LMFDB - Embedded newform 2475.4.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2475.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:	$146.029727264$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.3368.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 15x + 11 $ x^3 - x^2 - 15*x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 825)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.03932$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.03932 q^{2} +8.31608 q^{4} -6.49923 q^{7} -1.27677 q^{8} +O(q^{10})$	$q-4.03932 q^{2} +8.31608 q^{4} -6.49923 q^{7} -1.27677 q^{8} -11.0000 q^{11} +30.5385 q^{13} +26.2524 q^{14} -61.3714 q^{16} +43.1314 q^{17} -6.46044 q^{19} +44.4325 q^{22} -108.104 q^{23} -123.355 q^{26} -54.0481 q^{28} +274.857 q^{29} -68.5519 q^{31} +258.113 q^{32} -174.221 q^{34} -402.200 q^{37} +26.0958 q^{38} +268.450 q^{41} -30.5334 q^{43} -91.4769 q^{44} +436.666 q^{46} -31.5850 q^{47} -300.760 q^{49} +253.961 q^{52} -252.497 q^{53} +8.29800 q^{56} -1110.24 q^{58} +558.331 q^{59} +335.270 q^{61} +276.903 q^{62} -551.628 q^{64} +28.7521 q^{67} +358.684 q^{68} -89.0081 q^{71} -717.622 q^{73} +1624.61 q^{74} -53.7256 q^{76} +71.4915 q^{77} +200.702 q^{79} -1084.36 q^{82} -243.681 q^{83} +123.334 q^{86} +14.0444 q^{88} +312.832 q^{89} -198.477 q^{91} -899.002 q^{92} +127.582 q^{94} +27.9996 q^{97} +1214.87 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 7 q^{4} + 16 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q - q^2 + 7 * q^4 + 16 * q^7 + 3 * q^8	$ 3 q - q^{2} + 7 q^{4} + 16 q^{7} + 3 q^{8} - 33 q^{11} + 45 q^{13} + 116 q^{14} - 85 q^{16} + 58 q^{17} - 169 q^{19} + 11 q^{22} - 155 q^{23} - 167 q^{26} + 100 q^{28} + 277 q^{29} - 173 q^{31} - 97 q^{32} - 146 q^{34} + 60 q^{37} - 169 q^{38} - 44 q^{41} - 109 q^{43} - 77 q^{44} + 425 q^{46} - 270 q^{47} - 427 q^{49} + 45 q^{52} + 148 q^{53} - 168 q^{56} - 783 q^{58} + 684 q^{59} - 1038 q^{61} - 953 q^{62} - 1129 q^{64} + 314 q^{67} + 366 q^{68} + 1459 q^{71} - 1170 q^{73} + 1764 q^{74} + 211 q^{76} - 176 q^{77} - 506 q^{79} - 1040 q^{82} + 347 q^{83} + 527 q^{86} - 33 q^{88} + 607 q^{89} - 398 q^{91} - 687 q^{92} - 1822 q^{94} + 1263 q^{97} + 2273 q^{98}+O(q^{100}) $ 3 * q - q^2 + 7 * q^4 + 16 * q^7 + 3 * q^8 - 33 * q^11 + 45 * q^13 + 116 * q^14 - 85 * q^16 + 58 * q^17 - 169 * q^19 + 11 * q^22 - 155 * q^23 - 167 * q^26 + 100 * q^28 + 277 * q^29 - 173 * q^31 - 97 * q^32 - 146 * q^34 + 60 * q^37 - 169 * q^38 - 44 * q^41 - 109 * q^43 - 77 * q^44 + 425 * q^46 - 270 * q^47 - 427 * q^49 + 45 * q^52 + 148 * q^53 - 168 * q^56 - 783 * q^58 + 684 * q^59 - 1038 * q^61 - 953 * q^62 - 1129 * q^64 + 314 * q^67 + 366 * q^68 + 1459 * q^71 - 1170 * q^73 + 1764 * q^74 + 211 * q^76 - 176 * q^77 - 506 * q^79 - 1040 * q^82 + 347 * q^83 + 527 * q^86 - 33 * q^88 + 607 * q^89 - 398 * q^91 - 687 * q^92 - 1822 * q^94 + 1263 * q^97 + 2273 * q^98

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 $

$2$	−4.03932	−1.42811	−0.714057	−	0.700087i	$-0.753144\pi$
$2$	−4.03932	−1.42811	−0.714057	+	0.700087i	$0.753144\pi$
$3$	0	0
$3$	0	0
$4$	8.31608	1.03951
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−6.49923	−0.350925	−0.175463	−	0.984486i	$-0.556142\pi$
$7$	−6.49923	−0.350925	−0.175463	+	0.984486i	$0.556142\pi$
$8$	−1.27677	−0.0564257
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	30.5385	0.651528	0.325764	−	0.945451i	$-0.394378\pi$
$13$	30.5385	0.651528	0.325764	+	0.945451i	$0.394378\pi$
$14$	26.2524	0.501161
$15$	0	0
$16$	−61.3714	−0.958928
$17$	43.1314	0.615347	0.307674	−	0.951492i	$-0.400450\pi$
$17$	43.1314	0.615347	0.307674	+	0.951492i	$0.400450\pi$
$18$	0	0
$19$	−6.46044	−0.0780066	−0.0390033	−	0.999239i	$-0.512418\pi$
$19$	−6.46044	−0.0780066	−0.0390033	+	0.999239i	$0.512418\pi$
$20$	0	0
$21$	0	0
$22$	44.4325	0.430593
$23$	−108.104	−0.980054	−0.490027	−	0.871707i	$-0.663013\pi$
$23$	−108.104	−0.980054	−0.490027	+	0.871707i	$0.663013\pi$
$24$	0	0
$25$	0	0
$26$	−123.355	−0.930457
$27$	0	0
$28$	−54.0481	−0.364791
$29$	274.857	1.75999	0.879995	−	0.474984i	$-0.157546\pi$
$29$	274.857	1.75999	0.879995	+	0.474984i	$0.157546\pi$
$30$	0	0
$31$	−68.5519	−0.397170	−0.198585	−	0.980084i	$-0.563635\pi$
$31$	−68.5519	−0.397170	−0.198585	+	0.980084i	$0.563635\pi$
$32$	258.113	1.42588
$33$	0	0
$34$	−174.221	−0.878786
$35$	0	0
$36$	0	0
$37$	−402.200	−1.78706	−0.893530	−	0.449004i	$-0.851779\pi$
$37$	−402.200	−1.78706	−0.893530	+	0.449004i	$0.851779\pi$
$38$	26.0958	0.111402
$39$	0	0
$40$	0	0
$41$	268.450	1.02256	0.511280	−	0.859414i	$-0.329172\pi$
$41$	268.450	1.02256	0.511280	+	0.859414i	$0.329172\pi$
$42$	0	0
$43$	−30.5334	−0.108286	−0.0541431	−	0.998533i	$-0.517243\pi$
$43$	−30.5334	−0.108286	−0.0541431	+	0.998533i	$0.517243\pi$
$44$	−91.4769	−0.313424
$45$	0	0
$46$	436.666	1.39963
$47$	−31.5850	−0.0980243	−0.0490121	−	0.998798i	$-0.515607\pi$
$47$	−31.5850	−0.0980243	−0.0490121	+	0.998798i	$0.515607\pi$
$48$	0	0
