LMFDB - Embedded newform 1521.4.a.bn.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, 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("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 100 x^{16} + 4066 x^{14} - 86197 x^{12} + 1016064 x^{10} - 6594119 x^{8} + 22251817 x^{6} + \cdots - 6889792 $ x^18 - 100x^16 + 4066x^14 - 86197x^12 + 1016064x^10 - 6594119x^8 + 22251817x^6 - 37096332x^4 + 27824160x^2 - 6889792
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 7\cdot 13^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.56788$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.56788 q^{2} +12.8655 q^{4} +1.42360 q^{5} +3.82462 q^{7} -22.2252 q^{8} +O(q^{10})$	$q-4.56788 q^{2} +12.8655 q^{4} +1.42360 q^{5} +3.82462 q^{7} -22.2252 q^{8} -6.50284 q^{10} -18.2903 q^{11} -17.4704 q^{14} -1.40218 q^{16} -93.1475 q^{17} +74.6094 q^{19} +18.3154 q^{20} +83.5478 q^{22} +7.35466 q^{23} -122.973 q^{25} +49.2058 q^{28} -211.812 q^{29} -183.636 q^{31} +184.207 q^{32} +425.487 q^{34} +5.44473 q^{35} +289.095 q^{37} -340.807 q^{38} -31.6398 q^{40} -131.331 q^{41} +394.550 q^{43} -235.314 q^{44} -33.5952 q^{46} -201.122 q^{47} -328.372 q^{49} +561.728 q^{50} -16.7015 q^{53} -26.0380 q^{55} -85.0031 q^{56} +967.531 q^{58} +446.118 q^{59} +475.647 q^{61} +838.830 q^{62} -830.217 q^{64} +252.055 q^{67} -1198.39 q^{68} -24.8709 q^{70} -295.597 q^{71} +892.965 q^{73} -1320.55 q^{74} +959.890 q^{76} -69.9534 q^{77} +170.848 q^{79} -1.99615 q^{80} +599.903 q^{82} -1331.42 q^{83} -132.605 q^{85} -1802.26 q^{86} +406.505 q^{88} +1500.50 q^{89} +94.6217 q^{92} +918.701 q^{94} +106.214 q^{95} +472.304 q^{97} +1499.97 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 56 q^{4} + 94 q^{7}+O(q^{10}) $ 18 * q + 56 * q^4 + 94 * q^7	$ 18 q + 56 q^{4} + 94 q^{7} + 156 q^{10} + 88 q^{16} + 448 q^{19} - 256 q^{22} - 16 q^{25} + 1300 q^{28} + 818 q^{31} + 900 q^{34} + 524 q^{37} + 800 q^{40} + 752 q^{43} + 1650 q^{46} + 2948 q^{49} - 464 q^{55} + 1626 q^{58} - 1784 q^{61} - 4274 q^{64} + 2872 q^{67} + 5772 q^{70} + 3078 q^{73} + 5726 q^{76} + 3326 q^{79} - 1038 q^{82} + 2340 q^{85} + 780 q^{88} + 1220 q^{94} + 4630 q^{97}+O(q^{100}) $ 18 * q + 56 * q^4 + 94 * q^7 + 156 * q^10 + 88 * q^16 + 448 * q^19 - 256 * q^22 - 16 * q^25 + 1300 * q^28 + 818 * q^31 + 900 * q^34 + 524 * q^37 + 800 * q^40 + 752 * q^43 + 1650 * q^46 + 2948 * q^49 - 464 * q^55 + 1626 * q^58 - 1784 * q^61 - 4274 * q^64 + 2872 * q^67 + 5772 * q^70 + 3078 * q^73 + 5726 * q^76 + 3326 * q^79 - 1038 * q^82 + 2340 * q^85 + 780 * q^88 + 1220 * q^94 + 4630 * 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 $

$2$	−4.56788	−1.61499	−0.807495	−	0.589874i	$-0.799177\pi$
$2$	−4.56788	−1.61499	−0.807495	+	0.589874i	$0.799177\pi$
$3$	0	0
$3$	0	0
$4$	12.8655	1.60819
$5$	1.42360	0.127331	0.0636653	−	0.997971i	$-0.479721\pi$
$5$	1.42360	0.127331	0.0636653	+	0.997971i	$0.479721\pi$
$6$	0	0
$7$	3.82462	0.206510	0.103255	−	0.994655i	$-0.467074\pi$
$7$	3.82462	0.206510	0.103255	+	0.994655i	$0.467074\pi$
$8$	−22.2252	−0.982225
$9$	0	0
$10$	−6.50284	−0.205638
$11$	−18.2903	−0.501339	−0.250669	−	0.968073i	$-0.580651\pi$
$11$	−18.2903	−0.501339	−0.250669	+	0.968073i	$0.580651\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−17.4704	−0.333512
$15$	0	0
$16$	−1.40218	−0.0219091
$17$	−93.1475	−1.32892	−0.664459	−	0.747325i	$-0.731338\pi$
$17$	−93.1475	−1.32892	−0.664459	+	0.747325i	$0.731338\pi$
$18$	0	0
$19$	74.6094	0.900872	0.450436	−	0.892809i	$-0.351268\pi$
$19$	74.6094	0.900872	0.450436	+	0.892809i	$0.351268\pi$
$20$	18.3154	0.204772
$21$	0	0
$22$	83.5478	0.809657
$23$	7.35466	0.0666762	0.0333381	−	0.999444i	$-0.489386\pi$
$23$	7.35466	0.0666762	0.0333381	+	0.999444i	$0.489386\pi$
$24$	0	0
$25$	−122.973	−0.983787
$26$	0	0
$27$	0	0
$28$	49.2058	0.332108
$29$	−211.812	−1.35629	−0.678146	−	0.734927i	$-0.737216\pi$
$29$	−211.812	−1.35629	−0.678146	+	0.734927i	$0.737216\pi$
$30$	0	0
$31$	−183.636	−1.06394	−0.531969	−	0.846764i	$-0.678548\pi$
$31$	−183.636	−1.06394	−0.531969	+	0.846764i	$0.678548\pi$
$32$	184.207	1.01761
$33$	0	0
$34$	425.487	2.14619
$35$	5.44473	0.0262951
$36$	0	0
$37$	289.095	1.28451	0.642255	−	0.766491i	$-0.277999\pi$
$37$	289.095	1.28451	0.642255	+	0.766491i	$0.277999\pi$
$38$	−340.807	−1.45490
$39$	0	0
$40$	−31.6398	−0.125067
$41$	−131.331	−0.500254	−0.250127	−	0.968213i	$-0.580472\pi$
$41$	−131.331	−0.500254	−0.250127	+	0.968213i	$0.580472\pi$
$42$	0	0
$43$	394.550	1.39926	0.699632	−	0.714503i	$-0.253347\pi$
$43$	394.550	1.39926	0.699632	+	0.714503i	$0.253347\pi$
$44$	−235.314	−0.806249
$45$	0	0
$46$	−33.5952	−0.107681
$47$	−201.122	−0.624184	−0.312092	−	0.950052i	$-0.601030\pi$
$47$	−201.122	−0.624184	−0.312092	+	0.950052i	$0.601030\pi$
$48$	0	0
$49$	−328.372	−0.957354
$50$	561.728	1.58881
$51$	0	0
$52$	0	0
$53$	−16.7015	−0.0432853	−0.0216426	−	0.999766i	$-0.506890\pi$
$53$	−16.7015	−0.0432853	−0.0216426	+	0.999766i	$0.506890\pi$
$54$	0	0
$55$	−26.0380	−0.0638358
