LMFDB - Embedded newform 7200.2.f.d.6049.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7200,2,Mod(6049,7200)]

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([0, 0, 0, 1]))

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

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

chi := DirichletCharacter("7200.6049");

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.f (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:	$57.4922894553$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 1$ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 2400)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			6049.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	7200.6049
Dual form			7200.2.f.d.6049.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{7} -4.00000 q^{11} +3.00000i q^{13} +4.00000i q^{17} -1.00000 q^{19} -8.00000 q^{29} -1.00000 q^{31} +2.00000i q^{37} -2.00000 q^{41} -11.0000i q^{43} -2.00000i q^{47} +6.00000 q^{49} -10.0000i q^{53} +6.00000 q^{59} +11.0000 q^{61} -9.00000i q^{67} +6.00000 q^{71} +14.0000i q^{73} +4.00000i q^{77} +16.0000 q^{79} -2.00000i q^{83} +3.00000 q^{91} -11.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 8 q^{11} - 2 q^{19} - 16 q^{29} - 2 q^{31} - 4 q^{41} + 12 q^{49} + 12 q^{59} + 22 q^{61} + 12 q^{71} + 32 q^{79} + 6 q^{91}+O(q^{100})$ 2 * q - 8 * q^11 - 2 * q^19 - 16 * q^29 - 2 * q^31 - 4 * q^41 + 12 * q^49 + 12 * q^59 + 22 * q^61 + 12 * q^71 + 32 * q^79 + 6 * q^91

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
$7$	− 1.00000i	− 0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	− 1.00000i	− 0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	3.00000i	0.832050i	0.909353	+	0.416025i	$0.136577\pi$
$13$	3.00000i	0.832050i	−0.909353	+	0.416025i	$0.863423\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.00000i	0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$	4.00000i	0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	− 11.0000i	− 1.67748i	−0.544529	−	0.838742i	$-0.683292\pi$
$43$	− 11.0000i	− 1.67748i	0.544529	−	0.838742i	$-0.316708\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2.00000i	− 0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$	− 2.00000i	− 0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 10.0000i	− 1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	− 10.0000i	− 1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	11.0000	1.40841	0.704203	−	0.709999i	$-0.251305\pi$
$61$	11.0000	1.40841	0.704203	+	0.709999i	$0.251305\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 9.00000i	− 1.09952i	−0.835321	−	0.549762i	$-0.814718\pi$
$67$	− 9.00000i	− 1.09952i	0.835321	−	0.549762i	$-0.185282\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	14.0000i	1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.00000i	0.455842i
$78$	0	0
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 2.00000i	− 0.219529i	−0.993958	−	0.109764i	$-0.964990\pi$
$83$	− 2.00000i	− 0.219529i	0.993958	−	0.109764i	$-0.0350096\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	3.00000	0.314485
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 11.0000i	− 1.11688i	−0.829545	−	0.558440i	$-0.811400\pi$
$97$	− 11.0000i	− 1.11688i	0.829545	−	0.558440i	$-0.188600\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.00000i	0.580042i	0.957020	+	0.290021i	$0.0936623\pi$
$107$	6.00000i	0.580042i	−0.957020	+	0.290021i	$0.906338\pi$
$108$	0	0
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 12.0000i	− 1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$	− 12.0000i	− 1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−14.0000	−1.22319	−0.611593	−	0.791173i	$-0.709471\pi$
