LMFDB - Embedded newform 1225.4.a.m.1.1

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:	$1$
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			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-5.41421 q^{2} -4.65685 q^{3} +21.3137 q^{4} +25.2132 q^{6} -72.0833 q^{8} -5.31371 q^{9} +O(q^{10})$	\(q-5.41421 q^{2} -4.65685 q^{3} +21.3137 q^{4} +25.2132 q^{6} -72.0833 q^{8} -5.31371 q^{9} -52.2548 q^{11} -99.2548 q^{12} +30.6569 q^{13} +219.765 q^{16} +37.2254 q^{17} +28.7696 q^{18} -80.2254 q^{19} +282.919 q^{22} -25.8335 q^{23} +335.681 q^{24} -165.983 q^{26} +150.480 q^{27} +20.9411 q^{29} +314.558 q^{31} -613.186 q^{32} +243.343 q^{33} -201.546 q^{34} -113.255 q^{36} -197.147 q^{37} +434.357 q^{38} -142.765 q^{39} -11.3625 q^{41} +33.8335 q^{43} -1113.74 q^{44} +139.868 q^{46} -361.676 q^{47} -1023.41 q^{48} -173.353 q^{51} +653.411 q^{52} -153.019 q^{53} -814.732 q^{54} +373.598 q^{57} -113.380 q^{58} +616.000 q^{59} -15.2649 q^{61} -1703.09 q^{62} +1561.80 q^{64} -1317.51 q^{66} +166.510 q^{67} +793.411 q^{68} +120.303 q^{69} -952.000 q^{71} +383.029 q^{72} -148.489 q^{73} +1067.40 q^{74} -1709.90 q^{76} +772.958 q^{78} +857.725 q^{79} -557.294 q^{81} +61.5189 q^{82} +660.528 q^{83} -183.182 q^{86} -97.5198 q^{87} +3766.70 q^{88} +45.7746 q^{89} -550.607 q^{92} -1464.85 q^{93} +1958.19 q^{94} +2855.52 q^{96} +1682.13 q^{97} +277.667 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} + 2 q^{3} + 20 q^{4} + 8 q^{6} - 48 q^{8} + 12 q^{9}+O(q^{10}) $ 2 * q - 8 * q^2 + 2 * q^3 + 20 * q^4 + 8 * q^6 - 48 * q^8 + 12 * q^9	$ 2 q - 8 q^{2} + 2 q^{3} + 20 q^{4} + 8 q^{6} - 48 q^{8} + 12 q^{9} - 14 q^{11} - 108 q^{12} + 50 q^{13} + 168 q^{16} - 50 q^{17} - 16 q^{18} - 36 q^{19} + 184 q^{22} - 244 q^{23} + 496 q^{24} - 216 q^{26} + 86 q^{27} - 26 q^{29} + 120 q^{31} - 672 q^{32} + 498 q^{33} + 24 q^{34} - 136 q^{36} - 564 q^{37} + 320 q^{38} - 14 q^{39} + 328 q^{41} + 260 q^{43} - 1164 q^{44} + 704 q^{46} - 350 q^{47} - 1368 q^{48} - 754 q^{51} + 628 q^{52} + 56 q^{53} - 648 q^{54} + 668 q^{57} + 8 q^{58} + 1232 q^{59} - 336 q^{61} - 1200 q^{62} + 2128 q^{64} - 1976 q^{66} + 152 q^{67} + 908 q^{68} - 1332 q^{69} - 1904 q^{71} + 800 q^{72} + 676 q^{73} + 2016 q^{74} - 1768 q^{76} + 440 q^{78} + 1014 q^{79} - 1454 q^{81} - 816 q^{82} - 376 q^{83} - 768 q^{86} - 410 q^{87} + 4688 q^{88} + 216 q^{89} - 264 q^{92} - 2760 q^{93} + 1928 q^{94} + 2464 q^{96} + 2742 q^{97} + 940 q^{99}+O(q^{100}) $ 2 * q - 8 * q^2 + 2 * q^3 + 20 * q^4 + 8 * q^6 - 48 * q^8 + 12 * q^9 - 14 * q^11 - 108 * q^12 + 50 * q^13 + 168 * q^16 - 50 * q^17 - 16 * q^18 - 36 * q^19 + 184 * q^22 - 244 * q^23 + 496 * q^24 - 216 * q^26 + 86 * q^27 - 26 * q^29 + 120 * q^31 - 672 * q^32 + 498 * q^33 + 24 * q^34 - 136 * q^36 - 564 * q^37 + 320 * q^38 - 14 * q^39 + 328 * q^41 + 260 * q^43 - 1164 * q^44 + 704 * q^46 - 350 * q^47 - 1368 * q^48 - 754 * q^51 + 628 * q^52 + 56 * q^53 - 648 * q^54 + 668 * q^57 + 8 * q^58 + 1232 * q^59 - 336 * q^61 - 1200 * q^62 + 2128 * q^64 - 1976 * q^66 + 152 * q^67 + 908 * q^68 - 1332 * q^69 - 1904 * q^71 + 800 * q^72 + 676 * q^73 + 2016 * q^74 - 1768 * q^76 + 440 * q^78 + 1014 * q^79 - 1454 * q^81 - 816 * q^82 - 376 * q^83 - 768 * q^86 - 410 * q^87 + 4688 * q^88 + 216 * q^89 - 264 * q^92 - 2760 * q^93 + 1928 * q^94 + 2464 * q^96 + 2742 * q^97 + 940 * 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 $

$2$	−5.41421	−1.91421	−0.957107	−	0.289735i	$-0.906433\pi$
$2$	−5.41421	−1.91421	−0.957107	+	0.289735i	$0.906433\pi$
$3$	−4.65685	−0.896212	−0.448106	−	0.893980i	$-0.647901\pi$
$3$	−4.65685	−0.896212	−0.448106	+	0.893980i	$0.647901\pi$
$4$	21.3137	2.66421
$5$	0	0
$5$	0	0
$6$	25.2132	1.71554
$7$	0	0
$7$	0	0
$8$	−72.0833	−3.18566
$9$	−5.31371	−0.196804
$10$	0	0
$11$	−52.2548	−1.43231	−0.716156	−	0.697941i	$-0.754100\pi$
$11$	−52.2548	−1.43231	−0.716156	+	0.697941i	$0.754100\pi$
$12$	−99.2548	−2.38770
$13$	30.6569	0.654052	0.327026	−	0.945015i	$-0.393953\pi$
$13$	30.6569	0.654052	0.327026	+	0.945015i	$0.393953\pi$
$14$	0	0
$15$	0	0
$16$	219.765	3.43382
$17$	37.2254	0.531087	0.265544	−	0.964099i	$-0.414449\pi$
$17$	37.2254	0.531087	0.265544	+	0.964099i	$0.414449\pi$
$18$	28.7696	0.376725
$19$	−80.2254	−0.968683	−0.484341	−	0.874879i	$-0.660941\pi$
$19$	−80.2254	−0.968683	−0.484341	+	0.874879i	$0.660941\pi$
$20$	0	0
$21$	0	0
$22$	282.919	2.74175
$23$	−25.8335	−0.234202	−0.117101	−	0.993120i	$-0.537360\pi$
$23$	−25.8335	−0.234202	−0.117101	+	0.993120i	$0.537360\pi$
$24$	335.681	2.85503
$25$	0	0
$26$	−165.983	−1.25200
$27$	150.480	1.07259
$28$	0	0
$29$	20.9411	0.134092	0.0670460	−	0.997750i	$-0.478643\pi$
$29$	20.9411	0.134092	0.0670460	+	0.997750i	$0.478643\pi$
$30$	0	0
$31$	314.558	1.82246	0.911232	−	0.411894i	$-0.135133\pi$
$31$	314.558	1.82246	0.911232	+	0.411894i	$0.135133\pi$
$32$	−613.186	−3.38741
$33$	243.343	1.28365
$34$	−201.546	−1.01661
$35$	0	0
$36$	−113.255	−0.524328
$37$	−197.147	−0.875968	−0.437984	−	0.898983i	$-0.644307\pi$
$37$	−197.147	−0.875968	−0.437984	+	0.898983i	$0.644307\pi$
$38$	434.357	1.85427
$39$	−142.765	−0.586170
$40$	0	0
$41$	−11.3625	−0.0432810	−0.0216405	−	0.999766i	$-0.506889\pi$
$41$	−11.3625	−0.0432810	−0.0216405	+	0.999766i	$0.506889\pi$
$42$	0	0
$43$	33.8335	0.119990	0.0599948	−	0.998199i	$-0.480892\pi$
$43$	33.8335	0.119990	0.0599948	+	0.998199i	$0.480892\pi$
$44$	−1113.74	−3.81598
$45$	0	0
$46$	139.868	0.448313
