LMFDB - Embedded newform 6672.2.a.bs.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6672,2,Mod(1,6672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6672, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6672.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6672 = 2^{4} \cdot 3 \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6672.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:	$53.2761882286$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 31 x^{10} + 142 x^{9} + 397 x^{8} - 1508 x^{7} - 2549 x^{6} + 7294 x^{5} + \cdots - 4912 $ x^12 - 5x^11 - 31x^10 + 142x^9 + 397x^8 - 1508x^7 - 2549x^6 + 7294x^5 + 7482x^4 - 15919x^3 - 5172x^2 + 14748*x - 4912
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 3336)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-2.57593$ of defining polynomial
Character	$\chi$	$=$	6672.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.57593 q^{5} -5.14091 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.57593 q^{5} -5.14091 q^{7} +1.00000 q^{9} +1.14397 q^{11} +5.91156 q^{13} +3.57593 q^{15} -6.96589 q^{17} +1.28336 q^{19} -5.14091 q^{21} -1.91028 q^{23} +7.78726 q^{25} +1.00000 q^{27} +4.18818 q^{29} -5.75316 q^{31} +1.14397 q^{33} -18.3835 q^{35} +11.1001 q^{37} +5.91156 q^{39} -5.06591 q^{41} +0.490559 q^{43} +3.57593 q^{45} +6.81454 q^{47} +19.4289 q^{49} -6.96589 q^{51} -1.29923 q^{53} +4.09075 q^{55} +1.28336 q^{57} +3.39330 q^{59} -4.76102 q^{61} -5.14091 q^{63} +21.1393 q^{65} +11.9692 q^{67} -1.91028 q^{69} +10.1230 q^{71} +0.758422 q^{73} +7.78726 q^{75} -5.88103 q^{77} +12.9499 q^{79} +1.00000 q^{81} +12.2005 q^{83} -24.9095 q^{85} +4.18818 q^{87} +2.42354 q^{89} -30.3908 q^{91} -5.75316 q^{93} +4.58919 q^{95} +0.209005 q^{97} +1.14397 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{3} + 7 q^{5} + 2 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q + 12 * q^3 + 7 * q^5 + 2 * q^7 + 12 * q^9	$ 12 q + 12 q^{3} + 7 q^{5} + 2 q^{7} + 12 q^{9} + 10 q^{11} + 12 q^{13} + 7 q^{15} + 6 q^{17} + 5 q^{19} + 2 q^{21} + 8 q^{23} + 29 q^{25} + 12 q^{27} + 9 q^{29} + 10 q^{33} - 5 q^{35} + 23 q^{37} + 12 q^{39} + 19 q^{41} + 8 q^{43} + 7 q^{45} + 11 q^{47} + 22 q^{49} + 6 q^{51} + 19 q^{53} - 26 q^{55} + 5 q^{57} + 3 q^{59} + 27 q^{61} + 2 q^{63} + 12 q^{65} + 22 q^{67} + 8 q^{69} - q^{71} + 50 q^{73} + 29 q^{75} + q^{77} - 9 q^{79} + 12 q^{81} + 21 q^{83} + 35 q^{85} + 9 q^{87} + 26 q^{89} - 10 q^{91} - 30 q^{95} + 41 q^{97} + 10 q^{99}+O(q^{100}) $ 12 * q + 12 * q^3 + 7 * q^5 + 2 * q^7 + 12 * q^9 + 10 * q^11 + 12 * q^13 + 7 * q^15 + 6 * q^17 + 5 * q^19 + 2 * q^21 + 8 * q^23 + 29 * q^25 + 12 * q^27 + 9 * q^29 + 10 * q^33 - 5 * q^35 + 23 * q^37 + 12 * q^39 + 19 * q^41 + 8 * q^43 + 7 * q^45 + 11 * q^47 + 22 * q^49 + 6 * q^51 + 19 * q^53 - 26 * q^55 + 5 * q^57 + 3 * q^59 + 27 * q^61 + 2 * q^63 + 12 * q^65 + 22 * q^67 + 8 * q^69 - q^71 + 50 * q^73 + 29 * q^75 + q^77 - 9 * q^79 + 12 * q^81 + 21 * q^83 + 35 * q^85 + 9 * q^87 + 26 * q^89 - 10 * q^91 - 30 * q^95 + 41 * q^97 + 10 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.57593	1.59920	0.799602	−	0.600531i	$-0.205044\pi$
$5$	3.57593	1.59920	0.799602	+	0.600531i	$0.205044\pi$
$6$	0	0
$7$	−5.14091	−1.94308	−0.971540	−	0.236875i	$-0.923877\pi$
$7$	−5.14091	−1.94308	−0.971540	+	0.236875i	$0.923877\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.14397	0.344919	0.172460	−	0.985017i	$-0.444829\pi$
$11$	1.14397	0.344919	0.172460	+	0.985017i	$0.444829\pi$
$12$	0	0
$13$	5.91156	1.63957	0.819785	−	0.572671i	$-0.194093\pi$
$13$	5.91156	1.63957	0.819785	+	0.572671i	$0.194093\pi$
$14$	0	0
$15$	3.57593	0.923301
$16$	0	0
$17$	−6.96589	−1.68948	−0.844738	−	0.535180i	$-0.820244\pi$
$17$	−6.96589	−1.68948	−0.844738	+	0.535180i	$0.820244\pi$
$18$	0	0
$19$	1.28336	0.294422	0.147211	−	0.989105i	$-0.452970\pi$
$19$	1.28336	0.294422	0.147211	+	0.989105i	$0.452970\pi$
$20$	0	0
$21$	−5.14091	−1.12184
$22$	0	0
$23$	−1.91028	−0.398321	−0.199160	−	0.979967i	$-0.563821\pi$
$23$	−1.91028	−0.398321	−0.199160	+	0.979967i	$0.563821\pi$
$24$	0	0
$25$	7.78726	1.55745
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	4.18818	0.777725	0.388863	−	0.921296i	$-0.372868\pi$
$29$	4.18818	0.777725	0.388863	+	0.921296i	$0.372868\pi$
$30$	0	0
$31$	−5.75316	−1.03330	−0.516649	−	0.856197i	$-0.672821\pi$
$31$	−5.75316	−1.03330	−0.516649	+	0.856197i	$0.672821\pi$
$32$	0	0
$33$	1.14397	0.199139
$34$	0	0
$35$	−18.3835	−3.10738
$36$	0	0
$37$	11.1001	1.82485	0.912426	−	0.409241i	$-0.134207\pi$
$37$	11.1001	1.82485	0.912426	+	0.409241i	$0.134207\pi$
$38$	0	0
$39$	5.91156	0.946607
$40$	0	0
$41$	−5.06591	−0.791163	−0.395581	−	0.918431i	$-0.629457\pi$
$41$	−5.06591	−0.791163	−0.395581	+	0.918431i	$0.629457\pi$
$42$	0	0
$43$	0.490559	0.0748096	0.0374048	−	0.999300i	$-0.488091\pi$
