LMFDB - Embedded newform 688.4.a.i.1.2

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			1.2
Root			$4.15653$ 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)

$f(q)$	$=$	$q-7.20925 q^{3} +1.36370 q^{5} -13.0131 q^{7} +24.9733 q^{9} +O(q^{10})$	$q-7.20925 q^{3} +1.36370 q^{5} -13.0131 q^{7} +24.9733 q^{9} -64.7677 q^{11} -19.2944 q^{13} -9.83123 q^{15} -54.1213 q^{17} +69.0659 q^{19} +93.8149 q^{21} -29.6031 q^{23} -123.140 q^{25} +14.6112 q^{27} +13.1279 q^{29} -185.439 q^{31} +466.927 q^{33} -17.7460 q^{35} -369.949 q^{37} +139.098 q^{39} -294.860 q^{41} +43.0000 q^{43} +34.0560 q^{45} -367.319 q^{47} -173.659 q^{49} +390.174 q^{51} +708.046 q^{53} -88.3236 q^{55} -497.914 q^{57} -116.159 q^{59} +218.910 q^{61} -324.980 q^{63} -26.3116 q^{65} +133.114 q^{67} +213.416 q^{69} +926.738 q^{71} +455.867 q^{73} +887.749 q^{75} +842.831 q^{77} +620.178 q^{79} -779.614 q^{81} +1317.85 q^{83} -73.8050 q^{85} -94.6419 q^{87} +509.295 q^{89} +251.080 q^{91} +1336.88 q^{93} +94.1850 q^{95} +965.870 q^{97} -1617.46 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−7.20925	−1.38742	−0.693710	−	0.720254i	$-0.744025\pi$
$3$	−7.20925	−1.38742	−0.693710	+	0.720254i	$0.744025\pi$
$4$	0	0
$5$	1.36370	0.121973	0.0609864	−	0.998139i	$-0.480575\pi$
$5$	1.36370	0.121973	0.0609864	+	0.998139i	$0.480575\pi$
$6$	0	0
$7$	−13.0131	−0.702643	−0.351321	−	0.936255i	$-0.614268\pi$
$7$	−13.0131	−0.702643	−0.351321	+	0.936255i	$0.614268\pi$
$8$	0	0
$9$	24.9733	0.924936
$10$	0	0
$11$	−64.7677	−1.77529	−0.887646	−	0.460527i	$-0.847661\pi$
$11$	−64.7677	−1.77529	−0.887646	+	0.460527i	$0.847661\pi$
$12$	0	0
$13$	−19.2944	−0.411638	−0.205819	−	0.978590i	$-0.565986\pi$
$13$	−19.2944	−0.411638	−0.205819	+	0.978590i	$0.565986\pi$
$14$	0	0
$15$	−9.83123	−0.169227
$16$	0	0
$17$	−54.1213	−0.772138	−0.386069	−	0.922470i	$-0.626167\pi$
$17$	−54.1213	−0.772138	−0.386069	+	0.922470i	$0.626167\pi$
$18$	0	0
$19$	69.0659	0.833938	0.416969	−	0.908921i	$-0.363092\pi$
$19$	69.0659	0.833938	0.416969	+	0.908921i	$0.363092\pi$
$20$	0	0
$21$	93.8149	0.974861
$22$	0	0
$23$	−29.6031	−0.268377	−0.134189	−	0.990956i	$-0.542843\pi$
$23$	−29.6031	−0.268377	−0.134189	+	0.990956i	$0.542843\pi$
$24$	0	0
$25$	−123.140	−0.985123
$26$	0	0
$27$	14.6112	0.104145
$28$	0	0
$29$	13.1279	0.0840614	0.0420307	−	0.999116i	$-0.486617\pi$
$29$	13.1279	0.0840614	0.0420307	+	0.999116i	$0.486617\pi$
$30$	0	0
$31$	−185.439	−1.07438	−0.537191	−	0.843461i	$-0.680514\pi$
$31$	−185.439	−1.07438	−0.537191	+	0.843461i	$0.680514\pi$
$32$	0	0
$33$	466.927	2.46308
$34$	0	0
$35$	−17.7460	−0.0857032
$36$	0	0
$37$	−369.949	−1.64376	−0.821881	−	0.569659i	$-0.807075\pi$
$37$	−369.949	−1.64376	−0.821881	+	0.569659i	$0.807075\pi$
$38$	0	0
$39$	139.098	0.571115
$40$	0	0
$41$	−294.860	−1.12315	−0.561577	−	0.827424i	$-0.689805\pi$
$41$	−294.860	−1.12315	−0.561577	+	0.827424i	$0.689805\pi$
$42$	0	0
$43$	43.0000	0.152499
$43$	43.0000	0.152499
$44$	0	0
$45$	34.0560	0.112817
$46$	0	0
$47$	−367.319	−1.13998	−0.569989	−	0.821652i	$-0.693053\pi$
$47$	−367.319	−1.13998	−0.569989	+	0.821652i	$0.693053\pi$
$48$	0	0
$49$	−173.659	−0.506293
$50$	0	0
$51$	390.174	1.07128
$52$	0	0
$53$	708.046	1.83505	0.917524	−	0.397680i	$-0.130185\pi$
$53$	708.046	1.83505	0.917524	+	0.397680i	$0.130185\pi$
$54$	0	0
$55$	−88.3236	−0.216537
$56$	0	0
$57$	−497.914	−1.15702
$58$	0	0
$59$	−116.159	−0.256316	−0.128158	−	0.991754i	$-0.540906\pi$
$59$	−116.159	−0.256316	−0.128158	+	0.991754i	$0.540906\pi$
$60$	0	0
$61$	218.910	0.459484	0.229742	−	0.973252i	$-0.426212\pi$
$61$	218.910	0.459484	0.229742	+	0.973252i	$0.426212\pi$
$62$	0	0
$63$	−324.980	−0.649899
$64$	0	0
$65$	−26.3116	−0.0502086
$66$	0	0
$67$	133.114	0.242723	0.121361	−	0.992608i	$-0.461274\pi$
$67$	133.114	0.242723	0.121361	+	0.992608i	$0.461274\pi$
$68$	0	0
$69$	213.416	0.372352
$70$	0	0
$71$	926.738	1.54906	0.774532	−	0.632535i	$-0.217985\pi$
$71$	926.738	1.54906	0.774532	+	0.632535i	$0.217985\pi$
$72$	0	0
$73$	455.867	0.730893	0.365446	−	0.930832i	$-0.380916\pi$
$73$	455.867	0.730893	0.365446	+	0.930832i	$0.380916\pi$
$74$	0	0
$75$	887.749	1.36678
$76$	0	0
$77$	842.831	1.24740
$78$	0	0
$79$	620.178	0.883234	0.441617	−	0.897204i	$-0.354405\pi$
$79$	620.178	0.883234	0.441617	+	0.897204i	$0.354405\pi$
$80$	0	0
$81$	−779.614	−1.06943
$82$	0	0
$83$	1317.85	1.74281	0.871405	−	0.490564i	$-0.163209\pi$
$83$	1317.85	1.74281	0.871405	+	0.490564i	$0.163209\pi$
$84$	0	0
$85$	−73.8050	−0.0941797
$86$	0	0
