LMFDB - Embedded newform 945.2.a.m.1.3

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:	$4$
Coefficient field:	4.4.144344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 5x + 9 $ x^4 - x^3 - 8x^2 + 5x + 9
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			1.3
Root			$-0.857589$ 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)

$f(q)$	$=$	$q+0.857589 q^{2} -1.26454 q^{4} -1.00000 q^{5} +1.00000 q^{7} -2.79964 q^{8} +O(q^{10})$	\(q+0.857589 q^{2} -1.26454 q^{4} -1.00000 q^{5} +1.00000 q^{7} -2.79964 q^{8} -0.857589 q^{10} -1.40695 q^{11} +5.12213 q^{13} +0.857589 q^{14} +0.128144 q^{16} +5.92177 q^{17} -2.51481 q^{19} +1.26454 q^{20} -1.20659 q^{22} +5.92177 q^{23} +1.00000 q^{25} +4.39268 q^{26} -1.26454 q^{28} -4.52908 q^{29} +9.63694 q^{31} +5.70916 q^{32} +5.07844 q^{34} -1.00000 q^{35} +0.284821 q^{37} -2.15668 q^{38} +2.79964 q^{40} +1.60732 q^{41} +12.6369 q^{43} +1.77915 q^{44} +5.07844 q^{46} -7.82304 q^{47} +1.00000 q^{49} +0.857589 q^{50} -6.47714 q^{52} -7.32872 q^{53} +1.40695 q^{55} -2.79964 q^{56} -3.88409 q^{58} -1.71518 q^{59} -3.92177 q^{61} +8.26454 q^{62} +4.63983 q^{64} -5.12213 q^{65} +15.4508 q^{67} -7.48831 q^{68} -0.857589 q^{70} +11.6278 q^{71} -7.12213 q^{73} +0.244260 q^{74} +3.18008 q^{76} -1.40695 q^{77} -7.35212 q^{79} -0.128144 q^{80} +1.37842 q^{82} -15.2443 q^{83} -5.92177 q^{85} +10.8373 q^{86} +3.93895 q^{88} -3.53822 q^{89} +5.12213 q^{91} -7.48831 q^{92} -6.70896 q^{94} +2.51481 q^{95} +12.1284 q^{97} +0.857589 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 9 q^{4} - 4 q^{5} + 4 q^{7} - 6 q^{8}+O(q^{10}) $ 4 * q - q^2 + 9 * q^4 - 4 * q^5 + 4 * q^7 - 6 * q^8	$ 4 q - q^{2} + 9 q^{4} - 4 q^{5} + 4 q^{7} - 6 q^{8} + q^{10} + 4 q^{11} + 2 q^{13} - q^{14} + 19 q^{16} + 4 q^{19} - 9 q^{20} + 10 q^{22} + 4 q^{25} + 22 q^{26} + 9 q^{28} + 10 q^{29} + 6 q^{31} - 23 q^{32} - 13 q^{34} - 4 q^{35} + 10 q^{37} + q^{38} + 6 q^{40} + 2 q^{41} + 18 q^{43} + 36 q^{44} - 13 q^{46} - 18 q^{47} + 4 q^{49} - q^{50} - 34 q^{52} + 4 q^{53} - 4 q^{55} - 6 q^{56} - 14 q^{58} + 2 q^{59} + 8 q^{61} + 19 q^{62} + 54 q^{64} - 2 q^{65} + 10 q^{67} - 13 q^{68} + q^{70} + 8 q^{71} - 10 q^{73} - 36 q^{74} - 5 q^{76} + 4 q^{77} + 12 q^{79} - 19 q^{80} + 24 q^{82} - 24 q^{83} + 16 q^{86} + 4 q^{88} + 8 q^{89} + 2 q^{91} - 13 q^{92} - 38 q^{94} - 4 q^{95} + 10 q^{97} - q^{98}+O(q^{100}) $ 4 * q - q^2 + 9 * q^4 - 4 * q^5 + 4 * q^7 - 6 * q^8 + q^10 + 4 * q^11 + 2 * q^13 - q^14 + 19 * q^16 + 4 * q^19 - 9 * q^20 + 10 * q^22 + 4 * q^25 + 22 * q^26 + 9 * q^28 + 10 * q^29 + 6 * q^31 - 23 * q^32 - 13 * q^34 - 4 * q^35 + 10 * q^37 + q^38 + 6 * q^40 + 2 * q^41 + 18 * q^43 + 36 * q^44 - 13 * q^46 - 18 * q^47 + 4 * q^49 - q^50 - 34 * q^52 + 4 * q^53 - 4 * q^55 - 6 * q^56 - 14 * q^58 + 2 * q^59 + 8 * q^61 + 19 * q^62 + 54 * q^64 - 2 * q^65 + 10 * q^67 - 13 * q^68 + q^70 + 8 * q^71 - 10 * q^73 - 36 * q^74 - 5 * q^76 + 4 * q^77 + 12 * q^79 - 19 * q^80 + 24 * q^82 - 24 * q^83 + 16 * q^86 + 4 * q^88 + 8 * q^89 + 2 * q^91 - 13 * q^92 - 38 * q^94 - 4 * q^95 + 10 * q^97 - 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 $

$2$	0.857589	0.606407	0.303204	−	0.952926i	$-0.401944\pi$
$2$	0.857589	0.606407	0.303204	+	0.952926i	$0.401944\pi$
$3$	0	0
$3$	0	0
$4$	−1.26454	−0.632270
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.79964	−0.989821
$9$	0	0
$10$	−0.857589	−0.271194
$11$	−1.40695	−0.424212	−0.212106	−	0.977247i	$-0.568032\pi$
$11$	−1.40695	−0.424212	−0.212106	+	0.977247i	$0.568032\pi$
$12$	0	0
$13$	5.12213	1.42062	0.710312	−	0.703887i	$-0.248554\pi$
$13$	5.12213	1.42062	0.710312	+	0.703887i	$0.248554\pi$
$14$	0.857589	0.229200
$15$	0	0
$16$	0.128144	0.0320359
$17$	5.92177	1.43624	0.718119	−	0.695920i	$-0.245003\pi$
$17$	5.92177	1.43624	0.718119	+	0.695920i	$0.245003\pi$
$18$	0	0
$19$	−2.51481	−0.576938	−0.288469	−	0.957489i	$-0.593146\pi$
$19$	−2.51481	−0.576938	−0.288469	+	0.957489i	$0.593146\pi$
$20$	1.26454	0.282760
$21$	0	0
$22$	−1.20659	−0.257245
$23$	5.92177	1.23477	0.617387	−	0.786660i	$-0.288191\pi$
$23$	5.92177	1.23477	0.617387	+	0.786660i	$0.288191\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.39268	0.861476
$27$	0	0
$28$	−1.26454	−0.238976
$29$	−4.52908	−0.841029	−0.420515	−	0.907286i	$-0.638150\pi$
$29$	−4.52908	−0.841029	−0.420515	+	0.907286i	$0.638150\pi$
$30$	0	0
$31$	9.63694	1.73085	0.865423	−	0.501042i	$-0.167050\pi$
$31$	9.63694	1.73085	0.865423	+	0.501042i	$0.167050\pi$
$32$	5.70916	1.00925
$33$	0	0
$34$	5.07844	0.870946
$35$	−1.00000	−0.169031
$36$	0	0
$37$	0.284821	0.0468243	0.0234122	−	0.999726i	$-0.492547\pi$
$37$	0.284821	0.0468243	0.0234122	+	0.999726i	$0.492547\pi$
$38$	−2.15668	−0.349859
$39$	0	0
$40$	2.79964	0.442661
$41$	1.60732	0.251021	0.125510	−	0.992092i	$-0.459943\pi$
$41$	1.60732	0.251021	0.125510	+	0.992092i	$0.459943\pi$
$42$	0	0
$43$	12.6369	1.92712	0.963558	−	0.267500i	$-0.0861975\pi$
$43$	12.6369	1.92712	0.963558	+	0.267500i	$0.0861975\pi$
$44$	1.77915	0.268216
$45$	0	0
$46$	5.07844	0.748776
$47$	−7.82304	−1.14111	−0.570554	−	0.821260i	$-0.693271\pi$
$47$	−7.82304	−1.14111	−0.570554	+	0.821260i	$0.693271\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.857589	0.121281
$51$	0	0
$52$	−6.47714	−0.898218
