# Properties

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

## $q$-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 $$-0.785416\pi$$
0.781248 0.624221i $$-0.214584\pi$$
$$3$$ − 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 $$0.0651807\pi$$
−0.979108 + 0.203343i $$0.934819\pi$$
$$14$$ −24.7179 −0.471867
$$15$$ 0 0
$$16$$ −79.7802 −1.24656
$$17$$ 122.498i 1.74766i 0.486236 + 0.873828i $$0.338370\pi$$
−0.486236 + 0.873828i $$0.661630\pi$$
$$18$$ 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 $$-0.596138\pi$$
0.297455 0.954736i $$-0.403862\pi$$
$$24$$ −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 $$0.0692242\pi$$
−0.976446 + 0.215764i $$0.930776\pi$$
$$38$$ 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 $$0.0242967\pi$$
−0.997088 + 0.0762562i $$0.975703\pi$$
$$44$$ 13.1284 0.0449814
$$45$$ 0 0
$$46$$ −743.735 −2.38387
$$47$$ 473.494i 1.46949i 0.678341 + 0.734747i $$0.262699\pi$$
−0.678341 + 0.734747i $$0.737301\pi$$
$$48$$ 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 $$0.0764980\pi$$
−0.971261 + 0.238019i $$0.923502\pi$$
$$54$$ 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 $$-0.908795\pi$$
0.959231 0.282624i $$-0.0912048\pi$$
$$68$$ − 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 $$-0.834047\pi$$
0.867145 0.498057i $$-0.165953\pi$$
$$74$$ 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 $$0.0866727\pi$$
−0.963157 + 0.268938i $$0.913327\pi$$
$$84$$ 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 $$-0.823739\pi$$
0.850563 0.525873i $$-0.176261\pi$$
$$98$$ 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 $$-0.874020\pi$$
0.922697 0.385526i $$-0.125980\pi$$
$$104$$ 237.685 0.224105
$$105$$ 0 0
$$106$$ 648.587 0.594305
$$107$$ − 769.712i − 0.695429i −0.937600 0.347714i $$-0.886958\pi$$
0.937600 0.347714i $$-0.113042\pi$$
$$108$$ − 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 $$0.153725\pi$$
−0.885633 + 0.464386i $$0.846275\pi$$
$$114$$ 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 $$-0.772508\pi$$
0.755299 0.655381i $$-0.227492\pi$$
$$128$$ 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 $$0.166620\pi$$
−0.866099 + 0.499872i $$0.833380\pi$$
$$138$$ 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 $$-0.951802\pi$$
0.988558 0.150840i $$-0.0481979\pi$$
$$158$$ − 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 $$-0.824639\pi$$
0.852047 0.523465i $$-0.175361\pi$$
$$164$$ 2194.72 1.04499
$$165$$ 0 0
$$166$$ 1436.19 0.671508
$$167$$ 799.502i 0.370463i 0.982695 + 0.185231i $$0.0593035\pi$$
−0.982695 + 0.185231i $$0.940697\pi$$
$$168$$ 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 $$0.102802\pi$$
−0.948299 + 0.317377i $$0.897198\pi$$
$$174$$ 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 $$0.272933\pi$$
−0.654372 + 0.756173i $$0.727067\pi$$
$$194$$ −3547.99 −1.31304
$$195$$ 0 0
$$196$$ 218.975 0.0798013
$$197$$ − 5140.23i − 1.85902i −0.368802 0.929508i $$-0.620232\pi$$
0.368802 0.929508i $$-0.379768\pi$$
$$198$$ − 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 $$-0.752264\pi$$
0.712119 0.702059i $$-0.247736\pi$$
$$224$$ 1273.74 0.379935
$$225$$ 0 0
$$226$$ 3939.49 1.15952
$$227$$ 5443.11i 1.59151i 0.605621 + 0.795754i $$0.292925\pi$$
−0.605621 + 0.795754i $$0.707075\pi$$
$$228$$ − 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 $$0.00821196\pi$$
−0.999667 + 0.0257958i $$0.991788\pi$$
$$234$$ −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 $$0.209578\pi$$
−0.790967 + 0.611859i $$0.790422\pi$$
$$258$$ − 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 $$-0.651924\pi$$
0.459367 0.888246i $$-0.348076\pi$$
$$264$$ 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 $$-0.614456\pi$$
0.351875 0.936047i $$-0.385544\pi$$
$$278$$ 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 $$-0.999505\pi$$
0.999999 0.00155613i $$-0.000495333\pi$$
