LMFDB - Embedded newform 4368.2.h.k.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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000i q^{5} -1.00000i q^{7} +1.00000 q^{9} +1.00000i q^{11} +(3.00000 - 2.00000i) q^{13} -1.00000i q^{15} +1.00000 q^{17} -1.00000i q^{19} -1.00000i q^{21} -3.00000 q^{23} +4.00000 q^{25} +1.00000 q^{27} +9.00000 q^{29} -4.00000i q^{31} +1.00000i q^{33} -1.00000 q^{35} +9.00000i q^{37} +(3.00000 - 2.00000i) q^{39} -8.00000i q^{41} -7.00000 q^{43} -1.00000i q^{45} +8.00000i q^{47} -1.00000 q^{49} +1.00000 q^{51} -10.0000 q^{53} +1.00000 q^{55} -1.00000i q^{57} -6.00000i q^{59} +11.0000 q^{61} -1.00000i q^{63} +(-2.00000 - 3.00000i) q^{65} -12.0000i q^{67} -3.00000 q^{69} +6.00000i q^{71} -11.0000i q^{73} +4.00000 q^{75} +1.00000 q^{77} +12.0000 q^{79} +1.00000 q^{81} +6.00000i q^{83} -1.00000i q^{85} +9.00000 q^{87} -12.0000i q^{89} +(-2.00000 - 3.00000i) q^{91} -4.00000i q^{93} -1.00000 q^{95} +2.00000i q^{97} +1.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{17} - 6 q^{23} + 8 q^{25} + 2 q^{27} + 18 q^{29} - 2 q^{35} + 6 q^{39} - 14 q^{43} - 2 q^{49} + 2 q^{51} - 20 q^{53} + 2 q^{55} + 22 q^{61} - 4 q^{65} - 6 q^{69} + 8 q^{75} + 2 q^{77} + 24 q^{79} + 2 q^{81} + 18 q^{87} - 4 q^{91} - 2 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^9 + 6 * q^13 + 2 * q^17 - 6 * q^23 + 8 * q^25 + 2 * q^27 + 18 * q^29 - 2 * q^35 + 6 * q^39 - 14 * q^43 - 2 * q^49 + 2 * q^51 - 20 * q^53 + 2 * q^55 + 22 * q^61 - 4 * q^65 - 6 * q^69 + 8 * q^75 + 2 * q^77 + 24 * q^79 + 2 * q^81 + 18 * q^87 - 4 * q^91 - 2 * 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	$-0.928217\pi$
$5$		−	1.00000i		−	0.447214i	0.974679	−	0.223607i	$-0.0717831\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			1.00000i			0.301511i	0.988571	+	0.150756i	$0.0481707\pi$
$11$			1.00000i			0.301511i	−0.988571	+	0.150756i	$0.951829\pi$
$12$		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
$17$	1.00000			0.242536			0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000			0.242536			0.121268	+	0.992620i	$0.461304\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i	−0.993399	−	0.114708i	$-0.963407\pi$
$19$		−	1.00000i		−	0.229416i	0.993399	−	0.114708i	$-0.0365932\pi$
$20$		0			0
$21$		−	1.00000i		−	0.218218i
$22$		0			0
$23$	−3.00000			−0.625543			−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000			−0.625543			−0.312772	+	0.949828i	$0.601257\pi$
$24$		0			0
$25$	4.00000			0.800000
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	9.00000			1.67126			0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000			1.67126			0.835629	+	0.549294i	$0.185103\pi$
$30$		0			0
$31$		−	4.00000i		−	0.718421i	−0.933257	−	0.359211i	$-0.883046\pi$
$31$		−	4.00000i		−	0.718421i	0.933257	−	0.359211i	$-0.116954\pi$
$32$		0			0
$33$			1.00000i			0.174078i
$34$		0			0
$35$	−1.00000			−0.169031
$36$		0			0
$37$			9.00000i			1.47959i	0.672832	+	0.739795i	$0.265078\pi$
$37$			9.00000i			1.47959i	−0.672832	+	0.739795i	$0.734922\pi$
$38$		0			0
$39$	3.00000	−	2.00000i	0.480384	−	0.320256i
$40$		0			0
$41$		−	8.00000i		−	1.24939i	−0.780869	−	0.624695i	$-0.785223\pi$
$41$		−	8.00000i		−	1.24939i	0.780869	−	0.624695i	$-0.214777\pi$
$42$		0			0
$43$	−7.00000			−1.06749			−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000			−1.06749			−0.533745	+	0.845645i	$0.679216\pi$
$44$		0			0
$45$		−	1.00000i		−	0.149071i
$46$		0			0
$47$			8.00000i			1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$			8.00000i			1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	1.00000			0.140028
$52$		0			0
$53$	−10.0000			−1.37361			−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000			−1.37361			−0.686803	+	0.726844i	$0.740986\pi$
$54$		0			0
$55$	1.00000			0.134840
$56$		0			0
$57$		−	1.00000i		−	0.132453i
$58$		0			0
$59$		−	6.00000i		−	0.781133i	−0.920575	−	0.390567i	$-0.872279\pi$
$59$		−	6.00000i		−	0.781133i	0.920575	−	0.390567i	$-0.127721\pi$
$60$		0			0
$61$	11.0000			1.40841			0.704203	−	0.709999i	$-0.251305\pi$
$61$	11.0000			1.40841			0.704203	+	0.709999i	$0.251305\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	−2.00000	−	3.00000i	−0.248069	−	0.372104i
$66$		0			0
$67$		−	12.0000i		−	1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$		−	12.0000i		−	1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$		0			0
$69$	−3.00000			−0.361158
$70$		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
$73$		−	11.0000i		−	1.28745i	−0.765256	−	0.643726i	$-0.777388\pi$
$73$		−	11.0000i		−	1.28745i	0.765256	−	0.643726i	$-0.222612\pi$
$74$		0			0
$75$	4.00000			0.461880
$76$		0			0
$77$	1.00000			0.113961
$78$		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
$83$			6.00000i			0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$			6.00000i			0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$		0			0
$85$		−	1.00000i		−	0.108465i
$86$		0			0
$87$	9.00000			0.964901
$88$		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
$93$		−	4.00000i		−	0.414781i
$94$		0			0
