LMFDB - Embedded newform 945.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [945,2,Mod(1,945)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(945, 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("945.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 945 = 3^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	945.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:	$7.54586299101$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	945.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} -1.00000 q^{7} -4.41421 q^{8} +O(q^{10})$	\(q-2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} -1.00000 q^{7} -4.41421 q^{8} +2.41421 q^{10} -0.414214 q^{11} +2.41421 q^{13} +2.41421 q^{14} +3.00000 q^{16} -4.82843 q^{17} +1.58579 q^{19} -3.82843 q^{20} +1.00000 q^{22} +0.828427 q^{23} +1.00000 q^{25} -5.82843 q^{26} -3.82843 q^{28} +4.00000 q^{29} +6.00000 q^{31} +1.58579 q^{32} +11.6569 q^{34} +1.00000 q^{35} -8.48528 q^{37} -3.82843 q^{38} +4.41421 q^{40} +7.82843 q^{41} -6.65685 q^{43} -1.58579 q^{44} -2.00000 q^{46} -7.48528 q^{47} +1.00000 q^{49} -2.41421 q^{50} +9.24264 q^{52} +10.8995 q^{53} +0.414214 q^{55} +4.41421 q^{56} -9.65685 q^{58} +6.48528 q^{59} -8.48528 q^{61} -14.4853 q^{62} -9.82843 q^{64} -2.41421 q^{65} +7.82843 q^{67} -18.4853 q^{68} -2.41421 q^{70} -2.00000 q^{71} -2.07107 q^{73} +20.4853 q^{74} +6.07107 q^{76} +0.414214 q^{77} +14.8284 q^{79} -3.00000 q^{80} -18.8995 q^{82} +7.48528 q^{83} +4.82843 q^{85} +16.0711 q^{86} +1.82843 q^{88} +8.65685 q^{89} -2.41421 q^{91} +3.17157 q^{92} +18.0711 q^{94} -1.58579 q^{95} +14.1421 q^{97} -2.41421 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{8} + 2 q^{10} + 2 q^{11} + 2 q^{13} + 2 q^{14} + 6 q^{16} - 4 q^{17} + 6 q^{19} - 2 q^{20} + 2 q^{22} - 4 q^{23} + 2 q^{25} - 6 q^{26} - 2 q^{28} + 8 q^{29} + 12 q^{31} + 6 q^{32} + 12 q^{34} + 2 q^{35} - 2 q^{38} + 6 q^{40} + 10 q^{41} - 2 q^{43} - 6 q^{44} - 4 q^{46} + 2 q^{47} + 2 q^{49} - 2 q^{50} + 10 q^{52} + 2 q^{53} - 2 q^{55} + 6 q^{56} - 8 q^{58} - 4 q^{59} - 12 q^{62} - 14 q^{64} - 2 q^{65} + 10 q^{67} - 20 q^{68} - 2 q^{70} - 4 q^{71} + 10 q^{73} + 24 q^{74} - 2 q^{76} - 2 q^{77} + 24 q^{79} - 6 q^{80} - 18 q^{82} - 2 q^{83} + 4 q^{85} + 18 q^{86} - 2 q^{88} + 6 q^{89} - 2 q^{91} + 12 q^{92} + 22 q^{94} - 6 q^{95} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^8 + 2 * q^10 + 2 * q^11 + 2 * q^13 + 2 * q^14 + 6 * q^16 - 4 * q^17 + 6 * q^19 - 2 * q^20 + 2 * q^22 - 4 * q^23 + 2 * q^25 - 6 * q^26 - 2 * q^28 + 8 * q^29 + 12 * q^31 + 6 * q^32 + 12 * q^34 + 2 * q^35 - 2 * q^38 + 6 * q^40 + 10 * q^41 - 2 * q^43 - 6 * q^44 - 4 * q^46 + 2 * q^47 + 2 * q^49 - 2 * q^50 + 10 * q^52 + 2 * q^53 - 2 * q^55 + 6 * q^56 - 8 * q^58 - 4 * q^59 - 12 * q^62 - 14 * q^64 - 2 * q^65 + 10 * q^67 - 20 * q^68 - 2 * q^70 - 4 * q^71 + 10 * q^73 + 24 * q^74 - 2 * q^76 - 2 * q^77 + 24 * q^79 - 6 * q^80 - 18 * q^82 - 2 * q^83 + 4 * q^85 + 18 * q^86 - 2 * q^88 + 6 * q^89 - 2 * q^91 + 12 * q^92 + 22 * q^94 - 6 * q^95 - 2 * 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 $

$2$	−2.41421	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$2$	−2.41421	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−4.41421	−1.56066
$9$	0	0
$10$	2.41421	0.763441
$11$	−0.414214	−0.124890	−0.0624450	−	0.998048i	$-0.519890\pi$
$11$	−0.414214	−0.124890	−0.0624450	+	0.998048i	$0.519890\pi$
$12$	0	0
$13$	2.41421	0.669582	0.334791	−	0.942292i	$-0.391334\pi$
$13$	2.41421	0.669582	0.334791	+	0.942292i	$0.391334\pi$
$14$	2.41421	0.645226
$15$	0	0
$16$	3.00000	0.750000
$17$	−4.82843	−1.17107	−0.585533	−	0.810649i	$-0.699115\pi$
$17$	−4.82843	−1.17107	−0.585533	+	0.810649i	$0.699115\pi$
$18$	0	0
$19$	1.58579	0.363804	0.181902	−	0.983317i	$-0.441775\pi$
$19$	1.58579	0.363804	0.181902	+	0.983317i	$0.441775\pi$
$20$	−3.82843	−0.856062
$21$	0	0
$22$	1.00000	0.213201
$23$	0.828427	0.172739	0.0863695	−	0.996263i	$-0.472473\pi$
$23$	0.828427	0.172739	0.0863695	+	0.996263i	$0.472473\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−5.82843	−1.14305
$27$	0	0
$28$	−3.82843	−0.723505
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	1.58579	0.280330
$33$	0	0
$34$	11.6569	1.99913
$35$	1.00000	0.169031
$36$	0	0
$37$	−8.48528	−1.39497	−0.697486	−	0.716599i	$-0.745698\pi$
$37$	−8.48528	−1.39497	−0.697486	+	0.716599i	$0.745698\pi$
$38$	−3.82843	−0.621053
$39$	0	0
$40$	4.41421	0.697948
$41$	7.82843	1.22259	0.611297	−	0.791401i	$-0.290648\pi$
$41$	7.82843	1.22259	0.611297	+	0.791401i	$0.290648\pi$
$42$	0	0
$43$	−6.65685	−1.01516	−0.507580	−	0.861604i	$-0.669460\pi$
$43$	−6.65685	−1.01516	−0.507580	+	0.861604i	$0.669460\pi$
$44$	−1.58579	−0.239066
$45$	0	0
$46$	−2.00000	−0.294884
$47$	−7.48528	−1.09184	−0.545920	−	0.837837i	$-0.683820\pi$
$47$	−7.48528	−1.09184	−0.545920	+	0.837837i	$0.683820\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.41421	−0.341421
$51$	0	0
$52$	9.24264	1.28172
$53$	10.8995	1.49716	0.748580	−	0.663044i	$-0.230736\pi$
$53$	10.8995	1.49716	0.748580	+	0.663044i	$0.230736\pi$
$54$	0	0
$55$	0.414214	0.0558525
$56$	4.41421	0.589874
$57$	0	0
$58$	−9.65685	−1.26801
