LMFDB - Embedded newform 1225.4.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,4,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1225.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:	$72.2773397570$
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:	$\Z[a_1, a_2, a_3]$
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	$\chi$	$=$	1225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.85474 q^{2} +8.98858 q^{3} +0.149548 q^{4} -25.6601 q^{6} +22.4110 q^{8} +53.7945 q^{9} +37.4408 q^{11} +1.34423 q^{12} +3.96370 q^{13} -65.1740 q^{16} +51.6780 q^{17} -153.569 q^{18} -25.9323 q^{19} -106.884 q^{22} +173.454 q^{23} +201.443 q^{24} -11.3154 q^{26} +240.845 q^{27} -245.676 q^{29} +172.074 q^{31} +6.76690 q^{32} +336.539 q^{33} -147.527 q^{34} +8.04488 q^{36} +250.699 q^{37} +74.0300 q^{38} +35.6281 q^{39} +48.8649 q^{41} +143.612 q^{43} +5.59920 q^{44} -495.167 q^{46} -36.6415 q^{47} -585.822 q^{48} +464.511 q^{51} +0.592765 q^{52} -645.286 q^{53} -687.549 q^{54} -233.094 q^{57} +701.343 q^{58} -395.495 q^{59} -47.5130 q^{61} -491.228 q^{62} +502.074 q^{64} -960.733 q^{66} -263.189 q^{67} +7.72835 q^{68} +1559.11 q^{69} -268.177 q^{71} +1205.59 q^{72} +199.757 q^{73} -715.680 q^{74} -3.87813 q^{76} -101.709 q^{78} +473.640 q^{79} +712.399 q^{81} -139.497 q^{82} -72.7028 q^{83} -409.975 q^{86} -2208.28 q^{87} +839.086 q^{88} +1552.25 q^{89} +25.9398 q^{92} +1546.70 q^{93} +104.602 q^{94} +60.8248 q^{96} +243.338 q^{97} +2014.11 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{2} + 10 q^{3} + 18 q^{4} - 6 q^{6} - 42 q^{8} + 23 q^{9} + 42 q^{11} + 136 q^{12} + 34 q^{13} + 74 q^{16} + 238 q^{17} + 2 q^{18} + 36 q^{19} - 358 q^{22} - 152 q^{23} + 36 q^{24} + 310 q^{26}+ \cdots + 2652 q^{99}+O(q^{100}) $ 5 * q - 4 * q^2 + 10 * q^3 + 18 * q^4 - 6 * q^6 - 42 * q^8 + 23 * q^9 + 42 * q^11 + 136 * q^12 + 34 * q^13 + 74 * q^16 + 238 * q^17 + 2 * q^18 + 36 * q^19 - 358 * q^22 - 152 * q^23 + 36 * q^24 + 310 * q^26 + 334 * q^27 - 44 * q^29 - 60 * q^31 - 710 * q^32 + 426 * q^33 + 482 * q^34 - 210 * q^36 - 312 * q^37 + 280 * q^38 - 106 * q^39 + 426 * q^41 - 304 * q^43 + 712 * q^44 + 88 * q^46 + 370 * q^47 - 696 * q^48 + 638 * q^51 - 1156 * q^52 - 976 * q^53 - 498 * q^54 + 588 * q^57 + 2722 * q^58 + 432 * q^59 + 442 * q^61 - 956 * q^62 + 1362 * q^64 - 574 * q^66 + 804 * q^67 - 420 * q^68 + 2404 * q^69 + 440 * q^71 + 3150 * q^72 + 564 * q^73 - 1512 * q^74 + 1336 * q^76 + 2742 * q^78 + 1790 * q^79 - 151 * q^81 - 3480 * q^82 + 1656 * q^83 + 1216 * q^86 - 1674 * q^87 + 1092 * q^88 - 746 * q^89 - 572 * q^92 - 676 * q^93 + 826 * q^94 - 2040 * q^96 + 518 * q^97 + 2652 * q^99

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	$0.668379\pi$
$3$	8.98858	1.72985	0.864926	−	0.501899i	$-0.167365\pi$
$3$	8.98858	1.72985	0.864926	+	0.501899i	$0.167365\pi$
$4$	0.149548	0.0186935
$5$	0	0
$5$	0	0
$6$	−25.6601	−1.74595
$7$	0	0
$7$	0	0
$8$	22.4110	0.990436
$9$	53.7945	1.99239
$10$	0	0
$11$	37.4408	1.02626	0.513128	−	0.858312i	$-0.328487\pi$
$11$	37.4408	1.02626	0.513128	+	0.858312i	$0.328487\pi$
$12$	1.34423	0.0323371
$13$	3.96370	0.0845641	0.0422821	−	0.999106i	$-0.486537\pi$
$13$	3.96370	0.0845641	0.0422821	+	0.999106i	$0.486537\pi$
$14$	0	0
$15$	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	$0.379824\pi$
$18$	−153.569	−2.01093
$19$	−25.9323	−0.313120	−0.156560	−	0.987668i	$-0.550040\pi$
$19$	−25.9323	−0.313120	−0.156560	+	0.987668i	$0.550040\pi$
$20$	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	$0.212017\pi$
$24$	201.443	1.71331
$25$	0	0
$26$	−11.3154	−0.0853509
$27$	240.845	1.71669
$28$	0	0
$29$	−245.676	−1.57314	−0.786568	−	0.617503i	$-0.788144\pi$
$29$	−245.676	−1.57314	−0.786568	+	0.617503i	$0.788144\pi$
$30$	0	0
$31$	172.074	0.996951	0.498475	−	0.866904i	$-0.333893\pi$
$31$	172.074	0.996951	0.498475	+	0.866904i	$0.333893\pi$
$32$	6.76690	0.0373822
$33$	336.539	1.77527