$95$	−1.00000			−0.102598
$96$		0			0
$97$			2.00000i			0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$			2.00000i			0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$		0			0
$99$			1.00000i			0.100504i
$100$		0			0
$101$	−2.00000			−0.199007			−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000			−0.199007			−0.0995037	+	0.995037i	$0.531726\pi$
$102$		0			0
$103$	5.00000			0.492665			0.246332	−	0.969185i	$-0.420775\pi$
$103$	5.00000			0.492665			0.246332	+	0.969185i	$0.420775\pi$
$104$		0			0
$105$	−1.00000			−0.0975900
$106$		0			0
$107$	2.00000			0.193347			0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000			0.193347			0.0966736	+	0.995316i	$0.469180\pi$
$108$		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
$111$			9.00000i			0.854242i
$112$		0			0
$113$	−2.00000			−0.188144			−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000			−0.188144			−0.0940721	+	0.995565i	$0.529988\pi$
$114$		0			0
$115$			3.00000i			0.279751i
$116$		0			0
$117$	3.00000	−	2.00000i	0.277350	−	0.184900i
$118$		0			0
$119$		−	1.00000i		−	0.0916698i
$120$		0			0
$121$	10.0000			0.909091
$122$		0			0
$123$		−	8.00000i		−	0.721336i
$124$		0			0
$125$		−	9.00000i		−	0.804984i
$126$		0			0
$127$	−16.0000			−1.41977			−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000			−1.41977			−0.709885	+	0.704317i	$0.751253\pi$
$128$		0			0
$129$	−7.00000			−0.616316
$130$		0			0
$131$	3.00000			0.262111			0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000			0.262111			0.131056	+	0.991375i	$0.458163\pi$
$132$		0			0
$133$	−1.00000			−0.0867110
$134$		0			0
$135$		−	1.00000i		−	0.0860663i
$136$		0			0
$137$			15.0000i			1.28154i	0.767734	+	0.640768i	$0.221384\pi$
$137$			15.0000i			1.28154i	−0.767734	+	0.640768i	$0.778616\pi$
$138$		0			0
$139$	12.0000			1.01783			0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000			1.01783			0.508913	+	0.860818i	$0.330047\pi$
$140$		0			0
$141$			8.00000i			0.673722i
$142$		0			0
$143$	2.00000	+	3.00000i	0.167248	+	0.250873i
$144$		0			0
$145$		−	9.00000i		−	0.747409i
$146$		0			0
$147$	−1.00000			−0.0824786
$148$		0			0
$149$		−	6.00000i		−	0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$		−	6.00000i		−	0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$		0			0
$151$			1.00000i			0.0813788i	0.999172	+	0.0406894i	$0.0129554\pi$
$151$			1.00000i			0.0813788i	−0.999172	+	0.0406894i	$0.987045\pi$
$152$		0			0
$153$	1.00000			0.0808452
$154$		0			0
$155$	−4.00000			−0.321288
$156$		0			0
$157$	−21.0000			−1.67598			−0.837991	−	0.545684i	$-0.816270\pi$
$157$	−21.0000			−1.67598			−0.837991	+	0.545684i	$0.816270\pi$
$158$		0			0
$159$	−10.0000			−0.793052
$160$		0			0
$161$			3.00000i			0.236433i
$162$		0			0
$163$		−	20.0000i		−	1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$		−	20.0000i		−	1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$		0			0
$165$	1.00000			0.0778499
$166$		0			0
$167$		−	11.0000i		−	0.851206i	−0.904910	−	0.425603i	$-0.860062\pi$
$167$		−	11.0000i		−	0.851206i	0.904910	−	0.425603i	$-0.139938\pi$
$168$		0			0
$169$	5.00000	−	12.0000i	0.384615	−	0.923077i
$170$		0			0
$171$		−	1.00000i		−	0.0764719i
$172$		0			0
$173$	14.0000			1.06440			0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000			1.06440			0.532200	+	0.846619i	$0.321365\pi$
$174$		0			0
$175$		−	4.00000i		−	0.302372i
$176$		0			0
$177$		−	6.00000i		−	0.450988i
$178$		0			0
$179$	18.0000			1.34538			0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000			1.34538			0.672692	+	0.739923i	$0.265138\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$	11.0000			0.813143
$184$		0			0
$185$	9.00000			0.661693
$186$		0			0
$187$			1.00000i			0.0731272i
$188$		0			0
$189$		−	1.00000i		−	0.0727393i
$190$		0			0
$191$	11.0000			0.795932			0.397966	−	0.917400i	$-0.369716\pi$
$191$	11.0000			0.795932			0.397966	+	0.917400i	$0.369716\pi$
$192$		0			0
$193$			12.0000i			0.863779i	0.901927	+	0.431889i	$0.142153\pi$
$193$			12.0000i			0.863779i	−0.901927	+	0.431889i	$0.857847\pi$
$194$		0			0
$195$	−2.00000	−	3.00000i	−0.143223	−	0.214834i
$196$		0			0
$197$			2.00000i			0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$			2.00000i			0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$		0			0
$199$	−5.00000			−0.354441			−0.177220	−	0.984171i	$-0.556711\pi$
$199$	−5.00000			−0.354441			−0.177220	+	0.984171i	$0.556711\pi$
$200$		0			0
$201$		−	12.0000i		−	0.846415i
$202$		0			0
$203$		−	9.00000i		−	0.631676i
$204$		0			0
$205$	−8.00000			−0.558744
$206$		0			0
$207$	−3.00000			−0.208514
$208$		0			0
$209$	1.00000			0.0691714
$210$		0			0
$211$	5.00000			0.344214			0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000			0.344214			0.172107	+	0.985078i	$0.444942\pi$
$212$		0			0
$213$			6.00000i			0.411113i
$214$		0			0
$215$			7.00000i			0.477396i
$216$		0			0
$217$	−4.00000			−0.271538
$218$		0			0
$219$		−	11.0000i		−	0.743311i
$220$		0			0
$221$	3.00000	−	2.00000i	0.201802	−	0.134535i
$222$		0			0
