# Properties

 Label 3822.2.a.q.1.1 Level $3822$ Weight $2$ Character 3822.1 Self dual yes Analytic conductor $30.519$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3822,2,Mod(1,3822)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3822, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3822.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3822.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: $$30.5188236525$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3822.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +3.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} +3.00000 q^{19} +2.00000 q^{20} -3.00000 q^{22} -1.00000 q^{24} -1.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} +3.00000 q^{29} -2.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -3.00000 q^{38} +1.00000 q^{39} -2.00000 q^{40} -2.00000 q^{41} +6.00000 q^{43} +3.00000 q^{44} +2.00000 q^{45} +9.00000 q^{47} +1.00000 q^{48} +1.00000 q^{50} -1.00000 q^{51} +1.00000 q^{52} +1.00000 q^{53} -1.00000 q^{54} +6.00000 q^{55} +3.00000 q^{57} -3.00000 q^{58} -11.0000 q^{59} +2.00000 q^{60} -11.0000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -3.00000 q^{66} -7.00000 q^{67} -1.00000 q^{68} +15.0000 q^{71} -1.00000 q^{72} +12.0000 q^{73} +2.00000 q^{74} -1.00000 q^{75} +3.00000 q^{76} -1.00000 q^{78} +2.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -2.00000 q^{85} -6.00000 q^{86} +3.00000 q^{87} -3.00000 q^{88} -10.0000 q^{89} -2.00000 q^{90} +4.00000 q^{93} -9.00000 q^{94} +6.00000 q^{95} -1.00000 q^{96} +12.0000 q^{97} +3.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ −2.00000 −0.632456
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ 2.00000 0.516398
$$16$$ 1.00000 0.250000
$$17$$ −1.00000 −0.242536 −0.121268 0.992620i $$-0.538696\pi$$
−0.121268 + 0.992620i $$0.538696\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 3.00000 0.688247 0.344124 0.938924i $$-0.388176\pi$$
0.344124 + 0.938924i $$0.388176\pi$$
$$20$$ 2.00000 0.447214
$$21$$ 0 0
$$22$$ −3.00000 −0.639602
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ −1.00000 −0.200000
$$26$$ −1.00000 −0.196116
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ −2.00000 −0.365148
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 3.00000 0.522233
$$34$$ 1.00000 0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ −3.00000 −0.486664
$$39$$ 1.00000 0.160128
$$40$$ −2.00000 −0.316228
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ 6.00000 0.914991 0.457496 0.889212i $$-0.348747\pi$$
0.457496 + 0.889212i $$0.348747\pi$$
$$44$$ 3.00000 0.452267
$$45$$ 2.00000 0.298142
$$46$$ 0 0
$$47$$ 9.00000 1.31278 0.656392 0.754420i $$-0.272082\pi$$
0.656392 + 0.754420i $$0.272082\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 0 0
$$50$$ 1.00000 0.141421
$$51$$ −1.00000 −0.140028
$$52$$ 1.00000 0.138675
$$53$$ 1.00000 0.137361 0.0686803 0.997639i $$-0.478121\pi$$
0.0686803 + 0.997639i $$0.478121\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 6.00000 0.809040
$$56$$ 0 0
$$57$$ 3.00000 0.397360
$$58$$ −3.00000 −0.393919
$$59$$ −11.0000 −1.43208 −0.716039 0.698060i $$-0.754047\pi$$
−0.716039 + 0.698060i $$0.754047\pi$$
$$60$$ 2.00000 0.258199
$$61$$ −11.0000 −1.40841 −0.704203 0.709999i $$-0.748695\pi$$
−0.704203 + 0.709999i $$0.748695\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 2.00000 0.248069
$$66$$ −3.00000 −0.369274
$$67$$ −7.00000 −0.855186 −0.427593 0.903971i $$-0.640638\pi$$
−0.427593 + 0.903971i $$0.640638\pi$$
$$68$$ −1.00000 −0.121268
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 15.0000 1.78017 0.890086 0.455792i $$-0.150644\pi$$
0.890086 + 0.455792i $$0.150644\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 12.0000 1.40449 0.702247 0.711934i $$-0.252180\pi$$
0.702247 + 0.711934i $$0.252180\pi$$
$$74$$ 2.00000 0.232495
$$75$$ −1.00000 −0.115470
$$76$$ 3.00000 0.344124
$$77$$ 0 0
$$78$$ −1.00000 −0.113228
$$79$$ 2.00000 0.225018 0.112509 0.993651i $$-0.464111\pi$$
0.112509 + 0.993651i $$0.464111\pi$$
$$80$$ 2.00000 0.223607
$$81$$ 1.00000 0.111111
$$82$$ 2.00000 0.220863
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ −2.00000 −0.216930
$$86$$ −6.00000 −0.646997
$$87$$ 3.00000 0.321634
$$88$$ −3.00000 −0.319801
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ −2.00000 −0.210819
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 4.00000 0.414781
$$94$$ −9.00000 −0.928279
$$95$$ 6.00000 0.615587
$$96$$ −1.00000 −0.102062
$$97$$ 12.0000 1.21842 0.609208 0.793011i $$-0.291488\pi$$
