LMFDB - Embedded newform 1575.4.a.bn.1.2

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:	$5$
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			1.2
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	$-0.668379\pi$
$2$	−2.85474	−1.00930	−0.504652	+	0.863323i	$0.668379\pi$
$3$	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	$-0.486537\pi$
$13$	3.96370	0.0845641	0.0422821	+	0.999106i	$0.486537\pi$
$14$	−19.9832	−0.381481
$15$	0	0
$16$	−65.1740	−1.01834
$17$	−51.6780	−0.737279	−0.368640	−	0.929572i	$-0.620176\pi$
$17$	−51.6780	−0.737279	−0.368640	+	0.929572i	$0.620176\pi$
$18$	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	$-0.212017\pi$
$23$	173.454	1.57251	0.786255	+	0.617902i	$0.212017\pi$
$24$	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	$-0.688030\pi$
$37$	−250.699	−1.11391	−0.556954	+	0.830543i	$0.688030\pi$
$38$	−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	$-0.581963\pi$
$43$	−143.612	−0.509317	−0.254658	+	0.967031i	$0.581963\pi$
$44$	−5.59920	−0.0191844
$45$	0	0
$46$	−495.167	−1.58714
$47$	36.6415	0.113717	0.0568587	−	0.998382i	$-0.481892\pi$
$47$	36.6415	0.113717	0.0568587	+	0.998382i	$0.481892\pi$
$48$	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	$-0.815226\pi$
$53$	−645.286	−1.67239	−0.836196	+	0.548430i	$0.815226\pi$
$54$	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	$-0.422868\pi$
$67$	263.189	0.479906	0.239953	+	0.970785i	$0.422868\pi$
$68$	−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	$-0.448807\pi$
$73$	199.757	0.320271	0.160136	+	0.987095i	$0.448807\pi$
$74$	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	$-0.484692\pi$
$83$	72.7028	0.0961466	0.0480733	+	0.998844i	$0.484692\pi$
$84$	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	$-0.459351\pi$
$97$	243.338	0.254714	0.127357	+	0.991857i	$0.459351\pi$
$98$	−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	$-0.350088\pi$
$103$	948.628	0.907486	0.453743	+	0.891133i	$0.350088\pi$
$104$	88.8306	0.0837554
$105$	0	0
$106$	1842.12	1.68795
$107$	−863.983	−0.780602	−0.390301	−	0.920687i	$-0.627629\pi$
$107$	−863.983	−0.780602	−0.390301	+	0.920687i	$0.627629\pi$
$108$	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	$-0.603289\pi$
$113$	−765.957	−0.637657	−0.318828	+	0.947812i	$0.603289\pi$
$114$	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	$-0.443543\pi$
$127$	505.042	0.352876	0.176438	+	0.984312i	$0.443543\pi$
$128$	−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	$-0.660822\pi$
$137$	−1552.28	−0.968032	−0.484016	+	0.875059i	$0.660822\pi$
$138$	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	$-0.629121\pi$
$157$	−1552.56	−0.789223	−0.394611	+	0.918848i	$0.629121\pi$
$158$	−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	$-0.289741\pi$
$163$	2553.65	1.22710	0.613550	+	0.789656i	$0.289741\pi$
$164$	7.30766	0.00347947
$165$	0	0
$166$	−207.548	−0.0970411
$167$	−3573.14	−1.65568	−0.827839	−	0.560966i	$-0.810430\pi$
$167$	−3573.14	−1.65568	−0.827839	+	0.560966i	$0.810430\pi$
$168$	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	$-0.663385\pi$
$173$	−2234.71	−0.982090	−0.491045	+	0.871134i	$0.663385\pi$
$174$	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	$-0.675223\pi$
$193$	−2805.09	−1.04619	−0.523095	+	0.852274i	$0.675223\pi$
$194$	−694.667	−0.257084
$195$	0	0
$196$	7.32786	0.00267050
