LMFDB - Embedded newform 144.7.e.b.17.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,7,Mod(17,144)]

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([0, 0, 1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("144.17");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	144.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$33.1277880413$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	144.17
Dual form			144.7.e.b.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.07107i q^{5} -60.0000 q^{7} +O(q^{10})$	$q+7.07107i q^{5} -60.0000 q^{7} -333.754i q^{11} +1192.00 q^{13} +4150.72i q^{17} -8432.00 q^{19} -6409.22i q^{23} +15575.0 q^{25} +33639.9i q^{29} -46892.0 q^{31} -424.264i q^{35} +10926.0 q^{37} +73427.4i q^{41} -59416.0 q^{43} +117385. i q^{47} -114049. q^{49} +82390.7i q^{53} +2360.00 q^{55} +281904. i q^{59} -339902. q^{61} +8428.71i q^{65} +148024. q^{67} +411598. i q^{71} -401552. q^{73} +20025.3i q^{77} -79156.0 q^{79} +100392. i q^{83} -29350.0 q^{85} +375877. i q^{89} -71520.0 q^{91} -59623.2i q^{95} +663920. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 120 q^{7}+O(q^{10}) $ 2 * q - 120 * q^7	$ 2 q - 120 q^{7} + 2384 q^{13} - 16864 q^{19} + 31150 q^{25} - 93784 q^{31} + 21852 q^{37} - 118832 q^{43} - 228098 q^{49} + 4720 q^{55} - 679804 q^{61} + 296048 q^{67} - 803104 q^{73} - 158312 q^{79} - 58700 q^{85} - 143040 q^{91} + 1327840 q^{97}+O(q^{100}) $ 2 * q - 120 * q^7 + 2384 * q^13 - 16864 * q^19 + 31150 * q^25 - 93784 * q^31 + 21852 * q^37 - 118832 * q^43 - 228098 * q^49 + 4720 * q^55 - 679804 * q^61 + 296048 * q^67 - 803104 * q^73 - 158312 * q^79 - 58700 * q^85 - 143040 * q^91 + 1327840 * 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
$5$	7.07107i	0.0565685i	0.999600	+	0.0282843i	$0.00900436\pi$
$5$	7.07107i	0.0565685i	−0.999600	+	0.0282843i	$0.990996\pi$
$6$	0	0
$7$	−60.0000	−0.174927	−0.0874636	−	0.996168i	$-0.527876\pi$
$7$	−60.0000	−0.174927	−0.0874636	+	0.996168i	$0.527876\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 333.754i	− 0.250755i	−0.992109	−	0.125377i	$-0.959986\pi$
$11$	− 333.754i	− 0.250755i	0.992109	−	0.125377i	$-0.0400141\pi$
$12$	0	0
$13$	1192.00	0.542558	0.271279	−	0.962501i	$-0.412553\pi$
$13$	1192.00	0.542558	0.271279	+	0.962501i	$0.412553\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4150.72i	0.844844i	0.906399	+	0.422422i	$0.138820\pi$
$17$	4150.72i	0.844844i	−0.906399	+	0.422422i	$0.861180\pi$
$18$	0	0
$19$	−8432.00	−1.22933	−0.614667	−	0.788787i	$-0.710710\pi$
$19$	−8432.00	−1.22933	−0.614667	+	0.788787i	$0.710710\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 6409.22i	− 0.526770i	−0.964691	−	0.263385i	$-0.915161\pi$
$23$	− 6409.22i	− 0.526770i	0.964691	−	0.263385i	$-0.0848390\pi$
$24$	0	0
$25$	15575.0	0.996800
$26$	0	0
$27$	0	0
$28$	0	0
$29$	33639.9i	1.37931i	0.724140	+	0.689653i	$0.242237\pi$
$29$	33639.9i	1.37931i	−0.724140	+	0.689653i	$0.757763\pi$
$30$	0	0
$31$	−46892.0	−1.57403	−0.787016	−	0.616932i	$-0.788375\pi$
$31$	−46892.0	−1.57403	−0.787016	+	0.616932i	$0.788375\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 424.264i	− 0.00989537i
$36$	0	0
$37$	10926.0	0.215703	0.107851	−	0.994167i	$-0.465603\pi$
$37$	10926.0	0.215703	0.107851	+	0.994167i	$0.465603\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	73427.4i	1.06538i	0.846309	+	0.532692i	$0.178820\pi$
$41$	73427.4i	1.06538i	−0.846309	+	0.532692i	$0.821180\pi$
$42$	0	0
$43$	−59416.0	−0.747305	−0.373653	−	0.927569i	$-0.621895\pi$
$43$	−59416.0	−0.747305	−0.373653	+	0.927569i	$0.621895\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	117385.i	1.13063i	0.824875	+	0.565315i	$0.191245\pi$
$47$	117385.i	1.13063i	−0.824875	+	0.565315i	$0.808755\pi$
$48$	0	0
$49$	−114049.	−0.969401
$50$	0	0
$51$	0	0
$52$	0	0
$53$	82390.7i	0.553414i	0.960954	+	0.276707i	$0.0892432\pi$
$53$	82390.7i	0.553414i	−0.960954	+	0.276707i	$0.910757\pi$
$54$	0	0
$55$	2360.00	0.0141848
$56$	0	0
$57$	0	0
$58$	0	0
$59$	281904.i	1.37260i	0.727318	+	0.686301i	$0.240767\pi$
$59$	281904.i	1.37260i	−0.727318	+	0.686301i	$0.759233\pi$
$60$	0	0
$61$	−339902.	−1.49749	−0.748745	−	0.662858i	$-0.769343\pi$
$61$	−339902.	−1.49749	−0.748745	+	0.662858i	$0.769343\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	8428.71i	0.0306917i
$66$	0	0
$67$	148024.	0.492162	0.246081	−	0.969249i	$-0.420857\pi$
$67$	148024.	0.492162	0.246081	+	0.969249i	$0.420857\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	411598.i	1.15000i	0.818153	+	0.575001i	$0.194998\pi$
$71$	411598.i	1.15000i	−0.818153	+	0.575001i	$0.805002\pi$
$72$	0	0
$73$	−401552.	−1.03222	−0.516111	−	0.856522i	$-0.672621\pi$
$73$	−401552.	−1.03222	−0.516111	+	0.856522i	$0.672621\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	20025.3i	0.0438638i
$78$	0	0
