LMFDB - Embedded newform 3072.2.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3072,2,Mod(1,3072)]

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, 0, 0]))

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

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

chi := DirichletCharacter("3072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3072 = 2^{10} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3072.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$24.5300435009$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1536)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.28825$ of defining polynomial
Character	$\chi$	$=$	3072.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.16228 q^{5} +4.57649 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.16228 q^{5} +4.57649 q^{7} +1.00000 q^{9} +2.47214 q^{11} -1.41421 q^{13} +3.16228 q^{15} -6.47214 q^{17} -2.47214 q^{19} -4.57649 q^{21} -5.65685 q^{23} +5.00000 q^{25} -1.00000 q^{27} -0.333851 q^{29} +10.2333 q^{31} -2.47214 q^{33} -14.4721 q^{35} +2.08191 q^{37} +1.41421 q^{39} +6.47214 q^{41} -10.4721 q^{43} -3.16228 q^{45} +13.9443 q^{49} +6.47214 q^{51} -5.32300 q^{53} -7.81758 q^{55} +2.47214 q^{57} +8.94427 q^{59} +10.5672 q^{61} +4.57649 q^{63} +4.47214 q^{65} -12.0000 q^{67} +5.65685 q^{69} -3.49613 q^{71} -14.9443 q^{73} -5.00000 q^{75} +11.3137 q^{77} +1.08036 q^{79} +1.00000 q^{81} -2.47214 q^{83} +20.4667 q^{85} +0.333851 q^{87} -10.0000 q^{89} -6.47214 q^{91} -10.2333 q^{93} +7.81758 q^{95} +4.94427 q^{97} +2.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{9} - 8 q^{11} - 8 q^{17} + 8 q^{19} + 20 q^{25} - 4 q^{27} + 8 q^{33} - 40 q^{35} + 8 q^{41} - 24 q^{43} + 20 q^{49} + 8 q^{51} - 8 q^{57} - 48 q^{67} - 24 q^{73} - 20 q^{75} + 4 q^{81} + 8 q^{83} - 40 q^{89} - 8 q^{91} - 16 q^{97} - 8 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^9 - 8 * q^11 - 8 * q^17 + 8 * q^19 + 20 * q^25 - 4 * q^27 + 8 * q^33 - 40 * q^35 + 8 * q^41 - 24 * q^43 + 20 * q^49 + 8 * q^51 - 8 * q^57 - 48 * q^67 - 24 * q^73 - 20 * q^75 + 4 * q^81 + 8 * q^83 - 40 * q^89 - 8 * q^91 - 16 * q^97 - 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−3.16228	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$5$	−3.16228	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$6$	0	0
$7$	4.57649	1.72975	0.864876	−	0.501986i	$-0.167397\pi$
$7$	4.57649	1.72975	0.864876	+	0.501986i	$0.167397\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.47214	0.745377	0.372689	−	0.927957i	$-0.378436\pi$
$11$	2.47214	0.745377	0.372689	+	0.927957i	$0.378436\pi$
$12$	0	0
$13$	−1.41421	−0.392232	−0.196116	−	0.980581i	$-0.562833\pi$
$13$	−1.41421	−0.392232	−0.196116	+	0.980581i	$0.562833\pi$
$14$	0	0
$15$	3.16228	0.816497
$16$	0	0
$17$	−6.47214	−1.56972	−0.784862	−	0.619671i	$-0.787266\pi$
$17$	−6.47214	−1.56972	−0.784862	+	0.619671i	$0.787266\pi$
$18$	0	0
$19$	−2.47214	−0.567147	−0.283573	−	0.958951i	$-0.591520\pi$
$19$	−2.47214	−0.567147	−0.283573	+	0.958951i	$0.591520\pi$
$20$	0	0
$21$	−4.57649	−0.998672
$22$	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
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.333851	−0.0619945	−0.0309972	−	0.999519i	$-0.509868\pi$
$29$	−0.333851	−0.0619945	−0.0309972	+	0.999519i	$0.509868\pi$
$30$	0	0
$31$	10.2333	1.83796	0.918982	−	0.394301i	$-0.129013\pi$
$31$	10.2333	1.83796	0.918982	+	0.394301i	$0.129013\pi$
$32$	0	0
$33$	−2.47214	−0.430344
$34$	0	0
$35$	−14.4721	−2.44624
$36$	0	0
$37$	2.08191	0.342265	0.171132	−	0.985248i	$-0.445257\pi$
$37$	2.08191	0.342265	0.171132	+	0.985248i	$0.445257\pi$
$38$	0	0
$39$	1.41421	0.226455
$40$	0	0
$41$	6.47214	1.01078	0.505389	−	0.862892i	$-0.331349\pi$
$41$	6.47214	1.01078	0.505389	+	0.862892i	$0.331349\pi$
$42$	0	0
$43$	−10.4721	−1.59699	−0.798493	−	0.602004i	$-0.794369\pi$
$43$	−10.4721	−1.59699	−0.798493	+	0.602004i	$0.794369\pi$
$44$	0	0
$45$	−3.16228	−0.471405
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	13.9443	1.99204
$50$	0	0
$51$	6.47214	0.906280
$52$	0	0
$53$	−5.32300	−0.731171	−0.365585	−	0.930778i	$-0.619131\pi$
$53$	−5.32300	−0.731171	−0.365585	+	0.930778i	$0.619131\pi$
$54$	0	0
$55$	−7.81758	−1.05412
$56$	0	0
$57$	2.47214	0.327442
$58$	0	0
$59$	8.94427	1.16445	0.582223	−	0.813029i	$-0.302183\pi$
$59$	8.94427	1.16445	0.582223	+	0.813029i	$0.302183\pi$
$60$	0	0
$61$	10.5672	1.35299	0.676495	−	0.736447i	$-0.263498\pi$
$61$	10.5672	1.35299	0.676495	+	0.736447i	$0.263498\pi$
$62$	0	0
$63$	4.57649	0.576584
$64$	0	0
$65$	4.47214	0.554700
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	5.65685	0.681005
$70$	0	0
$71$	−3.49613	−0.414914	−0.207457	−	0.978244i	$-0.566519\pi$
