LMFDB - Embedded newform 3344.2.a.v.1.5

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:	$6$
Coefficient field:	6.6.576096652.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 8x^{4} + 11x^{3} + 16x^{2} - 6x - 4 $ x^6 - 2x^5 - 8x^4 + 11x^3 + 16x^2 - 6*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1672)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.20649$ 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)

$f(q)$	$=$	$q+1.20649 q^{3} -0.300070 q^{5} +4.45862 q^{7} -1.54439 q^{9} -1.00000 q^{11} -4.45862 q^{13} -0.362031 q^{15} -0.906418 q^{17} -1.00000 q^{19} +5.37927 q^{21} -5.49644 q^{23} -4.90996 q^{25} -5.48275 q^{27} -10.0685 q^{29} -9.95506 q^{31} -1.20649 q^{33} -1.33790 q^{35} +2.59003 q^{37} -5.37927 q^{39} -0.952057 q^{41} -7.49233 q^{43} +0.463425 q^{45} +2.50425 q^{47} +12.8792 q^{49} -1.09358 q^{51} +10.4543 q^{53} +0.300070 q^{55} -1.20649 q^{57} +7.12003 q^{59} -3.21061 q^{61} -6.88582 q^{63} +1.33790 q^{65} -12.8594 q^{67} -6.63139 q^{69} +7.93927 q^{71} +9.14576 q^{73} -5.92381 q^{75} -4.45862 q^{77} +7.63920 q^{79} -1.98171 q^{81} +8.60704 q^{83} +0.271989 q^{85} -12.1475 q^{87} -2.31158 q^{89} -19.8792 q^{91} -12.0107 q^{93} +0.300070 q^{95} -0.355672 q^{97} +1.54439 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} + q^{5} + 4 q^{9} - 6 q^{11} - 7 q^{15} + 3 q^{17} - 6 q^{19} + 7 q^{21} - 7 q^{23} + 3 q^{25} - 13 q^{27} - 4 q^{29} - 7 q^{31} + 4 q^{33} - 6 q^{35} - 2 q^{37} - 7 q^{39} + 7 q^{41} - 21 q^{43}+ \cdots - 4 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 + q^5 + 4 * q^9 - 6 * q^11 - 7 * q^15 + 3 * q^17 - 6 * q^19 + 7 * q^21 - 7 * q^23 + 3 * q^25 - 13 * q^27 - 4 * q^29 - 7 * q^31 + 4 * q^33 - 6 * q^35 - 2 * q^37 - 7 * q^39 + 7 * q^41 - 21 * q^43 + 11 * q^45 - 16 * q^47 + 2 * q^49 - 15 * q^51 + 17 * q^53 - q^55 + 4 * q^57 - 19 * q^59 - 6 * q^61 - 2 * q^63 + 6 * q^65 - 14 * q^67 + q^69 - q^71 - 5 * q^73 - 18 * q^75 + 2 * q^81 - 11 * q^83 - 26 * q^85 - 14 * q^87 + 12 * q^89 - 44 * q^91 - 6 * q^93 - q^95 + 14 * q^97 - 4 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.20649	0.696566	0.348283	−	0.937389i	$-0.386765\pi$
$3$	1.20649	0.696566	0.348283	+	0.937389i	$0.386765\pi$
$4$	0	0
$5$	−0.300070	−0.134196	−0.0670978	−	0.997746i	$-0.521374\pi$
$5$	−0.300070	−0.134196	−0.0670978	+	0.997746i	$0.521374\pi$
$6$	0	0
$7$	4.45862	1.68520	0.842599	−	0.538541i	$-0.181024\pi$
$7$	4.45862	1.68520	0.842599	+	0.538541i	$0.181024\pi$
$8$	0	0
$9$	−1.54439	−0.514795
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−4.45862	−1.23660	−0.618299	−	0.785943i	$-0.712178\pi$
$13$	−4.45862	−1.23660	−0.618299	+	0.785943i	$0.712178\pi$
$14$	0	0
$15$	−0.362031	−0.0934761
$16$	0	0
$17$	−0.906418	−0.219839	−0.109919	−	0.993941i	$-0.535059\pi$
$17$	−0.906418	−0.219839	−0.109919	+	0.993941i	$0.535059\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	5.37927	1.17385
$22$	0	0
$23$	−5.49644	−1.14609	−0.573044	−	0.819525i	$-0.694238\pi$
$23$	−5.49644	−1.14609	−0.573044	+	0.819525i	$0.694238\pi$
$24$	0	0
$25$	−4.90996	−0.981992
$26$	0	0
$27$	−5.48275	−1.05516
$28$	0	0
$29$	−10.0685	−1.86967	−0.934837	−	0.355077i	$-0.884455\pi$
$29$	−10.0685	−1.86967	−0.934837	+	0.355077i	$0.884455\pi$
$30$	0	0
$31$	−9.95506	−1.78798	−0.893991	−	0.448086i	$-0.852106\pi$
$31$	−9.95506	−1.78798	−0.893991	+	0.448086i	$0.852106\pi$
$32$	0	0
$33$	−1.20649	−0.210023
$34$	0	0
$35$	−1.33790	−0.226146
$36$	0	0
$37$	2.59003	0.425798	0.212899	−	0.977074i	$-0.431710\pi$
$37$	2.59003	0.425798	0.212899	+	0.977074i	$0.431710\pi$
$38$	0	0
$39$	−5.37927	−0.861372
$40$	0	0
$41$	−0.952057	−0.148686	−0.0743431	−	0.997233i	$-0.523686\pi$
$41$	−0.952057	−0.148686	−0.0743431	+	0.997233i	$0.523686\pi$
$42$	0	0
$43$	−7.49233	−1.14257	−0.571284	−	0.820752i	$-0.693555\pi$
$43$	−7.49233	−1.14257	−0.571284	+	0.820752i	$0.693555\pi$
$44$	0	0
$45$	0.463425	0.0690833
$46$	0	0
$47$	2.50425	0.365283	0.182642	−	0.983180i	$-0.441535\pi$
$47$	2.50425	0.365283	0.182642	+	0.983180i	$0.441535\pi$
$48$	0	0
$49$	12.8792	1.83989
$50$	0	0
$51$	−1.09358	−0.153132
$52$	0	0
$53$	10.4543	1.43601	0.718007	−	0.696036i	$-0.245054\pi$
$53$	10.4543	1.43601	0.718007	+	0.696036i	$0.245054\pi$
$54$	0	0
$55$	0.300070	0.0404615
$56$	0	0
$57$	−1.20649	−0.159803
$58$	0	0
$59$	7.12003	0.926948	0.463474	−	0.886110i	$-0.346603\pi$
$59$	7.12003	0.926948	0.463474	+	0.886110i	$0.346603\pi$
$60$	0	0
$61$	−3.21061	−0.411076	−0.205538	−	0.978649i	$-0.565894\pi$
$61$	−3.21061	−0.411076	−0.205538	+	0.978649i	$0.565894\pi$
$62$	0	0
$63$	−6.88582	−0.867532
$64$	0	0
$65$	1.33790	0.165946
$66$	0	0
