LMFDB - Embedded newform 4304.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4304,2,Mod(1,4304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4304, 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("4304.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4304 = 2^{4} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4304.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:	$34.3676130300$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.4913.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + x + 1 $ x^4 - x^3 - 6*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 538)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.487928$ of defining polynomial
Character	$\chi$	$=$	4304.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.34415 q^{3} +3.90570 q^{5} +0.487928 q^{7} -1.19326 q^{9} +O(q^{10})$	$q-1.34415 q^{3} +3.90570 q^{5} +0.487928 q^{7} -1.19326 q^{9} -4.04948 q^{11} +3.53741 q^{13} -5.24985 q^{15} -6.37296 q^{17} -0.975857 q^{19} -0.655849 q^{21} -0.750146 q^{23} +10.2545 q^{25} +5.63637 q^{27} -4.08193 q^{29} +7.86089 q^{31} +5.44311 q^{33} +1.90570 q^{35} -4.22571 q^{37} -4.75481 q^{39} -2.07362 q^{41} -10.7341 q^{43} -4.66052 q^{45} -9.31636 q^{47} -6.76193 q^{49} +8.56622 q^{51} +11.1231 q^{53} -15.8161 q^{55} +1.31170 q^{57} +2.98052 q^{59} -7.53741 q^{61} -0.582225 q^{63} +13.8161 q^{65} -3.51207 q^{67} +1.00831 q^{69} -14.1349 q^{71} -9.75828 q^{73} -13.7836 q^{75} -1.97586 q^{77} -2.29222 q^{79} -3.99636 q^{81} +17.1324 q^{83} -24.8909 q^{85} +5.48673 q^{87} +1.86978 q^{89} +1.72600 q^{91} -10.5662 q^{93} -3.81141 q^{95} -9.63757 q^{97} +4.83208 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{3} + 5 q^{5} + q^{7} + 3 q^{9}+O(q^{10}) $ 4 * q - 3 * q^3 + 5 * q^5 + q^7 + 3 * q^9	$ 4 q - 3 q^{3} + 5 q^{5} + q^{7} + 3 q^{9} - 7 q^{11} + 4 q^{13} - 8 q^{15} + 4 q^{17} - 2 q^{19} - 5 q^{21} - 16 q^{23} - q^{25} - 6 q^{27} + q^{31} + q^{33} - 3 q^{35} - 2 q^{37} - 3 q^{39} - q^{41} - 9 q^{43} + 8 q^{45} - 13 q^{47} - 15 q^{49} - 3 q^{51} + 28 q^{53} - 13 q^{55} + 10 q^{57} - 19 q^{59} - 20 q^{61} - 12 q^{63} + 5 q^{65} - 15 q^{67} - 5 q^{69} - 15 q^{71} - 7 q^{73} - 12 q^{75} - 6 q^{77} + 17 q^{79} + 4 q^{81} - 6 q^{83} - 29 q^{85} + 34 q^{87} + 20 q^{89} + 18 q^{91} - 5 q^{93} + 6 q^{95} + 3 q^{97} + 16 q^{99}+O(q^{100}) $ 4 * q - 3 * q^3 + 5 * q^5 + q^7 + 3 * q^9 - 7 * q^11 + 4 * q^13 - 8 * q^15 + 4 * q^17 - 2 * q^19 - 5 * q^21 - 16 * q^23 - q^25 - 6 * q^27 + q^31 + q^33 - 3 * q^35 - 2 * q^37 - 3 * q^39 - q^41 - 9 * q^43 + 8 * q^45 - 13 * q^47 - 15 * q^49 - 3 * q^51 + 28 * q^53 - 13 * q^55 + 10 * q^57 - 19 * q^59 - 20 * q^61 - 12 * q^63 + 5 * q^65 - 15 * q^67 - 5 * q^69 - 15 * q^71 - 7 * q^73 - 12 * q^75 - 6 * q^77 + 17 * q^79 + 4 * q^81 - 6 * q^83 - 29 * q^85 + 34 * q^87 + 20 * q^89 + 18 * q^91 - 5 * q^93 + 6 * q^95 + 3 * q^97 + 16 * q^99

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	0
$2$	0	0
$3$	−1.34415	−0.776046	−0.388023	−	0.921650i	$-0.626842\pi$
$3$	−1.34415	−0.776046	−0.388023	+	0.921650i	$0.626842\pi$
$4$	0	0
$5$	3.90570	1.74668	0.873342	−	0.487108i	$-0.161948\pi$
$5$	3.90570	1.74668	0.873342	+	0.487108i	$0.161948\pi$
$6$	0	0
$7$	0.487928	0.184420	0.0922098	−	0.995740i	$-0.470607\pi$
$7$	0.487928	0.184420	0.0922098	+	0.995740i	$0.470607\pi$
$8$	0	0
$9$	−1.19326	−0.397753
$10$	0	0
$11$	−4.04948	−1.22096	−0.610482	−	0.792030i	$-0.709024\pi$
$11$	−4.04948	−1.22096	−0.610482	+	0.792030i	$0.709024\pi$
$12$	0	0
$13$	3.53741	0.981101	0.490550	−	0.871413i	$-0.336796\pi$
$13$	3.53741	0.981101	0.490550	+	0.871413i	$0.336796\pi$
$14$	0	0
$15$	−5.24985	−1.35551
$16$	0	0
$17$	−6.37296	−1.54567	−0.772835	−	0.634607i	$-0.781162\pi$
$17$	−6.37296	−1.54567	−0.772835	+	0.634607i	$0.781162\pi$
$18$	0	0
$19$	−0.975857	−0.223877	−0.111938	−	0.993715i	$-0.535706\pi$
$19$	−0.975857	−0.223877	−0.111938	+	0.993715i	$0.535706\pi$
$20$	0	0
$21$	−0.655849	−0.143118
$22$	0	0
$23$	−0.750146	−0.156416	−0.0782081	−	0.996937i	$-0.524920\pi$
$23$	−0.750146	−0.156416	−0.0782081	+	0.996937i	$0.524920\pi$
$24$	0	0
$25$	10.2545	2.05090
$26$	0	0
$27$	5.63637	1.08472
$28$	0	0
$29$	−4.08193	−0.757996	−0.378998	−	0.925397i	$-0.623731\pi$
$29$	−4.08193	−0.757996	−0.378998	+	0.925397i	$0.623731\pi$
$30$	0	0
$31$	7.86089	1.41186	0.705929	−	0.708283i	$-0.250530\pi$
$31$	7.86089	1.41186	0.705929	+	0.708283i	$0.250530\pi$
$32$	0	0
$33$	5.44311	0.947524
$34$	0	0
$35$	1.90570	0.322123
$36$	0	0
$37$	−4.22571	−0.694703	−0.347351	−	0.937735i	$-0.612919\pi$
$37$	−4.22571	−0.694703	−0.347351	+	0.937735i	$0.612919\pi$
$38$	0	0
$39$	−4.75481	−0.761379
$40$	0	0
$41$	−2.07362	−0.323846	−0.161923	−	0.986803i	$-0.551770\pi$
$41$	−2.07362	−0.323846	−0.161923	+	0.986803i	$0.551770\pi$
$42$	0	0
$43$	−10.7341	−1.63694	−0.818470	−	0.574549i	$-0.805178\pi$
