LMFDB - Embedded newform 7200.2.h.l.1151.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7200,2,Mod(1151,7200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7200.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7200.h (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:	$57.4922894553$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.426337261060096.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 $ x^12 - 4x^9 - 3x^8 + 4x^7 + 8x^6 + 8x^5 - 12x^4 - 32*x^3 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 2^{15} $
Twist minimal:	no (minimal twist has level 1440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.6
Root			$1.19252 - 0.760198i$ of defining polynomial
Character	$\chi$	$=$	7200.1151
Dual form			7200.2.h.l.1151.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.864641i q^{7} +O(q^{10})$	$q-0.864641i q^{7} +3.90543 q^{11} -1.13536 q^{13} -3.71400i q^{17} -1.72928i q^{19} -9.03365 q^{23} +1.26843i q^{29} -3.25240i q^{31} -6.38776 q^{37} -6.39665i q^{41} +4.77551i q^{43} +4.59958 q^{47} +6.25240 q^{49} +8.98801i q^{53} -8.50501 q^{59} -9.04623 q^{61} -11.0462i q^{67} -8.10243 q^{71} -4.47689 q^{73} -3.37680i q^{77} +14.2986i q^{79} -8.10243 q^{83} +3.56822i q^{89} +0.981678i q^{91} -10.9817 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 24 q^{13} - 16 q^{37} + 4 q^{49} - 8 q^{61} - 104 q^{73} - 40 q^{97}+O(q^{100}) $ 12 * q - 24 * q^13 - 16 * q^37 + 4 * q^49 - 8 * q^61 - 104 * q^73 - 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$6401$	$6751$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.864641i	− 0.326804i	−0.986560	−	0.163402i	$-0.947753\pi$
$7$	− 0.864641i	− 0.326804i	0.986560	−	0.163402i	$-0.0522467\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.90543	1.17753	0.588766	−	0.808304i	$-0.299614\pi$
$11$	3.90543	1.17753	0.588766	+	0.808304i	$0.299614\pi$
$12$	0	0
$13$	−1.13536	−0.314892	−0.157446	−	0.987528i	$-0.550326\pi$
$13$	−1.13536	−0.314892	−0.157446	+	0.987528i	$0.550326\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.71400i	− 0.900779i	−0.892832	−	0.450389i	$-0.851285\pi$
$17$	− 3.71400i	− 0.900779i	0.892832	−	0.450389i	$-0.148715\pi$
$18$	0	0
$19$	− 1.72928i	− 0.396724i	−0.980129	−	0.198362i	$-0.936438\pi$
$19$	− 1.72928i	− 0.396724i	0.980129	−	0.198362i	$-0.0635622\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−9.03365	−1.88365	−0.941823	−	0.336109i	$-0.890889\pi$
$23$	−9.03365	−1.88365	−0.941823	+	0.336109i	$0.890889\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.26843i	0.235542i	0.993041	+	0.117771i	$0.0375748\pi$
$29$	1.26843i	0.235542i	−0.993041	+	0.117771i	$0.962425\pi$
$30$	0	0
$31$	− 3.25240i	− 0.584148i	−0.956396	−	0.292074i	$-0.905655\pi$
$31$	− 3.25240i	− 0.584148i	0.956396	−	0.292074i	$-0.0943453\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.38776	−1.05014	−0.525070	−	0.851059i	$-0.675961\pi$
$37$	−6.38776	−1.05014	−0.525070	+	0.851059i	$0.675961\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 6.39665i	− 0.998989i	−0.866317	−	0.499494i	$-0.833519\pi$
$41$	− 6.39665i	− 0.998989i	0.866317	−	0.499494i	$-0.166481\pi$
$42$	0	0
$43$	4.77551i	0.728259i	0.931348	+	0.364129i	$0.118633\pi$
$43$	4.77551i	0.728259i	−0.931348	+	0.364129i	$0.881367\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.59958	0.670918	0.335459	−	0.942055i	$-0.391109\pi$
$47$	4.59958	0.670918	0.335459	+	0.942055i	$0.391109\pi$
$48$	0	0
$49$	6.25240	0.893199
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.98801i	1.23460i	0.786729	+	0.617299i	$0.211773\pi$
$53$	8.98801i	1.23460i	−0.786729	+	0.617299i	$0.788227\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.50501	−1.10726	−0.553629	−	0.832763i	$-0.686758\pi$
$59$	−8.50501	−1.10726	−0.553629	+	0.832763i	$0.686758\pi$
$60$	0	0
$61$	−9.04623	−1.15825	−0.579125	−	0.815238i	$-0.696606\pi$
$61$	−9.04623	−1.15825	−0.579125	+	0.815238i	$0.696606\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 11.0462i	− 1.34951i	−0.738041	−	0.674756i	$-0.764249\pi$
$67$	− 11.0462i	− 1.34951i	0.738041	−	0.674756i	$-0.235751\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.10243	−0.961581	−0.480791	−	0.876835i	$-0.659650\pi$
$71$	−8.10243	−0.961581	−0.480791	+	0.876835i	$0.659650\pi$
$72$	0	0
$73$	−4.47689	−0.523980	−0.261990	−	0.965071i	$-0.584379\pi$
$73$	−4.47689	−0.523980	−0.261990	+	0.965071i	$0.584379\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.37680i	− 0.384822i
$78$	0	0
