LMFDB - Embedded newform 3344.2.a.y.1.2

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:	6.6.57500224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} + 12x^{3} + 11x^{2} - 18x + 2 $ x^6 - 2x^5 - 7x^4 + 12x^3 + 11x^2 - 18*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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			1.2
Root			$1.64887$ 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)

$f(q)$	$=$	$q-0.648873 q^{3} +1.55597 q^{5} +3.00900 q^{7} -2.57896 q^{9} +O(q^{10})$	$q-0.648873 q^{3} +1.55597 q^{5} +3.00900 q^{7} -2.57896 q^{9} -1.00000 q^{11} -2.37832 q^{13} -1.00963 q^{15} -0.984536 q^{17} +1.00000 q^{19} -1.95246 q^{21} -3.60342 q^{23} -2.57896 q^{25} +3.62004 q^{27} +7.28857 q^{29} +10.1972 q^{31} +0.648873 q^{33} +4.68190 q^{35} -6.17499 q^{37} +1.54323 q^{39} +0.843023 q^{41} +9.86165 q^{43} -4.01278 q^{45} +13.7074 q^{47} +2.05406 q^{49} +0.638839 q^{51} +2.54950 q^{53} -1.55597 q^{55} -0.648873 q^{57} -9.68822 q^{59} +13.0762 q^{61} -7.76009 q^{63} -3.70059 q^{65} +11.0084 q^{67} +2.33817 q^{69} +9.22930 q^{71} -3.67776 q^{73} +1.67342 q^{75} -3.00900 q^{77} +4.14246 q^{79} +5.38794 q^{81} -4.87924 q^{83} -1.53191 q^{85} -4.72936 q^{87} -11.9154 q^{89} -7.15635 q^{91} -6.61670 q^{93} +1.55597 q^{95} -9.28693 q^{97} +2.57896 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9	$ 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9} - 6 q^{11} + 2 q^{13} - 10 q^{17} + 6 q^{19} + 10 q^{21} + 14 q^{23} + 2 q^{25} + 16 q^{27} + 6 q^{29} - 4 q^{31} - 4 q^{33} - 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41} + 26 q^{43} - 2 q^{45} + 24 q^{47} + 20 q^{49} + 14 q^{51} - 8 q^{53} + 4 q^{55} + 4 q^{57} + 4 q^{59} - 6 q^{61} + 40 q^{63} - 34 q^{65} + 44 q^{67} + 8 q^{69} + 16 q^{71} + 12 q^{73} + 28 q^{75} - 4 q^{77} - 6 q^{79} + 10 q^{81} + 28 q^{83} - 10 q^{85} + 24 q^{87} + 64 q^{91} - 14 q^{93} - 4 q^{95} + 4 q^{97} - 2 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9 - 6 * q^11 + 2 * q^13 - 10 * q^17 + 6 * q^19 + 10 * q^21 + 14 * q^23 + 2 * q^25 + 16 * q^27 + 6 * q^29 - 4 * q^31 - 4 * q^33 - 2 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^41 + 26 * q^43 - 2 * q^45 + 24 * q^47 + 20 * q^49 + 14 * q^51 - 8 * q^53 + 4 * q^55 + 4 * q^57 + 4 * q^59 - 6 * q^61 + 40 * q^63 - 34 * q^65 + 44 * q^67 + 8 * q^69 + 16 * q^71 + 12 * q^73 + 28 * q^75 - 4 * q^77 - 6 * q^79 + 10 * q^81 + 28 * q^83 - 10 * q^85 + 24 * q^87 + 64 * q^91 - 14 * q^93 - 4 * q^95 + 4 * q^97 - 2 * 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
$3$	−0.648873	−0.374627	−0.187314	−	0.982300i	$-0.559978\pi$
$3$	−0.648873	−0.374627	−0.187314	+	0.982300i	$0.559978\pi$
$4$	0	0
$5$	1.55597	0.695850	0.347925	−	0.937522i	$-0.386886\pi$
$5$	1.55597	0.695850	0.347925	+	0.937522i	$0.386886\pi$
$6$	0	0
$7$	3.00900	1.13729	0.568647	−	0.822582i	$-0.307467\pi$
$7$	3.00900	1.13729	0.568647	+	0.822582i	$0.307467\pi$
$8$	0	0
$9$	−2.57896	−0.859655
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.37832	−0.659627	−0.329813	−	0.944046i	$-0.606986\pi$
$13$	−2.37832	−0.659627	−0.329813	+	0.944046i	$0.606986\pi$
$14$	0	0
$15$	−1.00963	−0.260684
$16$	0	0
$17$	−0.984536	−0.238785	−0.119393	−	0.992847i	$-0.538095\pi$
$17$	−0.984536	−0.238785	−0.119393	+	0.992847i	$0.538095\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.95246	−0.426061
$22$	0	0
$23$	−3.60342	−0.751366	−0.375683	−	0.926748i	$-0.622592\pi$
$23$	−3.60342	−0.751366	−0.375683	+	0.926748i	$0.622592\pi$
$24$	0	0
$25$	−2.57896	−0.515793
$26$	0	0
$27$	3.62004	0.696677
$28$	0	0
$29$	7.28857	1.35345	0.676727	−	0.736234i	$-0.263398\pi$
$29$	7.28857	1.35345	0.676727	+	0.736234i	$0.263398\pi$
$30$	0	0
$31$	10.1972	1.83148	0.915738	−	0.401777i	$-0.131607\pi$
$31$	10.1972	1.83148	0.915738	+	0.401777i	$0.131607\pi$
$32$	0	0
$33$	0.648873	0.112954
$34$	0	0
$35$	4.68190	0.791386
$36$	0	0
$37$	−6.17499	−1.01516	−0.507581	−	0.861604i	$-0.669460\pi$
$37$	−6.17499	−1.01516	−0.507581	+	0.861604i	$0.669460\pi$
$38$	0	0
$39$	1.54323	0.247114
$40$	0	0
$41$	0.843023	0.131658	0.0658291	−	0.997831i	$-0.479031\pi$
$41$	0.843023	0.131658	0.0658291	+	0.997831i	$0.479031\pi$
$42$	0	0
$43$	9.86165	1.50389	0.751943	−	0.659228i	$-0.229117\pi$
$43$	9.86165	1.50389	0.751943	+	0.659228i	$0.229117\pi$
$44$	0	0
$45$	−4.01278	−0.598191
$46$	0	0
$47$	13.7074	1.99943	0.999717	−	0.0237903i	$-0.00757340\pi$
$47$	13.7074	1.99943	0.999717	+	0.0237903i	$0.00757340\pi$
$48$	0	0
$49$	2.05406	0.293437
$50$	0	0
$51$	0.638839	0.0894554
$52$	0	0
$53$	2.54950	0.350201	0.175100	−	0.984551i	$-0.443975\pi$
$53$	2.54950	0.350201	0.175100	+	0.984551i	$0.443975\pi$
$54$	0	0
$55$	−1.55597	−0.209807
$56$	0	0
$57$	−0.648873	−0.0859453
$58$	0	0
$59$	−9.68822	−1.26130	−0.630649	−	0.776068i	$-0.717211\pi$
