LMFDB - Embedded newform 245.4.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [245,4,Mod(1,245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, 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("245.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 245 = 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	245.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:	$14.4554679514$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	245.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.58579 q^{2} +5.24264 q^{3} -5.48528 q^{4} -5.00000 q^{5} -8.31371 q^{6} +21.3848 q^{8} +0.485281 q^{9} +7.92893 q^{10} +28.1421 q^{11} -28.7574 q^{12} +3.85786 q^{13} -26.2132 q^{15} +9.97056 q^{16} -38.3675 q^{17} -0.769553 q^{18} -116.770 q^{19} +27.4264 q^{20} -44.6274 q^{22} -176.463 q^{23} +112.113 q^{24} +25.0000 q^{25} -6.11775 q^{26} -139.007 q^{27} -209.853 q^{29} +41.5685 q^{30} +207.421 q^{31} -186.889 q^{32} +147.539 q^{33} +60.8427 q^{34} -2.66190 q^{36} -15.6670 q^{37} +185.172 q^{38} +20.2254 q^{39} -106.924 q^{40} +10.5736 q^{41} -325.929 q^{43} -154.368 q^{44} -2.42641 q^{45} +279.833 q^{46} +188.510 q^{47} +52.2721 q^{48} -39.6447 q^{50} -201.147 q^{51} -21.1615 q^{52} -275.182 q^{53} +220.436 q^{54} -140.711 q^{55} -612.181 q^{57} +332.782 q^{58} -43.8192 q^{59} +143.787 q^{60} -855.102 q^{61} -328.926 q^{62} +216.602 q^{64} -19.2893 q^{65} -233.966 q^{66} -545.419 q^{67} +210.457 q^{68} -925.132 q^{69} +1026.97 q^{71} +10.3776 q^{72} +252.877 q^{73} +24.8444 q^{74} +131.066 q^{75} +640.514 q^{76} -32.0732 q^{78} -922.642 q^{79} -49.8528 q^{80} -741.867 q^{81} -16.7675 q^{82} +960.071 q^{83} +191.838 q^{85} +516.854 q^{86} -1100.18 q^{87} +601.813 q^{88} +133.205 q^{89} +3.84776 q^{90} +967.949 q^{92} +1087.44 q^{93} -298.936 q^{94} +583.848 q^{95} -979.794 q^{96} +1021.11 q^{97} +13.6569 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{2} + 2 q^{3} + 6 q^{4} - 10 q^{5} + 6 q^{6} + 6 q^{8} - 16 q^{9} + 30 q^{10} + 28 q^{11} - 66 q^{12} + 36 q^{13} - 10 q^{15} - 14 q^{16} + 76 q^{17} + 72 q^{18} - 160 q^{19} - 30 q^{20} - 44 q^{22}+ \cdots + 16 q^{99}+O(q^{100}) $ 2 * q - 6 * q^2 + 2 * q^3 + 6 * q^4 - 10 * q^5 + 6 * q^6 + 6 * q^8 - 16 * q^9 + 30 * q^10 + 28 * q^11 - 66 * q^12 + 36 * q^13 - 10 * q^15 - 14 * q^16 + 76 * q^17 + 72 * q^18 - 160 * q^19 - 30 * q^20 - 44 * q^22 - 22 * q^23 + 162 * q^24 + 50 * q^25 - 148 * q^26 + 2 * q^27 - 250 * q^29 - 30 * q^30 + 132 * q^31 + 42 * q^32 + 148 * q^33 - 444 * q^34 - 192 * q^36 - 416 * q^37 + 376 * q^38 - 84 * q^39 - 30 * q^40 + 106 * q^41 - 666 * q^43 - 156 * q^44 + 80 * q^45 - 402 * q^46 + 196 * q^47 + 130 * q^48 - 150 * q^50 - 572 * q^51 + 348 * q^52 - 952 * q^53 - 402 * q^54 - 140 * q^55 - 472 * q^57 + 510 * q^58 - 840 * q^59 + 330 * q^60 - 98 * q^61 + 4 * q^62 - 602 * q^64 - 180 * q^65 - 236 * q^66 - 1286 * q^67 + 1524 * q^68 - 1426 * q^69 + 1064 * q^71 + 264 * q^72 + 172 * q^73 + 1792 * q^74 + 50 * q^75 + 144 * q^76 + 428 * q^78 - 1240 * q^79 + 70 * q^80 - 754 * q^81 - 438 * q^82 + 1906 * q^83 - 380 * q^85 + 2018 * q^86 - 970 * q^87 + 604 * q^88 - 650 * q^89 - 360 * q^90 + 2742 * q^92 + 1332 * q^93 - 332 * q^94 + 800 * q^95 - 1722 * q^96 + 628 * q^97 + 16 * 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 $

$2$	−1.58579	−0.560660	−0.280330	−	0.959904i	$-0.590444\pi$
$2$	−1.58579	−0.560660	−0.280330	+	0.959904i	$0.590444\pi$
$3$	5.24264	1.00895	0.504473	−	0.863427i	$-0.331687\pi$
$3$	5.24264	1.00895	0.504473	+	0.863427i	$0.331687\pi$
$4$	−5.48528	−0.685660
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−8.31371	−0.565676
$7$	0	0
$7$	0	0
$8$	21.3848	0.945083
$9$	0.485281	0.0179734
$10$	7.92893	0.250735
$11$	28.1421	0.771379	0.385690	−	0.922629i	$-0.373964\pi$
$11$	28.1421	0.771379	0.385690	+	0.922629i	$0.373964\pi$
$12$	−28.7574	−0.691795
$13$	3.85786	0.0823061	0.0411530	−	0.999153i	$-0.486897\pi$
$13$	3.85786	0.0823061	0.0411530	+	0.999153i	$0.486897\pi$
$14$	0	0
$15$	−26.2132	−0.451215
$16$	9.97056	0.155790
$17$	−38.3675	−0.547382	−0.273691	−	0.961818i	$-0.588245\pi$
$17$	−38.3675	−0.547382	−0.273691	+	0.961818i	$0.588245\pi$
$18$	−0.769553	−0.0100770
$19$	−116.770	−1.40994	−0.704968	−	0.709239i	$-0.749039\pi$
$19$	−116.770	−1.40994	−0.704968	+	0.709239i	$0.749039\pi$
$20$	27.4264	0.306637
$21$	0	0
$22$	−44.6274	−0.432482
$23$	−176.463	−1.59979	−0.799893	−	0.600143i	$-0.795110\pi$
$23$	−176.463	−1.59979	−0.799893	+	0.600143i	$0.795110\pi$
$24$	112.113	0.953538
$25$	25.0000	0.200000
$26$	−6.11775	−0.0461457
$27$	−139.007	−0.990812
$28$	0	0
$29$	−209.853	−1.34375	−0.671874	−	0.740666i	$-0.734510\pi$
$29$	−209.853	−1.34375	−0.671874	+	0.740666i	$0.734510\pi$
$30$	41.5685	0.252978
$31$	207.421	1.20174	0.600871	−	0.799346i	$-0.294821\pi$
$31$	207.421	1.20174	0.600871	+	0.799346i	$0.294821\pi$
$32$	−186.889	−1.03243
$33$	147.539	0.778281
$34$	60.8427	0.306895
$35$	0	0
$36$	−2.66190	−0.0123236
$37$	−15.6670	−0.0696117	−0.0348058	−	0.999394i	$-0.511081\pi$
$37$	−15.6670	−0.0696117	−0.0348058	+	0.999394i	$0.511081\pi$
$38$	185.172	0.790495
$39$	20.2254	0.0830424
$40$	−106.924	−0.422654
$41$	10.5736	0.0402760	0.0201380	−	0.999797i	$-0.493589\pi$
$41$	10.5736	0.0402760	0.0201380	+	0.999797i	$0.493589\pi$
$42$	0	0
$43$	−325.929	−1.15590	−0.577950	−	0.816072i	$-0.696147\pi$
$43$	−325.929	−1.15590	−0.577950	+	0.816072i	$0.696147\pi$
$44$	−154.368	−0.528904
$45$	−2.42641	−0.00803794
$46$	279.833	0.896936
$47$	188.510	0.585042	0.292521	−	0.956259i	$-0.405506\pi$
$47$	188.510	0.585042	0.292521	+	0.956259i	$0.405506\pi$
$48$	52.2721	0.157184
$49$	0	0
$50$	−39.6447	−0.112132
$51$	−201.147	−0.552279
$52$	−21.1615	−0.0564340
$53$	−275.182	−0.713191	−0.356595	−	0.934259i	$-0.616063\pi$
$53$	−275.182	−0.713191	−0.356595	+	0.934259i	$0.616063\pi$
