# Properties

 Label 275.6.b.b.199.2 Level $275$ Weight $6$ Character 275.199 Analytic conductor $44.106$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [275,6,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, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("275.199");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$275 = 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$44.1055504486$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.11877512256.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 34x^{4} - 154x^{3} + 569x^{2} - 6512x + 17216$$ x^6 - 2*x^5 + 34*x^4 - 154*x^3 + 569*x^2 - 6512*x + 17216 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 11) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.2 Root $$3.64914 - 0.444721i$$ of defining polynomial Character $$\chi$$ $$=$$ 275.199 Dual form 275.6.b.b.199.5

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-8.18772i q^{2} -3.48600i q^{3} -35.0388 q^{4} -28.5424 q^{6} -145.071i q^{7} +24.8808i q^{8} +230.848 q^{9} +O(q^{10})$$ $$q-8.18772i q^{2} -3.48600i q^{3} -35.0388 q^{4} -28.5424 q^{6} -145.071i q^{7} +24.8808i q^{8} +230.848 q^{9} +121.000 q^{11} +122.145i q^{12} +615.772i q^{13} -1187.80 q^{14} -917.524 q^{16} -1840.68i q^{17} -1890.12i q^{18} -366.633 q^{19} -505.718 q^{21} -990.714i q^{22} -4516.38i q^{23} +86.7344 q^{24} +5041.77 q^{26} -1651.83i q^{27} +5083.12i q^{28} +1717.00 q^{29} -2650.54 q^{31} +8308.62i q^{32} -421.806i q^{33} -15070.9 q^{34} -8088.63 q^{36} -9660.61i q^{37} +3001.89i q^{38} +2146.58 q^{39} -11154.8 q^{41} +4140.68i q^{42} +8368.48i q^{43} -4239.69 q^{44} -36978.9 q^{46} +2221.22i q^{47} +3198.49i q^{48} -4238.64 q^{49} -6416.60 q^{51} -21575.9i q^{52} +23707.9i q^{53} -13524.8 q^{54} +3609.48 q^{56} +1278.08i q^{57} -14058.3i q^{58} -19517.8 q^{59} +20937.3 q^{61} +21701.9i q^{62} -33489.4i q^{63} +38667.9 q^{64} -3453.63 q^{66} +51707.7i q^{67} +64495.1i q^{68} -15744.1 q^{69} -1398.38 q^{71} +5743.67i q^{72} +72466.6i q^{73} -79098.4 q^{74} +12846.4 q^{76} -17553.6i q^{77} -17575.6i q^{78} -64632.2 q^{79} +50337.7 q^{81} +91332.4i q^{82} -96790.3i q^{83} +17719.8 q^{84} +68518.8 q^{86} -5985.47i q^{87} +3010.57i q^{88} +47614.1 q^{89} +89330.7 q^{91} +158249. i q^{92} +9239.79i q^{93} +18186.7 q^{94} +28963.9 q^{96} +38399.6i q^{97} +34704.8i q^{98} +27932.6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 168 q^{4} - 412 q^{6} + 14 q^{9}+O(q^{10})$$ 6 * q - 168 * q^4 - 412 * q^6 + 14 * q^9 $$6 q - 168 q^{4} - 412 q^{6} + 14 q^{9} + 726 q^{11} + 2040 q^{14} + 3984 q^{16} - 2760 q^{19} - 1816 q^{21} + 23496 q^{24} + 24264 q^{26} + 6852 q^{29} - 8196 q^{31} - 50640 q^{34} + 9512 q^{36} + 13120 q^{39} + 11988 q^{41} - 20328 q^{44} - 33612 q^{46} - 97062 q^{49} - 45448 q^{51} - 37628 q^{54} - 84624 q^{56} + 7476 q^{59} + 36972 q^{61} + 40704 q^{64} - 49852 q^{66} - 70084 q^{69} + 78564 q^{71} - 306588 q^{74} + 207840 q^{76} - 250296 q^{79} - 173834 q^{81} - 687232 q^{84} + 486120 q^{86} + 213648 q^{89} - 219264 q^{91} + 149856 q^{94} - 152912 q^{96} + 1694 q^{99}+O(q^{100})$$ 6 * q - 168 * q^4 - 412 * q^6 + 14 * q^9 + 726 * q^11 + 2040 * q^14 + 3984 * q^16 - 2760 * q^19 - 1816 * q^21 + 23496 * q^24 + 24264 * q^26 + 6852 * q^29 - 8196 * q^31 - 50640 * q^34 + 9512 * q^36 + 13120 * q^39 + 11988 * q^41 - 20328 * q^44 - 33612 * q^46 - 97062 * q^49 - 45448 * q^51 - 37628 * q^54 - 84624 * q^56 + 7476 * q^59 + 36972 * q^61 + 40704 * q^64 - 49852 * q^66 - 70084 * q^69 + 78564 * q^71 - 306588 * q^74 + 207840 * q^76 - 250296 * q^79 - 173834 * q^81 - 687232 * q^84 + 486120 * q^86 + 213648 * q^89 - 219264 * q^91 + 149856 * q^94 - 152912 * q^96 + 1694 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 8.18772i − 1.44740i −0.690116 0.723699i $$-0.742440\pi$$
0.690116 0.723699i $$-0.257560\pi$$
$$3$$ − 3.48600i − 0.223627i −0.993729 0.111814i $$-0.964334\pi$$
0.993729 0.111814i $$-0.0356659\pi$$
$$4$$ −35.0388 −1.09496
$$5$$ 0 0
$$6$$ −28.5424 −0.323678
$$7$$ − 145.071i − 1.11902i −0.828825 0.559508i $$-0.810990\pi$$
0.828825 0.559508i $$-0.189010\pi$$
$$8$$ 24.8808i 0.137448i
$$9$$ 230.848 0.949991
$$10$$ 0 0
$$11$$ 121.000 0.301511
$$12$$ 122.145i 0.244863i
$$13$$ 615.772i 1.01056i 0.862956 + 0.505279i $$0.168610\pi$$
−0.862956 + 0.505279i $$0.831390\pi$$
$$14$$ −1187.80 −1.61966
$$15$$ 0 0
$$16$$ −917.524 −0.896020
$$17$$ − 1840.68i − 1.54474i −0.635174 0.772369i $$-0.719071\pi$$
0.635174 0.772369i $$-0.280929\pi$$
$$18$$ − 1890.12i − 1.37502i
$$19$$ −366.633 −0.232996 −0.116498 0.993191i $$-0.537167\pi$$
−0.116498 + 0.993191i $$0.537167\pi$$
$$20$$ 0 0
$$21$$ −505.718 −0.250242
$$22$$ − 990.714i − 0.436407i
$$23$$ − 4516.38i − 1.78021i −0.455757 0.890104i $$-0.650631\pi$$
0.455757 0.890104i $$-0.349369\pi$$
$$24$$ 86.7344 0.0307371
$$25$$ 0 0
$$26$$ 5041.77 1.46268
$$27$$ − 1651.83i − 0.436071i
$$28$$ 5083.12i 1.22528i
$$29$$ 1717.00 0.379119 0.189560 0.981869i $$-0.439294\pi$$
0.189560 + 0.981869i $$0.439294\pi$$
$$30$$ 0 0
$$31$$ −2650.54 −0.495371 −0.247685 0.968841i $$-0.579670\pi$$
−0.247685 + 0.968841i $$0.579670\pi$$
$$32$$ 8308.62i 1.43435i
$$33$$ − 421.806i − 0.0674261i
$$34$$ −15070.9 −2.23585
$$35$$ 0 0
$$36$$ −8088.63 −1.04020
$$37$$ − 9660.61i − 1.16011i −0.814576 0.580057i $$-0.803030\pi$$
