LMFDB - Embedded newform 3072.2.d.b.1537.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3072,2,Mod(1537,3072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3072, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3072.1537");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3072 = 2^{10} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3072.d (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:	$24.5300435009$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1536)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1537.2
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3072.1537
Dual form			3072.2.d.b.1537.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +1.41421i q^{5} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +1.41421i q^{5} -1.00000 q^{9} -4.00000i q^{11} +1.41421i q^{13} +1.41421 q^{15} +4.00000i q^{19} +5.65685 q^{23} +3.00000 q^{25} +1.00000i q^{27} +7.07107i q^{29} -5.65685 q^{31} -4.00000 q^{33} +4.24264i q^{37} +1.41421 q^{39} -12.0000i q^{43} -1.41421i q^{45} +11.3137 q^{47} -7.00000 q^{49} +1.41421i q^{53} +5.65685 q^{55} +4.00000 q^{57} +4.00000i q^{59} -12.7279i q^{61} -2.00000 q^{65} +4.00000i q^{67} -5.65685i q^{69} +5.65685 q^{71} +10.0000 q^{73} -3.00000i q^{75} +16.9706 q^{79} +1.00000 q^{81} +12.0000i q^{83} +7.07107 q^{87} -6.00000 q^{89} +5.65685i q^{93} -5.65685 q^{95} +8.00000 q^{97} +4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} + 12 q^{25} - 16 q^{33} - 28 q^{49} + 16 q^{57} - 8 q^{65} + 40 q^{73} + 4 q^{81} - 24 q^{89} + 32 q^{97}+O(q^{100}) $ 4 * q - 4 * q^9 + 12 * q^25 - 16 * q^33 - 28 * q^49 + 16 * q^57 - 8 * q^65 + 40 * q^73 + 4 * q^81 - 24 * q^89 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times$.

$n$	$1025$	$2047$	$2053$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	1.41421i	0.632456i	0.948683	+	0.316228i	$0.102416\pi$
$5$	1.41421i	0.632456i	−0.948683	+	0.316228i	$0.897584\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 4.00000i	− 1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$11$	− 4.00000i	− 1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$	0	0
$13$	1.41421i	0.392232i	0.980581	+	0.196116i	$0.0628330\pi$
$13$	1.41421i	0.392232i	−0.980581	+	0.196116i	$0.937167\pi$
$14$	0	0
$15$	1.41421	0.365148
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.65685	1.17954	0.589768	−	0.807573i	$-0.299219\pi$
$23$	5.65685	1.17954	0.589768	+	0.807573i	$0.299219\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	7.07107i	1.31306i	0.754298	+	0.656532i	$0.227977\pi$
$29$	7.07107i	1.31306i	−0.754298	+	0.656532i	$0.772023\pi$
$30$	0	0
$31$	−5.65685	−1.01600	−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.24264i	0.697486i	0.937218	+	0.348743i	$0.113391\pi$
$37$	4.24264i	0.697486i	−0.937218	+	0.348743i	$0.886609\pi$
$38$	0	0
$39$	1.41421	0.226455
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	− 12.0000i	− 1.82998i	−0.403473	−	0.914991i	$-0.632197\pi$
$43$	− 12.0000i	− 1.82998i	0.403473	−	0.914991i	$-0.367803\pi$
$44$	0	0
$45$	− 1.41421i	− 0.210819i
$46$	0	0
$47$	11.3137	1.65027	0.825137	−	0.564933i	$-0.191098\pi$
$47$	11.3137	1.65027	0.825137	+	0.564933i	$0.191098\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.41421i	0.194257i	0.995272	+	0.0971286i	$0.0309658\pi$
$53$	1.41421i	0.194257i	−0.995272	+	0.0971286i	$0.969034\pi$
$54$	0	0
$55$	5.65685	0.762770
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	4.00000i	0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$	4.00000i	0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$	0	0
$61$	− 12.7279i	− 1.62964i	−0.579712	−	0.814822i	$-0.696835\pi$
$61$	− 12.7279i	− 1.62964i	0.579712	−	0.814822i	$-0.303165\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	0	0
$69$	− 5.65685i	− 0.681005i
$70$	0	0
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	− 3.00000i	− 0.346410i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	16.9706	1.90934	0.954669	−	0.297670i	$-0.0962096\pi$
$79$	16.9706	1.90934	0.954669	+	0.297670i	$0.0962096\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.07107	0.758098
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	5.65685i	0.586588i
$94$	0	0
$95$	−5.65685	−0.580381
