# Properties

 Label 441.6.a.n.1.1 Level $441$ Weight $6$ Character 441.1 Self dual yes Analytic conductor $70.729$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 441.a (trivial)

## 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: yes Analytic conductor: $$70.7292645375$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{37})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 9$$ x^2 - x - 9 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 7) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.54138$$ of defining polynomial Character $$\chi$$ $$=$$ 441.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-7.08276 q^{2} +18.1655 q^{4} +79.8276 q^{5} +97.9863 q^{8} +O(q^{10})$$ $$q-7.08276 q^{2} +18.1655 q^{4} +79.8276 q^{5} +97.9863 q^{8} -565.400 q^{10} -351.904 q^{11} -291.683 q^{13} -1275.31 q^{16} +370.075 q^{17} +1504.93 q^{19} +1450.11 q^{20} +2492.45 q^{22} +425.711 q^{23} +3247.45 q^{25} +2065.92 q^{26} +7783.93 q^{29} -2575.18 q^{31} +5897.16 q^{32} -2621.16 q^{34} +739.618 q^{37} -10659.0 q^{38} +7822.01 q^{40} -7029.84 q^{41} +1835.23 q^{43} -6392.51 q^{44} -3015.21 q^{46} +1532.68 q^{47} -23000.9 q^{50} -5298.57 q^{52} +9537.46 q^{53} -28091.6 q^{55} -55131.8 q^{58} +29674.1 q^{59} -46510.8 q^{61} +18239.4 q^{62} -958.246 q^{64} -23284.3 q^{65} +26746.1 q^{67} +6722.61 q^{68} +14388.8 q^{71} -70095.1 q^{73} -5238.54 q^{74} +27337.8 q^{76} -27085.8 q^{79} -101805. q^{80} +49790.7 q^{82} +79755.4 q^{83} +29542.2 q^{85} -12998.5 q^{86} -34481.7 q^{88} -43577.3 q^{89} +7733.26 q^{92} -10855.6 q^{94} +120135. q^{95} +103374. q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 12 q^{4} + 38 q^{5} - 96 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 12 * q^4 + 38 * q^5 - 96 * q^8 $$2 q - 2 q^{2} + 12 q^{4} + 38 q^{5} - 96 q^{8} - 778 q^{10} - 424 q^{11} - 924 q^{13} - 2064 q^{16} + 2346 q^{17} - 360 q^{19} + 1708 q^{20} + 2126 q^{22} + 12 q^{23} + 1872 q^{25} - 1148 q^{26} + 7052 q^{29} + 3548 q^{31} + 8096 q^{32} + 7422 q^{34} + 11090 q^{37} - 20138 q^{38} + 15936 q^{40} - 3500 q^{41} - 12680 q^{43} - 5948 q^{44} - 5118 q^{46} + 22956 q^{47} - 29992 q^{50} - 1400 q^{52} - 3042 q^{53} - 25076 q^{55} - 58852 q^{58} + 65808 q^{59} - 42486 q^{61} + 49362 q^{62} + 35456 q^{64} + 3164 q^{65} + 42312 q^{67} - 5460 q^{68} + 2208 q^{71} - 50506 q^{73} + 47370 q^{74} + 38836 q^{76} + 9004 q^{79} - 68816 q^{80} + 67732 q^{82} + 104328 q^{83} - 53106 q^{85} - 86776 q^{86} - 20496 q^{88} + 26666 q^{89} + 10284 q^{92} + 98034 q^{94} + 198140 q^{95} + 209132 q^{97}+O(q^{100})$$ 2 * q - 2 * q^2 + 12 * q^4 + 38 * q^5 - 96 * q^8 - 778 * q^10 - 424 * q^11 - 924 * q^13 - 2064 * q^16 + 2346 * q^17 - 360 * q^19 + 1708 * q^20 + 2126 * q^22 + 12 * q^23 + 1872 * q^25 - 1148 * q^26 + 7052 * q^29 + 3548 * q^31 + 8096 * q^32 + 7422 * q^34 + 11090 * q^37 - 20138 * q^38 + 15936 * q^40 - 3500 * q^41 - 12680 * q^43 - 5948 * q^44 - 5118 * q^46 + 22956 * q^47 - 29992 * q^50 - 1400 * q^52 - 3042 * q^53 - 25076 * q^55 - 58852 * q^58 + 65808 * q^59 - 42486 * q^61 + 49362 * q^62 + 35456 * q^64 + 3164 * q^65 + 42312 * q^67 - 5460 * q^68 + 2208 * q^71 - 50506 * q^73 + 47370 * q^74 + 38836 * q^76 + 9004 * q^79 - 68816 * q^80 + 67732 * q^82 + 104328 * q^83 - 53106 * q^85 - 86776 * q^86 - 20496 * q^88 + 26666 * q^89 + 10284 * q^92 + 98034 * q^94 + 198140 * q^95 + 209132 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −7.08276 −1.25207 −0.626034 0.779796i $$-0.715323\pi$$
−0.626034 + 0.779796i $$0.715323\pi$$
$$3$$ 0 0
$$4$$ 18.1655 0.567673
$$5$$ 79.8276 1.42800 0.714000 0.700146i $$-0.246882\pi$$
0.714000 + 0.700146i $$0.246882\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 97.9863 0.541303
$$9$$ 0 0
$$10$$ −565.400 −1.78795
$$11$$ −351.904 −0.876884 −0.438442 0.898760i $$-0.644469\pi$$
−0.438442 + 0.898760i $$0.644469\pi$$
$$12$$ 0 0
$$13$$ −291.683 −0.478688 −0.239344 0.970935i $$-0.576932\pi$$
−0.239344 + 0.970935i $$0.576932\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −1275.31 −1.24542
$$17$$ 370.075 0.310576 0.155288 0.987869i $$-0.450369\pi$$
0.155288 + 0.987869i $$0.450369\pi$$
$$18$$ 0 0
$$19$$ 1504.93 0.956381 0.478190 0.878256i $$-0.341293\pi$$
0.478190 + 0.878256i $$0.341293\pi$$
$$20$$ 1450.11 0.810637
$$21$$ 0 0
$$22$$ 2492.45 1.09792
$$23$$ 425.711 0.167801 0.0839006 0.996474i $$-0.473262\pi$$
0.0839006 + 0.996474i $$0.473262\pi$$
$$24$$ 0 0
$$25$$ 3247.45 1.03918
$$26$$ 2065.92 0.599349
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 7783.93 1.71872 0.859358 0.511374i $$-0.170863\pi$$
0.859358 + 0.511374i $$0.170863\pi$$
$$30$$ 0 0
$$31$$ −2575.18 −0.481285 −0.240643 0.970614i $$-0.577358\pi$$
−0.240643 + 0.970614i $$0.577358\pi$$
$$32$$ 5897.16 1.01805
$$33$$ 0 0
$$34$$ −2621.16 −0.388862
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 739.618 0.0888184 0.0444092 0.999013i $$-0.485859\pi$$
0.0444092 + 0.999013i $$0.485859\pi$$
