LMFDB - Embedded newform 1323.4.a.bm.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,4,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, 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("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1323.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:	$78.0595269376$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 54x^{6} + 887x^{4} - 4176x^{2} + 3136 $ x^8 - 54x^6 + 887x^4 - 4176*x^2 + 3136
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 7 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.67291$ of defining polynomial
Character	$\chi$	$=$	1323.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.67291 q^{2} +13.8361 q^{4} -15.5408 q^{5} -27.2713 q^{8} +O(q^{10})$	$q-4.67291 q^{2} +13.8361 q^{4} -15.5408 q^{5} -27.2713 q^{8} +72.6208 q^{10} -63.4897 q^{11} +53.1931 q^{13} +16.7479 q^{16} -68.4660 q^{17} -86.0311 q^{19} -215.024 q^{20} +296.682 q^{22} -46.9229 q^{23} +116.517 q^{25} -248.567 q^{26} +169.768 q^{29} -141.842 q^{31} +139.909 q^{32} +319.935 q^{34} -411.923 q^{37} +402.015 q^{38} +423.819 q^{40} -49.0000 q^{41} -356.204 q^{43} -878.447 q^{44} +219.266 q^{46} -387.186 q^{47} -544.475 q^{50} +735.983 q^{52} +184.468 q^{53} +986.683 q^{55} -793.308 q^{58} -627.676 q^{59} +821.903 q^{61} +662.816 q^{62} -787.766 q^{64} -826.666 q^{65} -95.9610 q^{67} -947.298 q^{68} -733.093 q^{71} -750.676 q^{73} +1924.88 q^{74} -1190.33 q^{76} +23.5747 q^{79} -260.276 q^{80} +228.972 q^{82} -592.109 q^{83} +1064.02 q^{85} +1664.51 q^{86} +1731.45 q^{88} +864.292 q^{89} -649.227 q^{92} +1809.29 q^{94} +1337.00 q^{95} +614.268 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 44 q^{4}+O(q^{10}) $ 8 * q + 44 * q^4	$ 8 q + 44 q^{4} + 132 q^{10} + 336 q^{13} + 204 q^{16} - 288 q^{19} + 484 q^{22} + 152 q^{25} + 120 q^{31} + 1008 q^{34} + 592 q^{37} + 1620 q^{40} - 1872 q^{43} - 1644 q^{46} + 2400 q^{52} + 1344 q^{55} - 1200 q^{58} + 2400 q^{61} - 1388 q^{64} + 1824 q^{73} + 2844 q^{76} + 2368 q^{79} + 2436 q^{82} + 3512 q^{85} + 3780 q^{88} + 4368 q^{94} + 5712 q^{97}+O(q^{100}) $ 8 * q + 44 * q^4 + 132 * q^10 + 336 * q^13 + 204 * q^16 - 288 * q^19 + 484 * q^22 + 152 * q^25 + 120 * q^31 + 1008 * q^34 + 592 * q^37 + 1620 * q^40 - 1872 * q^43 - 1644 * q^46 + 2400 * q^52 + 1344 * q^55 - 1200 * q^58 + 2400 * q^61 - 1388 * q^64 + 1824 * q^73 + 2844 * q^76 + 2368 * q^79 + 2436 * q^82 + 3512 * q^85 + 3780 * q^88 + 4368 * q^94 + 5712 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−4.67291	−1.65212	−0.826061	−	0.563581i	$-0.809423\pi$
$2$	−4.67291	−1.65212	−0.826061	+	0.563581i	$0.809423\pi$
$3$	0	0
$3$	0	0
$4$	13.8361	1.72951
$5$	−15.5408	−1.39001	−0.695007	−	0.719003i	$-0.744599\pi$
$5$	−15.5408	−1.39001	−0.695007	+	0.719003i	$0.744599\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−27.2713	−1.20523
$9$	0	0
$10$	72.6208	2.29647
$11$	−63.4897	−1.74026	−0.870131	−	0.492821i	$-0.835966\pi$
$11$	−63.4897	−1.74026	−0.870131	+	0.492821i	$0.835966\pi$
$12$	0	0
$13$	53.1931	1.13486	0.567428	−	0.823423i	$-0.307939\pi$
$13$	53.1931	1.13486	0.567428	+	0.823423i	$0.307939\pi$
$14$	0	0
$15$	0	0
$16$	16.7479	0.261686
$17$	−68.4660	−0.976790	−0.488395	−	0.872623i	$-0.662418\pi$
$17$	−68.4660	−0.976790	−0.488395	+	0.872623i	$0.662418\pi$
$18$	0	0
$19$	−86.0311	−1.03878	−0.519392	−	0.854536i	$-0.673842\pi$
$19$	−86.0311	−1.03878	−0.519392	+	0.854536i	$0.673842\pi$
$20$	−215.024	−2.40404
$21$	0	0
$22$	296.682	2.87512
$23$	−46.9229	−0.425395	−0.212698	−	0.977118i	$-0.568225\pi$
$23$	−46.9229	−0.425395	−0.212698	+	0.977118i	$0.568225\pi$
$24$	0	0
$25$	116.517	0.932140
$26$	−248.567	−1.87492
$27$	0	0
$28$	0	0
$29$	169.768	1.08707	0.543535	−	0.839386i	$-0.317085\pi$
$29$	169.768	1.08707	0.543535	+	0.839386i	$0.317085\pi$
$30$	0	0
$31$	−141.842	−0.821794	−0.410897	−	0.911682i	$-0.634784\pi$
$31$	−141.842	−0.821794	−0.410897	+	0.911682i	$0.634784\pi$
$32$	139.909	0.772897
$33$	0	0
$34$	319.935	1.61378
$35$	0	0
$36$	0	0
$37$	−411.923	−1.83026	−0.915132	−	0.403155i	$-0.867914\pi$
$37$	−411.923	−1.83026	−0.915132	+	0.403155i	$0.867914\pi$
$38$	402.015	1.71620
$39$	0	0
$40$	423.819	1.67529
$41$	−49.0000	−0.186647	−0.0933234	−	0.995636i	$-0.529749\pi$
$41$	−49.0000	−0.186647	−0.0933234	+	0.995636i	$0.529749\pi$
$42$	0	0
$43$	−356.204	−1.26327	−0.631636	−	0.775265i	$-0.717616\pi$
$43$	−356.204	−1.26327	−0.631636	+	0.775265i	$0.717616\pi$
$44$	−878.447	−3.00979
$45$	0	0
$46$	219.266	0.702805
$47$	−387.186	−1.20164	−0.600819	−	0.799385i	$-0.705159\pi$
$47$	−387.186	−1.20164	−0.600819	+	0.799385i	$0.705159\pi$
$48$	0	0
$49$	0	0
$50$	−544.475	−1.54001
$51$	0	0
$52$	735.983	1.96274
$53$	184.468	0.478088	0.239044	−	0.971009i	$-0.423166\pi$
$53$	184.468	0.478088	0.239044	+	0.971009i	$0.423166\pi$
$54$	0	0
$55$	986.683	2.41899
$56$	0	0
$57$	0	0
$58$	−793.308	−1.79597
