LMFDB - Embedded newform 225.8.b.f.199.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [225,8,Mod(199,225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("225.199"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	225.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,184,0,0,0,0,0,0,1896] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$70.2866307339$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3)
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			225.8.b.f.199.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+6.00000i q^{2} +92.0000 q^{4} +64.0000i q^{7} +1320.00i q^{8} +948.000 q^{11} -5098.00i q^{13} -384.000 q^{14} +3856.00 q^{16} +28386.0i q^{17} +8620.00 q^{19} +5688.00i q^{22} +15288.0i q^{23} +30588.0 q^{26} +5888.00i q^{28} +36510.0 q^{29} -276808. q^{31} +192096. i q^{32} -170316. q^{34} -268526. i q^{37} +51720.0i q^{38} +629718. q^{41} +685772. i q^{43} +87216.0 q^{44} -91728.0 q^{46} +583296. i q^{47} +819447. q^{49} -469016. i q^{52} +428058. i q^{53} -84480.0 q^{56} +219060. i q^{58} +1.30638e6 q^{59} +300662. q^{61} -1.66085e6i q^{62} -659008. q^{64} +507244. i q^{67} +2.61151e6i q^{68} -5.56063e6 q^{71} +1.36908e6i q^{73} +1.61116e6 q^{74} +793040. q^{76} +60672.0i q^{77} +6.91372e6 q^{79} +3.77831e6i q^{82} +4.37675e6i q^{83} -4.11463e6 q^{86} +1.25136e6i q^{88} -8.52831e6 q^{89} +326272. q^{91} +1.40650e6i q^{92} -3.49978e6 q^{94} +8.82681e6i q^{97} +4.91668e6i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 184 q^{4} + 1896 q^{11} - 768 q^{14} + 7712 q^{16} + 17240 q^{19} + 61176 q^{26} + 73020 q^{29} - 553616 q^{31} - 340632 q^{34} + 1259436 q^{41} + 174432 q^{44} - 183456 q^{46} + 1638894 q^{49} - 168960 q^{56}+ \cdots - 6999552 q^{94}+O(q^{100}) $ 2 * q + 184 * q^4 + 1896 * q^11 - 768 * q^14 + 7712 * q^16 + 17240 * q^19 + 61176 * q^26 + 73020 * q^29 - 553616 * q^31 - 340632 * q^34 + 1259436 * q^41 + 174432 * q^44 - 183456 * q^46 + 1638894 * q^49 - 168960 * q^56 + 2612760 * q^59 + 601324 * q^61 - 1318016 * q^64 - 11121264 * q^71 + 3222312 * q^74 + 1586080 * q^76 + 13827440 * q^79 - 8229264 * q^86 - 17056620 * q^89 + 652544 * q^91 - 6999552 * q^94

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 $

$2$	6.00000i	0.530330i	0.964203	+	0.265165i	$0.0854264\pi$
$2$	6.00000i	0.530330i	−0.964203	+	0.265165i	$0.914574\pi$
$3$	0	0
$3$	0	0
$4$	92.0000	0.718750
$5$	0	0
$5$	0	0
$6$	0	0
$7$	64.0000i	0.0705240i	0.999378	+	0.0352620i	$0.0112266\pi$
$7$	64.0000i	0.0705240i	−0.999378	+	0.0352620i	$0.988773\pi$
$8$	1320.00i	0.911505i
$9$	0	0
$10$	0	0
$11$	948.000	0.214750	0.107375	−	0.994219i	$-0.465755\pi$
$11$	948.000	0.214750	0.107375	+	0.994219i	$0.465755\pi$
$12$	0	0
$13$	− 5098.00i	− 0.643573i	−0.946812	−	0.321787i	$-0.895717\pi$
$13$	− 5098.00i	− 0.643573i	0.946812	−	0.321787i	$-0.104283\pi$
$14$	−384.000	−0.0374010
$15$	0	0
$16$	3856.00	0.235352
$17$	28386.0i	1.40131i	0.713502	+	0.700653i	$0.247108\pi$
$17$	28386.0i	1.40131i	−0.713502	+	0.700653i	$0.752892\pi$
$18$	0	0
$19$	8620.00	0.288317	0.144158	−	0.989555i	$-0.453953\pi$
$19$	8620.00	0.288317	0.144158	+	0.989555i	$0.453953\pi$
$20$	0	0
$21$	0	0
$22$	5688.00i	0.113889i
$23$	15288.0i	0.262001i	0.991382	+	0.131001i	$0.0418190\pi$
$23$	15288.0i	0.262001i	−0.991382	+	0.131001i	$0.958181\pi$
$24$	0	0
$25$	0	0
$26$	30588.0	0.341306
$27$	0	0
$28$	5888.00i	0.0506891i
$29$	36510.0	0.277983	0.138992	−	0.990294i	$-0.455614\pi$
$29$	36510.0	0.277983	0.138992	+	0.990294i	$0.455614\pi$
$30$	0	0
$31$	−276808.	−1.66883	−0.834416	−	0.551135i	$-0.814195\pi$
$31$	−276808.	−1.66883	−0.834416	+	0.551135i	$0.814195\pi$
$32$	192096.i	1.03632i
$33$	0	0
$34$	−170316.	−0.743155
$35$	0	0
$36$	0	0
$37$	− 268526.i	− 0.871526i	−0.900061	−	0.435763i	$-0.856479\pi$
$37$	− 268526.i	− 0.871526i	0.900061	−	0.435763i	$-0.143521\pi$
$38$	51720.0i	0.152903i
$39$	0	0
$40$	0	0
$41$	629718.	1.42693	0.713465	−	0.700691i	$-0.247125\pi$
$41$	629718.	1.42693	0.713465	+	0.700691i	$0.247125\pi$
$42$	0	0
$43$	685772.i	1.31535i	0.753303	+	0.657673i	$0.228459\pi$
$43$	685772.i	1.31535i	−0.753303	+	0.657673i	$0.771541\pi$
$44$	87216.0	0.154352
$45$	0	0
$46$	−91728.0	−0.138947
$47$	583296.i	0.819495i	0.912199	+	0.409748i	$0.134383\pi$
$47$	583296.i	0.819495i	−0.912199	+	0.409748i	$0.865617\pi$
$48$	0	0
$49$	819447.	0.995026
$50$	0	0
$51$	0	0
$52$	− 469016.i	− 0.462568i
$53$	428058.i	0.394945i	0.980308	+	0.197473i	$0.0632734\pi$
$53$	428058.i	0.394945i	−0.980308	+	0.197473i	$0.936727\pi$
$54$	0	0
$55$	0	0
$56$	−84480.0	−0.0642830
$57$	0	0
$58$	219060.i	0.147423i
$59$	1.30638e6	0.828109	0.414054	−	0.910252i	$-0.364112\pi$
$59$	1.30638e6	0.828109	0.414054	+	0.910252i	$0.364112\pi$
$60$	0	0
$61$	300662.	0.169599	0.0847997	−	0.996398i	$-0.472975\pi$
$61$	300662.	0.169599	0.0847997	+	0.996398i	$0.472975\pi$
$62$	− 1.66085e6i	− 0.885032i
$63$	0	0
$64$	−659008.	−0.314240
$65$	0	0
$66$	0	0
$67$	507244.i	0.206042i	0.994679	+	0.103021i	$0.0328508\pi$
$67$	507244.i	0.206042i	−0.994679	+	0.103021i	$0.967149\pi$
$68$	2.61151e6i	1.00719i
$69$	0	0
$70$	0	0
$71$	−5.56063e6	−1.84383	−0.921913	−	0.387397i	$-0.873374\pi$
$71$	−5.56063e6	−1.84383	−0.921913	+	0.387397i	$0.873374\pi$
$72$	0	0
