LMFDB - Embedded newform 9054.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9054,2,Mod(1,9054)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9054, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9054.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9054 = 2 \cdot 3^{2} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9054.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:	$72.2965539901$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1006)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	9054.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.23607 q^{5} -4.23607 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.23607 q^{5} -4.23607 q^{7} +1.00000 q^{8} -1.23607 q^{10} +4.23607 q^{11} -5.47214 q^{13} -4.23607 q^{14} +1.00000 q^{16} -2.00000 q^{17} -7.70820 q^{19} -1.23607 q^{20} +4.23607 q^{22} -2.23607 q^{23} -3.47214 q^{25} -5.47214 q^{26} -4.23607 q^{28} +2.76393 q^{29} +7.70820 q^{31} +1.00000 q^{32} -2.00000 q^{34} +5.23607 q^{35} -4.47214 q^{37} -7.70820 q^{38} -1.23607 q^{40} +5.23607 q^{41} +10.2361 q^{43} +4.23607 q^{44} -2.23607 q^{46} -4.23607 q^{47} +10.9443 q^{49} -3.47214 q^{50} -5.47214 q^{52} +0.472136 q^{53} -5.23607 q^{55} -4.23607 q^{56} +2.76393 q^{58} -8.94427 q^{59} +7.47214 q^{61} +7.70820 q^{62} +1.00000 q^{64} +6.76393 q^{65} +4.70820 q^{67} -2.00000 q^{68} +5.23607 q^{70} -0.763932 q^{71} -7.52786 q^{73} -4.47214 q^{74} -7.70820 q^{76} -17.9443 q^{77} +12.9443 q^{79} -1.23607 q^{80} +5.23607 q^{82} -16.7082 q^{83} +2.47214 q^{85} +10.2361 q^{86} +4.23607 q^{88} +14.4721 q^{89} +23.1803 q^{91} -2.23607 q^{92} -4.23607 q^{94} +9.52786 q^{95} -10.0000 q^{97} +10.9443 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 4 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 - 4 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 4 q^{7} + 2 q^{8} + 2 q^{10} + 4 q^{11} - 2 q^{13} - 4 q^{14} + 2 q^{16} - 4 q^{17} - 2 q^{19} + 2 q^{20} + 4 q^{22} + 2 q^{25} - 2 q^{26} - 4 q^{28} + 10 q^{29} + 2 q^{31} + 2 q^{32} - 4 q^{34} + 6 q^{35} - 2 q^{38} + 2 q^{40} + 6 q^{41} + 16 q^{43} + 4 q^{44} - 4 q^{47} + 4 q^{49} + 2 q^{50} - 2 q^{52} - 8 q^{53} - 6 q^{55} - 4 q^{56} + 10 q^{58} + 6 q^{61} + 2 q^{62} + 2 q^{64} + 18 q^{65} - 4 q^{67} - 4 q^{68} + 6 q^{70} - 6 q^{71} - 24 q^{73} - 2 q^{76} - 18 q^{77} + 8 q^{79} + 2 q^{80} + 6 q^{82} - 20 q^{83} - 4 q^{85} + 16 q^{86} + 4 q^{88} + 20 q^{89} + 24 q^{91} - 4 q^{94} + 28 q^{95} - 20 q^{97} + 4 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 - 4 * q^7 + 2 * q^8 + 2 * q^10 + 4 * q^11 - 2 * q^13 - 4 * q^14 + 2 * q^16 - 4 * q^17 - 2 * q^19 + 2 * q^20 + 4 * q^22 + 2 * q^25 - 2 * q^26 - 4 * q^28 + 10 * q^29 + 2 * q^31 + 2 * q^32 - 4 * q^34 + 6 * q^35 - 2 * q^38 + 2 * q^40 + 6 * q^41 + 16 * q^43 + 4 * q^44 - 4 * q^47 + 4 * q^49 + 2 * q^50 - 2 * q^52 - 8 * q^53 - 6 * q^55 - 4 * q^56 + 10 * q^58 + 6 * q^61 + 2 * q^62 + 2 * q^64 + 18 * q^65 - 4 * q^67 - 4 * q^68 + 6 * q^70 - 6 * q^71 - 24 * q^73 - 2 * q^76 - 18 * q^77 + 8 * q^79 + 2 * q^80 + 6 * q^82 - 20 * q^83 - 4 * q^85 + 16 * q^86 + 4 * q^88 + 20 * q^89 + 24 * q^91 - 4 * q^94 + 28 * q^95 - 20 * q^97 + 4 * q^98

Display 100 coefficients 10 coefficients

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 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.23607	−0.552786	−0.276393	−	0.961045i	$-0.589139\pi$
$5$	−1.23607	−0.552786	−0.276393	+	0.961045i	$0.589139\pi$
$6$	0	0
$7$	−4.23607	−1.60108	−0.800542	−	0.599277i	$-0.795455\pi$
$7$	−4.23607	−1.60108	−0.800542	+	0.599277i	$0.795455\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.23607	−0.390879
$11$	4.23607	1.27722	0.638611	−	0.769529i	$-0.279509\pi$
$11$	4.23607	1.27722	0.638611	+	0.769529i	$0.279509\pi$
$12$	0	0
$13$	−5.47214	−1.51770	−0.758849	−	0.651267i	$-0.774238\pi$
$13$	−5.47214	−1.51770	−0.758849	+	0.651267i	$0.774238\pi$
$14$	−4.23607	−1.13214
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−7.70820	−1.76838	−0.884192	−	0.467124i	$-0.845290\pi$
$19$	−7.70820	−1.76838	−0.884192	+	0.467124i	$0.845290\pi$
$20$	−1.23607	−0.276393
$21$	0	0
$22$	4.23607	0.903133
$23$	−2.23607	−0.466252	−0.233126	−	0.972446i	$-0.574896\pi$
$23$	−2.23607	−0.466252	−0.233126	+	0.972446i	$0.574896\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	−5.47214	−1.07317
$27$	0	0
$28$	−4.23607	−0.800542
$29$	2.76393	0.513249	0.256625	−	0.966511i	$-0.417390\pi$
$29$	2.76393	0.513249	0.256625	+	0.966511i	$0.417390\pi$
$30$	0	0
$31$	7.70820	1.38443	0.692217	−	0.721689i	$-0.256634\pi$
$31$	7.70820	1.38443	0.692217	+	0.721689i	$0.256634\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−2.00000	−0.342997
$35$	5.23607	0.885057
$36$	0	0
$37$	−4.47214	−0.735215	−0.367607	−	0.929981i	$-0.619823\pi$
$37$	−4.47214	−0.735215	−0.367607	+	0.929981i	$0.619823\pi$
$38$	−7.70820	−1.25044
$39$	0	0
$40$	−1.23607	−0.195440
$41$	5.23607	0.817736	0.408868	−	0.912593i	$-0.365924\pi$
$41$	5.23607	0.817736	0.408868	+	0.912593i	$0.365924\pi$
$42$	0	0
$43$	10.2361	1.56099	0.780493	−	0.625165i	$-0.214968\pi$
$43$	10.2361	1.56099	0.780493	+	0.625165i	$0.214968\pi$
$44$	4.23607	0.638611
$45$	0	0
$46$	−2.23607	−0.329690
