LMFDB - Embedded newform 1323.4.a.y.1.1

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.3576.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 15x + 3 $ x^3 - x^2 - 15*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.197906$ 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)

$f(q)$	$=$	$q-5.48042 q^{2} +22.0350 q^{4} -4.88670 q^{5} -76.9175 q^{8} +O(q^{10})$	$q-5.48042 q^{2} +22.0350 q^{4} -4.88670 q^{5} -76.9175 q^{8} +26.7812 q^{10} -48.1483 q^{11} -52.1049 q^{13} +245.260 q^{16} -47.8197 q^{17} +82.1399 q^{19} -107.678 q^{20} +263.873 q^{22} +22.1748 q^{23} -101.120 q^{25} +285.557 q^{26} +255.087 q^{29} +342.205 q^{31} -728.787 q^{32} +262.072 q^{34} +39.8099 q^{37} -450.161 q^{38} +375.873 q^{40} +254.847 q^{41} -83.3891 q^{43} -1060.95 q^{44} -121.527 q^{46} +394.439 q^{47} +554.181 q^{50} -1148.13 q^{52} -322.679 q^{53} +235.286 q^{55} -1397.98 q^{58} -147.990 q^{59} +748.694 q^{61} -1875.43 q^{62} +2031.98 q^{64} +254.621 q^{65} +37.4698 q^{67} -1053.70 q^{68} -893.601 q^{71} -531.776 q^{73} -218.175 q^{74} +1809.95 q^{76} -452.631 q^{79} -1198.51 q^{80} -1396.67 q^{82} -239.571 q^{83} +233.680 q^{85} +457.007 q^{86} +3703.44 q^{88} +753.610 q^{89} +488.622 q^{92} -2161.69 q^{94} -401.393 q^{95} +7.48636 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 21 q^{4} + 2 q^{5} - 93 q^{8}+O(q^{10}) $ 3 * q - q^2 + 21 * q^4 + 2 * q^5 - 93 * q^8	$ 3 q - q^{2} + 21 q^{4} + 2 q^{5} - 93 q^{8} + 84 q^{10} - 116 q^{11} - 21 q^{13} + 225 q^{16} - 63 q^{17} + 66 q^{19} + 76 q^{20} + 48 q^{22} - 159 q^{23} + 27 q^{25} + 289 q^{26} - 173 q^{29} + 411 q^{31} - 845 q^{32} - 111 q^{34} + 540 q^{37} - 398 q^{38} + 384 q^{40} + 448 q^{41} - 99 q^{43} - 1312 q^{44} - 417 q^{46} + 792 q^{47} + 1129 q^{50} - 1461 q^{52} + 45 q^{53} - 540 q^{55} - 2547 q^{58} + 207 q^{59} - 114 q^{61} - 2319 q^{62} + 1377 q^{64} - 200 q^{65} - 669 q^{67} - 2565 q^{68} - 1029 q^{71} + 282 q^{73} + 1146 q^{74} + 2214 q^{76} + 684 q^{79} - 2960 q^{80} - 1758 q^{82} + 582 q^{83} - 2694 q^{85} - 551 q^{86} + 4152 q^{88} + 71 q^{89} + 1077 q^{92} - 630 q^{94} + 292 q^{95} - 492 q^{97}+O(q^{100}) $ 3 * q - q^2 + 21 * q^4 + 2 * q^5 - 93 * q^8 + 84 * q^10 - 116 * q^11 - 21 * q^13 + 225 * q^16 - 63 * q^17 + 66 * q^19 + 76 * q^20 + 48 * q^22 - 159 * q^23 + 27 * q^25 + 289 * q^26 - 173 * q^29 + 411 * q^31 - 845 * q^32 - 111 * q^34 + 540 * q^37 - 398 * q^38 + 384 * q^40 + 448 * q^41 - 99 * q^43 - 1312 * q^44 - 417 * q^46 + 792 * q^47 + 1129 * q^50 - 1461 * q^52 + 45 * q^53 - 540 * q^55 - 2547 * q^58 + 207 * q^59 - 114 * q^61 - 2319 * q^62 + 1377 * q^64 - 200 * q^65 - 669 * q^67 - 2565 * q^68 - 1029 * q^71 + 282 * q^73 + 1146 * q^74 + 2214 * q^76 + 684 * q^79 - 2960 * q^80 - 1758 * q^82 + 582 * q^83 - 2694 * q^85 - 551 * q^86 + 4152 * q^88 + 71 * q^89 + 1077 * q^92 - 630 * q^94 + 292 * q^95 - 492 * 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 $

$2$	−5.48042	−1.93762	−0.968810	−	0.247805i	$-0.920291\pi$
$2$	−5.48042	−1.93762	−0.968810	+	0.247805i	$0.920291\pi$
$3$	0	0
$3$	0	0
$4$	22.0350	2.75437
$5$	−4.88670	−0.437080	−0.218540	−	0.975828i	$-0.570129\pi$
$5$	−4.88670	−0.437080	−0.218540	+	0.975828i	$0.570129\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−76.9175	−3.39930
$9$	0	0
$10$	26.7812	0.846894
$11$	−48.1483	−1.31975	−0.659875	−	0.751375i	$-0.729391\pi$
$11$	−48.1483	−1.31975	−0.659875	+	0.751375i	$0.729391\pi$
$12$	0	0
$13$	−52.1049	−1.11164	−0.555819	−	0.831303i	$-0.687595\pi$
$13$	−52.1049	−1.11164	−0.555819	+	0.831303i	$0.687595\pi$
$14$	0	0
$15$	0	0
$16$	245.260	3.83219
$17$	−47.8197	−0.682234	−0.341117	−	0.940021i	$-0.610805\pi$
$17$	−47.8197	−0.682234	−0.341117	+	0.940021i	$0.610805\pi$
$18$	0	0
$19$	82.1399	0.991799	0.495899	−	0.868380i	$-0.334838\pi$
$19$	82.1399	0.991799	0.495899	+	0.868380i	$0.334838\pi$
$20$	−107.678	−1.20388
$21$	0	0
$22$	263.873	2.55717
$23$	22.1748	0.201034	0.100517	−	0.994935i	$-0.467950\pi$
$23$	22.1748	0.201034	0.100517	+	0.994935i	$0.467950\pi$
$24$	0	0
$25$	−101.120	−0.808961
$26$	285.557	2.15393
$27$	0	0
$28$	0	0
$29$	255.087	1.63339	0.816696	−	0.577068i	$-0.195803\pi$
$29$	255.087	1.63339	0.816696	+	0.577068i	$0.195803\pi$
$30$	0	0
$31$	342.205	1.98264	0.991321	−	0.131464i	$-0.0419678\pi$
$31$	342.205	1.98264	0.991321	+	0.131464i	$0.0419678\pi$
$32$	−728.787	−4.02602
$33$	0	0
$34$	262.072	1.32191
$35$	0	0
$36$	0	0
$37$	39.8099	0.176884	0.0884420	−	0.996081i	$-0.471811\pi$
$37$	39.8099	0.176884	0.0884420	+	0.996081i	$0.471811\pi$
$38$	−450.161	−1.92173
$39$	0	0
$40$	375.873	1.48577
$41$	254.847	0.970743	0.485371	−	0.874308i	$-0.338684\pi$
$41$	254.847	0.970743	0.485371	+	0.874308i	$0.338684\pi$
$42$	0	0
$43$	−83.3891	−0.295738	−0.147869	−	0.989007i	$-0.547241\pi$
$43$	−83.3891	−0.295738	−0.147869	+	0.989007i	$0.547241\pi$
$44$	−1060.95	−3.63508
$45$	0	0
$46$	−121.527	−0.389527
$47$	394.439	1.22415	0.612073	−	0.790801i	$-0.290336\pi$
$47$	394.439	1.22415	0.612073	+	0.790801i	$0.290336\pi$
$48$	0	0
$49$	0	0
$50$	554.181	1.56746
