LMFDB - Embedded newform 144.11.g.e.127.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,11,Mod(127,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 11, names="a")

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

chi := DirichletCharacter("144.127");

S:= CuspForms(chi, 11);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 11 $
Character orbit:	$[\chi]$	$=$	144.g (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:	$91.4914443850$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{2545})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 637x^{2} + 636x + 404496 $ x^4 - x^3 + 637x^2 + 636x + 404496
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.3
Root			$12.8620 + 22.2776i$ of defining polynomial
Character	$\chi$	$=$	144.127
Dual form			144.11.g.e.127.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+460.752 q^{5} -26520.8i q^{7} +O(q^{10})$	$q+460.752 q^{5} -26520.8i q^{7} -300389. i q^{11} +571995. q^{13} +2.15197e6 q^{17} +272767. i q^{19} -3.47555e6i q^{23} -9.55333e6 q^{25} +1.13627e7 q^{29} +244478. i q^{31} -1.22195e7i q^{35} -7.44420e7 q^{37} +5.95090e7 q^{41} +2.21456e8i q^{43} -1.99625e8i q^{47} -4.20876e8 q^{49} +6.65587e8 q^{53} -1.38405e8i q^{55} -7.46014e8i q^{59} +5.06208e8 q^{61} +2.63548e8 q^{65} -7.63047e8i q^{67} +1.74011e9i q^{71} -4.86481e8 q^{73} -7.96655e9 q^{77} +1.61904e9i q^{79} -4.02373e9i q^{83} +9.91524e8 q^{85} -2.61045e9 q^{89} -1.51697e10i q^{91} +1.25678e8i q^{95} +1.28953e10 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3000 q^{5}+O(q^{10}) $ 4 * q - 3000 * q^5	$ 4 q - 3000 q^{5} + 864136 q^{13} + 55128 q^{17} - 30948820 q^{25} + 80790312 q^{29} - 212511640 q^{37} + 544453272 q^{41} - 1845182972 q^{49} + 888804072 q^{53} - 208335256 q^{61} + 1075819920 q^{65} + 4881669448 q^{73} - 18371224512 q^{77} + 10313912880 q^{85} - 21540382536 q^{89} + 14965768456 q^{97}+O(q^{100}) $ 4 * q - 3000 * q^5 + 864136 * q^13 + 55128 * q^17 - 30948820 * q^25 + 80790312 * q^29 - 212511640 * q^37 + 544453272 * q^41 - 1845182972 * q^49 + 888804072 * q^53 - 208335256 * q^61 + 1075819920 * q^65 + 4881669448 * q^73 - 18371224512 * q^77 + 10313912880 * q^85 - 21540382536 * q^89 + 14965768456 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	460.752	0.147441	0.0737203	−	0.997279i	$-0.476513\pi$
$5$	460.752	0.147441	0.0737203	+	0.997279i	$0.476513\pi$
$6$	0	0
$7$	− 26520.8i	− 1.57796i	−0.614419	−	0.788980i	$-0.710610\pi$
$7$	− 26520.8i	− 1.57796i	0.614419	−	0.788980i	$-0.289390\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 300389.i	− 1.86518i	−0.360939	−	0.932590i	$-0.617544\pi$
$11$	− 300389.i	− 1.86518i	0.360939	−	0.932590i	$-0.382456\pi$
$12$	0	0
$13$	571995.	1.54055	0.770274	−	0.637713i	$-0.220119\pi$
$13$	571995.	1.54055	0.770274	+	0.637713i	$0.220119\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.15197e6	1.51562	0.757812	−	0.652473i	$-0.226268\pi$
$17$	2.15197e6	1.51562	0.757812	+	0.652473i	$0.226268\pi$
$18$	0	0
$19$	272767.i	0.110160i	0.998482	+	0.0550801i	$0.0175414\pi$
$19$	272767.i	0.110160i	−0.998482	+	0.0550801i	$0.982459\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 3.47555e6i	− 0.539989i	−0.962862	−	0.269994i	$-0.912978\pi$
$23$	− 3.47555e6i	− 0.539989i	0.962862	−	0.269994i	$-0.0870219\pi$
$24$	0	0
$25$	−9.55333e6	−0.978261
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.13627e7	0.553978	0.276989	−	0.960873i	$-0.410663\pi$
$29$	1.13627e7	0.553978	0.276989	+	0.960873i	$0.410663\pi$
$30$	0	0
$31$	244478.i	0.00853949i	0.999991	+	0.00426975i	$0.00135911\pi$
$31$	244478.i	0.00853949i	−0.999991	+	0.00426975i	$0.998641\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1.22195e7i	− 0.232655i
$36$	0	0
$37$	−7.44420e7	−1.07352	−0.536759	−	0.843736i	$-0.680352\pi$
$37$	−7.44420e7	−1.07352	−0.536759	+	0.843736i	$0.680352\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.95090e7	0.513646	0.256823	−	0.966458i	$-0.417324\pi$
$41$	5.95090e7	0.513646	0.256823	+	0.966458i	$0.417324\pi$
$42$	0	0
$43$	2.21456e8i	1.50642i	0.657783	+	0.753208i	$0.271495\pi$
$43$	2.21456e8i	1.50642i	−0.657783	+	0.753208i	$0.728505\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 1.99625e8i	− 0.870412i	−0.900331	−	0.435206i	$-0.856676\pi$
$47$	− 1.99625e8i	− 0.870412i	0.900331	−	0.435206i	$-0.143324\pi$
$48$	0	0
$49$	−4.20876e8	−1.48996
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.65587e8	1.59157	0.795784	−	0.605580i	$-0.207059\pi$
$53$	6.65587e8	1.59157	0.795784	+	0.605580i	$0.207059\pi$
$54$	0	0
$55$	− 1.38405e8i	− 0.275003i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 7.46014e8i	− 1.04349i	−0.853102	−	0.521743i	$-0.825282\pi$
$59$	− 7.46014e8i	− 1.04349i	0.853102	−	0.521743i	$-0.174718\pi$
$60$	0	0
$61$	5.06208e8	0.599350	0.299675	−	0.954041i	$-0.403122\pi$
$61$	5.06208e8	0.599350	0.299675	+	0.954041i	$0.403122\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.63548e8	0.227139
$66$	0	0
