LMFDB - Embedded newform 1323.4.a.bm.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,4,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, 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("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1323.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:	$78.0595269376$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 54x^{6} + 887x^{4} - 4176x^{2} + 3136 $ x^8 - 54x^6 + 887x^4 - 4176*x^2 + 3136
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 7 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.49657$ of defining polynomial
Character	$\chi$	$=$	1323.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.49657 q^{2} -1.76715 q^{4} +14.5857 q^{5} +24.3844 q^{8} +O(q^{10})$	$q-2.49657 q^{2} -1.76715 q^{4} +14.5857 q^{5} +24.3844 q^{8} -36.4142 q^{10} -0.0794330 q^{11} +86.9394 q^{13} -46.7400 q^{16} +93.0605 q^{17} -147.953 q^{19} -25.7751 q^{20} +0.198310 q^{22} +154.273 q^{23} +87.7423 q^{25} -217.050 q^{26} +205.403 q^{29} +271.281 q^{31} -78.3853 q^{32} -232.332 q^{34} +48.3171 q^{37} +369.375 q^{38} +355.663 q^{40} +52.3857 q^{41} +48.0508 q^{43} +0.140370 q^{44} -385.152 q^{46} -266.115 q^{47} -219.055 q^{50} -153.635 q^{52} +11.8224 q^{53} -1.15859 q^{55} -512.802 q^{58} -542.677 q^{59} +84.4131 q^{61} -677.270 q^{62} +569.614 q^{64} +1268.07 q^{65} -51.0586 q^{67} -164.452 q^{68} +495.085 q^{71} +71.1990 q^{73} -120.627 q^{74} +261.455 q^{76} +378.650 q^{79} -681.735 q^{80} -130.785 q^{82} -486.792 q^{83} +1357.35 q^{85} -119.962 q^{86} -1.93692 q^{88} -1168.36 q^{89} -272.623 q^{92} +664.374 q^{94} -2158.00 q^{95} +1310.92 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 44 q^{4}+O(q^{10}) $ 8 * q + 44 * q^4	$ 8 q + 44 q^{4} + 132 q^{10} + 336 q^{13} + 204 q^{16} - 288 q^{19} + 484 q^{22} + 152 q^{25} + 120 q^{31} + 1008 q^{34} + 592 q^{37} + 1620 q^{40} - 1872 q^{43} - 1644 q^{46} + 2400 q^{52} + 1344 q^{55} - 1200 q^{58} + 2400 q^{61} - 1388 q^{64} + 1824 q^{73} + 2844 q^{76} + 2368 q^{79} + 2436 q^{82} + 3512 q^{85} + 3780 q^{88} + 4368 q^{94} + 5712 q^{97}+O(q^{100}) $ 8 * q + 44 * q^4 + 132 * q^10 + 336 * q^13 + 204 * q^16 - 288 * q^19 + 484 * q^22 + 152 * q^25 + 120 * q^31 + 1008 * q^34 + 592 * q^37 + 1620 * q^40 - 1872 * q^43 - 1644 * q^46 + 2400 * q^52 + 1344 * q^55 - 1200 * q^58 + 2400 * q^61 - 1388 * q^64 + 1824 * q^73 + 2844 * q^76 + 2368 * q^79 + 2436 * q^82 + 3512 * q^85 + 3780 * q^88 + 4368 * q^94 + 5712 * q^97

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 $

$2$	−2.49657	−0.882670	−0.441335	−	0.897342i	$-0.645495\pi$
$2$	−2.49657	−0.882670	−0.441335	+	0.897342i	$0.645495\pi$
$3$	0	0
$3$	0	0
$4$	−1.76715	−0.220894
$5$	14.5857	1.30458	0.652292	−	0.757968i	$-0.273808\pi$
$5$	14.5857	1.30458	0.652292	+	0.757968i	$0.273808\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	24.3844	1.07765
$9$	0	0
$10$	−36.4142	−1.15152
$11$	−0.0794330	−0.00217727	−0.00108863	−	0.999999i	$-0.500347\pi$
$11$	−0.0794330	−0.00217727	−0.00108863	+	0.999999i	$0.500347\pi$
$12$	0	0
$13$	86.9394	1.85482	0.927409	−	0.374048i	$-0.122030\pi$
$13$	86.9394	1.85482	0.927409	+	0.374048i	$0.122030\pi$
$14$	0	0
$15$	0	0
$16$	−46.7400	−0.730312
$17$	93.0605	1.32768	0.663838	−	0.747877i	$-0.268926\pi$
$17$	93.0605	1.32768	0.663838	+	0.747877i	$0.268926\pi$
$18$	0	0
$19$	−147.953	−1.78646	−0.893230	−	0.449599i	$-0.851567\pi$
$19$	−147.953	−1.78646	−0.893230	+	0.449599i	$0.851567\pi$
$20$	−25.7751	−0.288175
$21$	0	0
$22$	0.198310	0.00192181
$23$	154.273	1.39861	0.699306	−	0.714823i	$-0.253493\pi$
$23$	154.273	1.39861	0.699306	+	0.714823i	$0.253493\pi$
$24$	0	0
$25$	87.7423	0.701938
$26$	−217.050	−1.63719
$27$	0	0
$28$	0	0
$29$	205.403	1.31525	0.657626	−	0.753344i	$-0.271561\pi$
$29$	205.403	1.31525	0.657626	+	0.753344i	$0.271561\pi$
$30$	0	0
$31$	271.281	1.57172	0.785862	−	0.618402i	$-0.212220\pi$
$31$	271.281	1.57172	0.785862	+	0.618402i	$0.212220\pi$
$32$	−78.3853	−0.433022
$33$	0	0
$34$	−232.332	−1.17190
$35$	0	0
$36$	0	0
$37$	48.3171	0.214683	0.107342	−	0.994222i	$-0.465766\pi$
$37$	48.3171	0.214683	0.107342	+	0.994222i	$0.465766\pi$
$38$	369.375	1.57685
$39$	0	0
$40$	355.663	1.40588
$41$	52.3857	0.199543	0.0997717	−	0.995010i	$-0.468189\pi$
$41$	52.3857	0.199543	0.0997717	+	0.995010i	$0.468189\pi$
$42$	0	0
$43$	48.0508	0.170411	0.0852056	−	0.996363i	$-0.472845\pi$
$43$	48.0508	0.170411	0.0852056	+	0.996363i	$0.472845\pi$
$44$	0.140370	0.000480945 0
$45$	0	0
$46$	−385.152	−1.23451
$47$	−266.115	−0.825891	−0.412945	−	0.910756i	$-0.635500\pi$
$47$	−266.115	−0.825891	−0.412945	+	0.910756i	$0.635500\pi$
$48$	0	0
$49$	0	0
$50$	−219.055	−0.619580
$51$	0	0
$52$	−153.635	−0.409718
$53$	11.8224	0.0306403	0.0153202	−	0.999883i	$-0.495123\pi$
$53$	11.8224	0.0306403	0.0153202	+	0.999883i	$0.495123\pi$
$54$	0	0
$55$	−1.15859	−0.00284043
$56$	0	0
$57$	0	0
$58$	−512.802	−1.16093