$54$	220.436	0.555509
$55$	−140.711	−0.344971
$56$	0	0
$57$	−612.181	−1.42255
$58$	332.782	0.753386
$59$	−43.8192	−0.0966911	−0.0483455	−	0.998831i	$-0.515395\pi$
$59$	−43.8192	−0.0966911	−0.0483455	+	0.998831i	$0.515395\pi$
$60$	143.787	0.309380
$61$	−855.102	−1.79483	−0.897414	−	0.441189i	$-0.854557\pi$
$61$	−855.102	−1.79483	−0.897414	+	0.441189i	$0.854557\pi$
$62$	−328.926	−0.673768
$63$	0	0
$64$	216.602	0.423051
$65$	−19.2893	−0.0368084
$66$	−233.966	−0.436351
$67$	−545.419	−0.994531	−0.497265	−	0.867598i	$-0.665662\pi$
$67$	−545.419	−0.994531	−0.497265	+	0.867598i	$0.665662\pi$
$68$	210.457	0.375318
$69$	−925.132	−1.61410
$70$	0	0
$71$	1026.97	1.71661	0.858306	−	0.513138i	$-0.171517\pi$
$71$	1026.97	1.71661	0.858306	+	0.513138i	$0.171517\pi$
$72$	10.3776	0.0169863
$73$	252.877	0.405439	0.202719	−	0.979237i	$-0.435022\pi$
$73$	252.877	0.405439	0.202719	+	0.979237i	$0.435022\pi$
$74$	24.8444	0.0390285
$75$	131.066	0.201789
$76$	640.514	0.966737
$77$	0	0
$78$	−32.0732	−0.0465586
$79$	−922.642	−1.31399	−0.656996	−	0.753894i	$-0.728173\pi$
$79$	−922.642	−1.31399	−0.656996	+	0.753894i	$0.728173\pi$
$80$	−49.8528	−0.0696714
$81$	−741.867	−1.01765
$82$	−16.7675	−0.0225812
$83$	960.071	1.26966	0.634828	−	0.772653i	$-0.281071\pi$
$83$	960.071	1.26966	0.634828	+	0.772653i	$0.281071\pi$
$84$	0	0
$85$	191.838	0.244797
$86$	516.854	0.648067
$87$	−1100.18	−1.35577
$88$	601.813	0.729017
$89$	133.205	0.158649	0.0793243	−	0.996849i	$-0.474724\pi$
$89$	133.205	0.158649	0.0793243	+	0.996849i	$0.474724\pi$
$90$	3.84776	0.00450655
$91$	0	0
$92$	967.949	1.09691
$93$	1087.44	1.21249
$94$	−298.936	−0.328010
$95$	583.848	0.630542
$96$	−979.794	−1.04166
$97$	1021.11	1.06884	0.534421	−	0.845218i	$-0.320530\pi$
$97$	1021.11	1.06884	0.534421	+	0.845218i	$0.320530\pi$
$98$	0	0
$99$	13.6569	0.0138643
$100$	−137.132	−0.137132
$101$	−963.558	−0.949284	−0.474642	−	0.880179i	$-0.657422\pi$
$101$	−963.558	−0.949284	−0.474642	+	0.880179i	$0.657422\pi$
$102$	318.976	0.309641
$103$	1798.30	1.72031	0.860155	−	0.510034i	$-0.170367\pi$
$103$	1798.30	1.72031	0.860155	+	0.510034i	$0.170367\pi$
$104$	82.4996	0.0777860
$105$	0	0
$106$	436.379	0.399858
$107$	279.003	0.252077	0.126038	−	0.992025i	$-0.459774\pi$
$107$	279.003	0.252077	0.126038	+	0.992025i	$0.459774\pi$
$108$	762.493	0.679361
$109$	781.073	0.686360	0.343180	−	0.939270i	$-0.388496\pi$
$109$	781.073	0.686360	0.343180	+	0.939270i	$0.388496\pi$
$110$	223.137	0.193412
$111$	−82.1362	−0.0702345
$112$	0	0
$113$	−587.239	−0.488874	−0.244437	−	0.969665i	$-0.578603\pi$
$113$	−587.239	−0.488874	−0.244437	+	0.969665i	$0.578603\pi$
$114$	970.788	0.797567
$115$	882.315	0.715446
$116$	1151.10	0.921354
$117$	1.87215	0.00147932
$118$	69.4879	0.0542108
$119$	0	0
$120$	−560.563	−0.426435
$121$	−539.020	−0.404974
$122$	1356.01	1.00629
$123$	55.4335	0.0406364
$124$	−1137.76	−0.823986
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1559.62	1.08972	0.544860	−	0.838527i	$-0.316583\pi$
$127$	1559.62	1.08972	0.544860	+	0.838527i	$0.316583\pi$
$128$	1151.63	0.795240
$129$	−1708.73	−1.16624
$130$	30.5887	0.0206370
$131$	1653.65	1.10290	0.551449	−	0.834209i	$-0.314075\pi$
$131$	1653.65	1.10290	0.551449	+	0.834209i	$0.314075\pi$
$132$	−809.294	−0.533636
$133$	0	0
$134$	864.918	0.557594
$135$	695.036	0.443105
$136$	−820.481	−0.517321
$137$	−296.658	−0.185001	−0.0925007	−	0.995713i	$-0.529486\pi$
$137$	−296.658	−0.185001	−0.0925007	+	0.995713i	$0.529486\pi$
$138$	1467.06	0.904961
$139$	−237.868	−0.145149	−0.0725745	−	0.997363i	$-0.523121\pi$
$139$	−237.868	−0.145149	−0.0725745	+	0.997363i	$0.523121\pi$
$140$	0	0
$141$	988.288	0.590276
$142$	−1628.56	−0.962436
$143$	108.569	0.0634892
$144$	4.83853	0.00280007
$145$	1049.26	0.600942
$146$	−401.009	−0.227313
$147$	0	0
$148$	85.9377	0.0477299
$149$	2737.11	1.50492	0.752459	−	0.658639i	$-0.228868\pi$
$149$	2737.11	1.50492	0.752459	+	0.658639i	$0.228868\pi$
$150$	−207.843	−0.113135
$151$	−3563.65	−1.92057	−0.960283	−	0.279030i	$-0.909987\pi$
$151$	−3563.65	−1.92057	−0.960283	+	0.279030i	$0.909987\pi$
$152$	−2497.09	−1.33251
$153$	−18.6190	−0.00983831
$154$	0	0
$155$	−1037.11	−0.537435
$156$	−110.942	−0.0569389
$157$	−1249.08	−0.634951	−0.317475	−	0.948266i	$-0.602835\pi$
$157$	−1249.08	−0.634951	−0.317475	+	0.948266i	$0.602835\pi$
$158$	1463.11	0.736703
$159$	−1442.68	−0.719571
$160$	934.447	0.461716
$161$	0	0
$162$	1176.44	0.570556
$163$	−1964.64	−0.944062	−0.472031	−	0.881582i	$-0.656479\pi$
$163$	−1964.64	−0.944062	−0.472031	+	0.881582i	$0.656479\pi$
$164$	−57.9991	−0.0276157
$165$	−737.696	−0.348058
$166$	−1522.47	−0.711846
$167$	939.402	0.435288	0.217644	−	0.976028i	$-0.430163\pi$
$167$	939.402	0.435288	0.217644	+	0.976028i	$0.430163\pi$
$168$	0	0
$169$	−2182.12	−0.993226
$170$	−304.214	−0.137248
$171$	−56.6661	−0.0253413
$172$	1787.81	0.792555
$173$	2702.43	1.18764	0.593821	−	0.804597i	$-0.297619\pi$
$173$	2702.43	1.18764	0.593821	+	0.804597i	$0.297619\pi$
$174$	1744.66	0.760126
$175$	0	0
$176$	280.593	0.120173
$177$	−229.728	−0.0975561
$178$	−211.235	−0.0889479
$179$	−1434.05	−0.598804	−0.299402	−	0.954127i	$-0.596787\pi$
$179$	−1434.05	−0.598804	−0.299402	+	0.954127i	$0.596787\pi$
$180$	13.3095	0.00551130
$181$	−2711.21	−1.11339	−0.556693	−	0.830718i	$-0.687930\pi$
$181$	−2711.21	−1.11339	−0.556693	+	0.830718i	$0.687930\pi$
$182$	0	0
$183$	−4482.99	−1.81089
$184$	−3773.62	−1.51193
$185$	78.3348	0.0311313
$186$	−1724.44	−0.679796
$187$	−1079.74	−0.422239
$188$	−1034.03	−0.401140
$189$	0	0
$190$	−925.858	−0.353520
$191$	−3494.60	−1.32388	−0.661938	−	0.749559i	$-0.730266\pi$
$191$	−3494.60	−1.32388	−0.661938	+	0.749559i	$0.730266\pi$
$192$	1135.57	0.426836
$193$	−1629.76	−0.607838	−0.303919	−	0.952698i	$-0.598295\pi$
