LMFDB - Embedded newform 225.10.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,10,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("225.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	225.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:	$115.883063137$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{79}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 79 $ x^2 - 79
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-8.88819$ of defining polynomial
Character	$\chi$	$=$	225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.223611 q^{2} -511.950 q^{4} -580.825 q^{7} -228.967 q^{8} +O(q^{10})$	$q+0.223611 q^{2} -511.950 q^{4} -580.825 q^{7} -228.967 q^{8} -47275.7 q^{11} +22367.7 q^{13} -129.879 q^{14} +262067. q^{16} +244880. q^{17} -248536. q^{19} -10571.4 q^{22} +990208. q^{23} +5001.67 q^{26} +297353. q^{28} +4.35002e6 q^{29} -5.60419e6 q^{31} +175832. q^{32} +54758.0 q^{34} -4.73703e6 q^{37} -55575.5 q^{38} -6.11568e6 q^{41} +3.89014e7 q^{43} +2.42028e7 q^{44} +221422. q^{46} +5.09738e7 q^{47} -4.00162e7 q^{49} -1.14511e7 q^{52} +1.16798e7 q^{53} +132990. q^{56} +972713. q^{58} +1.79748e8 q^{59} -1.24120e8 q^{61} -1.25316e6 q^{62} -1.34139e8 q^{64} -1.90142e8 q^{67} -1.25366e8 q^{68} +1.13551e8 q^{71} +1.03304e8 q^{73} -1.05925e6 q^{74} +1.27238e8 q^{76} +2.74589e7 q^{77} -1.94841e7 q^{79} -1.36753e6 q^{82} -5.52722e8 q^{83} +8.69878e6 q^{86} +1.08246e7 q^{88} -3.25690e8 q^{89} -1.29917e7 q^{91} -5.06937e8 q^{92} +1.13983e7 q^{94} +1.33561e8 q^{97} -8.94808e6 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 36 q^{2} + 256 q^{4} + 3318 q^{7} + 8928 q^{8}+O(q^{10}) $ 2 * q + 36 * q^2 + 256 * q^4 + 3318 * q^7 + 8928 * q^8	$ 2 q + 36 q^{2} + 256 q^{4} + 3318 q^{7} + 8928 q^{8} - 24228 q^{11} - 90934 q^{13} + 139356 q^{14} + 196480 q^{16} - 124236 q^{17} - 1348846 q^{19} + 813992 q^{22} - 330444 q^{23} - 4048524 q^{26} + 3291456 q^{28} + 4283172 q^{29} + 1176582 q^{31} - 6859008 q^{32} - 13150888 q^{34} + 5207668 q^{37} - 39420684 q^{38} + 28888488 q^{41} + 13370630 q^{43} + 41902272 q^{44} - 47026728 q^{46} + 66459060 q^{47} - 65169020 q^{49} - 98461184 q^{52} - 3809664 q^{53} + 35834400 q^{56} - 1418792 q^{58} + 115150284 q^{59} - 231494410 q^{61} + 241338204 q^{62} - 352239616 q^{64} - 319773234 q^{67} - 408829248 q^{68} - 111792024 q^{71} - 69141676 q^{73} + 354726216 q^{74} - 717744704 q^{76} + 117317844 q^{77} - 334446000 q^{79} + 1250955136 q^{82} - 149433012 q^{83} - 904698468 q^{86} + 221871552 q^{88} + 246671136 q^{89} - 454735218 q^{91} - 1521131328 q^{92} + 565405736 q^{94} + 696298546 q^{97} - 908823384 q^{98}+O(q^{100}) $ 2 * q + 36 * q^2 + 256 * q^4 + 3318 * q^7 + 8928 * q^8 - 24228 * q^11 - 90934 * q^13 + 139356 * q^14 + 196480 * q^16 - 124236 * q^17 - 1348846 * q^19 + 813992 * q^22 - 330444 * q^23 - 4048524 * q^26 + 3291456 * q^28 + 4283172 * q^29 + 1176582 * q^31 - 6859008 * q^32 - 13150888 * q^34 + 5207668 * q^37 - 39420684 * q^38 + 28888488 * q^41 + 13370630 * q^43 + 41902272 * q^44 - 47026728 * q^46 + 66459060 * q^47 - 65169020 * q^49 - 98461184 * q^52 - 3809664 * q^53 + 35834400 * q^56 - 1418792 * q^58 + 115150284 * q^59 - 231494410 * q^61 + 241338204 * q^62 - 352239616 * q^64 - 319773234 * q^67 - 408829248 * q^68 - 111792024 * q^71 - 69141676 * q^73 + 354726216 * q^74 - 717744704 * q^76 + 117317844 * q^77 - 334446000 * q^79 + 1250955136 * q^82 - 149433012 * q^83 - 904698468 * q^86 + 221871552 * q^88 + 246671136 * q^89 - 454735218 * q^91 - 1521131328 * q^92 + 565405736 * q^94 + 696298546 * q^97 - 908823384 * q^98

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 $

$2$	0.223611	0.00988231	0.00494116	−	0.999988i	$-0.498427\pi$
$2$	0.223611	0.00988231	0.00494116	+	0.999988i	$0.498427\pi$
$3$	0	0
$3$	0	0
$4$	−511.950	−0.999902
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−580.825	−0.0914332	−0.0457166	−	0.998954i	$-0.514557\pi$
$7$	−580.825	−0.0914332	−0.0457166	+	0.998954i	$0.514557\pi$
$8$	−228.967	−0.0197637
$9$	0	0
$10$	0	0
$11$	−47275.7	−0.973578	−0.486789	−	0.873520i	$-0.661832\pi$
$11$	−47275.7	−0.973578	−0.486789	+	0.873520i	$0.661832\pi$
$12$	0	0
$13$	22367.7	0.217208	0.108604	−	0.994085i	$-0.465362\pi$
$13$	22367.7	0.217208	0.108604	+	0.994085i	$0.465362\pi$
$14$	−129.879	−0.000903572 0
$15$	0	0
$16$	262067.	0.999707
$17$	244880.	0.711105	0.355552	−	0.934656i	$-0.384293\pi$
$17$	244880.	0.711105	0.355552	+	0.934656i	$0.384293\pi$
$18$	0	0
$19$	−248536.	−0.437521	−0.218760	−	0.975779i	$-0.570201\pi$
$19$	−248536.	−0.437521	−0.218760	+	0.975779i	$0.570201\pi$
$20$	0	0
$21$	0	0
$22$	−10571.4	−0.00962120
$23$	990208.	0.737821	0.368911	−	0.929465i	$-0.379731\pi$
$23$	990208.	0.737821	0.368911	+	0.929465i	$0.379731\pi$
$24$	0	0
$25$	0	0
$26$	5001.67	0.00214652
$27$	0	0
$28$	297353.	0.0914243
$29$	4.35002e6	1.14209	0.571045	−	0.820919i	$-0.306538\pi$
$29$	4.35002e6	1.14209	0.571045	+	0.820919i	$0.306538\pi$
$30$	0	0
$31$	−5.60419e6	−1.08990	−0.544948	−	0.838470i	$-0.683451\pi$
$31$	−5.60419e6	−1.08990	−0.544948	+	0.838470i	$0.683451\pi$
$32$	175832.	0.0296431
$33$	0	0
$34$	54758.0	0.00702736
$35$	0	0
$36$	0	0
$37$	−4.73703e6	−0.415526	−0.207763	−	0.978179i	$-0.566618\pi$
$37$	−4.73703e6	−0.415526	−0.207763	+	0.978179i	$0.566618\pi$
$38$	−55575.5	−0.00432372
$39$	0	0
$40$	0	0
$41$	−6.11568e6	−0.338000	−0.169000	−	0.985616i	$-0.554054\pi$
$41$	−6.11568e6	−0.338000	−0.169000	+	0.985616i	$0.554054\pi$