$73$	1.36908e6i	0.411907i	0.978562	+	0.205954i	$0.0660296\pi$
$73$	1.36908e6i	0.411907i	−0.978562	+	0.205954i	$0.933970\pi$
$74$	1.61116e6	0.462196
$75$	0	0
$76$	793040.	0.207228
$77$	60672.0i	0.0151451i
$78$	0	0
$79$	6.91372e6	1.57767	0.788836	−	0.614603i	$-0.210684\pi$
$79$	6.91372e6	1.57767	0.788836	+	0.614603i	$0.210684\pi$
$80$	0	0
$81$	0	0
$82$	3.77831e6i	0.756744i
$83$	4.37675e6i	0.840191i	0.907480	+	0.420096i	$0.138003\pi$
$83$	4.37675e6i	0.840191i	−0.907480	+	0.420096i	$0.861997\pi$
$84$	0	0
$85$	0	0
$86$	−4.11463e6	−0.697568
$87$	0	0
$88$	1.25136e6i	0.195746i
$89$	−8.52831e6	−1.28232	−0.641162	−	0.767405i	$-0.721547\pi$
$89$	−8.52831e6	−1.28232	−0.641162	+	0.767405i	$0.721547\pi$
$90$	0	0
$91$	326272.	0.0453874
$92$	1.40650e6i	0.188313i
$93$	0	0
$94$	−3.49978e6	−0.434603
$95$	0	0
$96$	0	0
$97$	8.82681e6i	0.981981i	0.871165	+	0.490990i	$0.163365\pi$
$97$	8.82681e6i	0.981981i	−0.871165	+	0.490990i	$0.836635\pi$
$98$	4.91668e6i	0.527692i
$99$	0	0
$100$	0	0
$101$	−1.19864e7	−1.15762	−0.578808	−	0.815464i	$-0.696482\pi$
$101$	−1.19864e7	−1.15762	−0.578808	+	0.815464i	$0.696482\pi$
$102$	0	0
$103$	7.20939e6i	0.650082i	0.945700	+	0.325041i	$0.105378\pi$
$103$	7.20939e6i	0.650082i	−0.945700	+	0.325041i	$0.894622\pi$
$104$	6.72936e6	0.586620
$105$	0	0
$106$	−2.56835e6	−0.209451
$107$	1.14261e7i	0.901683i	0.892604	+	0.450842i	$0.148876\pi$
$107$	1.14261e7i	0.901683i	−0.892604	+	0.450842i	$0.851124\pi$
$108$	0	0
$109$	−4.02095e6	−0.297397	−0.148698	−	0.988883i	$-0.547508\pi$
$109$	−4.02095e6	−0.297397	−0.148698	+	0.988883i	$0.547508\pi$
$110$	0	0
$111$	0	0
$112$	246784.i	0.0165979i
$113$	1.77063e7i	1.15439i	0.816605	+	0.577197i	$0.195853\pi$
$113$	1.77063e7i	1.15439i	−0.816605	+	0.577197i	$0.804147\pi$
$114$	0	0
$115$	0	0
$116$	3.35892e6	0.199801
$117$	0	0
$118$	7.83828e6i	0.439171i
$119$	−1.81670e6	−0.0988257
$120$	0	0
$121$	−1.85885e7	−0.953882
$122$	1.80397e6i	0.0899436i
$123$	0	0
$124$	−2.54663e7	−1.19947
$125$	0	0
$126$	0	0
$127$	− 1.67883e7i	− 0.727267i	−0.931542	−	0.363633i	$-0.881536\pi$
$127$	− 1.67883e7i	− 0.727267i	0.931542	−	0.363633i	$-0.118464\pi$
$128$	2.06342e7i	0.869668i
$129$	0	0
$130$	0	0
$131$	−1.68268e7	−0.653960	−0.326980	−	0.945031i	$-0.606031\pi$
$131$	−1.68268e7	−0.653960	−0.326980	+	0.945031i	$0.606031\pi$
$132$	0	0
$133$	551680.i	0.0203332i
$134$	−3.04346e6	−0.109270
$135$	0	0
$136$	−3.74695e7	−1.27730
$137$	2.80449e7i	0.931820i	0.884832	+	0.465910i	$0.154273\pi$
$137$	2.80449e7i	0.931820i	−0.884832	+	0.465910i	$0.845727\pi$
$138$	0	0
$139$	1.18273e7	0.373537	0.186769	−	0.982404i	$-0.440199\pi$
$139$	1.18273e7	0.373537	0.186769	+	0.982404i	$0.440199\pi$
$140$	0	0
$141$	0	0
$142$	− 3.33638e7i	− 0.977836i
$143$	− 4.83290e6i	− 0.138208i
$144$	0	0
$145$	0	0
$146$	−8.21449e6	−0.218447
$147$	0	0
$148$	− 2.47044e7i	− 0.626409i
$149$	2.07846e7	0.514743	0.257371	−	0.966313i	$-0.417144\pi$
$149$	2.07846e7	0.514743	0.257371	+	0.966313i	$0.417144\pi$
$150$	0	0
$151$	76112.0	0.00179901	0.000899505	−	1.00000i	$-0.499714\pi$
$151$	76112.0	0.00179901	0.000899505		1.00000i	$0.499714\pi$
$152$	1.13784e7i	0.262802i
$153$	0	0
$154$	−364032.	−0.00803188
$155$	0	0
$156$	0	0
$157$	3.21825e7i	0.663698i	0.943332	+	0.331849i	$0.107672\pi$
$157$	3.21825e7i	0.663698i	−0.943332	+	0.331849i	$0.892328\pi$
$158$	4.14823e7i	0.836687i
$159$	0	0
$160$	0	0
$161$	−978432.	−0.0184774
$162$	0	0
$163$	5.83435e7i	1.05520i	0.849492	+	0.527601i	$0.176908\pi$
$163$	5.83435e7i	1.05520i	−0.849492	+	0.527601i	$0.823092\pi$
$164$	5.79341e7	1.02561
$165$	0	0
$166$	−2.62605e7	−0.445579
$167$	− 2.58365e7i	− 0.429266i	−0.976695	−	0.214633i	$-0.931145\pi$
$167$	− 2.58365e7i	− 0.429266i	0.976695	−	0.214633i	$-0.0688555\pi$
$168$	0	0
$169$	3.67589e7	0.585813
$170$	0	0
$171$	0	0
$172$	6.30910e7i	0.945405i
$173$	− 6.35201e7i	− 0.932716i	−0.884596	−	0.466358i	$-0.845566\pi$
$173$	− 6.35201e7i	− 0.932716i	0.884596	−	0.466358i	$-0.154434\pi$
$174$	0	0
$175$	0	0
$176$	3.65549e6	0.0505418
$177$	0	0
$178$	− 5.11699e7i	− 0.680055i
$179$	−8.09559e7	−1.05503	−0.527513	−	0.849547i	$-0.676875\pi$
$179$	−8.09559e7	−1.05503	−0.527513	+	0.849547i	$0.676875\pi$
$180$	0	0
$181$	6.45032e7	0.808549	0.404274	−	0.914638i	$-0.367524\pi$
$181$	6.45032e7	0.808549	0.404274	+	0.914638i	$0.367524\pi$
$182$	1.95763e6i	0.0240703i
$183$	0	0
$184$	−2.01802e7	−0.238815
$185$	0	0
$186$	0	0
$187$	2.69099e7i	0.300931i
$188$	5.36632e7i	0.589012i
$189$	0	0
$190$	0	0
$191$	−5.68274e7	−0.590121	−0.295060	−	0.955479i	$-0.595340\pi$
$191$	−5.68274e7	−0.590121	−0.295060	+	0.955479i	$0.595340\pi$
$192$	0	0
$193$	1.16377e8i	1.16524i	0.812744	+	0.582621i	$0.197973\pi$
$193$	1.16377e8i	1.16524i	−0.812744	+	0.582621i	$0.802027\pi$
$194$	−5.29609e7	−0.520774
$195$	0	0
$196$	7.53891e7	0.715175
$197$	− 1.18816e8i	− 1.10724i	−0.832768	−	0.553622i	$-0.813245\pi$
$197$	− 1.18816e8i	− 1.10724i	0.832768	−	0.553622i	$-0.186755\pi$
$198$	0	0
$199$	9.50106e7	0.854646	0.427323	−	0.904099i	$-0.359457\pi$
$199$	9.50106e7	0.854646	0.427323	+	0.904099i	$0.359457\pi$
$200$	0	0
$201$	0	0
$202$	− 7.19185e7i	− 0.613919i
$203$	2.33664e6i	0.0196045i
$204$	0	0
$205$	0	0
$206$	−4.32564e7	−0.344758
$207$	0	0
$208$	− 1.96579e7i	− 0.151466i
$209$	8.17176e6	0.0619161
