LMFDB - Embedded newform 7448.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7448,2,Mod(1,7448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7448, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7448 = 2^{3} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7448.a (trivial)

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:	yes
Analytic conductor:	$59.4725794254$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	7448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.30278 q^{3} +1.00000 q^{5} -1.30278 q^{9} +O(q^{10})$	$q-1.30278 q^{3} +1.00000 q^{5} -1.30278 q^{9} -2.30278 q^{11} +3.60555 q^{13} -1.30278 q^{15} -1.69722 q^{17} -1.00000 q^{19} -0.394449 q^{23} -4.00000 q^{25} +5.60555 q^{27} -4.30278 q^{29} +8.30278 q^{31} +3.00000 q^{33} +3.60555 q^{37} -4.69722 q^{39} -0.302776 q^{41} +7.21110 q^{43} -1.30278 q^{45} +7.60555 q^{47} +2.21110 q^{51} -3.90833 q^{53} -2.30278 q^{55} +1.30278 q^{57} -5.60555 q^{59} -8.21110 q^{61} +3.60555 q^{65} -10.9083 q^{67} +0.513878 q^{69} +8.81665 q^{71} +5.90833 q^{73} +5.21110 q^{75} -14.0000 q^{79} -3.39445 q^{81} +10.5139 q^{83} -1.69722 q^{85} +5.60555 q^{87} -13.8167 q^{89} -10.8167 q^{93} -1.00000 q^{95} -16.8167 q^{97} +3.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 2 q^{5} + q^{9}+O(q^{10}) $ 2 * q + q^3 + 2 * q^5 + q^9	$ 2 q + q^{3} + 2 q^{5} + q^{9} - q^{11} + q^{15} - 7 q^{17} - 2 q^{19} - 8 q^{23} - 8 q^{25} + 4 q^{27} - 5 q^{29} + 13 q^{31} + 6 q^{33} - 13 q^{39} + 3 q^{41} + q^{45} + 8 q^{47} - 10 q^{51} + 3 q^{53} - q^{55} - q^{57} - 4 q^{59} - 2 q^{61} - 11 q^{67} - 17 q^{69} - 4 q^{71} + q^{73} - 4 q^{75} - 28 q^{79} - 14 q^{81} + 3 q^{83} - 7 q^{85} + 4 q^{87} - 6 q^{89} - 2 q^{95} - 12 q^{97} + 6 q^{99}+O(q^{100}) $ 2 * q + q^3 + 2 * q^5 + q^9 - q^11 + q^15 - 7 * q^17 - 2 * q^19 - 8 * q^23 - 8 * q^25 + 4 * q^27 - 5 * q^29 + 13 * q^31 + 6 * q^33 - 13 * q^39 + 3 * q^41 + q^45 + 8 * q^47 - 10 * q^51 + 3 * q^53 - q^55 - q^57 - 4 * q^59 - 2 * q^61 - 11 * q^67 - 17 * q^69 - 4 * q^71 + q^73 - 4 * q^75 - 28 * q^79 - 14 * q^81 + 3 * q^83 - 7 * q^85 + 4 * q^87 - 6 * q^89 - 2 * q^95 - 12 * q^97 + 6 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.30278	−0.752158	−0.376079	−	0.926588i	$-0.622728\pi$
$3$	−1.30278	−0.752158	−0.376079	+	0.926588i	$0.622728\pi$
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.30278	−0.434259
$10$	0	0
$11$	−2.30278	−0.694313	−0.347156	−	0.937807i	$-0.612853\pi$
$11$	−2.30278	−0.694313	−0.347156	+	0.937807i	$0.612853\pi$
$12$	0	0
$13$	3.60555	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$13$	3.60555	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$14$	0	0
$15$	−1.30278	−0.336375
$16$	0	0
$17$	−1.69722	−0.411637	−0.205819	−	0.978590i	$-0.565986\pi$
$17$	−1.69722	−0.411637	−0.205819	+	0.978590i	$0.565986\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.394449	−0.0822482	−0.0411241	−	0.999154i	$-0.513094\pi$
$23$	−0.394449	−0.0822482	−0.0411241	+	0.999154i	$0.513094\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	5.60555	1.07879
$28$	0	0
$29$	−4.30278	−0.799005	−0.399503	−	0.916732i	$-0.630817\pi$
$29$	−4.30278	−0.799005	−0.399503	+	0.916732i	$0.630817\pi$
$30$	0	0
$31$	8.30278	1.49122	0.745611	−	0.666381i	$-0.232158\pi$
$31$	8.30278	1.49122	0.745611	+	0.666381i	$0.232158\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.60555	0.592749	0.296374	−	0.955072i	$-0.404222\pi$
$37$	3.60555	0.592749	0.296374	+	0.955072i	$0.404222\pi$
$38$	0	0
$39$	−4.69722	−0.752158
$40$	0	0
$41$	−0.302776	−0.0472856	−0.0236428	−	0.999720i	$-0.507526\pi$
$41$	−0.302776	−0.0472856	−0.0236428	+	0.999720i	$0.507526\pi$
$42$	0	0
$43$	7.21110	1.09968	0.549841	−	0.835269i	$-0.314688\pi$
$43$	7.21110	1.09968	0.549841	+	0.835269i	$0.314688\pi$
$44$	0	0
$45$	−1.30278	−0.194206
$46$	0	0
$47$	7.60555	1.10938	0.554692	−	0.832056i	$-0.312836\pi$
$47$	7.60555	1.10938	0.554692	+	0.832056i	$0.312836\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.21110	0.309616
$52$	0	0
$53$	−3.90833	−0.536850	−0.268425	−	0.963301i	$-0.586503\pi$
$53$	−3.90833	−0.536850	−0.268425	+	0.963301i	$0.586503\pi$
$54$	0	0
$55$	−2.30278	−0.310506
$56$	0	0
$57$	1.30278	0.172557
$58$	0	0
$59$	−5.60555	−0.729781	−0.364890	−	0.931051i	$-0.618894\pi$
$59$	−5.60555	−0.729781	−0.364890	+	0.931051i	$0.618894\pi$
$60$	0	0
$61$	−8.21110	−1.05132	−0.525662	−	0.850694i	$-0.676182\pi$
$61$	−8.21110	−1.05132	−0.525662	+	0.850694i	$0.676182\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.60555	0.447214
$66$	0	0
$67$	−10.9083	−1.33266	−0.666332	−	0.745655i	$-0.732137\pi$
