LMFDB - Embedded newform 3344.2.a.z.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, 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("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.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:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.106392688.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 9x^{4} + 12x^{3} + 25x^{2} - 10x - 16 $ x^6 - 2x^5 - 9x^4 + 12x^3 + 25x^2 - 10*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1672)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.972165$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0278351 q^{3} -0.122408 q^{5} +0.877592 q^{7} -2.99923 q^{9} +O(q^{10})$	$q+0.0278351 q^{3} -0.122408 q^{5} +0.877592 q^{7} -2.99923 q^{9} +1.00000 q^{11} +1.68436 q^{13} -0.00340723 q^{15} +6.34040 q^{17} -1.00000 q^{19} +0.0244279 q^{21} +6.12835 q^{23} -4.98502 q^{25} -0.166989 q^{27} +3.64572 q^{29} -0.00340723 q^{31} +0.0278351 q^{33} -0.107424 q^{35} -8.21050 q^{37} +0.0468844 q^{39} -11.1723 q^{41} -0.0339670 q^{43} +0.367128 q^{45} +6.10979 q^{47} -6.22983 q^{49} +0.176486 q^{51} +7.24999 q^{53} -0.122408 q^{55} -0.0278351 q^{57} +14.2031 q^{59} -4.05412 q^{61} -2.63210 q^{63} -0.206179 q^{65} +8.58181 q^{67} +0.170583 q^{69} -0.0136263 q^{71} +3.73280 q^{73} -0.138759 q^{75} +0.877592 q^{77} +0.356156 q^{79} +8.99303 q^{81} +15.5146 q^{83} -0.776113 q^{85} +0.101479 q^{87} +5.31740 q^{89} +1.47818 q^{91} -9.48408e-5 q^{93} +0.122408 q^{95} +16.7120 q^{97} -2.99923 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} - 3 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q + 4 * q^3 - 3 * q^5 + 3 * q^7 + 6 * q^9	$ 6 q + 4 q^{3} - 3 q^{5} + 3 q^{7} + 6 q^{9} + 6 q^{11} - 3 q^{13} + q^{15} - q^{17} - 6 q^{19} + 5 q^{21} + 4 q^{23} + 5 q^{25} + 16 q^{27} - 7 q^{29} + q^{31} + 4 q^{33} + 32 q^{35} + 5 q^{37} + 13 q^{39} - 6 q^{41} + 16 q^{43} + 4 q^{47} - 7 q^{49} + 35 q^{51} - 11 q^{53} - 3 q^{55} - 4 q^{57} + 18 q^{59} + 16 q^{61} + 6 q^{63} + 10 q^{65} + 18 q^{67} - 2 q^{69} + 7 q^{71} - 21 q^{73} + 23 q^{75} + 3 q^{77} + 22 q^{79} - 10 q^{81} + 16 q^{83} + 3 q^{87} - 13 q^{89} + 7 q^{91} - 9 q^{93} + 3 q^{95} - 15 q^{97} + 6 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 - 3 * q^5 + 3 * q^7 + 6 * q^9 + 6 * q^11 - 3 * q^13 + q^15 - q^17 - 6 * q^19 + 5 * q^21 + 4 * q^23 + 5 * q^25 + 16 * q^27 - 7 * q^29 + q^31 + 4 * q^33 + 32 * q^35 + 5 * q^37 + 13 * q^39 - 6 * q^41 + 16 * q^43 + 4 * q^47 - 7 * q^49 + 35 * q^51 - 11 * q^53 - 3 * q^55 - 4 * q^57 + 18 * q^59 + 16 * q^61 + 6 * q^63 + 10 * q^65 + 18 * q^67 - 2 * q^69 + 7 * q^71 - 21 * q^73 + 23 * q^75 + 3 * q^77 + 22 * q^79 - 10 * q^81 + 16 * q^83 + 3 * q^87 - 13 * q^89 + 7 * q^91 - 9 * q^93 + 3 * q^95 - 15 * q^97 + 6 * 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
$3$	0.0278351	0.0160706	0.00803531	−	0.999968i	$-0.497442\pi$
$3$	0.0278351	0.0160706	0.00803531	+	0.999968i	$0.497442\pi$
$4$	0	0
$5$	−0.122408	−0.0547424	−0.0273712	−	0.999625i	$-0.508714\pi$
$5$	−0.122408	−0.0547424	−0.0273712	+	0.999625i	$0.508714\pi$
$6$	0	0
$7$	0.877592	0.331699	0.165849	−	0.986151i	$-0.446963\pi$
$7$	0.877592	0.331699	0.165849	+	0.986151i	$0.446963\pi$
$8$	0	0
$9$	−2.99923	−0.999742
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.68436	0.467157	0.233579	−	0.972338i	$-0.424956\pi$
$13$	1.68436	0.467157	0.233579	+	0.972338i	$0.424956\pi$
$14$	0	0
$15$	−0.00340723	−0.000879744 0
$16$	0	0
$17$	6.34040	1.53777	0.768886	−	0.639386i	$-0.220811\pi$
$17$	6.34040	1.53777	0.768886	+	0.639386i	$0.220811\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.0244279	0.00533060
$22$	0	0
$23$	6.12835	1.27785	0.638925	−	0.769269i	$-0.279379\pi$
$23$	6.12835	1.27785	0.638925	+	0.769269i	$0.279379\pi$
$24$	0	0
$25$	−4.98502	−0.997003
$26$	0	0
$27$	−0.166989	−0.0321371
$28$	0	0
$29$	3.64572	0.676994	0.338497	−	0.940968i	$-0.390082\pi$
$29$	3.64572	0.676994	0.338497	+	0.940968i	$0.390082\pi$
$30$	0	0
$31$	−0.00340723	−0.000611957 0	−0.000305979	−	1.00000i	$-0.500097\pi$
$31$	−0.00340723	−0.000611957 0	−0.000305979		1.00000i	$0.500097\pi$
$32$	0	0
$33$	0.0278351	0.00484547
$34$	0	0
$35$	−0.107424	−0.0181580
$36$	0	0
$37$	−8.21050	−1.34980	−0.674899	−	0.737911i	$-0.735813\pi$
$37$	−8.21050	−1.34980	−0.674899	+	0.737911i	$0.735813\pi$
$38$	0	0
$39$	0.0468844	0.00750751
$40$	0	0
$41$	−11.1723	−1.74483	−0.872414	−	0.488768i	$-0.837446\pi$
$41$	−11.1723	−1.74483	−0.872414	+	0.488768i	$0.837446\pi$
$42$	0	0
$43$	−0.0339670	−0.00517992	−0.00258996	−	0.999997i	$-0.500824\pi$
$43$	−0.0339670	−0.00517992	−0.00258996	+	0.999997i	$0.500824\pi$
$44$	0	0
$45$	0.367128	0.0547282
$46$	0	0
$47$	6.10979	0.891205	0.445602	−	0.895231i	$-0.352990\pi$
$47$	6.10979	0.891205	0.445602	+	0.895231i	$0.352990\pi$
$48$	0	0
$49$	−6.22983	−0.889976
$50$	0	0
$51$	0.176486	0.0247130
$52$	0	0