$193$	−1629.76	−0.607838	−0.303919	+	0.952698i	$0.598295\pi$
$194$	−1619.26	−0.599257
$195$	−101.127	−0.0371377
$196$	0	0
$197$	693.696	0.250882	0.125441	−	0.992101i	$-0.459965\pi$
$197$	693.696	0.250882	0.125441	+	0.992101i	$0.459965\pi$
$198$	−21.6569	−0.00777316
$199$	3805.61	1.35564	0.677821	−	0.735227i	$-0.262924\pi$
$199$	3805.61	1.35564	0.677821	+	0.735227i	$0.262924\pi$
$200$	534.619	0.189017
$201$	−2859.44	−1.00343
$202$	1528.00	0.532226
$203$	0	0
$204$	1103.35	0.378676
$205$	−52.8680	−0.0180120
$206$	−2851.72	−0.964509
$207$	−85.6342	−0.0287536
$208$	38.4651	0.0128225
$209$	−3286.14	−1.08760
$210$	0	0
$211$	−627.239	−0.204649	−0.102324	−	0.994751i	$-0.532628\pi$
$211$	−627.239	−0.204649	−0.102324	+	0.994751i	$0.532628\pi$
$212$	1509.45	0.489006
$213$	5384.06	1.73197
$214$	−442.439	−0.141330
$215$	1629.64	0.516934
$216$	−2972.64	−0.936400
$217$	0	0
$218$	−1238.62	−0.384815
$219$	1325.74	0.409066
$220$	771.838	0.236533
$221$	−148.017	−0.0450529
$222$	130.251	0.0393777
$223$	2000.99	0.600878	0.300439	−	0.953801i	$-0.402867\pi$
$223$	2000.99	0.600878	0.300439	+	0.953801i	$0.402867\pi$
$224$	0	0
$225$	12.1320	0.00359468
$226$	931.235	0.274092
$227$	1592.24	0.465554	0.232777	−	0.972530i	$-0.425219\pi$
$227$	1592.24	0.465554	0.232777	+	0.972530i	$0.425219\pi$
$228$	3357.98	0.975386
$229$	3925.16	1.13267	0.566336	−	0.824174i	$-0.308360\pi$
$229$	3925.16	1.13267	0.566336	+	0.824174i	$0.308360\pi$
$230$	−1399.16	−0.401122
$231$	0	0
$232$	−4487.66	−1.26995
$233$	2169.75	0.610065	0.305032	−	0.952342i	$-0.401333\pi$
$233$	2169.75	0.610065	0.305032	+	0.952342i	$0.401333\pi$
$234$	−2.96883	−0.000829395 0
$235$	−942.548	−0.261639
$236$	240.361	0.0662972
$237$	−4837.08	−1.32575
$238$	0	0
$239$	1057.70	0.286262	0.143131	−	0.989704i	$-0.454283\pi$
$239$	1057.70	0.286262	0.143131	+	0.989704i	$0.454283\pi$
$240$	−261.360	−0.0702948
$241$	−2399.24	−0.641281	−0.320640	−	0.947201i	$-0.603898\pi$
$241$	−2399.24	−0.641281	−0.320640	+	0.947201i	$0.603898\pi$
$242$	854.771	0.227053
$243$	−136.150	−0.0359424
$244$	4690.47	1.23064
$245$	0	0
$246$	−87.9058	−0.0227832
$247$	−450.481	−0.116046
$248$	4435.66	1.13574
$249$	5033.31	1.28102
$250$	198.223	0.0501470
$251$	3812.95	0.958848	0.479424	−	0.877583i	$-0.340846\pi$
$251$	3812.95	0.958848	0.479424	+	0.877583i	$0.340846\pi$
$252$	0	0
$253$	−4966.05	−1.23404
$254$	−2473.23	−0.610962
$255$	1005.74	0.246987
$256$	−3559.06	−0.868910
$257$	−3490.16	−0.847122	−0.423561	−	0.905868i	$-0.639220\pi$
$257$	−3490.16	−0.847122	−0.423561	+	0.905868i	$0.639220\pi$
$258$	2709.68	0.653865
$259$	0	0
$260$	105.807	0.0252381
$261$	−101.838	−0.0241517
$262$	−2622.33	−0.618351
$263$	7124.56	1.67041	0.835207	−	0.549936i	$-0.185348\pi$
$263$	7124.56	1.67041	0.835207	+	0.549936i	$0.185348\pi$
$264$	3155.09	0.735539
$265$	1375.91	0.318949
$266$	0	0
$267$	698.347	0.160068
$268$	2991.78	0.681910
$269$	427.943	0.0969968	0.0484984	−	0.998823i	$-0.484556\pi$
$269$	427.943	0.0969968	0.0484984	+	0.998823i	$0.484556\pi$
$270$	−1102.18	−0.248431
$271$	−8188.58	−1.83550	−0.917751	−	0.397157i	$-0.869997\pi$
$271$	−8188.58	−1.83550	−0.917751	+	0.397157i	$0.869997\pi$
$272$	−382.546	−0.0852767
$273$	0	0
$274$	470.436	0.103723
$275$	703.553	0.154276
$276$	5074.61	1.10672
$277$	−4169.96	−0.904507	−0.452254	−	0.891889i	$-0.649380\pi$
$277$	−4169.96	−0.904507	−0.452254	+	0.891889i	$0.649380\pi$
$278$	377.208	0.0813792
$279$	100.658	0.0215994
$280$	0	0
$281$	1284.60	0.272714	0.136357	−	0.990660i	$-0.456461\pi$
$281$	1284.60	0.272714	0.136357	+	0.990660i	$0.456461\pi$
$282$	−1567.21	−0.330944
$283$	−4789.35	−1.00600	−0.502998	−	0.864287i	$-0.667770\pi$
$283$	−4789.35	−1.00600	−0.502998	+	0.864287i	$0.667770\pi$
$284$	−5633.25	−1.17701
$285$	3060.90	0.636184
$286$	−172.167	−0.0355959
$287$	0	0
$288$	−90.6939	−0.0185562
$289$	−3440.93	−0.700373
$290$	−1663.91	−0.336924
$291$	5353.30	1.07840
$292$	−1387.10	−0.277993
$293$	−6983.27	−1.39238	−0.696189	−	0.717858i	$-0.745123\pi$
$293$	−6983.27	−1.39238	−0.696189	+	0.717858i	$0.745123\pi$
$294$	0	0
$295$	219.096	0.0432416
$296$	−335.034	−0.0657888
$297$	−3911.96	−0.764292
$298$	−4340.47	−0.843748
$299$	−680.770	−0.131672
$300$	−718.934	−0.138359
$301$	0	0
$302$	5651.18	1.07678
$303$	−5051.59	−0.957777
$304$	−1164.26	−0.219654
$305$	4275.51	0.802672
$306$	29.5258	0.00551595
$307$	2069.43	0.384719	0.192359	−	0.981325i	$-0.438386\pi$
$307$	2069.43	0.384719	0.192359	+	0.981325i	$0.438386\pi$
$308$	0	0
$309$	9427.84	1.73570
$310$	1644.63	0.301318
$311$	−2121.96	−0.386898	−0.193449	−	0.981110i	$-0.561967\pi$
$311$	−2121.96	−0.386898	−0.193449	+	0.981110i	$0.561967\pi$
$312$	432.516	0.0784820
$313$	−1528.80	−0.276079	−0.138039	−	0.990427i	$-0.544080\pi$
$313$	−1528.80	−0.276079	−0.138039	+	0.990427i	$0.544080\pi$
$314$	1980.77	0.355992
$315$	0	0
$316$	5060.95	0.900951
$317$	5142.44	0.911131	0.455565	−	0.890202i	$-0.349437\pi$
$317$	5142.44	0.911131	0.455565	+	0.890202i	$0.349437\pi$
$318$	2287.78	0.403435
$319$	−5905.71	−1.03654
$320$	−1083.01	−0.189194
$321$	1462.71	0.254332
$322$	0	0
$323$	4480.16	0.771773
$324$	4069.35	0.697762
$325$	96.4466	0.0164612
$326$	3115.49	0.529298
$327$	4094.89	0.692501
$328$	226.114	0.0380642
$329$	0	0
$330$	1169.83	0.195142
$331$	−5774.04	−0.958822	−0.479411	−	0.877591i	$-0.659150\pi$
$331$	−5774.04	−0.958822	−0.479411	+	0.877591i	$0.659150\pi$
$332$	−5266.26	−0.870553
$333$	−7.60288	−0.00125116
$334$	−1489.69	−0.244049
$335$	2727.10	0.444768
$336$	0	0
$337$	−484.761	−0.0783579	−0.0391790	−	0.999232i	$-0.512474\pi$
$337$	−484.761	−0.0783579	−0.0391790	+	0.999232i	$0.512474\pi$
$338$	3460.37	0.556862
$339$	−3078.68	−0.493248
$340$	−1052.28	−0.167847
$341$	5837.28	0.926998
$342$	89.8603	0.0142079
$343$	0	0
$344$	−6969.92	−1.09242