$42$	0	0
$43$	3.89014e7	1.73523	0.867614	−	0.497238i	$-0.165652\pi$
$43$	3.89014e7	1.73523	0.867614	+	0.497238i	$0.165652\pi$
$44$	2.42028e7	0.973483
$45$	0	0
$46$	221422.	0.00729138
$47$	5.09738e7	1.52372	0.761862	−	0.647739i	$-0.224285\pi$
$47$	5.09738e7	1.52372	0.761862	+	0.647739i	$0.224285\pi$
$48$	0	0
$49$	−4.00162e7	−0.991640
$50$	0	0
$51$	0	0
$52$	−1.14511e7	−0.217187
$53$	1.16798e7	0.203327	0.101664	−	0.994819i	$-0.467583\pi$
$53$	1.16798e7	0.203327	0.101664	+	0.994819i	$0.467583\pi$
$54$	0	0
$55$	0	0
$56$	132990.	0.00180706
$57$	0	0
$58$	972713.	0.0112865
$59$	1.79748e8	1.93121	0.965607	−	0.260006i	$-0.0837247\pi$
$59$	1.79748e8	1.93121	0.965607	+	0.260006i	$0.0837247\pi$
$60$	0	0
$61$	−1.24120e8	−1.14777	−0.573887	−	0.818935i	$-0.694565\pi$
$61$	−1.24120e8	−1.14777	−0.573887	+	0.818935i	$0.694565\pi$
$62$	−1.25316e6	−0.0107707
$63$	0	0
$64$	−1.34139e8	−0.999414
$65$	0	0
$66$	0	0
$67$	−1.90142e8	−1.15277	−0.576383	−	0.817180i	$-0.695536\pi$
$67$	−1.90142e8	−1.15277	−0.576383	+	0.817180i	$0.695536\pi$
$68$	−1.25366e8	−0.711035
$69$	0	0
$70$	0	0
$71$	1.13551e8	0.530309	0.265155	−	0.964206i	$-0.414577\pi$
$71$	1.13551e8	0.530309	0.265155	+	0.964206i	$0.414577\pi$
$72$	0	0
$73$	1.03304e8	0.425761	0.212881	−	0.977078i	$-0.431715\pi$
$73$	1.03304e8	0.425761	0.212881	+	0.977078i	$0.431715\pi$
$74$	−1.05925e6	−0.00410636
$75$	0	0
$76$	1.27238e8	0.437478
$77$	2.74589e7	0.0890174
$78$	0	0
$79$	−1.94841e7	−0.0562807	−0.0281403	−	0.999604i	$-0.508959\pi$
$79$	−1.94841e7	−0.0562807	−0.0281403	+	0.999604i	$0.508959\pi$
$80$	0	0
$81$	0	0
$82$	−1.36753e6	−0.00334023
$83$	−5.52722e8	−1.27837	−0.639183	−	0.769055i	$-0.720727\pi$
$83$	−5.52722e8	−1.27837	−0.639183	+	0.769055i	$0.720727\pi$
$84$	0	0
$85$	0	0
$86$	8.69878e6	0.0171481
$87$	0	0
$88$	1.08246e7	0.0192415
$89$	−3.25690e8	−0.550237	−0.275119	−	0.961410i	$-0.588717\pi$
$89$	−3.25690e8	−0.550237	−0.275119	+	0.961410i	$0.588717\pi$
$90$	0	0
$91$	−1.29917e7	−0.0198601
$92$	−5.06937e8	−0.737749
$93$	0	0
$94$	1.13983e7	0.0150579
$95$	0	0
$96$	0	0
$97$	1.33561e8	0.153182	0.0765910	−	0.997063i	$-0.475596\pi$
$97$	1.33561e8	0.153182	0.0765910	+	0.997063i	$0.475596\pi$
$98$	−8.94808e6	−0.00979969
$99$	0	0
$100$	0	0
$101$	−1.72460e8	−0.164908	−0.0824542	−	0.996595i	$-0.526276\pi$
$101$	−1.72460e8	−0.164908	−0.0824542	+	0.996595i	$0.526276\pi$
$102$	0	0
$103$	−1.72881e9	−1.51349	−0.756746	−	0.653710i	$-0.773212\pi$
$103$	−1.72881e9	−1.51349	−0.756746	+	0.653710i	$0.773212\pi$
$104$	−5.12146e6	−0.00429283
$105$	0	0
$106$	2.61174e6	0.00200934
$107$	−2.12656e9	−1.56838	−0.784190	−	0.620521i	$-0.786921\pi$
$107$	−2.12656e9	−1.56838	−0.784190	+	0.620521i	$0.786921\pi$
$108$	0	0
$109$	1.65766e9	1.12480	0.562402	−	0.826864i	$-0.309877\pi$
$109$	1.65766e9	1.12480	0.562402	+	0.826864i	$0.309877\pi$
$110$	0	0
$111$	0	0
$112$	−1.52215e8	−0.0914065
$113$	1.83857e9	1.06079	0.530394	−	0.847752i	$-0.322044\pi$
$113$	1.83857e9	1.06079	0.530394	+	0.847752i	$0.322044\pi$
$114$	0	0
$115$	0	0
$116$	−2.22699e9	−1.14198
$117$	0	0
$118$	4.01937e7	0.0190849
$119$	−1.42233e8	−0.0650186
$120$	0	0
$121$	−1.22956e8	−0.0521454
$122$	−2.77545e7	−0.0113427
$123$	0	0
$124$	2.86906e9	1.08979
$125$	0	0
$126$	0	0
$127$	−7.74572e7	−0.0264207	−0.0132104	−	0.999913i	$-0.504205\pi$
$127$	−7.74572e7	−0.0264207	−0.0132104	+	0.999913i	$0.504205\pi$
$128$	−1.20021e8	−0.0395196
$129$	0	0
$130$	0	0
$131$	−4.50478e9	−1.33645	−0.668225	−	0.743960i	$-0.732945\pi$
$131$	−4.50478e9	−1.33645	−0.668225	+	0.743960i	$0.732945\pi$
$132$	0	0
$133$	1.44356e8	0.0400039
$134$	−4.25178e7	−0.0113920
$135$	0	0
$136$	−5.60694e7	−0.0140540
$137$	−6.80091e9	−1.64939	−0.824697	−	0.565575i	$-0.808654\pi$
$137$	−6.80091e9	−1.64939	−0.824697	+	0.565575i	$0.808654\pi$
$138$	0	0
$139$	1.95054e9	0.443187	0.221594	−	0.975139i	$-0.428874\pi$
$139$	1.95054e9	0.443187	0.221594	+	0.975139i	$0.428874\pi$
$140$	0	0
$141$	0	0
$142$	2.53913e7	0.00524068
$143$	−1.05745e9	−0.211469
$144$	0	0
$145$	0	0
$146$	2.31000e7	0.00420751
$147$	0	0
$148$	2.42512e9	0.415486
$149$	1.06040e10	1.76251	0.881257	−	0.472637i	$-0.156698\pi$
$149$	1.06040e10	1.76251	0.881257	+	0.472637i	$0.156698\pi$
$150$	0	0
$151$	−6.54596e9	−1.02465	−0.512327	−	0.858791i	$-0.671216\pi$
$151$	−6.54596e9	−1.02465	−0.512327	+	0.858791i	$0.671216\pi$
$152$	5.69065e7	0.00864701
$153$	0	0
$154$	6.14012e6	0.000879698 0
$155$	0	0
$156$	0	0
$157$	−4.83783e9	−0.635480	−0.317740	−	0.948178i	$-0.602924\pi$
$157$	−4.83783e9	−0.635480	−0.317740	+	0.948178i	$0.602924\pi$
$158$	−4.35687e6	−0.000556183 0
$159$	0	0
$160$	0	0
$161$	−5.75137e8	−0.0674614
$162$	0	0
$163$	−5.42724e9	−0.602191	−0.301096	−	0.953594i	$-0.597352\pi$
$163$	−5.42724e9	−0.602191	−0.301096	+	0.953594i	$0.597352\pi$
$164$	3.13092e9	0.337967
$165$	0	0
$166$	−1.23595e8	−0.0126332
$167$	−1.36699e10	−1.36000	−0.680002	−	0.733210i	$-0.738021\pi$
$167$	−1.36699e10	−1.36000	−0.680002	+	0.733210i	$0.738021\pi$
$168$	0	0
$169$	−1.01042e10	−0.952821
$170$	0	0
$171$	0	0
$172$	−1.99156e10	−1.73506
$173$	−1.84099e10	−1.56259	−0.781294	−	0.624163i	$-0.785440\pi$
$173$	−1.84099e10	−1.56259	−0.781294	+	0.624163i	$0.785440\pi$
$174$	0	0
$175$	0	0
$176$	−1.23894e10	−0.973293
$177$	0	0
$178$	−7.28280e7	−0.00543762
$179$	2.21847e10	1.61516	0.807580	−	0.589758i	$-0.200777\pi$
$179$	2.21847e10	1.61516	0.807580	+	0.589758i	$0.200777\pi$
$180$	0	0