$87$	−94.6419	−0.116629
$88$	0	0
$89$	509.295	0.606575	0.303287	−	0.952899i	$-0.401916\pi$
$89$	509.295	0.606575	0.303287	+	0.952899i	$0.401916\pi$
$90$	0	0
$91$	251.080	0.289234
$92$	0	0
$93$	1336.88	1.49062
$94$	0	0
$95$	94.1850	0.101718
$96$	0	0
$97$	965.870	1.01102	0.505511	−	0.862820i	$-0.331304\pi$
$97$	965.870	1.01102	0.505511	+	0.862820i	$0.331304\pi$
$98$	0	0
$99$	−1617.46	−1.64203
$100$	0	0
$101$	−1501.80	−1.47955	−0.739776	−	0.672853i	$-0.765068\pi$
$101$	−1501.80	−1.47955	−0.739776	+	0.672853i	$0.765068\pi$
$102$	0	0
$103$	−1312.60	−1.25567	−0.627837	−	0.778345i	$-0.716060\pi$
$103$	−1312.60	−1.25567	−0.627837	+	0.778345i	$0.716060\pi$
$104$	0	0
$105$	127.935	0.118906
$106$	0	0
$107$	611.656	0.552627	0.276313	−	0.961068i	$-0.410887\pi$
$107$	611.656	0.552627	0.276313	+	0.961068i	$0.410887\pi$
$108$	0	0
$109$	−946.664	−0.831871	−0.415935	−	0.909394i	$-0.636546\pi$
$109$	−946.664	−0.831871	−0.415935	+	0.909394i	$0.636546\pi$
$110$	0	0
$111$	2667.05	2.28059
$112$	0	0
$113$	2109.50	1.75615	0.878076	−	0.478521i	$-0.158827\pi$
$113$	2109.50	1.75615	0.878076	+	0.478521i	$0.158827\pi$
$114$	0	0
$115$	−40.3696	−0.0327347
$116$	0	0
$117$	−481.843	−0.380739
$118$	0	0
$119$	704.287	0.542537
$120$	0	0
$121$	2863.86	2.15166
$122$	0	0
$123$	2125.72	1.55829
$124$	0	0
$125$	−338.388	−0.242131
$126$	0	0
$127$	−788.583	−0.550988	−0.275494	−	0.961303i	$-0.588841\pi$
$127$	−788.583	−0.550988	−0.275494	+	0.961303i	$0.588841\pi$
$128$	0	0
$129$	−309.998	−0.211580
$130$	0	0
$131$	−1873.57	−1.24957	−0.624787	−	0.780795i	$-0.714814\pi$
$131$	−1873.57	−1.24957	−0.624787	+	0.780795i	$0.714814\pi$
$132$	0	0
$133$	−898.764	−0.585960
$134$	0	0
$135$	19.9252	0.0127029
$136$	0	0
$137$	−1860.27	−1.16010	−0.580050	−	0.814581i	$-0.696967\pi$
$137$	−1860.27	−1.16010	−0.580050	+	0.814581i	$0.696967\pi$
$138$	0	0
$139$	2822.52	1.72232	0.861161	−	0.508333i	$-0.169738\pi$
$139$	2822.52	1.72232	0.861161	+	0.508333i	$0.169738\pi$
$140$	0	0
$141$	2648.09	1.58163
$142$	0	0
$143$	1249.65	0.730777
$144$	0	0
$145$	17.9024	0.0102532
$146$	0	0
$147$	1251.95	0.702442
$148$	0	0
$149$	1615.30	0.888126	0.444063	−	0.895996i	$-0.353537\pi$
$149$	1615.30	0.888126	0.444063	+	0.895996i	$0.353537\pi$
$150$	0	0
$151$	−1399.96	−0.754483	−0.377241	−	0.926115i	$-0.623127\pi$
$151$	−1399.96	−0.754483	−0.377241	+	0.926115i	$0.623127\pi$
$152$	0	0
$153$	−1351.59	−0.714178
$154$	0	0
$155$	−252.882	−0.131045
$156$	0	0
$157$	2933.25	1.49108	0.745539	−	0.666462i	$-0.232192\pi$
$157$	2933.25	1.49108	0.745539	+	0.666462i	$0.232192\pi$
$158$	0	0
$159$	−5104.48	−2.54598
$160$	0	0
$161$	385.229	0.188573
$162$	0	0
$163$	166.212	0.0798695	0.0399347	−	0.999202i	$-0.487285\pi$
$163$	166.212	0.0798695	0.0399347	+	0.999202i	$0.487285\pi$
$164$	0	0
$165$	636.746	0.300428
$166$	0	0
$167$	−2143.42	−0.993191	−0.496595	−	0.867982i	$-0.665417\pi$
$167$	−2143.42	−0.993191	−0.496595	+	0.867982i	$0.665417\pi$
$168$	0	0
$169$	−1824.73	−0.830554
$170$	0	0
$171$	1724.80	0.771339
$172$	0	0
$173$	−119.018	−0.0523050	−0.0261525	−	0.999658i	$-0.508326\pi$
$173$	−119.018	−0.0523050	−0.0261525	+	0.999658i	$0.508326\pi$
$174$	0	0
$175$	1602.44	0.692189
$176$	0	0
$177$	837.420	0.355618
$178$	0	0
$179$	−2572.05	−1.07399	−0.536995	−	0.843585i	$-0.680441\pi$
$179$	−2572.05	−1.07399	−0.536995	+	0.843585i	$0.680441\pi$
$180$	0	0
$181$	1429.30	0.586956	0.293478	−	0.955966i	$-0.405187\pi$
$181$	1429.30	0.586956	0.293478	+	0.955966i	$0.405187\pi$
$182$	0	0
$183$	−1578.18	−0.637498
$184$	0	0
$185$	−504.498	−0.200494
$186$	0	0
$187$	3505.31	1.37077
$188$	0	0
$189$	−190.137	−0.0731769
$190$	0	0
$191$	−411.149	−0.155758	−0.0778788	−	0.996963i	$-0.524815\pi$
$191$	−411.149	−0.155758	−0.0778788	+	0.996963i	$0.524815\pi$
$192$	0	0
$193$	−3108.57	−1.15938	−0.579689	−	0.814838i	$-0.696826\pi$
$193$	−3108.57	−1.15938	−0.579689	+	0.814838i	$0.696826\pi$
$194$	0	0
$195$	189.687	0.0696604
$196$	0	0
$197$	1960.75	0.709125	0.354563	−	0.935032i	$-0.384630\pi$
$197$	1960.75	0.709125	0.354563	+	0.935032i	$0.384630\pi$
$198$	0	0
$199$	−456.187	−0.162504	−0.0812519	−	0.996694i	$-0.525892\pi$
$199$	−456.187	−0.162504	−0.0812519	+	0.996694i	$0.525892\pi$
$200$	0	0
$201$	−959.650	−0.336759
$202$	0	0
$203$	−170.834	−0.0590651
$204$	0	0
$205$	−402.099	−0.136994
$206$	0	0
$207$	−739.286	−0.248232
$208$	0	0
$209$	−4473.24	−1.48048
$210$	0	0
$211$	1742.50	0.568524	0.284262	−	0.958747i	$-0.408251\pi$
$211$	1742.50	0.568524	0.284262	+	0.958747i	$0.408251\pi$
$212$	0	0
