LMFDB - Embedded newform 1344.4.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(1,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1344.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.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:	$79.2985670477$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.27492$ of defining polynomial
Character	$\chi$	$=$	1344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +4.54983 q^{5} +7.00000 q^{7} +9.00000 q^{9} +40.7492 q^{11} -53.2990 q^{13} -13.6495 q^{15} +4.54983 q^{17} -122.598 q^{19} -21.0000 q^{21} +131.347 q^{23} -104.299 q^{25} -27.0000 q^{27} +216.598 q^{29} -251.794 q^{31} -122.248 q^{33} +31.8488 q^{35} -11.8970 q^{37} +159.897 q^{39} -111.752 q^{41} -369.196 q^{43} +40.9485 q^{45} -262.694 q^{47} +49.0000 q^{49} -13.6495 q^{51} +567.100 q^{53} +185.402 q^{55} +367.794 q^{57} -839.890 q^{59} +485.794 q^{61} +63.0000 q^{63} -242.502 q^{65} +333.691 q^{67} -394.042 q^{69} +590.248 q^{71} +490.701 q^{73} +312.897 q^{75} +285.244 q^{77} +121.691 q^{79} +81.0000 q^{81} -609.608 q^{83} +20.7010 q^{85} -649.794 q^{87} +719.038 q^{89} -373.093 q^{91} +755.382 q^{93} -557.801 q^{95} -637.877 q^{97} +366.743 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 6 q^{5} + 14 q^{7} + 18 q^{9} + 6 q^{11} - 16 q^{13} + 18 q^{15} - 6 q^{17} - 64 q^{19} - 42 q^{21} + 6 q^{23} - 118 q^{25} - 54 q^{27} + 252 q^{29} + 40 q^{31} - 18 q^{33} - 42 q^{35} + 248 q^{37}+ \cdots + 54 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 6 * q^5 + 14 * q^7 + 18 * q^9 + 6 * q^11 - 16 * q^13 + 18 * q^15 - 6 * q^17 - 64 * q^19 - 42 * q^21 + 6 * q^23 - 118 * q^25 - 54 * q^27 + 252 * q^29 + 40 * q^31 - 18 * q^33 - 42 * q^35 + 248 * q^37 + 48 * q^39 - 450 * q^41 - 376 * q^43 - 54 * q^45 - 12 * q^47 + 98 * q^49 + 18 * q^51 + 1104 * q^53 + 552 * q^55 + 192 * q^57 - 804 * q^59 + 428 * q^61 + 126 * q^63 - 636 * q^65 - 148 * q^67 - 18 * q^69 + 954 * q^71 + 1072 * q^73 + 354 * q^75 + 42 * q^77 - 572 * q^79 + 162 * q^81 - 1944 * q^83 + 132 * q^85 - 756 * q^87 + 366 * q^89 - 112 * q^91 - 120 * q^93 - 1176 * q^95 + 808 * q^97 + 54 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	4.54983	0.406950	0.203475	−	0.979080i	$-0.434777\pi$
$5$	4.54983	0.406950	0.203475	+	0.979080i	$0.434777\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	40.7492	1.11694	0.558470	−	0.829525i	$-0.311389\pi$
$11$	40.7492	1.11694	0.558470	+	0.829525i	$0.311389\pi$
$12$	0	0
$13$	−53.2990	−1.13711	−0.568557	−	0.822644i	$-0.692498\pi$
$13$	−53.2990	−1.13711	−0.568557	+	0.822644i	$0.692498\pi$
$14$	0	0
$15$	−13.6495	−0.234952
$16$	0	0
$17$	4.54983	0.0649116	0.0324558	−	0.999473i	$-0.489667\pi$
$17$	4.54983	0.0649116	0.0324558	+	0.999473i	$0.489667\pi$
$18$	0	0
$19$	−122.598	−1.48031	−0.740156	−	0.672436i	$-0.765248\pi$
$19$	−122.598	−1.48031	−0.740156	+	0.672436i	$0.765248\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	131.347	1.19077	0.595387	−	0.803439i	$-0.296999\pi$
$23$	131.347	1.19077	0.595387	+	0.803439i	$0.296999\pi$
$24$	0	0
$25$	−104.299	−0.834392
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	216.598	1.38694	0.693470	−	0.720486i	$-0.256081\pi$
$29$	216.598	1.38694	0.693470	+	0.720486i	$0.256081\pi$
$30$	0	0
$31$	−251.794	−1.45882	−0.729412	−	0.684075i	$-0.760206\pi$
$31$	−251.794	−1.45882	−0.729412	+	0.684075i	$0.760206\pi$
$32$	0	0
$33$	−122.248	−0.644865
$34$	0	0
$35$	31.8488	0.153812
$36$	0	0
$37$	−11.8970	−0.0528610	−0.0264305	−	0.999651i	$-0.508414\pi$
$37$	−11.8970	−0.0528610	−0.0264305	+	0.999651i	$0.508414\pi$
$38$	0	0
$39$	159.897	0.656513
$40$	0	0
$41$	−111.752	−0.425678	−0.212839	−	0.977087i	$-0.568271\pi$
$41$	−111.752	−0.425678	−0.212839	+	0.977087i	$0.568271\pi$
$42$	0	0
$43$	−369.196	−1.30935	−0.654673	−	0.755912i	$-0.727194\pi$
$43$	−369.196	−1.30935	−0.654673	+	0.755912i	$0.727194\pi$
$44$	0	0
$45$	40.9485	0.135650
$46$	0	0
$47$	−262.694	−0.815275	−0.407637	−	0.913144i	$-0.633647\pi$
$47$	−262.694	−0.815275	−0.407637	+	0.913144i	$0.633647\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−13.6495	−0.0374767
$52$	0	0
$53$	567.100	1.46976	0.734879	−	0.678199i	$-0.237239\pi$
$53$	567.100	1.46976	0.734879	+	0.678199i	$0.237239\pi$
$54$	0	0
$55$	185.402	0.454538
$56$	0	0
$57$	367.794	0.854658
$58$	0	0
$59$	−839.890	−1.85330	−0.926648	−	0.375931i	$-0.877323\pi$
$59$	−839.890	−1.85330	−0.926648	+	0.375931i	$0.877323\pi$
$60$	0	0
$61$	485.794	1.01966	0.509832	−	0.860274i	$-0.329707\pi$
$61$	485.794	1.01966	0.509832	+	0.860274i	$0.329707\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	−242.502	−0.462748
$66$	0	0
$67$	333.691	0.608460	0.304230	−	0.952599i	$-0.401601\pi$
$67$	333.691	0.608460	0.304230	+	0.952599i	$0.401601\pi$
$68$	0	0
$69$	−394.042	−0.687493
$70$	0	0
$71$	590.248	0.986613	0.493306	−	0.869856i	$-0.335788\pi$
