LMFDB - Embedded newform 1225.4.a.bi.1.6

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:	$6$
Coefficient field:	6.6.1163891200.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28 $ x^6 - 2x^5 - 23x^4 + 12x^3 + 154x^2 + 152*x + 28
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2\cdot 7 $
Twist minimal:	no (minimal twist has level 245)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$4.10376$ 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)

$f(q)$	$=$	$q+5.51797 q^{2} -3.86039 q^{3} +22.4480 q^{4} -21.3015 q^{6} +79.7239 q^{8} -12.0974 q^{9} +O(q^{10})$	\(q+5.51797 q^{2} -3.86039 q^{3} +22.4480 q^{4} -21.3015 q^{6} +79.7239 q^{8} -12.0974 q^{9} +34.5211 q^{11} -86.6582 q^{12} -68.8935 q^{13} +260.330 q^{16} +91.4346 q^{17} -66.7530 q^{18} +11.8278 q^{19} +190.486 q^{22} +0.104165 q^{23} -307.765 q^{24} -380.152 q^{26} +150.931 q^{27} +190.863 q^{29} +159.802 q^{31} +798.703 q^{32} -133.265 q^{33} +504.534 q^{34} -271.562 q^{36} +177.908 q^{37} +65.2657 q^{38} +265.956 q^{39} +145.247 q^{41} -8.25729 q^{43} +774.930 q^{44} +0.574782 q^{46} -260.529 q^{47} -1004.98 q^{48} -352.973 q^{51} -1546.52 q^{52} -353.107 q^{53} +832.834 q^{54} -45.6601 q^{57} +1053.18 q^{58} +240.495 q^{59} +778.188 q^{61} +881.783 q^{62} +2324.58 q^{64} -735.352 q^{66} -151.945 q^{67} +2052.53 q^{68} -0.402119 q^{69} -311.449 q^{71} -964.450 q^{72} -639.888 q^{73} +981.690 q^{74} +265.512 q^{76} +1467.54 q^{78} +391.186 q^{79} -256.024 q^{81} +801.472 q^{82} -493.205 q^{83} -45.5635 q^{86} -736.807 q^{87} +2752.15 q^{88} +473.850 q^{89} +2.33831 q^{92} -616.898 q^{93} -1437.59 q^{94} -3083.31 q^{96} -839.005 q^{97} -417.615 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} - 16 q^{3} + 14 q^{4} + 24 q^{6} + 66 q^{8} + 70 q^{9}+O(q^{10}) $ 6 * q + 2 * q^2 - 16 * q^3 + 14 * q^4 + 24 * q^6 + 66 * q^8 + 70 * q^9	$ 6 q + 2 q^{2} - 16 q^{3} + 14 q^{4} + 24 q^{6} + 66 q^{8} + 70 q^{9} - 16 q^{11} - 160 q^{12} - 168 q^{13} + 298 q^{16} + 4 q^{17} - 354 q^{18} + 308 q^{19} + 236 q^{22} + 336 q^{23} - 92 q^{24} + 56 q^{26} - 964 q^{27} + 176 q^{29} + 392 q^{31} + 770 q^{32} - 188 q^{33} + 812 q^{34} + 230 q^{36} + 140 q^{37} - 20 q^{38} + 140 q^{39} + 656 q^{41} + 388 q^{43} - 160 q^{44} - 388 q^{46} - 628 q^{47} - 1396 q^{48} + 744 q^{51} - 1520 q^{52} + 676 q^{53} + 2284 q^{54} - 1468 q^{57} + 2012 q^{58} + 996 q^{59} + 740 q^{61} + 364 q^{62} + 1426 q^{64} - 3620 q^{66} - 1768 q^{67} + 2940 q^{68} - 1048 q^{69} - 224 q^{71} - 2858 q^{72} - 2640 q^{73} + 928 q^{74} - 1340 q^{76} - 8 q^{78} + 1636 q^{79} + 4442 q^{81} + 1756 q^{82} - 140 q^{83} + 1180 q^{86} - 1940 q^{87} + 5652 q^{88} - 1904 q^{89} + 1952 q^{92} + 1592 q^{93} - 3332 q^{94} - 6460 q^{96} - 516 q^{97} - 2804 q^{99}+O(q^{100}) $ 6 * q + 2 * q^2 - 16 * q^3 + 14 * q^4 + 24 * q^6 + 66 * q^8 + 70 * q^9 - 16 * q^11 - 160 * q^12 - 168 * q^13 + 298 * q^16 + 4 * q^17 - 354 * q^18 + 308 * q^19 + 236 * q^22 + 336 * q^23 - 92 * q^24 + 56 * q^26 - 964 * q^27 + 176 * q^29 + 392 * q^31 + 770 * q^32 - 188 * q^33 + 812 * q^34 + 230 * q^36 + 140 * q^37 - 20 * q^38 + 140 * q^39 + 656 * q^41 + 388 * q^43 - 160 * q^44 - 388 * q^46 - 628 * q^47 - 1396 * q^48 + 744 * q^51 - 1520 * q^52 + 676 * q^53 + 2284 * q^54 - 1468 * q^57 + 2012 * q^58 + 996 * q^59 + 740 * q^61 + 364 * q^62 + 1426 * q^64 - 3620 * q^66 - 1768 * q^67 + 2940 * q^68 - 1048 * q^69 - 224 * q^71 - 2858 * q^72 - 2640 * q^73 + 928 * q^74 - 1340 * q^76 - 8 * q^78 + 1636 * q^79 + 4442 * q^81 + 1756 * q^82 - 140 * q^83 + 1180 * q^86 - 1940 * q^87 + 5652 * q^88 - 1904 * q^89 + 1952 * q^92 + 1592 * q^93 - 3332 * q^94 - 6460 * q^96 - 516 * q^97 - 2804 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	5.51797	1.95090	0.975449	−	0.220225i	$-0.0706790\pi$
$2$	5.51797	1.95090	0.975449	+	0.220225i	$0.0706790\pi$
$3$	−3.86039	−0.742933	−0.371466	−	0.928446i	$-0.621145\pi$
$3$	−3.86039	−0.742933	−0.371466	+	0.928446i	$0.621145\pi$
$4$	22.4480	2.80600
$5$	0	0
$5$	0	0
$6$	−21.3015	−1.44939
$7$	0	0
$7$	0	0
$8$	79.7239	3.52333
$9$	−12.0974	−0.448051
$10$	0	0
$11$	34.5211	0.946227	0.473113	−	0.881002i	$-0.343130\pi$
$11$	34.5211	0.946227	0.473113	+	0.881002i	$0.343130\pi$
$12$	−86.6582	−2.08467
$13$	−68.8935	−1.46982	−0.734908	−	0.678167i	$-0.762775\pi$
$13$	−68.8935	−1.46982	−0.734908	+	0.678167i	$0.762775\pi$
$14$	0	0
$15$	0	0
$16$	260.330	4.06766
$17$	91.4346	1.30448	0.652239	−	0.758013i	$-0.273830\pi$
$17$	91.4346	1.30448	0.652239	+	0.758013i	$0.273830\pi$
$18$	−66.7530	−0.874102
$19$	11.8278	0.142815	0.0714077	−	0.997447i	$-0.477251\pi$
$19$	11.8278	0.142815	0.0714077	+	0.997447i	$0.477251\pi$
$20$	0	0
$21$	0	0
$22$	190.486	1.84599
$23$	0.104165	0.000944348 0	0.000472174	−	1.00000i	$-0.499850\pi$
$23$	0.104165	0.000944348 0	0.000472174		1.00000i	$0.499850\pi$
$24$	−307.765	−2.61760
$25$	0	0
$26$	−380.152	−2.86746
$27$	150.931	1.07580
$28$	0	0
$29$	190.863	1.22215	0.611076	−	0.791572i	$-0.290737\pi$
$29$	190.863	1.22215	0.611076	+	0.791572i	$0.290737\pi$
$30$	0	0
$31$	159.802	0.925847	0.462924	−	0.886398i	$-0.346800\pi$
$31$	159.802	0.925847	0.462924	+	0.886398i	$0.346800\pi$
$32$	798.703	4.41225
$33$	−133.265	−0.702983
$34$	504.534	2.54491
$35$	0	0
$36$	−271.562	−1.25723
$37$	177.908	0.790482	0.395241	−	0.918577i	$-0.370661\pi$
$37$	177.908	0.790482	0.395241	+	0.918577i	$0.370661\pi$
$38$	65.2657	0.278618
$39$	265.956	1.09197
$40$	0	0
