LMFDB - Embedded newform 2475.4.a.q.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q-0.732051 q^{2} -7.46410 q^{4} -16.9282 q^{7} +11.3205 q^{8} +O(q^{10})$	$q-0.732051 q^{2} -7.46410 q^{4} -16.9282 q^{7} +11.3205 q^{8} +11.0000 q^{11} -74.6410 q^{13} +12.3923 q^{14} +51.4256 q^{16} -82.7846 q^{17} -67.9230 q^{19} -8.05256 q^{22} +13.3538 q^{23} +54.6410 q^{26} +126.354 q^{28} -168.995 q^{29} -65.4974 q^{31} -128.210 q^{32} +60.6025 q^{34} -40.8564 q^{37} +49.7231 q^{38} -274.928 q^{41} +2.28719 q^{43} -82.1051 q^{44} -9.77568 q^{46} +71.8461 q^{47} -56.4359 q^{49} +557.128 q^{52} -149.005 q^{53} -191.636 q^{56} +123.713 q^{58} -545.631 q^{59} +101.303 q^{61} +47.9474 q^{62} -317.549 q^{64} -411.641 q^{67} +617.913 q^{68} +470.636 q^{71} -610.600 q^{73} +29.9090 q^{74} +506.985 q^{76} -186.210 q^{77} -978.225 q^{79} +201.261 q^{82} +26.1539 q^{83} -1.67434 q^{86} +124.526 q^{88} +352.887 q^{89} +1263.54 q^{91} -99.6743 q^{92} -52.5950 q^{94} -847.585 q^{97} +41.3140 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 8 q^{4} - 20 q^{7} - 12 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 - 8 * q^4 - 20 * q^7 - 12 * q^8	$ 2 q + 2 q^{2} - 8 q^{4} - 20 q^{7} - 12 q^{8} + 22 q^{11} - 80 q^{13} + 4 q^{14} - 8 q^{16} - 124 q^{17} + 72 q^{19} + 22 q^{22} - 98 q^{23} + 40 q^{26} + 128 q^{28} - 144 q^{29} - 34 q^{31} - 104 q^{32} - 52 q^{34} - 54 q^{37} + 432 q^{38} - 536 q^{41} + 60 q^{43} - 88 q^{44} - 314 q^{46} - 272 q^{47} - 390 q^{49} + 560 q^{52} - 492 q^{53} - 120 q^{56} + 192 q^{58} - 634 q^{59} + 840 q^{61} + 134 q^{62} + 224 q^{64} - 754 q^{67} + 640 q^{68} + 678 q^{71} + 400 q^{73} - 6 q^{74} + 432 q^{76} - 220 q^{77} + 316 q^{79} - 512 q^{82} + 468 q^{83} + 156 q^{86} - 132 q^{88} + 1842 q^{89} + 1280 q^{91} - 40 q^{92} - 992 q^{94} - 2194 q^{97} - 870 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 8 * q^4 - 20 * q^7 - 12 * q^8 + 22 * q^11 - 80 * q^13 + 4 * q^14 - 8 * q^16 - 124 * q^17 + 72 * q^19 + 22 * q^22 - 98 * q^23 + 40 * q^26 + 128 * q^28 - 144 * q^29 - 34 * q^31 - 104 * q^32 - 52 * q^34 - 54 * q^37 + 432 * q^38 - 536 * q^41 + 60 * q^43 - 88 * q^44 - 314 * q^46 - 272 * q^47 - 390 * q^49 + 560 * q^52 - 492 * q^53 - 120 * q^56 + 192 * q^58 - 634 * q^59 + 840 * q^61 + 134 * q^62 + 224 * q^64 - 754 * q^67 + 640 * q^68 + 678 * q^71 + 400 * q^73 - 6 * q^74 + 432 * q^76 - 220 * q^77 + 316 * q^79 - 512 * q^82 + 468 * q^83 + 156 * q^86 - 132 * q^88 + 1842 * q^89 + 1280 * q^91 - 40 * q^92 - 992 * q^94 - 2194 * q^97 - 870 * 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 $

$2$	−0.732051	−0.258819	−0.129410	−	0.991591i	$-0.541308\pi$
$2$	−0.732051	−0.258819	−0.129410	+	0.991591i	$0.541308\pi$
$3$	0	0
$3$	0	0
$4$	−7.46410	−0.933013
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−16.9282	−0.914037	−0.457019	−	0.889457i	$-0.651083\pi$
$7$	−16.9282	−0.914037	−0.457019	+	0.889457i	$0.651083\pi$
$8$	11.3205	0.500301
$9$	0	0
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−74.6410	−1.59244	−0.796219	−	0.605009i	$-0.793170\pi$
$13$	−74.6410	−1.59244	−0.796219	+	0.605009i	$0.793170\pi$
$14$	12.3923	0.236570
$15$	0	0
$16$	51.4256	0.803525
$17$	−82.7846	−1.18107	−0.590536	−	0.807011i	$-0.701084\pi$
$17$	−82.7846	−1.18107	−0.590536	+	0.807011i	$0.701084\pi$
$18$	0	0
$19$	−67.9230	−0.820138	−0.410069	−	0.912055i	$-0.634495\pi$
$19$	−67.9230	−0.820138	−0.410069	+	0.912055i	$0.634495\pi$
$20$	0	0
$21$	0	0
$22$	−8.05256	−0.0780369
$23$	13.3538	0.121064	0.0605319	−	0.998166i	$-0.480720\pi$
$23$	13.3538	0.121064	0.0605319	+	0.998166i	$0.480720\pi$
$24$	0	0
$25$	0	0
$26$	54.6410	0.412153
$27$	0	0
$28$	126.354	0.852808
$29$	−168.995	−1.08212	−0.541061	−	0.840983i	$-0.681977\pi$
$29$	−168.995	−1.08212	−0.541061	+	0.840983i	$0.681977\pi$
$30$	0	0
$31$	−65.4974	−0.379474	−0.189737	−	0.981835i	$-0.560763\pi$
$31$	−65.4974	−0.379474	−0.189737	+	0.981835i	$0.560763\pi$
$32$	−128.210	−0.708268
$33$	0	0
$34$	60.6025	0.305684
$35$	0	0
$36$	0	0
$37$	−40.8564	−0.181534	−0.0907669	−	0.995872i	$-0.528932\pi$
$37$	−40.8564	−0.181534	−0.0907669	+	0.995872i	$0.528932\pi$
$38$	49.7231	0.212267
$39$	0	0
$40$	0	0
$41$	−274.928	−1.04723	−0.523617	−	0.851954i	$-0.675418\pi$
$41$	−274.928	−1.04723	−0.523617	+	0.851954i	$0.675418\pi$
$42$	0	0
$43$	2.28719	0.00811146	0.00405573	−	0.999992i	$-0.498709\pi$
$43$	2.28719	0.00811146	0.00405573	+	0.999992i	$0.498709\pi$
$44$	−82.1051	−0.281314
$45$	0	0
$46$	−9.77568	−0.0313336
$47$	71.8461	0.222975	0.111488	−	0.993766i	$-0.464438\pi$
$47$	71.8461	0.222975	0.111488	+	0.993766i	$0.464438\pi$
$48$	0	0
$49$	−56.4359	−0.164536
$50$	0	0
$51$	0	0
$52$	557.128	1.48576
$53$	−149.005	−0.386178	−0.193089	−	0.981181i	$-0.561851\pi$
$53$	−149.005	−0.386178	−0.193089	+	0.981181i	$0.561851\pi$
$54$	0	0
$55$	0	0
$56$	−191.636	−0.457293
$57$	0	0
$58$	123.713	0.280074
$59$	−545.631	−1.20398	−0.601992	−	0.798502i	$-0.705626\pi$
$59$	−545.631	−1.20398	−0.601992	+	0.798502i	$0.705626\pi$
$60$	0	0
$61$	101.303	0.212631	0.106315	−	0.994332i	$-0.466095\pi$
