LMFDB - Embedded newform 3072.2.d.g.1537.3

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			1537.3
Root			$0.437016 - 0.437016i$ of defining polynomial
Character	$\chi$	$=$	3072.1537
Dual form			3072.2.d.g.1537.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +3.16228i q^{5} -4.57649 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +3.16228i q^{5} -4.57649 q^{7} -1.00000 q^{9} -2.47214i q^{11} -1.41421i q^{13} +3.16228 q^{15} -6.47214 q^{17} -2.47214i q^{19} +4.57649i q^{21} +5.65685 q^{23} -5.00000 q^{25} +1.00000i q^{27} -0.333851i q^{29} +10.2333 q^{31} -2.47214 q^{33} -14.4721i q^{35} -2.08191i q^{37} -1.41421 q^{39} -6.47214 q^{41} +10.4721i q^{43} -3.16228i q^{45} +13.9443 q^{49} +6.47214i q^{51} +5.32300i q^{53} +7.81758 q^{55} -2.47214 q^{57} -8.94427i q^{59} +10.5672i q^{61} +4.57649 q^{63} +4.47214 q^{65} -12.0000i q^{67} -5.65685i q^{69} +3.49613 q^{71} +14.9443 q^{73} +5.00000i q^{75} +11.3137i q^{77} +1.08036 q^{79} +1.00000 q^{81} -2.47214i q^{83} -20.4667i q^{85} -0.333851 q^{87} +10.0000 q^{89} +6.47214i q^{91} -10.2333i q^{93} +7.81758 q^{95} +4.94427 q^{97} +2.47214i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$0.250000\pi$
$5$	3.16228i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$6$	0	0
$7$	−4.57649	−1.72975	−0.864876	−	0.501986i	$-0.832603\pi$
$7$	−4.57649	−1.72975	−0.864876	+	0.501986i	$0.832603\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 2.47214i	− 0.745377i	−0.927957	−	0.372689i	$-0.878436\pi$
$11$	− 2.47214i	− 0.745377i	0.927957	−	0.372689i	$-0.121564\pi$
$12$	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
$17$	−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
$19$	− 2.47214i	− 0.567147i	−0.958951	−	0.283573i	$-0.908480\pi$
$19$	− 2.47214i	− 0.567147i	0.958951	−	0.283573i	$-0.0915200\pi$
$20$	0	0
$21$	4.57649i	0.998672i
$22$	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
$25$	−5.00000	−1.00000
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	− 0.333851i	− 0.0619945i	−0.999519	−	0.0309972i	$-0.990132\pi$
$29$	− 0.333851i	− 0.0619945i	0.999519	−	0.0309972i	$-0.00986831\pi$
$30$	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
$35$	− 14.4721i	− 2.44624i
$36$	0	0
$37$	− 2.08191i	− 0.342265i	−0.985248	−	0.171132i	$-0.945257\pi$
$37$	− 2.08191i	− 0.342265i	0.985248	−	0.171132i	$-0.0547426\pi$
$38$	0	0
$39$	−1.41421	−0.226455
$40$	0	0
$41$	−6.47214	−1.01078	−0.505389	−	0.862892i	$-0.668651\pi$
$41$	−6.47214	−1.01078	−0.505389	+	0.862892i	$0.668651\pi$
$42$	0	0
$43$	10.4721i	1.59699i	0.602004	+	0.798493i	$0.294369\pi$
$43$	10.4721i	1.59699i	−0.602004	+	0.798493i	$0.705631\pi$
$44$	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
$49$	13.9443	1.99204
$50$	0	0
$51$	6.47214i	0.906280i
$52$	0	0
$53$	5.32300i	0.731171i	0.930778	+	0.365585i	$0.119131\pi$
$53$	5.32300i	0.731171i	−0.930778	+	0.365585i	$0.880869\pi$
$54$	0	0
$55$	7.81758	1.05412
$56$	0	0
$57$	−2.47214	−0.327442
$58$	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
$61$	10.5672i	1.35299i	0.736447	+	0.676495i	$0.236502\pi$
$61$	10.5672i	1.35299i	−0.736447	+	0.676495i	$0.763498\pi$
$62$	0	0
$63$	4.57649	0.576584
$64$	0	0
$65$	4.47214	0.554700
$66$	0	0
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	0	0
$69$	− 5.65685i	− 0.681005i
$70$	0	0
$71$	3.49613	0.414914	0.207457	−	0.978244i	$-0.433481\pi$
$71$	3.49613	0.414914	0.207457	+	0.978244i	$0.433481\pi$
$72$	0	0
$73$	14.9443	1.74909	0.874547	−	0.484940i	$-0.161159\pi$
$73$	14.9443	1.74909	0.874547	+	0.484940i	$0.161159\pi$
$74$	0	0
$75$	5.00000i	0.577350i
$76$	0	0
$77$	11.3137i	1.28932i
$78$	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
$83$	− 2.47214i	− 0.271352i	−0.990753	−	0.135676i	$-0.956679\pi$
$83$	− 2.47214i	− 0.271352i	0.990753	−	0.135676i	$-0.0433206\pi$
$84$	0	0
$85$	− 20.4667i	− 2.21992i
$86$	0	0
$87$	−0.333851	−0.0357925
$88$	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
$91$	6.47214i	0.678464i
$92$	0	0
