LMFDB - Embedded newform 1088.4.a.bh.1.7

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			1.7
Root			$8.08970$ 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)

$f(q)$	$=$	$q+9.40441 q^{3} -11.8817 q^{5} -5.65405 q^{7} +61.4429 q^{9} +O(q^{10})$	$q+9.40441 q^{3} -11.8817 q^{5} -5.65405 q^{7} +61.4429 q^{9} +63.6610 q^{11} -7.43149 q^{13} -111.740 q^{15} -17.0000 q^{17} -82.1519 q^{19} -53.1730 q^{21} +176.898 q^{23} +16.1742 q^{25} +323.916 q^{27} +58.2314 q^{29} +294.904 q^{31} +598.694 q^{33} +67.1796 q^{35} -259.776 q^{37} -69.8888 q^{39} -360.950 q^{41} +332.395 q^{43} -730.045 q^{45} -4.62596 q^{47} -311.032 q^{49} -159.875 q^{51} +354.664 q^{53} -756.399 q^{55} -772.591 q^{57} +518.983 q^{59} +139.045 q^{61} -347.402 q^{63} +88.2985 q^{65} +498.087 q^{67} +1663.62 q^{69} +752.109 q^{71} +507.347 q^{73} +152.109 q^{75} -359.943 q^{77} +364.914 q^{79} +1387.28 q^{81} +1157.67 q^{83} +201.988 q^{85} +547.632 q^{87} -976.782 q^{89} +42.0180 q^{91} +2773.40 q^{93} +976.103 q^{95} -329.616 q^{97} +3911.52 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	9.40441	1.80988	0.904940	−	0.425540i	$-0.139916\pi$
$3$	9.40441	1.80988	0.904940	+	0.425540i	$0.139916\pi$
$4$	0	0
$5$	−11.8817	−1.06273	−0.531365	−	0.847143i	$-0.678321\pi$
$5$	−11.8817	−1.06273	−0.531365	+	0.847143i	$0.678321\pi$
$6$	0	0
$7$	−5.65405	−0.305290	−0.152645	−	0.988281i	$-0.548779\pi$
$7$	−5.65405	−0.305290	−0.152645	+	0.988281i	$0.548779\pi$
$8$	0	0
$9$	61.4429	2.27566
$10$	0	0
$11$	63.6610	1.74496	0.872478	−	0.488653i	$-0.162512\pi$
$11$	63.6610	1.74496	0.872478	+	0.488653i	$0.162512\pi$
$12$	0	0
$13$	−7.43149	−0.158548	−0.0792740	−	0.996853i	$-0.525260\pi$
$13$	−7.43149	−0.158548	−0.0792740	+	0.996853i	$0.525260\pi$
$14$	0	0
$15$	−111.740	−1.92341
$16$	0	0
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	−82.1519	−0.991945	−0.495972	−	0.868338i	$-0.665188\pi$
$19$	−82.1519	−0.991945	−0.495972	+	0.868338i	$0.665188\pi$
$20$	0	0
$21$	−53.1730	−0.552539
$22$	0	0
$23$	176.898	1.60373	0.801865	−	0.597505i	$-0.203841\pi$
$23$	176.898	1.60373	0.801865	+	0.597505i	$0.203841\pi$
$24$	0	0
$25$	16.1742	0.129394
$26$	0	0
$27$	323.916	2.30880
$28$	0	0
$29$	58.2314	0.372872	0.186436	−	0.982467i	$-0.440306\pi$
$29$	58.2314	0.372872	0.186436	+	0.982467i	$0.440306\pi$
$30$	0	0
$31$	294.904	1.70859	0.854297	−	0.519786i	$-0.173988\pi$
$31$	294.904	1.70859	0.854297	+	0.519786i	$0.173988\pi$
$32$	0	0
$33$	598.694	3.15816
$34$	0	0
$35$	67.1796	0.324441
$36$	0	0
$37$	−259.776	−1.15424	−0.577121	−	0.816658i	$-0.695824\pi$
$37$	−259.776	−1.15424	−0.577121	+	0.816658i	$0.695824\pi$
$38$	0	0
$39$	−69.8888	−0.286953
$40$	0	0
$41$	−360.950	−1.37490	−0.687450	−	0.726232i	$-0.741270\pi$
$41$	−360.950	−1.37490	−0.687450	+	0.726232i	$0.741270\pi$
$42$	0	0
$43$	332.395	1.17883	0.589417	−	0.807829i	$-0.299358\pi$
$43$	332.395	1.17883	0.589417	+	0.807829i	$0.299358\pi$
$44$	0	0
$45$	−730.045	−2.41842
$46$	0	0
$47$	−4.62596	−0.0143567	−0.00717835	−	0.999974i	$-0.502285\pi$
$47$	−4.62596	−0.0143567	−0.00717835	+	0.999974i	$0.502285\pi$
$48$	0	0
$49$	−311.032	−0.906798
$50$	0	0
$51$	−159.875	−0.438960
$52$	0	0
$53$	354.664	0.919185	0.459592	−	0.888130i	$-0.347996\pi$
$53$	354.664	0.919185	0.459592	+	0.888130i	$0.347996\pi$
$54$	0	0
$55$	−756.399	−1.85442
$56$	0	0
$57$	−772.591	−1.79530
$58$	0	0
$59$	518.983	1.14518	0.572592	−	0.819840i	$-0.305938\pi$
$59$	518.983	1.14518	0.572592	+	0.819840i	$0.305938\pi$
$60$	0	0
$61$	139.045	0.291850	0.145925	−	0.989296i	$-0.453384\pi$
$61$	139.045	0.291850	0.145925	+	0.989296i	$0.453384\pi$
$62$	0	0
$63$	−347.402	−0.694738
$64$	0	0
$65$	88.2985	0.168494
$66$	0	0
$67$	498.087	0.908224	0.454112	−	0.890945i	$-0.349957\pi$
$67$	498.087	0.908224	0.454112	+	0.890945i	$0.349957\pi$
$68$	0	0
$69$	1663.62	2.90256
$70$	0	0
$71$	752.109	1.25717	0.628584	−	0.777742i	$-0.283635\pi$
$71$	752.109	1.25717	0.628584	+	0.777742i	$0.283635\pi$
$72$	0	0
$73$	507.347	0.813431	0.406715	−	0.913555i	$-0.366674\pi$
$73$	507.347	0.813431	0.406715	+	0.913555i	$0.366674\pi$
$74$	0	0
$75$	152.109	0.234187
$76$	0	0
$77$	−359.943	−0.532718
$78$	0	0
$79$	364.914	0.519697	0.259848	−	0.965649i	$-0.416327\pi$
$79$	364.914	0.519697	0.259848	+	0.965649i	$0.416327\pi$
$80$	0	0
$81$	1387.28	1.90298
$82$	0	0
$83$	1157.67	1.53097	0.765487	−	0.643452i	$-0.222498\pi$
$83$	1157.67	1.53097	0.765487	+	0.643452i	$0.222498\pi$
$84$	0	0
$85$	201.988	0.257750
$86$	0	0
