LMFDB - Embedded newform 1521.4.a.bm.1.9

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:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 100 x^{16} + 4066 x^{14} - 86197 x^{12} + 1016064 x^{10} - 6594119 x^{8} + 22251817 x^{6} + \cdots - 6889792 $ x^18 - 100x^16 + 4066x^14 - 86197x^12 + 1016064x^10 - 6594119x^8 + 22251817x^6 - 37096332x^4 + 27824160x^2 - 6889792
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 7\cdot 13^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-0.685025$ 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)

$f(q)$	$=$	$q-0.685025 q^{2} -7.53074 q^{4} -9.28431 q^{5} -6.04730 q^{7} +10.6389 q^{8} +O(q^{10})$	$q-0.685025 q^{2} -7.53074 q^{4} -9.28431 q^{5} -6.04730 q^{7} +10.6389 q^{8} +6.35999 q^{10} -59.2146 q^{11} +4.14255 q^{14} +52.9580 q^{16} +117.407 q^{17} -49.2492 q^{19} +69.9177 q^{20} +40.5635 q^{22} -23.1522 q^{23} -38.8015 q^{25} +45.5406 q^{28} +145.028 q^{29} +94.9826 q^{31} -121.389 q^{32} -80.4268 q^{34} +56.1450 q^{35} +379.757 q^{37} +33.7370 q^{38} -98.7753 q^{40} +268.749 q^{41} -23.6647 q^{43} +445.930 q^{44} +15.8599 q^{46} -301.164 q^{47} -306.430 q^{49} +26.5800 q^{50} +391.713 q^{53} +549.767 q^{55} -64.3369 q^{56} -99.3480 q^{58} -510.800 q^{59} +520.734 q^{61} -65.0655 q^{62} -340.509 q^{64} +470.377 q^{67} -884.162 q^{68} -38.4607 q^{70} +466.583 q^{71} -314.174 q^{73} -260.143 q^{74} +370.883 q^{76} +358.088 q^{77} +47.9136 q^{79} -491.678 q^{80} -184.100 q^{82} -310.030 q^{83} -1090.04 q^{85} +16.2109 q^{86} -629.981 q^{88} -216.244 q^{89} +174.353 q^{92} +206.305 q^{94} +457.245 q^{95} -219.507 q^{97} +209.912 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 56 q^{4} - 94 q^{7}+O(q^{10}) $ 18 * q + 56 * q^4 - 94 * q^7	$ 18 q + 56 q^{4} - 94 q^{7} + 156 q^{10} + 88 q^{16} - 448 q^{19} - 256 q^{22} - 16 q^{25} - 1300 q^{28} - 818 q^{31} - 900 q^{34} - 524 q^{37} + 800 q^{40} + 752 q^{43} - 1650 q^{46} + 2948 q^{49} - 464 q^{55} - 1626 q^{58} - 1784 q^{61} - 4274 q^{64} - 2872 q^{67} - 5772 q^{70} - 3078 q^{73} - 5726 q^{76} + 3326 q^{79} - 1038 q^{82} - 2340 q^{85} + 780 q^{88} + 1220 q^{94} - 4630 q^{97}+O(q^{100}) $ 18 * q + 56 * q^4 - 94 * q^7 + 156 * q^10 + 88 * q^16 - 448 * q^19 - 256 * q^22 - 16 * q^25 - 1300 * q^28 - 818 * q^31 - 900 * q^34 - 524 * q^37 + 800 * q^40 + 752 * q^43 - 1650 * q^46 + 2948 * q^49 - 464 * q^55 - 1626 * q^58 - 1784 * q^61 - 4274 * q^64 - 2872 * q^67 - 5772 * q^70 - 3078 * q^73 - 5726 * q^76 + 3326 * q^79 - 1038 * q^82 - 2340 * q^85 + 780 * q^88 + 1220 * q^94 - 4630 * 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 $

$2$	−0.685025	−0.242193	−0.121096	−	0.992641i	$-0.538641\pi$
$2$	−0.685025	−0.242193	−0.121096	+	0.992641i	$0.538641\pi$
$3$	0	0
$3$	0	0
$4$	−7.53074	−0.941343
$5$	−9.28431	−0.830414	−0.415207	−	0.909727i	$-0.636291\pi$
$5$	−9.28431	−0.830414	−0.415207	+	0.909727i	$0.636291\pi$
$6$	0	0
$7$	−6.04730	−0.326523	−0.163262	−	0.986583i	$-0.552201\pi$
$7$	−6.04730	−0.326523	−0.163262	+	0.986583i	$0.552201\pi$
$8$	10.6389	0.470180
$9$	0	0
$10$	6.35999	0.201120
$11$	−59.2146	−1.62308	−0.811539	−	0.584298i	$-0.801370\pi$
$11$	−59.2146	−1.62308	−0.811539	+	0.584298i	$0.801370\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	4.14255	0.0790817
$15$	0	0
$16$	52.9580	0.827468
$17$	117.407	1.67502	0.837512	−	0.546420i	$-0.184010\pi$
$17$	117.407	1.67502	0.837512	+	0.546420i	$0.184010\pi$
$18$	0	0
$19$	−49.2492	−0.594660	−0.297330	−	0.954775i	$-0.596096\pi$
$19$	−49.2492	−0.594660	−0.297330	+	0.954775i	$0.596096\pi$
$20$	69.9177	0.781704
$21$	0	0
$22$	40.5635	0.393098
$23$	−23.1522	−0.209894	−0.104947	−	0.994478i	$-0.533467\pi$
$23$	−23.1522	−0.209894	−0.104947	+	0.994478i	$0.533467\pi$
$24$	0	0
$25$	−38.8015	−0.310412
$26$	0	0
$27$	0	0
$28$	45.5406	0.307370
$29$	145.028	0.928658	0.464329	−	0.885663i	$-0.346296\pi$
$29$	145.028	0.928658	0.464329	+	0.885663i	$0.346296\pi$
$30$	0	0
$31$	94.9826	0.550303	0.275151	−	0.961401i	$-0.411272\pi$
$31$	94.9826	0.550303	0.275151	+	0.961401i	$0.411272\pi$
$32$	−121.389	−0.670587
$33$	0	0
$34$	−80.4268	−0.405679
$35$	56.1450	0.271150
$36$	0	0
$37$	379.757	1.68734	0.843672	−	0.536860i	$-0.180390\pi$
$37$	379.757	1.68734	0.843672	+	0.536860i	$0.180390\pi$
$38$	33.7370	0.144023
$39$	0	0
$40$	−98.7753	−0.390444
$41$	268.749	1.02370	0.511848	−	0.859076i	$-0.328961\pi$
$41$	268.749	1.02370	0.511848	+	0.859076i	$0.328961\pi$
$42$	0	0
$43$	−23.6647	−0.0839264	−0.0419632	−	0.999119i	$-0.513361\pi$
$43$	−23.6647	−0.0839264	−0.0419632	+	0.999119i	$0.513361\pi$
$44$	445.930	1.52787
$45$	0	0
$46$	15.8599	0.0508350
$47$	−301.164	−0.934666	−0.467333	−	0.884081i	$-0.654785\pi$
$47$	−301.164	−0.934666	−0.467333	+	0.884081i	$0.654785\pi$
$48$	0	0
$49$	−306.430	−0.893383
$50$	26.5800	0.0751797
$51$	0	0
$52$	0	0
$53$	391.713	1.01521	0.507603	−	0.861591i	$-0.330532\pi$
$53$	391.713	1.01521	0.507603	+	0.861591i	$0.330532\pi$
$54$	0	0
$55$	549.767	1.34783
$56$	−64.3369	−0.153525
$57$	0	0
$58$	−99.3480	−0.224914
