# Properties

 Label 275.4.b.c.199.4 Level $275$ Weight $4$ Character 275.199 Analytic conductor $16.226$ 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] = [275,4,Mod(199,275)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(275, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("275.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$275 = 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 275.b (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: $$16.2255252516$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 11) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.4 Root $$-0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 275.199 Dual form 275.4.b.c.199.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.73205i q^{2} +7.92820i q^{3} +0.535898 q^{4} -21.6603 q^{6} +3.07180i q^{7} +23.3205i q^{8} -35.8564 q^{9} +O(q^{10})$$ $$q+2.73205i q^{2} +7.92820i q^{3} +0.535898 q^{4} -21.6603 q^{6} +3.07180i q^{7} +23.3205i q^{8} -35.8564 q^{9} -11.0000 q^{11} +4.24871i q^{12} -5.35898i q^{13} -8.39230 q^{14} -59.4256 q^{16} -41.2154i q^{17} -97.9615i q^{18} -139.923 q^{19} -24.3538 q^{21} -30.0526i q^{22} +111.354i q^{23} -184.890 q^{24} +14.6410 q^{26} -70.2154i q^{27} +1.64617i q^{28} +24.9948 q^{29} +31.4974 q^{31} +24.2102i q^{32} -87.2102i q^{33} +112.603 q^{34} -19.2154 q^{36} +13.1436i q^{37} -382.277i q^{38} +42.4871 q^{39} +261.072 q^{41} -66.5359i q^{42} +57.7128i q^{43} -5.89488 q^{44} -304.224 q^{46} -343.846i q^{47} -471.138i q^{48} +333.564 q^{49} +326.764 q^{51} -2.87187i q^{52} +342.995i q^{53} +191.832 q^{54} -71.6359 q^{56} -1109.34i q^{57} +68.2872i q^{58} -88.3693 q^{59} +738.697 q^{61} +86.0526i q^{62} -110.144i q^{63} -541.549 q^{64} +238.263 q^{66} +342.359i q^{67} -22.0873i q^{68} -882.836 q^{69} -207.364 q^{71} -836.190i q^{72} +1010.60i q^{73} -35.9090 q^{74} -74.9845 q^{76} -33.7898i q^{77} +116.077i q^{78} -1294.23 q^{79} -411.441 q^{81} +713.261i q^{82} -441.846i q^{83} -13.0512 q^{84} -157.674 q^{86} +198.164i q^{87} -256.526i q^{88} +1489.11 q^{89} +16.4617 q^{91} +59.6743i q^{92} +249.718i q^{93} +939.405 q^{94} -191.944 q^{96} +1346.42i q^{97} +911.314i q^{98} +394.420 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 16 q^{4} - 52 q^{6} - 88 q^{9}+O(q^{10})$$ 4 * q + 16 * q^4 - 52 * q^6 - 88 * q^9 $$4 q + 16 q^{4} - 52 q^{6} - 88 q^{9} - 44 q^{11} + 8 q^{14} - 16 q^{16} - 144 q^{19} + 152 q^{21} - 504 q^{24} - 80 q^{26} - 288 q^{29} - 68 q^{31} + 104 q^{34} - 160 q^{36} - 800 q^{39} + 1072 q^{41} - 176 q^{44} - 628 q^{46} + 780 q^{49} - 328 q^{51} + 220 q^{54} + 240 q^{56} - 1268 q^{59} + 1680 q^{61} - 448 q^{64} + 572 q^{66} - 1924 q^{69} - 1356 q^{71} - 12 q^{74} + 864 q^{76} - 632 q^{79} - 2588 q^{81} + 1472 q^{84} - 312 q^{86} + 3684 q^{89} + 2560 q^{91} + 1984 q^{94} - 1904 q^{96} + 968 q^{99}+O(q^{100})$$ 4 * q + 16 * q^4 - 52 * q^6 - 88 * q^9 - 44 * q^11 + 8 * q^14 - 16 * q^16 - 144 * q^19 + 152 * q^21 - 504 * q^24 - 80 * q^26 - 288 * q^29 - 68 * q^31 + 104 * q^34 - 160 * q^36 - 800 * q^39 + 1072 * q^41 - 176 * q^44 - 628 * q^46 + 780 * q^49 - 328 * q^51 + 220 * q^54 + 240 * q^56 - 1268 * q^59 + 1680 * q^61 - 448 * q^64 + 572 * q^66 - 1924 * q^69 - 1356 * q^71 - 12 * q^74 + 864 * q^76 - 632 * q^79 - 2588 * q^81 + 1472 * q^84 - 312 * q^86 + 3684 * q^89 + 2560 * q^91 + 1984 * q^94 - 1904 * q^96 + 968 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.73205i 0.965926i 0.875641 + 0.482963i $$0.160439\pi$$
−0.875641 + 0.482963i $$0.839561\pi$$
$$3$$ 7.92820i 1.52578i 0.646526 + 0.762892i $$0.276221\pi$$
−0.646526 + 0.762892i $$0.723779\pi$$
$$4$$ 0.535898 0.0669873
$$5$$ 0 0
$$6$$ −21.6603 −1.47379
$$7$$ 3.07180i 0.165861i 0.996555 + 0.0829307i $$0.0264280\pi$$
−0.996555 + 0.0829307i $$0.973572\pi$$
$$8$$ 23.3205i 1.03063i
$$9$$ −35.8564 −1.32802
$$10$$ 0 0
$$11$$ −11.0000 −0.301511
$$12$$ 4.24871i 0.102208i
$$13$$ − 5.35898i − 0.114332i −0.998365 0.0571659i $$-0.981794\pi$$
0.998365 0.0571659i $$-0.0182064\pi$$
$$14$$ −8.39230 −0.160210
$$15$$ 0 0
$$16$$ −59.4256 −0.928525
$$17$$ − 41.2154i − 0.588012i −0.955804 0.294006i $$-0.905011\pi$$
0.955804 0.294006i $$-0.0949885\pi$$
$$18$$ − 97.9615i − 1.28276i
$$19$$ −139.923 −1.68950 −0.844751 0.535159i $$-0.820252\pi$$
−0.844751 + 0.535159i $$0.820252\pi$$
$$20$$ 0 0
$$21$$ −24.3538 −0.253069
$$22$$ − 30.0526i − 0.291238i
$$23$$ 111.354i 1.00952i 0.863261 + 0.504758i $$0.168418\pi$$
−0.863261 + 0.504758i $$0.831582\pi$$
$$24$$ −184.890 −1.57252
$$25$$ 0 0
$$26$$ 14.6410 0.110436
$$27$$ − 70.2154i − 0.500480i
$$28$$ 1.64617i 0.0111106i
$$29$$ 24.9948 0.160049 0.0800246 0.996793i $$-0.474500\pi$$
0.0800246 + 0.996793i $$0.474500\pi$$
$$30$$ 0 0
$$31$$ 31.4974 0.182487 0.0912436 0.995829i $$-0.470916\pi$$
0.0912436 + 0.995829i $$0.470916\pi$$
$$32$$ 24.2102i 0.133744i
$$33$$ − 87.2102i − 0.460041i
$$34$$ 112.603 0.567976
$$35$$ 0 0
$$36$$ −19.2154 −0.0889601
$$37$$ 13.1436i 0.0583998i 0.999574 + 0.0291999i $$0.00929594\pi$$
−0.999574 + 0.0291999i $$0.990704\pi$$
$$38$$ − 382.277i − 1.63193i
$$39$$ 42.4871 0.174446
$$40$$ 0 0
$$41$$ 261.072 0.994453 0.497226 0.867621i $$-0.334352\pi$$
0.497226 + 0.867621i $$0.334352\pi$$
$$42$$ − 66.5359i − 0.244446i
