LMFDB - Embedded newform 1225.4.a.bh.1.4

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			1.4
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	$-0.331621\pi$
$2$	2.85474	1.00930	0.504652	+	0.863323i	$0.331621\pi$
$3$	−8.98858	−1.72985	−0.864926	−	0.501899i	$-0.832635\pi$
$3$	−8.98858	−1.72985	−0.864926	+	0.501899i	$0.832635\pi$
$4$	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	$-0.513463\pi$
$13$	−3.96370	−0.0845641	−0.0422821	+	0.999106i	$0.513463\pi$
$14$	0	0
$15$	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	$0.620176\pi$
$18$	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	$-0.787983\pi$
$23$	−173.454	−1.57251	−0.786255	+	0.617902i	$0.787983\pi$
$24$	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	$-0.688030\pi$
$37$	−250.699	−1.11391	−0.556954	+	0.830543i	$0.688030\pi$
$38$	−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	$-0.581963\pi$
$43$	−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	$0.481892\pi$
$48$	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	$-0.184774\pi$
$53$	645.286	1.67239	0.836196	+	0.548430i	$0.184774\pi$
$54$	−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	$-0.422868\pi$
$67$	263.189	0.479906	0.239953	+	0.970785i	$0.422868\pi$
$68$	−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	$-0.551193\pi$
$73$	−199.757	−0.320271	−0.160136	+	0.987095i	$0.551193\pi$
$74$	−715.680	−1.12427
$75$	0	0
$76$	−3.87813	−0.00585331
$77$	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	$-0.484692\pi$
$83$	72.7028	0.0961466	0.0480733	+	0.998844i	$0.484692\pi$
$84$	0	0
$85$	0	0
$86$	−409.975	−0.514055
$87$	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	$-0.540649\pi$
$97$	−243.338	−0.254714	−0.127357	+	0.991857i	$0.540649\pi$
$98$	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	$-0.649912\pi$
$103$	−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	$0.372371\pi$
$108$	−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	$-0.396711\pi$
$113$	765.957	0.637657	0.318828	+	0.947812i	$0.396711\pi$
$114$	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	$-0.443543\pi$
$127$	505.042	0.352876	0.176438	+	0.984312i	$0.443543\pi$
$128$	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	$-0.339178\pi$
$137$	1552.28	0.968032	0.484016	+	0.875059i	$0.339178\pi$
$138$	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	$-0.370879\pi$
$157$	1552.56	0.789223	0.394611	+	0.918848i	$0.370879\pi$
$158$	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	$-0.289741\pi$
$163$	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
$171$	−1395.01	−0.623856
$172$	−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	$0.663385\pi$
$174$	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	$-0.675223\pi$
$193$	−2805.09	−1.04619	−0.523095	+	0.852274i	$0.675223\pi$
$194$	−694.667	−0.257084
$195$	0	0
$196$	0	0
$197$	−3107.79	−1.12396	−0.561982	−	0.827149i	$-0.689961\pi$
$197$	−3107.79	−1.12396	−0.561982	+	0.827149i	$0.689961\pi$
$198$	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	$-0.247834\pi$
$223$	4741.40	1.42380	0.711901	+	0.702280i	$0.247834\pi$
$224$	0	0
$225$	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	$0.455141\pi$
$228$	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	$-0.430052\pi$
$233$	1550.56	0.435968	0.217984	+	0.975952i	$0.430052\pi$
$234$	−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	$-0.607295\pi$
$257$	−2725.22	−0.661459	−0.330729	+	0.943726i	$0.607295\pi$
$258$	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	$-0.384520\pi$
$263$	3027.26	0.709767	0.354884	+	0.934910i	$0.384520\pi$
$264$	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	$-0.530309\pi$
$277$	−876.614	−0.190147	−0.0950733	+	0.995470i	$0.530309\pi$
$278$	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	$-0.424061\pi$
$283$	2250.07	0.472625	0.236312	+	0.971677i	$0.424061\pi$
$284$	−40.1054	−0.00837963
$285$	0	0
$286$	−423.655	−0.0875919
$287$	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	$-0.700862\pi$