$53$	7.24999	0.995862	0.497931	−	0.867217i	$-0.334093\pi$
$53$	7.24999	0.995862	0.497931	+	0.867217i	$0.334093\pi$
$54$	0	0
$55$	−0.122408	−0.0165054
$56$	0	0
$57$	−0.0278351	−0.00368685
$58$	0	0
$59$	14.2031	1.84909	0.924543	−	0.381077i	$-0.124447\pi$
$59$	14.2031	1.84909	0.924543	+	0.381077i	$0.124447\pi$
$60$	0	0
$61$	−4.05412	−0.519077	−0.259538	−	0.965733i	$-0.583570\pi$
$61$	−4.05412	−0.519077	−0.259538	+	0.965733i	$0.583570\pi$
$62$	0	0
$63$	−2.63210	−0.331613
$64$	0	0
$65$	−0.206179	−0.0255733
$66$	0	0
$67$	8.58181	1.04843	0.524217	−	0.851585i	$-0.324358\pi$
$67$	8.58181	1.04843	0.524217	+	0.851585i	$0.324358\pi$
$68$	0	0
$69$	0.170583	0.0205358
$70$	0	0
$71$	−0.0136263	−0.00161714	−0.000808572	−	1.00000i	$-0.500257\pi$
$71$	−0.0136263	−0.00161714	−0.000808572		1.00000i	$0.500257\pi$
$72$	0	0
$73$	3.73280	0.436891	0.218446	−	0.975849i	$-0.429901\pi$
$73$	3.73280	0.436891	0.218446	+	0.975849i	$0.429901\pi$
$74$	0	0
$75$	−0.138759	−0.0160225
$76$	0	0
$77$	0.877592	0.100011
$78$	0	0
$79$	0.356156	0.0400707	0.0200353	−	0.999799i	$-0.493622\pi$
$79$	0.356156	0.0400707	0.0200353	+	0.999799i	$0.493622\pi$
$80$	0	0
$81$	8.99303	0.999225
$82$	0	0
$83$	15.5146	1.70294	0.851472	−	0.524400i	$-0.175710\pi$
$83$	15.5146	1.70294	0.851472	+	0.524400i	$0.175710\pi$
$84$	0	0
$85$	−0.776113	−0.0841813
$86$	0	0
$87$	0.101479	0.0108797
$88$	0	0
$89$	5.31740	0.563644	0.281822	−	0.959467i	$-0.409061\pi$
$89$	5.31740	0.563644	0.281822	+	0.959467i	$0.409061\pi$
$90$	0	0
$91$	1.47818	0.154956
$92$	0	0
$93$	−9.48408e−5 0	−9.83453e−6 0
$94$	0	0
$95$	0.122408	0.0125588
$96$	0	0
$97$	16.7120	1.69685	0.848424	−	0.529317i	$-0.177552\pi$
$97$	16.7120	1.69685	0.848424	+	0.529317i	$0.177552\pi$
$98$	0	0
$99$	−2.99923	−0.301433
$100$	0	0
$101$	−11.0126	−1.09579	−0.547896	−	0.836547i	$-0.684571\pi$
$101$	−11.0126	−1.09579	−0.547896	+	0.836547i	$0.684571\pi$
$102$	0	0
$103$	6.43605	0.634162	0.317081	−	0.948398i	$-0.397297\pi$
$103$	6.43605	0.634162	0.317081	+	0.948398i	$0.397297\pi$
$104$	0	0
$105$	−0.00299016	−0.000291810 0
$106$	0	0
$107$	−6.58366	−0.636467	−0.318233	−	0.948012i	$-0.603090\pi$
$107$	−6.58366	−0.636467	−0.318233	+	0.948012i	$0.603090\pi$
$108$	0	0
$109$	14.1528	1.35559	0.677794	−	0.735252i	$-0.262936\pi$
$109$	14.1528	1.35559	0.677794	+	0.735252i	$0.262936\pi$
$110$	0	0
$111$	−0.228540	−0.0216921
$112$	0	0
$113$	4.82796	0.454176	0.227088	−	0.973874i	$-0.427079\pi$
$113$	4.82796	0.454176	0.227088	+	0.973874i	$0.427079\pi$
$114$	0	0
$115$	−0.750157	−0.0699525
$116$	0	0
$117$	−5.05178	−0.467037
$118$	0	0
$119$	5.56428	0.510077
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−0.310984	−0.0280405
$124$	0	0
$125$	1.22224	0.109321
$126$	0	0
$127$	−1.05382	−0.0935114	−0.0467557	−	0.998906i	$-0.514888\pi$
$127$	−1.05382	−0.0935114	−0.0467557	+	0.998906i	$0.514888\pi$
$128$	0	0
$129$	−0.000945476 0	−8.32445e−5 0
$130$	0	0
$131$	−9.20418	−0.804173	−0.402086	−	0.915602i	$-0.631715\pi$
$131$	−9.20418	−0.804173	−0.402086	+	0.915602i	$0.631715\pi$
$132$	0	0
$133$	−0.877592	−0.0760969
$134$	0	0
$135$	0.0204408	0.00175926
$136$	0	0
$137$	−6.50230	−0.555529	−0.277764	−	0.960649i	$-0.589593\pi$
$137$	−6.50230	−0.555529	−0.277764	+	0.960649i	$0.589593\pi$
$138$	0	0
$139$	11.8310	1.00349	0.501747	−	0.865015i	$-0.332691\pi$
$139$	11.8310	1.00349	0.501747	+	0.865015i	$0.332691\pi$
$140$	0	0
$141$	0.170067	0.0143222
$142$	0	0
$143$	1.68436	0.140853
$144$	0	0
$145$	−0.446264	−0.0370602
$146$	0	0
$147$	−0.173408	−0.0143025
$148$	0	0
$149$	−18.2102	−1.49184	−0.745921	−	0.666035i	$-0.767990\pi$
$149$	−18.2102	−1.49184	−0.745921	+	0.666035i	$0.767990\pi$
$150$	0	0
$151$	14.9002	1.21257	0.606283	−	0.795249i	$-0.292660\pi$
$151$	14.9002	1.21257	0.606283	+	0.795249i	$0.292660\pi$
$152$	0	0
$153$	−19.0163	−1.53738
$154$	0	0
$155$	0.000417071 0	3.35000e−5 0
$156$	0	0
$157$	19.8091	1.58094	0.790471	−	0.612500i	$-0.209836\pi$
$157$	19.8091	1.58094	0.790471	+	0.612500i	$0.209836\pi$
$158$	0	0
$159$	0.201804	0.0160041
$160$	0	0
$161$	5.37820	0.423861
$162$	0	0
$163$	17.2833	1.35373	0.676865	−	0.736107i	$-0.263338\pi$
$163$	17.2833	1.35373	0.676865	+	0.736107i	$0.263338\pi$
$164$	0	0
$165$	−0.00340723	−0.000265253 0
$166$	0	0
$167$	11.9578	0.925320	0.462660	−	0.886536i	$-0.346895\pi$
$167$	11.9578	0.925320	0.462660	+	0.886536i	$0.346895\pi$
$168$	0	0
$169$	−10.1629	−0.781764
$170$	0	0
$171$	2.99923	0.229356
$172$	0	0
$173$	−12.0077	−0.912930	−0.456465	−	0.889741i	$-0.650885\pi$
$173$	−12.0077	−0.912930	−0.456465	+	0.889741i	$0.650885\pi$
$174$	0	0
$175$	−4.37481	−0.330705
$176$	0	0
$177$	0.395345	0.0297160
$178$	0	0
$179$	−12.4981	−0.934153	−0.467076	−	0.884217i	$-0.654693\pi$
$179$	−12.4981	−0.934153	−0.467076	+	0.884217i	$0.654693\pi$
$180$	0	0
$181$	4.84332	0.360001	0.180001	−	0.983666i	$-0.442390\pi$
$181$	4.84332	0.360001	0.180001	+	0.983666i	$0.442390\pi$
$182$	0	0
$183$	−0.112847	−0.00834189