$43$	0.490559	0.0748096	0.0374048	+	0.999300i	$0.488091\pi$
$44$	0	0
$45$	3.57593	0.533068
$46$	0	0
$47$	6.81454	0.994002	0.497001	−	0.867750i	$-0.334434\pi$
$47$	6.81454	0.994002	0.497001	+	0.867750i	$0.334434\pi$
$48$	0	0
$49$	19.4289	2.77556
$50$	0	0
$51$	−6.96589	−0.975419
$52$	0	0
$53$	−1.29923	−0.178462	−0.0892312	−	0.996011i	$-0.528441\pi$
$53$	−1.29923	−0.178462	−0.0892312	+	0.996011i	$0.528441\pi$
$54$	0	0
$55$	4.09075	0.551596
$56$	0	0
$57$	1.28336	0.169985
$58$	0	0
$59$	3.39330	0.441770	0.220885	−	0.975300i	$-0.429105\pi$
$59$	3.39330	0.441770	0.220885	+	0.975300i	$0.429105\pi$
$60$	0	0
$61$	−4.76102	−0.609586	−0.304793	−	0.952419i	$-0.598587\pi$
$61$	−4.76102	−0.609586	−0.304793	+	0.952419i	$0.598587\pi$
$62$	0	0
$63$	−5.14091	−0.647693
$64$	0	0
$65$	21.1393	2.62201
$66$	0	0
$67$	11.9692	1.46227	0.731137	−	0.682230i	$-0.238990\pi$
$67$	11.9692	1.46227	0.731137	+	0.682230i	$0.238990\pi$
$68$	0	0
$69$	−1.91028	−0.229970
$70$	0	0
$71$	10.1230	1.20138	0.600691	−	0.799482i	$-0.294892\pi$
$71$	10.1230	1.20138	0.600691	+	0.799482i	$0.294892\pi$
$72$	0	0
$73$	0.758422	0.0887666	0.0443833	−	0.999015i	$-0.485868\pi$
$73$	0.758422	0.0887666	0.0443833	+	0.999015i	$0.485868\pi$
$74$	0	0
$75$	7.78726	0.899195
$76$	0	0
$77$	−5.88103	−0.670206
$78$	0	0
$79$	12.9499	1.45698	0.728492	−	0.685055i	$-0.240222\pi$
$79$	12.9499	1.45698	0.728492	+	0.685055i	$0.240222\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.2005	1.33918	0.669589	−	0.742732i	$-0.266470\pi$
$83$	12.2005	1.33918	0.669589	+	0.742732i	$0.266470\pi$
$84$	0	0
$85$	−24.9095	−2.70182
$86$	0	0
$87$	4.18818	0.449020
$88$	0	0
$89$	2.42354	0.256895	0.128447	−	0.991716i	$-0.459001\pi$
$89$	2.42354	0.256895	0.128447	+	0.991716i	$0.459001\pi$
$90$	0	0
$91$	−30.3908	−3.18582
$92$	0	0
$93$	−5.75316	−0.596575
$94$	0	0
$95$	4.58919	0.470841
$96$	0	0
$97$	0.209005	0.0212212	0.0106106	−	0.999944i	$-0.496622\pi$
$97$	0.209005	0.0212212	0.0106106	+	0.999944i	$0.496622\pi$
$98$	0	0
$99$	1.14397	0.114973
$100$	0	0
$101$	10.6122	1.05596	0.527979	−	0.849258i	$-0.322950\pi$
$101$	10.6122	1.05596	0.527979	+	0.849258i	$0.322950\pi$
$102$	0	0
$103$	−8.72980	−0.860173	−0.430087	−	0.902788i	$-0.641517\pi$
$103$	−8.72980	−0.860173	−0.430087	+	0.902788i	$0.641517\pi$
$104$	0	0
$105$	−18.3835	−1.79405
$106$	0	0
$107$	−5.93653	−0.573906	−0.286953	−	0.957945i	$-0.592642\pi$
$107$	−5.93653	−0.573906	−0.286953	+	0.957945i	$0.592642\pi$
$108$	0	0
$109$	−16.5841	−1.58847	−0.794235	−	0.607611i	$-0.792128\pi$
$109$	−16.5841	−1.58847	−0.794235	+	0.607611i	$0.792128\pi$
$110$	0	0
$111$	11.1001	1.05358
$112$	0	0
$113$	−4.84219	−0.455514	−0.227757	−	0.973718i	$-0.573139\pi$
$113$	−4.84219	−0.455514	−0.227757	+	0.973718i	$0.573139\pi$
$114$	0	0
$115$	−6.83102	−0.636996
$116$	0	0
$117$	5.91156	0.546524
$118$	0	0
$119$	35.8110	3.28279
$120$	0	0
$121$	−9.69134	−0.881031
$122$	0	0
$123$	−5.06591	−0.456778
$124$	0	0
$125$	9.96704	0.891479
$126$	0	0
$127$	3.81411	0.338447	0.169224	−	0.985578i	$-0.445874\pi$
$127$	3.81411	0.338447	0.169224	+	0.985578i	$0.445874\pi$
$128$	0	0
$129$	0.490559	0.0431913
$130$	0	0
$131$	−6.21300	−0.542833	−0.271416	−	0.962462i	$-0.587492\pi$
$131$	−6.21300	−0.542833	−0.271416	+	0.962462i	$0.587492\pi$
$132$	0	0
$133$	−6.59762	−0.572086
$134$	0	0
$135$	3.57593	0.307767
$136$	0	0
$137$	20.1012	1.71737	0.858683	−	0.512508i	$-0.171283\pi$
$137$	20.1012	1.71737	0.858683	+	0.512508i	$0.171283\pi$
$138$	0	0
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	0	0
$141$	6.81454	0.573888
$142$	0	0
$143$	6.76263	0.565519
$144$	0	0
$145$	14.9766	1.24374
$146$	0	0
$147$	19.4289	1.60247
$148$	0	0
$149$	7.10234	0.581846	0.290923	−	0.956746i	$-0.406038\pi$
$149$	7.10234	0.581846	0.290923	+	0.956746i	$0.406038\pi$
$150$	0	0
$151$	−9.26760	−0.754187	−0.377093	−	0.926175i	$-0.623076\pi$
$151$	−9.26760	−0.754187	−0.377093	+	0.926175i	$0.623076\pi$
$152$	0	0
$153$	−6.96589	−0.563159
$154$	0	0
$155$	−20.5729	−1.65245
$156$	0	0
$157$	21.8356	1.74267	0.871337	−	0.490685i	$-0.163253\pi$
$157$	21.8356	1.74267	0.871337	+	0.490685i	$0.163253\pi$
$158$	0	0
$159$	−1.29923	−0.103035
$160$	0	0
$161$	9.82056	0.773969
$162$	0	0
$163$	16.8513	1.31990	0.659949	−	0.751310i	$-0.270578\pi$
$163$	16.8513	1.31990	0.659949	+	0.751310i	$0.270578\pi$
$164$	0	0
$165$	4.09075	0.318464
$166$	0	0
$167$	−17.6438	−1.36532	−0.682660	−	0.730736i	$-0.739177\pi$
$167$	−17.6438	−1.36532	−0.682660	+	0.730736i	$0.739177\pi$
$168$	0	0
$169$	21.9465	1.68819
$170$	0	0
$171$	1.28336	0.0981408
$172$	0	0
$173$	5.20575	0.395786	0.197893	−	0.980224i	$-0.436590\pi$
$173$	5.20575	0.395786	0.197893	+	0.980224i	$0.436590\pi$
$174$	0	0
$175$	−40.0336	−3.02625
$176$	0	0
$177$	3.39330	0.255056
$178$	0	0
$179$	10.1796	0.760859	0.380430	−	0.924810i	$-0.375776\pi$
$179$	10.1796	0.760859	0.380430	+	0.924810i	$0.375776\pi$
