LMFDB - Embedded newform 1088.4.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1088, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1088.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$64.1940780862$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 17)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1088.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{3} -6.00000 q^{5} -28.0000 q^{7} +37.0000 q^{9} +O(q^{10})$

$q+8.00000 q^{3} -6.00000 q^{5} -28.0000 q^{7} +37.0000 q^{9} +24.0000 q^{11} +58.0000 q^{13} -48.0000 q^{15} +17.0000 q^{17} -116.000 q^{19} -224.000 q^{21} -60.0000 q^{23} -89.0000 q^{25} +80.0000 q^{27} -30.0000 q^{29} -172.000 q^{31} +192.000 q^{33} +168.000 q^{35} +58.0000 q^{37} +464.000 q^{39} -342.000 q^{41} +148.000 q^{43} -222.000 q^{45} +288.000 q^{47} +441.000 q^{49} +136.000 q^{51} -318.000 q^{53} -144.000 q^{55} -928.000 q^{57} -252.000 q^{59} -110.000 q^{61} -1036.00 q^{63} -348.000 q^{65} +484.000 q^{67} -480.000 q^{69} -708.000 q^{71} +362.000 q^{73} -712.000 q^{75} -672.000 q^{77} -484.000 q^{79} -359.000 q^{81} -756.000 q^{83} -102.000 q^{85} -240.000 q^{87} -774.000 q^{89} -1624.00 q^{91} -1376.00 q^{93} +696.000 q^{95} -382.000 q^{97} +888.000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	8.00000	1.53960	0.769800	−	0.638285i	$-0.220356\pi$
$3$	8.00000	1.53960	0.769800	+	0.638285i	$0.220356\pi$
$4$	0	0
$5$	−6.00000	−0.536656	−0.268328	−	0.963328i	$-0.586471\pi$
$5$	−6.00000	−0.536656	−0.268328	+	0.963328i	$0.586471\pi$
$6$	0	0
$7$	−28.0000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−28.0000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	37.0000	1.37037
$10$	0	0
$11$	24.0000	0.657843	0.328921	−	0.944357i	$-0.393315\pi$
$11$	24.0000	0.657843	0.328921	+	0.944357i	$0.393315\pi$
$12$	0	0
$13$	58.0000	1.23741	0.618704	−	0.785624i	$-0.287658\pi$
$13$	58.0000	1.23741	0.618704	+	0.785624i	$0.287658\pi$
$14$	0	0
$15$	−48.0000	−0.826236
$16$	0	0
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	−116.000	−1.40064	−0.700322	−	0.713827i	$-0.746960\pi$
$19$	−116.000	−1.40064	−0.700322	+	0.713827i	$0.746960\pi$
$20$	0	0
$21$	−224.000	−2.32766
$22$	0	0
$23$	−60.0000	−0.543951	−0.271975	−	0.962304i	$-0.587677\pi$
$23$	−60.0000	−0.543951	−0.271975	+	0.962304i	$0.587677\pi$
$24$	0	0
$25$	−89.0000	−0.712000
$26$	0	0
$27$	80.0000	0.570222
$28$	0	0
$29$	−30.0000	−0.192099	−0.0960493	−	0.995377i	$-0.530621\pi$
$29$	−30.0000	−0.192099	−0.0960493	+	0.995377i	$0.530621\pi$
$30$	0	0
$31$	−172.000	−0.996520	−0.498260	−	0.867028i	$-0.666027\pi$
$31$	−172.000	−0.996520	−0.498260	+	0.867028i	$0.666027\pi$
$32$	0	0
$33$	192.000	1.01282
$34$	0	0
$35$	168.000	0.811348
$36$	0	0
$37$	58.0000	0.257707	0.128853	−	0.991664i	$-0.458870\pi$
$37$	58.0000	0.257707	0.128853	+	0.991664i	$0.458870\pi$
$38$	0	0
$39$	464.000	1.90511
$40$	0	0
$41$	−342.000	−1.30272	−0.651359	−	0.758770i	$-0.725801\pi$
$41$	−342.000	−1.30272	−0.651359	+	0.758770i	$0.725801\pi$
$42$	0	0
$43$	148.000	0.524879	0.262439	−	0.964948i	$-0.415473\pi$
$43$	148.000	0.524879	0.262439	+	0.964948i	$0.415473\pi$
$44$	0	0
$45$	−222.000	−0.735418
$46$	0	0
$47$	288.000	0.893811	0.446906	−	0.894581i	$-0.352526\pi$
$47$	288.000	0.893811	0.446906	+	0.894581i	$0.352526\pi$
$48$	0	0
$49$	441.000	1.28571
$50$	0	0
$51$	136.000	0.373408
$52$	0	0
$53$	−318.000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−318.000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	−144.000	−0.353036
$56$	0	0
$57$	−928.000	−2.15643
$58$	0	0
$59$	−252.000	−0.556061	−0.278031	−	0.960572i	$-0.589682\pi$
$59$	−252.000	−0.556061	−0.278031	+	0.960572i	$0.589682\pi$
$60$	0	0
$61$	−110.000	−0.230886	−0.115443	−	0.993314i	$-0.536829\pi$
$61$	−110.000	−0.230886	−0.115443	+	0.993314i	$0.536829\pi$
$62$	0	0
$63$	−1036.00	−2.07181
$64$	0	0
$65$	−348.000	−0.664063
$66$	0	0
$67$	484.000	0.882537	0.441269	−	0.897375i	$-0.354529\pi$
$67$	484.000	0.882537	0.441269	+	0.897375i	$0.354529\pi$
$68$	0	0
$69$	−480.000	−0.837467
$70$	0	0
$71$	−708.000	−1.18344	−0.591719	−	0.806144i	$-0.701551\pi$
$71$	−708.000	−1.18344	−0.591719	+	0.806144i	$0.701551\pi$
$72$	0	0
$73$	362.000	0.580396	0.290198	−	0.956967i	$-0.406279\pi$
$73$	362.000	0.580396	0.290198	+	0.956967i	$0.406279\pi$
$74$	0	0
$75$	−712.000	−1.09620
$76$	0	0
$77$	−672.000	−0.994565
$78$	0	0
$79$	−484.000	−0.689294	−0.344647	−	0.938732i	$-0.612001\pi$