$184$	0	0
$185$	1.00503	0.0738911
$186$	0	0
$187$	6.34040	0.463656
$188$	0	0
$189$	−0.146548	−0.0106598
$190$	0	0
$191$	9.43667	0.682813	0.341407	−	0.939916i	$-0.389097\pi$
$191$	9.43667	0.682813	0.341407	+	0.939916i	$0.389097\pi$
$192$	0	0
$193$	16.7679	1.20698	0.603489	−	0.797371i	$-0.293777\pi$
$193$	16.7679	1.20698	0.603489	+	0.797371i	$0.293777\pi$
$194$	0	0
$195$	−0.00573901	−0.000410979 0
$196$	0	0
$197$	26.4397	1.88375	0.941875	−	0.335963i	$-0.109062\pi$
$197$	26.4397	1.88375	0.941875	+	0.335963i	$0.109062\pi$
$198$	0	0
$199$	−19.2973	−1.36795	−0.683974	−	0.729506i	$-0.739750\pi$
$199$	−19.2973	−1.36795	−0.683974	+	0.729506i	$0.739750\pi$
$200$	0	0
$201$	0.238876	0.0168490
$202$	0	0
$203$	3.19946	0.224558
$204$	0	0
$205$	1.36758	0.0955160
$206$	0	0
$207$	−18.3803	−1.27752
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−9.97964	−0.687026	−0.343513	−	0.939148i	$-0.611617\pi$
$211$	−9.97964	−0.687026	−0.343513	+	0.939148i	$0.611617\pi$
$212$	0	0
$213$	−0.000379290 0	−2.59885e−5 0
$214$	0	0
$215$	0.00415782	0.000283561 0
$216$	0	0
$217$	−0.00299016	−0.000202985 0
$218$	0	0
$219$	0.103903	0.00702112
$220$	0	0
$221$	10.6795	0.718382
$222$	0	0
$223$	−16.2692	−1.08947	−0.544734	−	0.838609i	$-0.683369\pi$
$223$	−16.2692	−1.08947	−0.544734	+	0.838609i	$0.683369\pi$
$224$	0	0
$225$	14.9512	0.996746
$226$	0	0
$227$	25.1015	1.66605	0.833023	−	0.553238i	$-0.186608\pi$
$227$	25.1015	1.66605	0.833023	+	0.553238i	$0.186608\pi$
$228$	0	0
$229$	−3.06007	−0.202215	−0.101107	−	0.994876i	$-0.532239\pi$
$229$	−3.06007	−0.202215	−0.101107	+	0.994876i	$0.532239\pi$
$230$	0	0
$231$	0.0244279	0.00160724
$232$	0	0
$233$	−22.1783	−1.45295	−0.726473	−	0.687195i	$-0.758842\pi$
$233$	−22.1783	−1.45295	−0.726473	+	0.687195i	$0.758842\pi$
$234$	0	0
$235$	−0.747885	−0.0487866
$236$	0	0
$237$	0.00991364	0.000643960 0
$238$	0	0
$239$	−21.9023	−1.41674	−0.708370	−	0.705842i	$-0.750569\pi$
$239$	−21.9023	−1.41674	−0.708370	+	0.705842i	$0.750569\pi$
$240$	0	0
$241$	−2.89608	−0.186553	−0.0932765	−	0.995640i	$-0.529734\pi$
$241$	−2.89608	−0.186553	−0.0932765	+	0.995640i	$0.529734\pi$
$242$	0	0
$243$	0.751290	0.0481953
$244$	0	0
$245$	0.762579	0.0487194
$246$	0	0
$247$	−1.68436	−0.107173
$248$	0	0
$249$	0.431850	0.0273674
$250$	0	0
$251$	7.46432	0.471143	0.235572	−	0.971857i	$-0.424304\pi$
$251$	7.46432	0.471143	0.235572	+	0.971857i	$0.424304\pi$
$252$	0	0
$253$	6.12835	0.385286
$254$	0	0
$255$	−0.0216032	−0.00135285
$256$	0	0
$257$	5.24751	0.327331	0.163665	−	0.986516i	$-0.447668\pi$
$257$	5.24751	0.327331	0.163665	+	0.986516i	$0.447668\pi$
$258$	0	0
$259$	−7.20547	−0.447726
$260$	0	0
$261$	−10.9343	−0.676819
$262$	0	0
$263$	−18.6240	−1.14841	−0.574204	−	0.818712i	$-0.694688\pi$
$263$	−18.6240	−1.14841	−0.574204	+	0.818712i	$0.694688\pi$
$264$	0	0
$265$	−0.887454	−0.0545158
$266$	0	0
$267$	0.148011	0.00905810
$268$	0	0
$269$	−15.0400	−0.917005	−0.458503	−	0.888693i	$-0.651614\pi$
$269$	−15.0400	−0.917005	−0.458503	+	0.888693i	$0.651614\pi$
$270$	0	0
$271$	28.2895	1.71847	0.859234	−	0.511583i	$-0.170941\pi$
$271$	28.2895	1.71847	0.859234	+	0.511583i	$0.170941\pi$
$272$	0	0
$273$	0.0411454	0.00249023
$274$	0	0
$275$	−4.98502	−0.300608
$276$	0	0
$277$	22.3581	1.34337	0.671685	−	0.740837i	$-0.265571\pi$
$277$	22.3581	1.34337	0.671685	+	0.740837i	$0.265571\pi$
$278$	0	0
$279$	0.0102191	0.000611799 0
$280$	0	0
$281$	12.3487	0.736662	0.368331	−	0.929695i	$-0.379929\pi$
$281$	12.3487	0.736662	0.368331	+	0.929695i	$0.379929\pi$
$282$	0	0
$283$	12.3996	0.737082	0.368541	−	0.929611i	$-0.379857\pi$
$283$	12.3996	0.737082	0.368541	+	0.929611i	$0.379857\pi$
$284$	0	0
$285$	0.00340723	0.000201827 0
$286$	0	0
$287$	−9.80477	−0.578757
$288$	0	0
$289$	23.2006	1.36474
$290$	0	0
$291$	0.465181	0.0272694
$292$	0	0
$293$	−7.24377	−0.423186	−0.211593	−	0.977358i	$-0.567865\pi$
$293$	−7.24377	−0.423186	−0.211593	+	0.977358i	$0.567865\pi$
$294$	0	0
$295$	−1.73857	−0.101223
$296$	0	0
$297$	−0.166989	−0.00968970
$298$	0	0
$299$	10.3224	0.596957
$300$	0	0
$301$	−0.0298092	−0.00171817
$302$	0	0
$303$	−0.306536	−0.0176100
$304$	0	0
$305$	0.496255	0.0284155
$306$	0	0
$307$	18.1408	1.03535	0.517674	−	0.855578i	$-0.326798\pi$
$307$	18.1408	1.03535	0.517674	+	0.855578i	$0.326798\pi$
$308$	0	0
$309$	0.179148	0.0101914
$310$	0	0
$311$	−9.59406	−0.544029	−0.272015	−	0.962293i	$-0.587690\pi$
$311$	−9.59406	−0.544029	−0.272015	+	0.962293i	$0.587690\pi$
$312$	0	0
$313$	29.2280	1.65207	0.826033	−	0.563622i	$-0.190592\pi$
$313$	29.2280	1.65207	0.826033	+	0.563622i	$0.190592\pi$
$314$	0	0
$315$	0.322189	0.0181533
$316$	0	0
$317$	12.8728	0.723008	0.361504	−	0.932371i	$-0.382263\pi$
$317$	12.8728	0.723008	0.361504	+	0.932371i	$0.382263\pi$
$318$	0	0
$319$	3.64572	0.204121
$320$	0	0
$321$	−0.183257	−0.0102284
$322$	0	0
$323$	−6.34040	−0.352789
$324$	0	0
$325$	−8.39656	−0.465758
$326$	0	0
$327$	0.393944	0.0217851
