LMFDB - Embedded newform 4842.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4842,2,Mod(1,4842)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4842, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4842.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4842 = 2 \cdot 3^{2} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4842.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:	$38.6635646587$
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			1.1
Root			$2.90570$ of defining polynomial
Character	$\chi$	$=$	4842.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} -3.90570 q^{5} -0.487928 q^{7} +1.00000 q^{8} -3.90570 q^{10} -4.04948 q^{11} +3.53741 q^{13} -0.487928 q^{14} +1.00000 q^{16} +6.37296 q^{17} +0.975857 q^{19} -3.90570 q^{20} -4.04948 q^{22} -0.750146 q^{23} +10.2545 q^{25} +3.53741 q^{26} -0.487928 q^{28} +4.08193 q^{29} -7.86089 q^{31} +1.00000 q^{32} +6.37296 q^{34} +1.90570 q^{35} -4.22571 q^{37} +0.975857 q^{38} -3.90570 q^{40} +2.07362 q^{41} +10.7341 q^{43} -4.04948 q^{44} -0.750146 q^{46} -9.31636 q^{47} -6.76193 q^{49} +10.2545 q^{50} +3.53741 q^{52} -11.1231 q^{53} +15.8161 q^{55} -0.487928 q^{56} +4.08193 q^{58} +2.98052 q^{59} -7.53741 q^{61} -7.86089 q^{62} +1.00000 q^{64} -13.8161 q^{65} +3.51207 q^{67} +6.37296 q^{68} +1.90570 q^{70} -14.1349 q^{71} -9.75828 q^{73} -4.22571 q^{74} +0.975857 q^{76} +1.97586 q^{77} +2.29222 q^{79} -3.90570 q^{80} +2.07362 q^{82} +17.1324 q^{83} -24.8909 q^{85} +10.7341 q^{86} -4.04948 q^{88} -1.86978 q^{89} -1.72600 q^{91} -0.750146 q^{92} -9.31636 q^{94} -3.81141 q^{95} -9.63757 q^{97} -6.76193 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} - 5 q^{5} - q^{7} + 4 q^{8} - 5 q^{10} - 7 q^{11} + 4 q^{13} - q^{14} + 4 q^{16} - 4 q^{17} + 2 q^{19} - 5 q^{20} - 7 q^{22} - 16 q^{23} - q^{25} + 4 q^{26} - q^{28} - q^{31}+ \cdots - 15 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 - 5 * q^5 - q^7 + 4 * q^8 - 5 * q^10 - 7 * q^11 + 4 * q^13 - q^14 + 4 * q^16 - 4 * q^17 + 2 * q^19 - 5 * q^20 - 7 * q^22 - 16 * q^23 - q^25 + 4 * q^26 - q^28 - q^31 + 4 * q^32 - 4 * q^34 - 3 * q^35 - 2 * q^37 + 2 * q^38 - 5 * q^40 + q^41 + 9 * q^43 - 7 * q^44 - 16 * q^46 - 13 * q^47 - 15 * q^49 - q^50 + 4 * q^52 - 28 * q^53 + 13 * q^55 - q^56 - 19 * q^59 - 20 * q^61 - q^62 + 4 * q^64 - 5 * q^65 + 15 * q^67 - 4 * q^68 - 3 * q^70 - 15 * q^71 - 7 * q^73 - 2 * q^74 + 2 * q^76 + 6 * q^77 - 17 * q^79 - 5 * q^80 + q^82 - 6 * q^83 - 29 * q^85 + 9 * q^86 - 7 * q^88 - 20 * q^89 - 18 * q^91 - 16 * q^92 - 13 * q^94 + 6 * q^95 + 3 * q^97 - 15 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.90570	−1.74668	−0.873342	−	0.487108i	$-0.838052\pi$
$5$	−3.90570	−1.74668	−0.873342	+	0.487108i	$0.838052\pi$
$6$	0	0
$7$	−0.487928	−0.184420	−0.0922098	−	0.995740i	$-0.529393\pi$
$7$	−0.487928	−0.184420	−0.0922098	+	0.995740i	$0.529393\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.90570	−1.23509
$11$	−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	$0.336796\pi$
$14$	−0.487928	−0.130404
$15$	0	0
$16$	1.00000	0.250000
$17$	6.37296	1.54567	0.772835	−	0.634607i	$-0.218838\pi$
$17$	6.37296	1.54567	0.772835	+	0.634607i	$0.218838\pi$
$18$	0	0
$19$	0.975857	0.223877	0.111938	−	0.993715i	$-0.464294\pi$
$19$	0.975857	0.223877	0.111938	+	0.993715i	$0.464294\pi$
$20$	−3.90570	−0.873342
$21$	0	0
$22$	−4.04948	−0.863352
$23$	−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
$26$	3.53741	0.693743
$27$	0	0
$28$	−0.487928	−0.0922098
$29$	4.08193	0.757996	0.378998	−	0.925397i	$-0.376269\pi$
$29$	4.08193	0.757996	0.378998	+	0.925397i	$0.376269\pi$
$30$	0	0
$31$	−7.86089	−1.41186	−0.705929	−	0.708283i	$-0.749470\pi$
$31$	−7.86089	−1.41186	−0.705929	+	0.708283i	$0.749470\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.37296	1.09295
$35$	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	$0.612919\pi$
$38$	0.975857	0.158305
$39$	0	0
$40$	−3.90570	−0.617546
$41$	2.07362	0.323846	0.161923	−	0.986803i	$-0.448230\pi$
$41$	2.07362	0.323846	0.161923	+	0.986803i	$0.448230\pi$
$42$	0	0
$43$	10.7341	1.63694	0.818470	−	0.574549i	$-0.194822\pi$
$43$	10.7341	1.63694	0.818470	+	0.574549i	$0.194822\pi$
$44$	−4.04948	−0.610482
