LMFDB - Embedded newform 1344.4.a.bt.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(1,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, 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("1344.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.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:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.22700.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 28x + 12 $ x^3 - x^2 - 28*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 672)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$5.60988$ of defining polynomial
Character	$\chi$	$=$	1344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +21.0807 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +21.0807 q^{5} -7.00000 q^{7} +9.00000 q^{9} +59.9598 q^{11} +58.8791 q^{13} -63.2421 q^{15} +56.5244 q^{17} +17.6050 q^{19} +21.0000 q^{21} +140.992 q^{23} +319.395 q^{25} -27.0000 q^{27} -203.911 q^{29} -215.669 q^{31} -179.879 q^{33} -147.565 q^{35} +265.283 q^{37} -176.637 q^{39} +256.138 q^{41} -119.355 q^{43} +189.726 q^{45} +118.161 q^{47} +49.0000 q^{49} -169.573 q^{51} -106.912 q^{53} +1263.99 q^{55} -52.8151 q^{57} -846.323 q^{59} +875.316 q^{61} -63.0000 q^{63} +1241.21 q^{65} -530.484 q^{67} -422.976 q^{69} -629.282 q^{71} -843.806 q^{73} -958.186 q^{75} -419.718 q^{77} +332.743 q^{79} +81.0000 q^{81} -525.146 q^{83} +1191.57 q^{85} +611.734 q^{87} -362.945 q^{89} -412.154 q^{91} +647.008 q^{93} +371.127 q^{95} -272.550 q^{97} +539.638 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} + 10 q^{5} - 21 q^{7} + 27 q^{9}+O(q^{10}) $ 3 * q - 9 * q^3 + 10 * q^5 - 21 * q^7 + 27 * q^9	$ 3 q - 9 q^{3} + 10 q^{5} - 21 q^{7} + 27 q^{9} + 50 q^{13} - 30 q^{15} + 30 q^{17} - 140 q^{19} + 63 q^{21} + 56 q^{23} + 325 q^{25} - 81 q^{27} - 298 q^{29} - 80 q^{31} - 70 q^{35} - 10 q^{37} - 150 q^{39} + 390 q^{41} - 784 q^{43} + 90 q^{45} + 248 q^{47} + 147 q^{49} - 90 q^{51} - 10 q^{53} + 1360 q^{55} + 420 q^{57} - 1500 q^{59} + 810 q^{61} - 189 q^{63} + 860 q^{65} - 1272 q^{67} - 168 q^{69} + 160 q^{71} - 1170 q^{73} - 975 q^{75} + 840 q^{79} + 243 q^{81} - 1564 q^{83} + 2740 q^{85} + 894 q^{87} - 178 q^{89} - 350 q^{91} + 240 q^{93} + 2840 q^{95} - 130 q^{97}+O(q^{100}) $ 3 * q - 9 * q^3 + 10 * q^5 - 21 * q^7 + 27 * q^9 + 50 * q^13 - 30 * q^15 + 30 * q^17 - 140 * q^19 + 63 * q^21 + 56 * q^23 + 325 * q^25 - 81 * q^27 - 298 * q^29 - 80 * q^31 - 70 * q^35 - 10 * q^37 - 150 * q^39 + 390 * q^41 - 784 * q^43 + 90 * q^45 + 248 * q^47 + 147 * q^49 - 90 * q^51 - 10 * q^53 + 1360 * q^55 + 420 * q^57 - 1500 * q^59 + 810 * q^61 - 189 * q^63 + 860 * q^65 - 1272 * q^67 - 168 * q^69 + 160 * q^71 - 1170 * q^73 - 975 * q^75 + 840 * q^79 + 243 * q^81 - 1564 * q^83 + 2740 * q^85 + 894 * q^87 - 178 * q^89 - 350 * q^91 + 240 * q^93 + 2840 * q^95 - 130 * q^97

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
$5$	21.0807	1.88551	0.942757	−	0.333481i	$-0.108223\pi$
$5$	21.0807	1.88551	0.942757	+	0.333481i	$0.108223\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	59.9598	1.64350	0.821752	−	0.569845i	$-0.192997\pi$
$11$	59.9598	1.64350	0.821752	+	0.569845i	$0.192997\pi$
$12$	0	0
$13$	58.8791	1.25616	0.628081	−	0.778148i	$-0.283840\pi$
$13$	58.8791	1.25616	0.628081	+	0.778148i	$0.283840\pi$
$14$	0	0
$15$	−63.2421	−1.08860
$16$	0	0
$17$	56.5244	0.806422	0.403211	−	0.915107i	$-0.367894\pi$
$17$	56.5244	0.806422	0.403211	+	0.915107i	$0.367894\pi$
$18$	0	0
$19$	17.6050	0.212572	0.106286	−	0.994336i	$-0.466104\pi$
$19$	17.6050	0.212572	0.106286	+	0.994336i	$0.466104\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	140.992	1.27821	0.639105	−	0.769119i	$-0.279305\pi$
$23$	140.992	1.27821	0.639105	+	0.769119i	$0.279305\pi$
$24$	0	0
$25$	319.395	2.55516
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−203.911	−1.30570	−0.652851	−	0.757486i	$-0.726427\pi$
$29$	−203.911	−1.30570	−0.652851	+	0.757486i	$0.726427\pi$
$30$	0	0
$31$	−215.669	−1.24953	−0.624764	−	0.780814i	$-0.714805\pi$
$31$	−215.669	−1.24953	−0.624764	+	0.780814i	$0.714805\pi$
$32$	0	0
$33$	−179.879	−0.948878
$34$	0	0
$35$	−147.565	−0.712657
$36$	0	0
$37$	265.283	1.17871	0.589354	−	0.807875i	$-0.299382\pi$
$37$	265.283	1.17871	0.589354	+	0.807875i	$0.299382\pi$
$38$	0	0
$39$	−176.637	−0.725246
$40$	0	0
$41$	256.138	0.975658	0.487829	−	0.872939i	$-0.337789\pi$
$41$	256.138	0.975658	0.487829	+	0.872939i	$0.337789\pi$
$42$	0	0
$43$	−119.355	−0.423288	−0.211644	−	0.977347i	$-0.567882\pi$
$43$	−119.355	−0.423288	−0.211644	+	0.977347i	$0.567882\pi$
$44$	0	0
$45$	189.726	0.628505
$46$	0	0
$47$	118.161	0.366715	0.183358	−	0.983046i	$-0.441303\pi$
$47$	118.161	0.366715	0.183358	+	0.983046i	$0.441303\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−169.573	−0.465588
$52$	0	0
$53$	−106.912	−0.277085	−0.138543	−	0.990356i	$-0.544242\pi$
