LMFDB - Embedded newform 3072.2.d.g.1537.6

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:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 1536)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1537.6
Root			$1.14412 + 1.14412i$ of defining polynomial
Character	$\chi$	$=$	3072.1537
Dual form			3072.2.d.g.1537.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -3.16228i q^{5} -1.74806 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -3.16228i q^{5} -1.74806 q^{7} -1.00000 q^{9} -6.47214i q^{11} -1.41421i q^{13} +3.16228 q^{15} +2.47214 q^{17} -6.47214i q^{19} -1.74806i q^{21} -5.65685 q^{23} -5.00000 q^{25} -1.00000i q^{27} +5.99070i q^{29} -3.90879 q^{31} +6.47214 q^{33} +5.52786i q^{35} +10.5672i q^{37} +1.41421 q^{39} +2.47214 q^{41} -1.52786i q^{43} +3.16228i q^{45} -3.94427 q^{49} +2.47214i q^{51} +11.6476i q^{53} -20.4667 q^{55} +6.47214 q^{57} -8.94427i q^{59} -2.08191i q^{61} +1.74806 q^{63} -4.47214 q^{65} +12.0000i q^{67} -5.65685i q^{69} +9.15298 q^{71} -2.94427 q^{73} -5.00000i q^{75} +11.3137i q^{77} -7.40492 q^{79} +1.00000 q^{81} -6.47214i q^{83} -7.81758i q^{85} -5.99070 q^{87} +10.0000 q^{89} +2.47214i q^{91} -3.90879i q^{93} -20.4667 q^{95} -12.9443 q^{97} +6.47214i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} - 16 q^{17} - 40 q^{25} + 16 q^{33} - 16 q^{41} + 40 q^{49} + 16 q^{57} + 48 q^{73} + 8 q^{81} + 80 q^{89} - 32 q^{97}+O(q^{100}) $ 8 * q - 8 * q^9 - 16 * q^17 - 40 * q^25 + 16 * q^33 - 16 * q^41 + 40 * q^49 + 16 * q^57 + 48 * q^73 + 8 * q^81 + 80 * 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
$5$	− 3.16228i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$5$	− 3.16228i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$6$	0	0
$7$	−1.74806	−0.660706	−0.330353	−	0.943857i	$-0.607168\pi$
$7$	−1.74806	−0.660706	−0.330353	+	0.943857i	$0.607168\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 6.47214i	− 1.95142i	−0.219061	−	0.975711i	$-0.570299\pi$
$11$	− 6.47214i	− 1.95142i	0.219061	−	0.975711i	$-0.429701\pi$
$12$	0	0
$13$	− 1.41421i	− 0.392232i	−0.980581	−	0.196116i	$-0.937167\pi$
$13$	− 1.41421i	− 0.392232i	0.980581	−	0.196116i	$-0.0628330\pi$
$14$	0	0
$15$	3.16228	0.816497
$16$	0	0
$17$	2.47214	0.599581	0.299791	−	0.954005i	$-0.403083\pi$
$17$	2.47214	0.599581	0.299791	+	0.954005i	$0.403083\pi$
$18$	0	0
$19$	− 6.47214i	− 1.48481i	−0.669951	−	0.742405i	$-0.733685\pi$
$19$	− 6.47214i	− 1.48481i	0.669951	−	0.742405i	$-0.266315\pi$
$20$	0	0
$21$	− 1.74806i	− 0.381459i
$22$	0	0
$23$	−5.65685	−1.17954	−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685	−1.17954	−0.589768	+	0.807573i	$0.700781\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	5.99070i	1.11245i	0.831033	+	0.556223i	$0.187750\pi$
$29$	5.99070i	1.11245i	−0.831033	+	0.556223i	$0.812250\pi$
$30$	0	0
$31$	−3.90879	−0.702039	−0.351020	−	0.936368i	$-0.614165\pi$
$31$	−3.90879	−0.702039	−0.351020	+	0.936368i	$0.614165\pi$
$32$	0	0
$33$	6.47214	1.12665
$34$	0	0
$35$	5.52786i	0.934380i
$36$	0	0
$37$	10.5672i	1.73724i	0.495482	+	0.868618i	$0.334991\pi$
$37$	10.5672i	1.73724i	−0.495482	+	0.868618i	$0.665009\pi$
$38$	0	0
$39$	1.41421	0.226455
$40$	0	0
$41$	2.47214	0.386083	0.193041	−	0.981191i	$-0.438165\pi$
$41$	2.47214	0.386083	0.193041	+	0.981191i	$0.438165\pi$
$42$	0	0
$43$	− 1.52786i	− 0.232997i	−0.993191	−	0.116499i	$-0.962833\pi$
$43$	− 1.52786i	− 0.232997i	0.993191	−	0.116499i	$-0.0371670\pi$
$44$	0	0
$45$	3.16228i	0.471405i
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−3.94427	−0.563467
$50$	0	0
$51$	2.47214i	0.346168i
$52$	0	0
$53$	11.6476i	1.59992i	0.600056	+	0.799958i	$0.295145\pi$
$53$	11.6476i	1.59992i	−0.600056	+	0.799958i	$0.704855\pi$
$54$	0	0
$55$	−20.4667	−2.75973
$56$	0	0
$57$	6.47214	0.857255
$58$	0	0
$59$	− 8.94427i	− 1.16445i	−0.813029	−	0.582223i	$-0.802183\pi$
$59$	− 8.94427i	− 1.16445i	0.813029	−	0.582223i	$-0.197817\pi$
$60$	0	0
$61$	− 2.08191i	− 0.266562i	−0.991078	−	0.133281i	$-0.957449\pi$
$61$	− 2.08191i	− 0.266562i	0.991078	−	0.133281i	$-0.0425513\pi$
$62$	0	0
$63$	1.74806	0.220235
$64$	0	0
$65$	−4.47214	−0.554700
$66$	0	0
$67$	12.0000i	1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$	12.0000i	1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	0	0
$69$	− 5.65685i	− 0.681005i
$70$	0	0
$71$	9.15298	1.08626	0.543130	−	0.839649i	$-0.317239\pi$
$71$	9.15298	1.08626	0.543130	+	0.839649i	$0.317239\pi$
$72$	0	0
$73$	−2.94427	−0.344601	−0.172300	−	0.985044i	$-0.555120\pi$
$73$	−2.94427	−0.344601	−0.172300	+	0.985044i	$0.555120\pi$
$74$	0	0
$75$	− 5.00000i	− 0.577350i
$76$	0	0
$77$	11.3137i	1.28932i
$78$	0	0
$79$	−7.40492	−0.833118	−0.416559	−	0.909109i	$-0.636764\pi$
$79$	−7.40492	−0.833118	−0.416559	+	0.909109i	$0.636764\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 6.47214i	− 0.710409i	−0.934789	−	0.355205i	$-0.884411\pi$
