LMFDB - Embedded newform 1323.4.a.q.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-3.82843 q^{2} +6.65685 q^{4} +12.4142 q^{5} +5.14214 q^{8} -47.5269 q^{10} +39.4853 q^{11} -7.95837 q^{13} -72.9411 q^{16} -41.4558 q^{17} +112.113 q^{19} +82.6396 q^{20} -151.167 q^{22} -1.26703 q^{23} +29.1127 q^{25} +30.4680 q^{26} -37.1299 q^{29} +257.291 q^{31} +238.113 q^{32} +158.711 q^{34} -128.331 q^{37} -429.215 q^{38} +63.8356 q^{40} +130.823 q^{41} +493.966 q^{43} +262.848 q^{44} +4.85072 q^{46} -423.754 q^{47} -111.456 q^{50} -52.9777 q^{52} +577.941 q^{53} +490.179 q^{55} +142.149 q^{58} -294.715 q^{59} -67.5147 q^{61} -985.021 q^{62} -328.068 q^{64} -98.7969 q^{65} +987.776 q^{67} -275.966 q^{68} -715.663 q^{71} +340.260 q^{73} +491.306 q^{74} +746.318 q^{76} +81.6123 q^{79} -905.507 q^{80} -500.848 q^{82} +846.409 q^{83} -514.642 q^{85} -1891.11 q^{86} +203.039 q^{88} -1311.42 q^{89} -8.43442 q^{92} +1622.31 q^{94} +1391.79 q^{95} -715.980 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 22 q^{5} - 18 q^{8} - 30 q^{10} + 62 q^{11} - 64 q^{13} - 78 q^{16} - 32 q^{17} + 162 q^{19} + 38 q^{20} - 110 q^{22} + 170 q^{23} - 4 q^{25} - 72 q^{26} - 128 q^{29} + 178 q^{31}+ \cdots - 640 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 22 * q^5 - 18 * q^8 - 30 * q^10 + 62 * q^11 - 64 * q^13 - 78 * q^16 - 32 * q^17 + 162 * q^19 + 38 * q^20 - 110 * q^22 + 170 * q^23 - 4 * q^25 - 72 * q^26 - 128 * q^29 + 178 * q^31 + 414 * q^32 + 176 * q^34 - 350 * q^37 - 338 * q^38 - 158 * q^40 + 58 * q^41 + 756 * q^43 + 158 * q^44 + 318 * q^46 - 180 * q^47 - 172 * q^50 + 208 * q^52 + 1088 * q^53 + 706 * q^55 - 24 * q^58 + 508 * q^59 - 152 * q^61 - 1130 * q^62 + 34 * q^64 - 636 * q^65 + 468 * q^67 - 320 * q^68 + 14 * q^71 + 788 * q^73 + 86 * q^74 + 514 * q^76 - 476 * q^79 - 954 * q^80 - 634 * q^82 + 1492 * q^83 - 424 * q^85 - 1412 * q^86 - 318 * q^88 - 230 * q^89 - 806 * q^92 + 2068 * q^94 + 1870 * q^95 - 640 * q^97

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 $

$2$	−3.82843	−1.35355	−0.676777	−	0.736188i	$-0.736624\pi$
$2$	−3.82843	−1.35355	−0.676777	+	0.736188i	$0.736624\pi$
$3$	0	0
$3$	0	0
$4$	6.65685	0.832107
$5$	12.4142	1.11036	0.555181	−	0.831730i	$-0.312649\pi$
$5$	12.4142	1.11036	0.555181	+	0.831730i	$0.312649\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	5.14214	0.227252
$9$	0	0
$10$	−47.5269	−1.50293
$11$	39.4853	1.08230	0.541148	−	0.840927i	$-0.317990\pi$
$11$	39.4853	1.08230	0.541148	+	0.840927i	$0.317990\pi$
$12$	0	0
$13$	−7.95837	−0.169789	−0.0848944	−	0.996390i	$-0.527055\pi$
$13$	−7.95837	−0.169789	−0.0848944	+	0.996390i	$0.527055\pi$
$14$	0	0
$15$	0	0
$16$	−72.9411	−1.13971
$17$	−41.4558	−0.591442	−0.295721	−	0.955274i	$-0.595560\pi$
$17$	−41.4558	−0.591442	−0.295721	+	0.955274i	$0.595560\pi$
$18$	0	0
$19$	112.113	1.35371	0.676853	−	0.736118i	$-0.263343\pi$
$19$	112.113	1.35371	0.676853	+	0.736118i	$0.263343\pi$
$20$	82.6396	0.923939
$21$	0	0
$22$	−151.167	−1.46495
$23$	−1.26703	−0.0114867	−0.00574334	−	0.999984i	$-0.501828\pi$
$23$	−1.26703	−0.0114867	−0.00574334	+	0.999984i	$0.501828\pi$
$24$	0	0
$25$	29.1127	0.232902
$26$	30.4680	0.229818
$27$	0	0
$28$	0	0
$29$	−37.1299	−0.237754	−0.118877	−	0.992909i	$-0.537929\pi$
$29$	−37.1299	−0.237754	−0.118877	+	0.992909i	$0.537929\pi$
$30$	0	0
$31$	257.291	1.49067	0.745337	−	0.666688i	$-0.232289\pi$
$31$	257.291	1.49067	0.745337	+	0.666688i	$0.232289\pi$
$32$	238.113	1.31540
$33$	0	0
$34$	158.711	0.800549
$35$	0	0
$36$	0	0
$37$	−128.331	−0.570202	−0.285101	−	0.958497i	$-0.592027\pi$
$37$	−128.331	−0.570202	−0.285101	+	0.958497i	$0.592027\pi$
$38$	−429.215	−1.83231
$39$	0	0
$40$	63.8356	0.252332
$41$	130.823	0.498321	0.249161	−	0.968462i	$-0.419845\pi$
$41$	130.823	0.498321	0.249161	+	0.968462i	$0.419845\pi$
$42$	0	0
$43$	493.966	1.75184	0.875919	−	0.482458i	$-0.160256\pi$
$43$	493.966	1.75184	0.875919	+	0.482458i	$0.160256\pi$
$44$	262.848	0.900586
$45$	0	0
$46$	4.85072	0.0155478
$47$	−423.754	−1.31513	−0.657563	−	0.753399i	$-0.728413\pi$
$47$	−423.754	−1.31513	−0.657563	+	0.753399i	$0.728413\pi$
$48$	0	0
$49$	0	0
$50$	−111.456	−0.315245
$51$	0	0
$52$	−52.9777	−0.141282
$53$	577.941	1.49786	0.748928	−	0.662652i	$-0.230569\pi$
$53$	577.941	1.49786	0.748928	+	0.662652i	$0.230569\pi$
$54$	0	0
$55$	490.179	1.20174
$56$	0	0
$57$	0	0
$58$	142.149	0.321812
$59$	−294.715	−0.650315	−0.325158	−	0.945660i	$-0.605417\pi$
$59$	−294.715	−0.650315	−0.325158	+	0.945660i	$0.605417\pi$
$60$	0	0
$61$	−67.5147	−0.141711	−0.0708555	−	0.997487i	$-0.522573\pi$
$61$	−67.5147	−0.141711	−0.0708555	+	0.997487i	$0.522573\pi$
$62$	−985.021	−2.01771
$63$	0	0
$64$	−328.068	−0.640758
$65$	−98.7969	−0.188527
$66$	0	0
$67$	987.776	1.80113	0.900567	−	0.434717i	$-0.143151\pi$
$67$	987.776	1.80113	0.900567	+	0.434717i	$0.143151\pi$
