LMFDB - Embedded newform 1088.4.a.bg.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1088,4,Mod(1,1088)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1088, 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("1088.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1088 = 2^{6} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1088.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:	$64.1940780862$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 67x^{5} - 35x^{4} + 893x^{3} + 595x^{2} - 3064x - 2804 $ x^7 - x^6 - 67x^5 - 35x^4 + 893x^3 + 595x^2 - 3064*x - 2804
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 544)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.84285$ of defining polynomial
Character	$\chi$	$=$	1088.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.23405 q^{3} -14.8262 q^{5} +20.4601 q^{7} -16.5409 q^{9} +O(q^{10})$	$q-3.23405 q^{3} -14.8262 q^{5} +20.4601 q^{7} -16.5409 q^{9} +30.9879 q^{11} -45.3781 q^{13} +47.9486 q^{15} -17.0000 q^{17} -128.769 q^{19} -66.1691 q^{21} +134.304 q^{23} +94.8158 q^{25} +140.814 q^{27} -214.969 q^{29} -82.5014 q^{31} -100.216 q^{33} -303.346 q^{35} +350.245 q^{37} +146.755 q^{39} -226.817 q^{41} -126.462 q^{43} +245.239 q^{45} -172.240 q^{47} +75.6165 q^{49} +54.9789 q^{51} -619.590 q^{53} -459.432 q^{55} +416.444 q^{57} -282.706 q^{59} +730.472 q^{61} -338.429 q^{63} +672.784 q^{65} -876.169 q^{67} -434.346 q^{69} +467.038 q^{71} -474.843 q^{73} -306.639 q^{75} +634.016 q^{77} +1090.67 q^{79} -8.79336 q^{81} -79.7726 q^{83} +252.045 q^{85} +695.221 q^{87} +903.357 q^{89} -928.441 q^{91} +266.814 q^{93} +1909.15 q^{95} -289.745 q^{97} -512.568 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{7} + 67 q^{9}+O(q^{10}) $ 7 * q - 2 * q^7 + 67 * q^9	$ 7 q - 2 q^{7} + 67 q^{9} - 108 q^{11} + 34 q^{13} + 128 q^{15} - 119 q^{17} - 124 q^{19} + 296 q^{21} - 6 q^{23} + 197 q^{25} - 248 q^{29} + 50 q^{31} + 512 q^{33} - 640 q^{35} + 484 q^{37} + 1504 q^{39} - 366 q^{41} - 1412 q^{43} + 80 q^{45} + 1012 q^{47} + 1115 q^{49} + 146 q^{53} + 1024 q^{55} + 48 q^{57} - 2332 q^{59} + 548 q^{61} + 2838 q^{63} - 208 q^{65} - 924 q^{67} + 1672 q^{69} + 1286 q^{71} + 870 q^{73} - 3136 q^{75} + 1344 q^{77} + 1818 q^{79} + 3039 q^{81} - 1772 q^{83} + 384 q^{87} + 1706 q^{89} - 588 q^{91} + 5576 q^{93} + 2048 q^{95} + 1802 q^{97} - 5148 q^{99}+O(q^{100}) $ 7 * q - 2 * q^7 + 67 * q^9 - 108 * q^11 + 34 * q^13 + 128 * q^15 - 119 * q^17 - 124 * q^19 + 296 * q^21 - 6 * q^23 + 197 * q^25 - 248 * q^29 + 50 * q^31 + 512 * q^33 - 640 * q^35 + 484 * q^37 + 1504 * q^39 - 366 * q^41 - 1412 * q^43 + 80 * q^45 + 1012 * q^47 + 1115 * q^49 + 146 * q^53 + 1024 * q^55 + 48 * q^57 - 2332 * q^59 + 548 * q^61 + 2838 * q^63 - 208 * q^65 - 924 * q^67 + 1672 * q^69 + 1286 * q^71 + 870 * q^73 - 3136 * q^75 + 1344 * q^77 + 1818 * q^79 + 3039 * q^81 - 1772 * q^83 + 384 * q^87 + 1706 * q^89 - 588 * q^91 + 5576 * q^93 + 2048 * q^95 + 1802 * q^97 - 5148 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−3.23405	−0.622393	−0.311197	−	0.950346i	$-0.600730\pi$
$3$	−3.23405	−0.622393	−0.311197	+	0.950346i	$0.600730\pi$
$4$	0	0
$5$	−14.8262	−1.32609	−0.663047	−	0.748578i	$-0.730737\pi$
$5$	−14.8262	−1.32609	−0.663047	+	0.748578i	$0.730737\pi$
$6$	0	0
$7$	20.4601	1.10474	0.552371	−	0.833598i	$-0.313723\pi$
$7$	20.4601	1.10474	0.552371	+	0.833598i	$0.313723\pi$
$8$	0	0
$9$	−16.5409	−0.612627
$10$	0	0
$11$	30.9879	0.849381	0.424691	−	0.905339i	$-0.360383\pi$
$11$	30.9879	0.849381	0.424691	+	0.905339i	$0.360383\pi$
$12$	0	0
$13$	−45.3781	−0.968124	−0.484062	−	0.875034i	$-0.660839\pi$
$13$	−45.3781	−0.968124	−0.484062	+	0.875034i	$0.660839\pi$
$14$	0	0
$15$	47.9486	0.825352
$16$	0	0
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	−128.769	−1.55482	−0.777409	−	0.628996i	$-0.783466\pi$
$19$	−128.769	−1.55482	−0.777409	+	0.628996i	$0.783466\pi$
$20$	0	0
$21$	−66.1691	−0.687584
$22$	0	0
$23$	134.304	1.21758	0.608790	−	0.793331i	$-0.291655\pi$
$23$	134.304	1.21758	0.608790	+	0.793331i	$0.291655\pi$
$24$	0	0
$25$	94.8158	0.758526
$26$	0	0
$27$	140.814	1.00369
$28$	0	0
$29$	−214.969	−1.37651	−0.688254	−	0.725469i	$-0.741623\pi$
$29$	−214.969	−1.37651	−0.688254	+	0.725469i	$0.741623\pi$
$30$	0	0
$31$	−82.5014	−0.477990	−0.238995	−	0.971021i	$-0.576818\pi$
$31$	−82.5014	−0.477990	−0.238995	+	0.971021i	$0.576818\pi$
$32$	0	0
$33$	−100.216	−0.528649
$34$	0	0
$35$	−303.346	−1.46499
$36$	0	0
$37$	350.245	1.55621	0.778107	−	0.628132i	$-0.216180\pi$
$37$	350.245	1.55621	0.778107	+	0.628132i	$0.216180\pi$
$38$	0	0
$39$	146.755	0.602554
$40$	0	0
$41$	−226.817	−0.863973	−0.431986	−	0.901880i	$-0.642187\pi$
$41$	−226.817	−0.863973	−0.431986	+	0.901880i	$0.642187\pi$
$42$	0	0
$43$	−126.462	−0.448496	−0.224248	−	0.974532i	$-0.571993\pi$
$43$	−126.462	−0.448496	−0.224248	+	0.974532i	$0.571993\pi$