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	4.00000i	0.402015i
$100$	0	0
$101$	− 1.41421i	− 0.140720i	−0.997522	−	0.0703598i	$-0.977585\pi$
$101$	− 1.41421i	− 0.140720i	0.997522	−	0.0703598i	$-0.0224147\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	0	0
$109$	9.89949i	0.948200i	0.880471	+	0.474100i	$0.157226\pi$
$109$	9.89949i	0.948200i	−0.880471	+	0.474100i	$0.842774\pi$
$110$	0	0
$111$	4.24264	0.402694
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	8.00000i	0.746004i
$116$	0	0
$117$	− 1.41421i	− 0.130744i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.3137i	1.01193i
$126$	0	0
$127$	16.9706	1.50589	0.752947	−	0.658081i	$-0.228632\pi$
$127$	16.9706	1.50589	0.752947	+	0.658081i	$0.228632\pi$
$128$	0	0
$129$	−12.0000	−1.05654
$130$	0	0
$131$	4.00000i	0.349482i	0.984614	+	0.174741i	$0.0559088\pi$
$131$	4.00000i	0.349482i	−0.984614	+	0.174741i	$0.944091\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.41421	−0.121716
$136$	0	0
$137$	−16.0000	−1.36697	−0.683486	−	0.729964i	$-0.739537\pi$
$137$	−16.0000	−1.36697	−0.683486	+	0.729964i	$0.739537\pi$
$138$	0	0
$139$	− 4.00000i	− 0.339276i	−0.985506	−	0.169638i	$-0.945740\pi$
$139$	− 4.00000i	− 0.339276i	0.985506	−	0.169638i	$-0.0542598\pi$
$140$	0	0
$141$	− 11.3137i	− 0.952786i
$142$	0	0
$143$	5.65685	0.473050
$144$	0	0
$145$	−10.0000	−0.830455
$146$	0	0
$147$	7.00000i	0.577350i
$148$	0	0
$149$	21.2132i	1.73785i	0.494941	+	0.868927i	$0.335190\pi$
$149$	21.2132i	1.73785i	−0.494941	+	0.868927i	$0.664810\pi$
$150$	0	0
$151$	11.3137	0.920697	0.460348	−	0.887738i	$-0.347725\pi$
$151$	11.3137	0.920697	0.460348	+	0.887738i	$0.347725\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 8.00000i	− 0.642575i
$156$	0	0
$157$	− 12.7279i	− 1.01580i	−0.861416	−	0.507899i	$-0.830422\pi$
$157$	− 12.7279i	− 1.01580i	0.861416	−	0.507899i	$-0.169578\pi$
$158$	0	0
$159$	1.41421	0.112154
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	0	0
$165$	− 5.65685i	− 0.440386i
$166$	0	0
$167$	−5.65685	−0.437741	−0.218870	−	0.975754i	$-0.570237\pi$
$167$	−5.65685	−0.437741	−0.218870	+	0.975754i	$0.570237\pi$
$168$	0	0
$169$	11.0000	0.846154
$170$	0	0
$171$	− 4.00000i	− 0.305888i
$172$	0	0
$173$	7.07107i	0.537603i	0.963196	+	0.268802i	$0.0866276\pi$
$173$	7.07107i	0.537603i	−0.963196	+	0.268802i	$0.913372\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	4.00000i	0.298974i	0.988764	+	0.149487i	$0.0477622\pi$
$179$	4.00000i	0.298974i	−0.988764	+	0.149487i	$0.952238\pi$
$180$	0	0
$181$	− 18.3848i	− 1.36653i	−0.730171	−	0.683265i	$-0.760559\pi$
$181$	− 18.3848i	− 1.36653i	0.730171	−	0.683265i	$-0.239441\pi$
$182$	0	0
$183$	−12.7279	−0.940875
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.6274	1.63726	0.818631	−	0.574320i	$-0.194733\pi$
$191$	22.6274	1.63726	0.818631	+	0.574320i	$0.194733\pi$
$192$	0	0
$193$	24.0000	1.72756	0.863779	−	0.503871i	$-0.168091\pi$
$193$	24.0000	1.72756	0.863779	+	0.503871i	$0.168091\pi$
$194$	0	0
$195$	2.00000i	0.143223i
$196$	0	0
$197$	12.7279i	0.906827i	0.891300	+	0.453413i	$0.149794\pi$
$197$	12.7279i	0.906827i	−0.891300	+	0.453413i	$0.850206\pi$
$198$	0	0
$199$	22.6274	1.60402	0.802008	−	0.597314i	$-0.203765\pi$
$199$	22.6274	1.60402	0.802008	+	0.597314i	$0.203765\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−5.65685	−0.393179
$208$	0	0
$209$	16.0000	1.10674
$210$	0	0
$211$	4.00000i	0.275371i	0.990476	+	0.137686i	$0.0439664\pi$
$211$	4.00000i	0.275371i	−0.990476	+	0.137686i	$0.956034\pi$
$212$	0	0
$213$	− 5.65685i	− 0.387601i
$214$	0	0
$215$	16.9706	1.15738
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 10.0000i	− 0.675737i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−5.65685	−0.378811	−0.189405	−	0.981899i	$-0.560656\pi$
$223$	−5.65685	−0.378811	−0.189405	+	0.981899i	$0.560656\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	0	0
$227$	20.0000i	1.32745i	0.747978	+	0.663723i	$0.231025\pi$
$227$	20.0000i	1.32745i	−0.747978	+	0.663723i	$0.768975\pi$
$228$	0	0
$229$	− 4.24264i	− 0.280362i	−0.990126	−	0.140181i	$-0.955232\pi$
$229$	− 4.24264i	− 0.280362i	0.990126	−	0.140181i	$-0.0447684\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	0	0
$235$	16.0000i	1.04372i
$236$	0	0
$237$	− 16.9706i	− 1.10236i
$238$	0	0