$79$	−79156.0	−0.160547	−0.0802736	−	0.996773i	$-0.525579\pi$
$79$	−79156.0	−0.160547	−0.0802736	+	0.996773i	$0.525579\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	100392.i	0.175576i	0.996139	+	0.0877881i	$0.0279798\pi$
$83$	100392.i	0.175576i	−0.996139	+	0.0877881i	$0.972020\pi$
$84$	0	0
$85$	−29350.0	−0.0477916
$86$	0	0
$87$	0	0
$88$	0	0
$89$	375877.i	0.533182i	0.963810	+	0.266591i	$0.0858973\pi$
$89$	375877.i	0.533182i	−0.963810	+	0.266591i	$0.914103\pi$
$90$	0	0
$91$	−71520.0	−0.0949081
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 59623.2i	− 0.0695416i
$96$	0	0
$97$	663920.	0.727446	0.363723	−	0.931507i	$-0.381506\pi$
$97$	663920.	0.727446	0.363723	+	0.931507i	$0.381506\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 48607.9i	− 0.0471784i	−0.999722	−	0.0235892i	$-0.992491\pi$
$101$	− 48607.9i	− 0.0471784i	0.999722	−	0.0235892i	$-0.00750937\pi$
$102$	0	0
$103$	−803236.	−0.735075	−0.367537	−	0.930009i	$-0.619799\pi$
$103$	−803236.	−0.735075	−0.367537	+	0.930009i	$0.619799\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2.28953e6i	− 1.86894i	−0.356041	−	0.934470i	$-0.615874\pi$
$107$	− 2.28953e6i	− 1.86894i	0.356041	−	0.934470i	$-0.384126\pi$
$108$	0	0
$109$	1.51247e6	1.16791	0.583953	−	0.811788i	$-0.301505\pi$
$109$	1.51247e6	1.16791	0.583953	+	0.811788i	$0.301505\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 366314.i	− 0.253874i	−0.991911	−	0.126937i	$-0.959485\pi$
$113$	− 366314.i	− 0.253874i	0.991911	−	0.126937i	$-0.0405146\pi$
$114$	0	0
$115$	45320.0	0.0297986
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 249043.i	− 0.147786i
$120$	0	0
$121$	1.66017e6	0.937122
$122$	0	0
$123$	0	0
$124$	0	0
$125$	220617.i	0.112956i
$126$	0	0
$127$	−291676.	−0.142393	−0.0711966	−	0.997462i	$-0.522682\pi$
$127$	−291676.	−0.142393	−0.0711966	+	0.997462i	$0.522682\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 2.39644e6i	− 1.06599i	−0.846119	−	0.532994i	$-0.821067\pi$
$131$	− 2.39644e6i	− 1.06599i	0.846119	−	0.532994i	$-0.178933\pi$
$132$	0	0
$133$	505920.	0.215044
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.92156e6i	− 1.13620i	−0.822961	−	0.568098i	$-0.807679\pi$
$137$	− 2.92156e6i	− 1.13620i	0.822961	−	0.568098i	$-0.192321\pi$
$138$	0	0
$139$	3.19428e6	1.18940	0.594701	−	0.803947i	$-0.297270\pi$
$139$	3.19428e6	1.18940	0.594701	+	0.803947i	$0.297270\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 397835.i	− 0.136049i
$144$	0	0
$145$	−237870.	−0.0780253
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 3.28266e6i	− 0.992354i	−0.868221	−	0.496177i	$-0.834737\pi$
$149$	− 3.28266e6i	− 0.992354i	0.868221	−	0.496177i	$-0.165263\pi$
$150$	0	0
$151$	877012.	0.254727	0.127363	−	0.991856i	$-0.459349\pi$
$151$	877012.	0.254727	0.127363	+	0.991856i	$0.459349\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 331577.i	− 0.0890407i
$156$	0	0
$157$	−1.48503e6	−0.383738	−0.191869	−	0.981421i	$-0.561455\pi$
$157$	−1.48503e6	−0.383738	−0.191869	+	0.981421i	$0.561455\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	384553.i	0.0921464i
$162$	0	0
$163$	6.24487e6	1.44198	0.720992	−	0.692943i	$-0.243686\pi$
$163$	6.24487e6	1.44198	0.720992	+	0.692943i	$0.243686\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.12706e6i	1.53024i	0.643885	+	0.765122i	$0.277322\pi$
$167$	7.12706e6i	1.53024i	−0.643885	+	0.765122i	$0.722678\pi$
$168$	0	0
$169$	−3.40594e6	−0.705631
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 4.68076e6i	− 0.904019i	−0.892013	−	0.452010i	$-0.850707\pi$
$173$	− 4.68076e6i	− 0.904019i	0.892013	−	0.452010i	$-0.149293\pi$
$174$	0	0
$175$	−934500.	−0.174367
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 8.10114e6i	− 1.41249i	−0.707965	−	0.706247i	$-0.750387\pi$
$179$	− 8.10114e6i	− 1.41249i	0.707965	−	0.706247i	$-0.249613\pi$
$180$	0	0
$181$	−2.53774e6	−0.427967	−0.213984	−	0.976837i	$-0.568644\pi$
$181$	−2.53774e6	−0.427967	−0.213984	+	0.976837i	$0.568644\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	77258.5i	0.0122020i
$186$	0	0
$187$	1.38532e6	0.211848
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.30177e7i	1.86824i	0.356955	+	0.934122i	$0.383815\pi$
$191$	1.30177e7i	1.86824i	−0.356955	+	0.934122i	$0.616185\pi$
$192$	0	0
$193$	−588850.	−0.0819092	−0.0409546	−	0.999161i	$-0.513040\pi$
$193$	−588850.	−0.0819092	−0.0409546	+	0.999161i	$0.513040\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.28095e7i	1.67546i	0.546085	+	0.837730i	$0.316118\pi$
$197$	1.28095e7i	1.67546i	−0.546085	+	0.837730i	$0.683882\pi$
$198$	0	0
$199$	−1.10814e7	−1.40617	−0.703083	−	0.711108i	$-0.748194\pi$
$199$	−1.10814e7	−1.40617	−0.703083	+	0.711108i	$0.748194\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 2.01839e6i	− 0.241278i
$204$	0	0
$205$	−519210.	−0.0602673
