LMFDB - Embedded newform 3344.2.a.y.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.57500224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} + 12x^{3} + 11x^{2} - 18x + 2 $ x^6 - 2x^5 - 7x^4 + 12x^3 + 11x^2 - 18*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1672)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.121186$ 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)

$f(q)$	$=$	$q+0.878814 q^{3} -1.66503 q^{5} -1.34977 q^{7} -2.22769 q^{9} +O(q^{10})$	$q+0.878814 q^{3} -1.66503 q^{5} -1.34977 q^{7} -2.22769 q^{9} -1.00000 q^{11} +0.817610 q^{13} -1.46325 q^{15} +3.28589 q^{17} +1.00000 q^{19} -1.18620 q^{21} -3.16380 q^{23} -2.22769 q^{25} -4.59416 q^{27} -5.77623 q^{29} +8.86436 q^{31} -0.878814 q^{33} +2.24740 q^{35} +11.8579 q^{37} +0.718527 q^{39} +4.14292 q^{41} +9.25641 q^{43} +3.70916 q^{45} +1.15469 q^{47} -5.17813 q^{49} +2.88768 q^{51} -9.30068 q^{53} +1.66503 q^{55} +0.878814 q^{57} +9.09741 q^{59} -8.45177 q^{61} +3.00686 q^{63} -1.36134 q^{65} +1.83722 q^{67} -2.78040 q^{69} +4.39228 q^{71} +7.79105 q^{73} -1.95772 q^{75} +1.34977 q^{77} -0.830516 q^{79} +2.64564 q^{81} +11.5154 q^{83} -5.47109 q^{85} -5.07623 q^{87} -3.06603 q^{89} -1.10358 q^{91} +7.79013 q^{93} -1.66503 q^{95} +15.1880 q^{97} +2.22769 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9	$ 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9} - 6 q^{11} + 2 q^{13} - 10 q^{17} + 6 q^{19} + 10 q^{21} + 14 q^{23} + 2 q^{25} + 16 q^{27} + 6 q^{29} - 4 q^{31} - 4 q^{33} - 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41} + 26 q^{43} - 2 q^{45} + 24 q^{47} + 20 q^{49} + 14 q^{51} - 8 q^{53} + 4 q^{55} + 4 q^{57} + 4 q^{59} - 6 q^{61} + 40 q^{63} - 34 q^{65} + 44 q^{67} + 8 q^{69} + 16 q^{71} + 12 q^{73} + 28 q^{75} - 4 q^{77} - 6 q^{79} + 10 q^{81} + 28 q^{83} - 10 q^{85} + 24 q^{87} + 64 q^{91} - 14 q^{93} - 4 q^{95} + 4 q^{97} - 2 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9 - 6 * q^11 + 2 * q^13 - 10 * q^17 + 6 * q^19 + 10 * q^21 + 14 * q^23 + 2 * q^25 + 16 * q^27 + 6 * q^29 - 4 * q^31 - 4 * q^33 - 2 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^41 + 26 * q^43 - 2 * q^45 + 24 * q^47 + 20 * q^49 + 14 * q^51 - 8 * q^53 + 4 * q^55 + 4 * q^57 + 4 * q^59 - 6 * q^61 + 40 * q^63 - 34 * q^65 + 44 * q^67 + 8 * q^69 + 16 * q^71 + 12 * q^73 + 28 * q^75 - 4 * q^77 - 6 * q^79 + 10 * q^81 + 28 * q^83 - 10 * q^85 + 24 * q^87 + 64 * q^91 - 14 * q^93 - 4 * q^95 + 4 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.878814	0.507384	0.253692	−	0.967285i	$-0.418355\pi$
$3$	0.878814	0.507384	0.253692	+	0.967285i	$0.418355\pi$
$4$	0	0
$5$	−1.66503	−0.744623	−0.372311	−	0.928108i	$-0.621435\pi$
$5$	−1.66503	−0.744623	−0.372311	+	0.928108i	$0.621435\pi$
$6$	0	0
$7$	−1.34977	−0.510164	−0.255082	−	0.966919i	$-0.582103\pi$
$7$	−1.34977	−0.510164	−0.255082	+	0.966919i	$0.582103\pi$
$8$	0	0
$9$	−2.22769	−0.742562
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.817610	0.226764	0.113382	−	0.993551i	$-0.463832\pi$
$13$	0.817610	0.226764	0.113382	+	0.993551i	$0.463832\pi$
$14$	0	0
$15$	−1.46325	−0.377809
$16$	0	0
$17$	3.28589	0.796945	0.398472	−	0.917180i	$-0.369540\pi$
$17$	3.28589	0.796945	0.398472	+	0.917180i	$0.369540\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.18620	−0.258849
$22$	0	0
$23$	−3.16380	−0.659699	−0.329849	−	0.944034i	$-0.606998\pi$
$23$	−3.16380	−0.659699	−0.329849	+	0.944034i	$0.606998\pi$
$24$	0	0
$25$	−2.22769	−0.445537
$26$	0	0
$27$	−4.59416	−0.884147
$28$	0	0
$29$	−5.77623	−1.07262	−0.536309	−	0.844022i	$-0.680182\pi$
$29$	−5.77623	−1.07262	−0.536309	+	0.844022i	$0.680182\pi$
$30$	0	0
$31$	8.86436	1.59209	0.796043	−	0.605240i	$-0.206923\pi$
$31$	8.86436	1.59209	0.796043	+	0.605240i	$0.206923\pi$
$32$	0	0
$33$	−0.878814	−0.152982
$34$	0	0
$35$	2.24740	0.379880
$36$	0	0
$37$	11.8579	1.94943	0.974717	−	0.223441i	$-0.0717291\pi$
$37$	11.8579	1.94943	0.974717	+	0.223441i	$0.0717291\pi$
$38$	0	0
$39$	0.718527	0.115056
$40$	0	0
$41$	4.14292	0.647016	0.323508	−	0.946225i	$-0.395138\pi$
$41$	4.14292	0.647016	0.323508	+	0.946225i	$0.395138\pi$
$42$	0	0
$43$	9.25641	1.41159	0.705794	−	0.708417i	$-0.250590\pi$
$43$	9.25641	1.41159	0.705794	+	0.708417i	$0.250590\pi$
$44$	0	0
$45$	3.70916	0.552928
$46$	0	0
$47$	1.15469	0.168429	0.0842144	−	0.996448i	$-0.473162\pi$
$47$	1.15469	0.168429	0.0842144	+	0.996448i	$0.473162\pi$
$48$	0	0
$49$	−5.17813	−0.739732
$50$	0	0
$51$	2.88768	0.404357
$52$	0	0
$53$	−9.30068	−1.27755	−0.638773	−	0.769395i	$-0.720558\pi$
$53$	−9.30068	−1.27755	−0.638773	+	0.769395i	$0.720558\pi$
$54$	0	0
$55$	1.66503	0.224512
$56$	0	0
$57$	0.878814	0.116402
$58$	0	0
$59$	9.09741	1.18438	0.592191	−	0.805798i	$-0.298263\pi$