$67$	−12.8594	−1.57102	−0.785512	−	0.618846i	$-0.787600\pi$
$67$	−12.8594	−1.57102	−0.785512	+	0.618846i	$0.787600\pi$
$68$	0	0
$69$	−6.63139	−0.798326
$70$	0	0
$71$	7.93927	0.942219	0.471109	−	0.882075i	$-0.343854\pi$
$71$	7.93927	0.942219	0.471109	+	0.882075i	$0.343854\pi$
$72$	0	0
$73$	9.14576	1.07043	0.535215	−	0.844716i	$-0.320230\pi$
$73$	9.14576	1.07043	0.535215	+	0.844716i	$0.320230\pi$
$74$	0	0
$75$	−5.92381	−0.684022
$76$	0	0
$77$	−4.45862	−0.508106
$78$	0	0
$79$	7.63920	0.859478	0.429739	−	0.902953i	$-0.358606\pi$
$79$	7.63920	0.859478	0.429739	+	0.902953i	$0.358606\pi$
$80$	0	0
$81$	−1.98171	−0.220190
$82$	0	0
$83$	8.60704	0.944746	0.472373	−	0.881399i	$-0.343398\pi$
$83$	8.60704	0.944746	0.472373	+	0.881399i	$0.343398\pi$
$84$	0	0
$85$	0.271989	0.0295014
$86$	0	0
$87$	−12.1475	−1.30235
$88$	0	0
$89$	−2.31158	−0.245027	−0.122514	−	0.992467i	$-0.539096\pi$
$89$	−2.31158	−0.245027	−0.122514	+	0.992467i	$0.539096\pi$
$90$	0	0
$91$	−19.8792	−2.08391
$92$	0	0
$93$	−12.0107	−1.24545
$94$	0	0
$95$	0.300070	0.0307866
$96$	0	0
$97$	−0.355672	−0.0361130	−0.0180565	−	0.999837i	$-0.505748\pi$
$97$	−0.355672	−0.0361130	−0.0180565	+	0.999837i	$0.505748\pi$
$98$	0	0
$99$	1.54439	0.155217
$100$	0	0
$101$	5.62358	0.559567	0.279784	−	0.960063i	$-0.409737\pi$
$101$	5.62358	0.559567	0.279784	+	0.960063i	$0.409737\pi$
$102$	0	0
$103$	−4.74911	−0.467944	−0.233972	−	0.972243i	$-0.575172\pi$
$103$	−4.74911	−0.467944	−0.233972	+	0.972243i	$0.575172\pi$
$104$	0	0
$105$	−1.61416	−0.157526
$106$	0	0
$107$	−11.0545	−1.06868	−0.534339	−	0.845270i	$-0.679439\pi$
$107$	−11.0545	−1.06868	−0.534339	+	0.845270i	$0.679439\pi$
$108$	0	0
$109$	−10.7328	−1.02801	−0.514007	−	0.857786i	$-0.671840\pi$
$109$	−10.7328	−1.02801	−0.514007	+	0.857786i	$0.671840\pi$
$110$	0	0
$111$	3.12483	0.296596
$112$	0	0
$113$	13.1816	1.24002	0.620011	−	0.784593i	$-0.287128\pi$
$113$	13.1816	1.24002	0.620011	+	0.784593i	$0.287128\pi$
$114$	0	0
$115$	1.64932	0.153800
$116$	0	0
$117$	6.88582	0.636595
$118$	0	0
$119$	−4.04137	−0.370472
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.14865	−0.103570
$124$	0	0
$125$	2.97368	0.265974
$126$	0	0
$127$	3.81075	0.338149	0.169075	−	0.985603i	$-0.445922\pi$
$127$	3.81075	0.338149	0.169075	+	0.985603i	$0.445922\pi$
$128$	0	0
$129$	−9.03940	−0.795875
$130$	0	0
$131$	−14.9109	−1.30277	−0.651384	−	0.758748i	$-0.725811\pi$
$131$	−14.9109	−1.30277	−0.651384	+	0.758748i	$0.725811\pi$
$132$	0	0
$133$	−4.45862	−0.386611
$134$	0	0
$135$	1.64521	0.141597
$136$	0	0
$137$	9.46162	0.808360	0.404180	−	0.914679i	$-0.367557\pi$
$137$	9.46162	0.808360	0.404180	+	0.914679i	$0.367557\pi$
$138$	0	0
$139$	3.56589	0.302455	0.151228	−	0.988499i	$-0.451677\pi$
$139$	3.56589	0.302455	0.151228	+	0.988499i	$0.451677\pi$
$140$	0	0
$141$	3.02135	0.254444
$142$	0	0
$143$	4.45862	0.372848
$144$	0	0
$145$	3.02126	0.250902
$146$	0	0
$147$	15.5387	1.28161
$148$	0	0
$149$	15.6675	1.28353	0.641767	−	0.766900i	$-0.278202\pi$
$149$	15.6675	1.28353	0.641767	+	0.766900i	$0.278202\pi$
$150$	0	0
$151$	7.57387	0.616353	0.308177	−	0.951329i	$-0.400281\pi$
$151$	7.57387	0.616353	0.308177	+	0.951329i	$0.400281\pi$
$152$	0	0
$153$	1.39986	0.113172
$154$	0	0
$155$	2.98722	0.239939
$156$	0	0
$157$	1.36969	0.109313	0.0546566	−	0.998505i	$-0.482594\pi$
$157$	1.36969	0.109313	0.0546566	+	0.998505i	$0.482594\pi$
$158$	0	0
$159$	12.6130	1.00028
$160$	0	0
$161$	−24.5065	−1.93138
$162$	0	0
$163$	−5.34336	−0.418524	−0.209262	−	0.977860i	$-0.567106\pi$
$163$	−5.34336	−0.418524	−0.209262	+	0.977860i	$0.567106\pi$
$164$	0	0
$165$	0.362031	0.0281841
$166$	0	0
$167$	2.24395	0.173642	0.0868212	−	0.996224i	$-0.472329\pi$
$167$	2.24395	0.173642	0.0868212	+	0.996224i	$0.472329\pi$
$168$	0	0
$169$	6.87925	0.529173
$170$	0	0
$171$	1.54439	0.118102
$172$	0	0
$173$	13.3776	1.01708	0.508539	−	0.861039i	$-0.330186\pi$
$173$	13.3776	1.01708	0.508539	+	0.861039i	$0.330186\pi$
$174$	0	0
$175$	−21.8916	−1.65485
$176$	0	0
$177$	8.59023	0.645681
$178$	0	0
$179$	16.2310	1.21316	0.606580	−	0.795023i	$-0.292541\pi$
$179$	16.2310	1.21316	0.606580	+	0.795023i	$0.292541\pi$
$180$	0	0
$181$	−9.83857	−0.731295	−0.365648	−	0.930753i	$-0.619152\pi$
$181$	−9.83857	−0.731295	−0.365648	+	0.930753i	$0.619152\pi$
$182$	0	0
$183$	−3.87356	−0.286342
$184$	0	0
$185$	−0.777190	−0.0571401
$186$	0	0
$187$	0.906418	0.0662838
$188$	0	0
$189$	−24.4455	−1.77815
$190$	0	0
$191$	11.3838	0.823701	0.411851	−	0.911251i	$-0.364883\pi$
$191$	11.3838	0.823701	0.411851	+	0.911251i	$0.364883\pi$
$192$	0	0
$193$	13.9724	1.00576	0.502878	−	0.864357i	$-0.332274\pi$
$193$	13.9724	1.00576	0.502878	+	0.864357i	$0.332274\pi$