$44$	0	0
$45$	245.239	0.812401
$46$	0	0
$47$	−172.240	−0.534550	−0.267275	−	0.963620i	$-0.586123\pi$
$47$	−172.240	−0.534550	−0.267275	+	0.963620i	$0.586123\pi$
$48$	0	0
$49$	75.6165	0.220456
$50$	0	0
$51$	54.9789	0.150953
$52$	0	0
$53$	−619.590	−1.60580	−0.802899	−	0.596116i	$-0.796710\pi$
$53$	−619.590	−1.60580	−0.802899	+	0.596116i	$0.796710\pi$
$54$	0	0
$55$	−459.432	−1.12636
$56$	0	0
$57$	416.444	0.967708
$58$	0	0
$59$	−282.706	−0.623816	−0.311908	−	0.950112i	$-0.600968\pi$
$59$	−282.706	−0.623816	−0.311908	+	0.950112i	$0.600968\pi$
$60$	0	0
$61$	730.472	1.53324	0.766618	−	0.642104i	$-0.221938\pi$
$61$	730.472	1.53324	0.766618	+	0.642104i	$0.221938\pi$
$62$	0	0
$63$	−338.429	−0.676795
$64$	0	0
$65$	672.784	1.28382
$66$	0	0
$67$	−876.169	−1.59763	−0.798814	−	0.601578i	$-0.794539\pi$
$67$	−876.169	−1.59763	−0.798814	+	0.601578i	$0.794539\pi$
$68$	0	0
$69$	−434.346	−0.757814
$70$	0	0
$71$	467.038	0.780664	0.390332	−	0.920674i	$-0.372360\pi$
$71$	467.038	0.780664	0.390332	+	0.920674i	$0.372360\pi$
$72$	0	0
$73$	−474.843	−0.761317	−0.380659	−	0.924716i	$-0.624303\pi$
$73$	−474.843	−0.761317	−0.380659	+	0.924716i	$0.624303\pi$
$74$	0	0
$75$	−306.639	−0.472102
$76$	0	0
$77$	634.016	0.938348
$78$	0	0
$79$	1090.67	1.55329	0.776645	−	0.629938i	$-0.216920\pi$
$79$	1090.67	1.55329	0.776645	+	0.629938i	$0.216920\pi$
$80$	0	0
$81$	−8.79336	−0.0120622
$82$	0	0
$83$	−79.7726	−0.105496	−0.0527481	−	0.998608i	$-0.516798\pi$
$83$	−79.7726	−0.105496	−0.0527481	+	0.998608i	$0.516798\pi$
$84$	0	0
$85$	252.045	0.321625
$86$	0	0
$87$	695.221	0.856730
$88$	0	0
$89$	903.357	1.07591	0.537953	−	0.842975i	$-0.319198\pi$
$89$	903.357	1.07591	0.537953	+	0.842975i	$0.319198\pi$
$90$	0	0
$91$	−928.441	−1.06953
$92$	0	0
$93$	266.814	0.297498
$94$	0	0
$95$	1909.15	2.06183
$96$	0	0
$97$	−289.745	−0.303290	−0.151645	−	0.988435i	$-0.548457\pi$
$97$	−289.745	−0.303290	−0.151645	+	0.988435i	$0.548457\pi$
$98$	0	0
$99$	−512.568	−0.520354
$100$	0	0
$101$	1812.07	1.78522	0.892610	−	0.450829i	$-0.148872\pi$
$101$	1812.07	1.78522	0.892610	+	0.450829i	$0.148872\pi$
$102$	0	0
$103$	226.785	0.216949	0.108475	−	0.994099i	$-0.465403\pi$
$103$	226.785	0.216949	0.108475	+	0.994099i	$0.465403\pi$
$104$	0	0
$105$	981.035	0.911802
$106$	0	0
$107$	468.440	0.423232	0.211616	−	0.977353i	$-0.432127\pi$
$107$	468.440	0.423232	0.211616	+	0.977353i	$0.432127\pi$
$108$	0	0
$109$	1908.40	1.67699	0.838494	−	0.544910i	$-0.183436\pi$
$109$	1908.40	1.67699	0.838494	+	0.544910i	$0.183436\pi$
$110$	0	0
$111$	−1132.71	−0.968577
$112$	0	0
$113$	487.589	0.405916	0.202958	−	0.979187i	$-0.434944\pi$
$113$	487.589	0.405916	0.202958	+	0.979187i	$0.434944\pi$
$114$	0	0
$115$	−1991.22	−1.61463
$116$	0	0
$117$	750.595	0.593098
$118$	0	0
$119$	−347.822	−0.267939
$120$	0	0
$121$	−370.752	−0.278551
$122$	0	0
$123$	733.538	0.537731
$124$	0	0
$125$	447.517	0.320217
$126$	0	0
$127$	1123.73	0.785154	0.392577	−	0.919719i	$-0.371584\pi$
$127$	1123.73	0.785154	0.392577	+	0.919719i	$0.371584\pi$
$128$	0	0
$129$	408.986	0.279141
$130$	0	0
$131$	518.217	0.345624	0.172812	−	0.984955i	$-0.444715\pi$
$131$	518.217	0.345624	0.172812	+	0.984955i	$0.444715\pi$
$132$	0	0
$133$	−2634.62	−1.71767
$134$	0	0
$135$	−2087.73	−1.33099
$136$	0	0
$137$	372.303	0.232175	0.116087	−	0.993239i	$-0.462965\pi$
$137$	372.303	0.232175	0.116087	+	0.993239i	$0.462965\pi$
$138$	0	0
$139$	2403.34	1.46654	0.733269	−	0.679938i	$-0.237993\pi$
$139$	2403.34	1.46654	0.733269	+	0.679938i	$0.237993\pi$
$140$	0	0
$141$	557.034	0.332700
$142$	0	0
$143$	−1406.17	−0.822306
$144$	0	0
$145$	3187.17	1.82538
$146$	0	0
$147$	−244.548	−0.137211
$148$	0	0
$149$	−1027.59	−0.564987	−0.282494	−	0.959269i	$-0.591162\pi$
$149$	−1027.59	−0.564987	−0.282494	+	0.959269i	$0.591162\pi$
$150$	0	0
$151$	3152.28	1.69887	0.849433	−	0.527696i	$-0.176944\pi$
$151$	3152.28	1.69887	0.849433	+	0.527696i	$0.176944\pi$
$152$	0	0
$153$	281.196	0.148584
$154$	0	0
$155$	1223.18	0.633860
$156$	0	0
$157$	1158.66	0.588989	0.294495	−	0.955653i	$-0.404849\pi$
$157$	1158.66	0.588989	0.294495	+	0.955653i	$0.404849\pi$
$158$	0	0
$159$	2003.79	0.999437
$160$	0	0
$161$	2747.88	1.34511
$162$	0	0
$163$	402.407	0.193368	0.0966838	−	0.995315i	$-0.469176\pi$
$163$	402.407	0.193368	0.0966838	+	0.995315i	$0.469176\pi$
$164$	0	0
$165$	1485.83	0.701039
$166$	0	0
$167$	396.122	0.183550	0.0917749	−	0.995780i	$-0.470746\pi$
$167$	396.122	0.183550	0.0917749	+	0.995780i	$0.470746\pi$
$168$	0	0
$169$	−137.831	−0.0627361
$170$	0	0
$171$	2129.95	0.952522
$172$	0	0
$173$	−1119.31	−0.491906	−0.245953	−	0.969282i	$-0.579101\pi$
$173$	−1119.31	−0.491906	−0.245953	+	0.969282i	$0.579101\pi$
$174$	0	0
$175$	1939.94	0.837976
$176$	0	0
$177$	914.284	0.388259
$178$	0	0
$179$	4245.20	1.77263	0.886316	−	0.463082i	$-0.153256\pi$
$179$	4245.20	1.77263	0.886316	+	0.463082i	$0.153256\pi$
