LMFDB - Embedded newform 1152.5.b.k.703.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,5,Mod(703,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("1152.703");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1152.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$119.082197473$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.49787136.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.6
Root			$-0.228425 - 1.39564i$ of defining polynomial
Character	$\chi$	$=$	1152.703
Dual form			1152.5.b.k.703.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+23.7490i q^{5} +9.38251i q^{7} +O(q^{10})$	$q+23.7490i q^{5} +9.38251i q^{7} +112.001 q^{11} -56.5020i q^{13} +79.9921 q^{17} -211.297 q^{19} +217.355i q^{23} +60.9843 q^{25} +616.753i q^{29} +1111.81i q^{31} -222.825 q^{35} -802.980i q^{37} +2411.93 q^{41} +2130.38 q^{43} -3596.58i q^{47} +2312.97 q^{49} -833.725i q^{53} +2659.92i q^{55} -1309.43 q^{59} +4785.91i q^{61} +1341.87 q^{65} -4025.23 q^{67} -9487.13i q^{71} -266.031 q^{73} +1050.85i q^{77} +5756.11i q^{79} +7287.16 q^{83} +1899.73i q^{85} +1414.08 q^{89} +530.130 q^{91} -5018.09i q^{95} -3110.08 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 1392 q^{17} - 3576 q^{25} + 1008 q^{41} + 10376 q^{49} - 23808 q^{65} - 10256 q^{73} + 31632 q^{89} - 45200 q^{97}+O(q^{100}) $ 8 * q - 1392 * q^17 - 3576 * q^25 + 1008 * q^41 + 10376 * q^49 - 23808 * q^65 - 10256 * q^73 + 31632 * q^89 - 45200 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	23.7490i	0.949961i	0.879996	+	0.474980i	$0.157545\pi$
$5$	23.7490i	0.949961i	−0.879996	+	0.474980i	$0.842455\pi$
$6$	0	0
$7$	9.38251i	0.191480i	0.995406	+	0.0957399i	$0.0305217\pi$
$7$	9.38251i	0.191480i	−0.995406	+	0.0957399i	$0.969478\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	112.001	0.925631	0.462816	−	0.886455i	$-0.346839\pi$
$11$	112.001	0.925631	0.462816	+	0.886455i	$0.346839\pi$
$12$	0	0
$13$	− 56.5020i	− 0.334331i	−0.985929	−	0.167166i	$-0.946539\pi$
$13$	− 56.5020i	− 0.334331i	0.985929	−	0.167166i	$-0.0534615\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	79.9921	0.276789	0.138395	−	0.990377i	$-0.455806\pi$
$17$	79.9921	0.276789	0.138395	+	0.990377i	$0.455806\pi$
$18$	0	0
$19$	−211.297	−0.585309	−0.292655	−	0.956218i	$-0.594539\pi$
$19$	−211.297	−0.585309	−0.292655	+	0.956218i	$0.594539\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	217.355i	0.410880i	0.978670	+	0.205440i	$0.0658625\pi$
$23$	217.355i	0.410880i	−0.978670	+	0.205440i	$0.934138\pi$
$24$	0	0
$25$	60.9843	0.0975748
$26$	0	0
$27$	0	0
$28$	0	0
$29$	616.753i	0.733357i	0.930348	+	0.366678i	$0.119505\pi$
$29$	616.753i	0.733357i	−0.930348	+	0.366678i	$0.880495\pi$
$30$	0	0
$31$	1111.81i	1.15693i	0.815707	+	0.578465i	$0.196348\pi$
$31$	1111.81i	1.15693i	−0.815707	+	0.578465i	$0.803652\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−222.825	−0.181898
$36$	0	0
$37$	− 802.980i	− 0.586545i	−0.956029	−	0.293273i	$-0.905256\pi$
$37$	− 802.980i	− 0.586545i	0.956029	−	0.293273i	$-0.0947443\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2411.93	1.43482	0.717409	−	0.696652i	$-0.245328\pi$
$41$	2411.93	1.43482	0.717409	+	0.696652i	$0.245328\pi$
$42$	0	0
$43$	2130.38	1.15218	0.576090	−	0.817386i	$-0.304578\pi$
$43$	2130.38	1.15218	0.576090	+	0.817386i	$0.304578\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 3596.58i	− 1.62815i	−0.580761	−	0.814074i	$-0.697245\pi$
$47$	− 3596.58i	− 1.62815i	0.580761	−	0.814074i	$-0.302755\pi$
$48$	0	0
$49$	2312.97	0.963335
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 833.725i	− 0.296805i	−0.988927	−	0.148403i	$-0.952587\pi$
$53$	− 833.725i	− 0.296805i	0.988927	−	0.148403i	$-0.0474131\pi$
$54$	0	0
$55$	2659.92i	0.879313i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1309.43	−0.376165	−0.188083	−	0.982153i	$-0.560227\pi$
$59$	−1309.43	−0.376165	−0.188083	+	0.982153i	$0.560227\pi$
$60$	0	0
$61$	4785.91i	1.28619i	0.765786	+	0.643095i	$0.222350\pi$
$61$	4785.91i	1.28619i	−0.765786	+	0.643095i	$0.777650\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1341.87	0.317601
$66$	0	0
$67$	−4025.23	−0.896688	−0.448344	−	0.893861i	$-0.647986\pi$
$67$	−4025.23	−0.896688	−0.448344	+	0.893861i	$0.647986\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 9487.13i	− 1.88199i	−0.338416	−	0.940997i	$-0.609891\pi$
$71$	− 9487.13i	− 1.88199i	0.338416	−	0.940997i	$-0.390109\pi$
$72$	0	0
$73$	−266.031	−0.0499215	−0.0249607	−	0.999688i	$-0.507946\pi$
$73$	−266.031	−0.0499215	−0.0249607	+	0.999688i	$0.507946\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1050.85i	0.177240i
$78$	0	0