$194$	0	0
$195$	1.61416	0.115592
$196$	0	0
$197$	3.69954	0.263582	0.131791	−	0.991278i	$-0.457927\pi$
$197$	3.69954	0.263582	0.131791	+	0.991278i	$0.457927\pi$
$198$	0	0
$199$	10.2275	0.725005	0.362503	−	0.931983i	$-0.381922\pi$
$199$	10.2275	0.725005	0.362503	+	0.931983i	$0.381922\pi$
$200$	0	0
$201$	−15.5147	−1.09432
$202$	0	0
$203$	−44.8916	−3.15077
$204$	0	0
$205$	0.285684	0.0199530
$206$	0	0
$207$	8.48863	0.590001
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−10.8546	−0.747263	−0.373631	−	0.927577i	$-0.621888\pi$
$211$	−10.8546	−0.747263	−0.373631	+	0.927577i	$0.621888\pi$
$212$	0	0
$213$	9.57864	0.656318
$214$	0	0
$215$	2.24822	0.153328
$216$	0	0
$217$	−44.3858	−3.01310
$218$	0	0
$219$	11.0343	0.745626
$220$	0	0
$221$	4.04137	0.271852
$222$	0	0
$223$	−19.1770	−1.28419	−0.642094	−	0.766626i	$-0.721934\pi$
$223$	−19.1770	−1.28419	−0.642094	+	0.766626i	$0.721934\pi$
$224$	0	0
$225$	7.58287	0.505525
$226$	0	0
$227$	−20.8606	−1.38457	−0.692284	−	0.721625i	$-0.743395\pi$
$227$	−20.8606	−1.38457	−0.692284	+	0.721625i	$0.743395\pi$
$228$	0	0
$229$	−2.14539	−0.141771	−0.0708856	−	0.997484i	$-0.522583\pi$
$229$	−2.14539	−0.141771	−0.0708856	+	0.997484i	$0.522583\pi$
$230$	0	0
$231$	−5.37927	−0.353930
$232$	0	0
$233$	−6.23935	−0.408753	−0.204377	−	0.978892i	$-0.565517\pi$
$233$	−6.23935	−0.408753	−0.204377	+	0.978892i	$0.565517\pi$
$234$	0	0
$235$	−0.751452	−0.0490194
$236$	0	0
$237$	9.21661	0.598683
$238$	0	0
$239$	−8.17685	−0.528916	−0.264458	−	0.964397i	$-0.585193\pi$
$239$	−8.17685	−0.528916	−0.264458	+	0.964397i	$0.585193\pi$
$240$	0	0
$241$	−23.2629	−1.49849	−0.749247	−	0.662290i	$-0.769585\pi$
$241$	−23.2629	−1.49849	−0.749247	+	0.662290i	$0.769585\pi$
$242$	0	0
$243$	14.0573	0.901778
$244$	0	0
$245$	−3.86468	−0.246905
$246$	0	0
$247$	4.45862	0.283695
$248$	0	0
$249$	10.3843	0.658078
$250$	0	0
$251$	−31.0264	−1.95837	−0.979185	−	0.202971i	$-0.934940\pi$
$251$	−31.0264	−1.95837	−0.979185	+	0.202971i	$0.934940\pi$
$252$	0	0
$253$	5.49644	0.345558
$254$	0	0
$255$	0.328152	0.0205497
$256$	0	0
$257$	4.82242	0.300814	0.150407	−	0.988624i	$-0.451942\pi$
$257$	4.82242	0.300814	0.150407	+	0.988624i	$0.451942\pi$
$258$	0	0
$259$	11.5479	0.717553
$260$	0	0
$261$	15.5497	0.962500
$262$	0	0
$263$	−11.7540	−0.724782	−0.362391	−	0.932026i	$-0.618039\pi$
$263$	−11.7540	−0.724782	−0.362391	+	0.932026i	$0.618039\pi$
$264$	0	0
$265$	−3.13704	−0.192707
$266$	0	0
$267$	−2.78890	−0.170678
$268$	0	0
$269$	15.2976	0.932709	0.466354	−	0.884598i	$-0.345567\pi$
$269$	15.2976	0.932709	0.466354	+	0.884598i	$0.345567\pi$
$270$	0	0
$271$	12.2735	0.745564	0.372782	−	0.927919i	$-0.378404\pi$
$271$	12.2735	0.745564	0.372782	+	0.927919i	$0.378404\pi$
$272$	0	0
$273$	−23.9841	−1.45158
$274$	0	0
$275$	4.90996	0.296082
$276$	0	0
$277$	−23.1030	−1.38813	−0.694063	−	0.719915i	$-0.744181\pi$
$277$	−23.1030	−1.38813	−0.694063	+	0.719915i	$0.744181\pi$
$278$	0	0
$279$	15.3745	0.920445
$280$	0	0
$281$	−0.300280	−0.0179132	−0.00895659	−	0.999960i	$-0.502851\pi$
$281$	−0.300280	−0.0179132	−0.00895659	+	0.999960i	$0.502851\pi$
$282$	0	0
$283$	23.7954	1.41449	0.707245	−	0.706968i	$-0.249938\pi$
$283$	23.7954	1.41449	0.707245	+	0.706968i	$0.249938\pi$
$284$	0	0
$285$	0.362031	0.0214449
$286$	0	0
$287$	−4.24485	−0.250566
$288$	0	0
$289$	−16.1784	−0.951671
$290$	0	0
$291$	−0.429114	−0.0251551
$292$	0	0
$293$	−28.4717	−1.66333	−0.831667	−	0.555274i	$-0.812613\pi$
$293$	−28.4717	−1.66333	−0.831667	+	0.555274i	$0.812613\pi$
$294$	0	0
$295$	−2.13651	−0.124392
$296$	0	0
$297$	5.48275	0.318141
$298$	0	0
$299$	24.5065	1.41725
$300$	0	0
$301$	−33.4054	−1.92545
$302$	0	0
$303$	6.78479	0.389776
$304$	0	0
$305$	0.963408	0.0551646
$306$	0	0
$307$	−26.5049	−1.51271	−0.756357	−	0.654159i	$-0.773023\pi$
$307$	−26.5049	−1.51271	−0.756357	+	0.654159i	$0.773023\pi$
$308$	0	0
$309$	−5.72974	−0.325954
$310$	0	0
$311$	2.43757	0.138222	0.0691109	−	0.997609i	$-0.477984\pi$
$311$	2.43757	0.138222	0.0691109	+	0.997609i	$0.477984\pi$
$312$	0	0
$313$	−31.4171	−1.77580	−0.887899	−	0.460039i	$-0.847835\pi$
$313$	−31.4171	−1.77580	−0.887899	+	0.460039i	$0.847835\pi$
$314$	0	0
$315$	2.06623	0.116419
$316$	0	0
$317$	−0.332720	−0.0186874	−0.00934371	−	0.999956i	$-0.502974\pi$
$317$	−0.332720	−0.0186874	−0.00934371	+	0.999956i	$0.502974\pi$
$318$	0	0
$319$	10.0685	0.563728
$320$	0	0
$321$	−13.3371	−0.744405
$322$	0	0
$323$	0.906418	0.0504344
$324$	0	0
$325$	21.8916	1.21433
$326$	0	0
$327$	−12.9490	−0.716080
$328$	0	0
$329$	11.1655	0.615574
$330$	0	0
$331$	−8.58194	−0.471706	−0.235853	−	0.971789i	$-0.575788\pi$