$59$	9.09741	1.18438	0.592191	+	0.805798i	$0.298263\pi$
$60$	0	0
$61$	−8.45177	−1.08214	−0.541069	−	0.840978i	$-0.681980\pi$
$61$	−8.45177	−1.08214	−0.541069	+	0.840978i	$0.681980\pi$
$62$	0	0
$63$	3.00686	0.378829
$64$	0	0
$65$	−1.36134	−0.168854
$66$	0	0
$67$	1.83722	0.224452	0.112226	−	0.993683i	$-0.464202\pi$
$67$	1.83722	0.224452	0.112226	+	0.993683i	$0.464202\pi$
$68$	0	0
$69$	−2.78040	−0.334720
$70$	0	0
$71$	4.39228	0.521268	0.260634	−	0.965438i	$-0.416068\pi$
$71$	4.39228	0.521268	0.260634	+	0.965438i	$0.416068\pi$
$72$	0	0
$73$	7.79105	0.911874	0.455937	−	0.890012i	$-0.349304\pi$
$73$	7.79105	0.911874	0.455937	+	0.890012i	$0.349304\pi$
$74$	0	0
$75$	−1.95772	−0.226058
$76$	0	0
$77$	1.34977	0.153820
$78$	0	0
$79$	−0.830516	−0.0934404	−0.0467202	−	0.998908i	$-0.514877\pi$
$79$	−0.830516	−0.0934404	−0.0467202	+	0.998908i	$0.514877\pi$
$80$	0	0
$81$	2.64564	0.293960
$82$	0	0
$83$	11.5154	1.26398	0.631988	−	0.774978i	$-0.282239\pi$
$83$	11.5154	1.26398	0.631988	+	0.774978i	$0.282239\pi$
$84$	0	0
$85$	−5.47109	−0.593423
$86$	0	0
$87$	−5.07623	−0.544229
$88$	0	0
$89$	−3.06603	−0.324998	−0.162499	−	0.986709i	$-0.551956\pi$
$89$	−3.06603	−0.324998	−0.162499	+	0.986709i	$0.551956\pi$
$90$	0	0
$91$	−1.10358	−0.115687
$92$	0	0
$93$	7.79013	0.807798
$94$	0	0
$95$	−1.66503	−0.170828
$96$	0	0
$97$	15.1880	1.54211	0.771054	−	0.636770i	$-0.219730\pi$
$97$	15.1880	1.54211	0.771054	+	0.636770i	$0.219730\pi$
$98$	0	0
$99$	2.22769	0.223891
$100$	0	0
$101$	−13.8749	−1.38060	−0.690300	−	0.723523i	$-0.742522\pi$
$101$	−13.8749	−1.38060	−0.690300	+	0.723523i	$0.742522\pi$
$102$	0	0
$103$	1.16029	0.114326	0.0571632	−	0.998365i	$-0.481794\pi$
$103$	1.16029	0.114326	0.0571632	+	0.998365i	$0.481794\pi$
$104$	0	0
$105$	1.97505	0.192745
$106$	0	0
$107$	17.9608	1.73634	0.868168	−	0.496271i	$-0.165298\pi$
$107$	17.9608	1.73634	0.868168	+	0.496271i	$0.165298\pi$
$108$	0	0
$109$	13.3323	1.27700	0.638502	−	0.769620i	$-0.279554\pi$
$109$	13.3323	1.27700	0.638502	+	0.769620i	$0.279554\pi$
$110$	0	0
$111$	10.4209	0.989111
$112$	0	0
$113$	−12.8550	−1.20930	−0.604650	−	0.796491i	$-0.706687\pi$
$113$	−12.8550	−1.20930	−0.604650	+	0.796491i	$0.706687\pi$
$114$	0	0
$115$	5.26782	0.491227
$116$	0	0
$117$	−1.82138	−0.168386
$118$	0	0
$119$	−4.43518	−0.406573
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	3.64086	0.328285
$124$	0	0
$125$	12.0343	1.07638
$126$	0	0
$127$	16.2706	1.44378	0.721892	−	0.692006i	$-0.243273\pi$
$127$	16.2706	1.44378	0.721892	+	0.692006i	$0.243273\pi$
$128$	0	0
$129$	8.13466	0.716217
$130$	0	0
$131$	21.9345	1.91643	0.958215	−	0.286049i	$-0.0923420\pi$
$131$	21.9345	1.91643	0.958215	+	0.286049i	$0.0923420\pi$
$132$	0	0
$133$	−1.34977	−0.117040
$134$	0	0
$135$	7.64941	0.658356
$136$	0	0
$137$	15.1194	1.29174	0.645868	−	0.763449i	$-0.276496\pi$
$137$	15.1194	1.29174	0.645868	+	0.763449i	$0.276496\pi$
$138$	0	0
$139$	7.90172	0.670215	0.335108	−	0.942180i	$-0.391227\pi$
$139$	7.90172	0.670215	0.335108	+	0.942180i	$0.391227\pi$
$140$	0	0
$141$	1.01476	0.0854580
$142$	0	0
$143$	−0.817610	−0.0683720
$144$	0	0
$145$	9.61757	0.798696
$146$	0	0
$147$	−4.55061	−0.375328
$148$	0	0
$149$	−5.45246	−0.446683	−0.223342	−	0.974740i	$-0.571697\pi$
$149$	−5.45246	−0.446683	−0.223342	+	0.974740i	$0.571697\pi$
$150$	0	0
$151$	−1.05605	−0.0859404	−0.0429702	−	0.999076i	$-0.513682\pi$
$151$	−1.05605	−0.0859404	−0.0429702	+	0.999076i	$0.513682\pi$
$152$	0	0
$153$	−7.31992	−0.591781
$154$	0	0
$155$	−14.7594	−1.18550
$156$	0	0
$157$	−10.5592	−0.842713	−0.421357	−	0.906895i	$-0.638446\pi$
$157$	−10.5592	−0.842713	−0.421357	+	0.906895i	$0.638446\pi$
$158$	0	0
$159$	−8.17357	−0.648206
$160$	0	0
$161$	4.27040	0.336555
$162$	0	0
$163$	−21.0902	−1.65191	−0.825954	−	0.563737i	$-0.809363\pi$
$163$	−21.0902	−1.65191	−0.825954	+	0.563737i	$0.809363\pi$
$164$	0	0
$165$	1.46325	0.113914
$166$	0	0
$167$	0.799351	0.0618556	0.0309278	−	0.999522i	$-0.490154\pi$
$167$	0.799351	0.0618556	0.0309278	+	0.999522i	$0.490154\pi$
$168$	0	0
$169$	−12.3315	−0.948578
$170$	0	0
$171$	−2.22769	−0.170355
$172$	0	0
$173$	−1.01748	−0.0773579	−0.0386789	−	0.999252i	$-0.512315\pi$
$173$	−1.01748	−0.0773579	−0.0386789	+	0.999252i	$0.512315\pi$
$174$	0	0
$175$	3.00686	0.227297
$176$	0	0
$177$	7.99493	0.600936
$178$	0	0
$179$	17.5417	1.31113	0.655563	−	0.755141i	$-0.272431\pi$
$179$	17.5417	1.31113	0.655563	+	0.755141i	$0.272431\pi$
$180$	0	0
$181$	6.27163	0.466167	0.233083	−	0.972457i	$-0.425118\pi$
$181$	6.27163	0.466167	0.233083	+	0.972457i	$0.425118\pi$
$182$	0	0
$183$	−7.42754	−0.549059
$184$	0	0
$185$	−19.7438	−1.45159
$186$	0	0
$187$	−3.28589	−0.240288
$188$	0	0
$189$	6.20106	0.451060
$190$	0	0
$191$	−3.75077	−0.271396	−0.135698	−	0.990750i	$-0.543328\pi$