$34$	−147.527	−0.744139
$35$	0	0
$36$	8.04488	0.0372448
$37$	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
$39$	35.6281	0.146283
$40$	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	$0.518108\pi$
$48$	−585.822	−1.76159
$49$	0	0
$50$	0	0
$51$	464.511	1.27538
$52$	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	$0.815226\pi$
$54$	−687.549	−1.73266
$55$	0	0
$56$	0	0
$57$	−233.094	−0.541651
$58$	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	$-0.515879\pi$
$61$	−47.5130	−0.0997282	−0.0498641	+	0.998756i	$0.515879\pi$
$62$	−491.228	−1.00623
$63$	0	0
$64$	502.074	0.980614
$65$	0	0
$66$	−960.733	−1.79179
$67$	−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
$69$	1559.11	2.72021
$70$	0	0
$71$	−268.177	−0.448264	−0.224132	−	0.974559i	$-0.571955\pi$
$71$	−268.177	−0.448264	−0.224132	+	0.974559i	$0.571955\pi$
$72$	1205.59	1.97333
$73$	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
$77$	0	0
$78$	−101.709	−0.147644
$79$	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
$81$	712.399	0.977227
$82$	−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
$87$	−2208.28	−2.72129
$88$	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
$91$	0	0
$92$	25.9398	0.0293958
$93$	1546.70	1.72458
$94$	104.602	0.114775
$95$	0	0
$96$	60.8248	0.0646657
$97$	243.338	0.254714	0.127357	−	0.991857i	$-0.459351\pi$
$97$	243.338	0.254714	0.127357	+	0.991857i	$0.459351\pi$
$98$	0	0
$99$	2014.11	2.04470
$100$	0	0
$101$	1539.34	1.51653	0.758265	−	0.651946i	$-0.226047\pi$
$101$	1539.34	1.51653	0.758265	+	0.651946i	$0.226047\pi$
$102$	−1326.06	−1.28725
$103$	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	$0.627629\pi$
$108$	36.0179	0.0320910
$109$	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
$111$	2253.42	1.92690
$112$	0	0
$113$	−765.957	−0.637657	−0.318828	−	0.947812i	$-0.603289\pi$
$113$	−765.957	−0.637657	−0.318828	+	0.947812i	$0.603289\pi$
$114$	665.424	0.546690
$115$	0	0
$116$	−36.7405	−0.0294075
$117$	213.226	0.168485
$118$	1129.04	0.880816
$119$	0	0
$120$	0	0
$121$	70.8116	0.0532018
$122$	135.637	0.100656
$123$	439.226	0.321981
$124$	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
$129$	1290.87	0.881043
$130$	0	0
$131$	−672.930	−0.448811	−0.224405	−	0.974496i	$-0.572044\pi$
$131$	−672.930	−0.448811	−0.224405	+	0.974496i	$0.572044\pi$
$132$	50.3289	0.0331861
$133$	0	0
$134$	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	$0.660822\pi$
$138$	−4450.85	−2.74552
$139$	1072.02	0.654154	0.327077	−	0.944998i	$-0.393936\pi$
$139$	1072.02	0.654154	0.327077	+	0.944998i	$0.393936\pi$
$140$	0	0
$141$	−329.355	−0.196714
$142$	765.575	0.452434
$143$	148.404	0.0867845
$144$	−3506.01	−2.02894
$145$	0	0
$146$	−570.255	−0.323251
$147$	0	0
$148$	37.4915	0.0208229
$149$	645.936	0.355149	0.177574	−	0.984107i	$-0.443175\pi$
$149$	645.936	0.355149	0.177574	+	0.984107i	$0.443175\pi$
$150$	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
$153$	2779.99	1.46895
$154$	0	0
$155$	0	0
$156$	5.32811	0.00273455
$157$	−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
$159$	−5800.20	−2.89299
$160$	0	0
$161$	0	0
$162$	−2033.71	−0.986319
$163$	−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
$171$	−1395.01	−0.623856
$172$	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	$0.336615\pi$
$174$	6304.07	2.74661
$175$	0	0
$176$	−2440.17	−1.04508
$177$	−3554.94	−1.50964
$178$	−4431.26	−1.86594
$179$	−1830.53	−0.764361	−0.382180	−	0.924088i	$-0.624827\pi$
$179$	−1830.53	−0.764361	−0.382180	+	0.924088i	$0.624827\pi$
$180$	0	0
$181$	−2437.22	−1.00087	−0.500433	−	0.865775i	$-0.666826\pi$
$181$	−2437.22	−1.00087	−0.500433	+	0.865775i	$0.666826\pi$
$182$	0	0
$183$	−427.075	−0.172515
$184$	3887.29	1.55747
$185$	0	0
$186$	−4415.44	−1.74062
$187$	1934.86	0.756638
$188$	−5.47968	−0.00212578
$189$	0	0
$190$	0	0
$191$	5079.50	1.92429	0.962145	−	0.272538i	$-0.0878632\pi$