0.814576 0.580057i $$-0.196970\pi$$
$$38$$ 3001.89i 0.337238i
$$39$$ 2146.58 0.225988
$$40$$ 0 0
$$41$$ −11154.8 −1.03634 −0.518170 0.855278i $$-0.673386\pi$$
−0.518170 + 0.855278i $$0.673386\pi$$
$$42$$ 4140.68i 0.362200i
$$43$$ 8368.48i 0.690201i 0.938566 + 0.345100i $$0.112155\pi$$
−0.938566 + 0.345100i $$0.887845\pi$$
$$44$$ −4239.69 −0.330144
$$45$$ 0 0
$$46$$ −36978.9 −2.57667
$$47$$ 2221.22i 0.146672i 0.997307 + 0.0733360i $$0.0233645\pi$$
−0.997307 + 0.0733360i $$0.976635\pi$$
$$48$$ 3198.49i 0.200374i
$$49$$ −4238.64 −0.252195
$$50$$ 0 0
$$51$$ −6416.60 −0.345445
$$52$$ − 21575.9i − 1.10652i
$$53$$ 23707.9i 1.15932i 0.814859 + 0.579659i $$0.196814\pi$$
−0.814859 + 0.579659i $$0.803186\pi$$
$$54$$ −13524.8 −0.631168
$$55$$ 0 0
$$56$$ 3609.48 0.153807
$$57$$ 1278.08i 0.0521042i
$$58$$ − 14058.3i − 0.548737i
$$59$$ −19517.8 −0.729964 −0.364982 0.931015i $$-0.618925\pi$$
−0.364982 + 0.931015i $$0.618925\pi$$
$$60$$ 0 0
$$61$$ 20937.3 0.720436 0.360218 0.932868i $$-0.382702\pi$$
0.360218 + 0.932868i $$0.382702\pi$$
$$62$$ 21701.9i 0.716999i
$$63$$ − 33489.4i − 1.06305i
$$64$$ 38667.9 1.18005
$$65$$ 0 0
$$66$$ −3453.63 −0.0975924
$$67$$ 51707.7i 1.40724i 0.710577 + 0.703619i $$0.248434\pi$$
−0.710577 + 0.703619i $$0.751566\pi$$
$$68$$ 64495.1i 1.69143i
$$69$$ −15744.1 −0.398103
$$70$$ 0 0
$$71$$ −1398.38 −0.0329216 −0.0164608 0.999865i $$-0.505240\pi$$
−0.0164608 + 0.999865i $$0.505240\pi$$
$$72$$ 5743.67i 0.130574i
$$73$$ 72466.6i 1.59159i 0.605567 + 0.795794i $$0.292946\pi$$
−0.605567 + 0.795794i $$0.707054\pi$$
$$74$$ −79098.4 −1.67915
$$75$$ 0 0
$$76$$ 12846.4 0.255122
$$77$$ − 17553.6i − 0.337396i
$$78$$ − 17575.6i − 0.327095i
$$79$$ −64632.2 −1.16515 −0.582574 0.812777i $$-0.697955\pi$$
−0.582574 + 0.812777i $$0.697955\pi$$
$$80$$ 0 0
$$81$$ 50337.7 0.852474
$$82$$ 91332.4i 1.50000i
$$83$$ − 96790.3i − 1.54219i −0.636723 0.771093i $$-0.719710\pi$$
0.636723 0.771093i $$-0.280290\pi$$
$$84$$ 17719.8 0.274006
$$85$$ 0 0
$$86$$ 68518.8 0.998995
$$87$$ − 5985.47i − 0.0847814i
$$88$$ 3010.57i 0.0414422i
$$89$$ 47614.1 0.637178 0.318589 0.947893i $$-0.396791\pi$$
0.318589 + 0.947893i $$0.396791\pi$$
$$90$$ 0 0
$$91$$ 89330.7 1.13083
$$92$$ 158249.i 1.94926i
$$93$$ 9239.79i 0.110778i
$$94$$ 18186.7 0.212293
$$95$$ 0 0
$$96$$ 28963.9 0.320759
$$97$$ 38399.6i 0.414378i 0.978301 + 0.207189i $$0.0664315\pi$$
−0.978301 + 0.207189i $$0.933568\pi$$
$$98$$ 34704.8i 0.365027i
$$99$$ 27932.6 0.286433
$$100$$ 0 0
$$101$$ −41011.2 −0.400036 −0.200018 0.979792i $$-0.564100\pi$$
−0.200018 + 0.979792i $$0.564100\pi$$
$$102$$ 52537.3i 0.499997i
$$103$$ − 49634.4i − 0.460988i −0.973074 0.230494i $$-0.925966\pi$$
0.973074 0.230494i $$-0.0740343\pi$$
$$104$$ −15320.9 −0.138899
$$105$$ 0 0
$$106$$ 194113. 1.67800
$$107$$ 6791.34i 0.0573450i 0.999589 + 0.0286725i $$0.00912800\pi$$
−0.999589 + 0.0286725i $$0.990872\pi$$
$$108$$ 57878.3i 0.477481i
$$109$$ −96780.7 −0.780230 −0.390115 0.920766i $$-0.627565\pi$$
−0.390115 + 0.920766i $$0.627565\pi$$
$$110$$ 0 0
$$111$$ −33676.9 −0.259433
$$112$$ 133106.i 1.00266i
$$113$$ − 212938.i − 1.56876i −0.620281 0.784379i $$-0.712982\pi$$
0.620281 0.784379i $$-0.287018\pi$$
$$114$$ 10464.6 0.0754155
$$115$$ 0 0
$$116$$ −60161.7 −0.415121
$$117$$ 142149.i 0.960021i
$$118$$ 159806.i 1.05655i
$$119$$ −267029. −1.72859
$$120$$ 0 0
$$121$$ 14641.0 0.0909091
$$122$$ − 171428.i − 1.04276i
$$123$$ 38885.6i 0.231754i
$$124$$ 92871.8 0.542412
$$125$$ 0 0
$$126$$ −274202. −1.53866
$$127$$ 90363.9i 0.497148i 0.968613 + 0.248574i $$0.0799619\pi$$
−0.968613 + 0.248574i $$0.920038\pi$$
$$128$$ − 50726.1i − 0.273657i
$$129$$ 29172.5 0.154348
$$130$$ 0 0
$$131$$ 65299.5 0.332454 0.166227 0.986088i $$-0.446842\pi$$
0.166227 + 0.986088i $$0.446842\pi$$
$$132$$ 14779.6i 0.0738290i
$$133$$ 53187.9i 0.260726i
$$134$$ 423368. 2.03684
$$135$$ 0 0
$$136$$ 45797.4 0.212321
$$137$$ − 5322.74i − 0.0242289i −0.999927 0.0121145i $$-0.996144\pi$$
0.999927 0.0121145i $$-0.00385625\pi$$
$$138$$ 128908.i 0.576214i
$$139$$ −89967.1 −0.394954 −0.197477 0.980308i $$-0.563275\pi$$
−0.197477 + 0.980308i $$0.563275\pi$$
$$140$$ 0 0
$$141$$ 7743.18 0.0327998
$$142$$ 11449.6i 0.0476506i
$$143$$ 74508.4i 0.304695i
$$144$$ −211808. −0.851211
$$145$$ 0 0
$$146$$ 593336. 2.30366
$$147$$ 14775.9i 0.0563977i
$$148$$ 338496.i 1.27028i
$$149$$ −66489.8 −0.245352 −0.122676 0.992447i $$-0.539148\pi$$
−0.122676 + 0.992447i $$0.539148\pi$$
$$150$$ 0 0
$$151$$ −130866. −0.467074 −0.233537 0.972348i $$-0.575030\pi$$
−0.233537 + 0.972348i $$0.575030\pi$$
$$152$$ − 9122.12i − 0.0320248i
$$153$$ − 424916.i − 1.46749i
$$154$$ −143724. −0.488346
$$155$$ 0 0
$$156$$ −75213.6 −0.247448
$$157$$ 163297.i 0.528723i 0.964424 + 0.264362i $$0.0851612\pi$$
−0.964424 + 0.264362i $$0.914839\pi$$
$$158$$ 529191.i 1.68643i
$$159$$ 82645.6 0.259255
$$160$$ 0 0
$$161$$ −655197. −1.99208
$$162$$ − 412151.i − 1.23387i
$$163$$ − 535758.i − 1.57943i −0.613477 0.789713i $$-0.710229\pi$$
0.613477 0.789713i $$-0.289771\pi$$
$$164$$ 390851. 1.13475
$$165$$ 0 0
$$166$$ −792492. −2.23216
$$167$$ − 553587.i − 1.53601i −0.640443 0.768005i $$-0.721249\pi$$
0.640443 0.768005i $$-0.278751\pi$$
$$168$$ − 12582.7i − 0.0343953i
$$169$$ −7881.54 −0.0212273
$$170$$ 0 0
$$171$$ −84636.5 −0.221344
$$172$$ − 293221.i − 0.755744i