0.609208 + 0.793011i $$0.291488\pi$$
$$98$$ 0 0
$$99$$ 3.00000 0.301511
$$100$$ −1.00000 −0.100000
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 1.00000 0.0990148
$$103$$ −14.0000 −1.37946 −0.689730 0.724066i $$-0.742271\pi$$
−0.689730 + 0.724066i $$0.742271\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ −1.00000 −0.0971286
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −8.00000 −0.766261 −0.383131 0.923694i $$-0.625154\pi$$
−0.383131 + 0.923694i $$0.625154\pi$$
$$110$$ −6.00000 −0.572078
$$111$$ −2.00000 −0.189832
$$112$$ 0 0
$$113$$ 3.00000 0.282216 0.141108 0.989994i $$-0.454933\pi$$
0.141108 + 0.989994i $$0.454933\pi$$
$$114$$ −3.00000 −0.280976
$$115$$ 0 0
$$116$$ 3.00000 0.278543
$$117$$ 1.00000 0.0924500
$$118$$ 11.0000 1.01263
$$119$$ 0 0
$$120$$ −2.00000 −0.182574
$$121$$ −2.00000 −0.181818
$$122$$ 11.0000 0.995893
$$123$$ −2.00000 −0.180334
$$124$$ 4.00000 0.359211
$$125$$ −12.0000 −1.07331
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 6.00000 0.528271
$$130$$ −2.00000 −0.175412
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 3.00000 0.261116
$$133$$ 0 0
$$134$$ 7.00000 0.604708
$$135$$ 2.00000 0.172133
$$136$$ 1.00000 0.0857493
$$137$$ 18.0000 1.53784 0.768922 0.639343i $$-0.220793\pi$$
0.768922 + 0.639343i $$0.220793\pi$$
$$138$$ 0 0
$$139$$ 14.0000 1.18746 0.593732 0.804663i $$-0.297654\pi$$
0.593732 + 0.804663i $$0.297654\pi$$
$$140$$ 0 0
$$141$$ 9.00000 0.757937
$$142$$ −15.0000 −1.25877
$$143$$ 3.00000 0.250873
$$144$$ 1.00000 0.0833333
$$145$$ 6.00000 0.498273
$$146$$ −12.0000 −0.993127
$$147$$ 0 0
$$148$$ −2.00000 −0.164399
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ −17.0000 −1.38344 −0.691720 0.722166i $$-0.743147\pi$$
−0.691720 + 0.722166i $$0.743147\pi$$
$$152$$ −3.00000 −0.243332
$$153$$ −1.00000 −0.0808452
$$154$$ 0 0
$$155$$ 8.00000 0.642575
$$156$$ 1.00000 0.0800641
$$157$$ 11.0000 0.877896 0.438948 0.898513i $$-0.355351\pi$$
0.438948 + 0.898513i $$0.355351\pi$$
$$158$$ −2.00000 −0.159111
$$159$$ 1.00000 0.0793052
$$160$$ −2.00000 −0.158114
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ −25.0000 −1.95815 −0.979076 0.203497i $$-0.934769\pi$$
−0.979076 + 0.203497i $$0.934769\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 6.00000 0.467099
$$166$$ 0 0
$$167$$ 19.0000 1.47026 0.735132 0.677924i $$-0.237120\pi$$
0.735132 + 0.677924i $$0.237120\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 2.00000 0.153393
$$171$$ 3.00000 0.229416
$$172$$ 6.00000 0.457496
$$173$$ −7.00000 −0.532200 −0.266100 0.963945i $$-0.585735\pi$$
−0.266100 + 0.963945i $$0.585735\pi$$
$$174$$ −3.00000 −0.227429
$$175$$ 0 0
$$176$$ 3.00000 0.226134
$$177$$ −11.0000 −0.826811
$$178$$ 10.0000 0.749532
$$179$$ −6.00000 −0.448461 −0.224231 0.974536i $$-0.571987\pi$$
−0.224231 + 0.974536i $$0.571987\pi$$
$$180$$ 2.00000 0.149071
$$181$$ 5.00000 0.371647 0.185824 0.982583i $$-0.440505\pi$$
0.185824 + 0.982583i $$0.440505\pi$$
$$182$$ 0 0
$$183$$ −11.0000 −0.813143
$$184$$ 0 0
$$185$$ −4.00000 −0.294086
$$186$$ −4.00000 −0.293294
$$187$$ −3.00000 −0.219382
$$188$$ 9.00000 0.656392
$$189$$ 0 0
$$190$$ −6.00000 −0.435286
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ −12.0000 −0.861550
$$195$$ 2.00000 0.143223
$$196$$ 0 0
$$197$$ 8.00000 0.569976 0.284988 0.958531i $$-0.408010\pi$$
0.284988 + 0.958531i $$0.408010\pi$$
$$198$$ −3.00000 −0.213201
$$199$$ −12.0000 −0.850657 −0.425329 0.905039i $$-0.639842\pi$$
−0.425329 + 0.905039i $$0.639842\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ −7.00000 −0.493742
$$202$$ −14.0000 −0.985037
$$203$$ 0 0
$$204$$ −1.00000 −0.0700140
$$205$$ −4.00000 −0.279372
$$206$$ 14.0000 0.975426
$$207$$ 0 0
$$208$$ 1.00000 0.0693375
$$209$$ 9.00000 0.622543
$$210$$ 0 0
$$211$$ −16.0000 −1.10149 −0.550743 0.834675i $$-0.685655\pi$$
−0.550743 + 0.834675i $$0.685655\pi$$
$$212$$ 1.00000 0.0686803
$$213$$ 15.0000 1.02778
$$214$$ 4.00000 0.273434
$$215$$ 12.0000 0.818393
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ 8.00000 0.541828
$$219$$ 12.0000 0.810885
$$220$$ 6.00000 0.404520
$$221$$ −1.00000 −0.0672673
$$222$$ 2.00000 0.134231
$$223$$ 1.00000 0.0669650 0.0334825 0.999439i $$-0.489340\pi$$
0.0334825 + 0.999439i $$0.489340\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ −3.00000 −0.199557
$$227$$ −8.00000 −0.530979 −0.265489 0.964114i $$-0.585534\pi$$
−0.265489 + 0.964114i $$0.585534\pi$$
$$228$$ 3.00000 0.198680
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −3.00000 −0.196960
