LMFDB - Embedded newform 9054.2.a.y.1.2

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			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.23607 q^{5} +0.236068 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +3.23607 q^{5} +0.236068 q^{7} +1.00000 q^{8} +3.23607 q^{10} -0.236068 q^{11} +3.47214 q^{13} +0.236068 q^{14} +1.00000 q^{16} -2.00000 q^{17} +5.70820 q^{19} +3.23607 q^{20} -0.236068 q^{22} +2.23607 q^{23} +5.47214 q^{25} +3.47214 q^{26} +0.236068 q^{28} +7.23607 q^{29} -5.70820 q^{31} +1.00000 q^{32} -2.00000 q^{34} +0.763932 q^{35} +4.47214 q^{37} +5.70820 q^{38} +3.23607 q^{40} +0.763932 q^{41} +5.76393 q^{43} -0.236068 q^{44} +2.23607 q^{46} +0.236068 q^{47} -6.94427 q^{49} +5.47214 q^{50} +3.47214 q^{52} -8.47214 q^{53} -0.763932 q^{55} +0.236068 q^{56} +7.23607 q^{58} +8.94427 q^{59} -1.47214 q^{61} -5.70820 q^{62} +1.00000 q^{64} +11.2361 q^{65} -8.70820 q^{67} -2.00000 q^{68} +0.763932 q^{70} -5.23607 q^{71} -16.4721 q^{73} +4.47214 q^{74} +5.70820 q^{76} -0.0557281 q^{77} -4.94427 q^{79} +3.23607 q^{80} +0.763932 q^{82} -3.29180 q^{83} -6.47214 q^{85} +5.76393 q^{86} -0.236068 q^{88} +5.52786 q^{89} +0.819660 q^{91} +2.23607 q^{92} +0.236068 q^{94} +18.4721 q^{95} -10.0000 q^{97} -6.94427 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	3.23607	1.44721	0.723607	−	0.690212i	$-0.242483\pi$
$5$	3.23607	1.44721	0.723607	+	0.690212i	$0.242483\pi$
$6$	0	0
$7$	0.236068	0.0892253	0.0446127	−	0.999004i	$-0.485795\pi$
$7$	0.236068	0.0892253	0.0446127	+	0.999004i	$0.485795\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	3.23607	1.02333
$11$	−0.236068	−0.0711772	−0.0355886	−	0.999367i	$-0.511331\pi$
$11$	−0.236068	−0.0711772	−0.0355886	+	0.999367i	$0.511331\pi$
$12$	0	0
$13$	3.47214	0.962997	0.481499	−	0.876447i	$-0.340093\pi$
$13$	3.47214	0.962997	0.481499	+	0.876447i	$0.340093\pi$
$14$	0.236068	0.0630918
$15$	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
$19$	5.70820	1.30955	0.654776	−	0.755823i	$-0.272763\pi$
$19$	5.70820	1.30955	0.654776	+	0.755823i	$0.272763\pi$
$20$	3.23607	0.723607
$21$	0	0
$22$	−0.236068	−0.0503299
$23$	2.23607	0.466252	0.233126	−	0.972446i	$-0.425104\pi$
$23$	2.23607	0.466252	0.233126	+	0.972446i	$0.425104\pi$
$24$	0	0
$25$	5.47214	1.09443
$26$	3.47214	0.680942
$27$	0	0
$28$	0.236068	0.0446127
$29$	7.23607	1.34370	0.671852	−	0.740685i	$-0.265499\pi$
$29$	7.23607	1.34370	0.671852	+	0.740685i	$0.265499\pi$
$30$	0	0
$31$	−5.70820	−1.02522	−0.512612	−	0.858620i	$-0.671322\pi$
$31$	−5.70820	−1.02522	−0.512612	+	0.858620i	$0.671322\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0.763932	0.129128
$36$	0	0
$37$	4.47214	0.735215	0.367607	−	0.929981i	$-0.380177\pi$
$37$	4.47214	0.735215	0.367607	+	0.929981i	$0.380177\pi$
$38$	5.70820	0.925993
$39$	0	0
$40$	3.23607	0.511667
$41$	0.763932	0.119306	0.0596531	−	0.998219i	$-0.481001\pi$
$41$	0.763932	0.119306	0.0596531	+	0.998219i	$0.481001\pi$
$42$	0	0
$43$	5.76393	0.878991	0.439496	−	0.898245i	$-0.355157\pi$
$43$	5.76393	0.878991	0.439496	+	0.898245i	$0.355157\pi$
$44$	−0.236068	−0.0355886
$45$	0	0
$46$	2.23607	0.329690
$47$	0.236068	0.0344341	0.0172170	−	0.999852i	$-0.494519\pi$
$47$	0.236068	0.0344341	0.0172170	+	0.999852i	$0.494519\pi$
$48$	0	0
$49$	−6.94427	−0.992039
$50$	5.47214	0.773877
$51$	0	0
$52$	3.47214	0.481499
$53$	−8.47214	−1.16374	−0.581869	−	0.813283i	$-0.697678\pi$
$53$	−8.47214	−1.16374	−0.581869	+	0.813283i	$0.697678\pi$
$54$	0	0
$55$	−0.763932	−0.103009
$56$	0.236068	0.0315459
$57$	0	0
$58$	7.23607	0.950142
$59$	8.94427	1.16445	0.582223	−	0.813029i	$-0.302183\pi$
$59$	8.94427	1.16445	0.582223	+	0.813029i	$0.302183\pi$
$60$	0	0
$61$	−1.47214	−0.188488	−0.0942438	−	0.995549i	$-0.530043\pi$
$61$	−1.47214	−0.188488	−0.0942438	+	0.995549i	$0.530043\pi$
$62$	−5.70820	−0.724943
$63$	0	0
$64$	1.00000	0.125000
$65$	11.2361	1.39366
$66$	0	0
$67$	−8.70820	−1.06388	−0.531938	−	0.846783i	$-0.678536\pi$
$67$	−8.70820	−1.06388	−0.531938	+	0.846783i	$0.678536\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	0.763932	0.0913073
$71$	−5.23607	−0.621407	−0.310703	−	0.950507i	$-0.600565\pi$
$71$	−5.23607	−0.621407	−0.310703	+	0.950507i	$0.600565\pi$
$72$	0	0
$73$	−16.4721	−1.92792	−0.963959	−	0.266051i	$-0.914281\pi$
$73$	−16.4721	−1.92792	−0.963959	+	0.266051i	$0.914281\pi$
$74$	4.47214	0.519875
$75$	0	0
$76$	5.70820	0.654776
$77$	−0.0557281	−0.00635081
$78$	0	0
$79$	−4.94427	−0.556274	−0.278137	−	0.960541i	$-0.589717\pi$
$79$	−4.94427	−0.556274	−0.278137	+	0.960541i	$0.589717\pi$
$80$	3.23607	0.361803
$81$	0	0
$82$	0.763932	0.0843622
$83$	−3.29180	−0.361322	−0.180661	−	0.983545i	$-0.557824\pi$