$59$	−627.676	−1.38503	−0.692513	−	0.721406i	$-0.743496\pi$
$59$	−627.676	−1.38503	−0.692513	+	0.721406i	$0.743496\pi$
$60$	0	0
$61$	821.903	1.72515	0.862573	−	0.505933i	$-0.168852\pi$
$61$	821.903	1.72515	0.862573	+	0.505933i	$0.168852\pi$
$62$	662.816	1.35770
$63$	0	0
$64$	−787.766	−1.53860
$65$	−826.666	−1.57747
$66$	0	0
$67$	−95.9610	−0.174978	−0.0874888	−	0.996166i	$-0.527884\pi$
$67$	−95.9610	−0.174978	−0.0874888	+	0.996166i	$0.527884\pi$
$68$	−947.298	−1.68936
$69$	0	0
$70$	0	0
$71$	−733.093	−1.22538	−0.612691	−	0.790322i	$-0.709913\pi$
$71$	−733.093	−1.22538	−0.612691	+	0.790322i	$0.709913\pi$
$72$	0	0
$73$	−750.676	−1.20356	−0.601781	−	0.798661i	$-0.705542\pi$
$73$	−750.676	−1.20356	−0.601781	+	0.798661i	$0.705542\pi$
$74$	1924.88	3.02382
$75$	0	0
$76$	−1190.33	−1.79658
$77$	0	0
$78$	0	0
$79$	23.5747	0.0335742	0.0167871	−	0.999859i	$-0.494656\pi$
$79$	23.5747	0.0335742	0.0167871	+	0.999859i	$0.494656\pi$
$80$	−260.276	−0.363747
$81$	0	0
$82$	228.972	0.308363
$83$	−592.109	−0.783042	−0.391521	−	0.920169i	$-0.628051\pi$
$83$	−592.109	−0.783042	−0.391521	+	0.920169i	$0.628051\pi$
$84$	0	0
$85$	1064.02	1.35775
$86$	1664.51	2.08708
$87$	0	0
$88$	1731.45	2.09742
$89$	864.292	1.02938	0.514690	−	0.857376i	$-0.327907\pi$
$89$	864.292	1.02938	0.514690	+	0.857376i	$0.327907\pi$
$90$	0	0
$91$	0	0
$92$	−649.227	−0.735724
$93$	0	0
$94$	1809.29	1.98525
$95$	1337.00	1.44392
$96$	0	0
$97$	614.268	0.642984	0.321492	−	0.946912i	$-0.395816\pi$
$97$	614.268	0.642984	0.321492	+	0.946912i	$0.395816\pi$
$98$	0	0
$99$	0	0
$100$	1612.14	1.61214
$101$	−541.906	−0.533878	−0.266939	−	0.963713i	$-0.586012\pi$
$101$	−541.906	−0.533878	−0.266939	+	0.963713i	$0.586012\pi$
$102$	0	0
$103$	−984.892	−0.942177	−0.471089	−	0.882086i	$-0.656139\pi$
$103$	−984.892	−0.942177	−0.471089	+	0.882086i	$0.656139\pi$
$104$	−1450.65	−1.36777
$105$	0	0
$106$	−862.003	−0.789860
$107$	−1531.02	−1.38326	−0.691631	−	0.722251i	$-0.743108\pi$
$107$	−1531.02	−1.38326	−0.691631	+	0.722251i	$0.743108\pi$
$108$	0	0
$109$	−1794.60	−1.57699	−0.788494	−	0.615042i	$-0.789139\pi$
$109$	−1794.60	−1.57699	−0.788494	+	0.615042i	$0.789139\pi$
$110$	−4610.68	−3.99646
$111$	0	0
$112$	0	0
$113$	−677.500	−0.564016	−0.282008	−	0.959412i	$-0.591000\pi$
$113$	−677.500	−0.564016	−0.282008	+	0.959412i	$0.591000\pi$
$114$	0	0
$115$	729.220	0.591306
$116$	2348.91	1.88010
$117$	0	0
$118$	2933.07	2.28823
$119$	0	0
$120$	0	0
$121$	2699.95	2.02851
$122$	−3840.68	−2.85015
$123$	0	0
$124$	−1962.54	−1.42130
$125$	131.825	0.0943266
$126$	0	0
$127$	−2287.68	−1.59842	−0.799208	−	0.601055i	$-0.794747\pi$
$127$	−2287.68	−1.59842	−0.799208	+	0.601055i	$0.794747\pi$
$128$	2561.88	1.76907
$129$	0	0
$130$	3862.93	2.60616
$131$	122.720	0.0818480	0.0409240	−	0.999162i	$-0.486970\pi$
$131$	122.720	0.0818480	0.0409240	+	0.999162i	$0.486970\pi$
$132$	0	0
$133$	0	0
$134$	448.417	0.289084
$135$	0	0
$136$	1867.16	1.17726
$137$	−780.935	−0.487005	−0.243503	−	0.969900i	$-0.578296\pi$
$137$	−780.935	−0.487005	−0.243503	+	0.969900i	$0.578296\pi$
$138$	0	0
$139$	−2052.89	−1.25269	−0.626345	−	0.779546i	$-0.715450\pi$
$139$	−2052.89	−1.25269	−0.626345	+	0.779546i	$0.715450\pi$
$140$	0	0
$141$	0	0
$142$	3425.68	2.02448
$143$	−3377.22	−1.97495
$144$	0	0
$145$	−2638.33	−1.51104
$146$	3507.84	1.98843
$147$	0	0
$148$	−5699.39	−3.16545
$149$	−2327.88	−1.27992	−0.639958	−	0.768410i	$-0.721048\pi$
$149$	−2327.88	−1.27992	−0.639958	+	0.768410i	$0.721048\pi$
$150$	0	0
$151$	−2547.61	−1.37299	−0.686494	−	0.727135i	$-0.740851\pi$
$151$	−2547.61	−1.37299	−0.686494	+	0.727135i	$0.740851\pi$
$152$	2346.18	1.25198
$153$	0	0
$154$	0	0
$155$	2204.35	1.14231
$156$	0	0
$157$	2882.74	1.46540	0.732699	−	0.680553i	$-0.238260\pi$
$157$	2882.74	1.46540	0.732699	+	0.680553i	$0.238260\pi$
$158$	−110.162	−0.0554686
$159$	0	0
$160$	−2174.31	−1.07434
$161$	0	0
$162$	0	0
$163$	−889.835	−0.427591	−0.213795	−	0.976878i	$-0.568583\pi$
$163$	−889.835	−0.427591	−0.213795	+	0.976878i	$0.568583\pi$
$164$	−677.967	−0.322807
$165$	0	0
$166$	2766.87	1.29368
$167$	−980.968	−0.454548	−0.227274	−	0.973831i	$-0.572981\pi$
$167$	−980.968	−0.454548	−0.227274	+	0.973831i	$0.572981\pi$
$168$	0	0
$169$	632.510	0.287897
$170$	−4972.06	−2.24317
$171$	0	0
$172$	−4928.46	−2.18484
$173$	1016.91	0.446903	0.223451	−	0.974715i	$-0.428268\pi$
$173$	1016.91	0.446903	0.223451	+	0.974715i	$0.428268\pi$
$174$	0	0
$175$	0	0
$176$	−1063.32	−0.455402
$177$	0	0
$178$	−4038.76	−1.70066
$179$	4232.30	1.76725	0.883623	−	0.468200i	$-0.155097\pi$
$179$	4232.30	1.76725	0.883623	+	0.468200i	$0.155097\pi$
$180$	0	0
$181$	−1002.46	−0.411670	−0.205835	−	0.978587i	$-0.565991\pi$
$181$	−1002.46	−0.411670	−0.205835	+	0.978587i	$0.565991\pi$
$182$	0	0
$183$	0	0
$184$	1279.65	0.512701
$185$	6401.63	2.54409
$186$	0	0
$187$	4346.89	1.69987
$188$	−5357.13	−2.07824
$189$	0	0
$190$	−6247.65	−2.38554