$$233$$ 7.00000 0.458585 0.229293 0.973358i $$-0.426359\pi$$
0.229293 + 0.973358i $$0.426359\pi$$
$$234$$ −1.00000 −0.0653720
$$235$$ 18.0000 1.17419
$$236$$ −11.0000 −0.716039
$$237$$ 2.00000 0.129914
$$238$$ 0 0
$$239$$ −11.0000 −0.711531 −0.355765 0.934575i $$-0.615780\pi$$
−0.355765 + 0.934575i $$0.615780\pi$$
$$240$$ 2.00000 0.129099
$$241$$ −12.0000 −0.772988 −0.386494 0.922292i $$-0.626314\pi$$
−0.386494 + 0.922292i $$0.626314\pi$$
$$242$$ 2.00000 0.128565
$$243$$ 1.00000 0.0641500
$$244$$ −11.0000 −0.704203
$$245$$ 0 0
$$246$$ 2.00000 0.127515
$$247$$ 3.00000 0.190885
$$248$$ −4.00000 −0.254000
$$249$$ 0 0
$$250$$ 12.0000 0.758947
$$251$$ 10.0000 0.631194 0.315597 0.948893i $$-0.397795\pi$$
0.315597 + 0.948893i $$0.397795\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −8.00000 −0.501965
$$255$$ −2.00000 −0.125245
$$256$$ 1.00000 0.0625000
$$257$$ 26.0000 1.62184 0.810918 0.585160i $$-0.198968\pi$$
0.810918 + 0.585160i $$0.198968\pi$$
$$258$$ −6.00000 −0.373544
$$259$$ 0 0
$$260$$ 2.00000 0.124035
$$261$$ 3.00000 0.185695
$$262$$ −12.0000 −0.741362
$$263$$ 10.0000 0.616626 0.308313 0.951285i $$-0.400236\pi$$
0.308313 + 0.951285i $$0.400236\pi$$
$$264$$ −3.00000 −0.184637
$$265$$ 2.00000 0.122859
$$266$$ 0 0
$$267$$ −10.0000 −0.611990
$$268$$ −7.00000 −0.427593
$$269$$ 21.0000 1.28039 0.640196 0.768211i $$-0.278853\pi$$
0.640196 + 0.768211i $$0.278853\pi$$
$$270$$ −2.00000 −0.121716
$$271$$ −15.0000 −0.911185 −0.455593 0.890188i $$-0.650573\pi$$
−0.455593 + 0.890188i $$0.650573\pi$$
$$272$$ −1.00000 −0.0606339
$$273$$ 0 0
$$274$$ −18.0000 −1.08742
$$275$$ −3.00000 −0.180907
$$276$$ 0 0
$$277$$ 5.00000 0.300421 0.150210 0.988654i $$-0.452005\pi$$
0.150210 + 0.988654i $$0.452005\pi$$
$$278$$ −14.0000 −0.839664
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ −24.0000 −1.43172 −0.715860 0.698244i $$-0.753965\pi$$
−0.715860 + 0.698244i $$0.753965\pi$$
$$282$$ −9.00000 −0.535942
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 15.0000 0.890086
$$285$$ 6.00000 0.355409
$$286$$ −3.00000 −0.177394
$$287$$ 0 0
$$288$$ −1.00000 −0.0589256
$$289$$ −16.0000 −0.941176
$$290$$ −6.00000 −0.352332
$$291$$ 12.0000 0.703452
$$292$$ 12.0000 0.702247
$$293$$ 12.0000 0.701047 0.350524 0.936554i $$-0.386004\pi$$
0.350524 + 0.936554i $$0.386004\pi$$
$$294$$ 0 0
$$295$$ −22.0000 −1.28089
$$296$$ 2.00000 0.116248
$$297$$ 3.00000 0.174078
$$298$$ 0 0
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ 0 0
$$302$$ 17.0000 0.978240
$$303$$ 14.0000 0.804279
$$304$$ 3.00000 0.172062
$$305$$ −22.0000 −1.25972
$$306$$ 1.00000 0.0571662
$$307$$ 15.0000 0.856095 0.428048 0.903756i $$-0.359202\pi$$
0.428048 + 0.903756i $$0.359202\pi$$
$$308$$ 0 0
$$309$$ −14.0000 −0.796432
$$310$$ −8.00000 −0.454369
$$311$$ −14.0000 −0.793867 −0.396934 0.917847i $$-0.629926\pi$$
−0.396934 + 0.917847i $$0.629926\pi$$
$$312$$ −1.00000 −0.0566139
$$313$$ 14.0000 0.791327 0.395663 0.918396i $$-0.370515\pi$$
0.395663 + 0.918396i $$0.370515\pi$$
$$314$$ −11.0000 −0.620766
$$315$$ 0 0
$$316$$ 2.00000 0.112509
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ −1.00000 −0.0560772
$$319$$ 9.00000 0.503903
$$320$$ 2.00000 0.111803
$$321$$ −4.00000 −0.223258
$$322$$ 0 0
$$323$$ −3.00000 −0.166924
$$324$$ 1.00000 0.0555556
$$325$$ −1.00000 −0.0554700
$$326$$ 25.0000 1.38462
$$327$$ −8.00000 −0.442401
$$328$$ 2.00000 0.110432
$$329$$ 0 0
$$330$$ −6.00000 −0.330289
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ 0 0
$$333$$ −2.00000 −0.109599
$$334$$ −19.0000 −1.03963
$$335$$ −14.0000 −0.764902
$$336$$ 0 0
$$337$$ 7.00000 0.381314 0.190657 0.981657i $$-0.438938\pi$$
0.190657 + 0.981657i $$0.438938\pi$$
$$338$$ −1.00000 −0.0543928
$$339$$ 3.00000 0.162938
$$340$$ −2.00000 −0.108465
$$341$$ 12.0000 0.649836
$$342$$ −3.00000 −0.162221
$$343$$ 0 0
$$344$$ −6.00000 −0.323498
$$345$$ 0 0
$$346$$ 7.00000 0.376322
$$347$$ 2.00000 0.107366 0.0536828 0.998558i $$-0.482904\pi$$
0.0536828 + 0.998558i $$0.482904\pi$$
$$348$$ 3.00000 0.160817
$$349$$ −20.0000 −1.07058 −0.535288 0.844670i $$-0.679797\pi$$
−0.535288 + 0.844670i $$0.679797\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ −3.00000 −0.159901
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 11.0000 0.584643
$$355$$ 30.0000 1.59223
$$356$$ −10.0000 −0.529999
$$357$$ 0 0
$$358$$ 6.00000 0.317110
$$359$$ 16.0000 0.844448 0.422224 0.906492i $$-0.361250\pi$$
0.422224 + 0.906492i $$0.361250\pi$$
$$360$$ −2.00000 −0.105409
$$361$$ −10.0000 −0.526316