$191$	5079.50	1.92429	0.962145	+	0.272538i	$0.0878632\pi$
$192$	4512.93	1.69632
$193$	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
$196$	0	0
$197$	3107.79	1.12396	0.561982	−	0.827149i	$-0.310039\pi$
$197$	3107.79	1.12396	0.561982	+	0.827149i	$0.310039\pi$
$198$	−5749.76	−2.06373
$199$	2145.63	0.764321	0.382161	−	0.924096i	$-0.375180\pi$
$199$	2145.63	0.764321	0.382161	+	0.924096i	$0.375180\pi$
$200$	0	0
$201$	−2365.70	−0.830166
$202$	−4394.41	−1.53064
$203$	0	0
$204$	69.4669	0.0238414
$205$	0	0
$206$	−2708.09	−0.915929
$207$	9330.89	3.13305
$208$	−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
$213$	−2410.53	−0.775430
$214$	2466.45	0.787864
$215$	0	0
$216$	5397.57	1.70027
$217$	0	0
$218$	−2530.21	−0.786089
$219$	1795.53	0.554022
$220$	0	0
$221$	204.836	0.0623474
$222$	−6432.94	−1.94482
$223$	−4741.40	−1.42380	−0.711901	−	0.702280i	$-0.752166\pi$
$223$	−4741.40	−1.42380	−0.711901	+	0.702280i	$0.752166\pi$
$224$	0	0
$225$	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	$0.544859\pi$
$228$	−34.8589	−0.0101254
$229$	−744.006	−0.214696	−0.107348	−	0.994222i	$-0.534236\pi$
$229$	−744.006	−0.214696	−0.107348	+	0.994222i	$0.534236\pi$
$230$	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	$0.569948\pi$
$234$	−608.704	−0.170052
$235$	0	0
$236$	−59.1456	−0.0163138
$237$	4257.35	1.16685
$238$	0	0
$239$	2775.00	0.751045	0.375523	−	0.926813i	$-0.377463\pi$
$239$	2775.00	0.751045	0.375523	+	0.926813i	$0.377463\pi$
$240$	0	0
$241$	2550.20	0.681630	0.340815	−	0.940130i	$-0.389297\pi$
$241$	2550.20	0.681630	0.340815	+	0.940130i	$0.389297\pi$
$242$	−202.149	−0.0536968
$243$	−99.3558	−0.0262291
$244$	−7.10549	−0.00186427
$245$	0	0
$246$	−1253.88	−0.324977
$247$	−102.788	−0.0264787
$248$	3856.36	0.987416
$249$	−653.494	−0.166319
$250$	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	$0.392705\pi$
$258$	−3685.09	−0.889240
$259$	0	0
$260$	0	0
$261$	−13216.0	−3.13430
$262$	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	$0.615480\pi$
$264$	7542.19	1.75829
$265$	0	0
$266$	0	0
$267$	13952.5	3.19804
$268$	−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	$-0.242063\pi$
$271$	6464.45	1.44903	0.724516	+	0.689258i	$0.242063\pi$
$272$	−3368.06	−0.750804
$273$	0	0
$274$	4431.36	0.977038
$275$	0	0
$276$	233.162	0.0508503
$277$	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
$279$	9256.66	1.98631
$280$	0	0
$281$	6252.19	1.32731	0.663655	−	0.748038i	$-0.269004\pi$
$281$	6252.19	1.32731	0.663655	+	0.748038i	$0.269004\pi$
$282$	940.224	0.198544
$283$	−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
$287$	0	0
$288$	364.022	0.0744799
$289$	−2242.39	−0.456419
$290$	0	0
$291$	2187.26	0.440617
$292$	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
$297$	9017.41	1.76176
$298$	−1843.98	−0.358453
$299$	687.522	0.132978
$300$	0	0
$301$	0	0
$302$	−694.256	−0.132284
$303$	13836.4	2.62337
$304$	1690.11	0.318864
$305$	0	0
$306$	−7936.16	−1.48261
$307$	−9458.47	−1.75838	−0.879191	−	0.476469i	$-0.841916\pi$
$307$	−9458.47	−1.75838	−0.879191	+	0.476469i	$0.841916\pi$
$308$	0	0
$309$	8526.81	1.56982
$310$	0	0
$311$	7576.78	1.38148	0.690739	−	0.723104i	$-0.257285\pi$
$311$	7576.78	1.38148	0.690739	+	0.723104i	$0.257285\pi$
$312$	798.461	0.144884
$313$	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	$0.587908\pi$
$318$	16558.1	2.91991
$319$	−9198.31	−1.61444
$320$	0	0
$321$	−7765.98	−1.35033
$322$	0	0
$323$	−1340.13	−0.230857
$324$	106.538	0.0182678
$325$	0	0
$326$	7290.01	1.23852
$327$	7966.75	1.34728
$328$	1095.11	0.184352
$329$	0	0
$330$	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
$333$	13486.2	2.21934
$334$	−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
$339$	−6884.87	−1.10305
$340$	0	0
$341$	6442.60	1.02313
$342$	3982.41	0.629660
$343$	0	0
$344$	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	$0.720115\pi$
$348$	−330.245	−0.0508706
$349$	−7173.78	−1.10030	−0.550148	−	0.835067i	$-0.685429\pi$
