LMFDB - Embedded newform 128.4.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [128,4,Mod(1,128)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(128, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("128.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$128 = 2^{7}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	128.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$7.55224448073$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 3$ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2^{2}$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	128.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.92820 q^{3} +11.8564 q^{5} -9.85641 q^{7} +52.7128 q^{9} +39.0718 q^{11} -91.5692 q^{13} +105.856 q^{15} -37.1384 q^{17} -46.4974 q^{19} -88.0000 q^{21} +120.708 q^{23} +15.5744 q^{25} +229.569 q^{27} +27.2820 q^{29} +81.1487 q^{31} +348.841 q^{33} -116.862 q^{35} -10.9948 q^{37} -817.549 q^{39} -205.426 q^{41} +115.359 q^{43} +624.985 q^{45} +312.841 q^{47} -245.851 q^{49} -331.580 q^{51} -90.9948 q^{53} +463.251 q^{55} -415.138 q^{57} -550.631 q^{59} -630.974 q^{61} -519.559 q^{63} -1085.68 q^{65} -661.041 q^{67} +1077.70 q^{69} -494.985 q^{71} +566.267 q^{73} +139.051 q^{75} -385.108 q^{77} -49.4153 q^{79} +626.395 q^{81} +564.067 q^{83} -440.328 q^{85} +243.580 q^{87} +1089.67 q^{89} +902.543 q^{91} +724.513 q^{93} -551.292 q^{95} +464.605 q^{97} +2059.58 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 4 q^{3} - 4 q^{5} + 8 q^{7} + 50 q^{9} + 92 q^{11} - 100 q^{13} + 184 q^{15} + 92 q^{17} + 4 q^{19} - 176 q^{21} - 8 q^{23} + 142 q^{25} + 376 q^{27} - 84 q^{29} + 384 q^{31} + 88 q^{33} - 400 q^{35}+ \cdots + 1916 q^{99}+O(q^{100})$ 2 * q + 4 * q^3 - 4 * q^5 + 8 * q^7 + 50 * q^9 + 92 * q^11 - 100 * q^13 + 184 * q^15 + 92 * q^17 + 4 * q^19 - 176 * q^21 - 8 * q^23 + 142 * q^25 + 376 * q^27 - 84 * q^29 + 384 * q^31 + 88 * q^33 - 400 * q^35 + 172 * q^37 - 776 * q^39 - 300 * q^41 + 300 * q^43 + 668 * q^45 + 16 * q^47 - 270 * q^49 - 968 * q^51 + 12 * q^53 - 376 * q^55 - 664 * q^57 - 644 * q^59 - 292 * q^61 - 568 * q^63 - 952 * q^65 - 172 * q^67 + 1712 * q^69 - 408 * q^71 + 412 * q^73 - 484 * q^75 + 560 * q^77 + 400 * q^79 - 22 * q^81 + 948 * q^83 - 2488 * q^85 + 792 * q^87 + 572 * q^89 + 752 * q^91 - 768 * q^93 - 1352 * q^95 + 2204 * q^97 + 1916 * q^99

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
$3$	8.92820	1.71823	0.859117	−	0.511780i	$-0.171014\pi$
$3$	8.92820	1.71823	0.859117	+	0.511780i	$0.171014\pi$
$4$	0	0
$5$	11.8564	1.06047	0.530235	−	0.847851i	$-0.322104\pi$
$5$	11.8564	1.06047	0.530235	+	0.847851i	$0.322104\pi$
$6$	0	0
$7$	−9.85641	−0.532196	−0.266098	−	0.963946i	$-0.585734\pi$
$7$	−9.85641	−0.532196	−0.266098	+	0.963946i	$0.585734\pi$
$8$	0	0
$9$	52.7128	1.95233
$10$	0	0
$11$	39.0718	1.07096	0.535481	−	0.844547i	$-0.320130\pi$
$11$	39.0718	1.07096	0.535481	+	0.844547i	$0.320130\pi$
$12$	0	0
$13$	−91.5692	−1.95359	−0.976797	−	0.214166i	$-0.931297\pi$
$13$	−91.5692	−1.95359	−0.976797	+	0.214166i	$0.931297\pi$
$14$	0	0
$15$	105.856	1.82213
$16$	0	0
$17$	−37.1384	−0.529847	−0.264923	−	0.964269i	$-0.585347\pi$
$17$	−37.1384	−0.529847	−0.264923	+	0.964269i	$0.585347\pi$
$18$	0	0
$19$	−46.4974	−0.561434	−0.280717	−	0.959791i	$-0.590572\pi$
$19$	−46.4974	−0.561434	−0.280717	+	0.959791i	$0.590572\pi$
$20$	0	0
$21$	−88.0000	−0.914437
$22$	0	0
$23$	120.708	1.09432	0.547158	−	0.837029i	$-0.315710\pi$
$23$	120.708	1.09432	0.547158	+	0.837029i	$0.315710\pi$
$24$	0	0
$25$	15.5744	0.124595
$26$	0	0
$27$	229.569	1.63632
$28$	0	0
$29$	27.2820	0.174695	0.0873473	−	0.996178i	$-0.472161\pi$
$29$	27.2820	0.174695	0.0873473	+	0.996178i	$0.472161\pi$
$30$	0	0
$31$	81.1487	0.470153	0.235077	−	0.971977i	$-0.424466\pi$
$31$	81.1487	0.470153	0.235077	+	0.971977i	$0.424466\pi$
$32$	0	0
$33$	348.841	1.84016
$34$	0	0
$35$	−116.862	−0.564377
$36$	0	0
$37$	−10.9948	−0.0488525	−0.0244262	−	0.999702i	$-0.507776\pi$
$37$	−10.9948	−0.0488525	−0.0244262	+	0.999702i	$0.507776\pi$
$38$	0	0
$39$	−817.549	−3.35673
$40$	0	0
$41$	−205.426	−0.782490	−0.391245	−	0.920287i	$-0.627956\pi$
$41$	−205.426	−0.782490	−0.391245	+	0.920287i	$0.627956\pi$
$42$	0	0
$43$	115.359	0.409118	0.204559	−	0.978854i	$-0.434424\pi$
$43$	115.359	0.409118	0.204559	+	0.978854i	$0.434424\pi$
$44$	0	0
$45$	624.985	2.07038
$46$	0	0
$47$	312.841	0.970905	0.485453	−	0.874263i	$-0.338655\pi$
$47$	312.841	0.970905	0.485453	+	0.874263i	$0.338655\pi$
$48$	0	0
$49$	−245.851	−0.716767
$50$	0	0
$51$	−331.580	−0.910400
$52$	0	0
$53$	−90.9948	−0.235832	−0.117916	−	0.993024i	$-0.537621\pi$
