LMFDB - Embedded newform 2352.2.h.k.2255.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,2,Mod(2255,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2352.2255");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.h (of order $2$, degree $1$, 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:	$18.7808145554$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2255.2
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2352.2255
Dual form			2352.2.h.k.2255.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.41421i) q^{3} +2.00000i q^{5} +(-1.00000 - 2.82843i) q^{9} +O(q^{10})$	$q+(1.00000 - 1.41421i) q^{3} +2.00000i q^{5} +(-1.00000 - 2.82843i) q^{9} -4.24264 q^{11} +1.41421 q^{13} +(2.82843 + 2.00000i) q^{15} -2.00000i q^{17} -4.24264 q^{23} +1.00000 q^{25} +(-5.00000 - 1.41421i) q^{27} -8.48528i q^{29} -8.48528i q^{31} +(-4.24264 + 6.00000i) q^{33} +6.00000 q^{37} +(1.41421 - 2.00000i) q^{39} -10.0000i q^{41} +12.0000i q^{43} +(5.65685 - 2.00000i) q^{45} -6.00000 q^{47} +(-2.82843 - 2.00000i) q^{51} -5.65685i q^{53} -8.48528i q^{55} -12.0000 q^{59} +1.41421 q^{61} +2.82843i q^{65} -12.0000i q^{67} +(-4.24264 + 6.00000i) q^{69} +12.7279 q^{71} -9.89949 q^{73} +(1.00000 - 1.41421i) q^{75} -12.0000i q^{79} +(-7.00000 + 5.65685i) q^{81} -6.00000 q^{83} +4.00000 q^{85} +(-12.0000 - 8.48528i) q^{87} -2.00000i q^{89} +(-12.0000 - 8.48528i) q^{93} +7.07107 q^{97} +(4.24264 + 12.0000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 4 * q^9	$ 4 q + 4 q^{3} - 4 q^{9} + 4 q^{25} - 20 q^{27} + 24 q^{37} - 24 q^{47} - 48 q^{59} + 4 q^{75} - 28 q^{81} - 24 q^{83} + 16 q^{85} - 48 q^{87} - 48 q^{93}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^9 + 4 * q^25 - 20 * q^27 + 24 * q^37 - 24 * q^47 - 48 * q^59 + 4 * q^75 - 28 * q^81 - 24 * q^83 + 16 * q^85 - 48 * q^87 - 48 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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	−	1.41421i	0.577350	−	0.816497i
$3$	1.00000	−	1.41421i	0.577350	−	0.816497i
$4$		0			0
$5$			2.00000i			0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$			2.00000i			0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−1.00000	−	2.82843i	−0.333333	−	0.942809i
$10$		0			0
$11$	−4.24264			−1.27920			−0.639602	−	0.768706i	$-0.720901\pi$
$11$	−4.24264			−1.27920			−0.639602	+	0.768706i	$0.720901\pi$
$12$		0			0
$13$	1.41421			0.392232			0.196116	−	0.980581i	$-0.437167\pi$
$13$	1.41421			0.392232			0.196116	+	0.980581i	$0.437167\pi$
$14$		0			0
$15$	2.82843	+	2.00000i	0.730297	+	0.516398i
$16$		0			0
$17$		−	2.00000i		−	0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$		−	2.00000i		−	0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.24264			−0.884652			−0.442326	−	0.896854i	$-0.645847\pi$
$23$	−4.24264			−0.884652			−0.442326	+	0.896854i	$0.645847\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−5.00000	−	1.41421i	−0.962250	−	0.272166i
$28$		0			0
$29$		−	8.48528i		−	1.57568i	−0.615882	−	0.787839i	$-0.711200\pi$
$29$		−	8.48528i		−	1.57568i	0.615882	−	0.787839i	$-0.288800\pi$
$30$		0			0
$31$		−	8.48528i		−	1.52400i	−0.647576	−	0.762001i	$-0.724217\pi$
$31$		−	8.48528i		−	1.52400i	0.647576	−	0.762001i	$-0.275783\pi$
$32$		0			0