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.81422e6i	0.308261i
$210$	0	0
$211$	−454120.	−0.0483418	−0.0241709	−	0.999708i	$-0.507695\pi$
$211$	−454120.	−0.0483418	−0.0241709	+	0.999708i	$0.507695\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 420135.i	− 0.0422740i
$216$	0	0
$217$	2.81352e6	0.275341
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.94765e6i	0.458377i
$222$	0	0
$223$	−1.97745e7	−1.78317	−0.891583	−	0.452857i	$-0.850405\pi$
$223$	−1.97745e7	−1.78317	−0.891583	+	0.452857i	$0.850405\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.77079e6i	− 0.151388i	−0.997131	−	0.0756938i	$-0.975883\pi$
$227$	− 1.77079e6i	− 0.151388i	0.997131	−	0.0756938i	$-0.0241172\pi$
$228$	0	0
$229$	7.16433e6	0.596580	0.298290	−	0.954475i	$-0.403584\pi$
$229$	7.16433e6	0.596580	0.298290	+	0.954475i	$0.403584\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.31622e7i	1.83110i	0.402204	+	0.915550i	$0.368244\pi$
$233$	2.31622e7i	1.83110i	−0.402204	+	0.915550i	$0.631756\pi$
$234$	0	0
$235$	−830040.	−0.0639581
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.57748e7i	1.15550i	0.816214	+	0.577750i	$0.196069\pi$
$239$	1.57748e7i	1.15550i	−0.816214	+	0.577750i	$0.803931\pi$
$240$	0	0
$241$	−3.17674e6	−0.226950	−0.113475	−	0.993541i	$-0.536198\pi$
$241$	−3.17674e6	−0.226950	−0.113475	+	0.993541i	$0.536198\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 806448.i	− 0.0548376i
$246$	0	0
$247$	−1.00509e7	−0.666985
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 2.55789e7i	− 1.61756i	−0.588113	−	0.808779i	$-0.700129\pi$
$251$	− 2.55789e7i	− 1.61756i	0.588113	−	0.808779i	$-0.299871\pi$
$252$	0	0
$253$	−2.13910e6	−0.132090
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 7.72373e6i	− 0.455017i	−0.973776	−	0.227509i	$-0.926942\pi$
$257$	− 7.72373e6i	− 0.455017i	0.973776	−	0.227509i	$-0.0730580\pi$
$258$	0	0
$259$	−655560.	−0.0377323
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.22734e7i	1.77410i	0.461677	+	0.887048i	$0.347248\pi$
$263$	3.22734e7i	1.77410i	−0.461677	+	0.887048i	$0.652752\pi$
$264$	0	0
$265$	−582590.	−0.0313058
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.38305e7i	0.710529i	0.934766	+	0.355264i	$0.115609\pi$
$269$	1.38305e7i	0.710529i	−0.934766	+	0.355264i	$0.884391\pi$
$270$	0	0
$271$	5.09722e6	0.256109	0.128055	−	0.991767i	$-0.459127\pi$
$271$	5.09722e6	0.256109	0.128055	+	0.991767i	$0.459127\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 5.19822e6i	− 0.249952i
$276$	0	0
$277$	−2.08194e7	−0.979557	−0.489778	−	0.871847i	$-0.662922\pi$
$277$	−2.08194e7	−0.979557	−0.489778	+	0.871847i	$0.662922\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 2.77281e7i	− 1.24969i	−0.780750	−	0.624843i	$-0.785163\pi$
$281$	− 2.77281e7i	− 1.24969i	0.780750	−	0.624843i	$-0.214837\pi$
$282$	0	0
$283$	3.72363e7	1.64289	0.821443	−	0.570291i	$-0.193170\pi$
$283$	3.72363e7	1.64289	0.821443	+	0.570291i	$0.193170\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 4.40564e6i	− 0.186365i
$288$	0	0
$289$	6.90912e6	0.286239
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 2.71919e7i	− 1.08103i	−0.841336	−	0.540513i	$-0.818230\pi$
$293$	− 2.71919e7i	− 1.08103i	0.841336	−	0.540513i	$-0.181770\pi$
$294$	0	0
$295$	−1.99336e6	−0.0776461
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 7.63979e6i	− 0.285804i
$300$	0	0
$301$	3.56496e6	0.130724
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 2.40347e6i	− 0.0847109i
$306$	0	0
$307$	3.94664e7	1.36400	0.681998	−	0.731354i	$-0.261111\pi$
$307$	3.94664e7	1.36400	0.681998	+	0.731354i	$0.261111\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.35111e7i	0.449169i	0.974455	+	0.224585i	$0.0721025\pi$
$311$	1.35111e7i	0.449169i	−0.974455	+	0.224585i	$0.927897\pi$
$312$	0	0
$313$	−2.98350e7	−0.972956	−0.486478	−	0.873693i	$-0.661719\pi$
$313$	−2.98350e7	−0.972956	−0.486478	+	0.873693i	$0.661719\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 4.25213e7i	− 1.33484i	−0.744682	−	0.667420i	$-0.767399\pi$
$317$	− 4.25213e7i	− 1.33484i	0.744682	−	0.667420i	$-0.232601\pi$
$318$	0	0
$319$	1.12275e7	0.345867
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 3.49988e7i	− 1.03859i
$324$	0	0
$325$	1.85654e7	0.540822
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 7.04312e6i	− 0.197778i
$330$	0	0
$331$	−946136.	−0.0260897	−0.0130449	−	0.999915i	$-0.504152\pi$
$331$	−946136.	−0.0260897	−0.0130449	+	0.999915i	$0.504152\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.04669e6i	0.0278409i
$336$	0	0
$337$	3.59455e7	0.939193	0.469597	−	0.882881i	$-0.344399\pi$
$337$	3.59455e7	0.939193	0.469597	+	0.882881i	$0.344399\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.56504e7i	0.394696i
$342$	0	0
$343$	1.39019e7	0.344502
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.07914e7i	− 0.497617i	−0.968553	−	0.248809i	$-0.919961\pi$