$$38$$ −10659.0 −1.19745
$$39$$ 0 0
$$40$$ 7822.01 0.772981
$$41$$ −7029.84 −0.653109 −0.326554 0.945178i $$-0.605888\pi$$
−0.326554 + 0.945178i $$0.605888\pi$$
$$42$$ 0 0
$$43$$ 1835.23 0.151363 0.0756816 0.997132i $$-0.475887\pi$$
0.0756816 + 0.997132i $$0.475887\pi$$
$$44$$ −6392.51 −0.497783
$$45$$ 0 0
$$46$$ −3015.21 −0.210098
$$47$$ 1532.68 0.101206 0.0506032 0.998719i $$-0.483886\pi$$
0.0506032 + 0.998719i $$0.483886\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −23000.9 −1.30113
$$51$$ 0 0
$$52$$ −5298.57 −0.271738
$$53$$ 9537.46 0.466383 0.233192 0.972431i $$-0.425083\pi$$
0.233192 + 0.972431i $$0.425083\pi$$
$$54$$ 0 0
$$55$$ −28091.6 −1.25219
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −55131.8 −2.15195
$$59$$ 29674.1 1.10981 0.554903 0.831915i $$-0.312755\pi$$
0.554903 + 0.831915i $$0.312755\pi$$
$$60$$ 0 0
$$61$$ −46510.8 −1.60040 −0.800201 0.599732i $$-0.795274\pi$$
−0.800201 + 0.599732i $$0.795274\pi$$
$$62$$ 18239.4 0.602602
$$63$$ 0 0
$$64$$ −958.246 −0.0292434
$$65$$ −23284.3 −0.683566
$$66$$ 0 0
$$67$$ 26746.1 0.727902 0.363951 0.931418i $$-0.381428\pi$$
0.363951 + 0.931418i $$0.381428\pi$$
$$68$$ 6722.61 0.176305
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 14388.8 0.338748 0.169374 0.985552i $$-0.445825\pi$$
0.169374 + 0.985552i $$0.445825\pi$$
$$72$$ 0 0
$$73$$ −70095.1 −1.53950 −0.769752 0.638343i $$-0.779620\pi$$
−0.769752 + 0.638343i $$0.779620\pi$$
$$74$$ −5238.54 −0.111207
$$75$$ 0 0
$$76$$ 27337.8 0.542911
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −27085.8 −0.488285 −0.244143 0.969739i $$-0.578507\pi$$
−0.244143 + 0.969739i $$0.578507\pi$$
$$80$$ −101805. −1.77846
$$81$$ 0 0
$$82$$ 49790.7 0.817736
$$83$$ 79755.4 1.27076 0.635382 0.772198i $$-0.280843\pi$$
0.635382 + 0.772198i $$0.280843\pi$$
$$84$$ 0 0
$$85$$ 29542.2 0.443502
$$86$$ −12998.5 −0.189517
$$87$$ 0 0
$$88$$ −34481.7 −0.474660
$$89$$ −43577.3 −0.583157 −0.291579 0.956547i $$-0.594180\pi$$
−0.291579 + 0.956547i $$0.594180\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 7733.26 0.0952561
$$93$$ 0 0
$$94$$ −10855.6 −0.126717
$$95$$ 120135. 1.36571
$$96$$ 0 0
$$97$$ 103374. 1.11553 0.557765 0.829999i $$-0.311659\pi$$
0.557765 + 0.829999i $$0.311659\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 58991.6 0.589916
$$101$$ −28700.2 −0.279951 −0.139975 0.990155i $$-0.544702\pi$$
−0.139975 + 0.990155i $$0.544702\pi$$
$$102$$ 0 0
$$103$$ 29227.9 0.271459 0.135730 0.990746i $$-0.456662\pi$$
0.135730 + 0.990746i $$0.456662\pi$$
$$104$$ −28580.9 −0.259115
$$105$$ 0 0
$$106$$ −67551.6 −0.583944
$$107$$ −87858.4 −0.741863 −0.370932 0.928660i $$-0.620962\pi$$
−0.370932 + 0.928660i $$0.620962\pi$$
$$108$$ 0 0
$$109$$ 220628. 1.77867 0.889333 0.457260i $$-0.151169\pi$$
0.889333 + 0.457260i $$0.151169\pi$$
$$110$$ 198966. 1.56783
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −39665.6 −0.292225 −0.146113 0.989268i $$-0.546676\pi$$
−0.146113 + 0.989268i $$0.546676\pi$$
$$114$$ 0 0
$$115$$ 33983.5 0.239620
$$116$$ 141399. 0.975668
$$117$$ 0 0
$$118$$ −210174. −1.38955
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −37214.9 −0.231075
$$122$$ 329425. 2.00381
$$123$$ 0 0
$$124$$ −46779.4 −0.273212
$$125$$ 9774.87 0.0559546
$$126$$ 0 0
$$127$$ 51740.3 0.284655 0.142328 0.989820i $$-0.454541\pi$$
0.142328 + 0.989820i $$0.454541\pi$$
$$128$$ −181922. −0.981433
$$129$$ 0 0
$$130$$ 164917. 0.855871
$$131$$ 166674. 0.848572 0.424286 0.905528i $$-0.360525\pi$$
0.424286 + 0.905528i $$0.360525\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −189436. −0.911382
$$135$$ 0 0
$$136$$ 36262.3 0.168116
$$137$$ −28259.4 −0.128636 −0.0643178 0.997929i $$-0.520487\pi$$
−0.0643178 + 0.997929i $$0.520487\pi$$
$$138$$ 0 0
$$139$$ 336393. 1.47676 0.738380 0.674384i $$-0.235591\pi$$
0.738380 + 0.674384i $$0.235591\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −101912. −0.424136
$$143$$ 102644. 0.419753
$$144$$ 0 0
$$145$$ 621373. 2.45433
$$146$$ 496467. 1.92756
$$147$$ 0 0
$$148$$ 13435.5 0.0504198
$$149$$ 355381. 1.31138 0.655691 0.755030i $$-0.272378\pi$$
0.655691 + 0.755030i $$0.272378\pi$$
$$150$$ 0 0
$$151$$ −358797. −1.28058 −0.640290 0.768133i $$-0.721186\pi$$
−0.640290 + 0.768133i $$0.721186\pi$$
$$152$$ 147462. 0.517692
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −205570. −0.687275
$$156$$ 0 0
$$157$$ 458911. 1.48586 0.742932 0.669367i $$-0.233435\pi$$
0.742932 + 0.669367i $$0.233435\pi$$
$$158$$ 191842. 0.611366
$$159$$ 0 0
$$160$$ 470756. 1.45377
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 502441. 1.48121 0.740603 0.671943i $$-0.234540\pi$$
0.740603 + 0.671943i $$0.234540\pi$$
$$164$$ −127701. −0.370752
$$165$$ 0 0
$$166$$ −564889. −1.59108
$$167$$ 676652. 1.87748 0.938738 0.344632i $$-0.111996\pi$$
0.938738 + 0.344632i $$0.111996\pi$$
$$168$$ 0 0
$$169$$ −286214. −0.770858
$$170$$ −209241. −0.555295
$$171$$ 0 0
$$172$$ 33338.0 0.0859247
