LMFDB - Embedded newform 1668.2.a.h.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1668,2,Mod(1,1668)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1668.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1668, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1668 = 2^{2} \cdot 3 \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1668.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,0,7,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$13.3190470571$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 28x^{5} - x^{4} + 217x^{3} + 51x^{2} - 456x - 300 $ x^7 - 28x^5 - x^4 + 217x^3 + 51x^2 - 456x - 300
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-4.13540$ of defining polynomial
Character	$\chi$	$=$	1668.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.13540 q^{5} +2.03932 q^{7} +1.00000 q^{9} -1.63586 q^{11} -3.68630 q^{13} +4.13540 q^{15} +7.73194 q^{17} -7.03591 q^{19} +2.03932 q^{21} -2.51472 q^{23} +12.1015 q^{25} +1.00000 q^{27} +8.58681 q^{29} +6.49073 q^{31} -1.63586 q^{33} +8.43342 q^{35} +5.82512 q^{37} -3.68630 q^{39} -8.79292 q^{41} -8.37574 q^{43} +4.13540 q^{45} -3.49705 q^{47} -2.84115 q^{49} +7.73194 q^{51} -10.3021 q^{53} -6.76495 q^{55} -7.03591 q^{57} +10.6760 q^{59} -5.86912 q^{61} +2.03932 q^{63} -15.2443 q^{65} +12.6719 q^{67} -2.51472 q^{69} +2.43684 q^{71} +0.344793 q^{73} +12.1015 q^{75} -3.33606 q^{77} -16.7797 q^{79} +1.00000 q^{81} -0.835705 q^{83} +31.9746 q^{85} +8.58681 q^{87} -12.4414 q^{89} -7.51756 q^{91} +6.49073 q^{93} -29.0963 q^{95} +7.35440 q^{97} -1.63586 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{3} + q^{7} + 7 q^{9} + 8 q^{13} + 27 q^{17} + q^{21} - 8 q^{23} + 21 q^{25} + 7 q^{27} + 6 q^{29} + 7 q^{31} + 6 q^{35} + 14 q^{37} + 8 q^{39} + q^{41} + 27 q^{43} - 6 q^{47} + 32 q^{49} + 27 q^{51}+ \cdots + 7 q^{97}+O(q^{100}) $ 7 * q + 7 * q^3 + q^7 + 7 * q^9 + 8 * q^13 + 27 * q^17 + q^21 - 8 * q^23 + 21 * q^25 + 7 * q^27 + 6 * q^29 + 7 * q^31 + 6 * q^35 + 14 * q^37 + 8 * q^39 + q^41 + 27 * q^43 - 6 * q^47 + 32 * q^49 + 27 * q^51 + 18 * q^53 + 13 * q^55 + 17 * q^59 + 13 * q^61 + q^63 - 19 * q^65 + 31 * q^67 - 8 * q^69 + 6 * q^73 + 21 * q^75 - 34 * q^77 + 2 * q^79 + 7 * q^81 - 2 * q^83 + 37 * q^85 + 6 * q^87 + 13 * q^89 + 39 * q^91 + 7 * q^93 - 36 * q^95 + 7 * 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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	4.13540	1.84941	0.924703	−	0.380689i	$-0.124313\pi$
$5$	4.13540	1.84941	0.924703	+	0.380689i	$0.124313\pi$
$6$	0	0
$7$	2.03932	0.770792	0.385396	−	0.922751i	$-0.374065\pi$
$7$	2.03932	0.770792	0.385396	+	0.922751i	$0.374065\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.63586	−0.493232	−0.246616	−	0.969113i	$-0.579319\pi$
$11$	−1.63586	−0.493232	−0.246616	+	0.969113i	$0.579319\pi$
$12$	0	0
$13$	−3.68630	−1.02240	−0.511198	−	0.859463i	$-0.670798\pi$
$13$	−3.68630	−1.02240	−0.511198	+	0.859463i	$0.670798\pi$
$14$	0	0
$15$	4.13540	1.06776
$16$	0	0
$17$	7.73194	1.87527	0.937635	−	0.347621i	$-0.113010\pi$
$17$	7.73194	1.87527	0.937635	+	0.347621i	$0.113010\pi$
$18$	0	0
$19$	−7.03591	−1.61415	−0.807074	−	0.590451i	$-0.798950\pi$
$19$	−7.03591	−1.61415	−0.807074	+	0.590451i	$0.798950\pi$
$20$	0	0
$21$	2.03932	0.445017
$22$	0	0
$23$	−2.51472	−0.524355	−0.262177	−	0.965020i	$-0.584441\pi$
$23$	−2.51472	−0.524355	−0.262177	+	0.965020i	$0.584441\pi$
$24$	0	0
$25$	12.1015	2.42030
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.58681	1.59453	0.797265	−	0.603629i	$-0.206279\pi$
$29$	8.58681	1.59453	0.797265	+	0.603629i	$0.206279\pi$
$30$	0	0
$31$	6.49073	1.16577	0.582885	−	0.812555i	$-0.301924\pi$
$31$	6.49073	1.16577	0.582885	+	0.812555i	$0.301924\pi$
$32$	0	0
$33$	−1.63586	−0.284767
$34$	0	0
$35$	8.43342	1.42551
$36$	0	0
$37$	5.82512	0.957644	0.478822	−	0.877912i	$-0.341064\pi$
$37$	5.82512	0.957644	0.478822	+	0.877912i	$0.341064\pi$
$38$	0	0
$39$	−3.68630	−0.590281
$40$	0	0
$41$	−8.79292	−1.37322	−0.686611	−	0.727025i	$-0.740903\pi$
$41$	−8.79292	−1.37322	−0.686611	+	0.727025i	$0.740903\pi$
$42$	0	0
$43$	−8.37574	−1.27729	−0.638644	−	0.769502i	$-0.720504\pi$
$43$	−8.37574	−1.27729	−0.638644	+	0.769502i	$0.720504\pi$
$44$	0	0
$45$	4.13540	0.616469
$46$	0	0
$47$	−3.49705	−0.510097	−0.255048	−	0.966928i	$-0.582091\pi$
$47$	−3.49705	−0.510097	−0.255048	+	0.966928i	$0.582091\pi$
$48$	0	0
$49$	−2.84115	−0.405879
$50$	0	0
$51$	7.73194	1.08269
$52$	0	0
$53$	−10.3021	−1.41511	−0.707554	−	0.706660i	$-0.750201\pi$
$53$	−10.3021	−1.41511	−0.707554	+	0.706660i	$0.750201\pi$
$54$	0	0
$55$	−6.76495	−0.912186
$56$	0	0
$57$	−7.03591	−0.931928
$58$	0	0
$59$	10.6760	1.38990	0.694951	−	0.719057i	$-0.255426\pi$