$59$	−542.677	−1.19747	−0.598733	−	0.800948i	$-0.704329\pi$
$59$	−542.677	−1.19747	−0.598733	+	0.800948i	$0.704329\pi$
$60$	0	0
$61$	84.4131	0.177180	0.0885901	−	0.996068i	$-0.471764\pi$
$61$	84.4131	0.177180	0.0885901	+	0.996068i	$0.471764\pi$
$62$	−677.270	−1.38731
$63$	0	0
$64$	569.614	1.11253
$65$	1268.07	2.41977
$66$	0	0
$67$	−51.0586	−0.0931015	−0.0465507	−	0.998916i	$-0.514823\pi$
$67$	−51.0586	−0.0931015	−0.0465507	+	0.998916i	$0.514823\pi$
$68$	−164.452	−0.293275
$69$	0	0
$70$	0	0
$71$	495.085	0.827547	0.413773	−	0.910380i	$-0.364211\pi$
$71$	495.085	0.827547	0.413773	+	0.910380i	$0.364211\pi$
$72$	0	0
$73$	71.1990	0.114154	0.0570768	−	0.998370i	$-0.481822\pi$
$73$	71.1990	0.114154	0.0570768	+	0.998370i	$0.481822\pi$
$74$	−120.627	−0.189494
$75$	0	0
$76$	261.455	0.394618
$77$	0	0
$78$	0	0
$79$	378.650	0.539259	0.269629	−	0.962964i	$-0.413099\pi$
$79$	378.650	0.539259	0.269629	+	0.962964i	$0.413099\pi$
$80$	−681.735	−0.952753
$81$	0	0
$82$	−130.785	−0.176131
$83$	−486.792	−0.643763	−0.321882	−	0.946780i	$-0.604315\pi$
$83$	−486.792	−0.643763	−0.321882	+	0.946780i	$0.604315\pi$
$84$	0	0
$85$	1357.35	1.73206
$86$	−119.962	−0.150417
$87$	0	0
$88$	−1.93692	−0.00234633
$89$	−1168.36	−1.39152	−0.695762	−	0.718273i	$-0.744933\pi$
$89$	−1168.36	−1.39152	−0.695762	+	0.718273i	$0.744933\pi$
$90$	0	0
$91$	0	0
$92$	−272.623	−0.308945
$93$	0	0
$94$	664.374	0.728989
$95$	−2158.00	−2.33059
$96$	0	0
$97$	1310.92	1.37221	0.686104	−	0.727504i	$-0.259320\pi$
$97$	1310.92	1.37221	0.686104	+	0.727504i	$0.259320\pi$
$98$	0	0
$99$	0	0
$100$	−155.054	−0.155054
$101$	−1699.25	−1.67408	−0.837040	−	0.547142i	$-0.815716\pi$
$101$	−1699.25	−1.67408	−0.837040	+	0.547142i	$0.815716\pi$
$102$	0	0
$103$	−615.292	−0.588607	−0.294304	−	0.955712i	$-0.595088\pi$
$103$	−615.292	−0.588607	−0.294304	+	0.955712i	$0.595088\pi$
$104$	2119.96	1.99884
$105$	0	0
$106$	−29.5155	−0.0270453
$107$	−1692.95	−1.52957	−0.764783	−	0.644288i	$-0.777154\pi$
$107$	−1692.95	−1.52957	−0.764783	+	0.644288i	$0.777154\pi$
$108$	0	0
$109$	872.784	0.766950	0.383475	−	0.923551i	$-0.374727\pi$
$109$	872.784	0.766950	0.383475	+	0.923551i	$0.374727\pi$
$110$	2.89249	0.00250716
$111$	0	0
$112$	0	0
$113$	−139.808	−0.116389	−0.0581947	−	0.998305i	$-0.518534\pi$
$113$	−139.808	−0.116389	−0.0581947	+	0.998305i	$0.518534\pi$
$114$	0	0
$115$	2250.17	1.82461
$116$	−362.978	−0.290531
$117$	0	0
$118$	1354.83	1.05697
$119$	0	0
$120$	0	0
$121$	−1330.99	−0.999995
$122$	−210.743	−0.156392
$123$	0	0
$124$	−479.394	−0.347184
$125$	−543.429	−0.388846
$126$	0	0
$127$	222.475	0.155444	0.0777222	−	0.996975i	$-0.475235\pi$
$127$	222.475	0.155444	0.0777222	+	0.996975i	$0.475235\pi$
$128$	−794.997	−0.548972
$129$	0	0
$130$	−3165.82	−2.13585
$131$	1774.60	1.18357	0.591785	−	0.806096i	$-0.298423\pi$
$131$	1774.60	1.18357	0.591785	+	0.806096i	$0.298423\pi$
$132$	0	0
$133$	0	0
$134$	127.471	0.0821779
$135$	0	0
$136$	2269.22	1.43076
$137$	−1570.56	−0.979433	−0.489717	−	0.871882i	$-0.662900\pi$
$137$	−1570.56	−0.979433	−0.489717	+	0.871882i	$0.662900\pi$
$138$	0	0
$139$	−108.881	−0.0664400	−0.0332200	−	0.999448i	$-0.510576\pi$
$139$	−108.881	−0.0664400	−0.0332200	+	0.999448i	$0.510576\pi$
$140$	0	0
$141$	0	0
$142$	−1236.01	−0.730451
$143$	−6.90586	−0.00403844
$144$	0	0
$145$	2995.94	1.71586
$146$	−177.753	−0.100760
$147$	0	0
$148$	−85.3836	−0.0474222
$149$	2975.29	1.63588	0.817938	−	0.575307i	$-0.195117\pi$
$149$	2975.29	1.63588	0.817938	+	0.575307i	$0.195117\pi$
$150$	0	0
$151$	−3220.19	−1.73546	−0.867732	−	0.497032i	$-0.834423\pi$
$151$	−3220.19	−1.73546	−0.867732	+	0.497032i	$0.834423\pi$
$152$	−3607.74	−1.92517
$153$	0	0
$154$	0	0
$155$	3956.81	2.05044
$156$	0	0
$157$	−1640.62	−0.833988	−0.416994	−	0.908909i	$-0.636916\pi$
$157$	−1640.62	−0.833988	−0.416994	+	0.908909i	$0.636916\pi$
$158$	−945.325	−0.475987
$159$	0	0
$160$	−1143.30	−0.564913
$161$	0	0
$162$	0	0
$163$	−2135.64	−1.02624	−0.513118	−	0.858318i	$-0.671510\pi$
$163$	−2135.64	−1.02624	−0.513118	+	0.858318i	$0.671510\pi$
$164$	−92.5735	−0.0440779
$165$	0	0
$166$	1215.31	0.568231
$167$	162.606	0.0753464	0.0376732	−	0.999290i	$-0.488005\pi$
$167$	162.606	0.0753464	0.0376732	+	0.999290i	$0.488005\pi$
$168$	0	0
$169$	5361.46	2.44035
$170$	−3388.72	−1.52884
$171$	0	0
$172$	−84.9131	−0.0376428
$173$	4239.10	1.86296	0.931481	−	0.363789i	$-0.118517\pi$
$173$	4239.10	1.86296	0.931481	+	0.363789i	$0.118517\pi$
$174$	0	0
$175$	0	0
$176$	3.71270	0.00159009
$177$	0	0
$178$	2916.88	1.22826
$179$	−799.480	−0.333832	−0.166916	−	0.985971i	$-0.553381\pi$
$179$	−799.480	−0.333832	−0.166916	+	0.985971i	$0.553381\pi$
$180$	0	0
$181$	1993.81	0.818776	0.409388	−	0.912360i	$-0.365742\pi$
$181$	1993.81	0.818776	0.409388	+	0.912360i	$0.365742\pi$
$182$	0	0
$183$	0	0
$184$	3761.84	1.50721
$185$	704.738	0.280072
$186$	0	0
$187$	−7.39208	−0.00289071
$188$	470.266	0.182434