$59$	−9.68822	−1.26130	−0.630649	+	0.776068i	$0.717211\pi$
$60$	0	0
$61$	13.0762	1.67423	0.837116	−	0.547026i	$-0.184240\pi$
$61$	13.0762	1.67423	0.837116	+	0.547026i	$0.184240\pi$
$62$	0	0
$63$	−7.76009	−0.977680
$64$	0	0
$65$	−3.70059	−0.459001
$66$	0	0
$67$	11.0084	1.34489	0.672443	−	0.740149i	$-0.265245\pi$
$67$	11.0084	1.34489	0.672443	+	0.740149i	$0.265245\pi$
$68$	0	0
$69$	2.33817	0.281482
$70$	0	0
$71$	9.22930	1.09532	0.547658	−	0.836702i	$-0.315519\pi$
$71$	9.22930	1.09532	0.547658	+	0.836702i	$0.315519\pi$
$72$	0	0
$73$	−3.67776	−0.430449	−0.215225	−	0.976565i	$-0.569048\pi$
$73$	−3.67776	−0.430449	−0.215225	+	0.976565i	$0.569048\pi$
$74$	0	0
$75$	1.67342	0.193230
$76$	0	0
$77$	−3.00900	−0.342907
$78$	0	0
$79$	4.14246	0.466064	0.233032	−	0.972469i	$-0.425135\pi$
$79$	4.14246	0.466064	0.233032	+	0.972469i	$0.425135\pi$
$80$	0	0
$81$	5.38794	0.598660
$82$	0	0
$83$	−4.87924	−0.535566	−0.267783	−	0.963479i	$-0.586291\pi$
$83$	−4.87924	−0.535566	−0.267783	+	0.963479i	$0.586291\pi$
$84$	0	0
$85$	−1.53191	−0.166159
$86$	0	0
$87$	−4.72936	−0.507040
$88$	0	0
$89$	−11.9154	−1.26303	−0.631514	−	0.775365i	$-0.717566\pi$
$89$	−11.9154	−1.26303	−0.631514	+	0.775365i	$0.717566\pi$
$90$	0	0
$91$	−7.15635	−0.750190
$92$	0	0
$93$	−6.61670	−0.686120
$94$	0	0
$95$	1.55597	0.159639
$96$	0	0
$97$	−9.28693	−0.942945	−0.471472	−	0.881881i	$-0.656277\pi$
$97$	−9.28693	−0.942945	−0.471472	+	0.881881i	$0.656277\pi$
$98$	0	0
$99$	2.57896	0.259196
$100$	0	0
$101$	16.1668	1.60865	0.804327	−	0.594187i	$-0.202526\pi$
$101$	16.1668	1.60865	0.804327	+	0.594187i	$0.202526\pi$
$102$	0	0
$103$	−1.19210	−0.117461	−0.0587306	−	0.998274i	$-0.518705\pi$
$103$	−1.19210	−0.117461	−0.0587306	+	0.998274i	$0.518705\pi$
$104$	0	0
$105$	−3.03796	−0.296475
$106$	0	0
$107$	−6.77337	−0.654807	−0.327403	−	0.944885i	$-0.606174\pi$
$107$	−6.77337	−0.654807	−0.327403	+	0.944885i	$0.606174\pi$
$108$	0	0
$109$	5.54415	0.531034	0.265517	−	0.964106i	$-0.414457\pi$
$109$	5.54415	0.531034	0.265517	+	0.964106i	$0.414457\pi$
$110$	0	0
$111$	4.00679	0.380308
$112$	0	0
$113$	8.98597	0.845329	0.422664	−	0.906286i	$-0.361095\pi$
$113$	8.98597	0.845329	0.422664	+	0.906286i	$0.361095\pi$
$114$	0	0
$115$	−5.60681	−0.522838
$116$	0	0
$117$	6.13360	0.567051
$118$	0	0
$119$	−2.96247	−0.271569
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−0.547015	−0.0493227
$124$	0	0
$125$	−11.7926	−1.05476
$126$	0	0
$127$	−19.4102	−1.72238	−0.861188	−	0.508287i	$-0.830279\pi$
$127$	−19.4102	−1.72238	−0.861188	+	0.508287i	$0.830279\pi$
$128$	0	0
$129$	−6.39896	−0.563397
$130$	0	0
$131$	13.9306	1.21713	0.608563	−	0.793506i	$-0.291746\pi$
$131$	13.9306	1.21713	0.608563	+	0.793506i	$0.291746\pi$
$132$	0	0
$133$	3.00900	0.260913
$134$	0	0
$135$	5.63267	0.484783
$136$	0	0
$137$	−2.92919	−0.250258	−0.125129	−	0.992140i	$-0.539934\pi$
$137$	−2.92919	−0.250258	−0.125129	+	0.992140i	$0.539934\pi$
$138$	0	0
$139$	3.20274	0.271653	0.135826	−	0.990733i	$-0.456631\pi$
$139$	3.20274	0.271653	0.135826	+	0.990733i	$0.456631\pi$
$140$	0	0
$141$	−8.89438	−0.749042
$142$	0	0
$143$	2.37832	0.198885
$144$	0	0
$145$	11.3408	0.941801
$146$	0	0
$147$	−1.33283	−0.109930
$148$	0	0
$149$	−2.34696	−0.192270	−0.0961350	−	0.995368i	$-0.530648\pi$
$149$	−2.34696	−0.192270	−0.0961350	+	0.995368i	$0.530648\pi$
$150$	0	0
$151$	12.5033	1.01751	0.508754	−	0.860912i	$-0.330106\pi$
$151$	12.5033	1.01751	0.508754	+	0.860912i	$0.330106\pi$
$152$	0	0
$153$	2.53908	0.205273
$154$	0	0
$155$	15.8666	1.27443
$156$	0	0
$157$	18.3829	1.46712	0.733558	−	0.679627i	$-0.237858\pi$
$157$	18.3829	1.46712	0.733558	+	0.679627i	$0.237858\pi$
$158$	0	0
$159$	−1.65430	−0.131195
$160$	0	0
$161$	−10.8427	−0.854524
$162$	0	0
$163$	10.3211	0.808410	0.404205	−	0.914668i	$-0.367548\pi$
$163$	10.3211	0.808410	0.404205	+	0.914668i	$0.367548\pi$
$164$	0	0
$165$	1.00963	0.0785993
$166$	0	0
$167$	2.89364	0.223916	0.111958	−	0.993713i	$-0.464288\pi$
$167$	2.89364	0.223916	0.111958	+	0.993713i	$0.464288\pi$
$168$	0	0
$169$	−7.34360	−0.564892
$170$	0	0
$171$	−2.57896	−0.197218
$172$	0	0
$173$	25.2755	1.92166	0.960829	−	0.277142i	$-0.0893872\pi$
$173$	25.2755	1.92166	0.960829	+	0.277142i	$0.0893872\pi$
$174$	0	0
$175$	−7.76009	−0.586608
$176$	0	0
$177$	6.28643	0.472517
$178$	0	0
$179$	−11.2398	−0.840103	−0.420052	−	0.907500i	$-0.637988\pi$
$179$	−11.2398	−0.840103	−0.420052	+	0.907500i	$0.637988\pi$
$180$	0	0
$181$	−13.8790	−1.03162	−0.515810	−	0.856703i	$-0.672509\pi$
$181$	−13.8790	−1.03162	−0.515810	+	0.856703i	$0.672509\pi$
$182$	0	0
$183$	−8.48477	−0.627212
$184$	0	0
$185$	−9.60809	−0.706401
$186$	0	0
$187$	0.984536	0.0719964
$188$	0	0
$189$	10.8927	0.792326
$190$	0	0
$191$	−11.4623	−0.829386	−0.414693	−	0.909961i	$-0.636111\pi$
