LMFDB - Embedded newform 688.4.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [688,4,Mod(1,688)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(688, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("688.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 688 = 2^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	688.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:	$40.5933140839$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 32x^{4} - 16x^{3} + 251x^{2} + 276x + 60 $ x^6 - 32x^4 - 16x^3 + 251x^2 + 276x + 60
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.847740$ of defining polynomial
Character	$\chi$	$=$	688.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.49653 q^{3} +2.98245 q^{5} +26.0720 q^{7} +63.1842 q^{9} +O(q^{10})$	$q-9.49653 q^{3} +2.98245 q^{5} +26.0720 q^{7} +63.1842 q^{9} +36.8506 q^{11} +89.5430 q^{13} -28.3230 q^{15} -28.8042 q^{17} +58.8677 q^{19} -247.594 q^{21} -2.63139 q^{23} -116.105 q^{25} -343.624 q^{27} +173.812 q^{29} -57.9476 q^{31} -349.953 q^{33} +77.7586 q^{35} +52.0754 q^{37} -850.348 q^{39} +142.704 q^{41} +43.0000 q^{43} +188.444 q^{45} +106.853 q^{47} +336.750 q^{49} +273.540 q^{51} +244.652 q^{53} +109.905 q^{55} -559.039 q^{57} +127.799 q^{59} -443.613 q^{61} +1647.34 q^{63} +267.058 q^{65} +117.896 q^{67} +24.9891 q^{69} -816.799 q^{71} -620.953 q^{73} +1102.59 q^{75} +960.770 q^{77} -377.771 q^{79} +1557.27 q^{81} +1453.23 q^{83} -85.9072 q^{85} -1650.62 q^{87} +627.993 q^{89} +2334.57 q^{91} +550.302 q^{93} +175.570 q^{95} -817.163 q^{97} +2328.38 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 7 q^{3} + 43 q^{5} - 8 q^{7} + 81 q^{9}+O(q^{10}) $ 6 * q - 7 * q^3 + 43 * q^5 - 8 * q^7 + 81 * q^9	$ 6 q - 7 q^{3} + 43 q^{5} - 8 q^{7} + 81 q^{9} + 28 q^{11} + 56 q^{13} + 124 q^{15} + 19 q^{17} + 75 q^{19} - 18 q^{21} - 131 q^{23} + 105 q^{25} - 238 q^{27} + 515 q^{29} - 237 q^{31} + 540 q^{33} - 198 q^{35} + 269 q^{37} - 290 q^{39} + 471 q^{41} + 258 q^{43} + 334 q^{45} - 415 q^{47} + 350 q^{49} + 1241 q^{51} + 450 q^{53} + 1732 q^{55} - 1000 q^{57} - 356 q^{59} - 1328 q^{61} + 2290 q^{63} - 62 q^{65} + 632 q^{67} - 1130 q^{69} + 144 q^{71} + 864 q^{73} + 2494 q^{75} + 2660 q^{77} + 1613 q^{79} - 102 q^{81} + 682 q^{83} + 84 q^{85} - 449 q^{87} + 3378 q^{89} + 3900 q^{91} + 1879 q^{93} + 79 q^{95} - 55 q^{97} + 1612 q^{99}+O(q^{100}) $ 6 * q - 7 * q^3 + 43 * q^5 - 8 * q^7 + 81 * q^9 + 28 * q^11 + 56 * q^13 + 124 * q^15 + 19 * q^17 + 75 * q^19 - 18 * q^21 - 131 * q^23 + 105 * q^25 - 238 * q^27 + 515 * q^29 - 237 * q^31 + 540 * q^33 - 198 * q^35 + 269 * q^37 - 290 * q^39 + 471 * q^41 + 258 * q^43 + 334 * q^45 - 415 * q^47 + 350 * q^49 + 1241 * q^51 + 450 * q^53 + 1732 * q^55 - 1000 * q^57 - 356 * q^59 - 1328 * q^61 + 2290 * q^63 - 62 * q^65 + 632 * q^67 - 1130 * q^69 + 144 * q^71 + 864 * q^73 + 2494 * q^75 + 2660 * q^77 + 1613 * q^79 - 102 * q^81 + 682 * q^83 + 84 * q^85 - 449 * q^87 + 3378 * q^89 + 3900 * q^91 + 1879 * q^93 + 79 * q^95 - 55 * q^97 + 1612 * 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
$3$	−9.49653	−1.82761	−0.913804	−	0.406154i	$-0.866870\pi$
$3$	−9.49653	−1.82761	−0.913804	+	0.406154i	$0.866870\pi$
$4$	0	0
$5$	2.98245	0.266759	0.133379	−	0.991065i	$-0.457417\pi$
$5$	2.98245	0.266759	0.133379	+	0.991065i	$0.457417\pi$
$6$	0	0
$7$	26.0720	1.40776	0.703878	−	0.710321i	$-0.251450\pi$
$7$	26.0720	1.40776	0.703878	+	0.710321i	$0.251450\pi$
$8$	0	0
$9$	63.1842	2.34015
$10$	0	0
$11$	36.8506	1.01008	0.505040	−	0.863096i	$-0.331478\pi$
$11$	36.8506	1.01008	0.505040	+	0.863096i	$0.331478\pi$
$12$	0	0
$13$	89.5430	1.91037	0.955183	−	0.296016i	$-0.0956581\pi$
$13$	89.5430	1.91037	0.955183	+	0.296016i	$0.0956581\pi$
$14$	0	0
$15$	−28.3230	−0.487531
$16$	0	0
$17$	−28.8042	−0.410944	−0.205472	−	0.978663i	$-0.565873\pi$
$17$	−28.8042	−0.410944	−0.205472	+	0.978663i	$0.565873\pi$
$18$	0	0
$19$	58.8677	0.710799	0.355400	−	0.934714i	$-0.384345\pi$
$19$	58.8677	0.710799	0.355400	+	0.934714i	$0.384345\pi$
$20$	0	0
$21$	−247.594	−2.57283
$22$	0	0
$23$	−2.63139	−0.0238558	−0.0119279	−	0.999929i	$-0.503797\pi$
$23$	−2.63139	−0.0238558	−0.0119279	+	0.999929i	$0.503797\pi$
$24$	0	0
$25$	−116.105	−0.928840
$26$	0	0
$27$	−343.624	−2.44928
$28$	0	0
$29$	173.812	1.11297	0.556486	−	0.830857i	$-0.312149\pi$
$29$	173.812	1.11297	0.556486	+	0.830857i	$0.312149\pi$
$30$	0	0
$31$	−57.9476	−0.335732	−0.167866	−	0.985810i	$-0.553688\pi$
$31$	−57.9476	−0.335732	−0.167866	+	0.985810i	$0.553688\pi$
$32$	0	0
$33$	−349.953	−1.84603
$34$	0	0
$35$	77.7586	0.375531
$36$	0	0
$37$	52.0754	0.231382	0.115691	−	0.993285i	$-0.463092\pi$
$37$	52.0754	0.231382	0.115691	+	0.993285i	$0.463092\pi$
$38$	0	0
$39$	−850.348	−3.49140
$40$	0	0
$41$	142.704	0.543577	0.271788	−	0.962357i	$-0.412385\pi$
$41$	142.704	0.543577	0.271788	+	0.962357i	$0.412385\pi$
$42$	0	0
$43$	43.0000	0.152499