$71$	590.248	0.986613	0.493306	+	0.869856i	$0.335788\pi$
$72$	0	0
$73$	490.701	0.786743	0.393371	−	0.919380i	$-0.371309\pi$
$73$	490.701	0.786743	0.393371	+	0.919380i	$0.371309\pi$
$74$	0	0
$75$	312.897	0.481736
$76$	0	0
$77$	285.244	0.422164
$78$	0	0
$79$	121.691	0.173308	0.0866539	−	0.996238i	$-0.472383\pi$
$79$	121.691	0.173308	0.0866539	+	0.996238i	$0.472383\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−609.608	−0.806183	−0.403091	−	0.915160i	$-0.632064\pi$
$83$	−609.608	−0.806183	−0.403091	+	0.915160i	$0.632064\pi$
$84$	0	0
$85$	20.7010	0.0264157
$86$	0	0
$87$	−649.794	−0.800750
$88$	0	0
$89$	719.038	0.856381	0.428190	−	0.903689i	$-0.359151\pi$
$89$	719.038	0.856381	0.428190	+	0.903689i	$0.359151\pi$
$90$	0	0
$91$	−373.093	−0.429789
$92$	0	0
$93$	755.382	0.842252
$94$	0	0
$95$	−557.801	−0.602412
$96$	0	0
$97$	−637.877	−0.667697	−0.333849	−	0.942627i	$-0.608347\pi$
$97$	−637.877	−0.667697	−0.333849	+	0.942627i	$0.608347\pi$
$98$	0	0
$99$	366.743	0.372313
$100$	0	0
$101$	−671.148	−0.661205	−0.330603	−	0.943770i	$-0.607252\pi$
$101$	−671.148	−0.661205	−0.330603	+	0.943770i	$0.607252\pi$
$102$	0	0
$103$	−912.412	−0.872841	−0.436420	−	0.899743i	$-0.643754\pi$
$103$	−912.412	−0.872841	−0.436420	+	0.899743i	$0.643754\pi$
$104$	0	0
$105$	−95.5465	−0.0888037
$106$	0	0
$107$	116.736	0.105470	0.0527350	−	0.998609i	$-0.483206\pi$
$107$	116.736	0.105470	0.0527350	+	0.998609i	$0.483206\pi$
$108$	0	0
$109$	−837.176	−0.735660	−0.367830	−	0.929893i	$-0.619899\pi$
$109$	−837.176	−0.735660	−0.367830	+	0.929893i	$0.619899\pi$
$110$	0	0
$111$	35.6911	0.0305193
$112$	0	0
$113$	−1086.58	−0.904572	−0.452286	−	0.891873i	$-0.649391\pi$
$113$	−1086.58	−0.904572	−0.452286	+	0.891873i	$0.649391\pi$
$114$	0	0
$115$	597.608	0.484585
$116$	0	0
$117$	−479.691	−0.379038
$118$	0	0
$119$	31.8488	0.0245343
$120$	0	0
$121$	329.495	0.247554
$122$	0	0
$123$	335.257	0.245765
$124$	0	0
$125$	−1043.27	−0.746505
$126$	0	0
$127$	−537.113	−0.375284	−0.187642	−	0.982237i	$-0.560084\pi$
$127$	−537.113	−0.375284	−0.187642	+	0.982237i	$0.560084\pi$
$128$	0	0
$129$	1107.59	0.755951
$130$	0	0
$131$	−1497.39	−0.998683	−0.499341	−	0.866405i	$-0.666425\pi$
$131$	−1497.39	−0.998683	−0.499341	+	0.866405i	$0.666425\pi$
$132$	0	0
$133$	−858.186	−0.559505
$134$	0	0
$135$	−122.846	−0.0783175
$136$	0	0
$137$	−1380.09	−0.860650	−0.430325	−	0.902674i	$-0.641601\pi$
$137$	−1380.09	−0.860650	−0.430325	+	0.902674i	$0.641601\pi$
$138$	0	0
$139$	141.980	0.0866374	0.0433187	−	0.999061i	$-0.486207\pi$
$139$	141.980	0.0866374	0.0433187	+	0.999061i	$0.486207\pi$
$140$	0	0
$141$	788.083	0.470699
$142$	0	0
$143$	−2171.89	−1.27009
$144$	0	0
$145$	985.485	0.564414
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	1943.87	1.06878	0.534390	−	0.845238i	$-0.320542\pi$
$149$	1943.87	1.06878	0.534390	+	0.845238i	$0.320542\pi$
$150$	0	0
$151$	−2654.76	−1.43074	−0.715370	−	0.698746i	$-0.753742\pi$
$151$	−2654.76	−1.43074	−0.715370	+	0.698746i	$0.753742\pi$
$152$	0	0
$153$	40.9485	0.0216372
$154$	0	0
$155$	−1145.62	−0.593668
$156$	0	0
$157$	−1665.22	−0.846489	−0.423244	−	0.906016i	$-0.639109\pi$
$157$	−1665.22	−0.846489	−0.423244	+	0.906016i	$0.639109\pi$
$158$	0	0
$159$	−1701.30	−0.848565
$160$	0	0
$161$	919.430	0.450070
$162$	0	0
$163$	33.0732	0.0158926	0.00794629	−	0.999968i	$-0.497471\pi$
$163$	33.0732	0.0158926	0.00794629	+	0.999968i	$0.497471\pi$
$164$	0	0
$165$	−556.206	−0.262428
$166$	0	0
$167$	1654.48	0.766630	0.383315	−	0.923618i	$-0.374782\pi$
$167$	1654.48	0.766630	0.383315	+	0.923618i	$0.374782\pi$
$168$	0	0
$169$	643.784	0.293029
$170$	0	0
$171$	−1103.38	−0.493437
$172$	0	0
$173$	−64.1909	−0.0282101	−0.0141050	−	0.999901i	$-0.504490\pi$
$173$	−64.1909	−0.0282101	−0.0141050	+	0.999901i	$0.504490\pi$
$174$	0	0
$175$	−730.093	−0.315371
$176$	0	0
$177$	2519.67	1.07000
$178$	0	0
$179$	−3914.68	−1.63462	−0.817309	−	0.576200i	$-0.804535\pi$
$179$	−3914.68	−1.63462	−0.817309	+	0.576200i	$0.804535\pi$
$180$	0	0
$181$	2058.04	0.845156	0.422578	−	0.906327i	$-0.361125\pi$
$181$	2058.04	0.845156	0.422578	+	0.906327i	$0.361125\pi$
$182$	0	0
$183$	−1457.38	−0.588704
$184$	0	0
$185$	−54.1295	−0.0215118
$186$	0	0
$187$	185.402	0.0725023
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	428.048	0.162160	0.0810798	−	0.996708i	$-0.474163\pi$
$191$	428.048	0.162160	0.0810798	+	0.996708i	$0.474163\pi$
$192$	0	0
$193$	1604.93	0.598576	0.299288	−	0.954163i	$-0.403251\pi$
$193$	1604.93	0.598576	0.299288	+	0.954163i	$0.403251\pi$
$194$	0	0
$195$	727.505	0.267168
$196$	0	0
$197$	−3738.83	−1.35218	−0.676092	−	0.736817i	$-0.736328\pi$
$197$	−3738.83	−1.35218	−0.676092	+	0.736817i	$0.736328\pi$
$198$	0	0
$199$	−349.030	−0.124332	−0.0621660	−	0.998066i	$-0.519801\pi$
$199$	−349.030	−0.124332	−0.0621660	+	0.998066i	$0.519801\pi$
$200$	0	0
$201$	−1001.07	−0.351295