$$43$$ 57.7128i 0.204677i 0.994750 + 0.102339i $$0.0326325\pi$$
−0.994750 + 0.102339i $$0.967367\pi$$
$$44$$ −5.89488 −0.0201974
$$45$$ 0 0
$$46$$ −304.224 −0.975118
$$47$$ − 343.846i − 1.06713i −0.845759 0.533565i $$-0.820852\pi$$
0.845759 0.533565i $$-0.179148\pi$$
$$48$$ − 471.138i − 1.41673i
$$49$$ 333.564 0.972490
$$50$$ 0 0
$$51$$ 326.764 0.897179
$$52$$ − 2.87187i − 0.00765879i
$$53$$ 342.995i 0.888943i 0.895793 + 0.444471i $$0.146608\pi$$
−0.895793 + 0.444471i $$0.853392\pi$$
$$54$$ 191.832 0.483426
$$55$$ 0 0
$$56$$ −71.6359 −0.170942
$$57$$ − 1109.34i − 2.57782i
$$58$$ 68.2872i 0.154596i
$$59$$ −88.3693 −0.194995 −0.0974975 0.995236i $$-0.531084\pi$$
−0.0974975 + 0.995236i $$0.531084\pi$$
$$60$$ 0 0
$$61$$ 738.697 1.55050 0.775250 0.631654i $$-0.217624\pi$$
0.775250 + 0.631654i $$0.217624\pi$$
$$62$$ 86.0526i 0.176269i
$$63$$ − 110.144i − 0.220266i
$$64$$ −541.549 −1.05771
$$65$$ 0 0
$$66$$ 238.263 0.444365
$$67$$ 342.359i 0.624266i 0.950038 + 0.312133i $$0.101043\pi$$
−0.950038 + 0.312133i $$0.898957\pi$$
$$68$$ − 22.0873i − 0.0393893i
$$69$$ −882.836 −1.54030
$$70$$ 0 0
$$71$$ −207.364 −0.346614 −0.173307 0.984868i $$-0.555445\pi$$
−0.173307 + 0.984868i $$0.555445\pi$$
$$72$$ − 836.190i − 1.36869i
$$73$$ 1010.60i 1.62030i 0.586224 + 0.810149i $$0.300614\pi$$
−0.586224 + 0.810149i $$0.699386\pi$$
$$74$$ −35.9090 −0.0564099
$$75$$ 0 0
$$76$$ −74.9845 −0.113175
$$77$$ − 33.7898i − 0.0500091i
$$78$$ 116.077i 0.168502i
$$79$$ −1294.23 −1.84319 −0.921593 0.388157i $$-0.873112\pi$$
−0.921593 + 0.388157i $$0.873112\pi$$
$$80$$ 0 0
$$81$$ −411.441 −0.564391
$$82$$ 713.261i 0.960568i
$$83$$ − 441.846i − 0.584324i −0.956369 0.292162i $$-0.905625\pi$$
0.956369 0.292162i $$-0.0943747\pi$$
$$84$$ −13.0512 −0.0169524
$$85$$ 0 0
$$86$$ −157.674 −0.197703
$$87$$ 198.164i 0.244200i
$$88$$ − 256.526i − 0.310747i
$$89$$ 1489.11 1.77355 0.886773 0.462205i $$-0.152942\pi$$
0.886773 + 0.462205i $$0.152942\pi$$
$$90$$ 0 0
$$91$$ 16.4617 0.0189633
$$92$$ 59.6743i 0.0676248i
$$93$$ 249.718i 0.278436i
$$94$$ 939.405 1.03077
$$95$$ 0 0
$$96$$ −191.944 −0.204064
$$97$$ 1346.42i 1.40936i 0.709526 + 0.704679i $$0.248909\pi$$
−0.709526 + 0.704679i $$0.751091\pi$$
$$98$$ 911.314i 0.939353i
$$99$$ 394.420 0.400412
$$100$$ 0 0
$$101$$ −161.461 −0.159069 −0.0795347 0.996832i $$-0.525343\pi$$
−0.0795347 + 0.996832i $$0.525343\pi$$
$$102$$ 892.736i 0.866608i
$$103$$ 34.7592i 0.0332517i 0.999862 + 0.0166259i $$0.00529242\pi$$
−0.999862 + 0.0166259i $$0.994708\pi$$
$$104$$ 124.974 0.117834
$$105$$ 0 0
$$106$$ −937.079 −0.858653
$$107$$ 832.179i 0.751867i 0.926647 + 0.375934i $$0.122678\pi$$
−0.926647 + 0.375934i $$0.877322\pi$$
$$108$$ − 37.6283i − 0.0335258i
$$109$$ −1044.26 −0.917629 −0.458815 0.888532i $$-0.651726\pi$$
−0.458815 + 0.888532i $$0.651726\pi$$
$$110$$ 0 0
$$111$$ −104.205 −0.0891055
$$112$$ − 182.543i − 0.154007i
$$113$$ − 295.082i − 0.245654i −0.992428 0.122827i $$-0.960804\pi$$
0.992428 0.122827i $$-0.0391961\pi$$
$$114$$ 3030.77 2.48998
$$115$$ 0 0
$$116$$ 13.3947 0.0107213
$$117$$ 192.154i 0.151834i
$$118$$ − 241.429i − 0.188351i
$$119$$ 126.605 0.0975285
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 2018.16i 1.49767i
$$123$$ 2069.83i 1.51732i
$$124$$ 16.8794 0.0122243
$$125$$ 0 0
$$126$$ 300.918 0.212761
$$127$$ − 1317.60i − 0.920618i −0.887759 0.460309i $$-0.847739\pi$$
0.887759 0.460309i $$-0.152261\pi$$
$$128$$ − 1285.86i − 0.887928i
$$129$$ −457.559 −0.312293
$$130$$ 0 0
$$131$$ −1600.71 −1.06759 −0.533797 0.845612i $$-0.679235\pi$$
−0.533797 + 0.845612i $$0.679235\pi$$
$$132$$ − 46.7358i − 0.0308169i
$$133$$ − 429.815i − 0.280223i
$$134$$ −935.342 −0.602994
$$135$$ 0 0
$$136$$ 961.164 0.606023
$$137$$ 1611.68i 1.00507i 0.864556 + 0.502536i $$0.167600\pi$$
−0.864556 + 0.502536i $$0.832400\pi$$
$$138$$ − 2411.95i − 1.48782i
$$139$$ 31.8619 0.0194424 0.00972120 0.999953i $$-0.496906\pi$$
0.00972120 + 0.999953i $$0.496906\pi$$
$$140$$ 0 0
$$141$$ 2726.08 1.62821
$$142$$ − 566.529i − 0.334803i
$$143$$ 58.9488i 0.0344724i
$$144$$ 2130.79 1.23310
$$145$$ 0 0
$$146$$ −2761.01 −1.56509
$$147$$ 2644.56i 1.48381i
$$148$$ 7.04363i 0.00391205i
$$149$$ 2428.34 1.33515 0.667576 0.744542i $$-0.267332\pi$$
0.667576 + 0.744542i $$0.267332\pi$$
$$150$$ 0 0
$$151$$ −2576.68 −1.38866 −0.694328 0.719659i $$-0.744298\pi$$
−0.694328 + 0.719659i $$0.744298\pi$$
$$152$$ − 3263.08i − 1.74125i
$$153$$ 1477.84i 0.780889i
$$154$$ 92.3154 0.0483051
$$155$$ 0 0
$$156$$ 22.7688 0.0116856
$$157$$ 2475.94i 1.25861i 0.777158 + 0.629305i $$0.216660\pi$$
−0.777158 + 0.629305i $$0.783340\pi$$
$$158$$ − 3535.89i − 1.78038i
$$159$$ −2719.33 −1.35633
$$160$$ 0 0
$$161$$ −342.056 −0.167440
$$162$$ − 1124.08i − 0.545160i
$$163$$ 2725.11i 1.30949i 0.755850 + 0.654745i $$0.227224\pi$$
−0.755850 + 0.654745i $$0.772776\pi$$
$$164$$ 139.908 0.0666157
$$165$$ 0 0
$$166$$ 1207.15 0.564414
$$167$$ 2737.30i 1.26837i 0.773180 + 0.634187i $$0.218665\pi$$
−0.773180 + 0.634187i $$0.781335\pi$$
$$168$$ − 567.944i − 0.260820i
$$169$$ 2168.28 0.986928
$$170$$ 0 0
$$171$$ 5017.14 2.24368
$$172$$ 30.9282i 0.0137108i
$$173$$ − 2307.42i − 1.01404i −0.861933 0.507022i $$-0.830746\pi$$
0.861933 0.507022i $$-0.169254\pi$$
$$174$$ −541.395 −0.235879
$$175$$ 0 0
$$176$$ 653.682 0.279961
