LMFDB - Embedded newform 1152.4.d.q.577.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1152,4,Mod(577,1152)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1152.577"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1152, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1152.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$67.9702003266$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1439868559360000.260
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 28x^{6} + 1023x^{4} - 9212x^{2} + 48841 $ x^8 + 28x^6 + 1023x^4 - 9212*x^2 + 48841
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{26} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.4
Root			$2.97558 - 5.19407i$ of defining polynomial
Character	$\chi$	$=$	1152.577
Dual form			1152.4.d.q.577.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-11.8322i q^{5} +20.9762 q^{7} +49.6387i q^{11} +70.1997i q^{13} -118.659 q^{17} -133.866i q^{19} -45.2548 q^{23} -15.0000 q^{25} +130.154i q^{29} -188.786 q^{31} -248.193i q^{35} -210.599i q^{37} -118.659 q^{41} -401.597i q^{43} -316.784 q^{47} +97.0000 q^{49} -556.111i q^{53} +587.333 q^{55} +99.2774i q^{59} -350.999i q^{61} +830.614 q^{65} +267.731i q^{67} -905.097 q^{71} -350.000 q^{73} +1041.23i q^{77} -440.500 q^{79} +1240.97i q^{83} +1403.99i q^{85} +711.955 q^{89} +1472.52i q^{91} -1583.92 q^{95} +770.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 120 q^{25} + 776 q^{49} - 2800 q^{73} + 6160 q^{97}+O(q^{100}) $ 8 * q - 120 * q^25 + 776 * q^49 - 2800 * q^73 + 6160 * q^97

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
$5$	− 11.8322i	− 1.05830i	−0.848528	−	0.529150i	$-0.822511\pi$
$5$	− 11.8322i	− 1.05830i	0.848528	−	0.529150i	$-0.177489\pi$
$6$	0	0
$7$	20.9762	1.13261	0.566304	−	0.824197i	$-0.308373\pi$
$7$	20.9762	1.13261	0.566304	+	0.824197i	$0.308373\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	49.6387i	1.36060i	0.732933	+	0.680301i	$0.238151\pi$
$11$	49.6387i	1.36060i	−0.732933	+	0.680301i	$0.761849\pi$
$12$	0	0
$13$	70.1997i	1.49768i	0.662748	+	0.748842i	$0.269390\pi$
$13$	70.1997i	1.49768i	−0.662748	+	0.748842i	$0.730610\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−118.659	−1.69289	−0.846443	−	0.532479i	$-0.821261\pi$
$17$	−118.659	−1.69289	−0.846443	+	0.532479i	$0.821261\pi$
$18$	0	0
$19$	− 133.866i	− 1.61636i	−0.588934	−	0.808181i	$-0.700452\pi$
$19$	− 133.866i	− 1.61636i	0.588934	−	0.808181i	$-0.299548\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−45.2548	−0.410273	−0.205137	−	0.978733i	$-0.565764\pi$
$23$	−45.2548	−0.410273	−0.205137	+	0.978733i	$0.565764\pi$
$24$	0	0
$25$	−15.0000	−0.120000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	130.154i	0.833412i	0.909041	+	0.416706i	$0.136816\pi$
$29$	130.154i	0.833412i	−0.909041	+	0.416706i	$0.863184\pi$
$30$	0	0
$31$	−188.786	−1.09377	−0.546885	−	0.837207i	$-0.684187\pi$
$31$	−188.786	−1.09377	−0.546885	+	0.837207i	$0.684187\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 248.193i	− 1.19864i
$36$	0	0
$37$	− 210.599i	− 0.935737i	−0.883798	−	0.467869i	$-0.845022\pi$
$37$	− 210.599i	− 0.935737i	0.883798	−	0.467869i	$-0.154978\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−118.659	−0.451987	−0.225993	−	0.974129i	$-0.572563\pi$
$41$	−118.659	−0.451987	−0.225993	+	0.974129i	$0.572563\pi$
$42$	0	0
$43$	− 401.597i	− 1.42425i	−0.702050	−	0.712127i	$-0.747732\pi$
$43$	− 401.597i	− 1.42425i	0.702050	−	0.712127i	$-0.252268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−316.784	−0.983142	−0.491571	−	0.870838i	$-0.663577\pi$
$47$	−316.784	−0.983142	−0.491571	+	0.870838i	$0.663577\pi$
$48$	0	0
$49$	97.0000	0.282799
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 556.111i	− 1.44128i	−0.693310	−	0.720640i	$-0.743848\pi$
$53$	− 556.111i	− 1.44128i	0.693310	−	0.720640i	$-0.256152\pi$
$54$	0	0
$55$	587.333	1.43993
$56$	0	0
$57$	0	0
$58$	0	0
$59$	99.2774i	0.219065i	0.993983	+	0.109532i	$0.0349353\pi$
$59$	99.2774i	0.219065i	−0.993983	+	0.109532i	$0.965065\pi$
$60$	0	0
$61$	− 350.999i	− 0.736734i	−0.929680	−	0.368367i	$-0.879917\pi$
$61$	− 350.999i	− 0.736734i	0.929680	−	0.368367i	$-0.120083\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	830.614	1.58500
$66$	0	0
$67$	267.731i	0.488188i	0.969752	+	0.244094i	$0.0784905\pi$
$67$	267.731i	0.488188i	−0.969752	+	0.244094i	$0.921510\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−905.097	−1.51289	−0.756445	−	0.654057i	$-0.773066\pi$
$71$	−905.097	−1.51289	−0.756445	+	0.654057i	$0.773066\pi$
$72$	0	0
$73$	−350.000	−0.561156	−0.280578	−	0.959831i	$-0.590526\pi$
$73$	−350.000	−0.561156	−0.280578	+	0.959831i	$0.590526\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1041.23i	1.54103i