$47$	−4.23607	−0.617894	−0.308947	−	0.951079i	$-0.599977\pi$
$47$	−4.23607	−0.617894	−0.308947	+	0.951079i	$0.599977\pi$
$48$	0	0
$49$	10.9443	1.56347
$50$	−3.47214	−0.491034
$51$	0	0
$52$	−5.47214	−0.758849
$53$	0.472136	0.0648529	0.0324264	−	0.999474i	$-0.489677\pi$
$53$	0.472136	0.0648529	0.0324264	+	0.999474i	$0.489677\pi$
$54$	0	0
$55$	−5.23607	−0.706031
$56$	−4.23607	−0.566068
$57$	0	0
$58$	2.76393	0.362922
$59$	−8.94427	−1.16445	−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427	−1.16445	−0.582223	+	0.813029i	$0.697817\pi$
$60$	0	0
$61$	7.47214	0.956709	0.478354	−	0.878167i	$-0.341233\pi$
$61$	7.47214	0.956709	0.478354	+	0.878167i	$0.341233\pi$
$62$	7.70820	0.978943
$63$	0	0
$64$	1.00000	0.125000
$65$	6.76393	0.838963
$66$	0	0
$67$	4.70820	0.575199	0.287599	−	0.957751i	$-0.407143\pi$
$67$	4.70820	0.575199	0.287599	+	0.957751i	$0.407143\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	5.23607	0.625830
$71$	−0.763932	−0.0906621	−0.0453310	−	0.998972i	$-0.514434\pi$
$71$	−0.763932	−0.0906621	−0.0453310	+	0.998972i	$0.514434\pi$
$72$	0	0
$73$	−7.52786	−0.881070	−0.440535	−	0.897735i	$-0.645211\pi$
$73$	−7.52786	−0.881070	−0.440535	+	0.897735i	$0.645211\pi$
$74$	−4.47214	−0.519875
$75$	0	0
$76$	−7.70820	−0.884192
$77$	−17.9443	−2.04494
$78$	0	0
$79$	12.9443	1.45634	0.728172	−	0.685394i	$-0.240370\pi$
$79$	12.9443	1.45634	0.728172	+	0.685394i	$0.240370\pi$
$80$	−1.23607	−0.138197
$81$	0	0
$82$	5.23607	0.578227
$83$	−16.7082	−1.83396	−0.916982	−	0.398929i	$-0.869382\pi$
$83$	−16.7082	−1.83396	−0.916982	+	0.398929i	$0.869382\pi$
$84$	0	0
$85$	2.47214	0.268141
$86$	10.2361	1.10378
$87$	0	0
$88$	4.23607	0.451566
$89$	14.4721	1.53404	0.767022	−	0.641621i	$-0.221738\pi$
$89$	14.4721	1.53404	0.767022	+	0.641621i	$0.221738\pi$
$90$	0	0
$91$	23.1803	2.42996
$92$	−2.23607	−0.233126
$93$	0	0
$94$	−4.23607	−0.436917
$95$	9.52786	0.977538
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	10.9443	1.10554
$99$	0	0
$100$	−3.47214	−0.347214
$101$	9.70820	0.966002	0.483001	−	0.875620i	$-0.339547\pi$
$101$	9.70820	0.966002	0.483001	+	0.875620i	$0.339547\pi$
$102$	0	0
$103$	−0.944272	−0.0930419	−0.0465209	−	0.998917i	$-0.514813\pi$
$103$	−0.944272	−0.0930419	−0.0465209	+	0.998917i	$0.514813\pi$
$104$	−5.47214	−0.536587
$105$	0	0
$106$	0.472136	0.0458579
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	0	0
$109$	8.47214	0.811483	0.405742	−	0.913988i	$-0.367013\pi$
$109$	8.47214	0.811483	0.405742	+	0.913988i	$0.367013\pi$
$110$	−5.23607	−0.499239
$111$	0	0
$112$	−4.23607	−0.400271
$113$	8.52786	0.802234	0.401117	−	0.916027i	$-0.368622\pi$
$113$	8.52786	0.802234	0.401117	+	0.916027i	$0.368622\pi$
$114$	0	0
$115$	2.76393	0.257738
$116$	2.76393	0.256625
$117$	0	0
$118$	−8.94427	−0.823387
$119$	8.47214	0.776639
$120$	0	0
$121$	6.94427	0.631297
$122$	7.47214	0.676495
$123$	0	0
$124$	7.70820	0.692217
$125$	10.4721	0.936656
$126$	0	0
$127$	2.29180	0.203364	0.101682	−	0.994817i	$-0.467578\pi$
$127$	2.29180	0.203364	0.101682	+	0.994817i	$0.467578\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	6.76393	0.593236
$131$	21.1803	1.85053	0.925267	−	0.379315i	$-0.123840\pi$
$131$	21.1803	1.85053	0.925267	+	0.379315i	$0.123840\pi$
$132$	0	0
$133$	32.6525	2.83133
$134$	4.70820	0.406727
$135$	0	0
$136$	−2.00000	−0.171499
$137$	7.23607	0.618219	0.309110	−	0.951026i	$-0.399969\pi$
$137$	7.23607	0.618219	0.309110	+	0.951026i	$0.399969\pi$
$138$	0	0
$139$	−10.1803	−0.863485	−0.431743	−	0.901997i	$-0.642101\pi$
$139$	−10.1803	−0.863485	−0.431743	+	0.901997i	$0.642101\pi$
$140$	5.23607	0.442529
$141$	0	0
$142$	−0.763932	−0.0641078
$143$	−23.1803	−1.93844
$144$	0	0
$145$	−3.41641	−0.283717
$146$	−7.52786	−0.623010
$147$	0	0
$148$	−4.47214	−0.367607
$149$	12.7639	1.04566	0.522831	−	0.852436i	$-0.324876\pi$
$149$	12.7639	1.04566	0.522831	+	0.852436i	$0.324876\pi$
$150$	0	0
$151$	−18.4721	−1.50324	−0.751621	−	0.659596i	$-0.770728\pi$
$151$	−18.4721	−1.50324	−0.751621	+	0.659596i	$0.770728\pi$
$152$	−7.70820	−0.625218
$153$	0	0
$154$	−17.9443	−1.44599
$155$	−9.52786	−0.765296
$156$	0	0
$157$	12.4721	0.995385	0.497692	−	0.867354i	$-0.334181\pi$
$157$	12.4721	0.995385	0.497692	+	0.867354i	$0.334181\pi$
$158$	12.9443	1.02979
$159$	0	0
$160$	−1.23607	−0.0977198
$161$	9.47214	0.746509
$162$	0	0
$163$	−11.7082	−0.917057	−0.458529	−	0.888680i	$-0.651623\pi$
$163$	−11.7082	−0.917057	−0.458529	+	0.888680i	$0.651623\pi$
$164$	5.23607	0.408868
$165$	0	0
$166$	−16.7082	−1.29681
$167$	7.70820	0.596479	0.298239	−	0.954491i	$-0.403601\pi$
$167$	7.70820	0.596479	0.298239	+	0.954491i	$0.403601\pi$
$168$	0	0
$169$	16.9443	1.30341
$170$	2.47214	0.189604
$171$	0	0
$172$	10.2361	0.780493
$173$	−11.0000	−0.836315	−0.418157	−	0.908375i	$-0.637324\pi$
$173$	−11.0000	−0.836315	−0.418157	+	0.908375i	$0.637324\pi$
$174$	0	0
$175$	14.7082	1.11184
$176$	4.23607	0.319306
$177$	0	0
$178$	14.4721	1.08473
$179$	−5.70820	−0.426651	−0.213326	−	0.976981i	$-0.568430\pi$
$179$	−5.70820	−0.426651	−0.213326	+	0.976981i	$0.568430\pi$
$180$	0	0
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	23.1803	1.71824
$183$	0	0
$184$	−2.23607	−0.164845