$213$	−6681.08	−2.14920
$214$	0	0
$215$	58.6390	0.0186007
$216$	0	0
$217$	2413.14	0.754906
$218$	0	0
$219$	−3286.46	−1.01406
$220$	0	0
$221$	1044.24	0.317841
$222$	0	0
$223$	−3149.26	−0.945697	−0.472848	−	0.881144i	$-0.656774\pi$
$223$	−3149.26	−0.945697	−0.472848	+	0.881144i	$0.656774\pi$
$224$	0	0
$225$	−3075.22	−0.911175
$226$	0	0
$227$	−2108.03	−0.616365	−0.308182	−	0.951327i	$-0.599721\pi$
$227$	−2108.03	−0.616365	−0.308182	+	0.951327i	$0.599721\pi$
$228$	0	0
$229$	1529.66	0.441409	0.220704	−	0.975341i	$-0.429164\pi$
$229$	1529.66	0.441409	0.220704	+	0.975341i	$0.429164\pi$
$230$	0	0
$231$	−6076.18	−1.73066
$232$	0	0
$233$	2999.86	0.843466	0.421733	−	0.906720i	$-0.361422\pi$
$233$	2999.86	0.843466	0.421733	+	0.906720i	$0.361422\pi$
$234$	0	0
$235$	−500.912	−0.139046
$236$	0	0
$237$	−4471.02	−1.22542
$238$	0	0
$239$	−3421.34	−0.925976	−0.462988	−	0.886364i	$-0.653223\pi$
$239$	−3421.34	−0.925976	−0.462988	+	0.886364i	$0.653223\pi$
$240$	0	0
$241$	−4496.86	−1.20194	−0.600971	−	0.799271i	$-0.705219\pi$
$241$	−4496.86	−1.20194	−0.600971	+	0.799271i	$0.705219\pi$
$242$	0	0
$243$	5225.93	1.37960
$244$	0	0
$245$	−236.818	−0.0617540
$246$	0	0
$247$	−1332.58	−0.343280
$248$	0	0
$249$	−9500.74	−2.41801
$250$	0	0
$251$	−1478.05	−0.371688	−0.185844	−	0.982579i	$-0.559502\pi$
$251$	−1478.05	−0.371688	−0.185844	+	0.982579i	$0.559502\pi$
$252$	0	0
$253$	1917.33	0.476448
$254$	0	0
$255$	532.079	0.130667
$256$	0	0
$257$	677.467	0.164433	0.0822164	−	0.996614i	$-0.473800\pi$
$257$	677.467	0.164433	0.0822164	+	0.996614i	$0.473800\pi$
$258$	0	0
$259$	4814.19	1.15498
$260$	0	0
$261$	327.845	0.0777514
$262$	0	0
$263$	7753.56	1.81789	0.908944	−	0.416917i	$-0.136890\pi$
$263$	7753.56	1.81789	0.908944	+	0.416917i	$0.136890\pi$
$264$	0	0
$265$	965.560	0.223826
$266$	0	0
$267$	−3671.63	−0.841574
$268$	0	0
$269$	−300.965	−0.0682161	−0.0341081	−	0.999418i	$-0.510859\pi$
$269$	−300.965	−0.0682161	−0.0341081	+	0.999418i	$0.510859\pi$
$270$	0	0
$271$	2176.33	0.487833	0.243917	−	0.969796i	$-0.421568\pi$
$271$	2176.33	0.487833	0.243917	+	0.969796i	$0.421568\pi$
$272$	0	0
$273$	−1810.10	−0.401290
$274$	0	0
$275$	7975.52	1.74888
$276$	0	0
$277$	2947.64	0.639373	0.319687	−	0.947523i	$-0.396422\pi$
$277$	2947.64	0.639373	0.319687	+	0.947523i	$0.396422\pi$
$278$	0	0
$279$	−4631.02	−0.993734
$280$	0	0
$281$	459.209	0.0974879	0.0487440	−	0.998811i	$-0.484478\pi$
$281$	459.209	0.0974879	0.0487440	+	0.998811i	$0.484478\pi$
$282$	0	0
$283$	−6190.86	−1.30038	−0.650192	−	0.759770i	$-0.725312\pi$
$283$	−6190.86	−1.30038	−0.650192	+	0.759770i	$0.725312\pi$
$284$	0	0
$285$	−679.003	−0.141125
$286$	0	0
$287$	3837.05	0.789176
$288$	0	0
$289$	−1983.89	−0.403803
$290$	0	0
$291$	−6963.20	−1.40271
$292$	0	0
$293$	5693.95	1.13530	0.567652	−	0.823269i	$-0.307852\pi$
$293$	5693.95	1.13530	0.567652	+	0.823269i	$0.307852\pi$
$294$	0	0
$295$	−158.406	−0.0312635
$296$	0	0
$297$	−946.333	−0.184888
$298$	0	0
$299$	571.173	0.110474
$300$	0	0
$301$	−559.564	−0.107152
$302$	0	0
$303$	10826.9	2.05276
$304$	0	0
$305$	298.527	0.0560446
$306$	0	0
$307$	5108.18	0.949639	0.474819	−	0.880083i	$-0.342513\pi$
$307$	5108.18	0.949639	0.474819	+	0.880083i	$0.342513\pi$
$308$	0	0
$309$	9462.87	1.74215
$310$	0	0
$311$	1048.11	0.191102	0.0955511	−	0.995425i	$-0.469539\pi$
$311$	1048.11	0.191102	0.0955511	+	0.995425i	$0.469539\pi$
$312$	0	0
$313$	4279.92	0.772893	0.386446	−	0.922312i	$-0.373702\pi$
$313$	4279.92	0.772893	0.386446	+	0.922312i	$0.373702\pi$
$314$	0	0
$315$	−443.175	−0.0792700
$316$	0	0
$317$	5929.95	1.05066	0.525330	−	0.850899i	$-0.323942\pi$
$317$	5929.95	1.05066	0.525330	+	0.850899i	$0.323942\pi$
$318$	0	0
$319$	−850.261	−0.149234
$320$	0	0
$321$	−4409.58	−0.766725
$322$	0	0
$323$	−3737.94	−0.643915
$324$	0	0
$325$	2375.91	0.405514
$326$	0	0
$327$	6824.73	1.15415
$328$	0	0
$329$	4779.97	0.800997
$330$	0	0
$331$	−509.174	−0.0845521	−0.0422761	−	0.999106i	$-0.513461\pi$
$331$	−509.174	−0.0845521	−0.0422761	+	0.999106i	$0.513461\pi$
$332$	0	0
$333$	−9238.83	−1.52037
$334$	0	0
$335$	181.527	0.0296056
$336$	0	0
$337$	−10856.3	−1.75484	−0.877419	−	0.479724i	$-0.840737\pi$
$337$	−10856.3	−1.75484	−0.877419	+	0.479724i	$0.840737\pi$
$338$	0	0
$339$	−15207.9	−2.43652
$340$	0	0
$341$	12010.5	1.90734
$342$	0	0
$343$	6723.34	1.05839
$344$	0	0
$345$	291.035	0.0454168
$346$	0	0
$347$	−8973.74	−1.38829	−0.694143	−	0.719837i	$-0.744217\pi$
$347$	−8973.74	−1.38829	−0.694143	+	0.719837i	$0.744217\pi$
$348$	0	0
