LMFDB - Embedded newform 1225.4.a.bi.1.3

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			1.3
Root			$-2.05886$ 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)

$f(q)$	$=$	$q-0.644648 q^{2} +4.18687 q^{3} -7.58443 q^{4} -2.69906 q^{6} +10.0465 q^{8} -9.47008 q^{9} +O(q^{10})$	\(q-0.644648 q^{2} +4.18687 q^{3} -7.58443 q^{4} -2.69906 q^{6} +10.0465 q^{8} -9.47008 q^{9} -47.7013 q^{11} -31.7550 q^{12} -57.2256 q^{13} +54.1990 q^{16} +36.9686 q^{17} +6.10487 q^{18} +30.7659 q^{19} +30.7506 q^{22} -53.1282 q^{23} +42.0633 q^{24} +36.8904 q^{26} -152.696 q^{27} -195.663 q^{29} +257.870 q^{31} -115.311 q^{32} -199.719 q^{33} -23.8317 q^{34} +71.8252 q^{36} -346.423 q^{37} -19.8332 q^{38} -239.596 q^{39} +267.050 q^{41} +176.859 q^{43} +361.787 q^{44} +34.2490 q^{46} +311.598 q^{47} +226.924 q^{48} +154.783 q^{51} +434.024 q^{52} +492.270 q^{53} +98.4349 q^{54} +128.813 q^{57} +126.134 q^{58} +98.7653 q^{59} +82.1682 q^{61} -166.235 q^{62} -359.257 q^{64} +128.749 q^{66} -654.668 q^{67} -280.386 q^{68} -222.441 q^{69} +779.658 q^{71} -95.1409 q^{72} +829.673 q^{73} +223.321 q^{74} -233.342 q^{76} +154.455 q^{78} -769.426 q^{79} -383.625 q^{81} -172.153 q^{82} -613.203 q^{83} -114.012 q^{86} -819.215 q^{87} -479.230 q^{88} -457.666 q^{89} +402.947 q^{92} +1079.67 q^{93} -200.871 q^{94} -482.793 q^{96} +1412.11 q^{97} +451.736 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.644648	−0.227917	−0.113959	−	0.993485i	$-0.536353\pi$
$2$	−0.644648	−0.227917	−0.113959	+	0.993485i	$0.536353\pi$
$3$	4.18687	0.805764	0.402882	−	0.915252i	$-0.368008\pi$
$3$	4.18687	0.805764	0.402882	+	0.915252i	$0.368008\pi$
$4$	−7.58443	−0.948054
$5$	0	0
$5$	0	0
$6$	−2.69906	−0.183648
$7$	0	0
$7$	0	0
$8$	10.0465	0.443995
$9$	−9.47008	−0.350744
$10$	0	0
$11$	−47.7013	−1.30750	−0.653750	−	0.756711i	$-0.726805\pi$
$11$	−47.7013	−1.30750	−0.653750	+	0.756711i	$0.726805\pi$
$12$	−31.7550	−0.763908
$13$	−57.2256	−1.22089	−0.610443	−	0.792060i	$-0.709009\pi$
$13$	−57.2256	−1.22089	−0.610443	+	0.792060i	$0.709009\pi$
$14$	0	0
$15$	0	0
$16$	54.1990	0.846859
$17$	36.9686	0.527423	0.263712	−	0.964602i	$-0.415053\pi$
$17$	36.9686	0.527423	0.263712	+	0.964602i	$0.415053\pi$
$18$	6.10487	0.0799407
$19$	30.7659	0.371484	0.185742	−	0.982599i	$-0.440531\pi$
$19$	30.7659	0.371484	0.185742	+	0.982599i	$0.440531\pi$
$20$	0	0
$21$	0	0
$22$	30.7506	0.298002
$23$	−53.1282	−0.481652	−0.240826	−	0.970568i	$-0.577418\pi$
$23$	−53.1282	−0.481652	−0.240826	+	0.970568i	$0.577418\pi$
$24$	42.0633	0.357756
$25$	0	0
$26$	36.8904	0.278261
$27$	−152.696	−1.08838
$28$	0	0
$29$	−195.663	−1.25288	−0.626442	−	0.779468i	$-0.715490\pi$
$29$	−195.663	−1.25288	−0.626442	+	0.779468i	$0.715490\pi$
$30$	0	0
$31$	257.870	1.49403	0.747014	−	0.664809i	$-0.231487\pi$
$31$	257.870	1.49403	0.747014	+	0.664809i	$0.231487\pi$
$32$	−115.311	−0.637010
$33$	−199.719	−1.05354
$34$	−23.8317	−0.120209
$35$	0	0
$36$	71.8252	0.332524
$37$	−346.423	−1.53923	−0.769616	−	0.638507i	$-0.779553\pi$
$37$	−346.423	−1.53923	−0.769616	+	0.638507i	$0.779553\pi$
$38$	−19.8332	−0.0846676
$39$	−239.596	−0.983747
$40$	0	0
$41$	267.050	1.01722	0.508611	−	0.860996i	$-0.330159\pi$
$41$	267.050	1.01722	0.508611	+	0.860996i	$0.330159\pi$
$42$	0	0
$43$	176.859	0.627227	0.313614	−	0.949551i	$-0.398460\pi$
$43$	176.859	0.627227	0.313614	+	0.949551i	$0.398460\pi$
$44$	361.787	1.23958
$45$	0	0
$46$	34.2490	0.109777
$47$	311.598	0.967049	0.483524	−	0.875331i	$-0.339356\pi$
$47$	311.598	0.967049	0.483524	+	0.875331i	$0.339356\pi$
$48$	226.924	0.682369
$49$	0	0
$50$	0	0
$51$	154.783	0.424979
$52$	434.024	1.15747
$53$	492.270	1.27582	0.637910	−	0.770111i	$-0.279799\pi$
$53$	492.270	1.27582	0.637910	+	0.770111i	$0.279799\pi$
$54$	98.4349	0.248061
$55$	0	0
$56$	0	0
$57$	128.813	0.299328
$58$	126.134	0.285554
$59$	98.7653	0.217935	0.108967	−	0.994045i	$-0.465246\pi$
$59$	98.7653	0.217935	0.108967	+	0.994045i	$0.465246\pi$
$60$	0	0
$61$	82.1682	0.172468	0.0862340	−	0.996275i	$-0.472517\pi$
$61$	82.1682	0.172468	0.0862340	+	0.996275i	$0.472517\pi$
$62$	−166.235	−0.340515
$63$	0	0
$64$	−359.257	−0.701674
$65$	0	0
$66$	128.749	0.240119
$67$	−654.668	−1.19374	−0.596869	−	0.802339i	$-0.703589\pi$
$67$	−654.668	−1.19374	−0.596869	+	0.802339i	$0.703589\pi$
$68$	−280.386	−0.500026
$69$	−222.441	−0.388098
$70$	0	0
$71$	779.658	1.30322	0.651608	−	0.758556i	$-0.274095\pi$
$71$	779.658	1.30322	0.651608	+	0.758556i	$0.274095\pi$
$72$	−95.1409	−0.155729
$73$	829.673	1.33022	0.665109	−	0.746747i	$-0.268385\pi$
$73$	829.673	1.33022	0.665109	+	0.746747i	$0.268385\pi$
$74$	223.321	0.350818
$75$	0	0
$76$	−233.342	−0.352186
$77$	0	0
$78$	154.455	0.224213
$79$	−769.426	−1.09579	−0.547894	−	0.836548i	$-0.684570\pi$
$79$	−769.426	−1.09579	−0.547894	+	0.836548i	$0.684570\pi$
$80$	0	0
$81$	−383.625	−0.526235
$82$	−172.153	−0.231843
$83$	−613.203	−0.810937	−0.405469	−	0.914109i	$-0.632892\pi$
