# Properties

 Label 525.6.d.b.274.1 Level $525$ Weight $6$ Character 525.274 Analytic conductor $84.202$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,6,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, 6, names="a")

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

chi := DirichletCharacter("525.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$84.2015054018$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 274.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 525.274 Dual form 525.6.d.b.274.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-6.00000i q^{2} +9.00000i q^{3} -4.00000 q^{4} +54.0000 q^{6} +49.0000i q^{7} -168.000i q^{8} -81.0000 q^{9} +O(q^{10})$$ $$q-6.00000i q^{2} +9.00000i q^{3} -4.00000 q^{4} +54.0000 q^{6} +49.0000i q^{7} -168.000i q^{8} -81.0000 q^{9} +444.000 q^{11} -36.0000i q^{12} +442.000i q^{13} +294.000 q^{14} -1136.00 q^{16} -126.000i q^{17} +486.000i q^{18} -2684.00 q^{19} -441.000 q^{21} -2664.00i q^{22} -4200.00i q^{23} +1512.00 q^{24} +2652.00 q^{26} -729.000i q^{27} -196.000i q^{28} +5442.00 q^{29} +80.0000 q^{31} +1440.00i q^{32} +3996.00i q^{33} -756.000 q^{34} +324.000 q^{36} -5434.00i q^{37} +16104.0i q^{38} -3978.00 q^{39} +7962.00 q^{41} +2646.00i q^{42} +11524.0i q^{43} -1776.00 q^{44} -25200.0 q^{46} -13920.0i q^{47} -10224.0i q^{48} -2401.00 q^{49} +1134.00 q^{51} -1768.00i q^{52} +9594.00i q^{53} -4374.00 q^{54} +8232.00 q^{56} -24156.0i q^{57} -32652.0i q^{58} -27492.0 q^{59} +49478.0 q^{61} -480.000i q^{62} -3969.00i q^{63} -27712.0 q^{64} +23976.0 q^{66} -59356.0i q^{67} +504.000i q^{68} +37800.0 q^{69} +32040.0 q^{71} +13608.0i q^{72} +61846.0i q^{73} -32604.0 q^{74} +10736.0 q^{76} +21756.0i q^{77} +23868.0i q^{78} +65776.0 q^{79} +6561.00 q^{81} -47772.0i q^{82} -40188.0i q^{83} +1764.00 q^{84} +69144.0 q^{86} +48978.0i q^{87} -74592.0i q^{88} +7974.00 q^{89} -21658.0 q^{91} +16800.0i q^{92} +720.000i q^{93} -83520.0 q^{94} -12960.0 q^{96} -143662. i q^{97} +14406.0i q^{98} -35964.0 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4} + 108 q^{6} - 162 q^{9}+O(q^{10})$$ 2 * q - 8 * q^4 + 108 * q^6 - 162 * q^9 $$2 q - 8 q^{4} + 108 q^{6} - 162 q^{9} + 888 q^{11} + 588 q^{14} - 2272 q^{16} - 5368 q^{19} - 882 q^{21} + 3024 q^{24} + 5304 q^{26} + 10884 q^{29} + 160 q^{31} - 1512 q^{34} + 648 q^{36} - 7956 q^{39} + 15924 q^{41} - 3552 q^{44} - 50400 q^{46} - 4802 q^{49} + 2268 q^{51} - 8748 q^{54} + 16464 q^{56} - 54984 q^{59} + 98956 q^{61} - 55424 q^{64} + 47952 q^{66} + 75600 q^{69} + 64080 q^{71} - 65208 q^{74} + 21472 q^{76} + 131552 q^{79} + 13122 q^{81} + 3528 q^{84} + 138288 q^{86} + 15948 q^{89} - 43316 q^{91} - 167040 q^{94} - 25920 q^{96} - 71928 q^{99}+O(q^{100})$$ 2 * q - 8 * q^4 + 108 * q^6 - 162 * q^9 + 888 * q^11 + 588 * q^14 - 2272 * q^16 - 5368 * q^19 - 882 * q^21 + 3024 * q^24 + 5304 * q^26 + 10884 * q^29 + 160 * q^31 - 1512 * q^34 + 648 * q^36 - 7956 * q^39 + 15924 * q^41 - 3552 * q^44 - 50400 * q^46 - 4802 * q^49 + 2268 * q^51 - 8748 * q^54 + 16464 * q^56 - 54984 * q^59 + 98956 * q^61 - 55424 * q^64 + 47952 * q^66 + 75600 * q^69 + 64080 * q^71 - 65208 * q^74 + 21472 * q^76 + 131552 * q^79 + 13122 * q^81 + 3528 * q^84 + 138288 * q^86 + 15948 * q^89 - 43316 * q^91 - 167040 * q^94 - 25920 * q^96 - 71928 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 6.00000i − 1.06066i −0.847791 0.530330i $$-0.822068\pi$$
0.847791 0.530330i $$-0.177932\pi$$
$$3$$ 9.00000i 0.577350i
$$4$$ −4.00000 −0.125000
$$5$$ 0 0
$$6$$ 54.0000 0.612372
$$7$$ 49.0000i 0.377964i
$$8$$ − 168.000i − 0.928078i
$$9$$ −81.0000 −0.333333
$$10$$ 0 0
$$11$$ 444.000 1.10637 0.553186 0.833058i $$-0.313412\pi$$
0.553186 + 0.833058i $$0.313412\pi$$
$$12$$ − 36.0000i − 0.0721688i
$$13$$ 442.000i 0.725377i 0.931910 + 0.362689i $$0.118141\pi$$
−0.931910 + 0.362689i $$0.881859\pi$$
$$14$$ 294.000 0.400892
$$15$$ 0 0
$$16$$ −1136.00 −1.10938
$$17$$ − 126.000i − 0.105742i −0.998601 0.0528711i $$-0.983163\pi$$
0.998601 0.0528711i $$-0.0168372\pi$$
$$18$$ 486.000i 0.353553i
$$19$$ −2684.00 −1.70568 −0.852842 0.522169i $$-0.825123\pi$$
−0.852842 + 0.522169i $$0.825123\pi$$
$$20$$ 0 0
$$21$$ −441.000 −0.218218
$$22$$ − 2664.00i − 1.17348i
$$23$$ − 4200.00i − 1.65550i −0.561096 0.827751i $$-0.689620\pi$$
0.561096 0.827751i $$-0.310380\pi$$
$$24$$ 1512.00 0.535826
$$25$$ 0 0
$$26$$ 2652.00 0.769379
$$27$$ − 729.000i − 0.192450i
$$28$$ − 196.000i − 0.0472456i
$$29$$ 5442.00 1.20161 0.600805 0.799396i $$-0.294847\pi$$
0.600805 + 0.799396i $$0.294847\pi$$
$$30$$ 0 0
$$31$$ 80.0000 0.0149515 0.00747577 0.999972i $$-0.497620\pi$$
0.00747577 + 0.999972i $$0.497620\pi$$
$$32$$ 1440.00i 0.248592i
$$33$$ 3996.00i 0.638764i
$$34$$ −756.000 −0.112157
$$35$$ 0 0
$$36$$ 324.000 0.0416667
$$37$$ − 5434.00i − 0.652552i −0.945274 0.326276i $$-0.894206\pi$$
0.945274 0.326276i $$-0.105794\pi$$
$$38$$ 16104.0i 1.80915i
$$39$$ −3978.00 −0.418797
$$40$$ 0 0
$$41$$ 7962.00 0.739712 0.369856 0.929089i $$-0.379407\pi$$
0.369856 + 0.929089i $$0.379407\pi$$
$$42$$ 2646.00i 0.231455i
$$43$$ 11524.0i 0.950456i 0.879863 + 0.475228i $$0.157634\pi$$
−0.879863 + 0.475228i $$0.842366\pi$$
$$44$$ −1776.00 −0.138297
$$45$$ 0 0
$$46$$ −25200.0 −1.75592
$$47$$ − 13920.0i − 0.919167i −0.888134 0.459584i $$-0.847999\pi$$