$202$	0	0
$203$	1516.19	0.524214
$204$	0	0
$205$	−508.455	−0.173230
$206$	0	0
$207$	1182.12	0.396924
$208$	0	0
$209$	−4995.77	−1.65342
$210$	0	0
$211$	−2588.58	−0.844574	−0.422287	−	0.906462i	$-0.638773\pi$
$211$	−2588.58	−0.844574	−0.422287	+	0.906462i	$0.638773\pi$
$212$	0	0
$213$	−1770.74	−0.569621
$214$	0	0
$215$	−1679.78	−0.532838
$216$	0	0
$217$	−1762.56	−0.551384
$218$	0	0
$219$	−1472.10	−0.454226
$220$	0	0
$221$	−242.502	−0.0738119
$222$	0	0
$223$	−3236.21	−0.971804	−0.485902	−	0.874013i	$-0.661509\pi$
$223$	−3236.21	−0.971804	−0.485902	+	0.874013i	$0.661509\pi$
$224$	0	0
$225$	−938.691	−0.278131
$226$	0	0
$227$	5631.62	1.64662	0.823312	−	0.567589i	$-0.192124\pi$
$227$	5631.62	1.64662	0.823312	+	0.567589i	$0.192124\pi$
$228$	0	0
$229$	−3770.25	−1.08797	−0.543985	−	0.839095i	$-0.683085\pi$
$229$	−3770.25	−1.08797	−0.543985	+	0.839095i	$0.683085\pi$
$230$	0	0
$231$	−855.733	−0.243736
$232$	0	0
$233$	−6560.90	−1.84472	−0.922358	−	0.386336i	$-0.873741\pi$
$233$	−6560.90	−1.84472	−0.922358	+	0.386336i	$0.873741\pi$
$234$	0	0
$235$	−1195.22	−0.331776
$236$	0	0
$237$	−365.073	−0.100059
$238$	0	0
$239$	−771.444	−0.208789	−0.104394	−	0.994536i	$-0.533290\pi$
$239$	−771.444	−0.208789	−0.104394	+	0.994536i	$0.533290\pi$
$240$	0	0
$241$	1252.10	0.334668	0.167334	−	0.985900i	$-0.446484\pi$
$241$	1252.10	0.334668	0.167334	+	0.985900i	$0.446484\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	222.942	0.0581357
$246$	0	0
$247$	6534.35	1.68328
$248$	0	0
$249$	1828.82	0.465450
$250$	0	0
$251$	−5166.27	−1.29917	−0.649586	−	0.760288i	$-0.725058\pi$
$251$	−5166.27	−1.29917	−0.649586	+	0.760288i	$0.725058\pi$
$252$	0	0
$253$	5352.29	1.33002
$254$	0	0
$255$	−62.1030	−0.0152511
$256$	0	0
$257$	2767.45	0.671707	0.335854	−	0.941914i	$-0.390975\pi$
$257$	2767.45	0.671707	0.335854	+	0.941914i	$0.390975\pi$
$258$	0	0
$259$	−83.2791	−0.0199796
$260$	0	0
$261$	1949.38	0.462313
$262$	0	0
$263$	−4101.78	−0.961699	−0.480849	−	0.876803i	$-0.659672\pi$
$263$	−4101.78	−0.961699	−0.480849	+	0.876803i	$0.659672\pi$
$264$	0	0
$265$	2580.21	0.598117
$266$	0	0
$267$	−2157.11	−0.494432
$268$	0	0
$269$	6950.84	1.57546	0.787732	−	0.616018i	$-0.211255\pi$
$269$	6950.84	1.57546	0.787732	+	0.616018i	$0.211255\pi$
$270$	0	0
$271$	−7140.29	−1.60052	−0.800262	−	0.599651i	$-0.795306\pi$
$271$	−7140.29	−1.60052	−0.800262	+	0.599651i	$0.795306\pi$
$272$	0	0
$273$	1119.28	0.248139
$274$	0	0
$275$	−4250.10	−0.931966
$276$	0	0
$277$	−1320.51	−0.286433	−0.143217	−	0.989691i	$-0.545745\pi$
$277$	−1320.51	−0.286433	−0.143217	+	0.989691i	$0.545745\pi$
$278$	0	0
$279$	−2266.15	−0.486275
$280$	0	0
$281$	−204.309	−0.0433738	−0.0216869	−	0.999765i	$-0.506904\pi$
$281$	−204.309	−0.0433738	−0.0216869	+	0.999765i	$0.506904\pi$
$282$	0	0
$283$	−975.794	−0.204964	−0.102482	−	0.994735i	$-0.532678\pi$
$283$	−975.794	−0.204964	−0.102482	+	0.994735i	$0.532678\pi$
$284$	0	0
$285$	1673.40	0.347803
$286$	0	0
$287$	−782.267	−0.160891
$288$	0	0
$289$	−4892.30	−0.995786
$290$	0	0
$291$	1913.63	0.385495
$292$	0	0
$293$	−607.919	−0.121212	−0.0606058	−	0.998162i	$-0.519303\pi$
$293$	−607.919	−0.121212	−0.0606058	+	0.998162i	$0.519303\pi$
$294$	0	0
$295$	−3821.36	−0.754198
$296$	0	0
$297$	−1100.23	−0.214955
$298$	0	0
$299$	−7000.67	−1.35405
$300$	0	0
$301$	−2584.37	−0.494886
$302$	0	0
$303$	2013.44	0.381747
$304$	0	0
$305$	2210.28	0.414952
$306$	0	0
$307$	−8037.08	−1.49414	−0.747069	−	0.664747i	$-0.768539\pi$
$307$	−8037.08	−1.49414	−0.747069	+	0.664747i	$0.768539\pi$
$308$	0	0
$309$	2737.24	0.503935
$310$	0	0
$311$	5311.60	0.968468	0.484234	−	0.874939i	$-0.339098\pi$
$311$	5311.60	0.968468	0.484234	+	0.874939i	$0.339098\pi$
$312$	0	0
$313$	−1531.61	−0.276587	−0.138293	−	0.990391i	$-0.544162\pi$
$313$	−1531.61	−0.276587	−0.138293	+	0.990391i	$0.544162\pi$
$314$	0	0
$315$	286.640	0.0512708
$316$	0	0
$317$	−4219.19	−0.747549	−0.373775	−	0.927520i	$-0.621937\pi$
$317$	−4219.19	−0.747549	−0.373775	+	0.927520i	$0.621937\pi$
$318$	0	0
$319$	8826.19	1.54913
$320$	0	0
$321$	−350.208	−0.0608931
$322$	0	0
$323$	−557.801	−0.0960893
$324$	0	0
$325$	5559.03	0.948799
$326$	0	0
$327$	2511.53	0.424733
$328$	0	0
$329$	−1838.86	−0.308145
$330$	0	0
$331$	−8298.19	−1.37797	−0.688987	−	0.724773i	$-0.741944\pi$
$331$	−8298.19	−1.37797	−0.688987	+	0.724773i	$0.741944\pi$
$332$	0	0
$333$	−107.073	−0.0176203
$334$	0	0
$335$	1518.24	0.247613
$336$	0	0
$337$	−4348.44	−0.702892	−0.351446	−	0.936208i	$-0.614310\pi$
$337$	−4348.44	−0.702892	−0.351446	+	0.936208i	$0.614310\pi$
$338$	0	0
$339$	3259.73	0.522255
$340$	0	0
$341$	−10260.4	−1.62942
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−1792.82	−0.279775
$346$	0	0
$347$	8345.54	1.29110	0.645550	−	0.763718i	$-0.276628\pi$
$347$	8345.54	1.29110	0.645550	+	0.763718i	$0.276628\pi$
$348$	0	0