$49$	−300.760	−0.876851
$50$	0	0
$51$	0	0
$52$	253.961	0.677271
$53$	−252.497	−0.654398	−0.327199	−	0.944956i	$-0.606105\pi$
$53$	−252.497	−0.654398	−0.327199	+	0.944956i	$0.606105\pi$
$54$	0	0
$55$	0	0
$56$	8.29800	0.0198012
$57$	0	0
$58$	−1110.24	−2.51347
$59$	558.331	1.23201	0.616004	−	0.787743i	$-0.288750\pi$
$59$	558.331	1.23201	0.616004	+	0.787743i	$0.288750\pi$
$60$	0	0
$61$	335.270	0.703719	0.351860	−	0.936053i	$-0.385549\pi$
$61$	335.270	0.703719	0.351860	+	0.936053i	$0.385549\pi$
$62$	276.903	0.567205
$63$	0	0
$64$	−551.628	−1.07740
$65$	0	0
$66$	0	0
$67$	28.7521	0.0524273	0.0262136	−	0.999656i	$-0.491655\pi$
$67$	28.7521	0.0524273	0.0262136	+	0.999656i	$0.491655\pi$
$68$	358.684	0.639660
$69$	0	0
$70$	0	0
$71$	−89.0081	−0.148779	−0.0743895	−	0.997229i	$-0.523701\pi$
$71$	−89.0081	−0.148779	−0.0743895	+	0.997229i	$0.523701\pi$
$72$	0	0
$73$	−717.622	−1.15057	−0.575283	−	0.817954i	$-0.695108\pi$
$73$	−717.622	−1.15057	−0.575283	+	0.817954i	$0.695108\pi$
$74$	1624.61	2.55213
$75$	0	0
$76$	−53.7256	−0.0810887
$77$	71.4915	0.105808
$78$	0	0
$79$	200.702	0.285832	0.142916	−	0.989735i	$-0.454352\pi$
$79$	200.702	0.285832	0.142916	+	0.989735i	$0.454352\pi$
$80$	0	0
$81$	0	0
$82$	−1084.36	−1.46033
$83$	−243.681	−0.322258	−0.161129	−	0.986933i	$-0.551514\pi$
$83$	−243.681	−0.322258	−0.161129	+	0.986933i	$0.551514\pi$
$84$	0	0
$85$	0	0
$86$	123.334	0.154645
$87$	0	0
$88$	14.0444	0.0170130
$89$	312.832	0.372585	0.186293	−	0.982494i	$-0.440353\pi$
$89$	312.832	0.372585	0.186293	+	0.982494i	$0.440353\pi$
$90$	0	0
$91$	−198.477	−0.228638
$92$	−899.002	−1.01878
$93$	0	0
$94$	127.582	0.139990
$95$	0	0
$96$	0	0
$97$	27.9996	0.0293085	0.0146543	−	0.999893i	$-0.495335\pi$
$97$	27.9996	0.0293085	0.0146543	+	0.999893i	$0.495335\pi$
$98$	1214.87	1.25224
$99$	0	0
$100$	0	0
$101$	228.278	0.224896	0.112448	−	0.993658i	$-0.464131\pi$
$101$	228.278	0.224896	0.112448	+	0.993658i	$0.464131\pi$
$102$	0	0
$103$	−800.328	−0.765618	−0.382809	−	0.923827i	$-0.625043\pi$
$103$	−800.328	−0.765618	−0.382809	+	0.923827i	$0.625043\pi$
$104$	−38.9906	−0.0367629
$105$	0	0
$106$	1019.91	0.934555
$107$	1046.89	0.945857	0.472929	−	0.881101i	$-0.343197\pi$
$107$	1046.89	0.945857	0.472929	+	0.881101i	$0.343197\pi$
$108$	0	0
$109$	317.023	0.278581	0.139291	−	0.990252i	$-0.455518\pi$
$109$	317.023	0.278581	0.139291	+	0.990252i	$0.455518\pi$
$110$	0	0
$111$	0	0
$112$	398.867	0.336512
$113$	−1333.82	−1.11040	−0.555198	−	0.831718i	$-0.687358\pi$
$113$	−1333.82	−1.11040	−0.555198	+	0.831718i	$0.687358\pi$
$114$	0	0
$115$	0	0
$116$	2285.74	1.82953
$117$	0	0
$118$	−2255.28	−1.75945
$119$	−280.321	−0.215941
$120$	0	0
$121$	121.000	0.0909091
$122$	−1354.26	−1.00499
$123$	0	0
$124$	−570.083	−0.412863
$125$	0	0
$126$	0	0
$127$	564.888	0.394691	0.197345	−	0.980334i	$-0.436768\pi$
$127$	564.888	0.394691	0.197345	+	0.980334i	$0.436768\pi$
$128$	163.299	0.112763
$129$	0	0
$130$	0	0
$131$	214.082	0.142782	0.0713908	−	0.997448i	$-0.477256\pi$
$131$	214.082	0.142782	0.0713908	+	0.997448i	$0.477256\pi$
$132$	0	0
$133$	41.9879	0.0273745
$134$	−116.139	−0.0748721
$135$	0	0
$136$	−55.0688	−0.0347214
$137$	65.2288	0.0406779	0.0203389	−	0.999793i	$-0.493525\pi$
$137$	65.2288	0.0406779	0.0203389	+	0.999793i	$0.493525\pi$
$138$	0	0
$139$	−407.084	−0.248406	−0.124203	−	0.992257i	$-0.539637\pi$
$139$	−407.084	−0.248406	−0.124203	+	0.992257i	$0.539637\pi$
$140$	0	0
$141$	0	0
$142$	359.532	0.212474
$143$	−335.924	−0.196443
$144$	0	0
$145$	0	0
$146$	2898.70	1.64314
$147$	0	0
$148$	−3344.73	−1.85767
$149$	−792.092	−0.435508	−0.217754	−	0.976004i	$-0.569873\pi$
$149$	−792.092	−0.435508	−0.217754	+	0.976004i	$0.569873\pi$
$150$	0	0
$151$	−2861.80	−1.54232	−0.771159	−	0.636642i	$-0.780323\pi$
$151$	−2861.80	−1.54232	−0.771159	+	0.636642i	$0.780323\pi$
$152$	8.24848	0.00440158
$153$	0	0
$154$	−288.777	−0.151106
$155$	0	0
$156$	0	0
$157$	2991.55	1.52071	0.760355	−	0.649507i	$-0.225025\pi$
$157$	2991.55	1.52071	0.760355	+	0.649507i	$0.225025\pi$
$158$	−810.698	−0.408201
$159$	0	0
$160$	0	0
$161$	702.592	0.343926
$162$	0	0
$163$	2907.60	1.39718	0.698592	−	0.715520i	$-0.253810\pi$
$163$	2907.60	1.39718	0.698592	+	0.715520i	$0.253810\pi$
$164$	2232.46	1.06296
$165$	0	0
$166$	984.304	0.460222
$167$	−1100.39	−0.509885	−0.254942	−	0.966956i	$-0.582056\pi$
$167$	−1100.39	−0.509885	−0.254942	+	0.966956i	$0.582056\pi$
$168$	0	0
$169$	−1264.40	−0.575511
$170$	0	0
$171$	0	0
$172$	−253.919	−0.112565
$173$	−3754.75	−1.65011	−0.825054	−	0.565054i	$-0.808855\pi$
$173$	−3754.75	−1.65011	−0.825054	+	0.565054i	$0.808855\pi$
$174$	0	0
$175$	0	0
$176$	675.086	0.289128
$177$	0	0
$178$	−1263.63	−0.532094
$179$	3218.79	1.34404	0.672021	−	0.740532i	$-0.265426\pi$
$179$	3218.79	1.34404	0.672021	+	0.740532i	$0.265426\pi$
$180$	0	0
$181$	−456.885	−0.187624	−0.0938121	−	0.995590i	$-0.529905\pi$
$181$	−456.885	−0.187624	−0.0938121	+	0.995590i	$0.529905\pi$
$182$	801.711	0.326521
$183$	0	0
