LMFDB - Embedded newform 1232.4.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1232,4,Mod(1,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1232.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1232 = 2^{4} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1232.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:	$72.6903531271$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 42x^{3} + 18x^{2} + 368x + 352 $ x^5 - x^4 - 42x^3 + 18x^2 + 368*x + 352
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.18888$ of defining polynomial
Character	$\chi$	$=$	1232.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.48496 q^{3} +7.60736 q^{5} -7.00000 q^{7} +15.0547 q^{9} -11.0000 q^{11} +0.174238 q^{13} -49.3335 q^{15} +128.863 q^{17} -141.685 q^{19} +45.3947 q^{21} +133.369 q^{23} -67.1280 q^{25} +77.4647 q^{27} -177.002 q^{29} -48.2757 q^{31} +71.3346 q^{33} -53.2515 q^{35} +161.625 q^{37} -1.12993 q^{39} -195.689 q^{41} +488.447 q^{43} +114.527 q^{45} +171.705 q^{47} +49.0000 q^{49} -835.671 q^{51} -431.477 q^{53} -83.6810 q^{55} +918.821 q^{57} -194.176 q^{59} +585.008 q^{61} -105.383 q^{63} +1.32549 q^{65} -155.905 q^{67} -864.891 q^{69} +374.994 q^{71} +210.419 q^{73} +435.323 q^{75} +77.0000 q^{77} +7.00618 q^{79} -908.833 q^{81} +93.6417 q^{83} +980.307 q^{85} +1147.85 q^{87} -307.119 q^{89} -1.21967 q^{91} +313.066 q^{93} -1077.85 q^{95} +965.991 q^{97} -165.602 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{3} - 24 q^{5} - 35 q^{7} + 63 q^{9} - 55 q^{11} - 50 q^{13} + 146 q^{15} + 222 q^{17} - 160 q^{19} + 14 q^{21} - 54 q^{23} + 125 q^{25} - 110 q^{27} + 14 q^{29} + 34 q^{31} + 22 q^{33} + 168 q^{35}+ \cdots - 693 q^{99}+O(q^{100}) $ 5 * q - 2 * q^3 - 24 * q^5 - 35 * q^7 + 63 * q^9 - 55 * q^11 - 50 * q^13 + 146 * q^15 + 222 * q^17 - 160 * q^19 + 14 * q^21 - 54 * q^23 + 125 * q^25 - 110 * q^27 + 14 * q^29 + 34 * q^31 + 22 * q^33 + 168 * q^35 + 1044 * q^37 + 124 * q^39 - 114 * q^41 - 672 * q^43 - 530 * q^45 + 292 * q^47 + 245 * q^49 - 768 * q^51 - 710 * q^53 + 264 * q^55 + 2012 * q^57 - 270 * q^59 + 138 * q^61 - 441 * q^63 - 196 * q^65 - 1942 * q^67 - 1306 * q^69 + 278 * q^71 - 338 * q^73 - 3960 * q^75 + 385 * q^77 - 576 * q^79 - 1439 * q^81 - 1644 * q^83 + 360 * q^85 + 1956 * q^87 - 3656 * q^89 + 350 * q^91 - 1442 * q^93 + 1276 * q^95 + 692 * q^97 - 693 * 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
$3$	−6.48496	−1.24803	−0.624016	−	0.781412i	$-0.714500\pi$
$3$	−6.48496	−1.24803	−0.624016	+	0.781412i	$0.714500\pi$
$4$	0	0
$5$	7.60736	0.680423	0.340212	−	0.940349i	$-0.389501\pi$
$5$	7.60736	0.680423	0.340212	+	0.940349i	$0.389501\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	15.0547	0.557582
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	0.174238	0.00371730	0.00185865	−	0.999998i	$-0.499408\pi$
$13$	0.174238	0.00371730	0.00185865	+	0.999998i	$0.499408\pi$
$14$	0	0
$15$	−49.3335	−0.849190
$16$	0	0
$17$	128.863	1.83846	0.919231	−	0.393719i	$-0.128812\pi$
$17$	128.863	1.83846	0.919231	+	0.393719i	$0.128812\pi$
$18$	0	0
$19$	−141.685	−1.71078	−0.855388	−	0.517988i	$-0.826681\pi$
$19$	−141.685	−1.71078	−0.855388	+	0.517988i	$0.826681\pi$
$20$	0	0
$21$	45.3947	0.471712
$22$	0	0
$23$	133.369	1.20910	0.604550	−	0.796567i	$-0.293353\pi$
$23$	133.369	1.20910	0.604550	+	0.796567i	$0.293353\pi$
$24$	0	0
$25$	−67.1280	−0.537024
$26$	0	0
$27$	77.4647	0.552151
$28$	0	0
$29$	−177.002	−1.13340	−0.566698	−	0.823925i	$-0.691779\pi$
$29$	−177.002	−1.13340	−0.566698	+	0.823925i	$0.691779\pi$
$30$	0	0
$31$	−48.2757	−0.279696	−0.139848	−	0.990173i	$-0.544661\pi$
$31$	−48.2757	−0.279696	−0.139848	+	0.990173i	$0.544661\pi$
$32$	0	0
$33$	71.3346	0.376296
$34$	0	0
$35$	−53.2515	−0.257176
$36$	0	0
$37$	161.625	0.718136	0.359068	−	0.933311i	$-0.383095\pi$
$37$	161.625	0.718136	0.359068	+	0.933311i	$0.383095\pi$
$38$	0	0
$39$	−1.12993	−0.00463931
$40$	0	0
$41$	−195.689	−0.745402	−0.372701	−	0.927952i	$-0.621568\pi$
$41$	−195.689	−0.745402	−0.372701	+	0.927952i	$0.621568\pi$
$42$	0	0
$43$	488.447	1.73227	0.866133	−	0.499814i	$-0.166598\pi$
$43$	488.447	1.73227	0.866133	+	0.499814i	$0.166598\pi$
$44$	0	0
$45$	114.527	0.379392
$46$	0	0
$47$	171.705	0.532889	0.266444	−	0.963850i	$-0.414151\pi$
$47$	171.705	0.532889	0.266444	+	0.963850i	$0.414151\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−835.671	−2.29446
$52$	0	0
$53$	−431.477	−1.11826	−0.559131	−	0.829079i	$-0.688865\pi$
$53$	−431.477	−1.11826	−0.559131	+	0.829079i	$0.688865\pi$
$54$	0	0
$55$	−83.6810	−0.205155
$56$	0	0
$57$	918.821	2.13510
$58$	0	0
$59$	−194.176	−0.428468	−0.214234	−	0.976782i	$-0.568725\pi$
$59$	−194.176	−0.428468	−0.214234	+	0.976782i	$0.568725\pi$
$60$	0	0
$61$	585.008	1.22791	0.613955	−	0.789341i	$-0.289577\pi$