$67$	−10.9083	−1.33266	−0.666332	+	0.745655i	$0.732137\pi$
$68$	0	0
$69$	0.513878	0.0618637
$70$	0	0
$71$	8.81665	1.04634	0.523172	−	0.852227i	$-0.324748\pi$
$71$	8.81665	1.04634	0.523172	+	0.852227i	$0.324748\pi$
$72$	0	0
$73$	5.90833	0.691517	0.345759	−	0.938323i	$-0.387622\pi$
$73$	5.90833	0.691517	0.345759	+	0.938323i	$0.387622\pi$
$74$	0	0
$75$	5.21110	0.601726
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	0	0
$81$	−3.39445	−0.377161
$82$	0	0
$83$	10.5139	1.15405	0.577024	−	0.816727i	$-0.304214\pi$
$83$	10.5139	1.15405	0.577024	+	0.816727i	$0.304214\pi$
$84$	0	0
$85$	−1.69722	−0.184090
$86$	0	0
$87$	5.60555	0.600978
$88$	0	0
$89$	−13.8167	−1.46456	−0.732281	−	0.681002i	$-0.761544\pi$
$89$	−13.8167	−1.46456	−0.732281	+	0.681002i	$0.761544\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−10.8167	−1.12163
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−16.8167	−1.70747	−0.853736	−	0.520706i	$-0.825669\pi$
$97$	−16.8167	−1.70747	−0.853736	+	0.520706i	$0.825669\pi$
$98$	0	0
$99$	3.00000	0.301511
$100$	0	0
$101$	−2.39445	−0.238257	−0.119128	−	0.992879i	$-0.538010\pi$
$101$	−2.39445	−0.238257	−0.119128	+	0.992879i	$0.538010\pi$
$102$	0	0
$103$	4.39445	0.432998	0.216499	−	0.976283i	$-0.430536\pi$
$103$	4.39445	0.432998	0.216499	+	0.976283i	$0.430536\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.8167	1.04569	0.522843	−	0.852429i	$-0.324872\pi$
$107$	10.8167	1.04569	0.522843	+	0.852429i	$0.324872\pi$
$108$	0	0
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	0	0
$111$	−4.69722	−0.445841
$112$	0	0
$113$	17.1194	1.61046	0.805230	−	0.592962i	$-0.202042\pi$
$113$	17.1194	1.61046	0.805230	+	0.592962i	$0.202042\pi$
$114$	0	0
$115$	−0.394449	−0.0367825
$116$	0	0
$117$	−4.69722	−0.434259
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.69722	−0.517929
$122$	0	0
$123$	0.394449	0.0355662
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−15.0278	−1.33350	−0.666749	−	0.745282i	$-0.732315\pi$
$127$	−15.0278	−1.33350	−0.666749	+	0.745282i	$0.732315\pi$
$128$	0	0
$129$	−9.39445	−0.827135
$130$	0	0
$131$	14.3028	1.24964	0.624820	−	0.780769i	$-0.285172\pi$
$131$	14.3028	1.24964	0.624820	+	0.780769i	$0.285172\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	5.60555	0.482449
$136$	0	0
$137$	0.788897	0.0674001	0.0337000	−	0.999432i	$-0.489271\pi$
$137$	0.788897	0.0674001	0.0337000	+	0.999432i	$0.489271\pi$
$138$	0	0
$139$	−4.39445	−0.372732	−0.186366	−	0.982480i	$-0.559671\pi$
$139$	−4.39445	−0.372732	−0.186366	+	0.982480i	$0.559671\pi$
$140$	0	0
$141$	−9.90833	−0.834432
$142$	0	0
$143$	−8.30278	−0.694313
$144$	0	0
$145$	−4.30278	−0.357326
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	0	0
$151$	−5.09167	−0.414354	−0.207177	−	0.978303i	$-0.566428\pi$
$151$	−5.09167	−0.414354	−0.207177	+	0.978303i	$0.566428\pi$
$152$	0	0
$153$	2.21110	0.178757
$154$	0	0
$155$	8.30278	0.666895
$156$	0	0
$157$	2.48612	0.198414	0.0992071	−	0.995067i	$-0.468369\pi$
$157$	2.48612	0.198414	0.0992071	+	0.995067i	$0.468369\pi$
$158$	0	0
$159$	5.09167	0.403796
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−0.513878	−0.0402500	−0.0201250	−	0.999797i	$-0.506406\pi$
$163$	−0.513878	−0.0402500	−0.0201250	+	0.999797i	$0.506406\pi$
$164$	0	0
$165$	3.00000	0.233550
$166$	0	0
$167$	21.4222	1.65770	0.828850	−	0.559471i	$-0.188996\pi$
$167$	21.4222	1.65770	0.828850	+	0.559471i	$0.188996\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	1.30278	0.0996257
$172$	0	0
$173$	−13.8167	−1.05046	−0.525230	−	0.850960i	$-0.676021\pi$
$173$	−13.8167	−1.05046	−0.525230	+	0.850960i	$0.676021\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.30278	0.548910
$178$	0	0
$179$	−8.30278	−0.620579	−0.310289	−	0.950642i	$-0.600426\pi$
$179$	−8.30278	−0.620579	−0.310289	+	0.950642i	$0.600426\pi$
$180$	0	0
$181$	−5.69722	−0.423471	−0.211736	−	0.977327i	$-0.567912\pi$
$181$	−5.69722	−0.423471	−0.211736	+	0.977327i	$0.567912\pi$
$182$	0	0
$183$	10.6972	0.790762
$184$	0	0
$185$	3.60555	0.265085
$186$	0	0
$187$	3.90833	0.285805
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−9.72498	−0.703675	−0.351837	−	0.936061i	$-0.614443\pi$
$191$	−9.72498	−0.703675	−0.351837	+	0.936061i	$0.614443\pi$
$192$	0	0
$193$	−15.6972	−1.12991	−0.564955	−	0.825121i	$-0.691107\pi$
$193$	−15.6972	−1.12991	−0.564955	+	0.825121i	$0.691107\pi$
$194$	0	0
$195$	−4.69722	−0.336375
$196$	0	0
$197$	3.90833	0.278457	0.139228	−	0.990260i	$-0.455538\pi$
$197$	3.90833	0.278457	0.139228	+	0.990260i	$0.455538\pi$
$198$	0	0
$199$	−5.60555	−0.397367	−0.198683	−	0.980064i	$-0.563667\pi$
$199$	−5.60555	−0.397367	−0.198683	+	0.980064i	$0.563667\pi$