$59$	10.6760	1.38990	0.694951	+	0.719057i	$0.255426\pi$
$60$	0	0
$61$	−5.86912	−0.751464	−0.375732	−	0.926728i	$-0.622609\pi$
$61$	−5.86912	−0.751464	−0.375732	+	0.926728i	$0.622609\pi$
$62$	0	0
$63$	2.03932	0.256931
$64$	0	0
$65$	−15.2443	−1.89083
$66$	0	0
$67$	12.6719	1.54812	0.774058	−	0.633114i	$-0.218224\pi$
$67$	12.6719	1.54812	0.774058	+	0.633114i	$0.218224\pi$
$68$	0	0
$69$	−2.51472	−0.302736
$70$	0	0
$71$	2.43684	0.289200	0.144600	−	0.989490i	$-0.453811\pi$
$71$	2.43684	0.289200	0.144600	+	0.989490i	$0.453811\pi$
$72$	0	0
$73$	0.344793	0.0403550	0.0201775	−	0.999796i	$-0.493577\pi$
$73$	0.344793	0.0403550	0.0201775	+	0.999796i	$0.493577\pi$
$74$	0	0
$75$	12.1015	1.39736
$76$	0	0
$77$	−3.33606	−0.380179
$78$	0	0
$79$	−16.7797	−1.88787	−0.943934	−	0.330136i	$-0.892906\pi$
$79$	−16.7797	−1.88787	−0.943934	+	0.330136i	$0.892906\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.835705	−0.0917305	−0.0458652	−	0.998948i	$-0.514604\pi$
$83$	−0.835705	−0.0917305	−0.0458652	+	0.998948i	$0.514604\pi$
$84$	0	0
$85$	31.9746	3.46814
$86$	0	0
$87$	8.58681	0.920602
$88$	0	0
$89$	−12.4414	−1.31878	−0.659391	−	0.751800i	$-0.729186\pi$
$89$	−12.4414	−1.31878	−0.659391	+	0.751800i	$0.729186\pi$
$90$	0	0
$91$	−7.51756	−0.788055
$92$	0	0
$93$	6.49073	0.673058
$94$	0	0
$95$	−29.0963	−2.98521
$96$	0	0
$97$	7.35440	0.746727	0.373363	−	0.927685i	$-0.378204\pi$
$97$	7.35440	0.746727	0.373363	+	0.927685i	$0.378204\pi$
$98$	0	0
$99$	−1.63586	−0.164411
$100$	0	0
$101$	14.7415	1.46684	0.733420	−	0.679776i	$-0.237923\pi$
$101$	14.7415	1.46684	0.733420	+	0.679776i	$0.237923\pi$
$102$	0	0
$103$	−8.92287	−0.879196	−0.439598	−	0.898195i	$-0.644879\pi$
$103$	−8.92287	−0.879196	−0.439598	+	0.898195i	$0.644879\pi$
$104$	0	0
$105$	8.43342	0.823018
$106$	0	0
$107$	−6.76418	−0.653918	−0.326959	−	0.945039i	$-0.606024\pi$
$107$	−6.76418	−0.653918	−0.326959	+	0.945039i	$0.606024\pi$
$108$	0	0
$109$	9.52098	0.911945	0.455972	−	0.889994i	$-0.349292\pi$
$109$	9.52098	0.911945	0.455972	+	0.889994i	$0.349292\pi$
$110$	0	0
$111$	5.82512	0.552896
$112$	0	0
$113$	−4.15857	−0.391205	−0.195602	−	0.980683i	$-0.562666\pi$
$113$	−4.15857	−0.391205	−0.195602	+	0.980683i	$0.562666\pi$
$114$	0	0
$115$	−10.3994	−0.969745
$116$	0	0
$117$	−3.68630	−0.340799
$118$	0	0
$119$	15.7679	1.44544
$120$	0	0
$121$	−8.32395	−0.756723
$122$	0	0
$123$	−8.79292	−0.792831
$124$	0	0
$125$	29.3676	2.62672
$126$	0	0
$127$	−0.451409	−0.0400560	−0.0200280	−	0.999799i	$-0.506376\pi$
$127$	−0.451409	−0.0400560	−0.0200280	+	0.999799i	$0.506376\pi$
$128$	0	0
$129$	−8.37574	−0.737443
$130$	0	0
$131$	−3.87376	−0.338452	−0.169226	−	0.985577i	$-0.554127\pi$
$131$	−3.87376	−0.338452	−0.169226	+	0.985577i	$0.554127\pi$
$132$	0	0
$133$	−14.3485	−1.24417
$134$	0	0
$135$	4.13540	0.355918
$136$	0	0
$137$	−9.97239	−0.851999	−0.425999	−	0.904723i	$-0.640077\pi$
$137$	−9.97239	−0.851999	−0.425999	+	0.904723i	$0.640077\pi$
$138$	0	0
$139$	1.00000	0.0848189
$139$	1.00000	0.0848189
$140$	0	0
$141$	−3.49705	−0.294504
$142$	0	0
$143$	6.03029	0.504278
$144$	0	0
$145$	35.5099	2.94893
$146$	0	0
$147$	−2.84115	−0.234334
$148$	0	0
$149$	5.00367	0.409916	0.204958	−	0.978771i	$-0.434294\pi$
$149$	5.00367	0.409916	0.204958	+	0.978771i	$0.434294\pi$
$150$	0	0
$151$	21.1194	1.71868	0.859338	−	0.511408i	$-0.170876\pi$
$151$	21.1194	1.71868	0.859338	+	0.511408i	$0.170876\pi$
$152$	0	0
$153$	7.73194	0.625090
$154$	0	0
$155$	26.8418	2.15598
$156$	0	0
$157$	16.4685	1.31433	0.657165	−	0.753747i	$-0.271755\pi$
$157$	16.4685	1.31433	0.657165	+	0.753747i	$0.271755\pi$
$158$	0	0
$159$	−10.3021	−0.817013
$160$	0	0
$161$	−5.12832	−0.404169
$162$	0	0
$163$	−2.51390	−0.196904	−0.0984519	−	0.995142i	$-0.531389\pi$
$163$	−2.51390	−0.196904	−0.0984519	+	0.995142i	$0.531389\pi$
$164$	0	0
$165$	−6.76495	−0.526651
$166$	0	0
$167$	−0.757415	−0.0586105	−0.0293052	−	0.999571i	$-0.509329\pi$
$167$	−0.757415	−0.0586105	−0.0293052	+	0.999571i	$0.509329\pi$
$168$	0	0
$169$	0.588815	0.0452935
$170$	0	0
$171$	−7.03591	−0.538049
$172$	0	0
$173$	−14.0438	−1.06773	−0.533867	−	0.845569i	$-0.679262\pi$
$173$	−14.0438	−1.06773	−0.533867	+	0.845569i	$0.679262\pi$
$174$	0	0
$175$	24.6789	1.86555
$176$	0	0
$177$	10.6760	0.802460
$178$	0	0
$179$	−2.27312	−0.169901	−0.0849504	−	0.996385i	$-0.527073\pi$
$179$	−2.27312	−0.169901	−0.0849504	+	0.996385i	$0.527073\pi$
$180$	0	0
$181$	0.399188	0.0296714	0.0148357	−	0.999890i	$-0.495277\pi$
$181$	0.399188	0.0296714	0.0148357	+	0.999890i	$0.495277\pi$
$182$	0	0
$183$	−5.86912	−0.433858
$184$	0	0
$185$	24.0892	1.77107
$186$	0	0
$187$	−12.6484	−0.924943
$188$	0	0
$189$	2.03932	0.148339
$190$	0	0
$191$	−18.1542	−1.31359	−0.656794	−	0.754070i	$-0.728088\pi$