$180$	0	0
$181$	−17.6754	−1.31380	−0.656900	−	0.753978i	$-0.728132\pi$
$181$	−17.6754	−1.31380	−0.656900	+	0.753978i	$0.728132\pi$
$182$	0	0
$183$	−4.76102	−0.351945
$184$	0	0
$185$	39.6933	2.91831
$186$	0	0
$187$	−7.96875	−0.582733
$188$	0	0
$189$	−5.14091	−0.373946
$190$	0	0
$191$	−14.2424	−1.03055	−0.515274	−	0.857026i	$-0.672310\pi$
$191$	−14.2424	−1.03055	−0.515274	+	0.857026i	$0.672310\pi$
$192$	0	0
$193$	−6.52707	−0.469829	−0.234914	−	0.972016i	$-0.575481\pi$
$193$	−6.52707	−0.469829	−0.234914	+	0.972016i	$0.575481\pi$
$194$	0	0
$195$	21.1393	1.51382
$196$	0	0
$197$	−12.5159	−0.891724	−0.445862	−	0.895102i	$-0.647103\pi$
$197$	−12.5159	−0.891724	−0.445862	+	0.895102i	$0.647103\pi$
$198$	0	0
$199$	11.5977	0.822138	0.411069	−	0.911604i	$-0.365156\pi$
$199$	11.5977	0.822138	0.411069	+	0.911604i	$0.365156\pi$
$200$	0	0
$201$	11.9692	0.844245
$202$	0	0
$203$	−21.5310	−1.51118
$204$	0	0
$205$	−18.1153	−1.26523
$206$	0	0
$207$	−1.91028	−0.132774
$208$	0	0
$209$	1.46812	0.101552
$210$	0	0
$211$	24.0992	1.65906	0.829528	−	0.558466i	$-0.188610\pi$
$211$	24.0992	1.65906	0.829528	+	0.558466i	$0.188610\pi$
$212$	0	0
$213$	10.1230	0.693618
$214$	0	0
$215$	1.75420	0.119636
$216$	0	0
$217$	29.5764	2.00778
$218$	0	0
$219$	0.758422	0.0512494
$220$	0	0
$221$	−41.1792	−2.77002
$222$	0	0
$223$	−1.87259	−0.125398	−0.0626991	−	0.998032i	$-0.519971\pi$
$223$	−1.87259	−0.125398	−0.0626991	+	0.998032i	$0.519971\pi$
$224$	0	0
$225$	7.78726	0.519151
$226$	0	0
$227$	−3.52966	−0.234272	−0.117136	−	0.993116i	$-0.537371\pi$
$227$	−3.52966	−0.234272	−0.117136	+	0.993116i	$0.537371\pi$
$228$	0	0
$229$	−26.0212	−1.71953	−0.859764	−	0.510691i	$-0.829389\pi$
$229$	−26.0212	−1.71953	−0.859764	+	0.510691i	$0.829389\pi$
$230$	0	0
$231$	−5.88103	−0.386943
$232$	0	0
$233$	1.51476	0.0992350	0.0496175	−	0.998768i	$-0.484200\pi$
$233$	1.51476	0.0992350	0.0496175	+	0.998768i	$0.484200\pi$
$234$	0	0
$235$	24.3683	1.58961
$236$	0	0
$237$	12.9499	0.841190
$238$	0	0
$239$	14.6220	0.945817	0.472908	−	0.881112i	$-0.343204\pi$
$239$	14.6220	0.945817	0.472908	+	0.881112i	$0.343204\pi$
$240$	0	0
$241$	14.5819	0.939303	0.469651	−	0.882852i	$-0.344380\pi$
$241$	14.5819	0.939303	0.469651	+	0.882852i	$0.344380\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	69.4765	4.43869
$246$	0	0
$247$	7.58664	0.482726
$248$	0	0
$249$	12.2005	0.773175
$250$	0	0
$251$	21.7149	1.37063	0.685317	−	0.728245i	$-0.259664\pi$
$251$	21.7149	1.37063	0.685317	+	0.728245i	$0.259664\pi$
$252$	0	0
$253$	−2.18530	−0.137388
$254$	0	0
$255$	−24.9095	−1.55989
$256$	0	0
$257$	−16.2562	−1.01403	−0.507016	−	0.861937i	$-0.669252\pi$
$257$	−16.2562	−1.01403	−0.507016	+	0.861937i	$0.669252\pi$
$258$	0	0
$259$	−57.0648	−3.54583
$260$	0	0
$261$	4.18818	0.259242
$262$	0	0
$263$	−1.63052	−0.100542	−0.0502710	−	0.998736i	$-0.516008\pi$
$263$	−1.63052	−0.100542	−0.0502710	+	0.998736i	$0.516008\pi$
$264$	0	0
$265$	−4.64594	−0.285398
$266$	0	0
$267$	2.42354	0.148318
$268$	0	0
$269$	−18.3524	−1.11896	−0.559482	−	0.828843i	$-0.689000\pi$
$269$	−18.3524	−1.11896	−0.559482	+	0.828843i	$0.689000\pi$
$270$	0	0
$271$	−3.41387	−0.207378	−0.103689	−	0.994610i	$-0.533065\pi$
$271$	−3.41387	−0.207378	−0.103689	+	0.994610i	$0.533065\pi$
$272$	0	0
$273$	−30.3908	−1.83933
$274$	0	0
$275$	8.90837	0.537195
$276$	0	0
$277$	25.7140	1.54500	0.772502	−	0.635012i	$-0.219005\pi$
$277$	25.7140	1.54500	0.772502	+	0.635012i	$0.219005\pi$
$278$	0	0
$279$	−5.75316	−0.344432
$280$	0	0
$281$	22.9950	1.37177	0.685884	−	0.727711i	$-0.259416\pi$
$281$	22.9950	1.37177	0.685884	+	0.727711i	$0.259416\pi$
$282$	0	0
$283$	1.59771	0.0949739	0.0474869	−	0.998872i	$-0.484879\pi$
$283$	1.59771	0.0949739	0.0474869	+	0.998872i	$0.484879\pi$
$284$	0	0
$285$	4.58919	0.271840
$286$	0	0
$287$	26.0434	1.53729
$288$	0	0
$289$	31.5236	1.85433
$290$	0	0
$291$	0.209005	0.0122521
$292$	0	0
$293$	14.7663	0.862658	0.431329	−	0.902195i	$-0.358045\pi$
$293$	14.7663	0.862658	0.431329	+	0.902195i	$0.358045\pi$
$294$	0	0
$295$	12.1342	0.706481
$296$	0	0
$297$	1.14397	0.0663797
$298$	0	0
$299$	−11.2927	−0.653075
$300$	0	0
$301$	−2.52192	−0.145361
$302$	0	0
$303$	10.6122	0.609657
$304$	0	0
$305$	−17.0251	−0.974853
$306$	0	0
$307$	2.12441	0.121247	0.0606234	−	0.998161i	$-0.480691\pi$
$307$	2.12441	0.121247	0.0606234	+	0.998161i	$0.480691\pi$
$308$	0	0
$309$	−8.72980	−0.496621
$310$	0	0
$311$	17.7172	1.00465	0.502325	−	0.864679i	$-0.332478\pi$
$311$	17.7172	1.00465	0.502325	+	0.864679i	$0.332478\pi$
$312$	0	0
$313$	10.4407	0.590144	0.295072	−	0.955475i	$-0.404656\pi$
$313$	10.4407	0.590144	0.295072	+	0.955475i	$0.404656\pi$
$314$	0	0
$315$	−18.3835	−1.03579
$316$	0	0
$317$	−28.8842	−1.62230	−0.811151	−	0.584837i	$-0.801158\pi$
$317$	−28.8842	−1.62230	−0.811151	+	0.584837i	$0.801158\pi$
$318$	0	0
$319$	4.79114	0.268252
$320$	0	0