$83$	−613.203	−0.810937	−0.405469	+	0.914109i	$0.632892\pi$
$84$	0	0
$85$	0	0
$86$	−114.012	−0.142956
$87$	−819.215	−1.00953
$88$	−479.230	−0.580524
$89$	−457.666	−0.545084	−0.272542	−	0.962144i	$-0.587864\pi$
$89$	−457.666	−0.545084	−0.272542	+	0.962144i	$0.587864\pi$
$90$	0	0
$91$	0	0
$92$	402.947	0.456632
$93$	1079.67	1.20383
$94$	−200.871	−0.220407
$95$	0	0
$96$	−482.793	−0.513280
$97$	1412.11	1.47813	0.739063	−	0.673636i	$-0.235268\pi$
$97$	1412.11	1.47813	0.739063	+	0.673636i	$0.235268\pi$
$98$	0	0
$99$	451.736	0.458597
$100$	0	0
$101$	1823.79	1.79677	0.898386	−	0.439208i	$-0.144741\pi$
$101$	1823.79	1.79677	0.898386	+	0.439208i	$0.144741\pi$
$102$	−99.7804	−0.0968601
$103$	−407.775	−0.390090	−0.195045	−	0.980794i	$-0.562485\pi$
$103$	−407.775	−0.390090	−0.195045	+	0.980794i	$0.562485\pi$
$104$	−574.915	−0.542068
$105$	0	0
$106$	−317.341	−0.290782
$107$	370.109	0.334390	0.167195	−	0.985924i	$-0.446529\pi$
$107$	370.109	0.334390	0.167195	+	0.985924i	$0.446529\pi$
$108$	1158.11	1.03184
$109$	975.570	0.857272	0.428636	−	0.903477i	$-0.358994\pi$
$109$	975.570	0.857272	0.428636	+	0.903477i	$0.358994\pi$
$110$	0	0
$111$	−1450.43	−1.24026
$112$	0	0
$113$	−1978.85	−1.64739	−0.823693	−	0.567036i	$-0.808090\pi$
$113$	−1978.85	−1.64739	−0.823693	+	0.567036i	$0.808090\pi$
$114$	−83.0391	−0.0682221
$115$	0	0
$116$	1483.99	1.18780
$117$	541.931	0.428219
$118$	−63.6689	−0.0496711
$119$	0	0
$120$	0	0
$121$	944.416	0.709554
$122$	−52.9695	−0.0393085
$123$	1118.10	0.819642
$124$	−1955.80	−1.41642
$125$	0	0
$126$	0	0
$127$	−1392.38	−0.972867	−0.486433	−	0.873718i	$-0.661702\pi$
$127$	−1392.38	−0.972867	−0.486433	+	0.873718i	$0.661702\pi$
$128$	1154.08	0.796933
$129$	740.487	0.505397
$130$	0	0
$131$	−1777.02	−1.18518	−0.592591	−	0.805504i	$-0.701895\pi$
$131$	−1777.02	−1.18518	−0.592591	+	0.805504i	$0.701895\pi$
$132$	1514.76	0.998809
$133$	0	0
$134$	422.031	0.272074
$135$	0	0
$136$	371.404	0.234174
$137$	1980.48	1.23506	0.617532	−	0.786545i	$-0.288132\pi$
$137$	1980.48	1.23506	0.617532	+	0.786545i	$0.288132\pi$
$138$	143.396	0.0884543
$139$	2182.09	1.33153	0.665763	−	0.746163i	$-0.268106\pi$
$139$	2182.09	1.33153	0.665763	+	0.746163i	$0.268106\pi$
$140$	0	0
$141$	1304.62	0.779213
$142$	−502.605	−0.297026
$143$	2729.74	1.59631
$144$	−513.269	−0.297031
$145$	0	0
$146$	−534.847	−0.303180
$147$	0	0
$148$	2627.42	1.45928
$149$	670.154	0.368464	0.184232	−	0.982883i	$-0.441020\pi$
$149$	670.154	0.368464	0.184232	+	0.982883i	$0.441020\pi$
$150$	0	0
$151$	3348.84	1.80480	0.902401	−	0.430898i	$-0.141803\pi$
$151$	3348.84	1.80480	0.902401	+	0.430898i	$0.141803\pi$
$152$	309.089	0.164937
$153$	−350.096	−0.184990
$154$	0	0
$155$	0	0
$156$	1817.20	0.932645
$157$	−2408.41	−1.22428	−0.612140	−	0.790749i	$-0.709691\pi$
$157$	−2408.41	−1.22428	−0.612140	+	0.790749i	$0.709691\pi$
$158$	496.009	0.249749
$159$	2061.07	1.02801
$160$	0	0
$161$	0	0
$162$	247.303	0.119938
$163$	3811.03	1.83131	0.915654	−	0.401968i	$-0.131674\pi$
$163$	3811.03	1.83131	0.915654	+	0.401968i	$0.131674\pi$
$164$	−2025.42	−0.964382
$165$	0	0
$166$	395.300	0.184827
$167$	−1207.15	−0.559354	−0.279677	−	0.960094i	$-0.590227\pi$
$167$	−1207.15	−0.559354	−0.279677	+	0.960094i	$0.590227\pi$
$168$	0	0
$169$	1077.77	0.490564
$170$	0	0
$171$	−291.356	−0.130296
$172$	−1341.38	−0.594645
$173$	−3243.28	−1.42533	−0.712665	−	0.701505i	$-0.752512\pi$
$173$	−3243.28	−1.42533	−0.712665	+	0.701505i	$0.752512\pi$
$174$	528.105	0.230089
$175$	0	0
$176$	−2585.36	−1.10727
$177$	413.518	0.175604
$178$	295.034	0.124234
$179$	859.728	0.358989	0.179495	−	0.983759i	$-0.442554\pi$
$179$	859.728	0.358989	0.179495	+	0.983759i	$0.442554\pi$
$180$	0	0
$181$	−290.504	−0.119298	−0.0596491	−	0.998219i	$-0.518998\pi$
$181$	−290.504	−0.119298	−0.0596491	+	0.998219i	$0.518998\pi$
$182$	0	0
$183$	344.028	0.138969
$184$	−533.751	−0.213851
$185$	0	0
$186$	−696.007	−0.274375
$187$	−1763.45	−0.689605
$188$	−2363.30	−0.916814
$189$	0	0
$190$	0	0
$191$	4895.61	1.85463	0.927315	−	0.374282i	$-0.122111\pi$
$191$	4895.61	1.85463	0.927315	+	0.374282i	$0.122111\pi$
$192$	−1504.16	−0.565384
$193$	3548.73	1.32354	0.661770	−	0.749707i	$-0.269806\pi$
$193$	3548.73	1.32354	0.661770	+	0.749707i	$0.269806\pi$
$194$	−910.315	−0.336891
$195$	0	0
$196$	0	0
$197$	650.107	0.235118	0.117559	−	0.993066i	$-0.462493\pi$
$197$	650.107	0.235118	0.117559	+	0.993066i	$0.462493\pi$
$198$	−291.210	−0.104522
$199$	4055.09	1.44451	0.722256	−	0.691626i	$-0.243105\pi$
$199$	4055.09	1.44451	0.722256	+	0.691626i	$0.243105\pi$
$200$	0	0
$201$	−2741.01	−0.961871
$202$	−1175.70	−0.409516
$203$	0	0
$204$	−1173.94	−0.402903
$205$	0	0
$206$	262.871	0.0889083
$207$	503.128	0.168936
$208$	−3101.57	−1.03392
$209$	−1467.58	−0.485714
$210$	0	0
$211$	−1569.67	−0.512134	−0.256067	−	0.966659i	$-0.582427\pi$
$211$	−1569.67	−0.512134	−0.256067	+	0.966659i	$0.582427\pi$
$212$	−3733.59	−1.20955
$213$	3264.33	1.05008
$214$	−238.590	−0.0762134
