LMFDB - Embedded newform 4368.2.h.g.337.1

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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2184)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} -4.00000i q^{11} +(2.00000 + 3.00000i) q^{13} +3.00000i q^{15} +6.00000 q^{17} +7.00000i q^{19} -1.00000i q^{21} -1.00000 q^{23} -4.00000 q^{25} -1.00000 q^{27} +1.00000 q^{29} +7.00000i q^{31} +4.00000i q^{33} +3.00000 q^{35} +12.0000i q^{37} +(-2.00000 - 3.00000i) q^{39} -6.00000i q^{41} -11.0000 q^{43} -3.00000i q^{45} -3.00000i q^{47} -1.00000 q^{49} -6.00000 q^{51} -3.00000 q^{53} -12.0000 q^{55} -7.00000i q^{57} +12.0000i q^{59} -4.00000 q^{61} +1.00000i q^{63} +(9.00000 - 6.00000i) q^{65} -6.00000i q^{67} +1.00000 q^{69} +12.0000i q^{71} +11.0000i q^{73} +4.00000 q^{75} +4.00000 q^{77} +15.0000 q^{79} +1.00000 q^{81} +15.0000i q^{83} -18.0000i q^{85} -1.00000 q^{87} -11.0000i q^{89} +(-3.00000 + 2.00000i) q^{91} -7.00000i q^{93} +21.0000 q^{95} -7.00000i q^{97} -4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{13} + 12 q^{17} - 2 q^{23} - 8 q^{25} - 2 q^{27} + 2 q^{29} + 6 q^{35} - 4 q^{39} - 22 q^{43} - 2 q^{49} - 12 q^{51} - 6 q^{53} - 24 q^{55} - 8 q^{61} + 18 q^{65} + 2 q^{69} + 8 q^{75} + 8 q^{77} + 30 q^{79} + 2 q^{81} - 2 q^{87} - 6 q^{91} + 42 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 + 4 * q^13 + 12 * q^17 - 2 * q^23 - 8 * q^25 - 2 * q^27 + 2 * q^29 + 6 * q^35 - 4 * q^39 - 22 * q^43 - 2 * q^49 - 12 * q^51 - 6 * q^53 - 24 * q^55 - 8 * q^61 + 18 * q^65 + 2 * q^69 + 8 * q^75 + 8 * q^77 + 30 * q^79 + 2 * q^81 - 2 * q^87 - 6 * q^91 + 42 * 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
$5$		−	3.00000i		−	1.34164i	−0.741620	−	0.670820i	$-0.765942\pi$
$5$		−	3.00000i		−	1.34164i	0.741620	−	0.670820i	$-0.234058\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	4.00000i		−	1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$11$		−	4.00000i		−	1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$		0			0
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$14$		0			0
$15$			3.00000i			0.774597i
$16$		0			0
$17$	6.00000			1.45521			0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000			1.45521			0.727607	+	0.685994i	$0.240633\pi$
$18$		0			0
$19$			7.00000i			1.60591i	0.596040	+	0.802955i	$0.296740\pi$
$19$			7.00000i			1.60591i	−0.596040	+	0.802955i	$0.703260\pi$
$20$		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	$-0.470403\pi$
$29$	1.00000			0.185695			0.0928477	+	0.995680i	$0.470403\pi$
$30$		0			0
$31$			7.00000i			1.25724i	0.777714	+	0.628619i	$0.216379\pi$
$31$			7.00000i			1.25724i	−0.777714	+	0.628619i	$0.783621\pi$
$32$		0			0
$33$			4.00000i			0.696311i
$34$		0			0
$35$	3.00000			0.507093
$36$		0			0
$37$			12.0000i			1.97279i	0.164399	+	0.986394i	$0.447432\pi$
$37$			12.0000i			1.97279i	−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$	−2.00000	−	3.00000i	−0.320256	−	0.480384i
$40$		0			0
$41$		−	6.00000i		−	0.937043i	−0.883452	−	0.468521i	$-0.844787\pi$
$41$		−	6.00000i		−	0.937043i	0.883452	−	0.468521i	$-0.155213\pi$
$42$		0			0
$43$	−11.0000			−1.67748			−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000			−1.67748			−0.838742	+	0.544529i	$0.816708\pi$
$44$		0			0
$45$		−	3.00000i		−	0.447214i
$46$		0			0
$47$		−	3.00000i		−	0.437595i	−0.975770	−	0.218797i	$-0.929787\pi$
$47$		−	3.00000i		−	0.437595i	0.975770	−	0.218797i	$-0.0702134\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−6.00000			−0.840168
$52$		0			0
$53$	−3.00000			−0.412082			−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000			−0.412082			−0.206041	+	0.978543i	$0.566058\pi$
$54$		0			0
$55$	−12.0000			−1.61808
$56$		0			0
$57$		−	7.00000i		−	0.927173i
$58$		0			0
$59$			12.0000i			1.56227i	0.624364	+	0.781133i	$0.285358\pi$
$59$			12.0000i			1.56227i	−0.624364	+	0.781133i	$0.714642\pi$
$60$		0			0
$61$	−4.00000			−0.512148			−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000			−0.512148			−0.256074	+	0.966657i	$0.582429\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$	9.00000	−	6.00000i	1.11631	−	0.744208i
$66$		0			0
$67$		−	6.00000i		−	0.733017i	−0.930415	−	0.366508i	$-0.880553\pi$
$67$		−	6.00000i		−	0.733017i	0.930415	−	0.366508i	$-0.119447\pi$
$68$		0			0
$69$	1.00000			0.120386
$70$		0			0
$71$			12.0000i			1.42414i	0.702109	+	0.712069i	$0.252242\pi$
$71$			12.0000i			1.42414i	−0.702109	+	0.712069i	$0.747758\pi$
$72$		0			0
$73$			11.0000i			1.28745i	0.765256	+	0.643726i	$0.222612\pi$
$73$			11.0000i			1.28745i	−0.765256	+	0.643726i	$0.777388\pi$
$74$		0			0
$75$	4.00000			0.461880
$76$		0			0
$77$	4.00000			0.455842
$78$		0			0
$79$	15.0000			1.68763			0.843816	−	0.536633i	$-0.180304\pi$
$79$	15.0000			1.68763			0.843816	+	0.536633i	$0.180304\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			15.0000i			1.64646i	0.567705	+	0.823232i	$0.307831\pi$
$83$			15.0000i			1.64646i	−0.567705	+	0.823232i	$0.692169\pi$
$84$		0			0
$85$		−	18.0000i		−	1.95237i
$86$		0			0
$87$	−1.00000			−0.107211
$88$		0			0
$89$		−	11.0000i		−	1.16600i	−0.812473	−	0.582999i	$-0.801879\pi$
