LMFDB - Embedded newform 336.4.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,4,Mod(1,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("336.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	336.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:	$19.8246417619$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{337}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 84 $ x^2 - x - 84
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-8.67878$ of defining polynomial
Character	$\chi$	$=$	336.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +15.3576 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +15.3576 q^{5} +7.00000 q^{7} +9.00000 q^{9} +5.35756 q^{11} +11.2849 q^{13} +46.0727 q^{15} +94.0727 q^{17} -20.0000 q^{19} +21.0000 q^{21} -102.788 q^{23} +110.855 q^{25} +27.0000 q^{27} +102.000 q^{29} -341.721 q^{31} +16.0727 q^{33} +107.503 q^{35} +288.715 q^{37} +33.8546 q^{39} -252.933 q^{41} +145.721 q^{43} +138.218 q^{45} +573.576 q^{47} +49.0000 q^{49} +282.218 q^{51} -234.436 q^{53} +82.2790 q^{55} -60.0000 q^{57} +151.006 q^{59} +243.721 q^{61} +63.0000 q^{63} +173.308 q^{65} -142.715 q^{67} -308.363 q^{69} +65.0668 q^{71} +380.424 q^{73} +332.564 q^{75} +37.5029 q^{77} +830.738 q^{79} +81.0000 q^{81} -469.163 q^{83} +1444.73 q^{85} +306.000 q^{87} -1554.65 q^{89} +78.9942 q^{91} -1025.16 q^{93} -307.151 q^{95} -1085.03 q^{97} +48.2180 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 6 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 - 6 * q^5 + 14 * q^7 + 18 * q^9	$ 2 q + 6 q^{3} - 6 q^{5} + 14 q^{7} + 18 q^{9} - 26 q^{11} + 96 q^{13} - 18 q^{15} + 78 q^{17} - 40 q^{19} + 42 q^{21} - 22 q^{23} + 442 q^{25} + 54 q^{27} + 204 q^{29} - 96 q^{31} - 78 q^{33} - 42 q^{35} + 504 q^{37} + 288 q^{39} - 102 q^{41} - 296 q^{43} - 54 q^{45} + 780 q^{47} + 98 q^{49} + 234 q^{51} + 192 q^{53} + 752 q^{55} - 120 q^{57} - 212 q^{59} - 100 q^{61} + 126 q^{63} - 1636 q^{65} - 212 q^{67} - 66 q^{69} + 534 q^{71} + 1128 q^{73} + 1326 q^{75} - 182 q^{77} - 468 q^{79} + 162 q^{81} + 824 q^{83} + 1788 q^{85} + 612 q^{87} - 2118 q^{89} + 672 q^{91} - 288 q^{93} + 120 q^{95} + 400 q^{97} - 234 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 6 * q^5 + 14 * q^7 + 18 * q^9 - 26 * q^11 + 96 * q^13 - 18 * q^15 + 78 * q^17 - 40 * q^19 + 42 * q^21 - 22 * q^23 + 442 * q^25 + 54 * q^27 + 204 * q^29 - 96 * q^31 - 78 * q^33 - 42 * q^35 + 504 * q^37 + 288 * q^39 - 102 * q^41 - 296 * q^43 - 54 * q^45 + 780 * q^47 + 98 * q^49 + 234 * q^51 + 192 * q^53 + 752 * q^55 - 120 * q^57 - 212 * q^59 - 100 * q^61 + 126 * q^63 - 1636 * q^65 - 212 * q^67 - 66 * q^69 + 534 * q^71 + 1128 * q^73 + 1326 * q^75 - 182 * q^77 - 468 * q^79 + 162 * q^81 + 824 * q^83 + 1788 * q^85 + 612 * q^87 - 2118 * q^89 + 672 * q^91 - 288 * q^93 + 120 * q^95 + 400 * q^97 - 234 * 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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	15.3576	1.37362	0.686811	−	0.726836i	$-0.259010\pi$
$5$	15.3576	1.37362	0.686811	+	0.726836i	$0.259010\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	5.35756	0.146851	0.0734257	−	0.997301i	$-0.476607\pi$
$11$	5.35756	0.146851	0.0734257	+	0.997301i	$0.476607\pi$
$12$	0	0
$13$	11.2849	0.240759	0.120379	−	0.992728i	$-0.461589\pi$
$13$	11.2849	0.240759	0.120379	+	0.992728i	$0.461589\pi$
$14$	0	0
$15$	46.0727	0.793061
$16$	0	0
$17$	94.0727	1.34212	0.671058	−	0.741405i	$-0.265840\pi$
$17$	94.0727	1.34212	0.671058	+	0.741405i	$0.265840\pi$
$18$	0	0
$19$	−20.0000	−0.241490	−0.120745	−	0.992684i	$-0.538528\pi$
$19$	−20.0000	−0.241490	−0.120745	+	0.992684i	$0.538528\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	−102.788	−0.931858	−0.465929	−	0.884822i	$-0.654280\pi$
$23$	−102.788	−0.931858	−0.465929	+	0.884822i	$0.654280\pi$
$24$	0	0
$25$	110.855	0.886837
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	102.000	0.653135	0.326568	−	0.945174i	$-0.394108\pi$
$29$	102.000	0.653135	0.326568	+	0.945174i	$0.394108\pi$
$30$	0	0
$31$	−341.721	−1.97984	−0.989918	−	0.141644i	$-0.954761\pi$
$31$	−341.721	−1.97984	−0.989918	+	0.141644i	$0.954761\pi$
$32$	0	0
$33$	16.0727	0.0847847
$34$	0	0
$35$	107.503	0.519180
$36$	0	0
$37$	288.715	1.28282	0.641412	−	0.767197i	$-0.278349\pi$
$37$	288.715	1.28282	0.641412	+	0.767197i	$0.278349\pi$
$38$	0	0
$39$	33.8546	0.139002
$40$	0	0
$41$	−252.933	−0.963452	−0.481726	−	0.876322i	$-0.659990\pi$
$41$	−252.933	−0.963452	−0.481726	+	0.876322i	$0.659990\pi$
$42$	0	0
$43$	145.721	0.516796	0.258398	−	0.966039i	$-0.416805\pi$
$43$	145.721	0.516796	0.258398	+	0.966039i	$0.416805\pi$
$44$	0	0
$45$	138.218	0.457874
$46$	0	0
$47$	573.576	1.78010	0.890049	−	0.455865i	$-0.150670\pi$
$47$	573.576	1.78010	0.890049	+	0.455865i	$0.150670\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	282.218	0.774871
$52$	0	0