$331$	−8.58194	−0.471706	−0.235853	+	0.971789i	$0.575788\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	0	0
$335$	3.85872	0.210824
$336$	0	0
$337$	−19.2032	−1.04606	−0.523032	−	0.852313i	$-0.675199\pi$
$337$	−19.2032	−1.04606	−0.523032	+	0.852313i	$0.675199\pi$
$338$	0	0
$339$	15.9035	0.863758
$340$	0	0
$341$	9.95506	0.539097
$342$	0	0
$343$	26.2133	1.41539
$344$	0	0
$345$	1.98988	0.107132
$346$	0	0
$347$	0.170099	0.00913141	0.00456570	−	0.999990i	$-0.498547\pi$
$347$	0.170099	0.00913141	0.00456570	+	0.999990i	$0.498547\pi$
$348$	0	0
$349$	29.4648	1.57722	0.788608	−	0.614897i	$-0.210802\pi$
$349$	29.4648	1.57722	0.788608	+	0.614897i	$0.210802\pi$
$350$	0	0
$351$	24.4455	1.30480
$352$	0	0
$353$	−9.14942	−0.486975	−0.243487	−	0.969904i	$-0.578291\pi$
$353$	−9.14942	−0.486975	−0.243487	+	0.969904i	$0.578291\pi$
$354$	0	0
$355$	−2.38234	−0.126442
$356$	0	0
$357$	−4.87586	−0.258058
$358$	0	0
$359$	26.3556	1.39100	0.695498	−	0.718528i	$-0.255184\pi$
$359$	26.3556	1.39100	0.695498	+	0.718528i	$0.255184\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	1.20649	0.0633242
$364$	0	0
$365$	−2.74437	−0.143647
$366$	0	0
$367$	18.0903	0.944308	0.472154	−	0.881516i	$-0.343477\pi$
$367$	18.0903	0.944308	0.472154	+	0.881516i	$0.343477\pi$
$368$	0	0
$369$	1.47034	0.0765430
$370$	0	0
$371$	46.6119	2.41997
$372$	0	0
$373$	28.4065	1.47083	0.735417	−	0.677615i	$-0.236986\pi$
$373$	28.4065	1.47083	0.735417	+	0.677615i	$0.236986\pi$
$374$	0	0
$375$	3.58772	0.185269
$376$	0	0
$377$	44.8916	2.31203
$378$	0	0
$379$	−35.9509	−1.84667	−0.923336	−	0.383992i	$-0.874549\pi$
$379$	−35.9509	−1.84667	−0.923336	+	0.383992i	$0.874549\pi$
$380$	0	0
$381$	4.59762	0.235543
$382$	0	0
$383$	22.1249	1.13053	0.565265	−	0.824910i	$-0.308774\pi$
$383$	22.1249	1.13053	0.565265	+	0.824910i	$0.308774\pi$
$384$	0	0
$385$	1.33790	0.0681856
$386$	0	0
$387$	11.5710	0.588189
$388$	0	0
$389$	6.88598	0.349133	0.174567	−	0.984645i	$-0.444148\pi$
$389$	6.88598	0.349133	0.174567	+	0.984645i	$0.444148\pi$
$390$	0	0
$391$	4.98207	0.251954
$392$	0	0
$393$	−17.9898	−0.907465
$394$	0	0
$395$	−2.29230	−0.115338
$396$	0	0
$397$	−31.5336	−1.58263	−0.791313	−	0.611411i	$-0.790602\pi$
$397$	−31.5336	−1.58263	−0.791313	+	0.611411i	$0.790602\pi$
$398$	0	0
$399$	−5.37927	−0.269300
$400$	0	0
$401$	−24.1324	−1.20511	−0.602557	−	0.798076i	$-0.705851\pi$
$401$	−24.1324	−1.20511	−0.602557	+	0.798076i	$0.705851\pi$
$402$	0	0
$403$	44.3858	2.21101
$404$	0	0
$405$	0.594653	0.0295485
$406$	0	0
$407$	−2.59003	−0.128383
$408$	0	0
$409$	0.675541	0.0334034	0.0167017	−	0.999861i	$-0.494683\pi$
$409$	0.675541	0.0334034	0.0167017	+	0.999861i	$0.494683\pi$
$410$	0	0
$411$	11.4153	0.563077
$412$	0	0
$413$	31.7455	1.56209
$414$	0	0
$415$	−2.58272	−0.126781
$416$	0	0
$417$	4.30221	0.210680
$418$	0	0
$419$	−0.903841	−0.0441555	−0.0220778	−	0.999756i	$-0.507028\pi$
$419$	−0.903841	−0.0441555	−0.0220778	+	0.999756i	$0.507028\pi$
$420$	0	0
$421$	4.48665	0.218666	0.109333	−	0.994005i	$-0.465129\pi$
$421$	4.48665	0.218666	0.109333	+	0.994005i	$0.465129\pi$
$422$	0	0
$423$	−3.86754	−0.188046
$424$	0	0
$425$	4.45047	0.215880
$426$	0	0
$427$	−14.3149	−0.692744
$428$	0	0
$429$	5.37927	0.259713
$430$	0	0
$431$	−38.7229	−1.86522	−0.932609	−	0.360889i	$-0.882473\pi$
$431$	−38.7229	−1.86522	−0.932609	+	0.360889i	$0.882473\pi$
$432$	0	0
$433$	1.96850	0.0946000	0.0473000	−	0.998881i	$-0.484938\pi$
$433$	1.96850	0.0946000	0.0473000	+	0.998881i	$0.484938\pi$
$434$	0	0
$435$	3.64511	0.174770
$436$	0	0
$437$	5.49644	0.262931
$438$	0	0
$439$	−11.3410	−0.541275	−0.270637	−	0.962681i	$-0.587234\pi$
$439$	−11.3410	−0.541275	−0.270637	+	0.962681i	$0.587234\pi$
$440$	0	0
$441$	−19.8905	−0.947168
$442$	0	0
$443$	−17.2122	−0.817775	−0.408888	−	0.912585i	$-0.634083\pi$
$443$	−17.2122	−0.817775	−0.408888	+	0.912585i	$0.634083\pi$
$444$	0	0
$445$	0.693638	0.0328816
$446$	0	0
$447$	18.9027	0.894066
$448$	0	0
$449$	−20.4686	−0.965972	−0.482986	−	0.875628i	$-0.660448\pi$
$449$	−20.4686	−0.965972	−0.482986	+	0.875628i	$0.660448\pi$
$450$	0	0
$451$	0.952057	0.0448306
$452$	0	0
$453$	9.13779	0.429331
$454$	0	0
$455$	5.96517	0.279652
$456$	0	0
$457$	36.7427	1.71875	0.859376	−	0.511345i	$-0.170853\pi$
$457$	36.7427	1.71875	0.859376	+	0.511345i	$0.170853\pi$
$458$	0	0
$459$	4.96966	0.231964
$460$	0	0
$461$	12.1826	0.567398	0.283699	−	0.958913i	$-0.408438\pi$
$461$	12.1826	0.567398	0.283699	+	0.958913i	$0.408438\pi$
$462$	0	0
$463$	10.6636	0.495579	0.247790	−	0.968814i	$-0.420296\pi$
$463$	10.6636	0.495579	0.247790	+	0.968814i	$0.420296\pi$
$464$	0	0
$465$	3.60404	0.167133
$466$	0	0
$467$	−24.8680	−1.15075	−0.575377	−	0.817888i	$-0.695145\pi$