$131$	−14.0000	−1.22319	−0.611593	+	0.791173i	$0.709471\pi$
$132$	0	0
$133$	1.00000i	0.0867110i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.00000i	− 0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$	− 2.00000i	− 0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 12.0000i	− 1.00349i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	−23.0000	−1.87171	−0.935857	−	0.352381i	$-0.885372\pi$
$151$	−23.0000	−1.87171	−0.935857	+	0.352381i	$0.885372\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 5.00000i	− 0.399043i	−0.979893	−	0.199522i	$-0.936061\pi$
$157$	− 5.00000i	− 0.399043i	0.979893	−	0.199522i	$-0.0639388\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 9.00000i	− 0.704934i	−0.935824	−	0.352467i	$-0.885343\pi$
$163$	− 9.00000i	− 0.704934i	0.935824	−	0.352467i	$-0.114657\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 22.0000i	− 1.70241i	−0.524832	−	0.851206i	$-0.675872\pi$
$167$	− 22.0000i	− 1.70241i	0.524832	−	0.851206i	$-0.324128\pi$
$168$	0	0
$169$	4.00000	0.307692
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 24.0000i	− 1.82469i	−0.409426	−	0.912343i	$-0.634271\pi$
$173$	− 24.0000i	− 1.82469i	0.409426	−	0.912343i	$-0.365729\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	0	0
$181$	−15.0000	−1.11494	−0.557471	−	0.830197i	$-0.688228\pi$
$181$	−15.0000	−1.11494	−0.557471	+	0.830197i	$0.688228\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 16.0000i	− 1.17004i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	0	0
$193$	13.0000i	0.935760i	0.883792	+	0.467880i	$0.154982\pi$
$193$	13.0000i	0.935760i	−0.883792	+	0.467880i	$0.845018\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2.00000i	− 0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	− 2.00000i	− 0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$	0	0
$199$	11.0000	0.779769	0.389885	−	0.920864i	$-0.372515\pi$
$199$	11.0000	0.779769	0.389885	+	0.920864i	$0.372515\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.00000i	0.561490i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.00000i	0.0678844i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	− 19.0000i	− 1.27233i	−0.771551	−	0.636167i	$-0.780519\pi$
$223$	− 19.0000i	− 1.27233i	0.771551	−	0.636167i	$-0.219481\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 18.0000i	− 1.19470i	−0.801980	−	0.597351i	$-0.796220\pi$
$227$	− 18.0000i	− 1.19470i	0.801980	−	0.597351i	$-0.203780\pi$
$228$	0	0
$229$	−5.00000	−0.330409	−0.165205	−	0.986259i	$-0.552828\pi$
$229$	−5.00000	−0.330409	−0.165205	+	0.986259i	$0.552828\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	26.0000i	1.70332i	0.524097	+	0.851658i	$0.324403\pi$
$233$	26.0000i	1.70332i	−0.524097	+	0.851658i	$0.675597\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.0000	−0.646846	−0.323423	−	0.946254i	$-0.604834\pi$
$239$	−10.0000	−0.646846	−0.323423	+	0.946254i	$0.604834\pi$
$240$	0	0
$241$	21.0000	1.35273	0.676364	−	0.736567i	$-0.263554\pi$
$241$	21.0000	1.35273	0.676364	+	0.736567i	$0.263554\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 3.00000i	− 0.190885i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−8.00000	−0.504956	−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000	−0.504956	−0.252478	+	0.967603i	$0.581245\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 8.00000i	− 0.499026i	−0.968371	−	0.249513i	$-0.919729\pi$
$257$	− 8.00000i	− 0.499026i	0.968371	−	0.249513i	$-0.0802706\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 26.0000i	− 1.60323i	−0.597841	−	0.801614i	$-0.703975\pi$