$53$	−106.912	−0.277085	−0.138543	+	0.990356i	$0.544242\pi$
$54$	0	0
$55$	1263.99	3.09885
$56$	0	0
$57$	−52.8151	−0.122729
$58$	0	0
$59$	−846.323	−1.86749	−0.933745	−	0.357939i	$-0.883479\pi$
$59$	−846.323	−1.86749	−0.933745	+	0.357939i	$0.883479\pi$
$60$	0	0
$61$	875.316	1.83726	0.918629	−	0.395121i	$-0.129297\pi$
$61$	875.316	1.83726	0.918629	+	0.395121i	$0.129297\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	1241.21	2.36851
$66$	0	0
$67$	−530.484	−0.967298	−0.483649	−	0.875262i	$-0.660689\pi$
$67$	−530.484	−0.967298	−0.483649	+	0.875262i	$0.660689\pi$
$68$	0	0
$69$	−422.976	−0.737975
$70$	0	0
$71$	−629.282	−1.05186	−0.525930	−	0.850528i	$-0.676283\pi$
$71$	−629.282	−1.05186	−0.525930	+	0.850528i	$0.676283\pi$
$72$	0	0
$73$	−843.806	−1.35288	−0.676439	−	0.736499i	$-0.736478\pi$
$73$	−843.806	−1.35288	−0.676439	+	0.736499i	$0.736478\pi$
$74$	0	0
$75$	−958.186	−1.47522
$76$	0	0
$77$	−419.718	−0.621186
$78$	0	0
$79$	332.743	0.473879	0.236940	−	0.971524i	$-0.423856\pi$
$79$	332.743	0.473879	0.236940	+	0.971524i	$0.423856\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−525.146	−0.694485	−0.347242	−	0.937775i	$-0.612882\pi$
$83$	−525.146	−0.694485	−0.347242	+	0.937775i	$0.612882\pi$
$84$	0	0
$85$	1191.57	1.52052
$86$	0	0
$87$	611.734	0.753848
$88$	0	0
$89$	−362.945	−0.432270	−0.216135	−	0.976363i	$-0.569345\pi$
$89$	−362.945	−0.432270	−0.216135	+	0.976363i	$0.569345\pi$
$90$	0	0
$91$	−412.154	−0.474785
$92$	0	0
$93$	647.008	0.721415
$94$	0	0
$95$	371.127	0.400808
$96$	0	0
$97$	−272.550	−0.285292	−0.142646	−	0.989774i	$-0.545561\pi$
$97$	−272.550	−0.285292	−0.142646	+	0.989774i	$0.545561\pi$
$98$	0	0
$99$	539.638	0.547835
$100$	0	0
$101$	−1502.54	−1.48028	−0.740138	−	0.672455i	$-0.765240\pi$
$101$	−1502.54	−1.48028	−0.740138	+	0.672455i	$0.765240\pi$
$102$	0	0
$103$	901.962	0.862844	0.431422	−	0.902150i	$-0.358012\pi$
$103$	901.962	0.862844	0.431422	+	0.902150i	$0.358012\pi$
$104$	0	0
$105$	442.694	0.411453
$106$	0	0
$107$	607.026	0.548443	0.274222	−	0.961666i	$-0.411580\pi$
$107$	607.026	0.548443	0.274222	+	0.961666i	$0.411580\pi$
$108$	0	0
$109$	−939.454	−0.825536	−0.412768	−	0.910836i	$-0.635438\pi$
$109$	−939.454	−0.825536	−0.412768	+	0.910836i	$0.635438\pi$
$110$	0	0
$111$	−795.848	−0.680528
$112$	0	0
$113$	−1951.34	−1.62449	−0.812243	−	0.583319i	$-0.801754\pi$
$113$	−1951.34	−1.62449	−0.812243	+	0.583319i	$0.801754\pi$
$114$	0	0
$115$	2972.21	2.41008
$116$	0	0
$117$	529.912	0.418721
$118$	0	0
$119$	−395.671	−0.304799
$120$	0	0
$121$	2264.17	1.70111
$122$	0	0
$123$	−768.413	−0.563297
$124$	0	0
$125$	4097.99	2.93228
$126$	0	0
$127$	114.724	0.0801584	0.0400792	−	0.999197i	$-0.487239\pi$
$127$	114.724	0.0801584	0.0400792	+	0.999197i	$0.487239\pi$
$128$	0	0
$129$	358.064	0.244386
$130$	0	0
$131$	849.486	0.566564	0.283282	−	0.959037i	$-0.408577\pi$
$131$	849.486	0.566564	0.283282	+	0.959037i	$0.408577\pi$
$132$	0	0
$133$	−123.235	−0.0803448
$134$	0	0
$135$	−569.179	−0.362867
$136$	0	0
$137$	1596.50	0.995607	0.497804	−	0.867290i	$-0.334140\pi$
$137$	1596.50	0.995607	0.497804	+	0.867290i	$0.334140\pi$
$138$	0	0
$139$	−492.305	−0.300408	−0.150204	−	0.988655i	$-0.547993\pi$
$139$	−492.305	−0.300408	−0.150204	+	0.988655i	$0.547993\pi$
$140$	0	0
$141$	−354.484	−0.211723
$142$	0	0
$143$	3530.38	2.06451
$144$	0	0
$145$	−4298.59	−2.46192
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	−2541.14	−1.39717	−0.698585	−	0.715527i	$-0.746187\pi$
$149$	−2541.14	−1.39717	−0.698585	+	0.715527i	$0.746187\pi$
$150$	0	0
$151$	1318.15	0.710393	0.355196	−	0.934792i	$-0.384414\pi$
$151$	1318.15	0.710393	0.355196	+	0.934792i	$0.384414\pi$
$152$	0	0
$153$	508.719	0.268807
$154$	0	0
$155$	−4546.46	−2.35600
$156$	0	0
$157$	−463.769	−0.235750	−0.117875	−	0.993028i	$-0.537608\pi$
$157$	−463.769	−0.235750	−0.117875	+	0.993028i	$0.537608\pi$
$158$	0	0
$159$	320.736	0.159975
$160$	0	0
$161$	−986.944	−0.483118
$162$	0	0
$163$	−2857.26	−1.37299	−0.686497	−	0.727133i	$-0.740852\pi$
$163$	−2857.26	−1.37299	−0.686497	+	0.727133i	$0.740852\pi$
$164$	0	0
$165$	−3791.98	−1.78912
$166$	0	0
$167$	−436.261	−0.202149	−0.101075	−	0.994879i	$-0.532228\pi$
$167$	−436.261	−0.202149	−0.101075	+	0.994879i	$0.532228\pi$
$168$	0	0
$169$	1269.75	0.577945
$170$	0	0
$171$	158.445	0.0708575
$172$	0	0
$173$	2335.25	1.02628	0.513138	−	0.858306i	$-0.328483\pi$
$173$	2335.25	1.02628	0.513138	+	0.858306i	$0.328483\pi$
$174$	0	0
$175$	−2235.77	−0.965761
$176$	0	0
$177$	2538.97	1.07820
$178$	0	0
$179$	−1980.67	−0.827054	−0.413527	−	0.910492i	$-0.635703\pi$
$179$	−1980.67	−0.827054	−0.413527	+	0.910492i	$0.635703\pi$
$180$	0	0
$181$	2454.90	1.00813	0.504063	−	0.863667i	$-0.331838\pi$
$181$	2454.90	1.00813	0.504063	+	0.863667i	$0.331838\pi$
$182$	0	0
$183$	−2625.95	−1.06074
$184$	0	0
$185$	5592.34	2.22247
