LMFDB - Embedded newform 3344.2.a.p.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:	$3$
Coefficient field:	3.3.469.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 4 $ x^3 - x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.772866$ 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)

$f(q)$	$=$	$q-0.772866 q^{3} +1.22713 q^{5} -3.40268 q^{7} -2.40268 q^{9} +O(q^{10})$	$q-0.772866 q^{3} +1.22713 q^{5} -3.40268 q^{7} -2.40268 q^{9} +1.00000 q^{11} -1.40268 q^{13} -0.948410 q^{15} -4.80536 q^{17} +1.00000 q^{19} +2.62981 q^{21} -6.80536 q^{23} -3.49414 q^{25} +4.17554 q^{27} +8.03249 q^{29} +4.94841 q^{31} -0.772866 q^{33} -4.17554 q^{35} +2.35109 q^{37} +1.08408 q^{39} +2.94841 q^{41} +9.12395 q^{43} -2.94841 q^{45} -5.25963 q^{47} +4.57822 q^{49} +3.71390 q^{51} -4.45427 q^{53} +1.22713 q^{55} -0.772866 q^{57} +14.5193 q^{59} +2.00000 q^{61} +8.17554 q^{63} -1.72128 q^{65} -3.40268 q^{67} +5.25963 q^{69} +7.57822 q^{71} -10.0650 q^{73} +2.70050 q^{75} -3.40268 q^{77} +3.71390 q^{79} +3.98090 q^{81} -10.6697 q^{83} -5.89682 q^{85} -6.20804 q^{87} +14.9883 q^{89} +4.77287 q^{91} -3.82446 q^{93} +1.22713 q^{95} +2.35109 q^{97} -2.40268 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} + 5 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) $ 3 * q - q^3 + 5 * q^5 - q^7 + 2 * q^9	$ 3 q - q^{3} + 5 q^{5} - q^{7} + 2 q^{9} + 3 q^{11} + 5 q^{13} + 9 q^{15} + 4 q^{17} + 3 q^{19} - 2 q^{23} + 4 q^{25} + 2 q^{27} + 7 q^{29} + 3 q^{31} - q^{33} - 2 q^{35} - 14 q^{37} - 2 q^{39} - 3 q^{41} + 5 q^{43} + 3 q^{45} - 6 q^{49} - 2 q^{51} - 16 q^{53} + 5 q^{55} - q^{57} + 12 q^{59} + 6 q^{61} + 14 q^{63} + 8 q^{65} - q^{67} + 3 q^{71} + 4 q^{73} + 41 q^{75} - q^{77} - 2 q^{79} - 17 q^{81} - 7 q^{83} + 6 q^{85} + 9 q^{87} + 16 q^{89} + 13 q^{91} - 22 q^{93} + 5 q^{95} - 14 q^{97} + 2 q^{99}+O(q^{100}) $ 3 * q - q^3 + 5 * q^5 - q^7 + 2 * q^9 + 3 * q^11 + 5 * q^13 + 9 * q^15 + 4 * q^17 + 3 * q^19 - 2 * q^23 + 4 * q^25 + 2 * q^27 + 7 * q^29 + 3 * q^31 - q^33 - 2 * q^35 - 14 * q^37 - 2 * q^39 - 3 * q^41 + 5 * q^43 + 3 * q^45 - 6 * q^49 - 2 * q^51 - 16 * q^53 + 5 * q^55 - q^57 + 12 * q^59 + 6 * q^61 + 14 * q^63 + 8 * q^65 - q^67 + 3 * q^71 + 4 * q^73 + 41 * q^75 - q^77 - 2 * q^79 - 17 * q^81 - 7 * q^83 + 6 * q^85 + 9 * q^87 + 16 * q^89 + 13 * q^91 - 22 * q^93 + 5 * q^95 - 14 * 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
$3$	−0.772866	−0.446214	−0.223107	−	0.974794i	$-0.571620\pi$
$3$	−0.772866	−0.446214	−0.223107	+	0.974794i	$0.571620\pi$
$4$	0	0
$5$	1.22713	0.548791	0.274396	−	0.961617i	$-0.411522\pi$
$5$	1.22713	0.548791	0.274396	+	0.961617i	$0.411522\pi$
$6$	0	0
$7$	−3.40268	−1.28609	−0.643046	−	0.765828i	$-0.722330\pi$
$7$	−3.40268	−1.28609	−0.643046	+	0.765828i	$0.722330\pi$
$8$	0	0
$9$	−2.40268	−0.800893
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.40268	−0.389033	−0.194517	−	0.980899i	$-0.562314\pi$
$13$	−1.40268	−0.389033	−0.194517	+	0.980899i	$0.562314\pi$
$14$	0	0
$15$	−0.948410	−0.244878
$16$	0	0
$17$	−4.80536	−1.16547	−0.582735	−	0.812662i	$-0.698018\pi$
$17$	−4.80536	−1.16547	−0.582735	+	0.812662i	$0.698018\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.62981	0.573872
$22$	0	0
$23$	−6.80536	−1.41902	−0.709508	−	0.704698i	$-0.751083\pi$
$23$	−6.80536	−1.41902	−0.709508	+	0.704698i	$0.751083\pi$
$24$	0	0
$25$	−3.49414	−0.698828
$26$	0	0
$27$	4.17554	0.803584
$28$	0	0
$29$	8.03249	1.49160	0.745798	−	0.666172i	$-0.232068\pi$
$29$	8.03249	1.49160	0.745798	+	0.666172i	$0.232068\pi$
$30$	0	0
$31$	4.94841	0.888761	0.444380	−	0.895838i	$-0.353424\pi$
$31$	4.94841	0.888761	0.444380	+	0.895838i	$0.353424\pi$
$32$	0	0
$33$	−0.772866	−0.134539
$34$	0	0
$35$	−4.17554	−0.705796
$36$	0	0
$37$	2.35109	0.386517	0.193258	−	0.981148i	$-0.438094\pi$
$37$	2.35109	0.386517	0.193258	+	0.981148i	$0.438094\pi$
$38$	0	0
$39$	1.08408	0.173592
$40$	0	0
$41$	2.94841	0.460464	0.230232	−	0.973136i	$-0.426051\pi$
$41$	2.94841	0.460464	0.230232	+	0.973136i	$0.426051\pi$
$42$	0	0
$43$	9.12395	1.39139	0.695695	−	0.718337i	$-0.255097\pi$
$43$	9.12395	1.39139	0.695695	+	0.718337i	$0.255097\pi$
$44$	0	0
$45$	−2.94841	−0.439523
$46$	0	0
$47$	−5.25963	−0.767195	−0.383598	−	0.923500i	$-0.625315\pi$
$47$	−5.25963	−0.767195	−0.383598	+	0.923500i	$0.625315\pi$
$48$	0	0
$49$	4.57822	0.654032
$50$	0	0
$51$	3.71390	0.520049
$52$	0	0
$53$	−4.45427	−0.611841	−0.305920	−	0.952057i	$-0.598964\pi$
$53$	−4.45427	−0.611841	−0.305920	+	0.952057i	$0.598964\pi$
$54$	0	0
$55$	1.22713	0.165467
$56$	0	0
$57$	−0.772866	−0.102369
$58$	0	0
$59$	14.5193	1.89025	0.945123	−	0.326715i	$-0.105942\pi$
$59$	14.5193	1.89025	0.945123	+	0.326715i	$0.105942\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	8.17554	1.03002