$47$	−361.676	−1.12247	−0.561233	−	0.827658i	$-0.689673\pi$
$47$	−361.676	−1.12247	−0.561233	+	0.827658i	$0.689673\pi$
$48$	−1023.41	−3.07743
$49$	0	0
$50$	0	0
$51$	−173.353	−0.475967
$52$	653.411	1.74254
$53$	−153.019	−0.396582	−0.198291	−	0.980143i	$-0.563539\pi$
$53$	−153.019	−0.396582	−0.198291	+	0.980143i	$0.563539\pi$
$54$	−814.732	−2.05317
$55$	0	0
$56$	0	0
$57$	373.598	0.868145
$58$	−113.380	−0.256681
$59$	616.000	1.35926	0.679630	−	0.733555i	$-0.262140\pi$
$59$	616.000	1.35926	0.679630	+	0.733555i	$0.262140\pi$
$60$	0	0
$61$	−15.2649	−0.0320406	−0.0160203	−	0.999872i	$-0.505100\pi$
$61$	−15.2649	−0.0320406	−0.0160203	+	0.999872i	$0.505100\pi$
$62$	−1703.09	−3.48858
$63$	0	0
$64$	1561.80	3.05040
$65$	0	0
$66$	−1317.51	−2.45719
$67$	166.510	0.303618	0.151809	−	0.988410i	$-0.451490\pi$
$67$	166.510	0.303618	0.151809	+	0.988410i	$0.451490\pi$
$68$	793.411	1.41493
$69$	120.303	0.209895
$70$	0	0
$71$	−952.000	−1.59129	−0.795645	−	0.605763i	$-0.792868\pi$
$71$	−952.000	−1.59129	−0.795645	+	0.605763i	$0.792868\pi$
$72$	383.029	0.626951
$73$	−148.489	−0.238074	−0.119037	−	0.992890i	$-0.537981\pi$
$73$	−148.489	−0.238074	−0.119037	+	0.992890i	$0.537981\pi$
$74$	1067.40	1.67679
$75$	0	0
$76$	−1709.90	−2.58078
$77$	0	0
$78$	772.958	1.12205
$79$	857.725	1.22154	0.610770	−	0.791808i	$-0.290860\pi$
$79$	857.725	1.22154	0.610770	+	0.791808i	$0.290860\pi$
$80$	0	0
$81$	−557.294	−0.764464
$82$	61.5189	0.0828491
$83$	660.528	0.873523	0.436761	−	0.899577i	$-0.356125\pi$
$83$	660.528	0.873523	0.436761	+	0.899577i	$0.356125\pi$
$84$	0	0
$85$	0	0
$86$	−183.182	−0.229686
$87$	−97.5198	−0.120175
$88$	3766.70	4.56286
$89$	45.7746	0.0545180	0.0272590	−	0.999628i	$-0.491322\pi$
$89$	45.7746	0.0545180	0.0272590	+	0.999628i	$0.491322\pi$
$90$	0	0
$91$	0	0
$92$	−550.607	−0.623965
$93$	−1464.85	−1.63331
$94$	1958.19	2.14864
$95$	0	0
$96$	2855.52	3.03583
$97$	1682.13	1.76076	0.880382	−	0.474265i	$-0.157286\pi$
$97$	1682.13	1.76076	0.880382	+	0.474265i	$0.157286\pi$
$98$	0	0
$99$	277.667	0.281885
$100$	0	0
$101$	434.167	0.427734	0.213867	−	0.976863i	$-0.431394\pi$
$101$	434.167	0.427734	0.213867	+	0.976863i	$0.431394\pi$
$102$	938.572	0.911102
$103$	345.577	0.330589	0.165295	−	0.986244i	$-0.447142\pi$
$103$	345.577	0.330589	0.165295	+	0.986244i	$0.447142\pi$
$104$	−2209.85	−2.08359
$105$	0	0
$106$	828.479	0.759142
$107$	−217.119	−0.196165	−0.0980825	−	0.995178i	$-0.531271\pi$
$107$	−217.119	−0.196165	−0.0980825	+	0.995178i	$0.531271\pi$
$108$	3207.29	2.85761
$109$	1734.41	1.52409	0.762047	−	0.647521i	$-0.224194\pi$
$109$	1734.41	1.52409	0.762047	+	0.647521i	$0.224194\pi$
$110$	0	0
$111$	918.086	0.785053
$112$	0	0
$113$	1854.20	1.54362	0.771809	−	0.635855i	$-0.219352\pi$
$113$	1854.20	1.54362	0.771809	+	0.635855i	$0.219352\pi$
$114$	−2022.74	−1.66181
$115$	0	0
$116$	446.333	0.357250
$117$	−162.902	−0.128720
$118$	−3335.16	−2.60191
$119$	0	0
$120$	0	0
$121$	1399.57	1.05152
$122$	82.6476	0.0613325
$123$	52.9134	0.0387890
$124$	6704.41	4.85543
$125$	0	0
$126$	0	0
$127$	−1394.51	−0.974352	−0.487176	−	0.873304i	$-0.661973\pi$
$127$	−1394.51	−0.974352	−0.487176	+	0.873304i	$0.661973\pi$
$128$	−3550.45	−2.45171
$129$	−157.558	−0.107536
$130$	0	0
$131$	−1762.42	−1.17544	−0.587722	−	0.809063i	$-0.699975\pi$
$131$	−1762.42	−1.17544	−0.587722	+	0.809063i	$0.699975\pi$
$132$	5186.54	3.41993
$133$	0	0
$134$	−901.519	−0.581189
$135$	0	0
$136$	−2683.33	−1.69186
$137$	922.949	0.575568	0.287784	−	0.957695i	$-0.407081\pi$
$137$	922.949	0.575568	0.287784	+	0.957695i	$0.407081\pi$
$138$	−651.345	−0.401784
$139$	196.039	0.119624	0.0598122	−	0.998210i	$-0.480950\pi$
$139$	196.039	0.119624	0.0598122	+	0.998210i	$0.480950\pi$
$140$	0	0
$141$	1684.27	1.00597
$142$	5154.33	3.04607
$143$	−1601.97	−0.936807
$144$	−1167.76	−0.675790
$145$	0	0
$146$	803.954	0.455724
$147$	0	0
$148$	−4201.94	−2.33376
$149$	780.372	0.429064	0.214532	−	0.976717i	$-0.431177\pi$
$149$	780.372	0.429064	0.214532	+	0.976717i	$0.431177\pi$
$150$	0	0
$151$	−2319.43	−1.25002	−0.625008	−	0.780618i	$-0.714904\pi$
$151$	−2319.43	−1.25002	−0.625008	+	0.780618i	$0.714904\pi$
$152$	5782.91	3.08589
$153$	−197.805	−0.104520
$154$	0	0
$155$	0	0
$156$	−3042.84	−1.56168
$157$	1022.90	0.519977	0.259989	−	0.965612i	$-0.416281\pi$
$157$	1022.90	0.519977	0.259989	+	0.965612i	$0.416281\pi$
$158$	−4643.91	−2.33829
$159$	712.589	0.355421
$160$	0	0
$161$	0	0
$162$	3017.31	1.46335
$163$	1350.63	0.649013	0.324507	−	0.945883i	$-0.394802\pi$
$163$	1350.63	0.649013	0.324507	+	0.945883i	$0.394802\pi$
$164$	−242.177	−0.115310
$165$	0	0
$166$	−3576.24	−1.67211
$167$	−1230.58	−0.570209	−0.285105	−	0.958496i	$-0.592028\pi$
$167$	−1230.58	−0.570209	−0.285105	+	0.958496i	$0.592028\pi$
$168$	0	0
$169$	−1257.16	−0.572215
$170$	0	0
$171$	426.294	0.190641
$172$	721.117	0.319678
$173$	−2487.65	−1.09325	−0.546626	−	0.837377i	$-0.684088\pi$
$173$	−2487.65	−1.09325	−0.546626	+	0.837377i	$0.684088\pi$
$174$	527.993	0.230040
$175$	0	0
$176$	−11483.8	−4.91830
$177$	−2868.62	−1.21819
$178$	−247.833	−0.104359
$179$	1621.18	0.676941	0.338471	−	0.940977i	$-0.390090\pi$
$179$	1621.18	0.676941	0.338471	+	0.940977i	$0.390090\pi$
$180$	0	0
$181$	−2593.69	−1.06512	−0.532561	−	0.846392i	$-0.678770\pi$
$181$	−2593.69	−1.06512	−0.532561	+	0.846392i	$0.678770\pi$
$182$	0	0
$183$	71.0866	0.0287151
$184$	1862.16	0.746089
$185$	0	0
$186$	7931.03	3.12651
$187$	−1945.21	−0.760682
$188$	−7708.66	−2.99049