$345$	4625.66	0.721847
$346$	−4285.48	−0.665864
$347$	−7798.21	−1.20643	−0.603213	−	0.797580i	$-0.706113\pi$
$347$	−7798.21	−1.20643	−0.603213	+	0.797580i	$0.706113\pi$
$348$	6034.81	0.929597
$349$	662.157	0.101560	0.0507800	−	0.998710i	$-0.483829\pi$
$349$	662.157	0.101560	0.0507800	+	0.998710i	$0.483829\pi$
$350$	0	0
$351$	−536.271	−0.0815499
$352$	−5259.47	−0.796394
$353$	−6626.14	−0.999076	−0.499538	−	0.866292i	$-0.666497\pi$
$353$	−6626.14	−0.999076	−0.499538	+	0.866292i	$0.666497\pi$
$354$	364.300	0.0546958
$355$	−5134.87	−0.767692
$356$	−730.668	−0.108779
$357$	0	0
$358$	2274.10	0.335725
$359$	10398.1	1.52866	0.764329	−	0.644827i	$-0.223070\pi$
$359$	10398.1	1.52866	0.764329	+	0.644827i	$0.223070\pi$
$360$	−51.8882	−0.00759652
$361$	6776.13	0.987918
$362$	4299.41	0.624231
$363$	−2825.89	−0.408597
$364$	0	0
$365$	−1264.39	−0.181318
$366$	7109.07	1.01529
$367$	−2246.31	−0.319499	−0.159750	−	0.987158i	$-0.551069\pi$
$367$	−2246.31	−0.319499	−0.159750	+	0.987158i	$0.551069\pi$
$368$	−1759.44	−0.249231
$369$	5.13117	0.000723897 0
$370$	−124.222	−0.0174541
$371$	0	0
$372$	−5964.89	−0.831358
$373$	192.025	0.0266560	0.0133280	−	0.999911i	$-0.495757\pi$
$373$	192.025	0.0266560	0.0133280	+	0.999911i	$0.495757\pi$
$374$	1712.24	0.236733
$375$	−655.330	−0.0902429
$376$	4031.24	0.552913
$377$	−809.584	−0.110599
$378$	0	0
$379$	−4565.76	−0.618805	−0.309403	−	0.950931i	$-0.600129\pi$
$379$	−4565.76	−0.618805	−0.309403	+	0.950931i	$0.600129\pi$
$380$	−3202.57	−0.432338
$381$	8176.55	1.09947
$382$	5541.69	0.742245
$383$	−1387.07	−0.185055	−0.0925273	−	0.995710i	$-0.529495\pi$
$383$	−1387.07	−0.185055	−0.0925273	+	0.995710i	$0.529495\pi$
$384$	6037.58	0.802355
$385$	0	0
$386$	2584.45	0.340791
$387$	−158.167	−0.0207754
$388$	−5601.06	−0.732862
$389$	4583.01	0.597346	0.298673	−	0.954355i	$-0.403456\pi$
$389$	4583.01	0.597346	0.298673	+	0.954355i	$0.403456\pi$
$390$	160.366	0.0208216
$391$	6770.45	0.875694
$392$	0	0
$393$	8669.47	1.11277
$394$	−1100.05	−0.140660
$395$	4613.21	0.587635
$396$	−74.9117	−0.00950620
$397$	8961.06	1.13285	0.566426	−	0.824112i	$-0.308326\pi$
$397$	8961.06	1.13285	0.566426	+	0.824112i	$0.308326\pi$
$398$	−6034.89	−0.760054
$399$	0	0
$400$	249.264	0.0311580
$401$	11629.3	1.44823	0.724113	−	0.689682i	$-0.242250\pi$
$401$	11629.3	1.44823	0.724113	+	0.689682i	$0.242250\pi$
$402$	4534.46	0.562582
$403$	800.203	0.0989106
$404$	5285.39	0.650886
$405$	3709.34	0.455107
$406$	0	0
$407$	−440.902	−0.0536970
$408$	−4301.49	−0.521949
$409$	−4320.13	−0.522290	−0.261145	−	0.965300i	$-0.584100\pi$
$409$	−4320.13	−0.522290	−0.261145	+	0.965300i	$0.584100\pi$
$410$	83.8373	0.0100986
$411$	−1555.27	−0.186656
$412$	−9864.19	−1.17955
$413$	0	0
$414$	135.798	0.0161210
$415$	−4800.36	−0.567808
$416$	−720.994	−0.0849751
$417$	−1247.06	−0.146448
$418$	5211.12	0.609771
$419$	−7824.02	−0.912240	−0.456120	−	0.889918i	$-0.650761\pi$
$419$	−7824.02	−0.912240	−0.456120	+	0.889918i	$0.650761\pi$
$420$	0	0
$421$	6944.28	0.803904	0.401952	−	0.915661i	$-0.368332\pi$
$421$	6944.28	0.803904	0.401952	+	0.915661i	$0.368332\pi$
$422$	994.667	0.114738
$423$	91.4802	0.0105152
$424$	−5884.70	−0.674024
$425$	−959.188	−0.109476
$426$	−8537.97	−0.971047
$427$	0	0
$428$	−1530.41	−0.172839
$429$	569.186	0.0640572
$430$	−2584.27	−0.289824
$431$	3257.36	0.364041	0.182020	−	0.983295i	$-0.441736\pi$
$431$	3257.36	0.364041	0.182020	+	0.983295i	$0.441736\pi$
$432$	−1385.98	−0.154359
$433$	16857.1	1.87090	0.935449	−	0.353461i	$-0.114995\pi$
$433$	16857.1	1.87090	0.935449	+	0.353461i	$0.114995\pi$
$434$	0	0
$435$	5500.91	0.606319
$436$	−4284.41	−0.470610
$437$	20605.5	2.25559
$438$	−2102.35	−0.229347
$439$	−7953.02	−0.864640	−0.432320	−	0.901720i	$-0.642305\pi$
$439$	−7953.02	−0.864640	−0.432320	+	0.901720i	$0.642305\pi$
$440$	−3009.07	−0.326026
$441$	0	0
$442$	234.723	0.0252593
$443$	−9709.76	−1.04137	−0.520683	−	0.853750i	$-0.674323\pi$
$443$	−9709.76	−1.04137	−0.520683	+	0.853750i	$0.674323\pi$
$444$	450.540	0.0481570
$445$	−666.026	−0.0709498
$446$	−3173.14	−0.336889
$447$	14349.7	1.51838
$448$	0	0
$449$	−11758.3	−1.23588	−0.617938	−	0.786227i	$-0.712032\pi$
$449$	−11758.3	−1.23588	−0.617938	+	0.786227i	$0.712032\pi$
$450$	−19.2388	−0.00201539
$451$	297.563	0.0310681
$452$	3221.17	0.335202
$453$	−18682.9	−1.93775
$454$	−2524.96	−0.261018
$455$	0	0
$456$	−13091.3	−1.34443
$457$	2750.18	0.281505	0.140753	−	0.990045i	$-0.455048\pi$
$457$	2750.18	0.281505	0.140753	+	0.990045i	$0.455048\pi$
$458$	−6224.47	−0.635044
$459$	5333.36	0.542353
$460$	−4839.75	−0.490553
$461$	3041.43	0.307275	0.153637	−	0.988127i	$-0.450901\pi$
$461$	3041.43	0.307275	0.153637	+	0.988127i	$0.450901\pi$
$462$	0	0
$463$	5422.89	0.544327	0.272163	−	0.962251i	$-0.412261\pi$
$463$	5422.89	0.544327	0.272163	+	0.962251i	$0.412261\pi$
$464$	−2092.35	−0.209343
$465$	−5437.18	−0.542243
$466$	−3440.76	−0.342039
$467$	5547.60	0.549705	0.274853	−	0.961486i	$-0.411371\pi$
$467$	5547.60	0.549705	0.274853	+	0.961486i	$0.411371\pi$
$468$	−10.2693	−0.00101431
$469$	0	0
$470$	1494.68	0.146690
$471$	−6548.47	−0.640632
$472$	−937.064	−0.0913810
$473$	−9172.34	−0.891637
$474$	7670.57	0.743294
$475$	−2919.24	−0.281987
$476$	0	0
$477$	−133.541	−0.0128185
$478$	−1677.28	−0.160496
$479$	−1128.55	−0.107651	−0.0538254	−	0.998550i	$-0.517141\pi$
$479$	−1128.55	−0.107651	−0.0538254	+	0.998550i	$0.517141\pi$
$480$	4898.97	0.465847
$481$	−60.4410	−0.00572946
$482$	3804.68	0.359540
$483$	0	0
$484$	2956.68	0.277674
$485$	−5105.53	−0.478001
$486$	215.904	0.0201515
$487$	4806.18	0.447205	0.223602	−	0.974680i	$-0.428218\pi$
$487$	4806.18	0.447205	0.223602	+	0.974680i	$0.428218\pi$
$488$	−18286.2	−1.69626
$489$	−10299.9	−0.952508
$490$	0	0
$491$	−12452.1	−1.14451	−0.572255	−	0.820076i	$-0.693931\pi$