$184$	138.024	0.0553002
$185$	0	0
$186$	0	0
$187$	−474.445	−0.185534
$188$	−262.663	−0.101897
$189$	0	0
$190$	0	0
$191$	4907.11	1.85899	0.929493	−	0.368841i	$-0.120245\pi$
$191$	4907.11	1.85899	0.929493	+	0.368841i	$0.120245\pi$
$192$	0	0
$193$	1462.17	0.545333	0.272667	−	0.962109i	$-0.412094\pi$
$193$	1462.17	0.545333	0.272667	+	0.962109i	$0.412094\pi$
$194$	−113.099	−0.0418559
$195$	0	0
$196$	−2501.15	−0.911496
$197$	−2210.10	−0.799307	−0.399653	−	0.916666i	$-0.630869\pi$
$197$	−2210.10	−0.799307	−0.399653	+	0.916666i	$0.630869\pi$
$198$	0	0
$199$	−395.272	−0.140804	−0.0704022	−	0.997519i	$-0.522428\pi$
$199$	−395.272	−0.140804	−0.0704022	+	0.997519i	$0.522428\pi$
$200$	0	0
$201$	0	0
$202$	−922.086	−0.321177
$203$	−1786.36	−0.617625
$204$	0	0
$205$	0	0
$206$	3232.78	1.09339
$207$	0	0
$208$	−1874.19	−0.624769
$209$	71.0648	0.0235199
$210$	0	0
$211$	1559.98	0.508974	0.254487	−	0.967076i	$-0.418093\pi$
$211$	1559.98	0.508974	0.254487	+	0.967076i	$0.418093\pi$
$212$	−2099.78	−0.680253
$213$	0	0
$214$	−4228.72	−1.35079
$215$	0	0
$216$	0	0
$217$	445.534	0.139377
$218$	−1280.56	−0.397846
$219$	0	0
$220$	0	0
$221$	1317.17	0.400916
$222$	0	0
$223$	756.215	0.227085	0.113542	−	0.993533i	$-0.463780\pi$
$223$	756.215	0.227085	0.113542	+	0.993533i	$0.463780\pi$
$224$	−1677.53	−0.500379
$225$	0	0
$226$	5387.70	1.58577
$227$	5758.69	1.68378	0.841890	−	0.539649i	$-0.181443\pi$
$227$	5758.69	1.68378	0.841890	+	0.539649i	$0.181443\pi$
$228$	0	0
$229$	−1912.63	−0.551921	−0.275961	−	0.961169i	$-0.588996\pi$
$229$	−1912.63	−0.551921	−0.275961	+	0.961169i	$0.588996\pi$
$230$	0	0
$231$	0	0
$232$	−350.929	−0.0993086
$233$	−1034.86	−0.290970	−0.145485	−	0.989360i	$-0.546474\pi$
$233$	−1034.86	−0.290970	−0.145485	+	0.989360i	$0.546474\pi$
$234$	0	0
$235$	0	0
$236$	4643.13	1.28069
$237$	0	0
$238$	1132.30	0.308388
$239$	−2647.23	−0.716465	−0.358232	−	0.933632i	$-0.616620\pi$
$239$	−2647.23	−0.716465	−0.358232	+	0.933632i	$0.616620\pi$
$240$	0	0
$241$	6734.00	1.79990	0.899948	−	0.435996i	$-0.143604\pi$
$241$	6734.00	1.79990	0.899948	+	0.435996i	$0.143604\pi$
$242$	−488.757	−0.129829
$243$	0	0
$244$	2788.13	0.731523
$245$	0	0
$246$	0	0
$247$	−197.292	−0.0508235
$248$	87.5248	0.0224106
$249$	0	0
$250$	0	0
$251$	5377.57	1.35231	0.676154	−	0.736761i	$-0.263646\pi$
$251$	5377.57	1.35231	0.676154	+	0.736761i	$0.263646\pi$
$252$	0	0
$253$	1189.14	0.295497
$254$	−2281.76	−0.563663
$255$	0	0
$256$	3753.41	0.916360
$257$	−5882.33	−1.42774	−0.713871	−	0.700278i	$-0.753060\pi$
$257$	−5882.33	−1.42774	−0.713871	+	0.700278i	$0.753060\pi$
$258$	0	0
$259$	2613.99	0.627124
$260$	0	0
$261$	0	0
$262$	−864.744	−0.203909
$263$	4001.67	0.938227	0.469113	−	0.883138i	$-0.344574\pi$
$263$	4001.67	0.938227	0.469113	+	0.883138i	$0.344574\pi$
$264$	0	0
$265$	0	0
$266$	−169.602	−0.0390939
$267$	0	0
$268$	239.105	0.0544987
$269$	1758.29	0.398530	0.199265	−	0.979946i	$-0.436144\pi$
$269$	1758.29	0.398530	0.199265	+	0.979946i	$0.436144\pi$
$270$	0	0
$271$	6018.44	1.34906	0.674528	−	0.738249i	$-0.264347\pi$
$271$	6018.44	1.34906	0.674528	+	0.738249i	$0.264347\pi$
$272$	−2647.03	−0.590074
$273$	0	0
$274$	−263.480	−0.0580927
$275$	0	0
$276$	0	0
$277$	2910.30	0.631275	0.315637	−	0.948880i	$-0.397782\pi$
$277$	2910.30	0.631275	0.315637	+	0.948880i	$0.397782\pi$
$278$	1644.34	0.354752
$279$	0	0
$280$	0	0
$281$	−6893.19	−1.46339	−0.731697	−	0.681631i	$-0.761271\pi$
$281$	−6893.19	−1.46339	−0.731697	+	0.681631i	$0.761271\pi$
$282$	0	0
$283$	3120.16	0.655385	0.327693	−	0.944784i	$-0.393729\pi$
$283$	3120.16	0.655385	0.327693	+	0.944784i	$0.393729\pi$
$284$	−740.199	−0.154657
$285$	0	0
$286$	1356.90	0.280543
$287$	−1744.72	−0.358842
$288$	0	0
$289$	−3052.68	−0.621348
$290$	0	0
$291$	0	0
$292$	−5967.81	−1.19603
$293$	−8869.62	−1.76849	−0.884247	−	0.467020i	$-0.845328\pi$
$293$	−8869.62	−1.76849	−0.884247	+	0.467020i	$0.845328\pi$
$294$	0	0
$295$	0	0
$296$	513.515	0.100836
$297$	0	0
$298$	3199.51	0.621955
$299$	−3301.34	−0.638533
$300$	0	0
$301$	198.444	0.0380003
$302$	11559.7	2.20261
$303$	0	0
$304$	396.486	0.0748028
$305$	0	0
$306$	0	0
$307$	−5713.42	−1.06216	−0.531078	−	0.847323i	$-0.678213\pi$
$307$	−5713.42	−1.06216	−0.531078	+	0.847323i	$0.678213\pi$
$308$	594.529	0.109988
$309$	0	0
$310$	0	0
$311$	−1984.01	−0.361746	−0.180873	−	0.983506i	$-0.557892\pi$
$311$	−1984.01	−0.361746	−0.180873	+	0.983506i	$0.557892\pi$
$312$	0	0
$313$	−2675.50	−0.483156	−0.241578	−	0.970381i	$-0.577665\pi$
$313$	−2675.50	−0.483156	−0.241578	+	0.970381i	$0.577665\pi$
$314$	−12083.8	−2.17175
$315$	0	0
$316$	1669.05	0.297125
$317$	7083.76	1.25509	0.627545	−	0.778580i	$-0.284060\pi$
$317$	7083.76	1.25509	0.627545	+	0.778580i	$0.284060\pi$
$318$	0	0
$319$	−3023.43	−0.530657
$320$	0	0
$321$	0	0
$322$	−2837.99	−0.491165
$323$	−278.648	−0.0480012
$324$	0	0
$325$	0	0
$326$	−11744.7	−1.99534
$327$	0	0
$328$	−342.749	−0.0576986
$329$	205.278	0.0343992
$330$	0	0
$331$	7812.39	1.29730	0.648652	−	0.761085i	$-0.275333\pi$