$83$	−3.29180	−0.361322	−0.180661	+	0.983545i	$0.557824\pi$
$84$	0	0
$85$	−6.47214	−0.702002
$86$	5.76393	0.621541
$87$	0	0
$88$	−0.236068	−0.0251649
$89$	5.52786	0.585952	0.292976	−	0.956120i	$-0.405354\pi$
$89$	5.52786	0.585952	0.292976	+	0.956120i	$0.405354\pi$
$90$	0	0
$91$	0.819660	0.0859237
$92$	2.23607	0.233126
$93$	0	0
$94$	0.236068	0.0243486
$95$	18.4721	1.89520
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	−6.94427	−0.701477
$99$	0	0
$100$	5.47214	0.547214
$101$	−3.70820	−0.368980	−0.184490	−	0.982834i	$-0.559063\pi$
$101$	−3.70820	−0.368980	−0.184490	+	0.982834i	$0.559063\pi$
$102$	0	0
$103$	16.9443	1.66957	0.834784	−	0.550577i	$-0.185592\pi$
$103$	16.9443	1.66957	0.834784	+	0.550577i	$0.185592\pi$
$104$	3.47214	0.340471
$105$	0	0
$106$	−8.47214	−0.822887
$107$	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
$109$	−0.472136	−0.0452224	−0.0226112	−	0.999744i	$-0.507198\pi$
$109$	−0.472136	−0.0452224	−0.0226112	+	0.999744i	$0.507198\pi$
$110$	−0.763932	−0.0728381
$111$	0	0
$112$	0.236068	0.0223063
$113$	17.4721	1.64364	0.821820	−	0.569747i	$-0.192959\pi$
$113$	17.4721	1.64364	0.821820	+	0.569747i	$0.192959\pi$
$114$	0	0
$115$	7.23607	0.674767
$116$	7.23607	0.671852
$117$	0	0
$118$	8.94427	0.823387
$119$	−0.472136	−0.0432806
$120$	0	0
$121$	−10.9443	−0.994934
$122$	−1.47214	−0.133281
$123$	0	0
$124$	−5.70820	−0.512612
$125$	1.52786	0.136656
$126$	0	0
$127$	15.7082	1.39388	0.696939	−	0.717131i	$-0.254545\pi$
$127$	15.7082	1.39388	0.696939	+	0.717131i	$0.254545\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	11.2361	0.985468
$131$	−1.18034	−0.103127	−0.0515634	−	0.998670i	$-0.516420\pi$
$131$	−1.18034	−0.103127	−0.0515634	+	0.998670i	$0.516420\pi$
$132$	0	0
$133$	1.34752	0.116845
$134$	−8.70820	−0.752274
$135$	0	0
$136$	−2.00000	−0.171499
$137$	2.76393	0.236139	0.118069	−	0.993005i	$-0.462329\pi$
$137$	2.76393	0.236139	0.118069	+	0.993005i	$0.462329\pi$
$138$	0	0
$139$	12.1803	1.03312	0.516561	−	0.856250i	$-0.327212\pi$
$139$	12.1803	1.03312	0.516561	+	0.856250i	$0.327212\pi$
$140$	0.763932	0.0645640
$141$	0	0
$142$	−5.23607	−0.439401
$143$	−0.819660	−0.0685434
$144$	0	0
$145$	23.4164	1.94463
$146$	−16.4721	−1.36324
$147$	0	0
$148$	4.47214	0.367607
$149$	17.2361	1.41203	0.706017	−	0.708195i	$-0.250490\pi$
$149$	17.2361	1.41203	0.706017	+	0.708195i	$0.250490\pi$
$150$	0	0
$151$	−9.52786	−0.775367	−0.387683	−	0.921793i	$-0.626725\pi$
$151$	−9.52786	−0.775367	−0.387683	+	0.921793i	$0.626725\pi$
$152$	5.70820	0.462996
$153$	0	0
$154$	−0.0557281	−0.00449070
$155$	−18.4721	−1.48372
$156$	0	0
$157$	3.52786	0.281554	0.140777	−	0.990041i	$-0.455040\pi$
$157$	3.52786	0.281554	0.140777	+	0.990041i	$0.455040\pi$
$158$	−4.94427	−0.393345
$159$	0	0
$160$	3.23607	0.255834
$161$	0.527864	0.0416015
$162$	0	0
$163$	1.70820	0.133797	0.0668984	−	0.997760i	$-0.478690\pi$
$163$	1.70820	0.133797	0.0668984	+	0.997760i	$0.478690\pi$
$164$	0.763932	0.0596531
$165$	0	0
$166$	−3.29180	−0.255493
$167$	−5.70820	−0.441714	−0.220857	−	0.975306i	$-0.570885\pi$
$167$	−5.70820	−0.441714	−0.220857	+	0.975306i	$0.570885\pi$
$168$	0	0
$169$	−0.944272	−0.0726363
$170$	−6.47214	−0.496390
$171$	0	0
$172$	5.76393	0.439496
$173$	−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
$175$	1.29180	0.0976506
$176$	−0.236068	−0.0177943
$177$	0	0
$178$	5.52786	0.414331
$179$	7.70820	0.576138	0.288069	−	0.957610i	$-0.406987\pi$
$179$	7.70820	0.576138	0.288069	+	0.957610i	$0.406987\pi$
$180$	0	0
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	0.819660	0.0607572
$183$	0	0
$184$	2.23607	0.164845
$185$	14.4721	1.06401
$186$	0	0
$187$	0.472136	0.0345260
$188$	0.236068	0.0172170
$189$	0	0
$190$	18.4721	1.34011
$191$	9.52786	0.689412	0.344706	−	0.938711i	$-0.387979\pi$
$191$	9.52786	0.689412	0.344706	+	0.938711i	$0.387979\pi$
$192$	0	0
$193$	−11.2361	−0.808790	−0.404395	−	0.914584i	$-0.632518\pi$
$193$	−11.2361	−0.808790	−0.404395	+	0.914584i	$0.632518\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	−6.94427	−0.496019
$197$	−6.41641	−0.457150	−0.228575	−	0.973526i	$-0.573407\pi$
$197$	−6.41641	−0.457150	−0.228575	+	0.973526i	$0.573407\pi$
$198$	0	0
$199$	−6.47214	−0.458798	−0.229399	−	0.973333i	$-0.573676\pi$
$199$	−6.47214	−0.458798	−0.229399	+	0.973333i	$0.573676\pi$
$200$	5.47214	0.386938
$201$	0	0
$202$	−3.70820	−0.260908
$203$	1.70820	0.119892
$204$	0	0
$205$	2.47214	0.172661
$206$	16.9443	1.18056
$207$	0	0
$208$	3.47214	0.240749
$209$	−1.34752	−0.0932102
$210$	0	0
$211$	9.70820	0.668340	0.334170	−	0.942513i	$-0.391544\pi$
$211$	9.70820	0.668340	0.334170	+	0.942513i	$0.391544\pi$
$212$	−8.47214	−0.581869
