LMFDB - Embedded newform 225.12.b.b.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,12,Mod(199,225)]

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, 1]))

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("225.199");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	225.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$172.877215626$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.12.b.b.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+56.0000i q^{2} -1088.00 q^{4} +27984.0i q^{7} +53760.0i q^{8} +O(q^{10})$	$q+56.0000i q^{2} -1088.00 q^{4} +27984.0i q^{7} +53760.0i q^{8} +112028. q^{11} +1.09692e6i q^{13} -1.56710e6 q^{14} -5.23878e6 q^{16} +249566. i q^{17} +1.37124e7 q^{19} +6.27357e6i q^{22} +4.13957e7i q^{23} -6.14276e7 q^{26} -3.04466e7i q^{28} -4.53385e6 q^{29} -2.65339e8 q^{31} -1.83271e8i q^{32} -1.39757e7 q^{34} -2.12137e8i q^{37} +7.67896e8i q^{38} +1.26697e9 q^{41} -1.41295e7i q^{43} -1.21886e8 q^{44} -2.31816e9 q^{46} +2.65727e9i q^{47} +1.19422e9 q^{49} -1.19345e9i q^{52} +2.40270e9i q^{53} -1.50442e9 q^{56} -2.53896e8i q^{58} +7.49874e9 q^{59} -4.06483e9 q^{61} -1.48590e10i q^{62} -4.65830e8 q^{64} +6.87151e9i q^{67} -2.71528e8i q^{68} +1.32837e10 q^{71} +2.88758e10i q^{73} +1.18797e10 q^{74} -1.49191e10 q^{76} +3.13499e9i q^{77} -2.71003e10 q^{79} +7.09503e10i q^{82} -3.43653e10i q^{83} +7.91255e8 q^{86} +6.02263e9i q^{88} -6.35004e10 q^{89} -3.06963e10 q^{91} -4.50386e10i q^{92} -1.48807e11 q^{94} +1.96345e10i q^{97} +6.68765e10i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2176 q^{4}+O(q^{10}) $ 2 * q - 2176 * q^4	$ 2 q - 2176 q^{4} + 224056 q^{11} - 3134208 q^{14} - 10477568 q^{16} + 27424840 q^{19} - 122855264 q^{26} - 9067700 q^{29} - 530678016 q^{31} - 27951392 q^{34} + 2533939916 q^{41} - 243772928 q^{44} - 4636321536 q^{46} + 2388444974 q^{49} - 3008839680 q^{56} + 14997474440 q^{59} - 8129657716 q^{61} - 931659776 q^{64} + 26567469296 q^{71} + 23759337952 q^{74} - 29838225920 q^{76} - 54200604480 q^{79} + 1582509376 q^{86} - 127000825260 q^{89} - 61392530496 q^{91} - 297614613632 q^{94}+O(q^{100}) $ 2 * q - 2176 * q^4 + 224056 * q^11 - 3134208 * q^14 - 10477568 * q^16 + 27424840 * q^19 - 122855264 * q^26 - 9067700 * q^29 - 530678016 * q^31 - 27951392 * q^34 + 2533939916 * q^41 - 243772928 * q^44 - 4636321536 * q^46 + 2388444974 * q^49 - 3008839680 * q^56 + 14997474440 * q^59 - 8129657716 * q^61 - 931659776 * q^64 + 26567469296 * q^71 + 23759337952 * q^74 - 29838225920 * q^76 - 54200604480 * q^79 + 1582509376 * q^86 - 127000825260 * q^89 - 61392530496 * q^91 - 297614613632 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	56.0000i	1.23744i	0.785613	+	0.618718i	$0.212348\pi$
$2$	56.0000i	1.23744i	−0.785613	+	0.618718i	$0.787652\pi$
$3$	0	0
$3$	0	0
$4$	−1088.00	−0.531250
$5$	0	0
$5$	0	0
$6$	0	0
$7$	27984.0i	0.629319i	0.949205	+	0.314659i	$0.101890\pi$
$7$	27984.0i	0.629319i	−0.949205	+	0.314659i	$0.898110\pi$
$8$	53760.0i	0.580049i
$9$	0	0
$10$	0	0
$11$	112028.	0.209733	0.104867	−	0.994486i	$-0.466558\pi$
$11$	112028.	0.209733	0.104867	+	0.994486i	$0.466558\pi$
$12$	0	0
$13$	1.09692e6i	0.819384i	0.912224	+	0.409692i	$0.134364\pi$
$13$	1.09692e6i	0.819384i	−0.912224	+	0.409692i	$0.865636\pi$
$14$	−1.56710e6	−0.778742
$15$	0	0
$16$	−5.23878e6	−1.24902
$17$	249566.i	0.0426301i	0.999773	+	0.0213150i	$0.00678530\pi$
$17$	249566.i	0.0426301i	−0.999773	+	0.0213150i	$0.993215\pi$
$18$	0	0
$19$	1.37124e7	1.27048	0.635242	−	0.772313i	$-0.280900\pi$
$19$	1.37124e7	1.27048	0.635242	+	0.772313i	$0.280900\pi$
$20$	0	0
$21$	0	0
$22$	6.27357e6i	0.259531i
$23$	4.13957e7i	1.34107i	0.741877	+	0.670537i	$0.233936\pi$
$23$	4.13957e7i	1.34107i	−0.741877	+	0.670537i	$0.766064\pi$
$24$	0	0
$25$	0	0
$26$	−6.14276e7	−1.01394
$27$	0	0
$28$	− 3.04466e7i	− 0.334326i
$29$	−4.53385e6	−0.0410467	−0.0205233	−	0.999789i	$-0.506533\pi$
$29$	−4.53385e6	−0.0410467	−0.0205233	+	0.999789i	$0.506533\pi$
$30$	0	0
$31$	−2.65339e8	−1.66461	−0.832304	−	0.554320i	$-0.812978\pi$
$31$	−2.65339e8	−1.66461	−0.832304	+	0.554320i	$0.812978\pi$
$32$	− 1.83271e8i	− 0.965539i
$33$	0	0
$34$	−1.39757e7	−0.0527521
$35$	0	0
$36$	0	0
$37$	− 2.12137e8i	− 0.502929i	−0.967866	−	0.251465i	$-0.919088\pi$
$37$	− 2.12137e8i	− 0.502929i	0.967866	−	0.251465i	$-0.0809122\pi$
$38$	7.67896e8i	1.57214i
$39$	0	0
$40$	0	0
$41$	1.26697e9	1.70787	0.853936	−	0.520379i	$-0.174209\pi$
$41$	1.26697e9	1.70787	0.853936	+	0.520379i	$0.174209\pi$
$42$	0	0
$43$	− 1.41295e7i	− 0.0146572i	−0.999973	−	0.00732861i	$-0.997667\pi$
$43$	− 1.41295e7i	− 0.0146572i	0.999973	−	0.00732861i	$-0.00233279\pi$
$44$	−1.21886e8	−0.111421
$45$	0	0
$46$	−2.31816e9	−1.65949
$47$	2.65727e9i	1.69004i	0.534732	+	0.845022i	$0.320413\pi$
$47$	2.65727e9i	1.69004i	−0.534732	+	0.845022i	$0.679587\pi$
$48$	0	0
$49$	1.19422e9	0.603958
$50$	0	0
$51$	0	0
$52$	− 1.19345e9i	− 0.435298i
$53$	2.40270e9i	0.789191i	0.918855	+	0.394596i	$0.129115\pi$
$53$	2.40270e9i	0.789191i	−0.918855	+	0.394596i	$0.870885\pi$
$54$	0	0
$55$	0	0
$56$	−1.50442e9	−0.365035
$57$	0	0
$58$	− 2.53896e8i	− 0.0507927i
$59$	7.49874e9	1.36553	0.682766	−	0.730637i	$-0.260777\pi$
$59$	7.49874e9	1.36553	0.682766	+	0.730637i	$0.260777\pi$
$60$	0	0
$61$	−4.06483e9	−0.616209	−0.308105	−	0.951352i	$-0.599695\pi$