$197$	3107.79	1.12396	0.561982	−	0.827149i	$-0.310039\pi$
$197$	3107.79	1.12396	0.561982	+	0.827149i	$0.310039\pi$
$198$	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	$-0.752166\pi$
$223$	−4741.40	−1.42380	−0.711901	+	0.702280i	$0.752166\pi$
$224$	47.3683	0.0141291
$225$	0	0
$226$	2186.61	0.643589
$227$	960.790	0.280925	0.140462	−	0.990086i	$-0.455141\pi$
$227$	960.790	0.280925	0.140462	+	0.990086i	$0.455141\pi$
$228$	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	$-0.569948\pi$
$233$	−1550.56	−0.435968	−0.217984	+	0.975952i	$0.569948\pi$
$234$	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	$-0.607295\pi$
$257$	−2725.22	−0.661459	−0.330729	+	0.943726i	$0.607295\pi$
$258$	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	$-0.615480\pi$
$263$	−3027.26	−0.709767	−0.354884	+	0.934910i	$0.615480\pi$
$264$	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	$-0.530309\pi$
$277$	−876.614	−0.190147	−0.0950733	+	0.995470i	$0.530309\pi$
$278$	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	$-0.575939\pi$
$283$	−2250.07	−0.472625	−0.236312	+	0.971677i	$0.575939\pi$
$284$	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	$-0.700862\pi$
$293$	−5917.86	−1.17995	−0.589975	+	0.807422i	$0.700862\pi$
$294$	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	$-0.841916\pi$
$307$	−9458.47	−1.75838	−0.879191	+	0.476469i	$0.841916\pi$
$308$	−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	$-0.189363\pi$
$313$	9172.41	1.65641	0.828204	+	0.560427i	$0.189363\pi$
$314$	4432.16	0.796565
$315$	0	0
$316$	70.8320	0.0126095
$317$	−3077.94	−0.545345	−0.272672	−	0.962107i	$-0.587908\pi$
$317$	−3077.94	−0.545345	−0.272672	+	0.962107i	$0.587908\pi$
$318$	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	$-0.401234\pi$
$337$	3777.84	0.610658	0.305329	+	0.952247i	$0.401234\pi$
$338$	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	$-0.720115\pi$
$347$	−8244.08	−1.27540	−0.637702	+	0.770283i	$0.720115\pi$
$348$	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	$-0.602338\pi$
$353$	−4191.51	−0.631987	−0.315994	+	0.948761i	$0.602338\pi$
$354$	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	$-0.539109\pi$
$367$	−1723.30	−0.245110	−0.122555	+	0.992462i	$0.539109\pi$
$368$	−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	$-0.437325\pi$
$373$	2818.55	0.391258	0.195629	+	0.980678i	$0.437325\pi$
$374$	−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	$-0.494520\pi$
$383$	258.055	0.0344282	0.0172141	+	0.999852i	$0.494520\pi$
$384$	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	$-0.426419\pi$
$397$	3624.55	0.458215	0.229107	+	0.973401i	$0.426419\pi$
$398$	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	$-0.654337\pi$
$433$	−8399.05	−0.932176	−0.466088	+	0.884738i	$0.654337\pi$
$434$	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	$-0.532515\pi$
$443$	−1901.57	−0.203942	−0.101971	+	0.994787i	$0.532515\pi$
$444$	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	$-0.694278\pi$
$457$	−11198.8	−1.14630	−0.573149	+	0.819451i	$0.694278\pi$
$458$	−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	$-0.493839\pi$
$463$	385.660	0.0387109	0.0193554	+	0.999813i	$0.493839\pi$
$464$	−16011.7	−1.60199
$465$	0	0
$466$	4426.44	0.440024
$467$	−5035.36	−0.498947	−0.249474	−	0.968382i	$-0.580258\pi$
$467$	−5035.36	−0.498947	−0.249474	+	0.968382i	$0.580258\pi$