$239$	−11.3137	−0.731823	−0.365911	−	0.930650i	$-0.619243\pi$
$239$	−11.3137	−0.731823	−0.365911	+	0.930650i	$0.619243\pi$
$240$	0	0
$241$	−24.0000	−1.54598	−0.772988	−	0.634421i	$-0.781239\pi$
$241$	−24.0000	−1.54598	−0.772988	+	0.634421i	$0.781239\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	− 9.89949i	− 0.632456i
$246$	0	0
$247$	−5.65685	−0.359937
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	− 20.0000i	− 1.26239i	−0.775625	−	0.631194i	$-0.782565\pi$
$251$	− 20.0000i	− 1.26239i	0.775625	−	0.631194i	$-0.217435\pi$
$252$	0	0
$253$	− 22.6274i	− 1.42257i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	− 7.07107i	− 0.437688i
$262$	0	0
$263$	16.9706	1.04645	0.523225	−	0.852195i	$-0.324729\pi$
$263$	16.9706	1.04645	0.523225	+	0.852195i	$0.324729\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	0	0
$267$	6.00000i	0.367194i
$268$	0	0
$269$	− 7.07107i	− 0.431131i	−0.976489	−	0.215565i	$-0.930841\pi$
$269$	− 7.07107i	− 0.431131i	0.976489	−	0.215565i	$-0.0691594\pi$
$270$	0	0
$271$	−5.65685	−0.343629	−0.171815	−	0.985129i	$-0.554963\pi$
$271$	−5.65685	−0.343629	−0.171815	+	0.985129i	$0.554963\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 12.0000i	− 0.723627i
$276$	0	0
$277$	− 15.5563i	− 0.934690i	−0.884075	−	0.467345i	$-0.845211\pi$
$277$	− 15.5563i	− 0.934690i	0.884075	−	0.467345i	$-0.154789\pi$
$278$	0	0
$279$	5.65685	0.338667
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	− 20.0000i	− 1.18888i	−0.804141	−	0.594438i	$-0.797374\pi$
$283$	− 20.0000i	− 1.18888i	0.804141	−	0.594438i	$-0.202626\pi$
$284$	0	0
$285$	5.65685i	0.335083i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	− 8.00000i	− 0.468968i
$292$	0	0
$293$	− 9.89949i	− 0.578335i	−0.957279	−	0.289167i	$-0.906622\pi$
$293$	− 9.89949i	− 0.578335i	0.957279	−	0.289167i	$-0.0933784\pi$
$294$	0	0
$295$	−5.65685	−0.329355
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	8.00000i	0.462652i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1.41421	−0.0812444
$304$	0	0
$305$	18.0000	1.03068
$306$	0	0
$307$	− 20.0000i	− 1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$	− 20.0000i	− 1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−28.2843	−1.60385	−0.801927	−	0.597422i	$-0.796192\pi$
$311$	−28.2843	−1.60385	−0.801927	+	0.597422i	$0.796192\pi$
$312$	0	0
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	29.6985i	1.66803i	0.551739	+	0.834017i	$0.313964\pi$
$317$	29.6985i	1.66803i	−0.551739	+	0.834017i	$0.686036\pi$
$318$	0	0
$319$	28.2843	1.58362
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	0	0
$324$	0	0
$325$	4.24264i	0.235339i
$326$	0	0
$327$	9.89949	0.547443
$328$	0	0
$329$	0	0
$330$	0	0
$331$	4.00000i	0.219860i	0.993939	+	0.109930i	$0.0350627\pi$
$331$	4.00000i	0.219860i	−0.993939	+	0.109930i	$0.964937\pi$
$332$	0	0
$333$	− 4.24264i	− 0.232495i
$334$	0	0
$335$	−5.65685	−0.309067
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	− 14.0000i	− 0.760376i
$340$	0	0
$341$	22.6274i	1.22534i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	8.00000	0.430706
$346$	0	0
$347$	− 4.00000i	− 0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$	− 4.00000i	− 0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$	0	0
$349$	1.41421i	0.0757011i	0.999283	+	0.0378506i	$0.0120511\pi$
$349$	1.41421i	0.0757011i	−0.999283	+	0.0378506i	$0.987949\pi$
$350$	0	0
$351$	−1.41421	−0.0754851
$352$	0	0
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	0	0
$355$	8.00000i	0.424596i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5.65685	0.298557	0.149279	−	0.988795i	$-0.452305\pi$
$359$	5.65685	0.298557	0.149279	+	0.988795i	$0.452305\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	5.00000i	0.262432i
$364$	0	0
$365$	14.1421i	0.740233i
$366$	0	0
$367$	−28.2843	−1.47643	−0.738213	−	0.674567i	$-0.764330\pi$
$367$	−28.2843	−1.47643	−0.738213	+	0.674567i	$0.764330\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 4.24264i	− 0.219676i	−0.993950	−	0.109838i	$-0.964967\pi$
$373$	− 4.24264i	− 0.219676i	0.993950	−	0.109838i	$-0.0350331\pi$
$374$	0	0
$375$	11.3137	0.584237
$376$	0	0
$377$	−10.0000	−0.515026
$378$	0	0
$379$	− 20.0000i	− 1.02733i	−0.857991	−	0.513665i	$-0.828287\pi$
$379$	− 20.0000i	− 1.02733i	0.857991	−	0.513665i	$-0.171713\pi$
