LMFDB - Embedded newform 225.9.c.e.26.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,9,Mod(26,225)]

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([1, 0]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("225.26");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	225.c (of order $2$, degree $1$, 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:	$91.6601872638$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} + 1007 x^{14} - 6180 x^{13} + 497360 x^{12} - 4408672 x^{11} + 150181132 x^{10} + \cdots + 13\!\cdots\!12 $ x^16 - 6x^15 + 1007x^14 - 6180x^13 + 497360x^12 - 4408672x^11 + 150181132x^10 - 1705592896x^9 + 31152107740x^8 - 389399613480x^7 + 4860614125132x^6 - 54757452157328x^5 + 510156800415536x^4 - 4019001017758304x^3 + 24939452087377328x^2 - 90270226672793536*x + 138676291658117312
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{42}\cdot 5^{12}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.12
Root			$3.96861 + 1.61014i$ of defining polynomial
Character	$\chi$	$=$	225.26
Dual form			225.9.c.e.26.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+13.0310i q^{2} +86.1932 q^{4} +1918.88 q^{7} +4459.12i q^{8} +O(q^{10})$	$q+13.0310i q^{2} +86.1932 q^{4} +1918.88 q^{7} +4459.12i q^{8} -7407.11i q^{11} -44920.9 q^{13} +25004.9i q^{14} -36041.3 q^{16} -122036. i q^{17} +146641. q^{19} +96522.1 q^{22} -345961. i q^{23} -585364. i q^{26} +165395. q^{28} +453681. i q^{29} -1.17653e6 q^{31} +671880. i q^{32} +1.59025e6 q^{34} -1.32581e6 q^{37} +1.91088e6i q^{38} -1.74823e6i q^{41} -1.07161e6 q^{43} -638443. i q^{44} +4.50822e6 q^{46} -3.34039e6i q^{47} -2.08269e6 q^{49} -3.87187e6 q^{52} -6.15049e6i q^{53} +8.55652e6i q^{56} -5.91192e6 q^{58} -1.31715e7i q^{59} +4.80050e6 q^{61} -1.53314e7i q^{62} -1.79818e7 q^{64} -3.65129e7 q^{67} -1.05187e7i q^{68} -798301. i q^{71} -7.40621e6 q^{73} -1.72767e7i q^{74} +1.26395e7 q^{76} -1.42134e7i q^{77} +6.47230e7 q^{79} +2.27811e7 q^{82} +2.15820e7i q^{83} -1.39641e7i q^{86} +3.30292e7 q^{88} +3.11064e7i q^{89} -8.61978e7 q^{91} -2.98195e7i q^{92} +4.35286e7 q^{94} -9.43227e7 q^{97} -2.71396e7i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 1572 q^{4}+O(q^{10}) $ 16 * q - 1572 * q^4	$ 16 q - 1572 q^{4} - 12844 q^{16} - 263600 q^{19} - 4312984 q^{31} - 2439352 q^{34} - 5376344 q^{46} + 7338968 q^{49} + 26097968 q^{61} + 123290756 q^{64} + 222695760 q^{76} + 262645864 q^{79} + 350561736 q^{91} + 789068384 q^{94}+O(q^{100}) $ 16 * q - 1572 * q^4 - 12844 * q^16 - 263600 * q^19 - 4312984 * q^31 - 2439352 * q^34 - 5376344 * q^46 + 7338968 * q^49 + 26097968 * q^61 + 123290756 * q^64 + 222695760 * q^76 + 262645864 * q^79 + 350561736 * q^91 + 789068384 * 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 $

$2$	13.0310i	0.814437i	0.913331	+	0.407219i	$0.133501\pi$
$2$	13.0310i	0.814437i	−0.913331	+	0.407219i	$0.866499\pi$
$3$	0	0
$3$	0	0
$4$	86.1932	0.336692
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1918.88	0.799201	0.399601	−	0.916689i	$-0.369149\pi$
$7$	1918.88	0.799201	0.399601	+	0.916689i	$0.369149\pi$
$8$	4459.12i	1.08865i
$9$	0	0
$10$	0	0
$11$	− 7407.11i	− 0.505916i	−0.967477	−	0.252958i	$-0.918597\pi$
$11$	− 7407.11i	− 0.505916i	0.967477	−	0.252958i	$-0.0814034\pi$
$12$	0	0
$13$	−44920.9	−1.57280	−0.786402	−	0.617715i	$-0.788059\pi$
$13$	−44920.9	−1.57280	−0.786402	+	0.617715i	$0.788059\pi$
$14$	25004.9i	0.650899i
$15$	0	0
$16$	−36041.3	−0.549946
$17$	− 122036.i	− 1.46114i	−0.682836	−	0.730572i	$-0.739254\pi$
$17$	− 122036.i	− 1.46114i	0.682836	−	0.730572i	$-0.260746\pi$
$18$	0	0
$19$	146641.	1.12523	0.562616	−	0.826718i	$-0.309795\pi$
$19$	146641.	1.12523	0.562616	+	0.826718i	$0.309795\pi$
$20$	0	0
$21$	0	0
$22$	96522.1	0.412037
$23$	− 345961.i	− 1.23628i	−0.786069	−	0.618139i	$-0.787887\pi$
$23$	− 345961.i	− 1.23628i	0.786069	−	0.618139i	$-0.212113\pi$
$24$	0	0
$25$	0	0
$26$	− 585364.i	− 1.28095i
$27$	0	0
$28$	165395.	0.269085
$29$	453681.i	0.641444i	0.947173	+	0.320722i	$0.103926\pi$
$29$	453681.i	0.641444i	−0.947173	+	0.320722i	$0.896074\pi$
$30$	0	0
$31$	−1.17653e6	−1.27396	−0.636982	−	0.770878i	$-0.719818\pi$
$31$	−1.17653e6	−1.27396	−0.636982	+	0.770878i	$0.719818\pi$
$32$	671880.i	0.640755i
$33$	0	0
$34$	1.59025e6	1.19001
$35$	0	0
$36$	0	0
$37$	−1.32581e6	−0.707417	−0.353709	−	0.935356i	$-0.615080\pi$
$37$	−1.32581e6	−0.707417	−0.353709	+	0.935356i	$0.615080\pi$
$38$	1.91088e6i	0.916430i
$39$	0	0
$40$	0	0
$41$	− 1.74823e6i	− 0.618675i	−0.950952	−	0.309337i	$-0.899893\pi$
$41$	− 1.74823e6i	− 0.618675i	0.950952	−	0.309337i	$-0.100107\pi$
$42$	0	0
$43$	−1.07161e6	−0.313445	−0.156722	−	0.987643i	$-0.550093\pi$
$43$	−1.07161e6	−0.313445	−0.156722	+	0.987643i	$0.550093\pi$
$44$	− 638443.i	− 0.170338i
$45$	0	0
$46$	4.50822e6	1.00687
$47$	− 3.34039e6i	− 0.684551i	−0.939600	−	0.342276i	$-0.888802\pi$
$47$	− 3.34039e6i	− 0.684551i	0.939600	−	0.342276i	$-0.111198\pi$
$48$	0	0
$49$	−2.08269e6	−0.361278
$50$	0	0
$51$	0	0
$52$	−3.87187e6	−0.529551
$53$	− 6.15049e6i	− 0.779482i	−0.920924	−	0.389741i	$-0.872564\pi$
$53$	− 6.15049e6i	− 0.779482i	0.920924	−	0.389741i	$-0.127436\pi$
$54$	0	0
$55$	0	0
$56$	8.55652e6i	0.870052i
$57$	0	0
$58$	−5.91192e6	−0.522416