$213$	0	0
$214$	8.00000	0.546869
$215$	18.6525	1.27209
$216$	0	0
$217$	−1.34752	−0.0914759
$218$	−0.472136	−0.0319771
$219$	0	0
$220$	−0.763932	−0.0515043
$221$	−6.94427	−0.467122
$222$	0	0
$223$	8.23607	0.551528	0.275764	−	0.961225i	$-0.411069\pi$
$223$	8.23607	0.551528	0.275764	+	0.961225i	$0.411069\pi$
$224$	0.236068	0.0157730
$225$	0	0
$226$	17.4721	1.16223
$227$	1.70820	0.113377	0.0566887	−	0.998392i	$-0.481946\pi$
$227$	1.70820	0.113377	0.0566887	+	0.998392i	$0.481946\pi$
$228$	0	0
$229$	−8.41641	−0.556172	−0.278086	−	0.960556i	$-0.589700\pi$
$229$	−8.41641	−0.556172	−0.278086	+	0.960556i	$0.589700\pi$
$230$	7.23607	0.477132
$231$	0	0
$232$	7.23607	0.475071
$233$	4.41641	0.289328	0.144664	−	0.989481i	$-0.453790\pi$
$233$	4.41641	0.289328	0.144664	+	0.989481i	$0.453790\pi$
$234$	0	0
$235$	0.763932	0.0498334
$236$	8.94427	0.582223
$237$	0	0
$238$	−0.472136	−0.0306040
$239$	−1.41641	−0.0916198	−0.0458099	−	0.998950i	$-0.514587\pi$
$239$	−1.41641	−0.0916198	−0.0458099	+	0.998950i	$0.514587\pi$
$240$	0	0
$241$	12.9443	0.833814	0.416907	−	0.908949i	$-0.363114\pi$
$241$	12.9443	0.833814	0.416907	+	0.908949i	$0.363114\pi$
$242$	−10.9443	−0.703524
$243$	0	0
$244$	−1.47214	−0.0942438
$245$	−22.4721	−1.43569
$246$	0	0
$247$	19.8197	1.26109
$248$	−5.70820	−0.362471
$249$	0	0
$250$	1.52786	0.0966306
$251$	−18.1803	−1.14753	−0.573766	−	0.819019i	$-0.694518\pi$
$251$	−18.1803	−1.14753	−0.573766	+	0.819019i	$0.694518\pi$
$252$	0	0
$253$	−0.527864	−0.0331865
$254$	15.7082	0.985620
$255$	0	0
$256$	1.00000	0.0625000
$257$	−27.9443	−1.74312	−0.871558	−	0.490293i	$-0.836890\pi$
$257$	−27.9443	−1.74312	−0.871558	+	0.490293i	$0.836890\pi$
$258$	0	0
$259$	1.05573	0.0655998
$260$	11.2361	0.696831
$261$	0	0
$262$	−1.18034	−0.0729216
$263$	24.1246	1.48759	0.743794	−	0.668409i	$-0.233025\pi$
$263$	24.1246	1.48759	0.743794	+	0.668409i	$0.233025\pi$
$264$	0	0
$265$	−27.4164	−1.68418
$266$	1.34752	0.0826220
$267$	0	0
$268$	−8.70820	−0.531938
$269$	0.944272	0.0575733	0.0287866	−	0.999586i	$-0.490836\pi$
$269$	0.944272	0.0575733	0.0287866	+	0.999586i	$0.490836\pi$
$270$	0	0
$271$	13.2918	0.807419	0.403710	−	0.914887i	$-0.367721\pi$
$271$	13.2918	0.807419	0.403710	+	0.914887i	$0.367721\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	2.76393	0.166975
$275$	−1.29180	−0.0778982
$276$	0	0
$277$	3.81966	0.229501	0.114751	−	0.993394i	$-0.463393\pi$
$277$	3.81966	0.229501	0.114751	+	0.993394i	$0.463393\pi$
$278$	12.1803	0.730528
$279$	0	0
$280$	0.763932	0.0456537
$281$	1.58359	0.0944692	0.0472346	−	0.998884i	$-0.484959\pi$
$281$	1.58359	0.0944692	0.0472346	+	0.998884i	$0.484959\pi$
$282$	0	0
$283$	−9.52786	−0.566373	−0.283186	−	0.959065i	$-0.591391\pi$
$283$	−9.52786	−0.566373	−0.283186	+	0.959065i	$0.591391\pi$
$284$	−5.23607	−0.310703
$285$	0	0
$286$	−0.819660	−0.0484675
$287$	0.180340	0.0106451
$288$	0	0
$289$	−13.0000	−0.764706
$290$	23.4164	1.37506
$291$	0	0
$292$	−16.4721	−0.963959
$293$	−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
$295$	28.9443	1.68520
$296$	4.47214	0.259938
$297$	0	0
$298$	17.2361	0.998459
$299$	7.76393	0.449000
$300$	0	0
$301$	1.36068	0.0784283
$302$	−9.52786	−0.548267
$303$	0	0
$304$	5.70820	0.327388
$305$	−4.76393	−0.272782
$306$	0	0
$307$	−17.4164	−0.994007	−0.497003	−	0.867749i	$-0.665566\pi$
$307$	−17.4164	−0.994007	−0.497003	+	0.867749i	$0.665566\pi$
$308$	−0.0557281	−0.00317540
$309$	0	0
$310$	−18.4721	−1.04915
$311$	28.1803	1.59796	0.798980	−	0.601357i	$-0.205373\pi$
$311$	28.1803	1.59796	0.798980	+	0.601357i	$0.205373\pi$
$312$	0	0
$313$	−13.0557	−0.737953	−0.368977	−	0.929439i	$-0.620292\pi$
$313$	−13.0557	−0.737953	−0.368977	+	0.929439i	$0.620292\pi$
$314$	3.52786	0.199089
$315$	0	0
$316$	−4.94427	−0.278137
$317$	25.9443	1.45718	0.728588	−	0.684952i	$-0.240177\pi$
$317$	25.9443	1.45718	0.728588	+	0.684952i	$0.240177\pi$
$318$	0	0
$319$	−1.70820	−0.0956411
$320$	3.23607	0.180902
$321$	0	0
$322$	0.527864	0.0294167
$323$	−11.4164	−0.635226
$324$	0	0
$325$	19.0000	1.05393
$326$	1.70820	0.0946087
$327$	0	0
$328$	0.763932	0.0421811
$329$	0.0557281	0.00307239
$330$	0	0
$331$	−6.94427	−0.381692	−0.190846	−	0.981620i	$-0.561123\pi$
$331$	−6.94427	−0.381692	−0.190846	+	0.981620i	$0.561123\pi$
$332$	−3.29180	−0.180661
$333$	0	0
$334$	−5.70820	−0.312339
$335$	−28.1803	−1.53966
$336$	0	0
$337$	−13.0557	−0.711191	−0.355595	−	0.934640i	$-0.615722\pi$
$337$	−13.0557	−0.711191	−0.355595	+	0.934640i	$0.615722\pi$
$338$	−0.944272	−0.0513616
$339$	0	0
$340$	−6.47214	−0.351001
$341$	1.34752	0.0729725
$342$	0	0
$343$	−3.29180	−0.177740
$344$	5.76393	0.310770
$345$	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	$0.395923\pi$