$$362$$ −5.00000 −0.262794
$$363$$ −2.00000 −0.104973
$$364$$ 0 0
$$365$$ 24.0000 1.25622
$$366$$ 11.0000 0.574979
$$367$$ −18.0000 −0.939592 −0.469796 0.882775i $$-0.655673\pi$$
−0.469796 + 0.882775i $$0.655673\pi$$
$$368$$ 0 0
$$369$$ −2.00000 −0.104116
$$370$$ 4.00000 0.207950
$$371$$ 0 0
$$372$$ 4.00000 0.207390
$$373$$ −11.0000 −0.569558 −0.284779 0.958593i $$-0.591920\pi$$
−0.284779 + 0.958593i $$0.591920\pi$$
$$374$$ 3.00000 0.155126
$$375$$ −12.0000 −0.619677
$$376$$ −9.00000 −0.464140
$$377$$ 3.00000 0.154508
$$378$$ 0 0
$$379$$ 36.0000 1.84920 0.924598 0.380945i $$-0.124401\pi$$
0.924598 + 0.380945i $$0.124401\pi$$
$$380$$ 6.00000 0.307794
$$381$$ 8.00000 0.409852
$$382$$ 12.0000 0.613973
$$383$$ 32.0000 1.63512 0.817562 0.575841i $$-0.195325\pi$$
0.817562 + 0.575841i $$0.195325\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 14.0000 0.712581
$$387$$ 6.00000 0.304997
$$388$$ 12.0000 0.609208
$$389$$ 13.0000 0.659126 0.329563 0.944134i $$-0.393099\pi$$
0.329563 + 0.944134i $$0.393099\pi$$
$$390$$ −2.00000 −0.101274
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 12.0000 0.605320
$$394$$ −8.00000 −0.403034
$$395$$ 4.00000 0.201262
$$396$$ 3.00000 0.150756
$$397$$ 20.0000 1.00377 0.501886 0.864934i $$-0.332640\pi$$
0.501886 + 0.864934i $$0.332640\pi$$
$$398$$ 12.0000 0.601506
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ 12.0000 0.599251 0.299626 0.954057i $$-0.403138\pi$$
0.299626 + 0.954057i $$0.403138\pi$$
$$402$$ 7.00000 0.349128
$$403$$ 4.00000 0.199254
$$404$$ 14.0000 0.696526
$$405$$ 2.00000 0.0993808
$$406$$ 0 0
$$407$$ −6.00000 −0.297409
$$408$$ 1.00000 0.0495074
$$409$$ 38.0000 1.87898 0.939490 0.342578i $$-0.111300\pi$$
0.939490 + 0.342578i $$0.111300\pi$$
$$410$$ 4.00000 0.197546
$$411$$ 18.0000 0.887875
$$412$$ −14.0000 −0.689730
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ 14.0000 0.685583
$$418$$ −9.00000 −0.440204
$$419$$ 6.00000 0.293119 0.146560 0.989202i $$-0.453180\pi$$
0.146560 + 0.989202i $$0.453180\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$422$$ 16.0000 0.778868
$$423$$ 9.00000 0.437595
$$424$$ −1.00000 −0.0485643
$$425$$ 1.00000 0.0485071
$$426$$ −15.0000 −0.726752
$$427$$ 0 0
$$428$$ −4.00000 −0.193347
$$429$$ 3.00000 0.144841
$$430$$ −12.0000 −0.578691
$$431$$ −40.0000 −1.92673 −0.963366 0.268190i $$-0.913575\pi$$
−0.963366 + 0.268190i $$0.913575\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 11.0000 0.528626 0.264313 0.964437i $$-0.414855\pi$$
0.264313 + 0.964437i $$0.414855\pi$$
$$434$$ 0 0
$$435$$ 6.00000 0.287678
$$436$$ −8.00000 −0.383131
$$437$$ 0 0
$$438$$ −12.0000 −0.573382
$$439$$ −16.0000 −0.763638 −0.381819 0.924237i $$-0.624702\pi$$
−0.381819 + 0.924237i $$0.624702\pi$$
$$440$$ −6.00000 −0.286039
$$441$$ 0 0
$$442$$ 1.00000 0.0475651
$$443$$ 6.00000 0.285069 0.142534 0.989790i $$-0.454475\pi$$
0.142534 + 0.989790i $$0.454475\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ −20.0000 −0.948091
$$446$$ −1.00000 −0.0473514
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −40.0000 −1.88772 −0.943858 0.330350i $$-0.892833\pi$$
−0.943858 + 0.330350i $$0.892833\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ −6.00000 −0.282529
$$452$$ 3.00000 0.141108
$$453$$ −17.0000 −0.798730
$$454$$ 8.00000 0.375459
$$455$$ 0 0
$$456$$ −3.00000 −0.140488
$$457$$ −32.0000 −1.49690 −0.748448 0.663193i $$-0.769201\pi$$
−0.748448 + 0.663193i $$0.769201\pi$$
$$458$$ −2.00000 −0.0934539
$$459$$ −1.00000 −0.0466760
$$460$$ 0 0
$$461$$ 20.0000 0.931493 0.465746 0.884918i $$-0.345786\pi$$
0.465746 + 0.884918i $$0.345786\pi$$
$$462$$ 0 0
$$463$$ −32.0000 −1.48717 −0.743583 0.668644i $$-0.766875\pi$$
−0.743583 + 0.668644i $$0.766875\pi$$
$$464$$ 3.00000 0.139272
$$465$$ 8.00000 0.370991
$$466$$ −7.00000 −0.324269
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 1.00000 0.0462250
$$469$$ 0 0
$$470$$ −18.0000 −0.830278
$$471$$ 11.0000 0.506853
$$472$$ 11.0000 0.506316
$$473$$ 18.0000 0.827641
$$474$$ −2.00000 −0.0918630
$$475$$ −3.00000 −0.137649
$$476$$ 0 0
$$477$$ 1.00000 0.0457869
$$478$$ 11.0000 0.503128
$$479$$ −15.0000 −0.685367 −0.342684 0.939451i $$-0.611336\pi$$
−0.342684 + 0.939451i $$0.611336\pi$$
$$480$$ −2.00000 −0.0912871
$$481$$ −2.00000 −0.0911922
$$482$$ 12.0000 0.546585
$$483$$ 0 0
$$484$$ −2.00000 −0.0909091
$$485$$ 24.0000 1.08978
$$486$$ −1.00000 −0.0453609
$$487$$ −33.0000 −1.49537 −0.747686 0.664052i $$-0.768835\pi$$
−0.747686 + 0.664052i $$0.768835\pi$$
$$488$$ 11.0000 0.497947
