LMFDB - Embedded newform 1521.4.a.bk.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, 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("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 70x^{8} + 1645x^{6} - 14700x^{4} + 44100x^{2} - 27648 $ x^10 - 70x^8 + 1645x^6 - 14700x^4 + 44100x^2 - 27648
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.27897$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.27897 q^{2} +2.75167 q^{4} +17.5414 q^{5} +26.6999 q^{7} +17.2091 q^{8} +O(q^{10})$	$q-3.27897 q^{2} +2.75167 q^{4} +17.5414 q^{5} +26.6999 q^{7} +17.2091 q^{8} -57.5178 q^{10} -21.4026 q^{11} -87.5483 q^{14} -78.4417 q^{16} -83.9630 q^{17} +77.1142 q^{19} +48.2682 q^{20} +70.1785 q^{22} -142.119 q^{23} +182.701 q^{25} +73.4693 q^{28} -134.223 q^{29} -122.559 q^{31} +119.535 q^{32} +275.312 q^{34} +468.354 q^{35} +222.587 q^{37} -252.855 q^{38} +301.873 q^{40} +198.321 q^{41} +154.656 q^{43} -58.8928 q^{44} +466.006 q^{46} -78.7956 q^{47} +369.884 q^{49} -599.072 q^{50} +477.088 q^{53} -375.431 q^{55} +459.482 q^{56} +440.114 q^{58} -42.9282 q^{59} +496.539 q^{61} +401.869 q^{62} +235.581 q^{64} +484.659 q^{67} -231.038 q^{68} -1535.72 q^{70} +382.432 q^{71} -193.622 q^{73} -729.858 q^{74} +212.193 q^{76} -571.447 q^{77} +1049.60 q^{79} -1375.98 q^{80} -650.289 q^{82} +861.900 q^{83} -1472.83 q^{85} -507.112 q^{86} -368.320 q^{88} +967.645 q^{89} -391.065 q^{92} +258.369 q^{94} +1352.69 q^{95} -591.470 q^{97} -1212.84 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 60 q^{4}+O(q^{10}) $ 10 * q + 60 * q^4	$ 10 q + 60 q^{4} + 80 q^{10} + 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} + 960 q^{25} - 990 q^{29} + 120 q^{35} - 1380 q^{38} + 2000 q^{40} - 740 q^{43} + 1550 q^{49} - 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} + 3140 q^{64} - 1200 q^{68} + 4380 q^{74} - 4320 q^{77} + 1100 q^{79} - 4780 q^{82} + 6340 q^{88} + 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100}) $ 10 * q + 60 * q^4 + 80 * q^10 + 60 * q^14 + 500 * q^16 - 210 * q^17 + 580 * q^22 + 120 * q^23 + 960 * q^25 - 990 * q^29 + 120 * q^35 - 1380 * q^38 + 2000 * q^40 - 740 * q^43 + 1550 * q^49 - 330 * q^53 + 520 * q^55 + 5340 * q^56 + 2750 * q^61 + 1560 * q^62 + 3140 * q^64 - 1200 * q^68 + 4380 * q^74 - 4320 * q^77 + 1100 * q^79 - 4780 * q^82 + 6340 * q^88 + 1740 * q^92 + 6460 * q^94 + 2760 * q^95

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 $

$2$	−3.27897	−1.15929	−0.579646	−	0.814868i	$-0.696809\pi$
$2$	−3.27897	−1.15929	−0.579646	+	0.814868i	$0.696809\pi$
$3$	0	0
$3$	0	0
$4$	2.75167	0.343959
$5$	17.5414	1.56895	0.784476	−	0.620160i	$-0.212932\pi$
$5$	17.5414	1.56895	0.784476	+	0.620160i	$0.212932\pi$
$6$	0	0
$7$	26.6999	1.44166	0.720829	−	0.693112i	$-0.243761\pi$
$7$	26.6999	1.44166	0.720829	+	0.693112i	$0.243761\pi$
$8$	17.2091	0.760544
$9$	0	0
$10$	−57.5178	−1.81887
$11$	−21.4026	−0.586647	−0.293324	−	0.956013i	$-0.594761\pi$
$11$	−21.4026	−0.586647	−0.293324	+	0.956013i	$0.594761\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−87.5483	−1.67130
$15$	0	0
$16$	−78.4417	−1.22565
$17$	−83.9630	−1.19788	−0.598942	−	0.800793i	$-0.704412\pi$
$17$	−83.9630	−1.19788	−0.598942	+	0.800793i	$0.704412\pi$
$18$	0	0
$19$	77.1142	0.931116	0.465558	−	0.885017i	$-0.345854\pi$
$19$	77.1142	0.931116	0.465558	+	0.885017i	$0.345854\pi$
$20$	48.2682	0.539654
$21$	0	0
$22$	70.1785	0.680096
$23$	−142.119	−1.28843	−0.644216	−	0.764844i	$-0.722816\pi$
$23$	−142.119	−1.28843	−0.644216	+	0.764844i	$0.722816\pi$
$24$	0	0
$25$	182.701	1.46161
$26$	0	0
$27$	0	0
$28$	73.4693	0.495871
$29$	−134.223	−0.859469	−0.429734	−	0.902955i	$-0.641393\pi$
$29$	−134.223	−0.859469	−0.429734	+	0.902955i	$0.641393\pi$
$30$	0	0
$31$	−122.559	−0.710074	−0.355037	−	0.934852i	$-0.615532\pi$
$31$	−122.559	−0.710074	−0.355037	+	0.934852i	$0.615532\pi$
$32$	119.535	0.660344
$33$	0	0
$34$	275.312	1.38870
$35$	468.354	2.26189
$36$	0	0
$37$	222.587	0.989003	0.494502	−	0.869177i	$-0.335351\pi$
$37$	222.587	0.989003	0.494502	+	0.869177i	$0.335351\pi$
$38$	−252.855	−1.07944
$39$	0	0
$40$	301.873	1.19326
$41$	198.321	0.755427	0.377713	−	0.925923i	$-0.376710\pi$
$41$	198.321	0.755427	0.377713	+	0.925923i	$0.376710\pi$
$42$	0	0
$43$	154.656	0.548483	0.274242	−	0.961661i	$-0.411573\pi$
$43$	154.656	0.548483	0.274242	+	0.961661i	$0.411573\pi$
$44$	−58.8928	−0.201782
$45$	0	0
$46$	466.006	1.49367
$47$	−78.7956	−0.244543	−0.122271	−	0.992497i	$-0.539018\pi$
$47$	−78.7956	−0.244543	−0.122271	+	0.992497i	$0.539018\pi$
$48$	0	0
$49$	369.884	1.07838
$50$	−599.072	−1.69443
$51$	0	0
$52$	0	0
$53$	477.088	1.23647	0.618237	−	0.785992i	$-0.287847\pi$
$53$	477.088	1.23647	0.618237	+	0.785992i	$0.287847\pi$
$54$	0	0
$55$	−375.431	−0.920421
$56$	459.482	1.09644
$57$	0	0
$58$	440.114	0.996376
$59$	−42.9282	−0.0947249	−0.0473625	−	0.998878i	$-0.515082\pi$
$59$	−42.9282	−0.0947249	−0.0473625	+	0.998878i	$0.515082\pi$
$60$	0	0
$61$	496.539	1.04222	0.521109	−	0.853490i	$-0.325519\pi$