$53$	−90.9948	−0.235832	−0.117916	+	0.993024i	$0.537621\pi$
$54$	0	0
$55$	463.251	1.13572
$56$	0	0
$57$	−415.138	−0.964674
$58$	0	0
$59$	−550.631	−1.21502	−0.607509	−	0.794313i	$-0.707831\pi$
$59$	−550.631	−1.21502	−0.607509	+	0.794313i	$0.707831\pi$
$60$	0	0
$61$	−630.974	−1.32439	−0.662196	−	0.749330i	$-0.730376\pi$
$61$	−630.974	−1.32439	−0.662196	+	0.749330i	$0.730376\pi$
$62$	0	0
$63$	−519.559	−1.03902
$64$	0	0
$65$	−1085.68	−2.07173
$66$	0	0
$67$	−661.041	−1.20536	−0.602679	−	0.797984i	$-0.705900\pi$
$67$	−661.041	−1.20536	−0.602679	+	0.797984i	$0.705900\pi$
$68$	0	0
$69$	1077.70	1.88029
$70$	0	0
$71$	−494.985	−0.827378	−0.413689	−	0.910418i	$-0.635760\pi$
$71$	−494.985	−0.827378	−0.413689	+	0.910418i	$0.635760\pi$
$72$	0	0
$73$	566.267	0.907897	0.453949	−	0.891028i	$-0.350015\pi$
$73$	566.267	0.907897	0.453949	+	0.891028i	$0.350015\pi$
$74$	0	0
$75$	139.051	0.214083
$76$	0	0
$77$	−385.108	−0.569962
$78$	0	0
$79$	−49.4153	−0.0703754	−0.0351877	−	0.999381i	$-0.511203\pi$
$79$	−49.4153	−0.0703754	−0.0351877	+	0.999381i	$0.511203\pi$
$80$	0	0
$81$	626.395	0.859252
$82$	0	0
$83$	564.067	0.745956	0.372978	−	0.927840i	$-0.378337\pi$
$83$	564.067	0.745956	0.372978	+	0.927840i	$0.378337\pi$
$84$	0	0
$85$	−440.328	−0.561886
$86$	0	0
$87$	243.580	0.300166
$88$	0	0
$89$	1089.67	1.29781	0.648904	−	0.760870i	$-0.275228\pi$
$89$	1089.67	1.29781	0.648904	+	0.760870i	$0.275228\pi$
$90$	0	0
$91$	902.543	1.03970
$92$	0	0
$93$	724.513	0.807833
$94$	0	0
$95$	−551.292	−0.595383
$96$	0	0
$97$	464.605	0.486325	0.243162	−	0.969986i	$-0.421815\pi$
$97$	464.605	0.486325	0.243162	+	0.969986i	$0.421815\pi$
$98$	0	0
$99$	2059.58	2.09087
$100$	0	0
$101$	965.005	0.950709	0.475354	−	0.879794i	$-0.342320\pi$
$101$	965.005	0.950709	0.475354	+	0.879794i	$0.342320\pi$
$102$	0	0
$103$	1828.99	1.74967	0.874836	−	0.484419i	$-0.160969\pi$
$103$	1828.99	1.74967	0.874836	+	0.484419i	$0.160969\pi$
$104$	0	0
$105$	−1043.36	−0.969732
$106$	0	0
$107$	−1726.59	−1.55996	−0.779980	−	0.625805i	$-0.784771\pi$
$107$	−1726.59	−1.55996	−0.779980	+	0.625805i	$0.784771\pi$
$108$	0	0
$109$	−423.015	−0.371720	−0.185860	−	0.982576i	$-0.559507\pi$
$109$	−423.015	−0.371720	−0.185860	+	0.982576i	$0.559507\pi$
$110$	0	0
$111$	−98.1642	−0.0839400
$112$	0	0
$113$	763.108	0.635284	0.317642	−	0.948211i	$-0.397109\pi$
$113$	763.108	0.635284	0.317642	+	0.948211i	$0.397109\pi$
$114$	0	0
$115$	1431.16	1.16049
$116$	0	0
$117$	−4826.87	−3.81405
$118$	0	0
$119$	366.052	0.281982
$120$	0	0
$121$	195.605	0.146961
$122$	0	0
$123$	−1834.08	−1.34450
$124$	0	0
$125$	−1297.39	−0.928340
$126$	0	0
$127$	2159.38	1.50877	0.754387	−	0.656430i	$-0.227934\pi$
$127$	2159.38	1.50877	0.754387	+	0.656430i	$0.227934\pi$
$128$	0	0
$129$	1029.95	0.702961
$130$	0	0
$131$	−710.723	−0.474016	−0.237008	−	0.971508i	$-0.576167\pi$
$131$	−710.723	−0.474016	−0.237008	+	0.971508i	$0.576167\pi$
$132$	0	0
$133$	458.297	0.298793
$134$	0	0
$135$	2721.87	1.73527
$136$	0	0
$137$	−987.128	−0.615592	−0.307796	−	0.951452i	$-0.599591\pi$
$137$	−987.128	−0.615592	−0.307796	+	0.951452i	$0.599591\pi$
$138$	0	0
$139$	2334.15	1.42432	0.712158	−	0.702019i	$-0.247718\pi$
$139$	2334.15	1.42432	0.712158	+	0.702019i	$0.247718\pi$
$140$	0	0
$141$	2793.11	1.66824
$142$	0	0
$143$	−3577.77	−2.09223
$144$	0	0
$145$	323.467	0.185258
$146$	0	0
$147$	−2195.01	−1.23157
$148$	0	0
$149$	1792.29	0.985435	0.492718	−	0.870189i	$-0.336004\pi$
$149$	1792.29	0.985435	0.492718	+	0.870189i	$0.336004\pi$
$150$	0	0
$151$	−2864.32	−1.54367	−0.771837	−	0.635820i	$-0.780662\pi$
$151$	−2864.32	−1.54367	−0.771837	+	0.635820i	$0.780662\pi$
$152$	0	0
$153$	−1957.67	−1.03443
$154$	0	0
$155$	962.133	0.498583
$156$	0	0
$157$	−2123.57	−1.07949	−0.539743	−	0.841830i	$-0.681479\pi$
$157$	−2123.57	−1.07949	−0.539743	+	0.841830i	$0.681479\pi$
$158$	0	0
$159$	−812.420	−0.405215
$160$	0	0
$161$	−1189.74	−0.582391
$162$	0	0
$163$	3164.21	1.52049	0.760245	−	0.649636i	$-0.225079\pi$
$163$	3164.21	1.52049	0.760245	+	0.649636i	$0.225079\pi$
$164$	0	0
$165$	4136.00	1.95144
$166$	0	0
$167$	1340.34	0.621069	0.310534	−	0.950562i	$-0.399492\pi$
$167$	1340.34	0.621069	0.310534	+	0.950562i	$0.399492\pi$
$168$	0	0
$169$	6187.92	2.81653
$170$	0	0
$171$	−2451.01	−1.09610
$172$	0	0
$173$	2802.13	1.23146	0.615729	−	0.787958i	$-0.288862\pi$
$173$	2802.13	1.23146	0.615729	+	0.787958i	$0.288862\pi$
$174$	0	0
$175$	−153.507	−0.0663089
$176$	0	0
$177$	−4916.14	−2.08768
$178$	0	0
$179$	1074.84	0.448810	0.224405	−	0.974496i	$-0.427956\pi$
$179$	1074.84	0.448810	0.224405	+	0.974496i	$0.427956\pi$
$180$	0	0