$191$	−18.1542	−1.31359	−0.656794	+	0.754070i	$0.728088\pi$
$192$	0	0
$193$	−6.03028	−0.434069	−0.217034	−	0.976164i	$-0.569638\pi$
$193$	−6.03028	−0.434069	−0.217034	+	0.976164i	$0.569638\pi$
$194$	0	0
$195$	−15.2443	−1.09167
$196$	0	0
$197$	3.04327	0.216824	0.108412	−	0.994106i	$-0.465423\pi$
$197$	3.04327	0.216824	0.108412	+	0.994106i	$0.465423\pi$
$198$	0	0
$199$	0.296733	0.0210349	0.0105174	−	0.999945i	$-0.496652\pi$
$199$	0.296733	0.0210349	0.0105174	+	0.999945i	$0.496652\pi$
$200$	0	0
$201$	12.6719	0.893806
$202$	0	0
$203$	17.5113	1.22905
$204$	0	0
$205$	−36.3622	−2.53965
$206$	0	0
$207$	−2.51472	−0.174785
$208$	0	0
$209$	11.5098	0.796148
$210$	0	0
$211$	7.68707	0.529200	0.264600	−	0.964358i	$-0.414760\pi$
$211$	7.68707	0.529200	0.264600	+	0.964358i	$0.414760\pi$
$212$	0	0
$213$	2.43684	0.166970
$214$	0	0
$215$	−34.6370	−2.36223
$216$	0	0
$217$	13.2367	0.898567
$218$	0	0
$219$	0.344793	0.0232989
$220$	0	0
$221$	−28.5023	−1.91727
$222$	0	0
$223$	−18.2622	−1.22293	−0.611465	−	0.791271i	$-0.709420\pi$
$223$	−18.2622	−1.22293	−0.611465	+	0.791271i	$0.709420\pi$
$224$	0	0
$225$	12.1015	0.806768
$226$	0	0
$227$	−22.4953	−1.49306	−0.746531	−	0.665350i	$-0.768282\pi$
$227$	−22.4953	−1.49306	−0.746531	+	0.665350i	$0.768282\pi$
$228$	0	0
$229$	22.9601	1.51724	0.758622	−	0.651531i	$-0.225873\pi$
$229$	22.9601	1.51724	0.758622	+	0.651531i	$0.225873\pi$
$230$	0	0
$231$	−3.33606	−0.219496
$232$	0	0
$233$	−15.9082	−1.04218	−0.521089	−	0.853502i	$-0.674474\pi$
$233$	−15.9082	−1.04218	−0.521089	+	0.853502i	$0.674474\pi$
$234$	0	0
$235$	−14.4617	−0.943376
$236$	0	0
$237$	−16.7797	−1.08996
$238$	0	0
$239$	−27.3799	−1.77106	−0.885528	−	0.464586i	$-0.846203\pi$
$239$	−27.3799	−1.77106	−0.885528	+	0.464586i	$0.846203\pi$
$240$	0	0
$241$	13.8747	0.893748	0.446874	−	0.894597i	$-0.352537\pi$
$241$	13.8747	0.893748	0.446874	+	0.894597i	$0.352537\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−11.7493	−0.750636
$246$	0	0
$247$	25.9365	1.65030
$248$	0	0
$249$	−0.835705	−0.0529606
$250$	0	0
$251$	9.45446	0.596760	0.298380	−	0.954447i	$-0.403554\pi$
$251$	9.45446	0.596760	0.298380	+	0.954447i	$0.403554\pi$
$252$	0	0
$253$	4.11373	0.258628
$254$	0	0
$255$	31.9746	2.00233
$256$	0	0
$257$	−28.3953	−1.77125	−0.885626	−	0.464399i	$-0.846270\pi$
$257$	−28.3953	−1.77125	−0.885626	+	0.464399i	$0.846270\pi$
$258$	0	0
$259$	11.8793	0.738144
$260$	0	0
$261$	8.58681	0.531510
$262$	0	0
$263$	19.8324	1.22292	0.611458	−	0.791277i	$-0.290583\pi$
$263$	19.8324	1.22292	0.611458	+	0.791277i	$0.290583\pi$
$264$	0	0
$265$	−42.6035	−2.61711
$266$	0	0
$267$	−12.4414	−0.761399
$268$	0	0
$269$	−3.92600	−0.239373	−0.119686	−	0.992812i	$-0.538189\pi$
$269$	−3.92600	−0.239373	−0.119686	+	0.992812i	$0.538189\pi$
$270$	0	0
$271$	12.4080	0.753735	0.376868	−	0.926267i	$-0.377001\pi$
$271$	12.4080	0.753735	0.376868	+	0.926267i	$0.377001\pi$
$272$	0	0
$273$	−7.51756	−0.454984
$274$	0	0
$275$	−19.7964	−1.19377
$276$	0	0
$277$	5.65186	0.339588	0.169794	−	0.985480i	$-0.445690\pi$
$277$	5.65186	0.339588	0.169794	+	0.985480i	$0.445690\pi$
$278$	0	0
$279$	6.49073	0.388590
$280$	0	0
$281$	−23.7644	−1.41766	−0.708832	−	0.705377i	$-0.750778\pi$
$281$	−23.7644	−1.41766	−0.708832	+	0.705377i	$0.750778\pi$
$282$	0	0
$283$	−6.66574	−0.396237	−0.198118	−	0.980178i	$-0.563483\pi$
$283$	−6.66574	−0.396237	−0.198118	+	0.980178i	$0.563483\pi$
$284$	0	0
$285$	−29.0963	−1.72351
$286$	0	0
$287$	−17.9316	−1.05847
$288$	0	0
$289$	42.7829	2.51664
$290$	0	0
$291$	7.35440	0.431123
$292$	0	0
$293$	9.13750	0.533819	0.266909	−	0.963722i	$-0.413998\pi$
$293$	9.13750	0.533819	0.266909	+	0.963722i	$0.413998\pi$
$294$	0	0
$295$	44.1497	2.57049
$296$	0	0
$297$	−1.63586	−0.0949225
$298$	0	0
$299$	9.27000	0.536098
$300$	0	0
$301$	−17.0809	−0.984524
$302$	0	0
$303$	14.7415	0.846880
$304$	0	0
$305$	−24.2712	−1.38976
$306$	0	0
$307$	−4.89637	−0.279450	−0.139725	−	0.990190i	$-0.544622\pi$
$307$	−4.89637	−0.279450	−0.139725	+	0.990190i	$0.544622\pi$
$308$	0	0
$309$	−8.92287	−0.507604
$310$	0	0
$311$	−9.23896	−0.523893	−0.261947	−	0.965082i	$-0.584364\pi$
$311$	−9.23896	−0.523893	−0.261947	+	0.965082i	$0.584364\pi$
$312$	0	0
$313$	20.8619	1.17919	0.589593	−	0.807701i	$-0.299288\pi$
$313$	20.8619	1.17919	0.589593	+	0.807701i	$0.299288\pi$
$314$	0	0
$315$	8.43342	0.475169
$316$	0	0
$317$	25.9505	1.45753	0.728763	−	0.684766i	$-0.240096\pi$
$317$	25.9505	1.45753	0.728763	+	0.684766i	$0.240096\pi$
$318$	0	0
$319$	−14.0468	−0.786473
$320$	0	0
$321$	−6.76418	−0.377540
$322$	0	0
$323$	−54.4012	−3.02696
$324$	0	0
$325$	−44.6099	−2.47451
$326$	0	0
$327$	9.52098	0.526512
$328$	0	0
$329$	−7.13161	−0.393178
$330$	0	0