$181$	−1.26287e10	−0.874593	−0.437297	−	0.899317i	$-0.644064\pi$
$181$	−1.26287e10	−0.874593	−0.437297	+	0.899317i	$0.644064\pi$
$182$	−2.90509e6	−0.000196263 0
$183$	0	0
$184$	−2.26725e8	−0.0145820
$185$	0	0
$186$	0	0
$187$	−1.15769e10	−0.692316
$188$	−2.60960e10	−1.52358
$189$	0	0
$190$	0	0
$191$	3.00606e9	0.163436	0.0817179	−	0.996655i	$-0.473959\pi$
$191$	3.00606e9	0.163436	0.0817179	+	0.996655i	$0.473959\pi$
$192$	0	0
$193$	3.17884e9	0.164915	0.0824576	−	0.996595i	$-0.473723\pi$
$193$	3.17884e9	0.164915	0.0824576	+	0.996595i	$0.473723\pi$
$194$	2.98658e7	0.00151379
$195$	0	0
$196$	2.04863e10	0.991543
$197$	−2.67639e10	−1.26605	−0.633026	−	0.774131i	$-0.718187\pi$
$197$	−2.67639e10	−1.26605	−0.633026	+	0.774131i	$0.718187\pi$
$198$	0	0
$199$	3.55476e8	0.0160684	0.00803419	−	0.999968i	$-0.497443\pi$
$199$	3.55476e8	0.0160684	0.00803419	+	0.999968i	$0.497443\pi$
$200$	0	0
$201$	0	0
$202$	−3.85640e7	−0.00162968
$203$	−2.52660e9	−0.104425
$204$	0	0
$205$	0	0
$206$	−3.86581e8	−0.0149568
$207$	0	0
$208$	5.86184e9	0.217145
$209$	1.17497e10	0.425961
$210$	0	0
$211$	−3.90979e10	−1.35795	−0.678973	−	0.734164i	$-0.737574\pi$
$211$	−3.90979e10	−1.35795	−0.678973	+	0.734164i	$0.737574\pi$
$212$	−5.97950e9	−0.203307
$213$	0	0
$214$	−4.75523e8	−0.0154992
$215$	0	0
$216$	0	0
$217$	3.25505e9	0.0996527
$218$	3.70672e8	0.0111157
$219$	0	0
$220$	0	0
$221$	5.47741e9	0.154458
$222$	0	0
$223$	−3.00688e10	−0.814225	−0.407112	−	0.913378i	$-0.633464\pi$
$223$	−3.00688e10	−0.814225	−0.407112	+	0.913378i	$0.633464\pi$
$224$	−1.02128e8	−0.00271036
$225$	0	0
$226$	4.11126e8	0.0104830
$227$	−5.17270e10	−1.29301	−0.646503	−	0.762911i	$-0.723769\pi$
$227$	−5.17270e10	−1.29301	−0.646503	+	0.762911i	$0.723769\pi$
$228$	0	0
$229$	4.39043e10	1.05499	0.527494	−	0.849559i	$-0.323132\pi$
$229$	4.39043e10	1.05499	0.527494	+	0.849559i	$0.323132\pi$
$230$	0	0
$231$	0	0
$232$	−9.96009e8	−0.0225719
$233$	5.20811e10	1.15765	0.578826	−	0.815451i	$-0.303511\pi$
$233$	5.20811e10	1.15765	0.578826	+	0.815451i	$0.303511\pi$
$234$	0	0
$235$	0	0
$236$	−9.20221e10	−1.93103
$237$	0	0
$238$	−3.18048e7	−0.000642534 0
$239$	−1.25028e9	−0.0247866	−0.0123933	−	0.999923i	$-0.503945\pi$
$239$	−1.25028e9	−0.0247866	−0.0123933	+	0.999923i	$0.503945\pi$
$240$	0	0
$241$	5.39086e9	0.102939	0.0514696	−	0.998675i	$-0.483609\pi$
$241$	5.39086e9	0.102939	0.0514696	+	0.998675i	$0.483609\pi$
$242$	−2.74944e7	−0.000515317 0
$243$	0	0
$244$	6.35431e10	1.14766
$245$	0	0
$246$	0	0
$247$	−5.55918e9	−0.0950331
$248$	1.28317e9	0.0215403
$249$	0	0
$250$	0	0
$251$	−3.79306e10	−0.603196	−0.301598	−	0.953435i	$-0.597520\pi$
$251$	−3.79306e10	−0.603196	−0.301598	+	0.953435i	$0.597520\pi$
$252$	0	0
$253$	−4.68128e10	−0.718327
$254$	−1.73203e7	−0.000261098 0
$255$	0	0
$256$	6.86524e10	0.999024
$257$	−1.23297e10	−0.176301	−0.0881504	−	0.996107i	$-0.528096\pi$
$257$	−1.23297e10	−0.176301	−0.0881504	+	0.996107i	$0.528096\pi$
$258$	0	0
$259$	2.75139e9	0.0379929
$260$	0	0
$261$	0	0
$262$	−1.00732e9	−0.0132072
$263$	8.86491e10	1.14255	0.571273	−	0.820760i	$-0.306450\pi$
$263$	8.86491e10	1.14255	0.571273	+	0.820760i	$0.306450\pi$
$264$	0	0
$265$	0	0
$266$	3.22796e7	0.000395331 0
$267$	0	0
$268$	9.73431e10	1.15265
$269$	−6.53334e10	−0.760764	−0.380382	−	0.924829i	$-0.624207\pi$
$269$	−6.53334e10	−0.760764	−0.380382	+	0.924829i	$0.624207\pi$
$270$	0	0
$271$	−4.43373e10	−0.499353	−0.249676	−	0.968329i	$-0.580324\pi$
$271$	−4.43373e10	−0.499353	−0.249676	+	0.968329i	$0.580324\pi$
$272$	6.41751e10	0.710896
$273$	0	0
$274$	−1.52076e9	−0.0162998
$275$	0	0
$276$	0	0
$277$	1.89388e11	1.93283	0.966416	−	0.256984i	$-0.0827288\pi$
$277$	1.89388e11	1.93283	0.966416	+	0.256984i	$0.0827288\pi$
$278$	4.36162e8	0.00437972
$279$	0	0
$280$	0	0
$281$	1.59371e11	1.52486	0.762432	−	0.647068i	$-0.224005\pi$
$281$	1.59371e11	1.52486	0.762432	+	0.647068i	$0.224005\pi$
$282$	0	0
$283$	−9.23708e10	−0.856043	−0.428022	−	0.903769i	$-0.640789\pi$
$283$	−9.23708e10	−0.856043	−0.428022	+	0.903769i	$0.640789\pi$
$284$	−5.81325e10	−0.530257
$285$	0	0
$286$	−2.36457e8	−0.00208980
$287$	3.55214e9	0.0309045
$288$	0	0
$289$	−5.86215e10	−0.494330
$290$	0	0
$291$	0	0
$292$	−5.28867e10	−0.425720
$293$	−1.47354e11	−1.16804	−0.584018	−	0.811740i	$-0.698520\pi$
$293$	−1.47354e11	−1.16804	−0.584018	+	0.811740i	$0.698520\pi$
$294$	0	0
$295$	0	0
$296$	1.08462e9	0.00821232
$297$	0	0
$298$	2.37118e9	0.0174177
$299$	2.21487e10	0.160261
$300$	0	0
$301$	−2.25949e10	−0.158658
$302$	−1.46375e9	−0.0101259
$303$	0	0
$304$	−6.51332e10	−0.437392
$305$	0	0
$306$	0	0
$307$	−1.85136e11	−1.18951	−0.594755	−	0.803907i	$-0.702751\pi$
$307$	−1.85136e11	−1.18951	−0.594755	+	0.803907i	$0.702751\pi$
$308$	−1.40576e10	−0.0890087
$309$	0	0
$310$	0	0
$311$	1.63232e11	0.989429	0.494715	−	0.869055i	$-0.335273\pi$
$311$	1.63232e11	0.989429	0.494715	+	0.869055i	$0.335273\pi$
$312$	0	0
$313$	1.58814e11	0.935272	0.467636	−	0.883921i	$-0.345106\pi$
$313$	1.58814e11	0.935272	0.467636	+	0.883921i	$0.345106\pi$
$314$	−1.08179e9	−0.00628001
$315$	0	0
$316$	9.97491e9	0.0562752
$317$	2.22998e11	1.24032	0.620161	−	0.784474i	$-0.287067\pi$
$317$	2.22998e11	1.24032	0.620161	+	0.784474i	$0.287067\pi$
$318$	0	0
$319$	−2.05650e11	−1.11191
$320$	0	0
$321$	0	0
$322$	−1.28607e8	−0.000666674 0
$323$	−6.08616e10	−0.311123
$324$	0	0
$325$	0	0
$326$	−1.21359e9	−0.00595104
$327$	0	0
$328$	1.40029e9	0.00668012
$329$	−2.96068e10	−0.139319
$330$	0	0