$79$	14.2986i	1.60872i	0.594142	+	0.804360i	$0.297492\pi$
$79$	14.2986i	1.60872i	−0.594142	+	0.804360i	$0.702508\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.10243	−0.889357	−0.444679	−	0.895690i	$-0.646682\pi$
$83$	−8.10243	−0.889357	−0.444679	+	0.895690i	$0.646682\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.56822i	0.378231i	0.981955	+	0.189115i	$0.0605620\pi$
$89$	3.56822i	0.378231i	−0.981955	+	0.189115i	$0.939438\pi$
$90$	0	0
$91$	0.981678i	0.102908i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.9817	−1.11502	−0.557510	−	0.830170i	$-0.688243\pi$
$97$	−10.9817	−1.11502	−0.557510	+	0.830170i	$0.688243\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 0.885578i	− 0.0881183i	−0.999029	−	0.0440591i	$-0.985971\pi$
$101$	− 0.885578i	− 0.0881183i	0.999029	−	0.0440591i	$-0.0140290\pi$
$102$	0	0
$103$	− 18.1816i	− 1.79149i	−0.444573	−	0.895743i	$-0.646645\pi$
$103$	− 18.1816i	− 1.79149i	0.444573	−	0.895743i	$-0.353355\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.44557	0.236423	0.118211	−	0.992988i	$-0.462284\pi$
$107$	2.44557	0.236423	0.118211	+	0.992988i	$0.462284\pi$
$108$	0	0
$109$	3.25240	0.311523	0.155762	−	0.987795i	$-0.450217\pi$
$109$	3.25240	0.311523	0.155762	+	0.987795i	$0.450217\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	17.5646i	1.65234i	0.563424	+	0.826168i	$0.309484\pi$
$113$	17.5646i	1.65234i	−0.563424	+	0.826168i	$0.690516\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.21128	−0.294378
$120$	0	0
$121$	4.25240	0.386581
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.18159i	0.193585i	0.995305	+	0.0967923i	$0.0308583\pi$
$127$	2.18159i	0.193585i	−0.995305	+	0.0967923i	$0.969142\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	22.2643	1.94524	0.972620	−	0.232400i	$-0.0746578\pi$
$131$	22.2643	1.94524	0.972620	+	0.232400i	$0.0746578\pi$
$132$	0	0
$133$	−1.49521	−0.129651
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.92529i	0.591667i	0.955240	+	0.295834i	$0.0955974\pi$
$137$	6.92529i	0.591667i	−0.955240	+	0.295834i	$0.904403\pi$
$138$	0	0
$139$	− 18.2986i	− 1.55207i	−0.630690	−	0.776035i	$-0.717228\pi$
$139$	− 18.2986i	− 1.55207i	0.630690	−	0.776035i	$-0.282772\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.43407	−0.370795
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.98801i	0.736326i	0.929761	+	0.368163i	$0.120013\pi$
$149$	8.98801i	0.736326i	−0.929761	+	0.368163i	$0.879987\pi$
$150$	0	0
$151$	− 6.29862i	− 0.512575i	−0.966601	−	0.256287i	$-0.917501\pi$
$151$	− 6.29862i	− 0.512575i	0.966601	−	0.256287i	$-0.0824994\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	11.1633	0.890926	0.445463	−	0.895300i	$-0.353039\pi$
$157$	11.1633	0.890926	0.445463	+	0.895300i	$0.353039\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.81086i	0.615582i
$162$	0	0
$163$	− 17.7293i	− 1.38866i	−0.719655	−	0.694332i	$-0.755700\pi$
$163$	− 17.7293i	− 1.38866i	0.719655	−	0.694332i	$-0.244300\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.51435	−0.117184	−0.0585920	−	0.998282i	$-0.518661\pi$
$167$	−1.51435	−0.117184	−0.0585920	+	0.998282i	$0.518661\pi$
$168$	0	0
$169$	−11.7110	−0.900843
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.4106i	1.17164i	0.810440	+	0.585822i	$0.199228\pi$
$173$	15.4106i	1.17164i	−0.810440	+	0.585822i	$0.800772\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.56229	0.714719	0.357359	−	0.933967i	$-0.383677\pi$
$179$	9.56229	0.714719	0.357359	+	0.933967i	$0.383677\pi$
$180$	0	0
$181$	14.2986	1.06281	0.531404	−	0.847118i	$-0.321665\pi$
$181$	14.2986	1.06281	0.531404	+	0.847118i	$0.321665\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 14.5048i	− 1.06070i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.1246	−1.38381	−0.691903	−	0.721991i	$-0.743227\pi$
$191$	−19.1246	−1.38381	−0.691903	+	0.721991i	$0.743227\pi$
$192$	0	0
$193$	−22.5048	−1.61993	−0.809965	−	0.586478i	$-0.800514\pi$
$193$	−22.5048	−1.61993	−0.809965	+	0.586478i	$0.800514\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.91923i	0.706716i	0.935488	+	0.353358i	$0.114960\pi$
$197$	9.91923i	0.706716i	−0.935488	+	0.353358i	$0.885040\pi$
$198$	0	0
$199$	3.66473i	0.259786i	0.991528	+	0.129893i	$0.0414634\pi$
$199$	3.66473i	0.259786i	−0.991528	+	0.129893i	$0.958537\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.09674	0.0769759
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 6.75359i	− 0.467156i
$210$	0	0
$211$	2.29862i	0.158244i	0.996865	+	0.0791219i	$0.0252116\pi$
