LMFDB - Embedded newform 88.6.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [88,6,Mod(1,88)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(88, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("88.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 88 = 2^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	88.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:	$14.1137761435$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 145x^{2} + 57x + 4950 $ x^4 - x^3 - 145x^2 + 57x + 4950
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-9.03563$ of defining polynomial
Character	$\chi$	$=$	88.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.76524 q^{3} +18.8067 q^{5} +185.753 q^{7} -220.292 q^{9} -121.000 q^{11} +450.936 q^{13} +89.6184 q^{15} +1878.09 q^{17} +271.092 q^{19} +885.156 q^{21} +2180.44 q^{23} -2771.31 q^{25} -2207.70 q^{27} +7291.56 q^{29} +5237.55 q^{31} -576.594 q^{33} +3493.39 q^{35} +8120.04 q^{37} +2148.82 q^{39} +13312.9 q^{41} -21476.7 q^{43} -4142.97 q^{45} -28445.2 q^{47} +17697.1 q^{49} +8949.53 q^{51} -5605.15 q^{53} -2275.61 q^{55} +1291.82 q^{57} -15408.5 q^{59} -20621.8 q^{61} -40919.9 q^{63} +8480.62 q^{65} +15477.3 q^{67} +10390.3 q^{69} -34594.0 q^{71} -72344.7 q^{73} -13205.9 q^{75} -22476.1 q^{77} -13843.5 q^{79} +43010.9 q^{81} -14365.9 q^{83} +35320.6 q^{85} +34746.0 q^{87} +37056.0 q^{89} +83762.7 q^{91} +24958.2 q^{93} +5098.34 q^{95} -2544.22 q^{97} +26655.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 5 q^{3} + 93 q^{5} - 94 q^{7} + 591 q^{9} - 484 q^{11} + 230 q^{13} + 3227 q^{15} + 1856 q^{17} - 40 q^{19} + 6114 q^{21} + 1515 q^{23} + 15147 q^{25} + 11605 q^{27} + 11390 q^{29} + 1339 q^{31} + 605 q^{33}+ \cdots - 71511 q^{99}+O(q^{100}) $ 4 * q - 5 * q^3 + 93 * q^5 - 94 * q^7 + 591 * q^9 - 484 * q^11 + 230 * q^13 + 3227 * q^15 + 1856 * q^17 - 40 * q^19 + 6114 * q^21 + 1515 * q^23 + 15147 * q^25 + 11605 * q^27 + 11390 * q^29 + 1339 * q^31 + 605 * q^33 - 5342 * q^35 + 8003 * q^37 - 24968 * q^39 + 26070 * q^41 + 8022 * q^43 + 50648 * q^45 - 31984 * q^47 + 13232 * q^49 - 69778 * q^51 + 9148 * q^53 - 11253 * q^55 - 51172 * q^57 - 101047 * q^59 + 9618 * q^61 - 78332 * q^63 - 22540 * q^65 - 101403 * q^67 - 73769 * q^69 - 18367 * q^71 + 73354 * q^73 - 53664 * q^75 + 11374 * q^77 - 92026 * q^79 + 20116 * q^81 + 62246 * q^83 - 246030 * q^85 - 55856 * q^87 + 138439 * q^89 + 848 * q^91 + 63791 * q^93 + 284820 * q^95 + 181869 * q^97 - 71511 * q^99

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
$3$	4.76524	0.305690	0.152845	−	0.988250i	$-0.451156\pi$
$3$	4.76524	0.305690	0.152845	+	0.988250i	$0.451156\pi$
$4$	0	0
$5$	18.8067	0.336424	0.168212	−	0.985751i	$-0.446201\pi$
$5$	18.8067	0.336424	0.168212	+	0.985751i	$0.446201\pi$
$6$	0	0
$7$	185.753	1.43282	0.716408	−	0.697682i	$-0.245785\pi$
$7$	185.753	1.43282	0.716408	+	0.697682i	$0.245785\pi$
$8$	0	0
$9$	−220.292	−0.906553
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	450.936	0.740043	0.370021	−	0.929023i	$-0.379350\pi$
$13$	450.936	0.740043	0.370021	+	0.929023i	$0.379350\pi$
$14$	0	0
$15$	89.6184	0.102842
$16$	0	0
$17$	1878.09	1.57613	0.788067	−	0.615589i	$-0.211082\pi$
$17$	1878.09	1.57613	0.788067	+	0.615589i	$0.211082\pi$
$18$	0	0
$19$	271.092	0.172279	0.0861395	−	0.996283i	$-0.472547\pi$
$19$	271.092	0.172279	0.0861395	+	0.996283i	$0.472547\pi$
$20$	0	0
$21$	885.156	0.437998
$22$	0	0
$23$	2180.44	0.859458	0.429729	−	0.902958i	$-0.358609\pi$
$23$	2180.44	0.859458	0.429729	+	0.902958i	$0.358609\pi$
$24$	0	0
$25$	−2771.31	−0.886819
$26$	0	0
$27$	−2207.70	−0.582815
$28$	0	0
$29$	7291.56	1.61000	0.804999	−	0.593276i	$-0.202166\pi$
$29$	7291.56	1.61000	0.804999	+	0.593276i	$0.202166\pi$
$30$	0	0
$31$	5237.55	0.978867	0.489434	−	0.872041i	$-0.337204\pi$
$31$	5237.55	0.978867	0.489434	+	0.872041i	$0.337204\pi$
$32$	0	0
$33$	−576.594	−0.0921691
$34$	0	0
$35$	3493.39	0.482034
$36$	0	0
$37$	8120.04	0.975110	0.487555	−	0.873092i	$-0.337889\pi$
$37$	8120.04	0.975110	0.487555	+	0.873092i	$0.337889\pi$
$38$	0	0
$39$	2148.82	0.226224
$40$	0	0
$41$	13312.9	1.23684	0.618419	−	0.785848i	$-0.287773\pi$
$41$	13312.9	1.23684	0.618419	+	0.785848i	$0.287773\pi$
$42$	0	0
$43$	−21476.7	−1.77132	−0.885658	−	0.464339i	$-0.846292\pi$
$43$	−21476.7	−1.77132	−0.885658	+	0.464339i	$0.846292\pi$
$44$	0	0
$45$	−4142.97	−0.304987
$46$	0	0
$47$	−28445.2	−1.87829	−0.939147	−	0.343515i	$-0.888382\pi$
$47$	−28445.2	−1.87829	−0.939147	+	0.343515i	$0.888382\pi$
$48$	0	0
$49$	17697.1	1.05296
$50$	0	0
$51$	8949.53	0.481809
$52$	0	0
$53$	−5605.15	−0.274093	−0.137046	−	0.990565i	$-0.543761\pi$
$53$	−5605.15	−0.274093	−0.137046	+	0.990565i	$0.543761\pi$
$54$	0	0
$55$	−2275.61	−0.101436
$56$	0	0
$57$	1291.82	0.0526640
$58$	0	0
$59$	−15408.5	−0.576275	−0.288137	−	0.957589i	$-0.593036\pi$
$59$	−15408.5	−0.576275	−0.288137	+	0.957589i	$0.593036\pi$