$59$	−510.800	−1.12713	−0.563563	−	0.826073i	$-0.690570\pi$
$59$	−510.800	−1.12713	−0.563563	+	0.826073i	$0.690570\pi$
$60$	0	0
$61$	520.734	1.09300	0.546501	−	0.837458i	$-0.315959\pi$
$61$	520.734	1.09300	0.546501	+	0.837458i	$0.315959\pi$
$62$	−65.0655	−0.133279
$63$	0	0
$64$	−340.509	−0.665057
$65$	0	0
$66$	0	0
$67$	470.377	0.857697	0.428849	−	0.903376i	$-0.358919\pi$
$67$	470.377	0.857697	0.428849	+	0.903376i	$0.358919\pi$
$68$	−884.162	−1.57677
$69$	0	0
$70$	−38.4607	−0.0656705
$71$	466.583	0.779905	0.389953	−	0.920835i	$-0.372491\pi$
$71$	466.583	0.779905	0.389953	+	0.920835i	$0.372491\pi$
$72$	0	0
$73$	−314.174	−0.503716	−0.251858	−	0.967764i	$-0.581042\pi$
$73$	−314.174	−0.503716	−0.251858	+	0.967764i	$0.581042\pi$
$74$	−260.143	−0.408663
$75$	0	0
$76$	370.883	0.559779
$77$	358.088	0.529973
$78$	0	0
$79$	47.9136	0.0682368	0.0341184	−	0.999418i	$-0.489138\pi$
$79$	47.9136	0.0682368	0.0341184	+	0.999418i	$0.489138\pi$
$80$	−491.678	−0.687141
$81$	0	0
$82$	−184.100	−0.247932
$83$	−310.030	−0.410003	−0.205001	−	0.978762i	$-0.565720\pi$
$83$	−310.030	−0.410003	−0.205001	+	0.978762i	$0.565720\pi$
$84$	0	0
$85$	−1090.04	−1.39096
$86$	16.2109	0.0203264
$87$	0	0
$88$	−629.981	−0.763138
$89$	−216.244	−0.257549	−0.128774	−	0.991674i	$-0.541104\pi$
$89$	−216.244	−0.257549	−0.128774	+	0.991674i	$0.541104\pi$
$90$	0	0
$91$	0	0
$92$	174.353	0.197583
$93$	0	0
$94$	206.305	0.226369
$95$	457.245	0.493814
$96$	0	0
$97$	−219.507	−0.229768	−0.114884	−	0.993379i	$-0.536650\pi$
$97$	−219.507	−0.229768	−0.114884	+	0.993379i	$0.536650\pi$
$98$	209.912	0.216371
$99$	0	0
$100$	292.204	0.292204
$101$	921.359	0.907710	0.453855	−	0.891076i	$-0.350048\pi$
$101$	921.359	0.907710	0.453855	+	0.891076i	$0.350048\pi$
$102$	0	0
$103$	−1277.55	−1.22214	−0.611072	−	0.791575i	$-0.709262\pi$
$103$	−1277.55	−1.22214	−0.611072	+	0.791575i	$0.709262\pi$
$104$	0	0
$105$	0	0
$106$	−268.333	−0.245876
$107$	1626.78	1.46978	0.734892	−	0.678184i	$-0.237233\pi$
$107$	1626.78	1.46978	0.734892	+	0.678184i	$0.237233\pi$
$108$	0	0
$109$	−1626.49	−1.42926	−0.714629	−	0.699504i	$-0.753404\pi$
$109$	−1626.49	−1.42926	−0.714629	+	0.699504i	$0.753404\pi$
$110$	−376.604	−0.326434
$111$	0	0
$112$	−320.253	−0.270188
$113$	2003.32	1.66776	0.833878	−	0.551949i	$-0.186116\pi$
$113$	2003.32	1.66776	0.833878	+	0.551949i	$0.186116\pi$
$114$	0	0
$115$	214.952	0.174299
$116$	−1092.17	−0.874185
$117$	0	0
$118$	349.911	0.272982
$119$	−709.995	−0.546934
$120$	0	0
$121$	2175.36	1.63438
$122$	−356.716	−0.264717
$123$	0	0
$124$	−715.290	−0.518023
$125$	1520.78	1.08818
$126$	0	0
$127$	−1966.75	−1.37418	−0.687091	−	0.726571i	$-0.741113\pi$
$127$	−1966.75	−1.37418	−0.687091	+	0.726571i	$0.741113\pi$
$128$	1204.37	0.831659
$129$	0	0
$130$	0	0
$131$	−2863.90	−1.91008	−0.955039	−	0.296480i	$-0.904187\pi$
$131$	−2863.90	−1.91008	−0.955039	+	0.296480i	$0.904187\pi$
$132$	0	0
$133$	297.825	0.194171
$134$	−322.220	−0.207728
$135$	0	0
$136$	1249.09	0.787562
$137$	1774.52	1.10663	0.553313	−	0.832974i	$-0.313363\pi$
$137$	1774.52	1.10663	0.553313	+	0.832974i	$0.313363\pi$
$138$	0	0
$139$	−1349.35	−0.823382	−0.411691	−	0.911324i	$-0.635062\pi$
$139$	−1349.35	−0.823382	−0.411691	+	0.911324i	$0.635062\pi$
$140$	−422.813	−0.255245
$141$	0	0
$142$	−319.621	−0.188888
$143$	0	0
$144$	0	0
$145$	−1346.49	−0.771171
$146$	215.217	0.121996
$147$	0	0
$148$	−2859.85	−1.58837
$149$	1267.51	0.696903	0.348452	−	0.937327i	$-0.386708\pi$
$149$	1267.51	0.696903	0.348452	+	0.937327i	$0.386708\pi$
$150$	0	0
$151$	−2928.66	−1.57835	−0.789176	−	0.614168i	$-0.789492\pi$
$151$	−2928.66	−1.57835	−0.789176	+	0.614168i	$0.789492\pi$
$152$	−523.960	−0.279597
$153$	0	0
$154$	−245.299	−0.128356
$155$	−881.849	−0.456979
$156$	0	0
$157$	2520.07	1.28104	0.640521	−	0.767940i	$-0.278718\pi$
$157$	2520.07	1.28104	0.640521	+	0.767940i	$0.278718\pi$
$158$	−32.8220	−0.0165265
$159$	0	0
$160$	1127.01	0.556865
$161$	140.008	0.0685354
$162$	0	0
$163$	−3596.35	−1.72815	−0.864073	−	0.503366i	$-0.832095\pi$
$163$	−3596.35	−1.72815	−0.864073	+	0.503366i	$0.832095\pi$
$164$	−2023.88	−0.963649
$165$	0	0
$166$	212.378	0.0992998
$167$	−1988.55	−0.921430	−0.460715	−	0.887548i	$-0.652407\pi$
$167$	−1988.55	−0.921430	−0.460715	+	0.887548i	$0.652407\pi$
$168$	0	0
$169$	0	0
$170$	746.707	0.336881
$171$	0	0
$172$	178.213	0.0790035
$173$	−584.054	−0.256675	−0.128338	−	0.991731i	$-0.540964\pi$
$173$	−584.054	−0.256675	−0.128338	+	0.991731i	$0.540964\pi$
$174$	0	0
$175$	234.644	0.101357
$176$	−3135.88	−1.34305
$177$	0	0
$178$	148.133	0.0623765
$179$	1253.49	0.523409	0.261704	−	0.965148i	$-0.415715\pi$
$179$	1253.49	0.523409	0.261704	+	0.965148i	$0.415715\pi$
$180$	0	0
$181$	−3426.69	−1.40720	−0.703601	−	0.710595i	$-0.748426\pi$
$181$	−3426.69	−1.40720	−0.703601	+	0.710595i	$0.748426\pi$
$182$	0	0
$183$	0	0
$184$	−246.315	−0.0986881
$185$	−3525.78	−1.40119
$186$	0	0
$187$	−6952.21	−2.71869
$188$	2267.99	0.879840
$189$	0	0
$190$	−313.225	−0.119598