$185$	5.52786	0.406417
$186$	0	0
$187$	−8.47214	−0.619544
$188$	−4.23607	−0.308947
$189$	0	0
$190$	9.52786	0.691224
$191$	18.4721	1.33660	0.668298	−	0.743893i	$-0.267023\pi$
$191$	18.4721	1.33660	0.668298	+	0.743893i	$0.267023\pi$
$192$	0	0
$193$	−6.76393	−0.486878	−0.243439	−	0.969916i	$-0.578276\pi$
$193$	−6.76393	−0.486878	−0.243439	+	0.969916i	$0.578276\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	10.9443	0.781734
$197$	20.4164	1.45461	0.727304	−	0.686315i	$-0.240773\pi$
$197$	20.4164	1.45461	0.727304	+	0.686315i	$0.240773\pi$
$198$	0	0
$199$	2.47214	0.175245	0.0876225	−	0.996154i	$-0.472073\pi$
$199$	2.47214	0.175245	0.0876225	+	0.996154i	$0.472073\pi$
$200$	−3.47214	−0.245517
$201$	0	0
$202$	9.70820	0.683067
$203$	−11.7082	−0.821755
$204$	0	0
$205$	−6.47214	−0.452034
$206$	−0.944272	−0.0657905
$207$	0	0
$208$	−5.47214	−0.379424
$209$	−32.6525	−2.25862
$210$	0	0
$211$	−3.70820	−0.255283	−0.127642	−	0.991820i	$-0.540741\pi$
$211$	−3.70820	−0.255283	−0.127642	+	0.991820i	$0.540741\pi$
$212$	0.472136	0.0324264
$213$	0	0
$214$	8.00000	0.546869
$215$	−12.6525	−0.862892
$216$	0	0
$217$	−32.6525	−2.21659
$218$	8.47214	0.573805
$219$	0	0
$220$	−5.23607	−0.353016
$221$	10.9443	0.736191
$222$	0	0
$223$	3.76393	0.252052	0.126026	−	0.992027i	$-0.459778\pi$
$223$	3.76393	0.252052	0.126026	+	0.992027i	$0.459778\pi$
$224$	−4.23607	−0.283034
$225$	0	0
$226$	8.52786	0.567265
$227$	−11.7082	−0.777101	−0.388550	−	0.921427i	$-0.627024\pi$
$227$	−11.7082	−0.777101	−0.388550	+	0.921427i	$0.627024\pi$
$228$	0	0
$229$	18.4164	1.21699	0.608495	−	0.793558i	$-0.291773\pi$
$229$	18.4164	1.21699	0.608495	+	0.793558i	$0.291773\pi$
$230$	2.76393	0.182248
$231$	0	0
$232$	2.76393	0.181461
$233$	−22.4164	−1.46855	−0.734274	−	0.678853i	$-0.762477\pi$
$233$	−22.4164	−1.46855	−0.734274	+	0.678853i	$0.762477\pi$
$234$	0	0
$235$	5.23607	0.341563
$236$	−8.94427	−0.582223
$237$	0	0
$238$	8.47214	0.549167
$239$	25.4164	1.64405	0.822025	−	0.569451i	$-0.192844\pi$
$239$	25.4164	1.64405	0.822025	+	0.569451i	$0.192844\pi$
$240$	0	0
$241$	−4.94427	−0.318489	−0.159244	−	0.987239i	$-0.550906\pi$
$241$	−4.94427	−0.318489	−0.159244	+	0.987239i	$0.550906\pi$
$242$	6.94427	0.446395
$243$	0	0
$244$	7.47214	0.478354
$245$	−13.5279	−0.864264
$246$	0	0
$247$	42.1803	2.68387
$248$	7.70820	0.489471
$249$	0	0
$250$	10.4721	0.662316
$251$	4.18034	0.263861	0.131930	−	0.991259i	$-0.457882\pi$
$251$	4.18034	0.263861	0.131930	+	0.991259i	$0.457882\pi$
$252$	0	0
$253$	−9.47214	−0.595508
$254$	2.29180	0.143800
$255$	0	0
$256$	1.00000	0.0625000
$257$	−10.0557	−0.627259	−0.313630	−	0.949545i	$-0.601545\pi$
$257$	−10.0557	−0.627259	−0.313630	+	0.949545i	$0.601545\pi$
$258$	0	0
$259$	18.9443	1.17714
$260$	6.76393	0.419481
$261$	0	0
$262$	21.1803	1.30853
$263$	−16.1246	−0.994286	−0.497143	−	0.867669i	$-0.665618\pi$
$263$	−16.1246	−0.994286	−0.497143	+	0.867669i	$0.665618\pi$
$264$	0	0
$265$	−0.583592	−0.0358498
$266$	32.6525	2.00205
$267$	0	0
$268$	4.70820	0.287599
$269$	−16.9443	−1.03311	−0.516555	−	0.856254i	$-0.672786\pi$
$269$	−16.9443	−1.03311	−0.516555	+	0.856254i	$0.672786\pi$
$270$	0	0
$271$	26.7082	1.62241	0.811204	−	0.584763i	$-0.198813\pi$
$271$	26.7082	1.62241	0.811204	+	0.584763i	$0.198813\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	7.23607	0.437147
$275$	−14.7082	−0.886938
$276$	0	0
$277$	26.1803	1.57302	0.786512	−	0.617575i	$-0.211885\pi$
$277$	26.1803	1.57302	0.786512	+	0.617575i	$0.211885\pi$
$278$	−10.1803	−0.610576
$279$	0	0
$280$	5.23607	0.312915
$281$	28.4164	1.69518	0.847590	−	0.530651i	$-0.178053\pi$
$281$	28.4164	1.69518	0.847590	+	0.530651i	$0.178053\pi$
$282$	0	0
$283$	−18.4721	−1.09805	−0.549027	−	0.835804i	$-0.685002\pi$
$283$	−18.4721	−1.09805	−0.549027	+	0.835804i	$0.685002\pi$
$284$	−0.763932	−0.0453310
$285$	0	0
$286$	−23.1803	−1.37068
$287$	−22.1803	−1.30926
$288$	0	0
$289$	−13.0000	−0.764706
$290$	−3.41641	−0.200618
$291$	0	0
$292$	−7.52786	−0.440535
$293$	−5.00000	−0.292103	−0.146052	−	0.989277i	$-0.546657\pi$
$293$	−5.00000	−0.292103	−0.146052	+	0.989277i	$0.546657\pi$
$294$	0	0
$295$	11.0557	0.643689
$296$	−4.47214	−0.259938
$297$	0	0
$298$	12.7639	0.739395
$299$	12.2361	0.707630
$300$	0	0
$301$	−43.3607	−2.49927
$302$	−18.4721	−1.06295
$303$	0	0
$304$	−7.70820	−0.442096
$305$	−9.23607	−0.528856
$306$	0	0
$307$	9.41641	0.537423	0.268711	−	0.963221i	$-0.413402\pi$
$307$	9.41641	0.537423	0.268711	+	0.963221i	$0.413402\pi$
$308$	−17.9443	−1.02247
$309$	0	0
$310$	−9.52786	−0.541146
$311$	5.81966	0.330003	0.165001	−	0.986293i	$-0.447237\pi$
$311$	5.81966	0.330003	0.165001	+	0.986293i	$0.447237\pi$
$312$	0	0
$313$	−30.9443	−1.74907	−0.874537	−	0.484959i	$-0.838834\pi$
$313$	−30.9443	−1.74907	−0.874537	+	0.484959i	$0.838834\pi$
$314$	12.4721	0.703843
$315$	0	0
$316$	12.9443	0.728172
$317$	8.05573	0.452455	0.226227	−	0.974075i	$-0.427361\pi$
$317$	8.05573	0.452455	0.226227	+	0.974075i	$0.427361\pi$
$318$	0	0
$319$	11.7082	0.655534
$320$	−1.23607	−0.0690983
$321$	0	0
$322$	9.47214	0.527861
$323$	15.4164	0.857792
$324$	0	0
$325$	19.0000	1.05393
$326$	−11.7082	−0.648457
$327$	0	0
$328$	5.23607	0.289113
$329$	17.9443	0.989300