$61$	496.539	1.04222	0.521109	+	0.853490i	$0.325519\pi$
$62$	401.869	0.823183
$63$	0	0
$64$	235.581	0.460119
$65$	0	0
$66$	0	0
$67$	484.659	0.883739	0.441869	−	0.897079i	$-0.354315\pi$
$67$	484.659	0.883739	0.441869	+	0.897079i	$0.354315\pi$
$68$	−231.038	−0.412022
$69$	0	0
$70$	−1535.72	−2.62219
$71$	382.432	0.639245	0.319622	−	0.947545i	$-0.396444\pi$
$71$	382.432	0.639245	0.319622	+	0.947545i	$0.396444\pi$
$72$	0	0
$73$	−193.622	−0.310435	−0.155217	−	0.987880i	$-0.549608\pi$
$73$	−193.622	−0.310435	−0.155217	+	0.987880i	$0.549608\pi$
$74$	−729.858	−1.14654
$75$	0	0
$76$	212.193	0.320265
$77$	−571.447	−0.845745
$78$	0	0
$79$	1049.60	1.49480	0.747399	−	0.664375i	$-0.231302\pi$
$79$	1049.60	1.49480	0.747399	+	0.664375i	$0.231302\pi$
$80$	−1375.98	−1.92299
$81$	0	0
$82$	−650.289	−0.875761
$83$	861.900	1.13983	0.569914	−	0.821704i	$-0.306976\pi$
$83$	861.900	1.13983	0.569914	+	0.821704i	$0.306976\pi$
$84$	0	0
$85$	−1472.83	−1.87942
$86$	−507.112	−0.635853
$87$	0	0
$88$	−368.320	−0.446171
$89$	967.645	1.15247	0.576237	−	0.817283i	$-0.304521\pi$
$89$	967.645	1.15247	0.576237	+	0.817283i	$0.304521\pi$
$90$	0	0
$91$	0	0
$92$	−391.065	−0.443167
$93$	0	0
$94$	258.369	0.283497
$95$	1352.69	1.46088
$96$	0	0
$97$	−591.470	−0.619120	−0.309560	−	0.950880i	$-0.600182\pi$
$97$	−591.470	−0.619120	−0.309560	+	0.950880i	$0.600182\pi$
$98$	−1212.84	−1.25016
$99$	0	0
$100$	502.733	0.502733
$101$	−255.110	−0.251331	−0.125665	−	0.992073i	$-0.540107\pi$
$101$	−255.110	−0.251331	−0.125665	+	0.992073i	$0.540107\pi$
$102$	0	0
$103$	−247.355	−0.236627	−0.118313	−	0.992976i	$-0.537749\pi$
$103$	−247.355	−0.236627	−0.118313	+	0.992976i	$0.537749\pi$
$104$	0	0
$105$	0	0
$106$	−1564.36	−1.43343
$107$	−683.484	−0.617522	−0.308761	−	0.951140i	$-0.599914\pi$
$107$	−683.484	−0.617522	−0.308761	+	0.951140i	$0.599914\pi$
$108$	0	0
$109$	−1697.76	−1.49189	−0.745946	−	0.666006i	$-0.768002\pi$
$109$	−1697.76	−1.49189	−0.745946	+	0.666006i	$0.768002\pi$
$110$	1231.03	1.06704
$111$	0	0
$112$	−2094.38	−1.76697
$113$	380.709	0.316939	0.158469	−	0.987364i	$-0.449344\pi$
$113$	380.709	0.316939	0.158469	+	0.987364i	$0.449344\pi$
$114$	0	0
$115$	−2492.97	−2.02149
$116$	−369.337	−0.295622
$117$	0	0
$118$	140.760	0.109814
$119$	−2241.80	−1.72694
$120$	0	0
$121$	−872.930	−0.655845
$122$	−1628.14	−1.20824
$123$	0	0
$124$	−337.243	−0.244236
$125$	1012.16	0.724241
$126$	0	0
$127$	123.231	0.0861025	0.0430513	−	0.999073i	$-0.486292\pi$
$127$	123.231	0.0861025	0.0430513	+	0.999073i	$0.486292\pi$
$128$	−1728.74	−1.19376
$129$	0	0
$130$	0	0
$131$	1218.41	0.812616	0.406308	−	0.913736i	$-0.366816\pi$
$131$	1218.41	0.812616	0.406308	+	0.913736i	$0.366816\pi$
$132$	0	0
$133$	2058.94	1.34235
$134$	−1589.18	−1.02451
$135$	0	0
$136$	−1444.93	−0.911042
$137$	2728.83	1.70175	0.850875	−	0.525369i	$-0.176073\pi$
$137$	2728.83	1.70175	0.850875	+	0.525369i	$0.176073\pi$
$138$	0	0
$139$	3112.78	1.89944	0.949722	−	0.313094i	$-0.101366\pi$
$139$	3112.78	1.89944	0.949722	+	0.313094i	$0.101366\pi$
$140$	1288.75	0.777998
$141$	0	0
$142$	−1253.99	−0.741071
$143$	0	0
$144$	0	0
$145$	−2354.46	−1.34846
$146$	634.881	0.359885
$147$	0	0
$148$	612.487	0.340176
$149$	1370.57	0.753567	0.376784	−	0.926301i	$-0.377030\pi$
$149$	1370.57	0.753567	0.376784	+	0.926301i	$0.377030\pi$
$150$	0	0
$151$	2847.56	1.53464	0.767321	−	0.641263i	$-0.221589\pi$
$151$	2847.56	1.53464	0.767321	+	0.641263i	$0.221589\pi$
$152$	1327.07	0.708154
$153$	0	0
$154$	1873.76	0.980466
$155$	−2149.86	−1.11407
$156$	0	0
$157$	3354.00	1.70496	0.852479	−	0.522761i	$-0.175098\pi$
$157$	3354.00	1.70496	0.852479	+	0.522761i	$0.175098\pi$
$158$	−3441.61	−1.73291
$159$	0	0
$160$	2096.81	1.03605
$161$	−3794.57	−1.85748
$162$	0	0
$163$	2196.18	1.05533	0.527664	−	0.849453i	$-0.323068\pi$
$163$	2196.18	1.05533	0.527664	+	0.849453i	$0.323068\pi$
$164$	545.713	0.259836
$165$	0	0
$166$	−2826.15	−1.32139
$167$	912.535	0.422839	0.211419	−	0.977395i	$-0.432191\pi$
$167$	912.535	0.422839	0.211419	+	0.977395i	$0.432191\pi$
$168$	0	0
$169$	0	0
$170$	4829.37	2.17880
$171$	0	0
$172$	425.562	0.188656
$173$	−899.636	−0.395364	−0.197682	−	0.980266i	$-0.563341\pi$
$173$	−899.636	−0.395364	−0.197682	+	0.980266i	$0.563341\pi$
$174$	0	0
$175$	4878.10	2.10714
$176$	1678.85	0.719025
$177$	0	0
$178$	−3172.88	−1.33605
$179$	−313.278	−0.130813	−0.0654064	−	0.997859i	$-0.520834\pi$
$179$	−313.278	−0.130813	−0.0654064	+	0.997859i	$0.520834\pi$
$180$	0	0
$181$	−2745.06	−1.12728	−0.563642	−	0.826019i	$-0.690600\pi$
$181$	−2745.06	−1.12728	−0.563642	+	0.826019i	$0.690600\pi$
$182$	0	0
$183$	0	0
$184$	−2445.75	−0.979909
$185$	3904.50	1.55170
$186$	0	0
$187$	1797.02	0.702735
$188$	−216.819	−0.0841126
$189$	0	0
$190$	−4435.44	−1.69358
$191$	−89.8679	−0.0340451	−0.0170226	−	0.999855i	$-0.505419\pi$
$191$	−89.8679	−0.0340451	−0.0170226	+	0.999855i	$0.505419\pi$
$192$	0	0
$193$	−848.954	−0.316627	−0.158314	−	0.987389i	$-0.550606\pi$
$193$	−848.954	−0.316627	−0.158314	+	0.987389i	$0.550606\pi$
$194$	1939.41	0.717741