$68$	−275.966	−0.492143
$69$	0	0
$70$	0	0
$71$	−715.663	−1.19625	−0.598124	−	0.801404i	$-0.704087\pi$
$71$	−715.663	−1.19625	−0.598124	+	0.801404i	$0.704087\pi$
$72$	0	0
$73$	340.260	0.545540	0.272770	−	0.962079i	$-0.412060\pi$
$73$	340.260	0.545540	0.272770	+	0.962079i	$0.412060\pi$
$74$	491.306	0.771799
$75$	0	0
$76$	746.318	1.12643
$77$	0	0
$78$	0	0
$79$	81.6123	0.116229	0.0581145	−	0.998310i	$-0.481491\pi$
$79$	81.6123	0.116229	0.0581145	+	0.998310i	$0.481491\pi$
$80$	−905.507	−1.26548
$81$	0	0
$82$	−500.848	−0.674505
$83$	846.409	1.11934	0.559672	−	0.828715i	$-0.310927\pi$
$83$	846.409	1.11934	0.559672	+	0.828715i	$0.310927\pi$
$84$	0	0
$85$	−514.642	−0.656714
$86$	−1891.11	−2.37121
$87$	0	0
$88$	203.039	0.245954
$89$	−1311.42	−1.56192	−0.780959	−	0.624582i	$-0.785269\pi$
$89$	−1311.42	−1.56192	−0.780959	+	0.624582i	$0.785269\pi$
$90$	0	0
$91$	0	0
$92$	−8.43442	−0.00955814
$93$	0	0
$94$	1622.31	1.78009
$95$	1391.79	1.50310
$96$	0	0
$97$	−715.980	−0.749451	−0.374725	−	0.927136i	$-0.622263\pi$
$97$	−715.980	−0.749451	−0.374725	+	0.927136i	$0.622263\pi$
$98$	0	0
$99$	0	0
$100$	193.799	0.193799
$101$	766.230	0.754878	0.377439	−	0.926034i	$-0.376805\pi$
$101$	766.230	0.754878	0.377439	+	0.926034i	$0.376805\pi$
$102$	0	0
$103$	1088.57	1.04136	0.520678	−	0.853753i	$-0.325679\pi$
$103$	1088.57	1.04136	0.520678	+	0.853753i	$0.325679\pi$
$104$	−40.9230	−0.0385849
$105$	0	0
$106$	−2212.61	−2.02743
$107$	−1146.87	−1.03619	−0.518095	−	0.855323i	$-0.673359\pi$
$107$	−1146.87	−1.03619	−0.518095	+	0.855323i	$0.673359\pi$
$108$	0	0
$109$	−1055.09	−0.927150	−0.463575	−	0.886058i	$-0.653434\pi$
$109$	−1055.09	−0.927150	−0.463575	+	0.886058i	$0.653434\pi$
$110$	−1876.61	−1.62662
$111$	0	0
$112$	0	0
$113$	514.296	0.428149	0.214075	−	0.976817i	$-0.431326\pi$
$113$	514.296	0.428149	0.214075	+	0.976817i	$0.431326\pi$
$114$	0	0
$115$	−15.7291	−0.0127544
$116$	−247.169	−0.197836
$117$	0	0
$118$	1128.29	0.880237
$119$	0	0
$120$	0	0
$121$	228.087	0.171365
$122$	258.475	0.191813
$123$	0	0
$124$	1712.75	1.24040
$125$	−1190.37	−0.851756
$126$	0	0
$127$	−1906.50	−1.33208	−0.666041	−	0.745915i	$-0.732012\pi$
$127$	−1906.50	−1.33208	−0.666041	+	0.745915i	$0.732012\pi$
$128$	−648.917	−0.448099
$129$	0	0
$130$	378.237	0.255181
$131$	−683.209	−0.455666	−0.227833	−	0.973700i	$-0.573164\pi$
$131$	−683.209	−0.455666	−0.227833	+	0.973700i	$0.573164\pi$
$132$	0	0
$133$	0	0
$134$	−3781.63	−2.43793
$135$	0	0
$136$	−213.172	−0.134407
$137$	2558.84	1.59574	0.797870	−	0.602829i	$-0.205960\pi$
$137$	2558.84	1.59574	0.797870	+	0.602829i	$0.205960\pi$
$138$	0	0
$139$	−2864.87	−1.74817	−0.874084	−	0.485774i	$-0.838538\pi$
$139$	−2864.87	−1.74817	−0.874084	+	0.485774i	$0.838538\pi$
$140$	0	0
$141$	0	0
$142$	2739.86	1.61919
$143$	−314.238	−0.183762
$144$	0	0
$145$	−460.939	−0.263992
$146$	−1302.66	−0.738417
$147$	0	0
$148$	−854.280	−0.474469
$149$	−1924.34	−1.05804	−0.529020	−	0.848609i	$-0.677440\pi$
$149$	−1924.34	−1.05804	−0.529020	+	0.848609i	$0.677440\pi$
$150$	0	0
$151$	2036.83	1.09771	0.548857	−	0.835916i	$-0.315063\pi$
$151$	2036.83	1.09771	0.548857	+	0.835916i	$0.315063\pi$
$152$	576.499	0.307633
$153$	0	0
$154$	0	0
$155$	3194.07	1.65519
$156$	0	0
$157$	1761.91	0.895639	0.447820	−	0.894124i	$-0.352201\pi$
$157$	1761.91	0.895639	0.447820	+	0.894124i	$0.352201\pi$
$158$	−312.447	−0.157322
$159$	0	0
$160$	2955.98	1.46057
$161$	0	0
$162$	0	0
$163$	3386.88	1.62749	0.813746	−	0.581221i	$-0.197425\pi$
$163$	3386.88	1.62749	0.813746	+	0.581221i	$0.197425\pi$
$164$	870.872	0.414657
$165$	0	0
$166$	−3240.42	−1.51509
$167$	1536.82	0.712114	0.356057	−	0.934464i	$-0.384121\pi$
$167$	1536.82	0.712114	0.356057	+	0.934464i	$0.384121\pi$
$168$	0	0
$169$	−2133.66	−0.971172
$170$	1970.27	0.888898
$171$	0	0
$172$	3288.26	1.45772
$173$	−3428.13	−1.50657	−0.753284	−	0.657696i	$-0.771531\pi$
$173$	−3428.13	−1.50657	−0.753284	+	0.657696i	$0.771531\pi$
$174$	0	0
$175$	0	0
$176$	−2880.10	−1.23350
$177$	0	0
$178$	5020.69	2.11414
$179$	−2456.95	−1.02593	−0.512964	−	0.858410i	$-0.671452\pi$
$179$	−2456.95	−1.02593	−0.512964	+	0.858410i	$0.671452\pi$
$180$	0	0
$181$	4389.03	1.80240	0.901199	−	0.433405i	$-0.142688\pi$
$181$	4389.03	1.80240	0.901199	+	0.433405i	$0.142688\pi$
$182$	0	0
$183$	0	0
$184$	−6.51523	−0.00261037
$185$	−1593.13	−0.633130
$186$	0	0
$187$	−1636.90	−0.640116
$188$	−2820.87	−1.09433
$189$	0	0
$190$	−5328.37	−2.03453
$191$	2997.74	1.13565	0.567824	−	0.823150i	$-0.307785\pi$
$191$	2997.74	1.13565	0.567824	+	0.823150i	$0.307785\pi$
$192$	0	0
$193$	−2029.00	−0.756738	−0.378369	−	0.925655i	$-0.623515\pi$
$193$	−2029.00	−0.756738	−0.378369	+	0.925655i	$0.623515\pi$
$194$	2741.08	1.01442
$195$	0	0
$196$	0	0
$197$	−1030.86	−0.372820	−0.186410	−	0.982472i	$-0.559685\pi$
$197$	−1030.86	−0.372820	−0.186410	+	0.982472i	$0.559685\pi$