$191$	−3.75077	−0.271396	−0.135698	+	0.990750i	$0.543328\pi$
$192$	0	0
$193$	7.51523	0.540958	0.270479	−	0.962726i	$-0.412818\pi$
$193$	7.51523	0.540958	0.270479	+	0.962726i	$0.412818\pi$
$194$	0	0
$195$	−1.19637	−0.0856736
$196$	0	0
$197$	−14.6280	−1.04220	−0.521102	−	0.853494i	$-0.674479\pi$
$197$	−14.6280	−1.04220	−0.521102	+	0.853494i	$0.674479\pi$
$198$	0	0
$199$	−14.1024	−0.999695	−0.499848	−	0.866113i	$-0.666611\pi$
$199$	−14.1024	−0.999695	−0.499848	+	0.866113i	$0.666611\pi$
$200$	0	0
$201$	1.61457	0.113883
$202$	0	0
$203$	7.79657	0.547212
$204$	0	0
$205$	−6.89808	−0.481783
$206$	0	0
$207$	7.04796	0.489867
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−25.3684	−1.74643	−0.873215	−	0.487335i	$-0.837969\pi$
$211$	−25.3684	−1.74643	−0.873215	+	0.487335i	$0.837969\pi$
$212$	0	0
$213$	3.86000	0.264483
$214$	0	0
$215$	−15.4122	−1.05110
$216$	0	0
$217$	−11.9648	−0.812225
$218$	0	0
$219$	6.84689	0.462670
$220$	0	0
$221$	2.68657	0.180718
$222$	0	0
$223$	−1.82632	−0.122299	−0.0611497	−	0.998129i	$-0.519477\pi$
$223$	−1.82632	−0.122299	−0.0611497	+	0.998129i	$0.519477\pi$
$224$	0	0
$225$	4.96258	0.330839
$226$	0	0
$227$	−8.64597	−0.573853	−0.286926	−	0.957953i	$-0.592634\pi$
$227$	−8.64597	−0.573853	−0.286926	+	0.957953i	$0.592634\pi$
$228$	0	0
$229$	5.13756	0.339499	0.169750	−	0.985487i	$-0.445704\pi$
$229$	5.13756	0.339499	0.169750	+	0.985487i	$0.445704\pi$
$230$	0	0
$231$	1.18620	0.0780459
$232$	0	0
$233$	−9.85540	−0.645649	−0.322824	−	0.946459i	$-0.604632\pi$
$233$	−9.85540	−0.645649	−0.322824	+	0.946459i	$0.604632\pi$
$234$	0	0
$235$	−1.92259	−0.125416
$236$	0	0
$237$	−0.729869	−0.0474101
$238$	0	0
$239$	1.77816	0.115019	0.0575097	−	0.998345i	$-0.481684\pi$
$239$	1.77816	0.115019	0.0575097	+	0.998345i	$0.481684\pi$
$240$	0	0
$241$	21.9545	1.41421	0.707106	−	0.707108i	$-0.250001\pi$
$241$	21.9545	1.41421	0.707106	+	0.707108i	$0.250001\pi$
$242$	0	0
$243$	16.1075	1.03330
$244$	0	0
$245$	8.62172	0.550822
$246$	0	0
$247$	0.817610	0.0520233
$248$	0	0
$249$	10.1199	0.641321
$250$	0	0
$251$	29.9550	1.89074	0.945371	−	0.325996i	$-0.105699\pi$
$251$	29.9550	1.89074	0.945371	+	0.325996i	$0.105699\pi$
$252$	0	0
$253$	3.16380	0.198907
$254$	0	0
$255$	−4.80807	−0.301093
$256$	0	0
$257$	11.8876	0.741531	0.370766	−	0.928727i	$-0.379095\pi$
$257$	11.8876	0.741531	0.370766	+	0.928727i	$0.379095\pi$
$258$	0	0
$259$	−16.0055	−0.994532
$260$	0	0
$261$	12.8676	0.796486
$262$	0	0
$263$	−11.8429	−0.730267	−0.365133	−	0.930955i	$-0.618977\pi$
$263$	−11.8429	−0.730267	−0.365133	+	0.930955i	$0.618977\pi$
$264$	0	0
$265$	15.4859	0.951290
$266$	0	0
$267$	−2.69447	−0.164899
$268$	0	0
$269$	31.3995	1.91446	0.957232	−	0.289322i	$-0.0934299\pi$
$269$	31.3995	1.91446	0.957232	+	0.289322i	$0.0934299\pi$
$270$	0	0
$271$	−21.7997	−1.32424	−0.662120	−	0.749398i	$-0.730343\pi$
$271$	−21.7997	−1.32424	−0.662120	+	0.749398i	$0.730343\pi$
$272$	0	0
$273$	−0.969845	−0.0586977
$274$	0	0
$275$	2.22769	0.134334
$276$	0	0
$277$	17.5190	1.05262	0.526308	−	0.850294i	$-0.323576\pi$
$277$	17.5190	1.05262	0.526308	+	0.850294i	$0.323576\pi$
$278$	0	0
$279$	−19.7470	−1.18222
$280$	0	0
$281$	−30.9711	−1.84758	−0.923790	−	0.382899i	$-0.874926\pi$
$281$	−30.9711	−1.84758	−0.923790	+	0.382899i	$0.874926\pi$
$282$	0	0
$283$	7.26977	0.432143	0.216072	−	0.976378i	$-0.430676\pi$
$283$	7.26977	0.432143	0.216072	+	0.976378i	$0.430676\pi$
$284$	0	0
$285$	−1.46325	−0.0866754
$286$	0	0
$287$	−5.59199	−0.330085
$288$	0	0
$289$	−6.20295	−0.364879
$290$	0	0
$291$	13.3474	0.782440
$292$	0	0
$293$	9.73955	0.568990	0.284495	−	0.958677i	$-0.408174\pi$
$293$	9.73955	0.568990	0.284495	+	0.958677i	$0.408174\pi$
$294$	0	0
$295$	−15.1474	−0.881918
$296$	0	0
$297$	4.59416	0.266580
$298$	0	0
$299$	−2.58676	−0.149596
$300$	0	0
$301$	−12.4940	−0.720142
$302$	0	0
$303$	−12.1934	−0.700494
$304$	0	0
$305$	14.0724	0.805785
$306$	0	0
$307$	28.4507	1.62376	0.811882	−	0.583821i	$-0.198443\pi$
$307$	28.4507	1.62376	0.811882	+	0.583821i	$0.198443\pi$
$308$	0	0
$309$	1.01968	0.0580074
$310$	0	0
$311$	−8.57536	−0.486264	−0.243132	−	0.969993i	$-0.578175\pi$
$311$	−8.57536	−0.486264	−0.243132	+	0.969993i	$0.578175\pi$
$312$	0	0
$313$	−31.5726	−1.78459	−0.892296	−	0.451452i	$-0.850906\pi$
$313$	−31.5726	−1.78459	−0.892296	+	0.451452i	$0.850906\pi$
$314$	0	0
$315$	−5.00650	−0.282084
$316$	0	0
$317$	−22.2389	−1.24906	−0.624532	−	0.780999i	$-0.714710\pi$
$317$	−22.2389	−1.24906	−0.624532	+	0.780999i	$0.714710\pi$
$318$	0	0
$319$	5.77623	0.323407
$320$	0	0
$321$	15.7842	0.880988
$322$	0	0
$323$	3.28589	0.182832
$324$	0	0
$325$	−1.82138	−0.101032
$326$	0	0
$327$	11.7166	0.647931
$328$	0	0
$329$	−1.55856	−0.0859263
$330$	0	0
$331$	24.4913	1.34617	0.673083	−	0.739567i	$-0.264970\pi$