$64$	0	0
$65$	−1.72128	−0.213498
$66$	0	0
$67$	−3.40268	−0.415703	−0.207852	−	0.978160i	$-0.566647\pi$
$67$	−3.40268	−0.415703	−0.207852	+	0.978160i	$0.566647\pi$
$68$	0	0
$69$	5.25963	0.633185
$70$	0	0
$71$	7.57822	0.899370	0.449685	−	0.893187i	$-0.351536\pi$
$71$	7.57822	0.899370	0.449685	+	0.893187i	$0.351536\pi$
$72$	0	0
$73$	−10.0650	−1.17802	−0.589009	−	0.808127i	$-0.700482\pi$
$73$	−10.0650	−1.17802	−0.589009	+	0.808127i	$0.700482\pi$
$74$	0	0
$75$	2.70050	0.311827
$76$	0	0
$77$	−3.40268	−0.387771
$78$	0	0
$79$	3.71390	0.417846	0.208923	−	0.977932i	$-0.433004\pi$
$79$	3.71390	0.417846	0.208923	+	0.977932i	$0.433004\pi$
$80$	0	0
$81$	3.98090	0.442322
$82$	0	0
$83$	−10.6697	−1.17115	−0.585575	−	0.810618i	$-0.699131\pi$
$83$	−10.6697	−1.17115	−0.585575	+	0.810618i	$0.699131\pi$
$84$	0	0
$85$	−5.89682	−0.639600
$86$	0	0
$87$	−6.20804	−0.665571
$88$	0	0
$89$	14.9883	1.58875	0.794377	−	0.607425i	$-0.207797\pi$
$89$	14.9883	1.58875	0.794377	+	0.607425i	$0.207797\pi$
$90$	0	0
$91$	4.77287	0.500332
$92$	0	0
$93$	−3.82446	−0.396578
$94$	0	0
$95$	1.22713	0.125901
$96$	0	0
$97$	2.35109	0.238717	0.119358	−	0.992851i	$-0.461916\pi$
$97$	2.35109	0.238717	0.119358	+	0.992851i	$0.461916\pi$
$98$	0	0
$99$	−2.40268	−0.241478
$100$	0	0
$101$	14.0650	1.39952	0.699759	−	0.714379i	$-0.253291\pi$
$101$	14.0650	1.39952	0.699759	+	0.714379i	$0.253291\pi$
$102$	0	0
$103$	15.5782	1.53497	0.767484	−	0.641068i	$-0.221508\pi$
$103$	15.5782	1.53497	0.767484	+	0.641068i	$0.221508\pi$
$104$	0	0
$105$	3.22713	0.314936
$106$	0	0
$107$	4.35109	0.420636	0.210318	−	0.977633i	$-0.432550\pi$
$107$	4.35109	0.420636	0.210318	+	0.977633i	$0.432550\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	−1.81708	−0.172469
$112$	0	0
$113$	−19.9618	−1.87785	−0.938924	−	0.344124i	$-0.888176\pi$
$113$	−19.9618	−1.87785	−0.938924	+	0.344124i	$0.888176\pi$
$114$	0	0
$115$	−8.35109	−0.778743
$116$	0	0
$117$	3.37019	0.311574
$118$	0	0
$119$	16.3511	1.49890
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.27872	−0.205466
$124$	0	0
$125$	−10.4235	−0.932302
$126$	0	0
$127$	−11.7139	−1.03944	−0.519720	−	0.854337i	$-0.673964\pi$
$127$	−11.7139	−1.03944	−0.519720	+	0.854337i	$0.673964\pi$
$128$	0	0
$129$	−7.05159	−0.620858
$130$	0	0
$131$	−1.85695	−0.162242	−0.0811211	−	0.996704i	$-0.525850\pi$
$131$	−1.85695	−0.162242	−0.0811211	+	0.996704i	$0.525850\pi$
$132$	0	0
$133$	−3.40268	−0.295050
$134$	0	0
$135$	5.12395	0.441000
$136$	0	0
$137$	10.5973	0.905390	0.452695	−	0.891665i	$-0.350463\pi$
$137$	10.5973	0.905390	0.452695	+	0.891665i	$0.350463\pi$
$138$	0	0
$139$	−14.7347	−1.24978	−0.624889	−	0.780713i	$-0.714856\pi$
$139$	−14.7347	−1.24978	−0.624889	+	0.780713i	$0.714856\pi$
$140$	0	0
$141$	4.06498	0.342333
$142$	0	0
$143$	−1.40268	−0.117298
$144$	0	0
$145$	9.85695	0.818575
$146$	0	0
$147$	−3.53835	−0.291838
$148$	0	0
$149$	18.3511	1.50338	0.751690	−	0.659517i	$-0.229239\pi$
$149$	18.3511	1.50338	0.751690	+	0.659517i	$0.229239\pi$
$150$	0	0
$151$	7.64891	0.622460	0.311230	−	0.950335i	$-0.399259\pi$
$151$	7.64891	0.622460	0.311230	+	0.950335i	$0.399259\pi$
$152$	0	0
$153$	11.5457	0.933417
$154$	0	0
$155$	6.07236	0.487744
$156$	0	0
$157$	8.38358	0.669083	0.334541	−	0.942381i	$-0.391419\pi$
$157$	8.38358	0.669083	0.334541	+	0.942381i	$0.391419\pi$
$158$	0	0
$159$	3.44255	0.273012
$160$	0	0
$161$	23.1564	1.82498
$162$	0	0
$163$	14.2479	1.11598	0.557991	−	0.829847i	$-0.311572\pi$
$163$	14.2479	1.11598	0.557991	+	0.829847i	$0.311572\pi$
$164$	0	0
$165$	−0.948410	−0.0738336
$166$	0	0
$167$	10.8703	0.841172	0.420586	−	0.907253i	$-0.361824\pi$
$167$	10.8703	0.841172	0.420586	+	0.907253i	$0.361824\pi$
$168$	0	0
$169$	−11.0325	−0.848653
$170$	0	0
$171$	−2.40268	−0.183737
$172$	0	0
$173$	3.39530	0.258140	0.129070	−	0.991635i	$-0.458801\pi$
$173$	3.39530	0.258140	0.129070	+	0.991635i	$0.458801\pi$
$174$	0	0
$175$	11.8894	0.898757
$176$	0	0
$177$	−11.2214	−0.843454
$178$	0	0
$179$	0.311217	0.0232614	0.0116307	−	0.999932i	$-0.496298\pi$
$179$	0.311217	0.0232614	0.0116307	+	0.999932i	$0.496298\pi$
$180$	0	0
$181$	−15.3246	−1.13907	−0.569535	−	0.821967i	$-0.692877\pi$
$181$	−15.3246	−1.13907	−0.569535	+	0.821967i	$0.692877\pi$
$182$	0	0
$183$	−1.54573	−0.114264
$184$	0	0
$185$	2.88510	0.212117
$186$	0	0
$187$	−4.80536	−0.351403
$188$	0	0
$189$	−14.2080	−1.03348
$190$	0	0
$191$	−26.8703	−1.94427	−0.972135	−	0.234422i	$-0.924680\pi$
$191$	−26.8703	−1.94427	−0.972135	+	0.234422i	$0.924680\pi$
$192$	0	0
$193$	18.5665	1.33645	0.668223	−	0.743961i	$-0.267055\pi$
$193$	18.5665	1.33645	0.668223	+	0.743961i	$0.267055\pi$
$194$	0	0
$195$	1.33031	0.0952658
$196$	0	0
$197$	14.9085	1.06219	0.531095	−	0.847312i	$-0.321781\pi$
