LMFDB - Embedded newform 153.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [153,4,Mod(1,153)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(153, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("153.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 153 = 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	153.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:	$9.02729223088$
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	$\chi$	$=$	153.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{2} +1.00000 q^{4} -6.00000 q^{5} -28.0000 q^{7} -21.0000 q^{8} +O(q^{10})$

\(q+3.00000 q^{2} +1.00000 q^{4} -6.00000 q^{5} -28.0000 q^{7} -21.0000 q^{8} -18.0000 q^{10} +24.0000 q^{11} -58.0000 q^{13} -84.0000 q^{14} -71.0000 q^{16} -17.0000 q^{17} +116.000 q^{19} -6.00000 q^{20} +72.0000 q^{22} +60.0000 q^{23} -89.0000 q^{25} -174.000 q^{26} -28.0000 q^{28} -30.0000 q^{29} -172.000 q^{31} -45.0000 q^{32} -51.0000 q^{34} +168.000 q^{35} -58.0000 q^{37} +348.000 q^{38} +126.000 q^{40} +342.000 q^{41} -148.000 q^{43} +24.0000 q^{44} +180.000 q^{46} -288.000 q^{47} +441.000 q^{49} -267.000 q^{50} -58.0000 q^{52} -318.000 q^{53} -144.000 q^{55} +588.000 q^{56} -90.0000 q^{58} -252.000 q^{59} +110.000 q^{61} -516.000 q^{62} +433.000 q^{64} +348.000 q^{65} -484.000 q^{67} -17.0000 q^{68} +504.000 q^{70} +708.000 q^{71} +362.000 q^{73} -174.000 q^{74} +116.000 q^{76} -672.000 q^{77} -484.000 q^{79} +426.000 q^{80} +1026.00 q^{82} -756.000 q^{83} +102.000 q^{85} -444.000 q^{86} -504.000 q^{88} +774.000 q^{89} +1624.00 q^{91} +60.0000 q^{92} -864.000 q^{94} -696.000 q^{95} -382.000 q^{97} +1323.00 q^{98} +O(q^{100})\)

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 $

$2$	3.00000	1.06066	0.530330	−	0.847791i	$-0.322068\pi$
$2$	3.00000	1.06066	0.530330	+	0.847791i	$0.322068\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.125000
$5$	−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	$0.772814\pi$
$8$	−21.0000	−0.928078
$9$	0	0
$10$	−18.0000	−0.569210
$11$	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	$-0.712342\pi$
$13$	−58.0000	−1.23741	−0.618704	+	0.785624i	$0.712342\pi$
$14$	−84.0000	−1.60357
$15$	0	0
$16$	−71.0000	−1.10938
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	116.000	1.40064	0.700322	−	0.713827i	$-0.253040\pi$
$19$	116.000	1.40064	0.700322	+	0.713827i	$0.253040\pi$
$20$	−6.00000	−0.0670820
$21$	0	0
$22$	72.0000	0.697748
$23$	60.0000	0.543951	0.271975	−	0.962304i	$-0.412323\pi$
$23$	60.0000	0.543951	0.271975	+	0.962304i	$0.412323\pi$
$24$	0	0
$25$	−89.0000	−0.712000
$26$	−174.000	−1.31247
$27$	0	0
$28$	−28.0000	−0.188982
$29$	−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	$0.666027\pi$
$32$	−45.0000	−0.248592
$33$	0	0
$34$	−51.0000	−0.257248
$35$	168.000	0.811348
$36$	0	0
$37$	−58.0000	−0.257707	−0.128853	−	0.991664i	$-0.541130\pi$
$37$	−58.0000	−0.257707	−0.128853	+	0.991664i	$0.541130\pi$
$38$	348.000	1.48561
$39$	0	0
$40$	126.000	0.498059
$41$	342.000	1.30272	0.651359	−	0.758770i	$-0.274199\pi$
$41$	342.000	1.30272	0.651359	+	0.758770i	$0.274199\pi$
$42$	0	0
$43$	−148.000	−0.524879	−0.262439	−	0.964948i	$-0.584527\pi$
$43$	−148.000	−0.524879	−0.262439	+	0.964948i	$0.584527\pi$
$44$	24.0000	0.0822304
$45$	0	0
$46$	180.000	0.576947
$47$	−288.000	−0.893811	−0.446906	−	0.894581i	$-0.647474\pi$
$47$	−288.000	−0.893811	−0.446906	+	0.894581i	$0.647474\pi$
$48$	0	0
$49$	441.000	1.28571
$50$	−267.000	−0.755190
$51$	0	0
$52$	−58.0000	−0.154676
$53$	−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
$56$	588.000	1.40312
$57$	0	0
$58$	−90.0000	−0.203751
$59$	−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	$-0.463171\pi$
$61$	110.000	0.230886	0.115443	+	0.993314i	$0.463171\pi$
$62$	−516.000	−1.05697
$63$	0	0
$64$	433.000	0.845703
$65$	348.000	0.664063
$66$	0	0
$67$	−484.000	−0.882537	−0.441269	−	0.897375i	$-0.645471\pi$
$67$	−484.000	−0.882537	−0.441269	+	0.897375i	$0.645471\pi$
$68$	−17.0000	−0.0303170