$331$	2.64535e11	1.21131	0.605657	−	0.795725i	$-0.292910\pi$
$331$	2.64535e11	1.21131	0.605657	+	0.795725i	$0.292910\pi$
$332$	2.82966e11	1.27824
$333$	0	0
$334$	−3.05674e9	−0.0134400
$335$	0	0
$336$	0	0
$337$	1.13648e11	0.479986	0.239993	−	0.970775i	$-0.422855\pi$
$337$	1.13648e11	0.479986	0.239993	+	0.970775i	$0.422855\pi$
$338$	−2.25941e9	−0.00941607
$339$	0	0
$340$	0	0
$341$	2.64942e11	1.06110
$342$	0	0
$343$	4.66808e10	0.182102
$344$	−8.90711e9	−0.0342945
$345$	0	0
$346$	−4.11667e9	−0.0154420
$347$	3.78341e11	1.40088	0.700441	−	0.713711i	$-0.252987\pi$
$347$	3.78341e11	1.40088	0.700441	+	0.713711i	$0.252987\pi$
$348$	0	0
$349$	−4.60370e11	−1.66109	−0.830545	−	0.556952i	$-0.811971\pi$
$349$	−4.60370e11	−1.66109	−0.830545	+	0.556952i	$0.811971\pi$
$350$	0	0
$351$	0	0
$352$	−8.31258e9	−0.0288598
$353$	1.01736e11	0.348728	0.174364	−	0.984681i	$-0.444213\pi$
$353$	1.01736e11	0.348728	0.174364	+	0.984681i	$0.444213\pi$
$354$	0	0
$355$	0	0
$356$	1.66737e11	0.550184
$357$	0	0
$358$	4.96075e9	0.0159615
$359$	−6.76144e10	−0.214839	−0.107420	−	0.994214i	$-0.534259\pi$
$359$	−6.76144e10	−0.214839	−0.107420	+	0.994214i	$0.534259\pi$
$360$	0	0
$361$	−2.60917e11	−0.808576
$362$	−2.82392e9	−0.00864300
$363$	0	0
$364$	6.65111e9	0.0198581
$365$	0	0
$366$	0	0
$367$	5.81071e11	1.67198	0.835991	−	0.548743i	$-0.184893\pi$
$367$	5.81071e11	1.67198	0.835991	+	0.548743i	$0.184893\pi$
$368$	2.59501e11	0.737605
$369$	0	0
$370$	0	0
$371$	−6.78395e9	−0.0185909
$372$	0	0
$373$	3.30933e11	0.885219	0.442610	−	0.896714i	$-0.354053\pi$
$373$	3.30933e11	0.885219	0.442610	+	0.896714i	$0.354053\pi$
$374$	−2.58872e9	−0.00684168
$375$	0	0
$376$	−1.16713e10	−0.0301144
$377$	9.72999e10	0.248071
$378$	0	0
$379$	5.19853e11	1.29421	0.647104	−	0.762401i	$-0.275980\pi$
$379$	5.19853e11	1.29421	0.647104	+	0.762401i	$0.275980\pi$
$380$	0	0
$381$	0	0
$382$	6.72188e8	0.00161512
$383$	4.51303e11	1.07170	0.535850	−	0.844313i	$-0.319991\pi$
$383$	4.51303e11	1.07170	0.535850	+	0.844313i	$0.319991\pi$
$384$	0	0
$385$	0	0
$386$	7.10824e8	0.00162974
$387$	0	0
$388$	−6.83767e10	−0.153167
$389$	−4.61900e11	−1.02276	−0.511381	−	0.859354i	$-0.670866\pi$
$389$	−4.61900e11	−1.02276	−0.511381	+	0.859354i	$0.670866\pi$
$390$	0	0
$391$	2.42482e11	0.524668
$392$	9.16239e9	0.0195984
$393$	0	0
$394$	−5.98471e9	−0.0125115
$395$	0	0
$396$	0	0
$397$	−4.81112e11	−0.972052	−0.486026	−	0.873944i	$-0.661554\pi$
$397$	−4.81112e11	−0.972052	−0.486026	+	0.873944i	$0.661554\pi$
$398$	7.94885e7	0.000158793 0
$399$	0	0
$400$	0	0
$401$	−4.12911e11	−0.797457	−0.398728	−	0.917069i	$-0.630548\pi$
$401$	−4.12911e11	−0.797457	−0.398728	+	0.917069i	$0.630548\pi$
$402$	0	0
$403$	−1.25353e11	−0.236734
$404$	8.82910e10	0.164892
$405$	0	0
$406$	−5.64976e8	−0.00103196
$407$	2.23947e11	0.404547
$408$	0	0
$409$	6.65886e11	1.17664	0.588322	−	0.808627i	$-0.299789\pi$
$409$	6.65886e11	1.17664	0.588322	+	0.808627i	$0.299789\pi$
$410$	0	0
$411$	0	0
$412$	8.85064e11	1.51334
$413$	−1.04402e11	−0.176577
$414$	0	0
$415$	0	0
$416$	3.93296e9	0.00643872
$417$	0	0
$418$	2.62737e9	0.00420948
$419$	−7.79778e11	−1.23597	−0.617985	−	0.786190i	$-0.712051\pi$
$419$	−7.79778e11	−1.23597	−0.617985	+	0.786190i	$0.712051\pi$
$420$	0	0
$421$	9.59110e11	1.48799	0.743994	−	0.668187i	$-0.232929\pi$
$421$	9.59110e11	1.48799	0.743994	+	0.668187i	$0.232929\pi$
$422$	−8.74272e9	−0.0134196
$423$	0	0
$424$	−2.67430e9	−0.00401849
$425$	0	0
$426$	0	0
$427$	7.20918e10	0.104945
$428$	1.08869e12	1.56823
$429$	0	0
$430$	0	0
$431$	−1.89582e10	−0.0264636	−0.0132318	−	0.999912i	$-0.504212\pi$
$431$	−1.89582e10	−0.0264636	−0.0132318	+	0.999912i	$0.504212\pi$
$432$	0	0
$433$	−8.39898e11	−1.14824	−0.574118	−	0.818773i	$-0.694655\pi$
$433$	−8.39898e11	−1.14824	−0.574118	+	0.818773i	$0.694655\pi$
$434$	7.27866e8	0.000984799 0
$435$	0	0
$436$	−8.48641e11	−1.12469
$437$	−2.46103e11	−0.322812
$438$	0	0
$439$	−3.74391e11	−0.481100	−0.240550	−	0.970637i	$-0.577328\pi$
$439$	−3.74391e11	−0.481100	−0.240550	+	0.970637i	$0.577328\pi$
$440$	0	0
$441$	0	0
$442$	1.22481e9	0.00152640
$443$	−7.04057e11	−0.868542	−0.434271	−	0.900782i	$-0.642994\pi$
$443$	−7.04057e11	−0.868542	−0.434271	+	0.900782i	$0.642994\pi$
$444$	0	0
$445$	0	0
$446$	−6.72372e9	−0.00804642
$447$	0	0
$448$	7.79113e10	0.0913797
$449$	−3.02133e11	−0.350824	−0.175412	−	0.984495i	$-0.556126\pi$
$449$	−3.02133e11	−0.350824	−0.175412	+	0.984495i	$0.556126\pi$
$450$	0	0
$451$	2.89123e11	0.329070
$452$	−9.41258e11	−1.06068
$453$	0	0
$454$	−1.15667e10	−0.0127779
$455$	0	0
$456$	0	0
$457$	−2.44866e11	−0.262607	−0.131303	−	0.991342i	$-0.541916\pi$
$457$	−2.44866e11	−0.262607	−0.131303	+	0.991342i	$0.541916\pi$
$458$	9.81749e9	0.0104257
$459$	0	0
$460$	0	0
$461$	5.68075e11	0.585803	0.292902	−	0.956143i	$-0.405379\pi$
$461$	5.68075e11	0.585803	0.292902	+	0.956143i	$0.405379\pi$
$462$	0	0
$463$	−3.27020e11	−0.330720	−0.165360	−	0.986233i	$-0.552879\pi$
$463$	−3.27020e11	−0.330720	−0.165360	+	0.986233i	$0.552879\pi$
$464$	1.14000e12	1.14175
$465$	0	0
$466$	1.16459e10	0.0114403
$467$	−7.11477e11	−0.692205	−0.346102	−	0.938197i	$-0.612495\pi$
$467$	−7.11477e11	−0.692205	−0.346102	+	0.938197i	$0.612495\pi$
$468$	0	0
$469$	1.10439e11	0.105401
$470$	0	0
$471$	0	0
$472$	−4.11563e10	−0.0381678
$473$	−1.83909e12	−1.68938
$474$	0	0
$475$	0	0
$476$	7.28160e10	0.0650123
$477$	0	0
$478$	−2.79577e8	−0.000244949 0
$479$	−3.36653e11	−0.292195	−0.146098	−	0.989270i	$-0.546671\pi$