$$489$$ −25.0000 −1.13054
$$490$$ 0 0
$$491$$ 22.0000 0.992846 0.496423 0.868081i $$-0.334646\pi$$
0.496423 + 0.868081i $$0.334646\pi$$
$$492$$ −2.00000 −0.0901670
$$493$$ −3.00000 −0.135113
$$494$$ −3.00000 −0.134976
$$495$$ 6.00000 0.269680
$$496$$ 4.00000 0.179605
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 28.0000 1.25345 0.626726 0.779240i $$-0.284395\pi$$
0.626726 + 0.779240i $$0.284395\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ 19.0000 0.848857
$$502$$ −10.0000 −0.446322
$$503$$ 20.0000 0.891756 0.445878 0.895094i $$-0.352892\pi$$
0.445878 + 0.895094i $$0.352892\pi$$
$$504$$ 0 0
$$505$$ 28.0000 1.24598
$$506$$ 0 0
$$507$$ 1.00000 0.0444116
$$508$$ 8.00000 0.354943
$$509$$ −12.0000 −0.531891 −0.265945 0.963988i $$-0.585684\pi$$
−0.265945 + 0.963988i $$0.585684\pi$$
$$510$$ 2.00000 0.0885615
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 3.00000 0.132453
$$514$$ −26.0000 −1.14681
$$515$$ −28.0000 −1.23383
$$516$$ 6.00000 0.264135
$$517$$ 27.0000 1.18746
$$518$$ 0 0
$$519$$ −7.00000 −0.307266
$$520$$ −2.00000 −0.0877058
$$521$$ 38.0000 1.66481 0.832405 0.554168i $$-0.186963\pi$$
0.832405 + 0.554168i $$0.186963\pi$$
$$522$$ −3.00000 −0.131306
$$523$$ −6.00000 −0.262362 −0.131181 0.991358i $$-0.541877\pi$$
−0.131181 + 0.991358i $$0.541877\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ −10.0000 −0.436021
$$527$$ −4.00000 −0.174243
$$528$$ 3.00000 0.130558
$$529$$ −23.0000 −1.00000
$$530$$ −2.00000 −0.0868744
$$531$$ −11.0000 −0.477359
$$532$$ 0 0
$$533$$ −2.00000 −0.0866296
$$534$$ 10.0000 0.432742
$$535$$ −8.00000 −0.345870
$$536$$ 7.00000 0.302354
$$537$$ −6.00000 −0.258919
$$538$$ −21.0000 −0.905374
$$539$$ 0 0
$$540$$ 2.00000 0.0860663
$$541$$ 22.0000 0.945854 0.472927 0.881102i $$-0.343197\pi$$
0.472927 + 0.881102i $$0.343197\pi$$
$$542$$ 15.0000 0.644305
$$543$$ 5.00000 0.214571
$$544$$ 1.00000 0.0428746
$$545$$ −16.0000 −0.685365
$$546$$ 0 0
$$547$$ 44.0000 1.88130 0.940652 0.339372i $$-0.110215\pi$$
0.940652 + 0.339372i $$0.110215\pi$$
$$548$$ 18.0000 0.768922
$$549$$ −11.0000 −0.469469
$$550$$ 3.00000 0.127920
$$551$$ 9.00000 0.383413
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −5.00000 −0.212430
$$555$$ −4.00000 −0.169791
$$556$$ 14.0000 0.593732
$$557$$ −22.0000 −0.932170 −0.466085 0.884740i $$-0.654336\pi$$
−0.466085 + 0.884740i $$0.654336\pi$$
$$558$$ −4.00000 −0.169334
$$559$$ 6.00000 0.253773
$$560$$ 0 0
$$561$$ −3.00000 −0.126660
$$562$$ 24.0000 1.01238
$$563$$ −24.0000 −1.01148 −0.505740 0.862686i $$-0.668780\pi$$
−0.505740 + 0.862686i $$0.668780\pi$$
$$564$$ 9.00000 0.378968
$$565$$ 6.00000 0.252422
$$566$$ −4.00000 −0.168133
$$567$$ 0 0
$$568$$ −15.0000 −0.629386
$$569$$ 39.0000 1.63497 0.817483 0.575953i $$-0.195369\pi$$
0.817483 + 0.575953i $$0.195369\pi$$
$$570$$ −6.00000 −0.251312
$$571$$ 44.0000 1.84134 0.920671 0.390339i $$-0.127642\pi$$
0.920671 + 0.390339i $$0.127642\pi$$
$$572$$ 3.00000 0.125436
$$573$$ −12.0000 −0.501307
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 18.0000 0.749350 0.374675 0.927156i $$-0.377754\pi$$
0.374675 + 0.927156i $$0.377754\pi$$
$$578$$ 16.0000 0.665512
$$579$$ −14.0000 −0.581820
$$580$$ 6.00000 0.249136
$$581$$ 0 0
$$582$$ −12.0000 −0.497416
$$583$$ 3.00000 0.124247
$$584$$ −12.0000 −0.496564
$$585$$ 2.00000 0.0826898
$$586$$ −12.0000 −0.495715
$$587$$ −39.0000 −1.60970 −0.804851 0.593477i $$-0.797755\pi$$
−0.804851 + 0.593477i $$0.797755\pi$$
$$588$$ 0 0
$$589$$ 12.0000 0.494451
$$590$$ 22.0000 0.905726
$$591$$ 8.00000 0.329076
$$592$$ −2.00000 −0.0821995
$$593$$ 34.0000 1.39621 0.698106 0.715994i $$-0.254026\pi$$
0.698106 + 0.715994i $$0.254026\pi$$
$$594$$ −3.00000 −0.123091
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −12.0000 −0.491127
$$598$$ 0 0
$$599$$ −30.0000 −1.22577 −0.612883 0.790173i $$-0.709990\pi$$
−0.612883 + 0.790173i $$0.709990\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ 35.0000 1.42768 0.713840 0.700309i $$-0.246954\pi$$
0.713840 + 0.700309i $$0.246954\pi$$
$$602$$ 0 0
$$603$$ −7.00000 −0.285062
$$604$$ −17.0000 −0.691720
$$605$$ −4.00000 −0.162623
$$606$$ −14.0000 −0.568711
$$607$$ −28.0000 −1.13648 −0.568242 0.822861i $$-0.692376\pi$$
−0.568242 + 0.822861i $$0.692376\pi$$
$$608$$ −3.00000 −0.121666
$$609$$ 0 0
$$610$$ 22.0000 0.890754
$$611$$ 9.00000 0.364101
$$612$$ −1.00000 −0.0404226
$$613$$ −16.0000 −0.646234 −0.323117 0.946359i $$-0.604731\pi$$
−0.323117 + 0.946359i $$0.604731\pi$$
$$614$$ −15.0000 −0.605351
$$615$$ −4.00000 −0.161296