$93$	− 10.2333i	− 1.06115i
$94$	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
$99$	2.47214i	0.248459i
$100$	0	0
$101$	5.99070i	0.596097i	0.954551	+	0.298049i	$0.0963358\pi$
$101$	5.99070i	0.596097i	−0.954551	+	0.298049i	$0.903664\pi$
$102$	0	0
$103$	4.57649	0.450935	0.225468	−	0.974251i	$-0.427609\pi$
$103$	4.57649	0.450935	0.225468	+	0.974251i	$0.427609\pi$
$104$	0	0
$105$	−14.4721	−1.41234
$106$	0	0
$107$	16.9443i	1.63806i	0.573747	+	0.819032i	$0.305489\pi$
$107$	16.9443i	1.63806i	−0.573747	+	0.819032i	$0.694511\pi$
$108$	0	0
$109$	8.40647i	0.805194i	0.915377	+	0.402597i	$0.131892\pi$
$109$	8.40647i	0.805194i	−0.915377	+	0.402597i	$0.868108\pi$
$110$	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
$115$	17.8885i	1.66812i
$116$	0	0
$117$	1.41421i	0.130744i
$118$	0	0
$119$	29.6197	2.71523
$120$	0	0
$121$	4.88854	0.444413
$122$	0	0
$123$	6.47214i	0.583573i
$124$	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
$131$	0.944272i	0.0825014i	0.999149	+	0.0412507i	$0.0131342\pi$
$131$	0.944272i	0.0825014i	−0.999149	+	0.0412507i	$0.986866\pi$
$132$	0	0
$133$	11.3137i	0.981023i
$134$	0	0
$135$	−3.16228	−0.272166
$136$	0	0
$137$	6.47214	0.552952	0.276476	−	0.961021i	$-0.410833\pi$
$137$	6.47214	0.552952	0.276476	+	0.961021i	$0.410833\pi$
$138$	0	0
$139$	− 0.944272i	− 0.0800921i	−0.999198	−	0.0400460i	$-0.987250\pi$
$139$	− 0.944272i	− 0.0800921i	0.999198	−	0.0400460i	$-0.0127505\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.49613	−0.292361
$144$	0	0
$145$	1.05573	0.0876734
$146$	0	0
$147$	− 13.9443i	− 1.15010i
$148$	0	0
$149$	15.1437i	1.24062i	0.784357	+	0.620310i	$0.212993\pi$
$149$	15.1437i	1.24062i	−0.784357	+	0.620310i	$0.787007\pi$
$150$	0	0
$151$	11.5687	0.941451	0.470726	−	0.882280i	$-0.343992\pi$
$151$	11.5687	0.941451	0.470726	+	0.882280i	$0.343992\pi$
$152$	0	0
$153$	6.47214	0.523241
$154$	0	0
$155$	32.3607i	2.59927i
$156$	0	0
$157$	− 0.746512i	− 0.0595782i	−0.999556	−	0.0297891i	$-0.990516\pi$
$157$	− 0.746512i	− 0.0595782i	0.999556	−	0.0297891i	$-0.00948357\pi$
$158$	0	0
$159$	5.32300	0.422142
$160$	0	0
$161$	−25.8885	−2.04030
$162$	0	0
$163$	− 5.52786i	− 0.432976i	−0.976285	−	0.216488i	$-0.930540\pi$
$163$	− 5.52786i	− 0.432976i	0.976285	−	0.216488i	$-0.0694602\pi$
$164$	0	0
$165$	− 7.81758i	− 0.608598i
$166$	0	0
$167$	7.81758	0.604943	0.302471	−	0.953159i	$-0.402188\pi$
$167$	7.81758	0.604943	0.302471	+	0.953159i	$0.402188\pi$
$168$	0	0
$169$	11.0000	0.846154
$170$	0	0
$171$	2.47214i	0.189049i
$172$	0	0
$173$	− 9.48683i	− 0.721271i	−0.932707	−	0.360635i	$-0.882560\pi$
$173$	− 9.48683i	− 0.721271i	0.932707	−	0.360635i	$-0.117440\pi$
$174$	0	0
$175$	22.8825	1.72975
$176$	0	0
$177$	−8.94427	−0.672293
$178$	0	0
$179$	− 12.0000i	− 0.896922i	−0.893802	−	0.448461i	$-0.851972\pi$
$179$	− 12.0000i	− 0.896922i	0.893802	−	0.448461i	$-0.148028\pi$
$180$	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
$183$	10.5672	0.781150
$184$	0	0
$185$	6.58359	0.484035
$186$	0	0
$187$	16.0000i	1.17004i
$188$	0	0
$189$	− 4.57649i	− 0.332891i
$190$	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
$195$	− 4.47214i	− 0.320256i
$196$	0	0
$197$	− 24.2967i	− 1.73107i	−0.500852	−	0.865533i	$-0.666980\pi$
$197$	− 24.2967i	− 1.73107i	0.500852	−	0.865533i	$-0.333020\pi$
$198$	0	0
$199$	13.7295	0.973257	0.486628	−	0.873609i	$-0.338227\pi$
$199$	13.7295	0.973257	0.486628	+	0.873609i	$0.338227\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	0	0
$203$	1.52786i	0.107235i
$204$	0	0
$205$	− 20.4667i	− 1.42946i
$206$	0	0
$207$	−5.65685	−0.393179
$208$	0	0
$209$	−6.11146	−0.422738
$210$	0	0
$211$	− 24.9443i	− 1.71723i	−0.512617	−	0.858617i	$-0.671324\pi$
$211$	− 24.9443i	− 1.71723i	0.512617	−	0.858617i	$-0.328676\pi$
$212$	0	0
$213$	− 3.49613i	− 0.239551i
$214$	0	0
$215$	−33.1158	−2.25848
$216$	0	0
$217$	−46.8328	−3.17922
$218$	0	0
$219$	− 14.9443i	− 1.00984i
$220$	0	0
$221$	9.15298i	0.615696i
$222$	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