$189$	0	0
$190$	5387.58	2.05714
$191$	3120.39	1.18211	0.591057	−	0.806630i	$-0.298711\pi$
$191$	3120.39	1.18211	0.591057	+	0.806630i	$0.298711\pi$
$192$	0	0
$193$	−53.5595	−0.0199756	−0.00998781	−	0.999950i	$-0.503179\pi$
$193$	−53.5595	−0.0199756	−0.00998781	+	0.999950i	$0.503179\pi$
$194$	−3272.81	−1.21121
$195$	0	0
$196$	0	0
$197$	−2174.32	−0.786366	−0.393183	−	0.919460i	$-0.628626\pi$
$197$	−2174.32	−0.786366	−0.393183	+	0.919460i	$0.628626\pi$
$198$	0	0
$199$	3765.55	1.34137	0.670686	−	0.741741i	$-0.266000\pi$
$199$	3765.55	1.34137	0.670686	+	0.741741i	$0.266000\pi$
$200$	2139.54	0.756441
$201$	0	0
$202$	4242.30	1.47766
$203$	0	0
$204$	0	0
$205$	764.082	0.260321
$206$	1536.12	0.519546
$207$	0	0
$208$	−4063.54	−1.35460
$209$	11.7524	0.00388961
$210$	0	0
$211$	−705.039	−0.230032	−0.115016	−	0.993364i	$-0.536692\pi$
$211$	−705.039	−0.230032	−0.115016	+	0.993364i	$0.536692\pi$
$212$	−20.8921	−0.00676826
$213$	0	0
$214$	4226.56	1.35010
$215$	700.854	0.222316
$216$	0	0
$217$	0	0
$218$	−2178.96	−0.676964
$219$	0	0
$220$	2.04740	0.000627434 0
$221$	8090.62	2.46260
$222$	0	0
$223$	2724.83	0.818243	0.409121	−	0.912480i	$-0.365835\pi$
$223$	2724.83	0.818243	0.409121	+	0.912480i	$0.365835\pi$
$224$	0	0
$225$	0	0
$226$	349.039	0.102733
$227$	617.477	0.180544	0.0902718	−	0.995917i	$-0.471226\pi$
$227$	617.477	0.180544	0.0902718	+	0.995917i	$0.471226\pi$
$228$	0	0
$229$	4558.34	1.31539	0.657694	−	0.753285i	$-0.271532\pi$
$229$	4558.34	1.31539	0.657694	+	0.753285i	$0.271532\pi$
$230$	−5617.71	−1.61052
$231$	0	0
$232$	5008.61	1.41738
$233$	2825.59	0.794465	0.397232	−	0.917718i	$-0.369971\pi$
$233$	2825.59	0.794465	0.397232	+	0.917718i	$0.369971\pi$
$234$	0	0
$235$	−3881.47	−1.07744
$236$	958.992	0.264513
$237$	0	0
$238$	0	0
$239$	−5809.06	−1.57220	−0.786102	−	0.618097i	$-0.787904\pi$
$239$	−5809.06	−1.57220	−0.786102	+	0.618097i	$0.787904\pi$
$240$	0	0
$241$	4726.10	1.26321	0.631607	−	0.775289i	$-0.282396\pi$
$241$	4726.10	1.26321	0.631607	+	0.775289i	$0.282396\pi$
$242$	3322.92	0.882666
$243$	0	0
$244$	−149.171	−0.0391380
$245$	0	0
$246$	0	0
$247$	−12862.9	−3.31356
$248$	6615.00	1.69376
$249$	0	0
$250$	1356.71	0.343223
$251$	3020.59	0.759595	0.379797	−	0.925070i	$-0.375994\pi$
$251$	3020.59	0.759595	0.379797	+	0.925070i	$0.375994\pi$
$252$	0	0
$253$	−12.2543	−0.00304515
$254$	−555.423	−0.137206
$255$	0	0
$256$	−2572.15	−0.627966
$257$	−4268.71	−1.03609	−0.518044	−	0.855354i	$-0.673340\pi$
$257$	−4268.71	−1.03609	−0.518044	+	0.855354i	$0.673340\pi$
$258$	0	0
$259$	0	0
$260$	−2240.87	−0.534512
$261$	0	0
$262$	−4430.41	−1.04470
$263$	997.520	0.233877	0.116939	−	0.993139i	$-0.462692\pi$
$263$	997.520	0.233877	0.116939	+	0.993139i	$0.462692\pi$
$264$	0	0
$265$	172.439	0.0399729
$266$	0	0
$267$	0	0
$268$	90.2282	0.0205655
$269$	4965.74	1.12553	0.562763	−	0.826618i	$-0.309738\pi$
$269$	4965.74	1.12553	0.562763	+	0.826618i	$0.309738\pi$
$270$	0	0
$271$	−2393.95	−0.536612	−0.268306	−	0.963334i	$-0.586464\pi$
$271$	−2393.95	−0.536612	−0.268306	+	0.963334i	$0.586464\pi$
$272$	−4349.64	−0.969617
$273$	0	0
$274$	3921.02	0.864516
$275$	−6.96964	−0.00152831
$276$	0	0
$277$	−1679.74	−0.364353	−0.182177	−	0.983266i	$-0.558314\pi$
$277$	−1679.74	−0.364353	−0.182177	+	0.983266i	$0.558314\pi$
$278$	271.829	0.0586446
$279$	0	0
$280$	0	0
$281$	7139.07	1.51559	0.757795	−	0.652492i	$-0.226277\pi$
$281$	7139.07	1.51559	0.757795	+	0.652492i	$0.226277\pi$
$282$	0	0
$283$	4403.14	0.924874	0.462437	−	0.886652i	$-0.346975\pi$
$283$	4403.14	0.924874	0.462437	+	0.886652i	$0.346975\pi$
$284$	−874.891	−0.182800
$285$	0	0
$286$	17.2409	0.00356461
$287$	0	0
$288$	0	0
$289$	3747.26	0.762723
$290$	−7479.57	−1.51454
$291$	0	0
$292$	−125.819	−0.0252158
$293$	3403.11	0.678539	0.339270	−	0.940689i	$-0.389820\pi$
$293$	3403.11	0.678539	0.339270	+	0.940689i	$0.389820\pi$
$294$	0	0
$295$	−7915.32	−1.56220
$296$	1178.18	0.231352
$297$	0	0
$298$	−7428.02	−1.44394
$299$	13412.4	2.59417
$300$	0	0
$301$	0	0
$302$	8039.41	1.53184
$303$	0	0
$304$	6915.32	1.30467
$305$	1231.22	0.231146
$306$	0	0
$307$	−641.152	−0.119194	−0.0595969	−	0.998223i	$-0.518982\pi$
$307$	−641.152	−0.119194	−0.0595969	+	0.998223i	$0.518982\pi$
$308$	0	0
$309$	0	0
$310$	−9878.45	−1.80987
$311$	−618.270	−0.112730	−0.0563648	−	0.998410i	$-0.517951\pi$
$311$	−618.270	−0.112730	−0.0563648	+	0.998410i	$0.517951\pi$
$312$	0	0
$313$	−4574.93	−0.826167	−0.413083	−	0.910693i	$-0.635548\pi$
$313$	−4574.93	−0.826167	−0.413083	+	0.910693i	$0.635548\pi$
$314$	4095.93	0.736136
$315$	0	0
$316$	−669.131	−0.119119
$317$	−4064.03	−0.720059	−0.360029	−	0.932941i	$-0.617233\pi$
$317$	−4064.03	−0.720059	−0.360029	+	0.932941i	$0.617233\pi$
$318$	0	0
$319$	−16.3158	−0.00286366
$320$	8308.21	1.45138
$321$	0	0
$322$	0	0
$323$	−13768.6	−2.37184
$324$	0	0
$325$	7628.26	1.30197
$326$	5331.78	0.905828
$327$	0	0
$328$	1277.39	0.215037
$329$	0	0
$330$	0	0