$61$	101.303	0.212631	0.106315	+	0.994332i	$0.466095\pi$
$62$	47.9474	0.0982150
$63$	0	0
$64$	−317.549	−0.620212
$65$	0	0
$66$	0	0
$67$	−411.641	−0.750596	−0.375298	−	0.926904i	$-0.622460\pi$
$67$	−411.641	−0.750596	−0.375298	+	0.926904i	$0.622460\pi$
$68$	617.913	1.10195
$69$	0	0
$70$	0	0
$71$	470.636	0.786679	0.393339	−	0.919393i	$-0.371320\pi$
$71$	470.636	0.786679	0.393339	+	0.919393i	$0.371320\pi$
$72$	0	0
$73$	−610.600	−0.978977	−0.489488	−	0.872010i	$-0.662816\pi$
$73$	−610.600	−0.978977	−0.489488	+	0.872010i	$0.662816\pi$
$74$	29.9090	0.0469844
$75$	0	0
$76$	506.985	0.765199
$77$	−186.210	−0.275593
$78$	0	0
$79$	−978.225	−1.39315	−0.696576	−	0.717483i	$-0.745294\pi$
$79$	−978.225	−1.39315	−0.696576	+	0.717483i	$0.745294\pi$
$80$	0	0
$81$	0	0
$82$	201.261	0.271044
$83$	26.1539	0.0345875	0.0172938	−	0.999850i	$-0.494495\pi$
$83$	26.1539	0.0345875	0.0172938	+	0.999850i	$0.494495\pi$
$84$	0	0
$85$	0	0
$86$	−1.67434	−0.00209940
$87$	0	0
$88$	124.526	0.150846
$89$	352.887	0.420292	0.210146	−	0.977670i	$-0.432606\pi$
$89$	352.887	0.420292	0.210146	+	0.977670i	$0.432606\pi$
$90$	0	0
$91$	1263.54	1.45555
$92$	−99.6743	−0.112954
$93$	0	0
$94$	−52.5950	−0.0577102
$95$	0	0
$96$	0	0
$97$	−847.585	−0.887208	−0.443604	−	0.896223i	$-0.646300\pi$
$97$	−847.585	−0.887208	−0.443604	+	0.896223i	$0.646300\pi$
$98$	41.3140	0.0425851
$99$	0	0
$100$	0	0
$101$	−1293.46	−1.27430	−0.637150	−	0.770740i	$-0.719887\pi$
$101$	−1293.46	−1.27430	−0.637150	+	0.770740i	$0.719887\pi$
$102$	0	0
$103$	1725.24	1.65042	0.825209	−	0.564828i	$-0.191057\pi$
$103$	1725.24	1.65042	0.825209	+	0.564828i	$0.191057\pi$
$104$	−844.974	−0.796697
$105$	0	0
$106$	109.079	0.0999502
$107$	−484.179	−0.437452	−0.218726	−	0.975786i	$-0.570190\pi$
$107$	−484.179	−0.437452	−0.218726	+	0.975786i	$0.570190\pi$
$108$	0	0
$109$	−64.2563	−0.0564645	−0.0282323	−	0.999601i	$-0.508988\pi$
$109$	−64.2563	−0.0564645	−0.0282323	+	0.999601i	$0.508988\pi$
$110$	0	0
$111$	0	0
$112$	−870.543	−0.734452
$113$	−2005.08	−1.66922	−0.834612	−	0.550839i	$-0.814308\pi$
$113$	−2005.08	−1.66922	−0.834612	+	0.550839i	$0.814308\pi$
$114$	0	0
$115$	0	0
$116$	1261.39	1.00963
$117$	0	0
$118$	399.429	0.311614
$119$	1401.39	1.07954
$120$	0	0
$121$	121.000	0.0909091
$122$	−74.1587	−0.0550329
$123$	0	0
$124$	488.879	0.354054
$125$	0	0
$126$	0	0
$127$	−109.605	−0.0765816	−0.0382908	−	0.999267i	$-0.512191\pi$
$127$	−109.605	−0.0765816	−0.0382908	+	0.999267i	$0.512191\pi$
$128$	1258.14	0.868791
$129$	0	0
$130$	0	0
$131$	−1156.71	−0.771469	−0.385734	−	0.922610i	$-0.626052\pi$
$131$	−1156.71	−0.771469	−0.385734	+	0.922610i	$0.626052\pi$
$132$	0	0
$133$	1149.82	0.749636
$134$	301.342	0.194269
$135$	0	0
$136$	−937.164	−0.590891
$137$	198.323	0.123678	0.0618391	−	0.998086i	$-0.480303\pi$
$137$	198.323	0.123678	0.0618391	+	0.998086i	$0.480303\pi$
$138$	0	0
$139$	−2900.14	−1.76969	−0.884844	−	0.465888i	$-0.845735\pi$
$139$	−2900.14	−1.76969	−0.884844	+	0.465888i	$0.845735\pi$
$140$	0	0
$141$	0	0
$142$	−344.529	−0.203607
$143$	−821.051	−0.480138
$144$	0	0
$145$	0	0
$146$	446.990	0.253378
$147$	0	0
$148$	304.956	0.169373
$149$	−3488.34	−1.91796	−0.958980	−	0.283472i	$-0.908514\pi$
$149$	−3488.34	−1.91796	−0.958980	+	0.283472i	$0.908514\pi$
$150$	0	0
$151$	−1163.32	−0.626953	−0.313477	−	0.949596i	$-0.601494\pi$
$151$	−1163.32	−0.626953	−0.313477	+	0.949596i	$0.601494\pi$
$152$	−768.923	−0.410315
$153$	0	0
$154$	136.315	0.0713286
$155$	0	0
$156$	0	0
$157$	−342.057	−0.173880	−0.0869398	−	0.996214i	$-0.527709\pi$
$157$	−342.057	−0.173880	−0.0869398	+	0.996214i	$0.527709\pi$
$158$	716.111	0.360574
$159$	0	0
$160$	0	0
$161$	−226.056	−0.110657
$162$	0	0
$163$	1394.89	0.670285	0.335142	−	0.942167i	$-0.391216\pi$
$163$	1394.89	0.670285	0.335142	+	0.942167i	$0.391216\pi$
$164$	2052.09	0.977082
$165$	0	0
$166$	−19.1460	−0.00895191
$167$	478.703	0.221815	0.110908	−	0.993831i	$-0.464624\pi$
$167$	478.703	0.221815	0.110908	+	0.993831i	$0.464624\pi$
$168$	0	0
$169$	3374.28	1.53586
$170$	0	0
$171$	0	0
$172$	−17.0718	−0.00756809
$173$	1808.58	0.794822	0.397411	−	0.917641i	$-0.369909\pi$
$173$	1808.58	0.794822	0.397411	+	0.917641i	$0.369909\pi$
$174$	0	0
$175$	0	0
$176$	565.682	0.242272
$177$	0	0
$178$	−258.331	−0.108780
$179$	4429.85	1.84973	0.924867	−	0.380292i	$-0.124176\pi$
$179$	4429.85	1.84973	0.924867	+	0.380292i	$0.124176\pi$
$180$	0	0
$181$	3409.17	1.40001	0.700005	−	0.714138i	$-0.253181\pi$
$181$	3409.17	1.40001	0.700005	+	0.714138i	$0.253181\pi$
$182$	−924.974	−0.376723
$183$	0	0
$184$	151.172	0.0605682
$185$	0	0
$186$	0	0
$187$	−910.631	−0.356106
$188$	−536.267	−0.208039
$189$	0	0
$190$	0	0
$191$	−2923.75	−1.10762	−0.553810	−	0.832643i	$-0.686827\pi$
$191$	−2923.75	−1.10762	−0.553810	+	0.832643i	$0.686827\pi$
$192$	0	0
$193$	2484.18	0.926505	0.463253	−	0.886226i	$-0.346682\pi$
$193$	2484.18	0.926505	0.463253	+	0.886226i	$0.346682\pi$
$194$	620.475	0.229626
$195$	0	0
$196$	421.244	0.153514
$197$	−5125.67	−1.85375	−0.926876	−	0.375369i	$-0.877516\pi$
$197$	−5125.67	−1.85375	−0.926876	+	0.375369i	$0.877516\pi$