$53$	−7.32872	−1.00668	−0.503338	−	0.864089i	$-0.667895\pi$
$53$	−7.32872	−1.00668	−0.503338	+	0.864089i	$0.667895\pi$
$54$	0	0
$55$	1.40695	0.189713
$56$	−2.79964	−0.374117
$57$	0	0
$58$	−3.88409	−0.510006
$59$	−1.71518	−0.223297	−0.111649	−	0.993748i	$-0.535613\pi$
$59$	−1.71518	−0.223297	−0.111649	+	0.993748i	$0.535613\pi$
$60$	0	0
$61$	−3.92177	−0.502131	−0.251065	−	0.967970i	$-0.580781\pi$
$61$	−3.92177	−0.502131	−0.251065	+	0.967970i	$0.580781\pi$
$62$	8.26454	1.04960
$63$	0	0
$64$	4.63983	0.579979
$65$	−5.12213	−0.635322
$66$	0	0
$67$	15.4508	1.88762	0.943811	−	0.330487i	$-0.107213\pi$
$67$	15.4508	1.88762	0.943811	+	0.330487i	$0.107213\pi$
$68$	−7.48831	−0.908091
$69$	0	0
$70$	−0.857589	−0.102502
$71$	11.6278	1.37997	0.689983	−	0.723825i	$-0.257618\pi$
$71$	11.6278	1.37997	0.689983	+	0.723825i	$0.257618\pi$
$72$	0	0
$73$	−7.12213	−0.833582	−0.416791	−	0.909002i	$-0.636845\pi$
$73$	−7.12213	−0.833582	−0.416791	+	0.909002i	$0.636845\pi$
$74$	0.244260	0.0283946
$75$	0	0
$76$	3.18008	0.364781
$77$	−1.40695	−0.160337
$78$	0	0
$79$	−7.35212	−0.827178	−0.413589	−	0.910464i	$-0.635725\pi$
$79$	−7.35212	−0.827178	−0.413589	+	0.910464i	$0.635725\pi$
$80$	−0.128144	−0.0143269
$81$	0	0
$82$	1.37842	0.152221
$83$	−15.2443	−1.67327	−0.836637	−	0.547757i	$-0.815482\pi$
$83$	−15.2443	−1.67327	−0.836637	+	0.547757i	$0.815482\pi$
$84$	0	0
$85$	−5.92177	−0.642306
$86$	10.8373	1.16862
$87$	0	0
$88$	3.93895	0.419893
$89$	−3.53822	−0.375051	−0.187525	−	0.982260i	$-0.560047\pi$
$89$	−3.53822	−0.375051	−0.187525	+	0.982260i	$0.560047\pi$
$90$	0	0
$91$	5.12213	0.536945
$92$	−7.48831	−0.780710
$93$	0	0
$94$	−6.70896	−0.691976
$95$	2.51481	0.258014
$96$	0	0
$97$	12.1284	1.23145	0.615724	−	0.787962i	$-0.288864\pi$
$97$	12.1284	1.23145	0.615724	+	0.787962i	$0.288864\pi$
$98$	0.857589	0.0866296
$99$	0	0
$100$	−1.26454	−0.126454
$101$	8.92177	0.887749	0.443874	−	0.896089i	$-0.353604\pi$
$101$	8.92177	0.887749	0.443874	+	0.896089i	$0.353604\pi$
$102$	0	0
$103$	9.93090	0.978521	0.489261	−	0.872138i	$-0.337267\pi$
$103$	9.93090	0.978521	0.489261	+	0.872138i	$0.337267\pi$
$104$	−14.3401	−1.40616
$105$	0	0
$106$	−6.28503	−0.610456
$107$	2.81390	0.272030	0.136015	−	0.990707i	$-0.456570\pi$
$107$	2.81390	0.272030	0.136015	+	0.990707i	$0.456570\pi$
$108$	0	0
$109$	4.38354	0.419867	0.209934	−	0.977716i	$-0.432675\pi$
$109$	4.38354	0.419867	0.209934	+	0.977716i	$0.432675\pi$
$110$	1.20659	0.115043
$111$	0	0
$112$	0.128144	0.0121084
$113$	−6.55249	−0.616406	−0.308203	−	0.951321i	$-0.599728\pi$
$113$	−6.55249	−0.616406	−0.308203	+	0.951321i	$0.599728\pi$
$114$	0	0
$115$	−5.92177	−0.552207
$116$	5.72721	0.531758
$117$	0	0
$118$	−1.47092	−0.135409
$119$	5.92177	0.542847
$120$	0	0
$121$	−9.02049	−0.820044
$122$	−3.36326	−0.304496
$123$	0	0
$124$	−12.1863	−1.09436
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−9.95139	−0.883043	−0.441522	−	0.897251i	$-0.645561\pi$
$127$	−9.95139	−0.883043	−0.441522	+	0.897251i	$0.645561\pi$
$128$	−7.43926	−0.657544
$129$	0	0
$130$	−4.39268	−0.385264
$131$	−16.9891	−1.48434	−0.742171	−	0.670211i	$-0.766204\pi$
$131$	−16.9891	−1.48434	−0.742171	+	0.670211i	$0.766204\pi$
$132$	0	0
$133$	−2.51481	−0.218062
$134$	13.2505	1.14467
$135$	0	0
$136$	−16.5788	−1.42162
$137$	7.32872	0.626134	0.313067	−	0.949731i	$-0.398643\pi$
$137$	7.32872	0.626134	0.313067	+	0.949731i	$0.398643\pi$
$138$	0	0
$139$	−9.84353	−0.834917	−0.417459	−	0.908696i	$-0.637079\pi$
$139$	−9.84353	−0.834917	−0.417459	+	0.908696i	$0.637079\pi$
$140$	1.26454	0.106873
$141$	0	0
$142$	9.97188	0.836822
$143$	−7.20659	−0.602645
$144$	0	0
$145$	4.52908	0.376120
$146$	−6.10786	−0.505490
$147$	0	0
$148$	−0.360168	−0.0296056
$149$	2.97037	0.243342	0.121671	−	0.992570i	$-0.461175\pi$
$149$	2.97037	0.243342	0.121671	+	0.992570i	$0.461175\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	7.04056	0.571065
$153$	0	0
$154$	−1.20659	−0.0972295
$155$	−9.63694	−0.774058
$156$	0	0
$157$	−17.3430	−1.38412	−0.692060	−	0.721840i	$-0.743297\pi$
$157$	−17.3430	−1.38412	−0.692060	+	0.721840i	$0.743297\pi$
$158$	−6.30510	−0.501607
$159$	0	0
$160$	−5.70916	−0.451349
$161$	5.92177	0.466700
$162$	0	0
$163$	−2.31336	−0.181196	−0.0905980	−	0.995888i	$-0.528878\pi$
$163$	−2.31336	−0.181196	−0.0905980	+	0.995888i	$0.528878\pi$
$164$	−2.03252	−0.158713
$165$	0	0
$166$	−13.0733	−1.01469
$167$	23.0673	1.78500	0.892501	−	0.451046i	$-0.148949\pi$
$167$	23.0673	1.78500	0.892501	+	0.451046i	$0.148949\pi$
$168$	0	0
$169$	13.2362	1.01817
$170$	−5.07844	−0.389499
$171$	0	0
$172$	−15.9799	−1.21846
$173$	−12.1660	−0.924966	−0.462483	−	0.886628i	$-0.653041\pi$
$173$	−12.1660	−0.924966	−0.462483	+	0.886628i	$0.653041\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−0.180292	−0.0135900
$177$	0	0
$178$	−3.03434	−0.227433
$179$	9.73859	0.727896	0.363948	−	0.931419i	$-0.381429\pi$
$179$	9.73859	0.727896	0.363948	+	0.931419i	$0.381429\pi$
$180$	0	0
$181$	5.96858	0.443641	0.221820	−	0.975088i	$-0.428800\pi$
$181$	5.96858	0.443641	0.221820	+	0.975088i	$0.428800\pi$
$182$	4.39268	0.325607
$183$	0	0
$184$	−16.5788	−1.22220
$185$	−0.284821	−0.0209405
$186$	0	0
$187$	−8.33163	−0.609269
$188$	9.89255	0.721489
$189$	0	0
$190$	2.15668	0.156462
$191$	1.62268	0.117413	0.0587064	−	0.998275i	$-0.481302\pi$