$350$	1.29180	0.0690494
$351$	0	0
$352$	−0.236068	−0.0125825
$353$	29.2361	1.55608	0.778039	−	0.628215i	$-0.216214\pi$
$353$	29.2361	1.55608	0.778039	+	0.628215i	$0.216214\pi$
$354$	0	0
$355$	−16.9443	−0.899309
$356$	5.52786	0.292976
$357$	0	0
$358$	7.70820	0.407391
$359$	−3.70820	−0.195712	−0.0978558	−	0.995201i	$-0.531198\pi$
$359$	−3.70820	−0.195712	−0.0978558	+	0.995201i	$0.531198\pi$
$360$	0	0
$361$	13.5836	0.714926
$362$	−20.0000	−1.05118
$363$	0	0
$364$	0.819660	0.0429619
$365$	−53.3050	−2.79011
$366$	0	0
$367$	2.23607	0.116722	0.0583609	−	0.998296i	$-0.481413\pi$
$367$	2.23607	0.116722	0.0583609	+	0.998296i	$0.481413\pi$
$368$	2.23607	0.116563
$369$	0	0
$370$	14.4721	0.752371
$371$	−2.00000	−0.103835
$372$	0	0
$373$	−36.3050	−1.87980	−0.939900	−	0.341451i	$-0.889082\pi$
$373$	−36.3050	−1.87980	−0.939900	+	0.341451i	$0.889082\pi$
$374$	0.472136	0.0244136
$375$	0	0
$376$	0.236068	0.0121743
$377$	25.1246	1.29398
$378$	0	0
$379$	6.70820	0.344577	0.172289	−	0.985047i	$-0.444884\pi$
$379$	6.70820	0.344577	0.172289	+	0.985047i	$0.444884\pi$
$380$	18.4721	0.947601
$381$	0	0
$382$	9.52786	0.487488
$383$	−4.94427	−0.252640	−0.126320	−	0.991990i	$-0.540317\pi$
$383$	−4.94427	−0.252640	−0.126320	+	0.991990i	$0.540317\pi$
$384$	0	0
$385$	−0.180340	−0.00919097
$386$	−11.2361	−0.571901
$387$	0	0
$388$	−10.0000	−0.507673
$389$	36.8328	1.86750	0.933749	−	0.357929i	$-0.116517\pi$
$389$	36.8328	1.86750	0.933749	+	0.357929i	$0.116517\pi$
$390$	0	0
$391$	−4.47214	−0.226166
$392$	−6.94427	−0.350739
$393$	0	0
$394$	−6.41641	−0.323254
$395$	−16.0000	−0.805047
$396$	0	0
$397$	−7.36068	−0.369422	−0.184711	−	0.982793i	$-0.559135\pi$
$397$	−7.36068	−0.369422	−0.184711	+	0.982793i	$0.559135\pi$
$398$	−6.47214	−0.324419
$399$	0	0
$400$	5.47214	0.273607
$401$	14.4164	0.719921	0.359961	−	0.932968i	$-0.382790\pi$
$401$	14.4164	0.719921	0.359961	+	0.932968i	$0.382790\pi$
$402$	0	0
$403$	−19.8197	−0.987288
$404$	−3.70820	−0.184490
$405$	0	0
$406$	1.70820	0.0847767
$407$	−1.05573	−0.0523305
$408$	0	0
$409$	−11.7082	−0.578933	−0.289467	−	0.957188i	$-0.593478\pi$
$409$	−11.7082	−0.578933	−0.289467	+	0.957188i	$0.593478\pi$
$410$	2.47214	0.122090
$411$	0	0
$412$	16.9443	0.834784
$413$	2.11146	0.103898
$414$	0	0
$415$	−10.6525	−0.522909
$416$	3.47214	0.170235
$417$	0	0
$418$	−1.34752	−0.0659096
$419$	21.7082	1.06052	0.530258	−	0.847837i	$-0.322095\pi$
$419$	21.7082	1.06052	0.530258	+	0.847837i	$0.322095\pi$
$420$	0	0
$421$	−22.9443	−1.11824	−0.559118	−	0.829088i	$-0.688860\pi$
$421$	−22.9443	−1.11824	−0.559118	+	0.829088i	$0.688860\pi$
$422$	9.70820	0.472588
$423$	0	0
$424$	−8.47214	−0.411443
$425$	−10.9443	−0.530875
$426$	0	0
$427$	−0.347524	−0.0168179
$428$	8.00000	0.386695
$429$	0	0
$430$	18.6525	0.899502
$431$	−32.6525	−1.57281	−0.786407	−	0.617708i	$-0.788061\pi$
$431$	−32.6525	−1.57281	−0.786407	+	0.617708i	$0.788061\pi$
$432$	0	0
$433$	11.5279	0.553994	0.276997	−	0.960871i	$-0.410661\pi$
$433$	11.5279	0.553994	0.276997	+	0.960871i	$0.410661\pi$
$434$	−1.34752	−0.0646832
$435$	0	0
$436$	−0.472136	−0.0226112
$437$	12.7639	0.610582
$438$	0	0
$439$	17.8885	0.853774	0.426887	−	0.904305i	$-0.359610\pi$
$439$	17.8885	0.853774	0.426887	+	0.904305i	$0.359610\pi$
$440$	−0.763932	−0.0364190
$441$	0	0
$442$	−6.94427	−0.330305
$443$	−0.819660	−0.0389432	−0.0194716	−	0.999810i	$-0.506198\pi$
$443$	−0.819660	−0.0389432	−0.0194716	+	0.999810i	$0.506198\pi$
$444$	0	0
$445$	17.8885	0.847998
$446$	8.23607	0.389989
$447$	0	0
$448$	0.236068	0.0111532
$449$	29.5967	1.39676	0.698378	−	0.715729i	$-0.253905\pi$
$449$	29.5967	1.39676	0.698378	+	0.715729i	$0.253905\pi$
$450$	0	0
$451$	−0.180340	−0.00849187
$452$	17.4721	0.821820
$453$	0	0
$454$	1.70820	0.0801700
$455$	2.65248	0.124350
$456$	0	0
$457$	22.3607	1.04599	0.522994	−	0.852336i	$-0.324815\pi$
$457$	22.3607	1.04599	0.522994	+	0.852336i	$0.324815\pi$
$458$	−8.41641	−0.393273
$459$	0	0
$460$	7.23607	0.337383
$461$	−42.0689	−1.95934	−0.979672	−	0.200608i	$-0.935708\pi$
$461$	−42.0689	−1.95934	−0.979672	+	0.200608i	$0.935708\pi$
$462$	0	0
$463$	−32.5967	−1.51490	−0.757450	−	0.652894i	$-0.773555\pi$
$463$	−32.5967	−1.51490	−0.757450	+	0.652894i	$0.773555\pi$
$464$	7.23607	0.335926
$465$	0	0
$466$	4.41641	0.204586
$467$	21.7082	1.00454	0.502268	−	0.864712i	$-0.332499\pi$
$467$	21.7082	1.00454	0.502268	+	0.864712i	$0.332499\pi$
$468$	0	0
$469$	−2.05573	−0.0949247
$470$	0.763932	0.0352376
$471$	0	0
$472$	8.94427	0.411693
$473$	−1.36068	−0.0625641
$474$	0	0
$475$	31.2361	1.43321
$476$	−0.472136	−0.0216403
$477$	0	0
$478$	−1.41641	−0.0647850
$479$	−4.94427	−0.225910	−0.112955	−	0.993600i	$-0.536032\pi$