$$173$$ 266973.i 0.678190i 0.940752 + 0.339095i $$0.110121\pi$$
−0.940752 + 0.339095i $$0.889879\pi$$
$$174$$ −49007.4 −0.122712
$$175$$ 0 0
$$176$$ −111020. −0.270160
$$177$$ 68039.2i 0.163240i
$$178$$ − 389851.i − 0.922250i
$$179$$ −3030.33 −0.00706900 −0.00353450 0.999994i $$-0.501125\pi$$
−0.00353450 + 0.999994i $$0.501125\pi$$
$$180$$ 0 0
$$181$$ 761242. 1.72714 0.863568 0.504233i $$-0.168225\pi$$
0.863568 + 0.504233i $$0.168225\pi$$
$$182$$ − 731415.i − 1.63676i
$$183$$ − 72987.3i − 0.161109i
$$184$$ 112371. 0.244686
$$185$$ 0 0
$$186$$ 75652.8 0.160340
$$187$$ − 222722.i − 0.465756i
$$188$$ − 77828.9i − 0.160600i
$$189$$ −239634. −0.487970
$$190$$ 0 0
$$191$$ −430653. −0.854170 −0.427085 0.904212i $$-0.640459\pi$$
−0.427085 + 0.904212i $$0.640459\pi$$
$$192$$ − 134796.i − 0.263891i
$$193$$ − 272285.i − 0.526175i −0.964772 0.263088i $$-0.915259\pi$$
0.964772 0.263088i $$-0.0847409\pi$$
$$194$$ 314405. 0.599770
$$195$$ 0 0
$$196$$ 148517. 0.276144
$$197$$ − 574550.i − 1.05478i −0.849623 0.527390i $$-0.823171\pi$$
0.849623 0.527390i $$-0.176829\pi$$
$$198$$ − 228704.i − 0.414583i
$$199$$ −269926. −0.483183 −0.241592 0.970378i $$-0.577669\pi$$
−0.241592 + 0.970378i $$0.577669\pi$$
$$200$$ 0 0
$$201$$ 180253. 0.314697
$$202$$ 335788.i 0.579011i
$$203$$ − 249088.i − 0.424240i
$$204$$ 224830. 0.378250
$$205$$ 0 0
$$206$$ −406393. −0.667234
$$207$$ − 1.04260e6i − 1.69118i
$$208$$ − 564985.i − 0.905480i
$$209$$ −44362.6 −0.0702509
$$210$$ 0 0
$$211$$ −753372. −1.16494 −0.582470 0.812853i $$-0.697913\pi$$
−0.582470 + 0.812853i $$0.697913\pi$$
$$212$$ − 830695.i − 1.26941i
$$213$$ 4874.77i 0.00736216i
$$214$$ 55605.6 0.0830011
$$215$$ 0 0
$$216$$ 41098.9 0.0599371
$$217$$ 384517.i 0.554327i
$$218$$ 792414.i 1.12930i
$$219$$ 252619. 0.355922
$$220$$ 0 0
$$221$$ 1.13344e6 1.56105
$$222$$ 275737.i 0.375503i
$$223$$ − 997692.i − 1.34349i −0.740783 0.671745i $$-0.765545\pi$$
0.740783 0.671745i $$-0.234455\pi$$
$$224$$ 1.20534e6 1.60506
$$225$$ 0 0
$$226$$ −1.74347e6 −2.27062
$$227$$ 495214.i 0.637864i 0.947778 + 0.318932i $$0.103324\pi$$
−0.947778 + 0.318932i $$0.896676\pi$$
$$228$$ − 44782.5i − 0.0570521i
$$229$$ 221893. 0.279611 0.139806 0.990179i $$-0.455352\pi$$
0.139806 + 0.990179i $$0.455352\pi$$
$$230$$ 0 0
$$231$$ −61191.9 −0.0754508
$$232$$ 42720.4i 0.0521093i
$$233$$ 619425.i 0.747479i 0.927534 + 0.373739i $$0.121925\pi$$
−0.927534 + 0.373739i $$0.878075\pi$$
$$234$$ 1.16388e6 1.38953
$$235$$ 0 0
$$236$$ 683881. 0.799283
$$237$$ 225308.i 0.260559i
$$238$$ 2.18636e6i 2.50195i
$$239$$ 295471. 0.334595 0.167298 0.985906i $$-0.446496\pi$$
0.167298 + 0.985906i $$0.446496\pi$$
$$240$$ 0 0
$$241$$ 693153. 0.768753 0.384376 0.923176i $$-0.374416\pi$$
0.384376 + 0.923176i $$0.374416\pi$$
$$242$$ − 119876.i − 0.131582i
$$243$$ − 576873.i − 0.626707i
$$244$$ −733616. −0.788850
$$245$$ 0 0
$$246$$ 318385. 0.335440
$$247$$ − 225762.i − 0.235456i
$$248$$ − 65947.5i − 0.0680878i
$$249$$ −337411. −0.344875
$$250$$ 0 0
$$251$$ 533816. 0.534820 0.267410 0.963583i $$-0.413832\pi$$
0.267410 + 0.963583i $$0.413832\pi$$
$$252$$ 1.17343e6i 1.16400i
$$253$$ − 546482.i − 0.536753i
$$254$$ 739874. 0.719571
$$255$$ 0 0
$$256$$ 822041. 0.783960
$$257$$ 652296.i 0.616044i 0.951379 + 0.308022i $$0.0996670\pi$$
−0.951379 + 0.308022i $$0.900333\pi$$
$$258$$ − 238857.i − 0.223402i
$$259$$ −1.40148e6 −1.29818
$$260$$ 0 0
$$261$$ 396366. 0.360160
$$262$$ − 534654.i − 0.481193i
$$263$$ − 622045.i − 0.554540i −0.960792 0.277270i $$-0.910570\pi$$
0.960792 0.277270i $$-0.0894296\pi$$
$$264$$ 10494.9 0.00926759
$$265$$ 0 0
$$266$$ 435488. 0.377374
$$267$$ − 165983.i − 0.142490i
$$268$$ − 1.81177e6i − 1.54087i
$$269$$ 482862. 0.406858 0.203429 0.979090i $$-0.434791\pi$$
0.203429 + 0.979090i $$0.434791\pi$$
$$270$$ 0 0
$$271$$ −1.10678e6 −0.915460 −0.457730 0.889091i $$-0.651337\pi$$
−0.457730 + 0.889091i $$0.651337\pi$$
$$272$$ 1.68887e6i 1.38412i
$$273$$ − 311407.i − 0.252884i
$$274$$ −43581.1 −0.0350689
$$275$$ 0 0
$$276$$ 551655. 0.435908
$$277$$ − 639062.i − 0.500430i −0.968190 0.250215i $$-0.919499\pi$$
0.968190 0.250215i $$-0.0805014\pi$$
$$278$$ 736626.i 0.571656i
$$279$$ −611872. −0.470598
$$280$$ 0 0
$$281$$ −257984. −0.194907 −0.0974534 0.995240i $$-0.531070\pi$$
−0.0974534 + 0.995240i $$0.531070\pi$$
$$282$$ − 63399.0i − 0.0474744i
$$283$$ 1.02991e6i 0.764425i 0.924074 + 0.382213i $$0.124838\pi$$
−0.924074 + 0.382213i $$0.875162\pi$$
$$284$$ 48997.7 0.0360479
$$285$$ 0 0
$$286$$ 610054. 0.441015
$$287$$ 1.61824e6i 1.15968i
$$288$$ 1.91803e6i 1.36262i
$$289$$ −1.96823e6 −1.38622
$$290$$ 0 0
$$291$$ 133861. 0.0926662
$$292$$ − 2.53914e6i − 1.74273i
$$293$$ 877712.i 0.597287i 0.954365 + 0.298644i $$0.0965342\pi$$
−0.954365 + 0.298644i $$0.903466\pi$$
$$294$$ 120981. 0.0816299
$$295$$ 0 0
$$296$$ 240363. 0.159455
$$297$$ − 199872.i − 0.131480i
$$298$$ 544400.i 0.355122i
$$299$$ 2.78106e6 1.79900
$$300$$ 0 0
$$301$$ 1.21403e6 0.772345
$$302$$ 1.07150e6i 0.676042i
$$303$$ 142965.i 0.0894589i
$$304$$ 336395. 0.208769
$$305$$ 0 0
$$306$$ −3.47909e6 −2.12404
$$307$$ 1.30925e6i 0.792826i 0.918072 + 0.396413i $$0.129745\pi$$
−0.918072 + 0.396413i $$0.870255\pi$$
$$308$$ 615057.i 0.369436i
$$309$$ −173026. −0.103089
$$310$$ 0 0
$$311$$ 3.35930e6 1.96946 0.984731 0.174083i $$-0.0556962\pi$$