$69$	0	0
$70$	504.000	0.860565
$71$	708.000	1.18344	0.591719	−	0.806144i	$-0.298449\pi$
$71$	708.000	1.18344	0.591719	+	0.806144i	$0.298449\pi$
$72$	0	0
$73$	362.000	0.580396	0.290198	−	0.956967i	$-0.406279\pi$
$73$	362.000	0.580396	0.290198	+	0.956967i	$0.406279\pi$
$74$	−174.000	−0.273339
$75$	0	0
$76$	116.000	0.175080
$77$	−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	$0.612001\pi$
$80$	426.000	0.595353
$81$	0	0
$82$	1026.00	1.38174
$83$	−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
$86$	−444.000	−0.556718
$87$	0	0
$88$	−504.000	−0.610529
$89$	774.000	0.921841	0.460920	−	0.887441i	$-0.347519\pi$
$89$	774.000	0.921841	0.460920	+	0.887441i	$0.347519\pi$
$90$	0	0
$91$	1624.00	1.87079
$92$	60.0000	0.0679938
$93$	0	0
$94$	−864.000	−0.948030
$95$	−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	$0.564071\pi$
$98$	1323.00	1.36371
$99$	0	0
$100$	−89.0000	−0.0890000
$101$	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	$0.535395\pi$
$104$	1218.00	1.14841
$105$	0	0
$106$	−954.000	−0.874157
$107$	−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	$-0.674475\pi$
$109$	−1186.00	−1.04219	−0.521093	+	0.853500i	$0.674475\pi$
$110$	−432.000	−0.374451
$111$	0	0
$112$	1988.00	1.67722
$113$	366.000	0.304694	0.152347	−	0.988327i	$-0.451317\pi$
$113$	366.000	0.304694	0.152347	+	0.988327i	$0.451317\pi$
$114$	0	0
$115$	−360.000	−0.291915
$116$	−30.0000	−0.0240123
$117$	0	0
$118$	−756.000	−0.589792
$119$	476.000	0.366679
$120$	0	0
$121$	−755.000	−0.567243
$122$	330.000	0.244892
$123$	0	0
$124$	−172.000	−0.124565
$125$	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	$0.552728\pi$
$128$	1659.00	1.14560
$129$	0	0
$130$	1044.00	0.704345
$131$	−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
$134$	−1452.00	−0.936072
$135$	0	0
$136$	357.000	0.225092
$137$	−1098.00	−0.684733	−0.342367	−	0.939566i	$-0.611229\pi$
$137$	−1098.00	−0.684733	−0.342367	+	0.939566i	$0.611229\pi$
$138$	0	0
$139$	2528.00	1.54261	0.771303	−	0.636468i	$-0.219605\pi$
$139$	2528.00	1.54261	0.771303	+	0.636468i	$0.219605\pi$
$140$	168.000	0.101419
$141$	0	0
$142$	2124.00	1.25523
$143$	−1392.00	−0.814020
$144$	0	0
$145$	180.000	0.103091
$146$	1086.00	0.615603
$147$	0	0
$148$	−58.0000	−0.0322133
$149$	−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	$0.854103\pi$
$152$	−2436.00	−1.29991
$153$	0	0
$154$	−2016.00	−1.05490
$155$	1032.00	0.534789
$156$	0	0
$157$	−2458.00	−1.24949	−0.624744	−	0.780829i	$-0.714797\pi$
$157$	−2458.00	−1.24949	−0.624744	+	0.780829i	$0.714797\pi$
$158$	−1452.00	−0.731107
$159$	0	0
$160$	270.000	0.133409
$161$	−1680.00	−0.822376
$162$	0	0
$163$	272.000	0.130704	0.0653518	−	0.997862i	$-0.479183\pi$
$163$	272.000	0.130704	0.0653518	+	0.997862i	$0.479183\pi$
$164$	342.000	0.162840
$165$	0	0
$166$	−2268.00	−1.06043
$167$	−3516.00	−1.62920	−0.814600	−	0.580024i	$-0.803043\pi$
$167$	−3516.00	−1.62920	−0.814600	+	0.580024i	$0.803043\pi$
$168$	0	0
$169$	1167.00	0.531179
$170$	306.000	0.138054
$171$	0	0
$172$	−148.000	−0.0656099
$173$	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
$176$	−1704.00	−0.729795
$177$	0	0
$178$	2322.00	0.977760
$179$	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	$-0.254202\pi$
$181$	3398.00	1.39542	0.697711	+	0.716379i	$0.254202\pi$
$182$	4872.00	1.98427
$183$	0	0
$184$	−1260.00	−0.504828
$185$	348.000	0.138300
$186$	0	0
$187$	−408.000	−0.159550
$188$	−288.000	−0.111726
$189$	0	0
$190$	−2088.00	−0.797260
$191$	2640.00	1.00012	0.500062	−	0.865990i	$-0.333311\pi$
$191$	2640.00	1.00012	0.500062	+	0.865990i	$0.333311\pi$
$192$	0	0
$193$	2882.00	1.07488	0.537438	−	0.843304i	$-0.319392\pi$
$193$	2882.00	1.07488	0.537438	+	0.843304i	$0.319392\pi$
$194$	−1146.00	−0.424113
$195$	0	0
$196$	441.000	0.160714
$197$	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	$0.694421\pi$
$200$	1869.00	0.660791
$201$	0	0
$202$	630.000	0.219439
$203$	840.000	0.290426
$204$	0	0
$205$	−2052.00	−0.699112
$206$	−696.000	−0.235401
$207$	0	0
$208$	4118.00	1.37275