$61$	585.008	1.22791	0.613955	+	0.789341i	$0.289577\pi$
$62$	0	0
$63$	−105.383	−0.210746
$64$	0	0
$65$	1.32549	0.00252934
$66$	0	0
$67$	−155.905	−0.284281	−0.142140	−	0.989847i	$-0.545398\pi$
$67$	−155.905	−0.284281	−0.142140	+	0.989847i	$0.545398\pi$
$68$	0	0
$69$	−864.891	−1.50899
$70$	0	0
$71$	374.994	0.626811	0.313405	−	0.949619i	$-0.398530\pi$
$71$	374.994	0.626811	0.313405	+	0.949619i	$0.398530\pi$
$72$	0	0
$73$	210.419	0.337365	0.168683	−	0.985670i	$-0.446049\pi$
$73$	210.419	0.337365	0.168683	+	0.985670i	$0.446049\pi$
$74$	0	0
$75$	435.323	0.670223
$76$	0	0
$77$	77.0000	0.113961
$78$	0	0
$79$	7.00618	0.00997793	0.00498897	−	0.999988i	$-0.498412\pi$
$79$	7.00618	0.00997793	0.00498897	+	0.999988i	$0.498412\pi$
$80$	0	0
$81$	−908.833	−1.24668
$82$	0	0
$83$	93.6417	0.123837	0.0619187	−	0.998081i	$-0.480278\pi$
$83$	93.6417	0.123837	0.0619187	+	0.998081i	$0.480278\pi$
$84$	0	0
$85$	980.307	1.25093
$86$	0	0
$87$	1147.85	1.41451
$88$	0	0
$89$	−307.119	−0.365781	−0.182891	−	0.983133i	$-0.558545\pi$
$89$	−307.119	−0.365781	−0.182891	+	0.983133i	$0.558545\pi$
$90$	0	0
$91$	−1.21967	−0.00140501
$92$	0	0
$93$	313.066	0.349069
$94$	0	0
$95$	−1077.85	−1.16405
$96$	0	0
$97$	965.991	1.01115	0.505575	−	0.862783i	$-0.331280\pi$
$97$	965.991	1.01115	0.505575	+	0.862783i	$0.331280\pi$
$98$	0	0
$99$	−165.602	−0.168117
$100$	0	0
$101$	−577.986	−0.569424	−0.284712	−	0.958613i	$-0.591898\pi$
$101$	−577.986	−0.569424	−0.284712	+	0.958613i	$0.591898\pi$
$102$	0	0
$103$	−133.232	−0.127453	−0.0637267	−	0.997967i	$-0.520299\pi$
$103$	−133.232	−0.127453	−0.0637267	+	0.997967i	$0.520299\pi$
$104$	0	0
$105$	345.334	0.320963
$106$	0	0
$107$	−8.33627	−0.00753175	−0.00376587	−	0.999993i	$-0.501199\pi$
$107$	−8.33627	−0.00753175	−0.00376587	+	0.999993i	$0.501199\pi$
$108$	0	0
$109$	−317.052	−0.278606	−0.139303	−	0.990250i	$-0.544486\pi$
$109$	−317.052	−0.278606	−0.139303	+	0.990250i	$0.544486\pi$
$110$	0	0
$111$	−1048.13	−0.896256
$112$	0	0
$113$	−2170.00	−1.80652	−0.903260	−	0.429093i	$-0.858833\pi$
$113$	−2170.00	−1.80652	−0.903260	+	0.429093i	$0.858833\pi$
$114$	0	0
$115$	1014.58	0.822700
$116$	0	0
$117$	2.62310	0.00207270
$118$	0	0
$119$	−902.040	−0.694873
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	1269.03	0.930285
$124$	0	0
$125$	−1461.59	−1.04583
$126$	0	0
$127$	−1301.24	−0.909184	−0.454592	−	0.890700i	$-0.650215\pi$
$127$	−1301.24	−0.909184	−0.454592	+	0.890700i	$0.650215\pi$
$128$	0	0
$129$	−3167.56	−2.16192
$130$	0	0
$131$	−2522.22	−1.68220	−0.841098	−	0.540883i	$-0.818090\pi$
$131$	−2522.22	−1.68220	−0.841098	+	0.540883i	$0.818090\pi$
$132$	0	0
$133$	991.794	0.646612
$134$	0	0
$135$	589.302	0.375696
$136$	0	0
$137$	2574.70	1.60563	0.802815	−	0.596228i	$-0.203334\pi$
$137$	2574.70	1.60563	0.802815	+	0.596228i	$0.203334\pi$
$138$	0	0
$139$	2741.49	1.67288	0.836438	−	0.548061i	$-0.184634\pi$
$139$	2741.49	1.67288	0.836438	+	0.548061i	$0.184634\pi$
$140$	0	0
$141$	−1113.50	−0.665062
$142$	0	0
$143$	−1.91662	−0.00112081
$144$	0	0
$145$	−1346.52	−0.771189
$146$	0	0
$147$	−317.763	−0.178290
$148$	0	0
$149$	−1021.98	−0.561907	−0.280954	−	0.959721i	$-0.590651\pi$
$149$	−1021.98	−0.561907	−0.280954	+	0.959721i	$0.590651\pi$
$150$	0	0
$151$	−1016.40	−0.547769	−0.273885	−	0.961763i	$-0.588309\pi$
$151$	−1016.40	−0.547769	−0.273885	+	0.961763i	$0.588309\pi$
$152$	0	0
$153$	1940.00	1.02509
$154$	0	0
$155$	−367.251	−0.190312
$156$	0	0
$157$	−1580.49	−0.803422	−0.401711	−	0.915767i	$-0.631584\pi$
$157$	−1580.49	−0.803422	−0.401711	+	0.915767i	$0.631584\pi$
$158$	0	0
$159$	2798.11	1.39563
$160$	0	0
$161$	−933.581	−0.456997
$162$	0	0
$163$	−768.358	−0.369218	−0.184609	−	0.982812i	$-0.559102\pi$
$163$	−768.358	−0.369218	−0.184609	+	0.982812i	$0.559102\pi$
$164$	0	0
$165$	542.668	0.256040
$166$	0	0
$167$	−3092.73	−1.43307	−0.716534	−	0.697552i	$-0.754273\pi$
$167$	−3092.73	−1.43307	−0.716534	+	0.697552i	$0.754273\pi$
$168$	0	0
$169$	−2196.97	−0.999986
$170$	0	0
$171$	−2133.03	−0.953898
$172$	0	0
$173$	−1327.46	−0.583380	−0.291690	−	0.956513i	$-0.594218\pi$
$173$	−1327.46	−0.583380	−0.291690	+	0.956513i	$0.594218\pi$
$174$	0	0
$175$	469.896	0.202976
$176$	0	0
$177$	1259.23	0.534741
$178$	0	0
$179$	−3141.37	−1.31171	−0.655857	−	0.754885i	$-0.727693\pi$
$179$	−3141.37	−1.31171	−0.655857	+	0.754885i	$0.727693\pi$
$180$	0	0
$181$	−3683.36	−1.51261	−0.756303	−	0.654221i	$-0.772997\pi$
$181$	−3683.36	−1.51261	−0.756303	+	0.654221i	$0.772997\pi$
$182$	0	0
$183$	−3793.75	−1.53247
$184$	0	0
$185$	1229.54	0.488636
$186$	0	0
$187$	−1417.49	−0.554317
$188$	0	0
$189$	−542.253	−0.208694
$190$	0	0
$191$	2862.38	1.08437	0.542185	−	0.840259i	$-0.317597\pi$
$191$	2862.38	1.08437	0.542185	+	0.840259i	$0.317597\pi$
$192$	0	0