$331$	10290.0	1.70873	0.854363	−	0.519677i	$-0.173948\pi$
$331$	10290.0	1.70873	0.854363	+	0.519677i	$0.173948\pi$
$332$	860.235	0.142203
$333$	0	0
$334$	−405.957	−0.0665060
$335$	−744.724	−0.121459
$336$	0	0
$337$	6307.22	1.01951	0.509757	−	0.860318i	$-0.329735\pi$
$337$	6307.22	1.01951	0.509757	+	0.860318i	$0.329735\pi$
$338$	−13385.2	−2.15403
$339$	0	0
$340$	−2398.65	−0.382602
$341$	−21.5486	−0.00342206
$342$	0	0
$343$	0	0
$344$	1171.69	0.183643
$345$	0	0
$346$	−10583.2	−1.64438
$347$	165.837	0.0256559	0.0128280	−	0.999918i	$-0.495917\pi$
$347$	165.837	0.0256559	0.0128280	+	0.999918i	$0.495917\pi$
$348$	0	0
$349$	−8456.39	−1.29702	−0.648510	−	0.761206i	$-0.724608\pi$
$349$	−8456.39	−1.29702	−0.648510	+	0.761206i	$0.724608\pi$
$350$	0	0
$351$	0	0
$352$	6.22638	0.000942805 0
$353$	5211.87	0.785836	0.392918	−	0.919574i	$-0.371466\pi$
$353$	5211.87	0.785836	0.392918	+	0.919574i	$0.371466\pi$
$354$	0	0
$355$	7221.16	1.07960
$356$	2064.66	0.307379
$357$	0	0
$358$	1995.96	0.294664
$359$	−6313.94	−0.928237	−0.464119	−	0.885773i	$-0.653629\pi$
$359$	−6313.94	−0.928237	−0.464119	+	0.885773i	$0.653629\pi$
$360$	0	0
$361$	15031.1	2.19144
$362$	−4977.67	−0.722709
$363$	0	0
$364$	0	0
$365$	1038.49	0.148923
$366$	0	0
$367$	−1375.97	−0.195708	−0.0978542	−	0.995201i	$-0.531198\pi$
$367$	−1375.97	−0.195708	−0.0978542	+	0.995201i	$0.531198\pi$
$368$	−7210.70	−1.02142
$369$	0	0
$370$	−1759.42	−0.247211
$371$	0	0
$372$	0	0
$373$	3951.97	0.548593	0.274296	−	0.961645i	$-0.411555\pi$
$373$	3951.97	0.548593	0.274296	+	0.961645i	$0.411555\pi$
$374$	18.4548	0.00255154
$375$	0	0
$376$	−6489.04	−0.890018
$377$	17857.6	2.43956
$378$	0	0
$379$	11050.5	1.49769	0.748845	−	0.662745i	$-0.230609\pi$
$379$	11050.5	1.49769	0.748845	+	0.662745i	$0.230609\pi$
$380$	3813.51	0.514812
$381$	0	0
$382$	−7790.27	−1.04342
$383$	803.663	0.107220	0.0536100	−	0.998562i	$-0.482927\pi$
$383$	803.663	0.107220	0.0536100	+	0.998562i	$0.482927\pi$
$384$	0	0
$385$	0	0
$386$	133.715	0.0176319
$387$	0	0
$388$	−2316.60	−0.303112
$389$	3215.21	0.419068	0.209534	−	0.977801i	$-0.432805\pi$
$389$	3215.21	0.419068	0.209534	+	0.977801i	$0.432805\pi$
$390$	0	0
$391$	14356.7	1.85690
$392$	0	0
$393$	0	0
$394$	5428.35	0.694102
$395$	5522.87	0.703508
$396$	0	0
$397$	12381.6	1.56528	0.782639	−	0.622476i	$-0.213873\pi$
$397$	12381.6	1.56528	0.782639	+	0.622476i	$0.213873\pi$
$398$	−9400.96	−1.18399
$399$	0	0
$400$	−4101.07	−0.512634
$401$	−10000.8	−1.24543	−0.622715	−	0.782449i	$-0.713970\pi$
$401$	−10000.8	−1.24543	−0.622715	+	0.782449i	$0.713970\pi$
$402$	0	0
$403$	23585.0	2.91526
$404$	3002.84	0.369794
$405$	0	0
$406$	0	0
$407$	−3.83797	−0.000467423 0
$408$	0	0
$409$	−13897.4	−1.68015	−0.840074	−	0.542472i	$-0.817488\pi$
$409$	−13897.4	−1.68015	−0.840074	+	0.542472i	$0.817488\pi$
$410$	−1907.58	−0.229778
$411$	0	0
$412$	1087.31	0.130020
$413$	0	0
$414$	0	0
$415$	−7100.20	−0.839843
$416$	−6814.77	−0.803177
$417$	0	0
$418$	−29.3406	−0.00343324
$419$	−4850.62	−0.565557	−0.282778	−	0.959185i	$-0.591256\pi$
$419$	−4850.62	−0.565557	−0.282778	+	0.959185i	$0.591256\pi$
$420$	0	0
$421$	−498.467	−0.0577050	−0.0288525	−	0.999584i	$-0.509185\pi$
$421$	−498.467	−0.0577050	−0.0288525	+	0.999584i	$0.509185\pi$
$422$	1760.18	0.203043
$423$	0	0
$424$	288.283	0.0330194
$425$	8165.34	0.931946
$426$	0	0
$427$	0	0
$428$	2991.70	0.337872
$429$	0	0
$430$	−1749.73	−0.196231
$431$	−15515.7	−1.73403	−0.867013	−	0.498286i	$-0.833963\pi$
$431$	−15515.7	−1.73403	−0.867013	+	0.498286i	$0.833963\pi$
$432$	0	0
$433$	−2259.28	−0.250748	−0.125374	−	0.992110i	$-0.540013\pi$
$433$	−2259.28	−0.250748	−0.125374	+	0.992110i	$0.540013\pi$
$434$	0	0
$435$	0	0
$436$	−1542.34	−0.169415
$437$	−22825.1	−2.49856
$438$	0	0
$439$	−4732.09	−0.514465	−0.257233	−	0.966349i	$-0.582811\pi$
$439$	−4732.09	−0.514465	−0.257233	+	0.966349i	$0.582811\pi$
$440$	−28.2514	−0.00306098
$441$	0	0
$442$	−20198.8	−2.17366
$443$	12706.8	1.36279	0.681396	−	0.731915i	$-0.261373\pi$
$443$	12706.8	1.36279	0.681396	+	0.731915i	$0.261373\pi$
$444$	0	0
$445$	−17041.3	−1.81536
$446$	−6802.72	−0.722238
$447$	0	0
$448$	0	0
$449$	−9176.08	−0.964468	−0.482234	−	0.876042i	$-0.660174\pi$
$449$	−9176.08	−0.964468	−0.482234	+	0.876042i	$0.660174\pi$
$450$	0	0
$451$	−4.16116	−0.000434460 0
$452$	247.061	0.0257097
$453$	0	0
$454$	−1541.57	−0.159360
$455$	0	0
$456$	0	0
$457$	−3666.07	−0.375255	−0.187627	−	0.982240i	$-0.560080\pi$
$457$	−3666.07	−0.375255	−0.187627	+	0.982240i	$0.560080\pi$
$458$	−11380.2	−1.16105
$459$	0	0
$460$	−3976.40	−0.403044
$461$	6409.12	0.647511	0.323756	−	0.946141i	$-0.395054\pi$
$461$	6409.12	0.647511	0.323756	+	0.946141i	$0.395054\pi$
$462$	0	0
$463$	−13191.9	−1.32414	−0.662071	−	0.749441i	$-0.730322\pi$
$463$	−13191.9	−1.32414	−0.662071	+	0.749441i	$0.730322\pi$
$464$	−9600.52	−0.960545
$465$	0	0
$466$	−7054.27	−0.701250