$186$	0	0
$187$	3389.19	1.32536
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	−1751.76	−0.663629	−0.331815	−	0.943345i	$-0.607661\pi$
$191$	−1751.76	−0.663629	−0.331815	+	0.943345i	$0.607661\pi$
$192$	0	0
$193$	2175.56	0.811401	0.405700	−	0.914006i	$-0.367028\pi$
$193$	2175.56	0.811401	0.405700	+	0.914006i	$0.367028\pi$
$194$	0	0
$195$	−3723.63	−1.36746
$196$	0	0
$197$	3426.13	1.23909	0.619547	−	0.784960i	$-0.287316\pi$
$197$	3426.13	1.23909	0.619547	+	0.784960i	$0.287316\pi$
$198$	0	0
$199$	5371.17	1.91333	0.956664	−	0.291195i	$-0.0940527\pi$
$199$	5371.17	1.91333	0.956664	+	0.291195i	$0.0940527\pi$
$200$	0	0
$201$	1591.45	0.558470
$202$	0	0
$203$	1427.38	0.493509
$204$	0	0
$205$	5399.56	1.83962
$206$	0	0
$207$	1268.93	0.426070
$208$	0	0
$209$	1055.59	0.349364
$210$	0	0
$211$	705.367	0.230140	0.115070	−	0.993357i	$-0.463291\pi$
$211$	705.367	0.230140	0.115070	+	0.993357i	$0.463291\pi$
$212$	0	0
$213$	1887.85	0.607292
$214$	0	0
$215$	−2516.07	−0.798116
$216$	0	0
$217$	1509.69	0.472277
$218$	0	0
$219$	2531.42	0.781084
$220$	0	0
$221$	3328.10	1.01300
$222$	0	0
$223$	69.2083	0.0207826	0.0103913	−	0.999946i	$-0.496692\pi$
$223$	69.2083	0.0207826	0.0103913	+	0.999946i	$0.496692\pi$
$224$	0	0
$225$	2874.56	0.851721
$226$	0	0
$227$	4763.74	1.39287	0.696433	−	0.717622i	$-0.254769\pi$
$227$	4763.74	1.39287	0.696433	+	0.717622i	$0.254769\pi$
$228$	0	0
$229$	−2049.55	−0.591432	−0.295716	−	0.955276i	$-0.595558\pi$
$229$	−2049.55	−0.591432	−0.295716	+	0.955276i	$0.595558\pi$
$230$	0	0
$231$	1259.16	0.358642
$232$	0	0
$233$	2326.53	0.654146	0.327073	−	0.944999i	$-0.393938\pi$
$233$	2326.53	0.654146	0.327073	+	0.944999i	$0.393938\pi$
$234$	0	0
$235$	2490.92	0.691446
$236$	0	0
$237$	−998.228	−0.273594
$238$	0	0
$239$	5835.81	1.57944	0.789722	−	0.613465i	$-0.210225\pi$
$239$	5835.81	1.57944	0.789722	+	0.613465i	$0.210225\pi$
$240$	0	0
$241$	−1887.18	−0.504414	−0.252207	−	0.967673i	$-0.581156\pi$
$241$	−1887.18	−0.504414	−0.252207	+	0.967673i	$0.581156\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	1032.95	0.269359
$246$	0	0
$247$	1036.57	0.267026
$248$	0	0
$249$	1575.44	0.400961
$250$	0	0
$251$	811.569	0.204087	0.102043	−	0.994780i	$-0.467462\pi$
$251$	811.569	0.204087	0.102043	+	0.994780i	$0.467462\pi$
$252$	0	0
$253$	8453.84	2.10075
$254$	0	0
$255$	−3574.72	−0.877872
$256$	0	0
$257$	3291.22	0.798835	0.399417	−	0.916769i	$-0.369212\pi$
$257$	3291.22	0.798835	0.399417	+	0.916769i	$0.369212\pi$
$258$	0	0
$259$	−1856.98	−0.445510
$260$	0	0
$261$	−1835.20	−0.435234
$262$	0	0
$263$	−1954.14	−0.458165	−0.229082	−	0.973407i	$-0.573573\pi$
$263$	−1954.14	−0.458165	−0.229082	+	0.973407i	$0.573573\pi$
$264$	0	0
$265$	−2253.78	−0.522448
$266$	0	0
$267$	1088.83	0.249571
$268$	0	0
$269$	1828.40	0.414422	0.207211	−	0.978296i	$-0.433561\pi$
$269$	1828.40	0.414422	0.207211	+	0.978296i	$0.433561\pi$
$270$	0	0
$271$	−2912.25	−0.652792	−0.326396	−	0.945233i	$-0.605834\pi$
$271$	−2912.25	−0.652792	−0.326396	+	0.945233i	$0.605834\pi$
$272$	0	0
$273$	1236.46	0.274117
$274$	0	0
$275$	19150.9	4.19942
$276$	0	0
$277$	−459.040	−0.0995705	−0.0497853	−	0.998760i	$-0.515854\pi$
$277$	−459.040	−0.0995705	−0.0497853	+	0.998760i	$0.515854\pi$
$278$	0	0
$279$	−1941.02	−0.416509
$280$	0	0
$281$	−5797.19	−1.23072	−0.615359	−	0.788247i	$-0.710989\pi$
$281$	−5797.19	−1.23072	−0.615359	+	0.788247i	$0.710989\pi$
$282$	0	0
$283$	−47.8839	−0.0100580	−0.00502898	−	0.999987i	$-0.501601\pi$
$283$	−47.8839	−0.0100580	−0.00502898	+	0.999987i	$0.501601\pi$
$284$	0	0
$285$	−1113.38	−0.231407
$286$	0	0
$287$	−1792.96	−0.368764
$288$	0	0
$289$	−1718.00	−0.349684
$290$	0	0
$291$	817.651	0.164713
$292$	0	0
$293$	−7073.78	−1.41043	−0.705213	−	0.708996i	$-0.749149\pi$
$293$	−7073.78	−1.41043	−0.705213	+	0.708996i	$0.749149\pi$
$294$	0	0
$295$	−17841.1	−3.52118
$296$	0	0
$297$	−1618.91	−0.316293
$298$	0	0
$299$	8301.47	1.60564
$300$	0	0
$301$	835.482	0.159988
$302$	0	0
$303$	4507.61	0.854638
$304$	0	0
$305$	18452.3	3.46418
$306$	0	0
$307$	−5652.53	−1.05084	−0.525418	−	0.850844i	$-0.676091\pi$
$307$	−5652.53	−1.05084	−0.525418	+	0.850844i	$0.676091\pi$
$308$	0	0
$309$	−2705.89	−0.498163
$310$	0	0
$311$	9073.01	1.65429	0.827144	−	0.561990i	$-0.189964\pi$
$311$	9073.01	1.65429	0.827144	+	0.561990i	$0.189964\pi$
$312$	0	0
$313$	−1543.32	−0.278701	−0.139351	−	0.990243i	$-0.544501\pi$
$313$	−1543.32	−0.278701	−0.139351	+	0.990243i	$0.544501\pi$
$314$	0	0
$315$	−1328.08	−0.237552
$316$	0	0
$317$	2259.71	0.400373	0.200186	−	0.979758i	$-0.435845\pi$
$317$	2259.71	0.400373	0.200186	+	0.979758i	$0.435845\pi$
$318$	0	0
$319$	−12226.5	−2.14593
$320$	0	0
$321$	−1821.08	−0.316644
$322$	0	0
$323$	995.114	0.171423
$324$	0	0
$325$	18805.7	3.20970
$326$	0	0
$327$	2818.36	0.476623
$328$	0	0
$329$	−827.130	−0.138605
$330$	0	0
