LMFDB - Embedded newform 4608.2.k.h.1153.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4608,2,Mod(1153,4608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4608, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 0]))

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

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

chi := DirichletCharacter("4608.1153");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4608 = 2^{9} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4608.k (of order $4$, degree $2$, 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:	$36.7950652514$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			1153.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4608.1153
Dual form			4608.2.k.h.3457.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.00000i) q^{5} +O(q^{10})$	$q+(-1.00000 + 1.00000i) q^{5} +(-1.00000 - 1.00000i) q^{13} +2.00000 q^{17} +3.00000i q^{25} +(-3.00000 - 3.00000i) q^{29} +(5.00000 - 5.00000i) q^{37} +10.0000i q^{41} +7.00000 q^{49} +(-9.00000 + 9.00000i) q^{53} +(1.00000 + 1.00000i) q^{61} +2.00000 q^{65} -6.00000i q^{73} +(-2.00000 + 2.00000i) q^{85} +16.0000i q^{89} -8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5}+O(q^{10}) $ 2 * q - 2 * q^5	$ 2 q - 2 q^{5} - 2 q^{13} + 4 q^{17} - 6 q^{29} + 10 q^{37} + 14 q^{49} - 18 q^{53} + 2 q^{61} + 4 q^{65} - 4 q^{85} - 16 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^13 + 4 * q^17 - 6 * q^29 + 10 * q^37 + 14 * q^49 - 18 * q^53 + 2 * q^61 + 4 * q^65 - 4 * q^85 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times$.

$n$	$2053$	$3583$	$4097$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.00000	+	1.00000i	−0.447214	+	0.447214i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−1.00000	+	1.00000i	−0.447214	+	0.447214i	0.447214	+	0.894427i	$0.352416\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$11$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$12$		0			0
$13$	−1.00000	−	1.00000i	−0.277350	−	0.277350i	0.554700	−	0.832050i	$-0.312833\pi$
$13$	−1.00000	−	1.00000i	−0.277350	−	0.277350i	−0.832050	+	0.554700i	$0.812833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.00000			0.485071			0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000			0.485071			0.242536	+	0.970143i	$0.422021\pi$
$18$		0			0
$19$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$19$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$			3.00000i			0.600000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.00000	−	3.00000i	−0.557086	−	0.557086i	0.371391	−	0.928477i	$-0.378881\pi$
$29$	−3.00000	−	3.00000i	−0.557086	−	0.557086i	−0.928477	+	0.371391i	$0.878881\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.00000	−	5.00000i	0.821995	−	0.821995i	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	5.00000	−	5.00000i	0.821995	−	0.821995i	0.986394	+	0.164399i	$0.0525685\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			10.0000i			1.56174i	0.624695	+	0.780869i	$0.285223\pi$
$41$			10.0000i			1.56174i	−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9.00000	+	9.00000i	−1.23625	+	1.23625i	−0.274721	+	0.961524i	$0.588586\pi$
$53$	−9.00000	+	9.00000i	−1.23625	+	1.23625i	−0.961524	+	0.274721i	$0.911414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$59$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$60$		0			0
$61$	1.00000	+	1.00000i	0.128037	+	0.128037i	0.768221	−	0.640184i	$-0.221142\pi$
$61$	1.00000	+	1.00000i	0.128037	+	0.128037i	−0.640184	+	0.768221i	$0.721142\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.00000			0.248069
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$	−2.00000	+	2.00000i	−0.216930	+	0.216930i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			16.0000i			1.69600i	0.529999	+	0.847998i	$0.322192\pi$
$89$			16.0000i			1.69600i	−0.529999	+	0.847998i	$0.677808\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−8.00000			−0.812277			−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000			−0.812277			−0.406138	+	0.913812i	$0.633125\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−9.00000	+	9.00000i	−0.895533	+	0.895533i	−0.995037	−	0.0995037i	$-0.968274\pi$