$$173$$ 249160. 0.632941 0.316470 0.948602i $$-0.397502\pi$$
0.316470 + 0.948602i $$0.397502\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 448786. 1.09209
$$177$$ 0 0
$$178$$ 308648. 0.730152
$$179$$ −139258. −0.324853 −0.162427 0.986721i $$-0.551932\pi$$
−0.162427 + 0.986721i $$0.551932\pi$$
$$180$$ 0 0
$$181$$ 306246. 0.694823 0.347412 0.937713i $$-0.387061\pi$$
0.347412 + 0.937713i $$0.387061\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 41713.8 0.0908312
$$185$$ 59041.9 0.126833
$$186$$ 0 0
$$187$$ −130231. −0.272339
$$188$$ 27842.0 0.0574521
$$189$$ 0 0
$$190$$ −850885. −1.70996
$$191$$ −227494. −0.451218 −0.225609 0.974218i $$-0.572437\pi$$
−0.225609 + 0.974218i $$0.572437\pi$$
$$192$$ 0 0
$$193$$ −672374. −1.29933 −0.649663 0.760223i $$-0.725090\pi$$
−0.649663 + 0.760223i $$0.725090\pi$$
$$194$$ −732172. −1.39672
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 1282.76 0.00235493 0.00117747 0.999999i $$-0.499625\pi$$
0.00117747 + 0.999999i $$0.499625\pi$$
$$198$$ 0 0
$$199$$ 368898. 0.660349 0.330175 0.943920i $$-0.392892\pi$$
0.330175 + 0.943920i $$0.392892\pi$$
$$200$$ 318206. 0.562513
$$201$$ 0 0
$$202$$ 203277. 0.350517
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −561175. −0.932640
$$206$$ −207014. −0.339885
$$207$$ 0 0
$$208$$ 371986. 0.596167
$$209$$ −529589. −0.838635
$$210$$ 0 0
$$211$$ 502168. 0.776503 0.388251 0.921553i $$-0.373079\pi$$
0.388251 + 0.921553i $$0.373079\pi$$
$$212$$ 173253. 0.264753
$$213$$ 0 0
$$214$$ 622280. 0.928863
$$215$$ 146502. 0.216147
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −1.56266e6 −2.22701
$$219$$ 0 0
$$220$$ −510299. −0.710834
$$221$$ −107945. −0.148669
$$222$$ 0 0
$$223$$ −1.17328e6 −1.57993 −0.789967 0.613149i $$-0.789902\pi$$
−0.789967 + 0.613149i $$0.789902\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 280942. 0.365886
$$227$$ 910159. 1.17234 0.586168 0.810189i $$-0.300636\pi$$
0.586168 + 0.810189i $$0.300636\pi$$
$$228$$ 0 0
$$229$$ 521924. 0.657686 0.328843 0.944385i $$-0.393341\pi$$
0.328843 + 0.944385i $$0.393341\pi$$
$$230$$ −240697. −0.300020
$$231$$ 0 0
$$232$$ 762719. 0.930346
$$233$$ 1.04279e6 1.25836 0.629182 0.777258i $$-0.283390\pi$$
0.629182 + 0.777258i $$0.283390\pi$$
$$234$$ 0 0
$$235$$ 122350. 0.144523
$$236$$ 539045. 0.630006
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 1.53447e6 1.73766 0.868830 0.495110i $$-0.164872\pi$$
0.868830 + 0.495110i $$0.164872\pi$$
$$240$$ 0 0
$$241$$ −1.00758e6 −1.11747 −0.558735 0.829346i $$-0.688713\pi$$
−0.558735 + 0.829346i $$0.688713\pi$$
$$242$$ 263584. 0.289322
$$243$$ 0 0
$$244$$ −844893. −0.908505
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −438961. −0.457808
$$248$$ −252332. −0.260521
$$249$$ 0 0
$$250$$ −69233.1 −0.0700590
$$251$$ −8511.89 −0.00852789 −0.00426394 0.999991i $$-0.501357\pi$$
−0.00426394 + 0.999991i $$0.501357\pi$$
$$252$$ 0 0
$$253$$ −149809. −0.147142
$$254$$ −366464. −0.356408
$$255$$ 0 0
$$256$$ 1.31917e6 1.25806
$$257$$ 527532. 0.498214 0.249107 0.968476i $$-0.419863\pi$$
0.249107 + 0.968476i $$0.419863\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −422972. −0.388042
$$261$$ 0 0
$$262$$ −1.18051e6 −1.06247
$$263$$ 352085. 0.313876 0.156938 0.987608i $$-0.449838\pi$$
0.156938 + 0.987608i $$0.449838\pi$$
$$264$$ 0 0
$$265$$ 761353. 0.665996
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 485856. 0.413210
$$269$$ −479540. −0.404058 −0.202029 0.979380i $$-0.564754\pi$$
−0.202029 + 0.979380i $$0.564754\pi$$
$$270$$ 0 0
$$271$$ −977611. −0.808617 −0.404308 0.914623i $$-0.632488\pi$$
−0.404308 + 0.914623i $$0.632488\pi$$
$$272$$ −471961. −0.386798
$$273$$ 0 0
$$274$$ 200155. 0.161061
$$275$$ −1.14279e6 −0.911243
$$276$$ 0 0
$$277$$ 968723. 0.758578 0.379289 0.925278i $$-0.376169\pi$$
0.379289 + 0.925278i $$0.376169\pi$$
$$278$$ −2.38259e6 −1.84900
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 318333. 0.240501 0.120250 0.992744i $$-0.461630\pi$$
0.120250 + 0.992744i $$0.461630\pi$$
$$282$$ 0 0
$$283$$ 1.77210e6 1.31529 0.657646 0.753327i $$-0.271552\pi$$
0.657646 + 0.753327i $$0.271552\pi$$
$$284$$ 261379. 0.192298
$$285$$ 0 0
$$286$$ −727004. −0.525559
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −1.28290e6 −0.903543
$$290$$ −4.40104e6 −3.07298
$$291$$ 0 0
$$292$$ −1.27331e6 −0.873934
$$293$$ −1.64148e6 −1.11703 −0.558516 0.829494i $$-0.688629\pi$$
−0.558516 + 0.829494i $$0.688629\pi$$
$$294$$ 0 0
$$295$$ 2.36881e6 1.58480
$$296$$ 72472.4 0.0480777
$$297$$ 0 0
$$298$$ −2.51708e6 −1.64194
$$299$$ −124172. −0.0803243
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 2.54128e6 1.60337
$$303$$ 0 0
$$304$$ −1.91925e6 −1.19110
$$305$$ −3.71285e6 −2.28537
$$306$$ 0 0
$$307$$ −466930. −0.282752 −0.141376 0.989956i $$-0.545153\pi$$
−0.141376 + 0.989956i $$0.545153\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 1.45600e6 0.860515
