LMFDB - Embedded newform 1449.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1449,2,Mod(1,1449)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1449, 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("1449.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1449 = 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1449.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:	$11.5703232529$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.24197.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - x + 2 $ x^4 - 6*x^2 - x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.27460$ of defining polynomial
Character	$\chi$	$=$	1449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.27460 q^{2} +3.17380 q^{4} -2.39532 q^{5} +1.00000 q^{7} -2.66992 q^{8} +O(q^{10})$	\(q-2.27460 q^{2} +3.17380 q^{4} -2.39532 q^{5} +1.00000 q^{7} -2.66992 q^{8} +5.44840 q^{10} +4.56912 q^{11} +5.05307 q^{13} -2.27460 q^{14} -0.274599 q^{16} +1.44840 q^{17} +1.98008 q^{19} -7.60227 q^{20} -10.3929 q^{22} +1.00000 q^{23} +0.737570 q^{25} -11.4937 q^{26} +3.17380 q^{28} -3.68985 q^{29} -3.44840 q^{31} +5.96444 q^{32} -3.29452 q^{34} -2.39532 q^{35} -3.99759 q^{37} -4.50388 q^{38} +6.39532 q^{40} -1.77848 q^{41} -1.25467 q^{43} +14.5015 q^{44} -2.27460 q^{46} -2.87928 q^{47} +1.00000 q^{49} -1.67768 q^{50} +16.0374 q^{52} +10.4128 q^{53} -10.9445 q^{55} -2.66992 q^{56} +8.39292 q^{58} +14.1315 q^{59} +8.37540 q^{61} +7.84372 q^{62} -13.0175 q^{64} -12.1037 q^{65} -7.73517 q^{67} +4.59692 q^{68} +5.44840 q^{70} +0.946925 q^{71} +3.86711 q^{73} +9.09292 q^{74} +6.28436 q^{76} +4.56912 q^{77} +4.23904 q^{79} +0.657752 q^{80} +4.04532 q^{82} -10.8281 q^{83} -3.46938 q^{85} +2.85388 q^{86} -12.1992 q^{88} -0.262430 q^{89} +5.05307 q^{91} +3.17380 q^{92} +6.54920 q^{94} -4.74292 q^{95} -7.20695 q^{97} -2.27460 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 5 q^{5} + 4 q^{7} + 3 q^{8}+O(q^{10}) $ 4 * q + 4 * q^4 - 5 * q^5 + 4 * q^7 + 3 * q^8	$ 4 q + 4 q^{4} - 5 q^{5} + 4 q^{7} + 3 q^{8} + 4 q^{10} + 5 q^{11} + 7 q^{13} + 8 q^{16} - 12 q^{17} + 3 q^{19} + q^{20} - q^{22} + 4 q^{23} + 7 q^{25} - 5 q^{26} + 4 q^{28} - 6 q^{29} + 4 q^{31} + 6 q^{32} - 9 q^{34} - 5 q^{35} + 20 q^{37} - 23 q^{38} + 21 q^{40} - 3 q^{41} + 9 q^{43} + 27 q^{44} - 7 q^{47} + 4 q^{49} - 3 q^{50} + 38 q^{52} + 6 q^{53} - 21 q^{55} + 3 q^{56} - 7 q^{58} + 2 q^{59} + 24 q^{61} + 9 q^{62} - 21 q^{64} + 14 q^{65} + q^{67} + 13 q^{68} + 4 q^{70} + 17 q^{71} + 16 q^{73} + 33 q^{74} - 25 q^{76} + 5 q^{77} - 10 q^{79} - 6 q^{80} - 7 q^{82} - 8 q^{83} + 17 q^{85} + 35 q^{86} - 12 q^{88} + 3 q^{89} + 7 q^{91} + 4 q^{92} + 8 q^{94} + 3 q^{95} - 2 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 - 5 * q^5 + 4 * q^7 + 3 * q^8 + 4 * q^10 + 5 * q^11 + 7 * q^13 + 8 * q^16 - 12 * q^17 + 3 * q^19 + q^20 - q^22 + 4 * q^23 + 7 * q^25 - 5 * q^26 + 4 * q^28 - 6 * q^29 + 4 * q^31 + 6 * q^32 - 9 * q^34 - 5 * q^35 + 20 * q^37 - 23 * q^38 + 21 * q^40 - 3 * q^41 + 9 * q^43 + 27 * q^44 - 7 * q^47 + 4 * q^49 - 3 * q^50 + 38 * q^52 + 6 * q^53 - 21 * q^55 + 3 * q^56 - 7 * q^58 + 2 * q^59 + 24 * q^61 + 9 * q^62 - 21 * q^64 + 14 * q^65 + q^67 + 13 * q^68 + 4 * q^70 + 17 * q^71 + 16 * q^73 + 33 * q^74 - 25 * q^76 + 5 * q^77 - 10 * q^79 - 6 * q^80 - 7 * q^82 - 8 * q^83 + 17 * q^85 + 35 * q^86 - 12 * q^88 + 3 * q^89 + 7 * q^91 + 4 * q^92 + 8 * q^94 + 3 * q^95 - 2 * q^97

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 $

$2$	−2.27460	−1.60838	−0.804192	−	0.594370i	$-0.797402\pi$
$2$	−2.27460	−1.60838	−0.804192	+	0.594370i	$0.797402\pi$
$3$	0	0
$3$	0	0
$4$	3.17380	1.58690
$5$	−2.39532	−1.07122	−0.535610	−	0.844465i	$-0.679918\pi$
$5$	−2.39532	−1.07122	−0.535610	+	0.844465i	$0.679918\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.66992	−0.943960
$9$	0	0
$10$	5.44840	1.72293
$11$	4.56912	1.37764	0.688821	−	0.724931i	$-0.258129\pi$
$11$	4.56912	1.37764	0.688821	+	0.724931i	$0.258129\pi$
$12$	0	0
$13$	5.05307	1.40147	0.700735	−	0.713421i	$-0.252855\pi$
$13$	5.05307	1.40147	0.700735	+	0.713421i	$0.252855\pi$
$14$	−2.27460	−0.607912
$15$	0	0
$16$	−0.274599	−0.0686497
$17$	1.44840	0.351288	0.175644	−	0.984454i	$-0.443799\pi$
$17$	1.44840	0.351288	0.175644	+	0.984454i	$0.443799\pi$
$18$	0	0
$19$	1.98008	0.454261	0.227130	−	0.973864i	$-0.427066\pi$
$19$	1.98008	0.454261	0.227130	+	0.973864i	$0.427066\pi$
$20$	−7.60227	−1.69992
$21$	0	0
$22$	−10.3929	−2.21578
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0.737570	0.147514
$26$	−11.4937	−2.25410
$27$	0	0
$28$	3.17380	0.599792
$29$	−3.68985	−0.685187	−0.342594	−	0.939484i	$-0.611305\pi$
$29$	−3.68985	−0.685187	−0.342594	+	0.939484i	$0.611305\pi$
$30$	0	0
$31$	−3.44840	−0.619350	−0.309675	−	0.950842i	$-0.600220\pi$
$31$	−3.44840	−0.619350	−0.309675	+	0.950842i	$0.600220\pi$
$32$	5.96444	1.05437
$33$	0	0
$34$	−3.29452	−0.565006
$35$	−2.39532	−0.404883
$36$	0	0
$37$	−3.99759	−0.657201	−0.328600	−	0.944469i	$-0.606577\pi$
$37$	−3.99759	−0.657201	−0.328600	+	0.944469i	$0.606577\pi$
$38$	−4.50388	−0.730625
$39$	0	0
$40$	6.39532	1.01119
$41$	−1.77848	−0.277751	−0.138876	−	0.990310i	$-0.544349\pi$
$41$	−1.77848	−0.277751	−0.138876	+	0.990310i	$0.544349\pi$
$42$	0	0
$43$	−1.25467	−0.191336	−0.0956680	−	0.995413i	$-0.530499\pi$