$467$	−24.8680	−1.15075	−0.575377	+	0.817888i	$0.695145\pi$
$468$	0	0
$469$	−57.3351	−2.64749
$470$	0	0
$471$	1.65251	0.0761438
$472$	0	0
$473$	7.49233	0.344497
$474$	0	0
$475$	4.90996	0.225284
$476$	0	0
$477$	−16.1455	−0.739254
$478$	0	0
$479$	39.6232	1.81043	0.905216	−	0.424952i	$-0.139709\pi$
$479$	39.6232	1.81043	0.905216	+	0.424952i	$0.139709\pi$
$480$	0	0
$481$	−11.5479	−0.526540
$482$	0	0
$483$	−29.5668	−1.34534
$484$	0	0
$485$	0.106727	0.00484620
$486$	0	0
$487$	−21.0909	−0.955718	−0.477859	−	0.878437i	$-0.658587\pi$
$487$	−21.0909	−0.955718	−0.477859	+	0.878437i	$0.658587\pi$
$488$	0	0
$489$	−6.44670	−0.291530
$490$	0	0
$491$	−14.0574	−0.634401	−0.317200	−	0.948359i	$-0.602743\pi$
$491$	−14.0574	−0.634401	−0.317200	+	0.948359i	$0.602743\pi$
$492$	0	0
$493$	9.12627	0.411027
$494$	0	0
$495$	−0.463425	−0.0208294
$496$	0	0
$497$	35.3982	1.58782
$498$	0	0
$499$	−42.0830	−1.88389	−0.941946	−	0.335763i	$-0.891006\pi$
$499$	−42.0830	−1.88389	−0.941946	+	0.335763i	$0.891006\pi$
$500$	0	0
$501$	2.70730	0.120953
$502$	0	0
$503$	−27.8209	−1.24047	−0.620235	−	0.784416i	$-0.712963\pi$
$503$	−27.8209	−1.24047	−0.620235	+	0.784416i	$0.712963\pi$
$504$	0	0
$505$	−1.68747	−0.0750914
$506$	0	0
$507$	8.29973	0.368604
$508$	0	0
$509$	21.5185	0.953791	0.476896	−	0.878960i	$-0.341762\pi$
$509$	21.5185	0.953791	0.476896	+	0.878960i	$0.341762\pi$
$510$	0	0
$511$	40.7774	1.80389
$512$	0	0
$513$	5.48275	0.242069
$514$	0	0
$515$	1.42507	0.0627959
$516$	0	0
$517$	−2.50425	−0.110137
$518$	0	0
$519$	16.1399	0.708462
$520$	0	0
$521$	−15.0058	−0.657415	−0.328708	−	0.944432i	$-0.606613\pi$
$521$	−15.0058	−0.657415	−0.328708	+	0.944432i	$0.606613\pi$
$522$	0	0
$523$	13.9657	0.610678	0.305339	−	0.952244i	$-0.401230\pi$
$523$	13.9657	0.610678	0.305339	+	0.952244i	$0.401230\pi$
$524$	0	0
$525$	−26.4120	−1.15271
$526$	0	0
$527$	9.02344	0.393067
$528$	0	0
$529$	7.21089	0.313517
$530$	0	0
$531$	−10.9961	−0.477189
$532$	0	0
$533$	4.24485	0.183865
$534$	0	0
$535$	3.31712	0.143412
$536$	0	0
$537$	19.5825	0.845046
$538$	0	0
$539$	−12.8792	−0.554749
$540$	0	0
$541$	22.2996	0.958735	0.479368	−	0.877614i	$-0.340866\pi$
$541$	22.2996	0.958735	0.479368	+	0.877614i	$0.340866\pi$
$542$	0	0
$543$	−11.8701	−0.509396
$544$	0	0
$545$	3.22059	0.137955
$546$	0	0
$547$	−26.4102	−1.12922	−0.564608	−	0.825359i	$-0.690973\pi$
$547$	−26.4102	−1.12922	−0.564608	+	0.825359i	$0.690973\pi$
$548$	0	0
$549$	4.95842	0.211620
$550$	0	0
$551$	10.0685	0.428933
$552$	0	0
$553$	34.0603	1.44839
$554$	0	0
$555$	−0.937670	−0.0398019
$556$	0	0
$557$	23.9318	1.01402	0.507012	−	0.861939i	$-0.330750\pi$
$557$	23.9318	1.01402	0.507012	+	0.861939i	$0.330750\pi$
$558$	0	0
$559$	33.4054	1.41290
$560$	0	0
$561$	1.09358	0.0461711
$562$	0	0
$563$	−7.95737	−0.335363	−0.167682	−	0.985841i	$-0.553628\pi$
$563$	−7.95737	−0.335363	−0.167682	+	0.985841i	$0.553628\pi$
$564$	0	0
$565$	−3.95541	−0.166406
$566$	0	0
$567$	−8.83569	−0.371064
$568$	0	0
$569$	33.1500	1.38972	0.694861	−	0.719144i	$-0.255466\pi$
$569$	33.1500	1.38972	0.694861	+	0.719144i	$0.255466\pi$
$570$	0	0
$571$	−3.92261	−0.164156	−0.0820781	−	0.996626i	$-0.526156\pi$
$571$	−3.92261	−0.164156	−0.0820781	+	0.996626i	$0.526156\pi$
$572$	0	0
$573$	13.7344	0.573763
$574$	0	0
$575$	26.9873	1.12545
$576$	0	0
$577$	−46.2687	−1.92619	−0.963096	−	0.269157i	$-0.913255\pi$
$577$	−46.2687	−1.92619	−0.963096	+	0.269157i	$0.913255\pi$
$578$	0	0
$579$	16.8576	0.700576
$580$	0	0
$581$	38.3755	1.59208
$582$	0	0
$583$	−10.4543	−0.432975
$584$	0	0
$585$	−2.06623	−0.0854282
$586$	0	0
$587$	−28.6664	−1.18319	−0.591595	−	0.806235i	$-0.701501\pi$
$587$	−28.6664	−1.18319	−0.591595	+	0.806235i	$0.701501\pi$
$588$	0	0
$589$	9.95506	0.410191
$590$	0	0
$591$	4.46346	0.183602
$592$	0	0
$593$	−9.07562	−0.372691	−0.186346	−	0.982484i	$-0.559664\pi$
$593$	−9.07562	−0.372691	−0.186346	+	0.982484i	$0.559664\pi$
$594$	0	0
$595$	1.21269	0.0497156
$596$	0	0
$597$	12.3393	0.505014
$598$	0	0
$599$	5.01718	0.204997	0.102498	−	0.994733i	$-0.467316\pi$
$599$	5.01718	0.204997	0.102498	+	0.994733i	$0.467316\pi$
$600$	0	0
$601$	17.1029	0.697643	0.348821	−	0.937189i	$-0.386582\pi$
$601$	17.1029	0.697643	0.348821	+	0.937189i	$0.386582\pi$
$602$	0	0
$603$	19.8599	0.808756
$604$	0	0
$605$	−0.300070	−0.0121996
$606$	0	0
$607$	−14.3143	−0.580999	−0.290499	−	0.956875i	$-0.593821\pi$
$607$	−14.3143	−0.580999	−0.290499	+	0.956875i	$0.593821\pi$
$608$	0	0
$609$	−54.1612	−2.19472
$610$	0	0
$611$	−11.1655	−0.451708
$612$	0	0
$613$	−43.1025	−1.74090	−0.870448	−	0.492261i	$-0.836171\pi$
$613$	−43.1025	−1.74090	−0.870448	+	0.492261i	$0.836171\pi$