$67$	− 7.63047e8i	− 0.565168i	−0.959243	−	0.282584i	$-0.908808\pi$
$67$	− 7.63047e8i	− 0.565168i	0.959243	−	0.282584i	$-0.0911916\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.74011e9i	0.964462i	0.876044	+	0.482231i	$0.160173\pi$
$71$	1.74011e9i	0.964462i	−0.876044	+	0.482231i	$0.839827\pi$
$72$	0	0
$73$	−4.86481e8	−0.234667	−0.117333	−	0.993093i	$-0.537435\pi$
$73$	−4.86481e8	−0.234667	−0.117333	+	0.993093i	$0.537435\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−7.96655e9	−2.94318
$78$	0	0
$79$	1.61904e9i	0.526166i	0.964773	+	0.263083i	$0.0847393\pi$
$79$	1.61904e9i	0.526166i	−0.964773	+	0.263083i	$0.915261\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 4.02373e9i	− 1.02150i	−0.859729	−	0.510750i	$-0.829368\pi$
$83$	− 4.02373e9i	− 1.02150i	0.859729	−	0.510750i	$-0.170632\pi$
$84$	0	0
$85$	9.91524e8	0.223465
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.61045e9	−0.467483	−0.233742	−	0.972299i	$-0.575097\pi$
$89$	−2.61045e9	−0.467483	−0.233742	+	0.972299i	$0.575097\pi$
$90$	0	0
$91$	− 1.51697e10i	− 2.43092i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.25678e8i	0.0162421i
$96$	0	0
$97$	1.28953e10	1.50166	0.750830	−	0.660496i	$-0.229654\pi$
$97$	1.28953e10	1.50166	0.750830	+	0.660496i	$0.229654\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.09086e10	1.03792	0.518958	−	0.854800i	$-0.326320\pi$
$101$	1.09086e10	1.03792	0.518958	+	0.854800i	$0.326320\pi$
$102$	0	0
$103$	3.44293e9i	0.296990i	0.988913	+	0.148495i	$0.0474429\pi$
$103$	3.44293e9i	0.296990i	−0.988913	+	0.148495i	$0.952557\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.67714e10i	− 1.19577i	−0.801580	−	0.597887i	$-0.796007\pi$
$107$	− 1.67714e10i	− 1.19577i	0.801580	−	0.597887i	$-0.203993\pi$
$108$	0	0
$109$	−1.34589e10	−0.874739	−0.437370	−	0.899282i	$-0.644090\pi$
$109$	−1.34589e10	−0.874739	−0.437370	+	0.899282i	$0.644090\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.36700e10	−1.82747	−0.913735	−	0.406310i	$-0.866815\pi$
$113$	−3.36700e10	−1.82747	−0.913735	+	0.406310i	$0.866815\pi$
$114$	0	0
$115$	− 1.60137e9i	− 0.0796163i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 5.70719e10i	− 2.39159i
$120$	0	0
$121$	−6.42961e10	−2.47889
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−8.90125e9	−0.291676
$126$	0	0
$127$	5.75809e8i	0.0174285i	0.999962	+	0.00871424i	$0.00277386\pi$
$127$	5.75809e8i	0.0174285i	−0.999962	+	0.00871424i	$0.997226\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.99650e10i	0.517504i	0.965944	+	0.258752i	$0.0833112\pi$
$131$	1.99650e10i	0.517504i	−0.965944	+	0.258752i	$0.916689\pi$
$132$	0	0
$133$	7.23400e9	0.173828
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.06890e10	0.843091	0.421545	−	0.906807i	$-0.361488\pi$
$137$	4.06890e10	0.843091	0.421545	+	0.906807i	$0.361488\pi$
$138$	0	0
$139$	− 1.67505e10i	− 0.322816i	−0.986888	−	0.161408i	$-0.948397\pi$
$139$	− 1.67505e10i	− 0.322816i	0.986888	−	0.161408i	$-0.0516035\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 1.71821e11i	− 2.87340i
$144$	0	0
$145$	5.23539e9	0.0816788
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.75416e10	−0.375023	−0.187511	−	0.982262i	$-0.560042\pi$
$149$	−2.75416e10	−0.375023	−0.187511	+	0.982262i	$0.560042\pi$
$150$	0	0
$151$	− 1.11816e11i	− 1.42435i	−0.702001	−	0.712176i	$-0.747710\pi$
$151$	− 1.11816e11i	− 1.42435i	0.702001	−	0.712176i	$-0.252290\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.12644e8i	0.00125907i
$156$	0	0
$157$	−1.18681e11	−1.24418	−0.622092	−	0.782944i	$-0.713717\pi$
$157$	−1.18681e11	−1.24418	−0.622092	+	0.782944i	$0.713717\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−9.21744e10	−0.852081
$162$	0	0
$163$	− 5.73474e10i	− 0.498397i	−0.968452	−	0.249199i	$-0.919833\pi$
$163$	− 5.73474e10i	− 0.498397i	0.968452	−	0.249199i	$-0.0801672\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1.54722e11i	− 1.19116i	−0.803296	−	0.595580i	$-0.796922\pi$
$167$	− 1.54722e11i	− 1.19116i	0.803296	−	0.595580i	$-0.203078\pi$
$168$	0	0
$169$	1.89320e11	1.37329
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2.91757e11	−1.88274	−0.941369	−	0.337378i	$-0.890460\pi$
$173$	−2.91757e11	−1.88274	−0.941369	+	0.337378i	$0.890460\pi$
$174$	0	0
$175$	2.53362e11i	1.54366i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.65669e11i	1.44569i	0.691010	+	0.722845i	$0.257166\pi$
$179$	2.65669e11i	1.44569i	−0.691010	+	0.722845i	$0.742834\pi$
$180$	0	0
$181$	2.43381e11	1.25283	0.626417	−	0.779488i	$-0.284521\pi$
$181$	2.43381e11	1.25283	0.626417	+	0.779488i	$0.284521\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3.42993e10	−0.158280
$186$	0	0
$187$	− 6.46428e11i	− 2.82691i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.38113e10i	0.369053i	0.982828	+	0.184526i	$0.0590751\pi$
$191$	9.38113e10i	0.369053i	−0.982828	+	0.184526i	$0.940925\pi$
$192$	0	0
$193$	−3.11495e11	−1.16323	−0.581614	−	0.813465i	$-0.697578\pi$
