LMFDB - Embedded newform 1575.4.a.bq.1.4

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:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 27x^{3} + 7x^{2} + 120x + 60 $ x^5 - x^4 - 27x^3 + 7x^2 + 120*x + 60
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
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			1.4
Root			$-1.85474$ 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)

$f(q)$	$=$	$q+2.85474 q^{2} +0.149548 q^{4} -7.00000 q^{7} -22.4110 q^{8} -37.4408 q^{11} -3.96370 q^{13} -19.9832 q^{14} -65.1740 q^{16} +51.6780 q^{17} +25.9323 q^{19} -106.884 q^{22} -173.454 q^{23} -11.3154 q^{26} -1.04684 q^{28} +245.676 q^{29} -172.074 q^{31} -6.76690 q^{32} +147.527 q^{34} +250.699 q^{37} +74.0300 q^{38} +48.8649 q^{41} +143.612 q^{43} -5.59920 q^{44} -495.167 q^{46} -36.6415 q^{47} +49.0000 q^{49} -0.592765 q^{52} +645.286 q^{53} +156.877 q^{56} +701.343 q^{58} -395.495 q^{59} +47.5130 q^{61} -491.228 q^{62} +502.074 q^{64} -263.189 q^{67} +7.72835 q^{68} +268.177 q^{71} -199.757 q^{73} +715.680 q^{74} +3.87813 q^{76} +262.085 q^{77} +473.640 q^{79} +139.497 q^{82} -72.7028 q^{83} +409.975 q^{86} +839.086 q^{88} +1552.25 q^{89} +27.7459 q^{91} -25.9398 q^{92} -104.602 q^{94} -243.338 q^{97} +139.882 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} + 18 q^{4} - 35 q^{7} + 42 q^{8} - 42 q^{11} - 34 q^{13} - 28 q^{14} + 74 q^{16} + 238 q^{17} - 36 q^{19} - 358 q^{22} + 152 q^{23} + 310 q^{26} - 126 q^{28} + 44 q^{29} + 60 q^{31} + 710 q^{32}+ \cdots + 196 q^{98}+O(q^{100}) $ 5 * q + 4 * q^2 + 18 * q^4 - 35 * q^7 + 42 * q^8 - 42 * q^11 - 34 * q^13 - 28 * q^14 + 74 * q^16 + 238 * q^17 - 36 * q^19 - 358 * q^22 + 152 * q^23 + 310 * q^26 - 126 * q^28 + 44 * q^29 + 60 * q^31 + 710 * q^32 - 482 * q^34 - 312 * q^37 + 280 * q^38 + 426 * q^41 - 304 * q^43 - 712 * q^44 + 88 * q^46 + 370 * q^47 + 245 * q^49 + 1156 * q^52 + 976 * q^53 - 294 * q^56 + 2722 * q^58 + 432 * q^59 - 442 * q^61 - 956 * q^62 + 1362 * q^64 + 804 * q^67 - 420 * q^68 - 440 * q^71 - 564 * q^73 + 1512 * q^74 - 1336 * q^76 + 294 * q^77 + 1790 * q^79 + 3480 * q^82 + 1656 * q^83 - 1216 * q^86 + 1092 * q^88 - 746 * q^89 + 238 * q^91 + 572 * q^92 - 826 * q^94 - 518 * q^97 + 196 * q^98

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 $

$2$	2.85474	1.00930	0.504652	−	0.863323i	$-0.331621\pi$
$2$	2.85474	1.00930	0.504652	+	0.863323i	$0.331621\pi$
$3$	0	0
$3$	0	0
$4$	0.149548	0.0186935
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−22.4110	−0.990436
$9$	0	0
$10$	0	0
$11$	−37.4408	−1.02626	−0.513128	−	0.858312i	$-0.671513\pi$
$11$	−37.4408	−1.02626	−0.513128	+	0.858312i	$0.671513\pi$
$12$	0	0
$13$	−3.96370	−0.0845641	−0.0422821	−	0.999106i	$-0.513463\pi$
$13$	−3.96370	−0.0845641	−0.0422821	+	0.999106i	$0.513463\pi$
$14$	−19.9832	−0.381481
$15$	0	0
$16$	−65.1740	−1.01834
$17$	51.6780	0.737279	0.368640	−	0.929572i	$-0.379824\pi$
$17$	51.6780	0.737279	0.368640	+	0.929572i	$0.379824\pi$
$18$	0	0
$19$	25.9323	0.313120	0.156560	−	0.987668i	$-0.449960\pi$
$19$	25.9323	0.313120	0.156560	+	0.987668i	$0.449960\pi$
$20$	0	0
$21$	0	0
$22$	−106.884	−1.03580
$23$	−173.454	−1.57251	−0.786255	−	0.617902i	$-0.787983\pi$
$23$	−173.454	−1.57251	−0.786255	+	0.617902i	$0.787983\pi$
$24$	0	0
$25$	0	0
$26$	−11.3154	−0.0853509
$27$	0	0
$28$	−1.04684	−0.00706549
$29$	245.676	1.57314	0.786568	−	0.617503i	$-0.211856\pi$
$29$	245.676	1.57314	0.786568	+	0.617503i	$0.211856\pi$
$30$	0	0
$31$	−172.074	−0.996951	−0.498475	−	0.866904i	$-0.666107\pi$
$31$	−172.074	−0.996951	−0.498475	+	0.866904i	$0.666107\pi$
$32$	−6.76690	−0.0373822
$33$	0	0
$34$	147.527	0.744139
$35$	0	0
$36$	0	0
$37$	250.699	1.11391	0.556954	−	0.830543i	$-0.311970\pi$
$37$	250.699	1.11391	0.556954	+	0.830543i	$0.311970\pi$
$38$	74.0300	0.316033
$39$	0	0
$40$	0	0
$41$	48.8649	0.186132	0.0930661	−	0.995660i	$-0.470333\pi$
$41$	48.8649	0.186132	0.0930661	+	0.995660i	$0.470333\pi$
$42$	0	0
$43$	143.612	0.509317	0.254658	−	0.967031i	$-0.418037\pi$
$43$	143.612	0.509317	0.254658	+	0.967031i	$0.418037\pi$
$44$	−5.59920	−0.0191844
$45$	0	0
$46$	−495.167	−1.58714
$47$	−36.6415	−0.113717	−0.0568587	−	0.998382i	$-0.518108\pi$
$47$	−36.6415	−0.113717	−0.0568587	+	0.998382i	$0.518108\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	−0.592765	−0.00158080
$53$	645.286	1.67239	0.836196	−	0.548430i	$-0.184774\pi$
$53$	645.286	1.67239	0.836196	+	0.548430i	$0.184774\pi$
$54$	0	0
$55$	0	0
$56$	156.877	0.374350
$57$	0	0
$58$	701.343	1.58777
$59$	−395.495	−0.872696	−0.436348	−	0.899778i	$-0.643728\pi$
$59$	−395.495	−0.872696	−0.436348	+	0.899778i	$0.643728\pi$
$60$	0	0
$61$	47.5130	0.0997282	0.0498641	−	0.998756i	$-0.484121\pi$
$61$	47.5130	0.0997282	0.0498641	+	0.998756i	$0.484121\pi$
$62$	−491.228	−1.00623
$63$	0	0
$64$	502.074	0.980614
$65$	0	0
$66$	0	0
$67$	−263.189	−0.479906	−0.239953	−	0.970785i	$-0.577132\pi$
$67$	−263.189	−0.479906	−0.239953	+	0.970785i	$0.577132\pi$