$43$	43.0000	0.152499
$44$	0	0
$45$	188.444	0.624257
$46$	0	0
$47$	106.853	0.331618	0.165809	−	0.986158i	$-0.446976\pi$
$47$	106.853	0.331618	0.165809	+	0.986158i	$0.446976\pi$
$48$	0	0
$49$	336.750	0.981779
$50$	0	0
$51$	273.540	0.751045
$52$	0	0
$53$	244.652	0.634067	0.317033	−	0.948414i	$-0.397313\pi$
$53$	244.652	0.634067	0.317033	+	0.948414i	$0.397313\pi$
$54$	0	0
$55$	109.905	0.269448
$56$	0	0
$57$	−559.039	−1.29906
$58$	0	0
$59$	127.799	0.281999	0.141000	−	0.990010i	$-0.454968\pi$
$59$	127.799	0.281999	0.141000	+	0.990010i	$0.454968\pi$
$60$	0	0
$61$	−443.613	−0.931128	−0.465564	−	0.885014i	$-0.654149\pi$
$61$	−443.613	−0.931128	−0.465564	+	0.885014i	$0.654149\pi$
$62$	0	0
$63$	1647.34	3.29437
$64$	0	0
$65$	267.058	0.509607
$66$	0	0
$67$	117.896	0.214975	0.107487	−	0.994206i	$-0.465720\pi$
$67$	117.896	0.214975	0.107487	+	0.994206i	$0.465720\pi$
$68$	0	0
$69$	24.9891	0.0435991
$70$	0	0
$71$	−816.799	−1.36530	−0.682650	−	0.730746i	$-0.739173\pi$
$71$	−816.799	−1.36530	−0.682650	+	0.730746i	$0.739173\pi$
$72$	0	0
$73$	−620.953	−0.995576	−0.497788	−	0.867299i	$-0.665854\pi$
$73$	−620.953	−0.995576	−0.497788	+	0.867299i	$0.665854\pi$
$74$	0	0
$75$	1102.59	1.69756
$76$	0	0
$77$	960.770	1.42195
$78$	0	0
$79$	−377.771	−0.538007	−0.269003	−	0.963139i	$-0.586694\pi$
$79$	−377.771	−0.538007	−0.269003	+	0.963139i	$0.586694\pi$
$80$	0	0
$81$	1557.27	2.13617
$82$	0	0
$83$	1453.23	1.92184	0.960919	−	0.276829i	$-0.0892835\pi$
$83$	1453.23	1.92184	0.960919	+	0.276829i	$0.0892835\pi$
$84$	0	0
$85$	−85.9072	−0.109623
$86$	0	0
$87$	−1650.62	−2.03408
$88$	0	0
$89$	627.993	0.747945	0.373973	−	0.927440i	$-0.377995\pi$
$89$	627.993	0.747945	0.373973	+	0.927440i	$0.377995\pi$
$90$	0	0
$91$	2334.57	2.68933
$92$	0	0
$93$	550.302	0.613587
$94$	0	0
$95$	175.570	0.189612
$96$	0	0
$97$	−817.163	−0.855365	−0.427682	−	0.903929i	$-0.640670\pi$
$97$	−817.163	−0.855365	−0.427682	+	0.903929i	$0.640670\pi$
$98$	0	0
$99$	2328.38	2.36374
$100$	0	0
$101$	513.438	0.505832	0.252916	−	0.967488i	$-0.418610\pi$
$101$	513.438	0.505832	0.252916	+	0.967488i	$0.418610\pi$
$102$	0	0
$103$	−689.788	−0.659872	−0.329936	−	0.944003i	$-0.607027\pi$
$103$	−689.788	−0.659872	−0.329936	+	0.944003i	$0.607027\pi$
$104$	0	0
$105$	−738.437	−0.686325
$106$	0	0
$107$	−320.710	−0.289759	−0.144879	−	0.989449i	$-0.546279\pi$
$107$	−320.710	−0.289759	−0.144879	+	0.989449i	$0.546279\pi$
$108$	0	0
$109$	−1691.51	−1.48640	−0.743200	−	0.669069i	$-0.766693\pi$
$109$	−1691.51	−1.48640	−0.743200	+	0.669069i	$0.766693\pi$
$110$	0	0
$111$	−494.535	−0.422876
$112$	0	0
$113$	856.360	0.712916	0.356458	−	0.934311i	$-0.383984\pi$
$113$	856.360	0.712916	0.356458	+	0.934311i	$0.383984\pi$
$114$	0	0
$115$	−7.84801	−0.00636374
$116$	0	0
$117$	5657.70	4.47055
$118$	0	0
$119$	−750.984	−0.578509
$120$	0	0
$121$	26.9674	0.0202610
$122$	0	0
$123$	−1355.19	−0.993445
$124$	0	0
$125$	−719.084	−0.514535
$126$	0	0
$127$	2233.72	1.56071	0.780357	−	0.625335i	$-0.215038\pi$
$127$	2233.72	1.56071	0.780357	+	0.625335i	$0.215038\pi$
$128$	0	0
$129$	−408.351	−0.278708
$130$	0	0
$131$	2051.51	1.36825	0.684126	−	0.729364i	$-0.260184\pi$
$131$	2051.51	1.36825	0.684126	+	0.729364i	$0.260184\pi$
$132$	0	0
$133$	1534.80	1.00063
$134$	0	0
$135$	−1024.84	−0.653367
$136$	0	0
$137$	2594.49	1.61797	0.808986	−	0.587827i	$-0.200017\pi$
$137$	2594.49	1.61797	0.808986	+	0.587827i	$0.200017\pi$
$138$	0	0
$139$	−1140.44	−0.695905	−0.347953	−	0.937512i	$-0.613123\pi$
$139$	−1140.44	−0.695905	−0.347953	+	0.937512i	$0.613123\pi$
$140$	0	0
$141$	−1014.73	−0.606068
$142$	0	0
$143$	3299.71	1.92962
$144$	0	0
$145$	518.388	0.296895
$146$	0	0
$147$	−3197.96	−1.79431
$148$	0	0
$149$	−2112.03	−1.16124	−0.580618	−	0.814176i	$-0.697189\pi$
$149$	−2112.03	−1.16124	−0.580618	+	0.814176i	$0.697189\pi$
$150$	0	0
$151$	1351.31	0.728265	0.364132	−	0.931347i	$-0.381366\pi$
$151$	1351.31	0.728265	0.364132	+	0.931347i	$0.381366\pi$
$152$	0	0
$153$	−1819.97	−0.961672
$154$	0	0
$155$	−172.826	−0.0895595
$156$	0	0
$157$	−1506.02	−0.765562	−0.382781	−	0.923839i	$-0.625034\pi$
$157$	−1506.02	−0.765562	−0.382781	+	0.923839i	$0.625034\pi$
$158$	0	0
$159$	−2323.35	−1.15883
$160$	0	0
$161$	−68.6057	−0.0335832
$162$	0	0
$163$	1258.30	0.604647	0.302323	−	0.953205i	$-0.402238\pi$
$163$	1258.30	0.604647	0.302323	+	0.953205i	$0.402238\pi$
$164$	0	0
$165$	−1043.72	−0.492445
$166$	0	0
$167$	−2764.50	−1.28098	−0.640489	−	0.767967i	$-0.721268\pi$
$167$	−2764.50	−1.28098	−0.640489	+	0.767967i	$0.721268\pi$
$168$	0	0
$169$	5820.95	2.64950
$170$	0	0
$171$	3719.51	1.66338
$172$	0	0
$173$	1004.21	0.441322	0.220661	−	0.975351i	$-0.429179\pi$
$173$	1004.21	0.441322	0.220661	+	0.975351i	$0.429179\pi$
$174$	0	0
$175$	−3027.09	−1.30758
$176$	0	0
$177$	−1213.64	−0.515384
$178$	0	0
$179$	−2666.39	−1.11338	−0.556692	−	0.830719i	$-0.687930\pi$
$179$	−2666.39	−1.11338	−0.556692	+	0.830719i	$0.687930\pi$
$180$	0	0