$89$		−	11.0000i		−	1.16600i	0.812473	−	0.582999i	$-0.198121\pi$
$90$		0			0
$91$	−3.00000	+	2.00000i	−0.314485	+	0.209657i
$92$		0			0
$93$		−	7.00000i		−	0.725866i
$94$		0			0
$95$	21.0000			2.15455
$96$		0			0
$97$		−	7.00000i		−	0.710742i	−0.934725	−	0.355371i	$-0.884354\pi$
$97$		−	7.00000i		−	0.710742i	0.934725	−	0.355371i	$-0.115646\pi$
$98$		0			0
$99$		−	4.00000i		−	0.402015i
$100$		0			0
$101$	−16.0000			−1.59206			−0.796030	−	0.605257i	$-0.793070\pi$
$101$	−16.0000			−1.59206			−0.796030	+	0.605257i	$0.793070\pi$
$102$		0			0
$103$	12.0000			1.18240			0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000			1.18240			0.591198	+	0.806527i	$0.298655\pi$
$104$		0			0
$105$	−3.00000			−0.292770
$106$		0			0
$107$	−4.00000			−0.386695			−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000			−0.386695			−0.193347	+	0.981130i	$0.561934\pi$
$108$		0			0
$109$			10.0000i			0.957826i	0.877862	+	0.478913i	$0.158969\pi$
$109$			10.0000i			0.957826i	−0.877862	+	0.478913i	$0.841031\pi$
$110$		0			0
$111$		−	12.0000i		−	1.13899i
$112$		0			0
$113$	11.0000			1.03479			0.517396	−	0.855746i	$-0.326901\pi$
$113$	11.0000			1.03479			0.517396	+	0.855746i	$0.326901\pi$
$114$		0			0
$115$			3.00000i			0.279751i
$116$		0			0
$117$	2.00000	+	3.00000i	0.184900	+	0.277350i
$118$		0			0
$119$			6.00000i			0.550019i
$120$		0			0
$121$	−5.00000			−0.454545
$122$		0			0
$123$			6.00000i			0.541002i
$124$		0			0
$125$		−	3.00000i		−	0.268328i
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$		0			0
$129$	11.0000			0.968496
$130$		0			0
$131$	−10.0000			−0.873704			−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000			−0.873704			−0.436852	+	0.899533i	$0.643907\pi$
$132$		0			0
$133$	−7.00000			−0.606977
$134$		0			0
$135$			3.00000i			0.258199i
$136$		0			0
$137$			4.00000i			0.341743i	0.985293	+	0.170872i	$0.0546583\pi$
$137$			4.00000i			0.341743i	−0.985293	+	0.170872i	$0.945342\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$			3.00000i			0.252646i
$142$		0			0
$143$	12.0000	−	8.00000i	1.00349	−	0.668994i
$144$		0			0
$145$		−	3.00000i		−	0.249136i
$146$		0			0
$147$	1.00000			0.0824786
$148$		0			0
$149$			8.00000i			0.655386i	0.944784	+	0.327693i	$0.106271\pi$
$149$			8.00000i			0.655386i	−0.944784	+	0.327693i	$0.893729\pi$
$150$		0			0
$151$			16.0000i			1.30206i	0.759051	+	0.651031i	$0.225663\pi$
$151$			16.0000i			1.30206i	−0.759051	+	0.651031i	$0.774337\pi$
$152$		0			0
$153$	6.00000			0.485071
$154$		0			0
$155$	21.0000			1.68676
$156$		0			0
$157$	18.0000			1.43656			0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000			1.43656			0.718278	+	0.695756i	$0.244931\pi$
$158$		0			0
$159$	3.00000			0.237915
$160$		0			0
$161$		−	1.00000i		−	0.0788110i
$162$		0			0
$163$		−	2.00000i		−	0.156652i	−0.996928	−	0.0783260i	$-0.975042\pi$
$163$		−	2.00000i		−	0.156652i	0.996928	−	0.0783260i	$-0.0249575\pi$
$164$		0			0
$165$	12.0000			0.934199
$166$		0			0
$167$		−	17.0000i		−	1.31550i	−0.753237	−	0.657750i	$-0.771508\pi$
$167$		−	17.0000i		−	1.31550i	0.753237	−	0.657750i	$-0.228492\pi$
$168$		0			0
$169$	−5.00000	+	12.0000i	−0.384615	+	0.923077i
$170$		0			0
$171$			7.00000i			0.535303i
$172$		0			0
$173$	12.0000			0.912343			0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000			0.912343			0.456172	+	0.889892i	$0.349220\pi$
$174$		0			0
$175$		−	4.00000i		−	0.302372i
$176$		0			0
$177$		−	12.0000i		−	0.901975i
$178$		0			0
$179$	−17.0000			−1.27064			−0.635320	−	0.772249i	$-0.719132\pi$
$179$	−17.0000			−1.27064			−0.635320	+	0.772249i	$0.719132\pi$
$180$		0			0
$181$	20.0000			1.48659			0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000			1.48659			0.743294	+	0.668965i	$0.233262\pi$
$182$		0			0
$183$	4.00000			0.295689
$184$		0			0
$185$	36.0000			2.64677
$186$		0			0
$187$		−	24.0000i		−	1.75505i
$188$		0			0
$189$		−	1.00000i		−	0.0727393i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$			14.0000i			1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$			14.0000i			1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$		0			0
$195$	−9.00000	+	6.00000i	−0.644503	+	0.429669i
$196$		0			0
$197$			4.00000i			0.284988i	0.989796	+	0.142494i	$0.0455122\pi$
$197$			4.00000i			0.284988i	−0.989796	+	0.142494i	$0.954488\pi$
$198$		0			0
$199$	10.0000			0.708881			0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000			0.708881			0.354441	+	0.935079i	$0.384671\pi$
$200$		0			0
$201$			6.00000i			0.423207i
$202$		0			0
$203$			1.00000i			0.0701862i
$204$		0			0
$205$	−18.0000			−1.25717
$206$		0			0
$207$	−1.00000			−0.0695048
$208$		0			0
$209$	28.0000			1.93680
$210$		0			0
$211$	23.0000			1.58339			0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000			1.58339			0.791693	+	0.610920i	$0.209200\pi$
$212$		0			0
$213$		−	12.0000i		−	0.822226i
$214$		0			0
$215$			33.0000i			2.25058i
$216$		0			0
$217$	−7.00000			−0.475191
$218$		0			0
$219$		−	11.0000i		−	0.743311i
$220$		0			0
$221$	12.0000	+	18.0000i	0.807207	+	1.21081i
$222$		0			0
$223$		−	15.0000i		−	1.00447i	−0.864730	−	0.502237i	$-0.832510\pi$