$180$	0	0
$181$	−2349.81	−0.964974	−0.482487	−	0.875903i	$-0.660266\pi$
$181$	−2349.81	−0.964974	−0.482487	+	0.875903i	$0.660266\pi$
$182$	0	0
$183$	−2362.38	−0.954276
$184$	0	0
$185$	−5192.80	−2.06369
$186$	0	0
$187$	−526.794	−0.206005
$188$	0	0
$189$	2881.06	1.10882
$190$	0	0
$191$	−4004.90	−1.51720	−0.758598	−	0.651559i	$-0.774115\pi$
$191$	−4004.90	−1.51720	−0.758598	+	0.651559i	$0.774115\pi$
$192$	0	0
$193$	2939.05	1.09615	0.548076	−	0.836429i	$-0.315360\pi$
$193$	2939.05	1.09615	0.548076	+	0.836429i	$0.315360\pi$
$194$	0	0
$195$	−2175.82	−0.799043
$196$	0	0
$197$	−1408.64	−0.509449	−0.254724	−	0.967014i	$-0.581985\pi$
$197$	−1408.64	−0.509449	−0.254724	+	0.967014i	$0.581985\pi$
$198$	0	0
$199$	−1138.15	−0.405435	−0.202718	−	0.979237i	$-0.564977\pi$
$199$	−1138.15	−0.405435	−0.202718	+	0.979237i	$0.564977\pi$
$200$	0	0
$201$	2833.58	0.994353
$202$	0	0
$203$	−4398.29	−1.52069
$204$	0	0
$205$	3362.83	1.14571
$206$	0	0
$207$	−2221.51	−0.745922
$208$	0	0
$209$	−3990.26	−1.32063
$210$	0	0
$211$	4428.15	1.44477	0.722385	−	0.691491i	$-0.243046\pi$
$211$	4428.15	1.44477	0.722385	+	0.691491i	$0.243046\pi$
$212$	0	0
$213$	−1510.42	−0.485880
$214$	0	0
$215$	1874.96	0.594748
$216$	0	0
$217$	−1687.99	−0.528056
$218$	0	0
$219$	1535.67	0.473839
$220$	0	0
$221$	771.427	0.234805
$222$	0	0
$223$	−3443.70	−1.03411	−0.517056	−	0.855951i	$-0.672972\pi$
$223$	−3443.70	−1.03411	−0.517056	+	0.855951i	$0.672972\pi$
$224$	0	0
$225$	−1568.34	−0.464693
$226$	0	0
$227$	−5689.39	−1.66352	−0.831758	−	0.555139i	$-0.812665\pi$
$227$	−5689.39	−1.66352	−0.831758	+	0.555139i	$0.812665\pi$
$228$	0	0
$229$	3137.13	0.905272	0.452636	−	0.891695i	$-0.350484\pi$
$229$	3137.13	0.905272	0.452636	+	0.891695i	$0.350484\pi$
$230$	0	0
$231$	−2050.44	−0.584021
$232$	0	0
$233$	4024.42	1.13154	0.565769	−	0.824564i	$-0.308580\pi$
$233$	4024.42	1.13154	0.565769	+	0.824564i	$0.308580\pi$
$234$	0	0
$235$	2553.67	0.708864
$236$	0	0
$237$	−3527.28	−0.966757
$238$	0	0
$239$	−2002.37	−0.541936	−0.270968	−	0.962588i	$-0.587344\pi$
$239$	−2002.37	−0.541936	−0.270968	+	0.962588i	$0.587344\pi$
$240$	0	0
$241$	−4487.69	−1.19949	−0.599746	−	0.800190i	$-0.704732\pi$
$241$	−4487.69	−1.19949	−0.599746	+	0.800190i	$0.704732\pi$
$242$	0	0
$243$	−3773.53	−0.996181
$244$	0	0
$245$	−1121.10	−0.292346
$246$	0	0
$247$	5843.27	1.50526
$248$	0	0
$249$	257.989	0.0656601
$250$	0	0
$251$	1958.20	0.492432	0.246216	−	0.969215i	$-0.420813\pi$
$251$	1958.20	0.492432	0.246216	+	0.969215i	$0.420813\pi$
$252$	0	0
$253$	4161.80	1.03419
$254$	0	0
$255$	−815.127	−0.200177
$256$	0	0
$257$	331.844	0.0805442	0.0402721	−	0.999189i	$-0.487178\pi$
$257$	331.844	0.0805442	0.0402721	+	0.999189i	$0.487178\pi$
$258$	0	0
$259$	7166.05	1.71922
$260$	0	0
$261$	3555.79	0.843286
$262$	0	0
$263$	−2516.26	−0.589960	−0.294980	−	0.955503i	$-0.595313\pi$
$263$	−2516.26	−0.589960	−0.294980	+	0.955503i	$0.595313\pi$
$264$	0	0
$265$	9186.16	2.12944
$266$	0	0
$267$	−2921.50	−0.669637
$268$	0	0
$269$	3807.69	0.863044	0.431522	−	0.902102i	$-0.357977\pi$
$269$	3807.69	0.863044	0.431522	+	0.902102i	$0.357977\pi$
$270$	0	0
$271$	−3855.07	−0.864128	−0.432064	−	0.901843i	$-0.642215\pi$
$271$	−3855.07	−0.864128	−0.432064	+	0.901843i	$0.642215\pi$
$272$	0	0
$273$	3002.62	0.665667
$274$	0	0
$275$	2938.14	0.644278
$276$	0	0
$277$	6652.57	1.44301	0.721506	−	0.692408i	$-0.243450\pi$
$277$	6652.57	1.44301	0.721506	+	0.692408i	$0.243450\pi$
$278$	0	0
$279$	1364.65	0.292829
$280$	0	0
$281$	−2986.58	−0.634038	−0.317019	−	0.948419i	$-0.602682\pi$
$281$	−2986.58	−0.634038	−0.317019	+	0.948419i	$0.602682\pi$
$282$	0	0
$283$	2793.41	0.586752	0.293376	−	0.955997i	$-0.405221\pi$
$283$	2793.41	0.586752	0.293376	+	0.955997i	$0.405221\pi$
$284$	0	0
$285$	−6174.28	−1.28327
$286$	0	0
$287$	−4640.71	−0.954467
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	937.050	0.188766
$292$	0	0
$293$	2906.62	0.579544	0.289772	−	0.957096i	$-0.406421\pi$
$293$	2906.62	0.579544	0.289772	+	0.957096i	$0.406421\pi$
$294$	0	0
$295$	4191.45	0.827239
$296$	0	0
$297$	4363.51	0.852514
$298$	0	0
$299$	−6094.46	−1.17877
$300$	0	0
$301$	−2587.44	−0.495473
$302$	0	0
$303$	−5860.31	−1.11111
$304$	0	0
$305$	−10830.1	−2.03322
$306$	0	0
$307$	8551.47	1.58977	0.794883	−	0.606763i	$-0.207532\pi$
$307$	8551.47	1.58977	0.794883	+	0.606763i	$0.207532\pi$
$308$	0	0
$309$	−733.434	−0.135028
$310$	0	0
$311$	257.113	0.0468796	0.0234398	−	0.999725i	$-0.492538\pi$
$311$	257.113	0.0468796	0.0234398	+	0.999725i	$0.492538\pi$
$312$	0	0
$313$	2077.26	0.375123	0.187561	−	0.982253i	$-0.439942\pi$
$313$	2077.26	0.375123	0.187561	+	0.982253i	$0.439942\pi$
$314$	0	0
$315$	5017.61	0.897494
$316$	0	0
$317$	−3599.66	−0.637782	−0.318891	−	0.947791i	$-0.603310\pi$
$317$	−3599.66	−0.637782	−0.318891	+	0.947791i	$0.603310\pi$
$318$	0	0
$319$	−6661.43	−1.16918
$320$	0	0
$321$	−1514.96	−0.263417