$227$	5.52786i	0.366897i	0.983029	+	0.183449i	$0.0587261\pi$
$227$	5.52786i	0.366897i	−0.983029	+	0.183449i	$0.941274\pi$
$228$	0	0
$229$	11.2349i	0.742423i	0.928548	+	0.371211i	$0.121057\pi$
$229$	11.2349i	0.742423i	−0.928548	+	0.371211i	$0.878943\pi$
$230$	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
$237$	− 1.08036i	− 0.0701771i
$238$	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
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	44.0957i	2.81717i
$246$	0	0
$247$	−3.49613	−0.222453
$248$	0	0
$249$	−2.47214	−0.156665
$250$	0	0
$251$	− 15.4164i	− 0.973075i	−0.873660	−	0.486538i	$-0.838260\pi$
$251$	− 15.4164i	− 0.973075i	0.873660	−	0.486538i	$-0.161740\pi$
$252$	0	0
$253$	− 13.9845i	− 0.879199i
$254$	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
$259$	9.52786i	0.592033i
$260$	0	0
$261$	0.333851i	0.0206648i
$262$	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
$265$	−16.8328	−1.03403
$266$	0	0
$267$	− 10.0000i	− 0.611990i
$268$	0	0
$269$	11.6476i	0.710164i	0.934835	+	0.355082i	$0.115547\pi$
$269$	11.6476i	0.710164i	−0.934835	+	0.355082i	$0.884453\pi$
$270$	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
$275$	12.3607i	0.745377i
$276$	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
$279$	−10.2333	−0.612654
$280$	0	0
$281$	31.8885	1.90231	0.951156	−	0.308712i	$-0.0998980\pi$
$281$	31.8885	1.90231	0.951156	+	0.308712i	$0.0998980\pi$
$282$	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
$285$	− 7.81758i	− 0.463073i
$286$	0	0
$287$	29.6197	1.74839
$288$	0	0
$289$	24.8885	1.46403
$290$	0	0
$291$	− 4.94427i	− 0.289838i
$292$	0	0
$293$	16.6367i	0.971927i	0.873979	+	0.485964i	$0.161531\pi$
$293$	16.6367i	0.971927i	−0.873979	+	0.485964i	$0.838469\pi$
$294$	0	0
$295$	28.2843	1.64677
$296$	0	0
$297$	2.47214	0.143448
$298$	0	0
$299$	− 8.00000i	− 0.462652i
$300$	0	0
$301$	− 47.9256i	− 2.76239i
$302$	0	0
$303$	5.99070	0.344157
$304$	0	0
$305$	−33.4164	−1.91342
$306$	0	0
$307$	− 29.8885i	− 1.70583i	−0.522050	−	0.852915i	$-0.674833\pi$
$307$	− 29.8885i	− 1.70583i	0.522050	−	0.852915i	$-0.325167\pi$
$308$	0	0
$309$	− 4.57649i	− 0.260347i
$310$	0	0
$311$	−30.9551	−1.75530	−0.877651	−	0.479301i	$-0.840890\pi$
$311$	−30.9551	−1.75530	−0.877651	+	0.479301i	$0.840890\pi$
$312$	0	0
$313$	−3.05573	−0.172720	−0.0863600	−	0.996264i	$-0.527524\pi$
$313$	−3.05573	−0.172720	−0.0863600	+	0.996264i	$0.527524\pi$
$314$	0	0
$315$	14.4721i	0.815412i
$316$	0	0
$317$	10.9799i	0.616690i	0.951275	+	0.308345i	$0.0997752\pi$
$317$	10.9799i	0.616690i	−0.951275	+	0.308345i	$0.900225\pi$
$318$	0	0
$319$	−0.825324	−0.0462093
$320$	0	0
$321$	16.9443	0.945737
$322$	0	0
$323$	16.0000i	0.890264i
$324$	0	0
$325$	7.07107i	0.392232i
$326$	0	0
$327$	8.40647	0.464879
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0.944272i	0.0519019i	0.999663	+	0.0259509i	$0.00826137\pi$
$331$	0.944272i	0.0519019i	−0.999663	+	0.0259509i	$0.991739\pi$
$332$	0	0
$333$	2.08191i	0.114088i
$334$	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
$339$	2.00000i	0.108625i
$340$	0	0
$341$	− 25.2982i	− 1.36998i
$342$	0	0
$343$	−31.7804	−1.71598
$344$	0	0
$345$	17.8885	0.963087
$346$	0	0
$347$	13.5279i	0.726214i	0.931747	+	0.363107i	$0.118284\pi$
$347$	13.5279i	0.726214i	−0.931747	+	0.363107i	$0.881716\pi$
$348$	0	0
$349$	− 21.8809i	− 1.17126i	−0.810579	−	0.585629i	$-0.800848\pi$
$349$	− 21.8809i	− 1.17126i	0.810579	−	0.585629i	$-0.199152\pi$
$350$	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
$355$	11.0557i	0.586777i
$356$	0	0
$357$	− 29.6197i	− 1.56764i
$358$	0	0
$359$	26.1235	1.37875	0.689374	−	0.724406i	$-0.257886\pi$
$359$	26.1235	1.37875	0.689374	+	0.724406i	$0.257886\pi$
$360$	0	0
$361$	12.8885	0.678344
$362$	0	0
$363$	− 4.88854i	− 0.256582i
$364$	0	0
$365$	47.2579i	2.47359i
$366$	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
$371$	− 24.3607i	− 1.26474i
$372$	0	0
$373$	− 9.23179i	− 0.478004i	−0.971019	−	0.239002i	$-0.923180\pi$