$83$	− 6.47214i	− 0.710409i	0.934789	−	0.355205i	$-0.115589\pi$
$84$	0	0
$85$	− 7.81758i	− 0.847936i
$86$	0	0
$87$	−5.99070	−0.642271
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	2.47214i	0.259150i
$92$	0	0
$93$	− 3.90879i	− 0.405323i
$94$	0	0
$95$	−20.4667	−2.09984
$96$	0	0
$97$	−12.9443	−1.31429	−0.657146	−	0.753763i	$-0.728236\pi$
$97$	−12.9443	−1.31429	−0.657146	+	0.753763i	$0.728236\pi$
$98$	0	0
$99$	6.47214i	0.650474i
$100$	0	0
$101$	− 0.333851i	− 0.0332194i	−0.999862	−	0.0166097i	$-0.994713\pi$
$101$	− 0.333851i	− 0.0332194i	0.999862	−	0.0166097i	$-0.00528727\pi$
$102$	0	0
$103$	1.74806	0.172242	0.0861209	−	0.996285i	$-0.472553\pi$
$103$	1.74806	0.172242	0.0861209	+	0.996285i	$0.472553\pi$
$104$	0	0
$105$	−5.52786	−0.539464
$106$	0	0
$107$	0.944272i	0.0912862i	0.998958	+	0.0456431i	$0.0145337\pi$
$107$	0.944272i	0.0912862i	−0.998958	+	0.0456431i	$0.985466\pi$
$108$	0	0
$109$	− 16.8918i	− 1.61794i	−0.587852	−	0.808968i	$-0.700026\pi$
$109$	− 16.8918i	− 1.61794i	0.587852	−	0.808968i	$-0.299974\pi$
$110$	0	0
$111$	−10.5672	−1.00299
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	17.8885i	1.66812i
$116$	0	0
$117$	1.41421i	0.130744i
$118$	0	0
$119$	−4.32145	−0.396147
$120$	0	0
$121$	−30.8885	−2.80805
$122$	0	0
$123$	2.47214i	0.222905i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10.9010	−0.967311	−0.483656	−	0.875258i	$-0.660691\pi$
$127$	−10.9010	−0.967311	−0.483656	+	0.875258i	$0.660691\pi$
$128$	0	0
$129$	1.52786	0.134521
$130$	0	0
$131$	16.9443i	1.48043i	0.672371	+	0.740214i	$0.265276\pi$
$131$	16.9443i	1.48043i	−0.672371	+	0.740214i	$0.734724\pi$
$132$	0	0
$133$	11.3137i	0.981023i
$134$	0	0
$135$	−3.16228	−0.272166
$136$	0	0
$137$	−2.47214	−0.211209	−0.105604	−	0.994408i	$-0.533678\pi$
$137$	−2.47214	−0.211209	−0.105604	+	0.994408i	$0.533678\pi$
$138$	0	0
$139$	− 16.9443i	− 1.43719i	−0.695427	−	0.718597i	$-0.744785\pi$
$139$	− 16.9443i	− 1.43719i	0.695427	−	0.718597i	$-0.255215\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−9.15298	−0.765411
$144$	0	0
$145$	18.9443	1.57324
$146$	0	0
$147$	− 3.94427i	− 0.325318i
$148$	0	0
$149$	− 3.82998i	− 0.313764i	−0.987617	−	0.156882i	$-0.949856\pi$
$149$	− 3.82998i	− 0.313764i	0.987617	−	0.156882i	$-0.0501442\pi$
$150$	0	0
$151$	20.0540	1.63197	0.815987	−	0.578070i	$-0.196194\pi$
$151$	20.0540	1.63197	0.815987	+	0.578070i	$0.196194\pi$
$152$	0	0
$153$	−2.47214	−0.199860
$154$	0	0
$155$	12.3607i	0.992834i
$156$	0	0
$157$	− 13.3956i	− 1.06909i	−0.845141	−	0.534544i	$-0.820484\pi$
$157$	− 13.3956i	− 1.06909i	0.845141	−	0.534544i	$-0.179516\pi$
$158$	0	0
$159$	−11.6476	−0.923712
$160$	0	0
$161$	9.88854	0.779326
$162$	0	0
$163$	14.4721i	1.13355i	0.823874	+	0.566773i	$0.191808\pi$
$163$	14.4721i	1.13355i	−0.823874	+	0.566773i	$0.808192\pi$
$164$	0	0
$165$	− 20.4667i	− 1.59333i
$166$	0	0
$167$	−20.4667	−1.58376	−0.791880	−	0.610677i	$-0.790898\pi$
$167$	−20.4667	−1.58376	−0.791880	+	0.610677i	$0.790898\pi$
$168$	0	0
$169$	11.0000	0.846154
$170$	0	0
$171$	6.47214i	0.494937i
$172$	0	0
$173$	9.48683i	0.721271i	0.932707	+	0.360635i	$0.117440\pi$
$173$	9.48683i	0.721271i	−0.932707	+	0.360635i	$0.882560\pi$
$174$	0	0
$175$	8.74032	0.660706
$176$	0	0
$177$	8.94427	0.672293
$178$	0	0
$179$	12.0000i	0.896922i	0.893802	+	0.448461i	$0.148028\pi$
$179$	12.0000i	0.896922i	−0.893802	+	0.448461i	$0.851972\pi$
$180$	0	0
$181$	− 15.5563i	− 1.15629i	−0.815933	−	0.578147i	$-0.803776\pi$
$181$	− 15.5563i	− 1.15629i	0.815933	−	0.578147i	$-0.196224\pi$
$182$	0	0
$183$	2.08191	0.153900
$184$	0	0
$185$	33.4164	2.45682
$186$	0	0
$187$	− 16.0000i	− 1.17004i
$188$	0	0
$189$	1.74806i	0.127153i
$190$	0	0
$191$	14.8098	1.07160	0.535801	−	0.844344i	$-0.320010\pi$
$191$	14.8098	1.07160	0.535801	+	0.844344i	$0.320010\pi$
$192$	0	0
$193$	4.94427	0.355896	0.177948	−	0.984040i	$-0.443054\pi$
$193$	4.94427	0.355896	0.177948	+	0.984040i	$0.443054\pi$
$194$	0	0
$195$	− 4.47214i	− 0.320256i
$196$	0	0
$197$	7.32611i	0.521964i	0.965344	+	0.260982i	$0.0840462\pi$
$197$	7.32611i	0.521964i	−0.965344	+	0.260982i	$0.915954\pi$
$198$	0	0
$199$	5.24419	0.371751	0.185875	−	0.982573i	$-0.440488\pi$
$199$	5.24419	0.371751	0.185875	+	0.982573i	$0.440488\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	0	0
$203$	− 10.4721i	− 0.735000i
$204$	0	0
$205$	− 7.81758i	− 0.546003i
$206$	0	0
$207$	5.65685	0.393179
$208$	0	0
$209$	−41.8885	−2.89749
$210$	0	0
$211$	7.05573i	0.485736i	0.970059	+	0.242868i	$0.0780882\pi$
$211$	7.05573i	0.485736i	−0.970059	+	0.242868i	$0.921912\pi$
$212$	0	0
$213$	9.15298i	0.627152i
$214$	0	0
$215$	−4.83153	−0.329508
$216$	0	0
$217$	6.83282	0.463842
$218$	0	0
$219$	− 2.94427i	− 0.198955i
$220$	0	0
$221$	− 3.49613i	− 0.235175i
$222$	0	0