$43$	−10.7341	−1.63694	−0.818470	+	0.574549i	$0.805178\pi$
$44$	0	0
$45$	−4.66052	−0.694749
$46$	0	0
$47$	−9.31636	−1.35893	−0.679466	−	0.733707i	$-0.737788\pi$
$47$	−9.31636	−1.35893	−0.679466	+	0.733707i	$0.737788\pi$
$48$	0	0
$49$	−6.76193	−0.965989
$50$	0	0
$51$	8.56622	1.19951
$52$	0	0
$53$	11.1231	1.52788	0.763938	−	0.645290i	$-0.223263\pi$
$53$	11.1231	1.52788	0.763938	+	0.645290i	$0.223263\pi$
$54$	0	0
$55$	−15.8161	−2.13264
$56$	0	0
$57$	1.31170	0.173739
$58$	0	0
$59$	2.98052	0.388031	0.194015	−	0.980998i	$-0.437849\pi$
$59$	2.98052	0.388031	0.194015	+	0.980998i	$0.437849\pi$
$60$	0	0
$61$	−7.53741	−0.965066	−0.482533	−	0.875878i	$-0.660283\pi$
$61$	−7.53741	−0.965066	−0.482533	+	0.875878i	$0.660283\pi$
$62$	0	0
$63$	−0.582225	−0.0733534
$64$	0	0
$65$	13.8161	1.71367
$66$	0	0
$67$	−3.51207	−0.429068	−0.214534	−	0.976717i	$-0.568823\pi$
$67$	−3.51207	−0.429068	−0.214534	+	0.976717i	$0.568823\pi$
$68$	0	0
$69$	1.00831	0.121386
$70$	0	0
$71$	−14.1349	−1.67750	−0.838751	−	0.544515i	$-0.816714\pi$
$71$	−14.1349	−1.67750	−0.838751	+	0.544515i	$0.816714\pi$
$72$	0	0
$73$	−9.75828	−1.14212	−0.571060	−	0.820908i	$-0.693468\pi$
$73$	−9.75828	−1.14212	−0.571060	+	0.820908i	$0.693468\pi$
$74$	0	0
$75$	−13.7836	−1.59160
$76$	0	0
$77$	−1.97586	−0.225170
$78$	0	0
$79$	−2.29222	−0.257895	−0.128948	−	0.991651i	$-0.541160\pi$
$79$	−2.29222	−0.257895	−0.128948	+	0.991651i	$0.541160\pi$
$80$	0	0
$81$	−3.99636	−0.444040
$82$	0	0
$83$	17.1324	1.88053	0.940265	−	0.340444i	$-0.110578\pi$
$83$	17.1324	1.88053	0.940265	+	0.340444i	$0.110578\pi$
$84$	0	0
$85$	−24.8909	−2.69980
$86$	0	0
$87$	5.48673	0.588240
$88$	0	0
$89$	1.86978	0.198196	0.0990981	−	0.995078i	$-0.468404\pi$
$89$	1.86978	0.198196	0.0990981	+	0.995078i	$0.468404\pi$
$90$	0	0
$91$	1.72600	0.180934
$92$	0	0
$93$	−10.5662	−1.09567
$94$	0	0
$95$	−3.81141	−0.391042
$96$	0	0
$97$	−9.63757	−0.978547	−0.489273	−	0.872131i	$-0.662738\pi$
$97$	−9.63757	−0.978547	−0.489273	+	0.872131i	$0.662738\pi$
$98$	0	0
$99$	4.83208	0.485642
$100$	0	0
$101$	5.91748	0.588812	0.294406	−	0.955680i	$-0.404878\pi$
$101$	5.91748	0.588812	0.294406	+	0.955680i	$0.404878\pi$
$102$	0	0
$103$	−0.480814	−0.0473760	−0.0236880	−	0.999719i	$-0.507541\pi$
$103$	−0.480814	−0.0473760	−0.0236880	+	0.999719i	$0.507541\pi$
$104$	0	0
$105$	−2.56155	−0.249982
$106$	0	0
$107$	−3.82496	−0.369773	−0.184887	−	0.982760i	$-0.559192\pi$
$107$	−3.82496	−0.369773	−0.184887	+	0.982760i	$0.559192\pi$
$108$	0	0
$109$	−17.0866	−1.63660	−0.818300	−	0.574792i	$-0.805083\pi$
$109$	−17.0866	−1.63660	−0.818300	+	0.574792i	$0.805083\pi$
$110$	0	0
$111$	5.67999	0.539121
$112$	0	0
$113$	13.3194	1.25299	0.626493	−	0.779427i	$-0.284490\pi$
$113$	13.3194	1.25299	0.626493	+	0.779427i	$0.284490\pi$
$114$	0	0
$115$	−2.92985	−0.273210
$116$	0	0
$117$	−4.22105	−0.390236
$118$	0	0
$119$	−3.10955	−0.285052
$120$	0	0
$121$	5.39830	0.490754
$122$	0	0
$123$	2.78726	0.251319
$124$	0	0
$125$	20.5226	1.83560
$126$	0	0
$127$	−3.18962	−0.283033	−0.141516	−	0.989936i	$-0.545198\pi$
$127$	−3.18962	−0.283033	−0.141516	+	0.989936i	$0.545198\pi$
$128$	0	0
$129$	14.4283	1.27034
$130$	0	0
$131$	11.5975	1.01328	0.506638	−	0.862159i	$-0.330888\pi$
$131$	11.5975	1.01328	0.506638	+	0.862159i	$0.330888\pi$
$132$	0	0
$133$	−0.476148	−0.0412873
$134$	0	0
$135$	22.0140	1.89466
$136$	0	0
$137$	−7.33059	−0.626295	−0.313147	−	0.949705i	$-0.601383\pi$
$137$	−7.33059	−0.626295	−0.313147	+	0.949705i	$0.601383\pi$
$138$	0	0
$139$	13.1927	1.11899	0.559494	−	0.828834i	$-0.310995\pi$
$139$	13.1927	1.11899	0.559494	+	0.828834i	$0.310995\pi$
$140$	0	0
$141$	12.5226	1.05459
$142$	0	0
$143$	−14.3247	−1.19789
$144$	0	0
$145$	−15.9428	−1.32398
$146$	0	0
$147$	9.08905	0.749652
$148$	0	0
$149$	19.2132	1.57400	0.787002	−	0.616950i	$-0.211632\pi$
$149$	19.2132	1.57400	0.787002	+	0.616950i	$0.211632\pi$
$150$	0	0
$151$	3.63637	0.295924	0.147962	−	0.988993i	$-0.452729\pi$
$151$	3.63637	0.295924	0.147962	+	0.988993i	$0.452729\pi$
$152$	0	0
$153$	7.60459	0.614795
$154$	0	0
$155$	30.7023	2.46607
$156$	0	0
$157$	−15.0118	−1.19807	−0.599035	−	0.800723i	$-0.704449\pi$
$157$	−15.0118	−1.19807	−0.599035	+	0.800723i	$0.704449\pi$
$158$	0	0
$159$	−14.9511	−1.18570
$160$	0	0
$161$	−0.366017	−0.0288462
$162$	0	0
$163$	4.77531	0.374031	0.187016	−	0.982357i	$-0.440118\pi$
$163$	4.77531	0.374031	0.187016	+	0.982357i	$0.440118\pi$
$164$	0	0
$165$	21.2592	1.65503
$166$	0	0
$167$	3.94163	0.305012	0.152506	−	0.988303i	$-0.451266\pi$
$167$	3.94163	0.305012	0.152506	+	0.988303i	$0.451266\pi$
$168$	0	0
$169$	−0.486734	−0.0374411
$170$	0	0
$171$	1.16445	0.0890477
$172$	0	0
$173$	−24.0607	−1.82930	−0.914650	−	0.404247i	$-0.867534\pi$
$173$	−24.0607	−1.82930	−0.914650	+	0.404247i	$0.867534\pi$
$174$	0	0
$175$	5.00347	0.378227
$176$	0	0
$177$	−4.00627	−0.301130
$178$	0	0