$60$	0	0
$61$	−20621.8	−0.709581	−0.354790	−	0.934946i	$-0.615448\pi$
$61$	−20621.8	−0.709581	−0.354790	+	0.934946i	$0.615448\pi$
$62$	0	0
$63$	−40919.9	−1.29892
$64$	0	0
$65$	8480.62	0.248968
$66$	0	0
$67$	15477.3	0.421219	0.210609	−	0.977570i	$-0.432455\pi$
$67$	15477.3	0.421219	0.210609	+	0.977570i	$0.432455\pi$
$68$	0	0
$69$	10390.3	0.262728
$70$	0	0
$71$	−34594.0	−0.814431	−0.407216	−	0.913332i	$-0.633500\pi$
$71$	−34594.0	−0.814431	−0.407216	+	0.913332i	$0.633500\pi$
$72$	0	0
$73$	−72344.7	−1.58891	−0.794456	−	0.607322i	$-0.792244\pi$
$73$	−72344.7	−1.58891	−0.794456	+	0.607322i	$0.792244\pi$
$74$	0	0
$75$	−13205.9	−0.271092
$76$	0	0
$77$	−22476.1	−0.432010
$78$	0	0
$79$	−13843.5	−0.249562	−0.124781	−	0.992184i	$-0.539823\pi$
$79$	−13843.5	−0.249562	−0.124781	+	0.992184i	$0.539823\pi$
$80$	0	0
$81$	43010.9	0.728393
$82$	0	0
$83$	−14365.9	−0.228895	−0.114447	−	0.993429i	$-0.536510\pi$
$83$	−14365.9	−0.228895	−0.114447	+	0.993429i	$0.536510\pi$
$84$	0	0
$85$	35320.6	0.530250
$86$	0	0
$87$	34746.0	0.492161
$88$	0	0
$89$	37056.0	0.495888	0.247944	−	0.968774i	$-0.420245\pi$
$89$	37056.0	0.495888	0.247944	+	0.968774i	$0.420245\pi$
$90$	0	0
$91$	83762.7	1.06034
$92$	0	0
$93$	24958.2	0.299230
$94$	0	0
$95$	5098.34	0.0579588
$96$	0	0
$97$	−2544.22	−0.0274552	−0.0137276	−	0.999906i	$-0.504370\pi$
$97$	−2544.22	−0.0274552	−0.0137276	+	0.999906i	$0.504370\pi$
$98$	0	0
$99$	26655.4	0.273336
$100$	0	0
$101$	−43827.3	−0.427506	−0.213753	−	0.976888i	$-0.568569\pi$
$101$	−43827.3	−0.427506	−0.213753	+	0.976888i	$0.568569\pi$
$102$	0	0
$103$	−112331.	−1.04329	−0.521645	−	0.853163i	$-0.674681\pi$
$103$	−112331.	−1.04329	−0.521645	+	0.853163i	$0.674681\pi$
$104$	0	0
$105$	16646.9	0.147353
$106$	0	0
$107$	−9509.43	−0.0802962	−0.0401481	−	0.999194i	$-0.512783\pi$
$107$	−9509.43	−0.0802962	−0.0401481	+	0.999194i	$0.512783\pi$
$108$	0	0
$109$	−87901.4	−0.708646	−0.354323	−	0.935123i	$-0.615289\pi$
$109$	−87901.4	−0.708646	−0.354323	+	0.935123i	$0.615289\pi$
$110$	0	0
$111$	38693.9	0.298082
$112$	0	0
$113$	222724.	1.64086	0.820430	−	0.571747i	$-0.193734\pi$
$113$	222724.	1.64086	0.820430	+	0.571747i	$0.193734\pi$
$114$	0	0
$115$	41006.8	0.289142
$116$	0	0
$117$	−99337.9	−0.670889
$118$	0	0
$119$	348860.	2.25831
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	63439.2	0.378090
$124$	0	0
$125$	−110890.	−0.634772
$126$	0	0
$127$	281292.	1.54756	0.773781	−	0.633453i	$-0.218363\pi$
$127$	281292.	1.54756	0.773781	+	0.633453i	$0.218363\pi$
$128$	0	0
$129$	−102341.	−0.541474
$130$	0	0
$131$	56794.9	0.289155	0.144578	−	0.989493i	$-0.453818\pi$
$131$	56794.9	0.289155	0.144578	+	0.989493i	$0.453818\pi$
$132$	0	0
$133$	50356.1	0.246844
$134$	0	0
$135$	−41519.5	−0.196073
$136$	0	0
$137$	97777.6	0.445080	0.222540	−	0.974924i	$-0.428565\pi$
$137$	97777.6	0.445080	0.222540	+	0.974924i	$0.428565\pi$
$138$	0	0
$139$	−136419.	−0.598875	−0.299438	−	0.954116i	$-0.596799\pi$
$139$	−136419.	−0.598875	−0.299438	+	0.954116i	$0.596799\pi$
$140$	0	0
$141$	−135548.	−0.574176
$142$	0	0
$143$	−54563.3	−0.223131
$144$	0	0
$145$	137130.	0.541642
$146$	0	0
$147$	84330.9	0.321879
$148$	0	0
$149$	−32856.5	−0.121243	−0.0606213	−	0.998161i	$-0.519308\pi$
$149$	−32856.5	−0.121243	−0.0606213	+	0.998161i	$0.519308\pi$
$150$	0	0
$151$	−96344.0	−0.343861	−0.171930	−	0.985109i	$-0.555000\pi$
$151$	−96344.0	−0.343861	−0.171930	+	0.985109i	$0.555000\pi$
$152$	0	0
$153$	−413728.	−1.42885
$154$	0	0
$155$	98500.9	0.329315
$156$	0	0
$157$	−464474.	−1.50388	−0.751938	−	0.659234i	$-0.770881\pi$
$157$	−464474.	−1.50388	−0.751938	+	0.659234i	$0.770881\pi$
$158$	0	0
$159$	−26709.9	−0.0837874
$160$	0	0
$161$	405023.	1.23144
$162$	0	0
$163$	−553151.	−1.63070	−0.815351	−	0.578967i	$-0.803456\pi$
$163$	−553151.	−1.63070	−0.815351	+	0.578967i	$0.803456\pi$
$164$	0	0
$165$	−10843.8	−0.0310079
$166$	0	0
$167$	527319.	1.46313	0.731564	−	0.681773i	$-0.238791\pi$
$167$	527319.	1.46313	0.731564	+	0.681773i	$0.238791\pi$
$168$	0	0
$169$	−167949.	−0.452336
$170$	0	0
$171$	−59719.5	−0.156180
$172$	0	0
$173$	−267245.	−0.678881	−0.339440	−	0.940628i	$-0.610238\pi$
$173$	−267245.	−0.678881	−0.339440	+	0.940628i	$0.610238\pi$
$174$	0	0
$175$	−514778.	−1.27065
$176$	0	0
$177$	−73425.1	−0.176162
$178$	0	0
$179$	225715.	0.526537	0.263268	−	0.964723i	$-0.415200\pi$
$179$	225715.	0.526537	0.263268	+	0.964723i	$0.415200\pi$
$180$	0	0
$181$	3570.84	0.00810165	0.00405083	−	0.999992i	$-0.498711\pi$
$181$	3570.84	0.00810165	0.00405083	+	0.999992i	$0.498711\pi$
$182$	0	0
$183$	−98267.8	−0.216912
$184$	0	0
$185$	152711.	0.328051
$186$	0	0
$187$	−227248.	−0.475222
$188$	0	0
$189$	−410086.	−0.835066
$190$	0	0
$191$	−505611.	−1.00284	−0.501421	−	0.865203i	$-0.667189\pi$
$191$	−505611.	−1.00284	−0.501421	+	0.865203i	$0.667189\pi$
$192$	0	0
$193$	642337.	1.24128	0.620640	−	0.784096i	$-0.286873\pi$