$293$	−5917.86	−1.17995	−0.589975	+	0.807422i	$0.700862\pi$
$294$	0	0
$295$	0	0
$296$	5618.41	1.10325
$297$	−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	$-0.158084\pi$
$307$	9458.47	1.75838	0.879191	+	0.476469i	$0.158084\pi$
$308$	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	$-0.810637\pi$
$313$	−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	$0.412092\pi$
$318$	−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	$-0.401234\pi$
$337$	3777.84	0.610658	0.305329	+	0.952247i	$0.401234\pi$
$338$	−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	$-0.279885\pi$
$347$	8244.08	1.27540	0.637702	+	0.770283i	$0.279885\pi$
$348$	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	$-0.602338\pi$
$353$	−4191.51	−0.631987	−0.315994	+	0.948761i	$0.602338\pi$
$354$	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	$-0.460891\pi$
$367$	1723.30	0.245110	0.122555	+	0.992462i	$0.460891\pi$
$368$	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	$-0.437325\pi$
$373$	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
$381$	−4539.61	−0.610423
$382$	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	$0.494520\pi$
$384$	−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	$-0.573581\pi$
$397$	−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	$-0.370429\pi$
$401$	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	$-0.345663\pi$
$433$	8399.05	0.932176	0.466088	+	0.884738i	$0.345663\pi$
$434$	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	$-0.467485\pi$
$443$	1901.57	0.203942	0.101971	+	0.994787i	$0.467485\pi$
$444$	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	$-0.694278\pi$
$457$	−11198.8	−1.14630	−0.573149	+	0.819451i	$0.694278\pi$
$458$	−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	$-0.493839\pi$
$463$	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	$0.580258\pi$
$468$	−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	$-0.486809\pi$
$487$	890.476	0.0828569	0.0414284	+	0.999141i	$0.486809\pi$
$488$	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	$-0.501023\pi$
$503$	−72.5340	−0.00642969	−0.00321484	+	0.999995i	$0.501023\pi$
$504$	0	0
$505$	0	0
$506$	−18539.4	−1.62881
$507$	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	$-0.619684\pi$
$523$	−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
$528$	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	$-0.572492\pi$
$547$	−5776.83	−0.451553	−0.225776	+	0.974179i	$0.572492\pi$
$548$	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	$-0.214146\pi$
$557$	20562.6	1.56421	0.782106	+	0.623145i	$0.214146\pi$
$558$	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	$-0.144549\pi$
$563$	24009.5	1.79730	0.898650	+	0.438666i	$0.144549\pi$
$564$	−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	$-0.696660\pi$
$577$	−16057.2	−1.15853	−0.579265	+	0.815139i	$0.696660\pi$
$578$	−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	$-0.597838\pi$
$587$	−8605.63	−0.605098	−0.302549	+	0.953134i	$0.597838\pi$
$588$	0	0
$589$	−4462.28	−0.312165
$590$	0	0
$591$	27934.6	1.94429
$592$	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	$0.251033\pi$
$594$	−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	$-0.300608\pi$
$607$	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
$612$	−415.743	−0.0274598
$613$	−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
$616$	0	0
$617$	13218.9	0.862516	0.431258	−	0.902229i	$-0.358070\pi$
$617$	13218.9	0.862516	0.431258	+	0.902229i	$0.358070\pi$
$618$	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	$-0.176976\pi$
$643$	27698.0	1.69876	0.849380	+	0.527782i	$0.176976\pi$
$644$	0	0
$645$	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	$0.391891\pi$
$648$	−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	$-0.620520\pi$
$653$	−12336.2	−0.739285	−0.369642	+	0.929174i	$0.620520\pi$
$654$	−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	$-0.164488\pi$
$673$	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	$0.562900\pi$