$349$	−7173.78	−1.10030	−0.550148	+	0.835067i	$0.685429\pi$
$350$	0	0
$351$	954.637	0.145170
$352$	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	$0.397662\pi$
$354$	10148.4	1.52368
$355$	0	0
$356$	232.136	0.0345594
$357$	0	0
$358$	5225.70	0.771472
$359$	−3136.29	−0.461078	−0.230539	−	0.973063i	$-0.574049\pi$
$359$	−3136.29	−0.461078	−0.230539	+	0.973063i	$0.574049\pi$
$360$	0	0
$361$	−6186.52	−0.901956
$362$	6957.62	1.01018
$363$	636.496	0.0920313
$364$	0	0
$365$	0	0
$366$	1219.19	0.174120
$367$	−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
$369$	2628.67	0.370848
$370$	0	0
$371$	0	0
$372$	231.307	0.0322384
$373$	−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
$381$	−4539.61	−0.610423
$382$	−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	$0.505480\pi$
$384$	−13369.9	−1.77677
$385$	0	0
$386$	−8007.81	−1.05592
$387$	7725.54	1.01476
$388$	36.3908	0.00476150
$389$	4573.87	0.596156	0.298078	−	0.954542i	$-0.403654\pi$
$389$	4573.87	0.596156	0.298078	+	0.954542i	$0.403654\pi$
$390$	0	0
$391$	8963.77	1.15938
$392$	0	0
$393$	−6048.69	−0.776376
$394$	−8871.94	−1.13442
$395$	0	0
$396$	301.206	0.0382227
$397$	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	$-0.370429\pi$
$401$	6358.32	0.791819	0.395910	+	0.918289i	$0.370429\pi$
$402$	6753.45	0.837890
$403$	682.052	0.0843063
$404$	230.205	0.0283493
$405$	0	0
$406$	0	0
$407$	9386.35	1.14316
$408$	10410.2	1.26319
$409$	6536.39	0.790228	0.395114	−	0.918632i	$-0.370705\pi$
$409$	6536.39	0.790228	0.395114	+	0.918632i	$0.370705\pi$
$410$	0	0
$411$	−13952.8	−1.67455
$412$	141.866	0.0169641
$413$	0	0
$414$	−26637.3	−3.16220
$415$	0	0
$416$	26.8220	0.00316119
$417$	9635.91	1.13159
$418$	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
$423$	−1971.11	−0.226569
$424$	−14461.5	−1.65640
$425$	0	0
$426$	6881.43	0.782645
$427$	0	0
$428$	−129.207	−0.0145922
$429$	1333.94	0.150124
$430$	0	0
$431$	−14367.6	−1.60571	−0.802856	−	0.596173i	$-0.796687\pi$
$431$	−14367.6	−1.60571	−0.802856	+	0.596173i	$0.796687\pi$
$432$	−15696.8	−1.74818
$433$	−8399.05	−0.932176	−0.466088	−	0.884738i	$-0.654337\pi$
$433$	−8399.05	−0.932176	−0.466088	+	0.884738i	$0.654337\pi$
$434$	0	0
$435$	0	0
$436$	132.547	0.0145593
$437$	−4498.07	−0.492384
$438$	−5125.78	−0.559176
$439$	17860.8	1.94180	0.970901	−	0.239482i	$-0.0769777\pi$
$439$	17860.8	1.94180	0.970901	+	0.239482i	$0.0769777\pi$
$440$	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	$0.532515\pi$
$444$	336.996	0.0360205
$445$	0	0
$446$	13535.5	1.43705
$447$	5806.05	0.614355
$448$	0	0
$449$	−5185.68	−0.545050	−0.272525	−	0.962149i	$-0.587859\pi$
$449$	−5185.68	−0.545050	−0.272525	+	0.962149i	$0.587859\pi$
$450$	0	0
$451$	1829.54	0.191019
$452$	−114.548	−0.0119201
$453$	2185.97	0.226723
$454$	2742.81	0.283538
$455$	0	0
$456$	−5223.88	−0.536471
$457$	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
$459$	12446.4	1.26568
$460$	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	$0.419742\pi$
$468$	31.8875	0.00314957
$469$	0	0
$470$	0	0
$471$	−13955.3	−1.36524
$472$	−8863.45	−0.864350
$473$	5376.94	0.522689
$474$	−12153.6	−1.17771
$475$	0	0
$476$	0	0
$477$	−34712.8	−3.33206
$478$	−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
$486$	283.635	0.0264731
$487$	−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
$489$	−22953.7	−2.12270
$490$	0	0
$491$	1562.48	0.143613	0.0718063	−	0.997419i	$-0.477124\pi$
$491$	1562.48	0.143613	0.0718063	+	0.997419i	$0.477124\pi$
$492$	65.6855	0.00601897
$493$	−12696.1	−1.15984
$494$	293.433	0.0267250
$495$	0	0
$496$	−11214.8	−1.01524
$497$	0	0
$498$	1865.56	0.167867
$499$	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
$501$	32117.5	2.86408
$502$	−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
$507$	−19606.7	−1.71748
$508$	−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
$511$	0	0
$512$	11243.9	0.970537
$513$	−6245.65	−0.537529