$43$	−1.25467	−0.191336	−0.0956680	+	0.995413i	$0.530499\pi$
$44$	14.5015	2.18618
$45$	0	0
$46$	−2.27460	−0.335371
$47$	−2.87928	−0.419986	−0.209993	−	0.977703i	$-0.567344\pi$
$47$	−2.87928	−0.419986	−0.209993	+	0.977703i	$0.567344\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.67768	−0.237259
$51$	0	0
$52$	16.0374	2.22399
$53$	10.4128	1.43031	0.715157	−	0.698964i	$-0.246355\pi$
$53$	10.4128	1.43031	0.715157	+	0.698964i	$0.246355\pi$
$54$	0	0
$55$	−10.9445	−1.47576
$56$	−2.66992	−0.356783
$57$	0	0
$58$	8.39292	1.10204
$59$	14.1315	1.83977	0.919885	−	0.392188i	$-0.128282\pi$
$59$	14.1315	1.83977	0.919885	+	0.392188i	$0.128282\pi$
$60$	0	0
$61$	8.37540	1.07236	0.536180	−	0.844104i	$-0.319867\pi$
$61$	8.37540	1.07236	0.536180	+	0.844104i	$0.319867\pi$
$62$	7.84372	0.996153
$63$	0	0
$64$	−13.0175	−1.62719
$65$	−12.1037	−1.50128
$66$	0	0
$67$	−7.73517	−0.945001	−0.472500	−	0.881330i	$-0.656648\pi$
$67$	−7.73517	−0.945001	−0.472500	+	0.881330i	$0.656648\pi$
$68$	4.59692	0.557459
$69$	0	0
$70$	5.44840	0.651208
$71$	0.946925	0.112379	0.0561897	−	0.998420i	$-0.482105\pi$
$71$	0.946925	0.112379	0.0561897	+	0.998420i	$0.482105\pi$
$72$	0	0
$73$	3.86711	0.452611	0.226305	−	0.974056i	$-0.427335\pi$
$73$	3.86711	0.452611	0.226305	+	0.974056i	$0.427335\pi$
$74$	9.09292	1.05703
$75$	0	0
$76$	6.28436	0.720866
$77$	4.56912	0.520700
$78$	0	0
$79$	4.23904	0.476930	0.238465	−	0.971151i	$-0.423356\pi$
$79$	4.23904	0.476930	0.238465	+	0.971151i	$0.423356\pi$
$80$	0.657752	0.0735389
$81$	0	0
$82$	4.04532	0.446731
$83$	−10.8281	−1.18854	−0.594269	−	0.804267i	$-0.702558\pi$
$83$	−10.8281	−1.18854	−0.594269	+	0.804267i	$0.702558\pi$
$84$	0	0
$85$	−3.46938	−0.376307
$86$	2.85388	0.307742
$87$	0	0
$88$	−12.1992	−1.30044
$89$	−0.262430	−0.0278175	−0.0139087	−	0.999903i	$-0.504427\pi$
$89$	−0.262430	−0.0278175	−0.0139087	+	0.999903i	$0.504427\pi$
$90$	0	0
$91$	5.05307	0.529706
$92$	3.17380	0.330891
$93$	0	0
$94$	6.54920	0.675498
$95$	−4.74292	−0.486613
$96$	0	0
$97$	−7.20695	−0.731755	−0.365877	−	0.930663i	$-0.619231\pi$
$97$	−7.20695	−0.731755	−0.365877	+	0.930663i	$0.619231\pi$
$98$	−2.27460	−0.229769
$99$	0	0
$100$	2.34090	0.234090
$101$	8.26684	0.822582	0.411291	−	0.911504i	$-0.365078\pi$
$101$	8.26684	0.822582	0.411291	+	0.911504i	$0.365078\pi$
$102$	0	0
$103$	−19.2619	−1.89793	−0.948966	−	0.315378i	$-0.897869\pi$
$103$	−19.2619	−1.89793	−0.948966	+	0.315378i	$0.897869\pi$
$104$	−13.4913	−1.32293
$105$	0	0
$106$	−23.6850	−2.30049
$107$	11.0531	1.06854	0.534271	−	0.845314i	$-0.320586\pi$
$107$	11.0531	1.06854	0.534271	+	0.845314i	$0.320586\pi$
$108$	0	0
$109$	12.9047	1.23604	0.618022	−	0.786161i	$-0.287934\pi$
$109$	12.9047	1.23604	0.618022	+	0.786161i	$0.287934\pi$
$110$	24.8944	2.37359
$111$	0	0
$112$	−0.274599	−0.0259471
$113$	1.25708	0.118256	0.0591281	−	0.998250i	$-0.481168\pi$
$113$	1.25708	0.118256	0.0591281	+	0.998250i	$0.481168\pi$
$114$	0	0
$115$	−2.39532	−0.223365
$116$	−11.7108	−1.08732
$117$	0	0
$118$	−32.1436	−2.95906
$119$	1.44840	0.132774
$120$	0	0
$121$	9.87687	0.897897
$122$	−19.0507	−1.72477
$123$	0	0
$124$	−10.9445	−0.982847
$125$	10.2099	0.913201
$126$	0	0
$127$	19.3385	1.71601	0.858007	−	0.513638i	$-0.171703\pi$
$127$	19.3385	1.71601	0.858007	+	0.513638i	$0.171703\pi$
$128$	17.6807	1.56277
$129$	0	0
$130$	27.5312	2.41464
$131$	−19.4659	−1.70075	−0.850373	−	0.526181i	$-0.823623\pi$
$131$	−19.4659	−1.70075	−0.850373	+	0.526181i	$0.823623\pi$
$132$	0	0
$133$	1.98008	0.171694
$134$	17.5944	1.51992
$135$	0	0
$136$	−3.86711	−0.331602
$137$	−20.4980	−1.75126	−0.875632	−	0.482980i	$-0.839554\pi$
$137$	−20.4980	−1.75126	−0.875632	+	0.482980i	$0.839554\pi$
$138$	0	0
$139$	21.0507	1.78549	0.892747	−	0.450558i	$-0.148775\pi$
$139$	21.0507	1.78549	0.892747	+	0.450558i	$0.148775\pi$
$140$	−7.60227	−0.642509
$141$	0	0
$142$	−2.15387	−0.180749
$143$	23.0881	1.93072
$144$	0	0
$145$	8.83837	0.733987
$146$	−8.79612	−0.727972
$147$	0	0
$148$	−12.6876	−1.04291
$149$	19.1159	1.56604	0.783018	−	0.621999i	$-0.213679\pi$
$149$	19.1159	1.56604	0.783018	+	0.621999i	$0.213679\pi$
$150$	0	0
$151$	18.2542	1.48550	0.742751	−	0.669568i	$-0.233521\pi$
$151$	18.2542	1.48550	0.742751	+	0.669568i	$0.233521\pi$
$152$	−5.28665	−0.428804
$153$	0	0
$154$	−10.3929	−0.837485
$155$	8.26002	0.663461
$156$	0	0
$157$	6.63783	0.529756	0.264878	−	0.964282i	$-0.414668\pi$
$157$	6.63783	0.529756	0.264878	+	0.964282i	$0.414668\pi$
$158$	−9.64212	−0.767086
$159$	0	0
$160$	−14.2868	−1.12947
$161$	1.00000	0.0788110
$162$	0	0
$163$	21.1080	1.65331	0.826655	−	0.562710i	$-0.190241\pi$
$163$	21.1080	1.65331	0.826655	+	0.562710i	$0.190241\pi$
$164$	−5.64453	−0.440763
$165$	0	0
$166$	24.6296	1.91162
$167$	1.01217	0.0783240	0.0391620	−	0.999233i	$-0.487531\pi$
$167$	1.01217	0.0783240	0.0391620	+	0.999233i	$0.487531\pi$
$168$	0	0
$169$	12.5336	0.964120
$170$	7.89144	0.605246
$171$	0	0
$172$	−3.98208	−0.303631
$173$	−15.2069	−1.15616	−0.578081	−	0.815979i	$-0.696198\pi$
$173$	−15.2069	−1.15616	−0.578081	+	0.815979i	$0.696198\pi$
$174$	0	0
$175$	0.737570	0.0557551
$176$	−1.25467	−0.0945746
$177$	0	0
$178$	0.596922	0.0447412
$179$	−18.1491	−1.35652	−0.678262	−	0.734820i	$-0.737267\pi$
$179$	−18.1491	−1.35652	−0.678262	+	0.734820i	$0.737267\pi$