$78$	0	0
$79$	−440.500	−0.627343	−0.313671	−	0.949532i	$-0.601559\pi$
$79$	−440.500	−0.627343	−0.313671	+	0.949532i	$0.601559\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1240.97i	1.64113i	0.571553	+	0.820565i	$0.306341\pi$
$83$	1240.97i	1.64113i	−0.571553	+	0.820565i	$0.693659\pi$
$84$	0	0
$85$	1403.99i	1.79158i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	711.955	0.847945	0.423972	−	0.905675i	$-0.360635\pi$
$89$	711.955	0.847945	0.423972	+	0.905675i	$0.360635\pi$
$90$	0	0
$91$	1472.52i	1.69629i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1583.92	−1.71060
$96$	0	0
$97$	770.000	0.805996	0.402998	−	0.915201i	$-0.367968\pi$
$97$	770.000	0.805996	0.402998	+	0.915201i	$0.367968\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 366.797i	− 0.361363i	−0.983542	−	0.180681i	$-0.942170\pi$
$101$	− 366.797i	− 0.361363i	0.983542	−	0.180681i	$-0.0578303\pi$
$102$	0	0
$103$	−1279.55	−1.22405	−0.612027	−	0.790837i	$-0.709645\pi$
$103$	−1279.55	−1.22405	−0.612027	+	0.790837i	$0.709645\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1489.16i	− 1.34544i	−0.739895	−	0.672722i	$-0.765125\pi$
$107$	− 1489.16i	− 1.34544i	0.739895	−	0.672722i	$-0.234875\pi$
$108$	0	0
$109$	350.999i	0.308436i	0.988037	+	0.154218i	$0.0492859\pi$
$109$	350.999i	0.308436i	−0.988037	+	0.154218i	$0.950714\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1661.23	−1.38297	−0.691483	−	0.722392i	$-0.743042\pi$
$113$	−1661.23	−1.38297	−0.691483	+	0.722392i	$0.743042\pi$
$114$	0	0
$115$	535.462i	0.434192i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2489.02	−1.91738
$120$	0	0
$121$	−1133.00	−0.851240
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1301.54i	− 0.931304i
$126$	0	0
$127$	2202.50	1.53890	0.769449	−	0.638708i	$-0.220531\pi$
$127$	2202.50	1.53890	0.769449	+	0.638708i	$0.220531\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 1786.99i	− 1.19183i	−0.803046	−	0.595917i	$-0.796789\pi$
$131$	− 1786.99i	− 1.19183i	0.803046	−	0.595917i	$-0.203211\pi$
$132$	0	0
$133$	− 2807.99i	− 1.83070i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−830.614	−0.517987	−0.258993	−	0.965879i	$-0.583391\pi$
$137$	−830.614	−0.517987	−0.258993	+	0.965879i	$0.583391\pi$
$138$	0	0
$139$	267.731i	0.163372i	0.996658	+	0.0816858i	$0.0260304\pi$
$139$	267.731i	0.163372i	−0.996658	+	0.0816858i	$0.973970\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3484.62	−2.03775
$144$	0	0
$145$	1540.00	0.882000
$146$	0	0
$147$	0	0
$148$	0	0
$149$	650.769i	0.357806i	0.983867	+	0.178903i	$0.0572548\pi$
$149$	650.769i	0.357806i	−0.983867	+	0.178903i	$0.942745\pi$
$150$	0	0
$151$	440.500	0.237400	0.118700	−	0.992930i	$-0.462127\pi$
$151$	440.500	0.237400	0.118700	+	0.992930i	$0.462127\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2233.74i	1.15754i
$156$	0	0
$157$	210.599i	0.107055i	0.998566	+	0.0535275i	$0.0170465\pi$
$157$	210.599i	0.107055i	−0.998566	+	0.0535275i	$0.982954\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−949.273	−0.464678
$162$	0	0
$163$	− 133.866i	− 0.0643262i	−0.999483	−	0.0321631i	$-0.989760\pi$
$163$	− 133.866i	− 0.0643262i	0.999483	−	0.0321631i	$-0.0102396\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3484.62	−1.61466	−0.807330	−	0.590100i	$-0.799088\pi$
$167$	−3484.62	−1.61466	−0.807330	+	0.590100i	$0.799088\pi$
$168$	0	0
$169$	−2731.00	−1.24306
$170$	0	0
$171$	0	0
$172$	0	0
$173$	911.076i	0.400392i	0.979756	+	0.200196i	$0.0641579\pi$
$173$	911.076i	0.400392i	−0.979756	+	0.200196i	$0.935842\pi$
$174$	0	0
$175$	−314.643	−0.135913
$176$	0	0
$177$	0	0
$178$	0	0
$179$	893.497i	0.373090i	0.982446	+	0.186545i	$0.0597290\pi$
$179$	893.497i	0.373090i	−0.982446	+	0.186545i	$0.940271\pi$
$180$	0	0
$181$	− 350.999i	− 0.144141i	−0.997400	−	0.0720705i	$-0.977039\pi$
$181$	− 350.999i	− 0.144141i	0.997400	−	0.0720705i	$-0.0229607\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2491.84	−0.990291
$186$	0	0
$187$	− 5890.09i	− 2.30335i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3846.66	−1.45725	−0.728625	−	0.684913i	$-0.759840\pi$
$191$	−3846.66	−1.45725	−0.728625	+	0.684913i	$0.759840\pi$
$192$	0	0
$193$	4230.00	1.57763	0.788814	−	0.614632i	$-0.210696\pi$
$193$	4230.00	1.57763	0.788814	+	0.614632i	$0.210696\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4082.10i	1.47633i	0.674620	+	0.738166i	$0.264308\pi$
$197$	4082.10i	1.47633i	−0.674620	+	0.738166i	$0.735692\pi$
$198$	0	0
$199$	−4803.54	−1.71113	−0.855563	−	0.517698i	$-0.826789\pi$
$199$	−4803.54	−1.71113	−0.855563	+	0.517698i	$0.826789\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2730.13i	0.943928i