$198$	0	0
$199$	−7.69219	−0.00274013	−0.00137006	−	0.999999i	$-0.500436\pi$
$199$	−7.69219	−0.00274013	−0.00137006	+	0.999999i	$0.500436\pi$
$200$	0	0
$201$	0	0
$202$	946.879	0.329813
$203$	2860.78	0.989100
$204$	0	0
$205$	0	0
$206$	−1262.96	−0.427160
$207$	0	0
$208$	−3838.46	−1.27956
$209$	−747.154	−0.247281
$210$	0	0
$211$	3107.34	1.01383	0.506915	−	0.861996i	$-0.330786\pi$
$211$	3107.34	1.01383	0.506915	+	0.861996i	$0.330786\pi$
$212$	1112.19	0.360309
$213$	0	0
$214$	354.444	0.113221
$215$	0	0
$216$	0	0
$217$	1108.75	0.346853
$218$	47.0388	0.0146141
$219$	0	0
$220$	0	0
$221$	6179.13	1.88078
$222$	0	0
$223$	12.3185	0.00369913	0.00184957	−	0.999998i	$-0.499411\pi$
$223$	12.3185	0.00369913	0.00184957	+	0.999998i	$0.499411\pi$
$224$	2170.37	0.647383
$225$	0	0
$226$	1467.82	0.432027
$227$	4615.90	1.34964	0.674820	−	0.737983i	$-0.264221\pi$
$227$	4615.90	1.34964	0.674820	+	0.737983i	$0.264221\pi$
$228$	0	0
$229$	5074.63	1.46437	0.732186	−	0.681105i	$-0.238500\pi$
$229$	5074.63	1.46437	0.732186	+	0.681105i	$0.238500\pi$
$230$	0	0
$231$	0	0
$232$	−1913.11	−0.541386
$233$	211.683	0.0595184	0.0297592	−	0.999557i	$-0.490526\pi$
$233$	211.683	0.0595184	0.0297592	+	0.999557i	$0.490526\pi$
$234$	0	0
$235$	0	0
$236$	4072.64	1.12333
$237$	0	0
$238$	−1025.89	−0.279406
$239$	−4312.49	−1.16716	−0.583581	−	0.812055i	$-0.698349\pi$
$239$	−4312.49	−1.16716	−0.583581	+	0.812055i	$0.698349\pi$
$240$	0	0
$241$	−996.584	−0.266372	−0.133186	−	0.991091i	$-0.542521\pi$
$241$	−996.584	−0.266372	−0.133186	+	0.991091i	$0.542521\pi$
$242$	−88.5781	−0.0235290
$243$	0	0
$244$	−756.133	−0.198387
$245$	0	0
$246$	0	0
$247$	5069.85	1.30602
$248$	−741.464	−0.189851
$249$	0	0
$250$	0	0
$251$	276.892	0.0696306	0.0348153	−	0.999394i	$-0.488916\pi$
$251$	276.892	0.0696306	0.0348153	+	0.999394i	$0.488916\pi$
$252$	0	0
$253$	146.892	0.0365021
$254$	80.2364	0.0198208
$255$	0	0
$256$	1619.36	0.395352
$257$	−3235.18	−0.785233	−0.392617	−	0.919702i	$-0.628430\pi$
$257$	−3235.18	−0.785233	−0.392617	+	0.919702i	$0.628430\pi$
$258$	0	0
$259$	691.626	0.165929
$260$	0	0
$261$	0	0
$262$	846.772	0.199671
$263$	207.944	0.0487544	0.0243772	−	0.999703i	$-0.492240\pi$
$263$	207.944	0.0487544	0.0243772	+	0.999703i	$0.492240\pi$
$264$	0	0
$265$	0	0
$266$	−841.723	−0.194020
$267$	0	0
$268$	3072.53	0.700316
$269$	−5033.04	−1.14078	−0.570390	−	0.821374i	$-0.693208\pi$
$269$	−5033.04	−1.14078	−0.570390	+	0.821374i	$0.693208\pi$
$270$	0	0
$271$	1487.01	0.333319	0.166660	−	0.986015i	$-0.446702\pi$
$271$	1487.01	0.333319	0.166660	+	0.986015i	$0.446702\pi$
$272$	−4257.25	−0.949021
$273$	0	0
$274$	−145.183	−0.0320102
$275$	0	0
$276$	0	0
$277$	235.836	0.0511552	0.0255776	−	0.999673i	$-0.491858\pi$
$277$	235.836	0.0511552	0.0255776	+	0.999673i	$0.491858\pi$
$278$	2123.05	0.458029
$279$	0	0
$280$	0	0
$281$	4915.01	1.04343	0.521717	−	0.853118i	$-0.325292\pi$
$281$	4915.01	1.04343	0.521717	+	0.853118i	$0.325292\pi$
$282$	0	0
$283$	5199.56	1.09216	0.546081	−	0.837733i	$-0.316119\pi$
$283$	5199.56	1.09216	0.546081	+	0.837733i	$0.316119\pi$
$284$	−3512.87	−0.733981
$285$	0	0
$286$	601.051	0.124269
$287$	4654.04	0.957210
$288$	0	0
$289$	1940.29	0.394930
$290$	0	0
$291$	0	0
$292$	4557.58	0.913398
$293$	−8880.92	−1.77075	−0.885373	−	0.464881i	$-0.846097\pi$
$293$	−8880.92	−1.77075	−0.885373	+	0.464881i	$0.846097\pi$
$294$	0	0
$295$	0	0
$296$	−462.515	−0.0908215
$297$	0	0
$298$	2553.64	0.496405
$299$	−996.743	−0.192786
$300$	0	0
$301$	−38.7180	−0.00741417
$302$	851.612	0.162267
$303$	0	0
$304$	−3492.99	−0.659001
$305$	0	0
$306$	0	0
$307$	1497.93	0.278474	0.139237	−	0.990259i	$-0.455535\pi$
$307$	1497.93	0.278474	0.139237	+	0.990259i	$0.455535\pi$
$308$	1389.89	0.257131
$309$	0	0
$310$	0	0
$311$	7484.71	1.36469	0.682345	−	0.731030i	$-0.260960\pi$
$311$	7484.71	1.36469	0.682345	+	0.731030i	$0.260960\pi$
$312$	0	0
$313$	658.363	0.118891	0.0594455	−	0.998232i	$-0.481067\pi$
$313$	658.363	0.118891	0.0594455	+	0.998232i	$0.481067\pi$
$314$	250.403	0.0450034
$315$	0	0
$316$	7301.57	1.29983
$317$	233.708	0.0414080	0.0207040	−	0.999786i	$-0.493409\pi$
$317$	233.708	0.0414080	0.0207040	+	0.999786i	$0.493409\pi$
$318$	0	0
$319$	−1858.94	−0.326272
$320$	0	0
$321$	0	0
$322$	165.485	0.0286401
$323$	5622.98	0.968641
$324$	0	0
$325$	0	0
$326$	−1021.13	−0.173482
$327$	0	0
$328$	−3112.33	−0.523931
$329$	−1216.23	−0.203808
$330$	0	0
$331$	8532.95	1.41696	0.708480	−	0.705731i	$-0.249381\pi$
$331$	8532.95	1.41696	0.708480	+	0.705731i	$0.249381\pi$
$332$	−195.215	−0.0322706
$333$	0	0
$334$	−350.435	−0.0574100
$335$	0	0
$336$	0	0
$337$	−11691.2	−1.88979	−0.944895	−	0.327373i	$-0.893837\pi$
$337$	−11691.2	−1.88979	−0.944895	+	0.327373i	$0.893837\pi$
$338$	−2470.15	−0.397509
$339$	0	0
$340$	0	0
$341$	−720.472	−0.114416
$342$	0	0
$343$	6761.73	1.06443
$344$	25.8921	0.00405817
$345$	0	0
$346$	−1323.98	−0.205715
$347$	4598.79	0.711459	0.355729	−	0.934589i	$-0.384232\pi$
$347$	4598.79	0.711459	0.355729	+	0.934589i	$0.384232\pi$
$348$	0	0
$349$	6720.27	1.03074	0.515369	−	0.856968i	$-0.327655\pi$