$$311$$ 2.43796e6 1.42931 0.714654 0.699478i $$-0.246584\pi$$
0.714654 + 0.699478i $$0.246584\pi$$
$$312$$ 0 0
$$313$$ 2.42094e6 1.39676 0.698381 0.715726i $$-0.253904\pi$$
0.698381 + 0.715726i $$0.253904\pi$$
$$314$$ −3.25035e6 −1.86040
$$315$$ 0 0
$$316$$ −492028. −0.277186
$$317$$ −1.87611e6 −1.04860 −0.524301 0.851533i $$-0.675673\pi$$
−0.524301 + 0.851533i $$0.675673\pi$$
$$318$$ 0 0
$$319$$ −2.73919e6 −1.50711
$$320$$ −76494.5 −0.0417595
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 556936. 0.297029
$$324$$ 0 0
$$325$$ −947225. −0.497445
$$326$$ −3.55867e6 −1.85457
$$327$$ 0 0
$$328$$ −688828. −0.353530
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −1.08310e6 −0.543373 −0.271686 0.962386i $$-0.587581\pi$$
−0.271686 + 0.962386i $$0.587581\pi$$
$$332$$ 1.44880e6 0.721378
$$333$$ 0 0
$$334$$ −4.79257e6 −2.35073
$$335$$ 2.13507e6 1.03944
$$336$$ 0 0
$$337$$ −2.59465e6 −1.24453 −0.622263 0.782809i $$-0.713786\pi$$
−0.622263 + 0.782809i $$0.713786\pi$$
$$338$$ 2.02719e6 0.965166
$$339$$ 0 0
$$340$$ 536650. 0.251764
$$341$$ 906213. 0.422031
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 179828. 0.0819333
$$345$$ 0 0
$$346$$ −1.76474e6 −0.792485
$$347$$ 1.87051e6 0.833943 0.416972 0.908920i $$-0.363091\pi$$
0.416972 + 0.908920i $$0.363091\pi$$
$$348$$ 0 0
$$349$$ −1.61685e6 −0.710568 −0.355284 0.934758i $$-0.615616\pi$$
−0.355284 + 0.934758i $$0.615616\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −2.07523e6 −0.892709
$$353$$ 578305. 0.247013 0.123507 0.992344i $$-0.460586\pi$$
0.123507 + 0.992344i $$0.460586\pi$$
$$354$$ 0 0
$$355$$ 1.14862e6 0.483733
$$356$$ −791605. −0.331042
$$357$$ 0 0
$$358$$ 986330. 0.406738
$$359$$ 1.96818e6 0.805988 0.402994 0.915203i $$-0.367970\pi$$
0.402994 + 0.915203i $$0.367970\pi$$
$$360$$ 0 0
$$361$$ −211299. −0.0853355
$$362$$ −2.16907e6 −0.869965
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −5.59553e6 −2.19841
$$366$$ 0 0
$$367$$ 2.17452e6 0.842749 0.421375 0.906887i $$-0.361548\pi$$
0.421375 + 0.906887i $$0.361548\pi$$
$$368$$ −542913. −0.208983
$$369$$ 0 0
$$370$$ −418180. −0.158803
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 1.38476e6 0.515349 0.257675 0.966232i $$-0.417044\pi$$
0.257675 + 0.966232i $$0.417044\pi$$
$$374$$ 922394. 0.340987
$$375$$ 0 0
$$376$$ 150182. 0.0547833
$$377$$ −2.27044e6 −0.822728
$$378$$ 0 0
$$379$$ 3.37190e6 1.20580 0.602902 0.797815i $$-0.294011\pi$$
0.602902 + 0.797815i $$0.294011\pi$$
$$380$$ 2.18231e6 0.775277
$$381$$ 0 0
$$382$$ 1.61128e6 0.564955
$$383$$ −3.28060e6 −1.14276 −0.571382 0.820685i $$-0.693592\pi$$
−0.571382 + 0.820685i $$0.693592\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 4.76227e6 1.62684
$$387$$ 0 0
$$388$$ 1.87784e6 0.633256
$$389$$ 2.94810e6 0.987797 0.493899 0.869520i $$-0.335571\pi$$
0.493899 + 0.869520i $$0.335571\pi$$
$$390$$ 0 0
$$391$$ 157545. 0.0521150
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −9085.46 −0.00294854
$$395$$ −2.16219e6 −0.697271
$$396$$ 0 0
$$397$$ −69270.2 −0.0220582 −0.0110291 0.999939i $$-0.503511\pi$$
−0.0110291 + 0.999939i $$0.503511\pi$$
$$398$$ −2.61282e6 −0.826802
$$399$$ 0 0
$$400$$ −4.14151e6 −1.29422
$$401$$ 3.34786e6 1.03970 0.519848 0.854259i $$-0.325988\pi$$
0.519848 + 0.854259i $$0.325988\pi$$
$$402$$ 0 0
$$403$$ 751134. 0.230385
$$404$$ −521354. −0.158920
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −260274. −0.0778834
$$408$$ 0 0
$$409$$ 2.91217e6 0.860812 0.430406 0.902636i $$-0.358370\pi$$
0.430406 + 0.902636i $$0.358370\pi$$
$$410$$ 3.97467e6 1.16773
$$411$$ 0 0
$$412$$ 530940. 0.154100
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 6.36669e6 1.81465
$$416$$ −1.72010e6 −0.487327
$$417$$ 0 0
$$418$$ 3.75095e6 1.05003
$$419$$ 4.62361e6 1.28661 0.643304 0.765611i $$-0.277563\pi$$
0.643304 + 0.765611i $$0.277563\pi$$
$$420$$ 0 0
$$421$$ −2.63042e6 −0.723303 −0.361652 0.932313i $$-0.617787\pi$$
−0.361652 + 0.932313i $$0.617787\pi$$
$$422$$ −3.55674e6 −0.972234
$$423$$ 0 0
$$424$$ 934541. 0.252455
$$425$$ 1.20180e6 0.322746
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −1.59599e6 −0.421135
$$429$$ 0 0
$$430$$ −1.03764e6 −0.270630
$$431$$ −7.54128e6 −1.95547 −0.977736 0.209837i $$-0.932707\pi$$
−0.977736 + 0.209837i $$0.932707\pi$$
$$432$$ 0 0
$$433$$ 5.83558e6 1.49577 0.747883 0.663830i $$-0.231070\pi$$
0.747883 + 0.663830i $$0.231070\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 4.00782e6 1.00970
$$437$$ 640663. 0.160482
$$438$$ 0 0
$$439$$ 168104. 0.0416310 0.0208155 0.999783i $$-0.493374\pi$$
0.0208155 + 0.999783i $$0.493374\pi$$
$$440$$ −2.75259e6 −0.677814
$$441$$ 0 0
$$442$$ 764546. 0.186143
$$443$$ 2.84151e6 0.687924 0.343962 0.938984i $$-0.388231\pi$$
0.343962 + 0.938984i $$0.388231\pi$$
$$444$$ 0 0
$$445$$ −3.47867e6 −0.832748
$$446$$ 8.31005e6 1.97818
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 1.41567e6 0.331396 0.165698 0.986177i $$-0.447012\pi$$
0.165698 + 0.986177i $$0.447012\pi$$
