LMFDB - Embedded newform 4368.2.h.h.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(337,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4368.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.h (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:	$34.8786556029$
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:	no (minimal twist has level 546)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4368.337
Dual form			4368.2.h.h.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000i q^{5} -1.00000i q^{7} +1.00000 q^{9} +3.00000i q^{11} +(3.00000 + 2.00000i) q^{13} -1.00000i q^{15} +7.00000 q^{17} +3.00000i q^{19} +1.00000i q^{21} -1.00000 q^{23} +4.00000 q^{25} -1.00000 q^{27} -1.00000 q^{29} -8.00000i q^{31} -3.00000i q^{33} +1.00000 q^{35} +1.00000i q^{37} +(-3.00000 - 2.00000i) q^{39} -4.00000i q^{41} +5.00000 q^{43} +1.00000i q^{45} -1.00000 q^{49} -7.00000 q^{51} -6.00000 q^{53} -3.00000 q^{55} -3.00000i q^{57} +10.0000i q^{59} -13.0000 q^{61} -1.00000i q^{63} +(-2.00000 + 3.00000i) q^{65} -8.00000i q^{67} +1.00000 q^{69} +6.00000i q^{71} +13.0000i q^{73} -4.00000 q^{75} +3.00000 q^{77} +12.0000 q^{79} +1.00000 q^{81} +2.00000i q^{83} +7.00000i q^{85} +1.00000 q^{87} -12.0000i q^{89} +(2.00000 - 3.00000i) q^{91} +8.00000i q^{93} -3.00000 q^{95} -6.00000i q^{97} +3.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{9} + 6 q^{13} + 14 q^{17} - 2 q^{23} + 8 q^{25} - 2 q^{27} - 2 q^{29} + 2 q^{35} - 6 q^{39} + 10 q^{43} - 2 q^{49} - 14 q^{51} - 12 q^{53} - 6 q^{55} - 26 q^{61} - 4 q^{65} + 2 q^{69} - 8 q^{75} + 6 q^{77} + 24 q^{79} + 2 q^{81} + 2 q^{87} + 4 q^{91} - 6 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 + 6 * q^13 + 14 * q^17 - 2 * q^23 + 8 * q^25 - 2 * q^27 - 2 * q^29 + 2 * q^35 - 6 * q^39 + 10 * q^43 - 2 * q^49 - 14 * q^51 - 12 * q^53 - 6 * q^55 - 26 * q^61 - 4 * q^65 + 2 * q^69 - 8 * q^75 + 6 * q^77 + 24 * q^79 + 2 * q^81 + 2 * q^87 + 4 * q^91 - 6 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$1457$	$2017$	$3823$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$			1.00000i			0.447214i	0.974679	+	0.223607i	$0.0717831\pi$
$5$			1.00000i			0.447214i	−0.974679	+	0.223607i	$0.928217\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			3.00000i			0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$			3.00000i			0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$		0			0
$13$	3.00000	+	2.00000i	0.832050	+	0.554700i
$13$	3.00000	+	2.00000i	0.832050	+	0.554700i
$14$		0			0
$15$		−	1.00000i		−	0.258199i
$16$		0			0
$17$	7.00000			1.69775			0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000			1.69775			0.848875	+	0.528594i	$0.177281\pi$
$18$		0			0
$19$			3.00000i			0.688247i	0.938924	+	0.344124i	$0.111824\pi$
$19$			3.00000i			0.688247i	−0.938924	+	0.344124i	$0.888176\pi$
$20$		0			0
$21$			1.00000i			0.218218i
$22$		0			0
$23$	−1.00000			−0.208514			−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000			−0.208514			−0.104257	+	0.994550i	$0.533247\pi$
$24$		0			0
$25$	4.00000			0.800000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	−1.00000			−0.185695			−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000			−0.185695			−0.0928477	+	0.995680i	$0.529597\pi$
$30$		0			0
$31$		−	8.00000i		−	1.43684i	−0.695608	−	0.718421i	$-0.744865\pi$
$31$		−	8.00000i		−	1.43684i	0.695608	−	0.718421i	$-0.255135\pi$
$32$		0			0
$33$		−	3.00000i		−	0.522233i
$34$		0			0
$35$	1.00000			0.169031
$36$		0			0
$37$			1.00000i			0.164399i	0.996616	+	0.0821995i	$0.0261945\pi$
$37$			1.00000i			0.164399i	−0.996616	+	0.0821995i	$0.973806\pi$
$38$		0			0
$39$	−3.00000	−	2.00000i	−0.480384	−	0.320256i
$40$		0			0
$41$		−	4.00000i		−	0.624695i	−0.949968	−	0.312348i	$-0.898885\pi$
$41$		−	4.00000i		−	0.624695i	0.949968	−	0.312348i	$-0.101115\pi$
$42$		0			0
$43$	5.00000			0.762493			0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000			0.762493			0.381246	+	0.924473i	$0.375495\pi$
$44$		0			0
$45$			1.00000i			0.149071i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−7.00000			−0.980196
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$	−3.00000			−0.404520
$56$		0			0
$57$		−	3.00000i		−	0.397360i
$58$		0			0
$59$			10.0000i			1.30189i	0.759125	+	0.650945i	$0.225627\pi$
$59$			10.0000i			1.30189i	−0.759125	+	0.650945i	$0.774373\pi$
$60$		0			0
$61$	−13.0000			−1.66448			−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000			−1.66448			−0.832240	+	0.554416i	$0.812942\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	−2.00000	+	3.00000i	−0.248069	+	0.372104i
$66$		0			0
$67$		−	8.00000i		−	0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$		−	8.00000i		−	0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$		0			0
$69$	1.00000			0.120386
$70$		0			0
$71$			6.00000i			0.712069i	0.934473	+	0.356034i	$0.115871\pi$
$71$			6.00000i			0.712069i	−0.934473	+	0.356034i	$0.884129\pi$
$72$		0			0
$73$			13.0000i			1.52153i	0.649025	+	0.760767i	$0.275177\pi$
$73$			13.0000i			1.52153i	−0.649025	+	0.760767i	$0.724823\pi$
$74$		0			0
$75$	−4.00000			−0.461880
$76$		0			0
$77$	3.00000			0.341882
$78$		0			0
$79$	12.0000			1.35011			0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000			1.35011			0.675053	+	0.737769i	$0.264121\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			2.00000i			0.219529i	0.993958	+	0.109764i	$0.0350096\pi$