$101$	−9.00000	+	9.00000i	−0.895533	+	0.895533i	0.0995037	+	0.995037i	$0.468274\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$	−7.00000	−	7.00000i	−0.670478	−	0.670478i	0.287348	−	0.957826i	$-0.407226\pi$
$109$	−7.00000	−	7.00000i	−0.670478	−	0.670478i	−0.957826	+	0.287348i	$0.907226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	16.0000			1.50515			0.752577	−	0.658505i	$-0.228811\pi$
$113$	16.0000			1.50515			0.752577	+	0.658505i	$0.228811\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$			11.0000i			1.00000i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.00000	−	8.00000i	−0.715542	−	0.715542i
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$131$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$			22.0000i			1.87959i	0.341743	+	0.939793i	$0.388983\pi$
$137$			22.0000i			1.87959i	−0.341743	+	0.939793i	$0.611017\pi$
$138$		0			0
$139$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$139$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	6.00000			0.498273
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−17.0000	+	17.0000i	−1.39269	+	1.39269i	−0.573462	+	0.819232i	$0.694400\pi$
$149$	−17.0000	+	17.0000i	−1.39269	+	1.39269i	−0.819232	+	0.573462i	$0.805600\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	17.0000	+	17.0000i	1.35675	+	1.35675i	0.877896	+	0.478852i	$0.158947\pi$
$157$	17.0000	+	17.0000i	1.35675	+	1.35675i	0.478852	+	0.877896i	$0.341053\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$		−	11.0000i		−	0.846154i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−11.0000	−	11.0000i	−0.836315	−	0.836315i	0.152057	−	0.988372i	$-0.451410\pi$
$173$	−11.0000	−	11.0000i	−0.836315	−	0.836315i	−0.988372	+	0.152057i	$0.951410\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$179$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$180$		0			0
$181$	−19.0000	+	19.0000i	−1.41226	+	1.41226i	−0.668965	+	0.743294i	$0.733262\pi$
$181$	−19.0000	+	19.0000i	−1.41226	+	1.41226i	−0.743294	+	0.668965i	$0.766738\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			10.0000i			0.735215i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	−24.0000			−1.72756			−0.863779	−	0.503871i	$-0.831909\pi$
$193$	−24.0000			−1.72756			−0.863779	+	0.503871i	$0.831909\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−15.0000	+	15.0000i	−1.06871	+	1.06871i	−0.0712470	+	0.997459i	$0.522698\pi$
$197$	−15.0000	+	15.0000i	−1.06871	+	1.06871i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−10.0000	−	10.0000i	−0.698430	−	0.698430i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$211$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.00000	−	2.00000i	−0.134535	−	0.134535i
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$	13.0000	−	13.0000i	0.859064	−	0.859064i	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	13.0000	−	13.0000i	0.859064	−	0.859064i	0.991228	+	0.132164i	$0.0421925\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			16.0000i			1.04819i	0.851658	+	0.524097i	$0.175597\pi$
$233$			16.0000i			1.04819i	−0.851658	+	0.524097i	$0.824403\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	8.00000			0.515325			0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000			0.515325			0.257663	+	0.966235i	$0.417048\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−7.00000	+	7.00000i	−0.447214	+	0.447214i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$251$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	32.0000			1.99611			0.998053	−	0.0623783i	$-0.0198685\pi$
$257$	32.0000			1.99611			0.998053	+	0.0623783i	$0.0198685\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		−	18.0000i		−	1.10573i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3.00000	−	3.00000i	−0.182913	−	0.182913i	0.609711	−	0.792624i	$-0.291286\pi$
$269$	−3.00000	−	3.00000i	−0.182913	−	0.182913i	−0.792624	+	0.609711i	$0.791286\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−5.00000	+	5.00000i	−0.300421	+	0.300421i	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−5.00000	+	5.00000i	−0.300421	+	0.300421i	0.540758	+	0.841178i	$0.318138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			32.0000i			1.90896i	0.298275	+	0.954480i	$0.403589\pi$