$180$	0	0
$181$	10.5866	0.786899	0.393449	−	0.919346i	$-0.371282\pi$
$181$	10.5866	0.786899	0.393449	+	0.919346i	$0.371282\pi$
$182$	−11.4937	−0.851971
$183$	0	0
$184$	−2.66992	−0.196829
$185$	9.57553	0.704007
$186$	0	0
$187$	6.61790	0.483949
$188$	−9.13824	−0.666475
$189$	0	0
$190$	10.7882	0.782661
$191$	19.8004	1.43271	0.716354	−	0.697737i	$-0.245810\pi$
$191$	19.8004	1.43271	0.716354	+	0.697737i	$0.245810\pi$
$192$	0	0
$193$	22.4956	1.61927	0.809635	−	0.586934i	$-0.199665\pi$
$193$	22.4956	1.61927	0.809635	+	0.586934i	$0.199665\pi$
$194$	16.3929	1.17694
$195$	0	0
$196$	3.17380	0.226700
$197$	−1.67338	−0.119224	−0.0596118	−	0.998222i	$-0.518986\pi$
$197$	−1.67338	−0.119224	−0.0596118	+	0.998222i	$0.518986\pi$
$198$	0	0
$199$	5.82380	0.412838	0.206419	−	0.978464i	$-0.433819\pi$
$199$	5.82380	0.412838	0.206419	+	0.978464i	$0.433819\pi$
$200$	−1.96925	−0.139247
$201$	0	0
$202$	−18.8038	−1.32303
$203$	−3.68985	−0.258976
$204$	0	0
$205$	4.26002	0.297533
$206$	43.8131	3.05260
$207$	0	0
$208$	−1.38757	−0.0962105
$209$	9.04721	0.625808
$210$	0	0
$211$	−14.2809	−0.983138	−0.491569	−	0.870839i	$-0.663576\pi$
$211$	−14.2809	−0.983138	−0.491569	+	0.870839i	$0.663576\pi$
$212$	33.0483	2.26976
$213$	0	0
$214$	−25.1413	−1.71862
$215$	3.00535	0.204963
$216$	0	0
$217$	−3.44840	−0.234092
$218$	−29.3529	−1.98803
$219$	0	0
$220$	−34.7357	−2.34188
$221$	7.31886	0.492320
$222$	0	0
$223$	−20.0031	−1.33950	−0.669752	−	0.742585i	$-0.733600\pi$
$223$	−20.0031	−1.33950	−0.669752	+	0.742585i	$0.733600\pi$
$224$	5.96444	0.398516
$225$	0	0
$226$	−2.85935	−0.190201
$227$	−18.4547	−1.22488	−0.612441	−	0.790517i	$-0.709812\pi$
$227$	−18.4547	−1.22488	−0.612441	+	0.790517i	$0.709812\pi$
$228$	0	0
$229$	−17.2299	−1.13859	−0.569293	−	0.822135i	$-0.692783\pi$
$229$	−17.2299	−1.13859	−0.569293	+	0.822135i	$0.692783\pi$
$230$	5.44840	0.359257
$231$	0	0
$232$	9.85160	0.646789
$233$	16.2844	1.06682	0.533412	−	0.845856i	$-0.320910\pi$
$233$	16.2844	1.06682	0.533412	+	0.845856i	$0.320910\pi$
$234$	0	0
$235$	6.89679	0.449897
$236$	44.8507	2.91953
$237$	0	0
$238$	−3.29452	−0.213552
$239$	26.1539	1.69175	0.845877	−	0.533378i	$-0.179078\pi$
$239$	26.1539	1.69175	0.845877	+	0.533378i	$0.179078\pi$
$240$	0	0
$241$	8.67527	0.558823	0.279412	−	0.960171i	$-0.409861\pi$
$241$	8.67527	0.558823	0.279412	+	0.960171i	$0.409861\pi$
$242$	−22.4659	−1.44416
$243$	0	0
$244$	26.5818	1.70173
$245$	−2.39532	−0.153032
$246$	0	0
$247$	10.0055	0.636633
$248$	9.20695	0.584642
$249$	0	0
$250$	−23.2234	−1.46878
$251$	−9.22888	−0.582522	−0.291261	−	0.956644i	$-0.594075\pi$
$251$	−9.22888	−0.582522	−0.291261	+	0.956644i	$0.594075\pi$
$252$	0	0
$253$	4.56912	0.287258
$254$	−43.9873	−2.76001
$255$	0	0
$256$	−14.1816	−0.886347
$257$	−17.7606	−1.10787	−0.553937	−	0.832559i	$-0.686875\pi$
$257$	−17.7606	−1.10787	−0.553937	+	0.832559i	$0.686875\pi$
$258$	0	0
$259$	−3.99759	−0.248398
$260$	−38.4148	−2.38239
$261$	0	0
$262$	44.2771	2.73545
$263$	10.2721	0.633403	0.316702	−	0.948525i	$-0.397425\pi$
$263$	10.2721	0.633403	0.316702	+	0.948525i	$0.397425\pi$
$264$	0	0
$265$	−24.9421	−1.53218
$266$	−4.50388	−0.276150
$267$	0	0
$268$	−24.5499	−1.49962
$269$	17.5312	1.06889	0.534447	−	0.845202i	$-0.320520\pi$
$269$	17.5312	1.06889	0.534447	+	0.845202i	$0.320520\pi$
$270$	0	0
$271$	21.1602	1.28539	0.642695	−	0.766123i	$-0.277816\pi$
$271$	21.1602	1.28539	0.642695	+	0.766123i	$0.277816\pi$
$272$	−0.397728	−0.0241158
$273$	0	0
$274$	46.6247	2.81670
$275$	3.37005	0.203222
$276$	0	0
$277$	2.92941	0.176011	0.0880055	−	0.996120i	$-0.471951\pi$
$277$	2.92941	0.176011	0.0880055	+	0.996120i	$0.471951\pi$
$278$	−47.8818	−2.87176
$279$	0	0
$280$	6.39532	0.382194
$281$	26.8655	1.60266	0.801332	−	0.598220i	$-0.204125\pi$
$281$	26.8655	1.60266	0.801332	+	0.598220i	$0.204125\pi$
$282$	0	0
$283$	−22.0584	−1.31124	−0.655619	−	0.755092i	$-0.727592\pi$
$283$	−22.0584	−1.31124	−0.655619	+	0.755092i	$0.727592\pi$
$284$	3.00535	0.178335
$285$	0	0
$286$	−52.5162	−3.10535
$287$	−1.77848	−0.104980
$288$	0	0
$289$	−14.9021	−0.876597
$290$	−20.1037	−1.18053
$291$	0	0
$292$	12.2734	0.718248
$293$	−23.7537	−1.38771	−0.693854	−	0.720116i	$-0.744089\pi$
$293$	−23.7537	−1.38771	−0.693854	+	0.720116i	$0.744089\pi$
$294$	0	0
$295$	−33.8496	−1.97080
$296$	10.6733	0.620371
$297$	0	0
$298$	−43.4810	−2.51879
$299$	5.05307	0.292227
$300$	0	0
$301$	−1.25467	−0.0723182
$302$	−41.5209	−2.38926
$303$	0	0
$304$	−0.543726	−0.0311848
$305$	−20.0618	−1.14873
$306$	0	0
$307$	9.61015	0.548480	0.274240	−	0.961661i	$-0.411574\pi$
$307$	9.61015	0.548480	0.274240	+	0.961661i	$0.411574\pi$
$308$	14.5015	0.826298
$309$	0	0
$310$	−18.7882	−1.06710
$311$	−13.5642	−0.769155	−0.384577	−	0.923093i	$-0.625653\pi$
$311$	−13.5642	−0.769155	−0.384577	+	0.923093i	$0.625653\pi$
$312$	0	0
$313$	1.23168	0.0696189	0.0348095	−	0.999394i	$-0.488918\pi$
$313$	1.23168	0.0696189	0.0348095	+	0.999394i	$0.488918\pi$
$314$	−15.0984	−0.852052
$315$	0	0
$316$	13.4539	0.756839
$317$	−16.0584	−0.901931	−0.450965	−	0.892541i	$-0.648920\pi$
$317$	−16.0584	−0.901931	−0.450965	+	0.892541i	$0.648920\pi$
$318$	0	0
$319$	−16.8594	−0.943942
$320$	31.1812	1.74308
$321$	0	0
$322$	−2.27460	−0.126758
$323$	2.86794	0.159576
$324$	0	0
$325$	3.72700	0.206737