$181$	3016.21	1.23863	0.619317	−	0.785141i	$-0.287409\pi$
$181$	3016.21	1.23863	0.619317	+	0.785141i	$0.287409\pi$
$182$	0	0
$183$	4212.79	1.70174
$184$	0	0
$185$	155.312	0.0617232
$186$	0	0
$187$	−1061.45	−0.415086
$188$	0	0
$189$	−8958.98	−3.44799
$190$	0	0
$191$	−1413.15	−0.535352	−0.267676	−	0.963509i	$-0.586256\pi$
$191$	−1413.15	−0.535352	−0.267676	+	0.963509i	$0.586256\pi$
$192$	0	0
$193$	−1246.60	−0.464934	−0.232467	−	0.972604i	$-0.574680\pi$
$193$	−1246.60	−0.464934	−0.232467	+	0.972604i	$0.574680\pi$
$194$	0	0
$195$	−2536.12	−0.931362
$196$	0	0
$197$	−4931.17	−1.78341	−0.891704	−	0.452619i	$-0.850490\pi$
$197$	−4931.17	−1.78341	−0.891704	+	0.452619i	$0.850490\pi$
$198$	0	0
$199$	−552.461	−0.196799	−0.0983993	−	0.995147i	$-0.531372\pi$
$199$	−552.461	−0.196799	−0.0983993	+	0.995147i	$0.531372\pi$
$200$	0	0
$201$	−1119.60	−0.392889
$202$	0	0
$203$	4531.64	1.56679
$204$	0	0
$205$	425.609	0.145004
$206$	0	0
$207$	−166.262	−0.0558263
$208$	0	0
$209$	2169.31	0.717964
$210$	0	0
$211$	−2302.22	−0.751145	−0.375572	−	0.926793i	$-0.622554\pi$
$211$	−2302.22	−0.751145	−0.375572	+	0.926793i	$0.622554\pi$
$212$	0	0
$213$	7756.76	2.49523
$214$	0	0
$215$	128.246	0.0406803
$216$	0	0
$217$	−1510.81	−0.472629
$218$	0	0
$219$	5896.90	1.81952
$220$	0	0
$221$	−2579.21	−0.785053
$222$	0	0
$223$	−2558.41	−0.768269	−0.384135	−	0.923277i	$-0.625500\pi$
$223$	−2558.41	−0.768269	−0.384135	+	0.923277i	$0.625500\pi$
$224$	0	0
$225$	−7336.00	−2.17363
$226$	0	0
$227$	−3622.76	−1.05926	−0.529628	−	0.848230i	$-0.677669\pi$
$227$	−3622.76	−1.05926	−0.529628	+	0.848230i	$0.677669\pi$
$228$	0	0
$229$	1155.46	0.333428	0.166714	−	0.986005i	$-0.446684\pi$
$229$	1155.46	0.333428	0.166714	+	0.986005i	$0.446684\pi$
$230$	0	0
$231$	−9123.98	−2.59876
$232$	0	0
$233$	527.800	0.148401	0.0742003	−	0.997243i	$-0.476360\pi$
$233$	527.800	0.148401	0.0742003	+	0.997243i	$0.476360\pi$
$234$	0	0
$235$	318.683	0.0884620
$236$	0	0
$237$	3587.51	0.983266
$238$	0	0
$239$	1341.41	0.363048	0.181524	−	0.983387i	$-0.441897\pi$
$239$	1341.41	0.363048	0.181524	+	0.983387i	$0.441897\pi$
$240$	0	0
$241$	−3738.93	−0.999361	−0.499680	−	0.866210i	$-0.666549\pi$
$241$	−3738.93	−0.999361	−0.499680	+	0.866210i	$0.666549\pi$
$242$	0	0
$243$	−5510.79	−1.45480
$244$	0	0
$245$	1004.34	0.261898
$246$	0	0
$247$	5271.19	1.35789
$248$	0	0
$249$	−13800.6	−3.51237
$250$	0	0
$251$	1741.62	0.437969	0.218985	−	0.975728i	$-0.429726\pi$
$251$	1741.62	0.437969	0.218985	+	0.975728i	$0.429726\pi$
$252$	0	0
$253$	−96.9684	−0.0240963
$254$	0	0
$255$	815.821	0.200348
$256$	0	0
$257$	4121.68	1.00040	0.500200	−	0.865910i	$-0.333260\pi$
$257$	4121.68	1.00040	0.500200	+	0.865910i	$0.333260\pi$
$258$	0	0
$259$	1357.71	0.325730
$260$	0	0
$261$	10982.2	2.60452
$262$	0	0
$263$	−4315.46	−1.01180	−0.505898	−	0.862593i	$-0.668839\pi$
$263$	−4315.46	−1.01180	−0.505898	+	0.862593i	$0.668839\pi$
$264$	0	0
$265$	729.663	0.169143
$266$	0	0
$267$	−5963.76	−1.36695
$268$	0	0
$269$	5195.44	1.17759	0.588794	−	0.808283i	$-0.299603\pi$
$269$	5195.44	1.17759	0.588794	+	0.808283i	$0.299603\pi$
$270$	0	0
$271$	7874.15	1.76502	0.882510	−	0.470294i	$-0.155852\pi$
$271$	7874.15	1.76502	0.882510	+	0.470294i	$0.155852\pi$
$272$	0	0
$273$	−22170.3	−4.91504
$274$	0	0
$275$	−4278.54	−0.938202
$276$	0	0
$277$	−175.471	−0.0380615	−0.0190307	−	0.999819i	$-0.506058\pi$
$277$	−175.471	−0.0380615	−0.0190307	+	0.999819i	$0.506058\pi$
$278$	0	0
$279$	−3661.37	−0.785665
$280$	0	0
$281$	7263.01	1.54190	0.770951	−	0.636894i	$-0.219781\pi$
$281$	7263.01	1.54190	0.770951	+	0.636894i	$0.219781\pi$
$282$	0	0
$283$	−2314.68	−0.486196	−0.243098	−	0.970002i	$-0.578164\pi$
$283$	−2314.68	−0.486196	−0.243098	+	0.970002i	$0.578164\pi$
$284$	0	0
$285$	−1667.31	−0.346536
$286$	0	0
$287$	3720.58	0.765224
$288$	0	0
$289$	−4083.32	−0.831125
$290$	0	0
$291$	7760.22	1.56327
$292$	0	0
$293$	8723.52	1.73936	0.869681	−	0.493614i	$-0.164324\pi$
$293$	8723.52	1.73936	0.869681	+	0.493614i	$0.164324\pi$
$294$	0	0
$295$	381.153	0.0752258
$296$	0	0
$297$	−12662.8	−2.47397
$298$	0	0
$299$	−235.623	−0.0455733
$300$	0	0
$301$	1121.10	0.214681
$302$	0	0
$303$	−4875.88	−0.924462
$304$	0	0
$305$	−1323.06	−0.248387
$306$	0	0
$307$	−1579.68	−0.293672	−0.146836	−	0.989161i	$-0.546909\pi$
$307$	−1579.68	−0.293672	−0.146836	+	0.989161i	$0.546909\pi$
$308$	0	0
$309$	6550.59	1.20599
$310$	0	0
$311$	5604.24	1.02182	0.510912	−	0.859633i	$-0.329308\pi$
$311$	5604.24	1.02182	0.510912	+	0.859633i	$0.329308\pi$
$312$	0	0
$313$	3429.86	0.619384	0.309692	−	0.950837i	$-0.399774\pi$
$313$	3429.86	0.619384	0.309692	+	0.950837i	$0.399774\pi$
$314$	0	0
$315$	4913.11	0.878802
$316$	0	0
$317$	4493.00	0.796064	0.398032	−	0.917372i	$-0.369693\pi$
$317$	4493.00	0.796064	0.398032	+	0.917372i	$0.369693\pi$
$318$	0	0
$319$	6405.09	1.12419
$320$	0	0
$321$	3045.63	0.529566
$322$	0	0
$323$	−1695.64	−0.292098
$324$	0	0