$193$	−3.11495e11	−1.16323	−0.581614	+	0.813465i	$0.697578\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.45186e10	−0.0826352	−0.0413176	−	0.999146i	$-0.513156\pi$
$197$	−2.45186e10	−0.0826352	−0.0413176	+	0.999146i	$0.513156\pi$
$198$	0	0
$199$	2.81079e11i	0.900664i	0.892861	+	0.450332i	$0.148694\pi$
$199$	2.81079e11i	0.900664i	−0.892861	+	0.450332i	$0.851306\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 3.01348e11i	− 0.874155i
$204$	0	0
$205$	2.74189e10	0.0757322
$206$	0	0
$207$	0	0
$208$	0	0
$209$	8.19363e10	0.205468
$210$	0	0
$211$	2.27760e11i	0.544585i	0.962214	+	0.272293i	$0.0877819\pi$
$211$	2.27760e11i	0.544585i	−0.962214	+	0.272293i	$0.912218\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.02036e11i	0.222107i
$216$	0	0
$217$	6.48376e9	0.0134750
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.23092e12	2.33489
$222$	0	0
$223$	6.95984e11i	1.26204i	0.775765	+	0.631022i	$0.217364\pi$
$223$	6.95984e11i	1.26204i	−0.775765	+	0.631022i	$0.782636\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.34405e11i	0.720718i	0.932814	+	0.360359i	$0.117346\pi$
$227$	4.34405e11i	0.720718i	−0.932814	+	0.360359i	$0.882654\pi$
$228$	0	0
$229$	−3.96862e11	−0.630176	−0.315088	−	0.949063i	$-0.602034\pi$
$229$	−3.96862e11	−0.630176	−0.315088	+	0.949063i	$0.602034\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.02024e11	1.02229	0.511143	−	0.859496i	$-0.329222\pi$
$233$	7.02024e11	1.02229	0.511143	+	0.859496i	$0.329222\pi$
$234$	0	0
$235$	− 9.19774e10i	− 0.128334i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.74178e10i	0.0351596i	0.999845	+	0.0175798i	$0.00559611\pi$
$239$	2.74178e10i	0.0351596i	−0.999845	+	0.0175798i	$0.994404\pi$
$240$	0	0
$241$	6.55046e11	0.805725	0.402862	−	0.915261i	$-0.368015\pi$
$241$	6.55046e11	0.805725	0.402862	+	0.915261i	$0.368015\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.93919e11	−0.219680
$246$	0	0
$247$	1.56022e11i	0.169707i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7.33706e11i	0.736467i	0.929733	+	0.368233i	$0.120037\pi$
$251$	7.33706e11i	0.736467i	−0.929733	+	0.368233i	$0.879963\pi$
$252$	0	0
$253$	−1.04402e12	−1.00718
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.26355e10	0.0291088	0.0145544	−	0.999894i	$-0.495367\pi$
$257$	3.26355e10	0.0291088	0.0145544	+	0.999894i	$0.495367\pi$
$258$	0	0
$259$	1.97426e12i	1.69397i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.14043e12i	1.70107i	0.525921	+	0.850533i	$0.323721\pi$
$263$	2.14043e12i	1.70107i	−0.525921	+	0.850533i	$0.676279\pi$
$264$	0	0
$265$	3.06670e11	0.234662
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7.80162e11	−0.553890	−0.276945	−	0.960886i	$-0.589322\pi$
$269$	−7.80162e11	−0.553890	−0.276945	+	0.960886i	$0.589322\pi$
$270$	0	0
$271$	− 7.92563e11i	− 0.542235i	−0.962546	−	0.271117i	$-0.912607\pi$
$271$	− 7.92563e11i	− 0.542235i	0.962546	−	0.271117i	$-0.0873932\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.86972e12i	1.82463i
$276$	0	0
$277$	1.11288e12	0.682414	0.341207	−	0.939988i	$-0.389164\pi$
$277$	1.11288e12	0.682414	0.341207	+	0.939988i	$0.389164\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.34971e12	1.34117	0.670584	−	0.741834i	$-0.266044\pi$
$281$	2.34971e12	1.34117	0.670584	+	0.741834i	$0.266044\pi$
$282$	0	0
$283$	− 6.45442e11i	− 0.355570i	−0.984069	−	0.177785i	$-0.943107\pi$
$283$	− 6.45442e11i	− 0.355570i	0.984069	−	0.177785i	$-0.0568931\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.57823e12i	− 0.810512i
$288$	0	0
$289$	2.61498e12	1.29712
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.61003e12	−0.745581	−0.372790	−	0.927916i	$-0.621599\pi$
$293$	−1.61003e12	−0.745581	−0.372790	+	0.927916i	$0.621599\pi$
$294$	0	0
$295$	− 3.43727e11i	− 0.153852i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 1.98800e12i	− 0.831879i
$300$	0	0
$301$	5.87318e12	2.37706
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.33236e11	0.0883684
$306$	0	0
$307$	3.29596e11i	0.120862i	0.998172	+	0.0604310i	$0.0192475\pi$
$307$	3.29596e11i	0.120862i	−0.998172	+	0.0604310i	$0.980752\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.65237e12i	1.25537i	0.778466	+	0.627687i	$0.215998\pi$
$311$	3.65237e12i	1.25537i	−0.778466	+	0.627687i	$0.784002\pi$
$312$	0	0
$313$	−4.81641e12	−1.60325	−0.801627	−	0.597825i	$-0.796032\pi$
$313$	−4.81641e12	−1.60325	−0.801627	+	0.597825i	$0.796032\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.50894e11	0.0471386	0.0235693	−	0.999722i	$-0.492497\pi$
$317$	1.50894e11	0.0471386	0.0235693	+	0.999722i	$0.492497\pi$
$318$	0	0
$319$	− 3.41324e12i	− 1.03327i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.86987e11i	0.166961i
$324$	0	0
$325$	−5.46446e12	−1.50706
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−5.29420e12	−1.37347
$330$	0	0
$331$	3.24447e12i	0.816589i	0.912850	+	0.408295i	$0.133876\pi$
$331$	3.24447e12i	0.816589i	−0.912850	+	0.408295i	$0.866124\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 3.51575e11i	− 0.0833287i