$59$	− 1.31715e7i	− 1.08699i	−0.839412	−	0.543495i	$-0.817101\pi$
$59$	− 1.31715e7i	− 1.08699i	0.839412	−	0.543495i	$-0.182899\pi$
$60$	0	0
$61$	4.80050e6	0.346711	0.173355	−	0.984859i	$-0.444539\pi$
$61$	4.80050e6	0.346711	0.173355	+	0.984859i	$0.444539\pi$
$62$	− 1.53314e7i	− 1.03756i
$63$	0	0
$64$	−1.79818e7	−1.07180
$65$	0	0
$66$	0	0
$67$	−3.65129e7	−1.81195	−0.905976	−	0.423329i	$-0.860861\pi$
$67$	−3.65129e7	−1.81195	−0.905976	+	0.423329i	$0.860861\pi$
$68$	− 1.05187e7i	− 0.491956i
$69$	0	0
$70$	0	0
$71$	− 798301.i	− 0.0314147i	−0.999877	−	0.0157074i	$-0.995000\pi$
$71$	− 798301.i	− 0.0314147i	0.999877	−	0.0157074i	$-0.00500001\pi$
$72$	0	0
$73$	−7.40621e6	−0.260798	−0.130399	−	0.991462i	$-0.541626\pi$
$73$	−7.40621e6	−0.260798	−0.130399	+	0.991462i	$0.541626\pi$
$74$	− 1.72767e7i	− 0.576147i
$75$	0	0
$76$	1.26395e7	0.378857
$77$	− 1.42134e7i	− 0.404329i
$78$	0	0
$79$	6.47230e7	1.66169	0.830845	−	0.556503i	$-0.187857\pi$
$79$	6.47230e7	1.66169	0.830845	+	0.556503i	$0.187857\pi$
$80$	0	0
$81$	0	0
$82$	2.27811e7	0.503872
$83$	2.15820e7i	0.454757i	0.973806	+	0.227378i	$0.0730154\pi$
$83$	2.15820e7i	0.454757i	−0.973806	+	0.227378i	$0.926985\pi$
$84$	0	0
$85$	0	0
$86$	− 1.39641e7i	− 0.255281i
$87$	0	0
$88$	3.30292e7	0.550766
$89$	3.11064e7i	0.495781i	0.968788	+	0.247891i	$0.0797374\pi$
$89$	3.11064e7i	0.495781i	−0.968788	+	0.247891i	$0.920263\pi$
$90$	0	0
$91$	−8.61978e7	−1.25699
$92$	− 2.98195e7i	− 0.416245i
$93$	0	0
$94$	4.35286e7	0.557524
$95$	0	0
$96$	0	0
$97$	−9.43227e7	−1.06544	−0.532720	−	0.846291i	$-0.678830\pi$
$97$	−9.43227e7	−1.06544	−0.532720	+	0.846291i	$0.678830\pi$
$98$	− 2.71396e7i	− 0.294238i
$99$	0	0
$100$	0	0
$101$	− 1.68846e8i	− 1.62258i	−0.584647	−	0.811288i	$-0.698767\pi$
$101$	− 1.68846e8i	− 1.62258i	0.584647	−	0.811288i	$-0.301233\pi$
$102$	0	0
$103$	−1.85101e8	−1.64460	−0.822298	−	0.569057i	$-0.807308\pi$
$103$	−1.85101e8	−1.64460	−0.822298	+	0.569057i	$0.807308\pi$
$104$	− 2.00307e8i	− 1.71224i
$105$	0	0
$106$	8.01470e7	0.634839
$107$	6.02130e7i	0.459362i	0.973266	+	0.229681i	$0.0737683\pi$
$107$	6.02130e7i	0.459362i	−0.973266	+	0.229681i	$0.926232\pi$
$108$	0	0
$109$	2.23264e8	1.58166	0.790830	−	0.612036i	$-0.209649\pi$
$109$	2.23264e8	1.58166	0.790830	+	0.612036i	$0.209649\pi$
$110$	0	0
$111$	0	0
$112$	−6.91590e7	−0.439518
$113$	− 3.21005e7i	− 0.196879i	−0.995143	−	0.0984393i	$-0.968615\pi$
$113$	− 3.21005e7i	− 0.196879i	0.995143	−	0.0984393i	$-0.0313850\pi$
$114$	0	0
$115$	0	0
$116$	3.91042e7i	0.215969i
$117$	0	0
$118$	1.71637e8	0.885286
$119$	− 2.34173e8i	− 1.16775i
$120$	0	0
$121$	1.59494e8	0.744049
$122$	6.25553e7i	0.282374i
$123$	0	0
$124$	−1.01409e8	−0.428934
$125$	0	0
$126$	0	0
$127$	4.80589e8	1.84739	0.923696	−	0.383126i	$-0.125153\pi$
$127$	4.80589e8	1.84739	0.923696	+	0.383126i	$0.125153\pi$
$128$	− 6.23198e7i	− 0.232159i
$129$	0	0
$130$	0	0
$131$	− 2.98566e7i	− 0.101381i	−0.998714	−	0.0506904i	$-0.983858\pi$
$131$	− 2.98566e7i	− 0.101381i	0.998714	−	0.0506904i	$-0.0161422\pi$
$132$	0	0
$133$	2.81387e8	0.899286
$134$	− 4.75799e8i	− 1.47572i
$135$	0	0
$136$	5.44174e8	1.59068
$137$	− 2.90456e8i	− 0.824515i	−0.911067	−	0.412257i	$-0.864740\pi$
$137$	− 2.90456e8i	− 0.824515i	0.911067	−	0.412257i	$-0.135260\pi$
$138$	0	0
$139$	3.12159e8	0.836212	0.418106	−	0.908398i	$-0.362694\pi$
$139$	3.12159e8	0.836212	0.418106	+	0.908398i	$0.362694\pi$
$140$	0	0
$141$	0	0
$142$	1.04027e7	0.0255853
$143$	3.32734e8i	0.795707i
$144$	0	0
$145$	0	0
$146$	− 9.65102e7i	− 0.212404i
$147$	0	0
$148$	−1.14276e8	−0.238182
$149$	− 8.72146e8i	− 1.76947i	−0.466091	−	0.884737i	$-0.654338\pi$
$149$	− 8.72146e8i	− 1.76947i	0.466091	−	0.884737i	$-0.345662\pi$
$150$	0	0
$151$	−2.42967e8	−0.467347	−0.233673	−	0.972315i	$-0.575075\pi$
$151$	−2.42967e8	−0.467347	−0.233673	+	0.972315i	$0.575075\pi$
$152$	6.53891e8i	1.22499i
$153$	0	0
$154$	1.85214e8	0.329300
$155$	0	0
$156$	0	0
$157$	5.05078e8	0.831304	0.415652	−	0.909524i	$-0.363553\pi$
$157$	5.05078e8	0.831304	0.415652	+	0.909524i	$0.363553\pi$
$158$	8.43405e8i	1.35334i
$159$	0	0
$160$	0	0
$161$	− 6.63859e8i	− 0.988035i
$162$	0	0
$163$	1.92088e8	0.272114	0.136057	−	0.990701i	$-0.456557\pi$
$163$	1.92088e8	0.272114	0.136057	+	0.990701i	$0.456557\pi$
$164$	− 1.50685e8i	− 0.208303i
$165$	0	0
$166$	−2.81235e8	−0.370371
$167$	1.26266e9i	1.62338i	0.584088	+	0.811690i	$0.301452\pi$
$167$	1.26266e9i	1.62338i	−0.584088	+	0.811690i	$0.698548\pi$
$168$	0	0
$169$	1.20215e9	1.47371
$170$	0	0
$171$	0	0
$172$	−9.23651e7	−0.105534
$173$	− 1.67194e9i	− 1.86653i	−0.359189	−	0.933265i	$-0.616946\pi$
$173$	− 1.67194e9i	− 1.86653i	0.359189	−	0.933265i	$-0.383054\pi$
$174$	0	0
$175$	0	0
$176$	2.66962e8i	0.278227i
$177$	0	0
$178$	−4.05348e8	−0.403783
$179$	2.38184e8i	0.232007i	0.993249	+	0.116003i	$0.0370083\pi$
$179$	2.38184e8i	0.232007i	−0.993249	+	0.116003i	$0.962992\pi$
$180$	0	0
$181$	−3.21450e8	−0.299502	−0.149751	−	0.988724i	$-0.547847\pi$
$181$	−3.21450e8	−0.299502	−0.149751	+	0.988724i	$0.547847\pi$
$182$	− 1.12324e9i	− 1.02374i
$183$	0	0
$184$	1.54268e9	1.34588
$185$	0	0
$186$	0	0
$187$	−9.03936e8	−0.739216
$188$	− 2.87919e8i	− 0.230483i
$189$	0	0
$190$	0	0
$191$	− 8.26431e8i	− 0.620973i	−0.950578	−	0.310487i	$-0.899508\pi$