$209$	2784.00	0.921403
$210$	0	0
$211$	−2080.00	−0.678640	−0.339320	−	0.940671i	$-0.610197\pi$
$211$	−2080.00	−0.678640	−0.339320	+	0.940671i	$0.610197\pi$
$212$	−318.000	−0.103020
$213$	0	0
$214$	−1296.00	−0.413985
$215$	888.000	0.281680
$216$	0	0
$217$	4816.00	1.50660
$218$	−3558.00	−1.10540
$219$	0	0
$220$	−144.000	−0.0441294
$221$	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	$0.253059\pi$
$224$	1260.00	0.375836
$225$	0	0
$226$	1098.00	0.323176
$227$	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	$-0.554739\pi$
$229$	−1186.00	−0.342241	−0.171120	+	0.985250i	$0.554739\pi$
$230$	−1080.00	−0.309622
$231$	0	0
$232$	630.000	0.178282
$233$	5334.00	1.49975	0.749875	−	0.661579i	$-0.230113\pi$
$233$	5334.00	1.49975	0.749875	+	0.661579i	$0.230113\pi$
$234$	0	0
$235$	1728.00	0.479669
$236$	−252.000	−0.0695076
$237$	0	0
$238$	1428.00	0.388922
$239$	−5328.00	−1.44201	−0.721003	−	0.692931i	$-0.756319\pi$
$239$	−5328.00	−1.44201	−0.721003	+	0.692931i	$0.756319\pi$
$240$	0	0
$241$	5618.00	1.50161	0.750803	−	0.660526i	$-0.229667\pi$
$241$	5618.00	1.50161	0.750803	+	0.660526i	$0.229667\pi$
$242$	−2265.00	−0.601652
$243$	0	0
$244$	110.000	0.0288608
$245$	−2646.00	−0.689987
$246$	0	0
$247$	−6728.00	−1.73317
$248$	3612.00	0.924848
$249$	0	0
$250$	3852.00	0.974487
$251$	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
$254$	−1416.00	−0.349794
$255$	0	0
$256$	1513.00	0.369385
$257$	1902.00	0.461648	0.230824	−	0.972996i	$-0.425858\pi$
$257$	1902.00	0.461648	0.230824	+	0.972996i	$0.425858\pi$
$258$	0	0
$259$	1624.00	0.389616
$260$	348.000	0.0830079
$261$	0	0
$262$	−8280.00	−1.95244
$263$	5472.00	1.28296	0.641479	−	0.767141i	$-0.278321\pi$
$263$	5472.00	1.28296	0.641479	+	0.767141i	$0.278321\pi$
$264$	0	0
$265$	1908.00	0.442292
$266$	−9744.00	−2.24603
$267$	0	0
$268$	−484.000	−0.110317
$269$	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	$0.490295\pi$
$272$	1207.00	0.269063
$273$	0	0
$274$	−3294.00	−0.726269
$275$	−2136.00	−0.468384
$276$	0	0
$277$	3830.00	0.830767	0.415383	−	0.909646i	$-0.363647\pi$
$277$	3830.00	0.830767	0.415383	+	0.909646i	$0.363647\pi$
$278$	7584.00	1.63618
$279$	0	0
$280$	−3528.00	−0.752994
$281$	−8874.00	−1.88391	−0.941955	−	0.335740i	$-0.891014\pi$
$281$	−8874.00	−1.88391	−0.941955	+	0.335740i	$0.891014\pi$
$282$	0	0
$283$	−2632.00	−0.552849	−0.276424	−	0.961036i	$-0.589150\pi$
$283$	−2632.00	−0.552849	−0.276424	+	0.961036i	$0.589150\pi$
$284$	708.000	0.147930
$285$	0	0
$286$	−4176.00	−0.863399
$287$	−9576.00	−1.96952
$288$	0	0
$289$	289.000	0.0588235
$290$	540.000	0.109344
$291$	0	0
$292$	362.000	0.0725495
$293$	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
$296$	1218.00	0.239172
$297$	0	0
$298$	−4842.00	−0.941240
$299$	−3480.00	−0.673089
$300$	0	0
$301$	4144.00	0.793542
$302$	−9984.00	−1.90237
$303$	0	0
$304$	−8236.00	−1.55384
$305$	−660.000	−0.123907
$306$	0	0
$307$	−8980.00	−1.66943	−0.834716	−	0.550681i	$-0.814368\pi$
$307$	−8980.00	−1.66943	−0.834716	+	0.550681i	$0.814368\pi$
$308$	−672.000	−0.124321
$309$	0	0
$310$	3096.00	0.567229
$311$	3972.00	0.724217	0.362108	−	0.932136i	$-0.382057\pi$
$311$	3972.00	0.724217	0.362108	+	0.932136i	$0.382057\pi$
$312$	0	0
$313$	4730.00	0.854171	0.427085	−	0.904211i	$-0.359540\pi$
$313$	4730.00	0.854171	0.427085	+	0.904211i	$0.359540\pi$
$314$	−7374.00	−1.32528
$315$	0	0
$316$	−484.000	−0.0861618
$317$	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
$320$	−2598.00	−0.453852
$321$	0	0
$322$	−5040.00	−0.872262
$323$	−1972.00	−0.339706
$324$	0	0
$325$	5162.00	0.881035
$326$	816.000	0.138632
$327$	0	0
$328$	−7182.00	−1.20902
$329$	8064.00	1.35132
$330$	0	0
$331$	−4564.00	−0.757886	−0.378943	−	0.925420i	$-0.623712\pi$
$331$	−4564.00	−0.757886	−0.378943	+	0.925420i	$0.623712\pi$
$332$	−756.000	−0.124973
$333$	0	0
$334$	−10548.0	−1.72803
$335$	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	$0.481415\pi$
$338$	3501.00	0.563400
$339$	0	0
$340$	102.000	0.0162698
$341$	−4128.00	−0.655553