$215$	0	0
$216$	−1534.05	−0.483236
$217$	0	0
$218$	−628.900	−0.195387
$219$	3473.74	1.07184
$220$	0	0
$221$	−2115.55	−0.643924
$222$	935.017	0.282677
$223$	−4723.86	−1.41853	−0.709267	−	0.704940i	$-0.750974\pi$
$223$	−4723.86	−1.41853	−0.709267	+	0.704940i	$0.750974\pi$
$224$	0	0
$225$	0	0
$226$	1275.66	0.375468
$227$	−3685.02	−1.07746	−0.538730	−	0.842479i	$-0.681096\pi$
$227$	−3685.02	−1.07746	−0.538730	+	0.842479i	$0.681096\pi$
$228$	−976.973	−0.283779
$229$	3356.78	0.968656	0.484328	−	0.874887i	$-0.339064\pi$
$229$	3356.78	0.968656	0.484328	+	0.874887i	$0.339064\pi$
$230$	0	0
$231$	0	0
$232$	−1965.72	−0.556275
$233$	−2314.01	−0.650627	−0.325314	−	0.945606i	$-0.605470\pi$
$233$	−2314.01	−0.650627	−0.325314	+	0.945606i	$0.605470\pi$
$234$	−349.355	−0.0975985
$235$	0	0
$236$	−749.079	−0.206614
$237$	−3221.49	−0.882946
$238$	0	0
$239$	941.179	0.254727	0.127364	−	0.991856i	$-0.459348\pi$
$239$	941.179	0.254727	0.127364	+	0.991856i	$0.459348\pi$
$240$	0	0
$241$	5638.24	1.50702	0.753509	−	0.657438i	$-0.228360\pi$
$241$	5638.24	1.50702	0.753509	+	0.657438i	$0.228360\pi$
$242$	−608.816	−0.161720
$243$	2516.59	0.664360
$244$	−623.199	−0.163509
$245$	0	0
$246$	−720.783	−0.186811
$247$	−1760.60	−0.453539
$248$	2590.69	0.663341
$249$	−2567.40	−0.653424
$250$	0	0
$251$	365.822	0.0919940	0.0459970	−	0.998942i	$-0.485354\pi$
$251$	365.822	0.0919940	0.0459970	+	0.998942i	$0.485354\pi$
$252$	0	0
$253$	2534.28	0.629759
$254$	897.598	0.221733
$255$	0	0
$256$	2130.08	0.520039
$257$	6276.46	1.52340	0.761702	−	0.647928i	$-0.224364\pi$
$257$	6276.46	1.52340	0.761702	+	0.647928i	$0.224364\pi$
$258$	−477.353	−0.115189
$259$	0	0
$260$	0	0
$261$	1852.94	0.439442
$262$	1145.55	0.270124
$263$	4225.97	0.990817	0.495408	−	0.868660i	$-0.335018\pi$
$263$	4225.97	0.990817	0.495408	+	0.868660i	$0.335018\pi$
$264$	−2006.48	−0.467765
$265$	0	0
$266$	0	0
$267$	−1916.19	−0.439210
$268$	4965.28	1.13173
$269$	981.591	0.222486	0.111243	−	0.993793i	$-0.464517\pi$
$269$	981.591	0.222486	0.111243	+	0.993793i	$0.464517\pi$
$270$	0	0
$271$	3884.42	0.870708	0.435354	−	0.900259i	$-0.356623\pi$
$271$	3884.42	0.870708	0.435354	+	0.900259i	$0.356623\pi$
$272$	2003.66	0.446653
$273$	0	0
$274$	−1276.71	−0.281493
$275$	0	0
$276$	1687.09	0.367938
$277$	−3614.93	−0.784116	−0.392058	−	0.919941i	$-0.628237\pi$
$277$	−3614.93	−0.784116	−0.392058	+	0.919941i	$0.628237\pi$
$278$	−1406.68	−0.303478
$279$	−2442.05	−0.524021
$280$	0	0
$281$	72.6835	0.0154304	0.00771518	−	0.999970i	$-0.497544\pi$
$281$	72.6835	0.0154304	0.00771518	+	0.999970i	$0.497544\pi$
$282$	−841.023	−0.177596
$283$	−1743.04	−0.366125	−0.183062	−	0.983101i	$-0.558601\pi$
$283$	−1743.04	−0.366125	−0.183062	+	0.983101i	$0.558601\pi$
$284$	−5913.26	−1.23552
$285$	0	0
$286$	−1759.72	−0.363827
$287$	0	0
$288$	1092.01	0.223427
$289$	−3546.32	−0.721825
$290$	0	0
$291$	5912.34	1.19102
$292$	−6292.59	−1.26112
$293$	4989.29	0.994804	0.497402	−	0.867520i	$-0.334287\pi$
$293$	4989.29	0.994804	0.497402	+	0.867520i	$0.334287\pi$
$294$	0	0
$295$	0	0
$296$	−3480.33	−0.683412
$297$	7283.78	1.42306
$298$	−432.013	−0.0839794
$299$	3040.29	0.588042
$300$	0	0
$301$	0	0
$302$	−2158.82	−0.411346
$303$	7635.98	1.44777
$304$	1667.48	0.314594
$305$	0	0
$306$	225.688	0.0421626
$307$	−1664.61	−0.309461	−0.154731	−	0.987957i	$-0.549451\pi$
$307$	−1664.61	−0.309461	−0.154731	+	0.987957i	$0.549451\pi$
$308$	0	0
$309$	−1707.30	−0.314320
$310$	0	0
$311$	545.623	0.0994838	0.0497419	−	0.998762i	$-0.484160\pi$
$311$	545.623	0.0994838	0.0497419	+	0.998762i	$0.484160\pi$
$312$	−2407.10	−0.436779
$313$	−213.564	−0.0385667	−0.0192833	−	0.999814i	$-0.506138\pi$
$313$	−213.564	−0.0385667	−0.0192833	+	0.999814i	$0.506138\pi$
$314$	1552.58	0.279035
$315$	0	0
$316$	5835.66	1.03887
$317$	2683.93	0.475535	0.237767	−	0.971322i	$-0.423584\pi$
$317$	2683.93	0.475535	0.237767	+	0.971322i	$0.423584\pi$
$318$	−1328.67	−0.234302
$319$	9333.37	1.63815
$320$	0	0
$321$	1549.60	0.269440
$322$	0	0
$323$	1137.37	0.195929
$324$	2909.58	0.498899
$325$	0	0
$326$	−2456.77	−0.417387
$327$	4084.59	0.690760
$328$	2682.91	0.451642
$329$	0	0
$330$	0	0
$331$	−5067.52	−0.841500	−0.420750	−	0.907177i	$-0.638233\pi$
$331$	−5067.52	−0.841500	−0.420750	+	0.907177i	$0.638233\pi$
$332$	4650.80	0.768812
$333$	3280.66	0.539876
$334$	778.187	0.127487
$335$	0	0
$336$	0	0
$337$	9353.21	1.51187	0.755937	−	0.654644i	$-0.227182\pi$
$337$	9353.21	1.51187	0.755937	+	0.654644i	$0.227182\pi$
$338$	−694.782	−0.111808
$339$	−8285.20	−1.32741
$340$	0	0
$341$	−12300.7	−1.95344
$342$	187.822	0.0296966
$343$	0	0
$344$	1776.81	0.278486
$345$	0	0
$346$	2090.77	0.324858
$347$	−2349.34	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$347$	−2349.34	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$348$	6213.28	0.957088
$349$	−10472.6	−1.60626	−0.803128	−	0.595806i	$-0.796833\pi$
$349$	−10472.6	−1.60626	−0.803128	+	0.595806i	$0.796833\pi$
