LMFDB - Embedded newform 3344.2.a.j.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
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			1.1
Root			$2.30278$ 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)

$f(q)$	$=$	$q-2.30278 q^{3} -2.30278 q^{5} +1.30278 q^{7} +2.30278 q^{9} +O(q^{10})$	$q-2.30278 q^{3} -2.30278 q^{5} +1.30278 q^{7} +2.30278 q^{9} +1.00000 q^{11} -5.30278 q^{13} +5.30278 q^{15} +4.60555 q^{17} +1.00000 q^{19} -3.00000 q^{21} -2.60555 q^{23} +0.302776 q^{25} +1.60555 q^{27} -5.69722 q^{29} +7.90833 q^{31} -2.30278 q^{33} -3.00000 q^{35} +8.00000 q^{37} +12.2111 q^{39} -7.30278 q^{41} +2.30278 q^{43} -5.30278 q^{45} +11.2111 q^{47} -5.30278 q^{49} -10.6056 q^{51} +6.60555 q^{53} -2.30278 q^{55} -2.30278 q^{57} +4.00000 q^{59} -2.00000 q^{61} +3.00000 q^{63} +12.2111 q^{65} -12.5139 q^{67} +6.00000 q^{69} +12.9083 q^{71} -9.39445 q^{73} -0.697224 q^{75} +1.30278 q^{77} +11.8167 q^{79} -10.6056 q^{81} -10.9083 q^{83} -10.6056 q^{85} +13.1194 q^{87} +13.8167 q^{89} -6.90833 q^{91} -18.2111 q^{93} -2.30278 q^{95} +14.4222 q^{97} +2.30278 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - q^{5} - q^{7} + q^{9}+O(q^{10}) $ 2 * q - q^3 - q^5 - q^7 + q^9	$ 2 q - q^{3} - q^{5} - q^{7} + q^{9} + 2 q^{11} - 7 q^{13} + 7 q^{15} + 2 q^{17} + 2 q^{19} - 6 q^{21} + 2 q^{23} - 3 q^{25} - 4 q^{27} - 15 q^{29} + 5 q^{31} - q^{33} - 6 q^{35} + 16 q^{37} + 10 q^{39} - 11 q^{41} + q^{43} - 7 q^{45} + 8 q^{47} - 7 q^{49} - 14 q^{51} + 6 q^{53} - q^{55} - q^{57} + 8 q^{59} - 4 q^{61} + 6 q^{63} + 10 q^{65} - 7 q^{67} + 12 q^{69} + 15 q^{71} - 26 q^{73} - 5 q^{75} - q^{77} + 2 q^{79} - 14 q^{81} - 11 q^{83} - 14 q^{85} + q^{87} + 6 q^{89} - 3 q^{91} - 22 q^{93} - q^{95} + q^{99}+O(q^{100}) $ 2 * q - q^3 - q^5 - q^7 + q^9 + 2 * q^11 - 7 * q^13 + 7 * q^15 + 2 * q^17 + 2 * q^19 - 6 * q^21 + 2 * q^23 - 3 * q^25 - 4 * q^27 - 15 * q^29 + 5 * q^31 - q^33 - 6 * q^35 + 16 * q^37 + 10 * q^39 - 11 * q^41 + q^43 - 7 * q^45 + 8 * q^47 - 7 * q^49 - 14 * q^51 + 6 * q^53 - q^55 - q^57 + 8 * q^59 - 4 * q^61 + 6 * q^63 + 10 * q^65 - 7 * q^67 + 12 * q^69 + 15 * q^71 - 26 * q^73 - 5 * q^75 - q^77 + 2 * q^79 - 14 * q^81 - 11 * q^83 - 14 * q^85 + q^87 + 6 * q^89 - 3 * q^91 - 22 * q^93 - q^95 + 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
$3$	−2.30278	−1.32951	−0.664754	−	0.747062i	$-0.731464\pi$
$3$	−2.30278	−1.32951	−0.664754	+	0.747062i	$0.731464\pi$
$4$	0	0
$5$	−2.30278	−1.02983	−0.514916	−	0.857240i	$-0.672177\pi$
$5$	−2.30278	−1.02983	−0.514916	+	0.857240i	$0.672177\pi$
$6$	0	0
$7$	1.30278	0.492403	0.246201	−	0.969219i	$-0.420818\pi$
$7$	1.30278	0.492403	0.246201	+	0.969219i	$0.420818\pi$
$8$	0	0
$9$	2.30278	0.767592
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.30278	−1.47073	−0.735363	−	0.677674i	$-0.762988\pi$
$13$	−5.30278	−1.47073	−0.735363	+	0.677674i	$0.762988\pi$
$14$	0	0
$15$	5.30278	1.36917
$16$	0	0
$17$	4.60555	1.11701	0.558505	−	0.829501i	$-0.311375\pi$
$17$	4.60555	1.11701	0.558505	+	0.829501i	$0.311375\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−3.00000	−0.654654
$22$	0	0
$23$	−2.60555	−0.543295	−0.271647	−	0.962397i	$-0.587568\pi$
$23$	−2.60555	−0.543295	−0.271647	+	0.962397i	$0.587568\pi$
$24$	0	0
$25$	0.302776	0.0605551
$26$	0	0
$27$	1.60555	0.308988
$28$	0	0
$29$	−5.69722	−1.05795	−0.528974	−	0.848638i	$-0.677423\pi$
$29$	−5.69722	−1.05795	−0.528974	+	0.848638i	$0.677423\pi$
$30$	0	0
$31$	7.90833	1.42038	0.710189	−	0.704011i	$-0.248610\pi$
$31$	7.90833	1.42038	0.710189	+	0.704011i	$0.248610\pi$
$32$	0	0
$33$	−2.30278	−0.400862
$34$	0	0
$35$	−3.00000	−0.507093
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	12.2111	1.95534
$40$	0	0
$41$	−7.30278	−1.14050	−0.570251	−	0.821471i	$-0.693154\pi$
$41$	−7.30278	−1.14050	−0.570251	+	0.821471i	$0.693154\pi$
$42$	0	0
$43$	2.30278	0.351170	0.175585	−	0.984464i	$-0.443818\pi$
$43$	2.30278	0.351170	0.175585	+	0.984464i	$0.443818\pi$
$44$	0	0
$45$	−5.30278	−0.790491
$46$	0	0
$47$	11.2111	1.63531	0.817654	−	0.575710i	$-0.195274\pi$
$47$	11.2111	1.63531	0.817654	+	0.575710i	$0.195274\pi$
$48$	0	0
$49$	−5.30278	−0.757539
$50$	0	0
$51$	−10.6056	−1.48507
$52$	0	0
$53$	6.60555	0.907342	0.453671	−	0.891169i	$-0.350114\pi$
$53$	6.60555	0.907342	0.453671	+	0.891169i	$0.350114\pi$
$54$	0	0
$55$	−2.30278	−0.310506
$56$	0	0
$57$	−2.30278	−0.305010
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	3.00000	0.377964
$64$	0	0
$65$	12.2111	1.51460
$66$	0	0
$67$	−12.5139	−1.52881	−0.764407	−	0.644734i	$-0.776968\pi$