$223$		−	15.0000i		−	1.00447i	0.864730	−	0.502237i	$-0.167490\pi$
$224$		0			0
$225$	−4.00000			−0.266667
$226$		0			0
$227$			4.00000i			0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$			4.00000i			0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$		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
$231$	−4.00000			−0.263181
$232$		0			0
$233$	−17.0000			−1.11371			−0.556854	−	0.830611i	$-0.687992\pi$
$233$	−17.0000			−1.11371			−0.556854	+	0.830611i	$0.687992\pi$
$234$		0			0
$235$	−9.00000			−0.587095
$236$		0			0
$237$	−15.0000			−0.974355
$238$		0			0
$239$		−	18.0000i		−	1.16432i	−0.813073	−	0.582162i	$-0.802207\pi$
$239$		−	18.0000i		−	1.16432i	0.813073	−	0.582162i	$-0.197793\pi$
$240$		0			0
$241$			19.0000i			1.22390i	0.790897	+	0.611949i	$0.209614\pi$
$241$			19.0000i			1.22390i	−0.790897	+	0.611949i	$0.790386\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$			3.00000i			0.191663i
$246$		0			0
$247$	−21.0000	+	14.0000i	−1.33620	+	0.890799i
$248$		0			0
$249$		−	15.0000i		−	0.950586i
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$			4.00000i			0.251478i
$254$		0			0
$255$			18.0000i			1.12720i
$256$		0			0
$257$	−8.00000			−0.499026			−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000			−0.499026			−0.249513	+	0.968371i	$0.580271\pi$
$258$		0			0
$259$	−12.0000			−0.745644
$260$		0			0
$261$	1.00000			0.0618984
$262$		0			0
$263$	19.0000			1.17159			0.585795	−	0.810459i	$-0.300782\pi$
$263$	19.0000			1.17159			0.585795	+	0.810459i	$0.300782\pi$
$264$		0			0
$265$			9.00000i			0.552866i
$266$		0			0
$267$			11.0000i			0.673189i
$268$		0			0
$269$	−10.0000			−0.609711			−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000			−0.609711			−0.304855	+	0.952399i	$0.598608\pi$
$270$		0			0
$271$		−	28.0000i		−	1.70088i	−0.526073	−	0.850439i	$-0.676336\pi$
$271$		−	28.0000i		−	1.70088i	0.526073	−	0.850439i	$-0.323664\pi$
$272$		0			0
$273$	3.00000	−	2.00000i	0.181568	−	0.121046i
$274$		0			0
$275$			16.0000i			0.964836i
$276$		0			0
$277$	−1.00000			−0.0600842			−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000			−0.0600842			−0.0300421	+	0.999549i	$0.509564\pi$
$278$		0			0
$279$			7.00000i			0.419079i
$280$		0			0
$281$			22.0000i			1.31241i	0.754583	+	0.656205i	$0.227839\pi$
$281$			22.0000i			1.31241i	−0.754583	+	0.656205i	$0.772161\pi$
$282$		0			0
$283$	28.0000			1.66443			0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000			1.66443			0.832214	+	0.554455i	$0.187073\pi$
$284$		0			0
$285$	−21.0000			−1.24393
$286$		0			0
$287$	6.00000			0.354169
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$			7.00000i			0.410347i
$292$		0			0
$293$		−	25.0000i		−	1.46052i	−0.683172	−	0.730258i	$-0.739400\pi$
$293$		−	25.0000i		−	1.46052i	0.683172	−	0.730258i	$-0.260600\pi$
$294$		0			0
$295$	36.0000			2.09600
$296$		0			0
$297$			4.00000i			0.232104i
$298$		0			0
$299$	−2.00000	−	3.00000i	−0.115663	−	0.173494i
$300$		0			0
$301$		−	11.0000i		−	0.634029i
$302$		0			0
$303$	16.0000			0.919176
$304$		0			0
$305$			12.0000i			0.687118i
$306$		0			0
$307$		−	25.0000i		−	1.42683i	−0.700744	−	0.713413i	$-0.747149\pi$
$307$		−	25.0000i		−	1.42683i	0.700744	−	0.713413i	$-0.252851\pi$
$308$		0			0
$309$	−12.0000			−0.682656
$310$		0			0
$311$	−2.00000			−0.113410			−0.0567048	−	0.998391i	$-0.518059\pi$
$311$	−2.00000			−0.113410			−0.0567048	+	0.998391i	$0.518059\pi$
$312$		0			0
$313$	−28.0000			−1.58265			−0.791327	−	0.611393i	$-0.790609\pi$
$313$	−28.0000			−1.58265			−0.791327	+	0.611393i	$0.790609\pi$
$314$		0			0
$315$	3.00000			0.169031
$316$		0			0
$317$			18.0000i			1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$			18.0000i			1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$		0			0
$319$		−	4.00000i		−	0.223957i
$320$		0			0
$321$	4.00000			0.223258
$322$		0			0
$323$			42.0000i			2.33694i
$324$		0			0
$325$	−8.00000	−	12.0000i	−0.443760	−	0.665640i
$326$		0			0
$327$		−	10.0000i		−	0.553001i
$328$		0			0
$329$	3.00000			0.165395
$330$		0			0
$331$			16.0000i			0.879440i	0.898135	+	0.439720i	$0.144922\pi$
$331$			16.0000i			0.879440i	−0.898135	+	0.439720i	$0.855078\pi$
$332$		0			0
$333$			12.0000i			0.657596i
$334$		0			0
$335$	−18.0000			−0.983445
$336$		0			0
$337$	31.0000			1.68868			0.844339	−	0.535810i	$-0.179994\pi$
$337$	31.0000			1.68868			0.844339	+	0.535810i	$0.179994\pi$
$338$		0			0
$339$	−11.0000			−0.597438
$340$		0			0
$341$	28.0000			1.51629
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$		−	3.00000i		−	0.161515i
$346$		0			0
$347$	−12.0000			−0.644194			−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000			−0.644194			−0.322097	+	0.946707i	$0.604388\pi$
$348$		0			0
$349$			23.0000i			1.23116i	0.788074	+	0.615581i	$0.211079\pi$
$349$			23.0000i			1.23116i	−0.788074	+	0.615581i	$0.788921\pi$
$350$		0			0
$351$	−2.00000	−	3.00000i	−0.106752	−	0.160128i
$352$		0			0
$353$		−	10.0000i		−	0.532246i	−0.963939	−	0.266123i	$-0.914257\pi$
$353$		−	10.0000i		−	0.532246i	0.963939	−	0.266123i	$-0.0857428\pi$
$354$		0			0
$355$	36.0000			1.91068
$356$		0			0
$357$		−	6.00000i		−	0.317554i