0.984731 + 0.174083i $$0.0556962\pi$$
$$312$$ 53408.6i 0.0310616i
$$313$$ − 3.00640e6i − 1.73454i −0.497835 0.867272i $$-0.665871\pi$$
0.497835 0.867272i $$-0.334129\pi$$
$$314$$ 1.33703e6 0.765273
$$315$$ 0 0
$$316$$ 2.26464e6 1.27579
$$317$$ − 2.10147e6i − 1.17456i −0.809385 0.587279i $$-0.800199\pi$$
0.809385 0.587279i $$-0.199801\pi$$
$$318$$ − 676679.i − 0.375245i
$$319$$ 207757. 0.114309
$$320$$ 0 0
$$321$$ 23674.6 0.0128239
$$322$$ 5.36457e6i 2.88333i
$$323$$ 674853.i 0.359918i
$$324$$ −1.76377e6 −0.933426
$$325$$ 0 0
$$326$$ −4.38663e6 −2.28606
$$327$$ 337378.i 0.174481i
$$328$$ − 277540.i − 0.142443i
$$329$$ 322235. 0.164128
$$330$$ 0 0
$$331$$ 1.23338e6 0.618766 0.309383 0.950938i $$-0.399878\pi$$
0.309383 + 0.950938i $$0.399878\pi$$
$$332$$ 3.39142e6i 1.68864i
$$333$$ − 2.23013e6i − 1.10210i
$$334$$ −4.53261e6 −2.22322
$$335$$ 0 0
$$336$$ 464009. 0.224222
$$337$$ − 679319.i − 0.325836i −0.986640 0.162918i $$-0.947909\pi$$
0.986640 0.162918i $$-0.0520906\pi$$
$$338$$ 64531.9i 0.0307243i
$$339$$ −742301. −0.350817
$$340$$ 0 0
$$341$$ −320715. −0.149360
$$342$$ 692980.i 0.320373i
$$343$$ − 1.82331e6i − 0.836805i
$$344$$ −208214. −0.0948668
$$345$$ 0 0
$$346$$ 2.18590e6 0.981611
$$347$$ − 2.67540e6i − 1.19279i −0.802690 0.596397i $$-0.796598\pi$$
0.802690 0.596397i $$-0.203402\pi$$
$$348$$ 209724.i 0.0928324i
$$349$$ 2.37636e6 1.04435 0.522177 0.852837i $$-0.325120\pi$$
0.522177 + 0.852837i $$0.325120\pi$$
$$350$$ 0 0
$$351$$ 1.01715e6 0.440675
$$352$$ 1.00534e6i 0.432472i
$$353$$ 638696.i 0.272808i 0.990653 + 0.136404i $$0.0435545\pi$$
−0.990653 + 0.136404i $$0.956445\pi$$
$$354$$ 557086. 0.236273
$$355$$ 0 0
$$356$$ −1.66834e6 −0.697686
$$357$$ 930864.i 0.386559i
$$358$$ 24811.5i 0.0102317i
$$359$$ −1.50842e6 −0.617712 −0.308856 0.951109i $$-0.599946\pi$$
−0.308856 + 0.951109i $$0.599946\pi$$
$$360$$ 0 0
$$361$$ −2.34168e6 −0.945713
$$362$$ − 6.23284e6i − 2.49985i
$$363$$ − 51038.5i − 0.0203297i
$$364$$ −3.13004e6 −1.23822
$$365$$ 0 0
$$366$$ −597600. −0.233189
$$367$$ − 1.77368e6i − 0.687403i −0.939079 0.343701i $$-0.888319\pi$$
0.939079 0.343701i $$-0.111681\pi$$
$$368$$ 4.14389e6i 1.59510i
$$369$$ −2.57506e6 −0.984513
$$370$$ 0 0
$$371$$ 3.43933e6 1.29729
$$372$$ − 323751.i − 0.121298i
$$373$$ 2.27176e6i 0.845456i 0.906257 + 0.422728i $$0.138927\pi$$
−0.906257 + 0.422728i $$0.861073\pi$$
$$374$$ −1.82358e6 −0.674135
$$375$$ 0 0
$$376$$ −55265.7 −0.0201598
$$377$$ 1.05728e6i 0.383122i
$$378$$ 1.96205e6i 0.706287i
$$379$$ 4.42409e6 1.58207 0.791035 0.611771i $$-0.209543\pi$$
0.791035 + 0.611771i $$0.209543\pi$$
$$380$$ 0 0
$$381$$ 315009. 0.111176
$$382$$ 3.52607e6i 1.23632i
$$383$$ 2.37588e6i 0.827615i 0.910364 + 0.413807i $$0.135801\pi$$
−0.910364 + 0.413807i $$0.864199\pi$$
$$384$$ −176831. −0.0611971
$$385$$ 0 0
$$386$$ −2.22939e6 −0.761586
$$387$$ 1.93185e6i 0.655684i
$$388$$ − 1.34547e6i − 0.453728i
$$389$$ 2.65905e6 0.890949 0.445474 0.895295i $$-0.353035\pi$$
0.445474 + 0.895295i $$0.353035\pi$$
$$390$$ 0 0
$$391$$ −8.31319e6 −2.74996
$$392$$ − 105461.i − 0.0346638i
$$393$$ − 227634.i − 0.0743457i
$$394$$ −4.70425e6 −1.52669
$$395$$ 0 0
$$396$$ −978724. −0.313633
$$397$$ − 2.15712e6i − 0.686907i −0.939170 0.343453i $$-0.888403\pi$$
0.939170 0.343453i $$-0.111597\pi$$
$$398$$ 2.21008e6i 0.699359i
$$399$$ 185413. 0.0583054
$$400$$ 0 0
$$401$$ −2.43031e6 −0.754744 −0.377372 0.926062i $$-0.623172\pi$$
−0.377372 + 0.926062i $$0.623172\pi$$
$$402$$ − 1.47586e6i − 0.455492i
$$403$$ − 1.63213e6i − 0.500601i
$$404$$ 1.43698e6 0.438024
$$405$$ 0 0
$$406$$ −2.03946e6 −0.614045
$$407$$ − 1.16893e6i − 0.349787i
$$408$$ − 159650.i − 0.0474808i
$$409$$ −6.12831e6 −1.81148 −0.905738 0.423839i $$-0.860682\pi$$
−0.905738 + 0.423839i $$0.860682\pi$$
$$410$$ 0 0
$$411$$ −18555.1 −0.00541824
$$412$$ 1.73913e6i 0.504765i
$$413$$ 2.83147e6i 0.816841i
$$414$$ −8.53649e6 −2.44781
$$415$$ 0 0
$$416$$ −5.11621e6 −1.44949
$$417$$ 313625.i 0.0883225i
$$418$$ 363229.i 0.101681i
$$419$$ 375626. 0.104525 0.0522626 0.998633i $$-0.483357\pi$$
0.0522626 + 0.998633i $$0.483357\pi$$
$$420$$ 0 0
$$421$$ 3.52333e6 0.968831 0.484416 0.874838i $$-0.339032\pi$$
0.484416 + 0.874838i $$0.339032\pi$$
$$422$$ 6.16840e6i 1.68613i
$$423$$ 512764.i 0.139337i
$$424$$ −589870. −0.159346
$$425$$ 0 0
$$426$$ 39913.3 0.0106560
$$427$$ − 3.03739e6i − 0.806178i
$$428$$ − 237960.i − 0.0627906i
$$429$$ 259736. 0.0681380
$$430$$ 0 0
$$431$$ 3.15287e6 0.817548 0.408774 0.912636i $$-0.365956\pi$$
0.408774 + 0.912636i $$0.365956\pi$$
$$432$$ 1.51560e6i 0.390728i
$$433$$ 1.62168e6i 0.415667i 0.978164 + 0.207833i $$0.0666412\pi$$
−0.978164 + 0.207833i $$0.933359\pi$$
$$434$$ 3.14832e6 0.802333
$$435$$ 0 0
$$436$$ 3.39108e6 0.854322
$$437$$ 1.65586e6i 0.414781i
$$438$$ − 2.06837e6i − 0.515161i
$$439$$ 2.48145e6 0.614533 0.307266 0.951624i $$-0.400586\pi$$
0.307266 + 0.951624i $$0.400586\pi$$
$$440$$ 0 0
$$441$$ −978482. −0.239583
$$442$$ − 9.28026e6i − 2.25946i
$$443$$ 3.75466e6i 0.908994i 0.890748 + 0.454497i $$0.150181\pi$$
−0.890748 + 0.454497i $$0.849819\pi$$
$$444$$ 1.18000e6 0.284069
$$445$$ 0 0
$$446$$ −8.16882e6 −1.94456
$$447$$ 231783.i 0.0548673i
$$448$$ − 5.60960e6i − 1.32049i
$$449$$ 4.80916e6 1.12578 0.562890 0.826532i $$-0.309689\pi$$