$223$		−	8.00000i		−	0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$		−	8.00000i		−	0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$		0			0
$225$	4.00000			0.266667
$226$		0			0
$227$		−	12.0000i		−	0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$		−	12.0000i		−	0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$		0			0
$229$		−	2.00000i		−	0.132164i	−0.997814	−	0.0660819i	$-0.978950\pi$
$229$		−	2.00000i		−	0.132164i	0.997814	−	0.0660819i	$-0.0210498\pi$
$230$		0			0
$231$	1.00000			0.0657952
$232$		0			0
$233$	−18.0000			−1.17922			−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000			−1.17922			−0.589610	+	0.807688i	$0.700718\pi$
$234$		0			0
$235$	8.00000			0.521862
$236$		0			0
$237$	12.0000			0.779484
$238$		0			0
$239$		−	8.00000i		−	0.517477i	−0.965947	−	0.258738i	$-0.916693\pi$
$239$		−	8.00000i		−	0.517477i	0.965947	−	0.258738i	$-0.0833068\pi$
$240$		0			0
$241$		−	10.0000i		−	0.644157i	−0.946713	−	0.322078i	$-0.895619\pi$
$241$		−	10.0000i		−	0.644157i	0.946713	−	0.322078i	$-0.104381\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$			1.00000i			0.0638877i
$246$		0			0
$247$	−2.00000	−	3.00000i	−0.127257	−	0.190885i
$248$		0			0
$249$			6.00000i			0.380235i
$250$		0			0
$251$	−11.0000			−0.694314			−0.347157	−	0.937807i	$-0.612853\pi$
$251$	−11.0000			−0.694314			−0.347157	+	0.937807i	$0.612853\pi$
$252$		0			0
$253$		−	3.00000i		−	0.188608i
$254$		0			0
$255$		−	1.00000i		−	0.0626224i
$256$		0			0
$257$	14.0000			0.873296			0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000			0.873296			0.436648	+	0.899632i	$0.356166\pi$
$258$		0			0
$259$	9.00000			0.559233
$260$		0			0
$261$	9.00000			0.557086
$262$		0			0
$263$	−16.0000			−0.986602			−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000			−0.986602			−0.493301	+	0.869859i	$0.664210\pi$
$264$		0			0
$265$			10.0000i			0.614295i
$266$		0			0
$267$		−	12.0000i		−	0.734388i
$268$		0			0
$269$	−20.0000			−1.21942			−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000			−1.21942			−0.609711	+	0.792624i	$0.708714\pi$
$270$		0			0
$271$		−	4.00000i		−	0.242983i	−0.992592	−	0.121491i	$-0.961232\pi$
$271$		−	4.00000i		−	0.242983i	0.992592	−	0.121491i	$-0.0387677\pi$
$272$		0			0
$273$	−2.00000	−	3.00000i	−0.121046	−	0.181568i
$274$		0			0
$275$			4.00000i			0.241209i
$276$		0			0
$277$	16.0000			0.961347			0.480673	−	0.876900i	$-0.340392\pi$
$277$	16.0000			0.961347			0.480673	+	0.876900i	$0.340392\pi$
$278$		0			0
$279$		−	4.00000i		−	0.239474i
$280$		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
$283$	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
$285$	−1.00000			−0.0592349
$286$		0			0
$287$	−8.00000			−0.472225
$288$		0			0
$289$	−16.0000			−0.941176
$290$		0			0
$291$			2.00000i			0.117242i
$292$		0			0
$293$			26.0000i			1.51894i	0.650545	+	0.759468i	$0.274541\pi$
$293$			26.0000i			1.51894i	−0.650545	+	0.759468i	$0.725459\pi$
$294$		0			0
$295$	−6.00000			−0.349334
$296$		0			0
$297$			1.00000i			0.0580259i
$298$		0			0
$299$	−9.00000	+	6.00000i	−0.520483	+	0.346989i
$300$		0			0
$301$			7.00000i			0.403473i
$302$		0			0
$303$	−2.00000			−0.114897
$304$		0			0
$305$		−	11.0000i		−	0.629858i
$306$		0			0
$307$		−	8.00000i		−	0.456584i	−0.973593	−	0.228292i	$-0.926686\pi$
$307$		−	8.00000i		−	0.456584i	0.973593	−	0.228292i	$-0.0733141\pi$
$308$		0			0
$309$	5.00000			0.284440
$310$		0			0
$311$	−8.00000			−0.453638			−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000			−0.453638			−0.226819	+	0.973937i	$0.572833\pi$
$312$		0			0
$313$	−14.0000			−0.791327			−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000			−0.791327			−0.395663	+	0.918396i	$0.629485\pi$
$314$		0			0
$315$	−1.00000			−0.0563436
$316$		0			0
$317$			12.0000i			0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$			12.0000i			0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$		0			0
$319$			9.00000i			0.503903i
$320$		0			0
$321$	2.00000			0.111629
$322$		0			0
$323$		−	1.00000i		−	0.0556415i
$324$		0			0
$325$	12.0000	−	8.00000i	0.665640	−	0.443760i
$326$		0			0
$327$		−	7.00000i		−	0.387101i
$328$		0			0
$329$	8.00000			0.441054
$330$		0			0
$331$		−	8.00000i		−	0.439720i	−0.975531	−	0.219860i	$-0.929440\pi$
$331$		−	8.00000i		−	0.439720i	0.975531	−	0.219860i	$-0.0705600\pi$
$332$		0			0
$333$			9.00000i			0.493197i
$334$		0			0
$335$	−12.0000			−0.655630
$336$		0			0
$337$	17.0000			0.926049			0.463025	−	0.886345i	$-0.346764\pi$
$337$	17.0000			0.926049			0.463025	+	0.886345i	$0.346764\pi$
$338$		0			0
$339$	−2.00000			−0.108625
$340$		0			0
$341$	4.00000			0.216612
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$			3.00000i			0.161515i
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$			4.00000i			0.214115i	0.994253	+	0.107058i	$0.0341429\pi$
$349$			4.00000i			0.214115i	−0.994253	+	0.107058i	$0.965857\pi$
$350$		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
$357$		−	1.00000i		−	0.0529256i
$358$		0			0
$359$			30.0000i			1.58334i	0.610949	+	0.791670i	$0.290788\pi$