$331$	7812.39	1.29730	0.648652	+	0.761085i	$0.275333\pi$
$332$	−2026.47	−0.334991
$333$	0	0
$334$	4444.82	0.728173
$335$	0	0
$336$	0	0
$337$	−11866.4	−1.91811	−0.959056	−	0.283218i	$-0.908598\pi$
$337$	−11866.4	−1.91811	−0.959056	+	0.283218i	$0.908598\pi$
$338$	5107.30	0.821895
$339$	0	0
$340$	0	0
$341$	754.071	0.119751
$342$	0	0
$343$	4183.94	0.658635
$344$	38.9841	0.00611012
$345$	0	0
$346$	15166.6	2.35654
$347$	−3485.50	−0.539225	−0.269613	−	0.962969i	$-0.586896\pi$
$347$	−3485.50	−0.539225	−0.269613	+	0.962969i	$0.586896\pi$
$348$	0	0
$349$	−6366.76	−0.976518	−0.488259	−	0.872699i	$-0.662368\pi$
$349$	−6366.76	−0.976518	−0.488259	+	0.872699i	$0.662368\pi$
$350$	0	0
$351$	0	0
$352$	−2839.24	−0.429920
$353$	−8223.15	−1.23987	−0.619935	−	0.784653i	$-0.712841\pi$
$353$	−8223.15	−1.23987	−0.619935	+	0.784653i	$0.712841\pi$
$354$	0	0
$355$	0	0
$356$	2601.54	0.387306
$357$	0	0
$358$	−13001.7	−1.91945
$359$	9377.54	1.37863	0.689314	−	0.724462i	$-0.257912\pi$
$359$	9377.54	1.37863	0.689314	+	0.724462i	$0.257912\pi$
$360$	0	0
$361$	−6817.26	−0.993915
$362$	1845.50	0.267949
$363$	0	0
$364$	−1650.55	−0.237671
$365$	0	0
$366$	0	0
$367$	−310.895	−0.0442196	−0.0221098	−	0.999756i	$-0.507038\pi$
$367$	−310.895	−0.0442196	−0.0221098	+	0.999756i	$0.507038\pi$
$368$	6634.49	0.939801
$369$	0	0
$370$	0	0
$371$	1641.03	0.229645
$372$	0	0
$373$	−4738.51	−0.657777	−0.328889	−	0.944369i	$-0.606674\pi$
$373$	−4738.51	−0.657777	−0.328889	+	0.944369i	$0.606674\pi$
$374$	1916.44	0.264964
$375$	0	0
$376$	40.3267	0.00553109
$377$	8393.74	1.14668
$378$	0	0
$379$	−4735.47	−0.641806	−0.320903	−	0.947112i	$-0.603986\pi$
$379$	−4735.47	−0.641806	−0.320903	+	0.947112i	$0.603986\pi$
$380$	0	0
$381$	0	0
$382$	−19821.4	−2.65484
$383$	−3632.14	−0.484579	−0.242289	−	0.970204i	$-0.577898\pi$
$383$	−3632.14	−0.484579	−0.242289	+	0.970204i	$0.577898\pi$
$384$	0	0
$385$	0	0
$386$	−5906.17	−0.778798
$387$	0	0
$388$	232.847	0.0304665
$389$	−6557.50	−0.854701	−0.427350	−	0.904086i	$-0.640553\pi$
$389$	−6557.50	−0.854701	−0.427350	+	0.904086i	$0.640553\pi$
$390$	0	0
$391$	−4662.68	−0.603073
$392$	384.001	0.0494769
$393$	0	0
$394$	8927.31	1.14150
$395$	0	0
$396$	0	0
$397$	4084.97	0.516420	0.258210	−	0.966089i	$-0.416867\pi$
$397$	4084.97	0.516420	0.258210	+	0.966089i	$0.416867\pi$
$398$	1596.63	0.201085
$399$	0	0
$400$	0	0
$401$	6676.04	0.831386	0.415693	−	0.909505i	$-0.363539\pi$
$401$	6676.04	0.831386	0.415693	+	0.909505i	$0.363539\pi$
$402$	0	0
$403$	−2093.47	−0.258768
$404$	1898.38	0.233782
$405$	0	0
$406$	7215.67	0.882039
$407$	4424.20	0.538819
$408$	0	0
$409$	1718.56	0.207769	0.103884	−	0.994589i	$-0.466873\pi$
$409$	1718.56	0.207769	0.103884	+	0.994589i	$0.466873\pi$
$410$	0	0
$411$	0	0
$412$	−6655.60	−0.795868
$413$	−3628.72	−0.432343
$414$	0	0
$415$	0	0
$416$	7882.39	0.929004
$417$	0	0
$418$	−287.053	−0.0335891
$419$	−5732.23	−0.668348	−0.334174	−	0.942511i	$-0.608457\pi$
$419$	−5732.23	−0.668348	−0.334174	+	0.942511i	$0.608457\pi$
$420$	0	0
$421$	−9647.38	−1.11683	−0.558414	−	0.829563i	$-0.688590\pi$
$421$	−9647.38	−1.11683	−0.558414	+	0.829563i	$0.688590\pi$
$422$	−6301.26	−0.726873
$423$	0	0
$424$	322.379	0.0369248
$425$	0	0
$426$	0	0
$427$	−2178.99	−0.246953
$428$	8706.03	0.983229
$429$	0	0
$430$	0	0
$431$	3054.57	0.341376	0.170688	−	0.985325i	$-0.445401\pi$
$431$	3054.57	0.341376	0.170688	+	0.985325i	$0.445401\pi$
$432$	0	0
$433$	11022.8	1.22337	0.611686	−	0.791100i	$-0.290491\pi$
$433$	11022.8	1.22337	0.611686	+	0.791100i	$0.290491\pi$
$434$	−1799.65	−0.199046
$435$	0	0
$436$	2636.39	0.289588
$437$	698.399	0.0764507
$438$	0	0
$439$	−7589.30	−0.825097	−0.412548	−	0.910936i	$-0.635361\pi$
$439$	−7589.30	−0.825097	−0.412548	+	0.910936i	$0.635361\pi$
$440$	0	0
$441$	0	0
$442$	−5320.47	−0.572554
$443$	−4043.30	−0.433641	−0.216821	−	0.976211i	$-0.569569\pi$
$443$	−4043.30	−0.433641	−0.216821	+	0.976211i	$0.569569\pi$
$444$	0	0
$445$	0	0
$446$	−3054.59	−0.324303
$447$	0	0
$448$	3585.16	0.378086
$449$	−14619.3	−1.53658	−0.768291	−	0.640101i	$-0.778893\pi$
$449$	−14619.3	−1.53658	−0.768291	+	0.640101i	$0.778893\pi$
$450$	0	0
$451$	−2952.96	−0.308313
$452$	−11092.1	−1.15427
$453$	0	0
$454$	−23261.2	−2.40463
$455$	0	0
$456$	0	0
$457$	−10910.8	−1.11682	−0.558411	−	0.829565i	$-0.688589\pi$
$457$	−10910.8	−1.11682	−0.558411	+	0.829565i	$0.688589\pi$
$458$	7725.71	0.788207
$459$	0	0
$460$	0	0
$461$	−16371.0	−1.65395	−0.826977	−	0.562236i	$-0.809941\pi$
$461$	−16371.0	−1.65395	−0.826977	+	0.562236i	$0.809941\pi$
$462$	0	0
$463$	−16023.5	−1.60837	−0.804186	−	0.594377i	$-0.797399\pi$
$463$	−16023.5	−1.60837	−0.804186	+	0.594377i	$0.797399\pi$
$464$	−16868.4	−1.68770
$465$	0	0
$466$	4180.13	0.415538
$467$	15097.3	1.49597	0.747985	−	0.663716i	$-0.231022\pi$
$467$	15097.3	1.49597	0.747985	+	0.663716i	$0.231022\pi$
$468$	0	0
$469$	−186.866	−0.0183981
$470$	0	0
$471$	0	0
$472$	−712.859	−0.0695170
$473$	335.868	0.0326495
$474$	0	0