$210$	0	0
$211$	1.79246e8	1.31360	0.656798	−	0.754067i	$-0.271910\pi$
$211$	1.79246e8	1.31360	0.656798	+	0.754067i	$0.271910\pi$
$212$	3.93813e7i	0.283867i
$213$	0	0
$214$	−6.85565e7	−0.478190
$215$	0	0
$216$	0	0
$217$	− 1.77157e7i	− 0.117693i
$218$	− 2.41257e7i	− 0.157718i
$219$	0	0
$220$	0	0
$221$	1.44712e8	0.901843
$222$	0	0
$223$	− 2.06537e8i	− 1.24718i	−0.781750	−	0.623592i	$-0.785673\pi$
$223$	− 2.06537e8i	− 1.24718i	0.781750	−	0.623592i	$-0.214327\pi$
$224$	−1.22941e7	−0.0730853
$225$	0	0
$226$	−1.06238e8	−0.612209
$227$	4.33954e7i	0.246237i	0.992392	+	0.123118i	$0.0392895\pi$
$227$	4.33954e7i	0.246237i	−0.992392	+	0.123118i	$0.960710\pi$
$228$	0	0
$229$	3.61931e7	0.199160	0.0995799	−	0.995030i	$-0.468250\pi$
$229$	3.61931e7	0.199160	0.0995799	+	0.995030i	$0.468250\pi$
$230$	0	0
$231$	0	0
$232$	4.81932e7i	0.253383i
$233$	− 9.22347e7i	− 0.477693i	−0.971057	−	0.238846i	$-0.923231\pi$
$233$	− 9.22347e7i	− 0.477693i	0.971057	−	0.238846i	$-0.0767692\pi$
$234$	0	0
$235$	0	0
$236$	1.20187e8	0.595203
$237$	0	0
$238$	− 1.09002e7i	− 0.0524102i
$239$	4.98468e7	0.236181	0.118090	−	0.993003i	$-0.462323\pi$
$239$	4.98468e7	0.236181	0.118090	+	0.993003i	$0.462323\pi$
$240$	0	0
$241$	1.99374e8	0.917506	0.458753	−	0.888564i	$-0.348296\pi$
$241$	1.99374e8	0.917506	0.458753	+	0.888564i	$0.348296\pi$
$242$	− 1.11531e8i	− 0.505872i
$243$	0	0
$244$	2.76609e7	0.121900
$245$	0	0
$246$	0	0
$247$	− 4.39448e7i	− 0.185553i
$248$	− 3.65387e8i	− 1.52115i
$249$	0	0
$250$	0	0
$251$	3.94678e8	1.57538	0.787689	−	0.616073i	$-0.211277\pi$
$251$	3.94678e8	1.57538	0.787689	+	0.616073i	$0.211277\pi$
$252$	0	0
$253$	1.44930e7i	0.0562649i
$254$	1.00730e8	0.385691
$255$	0	0
$256$	−2.08158e8	−0.775451
$257$	− 1.42885e8i	− 0.525076i	−0.964922	−	0.262538i	$-0.915441\pi$
$257$	− 1.42885e8i	− 0.525076i	0.964922	−	0.262538i	$-0.0845594\pi$
$258$	0	0
$259$	1.71857e7	0.0614635
$260$	0	0
$261$	0	0
$262$	− 1.00961e8i	− 0.346815i
$263$	− 4.40241e8i	− 1.49226i	−0.665799	−	0.746131i	$-0.731909\pi$
$263$	− 4.40241e8i	− 1.49226i	0.665799	−	0.746131i	$-0.268091\pi$
$264$	0	0
$265$	0	0
$266$	−3.31008e6	−0.0107833
$267$	0	0
$268$	4.66664e7i	0.148092i
$269$	2.75405e8	0.862657	0.431329	−	0.902195i	$-0.358045\pi$
$269$	2.75405e8	0.862657	0.431329	+	0.902195i	$0.358045\pi$
$270$	0	0
$271$	−4.24670e8	−1.29616	−0.648080	−	0.761572i	$-0.724428\pi$
$271$	−4.24670e8	−1.29616	−0.648080	+	0.761572i	$0.724428\pi$
$272$	1.09456e8i	0.329800i
$273$	0	0
$274$	−1.68269e8	−0.494172
$275$	0	0
$276$	0	0
$277$	− 5.16158e8i	− 1.45916i	−0.683894	−	0.729581i	$-0.739715\pi$
$277$	− 5.16158e8i	− 1.45916i	0.683894	−	0.729581i	$-0.260285\pi$
$278$	7.09638e7i	0.198098i
$279$	0	0
$280$	0	0
$281$	3.11043e8	0.836273	0.418137	−	0.908384i	$-0.362683\pi$
$281$	3.11043e8	0.836273	0.418137	+	0.908384i	$0.362683\pi$
$282$	0	0
$283$	− 5.94308e8i	− 1.55869i	−0.626596	−	0.779344i	$-0.715552\pi$
$283$	− 5.94308e8i	− 1.55869i	0.626596	−	0.779344i	$-0.284448\pi$
$284$	−5.11578e8	−1.32525
$285$	0	0
$286$	2.89974e7	0.0732957
$287$	4.03020e7i	0.100633i
$288$	0	0
$289$	−3.95426e8	−0.963658
$290$	0	0
$291$	0	0
$292$	1.25956e8i	0.296058i
$293$	− 1.15515e8i	− 0.268288i	−0.990962	−	0.134144i	$-0.957172\pi$
$293$	− 1.15515e8i	− 0.268288i	0.990962	−	0.134144i	$-0.0428284\pi$
$294$	0	0
$295$	0	0
$296$	3.54454e8	0.794400
$297$	0	0
$298$	1.24708e8i	0.272984i
$299$	7.79382e7	0.168617
$300$	0	0
$301$	−4.38894e7	−0.0927635
$302$	456672.i	0 0.000954070i
$303$	0	0
$304$	3.32387e7	0.0678558
$305$	0	0
$306$	0	0
$307$	2.60600e8i	0.514032i	0.966407	+	0.257016i	$0.0827392\pi$
$307$	2.60600e8i	0.514032i	−0.966407	+	0.257016i	$0.917261\pi$
$308$	5.58182e6i	0.0108855i
$309$	0	0
$310$	0	0
$311$	−5.76795e8	−1.08733	−0.543663	−	0.839303i	$-0.682963\pi$
$311$	−5.76795e8	−1.08733	−0.543663	+	0.839303i	$0.682963\pi$
$312$	0	0
$313$	− 4.60074e8i	− 0.848053i	−0.905650	−	0.424026i	$-0.860616\pi$
$313$	− 4.60074e8i	− 0.848053i	0.905650	−	0.424026i	$-0.139384\pi$
$314$	−1.93095e8	−0.351979
$315$	0	0
$316$	6.36062e8	1.13395
$317$	6.25561e7i	0.110297i	0.998478	+	0.0551483i	$0.0175632\pi$
$317$	6.25561e7i	0.110297i	−0.998478	+	0.0551483i	$0.982437\pi$
$318$	0	0
$319$	3.46115e7	0.0596970
$320$	0	0
$321$	0	0
$322$	− 5.87059e6i	− 0.00979910i
$323$	2.44687e8i	0.404020i
$324$	0	0
$325$	0	0
$326$	−3.50061e8	−0.559606
$327$	0	0
$328$	8.31228e8i	1.30065i
$329$	−3.73309e7	−0.0577941
$330$	0	0
$331$	6.84236e8	1.03707	0.518535	−	0.855057i	$-0.326478\pi$
$331$	6.84236e8	1.03707	0.518535	+	0.855057i	$0.326478\pi$
$332$	4.02661e8i	0.603888i
$333$	0	0
$334$	1.55019e8	0.227652
$335$	0	0
$336$	0	0
$337$	6.26313e8i	0.891429i	0.895175	+	0.445714i	$0.147050\pi$
$337$	6.26313e8i	0.891429i	−0.895175	+	0.445714i	$0.852950\pi$
$338$	2.20553e8i	0.310674i
$339$	0	0
$340$	0	0
$341$	−2.62414e8	−0.358382
$342$	0	0
$343$	1.05151e8i	0.140697i
$344$	−9.05219e8	−1.19894
$345$	0	0
$346$	3.81120e8	0.494647
$347$	− 1.25340e9i	− 1.61041i	−0.593000	−	0.805203i	$-0.702057\pi$
$347$	− 1.25340e9i	− 1.61041i	0.593000	−	0.805203i	$-0.297943\pi$
$348$	0	0
$349$	−2.65350e8	−0.334142	−0.167071	−	0.985945i	$-0.553431\pi$
$349$	−2.65350e8	−0.334142	−0.167071	+	0.985945i	$0.553431\pi$
$350$	0	0
$351$	0	0
$352$	1.82107e8i	0.222550i