$79$	−484.000	−0.689294	−0.344647	+	0.938732i	$0.612001\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	0	0
$83$	−756.000	−0.999780	−0.499890	−	0.866089i	$-0.666626\pi$
$83$	−756.000	−0.999780	−0.499890	+	0.866089i	$0.666626\pi$
$84$	0	0
$85$	−102.000	−0.130158
$86$	0	0
$87$	−240.000	−0.295755
$88$	0	0
$89$	−774.000	−0.921841	−0.460920	−	0.887441i	$-0.652481\pi$
$89$	−774.000	−0.921841	−0.460920	+	0.887441i	$0.652481\pi$
$90$	0	0
$91$	−1624.00	−1.87079
$92$	0	0
$93$	−1376.00	−1.53424
$94$	0	0
$95$	696.000	0.751664
$96$	0	0
$97$	−382.000	−0.399858	−0.199929	−	0.979810i	$-0.564071\pi$
$97$	−382.000	−0.399858	−0.199929	+	0.979810i	$0.564071\pi$
$98$	0	0
$99$	888.000	0.901488
$100$	0	0
$101$	210.000	0.206889	0.103444	−	0.994635i	$-0.467014\pi$
$101$	210.000	0.206889	0.103444	+	0.994635i	$0.467014\pi$
$102$	0	0
$103$	−232.000	−0.221938	−0.110969	−	0.993824i	$-0.535395\pi$
$103$	−232.000	−0.221938	−0.110969	+	0.993824i	$0.535395\pi$
$104$	0	0
$105$	1344.00	1.24915
$106$	0	0
$107$	−432.000	−0.390309	−0.195154	−	0.980773i	$-0.562521\pi$
$107$	−432.000	−0.390309	−0.195154	+	0.980773i	$0.562521\pi$
$108$	0	0
$109$	1186.00	1.04219	0.521093	−	0.853500i	$-0.325525\pi$
$109$	1186.00	1.04219	0.521093	+	0.853500i	$0.325525\pi$
$110$	0	0
$111$	464.000	0.396765
$112$	0	0
$113$	−366.000	−0.304694	−0.152347	−	0.988327i	$-0.548683\pi$
$113$	−366.000	−0.304694	−0.152347	+	0.988327i	$0.548683\pi$
$114$	0	0
$115$	360.000	0.291915
$116$	0	0
$117$	2146.00	1.69571
$118$	0	0
$119$	−476.000	−0.366679
$120$	0	0
$121$	−755.000	−0.567243
$122$	0	0
$123$	−2736.00	−2.00567
$124$	0	0
$125$	1284.00	0.918756
$126$	0	0
$127$	−472.000	−0.329789	−0.164895	−	0.986311i	$-0.552728\pi$
$127$	−472.000	−0.329789	−0.164895	+	0.986311i	$0.552728\pi$
$128$	0	0
$129$	1184.00	0.808104
$130$	0	0
$131$	−2760.00	−1.84078	−0.920391	−	0.391000i	$-0.872129\pi$
$131$	−2760.00	−1.84078	−0.920391	+	0.391000i	$0.872129\pi$
$132$	0	0
$133$	3248.00	2.11757
$134$	0	0
$135$	−480.000	−0.306013
$136$	0	0
$137$	1098.00	0.684733	0.342367	−	0.939566i	$-0.388771\pi$
$137$	1098.00	0.684733	0.342367	+	0.939566i	$0.388771\pi$
$138$	0	0
$139$	−2528.00	−1.54261	−0.771303	−	0.636468i	$-0.780395\pi$
$139$	−2528.00	−1.54261	−0.771303	+	0.636468i	$0.780395\pi$
$140$	0	0
$141$	2304.00	1.37611
$142$	0	0
$143$	1392.00	0.814020
$144$	0	0
$145$	180.000	0.103091
$146$	0	0
$147$	3528.00	1.97949
$148$	0	0
$149$	−1614.00	−0.887410	−0.443705	−	0.896173i	$-0.646336\pi$
$149$	−1614.00	−0.887410	−0.443705	+	0.896173i	$0.646336\pi$
$150$	0	0
$151$	−3328.00	−1.79357	−0.896784	−	0.442468i	$-0.854103\pi$
$151$	−3328.00	−1.79357	−0.896784	+	0.442468i	$0.854103\pi$
$152$	0	0
$153$	629.000	0.332364
$154$	0	0
$155$	1032.00	0.534789
$156$	0	0
$157$	2458.00	1.24949	0.624744	−	0.780829i	$-0.285203\pi$
$157$	2458.00	1.24949	0.624744	+	0.780829i	$0.285203\pi$
$158$	0	0
$159$	−2544.00	−1.26888
$160$	0	0
$161$	1680.00	0.822376
$162$	0	0
$163$	−272.000	−0.130704	−0.0653518	−	0.997862i	$-0.520817\pi$
$163$	−272.000	−0.130704	−0.0653518	+	0.997862i	$0.520817\pi$
$164$	0	0
$165$	−1152.00	−0.543534
$166$	0	0
$167$	3516.00	1.62920	0.814600	−	0.580024i	$-0.196957\pi$
$167$	3516.00	1.62920	0.814600	+	0.580024i	$0.196957\pi$
$168$	0	0
$169$	1167.00	0.531179
$170$	0	0
$171$	−4292.00	−1.91940
$172$	0	0
$173$	1842.00	0.809507	0.404753	−	0.914426i	$-0.367357\pi$
$173$	1842.00	0.809507	0.404753	+	0.914426i	$0.367357\pi$
$174$	0	0
$175$	2492.00	1.07644
$176$	0	0
$177$	−2016.00	−0.856112
$178$	0	0
$179$	3516.00	1.46815	0.734073	−	0.679070i	$-0.237617\pi$
$179$	3516.00	1.46815	0.734073	+	0.679070i	$0.237617\pi$
$180$	0	0
$181$	−3398.00	−1.39542	−0.697711	−	0.716379i	$-0.745798\pi$
$181$	−3398.00	−1.39542	−0.697711	+	0.716379i	$0.745798\pi$
$182$	0	0
$183$	−880.000	−0.355473
$184$	0	0
$185$	−348.000	−0.138300
$186$	0	0
$187$	408.000	0.159550
$188$	0	0
$189$	−2240.00	−0.862095
$190$	0	0
$191$	−2640.00	−1.00012	−0.500062	−	0.865990i	$-0.666689\pi$
$191$	−2640.00	−1.00012	−0.500062	+	0.865990i	$0.666689\pi$
$192$	0	0
$193$	2882.00	1.07488	0.537438	−	0.843304i	$-0.319392\pi$
$193$	2882.00	1.07488	0.537438	+	0.843304i	$0.319392\pi$
$194$	0	0
$195$	−2784.00	−1.02239
$196$	0	0
$197$	42.0000	0.0151897	0.00759486	−	0.999971i	$-0.497582\pi$
$197$	42.0000	0.0151897	0.00759486	+	0.999971i	$0.497582\pi$
$198$	0	0
$199$	−3220.00	−1.14703	−0.573517	−	0.819194i	$-0.694421\pi$
$199$	−3220.00	−1.14703	−0.573517	+	0.819194i	$0.694421\pi$
$200$	0	0