$79$	5756.11i	0.922306i	0.887321	+	0.461153i	$0.152564\pi$
$79$	5756.11i	0.922306i	−0.887321	+	0.461153i	$0.847436\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7287.16	1.05780	0.528898	−	0.848685i	$-0.322605\pi$
$83$	7287.16	1.05780	0.528898	+	0.848685i	$0.322605\pi$
$84$	0	0
$85$	1899.73i	0.262939i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1414.08	0.178523	0.0892614	−	0.996008i	$-0.471549\pi$
$89$	1414.08	0.178523	0.0892614	+	0.996008i	$0.471549\pi$
$90$	0	0
$91$	530.130	0.0640177
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 5018.09i	− 0.556021i
$96$	0	0
$97$	−3110.08	−0.330543	−0.165271	−	0.986248i	$-0.552850\pi$
$97$	−3110.08	−0.330543	−0.165271	+	0.986248i	$0.552850\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 488.501i	− 0.0478876i	−0.999713	−	0.0239438i	$-0.992378\pi$
$101$	− 488.501i	− 0.0478876i	0.999713	−	0.0239438i	$-0.00762227\pi$
$102$	0	0
$103$	20097.0i	1.89434i	0.320739	+	0.947168i	$0.396069\pi$
$103$	20097.0i	1.89434i	−0.320739	+	0.947168i	$0.603931\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12383.0	−1.08158	−0.540790	−	0.841158i	$-0.681874\pi$
$107$	−12383.0	−1.08158	−0.540790	+	0.841158i	$0.681874\pi$
$108$	0	0
$109$	15542.1i	1.30814i	0.756433	+	0.654072i	$0.226941\pi$
$109$	15542.1i	1.30814i	−0.756433	+	0.654072i	$0.773059\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−11653.6	−0.912651	−0.456325	−	0.889813i	$-0.650835\pi$
$113$	−11653.6	−0.912651	−0.456325	+	0.889813i	$0.650835\pi$
$114$	0	0
$115$	−5161.98	−0.390319
$116$	0	0
$117$	0	0
$118$	0	0
$119$	750.527i	0.0529996i
$120$	0	0
$121$	−2096.69	−0.143206
$122$	0	0
$123$	0	0
$124$	0	0
$125$	16291.5i	1.04265i
$126$	0	0
$127$	4724.04i	0.292891i	0.989219	+	0.146445i	$0.0467833\pi$
$127$	4724.04i	0.292891i	−0.989219	+	0.146445i	$0.953217\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9632.86	0.561323	0.280661	−	0.959807i	$-0.409446\pi$
$131$	9632.86	0.561323	0.280661	+	0.959807i	$0.409446\pi$
$132$	0	0
$133$	− 1982.49i	− 0.112075i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16407.6	−0.874185	−0.437093	−	0.899417i	$-0.643992\pi$
$137$	−16407.6	−0.874185	−0.437093	+	0.899417i	$0.643992\pi$
$138$	0	0
$139$	25544.1	1.32209	0.661046	−	0.750346i	$-0.270113\pi$
$139$	25544.1	1.32209	0.661046	+	0.750346i	$0.270113\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 6328.30i	− 0.309467i
$144$	0	0
$145$	−14647.3	−0.696660
$146$	0	0
$147$	0	0
$148$	0	0
$149$	18716.6i	0.843054i	0.906816	+	0.421527i	$0.138506\pi$
$149$	18716.6i	0.843054i	−0.906816	+	0.421527i	$0.861494\pi$
$150$	0	0
$151$	− 28860.0i	− 1.26574i	−0.774260	−	0.632868i	$-0.781878\pi$
$151$	− 28860.0i	− 1.26574i	0.774260	−	0.632868i	$-0.218122\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−26404.4	−1.09904
$156$	0	0
$157$	23963.1i	0.972172i	0.873911	+	0.486086i	$0.161576\pi$
$157$	23963.1i	0.972172i	−0.873911	+	0.486086i	$0.838424\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2039.34	−0.0786752
$162$	0	0
$163$	−24424.8	−0.919297	−0.459648	−	0.888101i	$-0.652025\pi$
$163$	−24424.8	−0.919297	−0.459648	+	0.888101i	$0.652025\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3088.40i	0.110739i	0.998466	+	0.0553696i	$0.0176337\pi$
$167$	3088.40i	0.110739i	−0.998466	+	0.0553696i	$0.982366\pi$
$168$	0	0
$169$	25368.5	0.888223
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18621.0i	0.622171i	0.950382	+	0.311085i	$0.100693\pi$
$173$	18621.0i	0.622171i	−0.950382	+	0.311085i	$0.899307\pi$
$174$	0	0
$175$	572.185i	0.0186836i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	50739.3	1.58357	0.791787	−	0.610797i	$-0.209151\pi$
$179$	50739.3	1.58357	0.791787	+	0.610797i	$0.209151\pi$
$180$	0	0
$181$	58050.5i	1.77194i	0.463742	+	0.885970i	$0.346506\pi$
$181$	58050.5i	1.77194i	−0.463742	+	0.885970i	$0.653494\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	19070.0	0.557195
$186$	0	0
$187$	8959.23	0.256205
$188$	0	0
$189$	0	0
$190$	0	0
$191$	21285.6i	0.583471i	0.956499	+	0.291736i	$0.0942327\pi$
$191$	21285.6i	0.583471i	−0.956499	+	0.291736i	$0.905767\pi$
$192$	0	0
$193$	−26074.3	−0.700001	−0.350001	−	0.936749i	$-0.613819\pi$
$193$	−26074.3	−0.700001	−0.350001	+	0.936749i	$0.613819\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4731.26i	0.121911i	0.998140	+	0.0609557i	$0.0194149\pi$
$197$	4731.26i	0.121911i	−0.998140	+	0.0609557i	$0.980585\pi$
$198$	0	0
$199$	− 38561.7i	− 0.973756i	−0.873470	−	0.486878i	$-0.838136\pi$
$199$	− 38561.7i	− 0.973756i	0.873470	−	0.486878i	$-0.161864\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5786.69	−0.140423
$204$	0	0
$205$	57280.9i	1.36302i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−23665.5	−0.541780
