# Properties

 Label 525.6.d.d.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: $$\Z[a_1, a_2]$$ 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.d.274.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -9.00000i q^{3} +31.0000 q^{4} -9.00000 q^{6} +49.0000i q^{7} -63.0000i q^{8} -81.0000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -9.00000i q^{3} +31.0000 q^{4} -9.00000 q^{6} +49.0000i q^{7} -63.0000i q^{8} -81.0000 q^{9} -340.000 q^{11} -279.000i q^{12} +454.000i q^{13} +49.0000 q^{14} +929.000 q^{16} +798.000i q^{17} +81.0000i q^{18} -892.000 q^{19} +441.000 q^{21} +340.000i q^{22} -3192.00i q^{23} -567.000 q^{24} +454.000 q^{26} +729.000i q^{27} +1519.00i q^{28} +8242.00 q^{29} -2496.00 q^{31} -2945.00i q^{32} +3060.00i q^{33} +798.000 q^{34} -2511.00 q^{36} -9798.00i q^{37} +892.000i q^{38} +4086.00 q^{39} +19834.0 q^{41} -441.000i q^{42} -17236.0i q^{43} -10540.0 q^{44} -3192.00 q^{46} -8928.00i q^{47} -8361.00i q^{48} -2401.00 q^{49} +7182.00 q^{51} +14074.0i q^{52} +150.000i q^{53} +729.000 q^{54} +3087.00 q^{56} +8028.00i q^{57} -8242.00i q^{58} +42396.0 q^{59} +14758.0 q^{61} +2496.00i q^{62} -3969.00i q^{63} +26783.0 q^{64} +3060.00 q^{66} +1676.00i q^{67} +24738.0i q^{68} -28728.0 q^{69} +14568.0 q^{71} +5103.00i q^{72} +78378.0i q^{73} -9798.00 q^{74} -27652.0 q^{76} -16660.0i q^{77} -4086.00i q^{78} +2272.00 q^{79} +6561.00 q^{81} -19834.0i q^{82} -37764.0i q^{83} +13671.0 q^{84} -17236.0 q^{86} -74178.0i q^{87} +21420.0i q^{88} +117286. q^{89} -22246.0 q^{91} -98952.0i q^{92} +22464.0i q^{93} -8928.00 q^{94} -26505.0 q^{96} -10002.0i q^{97} +2401.00i q^{98} +27540.0 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 62 q^{4} - 18 q^{6} - 162 q^{9}+O(q^{10})$$ 2 * q + 62 * q^4 - 18 * q^6 - 162 * q^9 $$2 q + 62 q^{4} - 18 q^{6} - 162 q^{9} - 680 q^{11} + 98 q^{14} + 1858 q^{16} - 1784 q^{19} + 882 q^{21} - 1134 q^{24} + 908 q^{26} + 16484 q^{29} - 4992 q^{31} + 1596 q^{34} - 5022 q^{36} + 8172 q^{39} + 39668 q^{41} - 21080 q^{44} - 6384 q^{46} - 4802 q^{49} + 14364 q^{51} + 1458 q^{54} + 6174 q^{56} + 84792 q^{59} + 29516 q^{61} + 53566 q^{64} + 6120 q^{66} - 57456 q^{69} + 29136 q^{71} - 19596 q^{74} - 55304 q^{76} + 4544 q^{79} + 13122 q^{81} + 27342 q^{84} - 34472 q^{86} + 234572 q^{89} - 44492 q^{91} - 17856 q^{94} - 53010 q^{96} + 55080 q^{99}+O(q^{100})$$ 2 * q + 62 * q^4 - 18 * q^6 - 162 * q^9 - 680 * q^11 + 98 * q^14 + 1858 * q^16 - 1784 * q^19 + 882 * q^21 - 1134 * q^24 + 908 * q^26 + 16484 * q^29 - 4992 * q^31 + 1596 * q^34 - 5022 * q^36 + 8172 * q^39 + 39668 * q^41 - 21080 * q^44 - 6384 * q^46 - 4802 * q^49 + 14364 * q^51 + 1458 * q^54 + 6174 * q^56 + 84792 * q^59 + 29516 * q^61 + 53566 * q^64 + 6120 * q^66 - 57456 * q^69 + 29136 * q^71 - 19596 * q^74 - 55304 * q^76 + 4544 * q^79 + 13122 * q^81 + 27342 * q^84 - 34472 * q^86 + 234572 * q^89 - 44492 * q^91 - 17856 * q^94 - 53010 * q^96 + 55080 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.176777i −0.996086 0.0883883i $$-0.971828\pi$$
0.996086 0.0883883i $$-0.0281716\pi$$
$$3$$ − 9.00000i − 0.577350i
$$4$$ 31.0000 0.968750
$$5$$ 0 0
$$6$$ −9.00000 −0.102062
$$7$$ 49.0000i 0.377964i
$$8$$ − 63.0000i − 0.348029i
$$9$$ −81.0000 −0.333333
$$10$$ 0 0
$$11$$ −340.000 −0.847222 −0.423611 0.905844i $$-0.639238\pi$$
−0.423611 + 0.905844i $$0.639238\pi$$
$$12$$ − 279.000i − 0.559308i
$$13$$ 454.000i 0.745071i 0.928018 + 0.372535i $$0.121511\pi$$
−0.928018 + 0.372535i $$0.878489\pi$$
$$14$$ 49.0000 0.0668153
$$15$$ 0 0
$$16$$ 929.000 0.907227
$$17$$ 798.000i 0.669700i 0.942271 + 0.334850i $$0.108686\pi$$
−0.942271 + 0.334850i $$0.891314\pi$$
$$18$$ 81.0000i 0.0589256i
$$19$$ −892.000 −0.566867 −0.283433 0.958992i $$-0.591473\pi$$
−0.283433 + 0.958992i $$0.591473\pi$$
$$20$$ 0 0
$$21$$ 441.000 0.218218
$$22$$ 340.000i 0.149769i
$$23$$ − 3192.00i − 1.25818i −0.777332 0.629091i $$-0.783427\pi$$
0.777332 0.629091i $$-0.216573\pi$$
$$24$$ −567.000 −0.200935
$$25$$ 0 0
$$26$$ 454.000 0.131711
$$27$$ 729.000i 0.192450i
$$28$$ 1519.00i 0.366153i
$$29$$ 8242.00 1.81986 0.909929 0.414764i $$-0.136136\pi$$
0.909929 + 0.414764i $$0.136136\pi$$
$$30$$ 0 0
$$31$$ −2496.00 −0.466488 −0.233244 0.972418i $$-0.574934\pi$$
−0.233244 + 0.972418i $$0.574934\pi$$
$$32$$ − 2945.00i − 0.508406i
$$33$$ 3060.00i 0.489144i
$$34$$ 798.000 0.118387
$$35$$ 0 0
$$36$$ −2511.00 −0.322917
$$37$$ − 9798.00i − 1.17661i −0.808639 0.588306i $$-0.799795\pi$$
0.808639 0.588306i $$-0.200205\pi$$
$$38$$ 892.000i 0.100209i
$$39$$ 4086.00 0.430167
$$40$$ 0 0
$$41$$ 19834.0 1.84268 0.921342 0.388754i $$-0.127094\pi$$
0.921342 + 0.388754i $$0.127094\pi$$
$$42$$ − 441.000i − 0.0385758i
$$43$$ − 17236.0i − 1.42156i −0.703414 0.710780i $$-0.748342\pi$$
0.703414 0.710780i $$-0.251658\pi$$
$$44$$ −10540.0 −0.820746
$$45$$ 0 0
$$46$$ −3192.00 −0.222417
$$47$$ − 8928.00i − 0.589535i −0.955569 0.294767i $$-0.904758\pi$$
0.955569 0.294767i $$-0.0952422\pi$$
$$48$$ − 8361.00i − 0.523788i
$$49$$ −2401.00 −0.142857
$$50$$ 0 0
$$51$$ 7182.00 0.386652
$$52$$ 14074.0i 0.721787i
$$53$$ 150.000i 0.00733502i 0.999993 + 0.00366751i $$0.00116741\pi$$
−0.999993 + 0.00366751i $$0.998833\pi$$
$$54$$ 729.000 0.0340207
$$55$$ 0 0