$350$	0	0
$351$	8738.10	1.32879
$352$	5500.49	0.832889
$353$	7174.63	1.08178	0.540888	−	0.841095i	$-0.318088\pi$
$353$	7174.63	1.08178	0.540888	+	0.841095i	$0.318088\pi$
$354$	−266.574	−0.0400232
$355$	0	0
$356$	3471.14	0.516769
$357$	0	0
$358$	−554.222	−0.0818199
$359$	−7534.13	−1.10762	−0.553811	−	0.832643i	$-0.686827\pi$
$359$	−7534.13	−1.10762	−0.553811	+	0.832643i	$0.686827\pi$
$360$	0	0
$361$	−5912.46	−0.862000
$362$	187.273	0.0271902
$363$	3954.15	0.571733
$364$	0	0
$365$	0	0
$366$	−221.777	−0.0316734
$367$	−5453.06	−0.775606	−0.387803	−	0.921742i	$-0.626766\pi$
$367$	−5453.06	−0.775606	−0.387803	+	0.921742i	$0.626766\pi$
$368$	−2879.49	−0.407891
$369$	−2528.98	−0.356785
$370$	0	0
$371$	0	0
$372$	−8188.68	−1.14130
$373$	−8231.29	−1.14263	−0.571314	−	0.820732i	$-0.693566\pi$
$373$	−8231.29	−1.14263	−0.571314	+	0.820732i	$0.693566\pi$
$374$	1136.80	0.157173
$375$	0	0
$376$	3130.46	0.429365
$377$	11196.9	1.52963
$378$	0	0
$379$	1670.06	0.226346	0.113173	−	0.993575i	$-0.463898\pi$
$379$	1670.06	0.226346	0.113173	+	0.993575i	$0.463898\pi$
$380$	0	0
$381$	−5829.74	−0.783901
$382$	−3155.95	−0.422703
$383$	−3220.14	−0.429612	−0.214806	−	0.976657i	$-0.568912\pi$
$383$	−3220.14	−0.429612	−0.214806	+	0.976657i	$0.568912\pi$
$384$	4832.00	0.642140
$385$	0	0
$386$	−2287.68	−0.301658
$387$	−1674.87	−0.219996
$388$	−10710.1	−1.40134
$389$	3522.23	0.459085	0.229543	−	0.973299i	$-0.426277\pi$
$389$	3522.23	0.459085	0.229543	+	0.973299i	$0.426277\pi$
$390$	0	0
$391$	−1964.07	−0.254034
$392$	0	0
$393$	−7440.15	−0.954977
$394$	−419.090	−0.0535875
$395$	0	0
$396$	−3426.16	−0.434775
$397$	5455.16	0.689639	0.344819	−	0.938669i	$-0.387940\pi$
$397$	5455.16	0.689639	0.344819	+	0.938669i	$0.387940\pi$
$398$	−2614.11	−0.329229
$399$	0	0
$400$	0	0
$401$	−1161.80	−0.144682	−0.0723409	−	0.997380i	$-0.523047\pi$
$401$	−1161.80	−0.144682	−0.0723409	+	0.997380i	$0.523047\pi$
$402$	1766.99	0.219227
$403$	−14756.8	−1.82404
$404$	−13832.4	−1.70344
$405$	0	0
$406$	0	0
$407$	16524.8	2.01255
$408$	1555.02	0.188689
$409$	−7398.65	−0.894473	−0.447237	−	0.894416i	$-0.647592\pi$
$409$	−7398.65	−0.894473	−0.447237	+	0.894416i	$0.647592\pi$
$410$	0	0
$411$	8292.02	0.995171
$412$	3092.74	0.369826
$413$	0	0
$414$	−324.341	−0.0385036
$415$	0	0
$416$	6598.74	0.777716
$417$	9136.12	1.07290
$418$	946.069	0.110703
$419$	2134.46	0.248867	0.124433	−	0.992228i	$-0.460289\pi$
$419$	2134.46	0.248867	0.124433	+	0.992228i	$0.460289\pi$
$420$	0	0
$421$	−3902.36	−0.451756	−0.225878	−	0.974156i	$-0.572525\pi$
$421$	−3902.36	−0.451756	−0.225878	+	0.974156i	$0.572525\pi$
$422$	1011.88	0.116724
$423$	−2950.86	−0.339186
$424$	4945.58	0.566459
$425$	0	0
$426$	−2104.34	−0.239333
$427$	0	0
$428$	−2807.06	−0.317020
$429$	11429.1	1.28625
$430$	0	0
$431$	−3618.13	−0.404360	−0.202180	−	0.979348i	$-0.564803\pi$
$431$	−3618.13	−0.404360	−0.202180	+	0.979348i	$0.564803\pi$
$432$	−8275.95	−0.921706
$433$	−4222.37	−0.468624	−0.234312	−	0.972161i	$-0.575284\pi$
$433$	−4222.37	−0.468624	−0.234312	+	0.972161i	$0.575284\pi$
$434$	0	0
$435$	0	0
$436$	−7399.14	−0.812740
$437$	−1634.54	−0.178926
$438$	−2239.34	−0.244291
$439$	13518.0	1.46965	0.734826	−	0.678256i	$-0.237264\pi$
$439$	13518.0	1.46965	0.734826	+	0.678256i	$0.237264\pi$
$440$	0	0
$441$	0	0
$442$	1363.78	0.146762
$443$	−16591.2	−1.77939	−0.889695	−	0.456555i	$-0.849083\pi$
$443$	−16591.2	−1.77939	−0.889695	+	0.456555i	$0.849083\pi$
$444$	11000.7	1.17583
$445$	0	0
$446$	3045.23	0.323309
$447$	2805.85	0.296895
$448$	0	0
$449$	8354.32	0.878095	0.439048	−	0.898464i	$-0.355316\pi$
$449$	8354.32	0.878095	0.439048	+	0.898464i	$0.355316\pi$
$450$	0	0
$451$	−12738.6	−1.33002
$452$	15008.5	1.56181
$453$	14021.2	1.45424
$454$	2375.54	0.245572
$455$	0	0
$456$	1294.12	0.132900
$457$	1280.34	0.131054	0.0655269	−	0.997851i	$-0.479127\pi$
$457$	1280.34	0.131054	0.0655269	+	0.997851i	$0.479127\pi$
$458$	−2163.94	−0.220774
$459$	−5644.94	−0.574038
$460$	0	0
$461$	6986.72	0.705865	0.352932	−	0.935649i	$-0.385185\pi$
$461$	6986.72	0.705865	0.352932	+	0.935649i	$0.385185\pi$
$462$	0	0
$463$	5587.32	0.560831	0.280416	−	0.959879i	$-0.409528\pi$
$463$	5587.32	0.560831	0.280416	+	0.959879i	$0.409528\pi$
$464$	−10604.7	−1.06102
$465$	0	0
$466$	1491.73	0.148289
$467$	−9933.83	−0.984332	−0.492166	−	0.870501i	$-0.663795\pi$
$467$	−9933.83	−0.984332	−0.492166	+	0.870501i	$0.663795\pi$
$468$	−4110.24	−0.405974
$469$	0	0
$470$	0	0
$471$	−10083.7	−0.986481
$472$	992.243	0.0967620
$473$	−8436.41	−0.820099
$474$	2076.73	0.201239
$475$	0	0
$476$	0	0
$477$	−4661.84	−0.447486
$478$	−606.729	−0.0580568
$479$	−5068.62	−0.483489	−0.241744	−	0.970340i	$-0.577720\pi$
$479$	−5068.62	−0.483489	−0.241744	+	0.970340i	$0.577720\pi$
$480$	0	0
$481$	19824.3	1.87923
$482$	−3634.68	−0.343476
$483$	0	0
$484$	−7162.86	−0.672695
$485$	0	0
$486$	−1622.32	−0.151419
$487$	−264.353	−0.0245975	−0.0122988	−	0.999924i	$-0.503915\pi$