$380$	0	0
$381$	− 16.9706i	− 0.869428i
$382$	0	0
$383$	11.3137	0.578103	0.289052	−	0.957313i	$-0.406660\pi$
$383$	11.3137	0.578103	0.289052	+	0.957313i	$0.406660\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	12.0000i	0.609994i
$388$	0	0
$389$	32.5269i	1.64918i	0.565731	+	0.824590i	$0.308594\pi$
$389$	32.5269i	1.64918i	−0.565731	+	0.824590i	$0.691406\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	24.0000i	1.20757i
$396$	0	0
$397$	21.2132i	1.06466i	0.846537	+	0.532330i	$0.178683\pi$
$397$	21.2132i	1.06466i	−0.846537	+	0.532330i	$0.821317\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	16.0000	0.799002	0.399501	−	0.916733i	$-0.369183\pi$
$401$	16.0000	0.799002	0.399501	+	0.916733i	$0.369183\pi$
$402$	0	0
$403$	− 8.00000i	− 0.398508i
$404$	0	0
$405$	1.41421i	0.0702728i
$406$	0	0
$407$	16.9706	0.841200
$408$	0	0
$409$	8.00000	0.395575	0.197787	−	0.980245i	$-0.436624\pi$
$409$	8.00000	0.395575	0.197787	+	0.980245i	$0.436624\pi$
$410$	0	0
$411$	16.0000i	0.789222i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−16.9706	−0.833052
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	− 36.0000i	− 1.75872i	−0.476162	−	0.879358i	$-0.657972\pi$
$419$	− 36.0000i	− 1.75872i	0.476162	−	0.879358i	$-0.342028\pi$
$420$	0	0
$421$	− 4.24264i	− 0.206774i	−0.994641	−	0.103387i	$-0.967032\pi$
$421$	− 4.24264i	− 0.206774i	0.994641	−	0.103387i	$-0.0329680\pi$
$422$	0	0
$423$	−11.3137	−0.550091
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 5.65685i	− 0.273115i
$430$	0	0
$431$	−22.6274	−1.08992	−0.544962	−	0.838461i	$-0.683456\pi$
$431$	−22.6274	−1.08992	−0.544962	+	0.838461i	$0.683456\pi$
$432$	0	0
$433$	−8.00000	−0.384455	−0.192228	−	0.981350i	$-0.561571\pi$
$433$	−8.00000	−0.384455	−0.192228	+	0.981350i	$0.561571\pi$
$434$	0	0
$435$	10.0000i	0.479463i
$436$	0	0
$437$	22.6274i	1.08242i
$438$	0	0
$439$	−33.9411	−1.61992	−0.809961	−	0.586484i	$-0.800512\pi$
$439$	−33.9411	−1.61992	−0.809961	+	0.586484i	$0.800512\pi$
$440$	0	0
$441$	7.00000	0.333333
$442$	0	0
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	0	0
$445$	− 8.48528i	− 0.402241i
$446$	0	0
$447$	21.2132	1.00335
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 11.3137i	− 0.531564i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	4.24264i	0.197599i	0.995107	+	0.0987997i	$0.0315003\pi$
$461$	4.24264i	0.197599i	−0.995107	+	0.0987997i	$0.968500\pi$
$462$	0	0
$463$	−16.9706	−0.788689	−0.394344	−	0.918963i	$-0.629028\pi$
$463$	−16.9706	−0.788689	−0.394344	+	0.918963i	$0.629028\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	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
$469$	0	0
$470$	0	0
$471$	−12.7279	−0.586472
$472$	0	0
$473$	−48.0000	−2.20704
$474$	0	0
$475$	12.0000i	0.550598i
$476$	0	0
$477$	− 1.41421i	− 0.0647524i
$478$	0	0
$479$	−22.6274	−1.03387	−0.516937	−	0.856024i	$-0.672928\pi$
$479$	−22.6274	−1.03387	−0.516937	+	0.856024i	$0.672928\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	11.3137i	0.513729i
$486$	0	0
$487$	−33.9411	−1.53802	−0.769010	−	0.639237i	$-0.779250\pi$
$487$	−33.9411	−1.53802	−0.769010	+	0.639237i	$0.779250\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	28.0000i	1.26362i	0.775122	+	0.631811i	$0.217688\pi$
$491$	28.0000i	1.26362i	−0.775122	+	0.631811i	$0.782312\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−5.65685	−0.254257
$496$	0	0
$497$	0	0
$498$	0	0
$499$	20.0000i	0.895323i	0.894203	+	0.447661i	$0.147743\pi$
$499$	20.0000i	0.895323i	−0.894203	+	0.447661i	$0.852257\pi$
$500$	0	0
$501$	5.65685i	0.252730i
$502$	0	0
$503$	−28.2843	−1.26113	−0.630567	−	0.776135i	$-0.717177\pi$
$503$	−28.2843	−1.26113	−0.630567	+	0.776135i	$0.717177\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	0	0
$507$	− 11.0000i	− 0.488527i
$508$	0	0
$509$	26.8701i	1.19099i	0.803357	+	0.595497i	$0.203045\pi$
$509$	26.8701i	1.19099i	−0.803357	+	0.595497i	$0.796955\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 45.2548i	− 1.99031i
$518$	0	0
$519$	7.07107	0.310385
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	− 4.00000i	− 0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$	− 4.00000i	− 0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	− 4.00000i	− 0.173585i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	16.9706	0.733701