$71$	−3.49613	−0.414914	−0.207457	+	0.978244i	$0.566519\pi$
$72$	0	0
$73$	−14.9443	−1.74909	−0.874547	−	0.484940i	$-0.838841\pi$
$73$	−14.9443	−1.74909	−0.874547	+	0.484940i	$0.838841\pi$
$74$	0	0
$75$	−5.00000	−0.577350
$76$	0	0
$77$	11.3137	1.28932
$78$	0	0
$79$	1.08036	0.121550	0.0607752	−	0.998151i	$-0.480643\pi$
$79$	1.08036	0.121550	0.0607752	+	0.998151i	$0.480643\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.47214	−0.271352	−0.135676	−	0.990753i	$-0.543321\pi$
$83$	−2.47214	−0.271352	−0.135676	+	0.990753i	$0.543321\pi$
$84$	0	0
$85$	20.4667	2.21992
$86$	0	0
$87$	0.333851	0.0357925
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−6.47214	−0.678464
$92$	0	0
$93$	−10.2333	−1.06115
$94$	0	0
$95$	7.81758	0.802067
$96$	0	0
$97$	4.94427	0.502015	0.251007	−	0.967985i	$-0.419238\pi$
$97$	4.94427	0.502015	0.251007	+	0.967985i	$0.419238\pi$
$98$	0	0
$99$	2.47214	0.248459
$100$	0	0
$101$	−5.99070	−0.596097	−0.298049	−	0.954551i	$-0.596336\pi$
$101$	−5.99070	−0.596097	−0.298049	+	0.954551i	$0.596336\pi$
$102$	0	0
$103$	−4.57649	−0.450935	−0.225468	−	0.974251i	$-0.572391\pi$
$103$	−4.57649	−0.450935	−0.225468	+	0.974251i	$0.572391\pi$
$104$	0	0
$105$	14.4721	1.41234
$106$	0	0
$107$	−16.9443	−1.63806	−0.819032	−	0.573747i	$-0.805489\pi$
$107$	−16.9443	−1.63806	−0.819032	+	0.573747i	$0.805489\pi$
$108$	0	0
$109$	8.40647	0.805194	0.402597	−	0.915377i	$-0.368108\pi$
$109$	8.40647	0.805194	0.402597	+	0.915377i	$0.368108\pi$
$110$	0	0
$111$	−2.08191	−0.197607
$112$	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
$115$	17.8885	1.66812
$116$	0	0
$117$	−1.41421	−0.130744
$118$	0	0
$119$	−29.6197	−2.71523
$120$	0	0
$121$	−4.88854	−0.444413
$122$	0	0
$123$	−6.47214	−0.583573
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−8.07262	−0.716329	−0.358165	−	0.933658i	$-0.616597\pi$
$127$	−8.07262	−0.716329	−0.358165	+	0.933658i	$0.616597\pi$
$128$	0	0
$129$	10.4721	0.922020
$130$	0	0
$131$	0.944272	0.0825014	0.0412507	−	0.999149i	$-0.486866\pi$
$131$	0.944272	0.0825014	0.0412507	+	0.999149i	$0.486866\pi$
$132$	0	0
$133$	−11.3137	−0.981023
$134$	0	0
$135$	3.16228	0.272166
$136$	0	0
$137$	−6.47214	−0.552952	−0.276476	−	0.961021i	$-0.589167\pi$
$137$	−6.47214	−0.552952	−0.276476	+	0.961021i	$0.589167\pi$
$138$	0	0
$139$	0.944272	0.0800921	0.0400460	−	0.999198i	$-0.487250\pi$
$139$	0.944272	0.0800921	0.0400460	+	0.999198i	$0.487250\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.49613	−0.292361
$144$	0	0
$145$	1.05573	0.0876734
$146$	0	0
$147$	−13.9443	−1.15010
$148$	0	0
$149$	−15.1437	−1.24062	−0.620310	−	0.784357i	$-0.712993\pi$
$149$	−15.1437	−1.24062	−0.620310	+	0.784357i	$0.712993\pi$
$150$	0	0
$151$	−11.5687	−0.941451	−0.470726	−	0.882280i	$-0.656008\pi$
$151$	−11.5687	−0.941451	−0.470726	+	0.882280i	$0.656008\pi$
$152$	0	0
$153$	−6.47214	−0.523241
$154$	0	0
$155$	−32.3607	−2.59927
$156$	0	0
$157$	−0.746512	−0.0595782	−0.0297891	−	0.999556i	$-0.509484\pi$
$157$	−0.746512	−0.0595782	−0.0297891	+	0.999556i	$0.509484\pi$
$158$	0	0
$159$	5.32300	0.422142
$160$	0	0
$161$	−25.8885	−2.04030
$162$	0	0
$163$	−5.52786	−0.432976	−0.216488	−	0.976285i	$-0.569460\pi$
$163$	−5.52786	−0.432976	−0.216488	+	0.976285i	$0.569460\pi$
$164$	0	0
$165$	7.81758	0.608598
$166$	0	0
$167$	−7.81758	−0.604943	−0.302471	−	0.953159i	$-0.597812\pi$
$167$	−7.81758	−0.604943	−0.302471	+	0.953159i	$0.597812\pi$
$168$	0	0
$169$	−11.0000	−0.846154
$170$	0	0
$171$	−2.47214	−0.189049
$172$	0	0
$173$	−9.48683	−0.721271	−0.360635	−	0.932707i	$-0.617440\pi$
$173$	−9.48683	−0.721271	−0.360635	+	0.932707i	$0.617440\pi$
$174$	0	0
$175$	22.8825	1.72975
$176$	0	0
$177$	−8.94427	−0.672293
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	15.5563	1.15629	0.578147	−	0.815933i	$-0.303776\pi$
$181$	15.5563	1.15629	0.578147	+	0.815933i	$0.303776\pi$
$182$	0	0
$183$	−10.5672	−0.781150
$184$	0	0
$185$	−6.58359	−0.484035
$186$	0	0
$187$	−16.0000	−1.17004
$188$	0	0
$189$	−4.57649	−0.332891
$190$	0	0
$191$	−2.16073	−0.156345	−0.0781723	−	0.996940i	$-0.524908\pi$
$191$	−2.16073	−0.156345	−0.0781723	+	0.996940i	$0.524908\pi$
$192$	0	0
$193$	−12.9443	−0.931749	−0.465875	−	0.884851i	$-0.654260\pi$
$193$	−12.9443	−0.931749	−0.465875	+	0.884851i	$0.654260\pi$
$194$	0	0
$195$	−4.47214	−0.320256
$196$	0	0
$197$	24.2967	1.73107	0.865533	−	0.500852i	$-0.166980\pi$
$197$	24.2967	1.73107	0.865533	+	0.500852i	$0.166980\pi$
$198$	0	0
$199$	−13.7295	−0.973257	−0.486628	−	0.873609i	$-0.661773\pi$
$199$	−13.7295	−0.973257	−0.486628	+	0.873609i	$0.661773\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	0	0
$203$	−1.52786	−0.107235
$204$	0	0
$205$	−20.4667	−1.42946
$206$	0	0
$207$	−5.65685	−0.393179
$208$	0	0
$209$	−6.11146	−0.422738
$210$	0	0