$189$	0	0
$190$	0	0
$191$	−1823.08	−0.690645	−0.345323	−	0.938484i	$-0.612231\pi$
$191$	−1823.08	−0.690645	−0.345323	+	0.938484i	$0.612231\pi$
$192$	−7273.09	−2.73380
$193$	1541.03	0.574744	0.287372	−	0.957819i	$-0.407218\pi$
$193$	1541.03	0.574744	0.287372	+	0.957819i	$0.407218\pi$
$194$	−9107.39	−3.37048
$195$	0	0
$196$	0	0
$197$	−701.243	−0.253612	−0.126806	−	0.991928i	$-0.540473\pi$
$197$	−701.243	−0.253612	−0.126806	+	0.991928i	$0.540473\pi$
$198$	−1503.35	−0.539587
$199$	−3294.96	−1.17374	−0.586868	−	0.809682i	$-0.699639\pi$
$199$	−3294.96	−1.17374	−0.586868	+	0.809682i	$0.699639\pi$
$200$	0	0
$201$	−775.411	−0.272106
$202$	−2350.67	−0.818775
$203$	0	0
$204$	−3694.80	−1.26808
$205$	0	0
$206$	−1871.03	−0.632819
$207$	137.272	0.0460920
$208$	6737.29	2.24590
$209$	4192.16	1.38746
$210$	0	0
$211$	4082.35	1.33195	0.665974	−	0.745975i	$-0.268016\pi$
$211$	4082.35	1.33195	0.665974	+	0.745975i	$0.268016\pi$
$212$	−3261.41	−1.05658
$213$	4433.33	1.42613
$214$	1175.53	0.375502
$215$	0	0
$216$	−10847.1	−3.41691
$217$	0	0
$218$	−9390.46	−2.91744
$219$	691.494	0.213364
$220$	0	0
$221$	1141.21	0.347359
$222$	−4970.71	−1.50276
$223$	747.161	0.224366	0.112183	−	0.993688i	$-0.464216\pi$
$223$	747.161	0.224366	0.112183	+	0.993688i	$0.464216\pi$
$224$	0	0
$225$	0	0
$226$	−10039.1	−2.95481
$227$	1665.67	0.487025	0.243513	−	0.969898i	$-0.421700\pi$
$227$	1665.67	0.487025	0.243513	+	0.969898i	$0.421700\pi$
$228$	7962.76	2.31292
$229$	6628.35	1.91272	0.956362	−	0.292183i	$-0.0943816\pi$
$229$	6628.35	1.91272	0.956362	+	0.292183i	$0.0943816\pi$
$230$	0	0
$231$	0	0
$232$	−1509.50	−0.427172
$233$	432.431	0.121586	0.0607929	−	0.998150i	$-0.480637\pi$
$233$	432.431	0.121586	0.0607929	+	0.998150i	$0.480637\pi$
$234$	881.984	0.246398
$235$	0	0
$236$	13129.2	3.62136
$237$	−3994.30	−1.09476
$238$	0	0
$239$	5580.44	1.51033	0.755165	−	0.655535i	$-0.227557\pi$
$239$	5580.44	1.51033	0.755165	+	0.655535i	$0.227557\pi$
$240$	0	0
$241$	6296.87	1.68306	0.841529	−	0.540212i	$-0.181656\pi$
$241$	6296.87	1.68306	0.841529	+	0.540212i	$0.181656\pi$
$242$	−7577.56	−2.01283
$243$	−1467.73	−0.387468
$244$	−325.352	−0.0853629
$245$	0	0
$246$	−286.485	−0.0742504
$247$	−2459.46	−0.633569
$248$	−22674.4	−5.80575
$249$	−3075.98	−0.782862
$250$	0	0
$251$	−311.921	−0.0784393	−0.0392197	−	0.999231i	$-0.512487\pi$
$251$	−311.921	−0.0784393	−0.0392197	+	0.999231i	$0.512487\pi$
$252$	0	0
$253$	1349.92	0.335451
$254$	7550.17	1.86512
$255$	0	0
$256$	6728.46	1.64269
$257$	−7861.39	−1.90809	−0.954046	−	0.299659i	$-0.903127\pi$
$257$	−7861.39	−1.90809	−0.954046	+	0.299659i	$0.903127\pi$
$258$	853.050	0.205847
$259$	0	0
$260$	0	0
$261$	−111.275	−0.0263899
$262$	9542.11	2.25005
$263$	−5227.09	−1.22554	−0.612769	−	0.790262i	$-0.709944\pi$
$263$	−5227.09	−1.22554	−0.612769	+	0.790262i	$0.709944\pi$
$264$	−17541.0	−4.08929
$265$	0	0
$266$	0	0
$267$	−213.166	−0.0488596
$268$	3548.94	0.808903
$269$	−1281.71	−0.290510	−0.145255	−	0.989394i	$-0.546400\pi$
$269$	−1281.71	−0.290510	−0.145255	+	0.989394i	$0.546400\pi$
$270$	0	0
$271$	−4704.14	−1.05445	−0.527226	−	0.849725i	$-0.676768\pi$
$271$	−4704.14	−1.05445	−0.527226	+	0.849725i	$0.676768\pi$
$272$	8180.82	1.82366
$273$	0	0
$274$	−4997.04	−1.10176
$275$	0	0
$276$	2564.10	0.559205
$277$	−8958.56	−1.94321	−0.971603	−	0.236619i	$-0.923961\pi$
$277$	−8958.56	−1.94321	−0.971603	+	0.236619i	$0.923961\pi$
$278$	−1061.40	−0.228987
$279$	−1671.47	−0.358668
$280$	0	0
$281$	−370.904	−0.0787412	−0.0393706	−	0.999225i	$-0.512535\pi$
$281$	−370.904	−0.0787412	−0.0393706	+	0.999225i	$0.512535\pi$
$282$	−9119.02	−1.92564
$283$	−5822.26	−1.22296	−0.611479	−	0.791261i	$-0.709425\pi$
$283$	−5822.26	−1.22296	−0.611479	+	0.791261i	$0.709425\pi$
$284$	−20290.7	−4.23954
$285$	0	0
$286$	8673.40	1.79325
$287$	0	0
$288$	3258.29	0.666655
$289$	−3527.27	−0.717946
$290$	0	0
$291$	−7833.42	−1.57802
$292$	−3164.86	−0.634279
$293$	7443.79	1.48420	0.742100	−	0.670289i	$-0.233830\pi$
$293$	7443.79	1.48420	0.742100	+	0.670289i	$0.233830\pi$
$294$	0	0
$295$	0	0
$296$	14211.0	2.79053
$297$	−7863.32	−1.53628
$298$	−4225.10	−0.821320
$299$	−791.973	−0.153181
$300$	0	0
$301$	0	0
$302$	12557.9	2.39280
$303$	−2021.85	−0.383341
$304$	−17630.7	−3.32628
$305$	0	0
$306$	1070.96	0.200074
$307$	−761.674	−0.141600	−0.0707998	−	0.997491i	$-0.522555\pi$
$307$	−761.674	−0.141600	−0.0707998	+	0.997491i	$0.522555\pi$
$308$	0	0
$309$	−1609.30	−0.296278
$310$	0	0
$311$	−7718.69	−1.40735	−0.703677	−	0.710520i	$-0.748460\pi$
$311$	−7718.69	−1.40735	−0.703677	+	0.710520i	$0.748460\pi$
$312$	10290.9	1.86734
$313$	8556.00	1.54509	0.772546	−	0.634959i	$-0.218983\pi$
$313$	8556.00	1.54509	0.772546	+	0.634959i	$0.218983\pi$
$314$	−5538.21	−0.995348
$315$	0	0
$316$	18281.3	3.25444
$317$	7780.95	1.37862	0.689309	−	0.724468i	$-0.257914\pi$
$317$	7780.95	1.37862	0.689309	+	0.724468i	$0.257914\pi$
$318$	−3858.11	−0.680352
$319$	−1094.28	−0.192062
$320$	0	0
$321$	1011.09	0.175805
$322$	0	0
$323$	−2986.42	−0.514455
$324$	−11878.0	−2.03670
$325$	0	0
$326$	−7312.58	−1.24235
$327$	−8076.89	−1.36591
$328$	819.045	0.137879
$329$	0	0
$330$	0	0
$331$	−4932.12	−0.819015	−0.409507	−	0.912307i	$-0.634299\pi$
$331$	−4932.12	−0.819015	−0.409507	+	0.912307i	$0.634299\pi$
$332$	14078.3	2.32725
$333$	1047.58	0.172394
$334$	6662.61	1.09150
$335$	0	0
$336$	0	0
$337$	7121.13	1.15108	0.575538	−	0.817775i	$-0.304793\pi$
$337$	7121.13	1.15108	0.575538	+	0.817775i	$0.304793\pi$