$514$	−7779.81	−0.667612
$515$	0	0
$516$	193.047	0.0164698
$517$	−1371.89	−0.116703
$518$	0	0
$519$	20086.8	1.69887
$520$	0	0
$521$	−4645.42	−0.390633	−0.195316	−	0.980740i	$-0.562573\pi$
$521$	−4645.42	−0.390633	−0.195316	+	0.980740i	$0.562573\pi$
$522$	37728.4	3.16346
$523$	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
$528$	−21933.6	−1.80784
$529$	17919.4	1.47279
$530$	0	0
$531$	−21275.5	−1.73875
$532$	0	0
$533$	193.686	0.0157401
$534$	−39830.7	−3.22780
$535$	0	0
$536$	−5898.34	−0.475316
$537$	−16453.9	−1.32223
$538$	−4117.86	−0.329988
$539$	0	0
$540$	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
$543$	−21907.1	−1.73135
$544$	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
$549$	−2555.94	−0.198697
$550$	0	0
$551$	6370.95	0.492580
$552$	34941.2	2.69419
$553$	0	0
$554$	−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	$0.785854\pi$
$558$	−26425.4	−2.00479
$559$	569.235	0.0430699
$560$	0	0
$561$	17391.7	1.30887
$562$	−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	$0.855451\pi$
$564$	−49.2545	−0.00367729
$565$	0	0
$566$	6423.37	0.477022
$567$	0	0
$568$	−6010.11	−0.443977
$569$	−24157.5	−1.77985	−0.889925	−	0.456107i	$-0.849243\pi$
$569$	−24157.5	−1.77985	−0.889925	+	0.456107i	$0.849243\pi$
$570$	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
$573$	45657.4	3.32874
$574$	0	0
$575$	0	0
$576$	27008.9	1.95377
$577$	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
$579$	25213.8	1.80976
$580$	0	0
$581$	0	0
$582$	−6244.07	−0.444717
$583$	−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
$591$	27934.6	1.94429
$592$	−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	$0.748967\pi$
$594$	−25742.4	−1.77815
$595$	0	0
$596$	96.5986	0.00663898
$597$	19286.2	1.32216
$598$	−1962.70	−0.134215
$599$	22635.7	1.54402	0.772010	−	0.635610i	$-0.219251\pi$
$599$	22635.7	1.54402	0.772010	+	0.635610i	$0.219251\pi$
$600$	0	0
$601$	22553.8	1.53077	0.765383	−	0.643575i	$-0.222550\pi$
$601$	22553.8	1.53077	0.765383	+	0.643575i	$0.222550\pi$
$602$	0	0
$603$	−14158.1	−0.956159
$604$	36.3692	0.00245007
$605$	0	0
$606$	−39499.4	−2.64778
$607$	−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
$612$	415.743	0.0274598
$613$	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
$616$	0	0
$617$	−13218.9	−0.862516	−0.431258	−	0.902229i	$-0.641930\pi$
$617$	−13218.9	−0.862516	−0.431258	+	0.902229i	$0.641930\pi$
$618$	−24341.8	−1.58442
$619$	−23438.9	−1.52195	−0.760976	−	0.648780i	$-0.775280\pi$
$619$	−23438.9	−1.52195	−0.760976	+	0.648780i	$0.775280\pi$
$620$	0	0
$621$	41775.5	2.69951
$622$	−21629.8	−1.39433
$623$	0	0
$624$	−2322.02	−0.148967
$625$	0	0
$626$	−26184.9	−1.67182
$627$	−8727.23	−0.555873
$628$	−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
$633$	25504.6	1.60145
$634$	8786.72	0.550419
$635$	0	0
$636$	−867.410	−0.0540802
$637$	0	0
$638$	26258.8	1.62946
$639$	−14426.4	−0.893116
$640$	0	0
$641$	23977.0	1.47743	0.738715	−	0.674017i	$-0.235433\pi$
$641$	23977.0	1.47743	0.738715	+	0.674017i	$0.235433\pi$
$642$	22169.9	1.36289
$643$	−27698.0	−1.69876	−0.849380	−	0.527782i	$-0.823024\pi$
$643$	−27698.0	−1.69876	−0.849380	+	0.527782i	$0.823024\pi$
$644$	0	0
$645$	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	$0.608109\pi$
$648$	15965.6	0.967881
$649$	−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	$0.379480\pi$
$654$	−22743.0	−1.35982
$655$	0	0
$656$	−3184.72	−0.189547
$657$	10745.8	0.638105
$658$	0	0
$659$	−25275.6	−1.49408	−0.747040	−	0.664779i	$-0.768526\pi$
$659$	−25275.6	−1.49408	−0.747040	+	0.664779i	$0.768526\pi$
$660$	0	0
$661$	4447.92	0.261731	0.130865	−	0.991400i	$-0.458224\pi$
$661$	4447.92	0.261731	0.130865	+	0.991400i	$0.458224\pi$
$662$	−9233.67	−0.542110
$663$	1841.19	0.107852
$664$	−1629.34	−0.0952270
$665$	0	0
$666$	−38499.7	−2.23999
$667$	−42613.6	−2.47377
$668$	534.357	0.0309505
$669$	−42618.5	−2.46297