$204$	0	0
$205$	1403.99i	0.478338i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6644.91	2.19923
$210$	0	0
$211$	− 2409.58i	− 0.786172i	−0.919502	−	0.393086i	$-0.871407\pi$
$211$	− 2409.58i	− 0.786172i	0.919502	−	0.393086i	$-0.128593\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4751.76	−1.50729
$216$	0	0
$217$	−3960.00	−1.23881
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 8329.84i	− 2.53541i
$222$	0	0
$223$	104.881	0.0314948	0.0157474	−	0.999876i	$-0.494987\pi$
$223$	104.881	0.0314948	0.0157474	+	0.999876i	$0.494987\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2233.74i	− 0.653122i	−0.945176	−	0.326561i	$-0.894110\pi$
$227$	− 2233.74i	− 0.653122i	0.945176	−	0.326561i	$-0.105890\pi$
$228$	0	0
$229$	6668.97i	1.92445i	0.272263	+	0.962223i	$0.412228\pi$
$229$	6668.97i	1.92445i	−0.272263	+	0.962223i	$0.587772\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4983.69	1.40125	0.700627	−	0.713528i	$-0.252904\pi$
$233$	4983.69	1.40125	0.700627	+	0.713528i	$0.252904\pi$
$234$	0	0
$235$	3748.24i	1.04046i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2489.02	−0.673645	−0.336822	−	0.941568i	$-0.609352\pi$
$239$	−2489.02	−0.673645	−0.336822	+	0.941568i	$0.609352\pi$
$240$	0	0
$241$	3402.00	0.909303	0.454652	−	0.890669i	$-0.349764\pi$
$241$	3402.00	0.909303	0.454652	+	0.890669i	$0.349764\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1147.72i	− 0.299286i
$246$	0	0
$247$	9397.33	2.42080
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 5509.90i	− 1.38558i	−0.721138	−	0.692792i	$-0.756380\pi$
$251$	− 5509.90i	− 1.38558i	0.721138	−	0.692792i	$-0.243620\pi$
$252$	0	0
$253$	− 2246.39i	− 0.558219i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4271.73	−1.03682	−0.518411	−	0.855132i	$-0.673476\pi$
$257$	−4271.73	−1.03682	−0.518411	+	0.855132i	$0.673476\pi$
$258$	0	0
$259$	− 4417.56i	− 1.05982i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1629.17	−0.381974	−0.190987	−	0.981593i	$-0.561169\pi$
$263$	−1629.17	−0.381974	−0.190987	+	0.981593i	$0.561169\pi$
$264$	0	0
$265$	−6580.00	−1.52531
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5998.90i	1.35970i	0.733351	+	0.679851i	$0.237955\pi$
$269$	5998.90i	1.35970i	−0.733351	+	0.679851i	$0.762045\pi$
$270$	0	0
$271$	−3419.12	−0.766408	−0.383204	−	0.923664i	$-0.625179\pi$
$271$	−3419.12	−0.766408	−0.383204	+	0.923664i	$0.625179\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 744.580i	− 0.163272i
$276$	0	0
$277$	− 4001.38i	− 0.867942i	−0.900927	−	0.433971i	$-0.857112\pi$
$277$	− 4001.38i	− 0.867942i	0.900927	−	0.433971i	$-0.142888\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3322.46	0.705342	0.352671	−	0.935747i	$-0.385273\pi$
$281$	3322.46	0.705342	0.352671	+	0.935747i	$0.385273\pi$
$282$	0	0
$283$	4283.70i	0.899786i	0.893082	+	0.449893i	$0.148538\pi$
$283$	4283.70i	0.899786i	−0.893082	+	0.449893i	$0.851462\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2489.02	−0.511923
$288$	0	0
$289$	9167.00	1.86587
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3892.78i	0.776173i	0.921623	+	0.388086i	$0.126864\pi$
$293$	3892.78i	0.776173i	−0.921623	+	0.388086i	$0.873136\pi$
$294$	0	0
$295$	1174.67	0.231836
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 3176.88i	− 0.614460i
$300$	0	0
$301$	− 8423.97i	− 1.61312i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4153.07	−0.779686
$306$	0	0
$307$	9906.05i	1.84159i	0.390046	+	0.920795i	$0.372459\pi$
$307$	9906.05i	1.84159i	−0.390046	+	0.920795i	$0.627541\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3167.84	0.577594	0.288797	−	0.957390i	$-0.406745\pi$
$311$	3167.84	0.577594	0.288797	+	0.957390i	$0.406745\pi$
$312$	0	0
$313$	−5390.00	−0.973357	−0.486679	−	0.873581i	$-0.661792\pi$
$313$	−5390.00	−0.973357	−0.486679	+	0.873581i	$0.661792\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	35.4965i	0.00628921i	0.999995	+	0.00314461i	$0.00100096\pi$
$317$	35.4965i	0.00628921i	−0.999995	+	0.00314461i	$0.998999\pi$
$318$	0	0
$319$	−6460.66	−1.13394
$320$	0	0
$321$	0	0
$322$	0	0
$323$	15884.4i	2.73632i
$324$	0	0
$325$	− 1053.00i	− 0.179722i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6644.91	−1.11351
$330$	0	0
$331$	− 2677.31i	− 0.444587i	−0.974980	−	0.222294i	$-0.928646\pi$
$331$	− 2677.31i	− 0.444587i	0.974980	−	0.222294i	$-0.0713544\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3167.84	0.516649
$336$	0	0
$337$	4310.00	0.696679	0.348339	−	0.937369i	$-0.386746\pi$
$337$	4310.00	0.696679	0.348339	+	0.937369i	$0.386746\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 9371.07i	− 1.48819i
$342$	0	0
$343$	−5160.14	−0.812307