$67$	−12.5139	−1.52881	−0.764407	+	0.644734i	$0.776968\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	12.9083	1.53194	0.765968	−	0.642878i	$-0.222260\pi$
$71$	12.9083	1.53194	0.765968	+	0.642878i	$0.222260\pi$
$72$	0	0
$73$	−9.39445	−1.09954	−0.549769	−	0.835317i	$-0.685284\pi$
$73$	−9.39445	−1.09954	−0.549769	+	0.835317i	$0.685284\pi$
$74$	0	0
$75$	−0.697224	−0.0805085
$76$	0	0
$77$	1.30278	0.148465
$78$	0	0
$79$	11.8167	1.32948	0.664739	−	0.747076i	$-0.268543\pi$
$79$	11.8167	1.32948	0.664739	+	0.747076i	$0.268543\pi$
$80$	0	0
$81$	−10.6056	−1.17839
$82$	0	0
$83$	−10.9083	−1.19734	−0.598672	−	0.800994i	$-0.704305\pi$
$83$	−10.9083	−1.19734	−0.598672	+	0.800994i	$0.704305\pi$
$84$	0	0
$85$	−10.6056	−1.15033
$86$	0	0
$87$	13.1194	1.40655
$88$	0	0
$89$	13.8167	1.46456	0.732281	−	0.681002i	$-0.238456\pi$
$89$	13.8167	1.46456	0.732281	+	0.681002i	$0.238456\pi$
$90$	0	0
$91$	−6.90833	−0.724189
$92$	0	0
$93$	−18.2111	−1.88840
$94$	0	0
$95$	−2.30278	−0.236260
$96$	0	0
$97$	14.4222	1.46435	0.732177	−	0.681115i	$-0.238505\pi$
$97$	14.4222	1.46435	0.732177	+	0.681115i	$0.238505\pi$
$98$	0	0
$99$	2.30278	0.231438
$100$	0	0
$101$	−2.60555	−0.259262	−0.129631	−	0.991562i	$-0.541379\pi$
$101$	−2.60555	−0.259262	−0.129631	+	0.991562i	$0.541379\pi$
$102$	0	0
$103$	−13.5139	−1.33156	−0.665781	−	0.746147i	$-0.731901\pi$
$103$	−13.5139	−1.33156	−0.665781	+	0.746147i	$0.731901\pi$
$104$	0	0
$105$	6.90833	0.674184
$106$	0	0
$107$	7.21110	0.697124	0.348562	−	0.937286i	$-0.386670\pi$
$107$	7.21110	0.697124	0.348562	+	0.937286i	$0.386670\pi$
$108$	0	0
$109$	−11.2111	−1.07383	−0.536914	−	0.843637i	$-0.680410\pi$
$109$	−11.2111	−1.07383	−0.536914	+	0.843637i	$0.680410\pi$
$110$	0	0
$111$	−18.4222	−1.74856
$112$	0	0
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	0	0
$115$	6.00000	0.559503
$116$	0	0
$117$	−12.2111	−1.12892
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	16.8167	1.51631
$124$	0	0
$125$	10.8167	0.967471
$126$	0	0
$127$	−14.6056	−1.29603	−0.648017	−	0.761626i	$-0.724401\pi$
$127$	−14.6056	−1.29603	−0.648017	+	0.761626i	$0.724401\pi$
$128$	0	0
$129$	−5.30278	−0.466883
$130$	0	0
$131$	−12.5139	−1.09334	−0.546671	−	0.837347i	$-0.684105\pi$
$131$	−12.5139	−1.09334	−0.546671	+	0.837347i	$0.684105\pi$
$132$	0	0
$133$	1.30278	0.112965
$134$	0	0
$135$	−3.69722	−0.318206
$136$	0	0
$137$	−21.9083	−1.87175	−0.935877	−	0.352326i	$-0.885391\pi$
$137$	−21.9083	−1.87175	−0.935877	+	0.352326i	$0.885391\pi$
$138$	0	0
$139$	−16.7250	−1.41859	−0.709297	−	0.704910i	$-0.750988\pi$
$139$	−16.7250	−1.41859	−0.709297	+	0.704910i	$0.750988\pi$
$140$	0	0
$141$	−25.8167	−2.17415
$142$	0	0
$143$	−5.30278	−0.443440
$144$	0	0
$145$	13.1194	1.08951
$146$	0	0
$147$	12.2111	1.00715
$148$	0	0
$149$	−21.2111	−1.73768	−0.868841	−	0.495092i	$-0.835134\pi$
$149$	−21.2111	−1.73768	−0.868841	+	0.495092i	$0.835134\pi$
$150$	0	0
$151$	−12.4222	−1.01090	−0.505452	−	0.862855i	$-0.668674\pi$
$151$	−12.4222	−1.01090	−0.505452	+	0.862855i	$0.668674\pi$
$152$	0	0
$153$	10.6056	0.857408
$154$	0	0
$155$	−18.2111	−1.46275
$156$	0	0
$157$	18.3028	1.46072	0.730360	−	0.683062i	$-0.239352\pi$
$157$	18.3028	1.46072	0.730360	+	0.683062i	$0.239352\pi$
$158$	0	0
$159$	−15.2111	−1.20632
$160$	0	0
$161$	−3.39445	−0.267520
$162$	0	0
$163$	−12.6056	−0.987343	−0.493671	−	0.869648i	$-0.664345\pi$
$163$	−12.6056	−0.987343	−0.493671	+	0.869648i	$0.664345\pi$
$164$	0	0
$165$	5.30278	0.412821
$166$	0	0
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	0	0
$169$	15.1194	1.16303
$170$	0	0
$171$	2.30278	0.176098
$172$	0	0
$173$	16.1194	1.22554	0.612769	−	0.790262i	$-0.290056\pi$
$173$	16.1194	1.22554	0.612769	+	0.790262i	$0.290056\pi$
$174$	0	0
$175$	0.394449	0.0298175
$176$	0	0
$177$	−9.21110	−0.692349
$178$	0	0
$179$	−4.69722	−0.351087	−0.175544	−	0.984472i	$-0.556168\pi$
$179$	−4.69722	−0.351087	−0.175544	+	0.984472i	$0.556168\pi$
$180$	0	0
$181$	11.3944	0.846943	0.423471	−	0.905909i	$-0.360811\pi$
$181$	11.3944	0.846943	0.423471	+	0.905909i	$0.360811\pi$
$182$	0	0
$183$	4.60555	0.340452
$184$	0	0
$185$	−18.4222	−1.35443
$186$	0	0
$187$	4.60555	0.336791
$188$	0	0
$189$	2.09167	0.152147
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	−6.11943	−0.440486	−0.220243	−	0.975445i	$-0.570685\pi$
$193$	−6.11943	−0.440486	−0.220243	+	0.975445i	$0.570685\pi$
$194$	0	0
$195$	−28.1194	−2.01367
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	−19.0278	−1.34884	−0.674421	−	0.738347i	$-0.735607\pi$
$199$	−19.0278	−1.34884	−0.674421	+	0.738347i	$0.735607\pi$