$56$	−85.0031	−0.202839
$57$	0	0
$58$	967.531	2.19040
$59$	446.118	0.984401	0.492201	−	0.870482i	$-0.336193\pi$
$59$	446.118	0.984401	0.492201	+	0.870482i	$0.336193\pi$
$60$	0	0
$61$	475.647	0.998366	0.499183	−	0.866497i	$-0.333634\pi$
$61$	475.647	0.998366	0.499183	+	0.866497i	$0.333634\pi$
$62$	838.830	1.71825
$63$	0	0
$64$	−830.217	−1.62152
$65$	0	0
$66$	0	0
$67$	252.055	0.459603	0.229801	−	0.973238i	$-0.426192\pi$
$67$	252.055	0.459603	0.229801	+	0.973238i	$0.426192\pi$
$68$	−1198.39	−2.13715
$69$	0	0
$70$	−24.8709	−0.0424663
$71$	−295.597	−0.494098	−0.247049	−	0.969003i	$-0.579461\pi$
$71$	−295.597	−0.494098	−0.247049	+	0.969003i	$0.579461\pi$
$72$	0	0
$73$	892.965	1.43169	0.715847	−	0.698257i	$-0.246041\pi$
$73$	892.965	1.43169	0.715847	+	0.698257i	$0.246041\pi$
$74$	−1320.55	−2.07447
$75$	0	0
$76$	959.890	1.44878
$77$	−69.9534	−0.103532
$78$	0	0
$79$	170.848	0.243315	0.121658	−	0.992572i	$-0.461179\pi$
$79$	170.848	0.243315	0.121658	+	0.992572i	$0.461179\pi$
$80$	−1.99615	−0.00278970
$81$	0	0
$82$	599.903	0.807905
$83$	−1331.42	−1.76075	−0.880375	−	0.474279i	$-0.842709\pi$
$83$	−1331.42	−1.76075	−0.880375	+	0.474279i	$0.842709\pi$
$84$	0	0
$85$	−132.605	−0.169212
$86$	−1802.26	−2.25980
$87$	0	0
$88$	406.505	0.492427
$89$	1500.50	1.78711	0.893554	−	0.448957i	$-0.148204\pi$
$89$	1500.50	1.78711	0.893554	+	0.448957i	$0.148204\pi$
$90$	0	0
$91$	0	0
$92$	94.6217	0.107228
$93$	0	0
$94$	918.701	1.00805
$95$	106.214	0.114709
$96$	0	0
$97$	472.304	0.494384	0.247192	−	0.968967i	$-0.420492\pi$
$97$	472.304	0.494384	0.247192	+	0.968967i	$0.420492\pi$
$98$	1499.97	1.54612
$99$	0	0
$100$	−1582.12	−1.58212
$101$	858.856	0.846133	0.423066	−	0.906099i	$-0.360954\pi$
$101$	858.856	0.846133	0.423066	+	0.906099i	$0.360954\pi$
$102$	0	0
$103$	−17.4895	−0.0167310	−0.00836548	−	0.999965i	$-0.502663\pi$
$103$	−17.4895	−0.0167310	−0.00836548	+	0.999965i	$0.502663\pi$
$104$	0	0
$105$	0	0
$106$	76.2902	0.0699053
$107$	−1674.26	−1.51268	−0.756341	−	0.654177i	$-0.773015\pi$
$107$	−1674.26	−1.51268	−0.756341	+	0.654177i	$0.773015\pi$
$108$	0	0
$109$	749.237	0.658385	0.329192	−	0.944263i	$-0.393224\pi$
$109$	749.237	0.658385	0.329192	+	0.944263i	$0.393224\pi$
$110$	118.939	0.103094
$111$	0	0
$112$	−5.36282	−0.00452445
$113$	−935.250	−0.778592	−0.389296	−	0.921113i	$-0.627282\pi$
$113$	−935.250	−0.778592	−0.389296	+	0.921113i	$0.627282\pi$
$114$	0	0
$115$	10.4701	0.00848993
$116$	−2725.07	−2.18118
$117$	0	0
$118$	−2037.82	−1.58980
$119$	−356.254	−0.274435
$120$	0	0
$121$	−996.466	−0.748660
$122$	−2172.70	−1.61235
$123$	0	0
$124$	−2362.58	−1.71102
$125$	−353.015	−0.252597
$126$	0	0
$127$	1383.93	0.966960	0.483480	−	0.875355i	$-0.339373\pi$
$127$	1383.93	0.966960	0.483480	+	0.875355i	$0.339373\pi$
$128$	2318.68	1.60113
$129$	0	0
$130$	0	0
$131$	−1104.68	−0.736764	−0.368382	−	0.929675i	$-0.620088\pi$
$131$	−1104.68	−0.736764	−0.368382	+	0.929675i	$0.620088\pi$
$132$	0	0
$133$	285.353	0.186039
$134$	−1151.36	−0.742254
$135$	0	0
$136$	2070.22	1.30530
$137$	1281.58	0.799215	0.399607	−	0.916686i	$-0.369146\pi$
$137$	1281.58	0.799215	0.399607	+	0.916686i	$0.369146\pi$
$138$	0	0
$139$	−1622.35	−0.989972	−0.494986	−	0.868901i	$-0.664827\pi$
$139$	−1622.35	−0.989972	−0.494986	+	0.868901i	$0.664827\pi$
$140$	70.0494	0.0422876
$141$	0	0
$142$	1350.25	0.797964
$143$	0	0
$144$	0	0
$145$	−301.535	−0.172698
$146$	−4078.96	−2.31217
$147$	0	0
$148$	3719.36	2.06574
$149$	3015.76	1.65812	0.829062	−	0.559157i	$-0.188875\pi$
$149$	3015.76	1.65812	0.829062	+	0.559157i	$0.188875\pi$
$150$	0	0
$151$	−3452.62	−1.86073	−0.930364	−	0.366636i	$-0.880509\pi$
$151$	−3452.62	−1.86073	−0.930364	+	0.366636i	$0.880509\pi$
$152$	−1658.21	−0.884859
$153$	0	0
$154$	319.539	0.167202
$155$	−261.425	−0.135472
$156$	0	0
$157$	−637.489	−0.324059	−0.162029	−	0.986786i	$-0.551804\pi$
$157$	−637.489	−0.324059	−0.162029	+	0.986786i	$0.551804\pi$
$158$	−780.414	−0.392952
$159$	0	0
$160$	262.237	0.129573
$161$	28.1288	0.0137693
$162$	0	0
$163$	730.216	0.350889	0.175445	−	0.984489i	$-0.443864\pi$
$163$	730.216	0.350889	0.175445	+	0.984489i	$0.443864\pi$
$164$	−1689.64	−0.804504
$165$	0	0
$166$	6081.76	2.84359
$167$	−1001.04	−0.463850	−0.231925	−	0.972734i	$-0.574502\pi$
$167$	−1001.04	−0.463850	−0.231925	+	0.972734i	$0.574502\pi$
$168$	0	0
$169$	0	0
$170$	605.723	0.273275
$171$	0	0
$172$	5076.10	2.25029
$173$	−1772.12	−0.778796	−0.389398	−	0.921070i	$-0.627317\pi$
$173$	−1772.12	−0.778796	−0.389398	+	0.921070i	$0.627317\pi$
$174$	0	0
$175$	−470.327	−0.203162
$176$	25.6463	0.0109839
$177$	0	0
$178$	−6854.10	−2.88616
$179$	3809.57	1.59073	0.795365	−	0.606131i	$-0.207279\pi$
$179$	3809.57	1.59073	0.795365	+	0.606131i	$0.207279\pi$
$180$	0	0
$181$	2025.78	0.831908	0.415954	−	0.909386i	$-0.363448\pi$
$181$	2025.78	0.831908	0.415954	+	0.909386i	$0.363448\pi$
$182$	0	0
$183$	0	0
$184$	−163.459	−0.0654911
$185$	411.555	0.163558
$186$	0	0
$187$	1703.69	0.666237
$188$	−2587.54	−1.00381
$189$	0	0