$670$	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	$0.437100\pi$
$678$	19654.5	1.11331
$679$	0	0
$680$	0	0
$681$	−8636.14	−0.485958
$682$	−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	$0.540525\pi$
$684$	−208.622	−0.0116621
$685$	0	0
$686$	0	0
$687$	−6687.56	−0.371392
$688$	−9359.77	−0.518660
$689$	−2557.72	−0.141424
$690$	0	0
$691$	−27235.2	−1.49939	−0.749694	−	0.661785i	$-0.769799\pi$
$691$	−27235.2	−1.49939	−0.749694	+	0.661785i	$0.769799\pi$
$692$	334.197	0.0183587
$693$	0	0
$694$	23534.7	1.28727
$695$	0	0
$696$	−49489.8	−2.69527
$697$	2525.24	0.137231
$698$	20479.3	1.11053
$699$	−13937.3	−0.754160
$700$	0	0
$701$	−17144.3	−0.923726	−0.461863	−	0.886951i	$-0.652819\pi$
$701$	−17144.3	−0.923726	−0.461863	+	0.886951i	$0.652819\pi$
$702$	−2725.24	−0.146521
$703$	−6501.19	−0.348787
$704$	18798.1	1.00636
$705$	0	0
$706$	−11965.7	−0.637867
$707$	0	0
$708$	−531.635	−0.0282204
$709$	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
$711$	25479.2	1.34395
$712$	34787.4	1.83106
$713$	29847.0	1.56771
$714$	0	0
$715$	0	0
$716$	−273.753	−0.0142886
$717$	24943.3	1.29920
$718$	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
$721$	0	0
$722$	17660.9	0.910347
$723$	22922.7	1.17912
$724$	−364.481	−0.0187097
$725$	0	0
$726$	−1817.03	−0.0928875
$727$	−29435.6	−1.50166	−0.750830	−	0.660496i	$-0.770346\pi$
$727$	−29435.6	−1.50166	−0.750830	+	0.660496i	$0.770346\pi$
$728$	0	0
$729$	−20127.8	−1.02260
$730$	0	0
$731$	7421.58	0.375509
$732$	−63.8683	−0.00322492
$733$	−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
$738$	−7504.16	−0.374298
$739$	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
$741$	−923.917	−0.0458042
$742$	0	0
$743$	−12271.9	−0.605940	−0.302970	−	0.953000i	$-0.597978\pi$
$743$	−12271.9	−0.605940	−0.302970	+	0.953000i	$0.597978\pi$
$744$	34663.2	1.70808
$745$	0	0
$746$	8046.24	0.394898
$747$	−3911.01	−0.191561
$748$	289.355	0.0141442
$749$	0	0
$750$	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
$753$	26363.5	1.27588
$754$	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
$759$	58374.2	2.79163
$760$	0	0
$761$	−14423.3	−0.687048	−0.343524	−	0.939144i	$-0.611621\pi$
$761$	−14423.3	−0.687048	−0.343524	+	0.939144i	$0.611621\pi$
$762$	12959.4	0.616102
$763$	0	0
$764$	759.630	0.0359718
$765$	0	0
$766$	736.681	0.0347485
$767$	−1567.63	−0.0737988
$768$	2064.01	0.0969775
$769$	−26772.8	−1.25546	−0.627731	−	0.778430i	$-0.716016\pi$
$769$	−26772.8	−1.25546	−0.627731	+	0.778430i	$0.716016\pi$
$770$	0	0
$771$	24495.9	1.14423
$772$	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	$0.277389\pi$
$774$	−22054.4	−1.02420
$775$	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
$783$	−59169.8	−2.70058
$784$	0	0
$785$	0	0
$786$	17267.4	0.783599
$787$	−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
$789$	−27210.8	−1.22779
$790$	0	0
$791$	0	0
$792$	45138.2	2.02515
$793$	−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
$801$	83502.3	3.68341
$802$	−18151.4	−0.799186
$803$	7479.06	0.328680
$804$	−353.786	−0.0155187
$805$	0	0
$806$	−1947.08	−0.0850906
$807$	12965.7	0.565568
$808$	34498.1	1.50203
$809$	−17924.8	−0.778989	−0.389495	−	0.921029i	$-0.627350\pi$
$809$	−17924.8	−0.778989	−0.389495	+	0.921029i	$0.627350\pi$
$810$	0	0
$811$	−28541.4	−1.23579	−0.617895	−	0.786261i	$-0.712014\pi$
$811$	−28541.4	−1.23579	−0.617895	+	0.786261i	$0.712014\pi$
$812$	0	0
$813$	58106.2	2.50661
$814$	−26795.6	−1.15379
$815$	0	0
$816$	−30274.1	−1.29878
$817$	−3724.19	−0.159477
$818$	−18659.7	−0.797580
$819$	0	0
$820$	0	0
$821$	18878.6	0.802517	0.401259	−	0.915965i	$-0.368573\pi$
$821$	18878.6	0.802517	0.401259	+	0.915965i	$0.368573\pi$
$822$	39831.6	1.69013
$823$	−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
$826$	0	0
$827$	−14649.8	−0.615989	−0.307994	−	0.951388i	$-0.599658\pi$
$827$	−14649.8	−0.615989	−0.307994	+	0.951388i	$0.599658\pi$
$828$	1395.42	0.0585678