$193$	642337.	1.24128	0.620640	+	0.784096i	$0.286873\pi$
$194$	0	0
$195$	40412.2	0.0761072
$196$	0	0
$197$	−809688.	−1.48646	−0.743228	−	0.669038i	$-0.766706\pi$
$197$	−809688.	−1.48646	−0.743228	+	0.669038i	$0.766706\pi$
$198$	0	0
$199$	−78134.0	−0.139864	−0.0699322	−	0.997552i	$-0.522278\pi$
$199$	−78134.0	−0.139864	−0.0699322	+	0.997552i	$0.522278\pi$
$200$	0	0
$201$	73753.0	0.128762
$202$	0	0
$203$	1.35443e6	2.30683
$204$	0	0
$205$	250372.	0.416103
$206$	0	0
$207$	−480334.	−0.779144
$208$	0	0
$209$	−32802.1	−0.0519441
$210$	0	0
$211$	−1.09424e6	−1.69202	−0.846012	−	0.533164i	$-0.821003\pi$
$211$	−1.09424e6	−1.69202	−0.846012	+	0.533164i	$0.821003\pi$
$212$	0	0
$213$	−164849.	−0.248964
$214$	0	0
$215$	−403905.	−0.595913
$216$	0	0
$217$	972889.	1.40254
$218$	0	0
$219$	−344740.	−0.485715
$220$	0	0
$221$	846897.	1.16641
$222$	0	0
$223$	1.11545e6	1.50206	0.751031	−	0.660267i	$-0.229557\pi$
$223$	1.11545e6	1.50206	0.751031	+	0.660267i	$0.229557\pi$
$224$	0	0
$225$	610498.	0.803949
$226$	0	0
$227$	−194126.	−0.250046	−0.125023	−	0.992154i	$-0.539900\pi$
$227$	−194126.	−0.250046	−0.125023	+	0.992154i	$0.539900\pi$
$228$	0	0
$229$	796852.	1.00413	0.502064	−	0.864830i	$-0.332574\pi$
$229$	796852.	1.00413	0.502064	+	0.864830i	$0.332574\pi$
$230$	0	0
$231$	−107104.	−0.132061
$232$	0	0
$233$	1.51904e6	1.83307	0.916533	−	0.399959i	$-0.130976\pi$
$233$	1.51904e6	1.83307	0.916533	+	0.399959i	$0.130976\pi$
$234$	0	0
$235$	−534959.	−0.631904
$236$	0	0
$237$	−65967.7	−0.0762887
$238$	0	0
$239$	−558399.	−0.632339	−0.316170	−	0.948703i	$-0.602397\pi$
$239$	−558399.	−0.632339	−0.316170	+	0.948703i	$0.602397\pi$
$240$	0	0
$241$	−311218.	−0.345161	−0.172581	−	0.984995i	$-0.555210\pi$
$241$	−311218.	−0.345161	−0.172581	+	0.984995i	$0.555210\pi$
$242$	0	0
$243$	741428.	0.805477
$244$	0	0
$245$	332824.	0.354241
$246$	0	0
$247$	122245.	0.127494
$248$	0	0
$249$	−68456.7	−0.0699710
$250$	0	0
$251$	−515782.	−0.516752	−0.258376	−	0.966044i	$-0.583187\pi$
$251$	−515782.	−0.516752	−0.258376	+	0.966044i	$0.583187\pi$
$252$	0	0
$253$	−263833.	−0.259136
$254$	0	0
$255$	168311.	0.162092
$256$	0	0
$257$	−677970.	−0.640292	−0.320146	−	0.947368i	$-0.603732\pi$
$257$	−677970.	−0.640292	−0.320146	+	0.947368i	$0.603732\pi$
$258$	0	0
$259$	1.50832e6	1.39715
$260$	0	0
$261$	−1.60628e6	−1.45955
$262$	0	0
$263$	665288.	0.593090	0.296545	−	0.955019i	$-0.404166\pi$
$263$	665288.	0.593090	0.296545	+	0.955019i	$0.404166\pi$
$264$	0	0
$265$	−105414.	−0.0922114
$266$	0	0
$267$	176581.	0.151588
$268$	0	0
$269$	2.24458e6	1.89128	0.945639	−	0.325219i	$-0.105438\pi$
$269$	2.24458e6	1.89128	0.945639	+	0.325219i	$0.105438\pi$
$270$	0	0
$271$	−77270.8	−0.0639134	−0.0319567	−	0.999489i	$-0.510174\pi$
$271$	−77270.8	−0.0639134	−0.0319567	+	0.999489i	$0.510174\pi$
$272$	0	0
$273$	399149.	0.324137
$274$	0	0
$275$	335328.	0.267386
$276$	0	0
$277$	−1.98599e6	−1.55517	−0.777584	−	0.628779i	$-0.783555\pi$
$277$	−1.98599e6	−1.55517	−0.777584	+	0.628779i	$0.783555\pi$
$278$	0	0
$279$	−1.15379e6	−0.887396
$280$	0	0
$281$	1.94967e6	1.47298	0.736489	−	0.676449i	$-0.236482\pi$
$281$	1.94967e6	1.47298	0.736489	+	0.676449i	$0.236482\pi$
$282$	0	0
$283$	−2.48694e6	−1.84586	−0.922930	−	0.384968i	$-0.874212\pi$
$283$	−2.48694e6	−1.84586	−0.922930	+	0.384968i	$0.874212\pi$
$284$	0	0
$285$	24294.8	0.0177175
$286$	0	0
$287$	2.47291e6	1.77216
$288$	0	0
$289$	2.10735e6	1.48420
$290$	0	0
$291$	−12123.8	−0.00839280
$292$	0	0
$293$	−1.07188e6	−0.729417	−0.364709	−	0.931122i	$-0.618831\pi$
$293$	−1.07188e6	−0.729417	−0.364709	+	0.931122i	$0.618831\pi$
$294$	0	0
$295$	−289782.	−0.193873
$296$	0	0
$297$	267132.	0.175725
$298$	0	0
$299$	983240.	0.636036
$300$	0	0
$301$	−3.98935e6	−2.53797
$302$	0	0
$303$	−208848.	−0.130684
$304$	0	0
$305$	−387828.	−0.238720
$306$	0	0
$307$	−1.96522e6	−1.19005	−0.595025	−	0.803707i	$-0.702858\pi$
$307$	−1.96522e6	−1.19005	−0.595025	+	0.803707i	$0.702858\pi$
$308$	0	0
$309$	−535282.	−0.318923
$310$	0	0
$311$	301849.	0.176966	0.0884828	−	0.996078i	$-0.471798\pi$
$311$	301849.	0.176966	0.0884828	+	0.996078i	$0.471798\pi$
$312$	0	0
$313$	1.02713e6	0.592605	0.296302	−	0.955094i	$-0.404246\pi$
$313$	1.02713e6	0.592605	0.296302	+	0.955094i	$0.404246\pi$
$314$	0	0
$315$	−769568.	−0.436989
$316$	0	0
$317$	1.00339e6	0.560815	0.280408	−	0.959881i	$-0.409530\pi$
$317$	1.00339e6	0.560815	0.280408	+	0.959881i	$0.409530\pi$
$318$	0	0
$319$	−882279.	−0.485433
$320$	0	0
$321$	−45314.7	−0.0245458
$322$	0	0
$323$	509134.	0.271535
$324$	0	0
$325$	−1.24968e6	−0.656284
$326$	0	0
$327$	−418871.	−0.216626
$328$	0	0
$329$	−5.28377e6	−2.69125
$330$	0	0
$331$	754787.	0.378665	0.189332	−	0.981913i	$-0.439368\pi$
$331$	754787.	0.378665	0.189332	+	0.981913i	$0.439368\pi$
$332$	0	0
$333$	−1.78878e6	−0.883990
$334$	0	0
$335$	291076.	0.141708
$336$	0	0
$337$	131673.	0.0631571	0.0315785	−	0.999501i	$-0.489947\pi$