$41$	145.247	0.553264	0.276632	−	0.960976i	$-0.410782\pi$
$41$	145.247	0.553264	0.276632	+	0.960976i	$0.410782\pi$
$42$	0	0
$43$	−8.25729	−0.0292843	−0.0146421	−	0.999893i	$-0.504661\pi$
$43$	−8.25729	−0.0292843	−0.0146421	+	0.999893i	$0.504661\pi$
$44$	774.930	2.65512
$45$	0	0
$46$	0.574782	0.00184233
$47$	−260.529	−0.808553	−0.404277	−	0.914637i	$-0.632477\pi$
$47$	−260.529	−0.808553	−0.404277	+	0.914637i	$0.632477\pi$
$48$	−1004.98	−3.02199
$49$	0	0
$50$	0	0
$51$	−352.973	−0.969140
$52$	−1546.52	−4.12431
$53$	−353.107	−0.915151	−0.457576	−	0.889171i	$-0.651282\pi$
$53$	−353.107	−0.915151	−0.457576	+	0.889171i	$0.651282\pi$
$54$	832.834	2.09879
$55$	0	0
$56$	0	0
$57$	−45.6601	−0.106102
$58$	1053.18	2.38429
$59$	240.495	0.530673	0.265337	−	0.964156i	$-0.414517\pi$
$59$	240.495	0.530673	0.265337	+	0.964156i	$0.414517\pi$
$60$	0	0
$61$	778.188	1.63339	0.816695	−	0.577069i	$-0.195804\pi$
$61$	778.188	1.63339	0.816695	+	0.577069i	$0.195804\pi$
$62$	881.783	1.80623
$63$	0	0
$64$	2324.58	4.54020
$65$	0	0
$66$	−735.352	−1.37145
$67$	−151.945	−0.277059	−0.138530	−	0.990358i	$-0.544238\pi$
$67$	−151.945	−0.277059	−0.138530	+	0.990358i	$0.544238\pi$
$68$	2052.53	3.66037
$69$	−0.402119	−0.000701587 0
$70$	0	0
$71$	−311.449	−0.520594	−0.260297	−	0.965529i	$-0.583820\pi$
$71$	−311.449	−0.520594	−0.260297	+	0.965529i	$0.583820\pi$
$72$	−964.450	−1.57863
$73$	−639.888	−1.02593	−0.512967	−	0.858408i	$-0.671454\pi$
$73$	−639.888	−1.02593	−0.512967	+	0.858408i	$0.671454\pi$
$74$	981.690	1.54215
$75$	0	0
$76$	265.512	0.400740
$77$	0	0
$78$	1467.54	2.13033
$79$	391.186	0.557112	0.278556	−	0.960420i	$-0.410144\pi$
$79$	391.186	0.557112	0.278556	+	0.960420i	$0.410144\pi$
$80$	0	0
$81$	−256.024	−0.351199
$82$	801.472	1.07936
$83$	−493.205	−0.652245	−0.326122	−	0.945328i	$-0.605742\pi$
$83$	−493.205	−0.652245	−0.326122	+	0.945328i	$0.605742\pi$
$84$	0	0
$85$	0	0
$86$	−45.5635	−0.0571307
$87$	−736.807	−0.907977
$88$	2752.15	3.33387
$89$	473.850	0.564359	0.282180	−	0.959362i	$-0.408943\pi$
$89$	473.850	0.564359	0.282180	+	0.959362i	$0.408943\pi$
$90$	0	0
$91$	0	0
$92$	2.33831	0.00264984
$93$	−616.898	−0.687842
$94$	−1437.59	−1.57741
$95$	0	0
$96$	−3083.31	−3.27801
$97$	−839.005	−0.878227	−0.439114	−	0.898431i	$-0.644707\pi$
$97$	−839.005	−0.878227	−0.439114	+	0.898431i	$0.644707\pi$
$98$	0	0
$99$	−417.615	−0.423958
$100$	0	0
$101$	887.440	0.874293	0.437146	−	0.899390i	$-0.355989\pi$
$101$	887.440	0.874293	0.437146	+	0.899390i	$0.355989\pi$
$102$	−1947.70	−1.89069
$103$	−619.087	−0.592238	−0.296119	−	0.955151i	$-0.595692\pi$
$103$	−619.087	−0.592238	−0.296119	+	0.955151i	$0.595692\pi$
$104$	−5492.46	−5.17865
$105$	0	0
$106$	−1948.44	−1.78537
$107$	2151.30	1.94368	0.971841	−	0.235639i	$-0.0757183\pi$
$107$	2151.30	1.94368	0.971841	+	0.235639i	$0.0757183\pi$
$108$	3388.11	3.01871
$109$	−407.076	−0.357714	−0.178857	−	0.983875i	$-0.557240\pi$
$109$	−407.076	−0.357714	−0.178857	+	0.983875i	$0.557240\pi$
$110$	0	0
$111$	−686.793	−0.587275
$112$	0	0
$113$	349.581	0.291025	0.145513	−	0.989356i	$-0.453517\pi$
$113$	349.581	0.291025	0.145513	+	0.989356i	$0.453517\pi$
$114$	−251.951	−0.206995
$115$	0	0
$116$	4284.50	3.42936
$117$	833.431	0.658553
$118$	1327.04	1.03529
$119$	0	0
$120$	0	0
$121$	−139.296	−0.104655
$122$	4294.02	3.18658
$123$	−560.712	−0.411038
$124$	3587.24	2.59793
$125$	0	0
$126$	0	0
$127$	−1183.78	−0.827114	−0.413557	−	0.910478i	$-0.635714\pi$
$127$	−1183.78	−0.827114	−0.413557	+	0.910478i	$0.635714\pi$
$128$	6437.36	4.44522
$129$	31.8764	0.0217563
$130$	0	0
$131$	223.357	0.148968	0.0744840	−	0.997222i	$-0.476269\pi$
$131$	223.357	0.148968	0.0744840	+	0.997222i	$0.476269\pi$
$132$	−2991.53	−1.97257
$133$	0	0
$134$	−838.426	−0.540515
$135$	0	0
$136$	7289.52	4.59611
$137$	−2036.66	−1.27010	−0.635050	−	0.772471i	$-0.719021\pi$
$137$	−2036.66	−1.27010	−0.635050	+	0.772471i	$0.719021\pi$
$138$	−2.21888	−0.00136872
$139$	2687.00	1.63963	0.819815	−	0.572629i	$-0.194076\pi$
$139$	2687.00	1.63963	0.819815	+	0.572629i	$0.194076\pi$
$140$	0	0
$141$	1005.74	0.600701
$142$	−1718.57	−1.01563
$143$	−2378.28	−1.39078
$144$	−3149.31	−1.82252
$145$	0	0
$146$	−3530.88	−2.00149
$147$	0	0
$148$	3993.68	2.21810
$149$	673.500	0.370304	0.185152	−	0.982710i	$-0.440722\pi$
$149$	673.500	0.370304	0.185152	+	0.982710i	$0.440722\pi$
$150$	0	0
$151$	−2125.18	−1.14533	−0.572664	−	0.819790i	$-0.694090\pi$
$151$	−2125.18	−1.14533	−0.572664	+	0.819790i	$0.694090\pi$
$152$	942.961	0.503186
$153$	−1106.12	−0.584473
$154$	0	0
$155$	0	0
$156$	5970.18	3.06409
$157$	2813.03	1.42997	0.714983	−	0.699142i	$-0.246435\pi$
$157$	2813.03	1.42997	0.714983	+	0.699142i	$0.246435\pi$
$158$	2158.55	1.08687
$159$	1363.13	0.679896
$160$	0	0
$161$	0	0
$162$	−1412.73	−0.685153
$163$	1344.42	0.646032	0.323016	−	0.946394i	$-0.395303\pi$
$163$	1344.42	0.646032	0.323016	+	0.946394i	$0.395303\pi$
$164$	3260.52	1.55246
$165$	0	0
$166$	−2721.49	−1.27246
$167$	−1451.24	−0.672456	−0.336228	−	0.941781i	$-0.609151\pi$
$167$	−1451.24	−0.672456	−0.336228	+	0.941781i	$0.609151\pi$
$168$	0	0
$169$	2549.31	1.16036
$170$	0	0
$171$	−143.086	−0.0639886
$172$	−185.360	−0.0821718
$173$	1979.16	0.869784	0.434892	−	0.900483i	$-0.356787\pi$
$173$	1979.16	0.869784	0.434892	+	0.900483i	$0.356787\pi$
$174$	−4065.68	−1.77137
$175$	0	0
$176$	8986.87	3.84893
$177$	−928.403	−0.394254
$178$	2614.69	1.10101
$179$	−4358.66	−1.82001	−0.910005	−	0.414598i	$-0.863922\pi$