$83$			2.00000i			0.219529i	−0.993958	+	0.109764i	$0.964990\pi$
$84$		0			0
$85$			7.00000i			0.759257i
$86$		0			0
$87$	1.00000			0.107211
$88$		0			0
$89$		−	12.0000i		−	1.27200i	−0.771690	−	0.635999i	$-0.780588\pi$
$89$		−	12.0000i		−	1.27200i	0.771690	−	0.635999i	$-0.219412\pi$
$90$		0			0
$91$	2.00000	−	3.00000i	0.209657	−	0.314485i
$92$		0			0
$93$			8.00000i			0.829561i
$94$		0			0
$95$	−3.00000			−0.307794
$96$		0			0
$97$		−	6.00000i		−	0.609208i	−0.952479	−	0.304604i	$-0.901476\pi$
$97$		−	6.00000i		−	0.609208i	0.952479	−	0.304604i	$-0.0985241\pi$
$98$		0			0
$99$			3.00000i			0.301511i
$100$		0			0
$101$	14.0000			1.39305			0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000			1.39305			0.696526	+	0.717532i	$0.254728\pi$
$102$		0			0
$103$	1.00000			0.0985329			0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000			0.0985329			0.0492665	+	0.998786i	$0.484312\pi$
$104$		0			0
$105$	−1.00000			−0.0975900
$106$		0			0
$107$	−2.00000			−0.193347			−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000			−0.193347			−0.0966736	+	0.995316i	$0.530820\pi$
$108$		0			0
$109$		−	7.00000i		−	0.670478i	−0.942133	−	0.335239i	$-0.891183\pi$
$109$		−	7.00000i		−	0.670478i	0.942133	−	0.335239i	$-0.108817\pi$
$110$		0			0
$111$		−	1.00000i		−	0.0949158i
$112$		0			0
$113$	−10.0000			−0.940721			−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000			−0.940721			−0.470360	+	0.882474i	$0.655876\pi$
$114$		0			0
$115$		−	1.00000i		−	0.0932505i
$116$		0			0
$117$	3.00000	+	2.00000i	0.277350	+	0.184900i
$118$		0			0
$119$		−	7.00000i		−	0.641689i
$120$		0			0
$121$	2.00000			0.181818
$122$		0			0
$123$			4.00000i			0.360668i
$124$		0			0
$125$			9.00000i			0.804984i
$126$		0			0
$127$	4.00000			0.354943			0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000			0.354943			0.177471	+	0.984126i	$0.443208\pi$
$128$		0			0
$129$	−5.00000			−0.440225
$130$		0			0
$131$	17.0000			1.48530			0.742648	−	0.669681i	$-0.233569\pi$
$131$	17.0000			1.48530			0.742648	+	0.669681i	$0.233569\pi$
$132$		0			0
$133$	3.00000			0.260133
$134$		0			0
$135$		−	1.00000i		−	0.0860663i
$136$		0			0
$137$			17.0000i			1.45241i	0.687479	+	0.726204i	$0.258717\pi$
$137$			17.0000i			1.45241i	−0.687479	+	0.726204i	$0.741283\pi$
$138$		0			0
$139$	8.00000			0.678551			0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000			0.678551			0.339276	+	0.940687i	$0.389818\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.00000	+	9.00000i	−0.501745	+	0.752618i
$144$		0			0
$145$		−	1.00000i		−	0.0830455i
$146$		0			0
$147$	1.00000			0.0824786
$148$		0			0
$149$			22.0000i			1.80231i	0.433497	+	0.901155i	$0.357280\pi$
$149$			22.0000i			1.80231i	−0.433497	+	0.901155i	$0.642720\pi$
$150$		0			0
$151$		−	11.0000i		−	0.895167i	−0.894242	−	0.447584i	$-0.852285\pi$
$151$		−	11.0000i		−	0.895167i	0.894242	−	0.447584i	$-0.147715\pi$
$152$		0			0
$153$	7.00000			0.565916
$154$		0			0
$155$	8.00000			0.642575
$156$		0			0
$157$	19.0000			1.51637			0.758183	−	0.652042i	$-0.226088\pi$
$157$	19.0000			1.51637			0.758183	+	0.652042i	$0.226088\pi$
$158$		0			0
$159$	6.00000			0.475831
$160$		0			0
$161$			1.00000i			0.0788110i
$162$		0			0
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$		0			0
$165$	3.00000			0.233550
$166$		0			0
$167$			15.0000i			1.16073i	0.814355	+	0.580367i	$0.197091\pi$
$167$			15.0000i			1.16073i	−0.814355	+	0.580367i	$0.802909\pi$
$168$		0			0
$169$	5.00000	+	12.0000i	0.384615	+	0.923077i
$170$		0			0
$171$			3.00000i			0.229416i
$172$		0			0
$173$	−6.00000			−0.456172			−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000			−0.456172			−0.228086	+	0.973641i	$0.573247\pi$
$174$		0			0
$175$		−	4.00000i		−	0.302372i
$176$		0			0
$177$		−	10.0000i		−	0.751646i
$178$		0			0
$179$	6.00000			0.448461			0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000			0.448461			0.224231	+	0.974536i	$0.428013\pi$
$180$		0			0
$181$	−26.0000			−1.93256			−0.966282	−	0.257485i	$-0.917106\pi$
$181$	−26.0000			−1.93256			−0.966282	+	0.257485i	$0.917106\pi$
$182$		0			0
$183$	13.0000			0.960988
$184$		0			0
$185$	−1.00000			−0.0735215
$186$		0			0
$187$			21.0000i			1.53567i
$188$		0			0
$189$			1.00000i			0.0727393i
$190$		0			0
$191$	9.00000			0.651217			0.325609	−	0.945505i	$-0.394431\pi$
$191$	9.00000			0.651217			0.325609	+	0.945505i	$0.394431\pi$
$192$		0			0
$193$			8.00000i			0.575853i	0.957653	+	0.287926i	$0.0929658\pi$
$193$			8.00000i			0.575853i	−0.957653	+	0.287926i	$0.907034\pi$
$194$		0			0
$195$	2.00000	−	3.00000i	0.143223	−	0.214834i
$196$		0			0
$197$		−	14.0000i		−	0.997459i	−0.866758	−	0.498729i	$-0.833800\pi$
$197$		−	14.0000i		−	0.997459i	0.866758	−	0.498729i	$-0.166200\pi$
$198$		0			0
$199$	−9.00000			−0.637993			−0.318997	−	0.947756i	$-0.603346\pi$
$199$	−9.00000			−0.637993			−0.318997	+	0.947756i	$0.603346\pi$
$200$		0			0
$201$			8.00000i			0.564276i
$202$		0			0
$203$			1.00000i			0.0701862i
$204$		0			0
$205$	4.00000			0.279372
$206$		0			0
$207$	−1.00000			−0.0695048
$208$		0			0
$209$	−9.00000			−0.622543
$210$		0			0
$211$	−15.0000			−1.03264			−0.516321	−	0.856395i	$-0.672699\pi$
$211$	−15.0000			−1.03264			−0.516321	+	0.856395i	$0.672699\pi$