$200$	0	0
$201$	14.2111	1.00237
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−0.302776	−0.0211468
$206$	0	0
$207$	0.513878	0.0357170
$208$	0	0
$209$	2.30278	0.159286
$210$	0	0
$211$	10.6972	0.736427	0.368214	−	0.929741i	$-0.379969\pi$
$211$	10.6972	0.736427	0.368214	+	0.929741i	$0.379969\pi$
$212$	0	0
$213$	−11.4861	−0.787016
$214$	0	0
$215$	7.21110	0.491793
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−7.69722	−0.520130
$220$	0	0
$221$	−6.11943	−0.411637
$222$	0	0
$223$	−8.21110	−0.549856	−0.274928	−	0.961465i	$-0.588654\pi$
$223$	−8.21110	−0.549856	−0.274928	+	0.961465i	$0.588654\pi$
$224$	0	0
$225$	5.21110	0.347407
$226$	0	0
$227$	−11.6972	−0.776372	−0.388186	−	0.921581i	$-0.626898\pi$
$227$	−11.6972	−0.776372	−0.388186	+	0.921581i	$0.626898\pi$
$228$	0	0
$229$	−3.21110	−0.212196	−0.106098	−	0.994356i	$-0.533836\pi$
$229$	−3.21110	−0.212196	−0.106098	+	0.994356i	$0.533836\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	29.5139	1.93352	0.966759	−	0.255688i	$-0.0823021\pi$
$233$	29.5139	1.93352	0.966759	+	0.255688i	$0.0823021\pi$
$234$	0	0
$235$	7.60555	0.496131
$236$	0	0
$237$	18.2389	1.18474
$238$	0	0
$239$	−14.0278	−0.907380	−0.453690	−	0.891160i	$-0.649893\pi$
$239$	−14.0278	−0.907380	−0.453690	+	0.891160i	$0.649893\pi$
$240$	0	0
$241$	0.183346	0.0118104	0.00590518	−	0.999983i	$-0.498120\pi$
$241$	0.183346	0.0118104	0.00590518	+	0.999983i	$0.498120\pi$
$242$	0	0
$243$	−12.3944	−0.795104
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.60555	−0.229416
$248$	0	0
$249$	−13.6972	−0.868026
$250$	0	0
$251$	−26.7250	−1.68687	−0.843433	−	0.537235i	$-0.819469\pi$
$251$	−26.7250	−1.68687	−0.843433	+	0.537235i	$0.819469\pi$
$252$	0	0
$253$	0.908327	0.0571060
$254$	0	0
$255$	2.21110	0.138465
$256$	0	0
$257$	−5.90833	−0.368551	−0.184276	−	0.982875i	$-0.558994\pi$
$257$	−5.90833	−0.368551	−0.184276	+	0.982875i	$0.558994\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	5.60555	0.346975
$262$	0	0
$263$	−4.69722	−0.289643	−0.144822	−	0.989458i	$-0.546261\pi$
$263$	−4.69722	−0.289643	−0.144822	+	0.989458i	$0.546261\pi$
$264$	0	0
$265$	−3.90833	−0.240087
$266$	0	0
$267$	18.0000	1.10158
$268$	0	0
$269$	2.51388	0.153274	0.0766369	−	0.997059i	$-0.475582\pi$
$269$	2.51388	0.153274	0.0766369	+	0.997059i	$0.475582\pi$
$270$	0	0
$271$	5.09167	0.309297	0.154649	−	0.987970i	$-0.450575\pi$
$271$	5.09167	0.309297	0.154649	+	0.987970i	$0.450575\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9.21110	0.555450
$276$	0	0
$277$	−27.8167	−1.67134	−0.835670	−	0.549231i	$-0.814921\pi$
$277$	−27.8167	−1.67134	−0.835670	+	0.549231i	$0.814921\pi$
$278$	0	0
$279$	−10.8167	−0.647576
$280$	0	0
$281$	19.4222	1.15863	0.579316	−	0.815103i	$-0.303320\pi$
$281$	19.4222	1.15863	0.579316	+	0.815103i	$0.303320\pi$
$282$	0	0
$283$	−22.5139	−1.33831	−0.669156	−	0.743122i	$-0.733344\pi$
$283$	−22.5139	−1.33831	−0.669156	+	0.743122i	$0.733344\pi$
$284$	0	0
$285$	1.30278	0.0771698
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−14.1194	−0.830555
$290$	0	0
$291$	21.9083	1.28429
$292$	0	0
$293$	−18.6333	−1.08857	−0.544285	−	0.838901i	$-0.683199\pi$
$293$	−18.6333	−1.08857	−0.544285	+	0.838901i	$0.683199\pi$
$294$	0	0
$295$	−5.60555	−0.326368
$296$	0	0
$297$	−12.9083	−0.749017
$298$	0	0
$299$	−1.42221	−0.0822482
$300$	0	0
$301$	0	0
$302$	0	0
$303$	3.11943	0.179207
$304$	0	0
$305$	−8.21110	−0.470166
$306$	0	0
$307$	14.1194	0.805838	0.402919	−	0.915236i	$-0.367996\pi$
$307$	14.1194	0.805838	0.402919	+	0.915236i	$0.367996\pi$
$308$	0	0
$309$	−5.72498	−0.325683
$310$	0	0
$311$	−7.33053	−0.415676	−0.207838	−	0.978163i	$-0.566643\pi$
$311$	−7.33053	−0.415676	−0.207838	+	0.978163i	$0.566643\pi$
$312$	0	0
$313$	27.8167	1.57229	0.786145	−	0.618042i	$-0.212074\pi$
$313$	27.8167	1.57229	0.786145	+	0.618042i	$0.212074\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.2111	−0.629678	−0.314839	−	0.949145i	$-0.601951\pi$
$317$	−11.2111	−0.629678	−0.314839	+	0.949145i	$0.601951\pi$
$318$	0	0
$319$	9.90833	0.554760
$320$	0	0
$321$	−14.0917	−0.786520
$322$	0	0
$323$	1.69722	0.0944361
$324$	0	0
$325$	−14.4222	−0.800000
$326$	0	0
$327$	9.11943	0.504306
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−9.48612	−0.521404	−0.260702	−	0.965419i	$-0.583954\pi$
$331$	−9.48612	−0.521404	−0.260702	+	0.965419i	$0.583954\pi$
$332$	0	0
$333$	−4.69722	−0.257406
$334$	0	0
$335$	−10.9083	−0.595986
$336$	0	0
$337$	−29.6972	−1.61771	−0.808855	−	0.588008i	$-0.799913\pi$
$337$	−29.6972	−1.61771	−0.808855	+	0.588008i	$0.799913\pi$
$338$	0	0
$339$	−22.3028	−1.21132
$340$	0	0