$61$	−4.06483e9	−0.616209	−0.308105	+	0.951352i	$0.599695\pi$
$62$	− 1.48590e10i	− 2.05985i
$63$	0	0
$64$	−4.65830e8	−0.0542297
$65$	0	0
$66$	0	0
$67$	6.87151e9i	0.621786i	0.950445	+	0.310893i	$0.100628\pi$
$67$	6.87151e9i	0.621786i	−0.950445	+	0.310893i	$0.899372\pi$
$68$	− 2.71528e8i	− 0.0226472i
$69$	0	0
$70$	0	0
$71$	1.32837e10	0.873774	0.436887	−	0.899516i	$-0.356081\pi$
$71$	1.32837e10	0.873774	0.436887	+	0.899516i	$0.356081\pi$
$72$	0	0
$73$	2.88758e10i	1.63027i	0.579273	+	0.815134i	$0.303337\pi$
$73$	2.88758e10i	1.63027i	−0.579273	+	0.815134i	$0.696663\pi$
$74$	1.18797e10	0.622343
$75$	0	0
$76$	−1.49191e10	−0.674945
$77$	3.13499e9i	0.131989i
$78$	0	0
$79$	−2.71003e10	−0.990889	−0.495445	−	0.868640i	$-0.664995\pi$
$79$	−2.71003e10	−0.990889	−0.495445	+	0.868640i	$0.664995\pi$
$80$	0	0
$81$	0	0
$82$	7.09503e10i	2.11338i
$83$	− 3.43653e10i	− 0.957613i	−0.877920	−	0.478807i	$-0.841070\pi$
$83$	− 3.43653e10i	− 0.957613i	0.877920	−	0.478807i	$-0.158930\pi$
$84$	0	0
$85$	0	0
$86$	7.91255e8	0.0181374
$87$	0	0
$88$	6.02263e9i	0.121655i
$89$	−6.35004e10	−1.20540	−0.602700	−	0.797968i	$-0.705909\pi$
$89$	−6.35004e10	−1.20540	−0.602700	+	0.797968i	$0.705909\pi$
$90$	0	0
$91$	−3.06963e10	−0.515653
$92$	− 4.50386e10i	− 0.712445i
$93$	0	0
$94$	−1.48807e11	−2.09132
$95$	0	0
$96$	0	0
$97$	1.96345e10i	0.232153i	0.993240	+	0.116077i	$0.0370319\pi$
$97$	1.96345e10i	0.232153i	−0.993240	+	0.116077i	$0.962968\pi$
$98$	6.68765e10i	0.747360i
$99$	0	0
$100$	0	0
$101$	1.21293e11	1.14834	0.574169	−	0.818737i	$-0.305325\pi$
$101$	1.21293e11	1.14834	0.574169	+	0.818737i	$0.305325\pi$
$102$	0	0
$103$	− 1.32353e11i	− 1.12494i	−0.826818	−	0.562469i	$-0.809851\pi$
$103$	− 1.32353e11i	− 1.12494i	0.826818	−	0.562469i	$-0.190149\pi$
$104$	−5.89705e10	−0.475282
$105$	0	0
$106$	−1.34551e11	−0.976574
$107$	− 1.93758e10i	− 0.133552i	−0.997768	−	0.0667759i	$-0.978729\pi$
$107$	− 1.93758e10i	− 0.133552i	0.997768	−	0.0667759i	$-0.0212713\pi$
$108$	0	0
$109$	−2.32688e11	−1.44853	−0.724265	−	0.689522i	$-0.757821\pi$
$109$	−2.32688e11	−1.44853	−0.724265	+	0.689522i	$0.757821\pi$
$110$	0	0
$111$	0	0
$112$	− 1.46602e11i	− 0.786034i
$113$	3.19485e11i	1.63124i	0.578586	+	0.815621i	$0.303605\pi$
$113$	3.19485e11i	1.63124i	−0.578586	+	0.815621i	$0.696395\pi$
$114$	0	0
$115$	0	0
$116$	4.93283e9	0.0218061
$117$	0	0
$118$	4.19929e11i	1.68976i
$119$	−6.98385e9	−0.0268279
$120$	0	0
$121$	−2.72761e11	−0.956012
$122$	− 2.27630e11i	− 0.762520i
$123$	0	0
$124$	2.88689e11	0.884323
$125$	0	0
$126$	0	0
$127$	− 5.61435e11i	− 1.50792i	−0.656918	−	0.753962i	$-0.728140\pi$
$127$	− 5.61435e11i	− 1.50792i	0.656918	−	0.753962i	$-0.271860\pi$
$128$	− 4.01426e11i	− 1.03264i
$129$	0	0
$130$	0	0
$131$	−4.57850e11	−1.03689	−0.518443	−	0.855112i	$-0.673488\pi$
$131$	−4.57850e11	−1.03689	−0.518443	+	0.855112i	$0.673488\pi$
$132$	0	0
$133$	3.83728e11i	0.799539i
$134$	−3.84805e11	−0.769421
$135$	0	0
$136$	−1.34167e10	−0.0247275
$137$	4.20737e10i	0.0744814i	0.999306	+	0.0372407i	$0.0118568\pi$
$137$	4.20737e10i	0.0744814i	−0.999306	+	0.0372407i	$0.988143\pi$
$138$	0	0
$139$	−8.13653e11	−1.33002	−0.665009	−	0.746835i	$-0.731572\pi$
$139$	−8.13653e11	−1.33002	−0.665009	+	0.746835i	$0.731572\pi$
$140$	0	0
$141$	0	0
$142$	7.43889e11i	1.08124i
$143$	1.22886e11i	0.171852i
$144$	0	0
$145$	0	0
$146$	−1.61705e12	−2.01735
$147$	0	0
$148$	2.30805e11i	0.267181i
$149$	−1.56119e11	−0.174153	−0.0870763	−	0.996202i	$-0.527752\pi$
$149$	−1.56119e11	−0.174153	−0.0870763	+	0.996202i	$0.527752\pi$
$150$	0	0
$151$	5.50139e11	0.570295	0.285147	−	0.958484i	$-0.407957\pi$
$151$	5.50139e11	0.570295	0.285147	+	0.958484i	$0.407957\pi$
$152$	7.37180e11i	0.736943i
$153$	0	0
$154$	−1.75560e11	−0.163328
$155$	0	0
$156$	0	0
$157$	− 1.87489e12i	− 1.56866i	−0.620346	−	0.784328i	$-0.713008\pi$
$157$	− 1.87489e12i	− 1.56866i	0.620346	−	0.784328i	$-0.286992\pi$
$158$	− 1.51762e12i	− 1.22616i
$159$	0	0
$160$	0	0
$161$	−1.15842e12	−0.843962
$162$	0	0
$163$	2.17731e12i	1.48214i	0.671430	+	0.741068i	$0.265680\pi$
$163$	2.17731e12i	1.48214i	−0.671430	+	0.741068i	$0.734320\pi$
$164$	−1.37846e12	−0.907307
$165$	0	0
$166$	1.92445e12	1.18499
$167$	1.18014e12i	0.703058i	0.936177	+	0.351529i	$0.114338\pi$
$167$	1.18014e12i	0.703058i	−0.936177	+	0.351529i	$0.885662\pi$
$168$	0	0
$169$	5.88923e11	0.328610
$170$	0	0
$171$	0	0
$172$	1.53729e10i	0.00778665i
$173$	4.99181e11i	0.244909i	0.992474	+	0.122454i	$0.0390765\pi$
$173$	4.99181e11i	0.244909i	−0.992474	+	0.122454i	$0.960923\pi$
$174$	0	0
$175$	0	0
$176$	−5.86890e11	−0.261961
$177$	0	0
$178$	− 3.55602e12i	− 1.49161i
$179$	3.21203e12	1.30643	0.653217	−	0.757171i	$-0.273419\pi$
$179$	3.21203e12	1.30643	0.653217	+	0.757171i	$0.273419\pi$
$180$	0	0
$181$	−2.93245e12	−1.12201	−0.561007	−	0.827811i	$-0.689586\pi$
$181$	−2.93245e12	−1.12201	−0.561007	+	0.827811i	$0.689586\pi$
$182$	− 1.71899e12i	− 0.638089i
$183$	0	0
$184$	−2.22543e12	−0.777887
$185$	0	0
$186$	0	0
$187$	2.79584e10i	0.00894094i
$188$	− 2.89111e12i	− 0.897836i
$189$	0	0
$190$	0	0
$191$	−2.28642e12	−0.650839	−0.325419	−	0.945570i	$-0.605505\pi$
$191$	−2.28642e12	−0.650839	−0.325419	+	0.945570i	$0.605505\pi$
$192$	0	0
$193$	− 5.86617e12i	− 1.57685i	−0.615133	−	0.788423i	$-0.710898\pi$
$193$	− 5.86617e12i	− 1.57685i	0.615133	−	0.788423i	$-0.289102\pi$