$191$	−3100.81	−1.17470	−0.587348	−	0.809335i	$-0.699828\pi$
$191$	−3100.81	−1.17470	−0.587348	+	0.809335i	$0.699828\pi$
$192$	0	0
$193$	−338.564	−0.126271	−0.0631357	−	0.998005i	$-0.520110\pi$
$193$	−338.564	−0.126271	−0.0631357	+	0.998005i	$0.520110\pi$
$194$	150.368	0.0556482
$195$	0	0
$196$	2307.65	0.840979
$197$	−1317.22	−0.476387	−0.238193	−	0.971218i	$-0.576555\pi$
$197$	−1317.22	−0.476387	−0.238193	+	0.971218i	$0.576555\pi$
$198$	0	0
$199$	3454.89	1.23071	0.615354	−	0.788251i	$-0.289013\pi$
$199$	3454.89	1.23071	0.615354	+	0.788251i	$0.289013\pi$
$200$	−412.808	−0.145950
$201$	0	0
$202$	−631.154	−0.219841
$203$	−877.029	−0.303228
$204$	0	0
$205$	−2495.15	−0.850092
$206$	875.155	0.295995
$207$	0	0
$208$	0	0
$209$	2916.27	0.965181
$210$	0	0
$211$	2391.57	0.780295	0.390148	−	0.920752i	$-0.372424\pi$
$211$	2391.57	0.780295	0.390148	+	0.920752i	$0.372424\pi$
$212$	−2949.89	−0.955657
$213$	0	0
$214$	−1114.39	−0.355971
$215$	219.711	0.0696937
$216$	0	0
$217$	−574.388	−0.179687
$218$	1114.18	0.346156
$219$	0	0
$220$	−4140.15	−1.26877
$221$	0	0
$222$	0	0
$223$	−1389.30	−0.417194	−0.208597	−	0.978002i	$-0.566890\pi$
$223$	−1389.30	−0.417194	−0.208597	+	0.978002i	$0.566890\pi$
$224$	734.076	0.218962
$225$	0	0
$226$	−1372.32	−0.403919
$227$	−4465.41	−1.30564	−0.652818	−	0.757514i	$-0.726414\pi$
$227$	−4465.41	−1.30564	−0.652818	+	0.757514i	$0.726414\pi$
$228$	0	0
$229$	−3605.58	−1.04045	−0.520226	−	0.854029i	$-0.674152\pi$
$229$	−3605.58	−1.04045	−0.520226	+	0.854029i	$0.674152\pi$
$230$	−147.248	−0.0422141
$231$	0	0
$232$	1542.95	0.436636
$233$	559.829	0.157406	0.0787031	−	0.996898i	$-0.474922\pi$
$233$	559.829	0.157406	0.0787031	+	0.996898i	$0.474922\pi$
$234$	0	0
$235$	2796.10	0.776159
$236$	3846.70	1.06101
$237$	0	0
$238$	486.365	0.132464
$239$	3411.14	0.923215	0.461607	−	0.887084i	$-0.347273\pi$
$239$	3411.14	0.923215	0.461607	+	0.887084i	$0.347273\pi$
$240$	0	0
$241$	−6188.58	−1.65411	−0.827057	−	0.562119i	$-0.809986\pi$
$241$	−6188.58	−1.65411	−0.827057	+	0.562119i	$0.809986\pi$
$242$	−1490.18	−0.395836
$243$	0	0
$244$	−3921.51	−1.02889
$245$	2844.99	0.741877
$246$	0	0
$247$	0	0
$248$	1010.52	0.258741
$249$	0	0
$250$	−1041.78	−0.263551
$251$	6493.27	1.63288	0.816438	−	0.577434i	$-0.195946\pi$
$251$	6493.27	1.63288	0.816438	+	0.577434i	$0.195946\pi$
$252$	0	0
$253$	1370.95	0.340675
$254$	1347.28	0.332817
$255$	0	0
$256$	1899.05	0.463635
$257$	−7490.26	−1.81801	−0.909007	−	0.416780i	$-0.863159\pi$
$257$	−7490.26	−1.81801	−0.909007	+	0.416780i	$0.863159\pi$
$258$	0	0
$259$	−2296.50	−0.550957
$260$	0	0
$261$	0	0
$262$	1961.84	0.462608
$263$	4040.55	0.947342	0.473671	−	0.880702i	$-0.342929\pi$
$263$	4040.55	0.947342	0.473671	+	0.880702i	$0.342929\pi$
$264$	0	0
$265$	−3636.79	−0.843041
$266$	−204.017	−0.0470267
$267$	0	0
$268$	−3542.29	−0.807387
$269$	7720.88	1.75000	0.875001	−	0.484121i	$-0.160860\pi$
$269$	7720.88	1.75000	0.875001	+	0.484121i	$0.160860\pi$
$270$	0	0
$271$	6584.34	1.47590	0.737952	−	0.674853i	$-0.235793\pi$
$271$	6584.34	1.47590	0.737952	+	0.674853i	$0.235793\pi$
$272$	6217.64	1.38603
$273$	0	0
$274$	−1215.59	−0.268017
$275$	2297.62	0.503824
$276$	0	0
$277$	3790.56	0.822213	0.411106	−	0.911587i	$-0.365142\pi$
$277$	3790.56	0.822213	0.411106	+	0.911587i	$0.365142\pi$
$278$	924.336	0.199417
$279$	0	0
$280$	597.324	0.127489
$281$	6408.89	1.36058	0.680288	−	0.732945i	$-0.261855\pi$
$281$	6408.89	1.36058	0.680288	+	0.732945i	$0.261855\pi$
$282$	0	0
$283$	−7328.79	−1.53940	−0.769702	−	0.638403i	$-0.779595\pi$
$283$	−7328.79	−1.53940	−0.769702	+	0.638403i	$0.779595\pi$
$284$	−3513.72	−0.734158
$285$	0	0
$286$	0	0
$287$	−1625.20	−0.334261
$288$	0	0
$289$	8871.42	1.80570
$290$	922.378	0.186772
$291$	0	0
$292$	2365.96	0.474169
$293$	527.824	0.105242	0.0526209	−	0.998615i	$-0.483243\pi$
$293$	527.824	0.105242	0.0526209	+	0.998615i	$0.483243\pi$
$294$	0	0
$295$	4742.42	0.935981
$296$	4040.22	0.793354
$297$	0	0
$298$	−868.277	−0.168785
$299$	0	0
$300$	0	0
$301$	143.108	0.0274039
$302$	2006.21	0.382266
$303$	0	0
$304$	−2608.14	−0.492063
$305$	−4834.66	−0.907645
$306$	0	0
$307$	−827.031	−0.153750	−0.0768749	−	0.997041i	$-0.524494\pi$
$307$	−827.031	−0.153750	−0.0768749	+	0.997041i	$0.524494\pi$
$308$	−2696.67	−0.498886
$309$	0	0
$310$	604.088	0.110677
$311$	−2022.53	−0.368769	−0.184384	−	0.982854i	$-0.559029\pi$
$311$	−2022.53	−0.368769	−0.184384	+	0.982854i	$0.559029\pi$
$312$	0	0
$313$	−6839.50	−1.23512	−0.617559	−	0.786525i	$-0.711878\pi$
$313$	−6839.50	−1.23512	−0.617559	+	0.786525i	$0.711878\pi$
$314$	−1726.31	−0.310260
$315$	0	0
$316$	−360.825	−0.0642342
$317$	−7535.31	−1.33510	−0.667548	−	0.744567i	$-0.732656\pi$
$317$	−7535.31	−1.33510	−0.667548	+	0.744567i	$0.732656\pi$
$318$	0	0
$319$	−8587.79	−1.50728
$320$	3161.39	0.552273
$321$	0	0
$322$	−95.9092	−0.0165988
$323$	−5782.21	−0.996070
$324$	0	0
$325$	0	0
$326$	2463.59	0.418545
$327$	0	0
$328$	2859.21	0.481321
$329$	1821.23	0.305190
$330$	0	0
$331$	−7616.11	−1.26471	−0.632355	−	0.774678i	$-0.717912\pi$