$353$	5.69636e8i	0.689264i	0.938738	+	0.344632i	$0.111996\pi$
$353$	5.69636e8i	0.689264i	−0.938738	+	0.344632i	$0.888004\pi$
$354$	0	0
$355$	0	0
$356$	−7.84605e8	−0.921671
$357$	0	0
$358$	− 4.85735e8i	− 0.559512i
$359$	9.32541e8	1.06374	0.531872	−	0.846825i	$-0.321489\pi$
$359$	9.32541e8	1.06374	0.531872	+	0.846825i	$0.321489\pi$
$360$	0	0
$361$	−8.19567e8	−0.916874
$362$	3.87019e8i	0.428798i
$363$	0	0
$364$	3.00170e7	0.0326222
$365$	0	0
$366$	0	0
$367$	8.52565e8i	0.900318i	0.892948	+	0.450159i	$0.148633\pi$
$367$	8.52565e8i	0.900318i	−0.892948	+	0.450159i	$0.851367\pi$
$368$	5.89505e7i	0.0616624i
$369$	0	0
$370$	0	0
$371$	−2.73957e7	−0.0278531
$372$	0	0
$373$	3.81183e8i	0.380323i	0.981753	+	0.190162i	$0.0609012\pi$
$373$	3.81183e8i	0.380323i	−0.981753	+	0.190162i	$0.939099\pi$
$374$	−1.61460e8	−0.159593
$375$	0	0
$376$	−7.69951e8	−0.746974
$377$	− 1.86128e8i	− 0.178903i
$378$	0	0
$379$	1.48353e9	1.39978	0.699889	−	0.714251i	$-0.253233\pi$
$379$	1.48353e9	1.39978	0.699889	+	0.714251i	$0.253233\pi$
$380$	0	0
$381$	0	0
$382$	− 3.40964e8i	− 0.312959i
$383$	7.61930e8i	0.692978i	0.938054	+	0.346489i	$0.112626\pi$
$383$	7.61930e8i	0.692978i	−0.938054	+	0.346489i	$0.887374\pi$
$384$	0	0
$385$	0	0
$386$	−6.98262e8	−0.617963
$387$	0	0
$388$	8.12067e8i	0.705799i
$389$	1.60902e9	1.38592	0.692959	−	0.720977i	$-0.256307\pi$
$389$	1.60902e9	1.38592	0.692959	+	0.720977i	$0.256307\pi$
$390$	0	0
$391$	−4.33965e8	−0.367144
$392$	1.08167e9i	0.906971i
$393$	0	0
$394$	7.12896e8	0.587205
$395$	0	0
$396$	0	0
$397$	− 1.88016e9i	− 1.50809i	−0.656822	−	0.754046i	$-0.728100\pi$
$397$	− 1.88016e9i	− 1.50809i	0.656822	−	0.754046i	$-0.271900\pi$
$398$	5.70064e8i	0.453245i
$399$	0	0
$400$	0	0
$401$	−2.68592e8	−0.208012	−0.104006	−	0.994577i	$-0.533166\pi$
$401$	−2.68592e8	−0.208012	−0.104006	+	0.994577i	$0.533166\pi$
$402$	0	0
$403$	1.41117e9i	1.07402i
$404$	−1.10275e9	−0.832037
$405$	0	0
$406$	−1.40198e7	−0.0103969
$407$	− 2.54563e8i	− 0.187161i
$408$	0	0
$409$	−8.99478e7	−0.0650069	−0.0325034	−	0.999472i	$-0.510348\pi$
$409$	−8.99478e7	−0.0650069	−0.0325034	+	0.999472i	$0.510348\pi$
$410$	0	0
$411$	0	0
$412$	6.63264e8i	0.467247i
$413$	8.36083e7i	0.0584015i
$414$	0	0
$415$	0	0
$416$	9.79305e8	0.666947
$417$	0	0
$418$	4.90306e7i	0.0328360i
$419$	1.69054e9	1.12273	0.561367	−	0.827567i	$-0.310276\pi$
$419$	1.69054e9	1.12273	0.561367	+	0.827567i	$0.310276\pi$
$420$	0	0
$421$	−1.13333e9	−0.740232	−0.370116	−	0.928985i	$-0.620682\pi$
$421$	−1.13333e9	−0.740232	−0.370116	+	0.928985i	$0.620682\pi$
$422$	1.07548e9i	0.696639i
$423$	0	0
$424$	−5.65037e8	−0.359995
$425$	0	0
$426$	0	0
$427$	1.92424e7i	0.0119608i
$428$	1.05120e9i	0.648085i
$429$	0	0
$430$	0	0
$431$	−2.19943e9	−1.32324	−0.661621	−	0.749839i	$-0.730131\pi$
$431$	−2.19943e9	−1.32324	−0.661621	+	0.749839i	$0.730131\pi$
$432$	0	0
$433$	− 1.51738e8i	− 0.0898227i	−0.998991	−	0.0449114i	$-0.985699\pi$
$433$	− 1.51738e8i	− 0.0898227i	0.998991	−	0.0449114i	$-0.0143005\pi$
$434$	1.06294e8	0.0624160
$435$	0	0
$436$	−3.69927e8	−0.213754
$437$	1.31783e8i	0.0755393i
$438$	0	0
$439$	−9.90763e8	−0.558912	−0.279456	−	0.960158i	$-0.590154\pi$
$439$	−9.90763e8	−0.558912	−0.279456	+	0.960158i	$0.590154\pi$
$440$	0	0
$441$	0	0
$442$	8.68271e8i	0.478275i
$443$	1.77376e9i	0.969351i	0.874694	+	0.484675i	$0.161062\pi$
$443$	1.77376e9i	0.969351i	−0.874694	+	0.484675i	$0.838938\pi$
$444$	0	0
$445$	0	0
$446$	1.23922e9	0.661419
$447$	0	0
$448$	− 4.21765e7i	− 0.0221614i
$449$	−2.77010e8	−0.144422	−0.0722110	−	0.997389i	$-0.523006\pi$
$449$	−2.77010e8	−0.144422	−0.0722110	+	0.997389i	$0.523006\pi$
$450$	0	0
$451$	5.96973e8	0.306434
$452$	1.62898e9i	0.829720i
$453$	0	0
$454$	−2.60372e8	−0.130587
$455$	0	0
$456$	0	0
$457$	− 2.94758e9i	− 1.44464i	−0.691559	−	0.722320i	$-0.743076\pi$
$457$	− 2.94758e9i	− 1.44464i	0.691559	−	0.722320i	$-0.256924\pi$
$458$	2.17159e8i	0.105620i
$459$	0	0
$460$	0	0
$461$	2.76687e9	1.31533	0.657667	−	0.753309i	$-0.271543\pi$
$461$	2.76687e9	1.31533	0.657667	+	0.753309i	$0.271543\pi$
$462$	0	0
$463$	4.63553e8i	0.217053i	0.994094	+	0.108527i	$0.0346132\pi$
$463$	4.63553e8i	0.217053i	−0.994094	+	0.108527i	$0.965387\pi$
$464$	1.40783e8	0.0654238
$465$	0	0
$466$	5.53408e8	0.253335
$467$	− 4.17922e8i	− 0.189883i	−0.995483	−	0.0949415i	$-0.969734\pi$
$467$	− 4.17922e8i	− 0.189883i	0.995483	−	0.0949415i	$-0.0302664\pi$
$468$	0	0
$469$	−3.24636e7	−0.0145309
$470$	0	0
$471$	0	0
$472$	1.72442e9i	0.754825i
$473$	6.50112e8i	0.282471i
$474$	0	0
$475$	0	0
$476$	−1.67137e8	−0.0710310
$477$	0	0
$478$	2.99081e8i	0.125254i
$479$	−1.50973e9	−0.627660	−0.313830	−	0.949479i	$-0.601612\pi$
$479$	−1.50973e9	−0.627660	−0.313830	+	0.949479i	$0.601612\pi$
$480$	0	0
$481$	−1.36895e9	−0.560891
$482$	1.19624e9i	0.486581i
$483$	0	0
$484$	−1.71014e9	−0.685603
$485$	0	0
$486$	0	0
$487$	− 9.29460e8i	− 0.364653i	−0.983238	−	0.182326i	$-0.941637\pi$
$487$	− 9.29460e8i	− 0.364653i	0.983238	−	0.182326i	$-0.0583627\pi$
$488$	3.96874e8i	0.154591i
$489$	0	0
$490$	0	0
$491$	−5.12803e9	−1.95508	−0.977541	−	0.210743i	$-0.932412\pi$
$491$	−5.12803e9	−1.95508	−0.977541	+	0.210743i	$0.932412\pi$
$492$	0	0
$493$	1.03637e9i	0.389540i
$494$	2.63669e8	0.0984043
$495$	0	0
$496$	−1.06737e9	−0.392762