$321$	−5.93653	−0.331345
$322$	0	0
$323$	−8.93972	−0.497419
$324$	0	0
$325$	46.0348	2.55355
$326$	0	0
$327$	−16.5841	−0.917104
$328$	0	0
$329$	−35.0329	−1.93143
$330$	0	0
$331$	−25.5544	−1.40460	−0.702300	−	0.711881i	$-0.747843\pi$
$331$	−25.5544	−1.40460	−0.702300	+	0.711881i	$0.747843\pi$
$332$	0	0
$333$	11.1001	0.608284
$334$	0	0
$335$	42.8011	2.33848
$336$	0	0
$337$	0.668171	0.0363976	0.0181988	−	0.999834i	$-0.494207\pi$
$337$	0.668171	0.0363976	0.0181988	+	0.999834i	$0.494207\pi$
$338$	0	0
$339$	−4.84219	−0.262991
$340$	0	0
$341$	−6.58142	−0.356404
$342$	0	0
$343$	−63.8960	−3.45006
$344$	0	0
$345$	−6.83102	−0.367770
$346$	0	0
$347$	−29.8061	−1.60008	−0.800038	−	0.599950i	$-0.795187\pi$
$347$	−29.8061	−1.60008	−0.800038	+	0.599950i	$0.795187\pi$
$348$	0	0
$349$	9.20759	0.492871	0.246436	−	0.969159i	$-0.420741\pi$
$349$	9.20759	0.492871	0.246436	+	0.969159i	$0.420741\pi$
$350$	0	0
$351$	5.91156	0.315536
$352$	0	0
$353$	13.3635	0.711266	0.355633	−	0.934626i	$-0.384265\pi$
$353$	13.3635	0.711266	0.355633	+	0.934626i	$0.384265\pi$
$354$	0	0
$355$	36.1992	1.92125
$356$	0	0
$357$	35.8110	1.89532
$358$	0	0
$359$	−9.06133	−0.478239	−0.239119	−	0.970990i	$-0.576859\pi$
$359$	−9.06133	−0.478239	−0.239119	+	0.970990i	$0.576859\pi$
$360$	0	0
$361$	−17.3530	−0.913315
$362$	0	0
$363$	−9.69134	−0.508663
$364$	0	0
$365$	2.71206	0.141956
$366$	0	0
$367$	−3.56806	−0.186251	−0.0931256	−	0.995654i	$-0.529686\pi$
$367$	−3.56806	−0.186251	−0.0931256	+	0.995654i	$0.529686\pi$
$368$	0	0
$369$	−5.06591	−0.263721
$370$	0	0
$371$	6.67920	0.346767
$372$	0	0
$373$	−12.3887	−0.641461	−0.320730	−	0.947171i	$-0.603928\pi$
$373$	−12.3887	−0.641461	−0.320730	+	0.947171i	$0.603928\pi$
$374$	0	0
$375$	9.96704	0.514696
$376$	0	0
$377$	24.7586	1.27514
$378$	0	0
$379$	−9.73668	−0.500140	−0.250070	−	0.968228i	$-0.580454\pi$
$379$	−9.73668	−0.500140	−0.250070	+	0.968228i	$0.580454\pi$
$380$	0	0
$381$	3.81411	0.195403
$382$	0	0
$383$	−31.7586	−1.62279	−0.811393	−	0.584501i	$-0.801290\pi$
$383$	−31.7586	−1.62279	−0.811393	+	0.584501i	$0.801290\pi$
$384$	0	0
$385$	−21.0301	−1.07180
$386$	0	0
$387$	0.490559	0.0249365
$388$	0	0
$389$	6.29837	0.319340	0.159670	−	0.987170i	$-0.448957\pi$
$389$	6.29837	0.319340	0.159670	+	0.987170i	$0.448957\pi$
$390$	0	0
$391$	13.3068	0.672953
$392$	0	0
$393$	−6.21300	−0.313405
$394$	0	0
$395$	46.3081	2.33001
$396$	0	0
$397$	−7.69501	−0.386201	−0.193101	−	0.981179i	$-0.561854\pi$
$397$	−7.69501	−0.386201	−0.193101	+	0.981179i	$0.561854\pi$
$398$	0	0
$399$	−6.59762	−0.330294
$400$	0	0
$401$	−18.2981	−0.913765	−0.456883	−	0.889527i	$-0.651034\pi$
$401$	−18.2981	−0.913765	−0.456883	+	0.889527i	$0.651034\pi$
$402$	0	0
$403$	−34.0101	−1.69416
$404$	0	0
$405$	3.57593	0.177689
$406$	0	0
$407$	12.6982	0.629427
$408$	0	0
$409$	18.7808	0.928652	0.464326	−	0.885664i	$-0.346297\pi$
$409$	18.7808	0.928652	0.464326	+	0.885664i	$0.346297\pi$
$410$	0	0
$411$	20.1012	0.991521
$412$	0	0
$413$	−17.4447	−0.858395
$414$	0	0
$415$	43.6281	2.14162
$416$	0	0
$417$	−1.00000	−0.0489702
$418$	0	0
$419$	11.2936	0.551731	0.275865	−	0.961196i	$-0.411036\pi$
$419$	11.2936	0.551731	0.275865	+	0.961196i	$0.411036\pi$
$420$	0	0
$421$	10.1673	0.495524	0.247762	−	0.968821i	$-0.420305\pi$
$421$	10.1673	0.495524	0.247762	+	0.968821i	$0.420305\pi$
$422$	0	0
$423$	6.81454	0.331334
$424$	0	0
$425$	−54.2452	−2.63128
$426$	0	0
$427$	24.4760	1.18448
$428$	0	0
$429$	6.76263	0.326503
$430$	0	0
$431$	−2.81849	−0.135762	−0.0678808	−	0.997693i	$-0.521624\pi$
$431$	−2.81849	−0.135762	−0.0678808	+	0.997693i	$0.521624\pi$
$432$	0	0
$433$	22.5628	1.08430	0.542151	−	0.840281i	$-0.317610\pi$
$433$	22.5628	1.08430	0.542151	+	0.840281i	$0.317610\pi$
$434$	0	0
$435$	14.9766	0.718074
$436$	0	0
$437$	−2.45157	−0.117274
$438$	0	0
$439$	−31.5391	−1.50528	−0.752640	−	0.658433i	$-0.771220\pi$
$439$	−31.5391	−1.50528	−0.752640	+	0.658433i	$0.771220\pi$
$440$	0	0
$441$	19.4289	0.925187
$442$	0	0
$443$	−10.9149	−0.518581	−0.259291	−	0.965799i	$-0.583489\pi$
$443$	−10.9149	−0.518581	−0.259291	+	0.965799i	$0.583489\pi$
$444$	0	0
$445$	8.66640	0.410827
$446$	0	0
$447$	7.10234	0.335929
$448$	0	0
$449$	10.6665	0.503383	0.251692	−	0.967807i	$-0.419013\pi$
$449$	10.6665	0.503383	0.251692	+	0.967807i	$0.419013\pi$
$450$	0	0
$451$	−5.79524	−0.272887
$452$	0	0
$453$	−9.26760	−0.435430
$454$	0	0
$455$	−108.675	−5.09477
$456$	0	0
$457$	−12.9798	−0.607168	−0.303584	−	0.952805i	$-0.598183\pi$
$457$	−12.9798	−0.607168	−0.303584	+	0.952805i	$0.598183\pi$
$458$	0	0
$459$	−6.96589	−0.325140
$460$	0	0
$461$	−25.6059	−1.19258	−0.596292	−	0.802767i	$-0.703360\pi$
$461$	−25.6059	−1.19258	−0.596292	+	0.802767i	$0.703360\pi$
$462$	0	0
$463$	38.6753	1.79739	0.898697	−	0.438569i	$-0.144515\pi$
$463$	38.6753	1.79739	0.898697	+	0.438569i	$0.144515\pi$
$464$	0	0
$465$	−20.5729	−0.954044