$223$	−3.90879	−0.261752	−0.130876	−	0.991399i	$-0.541779\pi$
$223$	−3.90879	−0.261752	−0.130876	+	0.991399i	$0.541779\pi$
$224$	0	0
$225$	5.00000	0.333333
$226$	0	0
$227$	− 14.4721i	− 0.960549i	−0.877118	−	0.480275i	$-0.840537\pi$
$227$	− 14.4721i	− 0.960549i	0.877118	−	0.480275i	$-0.159463\pi$
$228$	0	0
$229$	− 14.0633i	− 0.929331i	−0.885486	−	0.464665i	$-0.846175\pi$
$229$	− 14.0633i	− 0.929331i	0.885486	−	0.464665i	$-0.153825\pi$
$230$	0	0
$231$	−11.3137	−0.744387
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 7.40492i	− 0.481001i
$238$	0	0
$239$	−26.1235	−1.68979	−0.844896	−	0.534931i	$-0.820338\pi$
$239$	−26.1235	−1.68979	−0.844896	+	0.534931i	$0.820338\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	12.4729i	0.796863i
$246$	0	0
$247$	−9.15298	−0.582390
$248$	0	0
$249$	6.47214	0.410155
$250$	0	0
$251$	− 11.4164i	− 0.720597i	−0.932837	−	0.360299i	$-0.882675\pi$
$251$	− 11.4164i	− 0.720597i	0.932837	−	0.360299i	$-0.117325\pi$
$252$	0	0
$253$	36.6119i	2.30177i
$254$	0	0
$255$	7.81758	0.489556
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	− 18.4721i	− 1.14780i
$260$	0	0
$261$	− 5.99070i	− 0.370815i
$262$	0	0
$263$	−12.6491	−0.779978	−0.389989	−	0.920820i	$-0.627521\pi$
$263$	−12.6491	−0.779978	−0.389989	+	0.920820i	$0.627521\pi$
$264$	0	0
$265$	36.8328	2.26262
$266$	0	0
$267$	10.0000i	0.611990i
$268$	0	0
$269$	5.32300i	0.324549i	0.986746	+	0.162275i	$0.0518830\pi$
$269$	5.32300i	0.324549i	−0.986746	+	0.162275i	$0.948117\pi$
$270$	0	0
$271$	−10.9010	−0.662191	−0.331096	−	0.943597i	$-0.607418\pi$
$271$	−10.9010	−0.662191	−0.331096	+	0.943597i	$0.607418\pi$
$272$	0	0
$273$	−2.47214	−0.149620
$274$	0	0
$275$	32.3607i	1.95142i
$276$	0	0
$277$	− 18.3848i	− 1.10463i	−0.833634	−	0.552317i	$-0.813744\pi$
$277$	− 18.3848i	− 1.10463i	0.833634	−	0.552317i	$-0.186256\pi$
$278$	0	0
$279$	3.90879	0.234013
$280$	0	0
$281$	−3.88854	−0.231971	−0.115986	−	0.993251i	$-0.537003\pi$
$281$	−3.88854	−0.231971	−0.115986	+	0.993251i	$0.537003\pi$
$282$	0	0
$283$	− 26.8328i	− 1.59505i	−0.603289	−	0.797523i	$-0.706143\pi$
$283$	− 26.8328i	− 1.59505i	0.603289	−	0.797523i	$-0.293857\pi$
$284$	0	0
$285$	− 20.4667i	− 1.21234i
$286$	0	0
$287$	−4.32145	−0.255087
$288$	0	0
$289$	−10.8885	−0.640503
$290$	0	0
$291$	− 12.9443i	− 0.758807i
$292$	0	0
$293$	22.9613i	1.34141i	0.741723	+	0.670706i	$0.234009\pi$
$293$	22.9613i	1.34141i	−0.741723	+	0.670706i	$0.765991\pi$
$294$	0	0
$295$	−28.2843	−1.64677
$296$	0	0
$297$	−6.47214	−0.375551
$298$	0	0
$299$	8.00000i	0.462652i
$300$	0	0
$301$	2.67080i	0.153943i
$302$	0	0
$303$	0.333851	0.0191792
$304$	0	0
$305$	−6.58359	−0.376975
$306$	0	0
$307$	− 5.88854i	− 0.336077i	−0.985780	−	0.168038i	$-0.946257\pi$
$307$	− 5.88854i	− 0.336077i	0.985780	−	0.168038i	$-0.0537433\pi$
$308$	0	0
$309$	1.74806i	0.0994439i
$310$	0	0
$311$	−19.6414	−1.11376	−0.556880	−	0.830593i	$-0.688002\pi$
$311$	−19.6414	−1.11376	−0.556880	+	0.830593i	$0.688002\pi$
$312$	0	0
$313$	−20.9443	−1.18384	−0.591920	−	0.805997i	$-0.701630\pi$
$313$	−20.9443	−1.18384	−0.591920	+	0.805997i	$0.701630\pi$
$314$	0	0
$315$	− 5.52786i	− 0.311460i
$316$	0	0
$317$	17.3044i	0.971913i	0.873983	+	0.485956i	$0.161529\pi$
$317$	17.3044i	0.971913i	−0.873983	+	0.485956i	$0.838471\pi$
$318$	0	0
$319$	38.7727	2.17085
$320$	0	0
$321$	−0.944272	−0.0527041
$322$	0	0
$323$	− 16.0000i	− 0.890264i
$324$	0	0
$325$	7.07107i	0.392232i
$326$	0	0
$327$	16.8918	0.934116
$328$	0	0
$329$	0	0
$330$	0	0
$331$	16.9443i	0.931341i	0.884958	+	0.465671i	$0.154187\pi$
$331$	16.9443i	0.931341i	−0.884958	+	0.465671i	$0.845813\pi$
$332$	0	0
$333$	− 10.5672i	− 0.579079i
$334$	0	0
$335$	37.9473	2.07328
$336$	0	0
$337$	14.9443	0.814066	0.407033	−	0.913413i	$-0.366563\pi$
$337$	14.9443	0.814066	0.407033	+	0.913413i	$0.366563\pi$
$338$	0	0
$339$	− 2.00000i	− 0.108625i
$340$	0	0
$341$	25.2982i	1.36998i
$342$	0	0
$343$	19.1313	1.03299
$344$	0	0
$345$	−17.8885	−0.963087
$346$	0	0
$347$	− 22.4721i	− 1.20637i	−0.797602	−	0.603184i	$-0.793899\pi$
$347$	− 22.4721i	− 1.20637i	0.797602	−	0.603184i	$-0.206101\pi$
$348$	0	0
$349$	− 9.23179i	− 0.494167i	−0.968994	−	0.247083i	$-0.920528\pi$
$349$	− 9.23179i	− 0.494167i	0.968994	−	0.247083i	$-0.0794721\pi$
$350$	0	0
$351$	−1.41421	−0.0754851
$352$	0	0
$353$	−11.8885	−0.632763	−0.316382	−	0.948632i	$-0.602468\pi$
$353$	−11.8885	−0.632763	−0.316382	+	0.948632i	$0.602468\pi$
$354$	0	0
$355$	− 28.9443i	− 1.53620i
$356$	0	0
$357$	− 4.32145i	− 0.228715i
$358$	0	0
$359$	−13.4744	−0.711153	−0.355577	−	0.934647i	$-0.615716\pi$
$359$	−13.4744	−0.711153	−0.355577	+	0.934647i	$0.615716\pi$
$360$	0	0
$361$	−22.8885	−1.20466
$362$	0	0
$363$	− 30.8885i	− 1.62123i
$364$	0	0
$365$	9.31061i	0.487339i
$366$	0	0
$367$	−17.8933	−0.934023	−0.467011	−	0.884251i	$-0.654669\pi$
$367$	−17.8933	−0.934023	−0.467011	+	0.884251i	$0.654669\pi$