$51$	0	0
$52$	−1148.13	−3.06186
$53$	−322.679	−0.836289	−0.418144	−	0.908381i	$-0.637319\pi$
$53$	−322.679	−0.836289	−0.418144	+	0.908381i	$0.637319\pi$
$54$	0	0
$55$	235.286	0.576836
$56$	0	0
$57$	0	0
$58$	−1397.98	−3.16489
$59$	−147.990	−0.326554	−0.163277	−	0.986580i	$-0.552206\pi$
$59$	−147.990	−0.326554	−0.163277	+	0.986580i	$0.552206\pi$
$60$	0	0
$61$	748.694	1.57148	0.785741	−	0.618556i	$-0.212282\pi$
$61$	748.694	1.57148	0.785741	+	0.618556i	$0.212282\pi$
$62$	−1875.43	−3.84161
$63$	0	0
$64$	2031.98	3.96871
$65$	254.621	0.485875
$66$	0	0
$67$	37.4698	0.0683234	0.0341617	−	0.999416i	$-0.489124\pi$
$67$	37.4698	0.0683234	0.0341617	+	0.999416i	$0.489124\pi$
$68$	−1053.70	−1.87912
$69$	0	0
$70$	0	0
$71$	−893.601	−1.49367	−0.746837	−	0.665007i	$-0.768429\pi$
$71$	−893.601	−1.49367	−0.746837	+	0.665007i	$0.768429\pi$
$72$	0	0
$73$	−531.776	−0.852598	−0.426299	−	0.904582i	$-0.640183\pi$
$73$	−531.776	−0.852598	−0.426299	+	0.904582i	$0.640183\pi$
$74$	−218.175	−0.342734
$75$	0	0
$76$	1809.95	2.73178
$77$	0	0
$78$	0	0
$79$	−452.631	−0.644620	−0.322310	−	0.946634i	$-0.604459\pi$
$79$	−452.631	−0.644620	−0.322310	+	0.946634i	$0.604459\pi$
$80$	−1198.51	−1.67497
$81$	0	0
$82$	−1396.67	−1.88093
$83$	−239.571	−0.316824	−0.158412	−	0.987373i	$-0.550637\pi$
$83$	−239.571	−0.316824	−0.158412	+	0.987373i	$0.550637\pi$
$84$	0	0
$85$	233.680	0.298191
$86$	457.007	0.573027
$87$	0	0
$88$	3703.44	4.48623
$89$	753.610	0.897557	0.448778	−	0.893643i	$-0.351859\pi$
$89$	753.610	0.897557	0.448778	+	0.893643i	$0.351859\pi$
$90$	0	0
$91$	0	0
$92$	488.622	0.553721
$93$	0	0
$94$	−2161.69	−2.37193
$95$	−401.393	−0.433495
$96$	0	0
$97$	7.48636	0.00783634	0.00391817	−	0.999992i	$-0.498753\pi$
$97$	7.48636	0.00783634	0.00391817	+	0.999992i	$0.498753\pi$
$98$	0	0
$99$	0	0
$100$	−2228.18	−2.22818
$101$	1124.08	1.10742	0.553711	−	0.832709i	$-0.313211\pi$
$101$	1124.08	1.10742	0.553711	+	0.832709i	$0.313211\pi$
$102$	0	0
$103$	77.1780	0.0738308	0.0369154	−	0.999318i	$-0.488247\pi$
$103$	77.1780	0.0738308	0.0369154	+	0.999318i	$0.488247\pi$
$104$	4007.78	3.77880
$105$	0	0
$106$	1768.41	1.62041
$107$	−283.532	−0.256169	−0.128084	−	0.991763i	$-0.540883\pi$
$107$	−283.532	−0.256169	−0.128084	+	0.991763i	$0.540883\pi$
$108$	0	0
$109$	−1835.87	−1.61326	−0.806628	−	0.591059i	$-0.798710\pi$
$109$	−1835.87	−1.61326	−0.806628	+	0.591059i	$0.798710\pi$
$110$	−1289.47	−1.11769
$111$	0	0
$112$	0	0
$113$	−213.892	−0.178064	−0.0890320	−	0.996029i	$-0.528377\pi$
$113$	−213.892	−0.178064	−0.0890320	+	0.996029i	$0.528377\pi$
$114$	0	0
$115$	−108.362	−0.0878677
$116$	5620.82	4.49897
$117$	0	0
$118$	811.047	0.632737
$119$	0	0
$120$	0	0
$121$	987.256	0.741740
$122$	−4103.15	−3.04493
$123$	0	0
$124$	7540.48	5.46093
$125$	1104.98	0.790660
$126$	0	0
$127$	1747.93	1.22129	0.610644	−	0.791905i	$-0.290911\pi$
$127$	1747.93	1.22129	0.610644	+	0.791905i	$0.290911\pi$
$128$	−5305.79	−3.66383
$129$	0	0
$130$	−1395.43	−0.941440
$131$	−1432.14	−0.955162	−0.477581	−	0.878588i	$-0.658486\pi$
$131$	−1432.14	−0.955162	−0.477581	+	0.878588i	$0.658486\pi$
$132$	0	0
$133$	0	0
$134$	−205.350	−0.132385
$135$	0	0
$136$	3678.17	2.31912
$137$	1490.27	0.929359	0.464679	−	0.885479i	$-0.346170\pi$
$137$	1490.27	0.929359	0.464679	+	0.885479i	$0.346170\pi$
$138$	0	0
$139$	−1669.75	−1.01889	−0.509446	−	0.860503i	$-0.670150\pi$
$139$	−1669.75	−1.01889	−0.509446	+	0.860503i	$0.670150\pi$
$140$	0	0
$141$	0	0
$142$	4897.30	2.89417
$143$	2508.76	1.46708
$144$	0	0
$145$	−1246.53	−0.713923
$146$	2914.35	1.65201
$147$	0	0
$148$	877.210	0.487204
$149$	1587.35	0.872758	0.436379	−	0.899763i	$-0.356261\pi$
$149$	1587.35	0.872758	0.436379	+	0.899763i	$0.356261\pi$
$150$	0	0
$151$	848.437	0.457250	0.228625	−	0.973515i	$-0.426577\pi$
$151$	848.437	0.457250	0.228625	+	0.973515i	$0.426577\pi$
$152$	−6317.99	−3.37143
$153$	0	0
$154$	0	0
$155$	−1672.25	−0.866573
$156$	0	0
$157$	1661.23	0.844461	0.422230	−	0.906489i	$-0.361247\pi$
$157$	1661.23	0.844461	0.422230	+	0.906489i	$0.361247\pi$
$158$	2480.61	1.24903
$159$	0	0
$160$	3561.37	1.75969
$161$	0	0
$162$	0	0
$163$	−1016.91	−0.488655	−0.244327	−	0.969693i	$-0.578567\pi$
$163$	−1016.91	−0.488655	−0.244327	+	0.969693i	$0.578567\pi$
$164$	5615.55	2.67379
$165$	0	0
$166$	1312.95	0.613884
$167$	−1740.30	−0.806400	−0.403200	−	0.915112i	$-0.632102\pi$
$167$	−1740.30	−0.806400	−0.403200	+	0.915112i	$0.632102\pi$
$168$	0	0
$169$	517.921	0.235740
$170$	−1280.67	−0.577780
$171$	0	0
$172$	−1837.48	−0.814571
$173$	1456.36	0.640027	0.320014	−	0.947413i	$-0.396312\pi$
$173$	1456.36	0.640027	0.320014	+	0.947413i	$0.396312\pi$
$174$	0	0
$175$	0	0
$176$	−11808.8	−5.05753
$177$	0	0
$178$	−4130.10	−1.73912
$179$	−2896.58	−1.20950	−0.604749	−	0.796416i	$-0.706727\pi$
$179$	−2896.58	−1.20950	−0.604749	+	0.796416i	$0.706727\pi$
$180$	0	0
$181$	−4388.80	−1.80230	−0.901152	−	0.433503i	$-0.857277\pi$
$181$	−4388.80	−1.80230	−0.901152	+	0.433503i	$0.857277\pi$
$182$	0	0
$183$	0	0
$184$	−1705.63	−0.683374
$185$	−194.539	−0.0773124