$$177$$ − 700.610i − 0.297520i
$$178$$ 4068.33i 1.71311i
$$179$$ 1312.15 0.547905 0.273953 0.961743i $$-0.411669\pi$$
0.273953 + 0.961743i $$0.411669\pi$$
$$180$$ 0 0
$$181$$ −803.174 −0.329831 −0.164916 0.986308i $$-0.552735\pi$$
−0.164916 + 0.986308i $$0.552735\pi$$
$$182$$ 44.9742i 0.0183171i
$$183$$ 5856.54i 2.36573i
$$184$$ −2596.83 −1.04044
$$185$$ 0 0
$$186$$ −682.242 −0.268949
$$187$$ 453.369i 0.177292i
$$188$$ − 184.267i − 0.0714842i
$$189$$ 215.687 0.0830103
$$190$$ 0 0
$$191$$ 1718.25 0.650932 0.325466 0.945554i $$-0.394479\pi$$
0.325466 + 0.945554i $$0.394479\pi$$
$$192$$ − 4293.51i − 1.61384i
$$193$$ − 1340.18i − 0.499837i −0.968267 0.249919i $$-0.919596\pi$$
0.968267 0.249919i $$-0.0804038\pi$$
$$194$$ −3678.48 −1.36134
$$195$$ 0 0
$$196$$ 178.756 0.0651445
$$197$$ − 3518.33i − 1.27244i −0.771508 0.636220i $$-0.780497\pi$$
0.771508 0.636220i $$-0.219503\pi$$
$$198$$ 1077.58i 0.386768i
$$199$$ −823.692 −0.293417 −0.146709 0.989180i $$-0.546868\pi$$
−0.146709 + 0.989180i $$0.546868\pi$$
$$200$$ 0 0
$$201$$ −2714.29 −0.952494
$$202$$ − 441.121i − 0.153649i
$$203$$ 76.7791i 0.0265460i
$$204$$ 175.112 0.0600996
$$205$$ 0 0
$$206$$ −94.9639 −0.0321187
$$207$$ − 3992.75i − 1.34065i
$$208$$ 318.461i 0.106160i
$$209$$ 1539.15 0.509404
$$210$$ 0 0
$$211$$ −107.343 −0.0350228 −0.0175114 0.999847i $$-0.505574\pi$$
−0.0175114 + 0.999847i $$0.505574\pi$$
$$212$$ 183.810i 0.0595479i
$$213$$ − 1644.03i − 0.528858i
$$214$$ −2273.56 −0.726248
$$215$$ 0 0
$$216$$ 1637.46 0.515810
$$217$$ 96.7537i 0.0302676i
$$218$$ − 2852.96i − 0.886362i
$$219$$ −8012.24 −2.47222
$$220$$ 0 0
$$221$$ −220.873 −0.0672285
$$222$$ − 284.694i − 0.0860693i
$$223$$ 3933.68i 1.18125i 0.806946 + 0.590625i $$0.201119\pi$$
−0.806946 + 0.590625i $$0.798881\pi$$
$$224$$ −74.3689 −0.0221830
$$225$$ 0 0
$$226$$ 806.178 0.237284
$$227$$ − 1771.90i − 0.518085i −0.965866 0.259042i $$-0.916593\pi$$
0.965866 0.259042i $$-0.0834069\pi$$
$$228$$ − 594.493i − 0.172681i
$$229$$ −1915.37 −0.552713 −0.276356 0.961055i $$-0.589127\pi$$
−0.276356 + 0.961055i $$0.589127\pi$$
$$230$$ 0 0
$$231$$ 267.892 0.0763031
$$232$$ 582.892i 0.164952i
$$233$$ − 4396.32i − 1.23610i −0.786137 0.618052i $$-0.787922\pi$$
0.786137 0.618052i $$-0.212078\pi$$
$$234$$ −524.974 −0.146661
$$235$$ 0 0
$$236$$ −47.3570 −0.0130622
$$237$$ − 10260.9i − 2.81230i
$$238$$ 345.892i 0.0942053i
$$239$$ 4084.49 1.10546 0.552728 0.833362i $$-0.313587\pi$$
0.552728 + 0.833362i $$0.313587\pi$$
$$240$$ 0 0
$$241$$ 3908.58 1.04471 0.522353 0.852730i $$-0.325054\pi$$
0.522353 + 0.852730i $$0.325054\pi$$
$$242$$ 330.578i 0.0878114i
$$243$$ − 5157.80i − 1.36162i
$$244$$ 395.867 0.103864
$$245$$ 0 0
$$246$$ −5654.88 −1.46562
$$247$$ 749.845i 0.193164i
$$248$$ 734.536i 0.188077i
$$249$$ 3503.05 0.891552
$$250$$ 0 0
$$251$$ 1094.89 0.275335 0.137667 0.990479i $$-0.456040\pi$$
0.137667 + 0.990479i $$0.456040\pi$$
$$252$$ − 59.0258i − 0.0147551i
$$253$$ − 1224.89i − 0.304381i
$$254$$ 3599.76 0.889249
$$255$$ 0 0
$$256$$ −819.364 −0.200040
$$257$$ 783.179i 0.190091i 0.995473 + 0.0950454i $$0.0302996\pi$$
−0.995473 + 0.0950454i $$0.969700\pi$$
$$258$$ − 1250.07i − 0.301652i
$$259$$ −40.3744 −0.00968628
$$260$$ 0 0
$$261$$ −896.225 −0.212548
$$262$$ − 4373.23i − 1.03122i
$$263$$ − 6180.06i − 1.44897i −0.689292 0.724484i $$-0.742078\pi$$
0.689292 0.724484i $$-0.257922\pi$$
$$264$$ 2033.79 0.474132
$$265$$ 0 0
$$266$$ 1174.28 0.270675
$$267$$ 11806.0i 2.70605i
$$268$$ 183.470i 0.0418179i
$$269$$ −986.965 −0.223704 −0.111852 0.993725i $$-0.535678\pi$$
−0.111852 + 0.993725i $$0.535678\pi$$
$$270$$ 0 0
$$271$$ 4576.99 1.02595 0.512975 0.858404i $$-0.328543\pi$$
0.512975 + 0.858404i $$0.328543\pi$$
$$272$$ 2449.25i 0.545984i
$$273$$ 130.512i 0.0289338i
$$274$$ −4403.18 −0.970825
$$275$$ 0 0
$$276$$ −473.110 −0.103181
$$277$$ 567.836i 0.123169i 0.998102 + 0.0615847i $$0.0196154\pi$$
−0.998102 + 0.0615847i $$0.980385\pi$$
$$278$$ 87.0484i 0.0187799i
$$279$$ −1129.38 −0.242346
$$280$$ 0 0
$$281$$ 5311.01 1.12750 0.563752 0.825944i $$-0.309357\pi$$
0.563752 + 0.825944i $$0.309357\pi$$
$$282$$ 7447.79i 1.57273i
$$283$$ 4728.44i 0.993204i 0.867978 + 0.496602i $$0.165419\pi$$
−0.867978 + 0.496602i $$0.834581\pi$$
$$284$$ −111.126 −0.0232187
$$285$$ 0 0
$$286$$ −161.051 −0.0332977
$$287$$ 801.960i 0.164941i
$$288$$ − 868.092i − 0.177614i
$$289$$ 3214.29 0.654242
$$290$$ 0 0
$$291$$ −10674.7 −2.15038
$$292$$ 541.579i 0.108539i
$$293$$ − 2328.92i − 0.464358i −0.972673 0.232179i $$-0.925415\pi$$
0.972673 0.232179i $$-0.0745854\pi$$
$$294$$ −7225.08 −1.43325
$$295$$ 0 0
$$296$$ −306.515 −0.0601886
$$297$$ 772.369i 0.150900i
$$298$$ 6634.36i 1.28966i
$$299$$ 596.743 0.115420
$$300$$ 0 0
$$301$$ −177.282 −0.0339481
$$302$$ − 7039.61i − 1.34134i
$$303$$ − 1280.10i − 0.242705i
$$304$$ 8315.01 1.56875
$$305$$ 0 0
$$306$$ −4037.52 −0.754280
$$307$$ − 1678.07i − 0.311962i −0.987760 0.155981i $$-0.950146\pi$$
0.987760 0.155981i $$-0.0498539\pi$$
$$308$$ − 18.1079i − 0.00334997i
$$309$$ −275.578 −0.0507349
$$310$$ 0 0
$$311$$ 3572.71 0.651413 0.325707 0.945471i $$-0.394398\pi$$
0.325707 + 0.945471i $$0.394398\pi$$
$$312$$ 990.821i 0.179789i
$$313$$ − 7184.36i − 1.29739i −0.761047 0.648697i $$-0.775314\pi$$