$487$	−264.353	−0.0245975	−0.0122988	+	0.999924i	$0.503915\pi$
$488$	825.500	0.0765751
$489$	15956.3	1.47560
$490$	0	0
$491$	−7459.47	−0.685623	−0.342812	−	0.939404i	$-0.611379\pi$
$491$	−7459.47	−0.685623	−0.342812	+	0.939404i	$0.611379\pi$
$492$	−8480.17	−0.777064
$493$	−7233.37	−0.660800
$494$	1134.97	0.103370
$495$	0	0
$496$	13976.3	1.26523
$497$	0	0
$498$	1655.07	0.148927
$499$	7206.67	0.646523	0.323261	−	0.946310i	$-0.395221\pi$
$499$	7206.67	0.646523	0.323261	+	0.946310i	$0.395221\pi$
$500$	0	0
$501$	−5054.19	−0.450707
$502$	−235.827	−0.0209670
$503$	10886.7	0.965037	0.482519	−	0.875886i	$-0.339722\pi$
$503$	10886.7	0.965037	0.482519	+	0.875886i	$0.339722\pi$
$504$	0	0
$505$	0	0
$506$	−1633.72	−0.143533
$507$	4512.49	0.395279
$508$	10560.4	0.922330
$509$	−7504.97	−0.653540	−0.326770	−	0.945104i	$-0.605960\pi$
$509$	−7504.97	−0.653540	−0.326770	+	0.945104i	$0.605960\pi$
$510$	0	0
$511$	0	0
$512$	−10605.8	−0.915459
$513$	−4697.82	−0.404316
$514$	−4046.11	−0.347210
$515$	0	0
$516$	−5616.17	−0.479144
$517$	−14863.7	−1.26442
$518$	0	0
$519$	−13579.2	−1.14848
$520$	0	0
$521$	−22264.8	−1.87224	−0.936120	−	0.351681i	$-0.885610\pi$
$521$	−22264.8	−1.87224	−0.936120	+	0.351681i	$0.885610\pi$
$522$	−1194.50	−0.100156
$523$	−11286.7	−0.943655	−0.471828	−	0.881691i	$-0.656405\pi$
$523$	−11286.7	−0.943655	−0.471828	+	0.881691i	$0.656405\pi$
$524$	13477.7	1.12362
$525$	0	0
$526$	−2724.27	−0.225824
$527$	9533.09	0.787985
$528$	−10824.6	−0.892197
$529$	−9344.40	−0.768012
$530$	0	0
$531$	−935.316	−0.0764393
$532$	0	0
$533$	−15282.1	−1.24191
$534$	1235.27	0.100104
$535$	0	0
$536$	−6577.10	−0.530014
$537$	3599.57	0.289261
$538$	−632.781	−0.0507084
$539$	0	0
$540$	0	0
$541$	16406.8	1.30385	0.651924	−	0.758284i	$-0.273962\pi$
$541$	16406.8	1.30385	0.651924	+	0.758284i	$0.273962\pi$
$542$	−2504.08	−0.198450
$543$	−1216.30	−0.0961263
$544$	−4262.88	−0.335974
$545$	0	0
$546$	0	0
$547$	8692.48	0.679458	0.339729	−	0.940523i	$-0.389665\pi$
$547$	8692.48	0.679458	0.339729	+	0.940523i	$0.389665\pi$
$548$	−15020.8	−1.17091
$549$	−778.139	−0.0604921
$550$	0	0
$551$	−6019.74	−0.465426
$552$	−2234.75	−0.172314
$553$	0	0
$554$	2330.36	0.178714
$555$	0	0
$556$	−16549.9	−1.26236
$557$	13935.7	1.06010	0.530049	−	0.847967i	$-0.322174\pi$
$557$	13935.7	1.06010	0.530049	+	0.847967i	$0.322174\pi$
$558$	1574.26	0.119434
$559$	−10120.9	−0.765773
$560$	0	0
$561$	−7383.34	−0.555659
$562$	−46.8553	−0.00351685
$563$	−22263.6	−1.66661	−0.833304	−	0.552815i	$-0.813553\pi$
$563$	−22263.6	−1.66661	−0.833304	+	0.552815i	$0.813553\pi$
$564$	−9894.82	−0.738736
$565$	0	0
$566$	1123.65	0.0834462
$567$	0	0
$568$	7832.81	0.578622
$569$	9525.66	0.701822	0.350911	−	0.936409i	$-0.385872\pi$
$569$	9525.66	0.701822	0.350911	+	0.936409i	$0.385872\pi$
$570$	0	0
$571$	7669.99	0.562135	0.281068	−	0.959688i	$-0.409311\pi$
$571$	7669.99	0.562135	0.281068	+	0.959688i	$0.409311\pi$
$572$	−20703.5	−1.51339
$573$	20497.3	1.49439
$574$	0	0
$575$	0	0
$576$	3402.19	0.246108
$577$	3907.15	0.281901	0.140950	−	0.990017i	$-0.454984\pi$
$577$	3907.15	0.281901	0.140950	+	0.990017i	$0.454984\pi$
$578$	2286.13	0.164516
$579$	14858.1	1.06646
$580$	0	0
$581$	0	0
$582$	−3811.38	−0.271455
$583$	−23481.9	−1.66813
$584$	8335.28	0.590610
$585$	0	0
$586$	−3216.34	−0.226733
$587$	−19891.6	−1.39866	−0.699331	−	0.714798i	$-0.746519\pi$
$587$	−19891.6	−1.39866	−0.699331	+	0.714798i	$0.746519\pi$
$588$	0	0
$589$	7933.61	0.555007
$590$	0	0
$591$	2721.92	0.189450
$592$	−18775.8	−1.30351
$593$	4780.36	0.331038	0.165519	−	0.986207i	$-0.447070\pi$
$593$	4780.36	0.331038	0.165519	+	0.986207i	$0.447070\pi$
$594$	−4695.48	−0.324340
$595$	0	0
$596$	−5082.74	−0.349324
$597$	16978.2	1.16394
$598$	−1959.92	−0.134025
$599$	11085.6	0.756170	0.378085	−	0.925771i	$-0.376583\pi$
$599$	11085.6	0.756170	0.378085	+	0.925771i	$0.376583\pi$
$600$	0	0
$601$	1573.44	0.106792	0.0533958	−	0.998573i	$-0.482995\pi$
$601$	1573.44	0.106792	0.0533958	+	0.998573i	$0.482995\pi$
$602$	0	0
$603$	6199.76	0.418696
$604$	−25399.1	−1.71105
$605$	0	0
$606$	−4922.52	−0.329973
$607$	8542.73	0.571233	0.285617	−	0.958344i	$-0.407802\pi$
$607$	8542.73	0.571233	0.285617	+	0.958344i	$0.407802\pi$
$608$	−3547.65	−0.236639
$609$	0	0
$610$	0	0
$611$	−17831.4	−1.18066
$612$	2655.27	0.175381
$613$	−15068.2	−0.992817	−0.496409	−	0.868089i	$-0.665348\pi$
$613$	−15068.2	−0.992817	−0.496409	+	0.868089i	$0.665348\pi$
$614$	1073.09	0.0705316
$615$	0	0
$616$	0	0
$617$	−2524.58	−0.164725	−0.0823627	−	0.996602i	$-0.526247\pi$
$617$	−2524.58	−0.164725	−0.0823627	+	0.996602i	$0.526247\pi$
$618$	1100.61	0.0716391
$619$	21238.9	1.37910	0.689551	−	0.724237i	$-0.257808\pi$
$619$	21238.9	1.37910	0.689551	+	0.724237i	$0.257808\pi$
$620$	0	0
$621$	8112.44	0.524221
$622$	−351.735	−0.0226741
$623$	0	0
$624$	−12985.9	−0.833095
$625$	0	0
$626$	137.674	0.00879002
$627$	−6144.55	−0.391371
$628$	18266.4	1.16068
$629$	−12806.8	−0.811827