$195$	0	0
$196$	1017.80	0.370918
$197$	4343.86	1.57100	0.785501	−	0.618860i	$-0.212405\pi$
$197$	4343.86	1.57100	0.785501	+	0.618860i	$0.212405\pi$
$198$	0	0
$199$	−3328.41	−1.18565	−0.592825	−	0.805331i	$-0.701988\pi$
$199$	−3328.41	−1.18565	−0.592825	+	0.805331i	$0.701988\pi$
$200$	3144.13	1.11162
$201$	0	0
$202$	836.499	0.291366
$203$	−3583.74	−1.23906
$204$	0	0
$205$	3478.83	1.18523
$206$	811.069	0.274320
$207$	0	0
$208$	0	0
$209$	−1650.44	−0.546236
$210$	0	0
$211$	−4599.88	−1.50080	−0.750401	−	0.660983i	$-0.770139\pi$
$211$	−4599.88	−1.50080	−0.750401	+	0.660983i	$0.770139\pi$
$212$	1312.79	0.425296
$213$	0	0
$214$	2241.13	0.715889
$215$	2712.88	0.860544
$216$	0	0
$217$	−3272.32	−1.02368
$218$	5566.92	1.72954
$219$	0	0
$220$	−1033.06	−0.316587
$221$	0	0
$222$	0	0
$223$	2529.58	0.759611	0.379806	−	0.925066i	$-0.375991\pi$
$223$	2529.58	0.759611	0.379806	+	0.925066i	$0.375991\pi$
$224$	3191.57	0.951991
$225$	0	0
$226$	−1248.33	−0.367425
$227$	37.2670	0.0108965	0.00544823	−	0.999985i	$-0.498266\pi$
$227$	37.2670	0.0108965	0.00544823	+	0.999985i	$0.498266\pi$
$228$	0	0
$229$	4094.45	1.18152	0.590762	−	0.806846i	$-0.298827\pi$
$229$	4094.45	1.18152	0.590762	+	0.806846i	$0.298827\pi$
$230$	8174.40	2.34349
$231$	0	0
$232$	−2309.86	−0.653664
$233$	−1466.04	−0.412205	−0.206103	−	0.978530i	$-0.566078\pi$
$233$	−1466.04	−0.412205	−0.206103	+	0.978530i	$0.566078\pi$
$234$	0	0
$235$	−1382.19	−0.383676
$236$	−118.124	−0.0325815
$237$	0	0
$238$	7350.81	2.00203
$239$	−5520.53	−1.49412	−0.747058	−	0.664759i	$-0.768534\pi$
$239$	−5520.53	−1.49412	−0.747058	+	0.664759i	$0.768534\pi$
$240$	0	0
$241$	−2665.63	−0.712483	−0.356241	−	0.934394i	$-0.615942\pi$
$241$	−2665.63	−0.712483	−0.356241	+	0.934394i	$0.615942\pi$
$242$	2862.31	0.760316
$243$	0	0
$244$	1366.31	0.358480
$245$	6488.30	1.69193
$246$	0	0
$247$	0	0
$248$	−2109.14	−0.540042
$249$	0	0
$250$	−3318.84	−0.839607
$251$	−1579.21	−0.397127	−0.198564	−	0.980088i	$-0.563628\pi$
$251$	−1579.21	−0.397127	−0.198564	+	0.980088i	$0.563628\pi$
$252$	0	0
$253$	3041.72	0.755855
$254$	−404.073	−0.0998180
$255$	0	0
$256$	3783.86	0.923794
$257$	2663.91	0.646577	0.323288	−	0.946300i	$-0.395212\pi$
$257$	2663.91	0.646577	0.323288	+	0.946300i	$0.395212\pi$
$258$	0	0
$259$	5943.06	1.42581
$260$	0	0
$261$	0	0
$262$	−3995.12	−0.942059
$263$	2436.93	0.571360	0.285680	−	0.958325i	$-0.407781\pi$
$263$	2436.93	0.571360	0.285680	+	0.958325i	$0.407781\pi$
$264$	0	0
$265$	8368.80	1.93997
$266$	−6751.21	−1.55618
$267$	0	0
$268$	1333.62	0.303970
$269$	2683.18	0.608165	0.304083	−	0.952646i	$-0.401650\pi$
$269$	2683.18	0.608165	0.304083	+	0.952646i	$0.401650\pi$
$270$	0	0
$271$	3857.14	0.864592	0.432296	−	0.901732i	$-0.357704\pi$
$271$	3857.14	0.864592	0.432296	+	0.901732i	$0.357704\pi$
$272$	6586.20	1.46819
$273$	0	0
$274$	−8947.76	−1.97282
$275$	−3910.27	−0.857448
$276$	0	0
$277$	1104.91	0.239666	0.119833	−	0.992794i	$-0.461764\pi$
$277$	1104.91	0.239666	0.119833	+	0.992794i	$0.461764\pi$
$278$	−10206.7	−2.20201
$279$	0	0
$280$	8059.97	1.72027
$281$	4982.58	1.05778	0.528890	−	0.848691i	$-0.322609\pi$
$281$	4982.58	1.05778	0.528890	+	0.848691i	$0.322609\pi$
$282$	0	0
$283$	2584.86	0.542947	0.271473	−	0.962446i	$-0.412489\pi$
$283$	2584.86	0.542947	0.271473	+	0.962446i	$0.412489\pi$
$284$	1052.33	0.219874
$285$	0	0
$286$	0	0
$287$	5295.14	1.08907
$288$	0	0
$289$	2136.78	0.434924
$290$	7720.22	1.56326
$291$	0	0
$292$	−532.784	−0.106777
$293$	−92.5482	−0.0184530	−0.00922649	−	0.999957i	$-0.502937\pi$
$293$	−92.5482	−0.0184530	−0.00922649	+	0.999957i	$0.502937\pi$
$294$	0	0
$295$	−753.020	−0.148619
$296$	3830.54	0.752180
$297$	0	0
$298$	−4494.07	−0.873605
$299$	0	0
$300$	0	0
$301$	4129.29	0.790726
$302$	−9337.07	−1.77910
$303$	0	0
$304$	−6048.96	−1.14122
$305$	8709.99	1.63519
$306$	0	0
$307$	3979.46	0.739803	0.369901	−	0.929071i	$-0.379391\pi$
$307$	3979.46	0.739803	0.369901	+	0.929071i	$0.379391\pi$
$308$	−1572.43	−0.290901
$309$	0	0
$310$	7049.34	1.29153
$311$	−3450.91	−0.629207	−0.314604	−	0.949223i	$-0.601872\pi$
$311$	−3450.91	−0.629207	−0.314604	+	0.949223i	$0.601872\pi$
$312$	0	0
$313$	−6189.03	−1.11765	−0.558825	−	0.829285i	$-0.688748\pi$
$313$	−6189.03	−1.11765	−0.558825	+	0.829285i	$0.688748\pi$
$314$	−10997.7	−1.97654
$315$	0	0
$316$	2888.15	0.514149
$317$	5437.78	0.963459	0.481729	−	0.876320i	$-0.340009\pi$
$317$	5437.78	0.963459	0.481729	+	0.876320i	$0.340009\pi$
$318$	0	0
$319$	2872.72	0.504205
$320$	4132.42	0.721904
$321$	0	0
$322$	12442.3	2.15336
$323$	−6474.73	−1.11537
$324$	0	0
$325$	0	0
$326$	−7201.23	−1.22343
$327$	0	0
$328$	3412.93	0.574535
$329$	−2103.83	−0.352547
$330$	0	0
$331$	−6626.64	−1.10040	−0.550201	−	0.835032i	$-0.685449\pi$
$331$	−6626.64	−1.10040	−0.550201	+	0.835032i	$0.685449\pi$
$332$	2371.66	0.392054
$333$	0	0
$334$	−2992.18	−0.490194
$335$	8501.60	1.38654
$336$	0	0
$337$	−5538.63	−0.895277	−0.447638	−	0.894215i	$-0.647735\pi$
$337$	−5538.63	−0.895277	−0.447638	+	0.894215i	$0.647735\pi$
$338$	0	0
