829i 0.265919i
$$213$$ 1997.08i 0.642430i
$$214$$ −2717.95 −0.868203
$$215$$ 0 0
$$216$$ 336.660 0.106050
$$217$$ − 660.152i − 0.206516i
$$218$$ 2757.30i 0.856643i
$$219$$ −1863.86 −0.575106
$$220$$ 0 0
$$221$$ −2335.09 −0.710747
$$222$$ 1028.84i 0.311040i
$$223$$ 4675.85i 1.40412i 0.712119 + 0.702059i $$0.247736\pi$$
−0.712119 + 0.702059i $$0.752264\pi$$
$$224$$ 1273.74 0.379935
$$225$$ 0 0
$$226$$ 3939.49 1.15952
$$227$$ − 5443.11i − 1.59151i −0.605621 0.795754i $$-0.707075\pi$$
0.605621 0.795754i $$-0.292925\pi$$
$$228$$ 1440.35i 0.418375i
$$229$$ 536.303 0.154759 0.0773797 0.997002i $$-0.475345\pi$$
0.0773797 + 0.997002i $$0.475345\pi$$
$$230$$ 0 0
$$231$$ 61.6926 0.0175717
$$232$$ − 1190.70i − 0.336955i
$$233$$ − 183.490i − 0.0515916i −0.999667 0.0257958i $$-0.991788\pi$$
0.999667 0.0257958i $$-0.00821196\pi$$
$$234$$ −605.802 −0.169241
$$235$$ 0 0
$$236$$ −3399.09 −0.937550
$$237$$ 74.2878i 0.0203608i
$$238$$ 3027.90i 0.824661i
$$239$$ −643.218 −0.174085 −0.0870425 0.996205i $$-0.527742\pi$$
−0.0870425 + 0.996205i $$0.527742\pi$$
$$240$$ 0 0
$$241$$ −5755.61 −1.53839 −0.769194 0.639015i $$-0.779342\pi$$
−0.769194 + 0.639015i $$0.779342\pi$$
$$242$$ − 4669.46i − 1.24035i
$$243$$ 243.000i 0.0641500i
$$244$$ 888.175 0.233031
$$245$$ 0 0
$$246$$ 5202.55 1.34838
$$247$$ 2047.97i 0.527567i
$$248$$ − 1175.91i − 0.301089i
$$249$$ 1220.17 0.310543
$$250$$ 0 0
$$251$$ −5132.27 −1.29062 −0.645311 0.763920i $$-0.723272\pi$$
−0.645311 + 0.763920i $$0.723272\pi$$
$$252$$ 281.539i 0.0703781i
$$253$$ − 618.755i − 0.153758i
$$254$$ −6624.34 −1.63641
$$255$$ 0 0
$$256$$ 5121.09 1.25027
$$257$$ − 5041.74i − 1.22372i −0.790967 0.611859i $$-0.790422\pi$$
0.790967 0.611859i $$-0.209578\pi$$
$$258$$ 455.557i 0.109929i
$$259$$ 679.844 0.163102
$$260$$ 0 0
$$261$$ 859.448 0.203826
$$262$$ 1286.05i 0.303253i
$$263$$ 7577.00i 1.77649i 0.459367 + 0.888246i $$0.348076\pi$$
−0.459367 + 0.888246i $$0.651924\pi$$
$$264$$ 109.891 0.0256186
$$265$$ 0 0
$$266$$ 2655.59 0.612122
$$267$$ − 785.253i − 0.179988i
$$268$$ − 1385.32i − 0.315752i
$$269$$ −1023.10 −0.231893 −0.115947 0.993255i $$-0.536990\pi$$
−0.115947 + 0.993255i $$0.536990\pi$$
$$270$$ 0 0
$$271$$ −2251.98 −0.504790 −0.252395 0.967624i $$-0.581218\pi$$
−0.252395 + 0.967624i $$0.581218\pi$$
$$272$$ 9772.91i 2.17857i
$$273$$ 400.307i 0.0887462i
$$274$$ 5660.87 1.24812
$$275$$ 0 0
$$276$$ 2823.74 0.615829
$$277$$ 8630.72i 1.87209i 0.351875 + 0.936047i $$0.385544\pi$$
−0.351875 + 0.936047i $$0.614456\pi$$
$$278$$ − 8584.61i − 1.85205i
$$279$$ 848.767 0.182130
$$280$$ 0 0
$$281$$ −7521.62 −1.59680 −0.798402 0.602124i $$-0.794321\pi$$
−0.798402 + 0.602124i $$0.794321\pi$$
$$282$$ 5015.91i 1.05919i
$$283$$ 14.8169i 0.00311226i 0.999999 + 0.00155613i $$0.000495333\pi$$
−0.999999 + 0.00155613i $$0.999505\pi$$
$$284$$ −2974.89 −0.621576
$$285$$ 0 0
$$286$$ −197.743 −0.0408839
$$287$$ − 3437.79i − 0.707060i
$$288$$ 1637.67i 0.335071i
$$289$$ −10092.8 −2.05430
$$290$$ 0 0
$$291$$ −3014.32 −0.607226
$$292$$ − 2776.46i − 0.556438i
$$293$$ 6913.39i 1.37844i 0.724550 + 0.689222i $$0.242048\pi$$
−0.724550 + 0.689222i $$0.757952\pi$$
$$294$$ 519.076 0.102970
$$295$$ 0 0
$$296$$ 1210.98 0.237794
$$297$$ 79.3190i 0.0154968i
$$298$$ − 8268.39i − 1.60730i
$$299$$ 4014.94 0.776555
$$300$$ 0 0
$$301$$ 301.027 0.0576442
$$302$$ − 7429.74i − 1.41567i
$$303$$ − 386.615i − 0.0733018i
$$304$$ 8571.25 1.61709
$$305$$ 0 0
$$306$$ −3893.01 −0.727283
$$307$$ 7644.12i 1.42108i 0.703655 + 0.710542i $$0.251550\pi$$
−0.703655 + 0.710542i $$0.748450\pi$$
$$308$$ 91.8987i 0.0170014i
$$309$$ −2418.02 −0.445167
$$310$$ 0 0