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3722.90i	0.575953i	0.957637	+	0.287977i	$0.0929826\pi$
$347$	3722.90i	0.575953i	−0.957637	+	0.287977i	$0.907017\pi$
$348$	0	0
$349$	12285.0i	1.88424i	0.335282	+	0.942118i	$0.391168\pi$
$349$	12285.0i	1.88424i	−0.335282	+	0.942118i	$0.608832\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−711.955	−0.107347	−0.0536736	−	0.998559i	$-0.517093\pi$
$353$	−711.955	−0.107347	−0.0536736	+	0.998559i	$0.517093\pi$
$354$	0	0
$355$	10709.2i	1.60109i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	905.097	0.133062	0.0665309	−	0.997784i	$-0.478807\pi$
$359$	905.097	0.133062	0.0665309	+	0.997784i	$0.478807\pi$
$360$	0	0
$361$	−11061.0	−1.61263
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4141.26i	0.593872i
$366$	0	0
$367$	2831.78	0.402774	0.201387	−	0.979512i	$-0.435455\pi$
$367$	2831.78	0.402774	0.201387	+	0.979512i	$0.435455\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 11665.1i	− 1.63240i
$372$	0	0
$373$	− 7792.17i	− 1.08167i	−0.841128	−	0.540835i	$-0.818108\pi$
$373$	− 7792.17i	− 1.08167i	0.841128	−	0.540835i	$-0.181892\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9136.76	−1.24819
$378$	0	0
$379$	4953.03i	0.671293i	0.941988	+	0.335646i	$0.108955\pi$
$379$	4953.03i	0.671293i	−0.941988	+	0.335646i	$0.891045\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	11721.0	1.56375	0.781874	−	0.623437i	$-0.214264\pi$
$383$	11721.0	1.56375	0.781874	+	0.623437i	$0.214264\pi$
$384$	0	0
$385$	12320.0	1.63087
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 7111.13i	− 0.926860i	−0.886134	−	0.463430i	$-0.846619\pi$
$389$	− 7111.13i	− 0.926860i	0.886134	−	0.463430i	$-0.153381\pi$
$390$	0	0
$391$	5369.90	0.694546
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5212.06i	0.663917i
$396$	0	0
$397$	10319.4i	1.30457i	0.757974	+	0.652284i	$0.226189\pi$
$397$	10319.4i	1.30457i	−0.757974	+	0.652284i	$0.773811\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2491.84	0.310316	0.155158	−	0.987890i	$-0.450411\pi$
$401$	2491.84	0.310316	0.155158	+	0.987890i	$0.450411\pi$
$402$	0	0
$403$	− 13252.7i	− 1.63812i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10453.9	1.27317
$408$	0	0
$409$	−5306.00	−0.641479	−0.320739	−	0.947167i	$-0.603931\pi$
$409$	−5306.00	−0.641479	−0.320739	+	0.947167i	$0.603931\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2082.46i	0.248114i
$414$	0	0
$415$	14683.3	1.73681
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 9580.27i	− 1.11701i	−0.829502	−	0.558504i	$-0.811375\pi$
$419$	− 9580.27i	− 1.11701i	0.829502	−	0.558504i	$-0.188625\pi$
$420$	0	0
$421$	9476.96i	1.09710i	0.836118	+	0.548550i	$0.184820\pi$
$421$	9476.96i	1.09710i	−0.836118	+	0.548550i	$0.815180\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1779.89	0.203146
$426$	0	0
$427$	− 7362.61i	− 0.834430i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5656.85	0.632207	0.316103	−	0.948725i	$-0.397625\pi$
$431$	5656.85	0.632207	0.316103	+	0.948725i	$0.397625\pi$
$432$	0	0
$433$	2030.00	0.225302	0.112651	−	0.993635i	$-0.464066\pi$
$433$	2030.00	0.225302	0.112651	+	0.993635i	$0.464066\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6058.07i	0.663150i
$438$	0	0
$439$	−7824.11	−0.850625	−0.425313	−	0.905046i	$-0.639836\pi$
$439$	−7824.11	−0.850625	−0.425313	+	0.905046i	$0.639836\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 4219.29i	− 0.452516i	−0.974067	−	0.226258i	$-0.927351\pi$
$443$	− 4219.29i	− 0.452516i	0.974067	−	0.226258i	$-0.0726492\pi$
$444$	0	0
$445$	− 8423.97i	− 0.897380i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	17442.9	1.83337	0.916683	−	0.399615i	$-0.130856\pi$
$449$	17442.9	1.83337	0.916683	+	0.399615i	$0.130856\pi$
$450$	0	0
$451$	− 5890.09i	− 0.614974i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	17423.1	1.79518
$456$	0	0
$457$	−7530.00	−0.770763	−0.385381	−	0.922757i	$-0.625930\pi$
$457$	−7530.00	−0.770763	−0.385381	+	0.922757i	$0.625930\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	17002.8i	1.71779i	0.512154	+	0.858894i	$0.328848\pi$
$461$	17002.8i	1.71779i	−0.512154	+	0.858894i	$0.671152\pi$
$462$	0	0
$463$	−3670.83	−0.368462	−0.184231	−	0.982883i	$-0.558979\pi$
$463$	−3670.83	−0.368462	−0.184231	+	0.982883i	$0.558979\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 7197.61i	− 0.713203i	−0.934257	−	0.356601i	$-0.883935\pi$
$467$	− 7197.61i	− 0.713203i	0.934257	−	0.356601i	$-0.116065\pi$
$468$	0	0
$469$	5615.98i	0.552925i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	19934.7	1.93784
$474$	0	0
$475$	2007.98i	0.193963i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	12671.4	1.20870	0.604352	−	0.796718i	$-0.293432\pi$