$210$	0	0
$211$	−48656.4	−1.09289	−0.546444	−	0.837496i	$-0.684019\pi$
$211$	−48656.4	−1.09289	−0.546444	+	0.837496i	$0.684019\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	50594.5i	1.09453i
$216$	0	0
$217$	−10431.6	−0.221529
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 4519.71i	− 0.0925393i
$222$	0	0
$223$	− 73299.7i	− 1.47398i	−0.675902	−	0.736992i	$-0.736246\pi$
$223$	− 73299.7i	− 1.47398i	0.675902	−	0.736992i	$-0.263754\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−14996.0	−0.291020	−0.145510	−	0.989357i	$-0.546482\pi$
$227$	−14996.0	−0.291020	−0.145510	+	0.989357i	$0.546482\pi$
$228$	0	0
$229$	74694.9i	1.42436i	0.701996	+	0.712181i	$0.252292\pi$
$229$	74694.9i	1.42436i	−0.701996	+	0.712181i	$0.747708\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	69476.6	1.27975	0.639877	−	0.768477i	$-0.278985\pi$
$233$	69476.6	1.27975	0.639877	+	0.768477i	$0.278985\pi$
$234$	0	0
$235$	85415.2	1.54668
$236$	0	0
$237$	0	0
$238$	0	0
$239$	87478.2i	1.53145i	0.643166	+	0.765727i	$0.277620\pi$
$239$	87478.2i	1.53145i	−0.643166	+	0.765727i	$0.722380\pi$
$240$	0	0
$241$	74951.4	1.29046	0.645232	−	0.763987i	$-0.276761\pi$
$241$	74951.4	1.29046	0.645232	+	0.763987i	$0.276761\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	54930.7i	0.915131i
$246$	0	0
$247$	11938.7i	0.195687i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−43677.8	−0.693287	−0.346643	−	0.937997i	$-0.612679\pi$
$251$	−43677.8	−0.693287	−0.346643	+	0.937997i	$0.612679\pi$
$252$	0	0
$253$	24344.1i	0.380323i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1164.55	0.0176316	0.00881581	−	0.999961i	$-0.497194\pi$
$257$	1164.55	0.0176316	0.00881581	+	0.999961i	$0.497194\pi$
$258$	0	0
$259$	7533.97	0.112312
$260$	0	0
$261$	0	0
$262$	0	0
$263$	35001.0i	0.506021i	0.967464	+	0.253010i	$0.0814207\pi$
$263$	35001.0i	0.506021i	−0.967464	+	0.253010i	$0.918579\pi$
$264$	0	0
$265$	19800.2	0.281953
$266$	0	0
$267$	0	0
$268$	0	0
$269$	41077.9i	0.567681i	0.958872	+	0.283840i	$0.0916085\pi$
$269$	41077.9i	0.567681i	−0.958872	+	0.283840i	$0.908391\pi$
$270$	0	0
$271$	− 46419.1i	− 0.632060i	−0.948749	−	0.316030i	$-0.897650\pi$
$271$	− 46419.1i	− 0.632060i	0.948749	−	0.316030i	$-0.102350\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6830.32	0.0903183
$276$	0	0
$277$	150297.i	1.95881i	0.201907	+	0.979405i	$0.435286\pi$
$277$	150297.i	1.95881i	−0.201907	+	0.979405i	$0.564714\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	117793.	1.49179	0.745895	−	0.666063i	$-0.232022\pi$
$281$	117793.	1.49179	0.745895	+	0.666063i	$0.232022\pi$
$282$	0	0
$283$	−41243.6	−0.514973	−0.257486	−	0.966282i	$-0.582894\pi$
$283$	−41243.6	−0.514973	−0.257486	+	0.966282i	$0.582894\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	22629.9i	0.274739i
$288$	0	0
$289$	−77122.3	−0.923388
$290$	0	0
$291$	0	0
$292$	0	0
$293$	71937.8i	0.837957i	0.907996	+	0.418979i	$0.137612\pi$
$293$	71937.8i	0.837957i	−0.907996	+	0.418979i	$0.862388\pi$
$294$	0	0
$295$	− 31097.7i	− 0.357342i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	12281.0	0.137370
$300$	0	0
$301$	19988.3i	0.220619i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−113661.	−1.22183
$306$	0	0
$307$	−54572.7	−0.579027	−0.289514	−	0.957174i	$-0.593494\pi$
$307$	−54572.7	−0.579027	−0.289514	+	0.957174i	$0.593494\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 60945.8i	− 0.630119i	−0.949072	−	0.315060i	$-0.897975\pi$
$311$	− 60945.8i	− 0.630119i	0.949072	−	0.315060i	$-0.102025\pi$
$312$	0	0
$313$	102774.	1.04904	0.524521	−	0.851397i	$-0.324244\pi$
$313$	102774.	1.04904	0.524521	+	0.851397i	$0.324244\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	39779.2i	0.395857i	0.980217	+	0.197928i	$0.0634213\pi$
$317$	39779.2i	0.395857i	−0.980217	+	0.197928i	$0.936579\pi$
$318$	0	0
$319$	69077.2i	0.678818i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−16902.1	−0.162007
$324$	0	0
$325$	− 3445.73i	− 0.0326223i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	33744.9	0.311758
$330$	0	0
$331$	−87804.6	−0.801422	−0.400711	−	0.916204i	$-0.631237\pi$
$331$	−87804.6	−0.801422	−0.400711	+	0.916204i	$0.631237\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 95595.3i	− 0.851818i
$336$	0	0
$337$	−57972.0	−0.510456	−0.255228	−	0.966881i	$-0.582151\pi$
$337$	−57972.0	−0.510456	−0.255228	+	0.966881i	$0.582151\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	124524.i	1.07089i
$342$	0	0
$343$	44228.9i	0.375939i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	159188.	1.32206	0.661031	−	0.750359i	$-0.270119\pi$
$347$	159188.	1.32206	0.661031	+	0.750359i	$0.270119\pi$