$326$	−48.0123	−2.65916
$327$	0	0
$328$	4.74839	0.262186
$329$	−2.87928	−0.158740
$330$	0	0
$331$	−12.5424	−0.689391	−0.344696	−	0.938714i	$-0.612018\pi$
$331$	−12.5424	−0.689391	−0.344696	+	0.938714i	$0.612018\pi$
$332$	−34.3662	−1.88609
$333$	0	0
$334$	−2.30228	−0.125975
$335$	18.5282	1.01230
$336$	0	0
$337$	9.62661	0.524395	0.262197	−	0.965014i	$-0.415553\pi$
$337$	9.62661	0.524395	0.262197	+	0.965014i	$0.415553\pi$
$338$	−28.5088	−1.55068
$339$	0	0
$340$	−11.0111	−0.597161
$341$	−15.7561	−0.853243
$342$	0	0
$343$	1.00000	0.0539949
$344$	3.34988	0.180614
$345$	0	0
$346$	34.5897	1.85955
$347$	9.94199	0.533714	0.266857	−	0.963736i	$-0.414015\pi$
$347$	9.94199	0.533714	0.266857	+	0.963736i	$0.414015\pi$
$348$	0	0
$349$	−12.7274	−0.681283	−0.340641	−	0.940193i	$-0.610644\pi$
$349$	−12.7274	−0.681283	−0.340641	+	0.940193i	$0.610644\pi$
$350$	−1.67768	−0.0896756
$351$	0	0
$352$	27.2523	1.45255
$353$	1.67433	0.0891159	0.0445579	−	0.999007i	$-0.485812\pi$
$353$	1.67433	0.0891159	0.0445579	+	0.999007i	$0.485812\pi$
$354$	0	0
$355$	−2.26819	−0.120383
$356$	−0.832899	−0.0441436
$357$	0	0
$358$	41.2818	2.18181
$359$	36.9662	1.95100	0.975500	−	0.220001i	$-0.0706060\pi$
$359$	36.9662	1.95100	0.975500	+	0.220001i	$0.0706060\pi$
$360$	0	0
$361$	−15.0793	−0.793647
$362$	−24.0804	−1.26564
$363$	0	0
$364$	16.0374	0.840590
$365$	−9.26297	−0.484846
$366$	0	0
$367$	−22.5080	−1.17491	−0.587454	−	0.809258i	$-0.699870\pi$
$367$	−22.5080	−1.17491	−0.587454	+	0.809258i	$0.699870\pi$
$368$	−0.274599	−0.0143144
$369$	0	0
$370$	−21.7805	−1.13231
$371$	10.4128	0.540608
$372$	0	0
$373$	23.5896	1.22142	0.610711	−	0.791853i	$-0.290884\pi$
$373$	23.5896	1.22142	0.610711	+	0.791853i	$0.290884\pi$
$374$	−15.0531	−0.778376
$375$	0	0
$376$	7.68744	0.396449
$377$	−18.6451	−0.960270
$378$	0	0
$379$	6.93664	0.356311	0.178156	−	0.984002i	$-0.442987\pi$
$379$	6.93664	0.356311	0.178156	+	0.984002i	$0.442987\pi$
$380$	−15.0531	−0.772206
$381$	0	0
$382$	−45.0380	−2.30434
$383$	22.8281	1.16646	0.583230	−	0.812307i	$-0.301788\pi$
$383$	22.8281	1.16646	0.583230	+	0.812307i	$0.301788\pi$
$384$	0	0
$385$	−10.9445	−0.557784
$386$	−51.1685	−2.60441
$387$	0	0
$388$	−22.8734	−1.16122
$389$	−16.1350	−0.818077	−0.409039	−	0.912517i	$-0.634136\pi$
$389$	−16.1350	−0.818077	−0.409039	+	0.912517i	$0.634136\pi$
$390$	0	0
$391$	1.44840	0.0732486
$392$	−2.66992	−0.134851
$393$	0	0
$394$	3.80628	0.191757
$395$	−10.1539	−0.510897
$396$	0	0
$397$	18.5687	0.931938	0.465969	−	0.884801i	$-0.345706\pi$
$397$	18.5687	0.931938	0.465969	+	0.884801i	$0.345706\pi$
$398$	−13.2468	−0.664002
$399$	0	0
$400$	−0.202536	−0.0101268
$401$	7.22928	0.361013	0.180506	−	0.983574i	$-0.442226\pi$
$401$	7.22928	0.361013	0.180506	+	0.983574i	$0.442226\pi$
$402$	0	0
$403$	−17.4250	−0.868002
$404$	26.2373	1.30535
$405$	0	0
$406$	8.39292	0.416533
$407$	−18.2655	−0.905387
$408$	0	0
$409$	17.7912	0.879717	0.439859	−	0.898067i	$-0.355029\pi$
$409$	17.7912	0.879717	0.439859	+	0.898067i	$0.355029\pi$
$410$	−9.68985	−0.478547
$411$	0	0
$412$	−61.1334	−3.01183
$413$	14.1315	0.695368
$414$	0	0
$415$	25.9368	1.27319
$416$	30.1388	1.47768
$417$	0	0
$418$	−20.5788	−1.00654
$419$	16.7937	0.820426	0.410213	−	0.911990i	$-0.365454\pi$
$419$	16.7937	0.820426	0.410213	+	0.911990i	$0.365454\pi$
$420$	0	0
$421$	−11.1194	−0.541925	−0.270963	−	0.962590i	$-0.587342\pi$
$421$	−11.1194	−0.541925	−0.270963	+	0.962590i	$0.587342\pi$
$422$	32.4833	1.58126
$423$	0	0
$424$	−27.8015	−1.35016
$425$	1.06829	0.0518199
$426$	0	0
$427$	8.37540	0.405314
$428$	35.0802	1.69567
$429$	0	0
$430$	−6.83596	−0.329659
$431$	−35.8516	−1.72691	−0.863456	−	0.504424i	$-0.831705\pi$
$431$	−35.8516	−1.72691	−0.863456	+	0.504424i	$0.831705\pi$
$432$	0	0
$433$	13.2199	0.635310	0.317655	−	0.948206i	$-0.397105\pi$
$433$	13.2199	0.635310	0.317655	+	0.948206i	$0.397105\pi$
$434$	7.84372	0.376511
$435$	0	0
$436$	40.9568	1.96148
$437$	1.98008	0.0947199
$438$	0	0
$439$	−28.2898	−1.35020	−0.675100	−	0.737726i	$-0.735900\pi$
$439$	−28.2898	−1.35020	−0.675100	+	0.737726i	$0.735900\pi$
$440$	29.2210	1.39306
$441$	0	0
$442$	−16.6475	−0.791839
$443$	−23.7781	−1.12973	−0.564865	−	0.825183i	$-0.691072\pi$
$443$	−23.7781	−1.12973	−0.564865	+	0.825183i	$0.691072\pi$
$444$	0	0
$445$	0.628604	0.0297987
$446$	45.4989	2.15444
$447$	0	0
$448$	−13.0175	−0.615020
$449$	2.97578	0.140436	0.0702179	−	0.997532i	$-0.477631\pi$
$449$	2.97578	0.140436	0.0702179	+	0.997532i	$0.477631\pi$
$450$	0	0
$451$	−8.12607	−0.382642
$452$	3.98972	0.187661
$453$	0	0
$454$	41.9770	1.97008
$455$	−12.1037	−0.567432
$456$	0	0
$457$	−19.4381	−0.909277	−0.454638	−	0.890676i	$-0.650231\pi$
$457$	−19.4381	−0.909277	−0.454638	+	0.890676i	$0.650231\pi$
$458$	39.1912	1.83128
$459$	0	0
$460$	−7.60227	−0.354458
$461$	4.92877	0.229556	0.114778	−	0.993391i	$-0.463384\pi$
$461$	4.92877	0.229556	0.114778	+	0.993391i	$0.463384\pi$
$462$	0	0
$463$	3.27983	0.152427	0.0762133	−	0.997092i	$-0.475717\pi$
$463$	3.27983	0.152427	0.0762133	+	0.997092i	$0.475717\pi$
$464$	1.01323	0.0470379
$465$	0	0
$466$	−37.0404	−1.71586
$467$	−3.04520	−0.140915	−0.0704575	−	0.997515i	$-0.522446\pi$
$467$	−3.04520	−0.140915	−0.0704575	+	0.997515i	$0.522446\pi$
$468$	0	0
$469$	−7.73517	−0.357177
$470$	−15.6874	−0.723608