$211$	−24.9443	−1.71723	−0.858617	−	0.512617i	$-0.828676\pi$
$211$	−24.9443	−1.71723	−0.858617	+	0.512617i	$0.828676\pi$
$212$	0	0
$213$	3.49613	0.239551
$214$	0	0
$215$	33.1158	2.25848
$216$	0	0
$217$	46.8328	3.17922
$218$	0	0
$219$	14.9443	1.00984
$220$	0	0
$221$	9.15298	0.615696
$222$	0	0
$223$	10.2333	0.685275	0.342638	−	0.939468i	$-0.388680\pi$
$223$	10.2333	0.685275	0.342638	+	0.939468i	$0.388680\pi$
$224$	0	0
$225$	5.00000	0.333333
$226$	0	0
$227$	5.52786	0.366897	0.183449	−	0.983029i	$-0.441274\pi$
$227$	5.52786	0.366897	0.183449	+	0.983029i	$0.441274\pi$
$228$	0	0
$229$	−11.2349	−0.742423	−0.371211	−	0.928548i	$-0.621057\pi$
$229$	−11.2349	−0.742423	−0.371211	+	0.928548i	$0.621057\pi$
$230$	0	0
$231$	−11.3137	−0.744387
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−1.08036	−0.0701771
$238$	0	0
$239$	13.4744	0.871589	0.435794	−	0.900046i	$-0.356467\pi$
$239$	13.4744	0.871589	0.435794	+	0.900046i	$0.356467\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−44.0957	−2.81717
$246$	0	0
$247$	3.49613	0.222453
$248$	0	0
$249$	2.47214	0.156665
$250$	0	0
$251$	15.4164	0.973075	0.486538	−	0.873660i	$-0.338260\pi$
$251$	15.4164	0.973075	0.486538	+	0.873660i	$0.338260\pi$
$252$	0	0
$253$	−13.9845	−0.879199
$254$	0	0
$255$	−20.4667	−1.28167
$256$	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
$259$	9.52786	0.592033
$260$	0	0
$261$	−0.333851	−0.0206648
$262$	0	0
$263$	12.6491	0.779978	0.389989	−	0.920820i	$-0.372479\pi$
$263$	12.6491	0.779978	0.389989	+	0.920820i	$0.372479\pi$
$264$	0	0
$265$	16.8328	1.03403
$266$	0	0
$267$	10.0000	0.611990
$268$	0	0
$269$	11.6476	0.710164	0.355082	−	0.934835i	$-0.384453\pi$
$269$	11.6476	0.710164	0.355082	+	0.934835i	$0.384453\pi$
$270$	0	0
$271$	−8.07262	−0.490377	−0.245188	−	0.969475i	$-0.578850\pi$
$271$	−8.07262	−0.490377	−0.245188	+	0.969475i	$0.578850\pi$
$272$	0	0
$273$	6.47214	0.391711
$274$	0	0
$275$	12.3607	0.745377
$276$	0	0
$277$	18.3848	1.10463	0.552317	−	0.833634i	$-0.313744\pi$
$277$	18.3848	1.10463	0.552317	+	0.833634i	$0.313744\pi$
$278$	0	0
$279$	10.2333	0.612654
$280$	0	0
$281$	−31.8885	−1.90231	−0.951156	−	0.308712i	$-0.900102\pi$
$281$	−31.8885	−1.90231	−0.951156	+	0.308712i	$0.900102\pi$
$282$	0	0
$283$	26.8328	1.59505	0.797523	−	0.603289i	$-0.206143\pi$
$283$	26.8328	1.59505	0.797523	+	0.603289i	$0.206143\pi$
$284$	0	0
$285$	−7.81758	−0.463073
$286$	0	0
$287$	29.6197	1.74839
$288$	0	0
$289$	24.8885	1.46403
$290$	0	0
$291$	−4.94427	−0.289838
$292$	0	0
$293$	−16.6367	−0.971927	−0.485964	−	0.873979i	$-0.661531\pi$
$293$	−16.6367	−0.971927	−0.485964	+	0.873979i	$0.661531\pi$
$294$	0	0
$295$	−28.2843	−1.64677
$296$	0	0
$297$	−2.47214	−0.143448
$298$	0	0
$299$	8.00000	0.462652
$300$	0	0
$301$	−47.9256	−2.76239
$302$	0	0
$303$	5.99070	0.344157
$304$	0	0
$305$	−33.4164	−1.91342
$306$	0	0
$307$	−29.8885	−1.70583	−0.852915	−	0.522050i	$-0.825167\pi$
$307$	−29.8885	−1.70583	−0.852915	+	0.522050i	$0.825167\pi$
$308$	0	0
$309$	4.57649	0.260347
$310$	0	0
$311$	30.9551	1.75530	0.877651	−	0.479301i	$-0.159110\pi$
$311$	30.9551	1.75530	0.877651	+	0.479301i	$0.159110\pi$
$312$	0	0
$313$	3.05573	0.172720	0.0863600	−	0.996264i	$-0.472476\pi$
$313$	3.05573	0.172720	0.0863600	+	0.996264i	$0.472476\pi$
$314$	0	0
$315$	−14.4721	−0.815412
$316$	0	0
$317$	10.9799	0.616690	0.308345	−	0.951275i	$-0.400225\pi$
$317$	10.9799	0.616690	0.308345	+	0.951275i	$0.400225\pi$
$318$	0	0
$319$	−0.825324	−0.0462093
$320$	0	0
$321$	16.9443	0.945737
$322$	0	0
$323$	16.0000	0.890264
$324$	0	0
$325$	−7.07107	−0.392232
$326$	0	0
$327$	−8.40647	−0.464879
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−0.944272	−0.0519019	−0.0259509	−	0.999663i	$-0.508261\pi$
$331$	−0.944272	−0.0519019	−0.0259509	+	0.999663i	$0.508261\pi$
$332$	0	0
$333$	2.08191	0.114088
$334$	0	0
$335$	37.9473	2.07328
$336$	0	0
$337$	−2.94427	−0.160385	−0.0801924	−	0.996779i	$-0.525553\pi$
$337$	−2.94427	−0.160385	−0.0801924	+	0.996779i	$0.525553\pi$
$338$	0	0
$339$	2.00000	0.108625
$340$	0	0
$341$	25.2982	1.36998
$342$	0	0
$343$	31.7804	1.71598
$344$	0	0
$345$	−17.8885	−0.963087
$346$	0	0
$347$	−13.5279	−0.726214	−0.363107	−	0.931747i	$-0.618284\pi$
$347$	−13.5279	−0.726214	−0.363107	+	0.931747i	$0.618284\pi$
$348$	0	0
$349$	−21.8809	−1.17126	−0.585629	−	0.810579i	$-0.699152\pi$
$349$	−21.8809	−1.17126	−0.585629	+	0.810579i	$0.699152\pi$
$350$	0	0
$351$	1.41421	0.0754851
$352$	0	0
$353$	23.8885	1.27146	0.635729	−	0.771912i	$-0.280699\pi$
$353$	23.8885	1.27146	0.635729	+	0.771912i	$0.280699\pi$
$354$	0	0
$355$	11.0557	0.586777
$356$	0	0
$357$	29.6197	1.56764
$358$	0	0
$359$	−26.1235	−1.37875	−0.689374	−	0.724406i	$-0.742114\pi$
$359$	−26.1235	−1.37875	−0.689374	+	0.724406i	$0.742114\pi$