$359$			30.0000i			1.58334i	−0.610949	+	0.791670i	$0.709212\pi$
$360$		0			0
$361$	18.0000			0.947368
$362$		0			0
$363$	10.0000			0.524864
$364$		0			0
$365$	−11.0000			−0.575766
$366$		0			0
$367$	8.00000			0.417597			0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000			0.417597			0.208798	+	0.977959i	$0.433045\pi$
$368$		0			0
$369$		−	8.00000i		−	0.416463i
$370$		0			0
$371$			10.0000i			0.519174i
$372$		0			0
$373$	−14.0000			−0.724893			−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000			−0.724893			−0.362446	+	0.932005i	$0.618058\pi$
$374$		0			0
$375$		−	9.00000i		−	0.464758i
$376$		0			0
$377$	27.0000	−	18.0000i	1.39057	−	0.927047i
$378$		0			0
$379$			16.0000i			0.821865i	0.911666	+	0.410932i	$0.134797\pi$
$379$			16.0000i			0.821865i	−0.911666	+	0.410932i	$0.865203\pi$
$380$		0			0
$381$	−16.0000			−0.819705
$382$		0			0
$383$			35.0000i			1.78842i	0.447651	+	0.894208i	$0.352261\pi$
$383$			35.0000i			1.78842i	−0.447651	+	0.894208i	$0.647739\pi$
$384$		0			0
$385$		−	1.00000i		−	0.0509647i
$386$		0			0
$387$	−7.00000			−0.355830
$388$		0			0
$389$	2.00000			0.101404			0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000			0.101404			0.0507020	+	0.998714i	$0.483854\pi$
$390$		0			0
$391$	−3.00000			−0.151717
$392$		0			0
$393$	3.00000			0.151330
$394$		0			0
$395$		−	12.0000i		−	0.603786i
$396$		0			0
$397$		−	4.00000i		−	0.200754i	−0.994949	−	0.100377i	$-0.967995\pi$
$397$		−	4.00000i		−	0.200754i	0.994949	−	0.100377i	$-0.0320049\pi$
$398$		0			0
$399$	−1.00000			−0.0500626
$400$		0			0
$401$		−	14.0000i		−	0.699127i	−0.936913	−	0.349563i	$-0.886330\pi$
$401$		−	14.0000i		−	0.699127i	0.936913	−	0.349563i	$-0.113670\pi$
$402$		0			0
$403$	−8.00000	−	12.0000i	−0.398508	−	0.597763i
$404$		0			0
$405$		−	1.00000i		−	0.0496904i
$406$		0			0
$407$	−9.00000			−0.446113
$408$		0			0
$409$		−	27.0000i		−	1.33506i	−0.744581	−	0.667532i	$-0.767351\pi$
$409$		−	27.0000i		−	1.33506i	0.744581	−	0.667532i	$-0.232649\pi$
$410$		0			0
$411$			15.0000i			0.739895i
$412$		0			0
$413$	−6.00000			−0.295241
$414$		0			0
$415$	6.00000			0.294528
$416$		0			0
$417$	12.0000			0.587643
$418$		0			0
$419$	−23.0000			−1.12362			−0.561812	−	0.827265i	$-0.689895\pi$
$419$	−23.0000			−1.12362			−0.561812	+	0.827265i	$0.689895\pi$
$420$		0			0
$421$		−	22.0000i		−	1.07221i	−0.844150	−	0.536107i	$-0.819894\pi$
$421$		−	22.0000i		−	1.07221i	0.844150	−	0.536107i	$-0.180106\pi$
$422$		0			0
$423$			8.00000i			0.388973i
$424$		0			0
$425$	4.00000			0.194029
$426$		0			0
$427$		−	11.0000i		−	0.532327i
$428$		0			0
$429$	2.00000	+	3.00000i	0.0965609	+	0.144841i
$430$		0			0
$431$		−	18.0000i		−	0.867029i	−0.901146	−	0.433515i	$-0.857273\pi$
$431$		−	18.0000i		−	0.867029i	0.901146	−	0.433515i	$-0.142727\pi$
$432$		0			0
$433$	4.00000			0.192228			0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000			0.192228			0.0961139	+	0.995370i	$0.469359\pi$
$434$		0			0
$435$		−	9.00000i		−	0.431517i
$436$		0			0
$437$			3.00000i			0.143509i
$438$		0			0
$439$	33.0000			1.57500			0.787502	−	0.616312i	$-0.211374\pi$
$439$	33.0000			1.57500			0.787502	+	0.616312i	$0.211374\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	6.00000			0.285069			0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000			0.285069			0.142534	+	0.989790i	$0.454475\pi$
$444$		0			0
$445$	−12.0000			−0.568855
$446$		0			0
$447$		−	6.00000i		−	0.283790i
$448$		0			0
$449$			15.0000i			0.707894i	0.935266	+	0.353947i	$0.115161\pi$
$449$			15.0000i			0.707894i	−0.935266	+	0.353947i	$0.884839\pi$
$450$		0			0
$451$	8.00000			0.376705
$452$		0			0
$453$			1.00000i			0.0469841i
$454$		0			0
$455$	−3.00000	+	2.00000i	−0.140642	+	0.0937614i
$456$		0			0
$457$		−	4.00000i		−	0.187112i	−0.995614	−	0.0935561i	$-0.970177\pi$
$457$		−	4.00000i		−	0.187112i	0.995614	−	0.0935561i	$-0.0298234\pi$
$458$		0			0
$459$	1.00000			0.0466760
$460$		0			0
$461$		−	3.00000i		−	0.139724i	−0.997557	−	0.0698620i	$-0.977744\pi$
$461$		−	3.00000i		−	0.139724i	0.997557	−	0.0698620i	$-0.0222559\pi$
$462$		0			0
$463$			29.0000i			1.34774i	0.738848	+	0.673872i	$0.235370\pi$
$463$			29.0000i			1.34774i	−0.738848	+	0.673872i	$0.764630\pi$
$464$		0			0
$465$	−4.00000			−0.185496
$466$		0			0
$467$	37.0000			1.71216			0.856078	−	0.516847i	$-0.172894\pi$
$467$	37.0000			1.71216			0.856078	+	0.516847i	$0.172894\pi$
$468$		0			0
$469$	−12.0000			−0.554109
$470$		0			0
$471$	−21.0000			−0.967629
$472$		0			0
$473$		−	7.00000i		−	0.321860i
$474$		0			0
$475$		−	4.00000i		−	0.183533i
$476$		0			0
$477$	−10.0000			−0.457869
$478$		0			0
$479$			39.0000i			1.78196i	0.454047	+	0.890978i	$0.349980\pi$
$479$			39.0000i			1.78196i	−0.454047	+	0.890978i	$0.650020\pi$
$480$		0			0
$481$	18.0000	+	27.0000i	0.820729	+	1.23109i
$482$		0			0
$483$			3.00000i			0.136505i
$484$		0			0
$485$	2.00000			0.0908153
$486$		0			0
$487$			32.0000i			1.45006i	0.688718	+	0.725029i	$0.258174\pi$
$487$			32.0000i			1.45006i	−0.688718	+	0.725029i	$0.741826\pi$
$488$		0			0