$348$	0	0
$349$	17396.8i	0.142830i	0.997447	+	0.0714149i	$0.0227514\pi$
$349$	17396.8i	0.142830i	−0.997447	+	0.0714149i	$0.977249\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−150532.	−1.20803	−0.604017	−	0.796972i	$-0.706434\pi$
$353$	−150532.	−1.20803	−0.604017	+	0.796972i	$0.706434\pi$
$354$	0	0
$355$	225310.	1.78782
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 24818.5i	− 0.192569i	−0.995354	−	0.0962846i	$-0.969304\pi$
$359$	− 24818.5i	− 0.192569i	0.995354	−	0.0962846i	$-0.0306959\pi$
$360$	0	0
$361$	−85674.8	−0.657413
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 6317.99i	− 0.0474234i
$366$	0	0
$367$	89484.6i	0.664379i	0.943213	+	0.332190i	$0.107787\pi$
$367$	89484.6i	0.664379i	−0.943213	+	0.332190i	$0.892213\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7822.44	0.0568322
$372$	0	0
$373$	168536.i	1.21137i	0.795705	+	0.605684i	$0.207100\pi$
$373$	168536.i	1.21137i	−0.795705	+	0.605684i	$0.792900\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	34847.8	0.245184
$378$	0	0
$379$	−126865.	−0.883208	−0.441604	−	0.897210i	$-0.645590\pi$
$379$	−126865.	−0.883208	−0.441604	+	0.897210i	$0.645590\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 257277.i	− 1.75389i	−0.480587	−	0.876947i	$-0.659576\pi$
$383$	− 257277.i	− 1.75389i	0.480587	−	0.876947i	$-0.340424\pi$
$384$	0	0
$385$	−24956.8	−0.168371
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 150562.i	− 0.994983i	−0.867469	−	0.497492i	$-0.834255\pi$
$389$	− 150562.i	− 0.994983i	0.867469	−	0.497492i	$-0.165745\pi$
$390$	0	0
$391$	17386.7i	0.113727i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−136702.	−0.876155
$396$	0	0
$397$	185402.i	1.17634i	0.808736	+	0.588171i	$0.200152\pi$
$397$	185402.i	1.17634i	−0.808736	+	0.588171i	$0.799848\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−167681.	−1.04279	−0.521393	−	0.853317i	$-0.674587\pi$
$401$	−167681.	−1.04279	−0.521393	+	0.853317i	$0.674587\pi$
$402$	0	0
$403$	62819.4	0.386798
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 89934.9i	− 0.542925i
$408$	0	0
$409$	60544.6	0.361934	0.180967	−	0.983489i	$-0.442077\pi$
$409$	60544.6	0.361934	0.180967	+	0.983489i	$0.442077\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 12285.7i	− 0.0720280i
$414$	0	0
$415$	173063.i	1.00486i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−307306.	−1.75042	−0.875211	−	0.483741i	$-0.839277\pi$
$419$	−307306.	−1.75042	−0.875211	+	0.483741i	$0.839277\pi$
$420$	0	0
$421$	− 172189.i	− 0.971496i	−0.874099	−	0.485748i	$-0.838547\pi$
$421$	− 172189.i	− 0.971496i	0.874099	−	0.485748i	$-0.161453\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4878.26	0.0270077
$426$	0	0
$427$	−44903.9	−0.246279
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 28153.8i	− 0.151559i	−0.997125	−	0.0757797i	$-0.975855\pi$
$431$	− 28153.8i	− 0.151559i	0.997125	−	0.0757797i	$-0.0241446\pi$
$432$	0	0
$433$	−322037.	−1.71763	−0.858815	−	0.512285i	$-0.828799\pi$
$433$	−322037.	−1.71763	−0.858815	+	0.512285i	$0.828799\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 45926.4i	− 0.240492i
$438$	0	0
$439$	292438.i	1.51742i	0.651430	+	0.758709i	$0.274170\pi$
$439$	292438.i	1.51742i	−0.651430	+	0.758709i	$0.725830\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	227681.	1.16016	0.580082	−	0.814558i	$-0.303020\pi$
$443$	227681.	1.16016	0.580082	+	0.814558i	$0.303020\pi$
$444$	0	0
$445$	33583.0i	0.169590i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−159097.	−0.789169	−0.394585	−	0.918860i	$-0.629111\pi$
$449$	−159097.	−0.789169	−0.394585	+	0.918860i	$0.629111\pi$
$450$	0	0
$451$	270139.	1.32811
$452$	0	0
$453$	0	0
$454$	0	0
$455$	12590.1i	0.0608143i
$456$	0	0
$457$	340590.	1.63079	0.815397	−	0.578902i	$-0.196519\pi$
$457$	340590.	1.63079	0.815397	+	0.578902i	$0.196519\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	178561.i	0.840204i	0.907477	+	0.420102i	$0.138006\pi$
$461$	178561.i	0.840204i	−0.907477	+	0.420102i	$0.861994\pi$
$462$	0	0
$463$	− 4179.11i	− 0.0194950i	−0.999952	−	0.00974748i	$-0.996897\pi$
$463$	− 4179.11i	− 0.0194950i	0.999952	−	0.00974748i	$-0.00310277\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	160583.	0.736318	0.368159	−	0.929763i	$-0.379988\pi$
$467$	160583.	0.736318	0.368159	+	0.929763i	$0.379988\pi$
$468$	0	0
$469$	− 37766.8i	− 0.171698i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	238606.	1.06649
$474$	0	0
$475$	−12885.8	−0.0571114
$476$	0	0
$477$	0	0
$478$	0	0
$479$	153260.i	0.667970i	0.942578	+	0.333985i	$0.108393\pi$
$479$	153260.i	0.667970i	−0.942578	+	0.333985i	$0.891607\pi$
$480$	0	0
$481$	−45370.0	−0.196100