$338$	6806.52	1.09534
$339$	−8634.76	−1.38341
$340$	0	0
$341$	−16437.2	−2.61034
$342$	−2308.05	−0.364927
$343$	0	0
$344$	−2438.83	−0.382246
$345$	0	0
$346$	13468.7	2.09272
$347$	−9540.58	−1.47598	−0.737991	−	0.674811i	$-0.764225\pi$
$347$	−9540.58	−1.47598	−0.737991	+	0.674811i	$0.764225\pi$
$348$	−2078.51	−0.320172
$349$	−1281.65	−0.196576	−0.0982880	−	0.995158i	$-0.531337\pi$
$349$	−1281.65	−0.196576	−0.0982880	+	0.995158i	$0.531337\pi$
$350$	0	0
$351$	4613.25	0.701530
$352$	32041.9	4.85182
$353$	5798.07	0.874221	0.437110	−	0.899408i	$-0.356002\pi$
$353$	5798.07	0.874221	0.437110	+	0.899408i	$0.356002\pi$
$354$	15531.3	2.33187
$355$	0	0
$356$	975.627	0.145247
$357$	0	0
$358$	−8777.40	−1.29581
$359$	2267.29	0.333323	0.166662	−	0.986014i	$-0.446701\pi$
$359$	2267.29	0.333323	0.166662	+	0.986014i	$0.446701\pi$
$360$	0	0
$361$	−422.886	−0.0616541
$362$	14042.8	2.03887
$363$	−6517.58	−0.942381
$364$	0	0
$365$	0	0
$366$	−384.878	−0.0549669
$367$	−7372.85	−1.04866	−0.524332	−	0.851514i	$-0.675685\pi$
$367$	−7372.85	−1.04866	−0.524332	+	0.851514i	$0.675685\pi$
$368$	−5677.28	−0.804209
$369$	60.3769	0.00851788
$370$	0	0
$371$	0	0
$372$	−31221.4	−4.35150
$373$	−6447.14	−0.894961	−0.447480	−	0.894294i	$-0.647679\pi$
$373$	−6447.14	−0.894961	−0.447480	+	0.894294i	$0.647679\pi$
$374$	10531.8	1.45611
$375$	0	0
$376$	26070.8	3.57579
$377$	641.989	0.0877032
$378$	0	0
$379$	−4247.57	−0.575680	−0.287840	−	0.957678i	$-0.592937\pi$
$379$	−4247.57	−0.575680	−0.287840	+	0.957678i	$0.592937\pi$
$380$	0	0
$381$	6494.03	0.873226
$382$	9870.53	1.32204
$383$	−6681.86	−0.891454	−0.445727	−	0.895169i	$-0.647055\pi$
$383$	−6681.86	−0.891454	−0.445727	+	0.895169i	$0.647055\pi$
$384$	16533.9	2.19725
$385$	0	0
$386$	−8343.45	−1.10018
$387$	−179.781	−0.0236145
$388$	35852.4	4.69105
$389$	−6371.78	−0.830494	−0.415247	−	0.909709i	$-0.636305\pi$
$389$	−6371.78	−0.830494	−0.415247	+	0.909709i	$0.636305\pi$
$390$	0	0
$391$	−961.661	−0.124382
$392$	0	0
$393$	8207.33	1.05345
$394$	3796.68	0.485467
$395$	0	0
$396$	5918.11	0.751001
$397$	4247.93	0.537021	0.268510	−	0.963277i	$-0.413469\pi$
$397$	4247.93	0.537021	0.268510	+	0.963277i	$0.413469\pi$
$398$	17839.6	2.24678
$399$	0	0
$400$	0	0
$401$	−8833.62	−1.10008	−0.550038	−	0.835140i	$-0.685387\pi$
$401$	−8833.62	−1.10008	−0.550038	+	0.835140i	$0.685387\pi$
$402$	4198.24	0.520869
$403$	9643.37	1.19199
$404$	9253.70	1.13958
$405$	0	0
$406$	0	0
$407$	10301.9	1.25466
$408$	12495.9	1.51627
$409$	319.205	0.0385908	0.0192954	−	0.999814i	$-0.493858\pi$
$409$	319.205	0.0385908	0.0192954	+	0.999814i	$0.493858\pi$
$410$	0	0
$411$	−4298.04	−0.515831
$412$	7365.53	0.880761
$413$	0	0
$414$	−743.218	−0.0882298
$415$	0	0
$416$	−18798.3	−2.21554
$417$	−912.924	−0.107209
$418$	−22697.3	−2.65589
$419$	12789.2	1.49115	0.745577	−	0.666420i	$-0.232174\pi$
$419$	12789.2	1.49115	0.745577	+	0.666420i	$0.232174\pi$
$420$	0	0
$421$	−6747.40	−0.781112	−0.390556	−	0.920579i	$-0.627717\pi$
$421$	−6747.40	−0.781112	−0.390556	+	0.920579i	$0.627717\pi$
$422$	−22102.7	−2.54963
$423$	1921.84	0.220906
$424$	11030.1	1.26337
$425$	0	0
$426$	−24003.0	−2.72992
$427$	0	0
$428$	−4627.60	−0.522625
$429$	7460.14	0.839577
$430$	0	0
$431$	−5184.75	−0.579444	−0.289722	−	0.957111i	$-0.593563\pi$
$431$	−5184.75	−0.579444	−0.289722	+	0.957111i	$0.593563\pi$
$432$	33070.2	3.68308
$433$	−4242.03	−0.470806	−0.235403	−	0.971898i	$-0.575641\pi$
$433$	−4242.03	−0.470806	−0.235403	+	0.971898i	$0.575641\pi$
$434$	0	0
$435$	0	0
$436$	36966.7	4.06051
$437$	2072.50	0.226868
$438$	−3743.90	−0.408425
$439$	5434.12	0.590789	0.295394	−	0.955375i	$-0.404549\pi$
$439$	5434.12	0.590789	0.295394	+	0.955375i	$0.404549\pi$
$440$	0	0
$441$	0	0
$442$	−6178.77	−0.664919
$443$	11493.8	1.23270	0.616350	−	0.787472i	$-0.288611\pi$
$443$	11493.8	1.23270	0.616350	+	0.787472i	$0.288611\pi$
$444$	19567.8	2.09155
$445$	0	0
$446$	−4045.29	−0.429484
$447$	−3634.08	−0.384532
$448$	0	0
$449$	−16849.3	−1.77098	−0.885489	−	0.464661i	$-0.846176\pi$
$449$	−16849.3	−1.77098	−0.885489	+	0.464661i	$0.846176\pi$
$450$	0	0
$451$	593.745	0.0619919
$452$	39520.0	4.11253
$453$	10801.2	1.12028
$454$	−9018.32	−0.932270
$455$	0	0
$456$	−26930.2	−2.76561
$457$	−15348.5	−1.57106	−0.785528	−	0.618826i	$-0.787609\pi$
$457$	−15348.5	−1.57106	−0.785528	+	0.618826i	$0.787609\pi$
$458$	−35887.3	−3.66136
$459$	5601.69	0.569639
$460$	0	0
$461$	−14038.4	−1.41830	−0.709148	−	0.705059i	$-0.750920\pi$
$461$	−14038.4	−1.41830	−0.709148	+	0.705059i	$0.750920\pi$
$462$	0	0
$463$	8661.23	0.869377	0.434689	−	0.900581i	$-0.356858\pi$
$463$	8661.23	0.869377	0.434689	+	0.900581i	$0.356858\pi$
$464$	4602.12	0.460448
$465$	0	0
$466$	−2341.27	−0.232741
$467$	7014.71	0.695079	0.347539	−	0.937665i	$-0.387017\pi$
$467$	7014.71	0.695079	0.347539	+	0.937665i	$0.387017\pi$
$468$	−3472.04	−0.342938
$469$	0	0
$470$	0	0
$471$	−4763.50	−0.466010
$472$	−44403.3	−4.33014
$473$	−1767.96	−0.171863
$474$	21626.0	2.09560
$475$	0	0
$476$	0	0
$477$	813.100	0.0780488
$478$	−30213.7	−2.89109
$479$	−18134.7	−1.72984	−0.864922	−	0.501907i	$-0.832632\pi$
$479$	−18134.7	−1.72984	−0.864922	+	0.501907i	$0.832632\pi$
$480$	0	0
$481$	−6043.91	−0.572929
$482$	−34092.6	−3.22173
$483$	0	0
$484$	29830.0	2.80146
$485$	0	0
$486$	7946.59	0.741697
$487$	−16537.8	−1.53881	−0.769405	−	0.638761i	$-0.779447\pi$
$487$	−16537.8	−1.53881	−0.769405	+	0.638761i	$0.779447\pi$
$488$	1100.35	0.102070