$630$	0	0
$631$	−8885.83	−0.560601	−0.280300	−	0.959912i	$-0.590434\pi$
$631$	−8885.83	−0.560601	−0.280300	+	0.959912i	$0.590434\pi$
$632$	−7730.02	−0.486525
$633$	−6571.99	−0.412659
$634$	−1730.19	−0.108383
$635$	0	0
$636$	−15632.1	−0.974609
$637$	0	0
$638$	−6016.74	−0.373362
$639$	−7383.42	−0.457095
$640$	0	0
$641$	3655.63	0.225255	0.112628	−	0.993637i	$-0.464073\pi$
$641$	3655.63	0.225255	0.112628	+	0.993637i	$0.464073\pi$
$642$	−998.946	−0.0614100
$643$	4221.22	0.258894	0.129447	−	0.991586i	$-0.458680\pi$
$643$	4221.22	0.258894	0.129447	+	0.991586i	$0.458680\pi$
$644$	0	0
$645$	0	0
$646$	−733.205	−0.0446557
$647$	103.679	0.00629989	0.00314995	−	0.999995i	$-0.498997\pi$
$647$	103.679	0.00629989	0.00314995	+	0.999995i	$0.498997\pi$
$648$	−3854.08	−0.233646
$649$	−4711.24	−0.284949
$650$	0	0
$651$	0	0
$652$	−28904.5	−1.73618
$653$	−4476.11	−0.268245	−0.134122	−	0.990965i	$-0.542821\pi$
$653$	−4476.11	−0.268245	−0.134122	+	0.990965i	$0.542821\pi$
$654$	−2633.12	−0.157436
$655$	0	0
$656$	14473.8	0.861445
$657$	−7857.07	−0.466566
$658$	0	0
$659$	12022.0	0.710641	0.355321	−	0.934745i	$-0.384372\pi$
$659$	12022.0	0.710641	0.355321	+	0.934745i	$0.384372\pi$
$660$	0	0
$661$	−13451.0	−0.791504	−0.395752	−	0.918357i	$-0.629516\pi$
$661$	−13451.0	−0.791504	−0.395752	+	0.918357i	$0.629516\pi$
$662$	3266.77	0.191792
$663$	−8857.54	−0.518851
$664$	−6160.53	−0.360053
$665$	0	0
$666$	−2114.87	−0.123047
$667$	10395.2	0.603454
$668$	9155.55	0.530297
$669$	−19778.2	−1.14300
$670$	0	0
$671$	−3919.53	−0.225502
$672$	0	0
$673$	−9774.83	−0.559869	−0.279935	−	0.960019i	$-0.590313\pi$
$673$	−9774.83	−0.559869	−0.279935	+	0.960019i	$0.590313\pi$
$674$	−6029.53	−0.344583
$675$	0	0
$676$	−8174.27	−0.465081
$677$	23857.4	1.35438	0.677190	−	0.735808i	$-0.263198\pi$
$677$	23857.4	1.35438	0.677190	+	0.735808i	$0.263198\pi$
$678$	5341.04	0.302539
$679$	0	0
$680$	0	0
$681$	−15428.7	−0.868179
$682$	7929.65	0.445223
$683$	−11919.4	−0.667763	−0.333881	−	0.942615i	$-0.608358\pi$
$683$	−11919.4	−0.667763	−0.333881	+	0.942615i	$0.608358\pi$
$684$	2209.77	0.123527
$685$	0	0
$686$	0	0
$687$	14054.4	0.780508
$688$	9585.59	0.531173
$689$	−28170.5	−1.55763
$690$	0	0
$691$	8203.58	0.451634	0.225817	−	0.974170i	$-0.427495\pi$
$691$	8203.58	0.451634	0.225817	+	0.974170i	$0.427495\pi$
$692$	24598.4	1.35129
$693$	0	0
$694$	1514.50	0.0828380
$695$	0	0
$696$	−8230.22	−0.448227
$697$	9872.44	0.536507
$698$	6751.12	0.366094
$699$	−9688.49	−0.524252
$700$	0	0
$701$	449.084	0.0241964	0.0120982	−	0.999927i	$-0.496149\pi$
$701$	449.084	0.0241964	0.0120982	+	0.999927i	$0.496149\pi$
$702$	−5633.00	−0.302855
$703$	−10658.0	−0.571800
$704$	17137.0	0.917438
$705$	0	0
$706$	−4625.11	−0.246556
$707$	0	0
$708$	−3136.30	−0.166482
$709$	1897.64	0.100518	0.0502590	−	0.998736i	$-0.483995\pi$
$709$	1897.64	0.100518	0.0502590	+	0.998736i	$0.483995\pi$
$710$	0	0
$711$	7286.53	0.384341
$712$	−4597.93	−0.242015
$713$	−13700.2	−0.719601
$714$	0	0
$715$	0	0
$716$	−6520.54	−0.340341
$717$	3940.60	0.205250
$718$	4856.86	0.252446
$719$	−6494.70	−0.336873	−0.168436	−	0.985713i	$-0.553872\pi$
$719$	−6494.70	−0.336873	−0.168436	+	0.985713i	$0.553872\pi$
$720$	0	0
$721$	0	0
$722$	3811.45	0.196465
$723$	23606.6	1.21430
$724$	2203.31	0.113101
$725$	0	0
$726$	−2549.04	−0.130308
$727$	−18311.2	−0.934148	−0.467074	−	0.884218i	$-0.654692\pi$
$727$	−18311.2	−0.934148	−0.467074	+	0.884218i	$0.654692\pi$
$728$	0	0
$729$	20894.5	1.06155
$730$	0	0
$731$	6538.23	0.330814
$732$	−2609.25	−0.131750
$733$	21233.3	1.06994	0.534972	−	0.844870i	$-0.320322\pi$
$733$	21233.3	1.06994	0.534972	+	0.844870i	$0.320322\pi$
$734$	3515.30	0.176774
$735$	0	0
$736$	6126.27	0.306817
$737$	31228.5	1.56081
$738$	1630.30	0.0813175
$739$	−22023.5	−1.09628	−0.548138	−	0.836388i	$-0.684663\pi$
$739$	−22023.5	−1.09628	−0.548138	+	0.836388i	$0.684663\pi$
$740$	0	0
$741$	−7371.40	−0.365446
$742$	0	0
$743$	−9436.77	−0.465951	−0.232975	−	0.972483i	$-0.574846\pi$
$743$	−9436.77	−0.465951	−0.232975	+	0.972483i	$0.574846\pi$
$744$	10846.9	0.534497
$745$	0	0
$746$	5306.29	0.260425
$747$	5807.09	0.284431
$748$	13374.8	0.653783
$749$	0	0
$750$	0	0
$751$	−39161.8	−1.90284	−0.951421	−	0.307893i	$-0.900376\pi$
$751$	−39161.8	−1.90284	−0.951421	+	0.307893i	$0.900376\pi$
$752$	16888.3	0.818954
$753$	1531.65	0.0741255
$754$	−7218.07	−0.348629
$755$	0	0
$756$	0	0
$757$	−20340.6	−0.976607	−0.488303	−	0.872674i	$-0.662384\pi$
$757$	−20340.6	−0.976607	−0.488303	+	0.872674i	$0.662384\pi$
$758$	−1076.60	−0.0515883
$759$	10610.7	0.507438
$760$	0	0
$761$	−3307.90	−0.157570	−0.0787852	−	0.996892i	$-0.525104\pi$
$761$	−3307.90	−0.157570	−0.0787852	+	0.996892i	$0.525104\pi$
$762$	3758.13	0.178665
$763$	0	0
$764$	−37130.4	−1.75829
$765$	0	0
$766$	2075.85	0.0979160
$767$	−5651.91	−0.266074
$768$	8918.37	0.419029
$769$	−17234.0	−0.808159	−0.404079	−	0.914724i	$-0.632408\pi$
$769$	−17234.0	−0.808159	−0.404079	+	0.914724i	$0.632408\pi$