$191$	− 8.26431e8i	− 0.620973i	0.950578	−	0.310487i	$-0.100492\pi$
$192$	0	0
$193$	−3.42380e8	−0.246763	−0.123381	−	0.992359i	$-0.539374\pi$
$193$	−3.42380e8	−0.246763	−0.123381	+	0.992359i	$0.539374\pi$
$194$	− 1.22912e9i	− 0.867735i
$195$	0	0
$196$	−1.79514e8	−0.121639
$197$	− 1.16560e9i	− 0.773902i	−0.922100	−	0.386951i	$-0.873528\pi$
$197$	− 1.16560e9i	− 0.773902i	0.922100	−	0.386951i	$-0.126472\pi$
$198$	0	0
$199$	−3.49071e8	−0.222588	−0.111294	−	0.993788i	$-0.535500\pi$
$199$	−3.49071e8	−0.222588	−0.111294	+	0.993788i	$0.535500\pi$
$200$	0	0
$201$	0	0
$202$	2.20023e9	1.32149
$203$	8.70561e8i	0.512643i
$204$	0	0
$205$	0	0
$206$	− 2.41205e9i	− 1.33942i
$207$	0	0
$208$	1.61901e9	0.864958
$209$	− 1.08619e9i	− 0.569273i
$210$	0	0
$211$	1.88697e9	0.951995	0.475998	−	0.879447i	$-0.342087\pi$
$211$	1.88697e9	0.951995	0.475998	+	0.879447i	$0.342087\pi$
$212$	− 5.30130e8i	− 0.262446i
$213$	0	0
$214$	−7.84636e8	−0.374122
$215$	0	0
$216$	0	0
$217$	−2.25763e9	−1.01815
$218$	2.90935e9i	1.28816i
$219$	0	0
$220$	0	0
$221$	5.48197e9i	2.29809i
$222$	0	0
$223$	−3.00252e9	−1.21414	−0.607068	−	0.794650i	$-0.707654\pi$
$223$	−3.00252e9	−1.21414	−0.607068	+	0.794650i	$0.707654\pi$
$224$	1.28926e9i	0.512092i
$225$	0	0
$226$	4.18302e8	0.160345
$227$	− 7.55931e8i	− 0.284694i	−0.989817	−	0.142347i	$-0.954535\pi$
$227$	− 7.55931e8i	− 0.284694i	0.989817	−	0.142347i	$-0.0454649\pi$
$228$	0	0
$229$	−3.80872e9	−1.38496	−0.692479	−	0.721438i	$-0.743482\pi$
$229$	−3.80872e9	−1.38496	−0.692479	+	0.721438i	$0.743482\pi$
$230$	0	0
$231$	0	0
$232$	−2.02302e9	−0.698309
$233$	− 2.90545e9i	− 0.985801i	−0.870086	−	0.492900i	$-0.835937\pi$
$233$	− 2.90545e9i	− 0.985801i	0.870086	−	0.492900i	$-0.164063\pi$
$234$	0	0
$235$	0	0
$236$	− 1.13529e9i	− 0.365981i
$237$	0	0
$238$	3.05151e9	0.951057
$239$	− 3.35229e9i	− 1.02742i	−0.857963	−	0.513712i	$-0.828270\pi$
$239$	− 3.35229e9i	− 1.02742i	0.857963	−	0.513712i	$-0.171730\pi$
$240$	0	0
$241$	3.45448e9	1.02403	0.512017	−	0.858975i	$-0.328899\pi$
$241$	3.45448e9	1.02403	0.512017	+	0.858975i	$0.328899\pi$
$242$	2.07836e9i	0.605981i
$243$	0	0
$244$	4.13770e8	0.116735
$245$	0	0
$246$	0	0
$247$	−6.58726e9	−1.76977
$248$	− 5.24630e9i	− 1.38690i
$249$	0	0
$250$	0	0
$251$	− 1.34472e9i	− 0.338794i	−0.985548	−	0.169397i	$-0.945818\pi$
$251$	− 1.34472e9i	− 0.338794i	0.985548	−	0.169397i	$-0.0541820\pi$
$252$	0	0
$253$	−2.56258e9	−0.625453
$254$	6.26255e9i	1.50458i
$255$	0	0
$256$	−3.79126e9	−0.882722
$257$	− 5.38058e9i	− 1.23338i	−0.787207	−	0.616689i	$-0.788473\pi$
$257$	− 5.38058e9i	− 1.23338i	0.787207	−	0.616689i	$-0.211527\pi$
$258$	0	0
$259$	−2.54408e9	−0.565369
$260$	0	0
$261$	0	0
$262$	3.89062e8	0.0825683
$263$	1.59261e9i	0.332878i	0.986052	+	0.166439i	$0.0532269\pi$
$263$	1.59261e9i	0.332878i	−0.986052	+	0.166439i	$0.946773\pi$
$264$	0	0
$265$	0	0
$266$	3.66676e9i	0.732412i
$267$	0	0
$268$	−3.14716e9	−0.610070
$269$	1.03270e10i	1.97226i	0.165977	+	0.986130i	$0.446922\pi$
$269$	1.03270e10i	1.97226i	−0.165977	+	0.986130i	$0.553078\pi$
$270$	0	0
$271$	−7.47903e9	−1.38665	−0.693327	−	0.720623i	$-0.743856\pi$
$271$	−7.47903e9	−1.38665	−0.693327	+	0.720623i	$0.743856\pi$
$272$	4.39834e9i	0.803551i
$273$	0	0
$274$	3.78493e9	0.671516
$275$	0	0
$276$	0	0
$277$	8.73280e9	1.48332	0.741660	−	0.670777i	$-0.234039\pi$
$277$	8.73280e9	1.48332	0.741660	+	0.670777i	$0.234039\pi$
$278$	4.06774e9i	0.681042i
$279$	0	0
$280$	0	0
$281$	2.27147e9i	0.364319i	0.983269	+	0.182159i	$0.0583087\pi$
$281$	2.27147e9i	0.364319i	−0.983269	+	0.182159i	$0.941691\pi$
$282$	0	0
$283$	−4.90950e9	−0.765406	−0.382703	−	0.923872i	$-0.625007\pi$
$283$	−4.90950e9	−0.765406	−0.382703	+	0.923872i	$0.625007\pi$
$284$	− 6.88081e7i	− 0.0105771i
$285$	0	0
$286$	−4.33586e9	−0.648053
$287$	− 3.35464e9i	− 0.494446i
$288$	0	0
$289$	−7.91708e9	−1.13494
$290$	0	0
$291$	0	0
$292$	−6.38364e8	−0.0878086
$293$	5.24948e9i	0.712271i	0.934434	+	0.356136i	$0.115906\pi$
$293$	5.24948e9i	0.712271i	−0.934434	+	0.356136i	$0.884094\pi$
$294$	0	0
$295$	0	0
$296$	− 5.91196e9i	− 0.770131i
$297$	0	0
$298$	1.13649e10	1.44112
$299$	1.55409e10i	1.94442i
$300$	0	0
$301$	−2.05628e9	−0.250505
$302$	− 3.16610e9i	− 0.380625i
$303$	0	0
$304$	−5.28514e9	−0.618817
$305$	0	0
$306$	0	0
$307$	4.33898e9	0.488466	0.244233	−	0.969717i	$-0.421464\pi$
$307$	4.33898e9	0.488466	0.244233	+	0.969717i	$0.421464\pi$
$308$	− 1.22510e9i	− 0.136134i
$309$	0	0
$310$	0	0
$311$	1.02901e10i	1.09996i	0.835178	+	0.549980i	$0.185365\pi$
$311$	1.02901e10i	1.09996i	−0.835178	+	0.549980i	$0.814635\pi$
$312$	0	0
$313$	6.45458e9	0.672497	0.336249	−	0.941773i	$-0.390842\pi$
$313$	6.45458e9	0.672497	0.336249	+	0.941773i	$0.390842\pi$
$314$	6.58167e9i	0.677045i
$315$	0	0
$316$	5.57868e9	0.559478
$317$	1.40779e10i	1.39412i	0.717013	+	0.697059i	$0.245509\pi$
$317$	1.40779e10i	1.39412i	−0.717013	+	0.697059i	$0.754491\pi$
$318$	0	0
$319$	3.36047e9	0.324517
$320$	0	0
$321$	0	0
$322$	8.65074e9	0.804692
$323$	− 1.78955e10i	− 1.64413i
$324$	0	0
$325$	0	0
$326$	2.50310e9i	0.221620i
$327$	0	0
$328$	7.79555e9	0.673521
$329$	− 6.40982e9i	− 0.547094i
$330$	0	0
$331$	−1.04281e10	−0.868747	−0.434374	−	0.900733i	$-0.643030\pi$
$331$	−1.04281e10	−0.868747	−0.434374	+	0.900733i	$0.643030\pi$