$179$	−17.1745	−1.28368	−0.641839	−	0.766839i	$-0.721828\pi$
$179$	−17.1745	−1.28368	−0.641839	+	0.766839i	$0.721828\pi$
$180$	0	0
$181$	−7.28875	−0.541769	−0.270884	−	0.962612i	$-0.587316\pi$
$181$	−7.28875	−0.541769	−0.270884	+	0.962612i	$0.587316\pi$
$182$	0	0
$183$	10.1314	0.748936
$184$	0	0
$185$	−16.5044	−1.21343
$186$	0	0
$187$	25.8072	1.88721
$188$	0	0
$189$	2.75015	0.200044
$190$	0	0
$191$	6.55038	0.473969	0.236985	−	0.971513i	$-0.423841\pi$
$191$	6.55038	0.473969	0.236985	+	0.971513i	$0.423841\pi$
$192$	0	0
$193$	17.5467	1.26304	0.631521	−	0.775359i	$-0.282431\pi$
$193$	17.5467	1.26304	0.631521	+	0.775359i	$0.282431\pi$
$194$	0	0
$195$	−18.5709	−1.32989
$196$	0	0
$197$	−14.6051	−1.04057	−0.520286	−	0.853992i	$-0.674175\pi$
$197$	−14.6051	−1.04057	−0.520286	+	0.853992i	$0.674175\pi$
$198$	0	0
$199$	4.64162	0.329036	0.164518	−	0.986374i	$-0.447393\pi$
$199$	4.64162	0.329036	0.164518	+	0.986374i	$0.447393\pi$
$200$	0	0
$201$	4.72075	0.332976
$202$	0	0
$203$	−1.99169	−0.139789
$204$	0	0
$205$	−8.09896	−0.565656
$206$	0	0
$207$	0.895118	0.0622150
$208$	0	0
$209$	3.95171	0.273346
$210$	0	0
$211$	−15.7554	−1.08465	−0.542324	−	0.840169i	$-0.682455\pi$
$211$	−15.7554	−1.08465	−0.542324	+	0.840169i	$0.682455\pi$
$212$	0	0
$213$	18.9994	1.30182
$214$	0	0
$215$	−41.9244	−2.85922
$216$	0	0
$217$	3.83555	0.260374
$218$	0	0
$219$	13.1166	0.886338
$220$	0	0
$221$	−22.5438	−1.51646
$222$	0	0
$223$	−19.8791	−1.33120	−0.665602	−	0.746307i	$-0.731825\pi$
$223$	−19.8791	−1.33120	−0.665602	+	0.746307i	$0.731825\pi$
$224$	0	0
$225$	−12.2363	−0.815753
$226$	0	0
$227$	−27.7566	−1.84227	−0.921136	−	0.389242i	$-0.872737\pi$
$227$	−27.7566	−1.84227	−0.921136	+	0.389242i	$0.872737\pi$
$228$	0	0
$229$	−17.6723	−1.16782	−0.583909	−	0.811819i	$-0.698478\pi$
$229$	−17.6723	−1.16782	−0.583909	+	0.811819i	$0.698478\pi$
$230$	0	0
$231$	2.65585	0.174742
$232$	0	0
$233$	−1.67230	−0.109556	−0.0547779	−	0.998499i	$-0.517445\pi$
$233$	−1.67230	−0.109556	−0.0547779	+	0.998499i	$0.517445\pi$
$234$	0	0
$235$	−36.3870	−2.37362
$236$	0	0
$237$	3.08109	0.200138
$238$	0	0
$239$	24.0925	1.55841	0.779206	−	0.626768i	$-0.215623\pi$
$239$	24.0925	1.55841	0.779206	+	0.626768i	$0.215623\pi$
$240$	0	0
$241$	−18.2257	−1.17402	−0.587011	−	0.809579i	$-0.699695\pi$
$241$	−18.2257	−1.17402	−0.587011	+	0.809579i	$0.699695\pi$
$242$	0	0
$243$	−11.5374	−0.740125
$244$	0	0
$245$	−26.4101	−1.68728
$246$	0	0
$247$	−3.45200	−0.219646
$248$	0	0
$249$	−23.0286	−1.45938
$250$	0	0
$251$	1.21740	0.0768417	0.0384209	−	0.999262i	$-0.487767\pi$
$251$	1.21740	0.0768417	0.0384209	+	0.999262i	$0.487767\pi$
$252$	0	0
$253$	3.03770	0.190979
$254$	0	0
$255$	33.4571	2.09517
$256$	0	0
$257$	28.7806	1.79529	0.897644	−	0.440722i	$-0.145277\pi$
$257$	28.7806	1.79529	0.897644	+	0.440722i	$0.145277\pi$
$258$	0	0
$259$	−2.06184	−0.128117
$260$	0	0
$261$	4.87080	0.301495
$262$	0	0
$263$	4.35407	0.268483	0.134242	−	0.990949i	$-0.457140\pi$
$263$	4.35407	0.268483	0.134242	+	0.990949i	$0.457140\pi$
$264$	0	0
$265$	43.4436	2.66872
$266$	0	0
$267$	−2.51327	−0.153809
$268$	0	0
$269$	1.00000	0.0609711
$269$	1.00000	0.0609711
$270$	0	0
$271$	19.6940	1.19632	0.598162	−	0.801375i	$-0.295898\pi$
$271$	19.6940	1.19632	0.598162	+	0.801375i	$0.295898\pi$
$272$	0	0
$273$	−2.32001	−0.140413
$274$	0	0
$275$	−41.5255	−2.50408
$276$	0	0
$277$	−1.31009	−0.0787158	−0.0393579	−	0.999225i	$-0.512531\pi$
$277$	−1.31009	−0.0787158	−0.0393579	+	0.999225i	$0.512531\pi$
$278$	0	0
$279$	−9.38007	−0.561570
$280$	0	0
$281$	7.94749	0.474107	0.237054	−	0.971497i	$-0.423818\pi$
$281$	7.94749	0.474107	0.237054	+	0.971497i	$0.423818\pi$
$282$	0	0
$283$	−20.7544	−1.23372	−0.616861	−	0.787072i	$-0.711596\pi$
$283$	−20.7544	−1.23372	−0.616861	+	0.787072i	$0.711596\pi$
$284$	0	0
$285$	5.12311	0.303467
$286$	0	0
$287$	−1.01178	−0.0597235
$288$	0	0
$289$	23.6146	1.38910
$290$	0	0
$291$	12.9543	0.759397
$292$	0	0
$293$	−10.9994	−0.642593	−0.321296	−	0.946979i	$-0.604119\pi$
$293$	−10.9994	−0.642593	−0.321296	+	0.946979i	$0.604119\pi$
$294$	0	0
$295$	11.6410	0.677767
$296$	0	0
$297$	−22.8244	−1.32440
$298$	0	0
$299$	−2.65357	−0.153460
$300$	0	0
$301$	−5.23749	−0.301884
$302$	0	0
$303$	−7.95399	−0.456945
$304$	0	0
$305$	−29.4389	−1.68567
$306$	0	0
$307$	−15.1961	−0.867290	−0.433645	−	0.901084i	$-0.642773\pi$
$307$	−15.1961	−0.867290	−0.433645	+	0.901084i	$0.642773\pi$
$308$	0	0
$309$	0.646287	0.0367660
$310$	0	0
$311$	−9.02476	−0.511747	−0.255873	−	0.966710i	$-0.582363\pi$
$311$	−9.02476	−0.511747	−0.255873	+	0.966710i	$0.582363\pi$
$312$	0	0
$313$	34.3481	1.94147	0.970734	−	0.240159i	$-0.0771995\pi$
$313$	34.3481	1.94147	0.970734	+	0.240159i	$0.0771995\pi$
$314$	0	0
$315$	−2.27400	−0.128125
$316$	0	0
$317$	20.5386	1.15356	0.576781	−	0.816898i	$-0.304308\pi$
$317$	20.5386	1.15356	0.576781	+	0.816898i	$0.304308\pi$
$318$	0	0