$614$	0	0
$615$	0.344674	0.0138986
$616$	0	0
$617$	−2.43849	−0.0981699	−0.0490850	−	0.998795i	$-0.515631\pi$
$617$	−2.43849	−0.0981699	−0.0490850	+	0.998795i	$0.515631\pi$
$618$	0	0
$619$	−32.0264	−1.28725	−0.643624	−	0.765342i	$-0.722570\pi$
$619$	−32.0264	−1.28725	−0.643624	+	0.765342i	$0.722570\pi$
$620$	0	0
$621$	30.1356	1.20930
$622$	0	0
$623$	−10.3065	−0.412920
$624$	0	0
$625$	23.6575	0.946299
$626$	0	0
$627$	1.20649	0.0481825
$628$	0	0
$629$	−2.34764	−0.0936067
$630$	0	0
$631$	31.1109	1.23850	0.619252	−	0.785193i	$-0.287436\pi$
$631$	31.1109	1.23850	0.619252	+	0.785193i	$0.287436\pi$
$632$	0	0
$633$	−13.0960	−0.520518
$634$	0	0
$635$	−1.14349	−0.0453781
$636$	0	0
$637$	−57.4236	−2.27521
$638$	0	0
$639$	−12.2613	−0.485050
$640$	0	0
$641$	29.4226	1.16212	0.581062	−	0.813859i	$-0.302637\pi$
$641$	29.4226	1.16212	0.581062	+	0.813859i	$0.302637\pi$
$642$	0	0
$643$	21.7469	0.857616	0.428808	−	0.903396i	$-0.358934\pi$
$643$	21.7469	0.857616	0.428808	+	0.903396i	$0.358934\pi$
$644$	0	0
$645$	2.71246	0.106803
$646$	0	0
$647$	35.6511	1.40159	0.700795	−	0.713362i	$-0.252829\pi$
$647$	35.6511	1.40159	0.700795	+	0.713362i	$0.252829\pi$
$648$	0	0
$649$	−7.12003	−0.279485
$650$	0	0
$651$	−53.5509	−2.09883
$652$	0	0
$653$	−4.06119	−0.158927	−0.0794633	−	0.996838i	$-0.525321\pi$
$653$	−4.06119	−0.158927	−0.0794633	+	0.996838i	$0.525321\pi$
$654$	0	0
$655$	4.47431	0.174826
$656$	0	0
$657$	−14.1246	−0.551053
$658$	0	0
$659$	27.1510	1.05765	0.528827	−	0.848730i	$-0.322632\pi$
$659$	27.1510	1.05765	0.528827	+	0.848730i	$0.322632\pi$
$660$	0	0
$661$	−4.57876	−0.178093	−0.0890466	−	0.996027i	$-0.528382\pi$
$661$	−4.57876	−0.178093	−0.0890466	+	0.996027i	$0.528382\pi$
$662$	0	0
$663$	4.87586	0.189363
$664$	0	0
$665$	1.33790	0.0518815
$666$	0	0
$667$	55.3410	2.14281
$668$	0	0
$669$	−23.1368	−0.894522
$670$	0	0
$671$	3.21061	0.123944
$672$	0	0
$673$	49.8525	1.92167	0.960837	−	0.277115i	$-0.0893783\pi$
$673$	49.8525	1.92167	0.960837	+	0.277115i	$0.0893783\pi$
$674$	0	0
$675$	26.9201	1.03615
$676$	0	0
$677$	−9.80316	−0.376766	−0.188383	−	0.982096i	$-0.560325\pi$
$677$	−9.80316	−0.376766	−0.188383	+	0.982096i	$0.560325\pi$
$678$	0	0
$679$	−1.58580	−0.0608575
$680$	0	0
$681$	−25.1681	−0.964443
$682$	0	0
$683$	−45.2165	−1.73016	−0.865080	−	0.501634i	$-0.832732\pi$
$683$	−45.2165	−1.73016	−0.865080	+	0.501634i	$0.832732\pi$
$684$	0	0
$685$	−2.83915	−0.108478
$686$	0	0
$687$	−2.58839	−0.0987531
$688$	0	0
$689$	−46.6119	−1.77577
$690$	0	0
$691$	−20.4086	−0.776379	−0.388189	−	0.921580i	$-0.626899\pi$
$691$	−20.4086	−0.776379	−0.388189	+	0.921580i	$0.626899\pi$
$692$	0	0
$693$	6.88582	0.261571
$694$	0	0
$695$	−1.07002	−0.0405881
$696$	0	0
$697$	0.862961	0.0326870
$698$	0	0
$699$	−7.52770	−0.284724
$700$	0	0
$701$	0.444299	0.0167810	0.00839048	−	0.999965i	$-0.497329\pi$
$701$	0.444299	0.0167810	0.00839048	+	0.999965i	$0.497329\pi$
$702$	0	0
$703$	−2.59003	−0.0976847
$704$	0	0
$705$	−0.906618	−0.0341452
$706$	0	0
$707$	25.0734	0.942982
$708$	0	0
$709$	8.48823	0.318782	0.159391	−	0.987216i	$-0.449047\pi$
$709$	8.48823	0.318782	0.159391	+	0.987216i	$0.449047\pi$
$710$	0	0
$711$	−11.7979	−0.442455
$712$	0	0
$713$	54.7174	2.04918
$714$	0	0
$715$	−1.33790	−0.0500346
$716$	0	0
$717$	−9.86527	−0.368425
$718$	0	0
$719$	−21.0532	−0.785153	−0.392577	−	0.919719i	$-0.628416\pi$
$719$	−21.0532	−0.785153	−0.392577	+	0.919719i	$0.628416\pi$
$720$	0	0
$721$	−21.1744	−0.788578
$722$	0	0
$723$	−28.0664	−1.04380
$724$	0	0
$725$	49.4359	1.83600
$726$	0	0
$727$	36.8288	1.36591	0.682953	−	0.730463i	$-0.260696\pi$
$727$	36.8288	1.36591	0.682953	+	0.730463i	$0.260696\pi$
$728$	0	0
$729$	22.9051	0.848339
$730$	0	0
$731$	6.79118	0.251181
$732$	0	0
$733$	−3.67185	−0.135623	−0.0678114	−	0.997698i	$-0.521602\pi$
$733$	−3.67185	−0.135623	−0.0678114	+	0.997698i	$0.521602\pi$
$734$	0	0
$735$	−4.66269	−0.171986
$736$	0	0
$737$	12.8594	0.473682
$738$	0	0
$739$	−11.1902	−0.411637	−0.205818	−	0.978590i	$-0.565986\pi$
$739$	−11.1902	−0.411637	−0.205818	+	0.978590i	$0.565986\pi$
$740$	0	0
$741$	5.37927	0.197612
$742$	0	0
$743$	−44.3032	−1.62533	−0.812664	−	0.582732i	$-0.801984\pi$
$743$	−44.3032	−1.62533	−0.812664	+	0.582732i	$0.801984\pi$
$744$	0	0
$745$	−4.70136	−0.172245
$746$	0	0
$747$	−13.2926	−0.486351
$748$	0	0
$749$	−49.2877	−1.80093
$750$	0	0
$751$	−51.5836	−1.88231	−0.941156	−	0.337973i	$-0.890259\pi$
$751$	−51.5836	−1.88231	−0.941156	+	0.337973i	$0.890259\pi$
$752$	0	0
$753$	−37.4330	−1.36413
$754$	0	0
$755$	−2.27269	−0.0827118
$756$	0	0
$757$	10.0094	0.363799	0.181899	−	0.983317i	$-0.441775\pi$
$757$	10.0094	0.363799	0.181899	+	0.983317i	$0.441775\pi$