$337$	131673.	0.0631571	0.0315785	+	0.999501i	$0.489947\pi$
$338$	0	0
$339$	1.06133e6	0.501595
$340$	0	0
$341$	−633743.	−0.295140
$342$	0	0
$343$	165337.	0.0758813
$344$	0	0
$345$	195407.	0.0883880
$346$	0	0
$347$	4.10512e6	1.83021	0.915107	−	0.403212i	$-0.132106\pi$
$347$	4.10512e6	1.83021	0.915107	+	0.403212i	$0.132106\pi$
$348$	0	0
$349$	3.29395e6	1.44762	0.723808	−	0.690002i	$-0.242390\pi$
$349$	3.29395e6	1.44762	0.723808	+	0.690002i	$0.242390\pi$
$350$	0	0
$351$	−995532.	−0.431308
$352$	0	0
$353$	−2.30224e6	−0.983363	−0.491682	−	0.870775i	$-0.663618\pi$
$353$	−2.30224e6	−0.983363	−0.491682	+	0.870775i	$0.663618\pi$
$354$	0	0
$355$	−650598.	−0.273994
$356$	0	0
$357$	1.66240e6	0.690343
$358$	0	0
$359$	776222.	0.317870	0.158935	−	0.987289i	$-0.449194\pi$
$359$	776222.	0.317870	0.158935	+	0.987289i	$0.449194\pi$
$360$	0	0
$361$	−2.40261e6	−0.970320
$362$	0	0
$363$	69767.9	0.0277900
$364$	0	0
$365$	−1.36056e6	−0.534548
$366$	0	0
$367$	−2.51246e6	−0.973721	−0.486860	−	0.873480i	$-0.661858\pi$
$367$	−2.51246e6	−0.973721	−0.486860	+	0.873480i	$0.661858\pi$
$368$	0	0
$369$	−2.93273e6	−1.12126
$370$	0	0
$371$	−1.04117e6	−0.392724
$372$	0	0
$373$	−2.74730e6	−1.02243	−0.511216	−	0.859452i	$-0.670805\pi$
$373$	−2.74730e6	−1.02243	−0.511216	+	0.859452i	$0.670805\pi$
$374$	0	0
$375$	−528418.	−0.194043
$376$	0	0
$377$	3.28803e6	1.19147
$378$	0	0
$379$	−1.20072e6	−0.429381	−0.214690	−	0.976682i	$-0.568874\pi$
$379$	−1.20072e6	−0.429381	−0.214690	+	0.976682i	$0.568874\pi$
$380$	0	0
$381$	1.34042e6	0.473075
$382$	0	0
$383$	−2.18649e6	−0.761640	−0.380820	−	0.924649i	$-0.624358\pi$
$383$	−2.18649e6	−0.761640	−0.380820	+	0.924649i	$0.624358\pi$
$384$	0	0
$385$	−422701.	−0.145339
$386$	0	0
$387$	4.73115e6	1.60579
$388$	0	0
$389$	2.84332e6	0.952689	0.476344	−	0.879259i	$-0.341962\pi$
$389$	2.84332e6	0.952689	0.476344	+	0.879259i	$0.341962\pi$
$390$	0	0
$391$	4.09505e6	1.35462
$392$	0	0
$393$	270641.	0.0883919
$394$	0	0
$395$	−260351.	−0.0839588
$396$	0	0
$397$	6.04554e6	1.92512	0.962562	−	0.271062i	$-0.0873749\pi$
$397$	6.04554e6	1.92512	0.962562	+	0.271062i	$0.0873749\pi$
$398$	0	0
$399$	239959.	0.0754578
$400$	0	0
$401$	1.19358e6	0.370672	0.185336	−	0.982675i	$-0.440663\pi$
$401$	1.19358e6	0.370672	0.185336	+	0.982675i	$0.440663\pi$
$402$	0	0
$403$	2.36180e6	0.724404
$404$	0	0
$405$	808892.	0.245049
$406$	0	0
$407$	−982525.	−0.294007
$408$	0	0
$409$	3.11380e6	0.920412	0.460206	−	0.887812i	$-0.347776\pi$
$409$	3.11380e6	0.920412	0.460206	+	0.887812i	$0.347776\pi$
$410$	0	0
$411$	465934.	0.136057
$412$	0	0
$413$	−2.86217e6	−0.825695
$414$	0	0
$415$	−270174.	−0.0770058
$416$	0	0
$417$	−650067.	−0.183070
$418$	0	0
$419$	4.09395e6	1.13922	0.569609	−	0.821916i	$-0.307095\pi$
$419$	4.09395e6	1.13922	0.569609	+	0.821916i	$0.307095\pi$
$420$	0	0
$421$	3.07486e6	0.845513	0.422757	−	0.906243i	$-0.361063\pi$
$421$	3.07486e6	0.845513	0.422757	+	0.906243i	$0.361063\pi$
$422$	0	0
$423$	6.26625e6	1.70277
$424$	0	0
$425$	−5.20476e6	−1.39775
$426$	0	0
$427$	−3.83056e6	−1.01670
$428$	0	0
$429$	−260007.	−0.0682091
$430$	0	0
$431$	1.15761e6	0.300172	0.150086	−	0.988673i	$-0.452045\pi$
$431$	1.15761e6	0.300172	0.150086	+	0.988673i	$0.452045\pi$
$432$	0	0
$433$	−789612.	−0.202392	−0.101196	−	0.994866i	$-0.532267\pi$
$433$	−789612.	−0.202392	−0.101196	+	0.994866i	$0.532267\pi$
$434$	0	0
$435$	653458.	0.165575
$436$	0	0
$437$	591099.	0.148067
$438$	0	0
$439$	687808.	0.170336	0.0851679	−	0.996367i	$-0.472857\pi$
$439$	687808.	0.170336	0.0851679	+	0.996367i	$0.472857\pi$
$440$	0	0
$441$	−3.89854e6	−0.954564
$442$	0	0
$443$	−4.14991e6	−1.00468	−0.502342	−	0.864669i	$-0.667528\pi$
$443$	−4.14991e6	−1.00468	−0.502342	+	0.864669i	$0.667528\pi$
$444$	0	0
$445$	696901.	0.166829
$446$	0	0
$447$	−156569.	−0.0370627
$448$	0	0
$449$	5.35141e6	1.25272	0.626358	−	0.779536i	$-0.284545\pi$
$449$	5.35141e6	1.25272	0.626358	+	0.779536i	$0.284545\pi$
$450$	0	0
$451$	−1.61086e6	−0.372921
$452$	0	0
$453$	−459102.	−0.105115
$454$	0	0
$455$	1.57530e6	0.356726
$456$	0	0
$457$	3.23593e6	0.724784	0.362392	−	0.932026i	$-0.381960\pi$
$457$	3.23593e6	0.724784	0.362392	+	0.932026i	$0.381960\pi$
$458$	0	0
$459$	−4.14625e6	−0.918594
$460$	0	0
$461$	−3.96957e6	−0.869943	−0.434971	−	0.900444i	$-0.643242\pi$
$461$	−3.96957e6	−0.869943	−0.434971	+	0.900444i	$0.643242\pi$
$462$	0	0
$463$	−922430.	−0.199977	−0.0999887	−	0.994989i	$-0.531881\pi$
$463$	−922430.	−0.199977	−0.0999887	+	0.994989i	$0.531881\pi$
$464$	0	0
$465$	469381.	0.100668
$466$	0	0
$467$	−982248.	−0.208415	−0.104208	−	0.994556i	$-0.533231\pi$
$467$	−982248.	−0.208415	−0.104208	+	0.994556i	$0.533231\pi$
$468$	0	0
$469$	2.87495e6	0.603529
$470$	0	0
$471$	−2.21333e6	−0.459720
$472$	0	0
$473$	2.59868e6	0.534072
$474$	0	0
$475$	−751279.	−0.152780
$476$	0	0
$477$	1.23477e6	0.248480
$478$	0	0
$479$	−1.13568e6	−0.226161	−0.113081	−	0.993586i	$-0.536072\pi$