$489$		−	20.0000i		−	0.904431i
$490$		0			0
$491$	−12.0000			−0.541552			−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000			−0.541552			−0.270776	+	0.962642i	$0.587280\pi$
$492$		0			0
$493$	9.00000			0.405340
$494$		0			0
$495$	1.00000			0.0449467
$496$		0			0
$497$	6.00000			0.269137
$498$		0			0
$499$		−	10.0000i		−	0.447661i	−0.974628	−	0.223831i	$-0.928144\pi$
$499$		−	10.0000i		−	0.447661i	0.974628	−	0.223831i	$-0.0718563\pi$
$500$		0			0
$501$		−	11.0000i		−	0.491444i
$502$		0			0
$503$	38.0000			1.69434			0.847168	−	0.531325i	$-0.178306\pi$
$503$	38.0000			1.69434			0.847168	+	0.531325i	$0.178306\pi$
$504$		0			0
$505$			2.00000i			0.0889988i
$506$		0			0
$507$	5.00000	−	12.0000i	0.222058	−	0.532939i
$508$		0			0
$509$			37.0000i			1.64000i	0.572366	+	0.819998i	$0.306026\pi$
$509$			37.0000i			1.64000i	−0.572366	+	0.819998i	$0.693974\pi$
$510$		0			0
$511$	−11.0000			−0.486611
$512$		0			0
$513$		−	1.00000i		−	0.0441511i
$514$		0			0
$515$		−	5.00000i		−	0.220326i
$516$		0			0
$517$	−8.00000			−0.351840
$518$		0			0
$519$	14.0000			0.614532
$520$		0			0
$521$	−31.0000			−1.35813			−0.679067	−	0.734076i	$-0.737616\pi$
$521$	−31.0000			−1.35813			−0.679067	+	0.734076i	$0.737616\pi$
$522$		0			0
$523$	36.0000			1.57417			0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000			1.57417			0.787085	+	0.616844i	$0.211589\pi$
$524$		0			0
$525$		−	4.00000i		−	0.174574i
$526$		0			0
$527$		−	4.00000i		−	0.174243i
$528$		0			0
$529$	−14.0000			−0.608696
$530$		0			0
$531$		−	6.00000i		−	0.260378i
$532$		0			0
$533$	−16.0000	−	24.0000i	−0.693037	−	1.03956i
$534$		0			0
$535$		−	2.00000i		−	0.0864675i
$536$		0			0
$537$	18.0000			0.776757
$538$		0			0
$539$		−	1.00000i		−	0.0430730i
$540$		0			0
$541$		−	41.0000i		−	1.76273i	−0.472438	−	0.881364i	$-0.656626\pi$
$541$		−	41.0000i		−	1.76273i	0.472438	−	0.881364i	$-0.343374\pi$
$542$		0			0
$543$	−2.00000			−0.0858282
$544$		0			0
$545$	−7.00000			−0.299847
$546$		0			0
$547$	28.0000			1.19719			0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000			1.19719			0.598597	+	0.801050i	$0.295725\pi$
$548$		0			0
$549$	11.0000			0.469469
$550$		0			0
$551$		−	9.00000i		−	0.383413i
$552$		0			0
$553$		−	12.0000i		−	0.510292i
$554$		0			0
$555$	9.00000			0.382029
$556$		0			0
$557$			2.00000i			0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$			2.00000i			0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$		0			0
$559$	−21.0000	+	14.0000i	−0.888205	+	0.592137i
$560$		0			0
$561$			1.00000i			0.0422200i
$562$		0			0
$563$	−11.0000			−0.463595			−0.231797	−	0.972764i	$-0.574461\pi$
$563$	−11.0000			−0.463595			−0.231797	+	0.972764i	$0.574461\pi$
$564$		0			0
$565$			2.00000i			0.0841406i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	−30.0000			−1.25767			−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000			−1.25767			−0.628833	+	0.777541i	$0.716467\pi$
$570$		0			0
$571$	−32.0000			−1.33916			−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000			−1.33916			−0.669579	+	0.742741i	$0.733526\pi$
$572$		0			0
$573$	11.0000			0.459532
$574$		0			0
$575$	−12.0000			−0.500435
$576$		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
$579$			12.0000i			0.498703i
$580$		0			0
$581$	6.00000			0.248922
$582$		0			0
$583$		−	10.0000i		−	0.414158i
$584$		0			0
$585$	−2.00000	−	3.00000i	−0.0826898	−	0.124035i
$586$		0			0
$587$		−	16.0000i		−	0.660391i	−0.943913	−	0.330195i	$-0.892885\pi$
$587$		−	16.0000i		−	0.660391i	0.943913	−	0.330195i	$-0.107115\pi$
$588$		0			0
$589$	−4.00000			−0.164817
$590$		0			0
$591$			2.00000i			0.0822690i
$592$		0			0
$593$			28.0000i			1.14982i	0.818216	+	0.574911i	$0.194963\pi$
$593$			28.0000i			1.14982i	−0.818216	+	0.574911i	$0.805037\pi$
$594$		0			0
$595$	−1.00000			−0.0409960
$596$		0			0
$597$	−5.00000			−0.204636
$598$		0			0
$599$	−39.0000			−1.59350			−0.796748	−	0.604311i	$-0.793448\pi$
$599$	−39.0000			−1.59350			−0.796748	+	0.604311i	$0.793448\pi$
$600$		0			0
$601$	−16.0000			−0.652654			−0.326327	−	0.945257i	$-0.605811\pi$
$601$	−16.0000			−0.652654			−0.326327	+	0.945257i	$0.605811\pi$
$602$		0			0
$603$		−	12.0000i		−	0.488678i
$604$		0			0
$605$		−	10.0000i		−	0.406558i
$606$		0			0
$607$	−1.00000			−0.0405887			−0.0202944	−	0.999794i	$-0.506460\pi$
$607$	−1.00000			−0.0405887			−0.0202944	+	0.999794i	$0.506460\pi$
$608$		0			0
$609$		−	9.00000i		−	0.364698i
$610$		0			0
$611$	16.0000	+	24.0000i	0.647291	+	0.970936i
$612$		0			0
$613$			17.0000i			0.686624i	0.939222	+	0.343312i	$0.111549\pi$
$613$			17.0000i			0.686624i	−0.939222	+	0.343312i	$0.888451\pi$
$614$		0			0
$615$	−8.00000			−0.322591
$616$		0			0
$617$			19.0000i			0.764911i	0.923974	+	0.382456i	$0.124922\pi$
$617$			19.0000i			0.764911i	−0.923974	+	0.382456i	$0.875078\pi$
$618$		0			0
$619$			21.0000i			0.844061i	0.906582	+	0.422031i	$0.138683\pi$
$619$			21.0000i			0.844061i	−0.906582	+	0.422031i	$0.861317\pi$
$620$		0			0
$621$	−3.00000			−0.120386
$622$		0			0
$623$	−12.0000			−0.480770