$479$	−4.94427	−0.225910	−0.112955	+	0.993600i	$0.536032\pi$
$480$	0	0
$481$	15.5279	0.708010
$482$	12.9443	0.589595
$483$	0	0
$484$	−10.9443	−0.497467
$485$	−32.3607	−1.46942
$486$	0	0
$487$	−13.5967	−0.616127	−0.308064	−	0.951366i	$-0.599681\pi$
$487$	−13.5967	−0.616127	−0.308064	+	0.951366i	$0.599681\pi$
$488$	−1.47214	−0.0666405
$489$	0	0
$490$	−22.4721	−1.01519
$491$	−39.1246	−1.76567	−0.882835	−	0.469684i	$-0.844368\pi$
$491$	−39.1246	−1.76567	−0.882835	+	0.469684i	$0.844368\pi$
$492$	0	0
$493$	−14.4721	−0.651792
$494$	19.8197	0.891729
$495$	0	0
$496$	−5.70820	−0.256306
$497$	−1.23607	−0.0554452
$498$	0	0
$499$	−0.111456	−0.00498946	−0.00249473	−	0.999997i	$-0.500794\pi$
$499$	−0.111456	−0.00498946	−0.00249473	+	0.999997i	$0.500794\pi$
$500$	1.52786	0.0683282
$501$	0	0
$502$	−18.1803	−0.811428
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−12.0000	−0.533993
$506$	−0.527864	−0.0234664
$507$	0	0
$508$	15.7082	0.696939
$509$	20.8885	0.925868	0.462934	−	0.886393i	$-0.346797\pi$
$509$	20.8885	0.925868	0.462934	+	0.886393i	$0.346797\pi$
$510$	0	0
$511$	−3.88854	−0.172019
$512$	1.00000	0.0441942
$513$	0	0
$514$	−27.9443	−1.23257
$515$	54.8328	2.41622
$516$	0	0
$517$	−0.0557281	−0.00245092
$518$	1.05573	0.0463860
$519$	0	0
$520$	11.2361	0.492734
$521$	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
$523$	−32.3607	−1.41503	−0.707517	−	0.706696i	$-0.750185\pi$
$523$	−32.3607	−1.41503	−0.707517	+	0.706696i	$0.750185\pi$
$524$	−1.18034	−0.0515634
$525$	0	0
$526$	24.1246	1.05188
$527$	11.4164	0.497307
$528$	0	0
$529$	−18.0000	−0.782609
$530$	−27.4164	−1.19089
$531$	0	0
$532$	1.34752	0.0584226
$533$	2.65248	0.114891
$534$	0	0
$535$	25.8885	1.11926
$536$	−8.70820	−0.376137
$537$	0	0
$538$	0.944272	0.0407105
$539$	1.63932	0.0706105
$540$	0	0
$541$	−24.6525	−1.05989	−0.529946	−	0.848031i	$-0.677788\pi$
$541$	−24.6525	−1.05989	−0.529946	+	0.848031i	$0.677788\pi$
$542$	13.2918	0.570932
$543$	0	0
$544$	−2.00000	−0.0857493
$545$	−1.52786	−0.0654465
$546$	0	0
$547$	−8.36068	−0.357477	−0.178738	−	0.983897i	$-0.557202\pi$
$547$	−8.36068	−0.357477	−0.178738	+	0.983897i	$0.557202\pi$
$548$	2.76393	0.118069
$549$	0	0
$550$	−1.29180	−0.0550824
$551$	41.3050	1.75965
$552$	0	0
$553$	−1.16718	−0.0496337
$554$	3.81966	0.162282
$555$	0	0
$556$	12.1803	0.516561
$557$	−43.2492	−1.83253	−0.916264	−	0.400574i	$-0.868811\pi$
$557$	−43.2492	−1.83253	−0.916264	+	0.400574i	$0.868811\pi$
$558$	0	0
$559$	20.0132	0.846466
$560$	0.763932	0.0322820
$561$	0	0
$562$	1.58359	0.0667998
$563$	−22.2918	−0.939487	−0.469744	−	0.882803i	$-0.655654\pi$
$563$	−22.2918	−0.939487	−0.469744	+	0.882803i	$0.655654\pi$
$564$	0	0
$565$	56.5410	2.37870
$566$	−9.52786	−0.400486
$567$	0	0
$568$	−5.23607	−0.219701
$569$	23.3050	0.976994	0.488497	−	0.872565i	$-0.337545\pi$
$569$	23.3050	0.976994	0.488497	+	0.872565i	$0.337545\pi$
$570$	0	0
$571$	−19.8885	−0.832310	−0.416155	−	0.909294i	$-0.636623\pi$
$571$	−19.8885	−0.832310	−0.416155	+	0.909294i	$0.636623\pi$
$572$	−0.819660	−0.0342717
$573$	0	0
$574$	0.180340	0.00752724
$575$	12.2361	0.510279
$576$	0	0
$577$	22.3607	0.930887	0.465444	−	0.885078i	$-0.345895\pi$
$577$	22.3607	0.930887	0.465444	+	0.885078i	$0.345895\pi$
$578$	−13.0000	−0.540729
$579$	0	0
$580$	23.4164	0.972313
$581$	−0.777088	−0.0322390
$582$	0	0
$583$	2.00000	0.0828315
$584$	−16.4721	−0.681622
$585$	0	0
$586$	−5.00000	−0.206548
$587$	−38.8328	−1.60280	−0.801401	−	0.598128i	$-0.795912\pi$
$587$	−38.8328	−1.60280	−0.801401	+	0.598128i	$0.795912\pi$
$588$	0	0
$589$	−32.5836	−1.34258
$590$	28.9443	1.19162
$591$	0	0
$592$	4.47214	0.183804
$593$	−7.52786	−0.309132	−0.154566	−	0.987982i	$-0.549398\pi$
$593$	−7.52786	−0.309132	−0.154566	+	0.987982i	$0.549398\pi$
$594$	0	0
$595$	−1.52786	−0.0626363
$596$	17.2361	0.706017
$597$	0	0
$598$	7.76393	0.317491
$599$	−34.4721	−1.40849	−0.704247	−	0.709955i	$-0.748715\pi$
$599$	−34.4721	−1.40849	−0.704247	+	0.709955i	$0.748715\pi$
$600$	0	0
$601$	−0.527864	−0.0215320	−0.0107660	−	0.999942i	$-0.503427\pi$
$601$	−0.527864	−0.0215320	−0.0107660	+	0.999942i	$0.503427\pi$
$602$	1.36068	0.0554572
$603$	0	0
$604$	−9.52786	−0.387683
$605$	−35.4164	−1.43988
$606$	0	0
$607$	5.76393	0.233951	0.116975	−	0.993135i	$-0.462680\pi$
$607$	5.76393	0.233951	0.116975	+	0.993135i	$0.462680\pi$
$608$	5.70820	0.231498
$609$	0	0
$610$	−4.76393	−0.192886
$611$	0.819660	0.0331599
$612$	0	0
$613$	30.1803	1.21897	0.609486	−	0.792797i	$-0.291376\pi$
$613$	30.1803	1.21897	0.609486	+	0.792797i	$0.291376\pi$
$614$	−17.4164	−0.702869
$615$	0	0
$616$	−0.0557281	−0.00224535
$617$	−11.8885	−0.478615	−0.239307	−	0.970944i	$-0.576920\pi$