$368$	0	0
$369$	−2.47214	−0.128694
$370$	0	0
$371$	− 20.3607i	− 1.05707i
$372$	0	0
$373$	− 21.8809i	− 1.13295i	−0.824079	−	0.566475i	$-0.808307\pi$
$373$	− 21.8809i	− 1.13295i	0.824079	−	0.566475i	$-0.191693\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.47214	0.436337
$378$	0	0
$379$	27.4164i	1.40829i	0.710058	+	0.704143i	$0.248669\pi$
$379$	27.4164i	1.40829i	−0.710058	+	0.704143i	$0.751331\pi$
$380$	0	0
$381$	− 10.9010i	− 0.558478i
$382$	0	0
$383$	19.1313	0.977563	0.488782	−	0.872406i	$-0.337441\pi$
$383$	19.1313	0.977563	0.488782	+	0.872406i	$0.337441\pi$
$384$	0	0
$385$	35.7771	1.82337
$386$	0	0
$387$	1.52786i	0.0776657i
$388$	0	0
$389$	− 33.4497i	− 1.69596i	−0.530024	−	0.847982i	$-0.677817\pi$
$389$	− 33.4497i	− 1.69596i	0.530024	−	0.847982i	$-0.322183\pi$
$390$	0	0
$391$	−13.9845	−0.707227
$392$	0	0
$393$	−16.9443	−0.854725
$394$	0	0
$395$	23.4164i	1.17821i
$396$	0	0
$397$	− 9.07417i	− 0.455420i	−0.973729	−	0.227710i	$-0.926876\pi$
$397$	− 9.07417i	− 0.455420i	0.973729	−	0.227710i	$-0.0731238\pi$
$398$	0	0
$399$	−11.3137	−0.566394
$400$	0	0
$401$	13.5279	0.675549	0.337775	−	0.941227i	$-0.390326\pi$
$401$	13.5279	0.675549	0.337775	+	0.941227i	$0.390326\pi$
$402$	0	0
$403$	5.52786i	0.275363i
$404$	0	0
$405$	− 3.16228i	− 0.157135i
$406$	0	0
$407$	68.3923	3.39008
$408$	0	0
$409$	12.9443	0.640053	0.320027	−	0.947409i	$-0.396308\pi$
$409$	12.9443	0.640053	0.320027	+	0.947409i	$0.396308\pi$
$410$	0	0
$411$	− 2.47214i	− 0.121941i
$412$	0	0
$413$	15.6352i	0.769356i
$414$	0	0
$415$	−20.4667	−1.00467
$416$	0	0
$417$	16.9443	0.829765
$418$	0	0
$419$	− 1.52786i	− 0.0746410i	−0.999303	−	0.0373205i	$-0.988118\pi$
$419$	− 1.52786i	− 0.0746410i	0.999303	−	0.0373205i	$-0.0118823\pi$
$420$	0	0
$421$	− 7.07107i	− 0.344623i	−0.985043	−	0.172311i	$-0.944876\pi$
$421$	− 7.07107i	− 0.344623i	0.985043	−	0.172311i	$-0.0551235\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−12.3607	−0.599581
$426$	0	0
$427$	3.63932i	0.176119i
$428$	0	0
$429$	− 9.15298i	− 0.441910i
$430$	0	0
$431$	18.3060	0.881767	0.440884	−	0.897564i	$-0.354665\pi$
$431$	18.3060	0.881767	0.440884	+	0.897564i	$0.354665\pi$
$432$	0	0
$433$	24.0000	1.15337	0.576683	−	0.816968i	$-0.304347\pi$
$433$	24.0000	1.15337	0.576683	+	0.816968i	$0.304347\pi$
$434$	0	0
$435$	18.9443i	0.908308i
$436$	0	0
$437$	36.6119i	1.75139i
$438$	0	0
$439$	−6.06952	−0.289682	−0.144841	−	0.989455i	$-0.546267\pi$
$439$	−6.06952	−0.289682	−0.144841	+	0.989455i	$0.546267\pi$
$440$	0	0
$441$	3.94427	0.187822
$442$	0	0
$443$	11.4164i	0.542410i	0.962522	+	0.271205i	$0.0874221\pi$
$443$	11.4164i	0.542410i	−0.962522	+	0.271205i	$0.912578\pi$
$444$	0	0
$445$	− 31.6228i	− 1.49906i
$446$	0	0
$447$	3.82998	0.181152
$448$	0	0
$449$	−23.4164	−1.10509	−0.552544	−	0.833484i	$-0.686343\pi$
$449$	−23.4164	−1.10509	−0.552544	+	0.833484i	$0.686343\pi$
$450$	0	0
$451$	− 16.0000i	− 0.753411i
$452$	0	0
$453$	20.0540i	0.942220i
$454$	0	0
$455$	7.81758	0.366494
$456$	0	0
$457$	−3.05573	−0.142941	−0.0714705	−	0.997443i	$-0.522769\pi$
$457$	−3.05573	−0.142941	−0.0714705	+	0.997443i	$0.522769\pi$
$458$	0	0
$459$	− 2.47214i	− 0.115389i
$460$	0	0
$461$	8.81913i	0.410748i	0.978684	+	0.205374i	$0.0658411\pi$
$461$	8.81913i	0.410748i	−0.978684	+	0.205374i	$0.934159\pi$
$462$	0	0
$463$	7.40492	0.344136	0.172068	−	0.985085i	$-0.444955\pi$
$463$	7.40492	0.344136	0.172068	+	0.985085i	$0.444955\pi$
$464$	0	0
$465$	−12.3607	−0.573213
$466$	0	0
$467$	− 4.58359i	− 0.212103i	−0.994361	−	0.106052i	$-0.966179\pi$
$467$	− 4.58359i	− 0.212103i	0.994361	−	0.106052i	$-0.0338209\pi$
$468$	0	0
$469$	− 20.9768i	− 0.968617i
$470$	0	0
$471$	13.3956	0.617238
$472$	0	0
$473$	−9.88854	−0.454676
$474$	0	0
$475$	32.3607i	1.48481i
$476$	0	0
$477$	− 11.6476i	− 0.533305i
$478$	0	0
$479$	−3.49613	−0.159742	−0.0798711	−	0.996805i	$-0.525451\pi$
$479$	−3.49613	−0.159742	−0.0798711	+	0.996805i	$0.525451\pi$
$480$	0	0
$481$	14.9443	0.681400
$482$	0	0
$483$	9.88854i	0.449944i
$484$	0	0
$485$	40.9334i	1.85869i
$486$	0	0
$487$	−16.5579	−0.750310	−0.375155	−	0.926962i	$-0.622411\pi$
$487$	−16.5579	−0.750310	−0.375155	+	0.926962i	$0.622411\pi$
$488$	0	0
$489$	−14.4721	−0.654453
$490$	0	0
$491$	− 21.8885i	− 0.987816i	−0.869514	−	0.493908i	$-0.835568\pi$
$491$	− 21.8885i	− 0.987816i	0.869514	−	0.493908i	$-0.164432\pi$
$492$	0	0
$493$	14.8098i	0.667001i
$494$	0	0
$495$	20.4667	0.919909
$496$	0	0
$497$	−16.0000	−0.717698
$498$	0	0
$499$	− 24.9443i	− 1.11666i	−0.829619	−	0.558329i	$-0.811443\pi$
$499$	− 24.9443i	− 1.11666i	0.829619	−	0.558329i	$-0.188557\pi$
$500$	0	0
$501$	− 20.4667i	− 0.914384i
$502$	0	0
$503$	4.83153	0.215427	0.107714	−	0.994182i	$-0.465647\pi$
$503$	4.83153	0.215427	0.107714	+	0.994182i	$0.465647\pi$
$504$	0	0
$505$	−1.05573	−0.0469793
$506$	0	0
$507$	11.0000i	0.488527i