$331$	24.4913	1.34617	0.673083	+	0.739567i	$0.264970\pi$
$332$	0	0
$333$	−26.4158	−1.44758
$334$	0	0
$335$	−3.05901	−0.167132
$336$	0	0
$337$	21.3405	1.16249	0.581245	−	0.813729i	$-0.302566\pi$
$337$	21.3405	1.16249	0.581245	+	0.813729i	$0.302566\pi$
$338$	0	0
$339$	−11.2972	−0.613579
$340$	0	0
$341$	−8.86436	−0.480032
$342$	0	0
$343$	16.4376	0.887549
$344$	0	0
$345$	4.62944	0.249240
$346$	0	0
$347$	−18.3518	−0.985177	−0.492588	−	0.870262i	$-0.663949\pi$
$347$	−18.3518	−0.985177	−0.492588	+	0.870262i	$0.663949\pi$
$348$	0	0
$349$	26.1569	1.40015	0.700074	−	0.714071i	$-0.253150\pi$
$349$	26.1569	1.40015	0.700074	+	0.714071i	$0.253150\pi$
$350$	0	0
$351$	−3.75623	−0.200493
$352$	0	0
$353$	16.3613	0.870825	0.435413	−	0.900231i	$-0.356603\pi$
$353$	16.3613	0.870825	0.435413	+	0.900231i	$0.356603\pi$
$354$	0	0
$355$	−7.31326	−0.388148
$356$	0	0
$357$	−3.89770	−0.206288
$358$	0	0
$359$	−22.7523	−1.20082	−0.600410	−	0.799693i	$-0.704996\pi$
$359$	−22.7523	−1.20082	−0.600410	+	0.799693i	$0.704996\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.878814	0.0461258
$364$	0	0
$365$	−12.9723	−0.679002
$366$	0	0
$367$	−9.12886	−0.476523	−0.238261	−	0.971201i	$-0.576577\pi$
$367$	−9.12886	−0.476523	−0.238261	+	0.971201i	$0.576577\pi$
$368$	0	0
$369$	−9.22913	−0.480450
$370$	0	0
$371$	12.5538	0.651759
$372$	0	0
$373$	−19.8511	−1.02785	−0.513926	−	0.857835i	$-0.671809\pi$
$373$	−19.8511	−1.02785	−0.513926	+	0.857835i	$0.671809\pi$
$374$	0	0
$375$	10.5759	0.546137
$376$	0	0
$377$	−4.72270	−0.243231
$378$	0	0
$379$	−13.3337	−0.684906	−0.342453	−	0.939535i	$-0.611258\pi$
$379$	−13.3337	−0.684906	−0.342453	+	0.939535i	$0.611258\pi$
$380$	0	0
$381$	14.2989	0.732552
$382$	0	0
$383$	−13.9719	−0.713933	−0.356966	−	0.934117i	$-0.616189\pi$
$383$	−13.9719	−0.713933	−0.356966	+	0.934117i	$0.616189\pi$
$384$	0	0
$385$	−2.24740	−0.114538
$386$	0	0
$387$	−20.6204	−1.04819
$388$	0	0
$389$	−4.65375	−0.235955	−0.117977	−	0.993016i	$-0.537641\pi$
$389$	−4.65375	−0.235955	−0.117977	+	0.993016i	$0.537641\pi$
$390$	0	0
$391$	−10.3959	−0.525744
$392$	0	0
$393$	19.2764	0.972365
$394$	0	0
$395$	1.38283	0.0695778
$396$	0	0
$397$	5.85558	0.293883	0.146942	−	0.989145i	$-0.453057\pi$
$397$	5.85558	0.293883	0.146942	+	0.989145i	$0.453057\pi$
$398$	0	0
$399$	−1.18620	−0.0593840
$400$	0	0
$401$	−23.8011	−1.18857	−0.594286	−	0.804254i	$-0.702565\pi$
$401$	−23.8011	−1.18857	−0.594286	+	0.804254i	$0.702565\pi$
$402$	0	0
$403$	7.24759	0.361028
$404$	0	0
$405$	−4.40506	−0.218889
$406$	0	0
$407$	−11.8579	−0.587777
$408$	0	0
$409$	3.93836	0.194739	0.0973696	−	0.995248i	$-0.468957\pi$
$409$	3.93836	0.194739	0.0973696	+	0.995248i	$0.468957\pi$
$410$	0	0
$411$	13.2871	0.655406
$412$	0	0
$413$	−12.2794	−0.604229
$414$	0	0
$415$	−19.1734	−0.941185
$416$	0	0
$417$	6.94414	0.340056
$418$	0	0
$419$	−12.6606	−0.618509	−0.309255	−	0.950979i	$-0.600080\pi$
$419$	−12.6606	−0.618509	−0.309255	+	0.950979i	$0.600080\pi$
$420$	0	0
$421$	35.1684	1.71400	0.857000	−	0.515316i	$-0.172325\pi$
$421$	35.1684	1.71400	0.857000	+	0.515316i	$0.172325\pi$
$422$	0	0
$423$	−2.57228	−0.125069
$424$	0	0
$425$	−7.31992	−0.355068
$426$	0	0
$427$	11.4079	0.552068
$428$	0	0
$429$	−0.718527	−0.0346908
$430$	0	0
$431$	−35.9339	−1.73088	−0.865438	−	0.501016i	$-0.832960\pi$
$431$	−35.9339	−1.73088	−0.865438	+	0.501016i	$0.832960\pi$
$432$	0	0
$433$	35.6228	1.71192	0.855962	−	0.517039i	$-0.172966\pi$
$433$	35.6228	1.71192	0.855962	+	0.517039i	$0.172966\pi$
$434$	0	0
$435$	8.45206	0.405245
$436$	0	0
$437$	−3.16380	−0.151345
$438$	0	0
$439$	21.6735	1.03442	0.517210	−	0.855858i	$-0.326970\pi$
$439$	21.6735	1.03442	0.517210	+	0.855858i	$0.326970\pi$
$440$	0	0
$441$	11.5352	0.549297
$442$	0	0
$443$	38.3147	1.82039	0.910194	−	0.414182i	$-0.135932\pi$
$443$	38.3147	1.82039	0.910194	+	0.414182i	$0.135932\pi$
$444$	0	0
$445$	5.10502	0.242001
$446$	0	0
$447$	−4.79170	−0.226640
$448$	0	0
$449$	19.6768	0.928605	0.464303	−	0.885677i	$-0.346305\pi$
$449$	19.6768	0.928605	0.464303	+	0.885677i	$0.346305\pi$
$450$	0	0
$451$	−4.14292	−0.195083
$452$	0	0
$453$	−0.928074	−0.0436047
$454$	0	0
$455$	1.83750	0.0861431
$456$	0	0
$457$	−17.6713	−0.826629	−0.413315	−	0.910588i	$-0.635629\pi$
$457$	−17.6713	−0.826629	−0.413315	+	0.910588i	$0.635629\pi$
$458$	0	0
$459$	−15.0959	−0.704617
$460$	0	0
$461$	−32.7870	−1.52704	−0.763522	−	0.645782i	$-0.776532\pi$
$461$	−32.7870	−1.52704	−0.763522	+	0.645782i	$0.776532\pi$
$462$	0	0
$463$	28.7920	1.33808	0.669040	−	0.743227i	$-0.266706\pi$
$463$	28.7920	1.33808	0.669040	+	0.743227i	$0.266706\pi$
$464$	0	0
$465$	−12.9708	−0.601505
$466$	0	0
$467$	−14.0001	−0.647849	−0.323924	−	0.946083i	$-0.605002\pi$
$467$	−14.0001	−0.647849	−0.323924	+	0.946083i	$0.605002\pi$
$468$	0	0
$469$	−2.47981	−0.114507