$263$	− 26.0000i	− 1.60323i	0.597841	−	0.801614i	$-0.296025\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 7.00000i	− 0.420589i	−0.977638	−	0.210295i	$-0.932558\pi$
$277$	− 7.00000i	− 0.420589i	0.977638	−	0.210295i	$-0.0674423\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	0	0
$283$	− 17.0000i	− 1.01055i	−0.862960	−	0.505273i	$-0.831392\pi$
$283$	− 17.0000i	− 1.01055i	0.862960	−	0.505273i	$-0.168608\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.00000i	0.118056i
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−11.0000	−0.634029
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 23.0000i	− 1.31268i	−0.754466	−	0.656340i	$-0.772104\pi$
$307$	− 23.0000i	− 1.31268i	0.754466	−	0.656340i	$-0.227896\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−26.0000	−1.47432	−0.737162	−	0.675716i	$-0.763835\pi$
$311$	−26.0000	−1.47432	−0.737162	+	0.675716i	$0.763835\pi$
$312$	0	0
$313$	− 15.0000i	− 0.847850i	−0.905697	−	0.423925i	$-0.860652\pi$
$313$	− 15.0000i	− 0.847850i	0.905697	−	0.423925i	$-0.139348\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	28.0000i	1.57264i	0.617822	+	0.786318i	$0.288015\pi$
$317$	28.0000i	1.57264i	−0.617822	+	0.786318i	$0.711985\pi$
$318$	0	0
$319$	32.0000	1.79166
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 4.00000i	− 0.222566i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2.00000	−0.110264
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	13.0000i	0.708155i	0.935216	+	0.354078i	$0.115205\pi$
$337$	13.0000i	0.708155i	−0.935216	+	0.354078i	$0.884795\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.00000	0.216612
$342$	0	0
$343$	− 13.0000i	− 0.701934i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.0000i	1.61048i	0.592946	+	0.805242i	$0.297965\pi$
$347$	30.0000i	1.61048i	−0.592946	+	0.805242i	$0.702035\pi$
$348$	0	0
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	24.0000i	1.27739i	0.769460	+	0.638696i	$0.220526\pi$
$353$	24.0000i	1.27739i	−0.769460	+	0.638696i	$0.779474\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	26.0000	1.37223	0.686114	−	0.727494i	$-0.259315\pi$
$359$	26.0000	1.37223	0.686114	+	0.727494i	$0.259315\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 3.00000i	− 0.156599i	−0.996930	−	0.0782994i	$-0.975051\pi$
$367$	− 3.00000i	− 0.156599i	0.996930	−	0.0782994i	$-0.0249490\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−10.0000	−0.519174
$372$	0	0
$373$	− 19.0000i	− 0.983783i	−0.870657	−	0.491891i	$-0.836306\pi$
$373$	− 19.0000i	− 0.983783i	0.870657	−	0.491891i	$-0.163694\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 24.0000i	− 1.23606i
$378$	0	0
$379$	33.0000	1.69510	0.847548	−	0.530719i	$-0.178078\pi$
$379$	33.0000	1.69510	0.847548	+	0.530719i	$0.178078\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	8.00000i	0.408781i	0.978889	+	0.204390i	$0.0655212\pi$
$383$	8.00000i	0.408781i	−0.978889	+	0.204390i	$0.934479\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	23.0000i	1.15434i	0.816625	+	0.577168i	$0.195842\pi$
$397$	23.0000i	1.15434i	−0.816625	+	0.577168i	$0.804158\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	− 3.00000i	− 0.149441i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 8.00000i	− 0.396545i
$408$	0	0
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 6.00000i	− 0.295241i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 11.0000i	− 0.532327i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	1.00000i	0.0480569i	0.999711	+	0.0240285i	$0.00764923\pi$
$433$	1.00000i	0.0480569i	−0.999711	+	0.0240285i	$0.992351\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−7.00000	−0.334092	−0.167046	−	0.985949i	$-0.553423\pi$