$59$	6.48528	0.844312	0.422156	−	0.906523i	$-0.361273\pi$
$59$	6.48528	0.844312	0.422156	+	0.906523i	$0.361273\pi$
$60$	0	0
$61$	−8.48528	−1.08643	−0.543214	−	0.839594i	$-0.682793\pi$
$61$	−8.48528	−1.08643	−0.543214	+	0.839594i	$0.682793\pi$
$62$	−14.4853	−1.83963
$63$	0	0
$64$	−9.82843	−1.22855
$65$	−2.41421	−0.299446
$66$	0	0
$67$	7.82843	0.956395	0.478197	−	0.878252i	$-0.341290\pi$
$67$	7.82843	0.956395	0.478197	+	0.878252i	$0.341290\pi$
$68$	−18.4853	−2.24167
$69$	0	0
$70$	−2.41421	−0.288554
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	−2.07107	−0.242400	−0.121200	−	0.992628i	$-0.538674\pi$
$73$	−2.07107	−0.242400	−0.121200	+	0.992628i	$0.538674\pi$
$74$	20.4853	2.38137
$75$	0	0
$76$	6.07107	0.696399
$77$	0.414214	0.0472040
$78$	0	0
$79$	14.8284	1.66833	0.834164	−	0.551516i	$-0.185951\pi$
$79$	14.8284	1.66833	0.834164	+	0.551516i	$0.185951\pi$
$80$	−3.00000	−0.335410
$81$	0	0
$82$	−18.8995	−2.08710
$83$	7.48528	0.821616	0.410808	−	0.911722i	$-0.365247\pi$
$83$	7.48528	0.821616	0.410808	+	0.911722i	$0.365247\pi$
$84$	0	0
$85$	4.82843	0.523716
$86$	16.0711	1.73299
$87$	0	0
$88$	1.82843	0.194911
$89$	8.65685	0.917625	0.458812	−	0.888533i	$-0.348275\pi$
$89$	8.65685	0.917625	0.458812	+	0.888533i	$0.348275\pi$
$90$	0	0
$91$	−2.41421	−0.253078
$92$	3.17157	0.330659
$93$	0	0
$94$	18.0711	1.86389
$95$	−1.58579	−0.162698
$96$	0	0
$97$	14.1421	1.43592	0.717958	−	0.696086i	$-0.245077\pi$
$97$	14.1421	1.43592	0.717958	+	0.696086i	$0.245077\pi$
$98$	−2.41421	−0.243872
$99$	0	0
$100$	3.82843	0.382843
$101$	19.8284	1.97300	0.986501	−	0.163755i	$-0.0523605\pi$
$101$	19.8284	1.97300	0.986501	+	0.163755i	$0.0523605\pi$
$102$	0	0
$103$	10.8284	1.06696	0.533478	−	0.845814i	$-0.320885\pi$
$103$	10.8284	1.06696	0.533478	+	0.845814i	$0.320885\pi$
$104$	−10.6569	−1.04499
$105$	0	0
$106$	−26.3137	−2.55581
$107$	10.4853	1.01365	0.506825	−	0.862049i	$-0.330819\pi$
$107$	10.4853	1.01365	0.506825	+	0.862049i	$0.330819\pi$
$108$	0	0
$109$	3.48528	0.333829	0.166915	−	0.985971i	$-0.446620\pi$
$109$	3.48528	0.333829	0.166915	+	0.985971i	$0.446620\pi$
$110$	−1.00000	−0.0953463
$111$	0	0
$112$	−3.00000	−0.283473
$113$	−2.75736	−0.259391	−0.129695	−	0.991554i	$-0.541400\pi$
$113$	−2.75736	−0.259391	−0.129695	+	0.991554i	$0.541400\pi$
$114$	0	0
$115$	−0.828427	−0.0772512
$116$	15.3137	1.42184
$117$	0	0
$118$	−15.6569	−1.44133
$119$	4.82843	0.442621
$120$	0	0
$121$	−10.8284	−0.984402
$122$	20.4853	1.85465
$123$	0	0
$124$	22.9706	2.06282
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	16.6569	1.47806	0.739028	−	0.673674i	$-0.235285\pi$
$127$	16.6569	1.47806	0.739028	+	0.673674i	$0.235285\pi$
$128$	20.5563	1.81694
$129$	0	0
$130$	5.82843	0.511187
$131$	−12.1421	−1.06086	−0.530432	−	0.847728i	$-0.677970\pi$
$131$	−12.1421	−1.06086	−0.530432	+	0.847728i	$0.677970\pi$
$132$	0	0
$133$	−1.58579	−0.137505
$134$	−18.8995	−1.63267
$135$	0	0
$136$	21.3137	1.82764
$137$	8.41421	0.718875	0.359437	−	0.933169i	$-0.382969\pi$
$137$	8.41421	0.718875	0.359437	+	0.933169i	$0.382969\pi$
$138$	0	0
$139$	21.3137	1.80781	0.903903	−	0.427738i	$-0.140690\pi$
$139$	21.3137	1.80781	0.903903	+	0.427738i	$0.140690\pi$
$140$	3.82843	0.323561
$141$	0	0
$142$	4.82843	0.405193
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−4.00000	−0.332182
$146$	5.00000	0.413803
$147$	0	0
$148$	−32.4853	−2.67027
$149$	−17.7990	−1.45815	−0.729075	−	0.684434i	$-0.760049\pi$
$149$	−17.7990	−1.45815	−0.729075	+	0.684434i	$0.760049\pi$
$150$	0	0
$151$	−23.3137	−1.89724	−0.948621	−	0.316414i	$-0.897521\pi$
$151$	−23.3137	−1.89724	−0.948621	+	0.316414i	$0.897521\pi$
$152$	−7.00000	−0.567775
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	−6.00000	−0.481932
$156$	0	0
$157$	14.8284	1.18344	0.591719	−	0.806145i	$-0.298450\pi$
$157$	14.8284	1.18344	0.591719	+	0.806145i	$0.298450\pi$
$158$	−35.7990	−2.84801
$159$	0	0
$160$	−1.58579	−0.125367
$161$	−0.828427	−0.0652892
$162$	0	0
$163$	6.34315	0.496834	0.248417	−	0.968653i	$-0.420090\pi$
$163$	6.34315	0.496834	0.248417	+	0.968653i	$0.420090\pi$
$164$	29.9706	2.34031
$165$	0	0
$166$	−18.0711	−1.40259
$167$	11.3137	0.875481	0.437741	−	0.899101i	$-0.355779\pi$
$167$	11.3137	0.875481	0.437741	+	0.899101i	$0.355779\pi$
$168$	0	0
$169$	−7.17157	−0.551659
$170$	−11.6569	−0.894040
$171$	0	0
$172$	−25.4853	−1.94323
$173$	−15.3137	−1.16428	−0.582140	−	0.813089i	$-0.697784\pi$
$173$	−15.3137	−1.16428	−0.582140	+	0.813089i	$0.697784\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−1.24264	−0.0936676
$177$	0	0
$178$	−20.8995	−1.56648
$179$	13.7279	1.02607	0.513037	−	0.858367i	$-0.328521\pi$
$179$	13.7279	1.02607	0.513037	+	0.858367i	$0.328521\pi$
$180$	0	0
$181$	5.65685	0.420471	0.210235	−	0.977651i	$-0.432577\pi$
$181$	5.65685	0.420471	0.210235	+	0.977651i	$0.432577\pi$
$182$	5.82843	0.432032
$183$	0	0
$184$	−3.65685	−0.269587
$185$	8.48528	0.623850
$186$	0	0
$187$	2.00000	0.146254
$188$	−28.6569	−2.09002
$189$	0	0
$190$	3.82843	0.277743
$191$	20.0711	1.45229	0.726146	−	0.687541i	$-0.241310\pi$
$191$	20.0711	1.45229	0.726146	+	0.687541i	$0.241310\pi$
$192$	0	0
$193$	−5.51472	−0.396958	−0.198479	−	0.980105i	$-0.563600\pi$