$201$	3872.00	1.35876
$202$	0	0
$203$	840.000	0.290426
$204$	0	0
$205$	2052.00	0.699112
$206$	0	0
$207$	−2220.00	−0.745414
$208$	0	0
$209$	−2784.00	−0.921403
$210$	0	0
$211$	2080.00	0.678640	0.339320	−	0.940671i	$-0.389803\pi$
$211$	2080.00	0.678640	0.339320	+	0.940671i	$0.389803\pi$
$212$	0	0
$213$	−5664.00	−1.82202
$214$	0	0
$215$	−888.000	−0.281680
$216$	0	0
$217$	4816.00	1.50660
$218$	0	0
$219$	2896.00	0.893578
$220$	0	0
$221$	986.000	0.300116
$222$	0	0
$223$	4664.00	1.40056	0.700279	−	0.713869i	$-0.253059\pi$
$223$	4664.00	1.40056	0.700279	+	0.713869i	$0.253059\pi$
$224$	0	0
$225$	−3293.00	−0.975704
$226$	0	0
$227$	1440.00	0.421040	0.210520	−	0.977590i	$-0.432484\pi$
$227$	1440.00	0.421040	0.210520	+	0.977590i	$0.432484\pi$
$228$	0	0
$229$	1186.00	0.342241	0.171120	−	0.985250i	$-0.445261\pi$
$229$	1186.00	0.342241	0.171120	+	0.985250i	$0.445261\pi$
$230$	0	0
$231$	−5376.00	−1.53123
$232$	0	0
$233$	−5334.00	−1.49975	−0.749875	−	0.661579i	$-0.769887\pi$
$233$	−5334.00	−1.49975	−0.749875	+	0.661579i	$0.769887\pi$
$234$	0	0
$235$	−1728.00	−0.479669
$236$	0	0
$237$	−3872.00	−1.06124
$238$	0	0
$239$	5328.00	1.44201	0.721003	−	0.692931i	$-0.243681\pi$
$239$	5328.00	1.44201	0.721003	+	0.692931i	$0.243681\pi$
$240$	0	0
$241$	5618.00	1.50161	0.750803	−	0.660526i	$-0.229667\pi$
$241$	5618.00	1.50161	0.750803	+	0.660526i	$0.229667\pi$
$242$	0	0
$243$	−5032.00	−1.32841
$244$	0	0
$245$	−2646.00	−0.689987
$246$	0	0
$247$	−6728.00	−1.73317
$248$	0	0
$249$	−6048.00	−1.53926
$250$	0	0
$251$	2028.00	0.509985	0.254992	−	0.966943i	$-0.417927\pi$
$251$	2028.00	0.509985	0.254992	+	0.966943i	$0.417927\pi$
$252$	0	0
$253$	−1440.00	−0.357834
$254$	0	0
$255$	−816.000	−0.200392
$256$	0	0
$257$	−1902.00	−0.461648	−0.230824	−	0.972996i	$-0.574142\pi$
$257$	−1902.00	−0.461648	−0.230824	+	0.972996i	$0.574142\pi$
$258$	0	0
$259$	−1624.00	−0.389616
$260$	0	0
$261$	−1110.00	−0.263246
$262$	0	0
$263$	−5472.00	−1.28296	−0.641479	−	0.767141i	$-0.721679\pi$
$263$	−5472.00	−1.28296	−0.641479	+	0.767141i	$0.721679\pi$
$264$	0	0
$265$	1908.00	0.442292
$266$	0	0
$267$	−6192.00	−1.41927
$268$	0	0
$269$	3570.00	0.809170	0.404585	−	0.914500i	$-0.367416\pi$
$269$	3570.00	0.809170	0.404585	+	0.914500i	$0.367416\pi$
$270$	0	0
$271$	272.000	0.0609698	0.0304849	−	0.999535i	$-0.490295\pi$
$271$	272.000	0.0609698	0.0304849	+	0.999535i	$0.490295\pi$
$272$	0	0
$273$	−12992.0	−2.88026
$274$	0	0
$275$	−2136.00	−0.468384
$276$	0	0
$277$	−3830.00	−0.830767	−0.415383	−	0.909646i	$-0.636353\pi$
$277$	−3830.00	−0.830767	−0.415383	+	0.909646i	$0.636353\pi$
$278$	0	0
$279$	−6364.00	−1.36560
$280$	0	0
$281$	8874.00	1.88391	0.941955	−	0.335740i	$-0.108986\pi$
$281$	8874.00	1.88391	0.941955	+	0.335740i	$0.108986\pi$
$282$	0	0
$283$	2632.00	0.552849	0.276424	−	0.961036i	$-0.410850\pi$
$283$	2632.00	0.552849	0.276424	+	0.961036i	$0.410850\pi$
$284$	0	0
$285$	5568.00	1.15726
$286$	0	0
$287$	9576.00	1.96952
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	−3056.00	−0.615622
$292$	0	0
$293$	6402.00	1.27648	0.638240	−	0.769837i	$-0.279663\pi$
$293$	6402.00	1.27648	0.638240	+	0.769837i	$0.279663\pi$
$294$	0	0
$295$	1512.00	0.298414
$296$	0	0
$297$	1920.00	0.375117
$298$	0	0
$299$	−3480.00	−0.673089
$300$	0	0
$301$	−4144.00	−0.793542
$302$	0	0
$303$	1680.00	0.318526
$304$	0	0
$305$	660.000	0.123907
$306$	0	0
$307$	8980.00	1.66943	0.834716	−	0.550681i	$-0.185632\pi$
$307$	8980.00	1.66943	0.834716	+	0.550681i	$0.185632\pi$
$308$	0	0
$309$	−1856.00	−0.341696
$310$	0	0
$311$	−3972.00	−0.724217	−0.362108	−	0.932136i	$-0.617943\pi$
$311$	−3972.00	−0.724217	−0.362108	+	0.932136i	$0.617943\pi$
$312$	0	0
$313$	4730.00	0.854171	0.427085	−	0.904211i	$-0.359540\pi$
$313$	4730.00	0.854171	0.427085	+	0.904211i	$0.359540\pi$
$314$	0	0
$315$	6216.00	1.11185
$316$	0	0
$317$	2898.00	0.513463	0.256732	−	0.966483i	$-0.417354\pi$
$317$	2898.00	0.513463	0.256732	+	0.966483i	$0.417354\pi$
$318$	0	0
$319$	−720.000	−0.126371
$320$	0	0
$321$	−3456.00	−0.600919
$322$	0	0
$323$	−1972.00	−0.339706
$324$	0	0
$325$	−5162.00	−0.881035
$326$	0	0
$327$	9488.00	1.60455
$328$	0	0
$329$	−8064.00	−1.35132
$330$	0	0
$331$	4564.00	0.757886	0.378943	−	0.925420i	$-0.376288\pi$
$331$	4564.00	0.757886	0.378943	+	0.925420i	$0.376288\pi$
$332$	0	0
$333$	2146.00	0.353153
$334$	0	0
$335$	−2904.00	−0.473619
$336$	0	0
$337$	722.000	0.116706	0.0583529	−	0.998296i	$-0.481415\pi$