$336$	0	0
$337$	2.26171e11	0.0520339	0.0260170	−	0.999662i	$-0.491718\pi$
$337$	2.26171e11	0.0520339	0.0260170	+	0.999662i	$0.491718\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7.34386e10	0.0159277
$342$	0	0
$343$	3.67049e12i	0.773133i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 7.48815e12i	− 1.48843i	−0.667942	−	0.744213i	$-0.732825\pi$
$347$	− 7.48815e12i	− 1.48843i	0.667942	−	0.744213i	$-0.267175\pi$
$348$	0	0
$349$	−8.66532e12	−1.67362	−0.836811	−	0.547492i	$-0.815583\pi$
$349$	−8.66532e12	−1.67362	−0.836811	+	0.547492i	$0.815583\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7.70658e12	1.40601	0.703005	−	0.711185i	$-0.251841\pi$
$353$	7.70658e12	1.40601	0.703005	+	0.711185i	$0.251841\pi$
$354$	0	0
$355$	8.01759e11i	0.142201i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 1.24065e12i	− 0.208054i	−0.994574	−	0.104027i	$-0.966827\pi$
$359$	− 1.24065e12i	− 0.208054i	0.994574	−	0.104027i	$-0.0331728\pi$
$360$	0	0
$361$	6.05666e12	0.987865
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.24147e11	−0.0345994
$366$	0	0
$367$	− 9.99787e12i	− 1.50168i	−0.660485	−	0.750839i	$-0.729649\pi$
$367$	− 9.99787e12i	− 1.50168i	0.660485	−	0.750839i	$-0.270351\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 1.76519e13i	− 2.51143i
$372$	0	0
$373$	1.57643e12	0.218338	0.109169	−	0.994023i	$-0.465181\pi$
$373$	1.57643e12	0.218338	0.109169	+	0.994023i	$0.465181\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.49942e12	0.853430
$378$	0	0
$379$	2.26771e12i	0.289995i	0.989432	+	0.144998i	$0.0463174\pi$
$379$	2.26771e12i	0.289995i	−0.989432	+	0.144998i	$0.953683\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 1.17878e13i	− 1.43034i	−0.698951	−	0.715169i	$-0.746350\pi$
$383$	− 1.17878e13i	− 1.43034i	0.698951	−	0.715169i	$-0.253650\pi$
$384$	0	0
$385$	−3.67060e12	−0.433944
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.66987e13	−1.87471	−0.937356	−	0.348372i	$-0.886734\pi$
$389$	−1.66987e13	−1.87471	−0.937356	+	0.348372i	$0.886734\pi$
$390$	0	0
$391$	− 7.47929e12i	− 0.818420i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	7.45977e11i	0.0775782i
$396$	0	0
$397$	−9.31379e12	−0.944439	−0.472219	−	0.881481i	$-0.656547\pi$
$397$	−9.31379e12	−0.944439	−0.472219	+	0.881481i	$0.656547\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.49630e12	0.337199	0.168600	−	0.985685i	$-0.446076\pi$
$401$	3.49630e12	0.337199	0.168600	+	0.985685i	$0.446076\pi$
$402$	0	0
$403$	1.39840e11i	0.0131555i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.23616e13i	2.00230i
$408$	0	0
$409$	−2.14101e12	−0.187069	−0.0935345	−	0.995616i	$-0.529817\pi$
$409$	−2.14101e12	−0.187069	−0.0935345	+	0.995616i	$0.529817\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.97849e13	−1.64658
$414$	0	0
$415$	− 1.85394e12i	− 0.150611i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 7.93229e11i	− 0.0614226i	−0.999528	−	0.0307113i	$-0.990223\pi$
$419$	− 7.93229e11i	− 0.0614226i	0.999528	−	0.0307113i	$-0.00977725\pi$
$420$	0	0
$421$	6.83611e12	0.516891	0.258445	−	0.966026i	$-0.416790\pi$
$421$	6.83611e12	0.516891	0.258445	+	0.966026i	$0.416790\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.05585e13	−1.48268
$426$	0	0
$427$	− 1.34250e13i	− 0.945749i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 1.47056e13i	− 0.988771i	−0.869243	−	0.494385i	$-0.835393\pi$
$431$	− 1.47056e13i	− 0.988771i	0.869243	−	0.494385i	$-0.164607\pi$
$432$	0	0
$433$	2.02190e13	1.32837	0.664187	−	0.747567i	$-0.268778\pi$
$433$	2.02190e13	1.32837	0.664187	+	0.747567i	$0.268778\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.48018e11	0.0594852
$438$	0	0
$439$	− 7.17511e11i	− 0.0440054i	−0.999758	−	0.0220027i	$-0.992996\pi$
$439$	− 7.17511e11i	− 0.0440054i	0.999758	−	0.0220027i	$-0.00700424\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 6.63510e12i	− 0.388892i	−0.980913	−	0.194446i	$-0.937709\pi$
$443$	− 6.63510e12i	− 0.388892i	0.980913	−	0.194446i	$-0.0622909\pi$
$444$	0	0
$445$	−1.20277e12	−0.0689260
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1.67561e12	−0.0918208	−0.0459104	−	0.998946i	$-0.514619\pi$
$449$	−1.67561e12	−0.0918208	−0.0459104	+	0.998946i	$0.514619\pi$
$450$	0	0
$451$	− 1.78759e13i	− 0.958041i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 6.98949e12i	− 0.358417i
$456$	0	0
$457$	2.64438e13	1.32661	0.663305	−	0.748349i	$-0.269153\pi$
$457$	2.64438e13	1.32661	0.663305	+	0.748349i	$0.269153\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−8.10027e12	−0.389041	−0.194520	−	0.980898i	$-0.562315\pi$
$461$	−8.10027e12	−0.389041	−0.194520	+	0.980898i	$0.562315\pi$
$462$	0	0
$463$	3.41695e13i	1.60595i	0.596010	+	0.802977i	$0.296752\pi$
$463$	3.41695e13i	1.60595i	−0.596010	+	0.802977i	$0.703248\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.00344e13i	0.901971i	0.892531	+	0.450986i	$0.148927\pi$
$467$	2.00344e13i	0.901971i	−0.892531	+	0.450986i	$0.851073\pi$