$33$	−4.24264	+	6.00000i	−0.738549	+	1.04447i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	6.00000			0.986394			0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000			0.986394			0.493197	+	0.869918i	$0.335828\pi$
$38$		0			0
$39$	1.41421	−	2.00000i	0.226455	−	0.320256i
$40$		0			0
$41$		−	10.0000i		−	1.56174i	−0.624695	−	0.780869i	$-0.714777\pi$
$41$		−	10.0000i		−	1.56174i	0.624695	−	0.780869i	$-0.285223\pi$
$42$		0			0
$43$			12.0000i			1.82998i	0.403473	+	0.914991i	$0.367803\pi$
$43$			12.0000i			1.82998i	−0.403473	+	0.914991i	$0.632197\pi$
$44$		0			0
$45$	5.65685	−	2.00000i	0.843274	−	0.298142i
$46$		0			0
$47$	−6.00000			−0.875190			−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000			−0.875190			−0.437595	+	0.899172i	$0.644170\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−2.82843	−	2.00000i	−0.396059	−	0.280056i
$52$		0			0
$53$		−	5.65685i		−	0.777029i	−0.921443	−	0.388514i	$-0.872988\pi$
$53$		−	5.65685i		−	0.777029i	0.921443	−	0.388514i	$-0.127012\pi$
$54$		0			0
$55$		−	8.48528i		−	1.14416i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−12.0000			−1.56227			−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000			−1.56227			−0.781133	+	0.624364i	$0.785358\pi$
$60$		0			0
$61$	1.41421			0.181071			0.0905357	−	0.995893i	$-0.471142\pi$
$61$	1.41421			0.181071			0.0905357	+	0.995893i	$0.471142\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			2.82843i			0.350823i
$66$		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
$69$	−4.24264	+	6.00000i	−0.510754	+	0.722315i
$70$		0			0
$71$	12.7279			1.51053			0.755263	−	0.655422i	$-0.227509\pi$
$71$	12.7279			1.51053			0.755263	+	0.655422i	$0.227509\pi$
$72$		0			0
$73$	−9.89949			−1.15865			−0.579324	−	0.815097i	$-0.696683\pi$
$73$	−9.89949			−1.15865			−0.579324	+	0.815097i	$0.696683\pi$
$74$		0			0
$75$	1.00000	−	1.41421i	0.115470	−	0.163299i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	12.0000i		−	1.35011i	−0.737769	−	0.675053i	$-0.764121\pi$
$79$		−	12.0000i		−	1.35011i	0.737769	−	0.675053i	$-0.235879\pi$
$80$		0			0
$81$	−7.00000	+	5.65685i	−0.777778	+	0.628539i
$82$		0			0
$83$	−6.00000			−0.658586			−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000			−0.658586			−0.329293	+	0.944228i	$0.606810\pi$
$84$		0			0
$85$	4.00000			0.433861
$86$		0			0
$87$	−12.0000	−	8.48528i	−1.28654	−	0.909718i
$88$		0			0
$89$		−	2.00000i		−	0.212000i	−0.994366	−	0.106000i	$-0.966196\pi$
$89$		−	2.00000i		−	0.212000i	0.994366	−	0.106000i	$-0.0338043\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−12.0000	−	8.48528i	−1.24434	−	0.879883i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	7.07107			0.717958			0.358979	−	0.933346i	$-0.383125\pi$
$97$	7.07107			0.717958			0.358979	+	0.933346i	$0.383125\pi$
$98$		0			0
$99$	4.24264	+	12.0000i	0.426401	+	1.20605i
$100$		0			0
$101$		−	2.00000i		−	0.199007i	−0.995037	−	0.0995037i	$-0.968274\pi$
$101$		−	2.00000i		−	0.199007i	0.995037	−	0.0995037i	$-0.0317255\pi$
$102$		0			0
$103$		−	8.48528i		−	0.836080i	−0.908429	−	0.418040i	$-0.862717\pi$
$103$		−	8.48528i		−	0.836080i	0.908429	−	0.418040i	$-0.137283\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.24264			0.410152			0.205076	−	0.978746i	$-0.434256\pi$