$341$	−19.1194	−1.03538
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0.513878	0.0276663
$346$	0	0
$347$	−10.6972	−0.574257	−0.287129	−	0.957892i	$-0.592701\pi$
$347$	−10.6972	−0.574257	−0.287129	+	0.957892i	$0.592701\pi$
$348$	0	0
$349$	−6.51388	−0.348680	−0.174340	−	0.984686i	$-0.555779\pi$
$349$	−6.51388	−0.348680	−0.174340	+	0.984686i	$0.555779\pi$
$350$	0	0
$351$	20.2111	1.07879
$352$	0	0
$353$	−1.27502	−0.0678624	−0.0339312	−	0.999424i	$-0.510803\pi$
$353$	−1.27502	−0.0678624	−0.0339312	+	0.999424i	$0.510803\pi$
$354$	0	0
$355$	8.81665	0.467939
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10.3028	−0.543760	−0.271880	−	0.962331i	$-0.587645\pi$
$359$	−10.3028	−0.543760	−0.271880	+	0.962331i	$0.587645\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	7.42221	0.389565
$364$	0	0
$365$	5.90833	0.309256
$366$	0	0
$367$	−18.2111	−0.950612	−0.475306	−	0.879821i	$-0.657663\pi$
$367$	−18.2111	−0.950612	−0.475306	+	0.879821i	$0.657663\pi$
$368$	0	0
$369$	0.394449	0.0205342
$370$	0	0
$371$	0	0
$372$	0	0
$373$	19.9083	1.03081	0.515407	−	0.856945i	$-0.327641\pi$
$373$	19.9083	1.03081	0.515407	+	0.856945i	$0.327641\pi$
$374$	0	0
$375$	11.7250	0.605475
$376$	0	0
$377$	−15.5139	−0.799005
$378$	0	0
$379$	21.4222	1.10038	0.550192	−	0.835038i	$-0.314554\pi$
$379$	21.4222	1.10038	0.550192	+	0.835038i	$0.314554\pi$
$380$	0	0
$381$	19.5778	1.00300
$382$	0	0
$383$	32.4500	1.65812	0.829058	−	0.559163i	$-0.188877\pi$
$383$	32.4500	1.65812	0.829058	+	0.559163i	$0.188877\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.39445	−0.477547
$388$	0	0
$389$	−3.69722	−0.187457	−0.0937284	−	0.995598i	$-0.529879\pi$
$389$	−3.69722	−0.187457	−0.0937284	+	0.995598i	$0.529879\pi$
$390$	0	0
$391$	0.669468	0.0338565
$392$	0	0
$393$	−18.6333	−0.939926
$394$	0	0
$395$	−14.0000	−0.704416
$396$	0	0
$397$	4.60555	0.231146	0.115573	−	0.993299i	$-0.463130\pi$
$397$	4.60555	0.231146	0.115573	+	0.993299i	$0.463130\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12.9083	−0.644611	−0.322306	−	0.946636i	$-0.604458\pi$
$401$	−12.9083	−0.644611	−0.322306	+	0.946636i	$0.604458\pi$
$402$	0	0
$403$	29.9361	1.49122
$404$	0	0
$405$	−3.39445	−0.168672
$406$	0	0
$407$	−8.30278	−0.411553
$408$	0	0
$409$	38.7527	1.91620	0.958100	−	0.286435i	$-0.0924702\pi$
$409$	38.7527	1.91620	0.958100	+	0.286435i	$0.0924702\pi$
$410$	0	0
$411$	−1.02776	−0.0506955
$412$	0	0
$413$	0	0
$414$	0	0
$415$	10.5139	0.516106
$416$	0	0
$417$	5.72498	0.280354
$418$	0	0
$419$	2.81665	0.137603	0.0688013	−	0.997630i	$-0.478083\pi$
$419$	2.81665	0.137603	0.0688013	+	0.997630i	$0.478083\pi$
$420$	0	0
$421$	−15.1833	−0.739991	−0.369996	−	0.929034i	$-0.620641\pi$
$421$	−15.1833	−0.739991	−0.369996	+	0.929034i	$0.620641\pi$
$422$	0	0
$423$	−9.90833	−0.481759
$424$	0	0
$425$	6.78890	0.329310
$426$	0	0
$427$	0	0
$428$	0	0
$429$	10.8167	0.522233
$430$	0	0
$431$	−36.4222	−1.75440	−0.877198	−	0.480129i	$-0.840590\pi$
$431$	−36.4222	−1.75440	−0.877198	+	0.480129i	$0.840590\pi$
$432$	0	0
$433$	−7.78890	−0.374311	−0.187155	−	0.982330i	$-0.559927\pi$
$433$	−7.78890	−0.374311	−0.187155	+	0.982330i	$0.559927\pi$
$434$	0	0
$435$	5.60555	0.268766
$436$	0	0
$437$	0.394449	0.0188690
$438$	0	0
$439$	−5.21110	−0.248712	−0.124356	−	0.992238i	$-0.539687\pi$
$439$	−5.21110	−0.248712	−0.124356	+	0.992238i	$0.539687\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14.0917	0.669516	0.334758	−	0.942304i	$-0.391345\pi$
$443$	14.0917	0.669516	0.334758	+	0.942304i	$0.391345\pi$
$444$	0	0
$445$	−13.8167	−0.654972
$446$	0	0
$447$	−3.90833	−0.184858
$448$	0	0
$449$	−17.1194	−0.807916	−0.403958	−	0.914778i	$-0.632366\pi$
$449$	−17.1194	−0.807916	−0.403958	+	0.914778i	$0.632366\pi$
$450$	0	0
$451$	0.697224	0.0328310
$452$	0	0
$453$	6.63331	0.311660
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−24.9361	−1.16646	−0.583230	−	0.812307i	$-0.698212\pi$
$457$	−24.9361	−1.16646	−0.583230	+	0.812307i	$0.698212\pi$
$458$	0	0
$459$	−9.51388	−0.444070
$460$	0	0
$461$	−15.1194	−0.704182	−0.352091	−	0.935966i	$-0.614529\pi$
$461$	−15.1194	−0.704182	−0.352091	+	0.935966i	$0.614529\pi$
$462$	0	0
$463$	−12.2111	−0.567498	−0.283749	−	0.958899i	$-0.591578\pi$
$463$	−12.2111	−0.567498	−0.283749	+	0.958899i	$0.591578\pi$
$464$	0	0
$465$	−10.8167	−0.501610
$466$	0	0
$467$	29.1194	1.34749	0.673743	−	0.738966i	$-0.264685\pi$
$467$	29.1194	1.34749	0.673743	+	0.738966i	$0.264685\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−3.23886	−0.149239
$472$	0	0
$473$	−16.6056	−0.763524
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	5.09167	0.233132
$478$	0	0