$347$	− 2.07914e7i	− 0.497617i	0.968553	−	0.248809i	$-0.0800390\pi$
$348$	0	0
$349$	2.42551e7	0.570594	0.285297	−	0.958439i	$-0.407908\pi$
$349$	2.42551e7	0.570594	0.285297	+	0.958439i	$0.407908\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 2.37038e7i	− 0.538882i	−0.963017	−	0.269441i	$-0.913161\pi$
$353$	− 2.37038e7i	− 0.538882i	0.963017	−	0.269441i	$-0.0868390\pi$
$354$	0	0
$355$	−2.91044e6	−0.0650539
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 6.27957e7i	− 1.35721i	−0.734504	−	0.678604i	$-0.762585\pi$
$359$	− 6.27957e7i	− 1.35721i	0.734504	−	0.678604i	$-0.237415\pi$
$360$	0	0
$361$	2.40527e7	0.511261
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 2.83940e6i	− 0.0583913i
$366$	0	0
$367$	−5.44608e7	−1.10176	−0.550879	−	0.834585i	$-0.685707\pi$
$367$	−5.44608e7	−1.10176	−0.550879	+	0.834585i	$0.685707\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 4.94344e6i	− 0.0968072i
$372$	0	0
$373$	2.14491e7	0.413317	0.206659	−	0.978413i	$-0.433741\pi$
$373$	2.14491e7	0.413317	0.206659	+	0.978413i	$0.433741\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.00988e7i	0.748354i
$378$	0	0
$379$	−4.15260e7	−0.762785	−0.381393	−	0.924413i	$-0.624555\pi$
$379$	−4.15260e7	−0.762785	−0.381393	+	0.924413i	$0.624555\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 1.01905e6i	− 0.0181384i	−0.999959	−	0.00906919i	$-0.997113\pi$
$383$	− 1.01905e6i	− 0.0181384i	0.999959	−	0.00906919i	$-0.00288685\pi$
$384$	0	0
$385$	−141600.	−0.00248131
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5.27464e7i	0.896074i	0.894015	+	0.448037i	$0.147877\pi$
$389$	5.27464e7i	0.896074i	−0.894015	+	0.448037i	$0.852123\pi$
$390$	0	0
$391$	2.66028e7	0.445039
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 559717.i	− 0.00908192i
$396$	0	0
$397$	3.39246e7	0.542179	0.271090	−	0.962554i	$-0.412616\pi$
$397$	3.39246e7	0.542179	0.271090	+	0.962554i	$0.412616\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 5.18886e6i	− 0.0804710i	−0.999190	−	0.0402355i	$-0.987189\pi$
$401$	− 5.18886e6i	− 0.0804710i	0.999190	−	0.0402355i	$-0.0128108\pi$
$402$	0	0
$403$	−5.58953e7	−0.854004
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 3.64660e6i	− 0.0540885i
$408$	0	0
$409$	5.49754e7	0.803523	0.401762	−	0.915744i	$-0.368398\pi$
$409$	5.49754e7	0.803523	0.401762	+	0.915744i	$0.368398\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1.69142e7i	− 0.240105i
$414$	0	0
$415$	−709880.	−0.00993209
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.00169e7i	0.544003i	0.962297	+	0.272001i	$0.0876855\pi$
$419$	4.00169e7i	0.544003i	−0.962297	+	0.272001i	$0.912314\pi$
$420$	0	0
$421$	−1.22186e8	−1.63747	−0.818736	−	0.574170i	$-0.805325\pi$
$421$	−1.22186e8	−1.63747	−0.818736	+	0.574170i	$0.805325\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.46474e7i	0.842140i
$426$	0	0
$427$	2.03941e7	0.261952
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.58860e7i	0.573124i	0.958062	+	0.286562i	$0.0925124\pi$
$431$	4.58860e7i	0.573124i	−0.958062	+	0.286562i	$0.907488\pi$
$432$	0	0
$433$	−7.00530e7	−0.862905	−0.431453	−	0.902136i	$-0.641999\pi$
$433$	−7.00530e7	−0.862905	−0.431453	+	0.902136i	$0.641999\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.40425e7i	0.647577i
$438$	0	0
$439$	−7.22462e7	−0.853928	−0.426964	−	0.904269i	$-0.640417\pi$
$439$	−7.22462e7	−0.853928	−0.426964	+	0.904269i	$0.640417\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.34799e8i	1.55052i	0.631643	+	0.775259i	$0.282381\pi$
$443$	1.34799e8i	1.55052i	−0.631643	+	0.775259i	$0.717619\pi$
$444$	0	0
$445$	−2.65785e6	−0.0301613
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 2.34417e7i	− 0.258970i	−0.991581	−	0.129485i	$-0.958668\pi$
$449$	− 2.34417e7i	− 0.258970i	0.991581	−	0.129485i	$-0.0413324\pi$
$450$	0	0
$451$	2.45067e7	0.267150
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 505723.i	− 0.00536881i
$456$	0	0
$457$	−1.63670e8	−1.71483	−0.857415	−	0.514626i	$-0.827931\pi$
$457$	−1.63670e8	−1.71483	−0.857415	+	0.514626i	$0.827931\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 3.60679e7i	− 0.368144i	−0.982913	−	0.184072i	$-0.941072\pi$
$461$	− 3.60679e7i	− 0.368144i	0.982913	−	0.184072i	$-0.0589280\pi$
$462$	0	0
$463$	1.73563e8	1.74869	0.874347	−	0.485301i	$-0.161290\pi$
$463$	1.73563e8	1.74869	0.874347	+	0.485301i	$0.161290\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.67096e8i	1.64065i	0.571899	+	0.820324i	$0.306207\pi$
$467$	1.67096e8i	1.64065i	−0.571899	+	0.820324i	$0.693793\pi$
$468$	0	0
$469$	−8.88144e6	−0.0860924
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.98304e7i	0.187390i
$474$	0	0
$475$	−1.31328e8	−1.22540
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 1.19374e8i	− 1.08618i	−0.839674	−	0.543091i	$-0.817254\pi$