$322$	0	0
$323$	2189.07	0.377099
$324$	0	0
$325$	−4302.56	−0.734348
$326$	0	0
$327$	−6171.87	−1.04375
$328$	0	0
$329$	−3524.06	−0.590540
$330$	0	0
$331$	−8288.14	−1.37631	−0.688153	−	0.725565i	$-0.741578\pi$
$331$	−8288.14	−1.37631	−0.688153	+	0.725565i	$0.741578\pi$
$332$	0	0
$333$	−5793.37	−0.953378
$334$	0	0
$335$	12990.2	2.11861
$336$	0	0
$337$	5660.61	0.914994	0.457497	−	0.889211i	$-0.348746\pi$
$337$	5660.61	0.914994	0.457497	+	0.889211i	$0.348746\pi$
$338$	0	0
$339$	−1576.89	−0.252639
$340$	0	0
$341$	−2556.54	−0.405996
$342$	0	0
$343$	−5470.70	−0.861195
$344$	0	0
$345$	6439.70	1.00493
$346$	0	0
$347$	10900.4	1.68635	0.843173	−	0.537642i	$-0.180685\pi$
$347$	10900.4	1.68635	0.843173	+	0.537642i	$0.180685\pi$
$348$	0	0
$349$	10560.8	1.61978	0.809892	−	0.586579i	$-0.199526\pi$
$349$	10560.8	1.61978	0.809892	+	0.586579i	$0.199526\pi$
$350$	0	0
$351$	−6389.85	−0.971694
$352$	0	0
$353$	−7233.45	−1.09064	−0.545322	−	0.838226i	$-0.683593\pi$
$353$	−7233.45	−1.09064	−0.545322	+	0.838226i	$0.683593\pi$
$354$	0	0
$355$	−6924.39	−1.03523
$356$	0	0
$357$	1124.87	0.166764
$358$	0	0
$359$	6170.33	0.907124	0.453562	−	0.891225i	$-0.350153\pi$
$359$	6170.33	0.907124	0.453562	+	0.891225i	$0.350153\pi$
$360$	0	0
$361$	9722.34	1.41746
$362$	0	0
$363$	1199.03	0.173368
$364$	0	0
$365$	7040.11	1.00958
$366$	0	0
$367$	723.148	0.102856	0.0514278	−	0.998677i	$-0.483623\pi$
$367$	723.148	0.102856	0.0514278	+	0.998677i	$0.483623\pi$
$368$	0	0
$369$	3751.76	0.529293
$370$	0	0
$371$	−12676.9	−1.77399
$372$	0	0
$373$	−1008.46	−0.139989	−0.0699946	−	0.997547i	$-0.522298\pi$
$373$	−1008.46	−0.139989	−0.0699946	+	0.997547i	$0.522298\pi$
$374$	0	0
$375$	−1447.29	−0.199301
$376$	0	0
$377$	9754.88	1.33263
$378$	0	0
$379$	3084.08	0.417991	0.208996	−	0.977917i	$-0.432981\pi$
$379$	3084.08	0.417991	0.208996	+	0.977917i	$0.432981\pi$
$380$	0	0
$381$	−3634.19	−0.488675
$382$	0	0
$383$	4733.10	0.631462	0.315731	−	0.948849i	$-0.397750\pi$
$383$	4733.10	0.631462	0.315731	+	0.948849i	$0.397750\pi$
$384$	0	0
$385$	−9400.03	−1.24434
$386$	0	0
$387$	2091.80	0.274761
$388$	0	0
$389$	−10092.8	−1.31549	−0.657747	−	0.753239i	$-0.728490\pi$
$389$	−10092.8	−1.31549	−0.657747	+	0.753239i	$0.728490\pi$
$390$	0	0
$391$	−2283.17	−0.295306
$392$	0	0
$393$	−1675.94	−0.215114
$394$	0	0
$395$	−16170.5	−2.05981
$396$	0	0
$397$	3901.46	0.493221	0.246611	−	0.969115i	$-0.420683\pi$
$397$	3901.46	0.493221	0.246611	+	0.969115i	$0.420683\pi$
$398$	0	0
$399$	8520.49	1.06907
$400$	0	0
$401$	−15577.8	−1.93994	−0.969971	−	0.243222i	$-0.921796\pi$
$401$	−15577.8	−1.93994	−0.969971	+	0.243222i	$0.921796\pi$
$402$	0	0
$403$	3743.75	0.462753
$404$	0	0
$405$	130.372	0.0159956
$406$	0	0
$407$	10853.3	1.32182
$408$	0	0
$409$	−10459.9	−1.26457	−0.632285	−	0.774736i	$-0.717883\pi$
$409$	−10459.9	−1.26457	−0.632285	+	0.774736i	$0.717883\pi$
$410$	0	0
$411$	−1204.05	−0.144504
$412$	0	0
$413$	−5784.19	−0.689156
$414$	0	0
$415$	1182.72	0.139898
$416$	0	0
$417$	−7772.53	−0.912764
$418$	0	0
$419$	−5093.23	−0.593844	−0.296922	−	0.954902i	$-0.595960\pi$
$419$	−5093.23	−0.593844	−0.296922	+	0.954902i	$0.595960\pi$
$420$	0	0
$421$	1453.23	0.168233	0.0841167	−	0.996456i	$-0.473193\pi$
$421$	1453.23	0.168233	0.0841167	+	0.996456i	$0.473193\pi$
$422$	0	0
$423$	2849.01	0.327479
$424$	0	0
$425$	−1611.87	−0.183970
$426$	0	0
$427$	14945.5	1.69383
$428$	0	0
$429$	4547.62	0.511798
$430$	0	0
$431$	17568.3	1.96342	0.981712	−	0.190371i	$-0.0609692\pi$
$431$	17568.3	1.96342	0.981712	+	0.190371i	$0.0609692\pi$
$432$	0	0
$433$	4995.28	0.554407	0.277203	−	0.960811i	$-0.410592\pi$
$433$	4995.28	0.554407	0.277203	+	0.960811i	$0.410592\pi$
$434$	0	0
$435$	−10307.5	−1.13610
$436$	0	0
$437$	−17294.1	−1.89311
$438$	0	0
$439$	−8289.59	−0.901232	−0.450616	−	0.892718i	$-0.648796\pi$
$439$	−8289.59	−0.901232	−0.450616	+	0.892718i	$0.648796\pi$
$440$	0	0
$441$	−1250.77	−0.135057
$442$	0	0
$443$	−874.185	−0.0937557	−0.0468779	−	0.998901i	$-0.514927\pi$
$443$	−874.185	−0.0937557	−0.0468779	+	0.998901i	$0.514927\pi$
$444$	0	0
$445$	−13393.3	−1.42675
$446$	0	0
$447$	3323.26	0.351644
$448$	0	0
$449$	−6982.97	−0.733957	−0.366979	−	0.930229i	$-0.619608\pi$
$449$	−6982.97	−0.733957	−0.366979	+	0.930229i	$0.619608\pi$
$450$	0	0
$451$	−7028.58	−0.733842
$452$	0	0
$453$	−10194.6	−1.05736
$454$	0	0
$455$	13765.2	1.41829
$456$	0	0
$457$	10742.1	1.09955	0.549777	−	0.835312i	$-0.314713\pi$
$457$	10742.1	1.09955	0.549777	+	0.835312i	$0.314713\pi$
$458$	0	0
$459$	−2393.83	−0.243430
$460$	0	0
$461$	−9758.03	−0.985850	−0.492925	−	0.870072i	$-0.664072\pi$
$461$	−9758.03	−0.985850	−0.492925	+	0.870072i	$0.664072\pi$
$462$	0	0
$463$	1328.79	0.133379	0.0666893	−	0.997774i	$-0.478756\pi$
$463$	1328.79	0.133379	0.0666893	+	0.997774i	$0.478756\pi$
$464$	0	0
$465$	−3955.83	−0.394510
$466$	0	0
$467$	−7604.91	−0.753562	−0.376781	−	0.926302i	$-0.622969\pi$