$479$	−10.3028	−0.470746	−0.235373	−	0.971905i	$-0.575631\pi$
$479$	−10.3028	−0.470746	−0.235373	+	0.971905i	$0.575631\pi$
$480$	0	0
$481$	13.0000	0.592749
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−16.8167	−0.763605
$486$	0	0
$487$	−37.2389	−1.68745	−0.843727	−	0.536773i	$-0.819643\pi$
$487$	−37.2389	−1.68745	−0.843727	+	0.536773i	$0.819643\pi$
$488$	0	0
$489$	0.669468	0.0302744
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	0	0
$493$	7.30278	0.328900
$494$	0	0
$495$	3.00000	0.134840
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−18.3305	−0.820587	−0.410294	−	0.911953i	$-0.634574\pi$
$499$	−18.3305	−0.820587	−0.410294	+	0.911953i	$0.634574\pi$
$500$	0	0
$501$	−27.9083	−1.24685
$502$	0	0
$503$	−24.8444	−1.10776	−0.553879	−	0.832597i	$-0.686853\pi$
$503$	−24.8444	−1.10776	−0.553879	+	0.832597i	$0.686853\pi$
$504$	0	0
$505$	−2.39445	−0.106552
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8.78890	0.389561	0.194781	−	0.980847i	$-0.437601\pi$
$509$	8.78890	0.389561	0.194781	+	0.980847i	$0.437601\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−5.60555	−0.247491
$514$	0	0
$515$	4.39445	0.193643
$516$	0	0
$517$	−17.5139	−0.770259
$518$	0	0
$519$	18.0000	0.790112
$520$	0	0
$521$	13.4222	0.588037	0.294019	−	0.955800i	$-0.405007\pi$
$521$	13.4222	0.588037	0.294019	+	0.955800i	$0.405007\pi$
$522$	0	0
$523$	−5.42221	−0.237096	−0.118548	−	0.992948i	$-0.537824\pi$
$523$	−5.42221	−0.237096	−0.118548	+	0.992948i	$0.537824\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−14.0917	−0.613843
$528$	0	0
$529$	−22.8444	−0.993235
$530$	0	0
$531$	7.30278	0.316913
$532$	0	0
$533$	−1.09167	−0.0472856
$534$	0	0
$535$	10.8167	0.467645
$536$	0	0
$537$	10.8167	0.466773
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−39.2111	−1.68582	−0.842908	−	0.538057i	$-0.819159\pi$
$541$	−39.2111	−1.68582	−0.842908	+	0.538057i	$0.819159\pi$
$542$	0	0
$543$	7.42221	0.318517
$544$	0	0
$545$	−7.00000	−0.299847
$546$	0	0
$547$	−6.51388	−0.278513	−0.139257	−	0.990256i	$-0.544471\pi$
$547$	−6.51388	−0.278513	−0.139257	+	0.990256i	$0.544471\pi$
$548$	0	0
$549$	10.6972	0.456546
$550$	0	0
$551$	4.30278	0.183304
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−4.69722	−0.199386
$556$	0	0
$557$	35.7527	1.51489	0.757446	−	0.652898i	$-0.226447\pi$
$557$	35.7527	1.51489	0.757446	+	0.652898i	$0.226447\pi$
$558$	0	0
$559$	26.0000	1.09968
$560$	0	0
$561$	−5.09167	−0.214971
$562$	0	0
$563$	20.0278	0.844069	0.422035	−	0.906580i	$-0.361316\pi$
$563$	20.0278	0.844069	0.422035	+	0.906580i	$0.361316\pi$
$564$	0	0
$565$	17.1194	0.720220
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−14.0278	−0.588074	−0.294037	−	0.955794i	$-0.594999\pi$
$569$	−14.0278	−0.588074	−0.294037	+	0.955794i	$0.594999\pi$
$570$	0	0
$571$	−17.7889	−0.744442	−0.372221	−	0.928144i	$-0.621404\pi$
$571$	−17.7889	−0.744442	−0.372221	+	0.928144i	$0.621404\pi$
$572$	0	0
$573$	12.6695	0.529275
$574$	0	0
$575$	1.57779	0.0657986
$576$	0	0
$577$	−3.72498	−0.155073	−0.0775365	−	0.996990i	$-0.524705\pi$
$577$	−3.72498	−0.155073	−0.0775365	+	0.996990i	$0.524705\pi$
$578$	0	0
$579$	20.4500	0.849871
$580$	0	0
$581$	0	0
$582$	0	0
$583$	9.00000	0.372742
$584$	0	0
$585$	−4.69722	−0.194206
$586$	0	0
$587$	9.63331	0.397609	0.198805	−	0.980039i	$-0.436294\pi$
$587$	9.63331	0.397609	0.198805	+	0.980039i	$0.436294\pi$
$588$	0	0
$589$	−8.30278	−0.342110
$590$	0	0
$591$	−5.09167	−0.209443
$592$	0	0
$593$	−23.1833	−0.952026	−0.476013	−	0.879438i	$-0.657918\pi$
$593$	−23.1833	−0.952026	−0.476013	+	0.879438i	$0.657918\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7.30278	0.298883
$598$	0	0
$599$	14.3305	0.585530	0.292765	−	0.956184i	$-0.405425\pi$
$599$	14.3305	0.585530	0.292765	+	0.956184i	$0.405425\pi$
$600$	0	0
$601$	−30.1472	−1.22973	−0.614865	−	0.788633i	$-0.710789\pi$
$601$	−30.1472	−1.22973	−0.614865	+	0.788633i	$0.710789\pi$
$602$	0	0
$603$	14.2111	0.578721
$604$	0	0
$605$	−5.69722	−0.231625
$606$	0	0
$607$	8.81665	0.357857	0.178928	−	0.983862i	$-0.442737\pi$
$607$	8.81665	0.357857	0.178928	+	0.983862i	$0.442737\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	27.4222	1.10938
$612$	0	0
$613$	39.3583	1.58967	0.794833	−	0.606828i	$-0.207558\pi$
$613$	39.3583	1.58967	0.794833	+	0.606828i	$0.207558\pi$
$614$	0	0
$615$	0.394449	0.0159057
$616$	0	0
$617$	−42.5416	−1.71266	−0.856331	−	0.516428i	$-0.827262\pi$
$617$	−42.5416	−1.71266	−0.856331	+	0.516428i	$0.827262\pi$
$618$	0	0
$619$	46.5694	1.87178	0.935891	−	0.352290i	$-0.114597\pi$
$619$	46.5694	1.87178	0.935891	+	0.352290i	$0.114597\pi$
$620$	0	0
$621$	−2.21110	−0.0887285