$186$	0	0
$187$	2302.43	0.900378
$188$	8691.46	3.37175
$189$	0	0
$190$	2199.80	0.839949
$191$	−721.097	−0.273177	−0.136588	−	0.990628i	$-0.543614\pi$
$191$	−721.097	−0.273177	−0.136588	+	0.990628i	$0.543614\pi$
$192$	0	0
$193$	−148.803	−0.0554978	−0.0277489	−	0.999615i	$-0.508834\pi$
$193$	−148.803	−0.0554978	−0.0277489	+	0.999615i	$0.508834\pi$
$194$	−41.0284	−0.0151838
$195$	0	0
$196$	0	0
$197$	2542.02	0.919348	0.459674	−	0.888088i	$-0.347966\pi$
$197$	2542.02	0.919348	0.459674	+	0.888088i	$0.347966\pi$
$198$	0	0
$199$	−4455.19	−1.58703	−0.793517	−	0.608548i	$-0.791752\pi$
$199$	−4455.19	−1.58703	−0.793517	+	0.608548i	$0.791752\pi$
$200$	7777.91	2.74991
$201$	0	0
$202$	−6160.40	−2.14576
$203$	0	0
$204$	0	0
$205$	−1245.36	−0.424292
$206$	−422.968	−0.143056
$207$	0	0
$208$	−12779.2	−4.26001
$209$	−3954.89	−1.30893
$210$	0	0
$211$	2474.81	0.807454	0.403727	−	0.914879i	$-0.367715\pi$
$211$	2474.81	0.807454	0.403727	+	0.914879i	$0.367715\pi$
$212$	−7110.21	−2.30345
$213$	0	0
$214$	1553.87	0.496358
$215$	407.497	0.129261
$216$	0	0
$217$	0	0
$218$	10061.4	3.12588
$219$	0	0
$220$	5184.52	1.58882
$221$	2491.64	0.758397
$222$	0	0
$223$	2131.51	0.640073	0.320036	−	0.947405i	$-0.396305\pi$
$223$	2131.51	0.640073	0.320036	+	0.947405i	$0.396305\pi$
$224$	0	0
$225$	0	0
$226$	1172.22	0.345020
$227$	−3088.81	−0.903133	−0.451567	−	0.892237i	$-0.649135\pi$
$227$	−3088.81	−0.903133	−0.451567	+	0.892237i	$0.649135\pi$
$228$	0	0
$229$	−1487.09	−0.429125	−0.214563	−	0.976710i	$-0.568833\pi$
$229$	−1487.09	−0.429125	−0.214563	+	0.976710i	$0.568833\pi$
$230$	593.868	0.170254
$231$	0	0
$232$	−19620.6	−5.55240
$233$	−629.611	−0.177027	−0.0885133	−	0.996075i	$-0.528212\pi$
$233$	−629.611	−0.177027	−0.0885133	+	0.996075i	$0.528212\pi$
$234$	0	0
$235$	−1927.51	−0.535050
$236$	−3260.96	−0.899450
$237$	0	0
$238$	0	0
$239$	−0.670926	−0.000181584 0	−9.07921e−5	−	1.00000i	$-0.500029\pi$
$239$	−0.670926	−0.000181584 0	−9.07921e−5		1.00000i	$0.500029\pi$
$240$	0	0
$241$	−3617.99	−0.967033	−0.483517	−	0.875335i	$-0.660641\pi$
$241$	−3617.99	−0.967033	−0.483517	+	0.875335i	$0.660641\pi$
$242$	−5410.57	−1.43721
$243$	0	0
$244$	16497.4	4.32844
$245$	0	0
$246$	0	0
$247$	−4279.89	−1.10252
$248$	−26321.6	−6.73960
$249$	0	0
$250$	−6055.76	−1.53200
$251$	5128.36	1.28964	0.644819	−	0.764335i	$-0.276933\pi$
$251$	5128.36	1.28964	0.644819	+	0.764335i	$0.276933\pi$
$252$	0	0
$253$	−1067.68	−0.265314
$254$	−9579.38	−2.36639
$255$	0	0
$256$	12822.1	3.13040
$257$	755.313	0.183327	0.0916637	−	0.995790i	$-0.470782\pi$
$257$	755.313	0.183327	0.0916637	+	0.995790i	$0.470782\pi$
$258$	0	0
$259$	0	0
$260$	5610.57	1.33828
$261$	0	0
$262$	7848.70	1.85074
$263$	−1776.50	−0.416517	−0.208258	−	0.978074i	$-0.566779\pi$
$263$	−1776.50	−0.416517	−0.208258	+	0.978074i	$0.566779\pi$
$264$	0	0
$265$	1576.83	0.365525
$266$	0	0
$267$	0	0
$268$	825.647	0.188188
$269$	1191.15	0.269983	0.134991	−	0.990847i	$-0.456899\pi$
$269$	1191.15	0.269983	0.134991	+	0.990847i	$0.456899\pi$
$270$	0	0
$271$	3359.28	0.752995	0.376498	−	0.926418i	$-0.377128\pi$
$271$	3359.28	0.752995	0.376498	+	0.926418i	$0.377128\pi$
$272$	−11728.3	−2.61445
$273$	0	0
$274$	−8167.29	−1.80074
$275$	4868.76	1.06763
$276$	0	0
$277$	−8019.51	−1.73951	−0.869757	−	0.493480i	$-0.835725\pi$
$277$	−8019.51	−1.73951	−0.869757	+	0.493480i	$0.835725\pi$
$278$	9150.90	1.97422
$279$	0	0
$280$	0	0
$281$	−2091.27	−0.443966	−0.221983	−	0.975051i	$-0.571253\pi$
$281$	−2091.27	−0.443966	−0.221983	+	0.975051i	$0.571253\pi$
$282$	0	0
$283$	−5817.85	−1.22203	−0.611016	−	0.791618i	$-0.709239\pi$
$283$	−5817.85	−1.22203	−0.611016	+	0.791618i	$0.709239\pi$
$284$	−19690.5	−4.11413
$285$	0	0
$286$	−13749.1	−2.84265
$287$	0	0
$288$	0	0
$289$	−2626.28	−0.534557
$290$	6831.51	1.38331
$291$	0	0
$292$	−11717.7	−2.34837
$293$	−7318.66	−1.45925	−0.729626	−	0.683846i	$-0.760306\pi$
$293$	−7318.66	−1.45925	−0.729626	+	0.683846i	$0.760306\pi$
$294$	0	0
$295$	723.183	0.142730
$296$	−3062.08	−0.601282
$297$	0	0
$298$	−8699.35	−1.69107
$299$	−1155.42	−0.223477
$300$	0	0
$301$	0	0
$302$	−4649.79	−0.885978
$303$	0	0
$304$	20145.6	3.80076
$305$	−3658.64	−0.686863
$306$	0	0
$307$	−8679.36	−1.61354	−0.806771	−	0.590865i	$-0.798787\pi$
$307$	−8679.36	−1.61354	−0.806771	+	0.590865i	$0.798787\pi$
$308$	0	0
$309$	0	0
$310$	9164.65	1.67909
$311$	1740.98	0.317434	0.158717	−	0.987324i	$-0.449264\pi$
$311$	1740.98	0.317434	0.158717	+	0.987324i	$0.449264\pi$
$312$	0	0
$313$	4696.38	0.848099	0.424050	−	0.905639i	$-0.360608\pi$
$313$	4696.38	0.848099	0.424050	+	0.905639i	$0.360608\pi$
$314$	−9104.22	−1.63624
$315$	0	0
$316$	−9973.72	−1.77552
$317$	2685.20	0.475759	0.237880	−	0.971295i	$-0.423548\pi$
$317$	2685.20	0.475759	0.237880	+	0.971295i	$0.423548\pi$
$318$	0	0
$319$	−12282.0	−2.15567
$320$	−9929.67	−1.73464
$321$	0	0
$322$	0	0
$323$	−3927.90	−0.676639
$324$	0	0
$325$	5268.86	0.899272
$326$	5573.10	0.946827
$327$	0	0
$328$	−19602.2	−3.29985
$329$	0	0
$330$	0	0
$331$	−1029.60	−0.170973	−0.0854865	−	0.996339i	$-0.527244\pi$