0.888134 0.459584i $$-0.152001\pi$$
$$48$$ − 10224.0i − 0.640498i
$$49$$ −2401.00 −0.142857
$$50$$ 0 0
$$51$$ 1134.00 0.0610503
$$52$$ − 1768.00i − 0.0906721i
$$53$$ 9594.00i 0.469148i 0.972098 + 0.234574i $$0.0753695\pi$$
−0.972098 + 0.234574i $$0.924630\pi$$
$$54$$ −4374.00 −0.204124
$$55$$ 0 0
$$56$$ 8232.00 0.350780
$$57$$ − 24156.0i − 0.984777i
$$58$$ − 32652.0i − 1.27450i
$$59$$ −27492.0 −1.02820 −0.514098 0.857731i $$-0.671873\pi$$
−0.514098 + 0.857731i $$0.671873\pi$$
$$60$$ 0 0
$$61$$ 49478.0 1.70250 0.851251 0.524759i $$-0.175845\pi$$
0.851251 + 0.524759i $$0.175845\pi$$
$$62$$ − 480.000i − 0.0158585i
$$63$$ − 3969.00i − 0.125988i
$$64$$ −27712.0 −0.845703
$$65$$ 0 0
$$66$$ 23976.0 0.677512
$$67$$ − 59356.0i − 1.61539i −0.589600 0.807695i $$-0.700715\pi$$
0.589600 0.807695i $$-0.299285\pi$$
$$68$$ 504.000i 0.0132178i
$$69$$ 37800.0 0.955805
$$70$$ 0 0
$$71$$ 32040.0 0.754304 0.377152 0.926151i $$-0.376903\pi$$
0.377152 + 0.926151i $$0.376903\pi$$
$$72$$ 13608.0i 0.309359i
$$73$$ 61846.0i 1.35833i 0.733987 + 0.679164i $$0.237657\pi$$
−0.733987 + 0.679164i $$0.762343\pi$$
$$74$$ −32604.0 −0.692136
$$75$$ 0 0
$$76$$ 10736.0 0.213210
$$77$$ 21756.0i 0.418169i
$$78$$ 23868.0i 0.444201i
$$79$$ 65776.0 1.18577 0.592884 0.805288i $$-0.297989\pi$$
0.592884 + 0.805288i $$0.297989\pi$$
$$80$$ 0 0
$$81$$ 6561.00 0.111111
$$82$$ − 47772.0i − 0.784583i
$$83$$ − 40188.0i − 0.640326i −0.947362 0.320163i $$-0.896262\pi$$
0.947362 0.320163i $$-0.103738\pi$$
$$84$$ 1764.00 0.0272772
$$85$$ 0 0
$$86$$ 69144.0 1.00811
$$87$$ 48978.0i 0.693750i
$$88$$ − 74592.0i − 1.02680i
$$89$$ 7974.00 0.106709 0.0533545 0.998576i $$-0.483009\pi$$
0.0533545 + 0.998576i $$0.483009\pi$$
$$90$$ 0 0
$$91$$ −21658.0 −0.274167
$$92$$ 16800.0i 0.206938i
$$93$$ 720.000i 0.00863227i
$$94$$ −83520.0 −0.974924
$$95$$ 0 0
$$96$$ −12960.0 −0.143525
$$97$$ − 143662.i − 1.55029i −0.631784 0.775144i $$-0.717677\pi$$
0.631784 0.775144i $$-0.282323\pi$$
$$98$$ 14406.0i 0.151523i
$$99$$ −35964.0 −0.368791
$$100$$ 0 0
$$101$$ −2706.00 −0.0263952 −0.0131976 0.999913i $$-0.504201\pi$$
−0.0131976 + 0.999913i $$0.504201\pi$$
$$102$$ − 6804.00i − 0.0647536i
$$103$$ − 131768.i − 1.22382i −0.790928 0.611909i $$-0.790402\pi$$
0.790928 0.611909i $$-0.209598\pi$$
$$104$$ 74256.0 0.673206
$$105$$ 0 0
$$106$$ 57564.0 0.497607
$$107$$ − 128916.i − 1.08855i −0.838908 0.544274i $$-0.816805\pi$$
0.838908 0.544274i $$-0.183195\pi$$
$$108$$ 2916.00i 0.0240563i
$$109$$ 100978. 0.814068 0.407034 0.913413i $$-0.366563\pi$$
0.407034 + 0.913413i $$0.366563\pi$$
$$110$$ 0 0
$$111$$ 48906.0 0.376751
$$112$$ − 55664.0i − 0.419304i
$$113$$ − 220146.i − 1.62186i −0.585140 0.810932i $$-0.698960\pi$$
0.585140 0.810932i $$-0.301040\pi$$
$$114$$ −144936. −1.04451
$$115$$ 0 0
$$116$$ −21768.0 −0.150201
$$117$$ − 35802.0i − 0.241792i
$$118$$ 164952.i 1.09057i
$$119$$ 6174.00 0.0399668
$$120$$ 0 0
$$121$$ 36085.0 0.224059
$$122$$ − 296868.i − 1.80578i
$$123$$ 71658.0i 0.427073i
$$124$$ −320.000 −0.00186894
$$125$$ 0 0
$$126$$ −23814.0 −0.133631
$$127$$ − 74320.0i − 0.408880i −0.978879 0.204440i $$-0.934463\pi$$
0.978879 0.204440i $$-0.0655374\pi$$
$$128$$ 212352.i 1.14560i
$$129$$ −103716. −0.548746
$$130$$ 0 0
$$131$$ −155316. −0.790748 −0.395374 0.918520i $$-0.629385\pi$$
−0.395374 + 0.918520i $$0.629385\pi$$
$$132$$ − 15984.0i − 0.0798455i
$$133$$ − 131516.i − 0.644688i
$$134$$ −356136. −1.71338
$$135$$ 0 0
$$136$$ −21168.0 −0.0981369
$$137$$ − 264246.i − 1.20284i −0.798934 0.601419i $$-0.794602\pi$$
0.798934 0.601419i $$-0.205398\pi$$
$$138$$ − 226800.i − 1.01378i
$$139$$ −224612. −0.986043 −0.493022 0.870017i $$-0.664108\pi$$
−0.493022 + 0.870017i $$0.664108\pi$$
$$140$$ 0 0
$$141$$ 125280. 0.530682
$$142$$ − 192240.i − 0.800061i
$$143$$ 196248.i 0.802537i
$$144$$ 92016.0 0.369792
$$145$$ 0 0
$$146$$ 371076. 1.44072
$$147$$ − 21609.0i − 0.0824786i
$$148$$ 21736.0i 0.0815690i
$$149$$ 82074.0 0.302859 0.151429 0.988468i $$-0.451612\pi$$
0.151429 + 0.988468i $$0.451612\pi$$
$$150$$ 0 0
$$151$$ −287032. −1.02444 −0.512222 0.858853i $$-0.671177\pi$$
−0.512222 + 0.858853i $$0.671177\pi$$
$$152$$ 450912.i 1.58301i
$$153$$ 10206.0i 0.0352474i
$$154$$ 130536. 0.443536
$$155$$ 0 0
$$156$$ 15912.0 0.0523496
$$157$$ 129878.i 0.420520i 0.977646 + 0.210260i $$0.0674310\pi$$
−0.977646 + 0.210260i $$0.932569\pi$$
$$158$$ − 394656.i − 1.25770i
$$159$$ −86346.0 −0.270863
$$160$$ 0 0
$$161$$ 205800. 0.625721
$$162$$ − 39366.0i − 0.117851i
$$163$$ − 555284.i − 1.63699i −0.574513 0.818495i $$-0.694809\pi$$
0.574513 0.818495i $$-0.305191\pi$$
$$164$$ −31848.0 −0.0924640
$$165$$ 0 0
$$166$$ −241128. −0.679168
$$167$$ 43512.0i 0.120731i 0.998176 + 0.0603654i $$0.0192266\pi$$
−0.998176 + 0.0603654i $$0.980773\pi$$
$$168$$ 74088.0i 0.202523i
$$169$$ 175929. 0.473828
$$170$$ 0 0
$$171$$ 217404. 0.568561
$$172$$ − 46096.0i − 0.118807i
$$173$$ 18330.0i 0.0465637i 0.999729 + 0.0232818i $$0.00741151\pi$$
−0.999729 + 0.0232818i $$0.992588\pi$$
$$174$$ 293868. 0.735833
$$175$$ 0 0
$$176$$ −504384. −1.22738
$$177$$ − 247428.i − 0.593630i
$$178$$ − 47844.0i − 0.113182i
$$179$$ 153324. 0.357666 0.178833 0.983879i $$-0.442768\pi$$
0.178833 + 0.983879i $$0.442768\pi$$
$$180$$ 0 0