$622$	0	0
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−3.00000	−0.119808
$628$	0	0
$629$	−6.11943	−0.243998
$630$	0	0
$631$	−8.21110	−0.326879	−0.163439	−	0.986553i	$-0.552259\pi$
$631$	−8.21110	−0.326879	−0.163439	+	0.986553i	$0.552259\pi$
$632$	0	0
$633$	−13.9361	−0.553910
$634$	0	0
$635$	−15.0278	−0.596358
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−11.4861	−0.454384
$640$	0	0
$641$	−40.3028	−1.59186	−0.795932	−	0.605386i	$-0.793019\pi$
$641$	−40.3028	−1.59186	−0.795932	+	0.605386i	$0.793019\pi$
$642$	0	0
$643$	−14.5778	−0.574892	−0.287446	−	0.957797i	$-0.592806\pi$
$643$	−14.5778	−0.574892	−0.287446	+	0.957797i	$0.592806\pi$
$644$	0	0
$645$	−9.39445	−0.369906
$646$	0	0
$647$	0.816654	0.0321060	0.0160530	−	0.999871i	$-0.494890\pi$
$647$	0.816654	0.0321060	0.0160530	+	0.999871i	$0.494890\pi$
$648$	0	0
$649$	12.9083	0.506696
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−24.0278	−0.940279	−0.470139	−	0.882592i	$-0.655796\pi$
$653$	−24.0278	−0.940279	−0.470139	+	0.882592i	$0.655796\pi$
$654$	0	0
$655$	14.3028	0.558856
$656$	0	0
$657$	−7.69722	−0.300297
$658$	0	0
$659$	6.90833	0.269110	0.134555	−	0.990906i	$-0.457039\pi$
$659$	6.90833	0.269110	0.134555	+	0.990906i	$0.457039\pi$
$660$	0	0
$661$	16.7889	0.653012	0.326506	−	0.945195i	$-0.394129\pi$
$661$	16.7889	0.653012	0.326506	+	0.945195i	$0.394129\pi$
$662$	0	0
$663$	7.97224	0.309616
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.69722	0.0657168
$668$	0	0
$669$	10.6972	0.413579
$670$	0	0
$671$	18.9083	0.729948
$672$	0	0
$673$	−31.7250	−1.22291	−0.611454	−	0.791280i	$-0.709415\pi$
$673$	−31.7250	−1.22291	−0.611454	+	0.791280i	$0.709415\pi$
$674$	0	0
$675$	−22.4222	−0.863031
$676$	0	0
$677$	−17.4861	−0.672046	−0.336023	−	0.941854i	$-0.609082\pi$
$677$	−17.4861	−0.672046	−0.336023	+	0.941854i	$0.609082\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	15.2389	0.583954
$682$	0	0
$683$	−21.2389	−0.812682	−0.406341	−	0.913721i	$-0.633196\pi$
$683$	−21.2389	−0.812682	−0.406341	+	0.913721i	$0.633196\pi$
$684$	0	0
$685$	0.788897	0.0301422
$686$	0	0
$687$	4.18335	0.159605
$688$	0	0
$689$	−14.0917	−0.536850
$690$	0	0
$691$	5.39445	0.205215	0.102607	−	0.994722i	$-0.467282\pi$
$691$	5.39445	0.205215	0.102607	+	0.994722i	$0.467282\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.39445	−0.166691
$696$	0	0
$697$	0.513878	0.0194645
$698$	0	0
$699$	−38.4500	−1.45431
$700$	0	0
$701$	−27.6333	−1.04370	−0.521848	−	0.853039i	$-0.674757\pi$
$701$	−27.6333	−1.04370	−0.521848	+	0.853039i	$0.674757\pi$
$702$	0	0
$703$	−3.60555	−0.135986
$704$	0	0
$705$	−9.90833	−0.373169
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−21.2389	−0.797642	−0.398821	−	0.917029i	$-0.630581\pi$
$709$	−21.2389	−0.797642	−0.398821	+	0.917029i	$0.630581\pi$
$710$	0	0
$711$	18.2389	0.684011
$712$	0	0
$713$	−3.27502	−0.122650
$714$	0	0
$715$	−8.30278	−0.310506
$716$	0	0
$717$	18.2750	0.682493
$718$	0	0
$719$	−23.6333	−0.881374	−0.440687	−	0.897661i	$-0.645265\pi$
$719$	−23.6333	−0.881374	−0.440687	+	0.897661i	$0.645265\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−0.238859	−0.00888326
$724$	0	0
$725$	17.2111	0.639204
$726$	0	0
$727$	0.577795	0.0214292	0.0107146	−	0.999943i	$-0.496589\pi$
$727$	0.577795	0.0214292	0.0107146	+	0.999943i	$0.496589\pi$
$728$	0	0
$729$	26.3305	0.975205
$730$	0	0
$731$	−12.2389	−0.452671
$732$	0	0
$733$	−52.8444	−1.95185	−0.975926	−	0.218100i	$-0.930014\pi$
$733$	−52.8444	−1.95185	−0.975926	+	0.218100i	$0.930014\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	25.1194	0.925286
$738$	0	0
$739$	−24.8167	−0.912895	−0.456448	−	0.889750i	$-0.650878\pi$
$739$	−24.8167	−0.912895	−0.456448	+	0.889750i	$0.650878\pi$
$740$	0	0
$741$	4.69722	0.172557
$742$	0	0
$743$	23.2389	0.852551	0.426276	−	0.904593i	$-0.359825\pi$
$743$	23.2389	0.852551	0.426276	+	0.904593i	$0.359825\pi$
$744$	0	0
$745$	3.00000	0.109911
$746$	0	0
$747$	−13.6972	−0.501155
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−42.1472	−1.53797	−0.768986	−	0.639265i	$-0.779239\pi$
$751$	−42.1472	−1.53797	−0.768986	+	0.639265i	$0.779239\pi$
$752$	0	0
$753$	34.8167	1.26879
$754$	0	0
$755$	−5.09167	−0.185305
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	0	0
$759$	−1.18335	−0.0429527
$760$	0	0
$761$	−3.00000	−0.108750	−0.0543750	−	0.998521i	$-0.517317\pi$
$761$	−3.00000	−0.108750	−0.0543750	+	0.998521i	$0.517317\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.21110	0.0799426
$766$	0	0
$767$	−20.2111	−0.729781
$768$	0	0
$769$	−26.7889	−0.966032	−0.483016	−	0.875611i	$-0.660459\pi$