$331$	−1029.60	−0.170973	−0.0854865	+	0.996339i	$0.527244\pi$
$332$	−5278.95	−0.872650
$333$	0	0
$334$	9537.59	1.56250
$335$	−183.104	−0.0298628
$336$	0	0
$337$	−9328.16	−1.50782	−0.753912	−	0.656975i	$-0.771836\pi$
$337$	−9328.16	−1.50782	−0.753912	+	0.656975i	$0.771836\pi$
$338$	−2838.42	−0.456774
$339$	0	0
$340$	5149.14	0.821327
$341$	−16476.6	−2.61659
$342$	0	0
$343$	0	0
$344$	6414.08	1.00530
$345$	0	0
$346$	−7981.44	−1.24013
$347$	−11279.1	−1.74494	−0.872469	−	0.488669i	$-0.837483\pi$
$347$	−11279.1	−1.74494	−0.872469	+	0.488669i	$0.837483\pi$
$348$	0	0
$349$	−2248.87	−0.344927	−0.172463	−	0.985016i	$-0.555173\pi$
$349$	−2248.87	−0.344927	−0.172463	+	0.985016i	$0.555173\pi$
$350$	0	0
$351$	0	0
$352$	35089.9	5.31334
$353$	1286.51	0.193977	0.0969885	−	0.995286i	$-0.469079\pi$
$353$	1286.51	0.193977	0.0969885	+	0.995286i	$0.469079\pi$
$354$	0	0
$355$	4366.76	0.652855
$356$	16605.8	2.47220
$357$	0	0
$358$	15874.4	2.34355
$359$	742.339	0.109134	0.0545671	−	0.998510i	$-0.482622\pi$
$359$	742.339	0.109134	0.0545671	+	0.998510i	$0.482622\pi$
$360$	0	0
$361$	−112.042	−0.0163350
$362$	24052.5	3.49218
$363$	0	0
$364$	0	0
$365$	2598.63	0.372653
$366$	0	0
$367$	6174.33	0.878195	0.439097	−	0.898439i	$-0.355298\pi$
$367$	6174.33	0.878195	0.439097	+	0.898439i	$0.355298\pi$
$368$	5438.60	0.770399
$369$	0	0
$370$	1066.15	0.149802
$371$	0	0
$372$	0	0
$373$	11137.1	1.54600	0.773002	−	0.634403i	$-0.218754\pi$
$373$	11137.1	1.54600	0.773002	+	0.634403i	$0.218754\pi$
$374$	−12618.3	−1.74459
$375$	0	0
$376$	−30339.3	−4.16125
$377$	−13291.3	−1.81574
$378$	0	0
$379$	9922.21	1.34478	0.672388	−	0.740199i	$-0.265269\pi$
$379$	9922.21	1.34478	0.672388	+	0.740199i	$0.265269\pi$
$380$	−8844.68	−1.19401
$381$	0	0
$382$	3951.91	0.529312
$383$	−783.499	−0.104530	−0.0522649	−	0.998633i	$-0.516644\pi$
$383$	−783.499	−0.104530	−0.0522649	+	0.998633i	$0.516644\pi$
$384$	0	0
$385$	0	0
$386$	815.503	0.107534
$387$	0	0
$388$	164.962	0.0215842
$389$	10059.5	1.31115	0.655576	−	0.755129i	$-0.272426\pi$
$389$	10059.5	1.31115	0.655576	+	0.755129i	$0.272426\pi$
$390$	0	0
$391$	−1060.39	−0.137152
$392$	0	0
$393$	0	0
$394$	−13931.3	−1.78135
$395$	2211.87	0.281750
$396$	0	0
$397$	8168.09	1.03261	0.516303	−	0.856406i	$-0.327308\pi$
$397$	8168.09	1.03261	0.516303	+	0.856406i	$0.327308\pi$
$398$	24416.3	3.07507
$399$	0	0
$400$	−24800.7	−3.10009
$401$	4633.30	0.576998	0.288499	−	0.957480i	$-0.406844\pi$
$401$	4633.30	0.576998	0.288499	+	0.957480i	$0.406844\pi$
$402$	0	0
$403$	−17830.6	−2.20398
$404$	24769.0	3.05025
$405$	0	0
$406$	0	0
$407$	−1916.78	−0.233443
$408$	0	0
$409$	−4345.50	−0.525358	−0.262679	−	0.964883i	$-0.584606\pi$
$409$	−4345.50	−0.525358	−0.262679	+	0.964883i	$0.584606\pi$
$410$	6825.10	0.822117
$411$	0	0
$412$	1700.61	0.203357
$413$	0	0
$414$	0	0
$415$	1170.71	0.138477
$416$	37973.4	4.47548
$417$	0	0
$418$	21674.5	2.53620
$419$	13920.6	1.62307	0.811534	−	0.584305i	$-0.198633\pi$
$419$	13920.6	1.62307	0.811534	+	0.584305i	$0.198633\pi$
$420$	0	0
$421$	2507.79	0.290314	0.145157	−	0.989409i	$-0.453631\pi$
$421$	2507.79	0.290314	0.145157	+	0.989409i	$0.453631\pi$
$422$	−13563.0	−1.56454
$423$	0	0
$424$	24819.6	2.84280
$425$	4835.53	0.551901
$426$	0	0
$427$	0	0
$428$	−6247.62	−0.705584
$429$	0	0
$430$	−2233.26	−0.250458
$431$	16473.3	1.84105	0.920525	−	0.390683i	$-0.127761\pi$
$431$	16473.3	1.84105	0.920525	+	0.390683i	$0.127761\pi$
$432$	0	0
$433$	−4218.59	−0.468204	−0.234102	−	0.972212i	$-0.575215\pi$
$433$	−4218.59	−0.468204	−0.234102	+	0.972212i	$0.575215\pi$
$434$	0	0
$435$	0	0
$436$	−40453.4	−4.44351
$437$	1821.44	0.199385
$438$	0	0
$439$	−1032.82	−0.112287	−0.0561435	−	0.998423i	$-0.517880\pi$
$439$	−1032.82	−0.112287	−0.0561435	+	0.998423i	$0.517880\pi$
$440$	−18097.6	−1.96084
$441$	0	0
$442$	−13655.2	−1.46949
$443$	4650.00	0.498709	0.249355	−	0.968412i	$-0.419782\pi$
$443$	4650.00	0.498709	0.249355	+	0.968412i	$0.419782\pi$
$444$	0	0
$445$	−3682.67	−0.392304
$446$	−11681.5	−1.24022
$447$	0	0
$448$	0	0
$449$	490.593	0.0515646	0.0257823	−	0.999668i	$-0.491792\pi$
$449$	490.593	0.0515646	0.0257823	+	0.999668i	$0.491792\pi$
$450$	0	0
$451$	−12270.5	−1.28114
$452$	−4713.09	−0.490454
$453$	0	0
$454$	16927.9	1.74993
$455$	0	0
$456$	0	0
$457$	−9186.81	−0.940352	−0.470176	−	0.882573i	$-0.655810\pi$
$457$	−9186.81	−0.940352	−0.470176	+	0.882573i	$0.655810\pi$
$458$	8149.87	0.831482
$459$	0	0
$460$	−2387.75	−0.242020
$461$	1406.35	0.142083	0.0710414	−	0.997473i	$-0.477368\pi$
$461$	1406.35	0.142083	0.0710414	+	0.997473i	$0.477368\pi$
$462$	0	0
$463$	15311.7	1.53692	0.768460	−	0.639898i	$-0.221024\pi$
$463$	15311.7	1.53692	0.768460	+	0.639898i	$0.221024\pi$
$464$	62562.5	6.25947
$465$	0	0
$466$	3450.53	0.343010
$467$	1031.63	0.102223	0.0511115	−	0.998693i	$-0.483724\pi$
$467$	1031.63	0.102223	0.0511115	+	0.998693i	$0.483724\pi$
$468$	0	0
$469$	0	0
$470$	10563.5	1.03672
$471$	0	0
$472$	11383.0	1.11006
$473$	4015.04	0.390300
$474$	0	0
$475$	−8306.00	−0.802327
$476$	0	0
$477$	0	0