$349$	6720.27	1.03074	0.515369	+	0.856968i	$0.327655\pi$
$350$	0	0
$351$	0	0
$352$	−1410.31	−0.213551
$353$	5738.70	0.865270	0.432635	−	0.901569i	$-0.357584\pi$
$353$	5738.70	0.865270	0.432635	+	0.901569i	$0.357584\pi$
$354$	0	0
$355$	0	0
$356$	−2633.99	−0.392138
$357$	0	0
$358$	−3242.87	−0.478746
$359$	4115.27	0.605001	0.302501	−	0.953149i	$-0.402179\pi$
$359$	4115.27	0.605001	0.302501	+	0.953149i	$0.402179\pi$
$360$	0	0
$361$	−2245.46	−0.327374
$362$	−2495.69	−0.362349
$363$	0	0
$364$	−9431.18	−1.35804
$365$	0	0
$366$	0	0
$367$	−9662.99	−1.37440	−0.687199	−	0.726469i	$-0.741160\pi$
$367$	−9662.99	−1.37440	−0.687199	+	0.726469i	$0.741160\pi$
$368$	686.729	0.0972778
$369$	0	0
$370$	0	0
$371$	2522.39	0.352981
$372$	0	0
$373$	141.780	0.0196812	0.00984062	−	0.999952i	$-0.496868\pi$
$373$	141.780	0.0196812	0.00984062	+	0.999952i	$0.496868\pi$
$374$	666.628	0.0921671
$375$	0	0
$376$	813.334	0.111555
$377$	12613.9	1.72321
$378$	0	0
$379$	−2819.73	−0.382163	−0.191082	−	0.981574i	$-0.561200\pi$
$379$	−2819.73	−0.382163	−0.191082	+	0.981574i	$0.561200\pi$
$380$	0	0
$381$	0	0
$382$	2140.34	0.286673
$383$	−6337.84	−0.845557	−0.422778	−	0.906233i	$-0.638945\pi$
$383$	−6337.84	−0.845557	−0.422778	+	0.906233i	$0.638945\pi$
$384$	0	0
$385$	0	0
$386$	−1818.55	−0.239797
$387$	0	0
$388$	6326.46	0.827776
$389$	8805.25	1.14767	0.573836	−	0.818970i	$-0.305455\pi$
$389$	8805.25	1.14767	0.573836	+	0.818970i	$0.305455\pi$
$390$	0	0
$391$	−1105.49	−0.142985
$392$	−638.883	−0.0823176
$393$	0	0
$394$	3752.25	0.479786
$395$	0	0
$396$	0	0
$397$	−4315.26	−0.545534	−0.272767	−	0.962080i	$-0.587939\pi$
$397$	−4315.26	−0.545534	−0.272767	+	0.962080i	$0.587939\pi$
$398$	5.63108	0.000709197 0
$399$	0	0
$400$	0	0
$401$	−361.681	−0.0450411	−0.0225206	−	0.999746i	$-0.507169\pi$
$401$	−361.681	−0.0450411	−0.0225206	+	0.999746i	$0.507169\pi$
$402$	0	0
$403$	4888.79	0.604288
$404$	9654.53	1.18894
$405$	0	0
$406$	−2094.24	−0.255998
$407$	−449.420	−0.0547345
$408$	0	0
$409$	9220.50	1.11473	0.557365	−	0.830268i	$-0.311812\pi$
$409$	9220.50	1.11473	0.557365	+	0.830268i	$0.311812\pi$
$410$	0	0
$411$	0	0
$412$	−12877.4	−1.53986
$413$	9236.55	1.10049
$414$	0	0
$415$	0	0
$416$	9569.74	1.12787
$417$	0	0
$418$	546.954	0.0640010
$419$	14912.9	1.73876	0.869380	−	0.494144i	$-0.164519\pi$
$419$	14912.9	1.73876	0.869380	+	0.494144i	$0.164519\pi$
$420$	0	0
$421$	−13486.0	−1.56121	−0.780603	−	0.625027i	$-0.785088\pi$
$421$	−13486.0	−1.56121	−0.780603	+	0.625027i	$0.785088\pi$
$422$	−2274.73	−0.262399
$423$	0	0
$424$	−1686.81	−0.193205
$425$	0	0
$426$	0	0
$427$	−1714.87	−0.194352
$428$	3613.96	0.408148
$429$	0	0
$430$	0	0
$431$	−406.334	−0.0454116	−0.0227058	−	0.999742i	$-0.507228\pi$
$431$	−406.334	−0.0454116	−0.0227058	+	0.999742i	$0.507228\pi$
$432$	0	0
$433$	1766.69	0.196078	0.0980391	−	0.995183i	$-0.468743\pi$
$433$	1766.69	0.196078	0.0980391	+	0.995183i	$0.468743\pi$
$434$	−811.664	−0.0897722
$435$	0	0
$436$	479.615	0.0526821
$437$	−907.033	−0.0992889
$438$	0	0
$439$	7824.19	0.850634	0.425317	−	0.905044i	$-0.360163\pi$
$439$	7824.19	0.850634	0.425317	+	0.905044i	$0.360163\pi$
$440$	0	0
$441$	0	0
$442$	−4523.44	−0.486783
$443$	11667.9	1.25137	0.625686	−	0.780075i	$-0.284819\pi$
$443$	11667.9	1.25137	0.625686	+	0.780075i	$0.284819\pi$
$444$	0	0
$445$	0	0
$446$	−9.01776	−0.000957406 0
$447$	0	0
$448$	5375.53	0.566897
$449$	−16975.3	−1.78421	−0.892107	−	0.451825i	$-0.850773\pi$
$449$	−16975.3	−1.78421	−0.892107	+	0.451825i	$0.850773\pi$
$450$	0	0
$451$	−3024.21	−0.315753
$452$	14966.1	1.55741
$453$	0	0
$454$	−3379.07	−0.349312
$455$	0	0
$456$	0	0
$457$	16192.9	1.65748	0.828741	−	0.559632i	$-0.189057\pi$
$457$	16192.9	1.65748	0.828741	+	0.559632i	$0.189057\pi$
$458$	−3714.89	−0.379007
$459$	0	0
$460$	0	0
$461$	−8586.04	−0.867444	−0.433722	−	0.901047i	$-0.642800\pi$
$461$	−8586.04	−0.867444	−0.433722	+	0.901047i	$0.642800\pi$
$462$	0	0
$463$	7917.20	0.794694	0.397347	−	0.917668i	$-0.369931\pi$
$463$	7917.20	0.794694	0.397347	+	0.917668i	$0.369931\pi$
$464$	−8690.67	−0.869513
$465$	0	0
$466$	−154.962	−0.0154045
$467$	−15155.0	−1.50169	−0.750844	−	0.660480i	$-0.770353\pi$
$467$	−15155.0	−1.50169	−0.750844	+	0.660480i	$0.770353\pi$
$468$	0	0
$469$	6968.34	0.686073
$470$	0	0
$471$	0	0
$472$	−6176.82	−0.602354
$473$	25.1591	0.00244570
$474$	0	0
$475$	0	0
$476$	−10460.2	−1.00723
$477$	0	0
$478$	3156.96	0.302084
$479$	−10001.1	−0.953993	−0.476996	−	0.878905i	$-0.658275\pi$
$479$	−10001.1	−0.953993	−0.476996	+	0.878905i	$0.658275\pi$
$480$	0	0
$481$	3049.56	0.289081
$482$	729.550	0.0689421
$483$	0	0
$484$	−903.156	−0.0848193
$485$	0	0
$486$	0	0
$487$	−7044.54	−0.655480	−0.327740	−	0.944768i	$-0.606287\pi$
$487$	−7044.54	−0.655480	−0.327740	+	0.944768i	$0.606287\pi$
$488$	1146.80	0.106379
$489$	0	0
$490$	0	0
$491$	13326.4	1.22487	0.612437	−	0.790520i	$-0.290189\pi$
$491$	13326.4	1.22487	0.612437	+	0.790520i	$0.290189\pi$
$492$	0	0
$493$	13990.2	1.27806
$494$	−3711.38	−0.338022
$495$	0	0
$496$	−3368.25	−0.304917
$497$	−7967.02	−0.719054
$498$	0	0