$328$	0	0
$329$	5.36191	0.295611
$330$	0	0
$331$	−8.48289	−0.466262	−0.233131	−	0.972445i	$-0.574897\pi$
$331$	−8.48289	−0.466262	−0.233131	+	0.972445i	$0.574897\pi$
$332$	0	0
$333$	24.6251	1.34945
$334$	0	0
$335$	−1.05048	−0.0573938
$336$	0	0
$337$	−13.7212	−0.747442	−0.373721	−	0.927541i	$-0.621918\pi$
$337$	−13.7212	−0.747442	−0.373721	+	0.927541i	$0.621918\pi$
$338$	0	0
$339$	0.134387	0.00729890
$340$	0	0
$341$	−0.00340723	−0.000184512 0
$342$	0	0
$343$	−11.6104	−0.626903
$344$	0	0
$345$	−0.0208807	−0.00112418
$346$	0	0
$347$	13.8279	0.742318	0.371159	−	0.928569i	$-0.378960\pi$
$347$	13.8279	0.742318	0.371159	+	0.928569i	$0.378960\pi$
$348$	0	0
$349$	30.1644	1.61466	0.807331	−	0.590099i	$-0.200911\pi$
$349$	30.1644	1.61466	0.807331	+	0.590099i	$0.200911\pi$
$350$	0	0
$351$	−0.281270	−0.0150131
$352$	0	0
$353$	−32.9396	−1.75320	−0.876598	−	0.481223i	$-0.840193\pi$
$353$	−32.9396	−1.75320	−0.876598	+	0.481223i	$0.840193\pi$
$354$	0	0
$355$	0.00166796	8.85263e−5 0
$356$	0	0
$357$	0.154883	0.00819726
$358$	0	0
$359$	−35.8135	−1.89017	−0.945083	−	0.326830i	$-0.894020\pi$
$359$	−35.8135	−1.89017	−0.945083	+	0.326830i	$0.894020\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.0278351	0.00146097
$364$	0	0
$365$	−0.456923	−0.0239165
$366$	0	0
$367$	−29.6271	−1.54652	−0.773260	−	0.634089i	$-0.781375\pi$
$367$	−29.6271	−1.54652	−0.773260	+	0.634089i	$0.781375\pi$
$368$	0	0
$369$	33.5084	1.74438
$370$	0	0
$371$	6.36253	0.330326
$372$	0	0
$373$	3.64878	0.188927	0.0944633	−	0.995528i	$-0.469886\pi$
$373$	3.64878	0.188927	0.0944633	+	0.995528i	$0.469886\pi$
$374$	0	0
$375$	0.0340213	0.00175685
$376$	0	0
$377$	6.14071	0.316263
$378$	0	0
$379$	−2.07155	−0.106408	−0.0532041	−	0.998584i	$-0.516943\pi$
$379$	−2.07155	−0.106408	−0.0532041	+	0.998584i	$0.516943\pi$
$380$	0	0
$381$	−0.0293332	−0.00150279
$382$	0	0
$383$	−14.4545	−0.738592	−0.369296	−	0.929312i	$-0.620401\pi$
$383$	−14.4545	−0.738592	−0.369296	+	0.929312i	$0.620401\pi$
$384$	0	0
$385$	−0.107424	−0.00547483
$386$	0	0
$387$	0.101875	0.00517858
$388$	0	0
$389$	6.37041	0.322992	0.161496	−	0.986873i	$-0.448368\pi$
$389$	6.37041	0.322992	0.161496	+	0.986873i	$0.448368\pi$
$390$	0	0
$391$	38.8562	1.96504
$392$	0	0
$393$	−0.256199	−0.0129236
$394$	0	0
$395$	−0.0435962	−0.00219356
$396$	0	0
$397$	−13.9040	−0.697824	−0.348912	−	0.937155i	$-0.613449\pi$
$397$	−13.9040	−0.697824	−0.348912	+	0.937155i	$0.613449\pi$
$398$	0	0
$399$	−0.0244279	−0.00122292
$400$	0	0
$401$	−25.6324	−1.28002	−0.640011	−	0.768366i	$-0.721070\pi$
$401$	−25.6324	−1.28002	−0.640011	+	0.768366i	$0.721070\pi$
$402$	0	0
$403$	−0.00573901	−0.000285880 0
$404$	0	0
$405$	−1.10082	−0.0547000
$406$	0	0
$407$	−8.21050	−0.406979
$408$	0	0
$409$	2.60766	0.128941	0.0644703	−	0.997920i	$-0.479464\pi$
$409$	2.60766	0.128941	0.0644703	+	0.997920i	$0.479464\pi$
$410$	0	0
$411$	−0.180992	−0.00892769
$412$	0	0
$413$	12.4645	0.613340
$414$	0	0
$415$	−1.89910	−0.0932232
$416$	0	0
$417$	0.329318	0.0161268
$418$	0	0
$419$	4.20940	0.205643	0.102821	−	0.994700i	$-0.467213\pi$
$419$	4.20940	0.205643	0.102821	+	0.994700i	$0.467213\pi$
$420$	0	0
$421$	−34.8223	−1.69714	−0.848568	−	0.529086i	$-0.822535\pi$
$421$	−34.8223	−1.69714	−0.848568	+	0.529086i	$0.822535\pi$
$422$	0	0
$423$	−18.3246	−0.890974
$424$	0	0
$425$	−31.6070	−1.53316
$426$	0	0
$427$	−3.55787	−0.172177
$428$	0	0
$429$	0.0468844	0.00226360
$430$	0	0
$431$	21.1564	1.01907	0.509535	−	0.860450i	$-0.329818\pi$
$431$	21.1564	1.01907	0.509535	+	0.860450i	$0.329818\pi$
$432$	0	0
$433$	4.66913	0.224384	0.112192	−	0.993687i	$-0.464213\pi$
$433$	4.66913	0.224384	0.112192	+	0.993687i	$0.464213\pi$
$434$	0	0
$435$	−0.0124218	−0.000595581 0
$436$	0	0
$437$	−6.12835	−0.293159
$438$	0	0
$439$	−36.2453	−1.72989	−0.864947	−	0.501864i	$-0.832648\pi$
$439$	−36.2453	−1.72989	−0.864947	+	0.501864i	$0.832648\pi$
$440$	0	0
$441$	18.6847	0.889746
$442$	0	0
$443$	5.79016	0.275099	0.137549	−	0.990495i	$-0.456077\pi$
$443$	5.79016	0.275099	0.137549	+	0.990495i	$0.456077\pi$
$444$	0	0
$445$	−0.650891	−0.0308552
$446$	0	0
$447$	−0.506884	−0.0239748
$448$	0	0
$449$	−19.4020	−0.915635	−0.457818	−	0.889046i	$-0.651369\pi$
$449$	−19.4020	−0.915635	−0.457818	+	0.889046i	$0.651369\pi$
$450$	0	0
$451$	−11.1723	−0.526085
$452$	0	0
$453$	0.414750	0.0194867
$454$	0	0
$455$	−0.180941	−0.00848263
$456$	0	0
$457$	24.6795	1.15446	0.577228	−	0.816583i	$-0.304134\pi$
$457$	24.6795	1.15446	0.577228	+	0.816583i	$0.304134\pi$
$458$	0	0
$459$	−1.05878	−0.0494195
$460$	0	0
$461$	3.63779	0.169429	0.0847145	−	0.996405i	$-0.473002\pi$
$461$	3.63779	0.169429	0.0847145	+	0.996405i	$0.473002\pi$
$462$	0	0
$463$	−13.1814	−0.612592	−0.306296	−	0.951936i	$-0.599090\pi$
$463$	−13.1814	−0.612592	−0.306296	+	0.951936i	$0.599090\pi$
$464$	0	0
$465$	1.16092e−5 0	5.38365e−7 0
$466$	0	0
$467$	31.6892	1.46640	0.733200	−	0.680013i	$-0.238026\pi$
