LMFDB - Embedded newform 1575.4.a.ba.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,4,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, 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("1575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1575.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:	$92.9280082590$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.14360.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 17x - 14 $ x^3 - 17*x - 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.48565$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.48565 q^{2} +4.14976 q^{4} -7.00000 q^{7} -13.4206 q^{8} +O(q^{10})$	$q+3.48565 q^{2} +4.14976 q^{4} -7.00000 q^{7} -13.4206 q^{8} +6.90764 q^{11} +22.1364 q^{13} -24.3996 q^{14} -79.9776 q^{16} +88.3030 q^{17} +36.9560 q^{19} +24.0776 q^{22} -95.5283 q^{23} +77.1598 q^{26} -29.0483 q^{28} -269.029 q^{29} +197.114 q^{31} -171.409 q^{32} +307.793 q^{34} -2.14546 q^{37} +128.816 q^{38} -174.127 q^{41} +17.0345 q^{43} +28.6650 q^{44} -332.978 q^{46} -528.029 q^{47} +49.0000 q^{49} +91.8608 q^{52} -641.114 q^{53} +93.9441 q^{56} -937.742 q^{58} +642.975 q^{59} +142.967 q^{61} +687.070 q^{62} +42.3480 q^{64} -478.797 q^{67} +366.436 q^{68} -105.550 q^{71} -986.512 q^{73} -7.47834 q^{74} +153.358 q^{76} -48.3534 q^{77} -1099.86 q^{79} -606.947 q^{82} -1236.62 q^{83} +59.3763 q^{86} -92.7045 q^{88} +711.698 q^{89} -154.955 q^{91} -396.420 q^{92} -1840.52 q^{94} +636.553 q^{97} +170.797 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 13 q^{4} - 21 q^{7} - 15 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 13 * q^4 - 21 * q^7 - 15 * q^8	$ 3 q - 3 q^{2} + 13 q^{4} - 21 q^{7} - 15 q^{8} + 74 q^{11} - 44 q^{13} + 21 q^{14} - 79 q^{16} - 52 q^{17} + 168 q^{19} - 184 q^{22} - 124 q^{23} + 446 q^{26} - 91 q^{28} - 332 q^{29} + 320 q^{31} - 183 q^{32} + 582 q^{34} + 54 q^{37} - 460 q^{38} - 362 q^{41} + 16 q^{43} + 264 q^{44} - 336 q^{46} - 730 q^{47} + 147 q^{49} - 1202 q^{52} + 110 q^{53} + 105 q^{56} - 450 q^{58} + 180 q^{59} + 1222 q^{61} + 464 q^{62} - 391 q^{64} - 204 q^{67} + 918 q^{68} + 136 q^{71} - 310 q^{73} - 502 q^{74} + 1796 q^{76} - 518 q^{77} - 1034 q^{79} + 6 q^{82} - 1660 q^{83} - 764 q^{86} + 20 q^{88} - 242 q^{89} + 308 q^{91} + 96 q^{92} - 1108 q^{94} - 100 q^{97} - 147 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 13 * q^4 - 21 * q^7 - 15 * q^8 + 74 * q^11 - 44 * q^13 + 21 * q^14 - 79 * q^16 - 52 * q^17 + 168 * q^19 - 184 * q^22 - 124 * q^23 + 446 * q^26 - 91 * q^28 - 332 * q^29 + 320 * q^31 - 183 * q^32 + 582 * q^34 + 54 * q^37 - 460 * q^38 - 362 * q^41 + 16 * q^43 + 264 * q^44 - 336 * q^46 - 730 * q^47 + 147 * q^49 - 1202 * q^52 + 110 * q^53 + 105 * q^56 - 450 * q^58 + 180 * q^59 + 1222 * q^61 + 464 * q^62 - 391 * q^64 - 204 * q^67 + 918 * q^68 + 136 * q^71 - 310 * q^73 - 502 * q^74 + 1796 * q^76 - 518 * q^77 - 1034 * q^79 + 6 * q^82 - 1660 * q^83 - 764 * q^86 + 20 * q^88 - 242 * q^89 + 308 * q^91 + 96 * q^92 - 1108 * q^94 - 100 * q^97 - 147 * 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 $

$2$	3.48565	1.23236	0.616182	−	0.787604i	$-0.288679\pi$
$2$	3.48565	1.23236	0.616182	+	0.787604i	$0.288679\pi$
$3$	0	0
$3$	0	0
$4$	4.14976	0.518720
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−13.4206	−0.593112
$9$	0	0
$10$	0	0
$11$	6.90764	0.189339	0.0946696	−	0.995509i	$-0.469821\pi$
$11$	6.90764	0.189339	0.0946696	+	0.995509i	$0.469821\pi$
$12$	0	0
$13$	22.1364	0.472272	0.236136	−	0.971720i	$-0.424119\pi$
$13$	22.1364	0.472272	0.236136	+	0.971720i	$0.424119\pi$
$14$	−24.3996	−0.465790
$15$	0	0
$16$	−79.9776	−1.24965
$17$	88.3030	1.25980	0.629901	−	0.776676i	$-0.283096\pi$
$17$	88.3030	1.25980	0.629901	+	0.776676i	$0.283096\pi$
$18$	0	0
$19$	36.9560	0.446225	0.223113	−	0.974793i	$-0.428378\pi$
$19$	36.9560	0.446225	0.223113	+	0.974793i	$0.428378\pi$
$20$	0	0
$21$	0	0
$22$	24.0776	0.233335
$23$	−95.5283	−0.866045	−0.433022	−	0.901383i	$-0.642553\pi$
$23$	−95.5283	−0.866045	−0.433022	+	0.901383i	$0.642553\pi$
$24$	0	0
$25$	0	0
$26$	77.1598	0.582010
$27$	0	0
$28$	−29.0483	−0.196058
$29$	−269.029	−1.72267	−0.861336	−	0.508035i	$-0.830372\pi$
$29$	−269.029	−1.72267	−0.861336	+	0.508035i	$0.830372\pi$
$30$	0	0
$31$	197.114	1.14202	0.571012	−	0.820942i	$-0.306551\pi$
$31$	197.114	1.14202	0.571012	+	0.820942i	$0.306551\pi$
$32$	−171.409	−0.946911
$33$	0	0
$34$	307.793	1.55253
$35$	0	0
$36$	0	0
$37$	−2.14546	−0.00953276	−0.00476638	−	0.999989i	$-0.501517\pi$
$37$	−2.14546	−0.00953276	−0.00476638	+	0.999989i	$0.501517\pi$
$38$	128.816	0.549912
$39$	0	0
$40$	0	0
$41$	−174.127	−0.663271	−0.331636	−	0.943408i	$-0.607600\pi$
$41$	−174.127	−0.663271	−0.331636	+	0.943408i	$0.607600\pi$
$42$	0	0
$43$	17.0345	0.0604125	0.0302062	−	0.999544i	$-0.490384\pi$
$43$	17.0345	0.0604125	0.0302062	+	0.999544i	$0.490384\pi$
$44$	28.6650	0.0982140
$45$	0	0
$46$	−332.978	−1.06728
$47$	−528.029	−1.63874	−0.819371	−	0.573264i	$-0.805677\pi$