$467$	−7604.91	−0.753562	−0.376781	+	0.926302i	$0.622969\pi$
$468$	0	0
$469$	−17926.5	−1.76497
$470$	0	0
$471$	−3747.17	−0.366583
$472$	0	0
$473$	−3918.80	−0.380944
$474$	0	0
$475$	−12209.3	−1.17937
$476$	0	0
$477$	10248.6	0.983754
$478$	0	0
$479$	−14081.3	−1.34319	−0.671596	−	0.740917i	$-0.734391\pi$
$479$	−14081.3	−1.34319	−0.671596	+	0.740917i	$0.734391\pi$
$480$	0	0
$481$	−15893.4	−1.50661
$482$	0	0
$483$	−8886.78	−0.837189
$484$	0	0
$485$	4295.81	0.402191
$486$	0	0
$487$	1366.56	0.127155	0.0635777	−	0.997977i	$-0.479749\pi$
$487$	1366.56	0.127155	0.0635777	+	0.997977i	$0.479749\pi$
$488$	0	0
$489$	−1301.40	−0.120351
$490$	0	0
$491$	−2574.58	−0.236638	−0.118319	−	0.992976i	$-0.537750\pi$
$491$	−2574.58	−0.236638	−0.118319	+	0.992976i	$0.537750\pi$
$492$	0	0
$493$	3654.47	0.333852
$494$	0	0
$495$	7599.43	0.690038
$496$	0	0
$497$	9555.64	0.862433
$498$	0	0
$499$	11459.0	1.02801	0.514005	−	0.857787i	$-0.328161\pi$
$499$	11459.0	1.02801	0.514005	+	0.857787i	$0.328161\pi$
$500$	0	0
$501$	−1281.08	−0.114240
$502$	0	0
$503$	−13335.7	−1.18212	−0.591062	−	0.806626i	$-0.701291\pi$
$503$	−13335.7	−1.18212	−0.591062	+	0.806626i	$0.701291\pi$
$504$	0	0
$505$	−26866.0	−2.36737
$506$	0	0
$507$	445.753	0.0390465
$508$	0	0
$509$	−5458.78	−0.475356	−0.237678	−	0.971344i	$-0.576386\pi$
$509$	−5458.78	−0.475356	−0.237678	+	0.971344i	$0.576386\pi$
$510$	0	0
$511$	−9715.34	−0.841059
$512$	0	0
$513$	−18132.4	−1.56055
$514$	0	0
$515$	−3362.36	−0.287695
$516$	0	0
$517$	−5337.36	−0.454037
$518$	0	0
$519$	3619.91	0.306159
$520$	0	0
$521$	11374.8	0.956502	0.478251	−	0.878223i	$-0.341271\pi$
$521$	11374.8	0.956502	0.478251	+	0.878223i	$0.341271\pi$
$522$	0	0
$523$	−14397.3	−1.20373	−0.601865	−	0.798598i	$-0.705576\pi$
$523$	−14397.3	−1.20373	−0.601865	+	0.798598i	$0.705576\pi$
$524$	0	0
$525$	−6273.87	−0.521551
$526$	0	0
$527$	1402.52	0.115930
$528$	0	0
$529$	5870.58	0.482501
$530$	0	0
$531$	4676.21	0.382166
$532$	0	0
$533$	10292.5	0.836433
$534$	0	0
$535$	−6945.18	−0.561246
$536$	0	0
$537$	−13729.2	−1.10327
$538$	0	0
$539$	2343.20	0.187251
$540$	0	0
$541$	18827.7	1.49624	0.748122	−	0.663561i	$-0.230956\pi$
$541$	18827.7	1.49624	0.748122	+	0.663561i	$0.230956\pi$
$542$	0	0
$543$	7599.42	0.600593
$544$	0	0
$545$	−28294.3	−2.22385
$546$	0	0
$547$	14611.7	1.14214	0.571072	−	0.820900i	$-0.306528\pi$
$547$	14611.7	1.14214	0.571072	+	0.820900i	$0.306528\pi$
$548$	0	0
$549$	−12082.7	−0.939301
$550$	0	0
$551$	27681.3	2.14022
$552$	0	0
$553$	22315.2	1.71599
$554$	0	0
$555$	16793.8	1.28442
$556$	0	0
$557$	−6946.57	−0.528430	−0.264215	−	0.964464i	$-0.585113\pi$
$557$	−6946.57	−0.528430	−0.264215	+	0.964464i	$0.585113\pi$
$558$	0	0
$559$	5738.62	0.434200
$560$	0	0
$561$	1703.68	0.128216
$562$	0	0
$563$	9693.48	0.725634	0.362817	−	0.931861i	$-0.381815\pi$
$563$	9693.48	0.725634	0.362817	+	0.931861i	$0.381815\pi$
$564$	0	0
$565$	−7229.09	−0.538283
$566$	0	0
$567$	−179.913	−0.0133256
$568$	0	0
$569$	24145.1	1.77894	0.889468	−	0.456997i	$-0.151075\pi$
$569$	24145.1	1.77894	0.889468	+	0.456997i	$0.151075\pi$
$570$	0	0
$571$	13498.2	0.989283	0.494642	−	0.869097i	$-0.335299\pi$
$571$	13498.2	0.989283	0.494642	+	0.869097i	$0.335299\pi$
$572$	0	0
$573$	12952.1	0.944293
$574$	0	0
$575$	12734.1	0.923566
$576$	0	0
$577$	7282.64	0.525443	0.262721	−	0.964872i	$-0.415380\pi$
$577$	7282.64	0.525443	0.262721	+	0.964872i	$0.415380\pi$
$578$	0	0
$579$	−9505.03	−0.682238
$580$	0	0
$581$	−1632.16	−0.116546
$582$	0	0
$583$	−19199.8	−1.36393
$584$	0	0
$585$	−11128.5	−0.786504
$586$	0	0
$587$	−4429.97	−0.311489	−0.155745	−	0.987797i	$-0.549778\pi$
$587$	−4429.97	−0.311489	−0.155745	+	0.987797i	$0.549778\pi$
$588$	0	0
$589$	10623.6	0.743187
$590$	0	0
$591$	4555.61	0.317077
$592$	0	0
$593$	−9371.80	−0.648994	−0.324497	−	0.945887i	$-0.605195\pi$
$593$	−9371.80	−0.648994	−0.324497	+	0.945887i	$0.605195\pi$
$594$	0	0
$595$	5156.87	0.355313
$596$	0	0
$597$	3680.85	0.252340
$598$	0	0
$599$	−11990.3	−0.817879	−0.408940	−	0.912561i	$-0.634101\pi$
$599$	−11990.3	−0.817879	−0.408940	+	0.912561i	$0.634101\pi$
$600$	0	0
$601$	−1437.02	−0.0975331	−0.0487666	−	0.998810i	$-0.515529\pi$
$601$	−1437.02	−0.0975331	−0.0487666	+	0.998810i	$0.515529\pi$
$602$	0	0
$603$	14492.6	0.978750
$604$	0	0
$605$	5496.83	0.369385
$606$	0	0
$607$	26146.2	1.74834	0.874169	−	0.485621i	$-0.161407\pi$
$607$	26146.2	1.74834	0.874169	+	0.485621i	$0.161407\pi$
$608$	0	0
$609$	14224.3	0.946466
$610$	0	0
$611$	7815.94	0.517511
$612$	0	0
$613$	12373.7	0.815284	0.407642	−	0.913142i	$-0.366351\pi$
$613$	12373.7	0.815284	0.407642	+	0.913142i	$0.366351\pi$
$614$	0	0
$615$	−10875.6	−0.713082
$616$	0	0
$617$	7599.63	0.495866	0.247933	−	0.968777i	$-0.420249\pi$
$617$	7599.63	0.495866	0.247933	+	0.968777i	$0.420249\pi$
$618$	0	0
$619$	−15541.1	−1.00913	−0.504563	−	0.863375i	$-0.668346\pi$