$536$	0	0
$537$	4.00000	0.172613
$538$	0	0
$539$	28.0000i	1.20605i
$540$	0	0
$541$	21.2132i	0.912027i	0.889973	+	0.456013i	$0.150723\pi$
$541$	21.2132i	0.912027i	−0.889973	+	0.456013i	$0.849277\pi$
$542$	0	0
$543$	−18.3848	−0.788966
$544$	0	0
$545$	−14.0000	−0.599694
$546$	0	0
$547$	− 4.00000i	− 0.171028i	−0.996337	−	0.0855138i	$-0.972747\pi$
$547$	− 4.00000i	− 0.171028i	0.996337	−	0.0855138i	$-0.0272532\pi$
$548$	0	0
$549$	12.7279i	0.543214i
$550$	0	0
$551$	−28.2843	−1.20495
$552$	0	0
$553$	0	0
$554$	0	0
$555$	6.00000i	0.254686i
$556$	0	0
$557$	7.07107i	0.299611i	0.988716	+	0.149805i	$0.0478647\pi$
$557$	7.07107i	0.299611i	−0.988716	+	0.149805i	$0.952135\pi$
$558$	0	0
$559$	16.9706	0.717778
$560$	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
$565$	19.7990i	0.832950i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	32.0000	1.34151	0.670755	−	0.741679i	$-0.265970\pi$
$569$	32.0000	1.34151	0.670755	+	0.741679i	$0.265970\pi$
$570$	0	0
$571$	28.0000i	1.17176i	0.810397	+	0.585882i	$0.199252\pi$
$571$	28.0000i	1.17176i	−0.810397	+	0.585882i	$0.800748\pi$
$572$	0	0
$573$	− 22.6274i	− 0.945274i
$574$	0	0
$575$	16.9706	0.707721
$576$	0	0
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	0	0
$579$	− 24.0000i	− 0.997406i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	5.65685	0.234283
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	0	0
$589$	− 22.6274i	− 0.932346i
$590$	0	0
$591$	12.7279	0.523557
$592$	0	0
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 22.6274i	− 0.926079i
$598$	0	0
$599$	−5.65685	−0.231133	−0.115566	−	0.993300i	$-0.536868\pi$
$599$	−5.65685	−0.231133	−0.115566	+	0.993300i	$0.536868\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	− 4.00000i	− 0.162893i
$604$	0	0
$605$	− 7.07107i	− 0.287480i
$606$	0	0
$607$	−39.5980	−1.60723	−0.803616	−	0.595148i	$-0.797093\pi$
$607$	−39.5980	−1.60723	−0.803616	+	0.595148i	$0.797093\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	16.0000i	0.647291i
$612$	0	0
$613$	29.6985i	1.19951i	0.800184	+	0.599755i	$0.204735\pi$
$613$	29.6985i	1.19951i	−0.800184	+	0.599755i	$0.795265\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	36.0000i	1.44696i	0.690344	+	0.723481i	$0.257459\pi$
$619$	36.0000i	1.44696i	−0.690344	+	0.723481i	$0.742541\pi$
$620$	0	0
$621$	5.65685i	0.227002i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	− 16.0000i	− 0.638978i
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−33.9411	−1.35117	−0.675587	−	0.737280i	$-0.736110\pi$
$631$	−33.9411	−1.35117	−0.675587	+	0.737280i	$0.736110\pi$
$632$	0	0
$633$	4.00000	0.158986
$634$	0	0
$635$	24.0000i	0.952411i
$636$	0	0
$637$	− 9.89949i	− 0.392232i
$638$	0	0
$639$	−5.65685	−0.223782
$640$	0	0
$641$	32.0000	1.26392	0.631962	−	0.774999i	$-0.282250\pi$
$641$	32.0000	1.26392	0.631962	+	0.774999i	$0.282250\pi$
$642$	0	0
$643$	− 20.0000i	− 0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$643$	− 20.0000i	− 0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$644$	0	0
$645$	− 16.9706i	− 0.668215i
$646$	0	0
$647$	−28.2843	−1.11197	−0.555985	−	0.831193i	$-0.687659\pi$
$647$	−28.2843	−1.11197	−0.555985	+	0.831193i	$0.687659\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	0	0
$653$	26.8701i	1.05151i	0.850637	+	0.525753i	$0.176216\pi$
$653$	26.8701i	1.05151i	−0.850637	+	0.525753i	$0.823784\pi$
$654$	0	0
$655$	−5.65685	−0.221032
$656$	0	0
$657$	−10.0000	−0.390137
$658$	0	0
$659$	− 12.0000i	− 0.467454i	−0.972302	−	0.233727i	$-0.924908\pi$
$659$	− 12.0000i	− 0.467454i	0.972302	−	0.233727i	$-0.0750921\pi$
$660$	0	0
$661$	− 29.6985i	− 1.15514i	−0.816342	−	0.577569i	$-0.804002\pi$
$661$	− 29.6985i	− 1.15514i	0.816342	−	0.577569i	$-0.195998\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	40.0000i	1.54881i
$668$	0	0
$669$	5.65685i	0.218707i
$670$	0	0
$671$	−50.9117	−1.96542
$672$	0	0
$673$	−8.00000	−0.308377	−0.154189	−	0.988041i	$-0.549276\pi$
$673$	−8.00000	−0.308377	−0.154189	+	0.988041i	$0.549276\pi$
$674$	0	0
$675$	3.00000i	0.115470i
$676$	0	0
$677$	9.89949i	0.380468i	0.981739	+	0.190234i	$0.0609248\pi$
$677$	9.89949i	0.380468i	−0.981739	+	0.190234i	$0.939075\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	20.0000	0.766402