$466$	0	0
$467$	36.5033	1.68917	0.844586	−	0.535419i	$-0.179846\pi$
$467$	36.5033	1.68917	0.844586	+	0.535419i	$0.179846\pi$
$468$	0	0
$469$	−61.5327	−2.84132
$470$	0	0
$471$	21.8356	1.00613
$472$	0	0
$473$	0.561184	0.0258033
$474$	0	0
$475$	9.99384	0.458549
$476$	0	0
$477$	−1.29923	−0.0594875
$478$	0	0
$479$	5.42234	0.247753	0.123877	−	0.992298i	$-0.460467\pi$
$479$	5.42234	0.247753	0.123877	+	0.992298i	$0.460467\pi$
$480$	0	0
$481$	65.6191	2.99197
$482$	0	0
$483$	9.82056	0.446851
$484$	0	0
$485$	0.747386	0.0339370
$486$	0	0
$487$	−10.9279	−0.495190	−0.247595	−	0.968864i	$-0.579640\pi$
$487$	−10.9279	−0.495190	−0.247595	+	0.968864i	$0.579640\pi$
$488$	0	0
$489$	16.8513	0.762044
$490$	0	0
$491$	−12.1175	−0.546855	−0.273427	−	0.961893i	$-0.588157\pi$
$491$	−12.1175	−0.546855	−0.273427	+	0.961893i	$0.588157\pi$
$492$	0	0
$493$	−29.1744	−1.31395
$494$	0	0
$495$	4.09075	0.183865
$496$	0	0
$497$	−52.0415	−2.33438
$498$	0	0
$499$	−13.4624	−0.602661	−0.301330	−	0.953520i	$-0.597431\pi$
$499$	−13.4624	−0.602661	−0.301330	+	0.953520i	$0.597431\pi$
$500$	0	0
$501$	−17.6438	−0.788268
$502$	0	0
$503$	−25.9622	−1.15760	−0.578798	−	0.815471i	$-0.696478\pi$
$503$	−25.9622	−1.15760	−0.578798	+	0.815471i	$0.696478\pi$
$504$	0	0
$505$	37.9486	1.68869
$506$	0	0
$507$	21.9465	0.974679
$508$	0	0
$509$	36.8882	1.63504	0.817519	−	0.575901i	$-0.195349\pi$
$509$	36.8882	1.63504	0.817519	+	0.575901i	$0.195349\pi$
$510$	0	0
$511$	−3.89898	−0.172481
$512$	0	0
$513$	1.28336	0.0566616
$514$	0	0
$515$	−31.2171	−1.37559
$516$	0	0
$517$	7.79561	0.342851
$518$	0	0
$519$	5.20575	0.228507
$520$	0	0
$521$	−15.7055	−0.688071	−0.344035	−	0.938957i	$-0.611794\pi$
$521$	−15.7055	−0.688071	−0.344035	+	0.938957i	$0.611794\pi$
$522$	0	0
$523$	3.16573	0.138428	0.0692138	−	0.997602i	$-0.477951\pi$
$523$	3.16573	0.138428	0.0692138	+	0.997602i	$0.477951\pi$
$524$	0	0
$525$	−40.0336	−1.74721
$526$	0	0
$527$	40.0758	1.74573
$528$	0	0
$529$	−19.3508	−0.841341
$530$	0	0
$531$	3.39330	0.147257
$532$	0	0
$533$	−29.9474	−1.29717
$534$	0	0
$535$	−21.2286	−0.917793
$536$	0	0
$537$	10.1796	0.439282
$538$	0	0
$539$	22.2261	0.957344
$540$	0	0
$541$	−24.0888	−1.03566	−0.517830	−	0.855483i	$-0.673260\pi$
$541$	−24.0888	−1.03566	−0.517830	+	0.855483i	$0.673260\pi$
$542$	0	0
$543$	−17.6754	−0.758522
$544$	0	0
$545$	−59.3036	−2.54029
$546$	0	0
$547$	−35.7817	−1.52991	−0.764957	−	0.644081i	$-0.777240\pi$
$547$	−35.7817	−1.52991	−0.764957	+	0.644081i	$0.777240\pi$
$548$	0	0
$549$	−4.76102	−0.203195
$550$	0	0
$551$	5.37493	0.228980
$552$	0	0
$553$	−66.5745	−2.83104
$554$	0	0
$555$	39.6933	1.68489
$556$	0	0
$557$	−23.4510	−0.993649	−0.496824	−	0.867851i	$-0.665501\pi$
$557$	−23.4510	−0.993649	−0.496824	+	0.867851i	$0.665501\pi$
$558$	0	0
$559$	2.89997	0.122656
$560$	0	0
$561$	−7.96875	−0.336441
$562$	0	0
$563$	31.2925	1.31882	0.659411	−	0.751783i	$-0.270806\pi$
$563$	31.2925	1.31882	0.659411	+	0.751783i	$0.270806\pi$
$564$	0	0
$565$	−17.3153	−0.728460
$566$	0	0
$567$	−5.14091	−0.215898
$568$	0	0
$569$	−39.8067	−1.66878	−0.834392	−	0.551171i	$-0.814181\pi$
$569$	−39.8067	−1.66878	−0.834392	+	0.551171i	$0.814181\pi$
$570$	0	0
$571$	−43.6895	−1.82835	−0.914175	−	0.405320i	$-0.867160\pi$
$571$	−43.6895	−1.82835	−0.914175	+	0.405320i	$0.867160\pi$
$572$	0	0
$573$	−14.2424	−0.594987
$574$	0	0
$575$	−14.8758	−0.620365
$576$	0	0
$577$	−10.6935	−0.445177	−0.222589	−	0.974912i	$-0.571451\pi$
$577$	−10.6935	−0.445177	−0.222589	+	0.974912i	$0.571451\pi$
$578$	0	0
$579$	−6.52707	−0.271256
$580$	0	0
$581$	−62.7216	−2.60213
$582$	0	0
$583$	−1.48627	−0.0615551
$584$	0	0
$585$	21.1393	0.874003
$586$	0	0
$587$	24.1970	0.998719	0.499359	−	0.866395i	$-0.333569\pi$
$587$	24.1970	0.998719	0.499359	+	0.866395i	$0.333569\pi$
$588$	0	0
$589$	−7.38336	−0.304226
$590$	0	0
$591$	−12.5159	−0.514837
$592$	0	0
$593$	27.1526	1.11502	0.557511	−	0.830169i	$-0.311756\pi$
$593$	27.1526	1.11502	0.557511	+	0.830169i	$0.311756\pi$
$594$	0	0
$595$	128.057	5.24985
$596$	0	0
$597$	11.5977	0.474661
$598$	0	0
$599$	−39.1623	−1.60013	−0.800065	−	0.599913i	$-0.795202\pi$
$599$	−39.1623	−1.60013	−0.800065	+	0.599913i	$0.795202\pi$
$600$	0	0
$601$	22.4943	0.917560	0.458780	−	0.888550i	$-0.348287\pi$
$601$	22.4943	0.917560	0.458780	+	0.888550i	$0.348287\pi$
$602$	0	0
$603$	11.9692	0.487425
$604$	0	0
$605$	−34.6555	−1.40895
$606$	0	0
$607$	37.5097	1.52247	0.761235	−	0.648476i	$-0.224593\pi$
$607$	37.5097	1.52247	0.761235	+	0.648476i	$0.224593\pi$
$608$	0	0
$609$	−21.5310	−0.872481
$610$	0	0
$611$	40.2845	1.62974
$612$	0	0
$613$	31.0444	1.25387	0.626936	−	0.779071i	$-0.284309\pi$
$613$	31.0444	1.25387	0.626936	+	0.779071i	$0.284309\pi$
$614$	0	0
$615$	−18.1153	−0.730481
$616$	0	0
$617$	22.2354	0.895163	0.447581	−	0.894243i	$-0.352286\pi$
$617$	22.2354	0.895163	0.447581	+	0.894243i	$0.352286\pi$