$339$	0	0
$340$	−4052.74	−0.646443
$341$	2623.08	0.416563
$342$	0	0
$343$	717.813	0.112998
$344$	2661.49	0.417146
$345$	0	0
$346$	2949.88	0.458343
$347$	−9398.94	−1.45407	−0.727034	−	0.686602i	$-0.759102\pi$
$347$	−9398.94	−1.45407	−0.727034	+	0.686602i	$0.759102\pi$
$348$	0	0
$349$	−5757.33	−0.883045	−0.441522	−	0.897250i	$-0.645561\pi$
$349$	−5757.33	−0.883045	−0.441522	+	0.897250i	$0.645561\pi$
$350$	−15995.2	−2.44279
$351$	0	0
$352$	−2558.36	−0.387389
$353$	3457.82	0.521364	0.260682	−	0.965425i	$-0.416053\pi$
$353$	3457.82	0.521364	0.260682	+	0.965425i	$0.416053\pi$
$354$	0	0
$355$	6708.40	1.00294
$356$	2662.64	0.396403
$357$	0	0
$358$	1027.23	0.151650
$359$	−7168.96	−1.05394	−0.526968	−	0.849885i	$-0.676671\pi$
$359$	−7168.96	−1.05394	−0.526968	+	0.849885i	$0.676671\pi$
$360$	0	0
$361$	−912.407	−0.133023
$362$	9000.97	1.30685
$363$	0	0
$364$	0	0
$365$	−3396.40	−0.487057
$366$	0	0
$367$	3910.11	0.556147	0.278073	−	0.960560i	$-0.410304\pi$
$367$	3910.11	0.556147	0.278073	+	0.960560i	$0.410304\pi$
$368$	11148.1	1.57917
$369$	0	0
$370$	−12802.7	−1.79887
$371$	12738.2	1.78257
$372$	0	0
$373$	11377.6	1.57938	0.789691	−	0.613505i	$-0.210241\pi$
$373$	11377.6	1.57938	0.789691	+	0.613505i	$0.210241\pi$
$374$	−5892.39	−0.814675
$375$	0	0
$376$	−1356.00	−0.185986
$377$	0	0
$378$	0	0
$379$	4032.22	0.546494	0.273247	−	0.961944i	$-0.411902\pi$
$379$	4032.22	0.546494	0.273247	+	0.961944i	$0.411902\pi$
$380$	3722.16	0.502481
$381$	0	0
$382$	294.675	0.0394682
$383$	−1990.96	−0.265622	−0.132811	−	0.991141i	$-0.542400\pi$
$383$	−1990.96	−0.265622	−0.132811	+	0.991141i	$0.542400\pi$
$384$	0	0
$385$	−10024.0	−1.32693
$386$	2783.70	0.367063
$387$	0	0
$388$	−1627.53	−0.212952
$389$	11122.4	1.44969	0.724846	−	0.688911i	$-0.241911\pi$
$389$	11122.4	1.44969	0.724846	+	0.688911i	$0.241911\pi$
$390$	0	0
$391$	11932.8	1.54339
$392$	6365.39	0.820155
$393$	0	0
$394$	−14243.4	−1.82125
$395$	18411.4	2.34527
$396$	0	0
$397$	−10778.1	−1.36256	−0.681282	−	0.732021i	$-0.738577\pi$
$397$	−10778.1	−1.36256	−0.681282	+	0.732021i	$0.738577\pi$
$398$	10913.8	1.37452
$399$	0	0
$400$	−14331.4	−1.79142
$401$	−4711.58	−0.586746	−0.293373	−	0.955998i	$-0.594778\pi$
$401$	−4711.58	−0.586746	−0.293373	+	0.955998i	$0.594778\pi$
$402$	0	0
$403$	0	0
$404$	−701.978	−0.0864473
$405$	0	0
$406$	11751.0	1.43643
$407$	−4763.94	−0.580196
$408$	0	0
$409$	−1184.78	−0.143236	−0.0716179	−	0.997432i	$-0.522816\pi$
$409$	−1184.78	−0.143236	−0.0716179	+	0.997432i	$0.522816\pi$
$410$	−11407.0	−1.37403
$411$	0	0
$412$	−680.638	−0.0813899
$413$	−1146.18	−0.136561
$414$	0	0
$415$	15118.9	1.78834
$416$	0	0
$417$	0	0
$418$	5411.75	0.633248
$419$	6168.30	0.719191	0.359596	−	0.933108i	$-0.382915\pi$
$419$	6168.30	0.719191	0.359596	+	0.933108i	$0.382915\pi$
$420$	0	0
$421$	10328.8	1.19571	0.597857	−	0.801603i	$-0.296019\pi$
$421$	10328.8	1.19571	0.597857	+	0.801603i	$0.296019\pi$
$422$	15082.9	1.73987
$423$	0	0
$424$	8210.27	0.940392
$425$	−15340.1	−1.75084
$426$	0	0
$427$	13257.5	1.50252
$428$	−1880.72	−0.212402
$429$	0	0
$430$	−8895.46	−0.997622
$431$	−11312.5	−1.26427	−0.632137	−	0.774857i	$-0.717822\pi$
$431$	−11312.5	−1.26427	−0.632137	+	0.774857i	$0.717822\pi$
$432$	0	0
$433$	−10475.7	−1.16266	−0.581331	−	0.813667i	$-0.697468\pi$
$433$	−10475.7	−1.16266	−0.581331	+	0.813667i	$0.697468\pi$
$434$	10729.8	1.18675
$435$	0	0
$436$	−4671.68	−0.513149
$437$	−10959.4	−1.19968
$438$	0	0
$439$	−2040.62	−0.221853	−0.110926	−	0.993829i	$-0.535382\pi$
$439$	−2040.62	−0.221853	−0.110926	+	0.993829i	$0.535382\pi$
$440$	−6460.85	−0.700020
$441$	0	0
$442$	0	0
$443$	4089.28	0.438572	0.219286	−	0.975661i	$-0.429627\pi$
$443$	4089.28	0.438572	0.219286	+	0.975661i	$0.429627\pi$
$444$	0	0
$445$	16973.9	1.80818
$446$	−8294.43	−0.880611
$447$	0	0
$448$	6289.99	0.663335
$449$	−15217.9	−1.59951	−0.799753	−	0.600330i	$-0.795036\pi$
$449$	−15217.9	−1.59951	−0.799753	+	0.600330i	$0.795036\pi$
$450$	0	0
$451$	−4244.58	−0.443169
$452$	1047.58	0.109014
$453$	0	0
$454$	−122.197	−0.0126322
$455$	0	0
$456$	0	0
$457$	−876.316	−0.0896988	−0.0448494	−	0.998994i	$-0.514281\pi$
$457$	−876.316	−0.0896988	−0.0448494	+	0.998994i	$0.514281\pi$
$458$	−13425.6	−1.36973
$459$	0	0
$460$	−6859.84	−0.695308
$461$	−16293.4	−1.64611	−0.823057	−	0.567959i	$-0.807733\pi$
$461$	−16293.4	−1.64611	−0.823057	+	0.567959i	$0.807733\pi$
$462$	0	0
$463$	−11704.8	−1.17488	−0.587438	−	0.809269i	$-0.699863\pi$
$463$	−11704.8	−1.17488	−0.587438	+	0.809269i	$0.699863\pi$
$464$	10528.7	1.05341
$465$	0	0
$466$	4807.12	0.477866
$467$	15616.1	1.54738	0.773688	−	0.633567i	$-0.218410\pi$
$467$	15616.1	1.54738	0.773688	+	0.633567i	$0.218410\pi$
$468$	0	0
$469$	12940.3	1.27405
$470$	4532.15	0.444792
$471$	0	0
$472$	−738.757	−0.0720424
$473$	−3310.03	−0.321766
$474$	0	0
$475$	14088.8	1.36093
$476$	−6168.70	−0.593996
$477$	0	0
$478$	18101.7	1.73212
$479$	10376.0	0.989755	0.494877	−	0.868963i	$-0.335213\pi$
$479$	10376.0	0.989755	0.494877	+	0.868963i	$0.335213\pi$
$480$	0	0
$481$	0	0