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 73861.3i	− 0.314003i
$486$	0	0
$487$	113347.i	0.477917i	0.971030	+	0.238958i	$0.0768059\pi$
$487$	113347.i	0.477917i	−0.971030	+	0.238958i	$0.923194\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	248336.	1.03009	0.515046	−	0.857162i	$-0.327775\pi$
$491$	248336.	1.03009	0.515046	+	0.857162i	$0.327775\pi$
$492$	0	0
$493$	49335.4i	0.202985i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	89013.1	0.360364
$498$	0	0
$499$	−378612.	−1.52052	−0.760262	−	0.649617i	$-0.774929\pi$
$499$	−378612.	−1.52052	−0.760262	+	0.649617i	$0.774929\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 255639.i	− 1.01040i	−0.863004	−	0.505198i	$-0.831420\pi$
$503$	− 255639.i	− 1.01040i	0.863004	−	0.505198i	$-0.168580\pi$
$504$	0	0
$505$	11601.4	0.0454913
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 204106.i	− 0.787809i	−0.919151	−	0.393904i	$-0.871124\pi$
$509$	− 204106.i	− 0.787809i	0.919151	−	0.393904i	$-0.128876\pi$
$510$	0	0
$511$	− 2496.04i	− 0.00955895i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−477284.	−1.79954
$516$	0	0
$517$	− 402822.i	− 1.50707i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	295131.	1.08727	0.543637	−	0.839320i	$-0.317047\pi$
$521$	295131.	1.08727	0.543637	+	0.839320i	$0.317047\pi$
$522$	0	0
$523$	−212950.	−0.778527	−0.389263	−	0.921127i	$-0.627270\pi$
$523$	−212950.	−0.778527	−0.389263	+	0.921127i	$0.627270\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	88936.0i	0.320226i
$528$	0	0
$529$	232598.	0.831178
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 136279.i	− 0.479704i
$534$	0	0
$535$	− 294084.i	− 1.02746i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	259056.	0.891694
$540$	0	0
$541$	− 387756.i	− 1.32484i	−0.749132	−	0.662421i	$-0.769529\pi$
$541$	− 387756.i	− 1.32484i	0.749132	−	0.662421i	$-0.230471\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−369108.	−1.24268
$546$	0	0
$547$	241227.	0.806215	0.403107	−	0.915153i	$-0.367930\pi$
$547$	241227.	0.806215	0.403107	+	0.915153i	$0.367930\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 130318.i	− 0.429240i
$552$	0	0
$553$	−54006.8	−0.176603
$554$	0	0
$555$	0	0
$556$	0	0
$557$	86454.9i	0.278663i	0.990246	+	0.139331i	$0.0444954\pi$
$557$	86454.9i	0.278663i	−0.990246	+	0.139331i	$0.955505\pi$
$558$	0	0
$559$	− 120371.i	− 0.385210i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	99585.2	0.314179	0.157090	−	0.987584i	$-0.449789\pi$
$563$	99585.2	0.314179	0.157090	+	0.987584i	$0.449789\pi$
$564$	0	0
$565$	− 276762.i	− 0.866982i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	524085.	1.61874	0.809371	−	0.587298i	$-0.199808\pi$
$569$	524085.	1.61874	0.809371	+	0.587298i	$0.199808\pi$
$570$	0	0
$571$	−32749.2	−0.100445	−0.0502225	−	0.998738i	$-0.515993\pi$
$571$	−32749.2	−0.100445	−0.0502225	+	0.998738i	$0.515993\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	13255.3i	0.0400915i
$576$	0	0
$577$	−140276.	−0.421338	−0.210669	−	0.977557i	$-0.567564\pi$
$577$	−140276.	−0.421338	−0.210669	+	0.977557i	$0.567564\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	68371.8i	0.202547i
$582$	0	0
$583$	− 93378.4i	− 0.274732i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−303094.	−0.879634	−0.439817	−	0.898087i	$-0.644957\pi$
$587$	−303094.	−0.879634	−0.439817	+	0.898087i	$0.644957\pi$
$588$	0	0
$589$	− 234921.i	− 0.677161i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−117438.	−0.333963	−0.166981	−	0.985960i	$-0.553402\pi$
$593$	−117438.	−0.333963	−0.166981	+	0.985960i	$0.553402\pi$
$594$	0	0
$595$	−17824.3	−0.0503475
$596$	0	0
$597$	0	0
$598$	0	0
$599$	537187.i	1.49717i	0.663037	+	0.748587i	$0.269267\pi$
$599$	537187.i	1.49717i	−0.663037	+	0.748587i	$0.730733\pi$
$600$	0	0
$601$	382705.	1.05953	0.529767	−	0.848143i	$-0.322279\pi$
$601$	382705.	1.05953	0.529767	+	0.848143i	$0.322279\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 49794.2i	− 0.136040i
$606$	0	0
$607$	− 578934.i	− 1.57127i	−0.618688	−	0.785637i	$-0.712336\pi$
$607$	− 578934.i	− 1.57127i	0.618688	−	0.785637i	$-0.287664\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−203214.	−0.544341
$612$	0	0
$613$	− 287542.i	− 0.765209i	−0.923912	−	0.382605i	$-0.875027\pi$
$613$	− 287542.i	− 0.765209i	0.923912	−	0.382605i	$-0.124973\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	400806.	1.05284	0.526422	−	0.850223i	$-0.323533\pi$
$617$	400806.	1.05284	0.526422	+	0.850223i	$0.323533\pi$
$618$	0	0
$619$	233469.	0.609323	0.304661	−	0.952461i	$-0.401457\pi$
$619$	233469.	0.609323	0.304661	+	0.952461i	$0.401457\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	13267.6i	0.0341835i