$491$	−12452.1	−1.14451	−0.572255	+	0.820076i	$0.693931\pi$
$492$	−304.069	−0.0278627
$493$	8051.53	0.735543
$494$	714.367	0.0650625
$495$	−68.2843	−0.00620030
$496$	2068.11	0.187219
$497$	0	0
$498$	−7981.75	−0.718214
$499$	−11797.4	−1.05836	−0.529180	−	0.848509i	$-0.677501\pi$
$499$	−11797.4	−1.05836	−0.529180	+	0.848509i	$0.677501\pi$
$500$	685.660	0.0613273
$501$	4924.94	0.439182
$502$	−6046.52	−0.537588
$503$	−2900.55	−0.257116	−0.128558	−	0.991702i	$-0.541035\pi$
$503$	−2900.55	−0.257116	−0.128558	+	0.991702i	$0.541035\pi$
$504$	0	0
$505$	4817.79	0.424533
$506$	7875.09	0.691878
$507$	−11440.1	−1.00211
$508$	−8554.98	−0.747177
$509$	−9486.85	−0.826124	−0.413062	−	0.910703i	$-0.635541\pi$
$509$	−9486.85	−0.826124	−0.413062	+	0.910703i	$0.635541\pi$
$510$	−1594.88	−0.138476
$511$	0	0
$512$	−3569.14	−0.308076
$513$	16231.8	1.39698
$514$	5534.65	0.474947
$515$	−8991.50	−0.769346
$516$	9372.86	0.799645
$517$	5305.06	0.451289
$518$	0	0
$519$	14167.9	1.19827
$520$	−412.498	−0.0347870
$521$	20353.0	1.71148	0.855740	−	0.517406i	$-0.173102\pi$
$521$	20353.0	1.71148	0.855740	+	0.517406i	$0.173102\pi$
$522$	161.493	0.0135409
$523$	−9228.89	−0.771608	−0.385804	−	0.922581i	$-0.626076\pi$
$523$	−9228.89	−0.771608	−0.385804	+	0.922581i	$0.626076\pi$
$524$	−9070.71	−0.756213
$525$	0	0
$526$	−11298.0	−0.936535
$527$	−7958.25	−0.657811
$528$	1471.05	0.121248
$529$	18972.2	1.55932
$530$	−2181.90	−0.178822
$531$	−21.2646	−0.00173787
$532$	0	0
$533$	40.7915	0.00331496
$534$	−1107.43	−0.0897437
$535$	−1395.01	−0.112732
$536$	−11663.7	−0.939914
$537$	−7518.20	−0.604161
$538$	−678.626	−0.0543822
$539$	0	0
$540$	−3812.47	−0.303819
$541$	−697.274	−0.0554125	−0.0277062	−	0.999616i	$-0.508820\pi$
$541$	−697.274	−0.0554125	−0.0277062	+	0.999616i	$0.508820\pi$
$542$	12985.3	1.02909
$543$	−14213.9	−1.12335
$544$	7170.48	0.565132
$545$	−3905.37	−0.306950
$546$	0	0
$547$	−8032.62	−0.627879	−0.313940	−	0.949443i	$-0.601649\pi$
$547$	−8032.62	−0.627879	−0.313940	+	0.949443i	$0.601649\pi$
$548$	1627.25	0.126848
$549$	−414.965	−0.0322591
$550$	−1115.69	−0.0864963
$551$	24504.4	1.89460
$552$	−19783.7	−1.52546
$553$	0	0
$554$	6612.66	0.507121
$555$	410.681	0.0314098
$556$	1304.77	0.0995228
$557$	13586.7	1.03355	0.516776	−	0.856120i	$-0.327132\pi$
$557$	13586.7	1.03355	0.516776	+	0.856120i	$0.327132\pi$
$558$	−159.622	−0.0121099
$559$	−1257.39	−0.0951376
$560$	0	0
$561$	−5660.71	−0.426017
$562$	−2037.10	−0.152900
$563$	−17215.8	−1.28874	−0.644369	−	0.764715i	$-0.722880\pi$
$563$	−17215.8	−1.28874	−0.644369	+	0.764715i	$0.722880\pi$
$564$	−5421.04	−0.404729
$565$	2936.19	0.218631
$566$	7594.88	0.564022
$567$	0	0
$568$	21961.6	1.62234
$569$	−18441.5	−1.35871	−0.679357	−	0.733808i	$-0.737741\pi$
$569$	−18441.5	−1.35871	−0.679357	+	0.733808i	$0.737741\pi$
$570$	−4853.94	−0.356683
$571$	−10780.3	−0.790093	−0.395046	−	0.918661i	$-0.629271\pi$
$571$	−10780.3	−0.790093	−0.395046	+	0.918661i	$0.629271\pi$
$572$	−595.529	−0.0435320
$573$	−18320.9	−1.33572
$574$	0	0
$575$	−4411.57	−0.319957
$576$	105.113	0.00760366
$577$	12963.0	0.935279	0.467640	−	0.883919i	$-0.345105\pi$
$577$	12963.0	0.935279	0.467640	+	0.883919i	$0.345105\pi$
$578$	5456.58	0.392671
$579$	−8544.25	−0.613276
$580$	−5755.51	−0.412042
$581$	0	0
$582$	−8489.18	−0.604619
$583$	−7744.20	−0.550141
$584$	5407.72	0.383173
$585$	−9.36075	−0.000661571 0
$586$	11074.0	0.780651
$587$	16181.8	1.13781	0.568905	−	0.822403i	$-0.307367\pi$
$587$	16181.8	1.13781	0.568905	+	0.822403i	$0.307367\pi$
$588$	0	0
$589$	−24220.5	−1.69438
$590$	−347.439	−0.0242438
$591$	3636.80	0.253127
$592$	−156.208	−0.0108448
$593$	−747.630	−0.0517732	−0.0258866	−	0.999665i	$-0.508241\pi$
$593$	−747.630	−0.0517732	−0.0258866	+	0.999665i	$0.508241\pi$
$594$	6203.53	0.428508
$595$	0	0
$596$	−15013.8	−1.03186
$597$	19951.5	1.36777
$598$	1079.56	0.0738233
$599$	3849.86	0.262606	0.131303	−	0.991342i	$-0.458084\pi$
$599$	3849.86	0.262606	0.131303	+	0.991342i	$0.458084\pi$
$600$	2802.82	0.190708
$601$	−1808.82	−0.122768	−0.0613838	−	0.998114i	$-0.519551\pi$
$601$	−1808.82	−0.122768	−0.0613838	+	0.998114i	$0.519551\pi$
$602$	0	0
$603$	−264.682	−0.0178751
$604$	19547.6	1.31685
$605$	2695.10	0.181110
$606$	8010.74	0.536987
$607$	−536.092	−0.0358473	−0.0179237	−	0.999839i	$-0.505706\pi$
$607$	−536.092	−0.0358473	−0.0179237	+	0.999839i	$0.505706\pi$
$608$	21823.0	1.45566
$609$	0	0
$610$	−6780.04	−0.450026
$611$	727.245	0.0481525
$612$	102.131	0.00674573
$613$	14736.0	0.970931	0.485465	−	0.874256i	$-0.338650\pi$
$613$	14736.0	0.970931	0.485465	+	0.874256i	$0.338650\pi$
$614$	−3281.68	−0.215697
$615$	−277.168	−0.0181731
$616$	0	0
$617$	9604.91	0.626709	0.313354	−	0.949636i	$-0.398547\pi$
$617$	9604.91	0.626709	0.313354	+	0.949636i	$0.398547\pi$
$618$	−14950.5	−0.973138
$619$	8594.22	0.558047	0.279023	−	0.960284i	$-0.409989\pi$
$619$	8594.22	0.558047	0.279023	+	0.960284i	$0.409989\pi$
$620$	5688.82	0.368498
$621$	24529.6	1.58509
$622$	3364.98	0.216918
$623$	0	0
$624$	201.659	0.0129372
$625$	625.000	0.0400000
$626$	2424.34	0.154786
$627$	−17228.1	−1.09733
$628$	6851.55	0.435361
$629$	601.102	0.0381042
$630$	0	0
$631$	−14803.3	−0.933933	−0.466966	−	0.884275i	$-0.654653\pi$
$631$	−14803.3	−0.933933	−0.466966	+	0.884275i	$0.654653\pi$
$632$	−19730.5	−1.24183
$633$	−3288.39	−0.206480
$634$	−8154.82	−0.510835
$635$	−7798.12	−0.487337
$636$	7913.50	0.493381
$637$	0	0
$638$	9365.19	0.581146
$639$	498.372	0.0308533
$640$	−5758.15	−0.355642
$641$	−9290.31	−0.572458	−0.286229	−	0.958161i	$-0.592402\pi$
$641$	−9290.31	−0.572458	−0.286229	+	0.958161i	$0.592402\pi$
$642$	−2319.55	−0.142594
$643$	1861.78	0.114186	0.0570928	−	0.998369i	$-0.481817\pi$
$643$	1861.78	0.114186	0.0570928	+	0.998369i	$0.481817\pi$