$53$	−234.436	−0.607590	−0.303795	−	0.952737i	$-0.598254\pi$
$53$	−234.436	−0.607590	−0.303795	+	0.952737i	$0.598254\pi$
$54$	0	0
$55$	82.2790	0.201718
$56$	0	0
$57$	−60.0000	−0.139424
$58$	0	0
$59$	151.006	0.333208	0.166604	−	0.986024i	$-0.446720\pi$
$59$	151.006	0.333208	0.166604	+	0.986024i	$0.446720\pi$
$60$	0	0
$61$	243.721	0.511562	0.255781	−	0.966735i	$-0.417667\pi$
$61$	243.721	0.511562	0.255781	+	0.966735i	$0.417667\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	173.308	0.330711
$66$	0	0
$67$	−142.715	−0.260230	−0.130115	−	0.991499i	$-0.541535\pi$
$67$	−142.715	−0.260230	−0.130115	+	0.991499i	$0.541535\pi$
$68$	0	0
$69$	−308.363	−0.538009
$70$	0	0
$71$	65.0668	0.108761	0.0543804	−	0.998520i	$-0.482682\pi$
$71$	65.0668	0.108761	0.0543804	+	0.998520i	$0.482682\pi$
$72$	0	0
$73$	380.424	0.609936	0.304968	−	0.952363i	$-0.401354\pi$
$73$	380.424	0.609936	0.304968	+	0.952363i	$0.401354\pi$
$74$	0	0
$75$	332.564	0.512016
$76$	0	0
$77$	37.5029	0.0555046
$78$	0	0
$79$	830.738	1.18311	0.591553	−	0.806266i	$-0.298515\pi$
$79$	830.738	1.18311	0.591553	+	0.806266i	$0.298515\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−469.163	−0.620450	−0.310225	−	0.950663i	$-0.600404\pi$
$83$	−469.163	−0.620450	−0.310225	+	0.950663i	$0.600404\pi$
$84$	0	0
$85$	1444.73	1.84356
$86$	0	0
$87$	306.000	0.377088
$88$	0	0
$89$	−1554.65	−1.85161	−0.925803	−	0.378005i	$-0.876610\pi$
$89$	−1554.65	−1.85161	−0.925803	+	0.378005i	$0.876610\pi$
$90$	0	0
$91$	78.9942	0.0909982
$92$	0	0
$93$	−1025.16	−1.14306
$94$	0	0
$95$	−307.151	−0.331716
$96$	0	0
$97$	−1085.03	−1.13575	−0.567876	−	0.823114i	$-0.692235\pi$
$97$	−1085.03	−1.13575	−0.567876	+	0.823114i	$0.692235\pi$
$98$	0	0
$99$	48.2180	0.0489505
$100$	0	0
$101$	−1888.25	−1.86028	−0.930140	−	0.367206i	$-0.880314\pi$
$101$	−1888.25	−1.86028	−0.930140	+	0.367206i	$0.880314\pi$
$102$	0	0
$103$	1326.01	1.26850	0.634252	−	0.773127i	$-0.281308\pi$
$103$	1326.01	1.26850	0.634252	+	0.773127i	$0.281308\pi$
$104$	0	0
$105$	322.509	0.299749
$106$	0	0
$107$	−1398.08	−1.26316	−0.631579	−	0.775311i	$-0.717593\pi$
$107$	−1398.08	−1.26316	−0.631579	+	0.775311i	$0.717593\pi$
$108$	0	0
$109$	2088.33	1.83509	0.917547	−	0.397626i	$-0.130166\pi$
$109$	2088.33	1.83509	0.917547	+	0.397626i	$0.130166\pi$
$110$	0	0
$111$	866.145	0.740639
$112$	0	0
$113$	1442.87	1.20119	0.600593	−	0.799555i	$-0.294931\pi$
$113$	1442.87	1.20119	0.600593	+	0.799555i	$0.294931\pi$
$114$	0	0
$115$	−1578.57	−1.28002
$116$	0	0
$117$	101.564	0.0802529
$118$	0	0
$119$	658.509	0.507272
$120$	0	0
$121$	−1302.30	−0.978435
$122$	0	0
$123$	−758.799	−0.556249
$124$	0	0
$125$	−217.238	−0.155443
$126$	0	0
$127$	−1891.59	−1.32166	−0.660832	−	0.750534i	$-0.729796\pi$
$127$	−1891.59	−1.32166	−0.660832	+	0.750534i	$0.729796\pi$
$128$	0	0
$129$	437.163	0.298372
$130$	0	0
$131$	−765.407	−0.510488	−0.255244	−	0.966877i	$-0.582156\pi$
$131$	−765.407	−0.510488	−0.255244	+	0.966877i	$0.582156\pi$
$132$	0	0
$133$	−140.000	−0.0912747
$134$	0	0
$135$	414.654	0.264354
$136$	0	0
$137$	−346.459	−0.216059	−0.108029	−	0.994148i	$-0.534454\pi$
$137$	−346.459	−0.216059	−0.108029	+	0.994148i	$0.534454\pi$
$138$	0	0
$139$	−1079.44	−0.658684	−0.329342	−	0.944211i	$-0.606827\pi$
$139$	−1079.44	−0.658684	−0.329342	+	0.944211i	$0.606827\pi$
$140$	0	0
$141$	1720.73	1.02774
$142$	0	0
$143$	60.4594	0.0353557
$144$	0	0
$145$	1566.47	0.897161
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	−2922.44	−1.60681	−0.803407	−	0.595430i	$-0.796982\pi$
$149$	−2922.44	−1.60681	−0.803407	+	0.595430i	$0.796982\pi$
$150$	0	0
$151$	2222.88	1.19798	0.598992	−	0.800755i	$-0.295568\pi$
$151$	2222.88	1.19798	0.598992	+	0.800755i	$0.295568\pi$
$152$	0	0
$153$	846.654	0.447372
$154$	0	0
$155$	−5248.00	−2.71955
$156$	0	0
$157$	1362.27	0.692489	0.346244	−	0.938144i	$-0.387457\pi$
$157$	1362.27	0.692489	0.346244	+	0.938144i	$0.387457\pi$
$158$	0	0
$159$	−703.308	−0.350792
$160$	0	0
$161$	−719.515	−0.352209
$162$	0	0
$163$	−2267.03	−1.08937	−0.544685	−	0.838640i	$-0.683351\pi$
$163$	−2267.03	−1.08937	−0.544685	+	0.838640i	$0.683351\pi$
$164$	0	0
$165$	246.837	0.116462
$166$	0	0
$167$	343.901	0.159353	0.0796763	−	0.996821i	$-0.474611\pi$
$167$	343.901	0.159353	0.0796763	+	0.996821i	$0.474611\pi$
$168$	0	0
$169$	−2069.65	−0.942035
$170$	0	0
$171$	−180.000	−0.0804967
$172$	0	0
$173$	−2397.10	−1.05346	−0.526729	−	0.850033i	$-0.676582\pi$
$173$	−2397.10	−1.05346	−0.526729	+	0.850033i	$0.676582\pi$
$174$	0	0
$175$	775.982	0.335193
$176$	0	0
$177$	453.018	0.192378
$178$	0	0
$179$	−1354.50	−0.565586	−0.282793	−	0.959181i	$-0.591261\pi$
$179$	−1354.50	−0.565586	−0.282793	+	0.959181i	$0.591261\pi$
$180$	0	0
$181$	−3309.01	−1.35888	−0.679438	−	0.733733i	$-0.737776\pi$
$181$	−3309.01	−1.35888	−0.679438	+	0.733733i	$0.737776\pi$
$182$	0	0
$183$	731.163	0.295350
$184$	0	0
$185$	4433.96	1.76211
$186$	0	0
$187$	504.000	0.197092