$829$	34408.0	1.44154	0.720771	−	0.693173i	$-0.243788\pi$
$829$	34408.0	1.44154	0.720771	+	0.693173i	$0.243788\pi$
$830$	0	0
$831$	7879.51	0.328926
$832$	1990.07	0.0829248
$833$	0	0
$834$	−27508.0	−1.14212
$835$	0	0
$836$	−145.200	−0.00600700
$837$	41443.2	1.71145
$838$	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
$843$	56198.3	2.29605
$844$	424.335	0.0173060
$845$	0	0
$846$	5627.02	0.228677
$847$	0	0
$848$	42055.9	1.70307
$849$	−20224.9	−0.817571
$850$	0	0
$851$	43484.8	1.75163
$852$	−360.490	−0.0144955
$853$	−48456.3	−1.94503	−0.972516	−	0.232836i	$-0.925199\pi$
$853$	−48456.3	−1.94503	−0.972516	+	0.232836i	$0.925199\pi$
$854$	0	0
$855$	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	$0.242798\pi$
$858$	−3808.06	−0.151521
$859$	−28067.2	−1.11483	−0.557415	−	0.830234i	$-0.688207\pi$
$859$	−28067.2	−1.11483	−0.557415	+	0.830234i	$0.688207\pi$
$860$	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	$0.447462\pi$
$864$	1629.77	0.0641736
$865$	0	0
$866$	23977.1	0.940849
$867$	−20155.9	−0.789538
$868$	0	0
$869$	17733.4	0.692251
$870$	0	0
$871$	−1043.20	−0.0405828
$872$	19863.3	0.771395
$873$	13090.3	0.507489
$874$	12840.8	0.496965
$875$	0	0
$876$	268.519	0.0103566
$877$	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
$879$	53193.1	2.04114
$880$	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	$0.334683\pi$
$888$	50501.5	1.90847
$889$	0	0
$890$	0	0
$891$	26672.8	1.00289
$892$	−709.069	−0.0266159
$893$	950.199	0.0356072
$894$	−16574.8	−0.620070
$895$	0	0
$896$	0	0
$897$	6179.84	0.230032
$898$	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
$906$	−6240.37	−0.228833
$907$	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
$909$	82807.8	3.02152
$910$	0	0
$911$	−21189.3	−0.770619	−0.385310	−	0.922787i	$-0.625905\pi$
$911$	−21189.3	−0.770619	−0.385310	+	0.922787i	$0.625905\pi$
$912$	15191.7	0.551587
$913$	−2722.05	−0.0986710
$914$	−31969.7	−1.15696
$915$	0	0
$916$	−111.265	−0.00401342
$917$	0	0
$918$	−35531.1	−1.27745
$919$	−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
$921$	−85018.2	−3.04174
$922$	49303.4	1.76109
$923$	−1062.97	−0.0379070
$924$	0	0
$925$	0	0
$926$	1100.96	0.0390710
$927$	51031.0	1.80807
$928$	−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
$931$	0	0
$932$	−231.883	−0.00814978
$933$	68104.5	2.38975
$934$	−14374.6	−0.503589
$935$	0	0
$936$	4778.60	0.166873
$937$	28824.7	1.00498	0.502488	−	0.864584i	$-0.332418\pi$
$937$	28824.7	1.00498	0.502488	+	0.864584i	$0.332418\pi$
$938$	0	0
$939$	82446.9	2.86534
$940$	0	0
$941$	−42172.5	−1.46098	−0.730492	−	0.682922i	$-0.760709\pi$
$941$	−42172.5	−1.46098	−0.730492	+	0.682922i	$0.760709\pi$
$942$	39838.9	1.37794
$943$	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	$0.548942\pi$
$948$	636.679	0.0218126
$949$	791.778	0.0270835
$950$	0	0
$951$	−27666.3	−0.943366
$952$	0	0
$953$	40420.6	1.37392	0.686962	−	0.726693i	$-0.258944\pi$
$953$	40420.6	1.37392	0.686962	+	0.726693i	$0.258944\pi$
$954$	99096.2	3.36306
$955$	0	0
$956$	414.996	0.0140397
$957$	−82679.8	−2.79274
$958$	−24784.8	−0.835868
$959$	0	0
$960$	0	0
$961$	−181.408	−0.00608935
$962$	−2836.74	−0.0950730
$963$	−46477.6	−1.55526
$964$	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
$969$	−12045.8	−0.399348
$970$	0	0
$971$	59339.0	1.96115	0.980577	−	0.196136i	$-0.0628393\pi$
$971$	59339.0	1.96115	0.980577	+	0.196136i	$0.0628393\pi$
$972$	−14.8585	−0.000490315 0
$973$	0	0
$974$	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	$0.536731\pi$
$978$	65526.8	2.14245
$979$	58117.3	1.89728
$980$	0	0
$981$	47679.1	1.55176
$982$	−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	$0.508799\pi$
$984$	9843.50	0.318902
$985$	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
$993$	29073.6	0.929126
$994$	0	0
$995$	0	0
$996$	−97.7290	−0.00310910
$997$	−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
$999$	60379.4	1.91223