$331$	−524.151	−0.0870390	−0.0435195	−	0.999053i	$-0.513857\pi$
$331$	−524.151	−0.0870390	−0.0435195	+	0.999053i	$0.513857\pi$
$332$	0	0
$333$	2387.54	0.392903
$334$	0	0
$335$	−11183.0	−1.82385
$336$	0	0
$337$	4637.50	0.749616	0.374808	−	0.927103i	$-0.377709\pi$
$337$	4637.50	0.749616	0.374808	+	0.927103i	$0.377709\pi$
$338$	0	0
$339$	5854.03	0.937897
$340$	0	0
$341$	−12931.5	−2.05360
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−8916.62	−1.39146
$346$	0	0
$347$	−8430.95	−1.30431	−0.652157	−	0.758084i	$-0.726136\pi$
$347$	−8430.95	−1.30431	−0.652157	+	0.758084i	$0.726136\pi$
$348$	0	0
$349$	7859.51	1.20547	0.602736	−	0.797941i	$-0.294077\pi$
$349$	7859.51	1.20547	0.602736	+	0.797941i	$0.294077\pi$
$350$	0	0
$351$	−1589.74	−0.241749
$352$	0	0
$353$	−4156.41	−0.626695	−0.313347	−	0.949639i	$-0.601450\pi$
$353$	−4156.41	−0.626695	−0.313347	+	0.949639i	$0.601450\pi$
$354$	0	0
$355$	−13265.7	−1.98330
$356$	0	0
$357$	1187.01	0.175976
$358$	0	0
$359$	11281.2	1.65849	0.829243	−	0.558889i	$-0.188772\pi$
$359$	11281.2	1.65849	0.829243	+	0.558889i	$0.188772\pi$
$360$	0	0
$361$	−6549.06	−0.954813
$362$	0	0
$363$	−6792.52	−0.982134
$364$	0	0
$365$	−17788.0	−2.55087
$366$	0	0
$367$	−5447.87	−0.774868	−0.387434	−	0.921897i	$-0.626638\pi$
$367$	−5447.87	−0.774868	−0.387434	+	0.921897i	$0.626638\pi$
$368$	0	0
$369$	2305.24	0.325219
$370$	0	0
$371$	748.385	0.104728
$372$	0	0
$373$	−828.837	−0.115055	−0.0575276	−	0.998344i	$-0.518322\pi$
$373$	−828.837	−0.115055	−0.0575276	+	0.998344i	$0.518322\pi$
$374$	0	0
$375$	−12294.0	−1.69295
$376$	0	0
$377$	−12006.1	−1.64017
$378$	0	0
$379$	5435.31	0.736657	0.368329	−	0.929696i	$-0.379930\pi$
$379$	5435.31	0.736657	0.368329	+	0.929696i	$0.379930\pi$
$380$	0	0
$381$	−344.172	−0.0462795
$382$	0	0
$383$	182.241	0.0243136	0.0121568	−	0.999926i	$-0.496130\pi$
$383$	182.241	0.0243136	0.0121568	+	0.999926i	$0.496130\pi$
$384$	0	0
$385$	−8847.95	−1.17126
$386$	0	0
$387$	−1074.19	−0.141096
$388$	0	0
$389$	2185.13	0.284809	0.142404	−	0.989809i	$-0.454517\pi$
$389$	2185.13	0.284809	0.142404	+	0.989809i	$0.454517\pi$
$390$	0	0
$391$	7969.48	1.03078
$392$	0	0
$393$	−2548.46	−0.327106
$394$	0	0
$395$	7014.44	0.893506
$396$	0	0
$397$	−8926.17	−1.12844	−0.564221	−	0.825623i	$-0.690824\pi$
$397$	−8926.17	−1.12844	−0.564221	+	0.825623i	$0.690824\pi$
$398$	0	0
$399$	369.706	0.0463871
$400$	0	0
$401$	−3517.58	−0.438054	−0.219027	−	0.975719i	$-0.570288\pi$
$401$	−3517.58	−0.438054	−0.219027	+	0.975719i	$0.570288\pi$
$402$	0	0
$403$	−12698.4	−1.56961
$404$	0	0
$405$	1707.54	0.209502
$406$	0	0
$407$	15906.3	1.93721
$408$	0	0
$409$	−10647.5	−1.28725	−0.643625	−	0.765341i	$-0.722570\pi$
$409$	−10647.5	−1.28725	−0.643625	+	0.765341i	$0.722570\pi$
$410$	0	0
$411$	−4789.50	−0.574814
$412$	0	0
$413$	5924.26	0.705845
$414$	0	0
$415$	−11070.4	−1.30946
$416$	0	0
$417$	1476.91	0.173441
$418$	0	0
$419$	−4840.56	−0.564384	−0.282192	−	0.959358i	$-0.591061\pi$
$419$	−4840.56	−0.564384	−0.282192	+	0.959358i	$0.591061\pi$
$420$	0	0
$421$	−4441.72	−0.514195	−0.257097	−	0.966385i	$-0.582766\pi$
$421$	−4441.72	−0.514195	−0.257097	+	0.966385i	$0.582766\pi$
$422$	0	0
$423$	1063.45	0.122238
$424$	0	0
$425$	18053.6	2.06054
$426$	0	0
$427$	−6127.21	−0.694418
$428$	0	0
$429$	−10591.1	−1.19194
$430$	0	0
$431$	7883.82	0.881092	0.440546	−	0.897730i	$-0.354785\pi$
$431$	7883.82	0.881092	0.440546	+	0.897730i	$0.354785\pi$
$432$	0	0
$433$	−7256.81	−0.805405	−0.402702	−	0.915331i	$-0.631929\pi$
$433$	−7256.81	−0.805405	−0.402702	+	0.915331i	$0.631929\pi$
$434$	0	0
$435$	12895.8	1.42139
$436$	0	0
$437$	2482.17	0.271712
$438$	0	0
$439$	−4772.17	−0.518823	−0.259411	−	0.965767i	$-0.583529\pi$
$439$	−4772.17	−0.518823	−0.259411	+	0.965767i	$0.583529\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−15618.7	−1.67510	−0.837548	−	0.546364i	$-0.816012\pi$
$443$	−15618.7	−1.67510	−0.837548	+	0.546364i	$0.816012\pi$
$444$	0	0
$445$	−7651.12	−0.815051
$446$	0	0
$447$	7623.42	0.806657
$448$	0	0
$449$	11249.5	1.18239	0.591197	−	0.806527i	$-0.298656\pi$
$449$	11249.5	1.18239	0.591197	+	0.806527i	$0.298656\pi$
$450$	0	0
$451$	15358.0	1.60350
$452$	0	0
$453$	−3954.44	−0.410145
$454$	0	0
$455$	−8688.48	−0.895214
$456$	0	0
$457$	−871.153	−0.0891703	−0.0445851	−	0.999006i	$-0.514197\pi$
$457$	−871.153	−0.0891703	−0.0445851	+	0.999006i	$0.514197\pi$
$458$	0	0
$459$	−1526.16	−0.155196
$460$	0	0
$461$	−13914.9	−1.40582	−0.702908	−	0.711281i	$-0.748115\pi$
$461$	−13914.9	−1.40582	−0.702908	+	0.711281i	$0.748115\pi$
$462$	0	0
$463$	−7771.26	−0.780046	−0.390023	−	0.920805i	$-0.627533\pi$
$463$	−7771.26	−0.780046	−0.390023	+	0.920805i	$0.627533\pi$
$464$	0	0
$465$	13639.4	1.36024
$466$	0	0
$467$	11014.4	1.09140	0.545702	−	0.837979i	$-0.316263\pi$
$467$	11014.4	1.09140	0.545702	+	0.837979i	$0.316263\pi$
$468$	0	0
$469$	3713.39	0.365604