$200$	0	0
$201$	28.8167	2.03257
$202$	0	0
$203$	−7.42221	−0.520937
$204$	0	0
$205$	16.8167	1.17453
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−3.39445	−0.233683	−0.116842	−	0.993151i	$-0.537277\pi$
$211$	−3.39445	−0.233683	−0.116842	+	0.993151i	$0.537277\pi$
$212$	0	0
$213$	−29.7250	−2.03672
$214$	0	0
$215$	−5.30278	−0.361646
$216$	0	0
$217$	10.3028	0.699398
$218$	0	0
$219$	21.6333	1.46184
$220$	0	0
$221$	−24.4222	−1.64282
$222$	0	0
$223$	13.2111	0.884681	0.442340	−	0.896847i	$-0.354148\pi$
$223$	13.2111	0.884681	0.442340	+	0.896847i	$0.354148\pi$
$224$	0	0
$225$	0.697224	0.0464816
$226$	0	0
$227$	16.6056	1.10215	0.551075	−	0.834456i	$-0.314218\pi$
$227$	16.6056	1.10215	0.551075	+	0.834456i	$0.314218\pi$
$228$	0	0
$229$	−14.5139	−0.959104	−0.479552	−	0.877513i	$-0.659201\pi$
$229$	−14.5139	−0.959104	−0.479552	+	0.877513i	$0.659201\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	−16.0000	−1.04819	−0.524097	−	0.851658i	$-0.675597\pi$
$233$	−16.0000	−1.04819	−0.524097	+	0.851658i	$0.675597\pi$
$234$	0	0
$235$	−25.8167	−1.68409
$236$	0	0
$237$	−27.2111	−1.76755
$238$	0	0
$239$	6.11943	0.395833	0.197916	−	0.980219i	$-0.436583\pi$
$239$	6.11943	0.395833	0.197916	+	0.980219i	$0.436583\pi$
$240$	0	0
$241$	−14.7250	−0.948519	−0.474260	−	0.880385i	$-0.657284\pi$
$241$	−14.7250	−0.948519	−0.474260	+	0.880385i	$0.657284\pi$
$242$	0	0
$243$	19.6056	1.25770
$244$	0	0
$245$	12.2111	0.780139
$246$	0	0
$247$	−5.30278	−0.337408
$248$	0	0
$249$	25.1194	1.59188
$250$	0	0
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	−2.60555	−0.163810
$254$	0	0
$255$	24.4222	1.52938
$256$	0	0
$257$	−20.6056	−1.28534	−0.642669	−	0.766144i	$-0.722173\pi$
$257$	−20.6056	−1.28534	−0.642669	+	0.766144i	$0.722173\pi$
$258$	0	0
$259$	10.4222	0.647604
$260$	0	0
$261$	−13.1194	−0.812072
$262$	0	0
$263$	−13.5139	−0.833301	−0.416651	−	0.909067i	$-0.636796\pi$
$263$	−13.5139	−0.833301	−0.416651	+	0.909067i	$0.636796\pi$
$264$	0	0
$265$	−15.2111	−0.934411
$266$	0	0
$267$	−31.8167	−1.94715
$268$	0	0
$269$	−9.02776	−0.550432	−0.275216	−	0.961382i	$-0.588749\pi$
$269$	−9.02776	−0.550432	−0.275216	+	0.961382i	$0.588749\pi$
$270$	0	0
$271$	−17.3028	−1.05107	−0.525534	−	0.850772i	$-0.676135\pi$
$271$	−17.3028	−1.05107	−0.525534	+	0.850772i	$0.676135\pi$
$272$	0	0
$273$	15.9083	0.962816
$274$	0	0
$275$	0.302776	0.0182581
$276$	0	0
$277$	−23.2111	−1.39462	−0.697310	−	0.716770i	$-0.745620\pi$
$277$	−23.2111	−1.39462	−0.697310	+	0.716770i	$0.745620\pi$
$278$	0	0
$279$	18.2111	1.09027
$280$	0	0
$281$	−23.9361	−1.42791	−0.713954	−	0.700193i	$-0.753097\pi$
$281$	−23.9361	−1.42791	−0.713954	+	0.700193i	$0.753097\pi$
$282$	0	0
$283$	17.9083	1.06454	0.532270	−	0.846575i	$-0.321339\pi$
$283$	17.9083	1.06454	0.532270	+	0.846575i	$0.321339\pi$
$284$	0	0
$285$	5.30278	0.314109
$286$	0	0
$287$	−9.51388	−0.561586
$288$	0	0
$289$	4.21110	0.247712
$290$	0	0
$291$	−33.2111	−1.94687
$292$	0	0
$293$	2.51388	0.146862	0.0734312	−	0.997300i	$-0.476605\pi$
$293$	2.51388	0.146862	0.0734312	+	0.997300i	$0.476605\pi$
$294$	0	0
$295$	−9.21110	−0.536291
$296$	0	0
$297$	1.60555	0.0931635
$298$	0	0
$299$	13.8167	0.799038
$300$	0	0
$301$	3.00000	0.172917
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	4.60555	0.263713
$306$	0	0
$307$	3.02776	0.172803	0.0864016	−	0.996260i	$-0.472463\pi$
$307$	3.02776	0.172803	0.0864016	+	0.996260i	$0.472463\pi$
$308$	0	0
$309$	31.1194	1.77032
$310$	0	0
$311$	16.6056	0.941614	0.470807	−	0.882236i	$-0.343963\pi$
$311$	16.6056	0.941614	0.470807	+	0.882236i	$0.343963\pi$
$312$	0	0
$313$	−11.9361	−0.674667	−0.337334	−	0.941385i	$-0.609525\pi$
$313$	−11.9361	−0.674667	−0.337334	+	0.941385i	$0.609525\pi$
$314$	0	0
$315$	−6.90833	−0.389240
$316$	0	0
$317$	−1.81665	−0.102033	−0.0510167	−	0.998698i	$-0.516246\pi$
$317$	−1.81665	−0.102033	−0.0510167	+	0.998698i	$0.516246\pi$
$318$	0	0
$319$	−5.69722	−0.318983
$320$	0	0
$321$	−16.6056	−0.926831
$322$	0	0
$323$	4.60555	0.256260
$324$	0	0
$325$	−1.60555	−0.0890600
$326$	0	0
$327$	25.8167	1.42766
$328$	0	0
$329$	14.6056	0.805230
$330$	0	0
$331$	2.90833	0.159856	0.0799281	−	0.996801i	$-0.474531\pi$
$331$	2.90833	0.159856	0.0799281	+	0.996801i	$0.474531\pi$
$332$	0	0
$333$	18.4222	1.00953
$334$	0	0
$335$	28.8167	1.57442
$336$	0	0
$337$	−12.5139	−0.681674	−0.340837	−	0.940122i	$-0.610710\pi$
$337$	−12.5139	−0.681674	−0.340837	+	0.940122i	$0.610710\pi$
$338$	0	0
$339$	9.21110	0.500278
$340$	0	0
$341$	7.90833	0.428260
$342$	0	0
$343$	−16.0278	−0.865417