$$181$$ −382066. −0.866846 −0.433423 0.901191i $$-0.642694\pi$$
−0.433423 + 0.901191i $$0.642694\pi$$
$$182$$ 129948.i 0.290798i
$$183$$ 445302.i 0.982940i
$$184$$ −705600. −1.53643
$$185$$ 0 0
$$186$$ 4320.00 0.00915591
$$187$$ − 55944.0i − 0.116990i
$$188$$ 55680.0i 0.114896i
$$189$$ 35721.0 0.0727393
$$190$$ 0 0
$$191$$ −273408. −0.542285 −0.271143 0.962539i $$-0.587402\pi$$
−0.271143 + 0.962539i $$0.587402\pi$$
$$192$$ − 249408.i − 0.488267i
$$193$$ − 153602.i − 0.296827i −0.988925 0.148414i $$-0.952583\pi$$
0.988925 0.148414i $$-0.0474167\pi$$
$$194$$ −861972. −1.64433
$$195$$ 0 0
$$196$$ 9604.00 0.0178571
$$197$$ 154422.i 0.283494i 0.989903 + 0.141747i $$0.0452719\pi$$
−0.989903 + 0.141747i $$0.954728\pi$$
$$198$$ 215784.i 0.391162i
$$199$$ 366856. 0.656694 0.328347 0.944557i $$-0.393508\pi$$
0.328347 + 0.944557i $$0.393508\pi$$
$$200$$ 0 0
$$201$$ 534204. 0.932646
$$202$$ 16236.0i 0.0279963i
$$203$$ 266658.i 0.454166i
$$204$$ −4536.00 −0.00763128
$$205$$ 0 0
$$206$$ −790608. −1.29806
$$207$$ 340200.i 0.551834i
$$208$$ − 502112.i − 0.804715i
$$209$$ −1.19170e6 −1.88712
$$210$$ 0 0
$$211$$ 520244. 0.804453 0.402227 0.915540i $$-0.368236\pi$$
0.402227 + 0.915540i $$0.368236\pi$$
$$212$$ − 38376.0i − 0.0586435i
$$213$$ 288360.i 0.435498i
$$214$$ −773496. −1.15458
$$215$$ 0 0
$$216$$ −122472. −0.178609
$$217$$ 3920.00i 0.00565115i
$$218$$ − 605868.i − 0.863449i
$$219$$ −556614. −0.784231
$$220$$ 0 0
$$221$$ 55692.0 0.0767030
$$222$$ − 293436.i − 0.399605i
$$223$$ − 304736.i − 0.410357i −0.978725 0.205178i $$-0.934223\pi$$
0.978725 0.205178i $$-0.0657775\pi$$
$$224$$ −70560.0 −0.0939590
$$225$$ 0 0
$$226$$ −1.32088e6 −1.72025
$$227$$ 288588.i 0.371718i 0.982576 + 0.185859i $$0.0595068\pi$$
−0.982576 + 0.185859i $$0.940493\pi$$
$$228$$ 96624.0i 0.123097i
$$229$$ −772190. −0.973051 −0.486525 0.873666i $$-0.661736\pi$$
−0.486525 + 0.873666i $$0.661736\pi$$
$$230$$ 0 0
$$231$$ −195804. −0.241430
$$232$$ − 914256.i − 1.11519i
$$233$$ − 252234.i − 0.304378i −0.988351 0.152189i $$-0.951368\pi$$
0.988351 0.152189i $$-0.0486323\pi$$
$$234$$ −214812. −0.256460
$$235$$ 0 0
$$236$$ 109968. 0.128525
$$237$$ 591984.i 0.684603i
$$238$$ − 37044.0i − 0.0423912i
$$239$$ 1.45114e6 1.64329 0.821643 0.570002i $$-0.193058\pi$$
0.821643 + 0.570002i $$0.193058\pi$$
$$240$$ 0 0
$$241$$ −146398. −0.162365 −0.0811825 0.996699i $$-0.525870\pi$$
−0.0811825 + 0.996699i $$0.525870\pi$$
$$242$$ − 216510.i − 0.237651i
$$243$$ 59049.0i 0.0641500i
$$244$$ −197912. −0.212813
$$245$$ 0 0
$$246$$ 429948. 0.452979
$$247$$ − 1.18633e6i − 1.23726i
$$248$$ − 13440.0i − 0.0138762i
$$249$$ 361692. 0.369692
$$250$$ 0 0
$$251$$ 607860. 0.609003 0.304501 0.952512i $$-0.401510\pi$$
0.304501 + 0.952512i $$0.401510\pi$$
$$252$$ 15876.0i 0.0157485i
$$253$$ − 1.86480e6i − 1.83160i
$$254$$ −445920. −0.433683
$$255$$ 0 0
$$256$$ 387328. 0.369385
$$257$$ 95586.0i 0.0902737i 0.998981 + 0.0451369i $$0.0143724\pi$$
−0.998981 + 0.0451369i $$0.985628\pi$$
$$258$$ 622296.i 0.582033i
$$259$$ 266266. 0.246642
$$260$$ 0 0
$$261$$ −440802. −0.400537
$$262$$ 931896.i 0.838715i
$$263$$ 2.20034e6i 1.96156i 0.195121 + 0.980779i $$0.437490\pi$$
−0.195121 + 0.980779i $$0.562510\pi$$
$$264$$ 671328. 0.592823
$$265$$ 0 0
$$266$$ −789096. −0.683795
$$267$$ 71766.0i 0.0616085i
$$268$$ 237424.i 0.201924i
$$269$$ −1.77025e6 −1.49160 −0.745801 0.666169i $$-0.767933\pi$$
−0.745801 + 0.666169i $$0.767933\pi$$
$$270$$ 0 0
$$271$$ −223504. −0.184868 −0.0924341 0.995719i $$-0.529465\pi$$
−0.0924341 + 0.995719i $$0.529465\pi$$
$$272$$ 143136.i 0.117308i
$$273$$ − 194922.i − 0.158290i
$$274$$ −1.58548e6 −1.27580
$$275$$ 0 0
$$276$$ −151200. −0.119476
$$277$$ − 342778.i − 0.268419i −0.990953 0.134210i $$-0.957150\pi$$
0.990953 0.134210i $$-0.0428495\pi$$
$$278$$ 1.34767e6i 1.04586i
$$279$$ −6480.00 −0.00498384
$$280$$ 0 0
$$281$$ 480378. 0.362925 0.181463 0.983398i $$-0.441917\pi$$
0.181463 + 0.983398i $$0.441917\pi$$
$$282$$ − 751680.i − 0.562873i
$$283$$ 29980.0i 0.0222518i 0.999938 + 0.0111259i $$0.00354156\pi$$
−0.999938 + 0.0111259i $$0.996458\pi$$
$$284$$ −128160. −0.0942880
$$285$$ 0 0
$$286$$ 1.17749e6 0.851219
$$287$$ 390138.i 0.279585i
$$288$$ − 116640.i − 0.0828641i
$$289$$ 1.40398e6 0.988819
$$290$$ 0 0
$$291$$ 1.29296e6 0.895060
$$292$$ − 247384.i − 0.169791i
$$293$$ 198066.i 0.134785i 0.997727 + 0.0673924i $$0.0214679\pi$$
−0.997727 + 0.0673924i $$0.978532\pi$$
$$294$$ −129654. −0.0874818
$$295$$ 0 0
$$296$$ −912912. −0.605619
$$297$$ − 323676.i − 0.212921i
$$298$$ − 492444.i − 0.321230i
$$299$$ 1.85640e6 1.20086
$$300$$ 0 0
$$301$$ −564676. −0.359239
$$302$$ 1.72219e6i 1.08659i
$$303$$ − 24354.0i − 0.0152393i
$$304$$ 3.04902e6 1.89224
$$305$$ 0 0
$$306$$ 61236.0 0.0373855
$$307$$ − 1.04564e6i − 0.633191i −0.948561 0.316595i $$-0.897460\pi$$
0.948561 0.316595i $$-0.102540\pi$$
$$308$$ − 87024.0i − 0.0522712i
$$309$$ 1.18591e6 0.706572
$$310$$ 0 0
$$311$$ 1.83718e6 1.07708 0.538542 0.842598i $$-0.318975\pi$$
0.538542 + 0.842598i $$0.318975\pi$$
$$312$$ 668304.i 0.388676i
$$313$$ 365494.i 0.210872i 0.994426 + 0.105436i $$0.0336239\pi$$
−0.994426 + 0.105436i $$0.966376\pi$$
$$314$$ 779268. 0.446029
$$315$$ 0 0
$$316$$ −263104. −0.148221
$$317$$ − 28338.0i − 0.0158388i −0.999969 0.00791938i $$-0.997479\pi$$