$373$	− 9.23179i	− 0.478004i	0.971019	−	0.239002i	$-0.0768203\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.472136	−0.0243162
$378$	0	0
$379$	− 0.583592i	− 0.0299771i	−0.999888	−	0.0149886i	$-0.995229\pi$
$379$	− 0.583592i	− 0.0299771i	0.999888	−	0.0149886i	$-0.00477118\pi$
$380$	0	0
$381$	8.07262i	0.413573i
$382$	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
$387$	− 10.4721i	− 0.532329i
$388$	0	0
$389$	10.8222i	0.548709i	0.961629	+	0.274355i	$0.0884642\pi$
$389$	10.8222i	0.548709i	−0.961629	+	0.274355i	$0.911536\pi$
$390$	0	0
$391$	−36.6119	−1.85154
$392$	0	0
$393$	0.944272	0.0476322
$394$	0	0
$395$	3.41641i	0.171898i
$396$	0	0
$397$	28.8732i	1.44910i	0.689221	+	0.724551i	$0.257953\pi$
$397$	28.8732i	1.44910i	−0.689221	+	0.724551i	$0.742047\pi$
$398$	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
$403$	− 14.4721i	− 0.720908i
$404$	0	0
$405$	3.16228i	0.157135i
$406$	0	0
$407$	−5.14678	−0.255116
$408$	0	0
$409$	−4.94427	−0.244479	−0.122239	−	0.992501i	$-0.539008\pi$
$409$	−4.94427	−0.244479	−0.122239	+	0.992501i	$0.539008\pi$
$410$	0	0
$411$	− 6.47214i	− 0.319247i
$412$	0	0
$413$	40.9334i	2.01420i
$414$	0	0
$415$	7.81758	0.383750
$416$	0	0
$417$	−0.944272	−0.0462412
$418$	0	0
$419$	10.4721i	0.511597i	0.966730	+	0.255799i	$0.0823384\pi$
$419$	10.4721i	0.511597i	−0.966730	+	0.255799i	$0.917662\pi$
$420$	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
$425$	32.3607	1.56972
$426$	0	0
$427$	− 48.3607i	− 2.34034i
$428$	0	0
$429$	3.49613i	0.168795i
$430$	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
$435$	− 1.05573i	− 0.0506183i
$436$	0	0
$437$	− 13.9845i	− 0.668970i
$438$	0	0
$439$	25.0432	1.19525	0.597623	−	0.801777i	$-0.296112\pi$
$439$	25.0432	1.19525	0.597623	+	0.801777i	$0.296112\pi$
$440$	0	0
$441$	−13.9443	−0.664013
$442$	0	0
$443$	15.4164i	0.732456i	0.930525	+	0.366228i	$0.119351\pi$
$443$	15.4164i	0.732456i	−0.930525	+	0.366228i	$0.880649\pi$
$444$	0	0
$445$	31.6228i	1.49906i
$446$	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
$451$	16.0000i	0.753411i
$452$	0	0
$453$	− 11.5687i	− 0.543547i
$454$	0	0
$455$	−20.4667	−0.959493
$456$	0	0
$457$	−20.9443	−0.979732	−0.489866	−	0.871798i	$-0.662954\pi$
$457$	−20.9443	−0.979732	−0.489866	+	0.871798i	$0.662954\pi$
$458$	0	0
$459$	− 6.47214i	− 0.302093i
$460$	0	0
$461$	2.49458i	0.116184i	0.998311	+	0.0580920i	$0.0185017\pi$
$461$	2.49458i	0.116184i	−0.998311	+	0.0580920i	$0.981498\pi$
$462$	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
$467$	31.4164i	1.45378i	0.686755	+	0.726889i	$0.259035\pi$
$467$	31.4164i	1.45378i	−0.686755	+	0.726889i	$0.740965\pi$
$468$	0	0
$469$	54.9179i	2.53587i
$470$	0	0
$471$	−0.746512	−0.0343975
$472$	0	0
$473$	25.8885	1.19036
$474$	0	0
$475$	12.3607i	0.567147i
$476$	0	0
$477$	− 5.32300i	− 0.243724i
$478$	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
$483$	25.8885i	1.17797i
$484$	0	0
$485$	15.6352i	0.709956i
$486$	0	0
$487$	−2.41577	−0.109469	−0.0547344	−	0.998501i	$-0.517431\pi$
$487$	−2.41577	−0.109469	−0.0547344	+	0.998501i	$0.517431\pi$
$488$	0	0
$489$	−5.52786	−0.249979
$490$	0	0
$491$	− 13.8885i	− 0.626781i	−0.949624	−	0.313391i	$-0.898535\pi$
$491$	− 13.8885i	− 0.626781i	0.949624	−	0.313391i	$-0.101465\pi$
$492$	0	0
$493$	2.16073i	0.0973142i
$494$	0	0
$495$	−7.81758	−0.351374
$496$	0	0
$497$	−16.0000	−0.717698
$498$	0	0
$499$	7.05573i	0.315858i	0.987450	+	0.157929i	$0.0504817\pi$
$499$	7.05573i	0.315858i	−0.987450	+	0.157929i	$0.949518\pi$
$500$	0	0
$501$	− 7.81758i	− 0.349264i
$502$	0	0
$503$	33.1158	1.47656	0.738280	−	0.674494i	$-0.235638\pi$
$503$	33.1158	1.47656	0.738280	+	0.674494i	$0.235638\pi$
$504$	0	0
$505$	−18.9443	−0.843009
$506$	0	0
$507$	− 11.0000i	− 0.488527i
$508$	0	0
$509$	− 33.6073i	− 1.48962i	−0.667279	−	0.744808i	$-0.732541\pi$
$509$	− 33.6073i	− 1.48962i	0.667279	−	0.744808i	$-0.267459\pi$
$510$	0	0
$511$	−68.3923	−3.02550
$512$	0	0
$513$	2.47214	0.109147