$482$	8740.53	0.825976
$483$	0	0
$484$	−2402.01	−0.225584
$485$	−10375.2	−0.971369
$486$	0	0
$487$	3994.27	0.371658	0.185829	−	0.982582i	$-0.440503\pi$
$487$	3994.27	0.371658	0.185829	+	0.982582i	$0.440503\pi$
$488$	8545.01	0.792652
$489$	0	0
$490$	−21275.0	−1.96144
$491$	−12267.3	−1.12753	−0.563763	−	0.825936i	$-0.690647\pi$
$491$	−12267.3	−1.12753	−0.563763	+	0.825936i	$0.690647\pi$
$492$	0	0
$493$	11269.8	1.02954
$494$	0	0
$495$	0	0
$496$	9613.75	0.870303
$497$	10210.9	0.921573
$498$	0	0
$499$	3578.76	0.321056	0.160528	−	0.987031i	$-0.448680\pi$
$499$	3578.76	0.321056	0.160528	+	0.987031i	$0.448680\pi$
$500$	2785.12	0.249109
$501$	0	0
$502$	5178.19	0.460386
$503$	1296.82	0.114955	0.0574774	−	0.998347i	$-0.481694\pi$
$503$	1296.82	0.114955	0.0574774	+	0.998347i	$0.481694\pi$
$504$	0	0
$505$	−4474.99	−0.394325
$506$	−9973.72	−0.876257
$507$	0	0
$508$	339.092	0.0296157
$509$	4728.60	0.411771	0.205886	−	0.978576i	$-0.433993\pi$
$509$	4728.60	0.411771	0.205886	+	0.978576i	$0.433993\pi$
$510$	0	0
$511$	−5169.69	−0.447541
$512$	1422.78	0.122810
$513$	0	0
$514$	−8734.90	−0.749571
$515$	−4338.95	−0.371256
$516$	0	0
$517$	1686.43	0.143460
$518$	−19487.1	−1.65293
$519$	0	0
$520$	0	0
$521$	9220.74	0.775370	0.387685	−	0.921792i	$-0.373275\pi$
$521$	9220.74	0.775370	0.387685	+	0.921792i	$0.373275\pi$
$522$	0	0
$523$	−12102.7	−1.01188	−0.505939	−	0.862569i	$-0.668854\pi$
$523$	−12102.7	−1.01188	−0.505939	+	0.862569i	$0.668854\pi$
$524$	3352.65	0.279506
$525$	0	0
$526$	−7990.64	−0.662373
$527$	10290.4	0.850585
$528$	0	0
$529$	8030.91	0.660057
$530$	−27441.1	−2.24899
$531$	0	0
$532$	5665.52	0.461713
$533$	0	0
$534$	0	0
$535$	−11989.3	−0.968862
$536$	8340.56	0.672122
$537$	0	0
$538$	−8798.08	−0.705041
$539$	−7916.48	−0.632629
$540$	0	0
$541$	12801.3	1.01732	0.508659	−	0.860968i	$-0.330141\pi$
$541$	12801.3	1.01732	0.508659	+	0.860968i	$0.330141\pi$
$542$	−12647.5	−1.00231
$543$	0	0
$544$	−10036.5	−0.791015
$545$	−29781.2	−2.34071
$546$	0	0
$547$	400.693	0.0313207	0.0156603	−	0.999877i	$-0.495015\pi$
$547$	400.693	0.0313207	0.0156603	+	0.999877i	$0.495015\pi$
$548$	7508.84	0.585331
$549$	0	0
$550$	12821.7	0.994033
$551$	−10350.5	−0.800265
$552$	0	0
$553$	28024.2	2.15499
$554$	−3622.96	−0.277843
$555$	0	0
$556$	8565.35	0.653330
$557$	−14475.5	−1.10116	−0.550582	−	0.834781i	$-0.685594\pi$
$557$	−14475.5	−1.10116	−0.550582	+	0.834781i	$0.685594\pi$
$558$	0	0
$559$	0	0
$560$	−36738.5	−2.77229
$561$	0	0
$562$	−16337.8	−1.22628
$563$	14776.4	1.10613	0.553063	−	0.833139i	$-0.313459\pi$
$563$	14776.4	1.10613	0.553063	+	0.833139i	$0.313459\pi$
$564$	0	0
$565$	6678.17	0.497261
$566$	−8475.68	−0.629434
$567$	0	0
$568$	6581.33	0.486173
$569$	6868.88	0.506078	0.253039	−	0.967456i	$-0.418570\pi$
$569$	6868.88	0.506078	0.253039	+	0.967456i	$0.418570\pi$
$570$	0	0
$571$	3011.00	0.220677	0.110338	−	0.993894i	$-0.464807\pi$
$571$	3011.00	0.220677	0.110338	+	0.993894i	$0.464807\pi$
$572$	0	0
$573$	0	0
$574$	−17362.6	−1.26255
$575$	−25965.4	−1.88318
$576$	0	0
$577$	−23106.6	−1.66714	−0.833572	−	0.552411i	$-0.813708\pi$
$577$	−23106.6	−1.66714	−0.833572	+	0.552411i	$0.813708\pi$
$578$	−7006.45	−0.504204
$579$	0	0
$580$	−6478.70	−0.463816
$581$	23012.6	1.64324
$582$	0	0
$583$	−10210.9	−0.725374
$584$	−3332.07	−0.236099
$585$	0	0
$586$	303.463	0.0213924
$587$	−3024.81	−0.212687	−0.106343	−	0.994329i	$-0.533914\pi$
$587$	−3024.81	−0.212687	−0.106343	+	0.994329i	$0.533914\pi$
$588$	0	0
$589$	−9451.05	−0.661161
$590$	2469.13	0.172293
$591$	0	0
$592$	−17460.1	−1.21217
$593$	6396.07	0.442926	0.221463	−	0.975169i	$-0.428917\pi$
$593$	6396.07	0.442926	0.221463	+	0.975169i	$0.428917\pi$
$594$	0	0
$595$	−39324.4	−2.70948
$596$	3771.36	0.259196
$597$	0	0
$598$	0	0
$599$	−12095.9	−0.825084	−0.412542	−	0.910939i	$-0.635359\pi$
$599$	−12095.9	−0.825084	−0.412542	+	0.910939i	$0.635359\pi$
$600$	0	0
$601$	−11816.5	−0.802005	−0.401003	−	0.916077i	$-0.631338\pi$
$601$	−11816.5	−0.802005	−0.401003	+	0.916077i	$0.631338\pi$
$602$	−13539.8	−0.916682
$603$	0	0
$604$	7835.54	0.527853
$605$	−15312.4	−1.02899
$606$	0	0
$607$	−25164.0	−1.68266	−0.841329	−	0.540523i	$-0.818226\pi$
$607$	−25164.0	−1.68266	−0.841329	+	0.540523i	$0.818226\pi$
$608$	9217.85	0.614857
$609$	0	0
$610$	−28559.8	−1.89566
$611$	0	0
$612$	0	0
$613$	19583.0	1.29030	0.645148	−	0.764058i	$-0.276796\pi$
$613$	19583.0	1.29030	0.645148	+	0.764058i	$0.276796\pi$
$614$	−13048.5	−0.857648
$615$	0	0
$616$	−9834.10	−0.643226
$617$	−19677.1	−1.28390	−0.641952	−	0.766745i	$-0.721875\pi$
$617$	−19677.1	−1.28390	−0.641952	+	0.766745i	$0.721875\pi$
$618$	0	0
$619$	−4394.05	−0.285318	−0.142659	−	0.989772i	$-0.545565\pi$
$619$	−4394.05	−0.285318	−0.142659	+	0.989772i	$0.545565\pi$
$620$	−5915.71	−0.383194
$621$	0	0
$622$	11315.5	0.729435
$623$	25836.0	1.66147
$624$	0	0
$625$	−5082.96	−0.325310
$626$	20293.7	1.29568
$627$	0	0
$628$	9229.10	0.586435
$629$	−18689.1	−1.18471
$630$	0	0
$631$	−23887.7	−1.50706	−0.753529	−	0.657415i	$-0.771650\pi$