$439$	−7.00000	−0.334092	−0.167046	+	0.985949i	$0.553423\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	40.0000	1.88772	0.943858	−	0.330350i	$-0.107167\pi$
$449$	40.0000	1.88772	0.943858	+	0.330350i	$0.107167\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 6.00000i	− 0.280668i	−0.990104	−	0.140334i	$-0.955182\pi$
$457$	− 6.00000i	− 0.280668i	0.990104	−	0.140334i	$-0.0448177\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	− 20.0000i	− 0.929479i	−0.885448	−	0.464739i	$-0.846148\pi$
$463$	− 20.0000i	− 0.929479i	0.885448	−	0.464739i	$-0.153852\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	0	0
$469$	−9.00000	−0.415581
$470$	0	0
$471$	0	0
$472$	0	0
$473$	44.0000i	2.02312i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.00000	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 11.0000i	− 0.498458i	−0.968445	−	0.249229i	$-0.919823\pi$
$487$	− 11.0000i	− 0.498458i	0.968445	−	0.249229i	$-0.0801771\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	0	0
$493$	− 32.0000i	− 1.44121i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 6.00000i	− 0.269137i
$498$	0	0
$499$	−13.0000	−0.581960	−0.290980	−	0.956729i	$-0.593981\pi$
$499$	−13.0000	−0.581960	−0.290980	+	0.956729i	$0.593981\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 10.0000i	− 0.445878i	−0.974832	−	0.222939i	$-0.928435\pi$
$503$	− 10.0000i	− 0.445878i	0.974832	−	0.222939i	$-0.0715651\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	14.0000	0.619324
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.00000i	0.351840i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	− 17.0000i	− 0.743358i	−0.928361	−	0.371679i	$-0.878782\pi$
$523$	− 17.0000i	− 0.743358i	0.928361	−	0.371679i	$-0.121218\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 4.00000i	− 0.174243i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 6.00000i	− 0.259889i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−24.0000	−1.03375
$540$	0	0
$541$	−7.00000	−0.300954	−0.150477	−	0.988614i	$-0.548081\pi$
$541$	−7.00000	−0.300954	−0.150477	+	0.988614i	$0.548081\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.00000i	0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$	8.00000i	0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	8.00000	0.340811
$552$	0	0
$553$	− 16.0000i	− 0.680389i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.00000i	− 0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$	− 2.00000i	− 0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$	0	0
$559$	33.0000	1.39575
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	−31.0000	−1.29731	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	−31.0000	−1.29731	−0.648655	+	0.761083i	$0.724668\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	7.00000i	0.291414i	0.989328	+	0.145707i	$0.0465456\pi$
$577$	7.00000i	0.291414i	−0.989328	+	0.145707i	$0.953454\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.00000	−0.0829740
$582$	0	0
$583$	40.0000i	1.65663i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 26.0000i	− 1.07313i	−0.843857	−	0.536567i	$-0.819721\pi$
$587$	− 26.0000i	− 1.07313i	0.843857	−	0.536567i	$-0.180279\pi$
$588$	0	0
$589$	1.00000	0.0412043
$590$	0	0
$591$	0	0
$592$	0	0
$593$	30.0000i	1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$	30.0000i	1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−5.00000	−0.203954	−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 40.0000i	− 1.62355i	−0.583970	−	0.811775i	$-0.698502\pi$
$607$	− 40.0000i	− 1.62355i	0.583970	−	0.811775i	$-0.301498\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.00000	0.242734
$612$	0	0
$613$	− 38.0000i	− 1.53481i	−0.641165	−	0.767403i	$-0.721549\pi$