$190$	−485.173	−0.185253
$191$	979.007	0.370882	0.185441	−	0.982655i	$-0.440629\pi$
$191$	979.007	0.370882	0.185441	+	0.982655i	$0.440629\pi$
$192$	0	0
$193$	−647.578	−0.241522	−0.120761	−	0.992682i	$-0.538533\pi$
$193$	−647.578	−0.241522	−0.120761	+	0.992682i	$0.538533\pi$
$194$	−2157.43	−0.798424
$195$	0	0
$196$	−4224.69	−1.53961
$197$	−3358.09	−1.21449	−0.607244	−	0.794515i	$-0.707725\pi$
$197$	−3358.09	−1.21449	−0.607244	+	0.794515i	$0.707725\pi$
$198$	0	0
$199$	−3178.62	−1.13229	−0.566146	−	0.824305i	$-0.691566\pi$
$199$	−3178.62	−1.13229	−0.566146	+	0.824305i	$0.691566\pi$
$200$	2733.11	0.966300
$201$	0	0
$202$	−3923.15	−1.36650
$203$	−810.100	−0.280088
$204$	0	0
$205$	−186.962	−0.0636976
$206$	79.8898	0.0270203
$207$	0	0
$208$	0	0
$209$	−1364.63	−0.451642
$210$	0	0
$211$	789.477	0.257582	0.128791	−	0.991672i	$-0.458890\pi$
$211$	789.477	0.257582	0.128791	+	0.991672i	$0.458890\pi$
$212$	−214.873	−0.0696111
$213$	0	0
$214$	7647.83	2.44297
$215$	561.682	0.178169
$216$	0	0
$217$	−702.340	−0.219714
$218$	−3422.43	−1.06328
$219$	0	0
$220$	−334.993	−0.102660
$221$	0	0
$222$	0	0
$223$	−2349.00	−0.705383	−0.352691	−	0.935740i	$-0.614733\pi$
$223$	−2349.00	−0.705383	−0.352691	+	0.935740i	$0.614733\pi$
$224$	704.521	0.210146
$225$	0	0
$226$	4272.11	1.25742
$227$	−124.518	−0.0364078	−0.0182039	−	0.999834i	$-0.505795\pi$
$227$	−124.518	−0.0364078	−0.0182039	+	0.999834i	$0.505795\pi$
$228$	0	0
$229$	872.745	0.251846	0.125923	−	0.992040i	$-0.459811\pi$
$229$	872.745	0.251846	0.125923	+	0.992040i	$0.459811\pi$
$230$	−47.8262	−0.0137112
$231$	0	0
$232$	4707.56	1.33218
$233$	5501.06	1.54672	0.773361	−	0.633966i	$-0.218574\pi$
$233$	5501.06	1.54672	0.773361	+	0.633966i	$0.218574\pi$
$234$	0	0
$235$	−286.317	−0.0794777
$236$	5739.55	1.58311
$237$	0	0
$238$	1627.33	0.443210
$239$	−4035.74	−1.09226	−0.546130	−	0.837700i	$-0.683900\pi$
$239$	−4035.74	−1.09226	−0.546130	+	0.837700i	$0.683900\pi$
$240$	0	0
$241$	−3291.83	−0.879856	−0.439928	−	0.898033i	$-0.644996\pi$
$241$	−3291.83	−0.879856	−0.439928	+	0.898033i	$0.644996\pi$
$242$	4551.74	1.20908
$243$	0	0
$244$	6119.45	1.60557
$245$	−467.471	−0.121900
$246$	0	0
$247$	0	0
$248$	4081.36	1.04503
$249$	0	0
$250$	1612.53	0.407941
$251$	6212.66	1.56231	0.781155	−	0.624337i	$-0.214631\pi$
$251$	6212.66	1.56231	0.781155	+	0.624337i	$0.214631\pi$
$252$	0	0
$253$	−134.519	−0.0334274
$254$	−6321.63	−1.56163
$255$	0	0
$256$	−3949.72	−0.964286
$257$	−1856.15	−0.450519	−0.225260	−	0.974299i	$-0.572323\pi$
$257$	−1856.15	−0.450519	−0.225260	+	0.974299i	$0.572323\pi$
$258$	0	0
$259$	1105.68	0.265265
$260$	0	0
$261$	0	0
$262$	5046.03	1.18987
$263$	6004.40	1.40778	0.703892	−	0.710307i	$-0.251444\pi$
$263$	6004.40	1.40778	0.703892	+	0.710307i	$0.251444\pi$
$264$	0	0
$265$	−23.7762	−0.00551155
$266$	−1303.46	−0.300452
$267$	0	0
$268$	3242.82	0.739130
$269$	6213.11	1.40825	0.704126	−	0.710075i	$-0.251339\pi$
$269$	6213.11	1.40825	0.704126	+	0.710075i	$0.251339\pi$
$270$	0	0
$271$	5678.54	1.27287	0.636433	−	0.771332i	$-0.280409\pi$
$271$	5678.54	1.27287	0.636433	+	0.771332i	$0.280409\pi$
$272$	130.610	0.0291154
$273$	0	0
$274$	−5854.09	−1.29072
$275$	2249.22	0.493210
$276$	0	0
$277$	7111.40	1.54254	0.771268	−	0.636510i	$-0.219623\pi$
$277$	7111.40	1.54254	0.771268	+	0.636510i	$0.219623\pi$
$278$	7410.71	1.59879
$279$	0	0
$280$	−121.010	−0.0258277
$281$	4708.92	0.999683	0.499841	−	0.866117i	$-0.333392\pi$
$281$	4708.92	0.999683	0.499841	+	0.866117i	$0.333392\pi$
$282$	0	0
$283$	−3538.68	−0.743295	−0.371647	−	0.928374i	$-0.621207\pi$
$283$	−3538.68	−0.743295	−0.371647	+	0.928374i	$0.621207\pi$
$284$	−3803.02	−0.794605
$285$	0	0
$286$	0	0
$287$	−502.290	−0.103308
$288$	0	0
$289$	3763.46	0.766021
$290$	1377.38	0.278905
$291$	0	0
$292$	11488.5	2.30244
$293$	5172.71	1.03137	0.515687	−	0.856777i	$-0.327537\pi$
$293$	5172.71	1.03137	0.515687	+	0.856777i	$0.327537\pi$
$294$	0	0
$295$	635.094	0.125344
$296$	−6425.20	−1.26168
$297$	0	0
$298$	−13775.6	−2.67785
$299$	0	0
$300$	0	0
$301$	1509.01	0.288962
$302$	15771.1	3.00506
$303$	0	0
$304$	−104.616	−0.0197373
$305$	677.131	0.127123
$306$	0	0
$307$	8903.06	1.65513	0.827564	−	0.561371i	$-0.189726\pi$
$307$	8903.06	1.65513	0.827564	+	0.561371i	$0.189726\pi$
$308$	−899.988	−0.166499
$309$	0	0
$310$	1194.16	0.218786
$311$	9012.95	1.64334	0.821668	−	0.569967i	$-0.193044\pi$
$311$	9012.95	1.64334	0.821668	+	0.569967i	$0.193044\pi$
$312$	0	0
$313$	6177.27	1.11553	0.557763	−	0.830000i	$-0.311660\pi$
$313$	6177.27	1.11553	0.557763	+	0.830000i	$0.311660\pi$
$314$	2911.98	0.523351
$315$	0	0
$316$	2198.05	0.391298
$317$	−1524.80	−0.270162	−0.135081	−	0.990835i	$-0.543130\pi$
$317$	−1524.80	−0.270162	−0.135081	+	0.990835i	$0.543130\pi$
$318$	0	0
$319$	3874.09	0.679961
$320$	−1181.90	−0.206469
$321$	0	0
$322$	−128.489	−0.0222373
$323$	−6949.68	−1.19718
$324$	0	0
$325$	0	0
$326$	−3335.54	−0.566683
$327$	0	0
$328$	2918.85	0.491362
$329$	−769.215	−0.128900
$330$	0	0