$479$	−3.36653e11	−0.292195	−0.146098	+	0.989270i	$0.546671\pi$
$480$	0	0
$481$	−1.05957e11	−0.0902557
$482$	1.20546e9	0.00101728
$483$	0	0
$484$	6.29474e10	0.0521403
$485$	0	0
$486$	0	0
$487$	−1.66615e12	−1.34225	−0.671127	−	0.741342i	$-0.734190\pi$
$487$	−1.66615e12	−1.34225	−0.671127	+	0.741342i	$0.734190\pi$
$488$	2.84193e10	0.0226842
$489$	0	0
$490$	0	0
$491$	−7.80759e11	−0.606248	−0.303124	−	0.952951i	$-0.598030\pi$
$491$	−7.80759e11	−0.606248	−0.303124	+	0.952951i	$0.598030\pi$
$492$	0	0
$493$	1.06523e12	0.812145
$494$	−1.24310e9	−0.000939146 0
$495$	0	0
$496$	−1.46867e12	−1.08958
$497$	−6.59534e10	−0.0484879
$498$	0	0
$499$	−5.98100e11	−0.431838	−0.215919	−	0.976411i	$-0.569275\pi$
$499$	−5.98100e11	−0.431838	−0.215919	+	0.976411i	$0.569275\pi$
$500$	0	0
$501$	0	0
$502$	−8.48171e9	−0.00596097
$503$	−2.15422e12	−1.50050	−0.750248	−	0.661156i	$-0.770066\pi$
$503$	−2.15422e12	−1.50050	−0.750248	+	0.661156i	$0.770066\pi$
$504$	0	0
$505$	0	0
$506$	−1.04679e10	−0.00709873
$507$	0	0
$508$	3.96542e10	0.0264181
$509$	2.28931e12	1.51173	0.755866	−	0.654726i	$-0.227216\pi$
$509$	2.28931e12	1.51173	0.755866	+	0.654726i	$0.227216\pi$
$510$	0	0
$511$	−6.00018e10	−0.0389287
$512$	7.68022e10	0.0493923
$513$	0	0
$514$	−2.75706e9	−0.00174226
$515$	0	0
$516$	0	0
$517$	−2.40982e12	−1.48347
$518$	6.15241e8	0.000375458 0
$519$	0	0
$520$	0	0
$521$	−1.81335e12	−1.07823	−0.539114	−	0.842233i	$-0.681241\pi$
$521$	−1.81335e12	−1.07823	−0.539114	+	0.842233i	$0.681241\pi$
$522$	0	0
$523$	2.18455e11	0.127675	0.0638373	−	0.997960i	$-0.479666\pi$
$523$	2.18455e11	0.127675	0.0638373	+	0.997960i	$0.479666\pi$
$524$	2.30622e12	1.33632
$525$	0	0
$526$	1.98229e10	0.0112910
$527$	−1.37235e12	−0.775030
$528$	0	0
$529$	−8.20641e11	−0.455620
$530$	0	0
$531$	0	0
$532$	−7.39031e10	−0.0400000
$533$	−1.36794e11	−0.0734165
$534$	0	0
$535$	0	0
$536$	4.35361e10	0.0227829
$537$	0	0
$538$	−1.46093e10	−0.00751811
$539$	1.89180e12	0.965439
$540$	0	0
$541$	−1.00780e12	−0.505811	−0.252906	−	0.967491i	$-0.581386\pi$
$541$	−1.00780e12	−0.505811	−0.252906	+	0.967491i	$0.581386\pi$
$542$	−9.91431e9	−0.00493476
$543$	0	0
$544$	4.30578e10	0.0210793
$545$	0	0
$546$	0	0
$547$	1.06674e12	0.509468	0.254734	−	0.967011i	$-0.418012\pi$
$547$	1.06674e12	0.509468	0.254734	+	0.967011i	$0.418012\pi$
$548$	3.48173e12	1.64923
$549$	0	0
$550$	0	0
$551$	−1.08114e12	−0.499688
$552$	0	0
$553$	1.13169e10	0.00514593
$554$	4.23493e10	0.0191008
$555$	0	0
$556$	−9.98577e11	−0.443144
$557$	−2.02018e12	−0.889286	−0.444643	−	0.895708i	$-0.646669\pi$
$557$	−2.02018e12	−0.889286	−0.444643	+	0.895708i	$0.646669\pi$
$558$	0	0
$559$	8.70134e11	0.376906
$560$	0	0
$561$	0	0
$562$	3.56372e10	0.0150692
$563$	−5.92218e11	−0.248424	−0.124212	−	0.992256i	$-0.539640\pi$
$563$	−5.92218e11	−0.248424	−0.124212	+	0.992256i	$0.539640\pi$
$564$	0	0
$565$	0	0
$566$	−2.06551e10	−0.00845968
$567$	0	0
$568$	−2.59994e10	−0.0104808
$569$	1.47778e12	0.591025	0.295512	−	0.955339i	$-0.404510\pi$
$569$	1.47778e12	0.591025	0.295512	+	0.955339i	$0.404510\pi$
$570$	0	0
$571$	1.63251e11	0.0642677	0.0321339	−	0.999484i	$-0.489770\pi$
$571$	1.63251e11	0.0642677	0.0321339	+	0.999484i	$0.489770\pi$
$572$	5.41361e11	0.211449
$573$	0	0
$574$	7.94298e8	0.000305408 0
$575$	0	0
$576$	0	0
$577$	−1.06656e12	−0.400584	−0.200292	−	0.979736i	$-0.564189\pi$
$577$	−1.06656e12	−0.400584	−0.200292	+	0.979736i	$0.564189\pi$
$578$	−1.31084e10	−0.00488512
$579$	0	0
$580$	0	0
$581$	3.21035e11	0.116885
$582$	0	0
$583$	−5.52173e11	−0.197955
$584$	−2.36533e10	−0.00841460
$585$	0	0
$586$	−3.29499e10	−0.0115429
$587$	−2.49441e12	−0.867153	−0.433577	−	0.901117i	$-0.642749\pi$
$587$	−2.49441e12	−0.867153	−0.433577	+	0.901117i	$0.642749\pi$
$588$	0	0
$589$	1.39284e12	0.476852
$590$	0	0
$591$	0	0
$592$	−1.24142e12	−0.415405
$593$	−1.76944e12	−0.587610	−0.293805	−	0.955865i	$-0.594922\pi$
$593$	−1.76944e12	−0.587610	−0.293805	+	0.955865i	$0.594922\pi$
$594$	0	0
$595$	0	0
$596$	−5.42873e12	−1.76234
$597$	0	0
$598$	4.95269e9	0.00158375
$599$	−5.49741e12	−1.74477	−0.872383	−	0.488823i	$-0.837426\pi$
$599$	−5.49741e12	−1.74477	−0.872383	+	0.488823i	$0.837426\pi$
$600$	0	0
$601$	4.24474e11	0.132714	0.0663569	−	0.997796i	$-0.478862\pi$
$601$	4.24474e11	0.132714	0.0663569	+	0.997796i	$0.478862\pi$
$602$	−5.05247e9	−0.00156790
$603$	0	0
$604$	3.35120e12	1.02455
$605$	0	0
$606$	0	0
$607$	2.57263e12	0.769181	0.384590	−	0.923087i	$-0.374343\pi$
$607$	2.57263e12	0.769181	0.384590	+	0.923087i	$0.374343\pi$
$608$	−4.37006e10	−0.0129695
$609$	0	0
$610$	0	0
$611$	1.14017e12	0.330965
$612$	0	0
$613$	−3.01773e12	−0.863194	−0.431597	−	0.902066i	$-0.642050\pi$
$613$	−3.01773e12	−0.863194	−0.431597	+	0.902066i	$0.642050\pi$
$614$	−4.13985e10	−0.0117551
$615$	0	0
$616$	−6.28717e9	−0.00175931
$617$	−2.87854e12	−0.799631	−0.399816	−	0.916596i	$-0.630926\pi$
$617$	−2.87854e12	−0.799631	−0.399816	+	0.916596i	$0.630926\pi$
$618$	0	0
$619$	5.36553e12	1.46894	0.734471	−	0.678640i	$-0.237431\pi$
$619$	5.36553e12	1.46894	0.734471	+	0.678640i	$0.237431\pi$
$620$	0	0
$621$	0	0
$622$	3.65006e10	0.00977785
$623$	1.89169e11	0.0503100
$624$	0	0
$625$	0	0
$626$	3.55125e10	0.00924265
$627$	0	0
$628$	2.47672e12	0.635418
$629$	−1.16001e12	−0.295483
$630$	0	0
$631$	3.28798e12	0.825653	0.412826	−	0.910810i	$-0.364542\pi$
$631$	3.28798e12	0.825653	0.412826	+	0.910810i	$0.364542\pi$
$632$	4.46122e9	0.00111231
$633$	0	0
