# Properties

 Label 525.4.d.j.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: $$\Q(i, \sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 21x^{2} + 100$$ x^4 + 21*x^2 + 100 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$-2.70156i$$ of defining polynomial Character $$\chi$$ $$=$$ 525.274 Dual form 525.4.d.j.274.3

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.70156i q^{2} -3.00000i q^{3} +5.10469 q^{4} -5.10469 q^{6} +7.00000i q^{7} -22.2984i q^{8} -9.00000 q^{9} +O(q^{10})$$ $$q-1.70156i q^{2} -3.00000i q^{3} +5.10469 q^{4} -5.10469 q^{6} +7.00000i q^{7} -22.2984i q^{8} -9.00000 q^{9} +37.4031 q^{11} -15.3141i q^{12} -29.0156i q^{13} +11.9109 q^{14} +2.89531 q^{16} +58.4187i q^{17} +15.3141i q^{18} +54.5969 q^{19} +21.0000 q^{21} -63.6437i q^{22} -161.675i q^{23} -66.8953 q^{24} -49.3719 q^{26} +27.0000i q^{27} +35.7328i q^{28} -137.581 q^{29} +154.659 q^{31} -183.314i q^{32} -112.209i q^{33} +99.4031 q^{34} -45.9422 q^{36} -350.125i q^{37} -92.9000i q^{38} -87.0469 q^{39} +353.769 q^{41} -35.7328i q^{42} +518.156i q^{43} +190.931 q^{44} -275.100 q^{46} -542.219i q^{47} -8.68594i q^{48} -49.0000 q^{49} +175.256 q^{51} -148.116i q^{52} -305.309i q^{53} +45.9422 q^{54} +156.089 q^{56} -163.791i q^{57} +234.103i q^{58} -14.6813 q^{59} -171.069 q^{61} -263.163i q^{62} -63.0000i q^{63} -288.758 q^{64} -190.931 q^{66} +551.956i q^{67} +298.209i q^{68} -485.025 q^{69} -120.334 q^{71} +200.686i q^{72} -284.659i q^{73} -595.759 q^{74} +278.700 q^{76} +261.822i q^{77} +148.116i q^{78} -941.612 q^{79} +81.0000 q^{81} -601.959i q^{82} -377.150i q^{83} +107.198 q^{84} +881.675 q^{86} +412.744i q^{87} -834.031i q^{88} +677.725 q^{89} +203.109 q^{91} -825.300i q^{92} -463.978i q^{93} -922.619 q^{94} -549.942 q^{96} -1225.03i q^{97} +83.3765i q^{98} -336.628 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 18 q^{4} + 18 q^{6} - 36 q^{9}+O(q^{10})$$ 4 * q - 18 * q^4 + 18 * q^6 - 36 * q^9 $$4 q - 18 q^{4} + 18 q^{6} - 36 q^{9} + 124 q^{11} - 42 q^{14} + 50 q^{16} + 244 q^{19} + 84 q^{21} - 306 q^{24} - 428 q^{26} - 704 q^{29} + 132 q^{31} + 372 q^{34} + 162 q^{36} + 36 q^{39} + 32 q^{41} - 312 q^{44} - 1920 q^{46} - 196 q^{49} + 240 q^{51} - 162 q^{54} + 714 q^{56} - 1032 q^{59} - 1760 q^{61} + 958 q^{64} + 312 q^{66} - 96 q^{69} + 620 q^{71} - 2716 q^{74} - 1344 q^{76} - 3664 q^{79} + 324 q^{81} - 378 q^{84} + 2912 q^{86} - 1592 q^{89} - 84 q^{91} - 5176 q^{94} - 1854 q^{96} - 1116 q^{99}+O(q^{100})$$ 4 * q - 18 * q^4 + 18 * q^6 - 36 * q^9 + 124 * q^11 - 42 * q^14 + 50 * q^16 + 244 * q^19 + 84 * q^21 - 306 * q^24 - 428 * q^26 - 704 * q^29 + 132 * q^31 + 372 * q^34 + 162 * q^36 + 36 * q^39 + 32 * q^41 - 312 * q^44 - 1920 * q^46 - 196 * q^49 + 240 * q^51 - 162 * q^54 + 714 * q^56 - 1032 * q^59 - 1760 * q^61 + 958 * q^64 + 312 * q^66 - 96 * q^69 + 620 * q^71 - 2716 * q^74 - 1344 * q^76 - 3664 * q^79 + 324 * q^81 - 378 * q^84 + 2912 * q^86 - 1592 * q^89 - 84 * q^91 - 5176 * q^94 - 1854 * q^96 - 1116 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.70156i − 0.601593i −0.953688 0.300797i $$-0.902747\pi$$
0.953688 0.300797i $$-0.0972525\pi$$
$$3$$ − 3.00000i − 0.577350i
$$4$$ 5.10469 0.638086
$$5$$ 0 0
$$6$$ −5.10469 −0.347330
$$7$$ 7.00000i 0.377964i
$$8$$ − 22.2984i − 0.985461i
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ 37.4031 1.02522 0.512612 0.858620i $$-0.328678\pi$$
0.512612 + 0.858620i $$0.328678\pi$$
$$12$$ − 15.3141i − 0.368399i
$$13$$ − 29.0156i − 0.619037i −0.950893 0.309519i $$-0.899832\pi$$
0.950893 0.309519i $$-0.100168\pi$$
$$14$$ 11.9109 0.227381
$$15$$ 0 0
$$16$$ 2.89531 0.0452393
$$17$$ 58.4187i 0.833449i 0.909033 + 0.416724i $$0.136822\pi$$
−0.909033 + 0.416724i $$0.863178\pi$$
$$18$$ 15.3141i 0.200531i
$$19$$ 54.5969 0.659231 0.329615 0.944115i $$-0.393081\pi$$
0.329615 + 0.944115i $$0.393081\pi$$
$$20$$ 0 0
$$21$$ 21.0000 0.218218
$$22$$ − 63.6437i − 0.616768i
$$23$$ − 161.675i − 1.46572i −0.680379 0.732860i $$-0.738185\pi$$
0.680379 0.732860i $$-0.261815\pi$$
$$24$$ −66.8953 −0.568956
$$25$$ 0 0
$$26$$ −49.3719 −0.372409
$$27$$ 27.0000i 0.192450i
$$28$$ 35.7328i 0.241174i
$$29$$ −137.581 −0.880972 −0.440486 0.897759i $$-0.645194\pi$$
−0.440486 + 0.897759i $$0.645194\pi$$
$$30$$ 0 0
$$31$$ 154.659 0.896053 0.448026 0.894020i $$-0.352127\pi$$
0.448026 + 0.894020i $$0.352127\pi$$
$$32$$ − 183.314i − 1.01268i
$$33$$ − 112.209i − 0.591913i
$$34$$ 99.4031 0.501397
$$35$$ 0 0
$$36$$ −45.9422 −0.212695
$$37$$ − 350.125i − 1.55568i −0.628462 0.777840i $$-0.716315\pi$$
0.628462 0.777840i $$-0.283685\pi$$
$$38$$ − 92.9000i − 0.396589i
$$39$$ −87.0469 −0.357401
$$40$$ 0 0
$$41$$ 353.769 1.34755 0.673773 0.738938i $$-0.264673\pi$$
0.673773 + 0.738938i $$0.264673\pi$$
$$42$$ − 35.7328i − 0.131278i
$$43$$ 518.156i 1.83763i 0.394689 + 0.918815i $$0.370852\pi$$
−0.394689 + 0.918815i $$0.629148\pi$$
$$44$$ 190.931 0.654181
$$45$$ 0 0
$$46$$ −275.100 −0.881767
$$47$$ − 542.219i − 1.68278i −0.540427 0.841391i $$-0.681737\pi$$
0.540427 0.841391i $$-0.318263\pi$$
$$48$$ − 8.68594i − 0.0261189i