$325$	−10396.4	−1.77442
$326$	0	0
$327$	16063.5	2.71656
$328$	0	0
$329$	2785.86	0.466837
$330$	0	0
$331$	−4433.87	−0.736277	−0.368139	−	0.929771i	$-0.620005\pi$
$331$	−4433.87	−0.736277	−0.368139	+	0.929771i	$0.620005\pi$
$332$	0	0
$333$	3290.34	0.541470
$334$	0	0
$335$	351.620	0.0573463
$336$	0	0
$337$	6498.33	1.05040	0.525202	−	0.850977i	$-0.323990\pi$
$337$	6498.33	1.05040	0.525202	+	0.850977i	$0.323990\pi$
$338$	0	0
$339$	−8132.45	−1.30293
$340$	0	0
$341$	−2135.41	−0.339116
$342$	0	0
$343$	−162.945	−0.0256507
$344$	0	0
$345$	74.5289	0.0116304
$346$	0	0
$347$	11973.6	1.85237	0.926187	−	0.377064i	$-0.123066\pi$
$347$	11973.6	1.85237	0.926187	+	0.377064i	$0.123066\pi$
$348$	0	0
$349$	5611.30	0.860648	0.430324	−	0.902674i	$-0.358399\pi$
$349$	5611.30	0.860648	0.430324	+	0.902674i	$0.358399\pi$
$350$	0	0
$351$	−30769.1	−4.67902
$352$	0	0
$353$	−2022.00	−0.304874	−0.152437	−	0.988313i	$-0.548712\pi$
$353$	−2022.00	−0.304874	−0.152437	+	0.988313i	$0.548712\pi$
$354$	0	0
$355$	−2436.07	−0.364206
$356$	0	0
$357$	7131.74	1.05729
$358$	0	0
$359$	−10135.3	−1.49003	−0.745014	−	0.667049i	$-0.767557\pi$
$359$	−10135.3	−1.49003	−0.745014	+	0.667049i	$0.767557\pi$
$360$	0	0
$361$	−3393.59	−0.494765
$362$	0	0
$363$	−256.097	−0.0370292
$364$	0	0
$365$	−1851.96	−0.265579
$366$	0	0
$367$	4379.58	0.622922	0.311461	−	0.950259i	$-0.399182\pi$
$367$	4379.58	0.622922	0.311461	+	0.950259i	$0.399182\pi$
$368$	0	0
$369$	9016.64	1.27205
$370$	0	0
$371$	6378.57	0.892612
$372$	0	0
$373$	4392.98	0.609813	0.304906	−	0.952382i	$-0.401375\pi$
$373$	4392.98	0.609813	0.304906	+	0.952382i	$0.401375\pi$
$374$	0	0
$375$	6828.81	0.940369
$376$	0	0
$377$	15563.7	2.12618
$378$	0	0
$379$	878.629	0.119082	0.0595411	−	0.998226i	$-0.481036\pi$
$379$	878.629	0.119082	0.0595411	+	0.998226i	$0.481036\pi$
$380$	0	0
$381$	−21212.6	−2.85237
$382$	0	0
$383$	−5061.60	−0.675289	−0.337644	−	0.941274i	$-0.609630\pi$
$383$	−5061.60	−0.675289	−0.337644	+	0.941274i	$0.609630\pi$
$384$	0	0
$385$	2865.45	0.379317
$386$	0	0
$387$	2716.92	0.356870
$388$	0	0
$389$	−3417.17	−0.445392	−0.222696	−	0.974888i	$-0.571486\pi$
$389$	−3417.17	−0.445392	−0.222696	+	0.974888i	$0.571486\pi$
$390$	0	0
$391$	75.7952	0.00980339
$392$	0	0
$393$	−19482.2	−2.50063
$394$	0	0
$395$	−1126.68	−0.143518
$396$	0	0
$397$	7634.34	0.965130	0.482565	−	0.875860i	$-0.339705\pi$
$397$	7634.34	0.965130	0.482565	+	0.875860i	$0.339705\pi$
$398$	0	0
$399$	−14575.3	−1.82876
$400$	0	0
$401$	−8402.74	−1.04642	−0.523208	−	0.852205i	$-0.675265\pi$
$401$	−8402.74	−1.04642	−0.523208	+	0.852205i	$0.675265\pi$
$402$	0	0
$403$	−5188.80	−0.641372
$404$	0	0
$405$	4644.48	0.569842
$406$	0	0
$407$	1919.01	0.233714
$408$	0	0
$409$	−11792.8	−1.42571	−0.712857	−	0.701310i	$-0.752599\pi$
$409$	−11792.8	−1.42571	−0.712857	+	0.701310i	$0.752599\pi$
$410$	0	0
$411$	−24638.7	−2.95702
$412$	0	0
$413$	3331.97	0.396986
$414$	0	0
$415$	4334.19	0.512667
$416$	0	0
$417$	10830.2	1.27184
$418$	0	0
$419$	10631.9	1.23962	0.619811	−	0.784751i	$-0.287209\pi$
$419$	10631.9	1.23962	0.619811	+	0.784751i	$0.287209\pi$
$420$	0	0
$421$	−3136.52	−0.363099	−0.181550	−	0.983382i	$-0.558111\pi$
$421$	−3136.52	−0.363099	−0.181550	+	0.983382i	$0.558111\pi$
$422$	0	0
$423$	6751.39	0.776037
$424$	0	0
$425$	3344.31	0.381701
$426$	0	0
$427$	−11565.9	−1.31080
$428$	0	0
$429$	−31335.8	−3.52659
$430$	0	0
$431$	170.380	0.0190416	0.00952080	−	0.999955i	$-0.496969\pi$
$431$	170.380	0.0190416	0.00952080	+	0.999955i	$0.496969\pi$
$432$	0	0
$433$	2093.65	0.232365	0.116183	−	0.993228i	$-0.462934\pi$
$433$	2093.65	0.232365	0.116183	+	0.993228i	$0.462934\pi$
$434$	0	0
$435$	−4922.89	−0.542608
$436$	0	0
$437$	−154.904	−0.0169567
$438$	0	0
$439$	10860.1	1.18070	0.590348	−	0.807149i	$-0.298990\pi$
$439$	10860.1	1.18070	0.590348	+	0.807149i	$0.298990\pi$
$440$	0	0
$441$	21277.3	2.29751
$442$	0	0
$443$	−8256.28	−0.885480	−0.442740	−	0.896650i	$-0.645994\pi$
$443$	−8256.28	−0.885480	−0.442740	+	0.896650i	$0.645994\pi$
$444$	0	0
$445$	1872.96	0.199521
$446$	0	0
$447$	20057.0	2.12229
$448$	0	0
$449$	−6792.03	−0.713888	−0.356944	−	0.934126i	$-0.616181\pi$
$449$	−6792.03	−0.713888	−0.356944	+	0.934126i	$0.616181\pi$
$450$	0	0
$451$	5258.73	0.549056
$452$	0	0
$453$	−12832.8	−1.33098
$454$	0	0
$455$	6962.74	0.717403
$456$	0	0
$457$	4004.62	0.409908	0.204954	−	0.978772i	$-0.434295\pi$
$457$	4004.62	0.409908	0.204954	+	0.978772i	$0.434295\pi$
$458$	0	0
$459$	9897.82	1.00652
$460$	0	0
$461$	1164.36	0.117635	0.0588175	−	0.998269i	$-0.481267\pi$
$461$	1164.36	0.117635	0.0588175	+	0.998269i	$0.481267\pi$
$462$	0	0
$463$	−2566.99	−0.257664	−0.128832	−	0.991666i	$-0.541123\pi$
$463$	−2566.99	−0.257664	−0.128832	+	0.991666i	$0.541123\pi$
$464$	0	0
$465$	1641.25	0.163680
$466$	0	0
$467$	7654.43	0.758469	0.379234	−	0.925301i	$-0.376187\pi$
$467$	7654.43	0.758469	0.379234	+	0.925301i	$0.376187\pi$
$468$	0	0
$469$	3073.79	0.302632