$47$	−528.029	−1.63874	−0.819371	+	0.573264i	$0.805677\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	91.8608	0.244977
$53$	−641.114	−1.66158	−0.830790	−	0.556586i	$-0.812111\pi$
$53$	−641.114	−1.66158	−0.830790	+	0.556586i	$0.812111\pi$
$54$	0	0
$55$	0	0
$56$	93.9441	0.224175
$57$	0	0
$58$	−937.742	−2.12296
$59$	642.975	1.41878	0.709391	−	0.704815i	$-0.248970\pi$
$59$	642.975	1.41878	0.709391	+	0.704815i	$0.248970\pi$
$60$	0	0
$61$	142.967	0.300083	0.150042	−	0.988680i	$-0.452059\pi$
$61$	142.967	0.300083	0.150042	+	0.988680i	$0.452059\pi$
$62$	687.070	1.40739
$63$	0	0
$64$	42.3480	0.0827109
$65$	0	0
$66$	0	0
$67$	−478.797	−0.873050	−0.436525	−	0.899692i	$-0.643791\pi$
$67$	−478.797	−0.873050	−0.436525	+	0.899692i	$0.643791\pi$
$68$	366.436	0.653484
$69$	0	0
$70$	0	0
$71$	−105.550	−0.176430	−0.0882150	−	0.996101i	$-0.528116\pi$
$71$	−105.550	−0.176430	−0.0882150	+	0.996101i	$0.528116\pi$
$72$	0	0
$73$	−986.512	−1.58168	−0.790839	−	0.612024i	$-0.790356\pi$
$73$	−986.512	−1.58168	−0.790839	+	0.612024i	$0.790356\pi$
$74$	−7.47834	−0.0117478
$75$	0	0
$76$	153.358	0.231466
$77$	−48.3534	−0.0715635
$78$	0	0
$79$	−1099.86	−1.56638	−0.783190	−	0.621783i	$-0.786409\pi$
$79$	−1099.86	−1.56638	−0.783190	+	0.621783i	$0.786409\pi$
$80$	0	0
$81$	0	0
$82$	−606.947	−0.817391
$83$	−1236.62	−1.63538	−0.817691	−	0.575657i	$-0.804746\pi$
$83$	−1236.62	−1.63538	−0.817691	+	0.575657i	$0.804746\pi$
$84$	0	0
$85$	0	0
$86$	59.3763	0.0744501
$87$	0	0
$88$	−92.7045	−0.112299
$89$	711.698	0.847638	0.423819	−	0.905747i	$-0.360689\pi$
$89$	711.698	0.847638	0.423819	+	0.905747i	$0.360689\pi$
$90$	0	0
$91$	−154.955	−0.178502
$92$	−396.420	−0.449235
$93$	0	0
$94$	−1840.52	−2.01953
$95$	0	0
$96$	0	0
$97$	636.553	0.666311	0.333156	−	0.942872i	$-0.391887\pi$
$97$	636.553	0.666311	0.333156	+	0.942872i	$0.391887\pi$
$98$	170.797	0.176052
$99$	0	0
$100$	0	0
$101$	−1742.05	−1.71624	−0.858121	−	0.513448i	$-0.828368\pi$
$101$	−1742.05	−1.71624	−0.858121	+	0.513448i	$0.828368\pi$
$102$	0	0
$103$	−1454.62	−1.39154	−0.695769	−	0.718266i	$-0.744936\pi$
$103$	−1454.62	−1.39154	−0.695769	+	0.718266i	$0.744936\pi$
$104$	−297.083	−0.280110
$105$	0	0
$106$	−2234.70	−2.04767
$107$	−1181.67	−1.06763	−0.533813	−	0.845603i	$-0.679241\pi$
$107$	−1181.67	−1.06763	−0.533813	+	0.845603i	$0.679241\pi$
$108$	0	0
$109$	2204.43	1.93712	0.968559	−	0.248784i	$-0.0800310\pi$
$109$	2204.43	1.93712	0.968559	+	0.248784i	$0.0800310\pi$
$110$	0	0
$111$	0	0
$112$	559.843	0.472323
$113$	236.886	0.197207	0.0986034	−	0.995127i	$-0.468562\pi$
$113$	236.886	0.197207	0.0986034	+	0.995127i	$0.468562\pi$
$114$	0	0
$115$	0	0
$116$	−1116.41	−0.893585
$117$	0	0
$118$	2241.19	1.74846
$119$	−618.121	−0.476160
$120$	0	0
$121$	−1283.28	−0.964151
$122$	498.334	0.369811
$123$	0	0
$124$	817.976	0.592390
$125$	0	0
$126$	0	0
$127$	1667.21	1.16489	0.582446	−	0.812869i	$-0.302096\pi$
$127$	1667.21	1.16489	0.582446	+	0.812869i	$0.302096\pi$
$128$	1518.88	1.04884
$129$	0	0
$130$	0	0
$131$	−891.722	−0.594733	−0.297367	−	0.954763i	$-0.596108\pi$
$131$	−891.722	−0.594733	−0.297367	+	0.954763i	$0.596108\pi$
$132$	0	0
$133$	−258.692	−0.168657
$134$	−1668.92	−1.07591
$135$	0	0
$136$	−1185.08	−0.747203
$137$	−400.425	−0.249713	−0.124856	−	0.992175i	$-0.539847\pi$
$137$	−400.425	−0.249713	−0.124856	+	0.992175i	$0.539847\pi$
$138$	0	0
$139$	515.050	0.314287	0.157144	−	0.987576i	$-0.449771\pi$
$139$	515.050	0.314287	0.157144	+	0.987576i	$0.449771\pi$
$140$	0	0
$141$	0	0
$142$	−367.912	−0.217426
$143$	152.910	0.0894195
$144$	0	0
$145$	0	0
$146$	−3438.64	−1.94920
$147$	0	0
$148$	−8.90316	−0.00494483
$149$	−218.374	−0.120066	−0.0600332	−	0.998196i	$-0.519121\pi$
$149$	−218.374	−0.120066	−0.0600332	+	0.998196i	$0.519121\pi$
$150$	0	0
$151$	−175.011	−0.0943190	−0.0471595	−	0.998887i	$-0.515017\pi$
$151$	−175.011	−0.0943190	−0.0471595	+	0.998887i	$0.515017\pi$
$152$	−495.971	−0.264661
$153$	0	0
$154$	−168.543	−0.0881922
$155$	0	0
$156$	0	0
$157$	919.642	0.467487	0.233743	−	0.972298i	$-0.424902\pi$
$157$	919.642	0.467487	0.233743	+	0.972298i	$0.424902\pi$
$158$	−3833.73	−1.93035
$159$	0	0
$160$	0	0
$161$	668.698	0.327334
$162$	0	0
$163$	−2368.51	−1.13813	−0.569067	−	0.822291i	$-0.692695\pi$
$163$	−2368.51	−1.13813	−0.569067	+	0.822291i	$0.692695\pi$
$164$	−722.587	−0.344052
$165$	0	0
$166$	−4310.43	−2.01539
$167$	1079.37	0.500144	0.250072	−	0.968227i	$-0.419546\pi$
$167$	1079.37	0.500144	0.250072	+	0.968227i	$0.419546\pi$
$168$	0	0
$169$	−1706.98	−0.776959
$170$	0	0
$171$	0	0
$172$	70.6891	0.0313372
$173$	−881.271	−0.387294	−0.193647	−	0.981071i	$-0.562032\pi$
$173$	−881.271	−0.387294	−0.193647	+	0.981071i	$0.562032\pi$
$174$	0	0
$175$	0	0
$176$	−552.456	−0.236608
$177$	0	0
$178$	2480.73	1.04460
$179$	3377.72	1.41041	0.705203	−	0.709006i	$-0.250856\pi$
$179$	3377.72	1.41041	0.705203	+	0.709006i	$0.250856\pi$
$180$	0	0
$181$	1435.58	0.589533	0.294767	−	0.955569i	$-0.404758\pi$
$181$	1435.58	0.589533	0.294767	+	0.955569i	$0.404758\pi$
$182$	−540.118	−0.219979
$183$	0	0
$184$	1282.05	0.513661
$185$	0	0
$186$	0	0
$187$	609.965	0.238530