$$616$$ 0 0
$$617$$ −14.0000 −0.563619 −0.281809 0.959470i $$-0.590935\pi$$
−0.281809 + 0.959470i $$0.590935\pi$$
$$618$$ 14.0000 0.563163
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 8.00000 0.321288
$$621$$ 0 0
$$622$$ 14.0000 0.561349
$$623$$ 0 0
$$624$$ 1.00000 0.0400320
$$625$$ −19.0000 −0.760000
$$626$$ −14.0000 −0.559553
$$627$$ 9.00000 0.359425
$$628$$ 11.0000 0.438948
$$629$$ 2.00000 0.0797452
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ −2.00000 −0.0795557
$$633$$ −16.0000 −0.635943
$$634$$ −18.0000 −0.714871
$$635$$ 16.0000 0.634941
$$636$$ 1.00000 0.0396526
$$637$$ 0 0
$$638$$ −9.00000 −0.356313
$$639$$ 15.0000 0.593391
$$640$$ −2.00000 −0.0790569
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 4.00000 0.157867
$$643$$ −1.00000 −0.0394362 −0.0197181 0.999806i $$-0.506277\pi$$
−0.0197181 + 0.999806i $$0.506277\pi$$
$$644$$ 0 0
$$645$$ 12.0000 0.472500
$$646$$ 3.00000 0.118033
$$647$$ 6.00000 0.235884 0.117942 0.993020i $$-0.462370\pi$$
0.117942 + 0.993020i $$0.462370\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −33.0000 −1.29536
$$650$$ 1.00000 0.0392232
$$651$$ 0 0
$$652$$ −25.0000 −0.979076
$$653$$ −30.0000 −1.17399 −0.586995 0.809590i $$-0.699689\pi$$
−0.586995 + 0.809590i $$0.699689\pi$$
$$654$$ 8.00000 0.312825
$$655$$ 24.0000 0.937758
$$656$$ −2.00000 −0.0780869
$$657$$ 12.0000 0.468165
$$658$$ 0 0
$$659$$ 30.0000 1.16863 0.584317 0.811525i $$-0.301362\pi$$
0.584317 + 0.811525i $$0.301362\pi$$
$$660$$ 6.00000 0.233550
$$661$$ −38.0000 −1.47803 −0.739014 0.673690i $$-0.764708\pi$$
−0.739014 + 0.673690i $$0.764708\pi$$
$$662$$ 28.0000 1.08825
$$663$$ −1.00000 −0.0388368
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 2.00000 0.0774984
$$667$$ 0 0
$$668$$ 19.0000 0.735132
$$669$$ 1.00000 0.0386622
$$670$$ 14.0000 0.540867
$$671$$ −33.0000 −1.27395
$$672$$ 0 0
$$673$$ −34.0000 −1.31060 −0.655302 0.755367i $$-0.727459\pi$$
−0.655302 + 0.755367i $$0.727459\pi$$
$$674$$ −7.00000 −0.269630
$$675$$ −1.00000 −0.0384900
$$676$$ 1.00000 0.0384615
$$677$$ −1.00000 −0.0384331 −0.0192166 0.999815i $$-0.506117\pi$$
−0.0192166 + 0.999815i $$0.506117\pi$$
$$678$$ −3.00000 −0.115214
$$679$$ 0 0
$$680$$ 2.00000 0.0766965
$$681$$ −8.00000 −0.306561
$$682$$ −12.0000 −0.459504
$$683$$ −44.0000 −1.68361 −0.841807 0.539779i $$-0.818508\pi$$
−0.841807 + 0.539779i $$0.818508\pi$$
$$684$$ 3.00000 0.114708
$$685$$ 36.0000 1.37549
$$686$$ 0 0
$$687$$ 2.00000 0.0763048
$$688$$ 6.00000 0.228748
$$689$$ 1.00000 0.0380970
$$690$$ 0 0
$$691$$ −35.0000 −1.33146 −0.665731 0.746191i $$-0.731880\pi$$
−0.665731 + 0.746191i $$0.731880\pi$$
$$692$$ −7.00000 −0.266100
$$693$$ 0 0
$$694$$ −2.00000 −0.0759190
$$695$$ 28.0000 1.06210
$$696$$ −3.00000 −0.113715
$$697$$ 2.00000 0.0757554
$$698$$ 20.0000 0.757011
$$699$$ 7.00000 0.264764
$$700$$ 0 0
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ −1.00000 −0.0377426
$$703$$ −6.00000 −0.226294
$$704$$ 3.00000 0.113067
$$705$$ 18.0000 0.677919
$$706$$ −18.0000 −0.677439
$$707$$ 0 0
$$708$$ −11.0000 −0.413405
$$709$$ −20.0000 −0.751116 −0.375558 0.926799i $$-0.622549\pi$$
−0.375558 + 0.926799i $$0.622549\pi$$
$$710$$ −30.0000 −1.12588
$$711$$ 2.00000 0.0750059
$$712$$ 10.0000 0.374766
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 6.00000 0.224387
$$716$$ −6.00000 −0.224231
$$717$$ −11.0000 −0.410803
$$718$$ −16.0000 −0.597115
$$719$$ −38.0000 −1.41716 −0.708580 0.705630i $$-0.750664\pi$$
−0.708580 + 0.705630i $$0.750664\pi$$
$$720$$ 2.00000 0.0745356
$$721$$ 0 0
$$722$$ 10.0000 0.372161
$$723$$ −12.0000 −0.446285
$$724$$ 5.00000 0.185824
$$725$$ −3.00000 −0.111417
$$726$$ 2.00000 0.0742270
$$727$$ 12.0000 0.445055 0.222528 0.974926i $$-0.428569\pi$$
0.222528 + 0.974926i $$0.428569\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −24.0000 −0.888280
$$731$$ −6.00000 −0.221918
$$732$$ −11.0000 −0.406572
$$733$$ 8.00000 0.295487 0.147743 0.989026i $$-0.452799\pi$$
0.147743 + 0.989026i $$0.452799\pi$$
$$734$$ 18.0000 0.664392
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −21.0000 −0.773545
$$738$$ 2.00000 0.0736210
$$739$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$740$$ −4.00000 −0.147043
$$741$$ 3.00000 0.110208
$$742$$ 0 0
$$743$$ −35.0000 −1.28403 −0.642013 0.766694i $$-0.721900\pi$$
−0.642013 + 0.766694i $$0.721900\pi$$
$$744$$ −4.00000 −0.146647
$$745$$ 0 0
$$746$$ 11.0000 0.402739
$$747$$ 0 0
$$748$$ −3.00000 −0.109691
$$749$$ 0 0
$$750$$ 12.0000 0.438178
$$751$$ 28.0000 1.02173 0.510867 0.859660i $$-0.329324\pi$$