0.999969 0.00791938i $$-0.00252084\pi$$
$$318$$ 518076.i 0.287293i
$$319$$ 2.41625e6 1.32943
$$320$$ 0 0
$$321$$ 1.16024e6 0.628473
$$322$$ − 1.23480e6i − 0.663677i
$$323$$ 338184.i 0.180363i
$$324$$ −26244.0 −0.0138889
$$325$$ 0 0
$$326$$ −3.33170e6 −1.73629
$$327$$ 908802.i 0.470002i
$$328$$ − 1.33762e6i − 0.686510i
$$329$$ 682080. 0.347413
$$330$$ 0 0
$$331$$ 1.93392e6 0.970214 0.485107 0.874455i $$-0.338781\pi$$
0.485107 + 0.874455i $$0.338781\pi$$
$$332$$ 160752.i 0.0800408i
$$333$$ 440154.i 0.217517i
$$334$$ 261072. 0.128054
$$335$$ 0 0
$$336$$ 500976. 0.242085
$$337$$ − 1.88817e6i − 0.905664i −0.891596 0.452832i $$-0.850414\pi$$
0.891596 0.452832i $$-0.149586\pi$$
$$338$$ − 1.05557e6i − 0.502570i
$$339$$ 1.98131e6 0.936384
$$340$$ 0 0
$$341$$ 35520.0 0.0165420
$$342$$ − 1.30442e6i − 0.603050i
$$343$$ − 117649.i − 0.0539949i
$$344$$ 1.93603e6 0.882097
$$345$$ 0 0
$$346$$ 109980. 0.0493882
$$347$$ 2.91937e6i 1.30156i 0.759264 + 0.650782i $$0.225559\pi$$
−0.759264 + 0.650782i $$0.774441\pi$$
$$348$$ − 195912.i − 0.0867187i
$$349$$ 780682. 0.343092 0.171546 0.985176i $$-0.445124\pi$$
0.171546 + 0.985176i $$0.445124\pi$$
$$350$$ 0 0
$$351$$ 322218. 0.139599
$$352$$ 639360.i 0.275036i
$$353$$ − 1.33437e6i − 0.569954i −0.958534 0.284977i $$-0.908014\pi$$
0.958534 0.284977i $$-0.0919859\pi$$
$$354$$ −1.48457e6 −0.629639
$$355$$ 0 0
$$356$$ −31896.0 −0.0133386
$$357$$ 55566.0i 0.0230748i
$$358$$ − 919944.i − 0.379362i
$$359$$ −1.01743e6 −0.416648 −0.208324 0.978060i $$-0.566801\pi$$
−0.208324 + 0.978060i $$0.566801\pi$$
$$360$$ 0 0
$$361$$ 4.72776e6 1.90936
$$362$$ 2.29240e6i 0.919429i
$$363$$ 324765.i 0.129361i
$$364$$ 86632.0 0.0342709
$$365$$ 0 0
$$366$$ 2.67181e6 1.04257
$$367$$ 837680.i 0.324648i 0.986737 + 0.162324i $$0.0518990\pi$$
−0.986737 + 0.162324i $$0.948101\pi$$
$$368$$ 4.77120e6i 1.83657i
$$369$$ −644922. −0.246571
$$370$$ 0 0
$$371$$ −470106. −0.177321
$$372$$ − 2880.00i − 0.00107903i
$$373$$ 1.51993e6i 0.565655i 0.959171 + 0.282827i $$0.0912724\pi$$
−0.959171 + 0.282827i $$0.908728\pi$$
$$374$$ −335664. −0.124087
$$375$$ 0 0
$$376$$ −2.33856e6 −0.853059
$$377$$ 2.40536e6i 0.871620i
$$378$$ − 214326.i − 0.0771517i
$$379$$ −2.64465e6 −0.945737 −0.472869 0.881133i $$-0.656781\pi$$
−0.472869 + 0.881133i $$0.656781\pi$$
$$380$$ 0 0
$$381$$ 668880. 0.236067
$$382$$ 1.64045e6i 0.575180i
$$383$$ − 2.01336e6i − 0.701333i −0.936500 0.350667i $$-0.885955\pi$$
0.936500 0.350667i $$-0.114045\pi$$
$$384$$ −1.91117e6 −0.661410
$$385$$ 0 0
$$386$$ −921612. −0.314833
$$387$$ − 933444.i − 0.316819i
$$388$$ 574648.i 0.193786i
$$389$$ 726234. 0.243334 0.121667 0.992571i $$-0.461176\pi$$
0.121667 + 0.992571i $$0.461176\pi$$
$$390$$ 0 0
$$391$$ −529200. −0.175056
$$392$$ 403368.i 0.132583i
$$393$$ − 1.39784e6i − 0.456538i
$$394$$ 926532. 0.300691
$$395$$ 0 0
$$396$$ 143856. 0.0460988
$$397$$ 4.57578e6i 1.45710i 0.684993 + 0.728549i $$0.259805\pi$$
−0.684993 + 0.728549i $$0.740195\pi$$
$$398$$ − 2.20114e6i − 0.696529i
$$399$$ 1.18364e6 0.372211
$$400$$ 0 0
$$401$$ −33870.0 −0.0105185 −0.00525926 0.999986i $$-0.501674\pi$$
−0.00525926 + 0.999986i $$0.501674\pi$$
$$402$$ − 3.20522e6i − 0.989221i
$$403$$ 35360.0i 0.0108455i
$$404$$ 10824.0 0.00329940
$$405$$ 0 0
$$406$$ 1.59995e6 0.481716
$$407$$ − 2.41270e6i − 0.721966i
$$408$$ − 190512.i − 0.0566594i
$$409$$ 5.86178e6 1.73269 0.866346 0.499444i $$-0.166462\pi$$
0.866346 + 0.499444i $$0.166462\pi$$
$$410$$ 0 0
$$411$$ 2.37821e6 0.694459
$$412$$ 527072.i 0.152977i
$$413$$ − 1.34711e6i − 0.388622i
$$414$$ 2.04120e6 0.585308
$$415$$ 0 0
$$416$$ −636480. −0.180323
$$417$$ − 2.02151e6i − 0.569292i
$$418$$ 7.15018e6i 2.00159i
$$419$$ −302748. −0.0842454 −0.0421227 0.999112i $$-0.513412\pi$$
−0.0421227 + 0.999112i $$0.513412\pi$$
$$420$$ 0 0
$$421$$ −5.36708e6 −1.47582 −0.737909 0.674900i $$-0.764187\pi$$
−0.737909 + 0.674900i $$0.764187\pi$$
$$422$$ − 3.12146e6i − 0.853252i
$$423$$ 1.12752e6i 0.306389i
$$424$$ 1.61179e6 0.435406
$$425$$ 0 0
$$426$$ 1.73016e6 0.461915
$$427$$ 2.42442e6i 0.643485i
$$428$$ 515664.i 0.136068i
$$429$$ −1.76623e6 −0.463345
$$430$$ 0 0
$$431$$ 1.17706e6 0.305214 0.152607 0.988287i $$-0.451233\pi$$
0.152607 + 0.988287i $$0.451233\pi$$
$$432$$ 828144.i 0.213499i
$$433$$ 3.66249e6i 0.938766i 0.882995 + 0.469383i $$0.155524\pi$$
−0.882995 + 0.469383i $$0.844476\pi$$
$$434$$ 23520.0 0.00599395
$$435$$ 0 0
$$436$$ −403912. −0.101758
$$437$$ 1.12728e7i 2.82376i
$$438$$ 3.33968e6i 0.831802i
$$439$$ 2.53674e6 0.628225 0.314113 0.949386i $$-0.398293\pi$$
0.314113 + 0.949386i $$0.398293\pi$$
$$440$$ 0 0
$$441$$ 194481. 0.0476190
$$442$$ − 334152.i − 0.0813558i
$$443$$ − 6.01504e6i − 1.45623i −0.685457 0.728113i $$-0.740397\pi$$
0.685457 0.728113i $$-0.259603\pi$$
$$444$$ −195624. −0.0470939
$$445$$ 0 0
$$446$$ −1.82842e6 −0.435249
$$447$$ 738666.i 0.174856i
$$448$$ − 1.35789e6i − 0.319646i
$$449$$ −5.65965e6 −1.32487 −0.662436 0.749119i $$-0.730477\pi$$
−0.662436 + 0.749119i $$0.730477\pi$$
$$450$$ 0 0
$$451$$ 3.53513e6 0.818397
$$452$$ 880584.i 0.202733i
$$453$$ − 2.58329e6i − 0.591463i
$$454$$ 1.73153e6 0.394267
$$455$$ 0 0
$$456$$ −4.05821e6 −0.913949
$$457$$ − 6.46159e6i − 1.44727i −0.690184 0.723634i $$-0.742470\pi$$