$45$	0	0
$46$	−0.750146	−0.110603
$47$	−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
$50$	10.2545	1.45021
$51$	0	0
$52$	3.53741	0.490550
$53$	−11.1231	−1.52788	−0.763938	−	0.645290i	$-0.776737\pi$
$53$	−11.1231	−1.52788	−0.763938	+	0.645290i	$0.776737\pi$
$54$	0	0
$55$	15.8161	2.13264
$56$	−0.487928	−0.0652022
$57$	0	0
$58$	4.08193	0.535984
$59$	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	$0.660283\pi$
$62$	−7.86089	−0.998334
$63$	0	0
$64$	1.00000	0.125000
$65$	−13.8161	−1.71367
$66$	0	0
$67$	3.51207	0.429068	0.214534	−	0.976717i	$-0.431177\pi$
$67$	3.51207	0.429068	0.214534	+	0.976717i	$0.431177\pi$
$68$	6.37296	0.772835
$69$	0	0
$70$	1.90570	0.227775
$71$	−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	$0.693468\pi$
$74$	−4.22571	−0.491229
$75$	0	0
$76$	0.975857	0.111938
$77$	1.97586	0.225170
$78$	0	0
$79$	2.29222	0.257895	0.128948	−	0.991651i	$-0.458840\pi$
$79$	2.29222	0.257895	0.128948	+	0.991651i	$0.458840\pi$
$80$	−3.90570	−0.436671
$81$	0	0
$82$	2.07362	0.228994
$83$	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
$86$	10.7341	1.15749
$87$	0	0
$88$	−4.04948	−0.431676
$89$	−1.86978	−0.198196	−0.0990981	−	0.995078i	$-0.531596\pi$
$89$	−1.86978	−0.198196	−0.0990981	+	0.995078i	$0.531596\pi$
$90$	0	0
$91$	−1.72600	−0.180934
$92$	−0.750146	−0.0782081
$93$	0	0
$94$	−9.31636	−0.960910
$95$	−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	$0.662738\pi$
$98$	−6.76193	−0.683058
$99$	0	0
$100$	10.2545	1.02545
$101$	−5.91748	−0.588812	−0.294406	−	0.955680i	$-0.595122\pi$
$101$	−5.91748	−0.588812	−0.294406	+	0.955680i	$0.595122\pi$
$102$	0	0
$103$	0.480814	0.0473760	0.0236880	−	0.999719i	$-0.492459\pi$
$103$	0.480814	0.0473760	0.0236880	+	0.999719i	$0.492459\pi$
$104$	3.53741	0.346872
$105$	0	0
$106$	−11.1231	−1.08037
$107$	−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	$0.805083\pi$
$110$	15.8161	1.50800
$111$	0	0
$112$	−0.487928	−0.0461049
$113$	−13.3194	−1.25299	−0.626493	−	0.779427i	$-0.715510\pi$
$113$	−13.3194	−1.25299	−0.626493	+	0.779427i	$0.715510\pi$
$114$	0	0
$115$	2.92985	0.273210
$116$	4.08193	0.378998
$117$	0	0
$118$	2.98052	0.274379
$119$	−3.10955	−0.285052
$120$	0	0
$121$	5.39830	0.490754
$122$	−7.53741	−0.682405
$123$	0	0
$124$	−7.86089	−0.705929
$125$	−20.5226	−1.83560
$126$	0	0
$127$	3.18962	0.283033	0.141516	−	0.989936i	$-0.454802\pi$
$127$	3.18962	0.283033	0.141516	+	0.989936i	$0.454802\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−13.8161	−1.21175
$131$	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
$134$	3.51207	0.303397
$135$	0	0
$136$	6.37296	0.546477
$137$	7.33059	0.626295	0.313147	−	0.949705i	$-0.398617\pi$
$137$	7.33059	0.626295	0.313147	+	0.949705i	$0.398617\pi$
$138$	0	0
$139$	−13.1927	−1.11899	−0.559494	−	0.828834i	$-0.689005\pi$
$139$	−13.1927	−1.11899	−0.559494	+	0.828834i	$0.689005\pi$
$140$	1.90570	0.161061
$141$	0	0
$142$	−14.1349	−1.18617
$143$	−14.3247	−1.19789
$144$	0	0
$145$	−15.9428	−1.32398
$146$	−9.75828	−0.807601
$147$	0	0
$148$	−4.22571	−0.347351
$149$	−19.2132	−1.57400	−0.787002	−	0.616950i	$-0.788368\pi$
$149$	−19.2132	−1.57400	−0.787002	+	0.616950i	$0.788368\pi$
$150$	0	0
$151$	−3.63637	−0.295924	−0.147962	−	0.988993i	$-0.547271\pi$
$151$	−3.63637	−0.295924	−0.147962	+	0.988993i	$0.547271\pi$
$152$	0.975857	0.0791524
$153$	0	0
$154$	1.97586	0.159219
$155$	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	$0.704449\pi$
$158$	2.29222	0.182359
$159$	0	0
$160$	−3.90570	−0.308773
$161$	0.366017	0.0288462
$162$	0	0
$163$	−4.77531	−0.374031	−0.187016	−	0.982357i	$-0.559882\pi$
$163$	−4.77531	−0.374031	−0.187016	+	0.982357i	$0.559882\pi$
$164$	2.07362	0.161923
$165$	0	0
$166$	17.1324	1.32974
$167$	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
$170$	−24.8909	−1.90904
$171$	0	0
$172$	10.7341	0.818470
$173$	24.0607	1.82930	0.914650	−	0.404247i	$-0.132466\pi$
$173$	24.0607	1.82930	0.914650	+	0.404247i	$0.132466\pi$
$174$	0	0
$175$	−5.00347	−0.378227
$176$	−4.04948	−0.305241
$177$	0	0
$178$	−1.86978	−0.140146
$179$	−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	$0.587316\pi$
$182$	−1.72600	−0.127940
$183$	0	0
$184$	−0.750146	−0.0553015
$185$	16.5044	1.21343