$191$	−11.4623	−0.829386	−0.414693	+	0.909961i	$0.636111\pi$
$192$	0	0
$193$	−7.77127	−0.559388	−0.279694	−	0.960089i	$-0.590233\pi$
$193$	−7.77127	−0.559388	−0.279694	+	0.960089i	$0.590233\pi$
$194$	0	0
$195$	2.40121	0.171954
$196$	0	0
$197$	−21.8954	−1.55998	−0.779992	−	0.625789i	$-0.784777\pi$
$197$	−21.8954	−1.55998	−0.779992	+	0.625789i	$0.784777\pi$
$198$	0	0
$199$	5.30406	0.375995	0.187998	−	0.982169i	$-0.439800\pi$
$199$	5.30406	0.375995	0.187998	+	0.982169i	$0.439800\pi$
$200$	0	0
$201$	−7.14303	−0.503831
$202$	0	0
$203$	21.9313	1.53927
$204$	0	0
$205$	1.31172	0.0916143
$206$	0	0
$207$	9.29310	0.645915
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	21.0516	1.44925	0.724624	−	0.689144i	$-0.242013\pi$
$211$	21.0516	1.44925	0.724624	+	0.689144i	$0.242013\pi$
$212$	0	0
$213$	−5.98865	−0.410335
$214$	0	0
$215$	15.3444	1.04648
$216$	0	0
$217$	30.6834	2.08293
$218$	0	0
$219$	2.38640	0.161258
$220$	0	0
$221$	2.34154	0.157509
$222$	0	0
$223$	20.2686	1.35729	0.678645	−	0.734467i	$-0.262568\pi$
$223$	20.2686	1.35729	0.678645	+	0.734467i	$0.262568\pi$
$224$	0	0
$225$	6.65105	0.443404
$226$	0	0
$227$	−19.4348	−1.28993	−0.644966	−	0.764211i	$-0.723129\pi$
$227$	−19.4348	−1.28993	−0.644966	+	0.764211i	$0.723129\pi$
$228$	0	0
$229$	22.3074	1.47412	0.737058	−	0.675829i	$-0.236214\pi$
$229$	22.3074	1.47412	0.737058	+	0.675829i	$0.236214\pi$
$230$	0	0
$231$	1.95246	0.128462
$232$	0	0
$233$	1.60970	0.105455	0.0527274	−	0.998609i	$-0.483209\pi$
$233$	1.60970	0.105455	0.0527274	+	0.998609i	$0.483209\pi$
$234$	0	0
$235$	21.3283	1.39131
$236$	0	0
$237$	−2.68793	−0.174600
$238$	0	0
$239$	−24.8702	−1.60872	−0.804360	−	0.594142i	$-0.797492\pi$
$239$	−24.8702	−1.60872	−0.804360	+	0.594142i	$0.797492\pi$
$240$	0	0
$241$	−8.62638	−0.555674	−0.277837	−	0.960628i	$-0.589618\pi$
$241$	−8.62638	−0.555674	−0.277837	+	0.960628i	$0.589618\pi$
$242$	0	0
$243$	−14.3562	−0.920951
$244$	0	0
$245$	3.19605	0.204188
$246$	0	0
$247$	−2.37832	−0.151329
$248$	0	0
$249$	3.16601	0.200638
$250$	0	0
$251$	−17.2395	−1.08815	−0.544073	−	0.839038i	$-0.683119\pi$
$251$	−17.2395	−1.08815	−0.544073	+	0.839038i	$0.683119\pi$
$252$	0	0
$253$	3.60342	0.226545
$254$	0	0
$255$	0.994013	0.0622475
$256$	0	0
$257$	7.30942	0.455949	0.227975	−	0.973667i	$-0.426790\pi$
$257$	7.30942	0.455949	0.227975	+	0.973667i	$0.426790\pi$
$258$	0	0
$259$	−18.5805	−1.15454
$260$	0	0
$261$	−18.7970	−1.16350
$262$	0	0
$263$	16.9406	1.04460	0.522302	−	0.852761i	$-0.325073\pi$
$263$	16.9406	1.04460	0.522302	+	0.852761i	$0.325073\pi$
$264$	0	0
$265$	3.96694	0.243687
$266$	0	0
$267$	7.73157	0.473164
$268$	0	0
$269$	25.0938	1.52999	0.764997	−	0.644034i	$-0.222740\pi$
$269$	25.0938	1.52999	0.764997	+	0.644034i	$0.222740\pi$
$270$	0	0
$271$	−8.38851	−0.509565	−0.254783	−	0.966998i	$-0.582004\pi$
$271$	−8.38851	−0.509565	−0.254783	+	0.966998i	$0.582004\pi$
$272$	0	0
$273$	4.64357	0.281041
$274$	0	0
$275$	2.57896	0.155517
$276$	0	0
$277$	−0.197132	−0.0118445	−0.00592226	−	0.999982i	$-0.501885\pi$
$277$	−0.197132	−0.0118445	−0.00592226	+	0.999982i	$0.501885\pi$
$278$	0	0
$279$	−26.2983	−1.57444
$280$	0	0
$281$	17.3356	1.03416	0.517078	−	0.855939i	$-0.327020\pi$
$281$	17.3356	1.03416	0.517078	+	0.855939i	$0.327020\pi$
$282$	0	0
$283$	23.1466	1.37592	0.687962	−	0.725747i	$-0.258506\pi$
$283$	23.1466	1.37592	0.687962	+	0.725747i	$0.258506\pi$
$284$	0	0
$285$	−1.00963	−0.0598051
$286$	0	0
$287$	2.53665	0.149734
$288$	0	0
$289$	−16.0307	−0.942982
$290$	0	0
$291$	6.02604	0.353253
$292$	0	0
$293$	−21.8615	−1.27716	−0.638582	−	0.769554i	$-0.720479\pi$
$293$	−21.8615	−1.27716	−0.638582	+	0.769554i	$0.720479\pi$
$294$	0	0
$295$	−15.0746	−0.877675
$296$	0	0
$297$	−3.62004	−0.210056
$298$	0	0
$299$	8.57009	0.495621
$300$	0	0
$301$	29.6737	1.71036
$302$	0	0
$303$	−10.4902	−0.602645
$304$	0	0
$305$	20.3461	1.16501
$306$	0	0
$307$	9.95976	0.568434	0.284217	−	0.958760i	$-0.408266\pi$
$307$	9.95976	0.568434	0.284217	+	0.958760i	$0.408266\pi$
$308$	0	0
$309$	0.773522	0.0440041
$310$	0	0
$311$	24.9164	1.41288	0.706440	−	0.707773i	$-0.250300\pi$
$311$	24.9164	1.41288	0.706440	+	0.707773i	$0.250300\pi$
$312$	0	0
$313$	31.2798	1.76804	0.884019	−	0.467451i	$-0.154828\pi$
$313$	31.2798	1.76804	0.884019	+	0.467451i	$0.154828\pi$
$314$	0	0
$315$	−12.0745	−0.680319
$316$	0	0
$317$	−19.4148	−1.09044	−0.545222	−	0.838292i	$-0.683555\pi$
$317$	−19.4148	−1.09044	−0.545222	+	0.838292i	$0.683555\pi$
$318$	0	0
$319$	−7.28857	−0.408082
$320$	0	0
$321$	4.39506	0.245308
$322$	0	0
$323$	−0.984536	−0.0547811
$324$	0	0
$325$	6.13360	0.340231
$326$	0	0
$327$	−3.59745	−0.198940
$328$	0	0
$329$	41.2456	2.27394
$330$	0	0
$331$	14.3051	0.786278	0.393139	−	0.919479i	$-0.371389\pi$
$331$	14.3051	0.786278	0.393139	+	0.919479i	$0.371389\pi$
$332$	0	0