$337$	722.000	0.116706	0.0583529	+	0.998296i	$0.481415\pi$
$338$	0	0
$339$	−2928.00	−0.469107
$340$	0	0
$341$	−4128.00	−0.655553
$342$	0	0
$343$	−2744.00	−0.431959
$344$	0	0
$345$	2880.00	0.449432
$346$	0	0
$347$	−5544.00	−0.857687	−0.428844	−	0.903379i	$-0.641079\pi$
$347$	−5544.00	−0.857687	−0.428844	+	0.903379i	$0.641079\pi$
$348$	0	0
$349$	−11126.0	−1.70648	−0.853239	−	0.521519i	$-0.825365\pi$
$349$	−11126.0	−1.70648	−0.853239	+	0.521519i	$0.825365\pi$
$350$	0	0
$351$	4640.00	0.705598
$352$	0	0
$353$	7842.00	1.18240	0.591200	−	0.806525i	$-0.298654\pi$
$353$	7842.00	1.18240	0.591200	+	0.806525i	$0.298654\pi$
$354$	0	0
$355$	4248.00	0.635100
$356$	0	0
$357$	−3808.00	−0.564540
$358$	0	0
$359$	5040.00	0.740950	0.370475	−	0.928842i	$-0.379195\pi$
$359$	5040.00	0.740950	0.370475	+	0.928842i	$0.379195\pi$
$360$	0	0
$361$	6597.00	0.961802
$362$	0	0
$363$	−6040.00	−0.873327
$364$	0	0
$365$	−2172.00	−0.311473
$366$	0	0
$367$	−8404.00	−1.19533	−0.597664	−	0.801747i	$-0.703904\pi$
$367$	−8404.00	−1.19533	−0.597664	+	0.801747i	$0.703904\pi$
$368$	0	0
$369$	−12654.0	−1.78521
$370$	0	0
$371$	8904.00	1.24602
$372$	0	0
$373$	8098.00	1.12412	0.562062	−	0.827095i	$-0.310008\pi$
$373$	8098.00	1.12412	0.562062	+	0.827095i	$0.310008\pi$
$374$	0	0
$375$	10272.0	1.41452
$376$	0	0
$377$	−1740.00	−0.237704
$378$	0	0
$379$	−320.000	−0.0433702	−0.0216851	−	0.999765i	$-0.506903\pi$
$379$	−320.000	−0.0433702	−0.0216851	+	0.999765i	$0.506903\pi$
$380$	0	0
$381$	−3776.00	−0.507744
$382$	0	0
$383$	−10872.0	−1.45048	−0.725239	−	0.688497i	$-0.758271\pi$
$383$	−10872.0	−1.45048	−0.725239	+	0.688497i	$0.758271\pi$
$384$	0	0
$385$	4032.00	0.533740
$386$	0	0
$387$	5476.00	0.719278
$388$	0	0
$389$	−1374.00	−0.179086	−0.0895431	−	0.995983i	$-0.528541\pi$
$389$	−1374.00	−0.179086	−0.0895431	+	0.995983i	$0.528541\pi$
$390$	0	0
$391$	−1020.00	−0.131927
$392$	0	0
$393$	−22080.0	−2.83407
$394$	0	0
$395$	2904.00	0.369914
$396$	0	0
$397$	7522.00	0.950928	0.475464	−	0.879735i	$-0.342280\pi$
$397$	7522.00	0.950928	0.475464	+	0.879735i	$0.342280\pi$
$398$	0	0
$399$	25984.0	3.26022
$400$	0	0
$401$	2706.00	0.336986	0.168493	−	0.985703i	$-0.446110\pi$
$401$	2706.00	0.336986	0.168493	+	0.985703i	$0.446110\pi$
$402$	0	0
$403$	−9976.00	−1.23310
$404$	0	0
$405$	2154.00	0.264279
$406$	0	0
$407$	1392.00	0.169530
$408$	0	0
$409$	266.000	0.0321586	0.0160793	−	0.999871i	$-0.494882\pi$
$409$	266.000	0.0321586	0.0160793	+	0.999871i	$0.494882\pi$
$410$	0	0
$411$	8784.00	1.05422
$412$	0	0
$413$	7056.00	0.840685
$414$	0	0
$415$	4536.00	0.536539
$416$	0	0
$417$	−20224.0	−2.37500
$418$	0	0
$419$	−2688.00	−0.313407	−0.156703	−	0.987646i	$-0.550087\pi$
$419$	−2688.00	−0.313407	−0.156703	+	0.987646i	$0.550087\pi$
$420$	0	0
$421$	13810.0	1.59871	0.799357	−	0.600857i	$-0.205174\pi$
$421$	13810.0	1.59871	0.799357	+	0.600857i	$0.205174\pi$
$422$	0	0
$423$	10656.0	1.22485
$424$	0	0
$425$	−1513.00	−0.172685
$426$	0	0
$427$	3080.00	0.349067
$428$	0	0
$429$	11136.0	1.25327
$430$	0	0
$431$	3036.00	0.339302	0.169651	−	0.985504i	$-0.445736\pi$
$431$	3036.00	0.339302	0.169651	+	0.985504i	$0.445736\pi$
$432$	0	0
$433$	−11422.0	−1.26768	−0.633841	−	0.773463i	$-0.718523\pi$
$433$	−11422.0	−1.26768	−0.633841	+	0.773463i	$0.718523\pi$
$434$	0	0
$435$	1440.00	0.158719
$436$	0	0
$437$	6960.00	0.761881
$438$	0	0
$439$	−52.0000	−0.00565336	−0.00282668	−	0.999996i	$-0.500900\pi$
$439$	−52.0000	−0.00565336	−0.00282668	+	0.999996i	$0.500900\pi$
$440$	0	0
$441$	16317.0	1.76190
$442$	0	0
$443$	−3108.00	−0.333331	−0.166665	−	0.986014i	$-0.553300\pi$
$443$	−3108.00	−0.333331	−0.166665	+	0.986014i	$0.553300\pi$
$444$	0	0
$445$	4644.00	0.494712
$446$	0	0
$447$	−12912.0	−1.36626
$448$	0	0
$449$	6114.00	0.642622	0.321311	−	0.946974i	$-0.395876\pi$
$449$	6114.00	0.642622	0.321311	+	0.946974i	$0.395876\pi$
$450$	0	0
$451$	−8208.00	−0.856984
$452$	0	0
$453$	−26624.0	−2.76138
$454$	0	0
$455$	9744.00	1.00397
$456$	0	0
$457$	4106.00	0.420286	0.210143	−	0.977671i	$-0.432607\pi$
$457$	4106.00	0.420286	0.210143	+	0.977671i	$0.432607\pi$
$458$	0	0
$459$	1360.00	0.138299
$460$	0	0
$461$	−3366.00	−0.340066	−0.170033	−	0.985438i	$-0.554387\pi$
$461$	−3366.00	−0.340066	−0.170033	+	0.985438i	$0.554387\pi$
$462$	0	0
$463$	896.000	0.0899366	0.0449683	−	0.998988i	$-0.485681\pi$
$463$	896.000	0.0899366	0.0449683	+	0.998988i	$0.485681\pi$
$464$	0	0
$465$	8256.00	0.823361
$466$	0	0
$467$	10236.0	1.01427	0.507137	−	0.861866i	$-0.330704\pi$