$508$	0	0
$509$	− 39.9318i	− 1.76995i	−0.465641	−	0.884974i	$-0.654176\pi$
$509$	− 39.9318i	− 1.76995i	0.465641	−	0.884974i	$-0.345824\pi$
$510$	0	0
$511$	5.14678	0.227680
$512$	0	0
$513$	−6.47214	−0.285752
$514$	0	0
$515$	− 5.52786i	− 0.243587i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−9.48683	−0.416426
$520$	0	0
$521$	28.3607	1.24250	0.621252	−	0.783611i	$-0.286624\pi$
$521$	28.3607	1.24250	0.621252	+	0.783611i	$0.286624\pi$
$522$	0	0
$523$	16.3607i	0.715403i	0.933836	+	0.357701i	$0.116439\pi$
$523$	16.3607i	0.715403i	−0.933836	+	0.357701i	$0.883561\pi$
$524$	0	0
$525$	8.74032i	0.381459i
$526$	0	0
$527$	−9.66306	−0.420930
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	8.94427i	0.388148i
$532$	0	0
$533$	− 3.49613i	− 0.151434i
$534$	0	0
$535$	2.98605	0.129098
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	25.5279i	1.09956i
$540$	0	0
$541$	26.7124i	1.14846i	0.818695	+	0.574229i	$0.194698\pi$
$541$	26.7124i	1.14846i	−0.818695	+	0.574229i	$0.805302\pi$
$542$	0	0
$543$	15.5563	0.667587
$544$	0	0
$545$	−53.4164	−2.28811
$546$	0	0
$547$	32.3607i	1.38364i	0.722069	+	0.691821i	$0.243191\pi$
$547$	32.3607i	1.38364i	−0.722069	+	0.691821i	$0.756809\pi$
$548$	0	0
$549$	2.08191i	0.0888540i
$550$	0	0
$551$	38.7727	1.65177
$552$	0	0
$553$	12.9443	0.550446
$554$	0	0
$555$	33.4164i	1.41845i
$556$	0	0
$557$	− 15.8114i	− 0.669950i	−0.942227	−	0.334975i	$-0.891272\pi$
$557$	− 15.8114i	− 0.669950i	0.942227	−	0.334975i	$-0.108728\pi$
$558$	0	0
$559$	−2.16073	−0.0913890
$560$	0	0
$561$	16.0000	0.675521
$562$	0	0
$563$	1.52786i	0.0643918i	0.999482	+	0.0321959i	$0.0102500\pi$
$563$	1.52786i	0.0643918i	−0.999482	+	0.0321959i	$0.989750\pi$
$564$	0	0
$565$	6.32456i	0.266076i
$566$	0	0
$567$	−1.74806	−0.0734118
$568$	0	0
$569$	−7.41641	−0.310912	−0.155456	−	0.987843i	$-0.549685\pi$
$569$	−7.41641	−0.310912	−0.155456	+	0.987843i	$0.549685\pi$
$570$	0	0
$571$	8.94427i	0.374306i	0.982331	+	0.187153i	$0.0599260\pi$
$571$	8.94427i	0.374306i	−0.982331	+	0.187153i	$0.940074\pi$
$572$	0	0
$573$	14.8098i	0.618690i
$574$	0	0
$575$	28.2843	1.17954
$576$	0	0
$577$	26.9443	1.12170	0.560852	−	0.827916i	$-0.310474\pi$
$577$	26.9443	1.12170	0.560852	+	0.827916i	$0.310474\pi$
$578$	0	0
$579$	4.94427i	0.205477i
$580$	0	0
$581$	11.3137i	0.469372i
$582$	0	0
$583$	75.3846	3.12211
$584$	0	0
$585$	4.47214	0.184900
$586$	0	0
$587$	32.9443i	1.35976i	0.733325	+	0.679878i	$0.237967\pi$
$587$	32.9443i	1.35976i	−0.733325	+	0.679878i	$0.762033\pi$
$588$	0	0
$589$	25.2982i	1.04240i
$590$	0	0
$591$	−7.32611	−0.301356
$592$	0	0
$593$	7.88854	0.323944	0.161972	−	0.986795i	$-0.448215\pi$
$593$	7.88854	0.323944	0.161972	+	0.986795i	$0.448215\pi$
$594$	0	0
$595$	13.6656i	0.560236i
$596$	0	0
$597$	5.24419i	0.214631i
$598$	0	0
$599$	5.65685	0.231133	0.115566	−	0.993300i	$-0.463132\pi$
$599$	5.65685	0.231133	0.115566	+	0.993300i	$0.463132\pi$
$600$	0	0
$601$	6.94427	0.283263	0.141631	−	0.989919i	$-0.454765\pi$
$601$	6.94427	0.283263	0.141631	+	0.989919i	$0.454765\pi$
$602$	0	0
$603$	− 12.0000i	− 0.488678i
$604$	0	0
$605$	97.6782i	3.97118i
$606$	0	0
$607$	−44.8422	−1.82009	−0.910044	−	0.414512i	$-0.863952\pi$
$607$	−44.8422	−1.82009	−0.910044	+	0.414512i	$0.863952\pi$
$608$	0	0
$609$	10.4721	0.424352
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 10.5672i	− 0.426805i	−0.976964	−	0.213403i	$-0.931545\pi$
$613$	− 10.5672i	− 0.426805i	0.976964	−	0.213403i	$-0.0684546\pi$
$614$	0	0
$615$	7.81758	0.315235
$616$	0	0
$617$	15.8885	0.639649	0.319824	−	0.947477i	$-0.396376\pi$
$617$	15.8885	0.639649	0.319824	+	0.947477i	$0.396376\pi$
$618$	0	0
$619$	5.88854i	0.236681i	0.992973	+	0.118340i	$0.0377574\pi$
$619$	5.88854i	0.236681i	−0.992973	+	0.118340i	$0.962243\pi$
$620$	0	0
$621$	5.65685i	0.227002i
$622$	0	0
$623$	−17.4806	−0.700347
$624$	0	0
$625$	−25.0000	−1.00000
$626$	0	0
$627$	− 41.8885i	− 1.67287i
$628$	0	0
$629$	26.1235i	1.04161i
$630$	0	0
$631$	27.0463	1.07670	0.538348	−	0.842723i	$-0.319049\pi$
$631$	27.0463	1.07670	0.538348	+	0.842723i	$0.319049\pi$
$632$	0	0
$633$	−7.05573	−0.280440
$634$	0	0
$635$	34.4721i	1.36798i
$636$	0	0
$637$	5.57804i	0.221010i
$638$	0	0
$639$	−9.15298	−0.362086
$640$	0	0
$641$	8.58359	0.339032	0.169516	−	0.985527i	$-0.445780\pi$
$641$	8.58359	0.339032	0.169516	+	0.985527i	$0.445780\pi$
$642$	0	0
$643$	− 8.36068i	− 0.329713i	−0.986318	−	0.164857i	$-0.947284\pi$
$643$	− 8.36068i	− 0.329713i	0.986318	−	0.164857i	$-0.0527161\pi$
$644$	0	0
$645$	− 4.83153i	− 0.190241i
$646$	0	0
$647$	42.2688	1.66176	0.830879	−	0.556454i	$-0.187838\pi$
$647$	42.2688	1.66176	0.830879	+	0.556454i	$0.187838\pi$
$648$	0	0
$649$	−57.8885	−2.27232
$650$	0	0
$651$	6.83282i	0.267799i
$652$	0	0
$653$	38.4388i	1.50423i	0.659034	+	0.752113i	$0.270965\pi$
$653$	38.4388i	1.50423i	−0.659034	+	0.752113i	$0.729035\pi$