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	−2756.99	−1.04444	−0.522222	−	0.852809i	$-0.674897\pi$
$191$	−2756.99	−1.04444	−0.522222	+	0.852809i	$0.674897\pi$
$192$	0	0
$193$	−944.134	−0.352126	−0.176063	−	0.984379i	$-0.556336\pi$
$193$	−944.134	−0.352126	−0.176063	+	0.984379i	$0.556336\pi$
$194$	0	0
$195$	519.925	0.190936
$196$	0	0
$197$	891.285	0.322342	0.161171	−	0.986926i	$-0.448473\pi$
$197$	891.285	0.322342	0.161171	+	0.986926i	$0.448473\pi$
$198$	0	0
$199$	−1324.36	−0.471766	−0.235883	−	0.971781i	$-0.575798\pi$
$199$	−1324.36	−0.471766	−0.235883	+	0.971781i	$0.575798\pi$
$200$	0	0
$201$	−428.145	−0.150244
$202$	0	0
$203$	714.000	0.246862
$204$	0	0
$205$	−3884.44	−1.32342
$206$	0	0
$207$	−925.090	−0.310619
$208$	0	0
$209$	−107.151	−0.0354632
$210$	0	0
$211$	−1086.04	−0.354340	−0.177170	−	0.984180i	$-0.556694\pi$
$211$	−1086.04	−0.354340	−0.177170	+	0.984180i	$0.556694\pi$
$212$	0	0
$213$	195.201	0.0627931
$214$	0	0
$215$	2237.92	0.709883
$216$	0	0
$217$	−2392.05	−0.748307
$218$	0	0
$219$	1141.27	0.352147
$220$	0	0
$221$	1061.60	0.323126
$222$	0	0
$223$	5878.59	1.76529	0.882645	−	0.470040i	$-0.155761\pi$
$223$	5878.59	1.76529	0.882645	+	0.470040i	$0.155761\pi$
$224$	0	0
$225$	997.692	0.295612
$226$	0	0
$227$	−835.320	−0.244238	−0.122119	−	0.992515i	$-0.538969\pi$
$227$	−835.320	−0.244238	−0.122119	+	0.992515i	$0.538969\pi$
$228$	0	0
$229$	−1471.33	−0.424578	−0.212289	−	0.977207i	$-0.568092\pi$
$229$	−1471.33	−0.424578	−0.212289	+	0.977207i	$0.568092\pi$
$230$	0	0
$231$	112.509	0.0320456
$232$	0	0
$233$	−5125.32	−1.44108	−0.720538	−	0.693415i	$-0.756105\pi$
$233$	−5125.32	−1.44108	−0.720538	+	0.693415i	$0.756105\pi$
$234$	0	0
$235$	8808.72	2.44518
$236$	0	0
$237$	2492.22	0.683067
$238$	0	0
$239$	−1195.88	−0.323661	−0.161831	−	0.986819i	$-0.551740\pi$
$239$	−1195.88	−0.323661	−0.161831	+	0.986819i	$0.551740\pi$
$240$	0	0
$241$	4855.62	1.29783	0.648917	−	0.760859i	$-0.275222\pi$
$241$	4855.62	1.29783	0.648917	+	0.760859i	$0.275222\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	752.520	0.196232
$246$	0	0
$247$	−225.698	−0.0581409
$248$	0	0
$249$	−1407.49	−0.358217
$250$	0	0
$251$	5141.94	1.29305	0.646526	−	0.762892i	$-0.276221\pi$
$251$	5141.94	1.29305	0.646526	+	0.762892i	$0.276221\pi$
$252$	0	0
$253$	−550.692	−0.136845
$254$	0	0
$255$	4334.18	1.06438
$256$	0	0
$257$	5083.65	1.23389	0.616944	−	0.787007i	$-0.288370\pi$
$257$	5083.65	1.23389	0.616944	+	0.787007i	$0.288370\pi$
$258$	0	0
$259$	2021.01	0.484862
$260$	0	0
$261$	918.000	0.217712
$262$	0	0
$263$	−435.974	−0.102218	−0.0511090	−	0.998693i	$-0.516276\pi$
$263$	−435.974	−0.102218	−0.0511090	+	0.998693i	$0.516276\pi$
$264$	0	0
$265$	−3600.37	−0.834599
$266$	0	0
$267$	−4663.96	−1.06903
$268$	0	0
$269$	−212.084	−0.0480707	−0.0240353	−	0.999711i	$-0.507651\pi$
$269$	−212.084	−0.0480707	−0.0240353	+	0.999711i	$0.507651\pi$
$270$	0	0
$271$	8391.76	1.88104	0.940522	−	0.339733i	$-0.110337\pi$
$271$	8391.76	1.88104	0.940522	+	0.339733i	$0.110337\pi$
$272$	0	0
$273$	236.982	0.0525378
$274$	0	0
$275$	593.910	0.130233
$276$	0	0
$277$	7577.15	1.64356	0.821781	−	0.569804i	$-0.192981\pi$
$277$	7577.15	1.64356	0.821781	+	0.569804i	$0.192981\pi$
$278$	0	0
$279$	−3075.49	−0.659945
$280$	0	0
$281$	−2258.22	−0.479409	−0.239704	−	0.970846i	$-0.577051\pi$
$281$	−2258.22	−0.479409	−0.239704	+	0.970846i	$0.577051\pi$
$282$	0	0
$283$	1537.77	0.323006	0.161503	−	0.986872i	$-0.448366\pi$
$283$	1537.77	0.323006	0.161503	+	0.986872i	$0.448366\pi$
$284$	0	0
$285$	−921.454	−0.191516
$286$	0	0
$287$	−1770.53	−0.364151
$288$	0	0
$289$	3936.67	0.801276
$290$	0	0
$291$	−3255.09	−0.655727
$292$	0	0
$293$	2187.77	0.436215	0.218107	−	0.975925i	$-0.430012\pi$
$293$	2187.77	0.436215	0.218107	+	0.975925i	$0.430012\pi$
$294$	0	0
$295$	2319.08	0.457702
$296$	0	0
$297$	144.654	0.0282616
$298$	0	0
$299$	−1159.95	−0.224353
$300$	0	0
$301$	1020.05	0.195331
$302$	0	0
$303$	−5664.76	−1.07403
$304$	0	0
$305$	3742.96	0.702693
$306$	0	0
$307$	−8948.97	−1.66366	−0.831831	−	0.555028i	$-0.812707\pi$
$307$	−8948.97	−1.66366	−0.831831	+	0.555028i	$0.812707\pi$
$308$	0	0
$309$	3978.04	0.732371
$310$	0	0
$311$	8889.06	1.62075	0.810374	−	0.585913i	$-0.199264\pi$
$311$	8889.06	1.62075	0.810374	+	0.585913i	$0.199264\pi$
$312$	0	0
$313$	4656.88	0.840967	0.420483	−	0.907300i	$-0.361860\pi$
$313$	4656.88	0.840967	0.420483	+	0.907300i	$0.361860\pi$
$314$	0	0
$315$	967.526	0.173060
$316$	0	0
$317$	−9862.60	−1.74744	−0.873720	−	0.486428i	$-0.838299\pi$
$317$	−9862.60	−1.74744	−0.873720	+	0.486428i	$0.838299\pi$
$318$	0	0
$319$	546.471	0.0959138
$320$	0	0
$321$	−4194.25	−0.729285
$322$	0	0
$323$	−1881.45	−0.324108
$324$	0	0
$325$	1250.98	0.213514
$326$	0	0
$327$	6264.98	1.05949
$328$	0	0
$329$	4015.03	0.672814
$330$	0	0
$331$	−7084.92	−1.17650	−0.588251	−	0.808678i	$-0.700183\pi$
$331$	−7084.92	−1.17650	−0.588251	+	0.808678i	$0.700183\pi$