$330$	0	0
$331$	10.9443	0.601552	0.300776	−	0.953695i	$-0.402754\pi$
$331$	10.9443	0.601552	0.300776	+	0.953695i	$0.402754\pi$
$332$	−16.7082	−0.916982
$333$	0	0
$334$	7.70820	0.421774
$335$	−5.81966	−0.317962
$336$	0	0
$337$	−30.9443	−1.68564	−0.842821	−	0.538194i	$-0.819107\pi$
$337$	−30.9443	−1.68564	−0.842821	+	0.538194i	$0.819107\pi$
$338$	16.9443	0.921647
$339$	0	0
$340$	2.47214	0.134070
$341$	32.6525	1.76823
$342$	0	0
$343$	−16.7082	−0.902158
$344$	10.2361	0.551892
$345$	0	0
$346$	−11.0000	−0.591364
$347$	8.00000	0.429463	0.214731	−	0.976673i	$-0.431112\pi$
$347$	8.00000	0.429463	0.214731	+	0.976673i	$0.431112\pi$
$348$	0	0
$349$	12.0000	0.642345	0.321173	−	0.947021i	$-0.395923\pi$
$349$	12.0000	0.642345	0.321173	+	0.947021i	$0.395923\pi$
$350$	14.7082	0.786187
$351$	0	0
$352$	4.23607	0.225783
$353$	24.7639	1.31805	0.659026	−	0.752121i	$-0.270969\pi$
$353$	24.7639	1.31805	0.659026	+	0.752121i	$0.270969\pi$
$354$	0	0
$355$	0.944272	0.0501167
$356$	14.4721	0.767022
$357$	0	0
$358$	−5.70820	−0.301688
$359$	9.70820	0.512379	0.256190	−	0.966627i	$-0.417533\pi$
$359$	9.70820	0.512379	0.256190	+	0.966627i	$0.417533\pi$
$360$	0	0
$361$	40.4164	2.12718
$362$	−20.0000	−1.05118
$363$	0	0
$364$	23.1803	1.21498
$365$	9.30495	0.487043
$366$	0	0
$367$	−2.23607	−0.116722	−0.0583609	−	0.998296i	$-0.518587\pi$
$367$	−2.23607	−0.116722	−0.0583609	+	0.998296i	$0.518587\pi$
$368$	−2.23607	−0.116563
$369$	0	0
$370$	5.52786	0.287380
$371$	−2.00000	−0.103835
$372$	0	0
$373$	26.3050	1.36202	0.681009	−	0.732275i	$-0.261541\pi$
$373$	26.3050	1.36202	0.681009	+	0.732275i	$0.261541\pi$
$374$	−8.47214	−0.438084
$375$	0	0
$376$	−4.23607	−0.218459
$377$	−15.1246	−0.778957
$378$	0	0
$379$	−6.70820	−0.344577	−0.172289	−	0.985047i	$-0.555116\pi$
$379$	−6.70820	−0.344577	−0.172289	+	0.985047i	$0.555116\pi$
$380$	9.52786	0.488769
$381$	0	0
$382$	18.4721	0.945117
$383$	12.9443	0.661421	0.330711	−	0.943732i	$-0.392712\pi$
$383$	12.9443	0.661421	0.330711	+	0.943732i	$0.392712\pi$
$384$	0	0
$385$	22.1803	1.13041
$386$	−6.76393	−0.344275
$387$	0	0
$388$	−10.0000	−0.507673
$389$	−16.8328	−0.853458	−0.426729	−	0.904380i	$-0.640334\pi$
$389$	−16.8328	−0.853458	−0.426729	+	0.904380i	$0.640334\pi$
$390$	0	0
$391$	4.47214	0.226166
$392$	10.9443	0.552769
$393$	0	0
$394$	20.4164	1.02856
$395$	−16.0000	−0.805047
$396$	0	0
$397$	37.3607	1.87508	0.937539	−	0.347879i	$-0.113098\pi$
$397$	37.3607	1.87508	0.937539	+	0.347879i	$0.113098\pi$
$398$	2.47214	0.123917
$399$	0	0
$400$	−3.47214	−0.173607
$401$	−12.4164	−0.620046	−0.310023	−	0.950729i	$-0.600337\pi$
$401$	−12.4164	−0.620046	−0.310023	+	0.950729i	$0.600337\pi$
$402$	0	0
$403$	−42.1803	−2.10115
$404$	9.70820	0.483001
$405$	0	0
$406$	−11.7082	−0.581068
$407$	−18.9443	−0.939033
$408$	0	0
$409$	1.70820	0.0844652	0.0422326	−	0.999108i	$-0.486553\pi$
$409$	1.70820	0.0844652	0.0422326	+	0.999108i	$0.486553\pi$
$410$	−6.47214	−0.319636
$411$	0	0
$412$	−0.944272	−0.0465209
$413$	37.8885	1.86437
$414$	0	0
$415$	20.6525	1.01379
$416$	−5.47214	−0.268294
$417$	0	0
$418$	−32.6525	−1.59708
$419$	8.29180	0.405081	0.202540	−	0.979274i	$-0.435080\pi$
$419$	8.29180	0.405081	0.202540	+	0.979274i	$0.435080\pi$
$420$	0	0
$421$	−5.05573	−0.246401	−0.123201	−	0.992382i	$-0.539316\pi$
$421$	−5.05573	−0.246401	−0.123201	+	0.992382i	$0.539316\pi$
$422$	−3.70820	−0.180513
$423$	0	0
$424$	0.472136	0.0229289
$425$	6.94427	0.336847
$426$	0	0
$427$	−31.6525	−1.53177
$428$	8.00000	0.386695
$429$	0	0
$430$	−12.6525	−0.610157
$431$	−1.34752	−0.0649080	−0.0324540	−	0.999473i	$-0.510332\pi$
$431$	−1.34752	−0.0649080	−0.0324540	+	0.999473i	$0.510332\pi$
$432$	0	0
$433$	20.4721	0.983828	0.491914	−	0.870644i	$-0.336297\pi$
$433$	20.4721	0.983828	0.491914	+	0.870644i	$0.336297\pi$
$434$	−32.6525	−1.56737
$435$	0	0
$436$	8.47214	0.405742
$437$	17.2361	0.824513
$438$	0	0
$439$	−17.8885	−0.853774	−0.426887	−	0.904305i	$-0.640390\pi$
$439$	−17.8885	−0.853774	−0.426887	+	0.904305i	$0.640390\pi$
$440$	−5.23607	−0.249620
$441$	0	0
$442$	10.9443	0.520566
$443$	−23.1803	−1.10133	−0.550666	−	0.834726i	$-0.685626\pi$
$443$	−23.1803	−1.10133	−0.550666	+	0.834726i	$0.685626\pi$
$444$	0	0
$445$	−17.8885	−0.847998
$446$	3.76393	0.178227
$447$	0	0
$448$	−4.23607	−0.200135
$449$	−19.5967	−0.924828	−0.462414	−	0.886664i	$-0.653017\pi$
$449$	−19.5967	−0.924828	−0.462414	+	0.886664i	$0.653017\pi$
$450$	0	0
$451$	22.1803	1.04443
$452$	8.52786	0.401117
$453$	0	0
$454$	−11.7082	−0.549493
$455$	−28.6525	−1.34325
$456$	0	0
$457$	−22.3607	−1.04599	−0.522994	−	0.852336i	$-0.675185\pi$
$457$	−22.3607	−1.04599	−0.522994	+	0.852336i	$0.675185\pi$
$458$	18.4164	0.860542
$459$	0	0
$460$	2.76393	0.128869
$461$	16.0689	0.748403	0.374201	−	0.927348i	$-0.377917\pi$
$461$	16.0689	0.748403	0.374201	+	0.927348i	$0.377917\pi$
$462$	0	0
$463$	16.5967	0.771316	0.385658	−	0.922642i	$-0.373974\pi$
$463$	16.5967	0.771316	0.385658	+	0.922642i	$0.373974\pi$
$464$	2.76393	0.128312
$465$	0	0
$466$	−22.4164	−1.03842
$467$	8.29180	0.383699	0.191849	−	0.981424i	$-0.438552\pi$
$467$	8.29180	0.383699	0.191849	+	0.981424i	$0.438552\pi$
$468$	0	0
$469$	−19.9443	−0.920941
$470$	5.23607	0.241522