$471$	0	0
$472$	−37.7301	−1.73667
$473$	−5.73276	−0.263593
$474$	0	0
$475$	1.46045	0.0670098
$476$	4.59692	0.210700
$477$	0	0
$478$	−59.4896	−2.72099
$479$	−3.10080	−0.141679	−0.0708396	−	0.997488i	$-0.522568\pi$
$479$	−3.10080	−0.141679	−0.0708396	+	0.997488i	$0.522568\pi$
$480$	0	0
$481$	−20.2001	−0.921047
$482$	−19.7328	−0.898803
$483$	0	0
$484$	31.3472	1.42487
$485$	17.2630	0.783871
$486$	0	0
$487$	17.5634	0.795872	0.397936	−	0.917413i	$-0.369727\pi$
$487$	17.5634	0.795872	0.397936	+	0.917413i	$0.369727\pi$
$488$	−22.3617	−1.01226
$489$	0	0
$490$	5.44840	0.246134
$491$	−32.6646	−1.47413	−0.737066	−	0.675821i	$-0.763789\pi$
$491$	−32.6646	−1.47413	−0.737066	+	0.675821i	$0.763789\pi$
$492$	0	0
$493$	−5.34436	−0.240698
$494$	−22.7584	−1.02395
$495$	0	0
$496$	0.946925	0.0425182
$497$	0.946925	0.0424754
$498$	0	0
$499$	−16.9084	−0.756926	−0.378463	−	0.925616i	$-0.623547\pi$
$499$	−16.9084	−0.756926	−0.378463	+	0.925616i	$0.623547\pi$
$500$	32.4041	1.44916
$501$	0	0
$502$	20.9920	0.936919
$503$	9.22582	0.411359	0.205679	−	0.978619i	$-0.434060\pi$
$503$	9.22582	0.411359	0.205679	+	0.978619i	$0.434060\pi$
$504$	0	0
$505$	−19.8018	−0.881167
$506$	−10.3929	−0.462022
$507$	0	0
$508$	61.3765	2.72314
$509$	12.4362	0.551226	0.275613	−	0.961269i	$-0.411119\pi$
$509$	12.4362	0.551226	0.275613	+	0.961269i	$0.411119\pi$
$510$	0	0
$511$	3.86711	0.171071
$512$	−3.10414	−0.137185
$513$	0	0
$514$	40.3981	1.78189
$515$	46.1385	2.03310
$516$	0	0
$517$	−13.1558	−0.578590
$518$	9.09292	0.399520
$519$	0	0
$520$	32.3160	1.41715
$521$	26.2123	1.14838	0.574191	−	0.818721i	$-0.305317\pi$
$521$	26.2123	1.14838	0.574191	+	0.818721i	$0.305317\pi$
$522$	0	0
$523$	25.3309	1.10764	0.553822	−	0.832635i	$-0.313169\pi$
$523$	25.3309	1.10764	0.553822	+	0.832635i	$0.313169\pi$
$524$	−61.7809	−2.69891
$525$	0	0
$526$	−23.3648	−1.01876
$527$	−4.99465	−0.217570
$528$	0	0
$529$	1.00000	0.0434783
$530$	56.7333	2.46434
$531$	0	0
$532$	6.28436	0.272462
$533$	−8.98677	−0.389260
$534$	0	0
$535$	−26.4757	−1.14464
$536$	20.6523	0.892043
$537$	0	0
$538$	−39.8764	−1.71919
$539$	4.56912	0.196806
$540$	0	0
$541$	−16.2940	−0.700534	−0.350267	−	0.936650i	$-0.613909\pi$
$541$	−16.2940	−0.700534	−0.350267	+	0.936650i	$0.613909\pi$
$542$	−48.1309	−2.06740
$543$	0	0
$544$	8.63889	0.370389
$545$	−30.9109	−1.32408
$546$	0	0
$547$	13.1461	0.562087	0.281044	−	0.959695i	$-0.409319\pi$
$547$	13.1461	0.562087	0.281044	+	0.959695i	$0.409319\pi$
$548$	−65.0566	−2.77908
$549$	0	0
$550$	−7.66551	−0.326858
$551$	−7.30617	−0.311253
$552$	0	0
$553$	4.23904	0.180262
$554$	−6.66322	−0.283093
$555$	0	0
$556$	66.8106	2.83340
$557$	−1.28424	−0.0544150	−0.0272075	−	0.999630i	$-0.508661\pi$
$557$	−1.28424	−0.0544150	−0.0272075	+	0.999630i	$0.508661\pi$
$558$	0	0
$559$	−6.33996	−0.268152
$560$	0.657752	0.0277951
$561$	0	0
$562$	−61.1083	−2.57770
$563$	1.02181	0.0430642	0.0215321	−	0.999768i	$-0.493146\pi$
$563$	1.02181	0.0430642	0.0215321	+	0.999768i	$0.493146\pi$
$564$	0	0
$565$	−3.01111	−0.126678
$566$	50.1741	2.10897
$567$	0	0
$568$	−2.52822	−0.106082
$569$	14.9943	0.628592	0.314296	−	0.949325i	$-0.398232\pi$
$569$	14.9943	0.628592	0.314296	+	0.949325i	$0.398232\pi$
$570$	0	0
$571$	−15.3179	−0.641035	−0.320517	−	0.947243i	$-0.603857\pi$
$571$	−15.3179	−0.641035	−0.320517	+	0.947243i	$0.603857\pi$
$572$	73.2770	3.06387
$573$	0	0
$574$	4.04532	0.168848
$575$	0.737570	0.0307588
$576$	0	0
$577$	−36.4091	−1.51573	−0.757865	−	0.652411i	$-0.773757\pi$
$577$	−36.4091	−1.51573	−0.757865	+	0.652411i	$0.773757\pi$
$578$	33.8964	1.40990
$579$	0	0
$580$	28.0512	1.16476
$581$	−10.8281	−0.449225
$582$	0	0
$583$	47.5775	1.97046
$584$	−10.3249	−0.427246
$585$	0	0
$586$	54.0302	2.23197
$587$	32.1484	1.32691	0.663454	−	0.748217i	$-0.269090\pi$
$587$	32.1484	1.32691	0.663454	+	0.748217i	$0.269090\pi$
$588$	0	0
$589$	−6.82809	−0.281346
$590$	76.9943	3.16980
$591$	0	0
$592$	1.09773	0.0451166
$593$	−31.6719	−1.30061	−0.650305	−	0.759673i	$-0.725359\pi$
$593$	−31.6719	−1.30061	−0.650305	+	0.759673i	$0.725359\pi$
$594$	0	0
$595$	−3.46938	−0.142231
$596$	60.6701	2.48514
$597$	0	0
$598$	−11.4937	−0.470013
$599$	36.0114	1.47138	0.735692	−	0.677316i	$-0.236857\pi$
$599$	36.0114	1.47138	0.735692	+	0.677316i	$0.236857\pi$
$600$	0	0
$601$	−33.4295	−1.36362	−0.681810	−	0.731530i	$-0.738807\pi$
$601$	−33.4295	−1.36362	−0.681810	+	0.731530i	$0.738807\pi$
$602$	2.85388	0.116315
$603$	0	0
$604$	57.9350	2.35734
$605$	−23.6583	−0.961846
$606$	0	0
$607$	0.510803	0.0207329	0.0103664	−	0.999946i	$-0.496700\pi$
$607$	0.510803	0.0207329	0.0103664	+	0.999946i	$0.496700\pi$
$608$	11.8101	0.478961
$609$	0	0
$610$	45.6325	1.84761
$611$	−14.5492	−0.588598
$612$	0	0
$613$	−29.7444	−1.20137	−0.600683	−	0.799488i	$-0.705104\pi$
$613$	−29.7444	−1.20137	−0.600683	+	0.799488i	$0.705104\pi$
$614$	−21.8592	−0.882167
$615$	0	0
$616$	−12.1992	−0.491520
$617$	18.2312	0.733959	0.366980	−	0.930229i	$-0.380392\pi$
$617$	18.2312	0.733959	0.366980	+	0.930229i	$0.380392\pi$
$618$	0	0
$619$	31.3889	1.26163	0.630814	−	0.775934i	$-0.282721\pi$
$619$	31.3889	1.26163	0.630814	+	0.775934i	$0.282721\pi$
$620$	26.2157	1.05285
$621$	0	0
$622$	30.8531	1.23710
$623$	−0.262430	−0.0105140
$624$	0	0
$625$	−28.1438	−1.12575
$626$	−2.80159	−0.111974