$188$	−2191.19	−0.850049
$189$	0	0
$190$	0	0
$191$	1588.14	0.601642	0.300821	−	0.953681i	$-0.402739\pi$
$191$	1588.14	0.601642	0.300821	+	0.953681i	$0.402739\pi$
$192$	0	0
$193$	977.704	0.364646	0.182323	−	0.983239i	$-0.441638\pi$
$193$	977.704	0.364646	0.182323	+	0.983239i	$0.441638\pi$
$194$	2218.80	0.821138
$195$	0	0
$196$	203.338	0.0741029
$197$	359.682	0.130083	0.0650413	−	0.997883i	$-0.479282\pi$
$197$	359.682	0.130083	0.0650413	+	0.997883i	$0.479282\pi$
$198$	0	0
$199$	2818.38	1.00397	0.501983	−	0.864877i	$-0.332604\pi$
$199$	2818.38	1.00397	0.501983	+	0.864877i	$0.332604\pi$
$200$	0	0
$201$	0	0
$202$	−6072.18	−2.11503
$203$	1883.21	0.651109
$204$	0	0
$205$	0	0
$206$	−5070.31	−1.71488
$207$	0	0
$208$	−1770.42	−0.590174
$209$	255.278	0.0844879
$210$	0	0
$211$	−1009.64	−0.329415	−0.164708	−	0.986342i	$-0.552668\pi$
$211$	−1009.64	−0.329415	−0.164708	+	0.986342i	$0.552668\pi$
$212$	−2660.47	−0.861895
$213$	0	0
$214$	−4118.88	−1.31570
$215$	0	0
$216$	0	0
$217$	−1379.80	−0.431644
$218$	7683.86	2.38723
$219$	0	0
$220$	0	0
$221$	1954.71	0.594969
$222$	0	0
$223$	−1277.28	−0.383555	−0.191778	−	0.981438i	$-0.561425\pi$
$223$	−1277.28	−0.383555	−0.191778	+	0.981438i	$0.561425\pi$
$224$	1199.86	0.357899
$225$	0	0
$226$	825.702	0.243030
$227$	1399.87	0.409307	0.204654	−	0.978834i	$-0.434393\pi$
$227$	1399.87	0.409307	0.204654	+	0.978834i	$0.434393\pi$
$228$	0	0
$229$	3182.00	0.918222	0.459111	−	0.888379i	$-0.348168\pi$
$229$	3182.00	0.918222	0.459111	+	0.888379i	$0.348168\pi$
$230$	0	0
$231$	0	0
$232$	3610.53	1.02174
$233$	−3027.84	−0.851332	−0.425666	−	0.904880i	$-0.639960\pi$
$233$	−3027.84	−0.851332	−0.425666	+	0.904880i	$0.639960\pi$
$234$	0	0
$235$	0	0
$236$	2668.19	0.735951
$237$	0	0
$238$	−2154.55	−0.586802
$239$	4995.69	1.35207	0.676034	−	0.736870i	$-0.263697\pi$
$239$	4995.69	1.35207	0.676034	+	0.736870i	$0.263697\pi$
$240$	0	0
$241$	−3756.52	−1.00406	−0.502030	−	0.864850i	$-0.667413\pi$
$241$	−3756.52	−1.00406	−0.502030	+	0.864850i	$0.667413\pi$
$242$	−4473.08	−1.18818
$243$	0	0
$244$	593.280	0.155659
$245$	0	0
$246$	0	0
$247$	818.072	0.210740
$248$	−2645.39	−0.677347
$249$	0	0
$250$	0	0
$251$	−6565.46	−1.65103	−0.825514	−	0.564381i	$-0.809115\pi$
$251$	−6565.46	−1.65103	−0.825514	+	0.564381i	$0.809115\pi$
$252$	0	0
$253$	−659.875	−0.163976
$254$	5811.33	1.43557
$255$	0	0
$256$	4955.51	1.20984
$257$	−6879.44	−1.66976	−0.834879	−	0.550433i	$-0.814463\pi$
$257$	−6879.44	−1.66976	−0.834879	+	0.550433i	$0.814463\pi$
$258$	0	0
$259$	15.0182	0.00360304
$260$	0	0
$261$	0	0
$262$	−3108.23	−0.732928
$263$	3080.15	0.722169	0.361084	−	0.932533i	$-0.382407\pi$
$263$	3080.15	0.722169	0.361084	+	0.932533i	$0.382407\pi$
$264$	0	0
$265$	0	0
$266$	−901.709	−0.207847
$267$	0	0
$268$	−1986.89	−0.452869
$269$	−6710.33	−1.52095	−0.760476	−	0.649366i	$-0.775034\pi$
$269$	−6710.33	−1.52095	−0.760476	+	0.649366i	$0.775034\pi$
$270$	0	0
$271$	7842.95	1.75803	0.879014	−	0.476796i	$-0.158202\pi$
$271$	7842.95	1.75803	0.879014	+	0.476796i	$0.158202\pi$
$272$	−7062.26	−1.57431
$273$	0	0
$274$	−1395.74	−0.307737
$275$	0	0
$276$	0	0
$277$	−5446.87	−1.18148	−0.590742	−	0.806861i	$-0.701165\pi$
$277$	−5446.87	−1.18148	−0.590742	+	0.806861i	$0.701165\pi$
$278$	1795.28	0.387316
$279$	0	0
$280$	0	0
$281$	−2126.76	−0.451501	−0.225751	−	0.974185i	$-0.572483\pi$
$281$	−2126.76	−0.451501	−0.225751	+	0.974185i	$0.572483\pi$
$282$	0	0
$283$	3426.38	0.719707	0.359853	−	0.933009i	$-0.382827\pi$
$283$	3426.38	0.719707	0.359853	+	0.933009i	$0.382827\pi$
$284$	−438.009	−0.0915178
$285$	0	0
$286$	532.991	0.110197
$287$	1218.89	0.250693
$288$	0	0
$289$	2884.42	0.587099
$290$	0	0
$291$	0	0
$292$	−4093.79	−0.820448
$293$	−1749.82	−0.348894	−0.174447	−	0.984667i	$-0.555814\pi$
$293$	−1749.82	−0.348894	−0.174447	+	0.984667i	$0.555814\pi$
$294$	0	0
$295$	0	0
$296$	28.7934	0.00565399
$297$	0	0
$298$	−761.176	−0.147966
$299$	−2114.65	−0.409008
$300$	0	0
$301$	−119.241	−0.0228338
$302$	−610.026	−0.116235
$303$	0	0
$304$	−2955.65	−0.557625
$305$	0	0
$306$	0	0
$307$	−7970.33	−1.48173	−0.740864	−	0.671655i	$-0.765584\pi$
$307$	−7970.33	−1.48173	−0.740864	+	0.671655i	$0.765584\pi$
$308$	−200.655	−0.0371214
$309$	0	0
$310$	0	0
$311$	2560.72	0.466898	0.233449	−	0.972369i	$-0.424999\pi$
$311$	2560.72	0.466898	0.233449	+	0.972369i	$0.424999\pi$
$312$	0	0
$313$	4861.11	0.877848	0.438924	−	0.898524i	$-0.355360\pi$
$313$	4861.11	0.877848	0.438924	+	0.898524i	$0.355360\pi$
$314$	3205.55	0.576114
$315$	0	0
$316$	−4564.16	−0.812513
$317$	8166.16	1.44687	0.723434	−	0.690394i	$-0.242563\pi$
$317$	8166.16	1.44687	0.723434	+	0.690394i	$0.242563\pi$
$318$	0	0
$319$	−1858.36	−0.326169
$320$	0	0
$321$	0	0
$322$	2330.85	0.403395
$323$	3263.32	0.562155
$324$	0	0
$325$	0	0
$326$	−8255.79	−1.40259
$327$	0	0
$328$	2336.89	0.393394
$329$	3696.20	0.619386
$330$	0	0
$331$	2974.89	0.494002	0.247001	−	0.969015i	$-0.420555\pi$
$331$	2974.89	0.494002	0.247001	+	0.969015i	$0.420555\pi$
$332$	−5131.68	−0.848306
$333$	0	0
$334$	3762.30	0.616359
$335$	0	0
$336$	0	0