$$49$$ −49.0000 −0.142857
$$50$$ 0 0
$$51$$ 175.256 0.481192
$$52$$ − 148.116i − 0.394999i
$$53$$ − 305.309i − 0.791273i −0.918407 0.395637i $$-0.870524\pi$$
0.918407 0.395637i $$-0.129476\pi$$
$$54$$ 45.9422 0.115777
$$55$$ 0 0
$$56$$ 156.089 0.372469
$$57$$ − 163.791i − 0.380607i
$$58$$ 234.103i 0.529987i
$$59$$ −14.6813 −0.0323956 −0.0161978 0.999869i $$-0.505156\pi$$
−0.0161978 + 0.999869i $$0.505156\pi$$
$$60$$ 0 0
$$61$$ −171.069 −0.359067 −0.179534 0.983752i $$-0.557459\pi$$
−0.179534 + 0.983752i $$0.557459\pi$$
$$62$$ − 263.163i − 0.539059i
$$63$$ − 63.0000i − 0.125988i
$$64$$ −288.758 −0.563980
$$65$$ 0 0
$$66$$ −190.931 −0.356091
$$67$$ 551.956i 1.00645i 0.864155 + 0.503225i $$0.167853\pi$$
−0.864155 + 0.503225i $$0.832147\pi$$
$$68$$ 298.209i 0.531812i
$$69$$ −485.025 −0.846234
$$70$$ 0 0
$$71$$ −120.334 −0.201142 −0.100571 0.994930i $$-0.532067\pi$$
−0.100571 + 0.994930i $$0.532067\pi$$
$$72$$ 200.686i 0.328487i
$$73$$ − 284.659i − 0.456395i −0.973615 0.228198i $$-0.926717\pi$$
0.973615 0.228198i $$-0.0732832\pi$$
$$74$$ −595.759 −0.935887
$$75$$ 0 0
$$76$$ 278.700 0.420646
$$77$$ 261.822i 0.387498i
$$78$$ 148.116i 0.215010i
$$79$$ −941.612 −1.34101 −0.670504 0.741906i $$-0.733922\pi$$
−0.670504 + 0.741906i $$0.733922\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ − 601.959i − 0.810674i
$$83$$ − 377.150i − 0.498766i −0.968405 0.249383i $$-0.919772\pi$$
0.968405 0.249383i $$-0.0802278\pi$$
$$84$$ 107.198 0.139242
$$85$$ 0 0
$$86$$ 881.675 1.10551
$$87$$ 412.744i 0.508630i
$$88$$ − 834.031i − 1.01032i
$$89$$ 677.725 0.807176 0.403588 0.914941i $$-0.367763\pi$$
0.403588 + 0.914941i $$0.367763\pi$$
$$90$$ 0 0
$$91$$ 203.109 0.233974
$$92$$ − 825.300i − 0.935255i
$$93$$ − 463.978i − 0.517336i
$$94$$ −922.619 −1.01235
$$95$$ 0 0
$$96$$ −549.942 −0.584669
$$97$$ − 1225.03i − 1.28230i −0.767414 0.641151i $$-0.778457\pi$$
0.767414 0.641151i $$-0.221543\pi$$
$$98$$ 83.3765i 0.0859419i
$$99$$ −336.628 −0.341741
$$100$$ 0 0
$$101$$ −338.144 −0.333134 −0.166567 0.986030i $$-0.553268\pi$$
−0.166567 + 0.986030i $$0.553268\pi$$
$$102$$ − 298.209i − 0.289482i
$$103$$ 566.700i 0.542122i 0.962562 + 0.271061i $$0.0873746\pi$$
−0.962562 + 0.271061i $$0.912625\pi$$
$$104$$ −647.003 −0.610037
$$105$$ 0 0
$$106$$ −519.503 −0.476024
$$107$$ − 562.531i − 0.508242i −0.967172 0.254121i $$-0.918214\pi$$
0.967172 0.254121i $$-0.0817862\pi$$
$$108$$ 137.827i 0.122800i
$$109$$ −1830.79 −1.60879 −0.804396 0.594094i $$-0.797511\pi$$
−0.804396 + 0.594094i $$0.797511\pi$$
$$110$$ 0 0
$$111$$ −1050.37 −0.898173
$$112$$ 20.2672i 0.0170988i
$$113$$ 31.8032i 0.0264761i 0.999912 + 0.0132380i $$0.00421392\pi$$
−0.999912 + 0.0132380i $$0.995786\pi$$
$$114$$ −278.700 −0.228971
$$115$$ 0 0
$$116$$ −702.309 −0.562136
$$117$$ 261.141i 0.206346i
$$118$$ 24.9811i 0.0194890i
$$119$$ −408.931 −0.315014
$$120$$ 0 0
$$121$$ 67.9937 0.0510847
$$122$$ 291.084i 0.216012i
$$123$$ − 1061.31i − 0.778006i
$$124$$ 789.488 0.571759
$$125$$ 0 0
$$126$$ −107.198 −0.0757936
$$127$$ 2220.81i 1.55169i 0.630921 + 0.775847i $$0.282677\pi$$
−0.630921 + 0.775847i $$0.717323\pi$$
$$128$$ − 975.173i − 0.673390i
$$129$$ 1554.47 1.06096
$$130$$ 0 0
$$131$$ 646.512 0.431191 0.215596 0.976483i $$-0.430831\pi$$
0.215596 + 0.976483i $$0.430831\pi$$
$$132$$ − 572.794i − 0.377692i
$$133$$ 382.178i 0.249166i
$$134$$ 939.188 0.605474
$$135$$ 0 0
$$136$$ 1302.65 0.821331
$$137$$ − 896.009i − 0.558768i −0.960179 0.279384i $$-0.909870\pi$$
0.960179 0.279384i $$-0.0901303\pi$$
$$138$$ 825.300i 0.509088i
$$139$$ 2313.61 1.41178 0.705891 0.708320i $$-0.250547\pi$$
0.705891 + 0.708320i $$0.250547\pi$$
$$140$$ 0 0
$$141$$ −1626.66 −0.971554
$$142$$ 204.756i 0.121005i
$$143$$ − 1085.27i − 0.634652i
$$144$$ −26.0578 −0.0150798
$$145$$ 0 0
$$146$$ −484.366 −0.274564
$$147$$ 147.000i 0.0824786i
$$148$$ − 1787.28i − 0.992658i
$$149$$ −819.337 −0.450488 −0.225244 0.974302i $$-0.572318\pi$$
−0.225244 + 0.974302i $$0.572318\pi$$
$$150$$ 0 0
$$151$$ 534.744 0.288191 0.144095 0.989564i $$-0.453973\pi$$
0.144095 + 0.989564i $$0.453973\pi$$
$$152$$ − 1217.43i − 0.649646i
$$153$$ − 525.769i − 0.277816i
$$154$$ 445.506 0.233116
$$155$$ 0 0
$$156$$ −444.347 −0.228053
$$157$$ 1564.76i 0.795423i 0.917511 + 0.397711i $$0.130195\pi$$
−0.917511 + 0.397711i $$0.869805\pi$$
$$158$$ 1602.21i 0.806741i
$$159$$ −915.928 −0.456842
$$160$$ 0 0
$$161$$ 1131.72 0.553990
$$162$$ − 137.827i − 0.0668437i
$$163$$ 1114.31i 0.535455i 0.963495 + 0.267728i $$0.0862727\pi$$
−0.963495 + 0.267728i $$0.913727\pi$$
$$164$$ 1805.88 0.859850
$$165$$ 0 0
$$166$$ −641.744 −0.300054
$$167$$ − 1774.47i − 0.822231i −0.911583 0.411115i $$-0.865139\pi$$
0.911583 0.411115i $$-0.134861\pi$$
$$168$$ − 468.267i − 0.215045i
$$169$$ 1355.09 0.616793
$$170$$ 0 0
$$171$$ −491.372 −0.219744
$$172$$ 2645.02i 1.17257i
$$173$$ 4215.88i 1.85276i 0.376590 + 0.926380i $$0.377097\pi$$
−0.376590 + 0.926380i $$0.622903\pi$$
$$174$$ 702.309 0.305988
$$175$$ 0 0
$$176$$ 108.294 0.0463804
$$177$$ 44.0438i 0.0187036i
$$178$$ − 1153.19i − 0.485592i