$499$	−20069.1	−1.80044	−0.900218	−	0.435440i	$-0.856593\pi$
$499$	−20069.1	−1.80044	−0.900218	+	0.435440i	$0.856593\pi$
$500$	0	0
$501$	0	0
$502$	−202.699	−0.0180217
$503$	7782.35	0.689856	0.344928	−	0.938629i	$-0.387903\pi$
$503$	7782.35	0.689856	0.344928	+	0.938629i	$0.387903\pi$
$504$	0	0
$505$	0	0
$506$	−107.532	−0.00944744
$507$	0	0
$508$	818.102	0.0714516
$509$	1475.93	0.128526	0.0642628	−	0.997933i	$-0.479530\pi$
$509$	1475.93	0.128526	0.0642628	+	0.997933i	$0.479530\pi$
$510$	0	0
$511$	10336.4	0.894821
$512$	−11250.6	−0.971116
$513$	0	0
$514$	2368.32	0.203233
$515$	0	0
$516$	0	0
$517$	790.307	0.0672295
$518$	−506.305	−0.0429455
$519$	0	0
$520$	0	0
$521$	−7609.43	−0.639875	−0.319938	−	0.947439i	$-0.603662\pi$
$521$	−7609.43	−0.639875	−0.319938	+	0.947439i	$0.603662\pi$
$522$	0	0
$523$	−12452.9	−1.04116	−0.520581	−	0.853812i	$-0.674285\pi$
$523$	−12452.9	−1.04116	−0.520581	+	0.853812i	$0.674285\pi$
$524$	8633.82	0.719790
$525$	0	0
$526$	−152.226	−0.0126186
$527$	5422.18	0.448186
$528$	0	0
$529$	−11988.7	−0.985344
$530$	0	0
$531$	0	0
$532$	−8582.34	−0.699420
$533$	20520.9	1.66765
$534$	0	0
$535$	0	0
$536$	−4659.99	−0.375524
$537$	0	0
$538$	3684.44	0.295255
$539$	−620.795	−0.0496095
$540$	0	0
$541$	9312.17	0.740039	0.370020	−	0.929024i	$-0.379351\pi$
$541$	9312.17	0.740039	0.370020	+	0.929024i	$0.379351\pi$
$542$	−1088.57	−0.0862693
$543$	0	0
$544$	10613.8	0.836515
$545$	0	0
$546$	0	0
$547$	11018.6	0.861278	0.430639	−	0.902524i	$-0.358288\pi$
$547$	11018.6	0.861278	0.430639	+	0.902524i	$0.358288\pi$
$548$	−1480.31	−0.115393
$549$	0	0
$550$	0	0
$551$	11478.6	0.887490
$552$	0	0
$553$	16559.6	1.27339
$554$	−172.644	−0.0132399
$555$	0	0
$556$	21646.9	1.65114
$557$	−12018.4	−0.914250	−0.457125	−	0.889403i	$-0.651121\pi$
$557$	−12018.4	−0.914250	−0.457125	+	0.889403i	$0.651121\pi$
$558$	0	0
$559$	−170.718	−0.0129170
$560$	0	0
$561$	0	0
$562$	−3598.04	−0.270061
$563$	−8763.89	−0.656046	−0.328023	−	0.944670i	$-0.606382\pi$
$563$	−8763.89	−0.656046	−0.328023	+	0.944670i	$0.606382\pi$
$564$	0	0
$565$	0	0
$566$	−3806.34	−0.282672
$567$	0	0
$568$	5327.84	0.393576
$569$	10273.2	0.756895	0.378447	−	0.925623i	$-0.376458\pi$
$569$	10273.2	0.756895	0.378447	+	0.925623i	$0.376458\pi$
$570$	0	0
$571$	2602.62	0.190747	0.0953734	−	0.995442i	$-0.469596\pi$
$571$	2602.62	0.190747	0.0953734	+	0.995442i	$0.469596\pi$
$572$	6128.41	0.447975
$573$	0	0
$574$	−3406.99	−0.247744
$575$	0	0
$576$	0	0
$577$	19727.0	1.42331	0.711653	−	0.702532i	$-0.247947\pi$
$577$	19727.0	1.42331	0.711653	+	0.702532i	$0.247947\pi$
$578$	−1420.39	−0.102215
$579$	0	0
$580$	0	0
$581$	−442.739	−0.0316143
$582$	0	0
$583$	−1639.06	−0.116437
$584$	−6912.30	−0.489783
$585$	0	0
$586$	6501.28	0.458303
$587$	−10116.2	−0.711309	−0.355654	−	0.934618i	$-0.615742\pi$
$587$	−10116.2	−0.711309	−0.355654	+	0.934618i	$0.615742\pi$
$588$	0	0
$589$	4448.78	0.311221
$590$	0	0
$591$	0	0
$592$	−2101.07	−0.145867
$593$	3130.32	0.216774	0.108387	−	0.994109i	$-0.465431\pi$
$593$	3130.32	0.216774	0.108387	+	0.994109i	$0.465431\pi$
$594$	0	0
$595$	0	0
$596$	26037.3	1.78948
$597$	0	0
$598$	729.667	0.0498968
$599$	−10080.1	−0.687581	−0.343790	−	0.939046i	$-0.611711\pi$
$599$	−10080.1	−0.687581	−0.343790	+	0.939046i	$0.611711\pi$
$600$	0	0
$601$	4777.02	0.324224	0.162112	−	0.986772i	$-0.448169\pi$
$601$	4777.02	0.324224	0.162112	+	0.986772i	$0.448169\pi$
$602$	28.3435	0.00191893
$603$	0	0
$604$	8683.16	0.584955
$605$	0	0
$606$	0	0
$607$	2571.35	0.171941	0.0859703	−	0.996298i	$-0.472601\pi$
$607$	2571.35	0.171941	0.0859703	+	0.996298i	$0.472601\pi$
$608$	8708.43	0.580877
$609$	0	0
$610$	0	0
$611$	−5362.67	−0.355074
$612$	0	0
$613$	−12711.9	−0.837564	−0.418782	−	0.908087i	$-0.637543\pi$
$613$	−12711.9	−0.837564	−0.418782	+	0.908087i	$0.637543\pi$
$614$	−1096.56	−0.0720744
$615$	0	0
$616$	−2107.99	−0.137879
$617$	16236.1	1.05939	0.529693	−	0.848189i	$-0.322307\pi$
$617$	16236.1	1.05939	0.529693	+	0.848189i	$0.322307\pi$
$618$	0	0
$619$	12657.3	0.821874	0.410937	−	0.911664i	$-0.365202\pi$
$619$	12657.3	0.821874	0.410937	+	0.911664i	$0.365202\pi$
$620$	0	0
$621$	0	0
$622$	−5479.19	−0.353208
$623$	−5973.75	−0.384162
$624$	0	0
$625$	0	0
$626$	−481.955	−0.0307713
$627$	0	0
$628$	2553.15	0.162232
$629$	3382.28	0.214404
$630$	0	0
$631$	−3949.97	−0.249201	−0.124600	−	0.992207i	$-0.539765\pi$
$631$	−3949.97	−0.249201	−0.124600	+	0.992207i	$0.539765\pi$
$632$	−11074.0	−0.696994
$633$	0	0
$634$	−171.086	−0.0107172
$635$	0	0
$636$	0	0
$637$	4212.44	0.262014
$638$	1360.84	0.0844455
$639$	0	0
$640$	0	0
$641$	7398.27	0.455872	0.227936	−	0.973676i	$-0.426802\pi$
$641$	7398.27	0.455872	0.227936	+	0.973676i	$0.426802\pi$
$642$	0	0
$643$	12491.7	0.766134	0.383067	−	0.923721i	$-0.374868\pi$
$643$	12491.7	0.766134	0.383067	+	0.923721i	$0.374868\pi$
$644$	1687.31	0.103244
$645$	0	0
$646$	−4116.31	−0.250703
$647$	−10472.0	−0.636315	−0.318158	−	0.948038i	$-0.603064\pi$
$647$	−10472.0	−0.636315	−0.318158	+	0.948038i	$0.603064\pi$
$648$	0	0
$649$	−6001.94	−0.363015
$650$	0	0
$651$	0	0