$319$	16.5297	0.925486
$320$	0	0
$321$	5.14133	0.286961
$322$	0	0
$323$	6.21910	0.346040
$324$	0	0
$325$	36.2744	2.01214
$326$	0	0
$327$	22.9670	1.27008
$328$	0	0
$329$	−4.54572	−0.250614
$330$	0	0
$331$	8.30867	0.456686	0.228343	−	0.973581i	$-0.426669\pi$
$331$	8.30867	0.456686	0.228343	+	0.973581i	$0.426669\pi$
$332$	0	0
$333$	5.04237	0.276320
$334$	0	0
$335$	−13.7171	−0.749446
$336$	0	0
$337$	−17.9588	−0.978276	−0.489138	−	0.872206i	$-0.662689\pi$
$337$	−17.9588	−0.978276	−0.489138	+	0.872206i	$0.662689\pi$
$338$	0	0
$339$	−17.9033	−0.972375
$340$	0	0
$341$	−31.8325	−1.72383
$342$	0	0
$343$	−6.71483	−0.362567
$344$	0	0
$345$	3.93816	0.212023
$346$	0	0
$347$	−6.69769	−0.359551	−0.179775	−	0.983708i	$-0.557537\pi$
$347$	−6.69769	−0.359551	−0.179775	+	0.983708i	$0.557537\pi$
$348$	0	0
$349$	0.974251	0.0521505	0.0260752	−	0.999660i	$-0.491699\pi$
$349$	0.974251	0.0521505	0.0260752	+	0.999660i	$0.491699\pi$
$350$	0	0
$351$	19.9382	1.06422
$352$	0	0
$353$	15.8928	0.845886	0.422943	−	0.906156i	$-0.360997\pi$
$353$	15.8928	0.845886	0.422943	+	0.906156i	$0.360997\pi$
$354$	0	0
$355$	−55.2067	−2.93007
$356$	0	0
$357$	4.17970	0.221213
$358$	0	0
$359$	8.66535	0.457340	0.228670	−	0.973504i	$-0.426562\pi$
$359$	8.66535	0.457340	0.228670	+	0.973504i	$0.426562\pi$
$360$	0	0
$361$	−18.0477	−0.949879
$362$	0	0
$363$	−7.25613	−0.380848
$364$	0	0
$365$	−38.1130	−1.99492
$366$	0	0
$367$	−7.50802	−0.391915	−0.195958	−	0.980612i	$-0.562782\pi$
$367$	−7.50802	−0.391915	−0.195958	+	0.980612i	$0.562782\pi$
$368$	0	0
$369$	2.47437	0.128811
$370$	0	0
$371$	5.42728	0.281770
$372$	0	0
$373$	2.14911	0.111277	0.0556385	−	0.998451i	$-0.482281\pi$
$373$	2.14911	0.111277	0.0556385	+	0.998451i	$0.482281\pi$
$374$	0	0
$375$	−27.5855	−1.42451
$376$	0	0
$377$	−14.4395	−0.743671
$378$	0	0
$379$	−7.71644	−0.396367	−0.198183	−	0.980165i	$-0.563504\pi$
$379$	−7.71644	−0.396367	−0.198183	+	0.980165i	$0.563504\pi$
$380$	0	0
$381$	4.28732	0.219646
$382$	0	0
$383$	−1.88104	−0.0961165	−0.0480582	−	0.998845i	$-0.515303\pi$
$383$	−1.88104	−0.0961165	−0.0480582	+	0.998845i	$0.515303\pi$
$384$	0	0
$385$	−7.71711	−0.393300
$386$	0	0
$387$	12.8086	0.651098
$388$	0	0
$389$	−10.7025	−0.542640	−0.271320	−	0.962489i	$-0.587460\pi$
$389$	−10.7025	−0.542640	−0.271320	+	0.962489i	$0.587460\pi$
$390$	0	0
$391$	4.78065	0.241768
$392$	0	0
$393$	−15.5888	−0.786349
$394$	0	0
$395$	−8.95274	−0.450461
$396$	0	0
$397$	3.84774	0.193113	0.0965563	−	0.995328i	$-0.469217\pi$
$397$	3.84774	0.193113	0.0965563	+	0.995328i	$0.469217\pi$
$398$	0	0
$399$	0.640015	0.0320408
$400$	0	0
$401$	−0.717694	−0.0358399	−0.0179200	−	0.999839i	$-0.505704\pi$
$401$	−0.717694	−0.0358399	−0.0179200	+	0.999839i	$0.505704\pi$
$402$	0	0
$403$	27.8072	1.38517
$404$	0	0
$405$	−15.6086	−0.775597
$406$	0	0
$407$	17.1119	0.848207
$408$	0	0
$409$	−11.7906	−0.583006	−0.291503	−	0.956570i	$-0.594155\pi$
$409$	−11.7906	−0.583006	−0.291503	+	0.956570i	$0.594155\pi$
$410$	0	0
$411$	9.85342	0.486033
$412$	0	0
$413$	1.45428	0.0715605
$414$	0	0
$415$	66.9142	3.28469
$416$	0	0
$417$	−17.7329	−0.868386
$418$	0	0
$419$	7.07076	0.345429	0.172715	−	0.984972i	$-0.444746\pi$
$419$	7.07076	0.345429	0.172715	+	0.984972i	$0.444746\pi$
$420$	0	0
$421$	−4.96469	−0.241964	−0.120982	−	0.992655i	$-0.538604\pi$
$421$	−4.96469	−0.241964	−0.120982	+	0.992655i	$0.538604\pi$
$422$	0	0
$423$	11.1168	0.540519
$424$	0	0
$425$	−65.3516	−3.17002
$426$	0	0
$427$	−3.67772	−0.177977
$428$	0	0
$429$	19.2545	0.929617
$430$	0	0
$431$	−25.6140	−1.23378	−0.616892	−	0.787048i	$-0.711609\pi$
$431$	−25.6140	−1.23378	−0.616892	+	0.787048i	$0.711609\pi$
$432$	0	0
$433$	−20.5767	−0.988855	−0.494428	−	0.869219i	$-0.664622\pi$
$433$	−20.5767	−0.988855	−0.494428	+	0.869219i	$0.664622\pi$
$434$	0	0
$435$	21.4296	1.02747
$436$	0	0
$437$	0.732035	0.0350180
$438$	0	0
$439$	−15.6900	−0.748843	−0.374421	−	0.927259i	$-0.622159\pi$
$439$	−15.6900	−0.748843	−0.374421	+	0.927259i	$0.622159\pi$
$440$	0	0
$441$	8.06873	0.384225
$442$	0	0
$443$	−29.0018	−1.37792	−0.688959	−	0.724801i	$-0.741932\pi$
$443$	−29.0018	−1.37792	−0.688959	+	0.724801i	$0.741932\pi$
$444$	0	0
$445$	7.30281	0.346186
$446$	0	0
$447$	−25.8254	−1.22150
$448$	0	0
$449$	18.1978	0.858805	0.429403	−	0.903113i	$-0.358724\pi$
$449$	18.1978	0.858805	0.429403	+	0.903113i	$0.358724\pi$
$450$	0	0
$451$	8.39710	0.395404
$452$	0	0
$453$	−4.88783	−0.229650
$454$	0	0
$455$	6.74125	0.316035
$456$	0	0
$457$	32.1933	1.50594	0.752969	−	0.658056i	$-0.228621\pi$
$457$	32.1933	1.50594	0.752969	+	0.658056i	$0.228621\pi$
$458$	0	0
$459$	−35.9204	−1.67662
$460$	0	0
$461$	−26.9152	−1.25357	−0.626783	−	0.779194i	$-0.715629\pi$
$461$	−26.9152	−1.25357	−0.626783	+	0.779194i	$0.715629\pi$
$462$	0	0
$463$	24.2349	1.12629	0.563145	−	0.826358i	$-0.309591\pi$
$463$	24.2349	1.12629	0.563145	+	0.826358i	$0.309591\pi$