$468$	0	0
$469$	−2.02366e13	−0.891812
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.65229e13	2.80974
$474$	0	0
$475$	− 2.60584e12i	− 0.107765i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8.38613e12i	0.332571i	0.986078	+	0.166285i	$0.0531773\pi$
$479$	8.38613e12i	0.332571i	−0.986078	+	0.166285i	$0.946823\pi$
$480$	0	0
$481$	−4.25804e13	−1.65381
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.94152e12	0.221406
$486$	0	0
$487$	1.22059e13i	0.445580i	0.974866	+	0.222790i	$0.0715165\pi$
$487$	1.22059e13i	0.445580i	−0.974866	+	0.222790i	$0.928484\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.98580e13i	1.74714i	0.486700	+	0.873569i	$0.338200\pi$
$491$	4.98580e13i	1.74714i	−0.486700	+	0.873569i	$0.661800\pi$
$492$	0	0
$493$	2.44522e13	0.839622
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.61491e13	1.52188
$498$	0	0
$499$	3.39005e13i	1.09573i	0.836567	+	0.547864i	$0.184559\pi$
$499$	3.39005e13i	1.09573i	−0.836567	+	0.547864i	$0.815441\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 4.11458e13i	− 1.27787i	−0.769263	−	0.638933i	$-0.779376\pi$
$503$	− 4.11458e13i	− 1.27787i	0.769263	−	0.638933i	$-0.220624\pi$
$504$	0	0
$505$	5.02615e12	0.153031
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−5.62334e13	−1.64591	−0.822954	−	0.568108i	$-0.807675\pi$
$509$	−5.62334e13	−1.64591	−0.822954	+	0.568108i	$0.807675\pi$
$510$	0	0
$511$	1.29019e13i	0.370295i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.58634e12i	0.0437884i
$516$	0	0
$517$	−5.99650e13	−1.62347
$518$	0	0
$519$	0	0
$520$	0	0
$521$	3.69857e13	0.963486	0.481743	−	0.876313i	$-0.340004\pi$
$521$	3.69857e13	0.963486	0.481743	+	0.876313i	$0.340004\pi$
$522$	0	0
$523$	− 2.47557e13i	− 0.632656i	−0.948650	−	0.316328i	$-0.897550\pi$
$523$	− 2.47557e13i	− 0.632656i	0.948650	−	0.316328i	$-0.102450\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.26110e11i	0.0129427i
$528$	0	0
$529$	2.93470e13	0.708412
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.40389e13	0.791296
$534$	0	0
$535$	− 7.72743e12i	− 0.176306i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.26427e14i	2.77904i
$540$	0	0
$541$	1.33326e13	0.287693	0.143847	−	0.989600i	$-0.454053\pi$
$541$	1.33326e13	0.287693	0.143847	+	0.989600i	$0.454053\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.20123e12	−0.128972
$546$	0	0
$547$	7.37426e13i	1.50585i	0.658106	+	0.752925i	$0.271358\pi$
$547$	7.37426e13i	1.50585i	−0.658106	+	0.752925i	$0.728642\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.09938e12i	0.0610263i
$552$	0	0
$553$	4.29383e13	0.830269
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.31265e13	0.244835	0.122418	−	0.992479i	$-0.460935\pi$
$557$	1.31265e13	0.244835	0.122418	+	0.992479i	$0.460935\pi$
$558$	0	0
$559$	1.26672e14i	2.32071i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 8.96337e12i	− 0.158463i	−0.996856	−	0.0792317i	$-0.974753\pi$
$563$	− 8.96337e12i	− 0.158463i	0.996856	−	0.0792317i	$-0.0252467\pi$
$564$	0	0
$565$	−1.55135e13	−0.269443
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.45009e13	0.578454	0.289227	−	0.957260i	$-0.406602\pi$
$569$	3.45009e13	0.578454	0.289227	+	0.957260i	$0.406602\pi$
$570$	0	0
$571$	− 8.26686e13i	− 1.36195i	−0.732309	−	0.680973i	$-0.761557\pi$
$571$	− 8.26686e13i	− 1.36195i	0.732309	−	0.680973i	$-0.238443\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.32031e13i	0.528250i
$576$	0	0
$577$	7.28395e13	1.13891	0.569453	−	0.822024i	$-0.307155\pi$
$577$	7.28395e13	1.13891	0.569453	+	0.822024i	$0.307155\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.06712e14	−1.61189
$582$	0	0
$583$	− 1.99935e14i	− 2.96856i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.08055e13i	− 0.155044i	−0.996991	−	0.0775220i	$-0.975299\pi$
$587$	− 1.08055e13i	− 0.155044i	0.996991	−	0.0775220i	$-0.0247008\pi$
$588$	0	0
$589$	−6.66857e10	−0.000940712 0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	7.57687e13	1.03328	0.516638	−	0.856204i	$-0.327183\pi$
$593$	7.57687e13	1.03328	0.516638	+	0.856204i	$0.327183\pi$
$594$	0	0
$595$	− 2.62960e13i	− 0.352618i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.03062e13i	1.04139i	0.853742	+	0.520697i	$0.174328\pi$
$599$	8.03062e13i	1.04139i	−0.853742	+	0.520697i	$0.825672\pi$
$600$	0	0
$601$	2.41513e13	0.308013	0.154006	−	0.988070i	$-0.450782\pi$
$601$	2.41513e13	0.308013	0.154006	+	0.988070i	$0.450782\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.96245e13	−0.365489
$606$	0	0
$607$	8.40810e13i	1.02036i	0.860067	+	0.510181i	$0.170422\pi$
$607$	8.40810e13i	1.02036i	−0.860067	+	0.510181i	$0.829578\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 1.14184e14i	− 1.34091i
$612$	0	0
$613$	1.04158e14	1.20334	0.601672	−	0.798744i	$-0.294502\pi$
$613$	1.04158e14	1.20334	0.601672	+	0.798744i	$0.294502\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3.14098e13	−0.351268	−0.175634	−	0.984455i	$-0.556198\pi$