0.690184 0.723634i $$-0.257530\pi$$
$$458$$ 4.63314e6i 1.03208i
$$459$$ −91854.0 −0.0203501
$$460$$ 0 0
$$461$$ −3.37353e6 −0.739320 −0.369660 0.929167i $$-0.620526\pi$$
−0.369660 + 0.929167i $$0.620526\pi$$
$$462$$ 1.17482e6i 0.256075i
$$463$$ 4.54974e6i 0.986358i 0.869928 + 0.493179i $$0.164165\pi$$
−0.869928 + 0.493179i $$0.835835\pi$$
$$464$$ −6.18211e6 −1.33304
$$465$$ 0 0
$$466$$ −1.51340e6 −0.322842
$$467$$ 2.01136e6i 0.426773i 0.976968 + 0.213386i $$0.0684493\pi$$
−0.976968 + 0.213386i $$0.931551\pi$$
$$468$$ 143208.i 0.0302240i
$$469$$ 2.90844e6 0.610560
$$470$$ 0 0
$$471$$ −1.16890e6 −0.242787
$$472$$ 4.61866e6i 0.954247i
$$473$$ 5.11666e6i 1.05156i
$$474$$ 3.55190e6 0.726132
$$475$$ 0 0
$$476$$ −24696.0 −0.00499585
$$477$$ − 777114.i − 0.156383i
$$478$$ − 8.70682e6i − 1.74297i
$$479$$ 7.60402e6 1.51427 0.757137 0.653257i $$-0.226598\pi$$
0.757137 + 0.653257i $$0.226598\pi$$
$$480$$ 0 0
$$481$$ 2.40183e6 0.473347
$$482$$ 878388.i 0.172214i
$$483$$ 1.85220e6i 0.361260i
$$484$$ −144340. −0.0280074
$$485$$ 0 0
$$486$$ 354294. 0.0680414
$$487$$ 673112.i 0.128607i 0.997930 + 0.0643035i $$0.0204826\pi$$
−0.997930 + 0.0643035i $$0.979517\pi$$
$$488$$ − 8.31230e6i − 1.58005i
$$489$$ 4.99756e6 0.945117
$$490$$ 0 0
$$491$$ −2.47170e6 −0.462692 −0.231346 0.972872i $$-0.574313\pi$$
−0.231346 + 0.972872i $$0.574313\pi$$
$$492$$ − 286632.i − 0.0533841i
$$493$$ − 685692.i − 0.127061i
$$494$$ −7.11797e6 −1.31232
$$495$$ 0 0
$$496$$ −90880.0 −0.0165869
$$497$$ 1.56996e6i 0.285100i
$$498$$ − 2.17015e6i − 0.392118i
$$499$$ −6.08152e6 −1.09335 −0.546677 0.837343i $$-0.684108\pi$$
−0.546677 + 0.837343i $$0.684108\pi$$
$$500$$ 0 0
$$501$$ −391608. −0.0697039
$$502$$ − 3.64716e6i − 0.645945i
$$503$$ 846216.i 0.149129i 0.997216 + 0.0745644i $$0.0237566\pi$$
−0.997216 + 0.0745644i $$0.976243\pi$$
$$504$$ −666792. −0.116927
$$505$$ 0 0
$$506$$ −1.11888e7 −1.94271
$$507$$ 1.58336e6i 0.273565i
$$508$$ 297280.i 0.0511101i
$$509$$ 7.66785e6 1.31183 0.655917 0.754833i $$-0.272282\pi$$
0.655917 + 0.754833i $$0.272282\pi$$
$$510$$ 0 0
$$511$$ −3.03045e6 −0.513400
$$512$$ 4.47130e6i 0.753804i
$$513$$ 1.95664e6i 0.328259i
$$514$$ 573516. 0.0957498
$$515$$ 0 0
$$516$$ 414864. 0.0685933
$$517$$ − 6.18048e6i − 1.01694i
$$518$$ − 1.59760e6i − 0.261603i
$$519$$ −164970. −0.0268835
$$520$$ 0 0
$$521$$ −9.68938e6 −1.56387 −0.781937 0.623357i $$-0.785768\pi$$
−0.781937 + 0.623357i $$0.785768\pi$$
$$522$$ 2.64481e6i 0.424833i
$$523$$ 7.51678e6i 1.20165i 0.799381 + 0.600824i $$0.205161\pi$$
−0.799381 + 0.600824i $$0.794839\pi$$
$$524$$ 621264. 0.0988435
$$525$$ 0 0
$$526$$ 1.32021e7 2.08055
$$527$$ − 10080.0i − 0.00158101i
$$528$$ − 4.53946e6i − 0.708629i
$$529$$ −1.12037e7 −1.74069
$$530$$ 0 0
$$531$$ 2.22685e6 0.342732
$$532$$ 526064.i 0.0805860i
$$533$$ 3.51920e6i 0.536570i
$$534$$ 430596. 0.0653457
$$535$$ 0 0
$$536$$ −9.97181e6 −1.49921
$$537$$ 1.37992e6i 0.206499i
$$538$$ 1.06215e7i 1.58208i
$$539$$ −1.06604e6 −0.158053
$$540$$ 0 0
$$541$$ 7.34325e6 1.07869 0.539343 0.842086i $$-0.318673\pi$$
0.539343 + 0.842086i $$0.318673\pi$$
$$542$$ 1.34102e6i 0.196082i
$$543$$ − 3.43859e6i − 0.500474i
$$544$$ 181440. 0.0262867
$$545$$ 0 0
$$546$$ −1.16953e6 −0.167892
$$547$$ 2.18296e6i 0.311945i 0.987761 + 0.155973i $$0.0498512\pi$$
−0.987761 + 0.155973i $$0.950149\pi$$
$$548$$ 1.05698e6i 0.150355i
$$549$$ −4.00772e6 −0.567501
$$550$$ 0 0
$$551$$ −1.46063e7 −2.04957
$$552$$ − 6.35040e6i − 0.887061i
$$553$$ 3.22302e6i 0.448178i
$$554$$ −2.05667e6 −0.284702
$$555$$ 0 0
$$556$$ 898448. 0.123255
$$557$$ 1.25466e7i 1.71351i 0.515724 + 0.856755i $$0.327523\pi$$
−0.515724 + 0.856755i $$0.672477\pi$$
$$558$$ 38880.0i 0.00528617i
$$559$$ −5.09361e6 −0.689439
$$560$$ 0 0
$$561$$ 503496. 0.0675443
$$562$$ − 2.88227e6i − 0.384940i
$$563$$ − 5.15972e6i − 0.686050i −0.939326 0.343025i $$-0.888549\pi$$
0.939326 0.343025i $$-0.111451\pi$$
$$564$$ −501120. −0.0663352
$$565$$ 0 0
$$566$$ 179880. 0.0236016
$$567$$ 321489.i 0.0419961i
$$568$$ − 5.38272e6i − 0.700053i
$$569$$ −1.17452e7 −1.52083 −0.760414 0.649439i $$-0.775004\pi$$
−0.760414 + 0.649439i $$0.775004\pi$$
$$570$$ 0 0
$$571$$ −7.54728e6 −0.968725 −0.484362 0.874867i $$-0.660948\pi$$
−0.484362 + 0.874867i $$0.660948\pi$$
$$572$$ − 784992.i − 0.100317i
$$573$$ − 2.46067e6i − 0.313089i
$$574$$ 2.34083e6 0.296544
$$575$$ 0 0
$$576$$ 2.24467e6 0.281901
$$577$$ 9.28483e6i 1.16101i 0.814258 + 0.580503i $$0.197144\pi$$
−0.814258 + 0.580503i $$0.802856\pi$$
$$578$$ − 8.42389e6i − 1.04880i
$$579$$ 1.38242e6 0.171373
$$580$$ 0 0
$$581$$ 1.96921e6 0.242020
$$582$$ − 7.75775e6i − 0.949354i
$$583$$ 4.25974e6i 0.519053i
$$584$$ 1.03901e7 1.26063
$$585$$ 0 0
$$586$$ 1.18840e6 0.142961
$$587$$ 1.47623e6i 0.176831i 0.996084 + 0.0884155i $$0.0281803\pi$$
−0.996084 + 0.0884155i $$0.971820\pi$$
$$588$$ 86436.0i 0.0103098i
$$589$$ −214720. −0.0255026
$$590$$ 0 0
$$591$$ −1.38980e6 −0.163675
$$592$$ 6.17302e6i 0.723925i
$$593$$ 1.24007e7i 1.44813i 0.689729 + 0.724067i $$0.257730\pi$$
−0.689729 + 0.724067i $$0.742270\pi$$
$$594$$ −1.94206e6 −0.225837
$$595$$ 0 0
$$596$$ −328296. −0.0378573
$$597$$ 3.30170e6i 0.379142i
$$598$$ − 1.11384e7i − 1.27371i
$$599$$ 3.69127e6 0.420348 0.210174 0.977664i $$-0.432597\pi$$
0.210174 + 0.977664i $$0.432597\pi$$