$467$	−6765.27	−0.670363	−0.335181	−	0.942154i	$-0.608798\pi$
$467$	−6765.27	−0.670363	−0.335181	+	0.942154i	$0.608798\pi$
$468$	0	0
$469$	0	0
$470$	9690.35	0.951027
$471$	0	0
$472$	−13232.8	−1.29045
$473$	−3.81682	−0.000371031 0
$474$	0	0
$475$	−12981.7	−1.25399
$476$	0	0
$477$	0	0
$478$	14502.7	1.38774
$479$	11347.2	1.08239	0.541195	−	0.840897i	$-0.317972\pi$
$479$	11347.2	1.08239	0.541195	+	0.840897i	$0.317972\pi$
$480$	0	0
$481$	4200.66	0.398198
$482$	−11799.0	−1.11500
$483$	0	0
$484$	2352.07	0.220893
$485$	19120.7	1.79016
$486$	0	0
$487$	4958.91	0.461416	0.230708	−	0.973023i	$-0.425896\pi$
$487$	4958.91	0.461416	0.230708	+	0.973023i	$0.425896\pi$
$488$	2058.36	0.190938
$489$	0	0
$490$	0	0
$491$	−18475.0	−1.69810	−0.849048	−	0.528316i	$-0.822823\pi$
$491$	−18475.0	−1.69810	−0.849048	+	0.528316i	$0.822823\pi$
$492$	0	0
$493$	19114.9	1.74623
$494$	32113.2	2.92478
$495$	0	0
$496$	−12679.6	−1.14785
$497$	0	0
$498$	0	0
$499$	−2702.96	−0.242487	−0.121243	−	0.992623i	$-0.538688\pi$
$499$	−2702.96	−0.242487	−0.121243	+	0.992623i	$0.538688\pi$
$500$	960.322	0.0858938
$501$	0	0
$502$	−7541.12	−0.670471
$503$	−15684.2	−1.39031	−0.695154	−	0.718861i	$-0.744664\pi$
$503$	−15684.2	−1.39031	−0.695154	+	0.718861i	$0.744664\pi$
$504$	0	0
$505$	−24784.8	−2.18398
$506$	30.5938	0.00268787
$507$	0	0
$508$	−393.147	−0.0343367
$509$	−10016.1	−0.872214	−0.436107	−	0.899895i	$-0.643643\pi$
$509$	−10016.1	−0.872214	−0.436107	+	0.899895i	$0.643643\pi$
$510$	0	0
$511$	0	0
$512$	12781.5	1.10326
$513$	0	0
$514$	10657.1	0.914523
$515$	−8974.46	−0.767887
$516$	0	0
$517$	21.1383	0.00179819
$518$	0	0
$519$	0	0
$520$	30921.1	2.60765
$521$	9230.68	0.776206	0.388103	−	0.921616i	$-0.373130\pi$
$521$	9230.68	0.776206	0.388103	+	0.921616i	$0.373130\pi$
$522$	0	0
$523$	−7760.93	−0.648875	−0.324438	−	0.945907i	$-0.605175\pi$
$523$	−7760.93	−0.648875	−0.324438	+	0.945907i	$0.605175\pi$
$524$	−3135.99	−0.261443
$525$	0	0
$526$	−2490.38	−0.206436
$527$	25245.5	2.08674
$528$	0	0
$529$	11633.0	0.956115
$530$	−430.504	−0.0352829
$531$	0	0
$532$	0	0
$533$	4554.38	0.370117
$534$	0	0
$535$	−24692.8	−1.99545
$536$	−1245.03	−0.100330
$537$	0	0
$538$	−12397.3	−0.993467
$539$	0	0
$540$	0	0
$541$	−4005.35	−0.318306	−0.159153	−	0.987254i	$-0.550876\pi$
$541$	−4005.35	−0.318306	−0.159153	+	0.987254i	$0.550876\pi$
$542$	5976.65	0.473651
$543$	0	0
$544$	−7294.58	−0.574913
$545$	12730.2	1.00055
$546$	0	0
$547$	−6489.15	−0.507232	−0.253616	−	0.967305i	$-0.581620\pi$
$547$	−6489.15	−0.507232	−0.253616	+	0.967305i	$0.581620\pi$
$548$	2775.42	0.216351
$549$	0	0
$550$	17.4002	0.00134899
$551$	−30390.0	−2.34965
$552$	0	0
$553$	0	0
$554$	4193.59	0.321604
$555$	0	0
$556$	192.409	0.0146762
$557$	−23695.8	−1.80256	−0.901279	−	0.433240i	$-0.857370\pi$
$557$	−23695.8	−1.80256	−0.901279	+	0.433240i	$0.857370\pi$
$558$	0	0
$559$	4177.51	0.316082
$560$	0	0
$561$	0	0
$562$	−17823.2	−1.33777
$563$	13621.8	1.01970	0.509848	−	0.860265i	$-0.329702\pi$
$563$	13621.8	1.01970	0.509848	+	0.860265i	$0.329702\pi$
$564$	0	0
$565$	−2039.19	−0.151840
$566$	−10992.7	−0.816359
$567$	0	0
$568$	12072.3	0.891803
$569$	−4843.75	−0.356873	−0.178436	−	0.983951i	$-0.557104\pi$
$569$	−4843.75	−0.356873	−0.178436	+	0.983951i	$0.557104\pi$
$570$	0	0
$571$	22696.4	1.66342	0.831710	−	0.555210i	$-0.187362\pi$
$571$	22696.4	1.66342	0.831710	+	0.555210i	$0.187362\pi$
$572$	12.2037	0.000892067 0
$573$	0	0
$574$	0	0
$575$	13536.2	0.981739
$576$	0	0
$577$	−20143.2	−1.45333	−0.726665	−	0.686992i	$-0.758931\pi$
$577$	−20143.2	−1.45333	−0.726665	+	0.686992i	$0.758931\pi$
$578$	−9355.28	−0.673232
$579$	0	0
$580$	−5294.28	−0.379022
$581$	0	0
$582$	0	0
$583$	−0.939093	−6.67123e−5 0
$584$	1736.14	0.123017
$585$	0	0
$586$	−8496.10	−0.598926
$587$	1974.15	0.138811	0.0694055	−	0.997589i	$-0.477890\pi$
$587$	1974.15	0.138811	0.0694055	+	0.997589i	$0.477890\pi$
$588$	0	0
$589$	−40136.8	−2.80782
$590$	19761.1	1.37890
$591$	0	0
$592$	−2258.34	−0.156786
$593$	−23558.1	−1.63139	−0.815697	−	0.578479i	$-0.803647\pi$
$593$	−23558.1	−1.63139	−0.815697	+	0.578479i	$0.803647\pi$
$594$	0	0
$595$	0	0
$596$	−5257.79	−0.361355
$597$	0	0
$598$	−33484.9	−2.28980
$599$	−7564.46	−0.515986	−0.257993	−	0.966147i	$-0.583061\pi$
$599$	−7564.46	−0.515986	−0.257993	+	0.966147i	$0.583061\pi$
$600$	0	0
$601$	16361.7	1.11050	0.555248	−	0.831685i	$-0.312623\pi$
$601$	16361.7	1.11050	0.555248	+	0.831685i	$0.312623\pi$
$602$	0	0
$603$	0	0
$604$	5690.56	0.383353
$605$	−19413.5	−1.30458
$606$	0	0
$607$	27722.5	1.85374	0.926870	−	0.375383i	$-0.122489\pi$
$607$	27722.5	1.85374	0.926870	+	0.375383i	$0.122489\pi$
$608$	11597.3	0.773577
$609$	0	0
$610$	−3073.83	−0.204026
$611$	−23135.9	−1.53188
$612$	0	0
$613$	5301.97	0.349338	0.174669	−	0.984627i	$-0.444114\pi$
$613$	5301.97	0.349338	0.174669	+	0.984627i	$0.444114\pi$
$614$	1600.68	0.105209
$615$	0	0
$616$	0	0
$617$	4998.08	0.326118	0.163059	−	0.986616i	$-0.447864\pi$