$194$	−1.09953e12	−0.287275
$195$	0	0
$196$	−1.29931e12	−0.320853
$197$	− 5.41588e11i	− 0.130048i	−0.997884	−	0.0650242i	$-0.979288\pi$
$197$	− 5.41588e11i	− 0.130048i	0.997884	−	0.0650242i	$-0.0207125\pi$
$198$	0	0
$199$	6.12034e12	1.39022	0.695110	−	0.718903i	$-0.255356\pi$
$199$	6.12034e12	1.39022	0.695110	+	0.718903i	$0.255356\pi$
$200$	0	0
$201$	0	0
$202$	6.79243e12i	1.42100i
$203$	− 1.26875e11i	− 0.0258314i
$204$	0	0
$205$	0	0
$206$	7.41176e12	1.39204
$207$	0	0
$208$	− 5.74654e12i	− 1.02343i
$209$	1.53617e12	0.266463
$210$	0	0
$211$	1.92054e12	0.316133	0.158067	−	0.987428i	$-0.449474\pi$
$211$	1.92054e12	0.316133	0.158067	+	0.987428i	$0.449474\pi$
$212$	− 2.61414e12i	− 0.419258i
$213$	0	0
$214$	1.08505e12	0.165262
$215$	0	0
$216$	0	0
$217$	− 7.42525e12i	− 1.04757i
$218$	− 1.30305e13i	− 1.79246i
$219$	0	0
$220$	0	0
$221$	−2.73754e11	−0.0349304
$222$	0	0
$223$	− 8.97214e12i	− 1.08948i	−0.838605	−	0.544740i	$-0.816628\pi$
$223$	− 8.97214e12i	− 1.08948i	0.838605	−	0.544740i	$-0.183372\pi$
$224$	5.12867e12	0.607632
$225$	0	0
$226$	−1.78911e13	−2.01856
$227$	− 1.08223e13i	− 1.19173i	−0.803086	−	0.595864i	$-0.796810\pi$
$227$	− 1.08223e13i	− 1.19173i	0.803086	−	0.595864i	$-0.203190\pi$
$228$	0	0
$229$	8.41898e12	0.883414	0.441707	−	0.897159i	$-0.354373\pi$
$229$	8.41898e12	0.883414	0.441707	+	0.897159i	$0.354373\pi$
$230$	0	0
$231$	0	0
$232$	− 2.43740e11i	− 0.0238091i
$233$	− 1.69297e13i	− 1.61507i	−0.589817	−	0.807537i	$-0.700800\pi$
$233$	− 1.69297e13i	− 1.61507i	0.589817	−	0.807537i	$-0.299200\pi$
$234$	0	0
$235$	0	0
$236$	−8.15863e12	−0.725439
$237$	0	0
$238$	− 3.91096e11i	− 0.0331979i
$239$	6.99835e12	0.580506	0.290253	−	0.956950i	$-0.406260\pi$
$239$	6.99835e12	0.580506	0.290253	+	0.956950i	$0.406260\pi$
$240$	0	0
$241$	−9.06165e12	−0.717982	−0.358991	−	0.933341i	$-0.616879\pi$
$241$	−9.06165e12	−0.717982	−0.358991	+	0.933341i	$0.616879\pi$
$242$	− 1.52746e13i	− 1.18300i
$243$	0	0
$244$	4.42253e12	0.327361
$245$	0	0
$246$	0	0
$247$	1.50415e13i	1.04101i
$248$	− 1.42646e13i	− 0.965553i
$249$	0	0
$250$	0	0
$251$	−2.59291e13	−1.64279	−0.821394	−	0.570362i	$-0.806803\pi$
$251$	−2.59291e13	−1.64279	−0.821394	+	0.570362i	$0.806803\pi$
$252$	0	0
$253$	4.63748e12i	0.281267i
$254$	3.14404e13	1.86596
$255$	0	0
$256$	2.15259e13	1.22360
$257$	− 8.77582e12i	− 0.488265i	−0.969742	−	0.244133i	$-0.921497\pi$
$257$	− 8.77582e12i	− 0.488265i	0.969742	−	0.244133i	$-0.0785032\pi$
$258$	0	0
$259$	5.93644e12	0.316503
$260$	0	0
$261$	0	0
$262$	− 2.56396e13i	− 1.28308i
$263$	8.38157e12i	0.410742i	0.978684	+	0.205371i	$0.0658400\pi$
$263$	8.38157e12i	0.410742i	−0.978684	+	0.205371i	$0.934160\pi$
$264$	0	0
$265$	0	0
$266$	−2.14888e13	−0.989380
$267$	0	0
$268$	− 7.47621e12i	− 0.330324i
$269$	−3.22500e13	−1.39602	−0.698011	−	0.716087i	$-0.745932\pi$
$269$	−3.22500e13	−1.39602	−0.698011	+	0.716087i	$0.745932\pi$
$270$	0	0
$271$	−2.89199e12	−0.120189	−0.0600947	−	0.998193i	$-0.519140\pi$
$271$	−2.89199e12	−0.120189	−0.0600947	+	0.998193i	$0.519140\pi$
$272$	− 1.30742e12i	− 0.0532460i
$273$	0	0
$274$	−2.35613e12	−0.0921660
$275$	0	0
$276$	0	0
$277$	9.73968e12i	0.358844i	0.983772	+	0.179422i	$0.0574228\pi$
$277$	9.73968e12i	0.358844i	−0.983772	+	0.179422i	$0.942577\pi$
$278$	− 4.55645e13i	− 1.64581i
$279$	0	0
$280$	0	0
$281$	9.36930e12	0.319023	0.159512	−	0.987196i	$-0.449008\pi$
$281$	9.36930e12	0.319023	0.159512	+	0.987196i	$0.449008\pi$
$282$	0	0
$283$	− 4.70902e13i	− 1.54207i	−0.636791	−	0.771036i	$-0.719739\pi$
$283$	− 4.70902e13i	− 1.54207i	0.636791	−	0.771036i	$-0.280261\pi$
$284$	−1.44527e13	−0.464193
$285$	0	0
$286$	−6.88161e12	−0.212656
$287$	3.54549e13i	1.07480i
$288$	0	0
$289$	3.42096e13	0.998183
$290$	0	0
$291$	0	0
$292$	− 3.14169e13i	− 0.866080i
$293$	− 1.11851e13i	− 0.302600i	−0.988488	−	0.151300i	$-0.951654\pi$
$293$	− 1.11851e13i	− 0.302600i	0.988488	−	0.151300i	$-0.0483460\pi$
$294$	0	0
$295$	0	0
$296$	1.14045e13	0.291723
$297$	0	0
$298$	− 8.74264e12i	− 0.215503i
$299$	−4.54079e13	−1.09885
$300$	0	0
$301$	3.95401e11	0.00922406
$302$	3.08078e13i	0.705704i
$303$	0	0
$304$	−7.18364e13	−1.58686
$305$	0	0
$306$	0	0
$307$	4.85767e13i	1.01664i	0.861168	+	0.508320i	$0.169733\pi$
$307$	4.85767e13i	1.01664i	−0.861168	+	0.508320i	$0.830267\pi$
$308$	− 3.41087e12i	− 0.0701191i
$309$	0	0
$310$	0	0
$311$	−5.18050e13	−1.00969	−0.504846	−	0.863209i	$-0.668451\pi$
$311$	−5.18050e13	−1.00969	−0.504846	+	0.863209i	$0.668451\pi$
$312$	0	0
$313$	− 2.45673e13i	− 0.462236i	−0.972926	−	0.231118i	$-0.925762\pi$
$313$	− 2.45673e13i	− 0.462236i	0.972926	−	0.231118i	$-0.0742383\pi$
$314$	1.04994e14	1.94111
$315$	0	0
$316$	2.94851e13	0.526410
$317$	1.74942e13i	0.306950i	0.988153	+	0.153475i	$0.0490465\pi$
$317$	1.74942e13i	0.306950i	−0.988153	+	0.153475i	$0.950954\pi$
$318$	0	0
$319$	−5.07918e11	−0.00860885
$320$	0	0
$321$	0	0
$322$	− 6.48714e13i	− 1.04435i
$323$	3.42215e12i	0.0541609i
$324$	0	0
$325$	0	0
$326$	−1.21929e14	−1.83405
$327$	0	0
$328$	6.81123e13i	0.990648i
$329$	−7.43611e13	−1.06358
$330$	0	0
$331$	−6.84453e13	−0.946868	−0.473434	−	0.880829i	$-0.656986\pi$
$331$	−6.84453e13	−0.946868	−0.473434	+	0.880829i	$0.656986\pi$
$332$	3.73894e13i	0.508732i
$333$	0	0
$334$	−6.60876e13	−0.869990
$335$	0	0
$336$	0	0
$337$	− 7.69939e13i	− 0.964921i	−0.875918	−	0.482460i	$-0.839743\pi$