$193$	−5.51472	−0.396958	−0.198479	+	0.980105i	$0.563600\pi$
$194$	−34.1421	−2.45126
$195$	0	0
$196$	3.82843	0.273459
$197$	−12.8995	−0.919051	−0.459525	−	0.888165i	$-0.651981\pi$
$197$	−12.8995	−0.919051	−0.459525	+	0.888165i	$0.651981\pi$
$198$	0	0
$199$	−2.41421	−0.171139	−0.0855695	−	0.996332i	$-0.527271\pi$
$199$	−2.41421	−0.171139	−0.0855695	+	0.996332i	$0.527271\pi$
$200$	−4.41421	−0.312132
$201$	0	0
$202$	−47.8701	−3.36813
$203$	−4.00000	−0.280745
$204$	0	0
$205$	−7.82843	−0.546761
$206$	−26.1421	−1.82141
$207$	0	0
$208$	7.24264	0.502187
$209$	−0.656854	−0.0454356
$210$	0	0
$211$	2.48528	0.171094	0.0855469	−	0.996334i	$-0.472736\pi$
$211$	2.48528	0.171094	0.0855469	+	0.996334i	$0.472736\pi$
$212$	41.7279	2.86589
$213$	0	0
$214$	−25.3137	−1.73041
$215$	6.65685	0.453994
$216$	0	0
$217$	−6.00000	−0.407307
$218$	−8.41421	−0.569882
$219$	0	0
$220$	1.58579	0.106914
$221$	−11.6569	−0.784125
$222$	0	0
$223$	−5.65685	−0.378811	−0.189405	−	0.981899i	$-0.560656\pi$
$223$	−5.65685	−0.378811	−0.189405	+	0.981899i	$0.560656\pi$
$224$	−1.58579	−0.105955
$225$	0	0
$226$	6.65685	0.442807
$227$	−9.48528	−0.629560	−0.314780	−	0.949165i	$-0.601931\pi$
$227$	−9.48528	−0.629560	−0.314780	+	0.949165i	$0.601931\pi$
$228$	0	0
$229$	6.34315	0.419167	0.209583	−	0.977791i	$-0.432789\pi$
$229$	6.34315	0.419167	0.209583	+	0.977791i	$0.432789\pi$
$230$	2.00000	0.131876
$231$	0	0
$232$	−17.6569	−1.15923
$233$	14.5563	0.953618	0.476809	−	0.879007i	$-0.341793\pi$
$233$	14.5563	0.953618	0.476809	+	0.879007i	$0.341793\pi$
$234$	0	0
$235$	7.48528	0.488286
$236$	24.8284	1.61619
$237$	0	0
$238$	−11.6569	−0.755602
$239$	−24.6274	−1.59302	−0.796508	−	0.604629i	$-0.793322\pi$
$239$	−24.6274	−1.59302	−0.796508	+	0.604629i	$0.793322\pi$
$240$	0	0
$241$	−15.7990	−1.01770	−0.508851	−	0.860854i	$-0.669930\pi$
$241$	−15.7990	−1.01770	−0.508851	+	0.860854i	$0.669930\pi$
$242$	26.1421	1.68048
$243$	0	0
$244$	−32.4853	−2.07966
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	3.82843	0.243597
$248$	−26.4853	−1.68182
$249$	0	0
$250$	2.41421	0.152688
$251$	1.85786	0.117267	0.0586337	−	0.998280i	$-0.481326\pi$
$251$	1.85786	0.117267	0.0586337	+	0.998280i	$0.481326\pi$
$252$	0	0
$253$	−0.343146	−0.0215734
$254$	−40.2132	−2.52320
$255$	0	0
$256$	−29.9706	−1.87316
$257$	−6.34315	−0.395675	−0.197837	−	0.980235i	$-0.563392\pi$
$257$	−6.34315	−0.395675	−0.197837	+	0.980235i	$0.563392\pi$
$258$	0	0
$259$	8.48528	0.527250
$260$	−9.24264	−0.573204
$261$	0	0
$262$	29.3137	1.81101
$263$	−3.65685	−0.225491	−0.112746	−	0.993624i	$-0.535965\pi$
$263$	−3.65685	−0.225491	−0.112746	+	0.993624i	$0.535965\pi$
$264$	0	0
$265$	−10.8995	−0.669551
$266$	3.82843	0.234736
$267$	0	0
$268$	29.9706	1.83074
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	12.7574	0.774954	0.387477	−	0.921879i	$-0.373347\pi$
$271$	12.7574	0.774954	0.387477	+	0.921879i	$0.373347\pi$
$272$	−14.4853	−0.878299
$273$	0	0
$274$	−20.3137	−1.22720
$275$	−0.414214	−0.0249780
$276$	0	0
$277$	−4.34315	−0.260954	−0.130477	−	0.991451i	$-0.541651\pi$
$277$	−4.34315	−0.260954	−0.130477	+	0.991451i	$0.541651\pi$
$278$	−51.4558	−3.08612
$279$	0	0
$280$	−4.41421	−0.263800
$281$	−27.6569	−1.64987	−0.824935	−	0.565228i	$-0.808788\pi$
$281$	−27.6569	−1.64987	−0.824935	+	0.565228i	$0.808788\pi$
$282$	0	0
$283$	−12.4853	−0.742173	−0.371086	−	0.928598i	$-0.621015\pi$
$283$	−12.4853	−0.742173	−0.371086	+	0.928598i	$0.621015\pi$
$284$	−7.65685	−0.454351
$285$	0	0
$286$	2.41421	0.142755
$287$	−7.82843	−0.462097
$288$	0	0
$289$	6.31371	0.371395
$290$	9.65685	0.567070
$291$	0	0
$292$	−7.92893	−0.464006
$293$	18.4853	1.07992	0.539961	−	0.841690i	$-0.318439\pi$
$293$	18.4853	1.07992	0.539961	+	0.841690i	$0.318439\pi$
$294$	0	0
$295$	−6.48528	−0.377588
$296$	37.4558	2.17708
$297$	0	0
$298$	42.9706	2.48922
$299$	2.00000	0.115663
$300$	0	0
$301$	6.65685	0.383695
$302$	56.2843	3.23880
$303$	0	0
$304$	4.75736	0.272853
$305$	8.48528	0.485866
$306$	0	0
$307$	−14.4853	−0.826719	−0.413359	−	0.910568i	$-0.635645\pi$
$307$	−14.4853	−0.826719	−0.413359	+	0.910568i	$0.635645\pi$
$308$	1.58579	0.0903586
$309$	0	0
$310$	14.4853	0.822709
$311$	−14.3431	−0.813325	−0.406663	−	0.913578i	$-0.633308\pi$
$311$	−14.3431	−0.813325	−0.406663	+	0.913578i	$0.633308\pi$
$312$	0	0
$313$	27.3848	1.54788	0.773940	−	0.633259i	$-0.218283\pi$
$313$	27.3848	1.54788	0.773940	+	0.633259i	$0.218283\pi$
$314$	−35.7990	−2.02025
$315$	0	0
$316$	56.7696	3.19354
$317$	−28.4142	−1.59590	−0.797951	−	0.602723i	$-0.794082\pi$
$317$	−28.4142	−1.59590	−0.797951	+	0.602723i	$0.794082\pi$
$318$	0	0
$319$	−1.65685	−0.0927660
$320$	9.82843	0.549426
$321$	0	0
$322$	2.00000	0.111456
$323$	−7.65685	−0.426039
$324$	0	0
$325$	2.41421	0.133916
$326$	−15.3137	−0.848148
$327$	0	0
$328$	−34.5563	−1.90806
$329$	7.48528	0.412677
$330$	0	0
$331$	6.00000	0.329790	0.164895	−	0.986311i	$-0.447272\pi$
$331$	6.00000	0.329790	0.164895	+	0.986311i	$0.447272\pi$
$332$	28.6569	1.57275
$333$	0	0
$334$	−27.3137	−1.49454
$335$	−7.82843	−0.427713
$336$	0	0
$337$	28.6274	1.55943	0.779717	−	0.626132i	$-0.215363\pi$
$337$	28.6274	1.55943	0.779717	+	0.626132i	$0.215363\pi$
$338$	17.3137	0.941742
$339$	0	0
$340$	18.4853	1.00251