0.761047 0.648697i $$-0.224686\pi$$
$$314$$ −6764.40 −1.21572
$$315$$ 0 0
$$316$$ −693.573 −0.123470
$$317$$ − 15.7077i − 0.00278306i −0.999999 0.00139153i $$-0.999557\pi$$
0.999999 0.00139153i $$-0.000442938\pi$$
$$318$$ − 7429.36i − 1.31012i
$$319$$ −274.943 −0.0482566
$$320$$ 0 0
$$321$$ −6597.69 −1.14719
$$322$$ − 934.515i − 0.161734i
$$323$$ 5766.98i 0.993447i
$$324$$ −220.491 −0.0378070
$$325$$ 0 0
$$326$$ −7445.13 −1.26487
$$327$$ − 8279.08i − 1.40010i
$$328$$ 6088.33i 1.02491i
$$329$$ 1056.23 0.176996
$$330$$ 0 0
$$331$$ −1318.95 −0.219022 −0.109511 0.993986i $$-0.534928\pi$$
−0.109511 + 0.993986i $$0.534928\pi$$
$$332$$ − 236.785i − 0.0391423i
$$333$$ − 471.282i − 0.0775558i
$$334$$ −7478.43 −1.22515
$$335$$ 0 0
$$336$$ 1447.24 0.234981
$$337$$ − 239.183i − 0.0386621i −0.999813 0.0193310i $$-0.993846\pi$$
0.999813 0.0193310i $$-0.00615365\pi$$
$$338$$ 5923.85i 0.953299i
$$339$$ 2339.47 0.374816
$$340$$ 0 0
$$341$$ −346.472 −0.0550220
$$342$$ 13707.1i 2.16723i
$$343$$ 2078.27i 0.327160i
$$344$$ −1345.89 −0.210947
$$345$$ 0 0
$$346$$ 6303.98 0.979491
$$347$$ − 5862.79i − 0.907006i −0.891255 0.453503i $$-0.850174\pi$$
0.891255 0.453503i $$-0.149826\pi$$
$$348$$ 106.196i 0.0163583i
$$349$$ −3491.73 −0.535553 −0.267776 0.963481i $$-0.586289\pi$$
−0.267776 + 0.963481i $$0.586289\pi$$
$$350$$ 0 0
$$351$$ −376.283 −0.0572208
$$352$$ − 266.313i − 0.0403253i
$$353$$ 10916.7i 1.64600i 0.568043 + 0.822999i $$0.307701\pi$$
−0.568043 + 0.822999i $$0.692299\pi$$
$$354$$ 1914.10 0.287382
$$355$$ 0 0
$$356$$ 798.013 0.118805
$$357$$ 1003.75i 0.148807i
$$358$$ 3584.87i 0.529236i
$$359$$ 11500.7 1.69077 0.845384 0.534160i $$-0.179372\pi$$
0.845384 + 0.534160i $$0.179372\pi$$
$$360$$ 0 0
$$361$$ 12719.5 1.85442
$$362$$ − 2194.31i − 0.318592i
$$363$$ 959.313i 0.138708i
$$364$$ 8.82180 0.00127030
$$365$$ 0 0
$$366$$ −16000.4 −2.28512
$$367$$ 6767.01i 0.962493i 0.876585 + 0.481246i $$0.159816\pi$$
−0.876585 + 0.481246i $$0.840184\pi$$
$$368$$ − 6617.27i − 0.937362i
$$369$$ −9361.10 −1.32065
$$370$$ 0 0
$$371$$ −1053.61 −0.147441
$$372$$ 133.823i 0.0186517i
$$373$$ 5310.22i 0.737139i 0.929600 + 0.368569i $$0.120152\pi$$
−0.929600 + 0.368569i $$0.879848\pi$$
$$374$$ −1238.63 −0.171251
$$375$$ 0 0
$$376$$ 8018.67 1.09982
$$377$$ − 133.947i − 0.0182987i
$$378$$ 589.269i 0.0801818i
$$379$$ 838.267 0.113612 0.0568059 0.998385i $$-0.481908\pi$$
0.0568059 + 0.998385i $$0.481908\pi$$
$$380$$ 0 0
$$381$$ 10446.2 1.40466
$$382$$ 4694.34i 0.628752i
$$383$$ 2832.16i 0.377851i 0.981991 + 0.188925i $$0.0605004\pi$$
−0.981991 + 0.188925i $$0.939500\pi$$
$$384$$ 10194.5 1.35479
$$385$$ 0 0
$$386$$ 3661.45 0.482806
$$387$$ − 2069.37i − 0.271814i
$$388$$ 721.542i 0.0944091i
$$389$$ −3111.25 −0.405519 −0.202759 0.979229i $$-0.564991\pi$$
−0.202759 + 0.979229i $$0.564991\pi$$
$$390$$ 0 0
$$391$$ 4589.49 0.593608
$$392$$ 7778.88i 1.00228i
$$393$$ − 12690.8i − 1.62892i
$$394$$ 9612.25 1.22908
$$395$$ 0 0
$$396$$ 211.369 0.0268225
$$397$$ 14208.7i 1.79626i 0.439728 + 0.898131i $$0.355075\pi$$
−0.439728 + 0.898131i $$0.644925\pi$$
$$398$$ − 2250.37i − 0.283419i
$$399$$ 3407.66 0.427560
$$400$$ 0 0
$$401$$ −6261.68 −0.779784 −0.389892 0.920861i $$-0.627488\pi$$
−0.389892 + 0.920861i $$0.627488\pi$$
$$402$$ − 7415.58i − 0.920039i
$$403$$ − 168.794i − 0.0208641i
$$404$$ −86.5269 −0.0106556
$$405$$ 0 0
$$406$$ −209.764 −0.0256415
$$407$$ − 144.580i − 0.0176082i
$$408$$ 7620.30i 0.924660i
$$409$$ 4192.50 0.506860 0.253430 0.967354i $$-0.418441\pi$$
0.253430 + 0.967354i $$0.418441\pi$$
$$410$$ 0 0
$$411$$ −12777.7 −1.53352
$$412$$ 18.6274i 0.00222744i
$$413$$ − 271.453i − 0.0323421i
$$414$$ 10908.4 1.29497
$$415$$ 0 0
$$416$$ 129.742 0.0152912
$$417$$ 252.608i 0.0296649i
$$418$$ 4205.05i 0.492047i
$$419$$ 9287.15 1.08283 0.541416 0.840755i $$-0.317888\pi$$
0.541416 + 0.840755i $$0.317888\pi$$
$$420$$ 0 0
$$421$$ 13146.0 1.52185 0.760923 0.648842i $$-0.224746\pi$$
0.760923 + 0.648842i $$0.224746\pi$$
$$422$$ − 293.267i − 0.0338294i
$$423$$ 12329.1i 1.41716i
$$424$$ −7998.81 −0.916172
$$425$$ 0 0
$$426$$ 4491.56 0.510838
$$427$$ 2269.13i 0.257168i
$$428$$ 445.964i 0.0503656i
$$429$$ −467.358 −0.0525974
$$430$$ 0 0
$$431$$ 4909.67 0.548701 0.274351 0.961630i $$-0.411537\pi$$
0.274351 + 0.961630i $$0.411537\pi$$
$$432$$ 4172.59i 0.464708i
$$433$$ 11743.3i 1.30334i 0.758502 + 0.651671i $$0.225932\pi$$
−0.758502 + 0.651671i $$0.774068\pi$$
$$434$$ −264.336 −0.0292363
$$435$$ 0 0
$$436$$ −559.615 −0.0614695
$$437$$ − 15581.0i − 1.70558i
$$438$$ − 21889.8i − 2.38798i
$$439$$ 11824.2 1.28551 0.642754 0.766073i $$-0.277792\pi$$
0.642754 + 0.766073i $$0.277792\pi$$
$$440$$ 0 0
$$441$$ −11960.4 −1.29148
$$442$$ − 603.435i − 0.0649377i
$$443$$ − 10102.1i − 1.08344i −0.840558 0.541722i $$-0.817772\pi$$
0.840558 0.541722i $$-0.182228\pi$$
$$444$$ −55.8433 −0.00596894
$$445$$ 0 0
$$446$$ −10747.0 −1.14100
$$447$$ 19252.4i 2.03715i
$$448$$ − 1663.53i − 0.175434i
$$449$$ 345.254 0.0362885 0.0181443 0.999835i $$-0.494224\pi$$
0.0181443 + 0.999835i $$0.494224\pi$$
$$450$$ 0 0
$$451$$ −2871.79 −0.299839
$$452$$ − 158.134i − 0.0164557i
$$453$$ − 20428.4i − 2.11879i
$$454$$ 4840.93 0.500431
$$455$$ 0 0
$$456$$ 25870.3 2.65678