$197$	14.9085	1.06219	0.531095	+	0.847312i	$0.321781\pi$
$198$	0	0
$199$	−15.1564	−1.07441	−0.537206	−	0.843451i	$-0.680520\pi$
$199$	−15.1564	−1.07441	−0.537206	+	0.843451i	$0.680520\pi$
$200$	0	0
$201$	2.62981	0.185493
$202$	0	0
$203$	−27.3320	−1.91833
$204$	0	0
$205$	3.61810	0.252699
$206$	0	0
$207$	16.3511	1.13648
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−11.1564	−0.768041	−0.384021	−	0.923324i	$-0.625461\pi$
$211$	−11.1564	−0.768041	−0.384021	+	0.923324i	$0.625461\pi$
$212$	0	0
$213$	−5.85695	−0.401311
$214$	0	0
$215$	11.1963	0.763583
$216$	0	0
$217$	−16.8378	−1.14303
$218$	0	0
$219$	7.77888	0.525648
$220$	0	0
$221$	6.74037	0.453407
$222$	0	0
$223$	−1.61072	−0.107861	−0.0539307	−	0.998545i	$-0.517175\pi$
$223$	−1.61072	−0.107861	−0.0539307	+	0.998545i	$0.517175\pi$
$224$	0	0
$225$	8.39530	0.559687
$226$	0	0
$227$	16.7672	1.11288	0.556438	−	0.830889i	$-0.312168\pi$
$227$	16.7672	1.11288	0.556438	+	0.830889i	$0.312168\pi$
$228$	0	0
$229$	−8.55913	−0.565603	−0.282801	−	0.959178i	$-0.591264\pi$
$229$	−8.55913	−0.565603	−0.282801	+	0.959178i	$0.591264\pi$
$230$	0	0
$231$	2.62981	0.173029
$232$	0	0
$233$	5.44255	0.356553	0.178277	−	0.983980i	$-0.442948\pi$
$233$	5.44255	0.356553	0.178277	+	0.983980i	$0.442948\pi$
$234$	0	0
$235$	−6.45427	−0.421030
$236$	0	0
$237$	−2.87034	−0.186449
$238$	0	0
$239$	−11.2921	−0.730426	−0.365213	−	0.930924i	$-0.619004\pi$
$239$	−11.2921	−0.730426	−0.365213	+	0.930924i	$0.619004\pi$
$240$	0	0
$241$	11.4101	0.734987	0.367493	−	0.930026i	$-0.380216\pi$
$241$	11.4101	0.734987	0.367493	+	0.930026i	$0.380216\pi$
$242$	0	0
$243$	−15.6033	−1.00095
$244$	0	0
$245$	5.61810	0.358927
$246$	0	0
$247$	−1.40268	−0.0892503
$248$	0	0
$249$	8.24623	0.522584
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	−6.80536	−0.427849
$254$	0	0
$255$	4.55745	0.285399
$256$	0	0
$257$	24.2331	1.51162	0.755811	−	0.654790i	$-0.227243\pi$
$257$	24.2331	1.51162	0.755811	+	0.654790i	$0.227243\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	0	0
$261$	−19.2995	−1.19461
$262$	0	0
$263$	−20.5665	−1.26819	−0.634093	−	0.773257i	$-0.718626\pi$
$263$	−20.5665	−1.26819	−0.634093	+	0.773257i	$0.718626\pi$
$264$	0	0
$265$	−5.46599	−0.335773
$266$	0	0
$267$	−11.5839	−0.708925
$268$	0	0
$269$	−1.36281	−0.0830918	−0.0415459	−	0.999137i	$-0.513228\pi$
$269$	−1.36281	−0.0830918	−0.0415459	+	0.999137i	$0.513228\pi$
$270$	0	0
$271$	23.1963	1.40908	0.704538	−	0.709666i	$-0.251154\pi$
$271$	23.1963	1.40908	0.704538	+	0.709666i	$0.251154\pi$
$272$	0	0
$273$	−3.68878	−0.223255
$274$	0	0
$275$	−3.49414	−0.210705
$276$	0	0
$277$	31.0385	1.86492	0.932462	−	0.361269i	$-0.117656\pi$
$277$	31.0385	1.86492	0.932462	+	0.361269i	$0.117656\pi$
$278$	0	0
$279$	−11.8894	−0.711802
$280$	0	0
$281$	−2.77287	−0.165415	−0.0827076	−	0.996574i	$-0.526357\pi$
$281$	−2.77287	−0.165415	−0.0827076	+	0.996574i	$0.526357\pi$
$282$	0	0
$283$	9.23451	0.548935	0.274467	−	0.961596i	$-0.411498\pi$
$283$	9.23451	0.548935	0.274467	+	0.961596i	$0.411498\pi$
$284$	0	0
$285$	−0.948410	−0.0561790
$286$	0	0
$287$	−10.0325	−0.592199
$288$	0	0
$289$	6.09146	0.358321
$290$	0	0
$291$	−1.81708	−0.106519
$292$	0	0
$293$	17.7390	1.03632	0.518162	−	0.855283i	$-0.326616\pi$
$293$	17.7390	1.03632	0.518162	+	0.855283i	$0.326616\pi$
$294$	0	0
$295$	17.8171	1.03735
$296$	0	0
$297$	4.17554	0.242290
$298$	0	0
$299$	9.54573	0.552044
$300$	0	0
$301$	−31.0459	−1.78946
$302$	0	0
$303$	−10.8703	−0.624485
$304$	0	0
$305$	2.45427	0.140531
$306$	0	0
$307$	−20.9735	−1.19702	−0.598511	−	0.801115i	$-0.704241\pi$
$307$	−20.9735	−1.19702	−0.598511	+	0.801115i	$0.704241\pi$
$308$	0	0
$309$	−12.0399	−0.684924
$310$	0	0
$311$	3.36281	0.190687	0.0953436	−	0.995444i	$-0.469605\pi$
$311$	3.36281	0.190687	0.0953436	+	0.995444i	$0.469605\pi$
$312$	0	0
$313$	−4.94103	−0.279284	−0.139642	−	0.990202i	$-0.544595\pi$
$313$	−4.94103	−0.279284	−0.139642	+	0.990202i	$0.544595\pi$
$314$	0	0
$315$	10.0325	0.565267
$316$	0	0
$317$	14.2861	0.802388	0.401194	−	0.915993i	$-0.368595\pi$
$317$	14.2861	0.802388	0.401194	+	0.915993i	$0.368595\pi$
$318$	0	0
$319$	8.03249	0.449733
$320$	0	0
$321$	−3.36281	−0.187694
$322$	0	0
$323$	−4.80536	−0.267377
$324$	0	0
$325$	4.90116	0.271867
$326$	0	0
$327$	−4.63719	−0.256437
$328$	0	0
$329$	17.8968	0.986684
$330$	0	0
$331$	31.9293	1.75499	0.877497	−	0.479582i	$-0.159212\pi$
$331$	31.9293	1.75499	0.877497	+	0.479582i	$0.159212\pi$
$332$	0	0
$333$	−5.64891	−0.309558
$334$	0	0
$335$	−4.17554	−0.228134
$336$	0	0
$337$	−8.84523	−0.481830	−0.240915	−	0.970546i	$-0.577448\pi$
$337$	−8.84523	−0.481830	−0.240915	+	0.970546i	$0.577448\pi$
$338$	0	0
$339$	15.4278	0.837923
$340$	0	0
$341$	4.94841	0.267971
$342$	0	0