$332$	1.86022e9i	0.153113i
$333$	0	0
$334$	−1.64537e10	−1.32214
$335$	0	0
$336$	0	0
$337$	−4.20998e9	−0.326408	−0.163204	−	0.986592i	$-0.552183\pi$
$337$	−4.20998e9	−0.326408	−0.163204	+	0.986592i	$0.552183\pi$
$338$	1.56653e10i	1.20025i
$339$	0	0
$340$	0	0
$341$	8.71472e9i	0.644519i
$342$	0	0
$343$	−1.50584e10	−1.08793
$344$	− 4.77841e9i	− 0.341232i
$345$	0	0
$346$	2.17870e10	1.52017
$347$	1.61293e10i	1.11249i	0.831017	+	0.556246i	$0.187759\pi$
$347$	1.61293e10i	1.11249i	−0.831017	+	0.556246i	$0.812241\pi$
$348$	0	0
$349$	−1.41858e10	−0.956207	−0.478104	−	0.878303i	$-0.658676\pi$
$349$	−1.41858e10	−0.956207	−0.478104	+	0.878303i	$0.658676\pi$
$350$	0	0
$351$	0	0
$352$	4.97669e9	0.324168
$353$	7.51852e9i	0.484210i	0.970250	+	0.242105i	$0.0778378\pi$
$353$	7.51852e9i	0.484210i	−0.970250	+	0.242105i	$0.922162\pi$
$354$	0	0
$355$	0	0
$356$	2.68116e9i	0.166926i
$357$	0	0
$358$	−3.10378e9	−0.188955
$359$	1.71611e10i	1.03316i	0.856239	+	0.516581i	$0.172795\pi$
$359$	1.71611e10i	1.03316i	−0.856239	+	0.516581i	$0.827205\pi$
$360$	0	0
$361$	4.52011e9	0.266146
$362$	− 4.18881e9i	− 0.243925i
$363$	0	0
$364$	−7.42967e9	−0.423218
$365$	0	0
$366$	0	0
$367$	−6.96880e9	−0.384144	−0.192072	−	0.981381i	$-0.561521\pi$
$367$	−6.96880e9	−0.384144	−0.192072	+	0.981381i	$0.561521\pi$
$368$	1.24689e10i	0.679887i
$369$	0	0
$370$	0	0
$371$	− 1.18021e10i	− 0.622963i
$372$	0	0
$373$	−5.84873e9	−0.302152	−0.151076	−	0.988522i	$-0.548274\pi$
$373$	−5.84873e9	−0.302152	−0.151076	+	0.988522i	$0.548274\pi$
$374$	− 1.17792e10i	− 0.602045i
$375$	0	0
$376$	1.48952e10	0.745238
$377$	− 2.03798e10i	− 1.00887i
$378$	0	0
$379$	2.17986e10	1.05651	0.528253	−	0.849087i	$-0.322847\pi$
$379$	2.17986e10	1.05651	0.528253	+	0.849087i	$0.322847\pi$
$380$	0	0
$381$	0	0
$382$	1.07692e10	0.505744
$383$	− 1.63885e10i	− 0.761631i	−0.924651	−	0.380816i	$-0.875643\pi$
$383$	− 1.63885e10i	− 0.761631i	0.924651	−	0.380816i	$-0.124357\pi$
$384$	0	0
$385$	0	0
$386$	− 4.46155e9i	− 0.200973i
$387$	0	0
$388$	−8.12998e9	−0.358726
$389$	− 8.17971e9i	− 0.357223i	−0.983920	−	0.178612i	$-0.942839\pi$
$389$	− 8.17971e9i	− 0.357223i	0.983920	−	0.178612i	$-0.0571605\pi$
$390$	0	0
$391$	−4.22198e10	−1.80638
$392$	− 9.28697e9i	− 0.393305i
$393$	0	0
$394$	1.51890e10	0.630294
$395$	0	0
$396$	0	0
$397$	1.00785e10	0.405729	0.202864	−	0.979207i	$-0.434975\pi$
$397$	1.00785e10	0.405729	0.202864	+	0.979207i	$0.434975\pi$
$398$	− 4.54875e9i	− 0.181284i
$399$	0	0
$400$	0	0
$401$	1.95831e9i	0.0757364i	0.999283	+	0.0378682i	$0.0120567\pi$
$401$	1.95831e9i	0.0757364i	−0.999283	+	0.0378682i	$0.987943\pi$
$402$	0	0
$403$	5.28509e10	2.00370
$404$	− 1.45534e10i	− 0.546308i
$405$	0	0
$406$	−1.13443e10	−0.417515
$407$	9.82045e9i	0.357894i
$408$	0	0
$409$	1.16686e10	0.416990	0.208495	−	0.978023i	$-0.433143\pi$
$409$	1.16686e10	0.416990	0.208495	+	0.978023i	$0.433143\pi$
$410$	0	0
$411$	0	0
$412$	−1.59544e10	−0.553722
$413$	− 2.52745e10i	− 0.868724i
$414$	0	0
$415$	0	0
$416$	− 3.01815e10i	− 1.00778i
$417$	0	0
$418$	1.41541e10	0.463637
$419$	3.48051e10i	1.12924i	0.825351	+	0.564621i	$0.190978\pi$
$419$	3.48051e10i	1.12924i	−0.825351	+	0.564621i	$0.809022\pi$
$420$	0	0
$421$	1.15892e9	0.0368914	0.0184457	−	0.999830i	$-0.494128\pi$
$421$	1.15892e9	0.0368914	0.0184457	+	0.999830i	$0.494128\pi$
$422$	2.45891e10i	0.775340i
$423$	0	0
$424$	2.74258e10	0.848585
$425$	0	0
$426$	0	0
$427$	9.21159e9	0.277091
$428$	5.18995e9i	0.154664i
$429$	0	0
$430$	0	0
$431$	4.74653e9i	0.137552i	0.997632	+	0.0687760i	$0.0219094\pi$
$431$	4.74653e9i	0.137552i	−0.997632	+	0.0687760i	$0.978091\pi$
$432$	0	0
$433$	−5.86356e10	−1.66805	−0.834026	−	0.551725i	$-0.813970\pi$
$433$	−5.86356e10	−1.66805	−0.834026	+	0.551725i	$0.813970\pi$
$434$	− 2.94191e10i	− 0.829222i
$435$	0	0
$436$	1.92439e10	0.532532
$437$	− 5.07322e10i	− 1.39110i
$438$	0	0
$439$	5.06896e10	1.36477	0.682387	−	0.730991i	$-0.260942\pi$
$439$	5.06896e10	1.36477	0.682387	+	0.730991i	$0.260942\pi$
$440$	0	0
$441$	0	0
$442$	−7.14355e10	−1.87165
$443$	4.34873e10i	1.12914i	0.825385	+	0.564570i	$0.190958\pi$
$443$	4.34873e10i	1.12914i	−0.825385	+	0.564570i	$0.809042\pi$
$444$	0	0
$445$	0	0
$446$	− 3.91259e10i	− 0.988837i
$447$	0	0
$448$	−3.45050e10	−0.856585
$449$	− 7.22137e10i	− 1.77678i	−0.459087	−	0.888391i	$-0.651823\pi$
$449$	− 7.22137e10i	− 1.77678i	0.459087	−	0.888391i	$-0.348177\pi$
$450$	0	0
$451$	−1.29493e10	−0.312997
$452$	− 2.76685e9i	− 0.0662875i
$453$	0	0
$454$	9.85053e9	0.231866
$455$	0	0
$456$	0	0
$457$	−6.46241e10	−1.48160	−0.740798	−	0.671728i	$-0.765552\pi$
$457$	−6.46241e10	−1.48160	−0.740798	+	0.671728i	$0.765552\pi$
$458$	− 4.96314e10i	− 1.12796i
$459$	0	0
$460$	0	0
$461$	6.89816e10i	1.52732i	0.645620	+	0.763659i	$0.276599\pi$
$461$	6.89816e10i	1.52732i	−0.645620	+	0.763659i	$0.723401\pi$
$462$	0	0
$463$	−5.82360e10	−1.26727	−0.633633	−	0.773634i	$-0.718437\pi$
$463$	−5.82360e10	−1.26727	−0.633633	+	0.773634i	$0.718437\pi$
$464$	− 1.63513e10i	− 0.352760i
$465$	0	0
$466$	3.78608e10	0.802873
$467$	− 4.24626e10i	− 0.892768i	−0.894841	−	0.446384i	$-0.852712\pi$
$467$	− 4.24626e10i	− 0.892768i	0.894841	−	0.446384i	$-0.147288\pi$
$468$	0	0
$469$	−7.00639e10	−1.44811
$470$	0	0