$631$	−23887.7	−1.50706	−0.753529	+	0.657415i	$0.771650\pi$
$632$	18062.7	1.13686
$633$	0	0
$634$	−17830.4	−1.11693
$635$	2161.65	0.135091
$636$	0	0
$637$	0	0
$638$	−9419.57	−0.584521
$639$	0	0
$640$	−30324.6	−1.87295
$641$	5443.62	0.335429	0.167714	−	0.985836i	$-0.446361\pi$
$641$	5443.62	0.335429	0.167714	+	0.985836i	$0.446361\pi$
$642$	0	0
$643$	5839.00	0.358115	0.179057	−	0.983839i	$-0.442695\pi$
$643$	5839.00	0.358115	0.179057	+	0.983839i	$0.442695\pi$
$644$	−10441.4	−0.638896
$645$	0	0
$646$	21230.5	1.29304
$647$	−8708.86	−0.529182	−0.264591	−	0.964361i	$-0.585237\pi$
$647$	−8708.86	−0.529182	−0.264591	+	0.964361i	$0.585237\pi$
$648$	0	0
$649$	918.773	0.0555701
$650$	0	0
$651$	0	0
$652$	6043.17	0.362989
$653$	−2794.93	−0.167495	−0.0837475	−	0.996487i	$-0.526689\pi$
$653$	−2794.93	−0.167495	−0.0837475	+	0.996487i	$0.526689\pi$
$654$	0	0
$655$	21372.6	1.27495
$656$	−15556.6	−0.925890
$657$	0	0
$658$	6898.41	0.408705
$659$	−31389.9	−1.85550	−0.927752	−	0.373197i	$-0.878262\pi$
$659$	−31389.9	−1.85550	−0.927752	+	0.373197i	$0.878262\pi$
$660$	0	0
$661$	−20597.2	−1.21201	−0.606005	−	0.795461i	$-0.707229\pi$
$661$	−20597.2	−1.21201	−0.606005	+	0.795461i	$0.707229\pi$
$662$	21728.6	1.27569
$663$	0	0
$664$	14832.5	0.866889
$665$	36116.7	2.10608
$666$	0	0
$667$	19075.7	1.10737
$668$	2511.00	0.145439
$669$	0	0
$670$	−27876.5	−1.60741
$671$	−10627.2	−0.611414
$672$	0	0
$673$	−17935.8	−1.02730	−0.513651	−	0.857999i	$-0.671707\pi$
$673$	−17935.8	−1.02730	−0.513651	+	0.857999i	$0.671707\pi$
$674$	18161.0	1.03789
$675$	0	0
$676$	0	0
$677$	24104.0	1.36838	0.684188	−	0.729305i	$-0.260157\pi$
$677$	24104.0	1.36838	0.684188	+	0.729305i	$0.260157\pi$
$678$	0	0
$679$	−15792.2	−0.892560
$680$	−25346.1	−1.42938
$681$	0	0
$682$	−8601.02	−0.482918
$683$	−6744.25	−0.377835	−0.188918	−	0.981993i	$-0.560498\pi$
$683$	−6744.25	−0.377835	−0.188918	+	0.981993i	$0.560498\pi$
$684$	0	0
$685$	47867.5	2.66996
$686$	−2353.69	−0.130998
$687$	0	0
$688$	−12131.5	−0.672249
$689$	0	0
$690$	0	0
$691$	30844.4	1.69809	0.849043	−	0.528323i	$-0.177179\pi$
$691$	30844.4	1.69809	0.849043	+	0.528323i	$0.177179\pi$
$692$	−2475.50	−0.135989
$693$	0	0
$694$	30818.9	1.68569
$695$	54602.6	2.98014
$696$	0	0
$697$	−16651.6	−0.904913
$698$	18878.1	1.02371
$699$	0	0
$700$	13422.9	0.724769
$701$	21007.6	1.13188	0.565940	−	0.824447i	$-0.308514\pi$
$701$	21007.6	1.13188	0.565940	+	0.824447i	$0.308514\pi$
$702$	0	0
$703$	17164.6	0.920877
$704$	−5042.04	−0.269928
$705$	0	0
$706$	−11338.1	−0.604413
$707$	−6811.41	−0.362333
$708$	0	0
$709$	14763.4	0.782017	0.391008	−	0.920387i	$-0.372126\pi$
$709$	14763.4	0.782017	0.391008	+	0.920387i	$0.372126\pi$
$710$	−21996.7	−1.16271
$711$	0	0
$712$	16652.3	0.876507
$713$	17418.0	0.914882
$714$	0	0
$715$	0	0
$716$	−862.037	−0.0449942
$717$	0	0
$718$	23506.8	1.22182
$719$	26186.3	1.35825	0.679126	−	0.734022i	$-0.262359\pi$
$719$	26186.3	1.35825	0.679126	+	0.734022i	$0.262359\pi$
$720$	0	0
$721$	−6604.34	−0.341135
$722$	2991.76	0.154213
$723$	0	0
$724$	−7553.49	−0.387739
$725$	−24522.7	−1.25621
$726$	0	0
$727$	20044.0	1.02254	0.511272	−	0.859419i	$-0.329175\pi$
$727$	20044.0	1.02254	0.511272	+	0.859419i	$0.329175\pi$
$728$	0	0
$729$	0	0
$730$	11136.7	0.564641
$731$	−12985.4	−0.657019
$732$	0	0
$733$	−25555.5	−1.28774	−0.643871	−	0.765134i	$-0.722673\pi$
$733$	−25555.5	−1.28774	−0.643871	+	0.765134i	$0.722673\pi$
$734$	−12821.1	−0.644737
$735$	0	0
$736$	−16988.2	−0.850809
$737$	−10372.9	−0.518443
$738$	0	0
$739$	−11108.5	−0.552953	−0.276477	−	0.961021i	$-0.589167\pi$
$739$	−11108.5	−0.552953	−0.276477	+	0.961021i	$0.589167\pi$
$740$	10743.9	0.533720
$741$	0	0
$742$	−41768.2	−2.06652
$743$	28188.2	1.39182	0.695911	−	0.718128i	$-0.255001\pi$
$743$	28188.2	1.39182	0.695911	+	0.718128i	$0.255001\pi$
$744$	0	0
$745$	24041.7	1.18231
$746$	−37306.8	−1.83096
$747$	0	0
$748$	4944.82	0.241712
$749$	−18248.9	−0.890256
$750$	0	0
$751$	17176.6	0.834596	0.417298	−	0.908770i	$-0.362977\pi$
$751$	17176.6	0.834596	0.417298	+	0.908770i	$0.362977\pi$
$752$	6180.86	0.299724
$753$	0	0
$754$	0	0
$755$	49950.2	2.40778
$756$	0	0
$757$	−4409.96	−0.211734	−0.105867	−	0.994380i	$-0.533762\pi$
$757$	−4409.96	−0.211734	−0.105867	+	0.994380i	$0.533762\pi$
$758$	−13221.5	−0.633546
$759$	0	0
$760$	23278.6	1.11106
$761$	−32443.6	−1.54544	−0.772721	−	0.634746i	$-0.781105\pi$
$761$	−32443.6	−1.54544	−0.772721	+	0.634746i	$0.781105\pi$
$762$	0	0
$763$	−45330.1	−2.15080
$764$	−247.287	−0.0117101
$765$	0	0
$766$	6528.30	0.307934
$767$	0	0
$768$	0	0
$769$	−31994.1	−1.50031	−0.750155	−	0.661263i	$-0.770021\pi$
$769$	−31994.1	−1.50031	−0.750155	+	0.661263i	$0.770021\pi$
$770$	32868.4	1.53830
$771$	0	0
$772$	−2336.04	−0.108907
$773$	8979.39	0.417809	0.208904	−	0.977936i	$-0.433010\pi$
$773$	8979.39	0.417809	0.208904	+	0.977936i	$0.433010\pi$
$774$	0	0
$775$	−22391.7	−1.03785
$776$	−10178.7	−0.470868
$777$	0	0
$778$	−36470.2	−1.68062
$779$	15293.3	0.703390
$780$	0	0
$781$	−8185.04	−0.375011
$782$	−39127.2	−1.78924