0.562890 + 0.826532i $$0.309689\pi$$
$$450$$ 0 0
$$451$$ −1.34973e6 −0.312468
$$452$$ 7.46108e6i 1.71773i
$$453$$ 456200.i 0.104450i
$$454$$ 4.05467e6 0.923244
$$455$$ 0 0
$$456$$ −31799.7 −0.00716162
$$457$$ − 7.09951e6i − 1.59015i −0.606512 0.795075i $$-0.707432\pi$$
0.606512 0.795075i $$-0.292568\pi$$
$$458$$ − 1.81680e6i − 0.404709i
$$459$$ −3.04049e6 −0.673616
$$460$$ 0 0
$$461$$ 8.12745e6 1.78116 0.890578 0.454830i $$-0.150300\pi$$
0.890578 + 0.454830i $$0.150300\pi$$
$$462$$ 501022.i 0.109207i
$$463$$ − 2.67361e6i − 0.579623i −0.957084 0.289812i $$-0.906407\pi$$
0.957084 0.289812i $$-0.0935927\pi$$
$$464$$ −1.57539e6 −0.339699
$$465$$ 0 0
$$466$$ 5.07168e6 1.08190
$$467$$ 4.32733e6i 0.918180i 0.888390 + 0.459090i $$0.151825\pi$$
−0.888390 + 0.459090i $$0.848175\pi$$
$$468$$ − 4.98075e6i − 1.05119i
$$469$$ 7.50129e6 1.57472
$$470$$ 0 0
$$471$$ 569253. 0.118237
$$472$$ − 485618.i − 0.100332i
$$473$$ 1.01259e6i 0.208103i
$$474$$ 1.84476e6 0.377133
$$475$$ 0 0
$$476$$ 9.35637e6 1.89274
$$477$$ 5.47291e6i 1.10134i
$$478$$ − 2.41923e6i − 0.484293i
$$479$$ −1.55878e6 −0.310417 −0.155208 0.987882i $$-0.549605\pi$$
−0.155208 + 0.987882i $$0.549605\pi$$
$$480$$ 0 0
$$481$$ 5.94873e6 1.17236
$$482$$ − 5.67535e6i − 1.11269i
$$483$$ 2.28402e6i 0.445483i
$$484$$ −513003. −0.0995420
$$485$$ 0 0
$$486$$ −4.72328e6 −0.907095
$$487$$ 7.63818e6i 1.45938i 0.683779 + 0.729689i $$0.260335\pi$$
−0.683779 + 0.729689i $$0.739665\pi$$
$$488$$ 520935.i 0.0990225i
$$489$$ −1.86765e6 −0.353202
$$490$$ 0 0
$$491$$ 3.60872e6 0.675537 0.337768 0.941229i $$-0.390328\pi$$
0.337768 + 0.941229i $$0.390328\pi$$
$$492$$ − 1.36251e6i − 0.253761i
$$493$$ − 3.16045e6i − 0.585640i
$$494$$ −1.84848e6 −0.340798
$$495$$ 0 0
$$496$$ 2.43194e6 0.443862
$$497$$ 202865.i 0.0368397i
$$498$$ 2.76263e6i 0.499171i
$$499$$ 8.46131e6 1.52120 0.760599 0.649221i $$-0.224905\pi$$
0.760599 + 0.649221i $$0.224905\pi$$
$$500$$ 0 0
$$501$$ −1.92980e6 −0.343494
$$502$$ − 4.37074e6i − 0.774098i
$$503$$ − 8.28353e6i − 1.45981i −0.683550 0.729904i $$-0.739565\pi$$
0.683550 0.729904i $$-0.260435\pi$$
$$504$$ 833241. 0.146115
$$505$$ 0 0
$$506$$ −4.47444e6 −0.776896
$$507$$ 27475.1i 0.00474700i
$$508$$ − 3.16624e6i − 0.544358i
$$509$$ 7.60138e6 1.30046 0.650232 0.759736i $$-0.274672\pi$$
0.650232 + 0.759736i $$0.274672\pi$$
$$510$$ 0 0
$$511$$ 1.05128e7 1.78101
$$512$$ − 8.35388e6i − 1.40836i
$$513$$ 605618.i 0.101603i
$$514$$ 5.34082e6 0.891661
$$515$$ 0 0
$$516$$ −1.02217e6 −0.169005
$$517$$ 268768.i 0.0442233i
$$518$$ 1.14749e7i 1.87899i
$$519$$ 930667. 0.151662
$$520$$ 0 0
$$521$$ 9.60432e6 1.55015 0.775073 0.631872i $$-0.217713\pi$$
0.775073 + 0.631872i $$0.217713\pi$$
$$522$$ − 3.24534e6i − 0.521295i
$$523$$ − 9.97831e6i − 1.59515i −0.603217 0.797577i $$-0.706115\pi$$
0.603217 0.797577i $$-0.293885\pi$$
$$524$$ −2.28802e6 −0.364025
$$525$$ 0 0
$$526$$ −5.09313e6 −0.802640
$$527$$ 4.87879e6i 0.765218i
$$528$$ 387018.i 0.0604151i
$$529$$ −1.39613e7 −2.16914
$$530$$ 0 0
$$531$$ −4.50565e6 −0.693459
$$532$$ − 1.86364e6i − 0.285485i
$$533$$ − 6.86881e6i − 1.04728i
$$534$$ −1.35902e6 −0.206240
$$535$$ 0 0
$$536$$ −1.28653e6 −0.193422
$$537$$ 10563.8i 0.00158082i
$$538$$ − 3.95354e6i − 0.588885i
$$539$$ −512876. −0.0760397
$$540$$ 0 0
$$541$$ −4.34177e6 −0.637784 −0.318892 0.947791i $$-0.603311\pi$$
−0.318892 + 0.947791i $$0.603311\pi$$
$$542$$ 9.06203e6i 1.32503i
$$543$$ − 2.65369e6i − 0.386234i
$$544$$ 1.52935e7 2.21569
$$545$$ 0 0
$$546$$ −2.54971e6 −0.366024
$$547$$ − 1.14668e7i − 1.63860i −0.573363 0.819302i $$-0.694361\pi$$
0.573363 0.819302i $$-0.305639\pi$$
$$548$$ 186503.i 0.0265298i
$$549$$ 4.83332e6 0.684407
$$550$$ 0 0
$$551$$ −629511. −0.0883332
$$552$$ − 391726.i − 0.0547185i
$$553$$ 9.37627e6i 1.30382i
$$554$$ −5.23246e6 −0.724322
$$555$$ 0 0
$$556$$ 3.15234e6 0.432460
$$557$$ 1.57100e6i 0.214555i 0.994229 + 0.107278i $$0.0342133\pi$$
−0.994229 + 0.107278i $$0.965787\pi$$
$$558$$ 5.00983e6i 0.681142i
$$559$$ −5.15307e6 −0.697488
$$560$$ 0 0
$$561$$ −776409. −0.104156
$$562$$ 2.11230e6i 0.282108i
$$563$$ − 850908.i − 0.113139i −0.998399 0.0565694i $$-0.981984\pi$$
0.998399 0.0565694i $$-0.0180162\pi$$
$$564$$ −271312. −0.0359146
$$565$$ 0 0
$$566$$ 8.43265e6 1.10643
$$567$$ − 7.30255e6i − 0.953931i
$$568$$ − 34792.9i − 0.00452501i
$$569$$ 1.19642e7 1.54919 0.774595 0.632458i $$-0.217954\pi$$
0.774595 + 0.632458i $$0.217954\pi$$
$$570$$ 0 0
$$571$$ −7.97842e6 −1.02406 −0.512032 0.858967i $$-0.671107\pi$$
−0.512032 + 0.858967i $$0.671107\pi$$
$$572$$ − 2.61068e6i − 0.333629i
$$573$$ 1.50126e6i 0.191016i
$$574$$ 1.32497e7 1.67852
$$575$$ 0 0
$$576$$ 8.92640e6 1.12104
$$577$$ − 5.90743e6i − 0.738685i −0.929293 0.369342i $$-0.879583\pi$$
0.929293 0.369342i $$-0.120417\pi$$
$$578$$ 1.61153e7i 2.00641i
$$579$$ −949186. −0.117667
$$580$$ 0 0
$$581$$ −1.40415e7 −1.72573
$$582$$ − 1.09602e6i − 0.134125i
$$583$$ 2.86865e6i 0.349548i
$$584$$ −1.80303e6 −0.218761
$$585$$ 0 0
$$586$$ 7.18646e6 0.864512
$$587$$ − 1.34766e6i − 0.161430i −0.996737 0.0807151i $$-0.974280\pi$$
0.996737 0.0807151i $$-0.0257204\pi$$
$$588$$ − 517730.i − 0.0617533i
$$589$$ 971777. 0.115419
$$590$$ 0 0
$$591$$ −2.00288e6 −0.235877
$$592$$ 8.86385e6i 1.03948i
$$593$$ 1.05883e7i 1.23649i 0.785987 + 0.618243i $$0.212155\pi$$