$$311$$ 7593.99 1.38462 0.692308 0.721602i $$-0.256594\pi$$
0.692308 + 0.721602i $$0.256594\pi$$
$$312$$ 713.055i 0.129387i
$$313$$ − 9127.84i − 1.64836i −0.566329 0.824179i $$-0.691637\pi$$
0.566329 0.824179i $$-0.308363\pi$$
$$314$$ −2095.61 −0.376631
$$315$$ 0 0
$$316$$ −110.661 −0.0196999
$$317$$ 4929.81i 0.873456i 0.899593 + 0.436728i $$0.143863\pi$$
−0.899593 + 0.436728i $$0.856137\pi$$
$$318$$ 1945.76i 0.343122i
$$319$$ 280.537 0.0492385
$$320$$ 0 0
$$321$$ −2309.14 −0.401506
$$322$$ − 5206.15i − 0.901016i
$$323$$ 13160.7i 2.26712i
$$324$$ −361.979 −0.0620677
$$325$$ 0 0
$$326$$ −7693.30 −1.30703
$$327$$ 2342.57i 0.396160i
$$328$$ − 6123.62i − 1.03086i
$$329$$ 3314.46 0.555417
$$330$$ 0 0
$$331$$ 1221.67 0.202867 0.101433 0.994842i $$-0.467657\pi$$
0.101433 + 0.994842i $$0.467657\pi$$
$$332$$ 1817.60i 0.300463i
$$333$$ 874.086i 0.143843i
$$334$$ 2823.14 0.462502
$$335$$ 0 0
$$336$$ 1675.38 0.272023
$$337$$ 8744.83i 1.41354i 0.707446 + 0.706768i $$0.249847\pi$$
−0.707446 + 0.706768i $$0.750153\pi$$
$$338$$ 6474.79i 1.04196i
$$339$$ 3346.94 0.536226
$$340$$ 0 0
$$341$$ 277.051 0.0439975
$$342$$ 3414.33i 0.539841i
$$343$$ − 343.000i − 0.0539949i
$$344$$ 536.210 0.0840421
$$345$$ 0 0
$$346$$ 5100.21 0.792454
$$347$$ − 4589.56i − 0.710031i −0.934860 0.355015i $$-0.884476\pi$$
0.934860 0.355015i $$-0.115524\pi$$
$$348$$ 1280.25i 0.197209i
$$349$$ 3989.89 0.611960 0.305980 0.952038i $$-0.401016\pi$$
0.305980 + 0.952038i $$0.401016\pi$$
$$350$$ 0 0
$$351$$ −514.681 −0.0782668
$$352$$ 534.561i 0.0809437i
$$353$$ 2416.35i 0.364333i 0.983268 + 0.182166i $$0.0583109\pi$$
−0.983268 + 0.182166i $$0.941689\pi$$
$$354$$ −8057.49 −1.20975
$$355$$ 0 0
$$356$$ 1169.73 0.174145
$$357$$ 2572.46i 0.381370i
$$358$$ − 11806.3i − 1.74297i
$$359$$ 2756.24 0.405206 0.202603 0.979261i $$-0.435060\pi$$
0.202603 + 0.979261i $$0.435060\pi$$
$$360$$ 0 0
$$361$$ 4683.45 0.682818
$$362$$ 7951.44i 1.15447i
$$363$$ − 3967.11i − 0.573607i
$$364$$ −596.307 −0.0858654
$$365$$ 0 0
$$366$$ 2105.40 0.300687
$$367$$ − 11112.8i − 1.58061i −0.612711 0.790307i $$-0.709921\pi$$
0.612711 0.790307i $$-0.290079\pi$$
$$368$$ − 16803.5i − 2.38028i
$$369$$ 4420.02 0.623569
$$370$$ 0 0
$$371$$ 1285.74 0.179925
$$372$$ 1264.34i 0.176218i
$$373$$ 6091.09i 0.845535i 0.906238 + 0.422768i $$0.138941\pi$$
−0.906238 + 0.422768i $$0.861059\pi$$
$$374$$ −1270.74 −0.175691
$$375$$ 0 0
$$376$$ 5903.94 0.809767
$$377$$ 1820.33i 0.248679i
$$378$$ 667.383i 0.0908108i
$$379$$ −3984.29 −0.539998 −0.269999 0.962861i $$-0.587023\pi$$
−0.269999 + 0.962861i $$0.587023\pi$$
$$380$$ 0 0
$$381$$ −5627.95 −0.756768
$$382$$ − 3537.93i − 0.473865i
$$383$$ 318.475i 0.0424890i 0.999774 + 0.0212445i $$0.00676285\pi$$
−0.999774 + 0.0212445i $$0.993237\pi$$
$$384$$ 4321.63 0.574316
$$385$$ 0 0
$$386$$ 14318.6 1.88808
$$387$$ 387.035i 0.0508374i
$$388$$ − 4490.21i − 0.587515i
$$389$$ −3885.46 −0.506429 −0.253214 0.967410i $$-0.581488\pi$$
−0.253214 + 0.967410i $$0.581488\pi$$
$$390$$ 0 0
$$391$$ 25800.9 3.33710
$$392$$ − 610.975i − 0.0787216i
$$393$$ 1092.61i 0.140241i
$$394$$ −18150.8 −2.32088
$$395$$ 0 0
$$396$$ −118.156 −0.0149938
$$397$$ − 4806.04i − 0.607578i −0.952739 0.303789i $$-0.901748\pi$$
0.952739 0.303789i $$-0.0982518\pi$$
$$398$$ − 2067.94i − 0.260443i
$$399$$ 2256.15 0.283080
$$400$$ 0 0
$$401$$ 3618.59 0.450633 0.225316 0.974286i $$-0.427658\pi$$
0.225316 + 0.974286i $$0.427658\pi$$
$$402$$ − 3283.87i − 0.407424i
$$403$$ 1797.71i 0.222209i
$$404$$ 575.911 0.0709223
$$405$$ 0 0
$$406$$ 2360.42 0.288536
$$407$$ 285.315i 0.0347483i
$$408$$ 4582.24i 0.556016i
$$409$$ 2109.05 0.254978 0.127489 0.991840i $$-0.459308\pi$$