$181$	−571.036	−0.234502	−0.117251	−	0.993102i	$-0.537408\pi$
$181$	−571.036	−0.234502	−0.117251	+	0.993102i	$0.537408\pi$
$182$	0	0
$183$	−5633.47	−2.27562
$184$	0	0
$185$	−130.359	−0.0518065
$186$	0	0
$187$	−1451.07	−0.567446
$188$	0	0
$189$	−2262.73	−0.870842
$190$	0	0
$191$	−872.657	−0.330593	−0.165296	−	0.986244i	$-0.552858\pi$
$191$	−872.657	−0.330593	−0.165296	+	0.986244i	$0.552858\pi$
$192$	0	0
$193$	672.523	0.250825	0.125413	−	0.992105i	$-0.459975\pi$
$193$	672.523	0.250825	0.125413	+	0.992105i	$0.459975\pi$
$194$	0	0
$195$	−9693.19	−3.55971
$196$	0	0
$197$	3129.56	1.13184	0.565918	−	0.824461i	$-0.308522\pi$
$197$	3129.56	1.13184	0.565918	+	0.824461i	$0.308522\pi$
$198$	0	0
$199$	−1502.37	−0.535176	−0.267588	−	0.963533i	$-0.586227\pi$
$199$	−1502.37	−0.535176	−0.267588	+	0.963533i	$0.586227\pi$
$200$	0	0
$201$	−5901.91	−2.07109
$202$	0	0
$203$	−268.903	−0.0929718
$204$	0	0
$205$	−2435.61	−0.829807
$206$	0	0
$207$	6362.84	2.13646
$208$	0	0
$209$	−1816.74	−0.601275
$210$	0	0
$211$	−1773.70	−0.578703	−0.289352	−	0.957223i	$-0.593440\pi$
$211$	−1773.70	−0.578703	−0.289352	+	0.957223i	$0.593440\pi$
$212$	0	0
$213$	−4419.32	−1.42163
$214$	0	0
$215$	1367.74	0.433857
$216$	0	0
$217$	−799.835	−0.250214
$218$	0	0
$219$	5055.74	1.55998
$220$	0	0
$221$	3400.74	1.03511
$222$	0	0
$223$	5413.66	1.62568	0.812838	−	0.582490i	$-0.197922\pi$
$223$	5413.66	1.62568	0.812838	+	0.582490i	$0.197922\pi$
$224$	0	0
$225$	820.969	0.243250
$226$	0	0
$227$	3744.27	1.09478	0.547392	−	0.836876i	$-0.315621\pi$
$227$	3744.27	1.09478	0.547392	+	0.836876i	$0.315621\pi$
$228$	0	0
$229$	−680.739	−0.196439	−0.0982194	−	0.995165i	$-0.531315\pi$
$229$	−680.739	−0.196439	−0.0982194	+	0.995165i	$0.531315\pi$
$230$	0	0
$231$	−3438.32	−0.979328
$232$	0	0
$233$	−679.271	−0.190989	−0.0954947	−	0.995430i	$-0.530443\pi$
$233$	−679.271	−0.190989	−0.0954947	+	0.995430i	$0.530443\pi$
$234$	0	0
$235$	3709.17	1.02962
$236$	0	0
$237$	−441.190	−0.120921
$238$	0	0
$239$	−3901.27	−1.05587	−0.527934	−	0.849286i	$-0.677033\pi$
$239$	−3901.27	−1.05587	−0.527934	+	0.849286i	$0.677033\pi$
$240$	0	0
$241$	5859.72	1.56622	0.783108	−	0.621886i	$-0.213633\pi$
$241$	5859.72	1.56622	0.783108	+	0.621886i	$0.213633\pi$
$242$	0	0
$243$	−605.790	−0.159924
$244$	0	0
$245$	−2914.91	−0.760110
$246$	0	0
$247$	4257.73	1.09681
$248$	0	0
$249$	5036.10	1.28173
$250$	0	0
$251$	−3381.48	−0.850348	−0.425174	−	0.905112i	$-0.639787\pi$
$251$	−3381.48	−0.850348	−0.425174	+	0.905112i	$0.639787\pi$
$252$	0	0
$253$	4716.27	1.17197
$254$	0	0
$255$	−3931.34	−0.965452
$256$	0	0
$257$	3652.22	0.886455	0.443227	−	0.896409i	$-0.353833\pi$
$257$	3652.22	0.886455	0.443227	+	0.896409i	$0.353833\pi$
$258$	0	0
$259$	108.370	0.0259991
$260$	0	0
$261$	1438.11	0.341061
$262$	0	0
$263$	−2462.74	−0.577410	−0.288705	−	0.957418i	$-0.593225\pi$
$263$	−2462.74	−0.577410	−0.288705	+	0.957418i	$0.593225\pi$
$264$	0	0
$265$	−1078.87	−0.250093
$266$	0	0
$267$	9728.81	2.22994
$268$	0	0
$269$	−482.503	−0.109363	−0.0546816	−	0.998504i	$-0.517414\pi$
$269$	−482.503	−0.109363	−0.0546816	+	0.998504i	$0.517414\pi$
$270$	0	0
$271$	3602.48	0.807510	0.403755	−	0.914867i	$-0.367705\pi$
$271$	3602.48	0.807510	0.403755	+	0.914867i	$0.367705\pi$
$272$	0	0
$273$	8058.09	1.78644
$274$	0	0
$275$	608.519	0.133437
$276$	0	0
$277$	−6177.72	−1.34001	−0.670006	−	0.742356i	$-0.733709\pi$
$277$	−6177.72	−1.34001	−0.670006	+	0.742356i	$0.733709\pi$
$278$	0	0
$279$	4277.58	0.917892
$280$	0	0
$281$	−543.477	−0.115378	−0.0576888	−	0.998335i	$-0.518373\pi$
$281$	−543.477	−0.115378	−0.0576888	+	0.998335i	$0.518373\pi$
$282$	0	0
$283$	6582.11	1.38256	0.691282	−	0.722585i	$-0.257046\pi$
$283$	6582.11	1.38256	0.691282	+	0.722585i	$0.257046\pi$
$284$	0	0
$285$	−4922.05	−1.02301
$286$	0	0
$287$	2024.76	0.416438
$288$	0	0
$289$	−3533.74	−0.719262
$290$	0	0
$291$	4148.09	0.835620
$292$	0	0
$293$	−6753.68	−1.34660	−0.673301	−	0.739369i	$-0.735124\pi$
$293$	−6753.68	−1.34660	−0.673301	+	0.739369i	$0.735124\pi$
$294$	0	0
$295$	−6528.50	−1.28849
$296$	0	0
$297$	8969.68	1.75244
$298$	0	0
$299$	−11053.1	−2.13785
$300$	0	0
$301$	−1137.03	−0.217731
$302$	0	0
$303$	8615.76	1.63354
$304$	0	0
$305$	−7481.09	−1.40448
$306$	0	0
$307$	−1821.88	−0.338698	−0.169349	−	0.985556i	$-0.554167\pi$
$307$	−1821.88	−0.338698	−0.169349	+	0.985556i	$0.554167\pi$
$308$	0	0
$309$	16329.6	3.00635
$310$	0	0
$311$	10598.2	1.93237	0.966184	−	0.257852i	$-0.0830148\pi$
$311$	10598.2	1.93237	0.966184	+	0.257852i	$0.0830148\pi$
$312$	0	0
$313$	−7969.69	−1.43921	−0.719606	−	0.694382i	$-0.755678\pi$