−0.785987 + 0.618243i $$0.787845\pi$$
$$594$$ −1.63650e6 −0.190304
$$595$$ 0 0
$$596$$ 2.32972e6 0.268651
$$597$$ 940962.i 0.108053i
$$598$$ − 2.27705e7i − 2.60388i
$$599$$ −3.48377e6 −0.396718 −0.198359 0.980129i $$-0.563561\pi$$
−0.198359 + 0.980129i $$0.563561\pi$$
$$600$$ 0 0
$$601$$ −6.41433e6 −0.724378 −0.362189 0.932105i $$-0.617970\pi$$
−0.362189 + 0.932105i $$0.617970\pi$$
$$602$$ − 9.94010e6i − 1.11789i
$$603$$ 1.19366e7i 1.33686i
$$604$$ 4.58540e6 0.511428
$$605$$ 0 0
$$606$$ 1.17056e6 0.129483
$$607$$ 700912.i 0.0772132i 0.999254 + 0.0386066i $$0.0122919\pi$$
−0.999254 + 0.0386066i $$0.987708\pi$$
$$608$$ − 3.04622e6i − 0.334197i
$$609$$ −868320. −0.0948717
$$610$$ 0 0
$$611$$ −1.36776e6 −0.148221
$$612$$ 1.48885e7i 1.60684i
$$613$$ − 1.17591e7i − 1.26393i −0.774996 0.631966i $$-0.782248\pi$$
0.774996 0.631966i $$-0.217752\pi$$
$$614$$ 1.07198e7 1.14753
$$615$$ 0 0
$$616$$ 436747. 0.0463744
$$617$$ 1.00683e6i 0.106474i 0.998582 + 0.0532371i $$0.0169539\pi$$
−0.998582 + 0.0532371i $$0.983046\pi$$
$$618$$ 1.41669e6i 0.149212i
$$619$$ −1.27458e7 −1.33703 −0.668513 0.743700i $$-0.733069\pi$$
−0.668513 + 0.743700i $$0.733069\pi$$
$$620$$ 0 0
$$621$$ −7.46031e6 −0.776297
$$622$$ − 2.75050e7i − 2.85060i
$$623$$ − 6.90744e6i − 0.713012i
$$624$$ −1.96954e6 −0.202490
$$625$$ 0 0
$$626$$ −2.46155e7 −2.51058
$$627$$ 154648.i 0.0157100i
$$628$$ − 5.72172e6i − 0.578932i
$$629$$ −1.77821e7 −1.79207
$$630$$ 0 0
$$631$$ 1.41284e7 1.41260 0.706299 0.707913i $$-0.250363\pi$$
0.706299 + 0.707913i $$0.250363\pi$$
$$632$$ − 1.60810e6i − 0.160148i
$$633$$ 2.62626e6i 0.260512i
$$634$$ −1.72062e7 −1.70005
$$635$$ 0 0
$$636$$ −2.89580e6 −0.283874
$$637$$ − 2.61004e6i − 0.254858i
$$638$$ − 1.70106e6i − 0.165450i
$$639$$ −322814. −0.0312752
$$640$$ 0 0
$$641$$ −4.36680e6 −0.419777 −0.209888 0.977725i $$-0.567310\pi$$
−0.209888 + 0.977725i $$0.567310\pi$$
$$642$$ − 193841.i − 0.0185613i
$$643$$ 7.81597e6i 0.745513i 0.927929 + 0.372757i $$0.121587\pi$$
−0.927929 + 0.372757i $$0.878413\pi$$
$$644$$ 2.29573e7 2.18125
$$645$$ 0 0
$$646$$ 5.52551e6 0.520944
$$647$$ − 2.01624e7i − 1.89357i −0.321863 0.946786i $$-0.604309\pi$$
0.321863 0.946786i $$-0.395691\pi$$
$$648$$ 1.25244e6i 0.117171i
$$649$$ −2.36166e6 −0.220092
$$650$$ 0 0
$$651$$ 1.34043e6 0.123963
$$652$$ 1.87723e7i 1.72941i
$$653$$ 324619.i 0.0297914i 0.999889 + 0.0148957i $$0.00474163\pi$$
−0.999889 + 0.0148957i $$0.995258\pi$$
$$654$$ 2.76235e6 0.252543
$$655$$ 0 0
$$656$$ 1.02348e7 0.928581
$$657$$ 1.67288e7i 1.51199i
$$658$$ − 2.63837e6i − 0.237559i
$$659$$ 1.07107e7 0.960740 0.480370 0.877066i $$-0.340502\pi$$
0.480370 + 0.877066i $$0.340502\pi$$
$$660$$ 0 0
$$661$$ −1.11064e7 −0.988712 −0.494356 0.869260i $$-0.664596\pi$$
−0.494356 + 0.869260i $$0.664596\pi$$
$$662$$ − 1.00986e7i − 0.895601i
$$663$$ − 3.95116e6i − 0.349093i
$$664$$ 2.40822e6 0.211971
$$665$$ 0 0
$$666$$ −1.82597e7 −1.59517
$$667$$ − 7.75464e6i − 0.674912i
$$668$$ 1.93970e7i 1.68187i
$$669$$ −3.47795e6 −0.300441
$$670$$ 0 0
$$671$$ 2.53341e6 0.217219
$$672$$ − 4.20182e6i − 0.358934i
$$673$$ − 1.38137e7i − 1.17564i −0.808993 0.587818i $$-0.799987\pi$$
0.808993 0.587818i $$-0.200013\pi$$
$$674$$ −5.56208e6 −0.471615
$$675$$ 0 0
$$676$$ 276160. 0.0232431
$$677$$ − 2.29090e6i − 0.192103i −0.995376 0.0960514i $$-0.969379\pi$$
0.995376 0.0960514i $$-0.0306213\pi$$
$$678$$ 6.07775e6i 0.507772i
$$679$$ 5.57067e6 0.463695
$$680$$ 0 0
$$681$$ 1.72632e6 0.142644
$$682$$ 2.62593e6i 0.216183i
$$683$$ 4.40512e6i 0.361332i 0.983545 + 0.180666i $$0.0578253\pi$$
−0.983545 + 0.180666i $$0.942175\pi$$
$$684$$ 2.96556e6 0.242363
$$685$$ 0 0
$$686$$ −1.49287e7 −1.21119
$$687$$ − 773519.i − 0.0625287i
$$688$$ − 7.67828e6i − 0.618434i
$$689$$ −1.45986e7 −1.17156
$$690$$ 0 0
$$691$$ −5.86199e6 −0.467035 −0.233518 0.972353i $$-0.575024\pi$$
−0.233518 + 0.972353i $$0.575024\pi$$
$$692$$ − 9.35440e6i − 0.742592i
$$693$$ − 4.05221e6i − 0.320523i
$$694$$ −2.19054e7 −1.72645
$$695$$ 0 0
$$696$$ 148923. 0.0116530
$$697$$ 2.05324e7i 1.60087i
$$698$$ − 1.94569e7i − 1.51160i
$$699$$ 2.15932e6 0.167157
$$700$$ 0 0
$$701$$ −8.02106e6 −0.616505 −0.308253 0.951305i $$-0.599744\pi$$
−0.308253 + 0.951305i $$0.599744\pi$$
$$702$$ − 8.32816e6i − 0.637832i
$$703$$ 3.54190e6i 0.270301i
$$704$$ 4.67881e6 0.355799
$$705$$ 0 0
$$706$$ 5.22946e6 0.394862
$$707$$ 5.94954e6i 0.447646i
$$708$$ − 2.38401e6i − 0.178741i
$$709$$ −2.17891e7 −1.62788 −0.813941 0.580948i $$-0.802682\pi$$
−0.813941 + 0.580948i $$0.802682\pi$$
$$710$$ 0 0
$$711$$ −1.49202e7 −1.10688
$$712$$ 1.18468e6i 0.0875789i
$$713$$ 1.19709e7i 0.881863i
$$714$$ 7.62165e6 0.559505
$$715$$ 0 0
$$716$$ 106179. 0.00774029
$$717$$ − 1.03001e6i − 0.0748246i
$$718$$ 1.23505e7i 0.894076i
$$719$$ −1.03483e7 −0.746531 −0.373266 0.927724i $$-0.621762\pi$$
−0.373266 + 0.927724i $$0.621762\pi$$
$$720$$ 0 0
$$721$$ −7.20052e6 −0.515853
$$722$$ 1.91730e7i 1.36882i
$$723$$ − 2.41633e6i − 0.171914i
$$724$$ −2.66730e7 −1.89115
$$725$$ 0 0
$$726$$ −417889. −0.0294252
$$727$$ 2.03348e7i 1.42693i 0.700690 + 0.713466i $$0.252876\pi$$
−0.700690 + 0.713466i $$0.747124\pi$$
$$728$$ 2.22262e6i 0.155430i
$$729$$ 1.02211e7 0.712325
$$730$$ 0 0
$$731$$ 1.54037e7 1.06618
$$732$$ 2.55739e6i 0.176408i
$$733$$ 4.78280e6i 0.328793i 0.986394 + 0.164396i $$0.0525676\pi$$