$342$	0	0
$343$	−2744.00	−0.431959
$344$	3108.00	0.487128
$345$	0	0
$346$	5526.00	0.858612
$347$	−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	$-0.174635\pi$
$349$	11126.0	1.70648	0.853239	+	0.521519i	$0.174635\pi$
$350$	7476.00	1.14174
$351$	0	0
$352$	−1080.00	−0.163535
$353$	−7842.00	−1.18240	−0.591200	−	0.806525i	$-0.701346\pi$
$353$	−7842.00	−1.18240	−0.591200	+	0.806525i	$0.701346\pi$
$354$	0	0
$355$	−4248.00	−0.635100
$356$	774.000	0.115230
$357$	0	0
$358$	10548.0	1.55720
$359$	−5040.00	−0.740950	−0.370475	−	0.928842i	$-0.620805\pi$
$359$	−5040.00	−0.740950	−0.370475	+	0.928842i	$0.620805\pi$
$360$	0	0
$361$	6597.00	0.961802
$362$	10194.0	1.48007
$363$	0	0
$364$	1624.00	0.233848
$365$	−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	$0.703904\pi$
$368$	−4260.00	−0.603445
$369$	0	0
$370$	1044.00	0.146689
$371$	8904.00	1.24602
$372$	0	0
$373$	−8098.00	−1.12412	−0.562062	−	0.827095i	$-0.689992\pi$
$373$	−8098.00	−1.12412	−0.562062	+	0.827095i	$0.689992\pi$
$374$	−1224.00	−0.169229
$375$	0	0
$376$	6048.00	0.829526
$377$	1740.00	0.237704
$378$	0	0
$379$	320.000	0.0433702	0.0216851	−	0.999765i	$-0.493097\pi$
$379$	320.000	0.0433702	0.0216851	+	0.999765i	$0.493097\pi$
$380$	−696.000	−0.0939580
$381$	0	0
$382$	7920.00	1.06079
$383$	10872.0	1.45048	0.725239	−	0.688497i	$-0.241729\pi$
$383$	10872.0	1.45048	0.725239	+	0.688497i	$0.241729\pi$
$384$	0	0
$385$	4032.00	0.533740
$386$	8646.00	1.14008
$387$	0	0
$388$	−382.000	−0.0499822
$389$	−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
$392$	−9261.00	−1.19324
$393$	0	0
$394$	126.000	0.0161111
$395$	2904.00	0.369914
$396$	0	0
$397$	−7522.00	−0.950928	−0.475464	−	0.879735i	$-0.657720\pi$
$397$	−7522.00	−0.950928	−0.475464	+	0.879735i	$0.657720\pi$
$398$	−9660.00	−1.21661
$399$	0	0
$400$	6319.00	0.789875
$401$	−2706.00	−0.336986	−0.168493	−	0.985703i	$-0.553890\pi$
$401$	−2706.00	−0.336986	−0.168493	+	0.985703i	$0.553890\pi$
$402$	0	0
$403$	9976.00	1.23310
$404$	210.000	0.0258611
$405$	0	0
$406$	2520.00	0.308043
$407$	−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	$0.494882\pi$
$410$	−6156.00	−0.741520
$411$	0	0
$412$	−232.000	−0.0277423
$413$	7056.00	0.840685
$414$	0	0
$415$	4536.00	0.536539
$416$	2610.00	0.307610
$417$	0	0
$418$	8352.00	0.977296
$419$	−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	$-0.794826\pi$
$421$	−13810.0	−1.59871	−0.799357	+	0.600857i	$0.794826\pi$
$422$	−6240.00	−0.719807
$423$	0	0
$424$	6678.00	0.764888
$425$	1513.00	0.172685
$426$	0	0
$427$	−3080.00	−0.349067
$428$	−432.000	−0.0487886
$429$	0	0
$430$	2664.00	0.298766
$431$	−3036.00	−0.339302	−0.169651	−	0.985504i	$-0.554264\pi$
$431$	−3036.00	−0.339302	−0.169651	+	0.985504i	$0.554264\pi$
$432$	0	0
$433$	−11422.0	−1.26768	−0.633841	−	0.773463i	$-0.718523\pi$
$433$	−11422.0	−1.26768	−0.633841	+	0.773463i	$0.718523\pi$
$434$	14448.0	1.59799
$435$	0	0
$436$	−1186.00	−0.130273
$437$	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	$0.500900\pi$
$440$	3024.00	0.327644
$441$	0	0
$442$	2958.00	0.318321
$443$	−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
$446$	13992.0	1.48552
$447$	0	0
$448$	−12124.0	−1.27858
$449$	−6114.00	−0.642622	−0.321311	−	0.946974i	$-0.604124\pi$
$449$	−6114.00	−0.642622	−0.321311	+	0.946974i	$0.604124\pi$
$450$	0	0
$451$	8208.00	0.856984
$452$	366.000	0.0380867
$453$	0	0
$454$	4320.00	0.446581
$455$	−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	$0.432607\pi$
$458$	−3558.00	−0.363001
$459$	0	0
$460$	−360.000	−0.0364893
$461$	−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	$0.485681\pi$
$464$	2130.00	0.213109
$465$	0	0
$466$	16002.0	1.59073
$467$	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
$470$	5184.00	0.508766
$471$	0	0
$472$	5292.00	0.516068
$473$	−3552.00	−0.345288
$474$	0	0
$475$	−10324.0	−0.997258
$476$	476.000	0.0458349
$477$	0	0
$478$	−15984.0	−1.52948
$479$	−5172.00	−0.493350	−0.246675	−	0.969098i	$-0.579338\pi$
$479$	−5172.00	−0.493350	−0.246675	+	0.969098i	$0.579338\pi$