$514$	0	0
$515$	14.4721i	0.637719i
$516$	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	$-0.616673\pi$
$521$	−16.3607	−0.716774	−0.358387	+	0.933573i	$0.616673\pi$
$522$	0	0
$523$	28.3607i	1.24013i	0.784552	+	0.620063i	$0.212893\pi$
$523$	28.3607i	1.24013i	−0.784552	+	0.620063i	$0.787107\pi$
$524$	0	0
$525$	− 22.8825i	− 0.998672i
$526$	0	0
$527$	−66.2316	−2.88509
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	8.94427i	0.388148i
$532$	0	0
$533$	9.15298i	0.396460i
$534$	0	0
$535$	−53.5825	−2.31657
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	− 34.4721i	− 1.48482i
$540$	0	0
$541$	− 23.8840i	− 1.02685i	−0.858133	−	0.513427i	$-0.828376\pi$
$541$	− 23.8840i	− 1.02685i	0.858133	−	0.513427i	$-0.171624\pi$
$542$	0	0
$543$	−15.5563	−0.667587
$544$	0	0
$545$	−26.5836	−1.13872
$546$	0	0
$547$	12.3607i	0.528505i	0.964454	+	0.264252i	$0.0851251\pi$
$547$	12.3607i	0.528505i	−0.964454	+	0.264252i	$0.914875\pi$
$548$	0	0
$549$	− 10.5672i	− 0.450997i
$550$	0	0
$551$	−0.825324	−0.0351600
$552$	0	0
$553$	−4.94427	−0.210252
$554$	0	0
$555$	− 6.58359i	− 0.279458i
$556$	0	0
$557$	15.8114i	0.669950i	0.942227	+	0.334975i	$0.108728\pi$
$557$	15.8114i	0.669950i	−0.942227	+	0.334975i	$0.891272\pi$
$558$	0	0
$559$	14.8098	0.626389
$560$	0	0
$561$	16.0000	0.675521
$562$	0	0
$563$	− 10.4721i	− 0.441348i	−0.975348	−	0.220674i	$-0.929174\pi$
$563$	− 10.4721i	− 0.441348i	0.975348	−	0.220674i	$-0.0708257\pi$
$564$	0	0
$565$	− 6.32456i	− 0.266076i
$566$	0	0
$567$	−4.57649	−0.192195
$568$	0	0
$569$	19.4164	0.813978	0.406989	−	0.913433i	$-0.366579\pi$
$569$	19.4164	0.813978	0.406989	+	0.913433i	$0.366579\pi$
$570$	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
$573$	2.16073i	0.0902656i
$574$	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
$579$	12.9443i	0.537946i
$580$	0	0
$581$	11.3137i	0.469372i
$582$	0	0
$583$	13.1592	0.544998
$584$	0	0
$585$	−4.47214	−0.184900
$586$	0	0
$587$	− 15.0557i	− 0.621416i	−0.950505	−	0.310708i	$-0.899434\pi$
$587$	− 15.0557i	− 0.621416i	0.950505	−	0.310708i	$-0.100566\pi$
$588$	0	0
$589$	− 25.2982i	− 1.04240i
$590$	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
$595$	93.6656i	3.83992i
$596$	0	0
$597$	− 13.7295i	− 0.561910i
$598$	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
$601$	−10.9443	−0.446426	−0.223213	−	0.974770i	$-0.571655\pi$
$601$	−10.9443	−0.446426	−0.223213	+	0.974770i	$0.571655\pi$
$602$	0	0
$603$	12.0000i	0.488678i
$604$	0	0
$605$	15.4589i	0.628495i
$606$	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
$613$	2.08191i	0.0840877i	0.999116	+	0.0420439i	$0.0133869\pi$
$613$	2.08191i	0.0840877i	−0.999116	+	0.0420439i	$0.986613\pi$
$614$	0	0
$615$	−20.4667	−0.825297
$616$	0	0
$617$	−19.8885	−0.800683	−0.400341	−	0.916366i	$-0.631108\pi$
$617$	−19.8885	−0.800683	−0.400341	+	0.916366i	$0.631108\pi$
$618$	0	0
$619$	29.8885i	1.20132i	0.799504	+	0.600661i	$0.205096\pi$
$619$	29.8885i	1.20132i	−0.799504	+	0.600661i	$0.794904\pi$
$620$	0	0
$621$	5.65685i	0.227002i
$622$	0	0
$623$	−45.7649	−1.83353
$624$	0	0
$625$	−25.0000	−1.00000
$626$	0	0
$627$	6.11146i	0.244068i
$628$	0	0
$629$	13.4744i	0.537261i
$630$	0	0
$631$	29.8747	1.18929	0.594647	−	0.803987i	$-0.297292\pi$
$631$	29.8747	1.18929	0.594647	+	0.803987i	$0.297292\pi$
$632$	0	0
$633$	−24.9443	−0.991446
$634$	0	0
$635$	− 25.5279i	− 1.01304i
$636$	0	0
$637$	− 19.7202i	− 0.781342i
$638$	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
$643$	− 36.3607i	− 1.43393i	−0.697112	−	0.716963i	$-0.745532\pi$
$643$	− 36.3607i	− 1.43393i	0.697112	−	0.716963i	$-0.254468\pi$
$644$	0	0
$645$	33.1158i	1.30393i
$646$	0	0
$647$	8.32766	0.327394	0.163697	−	0.986511i	$-0.447658\pi$
$647$	8.32766	0.327394	0.163697	+	0.986511i	$0.447658\pi$
$648$	0	0
$649$	−22.1115	−0.867951
$650$	0	0
$651$	46.8328i	1.83552i
$652$	0	0
$653$	6.81603i	0.266732i	0.991067	+	0.133366i	$0.0425785\pi$
$653$	6.81603i	0.266732i	−0.991067	+	0.133366i	$0.957421\pi$
$654$	0	0