$313$	−7969.69	−1.43921	−0.719606	+	0.694382i	$0.755678\pi$
$314$	0	0
$315$	−6160.10	−1.10185
$316$	0	0
$317$	−7773.66	−1.37733	−0.688663	−	0.725082i	$-0.741802\pi$
$317$	−7773.66	−1.37733	−0.688663	+	0.725082i	$0.741802\pi$
$318$	0	0
$319$	1065.96	0.187092
$320$	0	0
$321$	−15415.3	−2.68038
$322$	0	0
$323$	1726.84	0.297474
$324$	0	0
$325$	−1426.13	−0.243408
$326$	0	0
$327$	−3776.77	−0.638703
$328$	0	0
$329$	−3083.49	−0.516712
$330$	0	0
$331$	−8212.87	−1.36381	−0.681903	−	0.731442i	$-0.738848\pi$
$331$	−8212.87	−1.36381	−0.681903	+	0.731442i	$0.738848\pi$
$332$	0	0
$333$	−579.569	−0.0953760
$334$	0	0
$335$	−7837.57	−1.27825
$336$	0	0
$337$	5260.17	0.850267	0.425133	−	0.905131i	$-0.360227\pi$
$337$	5260.17	0.850267	0.425133	+	0.905131i	$0.360227\pi$
$338$	0	0
$339$	6813.18	1.09157
$340$	0	0
$341$	3170.63	0.503516
$342$	0	0
$343$	5803.96	0.913657
$344$	0	0
$345$	12777.7	1.99399
$346$	0	0
$347$	−2162.12	−0.334492	−0.167246	−	0.985915i	$-0.553487\pi$
$347$	−2162.12	−0.334492	−0.167246	+	0.985915i	$0.553487\pi$
$348$	0	0
$349$	−2888.90	−0.443093	−0.221546	−	0.975150i	$-0.571110\pi$
$349$	−2888.90	−0.443093	−0.221546	+	0.975150i	$0.571110\pi$
$350$	0	0
$351$	−21021.5	−3.19670
$352$	0	0
$353$	11550.2	1.74152	0.870760	−	0.491709i	$-0.163628\pi$
$353$	11550.2	1.74152	0.870760	+	0.491709i	$0.163628\pi$
$354$	0	0
$355$	−5868.74	−0.877409
$356$	0	0
$357$	3268.18	0.484511
$358$	0	0
$359$	6523.97	0.959114	0.479557	−	0.877511i	$-0.340797\pi$
$359$	6523.97	0.959114	0.479557	+	0.877511i	$0.340797\pi$
$360$	0	0
$361$	−4696.99	−0.684792
$362$	0	0
$363$	1746.40	0.252514
$364$	0	0
$365$	6713.89	0.962797
$366$	0	0
$367$	−6856.96	−0.975287	−0.487644	−	0.873043i	$-0.662143\pi$
$367$	−6856.96	−0.975287	−0.487644	+	0.873043i	$0.662143\pi$
$368$	0	0
$369$	−10828.6	−1.52768
$370$	0	0
$371$	896.882	0.125509
$372$	0	0
$373$	5001.68	0.694309	0.347155	−	0.937808i	$-0.387148\pi$
$373$	5001.68	0.694309	0.347155	+	0.937808i	$0.387148\pi$
$374$	0	0
$375$	−11583.4	−1.59511
$376$	0	0
$377$	−2498.19	−0.341283
$378$	0	0
$379$	84.9775	0.0115172	0.00575858	−	0.999983i	$-0.498167\pi$
$379$	84.9775	0.0115172	0.00575858	+	0.999983i	$0.498167\pi$
$380$	0	0
$381$	19279.4	2.59243
$382$	0	0
$383$	−2596.76	−0.346444	−0.173222	−	0.984883i	$-0.555418\pi$
$383$	−2596.76	−0.346444	−0.173222	+	0.984883i	$0.555418\pi$
$384$	0	0
$385$	−4565.99	−0.604427
$386$	0	0
$387$	6080.90	0.798732
$388$	0	0
$389$	−9213.21	−1.20084	−0.600422	−	0.799683i	$-0.705001\pi$
$389$	−9213.21	−1.20084	−0.600422	+	0.799683i	$0.705001\pi$
$390$	0	0
$391$	−4482.89	−0.579820
$392$	0	0
$393$	−6345.48	−0.814471
$394$	0	0
$395$	−585.888	−0.0746310
$396$	0	0
$397$	6614.19	0.836163	0.418082	−	0.908409i	$-0.362703\pi$
$397$	6614.19	0.836163	0.418082	+	0.908409i	$0.362703\pi$
$398$	0	0
$399$	4091.77	0.513396
$400$	0	0
$401$	3434.94	0.427763	0.213881	−	0.976860i	$-0.431389\pi$
$401$	3434.94	0.427763	0.213881	+	0.976860i	$0.431389\pi$
$402$	0	0
$403$	−7430.73	−0.918489
$404$	0	0
$405$	7426.79	0.911210
$406$	0	0
$407$	−429.588	−0.0523192
$408$	0	0
$409$	−6435.82	−0.778071	−0.389035	−	0.921223i	$-0.627192\pi$
$409$	−6435.82	−0.778071	−0.389035	+	0.921223i	$0.627192\pi$
$410$	0	0
$411$	−8813.28	−1.05773
$412$	0	0
$413$	5427.24	0.646627
$414$	0	0
$415$	6687.80	0.791064
$416$	0	0
$417$	20839.8	2.44731
$418$	0	0
$419$	4222.04	0.492267	0.246134	−	0.969236i	$-0.420840\pi$
$419$	4222.04	0.492267	0.246134	+	0.969236i	$0.420840\pi$
$420$	0	0
$421$	−5164.92	−0.597917	−0.298958	−	0.954266i	$-0.596639\pi$
$421$	−5164.92	−0.597917	−0.298958	+	0.954266i	$0.596639\pi$
$422$	0	0
$423$	16490.7	1.89552
$424$	0	0
$425$	−578.408	−0.0660163
$426$	0	0
$427$	6219.14	0.704837
$428$	0	0
$429$	−31943.1	−3.59493
$430$	0	0
$431$	−7476.41	−0.835559	−0.417780	−	0.908548i	$-0.637192\pi$
$431$	−7476.41	−0.835559	−0.417780	+	0.908548i	$0.637192\pi$
$432$	0	0
$433$	−11289.7	−1.25299	−0.626497	−	0.779424i	$-0.715512\pi$
$433$	−11289.7	−1.25299	−0.626497	+	0.779424i	$0.715512\pi$
$434$	0	0
$435$	2887.98	0.318317
$436$	0	0
$437$	−5612.59	−0.614386
$438$	0	0
$439$	1800.79	0.195779	0.0978895	−	0.995197i	$-0.468791\pi$
$439$	1800.79	0.195779	0.0978895	+	0.995197i	$0.468791\pi$
$440$	0	0
$441$	−12959.5	−1.39936
$442$	0	0
$443$	−7020.76	−0.752972	−0.376486	−	0.926422i	$-0.622868\pi$
$443$	−7020.76	−0.752972	−0.376486	+	0.926422i	$0.622868\pi$
$444$	0	0
$445$	12919.6	1.37629
$446$	0	0
$447$	16001.9	1.69321
$448$	0	0
$449$	−3318.29	−0.348774	−0.174387	−	0.984677i	$-0.555794\pi$
$449$	−3318.29	−0.348774	−0.174387	+	0.984677i	$0.555794\pi$
$450$	0	0
$451$	−8026.35	−0.838018
$452$	0	0
$453$	−25573.2	−2.65239