$331$	8562.08	1.42180	0.710898	−	0.703295i	$-0.248289\pi$
$331$	8562.08	1.42180	0.710898	+	0.703295i	$0.248289\pi$
$332$	−17129.4	−2.83162
$333$	0	0
$334$	4572.64	0.749114
$335$	358.825	0.0585215
$336$	0	0
$337$	−4599.39	−0.743457	−0.371728	−	0.928342i	$-0.621235\pi$
$337$	−4599.39	−0.743457	−0.371728	+	0.928342i	$0.621235\pi$
$338$	0	0
$339$	0	0
$340$	−1706.03	−0.272125
$341$	3358.76	0.533393
$342$	0	0
$343$	−2567.75	−0.404213
$344$	−8768.97	−1.37439
$345$	0	0
$346$	8094.83	1.25775
$347$	7849.33	1.21433	0.607167	−	0.794574i	$-0.292306\pi$
$347$	7849.33	1.21433	0.607167	+	0.794574i	$0.292306\pi$
$348$	0	0
$349$	5989.62	0.918673	0.459336	−	0.888262i	$-0.348087\pi$
$349$	5989.62	0.918673	0.459336	+	0.888262i	$0.348087\pi$
$350$	2148.40	0.328105
$351$	0	0
$352$	−3369.19	−0.510166
$353$	−6425.03	−0.968752	−0.484376	−	0.874860i	$-0.660953\pi$
$353$	−6425.03	−0.968752	−0.484376	+	0.874860i	$0.660953\pi$
$354$	0	0
$355$	−420.813	−0.0629138
$356$	19304.7	2.87401
$357$	0	0
$358$	−17401.7	−2.56901
$359$	6077.13	0.893422	0.446711	−	0.894678i	$-0.352595\pi$
$359$	6077.13	0.893422	0.446711	+	0.894678i	$0.352595\pi$
$360$	0	0
$361$	−1292.44	−0.188429
$362$	−9253.54	−1.34352
$363$	0	0
$364$	0	0
$365$	1271.23	0.182299
$366$	0	0
$367$	−7171.22	−1.01998	−0.509992	−	0.860179i	$-0.670352\pi$
$367$	−7171.22	−1.01998	−0.509992	+	0.860179i	$0.670352\pi$
$368$	−10.3126	−0.00146082
$369$	0	0
$370$	−1879.94	−0.264144
$371$	−63.8767	−0.00893886
$372$	0	0
$373$	−323.348	−0.0448856	−0.0224428	−	0.999748i	$-0.507144\pi$
$373$	−323.348	−0.0448856	−0.0224428	+	0.999748i	$0.507144\pi$
$374$	−7782.27	−1.07597
$375$	0	0
$376$	4469.98	0.613089
$377$	0	0
$378$	0	0
$379$	2778.92	0.376632	0.188316	−	0.982108i	$-0.439697\pi$
$379$	2778.92	0.376632	0.188316	+	0.982108i	$0.439697\pi$
$380$	1366.50	0.184474
$381$	0	0
$382$	−4471.99	−0.598971
$383$	10057.2	1.34177	0.670883	−	0.741563i	$-0.265915\pi$
$383$	10057.2	1.34177	0.670883	+	0.741563i	$0.265915\pi$
$384$	0	0
$385$	−99.5856	−0.0131827
$386$	2958.06	0.390055
$387$	0	0
$388$	6076.45	0.795064
$389$	11319.2	1.47534	0.737670	−	0.675162i	$-0.235926\pi$
$389$	11319.2	1.47534	0.737670	+	0.675162i	$0.235926\pi$
$390$	0	0
$391$	−685.069	−0.0886072
$392$	7298.14	0.940337
$393$	0	0
$394$	15339.4	1.96139
$395$	243.219	0.0309815
$396$	0	0
$397$	12259.5	1.54984	0.774921	−	0.632058i	$-0.217790\pi$
$397$	12259.5	1.54984	0.774921	+	0.632058i	$0.217790\pi$
$398$	14519.5	1.82864
$399$	0	0
$400$	172.431	0.0215539
$401$	−11202.2	−1.39504	−0.697518	−	0.716567i	$-0.745712\pi$
$401$	−11202.2	−1.39504	−0.697518	+	0.716567i	$0.745712\pi$
$402$	0	0
$403$	0	0
$404$	11049.7	1.36074
$405$	0	0
$406$	3700.44	0.452339
$407$	−5287.62	−0.643975
$408$	0	0
$409$	769.899	0.0930784	0.0465392	−	0.998916i	$-0.485181\pi$
$409$	769.899	0.0930784	0.0465392	+	0.998916i	$0.485181\pi$
$410$	854.022	0.102871
$411$	0	0
$412$	−225.012	−0.0269066
$413$	1706.23	0.203289
$414$	0	0
$415$	−1895.41	−0.224197
$416$	0	0
$417$	0	0
$418$	6233.45	0.729397
$419$	8310.48	0.968958	0.484479	−	0.874803i	$-0.339009\pi$
$419$	8310.48	0.968958	0.484479	+	0.874803i	$0.339009\pi$
$420$	0	0
$421$	3440.71	0.398313	0.199157	−	0.979968i	$-0.436180\pi$
$421$	3440.71	0.398313	0.199157	+	0.979968i	$0.436180\pi$
$422$	−3606.24	−0.415993
$423$	0	0
$424$	371.193	0.0425159
$425$	11454.7	1.30737
$426$	0	0
$427$	1819.17	0.206173
$428$	−21540.3	−2.43269
$429$	0	0
$430$	−2565.70	−0.287742
$431$	−9740.73	−1.08862	−0.544309	−	0.838885i	$-0.683208\pi$
$431$	−9740.73	−1.08862	−0.544309	+	0.838885i	$0.683208\pi$
$432$	0	0
$433$	−16264.5	−1.80513	−0.902563	−	0.430557i	$-0.858317\pi$
$433$	−16264.5	−1.80513	−0.902563	+	0.430557i	$0.858317\pi$
$434$	3208.21	0.354836
$435$	0	0
$436$	9639.34	1.05881
$437$	548.727	0.0600668
$438$	0	0
$439$	−10934.6	−1.18879	−0.594395	−	0.804173i	$-0.702608\pi$
$439$	−10934.6	−1.18879	−0.594395	+	0.804173i	$0.702608\pi$
$440$	578.701	0.0627011
$441$	0	0
$442$	0	0
$443$	−16871.8	−1.80949	−0.904744	−	0.425957i	$-0.859938\pi$
$443$	−16871.8	−1.80949	−0.904744	+	0.425957i	$0.859938\pi$
$444$	0	0
$445$	2136.11	0.227554
$446$	10729.9	1.13919
$447$	0	0
$448$	−3175.27	−0.334860
$449$	592.053	0.0622287	0.0311144	−	0.999516i	$-0.490094\pi$
$449$	592.053	0.0622287	0.0311144	+	0.999516i	$0.490094\pi$
$450$	0	0
$451$	2402.07	0.250797
$452$	−12032.5	−1.25213
$453$	0	0
$454$	568.785	0.0587983
$455$	0	0
$456$	0	0
$457$	−14750.9	−1.50988	−0.754941	−	0.655792i	$-0.772335\pi$
$457$	−14750.9	−1.50988	−0.754941	+	0.655792i	$0.772335\pi$
$458$	−3986.60	−0.406728
$459$	0	0
$460$	134.704	0.0136534
$461$	2208.61	0.223135	0.111568	−	0.993757i	$-0.464413\pi$
$461$	2208.61	0.223135	0.111568	+	0.993757i	$0.464413\pi$
$462$	0	0
$463$	−3946.11	−0.396093	−0.198047	−	0.980193i	$-0.563460\pi$
$463$	−3946.11	−0.396093	−0.198047	+	0.980193i	$0.563460\pi$
$464$	296.999	0.0297151
$465$	0	0
$466$	−25128.2	−2.49794
$467$	13673.7	1.35491	0.677455	−	0.735564i	$-0.263083\pi$
$467$	13673.7	1.35491	0.677455	+	0.735564i	$0.263083\pi$
$468$	0	0