$349$	−5.70408	−0.000874877 0	−0.000437439	−	1.00000i	$-0.500139\pi$
$349$	−5.70408	−0.000874877 0	−0.000437439		1.00000i	$0.500139\pi$
$350$	0	0
$351$	−281.913	−0.0428702
$352$	0	0
$353$	−1504.40	−0.226830	−0.113415	−	0.993548i	$-0.536179\pi$
$353$	−1504.40	−0.226830	−0.113415	+	0.993548i	$0.536179\pi$
$354$	0	0
$355$	1263.79	0.188944
$356$	0	0
$357$	−5077.38	−0.752727
$358$	0	0
$359$	1296.86	0.190657	0.0953285	−	0.995446i	$-0.469610\pi$
$359$	1296.86	0.190657	0.0953285	+	0.995446i	$0.469610\pi$
$360$	0	0
$361$	−2088.90	−0.304548
$362$	0	0
$363$	−20646.3	−2.98526
$364$	0	0
$365$	621.664	0.0891490
$366$	0	0
$367$	−12908.3	−1.83599	−0.917995	−	0.396592i	$-0.870193\pi$
$367$	−12908.3	−1.83599	−0.917995	+	0.396592i	$0.870193\pi$
$368$	0	0
$369$	−7363.61	−1.03885
$370$	0	0
$371$	−9213.89	−1.28938
$372$	0	0
$373$	−1371.48	−0.190382	−0.0951908	−	0.995459i	$-0.530346\pi$
$373$	−1371.48	−0.190382	−0.0951908	+	0.995459i	$0.530346\pi$
$374$	0	0
$375$	2439.52	0.335937
$376$	0	0
$377$	−253.293	−0.0346029
$378$	0	0
$379$	458.768	0.0621776	0.0310888	−	0.999517i	$-0.490103\pi$
$379$	458.768	0.0621776	0.0310888	+	0.999517i	$0.490103\pi$
$380$	0	0
$381$	5685.09	0.764452
$382$	0	0
$383$	−9513.45	−1.26923	−0.634614	−	0.772829i	$-0.718841\pi$
$383$	−9513.45	−1.26923	−0.634614	+	0.772829i	$0.718841\pi$
$384$	0	0
$385$	1149.37	0.152148
$386$	0	0
$387$	1073.85	0.141051
$388$	0	0
$389$	5643.25	0.735538	0.367769	−	0.929917i	$-0.380122\pi$
$389$	5643.25	0.735538	0.367769	+	0.929917i	$0.380122\pi$
$390$	0	0
$391$	1602.16	0.207224
$392$	0	0
$393$	13507.0	1.73369
$394$	0	0
$395$	845.735	0.107730
$396$	0	0
$397$	−13816.5	−1.74668	−0.873340	−	0.487110i	$-0.838051\pi$
$397$	−13816.5	−1.74668	−0.873340	+	0.487110i	$0.838051\pi$
$398$	0	0
$399$	6479.41	0.812973
$400$	0	0
$401$	3544.45	0.441400	0.220700	−	0.975342i	$-0.429166\pi$
$401$	3544.45	0.441400	0.220700	+	0.975342i	$0.429166\pi$
$402$	0	0
$403$	3577.93	0.442256
$404$	0	0
$405$	−1063.16	−0.130441
$406$	0	0
$407$	23960.7	2.91816
$408$	0	0
$409$	−2736.88	−0.330880	−0.165440	−	0.986220i	$-0.552904\pi$
$409$	−2736.88	−0.330880	−0.165440	+	0.986220i	$0.552904\pi$
$410$	0	0
$411$	13411.2	1.60955
$412$	0	0
$413$	1511.59	0.180098
$414$	0	0
$415$	1797.15	0.212575
$416$	0	0
$417$	−20348.2	−2.38958
$418$	0	0
$419$	−9280.65	−1.08208	−0.541038	−	0.840998i	$-0.681968\pi$
$419$	−9280.65	−1.08208	−0.541038	+	0.840998i	$0.681968\pi$
$420$	0	0
$421$	−7308.00	−0.846010	−0.423005	−	0.906127i	$-0.639025\pi$
$421$	−7308.00	−0.846010	−0.423005	+	0.906127i	$0.639025\pi$
$422$	0	0
$423$	−9173.16	−1.05441
$424$	0	0
$425$	6664.51	0.760650
$426$	0	0
$427$	−2848.70	−0.322853
$428$	0	0
$429$	−9009.05	−1.01390
$430$	0	0
$431$	953.510	0.106564	0.0532818	−	0.998580i	$-0.483032\pi$
$431$	953.510	0.106564	0.0532818	+	0.998580i	$0.483032\pi$
$432$	0	0
$433$	3447.00	0.382569	0.191284	−	0.981535i	$-0.438735\pi$
$433$	3447.00	0.382569	0.191284	+	0.981535i	$0.438735\pi$
$434$	0	0
$435$	−129.063	−0.0142255
$436$	0	0
$437$	−2044.57	−0.223810
$438$	0	0
$439$	9245.65	1.00517	0.502586	−	0.864527i	$-0.332382\pi$
$439$	9245.65	1.00517	0.502586	+	0.864527i	$0.332382\pi$
$440$	0	0
$441$	−4336.82	−0.468289
$442$	0	0
$443$	3885.76	0.416745	0.208372	−	0.978050i	$-0.433183\pi$
$443$	3885.76	0.416745	0.208372	+	0.978050i	$0.433183\pi$
$444$	0	0
$445$	694.524	0.0739856
$446$	0	0
$447$	−11645.1	−1.23220
$448$	0	0
$449$	−14413.1	−1.51491	−0.757455	−	0.652887i	$-0.773557\pi$
$449$	−14413.1	−1.51491	−0.757455	+	0.652887i	$0.773557\pi$
$450$	0	0
$451$	19097.4	1.99393
$452$	0	0
$453$	10092.6	1.04679
$454$	0	0
$455$	342.397	0.0352787
$456$	0	0
$457$	−1765.21	−0.180685	−0.0903424	−	0.995911i	$-0.528796\pi$
$457$	−1765.21	−0.180685	−0.0903424	+	0.995911i	$0.528796\pi$
$458$	0	0
$459$	−790.776	−0.0804145
$460$	0	0
$461$	7626.64	0.770517	0.385258	−	0.922809i	$-0.374112\pi$
$461$	7626.64	0.770517	0.385258	+	0.922809i	$0.374112\pi$
$462$	0	0
$463$	5954.18	0.597655	0.298827	−	0.954307i	$-0.403405\pi$
$463$	5954.18	0.597655	0.298827	+	0.954307i	$0.403405\pi$
$464$	0	0
$465$	1823.09	0.181815
$466$	0	0
$467$	−153.388	−0.0151991	−0.00759954	−	0.999971i	$-0.502419\pi$
$467$	−153.388	−0.0151991	−0.00759954	+	0.999971i	$0.502419\pi$
$468$	0	0
$469$	−1732.22	−0.170547
$470$	0	0
$471$	−21146.6	−2.06875
$472$	0	0
$473$	−2785.01	−0.270729
$474$	0	0
$475$	−8504.80	−0.821531
$476$	0	0
$477$	17682.2	1.69730
$478$	0	0
$479$	11334.3	1.08117	0.540584	−	0.841290i	$-0.318203\pi$