$193$	2023.91	0.754839	0.377420	−	0.926042i	$-0.376811\pi$
$193$	2023.91	0.754839	0.377420	+	0.926042i	$0.376811\pi$
$194$	0	0
$195$	−8.59576	−0.00315669
$196$	0	0
$197$	−4767.82	−1.72433	−0.862165	−	0.506628i	$-0.830892\pi$
$197$	−4767.82	−1.72433	−0.862165	+	0.506628i	$0.830892\pi$
$198$	0	0
$199$	−3546.72	−1.26342	−0.631709	−	0.775205i	$-0.717646\pi$
$199$	−3546.72	−1.26342	−0.631709	+	0.775205i	$0.717646\pi$
$200$	0	0
$201$	1011.04	0.354791
$202$	0	0
$203$	1239.02	0.428384
$204$	0	0
$205$	−1488.68	−0.507189
$206$	0	0
$207$	2007.83	0.674173
$208$	0	0
$209$	1558.53	0.515818
$210$	0	0
$211$	−1453.49	−0.474230	−0.237115	−	0.971482i	$-0.576202\pi$
$211$	−1453.49	−0.474230	−0.237115	+	0.971482i	$0.576202\pi$
$212$	0	0
$213$	−2431.82	−0.782279
$214$	0	0
$215$	3715.79	1.17867
$216$	0	0
$217$	337.930	0.105715
$218$	0	0
$219$	−1364.56	−0.421042
$220$	0	0
$221$	22.4528	0.00683411
$222$	0	0
$223$	−6440.10	−1.93391	−0.966953	−	0.254955i	$-0.917939\pi$
$223$	−6440.10	−1.93391	−0.966953	+	0.254955i	$0.917939\pi$
$224$	0	0
$225$	−1010.59	−0.299435
$226$	0	0
$227$	1313.21	0.383970	0.191985	−	0.981398i	$-0.438508\pi$
$227$	1313.21	0.383970	0.191985	+	0.981398i	$0.438508\pi$
$228$	0	0
$229$	−3780.00	−1.09078	−0.545392	−	0.838181i	$-0.683619\pi$
$229$	−3780.00	−1.09078	−0.545392	+	0.838181i	$0.683619\pi$
$230$	0	0
$231$	−499.342	−0.142226
$232$	0	0
$233$	2322.65	0.653055	0.326527	−	0.945188i	$-0.394121\pi$
$233$	2322.65	0.653055	0.326527	+	0.945188i	$0.394121\pi$
$234$	0	0
$235$	1306.22	0.362590
$236$	0	0
$237$	−45.4348	−0.0124528
$238$	0	0
$239$	−2204.79	−0.596718	−0.298359	−	0.954454i	$-0.596439\pi$
$239$	−2204.79	−0.596718	−0.298359	+	0.954454i	$0.596439\pi$
$240$	0	0
$241$	−4610.39	−1.23229	−0.616144	−	0.787634i	$-0.711306\pi$
$241$	−4610.39	−1.23229	−0.616144	+	0.787634i	$0.711306\pi$
$242$	0	0
$243$	3802.20	1.00375
$244$	0	0
$245$	372.761	0.0972033
$246$	0	0
$247$	−24.6869	−0.00635946
$248$	0	0
$249$	−607.263	−0.154553
$250$	0	0
$251$	−4981.12	−1.25261	−0.626306	−	0.779577i	$-0.715434\pi$
$251$	−4981.12	−1.25261	−0.626306	+	0.779577i	$0.715434\pi$
$252$	0	0
$253$	−1467.06	−0.364557
$254$	0	0
$255$	−6357.25	−1.56120
$256$	0	0
$257$	2726.08	0.661665	0.330833	−	0.943689i	$-0.392670\pi$
$257$	2726.08	0.661665	0.330833	+	0.943689i	$0.392670\pi$
$258$	0	0
$259$	−1131.38	−0.271430
$260$	0	0
$261$	−2664.72	−0.631962
$262$	0	0
$263$	4228.84	0.991489	0.495744	−	0.868468i	$-0.334895\pi$
$263$	4228.84	0.991489	0.495744	+	0.868468i	$0.334895\pi$
$264$	0	0
$265$	−3282.40	−0.760892
$266$	0	0
$267$	1991.65	0.456507
$268$	0	0
$269$	−2396.04	−0.543081	−0.271541	−	0.962427i	$-0.587533\pi$
$269$	−2396.04	−0.543081	−0.271541	+	0.962427i	$0.587533\pi$
$270$	0	0
$271$	2461.68	0.551795	0.275897	−	0.961187i	$-0.411025\pi$
$271$	2461.68	0.551795	0.275897	+	0.961187i	$0.411025\pi$
$272$	0	0
$273$	7.90948	0.00175349
$274$	0	0
$275$	738.408	0.161919
$276$	0	0
$277$	5890.54	1.27772	0.638860	−	0.769323i	$-0.279406\pi$
$277$	5890.54	1.27772	0.638860	+	0.769323i	$0.279406\pi$
$278$	0	0
$279$	−726.777	−0.155953
$280$	0	0
$281$	−1964.26	−0.417004	−0.208502	−	0.978022i	$-0.566859\pi$
$281$	−1964.26	−0.417004	−0.208502	+	0.978022i	$0.566859\pi$
$282$	0	0
$283$	−8513.60	−1.78827	−0.894136	−	0.447796i	$-0.852209\pi$
$283$	−8513.60	−1.78827	−0.894136	+	0.447796i	$0.852209\pi$
$284$	0	0
$285$	6989.80	1.45277
$286$	0	0
$287$	1369.82	0.281735
$288$	0	0
$289$	11692.6	2.37994
$290$	0	0
$291$	−6264.42	−1.26195
$292$	0	0
$293$	5219.58	1.04072	0.520360	−	0.853947i	$-0.325798\pi$
$293$	5219.58	1.04072	0.520360	+	0.853947i	$0.325798\pi$
$294$	0	0
$295$	−1477.17	−0.291539
$296$	0	0
$297$	−852.111	−0.166480
$298$	0	0
$299$	23.2379	0.00449459
$300$	0	0
$301$	−3419.13	−0.654735
$302$	0	0
$303$	3748.22	0.710659
$304$	0	0
$305$	4450.37	0.835499
$306$	0	0
$307$	−2887.29	−0.536764	−0.268382	−	0.963313i	$-0.586489\pi$
$307$	−2887.29	−0.536764	−0.268382	+	0.963313i	$0.586489\pi$
$308$	0	0
$309$	864.001	0.159066
$310$	0	0
$311$	5876.90	1.07154	0.535769	−	0.844364i	$-0.320022\pi$
$311$	5876.90	1.07154	0.535769	+	0.844364i	$0.320022\pi$
$312$	0	0
$313$	−7683.14	−1.38747	−0.693733	−	0.720233i	$-0.744035\pi$
$313$	−7683.14	−1.38747	−0.693733	+	0.720233i	$0.744035\pi$
$314$	0	0
$315$	−801.687	−0.143397
$316$	0	0
$317$	−9642.33	−1.70841	−0.854207	−	0.519933i	$-0.825957\pi$
$317$	−9642.33	−1.70841	−0.854207	+	0.519933i	$0.825957\pi$
$318$	0	0
$319$	1947.03	0.341732
$320$	0	0
$321$	54.0604	0.00939986
$322$	0	0
$323$	−18257.9	−3.14519
$324$	0	0
$325$	−11.6962	−0.00199628
$326$	0	0
$327$	2056.07	0.347710
$328$	0	0
$329$	−1201.94	−0.201413
$330$	0	0
$331$	2069.87	0.343717	0.171859	−	0.985122i	$-0.445023\pi$
$331$	2069.87	0.343717	0.171859	+	0.985122i	$0.445023\pi$
$332$	0	0
$333$	2433.22	0.400420