$770$	0	0
$771$	26278.7	1.22750
$772$	−26915.1	−1.25479
$773$	−8726.33	−0.406034	−0.203017	−	0.979175i	$-0.565075\pi$
$773$	−8726.33	−0.406034	−0.203017	+	0.979175i	$0.565075\pi$
$774$	1079.70	0.0501409
$775$	0	0
$776$	14186.7	0.656281
$777$	0	0
$778$	−2270.60	−0.104634
$779$	8216.03	0.377882
$780$	0	0
$781$	−37190.7	−1.70395
$782$	1266.14	0.0578989
$783$	29876.8	1.36362
$784$	0	0
$785$	0	0
$786$	4796.28	0.217656
$787$	−10642.1	−0.482018	−0.241009	−	0.970523i	$-0.577478\pi$
$787$	−10642.1	−0.482018	−0.241009	+	0.970523i	$0.577478\pi$
$788$	−4930.69	−0.222904
$789$	17693.6	0.798365
$790$	0	0
$791$	0	0
$792$	4538.35	0.203615
$793$	−4702.12	−0.210564
$794$	−3516.66	−0.157181
$795$	0	0
$796$	−30755.5	−1.36947
$797$	29234.1	1.29928	0.649640	−	0.760242i	$-0.274920\pi$
$797$	29234.1	1.29928	0.649640	+	0.760242i	$0.274920\pi$
$798$	0	0
$799$	11519.3	0.510044
$800$	0	0
$801$	4334.14	0.191185
$802$	748.950	0.0329755
$803$	−39576.5	−1.73926
$804$	20789.0	0.911906
$805$	0	0
$806$	9512.93	0.415730
$807$	4109.80	0.179271
$808$	18322.7	0.797758
$809$	36211.2	1.57369	0.786846	−	0.617150i	$-0.211713\pi$
$809$	36211.2	1.57369	0.786846	+	0.617150i	$0.211713\pi$
$810$	0	0
$811$	27995.2	1.21214	0.606069	−	0.795412i	$-0.292746\pi$
$811$	27995.2	1.21214	0.606069	+	0.795412i	$0.292746\pi$
$812$	0	0
$813$	16263.6	0.701585
$814$	−10652.7	−0.458694
$815$	0	0
$816$	8389.07	0.359897
$817$	5441.23	0.233005
$818$	4769.52	0.203866
$819$	0	0
$820$	0	0
$821$	44805.7	1.90467	0.952333	−	0.305060i	$-0.0986766\pi$
$821$	44805.7	1.90467	0.952333	+	0.305060i	$0.0986766\pi$
$822$	−5345.44	−0.226817
$823$	15850.9	0.671360	0.335680	−	0.941976i	$-0.391034\pi$
$823$	15850.9	0.671360	0.335680	+	0.941976i	$0.391034\pi$
$824$	−4096.70	−0.173198
$825$	0	0
$826$	0	0
$827$	45013.9	1.89273	0.946363	−	0.323104i	$-0.104726\pi$
$827$	45013.9	1.89273	0.946363	+	0.323104i	$0.104726\pi$
$828$	−3815.94	−0.160161
$829$	−5655.83	−0.236954	−0.118477	−	0.992957i	$-0.537801\pi$
$829$	−5655.83	−0.236954	−0.118477	+	0.992957i	$0.537801\pi$
$830$	0	0
$831$	−15135.3	−0.631812
$832$	20558.7	0.856664
$833$	0	0
$834$	−5889.58	−0.244532
$835$	0	0
$836$	11130.7	0.460483
$837$	−39375.7	−1.62607
$838$	−1375.97	−0.0567211
$839$	22037.0	0.906797	0.453398	−	0.891308i	$-0.350211\pi$
$839$	22037.0	0.906797	0.453398	+	0.891308i	$0.350211\pi$
$840$	0	0
$841$	13894.9	0.569719
$842$	2515.65	0.102963
$843$	304.317	0.0124332
$844$	11905.0	0.485530
$845$	0	0
$846$	1902.27	0.0773065
$847$	0	0
$848$	26680.5	1.08044
$849$	−7297.91	−0.295010
$850$	0	0
$851$	18404.8	0.741374
$852$	−24758.1	−0.995537
$853$	−44486.7	−1.78569	−0.892846	−	0.450362i	$-0.851295\pi$
$853$	−44486.7	−1.78569	−0.892846	+	0.450362i	$0.851295\pi$
$854$	0	0
$855$	0	0
$856$	3718.29	0.148468
$857$	−38559.6	−1.53696	−0.768478	−	0.639877i	$-0.778985\pi$
$857$	−38559.6	−1.53696	−0.768478	+	0.639877i	$0.778985\pi$
$858$	−7367.72	−0.293158
$859$	−8426.01	−0.334682	−0.167341	−	0.985899i	$-0.553518\pi$
$859$	−8426.01	−0.334682	−0.167341	+	0.985899i	$0.553518\pi$
$860$	0	0
$861$	0	0
$862$	2332.42	0.0921608
$863$	20054.3	0.791029	0.395514	−	0.918460i	$-0.370566\pi$
$863$	20054.3	0.791029	0.395514	+	0.918460i	$0.370566\pi$
$864$	17607.5	0.693309
$865$	0	0
$866$	2721.94	0.106808
$867$	−14848.0	−0.581621
$868$	0	0
$869$	36702.6	1.43274
$870$	0	0
$871$	37463.8	1.45742
$872$	9801.04	0.380625
$873$	−13372.8	−0.518444
$874$	1053.70	0.0407803
$875$	0	0
$876$	−26346.3	−1.01616
$877$	−47082.1	−1.81283	−0.906414	−	0.422391i	$-0.861191\pi$
$877$	−47082.1	−1.81283	−0.906414	+	0.422391i	$0.861191\pi$
$878$	−8714.32	−0.334959
$879$	20889.5	0.801577
$880$	0	0
$881$	9467.24	0.362042	0.181021	−	0.983479i	$-0.442060\pi$
$881$	9467.24	0.362042	0.181021	+	0.983479i	$0.442060\pi$
$882$	0	0
$883$	3049.49	0.116221	0.0581106	−	0.998310i	$-0.481492\pi$
$883$	3049.49	0.116221	0.0581106	+	0.998310i	$0.481492\pi$
$884$	16045.2	0.610475
$885$	0	0
$886$	10695.5	0.405554
$887$	−19468.6	−0.736968	−0.368484	−	0.929634i	$-0.620123\pi$
$887$	−19468.6	−0.736968	−0.368484	+	0.929634i	$0.620123\pi$
$888$	−14571.7	−0.550669
$889$	0	0
$890$	0	0
$891$	18299.4	0.688052
$892$	35827.8	1.34485
$893$	9586.61	0.359243
$894$	−1808.79	−0.0676676
$895$	0	0
$896$	0	0
$897$	12729.3	0.473823
$898$	−5385.60	−0.200133
$899$	−50455.6	−1.87184
$900$	0	0
$901$	18198.5	0.672898
$902$	8211.92	0.303134
$903$	0	0
$904$	−19880.5	−0.731432
$905$	0	0
$906$	−9038.73	−0.331448
$907$	32014.2	1.17201	0.586006	−	0.810307i	$-0.300699\pi$
$907$	32014.2	1.17201	0.586006	+	0.810307i	$0.300699\pi$
$908$	27948.8	1.02149
$909$	−17271.4	−0.630206
$910$	0	0
$911$	20921.1	0.760866	0.380433	−	0.924809i	$-0.375775\pi$
$911$	20921.1	0.760866	0.380433	+	0.924809i	$0.375775\pi$
$912$	6981.54	0.253489
$913$	29250.6	1.06030
$914$	−825.366	−0.0298694
$915$	0	0
$916$	−25459.2	−0.918337
$917$	0	0
$918$	3639.00	0.130833
$919$	43467.4	1.56024	0.780118	−	0.625632i	$-0.215159\pi$