$634$	4.98649e10	0.0122573
$635$	0	0
$636$	0	0
$637$	−8.95071e11	−0.215392
$638$	−4.59857e10	−0.0109883
$639$	0	0
$640$	0	0
$641$	4.40972e12	1.03169	0.515846	−	0.856681i	$-0.327478\pi$
$641$	4.40972e12	1.03169	0.515846	+	0.856681i	$0.327478\pi$
$642$	0	0
$643$	−4.91168e12	−1.13313	−0.566566	−	0.824016i	$-0.691728\pi$
$643$	−4.91168e12	−1.13313	−0.566566	+	0.824016i	$0.691728\pi$
$644$	2.94442e11	0.0674548
$645$	0	0
$646$	−1.36093e10	−0.00307461
$647$	−2.80990e12	−0.630408	−0.315204	−	0.949024i	$-0.602073\pi$
$647$	−2.80990e12	−0.630408	−0.315204	+	0.949024i	$0.602073\pi$
$648$	0	0
$649$	−8.49772e12	−1.88019
$650$	0	0
$651$	0	0
$652$	2.77847e12	0.602133
$653$	−2.56025e12	−0.551028	−0.275514	−	0.961297i	$-0.588848\pi$
$653$	−2.56025e12	−0.551028	−0.275514	+	0.961297i	$0.588848\pi$
$654$	0	0
$655$	0	0
$656$	−1.60272e12	−0.337901
$657$	0	0
$658$	−6.62042e9	−0.00137679
$659$	−8.27996e12	−1.71019	−0.855094	−	0.518472i	$-0.826501\pi$
$659$	−8.27996e12	−1.71019	−0.855094	+	0.518472i	$0.826501\pi$
$660$	0	0
$661$	−3.18671e11	−0.0649286	−0.0324643	−	0.999473i	$-0.510336\pi$
$661$	−3.18671e11	−0.0649286	−0.0324643	+	0.999473i	$0.510336\pi$
$662$	5.91530e10	0.0119706
$663$	0	0
$664$	1.26555e11	0.0252652
$665$	0	0
$666$	0	0
$667$	4.30742e12	0.842658
$668$	6.99829e12	1.35987
$669$	0	0
$670$	0	0
$671$	5.86784e12	1.11745
$672$	0	0
$673$	−6.44097e12	−1.21027	−0.605136	−	0.796122i	$-0.706881\pi$
$673$	−6.44097e12	−1.21027	−0.605136	+	0.796122i	$0.706881\pi$
$674$	2.54130e10	0.00474337
$675$	0	0
$676$	5.17284e12	0.952728
$677$	7.45494e12	1.36394	0.681969	−	0.731381i	$-0.261124\pi$
$677$	7.45494e12	1.36394	0.681969	+	0.731381i	$0.261124\pi$
$678$	0	0
$679$	−7.75757e10	−0.0140059
$680$	0	0
$681$	0	0
$682$	5.92439e10	0.0104861
$683$	−1.88685e11	−0.0331776	−0.0165888	−	0.999862i	$-0.505281\pi$
$683$	−1.88685e11	−0.0331776	−0.0165888	+	0.999862i	$0.505281\pi$
$684$	0	0
$685$	0	0
$686$	1.04384e10	0.00179959
$687$	0	0
$688$	1.01948e13	1.73472
$689$	2.61251e11	0.0441644
$690$	0	0
$691$	−4.12572e12	−0.688412	−0.344206	−	0.938894i	$-0.611852\pi$
$691$	−4.12572e12	−0.688412	−0.344206	+	0.938894i	$0.611852\pi$
$692$	9.42497e12	1.56244
$693$	0	0
$694$	8.46014e10	0.0138439
$695$	0	0
$696$	0	0
$697$	−1.49761e12	−0.240354
$698$	−1.02944e11	−0.0164154
$699$	0	0
$700$	0	0
$701$	6.26038e12	0.979196	0.489598	−	0.871948i	$-0.337143\pi$
$701$	6.26038e12	0.979196	0.489598	+	0.871948i	$0.337143\pi$
$702$	0	0
$703$	1.17732e12	0.181801
$704$	6.34152e12	0.973008
$705$	0	0
$706$	2.27492e10	0.00344624
$707$	1.00169e11	0.0150781
$708$	0	0
$709$	−8.40293e12	−1.24888	−0.624442	−	0.781071i	$-0.714674\pi$
$709$	−8.40293e12	−1.24888	−0.624442	+	0.781071i	$0.714674\pi$
$710$	0	0
$711$	0	0
$712$	7.45723e10	0.0108747
$713$	−5.54931e12	−0.804148
$714$	0	0
$715$	0	0
$716$	−1.13575e13	−1.61500
$717$	0	0
$718$	−1.51193e10	−0.00212311
$719$	3.44185e11	0.0480300	0.0240150	−	0.999712i	$-0.492355\pi$
$719$	3.44185e11	0.0480300	0.0240150	+	0.999712i	$0.492355\pi$
$720$	0	0
$721$	1.00414e12	0.138383
$722$	−5.83440e10	−0.00799060
$723$	0	0
$724$	6.46528e12	0.874508
$725$	0	0
$726$	0	0
$727$	−8.67049e12	−1.15117	−0.575584	−	0.817743i	$-0.695225\pi$
$727$	−8.67049e12	−1.15117	−0.575584	+	0.817743i	$0.695225\pi$
$728$	2.97467e9	0.000392507 0
$729$	0	0
$730$	0	0
$731$	9.52617e12	1.23393
$732$	0	0
$733$	1.10927e13	1.41929	0.709644	−	0.704561i	$-0.248856\pi$
$733$	1.10927e13	1.41929	0.709644	+	0.704561i	$0.248856\pi$
$734$	1.29934e11	0.0165230
$735$	0	0
$736$	1.74110e11	0.0218713
$737$	8.98909e12	1.12231
$738$	0	0
$739$	−6.37409e12	−0.786173	−0.393087	−	0.919501i	$-0.628593\pi$
$739$	−6.37409e12	−0.786173	−0.393087	+	0.919501i	$0.628593\pi$
$740$	0	0
$741$	0	0
$742$	−1.51697e9	−0.000183721 0
$743$	6.36760e12	0.766525	0.383263	−	0.923639i	$-0.374800\pi$
$743$	6.36760e12	0.766525	0.383263	+	0.923639i	$0.374800\pi$
$744$	0	0
$745$	0	0
$746$	7.40004e10	0.00874801
$747$	0	0
$748$	5.92679e12	0.692249
$749$	1.23516e12	0.143402
$750$	0	0
$751$	−3.40024e12	−0.390058	−0.195029	−	0.980797i	$-0.562480\pi$
$751$	−3.40024e12	−0.390058	−0.195029	+	0.980797i	$0.562480\pi$
$752$	1.33586e13	1.52328
$753$	0	0
$754$	2.17573e10	0.00245152
$755$	0	0
$756$	0	0
$757$	1.38522e13	1.53316	0.766581	−	0.642148i	$-0.221956\pi$
$757$	1.38522e13	1.53316	0.766581	+	0.642148i	$0.221956\pi$
$758$	1.16245e11	0.0127898
$759$	0	0
$760$	0	0
$761$	1.25728e13	1.35894	0.679471	−	0.733703i	$-0.262210\pi$
$761$	1.25728e13	1.35894	0.679471	+	0.733703i	$0.262210\pi$
$762$	0	0
$763$	−9.62812e11	−0.102845
$764$	−1.53895e12	−0.163420
$765$	0	0
$766$	1.00916e11	0.0105909
$767$	4.02055e12	0.419475
$768$	0	0
$769$	−1.11440e13	−1.14914	−0.574569	−	0.818456i	$-0.694830\pi$
$769$	−1.11440e13	−1.14914	−0.574569	+	0.818456i	$0.694830\pi$
$770$	0	0
$771$	0	0
$772$	−1.62741e12	−0.164899
$773$	4.40857e12	0.444110	0.222055	−	0.975034i	$-0.428724\pi$
$773$	4.40857e12	0.444110	0.222055	+	0.975034i	$0.428724\pi$
$774$	0	0
$775$	0	0
$776$	−3.05811e10	−0.00302744
$777$	0	0
$778$	−1.03286e11	−0.0101073
$779$	1.51997e12	0.147882
$780$	0	0
$781$	−5.36821e12	−0.516297
$782$	5.42217e10	0.00518493
$783$	0	0
$784$	−1.04869e13	−0.991349
$785$	0	0
$786$	0	0
$787$	1.39083e13	1.29237	0.646186	−	0.763180i	$-0.276363\pi$
$787$	1.39083e13	1.29237	0.646186	+	0.763180i	$0.276363\pi$
$788$	1.37018e13	1.26593
$789$	0	0
$790$	0	0
$791$	−1.06789e12	−0.0969912
$792$	0	0
$793$	−2.77627e12	−0.249306