$479$	12671.4	1.20870	0.604352	+	0.796718i	$0.293432\pi$
$480$	0	0
$481$	14784.0	1.40144
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 9110.76i	− 0.852986i
$486$	0	0
$487$	−7194.83	−0.669464	−0.334732	−	0.942313i	$-0.608646\pi$
$487$	−7194.83	−0.669464	−0.334732	+	0.942313i	$0.608646\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6155.20i	0.565744i	0.959158	+	0.282872i	$0.0912871\pi$
$491$	6155.20i	0.565744i	−0.959158	+	0.282872i	$0.908713\pi$
$492$	0	0
$493$	− 15443.9i	− 1.41087i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−18985.5	−1.71351
$498$	0	0
$499$	− 11780.2i	− 1.05682i	−0.848989	−	0.528410i	$-0.822788\pi$
$499$	− 11780.2i	− 1.05682i	0.848989	−	0.528410i	$-0.177212\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6969.24	0.617780	0.308890	−	0.951098i	$-0.400043\pi$
$503$	6969.24	0.617780	0.308890	+	0.951098i	$0.400043\pi$
$504$	0	0
$505$	−4340.00	−0.382431
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 9524.89i	− 0.829437i	−0.909950	−	0.414718i	$-0.863880\pi$
$509$	− 9524.89i	− 0.829437i	0.909950	−	0.414718i	$-0.136120\pi$
$510$	0	0
$511$	−7341.66	−0.635569
$512$	0	0
$513$	0	0
$514$	0	0
$515$	15139.8i	1.29542i
$516$	0	0
$517$	− 15724.7i	− 1.33767i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	3203.80	0.269407	0.134703	−	0.990886i	$-0.456992\pi$
$521$	3203.80	0.269407	0.134703	+	0.990886i	$0.456992\pi$
$522$	0	0
$523$	− 9504.46i	− 0.794648i	−0.917678	−	0.397324i	$-0.869939\pi$
$523$	− 9504.46i	− 0.794648i	0.917678	−	0.397324i	$-0.130061\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	22401.1	1.85163
$528$	0	0
$529$	−10119.0	−0.831676
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 8329.84i	− 0.676933i
$534$	0	0
$535$	−17620.0	−1.42389
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4814.95i	0.384777i
$540$	0	0
$541$	− 13688.9i	− 1.08786i	−0.839130	−	0.543931i	$-0.816935\pi$
$541$	− 13688.9i	− 1.08786i	0.839130	−	0.543931i	$-0.183065\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4153.07	0.326418
$546$	0	0
$547$	− 3078.91i	− 0.240667i	−0.992734	−	0.120333i	$-0.961604\pi$
$547$	− 3078.91i	− 0.240667i	0.992734	−	0.120333i	$-0.0383963\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	17423.1	1.34710
$552$	0	0
$553$	−9240.00	−0.710533
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 21333.4i	− 1.62284i	−0.584460	−	0.811422i	$-0.698694\pi$
$557$	− 21333.4i	− 1.62284i	0.584460	−	0.811422i	$-0.301306\pi$
$558$	0	0
$559$	28192.0	2.13308
$560$	0	0
$561$	0	0
$562$	0	0
$563$	17621.7i	1.31913i	0.751650	+	0.659563i	$0.229259\pi$
$563$	17621.7i	1.31913i	−0.751650	+	0.659563i	$0.770741\pi$
$564$	0	0
$565$	19655.9i	1.46359i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2491.84	−0.183591	−0.0917957	−	0.995778i	$-0.529261\pi$
$569$	−2491.84	−0.183591	−0.0917957	+	0.995778i	$0.529261\pi$
$570$	0	0
$571$	− 7764.21i	− 0.569040i	−0.958670	−	0.284520i	$-0.908166\pi$
$571$	− 7764.21i	− 0.569040i	0.958670	−	0.284520i	$-0.0918342\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	678.823	0.0492328
$576$	0	0
$577$	20090.0	1.44949	0.724747	−	0.689015i	$-0.241957\pi$
$577$	20090.0	1.44949	0.724747	+	0.689015i	$0.241957\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	26030.8i	1.85876i
$582$	0	0
$583$	27604.6	1.96101
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 5956.64i	− 0.418836i	−0.977826	−	0.209418i	$-0.932843\pi$
$587$	− 5956.64i	− 0.418836i	0.977826	−	0.209418i	$-0.0671570\pi$
$588$	0	0
$589$	25271.9i	1.76793i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4034.41	−0.279382	−0.139691	−	0.990195i	$-0.544611\pi$
$593$	−4034.41	−0.279382	−0.139691	+	0.990195i	$0.544611\pi$
$594$	0	0
$595$	29450.4i	2.02916i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	16518.0	1.12672	0.563362	−	0.826210i	$-0.309508\pi$
$599$	16518.0	1.12672	0.563362	+	0.826210i	$0.309508\pi$
$600$	0	0
$601$	11438.0	0.776316	0.388158	−	0.921593i	$-0.373112\pi$
$601$	11438.0	0.776316	0.388158	+	0.921593i	$0.373112\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	13405.8i	0.900867i
$606$	0	0
$607$	−15753.1	−1.05338	−0.526688	−	0.850059i	$-0.676566\pi$
$607$	−15753.1	−1.05338	−0.526688	+	0.850059i	$0.676566\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 22238.1i	− 1.47244i
$612$	0	0
$613$	− 5545.78i	− 0.365403i	−0.983168	−	0.182701i	$-0.941516\pi$
$613$	− 5545.78i	− 0.365403i	0.983168	−	0.182701i	$-0.0584841\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−16612.3	−1.08393	−0.541965	−	0.840401i	$-0.682320\pi$
$617$	−16612.3	−1.08393	−0.541965	+	0.840401i	$0.682320\pi$
$618$	0	0
$619$	17670.3i	1.14738i	0.819073	+	0.573690i	$0.194488\pi$