$331$	33.5234	1.84261	0.921307	−	0.388836i	$-0.127123\pi$
$331$	33.5234	1.84261	0.921307	+	0.388836i	$0.127123\pi$
$332$	0	0
$333$	5.82512	0.319215
$334$	0	0
$335$	52.4033	2.86310
$336$	0	0
$337$	1.66843	0.0908850	0.0454425	−	0.998967i	$-0.485530\pi$
$337$	1.66843	0.0908850	0.0454425	+	0.998967i	$0.485530\pi$
$338$	0	0
$339$	−4.15857	−0.225862
$340$	0	0
$341$	−10.6180	−0.574995
$342$	0	0
$343$	−20.0693	−1.08364
$344$	0	0
$345$	−10.3994	−0.559883
$346$	0	0
$347$	−13.3632	−0.717372	−0.358686	−	0.933458i	$-0.616775\pi$
$347$	−13.3632	−0.717372	−0.358686	+	0.933458i	$0.616775\pi$
$348$	0	0
$349$	16.0944	0.861511	0.430756	−	0.902469i	$-0.358247\pi$
$349$	16.0944	0.861511	0.430756	+	0.902469i	$0.358247\pi$
$350$	0	0
$351$	−3.68630	−0.196760
$352$	0	0
$353$	33.2217	1.76821	0.884107	−	0.467285i	$-0.154768\pi$
$353$	33.2217	1.76821	0.884107	+	0.467285i	$0.154768\pi$
$354$	0	0
$355$	10.0773	0.534848
$356$	0	0
$357$	15.7679	0.834527
$358$	0	0
$359$	−13.2073	−0.697056	−0.348528	−	0.937298i	$-0.613318\pi$
$359$	−13.2073	−0.697056	−0.348528	+	0.937298i	$0.613318\pi$
$360$	0	0
$361$	30.5040	1.60547
$362$	0	0
$363$	−8.32395	−0.436894
$364$	0	0
$365$	1.42586	0.0746327
$366$	0	0
$367$	−8.06665	−0.421076	−0.210538	−	0.977586i	$-0.567522\pi$
$367$	−8.06665	−0.421076	−0.210538	+	0.977586i	$0.567522\pi$
$368$	0	0
$369$	−8.79292	−0.457741
$370$	0	0
$371$	−21.0094	−1.09075
$372$	0	0
$373$	−19.9397	−1.03244	−0.516219	−	0.856457i	$-0.672661\pi$
$373$	−19.9397	−1.03244	−0.516219	+	0.856457i	$0.672661\pi$
$374$	0	0
$375$	29.3676	1.51654
$376$	0	0
$377$	−31.6536	−1.63024
$378$	0	0
$379$	−8.49734	−0.436479	−0.218240	−	0.975895i	$-0.570031\pi$
$379$	−8.49734	−0.436479	−0.218240	+	0.975895i	$0.570031\pi$
$380$	0	0
$381$	−0.451409	−0.0231264
$382$	0	0
$383$	−6.64754	−0.339673	−0.169837	−	0.985472i	$-0.554324\pi$
$383$	−6.64754	−0.339673	−0.169837	+	0.985472i	$0.554324\pi$
$384$	0	0
$385$	−13.7959	−0.703106
$386$	0	0
$387$	−8.37574	−0.425763
$388$	0	0
$389$	7.43514	0.376976	0.188488	−	0.982075i	$-0.439641\pi$
$389$	7.43514	0.376976	0.188488	+	0.982075i	$0.439641\pi$
$390$	0	0
$391$	−19.4436	−0.983307
$392$	0	0
$393$	−3.87376	−0.195405
$394$	0	0
$395$	−69.3909	−3.49143
$396$	0	0
$397$	−23.0854	−1.15862	−0.579311	−	0.815107i	$-0.696678\pi$
$397$	−23.0854	−1.15862	−0.579311	+	0.815107i	$0.696678\pi$
$398$	0	0
$399$	−14.3485	−0.718323
$400$	0	0
$401$	−1.98546	−0.0991490	−0.0495745	−	0.998770i	$-0.515787\pi$
$401$	−1.98546	−0.0991490	−0.0495745	+	0.998770i	$0.515787\pi$
$402$	0	0
$403$	−23.9268	−1.19188
$404$	0	0
$405$	4.13540	0.205490
$406$	0	0
$407$	−9.52910	−0.472340
$408$	0	0
$409$	−18.8487	−0.932008	−0.466004	−	0.884783i	$-0.654307\pi$
$409$	−18.8487	−0.932008	−0.466004	+	0.884783i	$0.654307\pi$
$410$	0	0
$411$	−9.97239	−0.491902
$412$	0	0
$413$	21.7719	1.07133
$414$	0	0
$415$	−3.45597	−0.169647
$416$	0	0
$417$	1.00000	0.0489702
$418$	0	0
$419$	−22.7502	−1.11142	−0.555709	−	0.831377i	$-0.687553\pi$
$419$	−22.7502	−1.11142	−0.555709	+	0.831377i	$0.687553\pi$
$420$	0	0
$421$	−35.0392	−1.70771	−0.853854	−	0.520512i	$-0.825741\pi$
$421$	−35.0392	−1.70771	−0.853854	+	0.520512i	$0.825741\pi$
$422$	0	0
$423$	−3.49705	−0.170032
$424$	0	0
$425$	93.5682	4.53873
$426$	0	0
$427$	−11.9690	−0.579223
$428$	0	0
$429$	6.03029	0.291145
$430$	0	0
$431$	−4.78050	−0.230269	−0.115134	−	0.993350i	$-0.536730\pi$
$431$	−4.78050	−0.230269	−0.115134	+	0.993350i	$0.536730\pi$
$432$	0	0
$433$	10.3907	0.499346	0.249673	−	0.968330i	$-0.419677\pi$
$433$	10.3907	0.499346	0.249673	+	0.968330i	$0.419677\pi$
$434$	0	0
$435$	35.5099	1.70257
$436$	0	0
$437$	17.6933	0.846386
$438$	0	0
$439$	−37.8979	−1.80877	−0.904384	−	0.426719i	$-0.859669\pi$
$439$	−37.8979	−1.80877	−0.904384	+	0.426719i	$0.859669\pi$
$440$	0	0
$441$	−2.84115	−0.135293
$442$	0	0
$443$	37.2416	1.76940	0.884701	−	0.466158i	$-0.154362\pi$
$443$	37.2416	1.76940	0.884701	+	0.466158i	$0.154362\pi$
$444$	0	0
$445$	−51.4500	−2.43896
$446$	0	0
$447$	5.00367	0.236665
$448$	0	0
$449$	−5.90278	−0.278569	−0.139285	−	0.990252i	$-0.544480\pi$
$449$	−5.90278	−0.278569	−0.139285	+	0.990252i	$0.544480\pi$
$450$	0	0
$451$	14.3840	0.677317
$452$	0	0
$453$	21.1194	0.992278
$454$	0	0
$455$	−31.0881	−1.45743
$456$	0	0
$457$	−3.20091	−0.149732	−0.0748661	−	0.997194i	$-0.523853\pi$
$457$	−3.20091	−0.149732	−0.0748661	+	0.997194i	$0.523853\pi$
$458$	0	0
$459$	7.73194	0.360896
$460$	0	0
$461$	−29.7556	−1.38586	−0.692929	−	0.721006i	$-0.743680\pi$
$461$	−29.7556	−1.38586	−0.692929	+	0.721006i	$0.743680\pi$
$462$	0	0
$463$	2.46249	0.114441	0.0572207	−	0.998362i	$-0.481776\pi$
$463$	2.46249	0.114441	0.0572207	+	0.998362i	$0.481776\pi$
$464$	0	0
$465$	26.8418	1.24476
$466$	0	0