$468$	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	$-0.486809\pi$
$487$	890.476	0.0828569	0.0414284	+	0.999141i	$0.486809\pi$
$488$	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	$-0.501023\pi$
$503$	−72.5340	−0.00642969	−0.00321484	+	0.999995i	$0.501023\pi$
$504$	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	$-0.380316\pi$
$523$	8783.88	0.734402	0.367201	+	0.930142i	$0.380316\pi$
$524$	−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	$-0.572492\pi$
$547$	−5776.83	−0.451553	−0.225776	+	0.974179i	$0.572492\pi$
$548$	−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	$-0.785854\pi$
$557$	−20562.6	−1.56421	−0.782106	+	0.623145i	$0.785854\pi$
$558$	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	$-0.144549\pi$
$563$	24009.5	1.79730	0.898650	+	0.438666i	$0.144549\pi$
$564$	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	$-0.303340\pi$
$577$	16057.2	1.15853	0.579265	+	0.815139i	$0.303340\pi$
$578$	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	$-0.597838\pi$
$587$	−8605.63	−0.605098	−0.302549	+	0.953134i	$0.597838\pi$
$588$	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	$-0.251033\pi$
$593$	20355.6	1.40962	0.704809	+	0.709397i	$0.251033\pi$
$594$	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	$-0.699392\pi$
$607$	−17534.2	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$608$	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	$-0.611822\pi$
$613$	−10445.5	−0.688238	−0.344119	+	0.938926i	$0.611822\pi$
$614$	27001.5	1.77474
$615$	0	0
$616$	−5873.60	−0.384179
$617$	−13218.9	−0.862516	−0.431258	−	0.902229i	$-0.641930\pi$
$617$	−13218.9	−0.862516	−0.431258	+	0.902229i	$0.641930\pi$
$618$	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	$-0.823024\pi$
$643$	−27698.0	−1.69876	−0.849380	+	0.527782i	$0.823024\pi$
$644$	181.579	0.0111106
$645$	0	0
$646$	3825.72	0.233004
$647$	10965.1	0.666282	0.333141	−	0.942877i	$-0.391891\pi$
$647$	10965.1	0.666282	0.333141	+	0.942877i	$0.391891\pi$
$648$	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	$-0.379480\pi$
$653$	12336.2	0.739285	0.369642	+	0.929174i	$0.379480\pi$
$654$	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	$-0.164488\pi$
$673$	30358.9	1.73885	0.869427	+	0.494061i	$0.164488\pi$
$674$	−10784.7	−0.616340
$675$	0	0
$676$	−326.208	−0.0185599
$677$	−6916.48	−0.392647	−0.196324	−	0.980539i	$-0.562900\pi$
$677$	−6916.48	−0.392647	−0.196324	+	0.980539i	$0.562900\pi$
$678$	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	$-0.540525\pi$
$683$	−4532.72	−0.253938	−0.126969	+	0.991907i	$0.540525\pi$
$684$	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	$-0.770346\pi$
$727$	−29435.6	−1.50166	−0.750830	+	0.660496i	$0.770346\pi$
$728$	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	$-0.553078\pi$
$733$	−6587.69	−0.331954	−0.165977	+	0.986130i	$0.553078\pi$
$734$	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	$-0.597978\pi$
$743$	−12271.9	−0.605940	−0.302970	+	0.953000i	$0.597978\pi$
$744$	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	$-0.411427\pi$
$757$	11442.1	0.549368	0.274684	+	0.961535i	$0.411427\pi$
$758$	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	$-0.722611\pi$
$773$	−27669.3	−1.28745	−0.643724	+	0.765258i	$0.722611\pi$
$774$	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	$-0.579652\pi$
$787$	−10934.5	−0.495263	−0.247632	+	0.968854i	$0.579652\pi$