$619$	17670.3i	1.14738i	−0.819073	+	0.573690i	$0.805512\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	14934.1	0.960388
$624$	0	0
$625$	−17275.0	−1.10560
$626$	0	0
$627$	0	0
$628$	0	0
$629$	24989.5i	1.58410i
$630$	0	0
$631$	5726.50	0.361281	0.180640	−	0.983549i	$-0.442183\pi$
$631$	5726.50	0.361281	0.180640	+	0.983549i	$0.442183\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 26060.3i	− 1.62862i
$636$	0	0
$637$	6809.37i	0.423543i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−19104.1	−1.17717	−0.588586	−	0.808434i	$-0.700315\pi$
$641$	−19104.1	−1.17717	−0.588586	+	0.808434i	$0.700315\pi$
$642$	0	0
$643$	10307.7i	0.632184i	0.948729	+	0.316092i	$0.102371\pi$
$643$	10307.7i	0.632184i	−0.948729	+	0.316092i	$0.897629\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9186.73	−0.558219	−0.279109	−	0.960259i	$-0.590039\pi$
$647$	−9186.73	−0.558219	−0.279109	+	0.960259i	$0.590039\pi$
$648$	0	0
$649$	−4928.00	−0.298060
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 26728.8i	− 1.60181i	−0.598793	−	0.800904i	$-0.704353\pi$
$653$	− 26728.8i	− 1.60181i	0.598793	−	0.800904i	$-0.295647\pi$
$654$	0	0
$655$	−21144.0	−1.26132
$656$	0	0
$657$	0	0
$658$	0	0
$659$	19458.4i	1.15021i	0.818079	+	0.575107i	$0.195039\pi$
$659$	19458.4i	1.15021i	−0.818079	+	0.575107i	$0.804961\pi$
$660$	0	0
$661$	5966.98i	0.351117i	0.984469	+	0.175559i	$0.0561731\pi$
$661$	5966.98i	0.351117i	−0.984469	+	0.175559i	$0.943827\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−33224.6	−1.93743
$666$	0	0
$667$	− 5890.09i	− 0.341927i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	17423.1	1.00240
$672$	0	0
$673$	2270.00	0.130018	0.0650090	−	0.997885i	$-0.479292\pi$
$673$	2270.00	0.130018	0.0650090	+	0.997885i	$0.479292\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 16103.6i	− 0.914196i	−0.889416	−	0.457098i	$-0.848889\pi$
$677$	− 16103.6i	− 0.914196i	0.889416	−	0.457098i	$-0.151111\pi$
$678$	0	0
$679$	16151.7	0.912877
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 20600.1i	− 1.15408i	−0.816714	−	0.577042i	$-0.804207\pi$
$683$	− 20600.1i	− 1.15408i	0.816714	−	0.577042i	$-0.195793\pi$
$684$	0	0
$685$	9827.96i	0.548185i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	39038.9	2.15858
$690$	0	0
$691$	7362.61i	0.405335i	0.979248	+	0.202668i	$0.0649611\pi$
$691$	7362.61i	0.405335i	−0.979248	+	0.202668i	$0.935039\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3167.84	0.172896
$696$	0	0
$697$	14080.0	0.765162
$698$	0	0
$699$	0	0
$700$	0	0
$701$	13855.5i	0.746524i	0.927726	+	0.373262i	$0.121761\pi$
$701$	13855.5i	0.746524i	−0.927726	+	0.373262i	$0.878239\pi$
$702$	0	0
$703$	−28192.0	−1.51249
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 7694.00i	− 0.409282i
$708$	0	0
$709$	− 28430.9i	− 1.50599i	−0.658028	−	0.752993i	$-0.728609\pi$
$709$	− 28430.9i	− 1.50599i	0.658028	−	0.752993i	$-0.271391\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8543.46	0.448745
$714$	0	0
$715$	41230.6i	2.15656i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−28510.5	−1.47881	−0.739405	−	0.673261i	$-0.764893\pi$
$719$	−28510.5	−1.47881	−0.739405	+	0.673261i	$0.764893\pi$
$720$	0	0
$721$	−26840.0	−1.38637
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 1952.31i	− 0.100009i
$726$	0	0
$727$	−18689.8	−0.953460	−0.476730	−	0.879050i	$-0.658178\pi$
$727$	−18689.8	−0.953460	−0.476730	+	0.879050i	$0.658178\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	47653.1i	2.41110i
$732$	0	0
$733$	14110.1i	0.711010i	0.934674	+	0.355505i	$0.115691\pi$
$733$	14110.1i	0.711010i	−0.934674	+	0.355505i	$0.884309\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−13289.8	−0.664229
$738$	0	0
$739$	− 32930.9i	− 1.63922i	−0.572921	−	0.819610i	$-0.694190\pi$
$739$	− 32930.9i	− 1.63922i	0.572921	−	0.819610i	$-0.305810\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	17151.6	0.846878	0.423439	−	0.905925i	$-0.360823\pi$
$743$	17151.6	0.846878	0.423439	+	0.905925i	$0.360823\pi$
$744$	0	0
$745$	7700.00	0.378666
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 31236.9i	− 1.52386i
$750$	0	0
$751$	12480.8	0.606434	0.303217	−	0.952922i	$-0.401939\pi$
$751$	12480.8	0.606434	0.303217	+	0.952922i	$0.401939\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 5212.06i	− 0.251240i
$756$	0	0
$757$	− 12144.6i	− 0.583093i	−0.956557	−	0.291546i	$-0.905830\pi$
$757$	− 12144.6i	− 0.583093i	0.956557	−	0.291546i	$-0.0941698\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	36665.7	1.74656	0.873279	−	0.487221i	$-0.161989\pi$
$761$	36665.7	1.74656	0.873279	+	0.487221i	$0.161989\pi$
$762$	0	0
$763$	7362.61i	0.349337i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6969.24	−0.328090
$768$	0	0