$331$	−7616.11	−1.26471	−0.632355	+	0.774678i	$0.717912\pi$
$332$	2334.76	0.385953
$333$	0	0
$334$	1362.21	0.223164
$335$	−4367.13	−0.712244
$336$	0	0
$337$	−5959.66	−0.963334	−0.481667	−	0.876354i	$-0.659968\pi$
$337$	−5959.66	−0.963334	−0.481667	+	0.876354i	$0.659968\pi$
$338$	0	0
$339$	0	0
$340$	8208.84	1.30937
$341$	−5624.36	−0.893185
$342$	0	0
$343$	3927.30	0.618234
$344$	−251.768	−0.0394605
$345$	0	0
$346$	400.092	0.0621649
$347$	−9976.27	−1.54338	−0.771692	−	0.635997i	$-0.780589\pi$
$347$	−9976.27	−1.54338	−0.771692	+	0.635997i	$0.780589\pi$
$348$	0	0
$349$	5660.84	0.868246	0.434123	−	0.900854i	$-0.357058\pi$
$349$	5660.84	0.868246	0.434123	+	0.900854i	$0.357058\pi$
$350$	−160.737	−0.0245479
$351$	0	0
$352$	7188.01	1.08841
$353$	−682.431	−0.102896	−0.0514478	−	0.998676i	$-0.516384\pi$
$353$	−682.431	−0.102896	−0.0514478	+	0.998676i	$0.516384\pi$
$354$	0	0
$355$	−4331.91	−0.647644
$356$	1628.48	0.242442
$357$	0	0
$358$	−858.672	−0.126766
$359$	−8328.79	−1.22445	−0.612224	−	0.790685i	$-0.709725\pi$
$359$	−8328.79	−1.22445	−0.612224	+	0.790685i	$0.709725\pi$
$360$	0	0
$361$	−4433.51	−0.646379
$362$	2347.37	0.340815
$363$	0	0
$364$	0	0
$365$	2916.89	0.418293
$366$	0	0
$367$	559.900	0.0796364	0.0398182	−	0.999207i	$-0.487322\pi$
$367$	559.900	0.0796364	0.0398182	+	0.999207i	$0.487322\pi$
$368$	−1226.09	−0.173681
$369$	0	0
$370$	2415.25	0.339359
$371$	−2368.80	−0.331488
$372$	0	0
$373$	548.079	0.0760816	0.0380408	−	0.999276i	$-0.487888\pi$
$373$	548.079	0.0760816	0.0380408	+	0.999276i	$0.487888\pi$
$374$	4762.44	0.658449
$375$	0	0
$376$	−3204.07	−0.439461
$377$	0	0
$378$	0	0
$379$	−4680.27	−0.634325	−0.317163	−	0.948371i	$-0.602730\pi$
$379$	−4680.27	−0.634325	−0.317163	+	0.948371i	$0.602730\pi$
$380$	−3443.40	−0.464849
$381$	0	0
$382$	2124.13	0.284503
$383$	9174.13	1.22396	0.611979	−	0.790874i	$-0.290374\pi$
$383$	9174.13	1.22396	0.611979	+	0.790874i	$0.290374\pi$
$384$	0	0
$385$	−3324.60	−0.440097
$386$	231.925	0.0305821
$387$	0	0
$388$	1653.05	0.216291
$389$	−346.645	−0.0451816	−0.0225908	−	0.999745i	$-0.507191\pi$
$389$	−346.645	−0.0451816	−0.0225908	+	0.999745i	$0.507191\pi$
$390$	0	0
$391$	−2718.23	−0.351578
$392$	−3260.10	−0.420050
$393$	0	0
$394$	902.331	0.115378
$395$	−444.845	−0.0566648
$396$	0	0
$397$	−513.327	−0.0648946	−0.0324473	−	0.999473i	$-0.510330\pi$
$397$	−513.327	−0.0648946	−0.0324473	+	0.999473i	$0.510330\pi$
$398$	−2366.69	−0.298069
$399$	0	0
$400$	−2054.85	−0.256856
$401$	11168.4	1.39083	0.695413	−	0.718610i	$-0.255221\pi$
$401$	11168.4	1.39083	0.695413	+	0.718610i	$0.255221\pi$
$402$	0	0
$403$	0	0
$404$	−6938.52	−0.854466
$405$	0	0
$406$	600.787	0.0734398
$407$	−22487.2	−2.73869
$408$	0	0
$409$	−4286.40	−0.518212	−0.259106	−	0.965849i	$-0.583428\pi$
$409$	−4286.40	−0.518212	−0.259106	+	0.965849i	$0.583428\pi$
$410$	1709.24	0.205886
$411$	0	0
$412$	9620.91	1.15046
$413$	3088.96	0.368033
$414$	0	0
$415$	2878.42	0.340472
$416$	0	0
$417$	0	0
$418$	−1997.72	−0.233760
$419$	3822.03	0.445629	0.222814	−	0.974861i	$-0.428476\pi$
$419$	3822.03	0.445629	0.222814	+	0.974861i	$0.428476\pi$
$420$	0	0
$421$	−2751.33	−0.318508	−0.159254	−	0.987238i	$-0.550909\pi$
$421$	−2751.33	−0.318508	−0.159254	+	0.987238i	$0.550909\pi$
$422$	−1638.28	−0.188982
$423$	0	0
$424$	4167.41	0.477329
$425$	−4555.58	−0.519948
$426$	0	0
$427$	−3149.03	−0.356891
$428$	−12250.9	−1.38357
$429$	0	0
$430$	−150.507	−0.0168793
$431$	8079.52	0.902962	0.451481	−	0.892281i	$-0.350896\pi$
$431$	8079.52	0.902962	0.451481	+	0.892281i	$0.350896\pi$
$432$	0	0
$433$	3793.63	0.421040	0.210520	−	0.977590i	$-0.432484\pi$
$433$	3793.63	0.421040	0.210520	+	0.977590i	$0.432484\pi$
$434$	393.470	0.0435189
$435$	0	0
$436$	12248.6	1.34542
$437$	1140.23	0.124816
$438$	0	0
$439$	11525.0	1.25298	0.626491	−	0.779428i	$-0.284490\pi$
$439$	11525.0	1.25298	0.626491	+	0.779428i	$0.284490\pi$
$440$	5848.94	0.633721
$441$	0	0
$442$	0	0
$443$	−13127.4	−1.40791	−0.703954	−	0.710245i	$-0.748584\pi$
$443$	−13127.4	−1.40791	−0.703954	+	0.710245i	$0.748584\pi$
$444$	0	0
$445$	2007.68	0.213872
$446$	951.705	0.101042
$447$	0	0
$448$	2059.16	0.217157
$449$	−7763.80	−0.816028	−0.408014	−	0.912976i	$-0.633779\pi$
$449$	−7763.80	−0.816028	−0.408014	+	0.912976i	$0.633779\pi$
$450$	0	0
$451$	−15913.9	−1.66154
$452$	−15086.5	−1.56993
$453$	0	0
$454$	3058.92	0.316216
$455$	0	0
$456$	0	0
$457$	−291.436	−0.0298311	−0.0149156	−	0.999889i	$-0.504748\pi$
$457$	−291.436	−0.0298311	−0.0149156	+	0.999889i	$0.504748\pi$
$458$	2469.91	0.251990
$459$	0	0
$460$	−1618.75	−0.164075
$461$	1612.08	0.162868	0.0814338	−	0.996679i	$-0.474050\pi$
$461$	1612.08	0.162868	0.0814338	+	0.996679i	$0.474050\pi$
$462$	0	0
$463$	−916.747	−0.0920191	−0.0460095	−	0.998941i	$-0.514650\pi$
$463$	−916.747	−0.0920191	−0.0460095	+	0.998941i	$0.514650\pi$
$464$	7680.40	0.768435
$465$	0	0
$466$	−383.497	−0.0381227
$467$	3524.88	0.349277	0.174638	−	0.984633i	$-0.444124\pi$
$467$	3524.88	0.349277	0.174638	+	0.984633i	$0.444124\pi$