$333$	15.9251	0.872689
$334$	0	0
$335$	17.1287	0.935839
$336$	0	0
$337$	18.1133	0.986694	0.493347	−	0.869833i	$-0.335773\pi$
$337$	18.1133	0.986694	0.493347	+	0.869833i	$0.335773\pi$
$338$	0	0
$339$	−5.83075	−0.316683
$340$	0	0
$341$	−10.1972	−0.552211
$342$	0	0
$343$	−14.8823	−0.803569
$344$	0	0
$345$	3.63811	0.195869
$346$	0	0
$347$	5.68351	0.305107	0.152553	−	0.988295i	$-0.451250\pi$
$347$	5.68351	0.305107	0.152553	+	0.988295i	$0.451250\pi$
$348$	0	0
$349$	35.0341	1.87533	0.937666	−	0.347537i	$-0.112982\pi$
$349$	35.0341	1.87533	0.937666	+	0.347537i	$0.112982\pi$
$350$	0	0
$351$	−8.60961	−0.459547
$352$	0	0
$353$	3.60545	0.191899	0.0959494	−	0.995386i	$-0.469411\pi$
$353$	3.60545	0.191899	0.0959494	+	0.995386i	$0.469411\pi$
$354$	0	0
$355$	14.3605	0.762176
$356$	0	0
$357$	1.92227	0.101737
$358$	0	0
$359$	15.6880	0.827982	0.413991	−	0.910281i	$-0.364134\pi$
$359$	15.6880	0.827982	0.413991	+	0.910281i	$0.364134\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−0.648873	−0.0340570
$364$	0	0
$365$	−5.72248	−0.299528
$366$	0	0
$367$	−28.2412	−1.47418	−0.737088	−	0.675796i	$-0.763800\pi$
$367$	−28.2412	−1.47418	−0.737088	+	0.675796i	$0.763800\pi$
$368$	0	0
$369$	−2.17413	−0.113180
$370$	0	0
$371$	7.67144	0.398281
$372$	0	0
$373$	−16.6776	−0.863531	−0.431766	−	0.901986i	$-0.642109\pi$
$373$	−16.6776	−0.863531	−0.431766	+	0.901986i	$0.642109\pi$
$374$	0	0
$375$	7.65192	0.395143
$376$	0	0
$377$	−17.3345	−0.892775
$378$	0	0
$379$	−25.5322	−1.31150	−0.655751	−	0.754977i	$-0.727648\pi$
$379$	−25.5322	−1.31150	−0.655751	+	0.754977i	$0.727648\pi$
$380$	0	0
$381$	12.5947	0.645249
$382$	0	0
$383$	−23.8917	−1.22081	−0.610403	−	0.792091i	$-0.708993\pi$
$383$	−23.8917	−1.22081	−0.610403	+	0.792091i	$0.708993\pi$
$384$	0	0
$385$	−4.68190	−0.238612
$386$	0	0
$387$	−25.4328	−1.29282
$388$	0	0
$389$	−15.6454	−0.793255	−0.396628	−	0.917980i	$-0.629820\pi$
$389$	−15.6454	−0.793255	−0.396628	+	0.917980i	$0.629820\pi$
$390$	0	0
$391$	3.54770	0.179415
$392$	0	0
$393$	−9.03921	−0.455968
$394$	0	0
$395$	6.44554	0.324310
$396$	0	0
$397$	10.3915	0.521534	0.260767	−	0.965402i	$-0.416025\pi$
$397$	10.3915	0.521534	0.260767	+	0.965402i	$0.416025\pi$
$398$	0	0
$399$	−1.95246	−0.0977451
$400$	0	0
$401$	14.4409	0.721144	0.360572	−	0.932731i	$-0.382582\pi$
$401$	14.4409	0.721144	0.360572	+	0.932731i	$0.382582\pi$
$402$	0	0
$403$	−24.2522	−1.20809
$404$	0	0
$405$	8.38347	0.416578
$406$	0	0
$407$	6.17499	0.306083
$408$	0	0
$409$	−17.5325	−0.866925	−0.433462	−	0.901172i	$-0.642708\pi$
$409$	−17.5325	−0.866925	−0.433462	+	0.901172i	$0.642708\pi$
$410$	0	0
$411$	1.90067	0.0937534
$412$	0	0
$413$	−29.1518	−1.43447
$414$	0	0
$415$	−7.59194	−0.372674
$416$	0	0
$417$	−2.07817	−0.101768
$418$	0	0
$419$	−10.9059	−0.532787	−0.266394	−	0.963864i	$-0.585832\pi$
$419$	−10.9059	−0.532787	−0.266394	+	0.963864i	$0.585832\pi$
$420$	0	0
$421$	−4.00303	−0.195096	−0.0975480	−	0.995231i	$-0.531100\pi$
$421$	−4.00303	−0.195096	−0.0975480	+	0.995231i	$0.531100\pi$
$422$	0	0
$423$	−35.3510	−1.71882
$424$	0	0
$425$	2.53908	0.123164
$426$	0	0
$427$	39.3461	1.90409
$428$	0	0
$429$	−1.54323	−0.0745077
$430$	0	0
$431$	−36.1424	−1.74092	−0.870459	−	0.492241i	$-0.836178\pi$
$431$	−36.1424	−1.74092	−0.870459	+	0.492241i	$0.836178\pi$
$432$	0	0
$433$	−0.449187	−0.0215866	−0.0107933	−	0.999942i	$-0.503436\pi$
$433$	−0.449187	−0.0215866	−0.0107933	+	0.999942i	$0.503436\pi$
$434$	0	0
$435$	−7.35873	−0.352824
$436$	0	0
$437$	−3.60342	−0.172375
$438$	0	0
$439$	−24.6963	−1.17869	−0.589344	−	0.807882i	$-0.700614\pi$
$439$	−24.6963	−1.17869	−0.589344	+	0.807882i	$0.700614\pi$
$440$	0	0
$441$	−5.29735	−0.252255
$442$	0	0
$443$	37.3811	1.77603	0.888014	−	0.459817i	$-0.152085\pi$
$443$	37.3811	1.77603	0.888014	+	0.459817i	$0.152085\pi$
$444$	0	0
$445$	−18.5399	−0.878878
$446$	0	0
$447$	1.52288	0.0720296
$448$	0	0
$449$	−12.5090	−0.590337	−0.295168	−	0.955445i	$-0.595376\pi$
$449$	−12.5090	−0.590337	−0.295168	+	0.955445i	$0.595376\pi$
$450$	0	0
$451$	−0.843023	−0.0396964
$452$	0	0
$453$	−8.11308	−0.381186
$454$	0	0
$455$	−11.1351	−0.522020
$456$	0	0
$457$	−33.0973	−1.54823	−0.774114	−	0.633047i	$-0.781804\pi$
$457$	−33.0973	−1.54823	−0.774114	+	0.633047i	$0.781804\pi$
$458$	0	0
$459$	−3.56406	−0.166356
$460$	0	0
$461$	−20.5472	−0.956978	−0.478489	−	0.878093i	$-0.658815\pi$
$461$	−20.5472	−0.956978	−0.478489	+	0.878093i	$0.658815\pi$
$462$	0	0
$463$	−42.1889	−1.96068	−0.980342	−	0.197305i	$-0.936781\pi$
$463$	−42.1889	−1.96068	−0.980342	+	0.197305i	$0.936781\pi$
$464$	0	0
$465$	−10.2954	−0.477437
$466$	0	0
$467$	30.0985	1.39279	0.696397	−	0.717657i	$-0.254785\pi$
$467$	30.0985	1.39279	0.696397	+	0.717657i	$0.254785\pi$
$468$	0	0
$469$	33.1241	1.52953
$470$	0	0
$471$	−11.9282	−0.549621