$360$	0	0
$361$	−12.8885	−0.678344
$362$	0	0
$363$	4.88854	0.256582
$364$	0	0
$365$	47.2579	2.47359
$366$	0	0
$367$	−26.3786	−1.37695	−0.688475	−	0.725260i	$-0.741720\pi$
$367$	−26.3786	−1.37695	−0.688475	+	0.725260i	$0.741720\pi$
$368$	0	0
$369$	6.47214	0.336926
$370$	0	0
$371$	−24.3607	−1.26474
$372$	0	0
$373$	9.23179	0.478004	0.239002	−	0.971019i	$-0.423180\pi$
$373$	9.23179	0.478004	0.239002	+	0.971019i	$0.423180\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.472136	0.0243162
$378$	0	0
$379$	0.583592	0.0299771	0.0149886	−	0.999888i	$-0.495229\pi$
$379$	0.583592	0.0299771	0.0149886	+	0.999888i	$0.495229\pi$
$380$	0	0
$381$	8.07262	0.413573
$382$	0	0
$383$	−31.7804	−1.62390	−0.811951	−	0.583725i	$-0.801595\pi$
$383$	−31.7804	−1.62390	−0.811951	+	0.583725i	$0.801595\pi$
$384$	0	0
$385$	−35.7771	−1.82337
$386$	0	0
$387$	−10.4721	−0.532329
$388$	0	0
$389$	−10.8222	−0.548709	−0.274355	−	0.961629i	$-0.588464\pi$
$389$	−10.8222	−0.548709	−0.274355	+	0.961629i	$0.588464\pi$
$390$	0	0
$391$	36.6119	1.85154
$392$	0	0
$393$	−0.944272	−0.0476322
$394$	0	0
$395$	−3.41641	−0.171898
$396$	0	0
$397$	28.8732	1.44910	0.724551	−	0.689221i	$-0.242047\pi$
$397$	28.8732	1.44910	0.724551	+	0.689221i	$0.242047\pi$
$398$	0	0
$399$	11.3137	0.566394
$400$	0	0
$401$	22.4721	1.12220	0.561102	−	0.827746i	$-0.310377\pi$
$401$	22.4721	1.12220	0.561102	+	0.827746i	$0.310377\pi$
$402$	0	0
$403$	−14.4721	−0.720908
$404$	0	0
$405$	−3.16228	−0.157135
$406$	0	0
$407$	5.14678	0.255116
$408$	0	0
$409$	4.94427	0.244479	0.122239	−	0.992501i	$-0.460992\pi$
$409$	4.94427	0.244479	0.122239	+	0.992501i	$0.460992\pi$
$410$	0	0
$411$	6.47214	0.319247
$412$	0	0
$413$	40.9334	2.01420
$414$	0	0
$415$	7.81758	0.383750
$416$	0	0
$417$	−0.944272	−0.0462412
$418$	0	0
$419$	10.4721	0.511597	0.255799	−	0.966730i	$-0.417662\pi$
$419$	10.4721	0.511597	0.255799	+	0.966730i	$0.417662\pi$
$420$	0	0
$421$	7.07107	0.344623	0.172311	−	0.985043i	$-0.444876\pi$
$421$	7.07107	0.344623	0.172311	+	0.985043i	$0.444876\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−32.3607	−1.56972
$426$	0	0
$427$	48.3607	2.34034
$428$	0	0
$429$	3.49613	0.168795
$430$	0	0
$431$	6.99226	0.336805	0.168403	−	0.985718i	$-0.446139\pi$
$431$	6.99226	0.336805	0.168403	+	0.985718i	$0.446139\pi$
$432$	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
$435$	−1.05573	−0.0506183
$436$	0	0
$437$	13.9845	0.668970
$438$	0	0
$439$	−25.0432	−1.19525	−0.597623	−	0.801777i	$-0.703888\pi$
$439$	−25.0432	−1.19525	−0.597623	+	0.801777i	$0.703888\pi$
$440$	0	0
$441$	13.9443	0.664013
$442$	0	0
$443$	−15.4164	−0.732456	−0.366228	−	0.930525i	$-0.619351\pi$
$443$	−15.4164	−0.732456	−0.366228	+	0.930525i	$0.619351\pi$
$444$	0	0
$445$	31.6228	1.49906
$446$	0	0
$447$	15.1437	0.716272
$448$	0	0
$449$	3.41641	0.161230	0.0806151	−	0.996745i	$-0.474312\pi$
$449$	3.41641	0.161230	0.0806151	+	0.996745i	$0.474312\pi$
$450$	0	0
$451$	16.0000	0.753411
$452$	0	0
$453$	11.5687	0.543547
$454$	0	0
$455$	20.4667	0.959493
$456$	0	0
$457$	20.9443	0.979732	0.489866	−	0.871798i	$-0.337046\pi$
$457$	20.9443	0.979732	0.489866	+	0.871798i	$0.337046\pi$
$458$	0	0
$459$	6.47214	0.302093
$460$	0	0
$461$	2.49458	0.116184	0.0580920	−	0.998311i	$-0.481498\pi$
$461$	2.49458	0.116184	0.0580920	+	0.998311i	$0.481498\pi$
$462$	0	0
$463$	−1.08036	−0.0502087	−0.0251044	−	0.999685i	$-0.507992\pi$
$463$	−1.08036	−0.0502087	−0.0251044	+	0.999685i	$0.507992\pi$
$464$	0	0
$465$	32.3607	1.50069
$466$	0	0
$467$	31.4164	1.45378	0.726889	−	0.686755i	$-0.240965\pi$
$467$	31.4164	1.45378	0.726889	+	0.686755i	$0.240965\pi$
$468$	0	0
$469$	−54.9179	−2.53587
$470$	0	0
$471$	0.746512	0.0343975
$472$	0	0
$473$	−25.8885	−1.19036
$474$	0	0
$475$	−12.3607	−0.567147
$476$	0	0
$477$	−5.32300	−0.243724
$478$	0	0
$479$	−9.15298	−0.418210	−0.209105	−	0.977893i	$-0.567055\pi$
$479$	−9.15298	−0.418210	−0.209105	+	0.977893i	$0.567055\pi$
$480$	0	0
$481$	−2.94427	−0.134247
$482$	0	0
$483$	25.8885	1.17797
$484$	0	0
$485$	−15.6352	−0.709956
$486$	0	0
$487$	2.41577	0.109469	0.0547344	−	0.998501i	$-0.482569\pi$
$487$	2.41577	0.109469	0.0547344	+	0.998501i	$0.482569\pi$
$488$	0	0
$489$	5.52786	0.249979
$490$	0	0
$491$	13.8885	0.626781	0.313391	−	0.949624i	$-0.398535\pi$
$491$	13.8885	0.626781	0.313391	+	0.949624i	$0.398535\pi$
$492$	0	0
$493$	2.16073	0.0973142
$494$	0	0
$495$	−7.81758	−0.351374
$496$	0	0
$497$	−16.0000	−0.717698
$498$	0	0
$499$	7.05573	0.315858	0.157929	−	0.987450i	$-0.449518\pi$
$499$	7.05573	0.315858	0.157929	+	0.987450i	$0.449518\pi$
$500$	0	0
$501$	7.81758	0.349264
$502$	0	0
$503$	−33.1158	−1.47656	−0.738280	−	0.674494i	$-0.764362\pi$
$503$	−33.1158	−1.47656	−0.738280	+	0.674494i	$0.764362\pi$