$$450$$ 0 0
$$451$$ 2.47382e6 0.572701
$$452$$ −720547. −0.165888
$$453$$ 0 0
$$454$$ −6.44644e6 −1.46784
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 1.55727e6 0.348799 0.174399 0.984675i $$-0.444202\pi$$
0.174399 + 0.984675i $$0.444202\pi$$
$$458$$ −3.69667e6 −0.823468
$$459$$ 0 0
$$460$$ 617328. 0.136026
$$461$$ 4.45345e6 0.975987 0.487994 0.872847i $$-0.337729\pi$$
0.487994 + 0.872847i $$0.337729\pi$$
$$462$$ 0 0
$$463$$ 4.92263e6 1.06720 0.533599 0.845738i $$-0.320839\pi$$
0.533599 + 0.845738i $$0.320839\pi$$
$$464$$ −9.92693e6 −2.14052
$$465$$ 0 0
$$466$$ −7.38582e6 −1.57556
$$467$$ −5.09090e6 −1.08020 −0.540098 0.841602i $$-0.681613\pi$$
−0.540098 + 0.841602i $$0.681613\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −866579. −0.180952
$$471$$ 0 0
$$472$$ 2.90765e6 0.600741
$$473$$ −645825. −0.132728
$$474$$ 0 0
$$475$$ 4.88717e6 0.993856
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −1.08683e7 −2.17567
$$479$$ 8.30085e6 1.65304 0.826521 0.562907i $$-0.190317\pi$$
0.826521 + 0.562907i $$0.190317\pi$$
$$480$$ 0 0
$$481$$ −215734. −0.0425163
$$482$$ 7.13644e6 1.39915
$$483$$ 0 0
$$484$$ −676028. −0.131175
$$485$$ 8.25208e6 1.59298
$$486$$ 0 0
$$487$$ 8.63401e6 1.64964 0.824822 0.565392i $$-0.191275\pi$$
0.824822 + 0.565392i $$0.191275\pi$$
$$488$$ −4.55742e6 −0.866303
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −95039.5 −0.0177910 −0.00889550 0.999960i $$-0.502832\pi$$
−0.00889550 + 0.999960i $$0.502832\pi$$
$$492$$ 0 0
$$493$$ 2.88064e6 0.533792
$$494$$ 3.10905e6 0.573206
$$495$$ 0 0
$$496$$ 3.28415e6 0.599402
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 2.14203e6 0.385101 0.192551 0.981287i $$-0.438324\pi$$
0.192551 + 0.981287i $$0.438324\pi$$
$$500$$ 177566. 0.0317639
$$501$$ 0 0
$$502$$ 60287.7 0.0106775
$$503$$ −5.24794e6 −0.924844 −0.462422 0.886660i $$-0.653019\pi$$
−0.462422 + 0.886660i $$0.653019\pi$$
$$504$$ 0 0
$$505$$ −2.29107e6 −0.399769
$$506$$ 1.06106e6 0.184232
$$507$$ 0 0
$$508$$ 939889. 0.161591
$$509$$ 1.05891e7 1.81160 0.905802 0.423702i $$-0.139270\pi$$
0.905802 + 0.423702i $$0.139270\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −3.52190e6 −0.593747
$$513$$ 0 0
$$514$$ −3.73639e6 −0.623798
$$515$$ 2.33319e6 0.387644
$$516$$ 0 0
$$517$$ −539357. −0.0887462
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −2.28155e6 −0.370016
$$521$$ 4.54465e6 0.733510 0.366755 0.930318i $$-0.380469\pi$$
0.366755 + 0.930318i $$0.380469\pi$$
$$522$$ 0 0
$$523$$ −5.27197e6 −0.842789 −0.421394 0.906877i $$-0.638459\pi$$
−0.421394 + 0.906877i $$0.638459\pi$$
$$524$$ 3.02772e6 0.481711
$$525$$ 0 0
$$526$$ −2.49373e6 −0.392993
$$527$$ −953009. −0.149476
$$528$$ 0 0
$$529$$ −6.25511e6 −0.971843
$$530$$ −5.39248e6 −0.833871
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 2.05048e6 0.312635
$$534$$ 0 0
$$535$$ −7.01353e6 −1.05938
$$536$$ 2.62075e6 0.394015
$$537$$ 0 0
$$538$$ 3.39647e6 0.505908
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 5.93445e6 0.871741 0.435871 0.900009i $$-0.356441\pi$$
0.435871 + 0.900009i $$0.356441\pi$$
$$542$$ 6.92418e6 1.01244
$$543$$ 0 0
$$544$$ 2.18239e6 0.316181
$$545$$ 1.76122e7 2.53994
$$546$$ 0 0
$$547$$ −8.82017e6 −1.26040 −0.630200 0.776433i $$-0.717027\pi$$
−0.630200 + 0.776433i $$0.717027\pi$$
$$548$$ −513347. −0.0730230
$$549$$ 0 0
$$550$$ 8.09410e6 1.14094
$$551$$ 1.17142e7 1.64375
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −6.86124e6 −0.949791
$$555$$ 0 0
$$556$$ 6.11076e6 0.838317
$$557$$ 1.18224e6 0.161461 0.0807304 0.996736i $$-0.474275\pi$$
0.0807304 + 0.996736i $$0.474275\pi$$
$$558$$ 0 0
$$559$$ −535306. −0.0724556
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −2.25468e6 −0.301123
$$563$$ −4.07741e6 −0.542142 −0.271071 0.962559i $$-0.587378\pi$$
−0.271071 + 0.962559i $$0.587378\pi$$
$$564$$ 0 0
$$565$$ −3.16641e6 −0.417298
$$566$$ −1.25514e7 −1.64684
$$567$$ 0 0
$$568$$ 1.40990e6 0.183366
$$569$$ −8.17615e6 −1.05869 −0.529344 0.848407i $$-0.677562\pi$$
−0.529344 + 0.848407i $$0.677562\pi$$
$$570$$ 0 0
$$571$$ 3.30615e6 0.424357 0.212179 0.977231i $$-0.431944\pi$$
0.212179 + 0.977231i $$0.431944\pi$$
$$572$$ 1.86458e6 0.238282
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 1.38247e6 0.174376
$$576$$ 0 0
$$577$$ −7.14994e6 −0.894052 −0.447026 0.894521i $$-0.647517\pi$$
−0.447026 + 0.894521i $$0.647517\pi$$
$$578$$ 9.08648e6 1.13130
$$579$$ 0 0
$$580$$ 1.12876e7 1.39325
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −3.35627e6 −0.408964
$$584$$ −6.86836e6 −0.833338
$$585$$ 0 0
$$586$$ 1.16262e7 1.39860
$$587$$ −9.69191e6 −1.16095 −0.580476 0.814277i $$-0.697133\pi$$
−0.580476 + 0.814277i $$0.697133\pi$$
$$588$$ 0 0
$$589$$ −3.87545e6 −0.460292
$$590$$ −1.67777e7 −1.98428
$$591$$ 0 0
$$592$$ −943242. −0.110616
$$593$$ −6.63960e6 −0.775363 −0.387682 0.921793i $$-0.626724\pi$$