$198$	0	0
$199$	1674.99	0.596667	0.298334	−	0.954462i	$-0.403569\pi$
$199$	1674.99	0.596667	0.298334	+	0.954462i	$0.403569\pi$
$200$	149.701	0.0529275
$201$	0	0
$202$	−2933.45	−1.02177
$203$	0	0
$204$	0	0
$205$	1624.07	0.553317
$206$	−4167.50	−1.40953
$207$	0	0
$208$	580.492	0.193509
$209$	4426.80	1.46511
$210$	0	0
$211$	1693.66	0.552588	0.276294	−	0.961073i	$-0.410894\pi$
$211$	1693.66	0.552588	0.276294	+	0.961073i	$0.410894\pi$
$212$	3847.27	1.24638
$213$	0	0
$214$	4390.72	1.40254
$215$	6132.19	1.94517
$216$	0	0
$217$	0	0
$218$	4039.34	1.25495
$219$	0	0
$220$	3263.05	0.999976
$221$	329.921	0.100420
$222$	0	0
$223$	3754.40	1.12741	0.563707	−	0.825975i	$-0.309375\pi$
$223$	3754.40	1.12741	0.563707	+	0.825975i	$0.309375\pi$
$224$	0	0
$225$	0	0
$226$	−1968.94	−0.579523
$227$	3183.35	0.930778	0.465389	−	0.885106i	$-0.345914\pi$
$227$	3183.35	0.930778	0.465389	+	0.885106i	$0.345914\pi$
$228$	0	0
$229$	1956.24	0.564505	0.282253	−	0.959340i	$-0.408918\pi$
$229$	1956.24	0.564505	0.282253	+	0.959340i	$0.408918\pi$
$230$	60.2179	0.0172637
$231$	0	0
$232$	−190.927	−0.0540301
$233$	−4683.36	−1.31681	−0.658406	−	0.752663i	$-0.728769\pi$
$233$	−4683.36	−1.31681	−0.658406	+	0.752663i	$0.728769\pi$
$234$	0	0
$235$	−5260.58	−1.46026
$236$	−1961.87	−0.541132
$237$	0	0
$238$	0	0
$239$	5636.08	1.52539	0.762693	−	0.646760i	$-0.223877\pi$
$239$	5636.08	1.52539	0.762693	+	0.646760i	$0.223877\pi$
$240$	0	0
$241$	−2048.77	−0.547606	−0.273803	−	0.961786i	$-0.588282\pi$
$241$	−2048.77	−0.547606	−0.273803	+	0.961786i	$0.588282\pi$
$242$	−873.216	−0.231952
$243$	0	0
$244$	−449.436	−0.117919
$245$	0	0
$246$	0	0
$247$	−892.234	−0.229844
$248$	1323.03	0.338759
$249$	0	0
$250$	4557.23	1.15290
$251$	3702.47	0.931066	0.465533	−	0.885031i	$-0.345863\pi$
$251$	3702.47	0.931066	0.465533	+	0.885031i	$0.345863\pi$
$252$	0	0
$253$	−50.0289	−0.0124320
$254$	7298.89	1.80304
$255$	0	0
$256$	5108.88	1.24728
$257$	5968.05	1.44855	0.724274	−	0.689512i	$-0.242175\pi$
$257$	5968.05	1.44855	0.724274	+	0.689512i	$0.242175\pi$
$258$	0	0
$259$	0	0
$260$	−657.677	−0.156874
$261$	0	0
$262$	2615.61	0.616768
$263$	−3258.25	−0.763926	−0.381963	−	0.924178i	$-0.624752\pi$
$263$	−3258.25	−0.763926	−0.381963	+	0.924178i	$0.624752\pi$
$264$	0	0
$265$	7174.68	1.66316
$266$	0	0
$267$	0	0
$268$	6575.48	1.49874
$269$	1330.77	0.301630	0.150815	−	0.988562i	$-0.451810\pi$
$269$	1330.77	0.301630	0.150815	+	0.988562i	$0.451810\pi$
$270$	0	0
$271$	4821.61	1.08078	0.540391	−	0.841414i	$-0.318276\pi$
$271$	4821.61	1.08078	0.540391	+	0.841414i	$0.318276\pi$
$272$	3023.84	0.674070
$273$	0	0
$274$	−9796.33	−2.15992
$275$	1149.52	0.252069
$276$	0	0
$277$	2772.23	0.601324	0.300662	−	0.953731i	$-0.402792\pi$
$277$	2772.23	0.601324	0.300662	+	0.953731i	$0.402792\pi$
$278$	10968.0	2.36624
$279$	0	0
$280$	0	0
$281$	8873.74	1.88385	0.941927	−	0.335817i	$-0.109012\pi$
$281$	8873.74	1.88385	0.941927	+	0.335817i	$0.109012\pi$
$282$	0	0
$283$	7071.33	1.48532	0.742662	−	0.669666i	$-0.233563\pi$
$283$	7071.33	1.48532	0.742662	+	0.669666i	$0.233563\pi$
$284$	−4764.07	−0.995406
$285$	0	0
$286$	1203.04	0.248731
$287$	0	0
$288$	0	0
$289$	−3194.41	−0.650196
$290$	1764.67	0.357328
$291$	0	0
$292$	2265.06	0.453947
$293$	−3349.72	−0.667892	−0.333946	−	0.942592i	$-0.608380\pi$
$293$	−3349.72	−0.667892	−0.333946	+	0.942592i	$0.608380\pi$
$294$	0	0
$295$	−3658.65	−0.722085
$296$	−659.895	−0.129580
$297$	0	0
$298$	7367.19	1.43211
$299$	10.0835	0.00195031
$300$	0	0
$301$	0	0
$302$	−7797.85	−1.48581
$303$	0	0
$304$	−8177.63	−1.54283
$305$	−838.142	−0.157350
$306$	0	0
$307$	−812.535	−0.151055	−0.0755274	−	0.997144i	$-0.524064\pi$
$307$	−812.535	−0.151055	−0.0755274	+	0.997144i	$0.524064\pi$
$308$	0	0
$309$	0	0
$310$	−12228.3	−2.24038
$311$	−1128.24	−0.205712	−0.102856	−	0.994696i	$-0.532798\pi$
$311$	−1128.24	−0.205712	−0.102856	+	0.994696i	$0.532798\pi$
$312$	0	0
$313$	3454.94	0.623913	0.311956	−	0.950096i	$-0.399016\pi$
$313$	3454.94	0.623913	0.311956	+	0.950096i	$0.399016\pi$
$314$	−6745.33	−1.21230
$315$	0	0
$316$	543.281	0.0967150
$317$	2710.05	0.480163	0.240081	−	0.970753i	$-0.422826\pi$
$317$	2710.05	0.480163	0.240081	+	0.970753i	$0.422826\pi$
$318$	0	0
$319$	−1466.09	−0.257320
$320$	−4072.71	−0.711473
$321$	0	0
$322$	0	0
$323$	−4647.73	−0.800639
$324$	0	0
$325$	−231.690	−0.0395441
$326$	−12966.4	−2.20290
$327$	0	0
$328$	672.712	0.113245
$329$	0	0
$330$	0	0
$331$	−4511.86	−0.749228	−0.374614	−	0.927181i	$-0.622225\pi$
$331$	−4511.86	−0.749228	−0.374614	+	0.927181i	$0.622225\pi$
$332$	5634.42	0.931413
$333$	0	0
$334$	−5883.62	−0.963884
$335$	12262.5	1.99991
$336$	0	0
$337$	−159.558	−0.0257913	−0.0128956	−	0.999917i	$-0.504105\pi$
$337$	−159.558	−0.0257913	−0.0128956	+	0.999917i	$0.504105\pi$
$338$	8168.58	1.31453
$339$	0	0