$919$	43467.4	1.56024	0.780118	+	0.625632i	$0.215159\pi$
$920$	0	0
$921$	−6969.53	−0.249353
$922$	−4503.97	−0.160879
$923$	−44616.4	−1.59108
$924$	0	0
$925$	0	0
$926$	−3601.85	−0.127823
$927$	3861.66	0.136822
$928$	22562.1	0.798099
$929$	18542.1	0.654840	0.327420	−	0.944879i	$-0.393821\pi$
$929$	18542.1	0.654840	0.327420	+	0.944879i	$0.393821\pi$
$930$	0	0
$931$	0	0
$932$	17550.5	0.616830
$933$	2284.46	0.0801605
$934$	6403.82	0.224346
$935$	0	0
$936$	5444.50	0.190127
$937$	13842.5	0.482619	0.241309	−	0.970448i	$-0.422423\pi$
$937$	13842.5	0.482619	0.241309	+	0.970448i	$0.422423\pi$
$938$	0	0
$939$	−894.167	−0.0310757
$940$	0	0
$941$	53393.2	1.84970	0.924850	−	0.380333i	$-0.124191\pi$
$941$	53393.2	1.84970	0.924850	+	0.380333i	$0.124191\pi$
$942$	6500.44	0.224836
$943$	−14187.9	−0.489947
$944$	5352.98	0.184560
$945$	0	0
$946$	5438.52	0.186915
$947$	−22344.2	−0.766727	−0.383364	−	0.923598i	$-0.625234\pi$
$947$	−22344.2	−0.766727	−0.383364	+	0.923598i	$0.625234\pi$
$948$	24433.2	0.837081
$949$	−47478.5	−1.62404
$950$	0	0
$951$	11237.3	0.383169
$952$	0	0
$953$	8902.63	0.302607	0.151304	−	0.988487i	$-0.451653\pi$
$953$	8902.63	0.302607	0.151304	+	0.988487i	$0.451653\pi$
$954$	3005.25	0.101990
$955$	0	0
$956$	−7138.31	−0.241495
$957$	39077.6	1.31996
$958$	3267.48	0.110196
$959$	0	0
$960$	0	0
$961$	36706.0	1.23212
$962$	−12779.7	−0.428309
$963$	−3504.96	−0.117285
$964$	−42762.8	−1.42873
$965$	0	0
$966$	0	0
$967$	2225.57	0.0740119	0.0370059	−	0.999315i	$-0.488218\pi$
$967$	2225.57	0.0740119	0.0370059	+	0.999315i	$0.488218\pi$
$968$	9488.05	0.315039
$969$	4762.03	0.157873
$970$	0	0
$971$	11324.8	0.374285	0.187143	−	0.982333i	$-0.440077\pi$
$971$	11324.8	0.374285	0.187143	+	0.982333i	$0.440077\pi$
$972$	−19086.9	−0.629849
$973$	0	0
$974$	170.415	0.00560621
$975$	0	0
$976$	4453.43	0.146056
$977$	57695.9	1.88931	0.944654	−	0.328067i	$-0.106397\pi$
$977$	57695.9	1.88931	0.944654	+	0.328067i	$0.106397\pi$
$978$	−10286.2	−0.336315
$979$	21831.3	0.712697
$980$	0	0
$981$	−9238.73	−0.300683
$982$	4808.73	0.156266
$983$	59657.3	1.93568	0.967839	−	0.251570i	$-0.0809469\pi$
$983$	59657.3	1.93568	0.967839	+	0.251570i	$0.0809469\pi$
$984$	11233.0	0.363917
$985$	0	0
$986$	4662.98	0.150608
$987$	0	0
$988$	13353.1	0.429980
$989$	−9396.20	−0.302105
$990$	0	0
$991$	2890.60	0.0926568	0.0463284	−	0.998926i	$-0.485248\pi$
$991$	2890.60	0.0926568	0.0463284	+	0.998926i	$0.485248\pi$
$992$	−29735.3	−0.951710
$993$	−21217.1	−0.678050
$994$	0	0
$995$	0	0
$996$	19472.3	0.619481
$997$	−10619.6	−0.337337	−0.168669	−	0.985673i	$-0.553947\pi$
$997$	−10619.6	−0.337337	−0.168669	+	0.985673i	$0.553947\pi$
$998$	−4645.77	−0.147354
$999$	52897.3	1.67527

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.4.a.bi.1.3		6
5.4	even	2		245.4.a.p.1.4	yes	6
7.6	odd	2		1225.4.a.bj.1.3		6
15.14	odd	2		2205.4.a.ca.1.3		6
35.4	even	6		245.4.e.p.226.3		12
35.9	even	6		245.4.e.p.116.3		12
35.19	odd	6		245.4.e.q.116.3		12
35.24	odd	6		245.4.e.q.226.3		12
35.34	odd	2		245.4.a.o.1.4	✓	6
105.104	even	2		2205.4.a.bz.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.4.a.o.1.4	✓	6	35.34	odd	2
245.4.a.p.1.4	yes	6	5.4	even	2
245.4.e.p.116.3		12	35.9	even	6
245.4.e.p.226.3		12	35.4	even	6
245.4.e.q.116.3		12	35.19	odd	6
245.4.e.q.226.3		12	35.24	odd	6
1225.4.a.bi.1.3		6	1.1	even	1	trivial
1225.4.a.bj.1.3		6	7.6	odd	2
2205.4.a.bz.1.3		6	105.104	even	2
2205.4.a.ca.1.3		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)

\(f(q)\)	\(=\)	\(q-0.644648 q^{2} +4.18687 q^{3} -7.58443 q^{4} -2.69906 q^{6} +10.0465 q^{8} -9.47008 q^{9} +O(q^{10})\)	\(q-0.644648 q^{2} +4.18687 q^{3} -7.58443 q^{4} -2.69906 q^{6} +10.0465 q^{8} -9.47008 q^{9} -47.7013 q^{11} -31.7550 q^{12} -57.2256 q^{13} +54.1990 q^{16} +36.9686 q^{17} +6.10487 q^{18} +30.7659 q^{19} +30.7506 q^{22} -53.1282 q^{23} +42.0633 q^{24} +36.8904 q^{26} -152.696 q^{27} -195.663 q^{29} +257.870 q^{31} -115.311 q^{32} -199.719 q^{33} -23.8317 q^{34} +71.8252 q^{36} -346.423 q^{37} -19.8332 q^{38} -239.596 q^{39} +267.050 q^{41} +176.859 q^{43} +361.787 q^{44} +34.2490 q^{46} +311.598 q^{47} +226.924 q^{48} +154.783 q^{51} +434.024 q^{52} +492.270 q^{53} +98.4349 q^{54} +128.813 q^{57} +126.134 q^{58} +98.7653 q^{59} +82.1682 q^{61} -166.235 q^{62} -359.257 q^{64} +128.749 q^{66} -654.668 q^{67} -280.386 q^{68} -222.441 q^{69} +779.658 q^{71} -95.1409 q^{72} +829.673 q^{73} +223.321 q^{74} -233.342 q^{76} +154.455 q^{78} -769.426 q^{79} -383.625 q^{81} -172.153 q^{82} -613.203 q^{83} -114.012 q^{86} -819.215 q^{87} -479.230 q^{88} -457.666 q^{89} +402.947 q^{92} +1079.67 q^{93} -200.871 q^{94} -482.793 q^{96} +1412.11 q^{97} +451.736 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} - 16 q^{3} + 14 q^{4} + 24 q^{6} + 66 q^{8} + 70 q^{9}+O(q^{10}) \) 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

Introduction

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