$470$	0	0
$471$	14301.9	1.39915
$472$	0	0
$473$	1584.58	0.154036
$474$	0	0
$475$	−6834.84	−0.660218
$476$	0	0
$477$	15458.1	1.48381
$478$	0	0
$479$	8754.20	0.835051	0.417526	−	0.908665i	$-0.362897\pi$
$479$	8754.20	0.835051	0.417526	+	0.908665i	$0.362897\pi$
$480$	0	0
$481$	4662.98	0.442024
$482$	0	0
$483$	651.517	0.0613769
$484$	0	0
$485$	−2437.15	−0.228176
$486$	0	0
$487$	9406.39	0.875245	0.437623	−	0.899159i	$-0.355821\pi$
$487$	9406.39	0.875245	0.437623	+	0.899159i	$0.355821\pi$
$488$	0	0
$489$	−11949.5	−1.10506
$490$	0	0
$491$	−6362.18	−0.584768	−0.292384	−	0.956301i	$-0.594449\pi$
$491$	−6362.18	−0.584768	−0.292384	+	0.956301i	$0.594449\pi$
$492$	0	0
$493$	−5006.53	−0.457369
$494$	0	0
$495$	6944.27	0.630549
$496$	0	0
$497$	−21295.6	−1.92201
$498$	0	0
$499$	−11574.8	−1.03840	−0.519198	−	0.854654i	$-0.673769\pi$
$499$	−11574.8	−1.03840	−0.519198	+	0.854654i	$0.673769\pi$
$500$	0	0
$501$	26253.2	2.34113
$502$	0	0
$503$	−11443.5	−1.01439	−0.507196	−	0.861831i	$-0.669318\pi$
$503$	−11443.5	−1.01439	−0.507196	+	0.861831i	$0.669318\pi$
$504$	0	0
$505$	1531.31	0.134935
$506$	0	0
$507$	−55278.8	−4.84225
$508$	0	0
$509$	−17397.9	−1.51502	−0.757511	−	0.652822i	$-0.773585\pi$
$509$	−17397.9	−1.51502	−0.757511	+	0.652822i	$0.773585\pi$
$510$	0	0
$511$	−16189.5	−1.40153
$512$	0	0
$513$	−20228.4	−1.74095
$514$	0	0
$515$	−2057.26	−0.176027
$516$	0	0
$517$	3937.58	0.334961
$518$	0	0
$519$	−9536.53	−0.806565
$520$	0	0
$521$	−23236.4	−1.95394	−0.976972	−	0.213369i	$-0.931556\pi$
$521$	−23236.4	−1.95394	−0.976972	+	0.213369i	$0.931556\pi$
$522$	0	0
$523$	6523.69	0.545432	0.272716	−	0.962095i	$-0.412078\pi$
$523$	6523.69	0.545432	0.272716	+	0.962095i	$0.412078\pi$
$524$	0	0
$525$	28746.9	2.38975
$526$	0	0
$527$	1669.13	0.137967
$528$	0	0
$529$	−12160.1	−0.999431
$530$	0	0
$531$	8074.85	0.659922
$532$	0	0
$533$	12778.2	1.03843
$534$	0	0
$535$	−956.502	−0.0772957
$536$	0	0
$537$	25321.5	2.03483
$538$	0	0
$539$	12409.5	0.991675
$540$	0	0
$541$	−13311.4	−1.05786	−0.528930	−	0.848666i	$-0.677406\pi$
$541$	−13311.4	−1.05786	−0.528930	+	0.848666i	$0.677406\pi$
$542$	0	0
$543$	−28643.5	−2.26374
$544$	0	0
$545$	−5044.86	−0.396510
$546$	0	0
$547$	24529.8	1.91740	0.958700	−	0.284420i	$-0.0918010\pi$
$547$	24529.8	1.91740	0.958700	+	0.284420i	$0.0918010\pi$
$548$	0	0
$549$	−28029.3	−2.17898
$550$	0	0
$551$	10231.9	0.791099
$552$	0	0
$553$	−9849.25	−0.757383
$554$	0	0
$555$	−1474.93	−0.112806
$556$	0	0
$557$	−1732.87	−0.131821	−0.0659104	−	0.997826i	$-0.520995\pi$
$557$	−1732.87	−0.131821	−0.0659104	+	0.997826i	$0.520995\pi$
$558$	0	0
$559$	3850.35	0.291328
$560$	0	0
$561$	10080.1	0.758615
$562$	0	0
$563$	−13482.0	−1.00924	−0.504618	−	0.863343i	$-0.668366\pi$
$563$	−13482.0	−1.00924	−0.504618	+	0.863343i	$0.668366\pi$
$564$	0	0
$565$	2554.05	0.190177
$566$	0	0
$567$	40601.1	3.00721
$568$	0	0
$569$	23053.6	1.69852	0.849261	−	0.527973i	$-0.177048\pi$
$569$	23053.6	1.69852	0.849261	+	0.527973i	$0.177048\pi$
$570$	0	0
$571$	−8791.94	−0.644363	−0.322181	−	0.946678i	$-0.604416\pi$
$571$	−8791.94	−0.644363	−0.322181	+	0.946678i	$0.604416\pi$
$572$	0	0
$573$	13420.1	0.978414
$574$	0	0
$575$	305.518	0.0221582
$576$	0	0
$577$	−12464.3	−0.899302	−0.449651	−	0.893204i	$-0.648452\pi$
$577$	−12464.3	−0.899302	−0.449651	+	0.893204i	$0.648452\pi$
$578$	0	0
$579$	11838.4	0.849718
$580$	0	0
$581$	37888.6	2.70548
$582$	0	0
$583$	9015.58	0.640458
$584$	0	0
$585$	16873.8	1.19256
$586$	0	0
$587$	24395.5	1.71535	0.857674	−	0.514193i	$-0.171909\pi$
$587$	24395.5	1.71535	0.857674	+	0.514193i	$0.171909\pi$
$588$	0	0
$589$	−3411.24	−0.238638
$590$	0	0
$591$	46829.0	3.25937
$592$	0	0
$593$	−8779.99	−0.608012	−0.304006	−	0.952670i	$-0.598324\pi$
$593$	−8779.99	−0.608012	−0.304006	+	0.952670i	$0.598324\pi$
$594$	0	0
$595$	−2239.77	−0.154322
$596$	0	0
$597$	5246.46	0.359671
$598$	0	0
$599$	14440.3	0.985001	0.492501	−	0.870312i	$-0.336083\pi$
$599$	14440.3	0.985001	0.492501	+	0.870312i	$0.336083\pi$
$600$	0	0
$601$	7456.08	0.506056	0.253028	−	0.967459i	$-0.418573\pi$
$601$	7456.08	0.506056	0.253028	+	0.967459i	$0.418573\pi$
$602$	0	0
$603$	7449.16	0.503074
$604$	0	0
$605$	80.4290	0.00540480
$606$	0	0
$607$	6832.01	0.456842	0.228421	−	0.973563i	$-0.426644\pi$
$607$	6832.01	0.456842	0.228421	+	0.973563i	$0.426644\pi$
$608$	0	0
$609$	−43034.9	−2.86348
$610$	0	0
$611$	9567.89	0.633512
$612$	0	0
$613$	12239.8	0.806460	0.403230	−	0.915099i	$-0.367887\pi$
$613$	12239.8	0.806460	0.403230	+	0.915099i	$0.367887\pi$
$614$	0	0
$615$	−4041.81	−0.265010
$616$	0	0
$617$	4307.99	0.281091	0.140546	−	0.990074i	$-0.455114\pi$
$617$	4307.99	0.281091	0.140546	+	0.990074i	$0.455114\pi$
$618$	0	0
$619$	−21923.8	−1.42357	−0.711786	−	0.702396i	$-0.752113\pi$
$619$	−21923.8	−1.42357	−0.711786	+	0.702396i	$0.752113\pi$
$620$	0	0
$621$	904.210	0.0584295
$622$	0	0
$623$	16373.0	1.05292
$624$	0	0
$625$	12368.5	0.791583