$497$	− 3.55880e8i	− 0.130034i
$498$	0	0
$499$	4.10649e8	0.147951	0.0739757	−	0.997260i	$-0.476431\pi$
$499$	4.10649e8	0.147951	0.0739757	+	0.997260i	$0.476431\pi$
$500$	0	0
$501$	0	0
$502$	2.36807e9i	0.835470i
$503$	− 5.02041e9i	− 1.75894i	−0.475954	−	0.879470i	$-0.657897\pi$
$503$	− 5.02041e9i	− 1.75894i	0.475954	−	0.879470i	$-0.342103\pi$
$504$	0	0
$505$	0	0
$506$	−8.69581e7	−0.0298389
$507$	0	0
$508$	− 1.54452e9i	− 0.522723i
$509$	−3.24926e9	−1.09212	−0.546062	−	0.837745i	$-0.683874\pi$
$509$	−3.24926e9	−1.09212	−0.546062	+	0.837745i	$0.683874\pi$
$510$	0	0
$511$	−8.76212e7	−0.0290493
$512$	1.39223e9i	0.458423i
$513$	0	0
$514$	8.57312e8	0.278463
$515$	0	0
$516$	0	0
$517$	5.52965e8i	0.175987i
$518$	1.03114e8i	0.0325959i
$519$	0	0
$520$	0	0
$521$	2.10950e9	0.653503	0.326752	−	0.945110i	$-0.394046\pi$
$521$	2.10950e9	0.653503	0.326752	+	0.945110i	$0.394046\pi$
$522$	0	0
$523$	− 5.28911e9i	− 1.61669i	−0.588709	−	0.808345i	$-0.700364\pi$
$523$	− 5.28911e9i	− 1.61669i	0.588709	−	0.808345i	$-0.299636\pi$
$524$	−1.54806e9	−0.470034
$525$	0	0
$526$	2.64144e9	0.791391
$527$	− 7.85747e9i	− 2.33854i
$528$	0	0
$529$	3.17110e9	0.931355
$530$	0	0
$531$	0	0
$532$	5.07546e7i	0.0146145i
$533$	− 3.21030e9i	− 0.918334i
$534$	0	0
$535$	0	0
$536$	−6.69562e8	−0.187808
$537$	0	0
$538$	1.65243e9i	0.457493i
$539$	7.76836e8	0.213682
$540$	0	0
$541$	3.04614e9	0.827101	0.413551	−	0.910481i	$-0.364288\pi$
$541$	3.04614e9	0.827101	0.413551	+	0.910481i	$0.364288\pi$
$542$	− 2.54802e9i	− 0.687393i
$543$	0	0
$544$	−5.45284e9	−1.45220
$545$	0	0
$546$	0	0
$547$	4.85537e9i	1.26843i	0.773157	+	0.634215i	$0.218677\pi$
$547$	4.85537e9i	1.26843i	−0.773157	+	0.634215i	$0.781323\pi$
$548$	2.58013e9i	0.669746i
$549$	0	0
$550$	0	0
$551$	3.14716e8	0.0801472
$552$	0	0
$553$	4.42478e8i	0.111264i
$554$	3.09695e9	0.773838
$555$	0	0
$556$	1.08811e9	0.268480
$557$	1.27762e9i	0.313263i	0.987657	+	0.156631i	$0.0500635\pi$
$557$	1.27762e9i	0.313263i	−0.987657	+	0.156631i	$0.949937\pi$
$558$	0	0
$559$	3.49607e9	0.846522
$560$	0	0
$561$	0	0
$562$	1.86626e9i	0.443501i
$563$	− 4.71265e9i	− 1.11297i	−0.830856	−	0.556487i	$-0.812149\pi$
$563$	− 4.71265e9i	− 1.11297i	0.830856	−	0.556487i	$-0.187851\pi$
$564$	0	0
$565$	0	0
$566$	3.56585e9	0.826619
$567$	0	0
$568$	− 7.34003e9i	− 1.68066i
$569$	4.57800e9	1.04180	0.520898	−	0.853619i	$-0.325597\pi$
$569$	4.57800e9	1.04180	0.520898	+	0.853619i	$0.325597\pi$
$570$	0	0
$571$	4.95119e9	1.11297	0.556485	−	0.830858i	$-0.312150\pi$
$571$	4.95119e9	1.11297	0.556485	+	0.830858i	$0.312150\pi$
$572$	− 4.44627e8i	− 0.0993367i
$573$	0	0
$574$	−2.41812e8	−0.0533686
$575$	0	0
$576$	0	0
$577$	− 8.51847e9i	− 1.84606i	−0.384725	−	0.923031i	$-0.625704\pi$
$577$	− 8.51847e9i	− 1.84606i	0.384725	−	0.923031i	$-0.374296\pi$
$578$	− 2.37256e9i	− 0.511057i
$579$	0	0
$580$	0	0
$581$	−2.80112e8	−0.0592536
$582$	0	0
$583$	4.05799e8i	0.0848147i
$584$	−1.80719e9	−0.375455
$585$	0	0
$586$	6.93088e8	0.142281
$587$	− 5.62247e8i	− 0.114735i	−0.998353	−	0.0573673i	$-0.981729\pi$
$587$	− 5.62247e8i	− 0.114735i	0.998353	−	0.0573673i	$-0.0182706\pi$
$588$	0	0
$589$	−2.38608e9	−0.481152
$590$	0	0
$591$	0	0
$592$	− 1.03544e9i	− 0.205115i
$593$	− 3.62110e9i	− 0.713099i	−0.934277	−	0.356549i	$-0.883953\pi$
$593$	− 3.62110e9i	− 0.713099i	0.934277	−	0.356549i	$-0.116047\pi$
$594$	0	0
$595$	0	0
$596$	1.91219e9	0.369971
$597$	0	0
$598$	4.67629e8i	0.0894227i
$599$	−7.48104e9	−1.42222	−0.711112	−	0.703079i	$-0.751808\pi$
$599$	−7.48104e9	−1.42222	−0.711112	+	0.703079i	$0.751808\pi$
$600$	0	0
$601$	−5.81270e9	−1.09224	−0.546119	−	0.837707i	$-0.683895\pi$
$601$	−5.81270e9	−1.09224	−0.546119	+	0.837707i	$0.683895\pi$
$602$	− 2.63336e8i	− 0.0491953i
$603$	0	0
$604$	7.00230e6	0.00129304
$605$	0	0
$606$	0	0
$607$	− 3.84051e9i	− 0.696993i	−0.937310	−	0.348497i	$-0.886692\pi$
$607$	− 3.84051e9i	− 0.696993i	0.937310	−	0.348497i	$-0.113308\pi$
$608$	1.65587e9i	0.298788i
$609$	0	0
$610$	0	0
$611$	2.97364e9	0.527405
$612$	0	0
$613$	1.70484e9i	0.298932i	0.988767	+	0.149466i	$0.0477555\pi$
$613$	1.70484e9i	0.298932i	−0.988767	+	0.149466i	$0.952245\pi$
$614$	−1.56360e9	−0.272606
$615$	0	0
$616$	−8.00870e7	−0.0138048
$617$	− 2.80809e9i	− 0.481297i	−0.970612	−	0.240649i	$-0.922640\pi$
$617$	− 2.80809e9i	− 0.481297i	0.970612	−	0.240649i	$-0.0773601\pi$
$618$	0	0
$619$	2.54365e9	0.431063	0.215532	−	0.976497i	$-0.430852\pi$
$619$	2.54365e9	0.431063	0.215532	+	0.976497i	$0.430852\pi$
$620$	0	0
$621$	0	0
$622$	− 3.46077e9i	− 0.576642i
$623$	− 5.45812e8i	− 0.0904346i
$624$	0	0
$625$	0	0
$626$	2.76045e9	0.449748
$627$	0	0
$628$	2.96079e9i	0.477033i
$629$	7.62238e9	1.22127
$630$	0	0
$631$	−1.51146e8	−0.0239494	−0.0119747	−	0.999928i	$-0.503812\pi$
$631$	−1.51146e8	−0.0239494	−0.0119747	+	0.999928i	$0.503812\pi$
$632$	9.12611e9i	1.43806i
$633$	0	0
$634$	−3.75337e8	−0.0584936
$635$	0	0
$636$	0	0
$637$	− 4.17754e9i	− 0.640373i
$638$	2.07669e8i	0.0316591i
$639$	0	0
$640$	0	0
$641$	1.23625e10	1.85397	0.926987	−	0.375094i	$-0.122390\pi$
$641$	1.23625e10	1.85397	0.926987	+	0.375094i	$0.122390\pi$
$642$	0	0
$643$	2.86744e9i	0.425359i	0.977122	+	0.212680i	$0.0682191\pi$
$643$	2.86744e9i	0.425359i	−0.977122	+	0.212680i	$0.931781\pi$