$454$	0	0
$455$	10700.9	1.10256
$456$	0	0
$457$	4468.38	0.457378	0.228689	−	0.973500i	$-0.426556\pi$
$457$	4468.38	0.457378	0.228689	+	0.973500i	$0.426556\pi$
$458$	0	0
$459$	−8525.84	−0.866998
$460$	0	0
$461$	−1179.45	−0.119159	−0.0595795	−	0.998224i	$-0.518976\pi$
$461$	−1179.45	−0.119159	−0.0595795	+	0.998224i	$0.518976\pi$
$462$	0	0
$463$	−13763.1	−1.38148	−0.690742	−	0.723101i	$-0.742716\pi$
$463$	−13763.1	−1.38148	−0.690742	+	0.723101i	$0.742716\pi$
$464$	0	0
$465$	8590.11	0.856682
$466$	0	0
$467$	17014.1	1.68591	0.842955	−	0.537984i	$-0.180814\pi$
$467$	17014.1	1.68591	0.842955	+	0.537984i	$0.180814\pi$
$468$	0	0
$469$	6515.49	0.641487
$470$	0	0
$471$	−18959.7	−1.85481
$472$	0	0
$473$	4507.28	0.438150
$474$	0	0
$475$	−724.168	−0.0699518
$476$	0	0
$477$	−4796.59	−0.460421
$478$	0	0
$479$	18976.7	1.81017	0.905083	−	0.425236i	$-0.139809\pi$
$479$	18976.7	1.81017	0.905083	+	0.425236i	$0.139809\pi$
$480$	0	0
$481$	1006.79	0.0954379
$482$	0	0
$483$	−10622.3	−1.00068
$484$	0	0
$485$	5508.55	0.515733
$486$	0	0
$487$	−10579.6	−0.984408	−0.492204	−	0.870480i	$-0.663809\pi$
$487$	−10579.6	−0.984408	−0.492204	+	0.870480i	$0.663809\pi$
$488$	0	0
$489$	28250.7	2.61256
$490$	0	0
$491$	15200.9	1.39716	0.698579	−	0.715533i	$-0.253816\pi$
$491$	15200.9	1.39716	0.698579	+	0.715533i	$0.253816\pi$
$492$	0	0
$493$	−1013.21	−0.0925614
$494$	0	0
$495$	24419.3	2.21730
$496$	0	0
$497$	4878.77	0.440327
$498$	0	0
$499$	−2553.92	−0.229117	−0.114558	−	0.993417i	$-0.536545\pi$
$499$	−2553.92	−0.229117	−0.114558	+	0.993417i	$0.536545\pi$
$500$	0	0
$501$	11966.8	1.06714
$502$	0	0
$503$	−4863.70	−0.431137	−0.215568	−	0.976489i	$-0.569160\pi$
$503$	−4863.70	−0.431137	−0.215568	+	0.976489i	$0.569160\pi$
$504$	0	0
$505$	11441.5	1.00820
$506$	0	0
$507$	55247.0	4.83946
$508$	0	0
$509$	−17659.5	−1.53781	−0.768903	−	0.639366i	$-0.779197\pi$
$509$	−17659.5	−1.53781	−0.768903	+	0.639366i	$0.779197\pi$
$510$	0	0
$511$	−5581.35	−0.483179
$512$	0	0
$513$	−10674.4	−0.918685
$514$	0	0
$515$	21685.3	1.85547
$516$	0	0
$517$	12223.3	1.03980
$518$	0	0
$519$	25018.0	2.11593
$520$	0	0
$521$	−3778.19	−0.317707	−0.158853	−	0.987302i	$-0.550780\pi$
$521$	−3778.19	−0.317707	−0.158853	+	0.987302i	$0.550780\pi$
$522$	0	0
$523$	2792.67	0.233489	0.116745	−	0.993162i	$-0.462754\pi$
$523$	2792.67	0.233489	0.116745	+	0.993162i	$0.462754\pi$
$524$	0	0
$525$	−1370.54	−0.113934
$526$	0	0
$527$	−3013.74	−0.249109
$528$	0	0
$529$	2403.34	0.197529
$530$	0	0
$531$	−29025.3	−2.37211
$532$	0	0
$533$	18810.7	1.52867
$534$	0	0
$535$	−20471.1	−1.65429
$536$	0	0
$537$	9596.35	0.771160
$538$	0	0
$539$	−9605.85	−0.767631
$540$	0	0
$541$	2062.12	0.163877	0.0819383	−	0.996637i	$-0.473889\pi$
$541$	2062.12	0.163877	0.0819383	+	0.996637i	$0.473889\pi$
$542$	0	0
$543$	−5098.33	−0.402928
$544$	0	0
$545$	−5015.44	−0.394198
$546$	0	0
$547$	−16562.0	−1.29459	−0.647296	−	0.762239i	$-0.724100\pi$
$547$	−16562.0	−1.29459	−0.647296	+	0.762239i	$0.724100\pi$
$548$	0	0
$549$	−33260.4	−2.58565
$550$	0	0
$551$	−1268.54	−0.0980795
$552$	0	0
$553$	487.057	0.0374535
$554$	0	0
$555$	−1163.87	−0.0890157
$556$	0	0
$557$	9272.94	0.705399	0.352699	−	0.935737i	$-0.385264\pi$
$557$	9272.94	0.705399	0.352699	+	0.935737i	$0.385264\pi$
$558$	0	0
$559$	−10563.3	−0.799251
$560$	0	0
$561$	−12955.4	−0.975005
$562$	0	0
$563$	62.0181	0.00464254	0.00232127	−	0.999997i	$-0.499261\pi$
$563$	62.0181	0.00464254	0.00232127	+	0.999997i	$0.499261\pi$
$564$	0	0
$565$	9047.71	0.673699
$566$	0	0
$567$	−6174.00	−0.457290
$568$	0	0
$569$	10015.9	0.737938	0.368969	−	0.929442i	$-0.379711\pi$
$569$	10015.9	0.737938	0.368969	+	0.929442i	$0.379711\pi$
$570$	0	0
$571$	−15543.3	−1.13917	−0.569587	−	0.821931i	$-0.692897\pi$
$571$	−15543.3	−1.13917	−0.569587	+	0.821931i	$0.692897\pi$
$572$	0	0
$573$	−7791.26	−0.568036
$574$	0	0
$575$	1879.95	0.136346
$576$	0	0
$577$	3776.77	0.272494	0.136247	−	0.990675i	$-0.456496\pi$
$577$	3776.77	0.272494	0.136247	+	0.990675i	$0.456496\pi$
$578$	0	0
$579$	6004.42	0.430976
$580$	0	0
$581$	−5559.67	−0.396995
$582$	0	0
$583$	−3555.33	−0.252567
$584$	0	0
$585$	−57229.3	−4.04469
$586$	0	0
$587$	596.285	0.0419273	0.0209637	−	0.999780i	$-0.493327\pi$
$587$	596.285	0.0419273	0.0209637	+	0.999780i	$0.493327\pi$
$588$	0	0
$589$	−3773.21	−0.263960
$590$	0	0
$591$	27941.3	1.94476
$592$	0	0
$593$	20179.6	1.39743	0.698717	−	0.715398i	$-0.253755\pi$
$593$	20179.6	1.39743	0.698717	+	0.715398i	$0.253755\pi$
$594$	0	0
$595$	4340.06	0.299034
$596$	0	0
$597$	−13413.5	−0.919558
$598$	0	0
$599$	13076.3	0.891960	0.445980	−	0.895043i	$-0.352855\pi$