$179$	−4358.66	−1.82001	−0.910005	+	0.414598i	$0.863922\pi$
$180$	0	0
$181$	−377.923	−0.155198	−0.0775988	−	0.996985i	$-0.524725\pi$
$181$	−377.923	−0.155198	−0.0775988	+	0.996985i	$0.524725\pi$
$182$	0	0
$183$	−3004.11	−1.21350
$184$	8.30447	0.00332725
$185$	0	0
$186$	−3404.03	−1.34191
$187$	3156.42	1.23433
$188$	−5848.36	−2.26880
$189$	0	0
$190$	0	0
$191$	−2425.95	−0.919033	−0.459517	−	0.888169i	$-0.651977\pi$
$191$	−2425.95	−0.919033	−0.459517	+	0.888169i	$0.651977\pi$
$192$	−8973.80	−3.37306
$193$	622.923	0.232326	0.116163	−	0.993230i	$-0.462940\pi$
$193$	622.923	0.232326	0.116163	+	0.993230i	$0.462940\pi$
$194$	−4629.61	−1.71333
$195$	0	0
$196$	0	0
$197$	−2842.29	−1.02794	−0.513971	−	0.857807i	$-0.671826\pi$
$197$	−2842.29	−1.02794	−0.513971	+	0.857807i	$0.671826\pi$
$198$	−2304.39	−0.827099
$199$	−867.364	−0.308974	−0.154487	−	0.987995i	$-0.549372\pi$
$199$	−867.364	−0.308974	−0.154487	+	0.987995i	$0.549372\pi$
$200$	0	0
$201$	586.565	0.205836
$202$	4896.87	1.70566
$203$	0	0
$204$	−7923.55	−2.71941
$205$	0	0
$206$	−3416.11	−1.15540
$207$	−1.26013	−0.000423116 0
$208$	−17935.0	−5.97871
$209$	408.309	0.135136
$210$	0	0
$211$	−5975.92	−1.94976	−0.974880	−	0.222730i	$-0.928503\pi$
$211$	−5975.92	−1.94976	−0.974880	+	0.222730i	$0.928503\pi$
$212$	−7926.57	−2.56792
$213$	1202.31	0.386766
$214$	11870.8	3.79192
$215$	0	0
$216$	12032.8	3.79042
$217$	0	0
$218$	−2246.24	−0.697864
$219$	2470.22	0.762200
$220$	0	0
$221$	−6299.25	−1.91734
$222$	−3789.71	−1.14571
$223$	5181.58	1.55598	0.777992	−	0.628274i	$-0.216238\pi$
$223$	5181.58	1.55598	0.777992	+	0.628274i	$0.216238\pi$
$224$	0	0
$225$	0	0
$226$	1928.98	0.567761
$227$	−3753.06	−1.09735	−0.548677	−	0.836035i	$-0.684868\pi$
$227$	−3753.06	−1.09735	−0.548677	+	0.836035i	$0.684868\pi$
$228$	−1024.98	−0.297723
$229$	−6258.14	−1.80589	−0.902947	−	0.429752i	$-0.858601\pi$
$229$	−6258.14	−1.80589	−0.902947	+	0.429752i	$0.858601\pi$
$230$	0	0
$231$	0	0
$232$	15216.4	4.30605
$233$	−1779.96	−0.500469	−0.250234	−	0.968185i	$-0.580508\pi$
$233$	−1779.96	−0.500469	−0.250234	+	0.968185i	$0.580508\pi$
$234$	4598.85	1.28477
$235$	0	0
$236$	5398.63	1.48907
$237$	−1510.13	−0.413897
$238$	0	0
$239$	3519.46	0.952532	0.476266	−	0.879301i	$-0.341990\pi$
$239$	3519.46	0.952532	0.476266	+	0.879301i	$0.341990\pi$
$240$	0	0
$241$	362.930	0.0970058	0.0485029	−	0.998823i	$-0.484555\pi$
$241$	362.930	0.0970058	0.0485029	+	0.998823i	$0.484555\pi$
$242$	−768.629	−0.204171
$243$	−3086.79	−0.814887
$244$	17468.8	4.58330
$245$	0	0
$246$	−3093.99	−0.801894
$247$	−814.861	−0.209912
$248$	12740.0	3.26207
$249$	1903.97	0.484574
$250$	0	0
$251$	−5333.85	−1.34131	−0.670656	−	0.741768i	$-0.733988\pi$
$251$	−5333.85	−1.34131	−0.670656	+	0.741768i	$0.733988\pi$
$252$	0	0
$253$	3.59590	0.000893567 0
$254$	−6532.06	−1.61361
$255$	0	0
$256$	16924.5	4.13197
$257$	2438.78	0.591933	0.295966	−	0.955198i	$-0.404358\pi$
$257$	2438.78	0.591933	0.295966	+	0.955198i	$0.404358\pi$
$258$	175.893	0.0424442
$259$	0	0
$260$	0	0
$261$	−2308.95	−0.547587
$262$	1232.48	0.290621
$263$	−1526.25	−0.357843	−0.178921	−	0.983863i	$-0.557261\pi$
$263$	−1526.25	−0.357843	−0.178921	+	0.983863i	$0.557261\pi$
$264$	−10624.4	−2.47684
$265$	0	0
$266$	0	0
$267$	−1829.25	−0.419281
$268$	−3410.86	−0.777430
$269$	−7564.12	−1.71447	−0.857235	−	0.514925i	$-0.827820\pi$
$269$	−7564.12	−1.71447	−0.857235	+	0.514925i	$0.827820\pi$
$270$	0	0
$271$	4282.68	0.959980	0.479990	−	0.877274i	$-0.340640\pi$
$271$	4282.68	0.959980	0.479990	+	0.877274i	$0.340640\pi$
$272$	23803.2	5.30617
$273$	0	0
$274$	−11238.2	−2.47784
$275$	0	0
$276$	−9.02679	−0.00196866
$277$	4008.41	0.869465	0.434732	−	0.900560i	$-0.356843\pi$
$277$	4008.41	0.869465	0.434732	+	0.900560i	$0.356843\pi$
$278$	14826.8	3.19875
$279$	−1933.18	−0.414827
$280$	0	0
$281$	6935.44	1.47236	0.736181	−	0.676785i	$-0.236627\pi$
$281$	6935.44	1.47236	0.736181	+	0.676785i	$0.236627\pi$
$282$	5549.66	1.17191
$283$	2666.64	0.560124	0.280062	−	0.959982i	$-0.409645\pi$
$283$	2666.64	0.560124	0.280062	+	0.959982i	$0.409645\pi$
$284$	−6991.41	−1.46079
$285$	0	0
$286$	−13123.3	−2.71327
$287$	0	0
$288$	−9662.22	−1.97692
$289$	3447.28	0.701665
$290$	0	0
$291$	3238.89	0.652464
$292$	−14364.2	−2.87878
$293$	−5939.10	−1.18418	−0.592092	−	0.805870i	$-0.701698\pi$
$293$	−5939.10	−1.18418	−0.592092	+	0.805870i	$0.701698\pi$
$294$	0	0
$295$	0	0
$296$	14183.5	2.78513
$297$	5210.31	1.01795
$298$	3716.36	0.722425
$299$	−7.17632	−0.00138802
$300$	0	0
$301$	0	0
$302$	−11726.7	−2.23442
$303$	−3425.87	−0.649541
$304$	3079.14	0.580924
$305$	0	0
$306$	−6103.53	−1.14025
$307$	−10381.5	−1.92998	−0.964992	−	0.262278i	$-0.915526\pi$
$307$	−10381.5	−1.92998	−0.964992	+	0.262278i	$0.915526\pi$
$308$	0	0
$309$	2389.92	0.439993
$310$	0	0
$311$	4240.54	0.773181	0.386590	−	0.922252i	$-0.373653\pi$
$311$	4240.54	0.773181	0.386590	+	0.922252i	$0.373653\pi$
$312$	21203.0	3.84739
$313$	283.903	0.0512688	0.0256344	−	0.999671i	$-0.491839\pi$
$313$	283.903	0.0512688	0.0256344	+	0.999671i	$0.491839\pi$
$314$	15522.2	2.78972
$315$	0	0
$316$	8781.36	1.56326
$317$	−1739.49	−0.308201	−0.154100	−	0.988055i	$-0.549248\pi$
$317$	−1739.49	−0.308201	−0.154100	+	0.988055i	$0.549248\pi$
$318$	7521.73	1.32641
$319$	6588.80	1.15643
$320$	0	0
$321$	−8304.85	−1.44402
$322$	0	0
$323$	1081.47	0.186300
$324$	−5747.24	−0.985466
$325$	0	0
$326$	7418.48	1.26034
$327$	1571.47	0.265758
$328$	11579.7	1.94933
$329$	0	0
$330$	0	0