$617$	4998.08	0.326118	0.163059	+	0.986616i	$0.447864\pi$
$618$	0	0
$619$	4922.19	0.319612	0.159806	−	0.987148i	$-0.448913\pi$
$619$	4922.19	0.319612	0.159806	+	0.987148i	$0.448913\pi$
$620$	−6992.29	−0.452931
$621$	0	0
$622$	1543.55	0.0995030
$623$	0	0
$624$	0	0
$625$	−18894.1	−1.20922
$626$	11421.6	0.729233
$627$	0	0
$628$	2899.23	0.184223
$629$	4496.41	0.285030
$630$	0	0
$631$	14950.4	0.943213	0.471607	−	0.881809i	$-0.343674\pi$
$631$	14950.4	0.943213	0.471607	+	0.881809i	$0.343674\pi$
$632$	9233.13	0.581130
$633$	0	0
$634$	10146.1	0.635574
$635$	3244.95	0.202790
$636$	0	0
$637$	0	0
$638$	40.7334	0.00252767
$639$	0	0
$640$	−11595.6	−0.716180
$641$	5471.53	0.337149	0.168575	−	0.985689i	$-0.446084\pi$
$641$	5471.53	0.337149	0.168575	+	0.985689i	$0.446084\pi$
$642$	0	0
$643$	−4008.65	−0.245857	−0.122928	−	0.992416i	$-0.539229\pi$
$643$	−4008.65	−0.245857	−0.122928	+	0.992416i	$0.539229\pi$
$644$	0	0
$645$	0	0
$646$	34374.2	2.09355
$647$	−5301.48	−0.322137	−0.161069	−	0.986943i	$-0.551494\pi$
$647$	−5301.48	−0.322137	−0.161069	+	0.986943i	$0.551494\pi$
$648$	0	0
$649$	43.1065	0.00260721
$650$	−19044.5	−1.14921
$651$	0	0
$652$	3774.01	0.226689
$653$	15639.9	0.937271	0.468635	−	0.883392i	$-0.344746\pi$
$653$	15639.9	0.937271	0.468635	+	0.883392i	$0.344746\pi$
$654$	0	0
$655$	25883.8	1.54407
$656$	−2448.51	−0.145729
$657$	0	0
$658$	0	0
$659$	1496.34	0.0884507	0.0442254	−	0.999022i	$-0.485918\pi$
$659$	1496.34	0.0884507	0.0442254	+	0.999022i	$0.485918\pi$
$660$	0	0
$661$	−18248.7	−1.07382	−0.536908	−	0.843641i	$-0.680408\pi$
$661$	−18248.7	−1.07382	−0.536908	+	0.843641i	$0.680408\pi$
$662$	−25689.6	−1.50824
$663$	0	0
$664$	−11870.1	−0.693749
$665$	0	0
$666$	0	0
$667$	31688.0	1.83953
$668$	−287.350	−0.0166436
$669$	0	0
$670$	1859.25	0.107208
$671$	−6.70519	−0.000385769 0
$672$	0	0
$673$	−5442.26	−0.311715	−0.155857	−	0.987780i	$-0.549814\pi$
$673$	−5442.26	−0.311715	−0.155857	+	0.987780i	$0.549814\pi$
$674$	−15746.4	−0.899894
$675$	0	0
$676$	−9474.50	−0.539059
$677$	1298.38	0.0737085	0.0368542	−	0.999321i	$-0.488266\pi$
$677$	1298.38	0.0737085	0.0368542	+	0.999321i	$0.488266\pi$
$678$	0	0
$679$	0	0
$680$	33098.1	1.86655
$681$	0	0
$682$	53.7976	0.00302055
$683$	19142.2	1.07241	0.536204	−	0.844088i	$-0.319858\pi$
$683$	19142.2	1.07241	0.536204	+	0.844088i	$0.319858\pi$
$684$	0	0
$685$	−22907.8	−1.27775
$686$	0	0
$687$	0	0
$688$	−2245.89	−0.124453
$689$	1027.84	0.0568323
$690$	0	0
$691$	−789.395	−0.0434588	−0.0217294	−	0.999764i	$-0.506917\pi$
$691$	−789.395	−0.0434588	−0.0217294	+	0.999764i	$0.506917\pi$
$692$	−7491.12	−0.411517
$693$	0	0
$694$	−414.024	−0.0226457
$695$	−1588.10	−0.0866765
$696$	0	0
$697$	4875.04	0.264929
$698$	21111.9	1.14484
$699$	0	0
$700$	0	0
$701$	25438.5	1.37061	0.685306	−	0.728256i	$-0.259669\pi$
$701$	25438.5	1.37061	0.685306	+	0.728256i	$0.259669\pi$
$702$	0	0
$703$	−7148.66	−0.383523
$704$	−45.2462	−0.00242227
$705$	0	0
$706$	−13011.8	−0.693634
$707$	0	0
$708$	0	0
$709$	−10178.1	−0.539137	−0.269568	−	0.962981i	$-0.586881\pi$
$709$	−10178.1	−0.539137	−0.269568	+	0.962981i	$0.586881\pi$
$710$	−18028.1	−0.952934
$711$	0	0
$712$	−28489.6	−1.49957
$713$	41851.2	2.19823
$714$	0	0
$715$	−100.727	−0.00526848
$716$	1412.80	0.0737415
$717$	0	0
$718$	15763.2	0.819327
$719$	32744.0	1.69839	0.849196	−	0.528077i	$-0.177087\pi$
$719$	32744.0	1.69839	0.849196	+	0.528077i	$0.177087\pi$
$720$	0	0
$721$	0	0
$722$	−37526.1	−1.93432
$723$	0	0
$724$	−3523.36	−0.180863
$725$	18022.5	0.923226
$726$	0	0
$727$	−32742.7	−1.67037	−0.835185	−	0.549969i	$-0.814640\pi$
$727$	−32742.7	−1.67037	−0.835185	+	0.549969i	$0.814640\pi$
$728$	0	0
$729$	0	0
$730$	−2592.65	−0.131450
$731$	4471.63	0.226251
$732$	0	0
$733$	19002.5	0.957536	0.478768	−	0.877942i	$-0.341084\pi$
$733$	19002.5	0.957536	0.478768	+	0.877942i	$0.341084\pi$
$734$	3435.20	0.172746
$735$	0	0
$736$	−12092.7	−0.605629
$737$	4.05574	0.000202707 0
$738$	0	0
$739$	−9502.74	−0.473023	−0.236511	−	0.971629i	$-0.576004\pi$
$739$	−9502.74	−0.473023	−0.236511	+	0.971629i	$0.576004\pi$
$740$	−1245.38	−0.0618662
$741$	0	0
$742$	0	0
$743$	−4885.65	−0.241234	−0.120617	−	0.992699i	$-0.538487\pi$
$743$	−4885.65	−0.241234	−0.120617	+	0.992699i	$0.538487\pi$
$744$	0	0
$745$	43396.7	2.13414
$746$	−9866.35	−0.484226
$747$	0	0
$748$	13.0629	0.000638540 0
$749$	0	0
$750$	0	0
$751$	−20093.1	−0.976311	−0.488155	−	0.872757i	$-0.662330\pi$
$751$	−20093.1	−0.976311	−0.488155	+	0.872757i	$0.662330\pi$
$752$	12438.2	0.603158
$753$	0	0
$754$	−44582.7	−2.15332
$755$	−46968.6	−2.26406
$756$	0	0
$757$	−38123.9	−1.83043	−0.915216	−	0.402964i	$-0.867980\pi$
$757$	−38123.9	−1.83043	−0.915216	+	0.402964i	$0.867980\pi$
$758$	−27588.2	−1.32197
$759$	0	0
$760$	−52621.3	−2.51155
$761$	−6800.32	−0.323931	−0.161966	−	0.986796i	$-0.551783\pi$
$761$	−6800.32	−0.323931	−0.161966	+	0.986796i	$0.551783\pi$
$762$	0	0