$87$	547.632	0.674854
$88$	0	0
$89$	−976.782	−1.16336	−0.581678	−	0.813419i	$-0.697604\pi$
$89$	−976.782	−1.16336	−0.581678	+	0.813419i	$0.697604\pi$
$90$	0	0
$91$	42.0180	0.0484032
$92$	0	0
$93$	2773.40	3.09235
$94$	0	0
$95$	976.103	1.05417
$96$	0	0
$97$	−329.616	−0.345026	−0.172513	−	0.985007i	$-0.555189\pi$
$97$	−329.616	−0.345026	−0.172513	+	0.985007i	$0.555189\pi$
$98$	0	0
$99$	3911.52	3.97093
$100$	0	0
$101$	138.239	0.136191	0.0680955	−	0.997679i	$-0.478308\pi$
$101$	138.239	0.136191	0.0680955	+	0.997679i	$0.478308\pi$
$102$	0	0
$103$	−1537.92	−1.47122	−0.735611	−	0.677404i	$-0.763105\pi$
$103$	−1537.92	−1.47122	−0.735611	+	0.677404i	$0.763105\pi$
$104$	0	0
$105$	631.785	0.587199
$106$	0	0
$107$	−498.554	−0.450440	−0.225220	−	0.974308i	$-0.572310\pi$
$107$	−498.554	−0.450440	−0.225220	+	0.974308i	$0.572310\pi$
$108$	0	0
$109$	1225.57	1.07696	0.538480	−	0.842638i	$-0.318999\pi$
$109$	1225.57	1.07696	0.538480	+	0.842638i	$0.318999\pi$
$110$	0	0
$111$	−2443.04	−2.08904
$112$	0	0
$113$	722.458	0.601444	0.300722	−	0.953712i	$-0.402772\pi$
$113$	722.458	0.601444	0.300722	+	0.953712i	$0.402772\pi$
$114$	0	0
$115$	−2101.85	−1.70433
$116$	0	0
$117$	−456.612	−0.360802
$118$	0	0
$119$	96.1189	0.0740438
$120$	0	0
$121$	2721.72	2.04487
$122$	0	0
$123$	−3394.52	−2.48840
$124$	0	0
$125$	1293.03	0.925219
$126$	0	0
$127$	−1037.28	−0.724752	−0.362376	−	0.932032i	$-0.618034\pi$
$127$	−1037.28	−0.724752	−0.362376	+	0.932032i	$0.618034\pi$
$128$	0	0
$129$	3125.98	2.13355
$130$	0	0
$131$	2449.61	1.63376	0.816882	−	0.576804i	$-0.195700\pi$
$131$	2449.61	1.63376	0.816882	+	0.576804i	$0.195700\pi$
$132$	0	0
$133$	464.492	0.302831
$134$	0	0
$135$	−3848.66	−2.45363
$136$	0	0
$137$	488.832	0.304845	0.152422	−	0.988315i	$-0.451293\pi$
$137$	488.832	0.304845	0.152422	+	0.988315i	$0.451293\pi$
$138$	0	0
$139$	678.764	0.414187	0.207094	−	0.978321i	$-0.433600\pi$
$139$	678.764	0.414187	0.207094	+	0.978321i	$0.433600\pi$
$140$	0	0
$141$	−43.5044	−0.0259839
$142$	0	0
$143$	−473.096	−0.276659
$144$	0	0
$145$	−691.886	−0.396262
$146$	0	0
$147$	−2925.07	−1.64120
$148$	0	0
$149$	−698.017	−0.383784	−0.191892	−	0.981416i	$-0.561462\pi$
$149$	−698.017	−0.383784	−0.191892	+	0.981416i	$0.561462\pi$
$150$	0	0
$151$	−1667.97	−0.898924	−0.449462	−	0.893300i	$-0.648384\pi$
$151$	−1667.97	−0.898924	−0.449462	+	0.893300i	$0.648384\pi$
$152$	0	0
$153$	−1044.53	−0.551930
$154$	0	0
$155$	−3503.96	−1.81577
$156$	0	0
$157$	−558.574	−0.283943	−0.141972	−	0.989871i	$-0.545344\pi$
$157$	−558.574	−0.283943	−0.141972	+	0.989871i	$0.545344\pi$
$158$	0	0
$159$	3335.40	1.66361
$160$	0	0
$161$	−1000.19	−0.489603
$162$	0	0
$163$	−728.379	−0.350007	−0.175003	−	0.984568i	$-0.555994\pi$
$163$	−728.379	−0.350007	−0.175003	+	0.984568i	$0.555994\pi$
$164$	0	0
$165$	−7113.49	−3.35627
$166$	0	0
$167$	1621.64	0.751414	0.375707	−	0.926738i	$-0.377400\pi$
$167$	1621.64	0.751414	0.375707	+	0.926738i	$0.377400\pi$
$168$	0	0
$169$	−2141.77	−0.974863
$170$	0	0
$171$	−5047.66	−2.25733
$172$	0	0
$173$	−151.610	−0.0666282	−0.0333141	−	0.999445i	$-0.510606\pi$
$173$	−151.610	−0.0666282	−0.0333141	+	0.999445i	$0.510606\pi$
$174$	0	0
$175$	−91.4498	−0.0395026
$176$	0	0
$177$	4880.73	2.07265
$178$	0	0
$179$	324.602	0.135541	0.0677707	−	0.997701i	$-0.478411\pi$
$179$	324.602	0.135541	0.0677707	+	0.997701i	$0.478411\pi$
$180$	0	0
$181$	3428.48	1.40794	0.703968	−	0.710231i	$-0.251410\pi$
$181$	3428.48	1.40794	0.703968	+	0.710231i	$0.251410\pi$
$182$	0	0
$183$	1307.63	0.528214
$184$	0	0
$185$	3086.58	1.22665
$186$	0	0
$187$	−1082.24	−0.423214
$188$	0	0
$189$	−1831.44	−0.704854
$190$	0	0
$191$	9.36594	0.00354814	0.00177407	−	0.999998i	$-0.499435\pi$
$191$	9.36594	0.00354814	0.00177407	+	0.999998i	$0.499435\pi$
$192$	0	0
$193$	−2197.00	−0.819397	−0.409698	−	0.912221i	$-0.634366\pi$
$193$	−2197.00	−0.819397	−0.409698	+	0.912221i	$0.634366\pi$
$194$	0	0
$195$	830.396	0.304953
$196$	0	0
$197$	−4521.23	−1.63515	−0.817574	−	0.575824i	$-0.804681\pi$
$197$	−4521.23	−1.63515	−0.817574	+	0.575824i	$0.804681\pi$
$198$	0	0
$199$	−4665.37	−1.66191	−0.830954	−	0.556342i	$-0.812205\pi$
$199$	−4665.37	−1.66191	−0.830954	+	0.556342i	$0.812205\pi$
$200$	0	0
$201$	4684.21	1.64378
$202$	0	0
$203$	−329.243	−0.113834
$204$	0	0
$205$	4288.69	1.46115
$206$	0	0
$207$	10869.1	3.64955
$208$	0	0
$209$	−5229.88	−1.73090
$210$	0	0
$211$	2205.94	0.719731	0.359865	−	0.933004i	$-0.382823\pi$
$211$	2205.94	0.719731	0.359865	+	0.933004i	$0.382823\pi$