$470$	0	0
$471$	1391.31	0.136111
$472$	0	0
$473$	−7156.47	−0.695676
$474$	0	0
$475$	5622.97	0.543157
$476$	0	0
$477$	−962.209	−0.0923617
$478$	0	0
$479$	10472.7	0.998976	0.499488	−	0.866321i	$-0.333521\pi$
$479$	10472.7	0.998976	0.499488	+	0.866321i	$0.333521\pi$
$480$	0	0
$481$	15619.6	1.48065
$482$	0	0
$483$	2960.83	0.278928
$484$	0	0
$485$	−5745.55	−0.537921
$486$	0	0
$487$	17011.2	1.58285	0.791426	−	0.611265i	$-0.209339\pi$
$487$	17011.2	1.58285	0.791426	+	0.611265i	$0.209339\pi$
$488$	0	0
$489$	8571.78	0.792698
$490$	0	0
$491$	5398.64	0.496206	0.248103	−	0.968734i	$-0.420193\pi$
$491$	5398.64	0.496206	0.248103	+	0.968734i	$0.420193\pi$
$492$	0	0
$493$	−11526.0	−1.05295
$494$	0	0
$495$	11375.9	1.03295
$496$	0	0
$497$	4404.98	0.397566
$498$	0	0
$499$	−4638.55	−0.416132	−0.208066	−	0.978115i	$-0.566717\pi$
$499$	−4638.55	−0.416132	−0.208066	+	0.978115i	$0.566717\pi$
$500$	0	0
$501$	1308.78	0.116711
$502$	0	0
$503$	−18580.3	−1.64703	−0.823514	−	0.567296i	$-0.807989\pi$
$503$	−18580.3	−1.64703	−0.823514	+	0.567296i	$0.807989\pi$
$504$	0	0
$505$	−31674.5	−2.79108
$506$	0	0
$507$	−3809.24	−0.333677
$508$	0	0
$509$	3665.64	0.319207	0.159604	−	0.987181i	$-0.448978\pi$
$509$	3665.64	0.319207	0.159604	+	0.987181i	$0.448978\pi$
$510$	0	0
$511$	5906.64	0.511340
$512$	0	0
$513$	−475.336	−0.0409096
$514$	0	0
$515$	19014.0	1.62690
$516$	0	0
$517$	7084.93	0.602698
$518$	0	0
$519$	−7005.75	−0.592521
$520$	0	0
$521$	−22382.8	−1.88217	−0.941083	−	0.338175i	$-0.890190\pi$
$521$	−22382.8	−1.88217	−0.941083	+	0.338175i	$0.890190\pi$
$522$	0	0
$523$	−4410.88	−0.368785	−0.184392	−	0.982853i	$-0.559032\pi$
$523$	−4410.88	−0.368785	−0.184392	+	0.982853i	$0.559032\pi$
$524$	0	0
$525$	6707.30	0.557582
$526$	0	0
$527$	−12190.6	−1.00765
$528$	0	0
$529$	7711.73	0.633823
$530$	0	0
$531$	−7616.91	−0.622497
$532$	0	0
$533$	15081.1	1.22559
$534$	0	0
$535$	12796.5	1.03410
$536$	0	0
$537$	5942.02	0.477500
$538$	0	0
$539$	2938.03	0.234786
$540$	0	0
$541$	18888.9	1.50110	0.750551	−	0.660812i	$-0.229788\pi$
$541$	18888.9	1.50110	0.750551	+	0.660812i	$0.229788\pi$
$542$	0	0
$543$	−7364.69	−0.582042
$544$	0	0
$545$	−19804.3	−1.55656
$546$	0	0
$547$	−23481.7	−1.83548	−0.917739	−	0.397185i	$-0.869987\pi$
$547$	−23481.7	−1.83548	−0.917739	+	0.397185i	$0.869987\pi$
$548$	0	0
$549$	7877.85	0.612419
$550$	0	0
$551$	−3589.87	−0.277556
$552$	0	0
$553$	−2329.20	−0.179110
$554$	0	0
$555$	−16777.0	−1.28314
$556$	0	0
$557$	−15139.0	−1.15164	−0.575818	−	0.817578i	$-0.695316\pi$
$557$	−15139.0	−1.15164	−0.575818	+	0.817578i	$0.695316\pi$
$558$	0	0
$559$	−7027.48	−0.531719
$560$	0	0
$561$	−10167.6	−0.765196
$562$	0	0
$563$	−12120.5	−0.907317	−0.453659	−	0.891176i	$-0.649881\pi$
$563$	−12120.5	−0.907317	−0.453659	+	0.891176i	$0.649881\pi$
$564$	0	0
$565$	−41135.7	−3.06299
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	364.785	0.0268762	0.0134381	−	0.999910i	$-0.495722\pi$
$569$	364.785	0.0268762	0.0134381	+	0.999910i	$0.495722\pi$
$570$	0	0
$571$	7660.32	0.561426	0.280713	−	0.959792i	$-0.409429\pi$
$571$	7660.32	0.561426	0.280713	+	0.959792i	$0.409429\pi$
$572$	0	0
$573$	5255.29	0.383146
$574$	0	0
$575$	45032.2	3.26604
$576$	0	0
$577$	4279.93	0.308797	0.154398	−	0.988009i	$-0.450656\pi$
$577$	4279.93	0.308797	0.154398	+	0.988009i	$0.450656\pi$
$578$	0	0
$579$	−6526.69	−0.468463
$580$	0	0
$581$	3676.02	0.262491
$582$	0	0
$583$	−6410.43	−0.455391
$584$	0	0
$585$	11170.9	0.789504
$586$	0	0
$587$	18018.1	1.26693	0.633464	−	0.773773i	$-0.281633\pi$
$587$	18018.1	1.26693	0.633464	+	0.773773i	$0.281633\pi$
$588$	0	0
$589$	−3796.87	−0.265615
$590$	0	0
$591$	−10278.4	−0.715391
$592$	0	0
$593$	−5599.14	−0.387739	−0.193869	−	0.981027i	$-0.562104\pi$
$593$	−5599.14	−0.387739	−0.193869	+	0.981027i	$0.562104\pi$
$594$	0	0
$595$	−8341.01	−0.574702
$596$	0	0
$597$	−16113.5	−1.10466
$598$	0	0
$599$	18554.9	1.26566	0.632832	−	0.774289i	$-0.281893\pi$
$599$	18554.9	1.26566	0.632832	+	0.774289i	$0.281893\pi$
$600$	0	0
$601$	20293.5	1.37735	0.688677	−	0.725069i	$-0.258192\pi$
$601$	20293.5	1.37735	0.688677	+	0.725069i	$0.258192\pi$
$602$	0	0
$603$	−4774.36	−0.322433
$604$	0	0
$605$	47730.3	3.20746
$606$	0	0
$607$	−1947.81	−0.130245	−0.0651227	−	0.997877i	$-0.520744\pi$
$607$	−1947.81	−0.130245	−0.0651227	+	0.997877i	$0.520744\pi$
$608$	0	0
$609$	−4282.14	−0.284928
$610$	0	0
$611$	6957.23	0.460654
$612$	0	0
$613$	−15865.3	−1.04534	−0.522669	−	0.852536i	$-0.675064\pi$
$613$	−15865.3	−1.04534	−0.522669	+	0.852536i	$0.675064\pi$
$614$	0	0
$615$	−16198.7	−1.06210
$616$	0	0
$617$	−16375.8	−1.06850	−0.534250	−	0.845327i	$-0.679406\pi$
$617$	−16375.8	−1.06850	−0.534250	+	0.845327i	$0.679406\pi$
$618$	0	0
$619$	−11785.2	−0.765248	−0.382624	−	0.923904i	$-0.624980\pi$
$619$	−11785.2	−0.765248	−0.382624	+	0.923904i	$0.624980\pi$
$620$	0	0