$212$		0			0
$213$		−	6.00000i		−	0.411113i
$214$		0			0
$215$			5.00000i			0.340997i
$216$		0			0
$217$	−8.00000			−0.543075
$218$		0			0
$219$		−	13.0000i		−	0.878459i
$220$		0			0
$221$	21.0000	+	14.0000i	1.41261	+	0.941742i
$222$		0			0
$223$			4.00000i			0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$			4.00000i			0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$		0			0
$225$	4.00000			0.266667
$226$		0			0
$227$			20.0000i			1.32745i	0.747978	+	0.663723i	$0.231025\pi$
$227$			20.0000i			1.32745i	−0.747978	+	0.663723i	$0.768975\pi$
$228$		0			0
$229$		−	10.0000i		−	0.660819i	−0.943838	−	0.330409i	$-0.892813\pi$
$229$		−	10.0000i		−	0.660819i	0.943838	−	0.330409i	$-0.107187\pi$
$230$		0			0
$231$	−3.00000			−0.197386
$232$		0			0
$233$	18.0000			1.17922			0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−12.0000			−0.779484
$238$		0			0
$239$			8.00000i			0.517477i	0.965947	+	0.258738i	$0.0833068\pi$
$239$			8.00000i			0.517477i	−0.965947	+	0.258738i	$0.916693\pi$
$240$		0			0
$241$		−	26.0000i		−	1.67481i	−0.546585	−	0.837404i	$-0.684072\pi$
$241$		−	26.0000i		−	1.67481i	0.546585	−	0.837404i	$-0.315928\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$		−	1.00000i		−	0.0638877i
$246$		0			0
$247$	−6.00000	+	9.00000i	−0.381771	+	0.572656i
$248$		0			0
$249$		−	2.00000i		−	0.126745i
$250$		0			0
$251$	−17.0000			−1.07303			−0.536515	−	0.843891i	$-0.680260\pi$
$251$	−17.0000			−1.07303			−0.536515	+	0.843891i	$0.680260\pi$
$252$		0			0
$253$		−	3.00000i		−	0.188608i
$254$		0			0
$255$		−	7.00000i		−	0.438357i
$256$		0			0
$257$	−6.00000			−0.374270			−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000			−0.374270			−0.187135	+	0.982334i	$0.559920\pi$
$258$		0			0
$259$	1.00000			0.0621370
$260$		0			0
$261$	−1.00000			−0.0618984
$262$		0			0
$263$	24.0000			1.47990			0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000			1.47990			0.739952	+	0.672660i	$0.234848\pi$
$264$		0			0
$265$		−	6.00000i		−	0.368577i
$266$		0			0
$267$			12.0000i			0.734388i
$268$		0			0
$269$	−4.00000			−0.243884			−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000			−0.243884			−0.121942	+	0.992537i	$0.538912\pi$
$270$		0			0
$271$			20.0000i			1.21491i	0.794353	+	0.607457i	$0.207810\pi$
$271$			20.0000i			1.21491i	−0.794353	+	0.607457i	$0.792190\pi$
$272$		0			0
$273$	−2.00000	+	3.00000i	−0.121046	+	0.181568i
$274$		0			0
$275$			12.0000i			0.723627i
$276$		0			0
$277$	28.0000			1.68236			0.841178	−	0.540758i	$-0.181862\pi$
$277$	28.0000			1.68236			0.841178	+	0.540758i	$0.181862\pi$
$278$		0			0
$279$		−	8.00000i		−	0.478947i
$280$		0			0
$281$			10.0000i			0.596550i	0.954480	+	0.298275i	$0.0964112\pi$
$281$			10.0000i			0.596550i	−0.954480	+	0.298275i	$0.903589\pi$
$282$		0			0
$283$	14.0000			0.832214			0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000			0.832214			0.416107	+	0.909316i	$0.363394\pi$
$284$		0			0
$285$	3.00000			0.177705
$286$		0			0
$287$	−4.00000			−0.236113
$288$		0			0
$289$	32.0000			1.88235
$290$		0			0
$291$			6.00000i			0.351726i
$292$		0			0
$293$			14.0000i			0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$			14.0000i			0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$		0			0
$295$	−10.0000			−0.582223
$296$		0			0
$297$		−	3.00000i		−	0.174078i
$298$		0			0
$299$	−3.00000	−	2.00000i	−0.173494	−	0.115663i
$300$		0			0
$301$		−	5.00000i		−	0.288195i
$302$		0			0
$303$	−14.0000			−0.804279
$304$		0			0
$305$		−	13.0000i		−	0.744378i
$306$		0			0
$307$			16.0000i			0.913168i	0.889680	+	0.456584i	$0.150927\pi$
$307$			16.0000i			0.913168i	−0.889680	+	0.456584i	$0.849073\pi$
$308$		0			0
$309$	−1.00000			−0.0568880
$310$		0			0
$311$	4.00000			0.226819			0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000			0.226819			0.113410	+	0.993548i	$0.463823\pi$
$312$		0			0
$313$	−26.0000			−1.46961			−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000			−1.46961			−0.734803	+	0.678280i	$0.762726\pi$
$314$		0			0
$315$	1.00000			0.0563436
$316$		0			0
$317$		−	24.0000i		−	1.34797i	−0.738743	−	0.673987i	$-0.764580\pi$
$317$		−	24.0000i		−	1.34797i	0.738743	−	0.673987i	$-0.235420\pi$
$318$		0			0
$319$		−	3.00000i		−	0.167968i
$320$		0			0
$321$	2.00000			0.111629
$322$		0			0
$323$			21.0000i			1.16847i
$324$		0			0
$325$	12.0000	+	8.00000i	0.665640	+	0.443760i
$326$		0			0
$327$			7.00000i			0.387101i
$328$		0			0
$329$		0			0
$330$		0			0
$331$			8.00000i			0.439720i	0.975531	+	0.219860i	$0.0705600\pi$
$331$			8.00000i			0.439720i	−0.975531	+	0.219860i	$0.929440\pi$
$332$		0			0
$333$			1.00000i			0.0547997i
$334$		0			0
$335$	8.00000			0.437087
$336$		0			0
$337$	−23.0000			−1.25289			−0.626445	−	0.779466i	$-0.715491\pi$
$337$	−23.0000			−1.25289			−0.626445	+	0.779466i	$0.715491\pi$
$338$		0			0
$339$	10.0000			0.543125
$340$		0			0
$341$	24.0000			1.29967
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$			1.00000i			0.0538382i
$346$		0			0
$347$	24.0000			1.28839			0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000			1.28839			0.644194	+	0.764862i	$0.277193\pi$
$348$		0			0
$349$			16.0000i			0.856460i	0.903670	+	0.428230i	$0.140863\pi$
$349$			16.0000i			0.856460i	−0.903670	+	0.428230i	$0.859137\pi$
$350$		0			0