$211$	2.29862i	0.158244i	−0.996865	+	0.0791219i	$0.974788\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2.81215	−0.190901
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.21673i	0.283648i
$222$	0	0
$223$	− 15.1354i	− 1.01354i	−0.862082	−	0.506769i	$-0.830840\pi$
$223$	− 15.1354i	− 1.01354i	0.862082	−	0.506769i	$-0.169160\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−26.1697	−1.73695	−0.868473	−	0.495737i	$-0.834898\pi$
$227$	−26.1697	−1.73695	−0.868473	+	0.495737i	$0.834898\pi$
$228$	0	0
$229$	−9.75719	−0.644773	−0.322387	−	0.946608i	$-0.604485\pi$
$229$	−9.75719	−0.644773	−0.322387	+	0.946608i	$0.604485\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 8.02202i	− 0.525540i	−0.964858	−	0.262770i	$-0.915364\pi$
$233$	− 8.02202i	− 0.525540i	0.964858	−	0.262770i	$-0.0846361\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	17.0100	1.10029	0.550144	−	0.835070i	$-0.314573\pi$
$239$	17.0100	1.10029	0.550144	+	0.835070i	$0.314573\pi$
$240$	0	0
$241$	−21.0096	−1.35335	−0.676673	−	0.736284i	$-0.736579\pi$
$241$	−21.0096	−1.35335	−0.676673	+	0.736284i	$0.736579\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.96336i	0.124925i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	8.46555	0.534341	0.267170	−	0.963649i	$-0.413911\pi$
$251$	8.46555	0.534341	0.267170	+	0.963649i	$0.413911\pi$
$252$	0	0
$253$	−35.2803	−2.21805
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.8164i	0.737089i	0.929610	+	0.368544i	$0.120144\pi$
$257$	11.8164i	0.737089i	−0.929610	+	0.368544i	$0.879856\pi$
$258$	0	0
$259$	5.52311i	0.343190i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7.93691	−0.489411	−0.244705	−	0.969597i	$-0.578691\pi$
$263$	−7.93691	−0.489411	−0.244705	+	0.969597i	$0.578691\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 29.9750i	− 1.82761i	−0.406154	−	0.913805i	$-0.633130\pi$
$269$	− 29.9750i	− 1.82761i	0.406154	−	0.913805i	$-0.366870\pi$
$270$	0	0
$271$	26.8401i	1.63042i	0.579167	+	0.815209i	$0.303378\pi$
$271$	26.8401i	1.63042i	−0.579167	+	0.815209i	$0.696622\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−24.9571	−1.49953	−0.749763	−	0.661706i	$-0.769833\pi$
$277$	−24.9571	−1.49953	−0.749763	+	0.661706i	$0.769833\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.12265i	0.0669716i	0.999439	+	0.0334858i	$0.0106609\pi$
$281$	1.12265i	0.0669716i	−0.999439	+	0.0334858i	$0.989339\pi$
$282$	0	0
$283$	− 25.9634i	− 1.54336i	−0.636010	−	0.771681i	$-0.719416\pi$
$283$	− 25.9634i	− 1.54336i	0.636010	−	0.771681i	$-0.280584\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.53080	−0.326473
$288$	0	0
$289$	3.20617	0.188598
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 16.6728i	− 0.974037i	−0.873392	−	0.487018i	$-0.838085\pi$
$293$	− 16.6728i	− 0.974037i	0.873392	−	0.487018i	$-0.161915\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10.2564	0.593145
$300$	0	0
$301$	4.12910	0.237997
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	21.0096i	1.19908i	0.800345	+	0.599540i	$0.204650\pi$
$307$	21.0096i	1.19908i	−0.800345	+	0.599540i	$0.795350\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−34.7463	−1.97028	−0.985141	−	0.171748i	$-0.945058\pi$
$311$	−34.7463	−1.97028	−0.985141	+	0.171748i	$0.945058\pi$
$312$	0	0
$313$	−9.49521	−0.536701	−0.268350	−	0.963321i	$-0.586479\pi$
$313$	−9.49521	−0.536701	−0.268350	+	0.963321i	$0.586479\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.2505i	0.912719i	0.889795	+	0.456360i	$0.150847\pi$
$317$	16.2505i	0.912719i	−0.889795	+	0.456360i	$0.849153\pi$
$318$	0	0
$319$	4.95377i	0.277358i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6.42256	−0.357361
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 3.97699i	− 0.219258i
$330$	0	0
$331$	− 20.7755i	− 1.14193i	−0.820976	−	0.570963i	$-0.806570\pi$
$331$	− 20.7755i	− 1.14193i	0.820976	−	0.570963i	$-0.193430\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	28.1204	1.53181	0.765907	−	0.642951i	$-0.222290\pi$
$337$	28.1204	1.53181	0.765907	+	0.642951i	$0.222290\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 12.7020i	− 0.687852i
$342$	0	0
$343$	− 11.4586i	− 0.618704i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−18.8330	−1.01101	−0.505504	−	0.862824i	$-0.668694\pi$
$347$	−18.8330	−1.01101	−0.505504	+	0.862824i	$0.668694\pi$
$348$	0	0
$349$	11.5510	0.618312	0.309156	−	0.951011i	$-0.399953\pi$