$467$	6.21888	0.287776	0.143888	−	0.989594i	$-0.454040\pi$
$467$	6.21888	0.287776	0.143888	+	0.989594i	$0.454040\pi$
$468$	0	0
$469$	25.8421	1.19328
$470$	0	0
$471$	16.4685	0.758829
$472$	0	0
$473$	13.7016	0.629999
$474$	0	0
$475$	−85.1452	−3.90673
$476$	0	0
$477$	−10.3021	−0.471702
$478$	0	0
$479$	20.9961	0.959336	0.479668	−	0.877450i	$-0.340757\pi$
$479$	20.9961	0.959336	0.479668	+	0.877450i	$0.340757\pi$
$480$	0	0
$481$	−21.4731	−0.979091
$482$	0	0
$483$	−5.12832	−0.233347
$484$	0	0
$485$	30.4134	1.38100
$486$	0	0
$487$	0.745285	0.0337721	0.0168860	−	0.999857i	$-0.494625\pi$
$487$	0.745285	0.0337721	0.0168860	+	0.999857i	$0.494625\pi$
$488$	0	0
$489$	−2.51390	−0.113682
$490$	0	0
$491$	28.5775	1.28969	0.644843	−	0.764315i	$-0.276923\pi$
$491$	28.5775	1.28969	0.644843	+	0.764315i	$0.276923\pi$
$492$	0	0
$493$	66.3927	2.99018
$494$	0	0
$495$	−6.76495	−0.304062
$496$	0	0
$497$	4.96951	0.222913
$498$	0	0
$499$	11.6230	0.520317	0.260158	−	0.965566i	$-0.416225\pi$
$499$	11.6230	0.520317	0.260158	+	0.965566i	$0.416225\pi$
$500$	0	0
$501$	−0.757415	−0.0338388
$502$	0	0
$503$	36.8362	1.64244	0.821222	−	0.570608i	$-0.193292\pi$
$503$	36.8362	1.64244	0.821222	+	0.570608i	$0.193292\pi$
$504$	0	0
$505$	60.9622	2.71278
$506$	0	0
$507$	0.588815	0.0261502
$508$	0	0
$509$	−23.3697	−1.03584	−0.517921	−	0.855428i	$-0.673294\pi$
$509$	−23.3697	−1.03584	−0.517921	+	0.855428i	$0.673294\pi$
$510$	0	0
$511$	0.703145	0.0311053
$512$	0	0
$513$	−7.03591	−0.310643
$514$	0	0
$515$	−36.8996	−1.62599
$516$	0	0
$517$	5.72069	0.251596
$518$	0	0
$519$	−14.0438	−0.616456
$520$	0	0
$521$	17.1878	0.753011	0.376505	−	0.926414i	$-0.377126\pi$
$521$	17.1878	0.753011	0.376505	+	0.926414i	$0.377126\pi$
$522$	0	0
$523$	−7.34770	−0.321292	−0.160646	−	0.987012i	$-0.551358\pi$
$523$	−7.34770	−0.321292	−0.160646	+	0.987012i	$0.551358\pi$
$524$	0	0
$525$	24.6789	1.07708
$526$	0	0
$527$	50.1859	2.18613
$528$	0	0
$529$	−16.6762	−0.725052
$530$	0	0
$531$	10.6760	0.463301
$532$	0	0
$533$	32.4133	1.40398
$534$	0	0
$535$	−27.9726	−1.20936
$536$	0	0
$537$	−2.27312	−0.0980923
$538$	0	0
$539$	4.64774	0.200192
$540$	0	0
$541$	−14.7652	−0.634806	−0.317403	−	0.948291i	$-0.602811\pi$
$541$	−14.7652	−0.634806	−0.317403	+	0.948291i	$0.602811\pi$
$542$	0	0
$543$	0.399188	0.0171308
$544$	0	0
$545$	39.3731	1.68656
$546$	0	0
$547$	27.4575	1.17400	0.587000	−	0.809587i	$-0.300309\pi$
$547$	27.4575	1.17400	0.587000	+	0.809587i	$0.300309\pi$
$548$	0	0
$549$	−5.86912	−0.250488
$550$	0	0
$551$	−60.4160	−2.57381
$552$	0	0
$553$	−34.2193	−1.45515
$554$	0	0
$555$	24.0892	1.02253
$556$	0	0
$557$	−34.6674	−1.46890	−0.734452	−	0.678660i	$-0.762561\pi$
$557$	−34.6674	−1.46890	−0.734452	+	0.678660i	$0.762561\pi$
$558$	0	0
$559$	30.8755	1.30589
$560$	0	0
$561$	−12.6484	−0.534016
$562$	0	0
$563$	−18.0843	−0.762163	−0.381082	−	0.924541i	$-0.624448\pi$
$563$	−18.0843	−0.762163	−0.381082	+	0.924541i	$0.624448\pi$
$564$	0	0
$565$	−17.1973	−0.723497
$566$	0	0
$567$	2.03932	0.0856436
$568$	0	0
$569$	−8.48675	−0.355783	−0.177892	−	0.984050i	$-0.556928\pi$
$569$	−8.48675	−0.355783	−0.177892	+	0.984050i	$0.556928\pi$
$570$	0	0
$571$	1.29952	0.0543830	0.0271915	−	0.999630i	$-0.491344\pi$
$571$	1.29952	0.0543830	0.0271915	+	0.999630i	$0.491344\pi$
$572$	0	0
$573$	−18.1542	−0.758401
$574$	0	0
$575$	−30.4319	−1.26910
$576$	0	0
$577$	0.624720	0.0260074	0.0130037	−	0.999915i	$-0.495861\pi$
$577$	0.624720	0.0260074	0.0130037	+	0.999915i	$0.495861\pi$
$578$	0	0
$579$	−6.03028	−0.250610
$580$	0	0
$581$	−1.70427	−0.0707052
$582$	0	0
$583$	16.8529	0.697976
$584$	0	0
$585$	−15.2443	−0.630275
$586$	0	0
$587$	0.207182	0.00855132	0.00427566	−	0.999991i	$-0.498639\pi$
$587$	0.207182	0.00855132	0.00427566	+	0.999991i	$0.498639\pi$
$588$	0	0
$589$	−45.6682	−1.88172
$590$	0	0
$591$	3.04327	0.125183
$592$	0	0
$593$	−2.81207	−0.115478	−0.0577389	−	0.998332i	$-0.518389\pi$
$593$	−2.81207	−0.115478	−0.0577389	+	0.998332i	$0.518389\pi$
$594$	0	0
$595$	65.2067	2.67321
$596$	0	0
$597$	0.296733	0.0121445
$598$	0	0
$599$	−21.4267	−0.875470	−0.437735	−	0.899104i	$-0.644219\pi$
$599$	−21.4267	−0.875470	−0.437735	+	0.899104i	$0.644219\pi$
$600$	0	0
$601$	−31.9035	−1.30137	−0.650685	−	0.759348i	$-0.725518\pi$
$601$	−31.9035	−1.30137	−0.650685	+	0.759348i	$0.725518\pi$
$602$	0	0
$603$	12.6719	0.516039
$604$	0	0
$605$	−34.4228	−1.39949
$606$	0	0
$607$	−43.7971	−1.77767	−0.888834	−	0.458229i	$-0.848484\pi$
$607$	−43.7971	−1.77767	−0.888834	+	0.458229i	$0.848484\pi$
$608$	0	0
$609$	17.5113	0.709593
$610$	0	0
$611$	12.8912	0.521521
$612$	0	0
$613$	35.8916	1.44965	0.724824	−	0.688934i	$-0.241921\pi$
$613$	35.8916	1.44965	0.724824	+	0.688934i	$0.241921\pi$
$614$	0	0
$615$	−36.3622	−1.46627