$470$	0	0
$471$	−9.27955	−0.427579
$472$	0	0
$473$	−9.25641	−0.425610
$474$	0	0
$475$	−2.22769	−0.102213
$476$	0	0
$477$	20.7190	0.948658
$478$	0	0
$479$	2.64958	0.121062	0.0605312	−	0.998166i	$-0.480721\pi$
$479$	2.64958	0.121062	0.0605312	+	0.998166i	$0.480721\pi$
$480$	0	0
$481$	9.69517	0.442062
$482$	0	0
$483$	3.75289	0.170762
$484$	0	0
$485$	−25.2884	−1.14829
$486$	0	0
$487$	24.0961	1.09190	0.545949	−	0.837819i	$-0.316169\pi$
$487$	24.0961	1.09190	0.545949	+	0.837819i	$0.316169\pi$
$488$	0	0
$489$	−18.5343	−0.838151
$490$	0	0
$491$	−39.1524	−1.76692	−0.883462	−	0.468503i	$-0.844794\pi$
$491$	−39.1524	−1.76692	−0.883462	+	0.468503i	$0.844794\pi$
$492$	0	0
$493$	−18.9800	−0.854818
$494$	0	0
$495$	−3.70916	−0.166714
$496$	0	0
$497$	−5.92856	−0.265932
$498$	0	0
$499$	13.7406	0.615116	0.307558	−	0.951529i	$-0.400488\pi$
$499$	13.7406	0.615116	0.307558	+	0.951529i	$0.400488\pi$
$500$	0	0
$501$	0.702481	0.0313845
$502$	0	0
$503$	−12.4371	−0.554542	−0.277271	−	0.960792i	$-0.589430\pi$
$503$	−12.4371	−0.554542	−0.277271	+	0.960792i	$0.589430\pi$
$504$	0	0
$505$	23.1020	1.02803
$506$	0	0
$507$	−10.8371	−0.481293
$508$	0	0
$509$	30.3295	1.34433	0.672166	−	0.740401i	$-0.265364\pi$
$509$	30.3295	1.34433	0.672166	+	0.740401i	$0.265364\pi$
$510$	0	0
$511$	−10.5161	−0.465205
$512$	0	0
$513$	−4.59416	−0.202837
$514$	0	0
$515$	−1.93191	−0.0851301
$516$	0	0
$517$	−1.15469	−0.0507832
$518$	0	0
$519$	−0.894179	−0.0392501
$520$	0	0
$521$	13.0422	0.571390	0.285695	−	0.958321i	$-0.407776\pi$
$521$	13.0422	0.571390	0.285695	+	0.958321i	$0.407776\pi$
$522$	0	0
$523$	27.7304	1.21257	0.606284	−	0.795248i	$-0.292660\pi$
$523$	27.7304	1.21257	0.606284	+	0.795248i	$0.292660\pi$
$524$	0	0
$525$	2.64247	0.115327
$526$	0	0
$527$	29.1273	1.26880
$528$	0	0
$529$	−12.9903	−0.564797
$530$	0	0
$531$	−20.2662	−0.879477
$532$	0	0
$533$	3.38730	0.146720
$534$	0	0
$535$	−29.9052	−1.29291
$536$	0	0
$537$	15.4159	0.665244
$538$	0	0
$539$	5.17813	0.223038
$540$	0	0
$541$	−19.6754	−0.845911	−0.422956	−	0.906150i	$-0.639007\pi$
$541$	−19.6754	−0.845911	−0.422956	+	0.906150i	$0.639007\pi$
$542$	0	0
$543$	5.51160	0.236525
$544$	0	0
$545$	−22.1987	−0.950886
$546$	0	0
$547$	−2.39756	−0.102512	−0.0512562	−	0.998686i	$-0.516323\pi$
$547$	−2.39756	−0.102512	−0.0512562	+	0.998686i	$0.516323\pi$
$548$	0	0
$549$	18.8279	0.803555
$550$	0	0
$551$	−5.77623	−0.246076
$552$	0	0
$553$	1.12100	0.0476699
$554$	0	0
$555$	−17.3511	−0.736515
$556$	0	0
$557$	43.3233	1.83567	0.917834	−	0.396963i	$-0.129936\pi$
$557$	43.3233	1.83567	0.917834	+	0.396963i	$0.129936\pi$
$558$	0	0
$559$	7.56813	0.320098
$560$	0	0
$561$	−2.88768	−0.121918
$562$	0	0
$563$	−24.0022	−1.01157	−0.505786	−	0.862659i	$-0.668797\pi$
$563$	−24.0022	−1.01157	−0.505786	+	0.862659i	$0.668797\pi$
$564$	0	0
$565$	21.4040	0.900472
$566$	0	0
$567$	−3.57100	−0.149968
$568$	0	0
$569$	15.2311	0.638519	0.319260	−	0.947667i	$-0.396566\pi$
$569$	15.2311	0.638519	0.319260	+	0.947667i	$0.396566\pi$
$570$	0	0
$571$	−25.6443	−1.07318	−0.536589	−	0.843843i	$-0.680288\pi$
$571$	−25.6443	−1.07318	−0.536589	+	0.843843i	$0.680288\pi$
$572$	0	0
$573$	−3.29623	−0.137702
$574$	0	0
$575$	7.04796	0.293920
$576$	0	0
$577$	13.6270	0.567301	0.283650	−	0.958928i	$-0.408455\pi$
$577$	13.6270	0.567301	0.283650	+	0.958928i	$0.408455\pi$
$578$	0	0
$579$	6.60449	0.274473
$580$	0	0
$581$	−15.5431	−0.644835
$582$	0	0
$583$	9.30068	0.385195
$584$	0	0
$585$	3.03264	0.125384
$586$	0	0
$587$	28.0144	1.15628	0.578139	−	0.815938i	$-0.303779\pi$
$587$	28.0144	1.15628	0.578139	+	0.815938i	$0.303779\pi$
$588$	0	0
$589$	8.86436	0.365250
$590$	0	0
$591$	−12.8553	−0.528797
$592$	0	0
$593$	−30.0774	−1.23513	−0.617566	−	0.786519i	$-0.711881\pi$
$593$	−30.0774	−1.23513	−0.617566	+	0.786519i	$0.711881\pi$
$594$	0	0
$595$	7.38470	0.302743
$596$	0	0
$597$	−12.3934	−0.507229
$598$	0	0
$599$	35.9805	1.47012	0.735062	−	0.678000i	$-0.237153\pi$
$599$	35.9805	1.47012	0.735062	+	0.678000i	$0.237153\pi$
$600$	0	0
$601$	−23.1308	−0.943526	−0.471763	−	0.881725i	$-0.656382\pi$
$601$	−23.1308	−0.943526	−0.471763	+	0.881725i	$0.656382\pi$
$602$	0	0
$603$	−4.09274	−0.166669
$604$	0	0
$605$	−1.66503	−0.0676930
$606$	0	0
$607$	−27.6325	−1.12157	−0.560784	−	0.827962i	$-0.689500\pi$
$607$	−27.6325	−1.12157	−0.560784	+	0.827962i	$0.689500\pi$
$608$	0	0
$609$	6.85173	0.277646
$610$	0	0
$611$	0.944085	0.0381936
$612$	0	0
$613$	−0.515759	−0.0208313	−0.0104156	−	0.999946i	$-0.503315\pi$
$613$	−0.515759	−0.0208313	−0.0104156	+	0.999946i	$0.503315\pi$
$614$	0	0
$615$	−6.06213	−0.244449
$616$	0	0
$617$	−30.7387	−1.23750	−0.618748	−	0.785590i	$-0.712360\pi$
$617$	−30.7387	−1.23750	−0.618748	+	0.785590i	$0.712360\pi$
$618$	0	0
$619$	20.6137	0.828536	0.414268	−	0.910155i	$-0.364038\pi$