$479$	11334.3	1.08117	0.540584	+	0.841290i	$0.318203\pi$
$480$	0	0
$481$	7137.92	0.676635
$482$	0	0
$483$	−2777.21	−0.261630
$484$	0	0
$485$	1317.15	0.123317
$486$	0	0
$487$	2845.89	0.264804	0.132402	−	0.991196i	$-0.457731\pi$
$487$	2845.89	0.264804	0.132402	+	0.991196i	$0.457731\pi$
$488$	0	0
$489$	−1198.26	−0.110813
$490$	0	0
$491$	8345.94	0.767102	0.383551	−	0.923520i	$-0.374701\pi$
$491$	8345.94	0.767102	0.383551	+	0.923520i	$0.374701\pi$
$492$	0	0
$493$	−710.496	−0.0649070
$494$	0	0
$495$	−2205.73	−0.200283
$496$	0	0
$497$	−12059.8	−1.08844
$498$	0	0
$499$	−17591.1	−1.57813	−0.789064	−	0.614311i	$-0.789434\pi$
$499$	−17591.1	−1.57813	−0.789064	+	0.614311i	$0.789434\pi$
$500$	0	0
$501$	15452.5	1.37797
$502$	0	0
$503$	7975.83	0.707008	0.353504	−	0.935433i	$-0.384990\pi$
$503$	7975.83	0.707008	0.353504	+	0.935433i	$0.384990\pi$
$504$	0	0
$505$	−2048.00	−0.180465
$506$	0	0
$507$	13154.9	1.15233
$508$	0	0
$509$	−9016.94	−0.785204	−0.392602	−	0.919708i	$-0.628425\pi$
$509$	−9016.94	−0.785204	−0.392602	+	0.919708i	$0.628425\pi$
$510$	0	0
$511$	−5932.25	−0.513556
$512$	0	0
$513$	1009.14	0.0868507
$514$	0	0
$515$	−1789.99	−0.153158
$516$	0	0
$517$	23790.4	2.02379
$518$	0	0
$519$	858.030	0.0725691
$520$	0	0
$521$	−3086.13	−0.259512	−0.129756	−	0.991546i	$-0.541419\pi$
$521$	−3086.13	−0.259512	−0.129756	+	0.991546i	$0.541419\pi$
$522$	0	0
$523$	−8338.32	−0.697149	−0.348575	−	0.937281i	$-0.613334\pi$
$523$	−8338.32	−0.697149	−0.348575	+	0.937281i	$0.613334\pi$
$524$	0	0
$525$	−11552.4	−0.960358
$526$	0	0
$527$	10036.2	0.829570
$528$	0	0
$529$	−11290.7	−0.927974
$530$	0	0
$531$	−2900.87	−0.237076
$532$	0	0
$533$	5689.13	0.462333
$534$	0	0
$535$	834.114	0.0674054
$536$	0	0
$537$	18542.6	1.49008
$538$	0	0
$539$	11247.5	0.898818
$540$	0	0
$541$	−14924.6	−1.18606	−0.593029	−	0.805181i	$-0.702068\pi$
$541$	−14924.6	−1.18606	−0.593029	+	0.805181i	$0.702068\pi$
$542$	0	0
$543$	−10304.2	−0.814355
$544$	0	0
$545$	−1290.96	−0.101466
$546$	0	0
$547$	−10658.3	−0.833120	−0.416560	−	0.909108i	$-0.636764\pi$
$547$	−10658.3	−0.833120	−0.416560	+	0.909108i	$0.636764\pi$
$548$	0	0
$549$	5466.90	0.424994
$550$	0	0
$551$	906.687	0.0701020
$552$	0	0
$553$	−8070.45	−0.620598
$554$	0	0
$555$	3637.05	0.278170
$556$	0	0
$557$	4303.28	0.327354	0.163677	−	0.986514i	$-0.447665\pi$
$557$	4303.28	0.327354	0.163677	+	0.986514i	$0.447665\pi$
$558$	0	0
$559$	−829.657	−0.0627742
$560$	0	0
$561$	−25270.7	−1.90183
$562$	0	0
$563$	13280.8	0.994173	0.497086	−	0.867701i	$-0.334403\pi$
$563$	13280.8	0.994173	0.497086	+	0.867701i	$0.334403\pi$
$564$	0	0
$565$	2876.72	0.214203
$566$	0	0
$567$	10145.2	0.751427
$568$	0	0
$569$	11641.6	0.857720	0.428860	−	0.903371i	$-0.358915\pi$
$569$	11641.6	0.857720	0.428860	+	0.903371i	$0.358915\pi$
$570$	0	0
$571$	−21799.0	−1.59765	−0.798825	−	0.601563i	$-0.794545\pi$
$571$	−21799.0	−1.59765	−0.798825	+	0.601563i	$0.794545\pi$
$572$	0	0
$573$	2964.07	0.216101
$574$	0	0
$575$	3645.34	0.264384
$576$	0	0
$577$	6854.44	0.494548	0.247274	−	0.968946i	$-0.420465\pi$
$577$	6854.44	0.494548	0.247274	+	0.968946i	$0.420465\pi$
$578$	0	0
$579$	22410.5	1.60855
$580$	0	0
$581$	−17149.4	−1.22457
$582$	0	0
$583$	−45858.5	−3.25775
$584$	0	0
$585$	−657.088	−0.0464397
$586$	0	0
$587$	16741.7	1.17718	0.588589	−	0.808433i	$-0.299684\pi$
$587$	16741.7	1.17718	0.588589	+	0.808433i	$0.299684\pi$
$588$	0	0
$589$	−12807.5	−0.895967
$590$	0	0
$591$	−14135.5	−0.983855
$592$	0	0
$593$	21885.0	1.51553	0.757764	−	0.652529i	$-0.226292\pi$
$593$	21885.0	1.51553	0.757764	+	0.652529i	$0.226292\pi$
$594$	0	0
$595$	960.434	0.0661747
$596$	0	0
$597$	3288.77	0.225461
$598$	0	0
$599$	14440.9	0.985039	0.492519	−	0.870301i	$-0.336076\pi$
$599$	14440.9	0.985039	0.492519	+	0.870301i	$0.336076\pi$
$600$	0	0
$601$	2570.21	0.174444	0.0872221	−	0.996189i	$-0.472201\pi$
$601$	2570.21	0.174444	0.0872221	+	0.996189i	$0.472201\pi$
$602$	0	0
$603$	3324.28	0.224503
$604$	0	0
$605$	3905.44	0.262444
$606$	0	0
$607$	11683.5	0.781249	0.390624	−	0.920550i	$-0.372259\pi$
$607$	11683.5	0.781249	0.390624	+	0.920550i	$0.372259\pi$
$608$	0	0
$609$	1231.59	0.0819482
$610$	0	0
$611$	7087.18	0.469258
$612$	0	0
$613$	12739.0	0.839356	0.419678	−	0.907673i	$-0.362143\pi$
$613$	12739.0	0.839356	0.419678	+	0.907673i	$0.362143\pi$
$614$	0	0
$615$	2898.83	0.190069
$616$	0	0
$617$	4380.70	0.285835	0.142918	−	0.989735i	$-0.454352\pi$
$617$	4380.70	0.285835	0.142918	+	0.989735i	$0.454352\pi$
$618$	0	0
$619$	5157.68	0.334902	0.167451	−	0.985880i	$-0.446446\pi$