$613$	− 38.0000i	− 1.53481i	0.641165	−	0.767403i	$-0.278451\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	0	0
$619$	23.0000	0.924448	0.462224	−	0.886763i	$-0.347052\pi$
$619$	23.0000	0.924448	0.462224	+	0.886763i	$0.347052\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−8.00000	−0.318981
$630$	0	0
$631$	15.0000	0.597141	0.298570	−	0.954388i	$-0.403490\pi$
$631$	15.0000	0.597141	0.298570	+	0.954388i	$0.403490\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	18.0000i	0.713186i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	12.0000i	0.473234i	0.971603	+	0.236617i	$0.0760386\pi$
$643$	12.0000i	0.473234i	−0.971603	+	0.236617i	$0.923961\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 12.0000i	− 0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$	− 12.0000i	− 0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 30.0000i	− 1.17399i	−0.809590	−	0.586995i	$-0.800311\pi$
$653$	− 30.0000i	− 1.17399i	0.809590	−	0.586995i	$-0.199689\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−14.0000	−0.545363	−0.272681	−	0.962104i	$-0.587910\pi$
$659$	−14.0000	−0.545363	−0.272681	+	0.962104i	$0.587910\pi$
$660$	0	0
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−44.0000	−1.69860
$672$	0	0
$673$	2.00000i	0.0770943i	0.999257	+	0.0385472i	$0.0122730\pi$
$673$	2.00000i	0.0770943i	−0.999257	+	0.0385472i	$0.987727\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 44.0000i	− 1.69106i	−0.533930	−	0.845529i	$-0.679285\pi$
$677$	− 44.0000i	− 1.69106i	0.533930	−	0.845529i	$-0.320715\pi$
$678$	0	0
$679$	−11.0000	−0.422141
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 8.00000i	− 0.306111i	−0.988218	−	0.153056i	$-0.951089\pi$
$683$	− 8.00000i	− 0.306111i	0.988218	−	0.153056i	$-0.0489114\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	30.0000	1.14291
$690$	0	0
$691$	−12.0000	−0.456502	−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000	−0.456502	−0.228251	+	0.973602i	$0.573301\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 8.00000i	− 0.303022i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	0	0
$703$	− 2.00000i	− 0.0754314i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.0000i	0.526524i
$708$	0	0
$709$	−17.0000	−0.638448	−0.319224	−	0.947679i	$-0.603422\pi$
$709$	−17.0000	−0.638448	−0.319224	+	0.947679i	$0.603422\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−32.0000	−1.19340	−0.596699	−	0.802465i	$-0.703521\pi$
$719$	−32.0000	−1.19340	−0.596699	+	0.802465i	$0.703521\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	23.0000i	0.853023i	0.904482	+	0.426511i	$0.140258\pi$
$727$	23.0000i	0.853023i	−0.904482	+	0.426511i	$0.859742\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	44.0000	1.62740
$732$	0	0
$733$	− 6.00000i	− 0.221615i	−0.993842	−	0.110808i	$-0.964656\pi$
$733$	− 6.00000i	− 0.221615i	0.993842	−	0.110808i	$-0.0353437\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	36.0000i	1.32608i
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	36.0000i	1.32071i	0.750953	+	0.660356i	$0.229595\pi$
$743$	36.0000i	1.32071i	−0.750953	+	0.660356i	$0.770405\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	6.00000	0.219235
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 11.0000i	− 0.399802i	−0.979816	−	0.199901i	$-0.935938\pi$
$757$	− 11.0000i	− 0.399802i	0.979816	−	0.199901i	$-0.0640620\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	20.0000	0.724999	0.362500	−	0.931984i	$-0.381923\pi$
$761$	20.0000	0.724999	0.362500	+	0.931984i	$0.381923\pi$
$762$	0	0
$763$	− 11.0000i	− 0.398227i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18.0000i	0.649942i