$621$	−3806.78	−0.245992
$622$	0	0
$623$	2540.61	0.163383
$624$	0	0
$625$	46464.0	2.97369
$626$	0	0
$627$	−3166.78	−0.201705
$628$	0	0
$629$	14994.9	0.950536
$630$	0	0
$631$	24939.2	1.57340	0.786700	−	0.617335i	$-0.211788\pi$
$631$	24939.2	1.57340	0.786700	+	0.617335i	$0.211788\pi$
$632$	0	0
$633$	−2116.10	−0.132871
$634$	0	0
$635$	2418.46	0.151140
$636$	0	0
$637$	2885.07	0.179452
$638$	0	0
$639$	−5663.54	−0.350620
$640$	0	0
$641$	24524.4	1.51116	0.755582	−	0.655055i	$-0.227354\pi$
$641$	24524.4	1.51116	0.755582	+	0.655055i	$0.227354\pi$
$642$	0	0
$643$	9821.96	0.602396	0.301198	−	0.953562i	$-0.402614\pi$
$643$	9821.96	0.602396	0.301198	+	0.953562i	$0.402614\pi$
$644$	0	0
$645$	7548.22	0.460792
$646$	0	0
$647$	18229.2	1.10767	0.553836	−	0.832626i	$-0.313163\pi$
$647$	18229.2	1.10767	0.553836	+	0.832626i	$0.313163\pi$
$648$	0	0
$649$	−50745.3	−3.06923
$650$	0	0
$651$	−4529.06	−0.272669
$652$	0	0
$653$	740.926	0.0444023	0.0222011	−	0.999754i	$-0.492933\pi$
$653$	740.926	0.0444023	0.0222011	+	0.999754i	$0.492933\pi$
$654$	0	0
$655$	17907.7	1.06826
$656$	0	0
$657$	−7594.26	−0.450959
$658$	0	0
$659$	−30783.6	−1.81966	−0.909832	−	0.414977i	$-0.863790\pi$
$659$	−30783.6	−1.81966	−0.909832	+	0.414977i	$0.863790\pi$
$660$	0	0
$661$	−33202.1	−1.95372	−0.976862	−	0.213870i	$-0.931393\pi$
$661$	−33202.1	−1.95372	−0.976862	+	0.213870i	$0.931393\pi$
$662$	0	0
$663$	−9984.31	−0.584854
$664$	0	0
$665$	−2597.89	−0.151491
$666$	0	0
$667$	−28749.8	−1.66896
$668$	0	0
$669$	−207.625	−0.0119989
$670$	0	0
$671$	52483.8	3.01954
$672$	0	0
$673$	7336.48	0.420209	0.210105	−	0.977679i	$-0.432620\pi$
$673$	7336.48	0.420209	0.210105	+	0.977679i	$0.432620\pi$
$674$	0	0
$675$	−8623.68	−0.491741
$676$	0	0
$677$	−7048.41	−0.400136	−0.200068	−	0.979782i	$-0.564116\pi$
$677$	−7048.41	−0.400136	−0.200068	+	0.979782i	$0.564116\pi$
$678$	0	0
$679$	1907.85	0.107830
$680$	0	0
$681$	−14291.2	−0.804172
$682$	0	0
$683$	−1207.69	−0.0676588	−0.0338294	−	0.999428i	$-0.510770\pi$
$683$	−1207.69	−0.0676588	−0.0338294	+	0.999428i	$0.510770\pi$
$684$	0	0
$685$	33655.3	1.87723
$686$	0	0
$687$	6148.64	0.341463
$688$	0	0
$689$	−6294.89	−0.348064
$690$	0	0
$691$	−26632.3	−1.46619	−0.733096	−	0.680125i	$-0.761925\pi$
$691$	−26632.3	−1.46619	−0.733096	+	0.680125i	$0.761925\pi$
$692$	0	0
$693$	−3777.47	−0.207062
$694$	0	0
$695$	−10378.1	−0.566424
$696$	0	0
$697$	14478.0	0.786792
$698$	0	0
$699$	−6979.59	−0.377672
$700$	0	0
$701$	−11589.9	−0.624457	−0.312229	−	0.950007i	$-0.601075\pi$
$701$	−11589.9	−0.624457	−0.312229	+	0.950007i	$0.601075\pi$
$702$	0	0
$703$	4670.32	0.250561
$704$	0	0
$705$	−7472.77	−0.399207
$706$	0	0
$707$	10517.7	0.559492
$708$	0	0
$709$	16310.5	0.863970	0.431985	−	0.901881i	$-0.357813\pi$
$709$	16310.5	0.863970	0.431985	+	0.901881i	$0.357813\pi$
$710$	0	0
$711$	2994.68	0.157960
$712$	0	0
$713$	−30407.6	−1.59716
$714$	0	0
$715$	74422.7	3.89266
$716$	0	0
$717$	−17507.4	−0.911892
$718$	0	0
$719$	10054.2	0.521499	0.260749	−	0.965407i	$-0.416030\pi$
$719$	10054.2	0.521499	0.260749	+	0.965407i	$0.416030\pi$
$720$	0	0
$721$	−6313.73	−0.326124
$722$	0	0
$723$	5661.54	0.291224
$724$	0	0
$725$	−65128.3	−3.33628
$726$	0	0
$727$	−459.295	−0.0234310	−0.0117155	−	0.999931i	$-0.503729\pi$
$727$	−459.295	−0.0234310	−0.0117155	+	0.999931i	$0.503729\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−6746.44	−0.341349
$732$	0	0
$733$	3350.49	0.168831	0.0844156	−	0.996431i	$-0.473098\pi$
$733$	3350.49	0.168831	0.0844156	+	0.996431i	$0.473098\pi$
$734$	0	0
$735$	−3098.86	−0.155515
$736$	0	0
$737$	−31807.7	−1.58976
$738$	0	0
$739$	26916.7	1.33985	0.669924	−	0.742430i	$-0.266327\pi$
$739$	26916.7	1.33985	0.669924	+	0.742430i	$0.266327\pi$
$740$	0	0
$741$	−3109.71	−0.154167
$742$	0	0
$743$	15425.4	0.761646	0.380823	−	0.924648i	$-0.375641\pi$
$743$	15425.4	0.761646	0.380823	+	0.924648i	$0.375641\pi$
$744$	0	0
$745$	−53569.0	−2.63438
$746$	0	0
$747$	−4726.31	−0.231495
$748$	0	0
$749$	−4249.18	−0.207292
$750$	0	0
$751$	35460.6	1.72300	0.861502	−	0.507754i	$-0.169524\pi$
$751$	35460.6	1.72300	0.861502	+	0.507754i	$0.169524\pi$
$752$	0	0
$753$	−2434.71	−0.117830
$754$	0	0
$755$	27787.4	1.33946
$756$	0	0
$757$	−2554.15	−0.122632	−0.0613159	−	0.998118i	$-0.519530\pi$
$757$	−2554.15	−0.122632	−0.0613159	+	0.998118i	$0.519530\pi$
$758$	0	0
$759$	−25361.5	−1.21287
$760$	0	0
$761$	−8054.37	−0.383667	−0.191833	−	0.981427i	$-0.561443\pi$
$761$	−8054.37	−0.383667	−0.191833	+	0.981427i	$0.561443\pi$
$762$	0	0
$763$	6576.18	0.312023
$764$	0	0
$765$	10724.2	0.506840
$766$	0	0
$767$	−49830.7	−2.34587
$768$	0	0
$769$	−13859.3	−0.649910	−0.324955	−	0.945729i	$-0.605349\pi$
$769$	−13859.3	−0.649910	−0.324955	+	0.945729i	$0.605349\pi$
$770$	0	0
$771$	−9873.65	−0.461208
$772$	0	0
$773$	−15449.0	−0.718839	−0.359420	−	0.933176i	$-0.617025\pi$