$471$	0	0
$472$	5.87331e10	1.18335
$473$	7.93750e9i	0.158577i
$474$	0	0
$475$	0	0
$476$	− 2.01841e10i	− 0.393172i
$477$	0	0
$478$	4.36837e10	0.836773
$479$	− 5.67715e10i	− 1.07842i	−0.842171	−	0.539210i	$-0.818723\pi$
$479$	− 5.67715e10i	− 1.07842i	0.842171	−	0.539210i	$-0.181277\pi$
$480$	0	0
$481$	5.95567e10	1.11263
$482$	4.50153e10i	0.834011i
$483$	0	0
$484$	1.37473e10	0.250515
$485$	0	0
$486$	0	0
$487$	2.01211e9	0.0357715	0.0178857	−	0.999840i	$-0.494306\pi$
$487$	2.01211e9	0.0357715	0.0178857	+	0.999840i	$0.494306\pi$
$488$	2.14060e10i	0.377447i
$489$	0	0
$490$	0	0
$491$	− 4.44540e10i	− 0.764866i	−0.923983	−	0.382433i	$-0.875086\pi$
$491$	− 4.44540e10i	− 0.764866i	0.923983	−	0.382433i	$-0.124914\pi$
$492$	0	0
$493$	5.53655e10	0.937242
$494$	− 8.58385e10i	− 1.44137i
$495$	0	0
$496$	4.24038e10	0.700612
$497$	− 1.53185e9i	− 0.0251067i
$498$	0	0
$499$	3.00108e10	0.484034	0.242017	−	0.970272i	$-0.422191\pi$
$499$	3.00108e10	0.484034	0.242017	+	0.970272i	$0.422191\pi$
$500$	0	0
$501$	0	0
$502$	1.75230e10	0.275926
$503$	− 2.63318e10i	− 0.411347i	−0.978621	−	0.205673i	$-0.934062\pi$
$503$	− 2.63318e10i	− 0.411347i	0.978621	−	0.205673i	$-0.0659385\pi$
$504$	0	0
$505$	0	0
$506$	− 3.33929e10i	− 0.509392i
$507$	0	0
$508$	4.14235e10	0.622002
$509$	− 1.71042e10i	− 0.254819i	−0.991850	−	0.127410i	$-0.959334\pi$
$509$	− 1.71042e10i	− 0.254819i	0.991850	−	0.127410i	$-0.0406663\pi$
$510$	0	0
$511$	−1.42116e10	−0.208430
$512$	− 6.53578e10i	− 0.951081i
$513$	0	0
$514$	7.01143e10	1.00451
$515$	0	0
$516$	0	0
$517$	−2.47427e10	−0.346325
$518$	− 3.31519e10i	− 0.460457i
$519$	0	0
$520$	0	0
$521$	1.00723e11i	1.36703i	0.729937	+	0.683514i	$0.239549\pi$
$521$	1.00723e11i	1.36703i	−0.729937	+	0.683514i	$0.760451\pi$
$522$	0	0
$523$	3.74201e10	0.500148	0.250074	−	0.968227i	$-0.419545\pi$
$523$	3.74201e10	0.500148	0.250074	+	0.968227i	$0.419545\pi$
$524$	− 2.57344e9i	− 0.0341341i
$525$	0	0
$526$	−2.07532e10	−0.271108
$527$	1.43580e11i	1.86145i
$528$	0	0
$529$	−4.13783e10	−0.528384
$530$	0	0
$531$	0	0
$532$	2.42537e10	0.302783
$533$	7.85319e10i	0.973054i
$534$	0	0
$535$	0	0
$536$	− 1.62815e11i	− 1.97258i
$537$	0	0
$538$	−1.34571e11	−1.60628
$539$	1.54267e10i	0.182776i
$540$	0	0
$541$	−5.67222e9	−0.0662162	−0.0331081	−	0.999452i	$-0.510541\pi$
$541$	−5.67222e9	−0.0662162	−0.0331081	+	0.999452i	$0.510541\pi$
$542$	− 9.74592e10i	− 1.12934i
$543$	0	0
$544$	8.19937e10	0.936235
$545$	0	0
$546$	0	0
$547$	1.13867e11	1.27189	0.635944	−	0.771735i	$-0.280611\pi$
$547$	1.13867e11	1.27189	0.635944	+	0.771735i	$0.280611\pi$
$548$	− 2.50354e10i	− 0.277608i
$549$	0	0
$550$	0	0
$551$	6.65284e10i	0.721773i
$552$	0	0
$553$	1.24196e11	1.32803
$554$	1.13797e11i	1.20807i
$555$	0	0
$556$	2.69060e10	0.281546
$557$	1.20812e11i	1.25513i	0.778564	+	0.627565i	$0.215948\pi$
$557$	1.20812e11i	1.25513i	−0.778564	+	0.627565i	$0.784052\pi$
$558$	0	0
$559$	4.81374e10	0.492987
$560$	0	0
$561$	0	0
$562$	−2.95995e10	−0.296715
$563$	1.33228e11i	1.32605i	0.748596	+	0.663027i	$0.230728\pi$
$563$	1.33228e11i	1.32605i	−0.748596	+	0.663027i	$0.769272\pi$
$564$	0	0
$565$	0	0
$566$	− 6.39757e10i	− 0.623375i
$567$	0	0
$568$	3.55972e9	0.0341997
$569$	− 9.24293e8i	− 0.00881781i	−0.999990	−	0.00440891i	$-0.998597\pi$
$569$	− 9.24293e8i	− 0.00881781i	0.999990	−	0.00440891i	$-0.00140340\pi$
$570$	0	0
$571$	−1.74424e11	−1.64082	−0.820411	−	0.571775i	$-0.806255\pi$
$571$	−1.74424e11	−1.64082	−0.820411	+	0.571775i	$0.806255\pi$
$572$	2.86794e10i	0.267908i
$573$	0	0
$574$	4.37143e10	0.402695
$575$	0	0
$576$	0	0
$577$	1.86635e11	1.68380	0.841898	−	0.539637i	$-0.181439\pi$
$577$	1.86635e11	1.68380	0.841898	+	0.539637i	$0.181439\pi$
$578$	− 1.03167e11i	− 0.924338i
$579$	0	0
$580$	0	0
$581$	4.14133e10i	0.363442i
$582$	0	0
$583$	−4.55574e10	−0.394353
$584$	− 3.30251e10i	− 0.283918i
$585$	0	0
$586$	−6.84059e10	−0.580100
$587$	− 4.03447e10i	− 0.339809i	−0.985461	−	0.169904i	$-0.945654\pi$
$587$	− 4.03447e10i	− 0.339809i	0.985461	−	0.169904i	$-0.0543459\pi$
$588$	0	0
$589$	−1.72528e11	−1.43351
$590$	0	0
$591$	0	0
$592$	4.77840e10	0.389041
$593$	− 1.32300e11i	− 1.06989i	−0.844886	−	0.534946i	$-0.820332\pi$
$593$	− 1.32300e11i	− 1.06989i	0.844886	−	0.534946i	$-0.179668\pi$
$594$	0	0
$595$	0	0
$596$	− 7.51730e10i	− 0.595768i
$597$	0	0
$598$	−2.02513e11	−1.58361
$599$	1.69478e11i	1.31646i	0.752818	+	0.658228i	$0.228694\pi$
$599$	1.69478e11i	1.31646i	−0.752818	+	0.658228i	$0.771306\pi$
$600$	0	0
$601$	1.97703e10	0.151535	0.0757677	−	0.997125i	$-0.475859\pi$
$601$	1.97703e10	0.151535	0.0757677	+	0.997125i	$0.475859\pi$
$602$	− 2.67954e10i	− 0.204021i
$603$	0	0
$604$	−2.09421e10	−0.157352
$605$	0	0
$606$	0	0
$607$	−1.84942e11	−1.36232	−0.681161	−	0.732133i	$-0.738525\pi$
$607$	−1.84942e11	−1.36232	−0.681161	+	0.732133i	$0.738525\pi$
$608$	9.85254e10i	0.720998i
$609$	0	0
$610$	0	0
$611$	1.50053e11i	1.07667i
$612$	0	0
$613$	1.98997e11	1.40930	0.704651	−	0.709554i	$-0.251104\pi$
$613$	1.98997e11	1.40930	0.704651	+	0.709554i	$0.251104\pi$
$614$	5.65413e10i	0.397825i
$615$	0	0
$616$	6.33791e10	0.440173
$617$	6.55893e10i	0.452577i	0.974060	+	0.226289i	$0.0726592\pi$
$617$	6.55893e10i	0.452577i	−0.974060	+	0.226289i	$0.927341\pi$
$618$	0	0