$349$	11.5510	0.618312	0.309156	+	0.951011i	$0.399953\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.61727i	0.139303i	0.997571	+	0.0696515i	$0.0221887\pi$
$353$	2.61727i	0.139303i	−0.997571	+	0.0696515i	$0.977811\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−20.8044	−1.09802	−0.549008	−	0.835817i	$-0.684994\pi$
$359$	−20.8044	−1.09802	−0.549008	+	0.835817i	$0.684994\pi$
$360$	0	0
$361$	16.0096	0.842610
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 15.3694i	− 0.802278i	−0.916017	−	0.401139i	$-0.868614\pi$
$367$	− 15.3694i	− 0.802278i	0.916017	−	0.401139i	$-0.131386\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7.77140	0.403471
$372$	0	0
$373$	−29.3049	−1.51735	−0.758675	−	0.651470i	$-0.774153\pi$
$373$	−29.3049	−1.51735	−0.758675	+	0.651470i	$0.774153\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1.44012i	− 0.0741702i
$378$	0	0
$379$	− 11.7938i	− 0.605808i	−0.953021	−	0.302904i	$-0.902044\pi$
$379$	− 11.7938i	− 0.605808i	0.953021	−	0.302904i	$-0.0979562\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−16.6790	−0.852257	−0.426128	−	0.904663i	$-0.640123\pi$
$383$	−16.6790	−0.852257	−0.426128	+	0.904663i	$0.640123\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	23.2214i	1.17737i	0.808361	+	0.588687i	$0.200355\pi$
$389$	23.2214i	1.17737i	−0.808361	+	0.588687i	$0.799645\pi$
$390$	0	0
$391$	33.5510i	1.69675i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2.02458	−0.101611	−0.0508054	−	0.998709i	$-0.516179\pi$
$397$	−2.02458	−0.101611	−0.0508054	+	0.998709i	$0.516179\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	28.9722i	1.44680i	0.690427	+	0.723402i	$0.257423\pi$
$401$	28.9722i	1.44680i	−0.690427	+	0.723402i	$0.742577\pi$
$402$	0	0
$403$	3.69264i	0.183943i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−24.9469	−1.23657
$408$	0	0
$409$	−25.8863	−1.27999	−0.639997	−	0.768377i	$-0.721065\pi$
$409$	−25.8863	−1.27999	−0.639997	+	0.768377i	$0.721065\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.35378i	0.361856i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12.7736	0.624030	0.312015	−	0.950077i	$-0.398996\pi$
$419$	12.7736	0.624030	0.312015	+	0.950077i	$0.398996\pi$
$420$	0	0
$421$	18.1695	0.885528	0.442764	−	0.896638i	$-0.353998\pi$
$421$	18.1695	0.885528	0.442764	+	0.896638i	$0.353998\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	7.82174i	0.378520i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.40611	−0.115898	−0.0579491	−	0.998320i	$-0.518456\pi$
$431$	−2.40611	−0.115898	−0.0579491	+	0.998320i	$0.518456\pi$
$432$	0	0
$433$	−31.4094	−1.50944	−0.754720	−	0.656047i	$-0.772227\pi$
$433$	−31.4094	−1.50944	−0.754720	+	0.656047i	$0.772227\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	15.6217i	0.747289i
$438$	0	0
$439$	− 18.1695i	− 0.867184i	−0.901109	−	0.433592i	$-0.857246\pi$
$439$	− 18.1695i	− 0.867184i	0.901109	−	0.433592i	$-0.142754\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−18.8330	−0.894783	−0.447392	−	0.894338i	$-0.647647\pi$
$443$	−18.8330	−0.894783	−0.447392	+	0.894338i	$0.647647\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	20.9216i	0.987353i	0.869646	+	0.493677i	$0.164347\pi$
$449$	20.9216i	0.987353i	−0.869646	+	0.493677i	$0.835653\pi$
$450$	0	0
$451$	− 24.9817i	− 1.17634i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−35.5789	−1.66431	−0.832156	−	0.554542i	$-0.812894\pi$
$457$	−35.5789	−1.66431	−0.832156	+	0.554542i	$0.812894\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	14.0617i	0.654920i	0.944865	+	0.327460i	$0.106193\pi$
$461$	14.0617i	0.654920i	−0.944865	+	0.327460i	$0.893807\pi$
$462$	0	0
$463$	− 33.0987i	− 1.53823i	−0.639112	−	0.769114i	$-0.720698\pi$
$463$	− 33.0987i	− 1.53823i	0.639112	−	0.769114i	$-0.279302\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	32.0092	1.48121	0.740604	−	0.671942i	$-0.234540\pi$
$467$	32.0092	1.48121	0.740604	+	0.671942i	$0.234540\pi$
$468$	0	0
$469$	−9.55102	−0.441025
$470$	0	0
$471$	0	0
$472$	0	0
$473$	18.6504i	0.857548i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−28.6153	−1.30747	−0.653733	−	0.756725i	$-0.726798\pi$
$479$	−28.6153	−1.30747	−0.653733	+	0.756725i	$0.726798\pi$
$480$	0	0
$481$	7.25240	0.330681
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 2.18159i	− 0.0988572i	−0.998778	−	0.0494286i	$-0.984260\pi$
$487$	− 2.18159i	− 0.0988572i	0.998778	−	0.0494286i	$-0.0157400\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	28.2127	1.27322	0.636611	−	0.771185i	$-0.280336\pi$