−0.986394 + 0.164396i $$0.947432\pi$$
$$734$$ −1.45224e7 −0.994946
$$735$$ 0 0
$$736$$ 3.75249e7 2.55344
$$737$$ 6.25663e6i 0.424298i
$$738$$ 2.10839e7i 1.42498i
$$739$$ 1.08737e7 0.732429 0.366215 0.930530i $$-0.380654\pi$$
0.366215 + 0.930530i $$0.380654\pi$$
$$740$$ 0 0
$$741$$ −787008. −0.0526543
$$742$$ − 2.81603e7i − 1.87770i
$$743$$ − 1.01036e7i − 0.671434i −0.941963 0.335717i $$-0.891021\pi$$
0.941963 0.335717i $$-0.108979\pi$$
$$744$$ −229893. −0.0152263
$$745$$ 0 0
$$746$$ 1.86006e7 1.22371
$$747$$ − 2.23438e7i − 1.46506i
$$748$$ 7.80390e6i 0.509986i
$$749$$ 985227. 0.0641700
$$750$$ 0 0
$$751$$ 9.91947e6 0.641784 0.320892 0.947116i $$-0.396017\pi$$
0.320892 + 0.947116i $$0.396017\pi$$
$$752$$ − 2.03802e6i − 0.131421i
$$753$$ − 1.86088e6i − 0.119600i
$$754$$ 8.65673e6 0.554530
$$755$$ 0 0
$$756$$ 8.39647e6 0.534309
$$757$$ 1.33506e7i 0.846759i 0.905952 + 0.423380i $$0.139156\pi$$
−0.905952 + 0.423380i $$0.860844\pi$$
$$758$$ − 3.62232e7i − 2.28988i
$$759$$ −1.90504e6 −0.120033
$$760$$ 0 0
$$761$$ 1.28109e7 0.801898 0.400949 0.916100i $$-0.368680\pi$$
0.400949 + 0.916100i $$0.368680\pi$$
$$762$$ − 2.57920e6i − 0.160916i
$$763$$ 1.40401e7i 0.873089i
$$764$$ 1.50896e7 0.935284
$$765$$ 0 0
$$766$$ 1.94531e7 1.19789
$$767$$ − 1.20185e7i − 0.737671i
$$768$$ − 2.86564e6i − 0.175315i
$$769$$ −1.90629e7 −1.16245 −0.581224 0.813743i $$-0.697426\pi$$
−0.581224 + 0.813743i $$0.697426\pi$$
$$770$$ 0 0
$$771$$ 2.27390e6 0.137764
$$772$$ 9.54054e6i 0.576142i
$$773$$ − 2.34434e6i − 0.141115i −0.997508 0.0705574i $$-0.977522\pi$$
0.997508 0.0705574i $$-0.0224778\pi$$
$$774$$ 1.58174e7 0.949037
$$775$$ 0 0
$$776$$ −955410. −0.0569555
$$777$$ 4.88555e6i 0.290309i
$$778$$ − 2.17716e7i − 1.28956i
$$779$$ 4.08972e6 0.241463
$$780$$ 0 0
$$781$$ −169204. −0.00992623
$$782$$ 6.80661e7i 3.98028i
$$783$$ − 2.83620e6i − 0.165323i
$$784$$ 3.88906e6 0.225972
$$785$$ 0 0
$$786$$ −1.86381e6 −0.107608
$$787$$ 1.52283e7i 0.876425i 0.898871 + 0.438212i $$0.144388\pi$$
−0.898871 + 0.438212i $$0.855612\pi$$
$$788$$ 2.01315e7i 1.15494i
$$789$$ −2.16845e6 −0.124010
$$790$$ 0 0
$$791$$ −3.08911e7 −1.75547
$$792$$ 694984.i 0.0393697i
$$793$$ 1.28926e7i 0.728042i
$$794$$ −1.76619e7 −0.994228
$$795$$ 0 0
$$796$$ 9.45788e6 0.529068
$$797$$ 2.70618e7i 1.50907i 0.656257 + 0.754537i $$0.272139\pi$$
−0.656257 + 0.754537i $$0.727861\pi$$
$$798$$ − 1.51811e6i − 0.0843911i
$$799$$ 4.08855e6 0.226570
$$800$$ 0 0
$$801$$ 1.09916e7 0.605313
$$802$$ 1.98987e7i 1.09242i
$$803$$ 8.76846e6i 0.479882i
$$804$$ −6.31585e6 −0.344581
$$805$$ 0 0
$$806$$ −1.33634e7 −0.724569
$$807$$ − 1.68326e6i − 0.0909844i
$$808$$ − 1.02039e6i − 0.0549842i
$$809$$ 2.25663e6 0.121224 0.0606121 0.998161i $$-0.480695\pi$$
0.0606121 + 0.998161i $$0.480695\pi$$
$$810$$ 0 0
$$811$$ 4.49121e6 0.239779 0.119890 0.992787i $$-0.461746\pi$$
0.119890 + 0.992787i $$0.461746\pi$$
$$812$$ 8.72773e6i 0.464527i
$$813$$ 3.85825e6i 0.204722i
$$814$$ −9.57091e6 −0.506282
$$815$$ 0 0
$$816$$ 5.88739e6 0.309526
$$817$$ − 3.06816e6i − 0.160814i
$$818$$ 5.01769e7i 2.62193i
$$819$$ 2.06218e7 1.07428
$$820$$ 0 0
$$821$$ 592581. 0.0306825 0.0153412 0.999882i $$-0.495117\pi$$
0.0153412 + 0.999882i $$0.495117\pi$$
$$822$$ 151924.i 0.00784236i
$$823$$ − 1.14748e7i − 0.590533i −0.955415 0.295266i $$-0.904592\pi$$
0.955415 0.295266i $$-0.0954084\pi$$
$$824$$ 1.23494e6 0.0633620
$$825$$ 0 0
$$826$$ 2.31833e7 1.18229
$$827$$ 8.47060e6i 0.430676i 0.976540 + 0.215338i $$0.0690853\pi$$
−0.976540 + 0.215338i $$0.930915\pi$$
$$828$$ 3.65313e7i 1.85178i
$$829$$ 1.58876e7 0.802919 0.401460 0.915877i $$-0.368503\pi$$
0.401460 + 0.915877i $$0.368503\pi$$
$$830$$ 0 0
$$831$$ −2.22777e6 −0.111910
$$832$$ 2.38106e7i 1.19251i
$$833$$ 7.80197e6i 0.389576i
$$834$$ 2.56788e6 0.127838
$$835$$ 0 0
$$836$$ 1.55441e6 0.0769221
$$837$$ 4.37825e6i 0.216017i
$$838$$ − 3.07552e6i − 0.151290i
$$839$$ −2.66963e7 −1.30932 −0.654659 0.755924i $$-0.727188\pi$$
−0.654659 + 0.755924i $$0.727188\pi$$
$$840$$ 0 0
$$841$$ −1.75631e7 −0.856268
$$842$$ − 2.88480e7i − 1.40228i
$$843$$ 899333.i 0.0435864i
$$844$$ 2.63972e7 1.27556
$$845$$ 0 0
$$846$$ 4.19837e6 0.201676
$$847$$ − 2.12399e6i − 0.101729i
$$848$$ − 2.17525e7i − 1.03877i
$$849$$ 3.59028e6 0.170946
$$850$$ 0 0
$$851$$ −4.36310e7 −2.06524
$$852$$ − 170806.i − 0.00806128i
$$853$$ 3.58773e6i 0.168829i 0.996431 + 0.0844146i $$0.0269020\pi$$
−0.996431 + 0.0844146i $$0.973098\pi$$
$$854$$ −2.48693e7 −1.16686
$$855$$ 0 0
$$856$$ −168974. −0.00788197
$$857$$ 6.00941e6i 0.279499i 0.990187 + 0.139749i $$0.0446297\pi$$
−0.990187 + 0.139749i $$0.955370\pi$$
$$858$$ − 2.12665e6i − 0.0986228i
$$859$$ 1.74629e7 0.807484 0.403742 0.914873i $$-0.367709\pi$$
0.403742 + 0.914873i $$0.367709\pi$$
$$860$$ 0 0
$$861$$ 5.64118e6 0.259336
$$862$$ − 2.58149e7i − 1.18332i
$$863$$ 2.34431e6i 0.107149i 0.998564 + 0.0535746i $$0.0170615\pi$$
−0.998564 + 0.0535746i $$0.982939\pi$$
$$864$$ 1.37245e7 0.625476
$$865$$ 0 0
$$866$$ 1.32779e7 0.601635
$$867$$ 6.86126e6i 0.309996i
$$868$$ − 1.34730e7i − 0.606968i
$$869$$ −7.82050e6 −0.351306
$$870$$ 0 0
$$871$$ −3.18401e7 −1.42210
$$872$$ − 2.40798e6i − 0.107241i
$$873$$ 8.86445e6i 0.393655i
$$874$$ 1.35577e7 0.600354
$$875$$ 0 0
$$876$$ −8.85145e6 −0.389721