$619$	−8.30730e10	−0.565845	−0.282923	−	0.959143i	$-0.591304\pi$
$619$	−8.30730e10	−0.565845	−0.282923	+	0.959143i	$0.591304\pi$
$620$	0	0
$621$	0	0
$622$	−1.34090e11	−0.895849
$623$	5.96896e10i	0.396229i
$624$	0	0
$625$	0	0
$626$	8.41096e10i	0.547707i
$627$	0	0
$628$	4.35343e10	0.279894
$629$	1.61797e11i	1.03364i
$630$	0	0
$631$	−3.46002e10	−0.218254	−0.109127	−	0.994028i	$-0.534805\pi$
$631$	−3.46002e10	−0.218254	−0.109127	+	0.994028i	$0.534805\pi$
$632$	2.88607e11i	1.80900i
$633$	0	0
$634$	−1.83449e11	−1.13542
$635$	0	0
$636$	0	0
$637$	9.35564e10	0.568219
$638$	4.37903e10i	0.264299i
$639$	0	0
$640$	0	0
$641$	2.85486e11i	1.69104i	0.533947	+	0.845518i	$0.320708\pi$
$641$	2.85486e11i	1.69104i	−0.533947	+	0.845518i	$0.679292\pi$
$642$	0	0
$643$	7.11481e10	0.416217	0.208108	−	0.978106i	$-0.433269\pi$
$643$	7.11481e10	0.416217	0.208108	+	0.978106i	$0.433269\pi$
$644$	− 5.72201e10i	− 0.332664i
$645$	0	0
$646$	2.33197e11	1.33904
$647$	− 2.20623e11i	− 1.25903i	−0.776990	−	0.629513i	$-0.783255\pi$
$647$	− 2.20623e11i	− 1.25903i	0.776990	−	0.629513i	$-0.216745\pi$
$648$	0	0
$649$	−9.75625e10	−0.549926
$650$	0	0
$651$	0	0
$652$	1.65567e10	0.0916186
$653$	1.22498e10i	0.0673718i	0.999432	+	0.0336859i	$0.0107246\pi$
$653$	1.22498e10i	0.0673718i	−0.999432	+	0.0336859i	$0.989275\pi$
$654$	0	0
$655$	0	0
$656$	6.30083e10i	0.340238i
$657$	0	0
$658$	8.35263e10	0.445574
$659$	7.01341e10i	0.371867i	0.982562	+	0.185934i	$0.0595309\pi$
$659$	7.01341e10i	0.371867i	−0.982562	+	0.185934i	$0.940469\pi$
$660$	0	0
$661$	−1.27159e11	−0.666105	−0.333053	−	0.942908i	$-0.608079\pi$
$661$	−1.27159e11	−0.666105	−0.333053	+	0.942908i	$0.608079\pi$
$662$	− 1.35889e11i	− 0.707540i
$663$	0	0
$664$	−9.62366e10	−0.495072
$665$	0	0
$666$	0	0
$667$	1.56956e11	0.793003
$668$	1.08833e11i	0.546579i
$669$	0	0
$670$	0	0
$671$	− 3.55578e10i	− 0.175406i
$672$	0	0
$673$	−1.08256e11	−0.527703	−0.263852	−	0.964563i	$-0.584993\pi$
$673$	−1.08256e11	−0.527703	−0.263852	+	0.964563i	$0.584993\pi$
$674$	− 5.48602e10i	− 0.265839i
$675$	0	0
$676$	1.03617e11	0.496188
$677$	− 1.06487e11i	− 0.506923i	−0.967346	−	0.253461i	$-0.918431\pi$
$677$	− 1.06487e11i	− 0.506923i	0.967346	−	0.253461i	$-0.0815691\pi$
$678$	0	0
$679$	−1.80994e11	−0.851502
$680$	0	0
$681$	0	0
$682$	−1.13561e11	−0.524920
$683$	− 3.30690e11i	− 1.51963i	−0.650139	−	0.759815i	$-0.725289\pi$
$683$	− 3.30690e11i	− 1.51963i	0.650139	−	0.759815i	$-0.274711\pi$
$684$	0	0
$685$	0	0
$686$	− 1.96226e11i	− 0.886054i
$687$	0	0
$688$	3.86220e10	0.172378
$689$	2.76285e11i	1.22597i
$690$	0	0
$691$	−1.23710e11	−0.542615	−0.271308	−	0.962493i	$-0.587456\pi$
$691$	−1.23710e11	−0.542615	−0.271308	+	0.962493i	$0.587456\pi$
$692$	− 1.44109e11i	− 0.628446i
$693$	0	0
$694$	−2.10181e11	−0.906055
$695$	0	0
$696$	0	0
$697$	−2.13347e11	−0.903973
$698$	− 1.84855e11i	− 0.778771i
$699$	0	0
$700$	0	0
$701$	− 7.23976e10i	− 0.299814i	−0.988700	−	0.149907i	$-0.952103\pi$
$701$	− 7.23976e10i	− 0.299814i	0.988700	−	0.149907i	$-0.0478974\pi$
$702$	0	0
$703$	−1.94419e11	−0.796008
$704$	1.33194e11i	0.542241i
$705$	0	0
$706$	−9.79737e10	−0.394358
$707$	− 3.23995e11i	− 1.29676i
$708$	0	0
$709$	2.92209e11	1.15640	0.578200	−	0.815895i	$-0.303755\pi$
$709$	2.92209e11	1.15640	0.578200	+	0.815895i	$0.303755\pi$
$710$	0	0
$711$	0	0
$712$	−1.38707e11	−0.539733
$713$	4.07035e11i	1.57497i
$714$	0	0
$715$	0	0
$716$	2.05298e10i	0.0781148i
$717$	0	0
$718$	−2.23627e11	−0.841445
$719$	3.76304e11i	1.40807i	0.710166	+	0.704034i	$0.248620\pi$
$719$	3.76304e11i	1.40807i	−0.710166	+	0.704034i	$0.751380\pi$
$720$	0	0
$721$	−3.55186e11	−1.31436
$722$	5.89015e10i	0.216759i
$723$	0	0
$724$	−2.77068e10	−0.100840
$725$	0	0
$726$	0	0
$727$	−2.88082e11	−1.03129	−0.515643	−	0.856804i	$-0.672447\pi$
$727$	−2.88082e11	−1.03129	−0.515643	+	0.856804i	$0.672447\pi$
$728$	− 3.84366e11i	− 1.36842i
$729$	0	0
$730$	0	0
$731$	1.30775e11i	0.457988i
$732$	0	0
$733$	−2.35415e11	−0.815488	−0.407744	−	0.913096i	$-0.633684\pi$
$733$	−2.35415e11	−0.815488	−0.407744	+	0.913096i	$0.633684\pi$
$734$	− 9.08104e10i	− 0.312861i
$735$	0	0
$736$	2.32445e11	0.792152
$737$	2.70455e11i	0.916695i
$738$	0	0
$739$	1.40233e11	0.470190	0.235095	−	0.971972i	$-0.424460\pi$
$739$	1.40233e11	0.470190	0.235095	+	0.971972i	$0.424460\pi$
$740$	0	0
$741$	0	0
$742$	1.53793e11	0.507364
$743$	9.10790e10i	0.298857i	0.988773	+	0.149428i	$0.0477433\pi$
$743$	9.10790e10i	0.298857i	−0.988773	+	0.149428i	$0.952257\pi$
$744$	0	0
$745$	0	0
$746$	− 7.62147e10i	− 0.246084i
$747$	0	0
$748$	−7.79131e10	−0.248888
$749$	1.15542e11i	0.367123i
$750$	0	0
$751$	−3.21915e11	−1.01200	−0.506001	−	0.862533i	$-0.668877\pi$
$751$	−3.21915e11	−1.01200	−0.506001	+	0.862533i	$0.668877\pi$
$752$	1.20392e11i	0.376466i
$753$	0	0
$754$	2.65569e11	0.821658
$755$	0	0
$756$	0	0
$757$	−3.25603e11	−0.991527	−0.495763	−	0.868458i	$-0.665112\pi$
$757$	−3.25603e11	−0.991527	−0.495763	+	0.868458i	$0.665112\pi$
$758$	2.84058e11i	0.860458i
$759$	0	0
$760$	0	0
$761$	4.36690e10i	0.130207i	0.997879	+	0.0651035i	$0.0207378\pi$
$761$	4.36690e10i	0.130207i	−0.997879	+	0.0651035i	$0.979262\pi$
$762$	0	0
$763$	4.28418e11	1.26406
$764$	− 7.12327e10i	− 0.209077i
$765$	0	0
$766$	2.13559e11	0.620301
$767$	5.91673e11i	1.70962i