$624$	0	0
$625$	−348791.	−0.892904
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 64232.1i	− 0.162349i
$630$	0	0
$631$	− 499538.i	− 1.25461i	−0.778773	−	0.627306i	$-0.784157\pi$
$631$	− 499538.i	− 1.25461i	0.778773	−	0.627306i	$-0.215843\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−112191.	−0.278235
$636$	0	0
$637$	− 130687.i	− 0.322073i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−405774.	−0.987571	−0.493786	−	0.869584i	$-0.664387\pi$
$641$	−405774.	−0.987571	−0.493786	+	0.869584i	$0.664387\pi$
$642$	0	0
$643$	550571.	1.33165	0.665827	−	0.746106i	$-0.268079\pi$
$643$	550571.	1.33165	0.665827	+	0.746106i	$0.268079\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	428220.i	1.02296i	0.859295	+	0.511480i	$0.170903\pi$
$647$	428220.i	1.02296i	−0.859295	+	0.511480i	$0.829097\pi$
$648$	0	0
$649$	−146658.	−0.348190
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 802147.i	− 1.88117i	−0.339561	−	0.940584i	$-0.610279\pi$
$653$	− 802147.i	− 1.88117i	0.339561	−	0.940584i	$-0.389721\pi$
$654$	0	0
$655$	228771.i	0.533234i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	662958.	1.52656	0.763282	−	0.646065i	$-0.223586\pi$
$659$	662958.	1.52656	0.763282	+	0.646065i	$0.223586\pi$
$660$	0	0
$661$	− 231396.i	− 0.529607i	−0.964302	−	0.264804i	$-0.914693\pi$
$661$	− 231396.i	− 0.529607i	0.964302	−	0.264804i	$-0.0853071\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	47082.2	0.106467
$666$	0	0
$667$	−134055.	−0.301321
$668$	0	0
$669$	0	0
$670$	0	0
$671$	536029.i	1.19054i
$672$	0	0
$673$	311867.	0.688557	0.344278	−	0.938868i	$-0.388124\pi$
$673$	311867.	0.688557	0.344278	+	0.938868i	$0.388124\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 655173.i	− 1.42948i	−0.699389	−	0.714741i	$-0.746545\pi$
$677$	− 655173.i	− 1.42948i	0.699389	−	0.714741i	$-0.253455\pi$
$678$	0	0
$679$	− 29180.3i	− 0.0632923i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−358002.	−0.767439	−0.383719	−	0.923450i	$-0.625357\pi$
$683$	−358002.	−0.767439	−0.383719	+	0.923450i	$0.625357\pi$
$684$	0	0
$685$	− 389664.i	− 0.830442i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−47107.1	−0.0992312
$690$	0	0
$691$	−826006.	−1.72992	−0.864962	−	0.501838i	$-0.832657\pi$
$691$	−826006.	−1.72992	−0.864962	+	0.501838i	$0.832657\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	606648.i	1.25593i
$696$	0	0
$697$	192935.	0.397142
$698$	0	0
$699$	0	0
$700$	0	0
$701$	229057.i	0.466130i	0.972461	+	0.233065i	$0.0748756\pi$
$701$	229057.i	0.466130i	−0.972461	+	0.233065i	$0.925124\pi$
$702$	0	0
$703$	169667.i	0.343310i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4583.37	0.00916950
$708$	0	0
$709$	− 6312.88i	− 0.0125584i	−0.999980	−	0.00627921i	$-0.998001\pi$
$709$	− 6312.88i	− 0.0125584i	0.999980	−	0.00627921i	$-0.00199875\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−241658.	−0.475359
$714$	0	0
$715$	150291.	0.293982
$716$	0	0
$717$	0	0
$718$	0	0
$719$	179483.i	0.347189i	0.984817	+	0.173594i	$0.0555382\pi$
$719$	179483.i	0.347189i	−0.984817	+	0.173594i	$0.944462\pi$
$720$	0	0
$721$	−188560.	−0.362727
$722$	0	0
$723$	0	0
$724$	0	0
$725$	37612.2i	0.0715571i
$726$	0	0
$727$	− 310931.i	− 0.588295i	−0.955760	−	0.294147i	$-0.904964\pi$
$727$	− 310931.i	− 0.588295i	0.955760	−	0.294147i	$-0.0950357\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	170414.	0.318911
$732$	0	0
$733$	− 602271.i	− 1.12094i	−0.828173	−	0.560472i	$-0.810620\pi$
$733$	− 602271.i	− 1.12094i	0.828173	−	0.560472i	$-0.189380\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−450832.	−0.830002
$738$	0	0
$739$	64180.3	0.117520	0.0587602	−	0.998272i	$-0.481285\pi$
$739$	64180.3	0.117520	0.0587602	+	0.998272i	$0.481285\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 525793.i	− 0.952440i	−0.879326	−	0.476220i	$-0.842007\pi$
$743$	− 525793.i	− 0.952440i	0.879326	−	0.476220i	$-0.157993\pi$
$744$	0	0
$745$	−444502.	−0.800868
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 116184.i	− 0.207101i
$750$	0	0
$751$	− 405381.i	− 0.718759i	−0.933191	−	0.359380i	$-0.882988\pi$
$751$	− 405381.i	− 0.718759i	0.933191	−	0.359380i	$-0.117012\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	685398.	1.20240
$756$	0	0
$757$	− 5071.58i	− 0.00885016i	−0.999990	−	0.00442508i	$-0.998591\pi$
$757$	− 5071.58i	− 0.00885016i	0.999990	−	0.00442508i	$-0.00140855\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	29104.4	0.0502561	0.0251280	−	0.999684i	$-0.492001\pi$
$761$	29104.4	0.0502561	0.0251280	+	0.999684i	$0.492001\pi$
$762$	0	0
$763$	−145823.	−0.250483
$764$	0	0
$765$	0	0
$766$	0	0
$767$	73985.4i	0.125764i
$768$	0	0