$655$	−2.98605	−0.116675
$656$	0	0
$657$	−14.9443	−0.583032
$658$	0	0
$659$	4.00000i	0.155818i	0.996960	+	0.0779089i	$0.0248243\pi$
$659$	4.00000i	0.155818i	−0.996960	+	0.0779089i	$0.975176\pi$
$660$	0	0
$661$	2.23954i	0.0871079i	0.999051	+	0.0435540i	$0.0138680\pi$
$661$	2.23954i	0.0871079i	−0.999051	+	0.0435540i	$0.986132\pi$
$662$	0	0
$663$	9.15298	0.355472
$664$	0	0
$665$	−35.7771	−1.38738
$666$	0	0
$667$	− 1.88854i	− 0.0731247i
$668$	0	0
$669$	− 10.2333i	− 0.395644i
$670$	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
$675$	− 5.00000i	− 0.192450i
$676$	0	0
$677$	44.7634i	1.72040i	0.509960	+	0.860198i	$0.329660\pi$
$677$	44.7634i	1.72040i	−0.509960	+	0.860198i	$0.670340\pi$
$678$	0	0
$679$	−22.6274	−0.868361
$680$	0	0
$681$	5.52786	0.211828
$682$	0	0
$683$	− 23.4164i	− 0.896004i	−0.894033	−	0.448002i	$-0.852136\pi$
$683$	− 23.4164i	− 0.896004i	0.894033	−	0.448002i	$-0.147864\pi$
$684$	0	0
$685$	20.4667i	0.781992i
$686$	0	0
$687$	11.2349	0.428638
$688$	0	0
$689$	7.52786	0.286789
$690$	0	0
$691$	34.4721i	1.31138i	0.755029	+	0.655691i	$0.227623\pi$
$691$	34.4721i	1.31138i	−0.755029	+	0.655691i	$0.772377\pi$
$692$	0	0
$693$	− 11.3137i	− 0.429772i
$694$	0	0
$695$	2.98605	0.113267
$696$	0	0
$697$	41.8885	1.58664
$698$	0	0
$699$	− 10.0000i	− 0.378235i
$700$	0	0
$701$	− 3.98760i	− 0.150610i	−0.997161	−	0.0753048i	$-0.976007\pi$
$701$	− 3.98760i	− 0.150610i	0.997161	−	0.0753048i	$-0.0239930\pi$
$702$	0	0
$703$	−5.14678	−0.194114
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 27.4164i	− 1.03110i
$708$	0	0
$709$	− 8.56409i	− 0.321631i	−0.986984	−	0.160816i	$-0.948588\pi$
$709$	− 8.56409i	− 0.321631i	0.986984	−	0.160816i	$-0.0514125\pi$
$710$	0	0
$711$	−1.08036	−0.0405168
$712$	0	0
$713$	57.8885	2.16794
$714$	0	0
$715$	− 11.0557i	− 0.413461i
$716$	0	0
$717$	− 13.4744i	− 0.503212i
$718$	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
$725$	1.66925i	0.0619945i
$726$	0	0
$727$	−11.5687	−0.429061	−0.214531	−	0.976717i	$-0.568822\pi$
$727$	−11.5687	−0.429061	−0.214531	+	0.976717i	$0.568822\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 67.7771i	− 2.50683i
$732$	0	0
$733$	− 5.73567i	− 0.211852i	−0.994374	−	0.105926i	$-0.966219\pi$
$733$	− 5.73567i	− 0.211852i	0.994374	−	0.105926i	$-0.0337806\pi$
$734$	0	0
$735$	44.0957	1.62649
$736$	0	0
$737$	−29.6656	−1.09275
$738$	0	0
$739$	15.0557i	0.553834i	0.960894	+	0.276917i	$0.0893127\pi$
$739$	15.0557i	0.553834i	−0.960894	+	0.276917i	$0.910687\pi$
$740$	0	0
$741$	3.49613i	0.128433i
$742$	0	0
$743$	−14.8098	−0.543320	−0.271660	−	0.962393i	$-0.587573\pi$
$743$	−14.8098	−0.543320	−0.271660	+	0.962393i	$0.587573\pi$
$744$	0	0
$745$	−47.8885	−1.75450
$746$	0	0
$747$	2.47214i	0.0904507i
$748$	0	0
$749$	− 77.5453i	− 2.83344i
$750$	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
$755$	36.5836i	1.33141i
$756$	0	0
$757$	− 11.2349i	− 0.408339i	−0.978936	−	0.204170i	$-0.934551\pi$
$757$	− 11.2349i	− 0.408339i	0.978936	−	0.204170i	$-0.0654494\pi$
$758$	0	0
$759$	−13.9845	−0.507606
$760$	0	0
$761$	−19.4164	−0.703844	−0.351922	−	0.936029i	$-0.614472\pi$
$761$	−19.4164	−0.703844	−0.351922	+	0.936029i	$0.614472\pi$
$762$	0	0
$763$	− 38.4721i	− 1.39278i
$764$	0	0
$765$	20.4667i	0.739975i
$766$	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
$771$	14.0000i	0.504198i
$772$	0	0
$773$	− 12.3153i	− 0.442949i	−0.975166	−	0.221475i	$-0.928913\pi$
$773$	− 12.3153i	− 0.442949i	0.975166	−	0.221475i	$-0.0710870\pi$
$774$	0	0
$775$	−51.1667	−1.83796
$776$	0	0
$777$	9.52786	0.341810
$778$	0	0
$779$	16.0000i	0.573259i
$780$	0	0
$781$	− 8.64290i	− 0.309267i
$782$	0	0
$783$	0.333851	0.0119308
$784$	0	0
$785$	2.36068	0.0842563
$786$	0	0
$787$	5.52786i	0.197047i	0.995135	+	0.0985235i	$0.0314120\pi$
$787$	5.52786i	0.197047i	−0.995135	+	0.0985235i	$0.968588\pi$
$788$	0	0
$789$	12.6491i	0.450320i
$790$	0	0
$791$	9.15298	0.325443
$792$	0	0
$793$	14.9443	0.530687
$794$	0	0
$795$	16.8328i	0.596998i