$191$	755.900	0.286361	0.143181	−	0.989697i	$-0.454267\pi$
$191$	755.900	0.286361	0.143181	+	0.989697i	$0.454267\pi$
$192$	0	0
$193$	4938.52	1.84188	0.920939	−	0.389707i	$-0.127424\pi$
$193$	4938.52	1.84188	0.920939	+	0.389707i	$0.127424\pi$
$194$	−2870.42	−1.06229
$195$	0	0
$196$	0	0
$197$	−2404.91	−0.869762	−0.434881	−	0.900488i	$-0.643210\pi$
$197$	−2404.91	−0.869762	−0.434881	+	0.900488i	$0.643210\pi$
$198$	0	0
$199$	−2727.55	−0.971613	−0.485807	−	0.874066i	$-0.661474\pi$
$199$	−2727.55	−0.971613	−0.485807	+	0.874066i	$0.661474\pi$
$200$	−3177.58	−1.12345
$201$	0	0
$202$	2532.28	0.882031
$203$	0	0
$204$	0	0
$205$	761.501	0.259442
$206$	4602.31	1.55659
$207$	0	0
$208$	890.873	0.296975
$209$	5462.09	1.80776
$210$	0	0
$211$	−1860.21	−0.606931	−0.303465	−	0.952842i	$-0.598144\pi$
$211$	−1860.21	−0.606931	−0.303465	+	0.952842i	$0.598144\pi$
$212$	2552.31	0.826856
$213$	0	0
$214$	7154.30	2.28532
$215$	5535.71	1.75597
$216$	0	0
$217$	0	0
$218$	8386.01	2.60538
$219$	0	0
$220$	13651.8	4.18366
$221$	−3641.92	−1.10852
$222$	0	0
$223$	−1199.45	−0.360185	−0.180093	−	0.983650i	$-0.557640\pi$
$223$	−1199.45	−0.360185	−0.180093	+	0.983650i	$0.557640\pi$
$224$	0	0
$225$	0	0
$226$	3165.89	0.931823
$227$	3209.78	0.938505	0.469252	−	0.883064i	$-0.344523\pi$
$227$	3209.78	0.938505	0.469252	+	0.883064i	$0.344523\pi$
$228$	0	0
$229$	878.498	0.253506	0.126753	−	0.991934i	$-0.459545\pi$
$229$	878.498	0.253506	0.126753	+	0.991934i	$0.459545\pi$
$230$	−3407.58	−0.976909
$231$	0	0
$232$	−4629.79	−1.31017
$233$	−399.636	−0.112365	−0.0561824	−	0.998421i	$-0.517893\pi$
$233$	−399.636	−0.112365	−0.0561824	+	0.998421i	$0.517893\pi$
$234$	0	0
$235$	6017.20	1.67029
$236$	−8684.56	−2.39541
$237$	0	0
$238$	0	0
$239$	−5624.75	−1.52232	−0.761161	−	0.648563i	$-0.775370\pi$
$239$	−5624.75	−1.52232	−0.761161	+	0.648563i	$0.775370\pi$
$240$	0	0
$241$	4800.67	1.28315	0.641573	−	0.767062i	$-0.278282\pi$
$241$	4800.67	1.28315	0.641573	+	0.767062i	$0.278282\pi$
$242$	−12616.6	−3.35134
$243$	0	0
$244$	11371.9	2.98365
$245$	0	0
$246$	0	0
$247$	−4576.27	−1.17887
$248$	3868.23	0.990454
$249$	0	0
$250$	−616.008	−0.155839
$251$	3247.06	0.816545	0.408273	−	0.912860i	$-0.366131\pi$
$251$	3247.06	0.816545	0.408273	+	0.912860i	$0.366131\pi$
$252$	0	0
$253$	2979.12	0.740299
$254$	10690.1	2.64078
$255$	0	0
$256$	−5669.31	−1.38411
$257$	6449.64	1.56544	0.782719	−	0.622376i	$-0.213832\pi$
$257$	6449.64	1.56544	0.782719	+	0.622376i	$0.213832\pi$
$258$	0	0
$259$	0	0
$260$	−11437.8	−2.72824
$261$	0	0
$262$	−573.459	−0.135223
$263$	5008.98	1.17440	0.587199	−	0.809443i	$-0.300231\pi$
$263$	5008.98	1.17440	0.587199	+	0.809443i	$0.300231\pi$
$264$	0	0
$265$	−2866.79	−0.664549
$266$	0	0
$267$	0	0
$268$	−1327.72	−0.302625
$269$	−3426.54	−0.776654	−0.388327	−	0.921522i	$-0.626947\pi$
$269$	−3426.54	−0.776654	−0.388327	+	0.921522i	$0.626947\pi$
$270$	0	0
$271$	933.312	0.209205	0.104603	−	0.994514i	$-0.466643\pi$
$271$	933.312	0.209205	0.104603	+	0.994514i	$0.466643\pi$
$272$	−1146.66	−0.255612
$273$	0	0
$274$	3649.23	0.804592
$275$	−7397.66	−1.62217
$276$	0	0
$277$	−1542.04	−0.334485	−0.167243	−	0.985916i	$-0.553486\pi$
$277$	−1542.04	−0.334485	−0.167243	+	0.985916i	$0.553486\pi$
$278$	9592.96	2.06960
$279$	0	0
$280$	0	0
$281$	5352.87	1.13639	0.568195	−	0.822894i	$-0.307642\pi$
$281$	5352.87	1.13639	0.568195	+	0.822894i	$0.307642\pi$
$282$	0	0
$283$	36.3436	0.00763394	0.00381697	−	0.999993i	$-0.498785\pi$
$283$	36.3436	0.00763394	0.00381697	+	0.999993i	$0.498785\pi$
$284$	−10143.1	−2.11931
$285$	0	0
$286$	15781.4	3.26285
$287$	0	0
$288$	0	0
$289$	−225.413	−0.0458809
$290$	12328.7	2.49643
$291$	0	0
$292$	−10386.4	−2.08157
$293$	5226.71	1.04214	0.521071	−	0.853513i	$-0.325532\pi$
$293$	5226.71	1.04214	0.521071	+	0.853513i	$0.325532\pi$
$294$	0	0
$295$	9754.62	1.92521
$296$	11233.7	2.20589
$297$	0	0
$298$	10878.0	2.11458
$299$	−2495.97	−0.482762
$300$	0	0
$301$	0	0
$302$	11904.7	2.26834
$303$	0	0
$304$	−1440.84	−0.271835
$305$	−12773.1	−2.39798
$306$	0	0
$307$	3761.08	0.699205	0.349603	−	0.936898i	$-0.386317\pi$
$307$	3761.08	0.699205	0.349603	+	0.936898i	$0.386317\pi$
$308$	0	0
$309$	0	0
$310$	−10300.7	−1.88723
$311$	3235.35	0.589903	0.294952	−	0.955512i	$-0.404696\pi$
$311$	3235.35	0.589903	0.294952	+	0.955512i	$0.404696\pi$
$312$	0	0
$313$	5297.18	0.956595	0.478298	−	0.878198i	$-0.341254\pi$
$313$	5297.18	0.956595	0.478298	+	0.878198i	$0.341254\pi$
$314$	−13470.8	−2.42102
$315$	0	0
$316$	326.181	0.0580668
$317$	−1524.55	−0.270117	−0.135058	−	0.990838i	$-0.543122\pi$
$317$	−1524.55	−0.270117	−0.135058	+	0.990838i	$0.543122\pi$
$318$	0	0
$319$	−10778.5	−1.89179
$320$	12242.5	2.13868
$321$	0	0
$322$	0	0
$323$	5890.20	1.01467
$324$	0	0
$325$	6197.93	1.05784
$326$	4158.12	0.706432
$327$	0	0
$328$	1336.30	0.224953
$329$	0	0
$330$	0	0
$331$	7950.08	1.32017	0.660084	−	0.751191i	$-0.270520\pi$