$619$	20.6137	0.828536	0.414268	+	0.910155i	$0.364038\pi$
$620$	0	0
$621$	14.5350	0.583271
$622$	0	0
$623$	4.13843	0.165803
$624$	0	0
$625$	−8.89899	−0.355960
$626$	0	0
$627$	−0.878814	−0.0350965
$628$	0	0
$629$	38.9639	1.55359
$630$	0	0
$631$	8.69161	0.346008	0.173004	−	0.984921i	$-0.444653\pi$
$631$	8.69161	0.346008	0.173004	+	0.984921i	$0.444653\pi$
$632$	0	0
$633$	−22.2941	−0.886110
$634$	0	0
$635$	−27.0910	−1.07507
$636$	0	0
$637$	−4.23369	−0.167745
$638$	0	0
$639$	−9.78461	−0.387073
$640$	0	0
$641$	−32.5723	−1.28653	−0.643265	−	0.765644i	$-0.722420\pi$
$641$	−32.5723	−1.28653	−0.643265	+	0.765644i	$0.722420\pi$
$642$	0	0
$643$	16.1815	0.638137	0.319069	−	0.947732i	$-0.396630\pi$
$643$	16.1815	0.638137	0.319069	+	0.947732i	$0.396630\pi$
$644$	0	0
$645$	−13.5444	−0.533311
$646$	0	0
$647$	34.0634	1.33917	0.669585	−	0.742735i	$-0.266472\pi$
$647$	34.0634	1.33917	0.669585	+	0.742735i	$0.266472\pi$
$648$	0	0
$649$	−9.09741	−0.357105
$650$	0	0
$651$	−10.5149	−0.412110
$652$	0	0
$653$	−32.8869	−1.28696	−0.643482	−	0.765461i	$-0.722511\pi$
$653$	−32.8869	−1.28696	−0.643482	+	0.765461i	$0.722511\pi$
$654$	0	0
$655$	−36.5216	−1.42702
$656$	0	0
$657$	−17.3560	−0.677122
$658$	0	0
$659$	13.3762	0.521062	0.260531	−	0.965465i	$-0.416102\pi$
$659$	13.3762	0.521062	0.260531	+	0.965465i	$0.416102\pi$
$660$	0	0
$661$	31.2873	1.21694	0.608468	−	0.793578i	$-0.291784\pi$
$661$	31.2873	1.21694	0.608468	+	0.793578i	$0.291784\pi$
$662$	0	0
$663$	2.36100	0.0916936
$664$	0	0
$665$	2.24740	0.0871504
$666$	0	0
$667$	18.2749	0.707605
$668$	0	0
$669$	−1.60500	−0.0620527
$670$	0	0
$671$	8.45177	0.326277
$672$	0	0
$673$	−13.9371	−0.537235	−0.268617	−	0.963247i	$-0.586567\pi$
$673$	−13.9371	−0.537235	−0.268617	+	0.963247i	$0.586567\pi$
$674$	0	0
$675$	10.2344	0.393920
$676$	0	0
$677$	9.98968	0.383935	0.191967	−	0.981401i	$-0.438513\pi$
$677$	9.98968	0.383935	0.191967	+	0.981401i	$0.438513\pi$
$678$	0	0
$679$	−20.5003	−0.786728
$680$	0	0
$681$	−7.59820	−0.291164
$682$	0	0
$683$	−22.6274	−0.865814	−0.432907	−	0.901439i	$-0.642512\pi$
$683$	−22.6274	−0.865814	−0.432907	+	0.901439i	$0.642512\pi$
$684$	0	0
$685$	−25.1742	−0.961856
$686$	0	0
$687$	4.51496	0.172256
$688$	0	0
$689$	−7.60433	−0.289702
$690$	0	0
$691$	32.3521	1.23073	0.615367	−	0.788241i	$-0.289008\pi$
$691$	32.3521	1.23073	0.615367	+	0.788241i	$0.289008\pi$
$692$	0	0
$693$	−3.00686	−0.114221
$694$	0	0
$695$	−13.1566	−0.499057
$696$	0	0
$697$	13.6132	0.515636
$698$	0	0
$699$	−8.66107	−0.327592
$700$	0	0
$701$	−42.2646	−1.59631	−0.798156	−	0.602451i	$-0.794191\pi$
$701$	−42.2646	−1.59631	−0.798156	+	0.602451i	$0.794191\pi$
$702$	0	0
$703$	11.8579	0.447231
$704$	0	0
$705$	−1.68960	−0.0636340
$706$	0	0
$707$	18.7278	0.704333
$708$	0	0
$709$	−29.8665	−1.12166	−0.560829	−	0.827931i	$-0.689518\pi$
$709$	−29.8665	−1.12166	−0.560829	+	0.827931i	$0.689518\pi$
$710$	0	0
$711$	1.85013	0.0693852
$712$	0	0
$713$	−28.0451	−1.05030
$714$	0	0
$715$	1.36134	0.0509113
$716$	0	0
$717$	1.56267	0.0583590
$718$	0	0
$719$	7.32162	0.273050	0.136525	−	0.990637i	$-0.456407\pi$
$719$	7.32162	0.273050	0.136525	+	0.990637i	$0.456407\pi$
$720$	0	0
$721$	−1.56612	−0.0583253
$722$	0	0
$723$	19.2939	0.717548
$724$	0	0
$725$	12.8676	0.477891
$726$	0	0
$727$	−50.2904	−1.86517	−0.932584	−	0.360952i	$-0.882452\pi$
$727$	−50.2904	−1.86517	−0.932584	+	0.360952i	$0.882452\pi$
$728$	0	0
$729$	6.21860	0.230318
$730$	0	0
$731$	30.4155	1.12496
$732$	0	0
$733$	35.2406	1.30164	0.650821	−	0.759232i	$-0.274425\pi$
$733$	35.2406	1.30164	0.650821	+	0.759232i	$0.274425\pi$
$734$	0	0
$735$	7.57689	0.279478
$736$	0	0
$737$	−1.83722	−0.0676747
$738$	0	0
$739$	18.9200	0.695985	0.347992	−	0.937497i	$-0.386864\pi$
$739$	18.9200	0.695985	0.347992	+	0.937497i	$0.386864\pi$
$740$	0	0
$741$	0.718527	0.0263958
$742$	0	0
$743$	−19.2446	−0.706016	−0.353008	−	0.935620i	$-0.614841\pi$
$743$	−19.2446	−0.706016	−0.353008	+	0.935620i	$0.614841\pi$
$744$	0	0
$745$	9.07850	0.332610
$746$	0	0
$747$	−25.6526	−0.938580
$748$	0	0
$749$	−24.2429	−0.885816
$750$	0	0
$751$	8.52278	0.311001	0.155500	−	0.987836i	$-0.450301\pi$
$751$	8.52278	0.311001	0.155500	+	0.987836i	$0.450301\pi$
$752$	0	0
$753$	26.3249	0.959332
$754$	0	0
$755$	1.75836	0.0639931
$756$	0	0
$757$	−24.8051	−0.901557	−0.450778	−	0.892636i	$-0.648854\pi$
$757$	−24.8051	−0.901557	−0.450778	+	0.892636i	$0.648854\pi$
$758$	0	0
$759$	2.78040	0.100922
$760$	0	0
$761$	−25.2826	−0.916492	−0.458246	−	0.888825i	$-0.651522\pi$
$761$	−25.2826	−0.916492	−0.458246	+	0.888825i	$0.651522\pi$
$762$	0	0
$763$	−17.9955	−0.651482
$764$	0	0
$765$	12.1879	0.440653
$766$	0	0
$767$	7.43813	0.268575
$768$	0	0
$769$	−28.5998	−1.03134	−0.515668	−	0.856789i	$-0.672456\pi$