$479$	−1.13568e6	−0.226161	−0.113081	+	0.993586i	$0.536072\pi$
$480$	0	0
$481$	3.66162e6	0.721624
$482$	0	0
$483$	1.93003e6	0.376440
$484$	0	0
$485$	−47848.3	−0.00923660
$486$	0	0
$487$	3.03084e6	0.579083	0.289542	−	0.957165i	$-0.406497\pi$
$487$	3.03084e6	0.579083	0.289542	+	0.957165i	$0.406497\pi$
$488$	0	0
$489$	−2.63590e6	−0.498490
$490$	0	0
$491$	−7.93792e6	−1.48595	−0.742973	−	0.669322i	$-0.766585\pi$
$491$	−7.93792e6	−1.48595	−0.742973	+	0.669322i	$0.766585\pi$
$492$	0	0
$493$	1.36942e7	2.53757
$494$	0	0
$495$	501300.	0.0919569
$496$	0	0
$497$	−6.42592e6	−1.16693
$498$	0	0
$499$	−4.33152e6	−0.778734	−0.389367	−	0.921083i	$-0.627306\pi$
$499$	−4.33152e6	−0.778734	−0.389367	+	0.921083i	$0.627306\pi$
$500$	0	0
$501$	2.51280e6	0.447264
$502$	0	0
$503$	−4.02378e6	−0.709112	−0.354556	−	0.935035i	$-0.615368\pi$
$503$	−4.02378e6	−0.709112	−0.354556	+	0.935035i	$0.615368\pi$
$504$	0	0
$505$	−824247.	−0.143823
$506$	0	0
$507$	−800319.	−0.138275
$508$	0	0
$509$	−1.35686e6	−0.232135	−0.116068	−	0.993241i	$-0.537029\pi$
$509$	−1.35686e6	−0.232135	−0.116068	+	0.993241i	$0.537029\pi$
$510$	0	0
$511$	−1.34382e7	−2.27662
$512$	0	0
$513$	−598489.	−0.100407
$514$	0	0
$515$	−2.11256e6	−0.350988
$516$	0	0
$517$	3.44186e6	0.566327
$518$	0	0
$519$	−1.27348e6	−0.207527
$520$	0	0
$521$	−1.13437e7	−1.83088	−0.915439	−	0.402456i	$-0.868157\pi$
$521$	−1.13437e7	−1.83088	−0.915439	+	0.402456i	$0.868157\pi$
$522$	0	0
$523$	−8.38569e6	−1.34055	−0.670277	−	0.742111i	$-0.733825\pi$
$523$	−8.38569e6	−1.34055	−0.670277	+	0.742111i	$0.733825\pi$
$524$	0	0
$525$	−2.45304e6	−0.388425
$526$	0	0
$527$	9.83657e6	1.54283
$528$	0	0
$529$	−1.68203e6	−0.261333
$530$	0	0
$531$	3.39437e6	0.522424
$532$	0	0
$533$	6.00327e6	0.915314
$534$	0	0
$535$	−178841.	−0.0270136
$536$	0	0
$537$	1.07559e6	0.160957
$538$	0	0
$539$	−2.14135e6	−0.317479
$540$	0	0
$541$	−1.12820e7	−1.65726	−0.828632	−	0.559794i	$-0.810880\pi$
$541$	−1.12820e7	−1.65726	−0.828632	+	0.559794i	$0.810880\pi$
$542$	0	0
$543$	17015.9	0.00247660
$544$	0	0
$545$	−1.65313e6	−0.238406
$546$	0	0
$547$	1.10783e6	0.158308	0.0791542	−	0.996862i	$-0.474778\pi$
$547$	1.10783e6	0.158308	0.0791542	+	0.996862i	$0.474778\pi$
$548$	0	0
$549$	4.54283e6	0.643273
$550$	0	0
$551$	1.97668e6	0.277369
$552$	0	0
$553$	−2.57147e6	−0.357576
$554$	0	0
$555$	727705.	0.100282
$556$	0	0
$557$	1.33004e6	0.181646	0.0908229	−	0.995867i	$-0.471050\pi$
$557$	1.33004e6	0.181646	0.0908229	+	0.995867i	$0.471050\pi$
$558$	0	0
$559$	−9.68461e6	−1.31085
$560$	0	0
$561$	−1.08289e6	−0.145271
$562$	0	0
$563$	5.71080e6	0.759322	0.379661	−	0.925126i	$-0.376041\pi$
$563$	5.71080e6	0.759322	0.379661	+	0.925126i	$0.376041\pi$
$564$	0	0
$565$	4.18871e6	0.552025
$566$	0	0
$567$	7.98939e6	1.04365
$568$	0	0
$569$	−1.89022e6	−0.244755	−0.122378	−	0.992484i	$-0.539052\pi$
$569$	−1.89022e6	−0.244755	−0.122378	+	0.992484i	$0.539052\pi$
$570$	0	0
$571$	8.50941e6	1.09222	0.546109	−	0.837714i	$-0.316108\pi$
$571$	8.50941e6	1.09222	0.546109	+	0.837714i	$0.316108\pi$
$572$	0	0
$573$	−2.40936e6	−0.306559
$574$	0	0
$575$	−6.04267e6	−0.762183
$576$	0	0
$577$	−637951.	−0.0797715	−0.0398857	−	0.999204i	$-0.512699\pi$
$577$	−637951.	−0.0797715	−0.0398857	+	0.999204i	$0.512699\pi$
$578$	0	0
$579$	3.06089e6	0.379447
$580$	0	0
$581$	−2.66850e6	−0.327964
$582$	0	0
$583$	678223.	0.0826420
$584$	0	0
$585$	−1.86822e6	−0.225703
$586$	0	0
$587$	1.28267e7	1.53646	0.768230	−	0.640174i	$-0.221138\pi$
$587$	1.28267e7	1.53646	0.768230	+	0.640174i	$0.221138\pi$
$588$	0	0
$589$	1.41986e6	0.168638
$590$	0	0
$591$	−3.85836e6	−0.454395
$592$	0	0
$593$	−6.57824e6	−0.768198	−0.384099	−	0.923292i	$-0.625488\pi$
$593$	−6.57824e6	−0.768198	−0.384099	+	0.923292i	$0.625488\pi$
$594$	0	0
$595$	6.56090e6	0.759750
$596$	0	0
$597$	−372327.	−0.0427552
$598$	0	0
$599$	−1.05032e7	−1.19607	−0.598034	−	0.801471i	$-0.704051\pi$
$599$	−1.05032e7	−1.19607	−0.598034	+	0.801471i	$0.704051\pi$
$600$	0	0
$601$	3.03229e6	0.342441	0.171220	−	0.985233i	$-0.445229\pi$
$601$	3.03229e6	0.342441	0.171220	+	0.985233i	$0.445229\pi$
$602$	0	0
$603$	−3.40953e6	−0.381857
$604$	0	0
$605$	275349.	0.0305840
$606$	0	0
$607$	2.47261e6	0.272385	0.136193	−	0.990682i	$-0.456513\pi$
$607$	2.47261e6	0.272385	0.136193	+	0.990682i	$0.456513\pi$
$608$	0	0
$609$	6.45417e6	0.705175
$610$	0	0
$611$	−1.28270e7	−1.39002
$612$	0	0
$613$	1.51835e7	1.63200	0.816002	−	0.578050i	$-0.196186\pi$
$613$	1.51835e7	1.63200	0.816002	+	0.578050i	$0.196186\pi$
$614$	0	0
$615$	1.19308e6	0.127199
$616$	0	0
$617$	−1.22428e7	−1.29470	−0.647348	−	0.762195i	$-0.724122\pi$
$617$	−1.22428e7	−1.29470	−0.647348	+	0.762195i	$0.724122\pi$
$618$	0	0
$619$	−2.05223e6	−0.215278	−0.107639	−	0.994190i	$-0.534329\pi$
$619$	−2.05223e6	−0.215278	−0.107639	+	0.994190i	$0.534329\pi$
$620$	0	0
$621$	−4.81376e6	−0.500905
$622$	0	0
$623$	6.88326e6	0.710516
$624$	0	0
$625$	6.57487e6	0.673266