$212$	0	0
$213$	7073.14	2.27532
$214$	0	0
$215$	−3949.42	−1.25278
$216$	0	0
$217$	−1667.41	−0.521617
$218$	0	0
$219$	4771.30	1.47221
$220$	0	0
$221$	126.335	0.0384535
$222$	0	0
$223$	4638.79	1.39299	0.696494	−	0.717562i	$-0.254742\pi$
$223$	4638.79	1.39299	0.696494	+	0.717562i	$0.254742\pi$
$224$	0	0
$225$	993.790	0.294456
$226$	0	0
$227$	−2081.66	−0.608653	−0.304327	−	0.952568i	$-0.598431\pi$
$227$	−2081.66	−0.608653	−0.304327	+	0.952568i	$0.598431\pi$
$228$	0	0
$229$	−5710.65	−1.64790	−0.823952	−	0.566659i	$-0.808236\pi$
$229$	−5710.65	−1.64790	−0.823952	+	0.566659i	$0.808236\pi$
$230$	0	0
$231$	−3385.05	−0.964155
$232$	0	0
$233$	−1144.56	−0.321813	−0.160906	−	0.986970i	$-0.551442\pi$
$233$	−1144.56	−0.321813	−0.160906	+	0.986970i	$0.551442\pi$
$234$	0	0
$235$	54.9641	0.0152573
$236$	0	0
$237$	3431.80	0.940589
$238$	0	0
$239$	−3402.77	−0.920948	−0.460474	−	0.887673i	$-0.652321\pi$
$239$	−3402.77	−0.920948	−0.460474	+	0.887673i	$0.652321\pi$
$240$	0	0
$241$	−2415.05	−0.645508	−0.322754	−	0.946483i	$-0.604609\pi$
$241$	−2415.05	−0.645508	−0.322754	+	0.946483i	$0.604609\pi$
$242$	0	0
$243$	4300.79	1.13537
$244$	0	0
$245$	3695.58	0.963681
$246$	0	0
$247$	610.511	0.157271
$248$	0	0
$249$	10887.2	2.77088
$250$	0	0
$251$	−2531.54	−0.636612	−0.318306	−	0.947988i	$-0.603114\pi$
$251$	−2531.54	−0.636612	−0.318306	+	0.947988i	$0.603114\pi$
$252$	0	0
$253$	11261.5	2.79844
$254$	0	0
$255$	1899.58	0.466496
$256$	0	0
$257$	−5679.11	−1.37842	−0.689208	−	0.724564i	$-0.742041\pi$
$257$	−5679.11	−1.37842	−0.689208	+	0.724564i	$0.742041\pi$
$258$	0	0
$259$	1468.79	0.352379
$260$	0	0
$261$	3577.91	0.848532
$262$	0	0
$263$	−7659.79	−1.79591	−0.897953	−	0.440092i	$-0.854946\pi$
$263$	−7659.79	−1.79591	−0.897953	+	0.440092i	$0.854946\pi$
$264$	0	0
$265$	−4214.00	−0.976844
$266$	0	0
$267$	−9186.06	−2.10553
$268$	0	0
$269$	4729.05	1.07188	0.535939	−	0.844257i	$-0.319958\pi$
$269$	4729.05	1.07188	0.535939	+	0.844257i	$0.319958\pi$
$270$	0	0
$271$	7218.50	1.61805	0.809027	−	0.587771i	$-0.199995\pi$
$271$	7218.50	1.61805	0.809027	+	0.587771i	$0.199995\pi$
$272$	0	0
$273$	395.155	0.0876039
$274$	0	0
$275$	1029.67	0.225786
$276$	0	0
$277$	2060.71	0.446990	0.223495	−	0.974705i	$-0.428253\pi$
$277$	2060.71	0.446990	0.223495	+	0.974705i	$0.428253\pi$
$278$	0	0
$279$	18119.8	3.88818
$280$	0	0
$281$	−9094.69	−1.93076	−0.965380	−	0.260847i	$-0.915998\pi$
$281$	−9094.69	−1.93076	−0.965380	+	0.260847i	$0.915998\pi$
$282$	0	0
$283$	−5109.74	−1.07329	−0.536647	−	0.843807i	$-0.680309\pi$
$283$	−5109.74	−1.07329	−0.536647	+	0.843807i	$0.680309\pi$
$284$	0	0
$285$	9179.67	1.90792
$286$	0	0
$287$	2040.83	0.419743
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	−3099.85	−0.624455
$292$	0	0
$293$	−6969.95	−1.38972	−0.694861	−	0.719144i	$-0.744534\pi$
$293$	−6969.95	−1.38972	−0.694861	+	0.719144i	$0.744534\pi$
$294$	0	0
$295$	−6166.39	−1.21702
$296$	0	0
$297$	20620.8	4.02875
$298$	0	0
$299$	−1314.62	−0.254268
$300$	0	0
$301$	−1879.38	−0.359886
$302$	0	0
$303$	1300.06	0.246489
$304$	0	0
$305$	−1652.08	−0.310158
$306$	0	0
$307$	−3060.43	−0.568952	−0.284476	−	0.958683i	$-0.591820\pi$
$307$	−3060.43	−0.568952	−0.284476	+	0.958683i	$0.591820\pi$
$308$	0	0
$309$	−14463.2	−2.66273
$310$	0	0
$311$	4660.64	0.849778	0.424889	−	0.905246i	$-0.360313\pi$
$311$	4660.64	0.849778	0.424889	+	0.905246i	$0.360313\pi$
$312$	0	0
$313$	7516.98	1.35746	0.678730	−	0.734388i	$-0.262531\pi$
$313$	7516.98	1.35746	0.678730	+	0.734388i	$0.262531\pi$
$314$	0	0
$315$	4127.71	0.738318
$316$	0	0
$317$	−935.210	−0.165699	−0.0828496	−	0.996562i	$-0.526402\pi$
$317$	−935.210	−0.165699	−0.0828496	+	0.996562i	$0.526402\pi$
$318$	0	0
$319$	3707.07	0.650646
$320$	0	0
$321$	−4688.61	−0.815241
$322$	0	0
$323$	1396.58	0.240582
$324$	0	0
$325$	−120.198	−0.0205151
$326$	0	0
$327$	11525.8	1.94917
$328$	0	0
$329$	26.1554	0.00438296
$330$	0	0
$331$	−2370.14	−0.393580	−0.196790	−	0.980446i	$-0.563052\pi$
$331$	−2370.14	−0.393580	−0.196790	+	0.980446i	$0.563052\pi$
$332$	0	0
$333$	−15961.4	−2.62667
$334$	0	0
$335$	−5918.10	−0.965196
$336$	0	0
$337$	8659.45	1.39973	0.699867	−	0.714273i	$-0.253243\pi$
$337$	8659.45	1.39973	0.699867	+	0.714273i	$0.253243\pi$
$338$	0	0
$339$	6794.29	1.08854
$340$	0	0
$341$	18773.9	2.98142
$342$	0	0
$343$	3697.93	0.582127
$344$	0	0
$345$	−19766.6	−3.08463
$346$	0	0
$347$	7495.57	1.15961	0.579803	−	0.814757i	$-0.303130\pi$
$347$	7495.57	1.15961	0.579803	+	0.814757i	$0.303130\pi$