$$457$$ − 10567.1i − 1.08164i −0.841138 0.540821i $$-0.818114\pi$$
0.841138 0.540821i $$-0.181886\pi$$
$$458$$ − 5232.89i − 0.533879i
$$459$$ −2893.95 −0.294288
$$460$$ 0 0
$$461$$ 4733.96 0.478270 0.239135 0.970986i $$-0.423136\pi$$
0.239135 + 0.970986i $$0.423136\pi$$
$$462$$ 731.895i 0.0737031i
$$463$$ − 3431.20i − 0.344409i −0.985061 0.172204i $$-0.944911\pi$$
0.985061 0.172204i $$-0.0550890\pi$$
$$464$$ −1485.33 −0.148610
$$465$$ 0 0
$$466$$ 12011.0 1.19399
$$467$$ 5116.96i 0.507034i 0.967331 + 0.253517i $$0.0815873\pi$$
−0.967331 + 0.253517i $$0.918413\pi$$
$$468$$ 102.975i 0.0101710i
$$469$$ −1051.66 −0.103542
$$470$$ 0 0
$$471$$ −19629.8 −1.92037
$$472$$ − 2060.82i − 0.200968i
$$473$$ − 634.841i − 0.0617125i
$$474$$ 28033.2 2.71648
$$475$$ 0 0
$$476$$ 67.8476 0.00653317
$$477$$ − 12298.6i − 1.18053i
$$478$$ 11159.0i 1.06779i
$$479$$ −11566.9 −1.10335 −0.551675 0.834059i $$-0.686011\pi$$
−0.551675 + 0.834059i $$0.686011\pi$$
$$480$$ 0 0
$$481$$ 70.4363 0.00667696
$$482$$ 10678.4i 1.00911i
$$483$$ − 2711.89i − 0.255477i
$$484$$ 64.8437 0.00608975
$$485$$ 0 0
$$486$$ 14091.4 1.31522
$$487$$ − 18326.5i − 1.70525i −0.522527 0.852623i $$-0.675010\pi$$
0.522527 0.852623i $$-0.324990\pi$$
$$488$$ 17226.8i 1.59799i
$$489$$ −21605.2 −1.99800
$$490$$ 0 0
$$491$$ −7617.58 −0.700156 −0.350078 0.936721i $$-0.613845\pi$$
−0.350078 + 0.936721i $$0.613845\pi$$
$$492$$ 1109.22i 0.101641i
$$493$$ − 1030.17i − 0.0941108i
$$494$$ −2048.62 −0.186582
$$495$$ 0 0
$$496$$ −1871.75 −0.169444
$$497$$ − 636.980i − 0.0574899i
$$498$$ 9570.50i 0.861173i
$$499$$ −12909.1 −1.15810 −0.579050 0.815292i $$-0.696576\pi$$
−0.579050 + 0.815292i $$0.696576\pi$$
$$500$$ 0 0
$$501$$ −21701.8 −1.93526
$$502$$ 2991.30i 0.265953i
$$503$$ − 10165.7i − 0.901121i −0.892746 0.450561i $$-0.851224\pi$$
0.892746 0.450561i $$-0.148776\pi$$
$$504$$ 2568.60 0.227013
$$505$$ 0 0
$$506$$ 3346.47 0.294009
$$507$$ 17190.6i 1.50584i
$$508$$ − 706.102i − 0.0616697i
$$509$$ −6449.93 −0.561666 −0.280833 0.959757i $$-0.590611\pi$$
−0.280833 + 0.959757i $$0.590611\pi$$
$$510$$ 0 0
$$511$$ −3104.36 −0.268745
$$512$$ − 12525.4i − 1.08115i
$$513$$ 9824.75i 0.845562i
$$514$$ −2139.68 −0.183614
$$515$$ 0 0
$$516$$ −245.205 −0.0209197
$$517$$ 3782.31i 0.321752i
$$518$$ − 110.305i − 0.00935623i
$$519$$ 18293.7 1.54721
$$520$$ 0 0
$$521$$ −19327.4 −1.62524 −0.812620 0.582794i $$-0.801959\pi$$
−0.812620 + 0.582794i $$0.801959\pi$$
$$522$$ − 2448.53i − 0.205305i
$$523$$ − 6259.09i − 0.523310i −0.965161 0.261655i $$-0.915732\pi$$
0.965161 0.261655i $$-0.0842682\pi$$
$$524$$ −857.819 −0.0715153
$$525$$ 0 0
$$526$$ 16884.2 1.39960
$$527$$ − 1298.18i − 0.107305i
$$528$$ 5182.52i 0.427160i
$$529$$ −232.675 −0.0191235
$$530$$ 0 0
$$531$$ 3168.61 0.258956
$$532$$ − 230.337i − 0.0187714i
$$533$$ − 1399.08i − 0.113698i
$$534$$ −32254.6 −2.61384
$$535$$ 0 0
$$536$$ −7983.99 −0.643387
$$537$$ 10403.0i 0.835985i
$$538$$ − 2696.44i − 0.216081i
$$539$$ −3669.20 −0.293217
$$540$$ 0 0
$$541$$ −14008.2 −1.11323 −0.556616 0.830770i $$-0.687900\pi$$
−0.556616 + 0.830770i $$0.687900\pi$$
$$542$$ 12504.6i 0.990991i
$$543$$ − 6367.72i − 0.503251i
$$544$$ 997.834 0.0786430
$$545$$ 0 0
$$546$$ −356.565 −0.0279479
$$547$$ − 4949.45i − 0.386879i −0.981112 0.193440i $$-0.938036\pi$$
0.981112 0.193440i $$-0.0619644\pi$$
$$548$$ 863.695i 0.0673270i
$$549$$ −26487.0 −2.05909
$$550$$ 0 0
$$551$$ −3497.35 −0.270404
$$552$$ − 20588.2i − 1.58748i
$$553$$ − 3975.60i − 0.305714i
$$554$$ −1551.36 −0.118973
$$555$$ 0 0
$$556$$ 17.0748 0.00130239
$$557$$ − 3801.58i − 0.289188i −0.989491 0.144594i $$-0.953812\pi$$
0.989491 0.144594i $$-0.0461877\pi$$
$$558$$ − 3085.54i − 0.234088i
$$559$$ 309.282 0.0234011
$$560$$ 0 0
$$561$$ −3594.40 −0.270510
$$562$$ 14510.0i 1.08908i
$$563$$ 9900.11i 0.741101i 0.928812 + 0.370551i $$0.120831\pi$$
−0.928812 + 0.370551i $$0.879169\pi$$
$$564$$ 1460.90 0.109069
$$565$$ 0 0
$$566$$ −12918.3 −0.959361
$$567$$ − 1263.86i − 0.0936107i
$$568$$ − 4835.84i − 0.357231i
$$569$$ −5329.16 −0.392636 −0.196318 0.980540i $$-0.562898\pi$$
−0.196318 + 0.980540i $$0.562898\pi$$
$$570$$ 0 0
$$571$$ −16962.6 −1.24319 −0.621597 0.783337i $$-0.713516\pi$$
−0.621597 + 0.783337i $$0.713516\pi$$
$$572$$ 31.5906i 0.00230921i
$$573$$ 13622.6i 0.993181i
$$574$$ −2190.99 −0.159321
$$575$$ 0 0
$$576$$ 19418.0 1.40466
$$577$$ − 15487.0i − 1.11738i −0.829375 0.558692i $$-0.811303\pi$$
0.829375 0.558692i $$-0.188697\pi$$
$$578$$ 8781.61i 0.631949i
$$579$$ 10625.3 0.762643
$$580$$ 0 0
$$581$$ 1357.26 0.0969169
$$582$$ − 29163.7i − 2.07710i
$$583$$ − 3772.94i − 0.268026i
$$584$$ −23567.7 −1.66993
$$585$$ 0 0
$$586$$ 6362.72 0.448535
$$587$$ 11084.2i 0.779373i 0.920948 + 0.389686i $$0.127417\pi$$
−0.920948 + 0.389686i $$0.872583\pi$$
$$588$$ 1417.22i 0.0993964i
$$589$$ −4407.22 −0.308313
$$590$$ 0 0
$$591$$ 27894.0 1.94147
$$592$$ − 781.066i − 0.0542257i
$$593$$ − 4349.68i − 0.301214i −0.988594 0.150607i $$-0.951877\pi$$
0.988594 0.150607i $$-0.0481228\pi$$
$$594$$ −2110.15 −0.145759
$$595$$ 0 0
$$596$$ 1301.34 0.0894382
$$597$$ − 6530.40i − 0.447691i
$$598$$ 1630.33i 0.111487i
$$599$$ −13183.9 −0.899299 −0.449650 0.893205i $$-0.648451\pi$$
−0.449650 + 0.893205i $$0.648451\pi$$
$$600$$ 0 0
$$601$$ −18765.0 −1.27361 −0.636806 0.771024i $$-0.719745\pi$$