$627$	0	0
$628$	21.0671	0.840670
$629$	−5.79011	−0.230867
$630$	0	0
$631$	42.9030	1.70794	0.853970	−	0.520322i	$-0.174188\pi$
$631$	42.9030	1.70794	0.853970	+	0.520322i	$0.174188\pi$
$632$	−11.3179	−0.450202
$633$	0	0
$634$	36.5265	1.45065
$635$	−46.3219	−1.83823
$636$	0	0
$637$	5.05307	0.200210
$638$	38.3483	1.51822
$639$	0	0
$640$	−42.3511	−1.67407
$641$	19.8505	0.784049	0.392025	−	0.919955i	$-0.371775\pi$
$641$	19.8505	0.784049	0.392025	+	0.919955i	$0.371775\pi$
$642$	0	0
$643$	26.5457	1.04686	0.523431	−	0.852068i	$-0.324652\pi$
$643$	26.5457	1.04686	0.523431	+	0.852068i	$0.324652\pi$
$644$	3.17380	0.125065
$645$	0	0
$646$	−6.52340	−0.256660
$647$	−30.0117	−1.17988	−0.589940	−	0.807447i	$-0.700849\pi$
$647$	−30.0117	−1.17988	−0.589940	+	0.807447i	$0.700849\pi$
$648$	0	0
$649$	64.5687	2.53454
$650$	−8.47742	−0.332512
$651$	0	0
$652$	66.9927	2.62364
$653$	−4.41083	−0.172609	−0.0863046	−	0.996269i	$-0.527506\pi$
$653$	−4.41083	−0.172609	−0.0863046	+	0.996269i	$0.527506\pi$
$654$	0	0
$655$	46.6271	1.82187
$656$	0.488367	0.0190675
$657$	0	0
$658$	6.54920	0.255314
$659$	30.2147	1.17700	0.588499	−	0.808498i	$-0.299719\pi$
$659$	30.2147	1.17700	0.588499	+	0.808498i	$0.299719\pi$
$660$	0	0
$661$	18.8322	0.732488	0.366244	−	0.930519i	$-0.380644\pi$
$661$	18.8322	0.732488	0.366244	+	0.930519i	$0.380644\pi$
$662$	28.5289	1.10881
$663$	0	0
$664$	28.9101	1.12193
$665$	−4.74292	−0.183923
$666$	0	0
$667$	−3.68985	−0.142871
$668$	3.21242	0.124292
$669$	0	0
$670$	−42.1443	−1.62817
$671$	38.2682	1.47733
$672$	0	0
$673$	−17.0208	−0.656102	−0.328051	−	0.944660i	$-0.606392\pi$
$673$	−17.0208	−0.656102	−0.328051	+	0.944660i	$0.606392\pi$
$674$	−21.8967	−0.843428
$675$	0	0
$676$	39.7790	1.52996
$677$	−33.4116	−1.28411	−0.642056	−	0.766657i	$-0.721918\pi$
$677$	−33.4116	−1.28411	−0.642056	+	0.766657i	$0.721918\pi$
$678$	0	0
$679$	−7.20695	−0.276577
$680$	9.26297	0.355219
$681$	0	0
$682$	35.8389	1.37234
$683$	9.95082	0.380758	0.190379	−	0.981711i	$-0.439028\pi$
$683$	9.95082	0.380758	0.190379	+	0.981711i	$0.439028\pi$
$684$	0	0
$685$	49.0993	1.87599
$686$	−2.27460	−0.0868446
$687$	0	0
$688$	0.344532	0.0131352
$689$	52.6169	2.00454
$690$	0	0
$691$	21.7795	0.828533	0.414266	−	0.910156i	$-0.364038\pi$
$691$	21.7795	0.828533	0.414266	+	0.910156i	$0.364038\pi$
$692$	−48.2638	−1.83471
$693$	0	0
$694$	−22.6140	−0.858417
$695$	−50.4231	−1.91266
$696$	0	0
$697$	−2.57594	−0.0975707
$698$	28.9497	1.09576
$699$	0	0
$700$	2.34090	0.0884777
$701$	−17.4292	−0.658291	−0.329146	−	0.944279i	$-0.606761\pi$
$701$	−17.4292	−0.658291	−0.329146	+	0.944279i	$0.606761\pi$
$702$	0	0
$703$	−7.91554	−0.298540
$704$	−59.4786	−2.24169
$705$	0	0
$706$	−3.80844	−0.143333
$707$	8.26684	0.310907
$708$	0	0
$709$	29.6266	1.11265	0.556325	−	0.830965i	$-0.312211\pi$
$709$	29.6266	1.11265	0.556325	+	0.830965i	$0.312211\pi$
$710$	5.15922	0.193622
$711$	0	0
$712$	0.700667	0.0262586
$713$	−3.44840	−0.129143
$714$	0	0
$715$	−55.3035	−2.06823
$716$	−57.6015	−2.15267
$717$	0	0
$718$	−84.0832	−3.13796
$719$	17.1492	0.639558	0.319779	−	0.947492i	$-0.396391\pi$
$719$	17.1492	0.639558	0.319779	+	0.947492i	$0.396391\pi$
$720$	0	0
$721$	−19.2619	−0.717351
$722$	34.2994	1.27649
$723$	0	0
$724$	33.5999	1.24873
$725$	−2.72152	−0.101075
$726$	0	0
$727$	48.6199	1.80321	0.901606	−	0.432557i	$-0.142389\pi$
$727$	48.6199	1.80321	0.901606	+	0.432557i	$0.142389\pi$
$728$	−13.4913	−0.500021
$729$	0	0
$730$	21.0695	0.779819
$731$	−1.81727	−0.0672141
$732$	0	0
$733$	−39.3216	−1.45238	−0.726189	−	0.687495i	$-0.758710\pi$
$733$	−39.3216	−1.45238	−0.726189	+	0.687495i	$0.758710\pi$
$734$	51.1967	1.88970
$735$	0	0
$736$	5.96444	0.219852
$737$	−35.3429	−1.30187
$738$	0	0
$739$	20.5818	0.757115	0.378557	−	0.925578i	$-0.376420\pi$
$739$	20.5818	0.757115	0.378557	+	0.925578i	$0.376420\pi$
$740$	30.3908	1.11719
$741$	0	0
$742$	−23.6850	−0.869505
$743$	38.0313	1.39523	0.697616	−	0.716472i	$-0.254244\pi$
$743$	38.0313	1.39523	0.697616	+	0.716472i	$0.254244\pi$
$744$	0	0
$745$	−45.7888	−1.67757
$746$	−53.6568	−1.96452
$747$	0	0
$748$	21.0039	0.767978
$749$	11.0531	0.403871
$750$	0	0
$751$	−38.3146	−1.39812	−0.699060	−	0.715063i	$-0.746398\pi$
$751$	−38.3146	−1.39812	−0.699060	+	0.715063i	$0.746398\pi$
$752$	0.790645	0.0288319
$753$	0	0
$754$	42.4100	1.54448
$755$	−43.7246	−1.59130
$756$	0	0
$757$	−25.4278	−0.924190	−0.462095	−	0.886830i	$-0.652902\pi$
$757$	−25.4278	−0.924190	−0.462095	+	0.886830i	$0.652902\pi$
$758$	−15.7781	−0.573086
$759$	0	0
$760$	12.6632	0.459343
$761$	−46.3205	−1.67912	−0.839558	−	0.543271i	$-0.817186\pi$
$761$	−46.3205	−1.67912	−0.839558	+	0.543271i	$0.817186\pi$
$762$	0	0
$763$	12.9047	0.467180
$764$	62.8425	2.27356
$765$	0	0
$766$	−51.9247	−1.87612
$767$	71.4078	2.57838
$768$	0	0
$769$	2.92806	0.105588	0.0527942	−	0.998605i	$-0.483187\pi$
$769$	2.92806	0.105588	0.0527942	+	0.998605i	$0.483187\pi$
$770$	24.8944	0.897132
$771$	0	0
$772$	71.3965	2.56962
$773$	−0.848652	−0.0305239	−0.0152619	−	0.999884i	$-0.504858\pi$
$773$	−0.848652	−0.0305239	−0.0152619	+	0.999884i	$0.504858\pi$
$774$	0	0
$775$	−2.54344	−0.0913629
$776$	19.2420	0.690747
$777$	0	0
$778$	36.7007	1.31578
$779$	−3.52152	−0.126171
$780$	0	0
$781$	4.32662	0.154818
$782$	−3.29452	−0.117812
$783$	0	0