$348$	0	0
$349$	−12872.5	−1.97436	−0.987179	−	0.159620i	$-0.948973\pi$
$349$	−12872.5	−1.97436	−0.987179	+	0.159620i	$0.948973\pi$
$350$	0	0
$351$	−2407.17	−0.366055
$352$	0	0
$353$	−4713.22	−0.710650	−0.355325	−	0.934743i	$-0.615630\pi$
$353$	−4713.22	−0.710650	−0.355325	+	0.934743i	$0.615630\pi$
$354$	0	0
$355$	−8936.31	−1.33603
$356$	0	0
$357$	903.942	0.134010
$358$	0	0
$359$	−11899.8	−1.74944	−0.874721	−	0.484627i	$-0.838955\pi$
$359$	−11899.8	−1.74944	−0.874721	+	0.484627i	$0.838955\pi$
$360$	0	0
$361$	−110.058	−0.0160458
$362$	0	0
$363$	25596.2	3.70097
$364$	0	0
$365$	−6028.13	−0.864457
$366$	0	0
$367$	−4751.89	−0.675876	−0.337938	−	0.941168i	$-0.609729\pi$
$367$	−4751.89	−0.675876	−0.337938	+	0.941168i	$0.609729\pi$
$368$	0	0
$369$	−22177.8	−3.12881
$370$	0	0
$371$	−2005.29	−0.280618
$372$	0	0
$373$	9195.52	1.27648	0.638239	−	0.769839i	$-0.279663\pi$
$373$	9195.52	1.27648	0.638239	+	0.769839i	$0.279663\pi$
$374$	0	0
$375$	12160.2	1.67453
$376$	0	0
$377$	−432.746	−0.0591181
$378$	0	0
$379$	378.188	0.0512565	0.0256282	−	0.999672i	$-0.491841\pi$
$379$	378.188	0.0512565	0.0256282	+	0.999672i	$0.491841\pi$
$380$	0	0
$381$	−9754.99	−1.31171
$382$	0	0
$383$	11822.4	1.57727	0.788636	−	0.614860i	$-0.210788\pi$
$383$	11822.4	1.57727	0.788636	+	0.614860i	$0.210788\pi$
$384$	0	0
$385$	4276.72	0.566135
$386$	0	0
$387$	20423.4	2.68263
$388$	0	0
$389$	−3859.66	−0.503065	−0.251533	−	0.967849i	$-0.580935\pi$
$389$	−3859.66	−0.503065	−0.251533	+	0.967849i	$0.580935\pi$
$390$	0	0
$391$	−3007.27	−0.388962
$392$	0	0
$393$	23037.1	2.95692
$394$	0	0
$395$	−4335.79	−0.552297
$396$	0	0
$397$	648.707	0.0820093	0.0410046	−	0.999159i	$-0.486944\pi$
$397$	648.707	0.0820093	0.0410046	+	0.999159i	$0.486944\pi$
$398$	0	0
$399$	4368.27	0.548088
$400$	0	0
$401$	13829.7	1.72225	0.861124	−	0.508395i	$-0.169761\pi$
$401$	13829.7	1.72225	0.861124	+	0.508395i	$0.169761\pi$
$402$	0	0
$403$	−2191.58	−0.270894
$404$	0	0
$405$	−16483.2	−2.02236
$406$	0	0
$407$	−16537.6	−2.01410
$408$	0	0
$409$	−8355.25	−1.01012	−0.505062	−	0.863083i	$-0.668530\pi$
$409$	−8355.25	−1.01012	−0.505062	+	0.863083i	$0.668530\pi$
$410$	0	0
$411$	4597.17	0.551732
$412$	0	0
$413$	−2934.36	−0.349614
$414$	0	0
$415$	−13755.1	−1.62701
$416$	0	0
$417$	6383.38	0.749629
$418$	0	0
$419$	7865.34	0.917057	0.458529	−	0.888680i	$-0.348377\pi$
$419$	7865.34	0.917057	0.458529	+	0.888680i	$0.348377\pi$
$420$	0	0
$421$	3082.19	0.356809	0.178404	−	0.983957i	$-0.442906\pi$
$421$	3082.19	0.356809	0.178404	+	0.983957i	$0.442906\pi$
$422$	0	0
$423$	−284.232	−0.0326710
$424$	0	0
$425$	−274.961	−0.0313825
$426$	0	0
$427$	−786.166	−0.0890990
$428$	0	0
$429$	−4449.19	−0.500720
$430$	0	0
$431$	−12290.8	−1.37362	−0.686809	−	0.726838i	$-0.740989\pi$
$431$	−12290.8	−1.37362	−0.686809	+	0.726838i	$0.740989\pi$
$432$	0	0
$433$	14820.9	1.64492	0.822458	−	0.568825i	$-0.192602\pi$
$433$	14820.9	1.64492	0.822458	+	0.568825i	$0.192602\pi$
$434$	0	0
$435$	−6506.78	−0.717187
$436$	0	0
$437$	−14532.5	−1.59081
$438$	0	0
$439$	4325.06	0.470214	0.235107	−	0.971969i	$-0.424456\pi$
$439$	4325.06	0.470214	0.235107	+	0.971969i	$0.424456\pi$
$440$	0	0
$441$	−19110.7	−2.06357
$442$	0	0
$443$	−2470.97	−0.265009	−0.132505	−	0.991182i	$-0.542302\pi$
$443$	−2470.97	−0.265009	−0.132505	+	0.991182i	$0.542302\pi$
$444$	0	0
$445$	11605.8	1.23633
$446$	0	0
$447$	−6564.44	−0.694603
$448$	0	0
$449$	−13276.8	−1.39548	−0.697742	−	0.716349i	$-0.745812\pi$
$449$	−13276.8	−1.39548	−0.697742	+	0.716349i	$0.745812\pi$
$450$	0	0
$451$	−22978.4	−2.39914
$452$	0	0
$453$	−15686.3	−1.62694
$454$	0	0
$455$	−499.245	−0.0514394
$456$	0	0
$457$	−187.458	−0.0191880	−0.00959400	−	0.999954i	$-0.503054\pi$
$457$	−187.458	−0.0191880	−0.00959400	+	0.999954i	$0.503054\pi$
$458$	0	0
$459$	−5506.56	−0.559966
$460$	0	0
$461$	−6606.63	−0.667465	−0.333732	−	0.942668i	$-0.608308\pi$
$461$	−6606.63	−0.667465	−0.333732	+	0.942668i	$0.608308\pi$
$462$	0	0
$463$	12486.6	1.25335	0.626675	−	0.779281i	$-0.284415\pi$
$463$	12486.6	1.25335	0.626675	+	0.779281i	$0.284415\pi$
$464$	0	0
$465$	−32952.7	−3.28633
$466$	0	0
$467$	8400.64	0.832410	0.416205	−	0.909271i	$-0.363360\pi$
$467$	8400.64	0.832410	0.416205	+	0.909271i	$0.363360\pi$
$468$	0	0
$469$	−2816.21	−0.277272
$470$	0	0
$471$	−5253.06	−0.513903
$472$	0	0
$473$	21160.6	2.05701
$474$	0	0
$475$	−1328.74	−0.128351
$476$	0	0
$477$	21791.6	2.09176
$478$	0	0