$472$	0	0
$473$	−9.86165	−0.453439
$474$	0	0
$475$	−2.57896	−0.118331
$476$	0	0
$477$	−6.57507	−0.301052
$478$	0	0
$479$	−9.08471	−0.415091	−0.207545	−	0.978225i	$-0.566547\pi$
$479$	−9.08471	−0.415091	−0.207545	+	0.978225i	$0.566547\pi$
$480$	0	0
$481$	14.6861	0.669629
$482$	0	0
$483$	7.03553	0.320128
$484$	0	0
$485$	−14.4502	−0.656148
$486$	0	0
$487$	−25.1126	−1.13796	−0.568980	−	0.822351i	$-0.692662\pi$
$487$	−25.1126	−1.13796	−0.568980	+	0.822351i	$0.692662\pi$
$488$	0	0
$489$	−6.69708	−0.302852
$490$	0	0
$491$	13.7076	0.618616	0.309308	−	0.950962i	$-0.399903\pi$
$491$	13.7076	0.618616	0.309308	+	0.950962i	$0.399903\pi$
$492$	0	0
$493$	−7.17586	−0.323185
$494$	0	0
$495$	4.01278	0.180361
$496$	0	0
$497$	27.7709	1.24570
$498$	0	0
$499$	7.34793	0.328938	0.164469	−	0.986382i	$-0.447409\pi$
$499$	7.34793	0.328938	0.164469	+	0.986382i	$0.447409\pi$
$500$	0	0
$501$	−1.87760	−0.0838852
$502$	0	0
$503$	16.8783	0.752565	0.376282	−	0.926505i	$-0.377202\pi$
$503$	16.8783	0.752565	0.376282	+	0.926505i	$0.377202\pi$
$504$	0	0
$505$	25.1550	1.11938
$506$	0	0
$507$	4.76506	0.211624
$508$	0	0
$509$	−30.4487	−1.34961	−0.674807	−	0.737995i	$-0.735773\pi$
$509$	−30.4487	−1.34961	−0.674807	+	0.737995i	$0.735773\pi$
$510$	0	0
$511$	−11.0664	−0.489547
$512$	0	0
$513$	3.62004	0.159829
$514$	0	0
$515$	−1.85487	−0.0817353
$516$	0	0
$517$	−13.7074	−0.602852
$518$	0	0
$519$	−16.4006	−0.719905
$520$	0	0
$521$	−7.59750	−0.332852	−0.166426	−	0.986054i	$-0.553223\pi$
$521$	−7.59750	−0.332852	−0.166426	+	0.986054i	$0.553223\pi$
$522$	0	0
$523$	13.2619	0.579904	0.289952	−	0.957041i	$-0.406361\pi$
$523$	13.2619	0.579904	0.289952	+	0.957041i	$0.406361\pi$
$524$	0	0
$525$	5.03532	0.219759
$526$	0	0
$527$	−10.0395	−0.437329
$528$	0	0
$529$	−10.0153	−0.435449
$530$	0	0
$531$	24.9856	1.08428
$532$	0	0
$533$	−2.00498	−0.0868452
$534$	0	0
$535$	−10.5392	−0.455647
$536$	0	0
$537$	7.29321	0.314725
$538$	0	0
$539$	−2.05406	−0.0884747
$540$	0	0
$541$	34.5111	1.48375	0.741874	−	0.670540i	$-0.233937\pi$
$541$	34.5111	1.48375	0.741874	+	0.670540i	$0.233937\pi$
$542$	0	0
$543$	9.00573	0.386473
$544$	0	0
$545$	8.62652	0.369520
$546$	0	0
$547$	−31.4643	−1.34532	−0.672659	−	0.739952i	$-0.734848\pi$
$547$	−31.4643	−1.34532	−0.672659	+	0.739952i	$0.734848\pi$
$548$	0	0
$549$	−33.7230	−1.43926
$550$	0	0
$551$	7.28857	0.310504
$552$	0	0
$553$	12.4647	0.530051
$554$	0	0
$555$	6.23443	0.264637
$556$	0	0
$557$	11.7346	0.497209	0.248604	−	0.968605i	$-0.420028\pi$
$557$	11.7346	0.497209	0.248604	+	0.968605i	$0.420028\pi$
$558$	0	0
$559$	−23.4541	−0.992004
$560$	0	0
$561$	−0.638839	−0.0269718
$562$	0	0
$563$	42.8769	1.80705	0.903523	−	0.428540i	$-0.140972\pi$
$563$	42.8769	1.80705	0.903523	+	0.428540i	$0.140972\pi$
$564$	0	0
$565$	13.9819	0.588222
$566$	0	0
$567$	16.2123	0.680853
$568$	0	0
$569$	−33.6183	−1.40935	−0.704677	−	0.709528i	$-0.748908\pi$
$569$	−33.6183	−1.40935	−0.704677	+	0.709528i	$0.748908\pi$
$570$	0	0
$571$	−2.83048	−0.118452	−0.0592260	−	0.998245i	$-0.518863\pi$
$571$	−2.83048	−0.118452	−0.0592260	+	0.998245i	$0.518863\pi$
$572$	0	0
$573$	7.43761	0.310711
$574$	0	0
$575$	9.29310	0.387549
$576$	0	0
$577$	−16.6118	−0.691558	−0.345779	−	0.938316i	$-0.612385\pi$
$577$	−16.6118	−0.691558	−0.345779	+	0.938316i	$0.612385\pi$
$578$	0	0
$579$	5.04257	0.209562
$580$	0	0
$581$	−14.6816	−0.609096
$582$	0	0
$583$	−2.54950	−0.105590
$584$	0	0
$585$	9.54368	0.394583
$586$	0	0
$587$	−2.95793	−0.122087	−0.0610433	−	0.998135i	$-0.519443\pi$
$587$	−2.95793	−0.122087	−0.0610433	+	0.998135i	$0.519443\pi$
$588$	0	0
$589$	10.1972	0.420169
$590$	0	0
$591$	14.2074	0.584412
$592$	0	0
$593$	−27.7969	−1.14148	−0.570740	−	0.821130i	$-0.693344\pi$
$593$	−27.7969	−1.14148	−0.570740	+	0.821130i	$0.693344\pi$
$594$	0	0
$595$	−4.60950	−0.188971
$596$	0	0
$597$	−3.44166	−0.140858
$598$	0	0
$599$	−18.7222	−0.764967	−0.382484	−	0.923962i	$-0.624931\pi$
$599$	−18.7222	−0.764967	−0.382484	+	0.923962i	$0.624931\pi$
$600$	0	0
$601$	26.0097	1.06096	0.530480	−	0.847698i	$-0.322012\pi$
$601$	26.0097	1.06096	0.530480	+	0.847698i	$0.322012\pi$
$602$	0	0
$603$	−28.3902	−1.15614
$604$	0	0
$605$	1.55597	0.0632591
$606$	0	0
$607$	5.46452	0.221798	0.110899	−	0.993832i	$-0.464627\pi$
$607$	5.46452	0.221798	0.110899	+	0.993832i	$0.464627\pi$
$608$	0	0
$609$	−14.2306	−0.576654
$610$	0	0
$611$	−32.6006	−1.31888
$612$	0	0
$613$	−7.86368	−0.317611	−0.158806	−	0.987310i	$-0.550764\pi$
$613$	−7.86368	−0.317611	−0.158806	+	0.987310i	$0.550764\pi$
$614$	0	0
$615$	−0.851138	−0.0343212
$616$	0	0
$617$	−18.7961	−0.756703	−0.378351	−	0.925662i	$-0.623509\pi$
$617$	−18.7961	−0.756703	−0.378351	+	0.925662i	$0.623509\pi$
$618$	0	0
$619$	9.67151	0.388731	0.194365	−	0.980929i	$-0.437735\pi$
$619$	9.67151	0.388731	0.194365	+	0.980929i	$0.437735\pi$