$337$	− 7.69939e13i	− 0.964921i	0.875918	−	0.482460i	$-0.160257\pi$
$338$	3.29797e13i	0.406635i
$339$	0	0
$340$	0	0
$341$	−2.97254e13	−0.349123
$342$	0	0
$343$	8.87526e13i	1.00940i
$344$	7.59605e11	0.00850190
$345$	0	0
$346$	−2.79541e13	−0.303059
$347$	− 1.29519e14i	− 1.38204i	−0.722836	−	0.691020i	$-0.757162\pi$
$347$	− 1.29519e14i	− 1.38204i	0.722836	−	0.691020i	$-0.242838\pi$
$348$	0	0
$349$	−3.64763e13	−0.377112	−0.188556	−	0.982062i	$-0.560381\pi$
$349$	−3.64763e13	−0.377112	−0.188556	+	0.982062i	$0.560381\pi$
$350$	0	0
$351$	0	0
$352$	− 2.05315e13i	− 0.202505i
$353$	1.03351e14i	1.00358i	0.864989	+	0.501790i	$0.167325\pi$
$353$	1.03351e14i	1.00358i	−0.864989	+	0.501790i	$0.832675\pi$
$354$	0	0
$355$	0	0
$356$	6.90884e13	0.640369
$357$	0	0
$358$	1.79874e14i	1.61663i
$359$	−1.52982e14	−1.35400	−0.677002	−	0.735981i	$-0.736721\pi$
$359$	−1.52982e14	−1.35400	−0.677002	+	0.735981i	$0.736721\pi$
$360$	0	0
$361$	7.15402e13	0.614130
$362$	− 1.64217e14i	− 1.38842i
$363$	0	0
$364$	3.33975e13	0.273941
$365$	0	0
$366$	0	0
$367$	1.49106e14i	1.16905i	0.811377	+	0.584523i	$0.198718\pi$
$367$	1.49106e14i	1.16905i	−0.811377	+	0.584523i	$0.801282\pi$
$368$	− 2.16863e14i	− 1.67503i
$369$	0	0
$370$	0	0
$371$	−6.72371e13	−0.496653
$372$	0	0
$373$	5.61185e13i	0.402446i	0.979545	+	0.201223i	$0.0644916\pi$
$373$	5.61185e13i	0.402446i	−0.979545	+	0.201223i	$0.935508\pi$
$374$	−1.56567e12	−0.0110638
$375$	0	0
$376$	−1.42855e14	−0.980308
$377$	− 4.97328e12i	− 0.0336330i
$378$	0	0
$379$	−7.57417e13	−0.497530	−0.248765	−	0.968564i	$-0.580025\pi$
$379$	−7.57417e13	−0.497530	−0.248765	+	0.968564i	$0.580025\pi$
$380$	0	0
$381$	0	0
$382$	− 1.28040e14i	− 0.805372i
$383$	2.02071e13i	0.125289i	0.998036	+	0.0626443i	$0.0199534\pi$
$383$	2.02071e13i	0.125289i	−0.998036	+	0.0626443i	$0.980047\pi$
$384$	0	0
$385$	0	0
$386$	3.28505e14	1.95125
$387$	0	0
$388$	− 2.13623e13i	− 0.123332i
$389$	1.96148e14	1.11651	0.558253	−	0.829671i	$-0.311472\pi$
$389$	1.96148e14	1.11651	0.558253	+	0.829671i	$0.311472\pi$
$390$	0	0
$391$	−1.03310e13	−0.0571701
$392$	6.42014e13i	0.350325i
$393$	0	0
$394$	3.03289e13	0.160927
$395$	0	0
$396$	0	0
$397$	3.12645e14i	1.59112i	0.605872	+	0.795562i	$0.292824\pi$
$397$	3.12645e14i	1.59112i	−0.605872	+	0.795562i	$0.707176\pi$
$398$	3.42739e14i	1.72031i
$399$	0	0
$400$	0	0
$401$	1.29292e14	0.622696	0.311348	−	0.950296i	$-0.399219\pi$
$401$	1.29292e14	0.622696	0.311348	+	0.950296i	$0.399219\pi$
$402$	0	0
$403$	− 2.91056e14i	− 1.36395i
$404$	−1.31967e14	−0.610055
$405$	0	0
$406$	7.10501e12	0.0319648
$407$	− 2.37653e13i	− 0.105481i
$408$	0	0
$409$	−1.70213e14	−0.735385	−0.367692	−	0.929948i	$-0.619852\pi$
$409$	−1.70213e14	−0.735385	−0.367692	+	0.929948i	$0.619852\pi$
$410$	0	0
$411$	0	0
$412$	1.44000e14i	0.597624i
$413$	2.09845e14i	0.859355i
$414$	0	0
$415$	0	0
$416$	2.01034e14	0.791147
$417$	0	0
$418$	8.60258e13i	0.329731i
$419$	7.47406e13	0.282735	0.141367	−	0.989957i	$-0.454850\pi$
$419$	7.47406e13	0.282735	0.141367	+	0.989957i	$0.454850\pi$
$420$	0	0
$421$	−1.85677e14	−0.684239	−0.342119	−	0.939656i	$-0.611145\pi$
$421$	−1.85677e14	−0.684239	−0.342119	+	0.939656i	$0.611145\pi$
$422$	1.07550e14i	0.391195i
$423$	0	0
$424$	−1.29169e14	−0.457769
$425$	0	0
$426$	0	0
$427$	− 1.13750e14i	− 0.387792i
$428$	2.10809e13i	0.0709494i
$429$	0	0
$430$	0	0
$431$	3.42728e14	1.11001	0.555003	−	0.831849i	$-0.312717\pi$
$431$	3.42728e14	1.11001	0.555003	+	0.831849i	$0.312717\pi$
$432$	0	0
$433$	− 4.77829e14i	− 1.50865i	−0.656500	−	0.754326i	$-0.727964\pi$
$433$	− 4.77829e14i	− 1.50865i	0.656500	−	0.754326i	$-0.272036\pi$
$434$	4.15814e14	1.29630
$435$	0	0
$436$	2.53164e14	0.769532
$437$	5.67636e14i	1.70381i
$438$	0	0
$439$	5.50255e14	1.61068	0.805339	−	0.592814i	$-0.201983\pi$
$439$	5.50255e14	1.61068	0.805339	+	0.592814i	$0.201983\pi$
$440$	0	0
$441$	0	0
$442$	− 1.53302e13i	− 0.0432242i
$443$	2.60772e13i	0.0726173i	0.999341	+	0.0363087i	$0.0115599\pi$
$443$	2.60772e13i	0.0726173i	−0.999341	+	0.0363087i	$0.988440\pi$
$444$	0	0
$445$	0	0
$446$	5.02440e14	1.34816
$447$	0	0
$448$	− 1.30358e13i	− 0.0341278i
$449$	−1.54203e14	−0.398784	−0.199392	−	0.979920i	$-0.563897\pi$
$449$	−1.54203e14	−0.398784	−0.199392	+	0.979920i	$0.563897\pi$
$450$	0	0
$451$	1.41936e14	0.358197
$452$	− 3.47599e14i	− 0.866598i
$453$	0	0
$454$	6.06048e14	1.47469
$455$	0	0
$456$	0	0
$457$	7.76543e13i	0.182233i	0.995840	+	0.0911163i	$0.0290435\pi$
$457$	7.76543e13i	0.182233i	−0.995840	+	0.0911163i	$0.970956\pi$
$458$	4.71463e14i	1.09317i
$459$	0	0
$460$	0	0
$461$	−6.59986e14	−1.47632	−0.738158	−	0.674627i	$-0.764304\pi$
$461$	−6.59986e14	−1.47632	−0.738158	+	0.674627i	$0.764304\pi$
$462$	0	0
$463$	2.96908e14i	0.648525i	0.945967	+	0.324262i	$0.105116\pi$
$463$	2.96908e14i	0.648525i	−0.945967	+	0.324262i	$0.894884\pi$
$464$	2.37519e13	0.0512683
$465$	0	0
$466$	9.48065e14	1.99855
$467$	8.37123e14i	1.74400i	0.489507	+	0.872000i	$0.337177\pi$
$467$	8.37123e14i	1.74400i	−0.489507	+	0.872000i	$0.662823\pi$
$468$	0	0
$469$	−1.92292e14	−0.391302
$470$	0	0
$471$	0	0
$472$	4.03132e14i	0.792075i
$473$	− 1.58291e12i	− 0.00307410i
$474$	0	0
$475$	0	0
$476$	7.59843e12	0.0142523
$477$	0	0
$478$	3.91907e14i	0.718340i
$479$	1.21855e14	0.220800	0.110400	−	0.993887i	$-0.464787\pi$