$471$	0	0
$472$	−8.94427	−0.411693
$473$	43.3607	1.99373
$474$	0	0
$475$	26.7639	1.22801
$476$	8.47214	0.388320
$477$	0	0
$478$	25.4164	1.16252
$479$	12.9443	0.591439	0.295719	−	0.955275i	$-0.404441\pi$
$479$	12.9443	0.591439	0.295719	+	0.955275i	$0.404441\pi$
$480$	0	0
$481$	24.4721	1.11583
$482$	−4.94427	−0.225205
$483$	0	0
$484$	6.94427	0.315649
$485$	12.3607	0.561270
$486$	0	0
$487$	35.5967	1.61304	0.806521	−	0.591205i	$-0.201348\pi$
$487$	35.5967	1.61304	0.806521	+	0.591205i	$0.201348\pi$
$488$	7.47214	0.338248
$489$	0	0
$490$	−13.5279	−0.611127
$491$	1.12461	0.0507530	0.0253765	−	0.999678i	$-0.491922\pi$
$491$	1.12461	0.0507530	0.0253765	+	0.999678i	$0.491922\pi$
$492$	0	0
$493$	−5.52786	−0.248962
$494$	42.1803	1.89778
$495$	0	0
$496$	7.70820	0.346109
$497$	3.23607	0.145157
$498$	0	0
$499$	−35.8885	−1.60659	−0.803296	−	0.595580i	$-0.796922\pi$
$499$	−35.8885	−1.60659	−0.803296	+	0.595580i	$0.796922\pi$
$500$	10.4721	0.468328
$501$	0	0
$502$	4.18034	0.186578
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−12.0000	−0.533993
$506$	−9.47214	−0.421088
$507$	0	0
$508$	2.29180	0.101682
$509$	−14.8885	−0.659923	−0.329962	−	0.943994i	$-0.607036\pi$
$509$	−14.8885	−0.659923	−0.329962	+	0.943994i	$0.607036\pi$
$510$	0	0
$511$	31.8885	1.41067
$512$	1.00000	0.0441942
$513$	0	0
$514$	−10.0557	−0.443539
$515$	1.16718	0.0514323
$516$	0	0
$517$	−17.9443	−0.789188
$518$	18.9443	0.832364
$519$	0	0
$520$	6.76393	0.296618
$521$	41.0000	1.79624	0.898121	−	0.439748i	$-0.144932\pi$
$521$	41.0000	1.79624	0.898121	+	0.439748i	$0.144932\pi$
$522$	0	0
$523$	12.3607	0.540495	0.270247	−	0.962791i	$-0.412895\pi$
$523$	12.3607	0.540495	0.270247	+	0.962791i	$0.412895\pi$
$524$	21.1803	0.925267
$525$	0	0
$526$	−16.1246	−0.703066
$527$	−15.4164	−0.671549
$528$	0	0
$529$	−18.0000	−0.782609
$530$	−0.583592	−0.0253496
$531$	0	0
$532$	32.6525	1.41566
$533$	−28.6525	−1.24108
$534$	0	0
$535$	−9.88854	−0.427519
$536$	4.70820	0.203363
$537$	0	0
$538$	−16.9443	−0.730519
$539$	46.3607	1.99690
$540$	0	0
$541$	6.65248	0.286012	0.143006	−	0.989722i	$-0.454323\pi$
$541$	6.65248	0.286012	0.143006	+	0.989722i	$0.454323\pi$
$542$	26.7082	1.14722
$543$	0	0
$544$	−2.00000	−0.0857493
$545$	−10.4721	−0.448577
$546$	0	0
$547$	36.3607	1.55467	0.777335	−	0.629087i	$-0.216571\pi$
$547$	36.3607	1.55467	0.777335	+	0.629087i	$0.216571\pi$
$548$	7.23607	0.309110
$549$	0	0
$550$	−14.7082	−0.627160
$551$	−21.3050	−0.907621
$552$	0	0
$553$	−54.8328	−2.33173
$554$	26.1803	1.11230
$555$	0	0
$556$	−10.1803	−0.431743
$557$	37.2492	1.57830	0.789150	−	0.614200i	$-0.210521\pi$
$557$	37.2492	1.57830	0.789150	+	0.614200i	$0.210521\pi$
$558$	0	0
$559$	−56.0132	−2.36910
$560$	5.23607	0.221264
$561$	0	0
$562$	28.4164	1.19867
$563$	−35.7082	−1.50492	−0.752461	−	0.658637i	$-0.771133\pi$
$563$	−35.7082	−1.50492	−0.752461	+	0.658637i	$0.771133\pi$
$564$	0	0
$565$	−10.5410	−0.443464
$566$	−18.4721	−0.776442
$567$	0	0
$568$	−0.763932	−0.0320539
$569$	−39.3050	−1.64775	−0.823875	−	0.566772i	$-0.808192\pi$
$569$	−39.3050	−1.64775	−0.823875	+	0.566772i	$0.808192\pi$
$570$	0	0
$571$	15.8885	0.664915	0.332457	−	0.943118i	$-0.392122\pi$
$571$	15.8885	0.664915	0.332457	+	0.943118i	$0.392122\pi$
$572$	−23.1803	−0.969219
$573$	0	0
$574$	−22.1803	−0.925789
$575$	7.76393	0.323778
$576$	0	0
$577$	−22.3607	−0.930887	−0.465444	−	0.885078i	$-0.654105\pi$
$577$	−22.3607	−0.930887	−0.465444	+	0.885078i	$0.654105\pi$
$578$	−13.0000	−0.540729
$579$	0	0
$580$	−3.41641	−0.141859
$581$	70.7771	2.93633
$582$	0	0
$583$	2.00000	0.0828315
$584$	−7.52786	−0.311505
$585$	0	0
$586$	−5.00000	−0.206548
$587$	14.8328	0.612216	0.306108	−	0.951997i	$-0.400973\pi$
$587$	14.8328	0.612216	0.306108	+	0.951997i	$0.400973\pi$
$588$	0	0
$589$	−59.4164	−2.44821
$590$	11.0557	0.455157
$591$	0	0
$592$	−4.47214	−0.183804
$593$	−16.4721	−0.676430	−0.338215	−	0.941069i	$-0.609823\pi$
$593$	−16.4721	−0.676430	−0.338215	+	0.941069i	$0.609823\pi$
$594$	0	0
$595$	−10.4721	−0.429316
$596$	12.7639	0.522831
$597$	0	0
$598$	12.2361	0.500370
$599$	−25.5279	−1.04304	−0.521520	−	0.853239i	$-0.674635\pi$
$599$	−25.5279	−1.04304	−0.521520	+	0.853239i	$0.674635\pi$
$600$	0	0
$601$	−9.47214	−0.386376	−0.193188	−	0.981162i	$-0.561883\pi$
$601$	−9.47214	−0.386376	−0.193188	+	0.981162i	$0.561883\pi$
$602$	−43.3607	−1.76725
$603$	0	0
$604$	−18.4721	−0.751621
$605$	−8.58359	−0.348973
$606$	0	0
$607$	10.2361	0.415469	0.207735	−	0.978185i	$-0.433391\pi$
$607$	10.2361	0.415469	0.207735	+	0.978185i	$0.433391\pi$
$608$	−7.70820	−0.312609
$609$	0	0
$610$	−9.23607	−0.373957
$611$	23.1803	0.937776
$612$	0	0
$613$	7.81966	0.315833	0.157917	−	0.987452i	$-0.449522\pi$
$613$	7.81966	0.315833	0.157917	+	0.987452i	$0.449522\pi$
$614$	9.41641	0.380015
$615$	0	0
$616$	−17.9443	−0.722995
$617$	23.8885	0.961717	0.480858	−	0.876798i	$-0.340325\pi$
$617$	23.8885	0.961717	0.480858	+	0.876798i	$0.340325\pi$
$618$	0	0
$619$	31.7082	1.27446	0.637230	−	0.770674i	$-0.280080\pi$
$619$	31.7082	1.27446	0.637230	+	0.770674i	$0.280080\pi$
$620$	−9.52786	−0.382648
$621$	0	0
$622$	5.81966	0.233347
$623$	−61.3050	−2.45613
$624$	0	0
$625$	4.41641	0.176656