$769$	−235095.	−0.397548	−0.198774	−	0.980045i	$-0.563696\pi$
$769$	−235095.	−0.397548	−0.198774	+	0.980045i	$0.563696\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	89776.8i	0.150247i	0.997174	+	0.0751234i	$0.0239351\pi$
$773$	89776.8i	0.150247i	−0.997174	+	0.0751234i	$0.976065\pi$
$774$	0	0
$775$	67802.8i	0.112887i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−509632.	−0.839812
$780$	0	0
$781$	− 1.06257e6i	− 1.74203i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−569099.	−0.923525
$786$	0	0
$787$	−406715.	−0.656660	−0.328330	−	0.944563i	$-0.606486\pi$
$787$	−406715.	−0.656660	−0.328330	+	0.944563i	$0.606486\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 109340.i	− 0.174754i
$792$	0	0
$793$	270414.	0.430013
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.02634e6i	− 1.61576i	−0.589350	−	0.807878i	$-0.700616\pi$
$797$	− 1.02634e6i	− 1.61576i	0.589350	−	0.807878i	$-0.299384\pi$
$798$	0	0
$799$	− 287698.i	− 0.450654i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−29795.9	−0.0462089
$804$	0	0
$805$	− 48432.3i	− 0.0747383i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−245306.	−0.374810	−0.187405	−	0.982283i	$-0.560008\pi$
$809$	−245306.	−0.374810	−0.187405	+	0.982283i	$0.560008\pi$
$810$	0	0
$811$	1.26760e6	1.92726	0.963628	−	0.267246i	$-0.0861137\pi$
$811$	1.26760e6	1.92726	0.963628	+	0.267246i	$0.0861137\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 580065.i	− 0.873296i
$816$	0	0
$817$	−450142.	−0.674382
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 163484.i	− 0.242542i	−0.992619	−	0.121271i	$-0.961303\pi$
$821$	− 163484.i	− 0.242542i	0.992619	−	0.121271i	$-0.0386971\pi$
$822$	0	0
$823$	913000.i	1.34794i	0.738758	+	0.673971i	$0.235413\pi$
$823$	913000.i	1.34794i	−0.738758	+	0.673971i	$0.764587\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−378235.	−0.553032	−0.276516	−	0.961009i	$-0.589180\pi$
$827$	−378235.	−0.553032	−0.276516	+	0.961009i	$0.589180\pi$
$828$	0	0
$829$	70195.3i	0.102141i	0.998695	+	0.0510704i	$0.0162633\pi$
$829$	70195.3i	0.102141i	−0.998695	+	0.0510704i	$0.983737\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	185019.	0.266641
$834$	0	0
$835$	−73346.5	−0.105198
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 410513.i	− 0.583180i	−0.956543	−	0.291590i	$-0.905816\pi$
$839$	− 410513.i	− 0.583180i	0.956543	−	0.291590i	$-0.0941843\pi$
$840$	0	0
$841$	326897.	0.462188
$842$	0	0
$843$	0	0
$844$	0	0
$845$	602478.i	0.843777i
$846$	0	0
$847$	− 19672.2i	− 0.0274211i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	174532.	0.240999
$852$	0	0
$853$	− 577725.i	− 0.794005i	−0.917817	−	0.397003i	$-0.870050\pi$
$853$	− 577725.i	− 0.794005i	0.917817	−	0.397003i	$-0.129950\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	388294.	0.528687	0.264344	−	0.964429i	$-0.414845\pi$
$857$	388294.	0.528687	0.264344	+	0.964429i	$0.414845\pi$
$858$	0	0
$859$	124443.	0.168650	0.0843248	−	0.996438i	$-0.473127\pi$
$859$	124443.	0.168650	0.0843248	+	0.996438i	$0.473127\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 652166.i	− 0.875662i	−0.899057	−	0.437831i	$-0.855747\pi$
$863$	− 652166.i	− 0.875662i	0.899057	−	0.437831i	$-0.144253\pi$
$864$	0	0
$865$	−442229.	−0.591038
$866$	0	0
$867$	0	0
$868$	0	0
$869$	644693.i	0.853716i
$870$	0	0
$871$	227434.i	0.299791i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−152855.	−0.199647
$876$	0	0
$877$	− 75235.9i	− 0.0978196i	−0.998803	−	0.0489098i	$-0.984425\pi$
$877$	− 75235.9i	− 0.0978196i	0.998803	−	0.0489098i	$-0.0155747\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	916209.	1.18044	0.590218	−	0.807244i	$-0.299042\pi$
$881$	916209.	1.18044	0.590218	+	0.807244i	$0.299042\pi$
$882$	0	0
$883$	−1.14981e6	−1.47471	−0.737354	−	0.675506i	$-0.763925\pi$
$883$	−1.14981e6	−1.47471	−0.737354	+	0.675506i	$0.763925\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.01630e6i	− 1.29174i	−0.763448	−	0.645870i	$-0.776495\pi$
$887$	− 1.01630e6i	− 1.29174i	0.763448	−	0.645870i	$-0.223505\pi$
$888$	0	0
$889$	−44323.3	−0.0560827
$890$	0	0
$891$	0	0
$892$	0	0
$893$	759945.i	0.952970i
$894$	0	0
$895$	1.20501e6i	1.50433i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−685711.	−0.848442
$900$	0	0
$901$	− 66691.5i	− 0.0821525i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.37864e6	−1.68327
$906$	0	0
$907$	1.00415e6	1.22063	0.610315	−	0.792159i	$-0.291043\pi$
$907$	1.00415e6	1.22063	0.610315	+	0.792159i	$0.291043\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 231087.i	− 0.278444i	−0.990261	−	0.139222i	$-0.955540\pi$
$911$	− 231087.i	− 0.278444i	0.990261	−	0.139222i	$-0.0444602\pi$