$620$	0	0
$621$	−13.0445	−0.523459
$622$	0	0
$623$	−35.8533	−1.43643
$624$	0	0
$625$	−5.45413	−0.218165
$626$	0	0
$627$	0.648873	0.0259135
$628$	0	0
$629$	6.07951	0.242406
$630$	0	0
$631$	43.9683	1.75035	0.875175	−	0.483806i	$-0.160746\pi$
$631$	43.9683	1.75035	0.875175	+	0.483806i	$0.160746\pi$
$632$	0	0
$633$	−13.6598	−0.542928
$634$	0	0
$635$	−30.2016	−1.19852
$636$	0	0
$637$	−4.88521	−0.193559
$638$	0	0
$639$	−23.8020	−0.941594
$640$	0	0
$641$	−11.5103	−0.454629	−0.227314	−	0.973821i	$-0.572995\pi$
$641$	−11.5103	−0.454629	−0.227314	+	0.973821i	$0.572995\pi$
$642$	0	0
$643$	17.9328	0.707199	0.353599	−	0.935397i	$-0.384958\pi$
$643$	17.9328	0.707199	0.353599	+	0.935397i	$0.384958\pi$
$644$	0	0
$645$	−9.95657	−0.392040
$646$	0	0
$647$	6.27370	0.246645	0.123322	−	0.992367i	$-0.460645\pi$
$647$	6.27370	0.246645	0.123322	+	0.992367i	$0.460645\pi$
$648$	0	0
$649$	9.68822	0.380296
$650$	0	0
$651$	−19.9096	−0.780320
$652$	0	0
$653$	42.0285	1.64470	0.822351	−	0.568981i	$-0.192662\pi$
$653$	42.0285	1.64470	0.822351	+	0.568981i	$0.192662\pi$
$654$	0	0
$655$	21.6756	0.846937
$656$	0	0
$657$	9.48481	0.370038
$658$	0	0
$659$	−41.5487	−1.61851	−0.809254	−	0.587458i	$-0.800129\pi$
$659$	−41.5487	−1.61851	−0.809254	+	0.587458i	$0.800129\pi$
$660$	0	0
$661$	−10.4271	−0.405567	−0.202783	−	0.979224i	$-0.564999\pi$
$661$	−10.4271	−0.405567	−0.202783	+	0.979224i	$0.564999\pi$
$662$	0	0
$663$	−1.51936	−0.0590072
$664$	0	0
$665$	4.68190	0.181556
$666$	0	0
$667$	−26.2638	−1.01694
$668$	0	0
$669$	−13.1518	−0.508477
$670$	0	0
$671$	−13.0762	−0.504800
$672$	0	0
$673$	2.99864	0.115589	0.0577945	−	0.998329i	$-0.481593\pi$
$673$	2.99864	0.115589	0.0577945	+	0.998329i	$0.481593\pi$
$674$	0	0
$675$	−9.33595	−0.359341
$676$	0	0
$677$	−41.4801	−1.59421	−0.797104	−	0.603842i	$-0.793636\pi$
$677$	−41.4801	−1.59421	−0.797104	+	0.603842i	$0.793636\pi$
$678$	0	0
$679$	−27.9443	−1.07241
$680$	0	0
$681$	12.6107	0.483244
$682$	0	0
$683$	−27.0700	−1.03580	−0.517902	−	0.855440i	$-0.673287\pi$
$683$	−27.0700	−1.03580	−0.517902	+	0.855440i	$0.673287\pi$
$684$	0	0
$685$	−4.55773	−0.174142
$686$	0	0
$687$	−14.4747	−0.552244
$688$	0	0
$689$	−6.06353	−0.231002
$690$	0	0
$691$	16.3633	0.622491	0.311245	−	0.950330i	$-0.399254\pi$
$691$	16.3633	0.622491	0.311245	+	0.950330i	$0.399254\pi$
$692$	0	0
$693$	7.76009	0.294782
$694$	0	0
$695$	4.98336	0.189030
$696$	0	0
$697$	−0.829987	−0.0314380
$698$	0	0
$699$	−1.04449	−0.0395062
$700$	0	0
$701$	−2.61332	−0.0987039	−0.0493520	−	0.998781i	$-0.515716\pi$
$701$	−2.61332	−0.0987039	−0.0493520	+	0.998781i	$0.515716\pi$
$702$	0	0
$703$	−6.17499	−0.232894
$704$	0	0
$705$	−13.8394	−0.521221
$706$	0	0
$707$	48.6458	1.82951
$708$	0	0
$709$	34.0692	1.27950	0.639748	−	0.768585i	$-0.279039\pi$
$709$	34.0692	1.27950	0.639748	+	0.768585i	$0.279039\pi$
$710$	0	0
$711$	−10.6833	−0.400654
$712$	0	0
$713$	−36.7449	−1.37611
$714$	0	0
$715$	3.70059	0.138394
$716$	0	0
$717$	16.1376	0.602670
$718$	0	0
$719$	3.55433	0.132554	0.0662771	−	0.997801i	$-0.478888\pi$
$719$	3.55433	0.132554	0.0662771	+	0.997801i	$0.478888\pi$
$720$	0	0
$721$	−3.58703	−0.133588
$722$	0	0
$723$	5.59743	0.208171
$724$	0	0
$725$	−18.7970	−0.698102
$726$	0	0
$727$	21.4519	0.795605	0.397803	−	0.917471i	$-0.369773\pi$
$727$	21.4519	0.795605	0.397803	+	0.917471i	$0.369773\pi$
$728$	0	0
$729$	−6.84847	−0.253647
$730$	0	0
$731$	−9.70915	−0.359106
$732$	0	0
$733$	−11.0026	−0.406392	−0.203196	−	0.979138i	$-0.565133\pi$
$733$	−11.0026	−0.406392	−0.203196	+	0.979138i	$0.565133\pi$
$734$	0	0
$735$	−2.07383	−0.0764945
$736$	0	0
$737$	−11.0084	−0.405498
$738$	0	0
$739$	13.2742	0.488300	0.244150	−	0.969738i	$-0.421491\pi$
$739$	13.2742	0.488300	0.244150	+	0.969738i	$0.421491\pi$
$740$	0	0
$741$	1.54323	0.0566919
$742$	0	0
$743$	−18.6244	−0.683264	−0.341632	−	0.939834i	$-0.610980\pi$
$743$	−18.6244	−0.683264	−0.341632	+	0.939834i	$0.610980\pi$
$744$	0	0
$745$	−3.65179	−0.133791
$746$	0	0
$747$	12.5834	0.460402
$748$	0	0
$749$	−20.3811	−0.744708
$750$	0	0
$751$	43.9807	1.60488	0.802440	−	0.596733i	$-0.203535\pi$
$751$	43.9807	1.60488	0.802440	+	0.596733i	$0.203535\pi$
$752$	0	0
$753$	11.1862	0.407649
$754$	0	0
$755$	19.4548	0.708032
$756$	0	0
$757$	2.80658	0.102007	0.0510035	−	0.998698i	$-0.483758\pi$
$757$	2.80658	0.102007	0.0510035	+	0.998698i	$0.483758\pi$
$758$	0	0
$759$	−2.33817	−0.0848700
$760$	0	0
$761$	−38.6635	−1.40155	−0.700776	−	0.713381i	$-0.747163\pi$
$761$	−38.6635	−1.40155	−0.700776	+	0.713381i	$0.747163\pi$
$762$	0	0
$763$	16.6823	0.603941
$764$	0	0
$765$	3.95073	0.142839
$766$	0	0
$767$	23.0417	0.831987
$768$	0	0
$769$	−49.0719	−1.76958	−0.884789	−	0.465991i	$-0.845698\pi$
$769$	−49.0719	−1.76958	−0.884789	+	0.465991i	$0.845698\pi$
$770$	0	0