$619$	−15541.1	−1.00913	−0.504563	+	0.863375i	$0.668346\pi$
$620$	0	0
$621$	18911.8	1.22207
$622$	0	0
$623$	18482.8	1.18860
$624$	0	0
$625$	−18486.9	−1.18316
$626$	0	0
$627$	12904.7	0.821953
$628$	0	0
$629$	−5954.16	−0.377437
$630$	0	0
$631$	21892.5	1.38118	0.690591	−	0.723246i	$-0.257351\pi$
$631$	21892.5	1.38118	0.690591	+	0.723246i	$0.257351\pi$
$632$	0	0
$633$	−14320.9	−0.899215
$634$	0	0
$635$	−16660.6	−1.04119
$636$	0	0
$637$	−3431.33	−0.213429
$638$	0	0
$639$	−7725.23	−0.478256
$640$	0	0
$641$	−3967.77	−0.244489	−0.122245	−	0.992500i	$-0.539009\pi$
$641$	−3967.77	−0.244489	−0.122245	+	0.992500i	$0.539009\pi$
$642$	0	0
$643$	16227.4	0.995248	0.497624	−	0.867393i	$-0.334206\pi$
$643$	16227.4	0.995248	0.497624	+	0.867393i	$0.334206\pi$
$644$	0	0
$645$	−6063.70	−0.370167
$646$	0	0
$647$	11820.5	0.718257	0.359129	−	0.933288i	$-0.383074\pi$
$647$	11820.5	0.718257	0.359129	+	0.933288i	$0.383074\pi$
$648$	0	0
$649$	−8760.45	−0.529858
$650$	0	0
$651$	5459.04	0.328658
$652$	0	0
$653$	−25352.4	−1.51932	−0.759661	−	0.650319i	$-0.774635\pi$
$653$	−25352.4	−1.51932	−0.759661	+	0.650319i	$0.774635\pi$
$654$	0	0
$655$	−7683.18	−0.458331
$656$	0	0
$657$	7854.34	0.466403
$658$	0	0
$659$	−30253.7	−1.78834	−0.894170	−	0.447727i	$-0.852234\pi$
$659$	−30253.7	−1.78834	−0.894170	+	0.447727i	$0.852234\pi$
$660$	0	0
$661$	−3347.84	−0.196998	−0.0984991	−	0.995137i	$-0.531404\pi$
$661$	−3347.84	−0.196998	−0.0984991	+	0.995137i	$0.531404\pi$
$662$	0	0
$663$	−2494.83	−0.146141
$664$	0	0
$665$	39061.4	2.27780
$666$	0	0
$667$	−28871.2	−1.67601
$668$	0	0
$669$	11137.1	0.643625
$670$	0	0
$671$	22635.8	1.30230
$672$	0	0
$673$	21301.8	1.22009	0.610047	−	0.792365i	$-0.291151\pi$
$673$	21301.8	1.22009	0.610047	+	0.792365i	$0.291151\pi$
$674$	0	0
$675$	13351.3	0.761324
$676$	0	0
$677$	3735.64	0.212071	0.106036	−	0.994362i	$-0.466184\pi$
$677$	3735.64	0.212071	0.106036	+	0.994362i	$0.466184\pi$
$678$	0	0
$679$	−5928.22	−0.335058
$680$	0	0
$681$	18399.8	1.03536
$682$	0	0
$683$	−18135.3	−1.01600	−0.508001	−	0.861356i	$-0.669615\pi$
$683$	−18135.3	−1.01600	−0.508001	+	0.861356i	$0.669615\pi$
$684$	0	0
$685$	−5519.83	−0.307886
$686$	0	0
$687$	−10145.6	−0.563435
$688$	0	0
$689$	28115.8	1.55461
$690$	0	0
$691$	−13009.6	−0.716220	−0.358110	−	0.933679i	$-0.616579\pi$
$691$	−13009.6	−0.716220	−0.358110	+	0.933679i	$0.616579\pi$
$692$	0	0
$693$	−10487.2	−0.574857
$694$	0	0
$695$	−35632.4	−1.94477
$696$	0	0
$697$	3855.89	0.209544
$698$	0	0
$699$	−13015.2	−0.704261
$700$	0	0
$701$	4773.31	0.257183	0.128592	−	0.991698i	$-0.458954\pi$
$701$	4773.31	0.257183	0.128592	+	0.991698i	$0.458954\pi$
$702$	0	0
$703$	−45100.5	−2.41963
$704$	0	0
$705$	−8258.69	−0.441192
$706$	0	0
$707$	37075.1	1.97221
$708$	0	0
$709$	−10193.9	−0.539972	−0.269986	−	0.962864i	$-0.587019\pi$
$709$	−10193.9	−0.539972	−0.269986	+	0.962864i	$0.587019\pi$
$710$	0	0
$711$	−18040.7	−0.951587
$712$	0	0
$713$	−11080.3	−0.581991
$714$	0	0
$715$	20848.1	1.09046
$716$	0	0
$717$	6475.77	0.337297
$718$	0	0
$719$	−12183.9	−0.631966	−0.315983	−	0.948765i	$-0.602334\pi$
$719$	−12183.9	−0.631966	−0.315983	+	0.948765i	$0.602334\pi$
$720$	0	0
$721$	4640.05	0.239673
$722$	0	0
$723$	14513.4	0.746556
$724$	0	0
$725$	−20382.5	−1.04412
$726$	0	0
$727$	−7370.89	−0.376026	−0.188013	−	0.982167i	$-0.560205\pi$
$727$	−7370.89	−0.376026	−0.188013	+	0.982167i	$0.560205\pi$
$728$	0	0
$729$	12441.2	0.632078
$730$	0	0
$731$	2149.86	0.108776
$732$	0	0
$733$	−4303.45	−0.216850	−0.108425	−	0.994105i	$-0.534581\pi$
$733$	−4303.45	−0.216850	−0.108425	+	0.994105i	$0.534581\pi$
$734$	0	0
$735$	3625.71	0.181954
$736$	0	0
$737$	−27150.6	−1.35700
$738$	0	0
$739$	−9493.89	−0.472583	−0.236291	−	0.971682i	$-0.575932\pi$
$739$	−9493.89	−0.472583	−0.236291	+	0.971682i	$0.575932\pi$
$740$	0	0
$741$	−18897.4	−0.936861
$742$	0	0
$743$	−13433.7	−0.663302	−0.331651	−	0.943402i	$-0.607606\pi$
$743$	−13433.7	−0.663302	−0.331651	+	0.943402i	$0.607606\pi$
$744$	0	0
$745$	15235.2	0.749226
$746$	0	0
$747$	1319.51	0.0646297
$748$	0	0
$749$	9584.34	0.467562
$750$	0	0
$751$	−25795.0	−1.25336	−0.626679	−	0.779277i	$-0.715586\pi$
$751$	−25795.0	−1.25336	−0.626679	+	0.779277i	$0.715586\pi$
$752$	0	0
$753$	−6332.92	−0.306487
$754$	0	0
$755$	−46736.3	−2.25286
$756$	0	0
$757$	183.635	0.00881683	0.00440842	−	0.999990i	$-0.498597\pi$
$757$	183.635	0.00881683	0.00440842	+	0.999990i	$0.498597\pi$
$758$	0	0
$759$	−13459.5	−0.643673
$760$	0	0
$761$	−34650.0	−1.65054	−0.825269	−	0.564739i	$-0.808977\pi$
$761$	−34650.0	−1.65054	−0.825269	+	0.564739i	$0.808977\pi$
$762$	0	0
$763$	39046.1	1.85264
$764$	0	0
$765$	−4169.06	−0.197036
$766$	0	0
$767$	12828.6	0.603931
$768$	0	0
$769$	10059.9	0.471743	0.235872	−	0.971784i	$-0.424206\pi$
$769$	10059.9	0.471743	0.235872	+	0.971784i	$0.424206\pi$
$770$	0	0
$771$	−1073.20	−0.0501302
$772$	0	0