$351$	−3.00000	−	2.00000i	−0.160128	−	0.106752i
$352$		0			0
$353$			16.0000i			0.851594i	0.904819	+	0.425797i	$0.140006\pi$
$353$			16.0000i			0.851594i	−0.904819	+	0.425797i	$0.859994\pi$
$354$		0			0
$355$	−6.00000			−0.318447
$356$		0			0
$357$			7.00000i			0.370479i
$358$		0			0
$359$		−	26.0000i		−	1.37223i	−0.727494	−	0.686114i	$-0.759315\pi$
$359$		−	26.0000i		−	1.37223i	0.727494	−	0.686114i	$-0.240685\pi$
$360$		0			0
$361$	10.0000			0.526316
$362$		0			0
$363$	−2.00000			−0.104973
$364$		0			0
$365$	−13.0000			−0.680451
$366$		0			0
$367$	−32.0000			−1.67039			−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000			−1.67039			−0.835193	+	0.549957i	$0.814644\pi$
$368$		0			0
$369$		−	4.00000i		−	0.208232i
$370$		0			0
$371$			6.00000i			0.311504i
$372$		0			0
$373$	6.00000			0.310668			0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000			0.310668			0.155334	+	0.987862i	$0.450355\pi$
$374$		0			0
$375$		−	9.00000i		−	0.464758i
$376$		0			0
$377$	−3.00000	−	2.00000i	−0.154508	−	0.103005i
$378$		0			0
$379$			4.00000i			0.205466i	0.994709	+	0.102733i	$0.0327588\pi$
$379$			4.00000i			0.205466i	−0.994709	+	0.102733i	$0.967241\pi$
$380$		0			0
$381$	−4.00000			−0.204926
$382$		0			0
$383$			17.0000i			0.868659i	0.900754	+	0.434330i	$0.143015\pi$
$383$			17.0000i			0.868659i	−0.900754	+	0.434330i	$0.856985\pi$
$384$		0			0
$385$			3.00000i			0.152894i
$386$		0			0
$387$	5.00000			0.254164
$388$		0			0
$389$	−18.0000			−0.912636			−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000			−0.912636			−0.456318	+	0.889817i	$0.650832\pi$
$390$		0			0
$391$	−7.00000			−0.354005
$392$		0			0
$393$	−17.0000			−0.857537
$394$		0			0
$395$			12.0000i			0.603786i
$396$		0			0
$397$		−	8.00000i		−	0.401508i	−0.979642	−	0.200754i	$-0.935661\pi$
$397$		−	8.00000i		−	0.401508i	0.979642	−	0.200754i	$-0.0643393\pi$
$398$		0			0
$399$	−3.00000			−0.150188
$400$		0			0
$401$			6.00000i			0.299626i	0.988714	+	0.149813i	$0.0478671\pi$
$401$			6.00000i			0.299626i	−0.988714	+	0.149813i	$0.952133\pi$
$402$		0			0
$403$	16.0000	−	24.0000i	0.797017	−	1.19553i
$404$		0			0
$405$			1.00000i			0.0496904i
$406$		0			0
$407$	−3.00000			−0.148704
$408$		0			0
$409$			29.0000i			1.43396i	0.697095	+	0.716979i	$0.254476\pi$
$409$			29.0000i			1.43396i	−0.697095	+	0.716979i	$0.745524\pi$
$410$		0			0
$411$		−	17.0000i		−	0.838548i
$412$		0			0
$413$	10.0000			0.492068
$414$		0			0
$415$	−2.00000			−0.0981761
$416$		0			0
$417$	−8.00000			−0.391762
$418$		0			0
$419$	11.0000			0.537385			0.268693	−	0.963226i	$-0.413408\pi$
$419$	11.0000			0.537385			0.268693	+	0.963226i	$0.413408\pi$
$420$		0			0
$421$			34.0000i			1.65706i	0.559946	+	0.828529i	$0.310822\pi$
$421$			34.0000i			1.65706i	−0.559946	+	0.828529i	$0.689178\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	28.0000			1.35820
$426$		0			0
$427$			13.0000i			0.629114i
$428$		0			0
$429$	6.00000	−	9.00000i	0.289683	−	0.434524i
$430$		0			0
$431$		−	14.0000i		−	0.674356i	−0.941441	−	0.337178i	$-0.890528\pi$
$431$		−	14.0000i		−	0.674356i	0.941441	−	0.337178i	$-0.109472\pi$
$432$		0			0
$433$	−8.00000			−0.384455			−0.192228	−	0.981350i	$-0.561571\pi$
$433$	−8.00000			−0.384455			−0.192228	+	0.981350i	$0.561571\pi$
$434$		0			0
$435$			1.00000i			0.0479463i
$436$		0			0
$437$		−	3.00000i		−	0.143509i
$438$		0			0
$439$	−27.0000			−1.28864			−0.644320	−	0.764756i	$-0.722859\pi$
$439$	−27.0000			−1.28864			−0.644320	+	0.764756i	$0.722859\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	34.0000			1.61539			0.807694	−	0.589601i	$-0.200715\pi$
$443$	34.0000			1.61539			0.807694	+	0.589601i	$0.200715\pi$
$444$		0			0
$445$	12.0000			0.568855
$446$		0			0
$447$		−	22.0000i		−	1.04056i
$448$		0			0
$449$			9.00000i			0.424736i	0.977190	+	0.212368i	$0.0681176\pi$
$449$			9.00000i			0.424736i	−0.977190	+	0.212368i	$0.931882\pi$
$450$		0			0
$451$	12.0000			0.565058
$452$		0			0
$453$			11.0000i			0.516825i
$454$		0			0
$455$	3.00000	+	2.00000i	0.140642	+	0.0937614i
$456$		0			0
$457$		−	28.0000i		−	1.30978i	−0.755722	−	0.654892i	$-0.772714\pi$
$457$		−	28.0000i		−	1.30978i	0.755722	−	0.654892i	$-0.227286\pi$
$458$		0			0
$459$	−7.00000			−0.326732
$460$		0			0
$461$			27.0000i			1.25752i	0.777601	+	0.628758i	$0.216436\pi$
$461$			27.0000i			1.25752i	−0.777601	+	0.628758i	$0.783564\pi$
$462$		0			0
$463$		−	15.0000i		−	0.697109i	−0.937288	−	0.348555i	$-0.886673\pi$
$463$		−	15.0000i		−	0.697109i	0.937288	−	0.348555i	$-0.113327\pi$
$464$		0			0
$465$	−8.00000			−0.370991
$466$		0			0
$467$	−9.00000			−0.416470			−0.208235	−	0.978079i	$-0.566772\pi$
$467$	−9.00000			−0.416470			−0.208235	+	0.978079i	$0.566772\pi$
$468$		0			0
$469$	−8.00000			−0.369406
$470$		0			0
$471$	−19.0000			−0.875474
$472$		0			0
$473$			15.0000i			0.689701i
$474$		0			0
$475$			12.0000i			0.550598i
$476$		0			0
$477$	−6.00000			−0.274721
$478$		0			0
$479$		−	3.00000i		−	0.137073i	−0.997649	−	0.0685367i	$-0.978167\pi$
$479$		−	3.00000i		−	0.137073i	0.997649	−	0.0685367i	$-0.0218330\pi$
$480$		0			0
$481$	−2.00000	+	3.00000i	−0.0911922	+	0.136788i
$482$		0			0
$483$		−	1.00000i		−	0.0455016i
$484$		0			0
$485$	6.00000			0.272446
$486$		0			0
$487$		−	24.0000i		−	1.08754i	−0.839233	−	0.543772i	$-0.816996\pi$
$487$		−	24.0000i		−	1.08754i	0.839233	−	0.543772i	$-0.183004\pi$