$332$	0	0
$333$	2598.44	0.427608
$334$	0	0
$335$	−2191.76	−0.357458
$336$	0	0
$337$	−4018.27	−0.649522	−0.324761	−	0.945796i	$-0.605284\pi$
$337$	−4018.27	−0.649522	−0.324761	+	0.945796i	$0.605284\pi$
$338$	0	0
$339$	4328.62	0.693505
$340$	0	0
$341$	−1830.79	−0.290742
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−4735.71	−0.739020
$346$	0	0
$347$	−6147.17	−0.951001	−0.475501	−	0.879715i	$-0.657733\pi$
$347$	−6147.17	−0.951001	−0.475501	+	0.879715i	$0.657733\pi$
$348$	0	0
$349$	−4131.26	−0.633642	−0.316821	−	0.948485i	$-0.602615\pi$
$349$	−4131.26	−0.633642	−0.316821	+	0.948485i	$0.602615\pi$
$350$	0	0
$351$	304.692	0.0463340
$352$	0	0
$353$	8158.65	1.23015	0.615073	−	0.788470i	$-0.289127\pi$
$353$	8158.65	1.23015	0.615073	+	0.788470i	$0.289127\pi$
$354$	0	0
$355$	999.268	0.149396
$356$	0	0
$357$	1975.53	0.292874
$358$	0	0
$359$	1921.16	0.282437	0.141219	−	0.989978i	$-0.454898\pi$
$359$	1921.16	0.282437	0.141219	+	0.989978i	$0.454898\pi$
$360$	0	0
$361$	−6459.00	−0.941682
$362$	0	0
$363$	−3906.89	−0.564900
$364$	0	0
$365$	5842.39	0.837821
$366$	0	0
$367$	−7432.53	−1.05715	−0.528577	−	0.848886i	$-0.677274\pi$
$367$	−7432.53	−1.05715	−0.528577	+	0.848886i	$0.677274\pi$
$368$	0	0
$369$	−2276.40	−0.321151
$370$	0	0
$371$	−1641.05	−0.229647
$372$	0	0
$373$	7088.33	0.983967	0.491983	−	0.870605i	$-0.336272\pi$
$373$	7088.33	0.983967	0.491983	+	0.870605i	$0.336272\pi$
$374$	0	0
$375$	−651.715	−0.0897451
$376$	0	0
$377$	1151.06	0.157248
$378$	0	0
$379$	382.035	0.0517779	0.0258889	−	0.999665i	$-0.491758\pi$
$379$	382.035	0.0517779	0.0258889	+	0.999665i	$0.491758\pi$
$380$	0	0
$381$	−5674.76	−0.763063
$382$	0	0
$383$	4476.41	0.597216	0.298608	−	0.954376i	$-0.403478\pi$
$383$	4476.41	0.597216	0.298608	+	0.954376i	$0.403478\pi$
$384$	0	0
$385$	575.953	0.0762423
$386$	0	0
$387$	1311.49	0.172265
$388$	0	0
$389$	9922.07	1.29324	0.646618	−	0.762814i	$-0.276183\pi$
$389$	9922.07	1.29324	0.646618	+	0.762814i	$0.276183\pi$
$390$	0	0
$391$	−9669.52	−1.25066
$392$	0	0
$393$	−2296.22	−0.294730
$394$	0	0
$395$	12758.1	1.62514
$396$	0	0
$397$	6158.29	0.778528	0.389264	−	0.921126i	$-0.372729\pi$
$397$	6158.29	0.778528	0.389264	+	0.921126i	$0.372729\pi$
$398$	0	0
$399$	−420.000	−0.0526975
$400$	0	0
$401$	3633.32	0.452467	0.226234	−	0.974073i	$-0.427359\pi$
$401$	3633.32	0.452467	0.226234	+	0.974073i	$0.427359\pi$
$402$	0	0
$403$	−3856.28	−0.476663
$404$	0	0
$405$	1243.96	0.152625
$406$	0	0
$407$	1546.81	0.188384
$408$	0	0
$409$	11822.1	1.42925	0.714625	−	0.699507i	$-0.246597\pi$
$409$	11822.1	1.42925	0.714625	+	0.699507i	$0.246597\pi$
$410$	0	0
$411$	−1039.38	−0.124741
$412$	0	0
$413$	1057.04	0.125941
$414$	0	0
$415$	−7205.20	−0.852263
$416$	0	0
$417$	−3238.33	−0.380291
$418$	0	0
$419$	−5897.62	−0.687632	−0.343816	−	0.939037i	$-0.611720\pi$
$419$	−5897.62	−0.687632	−0.343816	+	0.939037i	$0.611720\pi$
$420$	0	0
$421$	−2322.39	−0.268851	−0.134426	−	0.990924i	$-0.542919\pi$
$421$	−2322.39	−0.268851	−0.134426	+	0.990924i	$0.542919\pi$
$422$	0	0
$423$	5162.18	0.593366
$424$	0	0
$425$	10428.4	1.19024
$426$	0	0
$427$	1706.05	0.193352
$428$	0	0
$429$	181.378	0.0204126
$430$	0	0
$431$	10107.6	1.12962	0.564809	−	0.825222i	$-0.308950\pi$
$431$	10107.6	1.12962	0.564809	+	0.825222i	$0.308950\pi$
$432$	0	0
$433$	−9842.94	−1.09243	−0.546214	−	0.837645i	$-0.683932\pi$
$433$	−9842.94	−1.09243	−0.546214	+	0.837645i	$0.683932\pi$
$434$	0	0
$435$	4699.41	0.517976
$436$	0	0
$437$	2055.76	0.225035
$438$	0	0
$439$	−12710.0	−1.38181	−0.690903	−	0.722947i	$-0.742787\pi$
$439$	−12710.0	−1.38181	−0.690903	+	0.722947i	$0.742787\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	3491.32	0.374441	0.187221	−	0.982318i	$-0.440052\pi$
$443$	3491.32	0.374441	0.187221	+	0.982318i	$0.440052\pi$
$444$	0	0
$445$	−23875.7	−2.54341
$446$	0	0
$447$	−8767.31	−0.927695
$448$	0	0
$449$	−18017.1	−1.89372	−0.946859	−	0.321649i	$-0.895763\pi$
$449$	−18017.1	−1.89372	−0.946859	+	0.321649i	$0.895763\pi$
$450$	0	0
$451$	−1355.10	−0.141484
$452$	0	0
$453$	6668.65	0.691657
$454$	0	0
$455$	1213.16	0.124997
$456$	0	0
$457$	6112.98	0.625718	0.312859	−	0.949800i	$-0.398713\pi$
$457$	6112.98	0.625718	0.312859	+	0.949800i	$0.398713\pi$
$458$	0	0
$459$	2539.96	0.258290
$460$	0	0
$461$	−11066.0	−1.11799	−0.558997	−	0.829170i	$-0.688814\pi$
$461$	−11066.0	−1.11799	−0.558997	+	0.829170i	$0.688814\pi$
$462$	0	0
$463$	13088.9	1.31380	0.656902	−	0.753976i	$-0.271867\pi$
$463$	13088.9	1.31380	0.656902	+	0.753976i	$0.271867\pi$
$464$	0	0
$465$	−15744.0	−1.57013
$466$	0	0
$467$	5854.32	0.580098	0.290049	−	0.957012i	$-0.406328\pi$
$467$	5854.32	0.580098	0.290049	+	0.957012i	$0.406328\pi$
$468$	0	0
$469$	−999.006	−0.0983578
$470$	0	0
$471$	4086.80	0.399809
$472$	0	0
$473$	780.709	0.0758922
$474$	0	0
$475$	−2217.09	−0.214163
$476$	0	0
$477$	−2109.92	−0.202530
$478$	0	0