$191$	1.62268	0.117413	0.0587064	+	0.998275i	$0.481302\pi$
$192$	0	0
$193$	17.6746	1.27225	0.636123	−	0.771587i	$-0.280537\pi$
$193$	17.6746	1.27225	0.636123	+	0.771587i	$0.280537\pi$
$194$	10.4011	0.746759
$195$	0	0
$196$	−1.26454	−0.0903243
$197$	−19.5730	−1.39452	−0.697258	−	0.716820i	$-0.745597\pi$
$197$	−19.5730	−1.39452	−0.697258	+	0.716820i	$0.745597\pi$
$198$	0	0
$199$	−7.36639	−0.522190	−0.261095	−	0.965313i	$-0.584084\pi$
$199$	−7.36639	−0.522190	−0.261095	+	0.965313i	$0.584084\pi$
$200$	−2.79964	−0.197964
$201$	0	0
$202$	7.65121	0.538337
$203$	−4.52908	−0.317879
$204$	0	0
$205$	−1.60732	−0.112260
$206$	8.51664	0.593382
$207$	0	0
$208$	0.656368	0.0455109
$209$	3.53822	0.244744
$210$	0	0
$211$	26.4103	1.81816	0.909079	−	0.416623i	$-0.136786\pi$
$211$	26.4103	1.81816	0.909079	+	0.416623i	$0.136786\pi$
$212$	9.26746	0.636492
$213$	0	0
$214$	2.41317	0.164961
$215$	−12.6369	−0.861832
$216$	0	0
$217$	9.63694	0.654198
$218$	3.75928	0.254611
$219$	0	0
$220$	−1.77915	−0.119950
$221$	30.3320	2.04035
$222$	0	0
$223$	−6.17516	−0.413520	−0.206760	−	0.978392i	$-0.566292\pi$
$223$	−6.17516	−0.413520	−0.206760	+	0.978392i	$0.566292\pi$
$224$	5.70916	0.381460
$225$	0	0
$226$	−5.61934	−0.373793
$227$	−14.4428	−0.958602	−0.479301	−	0.877650i	$-0.659110\pi$
$227$	−14.4428	−0.958602	−0.479301	+	0.877650i	$0.659110\pi$
$228$	0	0
$229$	−19.1079	−1.26268	−0.631342	−	0.775505i	$-0.717495\pi$
$229$	−19.1079	−1.26268	−0.631342	+	0.775505i	$0.717495\pi$
$230$	−5.07844	−0.334863
$231$	0	0
$232$	12.6798	0.832468
$233$	6.15176	0.403015	0.201508	−	0.979487i	$-0.435416\pi$
$233$	6.15176	0.403015	0.201508	+	0.979487i	$0.435416\pi$
$234$	0	0
$235$	7.82304	0.510319
$236$	2.16891	0.141184
$237$	0	0
$238$	5.07844	0.329187
$239$	2.56964	0.166216	0.0831082	−	0.996541i	$-0.473515\pi$
$239$	2.56964	0.166216	0.0831082	+	0.996541i	$0.473515\pi$
$240$	0	0
$241$	−9.70604	−0.625221	−0.312610	−	0.949881i	$-0.601203\pi$
$241$	−9.70604	−0.625221	−0.312610	+	0.949881i	$0.601203\pi$
$242$	−7.73588	−0.497281
$243$	0	0
$244$	4.95923	0.317482
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−12.8812	−0.819611
$248$	−26.9799	−1.71323
$249$	0	0
$250$	−0.857589	−0.0542387
$251$	7.55871	0.477101	0.238551	−	0.971130i	$-0.423328\pi$
$251$	7.55871	0.477101	0.238551	+	0.971130i	$0.423328\pi$
$252$	0	0
$253$	−8.33163	−0.523805
$254$	−8.53421	−0.535484
$255$	0	0
$256$	−15.6595	−0.978718
$257$	14.9514	0.932642	0.466321	−	0.884616i	$-0.345579\pi$
$257$	14.9514	0.932642	0.466321	+	0.884616i	$0.345579\pi$
$258$	0	0
$259$	0.284821	0.0176979
$260$	6.47714	0.401695
$261$	0	0
$262$	−14.5696	−0.900116
$263$	0.215726	0.0133022	0.00665111	−	0.999978i	$-0.497883\pi$
$263$	0.215726	0.0133022	0.00665111	+	0.999978i	$0.497883\pi$
$264$	0	0
$265$	7.32872	0.450199
$266$	−2.15668	−0.132234
$267$	0	0
$268$	−19.5382	−1.19349
$269$	25.6465	1.56369	0.781847	−	0.623470i	$-0.214278\pi$
$269$	25.6465	1.56369	0.781847	+	0.623470i	$0.214278\pi$
$270$	0	0
$271$	−2.69982	−0.164002	−0.0820011	−	0.996632i	$-0.526131\pi$
$271$	−2.69982	−0.164002	−0.0820011	+	0.996632i	$0.526131\pi$
$272$	0.758836	0.0460112
$273$	0	0
$274$	6.28503	0.379692
$275$	−1.40695	−0.0848423
$276$	0	0
$277$	−25.1174	−1.50916	−0.754580	−	0.656208i	$-0.772159\pi$
$277$	−25.1174	−1.50916	−0.754580	+	0.656208i	$0.772159\pi$
$278$	−8.44171	−0.506300
$279$	0	0
$280$	2.79964	0.167310
$281$	23.1741	1.38245	0.691225	−	0.722640i	$-0.257072\pi$
$281$	23.1741	1.38245	0.691225	+	0.722640i	$0.257072\pi$
$282$	0	0
$283$	−22.3035	−1.32581	−0.662903	−	0.748705i	$-0.730676\pi$
$283$	−22.3035	−1.32581	−0.662903	+	0.748705i	$0.730676\pi$
$284$	−14.7038	−0.872512
$285$	0	0
$286$	−6.18029	−0.365448
$287$	1.60732	0.0948769
$288$	0	0
$289$	18.0673	1.06278
$290$	3.88409	0.228082
$291$	0	0
$292$	9.00622	0.527049
$293$	−2.49141	−0.145550	−0.0727748	−	0.997348i	$-0.523185\pi$
$293$	−2.49141	−0.145550	−0.0727748	+	0.997348i	$0.523185\pi$
$294$	0	0
$295$	1.71518	0.0998616
$296$	−0.797396	−0.0463477
$297$	0	0
$298$	2.54736	0.147565
$299$	30.3320	1.75415
$300$	0	0
$301$	12.6369	0.728381
$302$	6.86071	0.394790
$303$	0	0
$304$	−0.322257	−0.0184827
$305$	3.92177	0.224560
$306$	0	0
$307$	12.5005	0.713444	0.356722	−	0.934211i	$-0.383894\pi$
$307$	12.5005	0.713444	0.356722	+	0.934211i	$0.383894\pi$
$308$	1.77915	0.101376
$309$	0	0
$310$	−8.26454	−0.469394
$311$	6.37220	0.361334	0.180667	−	0.983544i	$-0.442174\pi$
$311$	6.37220	0.361334	0.180667	+	0.983544i	$0.442174\pi$
$312$	0	0
$313$	−19.3664	−1.09465	−0.547327	−	0.836919i	$-0.684354\pi$
$313$	−19.3664	−1.09465	−0.547327	+	0.836919i	$0.684354\pi$
$314$	−14.8732	−0.839341
$315$	0	0
$316$	9.29706	0.523000
$317$	−26.1894	−1.47094	−0.735472	−	0.677555i	$-0.763040\pi$
$317$	−26.1894	−1.47094	−0.735472	+	0.677555i	$0.763040\pi$
$318$	0	0
$319$	6.37220	0.356774
$320$	−4.63983	−0.259374
$321$	0	0
$322$	5.07844	0.283011
$323$	−14.8921	−0.828621
$324$	0	0
$325$	5.12213	0.284125
$326$	−1.98391	−0.109879
$327$	0	0
$328$	−4.49990	−0.248465
$329$	−7.82304	−0.431298
$330$	0	0
$331$	−4.21573	−0.231717	−0.115859	−	0.993266i	$-0.536962\pi$
$331$	−4.21573	−0.231717	−0.115859	+	0.993266i	$0.536962\pi$
$332$	19.2770	1.05796
$333$	0	0
$334$	19.7823	1.08244
$335$	−15.4508	−0.844170
$336$	0	0