$337$	−3496.34	−0.565157	−0.282578	−	0.959244i	$-0.591190\pi$
$337$	−3496.34	−0.565157	−0.282578	+	0.959244i	$0.591190\pi$
$338$	−5949.94	−0.957497
$339$	0	0
$340$	0	0
$341$	1361.59	0.216230
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−228.613	−0.0358314
$345$	0	0
$346$	−3071.80	−0.477287
$347$	3959.08	0.612491	0.306246	−	0.951953i	$-0.400927\pi$
$347$	3959.08	0.612491	0.306246	+	0.951953i	$0.400927\pi$
$348$	0	0
$349$	5581.65	0.856099	0.428050	−	0.903755i	$-0.359201\pi$
$349$	5581.65	0.856099	0.428050	+	0.903755i	$0.359201\pi$
$350$	0	0
$351$	0	0
$352$	−1184.03	−0.179287
$353$	−9896.43	−1.49216	−0.746082	−	0.665854i	$-0.768067\pi$
$353$	−9896.43	−1.49216	−0.746082	+	0.665854i	$0.768067\pi$
$354$	0	0
$355$	0	0
$356$	2953.38	0.439687
$357$	0	0
$358$	11773.5	1.73813
$359$	−11917.6	−1.75205	−0.876025	−	0.482265i	$-0.839814\pi$
$359$	−11917.6	−1.75205	−0.876025	+	0.482265i	$0.839814\pi$
$360$	0	0
$361$	−5493.26	−0.800883
$362$	5003.92	0.726519
$363$	0	0
$364$	−643.025	−0.0925926
$365$	0	0
$366$	0	0
$367$	7101.58	1.01008	0.505040	−	0.863096i	$-0.331478\pi$
$367$	7101.58	1.01008	0.505040	+	0.863096i	$0.331478\pi$
$368$	7640.12	1.08225
$369$	0	0
$370$	0	0
$371$	4487.80	0.628018
$372$	0	0
$373$	−294.316	−0.0408555	−0.0204277	−	0.999791i	$-0.506503\pi$
$373$	−294.316	−0.0408555	−0.0204277	+	0.999791i	$0.506503\pi$
$374$	2126.12	0.293955
$375$	0	0
$376$	7086.45	0.971957
$377$	−5955.34	−0.813569
$378$	0	0
$379$	−9436.57	−1.27896	−0.639478	−	0.768810i	$-0.720849\pi$
$379$	−9436.57	−1.27896	−0.639478	+	0.768810i	$0.720849\pi$
$380$	0	0
$381$	0	0
$382$	5535.70	0.741442
$383$	−3160.82	−0.421699	−0.210849	−	0.977519i	$-0.567623\pi$
$383$	−3160.82	−0.421699	−0.210849	+	0.977519i	$0.567623\pi$
$384$	0	0
$385$	0	0
$386$	3407.93	0.449376
$387$	0	0
$388$	2641.55	0.345629
$389$	−7822.76	−1.01961	−0.509807	−	0.860289i	$-0.670283\pi$
$389$	−7822.76	−1.01961	−0.509807	+	0.860289i	$0.670283\pi$
$390$	0	0
$391$	−8435.43	−1.09104
$392$	−657.609	−0.0847302
$393$	0	0
$394$	1253.73	0.160309
$395$	0	0
$396$	0	0
$397$	7935.18	1.00316	0.501581	−	0.865111i	$-0.332752\pi$
$397$	7935.18	1.00316	0.501581	+	0.865111i	$0.332752\pi$
$398$	9823.87	1.23725
$399$	0	0
$400$	0	0
$401$	488.380	0.0608193	0.0304097	−	0.999538i	$-0.490319\pi$
$401$	488.380	0.0608193	0.0304097	+	0.999538i	$0.490319\pi$
$402$	0	0
$403$	4363.39	0.539345
$404$	−7229.09	−0.890249
$405$	0	0
$406$	6564.20	0.802403
$407$	−14.8201	−0.00180492
$408$	0	0
$409$	11230.6	1.35775	0.678874	−	0.734254i	$-0.262468\pi$
$409$	11230.6	1.35775	0.678874	+	0.734254i	$0.262468\pi$
$410$	0	0
$411$	0	0
$412$	−6036.34	−0.721818
$413$	−4500.82	−0.536249
$414$	0	0
$415$	0	0
$416$	−3794.38	−0.447199
$417$	0	0
$418$	889.811	0.104120
$419$	7369.62	0.859259	0.429629	−	0.903005i	$-0.358644\pi$
$419$	7369.62	0.859259	0.429629	+	0.903005i	$0.358644\pi$
$420$	0	0
$421$	11972.5	1.38599	0.692997	−	0.720941i	$-0.256290\pi$
$421$	11972.5	1.38599	0.692997	+	0.720941i	$0.256290\pi$
$422$	−3519.26	−0.405959
$423$	0	0
$424$	8604.12	0.985503
$425$	0	0
$426$	0	0
$427$	−1000.77	−0.113421
$428$	−4903.63	−0.553799
$429$	0	0
$430$	0	0
$431$	3568.60	0.398825	0.199412	−	0.979916i	$-0.436097\pi$
$431$	3568.60	0.398825	0.199412	+	0.979916i	$0.436097\pi$
$432$	0	0
$433$	−2291.60	−0.254335	−0.127168	−	0.991881i	$-0.540589\pi$
$433$	−2291.60	−0.254335	−0.127168	+	0.991881i	$0.540589\pi$
$434$	−4809.49	−0.531943
$435$	0	0
$436$	9147.85	1.00482
$437$	−3530.34	−0.386451
$438$	0	0
$439$	−7329.66	−0.796870	−0.398435	−	0.917197i	$-0.630446\pi$
$439$	−7329.66	−0.796870	−0.398435	+	0.917197i	$0.630446\pi$
$440$	0	0
$441$	0	0
$442$	6813.44	0.733218
$443$	8297.38	0.889889	0.444944	−	0.895558i	$-0.353223\pi$
$443$	8297.38	0.889889	0.444944	+	0.895558i	$0.353223\pi$
$444$	0	0
$445$	0	0
$446$	−4452.14	−0.472679
$447$	0	0
$448$	−296.436	−0.0312618
$449$	−9758.62	−1.02570	−0.512848	−	0.858479i	$-0.671410\pi$
$449$	−9758.62	−1.02570	−0.512848	+	0.858479i	$0.671410\pi$
$450$	0	0
$451$	−1202.81	−0.125583
$452$	983.021	0.102295
$453$	0	0
$454$	4879.47	0.504416
$455$	0	0
$456$	0	0
$457$	11745.0	1.20220	0.601102	−	0.799172i	$-0.294728\pi$
$457$	11745.0	1.20220	0.601102	+	0.799172i	$0.294728\pi$
$458$	11091.4	1.13158
$459$	0	0
$460$	0	0
$461$	10748.6	1.08593	0.542963	−	0.839756i	$-0.317302\pi$
$461$	10748.6	1.08593	0.542963	+	0.839756i	$0.317302\pi$
$462$	0	0
$463$	9862.51	0.989957	0.494978	−	0.868905i	$-0.335176\pi$
$463$	9862.51	0.989957	0.494978	+	0.868905i	$0.335176\pi$
$464$	21516.3	2.15274
$465$	0	0
$466$	−10554.0	−1.04915
$467$	−4660.78	−0.461831	−0.230916	−	0.972974i	$-0.574172\pi$
$467$	−4660.78	−0.461831	−0.230916	+	0.972974i	$0.574172\pi$
$468$	0	0
$469$	3351.58	0.329982
$470$	0	0
$471$	0	0
$472$	−8629.10	−0.841497
$473$	117.668	0.0114384
$474$	0	0
$475$	0	0
$476$	−2565.05	−0.246994
$477$	0	0
$478$	17413.2	1.66624
$479$	16293.2	1.55419	0.777094	−	0.629385i	$-0.216693\pi$
$479$	16293.2	1.55419	0.777094	+	0.629385i	$0.216693\pi$
$480$	0	0
$481$	−47.4928	−0.00450205
$482$	−13093.9	−1.23737
$483$	0	0