$349$	9982.54	1.53110	0.765549	−	0.643378i	$-0.222468\pi$
$349$	9982.54	1.53110	0.765549	+	0.643378i	$0.222468\pi$
$350$	0	0
$351$	1439.07	0.218838
$352$	0	0
$353$	8801.59	1.32709	0.663543	−	0.748138i	$-0.269052\pi$
$353$	8801.59	1.32709	0.663543	+	0.748138i	$0.269052\pi$
$354$	0	0
$355$	2685.53	0.401502
$356$	0	0
$357$	−95.5465	−0.0141649
$358$	0	0
$359$	−524.039	−0.0770409	−0.0385205	−	0.999258i	$-0.512264\pi$
$359$	−524.039	−0.0770409	−0.0385205	+	0.999258i	$0.512264\pi$
$360$	0	0
$361$	8171.27	1.19132
$362$	0	0
$363$	−988.485	−0.142926
$364$	0	0
$365$	2232.61	0.320165
$366$	0	0
$367$	−6362.72	−0.904991	−0.452495	−	0.891767i	$-0.649466\pi$
$367$	−6362.72	−0.904991	−0.452495	+	0.891767i	$0.649466\pi$
$368$	0	0
$369$	−1005.77	−0.141893
$370$	0	0
$371$	3969.70	0.555516
$372$	0	0
$373$	11265.8	1.56387	0.781935	−	0.623361i	$-0.214233\pi$
$373$	11265.8	1.56387	0.781935	+	0.623361i	$0.214233\pi$
$374$	0	0
$375$	3129.82	0.430995
$376$	0	0
$377$	−11544.5	−1.57711
$378$	0	0
$379$	1151.71	0.156094	0.0780470	−	0.996950i	$-0.475132\pi$
$379$	1151.71	0.156094	0.0780470	+	0.996950i	$0.475132\pi$
$380$	0	0
$381$	1611.34	0.216670
$382$	0	0
$383$	−151.554	−0.0202195	−0.0101097	−	0.999949i	$-0.503218\pi$
$383$	−151.554	−0.0202195	−0.0101097	+	0.999949i	$0.503218\pi$
$384$	0	0
$385$	1297.81	0.171799
$386$	0	0
$387$	−3322.76	−0.436449
$388$	0	0
$389$	−4794.18	−0.624870	−0.312435	−	0.949939i	$-0.601145\pi$
$389$	−4794.18	−0.624870	−0.312435	+	0.949939i	$0.601145\pi$
$390$	0	0
$391$	597.608	0.0772950
$392$	0	0
$393$	4492.17	0.576590
$394$	0	0
$395$	553.674	0.0705275
$396$	0	0
$397$	4623.94	0.584556	0.292278	−	0.956333i	$-0.405587\pi$
$397$	4623.94	0.584556	0.292278	+	0.956333i	$0.405587\pi$
$398$	0	0
$399$	2574.56	0.323030
$400$	0	0
$401$	−3610.63	−0.449642	−0.224821	−	0.974400i	$-0.572180\pi$
$401$	−3610.63	−0.449642	−0.224821	+	0.974400i	$0.572180\pi$
$402$	0	0
$403$	13420.4	1.65885
$404$	0	0
$405$	368.537	0.0452166
$406$	0	0
$407$	−484.794	−0.0590426
$408$	0	0
$409$	8959.57	1.08318	0.541592	−	0.840641i	$-0.317822\pi$
$409$	8959.57	1.08318	0.541592	+	0.840641i	$0.317822\pi$
$410$	0	0
$411$	4140.27	0.496896
$412$	0	0
$413$	−5879.23	−0.700480
$414$	0	0
$415$	−2773.62	−0.328076
$416$	0	0
$417$	−425.940	−0.0500201
$418$	0	0
$419$	7078.28	0.825290	0.412645	−	0.910892i	$-0.364605\pi$
$419$	7078.28	0.825290	0.412645	+	0.910892i	$0.364605\pi$
$420$	0	0
$421$	−11551.5	−1.33725	−0.668626	−	0.743599i	$-0.733117\pi$
$421$	−11551.5	−1.33725	−0.668626	+	0.743599i	$0.733117\pi$
$422$	0	0
$423$	−2364.25	−0.271758
$424$	0	0
$425$	−474.543	−0.0541617
$426$	0	0
$427$	3400.56	0.385397
$428$	0	0
$429$	6515.67	0.733286
$430$	0	0
$431$	−4064.38	−0.454232	−0.227116	−	0.973868i	$-0.572930\pi$
$431$	−4064.38	−0.454232	−0.227116	+	0.973868i	$0.572930\pi$
$432$	0	0
$433$	17456.3	1.93740	0.968701	−	0.248229i	$-0.0798487\pi$
$433$	17456.3	1.93740	0.968701	+	0.248229i	$0.0798487\pi$
$434$	0	0
$435$	−2956.46	−0.325865
$436$	0	0
$437$	−16102.9	−1.76271
$438$	0	0
$439$	4595.39	0.499604	0.249802	−	0.968297i	$-0.419635\pi$
$439$	4595.39	0.499604	0.249802	+	0.968297i	$0.419635\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	306.214	0.0328412	0.0164206	−	0.999865i	$-0.494773\pi$
$443$	306.214	0.0328412	0.0164206	+	0.999865i	$0.494773\pi$
$444$	0	0
$445$	3271.50	0.348504
$446$	0	0
$447$	−5831.61	−0.617060
$448$	0	0
$449$	9229.22	0.970053	0.485026	−	0.874500i	$-0.338810\pi$
$449$	9229.22	0.970053	0.485026	+	0.874500i	$0.338810\pi$
$450$	0	0
$451$	−4553.82	−0.475457
$452$	0	0
$453$	7964.29	0.826038
$454$	0	0
$455$	−1697.51	−0.174902
$456$	0	0
$457$	−10992.2	−1.12515	−0.562577	−	0.826745i	$-0.690190\pi$
$457$	−10992.2	−1.12515	−0.562577	+	0.826745i	$0.690190\pi$
$458$	0	0
$459$	−122.846	−0.0124922
$460$	0	0
$461$	−7387.88	−0.746394	−0.373197	−	0.927752i	$-0.621739\pi$
$461$	−7387.88	−0.746394	−0.373197	+	0.927752i	$0.621739\pi$
$462$	0	0
$463$	10163.8	1.02020	0.510101	−	0.860114i	$-0.329608\pi$
$463$	10163.8	1.02020	0.510101	+	0.860114i	$0.329608\pi$
$464$	0	0
$465$	3436.86	0.342754
$466$	0	0
$467$	15814.6	1.56705	0.783524	−	0.621362i	$-0.213420\pi$
$467$	15814.6	1.56705	0.783524	+	0.621362i	$0.213420\pi$
$468$	0	0
$469$	2335.84	0.229976
$470$	0	0
$471$	4995.65	0.488720
$472$	0	0
$473$	−15044.4	−1.46246
$474$	0	0
$475$	12786.9	1.23516
$476$	0	0
$477$	5103.90	0.489919
$478$	0	0
$479$	−1444.85	−0.137823	−0.0689113	−	0.997623i	$-0.521953\pi$
$479$	−1444.85	−0.137823	−0.0689113	+	0.997623i	$0.521953\pi$
$480$	0	0
$481$	634.099	0.0601090
$482$	0	0
$483$	−2758.29	−0.259848
$484$	0	0
$485$	−2902.24	−0.271719
$486$	0	0
$487$	−489.402	−0.0455378	−0.0227689	−	0.999741i	$-0.507248\pi$
$487$	−489.402	−0.0455378	−0.0227689	+	0.999741i	$0.507248\pi$
$488$	0	0
$489$	−99.2195	−0.00917559
$490$	0	0
$491$	3941.30	0.362257	0.181129	−	0.983459i	$-0.442025\pi$