$479$	4482.33	0.427564	0.213782	−	0.976881i	$-0.431422\pi$
$479$	4482.33	0.427564	0.213782	+	0.976881i	$0.431422\pi$
$480$	0	0
$481$	3258.12	0.308851
$482$	0	0
$483$	−2158.54	−0.203348
$484$	0	0
$485$	−16663.4	−1.56009
$486$	0	0
$487$	−10179.2	−0.947157	−0.473579	−	0.880752i	$-0.657038\pi$
$487$	−10179.2	−0.947157	−0.473579	+	0.880752i	$0.657038\pi$
$488$	0	0
$489$	−6801.09	−0.628948
$490$	0	0
$491$	9562.11	0.878884	0.439442	−	0.898271i	$-0.355176\pi$
$491$	9562.11	0.878884	0.439442	+	0.898271i	$0.355176\pi$
$492$	0	0
$493$	9595.41	0.876584
$494$	0	0
$495$	740.511	0.0672394
$496$	0	0
$497$	455.468	0.0411077
$498$	0	0
$499$	−15747.5	−1.41273	−0.706366	−	0.707847i	$-0.749667\pi$
$499$	−15747.5	−1.41273	−0.706366	+	0.707847i	$0.749667\pi$
$500$	0	0
$501$	1031.70	0.0920023
$502$	0	0
$503$	3131.88	0.277622	0.138811	−	0.990319i	$-0.455672\pi$
$503$	3131.88	0.277622	0.138811	+	0.990319i	$0.455672\pi$
$504$	0	0
$505$	−28999.0	−2.55532
$506$	0	0
$507$	−6208.95	−0.543884
$508$	0	0
$509$	−5795.11	−0.504644	−0.252322	−	0.967643i	$-0.581194\pi$
$509$	−5795.11	−0.504644	−0.252322	+	0.967643i	$0.581194\pi$
$510$	0	0
$511$	2662.97	0.230534
$512$	0	0
$513$	−540.000	−0.0464748
$514$	0	0
$515$	20364.3	1.74244
$516$	0	0
$517$	3072.97	0.261410
$518$	0	0
$519$	−7191.31	−0.608214
$520$	0	0
$521$	−19405.1	−1.63177	−0.815887	−	0.578212i	$-0.803751\pi$
$521$	−19405.1	−1.63177	−0.815887	+	0.578212i	$0.803751\pi$
$522$	0	0
$523$	−12823.0	−1.07210	−0.536050	−	0.844186i	$-0.680084\pi$
$523$	−12823.0	−1.07210	−0.536050	+	0.844186i	$0.680084\pi$
$524$	0	0
$525$	2327.95	0.193524
$526$	0	0
$527$	−32146.6	−2.65717
$528$	0	0
$529$	−1601.67	−0.131640
$530$	0	0
$531$	1359.05	0.111069
$532$	0	0
$533$	−2854.32	−0.231959
$534$	0	0
$535$	−21471.2	−1.73510
$536$	0	0
$537$	−4063.49	−0.326541
$538$	0	0
$539$	262.520	0.0209788
$540$	0	0
$541$	−9652.27	−0.767068	−0.383534	−	0.923527i	$-0.625293\pi$
$541$	−9652.27	−0.767068	−0.383534	+	0.923527i	$0.625293\pi$
$542$	0	0
$543$	−9927.02	−0.784547
$544$	0	0
$545$	32071.6	2.52073
$546$	0	0
$547$	17898.0	1.39902	0.699510	−	0.714623i	$-0.253402\pi$
$547$	17898.0	1.39902	0.699510	+	0.714623i	$0.253402\pi$
$548$	0	0
$549$	2193.49	0.170521
$550$	0	0
$551$	−2040.00	−0.157726
$552$	0	0
$553$	5815.17	0.447172
$554$	0	0
$555$	13301.9	1.01736
$556$	0	0
$557$	−6879.73	−0.523345	−0.261673	−	0.965157i	$-0.584274\pi$
$557$	−6879.73	−0.523345	−0.261673	+	0.965157i	$0.584274\pi$
$558$	0	0
$559$	1644.44	0.124423
$560$	0	0
$561$	1512.00	0.113791
$562$	0	0
$563$	10399.4	0.778476	0.389238	−	0.921137i	$-0.372738\pi$
$563$	10399.4	0.778476	0.389238	+	0.921137i	$0.372738\pi$
$564$	0	0
$565$	22159.0	1.64998
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	12477.3	0.919288	0.459644	−	0.888103i	$-0.347977\pi$
$569$	12477.3	0.919288	0.459644	+	0.888103i	$0.347977\pi$
$570$	0	0
$571$	26473.9	1.94028	0.970140	−	0.242544i	$-0.0779819\pi$
$571$	26473.9	1.94028	0.970140	+	0.242544i	$0.0779819\pi$
$572$	0	0
$573$	−8270.97	−0.603010
$574$	0	0
$575$	−11394.5	−0.826406
$576$	0	0
$577$	5902.41	0.425859	0.212929	−	0.977068i	$-0.431700\pi$
$577$	5902.41	0.425859	0.212929	+	0.977068i	$0.431700\pi$
$578$	0	0
$579$	−2832.40	−0.203300
$580$	0	0
$581$	−3284.14	−0.234508
$582$	0	0
$583$	−1256.01	−0.0892254
$584$	0	0
$585$	1559.77	0.110237
$586$	0	0
$587$	−4069.31	−0.286130	−0.143065	−	0.989713i	$-0.545696\pi$
$587$	−4069.31	−0.286130	−0.143065	+	0.989713i	$0.545696\pi$
$588$	0	0
$589$	6834.42	0.478111
$590$	0	0
$591$	2673.85	0.186104
$592$	0	0
$593$	14686.0	1.01700	0.508500	−	0.861062i	$-0.330200\pi$
$593$	14686.0	1.01700	0.508500	+	0.861062i	$0.330200\pi$
$594$	0	0
$595$	10113.1	0.696800
$596$	0	0
$597$	−3973.08	−0.272374
$598$	0	0
$599$	−4360.73	−0.297454	−0.148727	−	0.988878i	$-0.547518\pi$
$599$	−4360.73	−0.297454	−0.148727	+	0.988878i	$0.547518\pi$
$600$	0	0
$601$	2145.85	0.145642	0.0728211	−	0.997345i	$-0.476800\pi$
$601$	2145.85	0.145642	0.0728211	+	0.997345i	$0.476800\pi$
$602$	0	0
$603$	−1284.44	−0.0867434
$604$	0	0
$605$	−20000.1	−1.34400
$606$	0	0
$607$	−6682.67	−0.446856	−0.223428	−	0.974720i	$-0.571725\pi$
$607$	−6682.67	−0.446856	−0.223428	+	0.974720i	$0.571725\pi$
$608$	0	0
$609$	2142.00	0.142526
$610$	0	0
$611$	6472.73	0.428574
$612$	0	0
$613$	1188.30	0.0782954	0.0391477	−	0.999233i	$-0.487536\pi$
$613$	1188.30	0.0782954	0.0391477	+	0.999233i	$0.487536\pi$
$614$	0	0
$615$	−11653.3	−0.764076
$616$	0	0
$617$	10887.3	0.710381	0.355191	−	0.934794i	$-0.384416\pi$
$617$	10887.3	0.710381	0.355191	+	0.934794i	$0.384416\pi$
$618$	0	0
$619$	21376.9	1.38806	0.694030	−	0.719946i	$-0.255833\pi$
$619$	21376.9	1.38806	0.694030	+	0.719946i	$0.255833\pi$
$620$	0	0
$621$	−2775.27	−0.179336
$622$	0	0
$623$	−10882.6	−0.699842
$624$	0	0
$625$	−17193.1	−1.10036
$626$	0	0
$627$	−321.454	−0.0204747
$628$	0	0
$629$	27160.2	1.72170
$630$	0	0
$631$	12455.9	0.785835	0.392917	−	0.919574i	$-0.371466\pi$