$626$	0	0
$627$	−156310.	−0.0158788
$628$	0	0
$629$	1.52501e7	1.53690
$630$	0	0
$631$	1.32105e7	1.32082	0.660412	−	0.750904i	$-0.270382\pi$
$631$	1.32105e7	1.32082	0.660412	+	0.750904i	$0.270382\pi$
$632$	0	0
$633$	−5.21431e6	−0.517235
$634$	0	0
$635$	5.29017e6	0.520638
$636$	0	0
$637$	7.98026e6	0.779235
$638$	0	0
$639$	7.62079e6	0.738326
$640$	0	0
$641$	−1.34860e7	−1.29640	−0.648198	−	0.761472i	$-0.724477\pi$
$641$	−1.34860e7	−1.29640	−0.648198	+	0.761472i	$0.724477\pi$
$642$	0	0
$643$	1.41694e7	1.35153	0.675763	−	0.737119i	$-0.263814\pi$
$643$	1.41694e7	1.35153	0.675763	+	0.737119i	$0.263814\pi$
$644$	0	0
$645$	−1.92470e6	−0.182165
$646$	0	0
$647$	−4.52666e6	−0.425125	−0.212563	−	0.977147i	$-0.568181\pi$
$647$	−4.52666e6	−0.425125	−0.212563	+	0.977147i	$0.568181\pi$
$648$	0	0
$649$	1.86442e6	0.173753
$650$	0	0
$651$	4.63605e6	0.428742
$652$	0	0
$653$	−2.91674e6	−0.267679	−0.133840	−	0.991003i	$-0.542731\pi$
$653$	−2.91674e6	−0.267679	−0.133840	+	0.991003i	$0.542731\pi$
$654$	0	0
$655$	1.06812e6	0.0972788
$656$	0	0
$657$	1.59370e7	1.44043
$658$	0	0
$659$	2.00854e7	1.80163	0.900816	−	0.434201i	$-0.142969\pi$
$659$	2.00854e7	1.80163	0.900816	+	0.434201i	$0.142969\pi$
$660$	0	0
$661$	−1.73147e7	−1.54138	−0.770691	−	0.637209i	$-0.780089\pi$
$661$	−1.73147e7	−1.54138	−0.770691	+	0.637209i	$0.780089\pi$
$662$	0	0
$663$	4.03567e6	0.356559
$664$	0	0
$665$	947030.	0.0830443
$666$	0	0
$667$	1.58988e7	1.38373
$668$	0	0
$669$	5.31539e6	0.459166
$670$	0	0
$671$	2.49524e6	0.213947
$672$	0	0
$673$	−1.72167e7	−1.46525	−0.732627	−	0.680631i	$-0.761706\pi$
$673$	−1.72167e7	−1.46525	−0.732627	+	0.680631i	$0.761706\pi$
$674$	0	0
$675$	6.11822e6	0.516851
$676$	0	0
$677$	−1.67698e7	−1.40623	−0.703113	−	0.711078i	$-0.748207\pi$
$677$	−1.67698e7	−1.40623	−0.703113	+	0.711078i	$0.748207\pi$
$678$	0	0
$679$	−472596.	−0.0393383
$680$	0	0
$681$	−925058.	−0.0764366
$682$	0	0
$683$	−2.26092e7	−1.85452	−0.927262	−	0.374412i	$-0.877844\pi$
$683$	−2.26092e7	−1.85452	−0.927262	+	0.374412i	$0.877844\pi$
$684$	0	0
$685$	1.83887e6	0.149736
$686$	0	0
$687$	3.79719e6	0.306952
$688$	0	0
$689$	−2.52756e6	−0.202840
$690$	0	0
$691$	−2.03037e7	−1.61763	−0.808817	−	0.588061i	$-0.799891\pi$
$691$	−2.03037e7	−1.61763	−0.808817	+	0.588061i	$0.799891\pi$
$692$	0	0
$693$	4.95131e6	0.391640
$694$	0	0
$695$	−2.56558e6	−0.201476
$696$	0	0
$697$	2.50028e7	1.94942
$698$	0	0
$699$	7.23857e6	0.560350
$700$	0	0
$701$	6.50043e6	0.499629	0.249814	−	0.968294i	$-0.419630\pi$
$701$	6.50043e6	0.499629	0.249814	+	0.968294i	$0.419630\pi$
$702$	0	0
$703$	2.20128e6	0.167991
$704$	0	0
$705$	−2.54921e6	−0.193167
$706$	0	0
$707$	−8.14105e6	−0.612536
$708$	0	0
$709$	−1.08788e7	−0.812769	−0.406385	−	0.913702i	$-0.633211\pi$
$709$	−1.08788e7	−0.812769	−0.406385	+	0.913702i	$0.633211\pi$
$710$	0	0
$711$	3.04962e6	0.226241
$712$	0	0
$713$	1.14202e7	0.841295
$714$	0	0
$715$	−1.02615e6	−0.0750668
$716$	0	0
$717$	−2.66091e6	−0.193300
$718$	0	0
$719$	−2.35143e6	−0.169633	−0.0848163	−	0.996397i	$-0.527030\pi$
$719$	−2.35143e6	−0.169633	−0.0848163	+	0.996397i	$0.527030\pi$
$720$	0	0
$721$	−2.08657e7	−1.49484
$722$	0	0
$723$	−1.48303e6	−0.105512
$724$	0	0
$725$	−2.02072e7	−1.42778
$726$	0	0
$727$	1.16270e7	0.815890	0.407945	−	0.913007i	$-0.366246\pi$
$727$	1.16270e7	0.815890	0.407945	+	0.913007i	$0.366246\pi$
$728$	0	0
$729$	−6.91856e6	−0.482166
$730$	0	0
$731$	−4.03350e7	−2.79183
$732$	0	0
$733$	7.78239e6	0.534999	0.267499	−	0.963558i	$-0.413803\pi$
$733$	7.78239e6	0.534999	0.267499	+	0.963558i	$0.413803\pi$
$734$	0	0
$735$	1.58598e6	0.108288
$736$	0	0
$737$	−1.87275e6	−0.127002
$738$	0	0
$739$	1.62063e7	1.09162	0.545811	−	0.837908i	$-0.316222\pi$
$739$	1.62063e7	1.09162	0.545811	+	0.837908i	$0.316222\pi$
$740$	0	0
$741$	582527.	0.0389736
$742$	0	0
$743$	1.16461e7	0.773940	0.386970	−	0.922092i	$-0.373522\pi$
$743$	1.16461e7	0.773940	0.386970	+	0.922092i	$0.373522\pi$
$744$	0	0
$745$	−617922.	−0.0407890
$746$	0	0
$747$	3.16469e6	0.207506
$748$	0	0
$749$	−1.76640e6	−0.115050
$750$	0	0
$751$	−2.83018e7	−1.83111	−0.915555	−	0.402194i	$-0.868248\pi$
$751$	−2.83018e7	−1.83111	−0.915555	+	0.402194i	$0.868248\pi$
$752$	0	0
$753$	−2.45783e6	−0.157966
$754$	0	0
$755$	−1.81191e6	−0.115683
$756$	0	0
$757$	−6.31471e6	−0.400510	−0.200255	−	0.979744i	$-0.564177\pi$
$757$	−6.31471e6	−0.400510	−0.200255	+	0.979744i	$0.564177\pi$
$758$	0	0
$759$	−1.25723e6	−0.0792154
$760$	0	0
$761$	−5.26676e6	−0.329672	−0.164836	−	0.986321i	$-0.552709\pi$
$761$	−5.26676e6	−0.329672	−0.164836	+	0.986321i	$0.552709\pi$
$762$	0	0
$763$	−1.63279e7	−1.01536
$764$	0	0
$765$	−7.78086e6	−0.480700
$766$	0	0
$767$	−6.94824e6	−0.426468
$768$	0	0
$769$	2.93559e7	1.79011	0.895056	−	0.445954i	$-0.147135\pi$
$769$	2.93559e7	1.79011	0.895056	+	0.445954i	$0.147135\pi$
$770$	0	0
$771$	−3.23069e6	−0.195731
$772$	0	0