$783$	0	0
$784$	−29014.4	−1.32172
$785$	58833.9	2.67500
$786$	0	0
$787$	−12570.5	−0.569366	−0.284683	−	0.958622i	$-0.591888\pi$
$787$	−12570.5	−0.569366	−0.284683	+	0.958622i	$0.591888\pi$
$788$	11952.9	0.540360
$789$	0	0
$790$	−60370.6	−2.71885
$791$	10164.9	0.456917
$792$	0	0
$793$	0	0
$794$	35341.2	1.57961
$795$	0	0
$796$	−9158.67	−0.407815
$797$	−37863.3	−1.68280	−0.841398	−	0.540416i	$-0.818267\pi$
$797$	−37863.3	−1.68280	−0.841398	+	0.540416i	$0.818267\pi$
$798$	0	0
$799$	6615.91	0.292934
$800$	21839.2	0.965165
$801$	0	0
$802$	15449.2	0.680210
$803$	4144.01	0.182116
$804$	0	0
$805$	−66562.1	−2.91429
$806$	0	0
$807$	0	0
$808$	−4390.22	−0.191148
$809$	−2503.79	−0.108812	−0.0544058	−	0.998519i	$-0.517326\pi$
$809$	−2503.79	−0.108812	−0.0544058	+	0.998519i	$0.517326\pi$
$810$	0	0
$811$	−5409.55	−0.234223	−0.117112	−	0.993119i	$-0.537364\pi$
$811$	−5409.55	−0.234223	−0.117112	+	0.993119i	$0.537364\pi$
$812$	−9861.27	−0.426186
$813$	0	0
$814$	15620.8	0.672617
$815$	38524.2	1.65576
$816$	0	0
$817$	11926.1	0.510702
$818$	3884.85	0.166052
$819$	0	0
$820$	9572.58	0.407669
$821$	31381.6	1.33401	0.667007	−	0.745051i	$-0.267575\pi$
$821$	31381.6	1.33401	0.667007	+	0.745051i	$0.267575\pi$
$822$	0	0
$823$	−33046.1	−1.39965	−0.699827	−	0.714312i	$-0.746740\pi$
$823$	−33046.1	−1.39965	−0.699827	+	0.714312i	$0.746740\pi$
$824$	−4256.76	−0.179965
$825$	0	0
$826$	3758.29	0.158314
$827$	33653.7	1.41506	0.707529	−	0.706684i	$-0.249810\pi$
$827$	33653.7	1.41506	0.707529	+	0.706684i	$0.249810\pi$
$828$	0	0
$829$	12898.5	0.540390	0.270195	−	0.962806i	$-0.412912\pi$
$829$	12898.5	0.540390	0.270195	+	0.962806i	$0.412912\pi$
$830$	−49574.6	−2.07320
$831$	0	0
$832$	0	0
$833$	−31056.6	−1.29177
$834$	0	0
$835$	16007.2	0.663414
$836$	−4541.47	−0.187883
$837$	0	0
$838$	−20225.7	−0.833753
$839$	395.829	0.0162879	0.00814395	−	0.999967i	$-0.497408\pi$
$839$	395.829	0.0162879	0.00814395	+	0.999967i	$0.497408\pi$
$840$	0	0
$841$	−6373.17	−0.261313
$842$	−33867.9	−1.38618
$843$	0	0
$844$	−12657.4	−0.516214
$845$	0	0
$846$	0	0
$847$	−23307.1	−0.945505
$848$	−37423.6	−1.51548
$849$	0	0
$850$	50299.9	2.02973
$851$	−31634.0	−1.27426
$852$	0	0
$853$	−21248.9	−0.852930	−0.426465	−	0.904504i	$-0.640241\pi$
$853$	−21248.9	−0.852930	−0.426465	+	0.904504i	$0.640241\pi$
$854$	−43471.1	−1.74186
$855$	0	0
$856$	−11762.2	−0.469653
$857$	9920.37	0.395418	0.197709	−	0.980261i	$-0.436650\pi$
$857$	9920.37	0.395418	0.197709	+	0.980261i	$0.436650\pi$
$858$	0	0
$859$	20946.3	0.831990	0.415995	−	0.909367i	$-0.363433\pi$
$859$	20946.3	0.831990	0.415995	+	0.909367i	$0.363433\pi$
$860$	7464.95	0.295991
$861$	0	0
$862$	37093.2	1.46566
$863$	11271.4	0.444594	0.222297	−	0.974979i	$-0.428645\pi$
$863$	11271.4	0.444594	0.222297	+	0.974979i	$0.428645\pi$
$864$	0	0
$865$	−15780.9	−0.620308
$866$	34349.7	1.34786
$867$	0	0
$868$	−9004.34	−0.352105
$869$	−22464.1	−0.876920
$870$	0	0
$871$	0	0
$872$	−29217.0	−1.13465
$873$	0	0
$874$	35935.6	1.39078
$875$	27024.5	1.04411
$876$	0	0
$877$	−9436.36	−0.363333	−0.181667	−	0.983360i	$-0.558149\pi$
$877$	−9436.36	−0.363333	−0.181667	+	0.983360i	$0.558149\pi$
$878$	6691.13	0.257192
$879$	0	0
$880$	29449.5	1.12811
$881$	20343.0	0.777949	0.388974	−	0.921249i	$-0.372829\pi$
$881$	20343.0	0.777949	0.388974	+	0.921249i	$0.372829\pi$
$882$	0	0
$883$	46521.9	1.77303	0.886515	−	0.462699i	$-0.153119\pi$
$883$	46521.9	1.77303	0.886515	+	0.462699i	$0.153119\pi$
$884$	0	0
$885$	0	0
$886$	−13408.6	−0.508433
$887$	19955.1	0.755384	0.377692	−	0.925931i	$-0.376718\pi$
$887$	19955.1	0.755384	0.377692	+	0.925931i	$0.376718\pi$
$888$	0	0
$889$	3290.27	0.124130
$890$	−55656.8	−2.09620
$891$	0	0
$892$	6960.57	0.261275
$893$	−6076.25	−0.227698
$894$	0	0
$895$	−5495.33	−0.205239
$896$	−46157.3	−1.72099
$897$	0	0
$898$	49899.1	1.85429
$899$	16450.3	0.610286
$900$	0	0
$901$	−40057.7	−1.48115
$902$	13917.9	0.513762
$903$	0	0
$904$	6551.67	0.241046
$905$	−48152.2	−1.76866
$906$	0	0
$907$	−2653.95	−0.0971587	−0.0485793	−	0.998819i	$-0.515469\pi$
$907$	−2653.95	−0.0971587	−0.0485793	+	0.998819i	$0.515469\pi$
$908$	102.546	0.00374793
$909$	0	0
$910$	0	0
$911$	−1797.50	−0.0653720	−0.0326860	−	0.999466i	$-0.510406\pi$
$911$	−1797.50	−0.0653720	−0.0326860	+	0.999466i	$0.510406\pi$
$912$	0	0
$913$	−18446.9	−0.668677
$914$	2873.42	0.103987
$915$	0	0
$916$	11266.6	0.406395
$917$	32531.3	1.17151
$918$	0	0
$919$	−48642.0	−1.74597	−0.872987	−	0.487743i	$-0.837820\pi$
$919$	−48642.0	−1.74597	−0.872987	+	0.487743i	$0.837820\pi$
$920$	−42901.9	−1.53743
$921$	0	0
$922$	53425.6	1.90833
$923$	0	0
$924$	0	0
$925$	40666.9	1.44554
$926$	38379.7	1.36202
$927$	0	0
$928$	−16044.4	−0.567545
$929$	37745.3	1.33303	0.666513	−	0.745493i	$-0.267786\pi$
$929$	37745.3	1.33303	0.666513	+	0.745493i	$0.267786\pi$
$930$	0	0
$931$	28523.3	1.00410
$932$	−4034.07	−0.141782
$933$	0	0
$934$	−51204.6	−1.79386
$935$	31522.3	1.10256
$936$	0	0
$937$	2705.50	0.0943273	0.0471637	−	0.998887i	$-0.484982\pi$
$937$	2705.50	0.0943273	0.0471637	+	0.998887i	$0.484982\pi$