$464$	0	0
$465$	−41.2685	−1.91378
$466$	0	0
$467$	17.7628	0.821963	0.410982	−	0.911644i	$-0.365186\pi$
$467$	17.7628	0.821963	0.410982	+	0.911644i	$0.365186\pi$
$468$	0	0
$469$	−1.71364	−0.0791285
$470$	0	0
$471$	20.1781	0.929758
$472$	0	0
$473$	43.4677	1.99865
$474$	0	0
$475$	−10.0069	−0.459150
$476$	0	0
$477$	−13.2727	−0.607717
$478$	0	0
$479$	−11.4930	−0.525130	−0.262565	−	0.964914i	$-0.584568\pi$
$479$	−11.4930	−0.525130	−0.262565	+	0.964914i	$0.584568\pi$
$480$	0	0
$481$	−14.9481	−0.681573
$482$	0	0
$483$	0.491983	0.0223860
$484$	0	0
$485$	−37.6415	−1.70921
$486$	0	0
$487$	−5.85514	−0.265322	−0.132661	−	0.991161i	$-0.542352\pi$
$487$	−5.85514	−0.265322	−0.132661	+	0.991161i	$0.542352\pi$
$488$	0	0
$489$	−6.41874	−0.290265
$490$	0	0
$491$	−18.6544	−0.841862	−0.420931	−	0.907093i	$-0.638297\pi$
$491$	−18.6544	−0.841862	−0.420931	+	0.907093i	$0.638297\pi$
$492$	0	0
$493$	26.0140	1.17161
$494$	0	0
$495$	18.8727	0.848263
$496$	0	0
$497$	−6.89681	−0.309364
$498$	0	0
$499$	−35.1119	−1.57183	−0.785913	−	0.618337i	$-0.787807\pi$
$499$	−35.1119	−1.57183	−0.785913	+	0.618337i	$0.787807\pi$
$500$	0	0
$501$	−5.29814	−0.236703
$502$	0	0
$503$	−29.7342	−1.32578	−0.662892	−	0.748715i	$-0.730671\pi$
$503$	−29.7342	−1.32578	−0.662892	+	0.748715i	$0.730671\pi$
$504$	0	0
$505$	23.1119	1.02847
$506$	0	0
$507$	0.654243	0.0290560
$508$	0	0
$509$	30.1456	1.33618	0.668090	−	0.744081i	$-0.267112\pi$
$509$	30.1456	1.33618	0.668090	+	0.744081i	$0.267112\pi$
$510$	0	0
$511$	−4.76134	−0.210629
$512$	0	0
$513$	−5.50029	−0.242844
$514$	0	0
$515$	−1.87792	−0.0827509
$516$	0	0
$517$	37.7264	1.65921
$518$	0	0
$519$	32.3412	1.41962
$520$	0	0
$521$	37.9804	1.66395	0.831975	−	0.554813i	$-0.187210\pi$
$521$	37.9804	1.66395	0.831975	+	0.554813i	$0.187210\pi$
$522$	0	0
$523$	−10.1012	−0.441696	−0.220848	−	0.975308i	$-0.570883\pi$
$523$	−10.1012	−0.441696	−0.220848	+	0.975308i	$0.570883\pi$
$524$	0	0
$525$	−6.72542	−0.293521
$526$	0	0
$527$	−50.0971	−2.18227
$528$	0	0
$529$	−22.4373	−0.975534
$530$	0	0
$531$	−3.55653	−0.154340
$532$	0	0
$533$	−7.33526	−0.317725
$534$	0	0
$535$	−14.9392	−0.645877
$536$	0	0
$537$	23.0851	0.996194
$538$	0	0
$539$	27.3823	1.17944
$540$	0	0
$541$	1.06610	0.0458352	0.0229176	−	0.999737i	$-0.492704\pi$
$541$	1.06610	0.0458352	0.0229176	+	0.999737i	$0.492704\pi$
$542$	0	0
$543$	9.79718	0.420437
$544$	0	0
$545$	−66.7352	−2.85862
$546$	0	0
$547$	−8.62634	−0.368836	−0.184418	−	0.982848i	$-0.559040\pi$
$547$	−8.62634	−0.368836	−0.184418	+	0.982848i	$0.559040\pi$
$548$	0	0
$549$	8.99408	0.383858
$550$	0	0
$551$	3.98338	0.169698
$552$	0	0
$553$	−1.11844	−0.0475609
$554$	0	0
$555$	22.1844	0.941674
$556$	0	0
$557$	39.3859	1.66884	0.834418	−	0.551132i	$-0.185804\pi$
$557$	39.3859	1.66884	0.834418	+	0.551132i	$0.185804\pi$
$558$	0	0
$559$	−37.9710	−1.60600
$560$	0	0
$561$	−34.6887	−1.46456
$562$	0	0
$563$	29.8436	1.25776	0.628879	−	0.777503i	$-0.283514\pi$
$563$	29.8436	1.25776	0.628879	+	0.777503i	$0.283514\pi$
$564$	0	0
$565$	52.0217	2.18857
$566$	0	0
$567$	−1.94994	−0.0818896
$568$	0	0
$569$	18.1766	0.762004	0.381002	−	0.924574i	$-0.375579\pi$
$569$	18.1766	0.762004	0.381002	+	0.924574i	$0.375579\pi$
$570$	0	0
$571$	34.4465	1.44154	0.720771	−	0.693173i	$-0.243788\pi$
$571$	34.4465	1.44154	0.720771	+	0.693173i	$0.243788\pi$
$572$	0	0
$573$	−8.80470	−0.367822
$574$	0	0
$575$	−7.69238	−0.320795
$576$	0	0
$577$	6.04440	0.251632	0.125816	−	0.992054i	$-0.459845\pi$
$577$	6.04440	0.251632	0.125816	+	0.992054i	$0.459845\pi$
$578$	0	0
$579$	−23.5855	−0.980178
$580$	0	0
$581$	8.35940	0.346806
$582$	0	0
$583$	−45.0428	−1.86548
$584$	0	0
$585$	−16.4862	−0.681618
$586$	0	0
$587$	38.6176	1.59392	0.796959	−	0.604034i	$-0.206441\pi$
$587$	38.6176	1.59392	0.796959	+	0.604034i	$0.206441\pi$
$588$	0	0
$589$	−7.67110	−0.316082
$590$	0	0
$591$	19.6315	0.807531
$592$	0	0
$593$	−4.06998	−0.167134	−0.0835671	−	0.996502i	$-0.526631\pi$
$593$	−4.06998	−0.167134	−0.0835671	+	0.996502i	$0.526631\pi$
$594$	0	0
$595$	−12.1450	−0.497895
$596$	0	0
$597$	−6.23904	−0.255347
$598$	0	0
$599$	18.2628	0.746199	0.373099	−	0.927791i	$-0.378295\pi$
$599$	18.2628	0.746199	0.373099	+	0.927791i	$0.378295\pi$
$600$	0	0
$601$	31.6805	1.29228	0.646138	−	0.763220i	$-0.276383\pi$
$601$	31.6805	1.29228	0.646138	+	0.763220i	$0.276383\pi$
$602$	0	0
$603$	4.19081	0.170663
$604$	0	0
$605$	21.0842	0.857193
$606$	0	0
$607$	−25.5355	−1.03645	−0.518226	−	0.855244i	$-0.673407\pi$
$607$	−25.5355	−1.03645	−0.518226	+	0.855244i	$0.673407\pi$
$608$	0	0
$609$	2.67713	0.108483
$610$	0	0
$611$	−32.9558	−1.33325
$612$	0	0
$613$	−17.6816	−0.714154	−0.357077	−	0.934075i	$-0.616227\pi$
$613$	−17.6816	−0.714154	−0.357077	+	0.934075i	$0.616227\pi$
$614$	0	0
$615$	10.8862	0.438975
$616$	0	0
$617$	32.8371	1.32197	0.660986	−	0.750398i	$-0.270138\pi$