$763$	0	0
$764$	−5514.21	−0.261122
$765$	0	0
$766$	−2006.40	−0.0946399
$767$	−47180.0	−2.22108
$768$	0	0
$769$	27880.3	1.30740	0.653699	−	0.756755i	$-0.273216\pi$
$769$	27880.3	1.30740	0.653699	+	0.756755i	$0.273216\pi$
$770$	0	0
$771$	0	0
$772$	94.6477	0.00441249
$773$	−40273.3	−1.87391	−0.936954	−	0.349452i	$-0.886368\pi$
$773$	−40273.3	−1.87391	−0.936954	+	0.349452i	$0.886368\pi$
$774$	0	0
$775$	23802.8	1.10325
$776$	31966.0	1.47875
$777$	0	0
$778$	−8026.98	−0.369899
$779$	−7750.63	−0.356476
$780$	0	0
$781$	−39.3261	−0.00180179
$782$	−35842.4	−1.63903
$783$	0	0
$784$	0	0
$785$	−23929.6	−1.08801
$786$	0	0
$787$	13454.8	0.609419	0.304710	−	0.952445i	$-0.401441\pi$
$787$	13454.8	0.609419	0.304710	+	0.952445i	$0.401441\pi$
$788$	3842.36	0.173704
$789$	0	0
$790$	−13788.2	−0.620965
$791$	0	0
$792$	0	0
$793$	7338.83	0.328637
$794$	−30911.5	−1.38162
$795$	0	0
$796$	−6654.30	−0.296301
$797$	−7642.86	−0.339679	−0.169839	−	0.985472i	$-0.554325\pi$
$797$	−7642.86	−0.339679	−0.169839	+	0.985472i	$0.554325\pi$
$798$	0	0
$799$	−24764.8	−1.09652
$800$	−6877.71	−0.303955
$801$	0	0
$802$	24967.7	1.09930
$803$	−5.65555	−0.000248543 0
$804$	0	0
$805$	0	0
$806$	−58881.5	−2.57321
$807$	0	0
$808$	−41435.2	−1.80407
$809$	18212.2	0.791480	0.395740	−	0.918363i	$-0.370488\pi$
$809$	18212.2	0.791480	0.395740	+	0.918363i	$0.370488\pi$
$810$	0	0
$811$	−29294.1	−1.26838	−0.634188	−	0.773179i	$-0.718666\pi$
$811$	−29294.1	−1.26838	−0.634188	+	0.773179i	$0.718666\pi$
$812$	0	0
$813$	0	0
$814$	9.58175	0.000412580 0
$815$	−31149.8	−1.33881
$816$	0	0
$817$	−7109.26	−0.304433
$818$	34695.7	1.48302
$819$	0	0
$820$	−1350.25	−0.0575033
$821$	−25827.3	−1.09790	−0.548951	−	0.835855i	$-0.684973\pi$
$821$	−25827.3	−1.09790	−0.548951	+	0.835855i	$0.684973\pi$
$822$	0	0
$823$	−19831.0	−0.839932	−0.419966	−	0.907540i	$-0.637958\pi$
$823$	−19831.0	−0.839932	−0.419966	+	0.907540i	$0.637958\pi$
$824$	−15003.5	−0.634310
$825$	0	0
$826$	0	0
$827$	3488.51	0.146684	0.0733418	−	0.997307i	$-0.476634\pi$
$827$	3488.51	0.146684	0.0733418	+	0.997307i	$0.476634\pi$
$828$	0	0
$829$	19751.2	0.827486	0.413743	−	0.910394i	$-0.364221\pi$
$829$	19751.2	0.827486	0.413743	+	0.910394i	$0.364221\pi$
$830$	17726.1	0.741304
$831$	0	0
$832$	49521.9	2.06354
$833$	0	0
$834$	0	0
$835$	2371.72	0.0982957
$836$	−20.7682	−0.000859190 0
$837$	0	0
$838$	12109.9	0.499200
$839$	−11436.6	−0.470604	−0.235302	−	0.971922i	$-0.575608\pi$
$839$	−11436.6	−0.470604	−0.235302	+	0.971922i	$0.575608\pi$
$840$	0	0
$841$	17801.3	0.729890
$842$	1244.46	0.0509345
$843$	0	0
$844$	1245.91	0.0508128
$845$	78200.5	3.18364
$846$	0	0
$847$	0	0
$848$	−552.581	−0.0223770
$849$	0	0
$850$	−20385.3	−0.822601
$851$	7454.00	0.300258
$852$	0	0
$853$	17685.8	0.709907	0.354953	−	0.934884i	$-0.384497\pi$
$853$	17685.8	0.709907	0.354953	+	0.934884i	$0.384497\pi$
$854$	0	0
$855$	0	0
$856$	−41281.5	−1.64833
$857$	18601.9	0.741458	0.370729	−	0.928741i	$-0.379108\pi$
$857$	18601.9	0.741458	0.370729	+	0.928741i	$0.379108\pi$
$858$	0	0
$859$	20873.8	0.829109	0.414554	−	0.910025i	$-0.363937\pi$
$859$	20873.8	0.829109	0.414554	+	0.910025i	$0.363937\pi$
$860$	−1238.52	−0.0491082
$861$	0	0
$862$	38736.0	1.53057
$863$	3891.74	0.153507	0.0767533	−	0.997050i	$-0.475545\pi$
$863$	3891.74	0.153507	0.0767533	+	0.997050i	$0.475545\pi$
$864$	0	0
$865$	61830.1	2.43039
$866$	5640.44	0.221328
$867$	0	0
$868$	0	0
$869$	−30.0773	−0.00117411
$870$	0	0
$871$	−4439.00	−0.172686
$872$	21282.3	0.826501
$873$	0	0
$874$	56984.4	2.20541
$875$	0	0
$876$	0	0
$877$	5848.66	0.225194	0.112597	−	0.993641i	$-0.464083\pi$
$877$	5848.66	0.225194	0.112597	+	0.993641i	$0.464083\pi$
$878$	11814.0	0.454103
$879$	0	0
$880$	54.1522	0.00207440
$881$	−48386.3	−1.85037	−0.925185	−	0.379517i	$-0.876090\pi$
$881$	−48386.3	−1.85037	−0.925185	+	0.379517i	$0.876090\pi$
$882$	0	0
$883$	21834.4	0.832149	0.416075	−	0.909331i	$-0.363406\pi$
$883$	21834.4	0.832149	0.416075	+	0.909331i	$0.363406\pi$
$884$	−14297.4	−0.543973
$885$	0	0
$886$	−31723.3	−1.20290
$887$	30545.0	1.15626	0.578129	−	0.815945i	$-0.303783\pi$
$887$	30545.0	1.15626	0.578129	+	0.815945i	$0.303783\pi$
$888$	0	0
$889$	0	0
$890$	42544.7	1.60236
$891$	0	0
$892$	−4815.19	−0.180745
$893$	39372.5	1.47542
$894$	0	0
$895$	−11661.0	−0.435512
$896$	0	0
$897$	0	0
$898$	22908.7	0.851307
$899$	55721.8	2.06721
$900$	0	0
$901$	1100.20	0.0406804
$902$	10.3886	0.000383484 0
$903$	0	0
$904$	−3409.12	−0.125427
$905$	29081.0	1.06816
$906$	0	0
$907$	31375.7	1.14864	0.574318	−	0.818632i	$-0.305267\pi$
$907$	31375.7	1.14864	0.574318	+	0.818632i	$0.305267\pi$
$908$	−1091.18	−0.0398810
$909$	0	0
$910$	0	0
$911$	−88.9027	−0.00323324	−0.00161662	−	0.999999i	$-0.500515\pi$
$911$	−88.9027	−0.00323324	−0.00161662	+	0.999999i	$0.500515\pi$
$912$	0	0
$913$	38.6674	0.00140165
$914$	9152.59	0.331226
$915$	0	0
$916$	−8055.28	−0.290561
$917$	0	0
$918$	0	0