$107$	4.24264			0.410152			0.205076	+	0.978746i	$0.434256\pi$
$108$		0			0
$109$	12.0000			1.14939			0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000			1.14939			0.574696	+	0.818367i	$0.305120\pi$
$110$		0			0
$111$	6.00000	−	8.48528i	0.569495	−	0.805387i
$112$		0			0
$113$			11.3137i			1.06430i	0.846649	+	0.532152i	$0.178617\pi$
$113$			11.3137i			1.06430i	−0.846649	+	0.532152i	$0.821383\pi$
$114$		0			0
$115$		−	8.48528i		−	0.791257i
$116$		0			0
$117$	−1.41421	−	4.00000i	−0.130744	−	0.369800i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	7.00000			0.636364
$122$		0			0
$123$	−14.1421	−	10.0000i	−1.27515	−	0.901670i
$124$		0			0
$125$			12.0000i			1.07331i
$126$		0			0
$127$			12.0000i			1.06483i	0.846484	+	0.532414i	$0.178715\pi$
$127$			12.0000i			1.06483i	−0.846484	+	0.532414i	$0.821285\pi$
$128$		0			0
$129$	16.9706	+	12.0000i	1.49417	+	1.05654i
$130$		0			0
$131$	18.0000			1.57267			0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000			1.57267			0.786334	+	0.617802i	$0.211977\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	2.82843	−	10.0000i	0.243432	−	0.860663i
$136$		0			0
$137$			8.48528i			0.724947i	0.931994	+	0.362473i	$0.118068\pi$
$137$			8.48528i			0.724947i	−0.931994	+	0.362473i	$0.881932\pi$
$138$		0			0
$139$		−	8.48528i		−	0.719712i	−0.933008	−	0.359856i	$-0.882826\pi$
$139$		−	8.48528i		−	0.719712i	0.933008	−	0.359856i	$-0.117174\pi$
$140$		0			0
$141$	−6.00000	+	8.48528i	−0.505291	+	0.714590i
$142$		0			0
$143$	−6.00000			−0.501745
$144$		0			0
$145$	16.9706			1.40933
$146$		0			0
$147$		0			0
$148$		0			0
$149$			11.3137i			0.926855i	0.886135	+	0.463428i	$0.153381\pi$
$149$			11.3137i			0.926855i	−0.886135	+	0.463428i	$0.846619\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	−5.65685	+	2.00000i	−0.457330	+	0.161690i
$154$		0			0
$155$	16.9706			1.36311
$156$		0			0
$157$	−18.3848			−1.46726			−0.733632	−	0.679546i	$-0.762177\pi$
$157$	−18.3848			−1.46726			−0.733632	+	0.679546i	$0.762177\pi$
$158$		0			0
$159$	−8.00000	−	5.65685i	−0.634441	−	0.448618i
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$	−12.0000	−	8.48528i	−0.934199	−	0.660578i
$166$		0			0
$167$	−18.0000			−1.39288			−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000			−1.39288			−0.696441	+	0.717614i	$0.745234\pi$
$168$		0			0
$169$	−11.0000			−0.846154
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	22.0000i		−	1.67263i	−0.548250	−	0.836315i	$-0.684706\pi$
$173$		−	22.0000i		−	1.67263i	0.548250	−	0.836315i	$-0.315294\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−12.0000	+	16.9706i	−0.901975	+	1.27559i
$178$		0			0
$179$	−4.24264			−0.317110			−0.158555	−	0.987350i	$-0.550683\pi$
$179$	−4.24264			−0.317110			−0.158555	+	0.987350i	$0.550683\pi$
$180$		0			0
$181$	−1.41421			−0.105118			−0.0525588	−	0.998618i	$-0.516738\pi$
$181$	−1.41421			−0.105118			−0.0525588	+	0.998618i	$0.516738\pi$
$182$		0			0
$183$	1.41421	−	2.00000i	0.104542	−	0.147844i
$184$		0			0
$185$			12.0000i			0.882258i
$186$		0			0
$187$			8.48528i			0.620505i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−12.7279			−0.920960			−0.460480	−	0.887670i	$-0.652323\pi$
$191$	−12.7279			−0.920960			−0.460480	+	0.887670i	$0.652323\pi$