$773$	−15449.0	−0.718839	−0.359420	+	0.933176i	$0.617025\pi$
$774$	0	0
$775$	−68883.8	−3.19275
$776$	0	0
$777$	5570.94	0.257215
$778$	0	0
$779$	4509.32	0.207398
$780$	0	0
$781$	−37731.6	−1.72874
$782$	0	0
$783$	5505.60	0.251283
$784$	0	0
$785$	−9776.58	−0.444511
$786$	0	0
$787$	−18954.3	−0.858510	−0.429255	−	0.903183i	$-0.641224\pi$
$787$	−18954.3	−0.858510	−0.429255	+	0.903183i	$0.641224\pi$
$788$	0	0
$789$	5862.42	0.264522
$790$	0	0
$791$	13659.4	0.613998
$792$	0	0
$793$	51537.8	2.30790
$794$	0	0
$795$	6761.35	0.301636
$796$	0	0
$797$	−3291.25	−0.146276	−0.0731381	−	0.997322i	$-0.523301\pi$
$797$	−3291.25	−0.146276	−0.0731381	+	0.997322i	$0.523301\pi$
$798$	0	0
$799$	6679.00	0.295727
$800$	0	0
$801$	−3266.50	−0.144090
$802$	0	0
$803$	−50594.4	−2.22346
$804$	0	0
$805$	−20805.4	−0.910926
$806$	0	0
$807$	−5485.20	−0.239266
$808$	0	0
$809$	14462.5	0.628523	0.314262	−	0.949336i	$-0.398243\pi$
$809$	14462.5	0.628523	0.314262	+	0.949336i	$0.398243\pi$
$810$	0	0
$811$	−3591.30	−0.155496	−0.0777482	−	0.996973i	$-0.524773\pi$
$811$	−3591.30	−0.155496	−0.0777482	+	0.996973i	$0.524773\pi$
$812$	0	0
$813$	8736.75	0.376890
$814$	0	0
$815$	−60233.0	−2.58880
$816$	0	0
$817$	−2101.24	−0.0899794
$818$	0	0
$819$	−3709.38	−0.158262
$820$	0	0
$821$	−31463.5	−1.33750	−0.668749	−	0.743488i	$-0.733170\pi$
$821$	−31463.5	−1.33750	−0.668749	+	0.743488i	$0.733170\pi$
$822$	0	0
$823$	24102.0	1.02083	0.510415	−	0.859928i	$-0.329492\pi$
$823$	24102.0	1.02083	0.510415	+	0.859928i	$0.329492\pi$
$824$	0	0
$825$	−57452.6	−2.42454
$826$	0	0
$827$	41861.5	1.76018	0.880089	−	0.474809i	$-0.157483\pi$
$827$	41861.5	1.76018	0.880089	+	0.474809i	$0.157483\pi$
$828$	0	0
$829$	−20987.3	−0.879274	−0.439637	−	0.898176i	$-0.644893\pi$
$829$	−20987.3	−0.879274	−0.439637	+	0.898176i	$0.644893\pi$
$830$	0	0
$831$	1377.12	0.0574871
$832$	0	0
$833$	2769.69	0.115203
$834$	0	0
$835$	−9196.69	−0.381155
$836$	0	0
$837$	5823.07	0.240472
$838$	0	0
$839$	−39884.2	−1.64119	−0.820594	−	0.571511i	$-0.806357\pi$
$839$	−39884.2	−1.64119	−0.820594	+	0.571511i	$0.806357\pi$
$840$	0	0
$841$	17190.8	0.704859
$842$	0	0
$843$	17391.6	0.710555
$844$	0	0
$845$	26767.1	1.08972
$846$	0	0
$847$	−15849.2	−0.642958
$848$	0	0
$849$	143.652	0.00580696
$850$	0	0
$851$	37402.7	1.50664
$852$	0	0
$853$	−11790.9	−0.473286	−0.236643	−	0.971597i	$-0.576047\pi$
$853$	−11790.9	−0.473286	−0.236643	+	0.971597i	$0.576047\pi$
$854$	0	0
$855$	3340.14	0.133603
$856$	0	0
$857$	27400.9	1.09218	0.546090	−	0.837727i	$-0.316116\pi$
$857$	27400.9	1.09218	0.546090	+	0.837727i	$0.316116\pi$
$858$	0	0
$859$	37420.4	1.48634	0.743172	−	0.669101i	$-0.233320\pi$
$859$	37420.4	1.48634	0.743172	+	0.669101i	$0.233320\pi$
$860$	0	0
$861$	5378.89	0.212906
$862$	0	0
$863$	−5362.62	−0.211525	−0.105762	−	0.994391i	$-0.533728\pi$
$863$	−5362.62	−0.211525	−0.105762	+	0.994391i	$0.533728\pi$
$864$	0	0
$865$	49228.7	1.93506
$866$	0	0
$867$	5153.99	0.201890
$868$	0	0
$869$	19951.2	0.778823
$870$	0	0
$871$	−31234.4	−1.21508
$872$	0	0
$873$	−2452.95	−0.0950972
$874$	0	0
$875$	−28685.9	−1.10830
$876$	0	0
$877$	−13764.7	−0.529989	−0.264995	−	0.964250i	$-0.585370\pi$
$877$	−13764.7	−0.529989	−0.264995	+	0.964250i	$0.585370\pi$
$878$	0	0
$879$	21221.3	0.814310
$880$	0	0
$881$	44291.4	1.69377	0.846887	−	0.531774i	$-0.178474\pi$
$881$	44291.4	1.69377	0.846887	+	0.531774i	$0.178474\pi$
$882$	0	0
$883$	27302.8	1.04056	0.520280	−	0.853996i	$-0.325828\pi$
$883$	27302.8	1.04056	0.520280	+	0.853996i	$0.325828\pi$
$884$	0	0
$885$	53523.2	2.03295
$886$	0	0
$887$	−14012.9	−0.530449	−0.265225	−	0.964187i	$-0.585446\pi$
$887$	−14012.9	−0.530449	−0.265225	+	0.964187i	$0.585446\pi$
$888$	0	0
$889$	−803.069	−0.0302970
$890$	0	0
$891$	4856.74	0.182612
$892$	0	0
$893$	2080.24	0.0779535
$894$	0	0
$895$	−41754.0	−1.55942
$896$	0	0
$897$	−24904.4	−0.927017
$898$	0	0
$899$	43977.4	1.63151
$900$	0	0
$901$	−6043.14	−0.223448
$902$	0	0
$903$	−2506.44	−0.0923690
$904$	0	0
$905$	51750.9	1.90084
$906$	0	0
$907$	−2318.04	−0.0848615	−0.0424308	−	0.999099i	$-0.513510\pi$
$907$	−2318.04	−0.0848615	−0.0424308	+	0.999099i	$0.513510\pi$
$908$	0	0
$909$	−13522.8	−0.493425
$910$	0	0
$911$	−1372.94	−0.0499313	−0.0249657	−	0.999688i	$-0.507948\pi$
$911$	−1372.94	−0.0499313	−0.0249657	+	0.999688i	$0.507948\pi$
$912$	0	0
$913$	−31487.6	−1.14139
$914$	0	0
$915$	−55356.8	−2.00004
$916$	0	0
$917$	−5946.40	−0.214141
$918$	0	0
$919$	2727.14	0.0978891	0.0489445	−	0.998801i	$-0.484414\pi$
$919$	2727.14	0.0978891	0.0489445	+	0.998801i	$0.484414\pi$
$920$	0	0
$921$	16957.6	0.606701
$922$	0	0
$923$	−37051.6	−1.32131
$924$	0	0
$925$	84730.1	3.01179
$926$	0	0
$927$	8117.66	0.287615
$928$	0	0
$929$	−22714.1	−0.802180	−0.401090	−	0.916039i	$-0.631369\pi$
$929$	−22714.1	−0.802180	−0.401090	+	0.916039i	$0.631369\pi$
$930$	0	0
$931$	862.647	0.0303675
$932$	0	0