$644$	0	0
$645$	8543.64	0.521559
$646$	−7104.58	−0.432703
$647$	−29527.4	−1.79419	−0.897095	−	0.441837i	$-0.854327\pi$
$647$	−29527.4	−1.79419	−0.897095	+	0.441837i	$0.854327\pi$
$648$	−15864.7	−0.961764
$649$	−1233.17	−0.0745855
$650$	−152.944	−0.00922915
$651$	0	0
$652$	10776.6	0.647306
$653$	5192.55	0.311180	0.155590	−	0.987822i	$-0.450272\pi$
$653$	5192.55	0.311180	0.155590	+	0.987822i	$0.450272\pi$
$654$	−6493.61	−0.388258
$655$	−8268.23	−0.493231
$656$	105.425	0.00627461
$657$	122.717	0.00728711
$658$	0	0
$659$	−11740.1	−0.693976	−0.346988	−	0.937870i	$-0.612795\pi$
$659$	−11740.1	−0.693976	−0.346988	+	0.937870i	$0.612795\pi$
$660$	4046.47	0.238649
$661$	−8792.50	−0.517381	−0.258690	−	0.965960i	$-0.583291\pi$
$661$	−8792.50	−0.517381	−0.258690	+	0.965960i	$0.583291\pi$
$662$	9156.39	0.537573
$663$	−775.999	−0.0454559
$664$	20530.9	1.19993
$665$	0	0
$666$	12.0565	0.000701474 0
$667$	37031.3	2.14971
$668$	−5152.88	−0.298460
$669$	10490.4	0.606254
$670$	−4324.59	−0.249364
$671$	−24064.4	−1.38449
$672$	0	0
$673$	30366.5	1.73929	0.869646	−	0.493676i	$-0.164347\pi$
$673$	30366.5	1.73929	0.869646	+	0.493676i	$0.164347\pi$
$674$	768.728	0.0439322
$675$	−3475.18	−0.198162
$676$	11969.5	0.681015
$677$	17416.9	0.988754	0.494377	−	0.869248i	$-0.335396\pi$
$677$	17416.9	0.988754	0.494377	+	0.869248i	$0.335396\pi$
$678$	4882.13	0.276544
$679$	0	0
$680$	4102.41	0.231353
$681$	8347.55	0.469719
$682$	−9256.68	−0.519731
$683$	8828.32	0.494592	0.247296	−	0.968940i	$-0.420458\pi$
$683$	8828.32	0.494592	0.247296	+	0.968940i	$0.420458\pi$
$684$	310.829	0.0173755
$685$	1483.29	0.0827351
$686$	0	0
$687$	20578.2	1.14281
$688$	−3249.69	−0.180078
$689$	−1061.61	−0.0586999
$690$	−7335.31	−0.404711
$691$	−10631.0	−0.585273	−0.292636	−	0.956224i	$-0.594533\pi$
$691$	−10631.0	−0.585273	−0.292636	+	0.956224i	$0.594533\pi$
$692$	−14823.6	−0.814319
$693$	0	0
$694$	12366.3	0.676395
$695$	1189.34	0.0649126
$696$	−23527.2	−1.28131
$697$	−405.683	−0.0220464
$698$	−1050.04	−0.0569407
$699$	11375.2	0.615523
$700$	0	0
$701$	30120.7	1.62289	0.811443	−	0.584431i	$-0.198682\pi$
$701$	30120.7	1.62289	0.811443	+	0.584431i	$0.198682\pi$
$702$	850.411	0.0457218
$703$	1829.42	0.0981480
$704$	6095.65	0.326333
$705$	−4941.44	−0.263979
$706$	10507.6	0.560142
$707$	0	0
$708$	1260.12	0.0668904
$709$	4414.94	0.233860	0.116930	−	0.993140i	$-0.462695\pi$
$709$	4414.94	0.233860	0.116930	+	0.993140i	$0.462695\pi$
$710$	8142.81	0.430415
$711$	−447.741	−0.0236169
$712$	2848.56	0.149936
$713$	−36602.2	−1.92253
$714$	0	0
$715$	−542.843	−0.0283932
$716$	7866.16	0.410576
$717$	5545.12	0.288823
$718$	−16489.1	−0.857058
$719$	−17136.0	−0.888822	−0.444411	−	0.895823i	$-0.646587\pi$
$719$	−17136.0	−0.888822	−0.444411	+	0.895823i	$0.646587\pi$
$720$	−24.1926	−0.00125223
$721$	0	0
$722$	−10745.5	−0.553886
$723$	−12578.4	−0.647018
$724$	14871.8	0.763405
$725$	−5246.32	−0.268750
$726$	4481.26	0.229084
$727$	3271.77	0.166909	0.0834546	−	0.996512i	$-0.473405\pi$
$727$	3271.77	0.166909	0.0834546	+	0.996512i	$0.473405\pi$
$728$	0	0
$729$	19316.6	0.981386
$730$	2005.05	0.101658
$731$	12505.1	0.632719
$732$	24590.5	1.24165
$733$	−4469.84	−0.225235	−0.112618	−	0.993638i	$-0.535923\pi$
$733$	−4469.84	−0.225235	−0.112618	+	0.993638i	$0.535923\pi$
$734$	3562.16	0.179130
$735$	0	0
$736$	32979.1	1.65166
$737$	−15349.3	−0.767161
$738$	−8.13694	−0.000405860 0
$739$	−2475.20	−0.123209	−0.0616046	−	0.998101i	$-0.519622\pi$
$739$	−2475.20	−0.123209	−0.0616046	+	0.998101i	$0.519622\pi$
$740$	−429.688	−0.0213455
$741$	−2361.71	−0.117084
$742$	0	0
$743$	9240.48	0.456259	0.228129	−	0.973631i	$-0.426739\pi$
$743$	9240.48	0.456259	0.228129	+	0.973631i	$0.426739\pi$
$744$	23254.6	1.14591
$745$	−13685.6	−0.673020
$746$	−304.511	−0.0149450
$747$	465.905	0.0228200
$748$	5922.70	0.289513
$749$	0	0
$750$	1039.21	0.0505956
$751$	5052.47	0.245496	0.122748	−	0.992438i	$-0.460829\pi$
$751$	5052.47	0.245496	0.122748	+	0.992438i	$0.460829\pi$
$752$	1879.55	0.0911437
$753$	19989.9	0.967427
$754$	1283.83	0.0620082
$755$	17818.2	0.858903
$756$	0	0
$757$	−37779.2	−1.81388	−0.906942	−	0.421256i	$-0.861589\pi$
$757$	−37779.2	−1.81388	−0.906942	+	0.421256i	$0.861589\pi$
$758$	7240.32	0.346939
$759$	−26035.2	−1.24508
$760$	12485.5	0.595914
$761$	−7502.70	−0.357388	−0.178694	−	0.983905i	$-0.557187\pi$
$761$	−7502.70	−0.357388	−0.178694	+	0.983905i	$0.557187\pi$
$762$	−12966.3	−0.616428
$763$	0	0
$764$	19168.9	0.907729
$765$	93.0952	0.00439982
$766$	2199.60	0.103753
$767$	−169.049	−0.00795826
$768$	−18658.9	−0.876684
$769$	−24209.4	−1.13526	−0.567628	−	0.823285i	$-0.692139\pi$
$769$	−24209.4	−1.13526	−0.567628	+	0.823285i	$0.692139\pi$
$770$	0	0
$771$	−18297.7	−0.854701
$772$	8939.70	0.416770
$773$	−7520.99	−0.349950	−0.174975	−	0.984573i	$-0.555984\pi$
$773$	−7520.99	−0.349950	−0.174975	+	0.984573i	$0.555984\pi$
$774$	250.819	0.0116480
$775$	5185.53	0.240348
$776$	21836.1	1.01014
$777$	0	0
$778$	−7267.67	−0.334908
$779$	−1234.67	−0.0567866
$780$	554.710	0.0254638
$781$	28901.3	1.32416
$782$	−10736.5	−0.490967
$783$	29171.0	1.33140
$784$	0	0
$785$	6245.39	0.283959
$786$	−13747.9	−0.623883
$787$	16121.4	0.730196	0.365098	−	0.930969i	$-0.381035\pi$
$787$	16121.4	0.730196	0.365098	+	0.930969i	$0.381035\pi$
$788$	−3805.12	−0.172020
$789$	37351.5	1.68536
$790$	−7315.56	−0.329463
$791$	0	0
$792$	292.049	0.0131029
$793$	−3298.87	−0.147725
$794$	−14210.3	−0.635146
$795$	7213.39	0.321802
$796$	−20874.9	−0.929510
$797$	32223.0	1.43212	0.716058	−	0.698041i	$-0.245944\pi$
$797$	32223.0	1.43212	0.716058	+	0.698041i	$0.245944\pi$
$798$	0	0
$799$	−7232.65	−0.320241
$800$	−4672.23	−0.206486
$801$	64.6420	0.00285145
$802$	−18441.5	−0.811962
$803$	7116.50	0.312747
$804$	15684.8	0.688011
$805$	0	0