$341$	−2.48528	−0.134586
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	29.3848	1.58432
$345$	0	0
$346$	36.9706	1.98755
$347$	−3.17157	−0.170259	−0.0851295	−	0.996370i	$-0.527130\pi$
$347$	−3.17157	−0.170259	−0.0851295	+	0.996370i	$0.527130\pi$
$348$	0	0
$349$	−25.7990	−1.38099	−0.690494	−	0.723338i	$-0.742607\pi$
$349$	−25.7990	−1.38099	−0.690494	+	0.723338i	$0.742607\pi$
$350$	2.41421	0.129045
$351$	0	0
$352$	−0.656854	−0.0350104
$353$	−26.1421	−1.39141	−0.695703	−	0.718330i	$-0.744907\pi$
$353$	−26.1421	−1.39141	−0.695703	+	0.718330i	$0.744907\pi$
$354$	0	0
$355$	2.00000	0.106149
$356$	33.1421	1.75653
$357$	0	0
$358$	−33.1421	−1.75162
$359$	−28.6985	−1.51465	−0.757324	−	0.653039i	$-0.773494\pi$
$359$	−28.6985	−1.51465	−0.757324	+	0.653039i	$0.773494\pi$
$360$	0	0
$361$	−16.4853	−0.867646
$362$	−13.6569	−0.717788
$363$	0	0
$364$	−9.24264	−0.484446
$365$	2.07107	0.108405
$366$	0	0
$367$	26.0000	1.35719	0.678594	−	0.734513i	$-0.262589\pi$
$367$	26.0000	1.35719	0.678594	+	0.734513i	$0.262589\pi$
$368$	2.48528	0.129554
$369$	0	0
$370$	−20.4853	−1.06498
$371$	−10.8995	−0.565874
$372$	0	0
$373$	18.9706	0.982259	0.491129	−	0.871087i	$-0.336584\pi$
$373$	18.9706	0.982259	0.491129	+	0.871087i	$0.336584\pi$
$374$	−4.82843	−0.249672
$375$	0	0
$376$	33.0416	1.70399
$377$	9.65685	0.497353
$378$	0	0
$379$	22.2843	1.14467	0.572333	−	0.820021i	$-0.306038\pi$
$379$	22.2843	1.14467	0.572333	+	0.820021i	$0.306038\pi$
$380$	−6.07107	−0.311439
$381$	0	0
$382$	−48.4558	−2.47922
$383$	−10.3431	−0.528510	−0.264255	−	0.964453i	$-0.585126\pi$
$383$	−10.3431	−0.528510	−0.264255	+	0.964453i	$0.585126\pi$
$384$	0	0
$385$	−0.414214	−0.0211103
$386$	13.3137	0.677650
$387$	0	0
$388$	54.1421	2.74865
$389$	17.3137	0.877840	0.438920	−	0.898526i	$-0.355361\pi$
$389$	17.3137	0.877840	0.438920	+	0.898526i	$0.355361\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	−4.41421	−0.222951
$393$	0	0
$394$	31.1421	1.56892
$395$	−14.8284	−0.746099
$396$	0	0
$397$	14.8284	0.744217	0.372109	−	0.928189i	$-0.378635\pi$
$397$	14.8284	0.744217	0.372109	+	0.928189i	$0.378635\pi$
$398$	5.82843	0.292153
$399$	0	0
$400$	3.00000	0.150000
$401$	−5.79899	−0.289588	−0.144794	−	0.989462i	$-0.546252\pi$
$401$	−5.79899	−0.289588	−0.144794	+	0.989462i	$0.546252\pi$
$402$	0	0
$403$	14.4853	0.721563
$404$	75.9117	3.77675
$405$	0	0
$406$	9.65685	0.479262
$407$	3.51472	0.174218
$408$	0	0
$409$	−3.31371	−0.163852	−0.0819262	−	0.996638i	$-0.526107\pi$
$409$	−3.31371	−0.163852	−0.0819262	+	0.996638i	$0.526107\pi$
$410$	18.8995	0.933380
$411$	0	0
$412$	41.4558	2.04238
$413$	−6.48528	−0.319120
$414$	0	0
$415$	−7.48528	−0.367438
$416$	3.82843	0.187704
$417$	0	0
$418$	1.58579	0.0775634
$419$	3.17157	0.154941	0.0774707	−	0.996995i	$-0.475316\pi$
$419$	3.17157	0.154941	0.0774707	+	0.996995i	$0.475316\pi$
$420$	0	0
$421$	−7.00000	−0.341159	−0.170580	−	0.985344i	$-0.554564\pi$
$421$	−7.00000	−0.341159	−0.170580	+	0.985344i	$0.554564\pi$
$422$	−6.00000	−0.292075
$423$	0	0
$424$	−48.1127	−2.33656
$425$	−4.82843	−0.234213
$426$	0	0
$427$	8.48528	0.410632
$428$	40.1421	1.94034
$429$	0	0
$430$	−16.0711	−0.775016
$431$	36.4142	1.75401	0.877005	−	0.480480i	$-0.159538\pi$
$431$	36.4142	1.75401	0.877005	+	0.480480i	$0.159538\pi$
$432$	0	0
$433$	−29.8701	−1.43546	−0.717732	−	0.696320i	$-0.754820\pi$
$433$	−29.8701	−1.43546	−0.717732	+	0.696320i	$0.754820\pi$
$434$	14.4853	0.695316
$435$	0	0
$436$	13.3431	0.639021
$437$	1.31371	0.0628432
$438$	0	0
$439$	10.7574	0.513421	0.256710	−	0.966488i	$-0.417361\pi$
$439$	10.7574	0.513421	0.256710	+	0.966488i	$0.417361\pi$
$440$	−1.82843	−0.0871668
$441$	0	0
$442$	28.1421	1.33858
$443$	−7.02944	−0.333979	−0.166989	−	0.985959i	$-0.553405\pi$
$443$	−7.02944	−0.333979	−0.166989	+	0.985959i	$0.553405\pi$
$444$	0	0
$445$	−8.65685	−0.410374
$446$	13.6569	0.646671
$447$	0	0
$448$	9.82843	0.464350
$449$	11.1716	0.527219	0.263610	−	0.964629i	$-0.415087\pi$
$449$	11.1716	0.527219	0.263610	+	0.964629i	$0.415087\pi$
$450$	0	0
$451$	−3.24264	−0.152690
$452$	−10.5563	−0.496529
$453$	0	0
$454$	22.8995	1.07473
$455$	2.41421	0.113180
$456$	0	0
$457$	−30.1421	−1.40999	−0.704995	−	0.709212i	$-0.749051\pi$
$457$	−30.1421	−1.40999	−0.704995	+	0.709212i	$0.749051\pi$
$458$	−15.3137	−0.715563
$459$	0	0
$460$	−3.17157	−0.147875
$461$	13.6274	0.634692	0.317346	−	0.948310i	$-0.397208\pi$
$461$	13.6274	0.634692	0.317346	+	0.948310i	$0.397208\pi$
$462$	0	0
$463$	22.6569	1.05295	0.526477	−	0.850190i	$-0.323513\pi$
$463$	22.6569	1.05295	0.526477	+	0.850190i	$0.323513\pi$
$464$	12.0000	0.557086
$465$	0	0
$466$	−35.1421	−1.62793
$467$	−1.65685	−0.0766701	−0.0383350	−	0.999265i	$-0.512205\pi$
$467$	−1.65685	−0.0766701	−0.0383350	+	0.999265i	$0.512205\pi$
$468$	0	0
$469$	−7.82843	−0.361483
$470$	−18.0711	−0.833556
$471$	0	0
$472$	−28.6274	−1.31768
$473$	2.75736	0.126784
$474$	0	0
$475$	1.58579	0.0727609
$476$	18.4853	0.847271
$477$	0	0
$478$	59.4558	2.71945
$479$	26.1421	1.19446	0.597232	−	0.802068i	$-0.296267\pi$
$479$	26.1421	1.19446	0.597232	+	0.802068i	$0.296267\pi$
$480$	0	0
$481$	−20.4853	−0.934048
$482$	38.1421	1.73733
$483$	0	0
$484$	−41.4558	−1.88436