$652$	−10411.6	−0.625384
$653$	6337.94	0.379820	0.189910	−	0.981801i	$-0.439180\pi$
$653$	6337.94	0.379820	0.189910	+	0.981801i	$0.439180\pi$
$654$	0	0
$655$	0	0
$656$	−14138.4	−0.841479
$657$	0	0
$658$	890.339	0.0527493
$659$	−15196.7	−0.898302	−0.449151	−	0.893456i	$-0.648274\pi$
$659$	−15196.7	−0.898302	−0.449151	+	0.893456i	$0.648274\pi$
$660$	0	0
$661$	2298.17	0.135232	0.0676161	−	0.997711i	$-0.478461\pi$
$661$	2298.17	0.135232	0.0676161	+	0.997711i	$0.478461\pi$
$662$	−6246.55	−0.366736
$663$	0	0
$664$	296.075	0.0173042
$665$	0	0
$666$	0	0
$667$	−2256.73	−0.131006
$668$	−3573.09	−0.206956
$669$	0	0
$670$	0	0
$671$	1114.33	0.0641106
$672$	0	0
$673$	−23199.6	−1.32880	−0.664398	−	0.747379i	$-0.731312\pi$
$673$	−23199.6	−1.32880	−0.664398	+	0.747379i	$0.731312\pi$
$674$	8558.54	0.489114
$675$	0	0
$676$	−25186.0	−1.43298
$677$	−2145.38	−0.121793	−0.0608963	−	0.998144i	$-0.519396\pi$
$677$	−2145.38	−0.121793	−0.0608963	+	0.998144i	$0.519396\pi$
$678$	0	0
$679$	14348.1	0.810941
$680$	0	0
$681$	0	0
$682$	527.422	0.0296129
$683$	−29544.6	−1.65519	−0.827593	−	0.561329i	$-0.810290\pi$
$683$	−29544.6	−1.65519	−0.827593	+	0.561329i	$0.810290\pi$
$684$	0	0
$685$	0	0
$686$	−4949.93	−0.275495
$687$	0	0
$688$	117.620	0.00651776
$689$	11121.9	0.614964
$690$	0	0
$691$	27803.1	1.53065	0.765325	−	0.643644i	$-0.222578\pi$
$691$	27803.1	1.53065	0.765325	+	0.643644i	$0.222578\pi$
$692$	−13499.5	−0.741579
$693$	0	0
$694$	−3366.55	−0.184139
$695$	0	0
$696$	0	0
$697$	22759.8	1.23686
$698$	−4919.58	−0.266775
$699$	0	0
$700$	0	0
$701$	19697.8	1.06130	0.530652	−	0.847590i	$-0.321947\pi$
$701$	19697.8	1.06130	0.530652	+	0.847590i	$0.321947\pi$
$702$	0	0
$703$	2775.09	0.148883
$704$	−3493.03	−0.187001
$705$	0	0
$706$	−4201.02	−0.223948
$707$	21896.0	1.16476
$708$	0	0
$709$	19122.5	1.01292	0.506460	−	0.862263i	$-0.330954\pi$
$709$	19122.5	1.01292	0.506460	+	0.862263i	$0.330954\pi$
$710$	0	0
$711$	0	0
$712$	3994.86	0.210272
$713$	−874.641	−0.0459405
$714$	0	0
$715$	0	0
$716$	−33064.8	−1.72582
$717$	0	0
$718$	−3012.59	−0.156586
$719$	−1837.44	−0.0953060	−0.0476530	−	0.998864i	$-0.515174\pi$
$719$	−1837.44	−0.0953060	−0.0476530	+	0.998864i	$0.515174\pi$
$720$	0	0
$721$	−29205.2	−1.50854
$722$	1643.79	0.0847307
$723$	0	0
$724$	−25446.4	−1.30623
$725$	0	0
$726$	0	0
$727$	7555.46	0.385442	0.192721	−	0.981254i	$-0.438269\pi$
$727$	7555.46	0.385442	0.192721	+	0.981254i	$0.438269\pi$
$728$	14303.9	0.728211
$729$	0	0
$730$	0	0
$731$	−189.344	−0.00958021
$732$	0	0
$733$	−11984.6	−0.603905	−0.301952	−	0.953323i	$-0.597638\pi$
$733$	−11984.6	−0.603905	−0.301952	+	0.953323i	$0.597638\pi$
$734$	7073.80	0.355720
$735$	0	0
$736$	−1712.10	−0.0857456
$737$	−4528.05	−0.226313
$738$	0	0
$739$	−27142.5	−1.35109	−0.675543	−	0.737321i	$-0.736091\pi$
$739$	−27142.5	−1.35109	−0.675543	+	0.737321i	$0.736091\pi$
$740$	0	0
$741$	0	0
$742$	−1846.52	−0.0913582
$743$	−29222.6	−1.44290	−0.721450	−	0.692467i	$-0.756524\pi$
$743$	−29222.6	−1.44290	−0.721450	+	0.692467i	$0.756524\pi$
$744$	0	0
$745$	0	0
$746$	−103.790	−0.00509388
$747$	0	0
$748$	6797.04	0.332252
$749$	8196.29	0.399847
$750$	0	0
$751$	−8859.39	−0.430471	−0.215236	−	0.976562i	$-0.569052\pi$
$751$	−8859.39	−0.430471	−0.215236	+	0.976562i	$0.569052\pi$
$752$	3694.73	0.179166
$753$	0	0
$754$	−9234.05	−0.446000
$755$	0	0
$756$	0	0
$757$	−35734.4	−1.71571	−0.857853	−	0.513896i	$-0.828202\pi$
$757$	−35734.4	−1.71571	−0.857853	+	0.513896i	$0.828202\pi$
$758$	2064.19	0.0989112
$759$	0	0
$760$	0	0
$761$	−34394.7	−1.63838	−0.819189	−	0.573524i	$-0.805576\pi$
$761$	−34394.7	−1.63838	−0.819189	+	0.573524i	$0.805576\pi$
$762$	0	0
$763$	1087.74	0.0516107
$764$	21823.2	1.03342
$765$	0	0
$766$	4639.62	0.218846
$767$	40726.4	1.91727
$768$	0	0
$769$	−11602.7	−0.544091	−0.272045	−	0.962284i	$-0.587700\pi$
$769$	−11602.7	−0.544091	−0.272045	+	0.962284i	$0.587700\pi$
$770$	0	0
$771$	0	0
$772$	−18542.2	−0.864441
$773$	−12680.6	−0.590026	−0.295013	−	0.955493i	$-0.595324\pi$
$773$	−12680.6	−0.590026	−0.295013	+	0.955493i	$0.595324\pi$
$774$	0	0
$775$	0	0
$776$	−9595.09	−0.443871
$777$	0	0
$778$	−6445.89	−0.297039
$779$	18674.0	0.858876
$780$	0	0
$781$	5176.99	0.237193
$782$	809.276	0.0370072
$783$	0	0
$784$	−2902.25	−0.132209
$785$	0	0
$786$	0	0
$787$	4417.61	0.200090	0.100045	−	0.994983i	$-0.468101\pi$
$787$	4417.61	0.200090	0.100045	+	0.994983i	$0.468101\pi$
$788$	38258.5	1.72957
$789$	0	0
$790$	0	0
$791$	33942.4	1.52573
$792$	0	0
$793$	−7561.33	−0.338601
$794$	3158.99	0.141194
$795$	0	0
$796$	57.4153	0.00255657
$797$	−27030.1	−1.20132	−0.600661	−	0.799504i	$-0.705096\pi$
$797$	−27030.1	−1.20132	−0.600661	+	0.799504i	$0.705096\pi$
$798$	0	0
$799$	−5947.75	−0.263350
$800$	0	0
$801$	0	0
$802$	264.769	0.0116575
$803$	−6716.60	−0.295173
$804$	0	0
$805$	0	0
$806$	−3578.85	−0.156401
$807$	0	0
$808$	−14642.6	−0.637533
$809$	−23647.0	−1.02767	−0.513835	−	0.857889i	$-0.671776\pi$
$809$	−23647.0	−1.02767	−0.513835	+	0.857889i	$0.671776\pi$
$810$	0	0
$811$	33486.1	1.44988	0.724941	−	0.688811i	$-0.241867\pi$