$192$		0			0
$193$	4.00000			0.287926			0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000			0.287926			0.143963	+	0.989583i	$0.454015\pi$
$194$		0			0
$195$	4.00000	+	2.82843i	0.286446	+	0.202548i
$196$		0			0
$197$			5.65685i			0.403034i	0.979485	+	0.201517i	$0.0645872\pi$
$197$			5.65685i			0.403034i	−0.979485	+	0.201517i	$0.935413\pi$
$198$		0			0
$199$		−	16.9706i		−	1.20301i	−0.798869	−	0.601506i	$-0.794568\pi$
$199$		−	16.9706i		−	1.20301i	0.798869	−	0.601506i	$-0.205432\pi$
$200$		0			0
$201$	−16.9706	−	12.0000i	−1.19701	−	0.846415i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	20.0000			1.39686
$206$		0			0
$207$	4.24264	+	12.0000i	0.294884	+	0.834058i
$208$		0			0
$209$		0			0
$210$		0			0
$211$			12.0000i			0.826114i	0.910705	+	0.413057i	$0.135539\pi$
$211$			12.0000i			0.826114i	−0.910705	+	0.413057i	$0.864461\pi$
$212$		0			0
$213$	12.7279	−	18.0000i	0.872103	−	1.23334i
$214$		0			0
$215$	−24.0000			−1.63679
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−9.89949	+	14.0000i	−0.668946	+	0.946032i
$220$		0			0
$221$		−	2.82843i		−	0.190261i
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	−1.00000	−	2.82843i	−0.0666667	−	0.188562i
$226$		0			0
$227$	12.0000			0.796468			0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000			0.796468			0.398234	+	0.917284i	$0.369623\pi$
$228$		0			0
$229$	7.07107			0.467269			0.233635	−	0.972324i	$-0.424938\pi$
$229$	7.07107			0.467269			0.233635	+	0.972324i	$0.424938\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	8.48528i		−	0.555889i	−0.960597	−	0.277945i	$-0.910347\pi$
$233$		−	8.48528i		−	0.555889i	0.960597	−	0.277945i	$-0.0896532\pi$
$234$		0			0
$235$		−	12.0000i		−	0.782794i
$236$		0			0
$237$	−16.9706	−	12.0000i	−1.10236	−	0.779484i
$238$		0			0
$239$	−21.2132			−1.37217			−0.686084	−	0.727522i	$-0.740672\pi$
$239$	−21.2132			−1.37217			−0.686084	+	0.727522i	$0.740672\pi$
$240$		0			0
$241$	−24.0416			−1.54866			−0.774329	−	0.632783i	$-0.781912\pi$
$241$	−24.0416			−1.54866			−0.774329	+	0.632783i	$0.781912\pi$
$242$		0			0
$243$	1.00000	+	15.5563i	0.0641500	+	0.997940i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−6.00000	+	8.48528i	−0.380235	+	0.537733i
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$	18.0000			1.13165
$254$		0			0
$255$	4.00000	−	5.65685i	0.250490	−	0.354246i
$256$		0			0
$257$			14.0000i			0.873296i	0.899632	+	0.436648i	$0.143834\pi$
$257$			14.0000i			0.873296i	−0.899632	+	0.436648i	$0.856166\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−24.0000	+	8.48528i	−1.48556	+	0.525226i
$262$		0			0
$263$	4.24264			0.261612			0.130806	−	0.991408i	$-0.458243\pi$
$263$	4.24264			0.261612			0.130806	+	0.991408i	$0.458243\pi$
$264$		0			0
$265$	11.3137			0.694996
$266$		0			0
$267$	−2.82843	−	2.00000i	−0.173097	−	0.122398i
$268$		0			0
$269$			10.0000i			0.609711i	0.952399	+	0.304855i	$0.0986081\pi$
$269$			10.0000i			0.609711i	−0.952399	+	0.304855i	$0.901392\pi$
$270$		0			0
$271$		−	8.48528i		−	0.515444i	−0.966219	−	0.257722i	$-0.917028\pi$
$271$		−	8.48528i		−	0.515444i	0.966219	−	0.257722i	$-0.0829719\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.24264			−0.255841
$276$		0			0