$478$	3.67696	0.000351841 0
$479$	8385.72	0.799902	0.399951	−	0.916536i	$-0.369027\pi$
$479$	8385.72	0.799902	0.399951	+	0.916536i	$0.369027\pi$
$480$	0	0
$481$	−2074.29	−0.196631
$482$	19828.1	1.87374
$483$	0	0
$484$	21754.1	2.04303
$485$	−36.5836	−0.00342511
$486$	0	0
$487$	3505.37	0.326167	0.163084	−	0.986612i	$-0.447856\pi$
$487$	3505.37	0.326167	0.163084	+	0.986612i	$0.447856\pi$
$488$	−57587.6	−5.34194
$489$	0	0
$490$	0	0
$491$	−20244.7	−1.86075	−0.930377	−	0.366604i	$-0.880520\pi$
$491$	−20244.7	−1.86075	−0.930377	+	0.366604i	$0.880520\pi$
$492$	0	0
$493$	−12198.2	−1.11436
$494$	23455.6	2.13627
$495$	0	0
$496$	83929.3	7.59786
$497$	0	0
$498$	0	0
$499$	−13708.4	−1.22981	−0.614903	−	0.788603i	$-0.710805\pi$
$499$	−13708.4	−1.22981	−0.614903	+	0.788603i	$0.710805\pi$
$500$	24348.2	2.17777
$501$	0	0
$502$	−28105.5	−2.49883
$503$	−10236.6	−0.907408	−0.453704	−	0.891152i	$-0.649898\pi$
$503$	−10236.6	−0.907408	−0.453704	+	0.891152i	$0.649898\pi$
$504$	0	0
$505$	−5493.02	−0.484032
$506$	5851.33	0.514078
$507$	0	0
$508$	38515.6	3.36388
$509$	−6167.82	−0.537100	−0.268550	−	0.963266i	$-0.586544\pi$
$509$	−6167.82	−0.537100	−0.268550	+	0.963266i	$0.586544\pi$
$510$	0	0
$511$	0	0
$512$	−27824.2	−2.40169
$513$	0	0
$514$	−4139.43	−0.355219
$515$	−377.146	−0.0322700
$516$	0	0
$517$	−18991.6	−1.61557
$518$	0	0
$519$	0	0
$520$	−19584.8	−1.65164
$521$	−17592.9	−1.47939	−0.739693	−	0.672945i	$-0.765029\pi$
$521$	−17592.9	−1.47939	−0.739693	+	0.672945i	$0.765029\pi$
$522$	0	0
$523$	−1393.68	−0.116522	−0.0582611	−	0.998301i	$-0.518556\pi$
$523$	−1393.68	−0.116522	−0.0582611	+	0.998301i	$0.518556\pi$
$524$	−31557.1	−2.63087
$525$	0	0
$526$	9735.98	0.807051
$527$	−16364.1	−1.35263
$528$	0	0
$529$	−11675.3	−0.959585
$530$	−8641.70	−0.708248
$531$	0	0
$532$	0	0
$533$	−13278.8	−1.07912
$534$	0	0
$535$	1385.54	0.111966
$536$	−2882.08	−0.232252
$537$	0	0
$538$	−6527.97	−0.523124
$539$	0	0
$540$	0	0
$541$	4668.44	0.371001	0.185501	−	0.982644i	$-0.440609\pi$
$541$	4668.44	0.371001	0.185501	+	0.982644i	$0.440609\pi$
$542$	−18410.3	−1.45902
$543$	0	0
$544$	34850.4	2.74669
$545$	8971.37	0.705122
$546$	0	0
$547$	−2132.82	−0.166715	−0.0833573	−	0.996520i	$-0.526564\pi$
$547$	−2132.82	−0.166715	−0.0833573	+	0.996520i	$0.526564\pi$
$548$	32838.0	2.55980
$549$	0	0
$550$	−26682.8	−2.06865
$551$	20952.8	1.62000
$552$	0	0
$553$	0	0
$554$	43950.2	3.37052
$555$	0	0
$556$	−36792.8	−2.80641
$557$	−18058.5	−1.37372	−0.686860	−	0.726790i	$-0.741011\pi$
$557$	−18058.5	−1.37372	−0.686860	+	0.726790i	$0.741011\pi$
$558$	0	0
$559$	4344.98	0.328753
$560$	0	0
$561$	0	0
$562$	11461.0	0.860237
$563$	2524.26	0.188961	0.0944803	−	0.995527i	$-0.469881\pi$
$563$	2524.26	0.188961	0.0944803	+	0.995527i	$0.469881\pi$
$564$	0	0
$565$	1045.22	0.0778282
$566$	31884.2	2.36783
$567$	0	0
$568$	68733.5	5.07745
$569$	16016.9	1.18008	0.590039	−	0.807375i	$-0.299113\pi$
$569$	16016.9	1.18008	0.590039	+	0.807375i	$0.299113\pi$
$570$	0	0
$571$	−594.146	−0.0435451	−0.0217725	−	0.999763i	$-0.506931\pi$
$571$	−594.146	−0.0435451	−0.0217725	+	0.999763i	$0.506931\pi$
$572$	55280.5	4.04090
$573$	0	0
$574$	0	0
$575$	−2242.32	−0.162628
$576$	0	0
$577$	−24131.3	−1.74107	−0.870535	−	0.492106i	$-0.836227\pi$
$577$	−24131.3	−1.74107	−0.870535	+	0.492106i	$0.836227\pi$
$578$	14393.1	1.03577
$579$	0	0
$580$	−27467.3	−1.96641
$581$	0	0
$582$	0	0
$583$	15536.4	1.10369
$584$	40902.8	2.89824
$585$	0	0
$586$	40109.3	2.82748
$587$	4584.64	0.322365	0.161183	−	0.986925i	$-0.448469\pi$
$587$	4584.64	0.322365	0.161183	+	0.986925i	$0.448469\pi$
$588$	0	0
$589$	28108.7	1.96638
$590$	−3963.34	−0.276556
$591$	0	0
$592$	9763.78	0.677853
$593$	−21869.8	−1.51447	−0.757237	−	0.653140i	$-0.773451\pi$
$593$	−21869.8	−1.51447	−0.757237	+	0.653140i	$0.773451\pi$
$594$	0	0
$595$	0	0
$596$	34977.2	2.40390
$597$	0	0
$598$	6332.17	0.433013
$599$	3243.90	0.221272	0.110636	−	0.993861i	$-0.464711\pi$
$599$	3243.90	0.221272	0.110636	+	0.993861i	$0.464711\pi$
$600$	0	0
$601$	2804.69	0.190359	0.0951795	−	0.995460i	$-0.469657\pi$
$601$	2804.69	0.190359	0.0951795	+	0.995460i	$0.469657\pi$
$602$	0	0
$603$	0	0
$604$	18695.3	1.25944
$605$	−4824.42	−0.324199
$606$	0	0
$607$	2049.42	0.137040	0.0685201	−	0.997650i	$-0.478172\pi$
$607$	2049.42	0.137040	0.0685201	+	0.997650i	$0.478172\pi$
$608$	−59862.5	−3.99300
$609$	0	0
$610$	20050.9	1.33088
$611$	−20552.2	−1.36081
$612$	0	0
$613$	−2812.79	−0.185330	−0.0926650	−	0.995697i	$-0.529539\pi$
$613$	−2812.79	−0.185330	−0.0926650	+	0.995697i	$0.529539\pi$
$614$	47566.5	3.12643
$615$	0	0
$616$	0	0
$617$	2316.37	0.151140	0.0755702	−	0.997140i	$-0.475922\pi$
$617$	2316.37	0.151140	0.0755702	+	0.997140i	$0.475922\pi$
$618$	0	0
$619$	−1268.27	−0.0823526	−0.0411763	−	0.999152i	$-0.513111\pi$
$619$	−1268.27	−0.0823526	−0.0411763	+	0.999152i	$0.513111\pi$
$620$	−36848.1	−2.38686
$621$	0	0
$622$	−9541.29	−0.615066
$623$	0	0
$624$	0	0
$625$	7240.31	0.463380
$626$	−25738.1	−1.64329
$627$	0	0
$628$	36605.1	2.32596
$629$	−1903.70	−0.120676