−0.636806 + 0.771024i $$0.719745\pi$$
$$602$$ − 484.344i − 0.0327913i
$$603$$ − 12275.8i − 0.829034i
$$604$$ −1380.84 −0.0930223
$$605$$ 0 0
$$606$$ 3497.29 0.234435
$$607$$ 21871.4i 1.46249i 0.682116 + 0.731244i $$0.261060\pi$$
−0.682116 + 0.731244i $$0.738940\pi$$
$$608$$ − 3387.57i − 0.225961i
$$609$$ −608.720 −0.0405034
$$610$$ 0 0
$$611$$ −1842.67 −0.122007
$$612$$ 791.970i 0.0523096i
$$613$$ 3527.85i 0.232445i 0.993223 + 0.116222i $$0.0370785\pi$$
−0.993223 + 0.116222i $$0.962921\pi$$
$$614$$ 4584.56 0.301332
$$615$$ 0 0
$$616$$ 787.994 0.0515409
$$617$$ − 22728.1i − 1.48298i −0.670963 0.741490i $$-0.734119\pi$$
0.670963 0.741490i $$-0.265881\pi$$
$$618$$ − 752.893i − 0.0490062i
$$619$$ 21443.3 1.39237 0.696187 0.717861i $$-0.254879\pi$$
0.696187 + 0.717861i $$0.254879\pi$$
$$620$$ 0 0
$$621$$ 7818.75 0.505243
$$622$$ 9760.81i 0.629217i
$$623$$ 4574.25i 0.294163i
$$624$$ −2524.82 −0.161977
$$625$$ 0 0
$$626$$ 19628.0 1.25319
$$627$$ 12202.7i 0.777240i
$$628$$ 1326.85i 0.0843109i
$$629$$ 541.718 0.0343398
$$630$$ 0 0
$$631$$ 21532.0 1.35844 0.679219 0.733936i $$-0.262319\pi$$
0.679219 + 0.733936i $$0.262319\pi$$
$$632$$ − 30182.0i − 1.89964i
$$633$$ − 851.038i − 0.0534372i
$$634$$ 42.9141 0.00268823
$$635$$ 0 0
$$636$$ −1457.29 −0.0908572
$$637$$ − 1787.56i − 0.111187i
$$638$$ − 751.159i − 0.0466123i
$$639$$ 7435.33 0.460309
$$640$$ 0 0
$$641$$ 20148.3 1.24151 0.620756 0.784004i $$-0.286826\pi$$
0.620756 + 0.784004i $$0.286826\pi$$
$$642$$ − 18025.2i − 1.10810i
$$643$$ − 28869.7i − 1.77062i −0.465000 0.885310i $$-0.653946\pi$$
0.465000 0.885310i $$-0.346054\pi$$
$$644$$ −183.307 −0.0112163
$$645$$ 0 0
$$646$$ −15755.7 −0.959597
$$647$$ − 1590.02i − 0.0966155i −0.998833 0.0483077i $$-0.984617\pi$$
0.998833 0.0483077i $$-0.0153828\pi$$
$$648$$ − 9595.02i − 0.581679i
$$649$$ 972.062 0.0587932
$$650$$ 0 0
$$651$$ −767.083 −0.0461818
$$652$$ 1460.38i 0.0877192i
$$653$$ − 20028.1i − 1.20024i −0.799909 0.600122i $$-0.795119\pi$$
0.799909 0.600122i $$-0.204881\pi$$
$$654$$ 22618.9 1.35240
$$655$$ 0 0
$$656$$ −15514.4 −0.923375
$$657$$ − 36236.5i − 2.15178i
$$658$$ 2885.66i 0.170965i
$$659$$ 10520.7 0.621897 0.310948 0.950427i $$-0.399353\pi$$
0.310948 + 0.950427i $$0.399353\pi$$
$$660$$ 0 0
$$661$$ 3295.83 0.193938 0.0969690 0.995287i $$-0.469085\pi$$
0.0969690 + 0.995287i $$0.469085\pi$$
$$662$$ − 3603.45i − 0.211559i
$$663$$ − 1751.12i − 0.102576i
$$664$$ 10304.1 0.602222
$$665$$ 0 0
$$666$$ 1287.57 0.0749132
$$667$$ 2783.27i 0.161572i
$$668$$ 1466.91i 0.0849649i
$$669$$ −31187.0 −1.80233
$$670$$ 0 0
$$671$$ −8125.67 −0.467493
$$672$$ − 589.612i − 0.0338464i
$$673$$ 1187.64i 0.0680239i 0.999421 + 0.0340119i $$0.0108284\pi$$
−0.999421 + 0.0340119i $$0.989172\pi$$
$$674$$ 653.460 0.0373447
$$675$$ 0 0
$$676$$ 1161.98 0.0661117
$$677$$ 13221.4i 0.750574i 0.926909 + 0.375287i $$0.122456\pi$$
−0.926909 + 0.375287i $$0.877544\pi$$
$$678$$ 6391.55i 0.362044i
$$679$$ −4135.91 −0.233758
$$680$$ 0 0
$$681$$ 14048.0 0.790485
$$682$$ − 946.578i − 0.0531471i
$$683$$ 13831.4i 0.774882i 0.921894 + 0.387441i $$0.126641\pi$$
−0.921894 + 0.387441i $$0.873359\pi$$
$$684$$ 2688.68 0.150298
$$685$$ 0 0
$$686$$ −5677.93 −0.316012
$$687$$ − 15185.4i − 0.843320i
$$688$$ − 3429.62i − 0.190048i
$$689$$ 1838.10 0.101635
$$690$$ 0 0
$$691$$ −9817.07 −0.540462 −0.270231 0.962796i $$-0.587100\pi$$
−0.270231 + 0.962796i $$0.587100\pi$$
$$692$$ − 1236.54i − 0.0679280i
$$693$$ 1211.58i 0.0664128i
$$694$$ 16017.4 0.876101
$$695$$ 0 0
$$696$$ −4621.29 −0.251680
$$697$$ − 10760.2i − 0.584750i
$$698$$ − 9539.58i − 0.517304i
$$699$$ 34854.9 1.88603
$$700$$ 0 0
$$701$$ 29949.8 1.61368 0.806838 0.590773i $$-0.201177\pi$$
0.806838 + 0.590773i $$0.201177\pi$$
$$702$$ − 1028.02i − 0.0552711i
$$703$$ − 1839.09i − 0.0986667i
$$704$$ 5957.03 0.318912
$$705$$ 0 0
$$706$$ −29825.0 −1.58991
$$707$$ − 495.976i − 0.0263835i
$$708$$ − 375.456i − 0.0199301i
$$709$$ −11307.5 −0.598959 −0.299479 0.954103i $$-0.596813\pi$$
−0.299479 + 0.954103i $$0.596813\pi$$
$$710$$ 0 0
$$711$$ 46406.3 2.44778
$$712$$ 34726.9i 1.82787i
$$713$$ 3507.36i 0.184224i
$$714$$ −2742.30 −0.143737
$$715$$ 0 0
$$716$$ 703.181 0.0367027
$$717$$ 32382.7i 1.68669i
$$718$$ 31420.6i 1.63316i
$$719$$ 32623.4 1.69214 0.846070 0.533071i $$-0.178962\pi$$
0.846070 + 0.533071i $$0.178962\pi$$
$$720$$ 0 0
$$721$$ −106.773 −0.00551518
$$722$$ 34750.2i 1.79123i
$$723$$ 30988.0i 1.59399i
$$724$$ −430.420 −0.0220945
$$725$$ 0 0
$$726$$ −2620.89 −0.133981
$$727$$ − 502.545i − 0.0256373i −0.999918 0.0128187i $$-0.995920\pi$$
0.999918 0.0128187i $$-0.00408042\pi$$
$$728$$ 383.895i 0.0195441i
$$729$$ 29783.2 1.51314
$$730$$ 0 0
$$731$$ 2378.66 0.120353
$$732$$ 3138.51i 0.158474i
$$733$$ − 8631.37i − 0.434935i −0.976068 0.217467i $$-0.930220\pi$$
0.976068 0.217467i $$-0.0697796\pi$$
$$734$$ −18487.8 −0.929697
$$735$$ 0 0
$$736$$ −2695.90 −0.135017
$$737$$ − 3765.95i − 0.188223i
$$738$$ − 25575.0i − 1.27565i
$$739$$ 18357.5 0.913792 0.456896 0.889520i $$-0.348961\pi$$
0.456896 + 0.889520i $$0.348961\pi$$
$$740$$ 0 0
$$741$$ −5944.93 −0.294726
$$742$$ − 2878.52i − 0.142417i
$$743$$ − 11182.6i − 0.552155i −0.961135 0.276078i $$-0.910965\pi$$
0.961135 0.276078i $$-0.0890347\pi$$