$631$	12455.9	0.785835	0.392917	+	0.919574i	$0.371466\pi$
$632$	0	0
$633$	−3258.11	−0.204578
$634$	0	0
$635$	−29050.2	−1.81547
$636$	0	0
$637$	552.959	0.0343941
$638$	0	0
$639$	585.602	0.0362536
$640$	0	0
$641$	29946.5	1.84527	0.922634	−	0.385677i	$-0.126032\pi$
$641$	29946.5	1.84527	0.922634	+	0.385677i	$0.126032\pi$
$642$	0	0
$643$	11044.4	0.677372	0.338686	−	0.940899i	$-0.390018\pi$
$643$	11044.4	0.677372	0.338686	+	0.940899i	$0.390018\pi$
$644$	0	0
$645$	6713.75	0.409851
$646$	0	0
$647$	15103.4	0.917740	0.458870	−	0.888503i	$-0.348254\pi$
$647$	15103.4	0.917740	0.458870	+	0.888503i	$0.348254\pi$
$648$	0	0
$649$	809.023	0.0489321
$650$	0	0
$651$	−7176.14	−0.432035
$652$	0	0
$653$	6435.57	0.385671	0.192836	−	0.981231i	$-0.438232\pi$
$653$	6435.57	0.385671	0.192836	+	0.981231i	$0.438232\pi$
$654$	0	0
$655$	−11754.8	−0.701217
$656$	0	0
$657$	3423.82	0.203312
$658$	0	0
$659$	−9050.40	−0.534983	−0.267491	−	0.963560i	$-0.586195\pi$
$659$	−9050.40	−0.534983	−0.267491	+	0.963560i	$0.586195\pi$
$660$	0	0
$661$	30776.6	1.81100	0.905500	−	0.424347i	$-0.139496\pi$
$661$	30776.6	1.81100	0.905500	+	0.424347i	$0.139496\pi$
$662$	0	0
$663$	3184.80	0.186557
$664$	0	0
$665$	−2150.06	−0.125377
$666$	0	0
$667$	−10484.4	−0.608629
$668$	0	0
$669$	17635.8	1.01919
$670$	0	0
$671$	1305.75	0.0751236
$672$	0	0
$673$	−2925.61	−0.167569	−0.0837845	−	0.996484i	$-0.526701\pi$
$673$	−2925.61	−0.167569	−0.0837845	+	0.996484i	$0.526701\pi$
$674$	0	0
$675$	2993.08	0.170672
$676$	0	0
$677$	−15315.5	−0.869460	−0.434730	−	0.900561i	$-0.643156\pi$
$677$	−15315.5	−0.869460	−0.434730	+	0.900561i	$0.643156\pi$
$678$	0	0
$679$	−7595.20	−0.429274
$680$	0	0
$681$	−2505.96	−0.141011
$682$	0	0
$683$	17825.3	0.998632	0.499316	−	0.866420i	$-0.333585\pi$
$683$	17825.3	0.998632	0.499316	+	0.866420i	$0.333585\pi$
$684$	0	0
$685$	−5320.77	−0.296783
$686$	0	0
$687$	−4413.99	−0.245130
$688$	0	0
$689$	−2645.58	−0.146283
$690$	0	0
$691$	21268.5	1.17090	0.585451	−	0.810708i	$-0.300918\pi$
$691$	21268.5	1.17090	0.585451	+	0.810708i	$0.300918\pi$
$692$	0	0
$693$	337.526	0.0185015
$694$	0	0
$695$	−16577.6	−0.904783
$696$	0	0
$697$	−23794.1	−1.29306
$698$	0	0
$699$	−15376.0	−0.832006
$700$	0	0
$701$	−28776.7	−1.55047	−0.775237	−	0.631671i	$-0.782369\pi$
$701$	−28776.7	−1.55047	−0.775237	+	0.631671i	$0.782369\pi$
$702$	0	0
$703$	−5774.30	−0.309789
$704$	0	0
$705$	26426.2	1.41173
$706$	0	0
$707$	−13217.8	−0.703119
$708$	0	0
$709$	18340.1	0.971477	0.485739	−	0.874104i	$-0.338551\pi$
$709$	18340.1	0.971477	0.485739	+	0.874104i	$0.338551\pi$
$710$	0	0
$711$	7476.65	0.394369
$712$	0	0
$713$	35124.7	1.84493
$714$	0	0
$715$	928.509	0.0485654
$716$	0	0
$717$	−3587.64	−0.186866
$718$	0	0
$719$	25009.8	1.29723	0.648615	−	0.761116i	$-0.275348\pi$
$719$	25009.8	1.29723	0.648615	+	0.761116i	$0.275348\pi$
$720$	0	0
$721$	9282.08	0.479449
$722$	0	0
$723$	14566.9	0.749305
$724$	0	0
$725$	11307.2	0.579225
$726$	0	0
$727$	25248.3	1.28804	0.644022	−	0.765007i	$-0.277264\pi$
$727$	25248.3	1.28804	0.644022	+	0.765007i	$0.277264\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	13708.4	0.693601
$732$	0	0
$733$	−27929.6	−1.40737	−0.703685	−	0.710513i	$-0.748463\pi$
$733$	−27929.6	−1.40737	−0.703685	+	0.710513i	$0.748463\pi$
$734$	0	0
$735$	2257.56	0.113294
$736$	0	0
$737$	−764.605	−0.0382152
$738$	0	0
$739$	−12899.0	−0.642078	−0.321039	−	0.947066i	$-0.604032\pi$
$739$	−12899.0	−0.642078	−0.321039	+	0.947066i	$0.604032\pi$
$740$	0	0
$741$	−677.093	−0.0335676
$742$	0	0
$743$	−18723.5	−0.924493	−0.462246	−	0.886751i	$-0.652956\pi$
$743$	−18723.5	−0.924493	−0.462246	+	0.886751i	$0.652956\pi$
$744$	0	0
$745$	−44881.5	−2.20715
$746$	0	0
$747$	−4222.47	−0.206817
$748$	0	0
$749$	−9786.59	−0.477429
$750$	0	0
$751$	29841.0	1.44995	0.724975	−	0.688776i	$-0.241851\pi$
$751$	29841.0	1.44995	0.724975	+	0.688776i	$0.241851\pi$
$752$	0	0
$753$	15425.8	0.746544
$754$	0	0
$755$	34138.1	1.64558
$756$	0	0
$757$	15296.7	0.734437	0.367218	−	0.930135i	$-0.380310\pi$
$757$	15296.7	0.734437	0.367218	+	0.930135i	$0.380310\pi$
$758$	0	0
$759$	−1652.08	−0.0790073
$760$	0	0
$761$	−24627.5	−1.17312	−0.586562	−	0.809905i	$-0.699519\pi$
$761$	−24627.5	−1.17312	−0.586562	+	0.809905i	$0.699519\pi$
$762$	0	0
$763$	14618.3	0.693601
$764$	0	0
$765$	13002.5	0.614520
$766$	0	0
$767$	1704.08	0.0802228
$768$	0	0
$769$	21931.8	1.02845	0.514226	−	0.857654i	$-0.328079\pi$
$769$	21931.8	1.02845	0.514226	+	0.857654i	$0.328079\pi$
$770$	0	0
$771$	15250.9	0.712386
$772$	0	0
$773$	4968.46	0.231181	0.115591	−	0.993297i	$-0.463124\pi$
$773$	4968.46	0.231181	0.115591	+	0.993297i	$0.463124\pi$
$774$	0	0
$775$	−37881.4	−1.75579
$776$	0	0
$777$	6063.02	0.279935
$778$	0	0
$779$	5058.66	0.232664
$780$	0	0
$781$	348.599	0.0159717
$782$	0	0
$783$	2754.00	0.125696
$784$	0	0
$785$	20921.1	0.951218
$786$	0	0