$912$	0	0
$913$	816172.	0.979129
$914$	0	0
$915$	0	0
$916$	0	0
$917$	90380.4i	0.107482i
$918$	0	0
$919$	545658.i	0.646084i	0.946385	+	0.323042i	$0.104706\pi$
$919$	545658.i	0.646084i	−0.946385	+	0.323042i	$0.895294\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−536041.	−0.629209
$924$	0	0
$925$	− 48969.2i	− 0.0572320i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	243550.	0.282200	0.141100	−	0.989995i	$-0.454936\pi$
$929$	243550.	0.282200	0.141100	+	0.989995i	$0.454936\pi$
$930$	0	0
$931$	−488722.	−0.563849
$932$	0	0
$933$	0	0
$934$	0	0
$935$	212773.i	0.243385i
$936$	0	0
$937$	479076.	0.545664	0.272832	−	0.962062i	$-0.412040\pi$
$937$	479076.	0.545664	0.272832	+	0.962062i	$0.412040\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1.25655e6i	− 1.41906i	−0.704675	−	0.709530i	$-0.748907\pi$
$941$	− 1.25655e6i	− 1.41906i	0.704675	−	0.709530i	$-0.251093\pi$
$942$	0	0
$943$	524246.i	0.589538i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.22304e6	1.36377	0.681886	−	0.731459i	$-0.261160\pi$
$947$	1.22304e6	1.36377	0.681886	+	0.731459i	$0.261160\pi$
$948$	0	0
$949$	15031.3i	0.0166903i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.12329e6	1.23682	0.618409	−	0.785856i	$-0.287777\pi$
$953$	1.12329e6	1.23682	0.618409	+	0.785856i	$0.287777\pi$
$954$	0	0
$955$	−505512.	−0.554275
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 153944.i	− 0.167389i
$960$	0	0
$961$	−312598.	−0.338485
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 619240.i	− 0.664974i
$966$	0	0
$967$	310108.i	0.331635i	0.986156	+	0.165818i	$0.0530263\pi$
$967$	310108.i	0.331635i	−0.986156	+	0.165818i	$0.946974\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	459519.	0.487376	0.243688	−	0.969854i	$-0.421643\pi$
$971$	459519.	0.487376	0.243688	+	0.969854i	$0.421643\pi$
$972$	0	0
$973$	239668.i	0.253154i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	645770.	0.676532	0.338266	−	0.941050i	$-0.390160\pi$
$977$	645770.	0.676532	0.338266	+	0.941050i	$0.390160\pi$
$978$	0	0
$979$	158379.	0.165246
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 828941.i	− 0.857860i	−0.903338	−	0.428930i	$-0.858891\pi$
$983$	− 828941.i	− 0.857860i	0.903338	−	0.428930i	$-0.141109\pi$
$984$	0	0
$985$	−112363.	−0.115811
$986$	0	0
$987$	0	0
$988$	0	0
$989$	463050.i	0.473407i
$990$	0	0
$991$	460245.i	0.468642i	0.972159	+	0.234321i	$0.0752867\pi$
$991$	460245.i	0.468642i	−0.972159	+	0.234321i	$0.924713\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	915803.	0.925030
$996$	0	0
$997$	1.82730e6i	1.83831i	0.393894	+	0.919156i	$0.371128\pi$
$997$	1.82730e6i	1.83831i	−0.393894	+	0.919156i	$0.628872\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.5.b.k.703.6		8
3.2	odd	2		384.5.b.c.319.2	✓	8
4.3	odd	2	inner	1152.5.b.k.703.5		8
8.3	odd	2	inner	1152.5.b.k.703.3		8
8.5	even	2	inner	1152.5.b.k.703.4		8
12.11	even	2		384.5.b.c.319.6	yes	8
24.5	odd	2		384.5.b.c.319.7	yes	8
24.11	even	2		384.5.b.c.319.3	yes	8
48.5	odd	4		768.5.g.c.511.4		4
48.11	even	4		768.5.g.c.511.2		4
48.29	odd	4		768.5.g.g.511.1		4
48.35	even	4		768.5.g.g.511.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.b.c.319.2	✓	8	3.2	odd	2
384.5.b.c.319.3	yes	8	24.11	even	2
384.5.b.c.319.6	yes	8	12.11	even	2
384.5.b.c.319.7	yes	8	24.5	odd	2
768.5.g.c.511.2		4	48.11	even	4
768.5.g.c.511.4		4	48.5	odd	4
768.5.g.g.511.1		4	48.29	odd	4
768.5.g.g.511.3		4	48.35	even	4
1152.5.b.k.703.3		8	8.3	odd	2	inner
1152.5.b.k.703.4		8	8.5	even	2	inner
1152.5.b.k.703.5		8	4.3	odd	2	inner
1152.5.b.k.703.6		8	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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1152.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+23.7490i q^{5} +9.38251i q^{7} +O(q^{10})\)	\(q+23.7490i q^{5} +9.38251i q^{7} +112.001 q^{11} -56.5020i q^{13} +79.9921 q^{17} -211.297 q^{19} +217.355i q^{23} +60.9843 q^{25} +616.753i q^{29} +1111.81i q^{31} -222.825 q^{35} -802.980i q^{37} +2411.93 q^{41} +2130.38 q^{43} -3596.58i q^{47} +2312.97 q^{49} -833.725i q^{53} +2659.92i q^{55} -1309.43 q^{59} +4785.91i q^{61} +1341.87 q^{65} -4025.23 q^{67} -9487.13i q^{71} -266.031 q^{73} +1050.85i q^{77} +5756.11i q^{79} +7287.16 q^{83} +1899.73i q^{85} +1414.08 q^{89} +530.130 q^{91} -5018.09i q^{95} -3110.08 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 1392 q^{17} - 3576 q^{25} + 1008 q^{41} + 10376 q^{49} - 23808 q^{65} - 10256 q^{73} + 31632 q^{89} - 45200 q^{97}+O(q^{100}) \) 8 * q - 1392 * q^17 - 3576 * q^25 + 1008 * q^41 + 10376 * q^49 - 23808 * q^65 - 10256 * q^73 + 31632 * q^89 - 45200 * q^97

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