$617$	−11.8885	−0.478615	−0.239307	+	0.970944i	$0.576920\pi$
$618$	0	0
$619$	18.2918	0.735209	0.367605	−	0.929982i	$-0.380178\pi$
$619$	18.2918	0.735209	0.367605	+	0.929982i	$0.380178\pi$
$620$	−18.4721	−0.741859
$621$	0	0
$622$	28.1803	1.12993
$623$	1.30495	0.0522818
$624$	0	0
$625$	−22.4164	−0.896656
$626$	−13.0557	−0.521812
$627$	0	0
$628$	3.52786	0.140777
$629$	−8.94427	−0.356631
$630$	0	0
$631$	7.76393	0.309077	0.154539	−	0.987987i	$-0.450611\pi$
$631$	7.76393	0.309077	0.154539	+	0.987987i	$0.450611\pi$
$632$	−4.94427	−0.196673
$633$	0	0
$634$	25.9443	1.03038
$635$	50.8328	2.01724
$636$	0	0
$637$	−24.1115	−0.955331
$638$	−1.70820	−0.0676284
$639$	0	0
$640$	3.23607	0.127917
$641$	−16.4164	−0.648409	−0.324205	−	0.945987i	$-0.605097\pi$
$641$	−16.4164	−0.648409	−0.324205	+	0.945987i	$0.605097\pi$
$642$	0	0
$643$	38.9443	1.53581	0.767906	−	0.640562i	$-0.221299\pi$
$643$	38.9443	1.53581	0.767906	+	0.640562i	$0.221299\pi$
$644$	0.527864	0.0208008
$645$	0	0
$646$	−11.4164	−0.449173
$647$	27.0557	1.06367	0.531835	−	0.846848i	$-0.321503\pi$
$647$	27.0557	1.06367	0.531835	+	0.846848i	$0.321503\pi$
$648$	0	0
$649$	−2.11146	−0.0828819
$650$	19.0000	0.745241
$651$	0	0
$652$	1.70820	0.0668984
$653$	−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
$655$	−3.81966	−0.149246
$656$	0.763932	0.0298265
$657$	0	0
$658$	0.0557281	0.00217251
$659$	46.0132	1.79242	0.896209	−	0.443632i	$-0.146310\pi$
$659$	46.0132	1.79242	0.896209	+	0.443632i	$0.146310\pi$
$660$	0	0
$661$	−2.63932	−0.102658	−0.0513288	−	0.998682i	$-0.516346\pi$
$661$	−2.63932	−0.102658	−0.0513288	+	0.998682i	$0.516346\pi$
$662$	−6.94427	−0.269897
$663$	0	0
$664$	−3.29180	−0.127746
$665$	4.36068	0.169100
$666$	0	0
$667$	16.1803	0.626505
$668$	−5.70820	−0.220857
$669$	0	0
$670$	−28.1803	−1.08870
$671$	0.347524	0.0134160
$672$	0	0
$673$	5.05573	0.194884	0.0974420	−	0.995241i	$-0.468934\pi$
$673$	5.05573	0.194884	0.0974420	+	0.995241i	$0.468934\pi$
$674$	−13.0557	−0.502888
$675$	0	0
$676$	−0.944272	−0.0363182
$677$	−36.4721	−1.40174	−0.700869	−	0.713290i	$-0.747204\pi$
$677$	−36.4721	−1.40174	−0.700869	+	0.713290i	$0.747204\pi$
$678$	0	0
$679$	−2.36068	−0.0905946
$680$	−6.47214	−0.248195
$681$	0	0
$682$	1.34752	0.0515994
$683$	46.1803	1.76704	0.883521	−	0.468392i	$-0.155166\pi$
$683$	46.1803	1.76704	0.883521	+	0.468392i	$0.155166\pi$
$684$	0	0
$685$	8.94427	0.341743
$686$	−3.29180	−0.125681
$687$	0	0
$688$	5.76393	0.219748
$689$	−29.4164	−1.12068
$690$	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
$695$	39.4164	1.49515
$696$	0	0
$697$	−1.52786	−0.0578720
$698$	12.0000	0.454207
$699$	0	0
$700$	1.29180	0.0488253
$701$	−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
$703$	25.5279	0.962802
$704$	−0.236068	−0.00889715
$705$	0	0
$706$	29.2361	1.10031
$707$	−0.875388	−0.0329224
$708$	0	0
$709$	30.0000	1.12667	0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000	1.12667	0.563337	+	0.826227i	$0.309517\pi$
$710$	−16.9443	−0.635907
$711$	0	0
$712$	5.52786	0.207165
$713$	−12.7639	−0.478013
$714$	0	0
$715$	−2.65248	−0.0991970
$716$	7.70820	0.288069
$717$	0	0
$718$	−3.70820	−0.138389
$719$	−25.8885	−0.965480	−0.482740	−	0.875764i	$-0.660358\pi$
$719$	−25.8885	−0.965480	−0.482740	+	0.875764i	$0.660358\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	13.5836	0.505529
$723$	0	0
$724$	−20.0000	−0.743294
$725$	39.5967	1.47059
$726$	0	0
$727$	−37.0689	−1.37481	−0.687404	−	0.726275i	$-0.741250\pi$
$727$	−37.0689	−1.37481	−0.687404	+	0.726275i	$0.741250\pi$
$728$	0.819660	0.0303786
$729$	0	0
$730$	−53.3050	−1.97290
$731$	−11.5279	−0.426373
$732$	0	0
$733$	−24.7639	−0.914677	−0.457338	−	0.889293i	$-0.651197\pi$
$733$	−24.7639	−0.914677	−0.457338	+	0.889293i	$0.651197\pi$
$734$	2.23607	0.0825348
$735$	0	0
$736$	2.23607	0.0824226
$737$	2.05573	0.0757237
$738$	0	0
$739$	−25.7639	−0.947742	−0.473871	−	0.880594i	$-0.657144\pi$
$739$	−25.7639	−0.947742	−0.473871	+	0.880594i	$0.657144\pi$
$740$	14.4721	0.532006
$741$	0	0
$742$	−2.00000	−0.0734223
$743$	49.7082	1.82362	0.911809	−	0.410616i	$-0.134686\pi$
$743$	49.7082	1.82362	0.911809	+	0.410616i	$0.134686\pi$
$744$	0	0
$745$	55.7771	2.04351
$746$	−36.3050	−1.32922
$747$	0	0
$748$	0.472136	0.0172630
$749$	1.88854	0.0690059
$750$	0	0
$751$	4.36068	0.159123	0.0795617	−	0.996830i	$-0.474648\pi$
$751$	4.36068	0.159123	0.0795617	+	0.996830i	$0.474648\pi$
$752$	0.236068	0.00860851
$753$	0	0
$754$	25.1246	0.914984
$755$	−30.8328	−1.12212
$756$	0	0
$757$	−2.36068	−0.0858004	−0.0429002	−	0.999079i	$-0.513660\pi$
$757$	−2.36068	−0.0858004	−0.0429002	+	0.999079i	$0.513660\pi$
$758$	6.70820	0.243653