$$179$$ 2430.70 1.01497 0.507483 0.861662i $$-0.330576\pi$$
0.507483 + 0.861662i $$0.330576\pi$$
$$180$$ 0 0
$$181$$ −2700.91 −1.10916 −0.554578 0.832132i $$-0.687120\pi$$
−0.554578 + 0.832132i $$0.687120\pi$$
$$182$$ − 345.603i − 0.140757i
$$183$$ 513.206i 0.207308i
$$184$$ −3605.10 −1.44441
$$185$$ 0 0
$$186$$ −789.488 −0.311226
$$187$$ 2185.04i 0.854472i
$$188$$ − 2767.86i − 1.07376i
$$189$$ −189.000 −0.0727393
$$190$$ 0 0
$$191$$ 3611.10 1.36801 0.684005 0.729478i $$-0.260237\pi$$
0.684005 + 0.729478i $$0.260237\pi$$
$$192$$ 866.273i 0.325614i
$$193$$ 4468.33i 1.66651i 0.552886 + 0.833257i $$0.313526\pi$$
−0.552886 + 0.833257i $$0.686474\pi$$
$$194$$ −2084.47 −0.771425
$$195$$ 0 0
$$196$$ −250.130 −0.0911551
$$197$$ − 434.422i − 0.157113i −0.996910 0.0785566i $$-0.974969\pi$$
0.996910 0.0785566i $$-0.0250311\pi$$
$$198$$ 572.794i 0.205589i
$$199$$ 468.915 0.167038 0.0835189 0.996506i $$-0.473384\pi$$
0.0835189 + 0.996506i $$0.473384\pi$$
$$200$$ 0 0
$$201$$ 1655.87 0.581074
$$202$$ 575.372i 0.200411i
$$203$$ − 963.069i − 0.332976i
$$204$$ 894.628 0.307042
$$205$$ 0 0
$$206$$ 964.275 0.326137
$$207$$ 1455.07i 0.488573i
$$208$$ − 84.0093i − 0.0280048i
$$209$$ 2042.09 0.675859
$$210$$ 0 0
$$211$$ 3735.51 1.21878 0.609392 0.792869i $$-0.291414\pi$$
0.609392 + 0.792869i $$0.291414\pi$$
$$212$$ − 1558.51i − 0.504900i
$$213$$ 361.003i 0.116129i
$$214$$ −957.182 −0.305755
$$215$$ 0 0
$$216$$ 602.058 0.189652
$$217$$ 1082.62i 0.338676i
$$218$$ 3115.21i 0.967838i
$$219$$ −853.978 −0.263500
$$220$$ 0 0
$$221$$ 1695.06 0.515936
$$222$$ 1787.28i 0.540334i
$$223$$ − 842.806i − 0.253087i −0.991961 0.126544i $$-0.959612\pi$$
0.991961 0.126544i $$-0.0403884\pi$$
$$224$$ 1283.20 0.382756
$$225$$ 0 0
$$226$$ 54.1152 0.0159278
$$227$$ 992.150i 0.290094i 0.989425 + 0.145047i $$0.0463333\pi$$
−0.989425 + 0.145047i $$0.953667\pi$$
$$228$$ − 836.100i − 0.242860i
$$229$$ 6411.39 1.85012 0.925059 0.379825i $$-0.124016\pi$$
0.925059 + 0.379825i $$0.124016\pi$$
$$230$$ 0 0
$$231$$ 785.466 0.223722
$$232$$ 3067.85i 0.868164i
$$233$$ 2274.35i 0.639476i 0.947506 + 0.319738i $$0.103595\pi$$
−0.947506 + 0.319738i $$0.896405\pi$$
$$234$$ 444.347 0.124136
$$235$$ 0 0
$$236$$ −74.9433 −0.0206712
$$237$$ 2824.84i 0.774232i
$$238$$ 695.822i 0.189510i
$$239$$ −2863.12 −0.774893 −0.387447 0.921892i $$-0.626643\pi$$
−0.387447 + 0.921892i $$0.626643\pi$$
$$240$$ 0 0
$$241$$ −5364.23 −1.43378 −0.716889 0.697187i $$-0.754435\pi$$
−0.716889 + 0.697187i $$0.754435\pi$$
$$242$$ − 115.696i − 0.0307322i
$$243$$ − 243.000i − 0.0641500i
$$244$$ −873.252 −0.229116
$$245$$ 0 0
$$246$$ −1805.88 −0.468043
$$247$$ − 1584.16i − 0.408088i
$$248$$ − 3448.66i − 0.883025i
$$249$$ −1131.45 −0.287963
$$250$$ 0 0
$$251$$ 5569.81 1.40065 0.700325 0.713824i $$-0.253039\pi$$
0.700325 + 0.713824i $$0.253039\pi$$
$$252$$ − 321.595i − 0.0803913i
$$253$$ − 6047.15i − 1.50269i
$$254$$ 3778.85 0.933489
$$255$$ 0 0
$$256$$ −3969.38 −0.969087
$$257$$ 2095.36i 0.508580i 0.967128 + 0.254290i $$0.0818418\pi$$
−0.967128 + 0.254290i $$0.918158\pi$$
$$258$$ − 2645.02i − 0.638264i
$$259$$ 2450.87 0.587992
$$260$$ 0 0
$$261$$ 1238.23 0.293657
$$262$$ − 1100.08i − 0.259402i
$$263$$ 7465.88i 1.75044i 0.483724 + 0.875220i $$0.339284\pi$$
−0.483724 + 0.875220i $$0.660716\pi$$
$$264$$ −2502.09 −0.583308
$$265$$ 0 0
$$266$$ 650.300 0.149896
$$267$$ − 2033.17i − 0.466023i
$$268$$ 2817.56i 0.642202i
$$269$$ 6521.38 1.47812 0.739062 0.673637i $$-0.235269\pi$$
0.739062 + 0.673637i $$0.235269\pi$$
$$270$$ 0 0
$$271$$ 2409.70 0.540144 0.270072 0.962840i $$-0.412952\pi$$
0.270072 + 0.962840i $$0.412952\pi$$
$$272$$ 169.141i 0.0377046i
$$273$$ − 609.328i − 0.135085i
$$274$$ −1524.62 −0.336151
$$275$$ 0 0
$$276$$ −2475.90 −0.539970
$$277$$ − 2219.83i − 0.481503i −0.970587 0.240752i $$-0.922606\pi$$
0.970587 0.240752i $$-0.0773939\pi$$
$$278$$ − 3936.75i − 0.849319i
$$279$$ −1391.93 −0.298684
$$280$$ 0 0
$$281$$ 5838.56 1.23950 0.619749 0.784800i $$-0.287234\pi$$
0.619749 + 0.784800i $$0.287234\pi$$
$$282$$ 2767.86i 0.584480i
$$283$$ 3645.04i 0.765636i 0.923824 + 0.382818i $$0.125046\pi$$
−0.923824 + 0.382818i $$0.874954\pi$$
$$284$$ −614.269 −0.128346
$$285$$ 0 0
$$286$$ −1846.66 −0.381802
$$287$$ 2476.38i 0.509325i
$$288$$ 1649.83i 0.337559i
$$289$$ 1500.25 0.305363
$$290$$ 0 0
$$291$$ −3675.10 −0.740338
$$292$$ − 1453.10i − 0.291219i
$$293$$ − 3777.91i − 0.753268i −0.926362 0.376634i $$-0.877081\pi$$
0.926362 0.376634i $$-0.122919\pi$$
$$294$$ 250.130 0.0496186
$$295$$ 0 0
$$296$$ −7807.24 −1.53306
$$297$$ 1009.88i 0.197304i
$$298$$ 1394.15i 0.271011i
$$299$$ −4691.10 −0.907336
$$300$$ 0 0
$$301$$ −3627.09 −0.694559
$$302$$ − 909.900i − 0.173374i
$$303$$ 1014.43i 0.192335i
$$304$$ 158.075 0.0298231
$$305$$ 0 0
$$306$$ −894.628 −0.167132
$$307$$ − 4799.64i − 0.892281i −0.894963 0.446140i $$-0.852798\pi$$
0.894963 0.446140i $$-0.147202\pi$$
$$308$$ 1336.52i 0.247257i
$$309$$ 1700.10 0.312994
$$310$$ 0 0
$$311$$ 580.113 0.105772 0.0528861 0.998601i $$-0.483158\pi$$
0.0528861 + 0.998601i $$0.483158\pi$$
$$312$$ 1941.01i 0.352205i