$475$	0	0
$476$	−2331.17	−0.224473
$477$	0	0
$478$	10693.0	1.02319
$479$	10459.0	0.997667	0.498834	−	0.866698i	$-0.333762\pi$
$479$	10459.0	0.997667	0.498834	+	0.866698i	$0.333762\pi$
$480$	0	0
$481$	−12282.6	−1.16432
$482$	−27200.8	−2.57046
$483$	0	0
$484$	1006.25	0.0945010
$485$	0	0
$486$	0	0
$487$	−7939.86	−0.738788	−0.369394	−	0.929273i	$-0.620435\pi$
$487$	−7939.86	−0.738788	−0.369394	+	0.929273i	$0.620435\pi$
$488$	−428.061	−0.0397078
$489$	0	0
$490$	0	0
$491$	−11492.7	−1.05633	−0.528167	−	0.849141i	$-0.677120\pi$
$491$	−11492.7	−1.05633	−0.528167	+	0.849141i	$0.677120\pi$
$492$	0	0
$493$	11855.0	1.08300
$494$	796.927	0.0725818
$495$	0	0
$496$	4207.13	0.380858
$497$	578.484	0.0522103
$498$	0	0
$499$	4813.54	0.431831	0.215915	−	0.976412i	$-0.430726\pi$
$499$	4813.54	0.431831	0.215915	+	0.976412i	$0.430726\pi$
$500$	0	0
$501$	0	0
$502$	−21721.7	−1.93125
$503$	−17756.3	−1.57399	−0.786993	−	0.616962i	$-0.788363\pi$
$503$	−17756.3	−1.57399	−0.786993	+	0.616962i	$0.788363\pi$
$504$	0	0
$505$	0	0
$506$	−4803.33	−0.422004
$507$	0	0
$508$	4697.66	0.410285
$509$	−3183.05	−0.277183	−0.138592	−	0.990350i	$-0.544258\pi$
$509$	−3183.05	−0.277183	−0.138592	+	0.990350i	$0.544258\pi$
$510$	0	0
$511$	4663.99	0.403763
$512$	−16467.6	−1.42143
$513$	0	0
$514$	23760.6	2.03898
$515$	0	0
$516$	0	0
$517$	347.435	0.0295554
$518$	−10558.7	−0.895605
$519$	0	0
$520$	0	0
$521$	−7246.60	−0.609365	−0.304683	−	0.952454i	$-0.598550\pi$
$521$	−7246.60	−0.609365	−0.304683	+	0.952454i	$0.598550\pi$
$522$	0	0
$523$	−17985.0	−1.50369	−0.751845	−	0.659340i	$-0.770836\pi$
$523$	−17985.0	−1.50369	−0.751845	+	0.659340i	$0.770836\pi$
$524$	1780.32	0.148423
$525$	0	0
$526$	−16164.0	−1.33990
$527$	−2956.74	−0.244398
$528$	0	0
$529$	−480.530	−0.0394945
$530$	0	0
$531$	0	0
$532$	349.175	0.0284561
$533$	8198.09	0.666226
$534$	0	0
$535$	0	0
$536$	−36.7097	−0.00295825
$537$	0	0
$538$	−7102.28	−0.569147
$539$	3308.36	0.264381
$540$	0	0
$541$	−14549.3	−1.15624	−0.578119	−	0.815952i	$-0.696213\pi$
$541$	−14549.3	−1.15624	−0.578119	+	0.815952i	$0.696213\pi$
$542$	−24310.4	−1.92661
$543$	0	0
$544$	11132.8	0.877414
$545$	0	0
$546$	0	0
$547$	5487.29	0.428920	0.214460	−	0.976733i	$-0.431201\pi$
$547$	5487.29	0.428920	0.214460	+	0.976733i	$0.431201\pi$
$548$	542.448	0.0422851
$549$	0	0
$550$	0	0
$551$	−1775.70	−0.137291
$552$	0	0
$553$	−1304.41	−0.100306
$554$	−11755.6	−0.901532
$555$	0	0
$556$	−3385.35	−0.258221
$557$	7366.60	0.560382	0.280191	−	0.959944i	$-0.409602\pi$
$557$	7366.60	0.560382	0.280191	+	0.959944i	$0.409602\pi$
$558$	0	0
$559$	−932.446	−0.0705515
$560$	0	0
$561$	0	0
$562$	27843.8	2.08989
$563$	4516.14	0.338068	0.169034	−	0.985610i	$-0.445935\pi$
$563$	4516.14	0.338068	0.169034	+	0.985610i	$0.445935\pi$
$564$	0	0
$565$	0	0
$566$	−12603.3	−0.935965
$567$	0	0
$568$	113.643	0.00839496
$569$	−2174.35	−0.160200	−0.0800999	−	0.996787i	$-0.525524\pi$
$569$	−2174.35	−0.160200	−0.0800999	+	0.996787i	$0.525524\pi$
$570$	0	0
$571$	−25339.7	−1.85715	−0.928575	−	0.371145i	$-0.878965\pi$
$571$	−25339.7	−1.85715	−0.928575	+	0.371145i	$0.878965\pi$
$572$	−2793.57	−0.204205
$573$	0	0
$574$	7047.48	0.512467
$575$	0	0
$576$	0	0
$577$	−21909.1	−1.58074	−0.790370	−	0.612630i	$-0.790112\pi$
$577$	−21909.1	−1.58074	−0.790370	+	0.612630i	$0.790112\pi$
$578$	12330.8	0.887356
$579$	0	0
$580$	0	0
$581$	1583.74	0.113089
$582$	0	0
$583$	2777.46	0.197308
$584$	916.237	0.0649215
$585$	0	0
$586$	35827.2	2.52561
$587$	922.235	0.0648462	0.0324231	−	0.999474i	$-0.489678\pi$
$587$	922.235	0.0648462	0.0324231	+	0.999474i	$0.489678\pi$
$588$	0	0
$589$	442.875	0.0309819
$590$	0	0
$591$	0	0
$592$	24683.6	1.71366
$593$	9075.86	0.628501	0.314250	−	0.949340i	$-0.398247\pi$
$593$	9075.86	0.628501	0.314250	+	0.949340i	$0.398247\pi$
$594$	0	0
$595$	0	0
$596$	−6587.10	−0.452715
$597$	0	0
$598$	13335.2	0.911898
$599$	−19987.0	−1.36335	−0.681675	−	0.731655i	$-0.738748\pi$
$599$	−19987.0	−1.36335	−0.681675	+	0.731655i	$0.738748\pi$
$600$	0	0
$601$	11116.7	0.754509	0.377255	−	0.926110i	$-0.376868\pi$
$601$	11116.7	0.754509	0.377255	+	0.926110i	$0.376868\pi$
$602$	−801.577	−0.0542688
$603$	0	0
$604$	−23799.0	−1.60326
$605$	0	0
$606$	0	0
$607$	19355.0	1.29423	0.647114	−	0.762393i	$-0.275976\pi$
$607$	19355.0	1.29423	0.647114	+	0.762393i	$0.275976\pi$
$608$	−1667.52	−0.111228
$609$	0	0
$610$	0	0
$611$	−964.559	−0.0638656
$612$	0	0
$613$	−12325.5	−0.812111	−0.406056	−	0.913848i	$-0.633096\pi$
$613$	−12325.5	−0.812111	−0.406056	+	0.913848i	$0.633096\pi$
$614$	23078.3	1.51688
$615$	0	0
$616$	−91.2780	−0.00597029
$617$	−28207.0	−1.84047	−0.920236	−	0.391363i	$-0.872004\pi$
$617$	−28207.0	−1.84047	−0.920236	+	0.391363i	$0.872004\pi$
$618$	0	0
$619$	17236.9	1.11924	0.559621	−	0.828748i	$-0.310947\pi$
$619$	17236.9	1.11924	0.559621	+	0.828748i	$0.310947\pi$
$620$	0	0
$621$	0	0
$622$	8014.06	0.516615
$623$	−2033.16	−0.130750
$624$	0	0
$625$	0	0
$626$	10807.2	0.690003
$627$	0	0
$628$	24878.0	1.58080