$759$	0	0
$760$	18.4721	0.670055
$761$	−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
$763$	−0.111456	−0.00403498
$764$	9.52786	0.344706
$765$	0	0
$766$	−4.94427	−0.178644
$767$	31.0557	1.12136
$768$	0	0
$769$	−25.7771	−0.929546	−0.464773	−	0.885430i	$-0.653864\pi$
$769$	−25.7771	−0.929546	−0.464773	+	0.885430i	$0.653864\pi$
$770$	−0.180340	−0.00649900
$771$	0	0
$772$	−11.2361	−0.404395
$773$	15.5967	0.560976	0.280488	−	0.959858i	$-0.409504\pi$
$773$	15.5967	0.560976	0.280488	+	0.959858i	$0.409504\pi$
$774$	0	0
$775$	−31.2361	−1.12203
$776$	−10.0000	−0.358979
$777$	0	0
$778$	36.8328	1.32052
$779$	4.36068	0.156238
$780$	0	0
$781$	1.23607	0.0442300
$782$	−4.47214	−0.159923
$783$	0	0
$784$	−6.94427	−0.248010
$785$	11.4164	0.407469
$786$	0	0
$787$	−6.06888	−0.216332	−0.108166	−	0.994133i	$-0.534498\pi$
$787$	−6.06888	−0.216332	−0.108166	+	0.994133i	$0.534498\pi$
$788$	−6.41641	−0.228575
$789$	0	0
$790$	−16.0000	−0.569254
$791$	4.12461	0.146654
$792$	0	0
$793$	−5.11146	−0.181513
$794$	−7.36068	−0.261221
$795$	0	0
$796$	−6.47214	−0.229399
$797$	−48.8328	−1.72975	−0.864874	−	0.501990i	$-0.832601\pi$
$797$	−48.8328	−1.72975	−0.864874	+	0.501990i	$0.832601\pi$
$798$	0	0
$799$	−0.472136	−0.0167030
$800$	5.47214	0.193469
$801$	0	0
$802$	14.4164	0.509061
$803$	3.88854	0.137224
$804$	0	0
$805$	1.70820	0.0602063
$806$	−19.8197	−0.698118
$807$	0	0
$808$	−3.70820	−0.130454
$809$	−16.1803	−0.568870	−0.284435	−	0.958695i	$-0.591806\pi$
$809$	−16.1803	−0.568870	−0.284435	+	0.958695i	$0.591806\pi$
$810$	0	0
$811$	−3.76393	−0.132170	−0.0660848	−	0.997814i	$-0.521051\pi$
$811$	−3.76393	−0.132170	−0.0660848	+	0.997814i	$0.521051\pi$
$812$	1.70820	0.0599462
$813$	0	0
$814$	−1.05573	−0.0370033
$815$	5.52786	0.193633
$816$	0	0
$817$	32.9017	1.15108
$818$	−11.7082	−0.409368
$819$	0	0
$820$	2.47214	0.0863307
$821$	−23.0557	−0.804650	−0.402325	−	0.915497i	$-0.631798\pi$
$821$	−23.0557	−0.804650	−0.402325	+	0.915497i	$0.631798\pi$
$822$	0	0
$823$	−1.81966	−0.0634294	−0.0317147	−	0.999497i	$-0.510097\pi$
$823$	−1.81966	−0.0634294	−0.0317147	+	0.999497i	$0.510097\pi$
$824$	16.9443	0.590282
$825$	0	0
$826$	2.11146	0.0734670
$827$	−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
$829$	3.59675	0.124920	0.0624601	−	0.998047i	$-0.480105\pi$
$829$	3.59675	0.124920	0.0624601	+	0.998047i	$0.480105\pi$
$830$	−10.6525	−0.369753
$831$	0	0
$832$	3.47214	0.120375
$833$	13.8885	0.481210
$834$	0	0
$835$	−18.4721	−0.639255
$836$	−1.34752	−0.0466051
$837$	0	0
$838$	21.7082	0.749897
$839$	14.5967	0.503936	0.251968	−	0.967736i	$-0.418922\pi$
$839$	14.5967	0.503936	0.251968	+	0.967736i	$0.418922\pi$
$840$	0	0
$841$	23.3607	0.805541
$842$	−22.9443	−0.790712
$843$	0	0
$844$	9.70820	0.334170
$845$	−3.05573	−0.105120
$846$	0	0
$847$	−2.58359	−0.0887733
$848$	−8.47214	−0.290934
$849$	0	0
$850$	−10.9443	−0.375385
$851$	10.0000	0.342796
$852$	0	0
$853$	−37.8328	−1.29537	−0.647685	−	0.761908i	$-0.724263\pi$
$853$	−37.8328	−1.29537	−0.647685	+	0.761908i	$0.724263\pi$
$854$	−0.347524	−0.0118920
$855$	0	0
$856$	8.00000	0.273434
$857$	−33.9443	−1.15951	−0.579757	−	0.814789i	$-0.696853\pi$
$857$	−33.9443	−1.15951	−0.579757	+	0.814789i	$0.696853\pi$
$858$	0	0
$859$	−27.7082	−0.945392	−0.472696	−	0.881226i	$-0.656719\pi$
$859$	−27.7082	−0.945392	−0.472696	+	0.881226i	$0.656719\pi$
$860$	18.6525	0.636044
$861$	0	0
$862$	−32.6525	−1.11215
$863$	−2.76393	−0.0940853	−0.0470427	−	0.998893i	$-0.514980\pi$
$863$	−2.76393	−0.0940853	−0.0470427	+	0.998893i	$0.514980\pi$
$864$	0	0
$865$	−35.5967	−1.21033
$866$	11.5279	0.391733
$867$	0	0
$868$	−1.34752	−0.0457380
$869$	1.16718	0.0395940
$870$	0	0
$871$	−30.2361	−1.02451
$872$	−0.472136	−0.0159885
$873$	0	0
$874$	12.7639	0.431746
$875$	0.360680	0.0121932
$876$	0	0
$877$	53.7771	1.81592	0.907962	−	0.419053i	$-0.137638\pi$
$877$	53.7771	1.81592	0.907962	+	0.419053i	$0.137638\pi$
$878$	17.8885	0.603709
$879$	0	0
$880$	−0.763932	−0.0257521
$881$	−23.8885	−0.804825	−0.402413	−	0.915458i	$-0.631828\pi$
$881$	−23.8885	−0.804825	−0.402413	+	0.915458i	$0.631828\pi$
$882$	0	0
$883$	50.4721	1.69852	0.849261	−	0.527973i	$-0.177048\pi$
$883$	50.4721	1.69852	0.849261	+	0.527973i	$0.177048\pi$
$884$	−6.94427	−0.233561
$885$	0	0
$886$	−0.819660	−0.0275370
$887$	−48.7214	−1.63590	−0.817952	−	0.575287i	$-0.804890\pi$
$887$	−48.7214	−1.63590	−0.817952	+	0.575287i	$0.804890\pi$
$888$	0	0
$889$	3.70820	0.124369
$890$	17.8885	0.599625
$891$	0	0
$892$	8.23607	0.275764
$893$	1.34752	0.0450932
$894$	0	0
$895$	24.9443	0.833795
$896$	0.236068	0.00788648
$897$	0	0
$898$	29.5967	0.987656
$899$	−41.3050	−1.37760
$900$	0	0
$901$	16.9443	0.564496