$616$	0	0
$617$	34.4334	1.38624	0.693119	−	0.720823i	$-0.256236\pi$
$617$	34.4334	1.38624	0.693119	+	0.720823i	$0.256236\pi$
$618$	0	0
$619$	32.5231	1.30721	0.653607	−	0.756834i	$-0.273255\pi$
$619$	32.5231	1.30721	0.653607	+	0.756834i	$0.273255\pi$
$620$	0	0
$621$	−2.51472	−0.100912
$622$	0	0
$623$	−25.3720	−1.01651
$624$	0	0
$625$	60.9393	2.43757
$626$	0	0
$627$	11.5098	0.459657
$628$	0	0
$629$	45.0395	1.79584
$630$	0	0
$631$	−36.2746	−1.44407	−0.722035	−	0.691857i	$-0.756793\pi$
$631$	−36.2746	−1.44407	−0.722035	+	0.691857i	$0.756793\pi$
$632$	0	0
$633$	7.68707	0.305534
$634$	0	0
$635$	−1.86675	−0.0740799
$636$	0	0
$637$	10.4734	0.414969
$638$	0	0
$639$	2.43684	0.0963999
$640$	0	0
$641$	47.3324	1.86952	0.934758	−	0.355284i	$-0.115616\pi$
$641$	47.3324	1.86952	0.934758	+	0.355284i	$0.115616\pi$
$642$	0	0
$643$	31.2994	1.23433	0.617164	−	0.786834i	$-0.288281\pi$
$643$	31.2994	1.23433	0.617164	+	0.786834i	$0.288281\pi$
$644$	0	0
$645$	−34.6370	−1.36383
$646$	0	0
$647$	−16.1848	−0.636291	−0.318146	−	0.948042i	$-0.603060\pi$
$647$	−16.1848	−0.636291	−0.318146	+	0.948042i	$0.603060\pi$
$648$	0	0
$649$	−17.4646	−0.685544
$650$	0	0
$651$	13.2367	0.518788
$652$	0	0
$653$	−3.84266	−0.150375	−0.0751874	−	0.997169i	$-0.523956\pi$
$653$	−3.84266	−0.150375	−0.0751874	+	0.997169i	$0.523956\pi$
$654$	0	0
$655$	−16.0195	−0.625936
$656$	0	0
$657$	0.344793	0.0134517
$658$	0	0
$659$	3.27229	0.127470	0.0637351	−	0.997967i	$-0.479699\pi$
$659$	3.27229	0.127470	0.0637351	+	0.997967i	$0.479699\pi$
$660$	0	0
$661$	−5.41315	−0.210547	−0.105274	−	0.994443i	$-0.533572\pi$
$661$	−5.41315	−0.210547	−0.105274	+	0.994443i	$0.533572\pi$
$662$	0	0
$663$	−28.5023	−1.10694
$664$	0	0
$665$	−59.3368	−2.30098
$666$	0	0
$667$	−21.5934	−0.836099
$668$	0	0
$669$	−18.2622	−0.706059
$670$	0	0
$671$	9.60108	0.370646
$672$	0	0
$673$	5.88545	0.226867	0.113434	−	0.993546i	$-0.463815\pi$
$673$	5.88545	0.226867	0.113434	+	0.993546i	$0.463815\pi$
$674$	0	0
$675$	12.1015	0.465788
$676$	0	0
$677$	30.2298	1.16183	0.580913	−	0.813966i	$-0.302696\pi$
$677$	30.2298	1.16183	0.580913	+	0.813966i	$0.302696\pi$
$678$	0	0
$679$	14.9980	0.575571
$680$	0	0
$681$	−22.4953	−0.862020
$682$	0	0
$683$	−32.8502	−1.25698	−0.628488	−	0.777819i	$-0.716326\pi$
$683$	−32.8502	−1.25698	−0.628488	+	0.777819i	$0.716326\pi$
$684$	0	0
$685$	−41.2398	−1.57569
$686$	0	0
$687$	22.9601	0.875981
$688$	0	0
$689$	37.9768	1.44680
$690$	0	0
$691$	12.7559	0.485256	0.242628	−	0.970119i	$-0.421990\pi$
$691$	12.7559	0.485256	0.242628	+	0.970119i	$0.421990\pi$
$692$	0	0
$693$	−3.33606	−0.126726
$694$	0	0
$695$	4.13540	0.156865
$696$	0	0
$697$	−67.9863	−2.57516
$698$	0	0
$699$	−15.9082	−0.601702
$700$	0	0
$701$	29.2069	1.10313	0.551565	−	0.834132i	$-0.314031\pi$
$701$	29.2069	1.10313	0.551565	+	0.834132i	$0.314031\pi$
$702$	0	0
$703$	−40.9850	−1.54578
$704$	0	0
$705$	−14.4617	−0.544658
$706$	0	0
$707$	30.0628	1.13063
$708$	0	0
$709$	13.2927	0.499217	0.249609	−	0.968347i	$-0.419698\pi$
$709$	13.2927	0.499217	0.249609	+	0.968347i	$0.419698\pi$
$710$	0	0
$711$	−16.7797	−0.629289
$712$	0	0
$713$	−16.3224	−0.611277
$714$	0	0
$715$	24.9376	0.932615
$716$	0	0
$717$	−27.3799	−1.02252
$718$	0	0
$719$	16.6704	0.621700	0.310850	−	0.950459i	$-0.399386\pi$
$719$	16.6704	0.621700	0.310850	+	0.950459i	$0.399386\pi$
$720$	0	0
$721$	−18.1966	−0.677678
$722$	0	0
$723$	13.8747	0.516005
$724$	0	0
$725$	103.913	3.85925
$726$	0	0
$727$	−13.9984	−0.519173	−0.259586	−	0.965720i	$-0.583586\pi$
$727$	−13.9984	−0.519173	−0.259586	+	0.965720i	$0.583586\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−64.7607	−2.39526
$732$	0	0
$733$	7.06846	0.261079	0.130540	−	0.991443i	$-0.458329\pi$
$733$	7.06846	0.261079	0.130540	+	0.991443i	$0.458329\pi$
$734$	0	0
$735$	−11.7493	−0.433380
$736$	0	0
$737$	−20.7295	−0.763580
$738$	0	0
$739$	49.4808	1.82018	0.910090	−	0.414411i	$-0.136013\pi$
$739$	49.4808	1.82018	0.910090	+	0.414411i	$0.136013\pi$
$740$	0	0
$741$	25.9365	0.952800
$742$	0	0
$743$	−3.84450	−0.141041	−0.0705206	−	0.997510i	$-0.522466\pi$
$743$	−3.84450	−0.141041	−0.0705206	+	0.997510i	$0.522466\pi$
$744$	0	0
$745$	20.6922	0.758102
$746$	0	0
$747$	−0.835705	−0.0305768
$748$	0	0
$749$	−13.7944	−0.504035
$750$	0	0
$751$	6.68623	0.243984	0.121992	−	0.992531i	$-0.461072\pi$
$751$	6.68623	0.243984	0.121992	+	0.992531i	$0.461072\pi$
$752$	0	0
$753$	9.45446	0.344540
$754$	0	0
$755$	87.3373	3.17853
$756$	0	0
$757$	16.0096	0.581878	0.290939	−	0.956742i	$-0.406032\pi$
$757$	16.0096	0.581878	0.290939	+	0.956742i	$0.406032\pi$
$758$	0	0
$759$	4.11373	0.149319
$760$	0	0
$761$	−37.2278	−1.34951	−0.674754	−	0.738043i	$-0.735750\pi$
$761$	−37.2278	−1.34951	−0.674754	+	0.738043i	$0.735750\pi$