$629$	−17347.4	−1.09966
$630$	0	0
$631$	−23582.7	−1.48782	−0.743908	−	0.668282i	$-0.767030\pi$
$631$	−23582.7	−1.48782	−0.743908	+	0.668282i	$0.767030\pi$
$632$	−256.250	−0.0161283
$633$	0	0
$634$	−28613.6	−1.79241
$635$	0	0
$636$	0	0
$637$	−9184.77	−0.571294
$638$	12212.6	0.757839
$639$	0	0
$640$	0	0
$641$	13003.1	0.801233	0.400617	−	0.916246i	$-0.368796\pi$
$641$	13003.1	0.801233	0.400617	+	0.916246i	$0.368796\pi$
$642$	0	0
$643$	−21269.1	−1.30446	−0.652232	−	0.758019i	$-0.726167\pi$
$643$	−21269.1	−1.30446	−0.652232	+	0.758019i	$0.726167\pi$
$644$	5842.82	0.357514
$645$	0	0
$646$	1125.55	0.0685511
$647$	−29247.2	−1.77716	−0.888582	−	0.458717i	$-0.848309\pi$
$647$	−29247.2	−1.77716	−0.888582	+	0.458717i	$0.848309\pi$
$648$	0	0
$649$	−6141.64	−0.371465
$650$	0	0
$651$	0	0
$652$	24179.9	1.45239
$653$	1678.30	0.100577	0.0502885	−	0.998735i	$-0.483986\pi$
$653$	1678.30	0.100577	0.0502885	+	0.998735i	$0.483986\pi$
$654$	0	0
$655$	0	0
$656$	−16475.2	−0.980561
$657$	0	0
$658$	−829.182	−0.0491260
$659$	7620.41	0.450454	0.225227	−	0.974306i	$-0.427688\pi$
$659$	7620.41	0.450454	0.225227	+	0.974306i	$0.427688\pi$
$660$	0	0
$661$	−22762.2	−1.33941	−0.669703	−	0.742629i	$-0.733579\pi$
$661$	−22762.2	−1.33941	−0.669703	+	0.742629i	$0.733579\pi$
$662$	−31556.7	−1.85270
$663$	0	0
$664$	311.124	0.0181837
$665$	0	0
$666$	0	0
$667$	−29713.2	−1.72488
$668$	−9150.93	−0.530030
$669$	0	0
$670$	0	0
$671$	−3687.96	−0.212179
$672$	0	0
$673$	698.528	0.0400093	0.0200047	−	0.999800i	$-0.493632\pi$
$673$	698.528	0.0400093	0.0200047	+	0.999800i	$0.493632\pi$
$674$	47932.1	2.73928
$675$	0	0
$676$	−10514.8	−0.598250
$677$	10860.1	0.616524	0.308262	−	0.951302i	$-0.400253\pi$
$677$	10860.1	0.616524	0.308262	+	0.951302i	$0.400253\pi$
$678$	0	0
$679$	−181.976	−0.0102851
$680$	0	0
$681$	0	0
$682$	−3045.93	−0.171019
$683$	6534.95	0.366110	0.183055	−	0.983103i	$-0.441401\pi$
$683$	6534.95	0.366110	0.183055	+	0.983103i	$0.441401\pi$
$684$	0	0
$685$	0	0
$686$	−16900.3	−0.940606
$687$	0	0
$688$	1873.88	0.103839
$689$	−7710.88	−0.426359
$690$	0	0
$691$	−10915.3	−0.600924	−0.300462	−	0.953794i	$-0.597141\pi$
$691$	−10915.3	−0.600924	−0.300462	+	0.953794i	$0.597141\pi$
$692$	−31224.9	−1.71530
$693$	0	0
$694$	14079.0	0.770076
$695$	0	0
$696$	0	0
$697$	11578.6	0.629229
$698$	25717.3	1.39458
$699$	0	0
$700$	0	0
$701$	−18405.4	−0.991674	−0.495837	−	0.868416i	$-0.665139\pi$
$701$	−18405.4	−0.991674	−0.495837	+	0.868416i	$0.665139\pi$
$702$	0	0
$703$	2598.39	0.139403
$704$	6067.91	0.324848
$705$	0	0
$706$	33215.9	1.77068
$707$	−1483.63	−0.0789217
$708$	0	0
$709$	−21436.2	−1.13547	−0.567737	−	0.823210i	$-0.692181\pi$
$709$	−21436.2	−1.13547	−0.567737	+	0.823210i	$0.692181\pi$
$710$	0	0
$711$	0	0
$712$	−399.413	−0.0210234
$713$	7410.73	0.389248
$714$	0	0
$715$	0	0
$716$	26767.7	1.39715
$717$	0	0
$718$	−37878.9	−1.96884
$719$	−28641.7	−1.48561	−0.742807	−	0.669506i	$-0.766506\pi$
$719$	−28641.7	−1.48561	−0.742807	+	0.669506i	$0.766506\pi$
$720$	0	0
$721$	5201.51	0.268675
$722$	27537.1	1.41942
$723$	0	0
$724$	−3799.49	−0.195037
$725$	0	0
$726$	0	0
$727$	−1723.13	−0.0879055	−0.0439527	−	0.999034i	$-0.513995\pi$
$727$	−1723.13	−0.0879055	−0.0439527	+	0.999034i	$0.513995\pi$
$728$	253.409	0.0129010
$729$	0	0
$730$	0	0
$731$	−1316.95	−0.0666335
$732$	0	0
$733$	−18813.4	−0.948005	−0.474003	−	0.880523i	$-0.657191\pi$
$733$	−18813.4	−0.948005	−0.474003	+	0.880523i	$0.657191\pi$
$734$	1255.80	0.0631506
$735$	0	0
$736$	−27903.0	−1.39744
$737$	−316.273	−0.0158074
$738$	0	0
$739$	−10893.3	−0.542242	−0.271121	−	0.962545i	$-0.587394\pi$
$739$	−10893.3	−0.542242	−0.271121	+	0.962545i	$0.587394\pi$
$740$	0	0
$741$	0	0
$742$	−6628.65	−0.327959
$743$	−11888.6	−0.587013	−0.293507	−	0.955957i	$-0.594822\pi$
$743$	−11888.6	−0.587013	−0.293507	+	0.955957i	$0.594822\pi$
$744$	0	0
$745$	0	0
$746$	19140.4	0.939381
$747$	0	0
$748$	−3945.53	−0.192865
$749$	−6803.98	−0.331925
$750$	0	0
$751$	−32134.8	−1.56140	−0.780702	−	0.624903i	$-0.785138\pi$
$751$	−32134.8	−1.56140	−0.780702	+	0.624903i	$0.785138\pi$
$752$	1938.41	0.0939982
$753$	0	0
$754$	−33905.0	−1.63759
$755$	0	0
$756$	0	0
$757$	17366.1	0.833794	0.416897	−	0.908954i	$-0.363118\pi$
$757$	17366.1	0.833794	0.416897	+	0.908954i	$0.363118\pi$
$758$	19128.1	0.916573
$759$	0	0
$760$	0	0
$761$	27393.9	1.30490	0.652450	−	0.757832i	$-0.273741\pi$
$761$	27393.9	1.30490	0.652450	+	0.757832i	$0.273741\pi$
$762$	0	0
$763$	−2060.41	−0.0977611
$764$	40808.0	1.93243
$765$	0	0
$766$	14671.4	0.692034
$767$	17050.6	0.802689
$768$	0	0
$769$	−12091.3	−0.567000	−0.283500	−	0.958972i	$-0.591496\pi$
$769$	−12091.3	−0.567000	−0.283500	+	0.958972i	$0.591496\pi$
$770$	0	0
$771$	0	0
$772$	12159.5	0.566879
$773$	−18895.8	−0.879216	−0.439608	−	0.898190i	$-0.644883\pi$
$773$	−18895.8	−0.879216	−0.439608	+	0.898190i	$0.644883\pi$
$774$	0	0
$775$	0	0
$776$	−35.7490	−0.00165375
$777$	0	0
$778$	26487.8	1.22061
$779$	−1734.31	−0.0797664