$644$	−9.00157e7	−0.0132806
$645$	0	0
$646$	−1.46812e9	−0.214264
$647$	− 4.10640e9i	− 0.596068i	−0.954555	−	0.298034i	$-0.903669\pi$
$647$	− 4.10640e9i	− 0.596068i	0.954555	−	0.298034i	$-0.0963309\pi$
$648$	0	0
$649$	1.23845e9	0.177837
$650$	0	0
$651$	0	0
$652$	5.36760e9i	0.758427i
$653$	− 6.91100e9i	− 0.971280i	−0.874159	−	0.485640i	$-0.838587\pi$
$653$	− 6.91100e9i	− 0.971280i	0.874159	−	0.485640i	$-0.161413\pi$
$654$	0	0
$655$	0	0
$656$	2.42819e9	0.335830
$657$	0	0
$658$	− 2.23986e8i	− 0.0306499i
$659$	3.42444e9	0.466112	0.233056	−	0.972463i	$-0.425127\pi$
$659$	3.42444e9	0.466112	0.233056	+	0.972463i	$0.425127\pi$
$660$	0	0
$661$	−6.76437e9	−0.911008	−0.455504	−	0.890234i	$-0.650541\pi$
$661$	−6.76437e9	−0.911008	−0.455504	+	0.890234i	$0.650541\pi$
$662$	4.10541e9i	0.549989i
$663$	0	0
$664$	−5.77731e9	−0.765839
$665$	0	0
$666$	0	0
$667$	5.58165e8i	0.0728320i
$668$	− 2.37696e9i	− 0.308535i
$669$	0	0
$670$	0	0
$671$	2.85028e8	0.0364215
$672$	0	0
$673$	− 1.74959e9i	− 0.221250i	−0.993862	−	0.110625i	$-0.964715\pi$
$673$	− 1.74959e9i	− 0.221250i	0.993862	−	0.110625i	$-0.0352853\pi$
$674$	−3.75788e9	−0.472752
$675$	0	0
$676$	3.38182e9	0.421053
$677$	8.30011e9i	1.02807i	0.857769	+	0.514036i	$0.171850\pi$
$677$	8.30011e9i	1.02807i	−0.857769	+	0.514036i	$0.828150\pi$
$678$	0	0
$679$	−5.64916e8	−0.0692532
$680$	0	0
$681$	0	0
$682$	− 1.57448e9i	− 0.190061i
$683$	1.21232e10i	1.45594i	0.685610	+	0.727969i	$0.259536\pi$
$683$	1.21232e10i	1.45594i	−0.685610	+	0.727969i	$0.740464\pi$
$684$	0	0
$685$	0	0
$686$	−6.30908e8	−0.0746160
$687$	0	0
$688$	2.64434e9i	0.309569i
$689$	2.18224e9	0.254176
$690$	0	0
$691$	8.21846e9	0.947583	0.473791	−	0.880637i	$-0.342885\pi$
$691$	8.21846e9	0.947583	0.473791	+	0.880637i	$0.342885\pi$
$692$	− 5.84385e9i	− 0.670390i
$693$	0	0
$694$	7.52038e9	0.854046
$695$	0	0
$696$	0	0
$697$	1.78752e10i	1.99957i
$698$	− 1.59210e9i	− 0.177205i
$699$	0	0
$700$	0	0
$701$	−4.72231e9	−0.517775	−0.258888	−	0.965907i	$-0.583356\pi$
$701$	−4.72231e9	−0.517775	−0.258888	+	0.965907i	$0.583356\pi$
$702$	0	0
$703$	− 2.31469e9i	− 0.251275i
$704$	−6.24740e8	−0.0674831
$705$	0	0
$706$	−3.41781e9	−0.365537
$707$	− 7.67131e8i	− 0.0816397i
$708$	0	0
$709$	−2.78975e9	−0.293970	−0.146985	−	0.989139i	$-0.546957\pi$
$709$	−2.78975e9	−0.293970	−0.146985	+	0.989139i	$0.546957\pi$
$710$	0	0
$711$	0	0
$712$	− 1.12574e10i	− 1.16885i
$713$	− 4.23184e9i	− 0.437236i
$714$	0	0
$715$	0	0
$716$	−7.44794e9	−0.758299
$717$	0	0
$718$	5.59524e9i	0.564136i
$719$	1.51985e9	0.152493	0.0762463	−	0.997089i	$-0.475706\pi$
$719$	1.51985e9	0.152493	0.0762463	+	0.997089i	$0.475706\pi$
$720$	0	0
$721$	−4.61401e8	−0.0458464
$722$	− 4.91740e9i	− 0.486246i
$723$	0	0
$724$	5.93429e9	0.581144
$725$	0	0
$726$	0	0
$727$	8.11761e9i	0.783534i	0.920065	+	0.391767i	$0.128136\pi$
$727$	8.11761e9i	0.783534i	−0.920065	+	0.391767i	$0.871864\pi$
$728$	4.30679e8i	0.0413708i
$729$	0	0
$730$	0	0
$731$	−1.94663e10	−1.84320
$732$	0	0
$733$	− 1.03241e10i	− 0.968249i	−0.874999	−	0.484124i	$-0.839138\pi$
$733$	− 1.03241e10i	− 0.968249i	0.874999	−	0.484124i	$-0.160862\pi$
$734$	−5.11539e9	−0.477466
$735$	0	0
$736$	−2.93676e9	−0.271517
$737$	4.80867e8i	0.0442475i
$738$	0	0
$739$	1.35365e10	1.23382	0.616908	−	0.787035i	$-0.288385\pi$
$739$	1.35365e10	1.23382	0.616908	+	0.787035i	$0.288385\pi$
$740$	0	0
$741$	0	0
$742$	− 1.64374e8i	− 0.0147713i
$743$	1.71936e10i	1.53782i	0.639356	+	0.768910i	$0.279201\pi$
$743$	1.71936e10i	1.53782i	−0.639356	+	0.768910i	$0.720799\pi$
$744$	0	0
$745$	0	0
$746$	−2.28710e9	−0.201697
$747$	0	0
$748$	2.47571e9i	0.216294i
$749$	−7.31269e8	−0.0635903
$750$	0	0
$751$	1.12478e10	0.969013	0.484506	−	0.874788i	$-0.338999\pi$
$751$	1.12478e10	0.969013	0.484506	+	0.874788i	$0.338999\pi$
$752$	2.24919e9i	0.192870i
$753$	0	0
$754$	1.11677e9	0.0948775
$755$	0	0
$756$	0	0
$757$	− 1.63068e10i	− 1.36626i	−0.730296	−	0.683131i	$-0.760618\pi$
$757$	− 1.63068e10i	− 1.36626i	0.730296	−	0.683131i	$-0.239382\pi$
$758$	8.90118e9i	0.742345i
$759$	0	0
$760$	0	0
$761$	−6.14069e9	−0.505093	−0.252546	−	0.967585i	$-0.581268\pi$
$761$	−6.14069e9	−0.505093	−0.252546	+	0.967585i	$0.581268\pi$
$762$	0	0
$763$	− 2.57341e8i	− 0.0209736i
$764$	−5.22812e9	−0.424149
$765$	0	0
$766$	−4.57158e9	−0.367507
$767$	− 6.65993e9i	− 0.532949i
$768$	0	0
$769$	−2.45069e10	−1.94333	−0.971664	−	0.236368i	$-0.924043\pi$
$769$	−2.45069e10	−1.94333	−0.971664	+	0.236368i	$0.924043\pi$
$770$	0	0
$771$	0	0
$772$	1.07067e10i	0.837518i
$773$	1.01722e10i	0.792110i	0.918227	+	0.396055i	$0.129621\pi$
$773$	1.01722e10i	0.792110i	−0.918227	+	0.396055i	$0.870379\pi$
$774$	0	0
$775$	0	0
$776$	−1.16514e10	−0.895080
$777$	0	0
$778$	9.65411e9i	0.734994i
$779$	5.42817e9	0.411408
$780$	0	0
$781$	−5.27148e9	−0.395962
$782$	− 2.60379e9i	− 0.194707i
$783$	0	0
$784$	3.15979e9	0.234181
$785$	0	0
$786$	0	0
$787$	9.79135e9i	0.716030i	0.933716	+	0.358015i	$0.116546\pi$
$787$	9.79135e9i	0.716030i	−0.933716	+	0.358015i	$0.883454\pi$
$788$	− 1.09311e10i	− 0.795832i
$789$	0	0
$790$	0	0
$791$	−1.13320e9	−0.0814124
$792$	0	0
$793$	− 1.53277e9i	− 0.109150i
$794$	1.12809e10	0.799786
$795$	0	0
$796$	8.74098e9	0.614277
$797$	− 9.75782e9i	− 0.682729i	−0.939931	−	0.341365i	$-0.889111\pi$