$281$			32.0000i			1.90896i	−0.298275	+	0.954480i	$0.596411\pi$
$282$		0			0
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−15.0000	+	15.0000i	−0.876309	+	0.876309i	−0.993151	−	0.116841i	$-0.962723\pi$
$293$	−15.0000	+	15.0000i	−0.876309	+	0.876309i	0.116841	+	0.993151i	$0.462723\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.00000			−0.114520
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$			24.0000i			1.35656i	0.734803	+	0.678280i	$0.237274\pi$
$313$			24.0000i			1.35656i	−0.734803	+	0.678280i	$0.762726\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.00000	−	3.00000i	−0.168497	−	0.168497i	0.617822	−	0.786318i	$-0.288015\pi$
$317$	−3.00000	−	3.00000i	−0.168497	−	0.168497i	−0.786318	+	0.617822i	$0.788015\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	3.00000	−	3.00000i	0.166410	−	0.166410i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$331$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	18.0000			0.980522			0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000			0.980522			0.490261	+	0.871576i	$0.336901\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$	23.0000	+	23.0000i	1.23116	+	1.23116i	0.963518	+	0.267644i	$0.0862451\pi$
$349$	23.0000	+	23.0000i	1.23116	+	1.23116i	0.267644	+	0.963518i	$0.413755\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−16.0000			−0.851594			−0.425797	−	0.904819i	$-0.640006\pi$
$353$	−16.0000			−0.851594			−0.425797	+	0.904819i	$0.640006\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$		−	19.0000i		−	1.00000i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6.00000	+	6.00000i	0.314054	+	0.314054i
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−11.0000	+	11.0000i	−0.569558	+	0.569558i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	−11.0000	+	11.0000i	−0.569558	+	0.569558i	0.362446	+	0.932005i	$0.381942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			6.00000i			0.309016i
$378$		0			0
$379$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$379$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7.00000	+	7.00000i	−0.354914	+	0.354914i	−0.861934	−	0.507020i	$-0.830747\pi$
$389$	−7.00000	+	7.00000i	−0.354914	+	0.354914i	0.507020	+	0.861934i	$0.330747\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−25.0000	−	25.0000i	−1.25471	−	1.25471i	−0.953583	−	0.301131i	$-0.902636\pi$
$397$	−25.0000	−	25.0000i	−1.25471	−	1.25471i	−0.301131	−	0.953583i	$-0.597364\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2.00000			−0.0998752			−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000			−0.0998752			−0.0499376	+	0.998752i	$0.515902\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			40.0000i			1.97787i	0.148340	+	0.988936i	$0.452607\pi$
$409$			40.0000i			1.97787i	−0.148340	+	0.988936i	$0.547393\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$419$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$420$		0			0
$421$	29.0000	−	29.0000i	1.41337	−	1.41337i	0.682318	−	0.731055i	$-0.260972\pi$
$421$	29.0000	−	29.0000i	1.41337	−	1.41337i	0.731055	−	0.682318i	$-0.239028\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			6.00000i			0.291043i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		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
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$	−16.0000	−	16.0000i	−0.758473	−	0.758473i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	14.0000			0.660701			0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000			0.660701			0.330350	+	0.943858i	$0.392833\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	8.00000i		−	0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$		−	8.00000i		−	0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−29.0000	−	29.0000i	−1.35066	−	1.35066i	−0.884918	−	0.465746i	$-0.845786\pi$
$461$	−29.0000	−	29.0000i	−1.35066	−	1.35066i	−0.465746	−	0.884918i	$-0.654214\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	−10.0000			−0.455961
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.00000	−	8.00000i	0.363261	−	0.363261i
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$491$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$492$		0			0