$469$	964.014	0.0949126
$470$	1307.86	0.128356
$471$	0	0
$472$	−9915.08	−0.966903
$473$	−7216.43	−0.701505
$474$	0	0
$475$	−9174.97	−0.886266
$476$	−4583.40	−0.441344
$477$	0	0
$478$	18434.8	1.76399
$479$	13890.4	1.32499	0.662495	−	0.749066i	$-0.269497\pi$
$479$	13890.4	1.32499	0.662495	+	0.749066i	$0.269497\pi$
$480$	0	0
$481$	0	0
$482$	15036.7	1.42096
$483$	0	0
$484$	−12820.1	−1.20399
$485$	672.372	0.0629502
$486$	0	0
$487$	−10512.6	−0.978180	−0.489090	−	0.872233i	$-0.662671\pi$
$487$	−10512.6	−0.978180	−0.489090	+	0.872233i	$0.662671\pi$
$488$	−10571.4	−0.980620
$489$	0	0
$490$	2135.35	0.196868
$491$	11387.5	1.04667	0.523333	−	0.852128i	$-0.324688\pi$
$491$	11387.5	1.04667	0.523333	+	0.852128i	$0.324688\pi$
$492$	0	0
$493$	19729.7	1.80240
$494$	0	0
$495$	0	0
$496$	257.492	0.0233099
$497$	−1130.55	−0.102036
$498$	0	0
$499$	12664.5	1.13615	0.568076	−	0.822976i	$-0.307688\pi$
$499$	12664.5	1.13615	0.568076	+	0.822976i	$0.307688\pi$
$500$	−4541.73	−0.406224
$501$	0	0
$502$	−28378.7	−2.52312
$503$	−9908.07	−0.878288	−0.439144	−	0.898417i	$-0.644718\pi$
$503$	−9908.07	−0.878288	−0.439144	+	0.898417i	$0.644718\pi$
$504$	0	0
$505$	1222.67	0.107739
$506$	614.466	0.0539849
$507$	0	0
$508$	17805.0	1.55506
$509$	1801.99	0.156919	0.0784596	−	0.996917i	$-0.475000\pi$
$509$	1801.99	0.156919	0.0784596	+	0.996917i	$0.475000\pi$
$510$	0	0
$511$	3415.26	0.295660
$512$	−507.602	−0.0438146
$513$	0	0
$514$	8478.67	0.727584
$515$	−24.8980	−0.00213036
$516$	0	0
$517$	3678.57	0.312927
$518$	−5050.61	−0.428400
$519$	0	0
$520$	0	0
$521$	−15680.2	−1.31855	−0.659273	−	0.751903i	$-0.729136\pi$
$521$	−15680.2	−1.31855	−0.659273	+	0.751903i	$0.729136\pi$
$522$	0	0
$523$	17866.4	1.49377	0.746884	−	0.664954i	$-0.231549\pi$
$523$	17866.4	1.49377	0.746884	+	0.664954i	$0.231549\pi$
$524$	−14212.3	−1.18486
$525$	0	0
$526$	−27427.4	−2.27356
$527$	17105.3	1.41389
$528$	0	0
$529$	−12112.9	−0.995554
$530$	108.607	0.00890109
$531$	0	0
$532$	3671.22	0.299187
$533$	0	0
$534$	0	0
$535$	−2383.48	−0.192611
$536$	−5601.97	−0.451433
$537$	0	0
$538$	−28380.7	−2.27431
$539$	6006.02	0.479958
$540$	0	0
$541$	−9640.89	−0.766163	−0.383082	−	0.923714i	$-0.625137\pi$
$541$	−9640.89	−0.766163	−0.383082	+	0.923714i	$0.625137\pi$
$542$	−25938.9	−2.05567
$543$	0	0
$544$	−17158.4	−1.35232
$545$	1066.61	0.0838325
$546$	0	0
$547$	2378.17	0.185892	0.0929461	−	0.995671i	$-0.470372\pi$
$547$	2378.17	0.185892	0.0929461	+	0.995671i	$0.470372\pi$
$548$	16488.2	1.28529
$549$	0	0
$550$	−10274.2	−0.796530
$551$	−15803.2	−1.22185
$552$	0	0
$553$	653.430	0.0502471
$554$	−32484.0	−2.49118
$555$	0	0
$556$	−20872.4	−1.59207
$557$	6200.98	0.471712	0.235856	−	0.971788i	$-0.424211\pi$
$557$	6200.98	0.471712	0.235856	+	0.971788i	$0.424211\pi$
$558$	0	0
$559$	0	0
$560$	−7.63451	−0.000576102 0
$561$	0	0
$562$	−21509.8	−1.61448
$563$	14559.7	1.08991	0.544953	−	0.838467i	$-0.316548\pi$
$563$	14559.7	1.08991	0.544953	+	0.838467i	$0.316548\pi$
$564$	0	0
$565$	−1331.42	−0.0991387
$566$	16164.3	1.20041
$567$	0	0
$568$	6569.72	0.485316
$569$	15706.3	1.15719	0.578595	−	0.815615i	$-0.303601\pi$
$569$	15706.3	1.15719	0.578595	+	0.815615i	$0.303601\pi$
$570$	0	0
$571$	17788.6	1.30373	0.651864	−	0.758336i	$-0.273987\pi$
$571$	17788.6	1.30373	0.651864	+	0.758336i	$0.273987\pi$
$572$	0	0
$573$	0	0
$574$	2294.40	0.166841
$575$	−904.428	−0.0655952
$576$	0	0
$577$	16136.7	1.16426	0.582132	−	0.813095i	$-0.302219\pi$
$577$	16136.7	1.16426	0.582132	+	0.813095i	$0.302219\pi$
$578$	−17191.0	−1.23712
$579$	0	0
$580$	−3879.41	−0.277731
$581$	−5092.17	−0.363613
$582$	0	0
$583$	305.474	0.0217006
$584$	−19846.3	−1.40625
$585$	0	0
$586$	−23628.3	−1.66566
$587$	−2446.59	−0.172030	−0.0860150	−	0.996294i	$-0.527413\pi$
$587$	−2446.59	−0.172030	−0.0860150	+	0.996294i	$0.527413\pi$
$588$	0	0
$589$	−13701.0	−0.958473
$590$	−2901.03	−0.202430
$591$	0	0
$592$	−405.364	−0.0281425
$593$	−1730.14	−0.119811	−0.0599057	−	0.998204i	$-0.519080\pi$
$593$	−1730.14	−0.119811	−0.0599057	+	0.998204i	$0.519080\pi$
$594$	0	0
$595$	−507.163	−0.0349440
$596$	38799.3	2.66658
$597$	0	0
$598$	0	0
$599$	23597.1	1.60960	0.804800	−	0.593546i	$-0.202273\pi$
$599$	23597.1	1.60960	0.804800	+	0.593546i	$0.202273\pi$
$600$	0	0
$601$	−20907.8	−1.41905	−0.709525	−	0.704681i	$-0.751090\pi$
$601$	−20907.8	−1.41905	−0.709525	+	0.704681i	$0.751090\pi$
$602$	−6892.96	−0.466671
$603$	0	0
$604$	−44419.8	−2.99241
$605$	−1418.57	−0.0953273
$606$	0	0
$607$	1496.15	0.100045	0.0500223	−	0.998748i	$-0.484071\pi$
$607$	1496.15	0.100045	0.0500223	+	0.998748i	$0.484071\pi$
$608$	13743.6	0.916735
$609$	0	0
$610$	−3093.05	−0.205302
$611$	0	0
$612$	0	0
$613$	−9923.66	−0.653855	−0.326927	−	0.945049i	$-0.606013\pi$
$613$	−9923.66	−0.653855	−0.326927	+	0.945049i	$0.606013\pi$
$614$	−40668.1	−2.67302
$615$	0	0
$616$	1554.73	0.101691
$617$	19544.5	1.27525	0.637626	−	0.770346i	$-0.279917\pi$
$617$	19544.5	1.27525	0.637626	+	0.770346i	$0.279917\pi$
$618$	0	0
$619$	−13948.1	−0.905691	−0.452846	−	0.891589i	$-0.649591\pi$