$$313$$ 6114.78i 1.10424i 0.833764 + 0.552121i $$0.186182\pi$$
−0.833764 + 0.552121i $$0.813818\pi$$
$$314$$ 2662.53 0.478521
$$315$$ 0 0
$$316$$ −4806.64 −0.855679
$$317$$ − 4300.63i − 0.761979i −0.924579 0.380989i $$-0.875583\pi$$
0.924579 0.380989i $$-0.124417\pi$$
$$318$$ 1558.51i 0.274833i
$$319$$ −5145.97 −0.903194
$$320$$ 0 0
$$321$$ −1687.59 −0.293434
$$322$$ − 1925.70i − 0.333277i
$$323$$ 3189.48i 0.549435i
$$324$$ 413.480 0.0708984
$$325$$ 0 0
$$326$$ 1896.06 0.322126
$$327$$ 5492.38i 0.928836i
$$328$$ − 7888.49i − 1.32795i
$$329$$ 3795.53 0.636032
$$330$$ 0 0
$$331$$ −6687.54 −1.11051 −0.555257 0.831679i $$-0.687380\pi$$
−0.555257 + 0.831679i $$0.687380\pi$$
$$332$$ − 1925.23i − 0.318256i
$$333$$ 3151.12i 0.518560i
$$334$$ −3019.37 −0.494648
$$335$$ 0 0
$$336$$ 60.8016 0.00987202
$$337$$ 5869.28i 0.948723i 0.880330 + 0.474362i $$0.157321\pi$$
−0.880330 + 0.474362i $$0.842679\pi$$
$$338$$ − 2305.78i − 0.371058i
$$339$$ 95.4097 0.0152860
$$340$$ 0 0
$$341$$ 5784.74 0.918655
$$342$$ 836.100i 0.132196i
$$343$$ − 343.000i − 0.0539949i
$$344$$ 11554.1 1.81091
$$345$$ 0 0
$$346$$ 7173.58 1.11461
$$347$$ 1937.22i 0.299699i 0.988709 + 0.149850i $$0.0478790\pi$$
−0.988709 + 0.149850i $$0.952121\pi$$
$$348$$ 2106.93i 0.324549i
$$349$$ 9748.82 1.49525 0.747625 0.664121i $$-0.231194\pi$$
0.747625 + 0.664121i $$0.231194\pi$$
$$350$$ 0 0
$$351$$ 783.422 0.119134
$$352$$ − 6856.52i − 1.03822i
$$353$$ 4576.61i 0.690052i 0.938593 + 0.345026i $$0.112130\pi$$
−0.938593 + 0.345026i $$0.887870\pi$$
$$354$$ 74.9433 0.0112520
$$355$$ 0 0
$$356$$ 3459.57 0.515048
$$357$$ 1226.79i 0.181873i
$$358$$ − 4135.98i − 0.610596i
$$359$$ −10849.9 −1.59509 −0.797546 0.603258i $$-0.793869\pi$$
−0.797546 + 0.603258i $$0.793869\pi$$
$$360$$ 0 0
$$361$$ −3878.18 −0.565415
$$362$$ 4595.77i 0.667261i
$$363$$ − 203.981i − 0.0294938i
$$364$$ 1036.81 0.149296
$$365$$ 0 0
$$366$$ 873.252 0.124715
$$367$$ 11467.7i 1.63108i 0.578699 + 0.815541i $$0.303561\pi$$
−0.578699 + 0.815541i $$0.696439\pi$$
$$368$$ − 468.100i − 0.0663081i
$$369$$ −3183.92 −0.449182
$$370$$ 0 0
$$371$$ 2137.17 0.299073
$$372$$ − 2368.46i − 0.330105i
$$373$$ − 539.982i − 0.0749576i −0.999297 0.0374788i $$-0.988067\pi$$
0.999297 0.0374788i $$-0.0119327\pi$$
$$374$$ 3717.99 0.514044
$$375$$ 0 0
$$376$$ −12090.6 −1.65832
$$377$$ 3992.01i 0.545355i
$$378$$ 321.595i 0.0437595i
$$379$$ −8577.57 −1.16253 −0.581267 0.813713i $$-0.697443\pi$$
−0.581267 + 0.813713i $$0.697443\pi$$
$$380$$ 0 0
$$381$$ 6662.44 0.895871
$$382$$ − 6144.51i − 0.822985i
$$383$$ − 8627.96i − 1.15109i −0.817770 0.575546i $$-0.804790\pi$$
0.817770 0.575546i $$-0.195210\pi$$
$$384$$ −2925.52 −0.388782
$$385$$ 0 0
$$386$$ 7603.13 1.00256
$$387$$ − 4663.41i − 0.612543i
$$388$$ − 6253.42i − 0.818219i
$$389$$ −9234.06 −1.20356 −0.601781 0.798661i $$-0.705542\pi$$
−0.601781 + 0.798661i $$0.705542\pi$$
$$390$$ 0 0
$$391$$ 9444.85 1.22160
$$392$$ 1092.62i 0.140780i
$$393$$ − 1939.54i − 0.248948i
$$394$$ −739.196 −0.0945182
$$395$$ 0 0
$$396$$ −1718.38 −0.218060
$$397$$ 11618.0i 1.46874i 0.678747 + 0.734372i $$0.262523\pi$$
−0.678747 + 0.734372i $$0.737477\pi$$
$$398$$ − 797.889i − 0.100489i
$$399$$ 1146.53 0.143856
$$400$$ 0 0
$$401$$ 11157.1 1.38942 0.694711 0.719289i $$-0.255532\pi$$
0.694711 + 0.719289i $$0.255532\pi$$
$$402$$ − 2817.56i − 0.349570i
$$403$$ − 4487.54i − 0.554690i
$$404$$ −1726.12 −0.212568
$$405$$ 0 0
$$406$$ −1638.72 −0.200316
$$407$$ − 13095.8i − 1.59492i
$$408$$ − 3907.94i − 0.474196i
$$409$$ 7428.08 0.898031 0.449015 0.893524i $$-0.351775\pi$$
0.449015 + 0.893524i $$0.351775\pi$$
$$410$$ 0 0
$$411$$ −2688.03 −0.322605
$$412$$ 2892.83i 0.345921i
$$413$$ − 102.769i − 0.0122444i
$$414$$ 2475.90 0.293922
$$415$$ 0 0
$$416$$ −5318.97 −0.626885
$$417$$ − 6940.83i − 0.815093i
$$418$$ − 3474.75i − 0.406592i
$$419$$ −9644.74 −1.12453 −0.562263 0.826959i $$-0.690069\pi$$
−0.562263 + 0.826959i $$0.690069\pi$$
$$420$$ 0 0
$$421$$ 9918.18 1.14818 0.574088 0.818793i $$-0.305357\pi$$
0.574088 + 0.818793i $$0.305357\pi$$
$$422$$ − 6356.21i − 0.733212i
$$423$$ 4879.97i 0.560927i
$$424$$ −6807.92 −0.779769
$$425$$ 0 0
$$426$$ 614.269 0.0698625
$$427$$ − 1197.48i − 0.135715i
$$428$$ − 2871.54i − 0.324302i
$$429$$ −3255.82 −0.366417
$$430$$ 0 0
$$431$$ −16324.1 −1.82437 −0.912185 0.409779i $$-0.865606\pi$$
−0.912185 + 0.409779i $$0.865606\pi$$
$$432$$ 78.1735i 0.00870630i
$$433$$ 5168.75i 0.573659i 0.957982 + 0.286829i $$0.0926012\pi$$
−0.957982 + 0.286829i $$0.907399\pi$$
$$434$$ 1842.14 0.203745
$$435$$ 0 0
$$436$$ −9345.63 −1.02655
$$437$$ − 8826.95i − 0.966248i
$$438$$ 1453.10i 0.158520i
$$439$$ −18339.0 −1.99378 −0.996892 0.0787782i $$-0.974898\pi$$
−0.996892 + 0.0787782i $$0.974898\pi$$
$$440$$ 0 0
$$441$$ 441.000 0.0476190
$$442$$ − 2884.24i − 0.310383i
$$443$$ 1613.28i 0.173023i 0.996251 + 0.0865113i $$0.0275719\pi$$
−0.996251 + 0.0865113i $$0.972428\pi$$
$$444$$ −5361.83 −0.573111
$$445$$ 0 0
$$446$$ −1434.09 −0.152256
$$447$$ 2458.01i 0.260089i
$$448$$ − 2021.30i − 0.213164i
$$449$$ 886.750 0.0932034 0.0466017 0.998914i $$-0.485161\pi$$