$624$		0			0
$625$	11.0000			0.440000
$626$		0			0
$627$	1.00000			0.0399362
$628$		0			0
$629$			9.00000i			0.358854i
$630$		0			0
$631$			29.0000i			1.15447i	0.816577	+	0.577236i	$0.195869\pi$
$631$			29.0000i			1.15447i	−0.816577	+	0.577236i	$0.804131\pi$
$632$		0			0
$633$	5.00000			0.198732
$634$		0			0
$635$			16.0000i			0.634941i
$636$		0			0
$637$	−3.00000	+	2.00000i	−0.118864	+	0.0792429i
$638$		0			0
$639$			6.00000i			0.237356i
$640$		0			0
$641$	−44.0000			−1.73790			−0.868948	−	0.494904i	$-0.835203\pi$
$641$	−44.0000			−1.73790			−0.868948	+	0.494904i	$0.835203\pi$
$642$		0			0
$643$		−	19.0000i		−	0.749287i	−0.927169	−	0.374643i	$-0.877765\pi$
$643$		−	19.0000i		−	0.749287i	0.927169	−	0.374643i	$-0.122235\pi$
$644$		0			0
$645$			7.00000i			0.275625i
$646$		0			0
$647$	−42.0000			−1.65119			−0.825595	−	0.564263i	$-0.809160\pi$
$647$	−42.0000			−1.65119			−0.825595	+	0.564263i	$0.809160\pi$
$648$		0			0
$649$	6.00000			0.235521
$650$		0			0
$651$	−4.00000			−0.156772
$652$		0			0
$653$	−13.0000			−0.508729			−0.254365	−	0.967108i	$-0.581866\pi$
$653$	−13.0000			−0.508729			−0.254365	+	0.967108i	$0.581866\pi$
$654$		0			0
$655$		−	3.00000i		−	0.117220i
$656$		0			0
$657$		−	11.0000i		−	0.429151i
$658$		0			0
$659$	30.0000			1.16863			0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000			1.16863			0.584317	+	0.811525i	$0.301362\pi$
$660$		0			0
$661$		−	26.0000i		−	1.01128i	−0.862744	−	0.505641i	$-0.831256\pi$
$661$		−	26.0000i		−	1.01128i	0.862744	−	0.505641i	$-0.168744\pi$
$662$		0			0
$663$	3.00000	−	2.00000i	0.116510	−	0.0776736i
$664$		0			0
$665$			1.00000i			0.0387783i
$666$		0			0
$667$	−27.0000			−1.04544
$668$		0			0
$669$		−	8.00000i		−	0.309298i
$670$		0			0
$671$			11.0000i			0.424650i
$672$		0			0
$673$	29.0000			1.11787			0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000			1.11787			0.558934	+	0.829212i	$0.311211\pi$
$674$		0			0
$675$	4.00000			0.153960
$676$		0			0
$677$	32.0000			1.22986			0.614930	−	0.788582i	$-0.289184\pi$
$677$	32.0000			1.22986			0.614930	+	0.788582i	$0.289184\pi$
$678$		0			0
$679$	2.00000			0.0767530
$680$		0			0
$681$		−	12.0000i		−	0.459841i
$682$		0			0
$683$			19.0000i			0.727015i	0.931591	+	0.363507i	$0.118421\pi$
$683$			19.0000i			0.727015i	−0.931591	+	0.363507i	$0.881579\pi$
$684$		0			0
$685$	15.0000			0.573121
$686$		0			0
$687$		−	2.00000i		−	0.0763048i
$688$		0			0
$689$	−30.0000	+	20.0000i	−1.14291	+	0.761939i
$690$		0			0
$691$		−	40.0000i		−	1.52167i	−0.648944	−	0.760836i	$-0.724789\pi$
$691$		−	40.0000i		−	1.52167i	0.648944	−	0.760836i	$-0.275211\pi$
$692$		0			0
$693$	1.00000			0.0379869
$694$		0			0
$695$		−	12.0000i		−	0.455186i
$696$		0			0
$697$		−	8.00000i		−	0.303022i
$698$		0			0
$699$	−18.0000			−0.680823
$700$		0			0
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$		0			0
$703$	9.00000			0.339441
$704$		0			0
$705$	8.00000			0.301297
$706$		0			0
$707$			2.00000i			0.0752177i
$708$		0			0
$709$			14.0000i			0.525781i	0.964826	+	0.262891i	$0.0846758\pi$
$709$			14.0000i			0.525781i	−0.964826	+	0.262891i	$0.915324\pi$
$710$		0			0
$711$	12.0000			0.450035
$712$		0			0
$713$			12.0000i			0.449404i
$714$		0			0
$715$	3.00000	−	2.00000i	0.112194	−	0.0747958i
$716$		0			0
$717$		−	8.00000i		−	0.298765i
$718$		0			0
$719$	8.00000			0.298350			0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000			0.298350			0.149175	+	0.988811i	$0.452338\pi$
$720$		0			0
$721$		−	5.00000i		−	0.186210i
$722$		0			0
$723$		−	10.0000i		−	0.371904i
$724$		0			0
$725$	36.0000			1.33701
$726$		0			0
$727$	−23.0000			−0.853023			−0.426511	−	0.904482i	$-0.640258\pi$
$727$	−23.0000			−0.853023			−0.426511	+	0.904482i	$0.640258\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−7.00000			−0.258904
$732$		0			0
$733$			40.0000i			1.47743i	0.674016	+	0.738717i	$0.264568\pi$
$733$			40.0000i			1.47743i	−0.674016	+	0.738717i	$0.735432\pi$
$734$		0			0
$735$			1.00000i			0.0368856i
$736$		0			0
$737$	12.0000			0.442026
$738$		0			0
$739$			8.00000i			0.294285i	0.989115	+	0.147142i	$0.0470076\pi$
$739$			8.00000i			0.294285i	−0.989115	+	0.147142i	$0.952992\pi$
$740$		0			0
$741$	−2.00000	−	3.00000i	−0.0734718	−	0.110208i
$742$		0			0
$743$		−	36.0000i		−	1.32071i	−0.750953	−	0.660356i	$-0.770405\pi$
$743$		−	36.0000i		−	1.32071i	0.750953	−	0.660356i	$-0.229595\pi$
$744$		0			0
$745$	−6.00000			−0.219823
$746$		0			0
$747$			6.00000i			0.219529i
$748$		0			0
$749$		−	2.00000i		−	0.0730784i
$750$		0			0
$751$	−20.0000			−0.729810			−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000			−0.729810			−0.364905	+	0.931045i	$0.618899\pi$
$752$		0			0
$753$	−11.0000			−0.400862
$754$		0			0
$755$	1.00000			0.0363937
$756$		0			0
$757$	38.0000			1.38113			0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000			1.38113			0.690567	+	0.723269i	$0.257361\pi$
$758$		0			0
$759$		−	3.00000i		−	0.108893i
$760$		0			0
$761$			42.0000i			1.52250i	0.648459	+	0.761249i	$0.275414\pi$
$761$			42.0000i			1.52250i	−0.648459	+	0.761249i	$0.724586\pi$