$68$	7.72835	0.0137824
$69$	0	0
$70$	0	0
$71$	268.177	0.448264	0.224132	−	0.974559i	$-0.428045\pi$
$71$	268.177	0.448264	0.224132	+	0.974559i	$0.428045\pi$
$72$	0	0
$73$	−199.757	−0.320271	−0.160136	−	0.987095i	$-0.551193\pi$
$73$	−199.757	−0.320271	−0.160136	+	0.987095i	$0.551193\pi$
$74$	715.680	1.12427
$75$	0	0
$76$	3.87813	0.00585331
$77$	262.085	0.387888
$78$	0	0
$79$	473.640	0.674540	0.337270	−	0.941408i	$-0.390497\pi$
$79$	473.640	0.674540	0.337270	+	0.941408i	$0.390497\pi$
$80$	0	0
$81$	0	0
$82$	139.497	0.187864
$83$	−72.7028	−0.0961466	−0.0480733	−	0.998844i	$-0.515308\pi$
$83$	−72.7028	−0.0961466	−0.0480733	+	0.998844i	$0.515308\pi$
$84$	0	0
$85$	0	0
$86$	409.975	0.514055
$87$	0	0
$88$	839.086	1.01644
$89$	1552.25	1.84874	0.924369	−	0.381500i	$-0.124592\pi$
$89$	1552.25	1.84874	0.924369	+	0.381500i	$0.124592\pi$
$90$	0	0
$91$	27.7459	0.0319622
$92$	−25.9398	−0.0293958
$93$	0	0
$94$	−104.602	−0.114775
$95$	0	0
$96$	0	0
$97$	−243.338	−0.254714	−0.127357	−	0.991857i	$-0.540649\pi$
$97$	−243.338	−0.254714	−0.127357	+	0.991857i	$0.540649\pi$
$98$	139.882	0.144186
$99$	0	0
$100$	0	0
$101$	1539.34	1.51653	0.758265	−	0.651946i	$-0.226047\pi$
$101$	1539.34	1.51653	0.758265	+	0.651946i	$0.226047\pi$
$102$	0	0
$103$	−948.628	−0.907486	−0.453743	−	0.891133i	$-0.649912\pi$
$103$	−948.628	−0.907486	−0.453743	+	0.891133i	$0.649912\pi$
$104$	88.8306	0.0837554
$105$	0	0
$106$	1842.12	1.68795
$107$	863.983	0.780602	0.390301	−	0.920687i	$-0.372371\pi$
$107$	863.983	0.780602	0.390301	+	0.920687i	$0.372371\pi$
$108$	0	0
$109$	886.319	0.778844	0.389422	−	0.921060i	$-0.372675\pi$
$109$	886.319	0.778844	0.389422	+	0.921060i	$0.372675\pi$
$110$	0	0
$111$	0	0
$112$	456.218	0.384898
$113$	765.957	0.637657	0.318828	−	0.947812i	$-0.396711\pi$
$113$	765.957	0.637657	0.318828	+	0.947812i	$0.396711\pi$
$114$	0	0
$115$	0	0
$116$	36.7405	0.0294075
$117$	0	0
$118$	−1129.04	−0.880816
$119$	−361.746	−0.278665
$120$	0	0
$121$	70.8116	0.0532018
$122$	135.637	0.100656
$123$	0	0
$124$	−25.7334	−0.0186365
$125$	0	0
$126$	0	0
$127$	−505.042	−0.352876	−0.176438	−	0.984312i	$-0.556457\pi$
$127$	−505.042	−0.352876	−0.176438	+	0.984312i	$0.556457\pi$
$128$	1487.43	1.02712
$129$	0	0
$130$	0	0
$131$	−672.930	−0.448811	−0.224405	−	0.974496i	$-0.572044\pi$
$131$	−672.930	−0.448811	−0.224405	+	0.974496i	$0.572044\pi$
$132$	0	0
$133$	−181.526	−0.118348
$134$	−751.337	−0.484371
$135$	0	0
$136$	−1158.16	−0.730228
$137$	1552.28	0.968032	0.484016	−	0.875059i	$-0.339178\pi$
$137$	1552.28	0.968032	0.484016	+	0.875059i	$0.339178\pi$
$138$	0	0
$139$	−1072.02	−0.654154	−0.327077	−	0.944998i	$-0.606064\pi$
$139$	−1072.02	−0.654154	−0.327077	+	0.944998i	$0.606064\pi$
$140$	0	0
$141$	0	0
$142$	765.575	0.452434
$143$	148.404	0.0867845
$144$	0	0
$145$	0	0
$146$	−570.255	−0.323251
$147$	0	0
$148$	37.4915	0.0208229
$149$	−645.936	−0.355149	−0.177574	−	0.984107i	$-0.556825\pi$
$149$	−645.936	−0.355149	−0.177574	+	0.984107i	$0.556825\pi$
$150$	0	0
$151$	243.194	0.131065	0.0655326	−	0.997850i	$-0.479125\pi$
$151$	243.194	0.131065	0.0655326	+	0.997850i	$0.479125\pi$
$152$	−581.169	−0.310125
$153$	0	0
$154$	748.186	0.391497
$155$	0	0
$156$	0	0
$157$	1552.56	0.789223	0.394611	−	0.918848i	$-0.370879\pi$
$157$	1552.56	0.789223	0.394611	+	0.918848i	$0.370879\pi$
$158$	1352.12	0.680815
$159$	0	0
$160$	0	0
$161$	1214.18	0.594353
$162$	0	0
$163$	−2553.65	−1.22710	−0.613550	−	0.789656i	$-0.710259\pi$
$163$	−2553.65	−1.22710	−0.613550	+	0.789656i	$0.710259\pi$
$164$	7.30766	0.00347947
$165$	0	0
$166$	−207.548	−0.0970411
$167$	3573.14	1.65568	0.827839	−	0.560966i	$-0.189570\pi$
$167$	3573.14	1.65568	0.827839	+	0.560966i	$0.189570\pi$
$168$	0	0
$169$	−2181.29	−0.992849
$170$	0	0
$171$	0	0
$172$	21.4769	0.00952093
$173$	2234.71	0.982090	0.491045	−	0.871134i	$-0.336615\pi$
$173$	2234.71	0.982090	0.491045	+	0.871134i	$0.336615\pi$
$174$	0	0
$175$	0	0
$176$	2440.17	1.04508
$177$	0	0
$178$	4431.26	1.86594
$179$	1830.53	0.764361	0.382180	−	0.924088i	$-0.375173\pi$
$179$	1830.53	0.764361	0.382180	+	0.924088i	$0.375173\pi$
$180$	0	0
$181$	2437.22	1.00087	0.500433	−	0.865775i	$-0.333174\pi$
$181$	2437.22	1.00087	0.500433	+	0.865775i	$0.333174\pi$
$182$	79.2075	0.0322596
$183$	0	0
$184$	3887.29	1.55747
$185$	0	0
$186$	0	0
$187$	−1934.86	−0.756638
$188$	−5.47968	−0.00212578
$189$	0	0
$190$	0	0
$191$	−5079.50	−1.92429	−0.962145	−	0.272538i	$-0.912137\pi$
$191$	−5079.50	−1.92429	−0.962145	+	0.272538i	$0.912137\pi$
$192$	0	0
$193$	2805.09	1.04619	0.523095	−	0.852274i	$-0.324777\pi$
$193$	2805.09	1.04619	0.523095	+	0.852274i	$0.324777\pi$
$194$	−694.667	−0.257084
$195$	0	0
$196$	7.32786	0.00267050
$197$	−3107.79	−1.12396	−0.561982	−	0.827149i	$-0.689961\pi$
$197$	−3107.79	−1.12396	−0.561982	+	0.827149i	$0.689961\pi$
$198$	0	0
$199$	−2145.63	−0.764321	−0.382161	−	0.924096i	$-0.624820\pi$
$199$	−2145.63	−0.764321	−0.382161	+	0.924096i	$0.624820\pi$