$343$	8.24053	0.444947
$344$	0	0
$345$	6.45427	0.347486
$346$	0	0
$347$	−5.27439	−0.283144	−0.141572	−	0.989928i	$-0.545216\pi$
$347$	−5.27439	−0.283144	−0.141572	+	0.989928i	$0.545216\pi$
$348$	0	0
$349$	−10.9085	−0.583921	−0.291960	−	0.956430i	$-0.594308\pi$
$349$	−10.9085	−0.583921	−0.291960	+	0.956430i	$0.594308\pi$
$350$	0	0
$351$	−5.85695	−0.312621
$352$	0	0
$353$	−22.1300	−1.17786	−0.588930	−	0.808184i	$-0.700451\pi$
$353$	−22.1300	−1.17786	−0.588930	+	0.808184i	$0.700451\pi$
$354$	0	0
$355$	9.29950	0.493566
$356$	0	0
$357$	−12.6372	−0.668831
$358$	0	0
$359$	16.3910	0.865082	0.432541	−	0.901614i	$-0.357617\pi$
$359$	16.3910	0.865082	0.432541	+	0.901614i	$0.357617\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−0.772866	−0.0405649
$364$	0	0
$365$	−12.3511	−0.646486
$366$	0	0
$367$	32.3511	1.68871	0.844357	−	0.535782i	$-0.179983\pi$
$367$	32.3511	1.68871	0.844357	+	0.535782i	$0.179983\pi$
$368$	0	0
$369$	−7.08408	−0.368783
$370$	0	0
$371$	15.1564	0.786883
$372$	0	0
$373$	17.5782	0.910166	0.455083	−	0.890449i	$-0.349610\pi$
$373$	17.5782	0.910166	0.455083	+	0.890449i	$0.349610\pi$
$374$	0	0
$375$	8.05593	0.416006
$376$	0	0
$377$	−11.2670	−0.580280
$378$	0	0
$379$	−8.93365	−0.458891	−0.229445	−	0.973322i	$-0.573691\pi$
$379$	−8.93365	−0.458891	−0.229445	+	0.973322i	$0.573691\pi$
$380$	0	0
$381$	9.05327	0.463813
$382$	0	0
$383$	16.0399	0.819599	0.409800	−	0.912176i	$-0.365599\pi$
$383$	16.0399	0.819599	0.409800	+	0.912176i	$0.365599\pi$
$384$	0	0
$385$	−4.17554	−0.212805
$386$	0	0
$387$	−21.9219	−1.11435
$388$	0	0
$389$	17.2271	0.873450	0.436725	−	0.899595i	$-0.356138\pi$
$389$	17.2271	0.873450	0.436725	+	0.899595i	$0.356138\pi$
$390$	0	0
$391$	32.7022	1.65382
$392$	0	0
$393$	1.43517	0.0723948
$394$	0	0
$395$	4.55745	0.229310
$396$	0	0
$397$	15.0784	0.756762	0.378381	−	0.925650i	$-0.376481\pi$
$397$	15.0784	0.756762	0.378381	+	0.925650i	$0.376481\pi$
$398$	0	0
$399$	2.62981	0.131655
$400$	0	0
$401$	−13.1564	−0.657002	−0.328501	−	0.944504i	$-0.606543\pi$
$401$	−13.1564	−0.657002	−0.328501	+	0.944504i	$0.606543\pi$
$402$	0	0
$403$	−6.94103	−0.345757
$404$	0	0
$405$	4.88510	0.242743
$406$	0	0
$407$	2.35109	0.116539
$408$	0	0
$409$	−16.3836	−0.810116	−0.405058	−	0.914291i	$-0.632749\pi$
$409$	−16.3836	−0.810116	−0.405058	+	0.914291i	$0.632749\pi$
$410$	0	0
$411$	−8.19030	−0.403998
$412$	0	0
$413$	−49.4044	−2.43103
$414$	0	0
$415$	−13.0931	−0.642717
$416$	0	0
$417$	11.3879	0.557669
$418$	0	0
$419$	−1.12966	−0.0551874	−0.0275937	−	0.999619i	$-0.508784\pi$
$419$	−1.12966	−0.0551874	−0.0275937	+	0.999619i	$0.508784\pi$
$420$	0	0
$421$	−2.63719	−0.128529	−0.0642645	−	0.997933i	$-0.520470\pi$
$421$	−2.63719	−0.128529	−0.0642645	+	0.997933i	$0.520470\pi$
$422$	0	0
$423$	12.6372	0.614441
$424$	0	0
$425$	16.7906	0.814464
$426$	0	0
$427$	−6.80536	−0.329334
$428$	0	0
$429$	1.08408	0.0523400
$430$	0	0
$431$	28.4958	1.37260	0.686298	−	0.727321i	$-0.259235\pi$
$431$	28.4958	1.37260	0.686298	+	0.727321i	$0.259235\pi$
$432$	0	0
$433$	−37.5075	−1.80250	−0.901249	−	0.433302i	$-0.857348\pi$
$433$	−37.5075	−1.80250	−0.901249	+	0.433302i	$0.857348\pi$
$434$	0	0
$435$	−7.61810	−0.365260
$436$	0	0
$437$	−6.80536	−0.325544
$438$	0	0
$439$	32.1300	1.53348	0.766740	−	0.641958i	$-0.221878\pi$
$439$	32.1300	1.53348	0.766740	+	0.641958i	$0.221878\pi$
$440$	0	0
$441$	−11.0000	−0.523810
$442$	0	0
$443$	−33.0385	−1.56971	−0.784853	−	0.619681i	$-0.787262\pi$
$443$	−33.0385	−1.56971	−0.784853	+	0.619681i	$0.787262\pi$
$444$	0	0
$445$	18.3926	0.871895
$446$	0	0
$447$	−14.1829	−0.670829
$448$	0	0
$449$	7.88206	0.371977	0.185989	−	0.982552i	$-0.440451\pi$
$449$	7.88206	0.371977	0.185989	+	0.982552i	$0.440451\pi$
$450$	0	0
$451$	2.94841	0.138835
$452$	0	0
$453$	−5.91158	−0.277750
$454$	0	0
$455$	5.85695	0.274578
$456$	0	0
$457$	24.9353	1.16643	0.583213	−	0.812320i	$-0.301795\pi$
$457$	24.9353	1.16643	0.583213	+	0.812320i	$0.301795\pi$
$458$	0	0
$459$	−20.0650	−0.936553
$460$	0	0
$461$	−26.6874	−1.24296	−0.621478	−	0.783431i	$-0.713468\pi$
$461$	−26.6874	−1.24296	−0.621478	+	0.783431i	$0.713468\pi$
$462$	0	0
$463$	32.7022	1.51980	0.759900	−	0.650041i	$-0.225248\pi$
$463$	32.7022	1.51980	0.759900	+	0.650041i	$0.225248\pi$
$464$	0	0
$465$	−4.69312	−0.217638
$466$	0	0
$467$	−11.3776	−0.526491	−0.263246	−	0.964729i	$-0.584793\pi$
$467$	−11.3776	−0.526491	−0.263246	+	0.964729i	$0.584793\pi$
$468$	0	0
$469$	11.5782	0.534633
$470$	0	0
$471$	−6.47938	−0.298554
$472$	0	0
$473$	9.12395	0.419520
$474$	0	0
$475$	−3.49414	−0.160322
$476$	0	0
$477$	10.7022	0.490019
$478$	0	0
$479$	12.5973	0.575586	0.287793	−	0.957693i	$-0.407078\pi$
$479$	12.5973	0.575586	0.287793	+	0.957693i	$0.407078\pi$
$480$	0	0
$481$	−3.29782	−0.150368