$344$	0	0
$345$	−13.8167	−0.743864
$346$	0	0
$347$	29.2111	1.56813	0.784067	−	0.620676i	$-0.213142\pi$
$347$	29.2111	1.56813	0.784067	+	0.620676i	$0.213142\pi$
$348$	0	0
$349$	−4.78890	−0.256344	−0.128172	−	0.991752i	$-0.540911\pi$
$349$	−4.78890	−0.256344	−0.128172	+	0.991752i	$0.540911\pi$
$350$	0	0
$351$	−8.51388	−0.454437
$352$	0	0
$353$	8.78890	0.467786	0.233893	−	0.972262i	$-0.424853\pi$
$353$	8.78890	0.467786	0.233893	+	0.972262i	$0.424853\pi$
$354$	0	0
$355$	−29.7250	−1.57764
$356$	0	0
$357$	−13.8167	−0.731255
$358$	0	0
$359$	−1.11943	−0.0590812	−0.0295406	−	0.999564i	$-0.509404\pi$
$359$	−1.11943	−0.0590812	−0.0295406	+	0.999564i	$0.509404\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−2.30278	−0.120864
$364$	0	0
$365$	21.6333	1.13234
$366$	0	0
$367$	24.4222	1.27483	0.637414	−	0.770521i	$-0.280004\pi$
$367$	24.4222	1.27483	0.637414	+	0.770521i	$0.280004\pi$
$368$	0	0
$369$	−16.8167	−0.875440
$370$	0	0
$371$	8.60555	0.446778
$372$	0	0
$373$	5.69722	0.294991	0.147496	−	0.989063i	$-0.452879\pi$
$373$	5.69722	0.294991	0.147496	+	0.989063i	$0.452879\pi$
$374$	0	0
$375$	−24.9083	−1.28626
$376$	0	0
$377$	30.2111	1.55595
$378$	0	0
$379$	−28.6972	−1.47408	−0.737039	−	0.675851i	$-0.763776\pi$
$379$	−28.6972	−1.47408	−0.737039	+	0.675851i	$0.763776\pi$
$380$	0	0
$381$	33.6333	1.72309
$382$	0	0
$383$	−12.0917	−0.617856	−0.308928	−	0.951085i	$-0.599970\pi$
$383$	−12.0917	−0.617856	−0.308928	+	0.951085i	$0.599970\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	0	0
$387$	5.30278	0.269555
$388$	0	0
$389$	22.9083	1.16150	0.580749	−	0.814083i	$-0.302760\pi$
$389$	22.9083	1.16150	0.580749	+	0.814083i	$0.302760\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	28.8167	1.45361
$394$	0	0
$395$	−27.2111	−1.36914
$396$	0	0
$397$	27.7250	1.39148	0.695738	−	0.718295i	$-0.255077\pi$
$397$	27.7250	1.39148	0.695738	+	0.718295i	$0.255077\pi$
$398$	0	0
$399$	−3.00000	−0.150188
$400$	0	0
$401$	7.81665	0.390345	0.195173	−	0.980769i	$-0.437473\pi$
$401$	7.81665	0.390345	0.195173	+	0.980769i	$0.437473\pi$
$402$	0	0
$403$	−41.9361	−2.08899
$404$	0	0
$405$	24.4222	1.21355
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	20.5416	1.01572	0.507859	−	0.861440i	$-0.330437\pi$
$409$	20.5416	1.01572	0.507859	+	0.861440i	$0.330437\pi$
$410$	0	0
$411$	50.4500	2.48851
$412$	0	0
$413$	5.21110	0.256422
$414$	0	0
$415$	25.1194	1.23306
$416$	0	0
$417$	38.5139	1.88603
$418$	0	0
$419$	−23.2111	−1.13394	−0.566968	−	0.823740i	$-0.691884\pi$
$419$	−23.2111	−1.13394	−0.566968	+	0.823740i	$0.691884\pi$
$420$	0	0
$421$	−29.0278	−1.41473	−0.707363	−	0.706850i	$-0.750115\pi$
$421$	−29.0278	−1.41473	−0.707363	+	0.706850i	$0.750115\pi$
$422$	0	0
$423$	25.8167	1.25525
$424$	0	0
$425$	1.39445	0.0676407
$426$	0	0
$427$	−2.60555	−0.126091
$428$	0	0
$429$	12.2111	0.589558
$430$	0	0
$431$	17.2111	0.829030	0.414515	−	0.910043i	$-0.363951\pi$
$431$	17.2111	0.829030	0.414515	+	0.910043i	$0.363951\pi$
$432$	0	0
$433$	5.81665	0.279531	0.139765	−	0.990185i	$-0.455365\pi$
$433$	5.81665	0.279531	0.139765	+	0.990185i	$0.455365\pi$
$434$	0	0
$435$	−30.2111	−1.44851
$436$	0	0
$437$	−2.60555	−0.124640
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	−12.2111	−0.581481
$442$	0	0
$443$	2.78890	0.132505	0.0662523	−	0.997803i	$-0.478896\pi$
$443$	2.78890	0.132505	0.0662523	+	0.997803i	$0.478896\pi$
$444$	0	0
$445$	−31.8167	−1.50825
$446$	0	0
$447$	48.8444	2.31026
$448$	0	0
$449$	7.81665	0.368891	0.184445	−	0.982843i	$-0.440951\pi$
$449$	7.81665	0.368891	0.184445	+	0.982843i	$0.440951\pi$
$450$	0	0
$451$	−7.30278	−0.343874
$452$	0	0
$453$	28.6056	1.34401
$454$	0	0
$455$	15.9083	0.745794
$456$	0	0
$457$	23.0278	1.07719	0.538597	−	0.842564i	$-0.318955\pi$
$457$	23.0278	1.07719	0.538597	+	0.842564i	$0.318955\pi$
$458$	0	0
$459$	7.39445	0.345143
$460$	0	0
$461$	5.21110	0.242705	0.121353	−	0.992609i	$-0.461277\pi$
$461$	5.21110	0.242705	0.121353	+	0.992609i	$0.461277\pi$
$462$	0	0
$463$	5.57779	0.259222	0.129611	−	0.991565i	$-0.458627\pi$
$463$	5.57779	0.259222	0.129611	+	0.991565i	$0.458627\pi$
$464$	0	0
$465$	41.9361	1.94474
$466$	0	0
$467$	29.0278	1.34324	0.671622	−	0.740894i	$-0.265598\pi$
$467$	29.0278	1.34324	0.671622	+	0.740894i	$0.265598\pi$
$468$	0	0
$469$	−16.3028	−0.752792
$470$	0	0
$471$	−42.1472	−1.94204
$472$	0	0
$473$	2.30278	0.105882
$474$	0	0
$475$	0.302776	0.0138923
$476$	0	0
$477$	15.2111	0.696469
$478$	0	0
$479$	3.72498	0.170199	0.0850994	−	0.996372i	$-0.472879\pi$
$479$	3.72498	0.170199	0.0850994	+	0.996372i	$0.472879\pi$