0.0466017 + 0.998914i $$0.485161\pi$$
$$450$$ 0 0
$$451$$ 13232.1 1.38154
$$452$$ 162.345i 0.0168940i
$$453$$ − 1604.23i − 0.166387i
$$454$$ 1688.21 0.174518
$$455$$ 0 0
$$456$$ −3652.28 −0.375073
$$457$$ 7391.22i 0.756557i 0.925692 + 0.378279i $$0.123484\pi$$
−0.925692 + 0.378279i $$0.876516\pi$$
$$458$$ − 10909.4i − 1.11302i
$$459$$ −1577.31 −0.160397
$$460$$ 0 0
$$461$$ −7133.35 −0.720679 −0.360340 0.932821i $$-0.617339\pi$$
−0.360340 + 0.932821i $$0.617339\pi$$
$$462$$ − 1336.52i − 0.134590i
$$463$$ − 14461.8i − 1.45162i −0.687897 0.725808i $$-0.741466\pi$$
0.687897 0.725808i $$-0.258534\pi$$
$$464$$ −398.341 −0.0398546
$$465$$ 0 0
$$466$$ 3869.95 0.384704
$$467$$ − 16393.5i − 1.62441i −0.583370 0.812206i $$-0.698266\pi$$
0.583370 0.812206i $$-0.301734\pi$$
$$468$$ 1333.04i 0.131666i
$$469$$ −3863.69 −0.380403
$$470$$ 0 0
$$471$$ 4694.28 0.459238
$$472$$ 327.370i 0.0319246i
$$473$$ 19380.7i 1.88398i
$$474$$ 4806.64 0.465772
$$475$$ 0 0
$$476$$ −2087.47 −0.201006
$$477$$ 2747.78i 0.263758i
$$478$$ 4871.77i 0.466171i
$$479$$ 12991.4 1.23923 0.619617 0.784904i $$-0.287288\pi$$
0.619617 + 0.784904i $$0.287288\pi$$
$$480$$ 0 0
$$481$$ −10159.1 −0.963025
$$482$$ 9127.57i 0.862551i
$$483$$ − 3395.17i − 0.319846i
$$484$$ 347.087 0.0325964
$$485$$ 0 0
$$486$$ −413.480 −0.0385922
$$487$$ − 12863.5i − 1.19692i −0.801152 0.598461i $$-0.795779\pi$$
0.801152 0.598461i $$-0.204221\pi$$
$$488$$ 3814.57i 0.353847i
$$489$$ 3342.92 0.309145
$$490$$ 0 0
$$491$$ −4898.10 −0.450200 −0.225100 0.974336i $$-0.572271\pi$$
−0.225100 + 0.974336i $$0.572271\pi$$
$$492$$ − 5417.63i − 0.496435i
$$493$$ − 8037.32i − 0.734245i
$$494$$ −2695.55 −0.245503
$$495$$ 0 0
$$496$$ 447.787 0.0405368
$$497$$ − 842.340i − 0.0760244i
$$498$$ 1925.23i 0.173236i
$$499$$ −10308.0 −0.924746 −0.462373 0.886686i $$-0.653002\pi$$
−0.462373 + 0.886686i $$0.653002\pi$$
$$500$$ 0 0
$$501$$ −5323.41 −0.474715
$$502$$ − 9477.37i − 0.842621i
$$503$$ 15119.6i 1.34026i 0.742244 + 0.670130i $$0.233762\pi$$
−0.742244 + 0.670130i $$0.766238\pi$$
$$504$$ −1404.80 −0.124156
$$505$$ 0 0
$$506$$ −10289.6 −0.904009
$$507$$ − 4065.28i − 0.356105i
$$508$$ 11336.6i 0.990114i
$$509$$ −14183.8 −1.23514 −0.617571 0.786515i $$-0.711883\pi$$
−0.617571 + 0.786515i $$0.711883\pi$$
$$510$$ 0 0
$$511$$ 1992.62 0.172501
$$512$$ − 1047.24i − 0.0903943i
$$513$$ 1474.12i 0.126869i
$$514$$ 3565.39 0.305958
$$515$$ 0 0
$$516$$ 7935.07 0.676981
$$517$$ − 20280.7i − 1.72523i
$$518$$ − 4170.32i − 0.353732i
$$519$$ 12647.6 1.06969
$$520$$ 0 0
$$521$$ −7464.08 −0.627653 −0.313827 0.949480i $$-0.601611\pi$$
−0.313827 + 0.949480i $$0.601611\pi$$
$$522$$ − 2106.93i − 0.176662i
$$523$$ 16642.9i 1.39148i 0.718295 + 0.695739i $$0.244923\pi$$
−0.718295 + 0.695739i $$0.755077\pi$$
$$524$$ 3300.24 0.275137
$$525$$ 0 0
$$526$$ 12703.7 1.05305
$$527$$ 9035.01i 0.746814i
$$528$$ − 324.881i − 0.0267777i
$$529$$ −13971.8 −1.14834
$$530$$ 0 0
$$531$$ 132.132 0.0107985
$$532$$ 1950.90i 0.158989i
$$533$$ − 10264.8i − 0.834181i
$$534$$ −3459.57 −0.280356
$$535$$ 0 0
$$536$$ 12307.8 0.991818
$$537$$ − 7292.09i − 0.585991i
$$538$$ − 11096.5i − 0.889230i
$$539$$ −1832.75 −0.146461
$$540$$ 0 0
$$541$$ 67.8755 0.00539408 0.00269704 0.999996i $$-0.499142\pi$$
0.00269704 + 0.999996i $$0.499142\pi$$
$$542$$ − 4100.26i − 0.324947i
$$543$$ 8102.74i 0.640372i
$$544$$ 10709.0 0.844014
$$545$$ 0 0
$$546$$ −1036.81 −0.0812662
$$547$$ − 9212.91i − 0.720138i −0.932926 0.360069i $$-0.882753\pi$$
0.932926 0.360069i $$-0.117247\pi$$
$$548$$ − 4573.85i − 0.356542i
$$549$$ 1539.62 0.119689
$$550$$ 0 0
$$551$$ −7511.51 −0.580764
$$552$$ 10815.3i 0.833931i
$$553$$ − 6591.29i − 0.506854i
$$554$$ −3777.17 −0.289669
$$555$$ 0 0
$$556$$ 11810.2 0.900838
$$557$$ 16699.6i 1.27035i 0.772370 + 0.635173i $$0.219071\pi$$
−0.772370 + 0.635173i $$0.780929\pi$$
$$558$$ 2368.46i 0.179686i
$$559$$ 15034.6 1.13756
$$560$$ 0 0
$$561$$ 6555.13 0.493329
$$562$$ − 9934.67i − 0.745674i
$$563$$ − 14772.8i − 1.10586i −0.833227 0.552931i $$-0.813509\pi$$
0.833227 0.552931i $$-0.186491\pi$$
$$564$$ −8303.57 −0.619935
$$565$$ 0 0
$$566$$ 6202.26 0.460601
$$567$$ 567.000i 0.0419961i
$$568$$ 2683.27i 0.198217i
$$569$$ −5663.76 −0.417289 −0.208644 0.977992i $$-0.566905\pi$$
−0.208644 + 0.977992i $$0.566905\pi$$
$$570$$ 0 0
$$571$$ 5579.58 0.408929 0.204464 0.978874i $$-0.434455\pi$$
0.204464 + 0.978874i $$0.434455\pi$$
$$572$$ − 5539.99i − 0.404962i
$$573$$ − 10833.3i − 0.789821i
$$574$$ 4213.72 0.306406
$$575$$ 0 0
$$576$$ 2598.82 0.187993
$$577$$ − 2301.23i − 0.166034i −0.996548 0.0830170i $$-0.973544\pi$$
0.996548 0.0830170i $$-0.0264556\pi$$
$$578$$ − 2552.77i − 0.183704i
$$579$$ 13405.0 0.962162
$$580$$ 0 0
$$581$$ 2640.05 0.188516
$$582$$ 6253.42i 0.445382i
$$583$$ − 11419.5i − 0.811232i
$$584$$ −6347.46 −0.449760
$$585$$ 0 0
$$586$$ −6428.34 −0.453161
$$587$$ − 16470.4i − 1.15810i −0.815291 0.579052i $$-0.803423\pi$$
0.815291 0.579052i $$-0.196577\pi$$
$$588$$ 750.389i 0.0526284i
$$589$$ 8443.92 0.590706
$$590$$ 0 0
$$591$$ −1303.27 −0.0907093
$$592$$ − 1013.72i − 0.0703779i