$488$		0			0
$489$		−	4.00000i		−	0.180886i
$490$		0			0
$491$	40.0000			1.80517			0.902587	−	0.430507i	$-0.141665\pi$
$491$	40.0000			1.80517			0.902587	+	0.430507i	$0.141665\pi$
$492$		0			0
$493$	−7.00000			−0.315264
$494$		0			0
$495$	−3.00000			−0.134840
$496$		0			0
$497$	6.00000			0.269137
$498$		0			0
$499$			14.0000i			0.626726i	0.949633	+	0.313363i	$0.101456\pi$
$499$			14.0000i			0.626726i	−0.949633	+	0.313363i	$0.898544\pi$
$500$		0			0
$501$		−	15.0000i		−	0.670151i
$502$		0			0
$503$	6.00000			0.267527			0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000			0.267527			0.133763	+	0.991013i	$0.457294\pi$
$504$		0			0
$505$			14.0000i			0.622992i
$506$		0			0
$507$	−5.00000	−	12.0000i	−0.222058	−	0.532939i
$508$		0			0
$509$			27.0000i			1.19675i	0.801215	+	0.598377i	$0.204187\pi$
$509$			27.0000i			1.19675i	−0.801215	+	0.598377i	$0.795813\pi$
$510$		0			0
$511$	13.0000			0.575086
$512$		0			0
$513$		−	3.00000i		−	0.132453i
$514$		0			0
$515$			1.00000i			0.0440653i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	6.00000			0.263371
$520$		0			0
$521$	15.0000			0.657162			0.328581	−	0.944476i	$-0.393430\pi$
$521$	15.0000			0.657162			0.328581	+	0.944476i	$0.393430\pi$
$522$		0			0
$523$	−16.0000			−0.699631			−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000			−0.699631			−0.349816	+	0.936819i	$0.613756\pi$
$524$		0			0
$525$			4.00000i			0.174574i
$526$		0			0
$527$		−	56.0000i		−	2.43940i
$528$		0			0
$529$	−22.0000			−0.956522
$530$		0			0
$531$			10.0000i			0.433963i
$532$		0			0
$533$	8.00000	−	12.0000i	0.346518	−	0.519778i
$534$		0			0
$535$		−	2.00000i		−	0.0864675i
$536$		0			0
$537$	−6.00000			−0.258919
$538$		0			0
$539$		−	3.00000i		−	0.129219i
$540$		0			0
$541$		−	25.0000i		−	1.07483i	−0.843317	−	0.537417i	$-0.819400\pi$
$541$		−	25.0000i		−	1.07483i	0.843317	−	0.537417i	$-0.180600\pi$
$542$		0			0
$543$	26.0000			1.11577
$544$		0			0
$545$	7.00000			0.299847
$546$		0			0
$547$	−28.0000			−1.19719			−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000			−1.19719			−0.598597	+	0.801050i	$0.704275\pi$
$548$		0			0
$549$	−13.0000			−0.554826
$550$		0			0
$551$		−	3.00000i		−	0.127804i
$552$		0			0
$553$		−	12.0000i		−	0.510292i
$554$		0			0
$555$	1.00000			0.0424476
$556$		0			0
$557$		−	30.0000i		−	1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$		−	30.0000i		−	1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$		0			0
$559$	15.0000	+	10.0000i	0.634432	+	0.422955i
$560$		0			0
$561$		−	21.0000i		−	0.886621i
$562$		0			0
$563$	−41.0000			−1.72794			−0.863972	−	0.503540i	$-0.832031\pi$
$563$	−41.0000			−1.72794			−0.863972	+	0.503540i	$0.832031\pi$
$564$		0			0
$565$		−	10.0000i		−	0.420703i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	−2.00000			−0.0838444			−0.0419222	−	0.999121i	$-0.513348\pi$
$569$	−2.00000			−0.0838444			−0.0419222	+	0.999121i	$0.513348\pi$
$570$		0			0
$571$	40.0000			1.67395			0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000			1.67395			0.836974	+	0.547243i	$0.184323\pi$
$572$		0			0
$573$	−9.00000			−0.375980
$574$		0			0
$575$	−4.00000			−0.166812
$576$		0			0
$577$		−	38.0000i		−	1.58196i	−0.611842	−	0.790980i	$-0.709571\pi$
$577$		−	38.0000i		−	1.58196i	0.611842	−	0.790980i	$-0.290429\pi$
$578$		0			0
$579$		−	8.00000i		−	0.332469i
$580$		0			0
$581$	2.00000			0.0829740
$582$		0			0
$583$		−	18.0000i		−	0.745484i
$584$		0			0
$585$	−2.00000	+	3.00000i	−0.0826898	+	0.124035i
$586$		0			0
$587$			16.0000i			0.660391i	0.943913	+	0.330195i	$0.107115\pi$
$587$			16.0000i			0.660391i	−0.943913	+	0.330195i	$0.892885\pi$
$588$		0			0
$589$	24.0000			0.988903
$590$		0			0
$591$			14.0000i			0.575883i
$592$		0			0
$593$			16.0000i			0.657041i	0.944497	+	0.328521i	$0.106550\pi$
$593$			16.0000i			0.657041i	−0.944497	+	0.328521i	$0.893450\pi$
$594$		0			0
$595$	7.00000			0.286972
$596$		0			0
$597$	9.00000			0.368345
$598$		0			0
$599$	3.00000			0.122577			0.0612883	−	0.998120i	$-0.480479\pi$
$599$	3.00000			0.122577			0.0612883	+	0.998120i	$0.480479\pi$
$600$		0			0
$601$	4.00000			0.163163			0.0815817	−	0.996667i	$-0.474003\pi$
$601$	4.00000			0.163163			0.0815817	+	0.996667i	$0.474003\pi$
$602$		0			0
$603$		−	8.00000i		−	0.325785i
$604$		0			0
$605$			2.00000i			0.0813116i
$606$		0			0
$607$	11.0000			0.446476			0.223238	−	0.974764i	$-0.428337\pi$
$607$	11.0000			0.446476			0.223238	+	0.974764i	$0.428337\pi$
$608$		0			0
$609$		−	1.00000i		−	0.0405220i
$610$		0			0
$611$		0			0
$612$		0			0
$613$		−	31.0000i		−	1.25208i	−0.779792	−	0.626039i	$-0.784675\pi$
$613$		−	31.0000i		−	1.25208i	0.779792	−	0.626039i	$-0.215325\pi$
$614$		0			0
$615$	−4.00000			−0.161296
$616$		0			0
$617$			37.0000i			1.48956i	0.667308	+	0.744782i	$0.267447\pi$
$617$			37.0000i			1.48956i	−0.667308	+	0.744782i	$0.732553\pi$
$618$		0			0
$619$		−	23.0000i		−	0.924448i	−0.886763	−	0.462224i	$-0.847052\pi$
$619$		−	23.0000i		−	0.924448i	0.886763	−	0.462224i	$-0.152948\pi$
$620$		0			0
$621$	1.00000			0.0401286
$622$		0			0
$623$	−12.0000			−0.480770
$624$		0			0
$625$	11.0000			0.440000
$626$		0			0
$627$	9.00000			0.359425
$628$		0			0
$629$			7.00000i			0.279108i
$630$		0			0
$631$		−	47.0000i		−	1.87104i	−0.353273	−	0.935520i	$-0.614931\pi$