$619$	−13948.1	−0.905691	−0.452846	+	0.891589i	$0.649591\pi$
$620$	−3363.37	−0.217865
$621$	0	0
$622$	−41170.1	−2.65397
$623$	5738.84	0.369056
$624$	0	0
$625$	14869.1	0.951624
$626$	−28217.0	−1.80156
$627$	0	0
$628$	−8201.64	−0.521149
$629$	−26928.5	−1.70701
$630$	0	0
$631$	25953.1	1.63736	0.818681	−	0.574249i	$-0.194706\pi$
$631$	25953.1	1.63736	0.818681	+	0.574249i	$0.194706\pi$
$632$	−3797.14	−0.238991
$633$	0	0
$634$	6965.12	0.436309
$635$	1970.16	0.123124
$636$	0	0
$637$	0	0
$638$	−17696.4	−1.09813
$639$	0	0
$640$	3300.87	0.203873
$641$	−3214.66	−0.198083	−0.0990417	−	0.995083i	$-0.531578\pi$
$641$	−3214.66	−0.198083	−0.0990417	+	0.995083i	$0.531578\pi$
$642$	0	0
$643$	1996.07	0.122422	0.0612111	−	0.998125i	$-0.480504\pi$
$643$	1996.07	0.122422	0.0612111	+	0.998125i	$0.480504\pi$
$644$	361.892	0.0221437
$645$	0	0
$646$	31745.3	1.93344
$647$	−5539.40	−0.336594	−0.168297	−	0.985736i	$-0.553827\pi$
$647$	−5539.40	−0.336594	−0.168297	+	0.985736i	$0.553827\pi$
$648$	0	0
$649$	−8159.63	−0.493518
$650$	0	0
$651$	0	0
$652$	9394.63	0.564298
$653$	−15628.0	−0.936553	−0.468277	−	0.883582i	$-0.655125\pi$
$653$	−15628.0	−0.936553	−0.468277	+	0.883582i	$0.655125\pi$
$654$	0	0
$655$	−1572.62	−0.0938126
$656$	184.150	0.0109601
$657$	0	0
$658$	3513.68	0.208173
$659$	17229.9	1.01849	0.509244	−	0.860622i	$-0.329925\pi$
$659$	17229.9	1.01849	0.509244	+	0.860622i	$0.329925\pi$
$660$	0	0
$661$	19010.9	1.11867	0.559333	−	0.828943i	$-0.311057\pi$
$661$	19010.9	1.11867	0.559333	+	0.828943i	$0.311057\pi$
$662$	−39110.6	−2.29619
$663$	0	0
$664$	29591.1	1.72945
$665$	406.228	0.0236885
$666$	0	0
$667$	−1557.80	−0.0904324
$668$	−12879.0	−0.745961
$669$	0	0
$670$	−1639.07	−0.0945117
$671$	−8699.71	−0.500519
$672$	0	0
$673$	−18524.2	−1.06100	−0.530501	−	0.847684i	$-0.677996\pi$
$673$	−18524.2	−1.06100	−0.530501	+	0.847684i	$0.677996\pi$
$674$	21009.5	1.20067
$675$	0	0
$676$	0	0
$677$	−27599.1	−1.56679	−0.783397	−	0.621522i	$-0.786515\pi$
$677$	−27599.1	−1.56679	−0.783397	+	0.621522i	$0.786515\pi$
$678$	0	0
$679$	1806.38	0.102095
$680$	2947.17	0.166204
$681$	0	0
$682$	−15342.4	−0.861425
$683$	26301.5	1.47350	0.736749	−	0.676167i	$-0.236360\pi$
$683$	26301.5	1.47350	0.736749	+	0.676167i	$0.236360\pi$
$684$	0	0
$685$	1824.45	0.101765
$686$	11729.2	0.652801
$687$	0	0
$688$	−553.232	−0.0306566
$689$	0	0
$690$	0	0
$691$	131.528	0.00724106	0.00362053	−	0.999993i	$-0.498848\pi$
$691$	131.528	0.00724106	0.00362053	+	0.999993i	$0.498848\pi$
$692$	−22799.3	−1.25245
$693$	0	0
$694$	−35854.8	−1.96114
$695$	−2309.58	−0.126054
$696$	0	0
$697$	12233.1	0.664796
$698$	−27359.9	−1.48365
$699$	0	0
$700$	−6051.01	−0.326724
$701$	−11604.4	−0.625239	−0.312620	−	0.949878i	$-0.601207\pi$
$701$	−11604.4	−0.625239	−0.312620	+	0.949878i	$0.601207\pi$
$702$	0	0
$703$	21569.2	1.15718
$704$	15184.9	0.812929
$705$	0	0
$706$	29348.8	1.56453
$707$	3284.80	0.174735
$708$	0	0
$709$	−2046.79	−0.108419	−0.0542094	−	0.998530i	$-0.517264\pi$
$709$	−2046.79	−0.108419	−0.0542094	+	0.998530i	$0.517264\pi$
$710$	1922.22	0.101605
$711$	0	0
$712$	−33348.9	−1.75534
$713$	−1350.58	−0.0709394
$714$	0	0
$715$	0	0
$716$	49012.2	2.55820
$717$	0	0
$718$	−27759.6	−1.44287
$719$	−34019.7	−1.76456	−0.882282	−	0.470721i	$-0.843994\pi$
$719$	−34019.7	−1.76456	−0.882282	+	0.470721i	$0.843994\pi$
$720$	0	0
$721$	−66.8906	−0.00345511
$722$	5903.69	0.304311
$723$	0	0
$724$	26062.8	1.33787
$725$	26047.2	1.33430
$726$	0	0
$727$	10854.7	0.553754	0.276877	−	0.960905i	$-0.410701\pi$
$727$	10854.7	0.553754	0.276877	+	0.960905i	$0.410701\pi$
$728$	0	0
$729$	0	0
$730$	−5806.81	−0.294410
$731$	−36751.4	−1.85951
$732$	0	0
$733$	−32912.5	−1.65846	−0.829229	−	0.558910i	$-0.811220\pi$
$733$	−32912.5	−1.65846	−0.829229	+	0.558910i	$0.811220\pi$
$734$	32757.3	1.64727
$735$	0	0
$736$	1354.78	0.0678503
$737$	−4610.15	−0.230417
$738$	0	0
$739$	26680.8	1.32810	0.664052	−	0.747687i	$-0.268836\pi$
$739$	26680.8	1.32810	0.664052	+	0.747687i	$0.268836\pi$
$740$	5294.88	0.263032
$741$	0	0
$742$	291.781	0.0144362
$743$	−31128.8	−1.53702	−0.768509	−	0.639838i	$-0.779001\pi$
$743$	−31128.8	−1.53702	−0.768509	+	0.639838i	$0.779001\pi$
$744$	0	0
$745$	4293.23	0.211130
$746$	1477.02	0.0724898
$747$	0	0
$748$	21918.9	1.07144
$749$	−6403.42	−0.312384
$750$	0	0
$751$	−26246.3	−1.27529	−0.637644	−	0.770331i	$-0.720091\pi$
$751$	−26246.3	−1.27529	−0.637644	+	0.770331i	$0.720091\pi$
$752$	282.010	0.0136753
$753$	0	0
$754$	0	0
$755$	−4915.15	−0.236928
$756$	0	0
$757$	7414.10	0.355971	0.177986	−	0.984033i	$-0.443042\pi$
$757$	7414.10	0.355971	0.177986	+	0.984033i	$0.443042\pi$
$758$	−12693.8	−0.608257
$759$	0	0
$760$	−2360.63	−0.112670
$761$	13424.0	0.639448	0.319724	−	0.947511i	$-0.396410\pi$
$761$	13424.0	0.639448	0.319724	+	0.947511i	$0.396410\pi$
$762$	0	0
$763$	2865.55	0.135963
$764$	12595.5	0.596450
$765$	0	0
$766$	−45939.9	−2.16694
$767$	0	0
$768$	0	0
$769$	11201.4	0.525269	0.262634	−	0.964895i	$-0.415409\pi$