$771$	−4.74289	−0.170811
$772$	0	0
$773$	13.3520	0.480237	0.240118	−	0.970744i	$-0.422814\pi$
$773$	13.3520	0.480237	0.240118	+	0.970744i	$0.422814\pi$
$774$	0	0
$775$	−26.2983	−0.944662
$776$	0	0
$777$	12.0564	0.432521
$778$	0	0
$779$	0.843023	0.0302044
$780$	0	0
$781$	−9.22930	−0.330250
$782$	0	0
$783$	26.3849	0.942920
$784$	0	0
$785$	28.6032	1.02089
$786$	0	0
$787$	0.302278	0.0107750	0.00538752	−	0.999985i	$-0.498285\pi$
$787$	0.302278	0.0107750	0.00538752	+	0.999985i	$0.498285\pi$
$788$	0	0
$789$	−10.9923	−0.391337
$790$	0	0
$791$	27.0387	0.961387
$792$	0	0
$793$	−31.0993	−1.10437
$794$	0	0
$795$	−2.57404	−0.0912919
$796$	0	0
$797$	44.7238	1.58420	0.792099	−	0.610393i	$-0.208988\pi$
$797$	44.7238	1.58420	0.792099	+	0.610393i	$0.208988\pi$
$798$	0	0
$799$	−13.4955	−0.477435
$800$	0	0
$801$	30.7293	1.08577
$802$	0	0
$803$	3.67776	0.129785
$804$	0	0
$805$	−16.8709	−0.594620
$806$	0	0
$807$	−16.2827	−0.573177
$808$	0	0
$809$	11.2857	0.396785	0.198392	−	0.980123i	$-0.436428\pi$
$809$	11.2857	0.396785	0.198392	+	0.980123i	$0.436428\pi$
$810$	0	0
$811$	−22.0389	−0.773889	−0.386944	−	0.922103i	$-0.626469\pi$
$811$	−22.0389	−0.773889	−0.386944	+	0.922103i	$0.626469\pi$
$812$	0	0
$813$	5.44308	0.190897
$814$	0	0
$815$	16.0593	0.562532
$816$	0	0
$817$	9.86165	0.345015
$818$	0	0
$819$	18.4560	0.644904
$820$	0	0
$821$	−53.0370	−1.85100	−0.925501	−	0.378744i	$-0.876356\pi$
$821$	−53.0370	−1.85100	−0.925501	+	0.378744i	$0.876356\pi$
$822$	0	0
$823$	2.62045	0.0913431	0.0456715	−	0.998957i	$-0.485457\pi$
$823$	2.62045	0.0913431	0.0456715	+	0.998957i	$0.485457\pi$
$824$	0	0
$825$	−1.67342	−0.0582610
$826$	0	0
$827$	25.8370	0.898440	0.449220	−	0.893421i	$-0.351702\pi$
$827$	25.8370	0.898440	0.449220	+	0.893421i	$0.351702\pi$
$828$	0	0
$829$	−20.1565	−0.700065	−0.350033	−	0.936738i	$-0.613830\pi$
$829$	−20.1565	−0.700065	−0.350033	+	0.936738i	$0.613830\pi$
$830$	0	0
$831$	0.127914	0.00443728
$832$	0	0
$833$	−2.02230	−0.0700685
$834$	0	0
$835$	4.50241	0.155812
$836$	0	0
$837$	36.9143	1.27595
$838$	0	0
$839$	13.8056	0.476623	0.238312	−	0.971189i	$-0.423406\pi$
$839$	13.8056	0.476623	0.238312	+	0.971189i	$0.423406\pi$
$840$	0	0
$841$	24.1233	0.831838
$842$	0	0
$843$	−11.2486	−0.387422
$844$	0	0
$845$	−11.4264	−0.393080
$846$	0	0
$847$	3.00900	0.103390
$848$	0	0
$849$	−15.0192	−0.515458
$850$	0	0
$851$	22.2511	0.762759
$852$	0	0
$853$	2.63274	0.0901432	0.0450716	−	0.998984i	$-0.485648\pi$
$853$	2.63274	0.0901432	0.0450716	+	0.998984i	$0.485648\pi$
$854$	0	0
$855$	−4.01278	−0.137234
$856$	0	0
$857$	−22.6437	−0.773494	−0.386747	−	0.922186i	$-0.626401\pi$
$857$	−22.6437	−0.773494	−0.386747	+	0.922186i	$0.626401\pi$
$858$	0	0
$859$	20.8614	0.711783	0.355892	−	0.934527i	$-0.384177\pi$
$859$	20.8614	0.711783	0.355892	+	0.934527i	$0.384177\pi$
$860$	0	0
$861$	−1.64597	−0.0560944
$862$	0	0
$863$	−13.2239	−0.450148	−0.225074	−	0.974342i	$-0.572262\pi$
$863$	−13.2239	−0.450148	−0.225074	+	0.974342i	$0.572262\pi$
$864$	0	0
$865$	39.3278	1.33719
$866$	0	0
$867$	10.4019	0.353266
$868$	0	0
$869$	−4.14246	−0.140523
$870$	0	0
$871$	−26.1814	−0.887123
$872$	0	0
$873$	23.9507	0.810607
$874$	0	0
$875$	−35.4840	−1.19958
$876$	0	0
$877$	−23.6565	−0.798824	−0.399412	−	0.916772i	$-0.630786\pi$
$877$	−23.6565	−0.798824	−0.399412	+	0.916772i	$0.630786\pi$
$878$	0	0
$879$	14.1854	0.478460
$880$	0	0
$881$	−27.2543	−0.918222	−0.459111	−	0.888379i	$-0.651832\pi$
$881$	−27.2543	−0.918222	−0.459111	+	0.888379i	$0.651832\pi$
$882$	0	0
$883$	36.0737	1.21398	0.606989	−	0.794710i	$-0.292377\pi$
$883$	36.0737	1.21398	0.606989	+	0.794710i	$0.292377\pi$
$884$	0	0
$885$	9.78148	0.328801
$886$	0	0
$887$	11.0861	0.372234	0.186117	−	0.982528i	$-0.440410\pi$
$887$	11.0861	0.372234	0.186117	+	0.982528i	$0.440410\pi$
$888$	0	0
$889$	−58.4052	−1.95885
$890$	0	0
$891$	−5.38794	−0.180503
$892$	0	0
$893$	13.7074	0.458702
$894$	0	0
$895$	−17.4888	−0.584586
$896$	0	0
$897$	−5.56090	−0.185673
$898$	0	0
$899$	74.3232	2.47882
$900$	0	0
$901$	−2.51008	−0.0836228
$902$	0	0
$903$	−19.2544	−0.640748
$904$	0	0
$905$	−21.5953	−0.717853
$906$	0	0
$907$	43.1209	1.43181	0.715903	−	0.698199i	$-0.246015\pi$
$907$	43.1209	1.43181	0.715903	+	0.698199i	$0.246015\pi$
$908$	0	0
$909$	−41.6935	−1.38289
$910$	0	0
$911$	11.8066	0.391171	0.195586	−	0.980687i	$-0.437339\pi$
$911$	11.8066	0.391171	0.195586	+	0.980687i	$0.437339\pi$
$912$	0	0
$913$	4.87924	0.161479
$914$	0	0
$915$	−13.2020	−0.436446
$916$	0	0
$917$	41.9172	1.38423
$918$	0	0
$919$	−6.44474	−0.212592	−0.106296	−	0.994335i	$-0.533899\pi$
$919$	−6.44474	−0.212592	−0.106296	+	0.994335i	$0.533899\pi$
$920$	0	0
$921$	−6.46262	−0.212951
$922$	0	0
$923$	−21.9502	−0.722500
$924$	0	0
$925$	15.9251	0.523614
$926$	0	0
$927$	3.07438	0.100976
$928$	0	0