$619$	5157.68	0.334902	0.167451	+	0.985880i	$0.446446\pi$
$620$	0	0
$621$	−432.536	−0.0279502
$622$	0	0
$623$	−6627.52	−0.426205
$624$	0	0
$625$	14931.1	0.955589
$626$	0	0
$627$	32248.7	2.05405
$628$	0	0
$629$	20022.1	1.26921
$630$	0	0
$631$	27679.9	1.74631	0.873154	−	0.487444i	$-0.162071\pi$
$631$	27679.9	1.74631	0.873154	+	0.487444i	$0.162071\pi$
$632$	0	0
$633$	−12562.1	−0.788783
$634$	0	0
$635$	−1075.39	−0.0672055
$636$	0	0
$637$	3350.63	0.208409
$638$	0	0
$639$	23143.7	1.43278
$640$	0	0
$641$	−16453.5	−1.01384	−0.506921	−	0.861992i	$-0.669217\pi$
$641$	−16453.5	−1.01384	−0.506921	+	0.861992i	$0.669217\pi$
$642$	0	0
$643$	29203.6	1.79110	0.895551	−	0.444958i	$-0.146781\pi$
$643$	29203.6	1.79110	0.895551	+	0.444958i	$0.146781\pi$
$644$	0	0
$645$	−422.743	−0.0258070
$646$	0	0
$647$	16835.6	1.02299	0.511495	−	0.859286i	$-0.329092\pi$
$647$	16835.6	1.02299	0.511495	+	0.859286i	$0.329092\pi$
$648$	0	0
$649$	7523.36	0.455035
$650$	0	0
$651$	−17396.9	−1.04737
$652$	0	0
$653$	−13735.1	−0.823117	−0.411558	−	0.911383i	$-0.635015\pi$
$653$	−13735.1	−0.823117	−0.411558	+	0.911383i	$0.635015\pi$
$654$	0	0
$655$	−2554.98	−0.152414
$656$	0	0
$657$	11384.5	0.676029
$658$	0	0
$659$	5708.76	0.337453	0.168727	−	0.985663i	$-0.446034\pi$
$659$	5708.76	0.337453	0.168727	+	0.985663i	$0.446034\pi$
$660$	0	0
$661$	18339.5	1.07916	0.539578	−	0.841936i	$-0.318584\pi$
$661$	18339.5	1.07916	0.539578	+	0.841936i	$0.318584\pi$
$662$	0	0
$663$	−7528.15	−0.440979
$664$	0	0
$665$	−1225.64	−0.0714712
$666$	0	0
$667$	−388.625	−0.0225602
$668$	0	0
$669$	22703.8	1.31208
$670$	0	0
$671$	−14178.3	−0.815719
$672$	0	0
$673$	−482.234	−0.0276207	−0.0138104	−	0.999905i	$-0.504396\pi$
$673$	−482.234	−0.0276207	−0.0138104	+	0.999905i	$0.504396\pi$
$674$	0	0
$675$	−1799.23	−0.102596
$676$	0	0
$677$	31086.0	1.76474	0.882372	−	0.470553i	$-0.155945\pi$
$677$	31086.0	1.76474	0.882372	+	0.470553i	$0.155945\pi$
$678$	0	0
$679$	−12569.0	−0.710388
$680$	0	0
$681$	15197.3	0.855157
$682$	0	0
$683$	−31381.0	−1.75807	−0.879034	−	0.476758i	$-0.841812\pi$
$683$	−31381.0	−1.75807	−0.879034	+	0.476758i	$0.841812\pi$
$684$	0	0
$685$	−2536.85	−0.141501
$686$	0	0
$687$	−11027.7	−0.612420
$688$	0	0
$689$	−13661.3	−0.755375
$690$	0	0
$691$	−19856.9	−1.09319	−0.546593	−	0.837398i	$-0.684076\pi$
$691$	−19856.9	−1.09319	−0.546593	+	0.837398i	$0.684076\pi$
$692$	0	0
$693$	21048.2	1.15376
$694$	0	0
$695$	3849.06	0.210076
$696$	0	0
$697$	15958.2	0.867230
$698$	0	0
$699$	−21626.8	−1.17024
$700$	0	0
$701$	2722.80	0.146703	0.0733514	−	0.997306i	$-0.476631\pi$
$701$	2722.80	0.146703	0.0733514	+	0.997306i	$0.476631\pi$
$702$	0	0
$703$	−25550.9	−1.37079
$704$	0	0
$705$	3611.20	0.192916
$706$	0	0
$707$	19543.1	1.03960
$708$	0	0
$709$	−1225.30	−0.0649043	−0.0324521	−	0.999473i	$-0.510332\pi$
$709$	−1225.30	−0.0649043	−0.0324521	+	0.999473i	$0.510332\pi$
$710$	0	0
$711$	15487.9	0.816935
$712$	0	0
$713$	5489.57	0.288339
$714$	0	0
$715$	1704.15	0.0891349
$716$	0	0
$717$	24665.3	1.28472
$718$	0	0
$719$	28875.1	1.49772	0.748858	−	0.662730i	$-0.230602\pi$
$719$	28875.1	1.49772	0.748858	+	0.662730i	$0.230602\pi$
$720$	0	0
$721$	17081.0	0.882290
$722$	0	0
$723$	32419.0	1.66760
$724$	0	0
$725$	−1616.57	−0.0828108
$726$	0	0
$727$	25072.0	1.27905	0.639525	−	0.768770i	$-0.279131\pi$
$727$	25072.0	1.27905	0.639525	+	0.768770i	$0.279131\pi$
$728$	0	0
$729$	−16625.5	−0.844660
$730$	0	0
$731$	−2327.22	−0.117750
$732$	0	0
$733$	1017.07	0.0512503	0.0256251	−	0.999672i	$-0.491842\pi$
$733$	1017.07	0.0512503	0.0256251	+	0.999672i	$0.491842\pi$
$734$	0	0
$735$	1707.28	0.0856787
$736$	0	0
$737$	−8621.47	−0.430904
$738$	0	0
$739$	1001.77	0.0498658	0.0249329	−	0.999689i	$-0.492063\pi$
$739$	1001.77	0.0498658	0.0249329	+	0.999689i	$0.492063\pi$
$740$	0	0
$741$	9606.92	0.476274
$742$	0	0
$743$	2115.35	0.104448	0.0522238	−	0.998635i	$-0.483369\pi$
$743$	2115.35	0.104448	0.0522238	+	0.998635i	$0.483369\pi$
$744$	0	0
$745$	2202.78	0.108327
$746$	0	0
$747$	32911.1	1.61199
$748$	0	0
$749$	−7959.56	−0.388299
$750$	0	0
$751$	−39172.1	−1.90334	−0.951671	−	0.307119i	$-0.900635\pi$
$751$	−39172.1	−1.90334	−0.951671	+	0.307119i	$0.900635\pi$
$752$	0	0
$753$	10655.6	0.515688
$754$	0	0
$755$	−1909.12	−0.0920263
$756$	0	0
$757$	5944.10	0.285392	0.142696	−	0.989767i	$-0.454423\pi$
$757$	5944.10	0.285392	0.142696	+	0.989767i	$0.454423\pi$
$758$	0	0
$759$	−13822.5	−0.661033
$760$	0	0
$761$	−1954.89	−0.0931205	−0.0465603	−	0.998915i	$-0.514826\pi$