$773$	2.09811e7	1.26293	0.631466	−	0.775404i	$-0.282454\pi$
$773$	2.09811e7	1.26293	0.631466	+	0.775404i	$0.282454\pi$
$774$	0	0
$775$	−1.45149e7	−0.868078
$776$	0	0
$777$	7.18750e6	0.427096
$778$	0	0
$779$	3.60902e6	0.213081
$780$	0	0
$781$	4.18587e6	0.245560
$782$	0	0
$783$	−1.60976e7	−0.938331
$784$	0	0
$785$	−8.73521e6	−0.505940
$786$	0	0
$787$	2.72734e7	1.56965	0.784825	−	0.619717i	$-0.212753\pi$
$787$	2.72734e7	1.56965	0.784825	+	0.619717i	$0.212753\pi$
$788$	0	0
$789$	3.17026e6	0.181302
$790$	0	0
$791$	4.13717e7	2.35105
$792$	0	0
$793$	−9.29912e6	−0.525120
$794$	0	0
$795$	−502324.	−0.0281881
$796$	0	0
$797$	−2.17249e7	−1.21147	−0.605734	−	0.795667i	$-0.707121\pi$
$797$	−2.17249e7	−1.21147	−0.605734	+	0.795667i	$0.707121\pi$
$798$	0	0
$799$	−5.34225e7	−2.96044
$800$	0	0
$801$	−8.16316e6	−0.449549
$802$	0	0
$803$	8.75371e6	0.479075
$804$	0	0
$805$	7.61713e6	0.414288
$806$	0	0
$807$	1.06960e7	0.578145
$808$	0	0
$809$	792354.	0.0425645	0.0212823	−	0.999774i	$-0.493225\pi$
$809$	792354.	0.0425645	0.0212823	+	0.999774i	$0.493225\pi$
$810$	0	0
$811$	−3.63960e7	−1.94313	−0.971563	−	0.236780i	$-0.923908\pi$
$811$	−3.63960e7	−1.94313	−0.971563	+	0.236780i	$0.923908\pi$
$812$	0	0
$813$	−368214.	−0.0195377
$814$	0	0
$815$	−1.04029e7	−0.548608
$816$	0	0
$817$	−5.82215e6	−0.305160
$818$	0	0
$819$	−1.84523e7	−0.961259
$820$	0	0
$821$	696484.	0.0360623	0.0180311	−	0.999837i	$-0.494260\pi$
$821$	696484.	0.0360623	0.0180311	+	0.999837i	$0.494260\pi$
$822$	0	0
$823$	2.23374e7	1.14956	0.574782	−	0.818307i	$-0.305087\pi$
$823$	2.23374e7	1.14956	0.574782	+	0.818307i	$0.305087\pi$
$824$	0	0
$825$	1.59792e6	0.0817373
$826$	0	0
$827$	−8.80259e6	−0.447555	−0.223778	−	0.974640i	$-0.571839\pi$
$827$	−8.80259e6	−0.447555	−0.223778	+	0.974640i	$0.571839\pi$
$828$	0	0
$829$	1.46640e7	0.741081	0.370540	−	0.928816i	$-0.379173\pi$
$829$	1.46640e7	0.741081	0.370540	+	0.928816i	$0.379173\pi$
$830$	0	0
$831$	−9.46371e6	−0.475400
$832$	0	0
$833$	3.32367e7	1.65961
$834$	0	0
$835$	9.91713e6	0.492232
$836$	0	0
$837$	−1.15629e7	−0.570498
$838$	0	0
$839$	2.34020e7	1.14775	0.573877	−	0.818942i	$-0.305439\pi$
$839$	2.34020e7	1.14775	0.573877	+	0.818942i	$0.305439\pi$
$840$	0	0
$841$	3.26557e7	1.59209
$842$	0	0
$843$	9.29067e6	0.450275
$844$	0	0
$845$	−3.15857e6	−0.152177
$846$	0	0
$847$	2.71961e6	0.130256
$848$	0	0
$849$	−1.18508e7	−0.564261
$850$	0	0
$851$	1.77053e7	0.838066
$852$	0	0
$853$	−2.88463e7	−1.35743	−0.678714	−	0.734403i	$-0.737462\pi$
$853$	−2.88463e7	−1.35743	−0.678714	+	0.734403i	$0.737462\pi$
$854$	0	0
$855$	−1.12313e6	−0.0525428
$856$	0	0
$857$	2.23189e7	1.03806	0.519028	−	0.854757i	$-0.326294\pi$
$857$	2.23189e7	1.03806	0.519028	+	0.854757i	$0.326294\pi$
$858$	0	0
$859$	6.62134e6	0.306170	0.153085	−	0.988213i	$-0.451079\pi$
$859$	6.62134e6	0.306170	0.153085	+	0.988213i	$0.451079\pi$
$860$	0	0
$861$	1.17840e7	0.541733
$862$	0	0
$863$	−1.43706e7	−0.656820	−0.328410	−	0.944535i	$-0.606513\pi$
$863$	−1.43706e7	−0.656820	−0.328410	+	0.944535i	$0.606513\pi$
$864$	0	0
$865$	−5.02598e6	−0.228392
$866$	0	0
$867$	1.00420e7	0.453705
$868$	0	0
$869$	1.67507e6	0.0752458
$870$	0	0
$871$	6.97927e6	0.311720
$872$	0	0
$873$	560472.	0.0248896
$874$	0	0
$875$	−2.05981e7	−0.909510
$876$	0	0
$877$	−1.18877e7	−0.521912	−0.260956	−	0.965351i	$-0.584038\pi$
$877$	−1.18877e7	−0.521912	−0.260956	+	0.965351i	$0.584038\pi$
$878$	0	0
$879$	−5.10775e6	−0.222976
$880$	0	0
$881$	3.95772e7	1.71793	0.858965	−	0.512035i	$-0.171108\pi$
$881$	3.95772e7	1.71793	0.858965	+	0.512035i	$0.171108\pi$
$882$	0	0
$883$	−2.94934e7	−1.27298	−0.636491	−	0.771284i	$-0.719615\pi$
$883$	−2.94934e7	−1.27298	−0.636491	+	0.771284i	$0.719615\pi$
$884$	0	0
$885$	−1.38088e6	−0.0592650
$886$	0	0
$887$	7.48463e6	0.319419	0.159710	−	0.987164i	$-0.448944\pi$
$887$	7.48463e6	0.319419	0.159710	+	0.987164i	$0.448944\pi$
$888$	0	0
$889$	5.22508e7	2.21737
$890$	0	0
$891$	−5.20431e6	−0.219619
$892$	0	0
$893$	−7.71125e6	−0.323591
$894$	0	0
$895$	4.24496e6	0.177140
$896$	0	0
$897$	4.68537e6	0.194430
$898$	0	0
$899$	3.81899e7	1.57597
$900$	0	0
$901$	−1.05269e7	−0.432007
$902$	0	0
$903$	−1.90102e7	−0.775832
$904$	0	0
$905$	67155.6	0.00272559
$906$	0	0
$907$	3.17456e7	1.28134	0.640671	−	0.767816i	$-0.278656\pi$
$907$	3.17456e7	1.28134	0.640671	+	0.767816i	$0.278656\pi$
$908$	0	0
$909$	9.65484e6	0.387557
$910$	0	0
$911$	−3.13372e7	−1.25102	−0.625510	−	0.780216i	$-0.715109\pi$
$911$	−3.13372e7	−1.25102	−0.625510	+	0.780216i	$0.715109\pi$
$912$	0	0
$913$	1.73827e6	0.0690144
$914$	0	0
$915$	−1.84809e6	−0.0729744
$916$	0	0
$917$	1.05498e7	0.414306
$918$	0	0
$919$	−2.95485e7	−1.15411	−0.577054	−	0.816706i	$-0.695798\pi$
$919$	−2.95485e7	−1.15411	−0.577054	+	0.816706i	$0.695798\pi$
$920$	0	0
$921$	−9.36475e6	−0.363787
$922$	0	0
$923$	−1.55997e7	−0.602714
$924$	0	0
$925$	−2.25031e7	−0.864746
$926$	0	0
$927$	2.47456e7	0.945797
$928$	0	0