$794$	−1.07582e11	−0.00960612
$795$	0	0
$796$	−1.81986e11	−0.0160668
$797$	1.11093e13	0.975266	0.487633	−	0.873049i	$-0.337861\pi$
$797$	1.11093e13	0.975266	0.487633	+	0.873049i	$0.337861\pi$
$798$	0	0
$799$	1.24825e13	1.08353
$800$	0	0
$801$	0	0
$802$	−9.23316e10	−0.00788072
$803$	−4.88379e12	−0.414512
$804$	0	0
$805$	0	0
$806$	−2.80303e10	−0.00233948
$807$	0	0
$808$	3.94876e10	0.00325919
$809$	−8.47032e12	−0.695235	−0.347617	−	0.937636i	$-0.613009\pi$
$809$	−8.47032e12	−0.695235	−0.347617	+	0.937636i	$0.613009\pi$
$810$	0	0
$811$	1.98968e13	1.61506	0.807532	−	0.589823i	$-0.200803\pi$
$811$	1.98968e13	1.61506	0.807532	+	0.589823i	$0.200803\pi$
$812$	1.29349e12	0.104415
$813$	0	0
$814$	5.00769e10	0.00399786
$815$	0	0
$816$	0	0
$817$	−9.66840e12	−0.759198
$818$	1.48900e11	0.0116280
$819$	0	0
$820$	0	0
$821$	7.94161e11	0.0610049	0.0305024	−	0.999535i	$-0.490289\pi$
$821$	7.94161e11	0.0610049	0.0305024	+	0.999535i	$0.490289\pi$
$822$	0	0
$823$	−1.36089e13	−1.03401	−0.517004	−	0.855983i	$-0.672953\pi$
$823$	−1.36089e13	−1.03401	−0.517004	+	0.855983i	$0.672953\pi$
$824$	3.95840e11	0.0299121
$825$	0	0
$826$	−2.33455e10	−0.00174499
$827$	−1.79571e13	−1.33494	−0.667470	−	0.744637i	$-0.732623\pi$
$827$	−1.79571e13	−1.33494	−0.667470	+	0.744637i	$0.732623\pi$
$828$	0	0
$829$	−3.69953e12	−0.272052	−0.136026	−	0.990705i	$-0.543433\pi$
$829$	−3.69953e12	−0.272052	−0.136026	+	0.990705i	$0.543433\pi$
$830$	0	0
$831$	0	0
$832$	−3.00038e12	−0.217081
$833$	−9.79919e12	−0.705160
$834$	0	0
$835$	0	0
$836$	−6.01527e12	−0.425919
$837$	0	0
$838$	−1.74367e11	−0.0122142
$839$	−2.50920e13	−1.74826	−0.874132	−	0.485688i	$-0.838569\pi$
$839$	−2.50920e13	−1.74826	−0.874132	+	0.485688i	$0.838569\pi$
$840$	0	0
$841$	4.41551e12	0.304368
$842$	2.14468e11	0.0147048
$843$	0	0
$844$	2.00162e13	1.35781
$845$	0	0
$846$	0	0
$847$	7.14160e10	0.00476782
$848$	3.06091e12	0.203268
$849$	0	0
$850$	0	0
$851$	−4.69065e12	−0.306584
$852$	0	0
$853$	6.79534e12	0.439481	0.219741	−	0.975558i	$-0.429479\pi$
$853$	6.79534e12	0.439481	0.219741	+	0.975558i	$0.429479\pi$
$854$	1.61205e10	0.00103710
$855$	0	0
$856$	4.86912e11	0.0309969
$857$	−1.31867e13	−0.835070	−0.417535	−	0.908661i	$-0.637106\pi$
$857$	−1.31867e13	−0.835070	−0.417535	+	0.908661i	$0.637106\pi$
$858$	0	0
$859$	1.27461e13	0.798747	0.399374	−	0.916788i	$-0.369228\pi$
$859$	1.27461e13	0.798747	0.399374	+	0.916788i	$0.369228\pi$
$860$	0	0
$861$	0	0
$862$	−4.23926e9	−0.000261522 0
$863$	−8.12433e12	−0.498585	−0.249292	−	0.968428i	$-0.580198\pi$
$863$	−8.12433e12	−0.498585	−0.249292	+	0.968428i	$0.580198\pi$
$864$	0	0
$865$	0	0
$866$	−1.87811e11	−0.0113472
$867$	0	0
$868$	−1.66642e12	−0.0996429
$869$	9.21126e11	0.0547937
$870$	0	0
$871$	−4.25304e12	−0.250390
$872$	−3.79550e11	−0.0222303
$873$	0	0
$874$	−5.50313e10	−0.00319013
$875$	0	0
$876$	0	0
$877$	6.46585e12	0.369086	0.184543	−	0.982824i	$-0.440920\pi$
$877$	6.46585e12	0.369086	0.184543	+	0.982824i	$0.440920\pi$
$878$	−8.37180e10	−0.00475438
$879$	0	0
$880$	0	0
$881$	1.69271e13	0.946653	0.473326	−	0.880887i	$-0.343053\pi$
$881$	1.69271e13	0.946653	0.473326	+	0.880887i	$0.343053\pi$
$882$	0	0
$883$	−3.56100e13	−1.97128	−0.985642	−	0.168846i	$-0.945996\pi$
$883$	−3.56100e13	−1.97128	−0.985642	+	0.168846i	$0.945996\pi$
$884$	−2.80416e12	−0.154443
$885$	0	0
$886$	−1.57435e11	−0.00858321
$887$	−2.18278e13	−1.18400	−0.592002	−	0.805936i	$-0.701662\pi$
$887$	−2.18278e13	−1.18400	−0.592002	+	0.805936i	$0.701662\pi$
$888$	0	0
$889$	4.49891e10	0.00241573
$890$	0	0
$891$	0	0
$892$	1.53937e13	0.814145
$893$	−1.26688e13	−0.666661
$894$	0	0
$895$	0	0
$896$	6.97112e10	0.00361340
$897$	0	0
$898$	−6.75602e10	−0.00346695
$899$	−2.43783e13	−1.24476
$900$	0	0
$901$	2.86016e12	0.144587
$902$	6.46511e10	0.00325197
$903$	0	0
$904$	−4.20972e11	−0.0209650
$905$	0	0
$906$	0	0
$907$	4.79737e12	0.235380	0.117690	−	0.993050i	$-0.462451\pi$
$907$	4.79737e12	0.235380	0.117690	+	0.993050i	$0.462451\pi$
$908$	2.64816e13	1.29288
$909$	0	0
$910$	0	0
$911$	−1.70045e13	−0.817957	−0.408979	−	0.912544i	$-0.634115\pi$
$911$	−1.70045e13	−0.817957	−0.408979	+	0.912544i	$0.634115\pi$
$912$	0	0
$913$	2.61303e13	1.24459
$914$	−5.47549e10	−0.00259516
$915$	0	0
$916$	−2.24768e13	−1.05488
$917$	2.61649e12	0.122196
$918$	0	0
$919$	−1.12820e12	−0.0521756	−0.0260878	−	0.999660i	$-0.508305\pi$
$919$	−1.12820e12	−0.0521756	−0.0260878	+	0.999660i	$0.508305\pi$
$920$	0	0
$921$	0	0
$922$	1.27028e11	0.00578909
$923$	2.53988e12	0.115188
$924$	0	0
$925$	0	0
$926$	−7.31253e10	−0.00326827
$927$	0	0
$928$	7.64873e11	0.0338550
$929$	3.97727e13	1.75192	0.875960	−	0.482384i	$-0.160229\pi$
$929$	3.97727e13	1.75192	0.875960	+	0.482384i	$0.160229\pi$
$930$	0	0
$931$	9.94549e12	0.433863
$932$	−2.66629e13	−1.15754
$933$	0	0
$934$	−1.59094e11	−0.00684058
$935$	0	0
$936$	0	0
$937$	−3.91551e13	−1.65944	−0.829718	−	0.558184i	$-0.811499\pi$
$937$	−3.91551e13	−1.65944	−0.829718	+	0.558184i	$0.811499\pi$
$938$	2.46954e10	0.00104161
$939$	0	0
$940$	0	0
$941$	−2.55120e13	−1.06070	−0.530348	−	0.847780i	$-0.677939\pi$
$941$	−2.55120e13	−1.06070	−0.530348	+	0.847780i	$0.677939\pi$
$942$	0	0
$943$	−6.05579e12	−0.249384
$944$	4.71061e13	1.93065
$945$	0	0
$946$	−4.11241e11	−0.0166950
$947$	1.92093e12	0.0776133	0.0388066	−	0.999247i	$-0.487644\pi$
$947$	1.92093e12	0.0776133	0.0388066	+	0.999247i	$0.487644\pi$
$948$	0	0
$949$	2.31068e12	0.0924789
$950$	0	0
$951$	0	0