0.127489 + 0.991840i $$0.459308\pi$$
$$410$$ 0 0
$$411$$ 4809.40 0.577202
$$412$$ − 3601.94i − 0.430716i
$$413$$ 5324.30i 0.634363i
$$414$$ 6693.62 0.794622
$$415$$ 0 0
$$416$$ −3468.63 −0.408806
$$417$$ − 7293.37i − 0.856494i
$$418$$ 1114.49i 0.130410i
$$419$$ 6905.91 0.805193 0.402597 0.915377i $$-0.368108\pi$$
0.402597 + 0.915377i $$0.368108\pi$$
$$420$$ 0 0
$$421$$ −9647.54 −1.11685 −0.558423 0.829556i $$-0.688593\pi$$
−0.558423 + 0.829556i $$0.688593\pi$$
$$422$$ − 3725.91i − 0.429797i
$$423$$ 4261.45i 0.489831i
$$424$$ 2290.25 0.262321
$$425$$ 0 0
$$426$$ −7051.94 −0.802037
$$427$$ − 1391.23i − 0.157673i
$$428$$ − 3439.74i − 0.388473i
$$429$$ −168.000 −0.0189070
$$430$$ 0 0
$$431$$ −13002.7 −1.45318 −0.726589 0.687073i $$-0.758895\pi$$
−0.726589 + 0.687073i $$0.758895\pi$$
$$432$$ 2154.06i 0.239902i
$$433$$ 7356.07i 0.816420i 0.912888 + 0.408210i $$0.133847\pi$$
−0.912888 + 0.408210i $$0.866153\pi$$
$$434$$ 2331.08 0.257824
$$435$$ 0 0
$$436$$ −3489.55 −0.383300
$$437$$ − 22628.4i − 2.47703i
$$438$$ − 6581.54i − 0.717987i
$$439$$ −6909.21 −0.751159 −0.375579 0.926790i $$-0.622556\pi$$
−0.375579 + 0.926790i $$0.622556\pi$$
$$440$$ 0 0
$$441$$ 441.000 0.0476190
$$442$$ − 8245.50i − 0.887327i
$$443$$ − 14812.6i − 1.58864i −0.607502 0.794318i $$-0.707828\pi$$
0.607502 0.794318i $$-0.292172\pi$$
$$444$$ −1302.06 −0.139173
$$445$$ 0 0
$$446$$ −16511.0 −1.75296
$$447$$ − 7024.72i − 0.743306i
$$448$$ 30.0561i 0.00316968i
$$449$$ 10654.5 1.11986 0.559932 0.828538i $$-0.310827\pi$$
0.559932 + 0.828538i $$0.310827\pi$$
$$450$$ 0 0
$$451$$ 1442.76 0.150636
$$452$$ 4985.68i 0.518820i
$$453$$ − 6312.21i − 0.654688i
$$454$$ 19220.3 1.98691
$$455$$ 0 0
$$456$$ 4018.81 0.412715
$$457$$ 5855.16i 0.599328i 0.954045 + 0.299664i $$0.0968745\pi$$
−0.954045 + 0.299664i $$0.903125\pi$$
$$458$$ 1893.76i 0.193208i
$$459$$ −3307.45 −0.336336
$$460$$ 0 0
$$461$$ 3204.74 0.323774 0.161887 0.986809i $$-0.448242\pi$$
0.161887 + 0.986809i $$0.448242\pi$$
$$462$$ 217.844i 0.0219373i
$$463$$ 371.658i 0.0373054i 0.999826 + 0.0186527i $$0.00593769\pi$$
−0.999826 + 0.0186527i $$0.994062\pi$$
$$464$$ 7618.54 0.762245
$$465$$ 0 0
$$466$$ 647.927 0.0644091
$$467$$ 19752.3i 1.95723i 0.205703 + 0.978614i $$0.434052\pi$$
−0.205703 + 0.978614i $$0.565948\pi$$
$$468$$ − 766.681i − 0.0757262i
$$469$$ −2169.95 −0.213643
$$470$$ 0 0
$$471$$ −1780.40 −0.174175
$$472$$ 9484.01i 0.924866i
$$473$$ 126.334i 0.0122809i
$$474$$ −262.320 −0.0254193
$$475$$ 0 0
$$476$$ −3832.00 −0.368990
$$477$$ 1653.09i 0.158679i
$$478$$ − 2271.28i − 0.217335i
$$479$$ 20762.0 1.98046 0.990232 0.139433i $$-0.0445279\pi$$
0.990232 + 0.139433i $$0.0445279\pi$$
$$480$$ 0 0
$$481$$ −1851.34 −0.175496
$$482$$ − 20323.8i − 1.92059i
$$483$$ − 4423.07i − 0.416681i
$$484$$ 5909.50 0.554987
$$485$$ 0 0
$$486$$ −858.064 −0.0800876
$$487$$ − 17647.6i − 1.64207i −0.570878 0.821035i $$-0.693397\pi$$
0.570878 0.821035i $$-0.306603\pi$$
$$488$$ − 2478.15i − 0.229878i
$$489$$ −6536.12 −0.604445
$$490$$ 0 0
$$491$$ −5637.46 −0.518157 −0.259078 0.965856i $$-0.583419\pi$$
−0.259078 + 0.965856i $$0.583419\pi$$
$$492$$ 6584.16i 0.603327i
$$493$$ 11697.9i 1.06865i
$$494$$ −7231.64 −0.658638
$$495$$ 0 0
$$496$$ 7523.86 0.681112
$$497$$ 4659.85i 0.420569i
$$498$$ 4308.58i 0.387695i
$$499$$ 17474.1 1.56764 0.783818 0.620991i $$-0.213270\pi$$
0.783818 + 0.620991i $$0.213270\pi$$
$$500$$ 0 0
$$501$$ 2398.51 0.213887
$$502$$ − 18122.7i − 1.61127i
$$503$$ 7444.81i 0.659936i 0.943992 + 0.329968i $$0.107038\pi$$
−0.943992 + 0.329968i $$0.892962\pi$$
$$504$$ 785.539 0.0694260
$$505$$ 0 0
$$506$$ 2184.90 0.191958