$773$	15739.0	0.732331	0.366165	−	0.930550i	$-0.380670\pi$
$773$	15739.0	0.732331	0.366165	+	0.930550i	$0.380670\pi$
$774$	0	0
$775$	−7822.44	−0.362568
$776$	0	0
$777$	−23175.4	−1.07003
$778$	0	0
$779$	29206.9	1.34332
$780$	0	0
$781$	14472.5	0.663082
$782$	0	0
$783$	−30270.6	−1.38159
$784$	0	0
$785$	−17178.5	−0.781055
$786$	0	0
$787$	18685.2	0.846322	0.423161	−	0.906055i	$-0.360921\pi$
$787$	18685.2	0.846322	0.423161	+	0.906055i	$0.360921\pi$
$788$	0	0
$789$	8137.72	0.367187
$790$	0	0
$791$	9976.13	0.448433
$792$	0	0
$793$	−33147.4	−1.48436
$794$	0	0
$795$	−29708.5	−1.32535
$796$	0	0
$797$	27018.3	1.20080	0.600399	−	0.799701i	$-0.295009\pi$
$797$	27018.3	1.20080	0.600399	+	0.799701i	$0.295009\pi$
$798$	0	0
$799$	2928.09	0.129647
$800$	0	0
$801$	−14942.3	−0.659128
$802$	0	0
$803$	−14714.4	−0.646649
$804$	0	0
$805$	−40740.5	−1.78375
$806$	0	0
$807$	−12314.3	−0.537153
$808$	0	0
$809$	11420.1	0.496305	0.248152	−	0.968721i	$-0.420177\pi$
$809$	11420.1	0.496305	0.248152	+	0.968721i	$0.420177\pi$
$810$	0	0
$811$	25685.8	1.11215	0.556073	−	0.831133i	$-0.312307\pi$
$811$	25685.8	1.11215	0.556073	+	0.831133i	$0.312307\pi$
$812$	0	0
$813$	12467.5	0.537827
$814$	0	0
$815$	−5966.16	−0.256424
$816$	0	0
$817$	16284.4	0.697330
$818$	0	0
$819$	15357.3	0.655221
$820$	0	0
$821$	5672.70	0.241143	0.120572	−	0.992705i	$-0.461527\pi$
$821$	5672.70	0.241143	0.120572	+	0.992705i	$0.461527\pi$
$822$	0	0
$823$	13327.2	0.564467	0.282233	−	0.959346i	$-0.408925\pi$
$823$	13327.2	0.564467	0.282233	+	0.959346i	$0.408925\pi$
$824$	0	0
$825$	−9502.09	−0.400994
$826$	0	0
$827$	23460.4	0.986454	0.493227	−	0.869901i	$-0.335817\pi$
$827$	23460.4	0.986454	0.493227	+	0.869901i	$0.335817\pi$
$828$	0	0
$829$	43695.9	1.83067	0.915333	−	0.402699i	$-0.131928\pi$
$829$	43695.9	1.83067	0.915333	+	0.402699i	$0.131928\pi$
$830$	0	0
$831$	−21514.8	−0.898121
$832$	0	0
$833$	−1285.48	−0.0534685
$834$	0	0
$835$	−5872.98	−0.243404
$836$	0	0
$837$	−11617.3	−0.479753
$838$	0	0
$839$	1284.76	0.0528663	0.0264331	−	0.999651i	$-0.491585\pi$
$839$	1284.76	0.0528663	0.0264331	+	0.999651i	$0.491585\pi$
$840$	0	0
$841$	21822.7	0.894777
$842$	0	0
$843$	9658.76	0.394621
$844$	0	0
$845$	2043.51	0.0831940
$846$	0	0
$847$	−7585.62	−0.307727
$848$	0	0
$849$	−9034.02	−0.365191
$850$	0	0
$851$	47039.3	1.89481
$852$	0	0
$853$	−20512.5	−0.823371	−0.411685	−	0.911326i	$-0.635060\pi$
$853$	−20512.5	−0.823371	−0.411685	+	0.911326i	$0.635060\pi$
$854$	0	0
$855$	−31579.0	−1.26313
$856$	0	0
$857$	31252.6	1.24570	0.622852	−	0.782340i	$-0.285974\pi$
$857$	31252.6	1.24570	0.622852	+	0.782340i	$0.285974\pi$
$858$	0	0
$859$	−34755.4	−1.38049	−0.690244	−	0.723577i	$-0.742497\pi$
$859$	−34755.4	−1.38049	−0.690244	+	0.723577i	$0.742497\pi$
$860$	0	0
$861$	15008.3	0.594054
$862$	0	0
$863$	48390.6	1.90873	0.954365	−	0.298642i	$-0.0965335\pi$
$863$	48390.6	1.90873	0.954365	+	0.298642i	$0.0965335\pi$
$864$	0	0
$865$	16595.1	0.652314
$866$	0	0
$867$	−934.641	−0.0366114
$868$	0	0
$869$	33797.5	1.31934
$870$	0	0
$871$	39758.9	1.54670
$872$	0	0
$873$	4792.65	0.185804
$874$	0	0
$875$	9156.24	0.353757
$876$	0	0
$877$	−258.763	−0.00996328	−0.00498164	−	0.999988i	$-0.501586\pi$
$877$	−258.763	−0.00996328	−0.00498164	+	0.999988i	$0.501586\pi$
$878$	0	0
$879$	−9400.15	−0.360704
$880$	0	0
$881$	33678.9	1.28793	0.643967	−	0.765053i	$-0.277287\pi$
$881$	33678.9	1.28793	0.643967	+	0.765053i	$0.277287\pi$
$882$	0	0
$883$	−21224.7	−0.808912	−0.404456	−	0.914557i	$-0.632539\pi$
$883$	−21224.7	−0.808912	−0.404456	+	0.914557i	$0.632539\pi$
$884$	0	0
$885$	−13555.3	−0.514868
$886$	0	0
$887$	−572.377	−0.0216669	−0.0108334	−	0.999941i	$-0.503448\pi$
$887$	−572.377	−0.0216669	−0.0108334	+	0.999941i	$0.503448\pi$
$888$	0	0
$889$	22991.6	0.867394
$890$	0	0
$891$	−272.487	−0.0102454
$892$	0	0
$893$	22179.1	0.831127
$894$	0	0
$895$	−62940.1	−2.35068
$896$	0	0
$897$	19709.8	0.733657
$898$	0	0
$899$	17735.2	0.657957
$900$	0	0
$901$	10533.0	0.389463
$902$	0	0
$903$	8367.90	0.308379
$904$	0	0
$905$	34838.8	1.27965
$906$	0	0
$907$	−43138.1	−1.57925	−0.789624	−	0.613591i	$-0.789724\pi$
$907$	−43138.1	−1.57925	−0.789624	+	0.613591i	$0.789724\pi$
$908$	0	0
$909$	−29973.2	−1.09367
$910$	0	0
$911$	32501.4	1.18202	0.591010	−	0.806664i	$-0.298729\pi$
$911$	32501.4	1.18202	0.591010	+	0.806664i	$0.298729\pi$
$912$	0	0
$913$	−2471.98	−0.0896065
$914$	0	0
$915$	35025.1	1.26546
$916$	0	0
$917$	10602.8	0.381826
$918$	0	0
$919$	50933.4	1.82822	0.914112	−	0.405461i	$-0.132889\pi$
$919$	50933.4	1.82822	0.914112	+	0.405461i	$0.132889\pi$
$920$	0	0
$921$	−27655.9	−0.989459
$922$	0	0
$923$	−21193.3	−0.755780
$924$	0	0
$925$	33208.7	1.18043
$926$	0	0
$927$	−3751.23	−0.132909
$928$	0	0
$929$	−4890.44	−0.172713	−0.0863563	−	0.996264i	$-0.527522\pi$
$929$	−4890.44	−0.172713	−0.0863563	+	0.996264i	$0.527522\pi$
$930$	0	0
$931$	−9737.03	−0.342769
$932$	0	0
$933$	−831.517	−0.0291776