$787$	−28716.4	−1.30067	−0.650337	−	0.759646i	$-0.725372\pi$
$787$	−28716.4	−1.30067	−0.650337	+	0.759646i	$0.725372\pi$
$788$	0	0
$789$	−1307.92	−0.0590155
$790$	0	0
$791$	10100.1	0.454006
$792$	0	0
$793$	2750.36	0.123163
$794$	0	0
$795$	−10801.1	−0.481856
$796$	0	0
$797$	−24209.0	−1.07594	−0.537971	−	0.842963i	$-0.680809\pi$
$797$	−24209.0	−1.07594	−0.537971	+	0.842963i	$0.680809\pi$
$798$	0	0
$799$	53957.8	2.38910
$800$	0	0
$801$	−13991.9	−0.617202
$802$	0	0
$803$	2038.15	0.0895699
$804$	0	0
$805$	−11050.0	−0.483802
$806$	0	0
$807$	−636.253	−0.0277536
$808$	0	0
$809$	11513.0	0.500342	0.250171	−	0.968202i	$-0.419513\pi$
$809$	11513.0	0.500342	0.250171	+	0.968202i	$0.419513\pi$
$810$	0	0
$811$	6959.83	0.301347	0.150674	−	0.988584i	$-0.451856\pi$
$811$	6959.83	0.301347	0.150674	+	0.988584i	$0.451856\pi$
$812$	0	0
$813$	25175.3	1.08602
$814$	0	0
$815$	−34816.0	−1.49638
$816$	0	0
$817$	−2914.42	−0.124801
$818$	0	0
$819$	710.947	0.0303327
$820$	0	0
$821$	−13997.5	−0.595027	−0.297514	−	0.954718i	$-0.596157\pi$
$821$	−13997.5	−0.595027	−0.297514	+	0.954718i	$0.596157\pi$
$822$	0	0
$823$	20721.4	0.877645	0.438823	−	0.898574i	$-0.355396\pi$
$823$	20721.4	0.877645	0.438823	+	0.898574i	$0.355396\pi$
$824$	0	0
$825$	1781.73	0.0751902
$826$	0	0
$827$	−19248.9	−0.809373	−0.404686	−	0.914456i	$-0.632619\pi$
$827$	−19248.9	−0.809373	−0.404686	+	0.914456i	$0.632619\pi$
$828$	0	0
$829$	25605.8	1.07277	0.536386	−	0.843973i	$-0.319789\pi$
$829$	25605.8	1.07277	0.536386	+	0.843973i	$0.319789\pi$
$830$	0	0
$831$	22731.4	0.948911
$832$	0	0
$833$	4609.56	0.191731
$834$	0	0
$835$	5281.49	0.218890
$836$	0	0
$837$	−9226.47	−0.381019
$838$	0	0
$839$	21217.0	0.873055	0.436528	−	0.899691i	$-0.356208\pi$
$839$	21217.0	0.873055	0.436528	+	0.899691i	$0.356208\pi$
$840$	0	0
$841$	−13985.0	−0.573414
$842$	0	0
$843$	−6774.65	−0.276787
$844$	0	0
$845$	−31784.8	−1.29400
$846$	0	0
$847$	−9116.08	−0.369814
$848$	0	0
$849$	4613.30	0.186488
$850$	0	0
$851$	−29676.4	−1.19541
$852$	0	0
$853$	−34300.0	−1.37680	−0.688400	−	0.725332i	$-0.741686\pi$
$853$	−34300.0	−1.37680	−0.688400	+	0.725332i	$0.741686\pi$
$854$	0	0
$855$	−2764.36	−0.110572
$856$	0	0
$857$	−11272.1	−0.449297	−0.224648	−	0.974440i	$-0.572123\pi$
$857$	−11272.1	−0.449297	−0.224648	+	0.974440i	$0.572123\pi$
$858$	0	0
$859$	24535.6	0.974555	0.487278	−	0.873247i	$-0.337990\pi$
$859$	24535.6	0.974555	0.487278	+	0.873247i	$0.337990\pi$
$860$	0	0
$861$	−5311.60	−0.210242
$862$	0	0
$863$	−2660.31	−0.104934	−0.0524670	−	0.998623i	$-0.516708\pi$
$863$	−2660.31	−0.104934	−0.0524670	+	0.998623i	$0.516708\pi$
$864$	0	0
$865$	−36813.6	−1.44705
$866$	0	0
$867$	11810.0	0.462617
$868$	0	0
$869$	4450.73	0.173741
$870$	0	0
$871$	−1610.52	−0.0626527
$872$	0	0
$873$	−9765.26	−0.378584
$874$	0	0
$875$	−1520.67	−0.0587519
$876$	0	0
$877$	34591.4	1.33189	0.665945	−	0.746001i	$-0.268028\pi$
$877$	34591.4	1.33189	0.665945	+	0.746001i	$0.268028\pi$
$878$	0	0
$879$	6563.31	0.251849
$880$	0	0
$881$	−6244.54	−0.238801	−0.119401	−	0.992846i	$-0.538097\pi$
$881$	−6244.54	−0.238801	−0.119401	+	0.992846i	$0.538097\pi$
$882$	0	0
$883$	2986.69	0.113828	0.0569139	−	0.998379i	$-0.481874\pi$
$883$	2986.69	0.113828	0.0569139	+	0.998379i	$0.481874\pi$
$884$	0	0
$885$	6957.24	0.264254
$886$	0	0
$887$	−39005.3	−1.47652	−0.738258	−	0.674518i	$-0.764351\pi$
$887$	−39005.3	−1.47652	−0.738258	+	0.674518i	$0.764351\pi$
$888$	0	0
$889$	−13241.1	−0.499542
$890$	0	0
$891$	433.962	0.0163168
$892$	0	0
$893$	−11471.5	−0.429876
$894$	0	0
$895$	−20801.8	−0.776901
$896$	0	0
$897$	−3479.84	−0.129530
$898$	0	0
$899$	−34855.5	−1.29310
$900$	0	0
$901$	−22054.0	−0.815456
$902$	0	0
$903$	3060.14	0.112774
$904$	0	0
$905$	−50818.3	−1.86658
$906$	0	0
$907$	−14724.9	−0.539065	−0.269532	−	0.962991i	$-0.586869\pi$
$907$	−14724.9	−0.539065	−0.269532	+	0.962991i	$0.586869\pi$
$908$	0	0
$909$	−16994.3	−0.620093
$910$	0	0
$911$	−1770.06	−0.0643739	−0.0321870	−	0.999482i	$-0.510247\pi$
$911$	−1770.06	−0.0643739	−0.0321870	+	0.999482i	$0.510247\pi$
$912$	0	0
$913$	−2513.57	−0.0911139
$914$	0	0
$915$	11228.9	0.405700
$916$	0	0
$917$	−5357.85	−0.192946
$918$	0	0
$919$	35597.8	1.27776	0.638881	−	0.769306i	$-0.279398\pi$
$919$	35597.8	1.27776	0.638881	+	0.769306i	$0.279398\pi$
$920$	0	0
$921$	−26846.9	−0.960516
$922$	0	0
$923$	734.272	0.0261851
$924$	0	0
$925$	32005.4	1.13766
$926$	0	0
$927$	11934.1	0.422834
$928$	0	0
$929$	14466.3	0.510899	0.255449	−	0.966822i	$-0.417777\pi$
$929$	14466.3	0.510899	0.255449	+	0.966822i	$0.417777\pi$
$930$	0	0
$931$	−980.000	−0.0344986
$932$	0	0
$933$	26667.2	0.935739
$934$	0	0
$935$	7740.21	0.270729
$936$	0	0
$937$	−16004.2	−0.557989	−0.278995	−	0.960293i	$-0.590001\pi$
$937$	−16004.2	−0.557989	−0.278995	+	0.960293i	$0.590001\pi$
$938$	0	0
$939$	13970.7	0.485532
$940$	0	0
$941$	37463.2	1.29784	0.648919	−	0.760857i	$-0.275221\pi$
$941$	37463.2	1.29784	0.648919	+	0.760857i	$0.275221\pi$