$334$	0	0
$335$	−1186.03	−0.193431
$336$	0	0
$337$	7547.59	1.22001	0.610005	−	0.792397i	$-0.291167\pi$
$337$	7547.59	1.22001	0.610005	+	0.792397i	$0.291167\pi$
$338$	0	0
$339$	14072.4	2.25459
$340$	0	0
$341$	531.033	0.0843315
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−6579.54	−1.02675
$346$	0	0
$347$	6353.22	0.982877	0.491439	−	0.870912i	$-0.336471\pi$
$347$	6353.22	0.982877	0.491439	+	0.870912i	$0.336471\pi$
$348$	0	0
$349$	−1352.47	−0.207439	−0.103719	−	0.994607i	$-0.533074\pi$
$349$	−1352.47	−0.207439	−0.103719	+	0.994607i	$0.533074\pi$
$350$	0	0
$351$	13.4973	0.00205251
$352$	0	0
$353$	4203.80	0.633840	0.316920	−	0.948452i	$-0.397351\pi$
$353$	4203.80	0.633840	0.316920	+	0.948452i	$0.397351\pi$
$354$	0	0
$355$	2852.71	0.426497
$356$	0	0
$357$	5849.70	0.867223
$358$	0	0
$359$	11100.7	1.63195	0.815976	−	0.578086i	$-0.196200\pi$
$359$	11100.7	1.63195	0.815976	+	0.578086i	$0.196200\pi$
$360$	0	0
$361$	13215.6	1.92675
$362$	0	0
$363$	−784.680	−0.113457
$364$	0	0
$365$	1600.73	0.229551
$366$	0	0
$367$	4963.35	0.705953	0.352977	−	0.935632i	$-0.385170\pi$
$367$	4963.35	0.705953	0.352977	+	0.935632i	$0.385170\pi$
$368$	0	0
$369$	−2946.04	−0.415623
$370$	0	0
$371$	3020.34	0.422663
$372$	0	0
$373$	−8882.54	−1.23303	−0.616515	−	0.787343i	$-0.711456\pi$
$373$	−8882.54	−1.23303	−0.616515	+	0.787343i	$0.711456\pi$
$374$	0	0
$375$	9478.34	1.30522
$376$	0	0
$377$	−30.8405	−0.00421317
$378$	0	0
$379$	1218.83	0.165190	0.0825952	−	0.996583i	$-0.473679\pi$
$379$	1218.83	0.165190	0.0825952	+	0.996583i	$0.473679\pi$
$380$	0	0
$381$	8438.49	1.13469
$382$	0	0
$383$	−112.614	−0.0150243	−0.00751214	−	0.999972i	$-0.502391\pi$
$383$	−112.614	−0.0150243	−0.00751214	+	0.999972i	$0.502391\pi$
$384$	0	0
$385$	585.767	0.0775414
$386$	0	0
$387$	7353.43	0.965880
$388$	0	0
$389$	12656.2	1.64961	0.824804	−	0.565419i	$-0.191286\pi$
$389$	12656.2	1.64961	0.824804	+	0.565419i	$0.191286\pi$
$390$	0	0
$391$	17186.3	2.22288
$392$	0	0
$393$	16356.5	2.09943
$394$	0	0
$395$	53.2985	0.00678922
$396$	0	0
$397$	3707.14	0.468655	0.234327	−	0.972158i	$-0.424711\pi$
$397$	3707.14	0.468655	0.234327	+	0.972158i	$0.424711\pi$
$398$	0	0
$399$	−6431.74	−0.806992
$400$	0	0
$401$	5671.06	0.706232	0.353116	−	0.935579i	$-0.385122\pi$
$401$	5671.06	0.706232	0.353116	+	0.935579i	$0.385122\pi$
$402$	0	0
$403$	−8.41146	−0.00103971
$404$	0	0
$405$	−6913.82	−0.848273
$406$	0	0
$407$	−1777.88	−0.216526
$408$	0	0
$409$	260.465	0.0314894	0.0157447	−	0.999876i	$-0.494988\pi$
$409$	260.465	0.0314894	0.0157447	+	0.999876i	$0.494988\pi$
$410$	0	0
$411$	−16696.8	−2.00388
$412$	0	0
$413$	1359.23	0.161946
$414$	0	0
$415$	712.366	0.0842619
$416$	0	0
$417$	−17778.4	−2.08780
$418$	0	0
$419$	−4153.96	−0.484330	−0.242165	−	0.970235i	$-0.577858\pi$
$419$	−4153.96	−0.484330	−0.242165	+	0.970235i	$0.577858\pi$
$420$	0	0
$421$	4240.11	0.490856	0.245428	−	0.969415i	$-0.421072\pi$
$421$	4240.11	0.490856	0.245428	+	0.969415i	$0.421072\pi$
$422$	0	0
$423$	2584.97	0.297129
$424$	0	0
$425$	−8650.31	−0.987298
$426$	0	0
$427$	−4095.05	−0.464107
$428$	0	0
$429$	12.4292	0.00139880
$430$	0	0
$431$	−1012.17	−0.113119	−0.0565595	−	0.998399i	$-0.518013\pi$
$431$	−1012.17	−0.113119	−0.0565595	+	0.998399i	$0.518013\pi$
$432$	0	0
$433$	1604.50	0.178077	0.0890386	−	0.996028i	$-0.471621\pi$
$433$	1604.50	0.178077	0.0890386	+	0.996028i	$0.471621\pi$
$434$	0	0
$435$	8732.13	0.962469
$436$	0	0
$437$	−18896.3	−2.06850
$438$	0	0
$439$	−10671.6	−1.16020	−0.580099	−	0.814546i	$-0.696986\pi$
$439$	−10671.6	−1.16020	−0.580099	+	0.814546i	$0.696986\pi$
$440$	0	0
$441$	737.681	0.0796546
$442$	0	0
$443$	−2193.74	−0.235277	−0.117638	−	0.993056i	$-0.537532\pi$
$443$	−2193.74	−0.235277	−0.117638	+	0.993056i	$0.537532\pi$
$444$	0	0
$445$	−2336.37	−0.248886
$446$	0	0
$447$	6627.52	0.701278
$448$	0	0
$449$	−7070.36	−0.743143	−0.371571	−	0.928404i	$-0.621181\pi$
$449$	−7070.36	−0.743143	−0.371571	+	0.928404i	$0.621181\pi$
$450$	0	0
$451$	2152.58	0.224747
$452$	0	0
$453$	6591.29	0.683633
$454$	0	0
$455$	−9.27844	−0.000956000 0
$456$	0	0
$457$	16245.8	1.66290	0.831451	−	0.555599i	$-0.187511\pi$
$457$	16245.8	1.66290	0.831451	+	0.555599i	$0.187511\pi$
$458$	0	0
$459$	9982.32	1.01511
$460$	0	0
$461$	−5235.18	−0.528908	−0.264454	−	0.964398i	$-0.585192\pi$
$461$	−5235.18	−0.528908	−0.264454	+	0.964398i	$0.585192\pi$
$462$	0	0
$463$	11093.9	1.11355	0.556777	−	0.830662i	$-0.312038\pi$
$463$	11093.9	1.11355	0.556777	+	0.830662i	$0.312038\pi$
$464$	0	0
$465$	2381.61	0.237515
$466$	0	0
$467$	7813.14	0.774195	0.387097	−	0.922039i	$-0.373478\pi$
$467$	7813.14	0.774195	0.387097	+	0.922039i	$0.373478\pi$
$468$	0	0
$469$	1091.33	0.107448
$470$	0	0
$471$	10249.4	1.00270
$472$	0	0
$473$	−5372.91	−0.522298
$474$	0	0