$$56$$ 3087.00 0.131543
$$57$$ 8028.00i 0.327281i
$$58$$ − 8242.00i − 0.321709i
$$59$$ 42396.0 1.58560 0.792802 0.609479i $$-0.208621\pi$$
0.792802 + 0.609479i $$0.208621\pi$$
$$60$$ 0 0
$$61$$ 14758.0 0.507812 0.253906 0.967229i $$-0.418285\pi$$
0.253906 + 0.967229i $$0.418285\pi$$
$$62$$ 2496.00i 0.0824642i
$$63$$ − 3969.00i − 0.125988i
$$64$$ 26783.0 0.817352
$$65$$ 0 0
$$66$$ 3060.00 0.0864692
$$67$$ 1676.00i 0.0456128i 0.999740 + 0.0228064i $$0.00726014\pi$$
−0.999740 + 0.0228064i $$0.992740\pi$$
$$68$$ 24738.0i 0.648772i
$$69$$ −28728.0 −0.726411
$$70$$ 0 0
$$71$$ 14568.0 0.342968 0.171484 0.985187i $$-0.445144\pi$$
0.171484 + 0.985187i $$0.445144\pi$$
$$72$$ 5103.00i 0.116010i
$$73$$ 78378.0i 1.72142i 0.509095 + 0.860710i $$0.329980\pi$$
−0.509095 + 0.860710i $$0.670020\pi$$
$$74$$ −9798.00 −0.207998
$$75$$ 0 0
$$76$$ −27652.0 −0.549152
$$77$$ − 16660.0i − 0.320220i
$$78$$ − 4086.00i − 0.0760435i
$$79$$ 2272.00 0.0409582 0.0204791 0.999790i $$-0.493481\pi$$
0.0204791 + 0.999790i $$0.493481\pi$$
$$80$$ 0 0
$$81$$ 6561.00 0.111111
$$82$$ − 19834.0i − 0.325743i
$$83$$ − 37764.0i − 0.601704i −0.953671 0.300852i $$-0.902729\pi$$
0.953671 0.300852i $$-0.0972710\pi$$
$$84$$ 13671.0 0.211399
$$85$$ 0 0
$$86$$ −17236.0 −0.251299
$$87$$ − 74178.0i − 1.05070i
$$88$$ 21420.0i 0.294858i
$$89$$ 117286. 1.56954 0.784768 0.619790i $$-0.212782\pi$$
0.784768 + 0.619790i $$0.212782\pi$$
$$90$$ 0 0
$$91$$ −22246.0 −0.281610
$$92$$ − 98952.0i − 1.21886i
$$93$$ 22464.0i 0.269327i
$$94$$ −8928.00 −0.104216
$$95$$ 0 0
$$96$$ −26505.0 −0.293528
$$97$$ − 10002.0i − 0.107934i −0.998543 0.0539669i $$-0.982813\pi$$
0.998543 0.0539669i $$-0.0171865\pi$$
$$98$$ 2401.00i 0.0252538i
$$99$$ 27540.0 0.282407
$$100$$ 0 0
$$101$$ −108770. −1.06098 −0.530488 0.847692i $$-0.677991\pi$$
−0.530488 + 0.847692i $$0.677991\pi$$
$$102$$ − 7182.00i − 0.0683510i
$$103$$ − 199192.i − 1.85003i −0.379930 0.925015i $$-0.624052\pi$$
0.379930 0.925015i $$-0.375948\pi$$
$$104$$ 28602.0 0.259306
$$105$$ 0 0
$$106$$ 150.000 0.00129666
$$107$$ 79972.0i 0.675272i 0.941277 + 0.337636i $$0.109627\pi$$
−0.941277 + 0.337636i $$0.890373\pi$$
$$108$$ 22599.0i 0.186436i
$$109$$ 46098.0 0.371634 0.185817 0.982584i $$-0.440507\pi$$
0.185817 + 0.982584i $$0.440507\pi$$
$$110$$ 0 0
$$111$$ −88182.0 −0.679317
$$112$$ 45521.0i 0.342899i
$$113$$ 262706.i 1.93541i 0.252078 + 0.967707i $$0.418886\pi$$
−0.252078 + 0.967707i $$0.581114\pi$$
$$114$$ 8028.00 0.0578556
$$115$$ 0 0
$$116$$ 255502. 1.76299
$$117$$ − 36774.0i − 0.248357i
$$118$$ − 42396.0i − 0.280298i
$$119$$ −39102.0 −0.253123
$$120$$ 0 0
$$121$$ −45451.0 −0.282215
$$122$$ − 14758.0i − 0.0897693i
$$123$$ − 178506.i − 1.06387i
$$124$$ −77376.0 −0.451910
$$125$$ 0 0
$$126$$ −3969.00 −0.0222718
$$127$$ − 196608.i − 1.08166i −0.841131 0.540831i $$-0.818110\pi$$
0.841131 0.540831i $$-0.181890\pi$$
$$128$$ − 121023.i − 0.652894i
$$129$$ −155124. −0.820738
$$130$$ 0 0
$$131$$ −77140.0 −0.392737 −0.196368 0.980530i $$-0.562915\pi$$
−0.196368 + 0.980530i $$0.562915\pi$$
$$132$$ 94860.0i 0.473858i
$$133$$ − 43708.0i − 0.214255i
$$134$$ 1676.00 0.00806329
$$135$$ 0 0
$$136$$ 50274.0 0.233075
$$137$$ − 208170.i − 0.947582i −0.880637 0.473791i $$-0.842885\pi$$
0.880637 0.473791i $$-0.157115\pi$$
$$138$$ 28728.0i 0.128413i
$$139$$ 275580. 1.20979 0.604896 0.796304i $$-0.293215\pi$$
0.604896 + 0.796304i $$0.293215\pi$$
$$140$$ 0 0
$$141$$ −80352.0 −0.340368
$$142$$ − 14568.0i − 0.0606288i
$$143$$ − 154360.i − 0.631240i
$$144$$ −75249.0 −0.302409
$$145$$ 0 0
$$146$$ 78378.0 0.304307
$$147$$ 21609.0i 0.0824786i
$$148$$ − 303738.i − 1.13984i
$$149$$ 296106. 1.09265 0.546326 0.837573i $$-0.316026\pi$$
0.546326 + 0.837573i $$0.316026\pi$$
$$150$$ 0 0
$$151$$ −426472. −1.52212 −0.761059 0.648683i $$-0.775320\pi$$
−0.761059 + 0.648683i $$0.775320\pi$$
$$152$$ 56196.0i 0.197286i
$$153$$ − 64638.0i − 0.223233i
$$154$$ −16660.0 −0.0566074
$$155$$ 0 0
$$156$$ 126666. 0.416724
$$157$$ − 178486.i − 0.577903i −0.957344 0.288952i $$-0.906693\pi$$
0.957344 0.288952i $$-0.0933067\pi$$
$$158$$ − 2272.00i − 0.00724045i
$$159$$ 1350.00 0.00423488
$$160$$ 0 0
$$161$$ 156408. 0.475548
$$162$$ − 6561.00i − 0.0196419i
$$163$$ 252772.i 0.745178i 0.927996 + 0.372589i $$0.121530\pi$$
−0.927996 + 0.372589i $$0.878470\pi$$
$$164$$ 614854. 1.78510
$$165$$ 0 0
$$166$$ −37764.0 −0.106367
$$167$$ − 508088.i − 1.40977i −0.709322 0.704884i $$-0.750999\pi$$
0.709322 0.704884i $$-0.249001\pi$$
$$168$$ − 27783.0i − 0.0759462i
$$169$$ 165177. 0.444870
$$170$$ 0 0
$$171$$ 72252.0 0.188956
$$172$$ − 534316.i − 1.37714i
$$173$$ − 221834.i − 0.563525i −0.959484 0.281762i $$-0.909081\pi$$
0.959484 0.281762i $$-0.0909190\pi$$
$$174$$ −74178.0 −0.185739
$$175$$ 0 0
$$176$$ −315860. −0.768622
$$177$$ − 381564.i − 0.915449i
$$178$$ − 117286.i − 0.277457i
$$179$$ 113564. 0.264916 0.132458 0.991189i $$-0.457713\pi$$
0.132458 + 0.991189i $$0.457713\pi$$
$$180$$ 0 0
$$181$$ 663118. 1.50451 0.752254 0.658873i $$-0.228967\pi$$
0.752254 + 0.658873i $$0.228967\pi$$
$$182$$ 22246.0i 0.0497821i
$$183$$ − 132822.i − 0.293185i
$$184$$ −201096. −0.437884
$$185$$ 0 0
$$186$$ 22464.0 0.0476107
$$187$$ − 271320.i − 0.567385i
$$188$$ − 276768.i − 0.571112i