$768$	0	0
$769$	9.00000	0.324548	0.162274	−	0.986746i	$-0.448117\pi$
$769$	9.00000	0.324548	0.162274	+	0.986746i	$0.448117\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 2.00000i	− 0.0719350i	−0.999353	−	0.0359675i	$-0.988549\pi$
$773$	− 2.00000i	− 0.0719350i	0.999353	−	0.0359675i	$-0.0114513\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.00000	0.0716574
$780$	0	0
$781$	−24.0000	−0.858788
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	53.0000i	1.88925i	0.328158	+	0.944623i	$0.393572\pi$
$787$	53.0000i	1.88925i	−0.328158	+	0.944623i	$0.606428\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	33.0000i	1.17186i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24.0000i	0.850124i	0.905164	+	0.425062i	$0.139748\pi$
$797$	24.0000i	0.850124i	−0.905164	+	0.425062i	$0.860252\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 56.0000i	− 1.97620i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	−43.0000	−1.50993	−0.754967	−	0.655763i	$-0.772347\pi$
$811$	−43.0000	−1.50993	−0.754967	+	0.655763i	$0.772347\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	11.0000i	0.384841i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−46.0000	−1.60541	−0.802706	−	0.596376i	$-0.796607\pi$
$821$	−46.0000	−1.60541	−0.802706	+	0.596376i	$0.796607\pi$
$822$	0	0
$823$	− 39.0000i	− 1.35945i	−0.733465	−	0.679727i	$-0.762098\pi$
$823$	− 39.0000i	− 1.35945i	0.733465	−	0.679727i	$-0.237902\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 10.0000i	− 0.347734i	−0.984769	−	0.173867i	$-0.944374\pi$
$827$	− 10.0000i	− 0.347734i	0.984769	−	0.173867i	$-0.0556263\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	24.0000i	0.831551i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 5.00000i	− 0.171802i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	17.0000i	0.582069i	0.956713	+	0.291034i	$0.0939994\pi$
$853$	17.0000i	0.582069i	−0.956713	+	0.291034i	$0.906001\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 24.0000i	− 0.819824i	−0.912125	−	0.409912i	$-0.865559\pi$
$857$	− 24.0000i	− 0.819824i	0.912125	−	0.409912i	$-0.134441\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	28.0000i	0.953131i	0.879139	+	0.476566i	$0.158119\pi$
$863$	28.0000i	0.953131i	−0.879139	+	0.476566i	$0.841881\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−64.0000	−2.17105
$870$	0	0
$871$	27.0000	0.914860
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 17.0000i	− 0.574049i	−0.957923	−	0.287025i	$-0.907334\pi$
$877$	− 17.0000i	− 0.574049i	0.957923	−	0.287025i	$-0.0926662\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	52.0000	1.75192	0.875962	−	0.482380i	$-0.160227\pi$
$881$	52.0000	1.75192	0.875962	+	0.482380i	$0.160227\pi$
$882$	0	0
$883$	25.0000i	0.841317i	0.907219	+	0.420658i	$0.138201\pi$
$883$	25.0000i	0.841317i	−0.907219	+	0.420658i	$0.861799\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	34.0000i	1.14161i	0.821086	+	0.570804i	$0.193368\pi$
$887$	34.0000i	1.14161i	−0.821086	+	0.570804i	$0.806632\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.00000i	0.0669274i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.00000	0.266815
$900$	0	0
$901$	40.0000	1.33259
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 20.0000i	− 0.664089i	−0.943264	−	0.332045i	$-0.892262\pi$
$907$	− 20.0000i	− 0.664089i	0.943264	−	0.332045i	$-0.107738\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	50.0000	1.65657	0.828287	−	0.560304i	$-0.189316\pi$
$911$	50.0000	1.65657	0.828287	+	0.560304i	$0.189316\pi$
$912$	0	0
$913$	8.00000i	0.264761i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14.0000i	0.462321i
$918$	0	0