$479$	−11626.9	−1.10908	−0.554539	−	0.832158i	$-0.687105\pi$
$479$	−11626.9	−1.10908	−0.554539	+	0.832158i	$0.687105\pi$
$480$	0	0
$481$	1930.53	0.183003
$482$	0	0
$483$	−9406.21	−0.886123
$484$	0	0
$485$	3916.40	0.366669
$486$	0	0
$487$	3484.72	0.324246	0.162123	−	0.986771i	$-0.448166\pi$
$487$	3484.72	0.324246	0.162123	+	0.986771i	$0.448166\pi$
$488$	0	0
$489$	−6849.98	−0.633470
$490$	0	0
$491$	−2322.10	−0.213431	−0.106716	−	0.994290i	$-0.534033\pi$
$491$	−2322.10	−0.213431	−0.106716	+	0.994290i	$0.534033\pi$
$492$	0	0
$493$	−989.933	−0.0904348
$494$	0	0
$495$	−46475.4	−4.22003
$496$	0	0
$497$	−4252.46	−0.383801
$498$	0	0
$499$	6840.81	0.613701	0.306851	−	0.951758i	$-0.400725\pi$
$499$	6840.81	0.613701	0.306851	+	0.951758i	$0.400725\pi$
$500$	0	0
$501$	15250.6	1.35997
$502$	0	0
$503$	−18043.3	−1.59943	−0.799713	−	0.600383i	$-0.795015\pi$
$503$	−18043.3	−1.59943	−0.799713	+	0.600383i	$0.795015\pi$
$504$	0	0
$505$	−1642.51	−0.144734
$506$	0	0
$507$	−20142.1	−1.76438
$508$	0	0
$509$	3438.65	0.299441	0.149721	−	0.988728i	$-0.452163\pi$
$509$	3438.65	0.299441	0.149721	+	0.988728i	$0.452163\pi$
$510$	0	0
$511$	−2868.57	−0.248332
$512$	0	0
$513$	−26610.3	−2.29020
$514$	0	0
$515$	18273.1	1.56351
$516$	0	0
$517$	−294.493	−0.0250518
$518$	0	0
$519$	−1425.80	−0.120589
$520$	0	0
$521$	−21851.4	−1.83748	−0.918741	−	0.394861i	$-0.870793\pi$
$521$	−21851.4	−1.83748	−0.918741	+	0.394861i	$0.870793\pi$
$522$	0	0
$523$	−2262.57	−0.189169	−0.0945845	−	0.995517i	$-0.530152\pi$
$523$	−2262.57	−0.189169	−0.0945845	+	0.995517i	$0.530152\pi$
$524$	0	0
$525$	−860.031	−0.0714949
$526$	0	0
$527$	−5013.37	−0.414395
$528$	0	0
$529$	19125.9	1.57195
$530$	0	0
$531$	31887.9	2.60606
$532$	0	0
$533$	2682.39	0.217987
$534$	0	0
$535$	5923.66	0.478695
$536$	0	0
$537$	3052.69	0.245314
$538$	0	0
$539$	−19800.6	−1.58232
$540$	0	0
$541$	8909.95	0.708075	0.354038	−	0.935231i	$-0.384809\pi$
$541$	8909.95	0.708075	0.354038	+	0.935231i	$0.384809\pi$
$542$	0	0
$543$	32242.8	2.54820
$544$	0	0
$545$	−14561.9	−1.14452
$546$	0	0
$547$	−11313.3	−0.884318	−0.442159	−	0.896937i	$-0.645787\pi$
$547$	−11313.3	−0.884318	−0.442159	+	0.896937i	$0.645787\pi$
$548$	0	0
$549$	8543.32	0.664153
$550$	0	0
$551$	−4783.82	−0.369869
$552$	0	0
$553$	−2063.24	−0.158658
$554$	0	0
$555$	29027.5	2.22008
$556$	0	0
$557$	−7987.08	−0.607583	−0.303791	−	0.952739i	$-0.598253\pi$
$557$	−7987.08	−0.607583	−0.303791	+	0.952739i	$0.598253\pi$
$558$	0	0
$559$	−2470.19	−0.186902
$560$	0	0
$561$	−10177.8	−0.765966
$562$	0	0
$563$	−25921.9	−1.94046	−0.970230	−	0.242186i	$-0.922136\pi$
$563$	−25921.9	−1.94046	−0.970230	+	0.242186i	$0.922136\pi$
$564$	0	0
$565$	−8584.01	−0.639172
$566$	0	0
$567$	−7843.73	−0.580962
$568$	0	0
$569$	2904.54	0.213998	0.106999	−	0.994259i	$-0.465876\pi$
$569$	2904.54	0.213998	0.106999	+	0.994259i	$0.465876\pi$
$570$	0	0
$571$	8173.15	0.599012	0.299506	−	0.954094i	$-0.403178\pi$
$571$	8173.15	0.599012	0.299506	+	0.954094i	$0.403178\pi$
$572$	0	0
$573$	88.0811	0.00642171
$574$	0	0
$575$	2861.18	0.207512
$576$	0	0
$577$	−21847.9	−1.57633	−0.788164	−	0.615466i	$-0.788968\pi$
$577$	−21847.9	−1.57633	−0.788164	+	0.615466i	$0.788968\pi$
$578$	0	0
$579$	−20661.5	−1.48301
$580$	0	0
$581$	−6545.53	−0.467391
$582$	0	0
$583$	22578.2	1.60394
$584$	0	0
$585$	5425.32	0.383435
$586$	0	0
$587$	16265.2	1.14367	0.571836	−	0.820368i	$-0.306231\pi$
$587$	16265.2	1.14367	0.571836	+	0.820368i	$0.306231\pi$
$588$	0	0
$589$	−24227.0	−1.69483
$590$	0	0
$591$	−42519.5	−2.95942
$592$	0	0
$593$	−6862.51	−0.475227	−0.237613	−	0.971360i	$-0.576365\pi$
$593$	−6862.51	−0.475227	−0.237613	+	0.971360i	$0.576365\pi$
$594$	0	0
$595$	−1142.05	−0.0786885
$596$	0	0
$597$	−43875.1	−3.00785
$598$	0	0
$599$	−15177.4	−1.03528	−0.517639	−	0.855599i	$-0.673189\pi$
$599$	−15177.4	−1.03528	−0.517639	+	0.855599i	$0.673189\pi$
$600$	0	0
$601$	−5858.73	−0.397641	−0.198821	−	0.980036i	$-0.563711\pi$
$601$	−5858.73	−0.397641	−0.198821	+	0.980036i	$0.563711\pi$
$602$	0	0
$603$	30603.9	2.06681
$604$	0	0
$605$	−32338.6	−2.17314
$606$	0	0
$607$	5476.15	0.366178	0.183089	−	0.983096i	$-0.441390\pi$
$607$	5476.15	0.366178	0.183089	+	0.983096i	$0.441390\pi$
$608$	0	0
$609$	−3096.34	−0.206026
$610$	0	0
$611$	34.3777	0.00227623
$612$	0	0
$613$	8574.21	0.564941	0.282471	−	0.959276i	$-0.408846\pi$
$613$	8574.21	0.564941	0.282471	+	0.959276i	$0.408846\pi$
$614$	0	0
$615$	40332.6	2.64450
$616$	0	0
$617$	15340.4	1.00094	0.500470	−	0.865754i	$-0.333161\pi$