$489$	−6289.67	−0.581654
$490$	0	0
$491$	220.608	0.0202768	0.0101384	−	0.999949i	$-0.496773\pi$
$491$	220.608	0.0202768	0.0101384	+	0.999949i	$0.496773\pi$
$492$	1127.78	0.103342
$493$	779.542	0.0712146
$494$	13316.0	1.21279
$495$	0	0
$496$	69128.8	6.25801
$497$	0	0
$498$	16654.0	1.49856
$499$	5939.04	0.532801	0.266401	−	0.963862i	$-0.414166\pi$
$499$	5939.04	0.532801	0.266401	+	0.963862i	$0.414166\pi$
$500$	0	0
$501$	5730.62	0.511029
$502$	1688.81	0.150150
$503$	−11604.8	−1.02869	−0.514345	−	0.857584i	$-0.671965\pi$
$503$	−11604.8	−1.02869	−0.514345	+	0.857584i	$0.671965\pi$
$504$	0	0
$505$	0	0
$506$	−7308.78	−0.642124
$507$	5854.40	0.512826
$508$	−29722.2	−2.59588
$509$	1867.67	0.162639	0.0813193	−	0.996688i	$-0.474087\pi$
$509$	1867.67	0.162639	0.0813193	+	0.996688i	$0.474087\pi$
$510$	0	0
$511$	0	0
$512$	−8025.75	−0.692757
$513$	−12072.3	−1.03900
$514$	42563.2	3.65250
$515$	0	0
$516$	−3358.14	−0.286499
$517$	18899.3	1.60772
$518$	0	0
$519$	11584.6	0.979786
$520$	0	0
$521$	−6117.21	−0.514395	−0.257197	−	0.966359i	$-0.582799\pi$
$521$	−6117.21	−0.514395	−0.257197	+	0.966359i	$0.582799\pi$
$522$	602.467	0.0505158
$523$	−16685.6	−1.39505	−0.697524	−	0.716561i	$-0.745715\pi$
$523$	−16685.6	−1.39505	−0.697524	+	0.716561i	$0.745715\pi$
$524$	−37563.7	−3.13164
$525$	0	0
$526$	28300.6	2.34594
$527$	11709.6	0.967887
$528$	53478.2	4.40784
$529$	−11499.6	−0.945149
$530$	0	0
$531$	−3273.24	−0.267508
$532$	0	0
$533$	−348.338	−0.0283081
$534$	1154.12	0.0935278
$535$	0	0
$536$	−12002.6	−0.967223
$537$	−7549.59	−0.606683
$538$	6939.44	0.556097
$539$	0	0
$540$	0	0
$541$	9309.03	0.739790	0.369895	−	0.929074i	$-0.379394\pi$
$541$	9309.03	0.739790	0.369895	+	0.929074i	$0.379394\pi$
$542$	25469.2	2.01845
$543$	12078.4	0.954575
$544$	−22826.1	−1.79901
$545$	0	0
$546$	0	0
$547$	−10894.7	−0.851598	−0.425799	−	0.904818i	$-0.640007\pi$
$547$	−10894.7	−0.851598	−0.425799	+	0.904818i	$0.640007\pi$
$548$	19671.5	1.53344
$549$	81.1134	0.00630571
$550$	0	0
$551$	−1680.01	−0.129893
$552$	−8671.81	−0.668654
$553$	0	0
$554$	48503.6	3.71971
$555$	0	0
$556$	4178.31	0.318705
$557$	7873.90	0.598973	0.299486	−	0.954101i	$-0.403185\pi$
$557$	7873.90	0.598973	0.299486	+	0.954101i	$0.403185\pi$
$558$	9049.71	0.686567
$559$	1037.23	0.0784796
$560$	0	0
$561$	9058.55	0.681733
$562$	2008.15	0.150728
$563$	21770.7	1.62971	0.814854	−	0.579666i	$-0.196817\pi$
$563$	21770.7	1.62971	0.814854	+	0.579666i	$0.196817\pi$
$564$	35898.1	2.68011
$565$	0	0
$566$	31522.9	2.34100
$567$	0	0
$568$	68623.3	5.06931
$569$	−12381.3	−0.912213	−0.456106	−	0.889925i	$-0.650756\pi$
$569$	−12381.3	−0.912213	−0.456106	+	0.889925i	$0.650756\pi$
$570$	0	0
$571$	−5768.38	−0.422765	−0.211383	−	0.977403i	$-0.567797\pi$
$571$	−5768.38	−0.422765	−0.211383	+	0.977403i	$0.567797\pi$
$572$	−34143.9	−2.49585
$573$	8489.81	0.618965
$574$	0	0
$575$	0	0
$576$	−8298.97	−0.600330
$577$	4733.38	0.341513	0.170757	−	0.985313i	$-0.445379\pi$
$577$	4733.38	0.341513	0.170757	+	0.985313i	$0.445379\pi$
$578$	19097.4	1.37430
$579$	−7176.34	−0.515093
$580$	0	0
$581$	0	0
$582$	42411.8	3.02066
$583$	7996.00	0.568028
$584$	10703.6	0.758422
$585$	0	0
$586$	−40302.3	−2.84108
$587$	−8441.67	−0.593569	−0.296785	−	0.954944i	$-0.595914\pi$
$587$	−8441.67	−0.593569	−0.296785	+	0.954944i	$0.595914\pi$
$588$	0	0
$589$	−25235.6	−1.76539
$590$	0	0
$591$	3265.59	0.227290
$592$	−43326.0	−3.00792
$593$	18939.9	1.31158	0.655791	−	0.754943i	$-0.272335\pi$
$593$	18939.9	1.31158	0.655791	+	0.754943i	$0.272335\pi$
$594$	42573.7	2.94077
$595$	0	0
$596$	16632.6	1.14312
$597$	15344.2	1.05192
$598$	4287.91	0.293220
$599$	22655.3	1.54536	0.772681	−	0.634794i	$-0.218915\pi$
$599$	22655.3	1.54536	0.772681	+	0.634794i	$0.218915\pi$
$600$	0	0
$601$	15947.4	1.08237	0.541187	−	0.840902i	$-0.317975\pi$
$601$	15947.4	1.08237	0.541187	+	0.840902i	$0.317975\pi$
$602$	0	0
$603$	−884.784	−0.0597532
$604$	−49435.6	−3.33031
$605$	0	0
$606$	10946.7	0.733796
$607$	−25993.2	−1.73811	−0.869053	−	0.494719i	$-0.835271\pi$
$607$	−25993.2	−1.73811	−0.869053	+	0.494719i	$0.835271\pi$
$608$	49193.1	3.28132
$609$	0	0
$610$	0	0
$611$	−11087.9	−0.734152
$612$	−4215.96	−0.278464
$613$	−665.408	−0.0438427	−0.0219213	−	0.999760i	$-0.506978\pi$
$613$	−665.408	−0.0438427	−0.0219213	+	0.999760i	$0.506978\pi$
$614$	4123.87	0.271052
$615$	0	0
$616$	0	0
$617$	−18401.3	−1.20066	−0.600330	−	0.799752i	$-0.704964\pi$
$617$	−18401.3	−1.20066	−0.600330	+	0.799752i	$0.704964\pi$
$618$	8713.10	0.567140
$619$	11150.6	0.724040	0.362020	−	0.932170i	$-0.382087\pi$
$619$	11150.6	0.724040	0.362020	+	0.932170i	$0.382087\pi$
$620$	0	0
$621$	−3887.43	−0.251203
$622$	41790.6	2.69397
$623$	0	0
$624$	−31374.6	−2.01280
$625$	0	0
$626$	−46324.0	−2.95764
$627$	−19522.3	−1.24345
$628$	21801.8	1.38533
$629$	−7338.88	−0.465215
$630$	0	0
$631$	5381.79	0.339534	0.169767	−	0.985484i	$-0.445699\pi$
$631$	5381.79	0.339534	0.169767	+	0.985484i	$0.445699\pi$
$632$	−61827.6	−3.89141
$633$	−19010.9	−1.19371
$634$	−42127.7	−2.63897
$635$	0	0
$636$	15187.9	0.946918
$637$	0	0
$638$	5924.64	0.367647
$639$	5058.65	0.313172
$640$	0	0
$641$	−19455.1	−1.19880	−0.599398	−	0.800451i	$-0.704593\pi$
$641$	−19455.1	−1.19880	−0.599398	+	0.800451i	$0.704593\pi$
$642$	−5474.26	−0.336529
$643$	−14695.8	−0.901317	−0.450658	−	0.892696i	$-0.648811\pi$
$643$	−14695.8	−0.901317	−0.450658	+	0.892696i	$0.648811\pi$
$644$	0	0
$645$	0	0
$646$	16169.1	0.984777