$784$	−0.274599	−0.00980709
$785$	−15.8997	−0.567486
$786$	0	0
$787$	31.3118	1.11614	0.558072	−	0.829793i	$-0.311541\pi$
$787$	31.3118	1.11614	0.558072	+	0.829793i	$0.311541\pi$
$788$	−5.31098	−0.189196
$789$	0	0
$790$	23.0960	0.821718
$791$	1.25708	0.0446966
$792$	0	0
$793$	42.3215	1.50288
$794$	−42.2364	−1.49891
$795$	0	0
$796$	18.4836	0.655132
$797$	−4.66393	−0.165205	−0.0826025	−	0.996583i	$-0.526323\pi$
$797$	−4.66393	−0.165205	−0.0826025	+	0.996583i	$0.526323\pi$
$798$	0	0
$799$	−4.17034	−0.147536
$800$	4.39920	0.155535
$801$	0	0
$802$	−16.4437	−0.580648
$803$	17.6693	0.623535
$804$	0	0
$805$	−2.39532	−0.0844240
$806$	39.6349	1.39608
$807$	0	0
$808$	−22.0718	−0.776484
$809$	−26.2106	−0.921516	−0.460758	−	0.887526i	$-0.652422\pi$
$809$	−26.2106	−0.921516	−0.460758	+	0.887526i	$0.652422\pi$
$810$	0	0
$811$	−35.2107	−1.23642	−0.618208	−	0.786015i	$-0.712141\pi$
$811$	−35.2107	−1.23642	−0.618208	+	0.786015i	$0.712141\pi$
$812$	−11.7108	−0.410969
$813$	0	0
$814$	41.5467	1.45621
$815$	−50.5606	−1.77106
$816$	0	0
$817$	−2.48435	−0.0869164
$818$	−40.4678	−1.41492
$819$	0	0
$820$	13.5205	0.472155
$821$	−30.9369	−1.07970	−0.539852	−	0.841760i	$-0.681520\pi$
$821$	−30.9369	−1.07970	−0.539852	+	0.841760i	$0.681520\pi$
$822$	0	0
$823$	−17.7770	−0.619668	−0.309834	−	0.950791i	$-0.600273\pi$
$823$	−17.7770	−0.619668	−0.309834	+	0.950791i	$0.600273\pi$
$824$	51.4278	1.79157
$825$	0	0
$826$	−32.1436	−1.11842
$827$	3.40509	0.118406	0.0592032	−	0.998246i	$-0.481144\pi$
$827$	3.40509	0.118406	0.0592032	+	0.998246i	$0.481144\pi$
$828$	0	0
$829$	0.475680	0.0165210	0.00826052	−	0.999966i	$-0.497371\pi$
$829$	0.475680	0.0165210	0.00826052	+	0.999966i	$0.497371\pi$
$830$	−58.9957	−2.04777
$831$	0	0
$832$	−65.7785	−2.28046
$833$	1.44840	0.0501840
$834$	0	0
$835$	−2.42447	−0.0839023
$836$	28.7140	0.993095
$837$	0	0
$838$	−38.1990	−1.31956
$839$	15.4804	0.534442	0.267221	−	0.963635i	$-0.413895\pi$
$839$	15.4804	0.534442	0.267221	+	0.963635i	$0.413895\pi$
$840$	0	0
$841$	−15.3850	−0.530519
$842$	25.2921	0.871624
$843$	0	0
$844$	−45.3247	−1.56014
$845$	−30.0219	−1.03279
$846$	0	0
$847$	9.87687	0.339373
$848$	−2.85935	−0.0981905
$849$	0	0
$850$	−2.42994	−0.0833463
$851$	−3.99759	−0.137036
$852$	0	0
$853$	23.6583	0.810044	0.405022	−	0.914307i	$-0.367264\pi$
$853$	23.6583	0.810044	0.405022	+	0.914307i	$0.367264\pi$
$854$	−19.0507	−0.651900
$855$	0	0
$856$	−29.5108	−1.00866
$857$	10.2346	0.349608	0.174804	−	0.984603i	$-0.444071\pi$
$857$	10.2346	0.349608	0.174804	+	0.984603i	$0.444071\pi$
$858$	0	0
$859$	−38.4043	−1.31034	−0.655169	−	0.755483i	$-0.727402\pi$
$859$	−38.4043	−1.31034	−0.655169	+	0.755483i	$0.727402\pi$
$860$	9.53838	0.325256
$861$	0	0
$862$	81.5480	2.77754
$863$	−33.5209	−1.14106	−0.570532	−	0.821275i	$-0.693263\pi$
$863$	−33.5209	−1.14106	−0.570532	+	0.821275i	$0.693263\pi$
$864$	0	0
$865$	36.4255	1.23851
$866$	−30.0701	−1.02182
$867$	0	0
$868$	−10.9445	−0.371481
$869$	19.3687	0.657038
$870$	0	0
$871$	−39.0864	−1.32439
$872$	−34.4545	−1.16678
$873$	0	0
$874$	−4.50388	−0.152346
$875$	10.2099	0.345157
$876$	0	0
$877$	−52.7392	−1.78088	−0.890438	−	0.455105i	$-0.849602\pi$
$877$	−52.7392	−1.78088	−0.890438	+	0.455105i	$0.849602\pi$
$878$	64.3480	2.17164
$879$	0	0
$880$	3.00535	0.101310
$881$	−19.1890	−0.646495	−0.323247	−	0.946314i	$-0.604775\pi$
$881$	−19.1890	−0.646495	−0.323247	+	0.946314i	$0.604775\pi$
$882$	0	0
$883$	−33.2985	−1.12059	−0.560293	−	0.828295i	$-0.689311\pi$
$883$	−33.2985	−1.12059	−0.560293	+	0.828295i	$0.689311\pi$
$884$	23.2286	0.781262
$885$	0	0
$886$	54.0856	1.81704
$887$	17.8750	0.600183	0.300092	−	0.953910i	$-0.402983\pi$
$887$	17.8750	0.600183	0.300092	+	0.953910i	$0.402983\pi$
$888$	0	0
$889$	19.3385	0.648592
$890$	−1.42982	−0.0479277
$891$	0	0
$892$	−63.4857	−2.12566
$893$	−5.70118	−0.190783
$894$	0	0
$895$	43.4729	1.45314
$896$	17.6807	0.590672
$897$	0	0
$898$	−6.76871	−0.225875
$899$	12.7241	0.424371
$900$	0	0
$901$	15.0819	0.502452
$902$	18.4836	0.615435
$903$	0	0
$904$	−3.35630	−0.111629
$905$	−25.3584	−0.842942
$906$	0	0
$907$	6.71166	0.222857	0.111428	−	0.993772i	$-0.464457\pi$
$907$	6.71166	0.222857	0.111428	+	0.993772i	$0.464457\pi$
$908$	−58.5715	−1.94376
$909$	0	0
$910$	27.5312	0.912649
$911$	14.5404	0.481744	0.240872	−	0.970557i	$-0.422567\pi$
$911$	14.5404	0.481744	0.240872	+	0.970557i	$0.422567\pi$
$912$	0	0
$913$	−49.4749	−1.63738
$914$	44.2139	1.46247
$915$	0	0
$916$	−54.6844	−1.80682
$917$	−19.4659	−0.642821
$918$	0	0
$919$	25.3532	0.836325	0.418162	−	0.908372i	$-0.362674\pi$
$919$	25.3532	0.836325	0.418162	+	0.908372i	$0.362674\pi$
$920$	6.39532	0.210848
$921$	0	0
$922$	−11.2110	−0.369214
$923$	4.78488	0.157496
$924$	0	0
$925$	−2.94851	−0.0969463
$926$	−7.46029	−0.245160
$927$	0	0
$928$	−22.0079	−0.722444
$929$	−46.6378	−1.53014	−0.765069	−	0.643948i	$-0.777295\pi$
$929$	−46.6378	−1.53014	−0.765069	+	0.643948i	$0.777295\pi$
$930$	0	0
$931$	1.98008	0.0648944
$932$	51.6833	1.69294
$933$	0	0
$934$	6.92660	0.226645
$935$	−15.8520	−0.518416
$936$	0	0
$937$	−34.8699	−1.13915	−0.569576	−	0.821939i	$-0.692892\pi$
$937$	−34.8699	−1.13915	−0.569576	+	0.821939i	$0.692892\pi$
$938$	17.5944	0.574477
$939$	0	0
$940$	21.8890	0.713942
$941$	12.0243	0.391982	0.195991	−	0.980606i	$-0.437208\pi$