$769$	−26.7889	−0.966032	−0.483016	+	0.875611i	$0.660459\pi$
$770$	0	0
$771$	7.69722	0.277209
$772$	0	0
$773$	−27.8167	−1.00050	−0.500248	−	0.865882i	$-0.666758\pi$
$773$	−27.8167	−1.00050	−0.500248	+	0.865882i	$0.666758\pi$
$774$	0	0
$775$	−33.2111	−1.19298
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.302776	0.0108481
$780$	0	0
$781$	−20.3028	−0.726490
$782$	0	0
$783$	−24.1194	−0.861958
$784$	0	0
$785$	2.48612	0.0887335
$786$	0	0
$787$	−11.6333	−0.414683	−0.207341	−	0.978269i	$-0.566481\pi$
$787$	−11.6333	−0.414683	−0.207341	+	0.978269i	$0.566481\pi$
$788$	0	0
$789$	6.11943	0.217857
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−29.6056	−1.05132
$794$	0	0
$795$	5.09167	0.180583
$796$	0	0
$797$	38.7250	1.37171	0.685855	−	0.727739i	$-0.259429\pi$
$797$	38.7250	1.37171	0.685855	+	0.727739i	$0.259429\pi$
$798$	0	0
$799$	−12.9083	−0.456664
$800$	0	0
$801$	18.0000	0.635999
$802$	0	0
$803$	−13.6056	−0.480129
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3.27502	−0.115286
$808$	0	0
$809$	−11.1833	−0.393186	−0.196593	−	0.980485i	$-0.562988\pi$
$809$	−11.1833	−0.393186	−0.196593	+	0.980485i	$0.562988\pi$
$810$	0	0
$811$	24.4222	0.857580	0.428790	−	0.903404i	$-0.358940\pi$
$811$	24.4222	0.857580	0.428790	+	0.903404i	$0.358940\pi$
$812$	0	0
$813$	−6.63331	−0.232640
$814$	0	0
$815$	−0.513878	−0.0180004
$816$	0	0
$817$	−7.21110	−0.252285
$818$	0	0
$819$	0	0
$820$	0	0
$821$	47.6333	1.66241	0.831207	−	0.555963i	$-0.187650\pi$
$821$	47.6333	1.66241	0.831207	+	0.555963i	$0.187650\pi$
$822$	0	0
$823$	−6.23886	−0.217473	−0.108736	−	0.994071i	$-0.534680\pi$
$823$	−6.23886	−0.217473	−0.108736	+	0.994071i	$0.534680\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	−9.60555	−0.334018	−0.167009	−	0.985955i	$-0.553411\pi$
$827$	−9.60555	−0.334018	−0.167009	+	0.985955i	$0.553411\pi$
$828$	0	0
$829$	42.0555	1.46065	0.730324	−	0.683101i	$-0.239369\pi$
$829$	42.0555	1.46065	0.730324	+	0.683101i	$0.239369\pi$
$830$	0	0
$831$	36.2389	1.25711
$832$	0	0
$833$	0	0
$834$	0	0
$835$	21.4222	0.741346
$836$	0	0
$837$	46.5416	1.60871
$838$	0	0
$839$	8.36669	0.288850	0.144425	−	0.989516i	$-0.453867\pi$
$839$	8.36669	0.288850	0.144425	+	0.989516i	$0.453867\pi$
$840$	0	0
$841$	−10.4861	−0.361590
$842$	0	0
$843$	−25.3028	−0.871474
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	29.3305	1.00662
$850$	0	0
$851$	−1.42221	−0.0487526
$852$	0	0
$853$	−7.48612	−0.256320	−0.128160	−	0.991754i	$-0.540907\pi$
$853$	−7.48612	−0.256320	−0.128160	+	0.991754i	$0.540907\pi$
$854$	0	0
$855$	1.30278	0.0445540
$856$	0	0
$857$	−39.7527	−1.35793	−0.678964	−	0.734172i	$-0.737571\pi$
$857$	−39.7527	−1.35793	−0.678964	+	0.734172i	$0.737571\pi$
$858$	0	0
$859$	−35.7527	−1.21987	−0.609934	−	0.792452i	$-0.708804\pi$
$859$	−35.7527	−1.21987	−0.609934	+	0.792452i	$0.708804\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−50.5139	−1.71951	−0.859756	−	0.510705i	$-0.829385\pi$
$863$	−50.5139	−1.71951	−0.859756	+	0.510705i	$0.829385\pi$
$864$	0	0
$865$	−13.8167	−0.469780
$866$	0	0
$867$	18.3944	0.624708
$868$	0	0
$869$	32.2389	1.09363
$870$	0	0
$871$	−39.3305	−1.33266
$872$	0	0
$873$	21.9083	0.741485
$874$	0	0
$875$	0	0
$876$	0	0
$877$	51.8722	1.75160	0.875799	−	0.482675i	$-0.160335\pi$
$877$	51.8722	1.75160	0.875799	+	0.482675i	$0.160335\pi$
$878$	0	0
$879$	24.2750	0.818776
$880$	0	0
$881$	29.9638	1.00951	0.504754	−	0.863263i	$-0.331583\pi$
$881$	29.9638	1.00951	0.504754	+	0.863263i	$0.331583\pi$
$882$	0	0
$883$	35.6611	1.20009	0.600045	−	0.799966i	$-0.295149\pi$
$883$	35.6611	1.20009	0.600045	+	0.799966i	$0.295149\pi$
$884$	0	0
$885$	7.30278	0.245480
$886$	0	0
$887$	37.7889	1.26883	0.634413	−	0.772994i	$-0.281242\pi$
$887$	37.7889	1.26883	0.634413	+	0.772994i	$0.281242\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	7.81665	0.261868
$892$	0	0
$893$	−7.60555	−0.254510
$894$	0	0
$895$	−8.30278	−0.277531
$896$	0	0
$897$	1.85281	0.0618637
$898$	0	0
$899$	−35.7250	−1.19149
$900$	0	0
$901$	6.63331	0.220988
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.69722	−0.189382
$906$	0	0
$907$	0.577795	0.0191854	0.00959268	−	0.999954i	$-0.496947\pi$
$907$	0.577795	0.0191854	0.00959268	+	0.999954i	$0.496947\pi$
$908$	0	0
$909$	3.11943	0.103465
$910$	0	0
$911$	12.7889	0.423715	0.211858	−	0.977301i	$-0.432049\pi$
$911$	12.7889	0.423715	0.211858	+	0.977301i	$0.432049\pi$
$912$	0	0
$913$	−24.2111	−0.801271
$914$	0	0
$915$	10.6972	0.353639
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−3.84441	−0.126815	−0.0634077	−	0.997988i	$-0.520197\pi$
$919$	−3.84441	−0.126815	−0.0634077	+	0.997988i	$0.520197\pi$
$920$	0	0