$331$	7950.08	1.32017	0.660084	+	0.751191i	$0.270520\pi$
$332$	−8192.46	−1.35428
$333$	0	0
$334$	4583.97	0.750969
$335$	1491.31	0.243221
$336$	0	0
$337$	55.9475	0.00904348	0.00452174	−	0.999990i	$-0.498561\pi$
$337$	55.9475	0.00904348	0.00452174	+	0.999990i	$0.498561\pi$
$338$	−2955.66	−0.475641
$339$	0	0
$340$	14721.8	2.34824
$341$	9005.53	1.43014
$342$	0	0
$343$	0	0
$344$	9714.16	1.52254
$345$	0	0
$346$	−4751.92	−0.738338
$347$	8902.88	1.37733	0.688663	−	0.725082i	$-0.258198\pi$
$347$	8902.88	1.37733	0.688663	+	0.725082i	$0.258198\pi$
$348$	0	0
$349$	10134.5	1.55441	0.777203	−	0.629250i	$-0.216638\pi$
$349$	10134.5	1.55441	0.777203	+	0.629250i	$0.216638\pi$
$350$	0	0
$351$	0	0
$352$	−8882.80	−1.34504
$353$	−13.7977	−0.00208039	−0.00104020	−	0.999999i	$-0.500331\pi$
$353$	−13.7977	−0.00208039	−0.00104020	+	0.999999i	$0.500331\pi$
$354$	0	0
$355$	11392.9	1.70330
$356$	11958.4	1.78032
$357$	0	0
$358$	−19777.1	−2.91970
$359$	−6930.38	−1.01886	−0.509431	−	0.860512i	$-0.670144\pi$
$359$	−6930.38	−1.01886	−0.509431	+	0.860512i	$0.670144\pi$
$360$	0	0
$361$	542.357	0.0790723
$362$	4684.40	0.680129
$363$	0	0
$364$	0	0
$365$	11666.1	1.67297
$366$	0	0
$367$	1430.43	0.203455	0.101727	−	0.994812i	$-0.467563\pi$
$367$	1430.43	0.203455	0.101727	+	0.994812i	$0.467563\pi$
$368$	−785.859	−0.111320
$369$	0	0
$370$	−29914.2	−4.20315
$371$	0	0
$372$	0	0
$373$	−1524.04	−0.211560	−0.105780	−	0.994390i	$-0.533734\pi$
$373$	−1524.04	−0.211560	−0.105780	+	0.994390i	$0.533734\pi$
$374$	−20312.6	−2.80839
$375$	0	0
$376$	10559.1	1.44825
$377$	9030.47	1.23367
$378$	0	0
$379$	−1058.04	−0.143398	−0.0716988	−	0.997426i	$-0.522842\pi$
$379$	−1058.04	−0.143398	−0.0716988	+	0.997426i	$0.522842\pi$
$380$	18498.7	2.49728
$381$	0	0
$382$	−3532.25	−0.473104
$383$	3781.23	0.504470	0.252235	−	0.967666i	$-0.418834\pi$
$383$	3781.23	0.504470	0.252235	+	0.967666i	$0.418834\pi$
$384$	0	0
$385$	0	0
$386$	−23077.2	−3.04301
$387$	0	0
$388$	8499.04	1.11204
$389$	−8396.19	−1.09435	−0.547177	−	0.837017i	$-0.684298\pi$
$389$	−8396.19	−1.09435	−0.547177	+	0.837017i	$0.684298\pi$
$390$	0	0
$391$	3212.62	0.415522
$392$	0	0
$393$	0	0
$394$	11237.9	1.43695
$395$	−366.370	−0.0466686
$396$	0	0
$397$	3124.08	0.394945	0.197472	−	0.980308i	$-0.436727\pi$
$397$	3124.08	0.394945	0.197472	+	0.980308i	$0.436727\pi$
$398$	12745.6	1.60522
$399$	0	0
$400$	1951.42	0.243928
$401$	6482.85	0.807327	0.403664	−	0.914908i	$-0.367737\pi$
$401$	6482.85	0.807327	0.403664	+	0.914908i	$0.367737\pi$
$402$	0	0
$403$	−7545.04	−0.932618
$404$	−7497.84	−0.923345
$405$	0	0
$406$	0	0
$407$	26152.9	3.18514
$408$	0	0
$409$	−13147.6	−1.58950	−0.794749	−	0.606938i	$-0.792398\pi$
$409$	−13147.6	−1.58950	−0.794749	+	0.606938i	$0.792398\pi$
$410$	−3558.42	−0.428629
$411$	0	0
$412$	−13627.0	−1.62950
$413$	0	0
$414$	0	0
$415$	9201.87	1.08844
$416$	7442.21	0.877126
$417$	0	0
$418$	−25523.9	−2.98663
$419$	1727.45	0.201411	0.100706	−	0.994916i	$-0.467890\pi$
$419$	1727.45	0.201411	0.100706	+	0.994916i	$0.467890\pi$
$420$	0	0
$421$	−4600.25	−0.532548	−0.266274	−	0.963897i	$-0.585793\pi$
$421$	−4600.25	−0.532548	−0.266274	+	0.963897i	$0.585793\pi$
$422$	8692.60	1.00272
$423$	0	0
$424$	−5030.69	−0.576208
$425$	−7977.48	−0.910505
$426$	0	0
$427$	0	0
$428$	−21183.2	−2.39236
$429$	0	0
$430$	−25867.9	−2.90107
$431$	5707.15	0.637828	0.318914	−	0.947784i	$-0.396682\pi$
$431$	5707.15	0.637828	0.318914	+	0.947784i	$0.396682\pi$
$432$	0	0
$433$	−3097.55	−0.343785	−0.171892	−	0.985116i	$-0.554988\pi$
$433$	−3097.55	−0.343785	−0.171892	+	0.985116i	$0.554988\pi$
$434$	0	0
$435$	0	0
$436$	−24830.2	−2.72741
$437$	4036.83	0.441894
$438$	0	0
$439$	−8161.15	−0.887267	−0.443634	−	0.896208i	$-0.646311\pi$
$439$	−8161.15	−0.887267	−0.443634	+	0.896208i	$0.646311\pi$
$440$	−26908.1	−2.91544
$441$	0	0
$442$	17018.3	1.83140
$443$	−10224.6	−1.09658	−0.548292	−	0.836287i	$-0.684722\pi$
$443$	−10224.6	−1.09658	−0.548292	+	0.836287i	$0.684722\pi$
$444$	0	0
$445$	−13431.8	−1.43085
$446$	5604.93	0.595070
$447$	0	0
$448$	0	0
$449$	−2365.73	−0.248654	−0.124327	−	0.992241i	$-0.539677\pi$
$449$	−2365.73	−0.248654	−0.124327	+	0.992241i	$0.539677\pi$
$450$	0	0
$451$	3111.00	0.324814
$452$	−9373.92	−0.975469
$453$	0	0
$454$	−14999.0	−1.55052
$455$	0	0
$456$	0	0
$457$	1172.54	0.120020	0.0600100	−	0.998198i	$-0.480887\pi$
$457$	1172.54	0.120020	0.0600100	+	0.998198i	$0.480887\pi$
$458$	−4105.14	−0.418822
$459$	0	0
$460$	10089.5	1.02267
$461$	2447.81	0.247302	0.123651	−	0.992326i	$-0.460540\pi$
$461$	2447.81	0.247302	0.123651	+	0.992326i	$0.460540\pi$
$462$	0	0
$463$	−9459.57	−0.949510	−0.474755	−	0.880118i	$-0.657463\pi$
$463$	−9459.57	−0.949510	−0.474755	+	0.880118i	$0.657463\pi$
$464$	2843.25	0.284471
$465$	0	0
$466$	1867.46	0.185640
$467$	−3585.45	−0.355278	−0.177639	−	0.984096i	$-0.556846\pi$