$796$	0	0
$797$	39.1065i	1.38522i	0.721311	+	0.692612i	$0.243540\pi$
$797$	39.1065i	1.38522i	−0.721311	+	0.692612i	$0.756460\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−10.0000	−0.353333
$802$	0	0
$803$	− 36.9443i	− 1.30374i
$804$	0	0
$805$	− 81.8668i	− 2.88542i
$806$	0	0
$807$	11.6476	0.410013
$808$	0	0
$809$	−3.41641	−0.120115	−0.0600573	−	0.998195i	$-0.519128\pi$
$809$	−3.41641	−0.120115	−0.0600573	+	0.998195i	$0.519128\pi$
$810$	0	0
$811$	− 33.3050i	− 1.16950i	−0.811215	−	0.584748i	$-0.801194\pi$
$811$	− 33.3050i	− 1.16950i	0.811215	−	0.584748i	$-0.198806\pi$
$812$	0	0
$813$	8.07262i	0.283119i
$814$	0	0
$815$	17.4806	0.612320
$816$	0	0
$817$	25.8885	0.905725
$818$	0	0
$819$	− 6.47214i	− 0.226155i
$820$	0	0
$821$	44.6057i	1.55675i	0.627799	+	0.778375i	$0.283956\pi$
$821$	44.6057i	1.55675i	−0.627799	+	0.778375i	$0.716044\pi$
$822$	0	0
$823$	4.57649	0.159526	0.0797632	−	0.996814i	$-0.474584\pi$
$823$	4.57649	0.159526	0.0797632	+	0.996814i	$0.474584\pi$
$824$	0	0
$825$	12.3607	0.430344
$826$	0	0
$827$	53.8885i	1.87389i	0.349479	+	0.936944i	$0.386359\pi$
$827$	53.8885i	1.87389i	−0.349479	+	0.936944i	$0.613641\pi$
$828$	0	0
$829$	19.7202i	0.684910i	0.939534	+	0.342455i	$0.111258\pi$
$829$	19.7202i	0.684910i	−0.939534	+	0.342455i	$0.888742\pi$
$830$	0	0
$831$	−18.3848	−0.637761
$832$	0	0
$833$	−90.2492	−3.12695
$834$	0	0
$835$	24.7214i	0.855518i
$836$	0	0
$837$	10.2333i	0.353716i
$838$	0	0
$839$	−44.4295	−1.53388	−0.766939	−	0.641721i	$-0.778221\pi$
$839$	−44.4295	−1.53388	−0.766939	+	0.641721i	$0.778221\pi$
$840$	0	0
$841$	28.8885	0.996157
$842$	0	0
$843$	− 31.8885i	− 1.09830i
$844$	0	0
$845$	34.7851i	1.19664i
$846$	0	0
$847$	−22.3724	−0.768724
$848$	0	0
$849$	−26.8328	−0.920900
$850$	0	0
$851$	− 11.7771i	− 0.403713i
$852$	0	0
$853$	17.7171i	0.606621i	0.952892	+	0.303311i	$0.0980920\pi$
$853$	17.7171i	0.606621i	−0.952892	+	0.303311i	$0.901908\pi$
$854$	0	0
$855$	−7.81758	−0.267356
$856$	0	0
$857$	−28.5836	−0.976397	−0.488198	−	0.872733i	$-0.662346\pi$
$857$	−28.5836	−0.976397	−0.488198	+	0.872733i	$0.662346\pi$
$858$	0	0
$859$	− 45.5279i	− 1.55339i	−0.629876	−	0.776695i	$-0.716894\pi$
$859$	− 45.5279i	− 1.55339i	0.629876	−	0.776695i	$-0.283106\pi$
$860$	0	0
$861$	− 29.6197i	− 1.00944i
$862$	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
$867$	− 24.8885i	− 0.845259i
$868$	0	0
$869$	− 2.67080i	− 0.0906008i
$870$	0	0
$871$	−16.9706	−0.575026
$872$	0	0
$873$	−4.94427	−0.167338
$874$	0	0
$875$	0	0
$876$	0	0
$877$	35.8654i	1.21109i	0.795811	+	0.605545i	$0.207045\pi$
$877$	35.8654i	1.21109i	−0.795811	+	0.605545i	$0.792955\pi$
$878$	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
$883$	− 18.4721i	− 0.621637i	−0.950469	−	0.310818i	$-0.899397\pi$
$883$	− 18.4721i	− 0.621637i	0.950469	−	0.310818i	$-0.100603\pi$
$884$	0	0
$885$	− 28.2843i	− 0.950765i
$886$	0	0
$887$	42.2688	1.41925	0.709623	−	0.704581i	$-0.248865\pi$
$887$	42.2688	1.41925	0.709623	+	0.704581i	$0.248865\pi$
$888$	0	0
$889$	36.9443	1.23907
$890$	0	0
$891$	− 2.47214i	− 0.0828197i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	37.9473	1.26844
$896$	0	0
$897$	−8.00000	−0.267112
$898$	0	0
$899$	− 3.41641i	− 0.113944i
$900$	0	0
$901$	− 34.4512i	− 1.14774i
$902$	0	0
$903$	−47.9256	−1.59487
$904$	0	0
$905$	49.1935	1.63525
$906$	0	0
$907$	− 51.1935i	− 1.69985i	−0.526902	−	0.849926i	$-0.676647\pi$
$907$	− 51.1935i	− 1.69985i	0.526902	−	0.849926i	$-0.323353\pi$
$908$	0	0
$909$	− 5.99070i	− 0.198699i
$910$	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
$915$	33.4164i	1.10471i
$916$	0	0
$917$	− 4.32145i	− 0.142707i
$918$	0	0
$919$	34.1962	1.12803	0.564014	−	0.825765i	$-0.309257\pi$
$919$	34.1962	1.12803	0.564014	+	0.825765i	$0.309257\pi$
$920$	0	0
$921$	−29.8885	−0.984861
$922$	0	0
$923$	− 4.94427i	− 0.162743i
$924$	0	0
$925$	10.4096i	0.342265i
$926$	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
$931$	− 34.4721i	− 1.12978i
$932$	0	0
$933$	30.9551i	1.01342i