$482$	0	0
$483$	−17.8968	−0.814334
$484$	0	0
$485$	2.88510	0.131006
$486$	0	0
$487$	−9.69616	−0.439375	−0.219688	−	0.975570i	$-0.570504\pi$
$487$	−9.69616	−0.439375	−0.219688	+	0.975570i	$0.570504\pi$
$488$	0	0
$489$	−11.0117	−0.497967
$490$	0	0
$491$	11.7538	0.530440	0.265220	−	0.964188i	$-0.414555\pi$
$491$	11.7538	0.530440	0.265220	+	0.964188i	$0.414555\pi$
$492$	0	0
$493$	−38.5990	−1.73841
$494$	0	0
$495$	−2.94841	−0.132521
$496$	0	0
$497$	−25.7863	−1.15667
$498$	0	0
$499$	−26.6640	−1.19364	−0.596822	−	0.802374i	$-0.703570\pi$
$499$	−26.6640	−1.19364	−0.596822	+	0.802374i	$0.703570\pi$
$500$	0	0
$501$	−8.40131	−0.375343
$502$	0	0
$503$	−11.1622	−0.497696	−0.248848	−	0.968543i	$-0.580052\pi$
$503$	−11.1622	−0.497696	−0.248848	+	0.968543i	$0.580052\pi$
$504$	0	0
$505$	17.2596	0.768043
$506$	0	0
$507$	8.52663	0.378681
$508$	0	0
$509$	−35.9618	−1.59398	−0.796989	−	0.603993i	$-0.793575\pi$
$509$	−35.9618	−1.59398	−0.796989	+	0.603993i	$0.793575\pi$
$510$	0	0
$511$	34.2479	1.51504
$512$	0	0
$513$	4.17554	0.184355
$514$	0	0
$515$	19.1166	0.842377
$516$	0	0
$517$	−5.25963	−0.231318
$518$	0	0
$519$	−2.62411	−0.115186
$520$	0	0
$521$	10.1300	0.443802	0.221901	−	0.975069i	$-0.428774\pi$
$521$	10.1300	0.443802	0.221901	+	0.975069i	$0.428774\pi$
$522$	0	0
$523$	7.93502	0.346974	0.173487	−	0.984836i	$-0.444497\pi$
$523$	7.93502	0.346974	0.173487	+	0.984836i	$0.444497\pi$
$524$	0	0
$525$	−9.18894	−0.401038
$526$	0	0
$527$	−23.7789	−1.03582
$528$	0	0
$529$	23.3129	1.01360
$530$	0	0
$531$	−34.8851	−1.51388
$532$	0	0
$533$	−4.13567	−0.179136
$534$	0	0
$535$	5.33937	0.230841
$536$	0	0
$537$	−0.240529	−0.0103796
$538$	0	0
$539$	4.57822	0.197198
$540$	0	0
$541$	10.8436	0.466201	0.233100	−	0.972453i	$-0.425113\pi$
$541$	10.8436	0.466201	0.233100	+	0.972453i	$0.425113\pi$
$542$	0	0
$543$	11.8439	0.508269
$544$	0	0
$545$	7.36281	0.315388
$546$	0	0
$547$	10.3129	0.440947	0.220474	−	0.975393i	$-0.429240\pi$
$547$	10.3129	0.440947	0.220474	+	0.975393i	$0.429240\pi$
$548$	0	0
$549$	−4.80536	−0.205088
$550$	0	0
$551$	8.03249	0.342196
$552$	0	0
$553$	−12.6372	−0.537388
$554$	0	0
$555$	−2.22980	−0.0946496
$556$	0	0
$557$	−14.4811	−0.613582	−0.306791	−	0.951777i	$-0.599255\pi$
$557$	−14.4811	−0.613582	−0.306791	+	0.951777i	$0.599255\pi$
$558$	0	0
$559$	−12.7980	−0.541297
$560$	0	0
$561$	3.71390	0.156801
$562$	0	0
$563$	−10.5340	−0.443956	−0.221978	−	0.975052i	$-0.571251\pi$
$563$	−10.5340	−0.443956	−0.221978	+	0.975052i	$0.571251\pi$
$564$	0	0
$565$	−24.4958	−1.03055
$566$	0	0
$567$	−13.5457	−0.568867
$568$	0	0
$569$	10.3762	0.434993	0.217496	−	0.976061i	$-0.430211\pi$
$569$	10.3762	0.434993	0.217496	+	0.976061i	$0.430211\pi$
$570$	0	0
$571$	−33.2847	−1.39292	−0.696461	−	0.717594i	$-0.745243\pi$
$571$	−33.2847	−1.39292	−0.696461	+	0.717594i	$0.745243\pi$
$572$	0	0
$573$	20.7672	0.867561
$574$	0	0
$575$	23.7789	0.991648
$576$	0	0
$577$	−2.93365	−0.122129	−0.0610647	−	0.998134i	$-0.519450\pi$
$577$	−2.93365	−0.122129	−0.0610647	+	0.998134i	$0.519450\pi$
$578$	0	0
$579$	−14.3494	−0.596341
$580$	0	0
$581$	36.3055	1.50621
$582$	0	0
$583$	−4.45427	−0.184477
$584$	0	0
$585$	4.13567	0.170989
$586$	0	0
$587$	−8.90854	−0.367695	−0.183847	−	0.982955i	$-0.558855\pi$
$587$	−8.90854	−0.367695	−0.183847	+	0.982955i	$0.558855\pi$
$588$	0	0
$589$	4.94841	0.203896
$590$	0	0
$591$	−11.5223	−0.473964
$592$	0	0
$593$	16.3129	0.669890	0.334945	−	0.942238i	$-0.391282\pi$
$593$	16.3129	0.669890	0.334945	+	0.942238i	$0.391282\pi$
$594$	0	0
$595$	20.0650	0.822584
$596$	0	0
$597$	11.7139	0.479418
$598$	0	0
$599$	−40.9826	−1.67450	−0.837251	−	0.546818i	$-0.815839\pi$
$599$	−40.9826	−1.67450	−0.837251	+	0.546818i	$0.815839\pi$
$600$	0	0
$601$	0.349413	0.0142528	0.00712642	−	0.999975i	$-0.497732\pi$
$601$	0.349413	0.0142528	0.00712642	+	0.999975i	$0.497732\pi$
$602$	0	0
$603$	8.17554	0.332934
$604$	0	0
$605$	1.22713	0.0498901
$606$	0	0
$607$	−5.56049	−0.225693	−0.112847	−	0.993612i	$-0.535997\pi$
$607$	−5.56049	−0.225693	−0.112847	+	0.993612i	$0.535997\pi$
$608$	0	0
$609$	21.1240	0.855986
$610$	0	0
$611$	7.37757	0.298464
$612$	0	0
$613$	−19.7407	−0.797319	−0.398659	−	0.917099i	$-0.630524\pi$
$613$	−19.7407	−0.797319	−0.398659	+	0.917099i	$0.630524\pi$
$614$	0	0
$615$	−2.79630	−0.112758
$616$	0	0
$617$	10.5973	0.426632	0.213316	−	0.976983i	$-0.431574\pi$
$617$	10.5973	0.426632	0.213316	+	0.976983i	$0.431574\pi$
$618$	0	0
$619$	−4.22112	−0.169661	−0.0848306	−	0.996395i	$-0.527035\pi$
$619$	−4.22112	−0.169661	−0.0848306	+	0.996395i	$0.527035\pi$
$620$	0	0
$621$	−28.4161	−1.14030
$622$	0	0
$623$	−51.0003	−2.04328
$624$	0	0
$625$	4.67973	0.187189
$626$	0	0
$627$	−0.772866	−0.0308653
$628$	0	0
$629$	−11.2978	−0.450474