$340$	−3425.89	−0.546457
$341$	10159.2	1.61335
$342$	0	0
$343$	0	0
$344$	2540.04	0.398109
$345$	0	0
$346$	13124.4	2.03922
$347$	−9581.15	−1.48226	−0.741128	−	0.671364i	$-0.765709\pi$
$347$	−9581.15	−1.48226	−0.741128	+	0.671364i	$0.765709\pi$
$348$	0	0
$349$	−3871.09	−0.593738	−0.296869	−	0.954918i	$-0.595942\pi$
$349$	−3871.09	−0.593738	−0.296869	+	0.954918i	$0.595942\pi$
$350$	0	0
$351$	0	0
$352$	9401.95	1.42365
$353$	11436.0	1.72430	0.862151	−	0.506651i	$-0.169117\pi$
$353$	11436.0	1.72430	0.862151	+	0.506651i	$0.169117\pi$
$354$	0	0
$355$	−8884.39	−1.32827
$356$	−8729.96	−1.29968
$357$	0	0
$358$	9406.25	1.38865
$359$	−10320.2	−1.51721	−0.758604	−	0.651552i	$-0.774118\pi$
$359$	−10320.2	−1.51721	−0.758604	+	0.651552i	$0.774118\pi$
$360$	0	0
$361$	5710.26	0.832520
$362$	−16803.1	−2.43964
$363$	0	0
$364$	0	0
$365$	4224.06	0.605746
$366$	0	0
$367$	−3489.21	−0.496281	−0.248140	−	0.968724i	$-0.579819\pi$
$367$	−3489.21	−0.496281	−0.248140	+	0.968724i	$0.579819\pi$
$368$	92.4184	0.0130914
$369$	0	0
$370$	6099.17	0.856976
$371$	0	0
$372$	0	0
$373$	1829.43	0.253952	0.126976	−	0.991906i	$-0.459473\pi$
$373$	1829.43	0.253952	0.126976	+	0.991906i	$0.459473\pi$
$374$	6266.74	0.866431
$375$	0	0
$376$	−2179.00	−0.298866
$377$	295.494	0.0403679
$378$	0	0
$379$	11638.7	1.57742	0.788709	−	0.614767i	$-0.210750\pi$
$379$	11638.7	1.57742	0.788709	+	0.614767i	$0.210750\pi$
$380$	9264.95	1.25074
$381$	0	0
$382$	−11476.6	−1.53716
$383$	6536.92	0.872118	0.436059	−	0.899918i	$-0.356374\pi$
$383$	6536.92	0.872118	0.436059	+	0.899918i	$0.356374\pi$
$384$	0	0
$385$	0	0
$386$	7767.87	1.02428
$387$	0	0
$388$	−4766.17	−0.623623
$389$	10675.0	1.39137	0.695685	−	0.718347i	$-0.255101\pi$
$389$	10675.0	1.39137	0.695685	+	0.718347i	$0.255101\pi$
$390$	0	0
$391$	52.5257	0.00679370
$392$	0	0
$393$	0	0
$394$	3946.56	0.504631
$395$	1013.15	0.129056
$396$	0	0
$397$	7093.58	0.896767	0.448384	−	0.893841i	$-0.352000\pi$
$397$	7093.58	0.896767	0.448384	+	0.893841i	$0.352000\pi$
$398$	−6412.57	−0.807621
$399$	0	0
$400$	−2123.51	−0.265439
$401$	−6753.23	−0.840999	−0.420499	−	0.907293i	$-0.638145\pi$
$401$	−6753.23	−0.840999	−0.420499	+	0.907293i	$0.638145\pi$
$402$	0	0
$403$	−2047.62	−0.253100
$404$	5100.68	0.628139
$405$	0	0
$406$	0	0
$407$	−5067.18	−0.617128
$408$	0	0
$409$	5081.62	0.614352	0.307176	−	0.951653i	$-0.400616\pi$
$409$	5081.62	0.614352	0.307176	+	0.951653i	$0.400616\pi$
$410$	−6217.63	−0.748944
$411$	0	0
$412$	7246.43	0.866519
$413$	0	0
$414$	0	0
$415$	10507.5	1.24287
$416$	−1894.99	−0.223340
$417$	0	0
$418$	−16947.7	−1.98311
$419$	−985.533	−0.114908	−0.0574540	−	0.998348i	$-0.518298\pi$
$419$	−985.533	−0.114908	−0.0574540	+	0.998348i	$0.518298\pi$
$420$	0	0
$421$	3342.33	0.386924	0.193462	−	0.981108i	$-0.438028\pi$
$421$	3342.33	0.386924	0.193462	+	0.981108i	$0.438028\pi$
$422$	−6484.04	−0.747957
$423$	0	0
$424$	2971.85	0.340391
$425$	−1206.89	−0.137748
$426$	0	0
$427$	0	0
$428$	−7634.57	−0.862221
$429$	0	0
$430$	−23476.7	−2.63290
$431$	−5799.54	−0.648153	−0.324076	−	0.946031i	$-0.605053\pi$
$431$	−5799.54	−0.648153	−0.324076	+	0.946031i	$0.605053\pi$
$432$	0	0
$433$	−15279.1	−1.69576	−0.847881	−	0.530187i	$-0.822122\pi$
$433$	−15279.1	−1.69576	−0.847881	+	0.530187i	$0.822122\pi$
$434$	0	0
$435$	0	0
$436$	−7023.58	−0.771488
$437$	−142.050	−0.0155496
$438$	0	0
$439$	15279.9	1.66120	0.830602	−	0.556866i	$-0.187996\pi$
$439$	15279.9	1.66120	0.830602	+	0.556866i	$0.187996\pi$
$440$	2520.57	0.273098
$441$	0	0
$442$	−1263.08	−0.135924
$443$	−7466.99	−0.800829	−0.400415	−	0.916334i	$-0.631134\pi$
$443$	−7466.99	−0.800829	−0.400415	+	0.916334i	$0.631134\pi$
$444$	0	0
$445$	−16280.3	−1.73429
$446$	−14373.4	−1.52601
$447$	0	0
$448$	0	0
$449$	−8177.88	−0.859550	−0.429775	−	0.902936i	$-0.641407\pi$
$449$	−8177.88	−0.859550	−0.429775	+	0.902936i	$0.641407\pi$
$450$	0	0
$451$	5165.60	0.539331
$452$	3423.59	0.356266
$453$	0	0
$454$	−12187.2	−1.25986
$455$	0	0
$456$	0	0
$457$	15392.7	1.57558	0.787792	−	0.615941i	$-0.211224\pi$
$457$	15392.7	1.57558	0.787792	+	0.615941i	$0.211224\pi$
$458$	−7489.31	−0.764088
$459$	0	0
$460$	−104.707	−0.0106130
$461$	−1000.16	−0.101046	−0.0505228	−	0.998723i	$-0.516089\pi$
$461$	−1000.16	−0.101046	−0.0505228	+	0.998723i	$0.516089\pi$
$462$	0	0
$463$	−10927.3	−1.09684	−0.548420	−	0.836203i	$-0.684770\pi$
$463$	−10927.3	−1.09684	−0.548420	+	0.836203i	$0.684770\pi$
$464$	2708.30	0.270969
$465$	0	0
$466$	17929.9	1.78238
$467$	7131.43	0.706645	0.353322	−	0.935502i	$-0.385052\pi$
$467$	7131.43	0.706645	0.353322	+	0.935502i	$0.385052\pi$
$468$	0	0
$469$	0	0
$470$	20139.7	1.97655
$471$	0	0
$472$	−1515.46	−0.147786
$473$	19504.4	1.89601
$474$	0	0
$475$	3263.90	0.315280
$476$	0	0
$477$	0	0
$478$	−21577.3	−2.06469
$479$	−1517.76	−0.144777	−0.0723887	−	0.997376i	$-0.523062\pi$