$467$	−3585.45	−0.355278	−0.177639	+	0.984096i	$0.556846\pi$
$468$	0	0
$469$	0	0
$470$	−28117.8	−2.75953
$471$	0	0
$472$	17117.6	1.66928
$473$	22615.3	2.19842
$474$	0	0
$475$	−10024.1	−0.968292
$476$	0	0
$477$	0	0
$478$	26283.9	2.51506
$479$	4209.89	0.401575	0.200788	−	0.979635i	$-0.435650\pi$
$479$	4209.89	0.401575	0.200788	+	0.979635i	$0.435650\pi$
$480$	0	0
$481$	−21911.5	−2.07708
$482$	−22433.1	−2.11991
$483$	0	0
$484$	37356.6	3.50832
$485$	−9546.23	−0.893757
$486$	0	0
$487$	1245.26	0.115868	0.0579342	−	0.998320i	$-0.481549\pi$
$487$	1245.26	0.115868	0.0579342	+	0.998320i	$0.481549\pi$
$488$	−22414.4	−2.07920
$489$	0	0
$490$	0	0
$491$	−4102.67	−0.377089	−0.188545	−	0.982065i	$-0.560377\pi$
$491$	−4102.67	−0.377089	−0.188545	+	0.982065i	$0.560377\pi$
$492$	0	0
$493$	−11623.3	−1.06184
$494$	21384.5	1.94764
$495$	0	0
$496$	−2375.56	−0.215052
$497$	0	0
$498$	0	0
$499$	12712.3	1.14044	0.570221	−	0.821491i	$-0.306858\pi$
$499$	12712.3	1.14044	0.570221	+	0.821491i	$0.306858\pi$
$500$	1823.94	0.163139
$501$	0	0
$502$	−15173.2	−1.34903
$503$	−1928.88	−0.170983	−0.0854915	−	0.996339i	$-0.527246\pi$
$503$	−1928.88	−0.170983	−0.0854915	+	0.996339i	$0.527246\pi$
$504$	0	0
$505$	8421.67	0.742098
$506$	−13921.1	−1.22306
$507$	0	0
$508$	−31652.4	−2.76447
$509$	−6517.96	−0.567591	−0.283795	−	0.958885i	$-0.591594\pi$
$509$	−6517.96	−0.567591	−0.283795	+	0.958885i	$0.591594\pi$
$510$	0	0
$511$	0	0
$512$	5997.08	0.517648
$513$	0	0
$514$	−30138.6	−2.58629
$515$	15306.0	1.30964
$516$	0	0
$517$	24582.4	2.09116
$518$	0	0
$519$	0	0
$520$	22544.3	1.90121
$521$	−2677.83	−0.225178	−0.112589	−	0.993642i	$-0.535914\pi$
$521$	−2677.83	−0.225178	−0.112589	+	0.993642i	$0.535914\pi$
$522$	0	0
$523$	−643.647	−0.0538140	−0.0269070	−	0.999638i	$-0.508566\pi$
$523$	−643.647	−0.0538140	−0.0269070	+	0.999638i	$0.508566\pi$
$524$	1697.96	0.141557
$525$	0	0
$526$	−23406.5	−1.94025
$527$	9711.37	0.802721
$528$	0	0
$529$	−9965.25	−0.819039
$530$	13396.2	1.09792
$531$	0	0
$532$	0	0
$533$	−2606.46	−0.211817
$534$	0	0
$535$	23793.3	1.92275
$536$	2616.98	0.210889
$537$	0	0
$538$	16011.9	1.28313
$539$	0	0
$540$	0	0
$541$	3974.55	0.315858	0.157929	−	0.987450i	$-0.449518\pi$
$541$	3974.55	0.315858	0.157929	+	0.987450i	$0.449518\pi$
$542$	−4361.28	−0.345633
$543$	0	0
$544$	−9579.02	−0.754958
$545$	27889.6	2.19204
$546$	0	0
$547$	−13685.7	−1.06976	−0.534879	−	0.844929i	$-0.679643\pi$
$547$	−13685.7	−1.06976	−0.534879	+	0.844929i	$0.679643\pi$
$548$	−10805.1	−0.842279
$549$	0	0
$550$	34568.6	2.68002
$551$	−14605.3	−1.12923
$552$	0	0
$553$	0	0
$554$	7205.83	0.552611
$555$	0	0
$556$	−28403.9	−2.16653
$557$	−14712.7	−1.11921	−0.559603	−	0.828761i	$-0.689046\pi$
$557$	−14712.7	−1.11921	−0.559603	+	0.828761i	$0.689046\pi$
$558$	0	0
$559$	−18947.6	−1.43363
$560$	0	0
$561$	0	0
$562$	−25013.5	−1.87745
$563$	10464.2	0.783325	0.391663	−	0.920109i	$-0.371900\pi$
$563$	10464.2	0.783325	0.391663	+	0.920109i	$0.371900\pi$
$564$	0	0
$565$	10528.9	0.783990
$566$	−169.830	−0.0126122
$567$	0	0
$568$	19992.4	1.47687
$569$	−23684.3	−1.74499	−0.872494	−	0.488625i	$-0.837499\pi$
$569$	−23684.3	−1.74499	−0.872494	+	0.488625i	$0.837499\pi$
$570$	0	0
$571$	−3879.52	−0.284331	−0.142165	−	0.989843i	$-0.545407\pi$
$571$	−3879.52	−0.284331	−0.142165	+	0.989843i	$0.545407\pi$
$572$	−46727.4	−3.41568
$573$	0	0
$574$	0	0
$575$	−5467.33	−0.396528
$576$	0	0
$577$	5894.79	0.425309	0.212654	−	0.977127i	$-0.431789\pi$
$577$	5894.79	0.425309	0.212654	+	0.977127i	$0.431789\pi$
$578$	1053.33	0.0758008
$579$	0	0
$580$	−36504.1	−2.61336
$581$	0	0
$582$	0	0
$583$	−11711.8	−0.831998
$584$	20471.9	1.45057
$585$	0	0
$586$	−24423.9	−1.72175
$587$	−6780.32	−0.476752	−0.238376	−	0.971173i	$-0.576615\pi$
$587$	−6780.32	−0.476752	−0.238376	+	0.971173i	$0.576615\pi$
$588$	0	0
$589$	12202.9	0.853667
$590$	−45582.4	−3.18067
$591$	0	0
$592$	−6898.84	−0.478954
$593$	−22883.7	−1.58469	−0.792344	−	0.610075i	$-0.791139\pi$
$593$	−22883.7	−1.58469	−0.792344	+	0.610075i	$0.791139\pi$
$594$	0	0
$595$	0	0
$596$	−32208.7	−2.21362
$597$	0	0
$598$	11663.5	0.797582
$599$	9410.57	0.641912	0.320956	−	0.947094i	$-0.395996\pi$
$599$	9410.57	0.641912	0.320956	+	0.947094i	$0.395996\pi$
$600$	0	0
$601$	−17409.2	−1.18159	−0.590796	−	0.806821i	$-0.701186\pi$
$601$	−17409.2	−1.18159	−0.590796	+	0.806821i	$0.701186\pi$
$602$	0	0
$603$	0	0
$604$	−35248.8	−2.37459
$605$	−41959.4	−2.81966
$606$	0	0
$607$	14517.2	0.970731	0.485366	−	0.874311i	$-0.338687\pi$
$607$	14517.2	0.970731	0.485366	+	0.874311i	$0.338687\pi$
$608$	−12036.5	−0.802873
$609$	0	0
$610$	59687.3	3.96175
$611$	−20595.7	−1.36368
$612$	0	0
$613$	−28807.4	−1.89807	−0.949037	−	0.315165i	$-0.897940\pi$
$613$	−28807.4	−1.89807	−0.949037	+	0.315165i	$0.897940\pi$
$614$	−17575.2	−1.15517
$615$	0	0
$616$	0	0
$617$	3925.52	0.256135	0.128068	−	0.991765i	$-0.459123\pi$
$617$	3925.52	0.256135	0.128068	+	0.991765i	$0.459123\pi$