$929$	1.10989e7	0.421932	0.210966	−	0.977493i	$-0.432339\pi$
$929$	1.10989e7	0.421932	0.210966	+	0.977493i	$0.432339\pi$
$930$	0	0
$931$	4.79754e6	0.181403
$932$	0	0
$933$	1.43838e6	0.0540967
$934$	0	0
$935$	−4.27379e6	−0.159876
$936$	0	0
$937$	2.66254e7	0.990711	0.495356	−	0.868690i	$-0.335038\pi$
$937$	2.66254e7	0.990711	0.495356	+	0.868690i	$0.335038\pi$
$938$	0	0
$939$	4.89453e6	0.181154
$940$	0	0
$941$	4.39026e7	1.61628	0.808140	−	0.588991i	$-0.200475\pi$
$941$	4.39026e7	1.61628	0.808140	+	0.588991i	$0.200475\pi$
$942$	0	0
$943$	2.90280e7	1.06301
$944$	0	0
$945$	−7.71237e6	−0.280936
$946$	0	0
$947$	3.01899e7	1.09392	0.546961	−	0.837158i	$-0.315785\pi$
$947$	3.01899e7	1.09392	0.546961	+	0.837158i	$0.315785\pi$
$948$	0	0
$949$	−3.26229e7	−1.17586
$950$	0	0
$951$	4.78137e6	0.171436
$952$	0	0
$953$	4.59444e6	0.163870	0.0819352	−	0.996638i	$-0.473890\pi$
$953$	4.59444e6	0.163870	0.0819352	+	0.996638i	$0.473890\pi$
$954$	0	0
$955$	−9.50886e6	−0.337381
$956$	0	0
$957$	−4.20427e6	−0.148392
$958$	0	0
$959$	1.81625e7	0.637717
$960$	0	0
$961$	−1.19723e6	−0.0418186
$962$	0	0
$963$	2.09486e6	0.0727928
$964$	0	0
$965$	1.20802e7	0.417597
$966$	0	0
$967$	2.93857e7	1.01058	0.505289	−	0.862950i	$-0.331386\pi$
$967$	2.93857e7	1.01058	0.505289	+	0.862950i	$0.331386\pi$
$968$	0	0
$969$	2.42614e6	0.0830056
$970$	0	0
$971$	−1.06536e7	−0.362618	−0.181309	−	0.983426i	$-0.558033\pi$
$971$	−1.06536e7	−0.362618	−0.181309	+	0.983426i	$0.558033\pi$
$972$	0	0
$973$	−2.53401e7	−0.858077
$974$	0	0
$975$	−5.95504e6	−0.200620
$976$	0	0
$977$	−1.75504e6	−0.0588235	−0.0294118	−	0.999567i	$-0.509363\pi$
$977$	−1.75504e6	−0.0588235	−0.0294118	+	0.999567i	$0.509363\pi$
$978$	0	0
$979$	−4.48378e6	−0.149516
$980$	0	0
$981$	1.93640e7	0.642425
$982$	0	0
$983$	2.00508e7	0.661833	0.330917	−	0.943660i	$-0.392642\pi$
$983$	2.00508e7	0.661833	0.330917	+	0.943660i	$0.392642\pi$
$984$	0	0
$985$	−1.52276e7	−0.500080
$986$	0	0
$987$	−2.51784e7	−0.822689
$988$	0	0
$989$	−4.68286e7	−1.52237
$990$	0	0
$991$	1.39910e7	0.452547	0.226274	−	0.974064i	$-0.427346\pi$
$991$	1.39910e7	0.452547	0.226274	+	0.974064i	$0.427346\pi$
$992$	0	0
$993$	3.59674e6	0.115754
$994$	0	0
$995$	−1.46944e6	−0.0470538
$996$	0	0
$997$	−3.99863e7	−1.27401	−0.637006	−	0.770859i	$-0.719827\pi$
$997$	−3.99863e7	−1.27401	−0.637006	+	0.770859i	$0.719827\pi$
$998$	0	0
$999$	−1.79266e7	−0.568309

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	88.6.a.d.1.3	✓	4
3.2	odd	2		792.6.a.k.1.3		4
4.3	odd	2		176.6.a.k.1.2		4
8.3	odd	2		704.6.a.v.1.3		4
8.5	even	2		704.6.a.w.1.2		4
11.10	odd	2		968.6.a.e.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.6.a.d.1.3	✓	4	1.1	even	1	trivial
176.6.a.k.1.2		4	4.3	odd	2
704.6.a.v.1.3		4	8.3	odd	2
704.6.a.w.1.2		4	8.5	even	2
792.6.a.k.1.3		4	3.2	odd	2
968.6.a.e.1.3		4	11.10	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:	\( N \)	\(=\)	\( 88 = 2^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	88.a (trivial)

\(f(q)\)	\(=\)	\(q+4.76524 q^{3} +18.8067 q^{5} +185.753 q^{7} -220.292 q^{9} -121.000 q^{11} +450.936 q^{13} +89.6184 q^{15} +1878.09 q^{17} +271.092 q^{19} +885.156 q^{21} +2180.44 q^{23} -2771.31 q^{25} -2207.70 q^{27} +7291.56 q^{29} +5237.55 q^{31} -576.594 q^{33} +3493.39 q^{35} +8120.04 q^{37} +2148.82 q^{39} +13312.9 q^{41} -21476.7 q^{43} -4142.97 q^{45} -28445.2 q^{47} +17697.1 q^{49} +8949.53 q^{51} -5605.15 q^{53} -2275.61 q^{55} +1291.82 q^{57} -15408.5 q^{59} -20621.8 q^{61} -40919.9 q^{63} +8480.62 q^{65} +15477.3 q^{67} +10390.3 q^{69} -34594.0 q^{71} -72344.7 q^{73} -13205.9 q^{75} -22476.1 q^{77} -13843.5 q^{79} +43010.9 q^{81} -14365.9 q^{83} +35320.6 q^{85} +34746.0 q^{87} +37056.0 q^{89} +83762.7 q^{91} +24958.2 q^{93} +5098.34 q^{95} -2544.22 q^{97} +26655.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 5 q^{3} + 93 q^{5} - 94 q^{7} + 591 q^{9} - 484 q^{11} + 230 q^{13} + 3227 q^{15} + 1856 q^{17} - 40 q^{19} + 6114 q^{21} + 1515 q^{23} + 15147 q^{25} + 11605 q^{27} + 11390 q^{29} + 1339 q^{31} + 605 q^{33}+ \cdots - 71511 q^{99}+O(q^{100}) \) 4 * q - 5 * q^3 + 93 * q^5 - 94 * q^7 + 591 * q^9 - 484 * q^11 + 230 * q^13 + 3227 * q^15 + 1856 * q^17 - 40 * q^19 + 6114 * q^21 + 1515 * q^23 + 15147 * q^25 + 11605 * q^27 + 11390 * q^29 + 1339 * q^31 + 605 * q^33 - 5342 * q^35 + 8003 * q^37 - 24968 * q^39 + 26070 * q^41 + 8022 * q^43 + 50648 * q^45 - 31984 * q^47 + 13232 * q^49 - 69778 * q^51 + 9148 * q^53 - 11253 * q^55 - 51172 * q^57 - 101047 * q^59 + 9618 * q^61 - 78332 * q^63 - 22540 * q^65 - 101403 * q^67 - 73769 * q^69 - 18367 * q^71 + 73354 * q^73 - 53664 * q^75 + 11374 * q^77 - 92026 * q^79 + 20116 * q^81 + 62246 * q^83 - 246030 * q^85 - 55856 * q^87 + 138439 * q^89 + 848 * q^91 + 63791 * q^93 + 284820 * q^95 + 181869 * q^97 - 71511 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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