$630$	0	0
$631$	−13.6905	−0.545009	−0.272504	−	0.962155i	$-0.587852\pi$
$631$	−13.6905	−0.545009	−0.272504	+	0.962155i	$0.587852\pi$
$632$	0	0
$633$	8.62243	0.342711
$634$	0	0
$635$	−14.3745	−0.570436
$636$	0	0
$637$	−6.42178	−0.254440
$638$	0	0
$639$	−18.2080	−0.720299
$640$	0	0
$641$	−42.9883	−1.69794	−0.848968	−	0.528445i	$-0.822775\pi$
$641$	−42.9883	−1.69794	−0.848968	+	0.528445i	$0.822775\pi$
$642$	0	0
$643$	−16.3658	−0.645406	−0.322703	−	0.946500i	$-0.604592\pi$
$643$	−16.3658	−0.645406	−0.322703	+	0.946500i	$0.604592\pi$
$644$	0	0
$645$	−8.65325	−0.340721
$646$	0	0
$647$	−39.9886	−1.57211	−0.786057	−	0.618154i	$-0.787881\pi$
$647$	−39.9886	−1.57211	−0.786057	+	0.618154i	$0.787881\pi$
$648$	0	0
$649$	14.5193	0.569931
$650$	0	0
$651$	13.0134	0.510035
$652$	0	0
$653$	−37.5976	−1.47131	−0.735655	−	0.677357i	$-0.763125\pi$
$653$	−37.5976	−1.47131	−0.735655	+	0.677357i	$0.763125\pi$
$654$	0	0
$655$	−2.27872	−0.0890371
$656$	0	0
$657$	24.1829	0.943466
$658$	0	0
$659$	11.5075	0.448270	0.224135	−	0.974558i	$-0.428044\pi$
$659$	11.5075	0.448270	0.224135	+	0.974558i	$0.428044\pi$
$660$	0	0
$661$	−29.2362	−1.13716	−0.568578	−	0.822629i	$-0.692506\pi$
$661$	−29.2362	−1.13716	−0.568578	+	0.822629i	$0.692506\pi$
$662$	0	0
$663$	−5.20940	−0.202316
$664$	0	0
$665$	−4.17554	−0.161921
$666$	0	0
$667$	−54.6640	−2.11660
$668$	0	0
$669$	1.24487	0.0481293
$670$	0	0
$671$	2.00000	0.0772091
$672$	0	0
$673$	−47.0784	−1.81474	−0.907369	−	0.420335i	$-0.861913\pi$
$673$	−47.0784	−1.81474	−0.907369	+	0.420335i	$0.861913\pi$
$674$	0	0
$675$	−14.5899	−0.561567
$676$	0	0
$677$	43.2539	1.66238	0.831192	−	0.555986i	$-0.187659\pi$
$677$	43.2539	1.66238	0.831192	+	0.555986i	$0.187659\pi$
$678$	0	0
$679$	−8.00000	−0.307012
$680$	0	0
$681$	−12.9588	−0.496581
$682$	0	0
$683$	−34.8556	−1.33371	−0.666856	−	0.745187i	$-0.732360\pi$
$683$	−34.8556	−1.33371	−0.666856	+	0.745187i	$0.732360\pi$
$684$	0	0
$685$	13.0043	0.496870
$686$	0	0
$687$	6.61505	0.252380
$688$	0	0
$689$	6.24791	0.238026
$690$	0	0
$691$	−28.4811	−1.08347	−0.541735	−	0.840549i	$-0.682232\pi$
$691$	−28.4811	−1.08347	−0.541735	+	0.840549i	$0.682232\pi$
$692$	0	0
$693$	8.17554	0.310563
$694$	0	0
$695$	−18.0814	−0.685867
$696$	0	0
$697$	−14.1682	−0.536657
$698$	0	0
$699$	−4.20636	−0.159099
$700$	0	0
$701$	12.8703	0.486106	0.243053	−	0.970013i	$-0.421851\pi$
$701$	12.8703	0.486106	0.243053	+	0.970013i	$0.421851\pi$
$702$	0	0
$703$	2.35109	0.0886730
$704$	0	0
$705$	4.98828	0.187870
$706$	0	0
$707$	−47.8586	−1.79991
$708$	0	0
$709$	1.41744	0.0532330	0.0266165	−	0.999646i	$-0.491527\pi$
$709$	1.41744	0.0532330	0.0266165	+	0.999646i	$0.491527\pi$
$710$	0	0
$711$	−8.92330	−0.334650
$712$	0	0
$713$	−33.6757	−1.26116
$714$	0	0
$715$	−1.72128	−0.0643721
$716$	0	0
$717$	8.72729	0.325927
$718$	0	0
$719$	31.1564	1.16194	0.580970	−	0.813925i	$-0.302673\pi$
$719$	31.1564	1.16194	0.580970	+	0.813925i	$0.302673\pi$
$720$	0	0
$721$	−53.0077	−1.97411
$722$	0	0
$723$	−8.81844	−0.327961
$724$	0	0
$725$	−28.0667	−1.04237
$726$	0	0
$727$	8.57221	0.317926	0.158963	−	0.987285i	$-0.449185\pi$
$727$	8.57221	0.317926	0.158963	+	0.987285i	$0.449185\pi$
$728$	0	0
$729$	0.116574	0.00431757
$730$	0	0
$731$	−43.8439	−1.62162
$732$	0	0
$733$	44.6787	1.65025	0.825123	−	0.564952i	$-0.191105\pi$
$733$	44.6787	1.65025	0.825123	+	0.564952i	$0.191105\pi$
$734$	0	0
$735$	−4.34203	−0.160158
$736$	0	0
$737$	−3.40268	−0.125339
$738$	0	0
$739$	−29.4898	−1.08480	−0.542400	−	0.840120i	$-0.682484\pi$
$739$	−29.4898	−1.08480	−0.542400	+	0.840120i	$0.682484\pi$
$740$	0	0
$741$	1.08408	0.0398248
$742$	0	0
$743$	13.2596	0.486449	0.243224	−	0.969970i	$-0.421795\pi$
$743$	13.2596	0.486449	0.243224	+	0.969970i	$0.421795\pi$
$744$	0	0
$745$	22.5193	0.825042
$746$	0	0
$747$	25.6358	0.937966
$748$	0	0
$749$	−14.8054	−0.540976
$750$	0	0
$751$	10.1829	0.371580	0.185790	−	0.982589i	$-0.440516\pi$
$751$	10.1829	0.371580	0.185790	+	0.982589i	$0.440516\pi$
$752$	0	0
$753$	−9.27439	−0.337977
$754$	0	0
$755$	9.38624	0.341600
$756$	0	0
$757$	40.1122	1.45790	0.728952	−	0.684565i	$-0.240008\pi$
$757$	40.1122	1.45790	0.728952	+	0.684565i	$0.240008\pi$
$758$	0	0
$759$	5.25963	0.190912
$760$	0	0
$761$	−38.7819	−1.40584	−0.702922	−	0.711267i	$-0.748122\pi$
$761$	−38.7819	−1.40584	−0.702922	+	0.711267i	$0.748122\pi$
$762$	0	0
$763$	−20.4161	−0.739111
$764$	0	0
$765$	14.1682	0.512251
$766$	0	0
$767$	−20.3658	−0.735368
$768$	0	0
$769$	8.15340	0.294019	0.147010	−	0.989135i	$-0.453035\pi$
$769$	8.15340	0.294019	0.147010	+	0.989135i	$0.453035\pi$
$770$	0	0
$771$	−18.7290	−0.674507
$772$	0	0
$773$	−24.3893	−0.877222	−0.438611	−	0.898677i	$-0.644529\pi$
$773$	−24.3893	−0.877222	−0.438611	+	0.898677i	$0.644529\pi$
$774$	0	0