$806$	−1268.95	−0.0554552
$807$	2243.55	0.0978646
$808$	−20605.5	−0.897151
$809$	−24544.4	−1.06667	−0.533334	−	0.845905i	$-0.679061\pi$
$809$	−24544.4	−1.06667	−0.533334	+	0.845905i	$0.679061\pi$
$810$	−5882.21	−0.255160
$811$	−18783.9	−0.813308	−0.406654	−	0.913582i	$-0.633305\pi$
$811$	−18783.9	−0.813308	−0.406654	+	0.913582i	$0.633305\pi$
$812$	0	0
$813$	−42929.8	−1.85192
$814$	699.176	0.0301058
$815$	9823.18	0.422197
$816$	−2005.55	−0.0860396
$817$	38058.6	1.62974
$818$	6850.80	0.292827
$819$	0	0
$820$	289.996	0.0123501
$821$	18085.1	0.768786	0.384393	−	0.923170i	$-0.374411\pi$
$821$	18085.1	0.768786	0.384393	+	0.923170i	$0.374411\pi$
$822$	2466.33	0.104651
$823$	7435.18	0.314914	0.157457	−	0.987526i	$-0.449670\pi$
$823$	7435.18	0.314914	0.157457	+	0.987526i	$0.449670\pi$
$824$	38456.3	1.62583
$825$	3688.48	0.155656
$826$	0	0
$827$	26487.2	1.11373	0.556863	−	0.830604i	$-0.312005\pi$
$827$	26487.2	1.11373	0.556863	+	0.830604i	$0.312005\pi$
$828$	469.728	0.0197152
$829$	19377.1	0.811813	0.405906	−	0.913915i	$-0.366956\pi$
$829$	19377.1	0.811813	0.405906	+	0.913915i	$0.366956\pi$
$830$	7612.34	0.318347
$831$	−21861.6	−0.912600
$832$	835.622	0.0348197
$833$	0	0
$834$	1977.56	0.0821073
$835$	−4697.01	−0.194667
$836$	18025.4	0.745721
$837$	−28833.1	−1.19070
$838$	12407.2	0.511456
$839$	4645.97	0.191176	0.0955880	−	0.995421i	$-0.469527\pi$
$839$	4645.97	0.191176	0.0955880	+	0.995421i	$0.469527\pi$
$840$	0	0
$841$	19649.2	0.805658
$842$	−11012.1	−0.450717
$843$	6734.69	0.275154
$844$	3440.58	0.140320
$845$	10910.6	0.444184
$846$	−145.068	−0.00589544
$847$	0	0
$848$	−2743.72	−0.111108
$849$	−25108.8	−1.01500
$850$	1521.07	0.0613791
$851$	2764.64	0.111364
$852$	−29533.1	−1.18754
$853$	−32566.5	−1.30722	−0.653609	−	0.756833i	$-0.726746\pi$
$853$	−32566.5	−1.30722	−0.653609	+	0.756833i	$0.726746\pi$
$854$	0	0
$855$	283.330	0.0113330
$856$	5966.42	0.238234
$857$	−134.941	−0.00537865	−0.00268932	−	0.999996i	$-0.500856\pi$
$857$	−134.941	−0.00537865	−0.00268932	+	0.999996i	$0.500856\pi$
$858$	−902.607	−0.0359143
$859$	10380.9	0.412331	0.206165	−	0.978517i	$-0.433902\pi$
$859$	10380.9	0.412331	0.206165	+	0.978517i	$0.433902\pi$
$860$	−8939.06	−0.354441
$861$	0	0
$862$	−5165.48	−0.204103
$863$	25349.3	0.999883	0.499941	−	0.866059i	$-0.333355\pi$
$863$	25349.3	0.999883	0.499941	+	0.866059i	$0.333355\pi$
$864$	25979.0	1.02294
$865$	−13512.2	−0.531130
$866$	−26731.7	−1.04894
$867$	−18039.6	−0.706639
$868$	0	0
$869$	−25965.1	−1.01359
$870$	−8723.28	−0.339939
$871$	−2104.15	−0.0818559
$872$	16703.1	0.648667
$873$	495.524	0.0192107
$874$	−32675.9	−1.26462
$875$	0	0
$876$	−7272.08	−0.280480
$877$	−22519.4	−0.867075	−0.433537	−	0.901136i	$-0.642735\pi$
$877$	−22519.4	−0.867075	−0.433537	+	0.901136i	$0.642735\pi$
$878$	12611.8	0.484769
$879$	−36610.8	−1.40484
$880$	−1402.96	−0.0537431
$881$	12419.9	0.474958	0.237479	−	0.971393i	$-0.423679\pi$
$881$	12419.9	0.474958	0.237479	+	0.971393i	$0.423679\pi$
$882$	0	0
$883$	−46011.8	−1.75359	−0.876794	−	0.480866i	$-0.840322\pi$
$883$	−46011.8	−1.75359	−0.876794	+	0.480866i	$0.840322\pi$
$884$	811.913	0.0308910
$885$	1148.64	0.0436284
$886$	15397.6	0.583852
$887$	28086.7	1.06320	0.531600	−	0.846995i	$-0.321591\pi$
$887$	28086.7	1.06320	0.531600	+	0.846995i	$0.321591\pi$
$888$	−1756.46	−0.0663774
$889$	0	0
$890$	1056.17	0.0397787
$891$	−20877.7	−0.784994
$892$	−10976.0	−0.411998
$893$	−22012.2	−0.824871
$894$	−22755.5	−0.851296
$895$	7170.24	0.267793
$896$	0	0
$897$	−3569.03	−0.132850
$898$	18646.2	0.692907
$899$	−43528.0	−1.61484
$900$	−66.5476	−0.00246473
$901$	10558.0	0.390388
$902$	−471.872	−0.0174187
$903$	0	0
$904$	−12558.0	−0.462026
$905$	13556.1	0.497921
$906$	29627.1	1.08642
$907$	−30458.8	−1.11507	−0.557534	−	0.830154i	$-0.688252\pi$
$907$	−30458.8	−1.11507	−0.557534	+	0.830154i	$0.688252\pi$
$908$	−8733.90	−0.319212
$909$	−467.597	−0.0170618
$910$	0	0
$911$	−26850.4	−0.976502	−0.488251	−	0.872703i	$-0.662365\pi$
$911$	−26850.4	−0.976502	−0.488251	+	0.872703i	$0.662365\pi$
$912$	−6103.79	−0.221619
$913$	27018.5	0.979387
$914$	−4361.20	−0.157829
$915$	22415.0	0.809853
$916$	−21530.6	−0.776628
$917$	0	0
$918$	−8457.57	−0.304076
$919$	−38095.8	−1.36743	−0.683714	−	0.729750i	$-0.739636\pi$
$919$	−38095.8	−1.36743	−0.683714	+	0.729750i	$0.739636\pi$
$920$	18868.1	0.676156
$921$	10849.3	0.388161
$922$	−4823.06	−0.172277
$923$	3961.93	0.141288
$924$	0	0
$925$	−391.674	−0.0139223
$926$	−8599.55	−0.305182
$927$	872.682	0.0309198
$928$	39219.3	1.38732
$929$	−36355.0	−1.28393	−0.641964	−	0.766735i	$-0.721880\pi$
$929$	−36355.0	−1.28393	−0.641964	+	0.766735i	$0.721880\pi$
$930$	8622.20	0.304014
$931$	0	0
$932$	−11901.7	−0.418297
$933$	−11124.7	−0.390360
$934$	−8797.31	−0.308198
$935$	5398.72	0.188831
$936$	40.0355	0.00139808
$937$	−37878.5	−1.32064	−0.660318	−	0.750986i	$-0.729579\pi$
$937$	−37878.5	−1.32064	−0.660318	+	0.750986i	$0.729579\pi$
$938$	0	0
$939$	−8014.93	−0.278549
$940$	5170.14	0.179395
$941$	−54125.6	−1.87507	−0.937536	−	0.347887i	$-0.886899\pi$
$941$	−54125.6	−1.87507	−0.937536	+	0.347887i	$0.886899\pi$
$942$	10384.5	0.359177
$943$	−1865.85	−0.0644330
$944$	−436.902	−0.0150635
$945$	0	0
$946$	14545.4	0.499906
$947$	−38345.9	−1.31581	−0.657906	−	0.753100i	$-0.728557\pi$
$947$	−38345.9	−1.31581	−0.657906	+	0.753100i	$0.728557\pi$
$948$	26532.7	0.909012
$949$	975.566	0.0333701
$950$	4629.29	0.158099
$951$	26960.0	0.919282
$952$	0	0
$953$	−15096.4	−0.513137	−0.256569	−	0.966526i	$-0.582592\pi$
$953$	−15096.4	−0.513137	−0.256569	+	0.966526i	$0.582592\pi$
$954$	211.767	0.00718680
$955$	17473.0	0.592055
$956$	−5801.76	−0.196279
$957$	−30961.5	−1.04581
$958$	1789.64	0.0603556
$959$	0	0
$960$	−5677.84	−0.190887
$961$	13232.6	0.444182
$962$	95.8465	0.00321228