$479$	1.21855e14	0.220800	0.110400	+	0.993887i	$0.464787\pi$
$480$	0	0
$481$	2.32698e14	0.412092
$482$	− 5.07452e14i	− 0.888457i
$483$	0	0
$484$	2.96764e14	0.507881
$485$	0	0
$486$	0	0
$487$	9.45308e13i	0.156374i	0.996939	+	0.0781870i	$0.0249131\pi$
$487$	9.45308e13i	0.156374i	−0.996939	+	0.0781870i	$0.975087\pi$
$488$	− 2.18525e14i	− 0.357431i
$489$	0	0
$490$	0	0
$491$	2.09824e14	0.331823	0.165911	−	0.986141i	$-0.446943\pi$
$491$	2.09824e14	0.331823	0.165911	+	0.986141i	$0.446943\pi$
$492$	0	0
$493$	− 1.13149e12i	− 0.00174982i
$494$	−8.42321e14	−1.28819
$495$	0	0
$496$	1.39005e15	2.07913
$497$	3.71732e14i	0.549882i
$498$	0	0
$499$	2.64936e14	0.383343	0.191671	−	0.981459i	$-0.438609\pi$
$499$	2.64936e14	0.383343	0.191671	+	0.981459i	$0.438609\pi$
$500$	0	0
$501$	0	0
$502$	− 1.45203e15i	− 2.03285i
$503$	− 3.64546e14i	− 0.504811i	−0.967622	−	0.252405i	$-0.918778\pi$
$503$	− 3.64546e14i	− 0.504811i	0.967622	−	0.252405i	$-0.0812217\pi$
$504$	0	0
$505$	0	0
$506$	−2.59699e14	−0.348051
$507$	0	0
$508$	6.10842e14i	0.801085i
$509$	2.36671e14	0.307042	0.153521	−	0.988145i	$-0.450939\pi$
$509$	2.36671e14	0.307042	0.153521	+	0.988145i	$0.450939\pi$
$510$	0	0
$511$	−8.08062e14	−1.02596
$512$	3.83327e14i	0.481487i
$513$	0	0
$514$	4.91446e14	0.604197
$515$	0	0
$516$	0	0
$517$	2.97689e14i	0.354458i
$518$	3.32441e14i	0.391652i
$519$	0	0
$520$	0	0
$521$	−2.57734e14	−0.294147	−0.147073	−	0.989126i	$-0.546985\pi$
$521$	−2.57734e14	−0.294147	−0.147073	+	0.989126i	$0.546985\pi$
$522$	0	0
$523$	1.48029e15i	1.65420i	0.562053	+	0.827101i	$0.310012\pi$
$523$	1.48029e15i	1.65420i	−0.562053	+	0.827101i	$0.689988\pi$
$524$	4.98141e14	0.550846
$525$	0	0
$526$	−4.69368e14	−0.508267
$527$	− 6.62196e13i	− 0.0709624i
$528$	0	0
$529$	−7.60797e14	−0.798477
$530$	0	0
$531$	0	0
$532$	− 4.17496e14i	− 0.424755i
$533$	1.38977e15i	1.39940i
$534$	0	0
$535$	0	0
$536$	−3.69413e14	−0.360666
$537$	0	0
$538$	− 1.80600e15i	− 1.72749i
$539$	1.33786e14	0.126670
$540$	0	0
$541$	−1.21529e15	−1.12744	−0.563720	−	0.825966i	$-0.690630\pi$
$541$	−1.21529e15	−1.12744	−0.563720	+	0.825966i	$0.690630\pi$
$542$	− 1.61951e14i	− 0.148727i
$543$	0	0
$544$	4.57383e13	0.0411610
$545$	0	0
$546$	0	0
$547$	− 4.93440e14i	− 0.430828i	−0.976523	−	0.215414i	$-0.930890\pi$
$547$	− 4.93440e14i	− 0.430828i	0.976523	−	0.215414i	$-0.0691101\pi$
$548$	− 4.57762e13i	− 0.0395682i
$549$	0	0
$550$	0	0
$551$	−6.21701e13	−0.0521492
$552$	0	0
$553$	− 7.58375e14i	− 0.623585i
$554$	−5.45422e14	−0.444047
$555$	0	0
$556$	8.85254e14	0.706572
$557$	− 5.15728e14i	− 0.407584i	−0.979014	−	0.203792i	$-0.934673\pi$
$557$	− 5.15728e14i	− 0.407584i	0.979014	−	0.203792i	$-0.0653266\pi$
$558$	0	0
$559$	1.54990e13	0.0120099
$560$	0	0
$561$	0	0
$562$	5.24681e14i	0.394771i
$563$	2.86863e14i	0.213737i	0.994273	+	0.106868i	$0.0340823\pi$
$563$	2.86863e14i	0.213737i	−0.994273	+	0.106868i	$0.965918\pi$
$564$	0	0
$565$	0	0
$566$	2.63705e15	1.90822
$567$	0	0
$568$	7.14134e14i	0.506832i
$569$	−5.24421e14	−0.368606	−0.184303	−	0.982869i	$-0.559003\pi$
$569$	−5.24421e14	−0.368606	−0.184303	+	0.982869i	$0.559003\pi$
$570$	0	0
$571$	−1.33323e15	−0.919194	−0.459597	−	0.888127i	$-0.652006\pi$
$571$	−1.33323e15	−0.919194	−0.459597	+	0.888127i	$0.652006\pi$
$572$	− 1.33700e14i	− 0.0912963i
$573$	0	0
$574$	−1.98547e15	−1.32999
$575$	0	0
$576$	0	0
$577$	1.21802e15i	0.792844i	0.918068	+	0.396422i	$0.129748\pi$
$577$	1.21802e15i	0.792844i	−0.918068	+	0.396422i	$0.870252\pi$
$578$	1.91574e15i	1.23519i
$579$	0	0
$580$	0	0
$581$	9.61677e14	0.602644
$582$	0	0
$583$	2.69170e14i	0.165519i
$584$	−1.55237e15	−0.945634
$585$	0	0
$586$	6.26367e14	0.374448
$587$	− 7.51065e14i	− 0.444803i	−0.974955	−	0.222402i	$-0.928610\pi$
$587$	− 7.51065e14i	− 0.444803i	0.974955	−	0.222402i	$-0.0713896\pi$
$588$	0	0
$589$	−3.63844e15	−2.11486
$590$	0	0
$591$	0	0
$592$	1.11134e15i	0.628170i
$593$	− 1.95401e15i	− 1.09427i	−0.837043	−	0.547137i	$-0.815718\pi$
$593$	− 1.95401e15i	− 1.09427i	0.837043	−	0.547137i	$-0.184282\pi$
$594$	0	0
$595$	0	0
$596$	1.69857e14	0.0925186
$597$	0	0
$598$	− 2.54284e15i	− 1.35976i
$599$	2.08899e15	1.10685	0.553425	−	0.832899i	$-0.313321\pi$
$599$	2.08899e15	1.10685	0.553425	+	0.832899i	$0.313321\pi$
$600$	0	0
$601$	−2.78205e15	−1.44729	−0.723645	−	0.690173i	$-0.757535\pi$
$601$	−2.78205e15	−1.44729	−0.723645	+	0.690173i	$0.757535\pi$
$602$	2.21425e13i	0.0114142i
$603$	0	0
$604$	−5.98551e14	−0.302969
$605$	0	0
$606$	0	0
$607$	2.48334e15i	1.22320i	0.791166	+	0.611601i	$0.209474\pi$
$607$	2.48334e15i	1.22320i	−0.791166	+	0.611601i	$0.790526\pi$
$608$	− 2.51309e15i	− 1.22670i
$609$	0	0
$610$	0	0
$611$	−2.91482e15	−1.38479
$612$	0	0
$613$	3.54510e15i	1.65423i	0.562032	+	0.827116i	$0.310020\pi$
$613$	3.54510e15i	1.65423i	−0.562032	+	0.827116i	$0.689980\pi$
$614$	−2.72030e15	−1.25803
$615$	0	0
$616$	−1.68537e14	−0.0765600
$617$	− 1.21585e14i	− 0.0547410i	−0.999625	−	0.0273705i	$-0.991287\pi$
$617$	− 1.21585e14i	− 0.0547410i	0.999625	−	0.0273705i	$-0.00871338\pi$
$618$	0	0
$619$	1.33849e15	0.591994	0.295997	−	0.955189i	$-0.404348\pi$
$619$	1.33849e15	0.591994	0.295997	+	0.955189i	$0.404348\pi$
$620$	0	0
$621$	0	0
$622$	− 2.90108e15i	− 1.24943i
$623$	− 1.77700e15i	− 0.758581i
$624$	0	0
$625$	0	0
$626$	1.37577e15	0.571988
$627$	0	0
$628$	2.03988e15i	0.833348i