$186$	0	0
$187$	−25.8072	−1.88721
$188$	−9.31636	−0.679466
$189$	0	0
$190$	−3.81141	−0.276509
$191$	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	$0.282431\pi$
$194$	−9.63757	−0.691937
$195$	0	0
$196$	−6.76193	−0.482995
$197$	14.6051	1.04057	0.520286	−	0.853992i	$-0.325825\pi$
$197$	14.6051	1.04057	0.520286	+	0.853992i	$0.325825\pi$
$198$	0	0
$199$	−4.64162	−0.329036	−0.164518	−	0.986374i	$-0.552607\pi$
$199$	−4.64162	−0.329036	−0.164518	+	0.986374i	$0.552607\pi$
$200$	10.2545	0.725104
$201$	0	0
$202$	−5.91748	−0.416353
$203$	−1.99169	−0.139789
$204$	0	0
$205$	−8.09896	−0.565656
$206$	0.480814	0.0334999
$207$	0	0
$208$	3.53741	0.245275
$209$	−3.95171	−0.273346
$210$	0	0
$211$	15.7554	1.08465	0.542324	−	0.840169i	$-0.317545\pi$
$211$	15.7554	1.08465	0.542324	+	0.840169i	$0.317545\pi$
$212$	−11.1231	−0.763938
$213$	0	0
$214$	−3.82496	−0.261469
$215$	−41.9244	−2.85922
$216$	0	0
$217$	3.83555	0.260374
$218$	−17.0866	−1.15725
$219$	0	0
$220$	15.8161	1.06632
$221$	22.5438	1.51646
$222$	0	0
$223$	19.8791	1.33120	0.665602	−	0.746307i	$-0.268175\pi$
$223$	19.8791	1.33120	0.665602	+	0.746307i	$0.268175\pi$
$224$	−0.487928	−0.0326011
$225$	0	0
$226$	−13.3194	−0.885995
$227$	−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	$0.698478\pi$
$230$	2.92985	0.193188
$231$	0	0
$232$	4.08193	0.267992
$233$	1.67230	0.109556	0.0547779	−	0.998499i	$-0.482555\pi$
$233$	1.67230	0.109556	0.0547779	+	0.998499i	$0.482555\pi$
$234$	0	0
$235$	36.3870	2.37362
$236$	2.98052	0.194015
$237$	0	0
$238$	−3.10955	−0.201562
$239$	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	$0.699695\pi$
$242$	5.39830	0.347016
$243$	0	0
$244$	−7.53741	−0.482533
$245$	26.4101	1.68728
$246$	0	0
$247$	3.45200	0.219646
$248$	−7.86089	−0.499167
$249$	0	0
$250$	−20.5226	−1.29796
$251$	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
$254$	3.18962	0.200134
$255$	0	0
$256$	1.00000	0.0625000
$257$	−28.7806	−1.79529	−0.897644	−	0.440722i	$-0.854723\pi$
$257$	−28.7806	−1.79529	−0.897644	+	0.440722i	$0.854723\pi$
$258$	0	0
$259$	2.06184	0.128117
$260$	−13.8161	−0.856836
$261$	0	0
$262$	11.5975	0.716494
$263$	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
$266$	−0.476148	−0.0291945
$267$	0	0
$268$	3.51207	0.214534
$269$	−1.00000	−0.0609711
$269$	−1.00000	−0.0609711
$270$	0	0
$271$	−19.6940	−1.19632	−0.598162	−	0.801375i	$-0.704102\pi$
$271$	−19.6940	−1.19632	−0.598162	+	0.801375i	$0.704102\pi$
$272$	6.37296	0.386417
$273$	0	0
$274$	7.33059	0.442857
$275$	−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	$0.512531\pi$
$278$	−13.1927	−0.791244
$279$	0	0
$280$	1.90570	0.113888
$281$	−7.94749	−0.474107	−0.237054	−	0.971497i	$-0.576182\pi$
$281$	−7.94749	−0.474107	−0.237054	+	0.971497i	$0.576182\pi$
$282$	0	0
$283$	20.7544	1.23372	0.616861	−	0.787072i	$-0.288404\pi$
$283$	20.7544	1.23372	0.616861	+	0.787072i	$0.288404\pi$
$284$	−14.1349	−0.838751
$285$	0	0
$286$	−14.3247	−0.847036
$287$	−1.01178	−0.0597235
$288$	0	0
$289$	23.6146	1.38910
$290$	−15.9428	−0.936195
$291$	0	0
$292$	−9.75828	−0.571060
$293$	10.9994	0.642593	0.321296	−	0.946979i	$-0.395881\pi$
$293$	10.9994	0.642593	0.321296	+	0.946979i	$0.395881\pi$
$294$	0	0
$295$	−11.6410	−0.677767
$296$	−4.22571	−0.245614
$297$	0	0
$298$	−19.2132	−1.11299
$299$	−2.65357	−0.153460
$300$	0	0
$301$	−5.23749	−0.301884
$302$	−3.63637	−0.209250
$303$	0	0
$304$	0.975857	0.0559692
$305$	29.4389	1.68567
$306$	0	0
$307$	15.1961	0.867290	0.433645	−	0.901084i	$-0.357227\pi$
$307$	15.1961	0.867290	0.433645	+	0.901084i	$0.357227\pi$
$308$	1.97586	0.112585
$309$	0	0
$310$	30.7023	1.74377
$311$	−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	$0.0771995\pi$
$314$	−15.0118	−0.847164
$315$	0	0
$316$	2.29222	0.128948
$317$	−20.5386	−1.15356	−0.576781	−	0.816898i	$-0.695692\pi$
$317$	−20.5386	−1.15356	−0.576781	+	0.816898i	$0.695692\pi$
$318$	0	0
$319$	−16.5297	−0.925486
$320$	−3.90570	−0.218335
$321$	0	0
$322$	0.366017	0.0203974
$323$	6.21910	0.346040
$324$	0	0
$325$	36.2744	2.01214
$326$	−4.77531	−0.264480
$327$	0	0
$328$	2.07362	0.114497
$329$	4.54572	0.250614
$330$	0	0
$331$	−8.30867	−0.456686	−0.228343	−	0.973581i	$-0.573331\pi$