$682$	0	0
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	0	0
$685$	− 22.6274i	− 0.864549i
$686$	0	0
$687$	−4.24264	−0.161867
$688$	0	0
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	− 20.0000i	− 0.760836i	−0.924815	−	0.380418i	$-0.875780\pi$
$691$	− 20.0000i	− 0.760836i	0.924815	−	0.380418i	$-0.124220\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.65685	0.214577
$696$	0	0
$697$	0	0
$698$	0	0
$699$	22.0000i	0.832116i
$700$	0	0
$701$	4.24264i	0.160242i	0.996785	+	0.0801212i	$0.0255307\pi$
$701$	4.24264i	0.160242i	−0.996785	+	0.0801212i	$0.974469\pi$
$702$	0	0
$703$	−16.9706	−0.640057
$704$	0	0
$705$	16.0000	0.602595
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 7.07107i	− 0.265560i	−0.991146	−	0.132780i	$-0.957610\pi$
$709$	− 7.07107i	− 0.265560i	0.991146	−	0.132780i	$-0.0423903\pi$
$710$	0	0
$711$	−16.9706	−0.636446
$712$	0	0
$713$	−32.0000	−1.19841
$714$	0	0
$715$	8.00000i	0.299183i
$716$	0	0
$717$	11.3137i	0.422518i
$718$	0	0
$719$	33.9411	1.26579	0.632895	−	0.774237i	$-0.281866\pi$
$719$	33.9411	1.26579	0.632895	+	0.774237i	$0.281866\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	24.0000i	0.892570i
$724$	0	0
$725$	21.2132i	0.787839i
$726$	0	0
$727$	33.9411	1.25881	0.629403	−	0.777079i	$-0.283299\pi$
$727$	33.9411	1.25881	0.629403	+	0.777079i	$0.283299\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	12.7279i	0.470117i	0.971981	+	0.235058i	$0.0755281\pi$
$733$	12.7279i	0.470117i	−0.971981	+	0.235058i	$0.924472\pi$
$734$	0	0
$735$	−9.89949	−0.365148
$736$	0	0
$737$	16.0000	0.589368
$738$	0	0
$739$	− 36.0000i	− 1.32428i	−0.749380	−	0.662141i	$-0.769648\pi$
$739$	− 36.0000i	− 1.32428i	0.749380	−	0.662141i	$-0.230352\pi$
$740$	0	0
$741$	5.65685i	0.207810i
$742$	0	0
$743$	−28.2843	−1.03765	−0.518825	−	0.854881i	$-0.673630\pi$
$743$	−28.2843	−1.03765	−0.518825	+	0.854881i	$0.673630\pi$
$744$	0	0
$745$	−30.0000	−1.09911
$746$	0	0
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	28.2843	1.03211	0.516054	−	0.856556i	$-0.327400\pi$
$751$	28.2843	1.03211	0.516054	+	0.856556i	$0.327400\pi$
$752$	0	0
$753$	−20.0000	−0.728841
$754$	0	0
$755$	16.0000i	0.582300i
$756$	0	0
$757$	26.8701i	0.976609i	0.872673	+	0.488304i	$0.162384\pi$
$757$	26.8701i	0.976609i	−0.872673	+	0.488304i	$0.837616\pi$
$758$	0	0
$759$	−22.6274	−0.821323
$760$	0	0
$761$	−48.0000	−1.74000	−0.869999	−	0.493053i	$-0.835881\pi$
$761$	−48.0000	−1.74000	−0.869999	+	0.493053i	$0.835881\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.65685	−0.204257
$768$	0	0
$769$	−8.00000	−0.288487	−0.144244	−	0.989542i	$-0.546075\pi$
$769$	−8.00000	−0.288487	−0.144244	+	0.989542i	$0.546075\pi$
$770$	0	0
$771$	− 2.00000i	− 0.0720282i
$772$	0	0
$773$	43.8406i	1.57684i	0.615139	+	0.788419i	$0.289100\pi$
$773$	43.8406i	1.57684i	−0.615139	+	0.788419i	$0.710900\pi$
$774$	0	0
$775$	−16.9706	−0.609601
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	− 22.6274i	− 0.809673i
$782$	0	0
$783$	−7.07107	−0.252699
$784$	0	0
$785$	18.0000	0.642448
$786$	0	0
$787$	− 52.0000i	− 1.85360i	−0.375555	−	0.926800i	$-0.622548\pi$
$787$	− 52.0000i	− 1.85360i	0.375555	−	0.926800i	$-0.377452\pi$
$788$	0	0
$789$	− 16.9706i	− 0.604168i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	18.0000	0.639199
$794$	0	0
$795$	2.00000i	0.0709327i
$796$	0	0
$797$	− 52.3259i	− 1.85348i	−0.375705	−	0.926739i	$-0.622599\pi$
$797$	− 52.3259i	− 1.85348i	0.375705	−	0.926739i	$-0.377401\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	− 40.0000i	− 1.41157i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−7.07107	−0.248913
$808$	0	0
$809$	−48.0000	−1.68759	−0.843795	−	0.536666i	$-0.819684\pi$
$809$	−48.0000	−1.68759	−0.843795	+	0.536666i	$0.819684\pi$
$810$	0	0
$811$	− 52.0000i	− 1.82597i	−0.407997	−	0.912983i	$-0.633772\pi$
$811$	− 52.0000i	− 1.82597i	0.407997	−	0.912983i	$-0.366228\pi$
$812$	0	0
$813$	5.65685i	0.198395i
$814$	0	0
$815$	−5.65685	−0.198151
$816$	0	0
$817$	48.0000	1.67931
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 43.8406i	− 1.53005i	−0.644002	−	0.765024i	$-0.722727\pi$
$821$	− 43.8406i	− 1.53005i	0.644002	−	0.765024i	$-0.277273\pi$
$822$	0	0
$823$	−22.6274	−0.788742	−0.394371	−	0.918951i	$-0.629038\pi$