$485$	−14.1421	−0.642161
$486$	0	0
$487$	30.3137	1.37365	0.686823	−	0.726825i	$-0.259005\pi$
$487$	30.3137	1.37365	0.686823	+	0.726825i	$0.259005\pi$
$488$	37.4558	1.69555
$489$	0	0
$490$	2.41421	0.109063
$491$	12.6274	0.569867	0.284934	−	0.958547i	$-0.408028\pi$
$491$	12.6274	0.569867	0.284934	+	0.958547i	$0.408028\pi$
$492$	0	0
$493$	−19.3137	−0.869846
$494$	−9.24264	−0.415846
$495$	0	0
$496$	18.0000	0.808224
$497$	2.00000	0.0897123
$498$	0	0
$499$	0.201010	0.00899845	0.00449922	−	0.999990i	$-0.498568\pi$
$499$	0.201010	0.00899845	0.00449922	+	0.999990i	$0.498568\pi$
$500$	−3.82843	−0.171212
$501$	0	0
$502$	−4.48528	−0.200188
$503$	35.6274	1.58855	0.794274	−	0.607560i	$-0.207851\pi$
$503$	35.6274	1.58855	0.794274	+	0.607560i	$0.207851\pi$
$504$	0	0
$505$	−19.8284	−0.882353
$506$	0.828427	0.0368281
$507$	0	0
$508$	63.7696	2.82932
$509$	−4.34315	−0.192507	−0.0962533	−	0.995357i	$-0.530686\pi$
$509$	−4.34315	−0.192507	−0.0962533	+	0.995357i	$0.530686\pi$
$510$	0	0
$511$	2.07107	0.0916186
$512$	31.2426	1.38074
$513$	0	0
$514$	15.3137	0.675459
$515$	−10.8284	−0.477158
$516$	0	0
$517$	3.10051	0.136360
$518$	−20.4853	−0.900072
$519$	0	0
$520$	10.6569	0.467334
$521$	−27.4853	−1.20415	−0.602076	−	0.798439i	$-0.705660\pi$
$521$	−27.4853	−1.20415	−0.602076	+	0.798439i	$0.705660\pi$
$522$	0	0
$523$	21.1716	0.925768	0.462884	−	0.886419i	$-0.346815\pi$
$523$	21.1716	0.925768	0.462884	+	0.886419i	$0.346815\pi$
$524$	−46.4853	−2.03072
$525$	0	0
$526$	8.82843	0.384938
$527$	−28.9706	−1.26198
$528$	0	0
$529$	−22.3137	−0.970161
$530$	26.3137	1.14299
$531$	0	0
$532$	−6.07107	−0.263214
$533$	18.8995	0.818628
$534$	0	0
$535$	−10.4853	−0.453318
$536$	−34.5563	−1.49261
$537$	0	0
$538$	−33.7990	−1.45718
$539$	−0.414214	−0.0178414
$540$	0	0
$541$	−27.6274	−1.18780	−0.593898	−	0.804541i	$-0.702412\pi$
$541$	−27.6274	−1.18780	−0.593898	+	0.804541i	$0.702412\pi$
$542$	−30.7990	−1.32293
$543$	0	0
$544$	−7.65685	−0.328285
$545$	−3.48528	−0.149293
$546$	0	0
$547$	−40.2843	−1.72243	−0.861216	−	0.508240i	$-0.830296\pi$
$547$	−40.2843	−1.72243	−0.861216	+	0.508240i	$0.830296\pi$
$548$	32.2132	1.37608
$549$	0	0
$550$	1.00000	0.0426401
$551$	6.34315	0.270227
$552$	0	0
$553$	−14.8284	−0.630569
$554$	10.4853	0.445477
$555$	0	0
$556$	81.5980	3.46053
$557$	−43.7990	−1.85582	−0.927911	−	0.372801i	$-0.878397\pi$
$557$	−43.7990	−1.85582	−0.927911	+	0.372801i	$0.878397\pi$
$558$	0	0
$559$	−16.0711	−0.679734
$560$	3.00000	0.126773
$561$	0	0
$562$	66.7696	2.81650
$563$	−27.9706	−1.17882	−0.589409	−	0.807835i	$-0.700639\pi$
$563$	−27.9706	−1.17882	−0.589409	+	0.807835i	$0.700639\pi$
$564$	0	0
$565$	2.75736	0.116003
$566$	30.1421	1.26697
$567$	0	0
$568$	8.82843	0.370433
$569$	−8.68629	−0.364148	−0.182074	−	0.983285i	$-0.558281\pi$
$569$	−8.68629	−0.364148	−0.182074	+	0.983285i	$0.558281\pi$
$570$	0	0
$571$	−43.7990	−1.83293	−0.916465	−	0.400114i	$-0.868970\pi$
$571$	−43.7990	−1.83293	−0.916465	+	0.400114i	$0.868970\pi$
$572$	−3.82843	−0.160075
$573$	0	0
$574$	18.8995	0.788850
$575$	0.828427	0.0345478
$576$	0	0
$577$	5.72792	0.238457	0.119228	−	0.992867i	$-0.461958\pi$
$577$	5.72792	0.238457	0.119228	+	0.992867i	$0.461958\pi$
$578$	−15.2426	−0.634010
$579$	0	0
$580$	−15.3137	−0.635867
$581$	−7.48528	−0.310542
$582$	0	0
$583$	−4.51472	−0.186981
$584$	9.14214	0.378304
$585$	0	0
$586$	−44.6274	−1.84354
$587$	−12.6863	−0.523619	−0.261810	−	0.965120i	$-0.584319\pi$
$587$	−12.6863	−0.523619	−0.261810	+	0.965120i	$0.584319\pi$
$588$	0	0
$589$	9.51472	0.392047
$590$	15.6569	0.644582
$591$	0	0
$592$	−25.4558	−1.04623
$593$	9.85786	0.404814	0.202407	−	0.979301i	$-0.435124\pi$
$593$	9.85786	0.404814	0.202407	+	0.979301i	$0.435124\pi$
$594$	0	0
$595$	−4.82843	−0.197946
$596$	−68.1421	−2.79121
$597$	0	0
$598$	−4.82843	−0.197449
$599$	−5.10051	−0.208401	−0.104200	−	0.994556i	$-0.533228\pi$
$599$	−5.10051	−0.208401	−0.104200	+	0.994556i	$0.533228\pi$
$600$	0	0
$601$	24.6274	1.00457	0.502287	−	0.864701i	$-0.332492\pi$
$601$	24.6274	1.00457	0.502287	+	0.864701i	$0.332492\pi$
$602$	−16.0711	−0.655008
$603$	0	0
$604$	−89.2548	−3.63173
$605$	10.8284	0.440238
$606$	0	0
$607$	40.4853	1.64325	0.821623	−	0.570031i	$-0.193069\pi$
$607$	40.4853	1.64325	0.821623	+	0.570031i	$0.193069\pi$
$608$	2.51472	0.101985
$609$	0	0
$610$	−20.4853	−0.829425
$611$	−18.0711	−0.731077
$612$	0	0
$613$	37.9411	1.53243	0.766214	−	0.642586i	$-0.222138\pi$
$613$	37.9411	1.53243	0.766214	+	0.642586i	$0.222138\pi$
$614$	34.9706	1.41130
$615$	0	0
$616$	−1.82843	−0.0736694
$617$	−24.7696	−0.997185	−0.498592	−	0.866837i	$-0.666150\pi$
$617$	−24.7696	−0.997185	−0.498592	+	0.866837i	$0.666150\pi$
$618$	0	0
$619$	−26.2132	−1.05360	−0.526799	−	0.849990i	$-0.676608\pi$
$619$	−26.2132	−1.05360	−0.526799	+	0.849990i	$0.676608\pi$
$620$	−22.9706	−0.922520
$621$	0	0
$622$	34.6274	1.38843
$623$	−8.65685	−0.346830
$624$	0	0
$625$	1.00000	0.0400000
$626$	−66.1127	−2.64239
$627$	0	0
$628$	56.7696	2.26535
$629$	40.9706	1.63360
$630$	0	0
$631$	1.37258	0.0546417	0.0273208	−	0.999627i	$-0.491302\pi$
$631$	1.37258	0.0546417	0.0273208	+	0.999627i	$0.491302\pi$
$632$	−65.4558	−2.60369
$633$	0	0
$634$	68.5980	2.72437