$617$	15340.4	1.00094	0.500470	+	0.865754i	$0.333161\pi$
$618$	0	0
$619$	−13102.2	−0.850763	−0.425382	−	0.905014i	$-0.639860\pi$
$619$	−13102.2	−0.850763	−0.425382	+	0.905014i	$0.639860\pi$
$620$	0	0
$621$	57300.0	3.70269
$622$	0	0
$623$	5522.78	0.355161
$624$	0	0
$625$	−17385.2	−1.11265
$626$	0	0
$627$	−49183.9	−3.13272
$628$	0	0
$629$	4416.20	0.279945
$630$	0	0
$631$	−1221.64	−0.0770727	−0.0385363	−	0.999257i	$-0.512270\pi$
$631$	−1221.64	−0.0770727	−0.0385363	+	0.999257i	$0.512270\pi$
$632$	0	0
$633$	20745.6	1.30263
$634$	0	0
$635$	12324.6	0.770215
$636$	0	0
$637$	2311.43	0.143771
$638$	0	0
$639$	46211.8	2.86089
$640$	0	0
$641$	12751.9	0.785759	0.392879	−	0.919590i	$-0.371479\pi$
$641$	12751.9	0.785759	0.392879	+	0.919590i	$0.371479\pi$
$642$	0	0
$643$	−9814.75	−0.601953	−0.300977	−	0.953631i	$-0.597313\pi$
$643$	−9814.75	−0.601953	−0.300977	+	0.953631i	$0.597313\pi$
$644$	0	0
$645$	−37141.9	−2.26738
$646$	0	0
$647$	−11367.5	−0.690730	−0.345365	−	0.938468i	$-0.612245\pi$
$647$	−11367.5	−0.690730	−0.345365	+	0.938468i	$0.612245\pi$
$648$	0	0
$649$	33039.0	1.99830
$650$	0	0
$651$	−15681.0	−0.944064
$652$	0	0
$653$	28698.5	1.71984	0.859921	−	0.510427i	$-0.170513\pi$
$653$	28698.5	1.71984	0.859921	+	0.510427i	$0.170513\pi$
$654$	0	0
$655$	−29105.4	−1.73625
$656$	0	0
$657$	31172.9	1.85110
$658$	0	0
$659$	21093.7	1.24688	0.623440	−	0.781871i	$-0.285735\pi$
$659$	21093.7	1.24688	0.623440	+	0.781871i	$0.285735\pi$
$660$	0	0
$661$	4059.93	0.238900	0.119450	−	0.992840i	$-0.461887\pi$
$661$	4059.93	0.238900	0.119450	+	0.992840i	$0.461887\pi$
$662$	0	0
$663$	1188.11	0.0695963
$664$	0	0
$665$	−5518.94	−0.321827
$666$	0	0
$667$	10301.0	0.597987
$668$	0	0
$669$	43625.1	2.52114
$670$	0	0
$671$	8851.73	0.509266
$672$	0	0
$673$	−12169.0	−0.697002	−0.348501	−	0.937308i	$-0.613309\pi$
$673$	−12169.0	−0.697002	−0.348501	+	0.937308i	$0.613309\pi$
$674$	0	0
$675$	5239.07	0.298744
$676$	0	0
$677$	−8946.47	−0.507889	−0.253944	−	0.967219i	$-0.581728\pi$
$677$	−8946.47	−0.507889	−0.253944	+	0.967219i	$0.581728\pi$
$678$	0	0
$679$	1863.67	0.105333
$680$	0	0
$681$	−19576.7	−1.10159
$682$	0	0
$683$	4967.22	0.278280	0.139140	−	0.990273i	$-0.455566\pi$
$683$	4967.22	0.278280	0.139140	+	0.990273i	$0.455566\pi$
$684$	0	0
$685$	−5808.14	−0.323967
$686$	0	0
$687$	−53705.3	−2.98251
$688$	0	0
$689$	−2635.68	−0.145735
$690$	0	0
$691$	8327.95	0.458481	0.229240	−	0.973370i	$-0.426376\pi$
$691$	8327.95	0.458481	0.229240	+	0.973370i	$0.426376\pi$
$692$	0	0
$693$	−22115.9	−1.21229
$694$	0	0
$695$	−8064.85	−0.440169
$696$	0	0
$697$	6136.14	0.333462
$698$	0	0
$699$	−10763.9	−0.582442
$700$	0	0
$701$	−892.094	−0.0480655	−0.0240328	−	0.999711i	$-0.507651\pi$
$701$	−892.094	−0.0480655	−0.0240328	+	0.999711i	$0.507651\pi$
$702$	0	0
$703$	21341.1	1.14494
$704$	0	0
$705$	516.905	0.0276139
$706$	0	0
$707$	−781.610	−0.0415778
$708$	0	0
$709$	−17934.3	−0.949980	−0.474990	−	0.879991i	$-0.657548\pi$
$709$	−17934.3	−0.949980	−0.474990	+	0.879991i	$0.657548\pi$
$710$	0	0
$711$	22421.4	1.18266
$712$	0	0
$713$	52168.0	2.74012
$714$	0	0
$715$	5621.17	0.294014
$716$	0	0
$717$	−32001.0	−1.66681
$718$	0	0
$719$	10742.1	0.557181	0.278591	−	0.960410i	$-0.410133\pi$
$719$	10742.1	0.557181	0.278591	+	0.960410i	$0.410133\pi$
$720$	0	0
$721$	8695.49	0.449150
$722$	0	0
$723$	−22712.2	−1.16829
$724$	0	0
$725$	941.846	0.0482473
$726$	0	0
$727$	−17505.8	−0.893060	−0.446530	−	0.894769i	$-0.647340\pi$
$727$	−17505.8	−0.893060	−0.446530	+	0.894769i	$0.647340\pi$
$728$	0	0
$729$	2989.93	0.151904
$730$	0	0
$731$	−5650.72	−0.285909
$732$	0	0
$733$	35040.9	1.76571	0.882854	−	0.469647i	$-0.155619\pi$
$733$	35040.9	1.76571	0.882854	+	0.469647i	$0.155619\pi$
$734$	0	0
$735$	34754.7	1.74415
$736$	0	0
$737$	31708.7	1.58481
$738$	0	0
$739$	−3604.25	−0.179411	−0.0897053	−	0.995968i	$-0.528593\pi$
$739$	−3604.25	−0.179411	−0.0897053	+	0.995968i	$0.528593\pi$
$740$	0	0
$741$	5741.50	0.284641
$742$	0	0
$743$	−23709.4	−1.17068	−0.585339	−	0.810789i	$-0.699039\pi$
$743$	−23709.4	−1.17068	−0.585339	+	0.810789i	$0.699039\pi$
$744$	0	0
$745$	8293.62	0.407859
$746$	0	0
$747$	71130.7	3.48398
$748$	0	0
$749$	2818.85	0.137515
$750$	0	0
$751$	19081.2	0.927142	0.463571	−	0.886060i	$-0.346568\pi$
$751$	19081.2	0.927142	0.463571	+	0.886060i	$0.346568\pi$
$752$	0	0
$753$	−23807.7	−1.15219
$754$	0	0
$755$	19818.3	0.955313
$756$	0	0
$757$	1901.77	0.0913089	0.0456545	−	0.998957i	$-0.485463\pi$