$$189$$ −35721.0 −0.0727393
$$190$$ 0 0
$$191$$ 505664. 1.00295 0.501474 0.865173i $$-0.332791\pi$$
0.501474 + 0.865173i $$0.332791\pi$$
$$192$$ − 241047.i − 0.471899i
$$193$$ − 432382.i − 0.835554i −0.908550 0.417777i $$-0.862809\pi$$
0.908550 0.417777i $$-0.137191\pi$$
$$194$$ −10002.0 −0.0190802
$$195$$ 0 0
$$196$$ −74431.0 −0.138393
$$197$$ 131962.i 0.242261i 0.992637 + 0.121130i $$0.0386519\pi$$
−0.992637 + 0.121130i $$0.961348\pi$$
$$198$$ − 27540.0i − 0.0499230i
$$199$$ −298536. −0.534397 −0.267199 0.963642i $$-0.586098\pi$$
−0.267199 + 0.963642i $$0.586098\pi$$
$$200$$ 0 0
$$201$$ 15084.0 0.0263346
$$202$$ 108770.i 0.187556i
$$203$$ 403858.i 0.687842i
$$204$$ 222642. 0.374569
$$205$$ 0 0
$$206$$ −199192. −0.327042
$$207$$ 258552.i 0.419394i
$$208$$ 421766.i 0.675948i
$$209$$ 303280. 0.480262
$$210$$ 0 0
$$211$$ −1.17062e6 −1.81013 −0.905065 0.425273i $$-0.860178\pi$$
−0.905065 + 0.425273i $$0.860178\pi$$
$$212$$ 4650.00i 0.00710581i
$$213$$ − 131112.i − 0.198013i
$$214$$ 79972.0 0.119372
$$215$$ 0 0
$$216$$ 45927.0 0.0669782
$$217$$ − 122304.i − 0.176316i
$$218$$ − 46098.0i − 0.0656963i
$$219$$ 705402. 0.993863
$$220$$ 0 0
$$221$$ −362292. −0.498974
$$222$$ 88182.0i 0.120087i
$$223$$ 399376.i 0.537799i 0.963168 + 0.268899i $$0.0866599\pi$$
−0.963168 + 0.268899i $$0.913340\pi$$
$$224$$ 144305. 0.192159
$$225$$ 0 0
$$226$$ 262706. 0.342136
$$227$$ − 707916.i − 0.911837i −0.890022 0.455918i $$-0.849311\pi$$
0.890022 0.455918i $$-0.150689\pi$$
$$228$$ 248868.i 0.317053i
$$229$$ 735778. 0.927167 0.463584 0.886053i $$-0.346563\pi$$
0.463584 + 0.886053i $$0.346563\pi$$
$$230$$ 0 0
$$231$$ −149940. −0.184879
$$232$$ − 519246.i − 0.633364i
$$233$$ − 208758.i − 0.251915i −0.992036 0.125957i $$-0.959800\pi$$
0.992036 0.125957i $$-0.0402002\pi$$
$$234$$ −36774.0 −0.0439037
$$235$$ 0 0
$$236$$ 1.31428e6 1.53605
$$237$$ − 20448.0i − 0.0236472i
$$238$$ 39102.0i 0.0447462i
$$239$$ −713376. −0.807837 −0.403919 0.914795i $$-0.632352\pi$$
−0.403919 + 0.914795i $$0.632352\pi$$
$$240$$ 0 0
$$241$$ −505246. −0.560351 −0.280176 0.959949i $$-0.590393\pi$$
−0.280176 + 0.959949i $$0.590393\pi$$
$$242$$ 45451.0i 0.0498890i
$$243$$ − 59049.0i − 0.0641500i
$$244$$ 457498. 0.491943
$$245$$ 0 0
$$246$$ −178506. −0.188068
$$247$$ − 404968.i − 0.422356i
$$248$$ 157248.i 0.162351i
$$249$$ −339876. −0.347394
$$250$$ 0 0
$$251$$ 317108. 0.317704 0.158852 0.987302i $$-0.449221\pi$$
0.158852 + 0.987302i $$0.449221\pi$$
$$252$$ − 123039.i − 0.122051i
$$253$$ 1.08528e6i 1.06596i
$$254$$ −196608. −0.191213
$$255$$ 0 0
$$256$$ 736033. 0.701936
$$257$$ 1.44285e6i 1.36266i 0.731977 + 0.681329i $$0.238598\pi$$
−0.731977 + 0.681329i $$0.761402\pi$$
$$258$$ 155124.i 0.145087i
$$259$$ 480102. 0.444717
$$260$$ 0 0
$$261$$ −667602. −0.606619
$$262$$ 77140.0i 0.0694267i
$$263$$ 271496.i 0.242033i 0.992651 + 0.121016i $$0.0386153\pi$$
−0.992651 + 0.121016i $$0.961385\pi$$
$$264$$ 192780. 0.170236
$$265$$ 0 0
$$266$$ −43708.0 −0.0378754
$$267$$ − 1.05557e6i − 0.906172i
$$268$$ 51956.0i 0.0441874i
$$269$$ −850614. −0.716724 −0.358362 0.933583i $$-0.616665\pi$$
−0.358362 + 0.933583i $$0.616665\pi$$
$$270$$ 0 0
$$271$$ −540128. −0.446759 −0.223380 0.974732i $$-0.571709\pi$$
−0.223380 + 0.974732i $$0.571709\pi$$
$$272$$ 741342.i 0.607570i
$$273$$ 200214.i 0.162588i
$$274$$ −208170. −0.167510
$$275$$ 0 0
$$276$$ −890568. −0.703711
$$277$$ − 513574.i − 0.402164i −0.979574 0.201082i $$-0.935554\pi$$
0.979574 0.201082i $$-0.0644458\pi$$
$$278$$ − 275580.i − 0.213863i
$$279$$ 202176. 0.155496
$$280$$ 0 0
$$281$$ −1.35642e6 −1.02478 −0.512388 0.858754i $$-0.671239\pi$$
−0.512388 + 0.858754i $$0.671239\pi$$
$$282$$ 80352.0i 0.0601692i
$$283$$ 286756.i 0.212837i 0.994321 + 0.106418i $$0.0339383\pi$$
−0.994321 + 0.106418i $$0.966062\pi$$
$$284$$ 451608. 0.332251
$$285$$ 0 0
$$286$$ −154360. −0.111589
$$287$$ 971866.i 0.696469i
$$288$$ 238545.i 0.169469i
$$289$$ 783053. 0.551501
$$290$$ 0 0
$$291$$ −90018.0 −0.0623156
$$292$$ 2.42972e6i 1.66763i
$$293$$ − 1.70727e6i − 1.16180i −0.813974 0.580901i $$-0.802700\pi$$
0.813974 0.580901i $$-0.197300\pi$$
$$294$$ 21609.0 0.0145803
$$295$$ 0 0
$$296$$ −617274. −0.409495
$$297$$ − 247860.i − 0.163048i
$$298$$ − 296106.i − 0.193155i
$$299$$ 1.44917e6 0.937434
$$300$$ 0 0
$$301$$ 844564. 0.537299
$$302$$ 426472.i 0.269075i
$$303$$ 978930.i 0.612555i
$$304$$ −828668. −0.514276
$$305$$ 0 0
$$306$$ −64638.0 −0.0394625
$$307$$ 546788.i 0.331111i 0.986201 + 0.165555i $$0.0529416\pi$$
−0.986201 + 0.165555i $$0.947058\pi$$
$$308$$ − 516460.i − 0.310213i
$$309$$ −1.79273e6 −1.06812
$$310$$ 0 0
$$311$$ 3.23426e6 1.89616 0.948079 0.318035i $$-0.103023\pi$$
0.948079 + 0.318035i $$0.103023\pi$$
$$312$$ − 257418.i − 0.149711i
$$313$$ 1.81313e6i 1.04609i 0.852306 + 0.523044i $$0.175204\pi$$
−0.852306 + 0.523044i $$0.824796\pi$$
$$314$$ −178486. −0.102160
$$315$$ 0 0
$$316$$ 70432.0 0.0396782
$$317$$ 1.27658e6i 0.713509i 0.934198 + 0.356754i $$0.116117\pi$$
−0.934198 + 0.356754i $$0.883883\pi$$
$$318$$ − 1350.00i 0 0.000748628i
$$319$$ −2.80228e6 −1.54182
$$320$$ 0 0
$$321$$ 719748. 0.389868
$$322$$ − 156408.i − 0.0840658i
$$323$$ − 711816.i − 0.379631i
$$324$$ 203391. 0.107639
$$325$$ 0 0