$762$	0	0
$763$	19.4164	0.702920
$764$	0	0
$765$	31.9746	1.15605
$766$	0	0
$767$	−39.3551	−1.42103
$768$	0	0
$769$	36.2843	1.30845	0.654223	−	0.756301i	$-0.272996\pi$
$769$	36.2843	1.30845	0.654223	+	0.756301i	$0.272996\pi$
$770$	0	0
$771$	−28.3953	−1.02263
$772$	0	0
$773$	−14.0944	−0.506939	−0.253469	−	0.967343i	$-0.581572\pi$
$773$	−14.0944	−0.506939	−0.253469	+	0.967343i	$0.581572\pi$
$774$	0	0
$775$	78.5478	2.82152
$776$	0	0
$777$	11.8793	0.426168
$778$	0	0
$779$	61.8661	2.21658
$780$	0	0
$781$	−3.98634	−0.142642
$782$	0	0
$783$	8.58681	0.306867
$784$	0	0
$785$	68.1039	2.43073
$786$	0	0
$787$	−36.6214	−1.30541	−0.652706	−	0.757611i	$-0.726366\pi$
$787$	−36.6214	−1.30541	−0.652706	+	0.757611i	$0.726366\pi$
$788$	0	0
$789$	19.8324	0.706050
$790$	0	0
$791$	−8.48067	−0.301538
$792$	0	0
$793$	21.6353	0.768294
$794$	0	0
$795$	−42.6035	−1.51099
$796$	0	0
$797$	−3.19893	−0.113312	−0.0566560	−	0.998394i	$-0.518044\pi$
$797$	−3.19893	−0.113312	−0.0566560	+	0.998394i	$0.518044\pi$
$798$	0	0
$799$	−27.0389	−0.956569
$800$	0	0
$801$	−12.4414	−0.439594
$802$	0	0
$803$	−0.564034	−0.0199043
$804$	0	0
$805$	−21.2077	−0.747472
$806$	0	0
$807$	−3.92600	−0.138202
$808$	0	0
$809$	−7.86490	−0.276515	−0.138258	−	0.990396i	$-0.544150\pi$
$809$	−7.86490	−0.276515	−0.138258	+	0.990396i	$0.544150\pi$
$810$	0	0
$811$	11.1924	0.393020	0.196510	−	0.980502i	$-0.437039\pi$
$811$	11.1924	0.393020	0.196510	+	0.980502i	$0.437039\pi$
$812$	0	0
$813$	12.4080	0.435169
$814$	0	0
$815$	−10.3960	−0.364155
$816$	0	0
$817$	58.9309	2.06173
$818$	0	0
$819$	−7.51756	−0.262685
$820$	0	0
$821$	13.3665	0.466493	0.233247	−	0.972418i	$-0.425065\pi$
$821$	13.3665	0.466493	0.233247	+	0.972418i	$0.425065\pi$
$822$	0	0
$823$	−18.3290	−0.638910	−0.319455	−	0.947601i	$-0.603500\pi$
$823$	−18.3290	−0.638910	−0.319455	+	0.947601i	$0.603500\pi$
$824$	0	0
$825$	−19.7964	−0.689224
$826$	0	0
$827$	44.6917	1.55408	0.777041	−	0.629450i	$-0.216720\pi$
$827$	44.6917	1.55408	0.777041	+	0.629450i	$0.216720\pi$
$828$	0	0
$829$	−53.5445	−1.85968	−0.929839	−	0.367967i	$-0.880054\pi$
$829$	−53.5445	−1.85968	−0.929839	+	0.367967i	$0.880054\pi$
$830$	0	0
$831$	5.65186	0.196061
$832$	0	0
$833$	−21.9676	−0.761133
$834$	0	0
$835$	−3.13221	−0.108395
$836$	0	0
$837$	6.49073	0.224353
$838$	0	0
$839$	15.4253	0.532542	0.266271	−	0.963898i	$-0.414208\pi$
$839$	15.4253	0.532542	0.266271	+	0.963898i	$0.414208\pi$
$840$	0	0
$841$	44.7333	1.54253
$842$	0	0
$843$	−23.7644	−0.818489
$844$	0	0
$845$	2.43499	0.0837660
$846$	0	0
$847$	−16.9752	−0.583276
$848$	0	0
$849$	−6.66574	−0.228767
$850$	0	0
$851$	−14.6485	−0.502145
$852$	0	0
$853$	−12.5262	−0.428888	−0.214444	−	0.976736i	$-0.568794\pi$
$853$	−12.5262	−0.428888	−0.214444	+	0.976736i	$0.568794\pi$
$854$	0	0
$855$	−29.0963	−0.995072
$856$	0	0
$857$	31.5113	1.07641	0.538203	−	0.842815i	$-0.319103\pi$
$857$	31.5113	1.07641	0.538203	+	0.842815i	$0.319103\pi$
$858$	0	0
$859$	52.2723	1.78351	0.891754	−	0.452521i	$-0.149475\pi$
$859$	52.2723	1.78351	0.891754	+	0.452521i	$0.149475\pi$
$860$	0	0
$861$	−17.9316	−0.611108
$862$	0	0
$863$	27.2627	0.928034	0.464017	−	0.885826i	$-0.346408\pi$
$863$	27.2627	0.928034	0.464017	+	0.885826i	$0.346408\pi$
$864$	0	0
$865$	−58.0769	−1.97467
$866$	0	0
$867$	42.7829	1.45298
$868$	0	0
$869$	27.4494	0.931156
$870$	0	0
$871$	−46.7124	−1.58279
$872$	0	0
$873$	7.35440	0.248909
$874$	0	0
$875$	59.8901	2.02466
$876$	0	0
$877$	−35.6330	−1.20324	−0.601620	−	0.798782i	$-0.705478\pi$
$877$	−35.6330	−1.20324	−0.601620	+	0.798782i	$0.705478\pi$
$878$	0	0
$879$	9.13750	0.308200
$880$	0	0
$881$	−1.55133	−0.0522655	−0.0261328	−	0.999658i	$-0.508319\pi$
$881$	−1.55133	−0.0522655	−0.0261328	+	0.999658i	$0.508319\pi$
$882$	0	0
$883$	25.8685	0.870544	0.435272	−	0.900299i	$-0.356652\pi$
$883$	25.8685	0.870544	0.435272	+	0.900299i	$0.356652\pi$
$884$	0	0
$885$	44.1497	1.48408
$886$	0	0
$887$	1.80183	0.0604995	0.0302498	−	0.999542i	$-0.490370\pi$
$887$	1.80183	0.0604995	0.0302498	+	0.999542i	$0.490370\pi$
$888$	0	0
$889$	−0.920569	−0.0308749
$890$	0	0
$891$	−1.63586	−0.0548035
$892$	0	0
$893$	24.6049	0.823371
$894$	0	0
$895$	−9.40025	−0.314216
$896$	0	0
$897$	9.27000	0.309516
$898$	0	0
$899$	55.7347	1.85886
$900$	0	0
$901$	−79.6555	−2.65371
$902$	0	0
$903$	−17.0809	−0.568415
$904$	0	0
$905$	1.65080	0.0548745
$906$	0	0
$907$	19.3776	0.643422	0.321711	−	0.946838i	$-0.395742\pi$
$907$	19.3776	0.643422	0.321711	+	0.946838i	$0.395742\pi$
$908$	0	0
$909$	14.7415	0.488946
$910$	0	0
$911$	−26.5972	−0.881205	−0.440603	−	0.897702i	$-0.645235\pi$
$911$	−26.5972	−0.881205	−0.440603	+	0.897702i	$0.645235\pi$
$912$	0	0
$913$	1.36710	0.0452444
$914$	0	0
$915$	−24.2712	−0.802380