$331$	−8.30867	−0.456686	−0.228343	+	0.973581i	$0.573331\pi$
$332$	17.1324	0.940265
$333$	0	0
$334$	3.94163	0.215676
$335$	−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	$0.662689\pi$
$338$	−0.486734	−0.0264748
$339$	0	0
$340$	−24.8909	−1.34990
$341$	31.8325	1.72383
$342$	0	0
$343$	6.71483	0.362567
$344$	10.7341	0.578746
$345$	0	0
$346$	24.0607	1.29351
$347$	−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	$0.491699\pi$
$350$	−5.00347	−0.267447
$351$	0	0
$352$	−4.04948	−0.215838
$353$	−15.8928	−0.845886	−0.422943	−	0.906156i	$-0.639003\pi$
$353$	−15.8928	−0.845886	−0.422943	+	0.906156i	$0.639003\pi$
$354$	0	0
$355$	55.2067	2.93007
$356$	−1.86978	−0.0990981
$357$	0	0
$358$	−17.1745	−0.907698
$359$	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
$362$	−7.28875	−0.383088
$363$	0	0
$364$	−1.72600	−0.0904671
$365$	38.1130	1.99492
$366$	0	0
$367$	7.50802	0.391915	0.195958	−	0.980612i	$-0.437218\pi$
$367$	7.50802	0.391915	0.195958	+	0.980612i	$0.437218\pi$
$368$	−0.750146	−0.0391040
$369$	0	0
$370$	16.5044	0.858022
$371$	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	$0.482281\pi$
$374$	−25.8072	−1.33446
$375$	0	0
$376$	−9.31636	−0.480455
$377$	14.4395	0.743671
$378$	0	0
$379$	7.71644	0.396367	0.198183	−	0.980165i	$-0.436496\pi$
$379$	7.71644	0.396367	0.198183	+	0.980165i	$0.436496\pi$
$380$	−3.81141	−0.195521
$381$	0	0
$382$	6.55038	0.335147
$383$	−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
$386$	17.5467	0.893106
$387$	0	0
$388$	−9.63757	−0.489273
$389$	10.7025	0.542640	0.271320	−	0.962489i	$-0.412540\pi$
$389$	10.7025	0.542640	0.271320	+	0.962489i	$0.412540\pi$
$390$	0	0
$391$	−4.78065	−0.241768
$392$	−6.76193	−0.341529
$393$	0	0
$394$	14.6051	0.735795
$395$	−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	$0.469217\pi$
$398$	−4.64162	−0.232663
$399$	0	0
$400$	10.2545	0.512726
$401$	0.717694	0.0358399	0.0179200	−	0.999839i	$-0.494296\pi$
$401$	0.717694	0.0358399	0.0179200	+	0.999839i	$0.494296\pi$
$402$	0	0
$403$	−27.8072	−1.38517
$404$	−5.91748	−0.294406
$405$	0	0
$406$	−1.99169	−0.0988460
$407$	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	$0.594155\pi$
$410$	−8.09896	−0.399979
$411$	0	0
$412$	0.480814	0.0236880
$413$	−1.45428	−0.0715605
$414$	0	0
$415$	−66.9142	−3.28469
$416$	3.53741	0.173436
$417$	0	0
$418$	−3.95171	−0.193285
$419$	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	$0.538604\pi$
$422$	15.7554	0.766962
$423$	0	0
$424$	−11.1231	−0.540186
$425$	65.3516	3.17002
$426$	0	0
$427$	3.67772	0.177977
$428$	−3.82496	−0.184887
$429$	0	0
$430$	−41.9244	−2.02177
$431$	−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	$0.664622\pi$
$434$	3.83555	0.184112
$435$	0	0
$436$	−17.0866	−0.818300
$437$	−0.732035	−0.0350180
$438$	0	0
$439$	15.6900	0.748843	0.374421	−	0.927259i	$-0.377841\pi$
$439$	15.6900	0.748843	0.374421	+	0.927259i	$0.377841\pi$
$440$	15.8161	0.754002
$441$	0	0
$442$	22.5438	1.07230
$443$	−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
$446$	19.8791	0.941303
$447$	0	0
$448$	−0.487928	−0.0230524
$449$	−18.1978	−0.858805	−0.429403	−	0.903113i	$-0.641276\pi$
$449$	−18.1978	−0.858805	−0.429403	+	0.903113i	$0.641276\pi$
$450$	0	0
$451$	−8.39710	−0.395404
$452$	−13.3194	−0.626493
$453$	0	0
$454$	−27.7566	−1.30268
$455$	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	$0.228621\pi$
$458$	−17.6723	−0.825772
$459$	0	0
$460$	2.92985	0.136605
$461$	26.9152	1.25357	0.626783	−	0.779194i	$-0.284371\pi$
$461$	26.9152	1.25357	0.626783	+	0.779194i	$0.284371\pi$
$462$	0	0
$463$	−24.2349	−1.12629	−0.563145	−	0.826358i	$-0.690409\pi$
$463$	−24.2349	−1.12629	−0.563145	+	0.826358i	$0.690409\pi$
$464$	4.08193	0.189499
$465$	0	0
$466$	1.67230	0.0774676
$467$	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
$470$	36.3870	1.67841
$471$	0	0
$472$	2.98052	0.137190
$473$	−43.4677	−1.99865
$474$	0	0
$475$	10.0069	0.459150
$476$	−3.10955	−0.142526
$477$	0	0
$478$	24.0925	1.10196
$479$	−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
$482$	−18.2257	−0.830158
$483$	0	0
$484$	5.39830	0.245377
$485$	37.6415	1.70921
$486$	0	0
$487$	5.85514	0.265322	0.132661	−	0.991161i	$-0.457648\pi$
$487$	5.85514	0.265322	0.132661	+	0.991161i	$0.457648\pi$