$$593$$ 13570.0i 0.939715i 0.882742 + 0.469858i $$0.155695\pi$$
−0.882742 + 0.469858i $$0.844305\pi$$
$$594$$ 1718.38 0.118697
$$595$$ 0 0
$$596$$ −4182.46 −0.287450
$$597$$ − 1406.75i − 0.0964393i
$$598$$ 7982.20i 0.545847i
$$599$$ −27814.1 −1.89725 −0.948625 0.316403i $$-0.897525\pi$$
−0.948625 + 0.316403i $$0.897525\pi$$
$$600$$ 0 0
$$601$$ 20646.1 1.40128 0.700641 0.713514i $$-0.252898\pi$$
0.700641 + 0.713514i $$0.252898\pi$$
$$602$$ 6171.72i 0.417842i
$$603$$ − 4967.61i − 0.335483i
$$604$$ 2729.70 0.183891
$$605$$ 0 0
$$606$$ 1726.12 0.115707
$$607$$ 3315.28i 0.221686i 0.993838 + 0.110843i $$0.0353550\pi$$
−0.993838 + 0.110843i $$0.964645\pi$$
$$608$$ − 10008.4i − 0.667588i
$$609$$ −2889.21 −0.192244
$$610$$ 0 0
$$611$$ −15732.8 −1.04170
$$612$$ − 2683.88i − 0.177271i
$$613$$ 11113.9i 0.732278i 0.930560 + 0.366139i $$0.119320\pi$$
−0.930560 + 0.366139i $$0.880680\pi$$
$$614$$ −8166.89 −0.536790
$$615$$ 0 0
$$616$$ 5838.22 0.381865
$$617$$ 7871.34i 0.513595i 0.966465 + 0.256797i $$0.0826673\pi$$
−0.966465 + 0.256797i $$0.917333\pi$$
$$618$$ − 2892.83i − 0.188295i
$$619$$ 19107.1 1.24068 0.620339 0.784334i $$-0.286995\pi$$
0.620339 + 0.784334i $$0.286995\pi$$
$$620$$ 0 0
$$621$$ 4365.22 0.282078
$$622$$ − 987.098i − 0.0636319i
$$623$$ 4744.07i 0.305084i
$$624$$ −252.028 −0.0161686
$$625$$ 0 0
$$626$$ 10404.7 0.664305
$$627$$ − 6126.28i − 0.390208i
$$628$$ 7987.60i 0.507548i
$$629$$ 20453.9 1.29658
$$630$$ 0 0
$$631$$ −25769.9 −1.62580 −0.812902 0.582401i $$-0.802113\pi$$
−0.812902 + 0.582401i $$0.802113\pi$$
$$632$$ 20996.5i 1.32151i
$$633$$ − 11206.5i − 0.703665i
$$634$$ −7317.79 −0.458401
$$635$$ 0 0
$$636$$ −4675.53 −0.291504
$$637$$ 1421.77i 0.0884339i
$$638$$ 8756.19i 0.543355i
$$639$$ 1083.01 0.0670472
$$640$$ 0 0
$$641$$ −1954.61 −0.120440 −0.0602202 0.998185i $$-0.519180\pi$$
−0.0602202 + 0.998185i $$0.519180\pi$$
$$642$$ 2871.54i 0.176528i
$$643$$ 19396.5i 1.18961i 0.803868 + 0.594807i $$0.202772\pi$$
−0.803868 + 0.594807i $$0.797228\pi$$
$$644$$ 5777.10 0.353493
$$645$$ 0 0
$$646$$ 5427.10 0.330536
$$647$$ 31264.3i 1.89973i 0.312661 + 0.949865i $$0.398780\pi$$
−0.312661 + 0.949865i $$0.601220\pi$$
$$648$$ − 1806.17i − 0.109496i
$$649$$ −549.126 −0.0332127
$$650$$ 0 0
$$651$$ 3247.85 0.195535
$$652$$ 5688.18i 0.341666i
$$653$$ − 6442.75i − 0.386102i −0.981189 0.193051i $$-0.938162\pi$$
0.981189 0.193051i $$-0.0618382\pi$$
$$654$$ 9345.63 0.558781
$$655$$ 0 0
$$656$$ 1024.27 0.0609620
$$657$$ 2561.93i 0.152132i
$$658$$ − 6458.33i − 0.382632i
$$659$$ 3584.70 0.211897 0.105949 0.994372i $$-0.466212\pi$$
0.105949 + 0.994372i $$0.466212\pi$$
$$660$$ 0 0
$$661$$ 6294.43 0.370386 0.185193 0.982702i $$-0.440709\pi$$
0.185193 + 0.982702i $$0.440709\pi$$
$$662$$ 11379.3i 0.668078i
$$663$$ − 5085.17i − 0.297876i
$$664$$ −8409.85 −0.491514
$$665$$ 0 0
$$666$$ 5361.83 0.311962
$$667$$ 22243.4i 1.29126i
$$668$$ − 9058.11i − 0.524654i
$$669$$ −2528.42 −0.146120
$$670$$ 0 0
$$671$$ −6398.51 −0.368125
$$672$$ − 3849.60i − 0.220984i
$$673$$ 10233.9i 0.586162i 0.956088 + 0.293081i $$0.0946806\pi$$
−0.956088 + 0.293081i $$0.905319\pi$$
$$674$$ 9986.94 0.570745
$$675$$ 0 0
$$676$$ 6917.33 0.393567
$$677$$ − 7100.75i − 0.403108i −0.979477 0.201554i $$-0.935401\pi$$
0.979477 0.201554i $$-0.0645992\pi$$
$$678$$ − 162.345i − 0.00919593i
$$679$$ 8575.24 0.484665
$$680$$ 0 0
$$681$$ 2976.45 0.167486
$$682$$ − 9843.10i − 0.552657i
$$683$$ 35274.6i 1.97620i 0.153813 + 0.988100i $$0.450845\pi$$
−0.153813 + 0.988100i $$0.549155\pi$$
$$684$$ −2508.30 −0.140215
$$685$$ 0 0
$$686$$ −583.636 −0.0324830
$$687$$ − 19234.2i − 1.06817i
$$688$$ 1500.22i 0.0831330i
$$689$$ −8858.74 −0.489828
$$690$$ 0 0
$$691$$ 4945.12 0.272245 0.136122 0.990692i $$-0.456536\pi$$
0.136122 + 0.990692i $$0.456536\pi$$
$$692$$ 21520.8i 1.18222i
$$693$$ − 2356.40i − 0.129166i
$$694$$ 3296.31 0.180297
$$695$$ 0 0
$$696$$ 9203.54 0.501235
$$697$$ 20666.7i 1.12311i
$$698$$ − 16588.2i − 0.899532i
$$699$$ 6823.06 0.369201
$$700$$ 0 0
$$701$$ −15300.4 −0.824379 −0.412190 0.911098i $$-0.635236\pi$$
−0.412190 + 0.911098i $$0.635236\pi$$
$$702$$ − 1333.04i − 0.0716701i
$$703$$ − 19115.7i − 1.02555i
$$704$$ −10800.4 −0.578206
$$705$$ 0 0
$$706$$ 7787.38 0.415130
$$707$$ − 2367.01i − 0.125913i
$$708$$ 224.830i 0.0119345i
$$709$$ 28297.4 1.49892 0.749458 0.662052i $$-0.230314\pi$$
0.749458 + 0.662052i $$0.230314\pi$$
$$710$$ 0 0
$$711$$ 8474.51 0.447003
$$712$$ − 15112.2i − 0.795441i
$$713$$ − 25004.5i − 1.31336i
$$714$$ 2087.47 0.109414
$$715$$ 0 0
$$716$$ 12407.9 0.647635
$$717$$ 8589.35i 0.447385i
$$718$$ 18461.9i 0.959597i
$$719$$ −8548.96 −0.443425 −0.221712 0.975112i $$-0.571165\pi$$
−0.221712 + 0.975112i $$0.571165\pi$$
$$720$$ 0 0
$$721$$ −3966.90 −0.204903
$$722$$ 6598.97i 0.340150i
$$723$$ 16092.7i 0.827792i
$$724$$ −13787.3 −0.707737
$$725$$ 0 0
$$726$$ −347.087 −0.0177432
$$727$$ 14345.3i 0.731827i 0.930649 + 0.365913i $$0.119243\pi$$
−0.930649 + 0.365913i $$0.880757\pi$$
$$728$$ − 4529.02i − 0.230572i
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ −30270.0 −1.53157