$631$		−	47.0000i		−	1.87104i	0.353273	−	0.935520i	$-0.385069\pi$
$632$		0			0
$633$	15.0000			0.596196
$634$		0			0
$635$			4.00000i			0.158735i
$636$		0			0
$637$	−3.00000	−	2.00000i	−0.118864	−	0.0792429i
$638$		0			0
$639$			6.00000i			0.237356i
$640$		0			0
$641$	12.0000			0.473972			0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000			0.473972			0.236986	+	0.971513i	$0.423841\pi$
$642$		0			0
$643$		−	39.0000i		−	1.53801i	−0.639243	−	0.769005i	$-0.720752\pi$
$643$		−	39.0000i		−	1.53801i	0.639243	−	0.769005i	$-0.279248\pi$
$644$		0			0
$645$		−	5.00000i		−	0.196875i
$646$		0			0
$647$	−22.0000			−0.864909			−0.432455	−	0.901656i	$-0.642352\pi$
$647$	−22.0000			−0.864909			−0.432455	+	0.901656i	$0.642352\pi$
$648$		0			0
$649$	−30.0000			−1.17760
$650$		0			0
$651$	8.00000			0.313545
$652$		0			0
$653$	−35.0000			−1.36966			−0.684828	−	0.728705i	$-0.740123\pi$
$653$	−35.0000			−1.36966			−0.684828	+	0.728705i	$0.740123\pi$
$654$		0			0
$655$			17.0000i			0.664245i
$656$		0			0
$657$			13.0000i			0.507178i
$658$		0			0
$659$	−26.0000			−1.01282			−0.506408	−	0.862294i	$-0.669027\pi$
$659$	−26.0000			−1.01282			−0.506408	+	0.862294i	$0.669027\pi$
$660$		0			0
$661$			10.0000i			0.388955i	0.980907	+	0.194477i	$0.0623011\pi$
$661$			10.0000i			0.388955i	−0.980907	+	0.194477i	$0.937699\pi$
$662$		0			0
$663$	−21.0000	−	14.0000i	−0.815572	−	0.543715i
$664$		0			0
$665$			3.00000i			0.116335i
$666$		0			0
$667$	1.00000			0.0387202
$668$		0			0
$669$		−	4.00000i		−	0.154649i
$670$		0			0
$671$		−	39.0000i		−	1.50558i
$672$		0			0
$673$	−3.00000			−0.115642			−0.0578208	−	0.998327i	$-0.518415\pi$
$673$	−3.00000			−0.115642			−0.0578208	+	0.998327i	$0.518415\pi$
$674$		0			0
$675$	−4.00000			−0.153960
$676$		0			0
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$	−6.00000			−0.230259
$680$		0			0
$681$		−	20.0000i		−	0.766402i
$682$		0			0
$683$			9.00000i			0.344375i	0.985064	+	0.172188i	$0.0550836\pi$
$683$			9.00000i			0.344375i	−0.985064	+	0.172188i	$0.944916\pi$
$684$		0			0
$685$	−17.0000			−0.649537
$686$		0			0
$687$			10.0000i			0.381524i
$688$		0			0
$689$	−18.0000	−	12.0000i	−0.685745	−	0.457164i
$690$		0			0
$691$		−	16.0000i		−	0.608669i	−0.952565	−	0.304334i	$-0.901566\pi$
$691$		−	16.0000i		−	0.608669i	0.952565	−	0.304334i	$-0.0984340\pi$
$692$		0			0
$693$	3.00000			0.113961
$694$		0			0
$695$			8.00000i			0.303457i
$696$		0			0
$697$		−	28.0000i		−	1.06058i
$698$		0			0
$699$	−18.0000			−0.680823
$700$		0			0
$701$	−2.00000			−0.0755390			−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000			−0.0755390			−0.0377695	+	0.999286i	$0.512025\pi$
$702$		0			0
$703$	−3.00000			−0.113147
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	14.0000i		−	0.526524i
$708$		0			0
$709$			30.0000i			1.12667i	0.826227	+	0.563337i	$0.190483\pi$
$709$			30.0000i			1.12667i	−0.826227	+	0.563337i	$0.809517\pi$
$710$		0			0
$711$	12.0000			0.450035
$712$		0			0
$713$			8.00000i			0.299602i
$714$		0			0
$715$	−9.00000	−	6.00000i	−0.336581	−	0.224387i
$716$		0			0
$717$		−	8.00000i		−	0.298765i
$718$		0			0
$719$	−20.0000			−0.745874			−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000			−0.745874			−0.372937	+	0.927857i	$0.621649\pi$
$720$		0			0
$721$		−	1.00000i		−	0.0372419i
$722$		0			0
$723$			26.0000i			0.966950i
$724$		0			0
$725$	−4.00000			−0.148556
$726$		0			0
$727$	−35.0000			−1.29808			−0.649039	−	0.760755i	$-0.724829\pi$
$727$	−35.0000			−1.29808			−0.649039	+	0.760755i	$0.724829\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	35.0000			1.29452
$732$		0			0
$733$			8.00000i			0.295487i	0.989026	+	0.147743i	$0.0472010\pi$
$733$			8.00000i			0.295487i	−0.989026	+	0.147743i	$0.952799\pi$
$734$		0			0
$735$			1.00000i			0.0368856i
$736$		0			0
$737$	24.0000			0.884051
$738$		0			0
$739$			16.0000i			0.588570i	0.955718	+	0.294285i	$0.0950814\pi$
$739$			16.0000i			0.588570i	−0.955718	+	0.294285i	$0.904919\pi$
$740$		0			0
$741$	6.00000	−	9.00000i	0.220416	−	0.330623i
$742$		0			0
$743$			40.0000i			1.46746i	0.679442	+	0.733729i	$0.262222\pi$
$743$			40.0000i			1.46746i	−0.679442	+	0.733729i	$0.737778\pi$
$744$		0			0
$745$	−22.0000			−0.806018
$746$		0			0
$747$			2.00000i			0.0731762i
$748$		0			0
$749$			2.00000i			0.0730784i
$750$		0			0
$751$	4.00000			0.145962			0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000			0.145962			0.0729810	+	0.997333i	$0.476749\pi$
$752$		0			0
$753$	17.0000			0.619514
$754$		0			0
$755$	11.0000			0.400331
$756$		0			0
$757$	26.0000			0.944986			0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000			0.944986			0.472493	+	0.881334i	$0.343354\pi$
$758$		0			0
$759$			3.00000i			0.108893i
$760$		0			0
$761$			30.0000i			1.08750i	0.839248	+	0.543750i	$0.182996\pi$
$761$			30.0000i			1.08750i	−0.839248	+	0.543750i	$0.817004\pi$
$762$		0			0
$763$	−7.00000			−0.253417
$764$		0			0
$765$			7.00000i			0.253086i
$766$		0			0
$767$	−20.0000	+	30.0000i	−0.722158	+	1.08324i
$768$		0			0
$769$		−	39.0000i		−	1.40638i	−0.711004	−	0.703188i	$-0.751759\pi$
$769$		−	39.0000i		−	1.40638i	0.711004	−	0.703188i	$-0.248241\pi$
$770$		0			0
$771$	6.00000			0.216085
$772$		0			0
$773$		−	21.0000i		−	0.755318i	−0.925945	−	0.377659i	$-0.876729\pi$