$758$	0	0
$759$	6.63139	0.240704
$760$	0	0
$761$	−28.5459	−1.03479	−0.517395	−	0.855747i	$-0.673098\pi$
$761$	−28.5459	−1.03479	−0.517395	+	0.855747i	$0.673098\pi$
$762$	0	0
$763$	−47.8534	−1.73241
$764$	0	0
$765$	−0.420056	−0.0151872
$766$	0	0
$767$	−31.7455	−1.14626
$768$	0	0
$769$	31.4295	1.13338	0.566688	−	0.823933i	$-0.308225\pi$
$769$	31.4295	1.13338	0.566688	+	0.823933i	$0.308225\pi$
$770$	0	0
$771$	5.81819	0.209537
$772$	0	0
$773$	26.9313	0.968651	0.484326	−	0.874888i	$-0.339065\pi$
$773$	26.9313	0.968651	0.484326	+	0.874888i	$0.339065\pi$
$774$	0	0
$775$	48.8789	1.75578
$776$	0	0
$777$	13.9324	0.499823
$778$	0	0
$779$	0.952057	0.0341110
$780$	0	0
$781$	−7.93927	−0.284090
$782$	0	0
$783$	55.2031	1.97280
$784$	0	0
$785$	−0.411003	−0.0146693
$786$	0	0
$787$	−0.399831	−0.0142524	−0.00712621	−	0.999975i	$-0.502268\pi$
$787$	−0.399831	−0.0142524	−0.00712621	+	0.999975i	$0.502268\pi$
$788$	0	0
$789$	−14.1810	−0.504859
$790$	0	0
$791$	58.7718	2.08968
$792$	0	0
$793$	14.3149	0.508335
$794$	0	0
$795$	−3.78480	−0.134233
$796$	0	0
$797$	−43.7666	−1.55029	−0.775146	−	0.631783i	$-0.782324\pi$
$797$	−43.7666	−1.55029	−0.775146	+	0.631783i	$0.782324\pi$
$798$	0	0
$799$	−2.26990	−0.0803033
$800$	0	0
$801$	3.56998	0.126139
$802$	0	0
$803$	−9.14576	−0.322747
$804$	0	0
$805$	7.35368	0.259183
$806$	0	0
$807$	18.4563	0.649693
$808$	0	0
$809$	23.7756	0.835907	0.417954	−	0.908468i	$-0.362747\pi$
$809$	23.7756	0.835907	0.417954	+	0.908468i	$0.362747\pi$
$810$	0	0
$811$	−3.18273	−0.111761	−0.0558804	−	0.998437i	$-0.517797\pi$
$811$	−3.18273	−0.111761	−0.0558804	+	0.998437i	$0.517797\pi$
$812$	0	0
$813$	14.8079	0.519335
$814$	0	0
$815$	1.60338	0.0561641
$816$	0	0
$817$	7.49233	0.262123
$818$	0	0
$819$	30.7012	1.07279
$820$	0	0
$821$	24.2241	0.845426	0.422713	−	0.906264i	$-0.361078\pi$
$821$	24.2241	0.845426	0.422713	+	0.906264i	$0.361078\pi$
$822$	0	0
$823$	−31.3149	−1.09157	−0.545784	−	0.837926i	$-0.683768\pi$
$823$	−31.3149	−1.09157	−0.545784	+	0.837926i	$0.683768\pi$
$824$	0	0
$825$	5.92381	0.206240
$826$	0	0
$827$	45.5689	1.58459	0.792293	−	0.610141i	$-0.208887\pi$
$827$	45.5689	1.58459	0.792293	+	0.610141i	$0.208887\pi$
$828$	0	0
$829$	29.9517	1.04026	0.520132	−	0.854086i	$-0.325883\pi$
$829$	29.9517	1.04026	0.520132	+	0.854086i	$0.325883\pi$
$830$	0	0
$831$	−27.8735	−0.966921
$832$	0	0
$833$	−11.6740	−0.404479
$834$	0	0
$835$	−0.673344	−0.0233020
$836$	0	0
$837$	54.5811	1.88660
$838$	0	0
$839$	−52.8466	−1.82447	−0.912233	−	0.409671i	$-0.865643\pi$
$839$	−52.8466	−1.82447	−0.912233	+	0.409671i	$0.865643\pi$
$840$	0	0
$841$	72.3747	2.49568
$842$	0	0
$843$	−0.362284	−0.0124777
$844$	0	0
$845$	−2.06426	−0.0710127
$846$	0	0
$847$	4.45862	0.153200
$848$	0	0
$849$	28.7089	0.985286
$850$	0	0
$851$	−14.2359	−0.488001
$852$	0	0
$853$	−34.4140	−1.17831	−0.589156	−	0.808019i	$-0.700540\pi$
$853$	−34.4140	−1.17831	−0.589156	+	0.808019i	$0.700540\pi$
$854$	0	0
$855$	−0.463425	−0.0158488
$856$	0	0
$857$	24.3068	0.830305	0.415152	−	0.909752i	$-0.363728\pi$
$857$	24.3068	0.830305	0.415152	+	0.909752i	$0.363728\pi$
$858$	0	0
$859$	35.1368	1.19885	0.599427	−	0.800430i	$-0.295395\pi$
$859$	35.1368	1.19885	0.599427	+	0.800430i	$0.295395\pi$
$860$	0	0
$861$	−5.12137	−0.174536
$862$	0	0
$863$	−26.1477	−0.890076	−0.445038	−	0.895512i	$-0.646810\pi$
$863$	−26.1477	−0.890076	−0.445038	+	0.895512i	$0.646810\pi$
$864$	0	0
$865$	−4.01421	−0.136487
$866$	0	0
$867$	−19.5191	−0.662902
$868$	0	0
$869$	−7.63920	−0.259142
$870$	0	0
$871$	57.3351	1.94272
$872$	0	0
$873$	0.549294	0.0185908
$874$	0	0
$875$	13.2585	0.448220
$876$	0	0
$877$	−27.0021	−0.911795	−0.455897	−	0.890032i	$-0.650682\pi$
$877$	−27.0021	−0.911795	−0.455897	+	0.890032i	$0.650682\pi$
$878$	0	0
$879$	−34.3508	−1.15862
$880$	0	0
$881$	43.1698	1.45443	0.727213	−	0.686412i	$-0.240815\pi$
$881$	43.1698	1.45443	0.727213	+	0.686412i	$0.240815\pi$
$882$	0	0
$883$	2.16553	0.0728760	0.0364380	−	0.999336i	$-0.488399\pi$
$883$	2.16553	0.0728760	0.0364380	+	0.999336i	$0.488399\pi$
$884$	0	0
$885$	−2.57767	−0.0866475
$886$	0	0
$887$	−24.6181	−0.826596	−0.413298	−	0.910596i	$-0.635623\pi$
$887$	−24.6181	−0.826596	−0.413298	+	0.910596i	$0.635623\pi$
$888$	0	0
$889$	16.9907	0.569848
$890$	0	0
$891$	1.98171	0.0663898
$892$	0	0
$893$	−2.50425	−0.0838017
$894$	0	0
$895$	−4.87043	−0.162801
$896$	0	0
$897$	29.5668	0.987208
$898$	0	0
$899$	100.233	3.34294
$900$	0	0
$901$	−9.47600	−0.315691
$902$	0	0
$903$	−40.3032	−1.34121
$904$	0	0
$905$	2.95226	0.0981366
$906$	0	0
$907$	−16.3510	−0.542926	−0.271463	−	0.962449i	$-0.587507\pi$