$468$	0	0
$469$	−2844.51	−0.280058
$470$	−1915.40	−0.187980
$471$	0	0
$472$	−5434.37	−0.529952
$473$	1401.30	0.136219
$474$	0	0
$475$	1910.95	0.184590
$476$	5346.79	0.514852
$477$	0	0
$478$	−2336.72	−0.223596
$479$	−12392.8	−1.18213	−0.591066	−	0.806623i	$-0.701293\pi$
$479$	−12392.8	−1.18213	−0.591066	+	0.806623i	$0.701293\pi$
$480$	0	0
$481$	0	0
$482$	4239.33	0.400615
$483$	0	0
$484$	−16382.1	−1.53851
$485$	2037.97	0.190803
$486$	0	0
$487$	14186.4	1.32002	0.660009	−	0.751258i	$-0.270552\pi$
$487$	14186.4	1.32002	0.660009	+	0.751258i	$0.270552\pi$
$488$	5540.06	0.513907
$489$	0	0
$490$	−1948.89	−0.179678
$491$	14559.4	1.33820	0.669101	−	0.743172i	$-0.266680\pi$
$491$	14559.4	1.33820	0.669101	+	0.743172i	$0.266680\pi$
$492$	0	0
$493$	17027.3	1.55552
$494$	0	0
$495$	0	0
$496$	5030.09	0.455358
$497$	−2821.57	−0.254657
$498$	0	0
$499$	8470.02	0.759860	0.379930	−	0.925015i	$-0.375948\pi$
$499$	8470.02	0.759860	0.379930	+	0.925015i	$0.375948\pi$
$500$	−11452.6	−1.02435
$501$	0	0
$502$	−4448.05	−0.395471
$503$	7649.38	0.678070	0.339035	−	0.940774i	$-0.389900\pi$
$503$	7649.38	0.678070	0.339035	+	0.940774i	$0.389900\pi$
$504$	0	0
$505$	−8554.19	−0.753775
$506$	−939.134	−0.0825091
$507$	0	0
$508$	14811.1	1.29358
$509$	−13019.8	−1.13378	−0.566889	−	0.823794i	$-0.691853\pi$
$509$	−13019.8	−1.13378	−0.566889	+	0.823794i	$0.691853\pi$
$510$	0	0
$511$	1899.90	0.164475
$512$	−10935.9	−0.943948
$513$	0	0
$514$	5131.02	0.440310
$515$	11861.2	1.01489
$516$	0	0
$517$	17833.3	1.51704
$518$	1573.16	0.133438
$519$	0	0
$520$	0	0
$521$	1003.37	0.0843734	0.0421867	−	0.999110i	$-0.486568\pi$
$521$	1003.37	0.0843734	0.0421867	+	0.999110i	$0.486568\pi$
$522$	0	0
$523$	−4216.27	−0.352514	−0.176257	−	0.984344i	$-0.556399\pi$
$523$	−4216.27	−0.352514	−0.176257	+	0.984344i	$0.556399\pi$
$524$	21567.3	1.79804
$525$	0	0
$526$	−2767.88	−0.229439
$527$	11151.6	0.921770
$528$	0	0
$529$	−11631.0	−0.955944
$530$	2491.29	0.204179
$531$	0	0
$532$	−2242.84	−0.182781
$533$	0	0
$534$	0	0
$535$	−15103.5	−1.22053
$536$	5004.32	0.403272
$537$	0	0
$538$	−5289.00	−0.423838
$539$	18145.1	1.45003
$540$	0	0
$541$	23139.4	1.83889	0.919447	−	0.393213i	$-0.128637\pi$
$541$	23139.4	1.83889	0.919447	+	0.393213i	$0.128637\pi$
$542$	−4510.44	−0.357454
$543$	0	0
$544$	−14251.9	−1.12325
$545$	15100.8	1.18688
$546$	0	0
$547$	−12532.9	−0.979646	−0.489823	−	0.871822i	$-0.662939\pi$
$547$	−12532.9	−0.979646	−0.489823	+	0.871822i	$0.662939\pi$
$548$	−13363.5	−1.04171
$549$	0	0
$550$	−1573.93	−0.122023
$551$	−7142.53	−0.552236
$552$	0	0
$553$	−289.748	−0.0222809
$554$	−2596.63	−0.199134
$555$	0	0
$556$	10161.6	0.775084
$557$	−6240.38	−0.474710	−0.237355	−	0.971423i	$-0.576280\pi$
$557$	−6240.38	−0.474710	−0.237355	+	0.971423i	$0.576280\pi$
$558$	0	0
$559$	0	0
$560$	2973.32	0.224368
$561$	0	0
$562$	−4390.25	−0.329522
$563$	6742.87	0.504756	0.252378	−	0.967629i	$-0.418787\pi$
$563$	6742.87	0.504756	0.252378	+	0.967629i	$0.418787\pi$
$564$	0	0
$565$	−18599.4	−1.38493
$566$	5020.41	0.372833
$567$	0	0
$568$	4963.96	0.366695
$569$	−2935.57	−0.216283	−0.108142	−	0.994135i	$-0.534490\pi$
$569$	−2935.57	−0.216283	−0.108142	+	0.994135i	$0.534490\pi$
$570$	0	0
$571$	9709.50	0.711611	0.355805	−	0.934560i	$-0.384207\pi$
$571$	9709.50	0.711611	0.355805	+	0.934560i	$0.384207\pi$
$572$	0	0
$573$	0	0
$574$	1113.31	0.0809556
$575$	898.342	0.0651538
$576$	0	0
$577$	18892.5	1.36309	0.681546	−	0.731775i	$-0.261308\pi$
$577$	18892.5	1.36309	0.681546	+	0.731775i	$0.261308\pi$
$578$	−6077.14	−0.437329
$579$	0	0
$580$	10140.1	0.725936
$581$	1874.84	0.133875
$582$	0	0
$583$	−23195.1	−1.64776
$584$	−3342.48	−0.236837
$585$	0	0
$586$	−361.573	−0.0254888
$587$	24642.6	1.73272	0.866362	−	0.499416i	$-0.166452\pi$
$587$	24642.6	1.73272	0.866362	+	0.499416i	$0.166452\pi$
$588$	0	0
$589$	−4677.82	−0.327243
$590$	−3248.68	−0.226688
$591$	0	0
$592$	20111.2	1.39622
$593$	−6433.11	−0.445491	−0.222745	−	0.974877i	$-0.571502\pi$
$593$	−6433.11	−0.445491	−0.222745	+	0.974877i	$0.571502\pi$
$594$	0	0
$595$	6591.82	0.454182
$596$	−9545.30	−0.656025
$597$	0	0
$598$	0	0
$599$	19055.2	1.29979	0.649897	−	0.760023i	$-0.274812\pi$
$599$	19055.2	1.29979	0.649897	+	0.760023i	$0.274812\pi$
$600$	0	0
$601$	−20141.7	−1.36705	−0.683524	−	0.729928i	$-0.739554\pi$
$601$	−20141.7	−1.36705	−0.683524	+	0.729928i	$0.739554\pi$
$602$	−98.0323	−0.00663704
$603$	0	0
$604$	22055.0	1.48577
$605$	−20196.8	−1.35722
$606$	0	0
$607$	−10627.2	−0.710620	−0.355310	−	0.934749i	$-0.615625\pi$
$607$	−10627.2	−0.710620	−0.355310	+	0.934749i	$0.615625\pi$
$608$	5978.32	0.398771
$609$	0	0
$610$	3311.86	0.219825
$611$	0	0
$612$	0	0
$613$	−18460.9	−1.21636	−0.608181	−	0.793798i	$-0.708101\pi$
$613$	−18460.9	−1.21636	−0.608181	+	0.793798i	$0.708101\pi$
$614$	566.537	0.0372371
$615$	0	0
$616$	3809.68	0.249182
$617$	−21119.7	−1.37804	−0.689018	−	0.724744i	$-0.741958\pi$
$617$	−21119.7	−1.37804	−0.689018	+	0.724744i	$0.741958\pi$
$618$	0	0
$619$	−571.628	−0.0371174	−0.0185587	−	0.999828i	$-0.505908\pi$