$479$	− 1.19374e8i	− 1.08618i	0.839674	−	0.543091i	$-0.182746\pi$
$480$	0	0
$481$	1.30238e7	0.117031
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.69462e6i	0.0411505i
$486$	0	0
$487$	1.12187e8	0.971302	0.485651	−	0.874153i	$-0.338583\pi$
$487$	1.12187e8	0.971302	0.485651	+	0.874153i	$0.338583\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.01786e8i	0.859893i	0.902854	+	0.429946i	$0.141468\pi$
$491$	1.01786e8i	0.859893i	−0.902854	+	0.429946i	$0.858532\pi$
$492$	0	0
$493$	−1.39630e8	−1.16530
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2.46959e7i	− 0.201167i
$498$	0	0
$499$	−1.01327e8	−0.815502	−0.407751	−	0.913093i	$-0.633687\pi$
$499$	−1.01327e8	−0.815502	−0.407751	+	0.913093i	$0.633687\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9.00591e7i	0.707658i	0.935310	+	0.353829i	$0.115121\pi$
$503$	9.00591e7i	0.707658i	−0.935310	+	0.353829i	$0.884879\pi$
$504$	0	0
$505$	343710.	0.00266881
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5.59180e7i	0.424032i	0.977266	+	0.212016i	$0.0680029\pi$
$509$	5.59180e7i	0.424032i	−0.977266	+	0.212016i	$0.931997\pi$
$510$	0	0
$511$	2.40931e7	0.180564
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 5.67974e6i	− 0.0415821i
$516$	0	0
$517$	3.91779e7	0.283511
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 3.75627e7i	− 0.265610i	−0.991142	−	0.132805i	$-0.957602\pi$
$521$	− 3.75627e7i	− 0.265610i	0.991142	−	0.132805i	$-0.0423984\pi$
$522$	0	0
$523$	1.79687e7	0.125607	0.0628033	−	0.998026i	$-0.479996\pi$
$523$	1.79687e7	0.125607	0.0628033	+	0.998026i	$0.479996\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.94635e8i	− 1.32981i
$528$	0	0
$529$	1.06958e8	0.722513
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8.75254e7i	0.578033i
$534$	0	0
$535$	1.61894e7	0.105723
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.80644e7i	0.243082i
$540$	0	0
$541$	1.92941e8	1.21852	0.609262	−	0.792969i	$-0.291466\pi$
$541$	1.92941e8	1.21852	0.609262	+	0.792969i	$0.291466\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.06948e7i	0.0660667i
$546$	0	0
$547$	−1.75297e8	−1.07106	−0.535530	−	0.844516i	$-0.679888\pi$
$547$	−1.75297e8	−1.07106	−0.535530	+	0.844516i	$0.679888\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 2.83652e8i	− 1.69563i
$552$	0	0
$553$	4.74936e6	0.0280840
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.57046e8i	− 1.48746i	−0.668481	−	0.743729i	$-0.733055\pi$
$557$	− 2.57046e8i	− 1.48746i	0.668481	−	0.743729i	$-0.266945\pi$
$558$	0	0
$559$	−7.08239e7	−0.405456
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 4.73682e6i	− 0.0265437i	−0.999912	−	0.0132719i	$-0.995775\pi$
$563$	− 4.73682e6i	− 0.0265437i	0.999912	−	0.0132719i	$-0.00422469\pi$
$564$	0	0
$565$	2.59023e6	0.0143613
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2.87773e8i	1.56212i	0.624457	+	0.781059i	$0.285320\pi$
$569$	2.87773e8i	1.56212i	−0.624457	+	0.781059i	$0.714680\pi$
$570$	0	0
$571$	8.67761e7	0.466114	0.233057	−	0.972463i	$-0.425127\pi$
$571$	8.67761e7	0.466114	0.233057	+	0.972463i	$0.425127\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 9.98235e7i	− 0.525085i
$576$	0	0
$577$	2.87028e8	1.49416	0.747080	−	0.664734i	$-0.231455\pi$
$577$	2.87028e8	1.49416	0.747080	+	0.664734i	$0.231455\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 6.02353e6i	− 0.0307130i
$582$	0	0
$583$	2.74982e7	0.138771
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 9.48551e7i	− 0.468972i	−0.972120	−	0.234486i	$-0.924659\pi$
$587$	− 9.48551e7i	− 0.468972i	0.972120	−	0.234486i	$-0.0753406\pi$
$588$	0	0
$589$	3.95393e8	1.93501
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 3.71287e8i	− 1.78051i	−0.455460	−	0.890256i	$-0.650525\pi$
$593$	− 3.71287e8i	− 1.78051i	0.455460	−	0.890256i	$-0.349475\pi$
$594$	0	0
$595$	1.76100e6	0.00836004
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 3.36966e8i	− 1.56786i	−0.620852	−	0.783928i	$-0.713213\pi$
$599$	− 3.36966e8i	− 1.56786i	0.620852	−	0.783928i	$-0.286787\pi$
$600$	0	0
$601$	−2.12985e8	−0.981126	−0.490563	−	0.871406i	$-0.663209\pi$
$601$	−2.12985e8	−0.981126	−0.490563	+	0.871406i	$0.663209\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.17392e7i	0.0530116i
$606$	0	0
$607$	2.22206e8	0.993549	0.496775	−	0.867880i	$-0.334518\pi$
$607$	2.22206e8	0.993549	0.496775	+	0.867880i	$0.334518\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.39923e8i	0.613432i
$612$	0	0
$613$	−2.56448e8	−1.11331	−0.556656	−	0.830743i	$-0.687916\pi$
$613$	−2.56448e8	−1.11331	−0.556656	+	0.830743i	$0.687916\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.02398e6i	0.0213891i	0.999943	+	0.0106945i	$0.00340424\pi$
$617$	5.02398e6i	0.0213891i	−0.999943	+	0.0106945i	$0.996596\pi$
$618$	0	0
$619$	−3.93392e8	−1.65864	−0.829322	−	0.558771i	$-0.811273\pi$