$618$	0	0
$619$	−22349.6	−1.45122	−0.725612	−	0.688104i	$-0.758443\pi$
$619$	−22349.6	−1.45122	−0.725612	+	0.688104i	$0.758443\pi$
$620$	30499.5	1.97563
$621$	0	0
$622$	−15118.5	−0.974592
$623$	0	0
$624$	0	0
$625$	−16613.4	−1.06326
$626$	−24753.2	−1.58041
$627$	0	0
$628$	39885.7	2.53442
$629$	28202.7	1.78778
$630$	0	0
$631$	18686.1	1.17889	0.589447	−	0.807807i	$-0.299346\pi$
$631$	18686.1	1.17889	0.589447	+	0.807807i	$0.299346\pi$
$632$	−642.913	−0.0404647
$633$	0	0
$634$	7124.06	0.446266
$635$	35552.4	2.22182
$636$	0	0
$637$	0	0
$638$	50366.9	3.12546
$639$	0	0
$640$	−39813.8	−2.45903
$641$	6863.23	0.422903	0.211452	−	0.977388i	$-0.432181\pi$
$641$	6863.23	0.422903	0.211452	+	0.977388i	$0.432181\pi$
$642$	0	0
$643$	13442.9	0.824472	0.412236	−	0.911077i	$-0.364748\pi$
$643$	13442.9	0.824472	0.412236	+	0.911077i	$0.364748\pi$
$644$	0	0
$645$	0	0
$646$	−27524.4	−1.67637
$647$	16015.4	0.973156	0.486578	−	0.873637i	$-0.338245\pi$
$647$	16015.4	0.973156	0.486578	+	0.873637i	$0.338245\pi$
$648$	0	0
$649$	39851.0	2.41031
$650$	−28962.3	−1.74769
$651$	0	0
$652$	−12311.8	−0.739521
$653$	−4455.45	−0.267007	−0.133503	−	0.991048i	$-0.542623\pi$
$653$	−4455.45	−0.267007	−0.133503	+	0.991048i	$0.542623\pi$
$654$	0	0
$655$	−1907.17	−0.113770
$656$	−820.647	−0.0488428
$657$	0	0
$658$	0	0
$659$	13973.4	0.825990	0.412995	−	0.910733i	$-0.364483\pi$
$659$	13973.4	0.825990	0.412995	+	0.910733i	$0.364483\pi$
$660$	0	0
$661$	14950.0	0.879710	0.439855	−	0.898069i	$-0.355030\pi$
$661$	14950.0	0.879710	0.439855	+	0.898069i	$0.355030\pi$
$662$	−37150.0	−2.18108
$663$	0	0
$664$	16147.6	0.943748
$665$	0	0
$666$	0	0
$667$	−7965.98	−0.462435
$668$	−13572.7	−0.786144
$669$	0	0
$670$	−6968.77	−0.401831
$671$	−52182.4	−3.00220
$672$	0	0
$673$	−15041.5	−0.861527	−0.430764	−	0.902465i	$-0.641756\pi$
$673$	−15041.5	−0.861527	−0.430764	+	0.902465i	$0.641756\pi$
$674$	−261.437	−0.0149409
$675$	0	0
$676$	8751.43	0.497920
$677$	28889.4	1.64005	0.820023	−	0.572330i	$-0.193960\pi$
$677$	28889.4	1.64005	0.820023	+	0.572330i	$0.193960\pi$
$678$	0	0
$679$	0	0
$680$	−29017.2	−1.63641
$681$	0	0
$682$	−42082.0	−2.36276
$683$	12029.7	0.673941	0.336971	−	0.941515i	$-0.390598\pi$
$683$	12029.7	0.673941	0.336971	+	0.941515i	$0.390598\pi$
$684$	0	0
$685$	12136.4	0.676945
$686$	0	0
$687$	0	0
$688$	−5965.67	−0.330580
$689$	9812.45	0.542561
$690$	0	0
$691$	10239.0	0.563689	0.281845	−	0.959460i	$-0.409054\pi$
$691$	10239.0	0.563689	0.281845	+	0.959460i	$0.409054\pi$
$692$	14070.0	0.772921
$693$	0	0
$694$	−41602.3	−2.27551
$695$	31903.6	1.74126
$696$	0	0
$697$	3354.83	0.182315
$698$	−47357.6	−2.56807
$699$	0	0
$700$	0	0
$701$	−21368.4	−1.15132	−0.575659	−	0.817690i	$-0.695254\pi$
$701$	−21368.4	−1.15132	−0.575659	+	0.817690i	$0.695254\pi$
$702$	0	0
$703$	35438.2	1.90125
$704$	50015.0	2.67757
$705$	0	0
$706$	64.4755	0.00343706
$707$	0	0
$708$	0	0
$709$	−1402.56	−0.0742935	−0.0371467	−	0.999310i	$-0.511827\pi$
$709$	−1402.56	−0.0742935	−0.0371467	+	0.999310i	$0.511827\pi$
$710$	−53237.9	−2.81406
$711$	0	0
$712$	−23570.4	−1.24064
$713$	6655.65	0.349588
$714$	0	0
$715$	52484.8	2.74520
$716$	58558.3	3.05646
$717$	0	0
$718$	32385.0	1.68328
$719$	−29924.0	−1.55212	−0.776061	−	0.630658i	$-0.782785\pi$
$719$	−29924.0	−1.55212	−0.776061	+	0.630658i	$0.782785\pi$
$720$	0	0
$721$	0	0
$722$	−2534.38	−0.130637
$723$	0	0
$724$	−13870.1	−0.711985
$725$	19780.9	1.01330
$726$	0	0
$727$	18010.8	0.918823	0.459411	−	0.888224i	$-0.348060\pi$
$727$	18010.8	0.918823	0.459411	+	0.888224i	$0.348060\pi$
$728$	0	0
$729$	0	0
$730$	−54514.7	−2.76395
$731$	24387.9	1.23395
$732$	0	0
$733$	37857.6	1.90764	0.953820	−	0.300378i	$-0.0971127\pi$
$733$	37857.6	1.90764	0.953820	+	0.300378i	$0.0971127\pi$
$734$	−6684.27	−0.336132
$735$	0	0
$736$	−6564.94	−0.328787
$737$	6092.54	0.304507
$738$	0	0
$739$	−23780.5	−1.18374	−0.591868	−	0.806035i	$-0.701609\pi$
$739$	−23780.5	−1.18374	−0.591868	+	0.806035i	$0.701609\pi$
$740$	88573.3	4.40002
$741$	0	0
$742$	0	0
$743$	−10018.2	−0.494659	−0.247330	−	0.968931i	$-0.579553\pi$
$743$	−10018.2	−0.494659	−0.247330	+	0.968931i	$0.579553\pi$
$744$	0	0
$745$	36177.2	1.77910
$746$	7121.71	0.349523
$747$	0	0
$748$	60143.7	2.93994
$749$	0	0
$750$	0	0
$751$	18282.2	0.888319	0.444159	−	0.895948i	$-0.353502\pi$
$751$	18282.2	0.888319	0.444159	+	0.895948i	$0.353502\pi$
$752$	−6484.55	−0.314451
$753$	0	0
$754$	−42198.5	−2.03817
$755$	39591.9	1.90847
$756$	0	0
$757$	19815.3	0.951388	0.475694	−	0.879611i	$-0.342197\pi$
$757$	19815.3	0.951388	0.475694	+	0.879611i	$0.342197\pi$
$758$	4944.10	0.236910
$759$	0	0
$760$	−36461.6	−1.74027
$761$	−37697.8	−1.79572	−0.897860	−	0.440282i	$-0.854879\pi$
$761$	−37697.8	−1.79572	−0.897860	+	0.440282i	$0.854879\pi$
$762$	0	0
$763$	0	0
$764$	10458.7	0.495264
$765$	0	0
$766$	−17669.3	−0.833446
$767$	−33388.1	−1.57180