$479$	−1517.76	−0.144777	−0.0723887	+	0.997376i	$0.523062\pi$
$480$	0	0
$481$	1021.31	0.0968139
$482$	7843.58	0.741214
$483$	0	0
$484$	1518.34	0.142594
$485$	−8888.33	−0.832161
$486$	0	0
$487$	−7631.37	−0.710083	−0.355042	−	0.934851i	$-0.615533\pi$
$487$	−7631.37	−0.710083	−0.355042	+	0.934851i	$0.615533\pi$
$488$	−347.170	−0.0322042
$489$	0	0
$490$	0	0
$491$	−504.070	−0.0463307	−0.0231653	−	0.999732i	$-0.507374\pi$
$491$	−504.070	−0.0463307	−0.0231653	+	0.999732i	$0.507374\pi$
$492$	0	0
$493$	1539.25	0.140618
$494$	3415.85	0.311106
$495$	0	0
$496$	−18767.1	−1.69893
$497$	0	0
$498$	0	0
$499$	−7564.63	−0.678636	−0.339318	−	0.940672i	$-0.610196\pi$
$499$	−7564.63	−0.678636	−0.339318	+	0.940672i	$0.610196\pi$
$500$	−7924.09	−0.708752
$501$	0	0
$502$	−14174.6	−1.26025
$503$	18408.9	1.63184	0.815919	−	0.578166i	$-0.196232\pi$
$503$	18408.9	1.63184	0.815919	+	0.578166i	$0.196232\pi$
$504$	0	0
$505$	9512.14	0.838187
$506$	191.532	0.0168274
$507$	0	0
$508$	−12691.3	−1.10843
$509$	6253.92	0.544598	0.272299	−	0.962213i	$-0.412216\pi$
$509$	6253.92	0.544598	0.272299	+	0.962213i	$0.412216\pi$
$510$	0	0
$511$	0	0
$512$	−14367.6	−1.24017
$513$	0	0
$514$	−22848.2	−1.96069
$515$	13513.7	1.15628
$516$	0	0
$517$	−16732.1	−1.42336
$518$	0	0
$519$	0	0
$520$	−508.027	−0.0428432
$521$	−17119.3	−1.43955	−0.719777	−	0.694205i	$-0.755756\pi$
$521$	−17119.3	−1.43955	−0.719777	+	0.694205i	$0.755756\pi$
$522$	0	0
$523$	−260.727	−0.0217988	−0.0108994	−	0.999941i	$-0.503469\pi$
$523$	−260.727	−0.0217988	−0.0108994	+	0.999941i	$0.503469\pi$
$524$	−4548.02	−0.379163
$525$	0	0
$526$	12474.0	1.03401
$527$	−10666.2	−0.881648
$528$	0	0
$529$	−12165.4	−0.999868
$530$	−27467.8	−2.25118
$531$	0	0
$532$	0	0
$533$	−1041.14	−0.0846094
$534$	0	0
$535$	−14237.5	−1.15055
$536$	5079.28	0.409312
$537$	0	0
$538$	−5094.75	−0.408272
$539$	0	0
$540$	0	0
$541$	1950.11	0.154976	0.0774878	−	0.996993i	$-0.475310\pi$
$541$	1950.11	0.154976	0.0774878	+	0.996993i	$0.475310\pi$
$542$	−18459.2	−1.46290
$543$	0	0
$544$	−9871.16	−0.777983
$545$	−13098.1	−1.02947
$546$	0	0
$547$	5871.97	0.458990	0.229495	−	0.973310i	$-0.426293\pi$
$547$	5871.97	0.458990	0.229495	+	0.973310i	$0.426293\pi$
$548$	17033.8	1.32783
$549$	0	0
$550$	−4400.87	−0.341188
$551$	−4162.74	−0.321849
$552$	0	0
$553$	0	0
$554$	−10613.3	−0.813925
$555$	0	0
$556$	−19071.0	−1.45466
$557$	13538.6	1.02989	0.514944	−	0.857224i	$-0.327813\pi$
$557$	13538.6	1.02989	0.514944	+	0.857224i	$0.327813\pi$
$558$	0	0
$559$	−3931.16	−0.297442
$560$	0	0
$561$	0	0
$562$	−33972.5	−2.54990
$563$	−6173.58	−0.462141	−0.231071	−	0.972937i	$-0.574223\pi$
$563$	−6173.58	−0.462141	−0.231071	+	0.972937i	$0.574223\pi$
$564$	0	0
$565$	6384.58	0.475400
$566$	−27072.1	−2.01047
$567$	0	0
$568$	−3680.04	−0.271850
$569$	3100.80	0.228457	0.114229	−	0.993454i	$-0.463560\pi$
$569$	3100.80	0.228457	0.114229	+	0.993454i	$0.463560\pi$
$570$	0	0
$571$	−17731.1	−1.29952	−0.649758	−	0.760141i	$-0.725130\pi$
$571$	−17731.1	−1.29952	−0.649758	+	0.760141i	$0.725130\pi$
$572$	−2091.84	−0.152909
$573$	0	0
$574$	0	0
$575$	−36.8866	−0.00267526
$576$	0	0
$577$	8204.13	0.591928	0.295964	−	0.955199i	$-0.404359\pi$
$577$	8204.13	0.591928	0.295964	+	0.955199i	$0.404359\pi$
$578$	12229.6	0.880075
$579$	0	0
$580$	−3068.40	−0.219670
$581$	0	0
$582$	0	0
$583$	22820.2	1.62112
$584$	1749.66	0.123975
$585$	0	0
$586$	12824.1	0.904028
$587$	−4420.79	−0.310844	−0.155422	−	0.987848i	$-0.549674\pi$
$587$	−4420.79	−0.310844	−0.155422	+	0.987848i	$0.549674\pi$
$588$	0	0
$589$	28845.6	2.01793
$590$	14006.9	0.977380
$591$	0	0
$592$	9360.60	0.649862
$593$	−12927.5	−0.895228	−0.447614	−	0.894227i	$-0.647726\pi$
$593$	−12927.5	−0.895228	−0.447614	+	0.894227i	$0.647726\pi$
$594$	0	0
$595$	0	0
$596$	−12810.0	−0.880402
$597$	0	0
$598$	−38.6038	−0.00263985
$599$	6766.30	0.461542	0.230771	−	0.973008i	$-0.425875\pi$
$599$	6766.30	0.461542	0.230771	+	0.973008i	$0.425875\pi$
$600$	0	0
$601$	13167.0	0.893667	0.446834	−	0.894617i	$-0.352552\pi$
$601$	13167.0	0.893667	0.446834	+	0.894617i	$0.352552\pi$
$602$	0	0
$603$	0	0
$604$	13558.9	0.913415
$605$	2831.53	0.190278
$606$	0	0
$607$	2850.33	0.190595	0.0952977	−	0.995449i	$-0.469620\pi$
$607$	2850.33	0.190595	0.0952977	+	0.995449i	$0.469620\pi$
$608$	26695.5	1.78066
$609$	0	0
$610$	3208.77	0.212982
$611$	3372.39	0.223294
$612$	0	0
$613$	18140.6	1.19526	0.597628	−	0.801774i	$-0.296110\pi$
$613$	18140.6	1.19526	0.597628	+	0.801774i	$0.296110\pi$
$614$	3110.73	0.204461
$615$	0	0
$616$	0	0
$617$	14183.6	0.925462	0.462731	−	0.886499i	$-0.346870\pi$
$617$	14183.6	0.925462	0.462731	+	0.886499i	$0.346870\pi$
$618$	0	0
$619$	−1427.97	−0.0927220	−0.0463610	−	0.998925i	$-0.514762\pi$
$619$	−1427.97	−0.0927220	−0.0463610	+	0.998925i	$0.514762\pi$
$620$	21262.5	1.37729
$621$	0	0
$622$	4319.38	0.278442