$358$		0			0
$359$		−	18.0000i		−	0.950004i	−0.879985	−	0.475002i	$-0.842447\pi$
$359$		−	18.0000i		−	0.950004i	0.879985	−	0.475002i	$-0.157553\pi$
$360$		0			0
$361$	−30.0000			−1.57895
$362$		0			0
$363$	5.00000			0.262432
$364$		0			0
$365$	33.0000			1.72730
$366$		0			0
$367$	2.00000			0.104399			0.0521996	−	0.998637i	$-0.483377\pi$
$367$	2.00000			0.104399			0.0521996	+	0.998637i	$0.483377\pi$
$368$		0			0
$369$		−	6.00000i		−	0.312348i
$370$		0			0
$371$		−	3.00000i		−	0.155752i
$372$		0			0
$373$	14.0000			0.724893			0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000			0.724893			0.362446	+	0.932005i	$0.381942\pi$
$374$		0			0
$375$			3.00000i			0.154919i
$376$		0			0
$377$	2.00000	+	3.00000i	0.103005	+	0.154508i
$378$		0			0
$379$		−	12.0000i		−	0.616399i	−0.951322	−	0.308199i	$-0.900274\pi$
$379$		−	12.0000i		−	0.616399i	0.951322	−	0.308199i	$-0.0997264\pi$
$380$		0			0
$381$	8.00000			0.409852
$382$		0			0
$383$			16.0000i			0.817562i	0.912633	+	0.408781i	$0.134046\pi$
$383$			16.0000i			0.817562i	−0.912633	+	0.408781i	$0.865954\pi$
$384$		0			0
$385$		−	12.0000i		−	0.611577i
$386$		0			0
$387$	−11.0000			−0.559161
$388$		0			0
$389$	30.0000			1.52106			0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000			1.52106			0.760530	+	0.649303i	$0.224939\pi$
$390$		0			0
$391$	−6.00000			−0.303433
$392$		0			0
$393$	10.0000			0.504433
$394$		0			0
$395$		−	45.0000i		−	2.26420i
$396$		0			0
$397$			7.00000i			0.351320i	0.984451	+	0.175660i	$0.0562059\pi$
$397$			7.00000i			0.351320i	−0.984451	+	0.175660i	$0.943794\pi$
$398$		0			0
$399$	7.00000			0.350438
$400$		0			0
$401$			14.0000i			0.699127i	0.936913	+	0.349563i	$0.113670\pi$
$401$			14.0000i			0.699127i	−0.936913	+	0.349563i	$0.886330\pi$
$402$		0			0
$403$	−21.0000	+	14.0000i	−1.04608	+	0.697390i
$404$		0			0
$405$		−	3.00000i		−	0.149071i
$406$		0			0
$407$	48.0000			2.37927
$408$		0			0
$409$			5.00000i			0.247234i	0.992330	+	0.123617i	$0.0394494\pi$
$409$			5.00000i			0.247234i	−0.992330	+	0.123617i	$0.960551\pi$
$410$		0			0
$411$		−	4.00000i		−	0.197305i
$412$		0			0
$413$	−12.0000			−0.590481
$414$		0			0
$415$	45.0000			2.20896
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$		−	26.0000i		−	1.26716i	−0.773676	−	0.633581i	$-0.781584\pi$
$421$		−	26.0000i		−	1.26716i	0.773676	−	0.633581i	$-0.218416\pi$
$422$		0			0
$423$		−	3.00000i		−	0.145865i
$424$		0			0
$425$	−24.0000			−1.16417
$426$		0			0
$427$		−	4.00000i		−	0.193574i
$428$		0			0
$429$	−12.0000	+	8.00000i	−0.579365	+	0.386244i
$430$		0			0
$431$			36.0000i			1.73406i	0.498257	+	0.867029i	$0.333974\pi$
$431$			36.0000i			1.73406i	−0.498257	+	0.867029i	$0.666026\pi$
$432$		0			0
$433$	14.0000			0.672797			0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000			0.672797			0.336399	+	0.941720i	$0.390791\pi$
$434$		0			0
$435$			3.00000i			0.143839i
$436$		0			0
$437$		−	7.00000i		−	0.334855i
$438$		0			0
$439$	24.0000			1.14546			0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000			1.14546			0.572729	+	0.819745i	$0.305885\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	−11.0000			−0.522626			−0.261313	−	0.965254i	$-0.584155\pi$
$443$	−11.0000			−0.522626			−0.261313	+	0.965254i	$0.584155\pi$
$444$		0			0
$445$	−33.0000			−1.56435
$446$		0			0
$447$		−	8.00000i		−	0.378387i
$448$		0			0
$449$		−	14.0000i		−	0.660701i	−0.943858	−	0.330350i	$-0.892833\pi$
$449$		−	14.0000i		−	0.660701i	0.943858	−	0.330350i	$-0.107167\pi$
$450$		0			0
$451$	−24.0000			−1.13012
$452$		0			0
$453$		−	16.0000i		−	0.751746i
$454$		0			0
$455$	6.00000	+	9.00000i	0.281284	+	0.421927i
$456$		0			0
$457$			10.0000i			0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$			10.0000i			0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$		0			0
$459$	−6.00000			−0.280056
$460$		0			0
$461$			18.0000i			0.838344i	0.907907	+	0.419172i	$0.137680\pi$
$461$			18.0000i			0.838344i	−0.907907	+	0.419172i	$0.862320\pi$
$462$		0			0
$463$			2.00000i			0.0929479i	0.998920	+	0.0464739i	$0.0147984\pi$
$463$			2.00000i			0.0929479i	−0.998920	+	0.0464739i	$0.985202\pi$
$464$		0			0
$465$	−21.0000			−0.973852
$466$		0			0
$467$	14.0000			0.647843			0.323921	−	0.946084i	$-0.394999\pi$
$467$	14.0000			0.647843			0.323921	+	0.946084i	$0.394999\pi$
$468$		0			0
$469$	6.00000			0.277054
$470$		0			0
$471$	−18.0000			−0.829396
$472$		0			0
$473$			44.0000i			2.02312i
$474$		0			0
$475$		−	28.0000i		−	1.28473i
$476$		0			0
$477$	−3.00000			−0.137361
$478$		0			0
$479$			7.00000i			0.319838i	0.987130	+	0.159919i	$0.0511233\pi$
$479$			7.00000i			0.319838i	−0.987130	+	0.159919i	$0.948877\pi$
$480$		0			0
$481$	−36.0000	+	24.0000i	−1.64146	+	1.09431i
$482$		0			0
$483$			1.00000i			0.0455016i
$484$		0			0
$485$	−21.0000			−0.953561
$486$		0			0
$487$			4.00000i			0.181257i	0.995885	+	0.0906287i	$0.0288876\pi$
$487$			4.00000i			0.181257i	−0.995885	+	0.0906287i	$0.971112\pi$
$488$		0			0
$489$			2.00000i			0.0904431i
$490$		0			0
$491$	24.0000			1.08310			0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000			1.08310			0.541552	+	0.840667i	$0.317837\pi$