$768$	0	0
$769$	−34511.3	−1.61835	−0.809174	−	0.587569i	$-0.800085\pi$
$769$	−34511.3	−1.61835	−0.809174	+	0.587569i	$0.800085\pi$
$770$	0	0
$771$	0	0
$772$	68329.6	3.18554
$773$	−12845.1	−0.597681	−0.298841	−	0.954303i	$-0.596600\pi$
$773$	−12845.1	−0.597681	−0.298841	+	0.954303i	$0.596600\pi$
$774$	0	0
$775$	−16527.1	−0.766027
$776$	−16751.9	−0.774946
$777$	0	0
$778$	39234.6	1.80801
$779$	4215.53	0.193886
$780$	0	0
$781$	46543.9	2.13249
$782$	−15012.3	−0.686493
$783$	0	0
$784$	0	0
$785$	−44800.1	−2.03692
$786$	0	0
$787$	−5906.18	−0.267513	−0.133756	−	0.991014i	$-0.542704\pi$
$787$	−5906.18	−0.267513	−0.133756	+	0.991014i	$0.542704\pi$
$788$	−33274.5	−1.50426
$789$	0	0
$790$	1712.01	0.0771022
$791$	0	0
$792$	0	0
$793$	43719.6	1.95779
$794$	−14598.5	−0.652497
$795$	0	0
$796$	−37738.5	−1.68041
$797$	11027.5	0.490106	0.245053	−	0.969510i	$-0.421195\pi$
$797$	11027.5	0.490106	0.245053	+	0.969510i	$0.421195\pi$
$798$	0	0
$799$	26509.1	1.17375
$800$	16301.9	0.720448
$801$	0	0
$802$	−30293.8	−1.33380
$803$	47660.2	2.09451
$804$	0	0
$805$	0	0
$806$	35257.2	1.54080
$807$	0	0
$808$	14778.5	0.643447
$809$	−33735.3	−1.46609	−0.733047	−	0.680178i	$-0.761902\pi$
$809$	−33735.3	−1.46609	−0.733047	+	0.680178i	$0.761902\pi$
$810$	0	0
$811$	38016.9	1.64606	0.823029	−	0.567999i	$-0.192282\pi$
$811$	38016.9	1.64606	0.823029	+	0.567999i	$0.192282\pi$
$812$	0	0
$813$	0	0
$814$	−122210.	−5.26223
$815$	13828.8	0.594357
$816$	0	0
$817$	30644.7	1.31227
$818$	61437.3	2.62605
$819$	0	0
$820$	10536.2	0.448706
$821$	485.730	0.0206481	0.0103241	−	0.999947i	$-0.496714\pi$
$821$	485.730	0.0206481	0.0103241	+	0.999947i	$0.496714\pi$
$822$	0	0
$823$	19018.1	0.805505	0.402753	−	0.915309i	$-0.368054\pi$
$823$	19018.1	0.805505	0.402753	+	0.915309i	$0.368054\pi$
$824$	26859.3	1.13554
$825$	0	0
$826$	0	0
$827$	24657.9	1.03681	0.518404	−	0.855136i	$-0.326526\pi$
$827$	24657.9	1.03681	0.518404	+	0.855136i	$0.326526\pi$
$828$	0	0
$829$	−33962.3	−1.42287	−0.711435	−	0.702752i	$-0.751955\pi$
$829$	−33962.3	−1.42287	−0.711435	+	0.702752i	$0.751955\pi$
$830$	−42999.5	−1.79823
$831$	0	0
$832$	−41903.7	−1.74609
$833$	0	0
$834$	0	0
$835$	15245.1	0.631828
$836$	75573.8	3.12652
$837$	0	0
$838$	−8072.20	−0.332756
$839$	−20209.3	−0.831586	−0.415793	−	0.909459i	$-0.636496\pi$
$839$	−20209.3	−0.831586	−0.415793	+	0.909459i	$0.636496\pi$
$840$	0	0
$841$	4432.03	0.181723
$842$	21496.5	0.879834
$843$	0	0
$844$	−25738.0	−1.04969
$845$	−9829.73	−0.400181
$846$	0	0
$847$	0	0
$848$	3089.45	0.125109
$849$	0	0
$850$	37278.0	1.50427
$851$	19328.6	0.778585
$852$	0	0
$853$	−2271.91	−0.0911944	−0.0455972	−	0.998960i	$-0.514519\pi$
$853$	−2271.91	−0.0911944	−0.0455972	+	0.998960i	$0.514519\pi$
$854$	0	0
$855$	0	0
$856$	41752.9	1.66715
$857$	804.766	0.0320774	0.0160387	−	0.999871i	$-0.494895\pi$
$857$	804.766	0.0320774	0.0160387	+	0.999871i	$0.494895\pi$
$858$	0	0
$859$	−36023.4	−1.43085	−0.715426	−	0.698689i	$-0.753767\pi$
$859$	−36023.4	−1.43085	−0.715426	+	0.698689i	$0.753767\pi$
$860$	76592.4	3.03695
$861$	0	0
$862$	−26669.0	−1.05377
$863$	13880.5	0.547504	0.273752	−	0.961800i	$-0.411735\pi$
$863$	13880.5	0.547504	0.273752	+	0.961800i	$0.411735\pi$
$864$	0	0
$865$	−15803.6	−0.621201
$866$	14474.6	0.567974
$867$	0	0
$868$	0	0
$869$	−1496.75	−0.0584279
$870$	0	0
$871$	−5104.47	−0.198574
$872$	48941.2	1.90064
$873$	0	0
$874$	−18863.7	−0.730062
$875$	0	0
$876$	0	0
$877$	−4779.13	−0.184013	−0.0920067	−	0.995758i	$-0.529328\pi$
$877$	−4779.13	−0.184013	−0.0920067	+	0.995758i	$0.529328\pi$
$878$	38136.3	1.46587
$879$	0	0
$880$	16524.9	0.633015
$881$	−19785.5	−0.756628	−0.378314	−	0.925677i	$-0.623496\pi$
$881$	−19785.5	−0.756628	−0.378314	+	0.925677i	$0.623496\pi$
$882$	0	0
$883$	−33555.9	−1.27887	−0.639437	−	0.768844i	$-0.720832\pi$
$883$	−33555.9	−1.27887	−0.639437	+	0.768844i	$0.720832\pi$
$884$	−50389.8	−1.91718
$885$	0	0
$886$	47778.7	1.81169
$887$	47403.0	1.79440	0.897202	−	0.441620i	$-0.145596\pi$
$887$	47403.0	1.79440	0.897202	+	0.441620i	$0.145596\pi$
$888$	0	0
$889$	0	0
$890$	62765.6	2.36394
$891$	0	0
$892$	−16595.7	−0.622943
$893$	33310.1	1.24824
$894$	0	0
$895$	−65773.5	−2.45650
$896$	0	0
$897$	0	0
$898$	11054.8	0.410807
$899$	−24080.2	−0.893349
$900$	0	0
$901$	−12629.8	−0.466992
$902$	−14537.4	−0.536633
$903$	0	0
$904$	18476.3	0.679771
$905$	15579.1	0.572227
$906$	0	0
$907$	−9765.54	−0.357508	−0.178754	−	0.983894i	$-0.557207\pi$
$907$	−9765.54	−0.357508	−0.178754	+	0.983894i	$0.557207\pi$
$908$	44410.7	1.62315
$909$	0	0
$910$	0	0
$911$	−9244.54	−0.336208	−0.168104	−	0.985769i	$-0.553764\pi$
$911$	−9244.54	−0.336208	−0.168104	+	0.985769i	$0.553764\pi$
$912$	0	0
$913$	37592.9	1.36270
$914$	−5479.17	−0.198288
$915$	0	0
$916$	12154.9	0.438439
$917$	0	0
$918$	0	0
$919$	40397.9	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$919$	40397.9	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$920$	−19886.8	−0.712661