$629$	5.29422e13	0.0214399
$630$	0	0
$631$	−2.23141e15	−0.888010	−0.444005	−	0.896024i	$-0.646443\pi$
$631$	−2.23141e15	−0.888010	−0.444005	+	0.896024i	$0.646443\pi$
$632$	− 1.45691e15i	− 0.574764i
$633$	0	0
$634$	−9.79674e14	−0.379831
$635$	0	0
$636$	0	0
$637$	1.30997e15i	0.494873i
$638$	− 2.84434e13i	− 0.0106529i
$639$	0	0
$640$	0	0
$641$	−2.53917e15	−0.926773	−0.463386	−	0.886156i	$-0.653366\pi$
$641$	−2.53917e15	−0.926773	−0.463386	+	0.886156i	$0.653366\pi$
$642$	0	0
$643$	1.40215e15i	0.503076i	0.967847	+	0.251538i	$0.0809363\pi$
$643$	1.40215e15i	0.503076i	−0.967847	+	0.251538i	$0.919064\pi$
$644$	1.26036e15	0.448355
$645$	0	0
$646$	−1.91641e14	−0.0670207
$647$	− 1.45024e15i	− 0.502882i	−0.967873	−	0.251441i	$-0.919096\pi$
$647$	− 1.45024e15i	− 0.502882i	0.967873	−	0.251441i	$-0.0809045\pi$
$648$	0	0
$649$	8.40069e14	0.286397
$650$	0	0
$651$	0	0
$652$	− 2.36891e15i	− 0.787384i
$653$	6.74967e14i	0.222464i	0.993794	+	0.111232i	$0.0354797\pi$
$653$	6.74967e14i	0.222464i	−0.993794	+	0.111232i	$0.964520\pi$
$654$	0	0
$655$	0	0
$656$	−6.63738e15	−2.13317
$657$	0	0
$658$	− 4.16422e15i	− 1.31611i
$659$	3.55334e15	1.11370	0.556849	−	0.830614i	$-0.312010\pi$
$659$	3.55334e15	1.11370	0.556849	+	0.830614i	$0.312010\pi$
$660$	0	0
$661$	4.08998e15	1.26070	0.630351	−	0.776310i	$-0.282911\pi$
$661$	4.08998e15	1.26070	0.630351	+	0.776310i	$0.282911\pi$
$662$	− 3.83294e15i	− 1.17169i
$663$	0	0
$664$	1.84748e15	0.555462
$665$	0	0
$666$	0	0
$667$	− 1.87682e14i	− 0.0550466i
$668$	− 1.28399e15i	− 0.373500i
$669$	0	0
$670$	0	0
$671$	−4.55375e14	−0.129239
$672$	0	0
$673$	2.31714e14i	0.0646948i	0.999477	+	0.0323474i	$0.0102983\pi$
$673$	2.31714e14i	0.0646948i	−0.999477	+	0.0323474i	$0.989702\pi$
$674$	4.31166e15	1.19403
$675$	0	0
$676$	−6.40748e14	−0.174574
$677$	4.21442e15i	1.13894i	0.822013	+	0.569469i	$0.192851\pi$
$677$	4.21442e15i	1.13894i	−0.822013	+	0.569469i	$0.807149\pi$
$678$	0	0
$679$	−5.49452e14	−0.146099
$680$	0	0
$681$	0	0
$682$	− 1.66462e15i	− 0.432018i
$683$	− 9.30314e14i	− 0.239506i	−0.992804	−	0.119753i	$-0.961790\pi$
$683$	− 9.30314e14i	− 0.239506i	0.992804	−	0.119753i	$-0.0382102\pi$
$684$	0	0
$685$	0	0
$686$	−4.97015e15	−1.24907
$687$	0	0
$688$	7.40216e13i	0.0183072i
$689$	−2.63557e15	−0.646650
$690$	0	0
$691$	−3.13309e15	−0.756561	−0.378280	−	0.925691i	$-0.623485\pi$
$691$	−3.13309e15	−0.756561	−0.378280	+	0.925691i	$0.623485\pi$
$692$	− 5.43109e14i	− 0.130108i
$693$	0	0
$694$	7.25305e15	1.71019
$695$	0	0
$696$	0	0
$697$	3.16193e14i	0.0728067i
$698$	− 2.04267e15i	− 0.466653i
$699$	0	0
$700$	0	0
$701$	6.21787e15	1.38737	0.693686	−	0.720278i	$-0.255986\pi$
$701$	6.21787e15	1.38737	0.693686	+	0.720278i	$0.255986\pi$
$702$	0	0
$703$	− 2.90891e15i	− 0.638964i
$704$	−5.21860e13	−0.0113738
$705$	0	0
$706$	−5.78764e15	−1.24187
$707$	3.39428e15i	0.722670i
$708$	0	0
$709$	1.38076e15	0.289444	0.144722	−	0.989472i	$-0.453771\pi$
$709$	1.38076e15	0.289444	0.144722	+	0.989472i	$0.453771\pi$
$710$	0	0
$711$	0	0
$712$	− 3.41378e15i	− 0.699191i
$713$	− 1.09839e16i	− 2.23236i
$714$	0	0
$715$	0	0
$716$	−3.49469e15	−0.694043
$717$	0	0
$718$	− 8.56698e15i	− 1.67550i
$719$	−1.22219e15	−0.237209	−0.118604	−	0.992942i	$-0.537842\pi$
$719$	−1.22219e15	−0.237209	−0.118604	+	0.992942i	$0.537842\pi$
$720$	0	0
$721$	3.70376e15	0.707945
$722$	4.00625e15i	0.759948i
$723$	0	0
$724$	3.19051e15	0.596070
$725$	0	0
$726$	0	0
$727$	− 7.75213e15i	− 1.41573i	−0.706346	−	0.707867i	$-0.749658\pi$
$727$	− 7.75213e15i	− 1.41573i	0.706346	−	0.707867i	$-0.250342\pi$
$728$	− 1.65023e15i	− 0.299104i
$729$	0	0
$730$	0	0
$731$	3.52625e12	0.000624839 0
$732$	0	0
$733$	− 8.29426e15i	− 1.44779i	−0.689910	−	0.723895i	$-0.742350\pi$
$733$	− 8.29426e15i	− 1.44779i	0.689910	−	0.723895i	$-0.257650\pi$
$734$	−8.34994e15	−1.44662
$735$	0	0
$736$	7.58665e15	1.29486
$737$	7.69802e14i	0.130409i
$738$	0	0
$739$	4.08381e14	0.0681587	0.0340793	−	0.999419i	$-0.489150\pi$
$739$	4.08381e14	0.0681587	0.0340793	+	0.999419i	$0.489150\pi$
$740$	0	0
$741$	0	0
$742$	− 3.76528e15i	− 0.614576i
$743$	1.87602e15i	0.303948i	0.988384	+	0.151974i	$0.0485630\pi$
$743$	1.87602e15i	0.303948i	−0.988384	+	0.151974i	$0.951437\pi$
$744$	0	0
$745$	0	0
$746$	−3.14264e15	−0.498002
$747$	0	0
$748$	− 3.04187e13i	− 0.00474987i
$749$	5.42214e14	0.0840467
$750$	0	0
$751$	9.97525e15	1.52372	0.761858	−	0.647744i	$-0.224287\pi$
$751$	9.97525e15	1.52372	0.761858	+	0.647744i	$0.224287\pi$
$752$	− 1.39209e16i	− 2.11090i
$753$	0	0
$754$	2.78504e14	0.0416187
$755$	0	0
$756$	0	0
$757$	1.09293e16i	1.59795i	0.601361	+	0.798977i	$0.294625\pi$
$757$	1.09293e16i	1.59795i	−0.601361	+	0.798977i	$0.705375\pi$
$758$	− 4.24153e15i	− 0.615662i
$759$	0	0
$760$	0	0
$761$	3.39091e15	0.481615	0.240808	−	0.970573i	$-0.422588\pi$
$761$	3.39091e15	0.481615	0.240808	+	0.970573i	$0.422588\pi$
$762$	0	0
$763$	− 6.51153e15i	− 0.911587i
$764$	2.48763e15	0.345758
$765$	0	0
$766$	−1.13160e15	−0.155037
$767$	8.22553e15i	1.11890i
$768$	0	0
$769$	6.40507e15	0.858873	0.429437	−	0.903097i	$-0.358712\pi$
$769$	6.40507e15	0.858873	0.429437	+	0.903097i	$0.358712\pi$
$770$	0	0
$771$	0	0
$772$	6.38239e15i	0.837699i
$773$	6.92865e15i	0.902945i	0.892285	+	0.451473i	$0.149101\pi$
$773$	6.92865e15i	0.902945i	−0.892285	+	0.451473i	$0.850899\pi$
$774$	0	0
$775$	0	0