$823$	−22.6274	−0.788742	−0.394371	+	0.918951i	$0.629038\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	28.0000i	0.973655i	0.873498	+	0.486828i	$0.161846\pi$
$827$	28.0000i	0.973655i	−0.873498	+	0.486828i	$0.838154\pi$
$828$	0	0
$829$	43.8406i	1.52265i	0.648372	+	0.761324i	$0.275450\pi$
$829$	43.8406i	1.52265i	−0.648372	+	0.761324i	$0.724550\pi$
$830$	0	0
$831$	−15.5563	−0.539644
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 8.00000i	− 0.276851i
$836$	0	0
$837$	− 5.65685i	− 0.195529i
$838$	0	0
$839$	39.5980	1.36707	0.683537	−	0.729916i	$-0.260441\pi$
$839$	39.5980	1.36707	0.683537	+	0.729916i	$0.260441\pi$
$840$	0	0
$841$	−21.0000	−0.724138
$842$	0	0
$843$	− 6.00000i	− 0.206651i
$844$	0	0
$845$	15.5563i	0.535155i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	24.0000i	0.822709i
$852$	0	0
$853$	− 4.24264i	− 0.145265i	−0.997359	−	0.0726326i	$-0.976860\pi$
$853$	− 4.24264i	− 0.145265i	0.997359	−	0.0726326i	$-0.0231401\pi$
$854$	0	0
$855$	5.65685	0.193460
$856$	0	0
$857$	32.0000	1.09310	0.546550	−	0.837427i	$-0.315941\pi$
$857$	32.0000	1.09310	0.546550	+	0.837427i	$0.315941\pi$
$858$	0	0
$859$	28.0000i	0.955348i	0.878537	+	0.477674i	$0.158520\pi$
$859$	28.0000i	0.955348i	−0.878537	+	0.477674i	$0.841480\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−45.2548	−1.54049	−0.770246	−	0.637747i	$-0.779867\pi$
$863$	−45.2548	−1.54049	−0.770246	+	0.637747i	$0.779867\pi$
$864$	0	0
$865$	−10.0000	−0.340010
$866$	0	0
$867$	17.0000i	0.577350i
$868$	0	0
$869$	− 67.8823i	− 2.30275i
$870$	0	0
$871$	−5.65685	−0.191675
$872$	0	0
$873$	−8.00000	−0.270759
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 46.6690i	− 1.57590i	−0.615738	−	0.787951i	$-0.711142\pi$
$877$	− 46.6690i	− 1.57590i	0.615738	−	0.787951i	$-0.288858\pi$
$878$	0	0
$879$	−9.89949	−0.333902
$880$	0	0
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	0	0
$883$	36.0000i	1.21150i	0.795656	+	0.605748i	$0.207126\pi$
$883$	36.0000i	1.21150i	−0.795656	+	0.605748i	$0.792874\pi$
$884$	0	0
$885$	5.65685i	0.190153i
$886$	0	0
$887$	16.9706	0.569816	0.284908	−	0.958555i	$-0.408037\pi$
$887$	16.9706	0.569816	0.284908	+	0.958555i	$0.408037\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 4.00000i	− 0.134005i
$892$	0	0
$893$	45.2548i	1.51440i
$894$	0	0
$895$	−5.65685	−0.189088
$896$	0	0
$897$	8.00000	0.267112
$898$	0	0
$899$	− 40.0000i	− 1.33407i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	26.0000	0.864269
$906$	0	0
$907$	4.00000i	0.132818i	0.997792	+	0.0664089i	$0.0211542\pi$
$907$	4.00000i	0.132818i	−0.997792	+	0.0664089i	$0.978846\pi$
$908$	0	0
$909$	1.41421i	0.0469065i
$910$	0	0
$911$	−22.6274	−0.749680	−0.374840	−	0.927090i	$-0.622302\pi$
$911$	−22.6274	−0.749680	−0.374840	+	0.927090i	$0.622302\pi$
$912$	0	0
$913$	48.0000	1.58857
$914$	0	0
$915$	− 18.0000i	− 0.595062i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	33.9411	1.11961	0.559807	−	0.828623i	$-0.310875\pi$
$919$	33.9411	1.11961	0.559807	+	0.828623i	$0.310875\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	0	0
$923$	8.00000i	0.263323i
$924$	0	0
$925$	12.7279i	0.418491i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	16.0000	0.524943	0.262471	−	0.964940i	$-0.415462\pi$
$929$	16.0000	0.524943	0.262471	+	0.964940i	$0.415462\pi$
$930$	0	0
$931$	− 28.0000i	− 0.917663i
$932$	0	0
$933$	28.2843i	0.925985i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	0	0
$939$	8.00000i	0.261070i
$940$	0	0
$941$	26.8701i	0.875939i	0.898990	+	0.437969i	$0.144302\pi$
$941$	26.8701i	0.875939i	−0.898990	+	0.437969i	$0.855698\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 20.0000i	− 0.649913i	−0.945729	−	0.324956i	$-0.894650\pi$
$947$	− 20.0000i	− 0.649913i	0.945729	−	0.324956i	$-0.105350\pi$
$948$	0	0
$949$	14.1421i	0.459073i
$950$	0	0
$951$	29.6985	0.963039
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	32.0000i	1.03550i
$956$	0	0
$957$	− 28.2843i	− 0.914301i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	12.0000i	0.386695i
$964$	0	0
$965$	33.9411i	1.09260i
$966$	0	0
$967$	−22.6274	−0.727649	−0.363824	−	0.931468i	$-0.618529\pi$
$967$	−22.6274	−0.727649	−0.363824	+	0.931468i	$0.618529\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 12.0000i	− 0.385098i	−0.981287	−	0.192549i	$-0.938325\pi$