$492$		0			0
$493$	6.00000			0.270226
$494$		0			0
$495$	−12.0000			−0.539360
$496$		0			0
$497$	−12.0000			−0.538274
$498$		0			0
$499$			2.00000i			0.0895323i	0.998997	+	0.0447661i	$0.0142543\pi$
$499$			2.00000i			0.0895323i	−0.998997	+	0.0447661i	$0.985746\pi$
$500$		0			0
$501$			17.0000i			0.759504i
$502$		0			0
$503$	−12.0000			−0.535054			−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000			−0.535054			−0.267527	+	0.963550i	$0.586206\pi$
$504$		0			0
$505$			48.0000i			2.13597i
$506$		0			0
$507$	5.00000	−	12.0000i	0.222058	−	0.532939i
$508$		0			0
$509$			1.00000i			0.0443242i	0.999754	+	0.0221621i	$0.00705500\pi$
$509$			1.00000i			0.0443242i	−0.999754	+	0.0221621i	$0.992945\pi$
$510$		0			0
$511$	−11.0000			−0.486611
$512$		0			0
$513$		−	7.00000i		−	0.309058i
$514$		0			0
$515$		−	36.0000i		−	1.58635i
$516$		0			0
$517$	−12.0000			−0.527759
$518$		0			0
$519$	−12.0000			−0.526742
$520$		0			0
$521$	−32.0000			−1.40195			−0.700973	−	0.713188i	$-0.747251\pi$
$521$	−32.0000			−1.40195			−0.700973	+	0.713188i	$0.747251\pi$
$522$		0			0
$523$	−36.0000			−1.57417			−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000			−1.57417			−0.787085	+	0.616844i	$0.788411\pi$
$524$		0			0
$525$			4.00000i			0.174574i
$526$		0			0
$527$			42.0000i			1.82955i
$528$		0			0
$529$	−22.0000			−0.956522
$530$		0			0
$531$			12.0000i			0.520756i
$532$		0			0
$533$	18.0000	−	12.0000i	0.779667	−	0.519778i
$534$		0			0
$535$			12.0000i			0.518805i
$536$		0			0
$537$	17.0000			0.733604
$538$		0			0
$539$			4.00000i			0.172292i
$540$		0			0
$541$		−	20.0000i		−	0.859867i	−0.902861	−	0.429934i	$-0.858537\pi$
$541$		−	20.0000i		−	0.859867i	0.902861	−	0.429934i	$-0.141463\pi$
$542$		0			0
$543$	−20.0000			−0.858282
$544$		0			0
$545$	30.0000			1.28506
$546$		0			0
$547$	1.00000			0.0427569			0.0213785	−	0.999771i	$-0.493195\pi$
$547$	1.00000			0.0427569			0.0213785	+	0.999771i	$0.493195\pi$
$548$		0			0
$549$	−4.00000			−0.170716
$550$		0			0
$551$			7.00000i			0.298210i
$552$		0			0
$553$			15.0000i			0.637865i
$554$		0			0
$555$	−36.0000			−1.52811
$556$		0			0
$557$			12.0000i			0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$			12.0000i			0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$		0			0
$559$	−22.0000	−	33.0000i	−0.930501	−	1.39575i
$560$		0			0
$561$			24.0000i			1.01328i
$562$		0			0
$563$	18.0000			0.758610			0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000			0.758610			0.379305	+	0.925272i	$0.376163\pi$
$564$		0			0
$565$		−	33.0000i		−	1.38832i
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	−21.0000			−0.880366			−0.440183	−	0.897908i	$-0.645086\pi$
$569$	−21.0000			−0.880366			−0.440183	+	0.897908i	$0.645086\pi$
$570$		0			0
$571$	−19.0000			−0.795125			−0.397563	−	0.917575i	$-0.630144\pi$
$571$	−19.0000			−0.795125			−0.397563	+	0.917575i	$0.630144\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.00000			0.166812
$576$		0			0
$577$			18.0000i			0.749350i	0.927156	+	0.374675i	$0.122246\pi$
$577$			18.0000i			0.749350i	−0.927156	+	0.374675i	$0.877754\pi$
$578$		0			0
$579$		−	14.0000i		−	0.581820i
$580$		0			0
$581$	−15.0000			−0.622305
$582$		0			0
$583$			12.0000i			0.496989i
$584$		0			0
$585$	9.00000	−	6.00000i	0.372104	−	0.248069i
$586$		0			0
$587$		−	45.0000i		−	1.85735i	−0.370896	−	0.928674i	$-0.620949\pi$
$587$		−	45.0000i		−	1.85735i	0.370896	−	0.928674i	$-0.379051\pi$
$588$		0			0
$589$	−49.0000			−2.01901
$590$		0			0
$591$		−	4.00000i		−	0.164538i
$592$		0			0
$593$		−	37.0000i		−	1.51941i	−0.650269	−	0.759704i	$-0.725344\pi$
$593$		−	37.0000i		−	1.51941i	0.650269	−	0.759704i	$-0.274656\pi$
$594$		0			0
$595$	18.0000			0.737928
$596$		0			0
$597$	−10.0000			−0.409273
$598$		0			0
$599$	5.00000			0.204294			0.102147	−	0.994769i	$-0.467429\pi$
$599$	5.00000			0.204294			0.102147	+	0.994769i	$0.467429\pi$
$600$		0			0
$601$	14.0000			0.571072			0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000			0.571072			0.285536	+	0.958368i	$0.407828\pi$
$602$		0			0
$603$		−	6.00000i		−	0.244339i
$604$		0			0
$605$			15.0000i			0.609837i
$606$		0			0
$607$	20.0000			0.811775			0.405887	−	0.913923i	$-0.366962\pi$
$607$	20.0000			0.811775			0.405887	+	0.913923i	$0.366962\pi$
$608$		0			0
$609$		−	1.00000i		−	0.0405220i
$610$		0			0
$611$	9.00000	−	6.00000i	0.364101	−	0.242734i
$612$		0			0
$613$		−	22.0000i		−	0.888572i	−0.895885	−	0.444286i	$-0.853457\pi$
$613$		−	22.0000i		−	0.888572i	0.895885	−	0.444286i	$-0.146543\pi$
$614$		0			0
$615$	18.0000			0.725830
$616$		0			0
$617$		−	28.0000i		−	1.12724i	−0.826035	−	0.563619i	$-0.809409\pi$
$617$		−	28.0000i		−	1.12724i	0.826035	−	0.563619i	$-0.190591\pi$
$618$		0			0
$619$		−	12.0000i		−	0.482321i	−0.970485	−	0.241160i	$-0.922472\pi$
$619$		−	12.0000i		−	0.482321i	0.970485	−	0.241160i	$-0.0775280\pi$
$620$		0			0
$621$	1.00000			0.0401286
$622$		0			0
$623$	11.0000			0.440706
$624$		0			0
$625$	−29.0000			−1.16000
$626$		0			0
$627$	−28.0000			−1.11821
$628$		0			0
$629$			72.0000i			2.87083i
$630$		0			0
$631$		−	28.0000i		−	1.11466i	−0.830290	−	0.557331i	$-0.811825\pi$