$467$	31.6892	1.46640	0.733200	+	0.680013i	$0.238026\pi$
$468$	0	0
$469$	7.53133	0.347764
$470$	0	0
$471$	0.551390	0.0254067
$472$	0	0
$473$	−0.0339670	−0.00156180
$474$	0	0
$475$	4.98502	0.228728
$476$	0	0
$477$	−21.7443	−0.995605
$478$	0	0
$479$	−41.2270	−1.88371	−0.941854	−	0.336022i	$-0.890918\pi$
$479$	−41.2270	−1.88371	−0.941854	+	0.336022i	$0.890918\pi$
$480$	0	0
$481$	−13.8294	−0.630568
$482$	0	0
$483$	0.149703	0.00681171
$484$	0	0
$485$	−2.04568	−0.0928895
$486$	0	0
$487$	−3.21468	−0.145671	−0.0728354	−	0.997344i	$-0.523205\pi$
$487$	−3.21468	−0.145671	−0.0728354	+	0.997344i	$0.523205\pi$
$488$	0	0
$489$	0.481082	0.0217553
$490$	0	0
$491$	5.92336	0.267317	0.133659	−	0.991027i	$-0.457327\pi$
$491$	5.92336	0.267317	0.133659	+	0.991027i	$0.457327\pi$
$492$	0	0
$493$	23.1153	1.04106
$494$	0	0
$495$	0.367128	0.0165012
$496$	0	0
$497$	−0.0119583	−0.000536404 0
$498$	0	0
$499$	13.8155	0.618468	0.309234	−	0.950986i	$-0.399927\pi$
$499$	13.8155	0.618468	0.309234	+	0.950986i	$0.399927\pi$
$500$	0	0
$501$	0.332846	0.0148705
$502$	0	0
$503$	−22.8789	−1.02012	−0.510061	−	0.860138i	$-0.670377\pi$
$503$	−22.8789	−1.02012	−0.510061	+	0.860138i	$0.670377\pi$
$504$	0	0
$505$	1.34802	0.0599862
$506$	0	0
$507$	−0.282887	−0.0125634
$508$	0	0
$509$	−37.4761	−1.66110	−0.830549	−	0.556946i	$-0.811973\pi$
$509$	−37.4761	−1.66110	−0.830549	+	0.556946i	$0.811973\pi$
$510$	0	0
$511$	3.27588	0.144916
$512$	0	0
$513$	0.166989	0.00737275
$514$	0	0
$515$	−0.787821	−0.0347156
$516$	0	0
$517$	6.10979	0.268708
$518$	0	0
$519$	−0.334237	−0.0146714
$520$	0	0
$521$	25.4757	1.11611	0.558056	−	0.829803i	$-0.311547\pi$
$521$	25.4757	1.11611	0.558056	+	0.829803i	$0.311547\pi$
$522$	0	0
$523$	−32.3594	−1.41498	−0.707488	−	0.706725i	$-0.750172\pi$
$523$	−32.3594	−1.41498	−0.707488	+	0.706725i	$0.750172\pi$
$524$	0	0
$525$	−0.121773	−0.00531463
$526$	0	0
$527$	−0.0216032	−0.000941051 0
$528$	0	0
$529$	14.5567	0.632900
$530$	0	0
$531$	−42.5983	−1.84861
$532$	0	0
$533$	−18.8183	−0.815109
$534$	0	0
$535$	0.805891	0.0348417
$536$	0	0
$537$	−0.347887	−0.0150124
$538$	0	0
$539$	−6.22983	−0.268338
$540$	0	0
$541$	18.2211	0.783388	0.391694	−	0.920096i	$-0.371889\pi$
$541$	18.2211	0.783388	0.391694	+	0.920096i	$0.371889\pi$
$542$	0	0
$543$	0.134815	0.00578545
$544$	0	0
$545$	−1.73241	−0.0742081
$546$	0	0
$547$	30.5039	1.30425	0.652127	−	0.758110i	$-0.273877\pi$
$547$	30.5039	1.30425	0.652127	+	0.758110i	$0.273877\pi$
$548$	0	0
$549$	12.1592	0.518943
$550$	0	0
$551$	−3.64572	−0.155313
$552$	0	0
$553$	0.312560	0.0132914
$554$	0	0
$555$	0.0279751	0.00118748
$556$	0	0
$557$	−22.4094	−0.949519	−0.474759	−	0.880116i	$-0.657465\pi$
$557$	−22.4094	−0.949519	−0.474759	+	0.880116i	$0.657465\pi$
$558$	0	0
$559$	−0.0572127	−0.00241984
$560$	0	0
$561$	0.176486	0.00745124
$562$	0	0
$563$	41.7915	1.76130	0.880652	−	0.473764i	$-0.157105\pi$
$563$	41.7915	1.76130	0.880652	+	0.473764i	$0.157105\pi$
$564$	0	0
$565$	−0.590979	−0.0248627
$566$	0	0
$567$	7.89221	0.331442
$568$	0	0
$569$	−11.4500	−0.480009	−0.240004	−	0.970772i	$-0.577149\pi$
$569$	−11.4500	−0.480009	−0.240004	+	0.970772i	$0.577149\pi$
$570$	0	0
$571$	−1.39669	−0.0584497	−0.0292248	−	0.999573i	$-0.509304\pi$
$571$	−1.39669	−0.0584497	−0.0292248	+	0.999573i	$0.509304\pi$
$572$	0	0
$573$	0.262671	0.0109732
$574$	0	0
$575$	−30.5499	−1.27402
$576$	0	0
$577$	−8.85413	−0.368602	−0.184301	−	0.982870i	$-0.559002\pi$
$577$	−8.85413	−0.368602	−0.184301	+	0.982870i	$0.559002\pi$
$578$	0	0
$579$	0.466736	0.0193969
$580$	0	0
$581$	13.6155	0.564864
$582$	0	0
$583$	7.24999	0.300264
$584$	0	0
$585$	0.618376	0.0255667
$586$	0	0
$587$	10.1705	0.419783	0.209891	−	0.977725i	$-0.432689\pi$
$587$	10.1705	0.419783	0.209891	+	0.977725i	$0.432689\pi$
$588$	0	0
$589$	0.00340723	0.000140393 0
$590$	0	0
$591$	0.735952	0.0302730
$592$	0	0
$593$	−5.01632	−0.205996	−0.102998	−	0.994682i	$-0.532843\pi$
$593$	−5.01632	−0.205996	−0.102998	+	0.994682i	$0.532843\pi$
$594$	0	0
$595$	−0.681111	−0.0279228
$596$	0	0
$597$	−0.537143	−0.0219838
$598$	0	0
$599$	−27.9408	−1.14163	−0.570816	−	0.821078i	$-0.693373\pi$
$599$	−27.9408	−1.14163	−0.570816	+	0.821078i	$0.693373\pi$
$600$	0	0
$601$	−28.9161	−1.17951	−0.589756	−	0.807581i	$-0.700776\pi$
$601$	−28.9161	−1.17951	−0.589756	+	0.807581i	$0.700776\pi$
$602$	0	0
$603$	−25.7388	−1.04816
$604$	0	0
$605$	−0.122408	−0.00497658
$606$	0	0
$607$	3.30116	0.133990	0.0669950	−	0.997753i	$-0.478659\pi$
$607$	3.30116	0.133990	0.0669950	+	0.997753i	$0.478659\pi$
$608$	0	0
$609$	0.0890574	0.00360879
$610$	0	0
$611$	10.2911	0.416333
$612$	0	0
$613$	−34.7837	−1.40490	−0.702451	−	0.711732i	$-0.747911\pi$
$613$	−34.7837	−1.40490	−0.702451	+	0.711732i	$0.747911\pi$
$614$	0	0
$615$	0.0380668	0.00153500
$616$	0	0
$617$	−0.452766	−0.0182277	−0.00911383	−	0.999958i	$-0.502901\pi$
$617$	−0.452766	−0.0182277	−0.00911383	+	0.999958i	$0.502901\pi$
$618$	0	0