$493$	−6.00000	−	6.00000i	−0.270226	−	0.270226i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$499$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		−	18.0000i		−	0.800989i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	27.0000	+	27.0000i	1.19675	+	1.19675i	0.975133	+	0.221621i	$0.0711348\pi$
$509$	27.0000	+	27.0000i	1.19675	+	1.19675i	0.221621	+	0.975133i	$0.428865\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	22.0000i		−	0.963837i	−0.876216	−	0.481919i	$-0.839940\pi$
$521$		−	22.0000i		−	0.963837i	0.876216	−	0.481919i	$-0.160060\pi$
$522$		0			0
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$	10.0000	−	10.0000i	0.433148	−	0.433148i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	31.0000	+	31.0000i	1.33279	+	1.33279i	0.902861	+	0.429934i	$0.141463\pi$
$541$	31.0000	+	31.0000i	1.33279	+	1.33279i	0.429934	+	0.902861i	$0.358537\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	14.0000			0.599694
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	5.00000	+	5.00000i	0.211857	+	0.211857i	0.805056	−	0.593199i	$-0.202135\pi$
$557$	5.00000	+	5.00000i	0.211857	+	0.211857i	−0.593199	+	0.805056i	$0.702135\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$	−16.0000	+	16.0000i	−0.673125	+	0.673125i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	26.0000i		−	1.08998i	−0.838444	−	0.544988i	$-0.816534\pi$
$569$		−	26.0000i		−	1.08998i	0.838444	−	0.544988i	$-0.183466\pi$
$570$		0			0
$571$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$571$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−2.00000			−0.0832611			−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000			−0.0832611			−0.0416305	+	0.999133i	$0.513255\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−16.0000			−0.657041			−0.328521	−	0.944497i	$-0.606550\pi$
$593$	−16.0000			−0.657041			−0.328521	+	0.944497i	$0.606550\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$			10.0000i			0.407909i	0.978980	+	0.203954i	$0.0653794\pi$
$601$			10.0000i			0.407909i	−0.978980	+	0.203954i	$0.934621\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−11.0000	−	11.0000i	−0.447214	−	0.447214i
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	35.0000	−	35.0000i	1.41364	−	1.41364i	0.686624	−	0.727013i	$-0.259092\pi$
$613$	35.0000	−	35.0000i	1.41364	−	1.41364i	0.727013	−	0.686624i	$-0.240908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	32.0000i		−	1.28827i	−0.764911	−	0.644136i	$-0.777217\pi$
$617$		−	32.0000i		−	1.28827i	0.764911	−	0.644136i	$-0.222783\pi$
$618$		0			0
$619$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$619$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	10.0000	−	10.0000i	0.398726	−	0.398726i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−7.00000	−	7.00000i	−0.277350	−	0.277350i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−50.0000			−1.97488			−0.987441	−	0.157991i	$-0.949498\pi$
$641$	−50.0000			−1.97488			−0.987441	+	0.157991i	$0.949498\pi$
$642$		0			0
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	35.0000	+	35.0000i	1.36966	+	1.36966i	0.860927	+	0.508729i	$0.169885\pi$
$653$	35.0000	+	35.0000i	1.36966	+	1.36966i	0.508729	+	0.860927i	$0.330115\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$659$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$660$		0			0
$661$	19.0000	−	19.0000i	0.739014	−	0.739014i	−0.233373	−	0.972387i	$-0.574976\pi$
$661$	19.0000	−	19.0000i	0.739014	−	0.739014i	0.972387	+	0.233373i	$0.0749763\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−24.0000			−0.925132			−0.462566	−	0.886585i	$-0.653071\pi$
$673$	−24.0000			−0.925132			−0.462566	+	0.886585i	$0.653071\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	25.0000	−	25.0000i	0.960828	−	0.960828i	−0.0384331	−	0.999261i	$-0.512237\pi$
$677$	25.0000	−	25.0000i	0.960828	−	0.960828i	0.999261	+	0.0384331i	$0.0122367\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		0			0
$685$	−22.0000	−	22.0000i	−0.840577	−	0.840577i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	18.0000			0.685745
$690$		0			0