$491$	3941.30	0.362257	0.181129	+	0.983459i	$0.442025\pi$
$492$	0	0
$493$	985.485	0.0900284
$494$	0	0
$495$	1668.62	0.151513
$496$	0	0
$497$	4131.73	0.372905
$498$	0	0
$499$	−11.0894	−0.000994850 0	−0.000497425	−	1.00000i	$-0.500158\pi$
$499$	−11.0894	−0.000994850 0	−0.000497425		1.00000i	$0.500158\pi$
$500$	0	0
$501$	−4963.43	−0.442614
$502$	0	0
$503$	7088.41	0.628343	0.314172	−	0.949366i	$-0.398273\pi$
$503$	7088.41	0.628343	0.314172	+	0.949366i	$0.398273\pi$
$504$	0	0
$505$	−3053.61	−0.269077
$506$	0	0
$507$	−1931.35	−0.169180
$508$	0	0
$509$	−17588.4	−1.53162	−0.765810	−	0.643067i	$-0.777662\pi$
$509$	−17588.4	−1.53162	−0.765810	+	0.643067i	$0.777662\pi$
$510$	0	0
$511$	3434.91	0.297361
$512$	0	0
$513$	3310.15	0.284886
$514$	0	0
$515$	−4151.32	−0.355202
$516$	0	0
$517$	−10704.6	−0.910613
$518$	0	0
$519$	192.573	0.0162871
$520$	0	0
$521$	−11646.6	−0.979360	−0.489680	−	0.871902i	$-0.662886\pi$
$521$	−11646.6	−0.979360	−0.489680	+	0.871902i	$0.662886\pi$
$522$	0	0
$523$	−8965.82	−0.749614	−0.374807	−	0.927103i	$-0.622291\pi$
$523$	−8965.82	−0.749614	−0.374807	+	0.927103i	$0.622291\pi$
$524$	0	0
$525$	2190.28	0.182079
$526$	0	0
$527$	−1145.62	−0.0946946
$528$	0	0
$529$	5085.08	0.417941
$530$	0	0
$531$	−7559.01	−0.617765
$532$	0	0
$533$	5956.30	0.484045
$534$	0	0
$535$	531.129	0.0429210
$536$	0	0
$537$	11744.0	0.943747
$538$	0	0
$539$	1996.71	0.159563
$540$	0	0
$541$	195.272	0.0155183	0.00775914	−	0.999970i	$-0.497530\pi$
$541$	195.272	0.0155183	0.00775914	+	0.999970i	$0.497530\pi$
$542$	0	0
$543$	−6174.13	−0.487951
$544$	0	0
$545$	−3809.01	−0.299376
$546$	0	0
$547$	1399.26	0.109375	0.0546874	−	0.998504i	$-0.482584\pi$
$547$	1399.26	0.109375	0.0546874	+	0.998504i	$0.482584\pi$
$548$	0	0
$549$	4372.15	0.339888
$550$	0	0
$551$	−26554.5	−2.05310
$552$	0	0
$553$	851.837	0.0655042
$554$	0	0
$555$	162.388	0.0124198
$556$	0	0
$557$	−43.0467	−0.00327459	−0.00163730	−	0.999999i	$-0.500521\pi$
$557$	−43.0467	−0.00327459	−0.00163730	+	0.999999i	$0.500521\pi$
$558$	0	0
$559$	19677.8	1.48888
$560$	0	0
$561$	−556.206	−0.0418592
$562$	0	0
$563$	19232.9	1.43973	0.719865	−	0.694114i	$-0.244203\pi$
$563$	19232.9	1.43973	0.719865	+	0.694114i	$0.244203\pi$
$564$	0	0
$565$	−4943.75	−0.368115
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	5163.98	0.380466	0.190233	−	0.981739i	$-0.439076\pi$
$569$	5163.98	0.380466	0.190233	+	0.981739i	$0.439076\pi$
$570$	0	0
$571$	10231.9	0.749899	0.374950	−	0.927045i	$-0.377660\pi$
$571$	10231.9	0.749899	0.374950	+	0.927045i	$0.377660\pi$
$572$	0	0
$573$	−1284.14	−0.0936229
$574$	0	0
$575$	−13699.4	−0.993572
$576$	0	0
$577$	16563.7	1.19507	0.597537	−	0.801842i	$-0.296146\pi$
$577$	16563.7	1.19507	0.597537	+	0.801842i	$0.296146\pi$
$578$	0	0
$579$	−4814.78	−0.345588
$580$	0	0
$581$	−4267.26	−0.304708
$582$	0	0
$583$	23108.8	1.64163
$584$	0	0
$585$	−2182.51	−0.154249
$586$	0	0
$587$	−16020.6	−1.12648	−0.563239	−	0.826294i	$-0.690445\pi$
$587$	−16020.6	−1.12648	−0.563239	+	0.826294i	$0.690445\pi$
$588$	0	0
$589$	30869.4	2.15951
$590$	0	0
$591$	11216.5	0.780684
$592$	0	0
$593$	−6771.14	−0.468900	−0.234450	−	0.972128i	$-0.575329\pi$
$593$	−6771.14	−0.468900	−0.234450	+	0.972128i	$0.575329\pi$
$594$	0	0
$595$	144.907	0.00998421
$596$	0	0
$597$	1047.09	0.0717831
$598$	0	0
$599$	11070.2	0.755120	0.377560	−	0.925985i	$-0.376763\pi$
$599$	11070.2	0.755120	0.377560	+	0.925985i	$0.376763\pi$
$600$	0	0
$601$	−24187.7	−1.64166	−0.820830	−	0.571173i	$-0.806489\pi$
$601$	−24187.7	−1.64166	−0.820830	+	0.571173i	$0.806489\pi$
$602$	0	0
$603$	3003.22	0.202820
$604$	0	0
$605$	1499.15	0.100742
$606$	0	0
$607$	−10074.1	−0.673631	−0.336816	−	0.941571i	$-0.609350\pi$
$607$	−10074.1	−0.673631	−0.336816	+	0.941571i	$0.609350\pi$
$608$	0	0
$609$	−4548.56	−0.302655
$610$	0	0
$611$	14001.3	0.927060
$612$	0	0
$613$	11114.6	0.732323	0.366161	−	0.930551i	$-0.380672\pi$
$613$	11114.6	0.732323	0.366161	+	0.930551i	$0.380672\pi$
$614$	0	0
$615$	1525.37	0.100014
$616$	0	0
$617$	20496.4	1.33737	0.668683	−	0.743548i	$-0.266858\pi$
$617$	20496.4	1.33737	0.668683	+	0.743548i	$0.266858\pi$
$618$	0	0
$619$	16714.4	1.08532	0.542658	−	0.839954i	$-0.317418\pi$
$619$	16714.4	1.08532	0.542658	+	0.839954i	$0.317418\pi$
$620$	0	0
$621$	−3546.37	−0.229164
$622$	0	0
$623$	5033.27	0.323682
$624$	0	0
$625$	8290.66	0.530602
$626$	0	0
$627$	14987.3	0.954602
$628$	0	0
$629$	−54.1295	−0.00343129
$630$	0	0
$631$	9168.53	0.578437	0.289218	−	0.957263i	$-0.406605\pi$
$631$	9168.53	0.578437	0.289218	+	0.957263i	$0.406605\pi$
$632$	0	0
$633$	7765.73	0.487615
$634$	0	0
$635$	−2443.77	−0.152722
$636$	0	0
$637$	−2611.65	−0.162445
$638$	0	0
$639$	5312.23	0.328871
$640$	0	0
$641$	−4273.37	−0.263319	−0.131660	−	0.991295i	$-0.542031\pi$
$641$	−4273.37	−0.263319	−0.131660	+	0.991295i	$0.542031\pi$
$642$	0	0
$643$	−2955.75	−0.181281	−0.0906404	−	0.995884i	$-0.528891\pi$