$277$	8.00000			0.480673			0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000			0.480673			0.240337	+	0.970690i	$0.422742\pi$
$278$		0			0
$279$	−24.0000	+	8.48528i	−1.43684	+	0.508001i
$280$		0			0
$281$			8.48528i			0.506189i	0.967442	+	0.253095i	$0.0814484\pi$
$281$			8.48528i			0.506189i	−0.967442	+	0.253095i	$0.918552\pi$
$282$		0			0
$283$			16.9706i			1.00880i	0.863472	+	0.504398i	$0.168285\pi$
$283$			16.9706i			1.00880i	−0.863472	+	0.504398i	$0.831715\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$	7.07107	−	10.0000i	0.414513	−	0.586210i
$292$		0			0
$293$			22.0000i			1.28525i	0.766179	+	0.642627i	$0.222155\pi$
$293$			22.0000i			1.28525i	−0.766179	+	0.642627i	$0.777845\pi$
$294$		0			0
$295$		−	24.0000i		−	1.39733i
$296$		0			0
$297$	21.2132	+	6.00000i	1.23091	+	0.348155i
$298$		0			0
$299$	−6.00000			−0.346989
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−2.82843	−	2.00000i	−0.162489	−	0.114897i
$304$		0			0
$305$			2.82843i			0.161955i
$306$		0			0
$307$			16.9706i			0.968561i	0.874913	+	0.484281i	$0.160919\pi$
$307$			16.9706i			0.968561i	−0.874913	+	0.484281i	$0.839081\pi$
$308$		0			0
$309$	−12.0000	−	8.48528i	−0.682656	−	0.482711i
$310$		0			0
$311$	−6.00000			−0.340229			−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000			−0.340229			−0.170114	+	0.985424i	$0.554414\pi$
$312$		0			0
$313$	15.5563			0.879297			0.439648	−	0.898170i	$-0.355103\pi$
$313$	15.5563			0.879297			0.439648	+	0.898170i	$0.355103\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	28.2843i		−	1.58860i	−0.607524	−	0.794301i	$-0.707837\pi$
$317$		−	28.2843i		−	1.58860i	0.607524	−	0.794301i	$-0.292163\pi$
$318$		0			0
$319$			36.0000i			2.01561i
$320$		0			0
$321$	4.24264	−	6.00000i	0.236801	−	0.334887i
$322$		0			0
$323$		0			0
$324$		0			0
$325$	1.41421			0.0784465
$326$		0			0
$327$	12.0000	−	16.9706i	0.663602	−	0.938474i
$328$		0			0
$329$		0			0
$330$		0			0
$331$			24.0000i			1.31916i	0.751635	+	0.659580i	$0.229266\pi$
$331$			24.0000i			1.31916i	−0.751635	+	0.659580i	$0.770734\pi$
$332$		0			0
$333$	−6.00000	−	16.9706i	−0.328798	−	0.929981i
$334$		0			0
$335$	24.0000			1.31126
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$		0			0
$339$	16.0000	+	11.3137i	0.869001	+	0.614476i
$340$		0			0
$341$			36.0000i			1.94951i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−12.0000	−	8.48528i	−0.646058	−	0.456832i
$346$		0			0
$347$	−21.2132			−1.13878			−0.569392	−	0.822066i	$-0.692821\pi$
$347$	−21.2132			−1.13878			−0.569392	+	0.822066i	$0.692821\pi$
$348$		0			0
$349$	32.5269			1.74113			0.870563	−	0.492057i	$-0.163755\pi$
$349$	32.5269			1.74113			0.870563	+	0.492057i	$0.163755\pi$
$350$		0			0
$351$	−7.07107	−	2.00000i	−0.377426	−	0.106752i
$352$		0			0
$353$		−	34.0000i		−	1.80964i	−0.425797	−	0.904819i	$-0.640006\pi$
$353$		−	34.0000i		−	1.80964i	0.425797	−	0.904819i	$-0.359994\pi$
$354$		0			0
$355$			25.4558i			1.35106i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	12.7279			0.671754			0.335877	−	0.941906i	$-0.390967\pi$
$359$	12.7279			0.671754			0.335877	+	0.941906i	$0.390967\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$	7.00000	−	9.89949i	0.367405	−	0.519589i
$364$		0			0
$365$		−	19.7990i		−	1.03633i