$691$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$691$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$			20.0000i			0.757554i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−21.0000	−	21.0000i	−0.793159	−	0.793159i	0.188847	−	0.982006i	$-0.439525\pi$
$701$	−21.0000	−	21.0000i	−0.793159	−	0.793159i	−0.982006	+	0.188847i	$0.939525\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−37.0000	+	37.0000i	−1.38956	+	1.38956i	−0.563337	+	0.826227i	$0.690483\pi$
$709$	−37.0000	+	37.0000i	−1.38956	+	1.38956i	−0.826227	+	0.563337i	$0.809517\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	9.00000	−	9.00000i	0.334252	−	0.334252i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	25.0000	+	25.0000i	0.923396	+	0.923396i	0.997268	−	0.0738717i	$-0.0235355\pi$
$733$	25.0000	+	25.0000i	0.923396	+	0.923396i	−0.0738717	+	0.997268i	$0.523536\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$739$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		−	34.0000i		−	1.24566i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−35.0000	+	35.0000i	−1.27210	+	1.27210i	−0.327111	+	0.944986i	$0.606075\pi$
$757$	−35.0000	+	35.0000i	−1.27210	+	1.27210i	−0.944986	+	0.327111i	$0.893925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	38.0000i		−	1.37750i	−0.724999	−	0.688749i	$-0.758160\pi$
$761$		−	38.0000i		−	1.37750i	0.724999	−	0.688749i	$-0.241840\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−24.0000			−0.865462			−0.432731	−	0.901523i	$-0.642450\pi$
$769$	−24.0000			−0.865462			−0.432731	+	0.901523i	$0.642450\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	39.0000	−	39.0000i	1.40273	−	1.40273i	0.611448	−	0.791285i	$-0.290588\pi$
$773$	39.0000	−	39.0000i	1.40273	−	1.40273i	0.791285	−	0.611448i	$-0.209412\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−34.0000			−1.21351
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	2.00000i		−	0.0710221i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	37.0000	+	37.0000i	1.31061	+	1.31061i	0.920967	+	0.389640i	$0.127401\pi$
$797$	37.0000	+	37.0000i	1.31061	+	1.31061i	0.389640	+	0.920967i	$0.372599\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	10.0000i		−	0.351581i	−0.984428	−	0.175791i	$-0.943752\pi$
$809$		−	10.0000i		−	0.351581i	0.984428	−	0.175791i	$-0.0562482\pi$
$810$		0			0
$811$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$811$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	39.0000	−	39.0000i	1.36111	−	1.36111i	0.488603	−	0.872506i	$-0.337507\pi$
$821$	39.0000	−	39.0000i	1.36111	−	1.36111i	0.872506	−	0.488603i	$-0.162493\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$	−17.0000	−	17.0000i	−0.590434	−	0.590434i	0.347314	−	0.937749i	$-0.387094\pi$
$829$	−17.0000	−	17.0000i	−0.590434	−	0.590434i	−0.937749	+	0.347314i	$0.887094\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	14.0000			0.485071
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$		−	11.0000i		−	0.379310i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	11.0000	+	11.0000i	0.378412	+	0.378412i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	5.00000	−	5.00000i	0.171197	−	0.171197i	−0.616308	−	0.787505i	$-0.711372\pi$
$853$	5.00000	−	5.00000i	0.171197	−	0.171197i	0.787505	+	0.616308i	$0.211372\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	58.0000i		−	1.98124i	−0.136637	−	0.990621i	$-0.543630\pi$
$857$		−	58.0000i		−	1.98124i	0.136637	−	0.990621i	$-0.456370\pi$
$858$		0			0
$859$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$859$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	22.0000			0.748022
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	23.0000	+	23.0000i	0.776655	+	0.776655i	0.979260	−	0.202606i	$-0.0649409\pi$
$877$	23.0000	+	23.0000i	0.776655	+	0.776655i	−0.202606	+	0.979260i	$0.564941\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−32.0000			−1.07811			−0.539054	−	0.842271i	$-0.681218\pi$
$881$	−32.0000			−1.07811			−0.539054	+	0.842271i	$0.681218\pi$
$882$		0			0
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−18.0000	+	18.0000i	−0.599667	+	0.599667i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	38.0000i		−	1.26316i
$906$		0			0
$907$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	15.0000	+	15.0000i	0.493197	+	0.493197i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−46.0000			−1.50921			−0.754606	−	0.656179i	$-0.772172\pi$