$811$	33486.1	1.44988	0.724941	+	0.688811i	$0.241867\pi$
$812$	−21353.1	−0.922843
$813$	0	0
$814$	328.999	0.0141663
$815$	0	0
$816$	0	0
$817$	−155.353	−0.00665251
$818$	−6749.88	−0.288513
$819$	0	0
$820$	0	0
$821$	−2605.69	−0.110766	−0.0553832	−	0.998465i	$-0.517638\pi$
$821$	−2605.69	−0.110766	−0.0553832	+	0.998465i	$0.517638\pi$
$822$	0	0
$823$	−31976.2	−1.35434	−0.677169	−	0.735828i	$-0.736793\pi$
$823$	−31976.2	−1.35434	−0.677169	+	0.735828i	$0.736793\pi$
$824$	19530.6	0.825705
$825$	0	0
$826$	−6761.62	−0.284827
$827$	−37759.0	−1.58768	−0.793839	−	0.608128i	$-0.791921\pi$
$827$	−37759.0	−1.58768	−0.793839	+	0.608128i	$0.791921\pi$
$828$	0	0
$829$	−1137.55	−0.0476584	−0.0238292	−	0.999716i	$-0.507586\pi$
$829$	−1137.55	−0.0476584	−0.0238292	+	0.999716i	$0.507586\pi$
$830$	0	0
$831$	0	0
$832$	23702.2	0.987649
$833$	4672.03	0.194329
$834$	0	0
$835$	0	0
$836$	5576.83	0.230716
$837$	0	0
$838$	−10917.0	−0.450024
$839$	37372.2	1.53782	0.768911	−	0.639356i	$-0.220799\pi$
$839$	37372.2	1.53782	0.768911	+	0.639356i	$0.220799\pi$
$840$	0	0
$841$	4170.26	0.170989
$842$	9872.44	0.404070
$843$	0	0
$844$	−23193.5	−0.945917
$845$	0	0
$846$	0	0
$847$	−2048.31	−0.0830943
$848$	−7662.68	−0.310304
$849$	0	0
$850$	0	0
$851$	−545.589	−0.0219772
$852$	0	0
$853$	22490.8	0.902780	0.451390	−	0.892327i	$-0.350928\pi$
$853$	22490.8	0.902780	0.451390	+	0.892327i	$0.350928\pi$
$854$	1255.37	0.0503021
$855$	0	0
$856$	−5481.16	−0.218858
$857$	43409.5	1.73027	0.865135	−	0.501539i	$-0.167233\pi$
$857$	43409.5	1.73027	0.865135	+	0.501539i	$0.167233\pi$
$858$	0	0
$859$	29533.2	1.17306	0.586532	−	0.809926i	$-0.300493\pi$
$859$	29533.2	1.17306	0.586532	+	0.809926i	$0.300493\pi$
$860$	0	0
$861$	0	0
$862$	297.457	0.0117534
$863$	14351.6	0.566090	0.283045	−	0.959107i	$-0.408655\pi$
$863$	14351.6	0.566090	0.283045	+	0.959107i	$0.408655\pi$
$864$	0	0
$865$	0	0
$866$	−1293.31	−0.0507488
$867$	0	0
$868$	−8275.85	−0.323618
$869$	−10760.5	−0.420051
$870$	0	0
$871$	30725.3	1.19528
$872$	−727.413	−0.0282492
$873$	0	0
$874$	663.994	0.0256979
$875$	0	0
$876$	0	0
$877$	43248.7	1.66523	0.832614	−	0.553854i	$-0.186844\pi$
$877$	43248.7	1.66523	0.832614	+	0.553854i	$0.186844\pi$
$878$	−5727.71	−0.220160
$879$	0	0
$880$	0	0
$881$	−3816.13	−0.145935	−0.0729675	−	0.997334i	$-0.523247\pi$
$881$	−3816.13	−0.145935	−0.0729675	+	0.997334i	$0.523247\pi$
$882$	0	0
$883$	−48787.6	−1.85938	−0.929690	−	0.368343i	$-0.879925\pi$
$883$	−48787.6	−1.85938	−0.929690	+	0.368343i	$0.879925\pi$
$884$	−46121.6	−1.75479
$885$	0	0
$886$	−8541.49	−0.323879
$887$	41495.1	1.57077	0.785384	−	0.619009i	$-0.212466\pi$
$887$	41495.1	1.57077	0.785384	+	0.619009i	$0.212466\pi$
$888$	0	0
$889$	1855.41	0.0699984
$890$	0	0
$891$	0	0
$892$	−91.9464	−0.00345134
$893$	−4880.01	−0.182870
$894$	0	0
$895$	0	0
$896$	−21298.1	−0.794107
$897$	0	0
$898$	12426.7	0.461788
$899$	11068.7	0.410637
$900$	0	0
$901$	12335.3	0.456104
$902$	2213.88	0.0817228
$903$	0	0
$904$	−22698.5	−0.835113
$905$	0	0
$906$	0	0
$907$	−21615.3	−0.791316	−0.395658	−	0.918398i	$-0.629483\pi$
$907$	−21615.3	−0.791316	−0.395658	+	0.918398i	$0.629483\pi$
$908$	−34453.6	−1.25923
$909$	0	0
$910$	0	0
$911$	−3646.35	−0.132611	−0.0663057	−	0.997799i	$-0.521121\pi$
$911$	−3646.35	−0.132611	−0.0663057	+	0.997799i	$0.521121\pi$
$912$	0	0
$913$	287.693	0.0104285
$914$	−11854.0	−0.428988
$915$	0	0
$916$	−37877.6	−1.36628
$917$	19581.1	0.705151
$918$	0	0
$919$	31280.0	1.12278	0.561388	−	0.827553i	$-0.310267\pi$
$919$	31280.0	1.12278	0.561388	+	0.827553i	$0.310267\pi$
$920$	0	0
$921$	0	0
$922$	6285.42	0.224511
$923$	−35128.7	−1.25274
$924$	0	0
$925$	0	0
$926$	−5795.79	−0.205682
$927$	0	0
$928$	21666.9	0.766433
$929$	−6557.92	−0.231602	−0.115801	−	0.993272i	$-0.536944\pi$
$929$	−6557.92	−0.231602	−0.115801	+	0.993272i	$0.536944\pi$
$930$	0	0
$931$	3833.30	0.134942
$932$	−1580.02	−0.0555314
$933$	0	0
$934$	11094.2	0.388665
$935$	0	0
$936$	0	0
$937$	24473.3	0.853265	0.426632	−	0.904425i	$-0.359700\pi$
$937$	24473.3	0.853265	0.426632	+	0.904425i	$0.359700\pi$
$938$	−5101.18	−0.177569
$939$	0	0
$940$	0	0
$941$	−15420.8	−0.534224	−0.267112	−	0.963665i	$-0.586069\pi$
$941$	−15420.8	−0.534224	−0.267112	+	0.963665i	$0.586069\pi$
$942$	0	0
$943$	−3671.34	−0.126782
$944$	−28059.4	−0.967432
$945$	0	0
$946$	−18.4177	−0.000632993 0
$947$	−33141.2	−1.13722	−0.568608	−	0.822609i	$-0.692518\pi$
$947$	−33141.2	−1.13722	−0.568608	+	0.822609i	$0.692518\pi$
$948$	0	0
$949$	45575.8	1.55896
$950$	0	0
$951$	0	0
$952$	15864.5	0.540096
$953$	−20735.4	−0.704813	−0.352406	−	0.935847i	$-0.614637\pi$
$953$	−20735.4	−0.704813	−0.352406	+	0.935847i	$0.614637\pi$
$954$	0	0
$955$	0	0
$956$	32188.9	1.08898
$957$	0	0
$958$	7321.32	0.246911
$959$	−3357.26	−0.113046
$960$	0	0
$961$	−25501.1	−0.856000
$962$	−2232.44	−0.0748198
$963$	0	0
$964$	7438.60	0.248528
$965$	0	0
$966$	0	0
$967$	8178.87	0.271990	0.135995	−	0.990710i	$-0.456577\pi$
$967$	8178.87	0.271990	0.135995	+	0.990710i	$0.456577\pi$
$968$	1369.78	0.0454819
$969$	0	0
$970$	0	0
$971$	20576.1	0.680039	0.340020	−	0.940418i	$-0.389566\pi$