$331$	5606.96	0.931077	0.465538	−	0.885028i	$-0.345861\pi$
$331$	5606.96	0.931077	0.465538	+	0.885028i	$0.345861\pi$
$332$	−11071.5	−1.83020
$333$	−2152.22	−0.354176
$334$	−8007.89	−1.31189
$335$	0	0
$336$	0	0
$337$	9427.44	1.52387	0.761937	−	0.647652i	$-0.224249\pi$
$337$	9427.44	1.52387	0.761937	+	0.647652i	$0.224249\pi$
$338$	14067.0	2.26375
$339$	−1349.52	−0.216212
$340$	0	0
$341$	5516.53	0.876062
$342$	−789.544	−0.124835
$343$	0	0
$344$	−658.303	−0.103178
$345$	0	0
$346$	10920.9	1.69686
$347$	−11634.2	−1.79988	−0.899939	−	0.436017i	$-0.856389\pi$
$347$	−11634.2	−1.79988	−0.899939	+	0.436017i	$0.856389\pi$
$348$	−16539.9	−2.54779
$349$	−1317.10	−0.202013	−0.101006	−	0.994886i	$-0.532206\pi$
$349$	−1317.10	−0.202013	−0.101006	+	0.994886i	$0.532206\pi$
$350$	0	0
$351$	−10398.2	−1.58124
$352$	27572.1	4.17499
$353$	−5848.82	−0.881874	−0.440937	−	0.897538i	$-0.645354\pi$
$353$	−5848.82	−0.881874	−0.440937	+	0.897538i	$0.645354\pi$
$354$	−5122.90	−0.769150
$355$	0	0
$356$	10637.0	1.58359
$357$	0	0
$358$	−24051.0	−3.55065
$359$	−12422.0	−1.82621	−0.913104	−	0.407727i	$-0.866321\pi$
$359$	−12422.0	−1.82621	−0.913104	+	0.407727i	$0.866321\pi$
$360$	0	0
$361$	−6719.10	−0.979604
$362$	−2085.37	−0.302775
$363$	537.735	0.0777515
$364$	0	0
$365$	0	0
$366$	−16576.6	−2.36741
$367$	6590.78	0.937427	0.468713	−	0.883350i	$-0.344718\pi$
$367$	6590.78	0.937427	0.468713	+	0.883350i	$0.344718\pi$
$368$	27.1174	0.00384128
$369$	−1757.11	−0.247891
$370$	0	0
$371$	0	0
$372$	−13848.1	−1.93009
$373$	−344.278	−0.0477910	−0.0238955	−	0.999714i	$-0.507607\pi$
$373$	−344.278	−0.0477910	−0.0238955	+	0.999714i	$0.507607\pi$
$374$	17417.0	2.40806
$375$	0	0
$376$	−20770.4	−2.84880
$377$	−13149.2	−1.79634
$378$	0	0
$379$	5241.23	0.710353	0.355177	−	0.934799i	$-0.384421\pi$
$379$	5241.23	0.710353	0.355177	+	0.934799i	$0.384421\pi$
$380$	0	0
$381$	4569.85	0.614490
$382$	−13386.3	−1.79294
$383$	−7597.37	−1.01360	−0.506798	−	0.862065i	$-0.669171\pi$
$383$	−7597.37	−1.01360	−0.506798	+	0.862065i	$0.669171\pi$
$384$	−24850.7	−3.30250
$385$	0	0
$386$	3437.27	0.453245
$387$	99.8916	0.0131209
$388$	−18834.0	−2.46431
$389$	3101.84	0.404291	0.202146	−	0.979355i	$-0.435209\pi$
$389$	3101.84	0.404291	0.202146	+	0.979355i	$0.435209\pi$
$390$	0	0
$391$	9.52432	0.00123188
$392$	0	0
$393$	−862.246	−0.110673
$394$	−15683.7	−2.00541
$395$	0	0
$396$	−9374.63	−1.18963
$397$	2932.06	0.370669	0.185335	−	0.982675i	$-0.440663\pi$
$397$	2932.06	0.370669	0.185335	+	0.982675i	$0.440663\pi$
$398$	−4786.09	−0.602777
$399$	0	0
$400$	0	0
$401$	89.2375	0.0111130	0.00555649	−	0.999985i	$-0.498231\pi$
$401$	89.2375	0.0111130	0.00555649	+	0.999985i	$0.498231\pi$
$402$	3236.65	0.401566
$403$	−11009.3	−1.36083
$404$	19921.3	2.45327
$405$	0	0
$406$	0	0
$407$	6141.56	0.747975
$408$	−28140.4	−3.41460
$409$	147.388	0.0178188	0.00890940	−	0.999960i	$-0.497164\pi$
$409$	147.388	0.0178188	0.00890940	+	0.999960i	$0.497164\pi$
$410$	0	0
$411$	7862.31	0.943599
$412$	−13897.3	−1.66182
$413$	0	0
$414$	−6.95336	−0.000825457 0
$415$	0	0
$416$	−55025.4	−6.48520
$417$	−10372.9	−1.21813
$418$	2253.04	0.263636
$419$	−3781.67	−0.440923	−0.220462	−	0.975396i	$-0.570756\pi$
$419$	−3781.67	−0.440923	−0.220462	+	0.975396i	$0.570756\pi$
$420$	0	0
$421$	−10899.2	−1.26175	−0.630874	−	0.775885i	$-0.717304\pi$
$421$	−10899.2	−1.26175	−0.630874	+	0.775885i	$0.717304\pi$
$422$	−32975.0	−3.80378
$423$	3151.71	0.362273
$424$	−28151.1	−3.22438
$425$	0	0
$426$	6634.33	0.754541
$427$	0	0
$428$	48292.4	5.45398
$429$	9181.08	1.03326
$430$	0	0
$431$	−11283.4	−1.26102	−0.630512	−	0.776180i	$-0.717155\pi$
$431$	−11283.4	−1.26102	−0.630512	+	0.776180i	$0.717155\pi$
$432$	39291.9	4.37600
$433$	8906.19	0.988462	0.494231	−	0.869331i	$-0.335450\pi$
$433$	8906.19	0.988462	0.494231	+	0.869331i	$0.335450\pi$
$434$	0	0
$435$	0	0
$436$	−9138.07	−1.00375
$437$	1.23205	0.000134867 0
$438$	13630.6	1.48697
$439$	8149.49	0.886000	0.443000	−	0.896522i	$-0.353914\pi$
$439$	8149.49	0.886000	0.443000	+	0.896522i	$0.353914\pi$
$440$	0	0
$441$	0	0
$442$	−34759.1	−3.74054
$443$	11472.8	1.23045	0.615223	−	0.788353i	$-0.289066\pi$
$443$	11472.8	1.23045	0.615223	+	0.788353i	$0.289066\pi$
$444$	−15417.2	−1.64790
$445$	0	0
$446$	28591.8	3.03557
$447$	−2599.97	−0.275111
$448$	0	0
$449$	1963.80	0.206409	0.103204	−	0.994660i	$-0.467090\pi$
$449$	1963.80	0.206409	0.103204	+	0.994660i	$0.467090\pi$
$450$	0	0
$451$	5014.10	0.523514
$452$	7847.42	0.816618
$453$	8204.02	0.850902
$454$	−20709.3	−2.14083
$455$	0	0
$456$	−3640.20	−0.373833
$457$	−5589.85	−0.572171	−0.286086	−	0.958204i	$-0.592354\pi$
$457$	−5589.85	−0.572171	−0.286086	+	0.958204i	$0.592354\pi$
$458$	−34532.3	−3.52311
$459$	13800.3	1.40336
$460$	0	0
$461$	−18790.9	−1.89844	−0.949219	−	0.314618i	$-0.898124\pi$
$461$	−18790.9	−1.89844	−0.949219	+	0.314618i	$0.898124\pi$
$462$	0	0
$463$	7892.22	0.792187	0.396094	−	0.918210i	$-0.370366\pi$
$463$	7892.22	0.792187	0.396094	+	0.918210i	$0.370366\pi$
$464$	49687.4	4.97130
$465$	0	0
$466$	−9821.78	−0.976363
$467$	−385.511	−0.0381998	−0.0190999	−	0.999818i	$-0.506080\pi$
$467$	−385.511	−0.0381998	−0.0190999	+	0.999818i	$0.506080\pi$
$468$	18708.9	1.84790
$469$	0	0
$470$	0	0
$471$	−10859.4	−1.06237
$472$	19173.2	1.86974
$473$	−285.050	−0.0277096
$474$	−8332.86	−0.807471
$475$	0	0
$476$	0	0
$477$	4271.67	0.410035
$478$	19420.3	1.85829
$479$	1694.46	0.161632	0.0808160	−	0.996729i	$-0.474247\pi$