$$507$$ 5500.89i 0.481860i
$$508$$ − 8383.53i − 0.732203i
$$509$$ 3384.48 0.294724 0.147362 0.989083i $$-0.452922\pi$$
0.147362 + 0.989083i $$0.452922\pi$$
$$510$$ 0 0
$$511$$ −4349.02 −0.376495
$$512$$ 6558.89i 0.566142i
$$513$$ 2900.77i 0.249653i
$$514$$ 17803.0 1.52774
$$515$$ 0 0
$$516$$ −576.536 −0.0491872
$$517$$ 1391.00i 0.118329i
$$518$$ 2400.62i 0.203624i
$$519$$ 4333.07 0.366476
$$520$$ 0 0
$$521$$ 2973.12 0.250009 0.125005 0.992156i $$-0.460105\pi$$
0.125005 + 0.992156i $$0.460105\pi$$
$$522$$ 3034.82i 0.254465i
$$523$$ 2689.02i 0.224823i 0.993662 + 0.112412i $$0.0358575\pi$$
−0.993662 + 0.112412i $$0.964142\pi$$
$$524$$ −1627.57 −0.135689
$$525$$ 0 0
$$526$$ −26755.3 −2.21785
$$527$$ 11552.5i 0.954903i
$$528$$ 703.121i 0.0579534i
$$529$$ −32194.9 −2.64608
$$530$$ 0 0
$$531$$ −6845.53 −0.559455
$$532$$ 3360.82i 0.273891i
$$533$$ 9361.72i 0.760790i
$$534$$ 2772.83 0.224704
$$535$$ 0 0
$$536$$ −3865.25 −0.311480
$$537$$ − 10030.5i − 0.806046i
$$538$$ − 3612.69i − 0.289506i
$$539$$ 143.949 0.0115034
$$540$$ 0 0
$$541$$ −14429.5 −1.14671 −0.573356 0.819306i $$-0.694359\pi$$
−0.573356 + 0.819306i $$0.694359\pi$$
$$542$$ − 7952.03i − 0.630201i
$$543$$ 6755.44i 0.533893i
$$544$$ −22290.1 −1.75677
$$545$$ 0 0
$$546$$ −1413.54 −0.110795
$$547$$ − 13811.2i − 1.07957i −0.841804 0.539784i $$-0.818506\pi$$
0.841804 0.539784i $$-0.181494\pi$$
$$548$$ 7164.19i 0.558466i
$$549$$ 1788.72 0.139054
$$550$$ 0 0
$$551$$ 10259.5 0.793229
$$552$$ − 7878.68i − 0.607498i
$$553$$ 173.338i 0.0133293i
$$554$$ −30476.2 −2.33720
$$555$$ 0 0
$$556$$ 10864.4 0.828692
$$557$$ 6033.26i 0.458954i 0.973314 + 0.229477i $$0.0737016\pi$$
−0.973314 + 0.229477i $$0.926298\pi$$
$$558$$ 2997.10i 0.227379i
$$559$$ −819.751 −0.0620246
$$560$$ 0 0
$$561$$ −1079.60 −0.0812493
$$562$$ − 26559.8i − 1.99352i
$$563$$ − 6958.47i − 0.520896i −0.965488 0.260448i $$-0.916130\pi$$
0.965488 0.260448i $$-0.0838703\pi$$
$$564$$ −6347.95 −0.473931
$$565$$ 0 0
$$566$$ −52.3202 −0.00388548
$$567$$ 567.000i 0.0419961i
$$568$$ 8300.44i 0.613166i
$$569$$ 13396.4 0.987009 0.493505 0.869743i $$-0.335716\pi$$
0.493505 + 0.869743i $$0.335716\pi$$
$$570$$ 0 0
$$571$$ −8055.84 −0.590414 −0.295207 0.955433i $$-0.595389\pi$$
−0.295207 + 0.955433i $$0.595389\pi$$
$$572$$ − 250.257i − 0.0182933i
$$573$$ − 3005.78i − 0.219142i
$$574$$ 12139.3 0.882724
$$575$$ 0 0
$$576$$ −38.6435 −0.00279539
$$577$$ 21456.9i 1.54812i 0.633114 + 0.774059i $$0.281777\pi$$
−0.633114 + 0.774059i $$0.718223\pi$$
$$578$$ − 35638.9i − 2.56468i
$$579$$ 12164.9 0.873153
$$580$$ 0 0
$$581$$ 2847.07 0.203298
$$582$$ − 10644.0i − 0.758087i
$$583$$ 539.596i 0.0383324i
$$584$$ −7746.76 −0.548910
$$585$$ 0 0
$$586$$ −24412.1 −1.72091
$$587$$ − 20156.3i − 1.41728i −0.705572 0.708638i $$-0.749310\pi$$
0.705572 0.708638i $$-0.250690\pi$$
$$588$$ 656.924i 0.0460733i
$$589$$ 10132.0 0.708797
$$590$$ 0 0
$$591$$ −15420.7 −1.07330
$$592$$ 7748.30i 0.537928i
$$593$$ 599.307i 0.0415018i 0.999785 + 0.0207509i $$0.00660570\pi$$
−0.999785 + 0.0207509i $$0.993394\pi$$
$$594$$ −280.086 −0.0193469
$$595$$ 0 0
$$596$$ 10464.2 0.719177
$$597$$ − 1756.89i − 0.120444i
$$598$$ 14177.3i 0.969485i
$$599$$ 5493.05 0.374691 0.187346 0.982294i $$-0.440012\pi$$
0.187346 + 0.982294i $$0.440012\pi$$
$$600$$ 0 0
$$601$$ 24292.8 1.64879 0.824396 0.566014i $$-0.191515\pi$$
0.824396 + 0.566014i $$0.191515\pi$$
$$602$$ 1062.97i 0.0719655i
$$603$$ − 2789.93i − 0.188416i
$$604$$ 9402.82 0.633436
$$605$$ 0 0
$$606$$ 1365.19 0.0915131
$$607$$ − 3029.50i − 0.202576i −0.994857 0.101288i $$-0.967704\pi$$
0.994857 0.101288i $$-0.0322964\pi$$
$$608$$ 19549.3i 1.30400i