$480$	0	0
$481$	−42.4222	−1.93429
$482$	0	0
$483$	7.81665	0.355670
$484$	0	0
$485$	−33.2111	−1.50804
$486$	0	0
$487$	7.33053	0.332178	0.166089	−	0.986111i	$-0.446886\pi$
$487$	7.33053	0.332178	0.166089	+	0.986111i	$0.446886\pi$
$488$	0	0
$489$	29.0278	1.31268
$490$	0	0
$491$	35.1194	1.58492	0.792459	−	0.609925i	$-0.208801\pi$
$491$	35.1194	1.58492	0.792459	+	0.609925i	$0.208801\pi$
$492$	0	0
$493$	−26.2389	−1.18174
$494$	0	0
$495$	−5.30278	−0.238342
$496$	0	0
$497$	16.8167	0.754330
$498$	0	0
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	0	0
$501$	−4.60555	−0.205761
$502$	0	0
$503$	9.27502	0.413553	0.206776	−	0.978388i	$-0.433703\pi$
$503$	9.27502	0.413553	0.206776	+	0.978388i	$0.433703\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	0	0
$507$	−34.8167	−1.54626
$508$	0	0
$509$	10.7889	0.478209	0.239105	−	0.970994i	$-0.423146\pi$
$509$	10.7889	0.478209	0.239105	+	0.970994i	$0.423146\pi$
$510$	0	0
$511$	−12.2389	−0.541415
$512$	0	0
$513$	1.60555	0.0708868
$514$	0	0
$515$	31.1194	1.37129
$516$	0	0
$517$	11.2111	0.493064
$518$	0	0
$519$	−37.1194	−1.62936
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	−9.81665	−0.429252	−0.214626	−	0.976696i	$-0.568853\pi$
$523$	−9.81665	−0.429252	−0.214626	+	0.976696i	$0.568853\pi$
$524$	0	0
$525$	−0.908327	−0.0396426
$526$	0	0
$527$	36.4222	1.58658
$528$	0	0
$529$	−16.2111	−0.704831
$530$	0	0
$531$	9.21110	0.399728
$532$	0	0
$533$	38.7250	1.67737
$534$	0	0
$535$	−16.6056	−0.717921
$536$	0	0
$537$	10.8167	0.466773
$538$	0	0
$539$	−5.30278	−0.228407
$540$	0	0
$541$	−38.2389	−1.64402	−0.822008	−	0.569475i	$-0.807146\pi$
$541$	−38.2389	−1.64402	−0.822008	+	0.569475i	$0.807146\pi$
$542$	0	0
$543$	−26.2389	−1.12602
$544$	0	0
$545$	25.8167	1.10586
$546$	0	0
$547$	18.4222	0.787677	0.393838	−	0.919180i	$-0.371147\pi$
$547$	18.4222	0.787677	0.393838	+	0.919180i	$0.371147\pi$
$548$	0	0
$549$	−4.60555	−0.196560
$550$	0	0
$551$	−5.69722	−0.242710
$552$	0	0
$553$	15.3944	0.654639
$554$	0	0
$555$	42.4222	1.80072
$556$	0	0
$557$	6.42221	0.272118	0.136059	−	0.990701i	$-0.456556\pi$
$557$	6.42221	0.272118	0.136059	+	0.990701i	$0.456556\pi$
$558$	0	0
$559$	−12.2111	−0.516475
$560$	0	0
$561$	−10.6056	−0.447767
$562$	0	0
$563$	15.2111	0.641072	0.320536	−	0.947236i	$-0.396137\pi$
$563$	15.2111	0.641072	0.320536	+	0.947236i	$0.396137\pi$
$564$	0	0
$565$	9.21110	0.387514
$566$	0	0
$567$	−13.8167	−0.580245
$568$	0	0
$569$	24.6972	1.03536	0.517681	−	0.855574i	$-0.326796\pi$
$569$	24.6972	1.03536	0.517681	+	0.855574i	$0.326796\pi$
$570$	0	0
$571$	−10.0917	−0.422323	−0.211162	−	0.977451i	$-0.567725\pi$
$571$	−10.0917	−0.422323	−0.211162	+	0.977451i	$0.567725\pi$
$572$	0	0
$573$	−13.8167	−0.577199
$574$	0	0
$575$	−0.788897	−0.0328993
$576$	0	0
$577$	−20.1472	−0.838738	−0.419369	−	0.907816i	$-0.637749\pi$
$577$	−20.1472	−0.838738	−0.419369	+	0.907816i	$0.637749\pi$
$578$	0	0
$579$	14.0917	0.585630
$580$	0	0
$581$	−14.2111	−0.589576
$582$	0	0
$583$	6.60555	0.273574
$584$	0	0
$585$	28.1194	1.16260
$586$	0	0
$587$	−12.8444	−0.530146	−0.265073	−	0.964228i	$-0.585396\pi$
$587$	−12.8444	−0.530146	−0.265073	+	0.964228i	$0.585396\pi$
$588$	0	0
$589$	7.90833	0.325857
$590$	0	0
$591$	13.8167	0.568341
$592$	0	0
$593$	−28.4222	−1.16716	−0.583580	−	0.812056i	$-0.698349\pi$
$593$	−28.4222	−1.16716	−0.583580	+	0.812056i	$0.698349\pi$
$594$	0	0
$595$	−13.8167	−0.566428
$596$	0	0
$597$	43.8167	1.79330
$598$	0	0
$599$	−34.9638	−1.42858	−0.714292	−	0.699848i	$-0.753251\pi$
$599$	−34.9638	−1.42858	−0.714292	+	0.699848i	$0.753251\pi$
$600$	0	0
$601$	−37.9083	−1.54631	−0.773156	−	0.634215i	$-0.781323\pi$
$601$	−37.9083	−1.54631	−0.773156	+	0.634215i	$0.781323\pi$
$602$	0	0
$603$	−28.8167	−1.17350
$604$	0	0
$605$	−2.30278	−0.0936211
$606$	0	0
$607$	−7.81665	−0.317268	−0.158634	−	0.987337i	$-0.550709\pi$
$607$	−7.81665	−0.317268	−0.158634	+	0.987337i	$0.550709\pi$
$608$	0	0
$609$	17.0917	0.692590
$610$	0	0
$611$	−59.4500	−2.40509
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	−38.7250	−1.56154
$616$	0	0
$617$	−37.5416	−1.51137	−0.755685	−	0.654936i	$-0.772696\pi$
$617$	−37.5416	−1.51137	−0.755685	+	0.654936i	$0.772696\pi$
$618$	0	0
$619$	−39.2111	−1.57603	−0.788014	−	0.615658i	$-0.788890\pi$
$619$	−39.2111	−1.57603	−0.788014	+	0.615658i	$0.788890\pi$
$620$	0	0
$621$	−4.18335	−0.167872
$622$	0	0
$623$	18.0000	0.721155
$624$	0	0
$625$	−26.4222	−1.05689
$626$	0	0
$627$	−2.30278	−0.0919640
$628$	0	0
$629$	36.8444	1.46908
$630$	0	0
$631$	46.6056	1.85534	0.927669	−	0.373404i	$-0.121809\pi$