$757$	1901.77	0.0913089	0.0456545	+	0.998957i	$0.485463\pi$
$758$	0	0
$759$	105908.	5.06484
$760$	0	0
$761$	−10900.5	−0.519242	−0.259621	−	0.965711i	$-0.583598\pi$
$761$	−10900.5	−0.519242	−0.259621	+	0.965711i	$0.583598\pi$
$762$	0	0
$763$	−6929.45	−0.328785
$764$	0	0
$765$	12410.8	0.586552
$766$	0	0
$767$	−3856.82	−0.181567
$768$	0	0
$769$	38602.1	1.81018	0.905089	−	0.425221i	$-0.139804\pi$
$769$	38602.1	1.81018	0.905089	+	0.425221i	$0.139804\pi$
$770$	0	0
$771$	−53408.6	−2.49477
$772$	0	0
$773$	−20427.7	−0.950496	−0.475248	−	0.879852i	$-0.657642\pi$
$773$	−20427.7	−0.950496	−0.475248	+	0.879852i	$0.657642\pi$
$774$	0	0
$775$	4769.84	0.221081
$776$	0	0
$777$	13813.1	0.637763
$778$	0	0
$779$	29652.7	1.36382
$780$	0	0
$781$	47880.0	2.19370
$782$	0	0
$783$	18862.0	0.860887
$784$	0	0
$785$	6636.80	0.301755
$786$	0	0
$787$	9093.46	0.411877	0.205938	−	0.978565i	$-0.433975\pi$
$787$	9093.46	0.411877	0.205938	+	0.978565i	$0.433975\pi$
$788$	0	0
$789$	−72035.8	−3.25037
$790$	0	0
$791$	−4084.82	−0.183615
$792$	0	0
$793$	−1033.31	−0.0462722
$794$	0	0
$795$	−39630.2	−1.76797
$796$	0	0
$797$	−40927.1	−1.81896	−0.909480	−	0.415747i	$-0.863520\pi$
$797$	−40927.1	−1.81896	−0.909480	+	0.415747i	$0.863520\pi$
$798$	0	0
$799$	78.6413	0.00348201
$800$	0	0
$801$	−60016.4	−2.64741
$802$	0	0
$803$	32298.2	1.41940
$804$	0	0
$805$	11883.9	0.520316
$806$	0	0
$807$	44473.9	1.93997
$808$	0	0
$809$	18649.8	0.810499	0.405249	−	0.914206i	$-0.367185\pi$
$809$	18649.8	0.810499	0.405249	+	0.914206i	$0.367185\pi$
$810$	0	0
$811$	6202.34	0.268549	0.134275	−	0.990944i	$-0.457130\pi$
$811$	6202.34	0.268549	0.134275	+	0.990944i	$0.457130\pi$
$812$	0	0
$813$	67885.7	2.92848
$814$	0	0
$815$	8654.37	0.371962
$816$	0	0
$817$	−27306.9	−1.16934
$818$	0	0
$819$	2581.71	0.110149
$820$	0	0
$821$	12080.3	0.513527	0.256763	−	0.966474i	$-0.417344\pi$
$821$	12080.3	0.513527	0.256763	+	0.966474i	$0.417344\pi$
$822$	0	0
$823$	8772.26	0.371545	0.185773	−	0.982593i	$-0.440521\pi$
$823$	8772.26	0.371545	0.185773	+	0.982593i	$0.440521\pi$
$824$	0	0
$825$	9683.40	0.408646
$826$	0	0
$827$	1972.34	0.0829323	0.0414662	−	0.999140i	$-0.486797\pi$
$827$	1972.34	0.0829323	0.0414662	+	0.999140i	$0.486797\pi$
$828$	0	0
$829$	−28724.2	−1.20342	−0.601709	−	0.798715i	$-0.705513\pi$
$829$	−28724.2	−1.20342	−0.601709	+	0.798715i	$0.705513\pi$
$830$	0	0
$831$	19379.8	0.808998
$832$	0	0
$833$	5287.54	0.219931
$834$	0	0
$835$	−19267.8	−0.798550
$836$	0	0
$837$	95524.1	3.94480
$838$	0	0
$839$	−5994.39	−0.246662	−0.123331	−	0.992366i	$-0.539358\pi$
$839$	−5994.39	−0.246662	−0.123331	+	0.992366i	$0.539358\pi$
$840$	0	0
$841$	−20998.1	−0.860966
$842$	0	0
$843$	−85530.2	−3.49444
$844$	0	0
$845$	25447.8	1.03601
$846$	0	0
$847$	−15388.8	−0.624279
$848$	0	0
$849$	−48054.1	−1.94253
$850$	0	0
$851$	−45953.9	−1.85109
$852$	0	0
$853$	−33969.0	−1.36351	−0.681756	−	0.731580i	$-0.738783\pi$
$853$	−33969.0	−1.36351	−0.681756	+	0.731580i	$0.738783\pi$
$854$	0	0
$855$	59974.6	2.39893
$856$	0	0
$857$	−3606.62	−0.143757	−0.0718785	−	0.997413i	$-0.522899\pi$
$857$	−3606.62	−0.143757	−0.0718785	+	0.997413i	$0.522899\pi$
$858$	0	0
$859$	−11345.7	−0.450651	−0.225326	−	0.974284i	$-0.572345\pi$
$859$	−11345.7	−0.450651	−0.225326	+	0.974284i	$0.572345\pi$
$860$	0	0
$861$	19192.8	0.759685
$862$	0	0
$863$	6462.09	0.254892	0.127446	−	0.991845i	$-0.459322\pi$
$863$	6462.09	0.254892	0.127446	+	0.991845i	$0.459322\pi$
$864$	0	0
$865$	1801.38	0.0708078
$866$	0	0
$867$	2717.87	0.106464
$868$	0	0
$869$	23230.8	0.906848
$870$	0	0
$871$	−3701.53	−0.143997
$872$	0	0
$873$	−20252.6	−0.785162
$874$	0	0
$875$	−7310.88	−0.282460
$876$	0	0
$877$	−2142.14	−0.0824798	−0.0412399	−	0.999149i	$-0.513131\pi$
$877$	−2142.14	−0.0824798	−0.0412399	+	0.999149i	$0.513131\pi$
$878$	0	0
$879$	−65548.3	−2.51523
$880$	0	0
$881$	−28071.9	−1.07352	−0.536758	−	0.843736i	$-0.680351\pi$
$881$	−28071.9	−1.07352	−0.536758	+	0.843736i	$0.680351\pi$
$882$	0	0
$883$	32271.5	1.22992	0.614961	−	0.788558i	$-0.289172\pi$
$883$	32271.5	1.22992	0.614961	+	0.788558i	$0.289172\pi$
$884$	0	0
$885$	−57991.3	−2.20266
$886$	0	0
$887$	30956.0	1.17182	0.585908	−	0.810377i	$-0.300738\pi$
$887$	30956.0	1.17182	0.585908	+	0.810377i	$0.300738\pi$
$888$	0	0
$889$	5864.82	0.221260
$890$	0	0
$891$	88315.3	3.32062
$892$	0	0
$893$	380.031	0.0142411
$894$	0	0
$895$	−3856.82	−0.144044
$896$	0	0
$897$	−12363.2	−0.460195
$898$	0	0
$899$	17172.7	0.637087
$900$	0	0
$901$	−6029.28	−0.222935
$902$	0	0