$929$	−46.0000			−1.50921			−0.754606	+	0.656179i	$0.772172\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			38.0000i			1.24141i	0.784046	+	0.620703i	$0.213153\pi$
$937$			38.0000i			1.24141i	−0.784046	+	0.620703i	$0.786847\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	19.0000	+	19.0000i	0.619382	+	0.619382i	0.945373	−	0.325991i	$-0.105698\pi$
$941$	19.0000	+	19.0000i	0.619382	+	0.619382i	−0.325991	+	0.945373i	$0.605698\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$		0			0
$949$	−6.00000	+	6.00000i	−0.194768	+	0.194768i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	26.0000i		−	0.842223i	−0.907009	−	0.421111i	$-0.861640\pi$
$953$		−	26.0000i		−	0.842223i	0.907009	−	0.421111i	$-0.138360\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	24.0000	−	24.0000i	0.772587	−	0.772587i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$971$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	62.0000			1.98356			0.991778	−	0.127971i	$-0.0408466\pi$
$977$	62.0000			1.98356			0.991778	+	0.127971i	$0.0408466\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		−	30.0000i		−	0.955879i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	37.0000	−	37.0000i	1.17180	−	1.17180i	0.190022	−	0.981780i	$-0.439144\pi$
$997$	37.0000	−	37.0000i	1.17180	−	1.17180i	0.981780	−	0.190022i	$-0.0608559\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.k.h.1153.1	✓	2
3.2	odd	2		4608.2.k.p.1153.1	yes	2
4.3	odd	2	CM	4608.2.k.h.1153.1	✓	2
8.3	odd	2		4608.2.k.q.1153.1	yes	2
8.5	even	2		4608.2.k.q.1153.1	yes	2
12.11	even	2		4608.2.k.p.1153.1	yes	2
16.3	odd	4		4608.2.k.q.3457.1	yes	2
16.5	even	4	inner	4608.2.k.h.3457.1	yes	2
16.11	odd	4	inner	4608.2.k.h.3457.1	yes	2
16.13	even	4		4608.2.k.q.3457.1	yes	2
24.5	odd	2		4608.2.k.i.1153.1	yes	2
24.11	even	2		4608.2.k.i.1153.1	yes	2
32.5	even	8		9216.2.a.j.1.1		2
32.11	odd	8		9216.2.a.j.1.2		2
32.21	even	8		9216.2.a.j.1.2		2
32.27	odd	8		9216.2.a.j.1.1		2
48.5	odd	4		4608.2.k.p.3457.1	yes	2
48.11	even	4		4608.2.k.p.3457.1	yes	2
48.29	odd	4		4608.2.k.i.3457.1	yes	2
48.35	even	4		4608.2.k.i.3457.1	yes	2
96.5	odd	8		9216.2.a.l.1.2		2
96.11	even	8		9216.2.a.l.1.1		2
96.53	odd	8		9216.2.a.l.1.1		2
96.59	even	8		9216.2.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4608.2.k.h.1153.1	✓	2	1.1	even	1	trivial
4608.2.k.h.1153.1	✓	2	4.3	odd	2	CM
4608.2.k.h.3457.1	yes	2	16.5	even	4	inner
4608.2.k.h.3457.1	yes	2	16.11	odd	4	inner
4608.2.k.i.1153.1	yes	2	24.5	odd	2
4608.2.k.i.1153.1	yes	2	24.11	even	2
4608.2.k.i.3457.1	yes	2	48.29	odd	4
4608.2.k.i.3457.1	yes	2	48.35	even	4
4608.2.k.p.1153.1	yes	2	3.2	odd	2
4608.2.k.p.1153.1	yes	2	12.11	even	2
4608.2.k.p.3457.1	yes	2	48.5	odd	4
4608.2.k.p.3457.1	yes	2	48.11	even	4
4608.2.k.q.1153.1	yes	2	8.3	odd	2
4608.2.k.q.1153.1	yes	2	8.5	even	2
4608.2.k.q.3457.1	yes	2	16.3	odd	4
4608.2.k.q.3457.1	yes	2	16.13	even	4
9216.2.a.j.1.1		2	32.5	even	8
9216.2.a.j.1.1		2	32.27	odd	8
9216.2.a.j.1.2		2	32.11	odd	8
9216.2.a.j.1.2		2	32.21	even	8
9216.2.a.l.1.1		2	96.11	even	8
9216.2.a.l.1.1		2	96.53	odd	8
9216.2.a.l.1.2		2	96.5	odd	8
9216.2.a.l.1.2		2	96.59	even	8

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:	\( N \)	\(=\)	\( 4608 = 2^{9} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4608.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{5} +O(q^{10})\)	\(q+(-1.00000 + 1.00000i) q^{5} +(-1.00000 - 1.00000i) q^{13} +2.00000 q^{17} +3.00000i q^{25} +(-3.00000 - 3.00000i) q^{29} +(5.00000 - 5.00000i) q^{37} +10.0000i q^{41} +7.00000 q^{49} +(-9.00000 + 9.00000i) q^{53} +(1.00000 + 1.00000i) q^{61} +2.00000 q^{65} -6.00000i q^{73} +(-2.00000 + 2.00000i) q^{85} +16.0000i q^{89} -8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5}+O(q^{10}) \) 2 * q - 2 * q^5	\( 2 q - 2 q^{5} - 2 q^{13} + 4 q^{17} - 6 q^{29} + 10 q^{37} + 14 q^{49} - 18 q^{53} + 2 q^{61} + 4 q^{65} - 4 q^{85} - 16 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^13 + 4 * q^17 - 6 * q^29 + 10 * q^37 + 14 * q^49 - 18 * q^53 + 2 * q^61 + 4 * q^65 - 4 * q^85 - 16 * q^97

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