$631$	46.6056	1.85534	0.927669	+	0.373404i	$0.121809\pi$
$632$	0	0
$633$	7.81665	0.310684
$634$	0	0
$635$	33.6333	1.33470
$636$	0	0
$637$	28.1194	1.11413
$638$	0	0
$639$	29.7250	1.17590
$640$	0	0
$641$	−25.8167	−1.01970	−0.509848	−	0.860264i	$-0.670298\pi$
$641$	−25.8167	−1.01970	−0.509848	+	0.860264i	$0.670298\pi$
$642$	0	0
$643$	−13.2111	−0.520995	−0.260498	−	0.965475i	$-0.583887\pi$
$643$	−13.2111	−0.520995	−0.260498	+	0.965475i	$0.583887\pi$
$644$	0	0
$645$	12.2111	0.480812
$646$	0	0
$647$	4.60555	0.181063	0.0905315	−	0.995894i	$-0.471143\pi$
$647$	4.60555	0.181063	0.0905315	+	0.995894i	$0.471143\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	−23.7250	−0.929855
$652$	0	0
$653$	24.7527	0.968649	0.484325	−	0.874888i	$-0.339065\pi$
$653$	24.7527	0.968649	0.484325	+	0.874888i	$0.339065\pi$
$654$	0	0
$655$	28.8167	1.12596
$656$	0	0
$657$	−21.6333	−0.843996
$658$	0	0
$659$	−6.23886	−0.243031	−0.121516	−	0.992590i	$-0.538775\pi$
$659$	−6.23886	−0.243031	−0.121516	+	0.992590i	$0.538775\pi$
$660$	0	0
$661$	−7.63331	−0.296901	−0.148451	−	0.988920i	$-0.547429\pi$
$661$	−7.63331	−0.296901	−0.148451	+	0.988920i	$0.547429\pi$
$662$	0	0
$663$	56.2389	2.18414
$664$	0	0
$665$	−3.00000	−0.116335
$666$	0	0
$667$	14.8444	0.574778
$668$	0	0
$669$	−30.4222	−1.17619
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	38.5694	1.48674	0.743370	−	0.668880i	$-0.233226\pi$
$673$	38.5694	1.48674	0.743370	+	0.668880i	$0.233226\pi$
$674$	0	0
$675$	0.486122	0.0187108
$676$	0	0
$677$	−25.3305	−0.973531	−0.486766	−	0.873533i	$-0.661823\pi$
$677$	−25.3305	−0.973531	−0.486766	+	0.873533i	$0.661823\pi$
$678$	0	0
$679$	18.7889	0.721052
$680$	0	0
$681$	−38.2389	−1.46532
$682$	0	0
$683$	−8.00000	−0.306111	−0.153056	−	0.988218i	$-0.548911\pi$
$683$	−8.00000	−0.306111	−0.153056	+	0.988218i	$0.548911\pi$
$684$	0	0
$685$	50.4500	1.92759
$686$	0	0
$687$	33.4222	1.27514
$688$	0	0
$689$	−35.0278	−1.33445
$690$	0	0
$691$	−24.7889	−0.943014	−0.471507	−	0.881862i	$-0.656290\pi$
$691$	−24.7889	−0.943014	−0.471507	+	0.881862i	$0.656290\pi$
$692$	0	0
$693$	3.00000	0.113961
$694$	0	0
$695$	38.5139	1.46091
$696$	0	0
$697$	−33.6333	−1.27395
$698$	0	0
$699$	36.8444	1.39358
$700$	0	0
$701$	−18.7889	−0.709647	−0.354823	−	0.934933i	$-0.615459\pi$
$701$	−18.7889	−0.709647	−0.354823	+	0.934933i	$0.615459\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	59.4500	2.23902
$706$	0	0
$707$	−3.39445	−0.127661
$708$	0	0
$709$	−40.9361	−1.53739	−0.768693	−	0.639617i	$-0.779093\pi$
$709$	−40.9361	−1.53739	−0.768693	+	0.639617i	$0.779093\pi$
$710$	0	0
$711$	27.2111	1.02050
$712$	0	0
$713$	−20.6056	−0.771684
$714$	0	0
$715$	12.2111	0.456669
$716$	0	0
$717$	−14.0917	−0.526263
$718$	0	0
$719$	40.2389	1.50066	0.750328	−	0.661066i	$-0.229896\pi$
$719$	40.2389	1.50066	0.750328	+	0.661066i	$0.229896\pi$
$720$	0	0
$721$	−17.6056	−0.655665
$722$	0	0
$723$	33.9083	1.26106
$724$	0	0
$725$	−1.72498	−0.0640642
$726$	0	0
$727$	−19.6333	−0.728159	−0.364080	−	0.931368i	$-0.618616\pi$
$727$	−19.6333	−0.728159	−0.364080	+	0.931368i	$0.618616\pi$
$728$	0	0
$729$	−13.3305	−0.493723
$730$	0	0
$731$	10.6056	0.392260
$732$	0	0
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	0	0
$735$	−28.1194	−1.03720
$736$	0	0
$737$	−12.5139	−0.460955
$738$	0	0
$739$	18.5416	0.682065	0.341033	−	0.940051i	$-0.389223\pi$
$739$	18.5416	0.682065	0.341033	+	0.940051i	$0.389223\pi$
$740$	0	0
$741$	12.2111	0.448586
$742$	0	0
$743$	−18.0000	−0.660356	−0.330178	−	0.943919i	$-0.607109\pi$
$743$	−18.0000	−0.660356	−0.330178	+	0.943919i	$0.607109\pi$
$744$	0	0
$745$	48.8444	1.78952
$746$	0	0
$747$	−25.1194	−0.919072
$748$	0	0
$749$	9.39445	0.343266
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	0	0
$753$	46.0555	1.67836
$754$	0	0
$755$	28.6056	1.04106
$756$	0	0
$757$	20.1194	0.731253	0.365627	−	0.930762i	$-0.380855\pi$
$757$	20.1194	0.731253	0.365627	+	0.930762i	$0.380855\pi$
$758$	0	0
$759$	6.00000	0.217786
$760$	0	0
$761$	4.60555	0.166951	0.0834756	−	0.996510i	$-0.473398\pi$
$761$	4.60555	0.166951	0.0834756	+	0.996510i	$0.473398\pi$
$762$	0	0
$763$	−14.6056	−0.528756
$764$	0	0
$765$	−24.4222	−0.882987
$766$	0	0
$767$	−21.2111	−0.765889
$768$	0	0
$769$	−24.0555	−0.867464	−0.433732	−	0.901042i	$-0.642804\pi$
$769$	−24.0555	−0.867464	−0.433732	+	0.901042i	$0.642804\pi$
$770$	0	0
$771$	47.4500	1.70887
$772$	0	0
$773$	−28.0555	−1.00909	−0.504543	−	0.863386i	$-0.668339\pi$
$773$	−28.0555	−1.00909	−0.504543	+	0.863386i	$0.668339\pi$
$774$	0	0