−0.387682 + 0.921793i $$0.626724\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 6.45569e6 0.744435
$$597$$ 0 0
$$598$$ 879484. 0.100571
$$599$$ 3.24191e6 0.369177 0.184588 0.982816i $$-0.440905\pi$$
0.184588 + 0.982816i $$0.440905\pi$$
$$600$$ 0 0
$$601$$ −5.65076e6 −0.638147 −0.319074 0.947730i $$-0.603372\pi$$
−0.319074 + 0.947730i $$0.603372\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −6.51774e6 −0.726950
$$605$$ −2.97078e6 −0.329975
$$606$$ 0 0
$$607$$ 235674. 0.0259621 0.0129811 0.999916i $$-0.495868\pi$$
0.0129811 + 0.999916i $$0.495868\pi$$
$$608$$ 8.87478e6 0.973641
$$609$$ 0 0
$$610$$ 2.62972e7 2.86144
$$611$$ −447057. −0.0484462
$$612$$ 0 0
$$613$$ 788877. 0.0847926 0.0423963 0.999101i $$-0.486501\pi$$
0.0423963 + 0.999101i $$0.486501\pi$$
$$614$$ 3.30716e6 0.354025
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −1.67739e7 −1.77387 −0.886935 0.461894i $$-0.847170\pi$$
−0.886935 + 0.461894i $$0.847170\pi$$
$$618$$ 0 0
$$619$$ 8.22300e6 0.862588 0.431294 0.902211i $$-0.358057\pi$$
0.431294 + 0.902211i $$0.358057\pi$$
$$620$$ −3.73429e6 −0.390147
$$621$$ 0 0
$$622$$ −1.72675e7 −1.78959
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −9.36798e6 −0.959281
$$626$$ −1.71469e7 −1.74884
$$627$$ 0 0
$$628$$ 8.33635e6 0.843484
$$629$$ 273714. 0.0275849
$$630$$ 0 0
$$631$$ −5.94507e6 −0.594406 −0.297203 0.954814i $$-0.596054\pi$$
−0.297203 + 0.954814i $$0.596054\pi$$
$$632$$ −2.65404e6 −0.264310
$$633$$ 0 0
$$634$$ 1.32881e7 1.31292
$$635$$ 4.13030e6 0.406488
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 1.94011e7 1.88701
$$639$$ 0 0
$$640$$ −1.45224e7 −1.40149
$$641$$ 1.06761e7 1.02628 0.513141 0.858304i $$-0.328482\pi$$
0.513141 + 0.858304i $$0.328482\pi$$
$$642$$ 0 0
$$643$$ −3.13159e6 −0.298701 −0.149351 0.988784i $$-0.547718\pi$$
−0.149351 + 0.988784i $$0.547718\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −3.94464e6 −0.371900
$$647$$ 4.93457e6 0.463435 0.231717 0.972783i $$-0.425566\pi$$
0.231717 + 0.972783i $$0.425566\pi$$
$$648$$ 0 0
$$649$$ −1.04424e7 −0.973170
$$650$$ 6.70897e6 0.622834
$$651$$ 0 0
$$652$$ 9.12710e6 0.840841
$$653$$ −5.72224e6 −0.525150 −0.262575 0.964912i $$-0.584572\pi$$
−0.262575 + 0.964912i $$0.584572\pi$$
$$654$$ 0 0
$$655$$ 1.33052e7 1.21176
$$656$$ 8.96523e6 0.813395
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −362477. −0.0325137 −0.0162569 0.999868i $$-0.505175\pi$$
−0.0162569 + 0.999868i $$0.505175\pi$$
$$660$$ 0 0
$$661$$ −1.91211e7 −1.70219 −0.851096 0.525011i $$-0.824061\pi$$
−0.851096 + 0.525011i $$0.824061\pi$$
$$662$$ 7.67133e6 0.680339
$$663$$ 0 0
$$664$$ 7.81494e6 0.687868
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 3.31370e6 0.288403
$$668$$ 1.22917e7 1.06579
$$669$$ 0 0
$$670$$ −1.51222e7 −1.30145
$$671$$ 1.63673e7 1.40337
$$672$$ 0 0
$$673$$ −573374. −0.0487978 −0.0243989 0.999702i $$-0.507767\pi$$
−0.0243989 + 0.999702i $$0.507767\pi$$
$$674$$ 1.83773e7 1.55823
$$675$$ 0 0
$$676$$ −5.19923e6 −0.437595
$$677$$ −1.16903e7 −0.980291 −0.490146 0.871641i $$-0.663056\pi$$
−0.490146 + 0.871641i $$0.663056\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 2.89473e6 0.240069
$$681$$ 0 0
$$682$$ −6.41849e6 −0.528411
$$683$$ −1.83674e7 −1.50659 −0.753297 0.657681i $$-0.771538\pi$$
−0.753297 + 0.657681i $$0.771538\pi$$
$$684$$ 0 0
$$685$$ −2.25588e6 −0.183692
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −2.34049e6 −0.188511
$$689$$ −2.78191e6 −0.223252
$$690$$ 0 0
$$691$$ 2.35611e7 1.87716 0.938579 0.345066i $$-0.112143\pi$$
0.938579 + 0.345066i $$0.112143\pi$$
$$692$$ 4.52612e6 0.359303
$$693$$ 0 0
$$694$$ −1.32484e7 −1.04415
$$695$$ 2.68535e7 2.10881
$$696$$ 0 0
$$697$$ −2.60157e6 −0.202840
$$698$$ 1.14517e7 0.889679
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −1.32980e7 −1.02210 −0.511048 0.859552i $$-0.670743\pi$$
−0.511048 + 0.859552i $$0.670743\pi$$
$$702$$ 0 0
$$703$$ 1.11307e6 0.0849442
$$704$$ 337210. 0.0256430
$$705$$ 0 0
$$706$$ −4.09600e6 −0.309277
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −6.85353e6 −0.512034 −0.256017 0.966672i $$-0.582410\pi$$
−0.256017 + 0.966672i $$0.582410\pi$$
$$710$$ −8.13540e6 −0.605666
$$711$$ 0 0
$$712$$ −4.26998e6 −0.315665
$$713$$ −1.09628e6 −0.0807602
$$714$$ 0 0
$$715$$ 8.19384e6 0.599408
$$716$$ −2.52969e6 −0.184410
$$717$$ 0 0
$$718$$ −1.39402e7 −1.00915
$$719$$ −2.65729e7 −1.91698 −0.958490 0.285127i $$-0.907964\pi$$
−0.958490 + 0.285127i $$0.907964\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 1.49658e6 0.106846
$$723$$ 0 0
$$724$$ 5.56312e6 0.394432
$$725$$ 2.52779e7 1.78606
$$726$$ 0 0
$$727$$ 2.16991e6 0.152267 0.0761335 0.997098i $$-0.475742\pi$$
0.0761335 + 0.997098i $$0.475742\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 3.96318e7 2.75256
$$731$$ 679174. 0.0470097
$$732$$ 0 0
$$733$$ −1.74653e7 −1.20065 −0.600324 0.799757i $$-0.704962\pi$$
−0.600324 + 0.799757i $$0.704962\pi$$
$$734$$ −1.54016e7 −1.05518
$$735$$ 0 0