$631$		−	28.0000i		−	1.11466i	0.830290	−	0.557331i	$-0.188175\pi$
$632$		0			0
$633$	−23.0000			−0.914168
$634$		0			0
$635$			24.0000i			0.952411i
$636$		0			0
$637$	−2.00000	−	3.00000i	−0.0792429	−	0.118864i
$638$		0			0
$639$			12.0000i			0.474713i
$640$		0			0
$641$	9.00000			0.355479			0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000			0.355479			0.177739	+	0.984078i	$0.443122\pi$
$642$		0			0
$643$		−	28.0000i		−	1.10421i	−0.833774	−	0.552106i	$-0.813824\pi$
$643$		−	28.0000i		−	1.10421i	0.833774	−	0.552106i	$-0.186176\pi$
$644$		0			0
$645$		−	33.0000i		−	1.29937i
$646$		0			0
$647$	30.0000			1.17942			0.589711	−	0.807614i	$-0.299242\pi$
$647$	30.0000			1.17942			0.589711	+	0.807614i	$0.299242\pi$
$648$		0			0
$649$	48.0000			1.88416
$650$		0			0
$651$	7.00000			0.274352
$652$		0			0
$653$	−18.0000			−0.704394			−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000			−0.704394			−0.352197	+	0.935926i	$0.614565\pi$
$654$		0			0
$655$			30.0000i			1.17220i
$656$		0			0
$657$			11.0000i			0.429151i
$658$		0			0
$659$	−13.0000			−0.506408			−0.253204	−	0.967413i	$-0.581484\pi$
$659$	−13.0000			−0.506408			−0.253204	+	0.967413i	$0.581484\pi$
$660$		0			0
$661$			7.00000i			0.272268i	0.990690	+	0.136134i	$0.0434678\pi$
$661$			7.00000i			0.272268i	−0.990690	+	0.136134i	$0.956532\pi$
$662$		0			0
$663$	−12.0000	−	18.0000i	−0.466041	−	0.699062i
$664$		0			0
$665$			21.0000i			0.814345i
$666$		0			0
$667$	−1.00000			−0.0387202
$668$		0			0
$669$			15.0000i			0.579934i
$670$		0			0
$671$			16.0000i			0.617673i
$672$		0			0
$673$	11.0000			0.424019			0.212009	−	0.977268i	$-0.431999\pi$
$673$	11.0000			0.424019			0.212009	+	0.977268i	$0.431999\pi$
$674$		0			0
$675$	4.00000			0.153960
$676$		0			0
$677$	14.0000			0.538064			0.269032	−	0.963131i	$-0.413296\pi$
$677$	14.0000			0.538064			0.269032	+	0.963131i	$0.413296\pi$
$678$		0			0
$679$	7.00000			0.268635
$680$		0			0
$681$		−	4.00000i		−	0.153280i
$682$		0			0
$683$		−	4.00000i		−	0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$		−	4.00000i		−	0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$		0			0
$685$	12.0000			0.458496
$686$		0			0
$687$			10.0000i			0.381524i
$688$		0			0
$689$	−6.00000	−	9.00000i	−0.228582	−	0.342873i
$690$		0			0
$691$		−	35.0000i		−	1.33146i	−0.746191	−	0.665731i	$-0.768120\pi$
$691$		−	35.0000i		−	1.33146i	0.746191	−	0.665731i	$-0.231880\pi$
$692$		0			0
$693$	4.00000			0.151947
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	36.0000i		−	1.36360i
$698$		0			0
$699$	17.0000			0.642999
$700$		0			0
$701$	5.00000			0.188847			0.0944237	−	0.995532i	$-0.469899\pi$
$701$	5.00000			0.188847			0.0944237	+	0.995532i	$0.469899\pi$
$702$		0			0
$703$	−84.0000			−3.16812
$704$		0			0
$705$	9.00000			0.338960
$706$		0			0
$707$		−	16.0000i		−	0.601742i
$708$		0			0
$709$			44.0000i			1.65245i	0.563337	+	0.826227i	$0.309517\pi$
$709$			44.0000i			1.65245i	−0.563337	+	0.826227i	$0.690483\pi$
$710$		0			0
$711$	15.0000			0.562544
$712$		0			0
$713$		−	7.00000i		−	0.262152i
$714$		0			0
$715$	−24.0000	−	36.0000i	−0.897549	−	1.34632i
$716$		0			0
$717$			18.0000i			0.672222i
$718$		0			0
$719$	−26.0000			−0.969636			−0.484818	−	0.874615i	$-0.661114\pi$
$719$	−26.0000			−0.969636			−0.484818	+	0.874615i	$0.661114\pi$
$720$		0			0
$721$			12.0000i			0.446903i
$722$		0			0
$723$		−	19.0000i		−	0.706618i
$724$		0			0
$725$	−4.00000			−0.148556
$726$		0			0
$727$	14.0000			0.519231			0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000			0.519231			0.259616	+	0.965712i	$0.416404\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−66.0000			−2.44110
$732$		0			0
$733$		−	35.0000i		−	1.29275i	−0.763018	−	0.646377i	$-0.776283\pi$
$733$		−	35.0000i		−	1.29275i	0.763018	−	0.646377i	$-0.223717\pi$
$734$		0			0
$735$		−	3.00000i		−	0.110657i
$736$		0			0
$737$	−24.0000			−0.884051
$738$		0			0
$739$			4.00000i			0.147142i	0.997290	+	0.0735712i	$0.0234396\pi$
$739$			4.00000i			0.147142i	−0.997290	+	0.0735712i	$0.976560\pi$
$740$		0			0
$741$	21.0000	−	14.0000i	0.771454	−	0.514303i
$742$		0			0
$743$			26.0000i			0.953847i	0.878945	+	0.476924i	$0.158248\pi$
$743$			26.0000i			0.953847i	−0.878945	+	0.476924i	$0.841752\pi$
$744$		0			0
$745$	24.0000			0.879292
$746$		0			0
$747$			15.0000i			0.548821i
$748$		0			0
$749$		−	4.00000i		−	0.146157i
$750$		0			0
$751$	25.0000			0.912263			0.456131	−	0.889912i	$-0.349235\pi$
$751$	25.0000			0.912263			0.456131	+	0.889912i	$0.349235\pi$
$752$		0			0
$753$	12.0000			0.437304
$754$		0			0
$755$	48.0000			1.74690
$756$		0			0
$757$	−47.0000			−1.70824			−0.854122	−	0.520073i	$-0.825905\pi$
$757$	−47.0000			−1.70824			−0.854122	+	0.520073i	$0.825905\pi$
$758$		0			0
$759$		−	4.00000i		−	0.145191i
$760$		0			0
$761$		−	33.0000i		−	1.19625i	−0.801403	−	0.598125i	$-0.795913\pi$
$761$		−	33.0000i		−	1.19625i	0.801403	−	0.598125i	$-0.204087\pi$
$762$		0			0
$763$	−10.0000			−0.362024
$764$		0			0
$765$		−	18.0000i		−	0.650791i
$766$		0			0
$767$	−36.0000	+	24.0000i	−1.29988	+	0.866590i
$768$		0			0