$775$	−17.2904	−0.621091
$776$	0	0
$777$	6.18292	0.221811
$778$	0	0
$779$	2.94841	0.105638
$780$	0	0
$781$	7.57822	0.271170
$782$	0	0
$783$	33.5400	1.19862
$784$	0	0
$785$	10.2878	0.367187
$786$	0	0
$787$	46.3779	1.65319	0.826596	−	0.562795i	$-0.190274\pi$
$787$	46.3779	1.65319	0.826596	+	0.562795i	$0.190274\pi$
$788$	0	0
$789$	15.8951	0.565882
$790$	0	0
$791$	67.9236	2.41509
$792$	0	0
$793$	−2.80536	−0.0996212
$794$	0	0
$795$	4.22447	0.149827
$796$	0	0
$797$	3.89682	0.138032	0.0690162	−	0.997616i	$-0.478014\pi$
$797$	3.89682	0.138032	0.0690162	+	0.997616i	$0.478014\pi$
$798$	0	0
$799$	25.2744	0.894144
$800$	0	0
$801$	−36.0120	−1.27242
$802$	0	0
$803$	−10.0650	−0.355186
$804$	0	0
$805$	28.4161	1.00153
$806$	0	0
$807$	1.05327	0.0370767
$808$	0	0
$809$	−43.3896	−1.52550	−0.762748	−	0.646695i	$-0.776151\pi$
$809$	−43.3896	−1.52550	−0.762748	+	0.646695i	$0.776151\pi$
$810$	0	0
$811$	12.9233	0.453798	0.226899	−	0.973918i	$-0.427141\pi$
$811$	12.9233	0.453798	0.226899	+	0.973918i	$0.427141\pi$
$812$	0	0
$813$	−17.9276	−0.628750
$814$	0	0
$815$	17.4841	0.612441
$816$	0	0
$817$	9.12395	0.319207
$818$	0	0
$819$	−11.4677	−0.400713
$820$	0	0
$821$	−36.5842	−1.27680	−0.638399	−	0.769705i	$-0.720403\pi$
$821$	−36.5842	−1.27680	−0.638399	+	0.769705i	$0.720403\pi$
$822$	0	0
$823$	21.6107	0.753302	0.376651	−	0.926355i	$-0.377076\pi$
$823$	21.6107	0.753302	0.376651	+	0.926355i	$0.377076\pi$
$824$	0	0
$825$	2.70050	0.0940194
$826$	0	0
$827$	34.8851	1.21307	0.606537	−	0.795055i	$-0.292558\pi$
$827$	34.8851	1.21307	0.606537	+	0.795055i	$0.292558\pi$
$828$	0	0
$829$	31.7407	1.10240	0.551200	−	0.834373i	$-0.314170\pi$
$829$	31.7407	1.10240	0.551200	+	0.834373i	$0.314170\pi$
$830$	0	0
$831$	−23.9886	−0.832155
$832$	0	0
$833$	−22.0000	−0.762255
$834$	0	0
$835$	13.3394	0.461628
$836$	0	0
$837$	20.6623	0.714194
$838$	0	0
$839$	−13.4603	−0.464701	−0.232350	−	0.972632i	$-0.574642\pi$
$839$	−13.4603	−0.464701	−0.232350	+	0.972632i	$0.574642\pi$
$840$	0	0
$841$	35.5209	1.22486
$842$	0	0
$843$	2.14305	0.0738106
$844$	0	0
$845$	−13.5384	−0.465733
$846$	0	0
$847$	−3.40268	−0.116917
$848$	0	0
$849$	−7.13704	−0.244943
$850$	0	0
$851$	−16.0000	−0.548473
$852$	0	0
$853$	36.2981	1.24282	0.621412	−	0.783484i	$-0.286559\pi$
$853$	36.2981	1.24282	0.621412	+	0.783484i	$0.286559\pi$
$854$	0	0
$855$	−2.94841	−0.100833
$856$	0	0
$857$	−43.8569	−1.49812	−0.749062	−	0.662499i	$-0.769496\pi$
$857$	−43.8569	−1.49812	−0.749062	+	0.662499i	$0.769496\pi$
$858$	0	0
$859$	41.1980	1.40566	0.702829	−	0.711359i	$-0.251920\pi$
$859$	41.1980	1.40566	0.702829	+	0.711359i	$0.251920\pi$
$860$	0	0
$861$	7.75377	0.264248
$862$	0	0
$863$	24.8720	0.846653	0.423327	−	0.905977i	$-0.360862\pi$
$863$	24.8720	0.846653	0.423327	+	0.905977i	$0.360862\pi$
$864$	0	0
$865$	4.16649	0.141665
$866$	0	0
$867$	−4.70788	−0.159888
$868$	0	0
$869$	3.71390	0.125985
$870$	0	0
$871$	4.77287	0.161722
$872$	0	0
$873$	−5.64891	−0.191187
$874$	0	0
$875$	35.4677	1.19903
$876$	0	0
$877$	39.8911	1.34703	0.673514	−	0.739175i	$-0.264784\pi$
$877$	39.8911	1.34703	0.673514	+	0.739175i	$0.264784\pi$
$878$	0	0
$879$	−13.7099	−0.462422
$880$	0	0
$881$	−2.88343	−0.0971451	−0.0485725	−	0.998820i	$-0.515467\pi$
$881$	−2.88343	−0.0971451	−0.0485725	+	0.998820i	$0.515467\pi$
$882$	0	0
$883$	4.27134	0.143742	0.0718711	−	0.997414i	$-0.477103\pi$
$883$	4.27134	0.143742	0.0718711	+	0.997414i	$0.477103\pi$
$884$	0	0
$885$	−13.7702	−0.462880
$886$	0	0
$887$	−22.3129	−0.749194	−0.374597	−	0.927188i	$-0.622219\pi$
$887$	−22.3129	−0.749194	−0.374597	+	0.927188i	$0.622219\pi$
$888$	0	0
$889$	39.8586	1.33682
$890$	0	0
$891$	3.98090	0.133365
$892$	0	0
$893$	−5.25963	−0.176007
$894$	0	0
$895$	0.381905	0.0127657
$896$	0	0
$897$	−7.37757	−0.246330
$898$	0	0
$899$	39.7481	1.32567
$900$	0	0
$901$	21.4044	0.713082
$902$	0	0
$903$	23.9943	0.798480
$904$	0	0
$905$	−18.8054	−0.625111
$906$	0	0
$907$	35.5578	1.18068	0.590338	−	0.807156i	$-0.298994\pi$
$907$	35.5578	1.18068	0.590338	+	0.807156i	$0.298994\pi$
$908$	0	0
$909$	−33.7936	−1.12086
$910$	0	0
$911$	49.1980	1.63000	0.815001	−	0.579459i	$-0.196736\pi$
$911$	49.1980	1.63000	0.815001	+	0.579459i	$0.196736\pi$
$912$	0	0
$913$	−10.6697	−0.353115
$914$	0	0
$915$	−1.89682	−0.0627069
$916$	0	0
$917$	6.31860	0.208658
$918$	0	0
$919$	8.47200	0.279466	0.139733	−	0.990189i	$-0.455376\pi$
$919$	8.47200	0.279466	0.139733	+	0.990189i	$0.455376\pi$
$920$	0	0
$921$	16.2097	0.534128
$922$	0	0
$923$	−10.6298	−0.349885
$924$	0	0
$925$	−8.21504	−0.270109
$926$	0	0
$927$	−37.4295	−1.22934
$928$	0	0
$929$	−16.0017	−0.524998	−0.262499	−	0.964932i	$-0.584547\pi$
$929$	−16.0017	−0.524998	−0.262499	+	0.964932i	$0.584547\pi$
$930$	0	0
$931$	4.57822	0.150045