$475$	9511.02	0.918728
$476$	0	0
$477$	−6495.76	−0.623523
$478$	0	0
$479$	−5298.54	−0.505420	−0.252710	−	0.967542i	$-0.581322\pi$
$479$	−5298.54	−0.505420	−0.252710	+	0.967542i	$0.581322\pi$
$480$	0	0
$481$	28.1612	0.00266953
$482$	0	0
$483$	6054.23	0.570346
$484$	0	0
$485$	7348.65	0.688010
$486$	0	0
$487$	−5580.89	−0.519290	−0.259645	−	0.965704i	$-0.583606\pi$
$487$	−5580.89	−0.519290	−0.259645	+	0.965704i	$0.583606\pi$
$488$	0	0
$489$	4982.77	0.460795
$490$	0	0
$491$	−16718.8	−1.53668	−0.768341	−	0.640041i	$-0.778917\pi$
$491$	−16718.8	−1.53668	−0.768341	+	0.640041i	$0.778917\pi$
$492$	0	0
$493$	−22809.0	−2.08371
$494$	0	0
$495$	−1259.79	−0.114391
$496$	0	0
$497$	−2624.96	−0.236912
$498$	0	0
$499$	−19594.4	−1.75784	−0.878922	−	0.476965i	$-0.841737\pi$
$499$	−19594.4	−1.75784	−0.878922	+	0.476965i	$0.841737\pi$
$500$	0	0
$501$	20056.2	1.78851
$502$	0	0
$503$	−9008.04	−0.798506	−0.399253	−	0.916841i	$-0.630731\pi$
$503$	−9008.04	−0.798506	−0.399253	+	0.916841i	$0.630731\pi$
$504$	0	0
$505$	−4396.95	−0.387449
$506$	0	0
$507$	14247.3	1.24801
$508$	0	0
$509$	−3379.88	−0.294324	−0.147162	−	0.989112i	$-0.547014\pi$
$509$	−3379.88	−0.294324	−0.147162	+	0.989112i	$0.547014\pi$
$510$	0	0
$511$	−1472.93	−0.127512
$512$	0	0
$513$	−10975.6	−0.944606
$514$	0	0
$515$	−1013.54	−0.0867222
$516$	0	0
$517$	−1888.76	−0.160672
$518$	0	0
$519$	8608.51	0.728076
$520$	0	0
$521$	−736.886	−0.0619646	−0.0309823	−	0.999520i	$-0.509864\pi$
$521$	−736.886	−0.0619646	−0.0309823	+	0.999520i	$0.509864\pi$
$522$	0	0
$523$	−9541.00	−0.797703	−0.398852	−	0.917015i	$-0.630591\pi$
$523$	−9541.00	−0.797703	−0.398852	+	0.917015i	$0.630591\pi$
$524$	0	0
$525$	−3047.26	−0.253320
$526$	0	0
$527$	−6220.95	−0.514210
$528$	0	0
$529$	5620.20	0.461922
$530$	0	0
$531$	−2923.27	−0.238906
$532$	0	0
$533$	−34.0964	−0.00277088
$534$	0	0
$535$	−63.4170	−0.00512478
$536$	0	0
$537$	20371.7	1.63706
$538$	0	0
$539$	−539.000	−0.0430730
$540$	0	0
$541$	16721.0	1.32882	0.664411	−	0.747368i	$-0.268683\pi$
$541$	16721.0	1.32882	0.664411	+	0.747368i	$0.268683\pi$
$542$	0	0
$543$	23886.4	1.88778
$544$	0	0
$545$	−2411.93	−0.189570
$546$	0	0
$547$	−8958.59	−0.700259	−0.350129	−	0.936701i	$-0.613862\pi$
$547$	−8958.59	−0.700259	−0.350129	+	0.936701i	$0.613862\pi$
$548$	0	0
$549$	8807.13	0.684661
$550$	0	0
$551$	25078.5	1.93899
$552$	0	0
$553$	−49.0432	−0.00377130
$554$	0	0
$555$	−7973.53	−0.609834
$556$	0	0
$557$	−14329.6	−1.09006	−0.545031	−	0.838416i	$-0.683482\pi$
$557$	−14329.6	−1.09006	−0.545031	+	0.838416i	$0.683482\pi$
$558$	0	0
$559$	85.1059	0.00643935
$560$	0	0
$561$	9192.38	0.691805
$562$	0	0
$563$	−15791.2	−1.18209	−0.591047	−	0.806637i	$-0.701285\pi$
$563$	−15791.2	−1.18209	−0.591047	+	0.806637i	$0.701285\pi$
$564$	0	0
$565$	−16508.0	−1.22920
$566$	0	0
$567$	6361.83	0.471202
$568$	0	0
$569$	−13996.4	−1.03121	−0.515605	−	0.856826i	$-0.672433\pi$
$569$	−13996.4	−1.03121	−0.515605	+	0.856826i	$0.672433\pi$
$570$	0	0
$571$	6642.54	0.486833	0.243417	−	0.969922i	$-0.421732\pi$
$571$	6642.54	0.486833	0.243417	+	0.969922i	$0.421732\pi$
$572$	0	0
$573$	−18562.4	−1.35333
$574$	0	0
$575$	−8952.78	−0.649316
$576$	0	0
$577$	−85.5448	−0.00617206	−0.00308603	−	0.999995i	$-0.500982\pi$
$577$	−85.5448	−0.00617206	−0.00308603	+	0.999995i	$0.500982\pi$
$578$	0	0
$579$	−13125.0	−0.942063
$580$	0	0
$581$	−655.492	−0.0468062
$582$	0	0
$583$	4746.25	0.337169
$584$	0	0
$585$	19.9549	0.00141031
$586$	0	0
$587$	−11205.0	−0.787869	−0.393934	−	0.919139i	$-0.628886\pi$
$587$	−11205.0	−0.787869	−0.393934	+	0.919139i	$0.628886\pi$
$588$	0	0
$589$	6839.93	0.478497
$590$	0	0
$591$	30919.1	2.15202
$592$	0	0
$593$	11664.6	0.807773	0.403886	−	0.914809i	$-0.367659\pi$
$593$	11664.6	0.807773	0.403886	+	0.914809i	$0.367659\pi$
$594$	0	0
$595$	−6862.15	−0.472808
$596$	0	0
$597$	23000.3	1.57679
$598$	0	0
$599$	20545.1	1.40142	0.700711	−	0.713445i	$-0.252866\pi$
$599$	20545.1	1.40142	0.700711	+	0.713445i	$0.252866\pi$
$600$	0	0
$601$	3885.01	0.263682	0.131841	−	0.991271i	$-0.457911\pi$
$601$	3885.01	0.263682	0.131841	+	0.991271i	$0.457911\pi$
$602$	0	0
$603$	−2347.11	−0.158510
$604$	0	0
$605$	920.491	0.0618567
$606$	0	0
$607$	−8439.80	−0.564351	−0.282175	−	0.959363i	$-0.591056\pi$
$607$	−8439.80	−0.564351	−0.282175	+	0.959363i	$0.591056\pi$
$608$	0	0
$609$	−8034.97	−0.534636
$610$	0	0
$611$	29.9175	0.00198091
$612$	0	0
$613$	17997.8	1.18584	0.592922	−	0.805260i	$-0.297974\pi$
$613$	17997.8	1.18584	0.592922	+	0.805260i	$0.297974\pi$
$614$	0	0
$615$	9654.01	0.632987
$616$	0	0
$617$	−26336.6	−1.71843	−0.859214	−	0.511616i	$-0.829047\pi$
$617$	−26336.6	−1.71843	−0.859214	+	0.511616i	$0.829047\pi$
$618$	0	0
$619$	−27875.5	−1.81003	−0.905016	−	0.425376i	$-0.860142\pi$
$619$	−27875.5	−1.81003	−0.905016	+	0.425376i	$0.860142\pi$