$769$	−28.5998	−1.03134	−0.515668	+	0.856789i	$0.672456\pi$
$770$	0	0
$771$	10.4470	0.376241
$772$	0	0
$773$	4.87958	0.175506	0.0877531	−	0.996142i	$-0.472031\pi$
$773$	4.87958	0.175506	0.0877531	+	0.996142i	$0.472031\pi$
$774$	0	0
$775$	−19.7470	−0.709333
$776$	0	0
$777$	−14.0658	−0.504609
$778$	0	0
$779$	4.14292	0.148436
$780$	0	0
$781$	−4.39228	−0.157168
$782$	0	0
$783$	26.5369	0.948353
$784$	0	0
$785$	17.5813	0.627503
$786$	0	0
$787$	−13.9311	−0.496591	−0.248295	−	0.968684i	$-0.579870\pi$
$787$	−13.9311	−0.496591	−0.248295	+	0.968684i	$0.579870\pi$
$788$	0	0
$789$	−10.4077	−0.370525
$790$	0	0
$791$	17.3513	0.616942
$792$	0	0
$793$	−6.91025	−0.245390
$794$	0	0
$795$	13.6092	0.482669
$796$	0	0
$797$	−5.83526	−0.206695	−0.103348	−	0.994645i	$-0.532955\pi$
$797$	−5.83526	−0.206695	−0.103348	+	0.994645i	$0.532955\pi$
$798$	0	0
$799$	3.79418	0.134228
$800$	0	0
$801$	6.83015	0.241331
$802$	0	0
$803$	−7.79105	−0.274940
$804$	0	0
$805$	−7.11033	−0.250606
$806$	0	0
$807$	27.5944	0.971367
$808$	0	0
$809$	−28.1396	−0.989337	−0.494669	−	0.869082i	$-0.664711\pi$
$809$	−28.1396	−0.989337	−0.494669	+	0.869082i	$0.664711\pi$
$810$	0	0
$811$	14.0776	0.494332	0.247166	−	0.968973i	$-0.420501\pi$
$811$	14.0776	0.494332	0.247166	+	0.968973i	$0.420501\pi$
$812$	0	0
$813$	−19.1579	−0.671898
$814$	0	0
$815$	35.1157	1.23005
$816$	0	0
$817$	9.25641	0.323841
$818$	0	0
$819$	2.45844	0.0859047
$820$	0	0
$821$	−9.01860	−0.314751	−0.157376	−	0.987539i	$-0.550303\pi$
$821$	−9.01860	−0.314751	−0.157376	+	0.987539i	$0.550303\pi$
$822$	0	0
$823$	−9.82391	−0.342440	−0.171220	−	0.985233i	$-0.554771\pi$
$823$	−9.82391	−0.342440	−0.171220	+	0.985233i	$0.554771\pi$
$824$	0	0
$825$	1.95772	0.0681591
$826$	0	0
$827$	1.62586	0.0565366	0.0282683	−	0.999600i	$-0.491001\pi$
$827$	1.62586	0.0565366	0.0282683	+	0.999600i	$0.491001\pi$
$828$	0	0
$829$	35.0893	1.21870	0.609351	−	0.792901i	$-0.291430\pi$
$829$	35.0893	1.21870	0.609351	+	0.792901i	$0.291430\pi$
$830$	0	0
$831$	15.3960	0.534080
$832$	0	0
$833$	−17.0147	−0.589526
$834$	0	0
$835$	−1.33094	−0.0460591
$836$	0	0
$837$	−40.7243	−1.40764
$838$	0	0
$839$	−8.70068	−0.300381	−0.150190	−	0.988657i	$-0.547989\pi$
$839$	−8.70068	−0.300381	−0.150190	+	0.988657i	$0.547989\pi$
$840$	0	0
$841$	4.36481	0.150511
$842$	0	0
$843$	−27.2178	−0.937432
$844$	0	0
$845$	20.5323	0.706333
$846$	0	0
$847$	−1.34977	−0.0463786
$848$	0	0
$849$	6.38878	0.219262
$850$	0	0
$851$	−37.5162	−1.28604
$852$	0	0
$853$	−11.2883	−0.386504	−0.193252	−	0.981149i	$-0.561903\pi$
$853$	−11.2883	−0.386504	−0.193252	+	0.981149i	$0.561903\pi$
$854$	0	0
$855$	3.70916	0.126850
$856$	0	0
$857$	41.3661	1.41304	0.706519	−	0.707694i	$-0.250264\pi$
$857$	41.3661	1.41304	0.706519	+	0.707694i	$0.250264\pi$
$858$	0	0
$859$	16.5001	0.562977	0.281489	−	0.959565i	$-0.409172\pi$
$859$	16.5001	0.562977	0.281489	+	0.959565i	$0.409172\pi$
$860$	0	0
$861$	−4.91432	−0.167479
$862$	0	0
$863$	11.0012	0.374485	0.187243	−	0.982314i	$-0.440045\pi$
$863$	11.0012	0.374485	0.187243	+	0.982314i	$0.440045\pi$
$864$	0	0
$865$	1.69414	0.0576024
$866$	0	0
$867$	−5.45124	−0.185134
$868$	0	0
$869$	0.830516	0.0281733
$870$	0	0
$871$	1.50213	0.0508976
$872$	0	0
$873$	−33.8341	−1.14511
$874$	0	0
$875$	−16.2435	−0.549130
$876$	0	0
$877$	52.0312	1.75697	0.878484	−	0.477771i	$-0.158555\pi$
$877$	52.0312	1.75697	0.878484	+	0.477771i	$0.158555\pi$
$878$	0	0
$879$	8.55925	0.288696
$880$	0	0
$881$	26.4976	0.892727	0.446364	−	0.894852i	$-0.352719\pi$
$881$	26.4976	0.892727	0.446364	+	0.894852i	$0.352719\pi$
$882$	0	0
$883$	−5.72152	−0.192545	−0.0962723	−	0.995355i	$-0.530692\pi$
$883$	−5.72152	−0.192545	−0.0962723	+	0.995355i	$0.530692\pi$
$884$	0	0
$885$	−13.3118	−0.447471
$886$	0	0
$887$	−29.8727	−1.00303	−0.501513	−	0.865150i	$-0.667223\pi$
$887$	−29.8727	−1.00303	−0.501513	+	0.865150i	$0.667223\pi$
$888$	0	0
$889$	−21.9616	−0.736567
$890$	0	0
$891$	−2.64564	−0.0886323
$892$	0	0
$893$	1.15469	0.0386402
$894$	0	0
$895$	−29.2074	−0.976294
$896$	0	0
$897$	−2.27328	−0.0759026
$898$	0	0
$899$	−51.2026	−1.70770
$900$	0	0
$901$	−30.5610	−1.01813
$902$	0	0
$903$	−10.9799	−0.365388
$904$	0	0
$905$	−10.4424	−0.347118
$906$	0	0
$907$	29.0932	0.966024	0.483012	−	0.875614i	$-0.339543\pi$
$907$	29.0932	0.966024	0.483012	+	0.875614i	$0.339543\pi$
$908$	0	0
$909$	30.9088	1.02518
$910$	0	0
$911$	−24.7300	−0.819341	−0.409670	−	0.912234i	$-0.634356\pi$
$911$	−24.7300	−0.819341	−0.409670	+	0.912234i	$0.634356\pi$
$912$	0	0
$913$	−11.5154	−0.381103
$914$	0	0
$915$	12.3670	0.408842
$916$	0	0
$917$	−29.6065	−0.977694
$918$	0	0
$919$	−41.1661	−1.35794	−0.678972	−	0.734164i	$-0.737574\pi$
$919$	−41.1661	−1.35794	−0.678972	+	0.734164i	$0.737574\pi$
$920$	0	0
$921$	25.0028	0.823872
$922$	0	0