$$326$$ 252772. 0.131730
$$327$$ − 414882.i − 0.214563i
$$328$$ − 1.24954e6i − 0.641307i
$$329$$ 437472. 0.222823
$$330$$ 0 0
$$331$$ −1.73621e6 −0.871029 −0.435515 0.900182i $$-0.643434\pi$$
−0.435515 + 0.900182i $$0.643434\pi$$
$$332$$ − 1.17068e6i − 0.582901i
$$333$$ 793638.i 0.392204i
$$334$$ −508088. −0.249214
$$335$$ 0 0
$$336$$ 409689. 0.197973
$$337$$ − 2.07215e6i − 0.993907i −0.867777 0.496953i $$-0.834452\pi$$
0.867777 0.496953i $$-0.165548\pi$$
$$338$$ − 165177.i − 0.0786426i
$$339$$ 2.36435e6 1.11741
$$340$$ 0 0
$$341$$ 848640. 0.395219
$$342$$ − 72252.0i − 0.0334029i
$$343$$ − 117649.i − 0.0539949i
$$344$$ −1.08587e6 −0.494744
$$345$$ 0 0
$$346$$ −221834. −0.0996180
$$347$$ 1.65146e6i 0.736282i 0.929770 + 0.368141i $$0.120006\pi$$
−0.929770 + 0.368141i $$0.879994\pi$$
$$348$$ − 2.29952e6i − 1.01786i
$$349$$ −1.26645e6 −0.556578 −0.278289 0.960497i $$-0.589767\pi$$
−0.278289 + 0.960497i $$0.589767\pi$$
$$350$$ 0 0
$$351$$ −330966. −0.143389
$$352$$ 1.00130e6i 0.430732i
$$353$$ 573218.i 0.244840i 0.992478 + 0.122420i $$0.0390656\pi$$
−0.992478 + 0.122420i $$0.960934\pi$$
$$354$$ −381564. −0.161830
$$355$$ 0 0
$$356$$ 3.63587e6 1.52049
$$357$$ 351918.i 0.146141i
$$358$$ − 113564.i − 0.0468310i
$$359$$ −4.46322e6 −1.82773 −0.913866 0.406016i $$-0.866918\pi$$
−0.913866 + 0.406016i $$0.866918\pi$$
$$360$$ 0 0
$$361$$ −1.68044e6 −0.678662
$$362$$ − 663118.i − 0.265962i
$$363$$ 409059.i 0.162937i
$$364$$ −689626. −0.272810
$$365$$ 0 0
$$366$$ −132822. −0.0518283
$$367$$ 4.50797e6i 1.74709i 0.486742 + 0.873546i $$0.338185\pi$$
−0.486742 + 0.873546i $$0.661815\pi$$
$$368$$ − 2.96537e6i − 1.14146i
$$369$$ −1.60655e6 −0.614228
$$370$$ 0 0
$$371$$ −7350.00 −0.00277238
$$372$$ 696384.i 0.260910i
$$373$$ 1.66535e6i 0.619774i 0.950773 + 0.309887i $$0.100291\pi$$
−0.950773 + 0.309887i $$0.899709\pi$$
$$374$$ −271320. −0.100300
$$375$$ 0 0
$$376$$ −562464. −0.205175
$$377$$ 3.74187e6i 1.35592i
$$378$$ 35721.0i 0.0128586i
$$379$$ 2.53232e6 0.905568 0.452784 0.891620i $$-0.350431\pi$$
0.452784 + 0.891620i $$0.350431\pi$$
$$380$$ 0 0
$$381$$ −1.76947e6 −0.624498
$$382$$ − 505664.i − 0.177298i
$$383$$ 796368.i 0.277407i 0.990334 + 0.138703i $$0.0442934\pi$$
−0.990334 + 0.138703i $$0.955707\pi$$
$$384$$ −1.08921e6 −0.376949
$$385$$ 0 0
$$386$$ −432382. −0.147706
$$387$$ 1.39612e6i 0.473853i
$$388$$ − 310062.i − 0.104561i
$$389$$ −1.94799e6 −0.652699 −0.326349 0.945249i $$-0.605819\pi$$
−0.326349 + 0.945249i $$0.605819\pi$$
$$390$$ 0 0
$$391$$ 2.54722e6 0.842605
$$392$$ 151263.i 0.0497184i
$$393$$ 694260.i 0.226747i
$$394$$ 131962. 0.0428261
$$395$$ 0 0
$$396$$ 853740. 0.273582
$$397$$ − 1.08116e6i − 0.344281i −0.985072 0.172140i $$-0.944932\pi$$
0.985072 0.172140i $$-0.0550683\pi$$
$$398$$ 298536.i 0.0944689i
$$399$$ −393372. −0.123700
$$400$$ 0 0
$$401$$ 2.76770e6 0.859524 0.429762 0.902942i $$-0.358598\pi$$
0.429762 + 0.902942i $$0.358598\pi$$
$$402$$ − 15084.0i − 0.00465534i
$$403$$ − 1.13318e6i − 0.347566i
$$404$$ −3.37187e6 −1.02782
$$405$$ 0 0
$$406$$ 403858. 0.121594
$$407$$ 3.33132e6i 0.996851i
$$408$$ − 452466.i − 0.134566i
$$409$$ −2.36350e6 −0.698630 −0.349315 0.937005i $$-0.613586\pi$$
−0.349315 + 0.937005i $$0.613586\pi$$
$$410$$ 0 0
$$411$$ −1.87353e6 −0.547087
$$412$$ − 6.17495e6i − 1.79222i
$$413$$ 2.07740e6i 0.599302i
$$414$$ 258552. 0.0741391
$$415$$ 0 0
$$416$$ 1.33703e6 0.378798
$$417$$ − 2.48022e6i − 0.698474i
$$418$$ − 303280.i − 0.0848991i
$$419$$ 2.98669e6 0.831104 0.415552 0.909569i $$-0.363588\pi$$
0.415552 + 0.909569i $$0.363588\pi$$
$$420$$ 0 0
$$421$$ −3.46331e6 −0.952326 −0.476163 0.879357i $$-0.657973\pi$$
−0.476163 + 0.879357i $$0.657973\pi$$
$$422$$ 1.17062e6i 0.319989i
$$423$$ 723168.i 0.196512i
$$424$$ 9450.00 0.00255280
$$425$$ 0 0
$$426$$ −131112. −0.0350041
$$427$$ 723142.i 0.191935i
$$428$$ 2.47913e6i 0.654169i
$$429$$ −1.38924e6 −0.364447
$$430$$ 0 0
$$431$$ 2.33693e6 0.605971 0.302986 0.952995i $$-0.402017\pi$$
0.302986 + 0.952995i $$0.402017\pi$$
$$432$$ 677241.i 0.174596i
$$433$$ − 3.50838e6i − 0.899264i −0.893214 0.449632i $$-0.851555\pi$$
0.893214 0.449632i $$-0.148445\pi$$
$$434$$ −122304. −0.0311685
$$435$$ 0 0
$$436$$ 1.42904e6 0.360021
$$437$$ 2.84726e6i 0.713221i
$$438$$ − 705402.i − 0.175692i
$$439$$ −3.54833e6 −0.878744 −0.439372 0.898305i $$-0.644799\pi$$
−0.439372 + 0.898305i $$0.644799\pi$$
$$440$$ 0 0
$$441$$ 194481. 0.0476190
$$442$$ 362292.i 0.0882070i
$$443$$ 1.76833e6i 0.428109i 0.976822 + 0.214055i $$0.0686670\pi$$
−0.976822 + 0.214055i $$0.931333\pi$$
$$444$$ −2.73364e6 −0.658088
$$445$$ 0 0
$$446$$ 399376. 0.0950703
$$447$$ − 2.66495e6i − 0.630842i
$$448$$ 1.31237e6i 0.308930i
$$449$$ 5.52579e6 1.29354 0.646768 0.762687i $$-0.276120\pi$$
0.646768 + 0.762687i $$0.276120\pi$$
$$450$$ 0 0
$$451$$ −6.74356e6 −1.56116
$$452$$ 8.14389e6i 1.87493i
$$453$$ 3.83825e6i 0.878795i
$$454$$ −707916. −0.161191
$$455$$ 0 0
$$456$$ 505764. 0.113903
$$457$$ 2.96226e6i 0.663488i 0.943369 + 0.331744i $$0.107637\pi$$
−0.943369 + 0.331744i $$0.892363\pi$$
$$458$$ − 735778.i − 0.163902i
$$459$$ −581742. −0.128884
$$460$$ 0 0
$$461$$ 2.11884e6 0.464350 0.232175 0.972674i $$-0.425416\pi$$
0.232175 + 0.972674i $$0.425416\pi$$