$200$	0	0
$201$	0	0
$202$	4394.41	1.53064
$203$	−1719.73	−0.594590
$204$	0	0
$205$	0	0
$206$	−2708.09	−0.915929
$207$	0	0
$208$	258.331	0.0861154
$209$	−970.925	−0.321341
$210$	0	0
$211$	2837.45	0.925772	0.462886	−	0.886418i	$-0.346814\pi$
$211$	2837.45	0.925772	0.462886	+	0.886418i	$0.346814\pi$
$212$	96.5013	0.0312629
$213$	0	0
$214$	2466.45	0.787864
$215$	0	0
$216$	0	0
$217$	1204.52	0.376812
$218$	2530.21	0.786089
$219$	0	0
$220$	0	0
$221$	−204.836	−0.0623474
$222$	0	0
$223$	4741.40	1.42380	0.711901	−	0.702280i	$-0.247834\pi$
$223$	4741.40	1.42380	0.711901	+	0.702280i	$0.247834\pi$
$224$	47.3683	0.0141291
$225$	0	0
$226$	2186.61	0.643589
$227$	−960.790	−0.280925	−0.140462	−	0.990086i	$-0.544859\pi$
$227$	−960.790	−0.280925	−0.140462	+	0.990086i	$0.544859\pi$
$228$	0	0
$229$	744.006	0.214696	0.107348	−	0.994222i	$-0.465764\pi$
$229$	744.006	0.214696	0.107348	+	0.994222i	$0.465764\pi$
$230$	0	0
$231$	0	0
$232$	−5505.86	−1.55809
$233$	1550.56	0.435968	0.217984	−	0.975952i	$-0.430052\pi$
$233$	1550.56	0.435968	0.217984	+	0.975952i	$0.430052\pi$
$234$	0	0
$235$	0	0
$236$	−59.1456	−0.0163138
$237$	0	0
$238$	−1032.69	−0.281258
$239$	−2775.00	−0.751045	−0.375523	−	0.926813i	$-0.622537\pi$
$239$	−2775.00	−0.751045	−0.375523	+	0.926813i	$0.622537\pi$
$240$	0	0
$241$	−2550.20	−0.681630	−0.340815	−	0.940130i	$-0.610703\pi$
$241$	−2550.20	−0.681630	−0.340815	+	0.940130i	$0.610703\pi$
$242$	202.149	0.0536968
$243$	0	0
$244$	7.10549	0.00186427
$245$	0	0
$246$	0	0
$247$	−102.788	−0.0264787
$248$	3856.36	0.987416
$249$	0	0
$250$	0	0
$251$	2933.00	0.737568	0.368784	−	0.929515i	$-0.379774\pi$
$251$	2933.00	0.737568	0.368784	+	0.929515i	$0.379774\pi$
$252$	0	0
$253$	6494.26	1.61380
$254$	−1441.76	−0.356159
$255$	0	0
$256$	229.626	0.0560611
$257$	2725.22	0.661459	0.330729	−	0.943726i	$-0.392705\pi$
$257$	2725.22	0.661459	0.330729	+	0.943726i	$0.392705\pi$
$258$	0	0
$259$	−1754.89	−0.421018
$260$	0	0
$261$	0	0
$262$	−1921.04	−0.452986
$263$	3027.26	0.709767	0.354884	−	0.934910i	$-0.384520\pi$
$263$	3027.26	0.709767	0.354884	+	0.934910i	$0.384520\pi$
$264$	0	0
$265$	0	0
$266$	−518.210	−0.119449
$267$	0	0
$268$	−39.3595	−0.00897114
$269$	1442.46	0.326946	0.163473	−	0.986548i	$-0.447730\pi$
$269$	1442.46	0.326946	0.163473	+	0.986548i	$0.447730\pi$
$270$	0	0
$271$	−6464.45	−1.44903	−0.724516	−	0.689258i	$-0.757937\pi$
$271$	−6464.45	−1.44903	−0.724516	+	0.689258i	$0.757937\pi$
$272$	−3368.06	−0.750804
$273$	0	0
$274$	4431.36	0.977038
$275$	0	0
$276$	0	0
$277$	876.614	0.190147	0.0950733	−	0.995470i	$-0.469691\pi$
$277$	876.614	0.190147	0.0950733	+	0.995470i	$0.469691\pi$
$278$	−3060.33	−0.660240
$279$	0	0
$280$	0	0
$281$	−6252.19	−1.32731	−0.663655	−	0.748038i	$-0.730996\pi$
$281$	−6252.19	−1.32731	−0.663655	+	0.748038i	$0.730996\pi$
$282$	0	0
$283$	2250.07	0.472625	0.236312	−	0.971677i	$-0.424061\pi$
$283$	2250.07	0.472625	0.236312	+	0.971677i	$0.424061\pi$
$284$	40.1054	0.00837963
$285$	0	0
$286$	423.655	0.0875919
$287$	−342.054	−0.0703513
$288$	0	0
$289$	−2242.39	−0.456419
$290$	0	0
$291$	0	0
$292$	−29.8733	−0.00598700
$293$	5917.86	1.17995	0.589975	−	0.807422i	$-0.299138\pi$
$293$	5917.86	1.17995	0.589975	+	0.807422i	$0.299138\pi$
$294$	0	0
$295$	0	0
$296$	−5618.41	−1.10325
$297$	0	0
$298$	−1843.98	−0.358453
$299$	687.522	0.132978
$300$	0	0
$301$	−1005.28	−0.192504
$302$	694.256	0.132284
$303$	0	0
$304$	−1690.11	−0.318864
$305$	0	0
$306$	0	0
$307$	9458.47	1.75838	0.879191	−	0.476469i	$-0.158084\pi$
$307$	9458.47	1.75838	0.879191	+	0.476469i	$0.158084\pi$
$308$	39.1944	0.00725100
$309$	0	0
$310$	0	0
$311$	7576.78	1.38148	0.690739	−	0.723104i	$-0.257285\pi$
$311$	7576.78	1.38148	0.690739	+	0.723104i	$0.257285\pi$
$312$	0	0
$313$	−9172.41	−1.65641	−0.828204	−	0.560427i	$-0.810637\pi$
$313$	−9172.41	−1.65641	−0.828204	+	0.560427i	$0.810637\pi$
$314$	4432.16	0.796565
$315$	0	0
$316$	70.8320	0.0126095
$317$	3077.94	0.545345	0.272672	−	0.962107i	$-0.412092\pi$
$317$	3077.94	0.545345	0.272672	+	0.962107i	$0.412092\pi$
$318$	0	0
$319$	−9198.31	−1.61444
$320$	0	0
$321$	0	0
$322$	3466.17	0.599882
$323$	1340.13	0.230857
$324$	0	0
$325$	0	0
$326$	−7290.01	−1.23852
$327$	0	0
$328$	−1095.11	−0.184352
$329$	256.491	0.0429811
$330$	0	0
$331$	3234.50	0.537113	0.268557	−	0.963264i	$-0.413453\pi$
$331$	3234.50	0.537113	0.268557	+	0.963264i	$0.413453\pi$
$332$	−10.8726	−0.00179732
$333$	0	0
$334$	10200.4	1.67108
$335$	0	0
$336$	0	0
$337$	−3777.84	−0.610658	−0.305329	−	0.952247i	$-0.598766\pi$
$337$	−3777.84	−0.610658	−0.305329	+	0.952247i	$0.598766\pi$
$338$	−6227.02	−1.00209
$339$	0	0
$340$	0	0
$341$	6442.60	1.02313
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−3218.49	−0.504446
$345$	0	0
$346$	6379.51	0.991227
$347$	8244.08	1.27540	0.637702	−	0.770283i	$-0.279885\pi$
$347$	8244.08	1.27540	0.637702	+	0.770283i	$0.279885\pi$
$348$	0	0
$349$	7173.78	1.10030	0.550148	−	0.835067i	$-0.314571\pi$