$626$	−30.9443	−1.23678
$627$	0	0
$628$	12.4721	0.497692
$629$	8.94427	0.356631
$630$	0	0
$631$	12.2361	0.487110	0.243555	−	0.969887i	$-0.421686\pi$
$631$	12.2361	0.487110	0.243555	+	0.969887i	$0.421686\pi$
$632$	12.9443	0.514895
$633$	0	0
$634$	8.05573	0.319934
$635$	−2.83282	−0.112417
$636$	0	0
$637$	−59.8885	−2.37287
$638$	11.7082	0.463532
$639$	0	0
$640$	−1.23607	−0.0488599
$641$	10.4164	0.411423	0.205712	−	0.978613i	$-0.434049\pi$
$641$	10.4164	0.411423	0.205712	+	0.978613i	$0.434049\pi$
$642$	0	0
$643$	21.0557	0.830357	0.415178	−	0.909740i	$-0.363719\pi$
$643$	21.0557	0.830357	0.415178	+	0.909740i	$0.363719\pi$
$644$	9.47214	0.373254
$645$	0	0
$646$	15.4164	0.606550
$647$	44.9443	1.76694	0.883471	−	0.468486i	$-0.155200\pi$
$647$	44.9443	1.76694	0.883471	+	0.468486i	$0.155200\pi$
$648$	0	0
$649$	−37.8885	−1.48726
$650$	19.0000	0.745241
$651$	0	0
$652$	−11.7082	−0.458529
$653$	−9.00000	−0.352197	−0.176099	−	0.984373i	$-0.556348\pi$
$653$	−9.00000	−0.352197	−0.176099	+	0.984373i	$0.556348\pi$
$654$	0	0
$655$	−26.1803	−1.02295
$656$	5.23607	0.204434
$657$	0	0
$658$	17.9443	0.699541
$659$	−30.0132	−1.16915	−0.584573	−	0.811341i	$-0.698738\pi$
$659$	−30.0132	−1.16915	−0.584573	+	0.811341i	$0.698738\pi$
$660$	0	0
$661$	−47.3607	−1.84212	−0.921058	−	0.389424i	$-0.872674\pi$
$661$	−47.3607	−1.84212	−0.921058	+	0.389424i	$0.872674\pi$
$662$	10.9443	0.425361
$663$	0	0
$664$	−16.7082	−0.648404
$665$	−40.3607	−1.56512
$666$	0	0
$667$	−6.18034	−0.239304
$668$	7.70820	0.298239
$669$	0	0
$670$	−5.81966	−0.224833
$671$	31.6525	1.22193
$672$	0	0
$673$	22.9443	0.884437	0.442218	−	0.896907i	$-0.354192\pi$
$673$	22.9443	0.884437	0.442218	+	0.896907i	$0.354192\pi$
$674$	−30.9443	−1.19193
$675$	0	0
$676$	16.9443	0.651703
$677$	−27.5279	−1.05798	−0.528991	−	0.848628i	$-0.677429\pi$
$677$	−27.5279	−1.05798	−0.528991	+	0.848628i	$0.677429\pi$
$678$	0	0
$679$	42.3607	1.62565
$680$	2.47214	0.0948021
$681$	0	0
$682$	32.6525	1.25033
$683$	23.8197	0.911434	0.455717	−	0.890125i	$-0.349383\pi$
$683$	23.8197	0.911434	0.455717	+	0.890125i	$0.349383\pi$
$684$	0	0
$685$	−8.94427	−0.341743
$686$	−16.7082	−0.637922
$687$	0	0
$688$	10.2361	0.390246
$689$	−2.58359	−0.0984270
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	−11.0000	−0.418157
$693$	0	0
$694$	8.00000	0.303676
$695$	12.5836	0.477323
$696$	0	0
$697$	−10.4721	−0.396660
$698$	12.0000	0.454207
$699$	0	0
$700$	14.7082	0.555918
$701$	−9.00000	−0.339925	−0.169963	−	0.985451i	$-0.554365\pi$
$701$	−9.00000	−0.339925	−0.169963	+	0.985451i	$0.554365\pi$
$702$	0	0
$703$	34.4721	1.30014
$704$	4.23607	0.159653
$705$	0	0
$706$	24.7639	0.932003
$707$	−41.1246	−1.54665
$708$	0	0
$709$	30.0000	1.12667	0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000	1.12667	0.563337	+	0.826227i	$0.309517\pi$
$710$	0.944272	0.0354379
$711$	0	0
$712$	14.4721	0.542366
$713$	−17.2361	−0.645496
$714$	0	0
$715$	28.6525	1.07154
$716$	−5.70820	−0.213326
$717$	0	0
$718$	9.70820	0.362307
$719$	9.88854	0.368780	0.184390	−	0.982853i	$-0.440969\pi$
$719$	9.88854	0.368780	0.184390	+	0.982853i	$0.440969\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	40.4164	1.50414
$723$	0	0
$724$	−20.0000	−0.743294
$725$	−9.59675	−0.356414
$726$	0	0
$727$	21.0689	0.781402	0.390701	−	0.920518i	$-0.372233\pi$
$727$	21.0689	0.781402	0.390701	+	0.920518i	$0.372233\pi$
$728$	23.1803	0.859121
$729$	0	0
$730$	9.30495	0.344392
$731$	−20.4721	−0.757189
$732$	0	0
$733$	−29.2361	−1.07986	−0.539929	−	0.841710i	$-0.681549\pi$
$733$	−29.2361	−1.07986	−0.539929	+	0.841710i	$0.681549\pi$
$734$	−2.23607	−0.0825348
$735$	0	0
$736$	−2.23607	−0.0824226
$737$	19.9443	0.734657
$738$	0	0
$739$	−30.2361	−1.11225	−0.556126	−	0.831098i	$-0.687713\pi$
$739$	−30.2361	−1.11225	−0.556126	+	0.831098i	$0.687713\pi$
$740$	5.52786	0.203208
$741$	0	0
$742$	−2.00000	−0.0734223
$743$	36.2918	1.33142	0.665708	−	0.746212i	$-0.268129\pi$
$743$	36.2918	1.33142	0.665708	+	0.746212i	$0.268129\pi$
$744$	0	0
$745$	−15.7771	−0.578028
$746$	26.3050	0.963093
$747$	0	0
$748$	−8.47214	−0.309772
$749$	−33.8885	−1.23826
$750$	0	0
$751$	−40.3607	−1.47278	−0.736391	−	0.676556i	$-0.763472\pi$
$751$	−40.3607	−1.47278	−0.736391	+	0.676556i	$0.763472\pi$
$752$	−4.23607	−0.154474
$753$	0	0
$754$	−15.1246	−0.550806
$755$	22.8328	0.830971
$756$	0	0
$757$	42.3607	1.53963	0.769813	−	0.638270i	$-0.220350\pi$
$757$	42.3607	1.53963	0.769813	+	0.638270i	$0.220350\pi$
$758$	−6.70820	−0.243653
$759$	0	0
$760$	9.52786	0.345612
$761$	−7.00000	−0.253750	−0.126875	−	0.991919i	$-0.540495\pi$
$761$	−7.00000	−0.253750	−0.126875	+	0.991919i	$0.540495\pi$
$762$	0	0
$763$	−35.8885	−1.29925
$764$	18.4721	0.668298
$765$	0	0
$766$	12.9443	0.467696
$767$	48.9443	1.76728
$768$	0	0
$769$	45.7771	1.65076	0.825382	−	0.564575i	$-0.190960\pi$
$769$	45.7771	1.65076	0.825382	+	0.564575i	$0.190960\pi$
$770$	22.1803	0.799324
$771$	0	0
$772$	−6.76393	−0.243439
$773$	−33.5967	−1.20839	−0.604196	−	0.796836i	$-0.706505\pi$
$773$	−33.5967	−1.20839	−0.604196	+	0.796836i	$0.706505\pi$
$774$	0	0
$775$	−26.7639	−0.961389
$776$	−10.0000	−0.358979
$777$	0	0
$778$	−16.8328	−0.603486
$779$	−40.3607	−1.44607
$780$	0	0