$971$	− 12.0000i	− 0.385098i	0.981287	−	0.192549i	$-0.0616755\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	4.24264	0.135873
$976$	0	0
$977$	−16.0000	−0.511885	−0.255943	−	0.966692i	$-0.582386\pi$
$977$	−16.0000	−0.511885	−0.255943	+	0.966692i	$0.582386\pi$
$978$	0	0
$979$	24.0000i	0.767043i
$980$	0	0
$981$	− 9.89949i	− 0.316067i
$982$	0	0
$983$	−28.2843	−0.902128	−0.451064	−	0.892492i	$-0.648955\pi$
$983$	−28.2843	−0.902128	−0.451064	+	0.892492i	$0.648955\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 67.8823i	− 2.15853i
$990$	0	0
$991$	−16.9706	−0.539088	−0.269544	−	0.962988i	$-0.586873\pi$
$991$	−16.9706	−0.539088	−0.269544	+	0.962988i	$0.586873\pi$
$992$	0	0
$993$	4.00000	0.126936
$994$	0	0
$995$	32.0000i	1.01447i
$996$	0	0
$997$	− 18.3848i	− 0.582252i	−0.956685	−	0.291126i	$-0.905970\pi$
$997$	− 18.3848i	− 0.582252i	0.956685	−	0.291126i	$-0.0940298\pi$
$998$	0	0
$999$	−4.24264	−0.134231

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3072.2.d.b.1537.2		4
4.3	odd	2	inner	3072.2.d.b.1537.4		4
8.3	odd	2	inner	3072.2.d.b.1537.1		4
8.5	even	2	inner	3072.2.d.b.1537.3		4
16.3	odd	4		3072.2.a.c.1.2		2
16.5	even	4		3072.2.a.c.1.1		2
16.11	odd	4		3072.2.a.e.1.1		2
16.13	even	4		3072.2.a.e.1.2		2
32.3	odd	8		1536.2.j.a.385.1	✓	4
32.5	even	8		1536.2.j.a.1153.2	yes	4
32.11	odd	8		1536.2.j.d.1153.2	yes	4
32.13	even	8		1536.2.j.d.385.1	yes	4
32.19	odd	8		1536.2.j.d.385.2	yes	4
32.21	even	8		1536.2.j.d.1153.1	yes	4
32.27	odd	8		1536.2.j.a.1153.1	yes	4
32.29	even	8		1536.2.j.a.385.2	yes	4
48.5	odd	4		9216.2.a.f.1.2		2
48.11	even	4		9216.2.a.r.1.2		2
48.29	odd	4		9216.2.a.r.1.1		2
48.35	even	4		9216.2.a.f.1.1		2
96.5	odd	8		4608.2.k.ba.1153.2		4
96.11	even	8		4608.2.k.z.1153.2		4
96.29	odd	8		4608.2.k.ba.3457.2		4
96.35	even	8		4608.2.k.ba.3457.1		4
96.53	odd	8		4608.2.k.z.1153.1		4
96.59	even	8		4608.2.k.ba.1153.1		4
96.77	odd	8		4608.2.k.z.3457.1		4
96.83	even	8		4608.2.k.z.3457.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.j.a.385.1	✓	4	32.3	odd	8
1536.2.j.a.385.2	yes	4	32.29	even	8
1536.2.j.a.1153.1	yes	4	32.27	odd	8
1536.2.j.a.1153.2	yes	4	32.5	even	8
1536.2.j.d.385.1	yes	4	32.13	even	8
1536.2.j.d.385.2	yes	4	32.19	odd	8
1536.2.j.d.1153.1	yes	4	32.21	even	8
1536.2.j.d.1153.2	yes	4	32.11	odd	8
3072.2.a.c.1.1		2	16.5	even	4
3072.2.a.c.1.2		2	16.3	odd	4
3072.2.a.e.1.1		2	16.11	odd	4
3072.2.a.e.1.2		2	16.13	even	4
3072.2.d.b.1537.1		4	8.3	odd	2	inner
3072.2.d.b.1537.2		4	1.1	even	1	trivial
3072.2.d.b.1537.3		4	8.5	even	2	inner
3072.2.d.b.1537.4		4	4.3	odd	2	inner
4608.2.k.z.1153.1		4	96.53	odd	8
4608.2.k.z.1153.2		4	96.11	even	8
4608.2.k.z.3457.1		4	96.77	odd	8
4608.2.k.z.3457.2		4	96.83	even	8
4608.2.k.ba.1153.1		4	96.59	even	8
4608.2.k.ba.1153.2		4	96.5	odd	8
4608.2.k.ba.3457.1		4	96.35	even	8
4608.2.k.ba.3457.2		4	96.29	odd	8
9216.2.a.f.1.1		2	48.35	even	4
9216.2.a.f.1.2		2	48.5	odd	4
9216.2.a.r.1.1		2	48.29	odd	4
9216.2.a.r.1.2		2	48.11	even	4

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:	\( N \)	\(=\)	\( 3072 = 2^{10} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3072.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +1.41421i q^{5} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +1.41421i q^{5} -1.00000 q^{9} -4.00000i q^{11} +1.41421i q^{13} +1.41421 q^{15} +4.00000i q^{19} +5.65685 q^{23} +3.00000 q^{25} +1.00000i q^{27} +7.07107i q^{29} -5.65685 q^{31} -4.00000 q^{33} +4.24264i q^{37} +1.41421 q^{39} -12.0000i q^{43} -1.41421i q^{45} +11.3137 q^{47} -7.00000 q^{49} +1.41421i q^{53} +5.65685 q^{55} +4.00000 q^{57} +4.00000i q^{59} -12.7279i q^{61} -2.00000 q^{65} +4.00000i q^{67} -5.65685i q^{69} +5.65685 q^{71} +10.0000 q^{73} -3.00000i q^{75} +16.9706 q^{79} +1.00000 q^{81} +12.0000i q^{83} +7.07107 q^{87} -6.00000 q^{89} +5.65685i q^{93} -5.65685 q^{95} +8.00000 q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} + 12 q^{25} - 16 q^{33} - 28 q^{49} + 16 q^{57} - 8 q^{65} + 40 q^{73} + 4 q^{81} - 24 q^{89} + 32 q^{97}+O(q^{100}) \) 4 * q - 4 * q^9 + 12 * q^25 - 16 * q^33 - 28 * q^49 + 16 * q^57 - 8 * q^65 + 40 * q^73 + 4 * q^81 - 24 * q^89 + 32 * q^97

Introduction

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