$599$	13076.3	0.891960	0.445980	+	0.895043i	$0.352855\pi$
$600$	0	0
$601$	−16866.9	−1.14479	−0.572393	−	0.819980i	$-0.693985\pi$
$601$	−16866.9	−1.14479	−0.572393	+	0.819980i	$0.693985\pi$
$602$	0	0
$603$	−34845.3	−2.35325
$604$	0	0
$605$	2319.18	0.155848
$606$	0	0
$607$	19953.0	1.33421	0.667107	−	0.744962i	$-0.267532\pi$
$607$	19953.0	1.33421	0.667107	+	0.744962i	$0.267532\pi$
$608$	0	0
$609$	−2400.82	−0.159747
$610$	0	0
$611$	−28646.6	−1.89676
$612$	0	0
$613$	7214.32	0.475340	0.237670	−	0.971346i	$-0.423616\pi$
$613$	7214.32	0.475340	0.237670	+	0.971346i	$0.423616\pi$
$614$	0	0
$615$	−21745.6	−1.42580
$616$	0	0
$617$	22212.9	1.44936	0.724681	−	0.689084i	$-0.241987\pi$
$617$	22212.9	1.44936	0.724681	+	0.689084i	$0.241987\pi$
$618$	0	0
$619$	4349.02	0.282394	0.141197	−	0.989982i	$-0.454905\pi$
$619$	4349.02	0.282394	0.141197	+	0.989982i	$0.454905\pi$
$620$	0	0
$621$	27710.8	1.79065
$622$	0	0
$623$	−10740.2	−0.690688
$624$	0	0
$625$	−17329.2	−1.10907
$626$	0	0
$627$	−16220.2	−1.03313
$628$	0	0
$629$	408.331	0.0258843
$630$	0	0
$631$	−10127.9	−0.638966	−0.319483	−	0.947592i	$-0.603509\pi$
$631$	−10127.9	−0.638966	−0.319483	+	0.947592i	$0.603509\pi$
$632$	0	0
$633$	−15835.9	−0.994347
$634$	0	0
$635$	25602.5	1.60001
$636$	0	0
$637$	22512.4	1.40027
$638$	0	0
$639$	−26092.0	−1.61531
$640$	0	0
$641$	−17624.6	−1.08601	−0.543003	−	0.839731i	$-0.682713\pi$
$641$	−17624.6	−1.08601	−0.543003	+	0.839731i	$0.682713\pi$
$642$	0	0
$643$	27238.1	1.67055	0.835277	−	0.549829i	$-0.185307\pi$
$643$	27238.1	1.67055	0.835277	+	0.549829i	$0.185307\pi$
$644$	0	0
$645$	12211.5	0.745468
$646$	0	0
$647$	−28702.1	−1.74404	−0.872021	−	0.489469i	$-0.837191\pi$
$647$	−28702.1	−1.74404	−0.872021	+	0.489469i	$0.837191\pi$
$648$	0	0
$649$	−21514.1	−1.30124
$650$	0	0
$651$	−7141.09	−0.429925
$652$	0	0
$653$	−7937.84	−0.475700	−0.237850	−	0.971302i	$-0.576443\pi$
$653$	−7937.84	−0.475700	−0.237850	+	0.971302i	$0.576443\pi$
$654$	0	0
$655$	−8426.62	−0.502680
$656$	0	0
$657$	29849.5	1.77251
$658$	0	0
$659$	3675.29	0.217252	0.108626	−	0.994083i	$-0.465355\pi$
$659$	3675.29	0.217252	0.108626	+	0.994083i	$0.465355\pi$
$660$	0	0
$661$	4205.49	0.247466	0.123733	−	0.992316i	$-0.460513\pi$
$661$	4205.49	0.247466	0.123733	+	0.992316i	$0.460513\pi$
$662$	0	0
$663$	30362.5	1.77855
$664$	0	0
$665$	5433.76	0.316860
$666$	0	0
$667$	3293.15	0.191171
$668$	0	0
$669$	48334.3	2.79329
$670$	0	0
$671$	−24653.3	−1.41838
$672$	0	0
$673$	−4634.55	−0.265452	−0.132726	−	0.991153i	$-0.542373\pi$
$673$	−4634.55	−0.265452	−0.132726	+	0.991153i	$0.542373\pi$
$674$	0	0
$675$	3575.40	0.203877
$676$	0	0
$677$	−5327.42	−0.302437	−0.151218	−	0.988500i	$-0.548320\pi$
$677$	−5327.42	−0.302437	−0.151218	+	0.988500i	$0.548320\pi$
$678$	0	0
$679$	−4579.34	−0.258820
$680$	0	0
$681$	33429.6	1.88109
$682$	0	0
$683$	−2816.05	−0.157764	−0.0788822	−	0.996884i	$-0.525135\pi$
$683$	−2816.05	−0.157764	−0.0788822	+	0.996884i	$0.525135\pi$
$684$	0	0
$685$	−11703.8	−0.652816
$686$	0	0
$687$	−6077.77	−0.337528
$688$	0	0
$689$	8332.33	0.460720
$690$	0	0
$691$	16910.1	0.930955	0.465477	−	0.885060i	$-0.345883\pi$
$691$	16910.1	0.930955	0.465477	+	0.885060i	$0.345883\pi$
$692$	0	0
$693$	−20300.1	−1.11275
$694$	0	0
$695$	27674.6	1.51044
$696$	0	0
$697$	7629.19	0.414600
$698$	0	0
$699$	−6064.67	−0.328164
$700$	0	0
$701$	−30588.4	−1.64808	−0.824042	−	0.566528i	$-0.808286\pi$
$701$	−30588.4	−1.64808	−0.824042	+	0.566528i	$0.808286\pi$
$702$	0	0
$703$	511.232	0.0274274
$704$	0	0
$705$	33116.2	1.76912
$706$	0	0
$707$	−9511.48	−0.505963
$708$	0	0
$709$	21242.9	1.12524	0.562619	−	0.826716i	$-0.309794\pi$
$709$	21242.9	1.12524	0.562619	+	0.826716i	$0.309794\pi$
$710$	0	0
$711$	−2604.82	−0.137396
$712$	0	0
$713$	9795.28	0.514496
$714$	0	0
$715$	−42419.5	−2.21874
$716$	0	0
$717$	−34831.3	−1.81423
$718$	0	0
$719$	−12452.7	−0.645907	−0.322954	−	0.946415i	$-0.604676\pi$
$719$	−12452.7	−0.645907	−0.322954	+	0.946415i	$0.604676\pi$
$720$	0	0
$721$	−18027.3	−0.931168
$722$	0	0
$723$	52316.8	2.69112
$724$	0	0
$725$	424.901	0.0217661
$726$	0	0
$727$	27178.3	1.38650	0.693251	−	0.720696i	$-0.256178\pi$
$727$	27178.3	1.38650	0.693251	+	0.720696i	$0.256178\pi$
$728$	0	0
$729$	−22321.3	−1.13404
$730$	0	0
$731$	−4284.25	−0.216770
$732$	0	0
$733$	−21999.7	−1.10856	−0.554282	−	0.832329i	$-0.687007\pi$
$733$	−21999.7	−1.10856	−0.554282	+	0.832329i	$0.687007\pi$
$734$	0	0
$735$	−26024.9	−1.30605
$736$	0	0
$737$	−25828.1	−1.29089
$738$	0	0
$739$	−13644.0	−0.679166	−0.339583	−	0.940576i	$-0.610286\pi$
$739$	−13644.0	−0.679166	−0.339583	+	0.940576i	$0.610286\pi$
$740$	0	0