$337$	−13.2739	−0.723075	−0.361537	−	0.932358i	$-0.617748\pi$
$337$	−13.2739	−0.723075	−0.361537	+	0.932358i	$0.617748\pi$
$338$	11.3512	0.617426
$339$	0	0
$340$	7.48831	0.406111
$341$	−13.5587	−0.734245
$342$	0	0
$343$	1.00000	0.0539949
$344$	−35.3788	−1.90750
$345$	0	0
$346$	−10.4335	−0.560906
$347$	−11.3835	−0.611101	−0.305550	−	0.952176i	$-0.598840\pi$
$347$	−11.3835	−0.611101	−0.305550	+	0.952176i	$0.598840\pi$
$348$	0	0
$349$	−16.1375	−0.863820	−0.431910	−	0.901917i	$-0.642160\pi$
$349$	−16.1375	−0.863820	−0.431910	+	0.901917i	$0.642160\pi$
$350$	0.857589	0.0458401
$351$	0	0
$352$	−8.03252	−0.428135
$353$	20.8139	1.10781	0.553906	−	0.832579i	$-0.313137\pi$
$353$	20.8139	1.10781	0.553906	+	0.832579i	$0.313137\pi$
$354$	0	0
$355$	−11.6278	−0.617140
$356$	4.47422	0.237133
$357$	0	0
$358$	8.35171	0.441401
$359$	−14.9942	−0.791363	−0.395682	−	0.918388i	$-0.629492\pi$
$359$	−14.9942	−0.791363	−0.395682	+	0.918388i	$0.629492\pi$
$360$	0	0
$361$	−12.6757	−0.667143
$362$	5.11859	0.269027
$363$	0	0
$364$	−6.47714	−0.339494
$365$	7.12213	0.372789
$366$	0	0
$367$	20.2157	1.05525	0.527626	−	0.849477i	$-0.323082\pi$
$367$	20.2157	1.05525	0.527626	+	0.849477i	$0.323082\pi$
$368$	0.758836	0.0395571
$369$	0	0
$370$	−0.244260	−0.0126985
$371$	−7.32872	−0.380488
$372$	0	0
$373$	31.9599	1.65482	0.827409	−	0.561599i	$-0.189814\pi$
$373$	31.9599	1.65482	0.827409	+	0.561599i	$0.189814\pi$
$374$	−7.14512	−0.369465
$375$	0	0
$376$	21.9017	1.12949
$377$	−23.1985	−1.19479
$378$	0	0
$379$	23.3807	1.20098	0.600492	−	0.799631i	$-0.294971\pi$
$379$	23.3807	1.20098	0.600492	+	0.799631i	$0.294971\pi$
$380$	−3.18008	−0.163135
$381$	0	0
$382$	1.39159	0.0712000
$383$	−13.8527	−0.707838	−0.353919	−	0.935276i	$-0.615151\pi$
$383$	−13.8527	−0.707838	−0.353919	+	0.935276i	$0.615151\pi$
$384$	0	0
$385$	1.40695	0.0717049
$386$	15.1576	0.771500
$387$	0	0
$388$	−15.3368	−0.778608
$389$	25.9594	1.31620	0.658098	−	0.752932i	$-0.271361\pi$
$389$	25.9594	1.31620	0.658098	+	0.752932i	$0.271361\pi$
$390$	0	0
$391$	35.0673	1.77343
$392$	−2.79964	−0.141403
$393$	0	0
$394$	−16.7856	−0.845645
$395$	7.35212	0.369925
$396$	0	0
$397$	−2.06910	−0.103845	−0.0519225	−	0.998651i	$-0.516535\pi$
$397$	−2.06910	−0.103845	−0.0519225	+	0.998651i	$0.516535\pi$
$398$	−6.31734	−0.316660
$399$	0	0
$400$	0.128144	0.00640718
$401$	11.2271	0.560653	0.280327	−	0.959905i	$-0.409557\pi$
$401$	11.2271	0.560653	0.280327	+	0.959905i	$0.409557\pi$
$402$	0	0
$403$	49.3617	2.45888
$404$	−11.2819	−0.561297
$405$	0	0
$406$	−3.88409	−0.192764
$407$	−0.400730	−0.0198634
$408$	0	0
$409$	−25.3807	−1.25499	−0.627496	−	0.778620i	$-0.715920\pi$
$409$	−25.3807	−1.25499	−0.627496	+	0.778620i	$0.715920\pi$
$410$	−1.37842	−0.0680752
$411$	0	0
$412$	−12.5580	−0.618690
$413$	−1.71518	−0.0843984
$414$	0	0
$415$	15.2443	0.748311
$416$	29.2431	1.43376
$417$	0	0
$418$	3.03434	0.148414
$419$	31.9880	1.56271	0.781357	−	0.624084i	$-0.214528\pi$
$419$	31.9880	1.56271	0.781357	+	0.624084i	$0.214528\pi$
$420$	0	0
$421$	15.1346	0.737615	0.368808	−	0.929506i	$-0.379766\pi$
$421$	15.1346	0.737615	0.368808	+	0.929506i	$0.379766\pi$
$422$	22.6492	1.10254
$423$	0	0
$424$	20.5177	0.996429
$425$	5.92177	0.287248
$426$	0	0
$427$	−3.92177	−0.189788
$428$	−3.55829	−0.171997
$429$	0	0
$430$	−10.8373	−0.522621
$431$	−5.90750	−0.284554	−0.142277	−	0.989827i	$-0.545442\pi$
$431$	−5.90750	−0.284554	−0.142277	+	0.989827i	$0.545442\pi$
$432$	0	0
$433$	−35.0940	−1.68651	−0.843255	−	0.537513i	$-0.819364\pi$
$433$	−35.0940	−1.68651	−0.843255	+	0.537513i	$0.819364\pi$
$434$	8.26454	0.396711
$435$	0	0
$436$	−5.54317	−0.265470
$437$	−14.8921	−0.712388
$438$	0	0
$439$	2.71226	0.129449	0.0647246	−	0.997903i	$-0.479383\pi$
$439$	2.71226	0.129449	0.0647246	+	0.997903i	$0.479383\pi$
$440$	−3.93895	−0.187782
$441$	0	0
$442$	26.0124	1.23729
$443$	−18.5382	−0.880777	−0.440389	−	0.897807i	$-0.645159\pi$
$443$	−18.5382	−0.880777	−0.440389	+	0.897807i	$0.645159\pi$
$444$	0	0
$445$	3.53822	0.167728
$446$	−5.29575	−0.250761
$447$	0	0
$448$	4.63983	0.219211
$449$	−36.3035	−1.71327	−0.856634	−	0.515924i	$-0.827449\pi$
$449$	−36.3035	−1.71327	−0.856634	+	0.515924i	$0.827449\pi$
$450$	0	0
$451$	−2.26141	−0.106486
$452$	8.28589	0.389735
$453$	0	0
$454$	−12.3860	−0.581303
$455$	−5.12213	−0.240129
$456$	0	0
$457$	−9.36126	−0.437901	−0.218951	−	0.975736i	$-0.570263\pi$
$457$	−9.36126	−0.437901	−0.218951	+	0.975736i	$0.570263\pi$
$458$	−16.3867	−0.765700
$459$	0	0
$460$	7.48831	0.349144
$461$	−15.4040	−0.717437	−0.358719	−	0.933446i	$-0.616786\pi$
$461$	−15.4040	−0.717437	−0.358719	+	0.933446i	$0.616786\pi$
$462$	0	0
$463$	14.6187	0.679387	0.339693	−	0.940536i	$-0.389677\pi$
$463$	14.6187	0.679387	0.339693	+	0.940536i	$0.389677\pi$
$464$	−0.580373	−0.0269431
$465$	0	0
$466$	5.27568	0.244391
$467$	4.05012	0.187417	0.0937085	−	0.995600i	$-0.470128\pi$
$467$	4.05012	0.187417	0.0937085	+	0.995600i	$0.470128\pi$
$468$	0	0
$469$	15.4508	0.713454
$470$	6.70896	0.309461
$471$	0	0
$472$	4.80187	0.221024
$473$	−17.7796	−0.817505
$474$	0	0
$475$	−2.51481	−0.115388
$476$	−7.48831	−0.343226
$477$	0	0
$478$	2.20370	0.100795
$479$	7.71518	0.352516	0.176258	−	0.984344i	$-0.443601\pi$
$479$	7.71518	0.352516	0.176258	+	0.984344i	$0.443601\pi$