$630$	0	0
$631$	−14026.2	−0.884906	−0.442453	−	0.896792i	$-0.645892\pi$
$631$	−14026.2	−0.884906	−0.442453	+	0.896792i	$0.645892\pi$
$632$	34815.3	2.19126
$633$	0	0
$634$	−14716.0	−0.921841
$635$	−8541.61	−0.533801
$636$	0	0
$637$	0	0
$638$	67310.3	4.17687
$639$	0	0
$640$	25927.8	1.60138
$641$	−7653.88	−0.471623	−0.235811	−	0.971799i	$-0.575775\pi$
$641$	−7653.88	−0.471623	−0.235811	+	0.971799i	$0.575775\pi$
$642$	0	0
$643$	20248.8	1.24189	0.620946	−	0.783853i	$-0.286749\pi$
$643$	20248.8	1.24189	0.620946	+	0.783853i	$0.286749\pi$
$644$	0	0
$645$	0	0
$646$	21526.5	1.31107
$647$	19614.3	1.19184	0.595918	−	0.803045i	$-0.296788\pi$
$647$	19614.3	1.19184	0.595918	+	0.803045i	$0.296788\pi$
$648$	0	0
$649$	7125.47	0.430969
$650$	−28875.5	−1.74245
$651$	0	0
$652$	−22407.6	−1.34594
$653$	−17911.4	−1.07340	−0.536698	−	0.843775i	$-0.680328\pi$
$653$	−17911.4	−1.07340	−0.536698	+	0.843775i	$0.680328\pi$
$654$	0	0
$655$	6998.41	0.417482
$656$	62503.9	3.72007
$657$	0	0
$658$	0	0
$659$	−29119.5	−1.72130	−0.860649	−	0.509199i	$-0.829942\pi$
$659$	−29119.5	−1.72130	−0.860649	+	0.509199i	$0.829942\pi$
$660$	0	0
$661$	3775.06	0.222137	0.111069	−	0.993813i	$-0.464573\pi$
$661$	3775.06	0.222137	0.111069	+	0.993813i	$0.464573\pi$
$662$	5642.65	0.331281
$663$	0	0
$664$	18427.2	1.07698
$665$	0	0
$666$	0	0
$667$	5656.50	0.328367
$668$	−38347.5	−2.22112
$669$	0	0
$670$	1003.49	0.0578627
$671$	−36048.3	−2.07396
$672$	0	0
$673$	−871.755	−0.0499312	−0.0249656	−	0.999688i	$-0.507948\pi$
$673$	−871.755	−0.0499312	−0.0249656	+	0.999688i	$0.507948\pi$
$674$	51122.2	2.92159
$675$	0	0
$676$	11412.4	0.649315
$677$	−18357.5	−1.04215	−0.521076	−	0.853510i	$-0.674469\pi$
$677$	−18357.5	−1.04215	−0.521076	+	0.853510i	$0.674469\pi$
$678$	0	0
$679$	0	0
$680$	−17974.1	−1.01364
$681$	0	0
$682$	90298.6	5.06996
$683$	−10877.1	−0.609372	−0.304686	−	0.952453i	$-0.598552\pi$
$683$	−10877.1	−0.609372	−0.304686	+	0.952453i	$0.598552\pi$
$684$	0	0
$685$	−7282.49	−0.406204
$686$	0	0
$687$	0	0
$688$	−20452.0	−1.13332
$689$	16813.1	0.929651
$690$	0	0
$691$	−6484.97	−0.357018	−0.178509	−	0.983938i	$-0.557127\pi$
$691$	−6484.97	−0.357018	−0.178509	+	0.983938i	$0.557127\pi$
$692$	32090.8	1.76287
$693$	0	0
$694$	61814.1	3.38103
$695$	8159.54	0.445337
$696$	0	0
$697$	−12186.7	−0.662274
$698$	12324.8	0.668337
$699$	0	0
$700$	0	0
$701$	−13195.6	−0.710971	−0.355485	−	0.934682i	$-0.615684\pi$
$701$	−13195.6	−0.710971	−0.355485	+	0.934682i	$0.615684\pi$
$702$	0	0
$703$	3269.98	0.175433
$704$	−97836.2	−5.23770
$705$	0	0
$706$	−7050.59	−0.375854
$707$	0	0
$708$	0	0
$709$	−22766.5	−1.20594	−0.602971	−	0.797763i	$-0.706017\pi$
$709$	−22766.5	−1.20594	−0.602971	+	0.797763i	$0.706017\pi$
$710$	−23931.7	−1.26498
$711$	0	0
$712$	−57965.8	−3.05107
$713$	7588.35	0.398578
$714$	0	0
$715$	−12259.6	−0.641233
$716$	−63825.9	−3.33141
$717$	0	0
$718$	−4068.33	−0.211460
$719$	−33782.8	−1.75228	−0.876138	−	0.482061i	$-0.839888\pi$
$719$	−33782.8	−1.75228	−0.876138	+	0.482061i	$0.839888\pi$
$720$	0	0
$721$	0	0
$722$	614.037	0.0316511
$723$	0	0
$724$	−96707.1	−4.96421
$725$	−25794.4	−1.32135
$726$	0	0
$727$	−4552.77	−0.232260	−0.116130	−	0.993234i	$-0.537049\pi$
$727$	−4552.77	−0.232260	−0.116130	+	0.993234i	$0.537049\pi$
$728$	0	0
$729$	0	0
$730$	−14241.6	−0.722060
$731$	3987.64	0.201762
$732$	0	0
$733$	35316.9	1.77962	0.889809	−	0.456333i	$-0.150837\pi$
$733$	35316.9	1.77962	0.889809	+	0.456333i	$0.150837\pi$
$734$	−33837.9	−1.70161
$735$	0	0
$736$	−16160.7	−0.809365
$737$	−1804.11	−0.0901698
$738$	0	0
$739$	−13483.7	−0.671187	−0.335593	−	0.942007i	$-0.608937\pi$
$739$	−13483.7	−0.671187	−0.335593	+	0.942007i	$0.608937\pi$
$740$	−4286.66	−0.212947
$741$	0	0
$742$	0	0
$743$	−31281.7	−1.54457	−0.772284	−	0.635277i	$-0.780886\pi$
$743$	−31281.7	−1.54457	−0.772284	+	0.635277i	$0.780886\pi$
$744$	0	0
$745$	−7756.91	−0.381465
$746$	−61036.2	−2.99557
$747$	0	0
$748$	50734.1	2.47997
$749$	0	0
$750$	0	0
$751$	−16411.5	−0.797424	−0.398712	−	0.917076i	$-0.630543\pi$
$751$	−16411.5	−0.797424	−0.398712	+	0.917076i	$0.630543\pi$
$752$	96740.2	4.69116
$753$	0	0
$754$	72841.6	3.51822
$755$	−4146.06	−0.199855
$756$	0	0
$757$	−25110.0	−1.20560	−0.602801	−	0.797892i	$-0.705949\pi$
$757$	−25110.0	−1.20560	−0.602801	+	0.797892i	$0.705949\pi$
$758$	−54377.9	−2.60566
$759$	0	0
$760$	30874.1	1.47358
$761$	15936.2	0.759114	0.379557	−	0.925168i	$-0.376076\pi$
$761$	15936.2	0.759114	0.379557	+	0.925168i	$0.376076\pi$
$762$	0	0
$763$	0	0
$764$	−15889.3	−0.752430
$765$	0	0
$766$	4293.90	0.202539
$767$	7711.01	0.363010
$768$	0	0
$769$	8200.46	0.384546	0.192273	−	0.981341i	$-0.438414\pi$
$769$	8200.46	0.384546	0.192273	+	0.981341i	$0.438414\pi$
$770$	0	0
$771$	0	0
$772$	−3278.87	−0.152862
$773$	−5286.07	−0.245959	−0.122980	−	0.992409i	$-0.539245\pi$
$773$	−5286.07	−0.245959	−0.122980	+	0.992409i	$0.539245\pi$
$774$	0	0
$775$	−34603.9	−1.60388
$776$	−575.832	−0.0266381
$777$	0	0
$778$	−55130.4	−2.54051
$779$	20933.1	0.962782
$780$	0	0
$781$	43025.3	1.97128