$504$	0	0
$505$	18.9443	0.843009
$506$	0	0
$507$	11.0000	0.488527
$508$	0	0
$509$	−33.6073	−1.48962	−0.744808	−	0.667279i	$-0.767459\pi$
$509$	−33.6073	−1.48962	−0.744808	+	0.667279i	$0.767459\pi$
$510$	0	0
$511$	−68.3923	−3.02550
$512$	0	0
$513$	2.47214	0.109147
$514$	0	0
$515$	14.4721	0.637719
$516$	0	0
$517$	0	0
$518$	0	0
$519$	9.48683	0.416426
$520$	0	0
$521$	16.3607	0.716774	0.358387	−	0.933573i	$-0.383327\pi$
$521$	16.3607	0.716774	0.358387	+	0.933573i	$0.383327\pi$
$522$	0	0
$523$	−28.3607	−1.24013	−0.620063	−	0.784552i	$-0.712893\pi$
$523$	−28.3607	−1.24013	−0.620063	+	0.784552i	$0.712893\pi$
$524$	0	0
$525$	−22.8825	−0.998672
$526$	0	0
$527$	−66.2316	−2.88509
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	8.94427	0.388148
$532$	0	0
$533$	−9.15298	−0.396460
$534$	0	0
$535$	53.5825	2.31657
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	34.4721	1.48482
$540$	0	0
$541$	−23.8840	−1.02685	−0.513427	−	0.858133i	$-0.671624\pi$
$541$	−23.8840	−1.02685	−0.513427	+	0.858133i	$0.671624\pi$
$542$	0	0
$543$	−15.5563	−0.667587
$544$	0	0
$545$	−26.5836	−1.13872
$546$	0	0
$547$	12.3607	0.528505	0.264252	−	0.964454i	$-0.414875\pi$
$547$	12.3607	0.528505	0.264252	+	0.964454i	$0.414875\pi$
$548$	0	0
$549$	10.5672	0.450997
$550$	0	0
$551$	0.825324	0.0351600
$552$	0	0
$553$	4.94427	0.210252
$554$	0	0
$555$	6.58359	0.279458
$556$	0	0
$557$	15.8114	0.669950	0.334975	−	0.942227i	$-0.391272\pi$
$557$	15.8114	0.669950	0.334975	+	0.942227i	$0.391272\pi$
$558$	0	0
$559$	14.8098	0.626389
$560$	0	0
$561$	16.0000	0.675521
$562$	0	0
$563$	−10.4721	−0.441348	−0.220674	−	0.975348i	$-0.570826\pi$
$563$	−10.4721	−0.441348	−0.220674	+	0.975348i	$0.570826\pi$
$564$	0	0
$565$	6.32456	0.266076
$566$	0	0
$567$	4.57649	0.192195
$568$	0	0
$569$	−19.4164	−0.813978	−0.406989	−	0.913433i	$-0.633421\pi$
$569$	−19.4164	−0.813978	−0.406989	+	0.913433i	$0.633421\pi$
$570$	0	0
$571$	−8.94427	−0.374306	−0.187153	−	0.982331i	$-0.559926\pi$
$571$	−8.94427	−0.374306	−0.187153	+	0.982331i	$0.559926\pi$
$572$	0	0
$573$	2.16073	0.0902656
$574$	0	0
$575$	−28.2843	−1.17954
$576$	0	0
$577$	9.05573	0.376995	0.188497	−	0.982074i	$-0.439638\pi$
$577$	9.05573	0.376995	0.188497	+	0.982074i	$0.439638\pi$
$578$	0	0
$579$	12.9443	0.537946
$580$	0	0
$581$	−11.3137	−0.469372
$582$	0	0
$583$	−13.1592	−0.544998
$584$	0	0
$585$	4.47214	0.184900
$586$	0	0
$587$	15.0557	0.621416	0.310708	−	0.950505i	$-0.399434\pi$
$587$	15.0557	0.621416	0.310708	+	0.950505i	$0.399434\pi$
$588$	0	0
$589$	−25.2982	−1.04240
$590$	0	0
$591$	−24.2967	−0.999431
$592$	0	0
$593$	−27.8885	−1.14525	−0.572623	−	0.819819i	$-0.694074\pi$
$593$	−27.8885	−1.14525	−0.572623	+	0.819819i	$0.694074\pi$
$594$	0	0
$595$	93.6656	3.83992
$596$	0	0
$597$	13.7295	0.561910
$598$	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
$601$	10.9443	0.446426	0.223213	−	0.974770i	$-0.428345\pi$
$601$	10.9443	0.446426	0.223213	+	0.974770i	$0.428345\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	0	0
$605$	15.4589	0.628495
$606$	0	0
$607$	25.8685	1.04997	0.524985	−	0.851111i	$-0.324071\pi$
$607$	25.8685	1.04997	0.524985	+	0.851111i	$0.324071\pi$
$608$	0	0
$609$	1.52786	0.0619122
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−2.08191	−0.0840877	−0.0420439	−	0.999116i	$-0.513387\pi$
$613$	−2.08191	−0.0840877	−0.0420439	+	0.999116i	$0.513387\pi$
$614$	0	0
$615$	20.4667	0.825297
$616$	0	0
$617$	19.8885	0.800683	0.400341	−	0.916366i	$-0.368892\pi$
$617$	19.8885	0.800683	0.400341	+	0.916366i	$0.368892\pi$
$618$	0	0
$619$	−29.8885	−1.20132	−0.600661	−	0.799504i	$-0.705096\pi$
$619$	−29.8885	−1.20132	−0.600661	+	0.799504i	$0.705096\pi$
$620$	0	0
$621$	5.65685	0.227002
$622$	0	0
$623$	−45.7649	−1.83353
$624$	0	0
$625$	−25.0000	−1.00000
$626$	0	0
$627$	6.11146	0.244068
$628$	0	0
$629$	−13.4744	−0.537261
$630$	0	0
$631$	−29.8747	−1.18929	−0.594647	−	0.803987i	$-0.702708\pi$
$631$	−29.8747	−1.18929	−0.594647	+	0.803987i	$0.702708\pi$
$632$	0	0
$633$	24.9443	0.991446
$634$	0	0
$635$	25.5279	1.01304
$636$	0	0
$637$	−19.7202	−0.781342
$638$	0	0
$639$	−3.49613	−0.138305
$640$	0	0
$641$	35.4164	1.39886	0.699432	−	0.714699i	$-0.253436\pi$
$641$	35.4164	1.39886	0.699432	+	0.714699i	$0.253436\pi$
$642$	0	0
$643$	−36.3607	−1.43393	−0.716963	−	0.697112i	$-0.754468\pi$
$643$	−36.3607	−1.43393	−0.716963	+	0.697112i	$0.754468\pi$
$644$	0	0
$645$	−33.1158	−1.30393
$646$	0	0
$647$	−8.32766	−0.327394	−0.163697	−	0.986511i	$-0.552342\pi$
$647$	−8.32766	−0.327394	−0.163697	+	0.986511i	$0.552342\pi$
$648$	0	0
$649$	22.1115	0.867951
$650$	0	0
$651$	−46.8328	−1.83552
$652$	0	0
$653$	6.81603	0.266732	0.133366	−	0.991067i	$-0.457421\pi$
$653$	6.81603	0.266732	0.133366	+	0.991067i	$0.457421\pi$