$768$	0	0
$769$	9.02382e10	0.258039	0.129019	−	0.991642i	$-0.458817\pi$
$769$	9.02382e10	0.258039	0.129019	+	0.991642i	$0.458817\pi$
$770$	0	0
$771$	0	0
$772$	−2.95108e10	−0.0830830
$773$	− 2.31272e11i	− 0.647747i	−0.946100	−	0.323873i	$-0.895015\pi$
$773$	− 2.31272e11i	− 0.647747i	0.946100	−	0.323873i	$-0.104985\pi$
$774$	0	0
$775$	0	0
$776$	− 4.20596e11i	− 1.15989i
$777$	0	0
$778$	1.06590e11	0.290936
$779$	− 2.56362e11i	− 0.696152i
$780$	0	0
$781$	−5.91311e9	−0.0158932
$782$	− 5.50166e11i	− 1.47118i
$783$	0	0
$784$	7.50629e10	0.198683
$785$	0	0
$786$	0	0
$787$	6.09405e11	1.58857	0.794286	−	0.607544i	$-0.207845\pi$
$787$	6.09405e11	1.58857	0.794286	+	0.607544i	$0.207845\pi$
$788$	− 1.00467e11i	− 0.260567i
$789$	0	0
$790$	0	0
$791$	− 6.15971e10i	− 0.157346i
$792$	0	0
$793$	−2.15643e11	−0.545308
$794$	1.31333e11i	0.330440i
$795$	0	0
$796$	−3.00876e10	−0.0749437
$797$	− 1.10929e11i	− 0.274923i	−0.990507	−	0.137461i	$-0.956106\pi$
$797$	− 1.10929e11i	− 0.274923i	0.990507	−	0.137461i	$-0.0438943\pi$
$798$	0	0
$799$	−4.07649e11	−1.00023
$800$	0	0
$801$	0	0
$802$	−2.55188e10	−0.0616825
$803$	5.48586e10i	0.131942i
$804$	0	0
$805$	0	0
$806$	6.88700e11i	1.63189i
$807$	0	0
$808$	7.52903e11	1.76642
$809$	− 4.36316e10i	− 0.101861i	−0.998702	−	0.0509303i	$-0.983781\pi$
$809$	− 4.36316e10i	− 0.101861i	0.998702	−	0.0509303i	$-0.0162186\pi$
$810$	0	0
$811$	8.84218e10	0.204398	0.102199	−	0.994764i	$-0.467412\pi$
$811$	8.84218e10	0.204398	0.102199	+	0.994764i	$0.467412\pi$
$812$	7.50364e10i	0.172603i
$813$	0	0
$814$	−1.27970e11	−0.291482
$815$	0	0
$816$	0	0
$817$	−1.57142e11	−0.352698
$818$	1.52054e11i	0.339612i
$819$	0	0
$820$	0	0
$821$	3.18098e10i	0.0700145i	0.999387	+	0.0350072i	$0.0111454\pi$
$821$	3.18098e10i	0.0700145i	−0.999387	+	0.0350072i	$0.988855\pi$
$822$	0	0
$823$	2.98230e11	0.650058	0.325029	−	0.945704i	$-0.394626\pi$
$823$	2.98230e11	0.650058	0.325029	+	0.945704i	$0.394626\pi$
$824$	− 8.25386e11i	− 1.79039i
$825$	0	0
$826$	3.29352e11	0.707521
$827$	− 4.80677e11i	− 1.02762i	−0.857905	−	0.513808i	$-0.828234\pi$
$827$	− 4.80677e11i	− 1.02762i	0.857905	−	0.513808i	$-0.171766\pi$
$828$	0	0
$829$	−1.96280e11	−0.415583	−0.207792	−	0.978173i	$-0.566628\pi$
$829$	−1.96280e11	−0.415583	−0.207792	+	0.978173i	$0.566628\pi$
$830$	0	0
$831$	0	0
$832$	8.07760e11	1.68573
$833$	2.54164e11i	0.527878i
$834$	0	0
$835$	0	0
$836$	− 9.36221e10i	− 0.191670i
$837$	0	0
$838$	−4.53545e11	−0.919696
$839$	− 9.37646e11i	− 1.89230i	−0.323721	−	0.946152i	$-0.604934\pi$
$839$	− 9.37646e11i	− 1.89230i	0.323721	−	0.946152i	$-0.395066\pi$
$840$	0	0
$841$	2.94420e11	0.588549
$842$	1.51019e10i	0.0300457i
$843$	0	0
$844$	1.62644e11	0.320529
$845$	0	0
$846$	0	0
$847$	3.06049e11	0.594645
$848$	2.21672e11i	0.428673i
$849$	0	0
$850$	0	0
$851$	4.58680e11i	0.874565i
$852$	0	0
$853$	−6.26886e11	−1.18411	−0.592056	−	0.805897i	$-0.701683\pi$
$853$	−6.26886e11	−1.18411	−0.592056	+	0.805897i	$0.701683\pi$
$854$	1.20036e11i	0.225674i
$855$	0	0
$856$	−2.68497e11	−0.500086
$857$	1.01708e11i	0.188552i	0.995546	+	0.0942759i	$0.0300536\pi$
$857$	1.01708e11i	0.188552i	−0.995546	+	0.0942759i	$0.969946\pi$
$858$	0	0
$859$	2.13233e11	0.391635	0.195817	−	0.980640i	$-0.437264\pi$
$859$	2.13233e11	0.391635	0.195817	+	0.980640i	$0.437264\pi$
$860$	0	0
$861$	0	0
$862$	−6.18520e10	−0.112027
$863$	− 5.24346e11i	− 0.945311i	−0.881247	−	0.472655i	$-0.843296\pi$
$863$	− 5.24346e11i	− 0.945311i	0.881247	−	0.472655i	$-0.156704\pi$
$864$	0	0
$865$	0	0
$866$	− 7.64080e11i	− 1.35852i
$867$	0	0
$868$	−1.94592e11	−0.342804
$869$	− 4.79411e11i	− 0.840676i
$870$	0	0
$871$	1.64019e12	2.84985
$872$	9.95561e11i	1.72188i
$873$	0	0
$874$	6.61091e11	1.13296
$875$	0	0
$876$	0	0
$877$	−1.14765e11	−0.194005	−0.0970023	−	0.995284i	$-0.530925\pi$
$877$	−1.14765e11	−0.194005	−0.0970023	+	0.995284i	$0.530925\pi$
$878$	6.60536e11i	1.11152i
$879$	0	0
$880$	0	0
$881$	− 7.06861e11i	− 1.17336i	−0.809820	−	0.586679i	$-0.800435\pi$
$881$	− 7.06861e11i	− 1.17336i	0.809820	−	0.586679i	$-0.199565\pi$
$882$	0	0
$883$	1.23511e11	0.203171	0.101586	−	0.994827i	$-0.467608\pi$
$883$	1.23511e11	0.203171	0.101586	+	0.994827i	$0.467608\pi$
$884$	4.72509e11i	0.773750i
$885$	0	0
$886$	−5.66683e11	−0.919613
$887$	7.99657e11i	1.29184i	0.763405	+	0.645921i	$0.223526\pi$
$887$	7.99657e11i	1.29184i	−0.763405	+	0.645921i	$0.776474\pi$
$888$	0	0
$889$	9.22194e11	1.47644
$890$	0	0
$891$	0	0
$892$	−2.58797e11	−0.408790
$893$	− 4.89839e11i	− 0.770279i
$894$	0	0
$895$	0	0
$896$	− 1.19584e11i	− 0.185542i
$897$	0	0
$898$	9.41016e11	1.44708
$899$	− 5.33771e11i	− 0.817177i
$900$	0	0
$901$	−7.50583e11	−1.13894
$902$	− 1.68742e11i	− 0.254917i
$903$	0	0
$904$	1.43140e11	0.214332
$905$	0	0
$906$	0	0
$907$	3.05478e11	0.451389	0.225694	−	0.974198i	$-0.427535\pi$
$907$	3.05478e11	0.451389	0.225694	+	0.974198i	$0.427535\pi$
$908$	− 6.51561e10i	− 0.0958543i
$909$	0	0
$910$	0	0
$911$	6.34054e11i	0.920562i	0.887773	+	0.460281i	$0.152251\pi$
$911$	6.34054e11i	0.920562i	−0.887773	+	0.460281i	$0.847749\pi$
$912$	0	0
$913$	1.59860e11	0.230069
$914$	− 8.42116e11i	− 1.20667i
$915$	0	0
$916$	−3.28286e11	−0.466305
$917$	− 5.72914e10i	− 0.0810237i
$918$	0	0
$919$	1.51061e10	0.0211783	0.0105892	−	0.999944i	$-0.496629\pi$