$484$	−5325.33	−0.500124
$485$	0	0
$486$	0	0
$487$	3515.00	0.327063	0.163531	−	0.986538i	$-0.447711\pi$
$487$	3515.00	0.327063	0.163531	+	0.986538i	$0.447711\pi$
$488$	−1918.70	−0.177983
$489$	0	0
$490$	0	0
$491$	2516.79	0.231326	0.115663	−	0.993288i	$-0.463101\pi$
$491$	2516.79	0.231326	0.115663	+	0.993288i	$0.463101\pi$
$492$	0	0
$493$	−23756.1	−2.17023
$494$	2851.51	0.259708
$495$	0	0
$496$	−15764.7	−1.42713
$497$	738.853	0.0666842
$498$	0	0
$499$	−8747.48	−0.784751	−0.392376	−	0.919805i	$-0.628347\pi$
$499$	−8747.48	−0.784751	−0.392376	+	0.919805i	$0.628347\pi$
$500$	0	0
$501$	0	0
$502$	−22884.9	−2.03467
$503$	11426.1	1.01285	0.506426	−	0.862284i	$-0.330966\pi$
$503$	11426.1	1.01285	0.506426	+	0.862284i	$0.330966\pi$
$504$	0	0
$505$	0	0
$506$	−2300.09	−0.202078
$507$	0	0
$508$	6918.54	0.604254
$509$	8078.44	0.703478	0.351739	−	0.936098i	$-0.385590\pi$
$509$	8078.44	0.703478	0.351739	+	0.936098i	$0.385590\pi$
$510$	0	0
$511$	6905.58	0.597818
$512$	5122.12	0.442125
$513$	0	0
$514$	−23979.3	−2.05775
$515$	0	0
$516$	0	0
$517$	−3647.43	−0.310278
$518$	52.3484	0.00444026
$519$	0	0
$520$	0	0
$521$	−7226.14	−0.607645	−0.303822	−	0.952729i	$-0.598263\pi$
$521$	−7226.14	−0.607645	−0.303822	+	0.952729i	$0.598263\pi$
$522$	0	0
$523$	−9333.06	−0.780318	−0.390159	−	0.920748i	$-0.627580\pi$
$523$	−9333.06	−0.780318	−0.390159	+	0.920748i	$0.627580\pi$
$524$	−3700.43	−0.308500
$525$	0	0
$526$	10736.3	0.889974
$527$	17405.8	1.43872
$528$	0	0
$529$	−3041.35	−0.249967
$530$	0	0
$531$	0	0
$532$	−1073.51	−0.0874859
$533$	−3854.55	−0.313244
$534$	0	0
$535$	0	0
$536$	6425.73	0.517816
$537$	0	0
$538$	−23389.9	−1.87437
$539$	338.474	0.0270484
$540$	0	0
$541$	15263.1	1.21296	0.606482	−	0.795097i	$-0.292580\pi$
$541$	15263.1	1.21296	0.606482	+	0.795097i	$0.292580\pi$
$542$	27337.8	2.16653
$543$	0	0
$544$	−15135.9	−1.19292
$545$	0	0
$546$	0	0
$547$	13226.0	1.03382	0.516912	−	0.856039i	$-0.327082\pi$
$547$	13226.0	1.03382	0.516912	+	0.856039i	$0.327082\pi$
$548$	−1661.67	−0.129531
$549$	0	0
$550$	0	0
$551$	−9942.24	−0.768700
$552$	0	0
$553$	7699.03	0.592036
$554$	−18985.9	−1.45602
$555$	0	0
$556$	2137.33	0.163027
$557$	6993.63	0.532010	0.266005	−	0.963972i	$-0.414296\pi$
$557$	6993.63	0.532010	0.266005	+	0.963972i	$0.414296\pi$
$558$	0	0
$559$	377.082	0.0285311
$560$	0	0
$561$	0	0
$562$	−7413.14	−0.556413
$563$	392.197	0.0293590	0.0146795	−	0.999892i	$-0.495327\pi$
$563$	392.197	0.0293590	0.0146795	+	0.999892i	$0.495327\pi$
$564$	0	0
$565$	0	0
$566$	11943.2	0.886940
$567$	0	0
$568$	1416.55	0.104643
$569$	−8811.72	−0.649221	−0.324610	−	0.945848i	$-0.605233\pi$
$569$	−8811.72	−0.649221	−0.324610	+	0.945848i	$0.605233\pi$
$570$	0	0
$571$	−24775.6	−1.81581	−0.907905	−	0.419175i	$-0.862319\pi$
$571$	−24775.6	−1.81581	−0.907905	+	0.419175i	$0.862319\pi$
$572$	634.541	0.0463837
$573$	0	0
$574$	4248.63	0.308945
$575$	0	0
$576$	0	0
$577$	8850.62	0.638572	0.319286	−	0.947658i	$-0.396557\pi$
$577$	8850.62	0.638572	0.319286	+	0.947658i	$0.396557\pi$
$578$	10054.1	0.723520
$579$	0	0
$580$	0	0
$581$	8656.35	0.618117
$582$	0	0
$583$	−4428.58	−0.314602
$584$	13239.6	0.938112
$585$	0	0
$586$	−6099.28	−0.429964
$587$	−46.0232	−0.00323608	−0.00161804	−	0.999999i	$-0.500515\pi$
$587$	−46.0232	−0.00323608	−0.00161804	+	0.999999i	$0.500515\pi$
$588$	0	0
$589$	7284.54	0.509600
$590$	0	0
$591$	0	0
$592$	171.589	0.0119126
$593$	−2729.93	−0.189047	−0.0945235	−	0.995523i	$-0.530133\pi$
$593$	−2729.93	−0.189047	−0.0945235	+	0.995523i	$0.530133\pi$
$594$	0	0
$595$	0	0
$596$	−906.200	−0.0622809
$597$	0	0
$598$	−7370.94	−0.504047
$599$	5505.07	0.375511	0.187756	−	0.982216i	$-0.439879\pi$
$599$	5505.07	0.375511	0.187756	+	0.982216i	$0.439879\pi$
$600$	0	0
$601$	−7446.97	−0.505438	−0.252719	−	0.967540i	$-0.581325\pi$
$601$	−7446.97	−0.505438	−0.252719	+	0.967540i	$0.581325\pi$
$602$	−415.634	−0.0281395
$603$	0	0
$604$	−726.253	−0.0489252
$605$	0	0
$606$	0	0
$607$	24071.4	1.60960	0.804799	−	0.593547i	$-0.202273\pi$
$607$	24071.4	1.60960	0.804799	+	0.593547i	$0.202273\pi$
$608$	−6334.59	−0.422536
$609$	0	0
$610$	0	0
$611$	−11688.7	−0.773932
$612$	0	0
$613$	4108.61	0.270710	0.135355	−	0.990797i	$-0.456782\pi$
$613$	4108.61	0.270710	0.135355	+	0.990797i	$0.456782\pi$
$614$	−27781.8	−1.82603
$615$	0	0
$616$	648.932	0.0424451
$617$	3542.46	0.231141	0.115571	−	0.993299i	$-0.463130\pi$
$617$	3542.46	0.231141	0.115571	+	0.993299i	$0.463130\pi$
$618$	0	0
$619$	6484.81	0.421077	0.210538	−	0.977586i	$-0.432478\pi$
$619$	6484.81	0.421077	0.210538	+	0.977586i	$0.432478\pi$
$620$	0	0
$621$	0	0
$622$	8925.79	0.575388
$623$	−4981.88	−0.320377
$624$	0	0
$625$	0	0
$626$	16944.1	1.08183
$627$	0	0
$628$	3816.29	0.242495
$629$	−189.451	−0.0120094
$630$	0	0
$631$	3250.84	0.205094	0.102547	−	0.994728i	$-0.467301\pi$
$631$	3250.84	0.205094	0.102547	+	0.994728i	$0.467301\pi$
$632$	14760.8	0.929038
$633$	0	0
$634$	28464.4	1.78307
$635$	0	0
$636$	0	0
$637$	1084.68	0.0674674
$638$	−6477.58	−0.401959
$639$	0	0
$640$	0	0
$641$	−2800.61	−0.172570	−0.0862852	−	0.996270i	$-0.527500\pi$