$942$	0	0
$943$	25998.4	0.897800
$944$	0	0
$945$	2902.58	0.0999163
$946$	0	0
$947$	−9667.92	−0.331748	−0.165874	−	0.986147i	$-0.553044\pi$
$947$	−9667.92	−0.331748	−0.165874	+	0.986147i	$0.553044\pi$
$948$	0	0
$949$	4293.04	0.146847
$950$	0	0
$951$	−29587.8	−1.00889
$952$	0	0
$953$	−46563.7	−1.58274	−0.791368	−	0.611340i	$-0.790631\pi$
$953$	−46563.7	−1.58274	−0.791368	+	0.611340i	$0.790631\pi$
$954$	0	0
$955$	−42340.7	−1.43467
$956$	0	0
$957$	1639.41	0.0553759
$958$	0	0
$959$	−2425.22	−0.0816625
$960$	0	0
$961$	86982.2	2.91975
$962$	0	0
$963$	−12582.8	−0.421053
$964$	0	0
$965$	−14499.6	−0.483687
$966$	0	0
$967$	−54214.1	−1.80290	−0.901451	−	0.432881i	$-0.857497\pi$
$967$	−54214.1	−1.80290	−0.901451	+	0.432881i	$0.857497\pi$
$968$	0	0
$969$	−5644.36	−0.187124
$970$	0	0
$971$	−7503.37	−0.247986	−0.123993	−	0.992283i	$-0.539570\pi$
$971$	−7503.37	−0.247986	−0.123993	+	0.992283i	$0.539570\pi$
$972$	0	0
$973$	−7556.09	−0.248959
$974$	0	0
$975$	3752.94	0.123272
$976$	0	0
$977$	−9824.14	−0.321701	−0.160851	−	0.986979i	$-0.551424\pi$
$977$	−9824.14	−0.321701	−0.160851	+	0.986979i	$0.551424\pi$
$978$	0	0
$979$	−8329.15	−0.271911
$980$	0	0
$981$	18794.9	0.611698
$982$	0	0
$983$	−18332.5	−0.594827	−0.297413	−	0.954749i	$-0.596124\pi$
$983$	−18332.5	−0.594827	−0.297413	+	0.954749i	$0.596124\pi$
$984$	0	0
$985$	13688.0	0.442776
$986$	0	0
$987$	12045.1	0.388449
$988$	0	0
$989$	−14978.3	−0.481581
$990$	0	0
$991$	−58520.5	−1.87585	−0.937924	−	0.346841i	$-0.887254\pi$
$991$	−58520.5	−1.87585	−0.937924	+	0.346841i	$0.887254\pi$
$992$	0	0
$993$	−21254.8	−0.679254
$994$	0	0
$995$	−20338.9	−0.648028
$996$	0	0
$997$	35134.7	1.11608	0.558038	−	0.829815i	$-0.311554\pi$
$997$	35134.7	1.11608	0.558038	+	0.829815i	$0.311554\pi$
$998$	0	0
$999$	7795.31	0.246880

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.4.a.o.1.2		2
3.2	odd	2		1008.4.a.bg.1.1		2
4.3	odd	2		168.4.a.g.1.2	✓	2
7.6	odd	2		2352.4.a.br.1.1		2
8.3	odd	2		1344.4.a.bq.1.1		2
8.5	even	2		1344.4.a.bi.1.1		2
12.11	even	2		504.4.a.n.1.1		2
28.27	even	2		1176.4.a.w.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.a.g.1.2	✓	2	4.3	odd	2
336.4.a.o.1.2		2	1.1	even	1	trivial
504.4.a.n.1.1		2	12.11	even	2
1008.4.a.bg.1.1		2	3.2	odd	2
1176.4.a.w.1.1		2	28.27	even	2
1344.4.a.bi.1.1		2	8.5	even	2
1344.4.a.bq.1.1		2	8.3	odd	2
2352.4.a.br.1.1		2	7.6	odd	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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	336.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +15.3576 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +15.3576 q^{5} +7.00000 q^{7} +9.00000 q^{9} +5.35756 q^{11} +11.2849 q^{13} +46.0727 q^{15} +94.0727 q^{17} -20.0000 q^{19} +21.0000 q^{21} -102.788 q^{23} +110.855 q^{25} +27.0000 q^{27} +102.000 q^{29} -341.721 q^{31} +16.0727 q^{33} +107.503 q^{35} +288.715 q^{37} +33.8546 q^{39} -252.933 q^{41} +145.721 q^{43} +138.218 q^{45} +573.576 q^{47} +49.0000 q^{49} +282.218 q^{51} -234.436 q^{53} +82.2790 q^{55} -60.0000 q^{57} +151.006 q^{59} +243.721 q^{61} +63.0000 q^{63} +173.308 q^{65} -142.715 q^{67} -308.363 q^{69} +65.0668 q^{71} +380.424 q^{73} +332.564 q^{75} +37.5029 q^{77} +830.738 q^{79} +81.0000 q^{81} -469.163 q^{83} +1444.73 q^{85} +306.000 q^{87} -1554.65 q^{89} +78.9942 q^{91} -1025.16 q^{93} -307.151 q^{95} -1085.03 q^{97} +48.2180 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 6 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 - 6 * q^5 + 14 * q^7 + 18 * q^9	\( 2 q + 6 q^{3} - 6 q^{5} + 14 q^{7} + 18 q^{9} - 26 q^{11} + 96 q^{13} - 18 q^{15} + 78 q^{17} - 40 q^{19} + 42 q^{21} - 22 q^{23} + 442 q^{25} + 54 q^{27} + 204 q^{29} - 96 q^{31} - 78 q^{33} - 42 q^{35} + 504 q^{37} + 288 q^{39} - 102 q^{41} - 296 q^{43} - 54 q^{45} + 780 q^{47} + 98 q^{49} + 234 q^{51} + 192 q^{53} + 752 q^{55} - 120 q^{57} - 212 q^{59} - 100 q^{61} + 126 q^{63} - 1636 q^{65} - 212 q^{67} - 66 q^{69} + 534 q^{71} + 1128 q^{73} + 1326 q^{75} - 182 q^{77} - 468 q^{79} + 162 q^{81} + 824 q^{83} + 1788 q^{85} + 612 q^{87} - 2118 q^{89} + 672 q^{91} - 288 q^{93} + 120 q^{95} + 400 q^{97} - 234 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 6 * q^5 + 14 * q^7 + 18 * q^9 - 26 * q^11 + 96 * q^13 - 18 * q^15 + 78 * q^17 - 40 * q^19 + 42 * q^21 - 22 * q^23 + 442 * q^25 + 54 * q^27 + 204 * q^29 - 96 * q^31 - 78 * q^33 - 42 * q^35 + 504 * q^37 + 288 * q^39 - 102 * q^41 - 296 * q^43 - 54 * q^45 + 780 * q^47 + 98 * q^49 + 234 * q^51 + 192 * q^53 + 752 * q^55 - 120 * q^57 - 212 * q^59 - 100 * q^61 + 126 * q^63 - 1636 * q^65 - 212 * q^67 - 66 * q^69 + 534 * q^71 + 1128 * q^73 + 1326 * q^75 - 182 * q^77 - 468 * q^79 + 162 * q^81 + 824 * q^83 + 1788 * q^85 + 612 * q^87 - 2118 * q^89 + 672 * q^91 - 288 * q^93 + 120 * q^95 + 400 * q^97 - 234 * q^99

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