$$877$$ − 1.98979e7i − 0.873591i −0.899561 0.436796i $$-0.856113\pi$$
0.899561 0.436796i $$-0.143887\pi$$
$$878$$ − 2.03175e7i − 0.889474i
$$879$$ 3.05971e6 0.133570
$$880$$ 0 0
$$881$$ 2.32718e7 1.01016 0.505081 0.863072i $$-0.331463\pi$$
0.505081 + 0.863072i $$0.331463\pi$$
$$882$$ 8.01154e6i 0.346772i
$$883$$ − 2.71777e7i − 1.17304i −0.809936 0.586518i $$-0.800498\pi$$
0.809936 0.586518i $$-0.199502\pi$$
$$884$$ −3.97142e7 −1.70929
$$885$$ 0 0
$$886$$ 3.07421e7 1.31568
$$887$$ 1.39671e7i 0.596069i 0.954555 + 0.298034i $$0.0963310\pi$$
−0.954555 + 0.298034i $$0.903669\pi$$
$$888$$ − 837907.i − 0.0356585i
$$889$$ 1.31092e7 0.556316
$$890$$ 0 0
$$891$$ 6.09086e6 0.257030
$$892$$ 3.49579e7i 1.47107i
$$893$$ − 814374.i − 0.0341740i
$$894$$ 1.89778e6 0.0794149
$$895$$ 0 0
$$896$$ −7.35889e6 −0.306226
$$897$$ − 9.69477e6i − 0.402306i
$$898$$ − 3.93761e7i − 1.62945i
$$899$$ −4.55099e6 −0.187805
$$900$$ 0 0
$$901$$ 4.36385e7 1.79084
$$902$$ 1.10512e7i 0.452266i
$$903$$ − 4.23209e6i − 0.172717i
$$904$$ 5.29805e6 0.215623
$$905$$ 0 0
$$906$$ 3.73524e6 0.151181
$$907$$ − 1.77875e7i − 0.717954i −0.933346 0.358977i $$-0.883126\pi$$
0.933346 0.358977i $$-0.116874\pi$$
$$908$$ − 1.73517e7i − 0.698437i
$$909$$ −9.46734e6 −0.380031
$$910$$ 0 0
$$911$$ 30398.8 0.00121356 0.000606780 1.00000i $$-0.499807\pi$$
0.000606780 1.00000i $$0.499807\pi$$
$$912$$ − 1.17267e6i − 0.0466864i
$$913$$ − 1.17116e7i − 0.464987i
$$914$$ −5.81288e7 −2.30158
$$915$$ 0 0
$$916$$ −7.77486e6 −0.306164
$$917$$ − 9.47308e6i − 0.372021i
$$918$$ 2.48947e7i 0.974990i
$$919$$ 4.10055e6 0.160160 0.0800798 0.996788i $$-0.474482\pi$$
0.0800798 + 0.996788i $$0.474482\pi$$
$$920$$ 0 0
$$921$$ 4.56406e6 0.177297
$$922$$ − 6.65453e7i − 2.57804i
$$923$$ − 861085.i − 0.0332692i
$$924$$ 2.14409e6 0.0826158
$$925$$ 0 0
$$926$$ −2.18908e7 −0.838946
$$927$$ − 1.14580e7i − 0.437935i
$$928$$ 1.42659e7i 0.543788i
$$929$$ 1.46532e7 0.557048 0.278524 0.960429i $$-0.410155\pi$$
0.278524 + 0.960429i $$0.410155\pi$$
$$930$$ 0 0
$$931$$ 1.55403e6 0.0587604
$$932$$ − 2.17039e7i − 0.818461i
$$933$$ − 1.17105e7i − 0.440425i
$$934$$ 3.54310e7 1.32897
$$935$$ 0 0
$$936$$ −3.53679e6 −0.131953
$$937$$ 3.97538e7i 1.47921i 0.673042 + 0.739604i $$0.264987\pi$$
−0.673042 + 0.739604i $$0.735013\pi$$
$$938$$ − 6.14185e7i − 2.27925i
$$939$$ −1.04803e7 −0.387891
$$940$$ 0 0
$$941$$ 5.32850e6 0.196169 0.0980847 0.995178i $$-0.468728\pi$$
0.0980847 + 0.995178i $$0.468728\pi$$
$$942$$ − 4.66088e6i − 0.171136i
$$943$$ 5.03793e7i 1.84490i
$$944$$ 1.79081e7 0.654062
$$945$$ 0 0
$$946$$ 8.29077e6 0.301208
$$947$$ − 3.11430e7i − 1.12846i −0.825618 0.564230i $$-0.809173\pi$$
0.825618 0.564230i $$-0.190827\pi$$
$$948$$ − 7.89452e6i − 0.285302i
$$949$$ −4.46229e7 −1.60839
$$950$$ 0 0
$$951$$ −7.32571e6 −0.262663
$$952$$ − 6.64389e6i − 0.237591i
$$953$$ 4.87227e7i 1.73780i 0.494990 + 0.868899i $$0.335172\pi$$
−0.494990 + 0.868899i $$0.664828\pi$$
$$954$$ 4.48106e7 1.59408
$$955$$ 0 0
$$956$$ −1.03529e7 −0.366369
$$957$$ − 724242.i − 0.0255625i
$$958$$ 1.27628e7i 0.449297i
$$959$$ −772177. −0.0271125
$$960$$ 0 0
$$961$$ −2.16038e7 −0.754608
$$962$$ − 4.87065e7i − 1.69687i
$$963$$ 1.56776e6i 0.0544773i
$$964$$ −2.42873e7 −0.841755
$$965$$ 0 0
$$966$$ 1.87009e7 0.644792
$$967$$ − 4.85436e7i − 1.66942i −0.550688 0.834711i $$-0.685635\pi$$
0.550688 0.834711i $$-0.314365\pi$$
$$968$$ 364279.i 0.0124953i
$$969$$ 2.35254e6 0.0804873
$$970$$ 0 0
$$971$$ −3.15035e7 −1.07229 −0.536143 0.844127i $$-0.680119\pi$$
−0.536143 + 0.844127i $$0.680119\pi$$
$$972$$ 2.02129e7i 0.686221i
$$973$$ 1.30516e7i 0.441960i
$$974$$ 6.25393e7 2.11230
$$975$$ 0 0
$$976$$ −1.92104e7 −0.645525
$$977$$ 5.35354e7i 1.79434i 0.441684 + 0.897170i $$0.354381\pi$$
−0.441684 + 0.897170i $$0.645619\pi$$
$$978$$ 1.52918e7i 0.511225i
$$979$$ 5.76131e6 0.192116
$$980$$ 0 0
$$981$$ −2.23416e7 −0.741211
$$982$$ − 2.95472e7i − 0.977771i
$$983$$ − 5.40925e7i − 1.78547i −0.450580 0.892736i $$-0.648783\pi$$
0.450580 0.892736i $$-0.351217\pi$$
$$984$$ −967505. −0.0318541
$$985$$ 0 0
$$986$$ −2.58769e7 −0.847655
$$987$$ − 1.12331e6i − 0.0367035i
$$988$$ 7.91044e6i 0.257815i
$$989$$ 3.77952e7 1.22870
$$990$$ 0 0
$$991$$ 2.31007e7 0.747208 0.373604 0.927588i $$-0.378122\pi$$
0.373604 + 0.927588i $$0.378122\pi$$
$$992$$ − 2.20223e7i − 0.710533i
$$993$$ − 4.29956e6i − 0.138373i
$$994$$ 1.66100e6 0.0533218
$$995$$ 0 0
$$996$$ 1.18225e7 0.377625
$$997$$ 4.54061e7i 1.44669i 0.690486 + 0.723346i $$0.257397\pi$$
−0.690486 + 0.723346i $$0.742603\pi$$
$$998$$ − 6.92788e7i − 2.20178i
$$999$$ −1.59577e7 −0.505891
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 275.6.b.b.199.2 6
5.2 odd 4 11.6.a.b.1.3 3
5.3 odd 4 275.6.a.b.1.1 3
5.4 even 2 inner 275.6.b.b.199.5 6
15.2 even 4 99.6.a.g.1.1 3
20.7 even 4 176.6.a.i.1.3 3
35.27 even 4 539.6.a.e.1.3 3
40.27 even 4 704.6.a.t.1.1 3
40.37 odd 4 704.6.a.q.1.3 3
55.32 even 4 121.6.a.d.1.1 3
165.32 odd 4 1089.6.a.r.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
11.6.a.b.1.3 3 5.2 odd 4
99.6.a.g.1.1 3 15.2 even 4
121.6.a.d.1.1 3 55.32 even 4
176.6.a.i.1.3 3 20.7 even 4
275.6.a.b.1.1 3 5.3 odd 4
275.6.b.b.199.2 6 1.1 even 1 trivial
275.6.b.b.199.5 6 5.4 even 2 inner
539.6.a.e.1.3 3 35.27 even 4
704.6.a.q.1.3 3 40.37 odd 4
704.6.a.t.1.1 3 40.27 even 4
1089.6.a.r.1.3 3 165.32 odd 4