$623$	0	0
$624$	0	0
$625$	−18416.5	−1.17866
$626$	−13227.0	−0.844499
$627$	0	0
$628$	11728.7	0.745267
$629$	5320.07	0.337242
$630$	0	0
$631$	12501.4	0.788703	0.394352	−	0.918960i	$-0.370969\pi$
$631$	12501.4	0.788703	0.394352	+	0.918960i	$0.370969\pi$
$632$	419.661	0.0264133
$633$	0	0
$634$	−10375.2	−0.649926
$635$	−23667.7	−1.47909
$636$	0	0
$637$	0	0
$638$	5612.80	0.348296
$639$	0	0
$640$	−8055.79	−0.497552
$641$	564.946	0.0348113	0.0174056	−	0.999849i	$-0.494459\pi$
$641$	564.946	0.0348113	0.0174056	+	0.999849i	$0.494459\pi$
$642$	0	0
$643$	8594.12	0.527090	0.263545	−	0.964647i	$-0.415108\pi$
$643$	8594.12	0.527090	0.263545	+	0.964647i	$0.415108\pi$
$644$	0	0
$645$	0	0
$646$	17793.5	1.08371
$647$	−17721.5	−1.07682	−0.538410	−	0.842683i	$-0.680975\pi$
$647$	−17721.5	−1.07682	−0.538410	+	0.842683i	$0.680975\pi$
$648$	0	0
$649$	−11636.9	−0.703834
$650$	887.007	0.0535250
$651$	0	0
$652$	22546.0	1.35425
$653$	−31721.2	−1.90099	−0.950496	−	0.310738i	$-0.899424\pi$
$653$	−31721.2	−1.90099	−0.950496	+	0.310738i	$0.899424\pi$
$654$	0	0
$655$	−8481.50	−0.505953
$656$	−9542.40	−0.567939
$657$	0	0
$658$	0	0
$659$	−31116.1	−1.83932	−0.919659	−	0.392718i	$-0.871535\pi$
$659$	−31116.1	−1.83932	−0.919659	+	0.392718i	$0.871535\pi$
$660$	0	0
$661$	22782.0	1.34057	0.670285	−	0.742104i	$-0.266172\pi$
$661$	22782.0	1.34057	0.670285	+	0.742104i	$0.266172\pi$
$662$	17273.3	1.01412
$663$	0	0
$664$	4352.35	0.254373
$665$	0	0
$666$	0	0
$667$	47.0447	0.00273100
$668$	10230.4	0.592555
$669$	0	0
$670$	−46945.9	−2.70698
$671$	−2665.84	−0.153373
$672$	0	0
$673$	17281.0	0.989799	0.494900	−	0.868950i	$-0.335205\pi$
$673$	17281.0	0.989799	0.494900	+	0.868950i	$0.335205\pi$
$674$	610.855	0.0349099
$675$	0	0
$676$	−14203.5	−0.808119
$677$	22273.3	1.26445	0.632224	−	0.774786i	$-0.282142\pi$
$677$	22273.3	1.26445	0.632224	+	0.774786i	$0.282142\pi$
$678$	0	0
$679$	0	0
$680$	−2646.36	−0.149240
$681$	0	0
$682$	−38893.8	−2.18376
$683$	−18801.5	−1.05332	−0.526662	−	0.850075i	$-0.676557\pi$
$683$	−18801.5	−1.05332	−0.526662	+	0.850075i	$0.676557\pi$
$684$	0	0
$685$	31766.0	1.77185
$686$	0	0
$687$	0	0
$688$	−36030.4	−1.99658
$689$	−4599.47	−0.254319
$690$	0	0
$691$	9169.88	0.504831	0.252416	−	0.967619i	$-0.418775\pi$
$691$	9169.88	0.504831	0.252416	+	0.967619i	$0.418775\pi$
$692$	−22820.6	−1.25363
$693$	0	0
$694$	36680.7	2.00631
$695$	−35565.2	−1.94110
$696$	0	0
$697$	−5423.39	−0.294728
$698$	14820.2	0.803656
$699$	0	0
$700$	0	0
$701$	446.611	0.0240631	0.0120316	−	0.999928i	$-0.496170\pi$
$701$	446.611	0.0240631	0.0120316	+	0.999928i	$0.496170\pi$
$702$	0	0
$703$	−14387.5	−0.771886
$704$	−12953.9	−0.693490
$705$	0	0
$706$	−43782.0	−2.33393
$707$	0	0
$708$	0	0
$709$	−33315.3	−1.76471	−0.882357	−	0.470581i	$-0.844044\pi$
$709$	−33315.3	−1.76471	−0.882357	+	0.470581i	$0.844044\pi$
$710$	34013.3	1.79788
$711$	0	0
$712$	−6743.52	−0.354950
$713$	−325.995	−0.0171229
$714$	0	0
$715$	−3901.02	−0.204042
$716$	−16355.5	−0.853681
$717$	0	0
$718$	39510.0	2.05362
$719$	−28284.2	−1.46707	−0.733535	−	0.679652i	$-0.762131\pi$
$719$	−28284.2	−1.46707	−0.733535	+	0.679652i	$0.762131\pi$
$720$	0	0
$721$	0	0
$722$	−21861.3	−1.12686
$723$	0	0
$724$	29217.1	1.49979
$725$	−1080.95	−0.0553732
$726$	0	0
$727$	−14352.0	−0.732170	−0.366085	−	0.930581i	$-0.619302\pi$
$727$	−14352.0	−0.732170	−0.366085	+	0.930581i	$0.619302\pi$
$728$	0	0
$729$	0	0
$730$	−16171.5	−0.819910
$731$	−20477.8	−1.03611
$732$	0	0
$733$	22682.3	1.14296	0.571481	−	0.820615i	$-0.306369\pi$
$733$	22682.3	1.14296	0.571481	+	0.820615i	$0.306369\pi$
$734$	13358.2	0.671743
$735$	0	0
$736$	−301.695	−0.0151096
$737$	39002.6	1.94936
$738$	0	0
$739$	−22632.9	−1.12661	−0.563304	−	0.826250i	$-0.690470\pi$
$739$	−22632.9	−1.12661	−0.563304	+	0.826250i	$0.690470\pi$
$740$	−10605.2	−0.526832
$741$	0	0
$742$	0	0
$743$	26923.7	1.32939	0.664693	−	0.747116i	$-0.268562\pi$
$743$	26923.7	1.32939	0.664693	+	0.747116i	$0.268562\pi$
$744$	0	0
$745$	−23889.1	−1.17481
$746$	−7003.83	−0.343738
$747$	0	0
$748$	−10896.6	−0.532645
$749$	0	0
$750$	0	0
$751$	3753.03	0.182357	0.0911785	−	0.995835i	$-0.470937\pi$
$751$	3753.03	0.182357	0.0911785	+	0.995835i	$0.470937\pi$
$752$	30909.1	1.49886
$753$	0	0
$754$	−1131.28	−0.0546401
$755$	25285.6	1.21886
$756$	0	0
$757$	−17863.5	−0.857673	−0.428837	−	0.903382i	$-0.641076\pi$
$757$	−17863.5	−0.857673	−0.428837	+	0.903382i	$0.641076\pi$
$758$	−44558.0	−2.13512
$759$	0	0
$760$	7156.78	0.341584
$761$	14086.4	0.671003	0.335502	−	0.942040i	$-0.391094\pi$
$761$	14086.4	0.671003	0.335502	+	0.942040i	$0.391094\pi$
$762$	0	0
$763$	0	0
$764$	19955.5	0.944981
$765$	0	0
$766$	−25026.1	−1.18046
$767$	2345.45	0.110416
$768$	0	0
$769$	3686.96	0.172894	0.0864469	−	0.996256i	$-0.472449\pi$
$769$	3686.96	0.172894	0.0864469	+	0.996256i	$0.472449\pi$
$770$	0	0