$923$	3.59117	0.118205
$924$	0	0
$925$	−26.4158	−0.868546
$926$	0	0
$927$	−2.58475	−0.0848945
$928$	0	0
$929$	−22.9588	−0.753253	−0.376627	−	0.926365i	$-0.622916\pi$
$929$	−22.9588	−0.753253	−0.376627	+	0.926365i	$0.622916\pi$
$930$	0	0
$931$	−5.17813	−0.169706
$932$	0	0
$933$	−7.53615	−0.246722
$934$	0	0
$935$	5.47109	0.178924
$936$	0	0
$937$	−17.1059	−0.558824	−0.279412	−	0.960171i	$-0.590139\pi$
$937$	−17.1059	−0.558824	−0.279412	+	0.960171i	$0.590139\pi$
$938$	0	0
$939$	−27.7465	−0.905472
$940$	0	0
$941$	38.7305	1.26258	0.631289	−	0.775548i	$-0.282526\pi$
$941$	38.7305	1.26258	0.631289	+	0.775548i	$0.282526\pi$
$942$	0	0
$943$	−13.1074	−0.426836
$944$	0	0
$945$	−10.3249	−0.335870
$946$	0	0
$947$	14.1337	0.459284	0.229642	−	0.973275i	$-0.426245\pi$
$947$	14.1337	0.459284	0.229642	+	0.973275i	$0.426245\pi$
$948$	0	0
$949$	6.37004	0.206780
$950$	0	0
$951$	−19.5439	−0.633755
$952$	0	0
$953$	7.68973	0.249095	0.124547	−	0.992214i	$-0.460252\pi$
$953$	7.68973	0.249095	0.124547	+	0.992214i	$0.460252\pi$
$954$	0	0
$955$	6.24513	0.202088
$956$	0	0
$957$	5.07623	0.164091
$958$	0	0
$959$	−20.4077	−0.658998
$960$	0	0
$961$	47.5769	1.53474
$962$	0	0
$963$	−40.0110	−1.28934
$964$	0	0
$965$	−12.5131	−0.402810
$966$	0	0
$967$	52.4613	1.68704	0.843520	−	0.537098i	$-0.180479\pi$
$967$	52.4613	1.68704	0.843520	+	0.537098i	$0.180479\pi$
$968$	0	0
$969$	2.88768	0.0927658
$970$	0	0
$971$	43.0603	1.38187	0.690936	−	0.722916i	$-0.257199\pi$
$971$	43.0603	1.38187	0.690936	+	0.722916i	$0.257199\pi$
$972$	0	0
$973$	−10.6655	−0.341920
$974$	0	0
$975$	−1.60065	−0.0512619
$976$	0	0
$977$	−34.1230	−1.09169	−0.545846	−	0.837885i	$-0.683792\pi$
$977$	−34.1230	−1.09169	−0.545846	+	0.837885i	$0.683792\pi$
$978$	0	0
$979$	3.06603	0.0979907
$980$	0	0
$981$	−29.7002	−0.948254
$982$	0	0
$983$	−18.0491	−0.575677	−0.287839	−	0.957679i	$-0.592937\pi$
$983$	−18.0491	−0.575677	−0.287839	+	0.957679i	$0.592937\pi$
$984$	0	0
$985$	24.3561	0.776049
$986$	0	0
$987$	−1.36969	−0.0435976
$988$	0	0
$989$	−29.2855	−0.931224
$990$	0	0
$991$	30.1800	0.958698	0.479349	−	0.877624i	$-0.340873\pi$
$991$	30.1800	0.958698	0.479349	+	0.877624i	$0.340873\pi$
$992$	0	0
$993$	21.5233	0.683023
$994$	0	0
$995$	23.4809	0.744396
$996$	0	0
$997$	−36.2635	−1.14848	−0.574238	−	0.818689i	$-0.694702\pi$
$997$	−36.2635	−1.14848	−0.574238	+	0.818689i	$0.694702\pi$
$998$	0	0
$999$	−54.4774	−1.72359

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.h.1.3	✓	6	4.3	odd	2
3344.2.a.y.1.4		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.878814 q^{3} -1.66503 q^{5} -1.34977 q^{7} -2.22769 q^{9} +O(q^{10})\)	\(q+0.878814 q^{3} -1.66503 q^{5} -1.34977 q^{7} -2.22769 q^{9} -1.00000 q^{11} +0.817610 q^{13} -1.46325 q^{15} +3.28589 q^{17} +1.00000 q^{19} -1.18620 q^{21} -3.16380 q^{23} -2.22769 q^{25} -4.59416 q^{27} -5.77623 q^{29} +8.86436 q^{31} -0.878814 q^{33} +2.24740 q^{35} +11.8579 q^{37} +0.718527 q^{39} +4.14292 q^{41} +9.25641 q^{43} +3.70916 q^{45} +1.15469 q^{47} -5.17813 q^{49} +2.88768 q^{51} -9.30068 q^{53} +1.66503 q^{55} +0.878814 q^{57} +9.09741 q^{59} -8.45177 q^{61} +3.00686 q^{63} -1.36134 q^{65} +1.83722 q^{67} -2.78040 q^{69} +4.39228 q^{71} +7.79105 q^{73} -1.95772 q^{75} +1.34977 q^{77} -0.830516 q^{79} +2.64564 q^{81} +11.5154 q^{83} -5.47109 q^{85} -5.07623 q^{87} -3.06603 q^{89} -1.10358 q^{91} +7.79013 q^{93} -1.66503 q^{95} +15.1880 q^{97} +2.22769 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9	\( 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9} - 6 q^{11} + 2 q^{13} - 10 q^{17} + 6 q^{19} + 10 q^{21} + 14 q^{23} + 2 q^{25} + 16 q^{27} + 6 q^{29} - 4 q^{31} - 4 q^{33} - 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41} + 26 q^{43} - 2 q^{45} + 24 q^{47} + 20 q^{49} + 14 q^{51} - 8 q^{53} + 4 q^{55} + 4 q^{57} + 4 q^{59} - 6 q^{61} + 40 q^{63} - 34 q^{65} + 44 q^{67} + 8 q^{69} + 16 q^{71} + 12 q^{73} + 28 q^{75} - 4 q^{77} - 6 q^{79} + 10 q^{81} + 28 q^{83} - 10 q^{85} + 24 q^{87} + 64 q^{91} - 14 q^{93} - 4 q^{95} + 4 q^{97} - 2 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9 - 6 * q^11 + 2 * q^13 - 10 * q^17 + 6 * q^19 + 10 * q^21 + 14 * q^23 + 2 * q^25 + 16 * q^27 + 6 * q^29 - 4 * q^31 - 4 * q^33 - 2 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^41 + 26 * q^43 - 2 * q^45 + 24 * q^47 + 20 * q^49 + 14 * q^51 - 8 * q^53 + 4 * q^55 + 4 * q^57 + 4 * q^59 - 6 * q^61 + 40 * q^63 - 34 * q^65 + 44 * q^67 + 8 * q^69 + 16 * q^71 + 12 * q^73 + 28 * q^75 - 4 * q^77 - 6 * q^79 + 10 * q^81 + 28 * q^83 - 10 * q^85 + 24 * q^87 + 64 * q^91 - 14 * q^93 - 4 * q^95 + 4 * q^97 - 2 * q^99

Introduction

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Database

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