$619$	−571.628	−0.0371174	−0.0185587	+	0.999828i	$0.505908\pi$
$620$	6640.97	0.430174
$621$	0	0
$622$	1385.48	0.0893132
$623$	1307.69	0.0840957
$624$	0	0
$625$	−9269.25	−0.593232
$626$	4685.23	0.299137
$627$	0	0
$628$	−18978.0	−1.20590
$629$	44586.2	2.82634
$630$	0	0
$631$	7543.32	0.475903	0.237952	−	0.971277i	$-0.423524\pi$
$631$	7543.32	0.475903	0.237952	+	0.971277i	$0.423524\pi$
$632$	509.751	0.0320835
$633$	0	0
$634$	5161.88	0.323351
$635$	18260.0	1.14114
$636$	0	0
$637$	0	0
$638$	5882.85	0.365054
$639$	0	0
$640$	−11181.8	−0.690621
$641$	−6800.31	−0.419027	−0.209513	−	0.977806i	$-0.567188\pi$
$641$	−6800.31	−0.419027	−0.209513	+	0.977806i	$0.567188\pi$
$642$	0	0
$643$	−11713.1	−0.718380	−0.359190	−	0.933265i	$-0.616947\pi$
$643$	−11713.1	−0.718380	−0.359190	+	0.933265i	$0.616947\pi$
$644$	−1054.37	−0.0645153
$645$	0	0
$646$	3960.96	0.241241
$647$	−12455.4	−0.756835	−0.378417	−	0.925635i	$-0.623532\pi$
$647$	−12455.4	−0.756835	−0.378417	+	0.925635i	$0.623532\pi$
$648$	0	0
$649$	30246.8	1.82941
$650$	0	0
$651$	0	0
$652$	27083.2	1.62678
$653$	−14073.0	−0.843369	−0.421685	−	0.906743i	$-0.638561\pi$
$653$	−14073.0	−0.843369	−0.421685	+	0.906743i	$0.638561\pi$
$654$	0	0
$655$	26589.4	1.58616
$656$	14232.4	0.847076
$657$	0	0
$658$	−1247.59	−0.0739149
$659$	8037.27	0.475095	0.237548	−	0.971376i	$-0.423656\pi$
$659$	8037.27	0.475095	0.237548	+	0.971376i	$0.423656\pi$
$660$	0	0
$661$	−1994.71	−0.117375	−0.0586877	−	0.998276i	$-0.518692\pi$
$661$	−1994.71	−0.117375	−0.0586877	+	0.998276i	$0.518692\pi$
$662$	5217.23	0.306304
$663$	0	0
$664$	−3298.40	−0.192775
$665$	−2765.10	−0.161242
$666$	0	0
$667$	−3357.73	−0.194920
$668$	14975.3	0.867382
$669$	0	0
$670$	2991.59	0.172500
$671$	−30835.0	−1.77403
$672$	0	0
$673$	13886.1	0.795348	0.397674	−	0.917527i	$-0.369817\pi$
$673$	13886.1	0.795348	0.397674	+	0.917527i	$0.369817\pi$
$674$	4082.52	0.233313
$675$	0	0
$676$	0	0
$677$	−11024.0	−0.625831	−0.312915	−	0.949781i	$-0.601306\pi$
$677$	−11024.0	−0.625831	−0.312915	+	0.949781i	$0.601306\pi$
$678$	0	0
$679$	1327.42	0.0750247
$680$	−11596.9	−0.654002
$681$	0	0
$682$	3852.83	0.216323
$683$	−22696.5	−1.27154	−0.635768	−	0.771880i	$-0.719317\pi$
$683$	−22696.5	−1.27154	−0.635768	+	0.771880i	$0.719317\pi$
$684$	0	0
$685$	−16475.2	−0.918957
$686$	−2690.30	−0.149732
$687$	0	0
$688$	−1253.24	−0.0694465
$689$	0	0
$690$	0	0
$691$	3516.06	0.193570	0.0967852	−	0.995305i	$-0.469144\pi$
$691$	3516.06	0.193570	0.0967852	+	0.995305i	$0.469144\pi$
$692$	4398.36	0.241619
$693$	0	0
$694$	6834.00	0.373797
$695$	12527.8	0.683748
$696$	0	0
$697$	31553.0	1.71471
$698$	−3877.82	−0.210283
$699$	0	0
$700$	−1767.05	−0.0954115
$701$	−26652.4	−1.43602	−0.718009	−	0.696034i	$-0.754946\pi$
$701$	−26652.4	−1.43602	−0.718009	+	0.696034i	$0.754946\pi$
$702$	0	0
$703$	−18702.7	−1.00340
$704$	20163.1	1.07944
$705$	0	0
$706$	467.482	0.0249206
$707$	−5571.73	−0.296388
$708$	0	0
$709$	8938.91	0.473495	0.236747	−	0.971571i	$-0.423919\pi$
$709$	8938.91	0.473495	0.236747	+	0.971571i	$0.423919\pi$
$710$	2967.46	0.156855
$711$	0	0
$712$	−2300.61	−0.121094
$713$	−2199.06	−0.115505
$714$	0	0
$715$	0	0
$716$	−9439.70	−0.492707
$717$	0	0
$718$	5705.43	0.296553
$719$	5294.09	0.274599	0.137299	−	0.990530i	$-0.456158\pi$
$719$	5294.09	0.274599	0.137299	+	0.990530i	$0.456158\pi$
$720$	0	0
$721$	7725.73	0.399059
$722$	3037.07	0.156548
$723$	0	0
$724$	25805.5	1.32466
$725$	−5627.32	−0.288267
$726$	0	0
$727$	30010.0	1.53096	0.765480	−	0.643459i	$-0.222501\pi$
$727$	30010.0	1.53096	0.765480	+	0.643459i	$0.222501\pi$
$728$	0	0
$729$	0	0
$730$	−1998.14	−0.101308
$731$	−2778.40	−0.140579
$732$	0	0
$733$	35185.8	1.77301	0.886505	−	0.462720i	$-0.153126\pi$
$733$	35185.8	1.77301	0.886505	+	0.462720i	$0.153126\pi$
$734$	−383.546	−0.0192874
$735$	0	0
$736$	2810.43	0.140752
$737$	−27853.2	−1.39211
$738$	0	0
$739$	16737.3	0.833140	0.416570	−	0.909104i	$-0.363232\pi$
$739$	16737.3	0.833140	0.416570	+	0.909104i	$0.363232\pi$
$740$	26551.8	1.31900
$741$	0	0
$742$	1622.69	0.0802842
$743$	−14564.0	−0.719115	−0.359558	−	0.933123i	$-0.617072\pi$
$743$	−14564.0	−0.719115	−0.359558	+	0.933123i	$0.617072\pi$
$744$	0	0
$745$	−11768.0	−0.578718
$746$	−375.448	−0.0184264
$747$	0	0
$748$	52355.3	2.55922
$749$	−9837.63	−0.479919
$750$	0	0
$751$	22935.6	1.11442	0.557212	−	0.830370i	$-0.311871\pi$
$751$	22935.6	1.11442	0.557212	+	0.830370i	$0.311871\pi$
$752$	−15949.0	−0.773406
$753$	0	0
$754$	0	0
$755$	27190.6	1.31069
$756$	0	0
$757$	15356.3	0.737296	0.368648	−	0.929569i	$-0.379821\pi$
$757$	15356.3	0.737296	0.368648	+	0.929569i	$0.379821\pi$
$758$	3206.10	0.153629
$759$	0	0
$760$	4864.61	0.232181
$761$	25838.1	1.23079	0.615394	−	0.788219i	$-0.288997\pi$
$761$	25838.1	1.23079	0.615394	+	0.788219i	$0.288997\pi$
$762$	0	0
$763$	9835.84	0.466686
$764$	23351.4	1.10579
$765$	0	0
$766$	−6284.51	−0.296434
$767$	0	0
$768$	0	0
$769$	11315.0	0.530597	0.265299	−	0.964166i	$-0.414529\pi$
$769$	11315.0	0.530597	0.265299	+	0.964166i	$0.414529\pi$