$762$		0			0
$763$	−7.00000			−0.253417
$764$		0			0
$765$		−	1.00000i		−	0.0361551i
$766$		0			0
$767$	−12.0000	−	18.0000i	−0.433295	−	0.649942i
$768$		0			0
$769$		−	31.0000i		−	1.11789i	−0.829205	−	0.558944i	$-0.811207\pi$
$769$		−	31.0000i		−	1.11789i	0.829205	−	0.558944i	$-0.188793\pi$
$770$		0			0
$771$	14.0000			0.504198
$772$		0			0
$773$		−	19.0000i		−	0.683383i	−0.939812	−	0.341691i	$-0.889000\pi$
$773$		−	19.0000i		−	0.683383i	0.939812	−	0.341691i	$-0.111000\pi$
$774$		0			0
$775$		−	16.0000i		−	0.574737i
$776$		0			0
$777$	9.00000			0.322873
$778$		0			0
$779$	−8.00000			−0.286630
$780$		0			0
$781$	−6.00000			−0.214697
$782$		0			0
$783$	9.00000			0.321634
$784$		0			0
$785$			21.0000i			0.749522i
$786$		0			0
$787$			53.0000i			1.88925i	0.328158	+	0.944623i	$0.393572\pi$
$787$			53.0000i			1.88925i	−0.328158	+	0.944623i	$0.606428\pi$
$788$		0			0
$789$	−16.0000			−0.569615
$790$		0			0
$791$			2.00000i			0.0711118i
$792$		0			0
$793$	33.0000	−	22.0000i	1.17186	−	0.781243i
$794$		0			0
$795$			10.0000i			0.354663i
$796$		0			0
$797$	2.00000			0.0708436			0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000			0.0708436			0.0354218	+	0.999372i	$0.488723\pi$
$798$		0			0
$799$			8.00000i			0.283020i
$800$		0			0
$801$		−	12.0000i		−	0.423999i
$802$		0			0
$803$	11.0000			0.388182
$804$		0			0
$805$	3.00000			0.105736
$806$		0			0
$807$	−20.0000			−0.704033
$808$		0			0
$809$	−4.00000			−0.140633			−0.0703163	−	0.997525i	$-0.522401\pi$
$809$	−4.00000			−0.140633			−0.0703163	+	0.997525i	$0.522401\pi$
$810$		0			0
$811$		−	21.0000i		−	0.737410i	−0.929547	−	0.368705i	$-0.879801\pi$
$811$		−	21.0000i		−	0.737410i	0.929547	−	0.368705i	$-0.120199\pi$
$812$		0			0
$813$		−	4.00000i		−	0.140286i
$814$		0			0
$815$	−20.0000			−0.700569
$816$		0			0
$817$			7.00000i			0.244899i
$818$		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
$823$	8.00000			0.278862			0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000			0.278862			0.139431	+	0.990232i	$0.455473\pi$
$824$		0			0
$825$			4.00000i			0.139262i
$826$		0			0
$827$		−	3.00000i		−	0.104320i	−0.998639	−	0.0521601i	$-0.983389\pi$
$827$		−	3.00000i		−	0.104320i	0.998639	−	0.0521601i	$-0.0166106\pi$
$828$		0			0
$829$	25.0000			0.868286			0.434143	−	0.900844i	$-0.357051\pi$
$829$	25.0000			0.868286			0.434143	+	0.900844i	$0.357051\pi$
$830$		0			0
$831$	16.0000			0.555034
$832$		0			0
$833$	−1.00000			−0.0346479
$834$		0			0
$835$	−11.0000			−0.380671
$836$		0			0
$837$		−	4.00000i		−	0.138260i
$838$		0			0
$839$			52.0000i			1.79524i	0.440771	+	0.897620i	$0.354705\pi$
$839$			52.0000i			1.79524i	−0.440771	+	0.897620i	$0.645295\pi$
$840$		0			0
$841$	52.0000			1.79310
$842$		0			0
$843$			22.0000i			0.757720i
$844$		0			0
$845$	−12.0000	−	5.00000i	−0.412813	−	0.172005i
$846$		0			0
$847$		−	10.0000i		−	0.343604i
$848$		0			0
$849$	14.0000			0.480479
$850$		0			0
$851$		−	27.0000i		−	0.925548i
$852$		0			0
$853$			56.0000i			1.91740i	0.284413	+	0.958702i	$0.408201\pi$
$853$			56.0000i			1.91740i	−0.284413	+	0.958702i	$0.591799\pi$
$854$		0			0
$855$	−1.00000			−0.0341993
$856$		0			0
$857$	−22.0000			−0.751506			−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000			−0.751506			−0.375753	+	0.926720i	$0.622616\pi$
$858$		0			0
$859$	−10.0000			−0.341196			−0.170598	−	0.985341i	$-0.554570\pi$
$859$	−10.0000			−0.341196			−0.170598	+	0.985341i	$0.554570\pi$
$860$		0			0
$861$	−8.00000			−0.272639
$862$		0			0
$863$			48.0000i			1.63394i	0.576681	+	0.816970i	$0.304348\pi$
$863$			48.0000i			1.63394i	−0.576681	+	0.816970i	$0.695652\pi$
$864$		0			0
$865$		−	14.0000i		−	0.476014i
$866$		0			0
$867$	−16.0000			−0.543388
$868$		0			0
$869$			12.0000i			0.407072i
$870$		0			0
$871$	−24.0000	−	36.0000i	−0.813209	−	1.21981i
$872$		0			0
$873$			2.00000i			0.0676897i
$874$		0			0
$875$	−9.00000			−0.304256
$876$		0			0
$877$			22.0000i			0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$			22.0000i			0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$		0			0
$879$			26.0000i			0.876958i
$880$		0			0
$881$	−59.0000			−1.98776			−0.993880	−	0.110463i	$-0.964767\pi$
$881$	−59.0000			−1.98776			−0.993880	+	0.110463i	$0.964767\pi$
$882$		0			0
$883$	−31.0000			−1.04323			−0.521617	−	0.853180i	$-0.674671\pi$
$883$	−31.0000			−1.04323			−0.521617	+	0.853180i	$0.674671\pi$
$884$		0			0
$885$	−6.00000			−0.201688
$886$		0			0
$887$	−2.00000			−0.0671534			−0.0335767	−	0.999436i	$-0.510690\pi$
$887$	−2.00000			−0.0671534			−0.0335767	+	0.999436i	$0.510690\pi$
$888$		0			0
$889$			16.0000i			0.536623i
$890$		0			0
$891$			1.00000i			0.0335013i
$892$		0			0
$893$	8.00000			0.267710
$894$		0			0
$895$		−	18.0000i		−	0.601674i
$896$		0			0
$897$	−9.00000	+	6.00000i	−0.300501	+	0.200334i
$898$		0			0
$899$		−	36.0000i		−	1.20067i
$900$		0			0
$901$	−10.0000			−0.333148
$902$		0			0
$903$			7.00000i			0.232945i
$904$		0			0
$905$			2.00000i			0.0664822i
$906$		0			0