$479$	1694.46	0.161632	0.0808160	+	0.996729i	$0.474247\pi$
$480$	0	0
$481$	−12256.7	−1.16186
$482$	2002.64	0.189248
$483$	0	0
$484$	−3126.91	−0.293662
$485$	0	0
$486$	−17032.8	−1.58976
$487$	−15711.2	−1.46189	−0.730945	−	0.682436i	$-0.760921\pi$
$487$	−15711.2	−1.46189	−0.730945	+	0.682436i	$0.760921\pi$
$488$	62040.2	5.75498
$489$	−5189.99	−0.479958
$490$	0	0
$491$	−2716.34	−0.249667	−0.124834	−	0.992178i	$-0.539840\pi$
$491$	−2716.34	−0.249667	−0.124834	+	0.992178i	$0.539840\pi$
$492$	−12586.9	−1.15337
$493$	17451.5	1.59427
$494$	−4496.38	−0.409518
$495$	0	0
$496$	41601.2	3.76603
$497$	0	0
$498$	10506.0	0.945355
$499$	4295.34	0.385342	0.192671	−	0.981263i	$-0.438285\pi$
$499$	4295.34	0.385342	0.192671	+	0.981263i	$0.438285\pi$
$500$	0	0
$501$	5602.34	0.499589
$502$	−29432.0	−2.61677
$503$	−6515.26	−0.577537	−0.288769	−	0.957399i	$-0.593246\pi$
$503$	−6515.26	−0.577537	−0.288769	+	0.957399i	$0.593246\pi$
$504$	0	0
$505$	0	0
$506$	19.8421	0.00174326
$507$	−9841.34	−0.862070
$508$	−26573.5	−2.32088
$509$	14391.8	1.25325	0.626624	−	0.779321i	$-0.284436\pi$
$509$	14391.8	1.25325	0.626624	+	0.779321i	$0.284436\pi$
$510$	0	0
$511$	0	0
$512$	41890.2	3.61583
$513$	1785.19	0.153641
$514$	13457.1	1.15480
$515$	0	0
$516$	715.562	0.0610481
$517$	−8993.73	−0.765075
$518$	0	0
$519$	−7640.32	−0.646191
$520$	0	0
$521$	2913.36	0.244984	0.122492	−	0.992469i	$-0.460911\pi$
$521$	2913.36	0.244984	0.122492	+	0.992469i	$0.460911\pi$
$522$	−12740.7	−1.06829
$523$	−16870.2	−1.41048	−0.705242	−	0.708967i	$-0.749162\pi$
$523$	−16870.2	−1.41048	−0.705242	+	0.708967i	$0.749162\pi$
$524$	5013.93	0.418005
$525$	0	0
$526$	−8421.82	−0.698115
$527$	14611.4	1.20775
$528$	−34692.8	−2.85949
$529$	−12167.0	−0.999999
$530$	0	0
$531$	−2909.35	−0.237769
$532$	0	0
$533$	−10006.6	−0.813197
$534$	−10093.7	−0.817974
$535$	0	0
$536$	−12113.6	−0.976172
$537$	16826.1	1.35214
$538$	−41738.6	−3.34476
$539$	0	0
$540$	0	0
$541$	−2229.59	−0.177186	−0.0885929	−	0.996068i	$-0.528237\pi$
$541$	−2229.59	−0.177186	−0.0885929	+	0.996068i	$0.528237\pi$
$542$	23631.7	1.87282
$543$	1458.93	0.115301
$544$	73029.1	5.75569
$545$	0	0
$546$	0	0
$547$	1218.73	0.0952639	0.0476319	−	0.998865i	$-0.484833\pi$
$547$	1218.73	0.0952639	0.0476319	+	0.998865i	$0.484833\pi$
$548$	−45719.1	−3.56391
$549$	−9414.04	−0.731843
$550$	0	0
$551$	2257.50	0.174542
$552$	−32.0585	−0.00247192
$553$	0	0
$554$	22118.3	1.69624
$555$	0	0
$556$	60317.9	4.60081
$557$	22734.5	1.72943	0.864714	−	0.502264i	$-0.167499\pi$
$557$	22734.5	1.72943	0.864714	+	0.502264i	$0.167499\pi$
$558$	−10667.3	−0.809285
$559$	568.873	0.0430425
$560$	0	0
$561$	−12185.0	−0.917026
$562$	38269.6	2.87243
$563$	−4302.11	−0.322047	−0.161023	−	0.986951i	$-0.551479\pi$
$563$	−4302.11	−0.322047	−0.161023	+	0.986951i	$0.551479\pi$
$564$	22576.9	1.68557
$565$	0	0
$566$	14714.4	1.09275
$567$	0	0
$568$	−24829.9	−1.83422
$569$	14866.9	1.09535	0.547675	−	0.836691i	$-0.315513\pi$
$569$	14866.9	1.09535	0.547675	+	0.836691i	$0.315513\pi$
$570$	0	0
$571$	16514.5	1.21035	0.605174	−	0.796093i	$-0.293103\pi$
$571$	16514.5	1.21035	0.605174	+	0.796093i	$0.293103\pi$
$572$	−53387.6	−3.90253
$573$	9365.10	0.682780
$574$	0	0
$575$	0	0
$576$	−28121.4	−2.03424
$577$	−1867.97	−0.134774	−0.0673870	−	0.997727i	$-0.521466\pi$
$577$	−1867.97	−0.134774	−0.0673870	+	0.997727i	$0.521466\pi$
$578$	19022.0	1.36888
$579$	−2404.73	−0.172603
$580$	0	0
$581$	0	0
$582$	17872.1	1.27289
$583$	−12189.6	−0.865941
$584$	−51014.3	−3.61471
$585$	0	0
$586$	−32771.8	−2.31022
$587$	−3926.15	−0.276064	−0.138032	−	0.990428i	$-0.544078\pi$
$587$	−3926.15	−0.276064	−0.138032	+	0.990428i	$0.544078\pi$
$588$	0	0
$589$	1890.11	0.132225
$590$	0	0
$591$	10972.3	0.763692
$592$	46314.7	3.21541
$593$	7554.18	0.523125	0.261562	−	0.965187i	$-0.415762\pi$
$593$	7554.18	0.523125	0.261562	+	0.965187i	$0.415762\pi$
$594$	28750.3	1.98593
$595$	0	0
$596$	15118.8	1.03907
$597$	3348.36	0.229547
$598$	−39.5988	−0.00270788
$599$	28210.9	1.92432	0.962160	−	0.272487i	$-0.0878461\pi$
$599$	28210.9	1.92432	0.962160	+	0.272487i	$0.0878461\pi$
$600$	0	0
$601$	−23181.4	−1.57336	−0.786679	−	0.617363i	$-0.788201\pi$
$601$	−23181.4	−1.57336	−0.786679	+	0.617363i	$0.788201\pi$
$602$	0	0
$603$	1838.13	0.124137
$604$	−47706.1	−3.21380
$605$	0	0
$606$	−18903.8	−1.26719
$607$	24863.6	1.66257	0.831287	−	0.555844i	$-0.187605\pi$
$607$	24863.6	1.66257	0.831287	+	0.555844i	$0.187605\pi$
$608$	9446.93	0.630137
$609$	0	0
$610$	0	0
$611$	17948.7	1.18843
$612$	−24830.2	−1.64003
$613$	12792.7	0.842893	0.421446	−	0.906853i	$-0.361523\pi$
$613$	12792.7	0.842893	0.421446	+	0.906853i	$0.361523\pi$
$614$	−57285.0	−3.76520
$615$	0	0
$616$	0	0
$617$	−19793.3	−1.29149	−0.645744	−	0.763554i	$-0.723453\pi$
$617$	−19793.3	−1.29149	−0.645744	+	0.763554i	$0.723453\pi$
$618$	13187.5	0.858381
$619$	−20951.7	−1.36045	−0.680226	−	0.733002i	$-0.738118\pi$
$619$	−20951.7	−1.36045	−0.680226	+	0.733002i	$0.738118\pi$
$620$	0	0
$621$	15.7218	0.00101593
$622$	23399.2	1.50840
$623$	0	0
$624$	69236.3	4.44178
$625$	0	0
$626$	1566.57	0.100020
$627$	−1576.23	−0.100397
$628$	63147.1	4.01249
$629$	16266.9	1.03117
$630$	0	0
$631$	23273.5	1.46831	0.734156	−	0.678981i	$-0.237578\pi$
$631$	23273.5	1.46831	0.734156	+	0.678981i	$0.237578\pi$
$632$	31186.9	1.96289
$633$	23069.4	1.44854
$634$	−9598.47	−0.601268
$635$	0	0
$636$	30599.6	1.90779
$637$	0	0
$638$	36356.8	2.25608
$639$	3767.71	0.233253
$640$	0	0