$938$	−42431.0	−1.47700
$939$	0	0
$940$	−3803.32	−0.131969
$941$	5189.27	0.179772	0.0898860	−	0.995952i	$-0.471350\pi$
$941$	5189.27	0.179772	0.0898860	+	0.995952i	$0.471350\pi$
$942$	0	0
$943$	−28185.2	−0.973316
$944$	3367.36	0.116100
$945$	0	0
$946$	10853.5	0.373021
$947$	−72.2711	−0.00247993	−0.00123997	−	0.999999i	$-0.500395\pi$
$947$	−72.2711	−0.00247993	−0.00123997	+	0.999999i	$0.500395\pi$
$948$	0	0
$949$	0	0
$950$	−46196.9	−1.57771
$951$	0	0
$952$	−38579.5	−1.31341
$953$	−44695.1	−1.51922	−0.759609	−	0.650380i	$-0.774610\pi$
$953$	−44695.1	−1.51922	−0.759609	+	0.650380i	$0.774610\pi$
$954$	0	0
$955$	−1576.41	−0.0534151
$956$	−15190.7	−0.513914
$957$	0	0
$958$	−34022.7	−1.14742
$959$	72859.5	2.45334
$960$	0	0
$961$	−14770.2	−0.495795
$962$	0	0
$963$	0	0
$964$	−7334.93	−0.245065
$965$	−14891.8	−0.496772
$966$	0	0
$967$	17936.9	0.596496	0.298248	−	0.954488i	$-0.403598\pi$
$967$	17936.9	0.596496	0.298248	+	0.954488i	$0.403598\pi$
$968$	−15022.4	−0.498799
$969$	0	0
$970$	34020.1	1.12610
$971$	40914.6	1.35223	0.676113	−	0.736798i	$-0.263663\pi$
$971$	40914.6	1.35223	0.676113	+	0.736798i	$0.263663\pi$
$972$	0	0
$973$	83111.0	2.73835
$974$	−13097.1	−0.430860
$975$	0	0
$976$	−38949.3	−1.27740
$977$	24118.1	0.789770	0.394885	−	0.918731i	$-0.370784\pi$
$977$	24118.1	0.789770	0.394885	+	0.918731i	$0.370784\pi$
$978$	0	0
$979$	−20710.1	−0.676096
$980$	17853.6	0.581953
$981$	0	0
$982$	40224.2	1.30713
$983$	−2928.61	−0.0950235	−0.0475118	−	0.998871i	$-0.515129\pi$
$983$	−2928.61	−0.0950235	−0.0475118	+	0.998871i	$0.515129\pi$
$984$	0	0
$985$	76197.5	2.46483
$986$	−36953.3	−1.19354
$987$	0	0
$988$	0	0
$989$	−21979.6	−0.706683
$990$	0	0
$991$	49809.9	1.59663	0.798317	−	0.602238i	$-0.205724\pi$
$991$	49809.9	1.59663	0.798317	+	0.602238i	$0.205724\pi$
$992$	−14650.1	−0.468893
$993$	0	0
$994$	−33481.3	−1.06837
$995$	−58384.9	−1.86023
$996$	0	0
$997$	−41151.7	−1.30721	−0.653604	−	0.756837i	$-0.726744\pi$
$997$	−41151.7	−1.30721	−0.653604	+	0.756837i	$0.726744\pi$
$998$	−11734.7	−0.372198
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bk.1.3		10
3.2	odd	2		507.4.a.r.1.8		10
13.2	odd	12		117.4.q.e.82.2		10
13.7	odd	12		117.4.q.e.10.2		10
13.12	even	2	inner	1521.4.a.bk.1.8		10
39.2	even	12		39.4.j.c.4.4	✓	10
39.5	even	4		507.4.b.i.337.3		10
39.8	even	4		507.4.b.i.337.8		10
39.20	even	12		39.4.j.c.10.4	yes	10
39.38	odd	2		507.4.a.r.1.3		10
156.59	odd	12		624.4.bv.h.49.1		10
156.119	odd	12		624.4.bv.h.433.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.j.c.4.4	✓	10	39.2	even	12
39.4.j.c.10.4	yes	10	39.20	even	12
117.4.q.e.10.2		10	13.7	odd	12
117.4.q.e.82.2		10	13.2	odd	12
507.4.a.r.1.3		10	39.38	odd	2
507.4.a.r.1.8		10	3.2	odd	2
507.4.b.i.337.3		10	39.5	even	4
507.4.b.i.337.8		10	39.8	even	4
624.4.bv.h.49.1		10	156.59	odd	12
624.4.bv.h.433.5		10	156.119	odd	12
1521.4.a.bk.1.3		10	1.1	even	1	trivial
1521.4.a.bk.1.8		10	13.12	even	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q-3.27897 q^{2} +2.75167 q^{4} +17.5414 q^{5} +26.6999 q^{7} +17.2091 q^{8} +O(q^{10})\)	\(q-3.27897 q^{2} +2.75167 q^{4} +17.5414 q^{5} +26.6999 q^{7} +17.2091 q^{8} -57.5178 q^{10} -21.4026 q^{11} -87.5483 q^{14} -78.4417 q^{16} -83.9630 q^{17} +77.1142 q^{19} +48.2682 q^{20} +70.1785 q^{22} -142.119 q^{23} +182.701 q^{25} +73.4693 q^{28} -134.223 q^{29} -122.559 q^{31} +119.535 q^{32} +275.312 q^{34} +468.354 q^{35} +222.587 q^{37} -252.855 q^{38} +301.873 q^{40} +198.321 q^{41} +154.656 q^{43} -58.8928 q^{44} +466.006 q^{46} -78.7956 q^{47} +369.884 q^{49} -599.072 q^{50} +477.088 q^{53} -375.431 q^{55} +459.482 q^{56} +440.114 q^{58} -42.9282 q^{59} +496.539 q^{61} +401.869 q^{62} +235.581 q^{64} +484.659 q^{67} -231.038 q^{68} -1535.72 q^{70} +382.432 q^{71} -193.622 q^{73} -729.858 q^{74} +212.193 q^{76} -571.447 q^{77} +1049.60 q^{79} -1375.98 q^{80} -650.289 q^{82} +861.900 q^{83} -1472.83 q^{85} -507.112 q^{86} -368.320 q^{88} +967.645 q^{89} -391.065 q^{92} +258.369 q^{94} +1352.69 q^{95} -591.470 q^{97} -1212.84 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 60 q^{4}+O(q^{10}) \) 10 * q + 60 * q^4	\( 10 q + 60 q^{4} + 80 q^{10} + 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} + 960 q^{25} - 990 q^{29} + 120 q^{35} - 1380 q^{38} + 2000 q^{40} - 740 q^{43} + 1550 q^{49} - 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} + 3140 q^{64} - 1200 q^{68} + 4380 q^{74} - 4320 q^{77} + 1100 q^{79} - 4780 q^{82} + 6340 q^{88} + 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100}) \) 10 * q + 60 * q^4 + 80 * q^10 + 60 * q^14 + 500 * q^16 - 210 * q^17 + 580 * q^22 + 120 * q^23 + 960 * q^25 - 990 * q^29 + 120 * q^35 - 1380 * q^38 + 2000 * q^40 - 740 * q^43 + 1550 * q^49 - 330 * q^53 + 520 * q^55 + 5340 * q^56 + 2750 * q^61 + 1560 * q^62 + 3140 * q^64 - 1200 * q^68 + 4380 * q^74 - 4320 * q^77 + 1100 * q^79 - 4780 * q^82 + 6340 * q^88 + 1740 * q^92 + 6460 * q^94 + 2760 * q^95

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