$480$	0	0
$481$	3364.00	0.318888
$482$	16854.0	1.59269
$483$	0	0
$484$	−755.000	−0.0709053
$485$	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	$0.746941\pi$
$488$	−2310.00	−0.214280
$489$	0	0
$490$	−7938.00	−0.731841
$491$	−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
$494$	−20184.0	−1.83830
$495$	0	0
$496$	12212.0	1.10551
$497$	−19824.0	−1.78919
$498$	0	0
$499$	−1168.00	−0.104783	−0.0523916	−	0.998627i	$-0.516684\pi$
$499$	−1168.00	−0.104783	−0.0523916	+	0.998627i	$0.516684\pi$
$500$	1284.00	0.114844
$501$	0	0
$502$	6084.00	0.540921
$503$	1740.00	0.154240	0.0771200	−	0.997022i	$-0.475428\pi$
$503$	1740.00	0.154240	0.0771200	+	0.997022i	$0.475428\pi$
$504$	0	0
$505$	−1260.00	−0.111028
$506$	4320.00	0.379540
$507$	0	0
$508$	−472.000	−0.0412236
$509$	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
$512$	−8733.00	−0.753804
$513$	0	0
$514$	5706.00	0.489651
$515$	1392.00	0.119105
$516$	0	0
$517$	−6912.00	−0.587987
$518$	4872.00	0.413250
$519$	0	0
$520$	−7308.00	−0.616302
$521$	−11658.0	−0.980319	−0.490160	−	0.871633i	$-0.663061\pi$
$521$	−11658.0	−0.980319	−0.490160	+	0.871633i	$0.663061\pi$
$522$	0	0
$523$	13700.0	1.14543	0.572714	−	0.819755i	$-0.305890\pi$
$523$	13700.0	1.14543	0.572714	+	0.819755i	$0.305890\pi$
$524$	−2760.00	−0.230098
$525$	0	0
$526$	16416.0	1.36078
$527$	2924.00	0.241692
$528$	0	0
$529$	−8567.00	−0.704118
$530$	5724.00	0.469122
$531$	0	0
$532$	−3248.00	−0.264697
$533$	−19836.0	−1.61199
$534$	0	0
$535$	2592.00	0.209462
$536$	10164.0	0.819063
$537$	0	0
$538$	10710.0	0.858254
$539$	10584.0	0.845798
$540$	0	0
$541$	17822.0	1.41632	0.708159	−	0.706053i	$-0.249526\pi$
$541$	17822.0	1.41632	0.708159	+	0.706053i	$0.249526\pi$
$542$	816.000	0.0646683
$543$	0	0
$544$	765.000	0.0602925
$545$	7116.00	0.559295
$546$	0	0
$547$	3800.00	0.297032	0.148516	−	0.988910i	$-0.452550\pi$
$547$	3800.00	0.297032	0.148516	+	0.988910i	$0.452550\pi$
$548$	−1098.00	−0.0855917
$549$	0	0
$550$	−6408.00	−0.496796
$551$	−3480.00	−0.269062
$552$	0	0
$553$	13552.0	1.04212
$554$	11490.0	0.881161
$555$	0	0
$556$	2528.00	0.192826
$557$	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
$560$	−11928.0	−0.900089
$561$	0	0
$562$	−26622.0	−1.99819
$563$	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
$566$	−7896.00	−0.586385
$567$	0	0
$568$	−14868.0	−1.09832
$569$	−21834.0	−1.60866	−0.804331	−	0.594181i	$-0.797476\pi$
$569$	−21834.0	−1.60866	−0.804331	+	0.594181i	$0.797476\pi$
$570$	0	0
$571$	−21208.0	−1.55434	−0.777169	−	0.629292i	$-0.783345\pi$
$571$	−21208.0	−1.55434	−0.777169	+	0.629292i	$0.783345\pi$
$572$	−1392.00	−0.101753
$573$	0	0
$574$	−28728.0	−2.08900
$575$	−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	$0.350704\pi$
$578$	867.000	0.0623918
$579$	0	0
$580$	180.000	0.0128864
$581$	21168.0	1.51153
$582$	0	0
$583$	−7632.00	−0.542170
$584$	−7602.00	−0.538652
$585$	0	0
$586$	19206.0	1.35391
$587$	−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
$590$	4536.00	0.316516
$591$	0	0
$592$	4118.00	0.285893
$593$	25038.0	1.73387	0.866937	−	0.498418i	$-0.166085\pi$
$593$	25038.0	1.73387	0.866937	+	0.498418i	$0.166085\pi$
$594$	0	0
$595$	−2856.00	−0.196781
$596$	−1614.00	−0.110926
$597$	0	0
$598$	−10440.0	−0.713919
$599$	−5784.00	−0.394537	−0.197269	−	0.980349i	$-0.563207\pi$
$599$	−5784.00	−0.394537	−0.197269	+	0.980349i	$0.563207\pi$
$600$	0	0
$601$	−4198.00	−0.284925	−0.142463	−	0.989800i	$-0.545502\pi$
$601$	−4198.00	−0.284925	−0.142463	+	0.989800i	$0.545502\pi$
$602$	12432.0	0.841679
$603$	0	0
$604$	−3328.00	−0.224196
$605$	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	$0.632851\pi$
$608$	−5220.00	−0.348189
$609$	0	0
$610$	−1980.00	−0.131423
$611$	16704.0	1.10601
$612$	0	0
$613$	7454.00	0.491133	0.245566	−	0.969380i	$-0.421026\pi$
$613$	7454.00	0.491133	0.245566	+	0.969380i	$0.421026\pi$
$614$	−26940.0	−1.77070
$615$	0	0
$616$	14112.0	0.923034
$617$	−28842.0	−1.88190	−0.940952	−	0.338539i	$-0.890067\pi$