$773$		−	21.0000i		−	0.755318i	0.925945	−	0.377659i	$-0.123271\pi$
$774$		0			0
$775$		−	32.0000i		−	1.14947i
$776$		0			0
$777$	−1.00000			−0.0358748
$778$		0			0
$779$	12.0000			0.429945
$780$		0			0
$781$	−18.0000			−0.644091
$782$		0			0
$783$	1.00000			0.0357371
$784$		0			0
$785$			19.0000i			0.678139i
$786$		0			0
$787$			17.0000i			0.605985i	0.952993	+	0.302992i	$0.0979856\pi$
$787$			17.0000i			0.605985i	−0.952993	+	0.302992i	$0.902014\pi$
$788$		0			0
$789$	−24.0000			−0.854423
$790$		0			0
$791$			10.0000i			0.355559i
$792$		0			0
$793$	−39.0000	−	26.0000i	−1.38493	−	0.923287i
$794$		0			0
$795$			6.00000i			0.212798i
$796$		0			0
$797$	−2.00000			−0.0708436			−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000			−0.0708436			−0.0354218	+	0.999372i	$0.511277\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		−	12.0000i		−	0.423999i
$802$		0			0
$803$	−39.0000			−1.37628
$804$		0			0
$805$	−1.00000			−0.0352454
$806$		0			0
$807$	4.00000			0.140807
$808$		0			0
$809$	52.0000			1.82822			0.914111	−	0.405463i	$-0.132890\pi$
$809$	52.0000			1.82822			0.914111	+	0.405463i	$0.132890\pi$
$810$		0			0
$811$			7.00000i			0.245803i	0.992419	+	0.122902i	$0.0392200\pi$
$811$			7.00000i			0.245803i	−0.992419	+	0.122902i	$0.960780\pi$
$812$		0			0
$813$		−	20.0000i		−	0.701431i
$814$		0			0
$815$	−4.00000			−0.140114
$816$		0			0
$817$			15.0000i			0.524784i
$818$		0			0
$819$	2.00000	−	3.00000i	0.0698857	−	0.104828i
$820$		0			0
$821$		−	36.0000i		−	1.25641i	−0.778048	−	0.628204i	$-0.783790\pi$
$821$		−	36.0000i		−	1.25641i	0.778048	−	0.628204i	$-0.216210\pi$
$822$		0			0
$823$	−12.0000			−0.418294			−0.209147	−	0.977884i	$-0.567069\pi$
$823$	−12.0000			−0.418294			−0.209147	+	0.977884i	$0.567069\pi$
$824$		0			0
$825$		−	12.0000i		−	0.417786i
$826$		0			0
$827$		−	9.00000i		−	0.312961i	−0.987681	−	0.156480i	$-0.949985\pi$
$827$		−	9.00000i		−	0.312961i	0.987681	−	0.156480i	$-0.0500148\pi$
$828$		0			0
$829$	−15.0000			−0.520972			−0.260486	−	0.965478i	$-0.583883\pi$
$829$	−15.0000			−0.520972			−0.260486	+	0.965478i	$0.583883\pi$
$830$		0			0
$831$	−28.0000			−0.971309
$832$		0			0
$833$	−7.00000			−0.242536
$834$		0			0
$835$	−15.0000			−0.519096
$836$		0			0
$837$			8.00000i			0.276520i
$838$		0			0
$839$		−	28.0000i		−	0.966667i	−0.875436	−	0.483334i	$-0.839426\pi$
$839$		−	28.0000i		−	0.966667i	0.875436	−	0.483334i	$-0.160574\pi$
$840$		0			0
$841$	−28.0000			−0.965517
$842$		0			0
$843$		−	10.0000i		−	0.344418i
$844$		0			0
$845$	−12.0000	+	5.00000i	−0.412813	+	0.172005i
$846$		0			0
$847$		−	2.00000i		−	0.0687208i
$848$		0			0
$849$	−14.0000			−0.480479
$850$		0			0
$851$		−	1.00000i		−	0.0342796i
$852$		0			0
$853$		−	32.0000i		−	1.09566i	−0.836590	−	0.547830i	$-0.815454\pi$
$853$		−	32.0000i		−	1.09566i	0.836590	−	0.547830i	$-0.184546\pi$
$854$		0			0
$855$	−3.00000			−0.102598
$856$		0			0
$857$	38.0000			1.29806			0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000			1.29806			0.649028	+	0.760765i	$0.275176\pi$
$858$		0			0
$859$	−22.0000			−0.750630			−0.375315	−	0.926897i	$-0.622466\pi$
$859$	−22.0000			−0.750630			−0.375315	+	0.926897i	$0.622466\pi$
$860$		0			0
$861$	4.00000			0.136320
$862$		0			0
$863$			28.0000i			0.953131i	0.879139	+	0.476566i	$0.158119\pi$
$863$			28.0000i			0.953131i	−0.879139	+	0.476566i	$0.841881\pi$
$864$		0			0
$865$		−	6.00000i		−	0.204006i
$866$		0			0
$867$	−32.0000			−1.08678
$868$		0			0
$869$			36.0000i			1.22122i
$870$		0			0
$871$	16.0000	−	24.0000i	0.542139	−	0.813209i
$872$		0			0
$873$		−	6.00000i		−	0.203069i
$874$		0			0
$875$	9.00000			0.304256
$876$		0			0
$877$		−	2.00000i		−	0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$		−	2.00000i		−	0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$		0			0
$879$		−	14.0000i		−	0.472208i
$880$		0			0
$881$	51.0000			1.71823			0.859117	−	0.511780i	$-0.171014\pi$
$881$	51.0000			1.71823			0.859117	+	0.511780i	$0.171014\pi$
$882$		0			0
$883$	−3.00000			−0.100958			−0.0504790	−	0.998725i	$-0.516075\pi$
$883$	−3.00000			−0.100958			−0.0504790	+	0.998725i	$0.516075\pi$
$884$		0			0
$885$	10.0000			0.336146
$886$		0			0
$887$	−42.0000			−1.41022			−0.705111	−	0.709097i	$-0.749103\pi$
$887$	−42.0000			−1.41022			−0.705111	+	0.709097i	$0.749103\pi$
$888$		0			0
$889$		−	4.00000i		−	0.134156i
$890$		0			0
$891$			3.00000i			0.100504i
$892$		0			0
$893$		0			0
$894$		0			0
$895$			6.00000i			0.200558i
$896$		0			0
$897$	3.00000	+	2.00000i	0.100167	+	0.0667781i
$898$		0			0
$899$			8.00000i			0.266815i
$900$		0			0
$901$	−42.0000			−1.39922
$902$		0			0
$903$			5.00000i			0.166390i
$904$		0			0
$905$		−	26.0000i		−	0.864269i
$906$		0			0
$907$	−28.0000			−0.929725			−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000			−0.929725			−0.464862	+	0.885383i	$0.653896\pi$
$908$		0			0
$909$	14.0000			0.464351
$910$		0			0
$911$	−57.0000			−1.88849			−0.944247	−	0.329238i	$-0.893208\pi$
$911$	−57.0000			−1.88849			−0.944247	+	0.329238i	$0.893208\pi$
$912$		0			0
$913$	−6.00000			−0.198571
$914$		0			0
$915$			13.0000i			0.429767i
$916$		0			0
$917$		−	17.0000i		−	0.561389i
$918$		0			0
$919$	30.0000			0.989609			0.494804	−	0.869004i	$-0.335240\pi$
$919$	30.0000			0.989609			0.494804	+	0.869004i	$0.335240\pi$
$920$		0			0
$921$		−	16.0000i		−	0.527218i