$654$	0	0
$655$	53.5825	2.09364
$656$	0	0
$657$	2.94427	0.114867
$658$	0	0
$659$	− 4.00000i	− 0.155818i	−0.996960	−	0.0779089i	$-0.975176\pi$
$659$	− 4.00000i	− 0.155818i	0.996960	−	0.0779089i	$-0.0248243\pi$
$660$	0	0
$661$	40.1869i	1.56309i	0.623850	+	0.781544i	$0.285568\pi$
$661$	40.1869i	1.56309i	−0.623850	+	0.781544i	$0.714432\pi$
$662$	0	0
$663$	3.49613	0.135778
$664$	0	0
$665$	35.7771	1.38738
$666$	0	0
$667$	− 33.8885i	− 1.31217i
$668$	0	0
$669$	− 3.90879i	− 0.151123i
$670$	0	0
$671$	−13.4744	−0.520175
$672$	0	0
$673$	−17.8885	−0.689553	−0.344776	−	0.938685i	$-0.612045\pi$
$673$	−17.8885	−0.689553	−0.344776	+	0.938685i	$0.612045\pi$
$674$	0	0
$675$	5.00000i	0.192450i
$676$	0	0
$677$	0.491473i	0.0188889i	0.999955	+	0.00944443i	$0.00300630\pi$
$677$	0.491473i	0.0188889i	−0.999955	+	0.00944443i	$0.996994\pi$
$678$	0	0
$679$	22.6274	0.868361
$680$	0	0
$681$	14.4721	0.554573
$682$	0	0
$683$	− 3.41641i	− 0.130725i	−0.997862	−	0.0653626i	$-0.979180\pi$
$683$	− 3.41641i	− 0.130725i	0.997862	−	0.0653626i	$-0.0208204\pi$
$684$	0	0
$685$	7.81758i	0.298694i
$686$	0	0
$687$	14.0633	0.536549
$688$	0	0
$689$	16.4721	0.627538
$690$	0	0
$691$	− 25.5279i	− 0.971126i	−0.874202	−	0.485563i	$-0.838615\pi$
$691$	− 25.5279i	− 0.971126i	0.874202	−	0.485563i	$-0.161385\pi$
$692$	0	0
$693$	− 11.3137i	− 0.429772i
$694$	0	0
$695$	−53.5825	−2.03250
$696$	0	0
$697$	6.11146	0.231488
$698$	0	0
$699$	10.0000i	0.378235i
$700$	0	0
$701$	− 35.6104i	− 1.34499i	−0.740104	−	0.672493i	$-0.765224\pi$
$701$	− 35.6104i	− 1.34499i	0.740104	−	0.672493i	$-0.234776\pi$
$702$	0	0
$703$	68.3923	2.57947
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.583592i	0.0219482i
$708$	0	0
$709$	− 33.8623i	− 1.27173i	−0.771802	−	0.635863i	$-0.780644\pi$
$709$	− 33.8623i	− 1.27173i	0.771802	−	0.635863i	$-0.219356\pi$
$710$	0	0
$711$	7.40492	0.277706
$712$	0	0
$713$	22.1115	0.828081
$714$	0	0
$715$	28.9443i	1.08245i
$716$	0	0
$717$	− 26.1235i	− 0.975602i
$718$	0	0
$719$	−21.8021	−0.813081	−0.406540	−	0.913633i	$-0.633265\pi$
$719$	−21.8021	−0.813081	−0.406540	+	0.913633i	$0.633265\pi$
$720$	0	0
$721$	−3.05573	−0.113801
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 29.9535i	− 1.11245i
$726$	0	0
$727$	−20.0540	−0.743763	−0.371881	−	0.928280i	$-0.621287\pi$
$727$	−20.0540	−0.743763	−0.371881	+	0.928280i	$0.621287\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 3.77709i	− 0.139701i
$732$	0	0
$733$	− 31.0339i	− 1.14626i	−0.819463	−	0.573131i	$-0.805728\pi$
$733$	− 31.0339i	− 1.14626i	0.819463	−	0.573131i	$-0.194272\pi$
$734$	0	0
$735$	−12.4729	−0.460069
$736$	0	0
$737$	77.6656	2.86085
$738$	0	0
$739$	− 32.9443i	− 1.21187i	−0.795512	−	0.605937i	$-0.792798\pi$
$739$	− 32.9443i	− 1.21187i	0.795512	−	0.605937i	$-0.207202\pi$
$740$	0	0
$741$	− 9.15298i	− 0.336243i
$742$	0	0
$743$	2.16073	0.0792694	0.0396347	−	0.999214i	$-0.487381\pi$
$743$	2.16073	0.0792694	0.0396347	+	0.999214i	$0.487381\pi$
$744$	0	0
$745$	−12.1115	−0.443729
$746$	0	0
$747$	6.47214i	0.236803i
$748$	0	0
$749$	− 1.65065i	− 0.0603134i
$750$	0	0
$751$	7.40492	0.270209	0.135105	−	0.990831i	$-0.456863\pi$
$751$	7.40492	0.270209	0.135105	+	0.990831i	$0.456863\pi$
$752$	0	0
$753$	11.4164	0.416037
$754$	0	0
$755$	− 63.4164i	− 2.30796i
$756$	0	0
$757$	14.0633i	0.511140i	0.966791	+	0.255570i	$0.0822632\pi$
$757$	14.0633i	0.511140i	−0.966791	+	0.255570i	$0.917737\pi$
$758$	0	0
$759$	−36.6119	−1.32893
$760$	0	0
$761$	7.41641	0.268845	0.134422	−	0.990924i	$-0.457082\pi$
$761$	7.41641	0.268845	0.134422	+	0.990924i	$0.457082\pi$
$762$	0	0
$763$	29.5279i	1.06898i
$764$	0	0
$765$	7.81758i	0.282645i
$766$	0	0
$767$	−12.6491	−0.456733
$768$	0	0
$769$	−30.8328	−1.11186	−0.555930	−	0.831229i	$-0.687638\pi$
$769$	−30.8328	−1.11186	−0.555930	+	0.831229i	$0.687638\pi$
$770$	0	0
$771$	− 14.0000i	− 0.504198i
$772$	0	0
$773$	6.65841i	0.239486i	0.992805	+	0.119743i	$0.0382071\pi$
$773$	6.65841i	0.239486i	−0.992805	+	0.119743i	$0.961793\pi$
$774$	0	0
$775$	19.5440	0.702039
$776$	0	0
$777$	18.4721	0.662684
$778$	0	0
$779$	− 16.0000i	− 0.573259i
$780$	0	0
$781$	− 59.2393i	− 2.11975i
$782$	0	0
$783$	5.99070	0.214090
$784$	0	0
$785$	−42.3607	−1.51192
$786$	0	0
$787$	− 14.4721i	− 0.515876i	−0.966161	−	0.257938i	$-0.916957\pi$
$787$	− 14.4721i	− 0.515876i	0.966161	−	0.257938i	$-0.0830430\pi$
$788$	0	0
$789$	− 12.6491i	− 0.450320i
$790$	0	0
$791$	3.49613	0.124308
$792$	0	0
$793$	−2.94427	−0.104554
$794$	0	0
$795$	36.8328i	1.30633i
$796$	0	0
$797$	− 5.16538i	− 0.182967i	−0.995807	−	0.0914836i	$-0.970839\pi$
$797$	− 5.16538i	− 0.182967i	0.995807	−	0.0914836i	$-0.0291609\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−10.0000	−0.353333
$802$	0	0
$803$	19.0557i	0.672462i
$804$	0	0
$805$	− 31.2703i	− 1.10213i
$806$	0	0
$807$	−5.32300	−0.187379