$$744$$ −5823.55 −0.286965
$$745$$ 0 0
$$746$$ −14507.8 −0.712021
$$747$$ 15843.0i 0.775991i
$$748$$ 242.960i 0.0118763i
$$749$$ −2556.29 −0.124706
$$750$$ 0 0
$$751$$ 16733.4 0.813063 0.406531 0.913637i $$-0.366738\pi$$
0.406531 + 0.913637i $$0.366738\pi$$
$$752$$ 20433.3i 0.990857i
$$753$$ 8680.53i 0.420101i
$$754$$ 365.950 0.0176752
$$755$$ 0 0
$$756$$ 115.587 0.00556064
$$757$$ − 24402.4i − 1.17163i −0.810446 0.585813i $$-0.800775\pi$$
0.810446 0.585813i $$-0.199225\pi$$
$$758$$ 2290.19i 0.109741i
$$759$$ 9711.19 0.464419
$$760$$ 0 0
$$761$$ 8469.33 0.403434 0.201717 0.979444i $$-0.435348\pi$$
0.201717 + 0.979444i $$0.435348\pi$$
$$762$$ 28539.7i 1.35680i
$$763$$ − 3207.74i − 0.152199i
$$764$$ 920.805 0.0436042
$$765$$ 0 0
$$766$$ −7737.62 −0.364976
$$767$$ 473.570i 0.0222941i
$$768$$ − 6496.08i − 0.305218i
$$769$$ −32834.7 −1.53973 −0.769864 0.638208i $$-0.779676\pi$$
−0.769864 + 0.638208i $$0.779676\pi$$
$$770$$ 0 0
$$771$$ −6209.20 −0.290038
$$772$$ − 718.202i − 0.0334827i
$$773$$ 35571.4i 1.65513i 0.561371 + 0.827564i $$0.310274\pi$$
−0.561371 + 0.827564i $$0.689726\pi$$
$$774$$ 5653.64 0.262553
$$775$$ 0 0
$$776$$ −31399.1 −1.45253
$$777$$ − 320.097i − 0.0147792i
$$778$$ − 8500.11i − 0.391701i
$$779$$ −36530.0 −1.68013
$$780$$ 0 0
$$781$$ 2281.01 0.104508
$$782$$ 12538.7i 0.573381i
$$783$$ − 1755.02i − 0.0801014i
$$784$$ −19822.3 −0.902982
$$785$$ 0 0
$$786$$ 34671.8 1.57341
$$787$$ 15729.6i 0.712452i 0.934400 + 0.356226i $$0.115937\pi$$
−0.934400 + 0.356226i $$0.884063\pi$$
$$788$$ − 1885.47i − 0.0852373i
$$789$$ 48996.7 2.21081
$$790$$ 0 0
$$791$$ 906.431 0.0407446
$$792$$ 9198.09i 0.412676i
$$793$$ − 3958.67i − 0.177272i
$$794$$ −38819.0 −1.73506
$$795$$ 0 0
$$796$$ −441.415 −0.0196552
$$797$$ 7888.07i 0.350577i 0.984517 + 0.175288i $$0.0560858\pi$$
−0.984517 + 0.175288i $$0.943914\pi$$
$$798$$ 9309.91i 0.412991i
$$799$$ −14171.8 −0.627485
$$800$$ 0 0
$$801$$ −53394.2 −2.35530
$$802$$ − 17107.2i − 0.753214i
$$803$$ − 11116.6i − 0.488538i
$$804$$ −1454.58 −0.0638050
$$805$$ 0 0
$$806$$ 461.154 0.0201532
$$807$$ − 7824.86i − 0.341323i
$$808$$ − 3765.36i − 0.163942i
$$809$$ −5896.97 −0.256275 −0.128138 0.991756i $$-0.540900\pi$$
−0.128138 + 0.991756i $$0.540900\pi$$
$$810$$ 0 0
$$811$$ 14197.9 0.614744 0.307372 0.951589i $$-0.400550\pi$$
0.307372 + 0.951589i $$0.400550\pi$$
$$812$$ 41.1458i 0.00177824i
$$813$$ 36287.3i 1.56538i
$$814$$ 394.999 0.0170082
$$815$$ 0 0
$$816$$ −19418.2 −0.833053
$$817$$ − 8075.35i − 0.345803i
$$818$$ 11454.1i 0.489589i
$$819$$ −590.258 −0.0251835
$$820$$ 0 0
$$821$$ −19841.7 −0.843459 −0.421729 0.906722i $$-0.638577\pi$$
−0.421729 + 0.906722i $$0.638577\pi$$
$$822$$ − 34909.3i − 1.48127i
$$823$$ 28202.2i 1.19449i 0.802058 + 0.597246i $$0.203738\pi$$
−0.802058 + 0.597246i $$0.796262\pi$$
$$824$$ −810.602 −0.0342702
$$825$$ 0 0
$$826$$ 741.622 0.0312401
$$827$$ 34031.0i 1.43092i 0.698651 + 0.715462i $$0.253784\pi$$
−0.698651 + 0.715462i $$0.746216\pi$$
$$828$$ − 2139.71i − 0.0898067i
$$829$$ −4931.55 −0.206610 −0.103305 0.994650i $$-0.532942\pi$$
−0.103305 + 0.994650i $$0.532942\pi$$
$$830$$ 0 0
$$831$$ −4501.92 −0.187930
$$832$$ 2902.15i 0.120930i
$$833$$ − 13748.0i − 0.571836i
$$834$$ −690.138 −0.0286541
$$835$$ 0 0
$$836$$ 824.830 0.0341236
$$837$$ − 2211.60i − 0.0913312i
$$838$$ 25373.0i 1.04594i
$$839$$ 38189.8 1.57146 0.785731 0.618568i $$-0.212287\pi$$
0.785731 + 0.618568i $$0.212287\pi$$
$$840$$ 0 0
$$841$$ −23764.3 −0.974384
$$842$$ 35915.6i 1.46999i
$$843$$ 42106.8i 1.72033i
$$844$$ −57.5250 −0.00234608
$$845$$ 0 0
$$846$$ −33683.7 −1.36888
$$847$$ 371.687i 0.0150783i
$$848$$ − 20382.7i − 0.825406i
$$849$$ −37488.0 −1.51541
$$850$$ 0 0
$$851$$ −1463.59 −0.0589556
$$852$$ − 881.030i − 0.0354268i
$$853$$ − 42966.8i − 1.72469i −0.506325 0.862343i $$-0.668997\pi$$
0.506325 0.862343i $$-0.331003\pi$$
$$854$$ −6199.37 −0.248405
$$855$$ 0 0
$$856$$ −19406.8 −0.774898
$$857$$ − 17281.5i − 0.688828i −0.938818 0.344414i $$-0.888078\pi$$
0.938818 0.344414i $$-0.111922\pi$$
$$858$$ − 1276.85i − 0.0508052i
$$859$$ −9316.75 −0.370062 −0.185031 0.982733i $$-0.559239\pi$$
−0.185031 + 0.982733i $$0.559239\pi$$
$$860$$ 0 0
$$861$$ −6358.10 −0.251665
$$862$$ 13413.5i 0.530005i
$$863$$ 9647.65i 0.380544i 0.981731 + 0.190272i $$0.0609370\pi$$
−0.981731 + 0.190272i $$0.939063\pi$$
$$864$$ 1699.93 0.0669361
$$865$$ 0 0
$$866$$ −32083.3 −1.25893
$$867$$ 25483.6i 0.998232i
$$868$$ 51.8501i 0.00202754i
$$869$$ 14236.5 0.555742
$$870$$ 0 0
$$871$$ 1834.70 0.0713735
$$872$$ − 24352.6i − 0.945737i
$$873$$ − 48277.6i − 1.87165i
$$874$$ 42568.0 1.64746
$$875$$ 0 0
$$876$$ −4293.75 −0.165608
$$877$$ 19728.7i 0.759624i 0.925064 + 0.379812i $$0.124011\pi$$
−0.925064 + 0.379812i $$0.875989\pi$$
$$878$$ 32304.3i 1.24171i
$$879$$ 18464.1 0.708509
$$880$$ 0 0
$$881$$ 19473.9 0.744712 0.372356 0.928090i $$-0.378550\pi$$
0.372356 + 0.928090i $$0.378550\pi$$
$$882$$ − 32676.4i − 1.24748i
$$883$$ − 49092.4i − 1.87100i −0.353329 0.935499i $$-0.614950\pi$$
0.353329 0.935499i $$-0.385050\pi$$
$$884$$ −118.365 −0.00450346
$$885$$ 0 0
$$886$$ 27599.5 1.04653
$$887$$ 9292.86i 0.351774i 0.984410 + 0.175887i $$0.0562793\pi$$
−0.984410 + 0.175887i $$0.943721\pi$$