$934$	0	0
$935$	7810.34	0.273182
$936$	0	0
$937$	−15637.6	−0.545206	−0.272603	−	0.962127i	$-0.587885\pi$
$937$	−15637.6	−0.545206	−0.272603	+	0.962127i	$0.587885\pi$
$938$	0	0
$939$	−6717.95	−0.233474
$940$	0	0
$941$	1284.81	0.0445098	0.0222549	−	0.999752i	$-0.492915\pi$
$941$	1284.81	0.0445098	0.0222549	+	0.999752i	$0.492915\pi$
$942$	0	0
$943$	−30462.5	−1.05196
$944$	0	0
$945$	−42715.2	−1.47040
$946$	0	0
$947$	−16603.8	−0.569748	−0.284874	−	0.958565i	$-0.591952\pi$
$947$	−16603.8	−0.569748	−0.284874	+	0.958565i	$0.591952\pi$
$948$	0	0
$949$	21547.4	0.737049
$950$	0	0
$951$	11641.5	0.396951
$952$	0	0
$953$	37812.9	1.28529	0.642643	−	0.766165i	$-0.277838\pi$
$953$	37812.9	1.28529	0.642643	+	0.766165i	$0.277838\pi$
$954$	0	0
$955$	59377.4	2.01195
$956$	0	0
$957$	21543.4	0.727690
$958$	0	0
$959$	7617.36	0.256494
$960$	0	0
$961$	−22984.5	−0.771526
$962$	0	0
$963$	−7748.43	−0.259283
$964$	0	0
$965$	−43574.9	−1.45360
$966$	0	0
$967$	14492.1	0.481937	0.240968	−	0.970533i	$-0.422535\pi$
$967$	14492.1	0.481937	0.240968	+	0.970533i	$0.422535\pi$
$968$	0	0
$969$	−7079.55	−0.234704
$970$	0	0
$971$	−2491.26	−0.0823360	−0.0411680	−	0.999152i	$-0.513108\pi$
$971$	−2491.26	−0.0823360	−0.0411680	+	0.999152i	$0.513108\pi$
$972$	0	0
$973$	49172.7	1.62015
$974$	0	0
$975$	13914.7	0.457053
$976$	0	0
$977$	2404.98	0.0787536	0.0393768	−	0.999224i	$-0.487463\pi$
$977$	2404.98	0.0787536	0.0393768	+	0.999224i	$0.487463\pi$
$978$	0	0
$979$	27993.1	0.913854
$980$	0	0
$981$	−31566.7	−1.02737
$982$	0	0
$983$	−48909.8	−1.58696	−0.793479	−	0.608597i	$-0.791732\pi$
$983$	−48909.8	−1.58696	−0.793479	+	0.608597i	$0.791732\pi$
$984$	0	0
$985$	20884.7	0.675577
$986$	0	0
$987$	11397.0	0.367548
$988$	0	0
$989$	−16984.4	−0.546080
$990$	0	0
$991$	−10826.6	−0.347042	−0.173521	−	0.984830i	$-0.555514\pi$
$991$	−10826.6	−0.347042	−0.173521	+	0.984830i	$0.555514\pi$
$992$	0	0
$993$	26804.3	0.856604
$994$	0	0
$995$	16874.5	0.537645
$996$	0	0
$997$	−2284.06	−0.0725547	−0.0362773	−	0.999342i	$-0.511550\pi$
$997$	−2284.06	−0.0725547	−0.0362773	+	0.999342i	$0.511550\pi$
$998$	0	0
$999$	49319.2	1.56195

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1088.4.a.bg.1.3		7
4.3	odd	2		1088.4.a.bh.1.5		7
8.3	odd	2		544.4.a.l.1.3	yes	7
8.5	even	2		544.4.a.k.1.5	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
544.4.a.k.1.5	✓	7	8.5	even	2
544.4.a.l.1.3	yes	7	8.3	odd	2
1088.4.a.bg.1.3		7	1.1	even	1	trivial
1088.4.a.bh.1.5		7	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1088 = 2^{6} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1088.a (trivial)

\(f(q)\)	\(=\)	\(q-3.23405 q^{3} -14.8262 q^{5} +20.4601 q^{7} -16.5409 q^{9} +O(q^{10})\)	\(q-3.23405 q^{3} -14.8262 q^{5} +20.4601 q^{7} -16.5409 q^{9} +30.9879 q^{11} -45.3781 q^{13} +47.9486 q^{15} -17.0000 q^{17} -128.769 q^{19} -66.1691 q^{21} +134.304 q^{23} +94.8158 q^{25} +140.814 q^{27} -214.969 q^{29} -82.5014 q^{31} -100.216 q^{33} -303.346 q^{35} +350.245 q^{37} +146.755 q^{39} -226.817 q^{41} -126.462 q^{43} +245.239 q^{45} -172.240 q^{47} +75.6165 q^{49} +54.9789 q^{51} -619.590 q^{53} -459.432 q^{55} +416.444 q^{57} -282.706 q^{59} +730.472 q^{61} -338.429 q^{63} +672.784 q^{65} -876.169 q^{67} -434.346 q^{69} +467.038 q^{71} -474.843 q^{73} -306.639 q^{75} +634.016 q^{77} +1090.67 q^{79} -8.79336 q^{81} -79.7726 q^{83} +252.045 q^{85} +695.221 q^{87} +903.357 q^{89} -928.441 q^{91} +266.814 q^{93} +1909.15 q^{95} -289.745 q^{97} -512.568 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{7} + 67 q^{9}+O(q^{10}) \) 7 * q - 2 * q^7 + 67 * q^9	\( 7 q - 2 q^{7} + 67 q^{9} - 108 q^{11} + 34 q^{13} + 128 q^{15} - 119 q^{17} - 124 q^{19} + 296 q^{21} - 6 q^{23} + 197 q^{25} - 248 q^{29} + 50 q^{31} + 512 q^{33} - 640 q^{35} + 484 q^{37} + 1504 q^{39} - 366 q^{41} - 1412 q^{43} + 80 q^{45} + 1012 q^{47} + 1115 q^{49} + 146 q^{53} + 1024 q^{55} + 48 q^{57} - 2332 q^{59} + 548 q^{61} + 2838 q^{63} - 208 q^{65} - 924 q^{67} + 1672 q^{69} + 1286 q^{71} + 870 q^{73} - 3136 q^{75} + 1344 q^{77} + 1818 q^{79} + 3039 q^{81} - 1772 q^{83} + 384 q^{87} + 1706 q^{89} - 588 q^{91} + 5576 q^{93} + 2048 q^{95} + 1802 q^{97} - 5148 q^{99}+O(q^{100}) \) 7 * q - 2 * q^7 + 67 * q^9 - 108 * q^11 + 34 * q^13 + 128 * q^15 - 119 * q^17 - 124 * q^19 + 296 * q^21 - 6 * q^23 + 197 * q^25 - 248 * q^29 + 50 * q^31 + 512 * q^33 - 640 * q^35 + 484 * q^37 + 1504 * q^39 - 366 * q^41 - 1412 * q^43 + 80 * q^45 + 1012 * q^47 + 1115 * q^49 + 146 * q^53 + 1024 * q^55 + 48 * q^57 - 2332 * q^59 + 548 * q^61 + 2838 * q^63 - 208 * q^65 - 924 * q^67 + 1672 * q^69 + 1286 * q^71 + 870 * q^73 - 3136 * q^75 + 1344 * q^77 + 1818 * q^79 + 3039 * q^81 - 1772 * q^83 + 384 * q^87 + 1706 * q^89 - 588 * q^91 + 5576 * q^93 + 2048 * q^95 + 1802 * q^97 - 5148 * q^99

Introduction

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