$467$	10236.0	1.01427	0.507137	+	0.861866i	$0.330704\pi$
$468$	0	0
$469$	−13552.0	−1.33427
$470$	0	0
$471$	19664.0	1.92371
$472$	0	0
$473$	3552.00	0.345288
$474$	0	0
$475$	10324.0	0.997258
$476$	0	0
$477$	−11766.0	−1.12941
$478$	0	0
$479$	5172.00	0.493350	0.246675	−	0.969098i	$-0.420662\pi$
$479$	5172.00	0.493350	0.246675	+	0.969098i	$0.420662\pi$
$480$	0	0
$481$	3364.00	0.318888
$482$	0	0
$483$	13440.0	1.26613
$484$	0	0
$485$	2292.00	0.214586
$486$	0	0
$487$	−15052.0	−1.40056	−0.700278	−	0.713870i	$-0.746941\pi$
$487$	−15052.0	−1.40056	−0.700278	+	0.713870i	$0.746941\pi$
$488$	0	0
$489$	−2176.00	−0.201231
$490$	0	0
$491$	−8700.00	−0.799645	−0.399822	−	0.916593i	$-0.630928\pi$
$491$	−8700.00	−0.799645	−0.399822	+	0.916593i	$0.630928\pi$
$492$	0	0
$493$	−510.000	−0.0465908
$494$	0	0
$495$	−5328.00	−0.483789
$496$	0	0
$497$	19824.0	1.78919
$498$	0	0
$499$	1168.00	0.104783	0.0523916	−	0.998627i	$-0.483316\pi$
$499$	1168.00	0.104783	0.0523916	+	0.998627i	$0.483316\pi$
$500$	0	0
$501$	28128.0	2.50832
$502$	0	0
$503$	−1740.00	−0.154240	−0.0771200	−	0.997022i	$-0.524572\pi$
$503$	−1740.00	−0.154240	−0.0771200	+	0.997022i	$0.524572\pi$
$504$	0	0
$505$	−1260.00	−0.111028
$506$	0	0
$507$	9336.00	0.817803
$508$	0	0
$509$	12570.0	1.09461	0.547304	−	0.836934i	$-0.315654\pi$
$509$	12570.0	1.09461	0.547304	+	0.836934i	$0.315654\pi$
$510$	0	0
$511$	−10136.0	−0.877476
$512$	0	0
$513$	−9280.00	−0.798678
$514$	0	0
$515$	1392.00	0.119105
$516$	0	0
$517$	6912.00	0.587987
$518$	0	0
$519$	14736.0	1.24632
$520$	0	0
$521$	11658.0	0.980319	0.490160	−	0.871633i	$-0.336939\pi$
$521$	11658.0	0.980319	0.490160	+	0.871633i	$0.336939\pi$
$522$	0	0
$523$	−13700.0	−1.14543	−0.572714	−	0.819755i	$-0.694110\pi$
$523$	−13700.0	−1.14543	−0.572714	+	0.819755i	$0.694110\pi$
$524$	0	0
$525$	19936.0	1.65729
$526$	0	0
$527$	−2924.00	−0.241692
$528$	0	0
$529$	−8567.00	−0.704118
$530$	0	0
$531$	−9324.00	−0.762010
$532$	0	0
$533$	−19836.0	−1.61199
$534$	0	0
$535$	2592.00	0.209462
$536$	0	0
$537$	28128.0	2.26036
$538$	0	0
$539$	10584.0	0.845798
$540$	0	0
$541$	−17822.0	−1.41632	−0.708159	−	0.706053i	$-0.750474\pi$
$541$	−17822.0	−1.41632	−0.708159	+	0.706053i	$0.750474\pi$
$542$	0	0
$543$	−27184.0	−2.14839
$544$	0	0
$545$	−7116.00	−0.559295
$546$	0	0
$547$	−3800.00	−0.297032	−0.148516	−	0.988910i	$-0.547450\pi$
$547$	−3800.00	−0.297032	−0.148516	+	0.988910i	$0.547450\pi$
$548$	0	0
$549$	−4070.00	−0.316400
$550$	0	0
$551$	3480.00	0.269062
$552$	0	0
$553$	13552.0	1.04212
$554$	0	0
$555$	−2784.00	−0.212927
$556$	0	0
$557$	10074.0	0.766336	0.383168	−	0.923679i	$-0.374833\pi$
$557$	10074.0	0.766336	0.383168	+	0.923679i	$0.374833\pi$
$558$	0	0
$559$	8584.00	0.649489
$560$	0	0
$561$	3264.00	0.245644
$562$	0	0
$563$	15948.0	1.19383	0.596917	−	0.802303i	$-0.296392\pi$
$563$	15948.0	1.19383	0.596917	+	0.802303i	$0.296392\pi$
$564$	0	0
$565$	2196.00	0.163516
$566$	0	0
$567$	10052.0	0.744523
$568$	0	0
$569$	21834.0	1.60866	0.804331	−	0.594181i	$-0.202524\pi$
$569$	21834.0	1.60866	0.804331	+	0.594181i	$0.202524\pi$
$570$	0	0
$571$	21208.0	1.55434	0.777169	−	0.629292i	$-0.216655\pi$
$571$	21208.0	1.55434	0.777169	+	0.629292i	$0.216655\pi$
$572$	0	0
$573$	−21120.0	−1.53979
$574$	0	0
$575$	5340.00	0.387293
$576$	0	0
$577$	12530.0	0.904039	0.452020	−	0.892008i	$-0.350704\pi$
$577$	12530.0	0.904039	0.452020	+	0.892008i	$0.350704\pi$
$578$	0	0
$579$	23056.0	1.65488
$580$	0	0
$581$	21168.0	1.51153
$582$	0	0
$583$	−7632.00	−0.542170
$584$	0	0
$585$	−12876.0	−0.910012
$586$	0	0
$587$	−2220.00	−0.156097	−0.0780487	−	0.996950i	$-0.524869\pi$
$587$	−2220.00	−0.156097	−0.0780487	+	0.996950i	$0.524869\pi$
$588$	0	0
$589$	19952.0	1.39577
$590$	0	0
$591$	336.000	0.0233861
$592$	0	0
$593$	−25038.0	−1.73387	−0.866937	−	0.498418i	$-0.833915\pi$
$593$	−25038.0	−1.73387	−0.866937	+	0.498418i	$0.833915\pi$
$594$	0	0
$595$	2856.00	0.196781
$596$	0	0
$597$	−25760.0	−1.76597
$598$	0	0
$599$	5784.00	0.394537	0.197269	−	0.980349i	$-0.436793\pi$
$599$	5784.00	0.394537	0.197269	+	0.980349i	$0.436793\pi$
$600$	0	0
$601$	−4198.00	−0.284925	−0.142463	−	0.989800i	$-0.545502\pi$
$601$	−4198.00	−0.284925	−0.142463	+	0.989800i	$0.545502\pi$
$602$	0	0
$603$	17908.0	1.20940
$604$	0	0
$605$	4530.00	0.304414
$606$	0	0
$607$	−12124.0	−0.810705	−0.405353	−	0.914160i	$-0.632851\pi$
$607$	−12124.0	−0.810705	−0.405353	+	0.914160i	$0.632851\pi$
$608$	0	0
$609$	6720.00	0.447140