$647$	−12694.8	−0.771383	−0.385691	−	0.922628i	$-0.626037\pi$
$647$	−12694.8	−0.771383	−0.385691	+	0.922628i	$0.626037\pi$
$648$	40171.6	2.43532
$649$	−32189.0	−1.94688
$650$	0	0
$651$	0	0
$652$	28786.8	1.72911
$653$	12385.6	0.742247	0.371124	−	0.928583i	$-0.378973\pi$
$653$	12385.6	0.742247	0.371124	+	0.928583i	$0.378973\pi$
$654$	43730.0	2.61465
$655$	0	0
$656$	−2497.07	−0.148619
$657$	789.030	0.0468539
$658$	0	0
$659$	−2072.18	−0.122489	−0.0612447	−	0.998123i	$-0.519507\pi$
$659$	−2072.18	−0.122489	−0.0612447	+	0.998123i	$0.519507\pi$
$660$	0	0
$661$	−1074.36	−0.0632193	−0.0316096	−	0.999500i	$-0.510063\pi$
$661$	−1074.36	−0.0632193	−0.0316096	+	0.999500i	$0.510063\pi$
$662$	26703.6	1.56777
$663$	−5314.47	−0.311307
$664$	−47613.0	−2.78275
$665$	0	0
$666$	−5671.84	−0.329999
$667$	−540.982	−0.0314047
$668$	−26228.2	−1.51916
$669$	−3479.42	−0.201080
$670$	0	0
$671$	797.667	0.0458921
$672$	0	0
$673$	−26195.2	−1.50037	−0.750186	−	0.661226i	$-0.770036\pi$
$673$	−26195.2	−1.50037	−0.750186	+	0.661226i	$0.770036\pi$
$674$	−38555.3	−2.20341
$675$	0	0
$676$	−26794.7	−1.52450
$677$	−4228.44	−0.240047	−0.120024	−	0.992771i	$-0.538297\pi$
$677$	−4228.44	−0.240047	−0.120024	+	0.992771i	$0.538297\pi$
$678$	46750.4	2.64814
$679$	0	0
$680$	0	0
$681$	−7756.80	−0.436478
$682$	88994.5	4.99674
$683$	−27525.5	−1.54207	−0.771036	−	0.636792i	$-0.780261\pi$
$683$	−27525.5	−1.54207	−0.771036	+	0.636792i	$0.780261\pi$
$684$	9085.91	0.507907
$685$	0	0
$686$	0	0
$687$	−30867.3	−1.71421
$688$	7435.40	0.412023
$689$	−4691.09	−0.259385
$690$	0	0
$691$	33324.4	1.83462	0.917309	−	0.398177i	$-0.130357\pi$
$691$	33324.4	1.83462	0.917309	+	0.398177i	$0.130357\pi$
$692$	−53021.1	−2.91266
$693$	0	0
$694$	51654.8	2.82534
$695$	0	0
$696$	7029.54	0.382836
$697$	−422.973	−0.0229860
$698$	6939.12	0.376289
$699$	−2013.77	−0.108967
$700$	0	0
$701$	−33262.9	−1.79219	−0.896094	−	0.443864i	$-0.853607\pi$
$701$	−33262.9	−1.79219	−0.896094	+	0.443864i	$0.853607\pi$
$702$	−24977.1	−1.34288
$703$	15816.2	0.848534
$704$	−81611.8	−4.36912
$705$	0	0
$706$	−31392.0	−1.67345
$707$	0	0
$708$	−61141.0	−3.24551
$709$	13703.0	0.725851	0.362926	−	0.931818i	$-0.381778\pi$
$709$	13703.0	0.725851	0.362926	+	0.931818i	$0.381778\pi$
$710$	0	0
$711$	−4557.70	−0.240404
$712$	−3299.58	−0.173676
$713$	−8126.14	−0.426825
$714$	0	0
$715$	0	0
$716$	34553.3	1.80352
$717$	−25987.3	−1.35358
$718$	−12275.6	−0.638052
$719$	8074.93	0.418838	0.209419	−	0.977826i	$-0.432843\pi$
$719$	8074.93	0.418838	0.209419	+	0.977826i	$0.432843\pi$
$720$	0	0
$721$	0	0
$722$	2289.59	0.118019
$723$	−29323.6	−1.50838
$724$	−55281.1	−2.83771
$725$	0	0
$726$	35287.6	1.80392
$727$	−3668.70	−0.187159	−0.0935794	−	0.995612i	$-0.529831\pi$
$727$	−3668.70	−0.187159	−0.0935794	+	0.995612i	$0.529831\pi$
$728$	0	0
$729$	21881.9	1.11172
$730$	0	0
$731$	1259.46	0.0637250
$732$	1515.12	0.0765033
$733$	−14980.3	−0.754857	−0.377428	−	0.926039i	$-0.623192\pi$
$733$	−14980.3	−0.754857	−0.377428	+	0.926039i	$0.623192\pi$
$734$	39918.2	2.00737
$735$	0	0
$736$	15840.7	0.793338
$737$	−8700.94	−0.434875
$738$	−326.894	−0.0163050
$739$	6530.59	0.325077	0.162538	−	0.986702i	$-0.448032\pi$
$739$	6530.59	0.325077	0.162538	+	0.986702i	$0.448032\pi$
$740$	0	0
$741$	11453.3	0.567812
$742$	0	0
$743$	−25952.0	−1.28141	−0.640704	−	0.767788i	$-0.721357\pi$
$743$	−25952.0	−1.28141	−0.640704	+	0.767788i	$0.721357\pi$
$744$	105591.	5.20318
$745$	0	0
$746$	34906.2	1.71315
$747$	−3509.85	−0.171913
$748$	−41459.6	−2.02662
$749$	0	0
$750$	0	0
$751$	−14093.9	−0.684813	−0.342407	−	0.939552i	$-0.611242\pi$
$751$	−14093.9	−0.684813	−0.342407	+	0.939552i	$0.611242\pi$
$752$	−79483.6	−3.85435
$753$	1452.57	0.0702983
$754$	−3475.87	−0.167883
$755$	0	0
$756$	0	0
$757$	2554.41	0.122644	0.0613220	−	0.998118i	$-0.480468\pi$
$757$	2554.41	0.122644	0.0613220	+	0.998118i	$0.480468\pi$
$758$	22997.2	1.10197
$759$	−6286.40	−0.300635
$760$	0	0
$761$	−2219.08	−0.105705	−0.0528527	−	0.998602i	$-0.516831\pi$
$761$	−2219.08	−0.105705	−0.0528527	+	0.998602i	$0.516831\pi$
$762$	−35160.1	−1.67154
$763$	0	0
$764$	−38856.5	−1.84003
$765$	0	0
$766$	36177.0	1.70643
$767$	18884.6	0.889028
$768$	−31333.5	−1.47220
$769$	22466.2	1.05352	0.526758	−	0.850015i	$-0.323408\pi$
$769$	22466.2	1.05352	0.526758	+	0.850015i	$0.323408\pi$
$770$	0	0
$771$	36609.3	1.71006
$772$	32845.0	1.53124
$773$	9674.79	0.450165	0.225083	−	0.974340i	$-0.427735\pi$
$773$	9674.79	0.450165	0.225083	+	0.974340i	$0.427735\pi$
$774$	973.374	0.0452031
$775$	0	0
$776$	−121253.	−5.60920
$777$	0	0
$778$	34498.2	1.58974
$779$	911.560	0.0419256
$780$	0	0
$781$	49746.6	2.27922
$782$	5206.64	0.238093
$783$	3151.23	0.143826
$784$	0	0
$785$	0	0
$786$	−44436.2	−2.01652
$787$	−20942.8	−0.948577	−0.474288	−	0.880370i	$-0.657295\pi$
$787$	−20942.8	−0.948577	−0.474288	+	0.880370i	$0.657295\pi$
$788$	−14946.1	−0.675676
$789$	24341.8	1.09834
$790$	0	0
$791$	0	0
$792$	−20015.1	−0.897989
$793$	−467.975	−0.0209562
$794$	−22999.2	−1.02797
$795$	0	0
$796$	−70227.8	−3.12708
$797$	23526.6	1.04561	0.522807	−	0.852451i	$-0.324885\pi$
$797$	23526.6	1.04561	0.522807	+	0.852451i	$0.324885\pi$
$798$	0	0
$799$	−13463.5	−0.596127
$800$	0	0
$801$	−243.233	−0.0107294
$802$	47827.1	2.10578
$803$	7759.29	0.340996
$804$	−16526.9	−0.724948
$805$	0	0
$806$	−52211.3	−2.28172
$807$	5968.72	0.260358
$808$	−31296.1	−1.36262
$809$	−18202.2	−0.791047	−0.395523	−	0.918456i	$-0.629437\pi$