$971$	20576.1	0.680039	0.340020	+	0.940418i	$0.389566\pi$
$972$	0	0
$973$	49094.1	1.61756
$974$	5156.96	0.169651
$975$	0	0
$976$	5209.55	0.170854
$977$	14541.9	0.476188	0.238094	−	0.971242i	$-0.423477\pi$
$977$	14541.9	0.476188	0.238094	+	0.971242i	$0.423477\pi$
$978$	0	0
$979$	3881.76	0.126723
$980$	0	0
$981$	0	0
$982$	−9755.62	−0.317021
$983$	−29285.7	−0.950223	−0.475111	−	0.879926i	$-0.657592\pi$
$983$	−29285.7	−0.950223	−0.475111	+	0.879926i	$0.657592\pi$
$984$	0	0
$985$	0	0
$986$	−10241.5	−0.330787
$987$	0	0
$988$	−37841.8	−1.21853
$989$	30.5427	0.000982004 0
$990$	0	0
$991$	−38085.9	−1.22083	−0.610413	−	0.792083i	$-0.708997\pi$
$991$	−38085.9	−1.22083	−0.610413	+	0.792083i	$0.708997\pi$
$992$	8397.44	0.268769
$993$	0	0
$994$	5832.26	0.186105
$995$	0	0
$996$	0	0
$997$	26803.6	0.851434	0.425717	−	0.904856i	$-0.360022\pi$
$997$	26803.6	0.851434	0.425717	+	0.904856i	$0.360022\pi$
$998$	14691.6	0.465987
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.q.1.1		2
3.2	odd	2		275.4.a.b.1.2		2
5.4	even	2		99.4.a.c.1.2		2
15.2	even	4		275.4.b.c.199.3		4
15.8	even	4		275.4.b.c.199.2		4
15.14	odd	2		11.4.a.a.1.1	✓	2
20.19	odd	2		1584.4.a.bc.1.2		2
55.54	odd	2		1089.4.a.v.1.1		2
60.59	even	2		176.4.a.i.1.1		2
105.104	even	2		539.4.a.e.1.1		2
120.29	odd	2		704.4.a.p.1.1		2
120.59	even	2		704.4.a.n.1.2		2
165.14	odd	10		121.4.c.c.9.1		8
165.29	even	10		121.4.c.f.27.2		8
165.59	odd	10		121.4.c.c.27.1		8
165.74	even	10		121.4.c.f.9.2		8
165.104	odd	10		121.4.c.c.3.2		8
165.119	odd	10		121.4.c.c.81.2		8
165.134	even	10		121.4.c.f.81.1		8
165.149	even	10		121.4.c.f.3.1		8
165.164	even	2		121.4.a.c.1.2		2
195.194	odd	2		1859.4.a.a.1.2		2
660.659	odd	2		1936.4.a.w.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.4.a.a.1.1	✓	2	15.14	odd	2
99.4.a.c.1.2		2	5.4	even	2
121.4.a.c.1.2		2	165.164	even	2
121.4.c.c.3.2		8	165.104	odd	10
121.4.c.c.9.1		8	165.14	odd	10
121.4.c.c.27.1		8	165.59	odd	10
121.4.c.c.81.2		8	165.119	odd	10
121.4.c.f.3.1		8	165.149	even	10
121.4.c.f.9.2		8	165.74	even	10
121.4.c.f.27.2		8	165.29	even	10
121.4.c.f.81.1		8	165.134	even	10
176.4.a.i.1.1		2	60.59	even	2
275.4.a.b.1.2		2	3.2	odd	2
275.4.b.c.199.2		4	15.8	even	4
275.4.b.c.199.3		4	15.2	even	4
539.4.a.e.1.1		2	105.104	even	2
704.4.a.n.1.2		2	120.59	even	2
704.4.a.p.1.1		2	120.29	odd	2
1089.4.a.v.1.1		2	55.54	odd	2
1584.4.a.bc.1.2		2	20.19	odd	2
1859.4.a.a.1.2		2	195.194	odd	2
1936.4.a.w.1.1		2	660.659	odd	2
2475.4.a.q.1.1		2	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-0.732051 q^{2} -7.46410 q^{4} -16.9282 q^{7} +11.3205 q^{8} +O(q^{10})\)	\(q-0.732051 q^{2} -7.46410 q^{4} -16.9282 q^{7} +11.3205 q^{8} +11.0000 q^{11} -74.6410 q^{13} +12.3923 q^{14} +51.4256 q^{16} -82.7846 q^{17} -67.9230 q^{19} -8.05256 q^{22} +13.3538 q^{23} +54.6410 q^{26} +126.354 q^{28} -168.995 q^{29} -65.4974 q^{31} -128.210 q^{32} +60.6025 q^{34} -40.8564 q^{37} +49.7231 q^{38} -274.928 q^{41} +2.28719 q^{43} -82.1051 q^{44} -9.77568 q^{46} +71.8461 q^{47} -56.4359 q^{49} +557.128 q^{52} -149.005 q^{53} -191.636 q^{56} +123.713 q^{58} -545.631 q^{59} +101.303 q^{61} +47.9474 q^{62} -317.549 q^{64} -411.641 q^{67} +617.913 q^{68} +470.636 q^{71} -610.600 q^{73} +29.9090 q^{74} +506.985 q^{76} -186.210 q^{77} -978.225 q^{79} +201.261 q^{82} +26.1539 q^{83} -1.67434 q^{86} +124.526 q^{88} +352.887 q^{89} +1263.54 q^{91} -99.6743 q^{92} -52.5950 q^{94} -847.585 q^{97} +41.3140 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 8 q^{4} - 20 q^{7} - 12 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 - 8 * q^4 - 20 * q^7 - 12 * q^8	\( 2 q + 2 q^{2} - 8 q^{4} - 20 q^{7} - 12 q^{8} + 22 q^{11} - 80 q^{13} + 4 q^{14} - 8 q^{16} - 124 q^{17} + 72 q^{19} + 22 q^{22} - 98 q^{23} + 40 q^{26} + 128 q^{28} - 144 q^{29} - 34 q^{31} - 104 q^{32} - 52 q^{34} - 54 q^{37} + 432 q^{38} - 536 q^{41} + 60 q^{43} - 88 q^{44} - 314 q^{46} - 272 q^{47} - 390 q^{49} + 560 q^{52} - 492 q^{53} - 120 q^{56} + 192 q^{58} - 634 q^{59} + 840 q^{61} + 134 q^{62} + 224 q^{64} - 754 q^{67} + 640 q^{68} + 678 q^{71} + 400 q^{73} - 6 q^{74} + 432 q^{76} - 220 q^{77} + 316 q^{79} - 512 q^{82} + 468 q^{83} + 156 q^{86} - 132 q^{88} + 1842 q^{89} + 1280 q^{91} - 40 q^{92} - 992 q^{94} - 2194 q^{97} - 870 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 8 * q^4 - 20 * q^7 - 12 * q^8 + 22 * q^11 - 80 * q^13 + 4 * q^14 - 8 * q^16 - 124 * q^17 + 72 * q^19 + 22 * q^22 - 98 * q^23 + 40 * q^26 + 128 * q^28 - 144 * q^29 - 34 * q^31 - 104 * q^32 - 52 * q^34 - 54 * q^37 + 432 * q^38 - 536 * q^41 + 60 * q^43 - 88 * q^44 - 314 * q^46 - 272 * q^47 - 390 * q^49 + 560 * q^52 - 492 * q^53 - 120 * q^56 + 192 * q^58 - 634 * q^59 + 840 * q^61 + 134 * q^62 + 224 * q^64 - 754 * q^67 + 640 * q^68 + 678 * q^71 + 400 * q^73 - 6 * q^74 + 432 * q^76 - 220 * q^77 + 316 * q^79 - 512 * q^82 + 468 * q^83 + 156 * q^86 - 132 * q^88 + 1842 * q^89 + 1280 * q^91 - 40 * q^92 - 992 * q^94 - 2194 * q^97 - 870 * q^98

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