$780$	0	0
$781$	979.089	0.0448586
$782$	18834.0	0.861258
$783$	0	0
$784$	18458.1	0.840838
$785$	0	0
$786$	0	0
$787$	23705.9	1.07373	0.536863	−	0.843669i	$-0.319609\pi$
$787$	23705.9	1.07373	0.536863	+	0.843669i	$0.319609\pi$
$788$	−18379.4	−0.830888
$789$	0	0
$790$	0	0
$791$	8668.77	0.389666
$792$	0	0
$793$	10238.6	0.458493
$794$	−16500.5	−0.737507
$795$	0	0
$796$	−3287.11	−0.146368
$797$	37248.0	1.65545	0.827725	−	0.561135i	$-0.189635\pi$
$797$	37248.0	1.65545	0.827725	+	0.561135i	$0.189635\pi$
$798$	0	0
$799$	−1362.30	−0.0603189
$800$	0	0
$801$	0	0
$802$	−26966.7	−1.18731
$803$	7893.84	0.346909
$804$	0	0
$805$	0	0
$806$	8456.21	0.369550
$807$	0	0
$808$	−291.458	−0.0126899
$809$	24861.0	1.08043	0.540214	−	0.841528i	$-0.318343\pi$
$809$	24861.0	1.08043	0.540214	+	0.841528i	$0.318343\pi$
$810$	0	0
$811$	4131.24	0.178875	0.0894375	−	0.995992i	$-0.471493\pi$
$811$	4131.24	0.178875	0.0894375	+	0.995992i	$0.471493\pi$
$812$	−14855.5	−0.642027
$813$	0	0
$814$	−17870.7	−0.769495
$815$	0	0
$816$	0	0
$817$	197.259	0.00844704
$818$	−6941.81	−0.296717
$819$	0	0
$820$	0	0
$821$	−25297.3	−1.07537	−0.537687	−	0.843145i	$-0.680702\pi$
$821$	−25297.3	−1.07537	−0.537687	+	0.843145i	$0.680702\pi$
$822$	0	0
$823$	−18155.5	−0.768969	−0.384484	−	0.923131i	$-0.625621\pi$
$823$	−18155.5	−0.768969	−0.384484	+	0.923131i	$0.625621\pi$
$824$	1021.83	0.0432005
$825$	0	0
$826$	14657.6	0.617435
$827$	29566.8	1.24321	0.621607	−	0.783329i	$-0.286480\pi$
$827$	29566.8	1.24321	0.621607	+	0.783329i	$0.286480\pi$
$828$	0	0
$829$	27249.4	1.14163	0.570814	−	0.821079i	$-0.306628\pi$
$829$	27249.4	1.14163	0.570814	+	0.821079i	$0.306628\pi$
$830$	0	0
$831$	0	0
$832$	−16845.9	−0.701956
$833$	−12972.2	−0.539568
$834$	0	0
$835$	0	0
$836$	590.981	0.0244492
$837$	0	0
$838$	23154.3	0.954478
$839$	15506.1	0.638058	0.319029	−	0.947745i	$-0.396643\pi$
$839$	15506.1	0.638058	0.319029	+	0.947745i	$0.396643\pi$
$840$	0	0
$841$	51157.5	2.09756
$842$	38968.8	1.59496
$843$	0	0
$844$	12972.9	0.529084
$845$	0	0
$846$	0	0
$847$	−786.406	−0.0319023
$848$	15496.1	0.627520
$849$	0	0
$850$	0	0
$851$	43479.4	1.75141
$852$	0	0
$853$	8698.16	0.349143	0.174572	−	0.984644i	$-0.444146\pi$
$853$	8698.16	0.349143	0.174572	+	0.984644i	$0.444146\pi$
$854$	8801.64	0.352677
$855$	0	0
$856$	−1336.64	−0.0533707
$857$	28468.8	1.13474	0.567372	−	0.823462i	$-0.307960\pi$
$857$	28468.8	1.13474	0.567372	+	0.823462i	$0.307960\pi$
$858$	0	0
$859$	−21154.3	−0.840251	−0.420125	−	0.907466i	$-0.638014\pi$
$859$	−21154.3	−0.840251	−0.420125	+	0.907466i	$0.638014\pi$
$860$	0	0
$861$	0	0
$862$	−12338.4	−0.487525
$863$	45959.5	1.81284	0.906420	−	0.422378i	$-0.138805\pi$
$863$	45959.5	1.81284	0.906420	+	0.422378i	$0.138805\pi$
$864$	0	0
$865$	0	0
$866$	−44524.5	−1.74712
$867$	0	0
$868$	3705.10	0.144884
$869$	−2207.72	−0.0861815
$870$	0	0
$871$	878.047	0.0341579
$872$	−404.765	−0.0157191
$873$	0	0
$874$	−2821.06	−0.109180
$875$	0	0
$876$	0	0
$877$	−7315.10	−0.281657	−0.140829	−	0.990034i	$-0.544977\pi$
$877$	−7315.10	−0.281657	−0.140829	+	0.990034i	$0.544977\pi$
$878$	30655.6	1.17833
$879$	0	0
$880$	0	0
$881$	−4652.19	−0.177907	−0.0889536	−	0.996036i	$-0.528352\pi$
$881$	−4652.19	−0.177907	−0.0889536	+	0.996036i	$0.528352\pi$
$882$	0	0
$883$	−38954.3	−1.48462	−0.742309	−	0.670058i	$-0.766269\pi$
$883$	−38954.3	−1.48462	−0.742309	+	0.670058i	$0.766269\pi$
$884$	10953.7	0.416756
$885$	0	0
$886$	16332.2	0.619289
$887$	16917.5	0.640398	0.320199	−	0.947350i	$-0.396250\pi$
$887$	16917.5	0.640398	0.320199	+	0.947350i	$0.396250\pi$
$888$	0	0
$889$	−3671.33	−0.138507
$890$	0	0
$891$	0	0
$892$	6288.75	0.236057
$893$	204.053	0.00764654
$894$	0	0
$895$	0	0
$896$	−1061.32	−0.0395715
$897$	0	0
$898$	59051.8	2.19441
$899$	−18842.0	−0.699016
$900$	0	0
$901$	−10890.5	−0.402682
$902$	11927.9	0.440306
$903$	0	0
$904$	1702.97	0.0626549
$905$	0	0
$906$	0	0
$907$	38985.4	1.42722	0.713610	−	0.700543i	$-0.247059\pi$
$907$	38985.4	1.42722	0.713610	+	0.700543i	$0.247059\pi$
$908$	47889.8	1.75031
$909$	0	0
$910$	0	0
$911$	−27759.8	−1.00957	−0.504787	−	0.863244i	$-0.668429\pi$
$911$	−27759.8	−1.00957	−0.504787	+	0.863244i	$0.668429\pi$
$912$	0	0
$913$	2680.49	0.0971646
$914$	44072.3	1.59495
$915$	0	0
$916$	−15905.6	−0.573728
$917$	−1391.37	−0.0501057
$918$	0	0
$919$	29705.5	1.06626	0.533130	−	0.846033i	$-0.321016\pi$
$919$	29705.5	1.06626	0.533130	+	0.846033i	$0.321016\pi$
$920$	0	0
$921$	0	0
$922$	66127.6	2.36204
$923$	−2718.18	−0.0969338
$924$	0	0
$925$	0	0
$926$	64724.1	2.29694
$927$	0	0
$928$	70944.1	2.50954
$929$	−2940.01	−0.103831	−0.0519153	−	0.998651i	$-0.516533\pi$
$929$	−2940.01	−0.103831	−0.0519153	+	0.998651i	$0.516533\pi$
$930$	0	0
$931$	1943.04	0.0684002
$932$	−8605.99	−0.302466
$933$	0	0
$934$	−60982.6	−2.13642
$935$	0	0
$936$	0	0
$937$	−20041.9	−0.698762	−0.349381	−	0.936981i	$-0.613608\pi$
$937$	−20041.9	−0.698762	−0.349381	+	0.936981i	$0.613608\pi$
$938$	754.813	0.0262745
$939$	0	0
$940$	0	0