$349$	7173.78	1.10030	0.550148	+	0.835067i	$0.314571\pi$
$350$	0	0
$351$	0	0
$352$	253.358	0.0383637
$353$	4191.51	0.631987	0.315994	−	0.948761i	$-0.397662\pi$
$353$	4191.51	0.631987	0.315994	+	0.948761i	$0.397662\pi$
$354$	0	0
$355$	0	0
$356$	232.136	0.0345594
$357$	0	0
$358$	5225.70	0.771472
$359$	3136.29	0.461078	0.230539	−	0.973063i	$-0.425951\pi$
$359$	3136.29	0.461078	0.230539	+	0.973063i	$0.425951\pi$
$360$	0	0
$361$	−6186.52	−0.901956
$362$	6957.62	1.01018
$363$	0	0
$364$	4.14936	0.000597487 0
$365$	0	0
$366$	0	0
$367$	1723.30	0.245110	0.122555	−	0.992462i	$-0.460891\pi$
$367$	1723.30	0.245110	0.122555	+	0.992462i	$0.460891\pi$
$368$	11304.7	1.60136
$369$	0	0
$370$	0	0
$371$	−4517.00	−0.632105
$372$	0	0
$373$	−2818.55	−0.391258	−0.195629	−	0.980678i	$-0.562675\pi$
$373$	−2818.55	−0.391258	−0.195629	+	0.980678i	$0.562675\pi$
$374$	−5523.53	−0.763677
$375$	0	0
$376$	821.174	0.112630
$377$	−973.788	−0.133031
$378$	0	0
$379$	−10466.1	−1.41849	−0.709246	−	0.704961i	$-0.750964\pi$
$379$	−10466.1	−1.41849	−0.709246	+	0.704961i	$0.750964\pi$
$380$	0	0
$381$	0	0
$382$	−14500.6	−1.94219
$383$	−258.055	−0.0344282	−0.0172141	−	0.999852i	$-0.505480\pi$
$383$	−258.055	−0.0344282	−0.0172141	+	0.999852i	$0.505480\pi$
$384$	0	0
$385$	0	0
$386$	8007.81	1.05592
$387$	0	0
$388$	−36.3908	−0.00476150
$389$	−4573.87	−0.596156	−0.298078	−	0.954542i	$-0.596346\pi$
$389$	−4573.87	−0.596156	−0.298078	+	0.954542i	$0.596346\pi$
$390$	0	0
$391$	−8963.77	−1.15938
$392$	−1098.14	−0.141491
$393$	0	0
$394$	−8871.94	−1.13442
$395$	0	0
$396$	0	0
$397$	−3624.55	−0.458215	−0.229107	−	0.973401i	$-0.573581\pi$
$397$	−3624.55	−0.458215	−0.229107	+	0.973401i	$0.573581\pi$
$398$	−6125.23	−0.771432
$399$	0	0
$400$	0	0
$401$	−6358.32	−0.791819	−0.395910	−	0.918289i	$-0.629571\pi$
$401$	−6358.32	−0.791819	−0.395910	+	0.918289i	$0.629571\pi$
$402$	0	0
$403$	682.052	0.0843063
$404$	230.205	0.0283493
$405$	0	0
$406$	−4909.40	−0.600121
$407$	−9386.35	−1.14316
$408$	0	0
$409$	−6536.39	−0.790228	−0.395114	−	0.918632i	$-0.629295\pi$
$409$	−6536.39	−0.790228	−0.395114	+	0.918632i	$0.629295\pi$
$410$	0	0
$411$	0	0
$412$	−141.866	−0.0169641
$413$	2768.47	0.329848
$414$	0	0
$415$	0	0
$416$	26.8220	0.00316119
$417$	0	0
$418$	−2771.74	−0.324331
$419$	−6333.56	−0.738460	−0.369230	−	0.929338i	$-0.620379\pi$
$419$	−6333.56	−0.738460	−0.369230	+	0.929338i	$0.620379\pi$
$420$	0	0
$421$	−8139.62	−0.942282	−0.471141	−	0.882058i	$-0.656158\pi$
$421$	−8139.62	−0.942282	−0.471141	+	0.882058i	$0.656158\pi$
$422$	8100.18	0.934385
$423$	0	0
$424$	−14461.5	−1.65640
$425$	0	0
$426$	0	0
$427$	−332.591	−0.0376937
$428$	129.207	0.0145922
$429$	0	0
$430$	0	0
$431$	14367.6	1.60571	0.802856	−	0.596173i	$-0.203313\pi$
$431$	14367.6	1.60571	0.802856	+	0.596173i	$0.203313\pi$
$432$	0	0
$433$	8399.05	0.932176	0.466088	−	0.884738i	$-0.345663\pi$
$433$	8399.05	0.932176	0.466088	+	0.884738i	$0.345663\pi$
$434$	3438.59	0.380318
$435$	0	0
$436$	132.547	0.0145593
$437$	−4498.07	−0.492384
$438$	0	0
$439$	−17860.8	−1.94180	−0.970901	−	0.239482i	$-0.923022\pi$
$439$	−17860.8	−1.94180	−0.970901	+	0.239482i	$0.923022\pi$
$440$	0	0
$441$	0	0
$442$	−584.754	−0.0629274
$443$	1901.57	0.203942	0.101971	−	0.994787i	$-0.467485\pi$
$443$	1901.57	0.203942	0.101971	+	0.994787i	$0.467485\pi$
$444$	0	0
$445$	0	0
$446$	13535.5	1.43705
$447$	0	0
$448$	−3514.52	−0.370637
$449$	5185.68	0.545050	0.272525	−	0.962149i	$-0.412141\pi$
$449$	5185.68	0.545050	0.272525	+	0.962149i	$0.412141\pi$
$450$	0	0
$451$	−1829.54	−0.191019
$452$	114.548	0.0119201
$453$	0	0
$454$	−2742.81	−0.283538
$455$	0	0
$456$	0	0
$457$	11198.8	1.14630	0.573149	−	0.819451i	$-0.305722\pi$
$457$	11198.8	1.14630	0.573149	+	0.819451i	$0.305722\pi$
$458$	2123.94	0.216693
$459$	0	0
$460$	0	0
$461$	−17270.7	−1.74485	−0.872427	−	0.488744i	$-0.837455\pi$
$461$	−17270.7	−1.74485	−0.872427	+	0.488744i	$0.837455\pi$
$462$	0	0
$463$	−385.660	−0.0387109	−0.0193554	−	0.999813i	$-0.506161\pi$
$463$	−385.660	−0.0387109	−0.0193554	+	0.999813i	$0.506161\pi$
$464$	−16011.7	−1.60199
$465$	0	0
$466$	4426.44	0.440024
$467$	5035.36	0.498947	0.249474	−	0.968382i	$-0.419742\pi$
$467$	5035.36	0.498947	0.249474	+	0.968382i	$0.419742\pi$
$468$	0	0
$469$	1842.33	0.181387
$470$	0	0
$471$	0	0
$472$	8863.45	0.864350
$473$	−5376.94	−0.522689
$474$	0	0
$475$	0	0
$476$	−54.0985	−0.00520924
$477$	0	0
$478$	−7921.91	−0.758033
$479$	8681.99	0.828163	0.414082	−	0.910240i	$-0.364103\pi$
$479$	8681.99	0.828163	0.414082	+	0.910240i	$0.364103\pi$
$480$	0	0
$481$	−993.695	−0.0941967
$482$	−7280.16	−0.687972
$483$	0	0
$484$	10.5898	0.000994530 0
$485$	0	0
$486$	0	0
$487$	−890.476	−0.0828569	−0.0414284	−	0.999141i	$-0.513191\pi$
$487$	−890.476	−0.0828569	−0.0414284	+	0.999141i	$0.513191\pi$
$488$	−1064.81	−0.0987744
$489$	0	0
$490$	0	0
$491$	−1562.48	−0.143613	−0.0718063	−	0.997419i	$-0.522876\pi$
$491$	−1562.48	−0.143613	−0.0718063	+	0.997419i	$0.522876\pi$