0.510867 + 0.859660i $$0.329324\pi$$
$$752$$ 9.00000 0.328196
$$753$$ 10.0000 0.364420
$$754$$ −3.00000 −0.109254
$$755$$ −34.0000 −1.23739
$$756$$ 0 0
$$757$$ 19.0000 0.690567 0.345283 0.938498i $$-0.387783\pi$$
0.345283 + 0.938498i $$0.387783\pi$$
$$758$$ −36.0000 −1.30758
$$759$$ 0 0
$$760$$ −6.00000 −0.217643
$$761$$ −28.0000 −1.01500 −0.507500 0.861652i $$-0.669430\pi$$
−0.507500 + 0.861652i $$0.669430\pi$$
$$762$$ −8.00000 −0.289809
$$763$$ 0 0
$$764$$ −12.0000 −0.434145
$$765$$ −2.00000 −0.0723102
$$766$$ −32.0000 −1.15621
$$767$$ −11.0000 −0.397187
$$768$$ 1.00000 0.0360844
$$769$$ 4.00000 0.144244 0.0721218 0.997396i $$-0.477023\pi$$
0.0721218 + 0.997396i $$0.477023\pi$$
$$770$$ 0 0
$$771$$ 26.0000 0.936367
$$772$$ −14.0000 −0.503871
$$773$$ −42.0000 −1.51064 −0.755318 0.655359i $$-0.772517\pi$$
−0.755318 + 0.655359i $$0.772517\pi$$
$$774$$ −6.00000 −0.215666
$$775$$ −4.00000 −0.143684
$$776$$ −12.0000 −0.430775
$$777$$ 0 0
$$778$$ −13.0000 −0.466073
$$779$$ −6.00000 −0.214972
$$780$$ 2.00000 0.0716115
$$781$$ 45.0000 1.61023
$$782$$ 0 0
$$783$$ 3.00000 0.107211
$$784$$ 0 0
$$785$$ 22.0000 0.785214
$$786$$ −12.0000 −0.428026
$$787$$ −7.00000 −0.249523 −0.124762 0.992187i $$-0.539817\pi$$
−0.124762 + 0.992187i $$0.539817\pi$$
$$788$$ 8.00000 0.284988
$$789$$ 10.0000 0.356009
$$790$$ −4.00000 −0.142314
$$791$$ 0 0
$$792$$ −3.00000 −0.106600
$$793$$ −11.0000 −0.390621
$$794$$ −20.0000 −0.709773
$$795$$ 2.00000 0.0709327
$$796$$ −12.0000 −0.425329
$$797$$ −34.0000 −1.20434 −0.602171 0.798367i $$-0.705697\pi$$
−0.602171 + 0.798367i $$0.705697\pi$$
$$798$$ 0 0
$$799$$ −9.00000 −0.318397
$$800$$ 1.00000 0.0353553
$$801$$ −10.0000 −0.353333
$$802$$ −12.0000 −0.423735
$$803$$ 36.0000 1.27041
$$804$$ −7.00000 −0.246871
$$805$$ 0 0
$$806$$ −4.00000 −0.140894
$$807$$ 21.0000 0.739235
$$808$$ −14.0000 −0.492518
$$809$$ 7.00000 0.246107 0.123053 0.992400i $$-0.460731\pi$$
0.123053 + 0.992400i $$0.460731\pi$$
$$810$$ −2.00000 −0.0702728
$$811$$ −12.0000 −0.421377 −0.210688 0.977553i $$-0.567571\pi$$
−0.210688 + 0.977553i $$0.567571\pi$$
$$812$$ 0 0
$$813$$ −15.0000 −0.526073
$$814$$ 6.00000 0.210300
$$815$$ −50.0000 −1.75142
$$816$$ −1.00000 −0.0350070
$$817$$ 18.0000 0.629740
$$818$$ −38.0000 −1.32864
$$819$$ 0 0
$$820$$ −4.00000 −0.139686
$$821$$ −6.00000 −0.209401 −0.104701 0.994504i $$-0.533388\pi$$
−0.104701 + 0.994504i $$0.533388\pi$$
$$822$$ −18.0000 −0.627822
$$823$$ 26.0000 0.906303 0.453152 0.891434i $$-0.350300\pi$$
0.453152 + 0.891434i $$0.350300\pi$$
$$824$$ 14.0000 0.487713
$$825$$ −3.00000 −0.104447
$$826$$ 0 0
$$827$$ −3.00000 −0.104320 −0.0521601 0.998639i $$-0.516611\pi$$
−0.0521601 + 0.998639i $$0.516611\pi$$
$$828$$ 0 0
$$829$$ −53.0000 −1.84077 −0.920383 0.391018i $$-0.872123\pi$$
−0.920383 + 0.391018i $$0.872123\pi$$
$$830$$ 0 0
$$831$$ 5.00000 0.173448
$$832$$ 1.00000 0.0346688
$$833$$ 0 0
$$834$$ −14.0000 −0.484780
$$835$$ 38.0000 1.31504
$$836$$ 9.00000 0.311272
$$837$$ 4.00000 0.138260
$$838$$ −6.00000 −0.207267
$$839$$ −23.0000 −0.794048 −0.397024 0.917808i $$-0.629957\pi$$
−0.397024 + 0.917808i $$0.629957\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 0 0
$$843$$ −24.0000 −0.826604
$$844$$ −16.0000 −0.550743
$$845$$ 2.00000 0.0688021
$$846$$ −9.00000 −0.309426
$$847$$ 0 0
$$848$$ 1.00000 0.0343401
$$849$$ 4.00000 0.137280
$$850$$ −1.00000 −0.0342997
$$851$$ 0 0
$$852$$ 15.0000 0.513892
$$853$$ 32.0000 1.09566 0.547830 0.836590i $$-0.315454\pi$$
0.547830 + 0.836590i $$0.315454\pi$$
$$854$$ 0 0
$$855$$ 6.00000 0.205196
$$856$$ 4.00000 0.136717
$$857$$ −7.00000 −0.239115 −0.119558 0.992827i $$-0.538148\pi$$
−0.119558 + 0.992827i $$0.538148\pi$$
$$858$$ −3.00000 −0.102418
$$859$$ −40.0000 −1.36478 −0.682391 0.730987i $$-0.739060\pi$$
−0.682391 + 0.730987i $$0.739060\pi$$
$$860$$ 12.0000 0.409197
$$861$$ 0 0
$$862$$ 40.0000 1.36241
$$863$$ 4.00000 0.136162 0.0680808 0.997680i $$-0.478312\pi$$
0.0680808 + 0.997680i $$0.478312\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −14.0000 −0.476014
$$866$$ −11.0000 −0.373795
$$867$$ −16.0000 −0.543388
$$868$$ 0 0
$$869$$ 6.00000 0.203536
$$870$$ −6.00000 −0.203419
$$871$$ −7.00000 −0.237186
$$872$$ 8.00000 0.270914
$$873$$ 12.0000 0.406138
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 12.0000 0.405442
$$877$$ −42.0000 −1.41824 −0.709120 0.705088i $$-0.750907\pi$$
−0.709120 + 0.705088i $$0.750907\pi$$
$$878$$ 16.0000 0.539974
$$879$$ 12.0000 0.404750
$$880$$ 6.00000 0.202260