$$284$$ −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 $$-0.757952\pi$$
0.724550 0.689222i $$-0.242048\pi$$
$$294$$ 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 $$-0.748450\pi$$
0.703655 0.710542i $$-0.251550\pi$$
$$308$$ − 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 $$0.308363\pi$$
−0.566329 + 0.824179i $$0.691637\pi$$
$$314$$ −2095.61 −0.376631
$$315$$ 0 0
$$316$$ −110.661 −0.0196999
$$317$$ − 4929.81i − 0.873456i −0.899593 0.436728i $$-0.856137\pi$$
0.899593 0.436728i $$-0.143863\pi$$
$$318$$ − 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 $$-0.750153\pi$$
0.707446 0.706768i $$-0.249847\pi$$
$$338$$ − 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 $$0.115524\pi$$
−0.934860 + 0.355015i $$0.884476\pi$$
$$348$$ − 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 $$-0.941689\pi$$
0.983268 0.182166i $$-0.0583109\pi$$
$$354$$ −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 $$0.290079\pi$$
−0.612711 + 0.790307i $$0.709921\pi$$
$$368$$ 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 $$-0.861059\pi$$
0.906238 0.422768i $$-0.138941\pi$$
$$374$$ −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 $$-0.993237\pi$$
0.999774 0.0212445i $$-0.00676285\pi$$
$$384$$ 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 $$0.0982518\pi$$
−0.952739 + 0.303789i $$0.901748\pi$$
$$398$$ 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 $$-0.866153\pi$$
0.912888 0.408210i $$-0.133847\pi$$
$$434$$ 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 $$0.292172\pi$$
−0.607502 + 0.794318i $$0.707828\pi$$
$$444$$ −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 $$-0.903125\pi$$
0.954045 0.299664i $$-0.0968745\pi$$
$$458$$ − 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 $$-0.994062\pi$$
0.999826 0.0186527i $$-0.00593769\pi$$
$$464$$ 7618.54 0.762245
$$465$$ 0 0
$$466$$ 647.927 0.0644091
$$467$$ − 19752.3i − 1.95723i −0.205703 0.978614i $$-0.565948\pi$$
0.205703 0.978614i $$-0.434052\pi$$
$$468$$ 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 $$0.306603\pi$$
−0.570878 + 0.821035i $$0.693397\pi$$
$$488$$ 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 $$-0.892962\pi$$
0.943992 0.329968i $$-0.107038\pi$$
$$504$$ 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 $$-0.964142\pi$$
0.993662 0.112412i $$-0.0358575\pi$$
$$524$$ −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 $$0.181494\pi$$
−0.841804 + 0.539784i $$0.818506\pi$$
$$548$$ − 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 $$-0.926298\pi$$
0.973314 0.229477i $$-0.0737016\pi$$
$$558$$ − 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 $$0.0838703\pi$$
−0.965488 + 0.260448i $$0.916130\pi$$
$$564$$ −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 $$-0.718223\pi$$
0.633114 0.774059i $$-0.281777\pi$$
$$578$$ 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 $$0.250690\pi$$
−0.705572 + 0.708638i $$0.749310\pi$$
$$588$$ − 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 $$-0.993394\pi$$
0.999785 0.0207509i $$-0.00660570\pi$$
$$594$$ −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 $$0.0322964\pi$$
−0.994857 + 0.101288i $$0.967704\pi$$
$$608$$ − 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 $$0.219878\pi$$
−0.770757 + 0.637129i $$0.780122\pi$$
$$614$$ −26992.4 −1.77414
$$615$$ 0 0
$$616$$ 256.412 0.0167713
$$617$$ − 5743.91i − 0.374783i −0.982285 0.187391i $$-0.939997\pi$$
0.982285 0.187391i $$-0.0600033\pi$$
$$618$$ 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 $$-0.985859\pi$$
0.999013 0.0444104i $$-0.0141409\pi$$
$$644$$ 6588.72 0.403155
$$645$$ 0 0
$$646$$ −46472.0 −2.83037
$$647$$ − 8732.95i − 0.530646i −0.964160 0.265323i $$-0.914521\pi$$
0.964160 0.265323i $$-0.0854785\pi$$
$$648$$ − 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 $$0.228835\pi$$
−0.752527 + 0.658562i $$0.771165\pi$$