$809$	−18202.2	−0.791047	−0.395523	+	0.918456i	$0.629437\pi$
$810$	0	0
$811$	2510.24	0.108689	0.0543443	−	0.998522i	$-0.482693\pi$
$811$	2510.24	0.108689	0.0543443	+	0.998522i	$0.482693\pi$
$812$	0	0
$813$	21906.5	0.945012
$814$	−55776.7	−2.40168
$815$	0	0
$816$	−38096.9	−1.63438
$817$	−2714.30	−0.116232
$818$	−1728.24	−0.0738711
$819$	0	0
$820$	0	0
$821$	17899.6	0.760903	0.380451	−	0.924801i	$-0.375769\pi$
$821$	17899.6	0.760903	0.380451	+	0.924801i	$0.375769\pi$
$822$	23270.5	0.987411
$823$	−14039.5	−0.594637	−0.297318	−	0.954778i	$-0.596092\pi$
$823$	−14039.5	−0.594637	−0.297318	+	0.954778i	$0.596092\pi$
$824$	−24910.3	−1.05315
$825$	0	0
$826$	0	0
$827$	−15127.4	−0.636073	−0.318036	−	0.948079i	$-0.603023\pi$
$827$	−15127.4	−0.636073	−0.318036	+	0.948079i	$0.603023\pi$
$828$	2925.77	0.122799
$829$	−21986.5	−0.921136	−0.460568	−	0.887624i	$-0.652354\pi$
$829$	−21986.5	−0.921136	−0.460568	+	0.887624i	$0.652354\pi$
$830$	0	0
$831$	41718.7	1.74152
$832$	47880.0	1.99512
$833$	0	0
$834$	4942.76	0.205220
$835$	0	0
$836$	89350.6	3.69648
$837$	47334.8	1.95476
$838$	−69243.5	−2.85439
$839$	2276.89	0.0936914	0.0468457	−	0.998902i	$-0.485083\pi$
$839$	2276.89	0.0936914	0.0468457	+	0.998902i	$0.485083\pi$
$840$	0	0
$841$	−23950.5	−0.982019
$842$	36531.8	1.49521
$843$	1727.25	0.0705688
$844$	87010.1	3.54859
$845$	0	0
$846$	−10405.3	−0.422861
$847$	0	0
$848$	−33628.2	−1.36179
$849$	27113.4	1.09603
$850$	0	0
$851$	5093.00	0.205154
$852$	94490.6	3.79952
$853$	13342.6	0.535570	0.267785	−	0.963479i	$-0.413708\pi$
$853$	13342.6	0.535570	0.267785	+	0.963479i	$0.413708\pi$
$854$	0	0
$855$	0	0
$856$	15650.6	0.624915
$857$	18690.9	0.745003	0.372502	−	0.928032i	$-0.378500\pi$
$857$	18690.9	0.745003	0.372502	+	0.928032i	$0.378500\pi$
$858$	−40390.8	−1.60713
$859$	−18318.9	−0.727628	−0.363814	−	0.931472i	$-0.618526\pi$
$859$	−18318.9	−0.727628	−0.363814	+	0.931472i	$0.618526\pi$
$860$	0	0
$861$	0	0
$862$	28071.3	1.10918
$863$	−38133.1	−1.50413	−0.752067	−	0.659087i	$-0.770943\pi$
$863$	−38133.1	−1.50413	−0.752067	+	0.659087i	$0.770943\pi$
$864$	−92272.3	−3.63330
$865$	0	0
$866$	22967.3	0.901223
$867$	16426.0	0.643432
$868$	0	0
$869$	−44820.3	−1.74962
$870$	0	0
$871$	5104.66	0.198582
$872$	−125022.	−4.85525
$873$	−8938.33	−0.346525
$874$	−11221.0	−0.434273
$875$	0	0
$876$	14738.3	0.568449
$877$	19707.5	0.758807	0.379404	−	0.925231i	$-0.376129\pi$
$877$	19707.5	0.758807	0.379404	+	0.925231i	$0.376129\pi$
$878$	−29421.5	−1.13090
$879$	−34664.6	−1.33016
$880$	0	0
$881$	14091.5	0.538883	0.269441	−	0.963017i	$-0.413161\pi$
$881$	14091.5	0.538883	0.269441	+	0.963017i	$0.413161\pi$
$882$	0	0
$883$	−3115.87	−0.118751	−0.0593757	−	0.998236i	$-0.518911\pi$
$883$	−3115.87	−0.118751	−0.0593757	+	0.998236i	$0.518911\pi$
$884$	24323.5	0.925438
$885$	0	0
$886$	−62229.8	−2.35965
$887$	38734.6	1.46627	0.733134	−	0.680084i	$-0.238057\pi$
$887$	38734.6	1.46627	0.733134	+	0.680084i	$0.238057\pi$
$888$	−66178.6	−2.50091
$889$	0	0
$890$	0	0
$891$	29121.3	1.09495
$892$	15924.8	0.597759
$893$	29015.6	1.08731
$894$	19675.7	0.736077
$895$	0	0
$896$	0	0
$897$	3688.10	0.137282
$898$	91225.8	3.39003
$899$	6587.21	0.244378
$900$	0	0
$901$	−5696.21	−0.210619
$902$	−3214.66	−0.118666
$903$	0	0
$904$	−133657.	−4.91744
$905$	0	0
$906$	−58480.2	−2.14445
$907$	−19242.9	−0.704464	−0.352232	−	0.935913i	$-0.614577\pi$
$907$	−19242.9	−0.704464	−0.352232	+	0.935913i	$0.614577\pi$
$908$	35501.7	1.29754
$909$	−2307.03	−0.0841799
$910$	0	0
$911$	34613.3	1.25882	0.629412	−	0.777072i	$-0.283296\pi$
$911$	34613.3	1.25882	0.629412	+	0.777072i	$0.283296\pi$
$912$	82103.6	2.98105
$913$	−34515.8	−1.25116
$914$	83100.1	3.00734
$915$	0	0
$916$	141275.	5.09591
$917$	0	0
$918$	−30328.7	−1.09041
$919$	25826.4	0.927022	0.463511	−	0.886091i	$-0.346589\pi$
$919$	25826.4	0.927022	0.463511	+	0.886091i	$0.346589\pi$
$920$	0	0
$921$	3547.01	0.126903
$922$	76007.0	2.71492
$923$	−29185.3	−1.04079
$924$	0	0
$925$	0	0
$926$	−46893.8	−1.66417
$927$	−1836.29	−0.0650613
$928$	−12840.8	−0.454224
$929$	−19451.6	−0.686960	−0.343480	−	0.939160i	$-0.611606\pi$
$929$	−19451.6	−0.686960	−0.343480	+	0.939160i	$0.611606\pi$
$930$	0	0
$931$	0	0
$932$	9216.70	0.323930
$933$	35944.8	1.26129
$934$	−37979.1	−1.33053
$935$	0	0
$936$	11742.5	0.410059
$937$	34469.1	1.20177	0.600884	−	0.799336i	$-0.294815\pi$
$937$	34469.1	1.20177	0.600884	+	0.799336i	$0.294815\pi$
$938$	0	0
$939$	−39844.1	−1.38473
$940$	0	0
$941$	−14156.4	−0.490419	−0.245209	−	0.969470i	$-0.578857\pi$
$941$	−14156.4	−0.490419	−0.245209	+	0.969470i	$0.578857\pi$
$942$	25790.6	0.892042
$943$	293.532	0.0101365
$944$	135375.	4.66746
$945$	0	0
$946$	9572.13	0.328982
$947$	38092.4	1.30711	0.653557	−	0.756877i	$-0.273276\pi$
$947$	38092.4	1.30711	0.653557	+	0.756877i	$0.273276\pi$
$948$	−85133.3	−2.91667
$949$	−4552.22	−0.155713
$950$	0	0
$951$	−36234.8	−1.23553
$952$	0	0
$953$	−5037.40	−0.171225	−0.0856126	−	0.996329i	$-0.527285\pi$
$953$	−5037.40	−0.171225	−0.0856126	+	0.996329i	$0.527285\pi$
$954$	−4402.30	−0.149402
$955$	0	0
$956$	118940.	4.02384
$957$	5095.88	0.172128
$958$	98185.1	3.31129
$959$	0	0
$960$	0	0
$961$	69156.0	2.32137
$962$	32723.0	1.09671
$963$	1153.71	0.0386060
$964$	134210.	4.48403
$965$	0	0
$966$	0	0
$967$	−11495.3	−0.382278	−0.191139	−	0.981563i	$-0.561218\pi$
$967$	−11495.3	−0.382278	−0.191139	+	0.981563i	$0.561218\pi$
$968$	−100885.	−3.34977
$969$	13907.3	0.461061
$970$	0	0