$491$	28.2127	1.27322	0.636611	+	0.771185i	$0.280336\pi$
$492$	0	0
$493$	4.71096	0.212171
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.00569i	0.314248i
$498$	0	0
$499$	0.234074i	0.0104786i	0.999986	+	0.00523930i	$0.00166773\pi$
$499$	0.234074i	0.0104786i	−0.999986	+	0.00523930i	$0.998332\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	15.3302	0.683538	0.341769	−	0.939784i	$-0.388974\pi$
$503$	15.3302	0.683538	0.341769	+	0.939784i	$0.388974\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 22.7473i	− 1.00826i	−0.863629	−	0.504128i	$-0.831814\pi$
$509$	− 22.7473i	− 1.00826i	0.863629	−	0.504128i	$-0.168186\pi$
$510$	0	0
$511$	3.87090i	0.171238i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	17.9634	0.790027
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 23.9898i	− 1.05101i	−0.850790	−	0.525506i	$-0.823876\pi$
$521$	− 23.9898i	− 1.05101i	0.850790	−	0.525506i	$-0.176124\pi$
$522$	0	0
$523$	− 28.3632i	− 1.24024i	−0.784509	−	0.620118i	$-0.787085\pi$
$523$	− 28.3632i	− 1.24024i	0.784509	−	0.620118i	$-0.212915\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.0794	−0.526188
$528$	0	0
$529$	58.6068	2.54812
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7.26249i	0.314574i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	24.4183	1.05177
$540$	0	0
$541$	−36.8034	−1.58230	−0.791151	−	0.611621i	$-0.790518\pi$
$541$	−36.8034	−1.58230	−0.791151	+	0.611621i	$0.790518\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 24.2341i	− 1.03617i	−0.855328	−	0.518087i	$-0.826644\pi$
$547$	− 24.2341i	− 1.03617i	0.855328	−	0.518087i	$-0.173356\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.19347	0.0934452
$552$	0	0
$553$	12.3632	0.525736
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 43.3515i	− 1.83686i	−0.395584	−	0.918430i	$-0.629458\pi$
$557$	− 43.3515i	− 1.83686i	0.395584	−	0.918430i	$-0.370542\pi$
$558$	0	0
$559$	− 5.42192i	− 0.229323i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2.62815	0.110763	0.0553817	−	0.998465i	$-0.482362\pi$
$563$	2.62815	0.110763	0.0553817	+	0.998465i	$0.482362\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 23.6588i	− 0.991828i	−0.868372	−	0.495914i	$-0.834833\pi$
$569$	− 23.6588i	− 0.991828i	0.868372	−	0.495914i	$-0.165167\pi$
$570$	0	0
$571$	− 35.6926i	− 1.49369i	−0.664998	−	0.746845i	$-0.731568\pi$
$571$	− 35.6926i	− 1.49369i	0.664998	−	0.746845i	$-0.268432\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0.646409	0.0269104	0.0134552	−	0.999909i	$-0.495717\pi$
$577$	0.646409	0.0269104	0.0134552	+	0.999909i	$0.495717\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.00569i	0.290645i
$582$	0	0
$583$	35.1020i	1.45378i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.3753	0.923528	0.461764	−	0.887003i	$-0.347217\pi$
$587$	22.3753	0.923528	0.461764	+	0.887003i	$0.347217\pi$
$588$	0	0
$589$	−5.62431	−0.231746
$590$	0	0
$591$	0	0
$592$	0	0
$593$	20.4325i	0.839061i	0.907741	+	0.419530i	$0.137805\pi$
$593$	20.4325i	0.839061i	−0.907741	+	0.419530i	$0.862195\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6.13100	−0.250506	−0.125253	−	0.992125i	$-0.539974\pi$
$599$	−6.13100	−0.250506	−0.125253	+	0.992125i	$0.539974\pi$
$600$	0	0
$601$	−1.79383	−0.0731720	−0.0365860	−	0.999331i	$-0.511648\pi$
$601$	−1.79383	−0.0731720	−0.0365860	+	0.999331i	$0.511648\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 16.6864i	− 0.677279i	−0.940916	−	0.338640i	$-0.890033\pi$
$607$	− 16.6864i	− 0.677279i	0.940916	−	0.338640i	$-0.109967\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.22218	−0.211267
$612$	0	0
$613$	23.5264	0.950224	0.475112	−	0.879925i	$-0.342408\pi$
$613$	23.5264	0.950224	0.475112	+	0.879925i	$0.342408\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	23.3127i	0.938535i	0.883056	+	0.469267i	$0.155482\pi$
$617$	23.3127i	0.938535i	−0.883056	+	0.469267i	$0.844518\pi$
$618$	0	0
$619$	32.3911i	1.30191i	0.759117	+	0.650954i	$0.225631\pi$
$619$	32.3911i	1.30191i	−0.759117	+	0.650954i	$0.774369\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.08523	0.123607
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	23.7242i	0.945944i
$630$	0	0
$631$	19.2524i	0.766426i	0.923660	+	0.383213i	$0.125182\pi$
$631$	19.2524i	0.766426i	−0.923660	+	0.383213i	$0.874818\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−7.09871	−0.281261
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 25.9435i	− 1.02471i	−0.858774	−	0.512354i	$-0.828774\pi$