$741$	38013.9	1.88458
$742$	0	0
$743$	13980.2	0.690286	0.345143	−	0.938550i	$-0.387830\pi$
$743$	13980.2	0.690286	0.345143	+	0.938550i	$0.387830\pi$
$744$	0	0
$745$	21250.1	1.04502
$746$	0	0
$747$	29733.5	1.45635
$748$	0	0
$749$	17018.0	0.830204
$750$	0	0
$751$	21138.7	1.02711	0.513557	−	0.858055i	$-0.328327\pi$
$751$	21138.7	1.02711	0.513557	+	0.858055i	$0.328327\pi$
$752$	0	0
$753$	−30190.6	−1.46110
$754$	0	0
$755$	−33960.5	−1.63702
$756$	0	0
$757$	−28328.1	−1.36011	−0.680054	−	0.733162i	$-0.738044\pi$
$757$	−28328.1	−1.36011	−0.680054	+	0.733162i	$0.738044\pi$
$758$	0	0
$759$	42107.8	2.01372
$760$	0	0
$761$	−1093.79	−0.0521023	−0.0260512	−	0.999661i	$-0.508293\pi$
$761$	−1093.79	−0.0521023	−0.0260512	+	0.999661i	$0.508293\pi$
$762$	0	0
$763$	4169.41	0.197828
$764$	0	0
$765$	−23210.9	−1.09699
$766$	0	0
$767$	50420.8	2.37365
$768$	0	0
$769$	−1659.21	−0.0778059	−0.0389029	−	0.999243i	$-0.512386\pi$
$769$	−1659.21	−0.0778059	−0.0389029	+	0.999243i	$0.512386\pi$
$770$	0	0
$771$	32607.7	1.52314
$772$	0	0
$773$	−7537.89	−0.350736	−0.175368	−	0.984503i	$-0.556112\pi$
$773$	−7537.89	−0.350736	−0.175368	+	0.984503i	$0.556112\pi$
$774$	0	0
$775$	1263.84	0.0585787
$776$	0	0
$777$	967.546	0.0446725
$778$	0	0
$779$	9551.76	0.439316
$780$	0	0
$781$	−19339.9	−0.886091
$782$	0	0
$783$	6263.11	0.285856
$784$	0	0
$785$	−25177.9	−1.14476
$786$	0	0
$787$	−9304.20	−0.421422	−0.210711	−	0.977548i	$-0.567578\pi$
$787$	−9304.20	−0.421422	−0.210711	+	0.977548i	$0.567578\pi$
$788$	0	0
$789$	−21987.8	−0.992126
$790$	0	0
$791$	−7521.50	−0.338096
$792$	0	0
$793$	57777.8	2.58733
$794$	0	0
$795$	−9632.39	−0.429718
$796$	0	0
$797$	−38619.5	−1.71640	−0.858201	−	0.513314i	$-0.828418\pi$
$797$	−38619.5	−1.71640	−0.858201	+	0.513314i	$0.828418\pi$
$798$	0	0
$799$	−11618.4	−0.514431
$800$	0	0
$801$	57439.7	2.53375
$802$	0	0
$803$	22125.1	0.972324
$804$	0	0
$805$	−14106.1	−0.617608
$806$	0	0
$807$	−4307.88	−0.187912
$808$	0	0
$809$	−18550.5	−0.806179	−0.403090	−	0.915160i	$-0.632064\pi$
$809$	−18550.5	−0.806179	−0.403090	+	0.915160i	$0.632064\pi$
$810$	0	0
$811$	−3704.80	−0.160411	−0.0802054	−	0.996778i	$-0.525558\pi$
$811$	−3704.80	−0.160411	−0.0802054	+	0.996778i	$0.525558\pi$
$812$	0	0
$813$	32163.7	1.38749
$814$	0	0
$815$	37516.2	1.61243
$816$	0	0
$817$	−5363.90	−0.229693
$818$	0	0
$819$	47575.6	2.02982
$820$	0	0
$821$	5544.86	0.235709	0.117854	−	0.993031i	$-0.462398\pi$
$821$	5544.86	0.235709	0.117854	+	0.993031i	$0.462398\pi$
$822$	0	0
$823$	−3915.39	−0.165835	−0.0829174	−	0.996556i	$-0.526424\pi$
$823$	−3915.39	−0.165835	−0.0829174	+	0.996556i	$0.526424\pi$
$824$	0	0
$825$	5432.98	0.229275
$826$	0	0
$827$	−35091.2	−1.47550	−0.737751	−	0.675073i	$-0.764112\pi$
$827$	−35091.2	−1.47550	−0.737751	+	0.675073i	$0.764112\pi$
$828$	0	0
$829$	3532.71	0.148005	0.0740025	−	0.997258i	$-0.476423\pi$
$829$	3532.71	0.148005	0.0740025	+	0.997258i	$0.476423\pi$
$830$	0	0
$831$	−55156.0	−2.30245
$832$	0	0
$833$	9130.53	0.379777
$834$	0	0
$835$	15891.6	0.658624
$836$	0	0
$837$	18629.3	0.769320
$838$	0	0
$839$	−30081.1	−1.23780	−0.618900	−	0.785470i	$-0.712421\pi$
$839$	−30081.1	−1.23780	−0.618900	+	0.785470i	$0.712421\pi$
$840$	0	0
$841$	−23644.7	−0.969482
$842$	0	0
$843$	−4852.27	−0.198246
$844$	0	0
$845$	73366.5	2.98685
$846$	0	0
$847$	−1927.97	−0.0782121
$848$	0	0
$849$	58766.4	2.37557
$850$	0	0
$851$	−1327.16	−0.0534601
$852$	0	0
$853$	11222.1	0.450453	0.225226	−	0.974306i	$-0.427688\pi$
$853$	11222.1	0.450453	0.225226	+	0.974306i	$0.427688\pi$
$854$	0	0
$855$	−29060.2	−1.16238
$856$	0	0
$857$	−27486.0	−1.09557	−0.547785	−	0.836619i	$-0.684529\pi$
$857$	−27486.0	−1.09557	−0.547785	+	0.836619i	$0.684529\pi$
$858$	0	0
$859$	14976.2	0.594855	0.297427	−	0.954744i	$-0.403871\pi$
$859$	14976.2	0.594855	0.297427	+	0.954744i	$0.403871\pi$
$860$	0	0
$861$	18077.5	0.715538
$862$	0	0
$863$	−9419.49	−0.371545	−0.185772	−	0.982593i	$-0.559479\pi$
$863$	−9419.49	−0.371545	−0.185772	+	0.982593i	$0.559479\pi$
$864$	0	0
$865$	33223.2	1.30592
$866$	0	0
$867$	−31549.9	−1.23586
$868$	0	0
$869$	−1930.75	−0.0753694
$870$	0	0
$871$	60531.0	2.35478
$872$	0	0
$873$	24490.7	0.949465
$874$	0	0
$875$	12787.6	0.494059
$876$	0	0
$877$	−21069.8	−0.811263	−0.405631	−	0.914037i	$-0.632948\pi$
$877$	−21069.8	−0.811263	−0.405631	+	0.914037i	$0.632948\pi$
$878$	0	0
$879$	−60298.2	−2.31378
$880$	0	0
$881$	−14511.7	−0.554952	−0.277476	−	0.960733i	$-0.589498\pi$
$881$	−14511.7	−0.554952	−0.277476	+	0.960733i	$0.589498\pi$
$882$	0	0
$883$	−26515.9	−1.01057	−0.505284	−	0.862953i	$-0.668612\pi$