$620$	0	0
$621$	10331.4	0.667606
$622$	0	0
$623$	2149.83	0.138252
$624$	0	0
$625$	−2727.83	−0.174581
$626$	0	0
$627$	−10107.0	−0.643757
$628$	0	0
$629$	20827.5	1.32027
$630$	0	0
$631$	−8242.15	−0.519992	−0.259996	−	0.965610i	$-0.583721\pi$
$631$	−8242.15	−0.519992	−0.259996	+	0.965610i	$0.583721\pi$
$632$	0	0
$633$	9425.84	0.591854
$634$	0	0
$635$	−9899.00	−0.618630
$636$	0	0
$637$	8.53766	0.000531043 0
$638$	0	0
$639$	5645.43	0.349499
$640$	0	0
$641$	−7001.97	−0.431453	−0.215726	−	0.976454i	$-0.569212\pi$
$641$	−7001.97	−0.431453	−0.215726	+	0.976454i	$0.569212\pi$
$642$	0	0
$643$	−2473.25	−0.151688	−0.0758442	−	0.997120i	$-0.524165\pi$
$643$	−2473.25	−0.151688	−0.0758442	+	0.997120i	$0.524165\pi$
$644$	0	0
$645$	−24096.8	−1.47102
$646$	0	0
$647$	−9153.21	−0.556182	−0.278091	−	0.960555i	$-0.589702\pi$
$647$	−9153.21	−0.556182	−0.278091	+	0.960555i	$0.589702\pi$
$648$	0	0
$649$	2135.94	0.129188
$650$	0	0
$651$	−2191.46	−0.131936
$652$	0	0
$653$	725.254	0.0434630	0.0217315	−	0.999764i	$-0.493082\pi$
$653$	725.254	0.0434630	0.0217315	+	0.999764i	$0.493082\pi$
$654$	0	0
$655$	−19187.5	−1.14461
$656$	0	0
$657$	3167.80	0.188109
$658$	0	0
$659$	−14332.9	−0.847242	−0.423621	−	0.905840i	$-0.639241\pi$
$659$	−14332.9	−0.847242	−0.423621	+	0.905840i	$0.639241\pi$
$660$	0	0
$661$	26773.3	1.57543	0.787715	−	0.616040i	$-0.211264\pi$
$661$	26773.3	1.57543	0.787715	+	0.616040i	$0.211264\pi$
$662$	0	0
$663$	−145.606	−0.00852919
$664$	0	0
$665$	7544.94	0.439970
$666$	0	0
$667$	−23606.6	−1.37039
$668$	0	0
$669$	41763.8	2.41358
$670$	0	0
$671$	−6435.08	−0.370229
$672$	0	0
$673$	−25423.9	−1.45619	−0.728096	−	0.685475i	$-0.759595\pi$
$673$	−25423.9	−1.45619	−0.728096	+	0.685475i	$0.759595\pi$
$674$	0	0
$675$	−5200.05	−0.296519
$676$	0	0
$677$	9287.48	0.527248	0.263624	−	0.964625i	$-0.415082\pi$
$677$	9287.48	0.527248	0.263624	+	0.964625i	$0.415082\pi$
$678$	0	0
$679$	−6761.94	−0.382179
$680$	0	0
$681$	−8516.15	−0.479206
$682$	0	0
$683$	31888.8	1.78652	0.893258	−	0.449544i	$-0.148414\pi$
$683$	31888.8	1.78652	0.893258	+	0.449544i	$0.148414\pi$
$684$	0	0
$685$	19586.7	1.09251
$686$	0	0
$687$	24513.2	1.36133
$688$	0	0
$689$	−75.1796	−0.00415692
$690$	0	0
$691$	18650.3	1.02676	0.513379	−	0.858162i	$-0.328394\pi$
$691$	18650.3	1.02676	0.513379	+	0.858162i	$0.328394\pi$
$692$	0	0
$693$	1159.21	0.0635424
$694$	0	0
$695$	20855.5	1.13826
$696$	0	0
$697$	−25217.0	−1.37039
$698$	0	0
$699$	−15062.3	−0.815033
$700$	0	0
$701$	8566.01	0.461532	0.230766	−	0.973009i	$-0.425877\pi$
$701$	8566.01	0.461532	0.230766	+	0.973009i	$0.425877\pi$
$702$	0	0
$703$	−22899.8	−1.22857
$704$	0	0
$705$	−8470.81	−0.452524
$706$	0	0
$707$	4045.90	0.215222
$708$	0	0
$709$	680.116	0.0360258	0.0180129	−	0.999838i	$-0.494266\pi$
$709$	680.116	0.0360258	0.0180129	+	0.999838i	$0.494266\pi$
$710$	0	0
$711$	105.476	0.00556352
$712$	0	0
$713$	−6438.47	−0.338180
$714$	0	0
$715$	−14.5804	−0.000762624 0
$716$	0	0
$717$	14297.9	0.744723
$718$	0	0
$719$	−12557.6	−0.651346	−0.325673	−	0.945482i	$-0.605591\pi$
$719$	−12557.6	−0.651346	−0.325673	+	0.945482i	$0.605591\pi$
$720$	0	0
$721$	932.621	0.0481728
$722$	0	0
$723$	29898.2	1.53793
$724$	0	0
$725$	11881.8	0.608661
$726$	0	0
$727$	−22253.6	−1.13527	−0.567635	−	0.823281i	$-0.692141\pi$
$727$	−22253.6	−1.13527	−0.567635	+	0.823281i	$0.692141\pi$
$728$	0	0
$729$	−118.633	−0.00602716
$730$	0	0
$731$	62942.6	3.18470
$732$	0	0
$733$	6852.72	0.345308	0.172654	−	0.984982i	$-0.444766\pi$
$733$	6852.72	0.345308	0.172654	+	0.984982i	$0.444766\pi$
$734$	0	0
$735$	−2417.34	−0.121313
$736$	0	0
$737$	1714.95	0.0857139
$738$	0	0
$739$	9133.20	0.454628	0.227314	−	0.973821i	$-0.427006\pi$
$739$	9133.20	0.454628	0.227314	+	0.973821i	$0.427006\pi$
$740$	0	0
$741$	160.093	0.00793681
$742$	0	0
$743$	33458.7	1.65206	0.826031	−	0.563624i	$-0.190593\pi$
$743$	33458.7	1.65206	0.826031	+	0.563624i	$0.190593\pi$
$744$	0	0
$745$	−7774.60	−0.382335
$746$	0	0
$747$	1409.75	0.0690496
$748$	0	0
$749$	58.3539	0.00284673
$750$	0	0
$751$	37530.0	1.82355	0.911776	−	0.410688i	$-0.134711\pi$
$751$	37530.0	1.82355	0.911776	+	0.410688i	$0.134711\pi$
$752$	0	0
$753$	32302.4	1.56330
$754$	0	0
$755$	−7732.10	−0.372715
$756$	0	0
$757$	32174.8	1.54480	0.772399	−	0.635138i	$-0.219057\pi$
$757$	32174.8	1.54480	0.772399	+	0.635138i	$0.219057\pi$
$758$	0	0
$759$	9513.80	0.454979
$760$	0	0
$761$	4457.43	0.212328	0.106164	−	0.994349i	$-0.466143\pi$
$761$	4457.43	0.212328	0.106164	+	0.994349i	$0.466143\pi$
$762$	0	0
$763$	2219.37	0.105303
$764$	0	0
$765$	14758.2	0.697497
$766$	0	0
$767$	−33.8329	−0.00159274
$768$	0	0
$769$	−38329.0	−1.79737	−0.898687	−	0.438591i	$-0.855478\pi$
$769$	−38329.0	−1.79737	−0.898687	+	0.438591i	$0.855478\pi$
$770$	0	0