$488$	−7.53741	−0.341202
$489$	0	0
$490$	26.4101	1.19309
$491$	−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
$494$	3.45200	0.155313
$495$	0	0
$496$	−7.86089	−0.352964
$497$	6.89681	0.309364
$498$	0	0
$499$	35.1119	1.57183	0.785913	−	0.618337i	$-0.212193\pi$
$499$	35.1119	1.57183	0.785913	+	0.618337i	$0.212193\pi$
$500$	−20.5226	−0.917798
$501$	0	0
$502$	1.21740	0.0543353
$503$	−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
$506$	3.03770	0.135042
$507$	0	0
$508$	3.18962	0.141516
$509$	−30.1456	−1.33618	−0.668090	−	0.744081i	$-0.732888\pi$
$509$	−30.1456	−1.33618	−0.668090	+	0.744081i	$0.732888\pi$
$510$	0	0
$511$	4.76134	0.210629
$512$	1.00000	0.0441942
$513$	0	0
$514$	−28.7806	−1.26946
$515$	−1.87792	−0.0827509
$516$	0	0
$517$	37.7264	1.65921
$518$	2.06184	0.0905922
$519$	0	0
$520$	−13.8161	−0.605875
$521$	−37.9804	−1.66395	−0.831975	−	0.554813i	$-0.812790\pi$
$521$	−37.9804	−1.66395	−0.831975	+	0.554813i	$0.812790\pi$
$522$	0	0
$523$	10.1012	0.441696	0.220848	−	0.975308i	$-0.429117\pi$
$523$	10.1012	0.441696	0.220848	+	0.975308i	$0.429117\pi$
$524$	11.5975	0.506638
$525$	0	0
$526$	4.35407	0.189846
$527$	−50.0971	−2.18227
$528$	0	0
$529$	−22.4373	−0.975534
$530$	43.4436	1.88707
$531$	0	0
$532$	−0.476148	−0.0206436
$533$	7.33526	0.317725
$534$	0	0
$535$	14.9392	0.645877
$536$	3.51207	0.151698
$537$	0	0
$538$	−1.00000	−0.0431131
$539$	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	$0.492704\pi$
$542$	−19.6940	−0.845929
$543$	0	0
$544$	6.37296	0.273238
$545$	66.7352	2.85862
$546$	0	0
$547$	8.62634	0.368836	0.184418	−	0.982848i	$-0.440960\pi$
$547$	8.62634	0.368836	0.184418	+	0.982848i	$0.440960\pi$
$548$	7.33059	0.313147
$549$	0	0
$550$	−41.5255	−1.77065
$551$	3.98338	0.169698
$552$	0	0
$553$	−1.11844	−0.0475609
$554$	−1.31009	−0.0556605
$555$	0	0
$556$	−13.1927	−0.559494
$557$	−39.3859	−1.66884	−0.834418	−	0.551132i	$-0.814196\pi$
$557$	−39.3859	−1.66884	−0.834418	+	0.551132i	$0.814196\pi$
$558$	0	0
$559$	37.9710	1.60600
$560$	1.90570	0.0805307
$561$	0	0
$562$	−7.94749	−0.335245
$563$	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
$566$	20.7544	0.872373
$567$	0	0
$568$	−14.1349	−0.593087
$569$	−18.1766	−0.762004	−0.381002	−	0.924574i	$-0.624421\pi$
$569$	−18.1766	−0.762004	−0.381002	+	0.924574i	$0.624421\pi$
$570$	0	0
$571$	−34.4465	−1.44154	−0.720771	−	0.693173i	$-0.756212\pi$
$571$	−34.4465	−1.44154	−0.720771	+	0.693173i	$0.756212\pi$
$572$	−14.3247	−0.598945
$573$	0	0
$574$	−1.01178	−0.0422309
$575$	−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	$0.459845\pi$
$578$	23.6146	0.982239
$579$	0	0
$580$	−15.9428	−0.661990
$581$	−8.35940	−0.346806
$582$	0	0
$583$	45.0428	1.86548
$584$	−9.75828	−0.403801
$585$	0	0
$586$	10.9994	0.454382
$587$	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
$590$	−11.6410	−0.479254
$591$	0	0
$592$	−4.22571	−0.173676
$593$	4.06998	0.167134	0.0835671	−	0.996502i	$-0.473369\pi$
$593$	4.06998	0.167134	0.0835671	+	0.996502i	$0.473369\pi$
$594$	0	0
$595$	12.1450	0.497895
$596$	−19.2132	−0.787002
$597$	0	0
$598$	−2.65357	−0.108513
$599$	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	$0.276383\pi$
$602$	−5.23749	−0.213464
$603$	0	0
$604$	−3.63637	−0.147962
$605$	−21.0842	−0.857193
$606$	0	0
$607$	25.5355	1.03645	0.518226	−	0.855244i	$-0.326593\pi$
$607$	25.5355	1.03645	0.518226	+	0.855244i	$0.326593\pi$
$608$	0.975857	0.0395762
$609$	0	0
$610$	29.4389	1.19195
$611$	−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	$0.616227\pi$
$614$	15.1961	0.613267
$615$	0	0
$616$	1.97586	0.0796095
$617$	−32.8371	−1.32197	−0.660986	−	0.750398i	$-0.729862\pi$
$617$	−32.8371	−1.32197	−0.660986	+	0.750398i	$0.729862\pi$
$618$	0	0
$619$	10.5824	0.425342	0.212671	−	0.977124i	$-0.431784\pi$
$619$	10.5824	0.425342	0.212671	+	0.977124i	$0.431784\pi$
$620$	30.7023	1.23303
$621$	0	0
$622$	−9.02476	−0.361860
$623$	0.912319	0.0365513
$624$	0	0
$625$	28.8826	1.15530
$626$	34.3481	1.37282
$627$	0	0
$628$	−15.0118	−0.599035
$629$	−26.9303	−1.07378
$630$	0	0
$631$	−0.902404	−0.0359241	−0.0179621	−	0.999839i	$-0.505718\pi$
$631$	−0.902404	−0.0359241	−0.0179621	+	0.999839i	$0.505718\pi$
$632$	2.29222	0.0911797