$480$	0	0
$481$	1.45889	0.0665197
$482$	−8.32380	−0.379139
$483$	0	0
$484$	11.4068	0.518490
$485$	−12.1284	−0.550720
$486$	0	0
$487$	24.5028	1.11033	0.555163	−	0.831742i	$-0.312656\pi$
$487$	24.5028	1.11033	0.555163	+	0.831742i	$0.312656\pi$
$488$	10.9795	0.497019
$489$	0	0
$490$	−0.857589	−0.0387419
$491$	35.6278	1.60786	0.803930	−	0.594724i	$-0.202739\pi$
$491$	35.6278	1.60786	0.803930	+	0.594724i	$0.202739\pi$
$492$	0	0
$493$	−26.8202	−1.20792
$494$	−11.0468	−0.497018
$495$	0	0
$496$	1.23491	0.0554492
$497$	11.6278	0.521578
$498$	0	0
$499$	26.4695	1.18494	0.592470	−	0.805593i	$-0.298153\pi$
$499$	26.4695	1.18494	0.592470	+	0.805593i	$0.298153\pi$
$500$	1.26454	0.0565520
$501$	0	0
$502$	6.48227	0.289318
$503$	25.1346	1.12070	0.560348	−	0.828257i	$-0.310667\pi$
$503$	25.1346	1.12070	0.560348	+	0.828257i	$0.310667\pi$
$504$	0	0
$505$	−8.92177	−0.397013
$506$	−7.14512	−0.317639
$507$	0	0
$508$	12.5839	0.558322
$509$	−5.41208	−0.239886	−0.119943	−	0.992781i	$-0.538271\pi$
$509$	−5.41208	−0.239886	−0.119943	+	0.992781i	$0.538271\pi$
$510$	0	0
$511$	−7.12213	−0.315064
$512$	1.44910	0.0640419
$513$	0	0
$514$	12.8222	0.565561
$515$	−9.93090	−0.437608
$516$	0	0
$517$	11.0066	0.484071
$518$	0.244260	0.0107322
$519$	0	0
$520$	14.3401	0.628855
$521$	−42.0392	−1.84177	−0.920885	−	0.389834i	$-0.872532\pi$
$521$	−42.0392	−1.84177	−0.920885	+	0.389834i	$0.872532\pi$
$522$	0	0
$523$	−23.5587	−1.03015	−0.515075	−	0.857145i	$-0.672236\pi$
$523$	−23.5587	−1.03015	−0.515075	+	0.857145i	$0.672236\pi$
$524$	21.4834	0.938505
$525$	0	0
$526$	0.185004	0.00806656
$527$	57.0677	2.48591
$528$	0	0
$529$	12.0673	0.524665
$530$	6.28503	0.273004
$531$	0	0
$532$	3.18008	0.137874
$533$	8.23288	0.356606
$534$	0	0
$535$	−2.81390	−0.121656
$536$	−43.2567	−1.86841
$537$	0	0
$538$	21.9942	0.948236
$539$	−1.40695	−0.0606017
$540$	0	0
$541$	−14.2647	−0.613289	−0.306645	−	0.951824i	$-0.599206\pi$
$541$	−14.2647	−0.613289	−0.306645	+	0.951824i	$0.599206\pi$
$542$	−2.31534	−0.0994522
$543$	0	0
$544$	33.8083	1.44952
$545$	−4.38354	−0.187770
$546$	0	0
$547$	30.4319	1.30117	0.650586	−	0.759432i	$-0.274523\pi$
$547$	30.4319	1.30117	0.650586	+	0.759432i	$0.274523\pi$
$548$	−9.26746	−0.395886
$549$	0	0
$550$	−1.20659	−0.0514490
$551$	11.3898	0.485222
$552$	0	0
$553$	−7.35212	−0.312644
$554$	−21.5404	−0.915165
$555$	0	0
$556$	12.4475	0.527893
$557$	−24.2443	−1.02726	−0.513631	−	0.858011i	$-0.671700\pi$
$557$	−24.2443	−1.02726	−0.513631	+	0.858011i	$0.671700\pi$
$558$	0	0
$559$	64.7281	2.73771
$560$	−0.128144	−0.00541506
$561$	0	0
$562$	19.8738	0.838327
$563$	−0.618665	−0.0260736	−0.0130368	−	0.999915i	$-0.504150\pi$
$563$	−0.618665	−0.0260736	−0.0130368	+	0.999915i	$0.504150\pi$
$564$	0	0
$565$	6.55249	0.275665
$566$	−19.1273	−0.803979
$567$	0	0
$568$	−32.5536	−1.36592
$569$	−15.1174	−0.633755	−0.316878	−	0.948466i	$-0.602634\pi$
$569$	−15.1174	−0.633755	−0.316878	+	0.948466i	$0.602634\pi$
$570$	0	0
$571$	−18.3942	−0.769773	−0.384887	−	0.922964i	$-0.625759\pi$
$571$	−18.3942	−0.769773	−0.384887	+	0.922964i	$0.625759\pi$
$572$	9.11302	0.381035
$573$	0	0
$574$	1.37842	0.0575340
$575$	5.92177	0.246955
$576$	0	0
$577$	−21.0530	−0.876449	−0.438225	−	0.898865i	$-0.644393\pi$
$577$	−21.0530	−0.876449	−0.438225	+	0.898865i	$0.644393\pi$
$578$	15.4943	0.644479
$579$	0	0
$580$	−5.72721	−0.237809
$581$	−15.2443	−0.632438
$582$	0	0
$583$	10.3111	0.427044
$584$	19.9394	0.825097
$585$	0	0
$586$	−2.13660	−0.0882623
$587$	8.12546	0.335374	0.167687	−	0.985840i	$-0.446370\pi$
$587$	8.12546	0.335374	0.167687	+	0.985840i	$0.446370\pi$
$588$	0	0
$589$	−24.2351	−0.998591
$590$	1.47092	0.0605568
$591$	0	0
$592$	0.0364980	0.00150006
$593$	31.2139	1.28180	0.640901	−	0.767623i	$-0.278561\pi$
$593$	31.2139	1.28180	0.640901	+	0.767623i	$0.278561\pi$
$594$	0	0
$595$	−5.92177	−0.242769
$596$	−3.75616	−0.153858
$597$	0	0
$598$	26.0124	1.06373
$599$	−3.58212	−0.146361	−0.0731806	−	0.997319i	$-0.523315\pi$
$599$	−3.58212	−0.146361	−0.0731806	+	0.997319i	$0.523315\pi$
$600$	0	0
$601$	−21.6775	−0.884244	−0.442122	−	0.896955i	$-0.645774\pi$
$601$	−21.6775	−0.884244	−0.442122	+	0.896955i	$0.645774\pi$
$602$	10.8373	0.441696
$603$	0	0
$604$	−10.1163	−0.411627
$605$	9.02049	0.366735
$606$	0	0
$607$	−12.0124	−0.487570	−0.243785	−	0.969829i	$-0.578389\pi$
$607$	−12.0124	−0.487570	−0.243785	+	0.969829i	$0.578389\pi$
$608$	−14.3575	−0.582273
$609$	0	0
$610$	3.36326	0.136175
$611$	−40.0706	−1.62108
$612$	0	0
$613$	−11.5280	−0.465611	−0.232806	−	0.972523i	$-0.574791\pi$
$613$	−11.5280	−0.465611	−0.232806	+	0.972523i	$0.574791\pi$
$614$	10.7203	0.432638
$615$	0	0
$616$	3.93895	0.158705
$617$	−37.0205	−1.49039	−0.745194	−	0.666847i	$-0.767643\pi$
$617$	−37.0205	−1.49039	−0.745194	+	0.666847i	$0.767643\pi$
$618$	0	0
$619$	11.0936	0.445889	0.222945	−	0.974831i	$-0.428433\pi$
$619$	11.0936	0.445889	0.222945	+	0.974831i	$0.428433\pi$
$620$	12.1863	0.489414
$621$	0	0
$622$	5.46473	0.219116
$623$	−3.53822	−0.141756
$624$	0	0
$625$	1.00000	0.0400000
$626$	−16.6084	−0.663806
$627$	0	0
$628$	21.9309	0.875138
$629$	1.68664	0.0672509
$630$	0	0
$631$	−22.3818	−0.891003	−0.445502	−	0.895281i	$-0.646975\pi$
$631$	−22.3818	−0.891003	−0.445502	+	0.895281i	$0.646975\pi$
$632$	20.5833	0.818758
$633$	0	0