$761$	−1954.89	−0.0931205	−0.0465603	+	0.998915i	$0.514826\pi$
$762$	0	0
$763$	12319.1	0.584508
$764$	0	0
$765$	−1843.15	−0.0871102
$766$	0	0
$767$	2241.21	0.105509
$768$	0	0
$769$	−39312.9	−1.84351	−0.921756	−	0.387770i	$-0.873245\pi$
$769$	−39312.9	−1.84351	−0.921756	+	0.387770i	$0.873245\pi$
$770$	0	0
$771$	−4884.03	−0.228138
$772$	0	0
$773$	−5102.38	−0.237413	−0.118706	−	0.992929i	$-0.537875\pi$
$773$	−5102.38	−0.237413	−0.118706	+	0.992929i	$0.537875\pi$
$774$	0	0
$775$	22835.0	1.05840
$776$	0	0
$777$	−34706.7	−1.60244
$778$	0	0
$779$	−20364.8	−0.936641
$780$	0	0
$781$	−60022.7	−2.75004
$782$	0	0
$783$	191.813	0.00875460
$784$	0	0
$785$	4000.07	0.181871
$786$	0	0
$787$	−21898.3	−0.991855	−0.495928	−	0.868364i	$-0.665172\pi$
$787$	−21898.3	−0.991855	−0.495928	+	0.868364i	$0.665172\pi$
$788$	0	0
$789$	−55897.3	−2.52218
$790$	0	0
$791$	−27451.2	−1.23395
$792$	0	0
$793$	−4223.73	−0.189141
$794$	0	0
$795$	−6960.96	−0.310541
$796$	0	0
$797$	18262.4	0.811652	0.405826	−	0.913950i	$-0.366984\pi$
$797$	18262.4	0.811652	0.405826	+	0.913950i	$0.366984\pi$
$798$	0	0
$799$	19879.8	0.880220
$800$	0	0
$801$	12718.8	0.561043
$802$	0	0
$803$	−29525.5	−1.29755
$804$	0	0
$805$	525.335	0.0230008
$806$	0	0
$807$	2169.73	0.0946444
$808$	0	0
$809$	26097.8	1.13418	0.567089	−	0.823656i	$-0.308069\pi$
$809$	26097.8	1.13418	0.567089	+	0.823656i	$0.308069\pi$
$810$	0	0
$811$	−1067.58	−0.0462243	−0.0231122	−	0.999733i	$-0.507357\pi$
$811$	−1067.58	−0.0462243	−0.0231122	+	0.999733i	$0.507357\pi$
$812$	0	0
$813$	−15689.7	−0.676830
$814$	0	0
$815$	226.663	0.00974190
$816$	0	0
$817$	2969.84	0.127174
$818$	0	0
$819$	6270.29	0.267523
$820$	0	0
$821$	−20127.0	−0.855585	−0.427793	−	0.903877i	$-0.640709\pi$
$821$	−20127.0	−0.855585	−0.427793	+	0.903877i	$0.640709\pi$
$822$	0	0
$823$	3011.70	0.127559	0.0637797	−	0.997964i	$-0.479685\pi$
$823$	3011.70	0.127559	0.0637797	+	0.997964i	$0.479685\pi$
$824$	0	0
$825$	−57497.5	−2.42643
$826$	0	0
$827$	13138.4	0.552437	0.276219	−	0.961095i	$-0.410919\pi$
$827$	13138.4	0.552437	0.276219	+	0.961095i	$0.410919\pi$
$828$	0	0
$829$	19550.4	0.819075	0.409538	−	0.912293i	$-0.365690\pi$
$829$	19550.4	0.819075	0.409538	+	0.912293i	$0.365690\pi$
$830$	0	0
$831$	−21250.2	−0.887079
$832$	0	0
$833$	9398.63	0.390928
$834$	0	0
$835$	−2922.98	−0.121142
$836$	0	0
$837$	−2709.48	−0.111892
$838$	0	0
$839$	−3227.69	−0.132816	−0.0664078	−	0.997793i	$-0.521154\pi$
$839$	−3227.69	−0.132816	−0.0664078	+	0.997793i	$0.521154\pi$
$840$	0	0
$841$	−24216.7	−0.992934
$842$	0	0
$843$	−3310.55	−0.135257
$844$	0	0
$845$	−2488.38	−0.101305
$846$	0	0
$847$	−37267.8	−1.51185
$848$	0	0
$849$	44631.5	1.80418
$850$	0	0
$851$	10951.6	0.441148
$852$	0	0
$853$	12419.8	0.498530	0.249265	−	0.968435i	$-0.419811\pi$
$853$	12419.8	0.498530	0.249265	+	0.968435i	$0.419811\pi$
$854$	0	0
$855$	2352.11	0.0940823
$856$	0	0
$857$	18592.3	0.741075	0.370537	−	0.928818i	$-0.379174\pi$
$857$	18592.3	0.741075	0.370537	+	0.928818i	$0.379174\pi$
$858$	0	0
$859$	−6443.09	−0.255920	−0.127960	−	0.991779i	$-0.540843\pi$
$859$	−6443.09	−0.255920	−0.127960	+	0.991779i	$0.540843\pi$
$860$	0	0
$861$	−27662.2	−1.09492
$862$	0	0
$863$	22085.1	0.871132	0.435566	−	0.900157i	$-0.356548\pi$
$863$	22085.1	0.871132	0.435566	+	0.900157i	$0.356548\pi$
$864$	0	0
$865$	−162.304	−0.00637979
$866$	0	0
$867$	14302.3	0.560245
$868$	0	0
$869$	−40167.5	−1.56800
$870$	0	0
$871$	−2568.34	−0.0999139
$872$	0	0
$873$	24120.9	0.935131
$874$	0	0
$875$	4403.49	0.170131
$876$	0	0
$877$	39786.3	1.53191	0.765956	−	0.642893i	$-0.222266\pi$
$877$	39786.3	1.53191	0.765956	+	0.642893i	$0.222266\pi$
$878$	0	0
$879$	−41049.1	−1.57514
$880$	0	0
$881$	38834.0	1.48508	0.742538	−	0.669804i	$-0.233622\pi$
$881$	38834.0	1.48508	0.742538	+	0.669804i	$0.233622\pi$
$882$	0	0
$883$	5483.65	0.208992	0.104496	−	0.994525i	$-0.466677\pi$
$883$	5483.65	0.208992	0.104496	+	0.994525i	$0.466677\pi$
$884$	0	0
$885$	1141.99	0.0433756
$886$	0	0
$887$	−21289.7	−0.805905	−0.402953	−	0.915221i	$-0.632016\pi$
$887$	−21289.7	−0.805905	−0.402953	+	0.915221i	$0.632016\pi$
$888$	0	0
$889$	10261.9	0.387148
$890$	0	0
$891$	50493.8	1.89855
$892$	0	0
$893$	−25369.2	−0.950671
$894$	0	0
$895$	−3507.50	−0.130997
$896$	0	0
$897$	−4117.73	−0.153274
$898$	0	0
$899$	−2434.41	−0.0903140
$900$	0	0
$901$	−38320.4	−1.41691
$902$	0	0
$903$	4034.04	0.148665
$904$	0	0
$905$	1949.13	0.0715927
$906$	0	0
$907$	−16875.4	−0.617794	−0.308897	−	0.951095i	$-0.599960\pi$