$633$	0	0
$634$	−20.5386	−0.815692
$635$	−12.4577	−0.494368
$636$	0	0
$637$	−23.9197	−0.947733
$638$	−16.5297	−0.654418
$639$	0	0
$640$	−3.90570	−0.154386
$641$	19.2444	0.760109	0.380055	−	0.924964i	$-0.375905\pi$
$641$	19.2444	0.760109	0.380055	+	0.924964i	$0.375905\pi$
$642$	0	0
$643$	34.2804	1.35189	0.675944	−	0.736953i	$-0.263736\pi$
$643$	34.2804	1.35189	0.675944	+	0.736953i	$0.263736\pi$
$644$	0.366017	0.0144231
$645$	0	0
$646$	6.21910	0.244687
$647$	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
$650$	36.2744	1.42280
$651$	0	0
$652$	−4.77531	−0.187016
$653$	21.5877	0.844794	0.422397	−	0.906411i	$-0.361189\pi$
$653$	21.5877	0.844794	0.422397	+	0.906411i	$0.361189\pi$
$654$	0	0
$655$	−45.2963	−1.76987
$656$	2.07362	0.0809614
$657$	0	0
$658$	4.54572	0.177211
$659$	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	$0.625281\pi$
$662$	−8.30867	−0.322926
$663$	0	0
$664$	17.1324	0.664868
$665$	1.85969	0.0721158
$666$	0	0
$667$	−3.06204	−0.118563
$668$	3.94163	0.152506
$669$	0	0
$670$	−13.7171	−0.529938
$671$	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	$0.582056\pi$
$674$	−17.9588	−0.691746
$675$	0	0
$676$	−0.486734	−0.0187205
$677$	12.7629	0.490520	0.245260	−	0.969457i	$-0.421127\pi$
$677$	12.7629	0.490520	0.245260	+	0.969457i	$0.421127\pi$
$678$	0	0
$679$	4.70244	0.180463
$680$	−24.8909	−0.954522
$681$	0	0
$682$	31.8325	1.21893
$683$	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
$686$	6.71483	0.256374
$687$	0	0
$688$	10.7341	0.409235
$689$	−39.3470	−1.49900
$690$	0	0
$691$	25.3081	0.962767	0.481384	−	0.876510i	$-0.340134\pi$
$691$	25.3081	0.962767	0.481384	+	0.876510i	$0.340134\pi$
$692$	24.0607	0.914650
$693$	0	0
$694$	−6.69769	−0.254241
$695$	51.5267	1.95452
$696$	0	0
$697$	13.2151	0.500559
$698$	0.974251	0.0368759
$699$	0	0
$700$	−5.00347	−0.189113
$701$	15.5107	0.585831	0.292916	−	0.956138i	$-0.405374\pi$
$701$	15.5107	0.585831	0.292916	+	0.956138i	$0.405374\pi$
$702$	0	0
$703$	−4.12369	−0.155528
$704$	−4.04948	−0.152621
$705$	0	0
$706$	−15.8928	−0.598132
$707$	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	$0.480596\pi$
$710$	55.2067	2.07187
$711$	0	0
$712$	−1.86978	−0.0700730
$713$	5.89681	0.220837
$714$	0	0
$715$	55.9479	2.09233
$716$	−17.1745	−0.641839
$717$	0	0
$718$	8.66535	0.323388
$719$	−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
$722$	−18.0477	−0.671666
$723$	0	0
$724$	−7.28875	−0.270884
$725$	41.8583	1.55458
$726$	0	0
$727$	28.4941	1.05679	0.528395	−	0.848999i	$-0.322794\pi$
$727$	28.4941	1.05679	0.528395	+	0.848999i	$0.322794\pi$
$728$	−1.72600	−0.0639699
$729$	0	0
$730$	38.1130	1.41062
$731$	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	$0.304811\pi$
$734$	7.50802	0.277126
$735$	0	0
$736$	−0.750146	−0.0276507
$737$	−14.2221	−0.523877
$738$	0	0
$739$	22.7409	0.836538	0.418269	−	0.908323i	$-0.362637\pi$
$739$	22.7409	0.836538	0.418269	+	0.908323i	$0.362637\pi$
$740$	16.5044	0.606713
$741$	0	0
$742$	5.42728	0.199242
$743$	−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
$746$	2.14911	0.0786847
$747$	0	0
$748$	−25.8072	−0.943604
$749$	1.86631	0.0681934
$750$	0	0
$751$	−48.0493	−1.75334	−0.876672	−	0.481089i	$-0.840241\pi$
$751$	−48.0493	−1.75334	−0.876672	+	0.481089i	$0.840241\pi$
$752$	−9.31636	−0.339733
$753$	0	0
$754$	14.4395	0.525854
$755$	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	$0.0962279\pi$
$758$	7.71644	0.280274
$759$	0	0
$760$	−3.81141	−0.138254
$761$	−50.4604	−1.82919	−0.914594	−	0.404373i	$-0.867490\pi$
$761$	−50.4604	−1.82919	−0.914594	+	0.404373i	$0.867490\pi$
$762$	0	0
$763$	8.33704	0.301821
$764$	6.55038	0.236985
$765$	0	0
$766$	−1.88104	−0.0679646
$767$	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	$0.745575\pi$
$770$	−7.71711	−0.278105
$771$	0	0
$772$	17.5467	0.631521
$773$	27.8089	1.00022	0.500108	−	0.865963i	$-0.333294\pi$
$773$	27.8089	1.00022	0.500108	+	0.865963i	$0.333294\pi$
$774$	0	0
$775$	−80.6096	−2.89558
$776$	−9.63757	−0.345968
$777$	0	0
$778$	10.7025	0.383704
$779$	2.02356	0.0725016
$780$	0	0
$781$	57.2390	2.04817
$782$	−4.78065	−0.170956
$783$	0	0
$784$	−6.76193	−0.241497
$785$	58.6316	2.09265
$786$	0	0
$787$	−35.0440	−1.24918	−0.624592	−	0.780951i	$-0.714735\pi$