$617$	32.8371	1.32197	0.660986	+	0.750398i	$0.270138\pi$
$618$	0	0
$619$	−10.5824	−0.425342	−0.212671	−	0.977124i	$-0.568216\pi$
$619$	−10.5824	−0.425342	−0.212671	+	0.977124i	$0.568216\pi$
$620$	0	0
$621$	−4.22810	−0.169668
$622$	0	0
$623$	0.912319	0.0365513
$624$	0	0
$625$	28.8826	1.15530
$626$	0	0
$627$	−5.31170	−0.212129
$628$	0	0
$629$	26.9303	1.07378
$630$	0	0
$631$	0.902404	0.0359241	0.0179621	−	0.999839i	$-0.494282\pi$
$631$	0.902404	0.0359241	0.0179621	+	0.999839i	$0.494282\pi$
$632$	0	0
$633$	21.1777	0.841737
$634$	0	0
$635$	−12.4577	−0.494368
$636$	0	0
$637$	−23.9197	−0.947733
$638$	0	0
$639$	16.8666	0.667231
$640$	0	0
$641$	−19.2444	−0.760109	−0.380055	−	0.924964i	$-0.624095\pi$
$641$	−19.2444	−0.760109	−0.380055	+	0.924964i	$0.624095\pi$
$642$	0	0
$643$	−34.2804	−1.35189	−0.675944	−	0.736953i	$-0.736264\pi$
$643$	−34.2804	−1.35189	−0.675944	+	0.736953i	$0.736264\pi$
$644$	0	0
$645$	56.3527	2.21888
$646$	0	0
$647$	30.2029	1.18740	0.593698	−	0.804688i	$-0.297667\pi$
$647$	30.2029	1.18740	0.593698	+	0.804688i	$0.297667\pi$
$648$	0	0
$649$	−12.0696	−0.473772
$650$	0	0
$651$	−5.15556	−0.202062
$652$	0	0
$653$	−21.5877	−0.844794	−0.422397	−	0.906411i	$-0.638811\pi$
$653$	−21.5877	−0.844794	−0.422397	+	0.906411i	$0.638811\pi$
$654$	0	0
$655$	45.2963	1.76987
$656$	0	0
$657$	11.6442	0.454282
$658$	0	0
$659$	21.3740	0.832612	0.416306	−	0.909225i	$-0.363324\pi$
$659$	21.3740	0.832612	0.416306	+	0.909225i	$0.363324\pi$
$660$	0	0
$661$	−19.7195	−0.767000	−0.383500	−	0.923541i	$-0.625281\pi$
$661$	−19.7195	−0.767000	−0.383500	+	0.923541i	$0.625281\pi$
$662$	0	0
$663$	30.3022	1.17684
$664$	0	0
$665$	−1.85969	−0.0721158
$666$	0	0
$667$	3.06204	0.118563
$668$	0	0
$669$	26.7205	1.03308
$670$	0	0
$671$	30.5226	1.17831
$672$	0	0
$673$	−13.2275	−0.509883	−0.254942	−	0.966956i	$-0.582056\pi$
$673$	−13.2275	−0.509883	−0.254942	+	0.966956i	$0.582056\pi$
$674$	0	0
$675$	57.7983	2.22466
$676$	0	0
$677$	−12.7629	−0.490520	−0.245260	−	0.969457i	$-0.578873\pi$
$677$	−12.7629	−0.490520	−0.245260	+	0.969457i	$0.578873\pi$
$678$	0	0
$679$	−4.70244	−0.180463
$680$	0	0
$681$	37.3091	1.42969
$682$	0	0
$683$	3.12345	0.119515	0.0597577	−	0.998213i	$-0.480967\pi$
$683$	3.12345	0.119515	0.0597577	+	0.998213i	$0.480967\pi$
$684$	0	0
$685$	−28.6311	−1.09394
$686$	0	0
$687$	23.7542	0.906280
$688$	0	0
$689$	39.3470	1.49900
$690$	0	0
$691$	−25.3081	−0.962767	−0.481384	−	0.876510i	$-0.659866\pi$
$691$	−25.3081	−0.962767	−0.481384	+	0.876510i	$0.659866\pi$
$692$	0	0
$693$	2.35771	0.0895619
$694$	0	0
$695$	51.5267	1.95452
$696$	0	0
$697$	13.2151	0.500559
$698$	0	0
$699$	2.24782	0.0850202
$700$	0	0
$701$	−15.5107	−0.585831	−0.292916	−	0.956138i	$-0.594626\pi$
$701$	−15.5107	−0.585831	−0.292916	+	0.956138i	$0.594626\pi$
$702$	0	0
$703$	4.12369	0.155528
$704$	0	0
$705$	48.9096	1.84204
$706$	0	0
$707$	2.88731	0.108588
$708$	0	0
$709$	3.24435	0.121844	0.0609220	−	0.998143i	$-0.480596\pi$
$709$	3.24435	0.121844	0.0609220	+	0.998143i	$0.480596\pi$
$710$	0	0
$711$	2.73521	0.102579
$712$	0	0
$713$	−5.89681	−0.220837
$714$	0	0
$715$	−55.9479	−2.09233
$716$	0	0
$717$	−32.3839	−1.20940
$718$	0	0
$719$	−28.5289	−1.06395	−0.531974	−	0.846761i	$-0.678549\pi$
$719$	−28.5289	−1.06395	−0.531974	+	0.846761i	$0.678549\pi$
$720$	0	0
$721$	−0.234603	−0.00873707
$722$	0	0
$723$	24.4981	0.911094
$724$	0	0
$725$	−41.8583	−1.55458
$726$	0	0
$727$	−28.4941	−1.05679	−0.528395	−	0.848999i	$-0.677206\pi$
$727$	−28.4941	−1.05679	−0.528395	+	0.848999i	$0.677206\pi$
$728$	0	0
$729$	27.4971	1.01841
$730$	0	0
$731$	68.4082	2.53017
$732$	0	0
$733$	31.1617	1.15098	0.575491	−	0.817808i	$-0.304811\pi$
$733$	31.1617	1.15098	0.575491	+	0.817808i	$0.304811\pi$
$734$	0	0
$735$	35.4991	1.30940
$736$	0	0
$737$	14.2221	0.523877
$738$	0	0
$739$	−22.7409	−0.836538	−0.418269	−	0.908323i	$-0.637363\pi$
$739$	−22.7409	−0.836538	−0.418269	+	0.908323i	$0.637363\pi$
$740$	0	0
$741$	4.64001	0.170455
$742$	0	0
$743$	−33.4594	−1.22751	−0.613753	−	0.789498i	$-0.710341\pi$
$743$	−33.4594	−1.22751	−0.613753	+	0.789498i	$0.710341\pi$
$744$	0	0
$745$	75.0410	2.74929
$746$	0	0
$747$	−20.4434	−0.747986
$748$	0	0
$749$	−1.86631	−0.0681934
$750$	0	0
$751$	48.0493	1.75334	0.876672	−	0.481089i	$-0.159759\pi$
$751$	48.0493	1.75334	0.876672	+	0.481089i	$0.159759\pi$
$752$	0	0
$753$	−1.63637	−0.0596327
$754$	0	0
$755$	14.2026	0.516885
$756$	0	0
$757$	52.5319	1.90930	0.954652	−	0.297725i	$-0.0962279\pi$
$757$	52.5319	1.90930	0.954652	+	0.297725i	$0.0962279\pi$
$758$	0	0
$759$	−4.08313	−0.148208
$760$	0	0
$761$	50.4604	1.82919	0.914594	−	0.404373i	$-0.132510\pi$
$761$	50.4604	1.82919	0.914594	+	0.404373i	$0.132510\pi$
$762$	0	0
$763$	−8.33704	−0.301821
$764$	0	0
$765$	29.7013	1.07385
$766$	0	0
$767$	10.5433	0.380698
$768$	0	0
$769$	−38.6684	−1.39442	−0.697209	−	0.716867i	$-0.745575\pi$