$643$	−2955.75	−0.181281	−0.0906404	+	0.995884i	$0.528891\pi$
$644$	0	0
$645$	5039.34	0.307634
$646$	0	0
$647$	22701.2	1.37941	0.689704	−	0.724091i	$-0.257741\pi$
$647$	22701.2	1.37941	0.689704	+	0.724091i	$0.257741\pi$
$648$	0	0
$649$	−34224.8	−2.07002
$650$	0	0
$651$	5287.67	0.318341
$652$	0	0
$653$	−1537.81	−0.0921582	−0.0460791	−	0.998938i	$-0.514673\pi$
$653$	−1537.81	−0.0921582	−0.0460791	+	0.998938i	$0.514673\pi$
$654$	0	0
$655$	−6812.87	−0.406414
$656$	0	0
$657$	4416.31	0.262248
$658$	0	0
$659$	−12338.1	−0.729323	−0.364661	−	0.931140i	$-0.618815\pi$
$659$	−12338.1	−0.729323	−0.364661	+	0.931140i	$0.618815\pi$
$660$	0	0
$661$	−1845.10	−0.108572	−0.0542859	−	0.998525i	$-0.517288\pi$
$661$	−1845.10	−0.108572	−0.0542859	+	0.998525i	$0.517288\pi$
$662$	0	0
$663$	727.505	0.0426153
$664$	0	0
$665$	−3904.60	−0.227690
$666$	0	0
$667$	28449.5	1.65153
$668$	0	0
$669$	9708.62	0.561072
$670$	0	0
$671$	19795.7	1.13890
$672$	0	0
$673$	23955.4	1.37208	0.686041	−	0.727563i	$-0.259347\pi$
$673$	23955.4	1.37208	0.686041	+	0.727563i	$0.259347\pi$
$674$	0	0
$675$	2816.07	0.160579
$676$	0	0
$677$	3678.26	0.208814	0.104407	−	0.994535i	$-0.466706\pi$
$677$	3678.26	0.208814	0.104407	+	0.994535i	$0.466706\pi$
$678$	0	0
$679$	−4465.14	−0.252366
$680$	0	0
$681$	−16894.9	−0.950679
$682$	0	0
$683$	−4390.87	−0.245991	−0.122996	−	0.992407i	$-0.539250\pi$
$683$	−4390.87	−0.245991	−0.122996	+	0.992407i	$0.539250\pi$
$684$	0	0
$685$	−6279.18	−0.350241
$686$	0	0
$687$	11310.7	0.628140
$688$	0	0
$689$	−30225.8	−1.67128
$690$	0	0
$691$	−10371.7	−0.570994	−0.285497	−	0.958380i	$-0.592159\pi$
$691$	−10371.7	−0.570994	−0.285497	+	0.958380i	$0.592159\pi$
$692$	0	0
$693$	2567.20	0.140721
$694$	0	0
$695$	645.986	0.0352570
$696$	0	0
$697$	−508.455	−0.0276314
$698$	0	0
$699$	19682.7	1.06505
$700$	0	0
$701$	−109.675	−0.00590922	−0.00295461	−	0.999996i	$-0.500940\pi$
$701$	−109.675	−0.00590922	−0.00295461	+	0.999996i	$0.500940\pi$
$702$	0	0
$703$	1458.55	0.0782508
$704$	0	0
$705$	3585.65	0.191551
$706$	0	0
$707$	−4698.03	−0.249912
$708$	0	0
$709$	−26918.8	−1.42589	−0.712944	−	0.701221i	$-0.752639\pi$
$709$	−26918.8	−1.42589	−0.712944	+	0.701221i	$0.752639\pi$
$710$	0	0
$711$	1095.22	0.0577693
$712$	0	0
$713$	−33072.4	−1.73713
$714$	0	0
$715$	−9881.74	−0.516862
$716$	0	0
$717$	2314.33	0.120544
$718$	0	0
$719$	15170.8	0.786889	0.393445	−	0.919348i	$-0.371283\pi$
$719$	15170.8	0.786889	0.393445	+	0.919348i	$0.371283\pi$
$720$	0	0
$721$	−6386.88	−0.329903
$722$	0	0
$723$	−3756.31	−0.193221
$724$	0	0
$725$	−22591.0	−1.15725
$726$	0	0
$727$	−33286.9	−1.69813	−0.849066	−	0.528288i	$-0.822834\pi$
$727$	−33286.9	−1.69813	−0.849066	+	0.528288i	$0.822834\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−1679.78	−0.0849917
$732$	0	0
$733$	−20544.0	−1.03521	−0.517607	−	0.855619i	$-0.673177\pi$
$733$	−20544.0	−1.03521	−0.517607	+	0.855619i	$0.673177\pi$
$734$	0	0
$735$	−668.826	−0.0335646
$736$	0	0
$737$	13597.6	0.679614
$738$	0	0
$739$	−34357.2	−1.71022	−0.855109	−	0.518449i	$-0.826510\pi$
$739$	−34357.2	−1.71022	−0.855109	+	0.518449i	$0.826510\pi$
$740$	0	0
$741$	−19603.1	−0.971844
$742$	0	0
$743$	8166.99	0.403254	0.201627	−	0.979462i	$-0.435377\pi$
$743$	8166.99	0.403254	0.201627	+	0.979462i	$0.435377\pi$
$744$	0	0
$745$	8844.29	0.434939
$746$	0	0
$747$	−5486.47	−0.268728
$748$	0	0
$749$	817.151	0.0398639
$750$	0	0
$751$	17080.1	0.829909	0.414954	−	0.909842i	$-0.363798\pi$
$751$	17080.1	0.829909	0.414954	+	0.909842i	$0.363798\pi$
$752$	0	0
$753$	15498.8	0.750077
$754$	0	0
$755$	−12078.7	−0.582239
$756$	0	0
$757$	16324.0	0.783758	0.391879	−	0.920017i	$-0.371825\pi$
$757$	16324.0	0.783758	0.391879	+	0.920017i	$0.371825\pi$
$758$	0	0
$759$	−16056.9	−0.767888
$760$	0	0
$761$	−32366.2	−1.54175	−0.770875	−	0.636986i	$-0.780181\pi$
$761$	−32366.2	−1.54175	−0.770875	+	0.636986i	$0.780181\pi$
$762$	0	0
$763$	−5860.23	−0.278053
$764$	0	0
$765$	186.309	0.00880525
$766$	0	0
$767$	44765.3	2.10741
$768$	0	0
$769$	−7948.44	−0.372728	−0.186364	−	0.982481i	$-0.559670\pi$
$769$	−7948.44	−0.372728	−0.186364	+	0.982481i	$0.559670\pi$
$770$	0	0
$771$	−8302.35	−0.387810
$772$	0	0
$773$	−17819.3	−0.829127	−0.414564	−	0.910020i	$-0.636066\pi$
$773$	−17819.3	−0.829127	−0.414564	+	0.910020i	$0.636066\pi$
$774$	0	0
$775$	26261.9	1.21723
$776$	0	0
$777$	249.837	0.0115352
$778$	0	0
$779$	13700.6	0.630136
$780$	0	0
$781$	24052.1	1.10199
$782$	0	0
$783$	−5848.15	−0.266917
$784$	0	0
$785$	−7576.46	−0.344478
$786$	0	0
$787$	−2912.38	−0.131912	−0.0659562	−	0.997823i	$-0.521010\pi$
$787$	−2912.38	−0.131912	−0.0659562	+	0.997823i	$0.521010\pi$
$788$	0	0
$789$	12305.3	0.555237
$790$	0	0
$791$	−7606.05	−0.341896
$792$	0	0
$793$	−25892.3	−1.15948
$794$	0	0
$795$	−7740.63	−0.345323
$796$	0	0
$797$	33789.1	1.50172	0.750861	−	0.660460i	$-0.229639\pi$