$$736$$ 2.51048e6 0.170829
$$737$$ −9.41203e6 −0.638285
$$738$$ 0 0
$$739$$ −1.36461e7 −0.919172 −0.459586 0.888133i $$-0.652002\pi$$
−0.459586 + 0.888133i $$0.652002\pi$$
$$740$$ 1.07253e6 0.0719994
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −1.48965e7 −0.989944 −0.494972 0.868909i $$-0.664822\pi$$
−0.494972 + 0.868909i $$0.664822\pi$$
$$744$$ 0 0
$$745$$ 2.83692e7 1.87265
$$746$$ −9.80791e6 −0.645252
$$747$$ 0 0
$$748$$ −2.36571e6 −0.154599
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −2.53463e7 −1.63989 −0.819944 0.572443i $$-0.805996\pi$$
−0.819944 + 0.572443i $$0.805996\pi$$
$$752$$ −1.95465e6 −0.126044
$$753$$ 0 0
$$754$$ 1.60810e7 1.03011
$$755$$ −2.86419e7 −1.82867
$$756$$ 0 0
$$757$$ −2.66725e7 −1.69170 −0.845852 0.533417i $$-0.820908\pi$$
−0.845852 + 0.533417i $$0.820908\pi$$
$$758$$ −2.38824e7 −1.50975
$$759$$ 0 0
$$760$$ 1.17715e7 0.739264
$$761$$ −579829. −0.0362943 −0.0181471 0.999835i $$-0.505777\pi$$
−0.0181471 + 0.999835i $$0.505777\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −4.13255e6 −0.256144
$$765$$ 0 0
$$766$$ 2.32357e7 1.43082
$$767$$ −8.65541e6 −0.531250
$$768$$ 0 0
$$769$$ 1.52438e7 0.929562 0.464781 0.885426i $$-0.346133\pi$$
0.464781 + 0.885426i $$0.346133\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −1.22140e7 −0.737591
$$773$$ 1.94926e7 1.17333 0.586665 0.809830i $$-0.300441\pi$$
0.586665 + 0.809830i $$0.300441\pi$$
$$774$$ 0 0
$$775$$ −8.36275e6 −0.500144
$$776$$ 1.01292e7 0.603839
$$777$$ 0 0
$$778$$ −2.08807e7 −1.23679
$$779$$ −1.05794e7 −0.624621
$$780$$ 0 0
$$781$$ −5.06345e6 −0.297043
$$782$$ −1.11585e6 −0.0652515
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 3.66337e7 2.12181
$$786$$ 0 0
$$787$$ 1.36277e7 0.784305 0.392153 0.919900i $$-0.371730\pi$$
0.392153 + 0.919900i $$0.371730\pi$$
$$788$$ 23302.0 0.00133683
$$789$$ 0 0
$$790$$ 1.53143e7 0.873031
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 1.35664e7 0.766093
$$794$$ 490624. 0.0276184
$$795$$ 0 0
$$796$$ 6.70123e6 0.374862
$$797$$ −3.62853e6 −0.202341 −0.101171 0.994869i $$-0.532259\pi$$
−0.101171 + 0.994869i $$0.532259\pi$$
$$798$$ 0 0
$$799$$ 567208. 0.0314323
$$800$$ 1.91507e7 1.05794
$$801$$ 0 0
$$802$$ −2.37121e7 −1.30177
$$803$$ 2.46667e7 1.34997
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −5.32010e6 −0.288458
$$807$$ 0 0
$$808$$ −2.81223e6 −0.151538
$$809$$ −2.98780e7 −1.60502 −0.802509 0.596640i $$-0.796502\pi$$
−0.802509 + 0.596640i $$0.796502\pi$$
$$810$$ 0 0
$$811$$ −1.02643e6 −0.0547995 −0.0273998 0.999625i $$-0.508723\pi$$
−0.0273998 + 0.999625i $$0.508723\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 1.84346e6 0.0975153
$$815$$ 4.01086e7 2.11516
$$816$$ 0 0
$$817$$ 2.76189e6 0.144761
$$818$$ −2.06262e7 −1.07779
$$819$$ 0 0
$$820$$ −1.01940e7 −0.529434
$$821$$ 1.15062e7 0.595766 0.297883 0.954602i $$-0.403719\pi$$
0.297883 + 0.954602i $$0.403719\pi$$
$$822$$ 0 0
$$823$$ −2.51210e7 −1.29282 −0.646408 0.762992i $$-0.723730\pi$$
−0.646408 + 0.762992i $$0.723730\pi$$
$$824$$ 2.86393e6 0.146942
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 2.14447e6 0.109032 0.0545162 0.998513i $$-0.482638\pi$$
0.0545162 + 0.998513i $$0.482638\pi$$
$$828$$ 0 0
$$829$$ 818065. 0.0413430 0.0206715 0.999786i $$-0.493420\pi$$
0.0206715 + 0.999786i $$0.493420\pi$$
$$830$$ −4.50937e7 −2.27207
$$831$$ 0 0
$$832$$ 279504. 0.0139984
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 5.40155e7 2.68104
$$836$$ −9.62025e6 −0.476070
$$837$$ 0 0
$$838$$ −3.27480e7 −1.61092
$$839$$ 2.11279e7 1.03622 0.518110 0.855314i $$-0.326636\pi$$
0.518110 + 0.855314i $$0.326636\pi$$
$$840$$ 0 0
$$841$$ 4.00785e7 1.95398
$$842$$ 1.86307e7 0.905624
$$843$$ 0 0
$$844$$ 9.12215e6 0.440799
$$845$$ −2.28478e7 −1.10079
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −1.21632e7 −0.580844
$$849$$ 0 0
$$850$$ −8.51207e6 −0.404099
$$851$$ 314863. 0.0149038
$$852$$ 0 0
$$853$$ −1.89000e7 −0.889386 −0.444693 0.895683i $$-0.646687\pi$$
−0.444693 + 0.895683i $$0.646687\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −8.60892e6 −0.401573
$$857$$ −3.72286e7 −1.73151 −0.865753 0.500471i $$-0.833160\pi$$
−0.865753 + 0.500471i $$0.833160\pi$$
$$858$$ 0 0
$$859$$ 3.02064e6 0.139674 0.0698371 0.997558i $$-0.477752\pi$$
0.0698371 + 0.997558i $$0.477752\pi$$
$$860$$ 2.66129e6 0.122700
$$861$$ 0 0
$$862$$ 5.34131e7 2.44838
$$863$$ 2.71843e7 1.24248 0.621242 0.783619i $$-0.286629\pi$$
0.621242 + 0.783619i $$0.286629\pi$$
$$864$$ 0 0
$$865$$ 1.98899e7 0.903839
$$866$$ −4.13320e7 −1.87280
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 9.53158e6 0.428169
$$870$$ 0 0
$$871$$ −7.80136e6 −0.348438
$$872$$ 2.16185e7 0.962797
$$873$$ 0 0
$$874$$ −4.53766e6 −0.200934
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 1.73868e7 0.763344 0.381672 0.924298i $$-0.375348\pi$$
0.381672 + 0.924298i $$0.375348\pi$$
$$878$$ −1.19064e6 −0.0521248
$$879$$ 0 0
$$880$$ 3.58255e7 1.55950