$941$	14446.1	0.500457	0.250229	−	0.968187i	$-0.419494\pi$
$941$	14446.1	0.500457	0.250229	+	0.968187i	$0.419494\pi$
$942$	0	0
$943$	−29020.6	−1.00216
$944$	−34265.6	−1.18141
$945$	0	0
$946$	−1356.68	−0.0466272
$947$	−3901.33	−0.133871	−0.0669357	−	0.997757i	$-0.521322\pi$
$947$	−3901.33	−0.133871	−0.0669357	+	0.997757i	$0.521322\pi$
$948$	0	0
$949$	−21915.1	−0.749626
$950$	0	0
$951$	0	0
$952$	357.904	0.0121846
$953$	−613.963	−0.0208691	−0.0104345	−	0.999946i	$-0.503321\pi$
$953$	−613.963	−0.0208691	−0.0104345	+	0.999946i	$0.503321\pi$
$954$	0	0
$955$	0	0
$956$	−22014.6	−0.744773
$957$	0	0
$958$	−42247.1	−1.42478
$959$	−423.937	−0.0142749
$960$	0	0
$961$	−25091.6	−0.842256
$962$	49613.3	1.66278
$963$	0	0
$964$	56000.5	1.87101
$965$	0	0
$966$	0	0
$967$	−20617.4	−0.685637	−0.342819	−	0.939402i	$-0.611382\pi$
$967$	−20617.4	−0.685637	−0.342819	+	0.939402i	$0.611382\pi$
$968$	−154.489	−0.00512961
$969$	0	0
$970$	0	0
$971$	−22816.0	−0.754069	−0.377035	−	0.926199i	$-0.623056\pi$
$971$	−22816.0	−0.754069	−0.377035	+	0.926199i	$0.623056\pi$
$972$	0	0
$973$	2645.73	0.0871719
$974$	32071.6	1.05507
$975$	0	0
$976$	−20576.0	−0.674816
$977$	−22650.2	−0.741702	−0.370851	−	0.928692i	$-0.620934\pi$
$977$	−22650.2	−0.741702	−0.370851	+	0.928692i	$0.620934\pi$
$978$	0	0
$979$	−3441.15	−0.112339
$980$	0	0
$981$	0	0
$982$	46422.8	1.50857
$983$	8963.01	0.290820	0.145410	−	0.989372i	$-0.453550\pi$
$983$	8963.01	0.290820	0.145410	+	0.989372i	$0.453550\pi$
$984$	0	0
$985$	0	0
$986$	−47886.0	−1.54665
$987$	0	0
$988$	−1640.70	−0.0528316
$989$	3300.78	0.106126
$990$	0	0
$991$	422.204	0.0135336	0.00676678	−	0.999977i	$-0.497846\pi$
$991$	422.204	0.0135336	0.00676678	+	0.999977i	$0.497846\pi$
$992$	−17694.1	−0.566319
$993$	0	0
$994$	−2336.68	−0.0745623
$995$	0	0
$996$	0	0
$997$	−12577.3	−0.399524	−0.199762	−	0.979844i	$-0.564017\pi$
$997$	−12577.3	−0.399524	−0.199762	+	0.979844i	$0.564017\pi$
$998$	−19443.4	−0.616704
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.v.1.1		3
3.2	odd	2		825.4.a.q.1.3	yes	3
5.4	even	2		2475.4.a.y.1.3		3
15.2	even	4		825.4.c.n.199.6		6
15.8	even	4		825.4.c.n.199.1		6
15.14	odd	2		825.4.a.o.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.4.a.o.1.1	✓	3	15.14	odd	2
825.4.a.q.1.3	yes	3	3.2	odd	2
825.4.c.n.199.1		6	15.8	even	4
825.4.c.n.199.6		6	15.2	even	4
2475.4.a.v.1.1		3	1.1	even	1	trivial
2475.4.a.y.1.3		3	5.4	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q-4.03932 q^{2} +8.31608 q^{4} -6.49923 q^{7} -1.27677 q^{8} +O(q^{10})\)	\(q-4.03932 q^{2} +8.31608 q^{4} -6.49923 q^{7} -1.27677 q^{8} -11.0000 q^{11} +30.5385 q^{13} +26.2524 q^{14} -61.3714 q^{16} +43.1314 q^{17} -6.46044 q^{19} +44.4325 q^{22} -108.104 q^{23} -123.355 q^{26} -54.0481 q^{28} +274.857 q^{29} -68.5519 q^{31} +258.113 q^{32} -174.221 q^{34} -402.200 q^{37} +26.0958 q^{38} +268.450 q^{41} -30.5334 q^{43} -91.4769 q^{44} +436.666 q^{46} -31.5850 q^{47} -300.760 q^{49} +253.961 q^{52} -252.497 q^{53} +8.29800 q^{56} -1110.24 q^{58} +558.331 q^{59} +335.270 q^{61} +276.903 q^{62} -551.628 q^{64} +28.7521 q^{67} +358.684 q^{68} -89.0081 q^{71} -717.622 q^{73} +1624.61 q^{74} -53.7256 q^{76} +71.4915 q^{77} +200.702 q^{79} -1084.36 q^{82} -243.681 q^{83} +123.334 q^{86} +14.0444 q^{88} +312.832 q^{89} -198.477 q^{91} -899.002 q^{92} +127.582 q^{94} +27.9996 q^{97} +1214.87 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 7 q^{4} + 16 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q - q^2 + 7 * q^4 + 16 * q^7 + 3 * q^8	\( 3 q - q^{2} + 7 q^{4} + 16 q^{7} + 3 q^{8} - 33 q^{11} + 45 q^{13} + 116 q^{14} - 85 q^{16} + 58 q^{17} - 169 q^{19} + 11 q^{22} - 155 q^{23} - 167 q^{26} + 100 q^{28} + 277 q^{29} - 173 q^{31} - 97 q^{32} - 146 q^{34} + 60 q^{37} - 169 q^{38} - 44 q^{41} - 109 q^{43} - 77 q^{44} + 425 q^{46} - 270 q^{47} - 427 q^{49} + 45 q^{52} + 148 q^{53} - 168 q^{56} - 783 q^{58} + 684 q^{59} - 1038 q^{61} - 953 q^{62} - 1129 q^{64} + 314 q^{67} + 366 q^{68} + 1459 q^{71} - 1170 q^{73} + 1764 q^{74} + 211 q^{76} - 176 q^{77} - 506 q^{79} - 1040 q^{82} + 347 q^{83} + 527 q^{86} - 33 q^{88} + 607 q^{89} - 398 q^{91} - 687 q^{92} - 1822 q^{94} + 1263 q^{97} + 2273 q^{98}+O(q^{100}) \) 3 * q - q^2 + 7 * q^4 + 16 * q^7 + 3 * q^8 - 33 * q^11 + 45 * q^13 + 116 * q^14 - 85 * q^16 + 58 * q^17 - 169 * q^19 + 11 * q^22 - 155 * q^23 - 167 * q^26 + 100 * q^28 + 277 * q^29 - 173 * q^31 - 97 * q^32 - 146 * q^34 + 60 * q^37 - 169 * q^38 - 44 * q^41 - 109 * q^43 - 77 * q^44 + 425 * q^46 - 270 * q^47 - 427 * q^49 + 45 * q^52 + 148 * q^53 - 168 * q^56 - 783 * q^58 + 684 * q^59 - 1038 * q^61 - 953 * q^62 - 1129 * q^64 + 314 * q^67 + 366 * q^68 + 1459 * q^71 - 1170 * q^73 + 1764 * q^74 + 211 * q^76 - 176 * q^77 - 506 * q^79 - 1040 * q^82 + 347 * q^83 + 527 * q^86 - 33 * q^88 + 607 * q^89 - 398 * q^91 - 687 * q^92 - 1822 * q^94 + 1263 * q^97 + 2273 * q^98

Introduction

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