$769$	11201.4	0.525269	0.262634	+	0.964895i	$0.415409\pi$
$770$	454.895	0.0212900
$771$	0	0
$772$	−8331.44	−0.388413
$773$	−7544.68	−0.351052	−0.175526	−	0.984475i	$-0.556163\pi$
$773$	−7544.68	−0.351052	−0.175526	+	0.984475i	$0.556163\pi$
$774$	0	0
$775$	22582.4	1.04669
$776$	−10497.1	−0.485596
$777$	0	0
$778$	−51704.9	−2.38266
$779$	−9798.50	−0.450665
$780$	0	0
$781$	5406.56	0.247710
$782$	3129.31	0.143100
$783$	0	0
$784$	460.438	0.0209748
$785$	−907.530	−0.0412626
$786$	0	0
$787$	21988.8	0.995952	0.497976	−	0.867191i	$-0.334077\pi$
$787$	21988.8	0.995952	0.497976	+	0.867191i	$0.334077\pi$
$788$	−43203.7	−1.95313
$789$	0	0
$790$	−1111.00	−0.0500348
$791$	−3576.98	−0.160787
$792$	0	0
$793$	0	0
$794$	−56000.0	−2.50298
$795$	0	0
$796$	−40894.6	−1.82094
$797$	18599.4	0.826632	0.413316	−	0.910588i	$-0.364370\pi$
$797$	18599.4	0.826632	0.413316	+	0.910588i	$0.364370\pi$
$798$	0	0
$799$	18734.0	0.829488
$800$	−22652.5	−1.00111
$801$	0	0
$802$	51170.2	2.25297
$803$	−16332.6	−0.717764
$804$	0	0
$805$	40.0442	0.00175326
$806$	0	0
$807$	0	0
$808$	−19088.3	−0.831093
$809$	17943.9	0.779818	0.389909	−	0.920853i	$-0.372506\pi$
$809$	17943.9	0.779818	0.389909	+	0.920853i	$0.372506\pi$
$810$	0	0
$811$	26483.5	1.14668	0.573342	−	0.819316i	$-0.305647\pi$
$811$	26483.5	1.14668	0.573342	+	0.819316i	$0.305647\pi$
$812$	−10422.4	−0.450436
$813$	0	0
$814$	24153.2	1.04001
$815$	1039.54	0.0446790
$816$	0	0
$817$	29437.2	1.26056
$818$	−3516.81	−0.150321
$819$	0	0
$820$	−2405.37	−0.102438
$821$	−5337.82	−0.226907	−0.113454	−	0.993543i	$-0.536191\pi$
$821$	−5337.82	−0.226907	−0.113454	+	0.993543i	$0.536191\pi$
$822$	0	0
$823$	−18329.3	−0.776330	−0.388165	−	0.921590i	$-0.626891\pi$
$823$	−18329.3	−0.776330	−0.388165	+	0.921590i	$0.626891\pi$
$824$	388.707	0.0164336
$825$	0	0
$826$	−7793.88	−0.328309
$827$	−22310.2	−0.938092	−0.469046	−	0.883174i	$-0.655402\pi$
$827$	−22310.2	−0.938092	−0.469046	+	0.883174i	$0.655402\pi$
$828$	0	0
$829$	−15181.1	−0.636022	−0.318011	−	0.948087i	$-0.603015\pi$
$829$	−15181.1	−0.636022	−0.318011	+	0.948087i	$0.603015\pi$
$830$	8658.00	0.362076
$831$	0	0
$832$	0	0
$833$	30587.1	1.27224
$834$	0	0
$835$	−1425.08	−0.0590624
$836$	−17556.7	−0.726327
$837$	0	0
$838$	−37961.3	−1.56486
$839$	−20959.7	−0.862467	−0.431234	−	0.902240i	$-0.641922\pi$
$839$	−20959.7	−0.862467	−0.431234	+	0.902240i	$0.641922\pi$
$840$	0	0
$841$	20475.2	0.839527
$842$	−15716.7	−0.643272
$843$	0	0
$844$	10157.1	0.414242
$845$	0	0
$846$	0	0
$847$	−3811.11	−0.154606
$848$	23.4185	0.000948342 0
$849$	0	0
$850$	−52323.5	−2.11139
$851$	2126.20	0.0856464
$852$	0	0
$853$	44482.9	1.78554	0.892769	−	0.450515i	$-0.148760\pi$
$853$	44482.9	1.78554	0.892769	+	0.450515i	$0.148760\pi$
$854$	−8309.75	−0.332967
$855$	0	0
$856$	37210.8	1.48580
$857$	17206.5	0.685836	0.342918	−	0.939365i	$-0.388585\pi$
$857$	17206.5	0.685836	0.342918	+	0.939365i	$0.388585\pi$
$858$	0	0
$859$	3736.65	0.148420	0.0742101	−	0.997243i	$-0.476356\pi$
$859$	3736.65	0.148420	0.0742101	+	0.997243i	$0.476356\pi$
$860$	7226.34	0.286530
$861$	0	0
$862$	44494.5	1.75811
$863$	−7263.00	−0.286483	−0.143242	−	0.989688i	$-0.545753\pi$
$863$	−7263.00	−0.286483	−0.143242	+	0.989688i	$0.545753\pi$
$864$	0	0
$865$	−2522.79	−0.0991646
$866$	74294.1	2.91526
$867$	0	0
$868$	−9035.99	−0.353343
$869$	−3124.86	−0.121983
$870$	0	0
$871$	0	0
$872$	−16652.0	−0.646682
$873$	0	0
$874$	−2506.52	−0.0970073
$875$	−1350.15	−0.0521638
$876$	0	0
$877$	9024.17	0.347463	0.173731	−	0.984793i	$-0.444418\pi$
$877$	9024.17	0.347463	0.173731	+	0.984793i	$0.444418\pi$
$878$	49947.8	1.91988
$879$	0	0
$880$	36.5101	0.00139858
$881$	−9585.56	−0.366567	−0.183284	−	0.983060i	$-0.558673\pi$
$881$	−9585.56	−0.366567	−0.183284	+	0.983060i	$0.558673\pi$
$882$	0	0
$883$	−7138.33	−0.272054	−0.136027	−	0.990705i	$-0.543433\pi$
$883$	−7138.33	−0.272054	−0.136027	+	0.990705i	$0.543433\pi$
$884$	0	0
$885$	0	0
$886$	77068.3	2.92230
$887$	34358.1	1.30060	0.650299	−	0.759678i	$-0.274644\pi$
$887$	34358.1	1.30060	0.650299	+	0.759678i	$0.274644\pi$
$888$	0	0
$889$	5293.01	0.199687
$890$	−9757.50	−0.367497
$891$	0	0
$892$	−30221.1	−1.13439
$893$	−15005.6	−0.562310
$894$	0	0
$895$	5423.30	0.202549
$896$	8868.07	0.330649
$897$	0	0
$898$	−2704.43	−0.100499
$899$	38896.4	1.44301
$900$	0	0
$901$	1555.70	0.0575226
$902$	−10972.4	−0.405034
$903$	0	0
$904$	20786.1	0.764753
$905$	2883.91	0.105927
$906$	0	0
$907$	−13184.2	−0.482661	−0.241330	−	0.970443i	$-0.577584\pi$
$907$	−13184.2	−0.482661	−0.241330	+	0.970443i	$0.577584\pi$
$908$	−1602.00	−0.0585508
$909$	0	0
$910$	0	0
$911$	10953.7	0.398368	0.199184	−	0.979962i	$-0.436171\pi$
$911$	10953.7	0.398368	0.199184	+	0.979962i	$0.436171\pi$
$912$	0	0
$913$	24352.0	0.882732
$914$	67380.2	2.43844
$915$	0	0
$916$	11228.3	0.405016
$917$	−4224.97	−0.152149
$918$	0	0
$919$	16852.4	0.604907	0.302453	−	0.953164i	$-0.402194\pi$
$919$	16852.4	0.604907	0.302453	+	0.953164i	$0.402194\pi$
$920$	−232.700	−0.00833902
$921$	0	0