$641$	−22117.7	−1.36287	−0.681434	−	0.731880i	$-0.738643\pi$
$641$	−22117.7	−1.36287	−0.681434	+	0.731880i	$0.738643\pi$
$642$	−45826.0	−2.81714
$643$	−20269.9	−1.24318	−0.621591	−	0.783342i	$-0.713513\pi$
$643$	−20269.9	−1.24318	−0.621591	+	0.783342i	$0.713513\pi$
$644$	0	0
$645$	0	0
$646$	5967.54	0.363451
$647$	3145.25	0.191117	0.0955583	−	0.995424i	$-0.469536\pi$
$647$	3145.25	0.191117	0.0955583	+	0.995424i	$0.469536\pi$
$648$	−20411.2	−1.23739
$649$	8302.13	0.502137
$650$	0	0
$651$	0	0
$652$	30179.6	1.81277
$653$	5953.35	0.356773	0.178386	−	0.983961i	$-0.442912\pi$
$653$	5953.35	0.356773	0.178386	+	0.983961i	$0.442912\pi$
$654$	8671.35	0.518466
$655$	0	0
$656$	37812.3	2.25049
$657$	7740.96	0.459671
$658$	0	0
$659$	26277.5	1.55330	0.776652	−	0.629930i	$-0.216916\pi$
$659$	26277.5	1.55330	0.776652	+	0.629930i	$0.216916\pi$
$660$	0	0
$661$	−24004.3	−1.41250	−0.706248	−	0.707964i	$-0.749614\pi$
$661$	−24004.3	−1.41250	−0.706248	+	0.707964i	$0.749614\pi$
$662$	30939.1	1.81644
$663$	24317.6	1.42446
$664$	−39320.3	−2.29808
$665$	0	0
$666$	−11875.9	−0.690962
$667$	19.8814	0.00115414
$668$	−32577.4	−1.88691
$669$	−20002.9	−1.15599
$670$	0	0
$671$	26863.9	1.54556
$672$	0	0
$673$	−9205.36	−0.527252	−0.263626	−	0.964625i	$-0.584918\pi$
$673$	−9205.36	−0.527252	−0.263626	+	0.964625i	$0.584918\pi$
$674$	52020.4	2.97292
$675$	0	0
$676$	57227.1	3.25598
$677$	18773.1	1.06575	0.532873	−	0.846195i	$-0.321112\pi$
$677$	18773.1	1.06575	0.532873	+	0.846195i	$0.321112\pi$
$678$	−7446.62	−0.421808
$679$	0	0
$680$	0	0
$681$	14488.3	0.815260
$682$	30440.1	1.70911
$683$	10222.2	0.572684	0.286342	−	0.958127i	$-0.407561\pi$
$683$	10222.2	0.572684	0.286342	+	0.958127i	$0.407561\pi$
$684$	−3212.00	−0.179552
$685$	0	0
$686$	0	0
$687$	24158.9	1.34166
$688$	−2149.62	−0.119118
$689$	24326.8	1.34510
$690$	0	0
$691$	22355.1	1.23072	0.615361	−	0.788246i	$-0.289010\pi$
$691$	22355.1	1.23072	0.615361	+	0.788246i	$0.289010\pi$
$692$	44428.2	2.44062
$693$	0	0
$694$	−64197.3	−3.51138
$695$	0	0
$696$	−58741.1	−3.19910
$697$	13280.6	0.721722
$698$	−7267.70	−0.394107
$699$	6871.35	0.371814
$700$	0	0
$701$	16217.3	0.873779	0.436890	−	0.899515i	$-0.356080\pi$
$701$	16217.3	0.873779	0.436890	+	0.899515i	$0.356080\pi$
$702$	−57376.9	−3.08483
$703$	2104.26	0.112893
$704$	80247.1	4.29606
$705$	0	0
$706$	−32273.7	−1.72045
$707$	0	0
$708$	−20840.8	−1.10628
$709$	3922.92	0.207798	0.103899	−	0.994588i	$-0.466868\pi$
$709$	3922.92	0.207798	0.103899	+	0.994588i	$0.466868\pi$
$710$	0	0
$711$	−4732.33	−0.249615
$712$	37777.1	1.98842
$713$	16.6458	0.000874322 0
$714$	0	0
$715$	0	0
$716$	−97843.4	−5.10695
$717$	−13586.5	−0.707667
$718$	−68544.3	−3.56275
$719$	23908.6	1.24011	0.620055	−	0.784558i	$-0.287110\pi$
$719$	23908.6	1.24011	0.620055	+	0.784558i	$0.287110\pi$
$720$	0	0
$721$	0	0
$722$	−37075.8	−1.91111
$723$	−1401.05	−0.0720688
$724$	−8483.62	−0.435485
$725$	0	0
$726$	2967.21	0.151685
$727$	26906.4	1.37263	0.686315	−	0.727305i	$-0.259227\pi$
$727$	26906.4	1.37263	0.686315	+	0.727305i	$0.259227\pi$
$728$	0	0
$729$	18828.9	0.956605
$730$	0	0
$731$	−755.001	−0.0382007
$732$	−67436.4	−3.40508
$733$	−27637.5	−1.39265	−0.696325	−	0.717726i	$-0.745183\pi$
$733$	−27637.5	−1.39265	−0.696325	+	0.717726i	$0.745183\pi$
$734$	36367.7	1.82882
$735$	0	0
$736$	83.1973	0.00416670
$737$	−5245.29	−0.262161
$738$	−9695.71	−0.483610
$739$	−14038.9	−0.698821	−0.349410	−	0.936970i	$-0.613618\pi$
$739$	−14038.9	−0.698821	−0.349410	+	0.936970i	$0.613618\pi$
$740$	0	0
$741$	3145.68	0.155951
$742$	0	0
$743$	−17698.7	−0.873893	−0.436946	−	0.899488i	$-0.643940\pi$
$743$	−17698.7	−0.873893	−0.436946	+	0.899488i	$0.643940\pi$
$744$	−49181.5	−2.42350
$745$	0	0
$746$	−1899.72	−0.0932354
$747$	5966.50	0.292239
$748$	70855.4	3.46354
$749$	0	0
$750$	0	0
$751$	17199.7	0.835719	0.417859	−	0.908512i	$-0.362780\pi$
$751$	17199.7	0.835719	0.417859	+	0.908512i	$0.362780\pi$
$752$	−67823.4	−3.28892
$753$	20590.7	0.996505
$754$	−72557.1	−3.50448
$755$	0	0
$756$	0	0
$757$	19200.7	0.921879	0.460939	−	0.887432i	$-0.347513\pi$
$757$	19200.7	0.921879	0.460939	+	0.887432i	$0.347513\pi$
$758$	28921.0	1.38583
$759$	−13.8816	−0.000663860 0
$760$	0	0
$761$	−27918.7	−1.32990	−0.664949	−	0.746889i	$-0.731547\pi$
$761$	−27918.7	−1.32990	−0.664949	+	0.746889i	$0.731547\pi$
$762$	25216.3	1.19881
$763$	0	0
$764$	−54457.7	−2.57881
$765$	0	0
$766$	−41922.1	−1.97742
$767$	−16568.5	−0.779992
$768$	−65335.3	−3.06977
$769$	−7436.29	−0.348712	−0.174356	−	0.984683i	$-0.555784\pi$
$769$	−7436.29	−0.348712	−0.174356	+	0.984683i	$0.555784\pi$
$770$	0	0
$771$	−9414.64	−0.439766
$772$	13983.4	0.651908
$773$	−4989.84	−0.232176	−0.116088	−	0.993239i	$-0.537035\pi$
$773$	−4989.84	−0.232176	−0.116088	+	0.993239i	$0.537035\pi$
$774$	551.199	0.0255975
$775$	0	0
$776$	−66888.7	−3.09429
$777$	0	0
$778$	17115.9	0.788731
$779$	1717.96	0.0790146
$780$	0	0
$781$	−10751.5	−0.492600
$782$	52.5550	0.00240328
$783$	28807.2	1.31480
$784$	0	0
$785$	0	0
$786$	−4757.85	−0.215912
$787$	2870.69	0.130024	0.0650122	−	0.997884i	$-0.479291\pi$
$787$	2870.69	0.130024	0.0650122	+	0.997884i	$0.479291\pi$
$788$	−63803.8	−2.88441
$789$	5891.93	0.265853
$790$	0	0
$791$	0	0
$792$	−33293.9	−1.49374
$793$	−53612.1	−2.40078
$794$	16179.0	0.723138
$795$	0	0
$796$	−19470.6	−0.866982
$797$	−4676.61	−0.207847	−0.103923	−	0.994585i	$-0.533140\pi$
$797$	−4676.61	−0.207847	−0.103923	+	0.994585i	$0.533140\pi$
$798$	0	0