$903$	−17674.5	−0.651351
$904$	0	0
$905$	−40736.0	−1.49626
$906$	0	0
$907$	−46253.3	−1.69329	−0.846646	−	0.532156i	$-0.821382\pi$
$907$	−46253.3	−1.69329	−0.846646	+	0.532156i	$0.821382\pi$
$908$	0	0
$909$	8493.80	0.309925
$910$	0	0
$911$	710.492	0.0258393	0.0129197	−	0.999917i	$-0.495887\pi$
$911$	710.492	0.0258393	0.0129197	+	0.999917i	$0.495887\pi$
$912$	0	0
$913$	73698.4	2.67148
$914$	0	0
$915$	−15536.9	−0.561348
$916$	0	0
$917$	−13850.2	−0.498772
$918$	0	0
$919$	19461.2	0.698548	0.349274	−	0.937021i	$-0.386428\pi$
$919$	19461.2	0.698548	0.349274	+	0.937021i	$0.386428\pi$
$920$	0	0
$921$	−28781.6	−1.02973
$922$	0	0
$923$	−5589.29	−0.199321
$924$	0	0
$925$	−4201.67	−0.149352
$926$	0	0
$927$	−94494.4	−3.34801
$928$	0	0
$929$	−19883.7	−0.702219	−0.351110	−	0.936334i	$-0.614196\pi$
$929$	−19883.7	−0.702219	−0.351110	+	0.936334i	$0.614196\pi$
$930$	0	0
$931$	25551.9	0.899493
$932$	0	0
$933$	43830.6	1.53800
$934$	0	0
$935$	12858.8	0.449762
$936$	0	0
$937$	15990.6	0.557514	0.278757	−	0.960362i	$-0.410078\pi$
$937$	15990.6	0.557514	0.278757	+	0.960362i	$0.410078\pi$
$938$	0	0
$939$	70692.8	2.45684
$940$	0	0
$941$	−34057.3	−1.17985	−0.589923	−	0.807459i	$-0.700842\pi$
$941$	−34057.3	−1.17985	−0.589923	+	0.807459i	$0.700842\pi$
$942$	0	0
$943$	−63851.3	−2.20497
$944$	0	0
$945$	21760.5	0.749069
$946$	0	0
$947$	−33033.0	−1.13350	−0.566752	−	0.823889i	$-0.691800\pi$
$947$	−33033.0	−1.13350	−0.566752	+	0.823889i	$0.691800\pi$
$948$	0	0
$949$	−3770.34	−0.128968
$950$	0	0
$951$	−8795.10	−0.299896
$952$	0	0
$953$	−29155.3	−0.991011	−0.495505	−	0.868605i	$-0.665017\pi$
$953$	−29155.3	−0.991011	−0.495505	+	0.868605i	$0.665017\pi$
$954$	0	0
$955$	−111.283	−0.00377072
$956$	0	0
$957$	34862.8	1.17759
$958$	0	0
$959$	−2763.88	−0.0930661
$960$	0	0
$961$	57177.6	1.91929
$962$	0	0
$963$	−30632.6	−1.02505
$964$	0	0
$965$	26104.1	0.870797
$966$	0	0
$967$	−39343.0	−1.30836	−0.654180	−	0.756339i	$-0.726986\pi$
$967$	−39343.0	−1.30836	−0.654180	+	0.756339i	$0.726986\pi$
$968$	0	0
$969$	13134.0	0.435424
$970$	0	0
$971$	17310.9	0.572125	0.286062	−	0.958211i	$-0.407654\pi$
$971$	17310.9	0.572125	0.286062	+	0.958211i	$0.407654\pi$
$972$	0	0
$973$	−3837.77	−0.126447
$974$	0	0
$975$	−1130.39	−0.0371298
$976$	0	0
$977$	45257.2	1.48199	0.740996	−	0.671509i	$-0.234354\pi$
$977$	45257.2	1.48199	0.740996	+	0.671509i	$0.234354\pi$
$978$	0	0
$979$	−62182.9	−2.03001
$980$	0	0
$981$	75302.8	2.45080
$982$	0	0
$983$	12052.6	0.391065	0.195532	−	0.980697i	$-0.437356\pi$
$983$	12052.6	0.391065	0.195532	+	0.980697i	$0.437356\pi$
$984$	0	0
$985$	53719.7	1.73772
$986$	0	0
$987$	245.976	0.00793263
$988$	0	0
$989$	58800.1	1.89053
$990$	0	0
$991$	30528.6	0.978580	0.489290	−	0.872121i	$-0.337256\pi$
$991$	30528.6	0.978580	0.489290	+	0.872121i	$0.337256\pi$
$992$	0	0
$993$	−22289.8	−0.712332
$994$	0	0
$995$	55432.4	1.76616
$996$	0	0
$997$	5307.64	0.168601	0.0843003	−	0.996440i	$-0.473135\pi$
$997$	5307.64	0.168601	0.0843003	+	0.996440i	$0.473135\pi$
$998$	0	0
$999$	−84145.6	−2.66491

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1088.4.a.bh.1.7		7
4.3	odd	2		1088.4.a.bg.1.1		7
8.3	odd	2		544.4.a.k.1.7	✓	7
8.5	even	2		544.4.a.l.1.1	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
544.4.a.k.1.7	✓	7	8.3	odd	2
544.4.a.l.1.1	yes	7	8.5	even	2
1088.4.a.bg.1.1		7	4.3	odd	2
1088.4.a.bh.1.7		7	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+9.40441 q^{3} -11.8817 q^{5} -5.65405 q^{7} +61.4429 q^{9} +O(q^{10})\)	\(q+9.40441 q^{3} -11.8817 q^{5} -5.65405 q^{7} +61.4429 q^{9} +63.6610 q^{11} -7.43149 q^{13} -111.740 q^{15} -17.0000 q^{17} -82.1519 q^{19} -53.1730 q^{21} +176.898 q^{23} +16.1742 q^{25} +323.916 q^{27} +58.2314 q^{29} +294.904 q^{31} +598.694 q^{33} +67.1796 q^{35} -259.776 q^{37} -69.8888 q^{39} -360.950 q^{41} +332.395 q^{43} -730.045 q^{45} -4.62596 q^{47} -311.032 q^{49} -159.875 q^{51} +354.664 q^{53} -756.399 q^{55} -772.591 q^{57} +518.983 q^{59} +139.045 q^{61} -347.402 q^{63} +88.2985 q^{65} +498.087 q^{67} +1663.62 q^{69} +752.109 q^{71} +507.347 q^{73} +152.109 q^{75} -359.943 q^{77} +364.914 q^{79} +1387.28 q^{81} +1157.67 q^{83} +201.988 q^{85} +547.632 q^{87} -976.782 q^{89} +42.0180 q^{91} +2773.40 q^{93} +976.103 q^{95} -329.616 q^{97} +3911.52 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{7} + 67 q^{9}+O(q^{10}) \) 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

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