$617$	−3.14098e13	−0.351268	−0.175634	+	0.984455i	$0.556198\pi$
$618$	0	0
$619$	6.07637e12i	0.0668638i	0.999441	+	0.0334319i	$0.0106437\pi$
$619$	6.07637e12i	0.0668638i	−0.999441	+	0.0334319i	$0.989356\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.92313e13i	0.737670i
$624$	0	0
$625$	8.91930e13	0.935256
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.60197e14	−1.62705
$630$	0	0
$631$	1.22040e14i	1.21998i	0.792408	+	0.609992i	$0.208827\pi$
$631$	1.22040e14i	1.21998i	−0.792408	+	0.609992i	$0.791173\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.65305e11i	0.00256967i
$636$	0	0
$637$	−2.40739e14	−2.29535
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3.26570e13	−0.301777	−0.150888	−	0.988551i	$-0.548213\pi$
$641$	−3.26570e13	−0.301777	−0.150888	+	0.988551i	$0.548213\pi$
$642$	0	0
$643$	− 5.82103e13i	− 0.529597i	−0.964304	−	0.264798i	$-0.914695\pi$
$643$	− 5.82103e13i	− 0.529597i	0.964304	−	0.264798i	$-0.0853054\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 3.61095e13i	− 0.318493i	−0.987239	−	0.159246i	$-0.949094\pi$
$647$	− 3.61095e13i	− 0.318493i	0.987239	−	0.159246i	$-0.0509065\pi$
$648$	0	0
$649$	−2.24094e14	−1.94629
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.25915e13	0.527168	0.263584	−	0.964636i	$-0.415095\pi$
$653$	6.25915e13	0.527168	0.263584	+	0.964636i	$0.415095\pi$
$654$	0	0
$655$	9.19892e12i	0.0763010i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 2.22237e14i	− 1.78809i	−0.447977	−	0.894045i	$-0.647855\pi$
$659$	− 2.22237e14i	− 1.78809i	0.447977	−	0.894045i	$-0.352145\pi$
$660$	0	0
$661$	4.78576e12	0.0379266	0.0189633	−	0.999820i	$-0.493963\pi$
$661$	4.78576e12	0.0379266	0.0189633	+	0.999820i	$0.493963\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.33308e12	0.0256293
$666$	0	0
$667$	− 3.94918e13i	− 0.299142i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1.52059e14i	− 1.11789i
$672$	0	0
$673$	−9.56981e13	−0.693151	−0.346575	−	0.938022i	$-0.612656\pi$
$673$	−9.56981e13	−0.693151	−0.346575	+	0.938022i	$0.612656\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−4.01902e13	−0.282603	−0.141302	−	0.989967i	$-0.545129\pi$
$677$	−4.01902e13	−0.282603	−0.141302	+	0.989967i	$0.545129\pi$
$678$	0	0
$679$	− 3.41992e14i	− 2.36956i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1.55432e14i	− 1.04577i	−0.852403	−	0.522886i	$-0.824856\pi$
$683$	− 1.55432e14i	− 1.04577i	0.852403	−	0.522886i	$-0.175144\pi$
$684$	0	0
$685$	1.87475e13	0.124306
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.80712e14	2.45189
$690$	0	0
$691$	2.63049e14i	1.66973i	0.550457	+	0.834864i	$0.314454\pi$
$691$	2.63049e14i	1.66973i	−0.550457	+	0.834864i	$0.685546\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 7.71784e12i	− 0.0475962i
$696$	0	0
$697$	1.28062e14	0.778494
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−5.80243e13	−0.342783	−0.171392	−	0.985203i	$-0.554826\pi$
$701$	−5.80243e13	−0.342783	−0.171392	+	0.985203i	$0.554826\pi$
$702$	0	0
$703$	− 2.03053e13i	− 0.118259i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 2.89304e14i	− 1.63779i
$708$	0	0
$709$	−7.54928e13	−0.421381	−0.210690	−	0.977553i	$-0.567571\pi$
$709$	−7.54928e13	−0.421381	−0.210690	+	0.977553i	$0.567571\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.49698e11	0.00461123
$714$	0	0
$715$	− 7.91668e13i	− 0.423656i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1.76096e14i	− 0.916444i	−0.888838	−	0.458222i	$-0.848486\pi$
$719$	− 1.76096e14i	− 0.916444i	0.888838	−	0.458222i	$-0.151514\pi$
$720$	0	0
$721$	9.13091e13	0.468638
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.08552e14	−0.541935
$726$	0	0
$727$	− 1.17702e14i	− 0.579578i	−0.957091	−	0.289789i	$-0.906415\pi$
$727$	− 1.17702e14i	− 0.579578i	0.957091	−	0.289789i	$-0.0935851\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	4.76566e14i	2.28316i
$732$	0	0
$733$	−2.19931e14	−1.03936	−0.519681	−	0.854360i	$-0.673949\pi$
$733$	−2.19931e14	−1.03936	−0.519681	+	0.854360i	$0.673949\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.29211e14	−1.05414
$738$	0	0
$739$	1.96933e14i	0.893505i	0.894658	+	0.446753i	$0.147420\pi$
$739$	1.96933e14i	0.893505i	−0.894658	+	0.446753i	$0.852580\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8.82148e13i	0.389581i	0.980845	+	0.194790i	$0.0624027\pi$
$743$	8.82148e13i	0.389581i	−0.980845	+	0.194790i	$0.937597\pi$
$744$	0	0
$745$	−1.26898e13	−0.0552936
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−4.44789e14	−1.88688
$750$	0	0
$751$	− 3.36605e14i	− 1.40903i	−0.709688	−	0.704516i	$-0.751164\pi$
$751$	− 3.36605e14i	− 1.40903i	0.709688	−	0.704516i	$-0.248836\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 5.15192e13i	− 0.210007i
$756$	0	0
$757$	−3.87411e13	−0.155845	−0.0779224	−	0.996959i	$-0.524829\pi$
$757$	−3.87411e13	−0.155845	−0.0779224	+	0.996959i	$0.524829\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−3.27720e14	−1.28404	−0.642022	−	0.766687i	$-0.721904\pi$