$619$	−3.93392e8	−1.65864	−0.829322	+	0.558771i	$0.811273\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 2.25526e7i	− 0.0932680i
$624$	0	0
$625$	2.41799e8	0.990410
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.53507e7i	0.182235i
$630$	0	0
$631$	3.39614e8	1.35175	0.675876	−	0.737015i	$-0.263765\pi$
$631$	3.39614e8	1.35175	0.675876	+	0.737015i	$0.263765\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 2.06246e6i	− 0.00805498i
$636$	0	0
$637$	−1.35946e8	−0.525956
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 3.07963e8i	− 1.16930i	−0.811287	−	0.584648i	$-0.801233\pi$
$641$	− 3.07963e8i	− 1.16930i	0.811287	−	0.584648i	$-0.198767\pi$
$642$	0	0
$643$	3.32651e8	1.25128	0.625642	−	0.780110i	$-0.284837\pi$
$643$	3.32651e8	1.25128	0.625642	+	0.780110i	$0.284837\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.26019e8i	0.465288i	0.972562	+	0.232644i	$0.0747377\pi$
$647$	1.26019e8i	0.465288i	−0.972562	+	0.232644i	$0.925262\pi$
$648$	0	0
$649$	9.40866e7	0.344186
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 2.22564e8i	− 0.799312i	−0.916665	−	0.399656i	$-0.869130\pi$
$653$	− 2.22564e8i	− 0.799312i	0.916665	−	0.399656i	$-0.130870\pi$
$654$	0	0
$655$	1.69454e7	0.0603013
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 7.53159e7i	− 0.263167i	−0.991305	−	0.131583i	$-0.957994\pi$
$659$	− 7.53159e7i	− 0.263167i	0.991305	−	0.131583i	$-0.0420061\pi$
$660$	0	0
$661$	−1.23651e8	−0.428148	−0.214074	−	0.976817i	$-0.568673\pi$
$661$	−1.23651e8	−0.428148	−0.214074	+	0.976817i	$0.568673\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.57739e6i	0.0121647i
$666$	0	0
$667$	2.15605e8	0.726578
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.13444e8i	0.375503i
$672$	0	0
$673$	−4.74800e8	−1.55763	−0.778817	−	0.627252i	$-0.784180\pi$
$673$	−4.74800e8	−1.55763	−0.778817	+	0.627252i	$0.784180\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.11211e8i	0.680691i	0.940300	+	0.340346i	$0.110544\pi$
$677$	2.11211e8i	0.680691i	−0.940300	+	0.340346i	$0.889456\pi$
$678$	0	0
$679$	−3.98352e7	−0.127250
$680$	0	0
$681$	0	0
$682$	0	0
$683$	5.41774e8i	1.70042i	0.526444	+	0.850210i	$0.323525\pi$
$683$	5.41774e8i	1.70042i	−0.526444	+	0.850210i	$0.676475\pi$
$684$	0	0
$685$	2.06586e7	0.0642730
$686$	0	0
$687$	0	0
$688$	0	0
$689$	9.82097e7i	0.300259i
$690$	0	0
$691$	3.54838e7	0.107546	0.0537731	−	0.998553i	$-0.482875\pi$
$691$	3.54838e7	0.107546	0.0537731	+	0.998553i	$0.482875\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.25870e7i	0.0672827i
$696$	0	0
$697$	−3.04776e8	−0.900084
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 5.20754e8i	− 1.51175i	−0.654718	−	0.755873i	$-0.727213\pi$
$701$	− 5.20754e8i	− 1.51175i	0.654718	−	0.755873i	$-0.272787\pi$
$702$	0	0
$703$	−9.21280e7	−0.265171
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.91648e6i	0.00825278i
$708$	0	0
$709$	2.48937e7	0.0698474	0.0349237	−	0.999390i	$-0.488881\pi$
$709$	2.48937e7	0.0698474	0.0349237	+	0.999390i	$0.488881\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.00541e8i	0.829154i
$714$	0	0
$715$	2.81312e6	0.00769609
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.42310e7i	0.0651905i	0.999469	+	0.0325953i	$0.0103772\pi$
$719$	2.42310e7i	0.0651905i	−0.999469	+	0.0325953i	$0.989623\pi$
$720$	0	0
$721$	4.81942e7	0.128585
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5.23941e8i	1.37489i
$726$	0	0
$727$	−1.45628e8	−0.379001	−0.189501	−	0.981881i	$-0.560687\pi$
$727$	−1.45628e8	−0.379001	−0.189501	+	0.981881i	$0.560687\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 2.46619e8i	− 0.631356i
$732$	0	0
$733$	−9.69774e7	−0.246240	−0.123120	−	0.992392i	$-0.539290\pi$
$733$	−9.69774e7	−0.246240	−0.123120	+	0.992392i	$0.539290\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 4.94037e7i	− 0.123412i
$738$	0	0
$739$	−2.60556e8	−0.645607	−0.322804	−	0.946466i	$-0.604625\pi$
$739$	−2.60556e8	−0.645607	−0.322804	+	0.946466i	$0.604625\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	6.35246e8i	1.54873i	0.632740	+	0.774365i	$0.281930\pi$
$743$	6.35246e8i	1.54873i	−0.632740	+	0.774365i	$0.718070\pi$
$744$	0	0
$745$	2.32119e7	0.0561360
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.37372e8i	0.326928i
$750$	0	0
$751$	−1.81893e8	−0.429434	−0.214717	−	0.976676i	$-0.568883\pi$
$751$	−1.81893e8	−0.429434	−0.214717	+	0.976676i	$0.568883\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6.20141e6i	0.0144095i
$756$	0	0
$757$	−2.52138e8	−0.581235	−0.290617	−	0.956839i	$-0.593861\pi$
$757$	−2.52138e8	−0.581235	−0.290617	+	0.956839i	$0.593861\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4.28380e8i	0.972020i	0.873953	+	0.486010i	$0.161548\pi$
$761$	4.28380e8i	0.972020i	−0.873953	+	0.486010i	$0.838452\pi$
$762$	0	0
$763$	−9.07483e7	−0.204298