$907$	−16875.4	−0.617794	−0.308897	+	0.951095i	$0.599960\pi$
$908$	0	0
$909$	−37504.9	−1.36849
$910$	0	0
$911$	−17586.7	−0.639597	−0.319799	−	0.947486i	$-0.603615\pi$
$911$	−17586.7	−0.639597	−0.319799	+	0.947486i	$0.603615\pi$
$912$	0	0
$913$	−85354.4	−3.09400
$914$	0	0
$915$	−2152.15	−0.0777574
$916$	0	0
$917$	24381.0	0.878005
$918$	0	0
$919$	43891.7	1.57547	0.787733	−	0.616017i	$-0.211255\pi$
$919$	43891.7	1.57547	0.787733	+	0.616017i	$0.211255\pi$
$920$	0	0
$921$	−36826.1	−1.31755
$922$	0	0
$923$	−17880.8	−0.637653
$924$	0	0
$925$	45555.6	1.61931
$926$	0	0
$927$	−32779.9	−1.16142
$928$	0	0
$929$	−41161.8	−1.45369	−0.726844	−	0.686803i	$-0.759013\pi$
$929$	−41161.8	−1.45369	−0.726844	+	0.686803i	$0.759013\pi$
$930$	0	0
$931$	−11993.9	−0.422217
$932$	0	0
$933$	−7556.08	−0.265139
$934$	0	0
$935$	4780.18	0.167197
$936$	0	0
$937$	41754.1	1.45576	0.727880	−	0.685705i	$-0.240506\pi$
$937$	41754.1	1.45576	0.727880	+	0.685705i	$0.240506\pi$
$938$	0	0
$939$	−30855.0	−1.07233
$940$	0	0
$941$	−37587.5	−1.30214	−0.651071	−	0.759016i	$-0.725680\pi$
$941$	−37587.5	−1.30214	−0.651071	+	0.759016i	$0.725680\pi$
$942$	0	0
$943$	8728.76	0.301429
$944$	0	0
$945$	−259.289	−0.00892559
$946$	0	0
$947$	25689.0	0.881500	0.440750	−	0.897630i	$-0.354713\pi$
$947$	25689.0	0.881500	0.440750	+	0.897630i	$0.354713\pi$
$948$	0	0
$949$	−8795.66	−0.300863
$950$	0	0
$951$	−42750.5	−1.45771
$952$	0	0
$953$	−10527.3	−0.357831	−0.178916	−	0.983864i	$-0.557259\pi$
$953$	−10527.3	−0.357831	−0.178916	+	0.983864i	$0.557259\pi$
$954$	0	0
$955$	−560.682	−0.0189982
$956$	0	0
$957$	6129.75	0.207050
$958$	0	0
$959$	24207.9	0.815136
$960$	0	0
$961$	4596.60	0.154295
$962$	0	0
$963$	15275.1	0.511144
$964$	0	0
$965$	−4239.15	−0.141413
$966$	0	0
$967$	−4292.71	−0.142755	−0.0713776	−	0.997449i	$-0.522740\pi$
$967$	−4292.71	−0.142755	−0.0713776	+	0.997449i	$0.522740\pi$
$968$	0	0
$969$	26947.7	0.893380
$970$	0	0
$971$	41459.8	1.37025	0.685123	−	0.728427i	$-0.259748\pi$
$971$	41459.8	1.37025	0.685123	+	0.728427i	$0.259748\pi$
$972$	0	0
$973$	−36729.8	−1.21018
$974$	0	0
$975$	−17128.6	−0.562618
$976$	0	0
$977$	3577.72	0.117156	0.0585779	−	0.998283i	$-0.481343\pi$
$977$	3577.72	0.117156	0.0585779	+	0.998283i	$0.481343\pi$
$978$	0	0
$979$	−32985.9	−1.07685
$980$	0	0
$981$	−23641.3	−0.769427
$982$	0	0
$983$	−50532.3	−1.63960	−0.819802	−	0.572647i	$-0.805917\pi$
$983$	−50532.3	−1.63960	−0.819802	+	0.572647i	$0.805917\pi$
$984$	0	0
$985$	2673.87	0.0864939
$986$	0	0
$987$	−34460.0	−1.11132
$988$	0	0
$989$	−1272.93	−0.0409271
$990$	0	0
$991$	−46280.6	−1.48350	−0.741752	−	0.670674i	$-0.766005\pi$
$991$	−46280.6	−1.48350	−0.741752	+	0.670674i	$0.766005\pi$
$992$	0	0
$993$	3670.76	0.117309
$994$	0	0
$995$	−622.101	−0.0198210
$996$	0	0
$997$	53334.8	1.69421	0.847107	−	0.531423i	$-0.178342\pi$
$997$	53334.8	1.69421	0.847107	+	0.531423i	$0.178342\pi$
$998$	0	0
$999$	−5405.39	−0.171190

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.4.a.i.1.2		6
4.3	odd	2		43.4.a.b.1.2	✓	6
12.11	even	2		387.4.a.h.1.5		6
20.19	odd	2		1075.4.a.b.1.5		6
28.27	even	2		2107.4.a.c.1.2		6
172.171	even	2		1849.4.a.c.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.a.b.1.2	✓	6	4.3	odd	2
387.4.a.h.1.5		6	12.11	even	2
688.4.a.i.1.2		6	1.1	even	1	trivial
1075.4.a.b.1.5		6	20.19	odd	2
1849.4.a.c.1.5		6	172.171	even	2
2107.4.a.c.1.2		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)

\(f(q)\)	\(=\)	\(q-7.20925 q^{3} +1.36370 q^{5} -13.0131 q^{7} +24.9733 q^{9} +O(q^{10})\)	\(q-7.20925 q^{3} +1.36370 q^{5} -13.0131 q^{7} +24.9733 q^{9} -64.7677 q^{11} -19.2944 q^{13} -9.83123 q^{15} -54.1213 q^{17} +69.0659 q^{19} +93.8149 q^{21} -29.6031 q^{23} -123.140 q^{25} +14.6112 q^{27} +13.1279 q^{29} -185.439 q^{31} +466.927 q^{33} -17.7460 q^{35} -369.949 q^{37} +139.098 q^{39} -294.860 q^{41} +43.0000 q^{43} +34.0560 q^{45} -367.319 q^{47} -173.659 q^{49} +390.174 q^{51} +708.046 q^{53} -88.3236 q^{55} -497.914 q^{57} -116.159 q^{59} +218.910 q^{61} -324.980 q^{63} -26.3116 q^{65} +133.114 q^{67} +213.416 q^{69} +926.738 q^{71} +455.867 q^{73} +887.749 q^{75} +842.831 q^{77} +620.178 q^{79} -779.614 q^{81} +1317.85 q^{83} -73.8050 q^{85} -94.6419 q^{87} +509.295 q^{89} +251.080 q^{91} +1336.88 q^{93} +94.1850 q^{95} +965.870 q^{97} -1617.46 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 7 q^{3} + 43 q^{5} - 8 q^{7} + 81 q^{9}+O(q^{10}) \) 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

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