$641$	− 25.9435i	− 1.02471i	0.858774	−	0.512354i	$-0.171226\pi$
$642$	0	0
$643$	− 11.2803i	− 0.444852i	−0.974950	−	0.222426i	$-0.928602\pi$
$643$	− 11.2803i	− 0.444852i	0.974950	−	0.222426i	$-0.0713975\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	11.6053	0.456250	0.228125	−	0.973632i	$-0.426740\pi$
$647$	11.6053	0.456250	0.228125	+	0.973632i	$0.426740\pi$
$648$	0	0
$649$	−33.2158	−1.30383
$650$	0	0
$651$	0	0
$652$	0	0
$653$	35.4145i	1.38588i	0.720996	+	0.692939i	$0.243684\pi$
$653$	35.4145i	1.38588i	−0.720996	+	0.692939i	$0.756316\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	12.0079	0.467760	0.233880	−	0.972265i	$-0.424858\pi$
$659$	12.0079	0.467760	0.233880	+	0.972265i	$0.424858\pi$
$660$	0	0
$661$	−3.08287	−0.119910	−0.0599549	−	0.998201i	$-0.519096\pi$
$661$	−3.08287	−0.119910	−0.0599549	+	0.998201i	$0.519096\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 11.4586i	− 0.443677i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−35.3294	−1.36388
$672$	0	0
$673$	25.7851	0.993942	0.496971	−	0.867767i	$-0.334445\pi$
$673$	25.7851	0.993942	0.496971	+	0.867767i	$0.334445\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.16019i	0.0830227i	0.999138	+	0.0415114i	$0.0132173\pi$
$677$	2.16019i	0.0830227i	−0.999138	+	0.0415114i	$0.986783\pi$
$678$	0	0
$679$	9.49521i	0.364393i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−28.3632	−1.08529	−0.542644	−	0.839963i	$-0.682577\pi$
$683$	−28.3632	−1.08529	−0.542644	+	0.839963i	$0.682577\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 10.2046i	− 0.388765i
$690$	0	0
$691$	− 37.9142i	− 1.44232i	−0.692766	−	0.721162i	$-0.743608\pi$
$691$	− 37.9142i	− 1.44232i	0.692766	−	0.721162i	$-0.256392\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−23.7572	−0.899868
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 12.9650i	− 0.489681i	−0.969563	−	0.244841i	$-0.921264\pi$
$701$	− 12.9650i	− 0.489681i	0.969563	−	0.244841i	$-0.0787356\pi$
$702$	0	0
$703$	11.0462i	0.416616i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−0.765707	−0.0287974
$708$	0	0
$709$	26.8401	1.00800	0.504000	−	0.863704i	$-0.331861\pi$
$709$	26.8401	1.00800	0.504000	+	0.863704i	$0.331861\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	29.3810i	1.10033i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−40.7342	−1.51913	−0.759564	−	0.650432i	$-0.774588\pi$
$719$	−40.7342	−1.51913	−0.759564	+	0.650432i	$0.774588\pi$
$720$	0	0
$721$	−15.7205	−0.585464
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 24.9205i	− 0.924248i	−0.886815	−	0.462124i	$-0.847087\pi$
$727$	− 24.9205i	− 0.924248i	0.886815	−	0.462124i	$-0.152913\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	17.7363	0.656000
$732$	0	0
$733$	−11.2278	−0.414709	−0.207354	−	0.978266i	$-0.566485\pi$
$733$	−11.2278	−0.414709	−0.207354	+	0.978266i	$0.566485\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 43.1403i	− 1.58909i
$738$	0	0
$739$	− 3.22449i	− 0.118615i	−0.998240	−	0.0593074i	$-0.981111\pi$
$739$	− 3.22449i	− 0.118615i	0.998240	−	0.0593074i	$-0.0188892\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−14.5645	−0.534318	−0.267159	−	0.963652i	$-0.586085\pi$
$743$	−14.5645	−0.534318	−0.267159	+	0.963652i	$0.586085\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 2.11454i	− 0.0772637i
$750$	0	0
$751$	52.8034i	1.92682i	0.268025	+	0.963412i	$0.413629\pi$
$751$	52.8034i	1.92682i	−0.268025	+	0.963412i	$0.586371\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−8.82800	−0.320859	−0.160429	−	0.987047i	$-0.551288\pi$
$757$	−8.82800	−0.320859	−0.160429	+	0.987047i	$0.551288\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 32.6577i	− 1.18384i	−0.805997	−	0.591920i	$-0.798370\pi$
$761$	− 32.6577i	− 1.18384i	0.805997	−	0.591920i	$-0.201630\pi$
$762$	0	0
$763$	− 2.81215i	− 0.101807i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.65625	0.348667
$768$	0	0
$769$	7.83048	0.282374	0.141187	−	0.989983i	$-0.454908\pi$
$769$	7.83048	0.282374	0.141187	+	0.989983i	$0.454908\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 8.14807i	− 0.293066i	−0.989206	−	0.146533i	$-0.953189\pi$
$773$	− 8.14807i	− 0.293066i	0.989206	−	0.146533i	$-0.0468114\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−11.0616	−0.396323
$780$	0	0
$781$	−31.6435	−1.13229
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	35.2803i	1.25761i	0.777564	+	0.628803i	$0.216455\pi$
$787$	35.2803i	1.25761i	−0.777564	+	0.628803i	$0.783545\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	15.1871	0.539989