$797$	− 9.75782e9i	− 0.682729i	0.939931	−	0.341365i	$-0.110889\pi$
$798$	0	0
$799$	−1.65574e10	−1.14836
$800$	0	0
$801$	0	0
$802$	− 1.61155e9i	− 0.110315i
$803$	1.29789e9i	0.0884572i
$804$	0	0
$805$	0	0
$806$	−8.46700e9	−0.569583
$807$	0	0
$808$	− 1.58221e10i	− 1.05517i
$809$	−2.78706e9	−0.185066	−0.0925330	−	0.995710i	$-0.529496\pi$
$809$	−2.78706e9	−0.185066	−0.0925330	+	0.995710i	$0.529496\pi$
$810$	0	0
$811$	−7.99983e9	−0.526633	−0.263316	−	0.964710i	$-0.584816\pi$
$811$	−7.99983e9	−0.526633	−0.263316	+	0.964710i	$0.584816\pi$
$812$	2.14971e8i	0.0140907i
$813$	0	0
$814$	1.52738e9	0.0992569
$815$	0	0
$816$	0	0
$817$	5.91135e9i	0.379236i
$818$	− 5.39687e8i	− 0.0344751i
$819$	0	0
$820$	0	0
$821$	1.02402e10	0.645813	0.322906	−	0.946431i	$-0.395340\pi$
$821$	1.02402e10	0.645813	0.322906	+	0.946431i	$0.395340\pi$
$822$	0	0
$823$	2.78682e10i	1.74265i	0.490707	+	0.871324i	$0.336738\pi$
$823$	2.78682e10i	1.74265i	−0.490707	+	0.871324i	$0.663262\pi$
$824$	−9.51640e9	−0.592553
$825$	0	0
$826$	−5.01650e8	−0.0309721
$827$	2.35125e10i	1.44554i	0.691090	+	0.722769i	$0.257131\pi$
$827$	2.35125e10i	1.44554i	−0.691090	+	0.722769i	$0.742869\pi$
$828$	0	0
$829$	1.28598e10	0.783960	0.391980	−	0.919974i	$-0.371790\pi$
$829$	1.28598e10	0.783960	0.391980	+	0.919974i	$0.371790\pi$
$830$	0	0
$831$	0	0
$832$	3.35962e9i	0.202236i
$833$	2.32608e10i	1.39434i
$834$	0	0
$835$	0	0
$836$	7.51802e8	0.0445022
$837$	0	0
$838$	1.01433e10i	0.595420i
$839$	−7.99832e9	−0.467554	−0.233777	−	0.972290i	$-0.575109\pi$
$839$	−7.99832e9	−0.467554	−0.233777	+	0.972290i	$0.575109\pi$
$840$	0	0
$841$	−1.59169e10	−0.922725
$842$	− 6.79996e9i	− 0.392567i
$843$	0	0
$844$	1.64907e10	0.944147
$845$	0	0
$846$	0	0
$847$	− 1.18966e9i	− 0.0672716i
$848$	1.65059e9i	0.0929510i
$849$	0	0
$850$	0	0
$851$	4.10523e9	0.228341
$852$	0	0
$853$	4.20827e9i	0.232157i	0.993240	+	0.116079i	$0.0370324\pi$
$853$	4.20827e9i	0.232157i	−0.993240	+	0.116079i	$0.962968\pi$
$854$	−1.15454e8	−0.00634318
$855$	0	0
$856$	−1.50824e10	−0.821888
$857$	3.19307e10i	1.73291i	0.499259	+	0.866453i	$0.333606\pi$
$857$	3.19307e10i	1.73291i	−0.499259	+	0.866453i	$0.666394\pi$
$858$	0	0
$859$	−2.18002e10	−1.17350	−0.586752	−	0.809767i	$-0.699594\pi$
$859$	−2.18002e10	−1.17350	−0.586752	+	0.809767i	$0.699594\pi$
$860$	0	0
$861$	0	0
$862$	− 1.31966e10i	− 0.701755i
$863$	− 1.04728e10i	− 0.554657i	−0.960775	−	0.277329i	$-0.910551\pi$
$863$	− 1.04728e10i	− 0.554657i	0.960775	−	0.277329i	$-0.0894490\pi$
$864$	0	0
$865$	0	0
$866$	9.10427e8	0.0476357
$867$	0	0
$868$	− 1.62985e9i	− 0.0845916i
$869$	6.55421e9	0.338806
$870$	0	0
$871$	2.58593e9	0.132603
$872$	− 5.30765e9i	− 0.271078i
$873$	0	0
$874$	−7.90695e8	−0.0400608
$875$	0	0
$876$	0	0
$877$	1.77787e10i	0.890024i	0.895525	+	0.445012i	$0.146801\pi$
$877$	1.77787e10i	0.890024i	−0.895525	+	0.445012i	$0.853199\pi$
$878$	− 5.94458e9i	− 0.296408i
$879$	0	0
$880$	0	0
$881$	7.64253e9	0.376549	0.188274	−	0.982116i	$-0.439711\pi$
$881$	7.64253e9	0.376549	0.188274	+	0.982116i	$0.439711\pi$
$882$	0	0
$883$	− 2.76375e10i	− 1.35094i	−0.737386	−	0.675472i	$-0.763940\pi$
$883$	− 2.76375e10i	− 1.35094i	0.737386	−	0.675472i	$-0.236060\pi$
$884$	1.33135e10	0.648200
$885$	0	0
$886$	−1.06425e10	−0.514076
$887$	3.23087e10i	1.55449i	0.629200	+	0.777243i	$0.283383\pi$
$887$	3.23087e10i	1.55449i	−0.629200	+	0.777243i	$0.716617\pi$
$888$	0	0
$889$	1.07445e9	0.0512897
$890$	0	0
$891$	0	0
$892$	− 1.90014e10i	− 0.896414i
$893$	5.02801e9i	0.236274i
$894$	0	0
$895$	0	0
$896$	−1.32059e9	−0.0613325
$897$	0	0
$898$	− 1.66206e9i	− 0.0765914i
$899$	−1.01063e10	−0.463908
$900$	0	0
$901$	−1.21509e10	−0.553439
$902$	3.58184e9i	0.162511i
$903$	0	0
$904$	−2.33723e10	−1.05223
$905$	0	0
$906$	0	0
$907$	− 2.27142e10i	− 1.01082i	−0.862880	−	0.505409i	$-0.831342\pi$
$907$	− 2.27142e10i	− 1.01082i	0.862880	−	0.505409i	$-0.168658\pi$
$908$	3.99238e9i	0.176983i
$909$	0	0
$910$	0	0
$911$	−7.50925e9	−0.329065	−0.164533	−	0.986372i	$-0.552612\pi$
$911$	−7.50925e9	−0.329065	−0.164533	+	0.986372i	$0.552612\pi$
$912$	0	0
$913$	4.14916e9i	0.180431i
$914$	1.76855e10	0.766136
$915$	0	0
$916$	3.32976e9	0.143146
$917$	− 1.07691e9i	− 0.0461199i
$918$	0	0
$919$	2.49374e10	1.05986	0.529928	−	0.848043i	$-0.322219\pi$
$919$	2.49374e10	1.05986	0.529928	+	0.848043i	$0.322219\pi$
$920$	0	0
$921$	0	0
$922$	1.66012e10i	0.697561i
$923$	2.83481e10i	1.18664i
$924$	0	0
$925$	0	0
$926$	−2.78132e9	−0.115110
$927$	0	0
$928$	7.01342e9i	0.288079i
$929$	−8.66205e9	−0.354459	−0.177229	−	0.984170i	$-0.556713\pi$
$929$	−8.66205e9	−0.354459	−0.177229	+	0.984170i	$0.556713\pi$
$930$	0	0
$931$	7.06363e9	0.286883
$932$	− 8.48559e9i	− 0.343342i
$933$	0	0
$934$	2.50753e9	0.100701
$935$	0	0
$936$	0	0
$937$	− 2.82655e10i	− 1.12245i	−0.827663	−	0.561226i	$-0.810330\pi$
$937$	− 2.82655e10i	− 1.12245i	0.827663	−	0.561226i	$-0.189670\pi$
$938$	− 1.94782e8i	− 0.00770616i
$939$	0	0
$940$	0	0
$941$	4.67082e10	1.82738	0.913691	−	0.406410i	$-0.133220\pi$
$941$	4.67082e10	1.82738	0.913691	+	0.406410i	$0.133220\pi$
$942$	0	0
$943$	9.62713e9i	0.373857i
$944$	5.03740e9	0.194897
$945$	0	0
$946$	−3.90067e9	−0.149803
$947$	− 4.67392e10i	− 1.78837i	−0.447701	−	0.894184i	$-0.647757\pi$