$963$	135.395	0.00453068
$964$	13160.5	0.439701
$965$	8148.81	0.271833
$966$	0	0
$967$	6917.06	0.230029	0.115014	−	0.993364i	$-0.463309\pi$
$967$	6917.06	0.230029	0.115014	+	0.993364i	$0.463309\pi$
$968$	−11526.8	−0.382734
$969$	23487.9	0.778678
$970$	8096.29	0.267996
$971$	−39517.7	−1.30606	−0.653029	−	0.757333i	$-0.726502\pi$
$971$	−39517.7	−1.30606	−0.653029	+	0.757333i	$0.726502\pi$
$972$	746.820	0.0246443
$973$	0	0
$974$	−7621.57	−0.250730
$975$	505.635	0.0166085
$976$	−8525.85	−0.279616
$977$	−38858.4	−1.27246	−0.636229	−	0.771500i	$-0.719507\pi$
$977$	−38858.4	−1.27246	−0.636229	+	0.771500i	$0.719507\pi$
$978$	16333.4	0.534034
$979$	3748.68	0.122378
$980$	0	0
$981$	379.040	0.0123362
$982$	19746.3	0.641681
$983$	510.230	0.0165552	0.00827762	−	0.999966i	$-0.497365\pi$
$983$	510.230	0.0165552	0.00827762	+	0.999966i	$0.497365\pi$
$984$	1185.43	0.0384047
$985$	−3468.48	−0.112198
$986$	−12768.0	−0.412390
$987$	0	0
$988$	2471.02	0.0795683
$989$	57514.4	1.84919
$990$	108.284	0.00347626
$991$	−36222.8	−1.16111	−0.580553	−	0.814223i	$-0.697164\pi$
$991$	−36222.8	−1.16111	−0.580553	+	0.814223i	$0.697164\pi$
$992$	−38764.9	−1.24071
$993$	−30271.2	−0.967400
$994$	0	0
$995$	−19028.1	−0.606262
$996$	−27609.1	−0.878341
$997$	20856.1	0.662505	0.331253	−	0.943542i	$-0.392529\pi$
$997$	20856.1	0.662505	0.331253	+	0.943542i	$0.392529\pi$
$998$	18708.1	0.593381
$999$	2177.82	0.0689721

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.4.a.h.1.2		2
3.2	odd	2		2205.4.a.bg.1.1		2
5.4	even	2		1225.4.a.v.1.1		2
7.2	even	3		245.4.e.l.116.1		4
7.3	odd	6		35.4.e.b.16.1	yes	4
7.4	even	3		245.4.e.l.226.1		4
7.5	odd	6		35.4.e.b.11.1	✓	4
7.6	odd	2		245.4.a.g.1.2		2
21.5	even	6		315.4.j.c.46.2		4
21.17	even	6		315.4.j.c.226.2		4
21.20	even	2		2205.4.a.bf.1.1		2
28.3	even	6		560.4.q.i.401.1		4
28.19	even	6		560.4.q.i.81.1		4
35.3	even	12		175.4.k.c.149.2		8
35.12	even	12		175.4.k.c.74.2		8
35.17	even	12		175.4.k.c.149.3		8
35.19	odd	6		175.4.e.c.151.2		4
35.24	odd	6		175.4.e.c.51.2		4
35.33	even	12		175.4.k.c.74.3		8
35.34	odd	2		1225.4.a.x.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.e.b.11.1	✓	4	7.5	odd	6
35.4.e.b.16.1	yes	4	7.3	odd	6
175.4.e.c.51.2		4	35.24	odd	6
175.4.e.c.151.2		4	35.19	odd	6
175.4.k.c.74.2		8	35.12	even	12
175.4.k.c.74.3		8	35.33	even	12
175.4.k.c.149.2		8	35.3	even	12
175.4.k.c.149.3		8	35.17	even	12
245.4.a.g.1.2		2	7.6	odd	2
245.4.a.h.1.2		2	1.1	even	1	trivial
245.4.e.l.116.1		4	7.2	even	3
245.4.e.l.226.1		4	7.4	even	3
315.4.j.c.46.2		4	21.5	even	6
315.4.j.c.226.2		4	21.17	even	6
560.4.q.i.81.1		4	28.19	even	6
560.4.q.i.401.1		4	28.3	even	6
1225.4.a.v.1.1		2	5.4	even	2
1225.4.a.x.1.1		2	35.34	odd	2
2205.4.a.bf.1.1		2	21.20	even	2
2205.4.a.bg.1.1		2	3.2	odd	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:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	245.a (trivial)

\(f(q)\)	\(=\)	\(q-1.58579 q^{2} +5.24264 q^{3} -5.48528 q^{4} -5.00000 q^{5} -8.31371 q^{6} +21.3848 q^{8} +0.485281 q^{9} +7.92893 q^{10} +28.1421 q^{11} -28.7574 q^{12} +3.85786 q^{13} -26.2132 q^{15} +9.97056 q^{16} -38.3675 q^{17} -0.769553 q^{18} -116.770 q^{19} +27.4264 q^{20} -44.6274 q^{22} -176.463 q^{23} +112.113 q^{24} +25.0000 q^{25} -6.11775 q^{26} -139.007 q^{27} -209.853 q^{29} +41.5685 q^{30} +207.421 q^{31} -186.889 q^{32} +147.539 q^{33} +60.8427 q^{34} -2.66190 q^{36} -15.6670 q^{37} +185.172 q^{38} +20.2254 q^{39} -106.924 q^{40} +10.5736 q^{41} -325.929 q^{43} -154.368 q^{44} -2.42641 q^{45} +279.833 q^{46} +188.510 q^{47} +52.2721 q^{48} -39.6447 q^{50} -201.147 q^{51} -21.1615 q^{52} -275.182 q^{53} +220.436 q^{54} -140.711 q^{55} -612.181 q^{57} +332.782 q^{58} -43.8192 q^{59} +143.787 q^{60} -855.102 q^{61} -328.926 q^{62} +216.602 q^{64} -19.2893 q^{65} -233.966 q^{66} -545.419 q^{67} +210.457 q^{68} -925.132 q^{69} +1026.97 q^{71} +10.3776 q^{72} +252.877 q^{73} +24.8444 q^{74} +131.066 q^{75} +640.514 q^{76} -32.0732 q^{78} -922.642 q^{79} -49.8528 q^{80} -741.867 q^{81} -16.7675 q^{82} +960.071 q^{83} +191.838 q^{85} +516.854 q^{86} -1100.18 q^{87} +601.813 q^{88} +133.205 q^{89} +3.84776 q^{90} +967.949 q^{92} +1087.44 q^{93} -298.936 q^{94} +583.848 q^{95} -979.794 q^{96} +1021.11 q^{97} +13.6569 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{2} + 2 q^{3} + 6 q^{4} - 10 q^{5} + 6 q^{6} + 6 q^{8} - 16 q^{9} + 30 q^{10} + 28 q^{11} - 66 q^{12} + 36 q^{13} - 10 q^{15} - 14 q^{16} + 76 q^{17} + 72 q^{18} - 160 q^{19} - 30 q^{20} - 44 q^{22}+ \cdots + 16 q^{99}+O(q^{100}) \) 2 * q - 6 * q^2 + 2 * q^3 + 6 * q^4 - 10 * q^5 + 6 * q^6 + 6 * q^8 - 16 * q^9 + 30 * q^10 + 28 * q^11 - 66 * q^12 + 36 * q^13 - 10 * q^15 - 14 * q^16 + 76 * q^17 + 72 * q^18 - 160 * q^19 - 30 * q^20 - 44 * q^22 - 22 * q^23 + 162 * q^24 + 50 * q^25 - 148 * q^26 + 2 * q^27 - 250 * q^29 - 30 * q^30 + 132 * q^31 + 42 * q^32 + 148 * q^33 - 444 * q^34 - 192 * q^36 - 416 * q^37 + 376 * q^38 - 84 * q^39 - 30 * q^40 + 106 * q^41 - 666 * q^43 - 156 * q^44 + 80 * q^45 - 402 * q^46 + 196 * q^47 + 130 * q^48 - 150 * q^50 - 572 * q^51 + 348 * q^52 - 952 * q^53 - 402 * q^54 - 140 * q^55 - 472 * q^57 + 510 * q^58 - 840 * q^59 + 330 * q^60 - 98 * q^61 + 4 * q^62 - 602 * q^64 - 180 * q^65 - 236 * q^66 - 1286 * q^67 + 1524 * q^68 - 1426 * q^69 + 1064 * q^71 + 264 * q^72 + 172 * q^73 + 1792 * q^74 + 50 * q^75 + 144 * q^76 + 428 * q^78 - 1240 * q^79 + 70 * q^80 - 754 * q^81 - 438 * q^82 + 1906 * q^83 - 380 * q^85 + 2018 * q^86 - 970 * q^87 + 604 * q^88 - 650 * q^89 - 360 * q^90 + 2742 * q^92 + 1332 * q^93 - 332 * q^94 + 800 * q^95 - 1722 * q^96 + 628 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

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Database

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