$781$	−3.23607	−0.115796
$782$	4.47214	0.159923
$783$	0	0
$784$	10.9443	0.390867
$785$	−15.4164	−0.550235
$786$	0	0
$787$	52.0689	1.85606	0.928028	−	0.372511i	$-0.121503\pi$
$787$	52.0689	1.85606	0.928028	+	0.372511i	$0.121503\pi$
$788$	20.4164	0.727304
$789$	0	0
$790$	−16.0000	−0.569254
$791$	−36.1246	−1.28444
$792$	0	0
$793$	−40.8885	−1.45199
$794$	37.3607	1.32588
$795$	0	0
$796$	2.47214	0.0876225
$797$	4.83282	0.171187	0.0855936	−	0.996330i	$-0.472721\pi$
$797$	4.83282	0.171187	0.0855936	+	0.996330i	$0.472721\pi$
$798$	0	0
$799$	8.47214	0.299723
$800$	−3.47214	−0.122759
$801$	0	0
$802$	−12.4164	−0.438439
$803$	−31.8885	−1.12532
$804$	0	0
$805$	−11.7082	−0.412660
$806$	−42.1803	−1.48574
$807$	0	0
$808$	9.70820	0.341533
$809$	6.18034	0.217289	0.108645	−	0.994081i	$-0.465349\pi$
$809$	6.18034	0.217289	0.108645	+	0.994081i	$0.465349\pi$
$810$	0	0
$811$	−8.23607	−0.289207	−0.144604	−	0.989490i	$-0.546191\pi$
$811$	−8.23607	−0.289207	−0.144604	+	0.989490i	$0.546191\pi$
$812$	−11.7082	−0.410877
$813$	0	0
$814$	−18.9443	−0.663996
$815$	14.4721	0.506937
$816$	0	0
$817$	−78.9017	−2.76042
$818$	1.70820	0.0597259
$819$	0	0
$820$	−6.47214	−0.226017
$821$	−40.9443	−1.42896	−0.714482	−	0.699653i	$-0.753338\pi$
$821$	−40.9443	−1.42896	−0.714482	+	0.699653i	$0.753338\pi$
$822$	0	0
$823$	−24.1803	−0.842874	−0.421437	−	0.906858i	$-0.638474\pi$
$823$	−24.1803	−0.842874	−0.421437	+	0.906858i	$0.638474\pi$
$824$	−0.944272	−0.0328953
$825$	0	0
$826$	37.8885	1.31831
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	0	0
$829$	−45.5967	−1.58364	−0.791820	−	0.610754i	$-0.790866\pi$
$829$	−45.5967	−1.58364	−0.791820	+	0.610754i	$0.790866\pi$
$830$	20.6525	0.716858
$831$	0	0
$832$	−5.47214	−0.189712
$833$	−21.8885	−0.758393
$834$	0	0
$835$	−9.52786	−0.329725
$836$	−32.6525	−1.12931
$837$	0	0
$838$	8.29180	0.286435
$839$	−34.5967	−1.19441	−0.597206	−	0.802088i	$-0.703723\pi$
$839$	−34.5967	−1.19441	−0.597206	+	0.802088i	$0.703723\pi$
$840$	0	0
$841$	−21.3607	−0.736575
$842$	−5.05573	−0.174232
$843$	0	0
$844$	−3.70820	−0.127642
$845$	−20.9443	−0.720505
$846$	0	0
$847$	−29.4164	−1.01076
$848$	0.472136	0.0162132
$849$	0	0
$850$	6.94427	0.238187
$851$	10.0000	0.342796
$852$	0	0
$853$	15.8328	0.542105	0.271053	−	0.962565i	$-0.412628\pi$
$853$	15.8328	0.542105	0.271053	+	0.962565i	$0.412628\pi$
$854$	−31.6525	−1.08313
$855$	0	0
$856$	8.00000	0.273434
$857$	−16.0557	−0.548453	−0.274227	−	0.961665i	$-0.588422\pi$
$857$	−16.0557	−0.548453	−0.274227	+	0.961665i	$0.588422\pi$
$858$	0	0
$859$	−14.2918	−0.487630	−0.243815	−	0.969822i	$-0.578399\pi$
$859$	−14.2918	−0.487630	−0.243815	+	0.969822i	$0.578399\pi$
$860$	−12.6525	−0.431446
$861$	0	0
$862$	−1.34752	−0.0458969
$863$	−7.23607	−0.246319	−0.123159	−	0.992387i	$-0.539303\pi$
$863$	−7.23607	−0.246319	−0.123159	+	0.992387i	$0.539303\pi$
$864$	0	0
$865$	13.5967	0.462303
$866$	20.4721	0.695671
$867$	0	0
$868$	−32.6525	−1.10830
$869$	54.8328	1.86008
$870$	0	0
$871$	−25.7639	−0.872978
$872$	8.47214	0.286903
$873$	0	0
$874$	17.2361	0.583019
$875$	−44.3607	−1.49966
$876$	0	0
$877$	−17.7771	−0.600290	−0.300145	−	0.953894i	$-0.597035\pi$
$877$	−17.7771	−0.600290	−0.300145	+	0.953894i	$0.597035\pi$
$878$	−17.8885	−0.603709
$879$	0	0
$880$	−5.23607	−0.176508
$881$	11.8885	0.400535	0.200268	−	0.979741i	$-0.435819\pi$
$881$	11.8885	0.400535	0.200268	+	0.979741i	$0.435819\pi$
$882$	0	0
$883$	41.5279	1.39752	0.698762	−	0.715354i	$-0.253735\pi$
$883$	41.5279	1.39752	0.698762	+	0.715354i	$0.253735\pi$
$884$	10.9443	0.368096
$885$	0	0
$886$	−23.1803	−0.778759
$887$	40.7214	1.36729	0.683645	−	0.729815i	$-0.260394\pi$
$887$	40.7214	1.36729	0.683645	+	0.729815i	$0.260394\pi$
$888$	0	0
$889$	−9.70820	−0.325603
$890$	−17.8885	−0.599625
$891$	0	0
$892$	3.76393	0.126026
$893$	32.6525	1.09267
$894$	0	0
$895$	7.05573	0.235847
$896$	−4.23607	−0.141517
$897$	0	0
$898$	−19.5967	−0.653952
$899$	21.3050	0.710560
$900$	0	0
$901$	−0.944272	−0.0314583
$902$	22.1803	0.738525
$903$	0	0
$904$	8.52786	0.283633
$905$	24.7214	0.821766
$906$	0	0
$907$	−2.94427	−0.0977629	−0.0488815	−	0.998805i	$-0.515566\pi$
$907$	−2.94427	−0.0977629	−0.0488815	+	0.998805i	$0.515566\pi$
$908$	−11.7082	−0.388550
$909$	0	0
$910$	−28.6525	−0.949820
$911$	−6.76393	−0.224099	−0.112050	−	0.993703i	$-0.535742\pi$
$911$	−6.76393	−0.224099	−0.112050	+	0.993703i	$0.535742\pi$
$912$	0	0
$913$	−70.7771	−2.34238
$914$	−22.3607	−0.739626
$915$	0	0
$916$	18.4164	0.608495
$917$	−89.7214	−2.96286
$918$	0	0
$919$	4.94427	0.163096	0.0815482	−	0.996669i	$-0.474014\pi$
$919$	4.94427	0.163096	0.0815482	+	0.996669i	$0.474014\pi$
$920$	2.76393	0.0911241
$921$	0	0
$922$	16.0689	0.529201
$923$	4.18034	0.137598
$924$	0	0
$925$	15.5279	0.510553
$926$	16.5967	0.545403
$927$	0	0
$928$	2.76393	0.0907305
$929$	−26.6525	−0.874439	−0.437220	−	0.899355i	$-0.644037\pi$
$929$	−26.6525	−0.874439	−0.437220	+	0.899355i	$0.644037\pi$
$930$	0	0
$931$	−84.3607	−2.76481
$932$	−22.4164	−0.734274
$933$	0	0
$934$	8.29180	0.271316
$935$	10.4721	0.342475
$936$	0	0
$937$	−16.0689	−0.524948	−0.262474	−	0.964939i	$-0.584538\pi$
$937$	−16.0689	−0.524948	−0.262474	+	0.964939i	$0.584538\pi$