$$654$$ −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 $$-0.716338\pi$$
0.628518 0.777795i $$-0.283662\pi$$
$$674$$ −30879.1 −1.76472
$$675$$ 0 0
$$676$$ −8194.26 −0.466219
$$677$$ − 1392.30i − 0.0790404i −0.999219 0.0395202i $$-0.987417\pi$$
0.999219 0.0395202i $$-0.0125829\pi$$
$$678$$ − 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 $$0.0781327\pi$$
−0.970025 + 0.243004i $$0.921867\pi$$
$$684$$ −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 $$-0.880923\pi$$
0.930841 0.365426i $$-0.119077\pi$$
$$728$$ − 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 $$-0.945782\pi$$
0.985529 0.169508i $$-0.0542179\pi$$
$$734$$ 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 $$-0.935951\pi$$
0.979824 0.199862i $$-0.0640494\pi$$
$$744$$ 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 $$-0.980171\pi$$
0.998060 0.0622541i $$-0.0198289\pi$$
$$758$$ 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 $$-0.794481\pi$$
0.798704 0.601724i $$-0.205519\pi$$
$$774$$ −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 $$-0.738069\pi$$
0.680113 0.733107i $$-0.261931\pi$$
$$788$$ 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 $$0.0143239\pi$$
−0.998988 + 0.0449845i $$0.985676\pi$$
$$798$$ − 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 $$0.0270530\pi$$
−0.996391 + 0.0848871i $$0.972947\pi$$
$$824$$ −10050.0 −0.424889
$$825$$ 0 0
$$826$$ −18800.8 −0.791966
$$827$$ − 45110.4i − 1.89679i −0.317096 0.948394i $$-0.602708\pi$$
0.317096 0.948394i $$-0.397292\pi$$
$$828$$ − 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 $$0.207016\pi$$
−0.795867 + 0.605472i $$0.792984\pi$$
$$854$$ 4912.61 0.196846
$$855$$ 0 0
$$856$$ −9597.44 −0.383217
$$857$$ − 13393.6i − 0.533857i −0.963716 0.266929i $$-0.913991\pi$$
0.963716 0.266929i $$-0.0860088\pi$$
$$858$$ 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 $$0.0597814\pi$$
−0.982416 + 0.186707i $$0.940219\pi$$
$$864$$ −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 $$0.258888\pi$$
−0.687090 + 0.726573i $$0.741112\pi$$
$$878$$ 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 $$-0.729481\pi$$
0.660087 0.751189i $$-0.270519\pi$$
$$884$$ 10435.2 0.397030
$$885$$ 0 0
$$886$$ 52305.0 1.98332
$$887$$ 46005.2i 1.74149i 0.491735 + 0.870745i $$0.336363\pi$$
−0.491735 + 0.870745i $$0.663637\pi$$
$$888$$ − 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 $$-0.983451\pi$$
0.998649 0.0519661i $$-0.0165488\pi$$
$$908$$ − 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 $$0.173788\pi$$
−0.854624 + 0.519247i $$0.826212\pi$$
$$938$$ 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 $$0.387735\pi$$
−0.345425 + 0.938446i $$0.612265\pi$$
$$948$$ 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 $$0.0411069\pi$$
−0.991673 + 0.128783i $$0.958893\pi$$
$$954$$ −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 $$0.320004\pi$$
−0.535817 + 0.844334i $$0.679996\pi$$
$$968$$ 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 $$0.276430\pi$$
−0.646025 + 0.763316i $$0.723570\pi$$
$$978$$ 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 $$0.0145903\pi$$
−0.998950 + 0.0458207i $$0.985410\pi$$
$$984$$ 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 $$0.123653\pi$$
−0.925491 + 0.378770i $$0.876347\pi$$
$$998$$ − 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.274.2 4
5.2 odd 4 525.4.a.k.1.2 2
5.3 odd 4 105.4.a.f.1.1 2
5.4 even 2 inner 525.4.d.h.274.3 4
15.2 even 4 1575.4.a.w.1.1 2
15.8 even 4 315.4.a.i.1.2 2
20.3 even 4 1680.4.a.bg.1.1 2
35.13 even 4 735.4.a.p.1.1 2
105.83 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 5.3 odd 4
315.4.a.i.1.2 2 15.8 even 4
525.4.a.k.1.2 2 5.2 odd 4
525.4.d.h.274.2 4 1.1 even 1 trivial
525.4.d.h.274.3 4 5.4 even 2 inner
735.4.a.p.1.1 2 35.13 even 4
1575.4.a.w.1.1 2 15.2 even 4
1680.4.a.bg.1.1 2 20.3 even 4
2205.4.a.z.1.2 2 105.83 odd 4