$952$	3.25665e10	0.00128501
$953$	−3.70640e13	−1.45558	−0.727788	−	0.685802i	$-0.759451\pi$
$953$	−3.70640e13	−1.45558	−0.727788	+	0.685802i	$0.759451\pi$
$954$	0	0
$955$	0	0
$956$	6.40081e11	0.0247842
$957$	0	0
$958$	−7.52794e10	−0.00288756
$959$	3.95014e12	0.150809
$960$	0	0
$961$	4.96727e12	0.187872
$962$	−2.36931e10	−0.000891935 0
$963$	0	0
$964$	−2.75985e12	−0.102929
$965$	0	0
$966$	0	0
$967$	−2.97114e13	−1.09271	−0.546353	−	0.837555i	$-0.683984\pi$
$967$	−2.97114e13	−1.09271	−0.546353	+	0.837555i	$0.683984\pi$
$968$	2.81529e10	0.00103058
$969$	0	0
$970$	0	0
$971$	−4.42639e13	−1.59795	−0.798975	−	0.601364i	$-0.794624\pi$
$971$	−4.42639e13	−1.59795	−0.798975	+	0.601364i	$0.794624\pi$
$972$	0	0
$973$	−1.13292e12	−0.0405221
$974$	−3.72571e11	−0.0132646
$975$	0	0
$976$	−3.25277e13	−1.14744
$977$	3.02165e12	0.106101	0.0530503	−	0.998592i	$-0.483106\pi$
$977$	3.02165e12	0.106101	0.0530503	+	0.998592i	$0.483106\pi$
$978$	0	0
$979$	1.53972e13	0.535699
$980$	0	0
$981$	0	0
$982$	−1.74586e11	−0.00599113
$983$	2.56151e13	0.874994	0.437497	−	0.899220i	$-0.355865\pi$
$983$	2.56151e13	0.874994	0.437497	+	0.899220i	$0.355865\pi$
$984$	0	0
$985$	0	0
$986$	2.38198e11	0.00802587
$987$	0	0
$988$	2.84602e12	0.0950238
$989$	3.85204e13	1.28029
$990$	0	0
$991$	4.40641e12	0.145129	0.0725644	−	0.997364i	$-0.476882\pi$
$991$	4.40641e12	0.145129	0.0725644	+	0.997364i	$0.476882\pi$
$992$	−9.85396e11	−0.0323079
$993$	0	0
$994$	−1.47479e10	−0.000479172 0
$995$	0	0
$996$	0	0
$997$	−4.49122e13	−1.43958	−0.719790	−	0.694192i	$-0.755762\pi$
$997$	−4.49122e13	−1.43958	−0.719790	+	0.694192i	$0.755762\pi$
$998$	−1.33742e11	−0.00426756
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.10.a.l.1.1		2
3.2	odd	2		75.10.a.e.1.2	✓	2
5.2	odd	4		225.10.b.j.199.3		4
5.3	odd	4		225.10.b.j.199.2		4
5.4	even	2		225.10.a.g.1.2		2
15.2	even	4		75.10.b.g.49.2		4
15.8	even	4		75.10.b.g.49.3		4
15.14	odd	2		75.10.a.h.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.10.a.e.1.2	✓	2	3.2	odd	2
75.10.a.h.1.1	yes	2	15.14	odd	2
75.10.b.g.49.2		4	15.2	even	4
75.10.b.g.49.3		4	15.8	even	4
225.10.a.g.1.2		2	5.4	even	2
225.10.a.l.1.1		2	1.1	even	1	trivial
225.10.b.j.199.2		4	5.3	odd	4
225.10.b.j.199.3		4	5.2	odd	4

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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

\(f(q)\)	\(=\)	\(q+0.223611 q^{2} -511.950 q^{4} -580.825 q^{7} -228.967 q^{8} +O(q^{10})\)	\(q+0.223611 q^{2} -511.950 q^{4} -580.825 q^{7} -228.967 q^{8} -47275.7 q^{11} +22367.7 q^{13} -129.879 q^{14} +262067. q^{16} +244880. q^{17} -248536. q^{19} -10571.4 q^{22} +990208. q^{23} +5001.67 q^{26} +297353. q^{28} +4.35002e6 q^{29} -5.60419e6 q^{31} +175832. q^{32} +54758.0 q^{34} -4.73703e6 q^{37} -55575.5 q^{38} -6.11568e6 q^{41} +3.89014e7 q^{43} +2.42028e7 q^{44} +221422. q^{46} +5.09738e7 q^{47} -4.00162e7 q^{49} -1.14511e7 q^{52} +1.16798e7 q^{53} +132990. q^{56} +972713. q^{58} +1.79748e8 q^{59} -1.24120e8 q^{61} -1.25316e6 q^{62} -1.34139e8 q^{64} -1.90142e8 q^{67} -1.25366e8 q^{68} +1.13551e8 q^{71} +1.03304e8 q^{73} -1.05925e6 q^{74} +1.27238e8 q^{76} +2.74589e7 q^{77} -1.94841e7 q^{79} -1.36753e6 q^{82} -5.52722e8 q^{83} +8.69878e6 q^{86} +1.08246e7 q^{88} -3.25690e8 q^{89} -1.29917e7 q^{91} -5.06937e8 q^{92} +1.13983e7 q^{94} +1.33561e8 q^{97} -8.94808e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 36 q^{2} + 256 q^{4} + 3318 q^{7} + 8928 q^{8}+O(q^{10}) \) 2 * q + 36 * q^2 + 256 * q^4 + 3318 * q^7 + 8928 * q^8	\( 2 q + 36 q^{2} + 256 q^{4} + 3318 q^{7} + 8928 q^{8} - 24228 q^{11} - 90934 q^{13} + 139356 q^{14} + 196480 q^{16} - 124236 q^{17} - 1348846 q^{19} + 813992 q^{22} - 330444 q^{23} - 4048524 q^{26} + 3291456 q^{28} + 4283172 q^{29} + 1176582 q^{31} - 6859008 q^{32} - 13150888 q^{34} + 5207668 q^{37} - 39420684 q^{38} + 28888488 q^{41} + 13370630 q^{43} + 41902272 q^{44} - 47026728 q^{46} + 66459060 q^{47} - 65169020 q^{49} - 98461184 q^{52} - 3809664 q^{53} + 35834400 q^{56} - 1418792 q^{58} + 115150284 q^{59} - 231494410 q^{61} + 241338204 q^{62} - 352239616 q^{64} - 319773234 q^{67} - 408829248 q^{68} - 111792024 q^{71} - 69141676 q^{73} + 354726216 q^{74} - 717744704 q^{76} + 117317844 q^{77} - 334446000 q^{79} + 1250955136 q^{82} - 149433012 q^{83} - 904698468 q^{86} + 221871552 q^{88} + 246671136 q^{89} - 454735218 q^{91} - 1521131328 q^{92} + 565405736 q^{94} + 696298546 q^{97} - 908823384 q^{98}+O(q^{100}) \) 2 * q + 36 * q^2 + 256 * q^4 + 3318 * q^7 + 8928 * q^8 - 24228 * q^11 - 90934 * q^13 + 139356 * q^14 + 196480 * q^16 - 124236 * q^17 - 1348846 * q^19 + 813992 * q^22 - 330444 * q^23 - 4048524 * q^26 + 3291456 * q^28 + 4283172 * q^29 + 1176582 * q^31 - 6859008 * q^32 - 13150888 * q^34 + 5207668 * q^37 - 39420684 * q^38 + 28888488 * q^41 + 13370630 * q^43 + 41902272 * q^44 - 47026728 * q^46 + 66459060 * q^47 - 65169020 * q^49 - 98461184 * q^52 - 3809664 * q^53 + 35834400 * q^56 - 1418792 * q^58 + 115150284 * q^59 - 231494410 * q^61 + 241338204 * q^62 - 352239616 * q^64 - 319773234 * q^67 - 408829248 * q^68 - 111792024 * q^71 - 69141676 * q^73 + 354726216 * q^74 - 717744704 * q^76 + 117317844 * q^77 - 334446000 * q^79 + 1250955136 * q^82 - 149433012 * q^83 - 904698468 * q^86 + 221871552 * q^88 + 246671136 * q^89 - 454735218 * q^91 - 1521131328 * q^92 + 565405736 * q^94 + 696298546 * q^97 - 908823384 * q^98

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