$775$	2.39445	0.0860111
$776$	0	0
$777$	−24.0000	−0.860995
$778$	0	0
$779$	−7.30278	−0.261649
$780$	0	0
$781$	12.9083	0.461896
$782$	0	0
$783$	−9.14719	−0.326894
$784$	0	0
$785$	−42.1472	−1.50430
$786$	0	0
$787$	35.0278	1.24860	0.624302	−	0.781183i	$-0.285383\pi$
$787$	35.0278	1.24860	0.624302	+	0.781183i	$0.285383\pi$
$788$	0	0
$789$	31.1194	1.10788
$790$	0	0
$791$	−5.21110	−0.185285
$792$	0	0
$793$	10.6056	0.376614
$794$	0	0
$795$	35.0278	1.24231
$796$	0	0
$797$	48.2389	1.70871	0.854354	−	0.519691i	$-0.173953\pi$
$797$	48.2389	1.70871	0.854354	+	0.519691i	$0.173953\pi$
$798$	0	0
$799$	51.6333	1.82666
$800$	0	0
$801$	31.8167	1.12419
$802$	0	0
$803$	−9.39445	−0.331523
$804$	0	0
$805$	7.81665	0.275501
$806$	0	0
$807$	20.7889	0.731804
$808$	0	0
$809$	−8.36669	−0.294157	−0.147079	−	0.989125i	$-0.546987\pi$
$809$	−8.36669	−0.294157	−0.147079	+	0.989125i	$0.546987\pi$
$810$	0	0
$811$	50.8444	1.78539	0.892694	−	0.450663i	$-0.148812\pi$
$811$	50.8444	1.78539	0.892694	+	0.450663i	$0.148812\pi$
$812$	0	0
$813$	39.8444	1.39740
$814$	0	0
$815$	29.0278	1.01680
$816$	0	0
$817$	2.30278	0.0805639
$818$	0	0
$819$	−15.9083	−0.555882
$820$	0	0
$821$	−33.0278	−1.15268	−0.576338	−	0.817211i	$-0.695519\pi$
$821$	−33.0278	−1.15268	−0.576338	+	0.817211i	$0.695519\pi$
$822$	0	0
$823$	−50.4222	−1.75761	−0.878804	−	0.477183i	$-0.841658\pi$
$823$	−50.4222	−1.75761	−0.878804	+	0.477183i	$0.841658\pi$
$824$	0	0
$825$	−0.697224	−0.0242742
$826$	0	0
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	0	0
$829$	−11.2111	−0.389378	−0.194689	−	0.980865i	$-0.562370\pi$
$829$	−11.2111	−0.389378	−0.194689	+	0.980865i	$0.562370\pi$
$830$	0	0
$831$	53.4500	1.85416
$832$	0	0
$833$	−24.4222	−0.846179
$834$	0	0
$835$	−4.60555	−0.159382
$836$	0	0
$837$	12.6972	0.438880
$838$	0	0
$839$	−14.7250	−0.508363	−0.254181	−	0.967157i	$-0.581806\pi$
$839$	−14.7250	−0.508363	−0.254181	+	0.967157i	$0.581806\pi$
$840$	0	0
$841$	3.45837	0.119254
$842$	0	0
$843$	55.1194	1.89841
$844$	0	0
$845$	−34.8167	−1.19773
$846$	0	0
$847$	1.30278	0.0447639
$848$	0	0
$849$	−41.2389	−1.41531
$850$	0	0
$851$	−20.8444	−0.714537
$852$	0	0
$853$	−13.5778	−0.464895	−0.232447	−	0.972609i	$-0.574673\pi$
$853$	−13.5778	−0.464895	−0.232447	+	0.972609i	$0.574673\pi$
$854$	0	0
$855$	−5.30278	−0.181351
$856$	0	0
$857$	−16.3305	−0.557840	−0.278920	−	0.960314i	$-0.589976\pi$
$857$	−16.3305	−0.557840	−0.278920	+	0.960314i	$0.589976\pi$
$858$	0	0
$859$	7.63331	0.260445	0.130223	−	0.991485i	$-0.458431\pi$
$859$	7.63331	0.260445	0.130223	+	0.991485i	$0.458431\pi$
$860$	0	0
$861$	21.9083	0.746634
$862$	0	0
$863$	7.06392	0.240459	0.120229	−	0.992746i	$-0.461637\pi$
$863$	7.06392	0.240459	0.120229	+	0.992746i	$0.461637\pi$
$864$	0	0
$865$	−37.1194	−1.26210
$866$	0	0
$867$	−9.69722	−0.329335
$868$	0	0
$869$	11.8167	0.400853
$870$	0	0
$871$	66.3583	2.24846
$872$	0	0
$873$	33.2111	1.12403
$874$	0	0
$875$	14.0917	0.476385
$876$	0	0
$877$	−45.9361	−1.55115	−0.775576	−	0.631255i	$-0.782540\pi$
$877$	−45.9361	−1.55115	−0.775576	+	0.631255i	$0.782540\pi$
$878$	0	0
$879$	−5.78890	−0.195255
$880$	0	0
$881$	16.5139	0.556367	0.278183	−	0.960528i	$-0.410268\pi$
$881$	16.5139	0.556367	0.278183	+	0.960528i	$0.410268\pi$
$882$	0	0
$883$	2.18335	0.0734754	0.0367377	−	0.999325i	$-0.488303\pi$
$883$	2.18335	0.0734754	0.0367377	+	0.999325i	$0.488303\pi$
$884$	0	0
$885$	21.2111	0.713003
$886$	0	0
$887$	−47.6333	−1.59937	−0.799685	−	0.600420i	$-0.795000\pi$
$887$	−47.6333	−1.59937	−0.799685	+	0.600420i	$0.795000\pi$
$888$	0	0
$889$	−19.0278	−0.638170
$890$	0	0
$891$	−10.6056	−0.355299
$892$	0	0
$893$	11.2111	0.375165
$894$	0	0
$895$	10.8167	0.361561
$896$	0	0
$897$	−31.8167	−1.06233
$898$	0	0
$899$	−45.0555	−1.50269
$900$	0	0
$901$	30.4222	1.01351
$902$	0	0
$903$	−6.90833	−0.229895
$904$	0	0
$905$	−26.2389	−0.872209
$906$	0	0
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	58.0555	1.92346	0.961732	−	0.273990i	$-0.0883436\pi$
$911$	58.0555	1.92346	0.961732	+	0.273990i	$0.0883436\pi$
$912$	0	0
$913$	−10.9083	−0.361013
$914$	0	0
$915$	−10.6056	−0.350609
$916$	0	0
$917$	−16.3028	−0.538365
$918$	0	0
$919$	−27.0917	−0.893672	−0.446836	−	0.894616i	$-0.647449\pi$
$919$	−27.0917	−0.893672	−0.446836	+	0.894616i	$0.647449\pi$
$920$	0	0
$921$	−6.97224	−0.229743
$922$	0	0
$923$	−68.4500	−2.25306
$924$	0	0
$925$	2.42221	0.0796416
$926$	0	0
$927$	−31.1194	−1.02210
$928$	0	0
$929$	−24.5139	−0.804274	−0.402137	−	0.915579i	$-0.631732\pi$
$929$	−24.5139	−0.804274	−0.402137	+	0.915579i	$0.631732\pi$
$930$	0	0