$776$	−1.05555e15	−0.134660
$777$	0	0
$778$	1.09843e16i	1.38161i
$779$	1.73732e16	2.16982
$780$	0	0
$781$	1.48815e15	0.183259
$782$	− 5.78534e14i	− 0.0707444i
$783$	0	0
$784$	−6.25627e15	−0.754358
$785$	0	0
$786$	0	0
$787$	− 2.81869e15i	− 0.332802i	−0.986058	−	0.166401i	$-0.946785\pi$
$787$	− 2.81869e15i	− 0.332802i	0.986058	−	0.166401i	$-0.0532146\pi$
$788$	5.89248e14i	0.0690882i
$789$	0	0
$790$	0	0
$791$	−8.94046e15	−1.02657
$792$	0	0
$793$	− 4.45880e15i	− 0.504912i
$794$	−1.75081e16	−1.96892
$795$	0	0
$796$	−6.65893e15	−0.738555
$797$	− 3.10085e15i	− 0.341554i	−0.985310	−	0.170777i	$-0.945372\pi$
$797$	− 3.10085e15i	− 0.341554i	0.985310	−	0.170777i	$-0.0546278\pi$
$798$	0	0
$799$	−6.63165e14	−0.0720467
$800$	0	0
$801$	0	0
$802$	7.24033e15i	0.770547i
$803$	3.23490e15i	0.341921i
$804$	0	0
$805$	0	0
$806$	1.62991e16	1.68780
$807$	0	0
$808$	6.52073e15i	0.666092i
$809$	1.32956e16	1.34894	0.674468	−	0.738304i	$-0.264373\pi$
$809$	1.32956e16	1.34894	0.674468	+	0.738304i	$0.264373\pi$
$810$	0	0
$811$	6.98127e15	0.698747	0.349373	−	0.936984i	$-0.386394\pi$
$811$	6.98127e15	0.698747	0.349373	+	0.936984i	$0.386394\pi$
$812$	1.38040e14i	0.0137230i
$813$	0	0
$814$	1.33086e15	0.130526
$815$	0	0
$816$	0	0
$817$	− 1.93750e14i	− 0.0186218i
$818$	− 9.53193e15i	− 0.909992i
$819$	0	0
$820$	0	0
$821$	−6.46007e15	−0.604436	−0.302218	−	0.953239i	$-0.597727\pi$
$821$	−6.46007e15	−0.604436	−0.302218	+	0.953239i	$0.597727\pi$
$822$	0	0
$823$	4.54918e15i	0.419985i	0.977703	+	0.209993i	$0.0673440\pi$
$823$	4.54918e15i	0.419985i	−0.977703	+	0.209993i	$0.932656\pi$
$824$	7.11529e15	0.652519
$825$	0	0
$826$	−1.17513e16	−1.06340
$827$	3.58820e15i	0.322549i	0.986910	+	0.161275i	$0.0515605\pi$
$827$	3.58820e15i	0.322549i	−0.986910	+	0.161275i	$0.948440\pi$
$828$	0	0
$829$	−1.35990e16	−1.20630	−0.603152	−	0.797626i	$-0.706089\pi$
$829$	−1.35990e16	−1.20630	−0.603152	+	0.797626i	$0.706089\pi$
$830$	0	0
$831$	0	0
$832$	− 5.10979e14i	− 0.0444350i
$833$	2.98037e14i	0.0257468i
$834$	0	0
$835$	0	0
$836$	−1.67136e15	−0.141558
$837$	0	0
$838$	4.18547e15i	0.349867i
$839$	−1.49622e16	−1.24253	−0.621263	−	0.783602i	$-0.713380\pi$
$839$	−1.49622e16	−1.24253	−0.621263	+	0.783602i	$0.713380\pi$
$840$	0	0
$841$	−1.21800e16	−0.998315
$842$	− 1.03979e16i	− 0.846702i
$843$	0	0
$844$	−2.08955e15	−0.167946
$845$	0	0
$846$	0	0
$847$	− 7.63295e15i	− 0.601636i
$848$	− 1.25872e16i	− 0.985718i
$849$	0	0
$850$	0	0
$851$	8.78156e15	0.674465
$852$	0	0
$853$	9.31290e15i	0.706098i	0.935605	+	0.353049i	$0.114855\pi$
$853$	9.31290e15i	0.706098i	−0.935605	+	0.353049i	$0.885145\pi$
$854$	6.37001e15	0.479868
$855$	0	0
$856$	1.04165e15	0.0774666
$857$	2.01081e16i	1.48585i	0.669373	+	0.742927i	$0.266563\pi$
$857$	2.01081e16i	1.48585i	−0.669373	+	0.742927i	$0.733437\pi$
$858$	0	0
$859$	1.78636e16	1.30319	0.651594	−	0.758568i	$-0.274100\pi$
$859$	1.78636e16	1.30319	0.651594	+	0.758568i	$0.274100\pi$
$860$	0	0
$861$	0	0
$862$	1.91928e16i	1.37356i
$863$	2.51194e16i	1.78628i	0.449776	+	0.893141i	$0.351504\pi$
$863$	2.51194e16i	1.78628i	−0.449776	+	0.893141i	$0.648496\pi$
$864$	0	0
$865$	0	0
$866$	2.67584e16	1.86686
$867$	0	0
$868$	8.07867e15i	0.556521i
$869$	−3.03599e15	−0.207822
$870$	0	0
$871$	−7.53752e15	−0.509481
$872$	− 1.25093e16i	− 0.840218i
$873$	0	0
$874$	−3.17876e16	−2.10836
$875$	0	0
$876$	0	0
$877$	1.05249e16i	0.685045i	0.939510	+	0.342522i	$0.111281\pi$
$877$	1.05249e16i	0.685045i	−0.939510	+	0.342522i	$0.888719\pi$
$878$	3.08143e16i	1.99311i
$879$	0	0
$880$	0	0
$881$	2.10050e16	1.33338	0.666691	−	0.745334i	$-0.267710\pi$
$881$	2.10050e16	1.33338	0.666691	+	0.745334i	$0.267710\pi$
$882$	0	0
$883$	1.55181e16i	0.972868i	0.873717	+	0.486434i	$0.161703\pi$
$883$	1.55181e16i	0.972868i	−0.873717	+	0.486434i	$0.838297\pi$
$884$	2.97845e14	0.0185568
$885$	0	0
$886$	−1.46032e15	−0.0898593
$887$	− 7.61183e15i	− 0.465489i	−0.972538	−	0.232744i	$-0.925229\pi$
$887$	− 7.61183e15i	− 0.465489i	0.972538	−	0.232744i	$-0.0747705\pi$
$888$	0	0
$889$	1.57112e16	0.948964
$890$	0	0
$891$	0	0
$892$	9.76169e15i	0.578786i
$893$	3.64376e16i	2.14717i
$894$	0	0
$895$	0	0
$896$	1.12335e16	0.649863
$897$	0	0
$898$	− 8.63537e15i	− 0.493470i
$899$	1.20301e15	0.0683266
$900$	0	0
$901$	−5.99632e14	−0.0336433
$902$	7.94842e15i	0.443246i
$903$	0	0
$904$	−1.71755e16	−0.946200
$905$	0	0
$906$	0	0
$907$	− 2.39103e16i	− 1.29344i	−0.762728	−	0.646719i	$-0.776141\pi$
$907$	− 2.39103e16i	− 1.29344i	0.762728	−	0.646719i	$-0.223859\pi$
$908$	1.17747e16i	0.633105i
$909$	0	0
$910$	0	0
$911$	−2.44923e16	−1.29324	−0.646618	−	0.762814i	$-0.723817\pi$
$911$	−2.44923e16	−1.29324	−0.646618	+	0.762814i	$0.723817\pi$
$912$	0	0
$913$	− 3.84987e15i	− 0.200843i
$914$	−4.34864e15	−0.225501
$915$	0	0
$916$	−9.15985e15	−0.469314
$917$	− 1.28125e16i	− 0.652532i
$918$	0	0
$919$	3.21575e16	1.61826	0.809128	−	0.587632i	$-0.199940\pi$
$919$	3.21575e16	1.61826	0.809128	+	0.587632i	$0.199940\pi$
$920$	0	0
$921$	0	0
$922$	− 3.69592e16i	− 1.82685i
$923$	1.45712e16i	0.715956i
$924$	0	0
$925$	0	0
$926$	−1.66268e16	−0.802508
$927$	0	0
$928$	8.30925e14i	0.0396322i
$929$	−1.27739e16	−0.605672	−0.302836	−	0.953043i	$-0.597933\pi$
$929$	−1.27739e16	−0.605672	−0.302836	+	0.953043i	$0.597933\pi$
$930$	0	0
$931$	1.63757e16	0.767319
$932$	1.84196e16i	0.858008i
$933$	0	0