$771$	0	0
$772$	−13506.7	−0.629687
$773$	20443.5	0.951233	0.475616	−	0.879653i	$-0.342225\pi$
$773$	20443.5	0.951233	0.475616	+	0.879653i	$0.342225\pi$
$774$	0	0
$775$	7490.45	0.347180
$776$	−3681.67	−0.170315
$777$	0	0
$778$	−40868.4	−1.88329
$779$	14667.0	0.674581
$780$	0	0
$781$	−28258.2	−1.29469
$782$	−201.091	−0.00919564
$783$	0	0
$784$	0	0
$785$	21872.7	0.994483
$786$	0	0
$787$	24461.5	1.10795	0.553977	−	0.832532i	$-0.313110\pi$
$787$	24461.5	1.10795	0.553977	+	0.832532i	$0.313110\pi$
$788$	−6862.26	−0.310226
$789$	0	0
$790$	−3878.78	−0.174685
$791$	0	0
$792$	0	0
$793$	537.307	0.0240610
$794$	−27157.3	−1.21382
$795$	0	0
$796$	11150.2	0.496491
$797$	−29004.8	−1.28909	−0.644543	−	0.764568i	$-0.722952\pi$
$797$	−29004.8	−1.28909	−0.644543	+	0.764568i	$0.722952\pi$
$798$	0	0
$799$	17567.1	0.777821
$800$	6932.10	0.306359
$801$	0	0
$802$	25854.3	1.13834
$803$	13435.3	0.590436
$804$	0	0
$805$	0	0
$806$	7839.16	0.342584
$807$	0	0
$808$	3940.06	0.171548
$809$	−4947.16	−0.214997	−0.107499	−	0.994205i	$-0.534284\pi$
$809$	−4947.16	−0.214997	−0.107499	+	0.994205i	$0.534284\pi$
$810$	0	0
$811$	12576.5	0.544539	0.272270	−	0.962221i	$-0.412226\pi$
$811$	12576.5	0.544539	0.272270	+	0.962221i	$0.412226\pi$
$812$	0	0
$813$	0	0
$814$	19399.3	0.835315
$815$	42045.5	1.80710
$816$	0	0
$817$	55379.8	2.37147
$818$	−19454.6	−0.831559
$819$	0	0
$820$	10811.2	0.460419
$821$	18247.0	0.775669	0.387835	−	0.921729i	$-0.373223\pi$
$821$	18247.0	0.775669	0.387835	+	0.921729i	$0.373223\pi$
$822$	0	0
$823$	−898.728	−0.0380652	−0.0190326	−	0.999819i	$-0.506059\pi$
$823$	−898.728	−0.0380652	−0.0190326	+	0.999819i	$0.506059\pi$
$824$	5597.56	0.236651
$825$	0	0
$826$	0	0
$827$	1079.33	0.0453833	0.0226916	−	0.999743i	$-0.492776\pi$
$827$	1079.33	0.0453833	0.0226916	+	0.999743i	$0.492776\pi$
$828$	0	0
$829$	−20197.3	−0.846176	−0.423088	−	0.906089i	$-0.639054\pi$
$829$	−20197.3	−0.846176	−0.423088	+	0.906089i	$0.639054\pi$
$830$	−40227.2	−1.68230
$831$	0	0
$832$	2610.89	0.108794
$833$	0	0
$834$	0	0
$835$	19078.5	0.790703
$836$	29468.6	1.21913
$837$	0	0
$838$	3773.04	0.155534
$839$	−15739.9	−0.647679	−0.323840	−	0.946112i	$-0.604974\pi$
$839$	−15739.9	−0.647679	−0.323840	+	0.946112i	$0.604974\pi$
$840$	0	0
$841$	−23010.4	−0.943473
$842$	−12795.9	−0.523723
$843$	0	0
$844$	11274.4	0.459812
$845$	−26487.8	−1.07835
$846$	0	0
$847$	0	0
$848$	−42155.7	−1.70711
$849$	0	0
$850$	4620.50	0.186449
$851$	162.599	0.00654972
$852$	0	0
$853$	15677.9	0.629310	0.314655	−	0.949206i	$-0.398111\pi$
$853$	15677.9	0.629310	0.314655	+	0.949206i	$0.398111\pi$
$854$	0	0
$855$	0	0
$856$	−5897.38	−0.235477
$857$	27079.5	1.07937	0.539684	−	0.841868i	$-0.318544\pi$
$857$	27079.5	1.07937	0.539684	+	0.841868i	$0.318544\pi$
$858$	0	0
$859$	−10171.2	−0.404000	−0.202000	−	0.979386i	$-0.564744\pi$
$859$	−10171.2	−0.404000	−0.202000	+	0.979386i	$0.564744\pi$
$860$	40821.1	1.61859
$861$	0	0
$862$	22203.1	0.877309
$863$	2410.93	0.0950974	0.0475487	−	0.998869i	$-0.484859\pi$
$863$	2410.93	0.0950974	0.0475487	+	0.998869i	$0.484859\pi$
$864$	0	0
$865$	−42557.6	−1.67283
$866$	58494.8	2.29530
$867$	0	0
$868$	0	0
$869$	3222.48	0.125794
$870$	0	0
$871$	−7861.08	−0.305813
$872$	−5425.42	−0.210697
$873$	0	0
$874$	543.827	0.0210472
$875$	0	0
$876$	0	0
$877$	30684.5	1.18146	0.590731	−	0.806869i	$-0.298839\pi$
$877$	30684.5	1.18146	0.590731	+	0.806869i	$0.298839\pi$
$878$	−58497.9	−2.24853
$879$	0	0
$880$	−35754.2	−1.36963
$881$	7294.27	0.278944	0.139472	−	0.990226i	$-0.455459\pi$
$881$	7294.27	0.278944	0.139472	+	0.990226i	$0.455459\pi$
$882$	0	0
$883$	−33437.6	−1.27437	−0.637183	−	0.770713i	$-0.719900\pi$
$883$	−33437.6	−1.27437	−0.637183	+	0.770713i	$0.719900\pi$
$884$	2196.24	0.0835604
$885$	0	0
$886$	28586.8	1.08396
$887$	−7420.79	−0.280908	−0.140454	−	0.990087i	$-0.544856\pi$
$887$	−7420.79	−0.280908	−0.140454	+	0.990087i	$0.544856\pi$
$888$	0	0
$889$	0	0
$890$	62328.0	2.34746
$891$	0	0
$892$	24992.5	0.938129
$893$	−47508.2	−1.78029
$894$	0	0
$895$	−30501.1	−1.13915
$896$	0	0
$897$	0	0
$898$	31308.4	1.16345
$899$	−9553.22	−0.354413
$900$	0	0
$901$	−23959.0	−0.885895
$902$	−19776.1	−0.730014
$903$	0	0
$904$	2644.58	0.0972979
$905$	54486.4	2.00131
$906$	0	0
$907$	−15062.0	−0.551406	−0.275703	−	0.961243i	$-0.588911\pi$
$907$	−15062.0	−0.551406	−0.275703	+	0.961243i	$0.588911\pi$
$908$	21191.1	0.774506
$909$	0	0
$910$	0	0
$911$	−34458.6	−1.25320	−0.626600	−	0.779341i	$-0.715554\pi$
$911$	−34458.6	−1.25320	−0.626600	+	0.779341i	$0.715554\pi$
$912$	0	0
$913$	33420.7	1.21146
$914$	−58930.0	−2.13264
$915$	0	0
$916$	13022.4	0.469729
$917$	0	0
$918$	0	0
$919$	1282.11	0.0460205	0.0230102	−	0.999735i	$-0.492675\pi$
$919$	1282.11	0.0460205	0.0230102	+	0.999735i	$0.492675\pi$
$920$	−80.8814	−0.00289846
$921$	0	0
$922$	3829.04	0.136771
$923$	5695.51	0.203109