$$609$$ 2005.38 0.133435
$$610$$ 0 0
$$611$$ −9025.87 −0.597623
$$612$$ − 4926.85i − 0.325419i
$$613$$ − 19339.6i − 1.27426i −0.770757 0.637129i $$-0.780122\pi$$
0.770757 0.637129i $$-0.219878\pi$$
$$614$$ −26992.4 −1.77414
$$615$$ 0 0
$$616$$ 256.412 0.0167713
$$617$$ 5743.91i 0.374783i 0.982285 + 0.187391i $$0.0600033\pi$$
−0.982285 + 0.187391i $$0.939997\pi$$
$$618$$ − 8538.35i − 0.555765i
$$619$$ 8243.35 0.535264 0.267632 0.963521i $$-0.413759\pi$$
0.267632 + 0.963521i $$0.413759\pi$$
$$620$$ 0 0
$$621$$ 5686.81 0.367478
$$622$$ 26815.4i 1.72861i
$$623$$ − 1832.26i − 0.117830i
$$624$$ −4562.37 −0.292694
$$625$$ 0 0
$$626$$ 32231.6 2.05788
$$627$$ 946.856i 0.0603091i
$$628$$ − 2652.13i − 0.168521i
$$629$$ −11897.1 −0.754162
$$630$$ 0 0
$$631$$ −4376.56 −0.276114 −0.138057 0.990424i $$-0.544086\pi$$
−0.138057 + 0.990424i $$0.544086\pi$$
$$632$$ 308.762i 0.0194334i
$$633$$ − 3165.48i − 0.198762i
$$634$$ −17407.8 −1.09046
$$635$$ 0 0
$$636$$ −2462.49 −0.153528
$$637$$ 934.051i 0.0580980i
$$638$$ 990.613i 0.0614714i
$$639$$ −5991.23 −0.370907
$$640$$ 0 0
$$641$$ 11836.6 0.729357 0.364678 0.931133i $$-0.381179\pi$$
0.364678 + 0.931133i $$0.381179\pi$$
$$642$$ − 8153.86i − 0.501257i
$$643$$ 1448.21i 0.0888209i 0.999013 + 0.0444104i $$0.0141409\pi$$
−0.999013 + 0.0444104i $$0.985859\pi$$
$$644$$ 6588.72 0.403155
$$645$$ 0 0
$$646$$ −46472.0 −2.83037
$$647$$ 8732.95i 0.530646i 0.964160 + 0.265323i $$0.0854785\pi$$
−0.964160 + 0.265323i $$0.914521\pi$$
$$648$$ 1009.98i 0.0612279i
$$649$$ −2234.49 −0.135149
$$650$$ 0 0
$$651$$ 1980.46 0.119232
$$652$$ − 9736.37i − 0.584824i
$$653$$ − 21978.4i − 1.31712i −0.752527 0.658562i $$-0.771165\pi$$
0.752527 0.658562i $$-0.228835\pi$$
$$654$$ −8271.91 −0.494583
$$655$$ 0 0
$$656$$ 39181.1 2.33196
$$657$$ − 5591.59i − 0.332038i
$$658$$ 11703.8i 0.693406i
$$659$$ 27761.7 1.64103 0.820516 0.571623i $$-0.193686\pi$$
0.820516 + 0.571623i $$0.193686\pi$$
$$660$$ 0 0
$$661$$ −8573.72 −0.504507 −0.252254 0.967661i $$-0.581172\pi$$
−0.252254 + 0.967661i $$0.581172\pi$$
$$662$$ 4313.86i 0.253267i
$$663$$ − 7005.27i − 0.410350i
$$664$$ 5071.39 0.296398
$$665$$ 0 0
$$666$$ −3086.51 −0.179579
$$667$$ − 20113.2i − 1.16760i
$$668$$ 3572.87i 0.206944i
$$669$$ −14027.6 −0.810668
$$670$$ 0 0
$$671$$ 583.868 0.0335916
$$672$$ 3821.22i 0.219356i
$$673$$ 27159.2i 1.55559i 0.628518 + 0.777795i $$0.283662\pi$$
−0.628518 + 0.777795i $$0.716338\pi$$
$$674$$ −30879.1 −1.76472
$$675$$ 0 0
$$676$$ −8194.26 −0.466219
$$677$$ 1392.30i 0.0790404i 0.999219 + 0.0395202i $$0.0125829\pi$$
−0.999219 + 0.0395202i $$0.987417\pi$$
$$678$$ 11818.5i 0.669448i
$$679$$ −7033.42 −0.397523
$$680$$ 0 0
$$681$$ 16329.3 0.918857
$$682$$ 978.302i 0.0549283i
$$683$$ − 8675.09i − 0.486007i −0.970025 0.243004i $$-0.921867\pi$$
0.970025 0.243004i $$-0.0781327\pi$$
$$684$$ −4321.05 −0.241549
$$685$$ 0 0
$$686$$ 1211.18 0.0674096
$$687$$ 1608.91i 0.0893504i
$$688$$ 3430.86i 0.190117i
$$689$$ −3501.30 −0.193598
$$690$$ 0 0
$$691$$ −21426.0 −1.17957 −0.589785 0.807561i $$-0.700787\pi$$
−0.589785 + 0.807561i $$0.700787\pi$$
$$692$$ 6454.65i 0.354579i
$$693$$ 185.078i 0.0101451i
$$694$$ 16206.3 0.886433
$$695$$ 0 0
$$696$$ 3572.11 0.194541
$$697$$ 60160.4i 3.26935i
$$698$$ 14088.8i 0.763997i
$$699$$ 550.470 0.0297864
$$700$$ 0 0
$$701$$ 24840.5 1.33839 0.669197 0.743085i $$-0.266638\pi$$
0.669197 + 0.743085i $$0.266638\pi$$
$$702$$ − 1817.40i − 0.0977116i
$$703$$ 10434.2i 0.559793i
$$704$$ −12.6139 −0.000675288 0
$$705$$ 0 0
$$706$$ −8532.45 −0.454848
$$707$$ − 902.101i − 0.0479873i
$$708$$ − 10197.3i − 0.541295i
$$709$$ −12525.0 −0.663450 −0.331725 0.943376i $$-0.607631\pi$$