$792$	0	0
$793$	10.2707	0.364724
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.67999i	0.165774i	0.996559	+	0.0828868i	$0.0264140\pi$
$797$	4.67999i	0.165774i	−0.996559	+	0.0828868i	$0.973586\pi$
$798$	0	0
$799$	− 17.0829i	− 0.604349i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−17.4842	−0.617003
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	19.2294i	0.676070i	0.941133	+	0.338035i	$0.109762\pi$
$809$	19.2294i	0.676070i	−0.941133	+	0.338035i	$0.890238\pi$
$810$	0	0
$811$	− 18.7110i	− 0.657031i	−0.944499	−	0.328515i	$-0.893452\pi$
$811$	− 18.7110i	− 0.657031i	0.944499	−	0.328515i	$-0.106548\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8.25820	0.288918
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 6.92529i	− 0.241694i	−0.992671	−	0.120847i	$-0.961439\pi$
$821$	− 6.92529i	− 0.241694i	0.992671	−	0.120847i	$-0.0385611\pi$
$822$	0	0
$823$	23.6035i	0.822767i	0.911462	+	0.411383i	$0.134954\pi$
$823$	23.6035i	0.822767i	−0.911462	+	0.411383i	$0.865046\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	29.3810	1.02168	0.510839	−	0.859676i	$-0.329335\pi$
$827$	29.3810	1.02168	0.510839	+	0.859676i	$0.329335\pi$
$828$	0	0
$829$	−16.2062	−0.562863	−0.281432	−	0.959581i	$-0.590809\pi$
$829$	−16.2062	−0.562863	−0.281432	+	0.959581i	$0.590809\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 23.2214i	− 0.804575i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−9.49073	−0.327656	−0.163828	−	0.986489i	$-0.552384\pi$
$839$	−9.49073	−0.327656	−0.163828	+	0.986489i	$0.552384\pi$
$840$	0	0
$841$	27.3911	0.944520
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 3.67680i	− 0.126336i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	57.7047	1.97809
$852$	0	0
$853$	32.3511	1.10768	0.553840	−	0.832623i	$-0.313162\pi$
$853$	32.3511	1.10768	0.553840	+	0.832623i	$0.313162\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 3.80529i	− 0.129986i	−0.997886	−	0.0649932i	$-0.979297\pi$
$857$	− 3.80529i	− 0.129986i	0.997886	−	0.0649932i	$-0.0207026\pi$
$858$	0	0
$859$	2.29862i	0.0784281i	0.999231	+	0.0392140i	$0.0124854\pi$
$859$	2.29862i	0.0784281i	−0.999231	+	0.0392140i	$0.987515\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.79936	0.333574	0.166787	−	0.985993i	$-0.446661\pi$
$863$	9.79936	0.333574	0.166787	+	0.985993i	$0.446661\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	55.8423i	1.89432i
$870$	0	0
$871$	12.5414i	0.424950i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−23.7047	−0.800451	−0.400225	−	0.916417i	$-0.631068\pi$
$877$	−23.7047	−0.800451	−0.400225	+	0.916417i	$0.631068\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	37.1660i	1.25215i	0.779762	+	0.626077i	$0.215340\pi$
$881$	37.1660i	1.25215i	−0.779762	+	0.626077i	$0.784660\pi$
$882$	0	0
$883$	17.7293i	0.596638i	0.954466	+	0.298319i	$0.0964259\pi$
$883$	17.7293i	0.596638i	−0.954466	+	0.298319i	$0.903574\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	34.6202	1.16243	0.581217	−	0.813749i	$-0.302577\pi$
$887$	34.6202	1.16243	0.581217	+	0.813749i	$0.302577\pi$
$888$	0	0
$889$	1.88629	0.0632641
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 7.95397i	− 0.266170i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.12544	0.137591
$900$	0	0
$901$	33.3815	1.11210
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	7.76593i	0.257863i	0.991653	+	0.128932i	$0.0411548\pi$
$907$	7.76593i	0.257863i	−0.991653	+	0.128932i	$0.958845\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	4.30802	0.142731	0.0713655	−	0.997450i	$-0.477264\pi$
$911$	4.30802	0.142731	0.0713655	+	0.997450i	$0.477264\pi$
$912$	0	0
$913$	−31.6435	−1.04725
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 19.2506i	− 0.635712i
$918$	0	0
$919$	27.2524i	0.898974i	0.893287	+	0.449487i	$0.148393\pi$
$919$	27.2524i	0.898974i	−0.893287	+	0.449487i	$0.851607\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	9.19917	0.302794
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	37.3662i	1.22595i	0.790104	+	0.612973i	$0.210027\pi$
$929$	37.3662i	1.22595i	−0.790104	+	0.612973i	$0.789973\pi$
$930$	0	0
$931$	− 10.8122i	− 0.354354i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	35.4586	1.15838	0.579190	−	0.815192i	$-0.303369\pi$
$937$	35.4586	1.15838	0.579190	+	0.815192i	$0.303369\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	11.1420i	0.363219i	0.983371	+	0.181610i	$0.0581307\pi$