$769$			19.0000i			0.685158i	0.939489	+	0.342579i	$0.111300\pi$
$769$			19.0000i			0.685158i	−0.939489	+	0.342579i	$0.888700\pi$
$770$		0			0
$771$	8.00000			0.288113
$772$		0			0
$773$			54.0000i			1.94225i	0.238581	+	0.971123i	$0.423318\pi$
$773$			54.0000i			1.94225i	−0.238581	+	0.971123i	$0.576682\pi$
$774$		0			0
$775$		−	28.0000i		−	1.00579i
$776$		0			0
$777$	12.0000			0.430498
$778$		0			0
$779$	42.0000			1.50481
$780$		0			0
$781$	48.0000			1.71758
$782$		0			0
$783$	−1.00000			−0.0357371
$784$		0			0
$785$		−	54.0000i		−	1.92734i
$786$		0			0
$787$		−	47.0000i		−	1.67537i	−0.546154	−	0.837685i	$-0.683909\pi$
$787$		−	47.0000i		−	1.67537i	0.546154	−	0.837685i	$-0.316091\pi$
$788$		0			0
$789$	−19.0000			−0.676418
$790$		0			0
$791$			11.0000i			0.391115i
$792$		0			0
$793$	−8.00000	−	12.0000i	−0.284088	−	0.426132i
$794$		0			0
$795$		−	9.00000i		−	0.319197i
$796$		0			0
$797$	−46.0000			−1.62940			−0.814702	−	0.579880i	$-0.803099\pi$
$797$	−46.0000			−1.62940			−0.814702	+	0.579880i	$0.803099\pi$
$798$		0			0
$799$		−	18.0000i		−	0.636794i
$800$		0			0
$801$		−	11.0000i		−	0.388666i
$802$		0			0
$803$	44.0000			1.55273
$804$		0			0
$805$	−3.00000			−0.105736
$806$		0			0
$807$	10.0000			0.352017
$808$		0			0
$809$	11.0000			0.386739			0.193370	−	0.981126i	$-0.438058\pi$
$809$	11.0000			0.386739			0.193370	+	0.981126i	$0.438058\pi$
$810$		0			0
$811$			12.0000i			0.421377i	0.977553	+	0.210688i	$0.0675706\pi$
$811$			12.0000i			0.421377i	−0.977553	+	0.210688i	$0.932429\pi$
$812$		0			0
$813$			28.0000i			0.982003i
$814$		0			0
$815$	−6.00000			−0.210171
$816$		0			0
$817$		−	77.0000i		−	2.69389i
$818$		0			0
$819$	−3.00000	+	2.00000i	−0.104828	+	0.0698857i
$820$		0			0
$821$			18.0000i			0.628204i	0.949389	+	0.314102i	$0.101703\pi$
$821$			18.0000i			0.628204i	−0.949389	+	0.314102i	$0.898297\pi$
$822$		0			0
$823$	−40.0000			−1.39431			−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000			−1.39431			−0.697156	+	0.716919i	$0.745552\pi$
$824$		0			0
$825$		−	16.0000i		−	0.557048i
$826$		0			0
$827$		−	42.0000i		−	1.46048i	−0.683189	−	0.730242i	$-0.739408\pi$
$827$		−	42.0000i		−	1.46048i	0.683189	−	0.730242i	$-0.260592\pi$
$828$		0			0
$829$	14.0000			0.486240			0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000			0.486240			0.243120	+	0.969996i	$0.421829\pi$
$830$		0			0
$831$	1.00000			0.0346896
$832$		0			0
$833$	−6.00000			−0.207888
$834$		0			0
$835$	−51.0000			−1.76493
$836$		0			0
$837$		−	7.00000i		−	0.241955i
$838$		0			0
$839$		−	8.00000i		−	0.276191i	−0.990419	−	0.138095i	$-0.955902\pi$
$839$		−	8.00000i		−	0.276191i	0.990419	−	0.138095i	$-0.0440980\pi$
$840$		0			0
$841$	−28.0000			−0.965517
$842$		0			0
$843$		−	22.0000i		−	0.757720i
$844$		0			0
$845$	36.0000	+	15.0000i	1.23844	+	0.516016i
$846$		0			0
$847$		−	5.00000i		−	0.171802i
$848$		0			0
$849$	−28.0000			−0.960958
$850$		0			0
$851$		−	12.0000i		−	0.411355i
$852$		0			0
$853$			25.0000i			0.855984i	0.903783	+	0.427992i	$0.140779\pi$
$853$			25.0000i			0.855984i	−0.903783	+	0.427992i	$0.859221\pi$
$854$		0			0
$855$	21.0000			0.718185
$856$		0			0
$857$	−40.0000			−1.36637			−0.683187	−	0.730243i	$-0.739407\pi$
$857$	−40.0000			−1.36637			−0.683187	+	0.730243i	$0.739407\pi$
$858$		0			0
$859$	2.00000			0.0682391			0.0341196	−	0.999418i	$-0.489137\pi$
$859$	2.00000			0.0682391			0.0341196	+	0.999418i	$0.489137\pi$
$860$		0			0
$861$	−6.00000			−0.204479
$862$		0			0
$863$			16.0000i			0.544646i	0.962206	+	0.272323i	$0.0877920\pi$
$863$			16.0000i			0.544646i	−0.962206	+	0.272323i	$0.912208\pi$
$864$		0			0
$865$		−	36.0000i		−	1.22404i
$866$		0			0
$867$	−19.0000			−0.645274
$868$		0			0
$869$		−	60.0000i		−	2.03536i
$870$		0			0
$871$	18.0000	−	12.0000i	0.609907	−	0.406604i
$872$		0			0
$873$		−	7.00000i		−	0.236914i
$874$		0			0
$875$	3.00000			0.101419
$876$		0			0
$877$		−	10.0000i		−	0.337676i	−0.985644	−	0.168838i	$-0.945999\pi$
$877$		−	10.0000i		−	0.337676i	0.985644	−	0.168838i	$-0.0540015\pi$
$878$		0			0
$879$			25.0000i			0.843229i
$880$		0			0
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$		0			0
$883$	−40.0000			−1.34611			−0.673054	−	0.739594i	$-0.735018\pi$
$883$	−40.0000			−1.34611			−0.673054	+	0.739594i	$0.735018\pi$
$884$		0			0
$885$	−36.0000			−1.21013
$886$		0			0
$887$	−10.0000			−0.335767			−0.167884	−	0.985807i	$-0.553693\pi$
$887$	−10.0000			−0.335767			−0.167884	+	0.985807i	$0.553693\pi$
$888$		0			0
$889$		−	8.00000i		−	0.268311i
$890$		0			0
$891$		−	4.00000i		−	0.134005i
$892$		0			0
$893$	21.0000			0.702738
$894$		0			0
$895$			51.0000i			1.70474i
$896$		0			0
$897$	2.00000	+	3.00000i	0.0667781	+	0.100167i
$898$		0			0
$899$			7.00000i			0.233463i
$900$		0			0
$901$	−18.0000			−0.599667
$902$		0			0
$903$			11.0000i			0.366057i
$904$		0			0
$905$		−	60.0000i		−	1.99447i
$906$		0			0
$907$	33.0000			1.09575			0.547874	−	0.836561i	$-0.315438\pi$
$907$	33.0000			1.09575			0.547874	+	0.836561i	$0.315438\pi$
$908$		0			0
$909$	−16.0000			−0.530687
$910$		0			0
$911$	39.0000			1.29213			0.646064	−	0.763283i	$-0.276414\pi$