$$462$$ 149940.i 0.0326823i
$$463$$ 3.19226e6i 0.692062i 0.938223 + 0.346031i $$0.112471\pi$$
−0.938223 + 0.346031i $$0.887529\pi$$
$$464$$ 7.65682e6 1.65102
$$465$$ 0 0
$$466$$ −208758. −0.0445326
$$467$$ 7.42621e6i 1.57571i 0.615863 + 0.787853i $$0.288807\pi$$
−0.615863 + 0.787853i $$0.711193\pi$$
$$468$$ − 1.13999e6i − 0.240596i
$$469$$ −82124.0 −0.0172400
$$470$$ 0 0
$$471$$ −1.60637e6 −0.333653
$$472$$ − 2.67095e6i − 0.551837i
$$473$$ 5.86024e6i 1.20438i
$$474$$ −20448.0 −0.00418028
$$475$$ 0 0
$$476$$ −1.21216e6 −0.245213
$$477$$ − 12150.0i − 0.00244501i
$$478$$ 713376.i 0.142807i
$$479$$ 3.39685e6 0.676453 0.338226 0.941065i $$-0.390173\pi$$
0.338226 + 0.941065i $$0.390173\pi$$
$$480$$ 0 0
$$481$$ 4.44829e6 0.876659
$$482$$ 505246.i 0.0990570i
$$483$$ − 1.40767e6i − 0.274558i
$$484$$ −1.40898e6 −0.273396
$$485$$ 0 0
$$486$$ −59049.0 −0.0113402
$$487$$ 3.71382e6i 0.709574i 0.934947 + 0.354787i $$0.115447\pi$$
−0.934947 + 0.354787i $$0.884553\pi$$
$$488$$ − 929754.i − 0.176733i
$$489$$ 2.27495e6 0.430229
$$490$$ 0 0
$$491$$ 5.57494e6 1.04361 0.521803 0.853066i $$-0.325260\pi$$
0.521803 + 0.853066i $$0.325260\pi$$
$$492$$ − 5.53369e6i − 1.03063i
$$493$$ 6.57712e6i 1.21876i
$$494$$ −404968. −0.0746626
$$495$$ 0 0
$$496$$ −2.31878e6 −0.423210
$$497$$ 713832.i 0.129630i
$$498$$ 339876.i 0.0614111i
$$499$$ −3.92698e6 −0.706004 −0.353002 0.935623i $$-0.614839\pi$$
−0.353002 + 0.935623i $$0.614839\pi$$
$$500$$ 0 0
$$501$$ −4.57279e6 −0.813930
$$502$$ − 317108.i − 0.0561627i
$$503$$ 6.42079e6i 1.13154i 0.824564 + 0.565768i $$0.191420\pi$$
−0.824564 + 0.565768i $$0.808580\pi$$
$$504$$ −250047. −0.0438475
$$505$$ 0 0
$$506$$ 1.08528e6 0.188437
$$507$$ − 1.48659e6i − 0.256846i
$$508$$ − 6.09485e6i − 1.04786i
$$509$$ −146278. −0.0250256 −0.0125128 0.999922i $$-0.503983\pi$$
−0.0125128 + 0.999922i $$0.503983\pi$$
$$510$$ 0 0
$$511$$ −3.84052e6 −0.650636
$$512$$ − 4.60877e6i − 0.776980i
$$513$$ − 650268.i − 0.109094i
$$514$$ 1.44285e6 0.240886
$$515$$ 0 0
$$516$$ −4.80884e6 −0.795090
$$517$$ 3.03552e6i 0.499467i
$$518$$ − 480102.i − 0.0786157i
$$519$$ −1.99651e6 −0.325351
$$520$$ 0 0
$$521$$ 7.70937e6 1.24430 0.622149 0.782899i $$-0.286260\pi$$
0.622149 + 0.782899i $$0.286260\pi$$
$$522$$ 667602.i 0.107236i
$$523$$ − 569420.i − 0.0910287i −0.998964 0.0455144i $$-0.985507\pi$$
0.998964 0.0455144i $$-0.0144927\pi$$
$$524$$ −2.39134e6 −0.380464
$$525$$ 0 0
$$526$$ 271496. 0.0427857
$$527$$ − 1.99181e6i − 0.312407i
$$528$$ 2.84274e6i 0.443764i
$$529$$ −3.75252e6 −0.583021
$$530$$ 0 0
$$531$$ −3.43408e6 −0.528535
$$532$$ − 1.35495e6i − 0.207560i
$$533$$ 9.00464e6i 1.37293i
$$534$$ −1.05557e6 −0.160190
$$535$$ 0 0
$$536$$ 105588. 0.0158746
$$537$$ − 1.02208e6i − 0.152949i
$$538$$ 850614.i 0.126700i
$$539$$ 816340. 0.121032
$$540$$ 0 0
$$541$$ −9.44802e6 −1.38787 −0.693933 0.720040i $$-0.744124\pi$$
−0.693933 + 0.720040i $$0.744124\pi$$
$$542$$ 540128.i 0.0789766i
$$543$$ − 5.96806e6i − 0.868628i
$$544$$ 2.35011e6 0.340479
$$545$$ 0 0
$$546$$ 200214. 0.0287417
$$547$$ 1.35321e6i 0.193374i 0.995315 + 0.0966869i $$0.0308245\pi$$
−0.995315 + 0.0966869i $$0.969175\pi$$
$$548$$ − 6.45327e6i − 0.917970i
$$549$$ −1.19540e6 −0.169271
$$550$$ 0 0
$$551$$ −7.35186e6 −1.03162
$$552$$ 1.80986e6i 0.252812i
$$553$$ 111328.i 0.0154807i
$$554$$ −513574. −0.0710933
$$555$$ 0 0
$$556$$ 8.54298e6 1.17199
$$557$$ − 8.19390e6i − 1.11906i −0.828811 0.559529i $$-0.810982\pi$$
0.828811 0.559529i $$-0.189018\pi$$
$$558$$ − 202176.i − 0.0274881i
$$559$$ 7.82514e6 1.05916
$$560$$ 0 0
$$561$$ −2.44188e6 −0.327580
$$562$$ 1.35642e6i 0.181157i
$$563$$ − 1.05796e7i − 1.40669i −0.710847 0.703347i $$-0.751688\pi$$
0.710847 0.703347i $$-0.248312\pi$$
$$564$$ −2.49091e6 −0.329732
$$565$$ 0 0
$$566$$ 286756. 0.0376246
$$567$$ 321489.i 0.0419961i
$$568$$ − 917784.i − 0.119363i
$$569$$ 1.20205e7 1.55648 0.778238 0.627969i $$-0.216114\pi$$
0.778238 + 0.627969i $$0.216114\pi$$
$$570$$ 0 0
$$571$$ −2.48948e6 −0.319534 −0.159767 0.987155i $$-0.551074\pi$$
−0.159767 + 0.987155i $$0.551074\pi$$
$$572$$ − 4.78516e6i − 0.611514i
$$573$$ − 4.55098e6i − 0.579053i
$$574$$ 971866. 0.123119
$$575$$ 0 0
$$576$$ −2.16942e6 −0.272451
$$577$$ − 8.21322e6i − 1.02701i −0.858087 0.513504i $$-0.828347\pi$$
0.858087 0.513504i $$-0.171653\pi$$
$$578$$ − 783053.i − 0.0974926i
$$579$$ −3.89144e6 −0.482407
$$580$$ 0 0
$$581$$ 1.85044e6 0.227423
$$582$$ 90018.0i 0.0110159i
$$583$$ − 51000.0i − 0.00621439i
$$584$$ 4.93781e6 0.599105
$$585$$ 0 0
$$586$$ −1.70727e6 −0.205380
$$587$$ 1.21827e6i 0.145931i 0.997334 + 0.0729655i $$0.0232463\pi$$
−0.997334 + 0.0729655i $$0.976754\pi$$
$$588$$ 669879.i 0.0799012i
$$589$$ 2.22643e6 0.264436
$$590$$ 0 0
$$591$$ 1.18766e6 0.139869
$$592$$ − 9.10234e6i − 1.06745i
$$593$$ − 8.42379e6i − 0.983718i −0.870675 0.491859i $$-0.836317\pi$$
0.870675 0.491859i $$-0.163683\pi$$
$$594$$ −247860. −0.0288231
$$595$$ 0 0
$$596$$ 9.17929e6 1.05851
$$597$$ 2.68682e6i 0.308534i
$$598$$ − 1.44917e6i − 0.165717i
$$599$$ −8.21254e6 −0.935212 −0.467606 0.883937i $$-0.654883\pi$$
−0.467606 + 0.883937i $$0.654883\pi$$
$$600$$ 0 0
$$601$$ 3.25478e6 0.367566 0.183783 0.982967i $$-0.441166\pi$$
0.183783 + 0.982967i $$0.441166\pi$$