$938$	−19.9443	−0.651204
$939$	0	0
$940$	5.23607	0.170782
$941$	−36.4164	−1.18714	−0.593570	−	0.804782i	$-0.702282\pi$
$941$	−36.4164	−1.18714	−0.593570	+	0.804782i	$0.702282\pi$
$942$	0	0
$943$	−11.7082	−0.381272
$944$	−8.94427	−0.291111
$945$	0	0
$946$	43.3607	1.40978
$947$	29.2361	0.950045	0.475022	−	0.879974i	$-0.342440\pi$
$947$	29.2361	0.950045	0.475022	+	0.879974i	$0.342440\pi$
$948$	0	0
$949$	41.1935	1.33720
$950$	26.7639	0.868337
$951$	0	0
$952$	8.47214	0.274584
$953$	−20.0557	−0.649669	−0.324834	−	0.945771i	$-0.605309\pi$
$953$	−20.0557	−0.649669	−0.324834	+	0.945771i	$0.605309\pi$
$954$	0	0
$955$	−22.8328	−0.738853
$956$	25.4164	0.822025
$957$	0	0
$958$	12.9443	0.418210
$959$	−30.6525	−0.989820
$960$	0	0
$961$	28.4164	0.916658
$962$	24.4721	0.789013
$963$	0	0
$964$	−4.94427	−0.159244
$965$	8.36068	0.269140
$966$	0	0
$967$	61.3050	1.97143	0.985717	−	0.168409i	$-0.0538630\pi$
$967$	61.3050	1.97143	0.985717	+	0.168409i	$0.0538630\pi$
$968$	6.94427	0.223197
$969$	0	0
$970$	12.3607	0.396878
$971$	57.0689	1.83143	0.915714	−	0.401831i	$-0.131626\pi$
$971$	57.0689	1.83143	0.915714	+	0.401831i	$0.131626\pi$
$972$	0	0
$973$	43.1246	1.38251
$974$	35.5967	1.14059
$975$	0	0
$976$	7.47214	0.239177
$977$	−39.8885	−1.27615	−0.638074	−	0.769975i	$-0.720269\pi$
$977$	−39.8885	−1.27615	−0.638074	+	0.769975i	$0.720269\pi$
$978$	0	0
$979$	61.3050	1.95931
$980$	−13.5279	−0.432132
$981$	0	0
$982$	1.12461	0.0358878
$983$	39.2361	1.25144	0.625718	−	0.780049i	$-0.284806\pi$
$983$	39.2361	1.25144	0.625718	+	0.780049i	$0.284806\pi$
$984$	0	0
$985$	−25.2361	−0.804088
$986$	−5.52786	−0.176043
$987$	0	0
$988$	42.1803	1.34194
$989$	−22.8885	−0.727813
$990$	0	0
$991$	−36.9574	−1.17399	−0.586996	−	0.809590i	$-0.699689\pi$
$991$	−36.9574	−1.17399	−0.586996	+	0.809590i	$0.699689\pi$
$992$	7.70820	0.244736
$993$	0	0
$994$	3.23607	0.102642
$995$	−3.05573	−0.0968731
$996$	0	0
$997$	−2.06888	−0.0655222	−0.0327611	−	0.999463i	$-0.510430\pi$
$997$	−2.06888	−0.0655222	−0.0327611	+	0.999463i	$0.510430\pi$
$998$	−35.8885	−1.13603
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9054.2.a.y.1.1		2
3.2	odd	2		1006.2.a.f.1.2	✓	2
12.11	even	2		8048.2.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.f.1.2	✓	2	3.2	odd	2
8048.2.a.l.1.1		2	12.11	even	2
9054.2.a.y.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9054 = 2 \cdot 3^{2} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9054.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.23607 q^{5} -4.23607 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.23607 q^{5} -4.23607 q^{7} +1.00000 q^{8} -1.23607 q^{10} +4.23607 q^{11} -5.47214 q^{13} -4.23607 q^{14} +1.00000 q^{16} -2.00000 q^{17} -7.70820 q^{19} -1.23607 q^{20} +4.23607 q^{22} -2.23607 q^{23} -3.47214 q^{25} -5.47214 q^{26} -4.23607 q^{28} +2.76393 q^{29} +7.70820 q^{31} +1.00000 q^{32} -2.00000 q^{34} +5.23607 q^{35} -4.47214 q^{37} -7.70820 q^{38} -1.23607 q^{40} +5.23607 q^{41} +10.2361 q^{43} +4.23607 q^{44} -2.23607 q^{46} -4.23607 q^{47} +10.9443 q^{49} -3.47214 q^{50} -5.47214 q^{52} +0.472136 q^{53} -5.23607 q^{55} -4.23607 q^{56} +2.76393 q^{58} -8.94427 q^{59} +7.47214 q^{61} +7.70820 q^{62} +1.00000 q^{64} +6.76393 q^{65} +4.70820 q^{67} -2.00000 q^{68} +5.23607 q^{70} -0.763932 q^{71} -7.52786 q^{73} -4.47214 q^{74} -7.70820 q^{76} -17.9443 q^{77} +12.9443 q^{79} -1.23607 q^{80} +5.23607 q^{82} -16.7082 q^{83} +2.47214 q^{85} +10.2361 q^{86} +4.23607 q^{88} +14.4721 q^{89} +23.1803 q^{91} -2.23607 q^{92} -4.23607 q^{94} +9.52786 q^{95} -10.0000 q^{97} +10.9443 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 4 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 - 4 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 4 q^{7} + 2 q^{8} + 2 q^{10} + 4 q^{11} - 2 q^{13} - 4 q^{14} + 2 q^{16} - 4 q^{17} - 2 q^{19} + 2 q^{20} + 4 q^{22} + 2 q^{25} - 2 q^{26} - 4 q^{28} + 10 q^{29} + 2 q^{31} + 2 q^{32} - 4 q^{34} + 6 q^{35} - 2 q^{38} + 2 q^{40} + 6 q^{41} + 16 q^{43} + 4 q^{44} - 4 q^{47} + 4 q^{49} + 2 q^{50} - 2 q^{52} - 8 q^{53} - 6 q^{55} - 4 q^{56} + 10 q^{58} + 6 q^{61} + 2 q^{62} + 2 q^{64} + 18 q^{65} - 4 q^{67} - 4 q^{68} + 6 q^{70} - 6 q^{71} - 24 q^{73} - 2 q^{76} - 18 q^{77} + 8 q^{79} + 2 q^{80} + 6 q^{82} - 20 q^{83} - 4 q^{85} + 16 q^{86} + 4 q^{88} + 20 q^{89} + 24 q^{91} - 4 q^{94} + 28 q^{95} - 20 q^{97} + 4 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 - 4 * q^7 + 2 * q^8 + 2 * q^10 + 4 * q^11 - 2 * q^13 - 4 * q^14 + 2 * q^16 - 4 * q^17 - 2 * q^19 + 2 * q^20 + 4 * q^22 + 2 * q^25 - 2 * q^26 - 4 * q^28 + 10 * q^29 + 2 * q^31 + 2 * q^32 - 4 * q^34 + 6 * q^35 - 2 * q^38 + 2 * q^40 + 6 * q^41 + 16 * q^43 + 4 * q^44 - 4 * q^47 + 4 * q^49 + 2 * q^50 - 2 * q^52 - 8 * q^53 - 6 * q^55 - 4 * q^56 + 10 * q^58 + 6 * q^61 + 2 * q^62 + 2 * q^64 + 18 * q^65 - 4 * q^67 - 4 * q^68 + 6 * q^70 - 6 * q^71 - 24 * q^73 - 2 * q^76 - 18 * q^77 + 8 * q^79 + 2 * q^80 + 6 * q^82 - 20 * q^83 - 4 * q^85 + 16 * q^86 + 4 * q^88 + 20 * q^89 + 24 * q^91 - 4 * q^94 + 28 * q^95 - 20 * q^97 + 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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