$619$	−30.7171	−1.23462	−0.617312	−	0.786718i	$-0.711778\pi$
$619$	−30.7171	−1.23462	−0.617312	+	0.786718i	$0.711778\pi$
$620$	0	0
$621$	−1.02337	−0.0410664
$622$	0	0
$623$	4.66651	0.186960
$624$	0	0
$625$	24.7755	0.991019
$626$	0	0
$627$	−0.0278351	−0.00111163
$628$	0	0
$629$	−52.0578	−2.07568
$630$	0	0
$631$	−15.0741	−0.600092	−0.300046	−	0.953925i	$-0.597002\pi$
$631$	−15.0741	−0.600092	−0.300046	+	0.953925i	$0.597002\pi$
$632$	0	0
$633$	−0.277784	−0.0110409
$634$	0	0
$635$	0.128996	0.00511904
$636$	0	0
$637$	−10.4933	−0.415759
$638$	0	0
$639$	0.0408683	0.00161673
$640$	0	0
$641$	35.2312	1.39155	0.695774	−	0.718261i	$-0.255061\pi$
$641$	35.2312	1.39155	0.695774	+	0.718261i	$0.255061\pi$
$642$	0	0
$643$	−26.1788	−1.03239	−0.516195	−	0.856471i	$-0.672652\pi$
$643$	−26.1788	−1.03239	−0.516195	+	0.856471i	$0.672652\pi$
$644$	0	0
$645$	0.000115733 0	4.55700e−6 0
$646$	0	0
$647$	24.2822	0.954633	0.477316	−	0.878732i	$-0.341610\pi$
$647$	24.2822	0.954633	0.477316	+	0.878732i	$0.341610\pi$
$648$	0	0
$649$	14.2031	0.557520
$650$	0	0
$651$	−8.32315e−5 0	−3.26210e−6 0
$652$	0	0
$653$	−50.2439	−1.96620	−0.983098	−	0.183078i	$-0.941394\pi$
$653$	−50.2439	−1.96620	−0.983098	+	0.183078i	$0.941394\pi$
$654$	0	0
$655$	1.12666	0.0440223
$656$	0	0
$657$	−11.1955	−0.436779
$658$	0	0
$659$	−9.25601	−0.360563	−0.180281	−	0.983615i	$-0.557701\pi$
$659$	−9.25601	−0.360563	−0.180281	+	0.983615i	$0.557701\pi$
$660$	0	0
$661$	22.8800	0.889927	0.444964	−	0.895549i	$-0.353217\pi$
$661$	22.8800	0.889927	0.444964	+	0.895549i	$0.353217\pi$
$662$	0	0
$663$	0.297266	0.0115448
$664$	0	0
$665$	0.107424	0.00416572
$666$	0	0
$667$	22.3423	0.865096
$668$	0	0
$669$	−0.452856	−0.0175084
$670$	0	0
$671$	−4.05412	−0.156508
$672$	0	0
$673$	−42.5307	−1.63944	−0.819718	−	0.572767i	$-0.805870\pi$
$673$	−42.5307	−1.63944	−0.819718	+	0.572767i	$0.805870\pi$
$674$	0	0
$675$	0.832444	0.0320408
$676$	0	0
$677$	29.8688	1.14795	0.573976	−	0.818872i	$-0.305400\pi$
$677$	29.8688	1.14795	0.573976	+	0.818872i	$0.305400\pi$
$678$	0	0
$679$	14.6663	0.562842
$680$	0	0
$681$	0.698704	0.0267744
$682$	0	0
$683$	45.1162	1.72632	0.863162	−	0.504928i	$-0.168481\pi$
$683$	45.1162	1.72632	0.863162	+	0.504928i	$0.168481\pi$
$684$	0	0
$685$	0.795931	0.0304109
$686$	0	0
$687$	−0.0851773	−0.00324972
$688$	0	0
$689$	12.2116	0.465224
$690$	0	0
$691$	14.2491	0.542062	0.271031	−	0.962571i	$-0.412635\pi$
$691$	14.2491	0.542062	0.271031	+	0.962571i	$0.412635\pi$
$692$	0	0
$693$	−2.63210	−0.0999851
$694$	0	0
$695$	−1.44821	−0.0549336
$696$	0	0
$697$	−70.8371	−2.68315
$698$	0	0
$699$	−0.617335	−0.0233497
$700$	0	0
$701$	4.86294	0.183671	0.0918354	−	0.995774i	$-0.470727\pi$
$701$	4.86294	0.183671	0.0918354	+	0.995774i	$0.470727\pi$
$702$	0	0
$703$	8.21050	0.309665
$704$	0	0
$705$	−0.0208175	−0.000784032 0
$706$	0	0
$707$	−9.66454	−0.363473
$708$	0	0
$709$	−25.0524	−0.940863	−0.470432	−	0.882436i	$-0.655902\pi$
$709$	−25.0524	−0.940863	−0.470432	+	0.882436i	$0.655902\pi$
$710$	0	0
$711$	−1.06819	−0.0400603
$712$	0	0
$713$	−0.0208807	−0.000781989 0
$714$	0	0
$715$	−0.206179	−0.00771064
$716$	0	0
$717$	−0.609652	−0.0227679
$718$	0	0
$719$	45.9734	1.71452	0.857260	−	0.514884i	$-0.172165\pi$
$719$	45.9734	1.71452	0.857260	+	0.514884i	$0.172165\pi$
$720$	0	0
$721$	5.64822	0.210351
$722$	0	0
$723$	−0.0806128	−0.00299802
$724$	0	0
$725$	−18.1740	−0.674965
$726$	0	0
$727$	−26.4967	−0.982709	−0.491354	−	0.870960i	$-0.663498\pi$
$727$	−26.4967	−0.982709	−0.491354	+	0.870960i	$0.663498\pi$
$728$	0	0
$729$	−26.9582	−0.998451
$730$	0	0
$731$	−0.215364	−0.00796554
$732$	0	0
$733$	−0.180886	−0.00668116	−0.00334058	−	0.999994i	$-0.501063\pi$
$733$	−0.180886	−0.00668116	−0.00334058	+	0.999994i	$0.501063\pi$
$734$	0	0
$735$	0.0212265	0.000782951 0
$736$	0	0
$737$	8.58181	0.316115
$738$	0	0
$739$	−45.3719	−1.66903	−0.834516	−	0.550984i	$-0.814253\pi$
$739$	−45.3719	−1.66903	−0.834516	+	0.550984i	$0.814253\pi$
$740$	0	0
$741$	−0.0468844	−0.00172234
$742$	0	0
$743$	48.6104	1.78334	0.891672	−	0.452683i	$-0.149533\pi$
$743$	48.6104	1.78334	0.891672	+	0.452683i	$0.149533\pi$
$744$	0	0
$745$	2.22907	0.0816669
$746$	0	0
$747$	−46.5317	−1.70250
$748$	0	0
$749$	−5.77777	−0.211115
$750$	0	0
$751$	−43.6401	−1.59245	−0.796225	−	0.605000i	$-0.793173\pi$
$751$	−43.6401	−1.59245	−0.796225	+	0.605000i	$0.793173\pi$
$752$	0	0
$753$	0.207770	0.00757157
$754$	0	0
$755$	−1.82390	−0.0663787
$756$	0	0
$757$	40.0661	1.45623	0.728114	−	0.685456i	$-0.240397\pi$
$757$	40.0661	1.45623	0.728114	+	0.685456i	$0.240397\pi$
$758$	0	0
$759$	0.170583	0.00619179
$760$	0	0
$761$	24.0137	0.870494	0.435247	−	0.900311i	$-0.356661\pi$
$761$	24.0137	0.870494	0.435247	+	0.900311i	$0.356661\pi$
$762$	0	0
$763$	12.4204	0.449647
$764$	0	0
$765$	2.32774	0.0841596
$766$	0	0
$767$	23.9231	0.863814
$768$	0	0
$769$	30.8574	1.11275	0.556374	−	0.830932i	$-0.312192\pi$