$$602$$ − 844564.i − 0.0949820i
$$603$$ − 135756.i − 0.0152043i
$$604$$ −1.32206e7 −1.47455
$$605$$ 0 0
$$606$$ 978930. 0.108285
$$607$$ − 7.82101e6i − 0.861571i −0.902454 0.430785i $$-0.858237\pi$$
0.902454 0.430785i $$-0.141763\pi$$
$$608$$ 2.62694e6i 0.288198i
$$609$$ 3.63472e6 0.397126
$$610$$ 0 0
$$611$$ 4.05331e6 0.439245
$$612$$ − 2.00378e6i − 0.216257i
$$613$$ − 9.51670e6i − 1.02290i −0.859312 0.511452i $$-0.829108\pi$$
0.859312 0.511452i $$-0.170892\pi$$
$$614$$ 546788. 0.0585326
$$615$$ 0 0
$$616$$ −1.04958e6 −0.111446
$$617$$ 7.04895e6i 0.745438i 0.927944 + 0.372719i $$0.121574\pi$$
−0.927944 + 0.372719i $$0.878426\pi$$
$$618$$ 1.79273e6i 0.188818i
$$619$$ 6.32174e6 0.663147 0.331574 0.943429i $$-0.392420\pi$$
0.331574 + 0.943429i $$0.392420\pi$$
$$620$$ 0 0
$$621$$ 2.32697e6 0.242137
$$622$$ − 3.23426e6i − 0.335197i
$$623$$ 5.74701e6i 0.593229i
$$624$$ 3.79589e6 0.390259
$$625$$ 0 0
$$626$$ 1.81313e6 0.184924
$$627$$ − 2.72952e6i − 0.277279i
$$628$$ − 5.53307e6i − 0.559844i
$$629$$ 7.81880e6 0.787977
$$630$$ 0 0
$$631$$ 8.61236e6 0.861090 0.430545 0.902569i $$-0.358321\pi$$
0.430545 + 0.902569i $$0.358321\pi$$
$$632$$ − 143136.i − 0.0142546i
$$633$$ 1.05356e7i 1.04508i
$$634$$ 1.27658e6 0.126132
$$635$$ 0 0
$$636$$ 41850.0 0.00410254
$$637$$ − 1.09005e6i − 0.106439i
$$638$$ 2.80228e6i 0.272559i
$$639$$ −1.18001e6 −0.114323
$$640$$ 0 0
$$641$$ −5.22829e6 −0.502590 −0.251295 0.967910i $$-0.580857\pi$$
−0.251295 + 0.967910i $$0.580857\pi$$
$$642$$ − 719748.i − 0.0689196i
$$643$$ 1.61373e7i 1.53923i 0.638508 + 0.769615i $$0.279552\pi$$
−0.638508 + 0.769615i $$0.720448\pi$$
$$644$$ 4.84865e6 0.460687
$$645$$ 0 0
$$646$$ −711816. −0.0671099
$$647$$ 1.58749e7i 1.49090i 0.666560 + 0.745451i $$0.267766\pi$$
−0.666560 + 0.745451i $$0.732234\pi$$
$$648$$ − 413343.i − 0.0386699i
$$649$$ −1.44146e7 −1.34336
$$650$$ 0 0
$$651$$ −1.10074e6 −0.101796
$$652$$ 7.83593e6i 0.721891i
$$653$$ − 5.94112e6i − 0.545237i −0.962122 0.272619i $$-0.912110\pi$$
0.962122 0.272619i $$-0.0878897\pi$$
$$654$$ −414882. −0.0379298
$$655$$ 0 0
$$656$$ 1.84258e7 1.67173
$$657$$ − 6.34862e6i − 0.573807i
$$658$$ − 437472.i − 0.0393900i
$$659$$ 7.64430e6 0.685684 0.342842 0.939393i $$-0.388610\pi$$
0.342842 + 0.939393i $$0.388610\pi$$
$$660$$ 0 0
$$661$$ −7.58688e6 −0.675398 −0.337699 0.941254i $$-0.609649\pi$$
−0.337699 + 0.941254i $$0.609649\pi$$
$$662$$ 1.73621e6i 0.153978i
$$663$$ 3.26063e6i 0.288083i
$$664$$ −2.37913e6 −0.209410
$$665$$ 0 0
$$666$$ 793638. 0.0693325
$$667$$ − 2.63085e7i − 2.28971i
$$668$$ − 1.57507e7i − 1.36571i
$$669$$ 3.59438e6 0.310498
$$670$$ 0 0
$$671$$ −5.01772e6 −0.430229
$$672$$ − 1.29874e6i − 0.110943i
$$673$$ − 2.06681e7i − 1.75899i −0.475910 0.879494i $$-0.657881\pi$$
0.475910 0.879494i $$-0.342119\pi$$
$$674$$ −2.07215e6 −0.175700
$$675$$ 0 0
$$676$$ 5.12049e6 0.430968
$$677$$ − 7.89541e6i − 0.662068i −0.943619 0.331034i $$-0.892602\pi$$
0.943619 0.331034i $$-0.107398\pi$$
$$678$$ − 2.36435e6i − 0.197532i
$$679$$ 490098. 0.0407951
$$680$$ 0 0
$$681$$ −6.37124e6 −0.526449
$$682$$ − 848640.i − 0.0698655i
$$683$$ − 1.96015e7i − 1.60782i −0.594750 0.803911i $$-0.702749\pi$$
0.594750 0.803911i $$-0.297251\pi$$
$$684$$ 2.23981e6 0.183051
$$685$$ 0 0
$$686$$ −117649. −0.00954504
$$687$$ − 6.62200e6i − 0.535300i
$$688$$ − 1.60122e7i − 1.28968i
$$689$$ −68100.0 −0.00546511
$$690$$ 0 0
$$691$$ −1.72710e7 −1.37601 −0.688005 0.725706i $$-0.741513\pi$$
−0.688005 + 0.725706i $$0.741513\pi$$
$$692$$ − 6.87685e6i − 0.545914i
$$693$$ 1.34946e6i 0.106740i
$$694$$ 1.65146e6 0.130158
$$695$$ 0 0
$$696$$ −4.67321e6 −0.365673
$$697$$ 1.58275e7i 1.23405i
$$698$$ 1.26645e6i 0.0983900i
$$699$$ −1.87882e6 −0.145443
$$700$$ 0 0
$$701$$ −5.36344e6 −0.412238 −0.206119 0.978527i $$-0.566083\pi$$
−0.206119 + 0.978527i $$0.566083\pi$$
$$702$$ 330966.i 0.0253478i
$$703$$ 8.73982e6i 0.666982i
$$704$$ −9.10622e6 −0.692479
$$705$$ 0 0
$$706$$ 573218. 0.0432821
$$707$$ − 5.32973e6i − 0.401011i
$$708$$ − 1.18285e7i − 0.886841i
$$709$$ 1.73733e7 1.29798 0.648988 0.760798i $$-0.275192\pi$$
0.648988 + 0.760798i $$0.275192\pi$$
$$710$$ 0 0
$$711$$ −184032. −0.0136527
$$712$$ − 7.38902e6i − 0.546244i
$$713$$ 7.96723e6i 0.586926i
$$714$$ 351918. 0.0258343
$$715$$ 0 0
$$716$$ 3.52048e6 0.256637
$$717$$ 6.42038e6i 0.466405i
$$718$$ 4.46322e6i 0.323100i
$$719$$ −424608. −0.0306313 −0.0153157 0.999883i $$-0.504875\pi$$
−0.0153157 + 0.999883i $$0.504875\pi$$
$$720$$ 0 0
$$721$$ 9.76041e6 0.699246
$$722$$ 1.68044e6i 0.119972i
$$723$$ 4.54721e6i 0.323519i
$$724$$ 2.05567e7 1.45749
$$725$$ 0 0
$$726$$ 409059. 0.0288034
$$727$$ − 2.18290e7i − 1.53179i −0.642968 0.765893i $$-0.722297\pi$$
0.642968 0.765893i $$-0.277703\pi$$
$$728$$ 1.40150e6i 0.0980086i
$$729$$ −531441. −0.0370370
$$730$$ 0 0
$$731$$ 1.37543e7 0.952020
$$732$$ − 4.11748e6i − 0.284023i
$$733$$ 2.17675e7i 1.49640i 0.663470 + 0.748202i $$0.269083\pi$$
−0.663470 + 0.748202i $$0.730917\pi$$
$$734$$ 4.50797e6 0.308845
$$735$$ 0 0
$$736$$ −9.40044e6 −0.639667
$$737$$ − 569840.i − 0.0386442i
$$738$$ 1.60655e6i 0.108581i
$$739$$ −6.21786e6 −0.418822 −0.209411 0.977828i $$-0.567155\pi$$
−0.209411 + 0.977828i $$0.567155\pi$$
$$740$$ 0 0
$$741$$ −3.64471e6 −0.243847