$635$	−16.6569	−0.661007
$636$	0	0
$637$	2.41421	0.0956546
$638$	4.00000	0.158362
$639$	0	0
$640$	−20.5563	−0.812561
$641$	0.485281	0.0191675	0.00958373	−	0.999954i	$-0.496949\pi$
$641$	0.485281	0.0191675	0.00958373	+	0.999954i	$0.496949\pi$
$642$	0	0
$643$	28.7696	1.13456	0.567280	−	0.823525i	$-0.307996\pi$
$643$	28.7696	1.13456	0.567280	+	0.823525i	$0.307996\pi$
$644$	−3.17157	−0.124977
$645$	0	0
$646$	18.4853	0.727294
$647$	25.3431	0.996342	0.498171	−	0.867079i	$-0.334005\pi$
$647$	25.3431	0.996342	0.498171	+	0.867079i	$0.334005\pi$
$648$	0	0
$649$	−2.68629	−0.105446
$650$	−5.82843	−0.228610
$651$	0	0
$652$	24.2843	0.951045
$653$	8.20101	0.320930	0.160465	−	0.987042i	$-0.448701\pi$
$653$	8.20101	0.320930	0.160465	+	0.987042i	$0.448701\pi$
$654$	0	0
$655$	12.1421	0.474432
$656$	23.4853	0.916946
$657$	0	0
$658$	−18.0711	−0.704484
$659$	12.3431	0.480821	0.240410	−	0.970671i	$-0.422718\pi$
$659$	12.3431	0.480821	0.240410	+	0.970671i	$0.422718\pi$
$660$	0	0
$661$	−8.20101	−0.318982	−0.159491	−	0.987199i	$-0.550985\pi$
$661$	−8.20101	−0.318982	−0.159491	+	0.987199i	$0.550985\pi$
$662$	−14.4853	−0.562986
$663$	0	0
$664$	−33.0416	−1.28226
$665$	1.58579	0.0614942
$666$	0	0
$667$	3.31371	0.128307
$668$	43.3137	1.67586
$669$	0	0
$670$	18.8995	0.730151
$671$	3.51472	0.135684
$672$	0	0
$673$	3.17157	0.122255	0.0611276	−	0.998130i	$-0.480530\pi$
$673$	3.17157	0.122255	0.0611276	+	0.998130i	$0.480530\pi$
$674$	−69.1127	−2.66212
$675$	0	0
$676$	−27.4558	−1.05599
$677$	−13.8579	−0.532601	−0.266300	−	0.963890i	$-0.585801\pi$
$677$	−13.8579	−0.532601	−0.266300	+	0.963890i	$0.585801\pi$
$678$	0	0
$679$	−14.1421	−0.542725
$680$	−21.3137	−0.817343
$681$	0	0
$682$	6.00000	0.229752
$683$	15.5147	0.593654	0.296827	−	0.954931i	$-0.404071\pi$
$683$	15.5147	0.593654	0.296827	+	0.954931i	$0.404071\pi$
$684$	0	0
$685$	−8.41421	−0.321491
$686$	2.41421	0.0921751
$687$	0	0
$688$	−19.9706	−0.761371
$689$	26.3137	1.00247
$690$	0	0
$691$	28.5563	1.08633	0.543167	−	0.839624i	$-0.317225\pi$
$691$	28.5563	1.08633	0.543167	+	0.839624i	$0.317225\pi$
$692$	−58.6274	−2.22868
$693$	0	0
$694$	7.65685	0.290650
$695$	−21.3137	−0.808475
$696$	0	0
$697$	−37.7990	−1.43174
$698$	62.2843	2.35749
$699$	0	0
$700$	−3.82843	−0.144701
$701$	−28.3431	−1.07051	−0.535253	−	0.844692i	$-0.679784\pi$
$701$	−28.3431	−1.07051	−0.535253	+	0.844692i	$0.679784\pi$
$702$	0	0
$703$	−13.4558	−0.507497
$704$	4.07107	0.153434
$705$	0	0
$706$	63.1127	2.37528
$707$	−19.8284	−0.745725
$708$	0	0
$709$	1.62742	0.0611189	0.0305595	−	0.999533i	$-0.490271\pi$
$709$	1.62742	0.0611189	0.0305595	+	0.999533i	$0.490271\pi$
$710$	−4.82843	−0.181208
$711$	0	0
$712$	−38.2132	−1.43210
$713$	4.97056	0.186149
$714$	0	0
$715$	1.00000	0.0373979
$716$	52.5563	1.96412
$717$	0	0
$718$	69.2843	2.58567
$719$	35.7990	1.33508	0.667539	−	0.744575i	$-0.267348\pi$
$719$	35.7990	1.33508	0.667539	+	0.744575i	$0.267348\pi$
$720$	0	0
$721$	−10.8284	−0.403272
$722$	39.7990	1.48117
$723$	0	0
$724$	21.6569	0.804871
$725$	4.00000	0.148556
$726$	0	0
$727$	48.0833	1.78331	0.891655	−	0.452716i	$-0.149545\pi$
$727$	48.0833	1.78331	0.891655	+	0.452716i	$0.149545\pi$
$728$	10.6569	0.394969
$729$	0	0
$730$	−5.00000	−0.185058
$731$	32.1421	1.18882
$732$	0	0
$733$	−41.2426	−1.52333	−0.761666	−	0.647970i	$-0.775618\pi$
$733$	−41.2426	−1.52333	−0.761666	+	0.647970i	$0.775618\pi$
$734$	−62.7696	−2.31687
$735$	0	0
$736$	1.31371	0.0484239
$737$	−3.24264	−0.119444
$738$	0	0
$739$	−35.1716	−1.29381	−0.646904	−	0.762572i	$-0.723936\pi$
$739$	−35.1716	−1.29381	−0.646904	+	0.762572i	$0.723936\pi$
$740$	32.4853	1.19418
$741$	0	0
$742$	26.3137	0.966007
$743$	−11.6569	−0.427649	−0.213824	−	0.976872i	$-0.568592\pi$
$743$	−11.6569	−0.427649	−0.213824	+	0.976872i	$0.568592\pi$
$744$	0	0
$745$	17.7990	0.652105
$746$	−45.7990	−1.67682
$747$	0	0
$748$	7.65685	0.279962
$749$	−10.4853	−0.383124
$750$	0	0
$751$	−20.8284	−0.760040	−0.380020	−	0.924978i	$-0.624083\pi$
$751$	−20.8284	−0.760040	−0.380020	+	0.924978i	$0.624083\pi$
$752$	−22.4558	−0.818880
$753$	0	0
$754$	−23.3137	−0.849035
$755$	23.3137	0.848473
$756$	0	0
$757$	−38.4853	−1.39877	−0.699386	−	0.714744i	$-0.746543\pi$
$757$	−38.4853	−1.39877	−0.699386	+	0.714744i	$0.746543\pi$
$758$	−53.7990	−1.95407
$759$	0	0
$760$	7.00000	0.253917
$761$	24.6274	0.892743	0.446372	−	0.894848i	$-0.352716\pi$
$761$	24.6274	0.892743	0.446372	+	0.894848i	$0.352716\pi$
$762$	0	0
$763$	−3.48528	−0.126176
$764$	76.8406	2.78000
$765$	0	0
$766$	24.9706	0.902223
$767$	15.6569	0.565336
$768$	0	0
$769$	46.7696	1.68655	0.843277	−	0.537480i	$-0.180624\pi$
$769$	46.7696	1.68655	0.843277	+	0.537480i	$0.180624\pi$
$770$	1.00000	0.0360375
$771$	0	0
$772$	−21.1127	−0.759863
$773$	44.1421	1.58768	0.793841	−	0.608125i	$-0.208078\pi$
$773$	44.1421	1.58768	0.793841	+	0.608125i	$0.208078\pi$
$774$	0	0
$775$	6.00000	0.215526
$776$	−62.4264	−2.24098
$777$	0	0
$778$	−41.7990	−1.49857
$779$	12.4142	0.444785
$780$	0	0
$781$	0.828427	0.0296435
$782$	9.65685	0.345328
$783$	0	0
$784$	3.00000	0.107143
$785$	−14.8284	−0.529249
$786$	0	0
$787$	3.79899	0.135419	0.0677097	−	0.997705i	$-0.478431\pi$