$902$	−0.180340	−0.00600466
$903$	0	0
$904$	17.4721	0.581115
$905$	−64.7214	−2.15141
$906$	0	0
$907$	14.9443	0.496216	0.248108	−	0.968732i	$-0.420191\pi$
$907$	14.9443	0.496216	0.248108	+	0.968732i	$0.420191\pi$
$908$	1.70820	0.0566887
$909$	0	0
$910$	2.65248	0.0879287
$911$	−11.2361	−0.372268	−0.186134	−	0.982524i	$-0.559596\pi$
$911$	−11.2361	−0.372268	−0.186134	+	0.982524i	$0.559596\pi$
$912$	0	0
$913$	0.777088	0.0257178
$914$	22.3607	0.739626
$915$	0	0
$916$	−8.41641	−0.278086
$917$	−0.278640	−0.00920152
$918$	0	0
$919$	−12.9443	−0.426992	−0.213496	−	0.976944i	$-0.568485\pi$
$919$	−12.9443	−0.426992	−0.213496	+	0.976944i	$0.568485\pi$
$920$	7.23607	0.238566
$921$	0	0
$922$	−42.0689	−1.38546
$923$	−18.1803	−0.598413
$924$	0	0
$925$	24.4721	0.804639
$926$	−32.5967	−1.07120
$927$	0	0
$928$	7.23607	0.237536
$929$	4.65248	0.152643	0.0763214	−	0.997083i	$-0.475682\pi$
$929$	4.65248	0.152643	0.0763214	+	0.997083i	$0.475682\pi$
$930$	0	0
$931$	−39.6393	−1.29913
$932$	4.41641	0.144664
$933$	0	0
$934$	21.7082	0.710314
$935$	1.52786	0.0499665
$936$	0	0
$937$	42.0689	1.37433	0.687165	−	0.726501i	$-0.258855\pi$
$937$	42.0689	1.37433	0.687165	+	0.726501i	$0.258855\pi$
$938$	−2.05573	−0.0671219
$939$	0	0
$940$	0.763932	0.0249167
$941$	−9.58359	−0.312416	−0.156208	−	0.987724i	$-0.549927\pi$
$941$	−9.58359	−0.312416	−0.156208	+	0.987724i	$0.549927\pi$
$942$	0	0
$943$	1.70820	0.0556268
$944$	8.94427	0.291111
$945$	0	0
$946$	−1.36068	−0.0442395
$947$	24.7639	0.804720	0.402360	−	0.915482i	$-0.368190\pi$
$947$	24.7639	0.804720	0.402360	+	0.915482i	$0.368190\pi$
$948$	0	0
$949$	−57.1935	−1.85658
$950$	31.2361	1.01343
$951$	0	0
$952$	−0.472136	−0.0153020
$953$	−37.9443	−1.22914	−0.614568	−	0.788864i	$-0.710670\pi$
$953$	−37.9443	−1.22914	−0.614568	+	0.788864i	$0.710670\pi$
$954$	0	0
$955$	30.8328	0.997726
$956$	−1.41641	−0.0458099
$957$	0	0
$958$	−4.94427	−0.159742
$959$	0.652476	0.0210695
$960$	0	0
$961$	1.58359	0.0510836
$962$	15.5279	0.500638
$963$	0	0
$964$	12.9443	0.416907
$965$	−36.3607	−1.17049
$966$	0	0
$967$	−1.30495	−0.0419644	−0.0209822	−	0.999780i	$-0.506679\pi$
$967$	−1.30495	−0.0419644	−0.0209822	+	0.999780i	$0.506679\pi$
$968$	−10.9443	−0.351762
$969$	0	0
$970$	−32.3607	−1.03904
$971$	−1.06888	−0.0343021	−0.0171511	−	0.999853i	$-0.505460\pi$
$971$	−1.06888	−0.0343021	−0.0171511	+	0.999853i	$0.505460\pi$
$972$	0	0
$973$	2.87539	0.0921807
$974$	−13.5967	−0.435668
$975$	0	0
$976$	−1.47214	−0.0471219
$977$	−4.11146	−0.131537	−0.0657686	−	0.997835i	$-0.520950\pi$
$977$	−4.11146	−0.131537	−0.0657686	+	0.997835i	$0.520950\pi$
$978$	0	0
$979$	−1.30495	−0.0417064
$980$	−22.4721	−0.717846
$981$	0	0
$982$	−39.1246	−1.24852
$983$	34.7639	1.10880	0.554399	−	0.832251i	$-0.312948\pi$
$983$	34.7639	1.10880	0.554399	+	0.832251i	$0.312948\pi$
$984$	0	0
$985$	−20.7639	−0.661594
$986$	−14.4721	−0.460887
$987$	0	0
$988$	19.8197	0.630547
$989$	12.8885	0.409832
$990$	0	0
$991$	56.9574	1.80931	0.904656	−	0.426142i	$-0.140128\pi$
$991$	56.9574	1.80931	0.904656	+	0.426142i	$0.140128\pi$
$992$	−5.70820	−0.181236
$993$	0	0
$994$	−1.23607	−0.0392057
$995$	−20.9443	−0.663978
$996$	0	0
$997$	56.0689	1.77572	0.887860	−	0.460114i	$-0.152192\pi$
$997$	56.0689	1.77572	0.887860	+	0.460114i	$0.152192\pi$
$998$	−0.111456	−0.00352808
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.23607 q^{5} +0.236068 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.23607 q^{5} +0.236068 q^{7} +1.00000 q^{8} +3.23607 q^{10} -0.236068 q^{11} +3.47214 q^{13} +0.236068 q^{14} +1.00000 q^{16} -2.00000 q^{17} +5.70820 q^{19} +3.23607 q^{20} -0.236068 q^{22} +2.23607 q^{23} +5.47214 q^{25} +3.47214 q^{26} +0.236068 q^{28} +7.23607 q^{29} -5.70820 q^{31} +1.00000 q^{32} -2.00000 q^{34} +0.763932 q^{35} +4.47214 q^{37} +5.70820 q^{38} +3.23607 q^{40} +0.763932 q^{41} +5.76393 q^{43} -0.236068 q^{44} +2.23607 q^{46} +0.236068 q^{47} -6.94427 q^{49} +5.47214 q^{50} +3.47214 q^{52} -8.47214 q^{53} -0.763932 q^{55} +0.236068 q^{56} +7.23607 q^{58} +8.94427 q^{59} -1.47214 q^{61} -5.70820 q^{62} +1.00000 q^{64} +11.2361 q^{65} -8.70820 q^{67} -2.00000 q^{68} +0.763932 q^{70} -5.23607 q^{71} -16.4721 q^{73} +4.47214 q^{74} +5.70820 q^{76} -0.0557281 q^{77} -4.94427 q^{79} +3.23607 q^{80} +0.763932 q^{82} -3.29180 q^{83} -6.47214 q^{85} +5.76393 q^{86} -0.236068 q^{88} +5.52786 q^{89} +0.819660 q^{91} +2.23607 q^{92} +0.236068 q^{94} +18.4721 q^{95} -10.0000 q^{97} -6.94427 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 4 q^{7} + 2 q^{8}+O(q^{10}) \) 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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