$771$	−17678.5	−0.825779
$772$	0	0
$773$	−9529.23	−0.443393	−0.221696	−	0.975116i	$-0.571159\pi$
$773$	−9529.23	−0.443393	−0.221696	+	0.975116i	$0.571159\pi$
$774$	0	0
$775$	3240.65	0.150203
$776$	0	0
$777$	7336.94	0.338753
$778$	0	0
$779$	27726.1	1.27521
$780$	0	0
$781$	−4124.93	−0.188991
$782$	0	0
$783$	−13711.4	−0.625806
$784$	0	0
$785$	−12023.4	−0.546667
$786$	0	0
$787$	−29705.1	−1.34545	−0.672726	−	0.739892i	$-0.734877\pi$
$787$	−29705.1	−1.34545	−0.672726	+	0.739892i	$0.734877\pi$
$788$	0	0
$789$	−27423.9	−1.23741
$790$	0	0
$791$	15190.0	0.682801
$792$	0	0
$793$	101.931	0.00456451
$794$	0	0
$795$	21286.2	0.949617
$796$	0	0
$797$	8596.99	0.382084	0.191042	−	0.981582i	$-0.438813\pi$
$797$	8596.99	0.382084	0.191042	+	0.981582i	$0.438813\pi$
$798$	0	0
$799$	22126.4	0.979695
$800$	0	0
$801$	−4623.59	−0.203953
$802$	0	0
$803$	−2314.61	−0.101719
$804$	0	0
$805$	−7102.09	−0.310951
$806$	0	0
$807$	15538.2	0.677783
$808$	0	0
$809$	16630.2	0.722727	0.361364	−	0.932425i	$-0.382311\pi$
$809$	16630.2	0.722727	0.361364	+	0.932425i	$0.382311\pi$
$810$	0	0
$811$	−18584.2	−0.804660	−0.402330	−	0.915495i	$-0.631799\pi$
$811$	−18584.2	−0.804660	−0.402330	+	0.915495i	$0.631799\pi$
$812$	0	0
$813$	−15963.9	−0.688657
$814$	0	0
$815$	−5845.18	−0.251224
$816$	0	0
$817$	−69205.5	−2.96352
$818$	0	0
$819$	−18.3617	−0.000783407 0
$820$	0	0
$821$	−4088.88	−0.173816	−0.0869079	−	0.996216i	$-0.527699\pi$
$821$	−4088.88	−0.173816	−0.0869079	+	0.996216i	$0.527699\pi$
$822$	0	0
$823$	12830.0	0.543410	0.271705	−	0.962381i	$-0.412413\pi$
$823$	12830.0	0.543410	0.271705	+	0.962381i	$0.412413\pi$
$824$	0	0
$825$	−4788.55	−0.202080
$826$	0	0
$827$	13768.7	0.578942	0.289471	−	0.957187i	$-0.406521\pi$
$827$	13768.7	0.578942	0.289471	+	0.957187i	$0.406521\pi$
$828$	0	0
$829$	−10907.8	−0.456987	−0.228494	−	0.973545i	$-0.573380\pi$
$829$	−10907.8	−0.456987	−0.228494	+	0.973545i	$0.573380\pi$
$830$	0	0
$831$	−38199.9	−1.59463
$832$	0	0
$833$	6314.28	0.262637
$834$	0	0
$835$	−23527.5	−0.975093
$836$	0	0
$837$	−3739.66	−0.154434
$838$	0	0
$839$	27174.8	1.11821	0.559105	−	0.829097i	$-0.311145\pi$
$839$	27174.8	1.11821	0.559105	+	0.829097i	$0.311145\pi$
$840$	0	0
$841$	6940.81	0.284588
$842$	0	0
$843$	12738.2	0.520434
$844$	0	0
$845$	−16713.1	−0.680414
$846$	0	0
$847$	−847.000	−0.0343604
$848$	0	0
$849$	55210.3	2.23182
$850$	0	0
$851$	21555.7	0.868298
$852$	0	0
$853$	−33975.1	−1.36376	−0.681878	−	0.731466i	$-0.738837\pi$
$853$	−33975.1	−1.36376	−0.681878	+	0.731466i	$0.738837\pi$
$854$	0	0
$855$	−16226.7	−0.649054
$856$	0	0
$857$	20423.7	0.814073	0.407037	−	0.913412i	$-0.366562\pi$
$857$	20423.7	0.814073	0.407037	+	0.913412i	$0.366562\pi$
$858$	0	0
$859$	36704.2	1.45789	0.728947	−	0.684570i	$-0.240010\pi$
$859$	36704.2	1.45789	0.728947	+	0.684570i	$0.240010\pi$
$860$	0	0
$861$	−8883.24	−0.351615
$862$	0	0
$863$	−26222.8	−1.03434	−0.517170	−	0.855883i	$-0.673015\pi$
$863$	−26222.8	−1.03434	−0.517170	+	0.855883i	$0.673015\pi$
$864$	0	0
$865$	−10098.5	−0.396945
$866$	0	0
$867$	−75826.4	−2.97024
$868$	0	0
$869$	−77.0680	−0.00300846
$870$	0	0
$871$	−27.1645	−0.00105676
$872$	0	0
$873$	14542.7	0.563799
$874$	0	0
$875$	10231.1	0.395285
$876$	0	0
$877$	44950.8	1.73076	0.865382	−	0.501113i	$-0.167076\pi$
$877$	44950.8	1.73076	0.865382	+	0.501113i	$0.167076\pi$
$878$	0	0
$879$	−33848.8	−1.29885
$880$	0	0
$881$	3896.87	0.149023	0.0745113	−	0.997220i	$-0.476260\pi$
$881$	3896.87	0.149023	0.0745113	+	0.997220i	$0.476260\pi$
$882$	0	0
$883$	−6257.41	−0.238481	−0.119240	−	0.992865i	$-0.538046\pi$
$883$	−6257.41	−0.238481	−0.119240	+	0.992865i	$0.538046\pi$
$884$	0	0
$885$	9579.38	0.363850
$886$	0	0
$887$	−40626.2	−1.53787	−0.768937	−	0.639324i	$-0.779214\pi$
$887$	−40626.2	−1.53787	−0.768937	+	0.639324i	$0.779214\pi$
$888$	0	0
$889$	9108.68	0.343639
$890$	0	0
$891$	9997.16	0.375889
$892$	0	0
$893$	−24328.0	−0.911653
$894$	0	0
$895$	−23897.5	−0.892521
$896$	0	0
$897$	−150.697	−0.00560938
$898$	0	0
$899$	8544.91	0.317006
$900$	0	0
$901$	−55601.4	−2.05588
$902$	0	0
$903$	22172.9	0.817129
$904$	0	0
$905$	−28020.6	−1.02921
$906$	0	0
$907$	−42366.9	−1.55101	−0.775506	−	0.631340i	$-0.782505\pi$
$907$	−42366.9	−1.55101	−0.775506	+	0.631340i	$0.782505\pi$
$908$	0	0
$909$	−8701.42	−0.317501
$910$	0	0
$911$	−26231.4	−0.953992	−0.476996	−	0.878906i	$-0.658274\pi$
$911$	−26231.4	−0.953992	−0.476996	+	0.878906i	$0.658274\pi$
$912$	0	0
$913$	−1030.06	−0.0373384
$914$	0	0
$915$	−28860.5	−1.04273
$916$	0	0
$917$	17655.6	0.635810
$918$	0	0
$919$	−2542.24	−0.0912522	−0.0456261	−	0.998959i	$-0.514528\pi$
$919$	−2542.24	−0.0912522	−0.0456261	+	0.998959i	$0.514528\pi$
$920$	0	0
$921$	18724.0	0.669898
$922$	0	0
$923$	65.3381	0.00233004
$924$	0	0
$925$	−10849.6	−0.385656
$926$	0	0