$922$		0			0
$923$	−12.0000	+	18.0000i	−0.394985	+	0.592477i
$924$		0			0
$925$			4.00000i			0.131519i
$926$		0			0
$927$	1.00000			0.0328443
$928$		0			0
$929$			42.0000i			1.37798i	0.724773	+	0.688988i	$0.241945\pi$
$929$			42.0000i			1.37798i	−0.724773	+	0.688988i	$0.758055\pi$
$930$		0			0
$931$		−	3.00000i		−	0.0983210i
$932$		0			0
$933$	−4.00000			−0.130954
$934$		0			0
$935$	−21.0000			−0.686773
$936$		0			0
$937$	4.00000			0.130674			0.0653372	−	0.997863i	$-0.479188\pi$
$937$	4.00000			0.130674			0.0653372	+	0.997863i	$0.479188\pi$
$938$		0			0
$939$	26.0000			0.848478
$940$		0			0
$941$		−	38.0000i		−	1.23876i	−0.785090	−	0.619382i	$-0.787383\pi$
$941$		−	38.0000i		−	1.23876i	0.785090	−	0.619382i	$-0.212617\pi$
$942$		0			0
$943$			4.00000i			0.130258i
$944$		0			0
$945$	−1.00000			−0.0325300
$946$		0			0
$947$		−	25.0000i		−	0.812391i	−0.913786	−	0.406195i	$-0.866855\pi$
$947$		−	25.0000i		−	0.812391i	0.913786	−	0.406195i	$-0.133145\pi$
$948$		0			0
$949$	−26.0000	+	39.0000i	−0.843996	+	1.26599i
$950$		0			0
$951$			24.0000i			0.778253i
$952$		0			0
$953$	−30.0000			−0.971795			−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000			−0.971795			−0.485898	+	0.874016i	$0.661507\pi$
$954$		0			0
$955$			9.00000i			0.291233i
$956$		0			0
$957$			3.00000i			0.0969762i
$958$		0			0
$959$	17.0000			0.548959
$960$		0			0
$961$	−33.0000			−1.06452
$962$		0			0
$963$	−2.00000			−0.0644491
$964$		0			0
$965$	−8.00000			−0.257529
$966$		0			0
$967$		−	43.0000i		−	1.38279i	−0.722478	−	0.691393i	$-0.756997\pi$
$967$		−	43.0000i		−	1.38279i	0.722478	−	0.691393i	$-0.243003\pi$
$968$		0			0
$969$		−	21.0000i		−	0.674617i
$970$		0			0
$971$	−12.0000			−0.385098			−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000			−0.385098			−0.192549	+	0.981287i	$0.561675\pi$
$972$		0			0
$973$		−	8.00000i		−	0.256468i
$974$		0			0
$975$	−12.0000	−	8.00000i	−0.384308	−	0.256205i
$976$		0			0
$977$		−	15.0000i		−	0.479893i	−0.970786	−	0.239946i	$-0.922870\pi$
$977$		−	15.0000i		−	0.479893i	0.970786	−	0.239946i	$-0.0771298\pi$
$978$		0			0
$979$	36.0000			1.15056
$980$		0			0
$981$		−	7.00000i		−	0.223493i
$982$		0			0
$983$		−	51.0000i		−	1.62665i	−0.581811	−	0.813324i	$-0.697656\pi$
$983$		−	51.0000i		−	1.62665i	0.581811	−	0.813324i	$-0.302344\pi$
$984$		0			0
$985$	14.0000			0.446077
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−5.00000			−0.158991
$990$		0			0
$991$	20.0000			0.635321			0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000			0.635321			0.317660	+	0.948205i	$0.397103\pi$
$992$		0			0
$993$		−	8.00000i		−	0.253872i
$994$		0			0
$995$		−	9.00000i		−	0.285319i
$996$		0			0
$997$	18.0000			0.570066			0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000			0.570066			0.285033	+	0.958518i	$0.407995\pi$
$998$		0			0
$999$		−	1.00000i		−	0.0316386i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.h.337.2		2
4.3	odd	2		546.2.c.c.337.1	✓	2
12.11	even	2		1638.2.c.e.883.2		2
13.12	even	2	inner	4368.2.h.h.337.1		2
28.27	even	2		3822.2.c.b.883.1		2
52.31	even	4		7098.2.a.o.1.1		1
52.47	even	4		7098.2.a.y.1.1		1
52.51	odd	2		546.2.c.c.337.2	yes	2
156.155	even	2		1638.2.c.e.883.1		2
364.363	even	2		3822.2.c.b.883.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.c.c.337.1	✓	2	4.3	odd	2
546.2.c.c.337.2	yes	2	52.51	odd	2
1638.2.c.e.883.1		2	156.155	even	2
1638.2.c.e.883.2		2	12.11	even	2
3822.2.c.b.883.1		2	28.27	even	2
3822.2.c.b.883.2		2	364.363	even	2
4368.2.h.h.337.1		2	13.12	even	2	inner
4368.2.h.h.337.2		2	1.1	even	1	trivial
7098.2.a.o.1.1		1	52.31	even	4
7098.2.a.y.1.1		1	52.47	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000i q^{5} -1.00000i q^{7} +1.00000 q^{9} +3.00000i q^{11} +(3.00000 + 2.00000i) q^{13} -1.00000i q^{15} +7.00000 q^{17} +3.00000i q^{19} +1.00000i q^{21} -1.00000 q^{23} +4.00000 q^{25} -1.00000 q^{27} -1.00000 q^{29} -8.00000i q^{31} -3.00000i q^{33} +1.00000 q^{35} +1.00000i q^{37} +(-3.00000 - 2.00000i) q^{39} -4.00000i q^{41} +5.00000 q^{43} +1.00000i q^{45} -1.00000 q^{49} -7.00000 q^{51} -6.00000 q^{53} -3.00000 q^{55} -3.00000i q^{57} +10.0000i q^{59} -13.0000 q^{61} -1.00000i q^{63} +(-2.00000 + 3.00000i) q^{65} -8.00000i q^{67} +1.00000 q^{69} +6.00000i q^{71} +13.0000i q^{73} -4.00000 q^{75} +3.00000 q^{77} +12.0000 q^{79} +1.00000 q^{81} +2.00000i q^{83} +7.00000i q^{85} +1.00000 q^{87} -12.0000i q^{89} +(2.00000 - 3.00000i) q^{91} +8.00000i q^{93} -3.00000 q^{95} -6.00000i q^{97} +3.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{9} + 6 q^{13} + 14 q^{17} - 2 q^{23} + 8 q^{25} - 2 q^{27} - 2 q^{29} + 2 q^{35} - 6 q^{39} + 10 q^{43} - 2 q^{49} - 14 q^{51} - 12 q^{53} - 6 q^{55} - 26 q^{61} - 4 q^{65} + 2 q^{69} - 8 q^{75} + 6 q^{77} + 24 q^{79} + 2 q^{81} + 2 q^{87} + 4 q^{91} - 6 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 + 6 * q^13 + 14 * q^17 - 2 * q^23 + 8 * q^25 - 2 * q^27 - 2 * q^29 + 2 * q^35 - 6 * q^39 + 10 * q^43 - 2 * q^49 - 14 * q^51 - 12 * q^53 - 6 * q^55 - 26 * q^61 - 4 * q^65 + 2 * q^69 - 8 * q^75 + 6 * q^77 + 24 * q^79 + 2 * q^81 + 2 * q^87 + 4 * q^91 - 6 * q^95

Introduction

L-functions

Modular forms

Varieties

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Database

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