$$732$$ 2619.76i 0.132280i
$$733$$ 22624.0i 1.14002i 0.821637 + 0.570012i $$0.193061\pi$$
−0.821637 + 0.570012i $$0.806939\pi$$
$$734$$ 19512.9 0.981248
$$735$$ 0 0
$$736$$ −29637.3 −1.48430
$$737$$ 20644.9i 1.03184i
$$738$$ 5417.63i 0.270225i
$$739$$ −14837.3 −0.738566 −0.369283 0.929317i $$-0.620397\pi$$
−0.369283 + 0.929317i $$0.620397\pi$$
$$740$$ 0 0
$$741$$ −4752.49 −0.235610
$$742$$ − 3636.52i − 0.179920i
$$743$$ − 13073.1i − 0.645497i −0.946485 0.322749i $$-0.895393\pi$$
0.946485 0.322749i $$-0.104607\pi$$
$$744$$ −10346.0 −0.509815
$$745$$ 0 0
$$746$$ −918.813 −0.0450940
$$747$$ 3394.35i 0.166255i
$$748$$ 11154.0i 0.545226i
$$749$$ 3937.72 0.192098
$$750$$ 0 0
$$751$$ 16213.3 0.787791 0.393895 0.919155i $$-0.371127\pi$$
0.393895 + 0.919155i $$0.371127\pi$$
$$752$$ − 1569.89i − 0.0761278i
$$753$$ − 16709.4i − 0.808665i
$$754$$ 6792.65 0.328082
$$755$$ 0 0
$$756$$ −964.786 −0.0464139
$$757$$ 19903.9i 0.955642i 0.878457 + 0.477821i $$0.158573\pi$$
−0.878457 + 0.477821i $$0.841427\pi$$
$$758$$ 14595.3i 0.699372i
$$759$$ −18141.4 −0.867580
$$760$$ 0 0
$$761$$ −30125.5 −1.43502 −0.717509 0.696549i $$-0.754718\pi$$
−0.717509 + 0.696549i $$0.754718\pi$$
$$762$$ − 11336.6i − 0.538950i
$$763$$ − 12815.6i − 0.608066i
$$764$$ 18433.5 0.872907
$$765$$ 0 0
$$766$$ −14681.0 −0.692489
$$767$$ 425.986i 0.0200541i
$$768$$ 11908.1i 0.559503i
$$769$$ 36049.1 1.69046 0.845230 0.534402i $$-0.179463\pi$$
0.845230 + 0.534402i $$0.179463\pi$$
$$770$$ 0 0
$$771$$ 6286.09 0.293629
$$772$$ 22809.4i 1.06338i
$$773$$ 9644.77i 0.448769i 0.974501 + 0.224384i $$0.0720371\pi$$
−0.974501 + 0.224384i $$0.927963\pi$$
$$774$$ −7935.07 −0.368502
$$775$$ 0 0
$$776$$ −27316.4 −1.26366
$$777$$ − 7352.62i − 0.339477i
$$778$$ 15712.3i 0.724054i
$$779$$ 19314.7 0.888344
$$780$$ 0 0
$$781$$ −4500.88 −0.206215
$$782$$ − 16071.0i − 0.734908i
$$783$$ − 3714.69i − 0.169543i
$$784$$ −141.870 −0.00646275
$$785$$ 0 0
$$786$$ −3300.24 −0.149766
$$787$$ − 25218.6i − 1.14224i −0.820865 0.571122i $$-0.806508\pi$$
0.820865 0.571122i $$-0.193492\pi$$
$$788$$ − 2217.59i − 0.100252i
$$789$$ 22397.6 1.01062
$$790$$ 0 0
$$791$$ −222.623 −0.0100070
$$792$$ 7506.28i 0.336773i
$$793$$ 4963.67i 0.222276i
$$794$$ 19768.8 0.883586
$$795$$ 0 0
$$796$$ 2393.67 0.106584
$$797$$ 32042.3i 1.42409i 0.702135 + 0.712044i $$0.252230\pi$$
−0.702135 + 0.712044i $$0.747770\pi$$
$$798$$ − 1950.90i − 0.0865427i
$$799$$ 31675.7 1.40251
$$800$$ 0 0
$$801$$ −6099.52 −0.269059
$$802$$ − 18984.5i − 0.835867i
$$803$$ − 10647.1i − 0.467908i
$$804$$ 8452.69 0.370775
$$805$$ 0 0
$$806$$ −7635.82 −0.333698
$$807$$ − 19564.1i − 0.853396i
$$808$$ 7540.07i 0.328291i
$$809$$ −3427.90 −0.148972 −0.0744860 0.997222i $$-0.523732\pi$$
−0.0744860 + 0.997222i $$0.523732\pi$$
$$810$$ 0 0
$$811$$ 23094.4 0.999943 0.499972 0.866042i $$-0.333344\pi$$
0.499972 + 0.866042i $$0.333344\pi$$
$$812$$ − 4916.16i − 0.212467i
$$813$$ − 7229.11i − 0.311852i
$$814$$ −22283.3 −0.959494
$$815$$ 0 0
$$816$$ 507.422 0.0217688
$$817$$ 28289.7i 1.21142i
$$818$$ − 12639.3i − 0.540249i
$$819$$ −1827.98 −0.0779914
$$820$$ 0 0
$$821$$ 474.741 0.0201810 0.0100905 0.999949i $$-0.496788\pi$$
0.0100905 + 0.999949i $$0.496788\pi$$
$$822$$ 4573.85i 0.194077i
$$823$$ − 24159.8i − 1.02328i −0.859201 0.511638i $$-0.829039\pi$$
0.859201 0.511638i $$-0.170961\pi$$
$$824$$ 12636.5 0.534240
$$825$$ 0 0
$$826$$ −174.868 −0.00736613
$$827$$ − 7566.35i − 0.318147i −0.987267 0.159074i $$-0.949149\pi$$
0.987267 0.159074i $$-0.0508507\pi$$
$$828$$ 7427.70i 0.311752i
$$829$$ 10580.9 0.443295 0.221648 0.975127i $$-0.428857\pi$$
0.221648 + 0.975127i $$0.428857\pi$$
$$830$$ 0 0
$$831$$ −6659.48 −0.277996
$$832$$ 8378.49i 0.349125i
$$833$$ − 2862.52i − 0.119064i
$$834$$ −11810.2 −0.490354
$$835$$ 0 0
$$836$$ 10424.2 0.431256
$$837$$ 4175.80i 0.172445i
$$838$$ 16411.1i 0.676507i
$$839$$ −15315.6 −0.630218 −0.315109 0.949055i $$-0.602041\pi$$
−0.315109 + 0.949055i $$0.602041\pi$$
$$840$$ 0 0
$$841$$ −5460.40 −0.223888
$$842$$ − 16876.4i − 0.690735i
$$843$$ − 17515.7i − 0.715625i
$$844$$ 19068.6 0.777688
$$845$$ 0 0
$$846$$ 8303.57 0.337450
$$847$$ 475.956i 0.0193082i
$$848$$ − 883.966i − 0.0357966i
$$849$$ 10935.1 0.442040
$$850$$ 0 0
$$851$$ −56606.4 −2.28019
$$852$$ 1842.81i 0.0741004i
$$853$$ − 18598.2i − 0.746528i −0.927725 0.373264i $$-0.878239\pi$$
0.927725 0.373264i $$-0.121761\pi$$
$$854$$ −2037.59 −0.0816450
$$855$$ 0 0
$$856$$ −12543.6 −0.500853
$$857$$ 41775.3i 1.66513i 0.553926 + 0.832566i $$0.313129\pi$$
−0.553926 + 0.832566i $$0.686871\pi$$
$$858$$ 5539.99i 0.220434i
$$859$$ −32414.7 −1.28752 −0.643758 0.765229i $$-0.722626\pi$$
−0.643758 + 0.765229i $$0.722626\pi$$
$$860$$ 0 0
$$861$$ 7429.14 0.294059
$$862$$ 27776.4i 1.09753i
$$863$$ 31299.7i 1.23459i 0.786730 + 0.617297i $$0.211772\pi$$
−0.786730 + 0.617297i $$0.788228\pi$$
$$864$$ 4949.48 0.194890
$$865$$ 0 0
$$866$$ 8794.94 0.345109
$$867$$ − 4500.75i − 0.176302i
$$868$$ 5526.41i 0.216104i
$$869$$ −35219.2 −1.37483
$$870$$ 0 0
$$871$$ 16015.4 0.623030
$$872$$ 40823.8i 1.58540i
$$873$$ 11025.3i 0.427434i