$799$	−23821.3	−1.05474
$800$	0	0
$801$	−5732.34	−0.252862
$802$	492.410	0.0216803
$803$	−22089.6	−0.970766
$804$	13167.2	0.577578
$805$	0	0
$806$	−60749.1	−2.65483
$807$	29200.5	1.27374
$808$	70750.2	3.08042
$809$	−15376.9	−0.668261	−0.334131	−	0.942527i	$-0.608443\pi$
$809$	−15376.9	−0.668261	−0.334131	+	0.942527i	$0.608443\pi$
$810$	0	0
$811$	36422.9	1.57704	0.788522	−	0.615007i	$-0.210847\pi$
$811$	36422.9	1.57704	0.788522	+	0.615007i	$0.210847\pi$
$812$	0	0
$813$	−16532.8	−0.713200
$814$	33889.0	1.45922
$815$	0	0
$816$	−91889.5	−3.94213
$817$	−97.6658	−0.00418225
$818$	813.285	0.0347627
$819$	0	0
$820$	0	0
$821$	−25479.0	−1.08310	−0.541550	−	0.840669i	$-0.682162\pi$
$821$	−25479.0	−1.08310	−0.541550	+	0.840669i	$0.682162\pi$
$822$	43384.0	1.84087
$823$	17933.3	0.759557	0.379779	−	0.925077i	$-0.376000\pi$
$823$	17933.3	0.759557	0.379779	+	0.925077i	$0.376000\pi$
$824$	−49356.1	−2.08665
$825$	0	0
$826$	0	0
$827$	−13021.6	−0.547529	−0.273764	−	0.961797i	$-0.588269\pi$
$827$	−13021.6	−0.547529	−0.273764	+	0.961797i	$0.588269\pi$
$828$	−28.2874	−0.00118727
$829$	−11397.6	−0.477509	−0.238754	−	0.971080i	$-0.576739\pi$
$829$	−11397.6	−0.477509	−0.238754	+	0.971080i	$0.576739\pi$
$830$	0	0
$831$	−15474.0	−0.645954
$832$	−160149.	−6.67326
$833$	0	0
$834$	−57237.2	−2.37646
$835$	0	0
$836$	9165.75	0.379191
$837$	24119.1	0.996031
$838$	−20867.2	−0.860196
$839$	−37681.2	−1.55053	−0.775267	−	0.631633i	$-0.782385\pi$
$839$	−37681.2	−1.55053	−0.775267	+	0.631633i	$0.782385\pi$
$840$	0	0
$841$	12039.8	0.493656
$842$	−60141.7	−2.46154
$843$	−26773.5	−1.09386
$844$	−134148.	−5.47104
$845$	0	0
$846$	17391.1	0.706758
$847$	0	0
$848$	−91924.4	−3.72252
$849$	−10294.3	−0.416135
$850$	0	0
$851$	18.5318	0.000746490 0
$852$	26989.6	1.08527
$853$	−21771.6	−0.873911	−0.436956	−	0.899483i	$-0.643943\pi$
$853$	−21771.6	−0.873911	−0.436956	+	0.899483i	$0.643943\pi$
$854$	0	0
$855$	0	0
$856$	171510.	6.84823
$857$	−29860.2	−1.19021	−0.595103	−	0.803649i	$-0.702889\pi$
$857$	−29860.2	−1.19021	−0.595103	+	0.803649i	$0.702889\pi$
$858$	50661.0	2.01578
$859$	18530.6	0.736039	0.368019	−	0.929818i	$-0.380036\pi$
$859$	18530.6	0.736039	0.368019	+	0.929818i	$0.380036\pi$
$860$	0	0
$861$	0	0
$862$	−62261.4	−2.46013
$863$	21296.5	0.840024	0.420012	−	0.907519i	$-0.362026\pi$
$863$	21296.5	0.840024	0.420012	+	0.907519i	$0.362026\pi$
$864$	120549.	4.74672
$865$	0	0
$866$	49144.1	1.92839
$867$	−13307.8	−0.521290
$868$	0	0
$869$	13504.2	0.527155
$870$	0	0
$871$	10468.0	0.407226
$872$	−32453.7	−1.26035
$873$	10149.8	0.393491
$874$	6.79843	0.000263112 0
$875$	0	0
$876$	55451.5	2.13874
$877$	18793.4	0.723614	0.361807	−	0.932253i	$-0.382160\pi$
$877$	18793.4	0.723614	0.361807	+	0.932253i	$0.382160\pi$
$878$	44968.7	1.72850
$879$	22927.3	0.879769
$880$	0	0
$881$	1638.62	0.0626634	0.0313317	−	0.999509i	$-0.490025\pi$
$881$	1638.62	0.0626634	0.0313317	+	0.999509i	$0.490025\pi$
$882$	0	0
$883$	35424.1	1.35008	0.675038	−	0.737783i	$-0.264127\pi$
$883$	35424.1	1.35008	0.675038	+	0.737783i	$0.264127\pi$
$884$	−141406.	−5.38008
$885$	0	0
$886$	63306.5	2.40048
$887$	5131.41	0.194246	0.0971229	−	0.995272i	$-0.469036\pi$
$887$	5131.41	0.194246	0.0971229	+	0.995272i	$0.469036\pi$
$888$	−54753.8	−2.06916
$889$	0	0
$890$	0	0
$891$	−8838.22	−0.332314
$892$	116316.	4.36610
$893$	−3081.49	−0.115474
$894$	−14346.6	−0.536713
$895$	0	0
$896$	0	0
$897$	27.7034	0.00103120
$898$	10836.2	0.402682
$899$	30500.3	1.13153
$900$	0	0
$901$	−32286.2	−1.19380
$902$	27667.7	1.02132
$903$	0	0
$904$	27870.0	1.02538
$905$	0	0
$906$	45269.6	1.66002
$907$	19934.6	0.729787	0.364893	−	0.931049i	$-0.381105\pi$
$907$	19934.6	0.729787	0.364893	+	0.931049i	$0.381105\pi$
$908$	−84248.8	−3.07918
$909$	−10735.7	−0.391728
$910$	0	0
$911$	48387.4	1.75976	0.879882	−	0.475193i	$-0.157622\pi$
$911$	48387.4	1.75976	0.879882	+	0.475193i	$0.157622\pi$
$912$	−11886.7	−0.431587
$913$	−17026.0	−0.617172
$914$	−30844.7	−1.11625
$915$	0	0
$916$	−140483.	−5.06735
$917$	0	0
$918$	76149.8	2.73782
$919$	−14431.7	−0.518019	−0.259009	−	0.965875i	$-0.583396\pi$
$919$	−14431.7	−0.518019	−0.259009	+	0.965875i	$0.583396\pi$
$920$	0	0
$921$	40076.8	1.43385
$922$	−103688.	−3.70366
$923$	21456.8	0.765177
$924$	0	0
$925$	0	0
$926$	43549.1	1.54548
$927$	7489.34	0.265353
$928$	152443.	5.39244
$929$	5549.82	0.196000	0.0979998	−	0.995186i	$-0.468756\pi$
$929$	5549.82	0.196000	0.0979998	+	0.995186i	$0.468756\pi$
$930$	0	0
$931$	0	0
$932$	−39956.7	−1.40432
$933$	−16370.2	−0.574421
$934$	−2127.24	−0.0745239
$935$	0	0
$936$	66444.3	2.32030
$937$	−26231.9	−0.914576	−0.457288	−	0.889319i	$-0.651179\pi$
$937$	−26231.9	−0.914576	−0.457288	+	0.889319i	$0.651179\pi$
$938$	0	0
$939$	−1095.98	−0.0380893
$940$	0	0
$941$	2836.80	0.0982753	0.0491376	−	0.998792i	$-0.484353\pi$
$941$	2836.80	0.0982753	0.0491376	+	0.998792i	$0.484353\pi$
$942$	−59921.9	−2.07257
$943$	15.1298	0.000522474 0
$944$	62607.9	2.15860
$945$	0	0
$946$	−1572.90	−0.0540586
$947$	40272.9	1.38193	0.690967	−	0.722886i	$-0.257185\pi$
$947$	40272.9	1.38193	0.690967	+	0.722886i	$0.257185\pi$
$948$	−33899.5	−1.16140
$949$	44084.1	1.50793
$950$	0	0
$951$	6715.12	0.228972
$952$	0	0
$953$	−17770.1	−0.604019	−0.302010	−	0.953305i	$-0.597657\pi$
$953$	−17770.1	−0.604019	−0.302010	+	0.953305i	$0.597657\pi$
$954$	23571.0	0.799936
$955$	0	0
$956$	79005.0	2.67281
$957$	−25435.4	−0.859152
$958$	9349.97	0.315328
$959$	0	0
$960$	0	0