$617$	−28842.0	−1.88190	−0.940952	+	0.338539i	$0.890067\pi$
$618$	0	0
$619$	−17224.0	−1.11840	−0.559201	−	0.829032i	$-0.688892\pi$
$619$	−17224.0	−1.11840	−0.559201	+	0.829032i	$0.688892\pi$
$620$	1032.00	0.0668486
$621$	0	0
$622$	11916.0	0.768148
$623$	−21672.0	−1.39369
$624$	0	0
$625$	3421.00	0.218944
$626$	14190.0	0.905985
$627$	0	0
$628$	−2458.00	−0.156186
$629$	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	$0.628448\pi$
$632$	10164.0	0.639719
$633$	0	0
$634$	8694.00	0.544610
$635$	2832.00	0.176983
$636$	0	0
$637$	−25578.0	−1.59095
$638$	−2160.00	−0.134036
$639$	0	0
$640$	−9954.00	−0.614791
$641$	25182.0	1.55168	0.775842	−	0.630927i	$-0.217325\pi$
$641$	25182.0	1.55168	0.775842	+	0.630927i	$0.217325\pi$
$642$	0	0
$643$	17048.0	1.04558	0.522790	−	0.852462i	$-0.324891\pi$
$643$	17048.0	1.04558	0.522790	+	0.852462i	$0.324891\pi$
$644$	−1680.00	−0.102797
$645$	0	0
$646$	−5916.00	−0.360313
$647$	−7128.00	−0.433123	−0.216562	−	0.976269i	$-0.569484\pi$
$647$	−7128.00	−0.433123	−0.216562	+	0.976269i	$0.569484\pi$
$648$	0	0
$649$	−6048.00	−0.365801
$650$	15486.0	0.934478
$651$	0	0
$652$	272.000	0.0163379
$653$	−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
$656$	−24282.0	−1.44520
$657$	0	0
$658$	24192.0	1.43329
$659$	−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	$-0.395984\pi$
$661$	10910.0	0.641982	0.320991	+	0.947082i	$0.395984\pi$
$662$	−13692.0	−0.803859
$663$	0	0
$664$	15876.0	0.927874
$665$	19488.0	1.13641
$666$	0	0
$667$	−1800.00	−0.104492
$668$	−3516.00	−0.203650
$669$	0	0
$670$	8712.00	0.502349
$671$	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	$0.802566\pi$
$674$	2166.00	0.123785
$675$	0	0
$676$	1167.00	0.0663974
$677$	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
$680$	−2142.00	−0.120797
$681$	0	0
$682$	−12384.0	−0.695319
$683$	−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
$686$	−8232.00	−0.458162
$687$	0	0
$688$	10508.0	0.582287
$689$	18444.0	1.01983
$690$	0	0
$691$	18320.0	1.00858	0.504288	−	0.863536i	$-0.331755\pi$
$691$	18320.0	1.00858	0.504288	+	0.863536i	$0.331755\pi$
$692$	1842.00	0.101188
$693$	0	0
$694$	−16632.0	−0.909715
$695$	−15168.0	−0.827849
$696$	0	0
$697$	−5814.00	−0.315955
$698$	33378.0	1.80999
$699$	0	0
$700$	2492.00	0.134555
$701$	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
$704$	10392.0	0.556340
$705$	0	0
$706$	−23526.0	−1.25413
$707$	−5880.00	−0.312787
$708$	0	0
$709$	22886.0	1.21227	0.606137	−	0.795361i	$-0.292718\pi$
$709$	22886.0	1.21227	0.606137	+	0.795361i	$0.292718\pi$
$710$	−12744.0	−0.673625
$711$	0	0
$712$	−16254.0	−0.855540
$713$	−10320.0	−0.542058
$714$	0	0
$715$	8352.00	0.436849
$716$	3516.00	0.183518
$717$	0	0
$718$	−15120.0	−0.785896
$719$	13452.0	0.697740	0.348870	−	0.937171i	$-0.386565\pi$
$719$	13452.0	0.697740	0.348870	+	0.937171i	$0.386565\pi$
$720$	0	0
$721$	6496.00	0.335539
$722$	19791.0	1.02015
$723$	0	0
$724$	3398.00	0.174428
$725$	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	$0.745242\pi$
$728$	−34104.0	−1.73623
$729$	0	0
$730$	−6516.00	−0.330367
$731$	2516.00	0.127302
$732$	0	0
$733$	24470.0	1.23304	0.616521	−	0.787338i	$-0.288541\pi$
$733$	24470.0	1.23304	0.616521	+	0.787338i	$0.288541\pi$
$734$	−25212.0	−1.26784
$735$	0	0
$736$	−2700.00	−0.135222
$737$	−11616.0	−0.580571
$738$	0	0
$739$	35252.0	1.75476	0.877379	−	0.479798i	$-0.159290\pi$
$739$	35252.0	1.75476	0.877379	+	0.479798i	$0.159290\pi$
$740$	348.000	0.0172875
$741$	0	0
$742$	26712.0	1.32160
$743$	−1548.00	−0.0764342	−0.0382171	−	0.999269i	$-0.512168\pi$
$743$	−1548.00	−0.0764342	−0.0382171	+	0.999269i	$0.512168\pi$
$744$	0	0
$745$	9684.00	0.476234
$746$	−24294.0	−1.19231
$747$	0	0
$748$	−408.000	−0.0199438
$749$	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	$0.477183\pi$
$752$	20448.0	0.991572
$753$	0	0
$754$	5220.00	0.252124
$755$	19968.0	0.962530
$756$	0	0
$757$	−754.000	−0.0362016	−0.0181008	−	0.999836i	$-0.505762\pi$
$757$	−754.000	−0.0362016	−0.0181008	+	0.999836i	$0.505762\pi$