$919$	−49.0000	−1.61636	−0.808180	−	0.588935i	$-0.799547\pi$
$919$	−49.0000	−1.61636	−0.808180	+	0.588935i	$0.799547\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	18.0000i	0.592477i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−36.0000	−1.18112	−0.590561	−	0.806993i	$-0.701093\pi$
$929$	−36.0000	−1.18112	−0.590561	+	0.806993i	$0.701093\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 53.0000i	− 1.73143i	−0.500533	−	0.865717i	$-0.666863\pi$
$937$	− 53.0000i	− 1.73143i	0.500533	−	0.865717i	$-0.333137\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 32.0000i	− 1.03986i	−0.854209	−	0.519930i	$-0.825958\pi$
$947$	− 32.0000i	− 1.03986i	0.854209	−	0.519930i	$-0.174042\pi$
$948$	0	0
$949$	−42.0000	−1.36338
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 8.00000i	− 0.259145i	−0.991570	−	0.129573i	$-0.958639\pi$
$953$	− 8.00000i	− 0.259145i	0.991570	−	0.129573i	$-0.0413606\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.00000	−0.0645834
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	32.0000i	1.02905i	0.857475	+	0.514525i	$0.172032\pi$
$967$	32.0000i	1.02905i	−0.857475	+	0.514525i	$0.827968\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	0	0
$973$	− 20.0000i	− 0.641171i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 44.0000i	− 1.40338i	−0.712481	−	0.701691i	$-0.752429\pi$
$983$	− 44.0000i	− 1.40338i	0.712481	−	0.701691i	$-0.247571\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	47.0000	1.49300	0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000	1.49300	0.746502	+	0.665383i	$0.231732\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	18.0000i	0.570066i	0.958518	+	0.285033i	$0.0920045\pi$
$997$	18.0000i	0.570066i	−0.958518	+	0.285033i	$0.907995\pi$
$998$	0	0
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7200.2.f.d.6049.1		2
3.2	odd	2		2400.2.f.p.1249.2		2
4.3	odd	2		7200.2.f.z.6049.2		2
5.2	odd	4		7200.2.a.bh.1.1		1
5.3	odd	4		7200.2.a.q.1.1		1
5.4	even	2	inner	7200.2.f.d.6049.2		2
12.11	even	2		2400.2.f.c.1249.1		2
15.2	even	4		2400.2.a.bd.1.1	yes	1
15.8	even	4		2400.2.a.h.1.1	yes	1
15.14	odd	2		2400.2.f.p.1249.1		2
20.3	even	4		7200.2.a.bk.1.1		1
20.7	even	4		7200.2.a.t.1.1		1
20.19	odd	2		7200.2.f.z.6049.1		2
24.5	odd	2		4800.2.f.g.3649.1		2
24.11	even	2		4800.2.f.bd.3649.2		2
60.23	odd	4		2400.2.a.ba.1.1	yes	1
60.47	odd	4		2400.2.a.e.1.1	✓	1
60.59	even	2		2400.2.f.c.1249.2		2
120.29	odd	2		4800.2.f.g.3649.2		2
120.53	even	4		4800.2.a.bv.1.1		1
120.59	even	2		4800.2.f.bd.3649.1		2
120.77	even	4		4800.2.a.v.1.1		1
120.83	odd	4		4800.2.a.y.1.1		1
120.107	odd	4		4800.2.a.by.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2400.2.a.e.1.1	✓	1	60.47	odd	4
2400.2.a.h.1.1	yes	1	15.8	even	4
2400.2.a.ba.1.1	yes	1	60.23	odd	4
2400.2.a.bd.1.1	yes	1	15.2	even	4
2400.2.f.c.1249.1		2	12.11	even	2
2400.2.f.c.1249.2		2	60.59	even	2
2400.2.f.p.1249.1		2	15.14	odd	2
2400.2.f.p.1249.2		2	3.2	odd	2
4800.2.a.v.1.1		1	120.77	even	4
4800.2.a.y.1.1		1	120.83	odd	4
4800.2.a.bv.1.1		1	120.53	even	4
4800.2.a.by.1.1		1	120.107	odd	4
4800.2.f.g.3649.1		2	24.5	odd	2
4800.2.f.g.3649.2		2	120.29	odd	2
4800.2.f.bd.3649.1		2	120.59	even	2
4800.2.f.bd.3649.2		2	24.11	even	2
7200.2.a.q.1.1		1	5.3	odd	4
7200.2.a.t.1.1		1	20.7	even	4
7200.2.a.bh.1.1		1	5.2	odd	4
7200.2.a.bk.1.1		1	20.3	even	4
7200.2.f.d.6049.1		2	1.1	even	1	trivial
7200.2.f.d.6049.2		2	5.4	even	2	inner
7200.2.f.z.6049.1		2	20.19	odd	2
7200.2.f.z.6049.2		2	4.3	odd	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	7200.2.f.d.6049.1

Level	$7200$
Weight	$2$
Character	7200.6049
Analytic conductor	$57.492$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$