$654$	0	0
$655$	−2.98605	−0.116675
$656$	0	0
$657$	−14.9443	−0.583032
$658$	0	0
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	0	0
$661$	−2.23954	−0.0871079	−0.0435540	−	0.999051i	$-0.513868\pi$
$661$	−2.23954	−0.0871079	−0.0435540	+	0.999051i	$0.513868\pi$
$662$	0	0
$663$	−9.15298	−0.355472
$664$	0	0
$665$	35.7771	1.38738
$666$	0	0
$667$	1.88854	0.0731247
$668$	0	0
$669$	−10.2333	−0.395644
$670$	0	0
$671$	26.1235	1.00849
$672$	0	0
$673$	17.8885	0.689553	0.344776	−	0.938685i	$-0.387955\pi$
$673$	17.8885	0.689553	0.344776	+	0.938685i	$0.387955\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	−44.7634	−1.72040	−0.860198	−	0.509960i	$-0.829660\pi$
$677$	−44.7634	−1.72040	−0.860198	+	0.509960i	$0.829660\pi$
$678$	0	0
$679$	22.6274	0.868361
$680$	0	0
$681$	−5.52786	−0.211828
$682$	0	0
$683$	23.4164	0.896004	0.448002	−	0.894033i	$-0.352136\pi$
$683$	23.4164	0.896004	0.448002	+	0.894033i	$0.352136\pi$
$684$	0	0
$685$	20.4667	0.781992
$686$	0	0
$687$	11.2349	0.428638
$688$	0	0
$689$	7.52786	0.286789
$690$	0	0
$691$	34.4721	1.31138	0.655691	−	0.755029i	$-0.272377\pi$
$691$	34.4721	1.31138	0.655691	+	0.755029i	$0.272377\pi$
$692$	0	0
$693$	11.3137	0.429772
$694$	0	0
$695$	−2.98605	−0.113267
$696$	0	0
$697$	−41.8885	−1.58664
$698$	0	0
$699$	10.0000	0.378235
$700$	0	0
$701$	−3.98760	−0.150610	−0.0753048	−	0.997161i	$-0.523993\pi$
$701$	−3.98760	−0.150610	−0.0753048	+	0.997161i	$0.523993\pi$
$702$	0	0
$703$	−5.14678	−0.194114
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−27.4164	−1.03110
$708$	0	0
$709$	8.56409	0.321631	0.160816	−	0.986984i	$-0.448588\pi$
$709$	8.56409	0.321631	0.160816	+	0.986984i	$0.448588\pi$
$710$	0	0
$711$	1.08036	0.0405168
$712$	0	0
$713$	−57.8885	−2.16794
$714$	0	0
$715$	11.0557	0.413461
$716$	0	0
$717$	−13.4744	−0.503212
$718$	0	0
$719$	−16.1452	−0.602116	−0.301058	−	0.953606i	$-0.597340\pi$
$719$	−16.1452	−0.602116	−0.301058	+	0.953606i	$0.597340\pi$
$720$	0	0
$721$	−20.9443	−0.780005
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.66925	−0.0619945
$726$	0	0
$727$	11.5687	0.429061	0.214531	−	0.976717i	$-0.431178\pi$
$727$	11.5687	0.429061	0.214531	+	0.976717i	$0.431178\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	67.7771	2.50683
$732$	0	0
$733$	−5.73567	−0.211852	−0.105926	−	0.994374i	$-0.533781\pi$
$733$	−5.73567	−0.211852	−0.105926	+	0.994374i	$0.533781\pi$
$734$	0	0
$735$	44.0957	1.62649
$736$	0	0
$737$	−29.6656	−1.09275
$738$	0	0
$739$	15.0557	0.553834	0.276917	−	0.960894i	$-0.410687\pi$
$739$	15.0557	0.553834	0.276917	+	0.960894i	$0.410687\pi$
$740$	0	0
$741$	−3.49613	−0.128433
$742$	0	0
$743$	14.8098	0.543320	0.271660	−	0.962393i	$-0.412427\pi$
$743$	14.8098	0.543320	0.271660	+	0.962393i	$0.412427\pi$
$744$	0	0
$745$	47.8885	1.75450
$746$	0	0
$747$	−2.47214	−0.0904507
$748$	0	0
$749$	−77.5453	−2.83344
$750$	0	0
$751$	−1.08036	−0.0394230	−0.0197115	−	0.999806i	$-0.506275\pi$
$751$	−1.08036	−0.0394230	−0.0197115	+	0.999806i	$0.506275\pi$
$752$	0	0
$753$	−15.4164	−0.561805
$754$	0	0
$755$	36.5836	1.33141
$756$	0	0
$757$	11.2349	0.408339	0.204170	−	0.978936i	$-0.434551\pi$
$757$	11.2349	0.408339	0.204170	+	0.978936i	$0.434551\pi$
$758$	0	0
$759$	13.9845	0.507606
$760$	0	0
$761$	19.4164	0.703844	0.351922	−	0.936029i	$-0.385528\pi$
$761$	19.4164	0.703844	0.351922	+	0.936029i	$0.385528\pi$
$762$	0	0
$763$	38.4721	1.39278
$764$	0	0
$765$	20.4667	0.739975
$766$	0	0
$767$	−12.6491	−0.456733
$768$	0	0
$769$	22.8328	0.823372	0.411686	−	0.911326i	$-0.364940\pi$
$769$	22.8328	0.823372	0.411686	+	0.911326i	$0.364940\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	0	0
$773$	12.3153	0.442949	0.221475	−	0.975166i	$-0.428913\pi$
$773$	12.3153	0.442949	0.221475	+	0.975166i	$0.428913\pi$
$774$	0	0
$775$	51.1667	1.83796
$776$	0	0
$777$	−9.52786	−0.341810
$778$	0	0
$779$	−16.0000	−0.573259
$780$	0	0
$781$	−8.64290	−0.309267
$782$	0	0
$783$	0.333851	0.0119308
$784$	0	0
$785$	2.36068	0.0842563
$786$	0	0
$787$	5.52786	0.197047	0.0985235	−	0.995135i	$-0.468588\pi$
$787$	5.52786	0.197047	0.0985235	+	0.995135i	$0.468588\pi$
$788$	0	0
$789$	−12.6491	−0.450320
$790$	0	0
$791$	−9.15298	−0.325443
$792$	0	0
$793$	−14.9443	−0.530687
$794$	0	0
$795$	−16.8328	−0.596998
$796$	0	0
$797$	39.1065	1.38522	0.692612	−	0.721311i	$-0.256460\pi$
$797$	39.1065	1.38522	0.692612	+	0.721311i	$0.256460\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−10.0000	−0.353333
$802$	0	0
$803$	−36.9443	−1.30374
$804$	0	0
$805$	81.8668	2.88542
$806$	0	0
$807$	−11.6476	−0.410013
$808$	0	0
$809$	3.41641	0.120115	0.0600573	−	0.998195i	$-0.480872\pi$
$809$	3.41641	0.120115	0.0600573	+	0.998195i	$0.480872\pi$
$810$	0	0