$932$	0	0
$933$	−2.59900	−0.0850874
$934$	0	0
$935$	−5.89682	−0.192847
$936$	0	0
$937$	−15.9766	−0.521932	−0.260966	−	0.965348i	$-0.584041\pi$
$937$	−15.9766	−0.521932	−0.260966	+	0.965348i	$0.584041\pi$
$938$	0	0
$939$	3.81875	0.124620
$940$	0	0
$941$	−21.0915	−0.687562	−0.343781	−	0.939050i	$-0.611708\pi$
$941$	−21.0915	−0.687562	−0.343781	+	0.939050i	$0.611708\pi$
$942$	0	0
$943$	−20.0650	−0.653406
$944$	0	0
$945$	−17.4352	−0.567166
$946$	0	0
$947$	−20.9735	−0.681548	−0.340774	−	0.940145i	$-0.610689\pi$
$947$	−20.9735	−0.681548	−0.340774	+	0.940145i	$0.610689\pi$
$948$	0	0
$949$	14.1179	0.458288
$950$	0	0
$951$	−11.0412	−0.358037
$952$	0	0
$953$	3.27439	0.106068	0.0530339	−	0.998593i	$-0.483111\pi$
$953$	3.27439	0.106068	0.0530339	+	0.998593i	$0.483111\pi$
$954$	0	0
$955$	−32.9735	−1.06700
$956$	0	0
$957$	−6.20804	−0.200677
$958$	0	0
$959$	−36.0593	−1.16441
$960$	0	0
$961$	−6.51324	−0.210104
$962$	0	0
$963$	−10.4543	−0.336884
$964$	0	0
$965$	22.7836	0.733430
$966$	0	0
$967$	29.2744	0.941401	0.470700	−	0.882293i	$-0.344001\pi$
$967$	29.2744	0.941401	0.470700	+	0.882293i	$0.344001\pi$
$968$	0	0
$969$	3.71390	0.119308
$970$	0	0
$971$	38.3836	1.23179	0.615894	−	0.787829i	$-0.288795\pi$
$971$	38.3836	1.23179	0.615894	+	0.787829i	$0.288795\pi$
$972$	0	0
$973$	50.1373	1.60733
$974$	0	0
$975$	−3.78794	−0.121311
$976$	0	0
$977$	10.9233	0.349467	0.174734	−	0.984616i	$-0.444094\pi$
$977$	10.9233	0.349467	0.174734	+	0.984616i	$0.444094\pi$
$978$	0	0
$979$	14.9883	0.479028
$980$	0	0
$981$	−14.4161	−0.460270
$982$	0	0
$983$	32.1196	1.02446	0.512228	−	0.858849i	$-0.328820\pi$
$983$	32.1196	1.02446	0.512228	+	0.858849i	$0.328820\pi$
$984$	0	0
$985$	18.2948	0.582920
$986$	0	0
$987$	−13.8318	−0.440272
$988$	0	0
$989$	−62.0918	−1.97440
$990$	0	0
$991$	−4.37620	−0.139015	−0.0695073	−	0.997581i	$-0.522143\pi$
$991$	−4.37620	−0.139015	−0.0695073	+	0.997581i	$0.522143\pi$
$992$	0	0
$993$	−24.6771	−0.783103
$994$	0	0
$995$	−18.5990	−0.589628
$996$	0	0
$997$	−8.37788	−0.265330	−0.132665	−	0.991161i	$-0.542353\pi$
$997$	−8.37788	−0.265330	−0.132665	+	0.991161i	$0.542353\pi$
$998$	0	0
$999$	9.81708	0.310599

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.p.1.2		3
4.3	odd	2		418.2.a.h.1.2	✓	3
12.11	even	2		3762.2.a.bd.1.2		3
44.43	even	2		4598.2.a.bm.1.2		3
76.75	even	2		7942.2.a.bc.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.h.1.2	✓	3	4.3	odd	2
3344.2.a.p.1.2		3	1.1	even	1	trivial
3762.2.a.bd.1.2		3	12.11	even	2
4598.2.a.bm.1.2		3	44.43	even	2
7942.2.a.bc.1.2		3	76.75	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q-0.772866 q^{3} +1.22713 q^{5} -3.40268 q^{7} -2.40268 q^{9} +O(q^{10})\)	\(q-0.772866 q^{3} +1.22713 q^{5} -3.40268 q^{7} -2.40268 q^{9} +1.00000 q^{11} -1.40268 q^{13} -0.948410 q^{15} -4.80536 q^{17} +1.00000 q^{19} +2.62981 q^{21} -6.80536 q^{23} -3.49414 q^{25} +4.17554 q^{27} +8.03249 q^{29} +4.94841 q^{31} -0.772866 q^{33} -4.17554 q^{35} +2.35109 q^{37} +1.08408 q^{39} +2.94841 q^{41} +9.12395 q^{43} -2.94841 q^{45} -5.25963 q^{47} +4.57822 q^{49} +3.71390 q^{51} -4.45427 q^{53} +1.22713 q^{55} -0.772866 q^{57} +14.5193 q^{59} +2.00000 q^{61} +8.17554 q^{63} -1.72128 q^{65} -3.40268 q^{67} +5.25963 q^{69} +7.57822 q^{71} -10.0650 q^{73} +2.70050 q^{75} -3.40268 q^{77} +3.71390 q^{79} +3.98090 q^{81} -10.6697 q^{83} -5.89682 q^{85} -6.20804 q^{87} +14.9883 q^{89} +4.77287 q^{91} -3.82446 q^{93} +1.22713 q^{95} +2.35109 q^{97} -2.40268 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} + 5 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) \) 3 * q - q^3 + 5 * q^5 - q^7 + 2 * q^9	\( 3 q - q^{3} + 5 q^{5} - q^{7} + 2 q^{9} + 3 q^{11} + 5 q^{13} + 9 q^{15} + 4 q^{17} + 3 q^{19} - 2 q^{23} + 4 q^{25} + 2 q^{27} + 7 q^{29} + 3 q^{31} - q^{33} - 2 q^{35} - 14 q^{37} - 2 q^{39} - 3 q^{41} + 5 q^{43} + 3 q^{45} - 6 q^{49} - 2 q^{51} - 16 q^{53} + 5 q^{55} - q^{57} + 12 q^{59} + 6 q^{61} + 14 q^{63} + 8 q^{65} - q^{67} + 3 q^{71} + 4 q^{73} + 41 q^{75} - q^{77} - 2 q^{79} - 17 q^{81} - 7 q^{83} + 6 q^{85} + 9 q^{87} + 16 q^{89} + 13 q^{91} - 22 q^{93} + 5 q^{95} - 14 q^{97} + 2 q^{99}+O(q^{100}) \) 3 * q - q^3 + 5 * q^5 - q^7 + 2 * q^9 + 3 * q^11 + 5 * q^13 + 9 * q^15 + 4 * q^17 + 3 * q^19 - 2 * q^23 + 4 * q^25 + 2 * q^27 + 7 * q^29 + 3 * q^31 - q^33 - 2 * q^35 - 14 * q^37 - 2 * q^39 - 3 * q^41 + 5 * q^43 + 3 * q^45 - 6 * q^49 - 2 * q^51 - 16 * q^53 + 5 * q^55 - q^57 + 12 * q^59 + 6 * q^61 + 14 * q^63 + 8 * q^65 - q^67 + 3 * q^71 + 4 * q^73 + 41 * q^75 - q^77 - 2 * q^79 - 17 * q^81 - 7 * q^83 + 6 * q^85 + 9 * q^87 + 16 * q^89 + 13 * q^91 - 22 * q^93 + 5 * q^95 - 14 * q^97 + 2 * q^99

Introduction

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