$626$	0	0
$627$	−20600.9	−1.31216
$628$	0	0
$629$	−1499.99	−0.0950850
$630$	0	0
$631$	−13249.3	−0.835891	−0.417945	−	0.908472i	$-0.637250\pi$
$631$	−13249.3	−0.835891	−0.417945	+	0.908472i	$0.637250\pi$
$632$	0	0
$633$	21863.1	1.37280
$634$	0	0
$635$	6661.97	0.416334
$636$	0	0
$637$	30153.6	1.87556
$638$	0	0
$639$	−51608.8	−3.19501
$640$	0	0
$641$	−6897.79	−0.425033	−0.212517	−	0.977157i	$-0.568166\pi$
$641$	−6897.79	−0.425033	−0.212517	+	0.977157i	$0.568166\pi$
$642$	0	0
$643$	570.068	0.0349631	0.0174816	−	0.999847i	$-0.494435\pi$
$643$	570.068	0.0349631	0.0174816	+	0.999847i	$0.494435\pi$
$644$	0	0
$645$	−1217.89	−0.0743477
$646$	0	0
$647$	−15788.1	−0.959341	−0.479670	−	0.877449i	$-0.659244\pi$
$647$	−15788.1	−0.959341	−0.479670	+	0.877449i	$0.659244\pi$
$648$	0	0
$649$	4709.46	0.284842
$650$	0	0
$651$	14347.5	0.863782
$652$	0	0
$653$	21960.3	1.31604	0.658019	−	0.753001i	$-0.271394\pi$
$653$	21960.3	1.31604	0.658019	+	0.753001i	$0.271394\pi$
$654$	0	0
$655$	6118.52	0.364993
$656$	0	0
$657$	−39234.4	−2.32980
$658$	0	0
$659$	12142.6	0.717766	0.358883	−	0.933383i	$-0.383158\pi$
$659$	12142.6	0.717766	0.358883	+	0.933383i	$0.383158\pi$
$660$	0	0
$661$	3554.01	0.209130	0.104565	−	0.994518i	$-0.466655\pi$
$661$	3554.01	0.209130	0.104565	+	0.994518i	$0.466655\pi$
$662$	0	0
$663$	24493.6	1.43477
$664$	0	0
$665$	4577.47	0.266927
$666$	0	0
$667$	−457.369	−0.0265508
$668$	0	0
$669$	24296.1	1.40410
$670$	0	0
$671$	−16347.4	−0.940514
$672$	0	0
$673$	20271.0	1.16106	0.580528	−	0.814240i	$-0.302846\pi$
$673$	20271.0	1.16106	0.580528	+	0.814240i	$0.302846\pi$
$674$	0	0
$675$	39896.5	2.27499
$676$	0	0
$677$	26668.2	1.51395	0.756973	−	0.653447i	$-0.226678\pi$
$677$	26668.2	1.51395	0.756973	+	0.653447i	$0.226678\pi$
$678$	0	0
$679$	−21305.1	−1.20415
$680$	0	0
$681$	34403.7	1.93591
$682$	0	0
$683$	5584.15	0.312843	0.156421	−	0.987690i	$-0.450004\pi$
$683$	5584.15	0.312843	0.156421	+	0.987690i	$0.450004\pi$
$684$	0	0
$685$	7737.95	0.431608
$686$	0	0
$687$	−10972.9	−0.609377
$688$	0	0
$689$	21906.9	1.21130
$690$	0	0
$691$	2702.39	0.148776	0.0743878	−	0.997229i	$-0.476300\pi$
$691$	2702.39	0.148776	0.0743878	+	0.997229i	$0.476300\pi$
$692$	0	0
$693$	60705.4	3.32757
$694$	0	0
$695$	−3401.31	−0.185639
$696$	0	0
$697$	−4110.48	−0.223379
$698$	0	0
$699$	−5012.27	−0.271218
$700$	0	0
$701$	−19885.1	−1.07140	−0.535700	−	0.844408i	$-0.679952\pi$
$701$	−19885.1	−1.07140	−0.535700	+	0.844408i	$0.679952\pi$
$702$	0	0
$703$	3065.56	0.164466
$704$	0	0
$705$	−3026.38	−0.161674
$706$	0	0
$707$	13386.4	0.712088
$708$	0	0
$709$	−7798.56	−0.413090	−0.206545	−	0.978437i	$-0.566222\pi$
$709$	−7798.56	−0.413090	−0.206545	+	0.978437i	$0.566222\pi$
$710$	0	0
$711$	−23869.1	−1.25902
$712$	0	0
$713$	152.483	0.00800916
$714$	0	0
$715$	9841.24	0.514744
$716$	0	0
$717$	−12738.7	−0.663509
$718$	0	0
$719$	−1118.09	−0.0579939	−0.0289970	−	0.999579i	$-0.509231\pi$
$719$	−1118.09	−0.0579939	−0.0289970	+	0.999579i	$0.509231\pi$
$720$	0	0
$721$	−17984.2	−0.928939
$722$	0	0
$723$	35506.9	1.82644
$724$	0	0
$725$	−20180.5	−1.03377
$726$	0	0
$727$	1741.54	0.0888449	0.0444225	−	0.999013i	$-0.485855\pi$
$727$	1741.54	0.0888449	0.0444225	+	0.999013i	$0.485855\pi$
$728$	0	0
$729$	10287.2	0.522642
$730$	0	0
$731$	−1238.58	−0.0626683
$732$	0	0
$733$	−9461.10	−0.476744	−0.238372	−	0.971174i	$-0.576614\pi$
$733$	−9461.10	−0.476744	−0.238372	+	0.971174i	$0.576614\pi$
$734$	0	0
$735$	−9537.77	−0.478648
$736$	0	0
$737$	4344.54	0.217141
$738$	0	0
$739$	−30605.3	−1.52346	−0.761728	−	0.647897i	$-0.775649\pi$
$739$	−30605.3	−1.52346	−0.761728	+	0.647897i	$0.775649\pi$
$740$	0	0
$741$	−50058.1	−2.48169
$742$	0	0
$743$	−14556.5	−0.718745	−0.359373	−	0.933194i	$-0.617009\pi$
$743$	−14556.5	−0.718745	−0.359373	+	0.933194i	$0.617009\pi$
$744$	0	0
$745$	−6299.03	−0.309770
$746$	0	0
$747$	91821.1	4.49740
$748$	0	0
$749$	−8361.55	−0.407910
$750$	0	0
$751$	17528.3	0.851685	0.425843	−	0.904797i	$-0.359978\pi$
$751$	17528.3	0.851685	0.425843	+	0.904797i	$0.359978\pi$
$752$	0	0
$753$	−16539.4	−0.800436
$754$	0	0
$755$	4030.22	0.194271
$756$	0	0
$757$	−37789.1	−1.81436	−0.907179	−	0.420744i	$-0.861769\pi$
$757$	−37789.1	−1.81436	−0.907179	+	0.420744i	$0.861769\pi$
$758$	0	0
$759$	920.864	0.0440385
$760$	0	0
$761$	3292.90	0.156856	0.0784281	−	0.996920i	$-0.475010\pi$
$761$	3292.90	0.156856	0.0784281	+	0.996920i	$0.475010\pi$
$762$	0	0
$763$	−44101.2	−2.09249
$764$	0	0
$765$	−5427.98	−0.256534
$766$	0	0
$767$	11443.5	0.538722
$768$	0	0
$769$	−28023.9	−1.31413	−0.657065	−	0.753834i	$-0.728202\pi$
$769$	−28023.9	−1.31413	−0.657065	+	0.753834i	$0.728202\pi$
$770$	0	0
$771$	−39141.6	−1.82834
$772$	0	0
$773$	22131.0	1.02975	0.514875	−	0.857265i	$-0.327838\pi$
$773$	22131.0	1.02975	0.514875	+	0.857265i	$0.327838\pi$
$774$	0	0
$775$	6728.01	0.311841
$776$	0	0
$777$	−12893.5	−0.595306
$778$	0	0
$779$	8400.67	0.386374
$780$	0	0