$811$	33.3050	1.16950	0.584748	−	0.811215i	$-0.301194\pi$
$811$	33.3050	1.16950	0.584748	+	0.811215i	$0.301194\pi$
$812$	0	0
$813$	8.07262	0.283119
$814$	0	0
$815$	17.4806	0.612320
$816$	0	0
$817$	25.8885	0.905725
$818$	0	0
$819$	−6.47214	−0.226155
$820$	0	0
$821$	−44.6057	−1.55675	−0.778375	−	0.627799i	$-0.783956\pi$
$821$	−44.6057	−1.55675	−0.778375	+	0.627799i	$0.783956\pi$
$822$	0	0
$823$	−4.57649	−0.159526	−0.0797632	−	0.996814i	$-0.525416\pi$
$823$	−4.57649	−0.159526	−0.0797632	+	0.996814i	$0.525416\pi$
$824$	0	0
$825$	−12.3607	−0.430344
$826$	0	0
$827$	−53.8885	−1.87389	−0.936944	−	0.349479i	$-0.886359\pi$
$827$	−53.8885	−1.87389	−0.936944	+	0.349479i	$0.886359\pi$
$828$	0	0
$829$	19.7202	0.684910	0.342455	−	0.939534i	$-0.388742\pi$
$829$	19.7202	0.684910	0.342455	+	0.939534i	$0.388742\pi$
$830$	0	0
$831$	−18.3848	−0.637761
$832$	0	0
$833$	−90.2492	−3.12695
$834$	0	0
$835$	24.7214	0.855518
$836$	0	0
$837$	−10.2333	−0.353716
$838$	0	0
$839$	44.4295	1.53388	0.766939	−	0.641721i	$-0.221779\pi$
$839$	44.4295	1.53388	0.766939	+	0.641721i	$0.221779\pi$
$840$	0	0
$841$	−28.8885	−0.996157
$842$	0	0
$843$	31.8885	1.09830
$844$	0	0
$845$	34.7851	1.19664
$846$	0	0
$847$	−22.3724	−0.768724
$848$	0	0
$849$	−26.8328	−0.920900
$850$	0	0
$851$	−11.7771	−0.403713
$852$	0	0
$853$	−17.7171	−0.606621	−0.303311	−	0.952892i	$-0.598092\pi$
$853$	−17.7171	−0.606621	−0.303311	+	0.952892i	$0.598092\pi$
$854$	0	0
$855$	7.81758	0.267356
$856$	0	0
$857$	28.5836	0.976397	0.488198	−	0.872733i	$-0.337654\pi$
$857$	28.5836	0.976397	0.488198	+	0.872733i	$0.337654\pi$
$858$	0	0
$859$	45.5279	1.55339	0.776695	−	0.629876i	$-0.216894\pi$
$859$	45.5279	1.55339	0.776695	+	0.629876i	$0.216894\pi$
$860$	0	0
$861$	−29.6197	−1.00944
$862$	0	0
$863$	−34.4512	−1.17273	−0.586366	−	0.810046i	$-0.699442\pi$
$863$	−34.4512	−1.17273	−0.586366	+	0.810046i	$0.699442\pi$
$864$	0	0
$865$	30.0000	1.02003
$866$	0	0
$867$	−24.8885	−0.845259
$868$	0	0
$869$	2.67080	0.0906008
$870$	0	0
$871$	16.9706	0.575026
$872$	0	0
$873$	4.94427	0.167338
$874$	0	0
$875$	0	0
$876$	0	0
$877$	35.8654	1.21109	0.605545	−	0.795811i	$-0.292955\pi$
$877$	35.8654	1.21109	0.605545	+	0.795811i	$0.292955\pi$
$878$	0	0
$879$	16.6367	0.561142
$880$	0	0
$881$	27.8885	0.939589	0.469794	−	0.882776i	$-0.344328\pi$
$881$	27.8885	0.939589	0.469794	+	0.882776i	$0.344328\pi$
$882$	0	0
$883$	−18.4721	−0.621637	−0.310818	−	0.950469i	$-0.600603\pi$
$883$	−18.4721	−0.621637	−0.310818	+	0.950469i	$0.600603\pi$
$884$	0	0
$885$	28.2843	0.950765
$886$	0	0
$887$	−42.2688	−1.41925	−0.709623	−	0.704581i	$-0.751135\pi$
$887$	−42.2688	−1.41925	−0.709623	+	0.704581i	$0.751135\pi$
$888$	0	0
$889$	−36.9443	−1.23907
$890$	0	0
$891$	2.47214	0.0828197
$892$	0	0
$893$	0	0
$894$	0	0
$895$	37.9473	1.26844
$896$	0	0
$897$	−8.00000	−0.267112
$898$	0	0
$899$	−3.41641	−0.113944
$900$	0	0
$901$	34.4512	1.14774
$902$	0	0
$903$	47.9256	1.59487
$904$	0	0
$905$	−49.1935	−1.63525
$906$	0	0
$907$	51.1935	1.69985	0.849926	−	0.526902i	$-0.176647\pi$
$907$	51.1935	1.69985	0.849926	+	0.526902i	$0.176647\pi$
$908$	0	0
$909$	−5.99070	−0.198699
$910$	0	0
$911$	−2.16073	−0.0715880	−0.0357940	−	0.999359i	$-0.511396\pi$
$911$	−2.16073	−0.0715880	−0.0357940	+	0.999359i	$0.511396\pi$
$912$	0	0
$913$	−6.11146	−0.202260
$914$	0	0
$915$	33.4164	1.10471
$916$	0	0
$917$	4.32145	0.142707
$918$	0	0
$919$	−34.1962	−1.12803	−0.564014	−	0.825765i	$-0.690743\pi$
$919$	−34.1962	−1.12803	−0.564014	+	0.825765i	$0.690743\pi$
$920$	0	0
$921$	29.8885	0.984861
$922$	0	0
$923$	4.94427	0.162743
$924$	0	0
$925$	10.4096	0.342265
$926$	0	0
$927$	−4.57649	−0.150312
$928$	0	0
$929$	6.47214	0.212344	0.106172	−	0.994348i	$-0.466141\pi$
$929$	6.47214	0.212344	0.106172	+	0.994348i	$0.466141\pi$
$930$	0	0
$931$	−34.4721	−1.12978
$932$	0	0
$933$	−30.9551	−1.01342
$934$	0	0
$935$	50.5964	1.65468
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	−3.05573	−0.0997199
$940$	0	0
$941$	25.1220	0.818954	0.409477	−	0.912321i	$-0.365711\pi$
$941$	25.1220	0.818954	0.409477	+	0.912321i	$0.365711\pi$
$942$	0	0
$943$	−36.6119	−1.19225
$944$	0	0
$945$	14.4721	0.470779
$946$	0	0
$947$	47.7771	1.55255	0.776273	−	0.630396i	$-0.217108\pi$
$947$	47.7771	1.55255	0.776273	+	0.630396i	$0.217108\pi$
$948$	0	0
$949$	21.1344	0.686051
$950$	0	0
$951$	−10.9799	−0.356046
$952$	0	0
$953$	−9.52786	−0.308638	−0.154319	−	0.988021i	$-0.549318\pi$
$953$	−9.52786	−0.308638	−0.154319	+	0.988021i	$0.549318\pi$
$954$	0	0
$955$	6.83282	0.221105
$956$	0	0
$957$	0.825324	0.0266789
$958$	0	0
$959$	−29.6197	−0.956469
$960$	0	0
$961$	73.7214	2.37811
$962$	0	0
$963$	−16.9443	−0.546022
$964$	0	0
$965$	40.9334	1.31769
$966$	0	0
$967$	18.5610	0.596882	0.298441	−	0.954428i	$-0.403533\pi$