$$874$$ −15019.6 −0.581288
$$875$$ 0 0
$$876$$ −4359.29 −0.168136
$$877$$ − 19973.0i − 0.769031i −0.923119 0.384515i $$-0.874369\pi$$
0.923119 0.384515i $$-0.125631\pi$$
$$878$$ 31204.9i 1.19945i
$$879$$ −11333.7 −0.434900
$$880$$ 0 0
$$881$$ 17367.9 0.664176 0.332088 0.943248i $$-0.392247\pi$$
0.332088 + 0.943248i $$0.392247\pi$$
$$882$$ − 750.389i − 0.0286473i
$$883$$ 14364.2i 0.547446i 0.961809 + 0.273723i $$0.0882551\pi$$
−0.961809 + 0.273723i $$0.911745\pi$$
$$884$$ 8652.73 0.329211
$$885$$ 0 0
$$886$$ 2745.09 0.104089
$$887$$ − 33738.1i − 1.27713i −0.769568 0.638564i $$-0.779529\pi$$
0.769568 0.638564i $$-0.220471\pi$$
$$888$$ 23421.7i 0.885114i
$$889$$ −15545.7 −0.586485
$$890$$ 0 0
$$891$$ 3029.65 0.113914
$$892$$ − 4302.26i − 0.161491i
$$893$$ − 29603.4i − 1.10934i
$$894$$ 4182.46 0.156468
$$895$$ 0 0
$$896$$ 6826.21 0.254518
$$897$$ 14073.3i 0.523850i
$$898$$ − 1508.86i − 0.0560705i
$$899$$ −21278.2 −0.789398
$$900$$ 0 0
$$901$$ 17835.8 0.659485
$$902$$ − 22515.2i − 0.831123i
$$903$$ 10881.3i 0.401004i
$$904$$ 709.162 0.0260911
$$905$$ 0 0
$$906$$ −2729.70 −0.100097
$$907$$ − 32998.6i − 1.20805i −0.796966 0.604024i $$-0.793563\pi$$
0.796966 0.604024i $$-0.206437\pi$$
$$908$$ 5064.62i 0.185105i
$$909$$ 3043.29 0.111045
$$910$$ 0 0
$$911$$ 33446.3 1.21638 0.608192 0.793790i $$-0.291895\pi$$
0.608192 + 0.793790i $$0.291895\pi$$
$$912$$ − 474.225i − 0.0172184i
$$913$$ − 14106.6i − 0.511347i
$$914$$ 12576.6 0.455139
$$915$$ 0 0
$$916$$ 32728.2 1.18053
$$917$$ 4525.59i 0.162975i
$$918$$ 2683.88i 0.0964939i
$$919$$ −41708.7 −1.49711 −0.748554 0.663074i $$-0.769252\pi$$
−0.748554 + 0.663074i $$0.769252\pi$$
$$920$$ 0 0
$$921$$ −14398.9 −0.515158
$$922$$ 12137.8i 0.433556i
$$923$$ 3491.58i 0.124514i
$$924$$ 4009.56 0.142754
$$925$$ 0 0
$$926$$ −24607.7 −0.873283
$$927$$ − 5100.30i − 0.180707i
$$928$$ 25220.6i 0.892140i
$$929$$ −49024.6 −1.73137 −0.865686 0.500587i $$-0.833117\pi$$
−0.865686 + 0.500587i $$0.833117\pi$$
$$930$$ 0 0
$$931$$ −2675.25 −0.0941758
$$932$$ 11609.9i 0.408040i
$$933$$ − 1740.34i − 0.0610677i
$$934$$ −27894.6 −0.977235
$$935$$ 0 0
$$936$$ 5823.03 0.203346
$$937$$ − 5447.58i − 0.189930i −0.995481 0.0949651i $$-0.969726\pi$$
0.995481 0.0949651i $$-0.0302739\pi$$
$$938$$ 6574.31i 0.228848i
$$939$$ 18344.4 0.637535
$$940$$ 0 0
$$941$$ 4125.75 0.142928 0.0714642 0.997443i $$-0.477233\pi$$
0.0714642 + 0.997443i $$0.477233\pi$$
$$942$$ − 7987.60i − 0.276274i
$$943$$ − 57195.5i − 1.97513i
$$944$$ −42.5069 −0.00146555
$$945$$ 0 0
$$946$$ 32977.4 1.13339
$$947$$ 17332.3i 0.594747i 0.954761 + 0.297374i $$0.0961107\pi$$
−0.954761 + 0.297374i $$0.903889\pi$$
$$948$$ 14419.9i 0.494026i
$$949$$ −8259.57 −0.282526
$$950$$ 0 0
$$951$$ −12901.9 −0.439929
$$952$$ 9118.53i 0.310434i
$$953$$ − 56839.4i − 1.93201i −0.258517 0.966007i $$-0.583234\pi$$
0.258517 0.966007i $$-0.416766\pi$$
$$954$$ 4675.53 0.158675
$$955$$ 0 0
$$956$$ −14615.3 −0.494449
$$957$$ 15437.9i 0.521459i
$$958$$ − 22105.7i − 0.745515i
$$959$$ 6272.07 0.211195
$$960$$ 0 0
$$961$$ −5871.48 −0.197089
$$962$$ 17286.3i 0.579349i
$$963$$ 5062.78i 0.169414i
$$964$$ −27382.7 −0.914873
$$965$$ 0 0
$$966$$ −5777.10 −0.192417
$$967$$ − 13284.2i − 0.441769i −0.975300 0.220884i $$-0.929106\pi$$
0.975300 0.220884i $$-0.0708943\pi$$
$$968$$ − 1516.15i − 0.0503420i
$$969$$ 9568.44 0.317216
$$970$$ 0 0
$$971$$ 12153.0 0.401658 0.200829 0.979626i $$-0.435636\pi$$
0.200829 + 0.979626i $$0.435636\pi$$
$$972$$ − 1240.44i − 0.0409332i
$$973$$ 16195.3i 0.533604i
$$974$$ −21888.1 −0.720060
$$975$$ 0 0
$$976$$ −495.298 −0.0162440
$$977$$ − 37999.3i − 1.24433i −0.782888 0.622163i $$-0.786254\pi$$
0.782888 0.622163i $$-0.213746\pi$$
$$978$$ − 5688.18i − 0.185980i
$$979$$ 25349.0 0.827537
$$980$$ 0 0
$$981$$ 16477.1 0.536264
$$982$$ 8334.42i 0.270837i
$$983$$ − 22375.0i − 0.725993i −0.931791 0.362996i $$-0.881754\pi$$
0.931791 0.362996i $$-0.118246\pi$$
$$984$$ −23665.5 −0.766695
$$985$$ 0 0
$$986$$ −13676.0 −0.441717
$$987$$ − 11386.6i − 0.367213i
$$988$$ − 8086.65i − 0.260395i
$$989$$ 83772.9 2.69345
$$990$$ 0 0
$$991$$ 18985.3 0.608564 0.304282 0.952582i $$-0.401583\pi$$
0.304282 + 0.952582i $$0.401583\pi$$
$$992$$ − 28351.2i − 0.907412i
$$993$$ 20062.6i 0.641156i
$$994$$ −1433.29 −0.0457358
$$995$$ 0 0
$$996$$ −5775.70 −0.183745
$$997$$ − 56476.5i − 1.79401i −0.442019 0.897006i $$-0.645738\pi$$
0.442019 0.897006i $$-0.354262\pi$$
$$998$$ 17539.6i 0.556321i
$$999$$ 9453.37 0.299391
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.j.274.2 4
5.2 odd 4 525.4.a.i.1.2 2
5.3 odd 4 105.4.a.g.1.1 2
5.4 even 2 inner 525.4.d.j.274.3 4
15.2 even 4 1575.4.a.y.1.1 2
15.8 even 4 315.4.a.g.1.2 2
20.3 even 4 1680.4.a.y.1.1 2
35.13 even 4 735.4.a.q.1.1 2
105.83 odd 4 2205.4.a.v.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.g.1.1 2 5.3 odd 4
315.4.a.g.1.2 2 15.8 even 4
525.4.a.i.1.2 2 5.2 odd 4
525.4.d.j.274.2 4 1.1 even 1 trivial
525.4.d.j.274.3 4 5.4 even 2 inner
735.4.a.q.1.1 2 35.13 even 4
1575.4.a.y.1.1 2 15.2 even 4
1680.4.a.y.1.1 2 20.3 even 4
2205.4.a.v.1.2 2 105.83 odd 4