$$881$$ 8.14472e6 0.353538 0.176769 0.984252i $$-0.443435\pi$$
0.176769 + 0.984252i $$0.443435\pi$$
$$882$$ 0 0
$$883$$ −3.10298e7 −1.33930 −0.669649 0.742678i $$-0.733555\pi$$
−0.669649 + 0.742678i $$0.733555\pi$$
$$884$$ −1.96087e6 −0.0843953
$$885$$ 0 0
$$886$$ −2.01258e7 −0.861327
$$887$$ 1.47028e7 0.627465 0.313733 0.949511i $$-0.398420\pi$$
0.313733 + 0.949511i $$0.398420\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 2.46386e7 1.04266
$$891$$ 0 0
$$892$$ −2.13132e7 −0.896885
$$893$$ 2.30657e6 0.0967918
$$894$$ 0 0
$$895$$ −1.11166e7 −0.463890
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −1.00269e7 −0.414930
$$899$$ −2.00450e7 −0.827193
$$900$$ 0 0
$$901$$ 3.52958e6 0.144848
$$902$$ −1.75215e7 −0.717060
$$903$$ 0 0
$$904$$ −3.88669e6 −0.158183
$$905$$ 2.44469e7 0.992208
$$906$$ 0 0
$$907$$ 1.30940e7 0.528512 0.264256 0.964453i $$-0.414874\pi$$
0.264256 + 0.964453i $$0.414874\pi$$
$$908$$ 1.65335e7 0.665504
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −2.68695e7 −1.07266 −0.536332 0.844007i $$-0.680191\pi$$
−0.536332 + 0.844007i $$0.680191\pi$$
$$912$$ 0 0
$$913$$ −2.80662e7 −1.11431
$$914$$ −1.10298e7 −0.436719
$$915$$ 0 0
$$916$$ 9.48103e6 0.373351
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 1.27317e6 0.0497277 0.0248638 0.999691i $$-0.492085\pi$$
0.0248638 + 0.999691i $$0.492085\pi$$
$$920$$ 3.32991e6 0.129707
$$921$$ 0 0
$$922$$ −3.15427e7 −1.22200
$$923$$ −4.19695e6 −0.162155
$$924$$ 0 0
$$925$$ 2.40187e6 0.0922986
$$926$$ −3.48658e7 −1.33620
$$927$$ 0 0
$$928$$ 4.59031e7 1.74973
$$929$$ 9.31705e6 0.354192 0.177096 0.984194i $$-0.443330\pi$$
0.177096 + 0.984194i $$0.443330\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 1.89428e7 0.714338
$$933$$ 0 0
$$934$$ 3.60577e7 1.35248
$$935$$ −1.03960e7 −0.388900
$$936$$ 0 0
$$937$$ 1.18158e7 0.439657 0.219829 0.975539i $$-0.429450\pi$$
0.219829 + 0.975539i $$0.429450\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 2.22256e6 0.0820416
$$941$$ 2.53529e7 0.933371 0.466685 0.884423i $$-0.345448\pi$$
0.466685 + 0.884423i $$0.345448\pi$$
$$942$$ 0 0
$$943$$ −2.99268e6 −0.109592
$$944$$ −3.78436e7 −1.38217
$$945$$ 0 0
$$946$$ 4.57422e6 0.166184
$$947$$ −2.64941e7 −0.960008 −0.480004 0.877266i $$-0.659365\pi$$
−0.480004 + 0.877266i $$0.659365\pi$$
$$948$$ 0 0
$$949$$ 2.04455e7 0.736941
$$950$$ −3.46147e7 −1.24437
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 1.88335e7 0.671735 0.335868 0.941909i $$-0.390970\pi$$
0.335868 + 0.941909i $$0.390970\pi$$
$$954$$ 0 0
$$955$$ −1.81603e7 −0.644339
$$956$$ 2.78745e7 0.986423
$$957$$ 0 0
$$958$$ −5.87929e7 −2.06972
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −2.19976e7 −0.768365
$$962$$ 1.52799e6 0.0532332
$$963$$ 0 0
$$964$$ −1.83032e7 −0.634357
$$965$$ −5.36740e7 −1.85544
$$966$$ 0 0
$$967$$ −3.14956e7 −1.08314 −0.541569 0.840656i $$-0.682169\pi$$
−0.541569 + 0.840656i $$0.682169\pi$$
$$968$$ −3.64655e6 −0.125082
$$969$$ 0 0
$$970$$ −5.84475e7 −1.99451
$$971$$ 6.85669e6 0.233381 0.116691 0.993168i $$-0.462771\pi$$
0.116691 + 0.993168i $$0.462771\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −6.11527e7 −2.06547
$$975$$ 0 0
$$976$$ 5.93157e7 1.99317
$$977$$ −2.81471e7 −0.943402 −0.471701 0.881759i $$-0.656360\pi$$
−0.471701 + 0.881759i $$0.656360\pi$$
$$978$$ 0 0
$$979$$ 1.53350e7 0.511361
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 673142. 0.0222755
$$983$$ −2.34916e7 −0.775406 −0.387703 0.921784i $$-0.626731\pi$$
−0.387703 + 0.921784i $$0.626731\pi$$
$$984$$ 0 0
$$985$$ 102399. 0.00336285
$$986$$ −2.04029e7 −0.668343
$$987$$ 0 0
$$988$$ −7.97395e6 −0.259885
$$989$$ 781278. 0.0253989
$$990$$ 0 0
$$991$$ −2.14412e6 −0.0693530 −0.0346765 0.999399i $$-0.511040\pi$$
−0.0346765 + 0.999399i $$0.511040\pi$$
$$992$$ −1.51862e7 −0.489971
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 2.94483e7 0.942979
$$996$$ 0 0
$$997$$ 2.50872e7 0.799307 0.399654 0.916666i $$-0.369130\pi$$
0.399654 + 0.916666i $$0.369130\pi$$
$$998$$ −1.51715e7 −0.482173
$$999$$ 0 0
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 441.6.a.n.1.1 2
3.2 odd 2 49.6.a.d.1.2 2
7.2 even 3 63.6.e.d.46.2 4
7.4 even 3 63.6.e.d.37.2 4
7.6 odd 2 441.6.a.m.1.1 2
12.11 even 2 784.6.a.ba.1.2 2
21.2 odd 6 7.6.c.a.4.1 yes 4
21.5 even 6 49.6.c.f.18.1 4
21.11 odd 6 7.6.c.a.2.1 4
21.17 even 6 49.6.c.f.30.1 4
21.20 even 2 49.6.a.e.1.2 2
84.11 even 6 112.6.i.c.65.1 4
84.23 even 6 112.6.i.c.81.1 4
84.83 odd 2 784.6.a.t.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.c.a.2.1 4 21.11 odd 6
7.6.c.a.4.1 yes 4 21.2 odd 6
49.6.a.d.1.2 2 3.2 odd 2
49.6.a.e.1.2 2 21.20 even 2
49.6.c.f.18.1 4 21.5 even 6
49.6.c.f.30.1 4 21.17 even 6
63.6.e.d.37.2 4 7.4 even 3
63.6.e.d.46.2 4 7.2 even 3
112.6.i.c.65.1 4 84.11 even 6
112.6.i.c.81.1 4 84.23 even 6
441.6.a.m.1.1 2 7.6 odd 2
441.6.a.n.1.1 2 1.1 even 1 trivial
784.6.a.t.1.1 2 84.83 odd 2
784.6.a.ba.1.2 2 12.11 even 2