$618$	0	0
$619$	−23.8610	−0.959057	−0.479528	−	0.877526i	$-0.659192\pi$
$619$	−23.8610	−0.959057	−0.479528	+	0.877526i	$0.659192\pi$
$620$	0	0
$621$	−1.91028	−0.0766568
$622$	0	0
$623$	−12.4592	−0.499167
$624$	0	0
$625$	−3.29488	−0.131795
$626$	0	0
$627$	1.46812	0.0586310
$628$	0	0
$629$	−77.3223	−3.08304
$630$	0	0
$631$	1.58919	0.0632646	0.0316323	−	0.999500i	$-0.489929\pi$
$631$	1.58919	0.0632646	0.0316323	+	0.999500i	$0.489929\pi$
$632$	0	0
$633$	24.0992	0.957856
$634$	0	0
$635$	13.6390	0.541246
$636$	0	0
$637$	114.855	4.55073
$638$	0	0
$639$	10.1230	0.400460
$640$	0	0
$641$	−44.8846	−1.77284	−0.886418	−	0.462885i	$-0.846814\pi$
$641$	−44.8846	−1.77284	−0.886418	+	0.462885i	$0.846814\pi$
$642$	0	0
$643$	45.9924	1.81376	0.906882	−	0.421384i	$-0.138456\pi$
$643$	45.9924	1.81376	0.906882	+	0.421384i	$0.138456\pi$
$644$	0	0
$645$	1.75420	0.0690717
$646$	0	0
$647$	25.6255	1.00744	0.503721	−	0.863866i	$-0.331964\pi$
$647$	25.6255	1.00744	0.503721	+	0.863866i	$0.331964\pi$
$648$	0	0
$649$	3.88183	0.152375
$650$	0	0
$651$	29.5764	1.15919
$652$	0	0
$653$	11.2140	0.438839	0.219420	−	0.975631i	$-0.429584\pi$
$653$	11.2140	0.438839	0.219420	+	0.975631i	$0.429584\pi$
$654$	0	0
$655$	−22.2172	−0.868100
$656$	0	0
$657$	0.758422	0.0295889
$658$	0	0
$659$	−7.43132	−0.289483	−0.144742	−	0.989469i	$-0.546235\pi$
$659$	−7.43132	−0.289483	−0.144742	+	0.989469i	$0.546235\pi$
$660$	0	0
$661$	13.2868	0.516796	0.258398	−	0.966039i	$-0.416805\pi$
$661$	13.2868	0.516796	0.258398	+	0.966039i	$0.416805\pi$
$662$	0	0
$663$	−41.1792	−1.59927
$664$	0	0
$665$	−23.5926	−0.914882
$666$	0	0
$667$	−8.00058	−0.309784
$668$	0	0
$669$	−1.87259	−0.0723987
$670$	0	0
$671$	−5.44645	−0.210258
$672$	0	0
$673$	−2.63835	−0.101701	−0.0508504	−	0.998706i	$-0.516193\pi$
$673$	−2.63835	−0.101701	−0.0508504	+	0.998706i	$0.516193\pi$
$674$	0	0
$675$	7.78726	0.299732
$676$	0	0
$677$	−7.94146	−0.305215	−0.152608	−	0.988287i	$-0.548767\pi$
$677$	−7.94146	−0.305215	−0.152608	+	0.988287i	$0.548767\pi$
$678$	0	0
$679$	−1.07447	−0.0412345
$680$	0	0
$681$	−3.52966	−0.135257
$682$	0	0
$683$	−39.8678	−1.52550	−0.762749	−	0.646695i	$-0.776151\pi$
$683$	−39.8678	−1.52550	−0.762749	+	0.646695i	$0.776151\pi$
$684$	0	0
$685$	71.8806	2.74642
$686$	0	0
$687$	−26.0212	−0.992770
$688$	0	0
$689$	−7.68045	−0.292602
$690$	0	0
$691$	34.7461	1.32180	0.660902	−	0.750472i	$-0.270174\pi$
$691$	34.7461	1.32180	0.660902	+	0.750472i	$0.270174\pi$
$692$	0	0
$693$	−5.88103	−0.223402
$694$	0	0
$695$	−3.57593	−0.135643
$696$	0	0
$697$	35.2886	1.33665
$698$	0	0
$699$	1.51476	0.0572933
$700$	0	0
$701$	37.5044	1.41652	0.708262	−	0.705950i	$-0.249480\pi$
$701$	37.5044	1.41652	0.708262	+	0.705950i	$0.249480\pi$
$702$	0	0
$703$	14.2454	0.537277
$704$	0	0
$705$	24.3683	0.917763
$706$	0	0
$707$	−54.5565	−2.05181
$708$	0	0
$709$	45.1317	1.69496	0.847478	−	0.530830i	$-0.178120\pi$
$709$	45.1317	1.69496	0.847478	+	0.530830i	$0.178120\pi$
$710$	0	0
$711$	12.9499	0.485661
$712$	0	0
$713$	10.9901	0.411584
$714$	0	0
$715$	24.1827	0.904381
$716$	0	0
$717$	14.6220	0.546068
$718$	0	0
$719$	−31.9768	−1.19254	−0.596268	−	0.802786i	$-0.703350\pi$
$719$	−31.9768	−1.19254	−0.596268	+	0.802786i	$0.703350\pi$
$720$	0	0
$721$	44.8791	1.67139
$722$	0	0
$723$	14.5819	0.542307
$724$	0	0
$725$	32.6144	1.21127
$726$	0	0
$727$	24.6150	0.912919	0.456460	−	0.889744i	$-0.349117\pi$
$727$	24.6150	0.912919	0.456460	+	0.889744i	$0.349117\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.41718	−0.126389
$732$	0	0
$733$	−36.7139	−1.35606	−0.678030	−	0.735034i	$-0.737166\pi$
$733$	−36.7139	−1.35606	−0.678030	+	0.735034i	$0.737166\pi$
$734$	0	0
$735$	69.4765	2.56268
$736$	0	0
$737$	13.6924	0.504367
$738$	0	0
$739$	24.6819	0.907937	0.453969	−	0.891018i	$-0.350008\pi$
$739$	24.6819	0.907937	0.453969	+	0.891018i	$0.350008\pi$
$740$	0	0
$741$	7.58664	0.278702
$742$	0	0
$743$	22.8870	0.839642	0.419821	−	0.907607i	$-0.362093\pi$
$743$	22.8870	0.839642	0.419821	+	0.907607i	$0.362093\pi$
$744$	0	0
$745$	25.3974	0.930490
$746$	0	0
$747$	12.2005	0.446393
$748$	0	0
$749$	30.5192	1.11515
$750$	0	0
$751$	26.9447	0.983226	0.491613	−	0.870814i	$-0.336408\pi$
$751$	26.9447	0.983226	0.491613	+	0.870814i	$0.336408\pi$
$752$	0	0
$753$	21.7149	0.791336
$754$	0	0
$755$	−33.1403	−1.20610
$756$	0	0
$757$	−10.7760	−0.391661	−0.195830	−	0.980638i	$-0.562740\pi$
$757$	−10.7760	−0.391661	−0.195830	+	0.980638i	$0.562740\pi$
$758$	0	0
$759$	−2.18530	−0.0793212
$760$	0	0
$761$	34.4366	1.24833	0.624163	−	0.781294i	$-0.285440\pi$
$761$	34.4366	1.24833	0.624163	+	0.781294i	$0.285440\pi$
$762$	0	0
$763$	85.2574	3.08653
$764$	0	0
$765$	−24.9095	−0.900605
$766$	0	0
$767$	20.0597	0.724314
$768$	0	0
$769$	20.6858	0.745950	0.372975	−	0.927841i	$-0.378338\pi$
$769$	20.6858	0.745950	0.372975	+	0.927841i	$0.378338\pi$
$770$	0	0