$921$	0	0
$922$	−11438.4	−0.408572
$923$	−38995.5	−1.39063
$924$	0	0
$925$	−47996.2	−1.70606
$926$	44203.7	1.56871
$927$	0	0
$928$	23752.1	0.840193
$929$	10318.3	0.364405	0.182202	−	0.983261i	$-0.441677\pi$
$929$	10318.3	0.364405	0.182202	+	0.983261i	$0.441677\pi$
$930$	0	0
$931$	0	0
$932$	−5529.38	−0.194336
$933$	0	0
$934$	16754.5	0.586963
$935$	−67554.2	−2.36284
$936$	0	0
$937$	21399.9	0.746109	0.373055	−	0.927809i	$-0.378310\pi$
$937$	21399.9	0.746109	0.373055	+	0.927809i	$0.378310\pi$
$938$	0	0
$939$	0	0
$940$	83254.3	2.88878
$941$	−11405.5	−0.395121	−0.197561	−	0.980291i	$-0.563302\pi$
$941$	−11405.5	−0.395121	−0.197561	+	0.980291i	$0.563302\pi$
$942$	0	0
$943$	2299.22	0.0793987
$944$	−10512.3	−0.362441
$945$	0	0
$946$	−105679.	−3.63206
$947$	−830.840	−0.0285097	−0.0142548	−	0.999898i	$-0.504538\pi$
$947$	−830.840	−0.0285097	−0.0142548	+	0.999898i	$0.504538\pi$
$948$	0	0
$949$	−39930.8	−1.36587
$950$	46841.8	1.59974
$951$	0	0
$952$	0	0
$953$	4680.16	0.159082	0.0795410	−	0.996832i	$-0.474655\pi$
$953$	4680.16	0.159082	0.0795410	+	0.996832i	$0.474655\pi$
$954$	0	0
$955$	−11747.3	−0.398046
$956$	−77824.4	−2.63287
$957$	0	0
$958$	−19672.4	−0.663451
$959$	0	0
$960$	0	0
$961$	−9671.76	−0.324654
$962$	102390.	3.43160
$963$	0	0
$964$	66422.3	2.21921
$965$	−76748.7	−2.56024
$966$	0	0
$967$	−43642.9	−1.45136	−0.725679	−	0.688034i	$-0.758474\pi$
$967$	−43642.9	−1.45136	−0.725679	+	0.688034i	$0.758474\pi$
$968$	−73631.1	−2.44483
$969$	0	0
$970$	44608.7	1.47660
$971$	7552.17	0.249599	0.124800	−	0.992182i	$-0.460171\pi$
$971$	7552.17	0.249599	0.124800	+	0.992182i	$0.460171\pi$
$972$	0	0
$973$	0	0
$974$	−5818.96	−0.191429
$975$	0	0
$976$	13765.1	0.451446
$977$	−42782.8	−1.40097	−0.700483	−	0.713669i	$-0.747032\pi$
$977$	−42782.8	−1.40097	−0.700483	+	0.713669i	$0.747032\pi$
$978$	0	0
$979$	−54873.7	−1.79139
$980$	0	0
$981$	0	0
$982$	19171.4	0.622997
$983$	−3491.65	−0.113292	−0.0566462	−	0.998394i	$-0.518041\pi$
$983$	−3491.65	−0.113292	−0.0566462	+	0.998394i	$0.518041\pi$
$984$	0	0
$985$	37374.4	1.20898
$986$	54314.6	1.75429
$987$	0	0
$988$	−63317.4	−2.03886
$989$	16714.1	0.537390
$990$	0	0
$991$	−43897.4	−1.40711	−0.703556	−	0.710640i	$-0.748406\pi$
$991$	−43897.4	−1.40711	−0.703556	+	0.710640i	$0.748406\pi$
$992$	−19845.0	−0.635162
$993$	0	0
$994$	0	0
$995$	42388.4	1.35056
$996$	0	0
$997$	−7995.39	−0.253979	−0.126989	−	0.991904i	$-0.540531\pi$
$997$	−7995.39	−0.253979	−0.126989	+	0.991904i	$0.540531\pi$
$998$	−59403.4	−1.88415
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.bm.1.2	yes	8
3.2	odd	2	inner	1323.4.a.bm.1.7	yes	8
7.6	odd	2		1323.4.a.bl.1.2	✓	8
21.20	even	2		1323.4.a.bl.1.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.4.a.bl.1.2	✓	8	7.6	odd	2
1323.4.a.bl.1.7	yes	8	21.20	even	2
1323.4.a.bm.1.2	yes	8	1.1	even	1	trivial
1323.4.a.bm.1.7	yes	8	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

\(f(q)\)	\(=\)	\(q-4.67291 q^{2} +13.8361 q^{4} -15.5408 q^{5} -27.2713 q^{8} +O(q^{10})\)	\(q-4.67291 q^{2} +13.8361 q^{4} -15.5408 q^{5} -27.2713 q^{8} +72.6208 q^{10} -63.4897 q^{11} +53.1931 q^{13} +16.7479 q^{16} -68.4660 q^{17} -86.0311 q^{19} -215.024 q^{20} +296.682 q^{22} -46.9229 q^{23} +116.517 q^{25} -248.567 q^{26} +169.768 q^{29} -141.842 q^{31} +139.909 q^{32} +319.935 q^{34} -411.923 q^{37} +402.015 q^{38} +423.819 q^{40} -49.0000 q^{41} -356.204 q^{43} -878.447 q^{44} +219.266 q^{46} -387.186 q^{47} -544.475 q^{50} +735.983 q^{52} +184.468 q^{53} +986.683 q^{55} -793.308 q^{58} -627.676 q^{59} +821.903 q^{61} +662.816 q^{62} -787.766 q^{64} -826.666 q^{65} -95.9610 q^{67} -947.298 q^{68} -733.093 q^{71} -750.676 q^{73} +1924.88 q^{74} -1190.33 q^{76} +23.5747 q^{79} -260.276 q^{80} +228.972 q^{82} -592.109 q^{83} +1064.02 q^{85} +1664.51 q^{86} +1731.45 q^{88} +864.292 q^{89} -649.227 q^{92} +1809.29 q^{94} +1337.00 q^{95} +614.268 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 44 q^{4}+O(q^{10}) \) 8 * q + 44 * q^4	\( 8 q + 44 q^{4} + 132 q^{10} + 336 q^{13} + 204 q^{16} - 288 q^{19} + 484 q^{22} + 152 q^{25} + 120 q^{31} + 1008 q^{34} + 592 q^{37} + 1620 q^{40} - 1872 q^{43} - 1644 q^{46} + 2400 q^{52} + 1344 q^{55} - 1200 q^{58} + 2400 q^{61} - 1388 q^{64} + 1824 q^{73} + 2844 q^{76} + 2368 q^{79} + 2436 q^{82} + 3512 q^{85} + 3780 q^{88} + 4368 q^{94} + 5712 q^{97}+O(q^{100}) \) 8 * q + 44 * q^4 + 132 * q^10 + 336 * q^13 + 204 * q^16 - 288 * q^19 + 484 * q^22 + 152 * q^25 + 120 * q^31 + 1008 * q^34 + 592 * q^37 + 1620 * q^40 - 1872 * q^43 - 1644 * q^46 + 2400 * q^52 + 1344 * q^55 - 1200 * q^58 + 2400 * q^61 - 1388 * q^64 + 1824 * q^73 + 2844 * q^76 + 2368 * q^79 + 2436 * q^82 + 3512 * q^85 + 3780 * q^88 + 4368 * q^94 + 5712 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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