$941$	11.1420i	0.363219i	−0.983371	+	0.181610i	$0.941869\pi$
$942$	0	0
$943$	57.7851i	1.88174i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−31.4956	−1.02347	−0.511734	−	0.859144i	$-0.670997\pi$
$947$	−31.4956	−1.02347	−0.511734	+	0.859144i	$0.670997\pi$
$948$	0	0
$949$	5.08287	0.164997
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 41.1062i	− 1.33156i	−0.746149	−	0.665779i	$-0.768099\pi$
$953$	− 41.1062i	− 1.33156i	0.746149	−	0.665779i	$-0.231901\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.98788	0.193359
$960$	0	0
$961$	20.4219	0.658772
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	18.4157i	0.592208i	0.955156	+	0.296104i	$0.0956875\pi$
$967$	18.4157i	0.592208i	−0.955156	+	0.296104i	$0.904313\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	11.3853	0.365371	0.182685	−	0.983171i	$-0.441521\pi$
$971$	11.3853	0.365371	0.182685	+	0.983171i	$0.441521\pi$
$972$	0	0
$973$	−15.8217	−0.507222
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 43.4822i	− 1.39112i	−0.718469	−	0.695559i	$-0.755157\pi$
$977$	− 43.4822i	− 1.39112i	0.718469	−	0.695559i	$-0.244843\pi$
$978$	0	0
$979$	13.9354i	0.445379i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−5.36529	−0.171126	−0.0855631	−	0.996333i	$-0.527269\pi$
$983$	−5.36529	−0.171126	−0.0855631	+	0.996333i	$0.527269\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 43.1403i	− 1.37178i
$990$	0	0
$991$	− 24.9325i	− 0.792008i	−0.918249	−	0.396004i	$-0.870397\pi$
$991$	− 24.9325i	− 0.792008i	0.918249	−	0.396004i	$-0.129603\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−10.5169	−0.333072	−0.166536	−	0.986035i	$-0.553258\pi$
$997$	−10.5169	−0.333072	−0.166536	+	0.986035i	$0.553258\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7200.2.h.l.1151.6		12
3.2	odd	2	inner	7200.2.h.l.1151.5		12
4.3	odd	2	inner	7200.2.h.l.1151.7		12
5.2	odd	4		1440.2.o.a.1439.9	yes	12
5.3	odd	4		1440.2.o.b.1439.3	yes	12
5.4	even	2		7200.2.h.m.1151.8		12
12.11	even	2	inner	7200.2.h.l.1151.8		12
15.2	even	4		1440.2.o.a.1439.4	yes	12
15.8	even	4		1440.2.o.b.1439.10	yes	12
15.14	odd	2		7200.2.h.m.1151.7		12
20.3	even	4		1440.2.o.a.1439.3	✓	12
20.7	even	4		1440.2.o.b.1439.9	yes	12
20.19	odd	2		7200.2.h.m.1151.5		12
40.3	even	4		2880.2.o.f.2879.10		12
40.13	odd	4		2880.2.o.e.2879.10		12
40.27	even	4		2880.2.o.e.2879.4		12
40.37	odd	4		2880.2.o.f.2879.4		12
60.23	odd	4		1440.2.o.a.1439.10	yes	12
60.47	odd	4		1440.2.o.b.1439.4	yes	12
60.59	even	2		7200.2.h.m.1151.6		12
120.53	even	4		2880.2.o.e.2879.3		12
120.77	even	4		2880.2.o.f.2879.9		12
120.83	odd	4		2880.2.o.f.2879.3		12
120.107	odd	4		2880.2.o.e.2879.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.2.o.a.1439.3	✓	12	20.3	even	4
1440.2.o.a.1439.4	yes	12	15.2	even	4
1440.2.o.a.1439.9	yes	12	5.2	odd	4
1440.2.o.a.1439.10	yes	12	60.23	odd	4
1440.2.o.b.1439.3	yes	12	5.3	odd	4
1440.2.o.b.1439.4	yes	12	60.47	odd	4
1440.2.o.b.1439.9	yes	12	20.7	even	4
1440.2.o.b.1439.10	yes	12	15.8	even	4
2880.2.o.e.2879.3		12	120.53	even	4
2880.2.o.e.2879.4		12	40.27	even	4
2880.2.o.e.2879.9		12	120.107	odd	4
2880.2.o.e.2879.10		12	40.13	odd	4
2880.2.o.f.2879.3		12	120.83	odd	4
2880.2.o.f.2879.4		12	40.37	odd	4
2880.2.o.f.2879.9		12	120.77	even	4
2880.2.o.f.2879.10		12	40.3	even	4
7200.2.h.l.1151.5		12	3.2	odd	2	inner
7200.2.h.l.1151.6		12	1.1	even	1	trivial
7200.2.h.l.1151.7		12	4.3	odd	2	inner
7200.2.h.l.1151.8		12	12.11	even	2	inner
7200.2.h.m.1151.5		12	20.19	odd	2
7200.2.h.m.1151.6		12	60.59	even	2
7200.2.h.m.1151.7		12	15.14	odd	2
7200.2.h.m.1151.8		12	5.4	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:	\( N \)	\(=\)	\( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7200.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.864641i q^{7} +O(q^{10})\)	\(q-0.864641i q^{7} +3.90543 q^{11} -1.13536 q^{13} -3.71400i q^{17} -1.72928i q^{19} -9.03365 q^{23} +1.26843i q^{29} -3.25240i q^{31} -6.38776 q^{37} -6.39665i q^{41} +4.77551i q^{43} +4.59958 q^{47} +6.25240 q^{49} +8.98801i q^{53} -8.50501 q^{59} -9.04623 q^{61} -11.0462i q^{67} -8.10243 q^{71} -4.47689 q^{73} -3.37680i q^{77} +14.2986i q^{79} -8.10243 q^{83} +3.56822i q^{89} +0.981678i q^{91} -10.9817 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 24 q^{13} - 16 q^{37} + 4 q^{49} - 8 q^{61} - 104 q^{73} - 40 q^{97}+O(q^{100}) \) 12 * q - 24 * q^13 - 16 * q^37 + 4 * q^49 - 8 * q^61 - 104 * q^73 - 40 * q^97

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