$919$	1.51061e10	0.0211783	0.0105892	+	0.999944i	$0.496629\pi$
$920$	0	0
$921$	0	0
$922$	−8.98899e11	−1.24390
$923$	3.58604e10i	0.0494092i
$924$	0	0
$925$	0	0
$926$	− 7.58873e11i	− 1.03211i
$927$	0	0
$928$	−3.04820e11	−0.411009
$929$	8.37421e10i	0.112430i	0.998419	+	0.0562149i	$0.0179032\pi$
$929$	8.37421e10i	0.112430i	−0.998419	+	0.0562149i	$0.982097\pi$
$930$	0	0
$931$	−3.05409e11	−0.406521
$932$	− 2.50430e11i	− 0.331911i
$933$	0	0
$934$	5.53329e11	0.727104
$935$	0	0
$936$	0	0
$937$	8.11620e11	1.05292	0.526458	−	0.850201i	$-0.323520\pi$
$937$	8.11620e11	1.05292	0.526458	+	0.850201i	$0.323520\pi$
$938$	− 9.13002e11i	− 1.17940i
$939$	0	0
$940$	0	0
$941$	− 7.45393e11i	− 0.950664i	−0.879807	−	0.475332i	$-0.842328\pi$
$941$	− 7.45393e11i	− 0.950664i	0.879807	−	0.475332i	$-0.157672\pi$
$942$	0	0
$943$	−6.04819e11	−0.764854
$944$	4.74716e11i	0.597787i
$945$	0	0
$946$	−1.03434e11	−0.129151
$947$	− 6.69046e11i	− 0.831872i	−0.909394	−	0.415936i	$-0.863454\pi$
$947$	− 6.69046e11i	− 0.831872i	0.909394	−	0.415936i	$-0.136546\pi$
$948$	0	0
$949$	3.32693e11	0.410184
$950$	0	0
$951$	0	0
$952$	1.04421e12	1.27127
$953$	6.00889e11i	0.728489i	0.931303	+	0.364244i	$0.118673\pi$
$953$	6.00889e11i	0.728489i	−0.931303	+	0.364244i	$0.881327\pi$
$954$	0	0
$955$	0	0
$956$	− 2.88945e11i	− 0.345926i
$957$	0	0
$958$	7.39789e11	0.878306
$959$	− 5.57351e11i	− 0.658953i
$960$	0	0
$961$	5.31339e11	0.622986
$962$	7.76083e11i	0.906166i
$963$	0	0
$964$	2.97752e11	0.344784
$965$	0	0
$966$	0	0
$967$	−1.25476e12	−1.43501	−0.717503	−	0.696555i	$-0.754715\pi$
$967$	−1.25476e12	−1.43501	−0.717503	+	0.696555i	$0.754715\pi$
$968$	7.11200e11i	0.810010i
$969$	0	0
$970$	0	0
$971$	− 9.39290e11i	− 1.05663i	−0.849049	−	0.528315i	$-0.822824\pi$
$971$	− 9.39290e11i	− 1.05663i	0.849049	−	0.528315i	$-0.177176\pi$
$972$	0	0
$973$	5.98996e11	0.668302
$974$	2.62198e10i	0.0291336i
$975$	0	0
$976$	−1.73016e11	−0.190672
$977$	− 3.92709e11i	− 0.431015i	−0.976502	−	0.215508i	$-0.930859\pi$
$977$	− 3.92709e11i	− 0.431015i	0.976502	−	0.215508i	$-0.0691406\pi$
$978$	0	0
$979$	2.30409e11	0.250824
$980$	0	0
$981$	0	0
$982$	5.79280e11	0.622935
$983$	1.01412e12i	1.08611i	0.839696	+	0.543056i	$0.182733\pi$
$983$	1.01412e12i	1.08611i	−0.839696	+	0.543056i	$0.817267\pi$
$984$	0	0
$985$	0	0
$986$	7.21468e11i	0.763325i
$987$	0	0
$988$	−5.67777e11	−0.595867
$989$	3.70734e11i	0.387505i
$990$	0	0
$991$	9.73017e11	1.00885	0.504424	−	0.863456i	$-0.331705\pi$
$991$	9.73017e11	1.00885	0.504424	+	0.863456i	$0.331705\pi$
$992$	− 7.90489e11i	− 0.816299i
$993$	0	0
$994$	1.99615e10	0.0204478
$995$	0	0
$996$	0	0
$997$	3.28222e11	0.332191	0.166095	−	0.986110i	$-0.446884\pi$
$997$	3.28222e11	0.332191	0.166095	+	0.986110i	$0.446884\pi$
$998$	3.91071e11i	0.394215i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.9.c.e.26.12		16
3.2	odd	2	inner	225.9.c.e.26.6		16
5.2	odd	4		45.9.d.a.44.6	yes	16
5.3	odd	4		45.9.d.a.44.12	yes	16
5.4	even	2	inner	225.9.c.e.26.5		16
15.2	even	4		45.9.d.a.44.11	yes	16
15.8	even	4		45.9.d.a.44.5	✓	16
15.14	odd	2	inner	225.9.c.e.26.11		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.9.d.a.44.5	✓	16	15.8	even	4
45.9.d.a.44.6	yes	16	5.2	odd	4
45.9.d.a.44.11	yes	16	15.2	even	4
45.9.d.a.44.12	yes	16	5.3	odd	4
225.9.c.e.26.5		16	5.4	even	2	inner
225.9.c.e.26.6		16	3.2	odd	2	inner
225.9.c.e.26.11		16	15.14	odd	2	inner
225.9.c.e.26.12		16	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	225.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+13.0310i q^{2} +86.1932 q^{4} +1918.88 q^{7} +4459.12i q^{8} +O(q^{10})\)	\(q+13.0310i q^{2} +86.1932 q^{4} +1918.88 q^{7} +4459.12i q^{8} -7407.11i q^{11} -44920.9 q^{13} +25004.9i q^{14} -36041.3 q^{16} -122036. i q^{17} +146641. q^{19} +96522.1 q^{22} -345961. i q^{23} -585364. i q^{26} +165395. q^{28} +453681. i q^{29} -1.17653e6 q^{31} +671880. i q^{32} +1.59025e6 q^{34} -1.32581e6 q^{37} +1.91088e6i q^{38} -1.74823e6i q^{41} -1.07161e6 q^{43} -638443. i q^{44} +4.50822e6 q^{46} -3.34039e6i q^{47} -2.08269e6 q^{49} -3.87187e6 q^{52} -6.15049e6i q^{53} +8.55652e6i q^{56} -5.91192e6 q^{58} -1.31715e7i q^{59} +4.80050e6 q^{61} -1.53314e7i q^{62} -1.79818e7 q^{64} -3.65129e7 q^{67} -1.05187e7i q^{68} -798301. i q^{71} -7.40621e6 q^{73} -1.72767e7i q^{74} +1.26395e7 q^{76} -1.42134e7i q^{77} +6.47230e7 q^{79} +2.27811e7 q^{82} +2.15820e7i q^{83} -1.39641e7i q^{86} +3.30292e7 q^{88} +3.11064e7i q^{89} -8.61978e7 q^{91} -2.98195e7i q^{92} +4.35286e7 q^{94} -9.43227e7 q^{97} -2.71396e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 1572 q^{4}+O(q^{10}) \) 16 * q - 1572 * q^4	\( 16 q - 1572 q^{4} - 12844 q^{16} - 263600 q^{19} - 4312984 q^{31} - 2439352 q^{34} - 5376344 q^{46} + 7338968 q^{49} + 26097968 q^{61} + 123290756 q^{64} + 222695760 q^{76} + 262645864 q^{79} + 350561736 q^{91} + 789068384 q^{94}+O(q^{100}) \) 16 * q - 1572 * q^4 - 12844 * q^16 - 263600 * q^19 - 4312984 * q^31 - 2439352 * q^34 - 5376344 * q^46 + 7338968 * q^49 + 26097968 * q^61 + 123290756 * q^64 + 222695760 * q^76 + 262645864 * q^79 + 350561736 * q^91 + 789068384 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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