$971$	−22352.7	−0.738757	−0.369379	−	0.929279i	$-0.620429\pi$
$971$	−22352.7	−0.738757	−0.369379	+	0.929279i	$0.620429\pi$
$972$	−31282.7	−1.03230
$973$	0	0
$974$	89539.4	2.94561
$975$	0	0
$976$	−3354.69	−0.110022
$977$	−14345.7	−0.469765	−0.234882	−	0.972024i	$-0.575470\pi$
$977$	−14345.7	−0.469765	−0.234882	+	0.972024i	$0.575470\pi$
$978$	34053.6	1.11341
$979$	−2391.94	−0.0780867
$980$	0	0
$981$	−9216.15	−0.299948
$982$	−1194.42	−0.0388141
$983$	34460.9	1.11814	0.559070	−	0.829120i	$-0.311158\pi$
$983$	34460.9	1.11814	0.559070	+	0.829120i	$0.311158\pi$
$984$	−3814.17	−0.123568
$985$	0	0
$986$	−4220.61	−0.136320
$987$	0	0
$988$	−52420.2	−1.68796
$989$	−874.036	−0.0281019
$990$	0	0
$991$	−35189.6	−1.12799	−0.563993	−	0.825780i	$-0.690735\pi$
$991$	−35189.6	−1.12799	−0.563993	+	0.825780i	$0.690735\pi$
$992$	−192883.	−6.17342
$993$	22968.2	0.734011
$994$	0	0
$995$	0	0
$996$	−65560.6	−2.08571
$997$	−50730.0	−1.61147	−0.805734	−	0.592277i	$-0.798229\pi$
$997$	−50730.0	−1.61147	−0.805734	+	0.592277i	$0.798229\pi$
$998$	−32155.2	−1.01990
$999$	−29666.8	−0.939554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.4.a.m.1.1		2
5.4	even	2		245.4.a.k.1.2		2
7.6	odd	2		175.4.a.c.1.1		2
15.14	odd	2		2205.4.a.u.1.1		2
21.20	even	2		1575.4.a.z.1.2		2
35.4	even	6		245.4.e.i.226.1		4
35.9	even	6		245.4.e.i.116.1		4
35.13	even	4		175.4.b.c.99.4		4
35.19	odd	6		245.4.e.h.116.1		4
35.24	odd	6		245.4.e.h.226.1		4
35.27	even	4		175.4.b.c.99.1		4
35.34	odd	2		35.4.a.b.1.2	✓	2
105.104	even	2		315.4.a.f.1.1		2
140.139	even	2		560.4.a.r.1.2		2
280.69	odd	2		2240.4.a.bn.1.2		2
280.139	even	2		2240.4.a.bo.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.a.b.1.2	✓	2	35.34	odd	2
175.4.a.c.1.1		2	7.6	odd	2
175.4.b.c.99.1		4	35.27	even	4
175.4.b.c.99.4		4	35.13	even	4
245.4.a.k.1.2		2	5.4	even	2
245.4.e.h.116.1		4	35.19	odd	6
245.4.e.h.226.1		4	35.24	odd	6
245.4.e.i.116.1		4	35.9	even	6
245.4.e.i.226.1		4	35.4	even	6
315.4.a.f.1.1		2	105.104	even	2
560.4.a.r.1.2		2	140.139	even	2
1225.4.a.m.1.1		2	1.1	even	1	trivial
1575.4.a.z.1.2		2	21.20	even	2
2205.4.a.u.1.1		2	15.14	odd	2
2240.4.a.bn.1.2		2	280.69	odd	2
2240.4.a.bo.1.1		2	280.139	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q-5.41421 q^{2} -4.65685 q^{3} +21.3137 q^{4} +25.2132 q^{6} -72.0833 q^{8} -5.31371 q^{9} +O(q^{10})\)	\(q-5.41421 q^{2} -4.65685 q^{3} +21.3137 q^{4} +25.2132 q^{6} -72.0833 q^{8} -5.31371 q^{9} -52.2548 q^{11} -99.2548 q^{12} +30.6569 q^{13} +219.765 q^{16} +37.2254 q^{17} +28.7696 q^{18} -80.2254 q^{19} +282.919 q^{22} -25.8335 q^{23} +335.681 q^{24} -165.983 q^{26} +150.480 q^{27} +20.9411 q^{29} +314.558 q^{31} -613.186 q^{32} +243.343 q^{33} -201.546 q^{34} -113.255 q^{36} -197.147 q^{37} +434.357 q^{38} -142.765 q^{39} -11.3625 q^{41} +33.8335 q^{43} -1113.74 q^{44} +139.868 q^{46} -361.676 q^{47} -1023.41 q^{48} -173.353 q^{51} +653.411 q^{52} -153.019 q^{53} -814.732 q^{54} +373.598 q^{57} -113.380 q^{58} +616.000 q^{59} -15.2649 q^{61} -1703.09 q^{62} +1561.80 q^{64} -1317.51 q^{66} +166.510 q^{67} +793.411 q^{68} +120.303 q^{69} -952.000 q^{71} +383.029 q^{72} -148.489 q^{73} +1067.40 q^{74} -1709.90 q^{76} +772.958 q^{78} +857.725 q^{79} -557.294 q^{81} +61.5189 q^{82} +660.528 q^{83} -183.182 q^{86} -97.5198 q^{87} +3766.70 q^{88} +45.7746 q^{89} -550.607 q^{92} -1464.85 q^{93} +1958.19 q^{94} +2855.52 q^{96} +1682.13 q^{97} +277.667 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} + 2 q^{3} + 20 q^{4} + 8 q^{6} - 48 q^{8} + 12 q^{9}+O(q^{10}) \) 2 * q - 8 * q^2 + 2 * q^3 + 20 * q^4 + 8 * q^6 - 48 * q^8 + 12 * q^9	\( 2 q - 8 q^{2} + 2 q^{3} + 20 q^{4} + 8 q^{6} - 48 q^{8} + 12 q^{9} - 14 q^{11} - 108 q^{12} + 50 q^{13} + 168 q^{16} - 50 q^{17} - 16 q^{18} - 36 q^{19} + 184 q^{22} - 244 q^{23} + 496 q^{24} - 216 q^{26} + 86 q^{27} - 26 q^{29} + 120 q^{31} - 672 q^{32} + 498 q^{33} + 24 q^{34} - 136 q^{36} - 564 q^{37} + 320 q^{38} - 14 q^{39} + 328 q^{41} + 260 q^{43} - 1164 q^{44} + 704 q^{46} - 350 q^{47} - 1368 q^{48} - 754 q^{51} + 628 q^{52} + 56 q^{53} - 648 q^{54} + 668 q^{57} + 8 q^{58} + 1232 q^{59} - 336 q^{61} - 1200 q^{62} + 2128 q^{64} - 1976 q^{66} + 152 q^{67} + 908 q^{68} - 1332 q^{69} - 1904 q^{71} + 800 q^{72} + 676 q^{73} + 2016 q^{74} - 1768 q^{76} + 440 q^{78} + 1014 q^{79} - 1454 q^{81} - 816 q^{82} - 376 q^{83} - 768 q^{86} - 410 q^{87} + 4688 q^{88} + 216 q^{89} - 264 q^{92} - 2760 q^{93} + 1928 q^{94} + 2464 q^{96} + 2742 q^{97} + 940 q^{99}+O(q^{100}) \) 2 * q - 8 * q^2 + 2 * q^3 + 20 * q^4 + 8 * q^6 - 48 * q^8 + 12 * q^9 - 14 * q^11 - 108 * q^12 + 50 * q^13 + 168 * q^16 - 50 * q^17 - 16 * q^18 - 36 * q^19 + 184 * q^22 - 244 * q^23 + 496 * q^24 - 216 * q^26 + 86 * q^27 - 26 * q^29 + 120 * q^31 - 672 * q^32 + 498 * q^33 + 24 * q^34 - 136 * q^36 - 564 * q^37 + 320 * q^38 - 14 * q^39 + 328 * q^41 + 260 * q^43 - 1164 * q^44 + 704 * q^46 - 350 * q^47 - 1368 * q^48 - 754 * q^51 + 628 * q^52 + 56 * q^53 - 648 * q^54 + 668 * q^57 + 8 * q^58 + 1232 * q^59 - 336 * q^61 - 1200 * q^62 + 2128 * q^64 - 1976 * q^66 + 152 * q^67 + 908 * q^68 - 1332 * q^69 - 1904 * q^71 + 800 * q^72 + 676 * q^73 + 2016 * q^74 - 1768 * q^76 + 440 * q^78 + 1014 * q^79 - 1454 * q^81 - 816 * q^82 - 376 * q^83 - 768 * q^86 - 410 * q^87 + 4688 * q^88 + 216 * q^89 - 264 * q^92 - 2760 * q^93 + 1928 * q^94 + 2464 * q^96 + 2742 * q^97 + 940 * q^99

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