$761$	−3.27720e14	−1.28404	−0.642022	+	0.766687i	$0.721904\pi$
$762$	0	0
$763$	3.56942e14i	1.38030i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 4.26716e14i	− 1.60754i
$768$	0	0
$769$	4.42644e13	0.164597	0.0822987	−	0.996608i	$-0.473774\pi$
$769$	4.42644e13	0.164597	0.0822987	+	0.996608i	$0.473774\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.63190e13	−0.0591286	−0.0295643	−	0.999563i	$-0.509412\pi$
$773$	−1.63190e13	−0.0591286	−0.0295643	+	0.999563i	$0.509412\pi$
$774$	0	0
$775$	− 2.33558e12i	− 0.00835386i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.62321e13i	0.0565833i
$780$	0	0
$781$	5.22710e14	1.79889
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−5.46827e13	−0.183443
$786$	0	0
$787$	− 1.99986e14i	− 0.662409i	−0.943559	−	0.331205i	$-0.892545\pi$
$787$	− 1.99986e14i	− 0.662409i	0.943559	−	0.331205i	$-0.107455\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	8.92953e14i	2.88367i
$792$	0	0
$793$	2.89549e14	0.923327
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.73280e14	1.47172	0.735862	−	0.677131i	$-0.236777\pi$
$797$	4.73280e14	1.47172	0.735862	+	0.677131i	$0.236777\pi$
$798$	0	0
$799$	− 4.29586e14i	− 1.31922i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.46134e14i	0.437696i
$804$	0	0
$805$	−4.24695e13	−0.125631
$806$	0	0
$807$	0	0
$808$	0	0
$809$	6.35889e14	1.83501	0.917506	−	0.397722i	$-0.130199\pi$
$809$	6.35889e14	1.83501	0.917506	+	0.397722i	$0.130199\pi$
$810$	0	0
$811$	− 4.34011e14i	− 1.23708i	−0.785754	−	0.618538i	$-0.787725\pi$
$811$	− 4.34011e14i	− 1.23708i	0.785754	−	0.618538i	$-0.212275\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 2.64229e13i	− 0.0734840i
$816$	0	0
$817$	−6.04059e13	−0.165947
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.44877e14	1.46077	0.730386	−	0.683035i	$-0.239340\pi$
$821$	5.44877e14	1.46077	0.730386	+	0.683035i	$0.239340\pi$
$822$	0	0
$823$	6.22248e14i	1.64803i	0.566571	+	0.824013i	$0.308270\pi$
$823$	6.22248e14i	1.64803i	−0.566571	+	0.824013i	$0.691730\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3.04907e14i	− 0.788206i	−0.919066	−	0.394103i	$-0.871055\pi$
$827$	− 3.04907e14i	− 0.788206i	0.919066	−	0.394103i	$-0.128945\pi$
$828$	0	0
$829$	2.75163e14	0.702776	0.351388	−	0.936230i	$-0.385710\pi$
$829$	2.75163e14	0.702776	0.351388	+	0.936230i	$0.385710\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−9.05712e14	−2.25822
$834$	0	0
$835$	− 7.12885e13i	− 0.175625i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 3.10972e11i	0 0.000748018i	−1.00000		0.000374009i	$-0.999881\pi$
$839$	− 3.10972e11i	0 0.000748018i	1.00000		0.000374009i	$-0.000119051\pi$
$840$	0	0
$841$	−2.91596e14	−0.693109
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8.72295e13	0.202479
$846$	0	0
$847$	1.70518e15i	3.91159i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.58727e14i	0.579688i
$852$	0	0
$853$	−3.31847e14	−0.734839	−0.367420	−	0.930055i	$-0.619759\pi$
$853$	−3.31847e14	−0.734839	−0.367420	+	0.930055i	$0.619759\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5.56969e14	1.20483	0.602417	−	0.798181i	$-0.294204\pi$
$857$	5.56969e14	1.20483	0.602417	+	0.798181i	$0.294204\pi$
$858$	0	0
$859$	− 2.41672e14i	− 0.516726i	−0.966048	−	0.258363i	$-0.916817\pi$
$859$	− 2.41672e14i	− 0.516726i	0.966048	−	0.258363i	$-0.0831831\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.91688e14i	0.400442i	0.979751	+	0.200221i	$0.0641661\pi$
$863$	1.91688e14i	0.400442i	−0.979751	+	0.200221i	$0.935834\pi$
$864$	0	0
$865$	−1.34427e14	−0.277592
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4.86343e14	0.981394
$870$	0	0
$871$	− 4.36459e14i	− 0.870669i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.36068e14i	0.460253i
$876$	0	0
$877$	−8.13362e14	−1.56778	−0.783891	−	0.620898i	$-0.786768\pi$
$877$	−8.13362e14	−1.56778	−0.783891	+	0.620898i	$0.786768\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−4.09108e13	−0.0770830	−0.0385415	−	0.999257i	$-0.512271\pi$
$881$	−4.09108e13	−0.0770830	−0.0385415	+	0.999257i	$0.512271\pi$
$882$	0	0
$883$	7.87473e14i	1.46701i	0.679686	+	0.733503i	$0.262116\pi$
$883$	7.87473e14i	1.46701i	−0.679686	+	0.733503i	$0.737884\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.49540e14i	1.54727i	0.633633	+	0.773634i	$0.281563\pi$
$887$	8.49540e14i	1.54727i	−0.633633	+	0.773634i	$0.718437\pi$
$888$	0	0
$889$	1.52709e13	0.0275015
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.44511e13	0.0958847
$894$	0	0
$895$	1.22407e14i	0.213153i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.77794e12i	0.00473069i
$900$	0	0
$901$	1.43232e15	2.41222
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.12138e14	0.184719
$906$	0	0
$907$	− 7.55390e14i	− 1.23065i	−0.788273	−	0.615325i	$-0.789025\pi$
$907$	− 7.55390e14i	− 1.23065i	0.788273	−	0.615325i	$-0.210975\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 5.75873e14i	− 0.917772i	−0.888495	−	0.458886i	$-0.848249\pi$