$770$	2277.44	0.106588
$771$	0	0
$772$	2549.64	0.118865
$773$	−1991.25	−0.0926524	−0.0463262	−	0.998926i	$-0.514751\pi$
$773$	−1991.25	−0.0926524	−0.0463262	+	0.998926i	$0.514751\pi$
$774$	0	0
$775$	−3685.47	−0.170821
$776$	−2335.32	−0.108032
$777$	0	0
$778$	237.461	0.0109427
$779$	−13235.7	−0.608752
$780$	0	0
$781$	−27628.5	−1.26585
$782$	1862.06	0.0851497
$783$	0	0
$784$	−16227.9	−0.739246
$785$	−23397.1	−1.06380
$786$	0	0
$787$	−13262.7	−0.600717	−0.300359	−	0.953826i	$-0.597106\pi$
$787$	−13262.7	−0.600717	−0.300359	+	0.953826i	$0.597106\pi$
$788$	9919.66	0.448443
$789$	0	0
$790$	304.730	0.0137238
$791$	−12114.7	−0.544561
$792$	0	0
$793$	0	0
$794$	351.642	0.0157170
$795$	0	0
$796$	−26017.9	−1.15852
$797$	36462.7	1.62055	0.810273	−	0.586052i	$-0.199319\pi$
$797$	36462.7	1.62055	0.810273	+	0.586052i	$0.199319\pi$
$798$	0	0
$799$	−35358.8	−1.56559
$800$	4710.09	0.208158
$801$	0	0
$802$	−7650.61	−0.336848
$803$	18603.7	0.817570
$804$	0	0
$805$	−1299.88	−0.0569128
$806$	0	0
$807$	0	0
$808$	9802.29	0.426786
$809$	−2469.45	−0.107319	−0.0536596	−	0.998559i	$-0.517089\pi$
$809$	−2469.45	−0.107319	−0.0536596	+	0.998559i	$0.517089\pi$
$810$	0	0
$811$	42081.3	1.82204	0.911019	−	0.412363i	$-0.135297\pi$
$811$	42081.3	1.82204	0.911019	+	0.412363i	$0.135297\pi$
$812$	6604.68	0.285442
$813$	0	0
$814$	15404.3	0.663292
$815$	33389.6	1.43508
$816$	0	0
$817$	1165.47	0.0499077
$818$	2936.29	0.125507
$819$	0	0
$820$	18790.3	0.800228
$821$	−43282.1	−1.83990	−0.919949	−	0.392038i	$-0.871770\pi$
$821$	−43282.1	−1.83990	−0.919949	+	0.392038i	$0.871770\pi$
$822$	0	0
$823$	−38193.7	−1.61768	−0.808838	−	0.588031i	$-0.799903\pi$
$823$	−38193.7	−1.61768	−0.808838	+	0.588031i	$0.799903\pi$
$824$	−13591.8	−0.574628
$825$	0	0
$826$	−2116.01	−0.0891350
$827$	35724.1	1.50211	0.751057	−	0.660238i	$-0.229544\pi$
$827$	35724.1	1.50211	0.751057	+	0.660238i	$0.229544\pi$
$828$	0	0
$829$	7345.94	0.307762	0.153881	−	0.988089i	$-0.450823\pi$
$829$	7345.94	0.307762	0.153881	+	0.988089i	$0.450823\pi$
$830$	−1971.79	−0.0824600
$831$	0	0
$832$	0	0
$833$	−35977.1	−1.49644
$834$	0	0
$835$	18462.3	0.765169
$836$	−21961.7	−0.908566
$837$	0	0
$838$	−2618.19	−0.107928
$839$	11691.7	0.481100	0.240550	−	0.970637i	$-0.422672\pi$
$839$	11691.7	0.481100	0.240550	+	0.970637i	$0.422672\pi$
$840$	0	0
$841$	−3355.80	−0.137595
$842$	1884.73	0.0771404
$843$	0	0
$844$	−18010.3	−0.734525
$845$	0	0
$846$	0	0
$847$	−13155.1	−0.533664
$848$	20744.3	0.840051
$849$	0	0
$850$	3120.68	0.125928
$851$	−8792.22	−0.354164
$852$	0	0
$853$	−15321.4	−0.615000	−0.307500	−	0.951548i	$-0.599492\pi$
$853$	−15321.4	−0.615000	−0.307500	+	0.951548i	$0.599492\pi$
$854$	2157.17	0.0864364
$855$	0	0
$856$	17307.2	0.691063
$857$	−32949.6	−1.31335	−0.656673	−	0.754176i	$-0.728037\pi$
$857$	−32949.6	−1.31335	−0.656673	+	0.754176i	$0.728037\pi$
$858$	0	0
$859$	−8132.72	−0.323032	−0.161516	−	0.986870i	$-0.551638\pi$
$859$	−8132.72	−0.323032	−0.161516	+	0.986870i	$0.551638\pi$
$860$	−1654.58	−0.0656056
$861$	0	0
$862$	−5534.67	−0.218691
$863$	−25770.4	−1.01649	−0.508247	−	0.861212i	$-0.669706\pi$
$863$	−25770.4	−1.01649	−0.508247	+	0.861212i	$0.669706\pi$
$864$	0	0
$865$	5422.54	0.213147
$866$	−2598.73	−0.101973
$867$	0	0
$868$	4325.57	0.169147
$869$	−2837.19	−0.110754
$870$	0	0
$871$	0	0
$872$	−17304.1	−0.672008
$873$	0	0
$874$	−781.086	−0.0302295
$875$	−9196.64	−0.355318
$876$	0	0
$877$	−31715.4	−1.22116	−0.610578	−	0.791956i	$-0.709063\pi$
$877$	−31715.4	−1.22116	−0.610578	+	0.791956i	$0.709063\pi$
$878$	−7894.93	−0.303464
$879$	0	0
$880$	29114.5	1.11528
$881$	−29755.9	−1.13791	−0.568957	−	0.822367i	$-0.692653\pi$
$881$	−29755.9	−1.13791	−0.568957	+	0.822367i	$0.692653\pi$
$882$	0	0
$883$	41818.4	1.59377	0.796886	−	0.604129i	$-0.206479\pi$
$883$	41818.4	1.59377	0.796886	+	0.604129i	$0.206479\pi$
$884$	0	0
$885$	0	0
$886$	8992.63	0.340986
$887$	6514.49	0.246601	0.123301	−	0.992369i	$-0.460652\pi$
$887$	6514.49	0.246601	0.123301	+	0.992369i	$0.460652\pi$
$888$	0	0
$889$	11893.5	0.448703
$890$	−1375.31	−0.0517984
$891$	0	0
$892$	10462.5	0.392723
$893$	14832.1	0.555809
$894$	0	0
$895$	−11637.8	−0.434646
$896$	−7283.19	−0.271556
$897$	0	0
$898$	5318.40	0.197636
$899$	13775.2	0.511043
$900$	0	0
$901$	45989.9	1.70049
$902$	10901.4	0.402413
$903$	0	0
$904$	21313.2	0.784145
$905$	31814.4	1.16856
$906$	0	0
$907$	−1750.33	−0.0640780	−0.0320390	−	0.999487i	$-0.510200\pi$
$907$	−1750.33	−0.0640780	−0.0320390	+	0.999487i	$0.510200\pi$
$908$	33627.8	1.22905
$909$	0	0
$910$	0	0
$911$	−14268.9	−0.518936	−0.259468	−	0.965752i	$-0.583547\pi$
$911$	−14268.9	−0.518936	−0.259468	+	0.965752i	$0.583547\pi$
$912$	0	0
$913$	18358.3	0.665467
$914$	199.641	0.00722489
$915$	0	0
$916$	27152.7	0.979421
$917$	17318.9	0.623685
$918$	0	0
$919$	−19466.2	−0.698729	−0.349364	−	0.936987i	$-0.613602\pi$
$919$	−19466.2	−0.698729	−0.349364	+	0.936987i	$0.613602\pi$
$920$	2286.87	0.0819520
$921$	0	0
$922$	−1104.31	−0.0394454
$923$	0	0
$924$	0	0