$610$	0	0
$611$	16704.0	1.10601
$612$	0	0
$613$	−7454.00	−0.491133	−0.245566	−	0.969380i	$-0.578974\pi$
$613$	−7454.00	−0.491133	−0.245566	+	0.969380i	$0.578974\pi$
$614$	0	0
$615$	16416.0	1.07635
$616$	0	0
$617$	28842.0	1.88190	0.940952	−	0.338539i	$-0.109933\pi$
$617$	28842.0	1.88190	0.940952	+	0.338539i	$0.109933\pi$
$618$	0	0
$619$	17224.0	1.11840	0.559201	−	0.829032i	$-0.311108\pi$
$619$	17224.0	1.11840	0.559201	+	0.829032i	$0.311108\pi$
$620$	0	0
$621$	−4800.00	−0.310173
$622$	0	0
$623$	21672.0	1.39369
$624$	0	0
$625$	3421.00	0.218944
$626$	0	0
$627$	−22272.0	−1.41859
$628$	0	0
$629$	986.000	0.0625030
$630$	0	0
$631$	−12448.0	−0.785336	−0.392668	−	0.919680i	$-0.628448\pi$
$631$	−12448.0	−0.785336	−0.392668	+	0.919680i	$0.628448\pi$
$632$	0	0
$633$	16640.0	1.04484
$634$	0	0
$635$	2832.00	0.176983
$636$	0	0
$637$	25578.0	1.59095
$638$	0	0
$639$	−26196.0	−1.62175
$640$	0	0
$641$	−25182.0	−1.55168	−0.775842	−	0.630927i	$-0.782675\pi$
$641$	−25182.0	−1.55168	−0.775842	+	0.630927i	$0.782675\pi$
$642$	0	0
$643$	−17048.0	−1.04558	−0.522790	−	0.852462i	$-0.675109\pi$
$643$	−17048.0	−1.04558	−0.522790	+	0.852462i	$0.675109\pi$
$644$	0	0
$645$	−7104.00	−0.433674
$646$	0	0
$647$	7128.00	0.433123	0.216562	−	0.976269i	$-0.430516\pi$
$647$	7128.00	0.433123	0.216562	+	0.976269i	$0.430516\pi$
$648$	0	0
$649$	−6048.00	−0.365801
$650$	0	0
$651$	38528.0	2.31956
$652$	0	0
$653$	−18462.0	−1.10639	−0.553196	−	0.833051i	$-0.686592\pi$
$653$	−18462.0	−1.10639	−0.553196	+	0.833051i	$0.686592\pi$
$654$	0	0
$655$	16560.0	0.987867
$656$	0	0
$657$	13394.0	0.795357
$658$	0	0
$659$	−28092.0	−1.66056	−0.830280	−	0.557347i	$-0.811819\pi$
$659$	−28092.0	−1.66056	−0.830280	+	0.557347i	$0.811819\pi$
$660$	0	0
$661$	−10910.0	−0.641982	−0.320991	−	0.947082i	$-0.604016\pi$
$661$	−10910.0	−0.641982	−0.320991	+	0.947082i	$0.604016\pi$
$662$	0	0
$663$	7888.00	0.462058
$664$	0	0
$665$	−19488.0	−1.13641
$666$	0	0
$667$	1800.00	0.104492
$668$	0	0
$669$	37312.0	2.15630
$670$	0	0
$671$	−2640.00	−0.151887
$672$	0	0
$673$	−28414.0	−1.62746	−0.813729	−	0.581244i	$-0.802566\pi$
$673$	−28414.0	−1.62746	−0.813729	+	0.581244i	$0.802566\pi$
$674$	0	0
$675$	−7120.00	−0.405998
$676$	0	0
$677$	6042.00	0.343003	0.171501	−	0.985184i	$-0.445138\pi$
$677$	6042.00	0.343003	0.171501	+	0.985184i	$0.445138\pi$
$678$	0	0
$679$	10696.0	0.604528
$680$	0	0
$681$	11520.0	0.648234
$682$	0	0
$683$	−34752.0	−1.94692	−0.973461	−	0.228851i	$-0.926503\pi$
$683$	−34752.0	−1.94692	−0.973461	+	0.228851i	$0.926503\pi$
$684$	0	0
$685$	−6588.00	−0.367466
$686$	0	0
$687$	9488.00	0.526914
$688$	0	0
$689$	−18444.0	−1.01983
$690$	0	0
$691$	−18320.0	−1.00858	−0.504288	−	0.863536i	$-0.668245\pi$
$691$	−18320.0	−1.00858	−0.504288	+	0.863536i	$0.668245\pi$
$692$	0	0
$693$	−24864.0	−1.36292
$694$	0	0
$695$	15168.0	0.827849
$696$	0	0
$697$	−5814.00	−0.315955
$698$	0	0
$699$	−42672.0	−2.30902
$700$	0	0
$701$	22890.0	1.23330	0.616650	−	0.787237i	$-0.288489\pi$
$701$	22890.0	1.23330	0.616650	+	0.787237i	$0.288489\pi$
$702$	0	0
$703$	−6728.00	−0.360955
$704$	0	0
$705$	−13824.0	−0.738499
$706$	0	0
$707$	−5880.00	−0.312787
$708$	0	0
$709$	−22886.0	−1.21227	−0.606137	−	0.795361i	$-0.707282\pi$
$709$	−22886.0	−1.21227	−0.606137	+	0.795361i	$0.707282\pi$
$710$	0	0
$711$	−17908.0	−0.944589
$712$	0	0
$713$	10320.0	0.542058
$714$	0	0
$715$	−8352.00	−0.436849
$716$	0	0
$717$	42624.0	2.22011
$718$	0	0
$719$	−13452.0	−0.697740	−0.348870	−	0.937171i	$-0.613435\pi$
$719$	−13452.0	−0.697740	−0.348870	+	0.937171i	$0.613435\pi$
$720$	0	0
$721$	6496.00	0.335539
$722$	0	0
$723$	44944.0	2.31187
$724$	0	0
$725$	2670.00	0.136774
$726$	0	0
$727$	−27304.0	−1.39292	−0.696458	−	0.717598i	$-0.745242\pi$
$727$	−27304.0	−1.39292	−0.696458	+	0.717598i	$0.745242\pi$
$728$	0	0
$729$	−30563.0	−1.55276
$730$	0	0
$731$	2516.00	0.127302
$732$	0	0
$733$	−24470.0	−1.23304	−0.616521	−	0.787338i	$-0.711459\pi$
$733$	−24470.0	−1.23304	−0.616521	+	0.787338i	$0.711459\pi$
$734$	0	0
$735$	−21168.0	−1.06230
$736$	0	0
$737$	11616.0	0.580571
$738$	0	0
$739$	−35252.0	−1.75476	−0.877379	−	0.479798i	$-0.840710\pi$
$739$	−35252.0	−1.75476	−0.877379	+	0.479798i	$0.840710\pi$
$740$	0	0
$741$	−53824.0	−2.66839
$742$	0	0
$743$	1548.00	0.0764342	0.0382171	−	0.999269i	$-0.487832\pi$
$743$	1548.00	0.0764342	0.0382171	+	0.999269i	$0.487832\pi$
$744$	0	0
$745$	9684.00	0.476234
$746$	0	0
$747$	−27972.0	−1.37007
$748$	0	0
$749$	12096.0	0.590091