$$742$$ 7350.00i 0 0.000490092i
$$743$$ 3.77647e6i 0.250966i 0.992096 + 0.125483i $$0.0400480\pi$$
−0.992096 + 0.125483i $$0.959952\pi$$
$$744$$ 1.41523e6 0.0937336
$$745$$ 0 0
$$746$$ 1.66535e6 0.109562
$$747$$ 3.05888e6i 0.200568i
$$748$$ − 8.41092e6i − 0.549654i
$$749$$ −3.91863e6 −0.255229
$$750$$ 0 0
$$751$$ −2.88795e6 −0.186849 −0.0934244 0.995626i $$-0.529781\pi$$
−0.0934244 + 0.995626i $$0.529781\pi$$
$$752$$ − 8.29411e6i − 0.534842i
$$753$$ − 2.85397e6i − 0.183427i
$$754$$ 3.74187e6 0.239696
$$755$$ 0 0
$$756$$ −1.10735e6 −0.0704662
$$757$$ − 1.25519e6i − 0.0796104i −0.999207 0.0398052i $$-0.987326\pi$$
0.999207 0.0398052i $$-0.0126737\pi$$
$$758$$ − 2.53232e6i − 0.160083i
$$759$$ 9.76752e6 0.615432
$$760$$ 0 0
$$761$$ −1.42623e7 −0.892746 −0.446373 0.894847i $$-0.647284\pi$$
−0.446373 + 0.894847i $$0.647284\pi$$
$$762$$ 1.76947e6i 0.110397i
$$763$$ 2.25880e6i 0.140465i
$$764$$ 1.56756e7 0.971606
$$765$$ 0 0
$$766$$ 796368. 0.0490390
$$767$$ 1.92478e7i 1.18139i
$$768$$ − 6.62430e6i − 0.405263i
$$769$$ 2.02261e7 1.23338 0.616689 0.787207i $$-0.288474\pi$$
0.616689 + 0.787207i $$0.288474\pi$$
$$770$$ 0 0
$$771$$ 1.29856e7 0.786732
$$772$$ − 1.34038e7i − 0.809443i
$$773$$ 2.62288e7i 1.57881i 0.613872 + 0.789406i $$0.289611\pi$$
−0.613872 + 0.789406i $$0.710389\pi$$
$$774$$ 1.39612e6 0.0837663
$$775$$ 0 0
$$776$$ −630126. −0.0375641
$$777$$ − 4.32092e6i − 0.256758i
$$778$$ 1.94799e6i 0.115382i
$$779$$ −1.76919e7 −1.04456
$$780$$ 0 0
$$781$$ −4.95312e6 −0.290570
$$782$$ − 2.54722e6i − 0.148953i
$$783$$ 6.00842e6i 0.350232i
$$784$$ −2.23053e6 −0.129604
$$785$$ 0 0
$$786$$ 694260. 0.0400835
$$787$$ 9.92829e6i 0.571397i 0.958320 + 0.285698i $$0.0922255\pi$$
−0.958320 + 0.285698i $$0.907774\pi$$
$$788$$ 4.09082e6i 0.234690i
$$789$$ 2.44346e6 0.139738
$$790$$ 0 0
$$791$$ −1.28726e7 −0.731518
$$792$$ − 1.73502e6i − 0.0982860i
$$793$$ 6.70013e6i 0.378356i
$$794$$ −1.08116e6 −0.0608608
$$795$$ 0 0
$$796$$ −9.25462e6 −0.517697
$$797$$ − 1.09033e7i − 0.608014i −0.952670 0.304007i $$-0.901675\pi$$
0.952670 0.304007i $$-0.0983246\pi$$
$$798$$ 393372.i 0.0218674i
$$799$$ 7.12454e6 0.394812
$$800$$ 0 0
$$801$$ −9.50017e6 −0.523179
$$802$$ − 2.76770e6i − 0.151944i
$$803$$ − 2.66485e7i − 1.45843i
$$804$$ 467604. 0.0255116
$$805$$ 0 0
$$806$$ −1.13318e6 −0.0614416
$$807$$ 7.65553e6i 0.413801i
$$808$$ 6.85251e6i 0.369251i
$$809$$ −6.06398e6 −0.325751 −0.162876 0.986647i $$-0.552077\pi$$
−0.162876 + 0.986647i $$0.552077\pi$$
$$810$$ 0 0
$$811$$ −8.59438e6 −0.458841 −0.229421 0.973327i $$-0.573683\pi$$
−0.229421 + 0.973327i $$0.573683\pi$$
$$812$$ 1.25196e7i 0.666347i
$$813$$ 4.86115e6i 0.257937i
$$814$$ 3.33132e6 0.176220
$$815$$ 0 0
$$816$$ 6.67208e6 0.350781
$$817$$ 1.53745e7i 0.805835i
$$818$$ 2.36350e6i 0.123501i
$$819$$ 1.80193e6 0.0938701
$$820$$ 0 0
$$821$$ −2.01396e6 −0.104278 −0.0521391 0.998640i $$-0.516604\pi$$
−0.0521391 + 0.998640i $$0.516604\pi$$
$$822$$ 1.87353e6i 0.0967122i
$$823$$ − 2.64679e7i − 1.36213i −0.732221 0.681067i $$-0.761516\pi$$
0.732221 0.681067i $$-0.238484\pi$$
$$824$$ −1.25491e7 −0.643864
$$825$$ 0 0
$$826$$ 2.07740e6 0.105943
$$827$$ 3.90229e6i 0.198407i 0.995067 + 0.0992033i $$0.0316294\pi$$
−0.995067 + 0.0992033i $$0.968371\pi$$
$$828$$ 8.01511e6i 0.406288i
$$829$$ 1.95595e7 0.988487 0.494244 0.869323i $$-0.335445\pi$$
0.494244 + 0.869323i $$0.335445\pi$$
$$830$$ 0 0
$$831$$ −4.62217e6 −0.232190
$$832$$ 1.21595e7i 0.608985i
$$833$$ − 1.91600e6i − 0.0956715i
$$834$$ −2.48022e6 −0.123474
$$835$$ 0 0
$$836$$ 9.40168e6 0.465254
$$837$$ − 1.81958e6i − 0.0897756i
$$838$$ − 2.98669e6i − 0.146920i
$$839$$ 2.45448e7 1.20380 0.601901 0.798570i $$-0.294410\pi$$
0.601901 + 0.798570i $$0.294410\pi$$
$$840$$ 0 0
$$841$$ 4.74194e7 2.31188
$$842$$ 3.46331e6i 0.168349i
$$843$$ 1.22078e7i 0.591655i
$$844$$ −3.62892e7 −1.75356
$$845$$ 0 0
$$846$$ 723168. 0.0347387
$$847$$ − 2.22710e6i − 0.106667i
$$848$$ 139350.i 0.00665453i
$$849$$ 2.58080e6 0.122881
$$850$$ 0 0
$$851$$ −3.12752e7 −1.48039
$$852$$ − 4.06447e6i − 0.191825i
$$853$$ 3.38305e7i 1.59197i 0.605314 + 0.795987i $$0.293048\pi$$
−0.605314 + 0.795987i $$0.706952\pi$$
$$854$$ 723142. 0.0339296
$$855$$ 0 0
$$856$$ 5.03824e6 0.235014
$$857$$ − 3.18009e7i − 1.47907i −0.673120 0.739534i $$-0.735046\pi$$
0.673120 0.739534i $$-0.264954\pi$$
$$858$$ 1.38924e6i 0.0644257i
$$859$$ −638420. −0.0295205 −0.0147602 0.999891i $$-0.504699\pi$$
−0.0147602 + 0.999891i $$0.504699\pi$$
$$860$$ 0 0
$$861$$ 8.74679e6 0.402106
$$862$$ − 2.33693e6i − 0.107122i
$$863$$ − 4.22256e6i − 0.192996i −0.995333 0.0964981i $$-0.969236\pi$$
0.995333 0.0964981i $$-0.0307642\pi$$
$$864$$ 2.14690e6 0.0978427
$$865$$ 0 0
$$866$$ −3.50838e6 −0.158969
$$867$$ − 7.04748e6i − 0.318409i
$$868$$ − 3.79142e6i − 0.170806i
$$869$$ −772480. −0.0347007
$$870$$ 0 0
$$871$$ −760904. −0.0339848
$$872$$ − 2.90417e6i − 0.129340i
$$873$$ 810162.i 0.0359779i
$$874$$ 2.84726e6 0.126081
$$875$$ 0 0
$$876$$ 2.18675e7 0.962805
$$877$$ 2.45043e7i 1.07583i 0.842999 + 0.537915i $$0.180788\pi$$
−0.842999 + 0.537915i $$0.819212\pi$$
$$878$$ 3.54833e6i 0.155341i
$$879$$ −1.53654e7 −0.670767
$$880$$ 0 0
$$881$$ −2.77630e7 −1.20511 −0.602555 0.798078i $$-0.705850\pi$$