$947$	− 4.67392e10i	− 1.78837i	0.447701	−	0.894184i	$-0.352243\pi$
$948$	0	0
$949$	6.97958e9	0.265093
$950$	0	0
$951$	0	0
$952$	− 2.39805e9i	− 0.0900801i
$953$	− 3.82420e10i	− 1.43125i	−0.698484	−	0.715625i	$-0.746142\pi$
$953$	− 3.82420e10i	− 1.43125i	0.698484	−	0.715625i	$-0.253858\pi$
$954$	0	0
$955$	0	0
$956$	4.58591e9	0.169755
$957$	0	0
$958$	− 9.05837e9i	− 0.332867i
$959$	−1.79487e9	−0.0657157
$960$	0	0
$961$	4.91101e10	1.78500
$962$	− 8.21367e9i	− 0.297457i
$963$	0	0
$964$	1.83424e10	0.659457
$965$	0	0
$966$	0	0
$967$	4.90012e10i	1.74267i	0.490692	+	0.871333i	$0.336744\pi$
$967$	4.90012e10i	1.74267i	−0.490692	+	0.871333i	$0.663256\pi$
$968$	− 2.45368e10i	− 0.869468i
$969$	0	0
$970$	0	0
$971$	−2.72929e10	−0.956713	−0.478357	−	0.878166i	$-0.658767\pi$
$971$	−2.72929e10	−0.956713	−0.478357	+	0.878166i	$0.658767\pi$
$972$	0	0
$973$	7.56947e8i	0.0263433i
$974$	5.57676e9	0.193386
$975$	0	0
$976$	1.15935e9	0.0399155
$977$	3.94482e9i	0.135331i	0.997708	+	0.0676653i	$0.0215550\pi$
$977$	3.94482e9i	0.135331i	−0.997708	+	0.0676653i	$0.978445\pi$
$978$	0	0
$979$	−8.08484e9	−0.275380
$980$	0	0
$981$	0	0
$982$	− 3.07682e10i	− 1.03684i
$983$	− 4.74320e8i	− 0.0159270i	−0.999968	−	0.00796351i	$-0.997465\pi$
$983$	− 4.74320e8i	− 0.0159270i	0.999968	−	0.00796351i	$-0.00253489\pi$
$984$	0	0
$985$	0	0
$986$	−6.21824e9	−0.206585
$987$	0	0
$988$	− 4.04292e9i	− 0.133366i
$989$	−1.04841e10	−0.344622
$990$	0	0
$991$	1.22197e10	0.398843	0.199421	−	0.979914i	$-0.436094\pi$
$991$	1.22197e10	0.398843	0.199421	+	0.979914i	$0.436094\pi$
$992$	− 5.31737e10i	− 1.72944i
$993$	0	0
$994$	2.13528e9	0.0689609
$995$	0	0
$996$	0	0
$997$	3.60690e10i	1.15266i	0.817217	+	0.576330i	$0.195516\pi$
$997$	3.60690e10i	1.15266i	−0.817217	+	0.576330i	$0.804484\pi$
$998$	2.46390e9i	0.0784631i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.8.b.f.199.2		2
3.2	odd	2		75.8.b.c.49.1		2
5.2	odd	4		9.8.a.a.1.1		1
5.3	odd	4		225.8.a.i.1.1		1
5.4	even	2	inner	225.8.b.f.199.1		2
15.2	even	4		3.8.a.a.1.1	✓	1
15.8	even	4		75.8.a.a.1.1		1
15.14	odd	2		75.8.b.c.49.2		2
20.7	even	4		144.8.a.b.1.1		1
35.27	even	4		441.8.a.a.1.1		1
40.27	even	4		576.8.a.x.1.1		1
40.37	odd	4		576.8.a.w.1.1		1
45.2	even	12		81.8.c.a.28.1		2
45.7	odd	12		81.8.c.c.28.1		2
45.22	odd	12		81.8.c.c.55.1		2
45.32	even	12		81.8.c.a.55.1		2
60.47	odd	4		48.8.a.g.1.1		1
105.2	even	12		147.8.e.b.67.1		2
105.17	odd	12		147.8.e.a.79.1		2
105.32	even	12		147.8.e.b.79.1		2
105.47	odd	12		147.8.e.a.67.1		2
105.62	odd	4		147.8.a.b.1.1		1
120.77	even	4		192.8.a.i.1.1		1
120.107	odd	4		192.8.a.a.1.1		1
165.32	odd	4		363.8.a.b.1.1		1
195.77	even	4		507.8.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.8.a.a.1.1	✓	1	15.2	even	4
9.8.a.a.1.1		1	5.2	odd	4
48.8.a.g.1.1		1	60.47	odd	4
75.8.a.a.1.1		1	15.8	even	4
75.8.b.c.49.1		2	3.2	odd	2
75.8.b.c.49.2		2	15.14	odd	2
81.8.c.a.28.1		2	45.2	even	12
81.8.c.a.55.1		2	45.32	even	12
81.8.c.c.28.1		2	45.7	odd	12
81.8.c.c.55.1		2	45.22	odd	12
144.8.a.b.1.1		1	20.7	even	4
147.8.a.b.1.1		1	105.62	odd	4
147.8.e.a.67.1		2	105.47	odd	12
147.8.e.a.79.1		2	105.17	odd	12
147.8.e.b.67.1		2	105.2	even	12
147.8.e.b.79.1		2	105.32	even	12
192.8.a.a.1.1		1	120.107	odd	4
192.8.a.i.1.1		1	120.77	even	4
225.8.a.i.1.1		1	5.3	odd	4
225.8.b.f.199.1		2	5.4	even	2	inner
225.8.b.f.199.2		2	1.1	even	1	trivial
363.8.a.b.1.1		1	165.32	odd	4
441.8.a.a.1.1		1	35.27	even	4
507.8.a.a.1.1		1	195.77	even	4
576.8.a.w.1.1		1	40.37	odd	4
576.8.a.x.1.1		1	40.27	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+6.00000i q^{2} +92.0000 q^{4} +64.0000i q^{7} +1320.00i q^{8} +948.000 q^{11} -5098.00i q^{13} -384.000 q^{14} +3856.00 q^{16} +28386.0i q^{17} +8620.00 q^{19} +5688.00i q^{22} +15288.0i q^{23} +30588.0 q^{26} +5888.00i q^{28} +36510.0 q^{29} -276808. q^{31} +192096. i q^{32} -170316. q^{34} -268526. i q^{37} +51720.0i q^{38} +629718. q^{41} +685772. i q^{43} +87216.0 q^{44} -91728.0 q^{46} +583296. i q^{47} +819447. q^{49} -469016. i q^{52} +428058. i q^{53} -84480.0 q^{56} +219060. i q^{58} +1.30638e6 q^{59} +300662. q^{61} -1.66085e6i q^{62} -659008. q^{64} +507244. i q^{67} +2.61151e6i q^{68} -5.56063e6 q^{71} +1.36908e6i q^{73} +1.61116e6 q^{74} +793040. q^{76} +60672.0i q^{77} +6.91372e6 q^{79} +3.77831e6i q^{82} +4.37675e6i q^{83} -4.11463e6 q^{86} +1.25136e6i q^{88} -8.52831e6 q^{89} +326272. q^{91} +1.40650e6i q^{92} -3.49978e6 q^{94} +8.82681e6i q^{97} +4.91668e6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 184 q^{4} + 1896 q^{11} - 768 q^{14} + 7712 q^{16} + 17240 q^{19} + 61176 q^{26} + 73020 q^{29} - 553616 q^{31} - 340632 q^{34} + 1259436 q^{41} + 174432 q^{44} - 183456 q^{46} + 1638894 q^{49} - 168960 q^{56}+ \cdots - 6999552 q^{94}+O(q^{100}) \) 2 * q + 184 * q^4 + 1896 * q^11 - 768 * q^14 + 7712 * q^16 + 17240 * q^19 + 61176 * q^26 + 73020 * q^29 - 553616 * q^31 - 340632 * q^34 + 1259436 * q^41 + 174432 * q^44 - 183456 * q^46 + 1638894 * q^49 - 168960 * q^56 + 2612760 * q^59 + 601324 * q^61 - 1318016 * q^64 - 11121264 * q^71 + 3222312 * q^74 + 1586080 * q^76 + 13827440 * q^79 - 8229264 * q^86 - 17056620 * q^89 + 652544 * q^91 - 6999552 * q^94

Introduction

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