$750$	0	0
$751$	2948.00	0.143241	0.0716205	−	0.997432i	$-0.477183\pi$
$751$	2948.00	0.143241	0.0716205	+	0.997432i	$0.477183\pi$
$752$	0	0
$753$	16224.0	0.785173
$754$	0	0
$755$	19968.0	0.962530
$756$	0	0
$757$	754.000	0.0362016	0.0181008	−	0.999836i	$-0.494238\pi$
$757$	754.000	0.0362016	0.0181008	+	0.999836i	$0.494238\pi$
$758$	0	0
$759$	−11520.0	−0.550922
$760$	0	0
$761$	−41574.0	−1.98036	−0.990182	−	0.139787i	$-0.955358\pi$
$761$	−41574.0	−1.98036	−0.990182	+	0.139787i	$0.955358\pi$
$762$	0	0
$763$	−33208.0	−1.57564
$764$	0	0
$765$	−3774.00	−0.178365
$766$	0	0
$767$	−14616.0	−0.688075
$768$	0	0
$769$	−15118.0	−0.708932	−0.354466	−	0.935069i	$-0.615337\pi$
$769$	−15118.0	−0.708932	−0.354466	+	0.935069i	$0.615337\pi$
$770$	0	0
$771$	−15216.0	−0.710753
$772$	0	0
$773$	−23550.0	−1.09578	−0.547888	−	0.836552i	$-0.684568\pi$
$773$	−23550.0	−1.09578	−0.547888	+	0.836552i	$0.684568\pi$
$774$	0	0
$775$	15308.0	0.709522
$776$	0	0
$777$	−12992.0	−0.599853
$778$	0	0
$779$	39672.0	1.82464
$780$	0	0
$781$	−16992.0	−0.778517
$782$	0	0
$783$	−2400.00	−0.109539
$784$	0	0
$785$	−14748.0	−0.670546
$786$	0	0
$787$	−5240.00	−0.237339	−0.118670	−	0.992934i	$-0.537863\pi$
$787$	−5240.00	−0.237339	−0.118670	+	0.992934i	$0.537863\pi$
$788$	0	0
$789$	−43776.0	−1.97524
$790$	0	0
$791$	10248.0	0.460654
$792$	0	0
$793$	−6380.00	−0.285700
$794$	0	0
$795$	15264.0	0.680954
$796$	0	0
$797$	−5526.00	−0.245597	−0.122799	−	0.992432i	$-0.539187\pi$
$797$	−5526.00	−0.245597	−0.122799	+	0.992432i	$0.539187\pi$
$798$	0	0
$799$	4896.00	0.216781
$800$	0	0
$801$	−28638.0	−1.26326
$802$	0	0
$803$	8688.00	0.381809
$804$	0	0
$805$	−10080.0	−0.441333
$806$	0	0
$807$	28560.0	1.24580
$808$	0	0
$809$	−438.000	−0.0190349	−0.00951747	−	0.999955i	$-0.503030\pi$
$809$	−438.000	−0.0190349	−0.00951747	+	0.999955i	$0.503030\pi$
$810$	0	0
$811$	30448.0	1.31834	0.659170	−	0.751994i	$-0.270908\pi$
$811$	30448.0	1.31834	0.659170	+	0.751994i	$0.270908\pi$
$812$	0	0
$813$	2176.00	0.0938692
$814$	0	0
$815$	1632.00	0.0701429
$816$	0	0
$817$	−17168.0	−0.735168
$818$	0	0
$819$	−60088.0	−2.56367
$820$	0	0
$821$	21930.0	0.932232	0.466116	−	0.884724i	$-0.345653\pi$
$821$	21930.0	0.932232	0.466116	+	0.884724i	$0.345653\pi$
$822$	0	0
$823$	−27436.0	−1.16204	−0.581020	−	0.813889i	$-0.697346\pi$
$823$	−27436.0	−1.16204	−0.581020	+	0.813889i	$0.697346\pi$
$824$	0	0
$825$	−17088.0	−0.721125
$826$	0	0
$827$	17832.0	0.749794	0.374897	−	0.927067i	$-0.377678\pi$
$827$	17832.0	0.749794	0.374897	+	0.927067i	$0.377678\pi$
$828$	0	0
$829$	4090.00	0.171353	0.0856765	−	0.996323i	$-0.472695\pi$
$829$	4090.00	0.171353	0.0856765	+	0.996323i	$0.472695\pi$
$830$	0	0
$831$	−30640.0	−1.27905
$832$	0	0
$833$	7497.00	0.311832
$834$	0	0
$835$	−21096.0	−0.874320
$836$	0	0
$837$	−13760.0	−0.568238
$838$	0	0
$839$	−2508.00	−0.103201	−0.0516006	−	0.998668i	$-0.516432\pi$
$839$	−2508.00	−0.103201	−0.0516006	+	0.998668i	$0.516432\pi$
$840$	0	0
$841$	−23489.0	−0.963098
$842$	0	0
$843$	70992.0	2.90047
$844$	0	0
$845$	−7002.00	−0.285061
$846$	0	0
$847$	21140.0	0.857590
$848$	0	0
$849$	21056.0	0.851166
$850$	0	0
$851$	−3480.00	−0.140180
$852$	0	0
$853$	42442.0	1.70362	0.851809	−	0.523852i	$-0.175506\pi$
$853$	42442.0	1.70362	0.851809	+	0.523852i	$0.175506\pi$
$854$	0	0
$855$	25752.0	1.03006
$856$	0	0
$857$	32730.0	1.30459	0.652296	−	0.757964i	$-0.273806\pi$
$857$	32730.0	1.30459	0.652296	+	0.757964i	$0.273806\pi$
$858$	0	0
$859$	6148.00	0.244199	0.122100	−	0.992518i	$-0.461037\pi$
$859$	6148.00	0.244199	0.122100	+	0.992518i	$0.461037\pi$
$860$	0	0
$861$	76608.0	3.03228
$862$	0	0
$863$	−22512.0	−0.887969	−0.443985	−	0.896034i	$-0.646436\pi$
$863$	−22512.0	−0.887969	−0.443985	+	0.896034i	$0.646436\pi$
$864$	0	0
$865$	−11052.0	−0.434427
$866$	0	0
$867$	2312.00	0.0905647
$868$	0	0
$869$	−11616.0	−0.453447
$870$	0	0
$871$	28072.0	1.09206
$872$	0	0
$873$	−14134.0	−0.547954
$874$	0	0
$875$	−35952.0	−1.38903
$876$	0	0
$877$	−9182.00	−0.353539	−0.176770	−	0.984252i	$-0.556565\pi$
$877$	−9182.00	−0.353539	−0.176770	+	0.984252i	$0.556565\pi$
$878$	0	0
$879$	51216.0	1.96527
$880$	0	0
$881$	28530.0	1.09103	0.545517	−	0.838100i	$-0.316334\pi$
$881$	28530.0	1.09103	0.545517	+	0.838100i	$0.316334\pi$
$882$	0	0
$883$	12436.0	0.473958	0.236979	−	0.971515i	$-0.423843\pi$
$883$	12436.0	0.473958	0.236979	+	0.971515i	$0.423843\pi$
$884$	0	0
$885$	12096.0	0.459438
$886$	0	0