$$600$$ 0 0
$$601$$ 9.12223e6 1.03018 0.515092 0.857135i $$-0.327758\pi$$
0.515092 + 0.857135i $$0.327758\pi$$
$$602$$ 3.38806e6i 0.381030i
$$603$$ 4.80784e6i 0.538464i
$$604$$ 1.14813e6 0.128055
$$605$$ 0 0
$$606$$ −146124. −0.0161637
$$607$$ − 5.67914e6i − 0.625620i −0.949816 0.312810i $$-0.898730\pi$$
0.949816 0.312810i $$-0.101270\pi$$
$$608$$ − 3.86496e6i − 0.424020i
$$609$$ −2.39992e6 −0.262213
$$610$$ 0 0
$$611$$ 6.15264e6 0.666743
$$612$$ − 40824.0i − 0.00440592i
$$613$$ 1.40106e7i 1.50593i 0.658060 + 0.752966i $$0.271377\pi$$
−0.658060 + 0.752966i $$0.728623\pi$$
$$614$$ −6.27382e6 −0.671600
$$615$$ 0 0
$$616$$ 3.65501e6 0.388094
$$617$$ − 253686.i − 0.0268277i −0.999910 0.0134139i $$-0.995730\pi$$
0.999910 0.0134139i $$-0.00426989\pi$$
$$618$$ − 7.11547e6i − 0.749433i
$$619$$ −4.30034e6 −0.451103 −0.225552 0.974231i $$-0.572418\pi$$
−0.225552 + 0.974231i $$0.572418\pi$$
$$620$$ 0 0
$$621$$ −3.06180e6 −0.318602
$$622$$ − 1.10231e7i − 1.14242i
$$623$$ 390726.i 0.0403322i
$$624$$ 4.51901e6 0.464603
$$625$$ 0 0
$$626$$ 2.19296e6 0.223664
$$627$$ − 1.07253e7i − 1.08953i
$$628$$ − 519512.i − 0.0525650i
$$629$$ −684684. −0.0690023
$$630$$ 0 0
$$631$$ 1.04150e7 1.04132 0.520662 0.853763i $$-0.325685\pi$$
0.520662 + 0.853763i $$0.325685\pi$$
$$632$$ − 1.10504e7i − 1.10048i
$$633$$ 4.68220e6i 0.464451i
$$634$$ −170028. −0.0167995
$$635$$ 0 0
$$636$$ 345384. 0.0338579
$$637$$ − 1.06124e6i − 0.103625i
$$638$$ − 1.44975e7i − 1.41007i
$$639$$ −2.59524e6 −0.251435
$$640$$ 0 0
$$641$$ 4.52714e6 0.435190 0.217595 0.976039i $$-0.430179\pi$$
0.217595 + 0.976039i $$0.430179\pi$$
$$642$$ − 6.96146e6i − 0.666596i
$$643$$ − 1.49687e7i − 1.42776i −0.700266 0.713882i $$-0.746935\pi$$
0.700266 0.713882i $$-0.253065\pi$$
$$644$$ −823200. −0.0782151
$$645$$ 0 0
$$646$$ 2.02910e6 0.191304
$$647$$ − 1.73020e7i − 1.62493i −0.583010 0.812465i $$-0.698125\pi$$
0.583010 0.812465i $$-0.301875\pi$$
$$648$$ − 1.10225e6i − 0.103120i
$$649$$ −1.22064e7 −1.13757
$$650$$ 0 0
$$651$$ −35280.0 −0.00326269
$$652$$ 2.22114e6i 0.204624i
$$653$$ − 4.07470e6i − 0.373949i −0.982365 0.186975i $$-0.940132\pi$$
0.982365 0.186975i $$-0.0598683\pi$$
$$654$$ 5.45281e6 0.498513
$$655$$ 0 0
$$656$$ −9.04483e6 −0.820618
$$657$$ − 5.00953e6i − 0.452776i
$$658$$ − 4.09248e6i − 0.368487i
$$659$$ 3.79475e6 0.340384 0.170192 0.985411i $$-0.445561\pi$$
0.170192 + 0.985411i $$0.445561\pi$$
$$660$$ 0 0
$$661$$ 1.64261e7 1.46228 0.731142 0.682225i $$-0.238988\pi$$
0.731142 + 0.682225i $$0.238988\pi$$
$$662$$ − 1.16035e7i − 1.02907i
$$663$$ 501228.i 0.0442845i
$$664$$ −6.75158e6 −0.594272
$$665$$ 0 0
$$666$$ 2.64092e6 0.230712
$$667$$ − 2.28564e7i − 1.98927i
$$668$$ − 174048.i − 0.0150913i
$$669$$ 2.74262e6 0.236920
$$670$$ 0 0
$$671$$ 2.19682e7 1.88360
$$672$$ − 635040.i − 0.0542473i
$$673$$ − 5.50675e6i − 0.468660i −0.972157 0.234330i $$-0.924710\pi$$
0.972157 0.234330i $$-0.0752896\pi$$
$$674$$ −1.13290e7 −0.960602
$$675$$ 0 0
$$676$$ −703716. −0.0592285
$$677$$ 1.83957e7i 1.54257i 0.636488 + 0.771286i $$0.280386\pi$$
−0.636488 + 0.771286i $$0.719614\pi$$
$$678$$ − 1.18879e7i − 0.993185i
$$679$$ 7.03944e6 0.585954
$$680$$ 0 0
$$681$$ −2.59729e6 −0.214612
$$682$$ − 213120.i − 0.0175454i
$$683$$ − 1.75835e6i − 0.144229i −0.997396 0.0721146i $$-0.977025\pi$$
0.997396 0.0721146i $$-0.0229747\pi$$
$$684$$ −869616. −0.0710702
$$685$$ 0 0
$$686$$ −705894. −0.0572703
$$687$$ − 6.94971e6i − 0.561791i
$$688$$ − 1.30913e7i − 1.05441i
$$689$$ −4.24055e6 −0.340309
$$690$$ 0 0
$$691$$ −5.36314e6 −0.427291 −0.213646 0.976911i $$-0.568534\pi$$
−0.213646 + 0.976911i $$0.568534\pi$$
$$692$$ − 73320.0i − 0.00582046i
$$693$$ − 1.76224e6i − 0.139390i
$$694$$ 1.75162e7 1.38052
$$695$$ 0 0
$$696$$ 8.22830e6 0.643854
$$697$$ − 1.00321e6i − 0.0782187i
$$698$$ − 4.68409e6i − 0.363904i
$$699$$ 2.27011e6 0.175733
$$700$$ 0 0
$$701$$ −2.12606e7 −1.63411 −0.817054 0.576561i $$-0.804394\pi$$
−0.817054 + 0.576561i $$0.804394\pi$$
$$702$$ − 1.93331e6i − 0.148067i
$$703$$ 1.45849e7i 1.11305i
$$704$$ −1.23041e7 −0.935662
$$705$$ 0 0
$$706$$ −8.00622e6 −0.604527
$$707$$ − 132594.i − 0.00997643i
$$708$$ 989712.i 0.0742037i
$$709$$ −2.07729e6 −0.155196 −0.0775980 0.996985i $$-0.524725\pi$$
−0.0775980 + 0.996985i $$0.524725\pi$$
$$710$$ 0 0
$$711$$ −5.32786e6 −0.395256
$$712$$ − 1.33963e6i − 0.0990343i
$$713$$ − 336000.i − 0.0247523i
$$714$$ 333396. 0.0244746
$$715$$ 0 0
$$716$$ −613296. −0.0447082
$$717$$ 1.30602e7i 0.948752i
$$718$$ 6.10459e6i 0.441922i
$$719$$ −4.23619e6 −0.305600 −0.152800 0.988257i $$-0.548829\pi$$
−0.152800 + 0.988257i $$0.548829\pi$$
$$720$$ 0 0
$$721$$ 6.45663e6 0.462560
$$722$$ − 2.83665e7i − 2.02518i
$$723$$ − 1.31758e6i − 0.0937415i
$$724$$ 1.52826e6 0.108356
$$725$$ 0 0
$$726$$ 1.94859e6 0.137208
$$727$$ 2.14524e7i 1.50536i 0.658389 + 0.752678i $$0.271238\pi$$
−0.658389 + 0.752678i $$0.728762\pi$$
$$728$$ 3.63854e6i 0.254448i
$$729$$ −531441. −0.0370370
$$730$$ 0 0
$$731$$ 1.45202e6 0.100503
$$732$$ − 1.78121e6i − 0.122867i
$$733$$ 1.48892e7i 1.02355i 0.859118 + 0.511777i $$0.171013\pi$$
−0.859118 + 0.511777i $$0.828987\pi$$
$$734$$ 5.02608e6 0.344341
$$735$$ 0 0
$$736$$ 6.04800e6 0.411545
$$737$$ − 2.63541e7i − 1.78722i
$$738$$ 3.86953e6i 0.261528i