$907$	12.0000			0.398453			0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000			0.398453			0.199227	+	0.979953i	$0.436157\pi$
$908$		0			0
$909$	−2.00000			−0.0663358
$910$		0			0
$911$	29.0000			0.960813			0.480406	−	0.877046i	$-0.340489\pi$
$911$	29.0000			0.960813			0.480406	+	0.877046i	$0.340489\pi$
$912$		0			0
$913$	−6.00000			−0.198571
$914$		0			0
$915$		−	11.0000i		−	0.363649i
$916$		0			0
$917$		−	3.00000i		−	0.0990687i
$918$		0			0
$919$	−34.0000			−1.12156			−0.560778	−	0.827966i	$-0.689498\pi$
$919$	−34.0000			−1.12156			−0.560778	+	0.827966i	$0.689498\pi$
$920$		0			0
$921$		−	8.00000i		−	0.263609i
$922$		0			0
$923$	12.0000	+	18.0000i	0.394985	+	0.592477i
$924$		0			0
$925$			36.0000i			1.18367i
$926$		0			0
$927$	5.00000			0.164222
$928$		0			0
$929$			26.0000i			0.853032i	0.904480	+	0.426516i	$0.140259\pi$
$929$			26.0000i			0.853032i	−0.904480	+	0.426516i	$0.859741\pi$
$930$		0			0
$931$			1.00000i			0.0327737i
$932$		0			0
$933$	−8.00000			−0.261908
$934$		0			0
$935$	1.00000			0.0327035
$936$		0			0
$937$	−20.0000			−0.653372			−0.326686	−	0.945133i	$-0.605932\pi$
$937$	−20.0000			−0.653372			−0.326686	+	0.945133i	$0.605932\pi$
$938$		0			0
$939$	−14.0000			−0.456873
$940$		0			0
$941$			30.0000i			0.977972i	0.872292	+	0.488986i	$0.162633\pi$
$941$			30.0000i			0.977972i	−0.872292	+	0.488986i	$0.837367\pi$
$942$		0			0
$943$			24.0000i			0.781548i
$944$		0			0
$945$	−1.00000			−0.0325300
$946$		0			0
$947$		−	35.0000i		−	1.13735i	−0.822563	−	0.568674i	$-0.807457\pi$
$947$		−	35.0000i		−	1.13735i	0.822563	−	0.568674i	$-0.192543\pi$
$948$		0			0
$949$	−22.0000	−	33.0000i	−0.714150	−	1.07123i
$950$		0			0
$951$			12.0000i			0.389127i
$952$		0			0
$953$	−42.0000			−1.36051			−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000			−1.36051			−0.680257	+	0.732974i	$0.738132\pi$
$954$		0			0
$955$		−	11.0000i		−	0.355952i
$956$		0			0
$957$			9.00000i			0.290929i
$958$		0			0
$959$	15.0000			0.484375
$960$		0			0
$961$	15.0000			0.483871
$962$		0			0
$963$	2.00000			0.0644491
$964$		0			0
$965$	12.0000			0.386294
$966$		0			0
$967$			9.00000i			0.289420i	0.989474	+	0.144710i	$0.0462250\pi$
$967$			9.00000i			0.289420i	−0.989474	+	0.144710i	$0.953775\pi$
$968$		0			0
$969$		−	1.00000i		−	0.0321246i
$970$		0			0
$971$	4.00000			0.128366			0.0641831	−	0.997938i	$-0.479556\pi$
$971$	4.00000			0.128366			0.0641831	+	0.997938i	$0.479556\pi$
$972$		0			0
$973$		−	12.0000i		−	0.384702i
$974$		0			0
$975$	12.0000	−	8.00000i	0.384308	−	0.256205i
$976$		0			0
$977$			39.0000i			1.24772i	0.781536	+	0.623860i	$0.214437\pi$
$977$			39.0000i			1.24772i	−0.781536	+	0.623860i	$0.785563\pi$
$978$		0			0
$979$	12.0000			0.383522
$980$		0			0
$981$		−	7.00000i		−	0.223493i
$982$		0			0
$983$		−	9.00000i		−	0.287055i	−0.989646	−	0.143528i	$-0.954155\pi$
$983$		−	9.00000i		−	0.287055i	0.989646	−	0.143528i	$-0.0458446\pi$
$984$		0			0
$985$	2.00000			0.0637253
$986$		0			0
$987$	8.00000			0.254643
$988$		0			0
$989$	21.0000			0.667761
$990$		0			0
$991$	12.0000			0.381193			0.190596	−	0.981669i	$-0.438958\pi$
$991$	12.0000			0.381193			0.190596	+	0.981669i	$0.438958\pi$
$992$		0			0
$993$		−	8.00000i		−	0.253872i
$994$		0			0
$995$			5.00000i			0.158511i
$996$		0			0
$997$	−14.0000			−0.443384			−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000			−0.443384			−0.221692	+	0.975117i	$0.571158\pi$
$998$		0			0
$999$			9.00000i			0.284747i

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000i q^{5} -1.00000i q^{7} +1.00000 q^{9} +1.00000i q^{11} +(3.00000 - 2.00000i) q^{13} -1.00000i q^{15} +1.00000 q^{17} -1.00000i q^{19} -1.00000i q^{21} -3.00000 q^{23} +4.00000 q^{25} +1.00000 q^{27} +9.00000 q^{29} -4.00000i q^{31} +1.00000i q^{33} -1.00000 q^{35} +9.00000i q^{37} +(3.00000 - 2.00000i) q^{39} -8.00000i q^{41} -7.00000 q^{43} -1.00000i q^{45} +8.00000i q^{47} -1.00000 q^{49} +1.00000 q^{51} -10.0000 q^{53} +1.00000 q^{55} -1.00000i q^{57} -6.00000i q^{59} +11.0000 q^{61} -1.00000i q^{63} +(-2.00000 - 3.00000i) q^{65} -12.0000i q^{67} -3.00000 q^{69} +6.00000i q^{71} -11.0000i q^{73} +4.00000 q^{75} +1.00000 q^{77} +12.0000 q^{79} +1.00000 q^{81} +6.00000i q^{83} -1.00000i q^{85} +9.00000 q^{87} -12.0000i q^{89} +(-2.00000 - 3.00000i) q^{91} -4.00000i q^{93} -1.00000 q^{95} +2.00000i q^{97} +1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{9} + 6 q^{13} + 2 q^{17} - 6 q^{23} + 8 q^{25} + 2 q^{27} + 18 q^{29} - 2 q^{35} + 6 q^{39} - 14 q^{43} - 2 q^{49} + 2 q^{51} - 20 q^{53} + 2 q^{55} + 22 q^{61} - 4 q^{65} - 6 q^{69} + 8 q^{75} + 2 q^{77} + 24 q^{79} + 2 q^{81} + 18 q^{87} - 4 q^{91} - 2 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^9 + 6 * q^13 + 2 * q^17 - 6 * q^23 + 8 * q^25 + 2 * q^27 + 18 * q^29 - 2 * q^35 + 6 * q^39 - 14 * q^43 - 2 * q^49 + 2 * q^51 - 20 * q^53 + 2 * q^55 + 22 * q^61 - 4 * q^65 - 6 * q^69 + 8 * q^75 + 2 * q^77 + 24 * q^79 + 2 * q^81 + 18 * q^87 - 4 * q^91 - 2 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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