$634$	−22.4598	−0.891992
$635$	9.95139	0.394909
$636$	0	0
$637$	5.12213	0.202946
$638$	5.46473	0.216351
$639$	0	0
$640$	7.43926	0.294063
$641$	20.3602	0.804178	0.402089	−	0.915601i	$-0.368284\pi$
$641$	20.3602	0.804178	0.402089	+	0.915601i	$0.368284\pi$
$642$	0	0
$643$	−18.4194	−0.726391	−0.363196	−	0.931713i	$-0.618314\pi$
$643$	−18.4194	−0.726391	−0.363196	+	0.931713i	$0.618314\pi$
$644$	−7.48831	−0.295081
$645$	0	0
$646$	−12.7713	−0.502482
$647$	−48.5467	−1.90857	−0.954283	−	0.298903i	$-0.903379\pi$
$647$	−48.5467	−1.90857	−0.954283	+	0.298903i	$0.903379\pi$
$648$	0	0
$649$	2.41317	0.0947253
$650$	4.39268	0.172295
$651$	0	0
$652$	2.92533	0.114565
$653$	46.7236	1.82844	0.914219	−	0.405221i	$-0.132806\pi$
$653$	46.7236	1.82844	0.914219	+	0.405221i	$0.132806\pi$
$654$	0	0
$655$	16.9891	0.663818
$656$	0.205967	0.00804167
$657$	0	0
$658$	−6.70896	−0.261542
$659$	35.9028	1.39857	0.699287	−	0.714841i	$-0.253501\pi$
$659$	35.9028	1.39857	0.699287	+	0.714841i	$0.253501\pi$
$660$	0	0
$661$	−0.813902	−0.0316571	−0.0158286	−	0.999875i	$-0.505039\pi$
$661$	−0.813902	−0.0316571	−0.0158286	+	0.999875i	$0.505039\pi$
$662$	−3.61536	−0.140515
$663$	0	0
$664$	42.6784	1.65624
$665$	2.51481	0.0975203
$666$	0	0
$667$	−26.8202	−1.03848
$668$	−29.1695	−1.12860
$669$	0	0
$670$	−13.2505	−0.511911
$671$	5.51773	0.213010
$672$	0	0
$673$	−14.7163	−0.567271	−0.283635	−	0.958932i	$-0.591541\pi$
$673$	−14.7163	−0.567271	−0.283635	+	0.958932i	$0.591541\pi$
$674$	−11.3835	−0.438478
$675$	0	0
$676$	−16.7377	−0.643759
$677$	40.1631	1.54359	0.771797	−	0.635869i	$-0.219358\pi$
$677$	40.1631	1.54359	0.771797	+	0.635869i	$0.219358\pi$
$678$	0	0
$679$	12.1284	0.465443
$680$	16.5788	0.635767
$681$	0	0
$682$	−11.6278	−0.445252
$683$	−30.1375	−1.15318	−0.576590	−	0.817034i	$-0.695617\pi$
$683$	−30.1375	−1.15318	−0.576590	+	0.817034i	$0.695617\pi$
$684$	0	0
$685$	−7.32872	−0.280016
$686$	0.857589	0.0327429
$687$	0	0
$688$	1.61934	0.0617369
$689$	−37.5386	−1.43011
$690$	0	0
$691$	−44.2019	−1.68152	−0.840759	−	0.541409i	$-0.817891\pi$
$691$	−44.2019	−1.68152	−0.840759	+	0.541409i	$0.817891\pi$
$692$	15.3844	0.584828
$693$	0	0
$694$	−9.76241	−0.370576
$695$	9.84353	0.373386
$696$	0	0
$697$	9.51815	0.360526
$698$	−13.8393	−0.523827
$699$	0	0
$700$	−1.26454	−0.0477951
$701$	41.3741	1.56268	0.781339	−	0.624106i	$-0.214537\pi$
$701$	41.3741	1.56268	0.781339	+	0.624106i	$0.214537\pi$
$702$	0	0
$703$	−0.716272	−0.0270147
$704$	−6.52802	−0.246034
$705$	0	0
$706$	17.8498	0.671785
$707$	8.92177	0.335538
$708$	0	0
$709$	−23.3229	−0.875910	−0.437955	−	0.898997i	$-0.644297\pi$
$709$	−23.3229	−0.875910	−0.437955	+	0.898997i	$0.644297\pi$
$710$	−9.97188	−0.374238
$711$	0	0
$712$	9.90573	0.371233
$713$	57.0677	2.13720
$714$	0	0
$715$	7.20659	0.269511
$716$	−12.3148	−0.460227
$717$	0	0
$718$	−12.8589	−0.479888
$719$	17.3835	0.648297	0.324148	−	0.946006i	$-0.394922\pi$
$719$	17.3835	0.648297	0.324148	+	0.946006i	$0.394922\pi$
$720$	0	0
$721$	9.93090	0.369846
$722$	−10.8706	−0.404560
$723$	0	0
$724$	−7.54751	−0.280501
$725$	−4.52908	−0.168206
$726$	0	0
$727$	−31.2052	−1.15734	−0.578669	−	0.815563i	$-0.696427\pi$
$727$	−31.2052	−1.15734	−0.578669	+	0.815563i	$0.696427\pi$
$728$	−14.3401	−0.531479
$729$	0	0
$730$	6.10786	0.226062
$731$	74.8330	2.76780
$732$	0	0
$733$	1.29729	0.0479166	0.0239583	−	0.999713i	$-0.492373\pi$
$733$	1.29729	0.0479166	0.0239583	+	0.999713i	$0.492373\pi$
$734$	17.3368	0.639913
$735$	0	0
$736$	33.8083	1.24619
$737$	−21.7386	−0.800751
$738$	0	0
$739$	−14.4286	−0.530763	−0.265382	−	0.964143i	$-0.585498\pi$
$739$	−14.4286	−0.530763	−0.265382	+	0.964143i	$0.585498\pi$
$740$	0.360168	0.0132400
$741$	0	0
$742$	−6.28503	−0.230731
$743$	15.2146	0.558171	0.279085	−	0.960266i	$-0.409969\pi$
$743$	15.2146	0.558171	0.279085	+	0.960266i	$0.409969\pi$
$744$	0	0
$745$	−2.97037	−0.108826
$746$	27.4084	1.00349
$747$	0	0
$748$	10.5357	0.385223
$749$	2.81390	0.102818
$750$	0	0
$751$	29.3521	1.07107	0.535537	−	0.844512i	$-0.320109\pi$
$751$	29.3521	1.07107	0.535537	+	0.844512i	$0.320109\pi$
$752$	−1.00247	−0.0365564
$753$	0	0
$754$	−19.8948	−0.724527
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−10.4007	−0.378021	−0.189010	−	0.981975i	$-0.560528\pi$
$757$	−10.4007	−0.378021	−0.189010	+	0.981975i	$0.560528\pi$
$758$	20.0510	0.728285
$759$	0	0
$760$	−7.04056	−0.255388
$761$	−48.7613	−1.76760	−0.883798	−	0.467868i	$-0.845022\pi$
$761$	−48.7613	−1.76760	−0.883798	+	0.467868i	$0.845022\pi$
$762$	0	0
$763$	4.38354	0.158695
$764$	−2.05194	−0.0742366
$765$	0	0
$766$	−11.8799	−0.429238
$767$	−8.78537	−0.317221
$768$	0	0
$769$	−40.1946	−1.44945	−0.724726	−	0.689037i	$-0.758034\pi$
$769$	−40.1946	−1.44945	−0.724726	+	0.689037i	$0.758034\pi$
$770$	1.20659	0.0434824
$771$	0	0
$772$	−22.3503	−0.804404
$773$	27.7814	0.999227	0.499614	−	0.866248i	$-0.333475\pi$
$773$	27.7814	0.999227	0.499614	+	0.866248i	$0.333475\pi$
$774$	0	0
$775$	9.63694	0.346169
$776$	−33.9550	−1.21891
$777$	0	0
$778$	22.2625	0.798151
$779$	−4.04210	−0.144823
$780$	0	0
$781$	−16.3598	−0.585398
$782$	30.0733	1.07542
$783$	0	0
$784$	0.128144	0.00457656
$785$	17.3430	0.618998
$786$	0	0
$787$	21.0300	0.749640	0.374820	−	0.927098i	$-0.377704\pi$
$787$	21.0300	0.749640	0.374820	+	0.927098i	$0.377704\pi$