$781$	−30099.6	−1.37906
$782$	0	0
$783$	−59726.2	−2.72598
$784$	0	0
$785$	−4491.63	−0.204220
$786$	0	0
$787$	−5390.88	−0.244173	−0.122087	−	0.992519i	$-0.538959\pi$
$787$	−5390.88	−0.244173	−0.122087	+	0.992519i	$0.538959\pi$
$788$	0	0
$789$	40981.9	1.84917
$790$	0	0
$791$	22327.0	1.00361
$792$	0	0
$793$	−39722.4	−1.77880
$794$	0	0
$795$	−6929.27	−0.309127
$796$	0	0
$797$	43815.4	1.94733	0.973664	−	0.227986i	$-0.0732142\pi$
$797$	43815.4	1.94733	0.973664	+	0.227986i	$0.0732142\pi$
$798$	0	0
$799$	−3077.80	−0.136276
$800$	0	0
$801$	39679.2	1.75031
$802$	0	0
$803$	−22882.5	−1.00561
$804$	0	0
$805$	−204.613	−0.00895860
$806$	0	0
$807$	−49338.6	−2.15217
$808$	0	0
$809$	−19752.4	−0.858415	−0.429208	−	0.903206i	$-0.641207\pi$
$809$	−19752.4	−0.858415	−0.429208	+	0.903206i	$0.641207\pi$
$810$	0	0
$811$	−21280.3	−0.921398	−0.460699	−	0.887556i	$-0.652401\pi$
$811$	−21280.3	−0.921398	−0.460699	+	0.887556i	$0.652401\pi$
$812$	0	0
$813$	−74777.1	−3.22577
$814$	0	0
$815$	3752.81	0.161295
$816$	0	0
$817$	2531.31	0.108396
$818$	0	0
$819$	147508.	6.29345
$820$	0	0
$821$	−19716.5	−0.838136	−0.419068	−	0.907955i	$-0.637643\pi$
$821$	−19716.5	−0.838136	−0.419068	+	0.907955i	$0.637643\pi$
$822$	0	0
$823$	−16269.0	−0.689065	−0.344532	−	0.938774i	$-0.611963\pi$
$823$	−16269.0	−0.689065	−0.344532	+	0.938774i	$0.611963\pi$
$824$	0	0
$825$	40631.3	1.71467
$826$	0	0
$827$	27410.0	1.15253	0.576263	−	0.817264i	$-0.304510\pi$
$827$	27410.0	1.15253	0.576263	+	0.817264i	$0.304510\pi$
$828$	0	0
$829$	−25392.8	−1.06385	−0.531923	−	0.846792i	$-0.678531\pi$
$829$	−25392.8	−1.06385	−0.531923	+	0.846792i	$0.678531\pi$
$830$	0	0
$831$	1666.37	0.0695615
$832$	0	0
$833$	−9699.82	−0.403456
$834$	0	0
$835$	−8244.99	−0.341712
$836$	0	0
$837$	19912.2	0.822302
$838$	0	0
$839$	20579.5	0.846823	0.423412	−	0.905937i	$-0.360832\pi$
$839$	20579.5	0.846823	0.423412	+	0.905937i	$0.360832\pi$
$840$	0	0
$841$	5821.76	0.238704
$842$	0	0
$843$	−68973.4	−2.81799
$844$	0	0
$845$	17360.7	0.706777
$846$	0	0
$847$	703.094	0.0285225
$848$	0	0
$849$	21981.4	0.888575
$850$	0	0
$851$	−137.031	−0.00551980
$852$	0	0
$853$	6873.98	0.275921	0.137961	−	0.990438i	$-0.455945\pi$
$853$	6873.98	0.275921	0.137961	+	0.990438i	$0.455945\pi$
$854$	0	0
$855$	11093.3	0.443721
$856$	0	0
$857$	−20623.8	−0.822049	−0.411025	−	0.911624i	$-0.634829\pi$
$857$	−20623.8	−0.822049	−0.411025	+	0.911624i	$0.634829\pi$
$858$	0	0
$859$	−40959.1	−1.62690	−0.813450	−	0.581635i	$-0.802413\pi$
$859$	−40959.1	−1.62690	−0.813450	+	0.581635i	$0.802413\pi$
$860$	0	0
$861$	−35332.7	−1.39853
$862$	0	0
$863$	−33784.0	−1.33258	−0.666292	−	0.745691i	$-0.732120\pi$
$863$	−33784.0	−1.33258	−0.666292	+	0.745691i	$0.732120\pi$
$864$	0	0
$865$	2995.01	0.117727
$866$	0	0
$867$	38777.4	1.51897
$868$	0	0
$869$	−13921.1	−0.543430
$870$	0	0
$871$	10556.8	0.410680
$872$	0	0
$873$	−51631.8	−2.00169
$874$	0	0
$875$	−18748.0	−0.724340
$876$	0	0
$877$	−37125.7	−1.42947	−0.714736	−	0.699395i	$-0.753453\pi$
$877$	−37125.7	−1.42947	−0.714736	+	0.699395i	$0.753453\pi$
$878$	0	0
$879$	−82843.2	−3.17887
$880$	0	0
$881$	−108.651	−0.00415499	−0.00207749	−	0.999998i	$-0.500661\pi$
$881$	−108.651	−0.00415499	−0.00207749	+	0.999998i	$0.500661\pi$
$882$	0	0
$883$	19622.4	0.747843	0.373921	−	0.927460i	$-0.378013\pi$
$883$	19622.4	0.747843	0.373921	+	0.927460i	$0.378013\pi$
$884$	0	0
$885$	−3619.64	−0.137483
$886$	0	0
$887$	26012.1	0.984668	0.492334	−	0.870406i	$-0.336144\pi$
$887$	26012.1	0.984668	0.492334	+	0.870406i	$0.336144\pi$
$888$	0	0
$889$	58237.6	2.19710
$890$	0	0
$891$	57386.2	2.15770
$892$	0	0
$893$	6290.17	0.235714
$894$	0	0
$895$	−7952.40	−0.297005
$896$	0	0
$897$	2237.60	0.0832902
$898$	0	0
$899$	−10072.0	−0.373660
$900$	0	0
$901$	−7047.01	−0.260566
$902$	0	0
$903$	−10646.5	−0.392353
$904$	0	0
$905$	8995.70	0.330417
$906$	0	0
$907$	42258.2	1.54703	0.773517	−	0.633776i	$-0.218496\pi$
$907$	42258.2	1.54703	0.773517	+	0.633776i	$0.218496\pi$
$908$	0	0
$909$	32441.2	1.18372
$910$	0	0
$911$	−6717.75	−0.244313	−0.122156	−	0.992511i	$-0.538981\pi$
$911$	−6717.75	−0.244313	−0.122156	+	0.992511i	$0.538981\pi$
$912$	0	0
$913$	53552.4	1.94121
$914$	0	0
$915$	12564.4	0.453954
$916$	0	0
$917$	53486.9	1.92616
$918$	0	0
$919$	−46824.2	−1.68073	−0.840364	−	0.542023i	$-0.817659\pi$
$919$	−46824.2	−1.68073	−0.840364	+	0.542023i	$0.817659\pi$
$920$	0	0
$921$	15001.5	0.536717
$922$	0	0
$923$	−73138.7	−2.60822
$924$	0	0
$925$	−6046.21	−0.214917
$926$	0	0
$927$	−43583.7	−1.54420
$928$	0	0
$929$	14300.0	0.505025	0.252513	−	0.967594i	$-0.418743\pi$
$929$	14300.0	0.505025	0.252513	+	0.967594i	$0.418743\pi$
$930$	0	0
$931$	19823.7	0.697848
$932$	0	0
$933$	−53220.8	−1.86749
$934$	0	0
$935$	−3165.73	−0.110728
$936$	0	0
$937$	44630.3	1.55604	0.778019	−	0.628241i	$-0.216225\pi$
$937$	44630.3	1.55604	0.778019	+	0.628241i	$0.216225\pi$
$938$	0	0
$939$	−32571.8	−1.13199
$940$	0	0