−0.331725 + 0.943376i $$0.607631\pi$$
$$710$$ 0 0
$$711$$ −222.863 −0.0117553
$$712$$ − 3263.74i − 0.171789i
$$713$$ − 19863.3i − 1.04332i
$$714$$ −9083.69 −0.476118
$$715$$ 0 0
$$716$$ 14941.6 0.779881
$$717$$ − 1929.65i − 0.100508i
$$718$$ 9732.66i 0.505877i
$$719$$ −28085.0 −1.45674 −0.728369 0.685185i $$-0.759721\pi$$
−0.728369 + 0.685185i $$0.759721\pi$$
$$720$$ 0 0
$$721$$ −5642.05 −0.291430
$$722$$ 16537.9i 0.852460i
$$723$$ − 17266.8i − 0.888189i
$$724$$ −10063.1 −0.516562
$$725$$ 0 0
$$726$$ 14008.4 0.716115
$$727$$ 14326.2i 0.730851i 0.930841 + 0.365426i $$0.119077\pi$$
−0.930841 + 0.365426i $$0.880923\pi$$
$$728$$ 1663.79i 0.0847037i
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ −5267.89 −0.266539
$$732$$ 2664.53i 0.134541i
$$733$$ 6727.85i 0.339016i 0.985529 + 0.169508i $$0.0542179\pi$$
−0.985529 + 0.169508i $$0.945782\pi$$
$$734$$ 39240.8 1.97331
$$735$$ 0 0
$$736$$ 38325.5 1.91943
$$737$$ − 910.677i − 0.0455159i
$$738$$ 15607.6i 0.778490i
$$739$$ 3418.51 0.170165 0.0850826 0.996374i $$-0.472885\pi$$
0.0850826 + 0.996374i $$0.472885\pi$$
$$740$$ 0 0
$$741$$ −6143.91 −0.304591
$$742$$ 4540.11i 0.224626i
$$743$$ 8095.50i 0.399724i 0.979824 + 0.199862i $$0.0640494\pi$$
−0.979824 + 0.199862i $$0.935951\pi$$
$$744$$ 3527.72 0.173834
$$745$$ 0 0
$$746$$ −21508.4 −1.05560
$$747$$ 3660.51i 0.179292i
$$748$$ − 1608.20i − 0.0786119i
$$749$$ −5387.99 −0.262847
$$750$$ 0 0
$$751$$ 13446.8 0.653371 0.326686 0.945133i $$-0.394068\pi$$
0.326686 + 0.945133i $$0.394068\pi$$
$$752$$ 37775.4i 1.83182i
$$753$$ − 15396.8i − 0.745141i
$$754$$ −6427.84 −0.310462
$$755$$ 0 0
$$756$$ −844.617 −0.0406328
$$757$$ 2593.24i 0.124508i 0.998060 + 0.0622541i $$0.0198289\pi$$
−0.998060 + 0.0622541i $$0.980171\pi$$
$$758$$ − 14069.0i − 0.674156i
$$759$$ 1856.26 0.0887722
$$760$$ 0 0
$$761$$ 27079.4 1.28992 0.644959 0.764217i $$-0.276875\pi$$
0.644959 + 0.764217i $$0.276875\pi$$
$$762$$ − 19873.0i − 0.944782i
$$763$$ 5465.99i 0.259348i
$$764$$ 4477.48 0.212028
$$765$$ 0 0
$$766$$ −1124.58 −0.0530451
$$767$$ − 14499.0i − 0.682568i
$$768$$ 15363.3i 0.721842i
$$769$$ −2138.72 −0.100292 −0.0501458 0.998742i $$-0.515969\pi$$
−0.0501458 + 0.998742i $$0.515969\pi$$
$$770$$ 0 0
$$771$$ 15125.2 0.706513
$$772$$ 18121.1i 0.844810i
$$773$$ 25864.0i 1.20345i 0.798704 + 0.601724i $$0.205519\pi$$
−0.798704 + 0.601724i $$0.794481\pi$$
$$774$$ −1366.67 −0.0634676
$$775$$ 0 0
$$776$$ −12528.4 −0.579566
$$777$$ 2039.53i 0.0941671i
$$778$$ − 13720.1i − 0.632247i
$$779$$ 52763.1 2.42675
$$780$$ 0 0
$$781$$ −1955.63 −0.0896006
$$782$$ 91106.2i 4.16618i
$$783$$ 2578.34i 0.117679i
$$784$$ 3909.23 0.178081
$$785$$ 0 0
$$786$$ −3858.14 −0.175083
$$787$$ 32371.3i 1.46621i 0.680113 + 0.733107i $$0.261931\pi$$
−0.680113 + 0.733107i $$0.738069\pi$$
$$788$$ − 22971.0i − 1.03846i
$$789$$ −22731.0 −1.02566
$$790$$ 0 0
$$791$$ 7809.52 0.351043
$$792$$ 329.673i 0.0147909i
$$793$$ 3788.57i 0.169654i
$$794$$ 16970.8 0.758526
$$795$$ 0 0
$$796$$ 2617.11 0.116534
$$797$$ − 2024.33i − 0.0899691i −0.998988 0.0449845i $$-0.985676\pi$$
0.998988 0.0449845i $$-0.0143239\pi$$
$$798$$ 7966.76i 0.353409i
$$799$$ −58002.1 −2.56817
$$800$$ 0 0
$$801$$ 2355.76 0.103916
$$802$$ 12777.7i 0.562589i
$$803$$ − 1825.18i − 0.0802109i
$$804$$ 4155.95 0.182300
$$805$$ 0 0
$$806$$ −6347.95 −0.277416
$$807$$ − 3069.29i − 0.133884i
$$808$$ − 1606.88i − 0.0699628i
$$809$$ −12391.7 −0.538526 −0.269263 0.963067i $$-0.586780\pi$$
−0.269263 + 0.963067i $$0.586780\pi$$
$$810$$ 0 0
$$811$$ 14654.5 0.634511 0.317256 0.948340i $$-0.397239\pi$$
0.317256 + 0.948340i $$0.397239\pi$$
$$812$$ 2987.26i 0.129104i