$788$	24.7508	0.881711
$789$	0	0
$790$	6.30510	0.224325
$791$	−6.55249	−0.232980
$792$	0	0
$793$	−20.0878	−0.713338
$794$	−1.77443	−0.0629723
$795$	0	0
$796$	9.31510	0.330165
$797$	20.5792	0.728953	0.364476	−	0.931213i	$-0.381248\pi$
$797$	20.5792	0.728953	0.364476	+	0.931213i	$0.381248\pi$
$798$	0	0
$799$	−46.3262	−1.63890
$800$	5.70916	0.201849
$801$	0	0
$802$	9.62822	0.339984
$803$	10.0205	0.353615
$804$	0	0
$805$	−5.92177	−0.208715
$806$	42.3320	1.49108
$807$	0	0
$808$	−24.9777	−0.878712
$809$	23.9313	0.841380	0.420690	−	0.907204i	$-0.361788\pi$
$809$	23.9313	0.841380	0.420690	+	0.907204i	$0.361788\pi$
$810$	0	0
$811$	14.6117	0.513088	0.256544	−	0.966533i	$-0.417416\pi$
$811$	14.6117	0.513088	0.256544	+	0.966533i	$0.417416\pi$
$812$	5.72721	0.200986
$813$	0	0
$814$	−0.343661	−0.0120453
$815$	2.31336	0.0810333
$816$	0	0
$817$	−31.7796	−1.11183
$818$	−21.7662	−0.761037
$819$	0	0
$820$	2.03252	0.0709785
$821$	−51.5467	−1.79899	−0.899496	−	0.436929i	$-0.856066\pi$
$821$	−51.5467	−1.79899	−0.899496	+	0.436929i	$0.856066\pi$
$822$	0	0
$823$	22.6552	0.789711	0.394856	−	0.918743i	$-0.370795\pi$
$823$	22.6552	0.789711	0.394856	+	0.918743i	$0.370795\pi$
$824$	−27.8029	−0.968560
$825$	0	0
$826$	−1.47092	−0.0511798
$827$	−33.7814	−1.17469	−0.587347	−	0.809335i	$-0.699828\pi$
$827$	−33.7814	−1.17469	−0.587347	+	0.809335i	$0.699828\pi$
$828$	0	0
$829$	17.3024	0.600938	0.300469	−	0.953792i	$-0.402857\pi$
$829$	17.3024	0.600938	0.300469	+	0.953792i	$0.402857\pi$
$830$	13.0733	0.453781
$831$	0	0
$832$	23.7658	0.823932
$833$	5.92177	0.205177
$834$	0	0
$835$	−23.0673	−0.798277
$836$	−4.47422	−0.154744
$837$	0	0
$838$	27.4325	0.947642
$839$	−27.5997	−0.952847	−0.476423	−	0.879216i	$-0.658067\pi$
$839$	−27.5997	−0.952847	−0.476423	+	0.879216i	$0.658067\pi$
$840$	0	0
$841$	−8.48743	−0.292670
$842$	12.9793	0.447295
$843$	0	0
$844$	−33.3969	−1.14957
$845$	−13.2362	−0.455340
$846$	0	0
$847$	−9.02049	−0.309948
$848$	−0.939128	−0.0322498
$849$	0	0
$850$	5.07844	0.174189
$851$	1.68664	0.0578174
$852$	0	0
$853$	−3.91221	−0.133952	−0.0669758	−	0.997755i	$-0.521335\pi$
$853$	−3.91221	−0.133952	−0.0669758	+	0.997755i	$0.521335\pi$
$854$	−3.36326	−0.115089
$855$	0	0
$856$	−7.87790	−0.269261
$857$	−29.4216	−1.00502	−0.502512	−	0.864570i	$-0.667591\pi$
$857$	−29.4216	−1.00502	−0.502512	+	0.864570i	$0.667591\pi$
$858$	0	0
$859$	−30.3744	−1.03636	−0.518181	−	0.855271i	$-0.673391\pi$
$859$	−30.3744	−1.03636	−0.518181	+	0.855271i	$0.673391\pi$
$860$	15.9799	0.544911
$861$	0	0
$862$	−5.06621	−0.172556
$863$	−34.0278	−1.15832	−0.579160	−	0.815214i	$-0.696620\pi$
$863$	−34.0278	−1.15832	−0.579160	+	0.815214i	$0.696620\pi$
$864$	0	0
$865$	12.1660	0.413657
$866$	−30.0963	−1.02271
$867$	0	0
$868$	−12.1863	−0.413630
$869$	10.3441	0.350899
$870$	0	0
$871$	79.1412	2.68160
$872$	−12.2723	−0.415593
$873$	0	0
$874$	−12.7713	−0.431997
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−35.8908	−1.21194	−0.605972	−	0.795486i	$-0.707216\pi$
$877$	−35.8908	−1.21194	−0.605972	+	0.795486i	$0.707216\pi$
$878$	2.32601	0.0784989
$879$	0	0
$880$	0.180292	0.00607763
$881$	−11.1741	−0.376464	−0.188232	−	0.982125i	$-0.560276\pi$
$881$	−11.1741	−0.376464	−0.188232	+	0.982125i	$0.560276\pi$
$882$	0	0
$883$	−55.7847	−1.87730	−0.938652	−	0.344865i	$-0.887925\pi$
$883$	−55.7847	−1.87730	−0.938652	+	0.344865i	$0.887925\pi$
$884$	−38.3561	−1.29006
$885$	0	0
$886$	−15.8982	−0.534110
$887$	−45.4198	−1.52505	−0.762524	−	0.646959i	$-0.776040\pi$
$887$	−45.4198	−1.52505	−0.762524	+	0.646959i	$0.776040\pi$
$888$	0	0
$889$	−9.95139	−0.333759
$890$	3.03434	0.101711
$891$	0	0
$892$	7.80874	0.261456
$893$	19.6735	0.658348
$894$	0	0
$895$	−9.73859	−0.325525
$896$	−7.43926	−0.248528
$897$	0	0
$898$	−31.1335	−1.03894
$899$	−43.6465	−1.45569
$900$	0	0
$901$	−43.3989	−1.44583
$902$	−1.93937	−0.0645738
$903$	0	0
$904$	18.3446	0.610131
$905$	−5.96858	−0.198402
$906$	0	0
$907$	−6.54957	−0.217475	−0.108737	−	0.994071i	$-0.534681\pi$
$907$	−6.54957	−0.217475	−0.108737	+	0.994071i	$0.534681\pi$
$908$	18.2635	0.606096
$909$	0	0
$910$	−4.39268	−0.145616
$911$	−19.3097	−0.639760	−0.319880	−	0.947458i	$-0.603643\pi$
$911$	−19.3097	−0.639760	−0.319880	+	0.947458i	$0.603643\pi$
$912$	0	0
$913$	21.4479	0.709823
$914$	−8.02812	−0.265547
$915$	0	0
$916$	24.1627	0.798357
$917$	−16.9891	−0.561028
$918$	0	0
$919$	26.5171	0.874717	0.437358	−	0.899287i	$-0.355914\pi$
$919$	26.5171	0.874717	0.437358	+	0.899287i	$0.355914\pi$
$920$	16.5788	0.546586
$921$	0	0
$922$	−13.2103	−0.435059
$923$	59.5591	1.96041
$924$	0	0
$925$	0.284821	0.00936487
$926$	12.5368	0.411985
$927$	0	0
$928$	−25.8573	−0.848806
$929$	20.6775	0.678407	0.339203	−	0.940713i	$-0.389842\pi$
$929$	20.6775	0.678407	0.339203	+	0.940713i	$0.389842\pi$
$930$	0	0
$931$	−2.51481	−0.0824197
$932$	−7.77915	−0.254814
$933$	0	0
$934$	3.47334	0.113651
$935$	8.33163	0.272474
$936$	0	0
$937$	−33.9075	−1.10771	−0.553855	−	0.832613i	$-0.686844\pi$
$937$	−33.9075	−1.10771	−0.553855	+	0.832613i	$0.686844\pi$
$938$	13.2505	0.432644
$939$	0	0
$940$	−9.89255	−0.322659
$941$	−36.3236	−1.18412	−0.592058	−	0.805896i	$-0.701684\pi$
$941$	−36.3236	−1.18412	−0.592058	+	0.805896i	$0.701684\pi$
$942$	0	0
$943$	9.51815	0.309954