$492$	0	0
$493$	12696.1	1.15984
$494$	−293.433	−0.0267250
$495$	0	0
$496$	11214.8	1.01524
$497$	−1877.24	−0.169428
$498$	0	0
$499$	8234.33	0.738716	0.369358	−	0.929287i	$-0.379578\pi$
$499$	8234.33	0.738716	0.369358	+	0.929287i	$0.379578\pi$
$500$	0	0
$501$	0	0
$502$	8372.97	0.744430
$503$	72.5340	0.00642969	0.00321484	−	0.999995i	$-0.498977\pi$
$503$	72.5340	0.00642969	0.00321484	+	0.999995i	$0.498977\pi$
$504$	0	0
$505$	0	0
$506$	18539.4	1.62881
$507$	0	0
$508$	−75.5281	−0.00659649
$509$	−7793.44	−0.678660	−0.339330	−	0.940667i	$-0.610200\pi$
$509$	−7793.44	−0.678660	−0.339330	+	0.940667i	$0.610200\pi$
$510$	0	0
$511$	1398.30	0.121051
$512$	−11243.9	−0.970537
$513$	0	0
$514$	7779.81	0.667612
$515$	0	0
$516$	0	0
$517$	1371.89	0.116703
$518$	−5009.76	−0.424935
$519$	0	0
$520$	0	0
$521$	−4645.42	−0.390633	−0.195316	−	0.980740i	$-0.562573\pi$
$521$	−4645.42	−0.390633	−0.195316	+	0.980740i	$0.562573\pi$
$522$	0	0
$523$	−8783.88	−0.734402	−0.367201	−	0.930142i	$-0.619684\pi$
$523$	−8783.88	−0.734402	−0.367201	+	0.930142i	$0.619684\pi$
$524$	−100.636	−0.00838986
$525$	0	0
$526$	8642.04	0.716371
$527$	−8892.46	−0.735031
$528$	0	0
$529$	17919.4	1.47279
$530$	0	0
$531$	0	0
$532$	−27.1469	−0.00221234
$533$	−193.686	−0.0157401
$534$	0	0
$535$	0	0
$536$	5898.34	0.475316
$537$	0	0
$538$	4117.86	0.329988
$539$	−1834.60	−0.146608
$540$	0	0
$541$	−7054.13	−0.560593	−0.280296	−	0.959913i	$-0.590433\pi$
$541$	−7054.13	−0.560593	−0.280296	+	0.959913i	$0.590433\pi$
$542$	−18454.3	−1.46251
$543$	0	0
$544$	−349.700	−0.0275611
$545$	0	0
$546$	0	0
$547$	5776.83	0.451553	0.225776	−	0.974179i	$-0.427508\pi$
$547$	5776.83	0.451553	0.225776	+	0.974179i	$0.427508\pi$
$548$	232.141	0.0180959
$549$	0	0
$550$	0	0
$551$	6370.95	0.492580
$552$	0	0
$553$	−3315.48	−0.254952
$554$	2502.51	0.191916
$555$	0	0
$556$	−160.318	−0.0122284
$557$	20562.6	1.56421	0.782106	−	0.623145i	$-0.214146\pi$
$557$	20562.6	1.56421	0.782106	+	0.623145i	$0.214146\pi$
$558$	0	0
$559$	−569.235	−0.0430699
$560$	0	0
$561$	0	0
$562$	−17848.4	−1.33966
$563$	−24009.5	−1.79730	−0.898650	−	0.438666i	$-0.855451\pi$
$563$	−24009.5	−1.79730	−0.898650	+	0.438666i	$0.855451\pi$
$564$	0	0
$565$	0	0
$566$	6423.37	0.477022
$567$	0	0
$568$	−6010.11	−0.443977
$569$	24157.5	1.77985	0.889925	−	0.456107i	$-0.150757\pi$
$569$	24157.5	1.77985	0.889925	+	0.456107i	$0.150757\pi$
$570$	0	0
$571$	−706.993	−0.0518157	−0.0259078	−	0.999664i	$-0.508248\pi$
$571$	−706.993	−0.0518157	−0.0259078	+	0.999664i	$0.508248\pi$
$572$	22.1936	0.00162231
$573$	0	0
$574$	−976.477	−0.0710059
$575$	0	0
$576$	0	0
$577$	−16057.2	−1.15853	−0.579265	−	0.815139i	$-0.696660\pi$
$577$	−16057.2	−1.15853	−0.579265	+	0.815139i	$0.696660\pi$
$578$	−6401.44	−0.460665
$579$	0	0
$580$	0	0
$581$	508.919	0.0363400
$582$	0	0
$583$	−24160.0	−1.71630
$584$	4476.76	0.317208
$585$	0	0
$586$	16894.0	1.19093
$587$	8605.63	0.605098	0.302549	−	0.953134i	$-0.402162\pi$
$587$	8605.63	0.605098	0.302549	+	0.953134i	$0.402162\pi$
$588$	0	0
$589$	−4462.28	−0.312165
$590$	0	0
$591$	0	0
$592$	−16339.0	−1.13434
$593$	−20355.6	−1.40962	−0.704809	−	0.709397i	$-0.748967\pi$
$593$	−20355.6	−1.40962	−0.704809	+	0.709397i	$0.748967\pi$
$594$	0	0
$595$	0	0
$596$	−96.5986	−0.00663898
$597$	0	0
$598$	1962.70	0.134215
$599$	−22635.7	−1.54402	−0.772010	−	0.635610i	$-0.780749\pi$
$599$	−22635.7	−1.54402	−0.772010	+	0.635610i	$0.780749\pi$
$600$	0	0
$601$	−22553.8	−1.53077	−0.765383	−	0.643575i	$-0.777450\pi$
$601$	−22553.8	−1.53077	−0.765383	+	0.643575i	$0.777450\pi$
$602$	−2869.83	−0.194295
$603$	0	0
$604$	36.3692	0.00245007
$605$	0	0
$606$	0	0
$607$	17534.2	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$607$	17534.2	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$608$	−175.481	−0.0117051
$609$	0	0
$610$	0	0
$611$	145.236	0.00961641
$612$	0	0
$613$	10445.5	0.688238	0.344119	−	0.938926i	$-0.388178\pi$
$613$	10445.5	0.688238	0.344119	+	0.938926i	$0.388178\pi$
$614$	27001.5	1.77474
$615$	0	0
$616$	−5873.60	−0.384179
$617$	13218.9	0.862516	0.431258	−	0.902229i	$-0.358070\pi$
$617$	13218.9	0.862516	0.431258	+	0.902229i	$0.358070\pi$
$618$	0	0
$619$	23438.9	1.52195	0.760976	−	0.648780i	$-0.224720\pi$
$619$	23438.9	1.52195	0.760976	+	0.648780i	$0.224720\pi$
$620$	0	0
$621$	0	0
$622$	21629.8	1.39433
$623$	−10865.7	−0.698757
$624$	0	0
$625$	0	0
$626$	−26184.9	−1.67182
$627$	0	0
$628$	232.183	0.0147534
$629$	12955.6	0.821262
$630$	0	0
$631$	−874.004	−0.0551403	−0.0275702	−	0.999620i	$-0.508777\pi$
$631$	−874.004	−0.0551403	−0.0275702	+	0.999620i	$0.508777\pi$
$632$	−10614.7	−0.668089
$633$	0	0
$634$	8786.72	0.550419
$635$	0	0
$636$	0	0
$637$	−194.222	−0.0120806
$638$	−26258.8	−1.62946
$639$	0	0
$640$	0	0
$641$	−23977.0	−1.47743	−0.738715	−	0.674017i	$-0.764567\pi$
$641$	−23977.0	−1.47743	−0.738715	+	0.674017i	$0.764567\pi$
$642$	0	0
$643$	27698.0	1.69876	0.849380	−	0.527782i	$-0.176976\pi$