$678$	−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	$-0.459475\pi$
$683$	4532.72	0.253938	0.126969	+	0.991907i	$0.459475\pi$
$684$	−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	$-0.229654\pi$
$727$	29435.6	1.50166	0.750830	+	0.660496i	$0.229654\pi$
$728$	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	$-0.446922\pi$
$733$	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
$738$	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	$-0.402022\pi$
$743$	12271.9	0.605940	0.302970	+	0.953000i	$0.402022\pi$
$744$	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	$-0.411427\pi$
$757$	11442.1	0.549368	0.274684	+	0.961535i	$0.411427\pi$
$758$	−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	$-0.722611\pi$
$773$	−27669.3	−1.28745	−0.643724	+	0.765258i	$0.722611\pi$
$774$	−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	$-0.420348\pi$
$787$	10934.5	0.495263	0.247632	+	0.968854i	$0.420348\pi$
$788$	−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	$-0.248525\pi$
$797$	31967.3	1.42075	0.710376	+	0.703823i	$0.248525\pi$
$798$	0	0
$799$	−1893.56	−0.0838415
$800$	0	0
$801$	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	$-0.0521078\pi$
$823$	46589.1	1.97326	0.986631	+	0.162971i	$0.0521078\pi$
$824$	21259.7	0.898807
$825$	0	0
$826$	0	0
$827$	14649.8	0.615989	0.307994	−	0.951388i	$-0.400342\pi$
$827$	14649.8	0.615989	0.307994	+	0.951388i	$0.400342\pi$
$828$	−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	$-0.0748007\pi$
$853$	48456.3	1.94503	0.972516	+	0.232836i	$0.0748007\pi$
$854$	0	0
$855$	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	$0.757202\pi$
$858$	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	$-0.552538\pi$
$863$	−8330.92	−0.328607	−0.164303	+	0.986410i	$0.552538\pi$
$864$	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	$-0.602698\pi$
$877$	−16469.6	−0.634137	−0.317068	+	0.948403i	$0.602698\pi$
$878$	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	$-0.413562\pi$
$883$	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	$0.665317\pi$
$888$	−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	$-0.590798\pi$
$907$	−15373.0	−0.562793	−0.281396	+	0.959592i	$0.590798\pi$
$908$	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	$-0.667582\pi$
$937$	−28824.7	−1.00498	−0.502488	+	0.864584i	$0.667582\pi$
$938$	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	$-0.451058\pi$
$947$	8926.33	0.306301	0.153150	+	0.988203i	$0.451058\pi$
$948$	−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	$-0.741056\pi$
$953$	−40420.6	−1.37392	−0.686962	+	0.726693i	$0.741056\pi$
$954$	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	$-0.690704\pi$
$967$	−33914.1	−1.12782	−0.563910	+	0.825836i	$0.690704\pi$
$968$	−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	$-0.463269\pi$
$977$	7032.14	0.230274	0.115137	+	0.993350i	$0.463269\pi$
$978$	−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	$-0.491201\pi$
$983$	1703.63	0.0552772	0.0276386	+	0.999618i	$0.491201\pi$
$984$	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	$-0.415022\pi$
$997$	16609.6	0.527615	0.263808	+	0.964575i	$0.415022\pi$
$998$	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.bh.1.4		5
5.2	odd	4		245.4.b.d.99.8		10
5.3	odd	4		245.4.b.d.99.3		10
5.4	even	2		1225.4.a.be.1.2		5
7.6	odd	2		175.4.a.j.1.4		5
21.20	even	2		1575.4.a.bn.1.2		5
35.2	odd	12		245.4.j.f.214.8		20
35.3	even	12		245.4.j.e.79.8		20
35.12	even	12		245.4.j.e.214.8		20
35.13	even	4		35.4.b.a.29.3	✓	10
35.17	even	12		245.4.j.e.79.3		20
35.18	odd	12		245.4.j.f.79.8		20
35.23	odd	12		245.4.j.f.214.3		20
35.27	even	4		35.4.b.a.29.8	yes	10
35.32	odd	12		245.4.j.f.79.3		20
35.33	even	12		245.4.j.e.214.3		20
35.34	odd	2		175.4.a.i.1.2		5
105.62	odd	4		315.4.d.c.64.3		10
105.83	odd	4		315.4.d.c.64.8		10
105.104	even	2		1575.4.a.bq.1.4		5
140.27	odd	4		560.4.g.f.449.10		10
140.83	odd	4		560.4.g.f.449.1		10

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