$934$	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
$939$	3.05573i	0.0997199i
$940$	0	0
$941$	25.1220i	0.818954i	0.912321	+	0.409477i	$0.134289\pi$
$941$	25.1220i	0.818954i	−0.912321	+	0.409477i	$0.865711\pi$
$942$	0	0
$943$	−36.6119	−1.19225
$944$	0	0
$945$	14.4721	0.470779
$946$	0	0
$947$	47.7771i	1.55255i	0.630396	+	0.776273i	$0.282892\pi$
$947$	47.7771i	1.55255i	−0.630396	+	0.776273i	$0.717108\pi$
$948$	0	0
$949$	− 21.1344i	− 0.686051i
$950$	0	0
$951$	10.9799	0.356046
$952$	0	0
$953$	9.52786	0.308638	0.154319	−	0.988021i	$-0.450682\pi$
$953$	9.52786	0.308638	0.154319	+	0.988021i	$0.450682\pi$
$954$	0	0
$955$	− 6.83282i	− 0.221105i
$956$	0	0
$957$	0.825324i	0.0266789i
$958$	0	0
$959$	−29.6197	−0.956469
$960$	0	0
$961$	73.7214	2.37811
$962$	0	0
$963$	− 16.9443i	− 0.546022i
$964$	0	0
$965$	− 40.9334i	− 1.31769i
$966$	0	0
$967$	−18.5610	−0.596882	−0.298441	−	0.954428i	$-0.596467\pi$
$967$	−18.5610	−0.596882	−0.298441	+	0.954428i	$0.596467\pi$
$968$	0	0
$969$	16.0000	0.513994
$970$	0	0
$971$	− 20.3607i	− 0.653405i	−0.945127	−	0.326703i	$-0.894062\pi$
$971$	− 20.3607i	− 0.653405i	0.945127	−	0.326703i	$-0.105938\pi$
$972$	0	0
$973$	4.32145i	0.138539i
$974$	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
$979$	− 24.7214i	− 0.790098i
$980$	0	0
$981$	− 8.40647i	− 0.268398i
$982$	0	0
$983$	−53.5825	−1.70902	−0.854508	−	0.519438i	$-0.826141\pi$
$983$	−53.5825	−1.70902	−0.854508	+	0.519438i	$0.826141\pi$
$984$	0	0
$985$	76.8328	2.44810
$986$	0	0
$987$	0	0
$988$	0	0
$989$	59.2393i	1.88370i
$990$	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
$995$	43.4164i	1.37639i
$996$	0	0
$997$	− 29.0308i	− 0.919414i	−0.888071	−	0.459707i	$-0.847954\pi$
$997$	− 29.0308i	− 0.919414i	0.888071	−	0.459707i	$-0.152046\pi$
$998$	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.d.g.1537.3		8
4.3	odd	2	inner	3072.2.d.g.1537.8		8
8.3	odd	2	inner	3072.2.d.g.1537.2		8
8.5	even	2	inner	3072.2.d.g.1537.5		8
16.3	odd	4		3072.2.a.k.1.3		4
16.5	even	4		3072.2.a.k.1.2		4
16.11	odd	4		3072.2.a.q.1.1		4
16.13	even	4		3072.2.a.q.1.4		4
32.3	odd	8		1536.2.j.g.385.1	✓	8
32.5	even	8		1536.2.j.g.1153.3	yes	8
32.11	odd	8		1536.2.j.h.1153.4	yes	8
32.13	even	8		1536.2.j.h.385.2	yes	8
32.19	odd	8		1536.2.j.h.385.4	yes	8
32.21	even	8		1536.2.j.h.1153.2	yes	8
32.27	odd	8		1536.2.j.g.1153.1	yes	8
32.29	even	8		1536.2.j.g.385.3	yes	8
48.5	odd	4		9216.2.a.bj.1.4		4
48.11	even	4		9216.2.a.bd.1.3		4
48.29	odd	4		9216.2.a.bd.1.2		4
48.35	even	4		9216.2.a.bj.1.1		4
96.5	odd	8		4608.2.k.bf.1153.3		8
96.11	even	8		4608.2.k.bg.1153.2		8
96.29	odd	8		4608.2.k.bf.3457.4		8
96.35	even	8		4608.2.k.bf.3457.3		8
96.53	odd	8		4608.2.k.bg.1153.1		8
96.59	even	8		4608.2.k.bf.1153.4		8
96.77	odd	8		4608.2.k.bg.3457.2		8
96.83	even	8		4608.2.k.bg.3457.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.j.g.385.1	✓	8	32.3	odd	8
1536.2.j.g.385.3	yes	8	32.29	even	8
1536.2.j.g.1153.1	yes	8	32.27	odd	8
1536.2.j.g.1153.3	yes	8	32.5	even	8
1536.2.j.h.385.2	yes	8	32.13	even	8
1536.2.j.h.385.4	yes	8	32.19	odd	8
1536.2.j.h.1153.2	yes	8	32.21	even	8
1536.2.j.h.1153.4	yes	8	32.11	odd	8
3072.2.a.k.1.2		4	16.5	even	4
3072.2.a.k.1.3		4	16.3	odd	4
3072.2.a.q.1.1		4	16.11	odd	4
3072.2.a.q.1.4		4	16.13	even	4
3072.2.d.g.1537.2		8	8.3	odd	2	inner
3072.2.d.g.1537.3		8	1.1	even	1	trivial
3072.2.d.g.1537.5		8	8.5	even	2	inner
3072.2.d.g.1537.8		8	4.3	odd	2	inner
4608.2.k.bf.1153.3		8	96.5	odd	8
4608.2.k.bf.1153.4		8	96.59	even	8
4608.2.k.bf.3457.3		8	96.35	even	8
4608.2.k.bf.3457.4		8	96.29	odd	8
4608.2.k.bg.1153.1		8	96.53	odd	8
4608.2.k.bg.1153.2		8	96.11	even	8
4608.2.k.bg.3457.1		8	96.83	even	8
4608.2.k.bg.3457.2		8	96.77	odd	8
9216.2.a.bd.1.2		4	48.29	odd	4
9216.2.a.bd.1.3		4	48.11	even	4
9216.2.a.bj.1.1		4	48.35	even	4
9216.2.a.bj.1.4		4	48.5	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)

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

Related objects

Downloads

Learn more