$643$	27698.0	1.69876	0.849380	+	0.527782i	$0.176976\pi$
$644$	181.579	0.0111106
$645$	0	0
$646$	3825.72	0.233004
$647$	−10965.1	−0.666282	−0.333141	−	0.942877i	$-0.608109\pi$
$647$	−10965.1	−0.666282	−0.333141	+	0.942877i	$0.608109\pi$
$648$	0	0
$649$	14807.6	0.895610
$650$	0	0
$651$	0	0
$652$	−381.894	−0.0229388
$653$	−12336.2	−0.739285	−0.369642	−	0.929174i	$-0.620520\pi$
$653$	−12336.2	−0.739285	−0.369642	+	0.929174i	$0.620520\pi$
$654$	0	0
$655$	0	0
$656$	−3184.72	−0.189547
$657$	0	0
$658$	732.215	0.0433810
$659$	25275.6	1.49408	0.747040	−	0.664779i	$-0.231474\pi$
$659$	25275.6	1.49408	0.747040	+	0.664779i	$0.231474\pi$
$660$	0	0
$661$	−4447.92	−0.261731	−0.130865	−	0.991400i	$-0.541776\pi$
$661$	−4447.92	−0.261731	−0.130865	+	0.991400i	$0.541776\pi$
$662$	9233.67	0.542110
$663$	0	0
$664$	1629.34	0.0952270
$665$	0	0
$666$	0	0
$667$	−42613.6	−2.47377
$668$	534.357	0.0309505
$669$	0	0
$670$	0	0
$671$	−1778.92	−0.102347
$672$	0	0
$673$	−30358.9	−1.73885	−0.869427	−	0.494061i	$-0.835512\pi$
$673$	−30358.9	−1.73885	−0.869427	+	0.494061i	$0.835512\pi$
$674$	−10784.7	−0.616340
$675$	0	0
$676$	−326.208	−0.0185599
$677$	6916.48	0.392647	0.196324	−	0.980539i	$-0.437100\pi$
$677$	6916.48	0.392647	0.196324	+	0.980539i	$0.437100\pi$
$678$	0	0
$679$	1703.37	0.0962728
$680$	0	0
$681$	0	0
$682$	18392.0	1.03265
$683$	4532.72	0.253938	0.126969	−	0.991907i	$-0.459475\pi$
$683$	4532.72	0.253938	0.126969	+	0.991907i	$0.459475\pi$
$684$	0	0
$685$	0	0
$686$	−979.176	−0.0544973
$687$	0	0
$688$	−9359.77	−0.518660
$689$	−2557.72	−0.141424
$690$	0	0
$691$	27235.2	1.49939	0.749694	−	0.661785i	$-0.230201\pi$
$691$	27235.2	1.49939	0.749694	+	0.661785i	$0.230201\pi$
$692$	334.197	0.0183587
$693$	0	0
$694$	23534.7	1.28727
$695$	0	0
$696$	0	0
$697$	2525.24	0.137231
$698$	20479.3	1.11053
$699$	0	0
$700$	0	0
$701$	17144.3	0.923726	0.461863	−	0.886951i	$-0.347181\pi$
$701$	17144.3	0.923726	0.461863	+	0.886951i	$0.347181\pi$
$702$	0	0
$703$	6501.19	0.348787
$704$	−18798.1	−1.00636
$705$	0	0
$706$	11965.7	0.637867
$707$	−10775.3	−0.573195
$708$	0	0
$709$	16724.1	0.885877	0.442939	−	0.896552i	$-0.353936\pi$
$709$	16724.1	0.885877	0.442939	+	0.896552i	$0.353936\pi$
$710$	0	0
$711$	0	0
$712$	−34787.4	−1.83106
$713$	29847.0	1.56771
$714$	0	0
$715$	0	0
$716$	273.753	0.0142886
$717$	0	0
$718$	8953.30	0.465368
$719$	−4308.66	−0.223485	−0.111743	−	0.993737i	$-0.535643\pi$
$719$	−4308.66	−0.223485	−0.111743	+	0.993737i	$0.535643\pi$
$720$	0	0
$721$	6640.39	0.342997
$722$	−17660.9	−0.910347
$723$	0	0
$724$	364.481	0.0187097
$725$	0	0
$726$	0	0
$727$	29435.6	1.50166	0.750830	−	0.660496i	$-0.229654\pi$
$727$	29435.6	1.50166	0.750830	+	0.660496i	$0.229654\pi$
$728$	−621.814	−0.0316566
$729$	0	0
$730$	0	0
$731$	7421.58	0.375509
$732$	0	0
$733$	6587.69	0.331954	0.165977	−	0.986130i	$-0.446922\pi$
$733$	6587.69	0.331954	0.165977	+	0.986130i	$0.446922\pi$
$734$	4919.57	0.247390
$735$	0	0
$736$	1173.75	0.0587839
$737$	9854.01	0.492506
$738$	0	0
$739$	3684.46	0.183404	0.0917018	−	0.995787i	$-0.470769\pi$
$739$	3684.46	0.183404	0.0917018	+	0.995787i	$0.470769\pi$
$740$	0	0
$741$	0	0
$742$	−12894.9	−0.637986
$743$	12271.9	0.605940	0.302970	−	0.953000i	$-0.402022\pi$
$743$	12271.9	0.605940	0.302970	+	0.953000i	$0.402022\pi$
$744$	0	0
$745$	0	0
$746$	−8046.24	−0.394898
$747$	0	0
$748$	−289.355	−0.0141442
$749$	−6047.88	−0.295040
$750$	0	0
$751$	−30871.0	−1.50000	−0.749999	−	0.661439i	$-0.769946\pi$
$751$	−30871.0	−1.50000	−0.749999	+	0.661439i	$0.769946\pi$
$752$	2388.08	0.115803
$753$	0	0
$754$	−2779.91	−0.134269
$755$	0	0
$756$	0	0
$757$	−11442.1	−0.549368	−0.274684	−	0.961535i	$-0.588573\pi$
$757$	−11442.1	−0.549368	−0.274684	+	0.961535i	$0.588573\pi$
$758$	−29878.1	−1.43169
$759$	0	0
$760$	0	0
$761$	−14423.3	−0.687048	−0.343524	−	0.939144i	$-0.611621\pi$
$761$	−14423.3	−0.687048	−0.343524	+	0.939144i	$0.611621\pi$
$762$	0	0
$763$	−6204.23	−0.294375
$764$	−759.630	−0.0359718
$765$	0	0
$766$	−736.681	−0.0347485
$767$	1567.63	0.0737988
$768$	0	0
$769$	26772.8	1.25546	0.627731	−	0.778430i	$-0.283984\pi$
$769$	26772.8	1.25546	0.627731	+	0.778430i	$0.283984\pi$
$770$	0	0
$771$	0	0
$772$	419.496	0.0195570
$773$	27669.3	1.28745	0.643724	−	0.765258i	$-0.277389\pi$
$773$	27669.3	1.28745	0.643724	+	0.765258i	$0.277389\pi$
$774$	0	0
$775$	0	0
$776$	5453.45	0.252278
$777$	0	0
$778$	−13057.2	−0.601702
$779$	1267.18	0.0582816
$780$	0	0
$781$	−10040.7	−0.460034
$782$	−25589.2	−1.17017
$783$	0	0
$784$	−3193.53	−0.145478
$785$	0	0
$786$	0	0
$787$	10934.5	0.495263	0.247632	−	0.968854i	$-0.420348\pi$
$787$	10934.5	0.495263	0.247632	+	0.968854i	$0.420348\pi$
$788$	−464.765	−0.0210109
$789$	0	0
$790$	0	0
$791$	−5361.70	−0.241012
$792$	0	0
$793$	−188.328	−0.00843343
$794$	−10347.2	−0.462478
$795$	0	0
$796$	−320.876	−0.0142879
$797$	−31967.3	−1.42075	−0.710376	−	0.703823i	$-0.751475\pi$
$797$	−31967.3	−1.42075	−0.710376	+	0.703823i	$0.751475\pi$