$931$	−5.30278	−0.173791
$932$	0	0
$933$	−38.2389	−1.25188
$934$	0	0
$935$	−10.6056	−0.346839
$936$	0	0
$937$	31.2111	1.01962	0.509811	−	0.860286i	$-0.329715\pi$
$937$	31.2111	1.01962	0.509811	+	0.860286i	$0.329715\pi$
$938$	0	0
$939$	27.4861	0.896976
$940$	0	0
$941$	58.8444	1.91827	0.959136	−	0.282944i	$-0.0913110\pi$
$941$	58.8444	1.91827	0.959136	+	0.282944i	$0.0913110\pi$
$942$	0	0
$943$	19.0278	0.619629
$944$	0	0
$945$	−4.81665	−0.156686
$946$	0	0
$947$	41.8167	1.35886	0.679429	−	0.733741i	$-0.262227\pi$
$947$	41.8167	1.35886	0.679429	+	0.733741i	$0.262227\pi$
$948$	0	0
$949$	49.8167	1.61712
$950$	0	0
$951$	4.18335	0.135654
$952$	0	0
$953$	−34.8444	−1.12872	−0.564361	−	0.825528i	$-0.690877\pi$
$953$	−34.8444	−1.12872	−0.564361	+	0.825528i	$0.690877\pi$
$954$	0	0
$955$	−13.8167	−0.447096
$956$	0	0
$957$	13.1194	0.424091
$958$	0	0
$959$	−28.5416	−0.921657
$960$	0	0
$961$	31.5416	1.01747
$962$	0	0
$963$	16.6056	0.535106
$964$	0	0
$965$	14.0917	0.453627
$966$	0	0
$967$	5.21110	0.167578	0.0837889	−	0.996484i	$-0.473298\pi$
$967$	5.21110	0.167578	0.0837889	+	0.996484i	$0.473298\pi$
$968$	0	0
$969$	−10.6056	−0.340699
$970$	0	0
$971$	0.724981	0.0232657	0.0116329	−	0.999932i	$-0.496297\pi$
$971$	0.724981	0.0232657	0.0116329	+	0.999932i	$0.496297\pi$
$972$	0	0
$973$	−21.7889	−0.698520
$974$	0	0
$975$	3.69722	0.118406
$976$	0	0
$977$	7.63331	0.244211	0.122106	−	0.992517i	$-0.461035\pi$
$977$	7.63331	0.244211	0.122106	+	0.992517i	$0.461035\pi$
$978$	0	0
$979$	13.8167	0.441582
$980$	0	0
$981$	−25.8167	−0.824262
$982$	0	0
$983$	38.1472	1.21671	0.608353	−	0.793666i	$-0.291830\pi$
$983$	38.1472	1.21671	0.608353	+	0.793666i	$0.291830\pi$
$984$	0	0
$985$	13.8167	0.440235
$986$	0	0
$987$	−33.6333	−1.07056
$988$	0	0
$989$	−6.00000	−0.190789
$990$	0	0
$991$	−53.1194	−1.68739	−0.843697	−	0.536819i	$-0.819626\pi$
$991$	−53.1194	−1.68739	−0.843697	+	0.536819i	$0.819626\pi$
$992$	0	0
$993$	−6.69722	−0.212530
$994$	0	0
$995$	43.8167	1.38908
$996$	0	0
$997$	−17.3944	−0.550888	−0.275444	−	0.961317i	$-0.588825\pi$
$997$	−17.3944	−0.550888	−0.275444	+	0.961317i	$0.588825\pi$
$998$	0	0
$999$	12.8444	0.406379

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.j.1.1		2
4.3	odd	2		1672.2.a.d.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.d.1.2	✓	2	4.3	odd	2
3344.2.a.j.1.1		2	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)

\(f(q)\)	\(=\)	\(q-2.30278 q^{3} -2.30278 q^{5} +1.30278 q^{7} +2.30278 q^{9} +O(q^{10})\)	\(q-2.30278 q^{3} -2.30278 q^{5} +1.30278 q^{7} +2.30278 q^{9} +1.00000 q^{11} -5.30278 q^{13} +5.30278 q^{15} +4.60555 q^{17} +1.00000 q^{19} -3.00000 q^{21} -2.60555 q^{23} +0.302776 q^{25} +1.60555 q^{27} -5.69722 q^{29} +7.90833 q^{31} -2.30278 q^{33} -3.00000 q^{35} +8.00000 q^{37} +12.2111 q^{39} -7.30278 q^{41} +2.30278 q^{43} -5.30278 q^{45} +11.2111 q^{47} -5.30278 q^{49} -10.6056 q^{51} +6.60555 q^{53} -2.30278 q^{55} -2.30278 q^{57} +4.00000 q^{59} -2.00000 q^{61} +3.00000 q^{63} +12.2111 q^{65} -12.5139 q^{67} +6.00000 q^{69} +12.9083 q^{71} -9.39445 q^{73} -0.697224 q^{75} +1.30278 q^{77} +11.8167 q^{79} -10.6056 q^{81} -10.9083 q^{83} -10.6056 q^{85} +13.1194 q^{87} +13.8167 q^{89} -6.90833 q^{91} -18.2111 q^{93} -2.30278 q^{95} +14.4222 q^{97} +2.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - q^{5} - q^{7} + q^{9}+O(q^{10}) \) 2 * q - q^3 - q^5 - q^7 + q^9	\( 2 q - q^{3} - q^{5} - q^{7} + q^{9} + 2 q^{11} - 7 q^{13} + 7 q^{15} + 2 q^{17} + 2 q^{19} - 6 q^{21} + 2 q^{23} - 3 q^{25} - 4 q^{27} - 15 q^{29} + 5 q^{31} - q^{33} - 6 q^{35} + 16 q^{37} + 10 q^{39} - 11 q^{41} + q^{43} - 7 q^{45} + 8 q^{47} - 7 q^{49} - 14 q^{51} + 6 q^{53} - q^{55} - q^{57} + 8 q^{59} - 4 q^{61} + 6 q^{63} + 10 q^{65} - 7 q^{67} + 12 q^{69} + 15 q^{71} - 26 q^{73} - 5 q^{75} - q^{77} + 2 q^{79} - 14 q^{81} - 11 q^{83} - 14 q^{85} + q^{87} + 6 q^{89} - 3 q^{91} - 22 q^{93} - q^{95} + q^{99}+O(q^{100}) \) 2 * q - q^3 - q^5 - q^7 + q^9 + 2 * q^11 - 7 * q^13 + 7 * q^15 + 2 * q^17 + 2 * q^19 - 6 * q^21 + 2 * q^23 - 3 * q^25 - 4 * q^27 - 15 * q^29 + 5 * q^31 - q^33 - 6 * q^35 + 16 * q^37 + 10 * q^39 - 11 * q^41 + q^43 - 7 * q^45 + 8 * q^47 - 7 * q^49 - 14 * q^51 + 6 * q^53 - q^55 - q^57 + 8 * q^59 - 4 * q^61 + 6 * q^63 + 10 * q^65 - 7 * q^67 + 12 * q^69 + 15 * q^71 - 26 * q^73 - 5 * q^75 - q^77 + 2 * q^79 - 14 * q^81 - 11 * q^83 - 14 * q^85 + q^87 + 6 * q^89 - 3 * q^91 - 22 * q^93 - q^95 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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