$788$	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	$-0.248525\pi$
$797$	31967.3	1.42075	0.710376	+	0.703823i	$0.248525\pi$
$798$	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	$-0.0521078\pi$
$823$	46589.1	1.97326	0.986631	+	0.162971i	$0.0521078\pi$
$824$	21259.7	0.898807
$825$	0	0
$826$	7903.26	0.332917
$827$	−14649.8	−0.615989	−0.307994	−	0.951388i	$-0.599658\pi$
$827$	−14649.8	−0.615989	−0.307994	+	0.951388i	$0.599658\pi$
$828$	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	$-0.925199\pi$
$853$	−48456.3	−1.94503	−0.972516	+	0.232836i	$0.925199\pi$
$854$	−949.462	−0.0380444
$855$	0	0
$856$	−19362.7	−0.773136
$857$	−36273.9	−1.44585	−0.722924	−	0.690928i	$-0.757202\pi$
$857$	−36273.9	−1.44585	−0.722924	+	0.690928i	$0.757202\pi$
$858$	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	$-0.447462\pi$
$863$	8330.92	0.328607	0.164303	+	0.986410i	$0.447462\pi$
$864$	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	$-0.602698\pi$
$877$	−16469.6	−0.634137	−0.317068	+	0.948403i	$0.602698\pi$
$878$	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	$-0.413562\pi$
$883$	14075.8	0.536453	0.268227	+	0.963356i	$0.413562\pi$
$884$	−30.6329	−0.00116549
$885$	0	0
$886$	5428.48	0.205839
$887$	−26222.9	−0.992649	−0.496324	−	0.868137i	$-0.665317\pi$
$887$	−26222.9	−0.992649	−0.496324	+	0.868137i	$0.665317\pi$
$888$	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	$-0.590798\pi$
$907$	−15373.0	−0.562793	−0.281396	+	0.959592i	$0.590798\pi$
$908$	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	$-0.332418\pi$
$937$	28824.7	1.00498	0.502488	+	0.864584i	$0.332418\pi$
$938$	−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	$-0.548942\pi$
$947$	−8926.33	−0.306301	−0.153150	+	0.988203i	$0.548942\pi$
$948$	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	$-0.258944\pi$
$953$	40420.6	1.37392	0.686962	+	0.726693i	$0.258944\pi$
$954$	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	$-0.690704\pi$
$967$	−33914.1	−1.12782	−0.563910	+	0.825836i	$0.690704\pi$
$968$	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	$-0.536731\pi$
$977$	−7032.14	−0.230274	−0.115137	+	0.993350i	$0.536731\pi$
$978$	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	$-0.491201\pi$
$983$	1703.63	0.0552772	0.0276386	+	0.999618i	$0.491201\pi$
$984$	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	$-0.584978\pi$
$997$	−16609.6	−0.527615	−0.263808	+	0.964575i	$0.584978\pi$
$998$	−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.bn.1.2		5
3.2	odd	2		175.4.a.j.1.4		5
5.2	odd	4		315.4.d.c.64.3		10
5.3	odd	4		315.4.d.c.64.8		10
5.4	even	2		1575.4.a.bq.1.4		5
15.2	even	4		35.4.b.a.29.8	yes	10
15.8	even	4		35.4.b.a.29.3	✓	10
15.14	odd	2		175.4.a.i.1.2		5
21.20	even	2		1225.4.a.bh.1.4		5
60.23	odd	4		560.4.g.f.449.1		10
60.47	odd	4		560.4.g.f.449.10		10
105.2	even	12		245.4.j.e.214.8		20
105.17	odd	12		245.4.j.f.79.3		20
105.23	even	12		245.4.j.e.214.3		20
105.32	even	12		245.4.j.e.79.3		20
105.38	odd	12		245.4.j.f.79.8		20
105.47	odd	12		245.4.j.f.214.8		20
105.53	even	12		245.4.j.e.79.8		20
105.62	odd	4		245.4.b.d.99.8		10
105.68	odd	12		245.4.j.f.214.3		20
105.83	odd	4		245.4.b.d.99.3		10
105.104	even	2		1225.4.a.be.1.2		5

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
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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