$933$	−27219.0	−0.955103
$934$	0	0
$935$	71446.4	2.49898
$936$	0	0
$937$	−18313.9	−0.638515	−0.319257	−	0.947668i	$-0.603433\pi$
$937$	−18313.9	−0.638515	−0.319257	+	0.947668i	$0.603433\pi$
$938$	0	0
$939$	4629.95	0.160908
$940$	0	0
$941$	35310.4	1.22326	0.611629	−	0.791145i	$-0.290515\pi$
$941$	35310.4	1.22326	0.611629	+	0.791145i	$0.290515\pi$
$942$	0	0
$943$	36113.3	1.24710
$944$	0	0
$945$	3984.25	0.137151
$946$	0	0
$947$	−18127.5	−0.622032	−0.311016	−	0.950405i	$-0.600669\pi$
$947$	−18127.5	−0.622032	−0.311016	+	0.950405i	$0.600669\pi$
$948$	0	0
$949$	−49682.5	−1.69943
$950$	0	0
$951$	−6779.14	−0.231155
$952$	0	0
$953$	−43051.2	−1.46334	−0.731672	−	0.681657i	$-0.761260\pi$
$953$	−43051.2	−1.46334	−0.731672	+	0.681657i	$0.761260\pi$
$954$	0	0
$955$	−36928.4	−1.25128
$956$	0	0
$957$	36679.4	1.23895
$958$	0	0
$959$	−11175.5	−0.376304
$960$	0	0
$961$	16722.3	0.561320
$962$	0	0
$963$	5463.24	0.182814
$964$	0	0
$965$	45862.3	1.52991
$966$	0	0
$967$	21926.8	0.729181	0.364590	−	0.931168i	$-0.381209\pi$
$967$	21926.8	0.729181	0.364590	+	0.931168i	$0.381209\pi$
$968$	0	0
$969$	−2985.34	−0.0989711
$970$	0	0
$971$	−19984.9	−0.660500	−0.330250	−	0.943893i	$-0.607133\pi$
$971$	−19984.9	−0.660500	−0.330250	+	0.943893i	$0.607133\pi$
$972$	0	0
$973$	3446.13	0.113544
$974$	0	0
$975$	−56417.1	−1.85312
$976$	0	0
$977$	29442.3	0.964119	0.482059	−	0.876139i	$-0.339889\pi$
$977$	29442.3	0.964119	0.482059	+	0.876139i	$0.339889\pi$
$978$	0	0
$979$	−21762.1	−0.710438
$980$	0	0
$981$	−8455.09	−0.275179
$982$	0	0
$983$	6103.54	0.198039	0.0990196	−	0.995085i	$-0.468429\pi$
$983$	6103.54	0.198039	0.0990196	+	0.995085i	$0.468429\pi$
$984$	0	0
$985$	72225.1	2.33633
$986$	0	0
$987$	2481.39	0.0800238
$988$	0	0
$989$	−16828.0	−0.541052
$990$	0	0
$991$	−42780.5	−1.37131	−0.685655	−	0.727926i	$-0.740484\pi$
$991$	−42780.5	−1.37131	−0.685655	+	0.727926i	$0.740484\pi$
$992$	0	0
$993$	1572.45	0.0502520
$994$	0	0
$995$	113228.	3.60761
$996$	0	0
$997$	−35513.3	−1.12810	−0.564051	−	0.825740i	$-0.690758\pi$
$997$	−35513.3	−1.12810	−0.564051	+	0.825740i	$0.690758\pi$
$998$	0	0
$999$	−7162.63	−0.226843

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.a.bt.1.3		3
4.3	odd	2		1344.4.a.bv.1.3		3
8.3	odd	2		672.4.a.o.1.1	✓	3
8.5	even	2		672.4.a.q.1.1	yes	3
24.5	odd	2		2016.4.a.y.1.3		3
24.11	even	2		2016.4.a.z.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.a.o.1.1	✓	3	8.3	odd	2
672.4.a.q.1.1	yes	3	8.5	even	2
1344.4.a.bt.1.3		3	1.1	even	1	trivial
1344.4.a.bv.1.3		3	4.3	odd	2
2016.4.a.y.1.3		3	24.5	odd	2
2016.4.a.z.1.3		3	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +21.0807 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +21.0807 q^{5} -7.00000 q^{7} +9.00000 q^{9} +59.9598 q^{11} +58.8791 q^{13} -63.2421 q^{15} +56.5244 q^{17} +17.6050 q^{19} +21.0000 q^{21} +140.992 q^{23} +319.395 q^{25} -27.0000 q^{27} -203.911 q^{29} -215.669 q^{31} -179.879 q^{33} -147.565 q^{35} +265.283 q^{37} -176.637 q^{39} +256.138 q^{41} -119.355 q^{43} +189.726 q^{45} +118.161 q^{47} +49.0000 q^{49} -169.573 q^{51} -106.912 q^{53} +1263.99 q^{55} -52.8151 q^{57} -846.323 q^{59} +875.316 q^{61} -63.0000 q^{63} +1241.21 q^{65} -530.484 q^{67} -422.976 q^{69} -629.282 q^{71} -843.806 q^{73} -958.186 q^{75} -419.718 q^{77} +332.743 q^{79} +81.0000 q^{81} -525.146 q^{83} +1191.57 q^{85} +611.734 q^{87} -362.945 q^{89} -412.154 q^{91} +647.008 q^{93} +371.127 q^{95} -272.550 q^{97} +539.638 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} + 10 q^{5} - 21 q^{7} + 27 q^{9}+O(q^{10}) \) 3 * q - 9 * q^3 + 10 * q^5 - 21 * q^7 + 27 * q^9	\( 3 q - 9 q^{3} + 10 q^{5} - 21 q^{7} + 27 q^{9} + 50 q^{13} - 30 q^{15} + 30 q^{17} - 140 q^{19} + 63 q^{21} + 56 q^{23} + 325 q^{25} - 81 q^{27} - 298 q^{29} - 80 q^{31} - 70 q^{35} - 10 q^{37} - 150 q^{39} + 390 q^{41} - 784 q^{43} + 90 q^{45} + 248 q^{47} + 147 q^{49} - 90 q^{51} - 10 q^{53} + 1360 q^{55} + 420 q^{57} - 1500 q^{59} + 810 q^{61} - 189 q^{63} + 860 q^{65} - 1272 q^{67} - 168 q^{69} + 160 q^{71} - 1170 q^{73} - 975 q^{75} + 840 q^{79} + 243 q^{81} - 1564 q^{83} + 2740 q^{85} + 894 q^{87} - 178 q^{89} - 350 q^{91} + 240 q^{93} + 2840 q^{95} - 130 q^{97}+O(q^{100}) \) 3 * q - 9 * q^3 + 10 * q^5 - 21 * q^7 + 27 * q^9 + 50 * q^13 - 30 * q^15 + 30 * q^17 - 140 * q^19 + 63 * q^21 + 56 * q^23 + 325 * q^25 - 81 * q^27 - 298 * q^29 - 80 * q^31 - 70 * q^35 - 10 * q^37 - 150 * q^39 + 390 * q^41 - 784 * q^43 + 90 * q^45 + 248 * q^47 + 147 * q^49 - 90 * q^51 - 10 * q^53 + 1360 * q^55 + 420 * q^57 - 1500 * q^59 + 810 * q^61 - 189 * q^63 + 860 * q^65 - 1272 * q^67 - 168 * q^69 + 160 * q^71 - 1170 * q^73 - 975 * q^75 + 840 * q^79 + 243 * q^81 - 1564 * q^83 + 2740 * q^85 + 894 * q^87 - 178 * q^89 - 350 * q^91 + 240 * q^93 + 2840 * q^95 - 130 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more