$787$	3.79899	0.135419	0.0677097	+	0.997705i	$0.478431\pi$
$788$	−49.3848	−1.75926
$789$	0	0
$790$	35.7990	1.27367
$791$	2.75736	0.0980404
$792$	0	0
$793$	−20.4853	−0.727454
$794$	−35.7990	−1.27046
$795$	0	0
$796$	−9.24264	−0.327597
$797$	−16.0000	−0.566749	−0.283375	−	0.959009i	$-0.591454\pi$
$797$	−16.0000	−0.566749	−0.283375	+	0.959009i	$0.591454\pi$
$798$	0	0
$799$	36.1421	1.27862
$800$	1.58579	0.0560660
$801$	0	0
$802$	14.0000	0.494357
$803$	0.857864	0.0302734
$804$	0	0
$805$	0.828427	0.0291982
$806$	−34.9706	−1.23179
$807$	0	0
$808$	−87.5269	−3.07919
$809$	−52.0833	−1.83115	−0.915575	−	0.402147i	$-0.868264\pi$
$809$	−52.0833	−1.83115	−0.915575	+	0.402147i	$0.868264\pi$
$810$	0	0
$811$	−19.3848	−0.680692	−0.340346	−	0.940300i	$-0.610544\pi$
$811$	−19.3848	−0.680692	−0.340346	+	0.940300i	$0.610544\pi$
$812$	−15.3137	−0.537406
$813$	0	0
$814$	−8.48528	−0.297409
$815$	−6.34315	−0.222191
$816$	0	0
$817$	−10.5563	−0.369320
$818$	8.00000	0.279713
$819$	0	0
$820$	−29.9706	−1.04662
$821$	−9.02944	−0.315130	−0.157565	−	0.987509i	$-0.550364\pi$
$821$	−9.02944	−0.315130	−0.157565	+	0.987509i	$0.550364\pi$
$822$	0	0
$823$	29.2843	1.02079	0.510393	−	0.859941i	$-0.329500\pi$
$823$	29.2843	1.02079	0.510393	+	0.859941i	$0.329500\pi$
$824$	−47.7990	−1.66516
$825$	0	0
$826$	15.6569	0.544772
$827$	−36.4853	−1.26872	−0.634359	−	0.773039i	$-0.718736\pi$
$827$	−36.4853	−1.26872	−0.634359	+	0.773039i	$0.718736\pi$
$828$	0	0
$829$	−21.1716	−0.735319	−0.367660	−	0.929960i	$-0.619841\pi$
$829$	−21.1716	−0.735319	−0.367660	+	0.929960i	$0.619841\pi$
$830$	18.0711	0.627256
$831$	0	0
$832$	−23.7279	−0.822618
$833$	−4.82843	−0.167295
$834$	0	0
$835$	−11.3137	−0.391527
$836$	−2.51472	−0.0869734
$837$	0	0
$838$	−7.65685	−0.264502
$839$	25.7990	0.890680	0.445340	−	0.895362i	$-0.353083\pi$
$839$	25.7990	0.890680	0.445340	+	0.895362i	$0.353083\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	16.8995	0.582395
$843$	0	0
$844$	9.51472	0.327510
$845$	7.17157	0.246710
$846$	0	0
$847$	10.8284	0.372069
$848$	32.6985	1.12287
$849$	0	0
$850$	11.6569	0.399827
$851$	−7.02944	−0.240966
$852$	0	0
$853$	−4.48528	−0.153573	−0.0767866	−	0.997048i	$-0.524466\pi$
$853$	−4.48528	−0.153573	−0.0767866	+	0.997048i	$0.524466\pi$
$854$	−20.4853	−0.700992
$855$	0	0
$856$	−46.2843	−1.58196
$857$	−17.3137	−0.591425	−0.295713	−	0.955277i	$-0.595557\pi$
$857$	−17.3137	−0.591425	−0.295713	+	0.955277i	$0.595557\pi$
$858$	0	0
$859$	−38.3553	−1.30867	−0.654334	−	0.756206i	$-0.727051\pi$
$859$	−38.3553	−1.30867	−0.654334	+	0.756206i	$0.727051\pi$
$860$	25.4853	0.869041
$861$	0	0
$862$	−87.9117	−2.99428
$863$	39.6569	1.34994	0.674968	−	0.737847i	$-0.264158\pi$
$863$	39.6569	1.34994	0.674968	+	0.737847i	$0.264158\pi$
$864$	0	0
$865$	15.3137	0.520682
$866$	72.1127	2.45049
$867$	0	0
$868$	−22.9706	−0.779672
$869$	−6.14214	−0.208358
$870$	0	0
$871$	18.8995	0.640385
$872$	−15.3848	−0.520994
$873$	0	0
$874$	−3.17157	−0.107280
$875$	1.00000	0.0338062
$876$	0	0
$877$	−34.8284	−1.17607	−0.588036	−	0.808835i	$-0.700099\pi$
$877$	−34.8284	−1.17607	−0.588036	+	0.808835i	$0.700099\pi$
$878$	−25.9706	−0.876464
$879$	0	0
$880$	1.24264	0.0418894
$881$	15.6569	0.527493	0.263746	−	0.964592i	$-0.415042\pi$
$881$	15.6569	0.527493	0.263746	+	0.964592i	$0.415042\pi$
$882$	0	0
$883$	2.34315	0.0788531	0.0394266	−	0.999222i	$-0.487447\pi$
$883$	2.34315	0.0788531	0.0394266	+	0.999222i	$0.487447\pi$
$884$	−44.6274	−1.50098
$885$	0	0
$886$	16.9706	0.570137
$887$	17.7696	0.596643	0.298322	−	0.954465i	$-0.403573\pi$
$887$	17.7696	0.596643	0.298322	+	0.954465i	$0.403573\pi$
$888$	0	0
$889$	−16.6569	−0.558653
$890$	20.8995	0.700553
$891$	0	0
$892$	−21.6569	−0.725125
$893$	−11.8701	−0.397216
$894$	0	0
$895$	−13.7279	−0.458874
$896$	−20.5563	−0.686739
$897$	0	0
$898$	−26.9706	−0.900019
$899$	24.0000	0.800445
$900$	0	0
$901$	−52.6274	−1.75327
$902$	7.82843	0.260658
$903$	0	0
$904$	12.1716	0.404820
$905$	−5.65685	−0.188040
$906$	0	0
$907$	−26.5147	−0.880407	−0.440203	−	0.897898i	$-0.645094\pi$
$907$	−26.5147	−0.880407	−0.440203	+	0.897898i	$0.645094\pi$
$908$	−36.3137	−1.20511
$909$	0	0
$910$	−5.82843	−0.193210
$911$	40.8406	1.35311	0.676555	−	0.736392i	$-0.263472\pi$
$911$	40.8406	1.35311	0.676555	+	0.736392i	$0.263472\pi$
$912$	0	0
$913$	−3.10051	−0.102612
$914$	72.7696	2.40700
$915$	0	0
$916$	24.2843	0.802375
$917$	12.1421	0.400969
$918$	0	0
$919$	36.7696	1.21292	0.606458	−	0.795116i	$-0.292590\pi$
$919$	36.7696	1.21292	0.606458	+	0.795116i	$0.292590\pi$
$920$	3.65685	0.120563
$921$	0	0
$922$	−32.8995	−1.08349
$923$	−4.82843	−0.158930
$924$	0	0
$925$	−8.48528	−0.278994
$926$	−54.6985	−1.79750
$927$	0	0
$928$	6.34315	0.208224
$929$	−10.0294	−0.329055	−0.164528	−	0.986372i	$-0.552610\pi$
$929$	−10.0294	−0.329055	−0.164528	+	0.986372i	$0.552610\pi$
$930$	0	0
$931$	1.58579	0.0519721
$932$	55.7279	1.82543
$933$	0	0
$934$	4.00000	0.130884
$935$	−2.00000	−0.0654070
$936$	0	0
$937$	18.8995	0.617420	0.308710	−	0.951156i	$-0.400103\pi$
$937$	18.8995	0.617420	0.308710	+	0.951156i	$0.400103\pi$
$938$	18.8995	0.617090
$939$	0	0
$940$	28.6569	0.934684
$941$	−3.68629	−0.120170	−0.0600848	−	0.998193i	$-0.519137\pi$