$907$	−16.3510	−0.542926	−0.271463	+	0.962449i	$0.587507\pi$
$908$	0	0
$909$	−8.68498	−0.288063
$910$	0	0
$911$	−14.8694	−0.492645	−0.246323	−	0.969188i	$-0.579222\pi$
$911$	−14.8694	−0.492645	−0.246323	+	0.969188i	$0.579222\pi$
$912$	0	0
$913$	−8.60704	−0.284852
$914$	0	0
$915$	1.16234	0.0384258
$916$	0	0
$917$	−66.4818	−2.19542
$918$	0	0
$919$	52.5052	1.73199	0.865993	−	0.500056i	$-0.166687\pi$
$919$	52.5052	1.73199	0.865993	+	0.500056i	$0.166687\pi$
$920$	0	0
$921$	−31.9778	−1.05371
$922$	0	0
$923$	−35.3982	−1.16514
$924$	0	0
$925$	−12.7169	−0.418130
$926$	0	0
$927$	7.33446	0.240895
$928$	0	0
$929$	31.4411	1.03155	0.515775	−	0.856724i	$-0.327504\pi$
$929$	31.4411	1.03155	0.515775	+	0.856724i	$0.327504\pi$
$930$	0	0
$931$	−12.8792	−0.422100
$932$	0	0
$933$	2.94089	0.0962806
$934$	0	0
$935$	−0.271989	−0.00889499
$936$	0	0
$937$	20.4998	0.669698	0.334849	−	0.942272i	$-0.391315\pi$
$937$	20.4998	0.669698	0.334849	+	0.942272i	$0.391315\pi$
$938$	0	0
$939$	−37.9043	−1.23696
$940$	0	0
$941$	−3.93869	−0.128398	−0.0641988	−	0.997937i	$-0.520449\pi$
$941$	−3.93869	−0.128398	−0.0641988	+	0.997937i	$0.520449\pi$
$942$	0	0
$943$	5.23293	0.170408
$944$	0	0
$945$	7.33536	0.238619
$946$	0	0
$947$	5.83812	0.189714	0.0948568	−	0.995491i	$-0.469761\pi$
$947$	5.83812	0.189714	0.0948568	+	0.995491i	$0.469761\pi$
$948$	0	0
$949$	−40.7774	−1.32369
$950$	0	0
$951$	−0.401423	−0.0130170
$952$	0	0
$953$	28.3109	0.917081	0.458541	−	0.888673i	$-0.348372\pi$
$953$	28.3109	0.917081	0.458541	+	0.888673i	$0.348372\pi$
$954$	0	0
$955$	−3.41593	−0.110537
$956$	0	0
$957$	12.1475	0.392674
$958$	0	0
$959$	42.1857	1.36225
$960$	0	0
$961$	68.1032	2.19688
$962$	0	0
$963$	17.0724	0.550150
$964$	0	0
$965$	−4.19271	−0.134968
$966$	0	0
$967$	22.3056	0.717300	0.358650	−	0.933472i	$-0.383237\pi$
$967$	22.3056	0.717300	0.358650	+	0.933472i	$0.383237\pi$
$968$	0	0
$969$	1.09358	0.0351309
$970$	0	0
$971$	−2.13336	−0.0684628	−0.0342314	−	0.999414i	$-0.510898\pi$
$971$	−2.13336	−0.0684628	−0.0342314	+	0.999414i	$0.510898\pi$
$972$	0	0
$973$	15.8989	0.509697
$974$	0	0
$975$	26.4120	0.845860
$976$	0	0
$977$	−36.1059	−1.15513	−0.577566	−	0.816344i	$-0.695997\pi$
$977$	−36.1059	−1.15513	−0.577566	+	0.816344i	$0.695997\pi$
$978$	0	0
$979$	2.31158	0.0738785
$980$	0	0
$981$	16.5756	0.529217
$982$	0	0
$983$	23.2216	0.740655	0.370327	−	0.928901i	$-0.379246\pi$
$983$	23.2216	0.740655	0.370327	+	0.928901i	$0.379246\pi$
$984$	0	0
$985$	−1.11012	−0.0353715
$986$	0	0
$987$	13.4710	0.428788
$988$	0	0
$989$	41.1811	1.30948
$990$	0	0
$991$	−55.9783	−1.77821	−0.889104	−	0.457705i	$-0.848672\pi$
$991$	−55.9783	−1.77821	−0.889104	+	0.457705i	$0.848672\pi$
$992$	0	0
$993$	−10.3540	−0.328575
$994$	0	0
$995$	−3.06896	−0.0972925
$996$	0	0
$997$	−45.5766	−1.44343	−0.721713	−	0.692192i	$-0.756645\pi$
$997$	−45.5766	−1.44343	−0.721713	+	0.692192i	$0.756645\pi$
$998$	0	0
$999$	−14.2005	−0.449283

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.20649 q^{3} -0.300070 q^{5} +4.45862 q^{7} -1.54439 q^{9} -1.00000 q^{11} -4.45862 q^{13} -0.362031 q^{15} -0.906418 q^{17} -1.00000 q^{19} +5.37927 q^{21} -5.49644 q^{23} -4.90996 q^{25} -5.48275 q^{27} -10.0685 q^{29} -9.95506 q^{31} -1.20649 q^{33} -1.33790 q^{35} +2.59003 q^{37} -5.37927 q^{39} -0.952057 q^{41} -7.49233 q^{43} +0.463425 q^{45} +2.50425 q^{47} +12.8792 q^{49} -1.09358 q^{51} +10.4543 q^{53} +0.300070 q^{55} -1.20649 q^{57} +7.12003 q^{59} -3.21061 q^{61} -6.88582 q^{63} +1.33790 q^{65} -12.8594 q^{67} -6.63139 q^{69} +7.93927 q^{71} +9.14576 q^{73} -5.92381 q^{75} -4.45862 q^{77} +7.63920 q^{79} -1.98171 q^{81} +8.60704 q^{83} +0.271989 q^{85} -12.1475 q^{87} -2.31158 q^{89} -19.8792 q^{91} -12.0107 q^{93} +0.300070 q^{95} -0.355672 q^{97} +1.54439 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} + q^{5} + 4 q^{9} - 6 q^{11} - 7 q^{15} + 3 q^{17} - 6 q^{19} + 7 q^{21} - 7 q^{23} + 3 q^{25} - 13 q^{27} - 4 q^{29} - 7 q^{31} + 4 q^{33} - 6 q^{35} - 2 q^{37} - 7 q^{39} + 7 q^{41} - 21 q^{43}+ \cdots - 4 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 + q^5 + 4 * q^9 - 6 * q^11 - 7 * q^15 + 3 * q^17 - 6 * q^19 + 7 * q^21 - 7 * q^23 + 3 * q^25 - 13 * q^27 - 4 * q^29 - 7 * q^31 + 4 * q^33 - 6 * q^35 - 2 * q^37 - 7 * q^39 + 7 * q^41 - 21 * q^43 + 11 * q^45 - 16 * q^47 + 2 * q^49 - 15 * q^51 + 17 * q^53 - q^55 + 4 * q^57 - 19 * q^59 - 6 * q^61 - 2 * q^63 + 6 * q^65 - 14 * q^67 + q^69 - q^71 - 5 * q^73 - 18 * q^75 + 2 * q^81 - 11 * q^83 - 26 * q^85 - 14 * q^87 + 12 * q^89 - 44 * q^91 - 6 * q^93 - q^95 + 14 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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