$797$	33789.1	1.50172	0.750861	+	0.660460i	$0.229639\pi$
$798$	0	0
$799$	−1195.22	−0.0529208
$800$	0	0
$801$	6471.34	0.285460
$802$	0	0
$803$	19995.7	0.878744
$804$	0	0
$805$	4183.26	0.183156
$806$	0	0
$807$	−20852.5	−0.909595
$808$	0	0
$809$	1252.13	0.0544159	0.0272079	−	0.999630i	$-0.491338\pi$
$809$	1252.13	0.0544159	0.0272079	+	0.999630i	$0.491338\pi$
$810$	0	0
$811$	31913.1	1.38178	0.690889	−	0.722961i	$-0.257219\pi$
$811$	31913.1	1.38178	0.690889	+	0.722961i	$0.257219\pi$
$812$	0	0
$813$	21420.9	0.924063
$814$	0	0
$815$	150.477	0.00646748
$816$	0	0
$817$	45262.7	1.93824
$818$	0	0
$819$	−3357.84	−0.143263
$820$	0	0
$821$	−30742.4	−1.30684	−0.653421	−	0.756995i	$-0.726667\pi$
$821$	−30742.4	−1.30684	−0.653421	+	0.756995i	$0.726667\pi$
$822$	0	0
$823$	−13822.6	−0.585449	−0.292724	−	0.956197i	$-0.594562\pi$
$823$	−13822.6	−0.585449	−0.292724	+	0.956197i	$0.594562\pi$
$824$	0	0
$825$	12750.3	0.538071
$826$	0	0
$827$	42107.1	1.77051	0.885253	−	0.465110i	$-0.153985\pi$
$827$	42107.1	1.77051	0.885253	+	0.465110i	$0.153985\pi$
$828$	0	0
$829$	38763.8	1.62403	0.812015	−	0.583636i	$-0.198371\pi$
$829$	38763.8	1.62403	0.812015	+	0.583636i	$0.198371\pi$
$830$	0	0
$831$	3961.54	0.165372
$832$	0	0
$833$	222.942	0.00927308
$834$	0	0
$835$	7527.59	0.311980
$836$	0	0
$837$	6798.44	0.280751
$838$	0	0
$839$	16896.3	0.695262	0.347631	−	0.937631i	$-0.386986\pi$
$839$	16896.3	0.695262	0.347631	+	0.937631i	$0.386986\pi$
$840$	0	0
$841$	22525.7	0.923601
$842$	0	0
$843$	612.927	0.0250419
$844$	0	0
$845$	2929.11	0.119248
$846$	0	0
$847$	2306.47	0.0935668
$848$	0	0
$849$	2927.38	0.118336
$850$	0	0
$851$	−1562.64	−0.0629455
$852$	0	0
$853$	−46429.3	−1.86367	−0.931833	−	0.362887i	$-0.881791\pi$
$853$	−46429.3	−1.86367	−0.931833	+	0.362887i	$0.881791\pi$
$854$	0	0
$855$	−5020.21	−0.200804
$856$	0	0
$857$	21206.4	0.845272	0.422636	−	0.906300i	$-0.361105\pi$
$857$	21206.4	0.845272	0.422636	+	0.906300i	$0.361105\pi$
$858$	0	0
$859$	−13876.2	−0.551163	−0.275581	−	0.961278i	$-0.588870\pi$
$859$	−13876.2	−0.551163	−0.275581	+	0.961278i	$0.588870\pi$
$860$	0	0
$861$	2346.80	0.0928906
$862$	0	0
$863$	14337.1	0.565515	0.282757	−	0.959191i	$-0.408751\pi$
$863$	14337.1	0.565515	0.282757	+	0.959191i	$0.408751\pi$
$864$	0	0
$865$	−292.058	−0.0114801
$866$	0	0
$867$	14676.9	0.574918
$868$	0	0
$869$	4958.81	0.193574
$870$	0	0
$871$	−17785.4	−0.691889
$872$	0	0
$873$	−5740.89	−0.222566
$874$	0	0
$875$	−7302.91	−0.282152
$876$	0	0
$877$	24369.3	0.938304	0.469152	−	0.883118i	$-0.344560\pi$
$877$	24369.3	0.938304	0.469152	+	0.883118i	$0.344560\pi$
$878$	0	0
$879$	1823.76	0.0699815
$880$	0	0
$881$	−26127.0	−0.999140	−0.499570	−	0.866273i	$-0.666509\pi$
$881$	−26127.0	−0.999140	−0.499570	+	0.866273i	$0.666509\pi$
$882$	0	0
$883$	15713.1	0.598855	0.299428	−	0.954119i	$-0.403204\pi$
$883$	15713.1	0.598855	0.299428	+	0.954119i	$0.403204\pi$
$884$	0	0
$885$	11464.1	0.435436
$886$	0	0
$887$	13139.5	0.497385	0.248692	−	0.968583i	$-0.419999\pi$
$887$	13139.5	0.497385	0.248692	+	0.968583i	$0.419999\pi$
$888$	0	0
$889$	−3759.79	−0.141844
$890$	0	0
$891$	3300.68	0.124104
$892$	0	0
$893$	32205.8	1.20686
$894$	0	0
$895$	−17811.1	−0.665207
$896$	0	0
$897$	21002.0	0.781758
$898$	0	0
$899$	−54538.1	−2.02330
$900$	0	0
$901$	2580.21	0.0954043
$902$	0	0
$903$	7753.12	0.285723
$904$	0	0
$905$	9363.76	0.343936
$906$	0	0
$907$	3799.71	0.139104	0.0695519	−	0.997578i	$-0.477843\pi$
$907$	3799.71	0.139104	0.0695519	+	0.997578i	$0.477843\pi$
$908$	0	0
$909$	−6040.33	−0.220402
$910$	0	0
$911$	−51528.4	−1.87400	−0.936998	−	0.349334i	$-0.886408\pi$
$911$	−51528.4	−1.87400	−0.936998	+	0.349334i	$0.886408\pi$
$912$	0	0
$913$	−24841.0	−0.900458
$914$	0	0
$915$	−6630.85	−0.239573
$916$	0	0
$917$	−10481.7	−0.377467
$918$	0	0
$919$	−16984.7	−0.609657	−0.304828	−	0.952407i	$-0.598599\pi$
$919$	−16984.7	−0.609657	−0.304828	+	0.952407i	$0.598599\pi$
$920$	0	0
$921$	24111.2	0.862641
$922$	0	0
$923$	−31459.6	−1.12189
$924$	0	0
$925$	1240.85	0.0441068
$926$	0	0
$927$	−8211.71	−0.290947
$928$	0	0
$929$	−5451.85	−0.192540	−0.0962699	−	0.995355i	$-0.530691\pi$
$929$	−5451.85	−0.192540	−0.0962699	+	0.995355i	$0.530691\pi$
$930$	0	0
$931$	−6007.30	−0.211473
$932$	0	0
$933$	−15934.8	−0.559145
$934$	0	0
$935$	843.548	0.0295048
$936$	0	0
$937$	42429.4	1.47930	0.739652	−	0.672989i	$-0.234990\pi$
$937$	42429.4	1.47930	0.739652	+	0.672989i	$0.234990\pi$
$938$	0	0
$939$	4594.82	0.159687
$940$	0	0
$941$	32977.9	1.14245	0.571226	−	0.820793i	$-0.306468\pi$
$941$	32977.9	1.14245	0.571226	+	0.820793i	$0.306468\pi$
$942$	0	0
$943$	−14678.4	−0.506886
$944$	0	0
$945$	−859.919	−0.0296012
$946$	0	0
$947$	23753.4	0.815082	0.407541	−	0.913187i	$-0.366386\pi$
$947$	23753.4	0.815082	0.407541	+	0.913187i	$0.366386\pi$
$948$	0	0
$949$	−26153.9	−0.894616
$950$	0	0