$927$	−2005.76	−0.0710657
$928$	0	0
$929$	−20097.8	−0.709782	−0.354891	−	0.934908i	$-0.615482\pi$
$929$	−20097.8	−0.709782	−0.354891	+	0.934908i	$0.615482\pi$
$930$	0	0
$931$	−6942.56	−0.244396
$932$	0	0
$933$	−38111.5	−1.33731
$934$	0	0
$935$	−10783.4	−0.377170
$936$	0	0
$937$	−202.788	−0.00707023	−0.00353511	−	0.999994i	$-0.501125\pi$
$937$	−202.788	−0.00707023	−0.00353511	+	0.999994i	$0.501125\pi$
$938$	0	0
$939$	49824.9	1.73160
$940$	0	0
$941$	12419.1	0.430235	0.215118	−	0.976588i	$-0.430987\pi$
$941$	12419.1	0.430235	0.215118	+	0.976588i	$0.430987\pi$
$942$	0	0
$943$	−26098.8	−0.901265
$944$	0	0
$945$	−4125.11	−0.142000
$946$	0	0
$947$	−369.282	−0.0126717	−0.00633583	−	0.999980i	$-0.502017\pi$
$947$	−369.282	−0.0126717	−0.00633583	+	0.999980i	$0.502017\pi$
$948$	0	0
$949$	36.6629	0.00125409
$950$	0	0
$951$	62530.1	2.13215
$952$	0	0
$953$	25994.1	0.883558	0.441779	−	0.897124i	$-0.354348\pi$
$953$	25994.1	0.883558	0.441779	+	0.897124i	$0.354348\pi$
$954$	0	0
$955$	21775.2	0.737830
$956$	0	0
$957$	−12626.4	−0.426492
$958$	0	0
$959$	−18022.9	−0.606871
$960$	0	0
$961$	−27460.5	−0.921770
$962$	0	0
$963$	−125.500	−0.00419957
$964$	0	0
$965$	15396.6	0.513610
$966$	0	0
$967$	53830.0	1.79013	0.895065	−	0.445937i	$-0.147129\pi$
$967$	53830.0	1.79013	0.895065	+	0.445937i	$0.147129\pi$
$968$	0	0
$969$	118402.	3.92530
$970$	0	0
$971$	1349.54	0.0446022	0.0223011	−	0.999751i	$-0.492901\pi$
$971$	1349.54	0.0446022	0.0223011	+	0.999751i	$0.492901\pi$
$972$	0	0
$973$	−19190.4	−0.632288
$974$	0	0
$975$	75.8497	0.00249142
$976$	0	0
$977$	6611.71	0.216507	0.108253	−	0.994123i	$-0.465474\pi$
$977$	6611.71	0.216507	0.108253	+	0.994123i	$0.465474\pi$
$978$	0	0
$979$	3378.31	0.110287
$980$	0	0
$981$	−4773.13	−0.155346
$982$	0	0
$983$	−40595.9	−1.31720	−0.658601	−	0.752493i	$-0.728851\pi$
$983$	−40595.9	−1.31720	−0.658601	+	0.752493i	$0.728851\pi$
$984$	0	0
$985$	−36270.5	−1.17327
$986$	0	0
$987$	7794.51	0.251370
$988$	0	0
$989$	65143.5	2.09448
$990$	0	0
$991$	48877.7	1.56675	0.783376	−	0.621548i	$-0.213496\pi$
$991$	48877.7	1.56675	0.783376	+	0.621548i	$0.213496\pi$
$992$	0	0
$993$	−13423.0	−0.428970
$994$	0	0
$995$	−26981.2	−0.859660
$996$	0	0
$997$	13127.4	0.417000	0.208500	−	0.978022i	$-0.433142\pi$
$997$	13127.4	0.417000	0.208500	+	0.978022i	$0.433142\pi$
$998$	0	0
$999$	12520.2	0.396520

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.4.a.y.1.2		5
4.3	odd	2		77.4.a.e.1.2	✓	5
12.11	even	2		693.4.a.o.1.4		5
20.19	odd	2		1925.4.a.r.1.4		5
28.27	even	2		539.4.a.h.1.2		5
44.43	even	2		847.4.a.f.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.e.1.2	✓	5	4.3	odd	2
539.4.a.h.1.2		5	28.27	even	2
693.4.a.o.1.4		5	12.11	even	2
847.4.a.f.1.4		5	44.43	even	2
1232.4.a.y.1.2		5	1.1	even	1	trivial
1925.4.a.r.1.4		5	20.19	odd	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:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1232.a (trivial)

\(f(q)\)	\(=\)	\(q-6.48496 q^{3} +7.60736 q^{5} -7.00000 q^{7} +15.0547 q^{9} -11.0000 q^{11} +0.174238 q^{13} -49.3335 q^{15} +128.863 q^{17} -141.685 q^{19} +45.3947 q^{21} +133.369 q^{23} -67.1280 q^{25} +77.4647 q^{27} -177.002 q^{29} -48.2757 q^{31} +71.3346 q^{33} -53.2515 q^{35} +161.625 q^{37} -1.12993 q^{39} -195.689 q^{41} +488.447 q^{43} +114.527 q^{45} +171.705 q^{47} +49.0000 q^{49} -835.671 q^{51} -431.477 q^{53} -83.6810 q^{55} +918.821 q^{57} -194.176 q^{59} +585.008 q^{61} -105.383 q^{63} +1.32549 q^{65} -155.905 q^{67} -864.891 q^{69} +374.994 q^{71} +210.419 q^{73} +435.323 q^{75} +77.0000 q^{77} +7.00618 q^{79} -908.833 q^{81} +93.6417 q^{83} +980.307 q^{85} +1147.85 q^{87} -307.119 q^{89} -1.21967 q^{91} +313.066 q^{93} -1077.85 q^{95} +965.991 q^{97} -165.602 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{3} - 24 q^{5} - 35 q^{7} + 63 q^{9} - 55 q^{11} - 50 q^{13} + 146 q^{15} + 222 q^{17} - 160 q^{19} + 14 q^{21} - 54 q^{23} + 125 q^{25} - 110 q^{27} + 14 q^{29} + 34 q^{31} + 22 q^{33} + 168 q^{35}+ \cdots - 693 q^{99}+O(q^{100}) \) 5 * q - 2 * q^3 - 24 * q^5 - 35 * q^7 + 63 * q^9 - 55 * q^11 - 50 * q^13 + 146 * q^15 + 222 * q^17 - 160 * q^19 + 14 * q^21 - 54 * q^23 + 125 * q^25 - 110 * q^27 + 14 * q^29 + 34 * q^31 + 22 * q^33 + 168 * q^35 + 1044 * q^37 + 124 * q^39 - 114 * q^41 - 672 * q^43 - 530 * q^45 + 292 * q^47 + 245 * q^49 - 768 * q^51 - 710 * q^53 + 264 * q^55 + 2012 * q^57 - 270 * q^59 + 138 * q^61 - 441 * q^63 - 196 * q^65 - 1942 * q^67 - 1306 * q^69 + 278 * q^71 - 338 * q^73 - 3960 * q^75 + 385 * q^77 - 576 * q^79 - 1439 * q^81 - 1644 * q^83 + 360 * q^85 + 1956 * q^87 - 3656 * q^89 + 350 * q^91 - 1442 * q^93 + 1276 * q^95 + 692 * q^97 - 693 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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