$769$	16646.0	0.780585	0.390293	−	0.920691i	$-0.372374\pi$
$769$	16646.0	0.780585	0.390293	+	0.920691i	$0.372374\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7395.10i	0.344092i	0.985089	+	0.172046i	$0.0550378\pi$
$773$	7395.10i	0.344092i	−0.985089	+	0.172046i	$0.944962\pi$
$774$	0	0
$775$	2831.78	0.131252
$776$	0	0
$777$	0	0
$778$	0	0
$779$	15884.4i	0.730574i
$780$	0	0
$781$	− 44927.8i	− 2.05844i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2491.84	0.113296
$786$	0	0
$787$	− 11110.8i	− 0.503251i	−0.967825	−	0.251626i	$-0.919035\pi$
$787$	− 11110.8i	− 0.503251i	0.967825	−	0.251626i	$-0.0809652\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−34846.2	−1.56636
$792$	0	0
$793$	24640.0	1.10339
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2614.91i	0.116217i	0.998310	+	0.0581084i	$0.0185069\pi$
$797$	2614.91i	0.116217i	−0.998310	+	0.0581084i	$0.981493\pi$
$798$	0	0
$799$	37589.3	1.66435
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 17373.5i	− 0.763511i
$804$	0	0
$805$	11232.0i	0.491769i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	39038.9	1.69658	0.848290	−	0.529532i	$-0.177632\pi$
$809$	39038.9	1.69658	0.848290	+	0.529532i	$0.177632\pi$
$810$	0	0
$811$	− 29584.3i	− 1.28094i	−0.767982	−	0.640472i	$-0.778739\pi$
$811$	− 29584.3i	− 1.28094i	0.767982	−	0.640472i	$-0.221261\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1583.92	−0.0680764
$816$	0	0
$817$	−53760.0	−2.30211
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21877.7i	0.930007i	0.885309	+	0.465003i	$0.153947\pi$
$821$	21877.7i	0.930007i	−0.885309	+	0.465003i	$0.846053\pi$
$822$	0	0
$823$	35093.1	1.48635	0.743177	−	0.669094i	$-0.233318\pi$
$823$	35093.1	1.48635	0.743177	+	0.669094i	$0.233318\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31272.4i	1.31493i	0.753485	+	0.657465i	$0.228371\pi$
$827$	31272.4i	1.31493i	−0.753485	+	0.657465i	$0.771629\pi$
$828$	0	0
$829$	− 22112.9i	− 0.926433i	−0.886245	−	0.463217i	$-0.846695\pi$
$829$	− 22112.9i	− 0.926433i	0.886245	−	0.463217i	$-0.153305\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−11509.9	−0.478746
$834$	0	0
$835$	41230.6i	1.70880i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−31678.4	−1.30353	−0.651764	−	0.758422i	$-0.725971\pi$
$839$	−31678.4	−1.30353	−0.651764	+	0.758422i	$0.725971\pi$
$840$	0	0
$841$	7449.00	0.305425
$842$	0	0
$843$	0	0
$844$	0	0
$845$	32313.6i	1.31553i
$846$	0	0
$847$	−23766.0	−0.964120
$848$	0	0
$849$	0	0
$850$	0	0
$851$	9530.63i	0.383908i
$852$	0	0
$853$	16918.1i	0.679092i	0.940589	+	0.339546i	$0.110273\pi$
$853$	16918.1i	0.679092i	−0.940589	+	0.339546i	$0.889727\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18154.9	−0.723638	−0.361819	−	0.932248i	$-0.617844\pi$
$857$	−18154.9	−0.723638	−0.361819	+	0.932248i	$0.617844\pi$
$858$	0	0
$859$	− 48325.5i	− 1.91949i	−0.280868	−	0.959746i	$-0.590622\pi$
$859$	− 48325.5i	− 1.91949i	0.280868	−	0.959746i	$-0.409378\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−41227.2	−1.62617	−0.813087	−	0.582142i	$-0.802215\pi$
$863$	−41227.2	−1.62617	−0.813087	+	0.582142i	$0.802215\pi$
$864$	0	0
$865$	10780.0	0.423735
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 21865.8i	− 0.853564i
$870$	0	0
$871$	−18794.7	−0.731151
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 27301.3i	− 1.05480i
$876$	0	0
$877$	24359.3i	0.937919i	0.883220	+	0.468960i	$0.155371\pi$
$877$	24359.3i	0.937919i	−0.883220	+	0.468960i	$0.844629\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	9018.10	0.344867	0.172433	−	0.985021i	$-0.444837\pi$
$881$	9018.10	0.344867	0.172433	+	0.985021i	$0.444837\pi$
$882$	0	0
$883$	− 44577.2i	− 1.69892i	−0.527655	−	0.849459i	$-0.676929\pi$
$883$	− 44577.2i	− 1.69892i	0.527655	−	0.849459i	$-0.323071\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1900.70	0.0719497	0.0359748	−	0.999353i	$-0.488546\pi$
$887$	1900.70	0.0719497	0.0359748	+	0.999353i	$0.488546\pi$
$888$	0	0
$889$	46200.0	1.74297
$890$	0	0
$891$	0	0
$892$	0	0
$893$	42406.5i	1.58911i
$894$	0	0
$895$	10572.0	0.394841
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 24571.2i	− 0.911562i
$900$	0	0
$901$	65987.7i	2.43992i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4153.07	−0.152544
$906$	0	0
$907$	− 33868.0i	− 1.23988i	−0.784650	−	0.619938i	$-0.787158\pi$
$907$	− 33868.0i	− 1.23988i	0.784650	−	0.619938i	$-0.212842\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−15160.4	−0.551356	−0.275678	−	0.961250i	$-0.588902\pi$
$911$	−15160.4	−0.551356	−0.275678	+	0.961250i	$0.588902\pi$
$912$	0	0
$913$	−61600.0	−2.23293
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 37484.3i	− 1.34988i
$918$	0	0