$$739$$ −6.99324e6 −0.471050 −0.235525 0.971868i $$-0.575681\pi$$
−0.235525 + 0.971868i $$0.575681\pi$$
$$740$$ 0 0
$$741$$ 1.06770e7 0.714335
$$742$$ 2.82064e6i 0.188078i
$$743$$ − 1.90428e6i − 0.126549i −0.997996 0.0632745i $$-0.979846\pi$$
0.997996 0.0632745i $$-0.0201544\pi$$
$$744$$ 120960. 0.00801142
$$745$$ 0 0
$$746$$ 9.11958e6 0.599968
$$747$$ 3.25523e6i 0.213442i
$$748$$ 223776.i 0.0146238i
$$749$$ 6.31688e6 0.411432
$$750$$ 0 0
$$751$$ 1.95361e7 1.26398 0.631988 0.774978i $$-0.282239\pi$$
0.631988 + 0.774978i $$0.282239\pi$$
$$752$$ 1.58131e7i 1.01970i
$$753$$ 5.47074e6i 0.351608i
$$754$$ 1.44322e7 0.924493
$$755$$ 0 0
$$756$$ −142884. −0.00909241
$$757$$ 1.25183e6i 0.0793973i 0.999212 + 0.0396986i $$0.0126398\pi$$
−0.999212 + 0.0396986i $$0.987360\pi$$
$$758$$ 1.58679e7i 1.00311i
$$759$$ 1.67832e7 1.05748
$$760$$ 0 0
$$761$$ 2.04472e7 1.27989 0.639944 0.768422i $$-0.278958\pi$$
0.639944 + 0.768422i $$0.278958\pi$$
$$762$$ − 4.01328e6i − 0.250387i
$$763$$ 4.94792e6i 0.307689i
$$764$$ 1.09363e6 0.0677857
$$765$$ 0 0
$$766$$ −1.20802e7 −0.743876
$$767$$ − 1.21515e7i − 0.745831i
$$768$$ 3.48595e6i 0.213264i
$$769$$ −2.21064e6 −0.134804 −0.0674020 0.997726i $$-0.521471\pi$$
−0.0674020 + 0.997726i $$0.521471\pi$$
$$770$$ 0 0
$$771$$ −860274. −0.0521196
$$772$$ 614408.i 0.0371034i
$$773$$ − 1.29151e7i − 0.777405i −0.921363 0.388703i $$-0.872923\pi$$
0.921363 0.388703i $$-0.127077\pi$$
$$774$$ −5.60066e6 −0.336037
$$775$$ 0 0
$$776$$ −2.41352e7 −1.43879
$$777$$ 2.39639e6i 0.142399i
$$778$$ − 4.35740e6i − 0.258095i
$$779$$ −2.13700e7 −1.26171
$$780$$ 0 0
$$781$$ 1.42258e7 0.834541
$$782$$ 3.17520e6i 0.185675i
$$783$$ − 3.96722e6i − 0.231250i
$$784$$ 2.72754e6 0.158482
$$785$$ 0 0
$$786$$ −8.38706e6 −0.484232
$$787$$ − 1.35499e7i − 0.779830i −0.920851 0.389915i $$-0.872504\pi$$
0.920851 0.389915i $$-0.127496\pi$$
$$788$$ − 617688.i − 0.0354367i
$$789$$ −1.98031e7 −1.13251
$$790$$ 0 0
$$791$$ 1.07872e7 0.613007
$$792$$ 6.04195e6i 0.342266i
$$793$$ 2.18693e7i 1.23496i
$$794$$ 2.74547e7 1.54549
$$795$$ 0 0
$$796$$ −1.46742e6 −0.0820867
$$797$$ − 2.45956e7i − 1.37155i −0.727813 0.685776i $$-0.759463\pi$$
0.727813 0.685776i $$-0.240537\pi$$
$$798$$ − 7.10186e6i − 0.394789i
$$799$$ −1.75392e6 −0.0971948
$$800$$ 0 0
$$801$$ −645894. −0.0355697
$$802$$ 203220.i 0.0111566i
$$803$$ 2.74596e7i 1.50282i
$$804$$ −2.13682e6 −0.116581
$$805$$ 0 0
$$806$$ 212160. 0.0115034
$$807$$ − 1.59322e7i − 0.861177i
$$808$$ 454608.i 0.0244968i
$$809$$ −1.55237e7 −0.833920 −0.416960 0.908925i $$-0.636905\pi$$
−0.416960 + 0.908925i $$0.636905\pi$$
$$810$$ 0 0
$$811$$ −2.66262e7 −1.42153 −0.710766 0.703429i $$-0.751651\pi$$
−0.710766 + 0.703429i $$0.751651\pi$$
$$812$$ − 1.06663e6i − 0.0567707i
$$813$$ − 2.01154e6i − 0.106734i
$$814$$ −1.44762e7 −0.765760
$$815$$ 0 0
$$816$$ −1.28822e6 −0.0677276
$$817$$ − 3.09304e7i − 1.62118i
$$818$$ − 3.51707e7i − 1.83780i
$$819$$ 1.75430e6 0.0913889
$$820$$ 0 0
$$821$$ −1.23891e7 −0.641477 −0.320739 0.947168i $$-0.603931\pi$$
−0.320739 + 0.947168i $$0.603931\pi$$
$$822$$ − 1.42693e7i − 0.736585i
$$823$$ 3.65630e6i 0.188166i 0.995564 + 0.0940831i $$0.0299919\pi$$
−0.995564 + 0.0940831i $$0.970008\pi$$
$$824$$ −2.21370e7 −1.13580
$$825$$ 0 0
$$826$$ −8.08265e6 −0.412196
$$827$$ 2.80463e7i 1.42597i 0.701178 + 0.712987i $$0.252658\pi$$
−0.701178 + 0.712987i $$0.747342\pi$$
$$828$$ − 1.36080e6i − 0.0689792i
$$829$$ −2.11153e7 −1.06712 −0.533558 0.845763i $$-0.679145\pi$$
−0.533558 + 0.845763i $$0.679145\pi$$
$$830$$ 0 0
$$831$$ 3.08500e6 0.154972
$$832$$ − 1.22487e7i − 0.613454i
$$833$$ 302526.i 0.0151060i
$$834$$ −1.21290e7 −0.603826
$$835$$ 0 0
$$836$$ 4.76678e6 0.235890
$$837$$ − 58320.0i − 0.00287742i
$$838$$ 1.81649e6i 0.0893557i
$$839$$ −1.33947e7 −0.656944 −0.328472 0.944514i $$-0.606534\pi$$
−0.328472 + 0.944514i $$0.606534\pi$$
$$840$$ 0 0
$$841$$ 9.10422e6 0.443867
$$842$$ 3.22025e7i 1.56534i
$$843$$ 4.32340e6i 0.209535i
$$844$$ −2.08098e6 −0.100557
$$845$$ 0 0
$$846$$ 6.76512e6 0.324975
$$847$$ 1.76816e6i 0.0846865i
$$848$$ − 1.08988e7i − 0.520461i
$$849$$ −269820. −0.0128471
$$850$$ 0 0
$$851$$ −2.28228e7 −1.08030
$$852$$ − 1.15344e6i − 0.0544372i
$$853$$ − 3.01513e7i − 1.41884i −0.704786 0.709420i $$-0.748957\pi$$
0.704786 0.709420i $$-0.251043\pi$$
$$854$$ 1.45465e7 0.682519
$$855$$ 0 0
$$856$$ −2.16579e7 −1.01026
$$857$$ 2.39894e7i 1.11575i 0.829925 + 0.557875i $$0.188383\pi$$
−0.829925 + 0.557875i $$0.811617\pi$$
$$858$$ 1.05974e7i 0.491452i
$$859$$ 8.87576e6 0.410414 0.205207 0.978719i $$-0.434213\pi$$
0.205207 + 0.978719i $$0.434213\pi$$
$$860$$ 0 0
$$861$$ −3.51124e6 −0.161418
$$862$$ − 7.06234e6i − 0.323728i
$$863$$ 8.71286e6i 0.398230i 0.979976 + 0.199115i $$0.0638067\pi$$
−0.979976 + 0.199115i $$0.936193\pi$$
$$864$$ 1.04976e6 0.0478416
$$865$$ 0 0
$$866$$ 2.19750e7 0.995711
$$867$$ 1.26358e7i 0.570895i
$$868$$ − 15680.0i 0 0.000706394i
$$869$$ 2.92045e7 1.31190
$$870$$ 0 0
$$871$$ 2.62354e7 1.17177
$$872$$ − 1.69643e7i − 0.755518i
$$873$$ 1.16366e7i 0.516763i
$$874$$ 6.76368e7 2.99505
$$875$$ 0 0
$$876$$ 2.22646e6 0.0980288
$$877$$ − 2.95788e7i − 1.29862i −0.760524 0.649310i $$-0.775058\pi$$
0.760524 0.649310i $$-0.224942\pi$$
$$878$$ − 1.52205e7i − 0.666333i
$$879$$ −1.78259e6 −0.0778180