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.36029e8i	0.744716i
$768$	0	0
$769$	5.67451e8	1.24781	0.623907	−	0.781499i	$-0.285545\pi$
$769$	5.67451e8	1.24781	0.623907	+	0.781499i	$0.285545\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7.50015e8i	1.62380i	0.583799	+	0.811898i	$0.301565\pi$
$773$	7.50015e8i	1.62380i	−0.583799	+	0.811898i	$0.698435\pi$
$774$	0	0
$775$	−7.30343e8	−1.56900
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 6.19140e8i	− 1.30971i
$780$	0	0
$781$	1.37373e8	0.288368
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1.05007e7i	− 0.0217075i
$786$	0	0
$787$	−3.62103e8	−0.742863	−0.371431	−	0.928460i	$-0.621133\pi$
$787$	−3.62103e8	−0.742863	−0.371431	+	0.928460i	$0.621133\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.19788e7i	0.0444094i
$792$	0	0
$793$	−4.05163e8	−0.812476
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 3.47987e8i	− 0.687367i	−0.939086	−	0.343683i	$-0.888325\pi$
$797$	− 3.47987e8i	− 0.687367i	0.939086	−	0.343683i	$-0.111675\pi$
$798$	0	0
$799$	−4.87233e8	−0.955205
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.34020e8i	0.258834i
$804$	0	0
$805$	−2.71920e6	−0.00521259
$806$	0	0
$807$	0	0
$808$	0	0
$809$	7.07604e8i	1.33642i	0.743971	+	0.668212i	$0.232940\pi$
$809$	7.07604e8i	1.33642i	−0.743971	+	0.668212i	$0.767060\pi$
$810$	0	0
$811$	5.04592e8	0.945971	0.472986	−	0.881070i	$-0.343176\pi$
$811$	5.04592e8	0.945971	0.472986	+	0.881070i	$0.343176\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4.41579e7i	0.0815710i
$816$	0	0
$817$	5.00996e8	0.918688
$818$	0	0
$819$	0	0
$820$	0	0
$821$	7.34934e7i	0.132806i	0.997793	+	0.0664032i	$0.0211524\pi$
$821$	7.34934e7i	0.132806i	−0.997793	+	0.0664032i	$0.978848\pi$
$822$	0	0
$823$	1.01114e8	0.181389	0.0906943	−	0.995879i	$-0.471091\pi$
$823$	1.01114e8	0.181389	0.0906943	+	0.995879i	$0.471091\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 5.38616e8i	− 0.952276i	−0.879371	−	0.476138i	$-0.842036\pi$
$827$	− 5.38616e8i	− 0.952276i	0.879371	−	0.476138i	$-0.157964\pi$
$828$	0	0
$829$	1.00076e9	1.75657	0.878286	−	0.478135i	$-0.158687\pi$
$829$	1.00076e9	1.75657	0.878286	+	0.478135i	$0.158687\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 4.73385e8i	− 0.818992i
$834$	0	0
$835$	−5.03959e7	−0.0865637
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.33521e8i	0.226080i	0.993590	+	0.113040i	$0.0360589\pi$
$839$	1.33521e8i	0.226080i	−0.993590	+	0.113040i	$0.963941\pi$
$840$	0	0
$841$	−5.36819e8	−0.902485
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 2.40837e7i	− 0.0399165i
$846$	0	0
$847$	−9.96101e7	−0.163928
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 7.00271e7i	− 0.113626i
$852$	0	0
$853$	−9.81323e8	−1.58112	−0.790560	−	0.612384i	$-0.790211\pi$
$853$	−9.81323e8	−1.58112	−0.790560	+	0.612384i	$0.790211\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8.20792e7i	0.130404i	0.997872	+	0.0652020i	$0.0207692\pi$
$857$	8.20792e7i	0.130404i	−0.997872	+	0.0652020i	$0.979231\pi$
$858$	0	0
$859$	−5.65250e6	−0.00891788	−0.00445894	−	0.999990i	$-0.501419\pi$
$859$	−5.65250e6	−0.00891788	−0.00445894	+	0.999990i	$0.501419\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4.57968e8i	0.712529i	0.934385	+	0.356265i	$0.115950\pi$
$863$	4.57968e8i	0.712529i	−0.934385	+	0.356265i	$0.884050\pi$
$864$	0	0
$865$	3.30979e7	0.0511391
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.64187e7i	0.0402579i
$870$	0	0
$871$	1.76445e8	0.267026
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 1.32370e7i	− 0.0197591i
$876$	0	0
$877$	3.88791e8	0.576391	0.288196	−	0.957572i	$-0.406945\pi$
$877$	3.88791e8	0.576391	0.288196	+	0.957572i	$0.406945\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 9.09556e8i	− 1.33015i	−0.746775	−	0.665076i	$-0.768399\pi$
$881$	− 9.09556e8i	− 1.33015i	0.746775	−	0.665076i	$-0.231601\pi$
$882$	0	0
$883$	1.17558e9	1.70753	0.853765	−	0.520659i	$-0.174314\pi$
$883$	1.17558e9	1.70753	0.853765	+	0.520659i	$0.174314\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.36296e8i	0.338599i	0.985565	+	0.169299i	$0.0541505\pi$
$887$	2.36296e8i	0.338599i	−0.985565	+	0.169299i	$0.945850\pi$
$888$	0	0
$889$	1.75006e7	0.0249084
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 9.89794e8i	− 1.38992i
$894$	0	0
$895$	5.72837e7	0.0799028
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1.57744e9i	− 2.17107i
$900$	0	0
$901$	−3.41980e8	−0.467549
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1.79445e7i	− 0.0242095i
$906$	0	0
$907$	1.19117e9	1.59643	0.798217	−	0.602370i	$-0.205777\pi$
$907$	1.19117e9	1.59643	0.798217	+	0.602370i	$0.205777\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	6.28443e8i	0.831210i	0.909545	+	0.415605i	$0.136430\pi$
$911$	6.28443e8i	0.831210i	−0.909545	+	0.415605i	$0.863570\pi$