$787$	−35.0440	−1.24918	−0.624592	+	0.780951i	$0.714735\pi$
$788$	14.6051	0.520286
$789$	0	0
$790$	−8.95274	−0.318524
$791$	6.49893	0.231075
$792$	0	0
$793$	−26.6629	−0.946827
$794$	3.84774	0.136551
$795$	0	0
$796$	−4.64162	−0.164518
$797$	−44.3028	−1.56929	−0.784643	−	0.619947i	$-0.787154\pi$
$797$	−44.3028	−1.56929	−0.784643	+	0.619947i	$0.787154\pi$
$798$	0	0
$799$	−59.3728	−2.10046
$800$	10.2545	0.362552
$801$	0	0
$802$	0.717694	0.0253426
$803$	39.5160	1.39449
$804$	0	0
$805$	−1.42956	−0.0503852
$806$	−27.8072	−0.979466
$807$	0	0
$808$	−5.91748	−0.208176
$809$	−17.8666	−0.628157	−0.314079	−	0.949397i	$-0.601696\pi$
$809$	−17.8666	−0.628157	−0.314079	+	0.949397i	$0.601696\pi$
$810$	0	0
$811$	7.94755	0.279076	0.139538	−	0.990217i	$-0.455438\pi$
$811$	7.94755	0.279076	0.139538	+	0.990217i	$0.455438\pi$
$812$	−1.99169	−0.0698947
$813$	0	0
$814$	17.1119	0.599773
$815$	18.6510	0.653314
$816$	0	0
$817$	10.4750	0.366473
$818$	−11.7906	−0.412247
$819$	0	0
$820$	−8.09896	−0.282828
$821$	−14.0212	−0.489342	−0.244671	−	0.969606i	$-0.578680\pi$
$821$	−14.0212	−0.489342	−0.244671	+	0.969606i	$0.578680\pi$
$822$	0	0
$823$	−56.0461	−1.95364	−0.976822	−	0.214052i	$-0.931334\pi$
$823$	−56.0461	−1.95364	−0.976822	+	0.214052i	$0.931334\pi$
$824$	0.480814	0.0167500
$825$	0	0
$826$	−1.45428	−0.0506009
$827$	−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	$0.460737\pi$
$830$	−66.9142	−2.32263
$831$	0	0
$832$	3.53741	0.122638
$833$	−43.0935	−1.49310
$834$	0	0
$835$	−15.3948	−0.532760
$836$	−3.95171	−0.136673
$837$	0	0
$838$	7.07076	0.244256
$839$	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
$842$	−4.96469	−0.171094
$843$	0	0
$844$	15.7554	0.542324
$845$	1.90104	0.0653977
$846$	0	0
$847$	−2.63398	−0.0905047
$848$	−11.1231	−0.381969
$849$	0	0
$850$	65.3516	2.24154
$851$	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	$0.643951\pi$
$854$	3.67772	0.125849
$855$	0	0
$856$	−3.82496	−0.130735
$857$	47.9302	1.63726	0.818632	−	0.574318i	$-0.194733\pi$
$857$	47.9302	1.63726	0.818632	+	0.574318i	$0.194733\pi$
$858$	0	0
$859$	−6.23647	−0.212786	−0.106393	−	0.994324i	$-0.533930\pi$
$859$	−6.23647	−0.212786	−0.106393	+	0.994324i	$0.533930\pi$
$860$	−41.9244	−1.42961
$861$	0	0
$862$	−25.6140	−0.872417
$863$	−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
$866$	−20.5767	−0.699226
$867$	0	0
$868$	3.83555	0.130187
$869$	−9.28231	−0.314881
$870$	0	0
$871$	12.4236	0.420959
$872$	−17.0866	−0.578625
$873$	0	0
$874$	−0.732035	−0.0247614
$875$	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	$0.248198\pi$
$878$	15.6900	0.529512
$879$	0	0
$880$	15.8161	0.533160
$881$	35.0575	1.18112	0.590558	−	0.806995i	$-0.298908\pi$
$881$	35.0575	1.18112	0.590558	+	0.806995i	$0.298908\pi$
$882$	0	0
$883$	49.3854	1.66195	0.830976	−	0.556308i	$-0.187783\pi$
$883$	49.3854	1.66195	0.830976	+	0.556308i	$0.187783\pi$
$884$	22.5438	0.758229
$885$	0	0
$886$	−29.0018	−0.974335
$887$	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
$890$	7.30281	0.244791
$891$	0	0
$892$	19.8791	0.665602
$893$	−9.09144	−0.304233
$894$	0	0
$895$	67.0783	2.24218
$896$	−0.487928	−0.0163005
$897$	0	0
$898$	−18.1978	−0.607267
$899$	−32.0876	−1.07018
$900$	0	0
$901$	−70.8871	−2.36159
$902$	−8.39710	−0.279593
$903$	0	0
$904$	−13.3194	−0.442997
$905$	28.4677	0.946298
$906$	0	0
$907$	−57.1062	−1.89618	−0.948090	−	0.318003i	$-0.896988\pi$
$907$	−57.1062	−1.89618	−0.948090	+	0.318003i	$0.896988\pi$
$908$	−27.7566	−0.921136
$909$	0	0
$910$	6.74125	0.223470
$911$	−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
$914$	32.1933	1.06486
$915$	0	0
$916$	−17.6723	−0.583909
$917$	−5.65874	−0.186868
$918$	0	0
$919$	−24.6902	−0.814453	−0.407227	−	0.913327i	$-0.633504\pi$
$919$	−24.6902	−0.814453	−0.407227	+	0.913327i	$0.633504\pi$
$920$	2.92985	0.0965942
$921$	0	0
$922$	26.9152	0.886405
$923$	−50.0009	−1.64580
$924$	0	0
$925$	−43.3326	−1.42477
$926$	−24.2349	−0.796407
$927$	0	0
$928$	4.08193	0.133996
$929$	−53.4317	−1.75304	−0.876519	−	0.481368i	$-0.840140\pi$
$929$	−53.4317	−1.75304	−0.876519	+	0.481368i	$0.840140\pi$
$930$	0	0
$931$	−6.59867	−0.216263
$932$	1.67230	0.0547779
$933$	0	0
$934$	17.7628	0.581216