$911$	− 5.75873e14i	− 0.917772i	0.888495	−	0.458886i	$-0.151751\pi$
$912$	0	0
$913$	−1.20868e15	−1.90528
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.29488e14	0.816600
$918$	0	0
$919$	7.56273e14i	1.15372i	0.816842	+	0.576861i	$0.195722\pi$
$919$	7.56273e14i	1.15372i	−0.816842	+	0.576861i	$0.804278\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	9.95334e14i	1.48580i
$924$	0	0
$925$	7.11169e14	1.05018
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.33715e15	1.93243	0.966213	−	0.257743i	$-0.0829789\pi$
$929$	1.33715e15	1.93243	0.966213	+	0.257743i	$0.0829789\pi$
$930$	0	0
$931$	− 1.14801e14i	− 0.164134i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 2.97843e14i	− 0.416801i
$936$	0	0
$937$	4.85659e14	0.672409	0.336205	−	0.941789i	$-0.390857\pi$
$937$	4.85659e14	0.672409	0.336205	+	0.941789i	$0.390857\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−6.85064e14	−0.928502	−0.464251	−	0.885704i	$-0.653677\pi$
$941$	−6.85064e14	−0.928502	−0.464251	+	0.885704i	$0.653677\pi$
$942$	0	0
$943$	− 2.06827e14i	− 0.277363i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.87637e14i	1.03413i	0.855945	+	0.517066i	$0.172976\pi$
$947$	7.87637e14i	1.03413i	−0.855945	+	0.517066i	$0.827024\pi$
$948$	0	0
$949$	−2.78265e14	−0.361516
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.41031e15	−1.79412	−0.897058	−	0.441913i	$-0.854300\pi$
$953$	−1.41031e15	−1.79412	−0.897058	+	0.441913i	$0.854300\pi$
$954$	0	0
$955$	4.32237e13i	0.0544133i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1.07910e15i	− 1.33036i
$960$	0	0
$961$	8.19569e14	0.999927
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.43522e14	−0.171507
$966$	0	0
$967$	− 1.23546e15i	− 1.46116i	−0.682829	−	0.730578i	$-0.739251\pi$
$967$	− 1.23546e15i	− 1.46116i	0.682829	−	0.730578i	$-0.260749\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1.55080e14i	− 0.179664i	−0.995957	−	0.0898320i	$-0.971367\pi$
$971$	− 1.55080e14i	− 0.179664i	0.995957	−	0.0898320i	$-0.0286330\pi$
$972$	0	0
$973$	−4.44237e14	−0.509390
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.96452e13	0.0670043	0.0335022	−	0.999439i	$-0.489334\pi$
$977$	5.96452e13	0.0670043	0.0335022	+	0.999439i	$0.489334\pi$
$978$	0	0
$979$	7.84152e14i	0.871940i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 5.84884e14i	− 0.637239i	−0.947883	−	0.318619i	$-0.896781\pi$
$983$	− 5.84884e14i	− 0.637239i	0.947883	−	0.318619i	$-0.103219\pi$
$984$	0	0
$985$	−1.12970e13	−0.0121838
$986$	0	0
$987$	0	0
$988$	0	0
$989$	7.69682e14	0.813448
$990$	0	0
$991$	− 7.53043e13i	− 0.0787865i	−0.999224	−	0.0393932i	$-0.987458\pi$
$991$	− 7.53043e13i	− 0.0787865i	0.999224	−	0.0393932i	$-0.0125425\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.29508e14i	0.132794i
$996$	0	0
$997$	−6.03411e13	−0.0612544	−0.0306272	−	0.999531i	$-0.509750\pi$
$997$	−6.03411e13	−0.0612544	−0.0306272	+	0.999531i	$0.509750\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.11.g.e.127.3		4
3.2	odd	2		48.11.g.c.31.3	yes	4
4.3	odd	2	inner	144.11.g.e.127.4		4
12.11	even	2		48.11.g.c.31.1	✓	4
24.5	odd	2		192.11.g.b.127.2		4
24.11	even	2		192.11.g.b.127.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.11.g.c.31.1	✓	4	12.11	even	2
48.11.g.c.31.3	yes	4	3.2	odd	2
144.11.g.e.127.3		4	1.1	even	1	trivial
144.11.g.e.127.4		4	4.3	odd	2	inner
192.11.g.b.127.2		4	24.5	odd	2
192.11.g.b.127.4		4	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	144.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+460.752 q^{5} -26520.8i q^{7} +O(q^{10})\)	\(q+460.752 q^{5} -26520.8i q^{7} -300389. i q^{11} +571995. q^{13} +2.15197e6 q^{17} +272767. i q^{19} -3.47555e6i q^{23} -9.55333e6 q^{25} +1.13627e7 q^{29} +244478. i q^{31} -1.22195e7i q^{35} -7.44420e7 q^{37} +5.95090e7 q^{41} +2.21456e8i q^{43} -1.99625e8i q^{47} -4.20876e8 q^{49} +6.65587e8 q^{53} -1.38405e8i q^{55} -7.46014e8i q^{59} +5.06208e8 q^{61} +2.63548e8 q^{65} -7.63047e8i q^{67} +1.74011e9i q^{71} -4.86481e8 q^{73} -7.96655e9 q^{77} +1.61904e9i q^{79} -4.02373e9i q^{83} +9.91524e8 q^{85} -2.61045e9 q^{89} -1.51697e10i q^{91} +1.25678e8i q^{95} +1.28953e10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3000 q^{5}+O(q^{10}) \) 4 * q - 3000 * q^5	\( 4 q - 3000 q^{5} + 864136 q^{13} + 55128 q^{17} - 30948820 q^{25} + 80790312 q^{29} - 212511640 q^{37} + 544453272 q^{41} - 1845182972 q^{49} + 888804072 q^{53} - 208335256 q^{61} + 1075819920 q^{65} + 4881669448 q^{73} - 18371224512 q^{77} + 10313912880 q^{85} - 21540382536 q^{89} + 14965768456 q^{97}+O(q^{100}) \) 4 * q - 3000 * q^5 + 864136 * q^13 + 55128 * q^17 - 30948820 * q^25 + 80790312 * q^29 - 212511640 * q^37 + 544453272 * q^41 - 1845182972 * q^49 + 888804072 * q^53 - 208335256 * q^61 + 1075819920 * q^65 + 4881669448 * q^73 - 18371224512 * q^77 + 10313912880 * q^85 - 21540382536 * q^89 + 14965768456 * q^97

Introduction

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