$769$	−38.6684	−1.39442	−0.697209	+	0.716867i	$0.745575\pi$
$770$	0	0
$771$	−38.6855	−1.39323
$772$	0	0
$773$	−27.8089	−1.00022	−0.500108	−	0.865963i	$-0.666706\pi$
$773$	−27.8089	−1.00022	−0.500108	+	0.865963i	$0.666706\pi$
$774$	0	0
$775$	80.6096	2.89558
$776$	0	0
$777$	2.77143	0.0994245
$778$	0	0
$779$	2.02356	0.0725016
$780$	0	0
$781$	57.2390	2.04817
$782$	0	0
$783$	−23.0073	−0.822214
$784$	0	0
$785$	−58.6316	−2.09265
$786$	0	0
$787$	35.0440	1.24918	0.624592	−	0.780951i	$-0.285265\pi$
$787$	35.0440	1.24918	0.624592	+	0.780951i	$0.285265\pi$
$788$	0	0
$789$	−5.85252	−0.208355
$790$	0	0
$791$	6.49893	0.231075
$792$	0	0
$793$	−26.6629	−0.946827
$794$	0	0
$795$	−58.3947	−2.07105
$796$	0	0
$797$	44.3028	1.56929	0.784643	−	0.619947i	$-0.212846\pi$
$797$	44.3028	1.56929	0.784643	+	0.619947i	$0.212846\pi$
$798$	0	0
$799$	59.3728	2.10046
$800$	0	0
$801$	−2.23113	−0.0788332
$802$	0	0
$803$	39.5160	1.39449
$804$	0	0
$805$	−1.42956	−0.0503852
$806$	0	0
$807$	−1.34415	−0.0473163
$808$	0	0
$809$	17.8666	0.628157	0.314079	−	0.949397i	$-0.398304\pi$
$809$	17.8666	0.628157	0.314079	+	0.949397i	$0.398304\pi$
$810$	0	0
$811$	−7.94755	−0.279076	−0.139538	−	0.990217i	$-0.544562\pi$
$811$	−7.94755	−0.279076	−0.139538	+	0.990217i	$0.544562\pi$
$812$	0	0
$813$	−26.4717	−0.928403
$814$	0	0
$815$	18.6510	0.653314
$816$	0	0
$817$	10.4750	0.366473
$818$	0	0
$819$	−2.05957	−0.0719671
$820$	0	0
$821$	14.0212	0.489342	0.244671	−	0.969606i	$-0.421320\pi$
$821$	14.0212	0.489342	0.244671	+	0.969606i	$0.421320\pi$
$822$	0	0
$823$	56.0461	1.95364	0.976822	−	0.214052i	$-0.0686663\pi$
$823$	56.0461	1.95364	0.976822	+	0.214052i	$0.0686663\pi$
$824$	0	0
$825$	55.8165	1.94328
$826$	0	0
$827$	−28.6027	−0.994612	−0.497306	−	0.867575i	$-0.665677\pi$
$827$	−28.6027	−0.994612	−0.497306	+	0.867575i	$0.665677\pi$
$828$	0	0
$829$	7.08499	0.246072	0.123036	−	0.992402i	$-0.460737\pi$
$829$	7.08499	0.246072	0.123036	+	0.992402i	$0.460737\pi$
$830$	0	0
$831$	1.76096	0.0610871
$832$	0	0
$833$	43.0935	1.49310
$834$	0	0
$835$	15.3948	0.532760
$836$	0	0
$837$	44.3069	1.53147
$838$	0	0
$839$	16.9860	0.586423	0.293211	−	0.956048i	$-0.405276\pi$
$839$	16.9860	0.586423	0.293211	+	0.956048i	$0.405276\pi$
$840$	0	0
$841$	−12.3378	−0.425442
$842$	0	0
$843$	−10.6826	−0.367929
$844$	0	0
$845$	−1.90104	−0.0653977
$846$	0	0
$847$	2.63398	0.0905047
$848$	0	0
$849$	27.8970	0.957424
$850$	0	0
$851$	3.16990	0.108663
$852$	0	0
$853$	−25.5249	−0.873957	−0.436979	−	0.899472i	$-0.643951\pi$
$853$	−25.5249	−0.873957	−0.436979	+	0.899472i	$0.643951\pi$
$854$	0	0
$855$	4.54800	0.155538
$856$	0	0
$857$	−47.9302	−1.63726	−0.818632	−	0.574318i	$-0.805267\pi$
$857$	−47.9302	−1.63726	−0.818632	+	0.574318i	$0.805267\pi$
$858$	0	0
$859$	6.23647	0.212786	0.106393	−	0.994324i	$-0.466070\pi$
$859$	6.23647	0.212786	0.106393	+	0.994324i	$0.466070\pi$
$860$	0	0
$861$	1.35999	0.0463482
$862$	0	0
$863$	−8.46337	−0.288097	−0.144048	−	0.989571i	$-0.546012\pi$
$863$	−8.46337	−0.288097	−0.144048	+	0.989571i	$0.546012\pi$
$864$	0	0
$865$	−93.9739	−3.19521
$866$	0	0
$867$	−31.7416	−1.07800
$868$	0	0
$869$	9.28231	0.314881
$870$	0	0
$871$	−12.4236	−0.420959
$872$	0	0
$873$	11.5001	0.389220
$874$	0	0
$875$	10.0136	0.338520
$876$	0	0
$877$	42.1172	1.42220	0.711098	−	0.703093i	$-0.248198\pi$
$877$	42.1172	1.42220	0.711098	+	0.703093i	$0.248198\pi$
$878$	0	0
$879$	14.7849	0.498681
$880$	0	0
$881$	−35.0575	−1.18112	−0.590558	−	0.806995i	$-0.701092\pi$
$881$	−35.0575	−1.18112	−0.590558	+	0.806995i	$0.701092\pi$
$882$	0	0
$883$	−49.3854	−1.66195	−0.830976	−	0.556308i	$-0.812217\pi$
$883$	−49.3854	−1.66195	−0.830976	+	0.556308i	$0.812217\pi$
$884$	0	0
$885$	−15.6473	−0.525979
$886$	0	0
$887$	20.5875	0.691261	0.345630	−	0.938371i	$-0.387665\pi$
$887$	20.5875	0.691261	0.345630	+	0.938371i	$0.387665\pi$
$888$	0	0
$889$	−1.55630	−0.0521968
$890$	0	0
$891$	16.1832	0.542157
$892$	0	0
$893$	9.09144	0.304233
$894$	0	0
$895$	−67.0783	−2.24218
$896$	0	0
$897$	3.56680	0.119092
$898$	0	0
$899$	−32.0876	−1.07018
$900$	0	0
$901$	−70.8871	−2.36159
$902$	0	0
$903$	7.03998	0.234276
$904$	0	0
$905$	−28.4677	−0.946298
$906$	0	0
$907$	57.1062	1.89618	0.948090	−	0.318003i	$-0.103012\pi$
$907$	57.1062	1.89618	0.948090	+	0.318003i	$0.103012\pi$
$908$	0	0
$909$	−7.06109	−0.234202
$910$	0	0
$911$	−1.11278	−0.0368680	−0.0184340	−	0.999830i	$-0.505868\pi$
$911$	−1.11278	−0.0368680	−0.0184340	+	0.999830i	$0.505868\pi$
$912$	0	0
$913$	−69.3775	−2.29606
$914$	0	0
$915$	39.5703	1.30815
$916$	0	0
$917$	5.65874	0.186868
$918$	0	0
$919$	24.6902	0.814453	0.407227	−	0.913327i	$-0.366496\pi$
$919$	24.6902	0.814453	0.407227	+	0.913327i	$0.366496\pi$
$920$	0	0
$921$	20.4259	0.673057
$922$	0	0
$923$	−50.0009	−1.64580
$924$	0	0
$925$	−43.3326	−1.42477
$926$	0	0
$927$	0.573736	0.0188440
$928$	0	0
$929$	53.4317	1.75304	0.876519	−	0.481368i	$-0.159860\pi$