$919$	−22290.5	−0.800104	−0.400052	−	0.916492i	$-0.631008\pi$
$919$	−22290.5	−0.800104	−0.400052	+	0.916492i	$0.631008\pi$
$920$	54869.0	1.96628
$921$	0	0
$922$	−16000.8	−0.571539
$923$	43042.4	1.53495
$924$	0	0
$925$	4239.45	0.150694
$926$	32934.4	1.16878
$927$	0	0
$928$	−16100.6	−0.569533
$929$	−48888.8	−1.72658	−0.863289	−	0.504710i	$-0.831599\pi$
$929$	−48888.8	−1.72658	−0.863289	+	0.504710i	$0.831599\pi$
$930$	0	0
$931$	0	0
$932$	−4993.24	−0.175492
$933$	0	0
$934$	16890.0	0.591709
$935$	−107.819	−0.00377117
$936$	0	0
$937$	23882.4	0.832662	0.416331	−	0.909213i	$-0.363316\pi$
$937$	23882.4	0.832662	0.416331	+	0.909213i	$0.363316\pi$
$938$	0	0
$939$	0	0
$940$	6859.15	0.238001
$941$	1236.76	0.0428451	0.0214226	−	0.999771i	$-0.493180\pi$
$941$	1236.76	0.0428451	0.0214226	+	0.999771i	$0.493180\pi$
$942$	0	0
$943$	8081.69	0.279084
$944$	25364.7	0.874524
$945$	0	0
$946$	9.52895	0.000327498 0
$947$	−26627.8	−0.913712	−0.456856	−	0.889541i	$-0.651025\pi$
$947$	−26627.8	−0.913712	−0.456856	+	0.889541i	$0.651025\pi$
$948$	0	0
$949$	6190.00	0.211734
$950$	32409.8	1.10685
$951$	0	0
$952$	0	0
$953$	−10274.3	−0.349232	−0.174616	−	0.984637i	$-0.555868\pi$
$953$	−10274.3	−0.349232	−0.174616	+	0.984637i	$0.555868\pi$
$954$	0	0
$955$	45513.1	1.54217
$956$	10265.5	0.347290
$957$	0	0
$958$	−28329.0	−0.955393
$959$	0	0
$960$	0	0
$961$	43802.1	1.47031
$962$	−10487.2	−0.351478
$963$	0	0
$964$	−8351.73	−0.279036
$965$	−781.202	−0.0260599
$966$	0	0
$967$	43332.4	1.44103	0.720515	−	0.693440i	$-0.243906\pi$
$967$	43332.4	1.44103	0.720515	+	0.693440i	$0.243906\pi$
$968$	−32455.4	−1.07764
$969$	0	0
$970$	−47736.2	−1.58012
$971$	45019.6	1.48790	0.743949	−	0.668237i	$-0.232951\pi$
$971$	45019.6	1.48790	0.743949	+	0.668237i	$0.232951\pi$
$972$	0	0
$973$	0	0
$974$	−12380.3	−0.407278
$975$	0	0
$976$	−3945.47	−0.129397
$977$	45287.6	1.48299	0.741494	−	0.670960i	$-0.234118\pi$
$977$	45287.6	1.48299	0.741494	+	0.670960i	$0.234118\pi$
$978$	0	0
$979$	92.8061	0.00302972
$980$	0	0
$981$	0	0
$982$	46124.1	1.49886
$983$	42418.0	1.37632	0.688161	−	0.725558i	$-0.258418\pi$
$983$	42418.0	1.37632	0.688161	+	0.725558i	$0.258418\pi$
$984$	0	0
$985$	−31714.0	−1.02588
$986$	−47721.6	−1.54134
$987$	0	0
$988$	22730.8	0.731945
$989$	7412.93	0.238339
$990$	0	0
$991$	−13307.1	−0.426552	−0.213276	−	0.976992i	$-0.568413\pi$
$991$	−13307.1	−0.426552	−0.213276	+	0.976992i	$0.568413\pi$
$992$	−21264.4	−0.680591
$993$	0	0
$994$	0	0
$995$	54923.2	1.74993
$996$	0	0
$997$	58374.0	1.85429	0.927143	−	0.374707i	$-0.122257\pi$
$997$	58374.0	1.85429	0.927143	+	0.374707i	$0.122257\pi$
$998$	6748.12	0.214036
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.bm.1.3	yes	8
3.2	odd	2	inner	1323.4.a.bm.1.6	yes	8
7.6	odd	2		1323.4.a.bl.1.3	✓	8
21.20	even	2		1323.4.a.bl.1.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.4.a.bl.1.3	✓	8	7.6	odd	2
1323.4.a.bl.1.6	yes	8	21.20	even	2
1323.4.a.bm.1.3	yes	8	1.1	even	1	trivial
1323.4.a.bm.1.6	yes	8	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

\(f(q)\)	\(=\)	\(q-2.49657 q^{2} -1.76715 q^{4} +14.5857 q^{5} +24.3844 q^{8} +O(q^{10})\)	\(q-2.49657 q^{2} -1.76715 q^{4} +14.5857 q^{5} +24.3844 q^{8} -36.4142 q^{10} -0.0794330 q^{11} +86.9394 q^{13} -46.7400 q^{16} +93.0605 q^{17} -147.953 q^{19} -25.7751 q^{20} +0.198310 q^{22} +154.273 q^{23} +87.7423 q^{25} -217.050 q^{26} +205.403 q^{29} +271.281 q^{31} -78.3853 q^{32} -232.332 q^{34} +48.3171 q^{37} +369.375 q^{38} +355.663 q^{40} +52.3857 q^{41} +48.0508 q^{43} +0.140370 q^{44} -385.152 q^{46} -266.115 q^{47} -219.055 q^{50} -153.635 q^{52} +11.8224 q^{53} -1.15859 q^{55} -512.802 q^{58} -542.677 q^{59} +84.4131 q^{61} -677.270 q^{62} +569.614 q^{64} +1268.07 q^{65} -51.0586 q^{67} -164.452 q^{68} +495.085 q^{71} +71.1990 q^{73} -120.627 q^{74} +261.455 q^{76} +378.650 q^{79} -681.735 q^{80} -130.785 q^{82} -486.792 q^{83} +1357.35 q^{85} -119.962 q^{86} -1.93692 q^{88} -1168.36 q^{89} -272.623 q^{92} +664.374 q^{94} -2158.00 q^{95} +1310.92 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 44 q^{4}+O(q^{10}) \) 8 * q + 44 * q^4	\( 8 q + 44 q^{4} + 132 q^{10} + 336 q^{13} + 204 q^{16} - 288 q^{19} + 484 q^{22} + 152 q^{25} + 120 q^{31} + 1008 q^{34} + 592 q^{37} + 1620 q^{40} - 1872 q^{43} - 1644 q^{46} + 2400 q^{52} + 1344 q^{55} - 1200 q^{58} + 2400 q^{61} - 1388 q^{64} + 1824 q^{73} + 2844 q^{76} + 2368 q^{79} + 2436 q^{82} + 3512 q^{85} + 3780 q^{88} + 4368 q^{94} + 5712 q^{97}+O(q^{100}) \) 8 * q + 44 * q^4 + 132 * q^10 + 336 * q^13 + 204 * q^16 - 288 * q^19 + 484 * q^22 + 152 * q^25 + 120 * q^31 + 1008 * q^34 + 592 * q^37 + 1620 * q^40 - 1872 * q^43 - 1644 * q^46 + 2400 * q^52 + 1344 * q^55 - 1200 * q^58 + 2400 * q^61 - 1388 * q^64 + 1824 * q^73 + 2844 * q^76 + 2368 * q^79 + 2436 * q^82 + 3512 * q^85 + 3780 * q^88 + 4368 * q^94 + 5712 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more