$808$	0	0
$809$	23.4164	0.823277	0.411639	−	0.911347i	$-0.364957\pi$
$809$	23.4164	0.823277	0.411639	+	0.911347i	$0.364957\pi$
$810$	0	0
$811$	− 29.3050i	− 1.02904i	−0.857480	−	0.514518i	$-0.827971\pi$
$811$	− 29.3050i	− 1.02904i	0.857480	−	0.514518i	$-0.172029\pi$
$812$	0	0
$813$	− 10.9010i	− 0.382316i
$814$	0	0
$815$	45.7649	1.60307
$816$	0	0
$817$	−9.88854	−0.345956
$818$	0	0
$819$	− 2.47214i	− 0.0863834i
$820$	0	0
$821$	− 50.2626i	− 1.75418i	−0.480329	−	0.877088i	$-0.659483\pi$
$821$	− 50.2626i	− 1.75418i	0.480329	−	0.877088i	$-0.340517\pi$
$822$	0	0
$823$	1.74806	0.0609337	0.0304668	−	0.999536i	$-0.490301\pi$
$823$	1.74806	0.0609337	0.0304668	+	0.999536i	$0.490301\pi$
$824$	0	0
$825$	−32.3607	−1.12665
$826$	0	0
$827$	− 18.1115i	− 0.629797i	−0.949125	−	0.314899i	$-0.898030\pi$
$827$	− 18.1115i	− 0.629797i	0.949125	−	0.314899i	$-0.101970\pi$
$828$	0	0
$829$	− 5.57804i	− 0.193733i	−0.995297	−	0.0968667i	$-0.969118\pi$
$829$	− 5.57804i	− 0.193733i	0.995297	−	0.0968667i	$-0.0308821\pi$
$830$	0	0
$831$	18.3848	0.637761
$832$	0	0
$833$	−9.75078	−0.337844
$834$	0	0
$835$	64.7214i	2.23978i
$836$	0	0
$837$	3.90879i	0.135108i
$838$	0	0
$839$	6.48218	0.223790	0.111895	−	0.993720i	$-0.464308\pi$
$839$	6.48218	0.223790	0.111895	+	0.993720i	$0.464308\pi$
$840$	0	0
$841$	−6.88854	−0.237536
$842$	0	0
$843$	− 3.88854i	− 0.133929i
$844$	0	0
$845$	− 34.7851i	− 1.19664i
$846$	0	0
$847$	53.9952	1.85530
$848$	0	0
$849$	26.8328	0.920900
$850$	0	0
$851$	− 59.7771i	− 2.04913i
$852$	0	0
$853$	30.3662i	1.03972i	0.854252	+	0.519859i	$0.174016\pi$
$853$	30.3662i	1.03972i	−0.854252	+	0.519859i	$0.825984\pi$
$854$	0	0
$855$	20.4667	0.699946
$856$	0	0
$857$	−55.4164	−1.89299	−0.946494	−	0.322721i	$-0.895403\pi$
$857$	−55.4164	−1.89299	−0.946494	+	0.322721i	$0.895403\pi$
$858$	0	0
$859$	54.4721i	1.85857i	0.369369	+	0.929283i	$0.379574\pi$
$859$	54.4721i	1.85857i	−0.369369	+	0.929283i	$0.620426\pi$
$860$	0	0
$861$	− 4.32145i	− 0.147275i
$862$	0	0
$863$	−28.7943	−0.980171	−0.490086	−	0.871674i	$-0.663034\pi$
$863$	−28.7943	−0.980171	−0.490086	+	0.871674i	$0.663034\pi$
$864$	0	0
$865$	30.0000	1.02003
$866$	0	0
$867$	− 10.8885i	− 0.369794i
$868$	0	0
$869$	47.9256i	1.62577i
$870$	0	0
$871$	16.9706	0.575026
$872$	0	0
$873$	12.9443	0.438097
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 27.3801i	− 0.924561i	−0.886734	−	0.462281i	$-0.847031\pi$
$877$	− 27.3801i	− 0.924561i	0.886734	−	0.462281i	$-0.152969\pi$
$878$	0	0
$879$	−22.9613	−0.774464
$880$	0	0
$881$	−7.88854	−0.265772	−0.132886	−	0.991131i	$-0.542424\pi$
$881$	−7.88854	−0.265772	−0.132886	+	0.991131i	$0.542424\pi$
$882$	0	0
$883$	9.52786i	0.320638i	0.987065	+	0.160319i	$0.0512523\pi$
$883$	9.52786i	0.320638i	−0.987065	+	0.160319i	$0.948748\pi$
$884$	0	0
$885$	− 28.2843i	− 0.950765i
$886$	0	0
$887$	8.32766	0.279615	0.139808	−	0.990179i	$-0.455352\pi$
$887$	8.32766	0.279615	0.139808	+	0.990179i	$0.455352\pi$
$888$	0	0
$889$	19.0557	0.639109
$890$	0	0
$891$	− 6.47214i	− 0.216825i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	37.9473	1.26844
$896$	0	0
$897$	−8.00000	−0.267112
$898$	0	0
$899$	− 23.4164i	− 0.780981i
$900$	0	0
$901$	28.7943i	0.959279i
$902$	0	0
$903$	−2.67080	−0.0888788
$904$	0	0
$905$	−49.1935	−1.63525
$906$	0	0
$907$	− 47.1935i	− 1.56703i	−0.621370	−	0.783517i	$-0.713424\pi$
$907$	− 47.1935i	− 1.56703i	0.621370	−	0.783517i	$-0.286576\pi$
$908$	0	0
$909$	0.333851i	0.0110731i
$910$	0	0
$911$	14.8098	0.490672	0.245336	−	0.969438i	$-0.421102\pi$
$911$	14.8098	0.490672	0.245336	+	0.969438i	$0.421102\pi$
$912$	0	0
$913$	−41.8885	−1.38631
$914$	0	0
$915$	− 6.58359i	− 0.217647i
$916$	0	0
$917$	− 29.6197i	− 0.978128i
$918$	0	0
$919$	−2.57339	−0.0848882	−0.0424441	−	0.999099i	$-0.513514\pi$
$919$	−2.57339	−0.0848882	−0.0424441	+	0.999099i	$0.513514\pi$
$920$	0	0
$921$	5.88854	0.194034
$922$	0	0
$923$	− 12.9443i	− 0.426066i
$924$	0	0
$925$	− 52.8360i	− 1.73724i
$926$	0	0
$927$	−1.74806	−0.0574140
$928$	0	0
$929$	−2.47214	−0.0811081	−0.0405541	−	0.999177i	$-0.512912\pi$
$929$	−2.47214	−0.0811081	−0.0405541	+	0.999177i	$0.512912\pi$
$930$	0	0
$931$	25.5279i	0.836642i
$932$	0	0
$933$	− 19.6414i	− 0.643029i
$934$	0	0
$935$	−50.5964	−1.65468
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	− 20.9443i	− 0.683490i
$940$	0	0
$941$	31.4465i	1.02513i	0.858649	+	0.512564i	$0.171304\pi$
$941$	31.4465i	1.02513i	−0.858649	+	0.512564i	$0.828696\pi$
$942$	0	0
$943$	−13.9845	−0.455398
$944$	0	0
$945$	5.52786	0.179821
$946$	0	0
$947$	23.7771i	0.772652i	0.922362	+	0.386326i	$0.126256\pi$
$947$	23.7771i	0.772652i	−0.922362	+	0.386326i	$0.873744\pi$
$948$	0	0
$949$	4.16383i	0.135164i
$950$	0	0
$951$	−17.3044	−0.561134
$952$	0	0
$953$	18.4721	0.598371	0.299186	−	0.954195i	$-0.403285\pi$
$953$	18.4721	0.598371	0.299186	+	0.954195i	$0.403285\pi$