$769$	30.8574	1.11275	0.556374	+	0.830932i	$0.312192\pi$
$770$	0	0
$771$	0.146065	0.00526041
$772$	0	0
$773$	−18.0873	−0.650554	−0.325277	−	0.945619i	$-0.605458\pi$
$773$	−18.0873	−0.650554	−0.325277	+	0.945619i	$0.605458\pi$
$774$	0	0
$775$	0.0169851	0.000610123 0
$776$	0	0
$777$	−0.200565	−0.00719523
$778$	0	0
$779$	11.1723	0.400291
$780$	0	0
$781$	−0.0136263	−0.000487587 0
$782$	0	0
$783$	−0.608797	−0.0217566
$784$	0	0
$785$	−2.42479	−0.0865445
$786$	0	0
$787$	44.3431	1.58066	0.790330	−	0.612681i	$-0.209909\pi$
$787$	44.3431	1.58066	0.790330	+	0.612681i	$0.209909\pi$
$788$	0	0
$789$	−0.518403	−0.0184556
$790$	0	0
$791$	4.23698	0.150650
$792$	0	0
$793$	−6.82860	−0.242491
$794$	0	0
$795$	−0.0247024	−0.000876103 0
$796$	0	0
$797$	−11.4047	−0.403976	−0.201988	−	0.979388i	$-0.564740\pi$
$797$	−11.4047	−0.403976	−0.201988	+	0.979388i	$0.564740\pi$
$798$	0	0
$799$	38.7385	1.37047
$800$	0	0
$801$	−15.9481	−0.563498
$802$	0	0
$803$	3.73280	0.131728
$804$	0	0
$805$	−0.658332	−0.0232032
$806$	0	0
$807$	−0.418641	−0.0147368
$808$	0	0
$809$	−11.4641	−0.403057	−0.201528	−	0.979483i	$-0.564591\pi$
$809$	−11.4641	−0.403057	−0.201528	+	0.979483i	$0.564591\pi$
$810$	0	0
$811$	33.7845	1.18633	0.593167	−	0.805080i	$-0.297878\pi$
$811$	33.7845	1.18633	0.593167	+	0.805080i	$0.297878\pi$
$812$	0	0
$813$	0.787443	0.0276168
$814$	0	0
$815$	−2.11560	−0.0741064
$816$	0	0
$817$	0.0339670	0.00118836
$818$	0	0
$819$	−4.43340	−0.154916
$820$	0	0
$821$	14.9537	0.521889	0.260944	−	0.965354i	$-0.415966\pi$
$821$	14.9537	0.521889	0.260944	+	0.965354i	$0.415966\pi$
$822$	0	0
$823$	18.9606	0.660925	0.330462	−	0.943819i	$-0.392795\pi$
$823$	18.9606	0.660925	0.330462	+	0.943819i	$0.392795\pi$
$824$	0	0
$825$	−0.138759	−0.00483095
$826$	0	0
$827$	−28.1451	−0.978699	−0.489350	−	0.872088i	$-0.662766\pi$
$827$	−28.1451	−0.978699	−0.489350	+	0.872088i	$0.662766\pi$
$828$	0	0
$829$	31.4452	1.09214	0.546069	−	0.837740i	$-0.316124\pi$
$829$	31.4452	1.09214	0.546069	+	0.837740i	$0.316124\pi$
$830$	0	0
$831$	0.622342	0.0215888
$832$	0	0
$833$	−39.4996	−1.36858
$834$	0	0
$835$	−1.46372	−0.0506542
$836$	0	0
$837$	0.000568971 0	1.96665e−5 0
$838$	0	0
$839$	−49.6194	−1.71305	−0.856526	−	0.516104i	$-0.827382\pi$
$839$	−49.6194	−1.71305	−0.856526	+	0.516104i	$0.827382\pi$
$840$	0	0
$841$	−15.7087	−0.541679
$842$	0	0
$843$	0.343728	0.0118386
$844$	0	0
$845$	1.24402	0.0427956
$846$	0	0
$847$	0.877592	0.0301544
$848$	0	0
$849$	0.345146	0.0118454
$850$	0	0
$851$	−50.3168	−1.72484
$852$	0	0
$853$	−27.9643	−0.957478	−0.478739	−	0.877957i	$-0.658906\pi$
$853$	−27.9643	−0.957478	−0.478739	+	0.877957i	$0.658906\pi$
$854$	0	0
$855$	−0.367128	−0.0125555
$856$	0	0
$857$	−1.38737	−0.0473916	−0.0236958	−	0.999719i	$-0.507543\pi$
$857$	−1.38737	−0.0473916	−0.0236958	+	0.999719i	$0.507543\pi$
$858$	0	0
$859$	−18.3432	−0.625864	−0.312932	−	0.949776i	$-0.601311\pi$
$859$	−18.3432	−0.625864	−0.312932	+	0.949776i	$0.601311\pi$
$860$	0	0
$861$	−0.272917	−0.00930098
$862$	0	0
$863$	45.2423	1.54006	0.770032	−	0.638005i	$-0.220240\pi$
$863$	45.2423	1.54006	0.770032	+	0.638005i	$0.220240\pi$
$864$	0	0
$865$	1.46984	0.0499760
$866$	0	0
$867$	0.645793	0.0219323
$868$	0	0
$869$	0.356156	0.0120818
$870$	0	0
$871$	14.4549	0.489784
$872$	0	0
$873$	−50.1231	−1.69641
$874$	0	0
$875$	1.07263	0.0362615
$876$	0	0
$877$	−42.8202	−1.44593	−0.722967	−	0.690882i	$-0.757222\pi$
$877$	−42.8202	−1.44593	−0.722967	+	0.690882i	$0.757222\pi$
$878$	0	0
$879$	−0.201631	−0.00680086
$880$	0	0
$881$	−32.4911	−1.09465	−0.547326	−	0.836920i	$-0.684354\pi$
$881$	−32.4911	−1.09465	−0.547326	+	0.836920i	$0.684354\pi$
$882$	0	0
$883$	−13.0319	−0.438558	−0.219279	−	0.975662i	$-0.570370\pi$
$883$	−13.0319	−0.438558	−0.219279	+	0.975662i	$0.570370\pi$
$884$	0	0
$885$	−0.0483933	−0.00162672
$886$	0	0
$887$	39.6956	1.33285	0.666424	−	0.745573i	$-0.267824\pi$
$887$	39.6956	1.33285	0.666424	+	0.745573i	$0.267824\pi$
$888$	0	0
$889$	−0.924824	−0.0310176
$890$	0	0
$891$	8.99303	0.301278
$892$	0	0
$893$	−6.10979	−0.204456
$894$	0	0
$895$	1.52986	0.0511377
$896$	0	0
$897$	0.287324	0.00959347
$898$	0	0
$899$	−0.0124218	−0.000414291 0
$900$	0	0
$901$	45.9678	1.53141
$902$	0	0
$903$	−0.000829742 0	−2.76121e−5 0
$904$	0	0
$905$	−0.592860	−0.0197073
$906$	0	0
$907$	0.181821	0.00603727	0.00301863	−	0.999995i	$-0.499039\pi$
$907$	0.181821	0.00603727	0.00301863	+	0.999995i	$0.499039\pi$
$908$	0	0
$909$	33.0292	1.09551
$910$	0	0
$911$	31.9193	1.05753	0.528767	−	0.848767i	$-0.322655\pi$
$911$	31.9193	1.05753	0.528767	+	0.848767i	$0.322655\pi$
$912$	0	0
$913$	15.5146	0.513457
$914$	0	0
$915$	0.0138133	0.000456655 0
$916$	0	0
$917$	−8.07751	−0.266743
$918$	0	0
$919$	−28.8673	−0.952246	−0.476123	−	0.879379i	$-0.657958\pi$
$919$	−28.8673	−0.952246	−0.476123	+	0.879379i	$0.657958\pi$
$920$	0	0
$921$	0.504950	0.0166387
$922$	0	0
$923$	−0.0229516	−0.000755461 0
$924$	0	0