−0.602555 + 0.798078i $$0.705850\pi$$
$$882$$ − 194481.i − 0.00841794i
$$883$$ 3.30170e7i 1.42507i 0.701638 + 0.712534i $$0.252452\pi$$
−0.701638 + 0.712534i $$0.747548\pi$$
$$884$$ −1.12311e7 −0.483381
$$885$$ 0 0
$$886$$ 1.76833e6 0.0756797
$$887$$ − 4.34462e6i − 0.185414i −0.995693 0.0927070i $$-0.970448\pi$$
0.995693 0.0927070i $$-0.0295520\pi$$
$$888$$ 5.55547e6i 0.236422i
$$889$$ 9.63379e6 0.408830
$$890$$ 0 0
$$891$$ −2.23074e6 −0.0941358
$$892$$ 1.23807e7i 0.520993i
$$893$$ 7.96378e6i 0.334188i
$$894$$ −2.66495e6 −0.111518
$$895$$ 0 0
$$896$$ 5.93013e6 0.246771
$$897$$ − 1.30425e7i − 0.541228i
$$898$$ − 5.52579e6i − 0.228667i
$$899$$ −2.05720e7 −0.848942
$$900$$ 0 0
$$901$$ −119700. −0.00491227
$$902$$ 6.74356e6i 0.275977i
$$903$$ − 7.60108e6i − 0.310210i
$$904$$ 1.65505e7 0.673580
$$905$$ 0 0
$$906$$ 3.83825e6 0.155350
$$907$$ − 1.96499e7i − 0.793128i −0.918007 0.396564i $$-0.870203\pi$$
0.918007 0.396564i $$-0.129797\pi$$
$$908$$ − 2.19454e7i − 0.883342i
$$909$$ 8.81037e6 0.353659
$$910$$ 0 0
$$911$$ −7.26518e6 −0.290035 −0.145018 0.989429i $$-0.546324\pi$$
−0.145018 + 0.989429i $$0.546324\pi$$
$$912$$ 7.45801e6i 0.296918i
$$913$$ 1.28398e7i 0.509777i
$$914$$ 2.96226e6 0.117289
$$915$$ 0 0
$$916$$ 2.28091e7 0.898193
$$917$$ − 3.77986e6i − 0.148440i
$$918$$ 581742.i 0.0227837i
$$919$$ −9.82532e6 −0.383758 −0.191879 0.981419i $$-0.561458\pi$$
−0.191879 + 0.981419i $$0.561458\pi$$
$$920$$ 0 0
$$921$$ 4.92109e6 0.191167
$$922$$ − 2.11884e6i − 0.0820863i
$$923$$ 6.61387e6i 0.255536i
$$924$$ −4.64814e6 −0.179102
$$925$$ 0 0
$$926$$ 3.19226e6 0.122340
$$927$$ 1.61346e7i 0.616677i
$$928$$ − 2.42727e7i − 0.925226i
$$929$$ −2.71152e7 −1.03080 −0.515399 0.856951i $$-0.672356\pi$$
−0.515399 + 0.856951i $$0.672356\pi$$
$$930$$ 0 0
$$931$$ 2.14169e6 0.0809809
$$932$$ − 6.47150e6i − 0.244042i
$$933$$ − 2.91084e7i − 1.09475i
$$934$$ 7.42621e6 0.278548
$$935$$ 0 0
$$936$$ −2.31676e6 −0.0864354
$$937$$ 4.53522e7i 1.68752i 0.536720 + 0.843761i $$0.319663\pi$$
−0.536720 + 0.843761i $$0.680337\pi$$
$$938$$ 82124.0i 0.00304764i
$$939$$ 1.63182e7 0.603959
$$940$$ 0 0
$$941$$ 4.65780e7 1.71477 0.857387 0.514672i $$-0.172086\pi$$
0.857387 + 0.514672i $$0.172086\pi$$
$$942$$ 1.60637e6i 0.0589820i
$$943$$ − 6.33101e7i − 2.31843i
$$944$$ 3.93859e7 1.43850
$$945$$ 0 0
$$946$$ 5.86024e6 0.212906
$$947$$ − 2.53799e7i − 0.919632i −0.888014 0.459816i $$-0.847915\pi$$
0.888014 0.459816i $$-0.152085\pi$$
$$948$$ − 633888.i − 0.0229082i
$$949$$ −3.55836e7 −1.28258
$$950$$ 0 0
$$951$$ 1.14892e7 0.411944
$$952$$ 2.46343e6i 0.0880942i
$$953$$ 1.52948e7i 0.545520i 0.962082 + 0.272760i $$0.0879365\pi$$
−0.962082 + 0.272760i $$0.912063\pi$$
$$954$$ −12150.0 −0.000432220 0
$$955$$ 0 0
$$956$$ −2.21147e7 −0.782592
$$957$$ 2.52205e7i 0.890173i
$$958$$ − 3.39685e6i − 0.119581i
$$959$$ 1.02003e7 0.358152
$$960$$ 0 0
$$961$$ −2.23991e7 −0.782389
$$962$$ − 4.44829e6i − 0.154973i
$$963$$ − 6.47773e6i − 0.225091i
$$964$$ −1.56626e7 −0.542840
$$965$$ 0 0
$$966$$ −1.40767e6 −0.0485354
$$967$$ 5.71465e6i 0.196527i 0.995160 + 0.0982637i $$0.0313289\pi$$
−0.995160 + 0.0982637i $$0.968671\pi$$
$$968$$ 2.86341e6i 0.0982190i
$$969$$ −6.40634e6 −0.219180
$$970$$ 0 0
$$971$$ 1.30250e7 0.443332 0.221666 0.975123i $$-0.428851\pi$$
0.221666 + 0.975123i $$0.428851\pi$$
$$972$$ − 1.83052e6i − 0.0621453i
$$973$$ 1.35034e7i 0.457258i
$$974$$ 3.71382e6 0.125436
$$975$$ 0 0
$$976$$ 1.37102e7 0.460700
$$977$$ 1.70360e7i 0.570992i 0.958380 + 0.285496i $$0.0921583\pi$$
−0.958380 + 0.285496i $$0.907842\pi$$
$$978$$ − 2.27495e6i − 0.0760544i
$$979$$ −3.98772e7 −1.32974
$$980$$ 0 0
$$981$$ −3.73394e6 −0.123878
$$982$$ − 5.57494e6i − 0.184485i
$$983$$ − 1.36985e7i − 0.452156i −0.974109 0.226078i $$-0.927410\pi$$
0.974109 0.226078i $$-0.0725903\pi$$
$$984$$ −1.12459e7 −0.370259
$$985$$ 0 0
$$986$$ 6.57712e6 0.215448
$$987$$ − 3.93725e6i − 0.128647i
$$988$$ − 1.25540e7i − 0.409157i
$$989$$ −5.50173e7 −1.78858
$$990$$ 0 0
$$991$$ −3.49088e7 −1.12915 −0.564574 0.825383i $$-0.690959\pi$$
−0.564574 + 0.825383i $$0.690959\pi$$
$$992$$ 7.35072e6i 0.237165i
$$993$$ 1.56259e7i 0.502889i
$$994$$ 713832. 0.0229155
$$995$$ 0 0
$$996$$ −1.05362e7 −0.336538
$$997$$ − 875662.i − 0.0278996i −0.999903 0.0139498i $$-0.995559\pi$$
0.999903 0.0139498i $$-0.00444051\pi$$
$$998$$ 3.92698e6i 0.124805i
$$999$$ 7.14274e6 0.226439
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.d.274.1 2
5.2 odd 4 21.6.a.b.1.1 1
5.3 odd 4 525.6.a.c.1.1 1
5.4 even 2 inner 525.6.d.d.274.2 2
15.2 even 4 63.6.a.c.1.1 1
20.7 even 4 336.6.a.l.1.1 1
35.2 odd 12 147.6.e.f.67.1 2
35.12 even 12 147.6.e.e.67.1 2
35.17 even 12 147.6.e.e.79.1 2
35.27 even 4 147.6.a.e.1.1 1
35.32 odd 12 147.6.e.f.79.1 2
60.47 odd 4 1008.6.a.t.1.1 1
105.62 odd 4 441.6.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.a.b.1.1 1 5.2 odd 4
63.6.a.c.1.1 1 15.2 even 4
147.6.a.e.1.1 1 35.27 even 4
147.6.e.e.67.1 2 35.12 even 12
147.6.e.e.79.1 2 35.17 even 12
147.6.e.f.67.1 2 35.2 odd 12
147.6.e.f.79.1 2 35.32 odd 12
336.6.a.l.1.1 1 20.7 even 4
441.6.a.d.1.1 1 105.62 odd 4
525.6.a.c.1.1 1 5.3 odd 4
525.6.d.d.274.1 2 1.1 even 1 trivial
525.6.d.d.274.2 2 5.4 even 2 inner
1008.6.a.t.1.1 1 60.47 odd 4