$782$	5811.40	0.265748
$783$	0	0
$784$	0	0
$785$	−8117.92	−0.369097
$786$	0	0
$787$	25535.9	1.15661	0.578307	−	0.815819i	$-0.303714\pi$
$787$	25535.9	1.15661	0.578307	+	0.815819i	$0.303714\pi$
$788$	56013.4	2.53223
$789$	0	0
$790$	−12122.0	−0.545925
$791$	0	0
$792$	0	0
$793$	−39010.6	−1.74692
$794$	−44764.5	−2.00080
$795$	0	0
$796$	−98169.9	−4.37128
$797$	−34167.6	−1.51854	−0.759271	−	0.650775i	$-0.774444\pi$
$797$	−34167.6	−1.51854	−0.759271	+	0.650775i	$0.774444\pi$
$798$	0	0
$799$	−18862.0	−0.835154
$800$	73695.1	3.25689
$801$	0	0
$802$	−25392.4	−1.11800
$803$	25604.1	1.12522
$804$	0	0
$805$	0	0
$806$	97719.0	4.27048
$807$	0	0
$808$	−86461.0	−3.76446
$809$	−13529.3	−0.587964	−0.293982	−	0.955811i	$-0.594981\pi$
$809$	−13529.3	−0.587964	−0.293982	+	0.955811i	$0.594981\pi$
$810$	0	0
$811$	−19721.0	−0.853883	−0.426941	−	0.904279i	$-0.640409\pi$
$811$	−19721.0	−0.853883	−0.426941	+	0.904279i	$0.640409\pi$
$812$	0	0
$813$	0	0
$814$	10504.7	0.452323
$815$	4969.35	0.213581
$816$	0	0
$817$	−6849.57	−0.293312
$818$	23815.2	1.01794
$819$	0	0
$820$	−27441.5	−1.16866
$821$	12517.9	0.532128	0.266064	−	0.963955i	$-0.414277\pi$
$821$	12517.9	0.532128	0.266064	+	0.963955i	$0.414277\pi$
$822$	0	0
$823$	21949.2	0.929649	0.464825	−	0.885403i	$-0.346117\pi$
$823$	21949.2	0.929649	0.464825	+	0.885403i	$0.346117\pi$
$824$	−5936.34	−0.250973
$825$	0	0
$826$	0	0
$827$	29064.6	1.22210	0.611049	−	0.791593i	$-0.290748\pi$
$827$	29064.6	1.22210	0.611049	+	0.791593i	$0.290748\pi$
$828$	0	0
$829$	45029.2	1.88653	0.943263	−	0.332047i	$-0.107739\pi$
$829$	45029.2	1.88653	0.943263	+	0.332047i	$0.107739\pi$
$830$	−6416.00	−0.268316
$831$	0	0
$832$	−105876.	−4.41177
$833$	0	0
$834$	0	0
$835$	8504.34	0.352461
$836$	−87145.9	−3.60527
$837$	0	0
$838$	−76290.7	−3.14489
$839$	28612.8	1.17738	0.588691	−	0.808358i	$-0.299643\pi$
$839$	28612.8	1.17738	0.588691	+	0.808358i	$0.299643\pi$
$840$	0	0
$841$	40680.2	1.66797
$842$	−13743.7	−0.562518
$843$	0	0
$844$	54532.3	2.22403
$845$	−2530.92	−0.103037
$846$	0	0
$847$	0	0
$848$	−79140.2	−3.20482
$849$	0	0
$850$	−26500.7	−1.06937
$851$	882.778	0.0355596
$852$	0	0
$853$	42950.0	1.72401	0.862005	−	0.506901i	$-0.169209\pi$
$853$	42950.0	1.72401	0.862005	+	0.506901i	$0.169209\pi$
$854$	0	0
$855$	0	0
$856$	21808.6	0.870796
$857$	−27834.6	−1.10946	−0.554732	−	0.832029i	$-0.687179\pi$
$857$	−27834.6	−1.10946	−0.554732	+	0.832029i	$0.687179\pi$
$858$	0	0
$859$	−19931.2	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$859$	−19931.2	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$860$	8979.19	0.356032
$861$	0	0
$862$	−90280.8	−3.56726
$863$	−25761.4	−1.01614	−0.508071	−	0.861315i	$-0.669641\pi$
$863$	−25761.4	−1.01614	−0.508071	+	0.861315i	$0.669641\pi$
$864$	0	0
$865$	−7116.78	−0.279743
$866$	23119.6	0.907202
$867$	0	0
$868$	0	0
$869$	21793.4	0.850738
$870$	0	0
$871$	−1952.36	−0.0759509
$872$	141211.	5.48395
$873$	0	0
$874$	−9982.24	−0.386332
$875$	0	0
$876$	0	0
$877$	1923.70	0.0740693	0.0370347	−	0.999314i	$-0.488209\pi$
$877$	1923.70	0.0740693	0.0370347	+	0.999314i	$0.488209\pi$
$878$	5660.30	0.217569
$879$	0	0
$880$	57706.3	2.21054
$881$	−3378.44	−0.129197	−0.0645985	−	0.997911i	$-0.520577\pi$
$881$	−3378.44	−0.129197	−0.0645985	+	0.997911i	$0.520577\pi$
$882$	0	0
$883$	27360.8	1.04277	0.521384	−	0.853322i	$-0.325416\pi$
$883$	27360.8	1.04277	0.521384	+	0.853322i	$0.325416\pi$
$884$	54903.2	2.08891
$885$	0	0
$886$	−25483.9	−0.966309
$887$	−38365.4	−1.45229	−0.726146	−	0.687540i	$-0.758691\pi$
$887$	−38365.4	−1.45229	−0.726146	+	0.687540i	$0.758691\pi$
$888$	0	0
$889$	0	0
$890$	20182.6	0.760136
$891$	0	0
$892$	46967.7	1.76300
$893$	32399.2	1.21411
$894$	0	0
$895$	14154.7	0.528647
$896$	0	0
$897$	0	0
$898$	−2688.65	−0.0999126
$899$	87292.0	3.23843
$900$	0	0
$901$	15430.4	0.570544
$902$	67247.2	2.48236
$903$	0	0
$904$	16452.0	0.605294
$905$	21446.8	0.787751
$906$	0	0
$907$	−44875.7	−1.64286	−0.821429	−	0.570311i	$-0.806823\pi$
$907$	−44875.7	−1.64286	−0.821429	+	0.570311i	$0.806823\pi$
$908$	−68061.7	−2.48756
$909$	0	0
$910$	0	0
$911$	−72.3353	−0.00263071	−0.00131535	−	0.999999i	$-0.500419\pi$
$911$	−72.3353	−0.00263071	−0.00131535	+	0.999999i	$0.500419\pi$
$912$	0	0
$913$	11534.9	0.418128
$914$	50347.5	1.82204
$915$	0	0
$916$	−32768.0	−1.18197
$917$	0	0
$918$	0	0
$919$	−5858.03	−0.210270	−0.105135	−	0.994458i	$-0.533528\pi$
$919$	−5858.03	−0.210270	−0.105135	+	0.994458i	$0.533528\pi$
$920$	8334.91	0.298689
$921$	0	0
$922$	−7707.38	−0.275302
$923$	46561.0	1.66043
$924$	0	0
$925$	−4025.58	−0.143092
$926$	−83914.3	−2.97797
$927$	0	0
$928$	−185904.	−6.57607
$929$	43678.3	1.54256	0.771280	−	0.636496i	$-0.219617\pi$
$929$	43678.3	1.54256	0.771280	+	0.636496i	$0.219617\pi$
$930$	0	0
$931$	0	0
$932$	−13873.5	−0.487597
$933$	0	0
$934$	−5653.77	−0.198069
$935$	−11251.3	−0.393537
$936$	0	0
$937$	37450.2	1.30570	0.652852	−	0.757485i	$-0.273572\pi$
$937$	37450.2	1.30570	0.652852	+	0.757485i	$0.273572\pi$
$938$	0	0
$939$	0	0
$940$	−42472.5	−1.47373