$922$	−10088.7	−0.360361
$923$	0	0
$924$	0	0
$925$	−35551.0	−1.26369
$926$	18025.3	0.639687
$927$	0	0
$928$	−39017.2	−1.38017
$929$	−20981.5	−0.740993	−0.370496	−	0.928834i	$-0.620812\pi$
$929$	−20981.5	−0.740993	−0.370496	+	0.928834i	$0.620812\pi$
$930$	0	0
$931$	−24499.7	−0.862453
$932$	70774.1	2.48743
$933$	0	0
$934$	−62459.8	−2.18817
$935$	2425.38	0.0848324
$936$	0	0
$937$	21492.1	0.749323	0.374662	−	0.927162i	$-0.377759\pi$
$937$	21492.1	0.749323	0.374662	+	0.927162i	$0.377759\pi$
$938$	−4403.50	−0.153283
$939$	0	0
$940$	−3683.62	−0.127815
$941$	26577.9	0.920740	0.460370	−	0.887727i	$-0.347717\pi$
$941$	26577.9	0.920740	0.460370	+	0.887727i	$0.347717\pi$
$942$	0	0
$943$	−965.893	−0.0333550
$944$	−625.539	−0.0215673
$945$	0	0
$946$	32963.8	1.13292
$947$	7432.76	0.255050	0.127525	−	0.991835i	$-0.459297\pi$
$947$	7432.76	0.255050	0.127525	+	0.991835i	$0.459297\pi$
$948$	0	0
$949$	0	0
$950$	41910.2	1.43131
$951$	0	0
$952$	7917.82	0.269557
$953$	27981.8	0.951121	0.475561	−	0.879683i	$-0.342245\pi$
$953$	27981.8	0.951121	0.475561	+	0.879683i	$0.342245\pi$
$954$	0	0
$955$	1393.71	0.0472246
$956$	−51922.0	−1.75657
$957$	0	0
$958$	−63449.9	−2.13985
$959$	4901.54	0.165046
$960$	0	0
$961$	3931.36	0.131965
$962$	0	0
$963$	0	0
$964$	−42351.2	−1.41498
$965$	−921.892	−0.0307531
$966$	0	0
$967$	43777.1	1.45582	0.727910	−	0.685673i	$-0.240492\pi$
$967$	43777.1	1.45582	0.727910	+	0.685673i	$0.240492\pi$
$968$	22146.7	0.735352
$969$	0	0
$970$	−3071.32	−0.101664
$971$	−4722.08	−0.156065	−0.0780323	−	0.996951i	$-0.524864\pi$
$971$	−4722.08	−0.156065	−0.0780323	+	0.996951i	$0.524864\pi$
$972$	0	0
$973$	−6204.88	−0.204439
$974$	48020.5	1.57975
$975$	0	0
$976$	−666.944	−0.0218733
$977$	22867.5	0.748821	0.374410	−	0.927263i	$-0.377845\pi$
$977$	22867.5	0.748821	0.374410	+	0.927263i	$0.377845\pi$
$978$	0	0
$979$	−27444.5	−0.895946
$980$	−6014.26	−0.196039
$981$	0	0
$982$	−52017.0	−1.69035
$983$	51887.9	1.68359	0.841794	−	0.539799i	$-0.181500\pi$
$983$	51887.9	1.68359	0.841794	+	0.539799i	$0.181500\pi$
$984$	0	0
$985$	−4780.58	−0.154642
$986$	−90123.1	−2.91086
$987$	0	0
$988$	0	0
$989$	2901.79	0.0932977
$990$	0	0
$991$	−26676.4	−0.855099	−0.427550	−	0.903992i	$-0.640623\pi$
$991$	−26676.4	−0.855099	−0.427550	+	0.903992i	$0.640623\pi$
$992$	−33827.1	−1.08267
$993$	0	0
$994$	5164.21	0.164788
$995$	−4525.08	−0.144176
$996$	0	0
$997$	−20542.0	−0.652529	−0.326264	−	0.945279i	$-0.605790\pi$
$997$	−20542.0	−0.652529	−0.326264	+	0.945279i	$0.605790\pi$
$998$	−57849.8	−1.83487
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bn.1.3	yes	18
3.2	odd	2	inner	1521.4.a.bn.1.16	yes	18
13.12	even	2		1521.4.a.bm.1.16	yes	18
39.38	odd	2		1521.4.a.bm.1.3	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1521.4.a.bm.1.3	✓	18	39.38	odd	2
1521.4.a.bm.1.16	yes	18	13.12	even	2
1521.4.a.bn.1.3	yes	18	1.1	even	1	trivial
1521.4.a.bn.1.16	yes	18	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q-4.56788 q^{2} +12.8655 q^{4} +1.42360 q^{5} +3.82462 q^{7} -22.2252 q^{8} +O(q^{10})\)	\(q-4.56788 q^{2} +12.8655 q^{4} +1.42360 q^{5} +3.82462 q^{7} -22.2252 q^{8} -6.50284 q^{10} -18.2903 q^{11} -17.4704 q^{14} -1.40218 q^{16} -93.1475 q^{17} +74.6094 q^{19} +18.3154 q^{20} +83.5478 q^{22} +7.35466 q^{23} -122.973 q^{25} +49.2058 q^{28} -211.812 q^{29} -183.636 q^{31} +184.207 q^{32} +425.487 q^{34} +5.44473 q^{35} +289.095 q^{37} -340.807 q^{38} -31.6398 q^{40} -131.331 q^{41} +394.550 q^{43} -235.314 q^{44} -33.5952 q^{46} -201.122 q^{47} -328.372 q^{49} +561.728 q^{50} -16.7015 q^{53} -26.0380 q^{55} -85.0031 q^{56} +967.531 q^{58} +446.118 q^{59} +475.647 q^{61} +838.830 q^{62} -830.217 q^{64} +252.055 q^{67} -1198.39 q^{68} -24.8709 q^{70} -295.597 q^{71} +892.965 q^{73} -1320.55 q^{74} +959.890 q^{76} -69.9534 q^{77} +170.848 q^{79} -1.99615 q^{80} +599.903 q^{82} -1331.42 q^{83} -132.605 q^{85} -1802.26 q^{86} +406.505 q^{88} +1500.50 q^{89} +94.6217 q^{92} +918.701 q^{94} +106.214 q^{95} +472.304 q^{97} +1499.97 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 56 q^{4} + 94 q^{7}+O(q^{10}) \) 18 * q + 56 * q^4 + 94 * q^7	\( 18 q + 56 q^{4} + 94 q^{7} + 156 q^{10} + 88 q^{16} + 448 q^{19} - 256 q^{22} - 16 q^{25} + 1300 q^{28} + 818 q^{31} + 900 q^{34} + 524 q^{37} + 800 q^{40} + 752 q^{43} + 1650 q^{46} + 2948 q^{49} - 464 q^{55} + 1626 q^{58} - 1784 q^{61} - 4274 q^{64} + 2872 q^{67} + 5772 q^{70} + 3078 q^{73} + 5726 q^{76} + 3326 q^{79} - 1038 q^{82} + 2340 q^{85} + 780 q^{88} + 1220 q^{94} + 4630 q^{97}+O(q^{100}) \) 18 * q + 56 * q^4 + 94 * q^7 + 156 * q^10 + 88 * q^16 + 448 * q^19 - 256 * q^22 - 16 * q^25 + 1300 * q^28 + 818 * q^31 + 900 * q^34 + 524 * q^37 + 800 * q^40 + 752 * q^43 + 1650 * q^46 + 2948 * q^49 - 464 * q^55 + 1626 * q^58 - 1784 * q^61 - 4274 * q^64 + 2872 * q^67 + 5772 * q^70 + 3078 * q^73 + 5726 * q^76 + 3326 * q^79 - 1038 * q^82 + 2340 * q^85 + 780 * q^88 + 1220 * q^94 + 4630 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more