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.4.a.be.1.2		5
5.2	odd	4		245.4.b.d.99.3		10
5.3	odd	4		245.4.b.d.99.8		10
5.4	even	2		1225.4.a.bh.1.4		5
7.6	odd	2		175.4.a.i.1.2		5
21.20	even	2		1575.4.a.bq.1.4		5
35.2	odd	12		245.4.j.f.214.3		20
35.3	even	12		245.4.j.e.79.3		20
35.12	even	12		245.4.j.e.214.3		20
35.13	even	4		35.4.b.a.29.8	yes	10
35.17	even	12		245.4.j.e.79.8		20
35.18	odd	12		245.4.j.f.79.3		20
35.23	odd	12		245.4.j.f.214.8		20
35.27	even	4		35.4.b.a.29.3	✓	10
35.32	odd	12		245.4.j.f.79.8		20
35.33	even	12		245.4.j.e.214.8		20
35.34	odd	2		175.4.a.j.1.4		5
105.62	odd	4		315.4.d.c.64.8		10
105.83	odd	4		315.4.d.c.64.3		10
105.104	even	2		1575.4.a.bn.1.2		5
140.27	odd	4		560.4.g.f.449.1		10
140.83	odd	4		560.4.g.f.449.10		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.b.a.29.3	✓	10	35.27	even	4
35.4.b.a.29.8	yes	10	35.13	even	4
175.4.a.i.1.2		5	7.6	odd	2
175.4.a.j.1.4		5	35.34	odd	2
245.4.b.d.99.3		10	5.2	odd	4
245.4.b.d.99.8		10	5.3	odd	4
245.4.j.e.79.3		20	35.3	even	12
245.4.j.e.79.8		20	35.17	even	12
245.4.j.e.214.3		20	35.12	even	12
245.4.j.e.214.8		20	35.33	even	12
245.4.j.f.79.3		20	35.18	odd	12
245.4.j.f.79.8		20	35.32	odd	12
245.4.j.f.214.3		20	35.2	odd	12
245.4.j.f.214.8		20	35.23	odd	12
315.4.d.c.64.3		10	105.83	odd	4
315.4.d.c.64.8		10	105.62	odd	4
560.4.g.f.449.1		10	140.27	odd	4
560.4.g.f.449.10		10	140.83	odd	4
1225.4.a.be.1.2		5	1.1	even	1	trivial
1225.4.a.bh.1.4		5	5.4	even	2
1575.4.a.bn.1.2		5	105.104	even	2
1575.4.a.bq.1.4		5	21.20	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:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q-2.85474 q^{2} +8.98858 q^{3} +0.149548 q^{4} -25.6601 q^{6} +22.4110 q^{8} +53.7945 q^{9} +37.4408 q^{11} +1.34423 q^{12} +3.96370 q^{13} -65.1740 q^{16} +51.6780 q^{17} -153.569 q^{18} -25.9323 q^{19} -106.884 q^{22} +173.454 q^{23} +201.443 q^{24} -11.3154 q^{26} +240.845 q^{27} -245.676 q^{29} +172.074 q^{31} +6.76690 q^{32} +336.539 q^{33} -147.527 q^{34} +8.04488 q^{36} +250.699 q^{37} +74.0300 q^{38} +35.6281 q^{39} +48.8649 q^{41} +143.612 q^{43} +5.59920 q^{44} -495.167 q^{46} -36.6415 q^{47} -585.822 q^{48} +464.511 q^{51} +0.592765 q^{52} -645.286 q^{53} -687.549 q^{54} -233.094 q^{57} +701.343 q^{58} -395.495 q^{59} -47.5130 q^{61} -491.228 q^{62} +502.074 q^{64} -960.733 q^{66} -263.189 q^{67} +7.72835 q^{68} +1559.11 q^{69} -268.177 q^{71} +1205.59 q^{72} +199.757 q^{73} -715.680 q^{74} -3.87813 q^{76} -101.709 q^{78} +473.640 q^{79} +712.399 q^{81} -139.497 q^{82} -72.7028 q^{83} -409.975 q^{86} -2208.28 q^{87} +839.086 q^{88} +1552.25 q^{89} +25.9398 q^{92} +1546.70 q^{93} +104.602 q^{94} +60.8248 q^{96} +243.338 q^{97} +2014.11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{2} + 10 q^{3} + 18 q^{4} - 6 q^{6} - 42 q^{8} + 23 q^{9} + 42 q^{11} + 136 q^{12} + 34 q^{13} + 74 q^{16} + 238 q^{17} + 2 q^{18} + 36 q^{19} - 358 q^{22} - 152 q^{23} + 36 q^{24} + 310 q^{26}+ \cdots + 2652 q^{99}+O(q^{100}) \) 5 * q - 4 * q^2 + 10 * q^3 + 18 * q^4 - 6 * q^6 - 42 * q^8 + 23 * q^9 + 42 * q^11 + 136 * q^12 + 34 * q^13 + 74 * q^16 + 238 * q^17 + 2 * q^18 + 36 * q^19 - 358 * q^22 - 152 * q^23 + 36 * q^24 + 310 * q^26 + 334 * q^27 - 44 * q^29 - 60 * q^31 - 710 * q^32 + 426 * q^33 + 482 * q^34 - 210 * q^36 - 312 * q^37 + 280 * q^38 - 106 * q^39 + 426 * q^41 - 304 * q^43 + 712 * q^44 + 88 * q^46 + 370 * q^47 - 696 * q^48 + 638 * q^51 - 1156 * q^52 - 976 * q^53 - 498 * q^54 + 588 * q^57 + 2722 * q^58 + 432 * q^59 + 442 * q^61 - 956 * q^62 + 1362 * q^64 - 574 * q^66 + 804 * q^67 - 420 * q^68 + 2404 * q^69 + 440 * q^71 + 3150 * q^72 + 564 * q^73 - 1512 * q^74 + 1336 * q^76 + 2742 * q^78 + 1790 * q^79 - 151 * q^81 - 3480 * q^82 + 1656 * q^83 + 1216 * q^86 - 1674 * q^87 + 1092 * q^88 - 746 * q^89 - 572 * q^92 - 676 * q^93 + 826 * q^94 - 2040 * q^96 + 518 * q^97 + 2652 * q^99

Introduction

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