$$813$$ − 6755.94i − 0.291441i
$$814$$ −1007.49 −0.0433813
$$815$$ 0 0
$$816$$ −29318.7 −1.25780
$$817$$ 4620.16i 0.197844i
$$818$$ 7447.33i 0.318325i
$$819$$ −1200.92 −0.0512376
$$820$$ 0 0
$$821$$ 23887.9 1.01546 0.507731 0.861516i $$-0.330485\pi$$
0.507731 + 0.861516i $$0.330485\pi$$
$$822$$ 16982.6i 0.720604i
$$823$$ − 4008.41i − 0.169774i −0.996391 0.0848871i $$-0.972947\pi$$
0.996391 0.0848871i $$-0.0270530\pi$$
$$824$$ −10050.0 −0.424889
$$825$$ 0 0
$$826$$ −18800.8 −0.791966
$$827$$ 45110.4i 1.89679i 0.317096 + 0.948394i $$0.397292\pi$$
−0.317096 + 0.948394i $$0.602708\pi$$
$$828$$ 8471.21i 0.355549i
$$829$$ −16165.4 −0.677260 −0.338630 0.940920i $$-0.609964\pi$$
−0.338630 + 0.940920i $$0.609964\pi$$
$$830$$ 0 0
$$831$$ −25892.2 −1.08085
$$832$$ − 81.8481i − 0.00341054i
$$833$$ 6002.41i 0.249665i
$$834$$ 25753.8 1.06928
$$835$$ 0 0
$$836$$ −1410.46 −0.0583514
$$837$$ 2546.30i 0.105153i
$$838$$ 24385.7i 1.00524i
$$839$$ 25244.4 1.03878 0.519388 0.854538i $$-0.326160\pi$$
0.519388 + 0.854538i $$0.326160\pi$$
$$840$$ 0 0
$$841$$ −15269.9 −0.626096
$$842$$ − 34066.7i − 1.39432i
$$843$$ − 22564.9i − 0.921916i
$$844$$ 4715.37 0.192310
$$845$$ 0 0
$$846$$ −15047.7 −0.611526
$$847$$ − 9256.59i − 0.375514i
$$848$$ 14653.8i 0.593412i
$$849$$ −44.4506 −0.00179687
$$850$$ 0 0
$$851$$ 20455.8 0.823990
$$852$$ − 8924.68i − 0.358867i
$$853$$ − 30168.1i − 1.21094i −0.795867 0.605472i $$-0.792984\pi$$
0.795867 0.605472i $$-0.207016\pi$$
$$854$$ 4912.61 0.196846
$$855$$ 0 0
$$856$$ −9597.44 −0.383217
$$857$$ 13393.6i 0.533857i 0.963716 + 0.266929i $$0.0860088\pi$$
−0.963716 + 0.266929i $$0.913991\pi$$
$$858$$ − 593.230i − 0.0236043i
$$859$$ −19060.4 −0.757081 −0.378541 0.925585i $$-0.623574\pi$$
−0.378541 + 0.925585i $$0.623574\pi$$
$$860$$ 0 0
$$861$$ 10313.4 0.408222
$$862$$ − 45914.3i − 1.81421i
$$863$$ − 9466.86i − 0.373413i −0.982416 0.186707i $$-0.940219\pi$$
0.982416 0.186707i $$-0.0597814\pi$$
$$864$$ −4913.00 −0.193453
$$865$$ 0 0
$$866$$ −25975.2 −1.01925
$$867$$ − 30278.3i − 1.18605i
$$868$$ 2950.13i 0.115362i
$$869$$ −72.7461 −0.00283975
$$870$$ 0 0
$$871$$ 5909.15 0.229878
$$872$$ 9736.39i 0.378115i
$$873$$ − 9042.97i − 0.350582i
$$874$$ 79903.8 3.09243
$$875$$ 0 0
$$876$$ 8329.37 0.321259
$$877$$ − 37740.6i − 1.45315i −0.687090 0.726573i $$-0.741112\pi$$
0.687090 0.726573i $$-0.258888\pi$$
$$878$$ − 24397.3i − 0.937778i
$$879$$ −20740.2 −0.795845
$$880$$ 0 0
$$881$$ 25991.5 0.993957 0.496979 0.867763i $$-0.334443\pi$$
0.496979 + 0.867763i $$0.334443\pi$$
$$882$$ 1557.23i 0.0594496i
$$883$$ 39420.3i 1.50238i 0.660087 + 0.751189i $$0.270519\pi$$
−0.660087 + 0.751189i $$0.729481\pi$$
$$884$$ 10435.2 0.397030
$$885$$ 0 0
$$886$$ 52305.0 1.98332
$$887$$ − 46005.2i − 1.74149i −0.491735 0.870745i $$-0.663637\pi$$
0.491735 0.870745i $$-0.336363\pi$$
$$888$$ 3632.95i 0.137290i
$$889$$ −13131.9 −0.495421
$$890$$ 0 0
$$891$$ −237.957 −0.00894710
$$892$$ − 20895.8i − 0.784353i
$$893$$ 50870.2i 1.90628i
$$894$$ 24805.2 0.927975
$$895$$ 0 0
$$896$$ 10083.8 0.375978
$$897$$ 12044.8i 0.448345i
$$898$$ 37622.6i 1.39809i
$$899$$ 9005.81 0.334105
$$900$$ 0 0
$$901$$ −22500.1 −0.831950
$$902$$ 5094.58i 0.188061i
$$903$$ 903.081i 0.0332809i
$$904$$ 13910.8 0.511801
$$905$$ 0 0
$$906$$ 22289.2 0.817340
$$907$$ 2838.97i 0.103932i 0.998649 + 0.0519661i $$0.0165488\pi$$
−0.998649 + 0.0519661i $$0.983451\pi$$
$$908$$ 24324.6i 0.889030i
$$909$$ 1159.84 0.0423208
$$910$$ 0 0
$$911$$ 39890.9 1.45076 0.725382 0.688347i $$-0.241663\pi$$
0.725382 + 0.688347i $$0.241663\pi$$
$$912$$ 25713.7i 0.933626i
$$913$$ 1194.85i 0.0433119i
$$914$$ −20675.3 −0.748226