$912$	0	0
$913$	3.35063e7	0.0440265
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.43786e8i	0.186470i
$918$	0	0
$919$	−6.28812e8	−0.810166	−0.405083	−	0.914280i	$-0.632757\pi$
$919$	−6.28812e8	−0.810166	−0.405083	+	0.914280i	$0.632757\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.90625e8i	0.623943i
$924$	0	0
$925$	1.70172e8	0.215013
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1.16872e9i	− 1.45768i	−0.684684	−	0.728840i	$-0.740060\pi$
$929$	− 1.16872e9i	− 1.45768i	0.684684	−	0.728840i	$-0.259940\pi$
$930$	0	0
$931$	9.61661e8	1.19172
$932$	0	0
$933$	0	0
$934$	0	0
$935$	9.79569e6i	0.0119840i
$936$	0	0
$937$	−6.10906e8	−0.742601	−0.371300	−	0.928513i	$-0.621088\pi$
$937$	−6.10906e8	−0.742601	−0.371300	+	0.928513i	$0.621088\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 9.94118e8i	− 1.19308i	−0.802584	−	0.596539i	$-0.796542\pi$
$941$	− 9.94118e8i	− 1.19308i	0.802584	−	0.596539i	$-0.203458\pi$
$942$	0	0
$943$	4.70612e8	0.561213
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 7.14448e8i	− 0.841242i	−0.907237	−	0.420621i	$-0.861812\pi$
$947$	− 7.14448e8i	− 0.841242i	0.907237	−	0.420621i	$-0.138188\pi$
$948$	0	0
$949$	−4.78650e8	−0.560040
$950$	0	0
$951$	0	0
$952$	0	0
$953$	7.94083e8i	0.917460i	0.888576	+	0.458730i	$0.151695\pi$
$953$	7.94083e8i	0.917460i	−0.888576	+	0.458730i	$0.848305\pi$
$954$	0	0
$955$	−9.20489e7	−0.105684
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.75294e8i	0.198752i
$960$	0	0
$961$	1.31136e9	1.47758
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 4.16380e6i	− 0.00463348i
$966$	0	0
$967$	4.36986e8	0.483268	0.241634	−	0.970367i	$-0.422317\pi$
$967$	4.36986e8	0.483268	0.241634	+	0.970367i	$0.422317\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1.37787e9i	− 1.50505i	−0.658566	−	0.752523i	$-0.728837\pi$
$971$	− 1.37787e9i	− 1.50505i	0.658566	−	0.752523i	$-0.271163\pi$
$972$	0	0
$973$	−1.91657e8	−0.208059
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.09340e9i	− 1.17245i	−0.810148	−	0.586226i	$-0.800613\pi$
$977$	− 1.09340e9i	− 1.17245i	0.810148	−	0.586226i	$-0.199387\pi$
$978$	0	0
$979$	1.25451e8	0.133698
$980$	0	0
$981$	0	0
$982$	0	0
$983$	7.63403e8i	0.803699i	0.915706	+	0.401850i	$0.131633\pi$
$983$	7.63403e8i	0.803699i	−0.915706	+	0.401850i	$0.868367\pi$
$984$	0	0
$985$	−9.05769e7	−0.0947783
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.80810e8i	0.393658i
$990$	0	0
$991$	−3.32088e7	−0.0341219	−0.0170609	−	0.999854i	$-0.505431\pi$
$991$	−3.32088e7	−0.0341219	−0.0170609	+	0.999854i	$0.505431\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 7.83575e7i	− 0.0795447i
$996$	0	0
$997$	1.26157e9	1.27299	0.636495	−	0.771281i	$-0.280384\pi$
$997$	1.26157e9	1.27299	0.636495	+	0.771281i	$0.280384\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.7.e.b.17.2		2
3.2	odd	2	inner	144.7.e.b.17.1		2
4.3	odd	2		72.7.e.a.17.2	yes	2
8.3	odd	2		576.7.e.h.449.1		2
8.5	even	2		576.7.e.e.449.1		2
12.11	even	2		72.7.e.a.17.1	✓	2
24.5	odd	2		576.7.e.e.449.2		2
24.11	even	2		576.7.e.h.449.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.7.e.a.17.1	✓	2	12.11	even	2
72.7.e.a.17.2	yes	2	4.3	odd	2
144.7.e.b.17.1		2	3.2	odd	2	inner
144.7.e.b.17.2		2	1.1	even	1	trivial
576.7.e.e.449.1		2	8.5	even	2
576.7.e.e.449.2		2	24.5	odd	2
576.7.e.h.449.1		2	8.3	odd	2
576.7.e.h.449.2		2	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:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	144.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+7.07107i q^{5} -60.0000 q^{7} +O(q^{10})\)	\(q+7.07107i q^{5} -60.0000 q^{7} -333.754i q^{11} +1192.00 q^{13} +4150.72i q^{17} -8432.00 q^{19} -6409.22i q^{23} +15575.0 q^{25} +33639.9i q^{29} -46892.0 q^{31} -424.264i q^{35} +10926.0 q^{37} +73427.4i q^{41} -59416.0 q^{43} +117385. i q^{47} -114049. q^{49} +82390.7i q^{53} +2360.00 q^{55} +281904. i q^{59} -339902. q^{61} +8428.71i q^{65} +148024. q^{67} +411598. i q^{71} -401552. q^{73} +20025.3i q^{77} -79156.0 q^{79} +100392. i q^{83} -29350.0 q^{85} +375877. i q^{89} -71520.0 q^{91} -59623.2i q^{95} +663920. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 120 q^{7}+O(q^{10}) \) 2 * q - 120 * q^7	\( 2 q - 120 q^{7} + 2384 q^{13} - 16864 q^{19} + 31150 q^{25} - 93784 q^{31} + 21852 q^{37} - 118832 q^{43} - 228098 q^{49} + 4720 q^{55} - 679804 q^{61} + 296048 q^{67} - 803104 q^{73} - 158312 q^{79} - 58700 q^{85} - 143040 q^{91} + 1327840 q^{97}+O(q^{100}) \) 2 * q - 120 * q^7 + 2384 * q^13 - 16864 * q^19 + 31150 * q^25 - 93784 * q^31 + 21852 * q^37 - 118832 * q^43 - 228098 * q^49 + 4720 * q^55 - 679804 * q^61 + 296048 * q^67 - 803104 * q^73 - 158312 * q^79 - 58700 * q^85 - 143040 * q^91 + 1327840 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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