$887$	7404.00	0.280273	0.140136	−	0.990132i	$-0.455246\pi$
$887$	7404.00	0.280273	0.140136	+	0.990132i	$0.455246\pi$
$888$	0	0
$889$	13216.0	0.498594
$890$	0	0
$891$	−8616.00	−0.323958
$892$	0	0
$893$	−33408.0	−1.25191
$894$	0	0
$895$	−21096.0	−0.787890
$896$	0	0
$897$	−27840.0	−1.03629
$898$	0	0
$899$	5160.00	0.191430
$900$	0	0
$901$	−5406.00	−0.199889
$902$	0	0
$903$	−33152.0	−1.22174
$904$	0	0
$905$	20388.0	0.748862
$906$	0	0
$907$	−15368.0	−0.562609	−0.281304	−	0.959619i	$-0.590767\pi$
$907$	−15368.0	−0.562609	−0.281304	+	0.959619i	$0.590767\pi$
$908$	0	0
$909$	7770.00	0.283514
$910$	0	0
$911$	27276.0	0.991980	0.495990	−	0.868328i	$-0.334805\pi$
$911$	27276.0	0.991980	0.495990	+	0.868328i	$0.334805\pi$
$912$	0	0
$913$	−18144.0	−0.657699
$914$	0	0
$915$	5280.00	0.190767
$916$	0	0
$917$	77280.0	2.78300
$918$	0	0
$919$	−46456.0	−1.66751	−0.833755	−	0.552134i	$-0.813814\pi$
$919$	−46456.0	−1.66751	−0.833755	+	0.552134i	$0.813814\pi$
$920$	0	0
$921$	71840.0	2.57026
$922$	0	0
$923$	−41064.0	−1.46440
$924$	0	0
$925$	−5162.00	−0.183487
$926$	0	0
$927$	−8584.00	−0.304138
$928$	0	0
$929$	13026.0	0.460031	0.230016	−	0.973187i	$-0.426122\pi$
$929$	13026.0	0.460031	0.230016	+	0.973187i	$0.426122\pi$
$930$	0	0
$931$	−51156.0	−1.80083
$932$	0	0
$933$	−31776.0	−1.11500
$934$	0	0
$935$	−2448.00	−0.0856237
$936$	0	0
$937$	26330.0	0.917997	0.458999	−	0.888437i	$-0.348208\pi$
$937$	26330.0	0.917997	0.458999	+	0.888437i	$0.348208\pi$
$938$	0	0
$939$	37840.0	1.31508
$940$	0	0
$941$	−28254.0	−0.978803	−0.489402	−	0.872058i	$-0.662785\pi$
$941$	−28254.0	−0.978803	−0.489402	+	0.872058i	$0.662785\pi$
$942$	0	0
$943$	20520.0	0.708614
$944$	0	0
$945$	13440.0	0.462649
$946$	0	0
$947$	−49272.0	−1.69073	−0.845367	−	0.534186i	$-0.820618\pi$
$947$	−49272.0	−1.69073	−0.845367	+	0.534186i	$0.820618\pi$
$948$	0	0
$949$	20996.0	0.718187
$950$	0	0
$951$	23184.0	0.790529
$952$	0	0
$953$	32922.0	1.11904	0.559522	−	0.828816i	$-0.310985\pi$
$953$	32922.0	1.11904	0.559522	+	0.828816i	$0.310985\pi$
$954$	0	0
$955$	15840.0	0.536723
$956$	0	0
$957$	−5760.00	−0.194560
$958$	0	0
$959$	−30744.0	−1.03522
$960$	0	0
$961$	−207.000	−0.00694841
$962$	0	0
$963$	−15984.0	−0.534867
$964$	0	0
$965$	−17292.0	−0.576839
$966$	0	0
$967$	−1168.00	−0.0388421	−0.0194211	−	0.999811i	$-0.506182\pi$
$967$	−1168.00	−0.0388421	−0.0194211	+	0.999811i	$0.506182\pi$
$968$	0	0
$969$	−15776.0	−0.523011
$970$	0	0
$971$	19812.0	0.654786	0.327393	−	0.944888i	$-0.393830\pi$
$971$	19812.0	0.654786	0.327393	+	0.944888i	$0.393830\pi$
$972$	0	0
$973$	70784.0	2.33220
$974$	0	0
$975$	−41296.0	−1.35644
$976$	0	0
$977$	−28494.0	−0.933064	−0.466532	−	0.884504i	$-0.654497\pi$
$977$	−28494.0	−0.933064	−0.466532	+	0.884504i	$0.654497\pi$
$978$	0	0
$979$	−18576.0	−0.606426
$980$	0	0
$981$	43882.0	1.42818
$982$	0	0
$983$	−42708.0	−1.38573	−0.692866	−	0.721067i	$-0.743652\pi$
$983$	−42708.0	−1.38573	−0.692866	+	0.721067i	$0.743652\pi$
$984$	0	0
$985$	−252.000	−0.00815166
$986$	0	0
$987$	−64512.0	−2.08049
$988$	0	0
$989$	−8880.00	−0.285508
$990$	0	0
$991$	−29500.0	−0.945609	−0.472804	−	0.881167i	$-0.656758\pi$
$991$	−29500.0	−0.945609	−0.472804	+	0.881167i	$0.656758\pi$
$992$	0	0
$993$	36512.0	1.16684
$994$	0	0
$995$	19320.0	0.615563
$996$	0	0
$997$	9322.00	0.296119	0.148060	−	0.988978i	$-0.452697\pi$
$997$	9322.00	0.296119	0.148060	+	0.988978i	$0.452697\pi$
$998$	0	0
$999$	4640.00	0.146950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1088.4.a.l.1.1		1
4.3	odd	2		1088.4.a.a.1.1		1
8.3	odd	2		272.4.a.d.1.1		1
8.5	even	2		17.4.a.a.1.1	✓	1
24.5	odd	2		153.4.a.d.1.1		1
24.11	even	2		2448.4.a.f.1.1		1
40.13	odd	4		425.4.b.c.324.2		2
40.29	even	2		425.4.a.d.1.1		1
40.37	odd	4		425.4.b.c.324.1		2
56.13	odd	2		833.4.a.a.1.1		1
88.21	odd	2		2057.4.a.d.1.1		1
136.13	even	4		289.4.b.a.288.1		2
136.21	even	4		289.4.b.a.288.2		2
136.101	even	2		289.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.4.a.a.1.1	✓	1	8.5	even	2
153.4.a.d.1.1		1	24.5	odd	2
272.4.a.d.1.1		1	8.3	odd	2
289.4.a.a.1.1		1	136.101	even	2
289.4.b.a.288.1		2	136.13	even	4
289.4.b.a.288.2		2	136.21	even	4
425.4.a.d.1.1		1	40.29	even	2
425.4.b.c.324.1		2	40.37	odd	4
425.4.b.c.324.2		2	40.13	odd	4
833.4.a.a.1.1		1	56.13	odd	2
1088.4.a.a.1.1		1	4.3	odd	2
1088.4.a.l.1.1		1	1.1	even	1	trivial
2057.4.a.d.1.1		1	88.21	odd	2
2448.4.a.f.1.1		1	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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