$$915$$ 0 0
$$916$$ −2396.67 −0.0864500
$$917$$ 2549.42i 0.0918094i
$$918$$ − 11679.0i − 0.419897i
$$919$$ 646.475 0.0232048 0.0116024 0.999933i $$-0.496307\pi$$
0.0116024 + 0.999933i $$0.496307\pi$$
$$920$$ 0 0
$$921$$ −22932.4 −0.820463
$$922$$ 11316.3i 0.404213i
$$923$$ − 12689.6i − 0.452528i
$$924$$ −275.696 −0.00981574
$$925$$ 0 0
$$926$$ −1312.37 −0.0465737
$$927$$ − 7254.07i − 0.257017i
$$928$$ 17376.4i 0.614665i
$$929$$ −51188.2 −1.80778 −0.903892 0.427760i $$-0.859303\pi$$
−0.903892 + 0.427760i $$0.859303\pi$$
$$930$$ 0 0
$$931$$ 5264.35 0.185319
$$932$$ 819.993i 0.0288195i
$$933$$ 22782.0i 0.799409i
$$934$$ −69747.8 −2.44349
$$935$$ 0 0
$$936$$ −2139.16 −0.0747017
$$937$$ − 29786.1i − 1.03849i −0.854624 0.519247i $$-0.826212\pi$$
0.854624 0.519247i $$-0.173788\pi$$
$$938$$ − 7662.36i − 0.266722i
$$939$$ 27383.5 0.951680
$$940$$ 0 0
$$941$$ −44817.4 −1.55261 −0.776304 0.630358i $$-0.782908\pi$$
−0.776304 + 0.630358i $$0.782908\pi$$
$$942$$ − 6286.82i − 0.217448i
$$943$$ − 103439.i − 3.57206i
$$944$$ −60682.0 −2.09219
$$945$$ 0 0
$$946$$ −446.103 −0.0153320
$$947$$ − 54697.1i − 1.87689i −0.345425 0.938446i $$-0.612265\pi$$
0.345425 0.938446i $$-0.387735\pi$$
$$948$$ − 331.983i − 0.0113737i
$$949$$ 11843.2 0.405105
$$950$$ 0 0
$$951$$ −14789.4 −0.504290
$$952$$ 10691.9i 0.363998i
$$953$$ − 7577.51i − 0.257565i −0.991673 0.128783i $$-0.958893\pi$$
0.991673 0.128783i $$-0.0411069\pi$$
$$954$$ −5837.29 −0.198102
$$955$$ 0 0
$$956$$ 2874.46 0.0972454
$$957$$ 841.612i 0.0284278i
$$958$$ 73313.4i 2.47249i
$$959$$ 11221.9 0.377868
$$960$$ 0 0
$$961$$ −20897.1 −0.701457
$$962$$ − 6537.32i − 0.219097i
$$963$$ − 6927.41i − 0.231810i
$$964$$ 25721.1 0.859357
$$965$$ 0 0
$$966$$ 15618.4 0.520202
$$967$$ − 50779.0i − 1.68867i −0.535817 0.844334i $$-0.679996\pi$$
0.535817 0.844334i $$-0.320004\pi$$
$$968$$ − 16488.5i − 0.547478i
$$969$$ −39482.0 −1.30892
$$970$$ 0 0
$$971$$ −15313.2 −0.506102 −0.253051 0.967453i $$-0.581434\pi$$
−0.253051 + 0.967453i $$0.581434\pi$$
$$972$$ − 1085.94i − 0.0358348i
$$973$$ − 17017.9i − 0.560707i
$$974$$ 62315.9 2.05003
$$975$$ 0 0
$$976$$ 15856.1 0.520021
$$977$$ − 46620.4i − 1.52663i −0.646025 0.763316i $$-0.723570\pi$$
0.646025 0.763316i $$-0.276430\pi$$
$$978$$ − 23079.9i − 0.754615i
$$979$$ 768.957 0.0251031
$$980$$ 0 0
$$981$$ −7027.70 −0.228723
$$982$$ − 19906.6i − 0.646889i
$$983$$ − 2824.37i − 0.0916414i −0.998950 0.0458207i $$-0.985410\pi$$
0.998950 0.0458207i $$-0.0145903\pi$$
$$984$$ 18370.9 0.595165
$$985$$ 0 0
$$986$$ −41306.6 −1.33415
$$987$$ 9943.38i 0.320670i
$$988$$ − 9152.11i − 0.294704i
$$989$$ 9057.59 0.291218
$$990$$ 0 0
$$991$$ 16951.4 0.543370 0.271685 0.962386i $$-0.412419\pi$$
0.271685 + 0.962386i $$0.412419\pi$$
$$992$$ 17160.5i 0.549239i
$$993$$ 3665.00i 0.117125i
$$994$$ −16454.5 −0.525056
$$995$$ 0 0
$$996$$ −5452.79 −0.173472
$$997$$ − 23847.8i − 0.757540i −0.925491 0.378770i $$-0.876347\pi$$
0.925491 0.378770i $$-0.123653\pi$$
$$998$$ 61703.4i 1.95710i
$$999$$ −2622.26 −0.0830476
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.274.3 4
5.2 odd 4 105.4.a.f.1.1 2
5.3 odd 4 525.4.a.k.1.2 2
5.4 even 2 inner 525.4.d.h.274.2 4
15.2 even 4 315.4.a.i.1.2 2
15.8 even 4 1575.4.a.w.1.1 2
20.7 even 4 1680.4.a.bg.1.1 2
35.27 even 4 735.4.a.p.1.1 2
105.62 odd 4 2205.4.a.z.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.f.1.1 2 5.2 odd 4
315.4.a.i.1.2 2 15.2 even 4
525.4.a.k.1.2 2 5.3 odd 4
525.4.d.h.274.2 4 5.4 even 2 inner
525.4.d.h.274.3 4 1.1 even 1 trivial
735.4.a.p.1.1 2 35.27 even 4
1575.4.a.w.1.1 2 15.8 even 4
1680.4.a.bg.1.1 2 20.7 even 4
2205.4.a.z.1.2 2 105.62 odd 4