$941$	−12069.3	−0.418116	−0.209058	−	0.977903i	$-0.567040\pi$
$941$	−12069.3	−0.418116	−0.209058	+	0.977903i	$0.567040\pi$
$942$	0	0
$943$	−375.511	−0.0129675
$944$	0	0
$945$	−26719.7	−0.919781
$946$	0	0
$947$	−23634.0	−0.810984	−0.405492	−	0.914099i	$-0.632900\pi$
$947$	−23634.0	−0.810984	−0.405492	+	0.914099i	$0.632900\pi$
$948$	0	0
$949$	−55602.0	−1.90192
$950$	0	0
$951$	−42668.0	−1.45489
$952$	0	0
$953$	−31376.9	−1.06652	−0.533262	−	0.845950i	$-0.679034\pi$
$953$	−31376.9	−1.06652	−0.533262	+	0.845950i	$0.679034\pi$
$954$	0	0
$955$	−4214.67	−0.142810
$956$	0	0
$957$	−60826.2	−2.05458
$958$	0	0
$959$	67643.6	2.27771
$960$	0	0
$961$	−26433.1	−0.887284
$962$	0	0
$963$	−20263.8	−0.678080
$964$	0	0
$965$	−3717.93	−0.124025
$966$	0	0
$967$	−32665.7	−1.08631	−0.543154	−	0.839633i	$-0.682770\pi$
$967$	−32665.7	−1.08631	−0.543154	+	0.839633i	$0.682770\pi$
$968$	0	0
$969$	16102.7	0.533842
$970$	0	0
$971$	41567.8	1.37381	0.686907	−	0.726745i	$-0.258968\pi$
$971$	41567.8	1.37381	0.686907	+	0.726745i	$0.258968\pi$
$972$	0	0
$973$	−29733.6	−0.979665
$974$	0	0
$975$	98729.6	3.24295
$976$	0	0
$977$	−25363.2	−0.830542	−0.415271	−	0.909698i	$-0.636313\pi$
$977$	−25363.2	−0.830542	−0.415271	+	0.909698i	$0.636313\pi$
$978$	0	0
$979$	23141.9	0.755484
$980$	0	0
$981$	−106877.	−3.47841
$982$	0	0
$983$	3961.23	0.128528	0.0642642	−	0.997933i	$-0.479530\pi$
$983$	3961.23	0.128528	0.0642642	+	0.997933i	$0.479530\pi$
$984$	0	0
$985$	−14707.0	−0.475740
$986$	0	0
$987$	−26456.0	−0.853196
$988$	0	0
$989$	−113.150	−0.00363798
$990$	0	0
$991$	−27940.2	−0.895610	−0.447805	−	0.894131i	$-0.647794\pi$
$991$	−27940.2	−0.895610	−0.447805	+	0.894131i	$0.647794\pi$
$992$	0	0
$993$	42106.4	1.34563
$994$	0	0
$995$	−1647.69	−0.0524978
$996$	0	0
$997$	−15531.3	−0.493363	−0.246681	−	0.969097i	$-0.579340\pi$
$997$	−15531.3	−0.493363	−0.246681	+	0.969097i	$0.579340\pi$
$998$	0	0
$999$	−17894.4	−0.566719

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.4.a.i.1.1		6
4.3	odd	2		43.4.a.b.1.4	✓	6
12.11	even	2		387.4.a.h.1.3		6
20.19	odd	2		1075.4.a.b.1.3		6
28.27	even	2		2107.4.a.c.1.4		6
172.171	even	2		1849.4.a.c.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.a.b.1.4	✓	6	4.3	odd	2
387.4.a.h.1.3		6	12.11	even	2
688.4.a.i.1.1		6	1.1	even	1	trivial
1075.4.a.b.1.3		6	20.19	odd	2
1849.4.a.c.1.3		6	172.171	even	2
2107.4.a.c.1.4		6	28.27	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:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	688.a (trivial)

\(f(q)\)	\(=\)	\(q-9.49653 q^{3} +2.98245 q^{5} +26.0720 q^{7} +63.1842 q^{9} +O(q^{10})\)	\(q-9.49653 q^{3} +2.98245 q^{5} +26.0720 q^{7} +63.1842 q^{9} +36.8506 q^{11} +89.5430 q^{13} -28.3230 q^{15} -28.8042 q^{17} +58.8677 q^{19} -247.594 q^{21} -2.63139 q^{23} -116.105 q^{25} -343.624 q^{27} +173.812 q^{29} -57.9476 q^{31} -349.953 q^{33} +77.7586 q^{35} +52.0754 q^{37} -850.348 q^{39} +142.704 q^{41} +43.0000 q^{43} +188.444 q^{45} +106.853 q^{47} +336.750 q^{49} +273.540 q^{51} +244.652 q^{53} +109.905 q^{55} -559.039 q^{57} +127.799 q^{59} -443.613 q^{61} +1647.34 q^{63} +267.058 q^{65} +117.896 q^{67} +24.9891 q^{69} -816.799 q^{71} -620.953 q^{73} +1102.59 q^{75} +960.770 q^{77} -377.771 q^{79} +1557.27 q^{81} +1453.23 q^{83} -85.9072 q^{85} -1650.62 q^{87} +627.993 q^{89} +2334.57 q^{91} +550.302 q^{93} +175.570 q^{95} -817.163 q^{97} +2328.38 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 7 q^{3} + 43 q^{5} - 8 q^{7} + 81 q^{9}+O(q^{10}) \) 6 * q - 7 * q^3 + 43 * q^5 - 8 * q^7 + 81 * q^9	\( 6 q - 7 q^{3} + 43 q^{5} - 8 q^{7} + 81 q^{9} + 28 q^{11} + 56 q^{13} + 124 q^{15} + 19 q^{17} + 75 q^{19} - 18 q^{21} - 131 q^{23} + 105 q^{25} - 238 q^{27} + 515 q^{29} - 237 q^{31} + 540 q^{33} - 198 q^{35} + 269 q^{37} - 290 q^{39} + 471 q^{41} + 258 q^{43} + 334 q^{45} - 415 q^{47} + 350 q^{49} + 1241 q^{51} + 450 q^{53} + 1732 q^{55} - 1000 q^{57} - 356 q^{59} - 1328 q^{61} + 2290 q^{63} - 62 q^{65} + 632 q^{67} - 1130 q^{69} + 144 q^{71} + 864 q^{73} + 2494 q^{75} + 2660 q^{77} + 1613 q^{79} - 102 q^{81} + 682 q^{83} + 84 q^{85} - 449 q^{87} + 3378 q^{89} + 3900 q^{91} + 1879 q^{93} + 79 q^{95} - 55 q^{97} + 1612 q^{99}+O(q^{100}) \) 6 * q - 7 * q^3 + 43 * q^5 - 8 * q^7 + 81 * q^9 + 28 * q^11 + 56 * q^13 + 124 * q^15 + 19 * q^17 + 75 * q^19 - 18 * q^21 - 131 * q^23 + 105 * q^25 - 238 * q^27 + 515 * q^29 - 237 * q^31 + 540 * q^33 - 198 * q^35 + 269 * q^37 - 290 * q^39 + 471 * q^41 + 258 * q^43 + 334 * q^45 - 415 * q^47 + 350 * q^49 + 1241 * q^51 + 450 * q^53 + 1732 * q^55 - 1000 * q^57 - 356 * q^59 - 1328 * q^61 + 2290 * q^63 - 62 * q^65 + 632 * q^67 - 1130 * q^69 + 144 * q^71 + 864 * q^73 + 2494 * q^75 + 2660 * q^77 + 1613 * q^79 - 102 * q^81 + 682 * q^83 + 84 * q^85 - 449 * q^87 + 3378 * q^89 + 3900 * q^91 + 1879 * q^93 + 79 * q^95 - 55 * q^97 + 1612 * q^99

Introduction

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