$954$	0	0
$955$	− 46.8328i	− 1.51547i
$956$	0	0
$957$	38.7727i	1.25334i
$958$	0	0
$959$	4.32145	0.139547
$960$	0	0
$961$	−15.7214	−0.507141
$962$	0	0
$963$	− 0.944272i	− 0.0304287i
$964$	0	0
$965$	− 15.6352i	− 0.503314i
$966$	0	0
$967$	−38.3600	−1.23357	−0.616787	−	0.787130i	$-0.711566\pi$
$967$	−38.3600	−1.23357	−0.616787	+	0.787130i	$0.711566\pi$
$968$	0	0
$969$	16.0000	0.513994
$970$	0	0
$971$	− 24.3607i	− 0.781771i	−0.920439	−	0.390886i	$-0.872169\pi$
$971$	− 24.3607i	− 0.781771i	0.920439	−	0.390886i	$-0.127831\pi$
$972$	0	0
$973$	29.6197i	0.949563i
$974$	0	0
$975$	−7.07107	−0.226455
$976$	0	0
$977$	39.4164	1.26104	0.630521	−	0.776172i	$-0.282841\pi$
$977$	39.4164	1.26104	0.630521	+	0.776172i	$0.282841\pi$
$978$	0	0
$979$	− 64.7214i	− 2.06850i
$980$	0	0
$981$	16.8918i	0.539312i
$982$	0	0
$983$	2.98605	0.0952402	0.0476201	−	0.998866i	$-0.484836\pi$
$983$	2.98605	0.0952402	0.0476201	+	0.998866i	$0.484836\pi$
$984$	0	0
$985$	23.1672	0.738168
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.64290i	0.274828i
$990$	0	0
$991$	14.3972	0.457341	0.228671	−	0.973504i	$-0.426562\pi$
$991$	14.3972	0.457341	0.228671	+	0.973504i	$0.426562\pi$
$992$	0	0
$993$	−16.9443	−0.537710
$994$	0	0
$995$	− 16.5836i	− 0.525735i
$996$	0	0
$997$	− 41.6799i	− 1.32002i	−0.751259	−	0.660008i	$-0.770553\pi$
$997$	− 41.6799i	− 1.32002i	0.751259	−	0.660008i	$-0.229447\pi$
$998$	0	0
$999$	10.5672	0.334331

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3072.2.d.g.1537.6		8
4.3	odd	2	inner	3072.2.d.g.1537.1		8
8.3	odd	2	inner	3072.2.d.g.1537.7		8
8.5	even	2	inner	3072.2.d.g.1537.4		8
16.3	odd	4		3072.2.a.q.1.2		4
16.5	even	4		3072.2.a.q.1.3		4
16.11	odd	4		3072.2.a.k.1.4		4
16.13	even	4		3072.2.a.k.1.1		4
32.3	odd	8		1536.2.j.g.385.4	yes	8
32.5	even	8		1536.2.j.g.1153.2	yes	8
32.11	odd	8		1536.2.j.h.1153.1	yes	8
32.13	even	8		1536.2.j.h.385.3	yes	8
32.19	odd	8		1536.2.j.h.385.1	yes	8
32.21	even	8		1536.2.j.h.1153.3	yes	8
32.27	odd	8		1536.2.j.g.1153.4	yes	8
32.29	even	8		1536.2.j.g.385.2	✓	8
48.5	odd	4		9216.2.a.bd.1.1		4
48.11	even	4		9216.2.a.bj.1.2		4
48.29	odd	4		9216.2.a.bj.1.3		4
48.35	even	4		9216.2.a.bd.1.4		4
96.5	odd	8		4608.2.k.bf.1153.1		8
96.11	even	8		4608.2.k.bg.1153.4		8
96.29	odd	8		4608.2.k.bf.3457.2		8
96.35	even	8		4608.2.k.bf.3457.1		8
96.53	odd	8		4608.2.k.bg.1153.3		8
96.59	even	8		4608.2.k.bf.1153.2		8
96.77	odd	8		4608.2.k.bg.3457.4		8
96.83	even	8		4608.2.k.bg.3457.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.j.g.385.2	✓	8	32.29	even	8
1536.2.j.g.385.4	yes	8	32.3	odd	8
1536.2.j.g.1153.2	yes	8	32.5	even	8
1536.2.j.g.1153.4	yes	8	32.27	odd	8
1536.2.j.h.385.1	yes	8	32.19	odd	8
1536.2.j.h.385.3	yes	8	32.13	even	8
1536.2.j.h.1153.1	yes	8	32.11	odd	8
1536.2.j.h.1153.3	yes	8	32.21	even	8
3072.2.a.k.1.1		4	16.13	even	4
3072.2.a.k.1.4		4	16.11	odd	4
3072.2.a.q.1.2		4	16.3	odd	4
3072.2.a.q.1.3		4	16.5	even	4
3072.2.d.g.1537.1		8	4.3	odd	2	inner
3072.2.d.g.1537.4		8	8.5	even	2	inner
3072.2.d.g.1537.6		8	1.1	even	1	trivial
3072.2.d.g.1537.7		8	8.3	odd	2	inner
4608.2.k.bf.1153.1		8	96.5	odd	8
4608.2.k.bf.1153.2		8	96.59	even	8
4608.2.k.bf.3457.1		8	96.35	even	8
4608.2.k.bf.3457.2		8	96.29	odd	8
4608.2.k.bg.1153.3		8	96.53	odd	8
4608.2.k.bg.1153.4		8	96.11	even	8
4608.2.k.bg.3457.3		8	96.83	even	8
4608.2.k.bg.3457.4		8	96.77	odd	8
9216.2.a.bd.1.1		4	48.5	odd	4
9216.2.a.bd.1.4		4	48.35	even	4
9216.2.a.bj.1.2		4	48.11	even	4
9216.2.a.bj.1.3		4	48.29	odd	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -3.16228i q^{5} -1.74806 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -3.16228i q^{5} -1.74806 q^{7} -1.00000 q^{9} -6.47214i q^{11} -1.41421i q^{13} +3.16228 q^{15} +2.47214 q^{17} -6.47214i q^{19} -1.74806i q^{21} -5.65685 q^{23} -5.00000 q^{25} -1.00000i q^{27} +5.99070i q^{29} -3.90879 q^{31} +6.47214 q^{33} +5.52786i q^{35} +10.5672i q^{37} +1.41421 q^{39} +2.47214 q^{41} -1.52786i q^{43} +3.16228i q^{45} -3.94427 q^{49} +2.47214i q^{51} +11.6476i q^{53} -20.4667 q^{55} +6.47214 q^{57} -8.94427i q^{59} -2.08191i q^{61} +1.74806 q^{63} -4.47214 q^{65} +12.0000i q^{67} -5.65685i q^{69} +9.15298 q^{71} -2.94427 q^{73} -5.00000i q^{75} +11.3137i q^{77} -7.40492 q^{79} +1.00000 q^{81} -6.47214i q^{83} -7.81758i q^{85} -5.99070 q^{87} +10.0000 q^{89} +2.47214i q^{91} -3.90879i q^{93} -20.4667 q^{95} -12.9443 q^{97} +6.47214i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} - 16 q^{17} - 40 q^{25} + 16 q^{33} - 16 q^{41} + 40 q^{49} + 16 q^{57} + 48 q^{73} + 8 q^{81} + 80 q^{89} - 32 q^{97}+O(q^{100}) \) 8 * q - 8 * q^9 - 16 * q^17 - 40 * q^25 + 16 * q^33 - 16 * q^41 + 40 * q^49 + 16 * q^57 + 48 * q^73 + 8 * q^81 + 80 * q^89 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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