$798$	0	0
$799$	−1893.56	−0.0838415
$800$	0	0
$801$	0	0
$802$	−18151.4	−0.799186
$803$	7479.06	0.328680
$804$	0	0
$805$	0	0
$806$	1947.08	0.0850906
$807$	0	0
$808$	−34498.1	−1.50203
$809$	17924.8	0.778989	0.389495	−	0.921029i	$-0.372650\pi$
$809$	17924.8	0.778989	0.389495	+	0.921029i	$0.372650\pi$
$810$	0	0
$811$	28541.4	1.23579	0.617895	−	0.786261i	$-0.287986\pi$
$811$	28541.4	1.23579	0.617895	+	0.786261i	$0.287986\pi$
$812$	−257.183	−0.0111150
$813$	0	0
$814$	−26795.6	−1.15379
$815$	0	0
$816$	0	0
$817$	3724.19	0.159477
$818$	−18659.7	−0.797580
$819$	0	0
$820$	0	0
$821$	−18878.6	−0.802517	−0.401259	−	0.915965i	$-0.631427\pi$
$821$	−18878.6	−0.802517	−0.401259	+	0.915965i	$0.631427\pi$
$822$	0	0
$823$	−46589.1	−1.97326	−0.986631	−	0.162971i	$-0.947892\pi$
$823$	−46589.1	−1.97326	−0.986631	+	0.162971i	$0.947892\pi$
$824$	21259.7	0.898807
$825$	0	0
$826$	7903.26	0.332917
$827$	14649.8	0.615989	0.307994	−	0.951388i	$-0.400342\pi$
$827$	14649.8	0.615989	0.307994	+	0.951388i	$0.400342\pi$
$828$	0	0
$829$	−34408.0	−1.44154	−0.720771	−	0.693173i	$-0.756212\pi$
$829$	−34408.0	−1.44154	−0.720771	+	0.693173i	$0.756212\pi$
$830$	0	0
$831$	0	0
$832$	−1990.07	−0.0829248
$833$	2532.22	0.105326
$834$	0	0
$835$	0	0
$836$	−145.200	−0.00600700
$837$	0	0
$838$	−18080.7	−0.745331
$839$	−25934.2	−1.06716	−0.533580	−	0.845749i	$-0.679154\pi$
$839$	−25934.2	−1.06716	−0.533580	+	0.845749i	$0.679154\pi$
$840$	0	0
$841$	35967.9	1.47476
$842$	−23236.5	−0.951048
$843$	0	0
$844$	424.335	0.0173060
$845$	0	0
$846$	0	0
$847$	−495.681	−0.0201084
$848$	−42055.9	−1.70307
$849$	0	0
$850$	0	0
$851$	−43484.8	−1.75163
$852$	0	0
$853$	48456.3	1.94503	0.972516	−	0.232836i	$-0.0748007\pi$
$853$	48456.3	1.94503	0.972516	+	0.232836i	$0.0748007\pi$
$854$	−949.462	−0.0380444
$855$	0	0
$856$	−19362.7	−0.773136
$857$	36273.9	1.44585	0.722924	−	0.690928i	$-0.242798\pi$
$857$	36273.9	1.44585	0.722924	+	0.690928i	$0.242798\pi$
$858$	0	0
$859$	28067.2	1.11483	0.557415	−	0.830234i	$-0.311793\pi$
$859$	28067.2	1.11483	0.557415	+	0.830234i	$0.311793\pi$
$860$	0	0
$861$	0	0
$862$	41015.7	1.62065
$863$	−8330.92	−0.328607	−0.164303	−	0.986410i	$-0.552538\pi$
$863$	−8330.92	−0.328607	−0.164303	+	0.986410i	$0.552538\pi$
$864$	0	0
$865$	0	0
$866$	23977.1	0.940849
$867$	0	0
$868$	180.134	0.00704395
$869$	−17733.4	−0.692251
$870$	0	0
$871$	1043.20	0.0405828
$872$	−19863.3	−0.771395
$873$	0	0
$874$	−12840.8	−0.496965
$875$	0	0
$876$	0	0
$877$	16469.6	0.634137	0.317068	−	0.948403i	$-0.397302\pi$
$877$	16469.6	0.634137	0.317068	+	0.948403i	$0.397302\pi$
$878$	−50988.0	−1.95987
$879$	0	0
$880$	0	0
$881$	46635.9	1.78343	0.891715	−	0.452597i	$-0.149502\pi$
$881$	46635.9	1.78343	0.891715	+	0.452597i	$0.149502\pi$
$882$	0	0
$883$	−14075.8	−0.536453	−0.268227	−	0.963356i	$-0.586438\pi$
$883$	−14075.8	−0.536453	−0.268227	+	0.963356i	$0.586438\pi$
$884$	−30.6329	−0.00116549
$885$	0	0
$886$	5428.48	0.205839
$887$	26222.9	0.992649	0.496324	−	0.868137i	$-0.334683\pi$
$887$	26222.9	0.992649	0.496324	+	0.868137i	$0.334683\pi$
$888$	0	0
$889$	3535.29	0.133374
$890$	0	0
$891$	0	0
$892$	709.069	0.0266159
$893$	−950.199	−0.0356072
$894$	0	0
$895$	0	0
$896$	−10412.0	−0.388215
$897$	0	0
$898$	14803.8	0.550121
$899$	−42274.6	−1.56834
$900$	0	0
$901$	33347.1	1.23302
$902$	−5222.87	−0.192796
$903$	0	0
$904$	−17165.9	−0.631558
$905$	0	0
$906$	0	0
$907$	15373.0	0.562793	0.281396	−	0.959592i	$-0.409202\pi$
$907$	15373.0	0.562793	0.281396	+	0.959592i	$0.409202\pi$
$908$	−143.685	−0.00525147
$909$	0	0
$910$	0	0
$911$	21189.3	0.770619	0.385310	−	0.922787i	$-0.374095\pi$
$911$	21189.3	0.770619	0.385310	+	0.922787i	$0.374095\pi$
$912$	0	0
$913$	2722.05	0.0986710
$914$	31969.7	1.15696
$915$	0	0
$916$	111.265	0.00401342
$917$	4710.51	0.169635
$918$	0	0
$919$	−18364.2	−0.659172	−0.329586	−	0.944126i	$-0.606909\pi$
$919$	−18364.2	−0.659172	−0.329586	+	0.944126i	$0.606909\pi$
$920$	0	0
$921$	0	0
$922$	−49303.4	−1.76109
$923$	−1062.97	−0.0379070
$924$	0	0
$925$	0	0
$926$	−1100.96	−0.0390710
$927$	0	0
$928$	−1662.47	−0.0588073
$929$	−15460.6	−0.546011	−0.273006	−	0.962012i	$-0.588018\pi$
$929$	−15460.6	−0.546011	−0.273006	+	0.962012i	$0.588018\pi$
$930$	0	0
$931$	1270.68	0.0447314
$932$	231.883	0.00814978
$933$	0	0
$934$	14374.6	0.503589
$935$	0	0
$936$	0	0
$937$	−28824.7	−1.00498	−0.502488	−	0.864584i	$-0.667582\pi$
$937$	−28824.7	−1.00498	−0.502488	+	0.864584i	$0.667582\pi$
$938$	5259.36	0.183075
$939$	0	0
$940$	0	0
$941$	−42172.5	−1.46098	−0.730492	−	0.682922i	$-0.760709\pi$
$941$	−42172.5	−1.46098	−0.730492	+	0.682922i	$0.760709\pi$
$942$	0	0
$943$	−8475.83	−0.292695
$944$	25776.0	0.888705
$945$	0	0
$946$	−15349.8	−0.527552
$947$	8926.33	0.306301	0.153150	−	0.988203i	$-0.451058\pi$
$947$	8926.33	0.306301	0.153150	+	0.988203i	$0.451058\pi$
$948$	0	0
$949$	791.778	0.0270835
$950$	0	0
$951$	0	0
$952$	8107.09	0.276000
$953$	−40420.6	−1.37392	−0.686962	−	0.726693i	$-0.741056\pi$