$$888$$ − 2430.12i − 0.0918348i
$$889$$ 4047.41 0.152695
$$890$$ 0 0
$$891$$ 4525.85 0.170170
$$892$$ 2108.05i 0.0791288i
$$893$$ 48112.0i 1.80292i
$$894$$ −52598.5 −1.96774
$$895$$ 0 0
$$896$$ 3949.89 0.147273
$$897$$ 4731.10i 0.176106i
$$898$$ 943.252i 0.0350520i
$$899$$ 787.273 0.0292069
$$900$$ 0 0
$$901$$ 14136.7 0.522709
$$902$$ − 7845.88i − 0.289622i
$$903$$ − 1405.53i − 0.0517974i
$$904$$ 6881.46 0.253179
$$905$$ 0 0
$$906$$ 55811.5 2.04659
$$907$$ 37688.7i 1.37975i 0.723928 + 0.689875i $$0.242335\pi$$
−0.723928 + 0.689875i $$0.757665\pi$$
$$908$$ − 949.559i − 0.0347051i
$$909$$ 5789.42 0.211246
$$910$$ 0 0
$$911$$ 33049.6 1.20196 0.600979 0.799265i $$-0.294778\pi$$
0.600979 + 0.799265i $$0.294778\pi$$
$$912$$ 65923.1i 2.39357i
$$913$$ 4860.31i 0.176180i
$$914$$ 28870.0 1.04479
$$915$$ 0 0
$$916$$ −1026.44 −0.0370247
$$917$$ − 4917.06i − 0.177073i
$$918$$ − 7906.43i − 0.284260i
$$919$$ 23148.0 0.830883 0.415442 0.909620i $$-0.363627\pi$$
0.415442 + 0.909620i $$0.363627\pi$$
$$920$$ 0 0
$$921$$ 13304.1 0.475986
$$922$$ 12933.4i 0.461973i
$$923$$ 1111.26i 0.0396290i
$$924$$ 143.563 0.00511134
$$925$$ 0 0
$$926$$ 9374.21 0.332673
$$927$$ − 1246.34i − 0.0441588i
$$928$$ 605.131i 0.0214056i
$$929$$ 23177.9 0.818561 0.409280 0.912409i $$-0.365780\pi$$
0.409280 + 0.912409i $$0.365780\pi$$
$$930$$ 0 0
$$931$$ −46673.3 −1.64302
$$932$$ − 2355.98i − 0.0828033i
$$933$$ 28325.1i 0.993916i
$$934$$ −13979.8 −0.489757
$$935$$ 0 0
$$936$$ −4481.13 −0.156485
$$937$$ − 34574.7i − 1.20545i −0.797950 0.602724i $$-0.794082\pi$$
0.797950 0.602724i $$-0.205918\pi$$
$$938$$ − 2873.18i − 0.100014i
$$939$$ 56959.1 1.97954
$$940$$ 0 0
$$941$$ 41831.2 1.44916 0.724578 0.689192i $$-0.242034\pi$$
0.724578 + 0.689192i $$0.242034\pi$$
$$942$$ − 53629.6i − 1.85493i
$$943$$ 29071.3i 1.00392i
$$944$$ 5251.40 0.181058
$$945$$ 0 0
$$946$$ 1734.42 0.0596097
$$947$$ 27231.2i 0.934419i 0.884147 + 0.467209i $$0.154741\pi$$
−0.884147 + 0.467209i $$0.845259\pi$$
$$948$$ − 5498.79i − 0.188389i
$$949$$ 5415.79 0.185252
$$950$$ 0 0
$$951$$ 124.534 0.00424635
$$952$$ 2952.50i 0.100516i
$$953$$ − 40939.4i − 1.39156i −0.718254 0.695781i $$-0.755058\pi$$
0.718254 0.695781i $$-0.244942\pi$$
$$954$$ 33600.3 1.14030
$$955$$ 0 0
$$956$$ 2188.87 0.0740515
$$957$$ − 2179.81i − 0.0736292i
$$958$$ − 31601.3i − 1.06575i
$$959$$ −4950.74 −0.166703
$$960$$ 0 0
$$961$$ −28798.9 −0.966698
$$962$$ 192.436i 0.00644945i
$$963$$ − 29839.0i − 0.998491i
$$964$$ 2094.60 0.0699820
$$965$$ 0 0
$$966$$ 7409.03 0.246772
$$967$$ − 46173.1i − 1.53550i −0.640750 0.767750i $$-0.721376\pi$$
0.640750 0.767750i $$-0.278624\pi$$
$$968$$ 2821.78i 0.0936937i
$$969$$ −45721.8 −1.51579
$$970$$ 0 0
$$971$$ −5153.91 −0.170337 −0.0851683 0.996367i $$-0.527143\pi$$
−0.0851683 + 0.996367i $$0.527143\pi$$
$$972$$ − 2764.06i − 0.0912111i
$$973$$ 97.8734i 0.00322474i
$$974$$ 50069.0 1.64714
$$975$$ 0 0
$$976$$ −43897.6 −1.43968
$$977$$ 9692.13i 0.317378i 0.987329 + 0.158689i $$0.0507268\pi$$
−0.987329 + 0.158689i $$0.949273\pi$$
$$978$$ − 59026.5i − 1.92992i
$$979$$ −16380.2 −0.534744
$$980$$ 0 0
$$981$$ 37443.3 1.21863
$$982$$ − 20811.6i − 0.676299i
$$983$$ − 32915.7i − 1.06800i −0.845483 0.534002i $$-0.820687\pi$$
0.845483 0.534002i $$-0.179313\pi$$
$$984$$ −48269.5 −1.56380
$$985$$ 0 0
$$986$$ 2814.48 0.0909041
$$987$$ 8373.97i 0.270057i
$$988$$ 401.841i 0.0129395i
$$989$$ −6426.54 −0.206625
$$990$$ 0 0
$$991$$ 29477.9 0.944901 0.472451 0.881357i $$-0.343370\pi$$
0.472451 + 0.881357i $$0.343370\pi$$
$$992$$ 762.560i 0.0244066i
$$993$$ − 10456.9i − 0.334180i
$$994$$ 1740.26 0.0555310
$$995$$ 0 0
$$996$$ 1877.28 0.0597227
$$997$$ − 31944.4i − 1.01473i −0.861731 0.507366i $$-0.830619\pi$$
0.861731 0.507366i $$-0.169381\pi$$
$$998$$ − 35268.4i − 1.11864i
$$999$$ 922.883 0.0292279
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 275.4.b.c.199.4 4
5.2 odd 4 275.4.a.b.1.1 2
5.3 odd 4 11.4.a.a.1.2 2
5.4 even 2 inner 275.4.b.c.199.1 4
15.2 even 4 2475.4.a.q.1.2 2
15.8 even 4 99.4.a.c.1.1 2
20.3 even 4 176.4.a.i.1.2 2
35.13 even 4 539.4.a.e.1.2 2
40.3 even 4 704.4.a.n.1.1 2
40.13 odd 4 704.4.a.p.1.2 2
55.3 odd 20 121.4.c.c.9.2 8
55.8 even 20 121.4.c.f.9.1 8
55.13 even 20 121.4.c.f.81.2 8
55.18 even 20 121.4.c.f.27.1 8
55.28 even 20 121.4.c.f.3.2 8
55.38 odd 20 121.4.c.c.3.1 8
55.43 even 4 121.4.a.c.1.1 2
55.48 odd 20 121.4.c.c.27.2 8
55.53 odd 20 121.4.c.c.81.1 8
60.23 odd 4 1584.4.a.bc.1.1 2
65.38 odd 4 1859.4.a.a.1.1 2
165.98 odd 4 1089.4.a.v.1.2 2
220.43 odd 4 1936.4.a.w.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
11.4.a.a.1.2 2 5.3 odd 4
99.4.a.c.1.1 2 15.8 even 4
121.4.a.c.1.1 2 55.43 even 4
121.4.c.c.3.1 8 55.38 odd 20
121.4.c.c.9.2 8 55.3 odd 20
121.4.c.c.27.2 8 55.48 odd 20
121.4.c.c.81.1 8 55.53 odd 20
121.4.c.f.3.2 8 55.28 even 20
121.4.c.f.9.1 8 55.8 even 20
121.4.c.f.27.1 8 55.18 even 20
121.4.c.f.81.2 8 55.13 even 20
176.4.a.i.1.2 2 20.3 even 4
275.4.a.b.1.1 2 5.2 odd 4
275.4.b.c.199.1 4 5.4 even 2 inner
275.4.b.c.199.4 4 1.1 even 1 trivial
539.4.a.e.1.2 2 35.13 even 4
704.4.a.n.1.1 2 40.3 even 4
704.4.a.p.1.2 2 40.13 odd 4
1089.4.a.v.1.2 2 165.98 odd 4
1584.4.a.bc.1.1 2 60.23 odd 4
1859.4.a.a.1.1 2 65.38 odd 4
1936.4.a.w.1.2 2 220.43 odd 4
2475.4.a.q.1.2 2 15.2 even 4