$925$	−14735.2	−0.523772
$926$	627.995	0.0222864
$927$	0	0
$928$	−17604.9	−0.622745
$929$	866.279	0.0305939	0.0152969	−	0.999883i	$-0.495131\pi$
$929$	866.279	0.0305939	0.0152969	+	0.999883i	$0.495131\pi$
$930$	0	0
$931$	15091.5	0.531259
$932$	−4215.93	−0.148173
$933$	0	0
$934$	−2414.63	−0.0845924
$935$	64546.5	2.25764
$936$	0	0
$937$	−33092.8	−1.15378	−0.576891	−	0.816821i	$-0.695734\pi$
$937$	−33092.8	−1.15378	−0.576891	+	0.816821i	$0.695734\pi$
$938$	1948.56	0.0678281
$939$	0	0
$940$	−21056.7	−0.730632
$941$	27089.5	0.938461	0.469231	−	0.883076i	$-0.344531\pi$
$941$	27089.5	0.938461	0.469231	+	0.883076i	$0.344531\pi$
$942$	0	0
$943$	−6222.14	−0.214868
$944$	−27050.9	−0.932661
$945$	0	0
$946$	−959.923	−0.0329913
$947$	45146.5	1.54917	0.774586	−	0.632469i	$-0.217958\pi$
$947$	45146.5	1.54917	0.774586	+	0.632469i	$0.217958\pi$
$948$	0	0
$949$	0	0
$950$	−1309.05	−0.0447064
$951$	0	0
$952$	−7553.60	−0.257157
$953$	−32600.5	−1.10812	−0.554058	−	0.832478i	$-0.686921\pi$
$953$	−32600.5	−1.10812	−0.554058	+	0.832478i	$0.686921\pi$
$954$	0	0
$955$	28788.9	0.975484
$956$	−25688.4	−0.869062
$957$	0	0
$958$	8489.38	0.286304
$959$	−10731.1	−0.361339
$960$	0	0
$961$	−20769.3	−0.697167
$962$	0	0
$963$	0	0
$964$	46604.6	1.55709
$965$	3143.34	0.104858
$966$	0	0
$967$	−13124.6	−0.436462	−0.218231	−	0.975897i	$-0.570029\pi$
$967$	−13124.6	−0.436462	−0.218231	+	0.975897i	$0.570029\pi$
$968$	23143.6	0.768454
$969$	0	0
$970$	−1396.06	−0.0462111
$971$	−7776.85	−0.257025	−0.128512	−	0.991708i	$-0.541020\pi$
$971$	−7776.85	−0.257025	−0.128512	+	0.991708i	$0.541020\pi$
$972$	0	0
$973$	8159.90	0.268853
$974$	−9718.07	−0.319699
$975$	0	0
$976$	27577.0	0.904425
$977$	25662.3	0.840339	0.420169	−	0.907446i	$-0.361971\pi$
$977$	25662.3	0.840339	0.420169	+	0.907446i	$0.361971\pi$
$978$	0	0
$979$	12804.8	0.418022
$980$	−21424.9	−0.698361
$981$	0	0
$982$	−9973.56	−0.324103
$983$	−1747.89	−0.0567132	−0.0283566	−	0.999598i	$-0.509027\pi$
$983$	−1747.89	−0.0567132	−0.0283566	+	0.999598i	$0.509027\pi$
$984$	0	0
$985$	12229.5	0.395598
$986$	−11664.2	−0.376737
$987$	0	0
$988$	0	0
$989$	547.891	0.0176157
$990$	0	0
$991$	−14983.4	−0.480286	−0.240143	−	0.970738i	$-0.577194\pi$
$991$	−14983.4	−0.480286	−0.240143	+	0.970738i	$0.577194\pi$
$992$	−11529.9	−0.369026
$993$	0	0
$994$	1932.85	0.0616762
$995$	−32076.3	−1.02200
$996$	0	0
$997$	−51834.8	−1.64656	−0.823282	−	0.567632i	$-0.807860\pi$
$997$	−51834.8	−1.64656	−0.823282	+	0.567632i	$0.807860\pi$
$998$	−5802.18	−0.184033
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bm.1.9	✓	18
3.2	odd	2	inner	1521.4.a.bm.1.10	yes	18
13.12	even	2		1521.4.a.bn.1.10	yes	18
39.38	odd	2		1521.4.a.bn.1.9	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1521.4.a.bm.1.9	✓	18	1.1	even	1	trivial
1521.4.a.bm.1.10	yes	18	3.2	odd	2	inner
1521.4.a.bn.1.9	yes	18	39.38	odd	2
1521.4.a.bn.1.10	yes	18	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q-0.685025 q^{2} -7.53074 q^{4} -9.28431 q^{5} -6.04730 q^{7} +10.6389 q^{8} +O(q^{10})\)	\(q-0.685025 q^{2} -7.53074 q^{4} -9.28431 q^{5} -6.04730 q^{7} +10.6389 q^{8} +6.35999 q^{10} -59.2146 q^{11} +4.14255 q^{14} +52.9580 q^{16} +117.407 q^{17} -49.2492 q^{19} +69.9177 q^{20} +40.5635 q^{22} -23.1522 q^{23} -38.8015 q^{25} +45.5406 q^{28} +145.028 q^{29} +94.9826 q^{31} -121.389 q^{32} -80.4268 q^{34} +56.1450 q^{35} +379.757 q^{37} +33.7370 q^{38} -98.7753 q^{40} +268.749 q^{41} -23.6647 q^{43} +445.930 q^{44} +15.8599 q^{46} -301.164 q^{47} -306.430 q^{49} +26.5800 q^{50} +391.713 q^{53} +549.767 q^{55} -64.3369 q^{56} -99.3480 q^{58} -510.800 q^{59} +520.734 q^{61} -65.0655 q^{62} -340.509 q^{64} +470.377 q^{67} -884.162 q^{68} -38.4607 q^{70} +466.583 q^{71} -314.174 q^{73} -260.143 q^{74} +370.883 q^{76} +358.088 q^{77} +47.9136 q^{79} -491.678 q^{80} -184.100 q^{82} -310.030 q^{83} -1090.04 q^{85} +16.2109 q^{86} -629.981 q^{88} -216.244 q^{89} +174.353 q^{92} +206.305 q^{94} +457.245 q^{95} -219.507 q^{97} +209.912 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 56 q^{4} - 94 q^{7}+O(q^{10}) \) 18 * q + 56 * q^4 - 94 * q^7	\( 18 q + 56 q^{4} - 94 q^{7} + 156 q^{10} + 88 q^{16} - 448 q^{19} - 256 q^{22} - 16 q^{25} - 1300 q^{28} - 818 q^{31} - 900 q^{34} - 524 q^{37} + 800 q^{40} + 752 q^{43} - 1650 q^{46} + 2948 q^{49} - 464 q^{55} - 1626 q^{58} - 1784 q^{61} - 4274 q^{64} - 2872 q^{67} - 5772 q^{70} - 3078 q^{73} - 5726 q^{76} + 3326 q^{79} - 1038 q^{82} - 2340 q^{85} + 780 q^{88} + 1220 q^{94} - 4630 q^{97}+O(q^{100}) \) 18 * q + 56 * q^4 - 94 * q^7 + 156 * q^10 + 88 * q^16 - 448 * q^19 - 256 * q^22 - 16 * q^25 - 1300 * q^28 - 818 * q^31 - 900 * q^34 - 524 * q^37 + 800 * q^40 + 752 * q^43 - 1650 * q^46 + 2948 * q^49 - 464 * q^55 - 1626 * q^58 - 1784 * q^61 - 4274 * q^64 - 2872 * q^67 - 5772 * q^70 - 3078 * q^73 - 5726 * q^76 + 3326 * q^79 - 1038 * q^82 - 2340 * q^85 + 780 * q^88 + 1220 * q^94 - 4630 * q^97

Introduction

L-functions

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