$758$	960.000	0.0460010
$759$	0	0
$760$	14616.0	0.697603
$761$	41574.0	1.98036	0.990182	−	0.139787i	$-0.0446419\pi$
$761$	41574.0	1.98036	0.990182	+	0.139787i	$0.0446419\pi$
$762$	0	0
$763$	33208.0	1.57564
$764$	2640.00	0.125016
$765$	0	0
$766$	32616.0	1.53846
$767$	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	$0.615337\pi$
$770$	12096.0	0.566116
$771$	0	0
$772$	2882.00	0.134359
$773$	−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
$776$	8022.00	0.371099
$777$	0	0
$778$	−4122.00	−0.189950
$779$	39672.0	1.82464
$780$	0	0
$781$	16992.0	0.778517
$782$	−3060.00	−0.139930
$783$	0	0
$784$	−31311.0	−1.42634
$785$	14748.0	0.670546
$786$	0	0
$787$	5240.00	0.237339	0.118670	−	0.992934i	$-0.462137\pi$
$787$	5240.00	0.237339	0.118670	+	0.992934i	$0.462137\pi$
$788$	42.0000	0.00189872
$789$	0	0
$790$	8712.00	0.392353
$791$	−10248.0	−0.460654
$792$	0	0
$793$	−6380.00	−0.285700
$794$	−22566.0	−1.00861
$795$	0	0
$796$	−3220.00	−0.143379
$797$	−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
$800$	4005.00	0.176998
$801$	0	0
$802$	−8118.00	−0.357427
$803$	8688.00	0.381809
$804$	0	0
$805$	10080.0	0.441333
$806$	29928.0	1.30790
$807$	0	0
$808$	−4410.00	−0.192009
$809$	438.000	0.0190349	0.00951747	−	0.999955i	$-0.496970\pi$
$809$	438.000	0.0190349	0.00951747	+	0.999955i	$0.496970\pi$
$810$	0	0
$811$	−30448.0	−1.31834	−0.659170	−	0.751994i	$-0.729092\pi$
$811$	−30448.0	−1.31834	−0.659170	+	0.751994i	$0.729092\pi$
$812$	840.000	0.0363032
$813$	0	0
$814$	−4176.00	−0.179814
$815$	−1632.00	−0.0701429
$816$	0	0
$817$	−17168.0	−0.735168
$818$	798.000	0.0341093
$819$	0	0
$820$	−2052.00	−0.0873890
$821$	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	$0.697346\pi$
$824$	4872.00	0.205976
$825$	0	0
$826$	21168.0	0.891681
$827$	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	$-0.527305\pi$
$829$	−4090.00	−0.171353	−0.0856765	+	0.996323i	$0.527305\pi$
$830$	13608.0	0.569085
$831$	0	0
$832$	−25114.0	−1.04648
$833$	−7497.00	−0.311832
$834$	0	0
$835$	21096.0	0.874320
$836$	2784.00	0.115175
$837$	0	0
$838$	−8064.00	−0.332418
$839$	2508.00	0.103201	0.0516006	−	0.998668i	$-0.483568\pi$
$839$	2508.00	0.103201	0.0516006	+	0.998668i	$0.483568\pi$
$840$	0	0
$841$	−23489.0	−0.963098
$842$	−41430.0	−1.69569
$843$	0	0
$844$	−2080.00	−0.0848300
$845$	−7002.00	−0.285061
$846$	0	0
$847$	21140.0	0.857590
$848$	22578.0	0.914306
$849$	0	0
$850$	4539.00	0.183160
$851$	−3480.00	−0.140180
$852$	0	0
$853$	−42442.0	−1.70362	−0.851809	−	0.523852i	$-0.824494\pi$
$853$	−42442.0	−1.70362	−0.851809	+	0.523852i	$0.824494\pi$
$854$	−9240.00	−0.370242
$855$	0	0
$856$	9072.00	0.362237
$857$	−32730.0	−1.30459	−0.652296	−	0.757964i	$-0.726194\pi$
$857$	−32730.0	−1.30459	−0.652296	+	0.757964i	$0.726194\pi$
$858$	0	0
$859$	−6148.00	−0.244199	−0.122100	−	0.992518i	$-0.538963\pi$
$859$	−6148.00	−0.244199	−0.122100	+	0.992518i	$0.538963\pi$
$860$	888.000	0.0352099
$861$	0	0
$862$	−9108.00	−0.359884
$863$	22512.0	0.887969	0.443985	−	0.896034i	$-0.353564\pi$
$863$	22512.0	0.887969	0.443985	+	0.896034i	$0.353564\pi$
$864$	0	0
$865$	−11052.0	−0.434427
$866$	−34266.0	−1.34458
$867$	0	0
$868$	4816.00	0.188325
$869$	−11616.0	−0.453447
$870$	0	0
$871$	28072.0	1.09206
$872$	24906.0	0.967229
$873$	0	0
$874$	20880.0	0.808097
$875$	−35952.0	−1.38903
$876$	0	0
$877$	9182.00	0.353539	0.176770	−	0.984252i	$-0.443435\pi$
$877$	9182.00	0.353539	0.176770	+	0.984252i	$0.443435\pi$
$878$	−156.000	−0.00599629
$879$	0	0
$880$	10224.0	0.391649
$881$	−28530.0	−1.09103	−0.545517	−	0.838100i	$-0.683666\pi$
$881$	−28530.0	−1.09103	−0.545517	+	0.838100i	$0.683666\pi$
$882$	0	0
$883$	−12436.0	−0.473958	−0.236979	−	0.971515i	$-0.576157\pi$
$883$	−12436.0	−0.473958	−0.236979	+	0.971515i	$0.576157\pi$
$884$	986.000	0.0375144
$885$	0	0
$886$	−9324.00	−0.353551
$887$	−7404.00	−0.280273	−0.140136	−	0.990132i	$-0.544754\pi$
$887$	−7404.00	−0.280273	−0.140136	+	0.990132i	$0.544754\pi$
$888$	0	0
$889$	13216.0	0.498594
$890$	−13932.0	−0.524721
$891$	0	0