$925$	40.9295	1.34575
$926$	0	0
$927$	−19.3031	−0.633999
$928$	0	0
$929$	24.5258	0.804666	0.402333	−	0.915493i	$-0.368199\pi$
$929$	24.5258	0.804666	0.402333	+	0.915493i	$0.368199\pi$
$930$	0	0
$931$	6.22983	0.204174
$932$	0	0
$933$	−0.267052	−0.00874289
$934$	0	0
$935$	−0.776113	−0.0253816
$936$	0	0
$937$	−10.6373	−0.347506	−0.173753	−	0.984789i	$-0.555589\pi$
$937$	−10.6373	−0.347506	−0.173753	+	0.984789i	$0.555589\pi$
$938$	0	0
$939$	0.813566	0.0265497
$940$	0	0
$941$	1.41861	0.0462453	0.0231226	−	0.999733i	$-0.492639\pi$
$941$	1.41861	0.0462453	0.0231226	+	0.999733i	$0.492639\pi$
$942$	0	0
$943$	−68.4681	−2.22963
$944$	0	0
$945$	0.0179387	0.000583544 0
$946$	0	0
$947$	5.25867	0.170884	0.0854418	−	0.996343i	$-0.472770\pi$
$947$	5.25867	0.170884	0.0854418	+	0.996343i	$0.472770\pi$
$948$	0	0
$949$	6.28738	0.204097
$950$	0	0
$951$	0.358316	0.0116192
$952$	0	0
$953$	−51.5273	−1.66913	−0.834567	−	0.550906i	$-0.814282\pi$
$953$	−51.5273	−1.66913	−0.834567	+	0.550906i	$0.814282\pi$
$954$	0	0
$955$	−1.15512	−0.0373788
$956$	0	0
$957$	0.101479	0.00328036
$958$	0	0
$959$	−5.70636	−0.184268
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	19.7459	0.636302
$964$	0	0
$965$	−2.05252	−0.0660728
$966$	0	0
$967$	−47.8627	−1.53916	−0.769581	−	0.638550i	$-0.779535\pi$
$967$	−47.8627	−1.53916	−0.769581	+	0.638550i	$0.779535\pi$
$968$	0	0
$969$	−0.176486	−0.00566954
$970$	0	0
$971$	−7.48025	−0.240053	−0.120026	−	0.992771i	$-0.538298\pi$
$971$	−7.48025	−0.240053	−0.120026	+	0.992771i	$0.538298\pi$
$972$	0	0
$973$	10.3828	0.332857
$974$	0	0
$975$	−0.233719	−0.00748501
$976$	0	0
$977$	−13.0292	−0.416841	−0.208420	−	0.978039i	$-0.566832\pi$
$977$	−13.0292	−0.416841	−0.208420	+	0.978039i	$0.566832\pi$
$978$	0	0
$979$	5.31740	0.169945
$980$	0	0
$981$	−42.4473	−1.35524
$982$	0	0
$983$	−48.1827	−1.53679	−0.768394	−	0.639977i	$-0.778944\pi$
$983$	−48.1827	−1.53679	−0.768394	+	0.639977i	$0.778944\pi$
$984$	0	0
$985$	−3.23642	−0.103121
$986$	0	0
$987$	0.149249	0.00475066
$988$	0	0
$989$	−0.208162	−0.00661916
$990$	0	0
$991$	54.1756	1.72095	0.860473	−	0.509496i	$-0.170168\pi$
$991$	54.1756	1.72095	0.860473	+	0.509496i	$0.170168\pi$
$992$	0	0
$993$	−0.236122	−0.00749312
$994$	0	0
$995$	2.36214	0.0748848
$996$	0	0
$997$	−17.3227	−0.548617	−0.274308	−	0.961642i	$-0.588449\pi$
$997$	−17.3227	−0.548617	−0.274308	+	0.961642i	$0.588449\pi$
$998$	0	0
$999$	1.37106	0.0433786

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.z.1.3		6
4.3	odd	2		1672.2.a.i.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.i.1.4	✓	6	4.3	odd	2
3344.2.a.z.1.3		6	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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0278351 q^{3} -0.122408 q^{5} +0.877592 q^{7} -2.99923 q^{9} +O(q^{10})\)	\(q+0.0278351 q^{3} -0.122408 q^{5} +0.877592 q^{7} -2.99923 q^{9} +1.00000 q^{11} +1.68436 q^{13} -0.00340723 q^{15} +6.34040 q^{17} -1.00000 q^{19} +0.0244279 q^{21} +6.12835 q^{23} -4.98502 q^{25} -0.166989 q^{27} +3.64572 q^{29} -0.00340723 q^{31} +0.0278351 q^{33} -0.107424 q^{35} -8.21050 q^{37} +0.0468844 q^{39} -11.1723 q^{41} -0.0339670 q^{43} +0.367128 q^{45} +6.10979 q^{47} -6.22983 q^{49} +0.176486 q^{51} +7.24999 q^{53} -0.122408 q^{55} -0.0278351 q^{57} +14.2031 q^{59} -4.05412 q^{61} -2.63210 q^{63} -0.206179 q^{65} +8.58181 q^{67} +0.170583 q^{69} -0.0136263 q^{71} +3.73280 q^{73} -0.138759 q^{75} +0.877592 q^{77} +0.356156 q^{79} +8.99303 q^{81} +15.5146 q^{83} -0.776113 q^{85} +0.101479 q^{87} +5.31740 q^{89} +1.47818 q^{91} -9.48408e-5 q^{93} +0.122408 q^{95} +16.7120 q^{97} -2.99923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} - 3 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q + 4 * q^3 - 3 * q^5 + 3 * q^7 + 6 * q^9	\( 6 q + 4 q^{3} - 3 q^{5} + 3 q^{7} + 6 q^{9} + 6 q^{11} - 3 q^{13} + q^{15} - q^{17} - 6 q^{19} + 5 q^{21} + 4 q^{23} + 5 q^{25} + 16 q^{27} - 7 q^{29} + q^{31} + 4 q^{33} + 32 q^{35} + 5 q^{37} + 13 q^{39} - 6 q^{41} + 16 q^{43} + 4 q^{47} - 7 q^{49} + 35 q^{51} - 11 q^{53} - 3 q^{55} - 4 q^{57} + 18 q^{59} + 16 q^{61} + 6 q^{63} + 10 q^{65} + 18 q^{67} - 2 q^{69} + 7 q^{71} - 21 q^{73} + 23 q^{75} + 3 q^{77} + 22 q^{79} - 10 q^{81} + 16 q^{83} + 3 q^{87} - 13 q^{89} + 7 q^{91} - 9 q^{93} + 3 q^{95} - 15 q^{97} + 6 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 - 3 * q^5 + 3 * q^7 + 6 * q^9 + 6 * q^11 - 3 * q^13 + q^15 - q^17 - 6 * q^19 + 5 * q^21 + 4 * q^23 + 5 * q^25 + 16 * q^27 - 7 * q^29 + q^31 + 4 * q^33 + 32 * q^35 + 5 * q^37 + 13 * q^39 - 6 * q^41 + 16 * q^43 + 4 * q^47 - 7 * q^49 + 35 * q^51 - 11 * q^53 - 3 * q^55 - 4 * q^57 + 18 * q^59 + 16 * q^61 + 6 * q^63 + 10 * q^65 + 18 * q^67 - 2 * q^69 + 7 * q^71 - 21 * q^73 + 23 * q^75 + 3 * q^77 + 22 * q^79 - 10 * q^81 + 16 * q^83 + 3 * q^87 - 13 * q^89 + 7 * q^91 - 9 * q^93 + 3 * q^95 - 15 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more