$771$	−16.2562	−0.585452
$772$	0	0
$773$	11.6083	0.417523	0.208761	−	0.977967i	$-0.433057\pi$
$773$	11.6083	0.417523	0.208761	+	0.977967i	$0.433057\pi$
$774$	0	0
$775$	−44.8013	−1.60931
$776$	0	0
$777$	−57.0648	−2.04719
$778$	0	0
$779$	−6.50137	−0.232936
$780$	0	0
$781$	11.5804	0.414379
$782$	0	0
$783$	4.18818	0.149673
$784$	0	0
$785$	78.0827	2.78689
$786$	0	0
$787$	27.0903	0.965666	0.482833	−	0.875713i	$-0.339608\pi$
$787$	27.0903	0.965666	0.482833	+	0.875713i	$0.339608\pi$
$788$	0	0
$789$	−1.63052	−0.0580479
$790$	0	0
$791$	24.8932	0.885101
$792$	0	0
$793$	−28.1451	−0.999460
$794$	0	0
$795$	−4.64594	−0.164774
$796$	0	0
$797$	−22.4208	−0.794187	−0.397093	−	0.917778i	$-0.629981\pi$
$797$	−22.4208	−0.794187	−0.397093	+	0.917778i	$0.629981\pi$
$798$	0	0
$799$	−47.4693	−1.67934
$800$	0	0
$801$	2.42354	0.0856315
$802$	0	0
$803$	0.867611	0.0306173
$804$	0	0
$805$	35.1176	1.23773
$806$	0	0
$807$	−18.3524	−0.646034
$808$	0	0
$809$	−23.5438	−0.827755	−0.413878	−	0.910333i	$-0.635826\pi$
$809$	−23.5438	−0.827755	−0.413878	+	0.910333i	$0.635826\pi$
$810$	0	0
$811$	40.4748	1.42126	0.710631	−	0.703565i	$-0.248410\pi$
$811$	40.4748	1.42126	0.710631	+	0.703565i	$0.248410\pi$
$812$	0	0
$813$	−3.41387	−0.119730
$814$	0	0
$815$	60.2592	2.11079
$816$	0	0
$817$	0.629563	0.0220256
$818$	0	0
$819$	−30.3908	−1.06194
$820$	0	0
$821$	−31.3079	−1.09265	−0.546326	−	0.837572i	$-0.683974\pi$
$821$	−31.3079	−1.09265	−0.546326	+	0.837572i	$0.683974\pi$
$822$	0	0
$823$	−30.9803	−1.07991	−0.539953	−	0.841695i	$-0.681558\pi$
$823$	−30.9803	−1.07991	−0.539953	+	0.841695i	$0.681558\pi$
$824$	0	0
$825$	8.90837	0.310150
$826$	0	0
$827$	44.3097	1.54080	0.770400	−	0.637560i	$-0.220056\pi$
$827$	44.3097	1.54080	0.770400	+	0.637560i	$0.220056\pi$
$828$	0	0
$829$	−3.36458	−0.116857	−0.0584283	−	0.998292i	$-0.518609\pi$
$829$	−3.36458	−0.116857	−0.0584283	+	0.998292i	$0.518609\pi$
$830$	0	0
$831$	25.7140	0.892009
$832$	0	0
$833$	−135.340	−4.68924
$834$	0	0
$835$	−63.0930	−2.18342
$836$	0	0
$837$	−5.75316	−0.198858
$838$	0	0
$839$	−1.39662	−0.0482168	−0.0241084	−	0.999709i	$-0.507675\pi$
$839$	−1.39662	−0.0482168	−0.0241084	+	0.999709i	$0.507675\pi$
$840$	0	0
$841$	−11.4592	−0.395144
$842$	0	0
$843$	22.9950	0.791991
$844$	0	0
$845$	78.4791	2.69976
$846$	0	0
$847$	49.8223	1.71191
$848$	0	0
$849$	1.59771	0.0548332
$850$	0	0
$851$	−21.2044	−0.726876
$852$	0	0
$853$	−42.9968	−1.47218	−0.736092	−	0.676882i	$-0.763331\pi$
$853$	−42.9968	−1.47218	−0.736092	+	0.676882i	$0.763331\pi$
$854$	0	0
$855$	4.58919	0.156947
$856$	0	0
$857$	13.4032	0.457845	0.228923	−	0.973445i	$-0.426480\pi$
$857$	13.4032	0.457845	0.228923	+	0.973445i	$0.426480\pi$
$858$	0	0
$859$	3.89335	0.132839	0.0664197	−	0.997792i	$-0.478842\pi$
$859$	3.89335	0.132839	0.0664197	+	0.997792i	$0.478842\pi$
$860$	0	0
$861$	26.0434	0.887556
$862$	0	0
$863$	−14.1403	−0.481341	−0.240671	−	0.970607i	$-0.577367\pi$
$863$	−14.1403	−0.481341	−0.240671	+	0.970607i	$0.577367\pi$
$864$	0	0
$865$	18.6154	0.632942
$866$	0	0
$867$	31.5236	1.07060
$868$	0	0
$869$	14.8143	0.502541
$870$	0	0
$871$	70.7568	2.39750
$872$	0	0
$873$	0.209005	0.00707374
$874$	0	0
$875$	−51.2396	−1.73222
$876$	0	0
$877$	−36.3943	−1.22895	−0.614475	−	0.788936i	$-0.710632\pi$
$877$	−36.3943	−1.22895	−0.614475	+	0.788936i	$0.710632\pi$
$878$	0	0
$879$	14.7663	0.498056
$880$	0	0
$881$	−12.1389	−0.408969	−0.204485	−	0.978870i	$-0.565552\pi$
$881$	−12.1389	−0.408969	−0.204485	+	0.978870i	$0.565552\pi$
$882$	0	0
$883$	16.2458	0.546715	0.273357	−	0.961913i	$-0.411866\pi$
$883$	16.2458	0.546715	0.273357	+	0.961913i	$0.411866\pi$
$884$	0	0
$885$	12.1342	0.407887
$886$	0	0
$887$	−57.4417	−1.92870	−0.964351	−	0.264626i	$-0.914751\pi$
$887$	−57.4417	−1.92870	−0.964351	+	0.264626i	$0.914751\pi$
$888$	0	0
$889$	−19.6080	−0.657630
$890$	0	0
$891$	1.14397	0.0383244
$892$	0	0
$893$	8.74548	0.292656
$894$	0	0
$895$	36.4015	1.21677
$896$	0	0
$897$	−11.2927	−0.377053
$898$	0	0
$899$	−24.0952	−0.803621
$900$	0	0
$901$	9.05026	0.301508
$902$	0	0
$903$	−2.52192	−0.0839242
$904$	0	0
$905$	−63.2058	−2.10103
$906$	0	0
$907$	25.1077	0.833688	0.416844	−	0.908978i	$-0.363136\pi$
$907$	25.1077	0.833688	0.416844	+	0.908978i	$0.363136\pi$
$908$	0	0
$909$	10.6122	0.351986
$910$	0	0
$911$	19.6956	0.652543	0.326271	−	0.945276i	$-0.394208\pi$
$911$	19.6956	0.652543	0.326271	+	0.945276i	$0.394208\pi$
$912$	0	0
$913$	13.9570	0.461908
$914$	0	0
$915$	−17.0251	−0.562832
$916$	0	0
$917$	31.9405	1.05477
$918$	0	0
$919$	−57.6651	−1.90220	−0.951099	−	0.308887i	$-0.900044\pi$
$919$	−57.6651	−1.90220	−0.951099	+	0.308887i	$0.900044\pi$
$920$	0	0
$921$	2.12441	0.0700018
$922$	0	0
$923$	59.8428	1.96975
$924$	0	0
$925$	86.4397	2.84212
$926$	0	0
$927$	−8.72980	−0.286724
$928$	0	0
$929$	−22.3222	−0.732369	−0.366184	−	0.930542i	$-0.619336\pi$