$934$	−4.68789e16	−2.15809
$935$	0	0
$936$	0	0
$937$	1.11554e15i	0.0504565i	0.999682	+	0.0252283i	$0.00803126\pi$
$937$	1.11554e15i	0.0504565i	−0.999682	+	0.0252283i	$0.991969\pi$
$938$	− 1.07684e16i	− 0.484211i
$939$	0	0
$940$	0	0
$941$	3.97991e16	1.75845	0.879226	−	0.476405i	$-0.158060\pi$
$941$	3.97991e16	1.75845	0.879226	+	0.476405i	$0.158060\pi$
$942$	0	0
$943$	5.24471e16i	2.29038i
$944$	−3.92843e16	−1.70558
$945$	0	0
$946$	8.86427e13	0.00380401
$947$	3.26979e16i	1.39507i	0.716552	+	0.697534i	$0.245719\pi$
$947$	3.26979e16i	1.39507i	−0.716552	+	0.697534i	$0.754281\pi$
$948$	0	0
$949$	−3.16745e16	−1.33581
$950$	0	0
$951$	0	0
$952$	− 3.75452e14i	− 0.0155615i
$953$	− 2.74452e15i	− 0.113098i	−0.998400	−	0.0565491i	$-0.981990\pi$
$953$	− 2.74452e15i	− 0.113098i	0.998400	−	0.0565491i	$-0.0180098\pi$
$954$	0	0
$955$	0	0
$956$	−7.61420e15	−0.308394
$957$	0	0
$958$	6.82390e15i	0.273226i
$959$	−1.17739e15	−0.0468725
$960$	0	0
$961$	4.49963e16	1.77092
$962$	1.30311e16i	0.509938i
$963$	0	0
$964$	9.85907e15	0.381428
$965$	0	0
$966$	0	0
$967$	7.81335e15i	0.297161i	0.988900	+	0.148580i	$0.0474704\pi$
$967$	7.81335e15i	0.297161i	−0.988900	+	0.148580i	$0.952530\pi$
$968$	− 1.46637e16i	− 0.554533i
$969$	0	0
$970$	0	0
$971$	−4.79623e16	−1.78318	−0.891588	−	0.452848i	$-0.850408\pi$
$971$	−4.79623e16	−1.78318	−0.891588	+	0.452848i	$0.850408\pi$
$972$	0	0
$973$	− 2.27693e16i	− 0.837005i
$974$	−5.29373e15	−0.193503
$975$	0	0
$976$	2.12948e16	0.769660
$977$	− 4.28901e16i	− 1.54148i	−0.637152	−	0.770738i	$-0.719888\pi$
$977$	− 4.28901e16i	− 1.54148i	0.637152	−	0.770738i	$-0.280112\pi$
$978$	0	0
$979$	−7.11382e15	−0.252812
$980$	0	0
$981$	0	0
$982$	1.17501e16i	0.410610i
$983$	− 1.90984e16i	− 0.663671i	−0.943337	−	0.331836i	$-0.892332\pi$
$983$	− 1.90984e16i	− 0.663671i	0.943337	−	0.331836i	$-0.107668\pi$
$984$	0	0
$985$	0	0
$986$	6.33637e13	0.00216530
$987$	0	0
$988$	− 1.63651e16i	− 0.553039i
$989$	5.84903e14	0.0196564
$990$	0	0
$991$	−7.64435e15	−0.254060	−0.127030	−	0.991899i	$-0.540544\pi$
$991$	−7.64435e15	−0.254060	−0.127030	+	0.991899i	$0.540544\pi$
$992$	4.86291e16i	1.60724i
$993$	0	0
$994$	−2.08170e16	−0.680445
$995$	0	0
$996$	0	0
$997$	4.72891e16i	1.52033i	0.649731	+	0.760164i	$0.274882\pi$
$997$	4.72891e16i	1.52033i	−0.649731	+	0.760164i	$0.725118\pi$
$998$	1.48364e16i	0.474362i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.12.b.b.199.2		2
3.2	odd	2		75.12.b.b.49.1		2
5.2	odd	4		225.12.a.a.1.1		1
5.3	odd	4		45.12.a.b.1.1		1
5.4	even	2	inner	225.12.b.b.199.1		2
15.2	even	4		75.12.a.b.1.1		1
15.8	even	4		15.12.a.a.1.1	✓	1
15.14	odd	2		75.12.b.b.49.2		2
60.23	odd	4		240.12.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.12.a.a.1.1	✓	1	15.8	even	4
45.12.a.b.1.1		1	5.3	odd	4
75.12.a.b.1.1		1	15.2	even	4
75.12.b.b.49.1		2	3.2	odd	2
75.12.b.b.49.2		2	15.14	odd	2
225.12.a.a.1.1		1	5.2	odd	4
225.12.b.b.199.1		2	5.4	even	2	inner
225.12.b.b.199.2		2	1.1	even	1	trivial
240.12.a.e.1.1		1	60.23	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:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+56.0000i q^{2} -1088.00 q^{4} +27984.0i q^{7} +53760.0i q^{8} +O(q^{10})\)	\(q+56.0000i q^{2} -1088.00 q^{4} +27984.0i q^{7} +53760.0i q^{8} +112028. q^{11} +1.09692e6i q^{13} -1.56710e6 q^{14} -5.23878e6 q^{16} +249566. i q^{17} +1.37124e7 q^{19} +6.27357e6i q^{22} +4.13957e7i q^{23} -6.14276e7 q^{26} -3.04466e7i q^{28} -4.53385e6 q^{29} -2.65339e8 q^{31} -1.83271e8i q^{32} -1.39757e7 q^{34} -2.12137e8i q^{37} +7.67896e8i q^{38} +1.26697e9 q^{41} -1.41295e7i q^{43} -1.21886e8 q^{44} -2.31816e9 q^{46} +2.65727e9i q^{47} +1.19422e9 q^{49} -1.19345e9i q^{52} +2.40270e9i q^{53} -1.50442e9 q^{56} -2.53896e8i q^{58} +7.49874e9 q^{59} -4.06483e9 q^{61} -1.48590e10i q^{62} -4.65830e8 q^{64} +6.87151e9i q^{67} -2.71528e8i q^{68} +1.32837e10 q^{71} +2.88758e10i q^{73} +1.18797e10 q^{74} -1.49191e10 q^{76} +3.13499e9i q^{77} -2.71003e10 q^{79} +7.09503e10i q^{82} -3.43653e10i q^{83} +7.91255e8 q^{86} +6.02263e9i q^{88} -6.35004e10 q^{89} -3.06963e10 q^{91} -4.50386e10i q^{92} -1.48807e11 q^{94} +1.96345e10i q^{97} +6.68765e10i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2176 q^{4}+O(q^{10}) \) 2 * q - 2176 * q^4	\( 2 q - 2176 q^{4} + 224056 q^{11} - 3134208 q^{14} - 10477568 q^{16} + 27424840 q^{19} - 122855264 q^{26} - 9067700 q^{29} - 530678016 q^{31} - 27951392 q^{34} + 2533939916 q^{41} - 243772928 q^{44} - 4636321536 q^{46} + 2388444974 q^{49} - 3008839680 q^{56} + 14997474440 q^{59} - 8129657716 q^{61} - 931659776 q^{64} + 26567469296 q^{71} + 23759337952 q^{74} - 29838225920 q^{76} - 54200604480 q^{79} + 1582509376 q^{86} - 127000825260 q^{89} - 61392530496 q^{91} - 297614613632 q^{94}+O(q^{100}) \) 2 * q - 2176 * q^4 + 224056 * q^11 - 3134208 * q^14 - 10477568 * q^16 + 27424840 * q^19 - 122855264 * q^26 - 9067700 * q^29 - 530678016 * q^31 - 27951392 * q^34 + 2533939916 * q^41 - 243772928 * q^44 - 4636321536 * q^46 + 2388444974 * q^49 - 3008839680 * q^56 + 14997474440 * q^59 - 8129657716 * q^61 - 931659776 * q^64 + 26567469296 * q^71 + 23759337952 * q^74 - 29838225920 * q^76 - 54200604480 * q^79 + 1582509376 * q^86 - 127000825260 * q^89 - 61392530496 * q^91 - 297614613632 * q^94

Introduction

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