$941$	12.0243	0.391982	0.195991	+	0.980606i	$0.437208\pi$
$942$	0	0
$943$	−1.77848	−0.0579152
$944$	−3.88050	−0.126300
$945$	0	0
$946$	13.0397	0.423958
$947$	18.4626	0.599953	0.299977	−	0.953947i	$-0.403021\pi$
$947$	18.4626	0.599953	0.299977	+	0.953947i	$0.403021\pi$
$948$	0	0
$949$	19.5408	0.634321
$950$	−3.32193	−0.107778
$951$	0	0
$952$	−3.86711	−0.125334
$953$	−8.91601	−0.288818	−0.144409	−	0.989518i	$-0.546128\pi$
$953$	−8.91601	−0.288818	−0.144409	+	0.989518i	$0.546128\pi$
$954$	0	0
$955$	−47.4284	−1.53475
$956$	83.0071	2.68464
$957$	0	0
$958$	7.05307	0.227875
$959$	−20.4980	−0.661915
$960$	0	0
$961$	−19.1086	−0.616405
$962$	45.9472	1.48140
$963$	0	0
$964$	27.5336	0.886796
$965$	−53.8842	−1.73459
$966$	0	0
$967$	−11.1091	−0.357244	−0.178622	−	0.983918i	$-0.557164\pi$
$967$	−11.1091	−0.357244	−0.178622	+	0.983918i	$0.557164\pi$
$968$	−26.3705	−0.847579
$969$	0	0
$970$	−39.2663	−1.26077
$971$	−18.9226	−0.607255	−0.303627	−	0.952791i	$-0.598198\pi$
$971$	−18.9226	−0.607255	−0.303627	+	0.952791i	$0.598198\pi$
$972$	0	0
$973$	21.0507	0.674853
$974$	−39.9496	−1.28007
$975$	0	0
$976$	−2.29987	−0.0736171
$977$	−17.0219	−0.544579	−0.272290	−	0.962215i	$-0.587781\pi$
$977$	−17.0219	−0.544579	−0.272290	+	0.962215i	$0.587781\pi$
$978$	0	0
$979$	−1.19907	−0.0383225
$980$	−7.60227	−0.242846
$981$	0	0
$982$	74.2988	2.37097
$983$	−37.7249	−1.20324	−0.601618	−	0.798784i	$-0.705477\pi$
$983$	−37.7249	−1.20324	−0.601618	+	0.798784i	$0.705477\pi$
$984$	0	0
$985$	4.00829	0.127715
$986$	12.1563	0.387135
$987$	0	0
$988$	31.7554	1.01027
$989$	−1.25467	−0.0398963
$990$	0	0
$991$	−51.0190	−1.62067	−0.810336	−	0.585965i	$-0.800715\pi$
$991$	−51.0190	−1.62067	−0.810336	+	0.585965i	$0.800715\pi$
$992$	−20.5678	−0.653027
$993$	0	0
$994$	−2.15387	−0.0683168
$995$	−13.9499	−0.442241
$996$	0	0
$997$	−43.6823	−1.38343	−0.691717	−	0.722169i	$-0.743145\pi$
$997$	−43.6823	−1.38343	−0.691717	+	0.722169i	$0.743145\pi$
$998$	38.4599	1.21743
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1449.2.a.p.1.1		4
3.2	odd	2		483.2.a.i.1.4	✓	4
12.11	even	2		7728.2.a.cd.1.3		4
21.20	even	2		3381.2.a.w.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.i.1.4	✓	4	3.2	odd	2
1449.2.a.p.1.1		4	1.1	even	1	trivial
3381.2.a.w.1.4		4	21.20	even	2
7728.2.a.cd.1.3		4	12.11	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:	\( N \)	\(=\)	\( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1449.a (trivial)

\(f(q)\)	\(=\)	\(q-2.27460 q^{2} +3.17380 q^{4} -2.39532 q^{5} +1.00000 q^{7} -2.66992 q^{8} +O(q^{10})\)	\(q-2.27460 q^{2} +3.17380 q^{4} -2.39532 q^{5} +1.00000 q^{7} -2.66992 q^{8} +5.44840 q^{10} +4.56912 q^{11} +5.05307 q^{13} -2.27460 q^{14} -0.274599 q^{16} +1.44840 q^{17} +1.98008 q^{19} -7.60227 q^{20} -10.3929 q^{22} +1.00000 q^{23} +0.737570 q^{25} -11.4937 q^{26} +3.17380 q^{28} -3.68985 q^{29} -3.44840 q^{31} +5.96444 q^{32} -3.29452 q^{34} -2.39532 q^{35} -3.99759 q^{37} -4.50388 q^{38} +6.39532 q^{40} -1.77848 q^{41} -1.25467 q^{43} +14.5015 q^{44} -2.27460 q^{46} -2.87928 q^{47} +1.00000 q^{49} -1.67768 q^{50} +16.0374 q^{52} +10.4128 q^{53} -10.9445 q^{55} -2.66992 q^{56} +8.39292 q^{58} +14.1315 q^{59} +8.37540 q^{61} +7.84372 q^{62} -13.0175 q^{64} -12.1037 q^{65} -7.73517 q^{67} +4.59692 q^{68} +5.44840 q^{70} +0.946925 q^{71} +3.86711 q^{73} +9.09292 q^{74} +6.28436 q^{76} +4.56912 q^{77} +4.23904 q^{79} +0.657752 q^{80} +4.04532 q^{82} -10.8281 q^{83} -3.46938 q^{85} +2.85388 q^{86} -12.1992 q^{88} -0.262430 q^{89} +5.05307 q^{91} +3.17380 q^{92} +6.54920 q^{94} -4.74292 q^{95} -7.20695 q^{97} -2.27460 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 5 q^{5} + 4 q^{7} + 3 q^{8}+O(q^{10}) \) 4 * q + 4 * q^4 - 5 * q^5 + 4 * q^7 + 3 * q^8	\( 4 q + 4 q^{4} - 5 q^{5} + 4 q^{7} + 3 q^{8} + 4 q^{10} + 5 q^{11} + 7 q^{13} + 8 q^{16} - 12 q^{17} + 3 q^{19} + q^{20} - q^{22} + 4 q^{23} + 7 q^{25} - 5 q^{26} + 4 q^{28} - 6 q^{29} + 4 q^{31} + 6 q^{32} - 9 q^{34} - 5 q^{35} + 20 q^{37} - 23 q^{38} + 21 q^{40} - 3 q^{41} + 9 q^{43} + 27 q^{44} - 7 q^{47} + 4 q^{49} - 3 q^{50} + 38 q^{52} + 6 q^{53} - 21 q^{55} + 3 q^{56} - 7 q^{58} + 2 q^{59} + 24 q^{61} + 9 q^{62} - 21 q^{64} + 14 q^{65} + q^{67} + 13 q^{68} + 4 q^{70} + 17 q^{71} + 16 q^{73} + 33 q^{74} - 25 q^{76} + 5 q^{77} - 10 q^{79} - 6 q^{80} - 7 q^{82} - 8 q^{83} + 17 q^{85} + 35 q^{86} - 12 q^{88} + 3 q^{89} + 7 q^{91} + 4 q^{92} + 8 q^{94} + 3 q^{95} - 2 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 - 5 * q^5 + 4 * q^7 + 3 * q^8 + 4 * q^10 + 5 * q^11 + 7 * q^13 + 8 * q^16 - 12 * q^17 + 3 * q^19 + q^20 - q^22 + 4 * q^23 + 7 * q^25 - 5 * q^26 + 4 * q^28 - 6 * q^29 + 4 * q^31 + 6 * q^32 - 9 * q^34 - 5 * q^35 + 20 * q^37 - 23 * q^38 + 21 * q^40 - 3 * q^41 + 9 * q^43 + 27 * q^44 - 7 * q^47 + 4 * q^49 - 3 * q^50 + 38 * q^52 + 6 * q^53 - 21 * q^55 + 3 * q^56 - 7 * q^58 + 2 * q^59 + 24 * q^61 + 9 * q^62 - 21 * q^64 + 14 * q^65 + q^67 + 13 * q^68 + 4 * q^70 + 17 * q^71 + 16 * q^73 + 33 * q^74 - 25 * q^76 + 5 * q^77 - 10 * q^79 - 6 * q^80 - 7 * q^82 - 8 * q^83 + 17 * q^85 + 35 * q^86 - 12 * q^88 + 3 * q^89 + 7 * q^91 + 4 * q^92 + 8 * q^94 + 3 * q^95 - 2 * q^97

Introduction

L-functions

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