$$880$$ 0 0
$$881$$ 2.45670e7 1.06638 0.533190 0.845995i $$-0.320993\pi$$
0.533190 + 0.845995i $$0.320993\pi$$
$$882$$ − 1.16689e6i − 0.0505076i
$$883$$ − 1.45682e7i − 0.628788i −0.949293 0.314394i $$-0.898199\pi$$
0.949293 0.314394i $$-0.101801\pi$$
$$884$$ −222768. −0.00958787
$$885$$ 0 0
$$886$$ −3.60902e7 −1.54456
$$887$$ 1.61714e7i 0.690141i 0.938577 + 0.345070i $$0.112145\pi$$
−0.938577 + 0.345070i $$0.887855\pi$$
$$888$$ − 8.21621e6i − 0.349654i
$$889$$ 3.64168e6 0.154542
$$890$$ 0 0
$$891$$ 2.91308e6 0.122930
$$892$$ 1.21894e6i 0.0512946i
$$893$$ 3.73613e7i 1.56781i
$$894$$ 4.43200e6 0.185462
$$895$$ 0 0
$$896$$ −1.04052e7 −0.432995
$$897$$ 1.67076e7i 0.693319i
$$898$$ 3.39579e7i 1.40524i
$$899$$ 435360. 0.0179659
$$900$$ 0 0
$$901$$ 1.20884e6 0.0496087
$$902$$ − 2.12108e7i − 0.868041i
$$903$$ − 5.08208e6i − 0.207407i
$$904$$ −3.69845e7 −1.50522
$$905$$ 0 0
$$906$$ −1.54997e7 −0.627341
$$907$$ 3.14446e7i 1.26919i 0.772844 + 0.634596i $$0.218833\pi$$
−0.772844 + 0.634596i $$0.781167\pi$$
$$908$$ − 1.15435e6i − 0.0464648i
$$909$$ 219186. 0.00879839
$$910$$ 0 0
$$911$$ 1.51427e7 0.604514 0.302257 0.953227i $$-0.402260\pi$$
0.302257 + 0.953227i $$0.402260\pi$$
$$912$$ 2.74412e7i 1.09249i
$$913$$ − 1.78435e7i − 0.708439i
$$914$$ −3.87695e7 −1.53506
$$915$$ 0 0
$$916$$ 3.08876e6 0.121631
$$917$$ − 7.61048e6i − 0.298875i
$$918$$ 551124.i 0.0215845i
$$919$$ −4.14876e7 −1.62043 −0.810214 0.586134i $$-0.800649\pi$$
−0.810214 + 0.586134i $$0.800649\pi$$
$$920$$ 0 0
$$921$$ 9.41072e6 0.365573
$$922$$ 2.02412e7i 0.784167i
$$923$$ 1.41617e7i 0.547155i
$$924$$ 783216. 0.0301788
$$925$$ 0 0
$$926$$ 2.72985e7 1.04619
$$927$$ 1.06732e7i 0.407939i
$$928$$ 7.83648e6i 0.298711i
$$929$$ 1.78495e7 0.678556 0.339278 0.940686i $$-0.389817\pi$$
0.339278 + 0.940686i $$0.389817\pi$$
$$930$$ 0 0
$$931$$ 6.44428e6 0.243669
$$932$$ 1.00894e6i 0.0380473i
$$933$$ 1.65346e7i 0.621855i
$$934$$ 1.20681e7 0.452661
$$935$$ 0 0
$$936$$ −6.01474e6 −0.224402
$$937$$ 2.96399e7i 1.10288i 0.834215 + 0.551439i $$0.185921\pi$$
−0.834215 + 0.551439i $$0.814079\pi$$
$$938$$ − 1.74507e7i − 0.647597i
$$939$$ −3.28945e6 −0.121747
$$940$$ 0 0
$$941$$ −3.22282e7 −1.18648 −0.593242 0.805024i $$-0.702152\pi$$
−0.593242 + 0.805024i $$0.702152\pi$$
$$942$$ 7.01341e6i 0.257515i
$$943$$ − 3.34404e7i − 1.22459i
$$944$$ 3.12309e7 1.14066
$$945$$ 0 0
$$946$$ 3.06999e7 1.11535
$$947$$ 4.84885e7i 1.75697i 0.477772 + 0.878484i $$0.341444\pi$$
−0.477772 + 0.878484i $$0.658556\pi$$
$$948$$ − 2.36794e6i − 0.0855754i
$$949$$ −2.73359e7 −0.985300
$$950$$ 0 0
$$951$$ 255042. 0.00914451
$$952$$ − 1.03723e6i − 0.0370923i
$$953$$ 2.03264e7i 0.724983i 0.931987 + 0.362491i $$0.118074\pi$$
−0.931987 + 0.362491i $$0.881926\pi$$
$$954$$ −4.66268e6 −0.165869
$$955$$ 0 0
$$956$$ −5.80454e6 −0.205411
$$957$$ 2.17462e7i 0.767546i
$$958$$ − 4.56241e7i − 1.60613i
$$959$$ 1.29481e7 0.454630
$$960$$ 0 0
$$961$$ −2.86228e7 −0.999776
$$962$$ − 1.44110e7i − 0.502060i
$$963$$ 1.04422e7i 0.362849i
$$964$$ 585592. 0.0202956
$$965$$ 0 0
$$966$$ 1.11132e7 0.383174
$$967$$ − 3.66292e6i − 0.125968i −0.998015 0.0629841i $$-0.979938\pi$$
0.998015 0.0629841i $$-0.0200618\pi$$
$$968$$ − 6.06228e6i − 0.207945i
$$969$$ −3.04366e6 −0.104132
$$970$$ 0 0
$$971$$ 1.48741e6 0.0506271 0.0253136 0.999680i $$-0.491942\pi$$
0.0253136 + 0.999680i $$0.491942\pi$$
$$972$$ − 236196.i − 0.00801875i
$$973$$ − 1.10060e7i − 0.372689i
$$974$$ 4.03867e6 0.136408
$$975$$ 0 0
$$976$$ −5.62070e7 −1.88871
$$977$$ 4.07930e7i 1.36725i 0.729831 + 0.683627i $$0.239599\pi$$
−0.729831 + 0.683627i $$0.760401\pi$$
$$978$$ − 2.99853e7i − 1.00245i
$$979$$ 3.54046e6 0.118060
$$980$$ 0 0
$$981$$ −8.17922e6 −0.271356
$$982$$ 1.48302e7i 0.490759i
$$983$$ 9.26326e6i 0.305759i 0.988245 + 0.152880i $$0.0488547\pi$$
−0.988245 + 0.152880i $$0.951145\pi$$
$$984$$ 1.20385e7 0.396357
$$985$$ 0 0
$$986$$ −4.11415e6 −0.134768
$$987$$ 6.13872e6i 0.200579i
$$988$$ 4.74531e6i 0.154658i
$$989$$ 4.84008e7 1.57348
$$990$$ 0 0
$$991$$ −5.22051e7 −1.68861 −0.844303 0.535866i $$-0.819985\pi$$
−0.844303 + 0.535866i $$0.819985\pi$$
$$992$$ 115200.i 0.00371684i
$$993$$ 1.74052e7i 0.560153i
$$994$$ 9.41976e6 0.302394
$$995$$ 0 0
$$996$$ −1.44677e6 −0.0462116
$$997$$ − 1.86609e7i − 0.594560i −0.954790 0.297280i $$-0.903921\pi$$
0.954790 0.297280i $$-0.0960795\pi$$
$$998$$ 3.64891e7i 1.15968i
$$999$$ −3.96139e6 −0.125584
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 525.6.d.b.274.1 2
5.2 odd 4 525.6.a.d.1.1 1
5.3 odd 4 21.6.a.a.1.1 1
5.4 even 2 inner 525.6.d.b.274.2 2
15.8 even 4 63.6.a.d.1.1 1
20.3 even 4 336.6.a.r.1.1 1
35.3 even 12 147.6.e.i.79.1 2
35.13 even 4 147.6.a.b.1.1 1
35.18 odd 12 147.6.e.j.79.1 2
35.23 odd 12 147.6.e.j.67.1 2
35.33 even 12 147.6.e.i.67.1 2
60.23 odd 4 1008.6.a.c.1.1 1
105.83 odd 4 441.6.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.a.a.1.1 1 5.3 odd 4
63.6.a.d.1.1 1 15.8 even 4
147.6.a.b.1.1 1 35.13 even 4
147.6.e.i.67.1 2 35.33 even 12
147.6.e.i.79.1 2 35.3 even 12
147.6.e.j.67.1 2 35.23 odd 12
147.6.e.j.79.1 2 35.18 odd 12
336.6.a.r.1.1 1 20.3 even 4
441.6.a.j.1.1 1 105.83 odd 4
525.6.a.d.1.1 1 5.2 odd 4
525.6.d.b.274.1 2 1.1 even 1 trivial
525.6.d.b.274.2 2 5.4 even 2 inner
1008.6.a.c.1.1 1 60.23 odd 4