$916$	0	0
$917$	−7.89986	−0.260876
$918$	0	0
$919$	43.3520	1.43005	0.715026	−	0.699098i	$-0.246415\pi$
$919$	43.3520	1.43005	0.715026	+	0.699098i	$0.246415\pi$
$920$	0	0
$921$	−4.89637	−0.161341
$922$	0	0
$923$	−8.98293	−0.295677
$924$	0	0
$925$	70.4928	2.31779
$926$	0	0
$927$	−8.92287	−0.293065
$928$	0	0
$929$	−34.3522	−1.12706	−0.563530	−	0.826096i	$-0.690557\pi$
$929$	−34.3522	−1.12706	−0.563530	+	0.826096i	$0.690557\pi$
$930$	0	0
$931$	19.9901	0.655149
$932$	0	0
$933$	−9.23896	−0.302470
$934$	0	0
$935$	−52.3062	−1.71059
$936$	0	0
$937$	33.7985	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$937$	33.7985	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$938$	0	0
$939$	20.8619	0.680803
$940$	0	0
$941$	18.7315	0.610629	0.305315	−	0.952252i	$-0.401238\pi$
$941$	18.7315	0.610629	0.305315	+	0.952252i	$0.401238\pi$
$942$	0	0
$943$	22.1117	0.720056
$944$	0	0
$945$	8.43342	0.274339
$946$	0	0
$947$	−38.7347	−1.25871	−0.629354	−	0.777118i	$-0.716681\pi$
$947$	−38.7347	−1.25871	−0.629354	+	0.777118i	$0.716681\pi$
$948$	0	0
$949$	−1.27101	−0.0412587
$950$	0	0
$951$	25.9505	0.841503
$952$	0	0
$953$	16.1394	0.522808	0.261404	−	0.965229i	$-0.415815\pi$
$953$	16.1394	0.522808	0.261404	+	0.965229i	$0.415815\pi$
$954$	0	0
$955$	−75.0747	−2.42936
$956$	0	0
$957$	−14.0468	−0.454070
$958$	0	0
$959$	−20.3369	−0.656714
$960$	0	0
$961$	11.1296	0.359020
$962$	0	0
$963$	−6.76418	−0.217973
$964$	0	0
$965$	−24.9376	−0.802770
$966$	0	0
$967$	−0.0947112	−0.00304571	−0.00152285	−	0.999999i	$-0.500485\pi$
$967$	−0.0947112	−0.00304571	−0.00152285	+	0.999999i	$0.500485\pi$
$968$	0	0
$969$	−54.4012	−1.74762
$970$	0	0
$971$	19.6252	0.629804	0.314902	−	0.949124i	$-0.398028\pi$
$971$	19.6252	0.629804	0.314902	+	0.949124i	$0.398028\pi$
$972$	0	0
$973$	2.03932	0.0653777
$974$	0	0
$975$	−44.6099	−1.42866
$976$	0	0
$977$	3.13026	0.100146	0.0500730	−	0.998746i	$-0.484055\pi$
$977$	3.13026	0.100146	0.0500730	+	0.998746i	$0.484055\pi$
$978$	0	0
$979$	20.3524	0.650465
$980$	0	0
$981$	9.52098	0.303982
$982$	0	0
$983$	3.63119	0.115817	0.0579085	−	0.998322i	$-0.481557\pi$
$983$	3.63119	0.115817	0.0579085	+	0.998322i	$0.481557\pi$
$984$	0	0
$985$	12.5851	0.400996
$986$	0	0
$987$	−7.13161	−0.227002
$988$	0	0
$989$	21.0626	0.669752
$990$	0	0
$991$	45.5001	1.44536	0.722679	−	0.691183i	$-0.242910\pi$
$991$	45.5001	1.44536	0.722679	+	0.691183i	$0.242910\pi$
$992$	0	0
$993$	33.5234	1.06383
$994$	0	0
$995$	1.22711	0.0389020
$996$	0	0
$997$	10.5032	0.332641	0.166320	−	0.986072i	$-0.446811\pi$
$997$	10.5032	0.332641	0.166320	+	0.986072i	$0.446811\pi$
$998$	0	0
$999$	5.82512	0.184299

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1668.2.a.h.1.7	✓	7
3.2	odd	2		5004.2.a.m.1.1		7
4.3	odd	2		6672.2.a.bn.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1668.2.a.h.1.7	✓	7	1.1	even	1	trivial
5004.2.a.m.1.1		7	3.2	odd	2
6672.2.a.bn.1.7		7	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1668 = 2^{2} \cdot 3 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1668.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.13540 q^{5} +2.03932 q^{7} +1.00000 q^{9} -1.63586 q^{11} -3.68630 q^{13} +4.13540 q^{15} +7.73194 q^{17} -7.03591 q^{19} +2.03932 q^{21} -2.51472 q^{23} +12.1015 q^{25} +1.00000 q^{27} +8.58681 q^{29} +6.49073 q^{31} -1.63586 q^{33} +8.43342 q^{35} +5.82512 q^{37} -3.68630 q^{39} -8.79292 q^{41} -8.37574 q^{43} +4.13540 q^{45} -3.49705 q^{47} -2.84115 q^{49} +7.73194 q^{51} -10.3021 q^{53} -6.76495 q^{55} -7.03591 q^{57} +10.6760 q^{59} -5.86912 q^{61} +2.03932 q^{63} -15.2443 q^{65} +12.6719 q^{67} -2.51472 q^{69} +2.43684 q^{71} +0.344793 q^{73} +12.1015 q^{75} -3.33606 q^{77} -16.7797 q^{79} +1.00000 q^{81} -0.835705 q^{83} +31.9746 q^{85} +8.58681 q^{87} -12.4414 q^{89} -7.51756 q^{91} +6.49073 q^{93} -29.0963 q^{95} +7.35440 q^{97} -1.63586 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{3} + q^{7} + 7 q^{9} + 8 q^{13} + 27 q^{17} + q^{21} - 8 q^{23} + 21 q^{25} + 7 q^{27} + 6 q^{29} + 7 q^{31} + 6 q^{35} + 14 q^{37} + 8 q^{39} + q^{41} + 27 q^{43} - 6 q^{47} + 32 q^{49} + 27 q^{51}+ \cdots + 7 q^{97}+O(q^{100}) \) 7 * q + 7 * q^3 + q^7 + 7 * q^9 + 8 * q^13 + 27 * q^17 + q^21 - 8 * q^23 + 21 * q^25 + 7 * q^27 + 6 * q^29 + 7 * q^31 + 6 * q^35 + 14 * q^37 + 8 * q^39 + q^41 + 27 * q^43 - 6 * q^47 + 32 * q^49 + 27 * q^51 + 18 * q^53 + 13 * q^55 + 17 * q^59 + 13 * q^61 + q^63 - 19 * q^65 + 31 * q^67 - 8 * q^69 + 6 * q^73 + 21 * q^75 - 34 * q^77 + 2 * q^79 + 7 * q^81 - 2 * q^83 + 37 * q^85 + 6 * q^87 + 13 * q^89 + 39 * q^91 + 7 * q^93 - 36 * q^95 + 7 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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