$883$	−26515.9	−1.01057	−0.505284	+	0.862953i	$0.668612\pi$
$884$	0	0
$885$	−58287.8	−2.21392
$886$	0	0
$887$	−29223.6	−1.10624	−0.553119	−	0.833102i	$-0.686563\pi$
$887$	−29223.6	−1.10624	−0.553119	+	0.833102i	$0.686563\pi$
$888$	0	0
$889$	−21283.8	−0.802964
$890$	0	0
$891$	24474.4	0.920227
$892$	0	0
$893$	−14546.3	−0.545099
$894$	0	0
$895$	12743.7	0.475949
$896$	0	0
$897$	−98684.4	−3.67333
$898$	0	0
$899$	2213.90	0.0821332
$900$	0	0
$901$	3379.41	0.124955
$902$	0	0
$903$	−10151.6	−0.374113
$904$	0	0
$905$	−6770.44	−0.248682
$906$	0	0
$907$	32139.3	1.17659	0.588296	−	0.808646i	$-0.299799\pi$
$907$	32139.3	1.17659	0.588296	+	0.808646i	$0.299799\pi$
$908$	0	0
$909$	50868.1	1.85609
$910$	0	0
$911$	21430.7	0.779396	0.389698	−	0.920943i	$-0.372579\pi$
$911$	21430.7	0.779396	0.389698	+	0.920943i	$0.372579\pi$
$912$	0	0
$913$	22039.1	0.798891
$914$	0	0
$915$	−66792.7	−2.41322
$916$	0	0
$917$	7005.17	0.252270
$918$	0	0
$919$	11676.1	0.419106	0.209553	−	0.977797i	$-0.432799\pi$
$919$	11676.1	0.419106	0.209553	+	0.977797i	$0.432799\pi$
$920$	0	0
$921$	−16266.1	−0.581962
$922$	0	0
$923$	45325.3	1.61636
$924$	0	0
$925$	−171.238	−0.00608677
$926$	0	0
$927$	96411.5	3.41593
$928$	0	0
$929$	41969.7	1.48222	0.741110	−	0.671383i	$-0.234300\pi$
$929$	41969.7	1.48222	0.741110	+	0.671383i	$0.234300\pi$
$930$	0	0
$931$	11431.4	0.402417
$932$	0	0
$933$	94622.6	3.32026
$934$	0	0
$935$	−17204.4	−0.601759
$936$	0	0
$937$	−45948.6	−1.60200	−0.801001	−	0.598663i	$-0.795699\pi$
$937$	−45948.6	−1.60200	−0.801001	+	0.598663i	$0.795699\pi$
$938$	0	0
$939$	−71155.0	−2.47290
$940$	0	0
$941$	−15000.1	−0.519648	−0.259824	−	0.965656i	$-0.583665\pi$
$941$	−15000.1	−0.519648	−0.259824	+	0.965656i	$0.583665\pi$
$942$	0	0
$943$	−24796.4	−0.856292
$944$	0	0
$945$	−26827.8	−0.923502
$946$	0	0
$947$	−37423.2	−1.28415	−0.642075	−	0.766642i	$-0.721926\pi$
$947$	−37423.2	−1.28415	−0.642075	+	0.766642i	$0.721926\pi$
$948$	0	0
$949$	−51852.6	−1.77366
$950$	0	0
$951$	−69404.8	−2.36657
$952$	0	0
$953$	−55403.0	−1.88319	−0.941594	−	0.336750i	$-0.890672\pi$
$953$	−55403.0	−1.88319	−0.941594	+	0.336750i	$0.890672\pi$
$954$	0	0
$955$	−10346.6	−0.350584
$956$	0	0
$957$	9517.09	0.321467
$958$	0	0
$959$	9729.54	0.327615
$960$	0	0
$961$	−23205.9	−0.778956
$962$	0	0
$963$	−91013.4	−3.04555
$964$	0	0
$965$	7973.70	0.265992
$966$	0	0
$967$	31440.5	1.04556	0.522780	−	0.852467i	$-0.324895\pi$
$967$	31440.5	1.04556	0.522780	+	0.852467i	$0.324895\pi$
$968$	0	0
$969$	15417.6	0.511129
$970$	0	0
$971$	−19010.3	−0.628290	−0.314145	−	0.949375i	$-0.601718\pi$
$971$	−19010.3	−0.628290	−0.314145	+	0.949375i	$0.601718\pi$
$972$	0	0
$973$	−23006.3	−0.758015
$974$	0	0
$975$	−12732.8	−0.418232
$976$	0	0
$977$	−8504.01	−0.278472	−0.139236	−	0.990259i	$-0.544465\pi$
$977$	−8504.01	−0.278472	−0.139236	+	0.990259i	$0.544465\pi$
$978$	0	0
$979$	42575.4	1.38990
$980$	0	0
$981$	−22298.3	−0.725720
$982$	0	0
$983$	−52828.3	−1.71410	−0.857051	−	0.515232i	$-0.827706\pi$
$983$	−52828.3	−1.71410	−0.857051	+	0.515232i	$0.827706\pi$
$984$	0	0
$985$	37105.3	1.20028
$986$	0	0
$987$	−27530.0	−0.887831
$988$	0	0
$989$	13924.7	0.447705
$990$	0	0
$991$	−13275.5	−0.425541	−0.212770	−	0.977102i	$-0.568249\pi$
$991$	−13275.5	−0.425541	−0.212770	+	0.977102i	$0.568249\pi$
$992$	0	0
$993$	−73326.1	−2.34334
$994$	0	0
$995$	−17812.7	−0.567538
$996$	0	0
$997$	−20858.1	−0.662570	−0.331285	−	0.943531i	$-0.607482\pi$
$997$	−20858.1	−0.662570	−0.331285	+	0.943531i	$0.607482\pi$
$998$	0	0
$999$	−2524.08	−0.0799382

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.4.a.g.1.2	yes	2
3.2	odd	2		1152.4.a.t.1.1		2
4.3	odd	2		128.4.a.e.1.1	✓	2
8.3	odd	2		128.4.a.h.1.2	yes	2
8.5	even	2		128.4.a.f.1.1	yes	2
12.11	even	2		1152.4.a.s.1.1		2
16.3	odd	4		256.4.b.i.129.1		4
16.5	even	4		256.4.b.h.129.1		4
16.11	odd	4		256.4.b.i.129.4		4
16.13	even	4		256.4.b.h.129.4		4
24.5	odd	2		1152.4.a.r.1.2		2
24.11	even	2		1152.4.a.q.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.4.a.e.1.1	✓	2	4.3	odd	2
128.4.a.f.1.1	yes	2	8.5	even	2
128.4.a.g.1.2	yes	2	1.1	even	1	trivial
128.4.a.h.1.2	yes	2	8.3	odd	2
256.4.b.h.129.1		4	16.5	even	4
256.4.b.h.129.4		4	16.13	even	4
256.4.b.i.129.1		4	16.3	odd	4
256.4.b.i.129.4		4	16.11	odd	4
1152.4.a.q.1.2		2	24.11	even	2
1152.4.a.r.1.2		2	24.5	odd	2
1152.4.a.s.1.1		2	12.11	even	2
1152.4.a.t.1.1		2	3.2	odd	2

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

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	128.4.a.g.1.2

Level	$128$
Weight	$4$
Character	128.1
Self dual	yes
Analytic conductor	$7.552$
Analytic rank	$0$
Dimension	$2$
CM	no
Inner twists	$1$