$919$	−44784.1	−1.60750	−0.803750	−	0.594967i	$-0.797165\pi$
$919$	−44784.1	−1.60750	−0.803750	+	0.594967i	$0.797165\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 63537.5i	− 2.26583i
$924$	0	0
$925$	3158.99i	0.112288i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	16730.9	0.590877	0.295438	−	0.955362i	$-0.404534\pi$
$929$	16730.9	0.590877	0.295438	+	0.955362i	$0.404534\pi$
$930$	0	0
$931$	− 12985.0i	− 0.457105i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−69692.4	−2.43763
$936$	0	0
$937$	−6090.00	−0.212328	−0.106164	−	0.994349i	$-0.533857\pi$
$937$	−6090.00	−0.212328	−0.106164	+	0.994349i	$0.533857\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 41684.7i	− 1.44408i	−0.691850	−	0.722042i	$-0.743204\pi$
$941$	− 41684.7i	− 1.44408i	0.691850	−	0.722042i	$-0.256796\pi$
$942$	0	0
$943$	5369.90	0.185438
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4467.48i	− 0.153298i	−0.997058	−	0.0766492i	$-0.975578\pi$
$947$	− 4467.48i	− 0.153298i	0.997058	−	0.0766492i	$-0.0244222\pi$
$948$	0	0
$949$	− 24569.9i	− 0.840435i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−12459.2	−0.423498	−0.211749	−	0.977324i	$-0.567916\pi$
$953$	−12459.2	−0.423498	−0.211749	+	0.977324i	$0.567916\pi$
$954$	0	0
$955$	45514.3i	1.54221i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−17423.1	−0.586675
$960$	0	0
$961$	5849.00	0.196334
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 50050.0i	− 1.66960i
$966$	0	0
$967$	−45665.1	−1.51861	−0.759303	−	0.650737i	$-0.774460\pi$
$967$	−45665.1	−1.51861	−0.759303	+	0.650737i	$0.774460\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	24869.0i	0.821919i	0.911654	+	0.410960i	$0.134806\pi$
$971$	24869.0i	0.821919i	−0.911654	+	0.410960i	$0.865194\pi$
$972$	0	0
$973$	5615.98i	0.185036i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−40700.1	−1.33277	−0.666383	−	0.745610i	$-0.732158\pi$
$977$	−40700.1	−1.33277	−0.666383	+	0.745610i	$0.732158\pi$
$978$	0	0
$979$	35340.5i	1.15372i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−13304.9	−0.431700	−0.215850	−	0.976427i	$-0.569252\pi$
$983$	−13304.9	−0.431700	−0.215850	+	0.976427i	$0.569252\pi$
$984$	0	0
$985$	48300.0	1.56240
$986$	0	0
$987$	0	0
$988$	0	0
$989$	18174.2i	0.584334i
$990$	0	0
$991$	17179.5	0.550681	0.275340	−	0.961347i	$-0.411209\pi$
$991$	17179.5	0.550681	0.275340	+	0.961347i	$0.411209\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	56836.3i	1.81089i
$996$	0	0
$997$	− 17058.5i	− 0.541875i	−0.962597	−	0.270937i	$-0.912666\pi$
$997$	− 17058.5i	− 0.541875i	0.962597	−	0.270937i	$-0.0873336\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.4.d.q.577.4	yes	8
3.2	odd	2	inner	1152.4.d.q.577.8	yes	8
4.3	odd	2	inner	1152.4.d.q.577.2	yes	8
8.3	odd	2	inner	1152.4.d.q.577.5	yes	8
8.5	even	2	inner	1152.4.d.q.577.7	yes	8
12.11	even	2	inner	1152.4.d.q.577.6	yes	8
16.3	odd	4		2304.4.a.cd.1.3		8
16.5	even	4		2304.4.a.cd.1.6		8
16.11	odd	4		2304.4.a.cd.1.8		8
16.13	even	4		2304.4.a.cd.1.1		8
24.5	odd	2	inner	1152.4.d.q.577.3	yes	8
24.11	even	2	inner	1152.4.d.q.577.1	✓	8
48.5	odd	4		2304.4.a.cd.1.2		8
48.11	even	4		2304.4.a.cd.1.4		8
48.29	odd	4		2304.4.a.cd.1.5		8
48.35	even	4		2304.4.a.cd.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.4.d.q.577.1	✓	8	24.11	even	2	inner
1152.4.d.q.577.2	yes	8	4.3	odd	2	inner
1152.4.d.q.577.3	yes	8	24.5	odd	2	inner
1152.4.d.q.577.4	yes	8	1.1	even	1	trivial
1152.4.d.q.577.5	yes	8	8.3	odd	2	inner
1152.4.d.q.577.6	yes	8	12.11	even	2	inner
1152.4.d.q.577.7	yes	8	8.5	even	2	inner
1152.4.d.q.577.8	yes	8	3.2	odd	2	inner
2304.4.a.cd.1.1		8	16.13	even	4
2304.4.a.cd.1.2		8	48.5	odd	4
2304.4.a.cd.1.3		8	16.3	odd	4
2304.4.a.cd.1.4		8	48.11	even	4
2304.4.a.cd.1.5		8	48.29	odd	4
2304.4.a.cd.1.6		8	16.5	even	4
2304.4.a.cd.1.7		8	48.35	even	4
2304.4.a.cd.1.8		8	16.11	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1152.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-11.8322i q^{5} +20.9762 q^{7} +49.6387i q^{11} +70.1997i q^{13} -118.659 q^{17} -133.866i q^{19} -45.2548 q^{23} -15.0000 q^{25} +130.154i q^{29} -188.786 q^{31} -248.193i q^{35} -210.599i q^{37} -118.659 q^{41} -401.597i q^{43} -316.784 q^{47} +97.0000 q^{49} -556.111i q^{53} +587.333 q^{55} +99.2774i q^{59} -350.999i q^{61} +830.614 q^{65} +267.731i q^{67} -905.097 q^{71} -350.000 q^{73} +1041.23i q^{77} -440.500 q^{79} +1240.97i q^{83} +1403.99i q^{85} +711.955 q^{89} +1472.52i q^{91} -1583.92 q^{95} +770.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 120 q^{25} + 776 q^{49} - 2800 q^{73} + 6160 q^{97}+O(q^{100}) \) 8 * q - 120 * q^25 + 776 * q^49 - 2800 * q^73 + 6160 * q^97

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