$967$	18.5610	0.596882	0.298441	+	0.954428i	$0.403533\pi$
$968$	0	0
$969$	−16.0000	−0.513994
$970$	0	0
$971$	20.3607	0.653405	0.326703	−	0.945127i	$-0.394062\pi$
$971$	20.3607	0.653405	0.326703	+	0.945127i	$0.394062\pi$
$972$	0	0
$973$	4.32145	0.138539
$974$	0	0
$975$	7.07107	0.226455
$976$	0	0
$977$	12.5836	0.402585	0.201292	−	0.979531i	$-0.435486\pi$
$977$	12.5836	0.402585	0.201292	+	0.979531i	$0.435486\pi$
$978$	0	0
$979$	−24.7214	−0.790098
$980$	0	0
$981$	8.40647	0.268398
$982$	0	0
$983$	53.5825	1.70902	0.854508	−	0.519438i	$-0.173859\pi$
$983$	53.5825	1.70902	0.854508	+	0.519438i	$0.173859\pi$
$984$	0	0
$985$	−76.8328	−2.44810
$986$	0	0
$987$	0	0
$988$	0	0
$989$	59.2393	1.88370
$990$	0	0
$991$	17.2256	0.547189	0.273595	−	0.961845i	$-0.411787\pi$
$991$	17.2256	0.547189	0.273595	+	0.961845i	$0.411787\pi$
$992$	0	0
$993$	0.944272	0.0299656
$994$	0	0
$995$	43.4164	1.37639
$996$	0	0
$997$	29.0308	0.919414	0.459707	−	0.888071i	$-0.347954\pi$
$997$	29.0308	0.919414	0.459707	+	0.888071i	$0.347954\pi$
$998$	0	0
$999$	−2.08191	−0.0658689

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3072.2.a.k.1.2		4
3.2	odd	2		9216.2.a.bj.1.4		4
4.3	odd	2		3072.2.a.q.1.1		4
8.3	odd	2	inner	3072.2.a.k.1.3		4
8.5	even	2		3072.2.a.q.1.4		4
12.11	even	2		9216.2.a.bd.1.3		4
16.3	odd	4		3072.2.d.g.1537.8		8
16.5	even	4		3072.2.d.g.1537.5		8
16.11	odd	4		3072.2.d.g.1537.2		8
16.13	even	4		3072.2.d.g.1537.3		8
24.5	odd	2		9216.2.a.bd.1.2		4
24.11	even	2		9216.2.a.bj.1.1		4
32.3	odd	8		1536.2.j.h.1153.4	yes	8
32.5	even	8		1536.2.j.g.385.3	yes	8
32.11	odd	8		1536.2.j.h.385.4	yes	8
32.13	even	8		1536.2.j.g.1153.3	yes	8
32.19	odd	8		1536.2.j.g.1153.1	yes	8
32.21	even	8		1536.2.j.h.385.2	yes	8
32.27	odd	8		1536.2.j.g.385.1	✓	8
32.29	even	8		1536.2.j.h.1153.2	yes	8
96.5	odd	8		4608.2.k.bf.3457.4		8
96.11	even	8		4608.2.k.bg.3457.1		8
96.29	odd	8		4608.2.k.bg.1153.1		8
96.35	even	8		4608.2.k.bg.1153.2		8
96.53	odd	8		4608.2.k.bg.3457.2		8
96.59	even	8		4608.2.k.bf.3457.3		8
96.77	odd	8		4608.2.k.bf.1153.3		8
96.83	even	8		4608.2.k.bf.1153.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.j.g.385.1	✓	8	32.27	odd	8
1536.2.j.g.385.3	yes	8	32.5	even	8
1536.2.j.g.1153.1	yes	8	32.19	odd	8
1536.2.j.g.1153.3	yes	8	32.13	even	8
1536.2.j.h.385.2	yes	8	32.21	even	8
1536.2.j.h.385.4	yes	8	32.11	odd	8
1536.2.j.h.1153.2	yes	8	32.29	even	8
1536.2.j.h.1153.4	yes	8	32.3	odd	8
3072.2.a.k.1.2		4	1.1	even	1	trivial
3072.2.a.k.1.3		4	8.3	odd	2	inner
3072.2.a.q.1.1		4	4.3	odd	2
3072.2.a.q.1.4		4	8.5	even	2
3072.2.d.g.1537.2		8	16.11	odd	4
3072.2.d.g.1537.3		8	16.13	even	4
3072.2.d.g.1537.5		8	16.5	even	4
3072.2.d.g.1537.8		8	16.3	odd	4
4608.2.k.bf.1153.3		8	96.77	odd	8
4608.2.k.bf.1153.4		8	96.83	even	8
4608.2.k.bf.3457.3		8	96.59	even	8
4608.2.k.bf.3457.4		8	96.5	odd	8
4608.2.k.bg.1153.1		8	96.29	odd	8
4608.2.k.bg.1153.2		8	96.35	even	8
4608.2.k.bg.3457.1		8	96.11	even	8
4608.2.k.bg.3457.2		8	96.53	odd	8
9216.2.a.bd.1.2		4	24.5	odd	2
9216.2.a.bd.1.3		4	12.11	even	2
9216.2.a.bj.1.1		4	24.11	even	2
9216.2.a.bj.1.4		4	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3072 = 2^{10} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3072.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.16228 q^{5} +4.57649 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.16228 q^{5} +4.57649 q^{7} +1.00000 q^{9} +2.47214 q^{11} -1.41421 q^{13} +3.16228 q^{15} -6.47214 q^{17} -2.47214 q^{19} -4.57649 q^{21} -5.65685 q^{23} +5.00000 q^{25} -1.00000 q^{27} -0.333851 q^{29} +10.2333 q^{31} -2.47214 q^{33} -14.4721 q^{35} +2.08191 q^{37} +1.41421 q^{39} +6.47214 q^{41} -10.4721 q^{43} -3.16228 q^{45} +13.9443 q^{49} +6.47214 q^{51} -5.32300 q^{53} -7.81758 q^{55} +2.47214 q^{57} +8.94427 q^{59} +10.5672 q^{61} +4.57649 q^{63} +4.47214 q^{65} -12.0000 q^{67} +5.65685 q^{69} -3.49613 q^{71} -14.9443 q^{73} -5.00000 q^{75} +11.3137 q^{77} +1.08036 q^{79} +1.00000 q^{81} -2.47214 q^{83} +20.4667 q^{85} +0.333851 q^{87} -10.0000 q^{89} -6.47214 q^{91} -10.2333 q^{93} +7.81758 q^{95} +4.94427 q^{97} +2.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{9} - 8 q^{11} - 8 q^{17} + 8 q^{19} + 20 q^{25} - 4 q^{27} + 8 q^{33} - 40 q^{35} + 8 q^{41} - 24 q^{43} + 20 q^{49} + 8 q^{51} - 8 q^{57} - 48 q^{67} - 24 q^{73} - 20 q^{75} + 4 q^{81} + 8 q^{83} - 40 q^{89} - 8 q^{91} - 16 q^{97} - 8 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^9 - 8 * q^11 - 8 * q^17 + 8 * q^19 + 20 * q^25 - 4 * q^27 + 8 * q^33 - 40 * q^35 + 8 * q^41 - 24 * q^43 + 20 * q^49 + 8 * q^51 - 8 * q^57 - 48 * q^67 - 24 * q^73 - 20 * q^75 + 4 * q^81 + 8 * q^83 - 40 * q^89 - 8 * q^91 - 16 * q^97 - 8 * q^99

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