$641$	−2800.61	−0.172570	−0.0862852	+	0.996270i	$0.527500\pi$
$642$	0	0
$643$	−18910.6	−1.15982	−0.579908	−	0.814682i	$-0.696911\pi$
$643$	−18910.6	−1.15982	−0.579908	+	0.814682i	$0.696911\pi$
$644$	2774.94	0.169795
$645$	0	0
$646$	11374.8	0.692780
$647$	−24522.7	−1.49009	−0.745043	−	0.667016i	$-0.767571\pi$
$647$	−24522.7	−1.49009	−0.745043	+	0.667016i	$0.767571\pi$
$648$	0	0
$649$	4441.43	0.268631
$650$	0	0
$651$	0	0
$652$	−9828.74	−0.590373
$653$	−15299.6	−0.916875	−0.458438	−	0.888727i	$-0.651591\pi$
$653$	−15299.6	−0.916875	−0.458438	+	0.888727i	$0.651591\pi$
$654$	0	0
$655$	0	0
$656$	13926.3	0.828857
$657$	0	0
$658$	12883.7	0.763309
$659$	2203.13	0.130230	0.0651151	−	0.997878i	$-0.479259\pi$
$659$	2203.13	0.130230	0.0651151	+	0.997878i	$0.479259\pi$
$660$	0	0
$661$	−3162.36	−0.186084	−0.0930421	−	0.995662i	$-0.529659\pi$
$661$	−3162.36	−0.186084	−0.0930421	+	0.995662i	$0.529659\pi$
$662$	10369.4	0.608790
$663$	0	0
$664$	16596.2	0.969965
$665$	0	0
$666$	0	0
$667$	25699.9	1.49191
$668$	4479.12	0.259435
$669$	0	0
$670$	0	0
$671$	987.565	0.0568175
$672$	0	0
$673$	4443.07	0.254484	0.127242	−	0.991872i	$-0.459387\pi$
$673$	4443.07	0.254484	0.127242	+	0.991872i	$0.459387\pi$
$674$	−12187.0	−0.696479
$675$	0	0
$676$	−7083.56	−0.403025
$677$	4456.32	0.252984	0.126492	−	0.991968i	$-0.459628\pi$
$677$	4456.32	0.252984	0.126492	+	0.991968i	$0.459628\pi$
$678$	0	0
$679$	−4455.87	−0.251842
$680$	0	0
$681$	0	0
$682$	4746.03	0.266474
$683$	−10046.8	−0.562858	−0.281429	−	0.959582i	$-0.590808\pi$
$683$	−10046.8	−0.562858	−0.281429	+	0.959582i	$0.590808\pi$
$684$	0	0
$685$	0	0
$686$	−1195.58	−0.0665414
$687$	0	0
$688$	−1362.38	−0.0754944
$689$	−14191.9	−0.784717
$690$	0	0
$691$	31811.2	1.75131	0.875655	−	0.482938i	$-0.160430\pi$
$691$	31811.2	1.75131	0.875655	+	0.482938i	$0.160430\pi$
$692$	−3657.06	−0.200897
$693$	0	0
$694$	13800.0	0.754812
$695$	0	0
$696$	0	0
$697$	−15376.0	−0.835590
$698$	19455.7	1.05503
$699$	0	0
$700$	0	0
$701$	13907.2	0.749312	0.374656	−	0.927164i	$-0.377761\pi$
$701$	13907.2	0.749312	0.374656	+	0.927164i	$0.377761\pi$
$702$	0	0
$703$	−79.2877	−0.00425376
$704$	292.524	0.0156604
$705$	0	0
$706$	−34495.5	−1.83889
$707$	12194.3	0.648678
$708$	0	0
$709$	−228.952	−0.0121276	−0.00606381	−	0.999982i	$-0.501930\pi$
$709$	−228.952	−0.0121276	−0.00606381	+	0.999982i	$0.501930\pi$
$710$	0	0
$711$	0	0
$712$	−9551.40	−0.502744
$713$	−18830.0	−0.989043
$714$	0	0
$715$	0	0
$716$	14016.7	0.731606
$717$	0	0
$718$	−41540.5	−2.15916
$719$	−36162.2	−1.87569	−0.937846	−	0.347052i	$-0.887183\pi$
$719$	−36162.2	−1.87569	−0.937846	+	0.347052i	$0.887183\pi$
$720$	0	0
$721$	10182.4	0.525952
$722$	−19147.6	−0.986979
$723$	0	0
$724$	5957.30	0.305803
$725$	0	0
$726$	0	0
$727$	−22268.3	−1.13602	−0.568010	−	0.823021i	$-0.692287\pi$
$727$	−22268.3	−1.13602	−0.568010	+	0.823021i	$0.692287\pi$
$728$	2079.58	0.105872
$729$	0	0
$730$	0	0
$731$	1504.20	0.0761077
$732$	0	0
$733$	2333.20	0.117570	0.0587848	−	0.998271i	$-0.481277\pi$
$733$	2333.20	0.117570	0.0587848	+	0.998271i	$0.481277\pi$
$734$	24753.6	1.24479
$735$	0	0
$736$	16374.4	0.820067
$737$	−3307.35	−0.165302
$738$	0	0
$739$	−4829.15	−0.240383	−0.120192	−	0.992751i	$-0.538351\pi$
$739$	−4829.15	−0.240383	−0.120192	+	0.992751i	$0.538351\pi$
$740$	0	0
$741$	0	0
$742$	15642.9	0.773947
$743$	−25459.0	−1.25707	−0.628533	−	0.777783i	$-0.716344\pi$
$743$	−25459.0	−1.25707	−0.628533	+	0.777783i	$0.716344\pi$
$744$	0	0
$745$	0	0
$746$	−1025.88	−0.0503488
$747$	0	0
$748$	2531.21	0.123730
$749$	8271.66	0.403525
$750$	0	0
$751$	−5707.08	−0.277303	−0.138651	−	0.990341i	$-0.544277\pi$
$751$	−5707.08	−0.277303	−0.138651	+	0.990341i	$0.544277\pi$
$752$	42230.4	2.04785
$753$	0	0
$754$	−20758.2	−1.00261
$755$	0	0
$756$	0	0
$757$	1900.91	0.0912677	0.0456339	−	0.998958i	$-0.485469\pi$
$757$	1900.91	0.0912677	0.0456339	+	0.998958i	$0.485469\pi$
$758$	−32892.6	−1.57614
$759$	0	0
$760$	0	0
$761$	11583.8	0.551791	0.275896	−	0.961188i	$-0.411026\pi$
$761$	11583.8	0.551791	0.275896	+	0.961188i	$0.411026\pi$
$762$	0	0
$763$	−15431.0	−0.732162
$764$	6590.40	0.312084
$765$	0	0
$766$	−11017.5	−0.519686
$767$	14233.1	0.670051
$768$	0	0
$769$	−26059.7	−1.22202	−0.611012	−	0.791622i	$-0.709237\pi$
$769$	−26059.7	−1.22202	−0.611012	+	0.791622i	$0.709237\pi$
$770$	0	0
$771$	0	0
$772$	4057.24	0.189149
$773$	−16213.6	−0.754413	−0.377206	−	0.926129i	$-0.623115\pi$
$773$	−16213.6	−0.754413	−0.377206	+	0.926129i	$0.623115\pi$
$774$	0	0
$775$	0	0
$776$	−8542.92	−0.395197
$777$	0	0
$778$	−27267.4	−1.25653
$779$	−6435.04	−0.295968
$780$	0	0
$781$	−729.103	−0.0334051
$782$	−29403.0	−1.34456
$783$	0	0
$784$	−3918.90	−0.178521
$785$	0	0
$786$	0	0
$787$	1371.34	0.0621131	0.0310565	−	0.999518i	$-0.490113\pi$
$787$	1371.34	0.0621131	0.0310565	+	0.999518i	$0.490113\pi$
$788$	1492.59	0.0674765
$789$	0	0
$790$	0	0
$791$	−1658.20	−0.0745371
$792$	0	0
$793$	3164.78	0.141721
$794$	27659.3	1.23626
$795$	0	0
$796$	11695.6	0.520778
$797$	7991.49	0.355173	0.177587	−	0.984105i	$-0.443171\pi$