$$881$$ 10.0000 0.336909 0.168454 0.985709i $$-0.446122\pi$$
0.168454 + 0.985709i $$0.446122\pi$$
$$882$$ 0 0
$$883$$ −14.0000 −0.471138 −0.235569 0.971858i $$-0.575695\pi$$
−0.235569 + 0.971858i $$0.575695\pi$$
$$884$$ −1.00000 −0.0336336
$$885$$ −22.0000 −0.739522
$$886$$ −6.00000 −0.201574
$$887$$ 6.00000 0.201460 0.100730 0.994914i $$-0.467882\pi$$
0.100730 + 0.994914i $$0.467882\pi$$
$$888$$ 2.00000 0.0671156
$$889$$ 0 0
$$890$$ 20.0000 0.670402
$$891$$ 3.00000 0.100504
$$892$$ 1.00000 0.0334825
$$893$$ 27.0000 0.903521
$$894$$ 0 0
$$895$$ −12.0000 −0.401116
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 40.0000 1.33482
$$899$$ 12.0000 0.400222
$$900$$ −1.00000 −0.0333333
$$901$$ −1.00000 −0.0333148
$$902$$ 6.00000 0.199778
$$903$$ 0 0
$$904$$ −3.00000 −0.0997785
$$905$$ 10.0000 0.332411
$$906$$ 17.0000 0.564787
$$907$$ −44.0000 −1.46100 −0.730498 0.682915i $$-0.760712\pi$$
−0.730498 + 0.682915i $$0.760712\pi$$
$$908$$ −8.00000 −0.265489
$$909$$ 14.0000 0.464351
$$910$$ 0 0
$$911$$ −14.0000 −0.463841 −0.231920 0.972735i $$-0.574501\pi$$
−0.231920 + 0.972735i $$0.574501\pi$$
$$912$$ 3.00000 0.0993399
$$913$$ 0 0
$$914$$ 32.0000 1.05847
$$915$$ −22.0000 −0.727298
$$916$$ 2.00000 0.0660819
$$917$$ 0 0
$$918$$ 1.00000 0.0330049
$$919$$ 46.0000 1.51740 0.758700 0.651440i $$-0.225835\pi$$
0.758700 + 0.651440i $$0.225835\pi$$
$$920$$ 0 0
$$921$$ 15.0000 0.494267
$$922$$ −20.0000 −0.658665
$$923$$ 15.0000 0.493731
$$924$$ 0 0
$$925$$ 2.00000 0.0657596
$$926$$ 32.0000 1.05159
$$927$$ −14.0000 −0.459820
$$928$$ −3.00000 −0.0984798
$$929$$ −34.0000 −1.11550 −0.557752 0.830008i $$-0.688336\pi$$
−0.557752 + 0.830008i $$0.688336\pi$$
$$930$$ −8.00000 −0.262330
$$931$$ 0 0
$$932$$ 7.00000 0.229293
$$933$$ −14.0000 −0.458339
$$934$$ 0 0
$$935$$ −6.00000 −0.196221
$$936$$ −1.00000 −0.0326860
$$937$$ 13.0000 0.424691 0.212346 0.977195i $$-0.431890\pi$$
0.212346 + 0.977195i $$0.431890\pi$$
$$938$$ 0 0
$$939$$ 14.0000 0.456873
$$940$$ 18.0000 0.587095
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ −11.0000 −0.358399
$$943$$ 0 0
$$944$$ −11.0000 −0.358020
$$945$$ 0 0
$$946$$ −18.0000 −0.585230
$$947$$ −33.0000 −1.07236 −0.536178 0.844105i $$-0.680132\pi$$
−0.536178 + 0.844105i $$0.680132\pi$$
$$948$$ 2.00000 0.0649570
$$949$$ 12.0000 0.389536
$$950$$ 3.00000 0.0973329
$$951$$ 18.0000 0.583690
$$952$$ 0 0
$$953$$ −33.0000 −1.06897 −0.534487 0.845176i $$-0.679495\pi$$
−0.534487 + 0.845176i $$0.679495\pi$$
$$954$$ −1.00000 −0.0323762
$$955$$ −24.0000 −0.776622
$$956$$ −11.0000 −0.355765
$$957$$ 9.00000 0.290929
$$958$$ 15.0000 0.484628
$$959$$ 0 0
$$960$$ 2.00000 0.0645497
$$961$$ −15.0000 −0.483871
$$962$$ 2.00000 0.0644826
$$963$$ −4.00000 −0.128898
$$964$$ −12.0000 −0.386494
$$965$$ −28.0000 −0.901352
$$966$$ 0 0
$$967$$ 21.0000 0.675314 0.337657 0.941269i $$-0.390366\pi$$
0.337657 + 0.941269i $$0.390366\pi$$
$$968$$ 2.00000 0.0642824
$$969$$ −3.00000 −0.0963739
$$970$$ −24.0000 −0.770594
$$971$$ −10.0000 −0.320915 −0.160458 0.987043i $$-0.551297\pi$$
−0.160458 + 0.987043i $$0.551297\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 0 0
$$974$$ 33.0000 1.05739
$$975$$ −1.00000 −0.0320256
$$976$$ −11.0000 −0.352101
$$977$$ 38.0000 1.21573 0.607864 0.794041i $$-0.292027\pi$$
0.607864 + 0.794041i $$0.292027\pi$$
$$978$$ 25.0000 0.799412
$$979$$ −30.0000 −0.958804
$$980$$ 0 0
$$981$$ −8.00000 −0.255420
$$982$$ −22.0000 −0.702048
$$983$$ −21.0000 −0.669796 −0.334898 0.942254i $$-0.608702\pi$$
−0.334898 + 0.942254i $$0.608702\pi$$
$$984$$ 2.00000 0.0637577
$$985$$ 16.0000 0.509802
$$986$$ 3.00000 0.0955395
$$987$$ 0 0
$$988$$ 3.00000 0.0954427
$$989$$ 0 0
$$990$$ −6.00000 −0.190693
$$991$$ −24.0000 −0.762385 −0.381193 0.924496i $$-0.624487\pi$$
−0.381193 + 0.924496i $$0.624487\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ −28.0000 −0.888553
$$994$$ 0 0
$$995$$ −24.0000 −0.760851
$$996$$ 0 0
$$997$$ −11.0000 −0.348373 −0.174187 0.984713i $$-0.555730\pi$$
−0.174187 + 0.984713i $$0.555730\pi$$
$$998$$ −28.0000 −0.886325
$$999$$ −2.00000 −0.0632772
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 3822.2.a.q.1.1 1
7.3 odd 6 546.2.i.g.79.1 2
7.5 odd 6 546.2.i.g.235.1 yes 2
7.6 odd 2 3822.2.a.c.1.1 1
21.5 even 6 1638.2.j.b.235.1 2
21.17 even 6 1638.2.j.b.1171.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.i.g.79.1 2 7.3 odd 6
546.2.i.g.235.1 yes 2 7.5 odd 6
1638.2.j.b.235.1 2 21.5 even 6
1638.2.j.b.1171.1 2 21.17 even 6
3822.2.a.c.1.1 1 7.6 odd 2
3822.2.a.q.1.1 1 1.1 even 1 trivial