$935$	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	$0.541797\pi$
$938$	−1.71364	−0.0559523
$939$	0	0
$940$	36.3870	1.18681
$941$	10.2934	0.335556	0.167778	−	0.985825i	$-0.446341\pi$
$941$	10.2934	0.335556	0.167778	+	0.985825i	$0.446341\pi$
$942$	0	0
$943$	−1.55552	−0.0506547
$944$	2.98052	0.0970077
$945$	0	0
$946$	−43.4677	−1.41326
$947$	−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
$950$	10.0069	0.324668
$951$	0	0
$952$	−3.10955	−0.100781
$953$	−59.0011	−1.91123	−0.955617	−	0.294612i	$-0.904809\pi$
$953$	−59.0011	−1.91123	−0.955617	+	0.294612i	$0.904809\pi$
$954$	0	0
$955$	−25.5839	−0.827874
$956$	24.0925	0.779206
$957$	0	0
$958$	−11.4930	−0.371323
$959$	−3.57680	−0.115501
$960$	0	0
$961$	30.7936	0.993341
$962$	−14.9481	−0.481945
$963$	0	0
$964$	−18.2257	−0.587011
$965$	−68.5324	−2.20614
$966$	0	0
$967$	−0.781747	−0.0251393	−0.0125696	−	0.999921i	$-0.504001\pi$
$967$	−0.781747	−0.0251393	−0.0125696	+	0.999921i	$0.504001\pi$
$968$	5.39830	0.173508
$969$	0	0
$970$	37.6415	1.20860
$971$	−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
$974$	5.85514	0.187611
$975$	0	0
$976$	−7.53741	−0.241267
$977$	−3.02820	−0.0968806	−0.0484403	−	0.998826i	$-0.515425\pi$
$977$	−3.02820	−0.0968806	−0.0484403	+	0.998826i	$0.515425\pi$
$978$	0	0
$979$	7.57164	0.241991
$980$	26.4101	0.843639
$981$	0	0
$982$	−18.6544	−0.595287
$983$	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
$986$	26.0140	0.828454
$987$	0	0
$988$	3.45200	0.109823
$989$	−8.05217	−0.256044
$990$	0	0
$991$	−30.3370	−0.963685	−0.481843	−	0.876258i	$-0.660032\pi$
$991$	−30.3370	−0.963685	−0.481843	+	0.876258i	$0.660032\pi$
$992$	−7.86089	−0.249583
$993$	0	0
$994$	6.89681	0.218754
$995$	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	$0.354091\pi$
$998$	35.1119	1.11145
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.c.1.3	✓	4	3.2	odd	2
4304.2.a.e.1.2		4	12.11	even	2
4842.2.a.j.1.1		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4842 = 2 \cdot 3^{2} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4842.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.90570 q^{5} -0.487928 q^{7} +1.00000 q^{8} -3.90570 q^{10} -4.04948 q^{11} +3.53741 q^{13} -0.487928 q^{14} +1.00000 q^{16} +6.37296 q^{17} +0.975857 q^{19} -3.90570 q^{20} -4.04948 q^{22} -0.750146 q^{23} +10.2545 q^{25} +3.53741 q^{26} -0.487928 q^{28} +4.08193 q^{29} -7.86089 q^{31} +1.00000 q^{32} +6.37296 q^{34} +1.90570 q^{35} -4.22571 q^{37} +0.975857 q^{38} -3.90570 q^{40} +2.07362 q^{41} +10.7341 q^{43} -4.04948 q^{44} -0.750146 q^{46} -9.31636 q^{47} -6.76193 q^{49} +10.2545 q^{50} +3.53741 q^{52} -11.1231 q^{53} +15.8161 q^{55} -0.487928 q^{56} +4.08193 q^{58} +2.98052 q^{59} -7.53741 q^{61} -7.86089 q^{62} +1.00000 q^{64} -13.8161 q^{65} +3.51207 q^{67} +6.37296 q^{68} +1.90570 q^{70} -14.1349 q^{71} -9.75828 q^{73} -4.22571 q^{74} +0.975857 q^{76} +1.97586 q^{77} +2.29222 q^{79} -3.90570 q^{80} +2.07362 q^{82} +17.1324 q^{83} -24.8909 q^{85} +10.7341 q^{86} -4.04948 q^{88} -1.86978 q^{89} -1.72600 q^{91} -0.750146 q^{92} -9.31636 q^{94} -3.81141 q^{95} -9.63757 q^{97} -6.76193 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} - 5 q^{5} - q^{7} + 4 q^{8} - 5 q^{10} - 7 q^{11} + 4 q^{13} - q^{14} + 4 q^{16} - 4 q^{17} + 2 q^{19} - 5 q^{20} - 7 q^{22} - 16 q^{23} - q^{25} + 4 q^{26} - q^{28} - q^{31}+ \cdots - 15 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 - 5 * q^5 - q^7 + 4 * q^8 - 5 * q^10 - 7 * q^11 + 4 * q^13 - q^14 + 4 * q^16 - 4 * q^17 + 2 * q^19 - 5 * q^20 - 7 * q^22 - 16 * q^23 - q^25 + 4 * q^26 - q^28 - q^31 + 4 * q^32 - 4 * q^34 - 3 * q^35 - 2 * q^37 + 2 * q^38 - 5 * q^40 + q^41 + 9 * q^43 - 7 * q^44 - 16 * q^46 - 13 * q^47 - 15 * q^49 - q^50 + 4 * q^52 - 28 * q^53 + 13 * q^55 - q^56 - 19 * q^59 - 20 * q^61 - q^62 + 4 * q^64 - 5 * q^65 + 15 * q^67 - 4 * q^68 - 3 * q^70 - 15 * q^71 - 7 * q^73 - 2 * q^74 + 2 * q^76 + 6 * q^77 - 17 * q^79 - 5 * q^80 + q^82 - 6 * q^83 - 29 * q^85 + 9 * q^86 - 7 * q^88 - 20 * q^89 - 18 * q^91 - 16 * q^92 - 13 * q^94 + 6 * q^95 + 3 * q^97 - 15 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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