$911$	39.0000			1.29213			0.646064	+	0.763283i	$0.276414\pi$
$912$		0			0
$913$	60.0000			1.98571
$914$		0			0
$915$		−	12.0000i		−	0.396708i
$916$		0			0
$917$		−	10.0000i		−	0.330229i
$918$		0			0
$919$	32.0000			1.05558			0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000			1.05558			0.527791	+	0.849374i	$0.323020\pi$
$920$		0			0
$921$			25.0000i			0.823778i
$922$		0			0
$923$	−36.0000	+	24.0000i	−1.18495	+	0.789970i
$924$		0			0
$925$		−	48.0000i		−	1.57823i
$926$		0			0
$927$	12.0000			0.394132
$928$		0			0
$929$			9.00000i			0.295280i	0.989041	+	0.147640i	$0.0471678\pi$
$929$			9.00000i			0.295280i	−0.989041	+	0.147640i	$0.952832\pi$
$930$		0			0
$931$		−	7.00000i		−	0.229416i
$932$		0			0
$933$	2.00000			0.0654771
$934$		0			0
$935$	−72.0000			−2.35465
$936$		0			0
$937$	−12.0000			−0.392023			−0.196011	−	0.980602i	$-0.562799\pi$
$937$	−12.0000			−0.392023			−0.196011	+	0.980602i	$0.562799\pi$
$938$		0			0
$939$	28.0000			0.913745
$940$		0			0
$941$		−	45.0000i		−	1.46696i	−0.679712	−	0.733479i	$-0.737895\pi$
$941$		−	45.0000i		−	1.46696i	0.679712	−	0.733479i	$-0.262105\pi$
$942$		0			0
$943$			6.00000i			0.195387i
$944$		0			0
$945$	−3.00000			−0.0975900
$946$		0			0
$947$		−	18.0000i		−	0.584921i	−0.956278	−	0.292461i	$-0.905526\pi$
$947$		−	18.0000i		−	0.584921i	0.956278	−	0.292461i	$-0.0944741\pi$
$948$		0			0
$949$	−33.0000	+	22.0000i	−1.07123	+	0.714150i
$950$		0			0
$951$		−	18.0000i		−	0.583690i
$952$		0			0
$953$	−11.0000			−0.356325			−0.178162	−	0.984001i	$-0.557015\pi$
$953$	−11.0000			−0.356325			−0.178162	+	0.984001i	$0.557015\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			4.00000i			0.129302i
$958$		0			0
$959$	−4.00000			−0.129167
$960$		0			0
$961$	−18.0000			−0.580645
$962$		0			0
$963$	−4.00000			−0.128898
$964$		0			0
$965$	42.0000			1.35203
$966$		0			0
$967$			12.0000i			0.385894i	0.981209	+	0.192947i	$0.0618045\pi$
$967$			12.0000i			0.385894i	−0.981209	+	0.192947i	$0.938195\pi$
$968$		0			0
$969$		−	42.0000i		−	1.34923i
$970$		0			0
$971$	24.0000			0.770197			0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000			0.770197			0.385098	+	0.922876i	$0.374168\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	8.00000	+	12.0000i	0.256205	+	0.384308i
$976$		0			0
$977$		−	12.0000i		−	0.383914i	−0.981403	−	0.191957i	$-0.938517\pi$
$977$		−	12.0000i		−	0.383914i	0.981403	−	0.191957i	$-0.0614834\pi$
$978$		0			0
$979$	−44.0000			−1.40625
$980$		0			0
$981$			10.0000i			0.319275i
$982$		0			0
$983$		−	21.0000i		−	0.669796i	−0.942254	−	0.334898i	$-0.891298\pi$
$983$		−	21.0000i		−	0.669796i	0.942254	−	0.334898i	$-0.108702\pi$
$984$		0			0
$985$	12.0000			0.382352
$986$		0			0
$987$	−3.00000			−0.0954911
$988$		0			0
$989$	11.0000			0.349780
$990$		0			0
$991$	24.0000			0.762385			0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000			0.762385			0.381193	+	0.924496i	$0.375513\pi$
$992$		0			0
$993$		−	16.0000i		−	0.507745i
$994$		0			0
$995$		−	30.0000i		−	0.951064i
$996$		0			0
$997$	−52.0000			−1.64686			−0.823428	−	0.567420i	$-0.807941\pi$
$997$	−52.0000			−1.64686			−0.823428	+	0.567420i	$0.807941\pi$
$998$		0			0
$999$		−	12.0000i		−	0.379663i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.g.337.1		2
4.3	odd	2		2184.2.h.b.337.1	✓	2
13.12	even	2	inner	4368.2.h.g.337.2		2
52.51	odd	2		2184.2.h.b.337.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2184.2.h.b.337.1	✓	2	4.3	odd	2
2184.2.h.b.337.2	yes	2	52.51	odd	2
4368.2.h.g.337.1		2	1.1	even	1	trivial
4368.2.h.g.337.2		2	13.12	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} -4.00000i q^{11} +(2.00000 + 3.00000i) q^{13} +3.00000i q^{15} +6.00000 q^{17} +7.00000i q^{19} -1.00000i q^{21} -1.00000 q^{23} -4.00000 q^{25} -1.00000 q^{27} +1.00000 q^{29} +7.00000i q^{31} +4.00000i q^{33} +3.00000 q^{35} +12.0000i q^{37} +(-2.00000 - 3.00000i) q^{39} -6.00000i q^{41} -11.0000 q^{43} -3.00000i q^{45} -3.00000i q^{47} -1.00000 q^{49} -6.00000 q^{51} -3.00000 q^{53} -12.0000 q^{55} -7.00000i q^{57} +12.0000i q^{59} -4.00000 q^{61} +1.00000i q^{63} +(9.00000 - 6.00000i) q^{65} -6.00000i q^{67} +1.00000 q^{69} +12.0000i q^{71} +11.0000i q^{73} +4.00000 q^{75} +4.00000 q^{77} +15.0000 q^{79} +1.00000 q^{81} +15.0000i q^{83} -18.0000i q^{85} -1.00000 q^{87} -11.0000i q^{89} +(-3.00000 + 2.00000i) q^{91} -7.00000i q^{93} +21.0000 q^{95} -7.00000i q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{9} + 4 q^{13} + 12 q^{17} - 2 q^{23} - 8 q^{25} - 2 q^{27} + 2 q^{29} + 6 q^{35} - 4 q^{39} - 22 q^{43} - 2 q^{49} - 12 q^{51} - 6 q^{53} - 24 q^{55} - 8 q^{61} + 18 q^{65} + 2 q^{69} + 8 q^{75} + 8 q^{77} + 30 q^{79} + 2 q^{81} - 2 q^{87} - 6 q^{91} + 42 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 + 4 * q^13 + 12 * q^17 - 2 * q^23 - 8 * q^25 - 2 * q^27 + 2 * q^29 + 6 * q^35 - 4 * q^39 - 22 * q^43 - 2 * q^49 - 12 * q^51 - 6 * q^53 - 24 * q^55 - 8 * q^61 + 18 * q^65 + 2 * q^69 + 8 * q^75 + 8 * q^77 + 30 * q^79 + 2 * q^81 - 2 * q^87 - 6 * q^91 + 42 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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