$941$	16223.4	0.562026	0.281013	−	0.959704i	$-0.409330\pi$
$941$	16223.4	0.562026	0.281013	+	0.959704i	$0.409330\pi$
$942$	0	0
$943$	5651.20	0.195152
$944$	−36296.0	−1.25142
$945$	0	0
$946$	−22004.1	−0.756252
$947$	22271.5	0.764231	0.382115	−	0.924115i	$-0.375196\pi$
$947$	22271.5	0.764231	0.382115	+	0.924115i	$0.375196\pi$
$948$	0	0
$949$	27708.1	0.947780
$950$	45520.3	1.55460
$951$	0	0
$952$	0	0
$953$	−54784.3	−1.86216	−0.931080	−	0.364816i	$-0.881132\pi$
$953$	−54784.3	−1.86216	−0.931080	+	0.364816i	$0.881132\pi$
$954$	0	0
$955$	3523.78	0.119400
$956$	−14.7838	−0.000500150 0
$957$	0	0
$958$	−45957.2	−1.54991
$959$	0	0
$960$	0	0
$961$	87313.5	2.93087
$962$	11368.0	0.380996
$963$	0	0
$964$	−79722.2	−2.66357
$965$	727.156	0.0242570
$966$	0	0
$967$	−49443.8	−1.64427	−0.822134	−	0.569295i	$-0.807216\pi$
$967$	−49443.8	−1.64427	−0.822134	+	0.569295i	$0.807216\pi$
$968$	−75937.2	−2.52140
$969$	0	0
$970$	200.493	0.00663655
$971$	−52102.4	−1.72198	−0.860991	−	0.508620i	$-0.830156\pi$
$971$	−52102.4	−1.72198	−0.860991	+	0.508620i	$0.830156\pi$
$972$	0	0
$973$	0	0
$974$	−19210.9	−0.631988
$975$	0	0
$976$	183625.	6.02221
$977$	18772.7	0.614732	0.307366	−	0.951591i	$-0.400552\pi$
$977$	18772.7	0.614732	0.307366	+	0.951591i	$0.400552\pi$
$978$	0	0
$979$	−36285.0	−1.18455
$980$	0	0
$981$	0	0
$982$	110949.	3.60543
$983$	−46338.0	−1.50351	−0.751756	−	0.659441i	$-0.770793\pi$
$983$	−46338.0	−1.50351	−0.751756	+	0.659441i	$0.770793\pi$
$984$	0	0
$985$	−12422.1	−0.401828
$986$	66851.0	2.15920
$987$	0	0
$988$	−94307.2	−3.03675
$989$	−1849.14	−0.0594532
$990$	0	0
$991$	43943.7	1.40860	0.704298	−	0.709905i	$-0.251262\pi$
$991$	43943.7	1.40860	0.704298	+	0.709905i	$0.251262\pi$
$992$	−249395.	−7.98216
$993$	0	0
$994$	0	0
$995$	21771.2	0.693660
$996$	0	0
$997$	−23286.8	−0.739721	−0.369860	−	0.929087i	$-0.620594\pi$
$997$	−23286.8	−0.739721	−0.369860	+	0.929087i	$0.620594\pi$
$998$	75127.9	2.38290
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.y.1.1		3
3.2	odd	2		1323.4.a.z.1.3		3
7.6	odd	2		189.4.a.j.1.1	✓	3
21.20	even	2		189.4.a.k.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.a.j.1.1	✓	3	7.6	odd	2
189.4.a.k.1.3	yes	3	21.20	even	2
1323.4.a.y.1.1		3	1.1	even	1	trivial
1323.4.a.z.1.3		3	3.2	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:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

\(f(q)\)	\(=\)	\(q-5.48042 q^{2} +22.0350 q^{4} -4.88670 q^{5} -76.9175 q^{8} +O(q^{10})\)	\(q-5.48042 q^{2} +22.0350 q^{4} -4.88670 q^{5} -76.9175 q^{8} +26.7812 q^{10} -48.1483 q^{11} -52.1049 q^{13} +245.260 q^{16} -47.8197 q^{17} +82.1399 q^{19} -107.678 q^{20} +263.873 q^{22} +22.1748 q^{23} -101.120 q^{25} +285.557 q^{26} +255.087 q^{29} +342.205 q^{31} -728.787 q^{32} +262.072 q^{34} +39.8099 q^{37} -450.161 q^{38} +375.873 q^{40} +254.847 q^{41} -83.3891 q^{43} -1060.95 q^{44} -121.527 q^{46} +394.439 q^{47} +554.181 q^{50} -1148.13 q^{52} -322.679 q^{53} +235.286 q^{55} -1397.98 q^{58} -147.990 q^{59} +748.694 q^{61} -1875.43 q^{62} +2031.98 q^{64} +254.621 q^{65} +37.4698 q^{67} -1053.70 q^{68} -893.601 q^{71} -531.776 q^{73} -218.175 q^{74} +1809.95 q^{76} -452.631 q^{79} -1198.51 q^{80} -1396.67 q^{82} -239.571 q^{83} +233.680 q^{85} +457.007 q^{86} +3703.44 q^{88} +753.610 q^{89} +488.622 q^{92} -2161.69 q^{94} -401.393 q^{95} +7.48636 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 21 q^{4} + 2 q^{5} - 93 q^{8}+O(q^{10}) \) 3 * q - q^2 + 21 * q^4 + 2 * q^5 - 93 * q^8	\( 3 q - q^{2} + 21 q^{4} + 2 q^{5} - 93 q^{8} + 84 q^{10} - 116 q^{11} - 21 q^{13} + 225 q^{16} - 63 q^{17} + 66 q^{19} + 76 q^{20} + 48 q^{22} - 159 q^{23} + 27 q^{25} + 289 q^{26} - 173 q^{29} + 411 q^{31} - 845 q^{32} - 111 q^{34} + 540 q^{37} - 398 q^{38} + 384 q^{40} + 448 q^{41} - 99 q^{43} - 1312 q^{44} - 417 q^{46} + 792 q^{47} + 1129 q^{50} - 1461 q^{52} + 45 q^{53} - 540 q^{55} - 2547 q^{58} + 207 q^{59} - 114 q^{61} - 2319 q^{62} + 1377 q^{64} - 200 q^{65} - 669 q^{67} - 2565 q^{68} - 1029 q^{71} + 282 q^{73} + 1146 q^{74} + 2214 q^{76} + 684 q^{79} - 2960 q^{80} - 1758 q^{82} + 582 q^{83} - 2694 q^{85} - 551 q^{86} + 4152 q^{88} + 71 q^{89} + 1077 q^{92} - 630 q^{94} + 292 q^{95} - 492 q^{97}+O(q^{100}) \) 3 * q - q^2 + 21 * q^4 + 2 * q^5 - 93 * q^8 + 84 * q^10 - 116 * q^11 - 21 * q^13 + 225 * q^16 - 63 * q^17 + 66 * q^19 + 76 * q^20 + 48 * q^22 - 159 * q^23 + 27 * q^25 + 289 * q^26 - 173 * q^29 + 411 * q^31 - 845 * q^32 - 111 * q^34 + 540 * q^37 - 398 * q^38 + 384 * q^40 + 448 * q^41 - 99 * q^43 - 1312 * q^44 - 417 * q^46 + 792 * q^47 + 1129 * q^50 - 1461 * q^52 + 45 * q^53 - 540 * q^55 - 2547 * q^58 + 207 * q^59 - 114 * q^61 - 2319 * q^62 + 1377 * q^64 - 200 * q^65 - 669 * q^67 - 2565 * q^68 - 1029 * q^71 + 282 * q^73 + 1146 * q^74 + 2214 * q^76 + 684 * q^79 - 2960 * q^80 - 1758 * q^82 + 582 * q^83 - 2694 * q^85 - 551 * q^86 + 4152 * q^88 + 71 * q^89 + 1077 * q^92 - 630 * q^94 + 292 * q^95 - 492 * q^97

Introduction

L-functions

Modular forms

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