$951$	12657.6	0.431598
$952$	0	0
$953$	−28074.3	−0.954267	−0.477134	−	0.878831i	$-0.658324\pi$
$953$	−28074.3	−0.954267	−0.477134	+	0.878831i	$0.658324\pi$
$954$	0	0
$955$	1947.55	0.0659908
$956$	0	0
$957$	−26478.6	−0.894389
$958$	0	0
$959$	−9660.63	−0.325295
$960$	0	0
$961$	33609.2	1.12817
$962$	0	0
$963$	1050.62	0.0351567
$964$	0	0
$965$	7302.15	0.243590
$966$	0	0
$967$	−11150.3	−0.370806	−0.185403	−	0.982663i	$-0.559359\pi$
$967$	−11150.3	−0.370806	−0.185403	+	0.982663i	$0.559359\pi$
$968$	0	0
$969$	1673.40	0.0554772
$970$	0	0
$971$	−6059.04	−0.200251	−0.100126	−	0.994975i	$-0.531924\pi$
$971$	−6059.04	−0.200251	−0.100126	+	0.994975i	$0.531924\pi$
$972$	0	0
$973$	993.861	0.0327459
$974$	0	0
$975$	−16677.1	−0.547789
$976$	0	0
$977$	5700.49	0.186668	0.0933341	−	0.995635i	$-0.470248\pi$
$977$	5700.49	0.186668	0.0933341	+	0.995635i	$0.470248\pi$
$978$	0	0
$979$	29300.2	0.956526
$980$	0	0
$981$	−7534.59	−0.245220
$982$	0	0
$983$	197.480	0.00640757	0.00320378	−	0.999995i	$-0.498980\pi$
$983$	197.480	0.00640757	0.00320378	+	0.999995i	$0.498980\pi$
$984$	0	0
$985$	−17011.0	−0.550271
$986$	0	0
$987$	5516.58	0.177908
$988$	0	0
$989$	−48492.9	−1.55913
$990$	0	0
$991$	20620.8	0.660990	0.330495	−	0.943808i	$-0.392784\pi$
$991$	20620.8	0.660990	0.330495	+	0.943808i	$0.392784\pi$
$992$	0	0
$993$	24894.6	0.795574
$994$	0	0
$995$	−1588.03	−0.0505969
$996$	0	0
$997$	−19326.8	−0.613928	−0.306964	−	0.951721i	$-0.599313\pi$
$997$	−19326.8	−0.613928	−0.306964	+	0.951721i	$0.599313\pi$
$998$	0	0
$999$	321.220	0.0101731

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.a.bg.1.2		2
4.3	odd	2		1344.4.a.bo.1.2		2
8.3	odd	2		336.4.a.m.1.1		2
8.5	even	2		21.4.a.c.1.2	✓	2
24.5	odd	2		63.4.a.e.1.1		2
24.11	even	2		1008.4.a.ba.1.2		2
40.13	odd	4		525.4.d.g.274.2		4
40.29	even	2		525.4.a.n.1.1		2
40.37	odd	4		525.4.d.g.274.3		4
56.5	odd	6		147.4.e.m.67.1		4
56.13	odd	2		147.4.a.i.1.2		2
56.27	even	2		2352.4.a.bz.1.2		2
56.37	even	6		147.4.e.l.67.1		4
56.45	odd	6		147.4.e.m.79.1		4
56.53	even	6		147.4.e.l.79.1		4
120.29	odd	2		1575.4.a.p.1.2		2
168.5	even	6		441.4.e.p.361.2		4
168.53	odd	6		441.4.e.q.226.2		4
168.101	even	6		441.4.e.p.226.2		4
168.125	even	2		441.4.a.r.1.1		2
168.149	odd	6		441.4.e.q.361.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.a.c.1.2	✓	2	8.5	even	2
63.4.a.e.1.1		2	24.5	odd	2
147.4.a.i.1.2		2	56.13	odd	2
147.4.e.l.67.1		4	56.37	even	6
147.4.e.l.79.1		4	56.53	even	6
147.4.e.m.67.1		4	56.5	odd	6
147.4.e.m.79.1		4	56.45	odd	6
336.4.a.m.1.1		2	8.3	odd	2
441.4.a.r.1.1		2	168.125	even	2
441.4.e.p.226.2		4	168.101	even	6
441.4.e.p.361.2		4	168.5	even	6
441.4.e.q.226.2		4	168.53	odd	6
441.4.e.q.361.2		4	168.149	odd	6
525.4.a.n.1.1		2	40.29	even	2
525.4.d.g.274.2		4	40.13	odd	4
525.4.d.g.274.3		4	40.37	odd	4
1008.4.a.ba.1.2		2	24.11	even	2
1344.4.a.bg.1.2		2	1.1	even	1	trivial
1344.4.a.bo.1.2		2	4.3	odd	2
1575.4.a.p.1.2		2	120.29	odd	2
2352.4.a.bz.1.2		2	56.27	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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +4.54983 q^{5} +7.00000 q^{7} +9.00000 q^{9} +40.7492 q^{11} -53.2990 q^{13} -13.6495 q^{15} +4.54983 q^{17} -122.598 q^{19} -21.0000 q^{21} +131.347 q^{23} -104.299 q^{25} -27.0000 q^{27} +216.598 q^{29} -251.794 q^{31} -122.248 q^{33} +31.8488 q^{35} -11.8970 q^{37} +159.897 q^{39} -111.752 q^{41} -369.196 q^{43} +40.9485 q^{45} -262.694 q^{47} +49.0000 q^{49} -13.6495 q^{51} +567.100 q^{53} +185.402 q^{55} +367.794 q^{57} -839.890 q^{59} +485.794 q^{61} +63.0000 q^{63} -242.502 q^{65} +333.691 q^{67} -394.042 q^{69} +590.248 q^{71} +490.701 q^{73} +312.897 q^{75} +285.244 q^{77} +121.691 q^{79} +81.0000 q^{81} -609.608 q^{83} +20.7010 q^{85} -649.794 q^{87} +719.038 q^{89} -373.093 q^{91} +755.382 q^{93} -557.801 q^{95} -637.877 q^{97} +366.743 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 6 q^{5} + 14 q^{7} + 18 q^{9} + 6 q^{11} - 16 q^{13} + 18 q^{15} - 6 q^{17} - 64 q^{19} - 42 q^{21} + 6 q^{23} - 118 q^{25} - 54 q^{27} + 252 q^{29} + 40 q^{31} - 18 q^{33} - 42 q^{35} + 248 q^{37}+ \cdots + 54 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 6 * q^5 + 14 * q^7 + 18 * q^9 + 6 * q^11 - 16 * q^13 + 18 * q^15 - 6 * q^17 - 64 * q^19 - 42 * q^21 + 6 * q^23 - 118 * q^25 - 54 * q^27 + 252 * q^29 + 40 * q^31 - 18 * q^33 - 42 * q^35 + 248 * q^37 + 48 * q^39 - 450 * q^41 - 376 * q^43 - 54 * q^45 - 12 * q^47 + 98 * q^49 + 18 * q^51 + 1104 * q^53 + 552 * q^55 + 192 * q^57 - 804 * q^59 + 428 * q^61 + 126 * q^63 - 636 * q^65 - 148 * q^67 - 18 * q^69 + 954 * q^71 + 1072 * q^73 + 354 * q^75 + 42 * q^77 - 572 * q^79 + 162 * q^81 - 1944 * q^83 + 132 * q^85 - 756 * q^87 + 366 * q^89 - 112 * q^91 - 120 * q^93 - 1176 * q^95 + 808 * q^97 + 54 * q^99

Introduction

L-functions

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