$929$	7.63030	0.250342	0.125171	−	0.992135i	$-0.460052\pi$
$929$	7.63030	0.250342	0.125171	+	0.992135i	$0.460052\pi$
$930$	0	0
$931$	2.05406	0.0673192
$932$	0	0
$933$	−16.1676	−0.529303
$934$	0	0
$935$	1.53191	0.0500987
$936$	0	0
$937$	−25.9049	−0.846276	−0.423138	−	0.906065i	$-0.639071\pi$
$937$	−25.9049	−0.846276	−0.423138	+	0.906065i	$0.639071\pi$
$938$	0	0
$939$	−20.2966	−0.662355
$940$	0	0
$941$	1.05992	0.0345524	0.0172762	−	0.999851i	$-0.494501\pi$
$941$	1.05992	0.0345524	0.0172762	+	0.999851i	$0.494501\pi$
$942$	0	0
$943$	−3.03777	−0.0989234
$944$	0	0
$945$	16.9487	0.551340
$946$	0	0
$947$	−22.5255	−0.731980	−0.365990	−	0.930619i	$-0.619270\pi$
$947$	−22.5255	−0.731980	−0.365990	+	0.930619i	$0.619270\pi$
$948$	0	0
$949$	8.74688	0.283936
$950$	0	0
$951$	12.5977	0.408510
$952$	0	0
$953$	−18.0244	−0.583867	−0.291934	−	0.956439i	$-0.594299\pi$
$953$	−18.0244	−0.583867	−0.291934	+	0.956439i	$0.594299\pi$
$954$	0	0
$955$	−17.8350	−0.577129
$956$	0	0
$957$	4.72936	0.152878
$958$	0	0
$959$	−8.81393	−0.284617
$960$	0	0
$961$	72.9833	2.35430
$962$	0	0
$963$	17.4683	0.562908
$964$	0	0
$965$	−12.0918	−0.389250
$966$	0	0
$967$	48.5167	1.56019	0.780095	−	0.625661i	$-0.215170\pi$
$967$	48.5167	1.56019	0.780095	+	0.625661i	$0.215170\pi$
$968$	0	0
$969$	0.638839	0.0205225
$970$	0	0
$971$	9.89132	0.317428	0.158714	−	0.987325i	$-0.449265\pi$
$971$	9.89132	0.317428	0.158714	+	0.987325i	$0.449265\pi$
$972$	0	0
$973$	9.63703	0.308949
$974$	0	0
$975$	−3.97993	−0.127460
$976$	0	0
$977$	−17.7584	−0.568140	−0.284070	−	0.958803i	$-0.591685\pi$
$977$	−17.7584	−0.568140	−0.284070	+	0.958803i	$0.591685\pi$
$978$	0	0
$979$	11.9154	0.380817
$980$	0	0
$981$	−14.2982	−0.456505
$982$	0	0
$983$	−27.9045	−0.890015	−0.445007	−	0.895527i	$-0.646799\pi$
$983$	−27.9045	−0.890015	−0.445007	+	0.895527i	$0.646799\pi$
$984$	0	0
$985$	−34.0686	−1.08552
$986$	0	0
$987$	−26.7632	−0.851881
$988$	0	0
$989$	−35.5357	−1.12997
$990$	0	0
$991$	−32.0158	−1.01701	−0.508507	−	0.861058i	$-0.669802\pi$
$991$	−32.0158	−1.01701	−0.508507	+	0.861058i	$0.669802\pi$
$992$	0	0
$993$	−9.28218	−0.294561
$994$	0	0
$995$	8.25295	0.261636
$996$	0	0
$997$	−16.5713	−0.524818	−0.262409	−	0.964957i	$-0.584517\pi$
$997$	−16.5713	−0.524818	−0.262409	+	0.964957i	$0.584517\pi$
$998$	0	0
$999$	−22.3537	−0.707241

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.y.1.2		6
4.3	odd	2		1672.2.a.h.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.h.1.5	✓	6	4.3	odd	2
3344.2.a.y.1.2		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)

\(f(q)\)	\(=\)	\(q-0.648873 q^{3} +1.55597 q^{5} +3.00900 q^{7} -2.57896 q^{9} +O(q^{10})\)	\(q-0.648873 q^{3} +1.55597 q^{5} +3.00900 q^{7} -2.57896 q^{9} -1.00000 q^{11} -2.37832 q^{13} -1.00963 q^{15} -0.984536 q^{17} +1.00000 q^{19} -1.95246 q^{21} -3.60342 q^{23} -2.57896 q^{25} +3.62004 q^{27} +7.28857 q^{29} +10.1972 q^{31} +0.648873 q^{33} +4.68190 q^{35} -6.17499 q^{37} +1.54323 q^{39} +0.843023 q^{41} +9.86165 q^{43} -4.01278 q^{45} +13.7074 q^{47} +2.05406 q^{49} +0.638839 q^{51} +2.54950 q^{53} -1.55597 q^{55} -0.648873 q^{57} -9.68822 q^{59} +13.0762 q^{61} -7.76009 q^{63} -3.70059 q^{65} +11.0084 q^{67} +2.33817 q^{69} +9.22930 q^{71} -3.67776 q^{73} +1.67342 q^{75} -3.00900 q^{77} +4.14246 q^{79} +5.38794 q^{81} -4.87924 q^{83} -1.53191 q^{85} -4.72936 q^{87} -11.9154 q^{89} -7.15635 q^{91} -6.61670 q^{93} +1.55597 q^{95} -9.28693 q^{97} +2.57896 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9	\( 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9} - 6 q^{11} + 2 q^{13} - 10 q^{17} + 6 q^{19} + 10 q^{21} + 14 q^{23} + 2 q^{25} + 16 q^{27} + 6 q^{29} - 4 q^{31} - 4 q^{33} - 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41} + 26 q^{43} - 2 q^{45} + 24 q^{47} + 20 q^{49} + 14 q^{51} - 8 q^{53} + 4 q^{55} + 4 q^{57} + 4 q^{59} - 6 q^{61} + 40 q^{63} - 34 q^{65} + 44 q^{67} + 8 q^{69} + 16 q^{71} + 12 q^{73} + 28 q^{75} - 4 q^{77} - 6 q^{79} + 10 q^{81} + 28 q^{83} - 10 q^{85} + 24 q^{87} + 64 q^{91} - 14 q^{93} - 4 q^{95} + 4 q^{97} - 2 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9 - 6 * q^11 + 2 * q^13 - 10 * q^17 + 6 * q^19 + 10 * q^21 + 14 * q^23 + 2 * q^25 + 16 * q^27 + 6 * q^29 - 4 * q^31 - 4 * q^33 - 2 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^41 + 26 * q^43 - 2 * q^45 + 24 * q^47 + 20 * q^49 + 14 * q^51 - 8 * q^53 + 4 * q^55 + 4 * q^57 + 4 * q^59 - 6 * q^61 + 40 * q^63 - 34 * q^65 + 44 * q^67 + 8 * q^69 + 16 * q^71 + 12 * q^73 + 28 * q^75 - 4 * q^77 - 6 * q^79 + 10 * q^81 + 28 * q^83 - 10 * q^85 + 24 * q^87 + 64 * q^91 - 14 * q^93 - 4 * q^95 + 4 * q^97 - 2 * q^99

Introduction

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Database

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