$929$	−22.3222	−0.732369	−0.366184	+	0.930542i	$0.619336\pi$
$930$	0	0
$931$	24.9343	0.817187
$932$	0	0
$933$	17.7172	0.580034
$934$	0	0
$935$	−28.4957	−0.931908
$936$	0	0
$937$	4.05119	0.132347	0.0661733	−	0.997808i	$-0.478921\pi$
$937$	4.05119	0.132347	0.0661733	+	0.997808i	$0.478921\pi$
$938$	0	0
$939$	10.4407	0.340720
$940$	0	0
$941$	12.1392	0.395726	0.197863	−	0.980230i	$-0.436600\pi$
$941$	12.1392	0.395726	0.197863	+	0.980230i	$0.436600\pi$
$942$	0	0
$943$	9.67730	0.315136
$944$	0	0
$945$	−18.3835	−0.598016
$946$	0	0
$947$	−22.9029	−0.744245	−0.372122	−	0.928184i	$-0.621370\pi$
$947$	−22.9029	−0.744245	−0.372122	+	0.928184i	$0.621370\pi$
$948$	0	0
$949$	4.48346	0.145539
$950$	0	0
$951$	−28.8842	−0.936636
$952$	0	0
$953$	−28.3319	−0.917760	−0.458880	−	0.888498i	$-0.651749\pi$
$953$	−28.3319	−0.917760	−0.458880	+	0.888498i	$0.651749\pi$
$954$	0	0
$955$	−50.9300	−1.64805
$956$	0	0
$957$	4.79114	0.154876
$958$	0	0
$959$	−103.339	−3.33698
$960$	0	0
$961$	2.09881	0.0677037
$962$	0	0
$963$	−5.93653	−0.191302
$964$	0	0
$965$	−23.3403	−0.751352
$966$	0	0
$967$	10.7311	0.345088	0.172544	−	0.985002i	$-0.444801\pi$
$967$	10.7311	0.345088	0.172544	+	0.985002i	$0.444801\pi$
$968$	0	0
$969$	−8.93972	−0.287185
$970$	0	0
$971$	17.9898	0.577319	0.288660	−	0.957432i	$-0.406790\pi$
$971$	17.9898	0.577319	0.288660	+	0.957432i	$0.406790\pi$
$972$	0	0
$973$	5.14091	0.164810
$974$	0	0
$975$	46.0348	1.47429
$976$	0	0
$977$	47.9787	1.53497	0.767487	−	0.641064i	$-0.221507\pi$
$977$	47.9787	1.53497	0.767487	+	0.641064i	$0.221507\pi$
$978$	0	0
$979$	2.77245	0.0886079
$980$	0	0
$981$	−16.5841	−0.529490
$982$	0	0
$983$	−6.45079	−0.205748	−0.102874	−	0.994694i	$-0.532804\pi$
$983$	−6.45079	−0.205748	−0.102874	+	0.994694i	$0.532804\pi$
$984$	0	0
$985$	−44.7561	−1.42605
$986$	0	0
$987$	−35.0329	−1.11511
$988$	0	0
$989$	−0.937105	−0.0297982
$990$	0	0
$991$	−53.2643	−1.69200	−0.845998	−	0.533186i	$-0.820995\pi$
$991$	−53.2643	−1.69200	−0.845998	+	0.533186i	$0.820995\pi$
$992$	0	0
$993$	−25.5544	−0.810946
$994$	0	0
$995$	41.4725	1.31477
$996$	0	0
$997$	−23.6384	−0.748636	−0.374318	−	0.927300i	$-0.622123\pi$
$997$	−23.6384	−0.748636	−0.374318	+	0.927300i	$0.622123\pi$
$998$	0	0
$999$	11.1001	0.351193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6672.2.a.bs.1.10		12
4.3	odd	2		3336.2.a.r.1.10	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3336.2.a.r.1.10	✓	12	4.3	odd	2
6672.2.a.bs.1.10		12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6672 = 2^{4} \cdot 3 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6672.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.57593 q^{5} -5.14091 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.57593 q^{5} -5.14091 q^{7} +1.00000 q^{9} +1.14397 q^{11} +5.91156 q^{13} +3.57593 q^{15} -6.96589 q^{17} +1.28336 q^{19} -5.14091 q^{21} -1.91028 q^{23} +7.78726 q^{25} +1.00000 q^{27} +4.18818 q^{29} -5.75316 q^{31} +1.14397 q^{33} -18.3835 q^{35} +11.1001 q^{37} +5.91156 q^{39} -5.06591 q^{41} +0.490559 q^{43} +3.57593 q^{45} +6.81454 q^{47} +19.4289 q^{49} -6.96589 q^{51} -1.29923 q^{53} +4.09075 q^{55} +1.28336 q^{57} +3.39330 q^{59} -4.76102 q^{61} -5.14091 q^{63} +21.1393 q^{65} +11.9692 q^{67} -1.91028 q^{69} +10.1230 q^{71} +0.758422 q^{73} +7.78726 q^{75} -5.88103 q^{77} +12.9499 q^{79} +1.00000 q^{81} +12.2005 q^{83} -24.9095 q^{85} +4.18818 q^{87} +2.42354 q^{89} -30.3908 q^{91} -5.75316 q^{93} +4.58919 q^{95} +0.209005 q^{97} +1.14397 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 7 q^{5} + 2 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q + 12 * q^3 + 7 * q^5 + 2 * q^7 + 12 * q^9	\( 12 q + 12 q^{3} + 7 q^{5} + 2 q^{7} + 12 q^{9} + 10 q^{11} + 12 q^{13} + 7 q^{15} + 6 q^{17} + 5 q^{19} + 2 q^{21} + 8 q^{23} + 29 q^{25} + 12 q^{27} + 9 q^{29} + 10 q^{33} - 5 q^{35} + 23 q^{37} + 12 q^{39} + 19 q^{41} + 8 q^{43} + 7 q^{45} + 11 q^{47} + 22 q^{49} + 6 q^{51} + 19 q^{53} - 26 q^{55} + 5 q^{57} + 3 q^{59} + 27 q^{61} + 2 q^{63} + 12 q^{65} + 22 q^{67} + 8 q^{69} - q^{71} + 50 q^{73} + 29 q^{75} + q^{77} - 9 q^{79} + 12 q^{81} + 21 q^{83} + 35 q^{85} + 9 q^{87} + 26 q^{89} - 10 q^{91} - 30 q^{95} + 41 q^{97} + 10 q^{99}+O(q^{100}) \) 12 * q + 12 * q^3 + 7 * q^5 + 2 * q^7 + 12 * q^9 + 10 * q^11 + 12 * q^13 + 7 * q^15 + 6 * q^17 + 5 * q^19 + 2 * q^21 + 8 * q^23 + 29 * q^25 + 12 * q^27 + 9 * q^29 + 10 * q^33 - 5 * q^35 + 23 * q^37 + 12 * q^39 + 19 * q^41 + 8 * q^43 + 7 * q^45 + 11 * q^47 + 22 * q^49 + 6 * q^51 + 19 * q^53 - 26 * q^55 + 5 * q^57 + 3 * q^59 + 27 * q^61 + 2 * q^63 + 12 * q^65 + 22 * q^67 + 8 * q^69 - q^71 + 50 * q^73 + 29 * q^75 + q^77 - 9 * q^79 + 12 * q^81 + 21 * q^83 + 35 * q^85 + 9 * q^87 + 26 * q^89 - 10 * q^91 - 30 * q^95 + 41 * q^97 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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