$929$	53.4317	1.75304	0.876519	+	0.481368i	$0.159860\pi$
$930$	0	0
$931$	6.59867	0.216263
$932$	0	0
$933$	12.1306	0.397139
$934$	0	0
$935$	100.795	3.29636
$936$	0	0
$937$	−8.01583	−0.261866	−0.130933	−	0.991391i	$-0.541797\pi$
$937$	−8.01583	−0.261866	−0.130933	+	0.991391i	$0.541797\pi$
$938$	0	0
$939$	−46.1690	−1.50667
$940$	0	0
$941$	−10.2934	−0.335556	−0.167778	−	0.985825i	$-0.553659\pi$
$941$	−10.2934	−0.335556	−0.167778	+	0.985825i	$0.553659\pi$
$942$	0	0
$943$	1.55552	0.0506547
$944$	0	0
$945$	10.7413	0.349413
$946$	0	0
$947$	−42.4848	−1.38057	−0.690285	−	0.723538i	$-0.742515\pi$
$947$	−42.4848	−1.38057	−0.690285	+	0.723538i	$0.742515\pi$
$948$	0	0
$949$	−34.5190	−1.12054
$950$	0	0
$951$	−27.6070	−0.895218
$952$	0	0
$953$	59.0011	1.91123	0.955617	−	0.294612i	$-0.0951905\pi$
$953$	59.0011	1.91123	0.955617	+	0.294612i	$0.0951905\pi$
$954$	0	0
$955$	25.5839	0.827874
$956$	0	0
$957$	−22.2184	−0.718220
$958$	0	0
$959$	−3.57680	−0.115501
$960$	0	0
$961$	30.7936	0.993341
$962$	0	0
$963$	4.56417	0.147078
$964$	0	0
$965$	68.5324	2.20614
$966$	0	0
$967$	0.781747	0.0251393	0.0125696	−	0.999921i	$-0.495999\pi$
$967$	0.781747	0.0251393	0.0125696	+	0.999921i	$0.495999\pi$
$968$	0	0
$969$	−8.35940	−0.268543
$970$	0	0
$971$	−51.5702	−1.65497	−0.827484	−	0.561489i	$-0.810229\pi$
$971$	−51.5702	−1.65497	−0.827484	+	0.561489i	$0.810229\pi$
$972$	0	0
$973$	6.43708	0.206363
$974$	0	0
$975$	−48.7583	−1.56152
$976$	0	0
$977$	3.02820	0.0968806	0.0484403	−	0.998826i	$-0.484575\pi$
$977$	3.02820	0.0968806	0.0484403	+	0.998826i	$0.484575\pi$
$978$	0	0
$979$	−7.57164	−0.241991
$980$	0	0
$981$	20.3887	0.650962
$982$	0	0
$983$	45.6956	1.45746	0.728732	−	0.684799i	$-0.240110\pi$
$983$	45.6956	1.45746	0.728732	+	0.684799i	$0.240110\pi$
$984$	0	0
$985$	−57.0432	−1.81755
$986$	0	0
$987$	6.11013	0.194488
$988$	0	0
$989$	8.05217	0.256044
$990$	0	0
$991$	30.3370	0.963685	0.481843	−	0.876258i	$-0.339968\pi$
$991$	30.3370	0.963685	0.481843	+	0.876258i	$0.339968\pi$
$992$	0	0
$993$	−11.1681	−0.354409
$994$	0	0
$995$	18.1288	0.574721
$996$	0	0
$997$	27.9443	0.885004	0.442502	−	0.896767i	$-0.354091\pi$
$997$	27.9443	0.885004	0.442502	+	0.896767i	$0.354091\pi$
$998$	0	0
$999$	−23.8177	−0.753558

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4304.2.a.e.1.2		4
4.3	odd	2		538.2.a.c.1.3	✓	4
12.11	even	2		4842.2.a.j.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.c.1.3	✓	4	4.3	odd	2
4304.2.a.e.1.2		4	1.1	even	1	trivial
4842.2.a.j.1.1		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:	\( N \)	\(=\)	\( 4304 = 2^{4} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4304.a (trivial)

\(f(q)\)	\(=\)	\(q-1.34415 q^{3} +3.90570 q^{5} +0.487928 q^{7} -1.19326 q^{9} +O(q^{10})\)	\(q-1.34415 q^{3} +3.90570 q^{5} +0.487928 q^{7} -1.19326 q^{9} -4.04948 q^{11} +3.53741 q^{13} -5.24985 q^{15} -6.37296 q^{17} -0.975857 q^{19} -0.655849 q^{21} -0.750146 q^{23} +10.2545 q^{25} +5.63637 q^{27} -4.08193 q^{29} +7.86089 q^{31} +5.44311 q^{33} +1.90570 q^{35} -4.22571 q^{37} -4.75481 q^{39} -2.07362 q^{41} -10.7341 q^{43} -4.66052 q^{45} -9.31636 q^{47} -6.76193 q^{49} +8.56622 q^{51} +11.1231 q^{53} -15.8161 q^{55} +1.31170 q^{57} +2.98052 q^{59} -7.53741 q^{61} -0.582225 q^{63} +13.8161 q^{65} -3.51207 q^{67} +1.00831 q^{69} -14.1349 q^{71} -9.75828 q^{73} -13.7836 q^{75} -1.97586 q^{77} -2.29222 q^{79} -3.99636 q^{81} +17.1324 q^{83} -24.8909 q^{85} +5.48673 q^{87} +1.86978 q^{89} +1.72600 q^{91} -10.5662 q^{93} -3.81141 q^{95} -9.63757 q^{97} +4.83208 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{3} + 5 q^{5} + q^{7} + 3 q^{9}+O(q^{10}) \) 4 * q - 3 * q^3 + 5 * q^5 + q^7 + 3 * q^9	\( 4 q - 3 q^{3} + 5 q^{5} + q^{7} + 3 q^{9} - 7 q^{11} + 4 q^{13} - 8 q^{15} + 4 q^{17} - 2 q^{19} - 5 q^{21} - 16 q^{23} - q^{25} - 6 q^{27} + q^{31} + q^{33} - 3 q^{35} - 2 q^{37} - 3 q^{39} - q^{41} - 9 q^{43} + 8 q^{45} - 13 q^{47} - 15 q^{49} - 3 q^{51} + 28 q^{53} - 13 q^{55} + 10 q^{57} - 19 q^{59} - 20 q^{61} - 12 q^{63} + 5 q^{65} - 15 q^{67} - 5 q^{69} - 15 q^{71} - 7 q^{73} - 12 q^{75} - 6 q^{77} + 17 q^{79} + 4 q^{81} - 6 q^{83} - 29 q^{85} + 34 q^{87} + 20 q^{89} + 18 q^{91} - 5 q^{93} + 6 q^{95} + 3 q^{97} + 16 q^{99}+O(q^{100}) \) 4 * q - 3 * q^3 + 5 * q^5 + q^7 + 3 * q^9 - 7 * q^11 + 4 * q^13 - 8 * q^15 + 4 * q^17 - 2 * q^19 - 5 * q^21 - 16 * q^23 - q^25 - 6 * q^27 + q^31 + q^33 - 3 * q^35 - 2 * q^37 - 3 * q^39 - q^41 - 9 * q^43 + 8 * q^45 - 13 * q^47 - 15 * q^49 - 3 * q^51 + 28 * q^53 - 13 * q^55 + 10 * q^57 - 19 * q^59 - 20 * q^61 - 12 * q^63 + 5 * q^65 - 15 * q^67 - 5 * q^69 - 15 * q^71 - 7 * q^73 - 12 * q^75 - 6 * q^77 + 17 * q^79 + 4 * q^81 - 6 * q^83 - 29 * q^85 + 34 * q^87 + 20 * q^89 + 18 * q^91 - 5 * q^93 + 6 * q^95 + 3 * q^97 + 16 * q^99

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