$944$	−0.219789	−0.00715353
$945$	0	0
$946$	−15.2476	−0.495741
$947$	−56.0791	−1.82232	−0.911162	−	0.412047i	$-0.864814\pi$
$947$	−56.0791	−1.82232	−0.911162	+	0.412047i	$0.864814\pi$
$948$	0	0
$949$	−36.4805	−1.18421
$950$	−2.15668	−0.0699719
$951$	0	0
$952$	−16.5788	−0.537321
$953$	6.61646	0.214328	0.107164	−	0.994241i	$-0.465823\pi$
$953$	6.61646	0.214328	0.107164	+	0.994241i	$0.465823\pi$
$954$	0	0
$955$	−1.62268	−0.0525086
$956$	−3.24942	−0.105094
$957$	0	0
$958$	6.61646	0.213768
$959$	7.32872	0.236657
$960$	0	0
$961$	61.8707	1.99583
$962$	1.25113	0.0403380
$963$	0	0
$964$	12.2737	0.395309
$965$	−17.6746	−0.568966
$966$	0	0
$967$	−2.09542	−0.0673842	−0.0336921	−	0.999432i	$-0.510727\pi$
$967$	−2.09542	−0.0673842	−0.0336921	+	0.999432i	$0.510727\pi$
$968$	25.2541	0.811697
$969$	0	0
$970$	−10.4011	−0.333961
$971$	−24.3013	−0.779867	−0.389933	−	0.920843i	$-0.627502\pi$
$971$	−24.3013	−0.779867	−0.389933	+	0.920843i	$0.627502\pi$
$972$	0	0
$973$	−9.84353	−0.315569
$974$	21.0133	0.673310
$975$	0	0
$976$	−0.502549	−0.0160862
$977$	−51.5134	−1.64806	−0.824030	−	0.566546i	$-0.808280\pi$
$977$	−51.5134	−1.64806	−0.824030	+	0.566546i	$0.808280\pi$
$978$	0	0
$979$	4.97810	0.159101
$980$	1.26454	0.0403943
$981$	0	0
$982$	30.5540	0.975018
$983$	39.9507	1.27423	0.637115	−	0.770769i	$-0.280128\pi$
$983$	39.9507	1.27423	0.637115	+	0.770769i	$0.280128\pi$
$984$	0	0
$985$	19.5730	0.623647
$986$	−23.0007	−0.732491
$987$	0	0
$988$	16.2888	0.518216
$989$	74.8330	2.37955
$990$	0	0
$991$	−1.19565	−0.0379812	−0.0189906	−	0.999820i	$-0.506045\pi$
$991$	−1.19565	−0.0379812	−0.0189906	+	0.999820i	$0.506045\pi$
$992$	55.0189	1.74685
$993$	0	0
$994$	9.97188	0.316289
$995$	7.36639	0.233530
$996$	0	0
$997$	55.0239	1.74262	0.871311	−	0.490730i	$-0.163270\pi$
$997$	55.0239	1.74262	0.871311	+	0.490730i	$0.163270\pi$
$998$	22.7000	0.718556
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	945.2.a.m.1.3	✓	4
3.2	odd	2		945.2.a.n.1.2	yes	4
5.4	even	2		4725.2.a.bx.1.2		4
7.6	odd	2		6615.2.a.be.1.3		4
15.14	odd	2		4725.2.a.bo.1.3		4
21.20	even	2		6615.2.a.bh.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
945.2.a.m.1.3	✓	4	1.1	even	1	trivial
945.2.a.n.1.2	yes	4	3.2	odd	2
4725.2.a.bo.1.3		4	15.14	odd	2
4725.2.a.bx.1.2		4	5.4	even	2
6615.2.a.be.1.3		4	7.6	odd	2
6615.2.a.bh.1.2		4	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)

\(f(q)\)	\(=\)	\(q+0.857589 q^{2} -1.26454 q^{4} -1.00000 q^{5} +1.00000 q^{7} -2.79964 q^{8} +O(q^{10})\)	\(q+0.857589 q^{2} -1.26454 q^{4} -1.00000 q^{5} +1.00000 q^{7} -2.79964 q^{8} -0.857589 q^{10} -1.40695 q^{11} +5.12213 q^{13} +0.857589 q^{14} +0.128144 q^{16} +5.92177 q^{17} -2.51481 q^{19} +1.26454 q^{20} -1.20659 q^{22} +5.92177 q^{23} +1.00000 q^{25} +4.39268 q^{26} -1.26454 q^{28} -4.52908 q^{29} +9.63694 q^{31} +5.70916 q^{32} +5.07844 q^{34} -1.00000 q^{35} +0.284821 q^{37} -2.15668 q^{38} +2.79964 q^{40} +1.60732 q^{41} +12.6369 q^{43} +1.77915 q^{44} +5.07844 q^{46} -7.82304 q^{47} +1.00000 q^{49} +0.857589 q^{50} -6.47714 q^{52} -7.32872 q^{53} +1.40695 q^{55} -2.79964 q^{56} -3.88409 q^{58} -1.71518 q^{59} -3.92177 q^{61} +8.26454 q^{62} +4.63983 q^{64} -5.12213 q^{65} +15.4508 q^{67} -7.48831 q^{68} -0.857589 q^{70} +11.6278 q^{71} -7.12213 q^{73} +0.244260 q^{74} +3.18008 q^{76} -1.40695 q^{77} -7.35212 q^{79} -0.128144 q^{80} +1.37842 q^{82} -15.2443 q^{83} -5.92177 q^{85} +10.8373 q^{86} +3.93895 q^{88} -3.53822 q^{89} +5.12213 q^{91} -7.48831 q^{92} -6.70896 q^{94} +2.51481 q^{95} +12.1284 q^{97} +0.857589 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 9 q^{4} - 4 q^{5} + 4 q^{7} - 6 q^{8}+O(q^{10}) \) 4 * q - q^2 + 9 * q^4 - 4 * q^5 + 4 * q^7 - 6 * q^8	\( 4 q - q^{2} + 9 q^{4} - 4 q^{5} + 4 q^{7} - 6 q^{8} + q^{10} + 4 q^{11} + 2 q^{13} - q^{14} + 19 q^{16} + 4 q^{19} - 9 q^{20} + 10 q^{22} + 4 q^{25} + 22 q^{26} + 9 q^{28} + 10 q^{29} + 6 q^{31} - 23 q^{32} - 13 q^{34} - 4 q^{35} + 10 q^{37} + q^{38} + 6 q^{40} + 2 q^{41} + 18 q^{43} + 36 q^{44} - 13 q^{46} - 18 q^{47} + 4 q^{49} - q^{50} - 34 q^{52} + 4 q^{53} - 4 q^{55} - 6 q^{56} - 14 q^{58} + 2 q^{59} + 8 q^{61} + 19 q^{62} + 54 q^{64} - 2 q^{65} + 10 q^{67} - 13 q^{68} + q^{70} + 8 q^{71} - 10 q^{73} - 36 q^{74} - 5 q^{76} + 4 q^{77} + 12 q^{79} - 19 q^{80} + 24 q^{82} - 24 q^{83} + 16 q^{86} + 4 q^{88} + 8 q^{89} + 2 q^{91} - 13 q^{92} - 38 q^{94} - 4 q^{95} + 10 q^{97} - q^{98}+O(q^{100}) \) 4 * q - q^2 + 9 * q^4 - 4 * q^5 + 4 * q^7 - 6 * q^8 + q^10 + 4 * q^11 + 2 * q^13 - q^14 + 19 * q^16 + 4 * q^19 - 9 * q^20 + 10 * q^22 + 4 * q^25 + 22 * q^26 + 9 * q^28 + 10 * q^29 + 6 * q^31 - 23 * q^32 - 13 * q^34 - 4 * q^35 + 10 * q^37 + q^38 + 6 * q^40 + 2 * q^41 + 18 * q^43 + 36 * q^44 - 13 * q^46 - 18 * q^47 + 4 * q^49 - q^50 - 34 * q^52 + 4 * q^53 - 4 * q^55 - 6 * q^56 - 14 * q^58 + 2 * q^59 + 8 * q^61 + 19 * q^62 + 54 * q^64 - 2 * q^65 + 10 * q^67 - 13 * q^68 + q^70 + 8 * q^71 - 10 * q^73 - 36 * q^74 - 5 * q^76 + 4 * q^77 + 12 * q^79 - 19 * q^80 + 24 * q^82 - 24 * q^83 + 16 * q^86 + 4 * q^88 + 8 * q^89 + 2 * q^91 - 13 * q^92 - 38 * q^94 - 4 * q^95 + 10 * q^97 - q^98

Introduction

L-functions

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Database

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