$797$	7991.49	0.355173	0.177587	+	0.984105i	$0.443171\pi$
$798$	0	0
$799$	−46626.5	−2.06449
$800$	0	0
$801$	0	0
$802$	1702.32	0.0749515
$803$	−6814.47	−0.299474
$804$	0	0
$805$	0	0
$806$	15209.3	0.664669
$807$	0	0
$808$	23379.3	1.01792
$809$	17661.4	0.767542	0.383771	−	0.923428i	$-0.374625\pi$
$809$	17661.4	0.767542	0.383771	+	0.923428i	$0.374625\pi$
$810$	0	0
$811$	−24180.6	−1.04697	−0.523486	−	0.852034i	$-0.675369\pi$
$811$	−24180.6	−1.04697	−0.523486	+	0.852034i	$0.675369\pi$
$812$	7814.85	0.337743
$813$	0	0
$814$	−51.6576	−0.00222432
$815$	0	0
$816$	0	0
$817$	629.526	0.0269576
$818$	39146.1	1.67324
$819$	0	0
$820$	0	0
$821$	23340.9	0.992208	0.496104	−	0.868263i	$-0.334763\pi$
$821$	23340.9	0.992208	0.496104	+	0.868263i	$0.334763\pi$
$822$	0	0
$823$	−20630.4	−0.873792	−0.436896	−	0.899512i	$-0.643922\pi$
$823$	−20630.4	−0.873792	−0.436896	+	0.899512i	$0.643922\pi$
$824$	19521.9	0.825337
$825$	0	0
$826$	−15688.3	−0.660854
$827$	24113.7	1.01393	0.506963	−	0.861968i	$-0.330768\pi$
$827$	24113.7	1.01393	0.506963	+	0.861968i	$0.330768\pi$
$828$	0	0
$829$	41738.9	1.74868	0.874338	−	0.485317i	$-0.161296\pi$
$829$	41738.9	1.74868	0.874338	+	0.485317i	$0.161296\pi$
$830$	0	0
$831$	0	0
$832$	937.432	0.0390620
$833$	4326.85	0.179972
$834$	0	0
$835$	0	0
$836$	1059.34	0.0438256
$837$	0	0
$838$	25687.9	1.05892
$839$	−30403.0	−1.25105	−0.625523	−	0.780205i	$-0.715114\pi$
$839$	−30403.0	−1.25105	−0.625523	+	0.780205i	$0.715114\pi$
$840$	0	0
$841$	47987.8	1.96760
$842$	41731.9	1.70805
$843$	0	0
$844$	−4189.77	−0.170874
$845$	0	0
$846$	0	0
$847$	8982.99	0.364415
$848$	51274.7	2.07639
$849$	0	0
$850$	0	0
$851$	204.952	0.00825579
$852$	0	0
$853$	−5900.43	−0.236843	−0.118421	−	0.992963i	$-0.537783\pi$
$853$	−5900.43	−0.236843	−0.118421	+	0.992963i	$0.537783\pi$
$854$	−3488.33	−0.139776
$855$	0	0
$856$	15858.7	0.633221
$857$	−18226.2	−0.726480	−0.363240	−	0.931696i	$-0.618330\pi$
$857$	−18226.2	−0.726480	−0.363240	+	0.931696i	$0.618330\pi$
$858$	0	0
$859$	−19944.2	−0.792186	−0.396093	−	0.918210i	$-0.629634\pi$
$859$	−19944.2	−0.792186	−0.396093	+	0.918210i	$0.629634\pi$
$860$	0	0
$861$	0	0
$862$	12438.9	0.491497
$863$	30830.3	1.21608	0.608038	−	0.793908i	$-0.291957\pi$
$863$	30830.3	1.21608	0.608038	+	0.793908i	$0.291957\pi$
$864$	0	0
$865$	0	0
$866$	−7987.70	−0.313433
$867$	0	0
$868$	−5725.83	−0.223903
$869$	−7597.44	−0.296577
$870$	0	0
$871$	−10598.8	−0.412317
$872$	−29584.7	−1.14893
$873$	0	0
$874$	−12305.5	−0.476248
$875$	0	0
$876$	0	0
$877$	45885.6	1.76676	0.883379	−	0.468659i	$-0.155263\pi$
$877$	45885.6	1.76676	0.883379	+	0.468659i	$0.155263\pi$
$878$	−25548.6	−0.982033
$879$	0	0
$880$	0	0
$881$	−41132.6	−1.57298	−0.786489	−	0.617605i	$-0.788103\pi$
$881$	−41132.6	−1.57298	−0.786489	+	0.617605i	$0.788103\pi$
$882$	0	0
$883$	−19850.0	−0.756520	−0.378260	−	0.925699i	$-0.623477\pi$
$883$	−19850.0	−0.756520	−0.378260	+	0.925699i	$0.623477\pi$
$884$	8111.58	0.308622
$885$	0	0
$886$	28921.8	1.09667
$887$	29029.3	1.09888	0.549441	−	0.835532i	$-0.314841\pi$
$887$	29029.3	1.09888	0.549441	+	0.835532i	$0.314841\pi$
$888$	0	0
$889$	−11670.5	−0.440288
$890$	0	0
$891$	0	0
$892$	−5300.40	−0.198958
$893$	−19513.8	−0.731248
$894$	0	0
$895$	0	0
$896$	−10632.2	−0.396425
$897$	0	0
$898$	−34015.2	−1.26403
$899$	−53029.4	−1.96733
$900$	0	0
$901$	−56612.3	−2.09326
$902$	−4192.57	−0.154764
$903$	0	0
$904$	−3179.15	−0.116966
$905$	0	0
$906$	0	0
$907$	21029.1	0.769856	0.384928	−	0.922947i	$-0.374226\pi$
$907$	21029.1	0.769856	0.384928	+	0.922947i	$0.374226\pi$
$908$	5809.14	0.212316
$909$	0	0
$910$	0	0
$911$	19225.5	0.699198	0.349599	−	0.936899i	$-0.386318\pi$
$911$	19225.5	0.699198	0.349599	+	0.936899i	$0.386318\pi$
$912$	0	0
$913$	−8542.13	−0.309642
$914$	40938.9	1.48155
$915$	0	0
$916$	13204.6	0.476300
$917$	6242.05	0.224788
$918$	0	0
$919$	21316.8	0.765154	0.382577	−	0.923924i	$-0.375037\pi$
$919$	21316.8	0.765154	0.382577	+	0.923924i	$0.375037\pi$
$920$	0	0
$921$	0	0
$922$	37465.9	1.33826
$923$	−2336.50	−0.0833229
$924$	0	0
$925$	0	0
$926$	34377.3	1.21999
$927$	0	0
$928$	46114.1	1.63122
$929$	−19989.6	−0.705959	−0.352979	−	0.935631i	$-0.614831\pi$
$929$	−19989.6	−0.705959	−0.352979	+	0.935631i	$0.614831\pi$
$930$	0	0
$931$	1810.84	0.0637465
$932$	−12564.8	−0.441603
$933$	0	0
$934$	−16245.9	−0.569144
$935$	0	0
$936$	0	0
$937$	−55676.4	−1.94116	−0.970580	−	0.240779i	$-0.922597\pi$
$937$	−55676.4	−1.94116	−0.970580	+	0.240779i	$0.922597\pi$
$938$	11682.4	0.406658
$939$	0	0
$940$	0	0
$941$	108.842	0.00377062	0.00188531	−	0.999998i	$-0.499400\pi$
$941$	108.842	0.00377062	0.00188531	+	0.999998i	$0.499400\pi$
$942$	0	0
$943$	16634.1	0.574422
$944$	−51423.6	−1.77298
$945$	0	0
$946$	410.150	0.0140963
$947$	−7785.95	−0.267169	−0.133585	−	0.991037i	$-0.542649\pi$
$947$	−7785.95	−0.267169	−0.133585	+	0.991037i	$0.542649\pi$
$948$	0	0
$949$	−21837.8	−0.746982
$950$	0	0
$951$	0	0
$952$	8295.55	0.282416
$953$	41445.5	1.40876	0.704381	−	0.709822i	$-0.251224\pi$
$953$	41445.5	1.40876	0.704381	+	0.709822i	$0.251224\pi$