$921$	−18.3944	−0.606118
$922$	0	0
$923$	31.7889	1.04634
$924$	0	0
$925$	−14.4222	−0.474199
$926$	0	0
$927$	−5.72498	−0.188033
$928$	0	0
$929$	−17.3028	−0.567686	−0.283843	−	0.958871i	$-0.591609\pi$
$929$	−17.3028	−0.567686	−0.283843	+	0.958871i	$0.591609\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	9.55004	0.312654
$934$	0	0
$935$	3.90833	0.127816
$936$	0	0
$937$	9.93608	0.324598	0.162299	−	0.986742i	$-0.448109\pi$
$937$	9.93608	0.324598	0.162299	+	0.986742i	$0.448109\pi$
$938$	0	0
$939$	−36.2389	−1.18261
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	0.119429	0.00388916
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−14.5139	−0.471638	−0.235819	−	0.971797i	$-0.575777\pi$
$947$	−14.5139	−0.471638	−0.235819	+	0.971797i	$0.575777\pi$
$948$	0	0
$949$	21.3028	0.691517
$950$	0	0
$951$	14.6056	0.473617
$952$	0	0
$953$	42.1194	1.36438	0.682191	−	0.731174i	$-0.261027\pi$
$953$	42.1194	1.36438	0.682191	+	0.731174i	$0.261027\pi$
$954$	0	0
$955$	−9.72498	−0.314693
$956$	0	0
$957$	−12.9083	−0.417267
$958$	0	0
$959$	0	0
$960$	0	0
$961$	37.9361	1.22374
$962$	0	0
$963$	−14.0917	−0.454098
$964$	0	0
$965$	−15.6972	−0.505312
$966$	0	0
$967$	−6.48612	−0.208580	−0.104290	−	0.994547i	$-0.533257\pi$
$967$	−6.48612	−0.208580	−0.104290	+	0.994547i	$0.533257\pi$
$968$	0	0
$969$	−2.21110	−0.0710308
$970$	0	0
$971$	−4.81665	−0.154574	−0.0772869	−	0.997009i	$-0.524626\pi$
$971$	−4.81665	−0.154574	−0.0772869	+	0.997009i	$0.524626\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	18.7889	0.601726
$976$	0	0
$977$	34.6056	1.10713	0.553565	−	0.832806i	$-0.313267\pi$
$977$	34.6056	1.10713	0.553565	+	0.832806i	$0.313267\pi$
$978$	0	0
$979$	31.8167	1.01686
$980$	0	0
$981$	9.11943	0.291161
$982$	0	0
$983$	43.6611	1.39257	0.696286	−	0.717765i	$-0.254835\pi$
$983$	43.6611	1.39257	0.696286	+	0.717765i	$0.254835\pi$
$984$	0	0
$985$	3.90833	0.124530
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.84441	−0.0904470
$990$	0	0
$991$	−22.2389	−0.706441	−0.353220	−	0.935540i	$-0.614913\pi$
$991$	−22.2389	−0.706441	−0.353220	+	0.935540i	$0.614913\pi$
$992$	0	0
$993$	12.3583	0.392178
$994$	0	0
$995$	−5.60555	−0.177708
$996$	0	0
$997$	13.7250	0.434675	0.217337	−	0.976097i	$-0.430263\pi$
$997$	13.7250	0.434675	0.217337	+	0.976097i	$0.430263\pi$
$998$	0	0
$999$	20.2111	0.639451

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bb.1.1		2
7.6	odd	2		1064.2.a.b.1.2	✓	2
21.20	even	2		9576.2.a.bs.1.2		2
28.27	even	2		2128.2.a.i.1.1		2
56.13	odd	2		8512.2.a.x.1.1		2
56.27	even	2		8512.2.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.a.b.1.2	✓	2	7.6	odd	2
2128.2.a.i.1.1		2	28.27	even	2
7448.2.a.bb.1.1		2	1.1	even	1	trivial
8512.2.a.r.1.2		2	56.27	even	2
8512.2.a.x.1.1		2	56.13	odd	2
9576.2.a.bs.1.2		2	21.20	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:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q-1.30278 q^{3} +1.00000 q^{5} -1.30278 q^{9} +O(q^{10})\)	\(q-1.30278 q^{3} +1.00000 q^{5} -1.30278 q^{9} -2.30278 q^{11} +3.60555 q^{13} -1.30278 q^{15} -1.69722 q^{17} -1.00000 q^{19} -0.394449 q^{23} -4.00000 q^{25} +5.60555 q^{27} -4.30278 q^{29} +8.30278 q^{31} +3.00000 q^{33} +3.60555 q^{37} -4.69722 q^{39} -0.302776 q^{41} +7.21110 q^{43} -1.30278 q^{45} +7.60555 q^{47} +2.21110 q^{51} -3.90833 q^{53} -2.30278 q^{55} +1.30278 q^{57} -5.60555 q^{59} -8.21110 q^{61} +3.60555 q^{65} -10.9083 q^{67} +0.513878 q^{69} +8.81665 q^{71} +5.90833 q^{73} +5.21110 q^{75} -14.0000 q^{79} -3.39445 q^{81} +10.5139 q^{83} -1.69722 q^{85} +5.60555 q^{87} -13.8167 q^{89} -10.8167 q^{93} -1.00000 q^{95} -16.8167 q^{97} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 2 q^{5} + q^{9}+O(q^{10}) \) 2 * q + q^3 + 2 * q^5 + q^9	\( 2 q + q^{3} + 2 q^{5} + q^{9} - q^{11} + q^{15} - 7 q^{17} - 2 q^{19} - 8 q^{23} - 8 q^{25} + 4 q^{27} - 5 q^{29} + 13 q^{31} + 6 q^{33} - 13 q^{39} + 3 q^{41} + q^{45} + 8 q^{47} - 10 q^{51} + 3 q^{53} - q^{55} - q^{57} - 4 q^{59} - 2 q^{61} - 11 q^{67} - 17 q^{69} - 4 q^{71} + q^{73} - 4 q^{75} - 28 q^{79} - 14 q^{81} + 3 q^{83} - 7 q^{85} + 4 q^{87} - 6 q^{89} - 2 q^{95} - 12 q^{97} + 6 q^{99}+O(q^{100}) \) 2 * q + q^3 + 2 * q^5 + q^9 - q^11 + q^15 - 7 * q^17 - 2 * q^19 - 8 * q^23 - 8 * q^25 + 4 * q^27 - 5 * q^29 + 13 * q^31 + 6 * q^33 - 13 * q^39 + 3 * q^41 + q^45 + 8 * q^47 - 10 * q^51 + 3 * q^53 - q^55 - q^57 - 4 * q^59 - 2 * q^61 - 11 * q^67 - 17 * q^69 - 4 * q^71 + q^73 - 4 * q^75 - 28 * q^79 - 14 * q^81 + 3 * q^83 - 7 * q^85 + 4 * q^87 - 6 * q^89 - 2 * q^95 - 12 * q^97 + 6 * q^99

Introduction

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