$953$	−40420.6	−1.37392	−0.686962	+	0.726693i	$0.741056\pi$
$954$	0	0
$955$	0	0
$956$	−414.996	−0.0140397
$957$	0	0
$958$	24784.8	0.835868
$959$	−10866.0	−0.365882
$960$	0	0
$961$	−181.408	−0.00608935
$962$	−2836.74	−0.0950730
$963$	0	0
$964$	−381.378	−0.0127421
$965$	0	0
$966$	0	0
$967$	33914.1	1.12782	0.563910	−	0.825836i	$-0.309296\pi$
$967$	33914.1	1.12782	0.563910	+	0.825836i	$0.309296\pi$
$968$	−1586.96	−0.0526930
$969$	0	0
$970$	0	0
$971$	59339.0	1.96115	0.980577	−	0.196136i	$-0.0628393\pi$
$971$	59339.0	1.96115	0.980577	+	0.196136i	$0.0628393\pi$
$972$	0	0
$973$	7504.12	0.247247
$974$	−2542.08	−0.0836277
$975$	0	0
$976$	−3096.62	−0.101558
$977$	7032.14	0.230274	0.115137	−	0.993350i	$-0.463269\pi$
$977$	7032.14	0.230274	0.115137	+	0.993350i	$0.463269\pi$
$978$	0	0
$979$	−58117.3	−1.89728
$980$	0	0
$981$	0	0
$982$	−4460.48	−0.144949
$983$	−1703.63	−0.0552772	−0.0276386	−	0.999618i	$-0.508799\pi$
$983$	−1703.63	−0.0552772	−0.0276386	+	0.999618i	$0.508799\pi$
$984$	0	0
$985$	0	0
$986$	36244.0	1.17063
$987$	0	0
$988$	−15.3718	−0.000494980 0
$989$	−24910.1	−0.800906
$990$	0	0
$991$	12740.3	0.408383	0.204192	−	0.978931i	$-0.434543\pi$
$991$	12740.3	0.408383	0.204192	+	0.978931i	$0.434543\pi$
$992$	1164.41	0.0372682
$993$	0	0
$994$	−5359.03	−0.171004
$995$	0	0
$996$	0	0
$997$	16609.6	0.527615	0.263808	−	0.964575i	$-0.415022\pi$
$997$	16609.6	0.527615	0.263808	+	0.964575i	$0.415022\pi$
$998$	23506.9	0.745589
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bq.1.4		5
3.2	odd	2		175.4.a.i.1.2		5
5.2	odd	4		315.4.d.c.64.8		10
5.3	odd	4		315.4.d.c.64.3		10
5.4	even	2		1575.4.a.bn.1.2		5
15.2	even	4		35.4.b.a.29.3	✓	10
15.8	even	4		35.4.b.a.29.8	yes	10
15.14	odd	2		175.4.a.j.1.4		5
21.20	even	2		1225.4.a.be.1.2		5
60.23	odd	4		560.4.g.f.449.10		10
60.47	odd	4		560.4.g.f.449.1		10
105.2	even	12		245.4.j.e.214.3		20
105.17	odd	12		245.4.j.f.79.8		20
105.23	even	12		245.4.j.e.214.8		20
105.32	even	12		245.4.j.e.79.8		20
105.38	odd	12		245.4.j.f.79.3		20
105.47	odd	12		245.4.j.f.214.3		20
105.53	even	12		245.4.j.e.79.3		20
105.62	odd	4		245.4.b.d.99.3		10
105.68	odd	12		245.4.j.f.214.8		20
105.83	odd	4		245.4.b.d.99.8		10
105.104	even	2		1225.4.a.bh.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.b.a.29.3	✓	10	15.2	even	4
35.4.b.a.29.8	yes	10	15.8	even	4
175.4.a.i.1.2		5	3.2	odd	2
175.4.a.j.1.4		5	15.14	odd	2
245.4.b.d.99.3		10	105.62	odd	4
245.4.b.d.99.8		10	105.83	odd	4
245.4.j.e.79.3		20	105.53	even	12
245.4.j.e.79.8		20	105.32	even	12
245.4.j.e.214.3		20	105.2	even	12
245.4.j.e.214.8		20	105.23	even	12
245.4.j.f.79.3		20	105.38	odd	12
245.4.j.f.79.8		20	105.17	odd	12
245.4.j.f.214.3		20	105.47	odd	12
245.4.j.f.214.8		20	105.68	odd	12
315.4.d.c.64.3		10	5.3	odd	4
315.4.d.c.64.8		10	5.2	odd	4
560.4.g.f.449.1		10	60.47	odd	4
560.4.g.f.449.10		10	60.23	odd	4
1225.4.a.be.1.2		5	21.20	even	2
1225.4.a.bh.1.4		5	105.104	even	2
1575.4.a.bn.1.2		5	5.4	even	2
1575.4.a.bq.1.4		5	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q+2.85474 q^{2} +0.149548 q^{4} -7.00000 q^{7} -22.4110 q^{8} -37.4408 q^{11} -3.96370 q^{13} -19.9832 q^{14} -65.1740 q^{16} +51.6780 q^{17} +25.9323 q^{19} -106.884 q^{22} -173.454 q^{23} -11.3154 q^{26} -1.04684 q^{28} +245.676 q^{29} -172.074 q^{31} -6.76690 q^{32} +147.527 q^{34} +250.699 q^{37} +74.0300 q^{38} +48.8649 q^{41} +143.612 q^{43} -5.59920 q^{44} -495.167 q^{46} -36.6415 q^{47} +49.0000 q^{49} -0.592765 q^{52} +645.286 q^{53} +156.877 q^{56} +701.343 q^{58} -395.495 q^{59} +47.5130 q^{61} -491.228 q^{62} +502.074 q^{64} -263.189 q^{67} +7.72835 q^{68} +268.177 q^{71} -199.757 q^{73} +715.680 q^{74} +3.87813 q^{76} +262.085 q^{77} +473.640 q^{79} +139.497 q^{82} -72.7028 q^{83} +409.975 q^{86} +839.086 q^{88} +1552.25 q^{89} +27.7459 q^{91} -25.9398 q^{92} -104.602 q^{94} -243.338 q^{97} +139.882 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 18 q^{4} - 35 q^{7} + 42 q^{8} - 42 q^{11} - 34 q^{13} - 28 q^{14} + 74 q^{16} + 238 q^{17} - 36 q^{19} - 358 q^{22} + 152 q^{23} + 310 q^{26} - 126 q^{28} + 44 q^{29} + 60 q^{31} + 710 q^{32}+ \cdots + 196 q^{98}+O(q^{100}) \) 5 * q + 4 * q^2 + 18 * q^4 - 35 * q^7 + 42 * q^8 - 42 * q^11 - 34 * q^13 - 28 * q^14 + 74 * q^16 + 238 * q^17 - 36 * q^19 - 358 * q^22 + 152 * q^23 + 310 * q^26 - 126 * q^28 + 44 * q^29 + 60 * q^31 + 710 * q^32 - 482 * q^34 - 312 * q^37 + 280 * q^38 + 426 * q^41 - 304 * q^43 - 712 * q^44 + 88 * q^46 + 370 * q^47 + 245 * q^49 + 1156 * q^52 + 976 * q^53 - 294 * q^56 + 2722 * q^58 + 432 * q^59 - 442 * q^61 - 956 * q^62 + 1362 * q^64 + 804 * q^67 - 420 * q^68 - 440 * q^71 - 564 * q^73 + 1512 * q^74 - 1336 * q^76 + 294 * q^77 + 1790 * q^79 + 3480 * q^82 + 1656 * q^83 - 1216 * q^86 + 1092 * q^88 - 746 * q^89 + 238 * q^91 + 572 * q^92 - 826 * q^94 - 518 * q^97 + 196 * q^98

Introduction

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