$954$	0	0
$955$	0	0
$956$	20730.9	0.701345
$957$	0	0
$958$	56792.5	1.91532
$959$	2802.98	0.0943825
$960$	0	0
$961$	9062.92	0.304217
$962$	−165.543	−0.00554817
$963$	0	0
$964$	−15588.7	−0.520826
$965$	0	0
$966$	0	0
$967$	−39155.0	−1.30211	−0.651055	−	0.759030i	$-0.725673\pi$
$967$	−39155.0	−1.30211	−0.651055	+	0.759030i	$0.725673\pi$
$968$	17222.4	0.571849
$969$	0	0
$970$	0	0
$971$	43440.8	1.43572	0.717859	−	0.696189i	$-0.245122\pi$
$971$	43440.8	1.43572	0.717859	+	0.696189i	$0.245122\pi$
$972$	0	0
$973$	−3605.35	−0.118789
$974$	12252.0	0.403061
$975$	0	0
$976$	−11434.2	−0.374999
$977$	11297.8	0.369957	0.184978	−	0.982743i	$-0.440778\pi$
$977$	11297.8	0.369957	0.184978	+	0.982743i	$0.440778\pi$
$978$	0	0
$979$	4916.15	0.160491
$980$	0	0
$981$	0	0
$982$	8772.66	0.285078
$983$	10865.9	0.352563	0.176282	−	0.984340i	$-0.443593\pi$
$983$	10865.9	0.352563	0.176282	+	0.984340i	$0.443593\pi$
$984$	0	0
$985$	0	0
$986$	−82805.5	−2.67451
$987$	0	0
$988$	3394.80	0.109315
$989$	−1627.28	−0.0523199
$990$	0	0
$991$	13884.1	0.445048	0.222524	−	0.974927i	$-0.428570\pi$
$991$	13884.1	0.445048	0.222524	+	0.974927i	$0.428570\pi$
$992$	−33787.1	−1.08139
$993$	0	0
$994$	2575.38	0.0821792
$995$	0	0
$996$	0	0
$997$	−31665.7	−1.00588	−0.502940	−	0.864321i	$-0.667748\pi$
$997$	−31665.7	−1.00588	−0.502940	+	0.864321i	$0.667748\pi$
$998$	−30490.7	−0.967099
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.ba.1.3		3
3.2	odd	2		175.4.a.f.1.1		3
5.4	even	2		315.4.a.p.1.1		3
15.2	even	4		175.4.b.e.99.2		6
15.8	even	4		175.4.b.e.99.5		6
15.14	odd	2		35.4.a.c.1.3	✓	3
21.20	even	2		1225.4.a.y.1.1		3
35.34	odd	2		2205.4.a.bm.1.1		3
60.59	even	2		560.4.a.u.1.2		3
105.44	odd	6		245.4.e.m.116.1		6
105.59	even	6		245.4.e.n.226.1		6
105.74	odd	6		245.4.e.m.226.1		6
105.89	even	6		245.4.e.n.116.1		6
105.104	even	2		245.4.a.l.1.3		3
120.29	odd	2		2240.4.a.bt.1.2		3
120.59	even	2		2240.4.a.bv.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.a.c.1.3	✓	3	15.14	odd	2
175.4.a.f.1.1		3	3.2	odd	2
175.4.b.e.99.2		6	15.2	even	4
175.4.b.e.99.5		6	15.8	even	4
245.4.a.l.1.3		3	105.104	even	2
245.4.e.m.116.1		6	105.44	odd	6
245.4.e.m.226.1		6	105.74	odd	6
245.4.e.n.116.1		6	105.89	even	6
245.4.e.n.226.1		6	105.59	even	6
315.4.a.p.1.1		3	5.4	even	2
560.4.a.u.1.2		3	60.59	even	2
1225.4.a.y.1.1		3	21.20	even	2
1575.4.a.ba.1.3		3	1.1	even	1	trivial
2205.4.a.bm.1.1		3	35.34	odd	2
2240.4.a.bt.1.2		3	120.29	odd	2
2240.4.a.bv.1.2		3	120.59	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q+3.48565 q^{2} +4.14976 q^{4} -7.00000 q^{7} -13.4206 q^{8} +O(q^{10})\)	\(q+3.48565 q^{2} +4.14976 q^{4} -7.00000 q^{7} -13.4206 q^{8} +6.90764 q^{11} +22.1364 q^{13} -24.3996 q^{14} -79.9776 q^{16} +88.3030 q^{17} +36.9560 q^{19} +24.0776 q^{22} -95.5283 q^{23} +77.1598 q^{26} -29.0483 q^{28} -269.029 q^{29} +197.114 q^{31} -171.409 q^{32} +307.793 q^{34} -2.14546 q^{37} +128.816 q^{38} -174.127 q^{41} +17.0345 q^{43} +28.6650 q^{44} -332.978 q^{46} -528.029 q^{47} +49.0000 q^{49} +91.8608 q^{52} -641.114 q^{53} +93.9441 q^{56} -937.742 q^{58} +642.975 q^{59} +142.967 q^{61} +687.070 q^{62} +42.3480 q^{64} -478.797 q^{67} +366.436 q^{68} -105.550 q^{71} -986.512 q^{73} -7.47834 q^{74} +153.358 q^{76} -48.3534 q^{77} -1099.86 q^{79} -606.947 q^{82} -1236.62 q^{83} +59.3763 q^{86} -92.7045 q^{88} +711.698 q^{89} -154.955 q^{91} -396.420 q^{92} -1840.52 q^{94} +636.553 q^{97} +170.797 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 13 q^{4} - 21 q^{7} - 15 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 13 * q^4 - 21 * q^7 - 15 * q^8	\( 3 q - 3 q^{2} + 13 q^{4} - 21 q^{7} - 15 q^{8} + 74 q^{11} - 44 q^{13} + 21 q^{14} - 79 q^{16} - 52 q^{17} + 168 q^{19} - 184 q^{22} - 124 q^{23} + 446 q^{26} - 91 q^{28} - 332 q^{29} + 320 q^{31} - 183 q^{32} + 582 q^{34} + 54 q^{37} - 460 q^{38} - 362 q^{41} + 16 q^{43} + 264 q^{44} - 336 q^{46} - 730 q^{47} + 147 q^{49} - 1202 q^{52} + 110 q^{53} + 105 q^{56} - 450 q^{58} + 180 q^{59} + 1222 q^{61} + 464 q^{62} - 391 q^{64} - 204 q^{67} + 918 q^{68} + 136 q^{71} - 310 q^{73} - 502 q^{74} + 1796 q^{76} - 518 q^{77} - 1034 q^{79} + 6 q^{82} - 1660 q^{83} - 764 q^{86} + 20 q^{88} - 242 q^{89} + 308 q^{91} + 96 q^{92} - 1108 q^{94} - 100 q^{97} - 147 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 13 * q^4 - 21 * q^7 - 15 * q^8 + 74 * q^11 - 44 * q^13 + 21 * q^14 - 79 * q^16 - 52 * q^17 + 168 * q^19 - 184 * q^22 - 124 * q^23 + 446 * q^26 - 91 * q^28 - 332 * q^29 + 320 * q^31 - 183 * q^32 + 582 * q^34 + 54 * q^37 - 460 * q^38 - 362 * q^41 + 16 * q^43 + 264 * q^44 - 336 * q^46 - 730 * q^47 + 147 * q^49 - 1202 * q^52 + 110 * q^53 + 105 * q^56 - 450 * q^58 + 180 * q^59 + 1222 * q^61 + 464 * q^62 - 391 * q^64 - 204 * q^67 + 918 * q^68 + 136 * q^71 - 310 * q^73 - 502 * q^74 + 1796 * q^76 - 518 * q^77 - 1034 * q^79 + 6 * q^82 - 1660 * q^83 - 764 * q^86 + 20 * q^88 - 242 * q^89 + 308 * q^91 + 96 * q^92 - 1108 * q^94 - 100 * q^97 - 147 * q^98

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