$924$	0	0
$925$	−3736.06	−0.132801
$926$	41834.5	1.48463
$927$	0	0
$928$	−8841.11	−0.312741
$929$	48599.8	1.71637	0.858185	−	0.513341i	$-0.171592\pi$
$929$	48599.8	1.71637	0.858185	+	0.513341i	$0.171592\pi$
$930$	0	0
$931$	0	0
$932$	−31176.5	−1.09573
$933$	0	0
$934$	−27302.2	−0.956482
$935$	−20320.8	−0.710760
$936$	0	0
$937$	12513.3	0.436278	0.218139	−	0.975918i	$-0.430001\pi$
$937$	12513.3	0.436278	0.218139	+	0.975918i	$0.430001\pi$
$938$	0	0
$939$	0	0
$940$	−35018.9	−1.21510
$941$	39362.5	1.36363	0.681817	−	0.731522i	$-0.261190\pi$
$941$	39362.5	1.36363	0.681817	+	0.731522i	$0.261190\pi$
$942$	0	0
$943$	−165.757	−0.00572405
$944$	21496.8	0.741168
$945$	0	0
$946$	−74671.0	−2.56635
$947$	33893.8	1.16304	0.581520	−	0.813532i	$-0.302458\pi$
$947$	33893.8	1.16304	0.581520	+	0.813532i	$0.302458\pi$
$948$	0	0
$949$	−2707.91	−0.0926266
$950$	−12495.6	−0.426749
$951$	0	0
$952$	0	0
$953$	30224.1	1.02734	0.513669	−	0.857988i	$-0.328286\pi$
$953$	30224.1	1.02734	0.513669	+	0.857988i	$0.328286\pi$
$954$	0	0
$955$	37214.6	1.26098
$956$	37518.5	1.26928
$957$	0	0
$958$	5810.65	0.195964
$959$	0	0
$960$	0	0
$961$	36407.9	1.22211
$962$	−3909.99	−0.131043
$963$	0	0
$964$	−13638.4	−0.455667
$965$	−25188.4	−0.840252
$966$	0	0
$967$	47216.8	1.57021	0.785103	−	0.619366i	$-0.212610\pi$
$967$	47216.8	1.57021	0.785103	+	0.619366i	$0.212610\pi$
$968$	1172.86	0.0389432
$969$	0	0
$970$	34028.3	1.12637
$971$	−32285.7	−1.06704	−0.533521	−	0.845787i	$-0.679132\pi$
$971$	−32285.7	−1.06704	−0.533521	+	0.845787i	$0.679132\pi$
$972$	0	0
$973$	0	0
$974$	29216.2	0.961136
$975$	0	0
$976$	4924.60	0.161509
$977$	3612.93	0.118309	0.0591545	−	0.998249i	$-0.481160\pi$
$977$	3612.93	0.118309	0.0591545	+	0.998249i	$0.481160\pi$
$978$	0	0
$979$	−51782.0	−1.69046
$980$	0	0
$981$	0	0
$982$	1929.80	0.0627111
$983$	−25683.7	−0.833350	−0.416675	−	0.909056i	$-0.636805\pi$
$983$	−25683.7	−0.833350	−0.416675	+	0.909056i	$0.636805\pi$
$984$	0	0
$985$	−12797.3	−0.413964
$986$	−5892.92	−0.190333
$987$	0	0
$988$	−5939.47	−0.191255
$989$	−625.868	−0.0201228
$990$	0	0
$991$	5948.30	0.190670	0.0953351	−	0.995445i	$-0.469608\pi$
$991$	5948.30	0.190670	0.0953351	+	0.995445i	$0.469608\pi$
$992$	61264.4	1.96083
$993$	0	0
$994$	0	0
$995$	20793.7	0.662516
$996$	0	0
$997$	31203.4	0.991196	0.495598	−	0.868552i	$-0.334949\pi$
$997$	31203.4	0.991196	0.495598	+	0.868552i	$0.334949\pi$
$998$	28960.7	0.918571
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.q.1.1	yes	2
3.2	odd	2		1323.4.a.v.1.2	yes	2
7.6	odd	2		1323.4.a.p.1.1	✓	2
21.20	even	2		1323.4.a.w.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.4.a.p.1.1	✓	2	7.6	odd	2
1323.4.a.q.1.1	yes	2	1.1	even	1	trivial
1323.4.a.v.1.2	yes	2	3.2	odd	2
1323.4.a.w.1.2	yes	2	21.20	even	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)

\(f(q)\)	\(=\)	\(q-3.82843 q^{2} +6.65685 q^{4} +12.4142 q^{5} +5.14214 q^{8} -47.5269 q^{10} +39.4853 q^{11} -7.95837 q^{13} -72.9411 q^{16} -41.4558 q^{17} +112.113 q^{19} +82.6396 q^{20} -151.167 q^{22} -1.26703 q^{23} +29.1127 q^{25} +30.4680 q^{26} -37.1299 q^{29} +257.291 q^{31} +238.113 q^{32} +158.711 q^{34} -128.331 q^{37} -429.215 q^{38} +63.8356 q^{40} +130.823 q^{41} +493.966 q^{43} +262.848 q^{44} +4.85072 q^{46} -423.754 q^{47} -111.456 q^{50} -52.9777 q^{52} +577.941 q^{53} +490.179 q^{55} +142.149 q^{58} -294.715 q^{59} -67.5147 q^{61} -985.021 q^{62} -328.068 q^{64} -98.7969 q^{65} +987.776 q^{67} -275.966 q^{68} -715.663 q^{71} +340.260 q^{73} +491.306 q^{74} +746.318 q^{76} +81.6123 q^{79} -905.507 q^{80} -500.848 q^{82} +846.409 q^{83} -514.642 q^{85} -1891.11 q^{86} +203.039 q^{88} -1311.42 q^{89} -8.43442 q^{92} +1622.31 q^{94} +1391.79 q^{95} -715.980 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 22 q^{5} - 18 q^{8} - 30 q^{10} + 62 q^{11} - 64 q^{13} - 78 q^{16} - 32 q^{17} + 162 q^{19} + 38 q^{20} - 110 q^{22} + 170 q^{23} - 4 q^{25} - 72 q^{26} - 128 q^{29} + 178 q^{31}+ \cdots - 640 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 22 * q^5 - 18 * q^8 - 30 * q^10 + 62 * q^11 - 64 * q^13 - 78 * q^16 - 32 * q^17 + 162 * q^19 + 38 * q^20 - 110 * q^22 + 170 * q^23 - 4 * q^25 - 72 * q^26 - 128 * q^29 + 178 * q^31 + 414 * q^32 + 176 * q^34 - 350 * q^37 - 338 * q^38 - 158 * q^40 + 58 * q^41 + 756 * q^43 + 158 * q^44 + 318 * q^46 - 180 * q^47 - 172 * q^50 + 208 * q^52 + 1088 * q^53 + 706 * q^55 - 24 * q^58 + 508 * q^59 - 152 * q^61 - 1130 * q^62 + 34 * q^64 - 636 * q^65 + 468 * q^67 - 320 * q^68 + 14 * q^71 + 788 * q^73 + 86 * q^74 + 514 * q^76 - 476 * q^79 - 954 * q^80 - 634 * q^82 + 1492 * q^83 - 424 * q^85 - 1412 * q^86 - 318 * q^88 - 230 * q^89 - 806 * q^92 + 2068 * q^94 + 1870 * q^95 - 640 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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