$941$	−3.68629	−0.120170	−0.0600848	+	0.998193i	$0.519137\pi$
$942$	0	0
$943$	6.48528	0.211190
$944$	19.4558	0.633234
$945$	0	0
$946$	−6.65685	−0.216433
$947$	9.45584	0.307274	0.153637	−	0.988127i	$-0.450901\pi$
$947$	9.45584	0.307274	0.153637	+	0.988127i	$0.450901\pi$
$948$	0	0
$949$	−5.00000	−0.162307
$950$	−3.82843	−0.124211
$951$	0	0
$952$	−21.3137	−0.690781
$953$	46.1421	1.49469	0.747345	−	0.664436i	$-0.231328\pi$
$953$	46.1421	1.49469	0.747345	+	0.664436i	$0.231328\pi$
$954$	0	0
$955$	−20.0711	−0.649485
$956$	−94.2843	−3.04937
$957$	0	0
$958$	−63.1127	−2.03908
$959$	−8.41421	−0.271709
$960$	0	0
$961$	5.00000	0.161290
$962$	49.4558	1.59452
$963$	0	0
$964$	−60.4853	−1.94810
$965$	5.51472	0.177525
$966$	0	0
$967$	13.8284	0.444692	0.222346	−	0.974968i	$-0.428629\pi$
$967$	13.8284	0.444692	0.222346	+	0.974968i	$0.428629\pi$
$968$	47.7990	1.53632
$969$	0	0
$970$	34.1421	1.09624
$971$	−13.4558	−0.431819	−0.215909	−	0.976413i	$-0.569272\pi$
$971$	−13.4558	−0.431819	−0.215909	+	0.976413i	$0.569272\pi$
$972$	0	0
$973$	−21.3137	−0.683286
$974$	−73.1838	−2.34496
$975$	0	0
$976$	−25.4558	−0.814822
$977$	43.5269	1.39255	0.696275	−	0.717775i	$-0.254840\pi$
$977$	43.5269	1.39255	0.696275	+	0.717775i	$0.254840\pi$
$978$	0	0
$979$	−3.58579	−0.114602
$980$	−3.82843	−0.122295
$981$	0	0
$982$	−30.4853	−0.972824
$983$	3.02944	0.0966240	0.0483120	−	0.998832i	$-0.484616\pi$
$983$	3.02944	0.0966240	0.0483120	+	0.998832i	$0.484616\pi$
$984$	0	0
$985$	12.8995	0.411012
$986$	46.6274	1.48492
$987$	0	0
$988$	14.6569	0.466297
$989$	−5.51472	−0.175358
$990$	0	0
$991$	23.1716	0.736069	0.368035	−	0.929812i	$-0.380031\pi$
$991$	23.1716	0.736069	0.368035	+	0.929812i	$0.380031\pi$
$992$	9.51472	0.302093
$993$	0	0
$994$	−4.82843	−0.153148
$995$	2.41421	0.0765357
$996$	0	0
$997$	−31.5147	−0.998081	−0.499041	−	0.866579i	$-0.666314\pi$
$997$	−31.5147	−0.998081	−0.499041	+	0.866579i	$0.666314\pi$
$998$	−0.485281	−0.0153613
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	945.2.a.b.1.1	✓	2
3.2	odd	2		945.2.a.k.1.2	yes	2
5.4	even	2		4725.2.a.bg.1.2		2
7.6	odd	2		6615.2.a.l.1.1		2
15.14	odd	2		4725.2.a.v.1.1		2
21.20	even	2		6615.2.a.w.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
945.2.a.b.1.1	✓	2	1.1	even	1	trivial
945.2.a.k.1.2	yes	2	3.2	odd	2
4725.2.a.v.1.1		2	15.14	odd	2
4725.2.a.bg.1.2		2	5.4	even	2
6615.2.a.l.1.1		2	7.6	odd	2
6615.2.a.w.1.2		2	21.20	even	2

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:	\( N \)	\(=\)	\( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	945.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} -1.00000 q^{7} -4.41421 q^{8} +O(q^{10})\)	\(q-2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} -1.00000 q^{7} -4.41421 q^{8} +2.41421 q^{10} -0.414214 q^{11} +2.41421 q^{13} +2.41421 q^{14} +3.00000 q^{16} -4.82843 q^{17} +1.58579 q^{19} -3.82843 q^{20} +1.00000 q^{22} +0.828427 q^{23} +1.00000 q^{25} -5.82843 q^{26} -3.82843 q^{28} +4.00000 q^{29} +6.00000 q^{31} +1.58579 q^{32} +11.6569 q^{34} +1.00000 q^{35} -8.48528 q^{37} -3.82843 q^{38} +4.41421 q^{40} +7.82843 q^{41} -6.65685 q^{43} -1.58579 q^{44} -2.00000 q^{46} -7.48528 q^{47} +1.00000 q^{49} -2.41421 q^{50} +9.24264 q^{52} +10.8995 q^{53} +0.414214 q^{55} +4.41421 q^{56} -9.65685 q^{58} +6.48528 q^{59} -8.48528 q^{61} -14.4853 q^{62} -9.82843 q^{64} -2.41421 q^{65} +7.82843 q^{67} -18.4853 q^{68} -2.41421 q^{70} -2.00000 q^{71} -2.07107 q^{73} +20.4853 q^{74} +6.07107 q^{76} +0.414214 q^{77} +14.8284 q^{79} -3.00000 q^{80} -18.8995 q^{82} +7.48528 q^{83} +4.82843 q^{85} +16.0711 q^{86} +1.82843 q^{88} +8.65685 q^{89} -2.41421 q^{91} +3.17157 q^{92} +18.0711 q^{94} -1.58579 q^{95} +14.1421 q^{97} -2.41421 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{8} + 2 q^{10} + 2 q^{11} + 2 q^{13} + 2 q^{14} + 6 q^{16} - 4 q^{17} + 6 q^{19} - 2 q^{20} + 2 q^{22} - 4 q^{23} + 2 q^{25} - 6 q^{26} - 2 q^{28} + 8 q^{29} + 12 q^{31} + 6 q^{32} + 12 q^{34} + 2 q^{35} - 2 q^{38} + 6 q^{40} + 10 q^{41} - 2 q^{43} - 6 q^{44} - 4 q^{46} + 2 q^{47} + 2 q^{49} - 2 q^{50} + 10 q^{52} + 2 q^{53} - 2 q^{55} + 6 q^{56} - 8 q^{58} - 4 q^{59} - 12 q^{62} - 14 q^{64} - 2 q^{65} + 10 q^{67} - 20 q^{68} - 2 q^{70} - 4 q^{71} + 10 q^{73} + 24 q^{74} - 2 q^{76} - 2 q^{77} + 24 q^{79} - 6 q^{80} - 18 q^{82} - 2 q^{83} + 4 q^{85} + 18 q^{86} - 2 q^{88} + 6 q^{89} - 2 q^{91} + 12 q^{92} + 22 q^{94} - 6 q^{95} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^8 + 2 * q^10 + 2 * q^11 + 2 * q^13 + 2 * q^14 + 6 * q^16 - 4 * q^17 + 6 * q^19 - 2 * q^20 + 2 * q^22 - 4 * q^23 + 2 * q^25 - 6 * q^26 - 2 * q^28 + 8 * q^29 + 12 * q^31 + 6 * q^32 + 12 * q^34 + 2 * q^35 - 2 * q^38 + 6 * q^40 + 10 * q^41 - 2 * q^43 - 6 * q^44 - 4 * q^46 + 2 * q^47 + 2 * q^49 - 2 * q^50 + 10 * q^52 + 2 * q^53 - 2 * q^55 + 6 * q^56 - 8 * q^58 - 4 * q^59 - 12 * q^62 - 14 * q^64 - 2 * q^65 + 10 * q^67 - 20 * q^68 - 2 * q^70 - 4 * q^71 + 10 * q^73 + 24 * q^74 - 2 * q^76 - 2 * q^77 + 24 * q^79 - 6 * q^80 - 18 * q^82 - 2 * q^83 + 4 * q^85 + 18 * q^86 - 2 * q^88 + 6 * q^89 - 2 * q^91 + 12 * q^92 + 22 * q^94 - 6 * q^95 - 2 * q^98

Introduction

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