$961$	−4254.35	−0.142807
$962$	−67632.0	−2.26668
$963$	−26025.1	−0.870869
$964$	8147.07	0.272199
$965$	0	0
$966$	0	0
$967$	3530.14	0.117396	0.0586978	−	0.998276i	$-0.481305\pi$
$967$	3530.14	0.117396	0.0586978	+	0.998276i	$0.481305\pi$
$968$	−11105.2	−0.368734
$969$	−4174.91	−0.138408
$970$	0	0
$971$	17650.7	0.583356	0.291678	−	0.956517i	$-0.405786\pi$
$971$	17650.7	0.583356	0.291678	+	0.956517i	$0.405786\pi$
$972$	−69292.3	−2.28658
$973$	0	0
$974$	−86693.8	−2.85200
$975$	0	0
$976$	202586.	6.64407
$977$	14477.6	0.474085	0.237042	−	0.971499i	$-0.423822\pi$
$977$	14477.6	0.474085	0.237042	+	0.971499i	$0.423822\pi$
$978$	−28638.2	−0.936349
$979$	16357.8	0.534012
$980$	0	0
$981$	4924.56	0.160274
$982$	−14988.7	−0.487075
$983$	8764.09	0.284365	0.142183	−	0.989840i	$-0.454588\pi$
$983$	8764.09	0.284365	0.142183	+	0.989840i	$0.454588\pi$
$984$	−44702.1	−1.44822
$985$	0	0
$986$	96296.9	3.11026
$987$	0	0
$988$	−18292.0	−0.589015
$989$	−0.860124	−2.76546e−5 0
$990$	0	0
$991$	33624.1	1.07781	0.538903	−	0.842368i	$-0.318839\pi$
$991$	33624.1	1.07781	0.538903	+	0.842368i	$0.318839\pi$
$992$	127634.	4.08507
$993$	−21645.1	−0.691727
$994$	0	0
$995$	0	0
$996$	42740.3	1.35972
$997$	−16631.3	−0.528302	−0.264151	−	0.964481i	$-0.585092\pi$
$997$	−16631.3	−0.528302	−0.264151	+	0.964481i	$0.585092\pi$
$998$	23701.6	0.751764
$999$	26851.8	0.850404

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.4.a.bi.1.6		6
5.4	even	2		245.4.a.p.1.1	yes	6
7.6	odd	2		1225.4.a.bj.1.6		6
15.14	odd	2		2205.4.a.ca.1.6		6
35.4	even	6		245.4.e.p.226.6		12
35.9	even	6		245.4.e.p.116.6		12
35.19	odd	6		245.4.e.q.116.6		12
35.24	odd	6		245.4.e.q.226.6		12
35.34	odd	2		245.4.a.o.1.1	✓	6
105.104	even	2		2205.4.a.bz.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.4.a.o.1.1	✓	6	35.34	odd	2
245.4.a.p.1.1	yes	6	5.4	even	2
245.4.e.p.116.6		12	35.9	even	6
245.4.e.p.226.6		12	35.4	even	6
245.4.e.q.116.6		12	35.19	odd	6
245.4.e.q.226.6		12	35.24	odd	6
1225.4.a.bi.1.6		6	1.1	even	1	trivial
1225.4.a.bj.1.6		6	7.6	odd	2
2205.4.a.bz.1.6		6	105.104	even	2
2205.4.a.ca.1.6		6	15.14	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:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q+5.51797 q^{2} -3.86039 q^{3} +22.4480 q^{4} -21.3015 q^{6} +79.7239 q^{8} -12.0974 q^{9} +O(q^{10})\)	\(q+5.51797 q^{2} -3.86039 q^{3} +22.4480 q^{4} -21.3015 q^{6} +79.7239 q^{8} -12.0974 q^{9} +34.5211 q^{11} -86.6582 q^{12} -68.8935 q^{13} +260.330 q^{16} +91.4346 q^{17} -66.7530 q^{18} +11.8278 q^{19} +190.486 q^{22} +0.104165 q^{23} -307.765 q^{24} -380.152 q^{26} +150.931 q^{27} +190.863 q^{29} +159.802 q^{31} +798.703 q^{32} -133.265 q^{33} +504.534 q^{34} -271.562 q^{36} +177.908 q^{37} +65.2657 q^{38} +265.956 q^{39} +145.247 q^{41} -8.25729 q^{43} +774.930 q^{44} +0.574782 q^{46} -260.529 q^{47} -1004.98 q^{48} -352.973 q^{51} -1546.52 q^{52} -353.107 q^{53} +832.834 q^{54} -45.6601 q^{57} +1053.18 q^{58} +240.495 q^{59} +778.188 q^{61} +881.783 q^{62} +2324.58 q^{64} -735.352 q^{66} -151.945 q^{67} +2052.53 q^{68} -0.402119 q^{69} -311.449 q^{71} -964.450 q^{72} -639.888 q^{73} +981.690 q^{74} +265.512 q^{76} +1467.54 q^{78} +391.186 q^{79} -256.024 q^{81} +801.472 q^{82} -493.205 q^{83} -45.5635 q^{86} -736.807 q^{87} +2752.15 q^{88} +473.850 q^{89} +2.33831 q^{92} -616.898 q^{93} -1437.59 q^{94} -3083.31 q^{96} -839.005 q^{97} -417.615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} - 16 q^{3} + 14 q^{4} + 24 q^{6} + 66 q^{8} + 70 q^{9}+O(q^{10}) \) 6 * q + 2 * q^2 - 16 * q^3 + 14 * q^4 + 24 * q^6 + 66 * q^8 + 70 * q^9	\( 6 q + 2 q^{2} - 16 q^{3} + 14 q^{4} + 24 q^{6} + 66 q^{8} + 70 q^{9} - 16 q^{11} - 160 q^{12} - 168 q^{13} + 298 q^{16} + 4 q^{17} - 354 q^{18} + 308 q^{19} + 236 q^{22} + 336 q^{23} - 92 q^{24} + 56 q^{26} - 964 q^{27} + 176 q^{29} + 392 q^{31} + 770 q^{32} - 188 q^{33} + 812 q^{34} + 230 q^{36} + 140 q^{37} - 20 q^{38} + 140 q^{39} + 656 q^{41} + 388 q^{43} - 160 q^{44} - 388 q^{46} - 628 q^{47} - 1396 q^{48} + 744 q^{51} - 1520 q^{52} + 676 q^{53} + 2284 q^{54} - 1468 q^{57} + 2012 q^{58} + 996 q^{59} + 740 q^{61} + 364 q^{62} + 1426 q^{64} - 3620 q^{66} - 1768 q^{67} + 2940 q^{68} - 1048 q^{69} - 224 q^{71} - 2858 q^{72} - 2640 q^{73} + 928 q^{74} - 1340 q^{76} - 8 q^{78} + 1636 q^{79} + 4442 q^{81} + 1756 q^{82} - 140 q^{83} + 1180 q^{86} - 1940 q^{87} + 5652 q^{88} - 1904 q^{89} + 1952 q^{92} + 1592 q^{93} - 3332 q^{94} - 6460 q^{96} - 516 q^{97} - 2804 q^{99}+O(q^{100}) \) 6 * q + 2 * q^2 - 16 * q^3 + 14 * q^4 + 24 * q^6 + 66 * q^8 + 70 * q^9 - 16 * q^11 - 160 * q^12 - 168 * q^13 + 298 * q^16 + 4 * q^17 - 354 * q^18 + 308 * q^19 + 236 * q^22 + 336 * q^23 - 92 * q^24 + 56 * q^26 - 964 * q^27 + 176 * q^29 + 392 * q^31 + 770 * q^32 - 188 * q^33 + 812 * q^34 + 230 * q^36 + 140 * q^37 - 20 * q^38 + 140 * q^39 + 656 * q^41 + 388 * q^43 - 160 * q^44 - 388 * q^46 - 628 * q^47 - 1396 * q^48 + 744 * q^51 - 1520 * q^52 + 676 * q^53 + 2284 * q^54 - 1468 * q^57 + 2012 * q^58 + 996 * q^59 + 740 * q^61 + 364 * q^62 + 1426 * q^64 - 3620 * q^66 - 1768 * q^67 + 2940 * q^68 - 1048 * q^69 - 224 * q^71 - 2858 * q^72 - 2640 * q^73 + 928 * q^74 - 1340 * q^76 - 8 * q^78 + 1636 * q^79 + 4442 * q^81 + 1756 * q^82 - 140 * q^83 + 1180 * q^86 - 1940 * q^87 + 5652 * q^88 - 1904 * q^89 + 1952 * q^92 + 1592 * q^93 - 3332 * q^94 - 6460 * q^96 - 516 * q^97 - 2804 * q^99

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