$892$	4664.00	0.175070
$893$	−33408.0	−1.25191
$894$	0	0
$895$	−21096.0	−0.787890
$896$	−46452.0	−1.73198
$897$	0	0
$898$	−18342.0	−0.681604
$899$	5160.00	0.191430
$900$	0	0
$901$	5406.00	0.199889
$902$	24624.0	0.908968
$903$	0	0
$904$	−7686.00	−0.282779
$905$	−20388.0	−0.748862
$906$	0	0
$907$	15368.0	0.562609	0.281304	−	0.959619i	$-0.409233\pi$
$907$	15368.0	0.562609	0.281304	+	0.959619i	$0.409233\pi$
$908$	1440.00	0.0526300
$909$	0	0
$910$	−29232.0	−1.06487
$911$	−27276.0	−0.991980	−0.495990	−	0.868328i	$-0.665195\pi$
$911$	−27276.0	−0.991980	−0.495990	+	0.868328i	$0.665195\pi$
$912$	0	0
$913$	−18144.0	−0.657699
$914$	12318.0	0.445780
$915$	0	0
$916$	−1186.00	−0.0427801
$917$	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	$0.813814\pi$
$920$	7560.00	0.270919
$921$	0	0
$922$	−10098.0	−0.360694
$923$	−41064.0	−1.46440
$924$	0	0
$925$	5162.00	0.183487
$926$	2688.00	0.0953922
$927$	0	0
$928$	1350.00	0.0477542
$929$	−13026.0	−0.460031	−0.230016	−	0.973187i	$-0.573878\pi$
$929$	−13026.0	−0.460031	−0.230016	+	0.973187i	$0.573878\pi$
$930$	0	0
$931$	51156.0	1.80083
$932$	5334.00	0.187469
$933$	0	0
$934$	30708.0	1.07580
$935$	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	$0.348208\pi$
$938$	40656.0	1.41521
$939$	0	0
$940$	1728.00	0.0599587
$941$	−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
$944$	17892.0	0.616880
$945$	0	0
$946$	−10656.0	−0.366233
$947$	−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
$950$	−30972.0	−1.05775
$951$	0	0
$952$	−9996.00	−0.340307
$953$	−32922.0	−1.11904	−0.559522	−	0.828816i	$-0.689015\pi$
$953$	−32922.0	−1.11904	−0.559522	+	0.828816i	$0.689015\pi$
$954$	0	0
$955$	−15840.0	−0.536723
$956$	−5328.00	−0.180251
$957$	0	0
$958$	−15516.0	−0.523277
$959$	30744.0	1.03522
$960$	0	0
$961$	−207.000	−0.00694841
$962$	10092.0	0.338232
$963$	0	0
$964$	5618.00	0.187701
$965$	−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	$0.506182\pi$
$968$	15855.0	0.526445
$969$	0	0
$970$	6876.00	0.227603
$971$	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
$974$	−45156.0	−1.48551
$975$	0	0
$976$	−7810.00	−0.256139
$977$	28494.0	0.933064	0.466532	−	0.884504i	$-0.345503\pi$
$977$	28494.0	0.933064	0.466532	+	0.884504i	$0.345503\pi$
$978$	0	0
$979$	18576.0	0.606426
$980$	−2646.00	−0.0862483
$981$	0	0
$982$	−26100.0	−0.848151
$983$	42708.0	1.38573	0.692866	−	0.721067i	$-0.256348\pi$
$983$	42708.0	1.38573	0.692866	+	0.721067i	$0.256348\pi$
$984$	0	0
$985$	−252.000	−0.00815166
$986$	1530.00	0.0494170
$987$	0	0
$988$	−6728.00	−0.216646
$989$	−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	$0.656758\pi$
$992$	7740.00	0.247727
$993$	0	0
$994$	−59472.0	−1.89772
$995$	19320.0	0.615563
$996$	0	0
$997$	−9322.00	−0.296119	−0.148060	−	0.988978i	$-0.547303\pi$
$997$	−9322.00	−0.296119	−0.148060	+	0.988978i	$0.547303\pi$
$998$	−3504.00	−0.111139
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.4.a.d.1.1		1
3.2	odd	2		17.4.a.a.1.1	✓	1
4.3	odd	2		2448.4.a.f.1.1		1
12.11	even	2		272.4.a.d.1.1		1
15.2	even	4		425.4.b.c.324.1		2
15.8	even	4		425.4.b.c.324.2		2
15.14	odd	2		425.4.a.d.1.1		1
21.20	even	2		833.4.a.a.1.1		1
24.5	odd	2		1088.4.a.l.1.1		1
24.11	even	2		1088.4.a.a.1.1		1
33.32	even	2		2057.4.a.d.1.1		1
51.38	odd	4		289.4.b.a.288.2		2
51.47	odd	4		289.4.b.a.288.1		2
51.50	odd	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	3.2	odd	2
153.4.a.d.1.1		1	1.1	even	1	trivial
272.4.a.d.1.1		1	12.11	even	2
289.4.a.a.1.1		1	51.50	odd	2
289.4.b.a.288.1		2	51.47	odd	4
289.4.b.a.288.2		2	51.38	odd	4
425.4.a.d.1.1		1	15.14	odd	2
425.4.b.c.324.1		2	15.2	even	4
425.4.b.c.324.2		2	15.8	even	4
833.4.a.a.1.1		1	21.20	even	2
1088.4.a.a.1.1		1	24.11	even	2
1088.4.a.l.1.1		1	24.5	odd	2
2057.4.a.d.1.1		1	33.32	even	2
2448.4.a.f.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 153 = 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	153.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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