$366$		0			0
$367$			16.9706i			0.885856i	0.896557	+	0.442928i	$0.146060\pi$
$367$			16.9706i			0.885856i	−0.896557	+	0.442928i	$0.853940\pi$
$368$		0			0
$369$	−28.2843	+	10.0000i	−1.47242	+	0.520579i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	22.0000			1.13912			0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000			1.13912			0.569558	+	0.821951i	$0.307114\pi$
$374$		0			0
$375$	16.9706	+	12.0000i	0.876356	+	0.619677i
$376$		0			0
$377$		−	12.0000i		−	0.618031i
$378$		0			0
$379$		−	24.0000i		−	1.23280i	−0.787434	−	0.616399i	$-0.788591\pi$
$379$		−	24.0000i		−	1.23280i	0.787434	−	0.616399i	$-0.211409\pi$
$380$		0			0
$381$	16.9706	+	12.0000i	0.869428	+	0.614779i
$382$		0			0
$383$	24.0000			1.22634			0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000			1.22634			0.613171	+	0.789950i	$0.289894\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	33.9411	−	12.0000i	1.72532	−	0.609994i
$388$		0			0
$389$			25.4558i			1.29066i	0.763903	+	0.645331i	$0.223281\pi$
$389$			25.4558i			1.29066i	−0.763903	+	0.645331i	$0.776719\pi$
$390$		0			0
$391$			8.48528i			0.429119i
$392$		0			0
$393$	18.0000	−	25.4558i	0.907980	−	1.28408i
$394$		0			0
$395$	24.0000			1.20757
$396$		0			0
$397$	−7.07107			−0.354887			−0.177443	−	0.984131i	$-0.556783\pi$
$397$	−7.07107			−0.354887			−0.177443	+	0.984131i	$0.556783\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	8.48528i		−	0.423735i	−0.977298	−	0.211867i	$-0.932046\pi$
$401$		−	8.48528i		−	0.423735i	0.977298	−	0.211867i	$-0.0679545\pi$
$402$		0			0
$403$		−	12.0000i		−	0.597763i
$404$		0			0
$405$	−11.3137	−	14.0000i	−0.562183	−	0.695666i
$406$		0			0
$407$	−25.4558			−1.26180
$408$		0			0
$409$	−26.8701			−1.32864			−0.664319	−	0.747449i	$-0.731279\pi$
$409$	−26.8701			−1.32864			−0.664319	+	0.747449i	$0.731279\pi$
$410$		0			0
$411$	12.0000	+	8.48528i	0.591916	+	0.418548i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		−	12.0000i		−	0.589057i
$416$		0			0
$417$	−12.0000	−	8.48528i	−0.587643	−	0.415526i
$418$		0			0
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$	8.00000			0.389896			0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000			0.389896			0.194948	+	0.980814i	$0.437546\pi$
$422$		0			0
$423$	6.00000	+	16.9706i	0.291730	+	0.825137i
$424$		0			0
$425$		−	2.00000i		−	0.0970143i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−6.00000	+	8.48528i	−0.289683	+	0.409673i
$430$		0			0
$431$	29.6985			1.43053			0.715263	−	0.698856i	$-0.246307\pi$
$431$	29.6985			1.43053			0.715263	+	0.698856i	$0.246307\pi$
$432$		0			0
$433$	−32.5269			−1.56314			−0.781572	−	0.623815i	$-0.785582\pi$
$433$	−32.5269			−1.56314			−0.781572	+	0.623815i	$0.785582\pi$
$434$		0			0
$435$	16.9706	−	24.0000i	0.813676	−	1.15071i
$436$		0			0
$437$		0			0
$438$		0			0
$439$		−	33.9411i		−	1.61992i	−0.586484	−	0.809961i	$-0.699488\pi$
$439$		−	33.9411i		−	1.61992i	0.586484	−	0.809961i	$-0.300512\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	38.1838			1.81417			0.907083	−	0.420952i	$-0.138304\pi$
$443$	38.1838			1.81417			0.907083	+	0.420952i	$0.138304\pi$
$444$		0			0
$445$	4.00000			0.189618
$446$		0			0
$447$	16.0000	+	11.3137i	0.756774	+	0.535120i
$448$		0			0
$449$		−	5.65685i		−	0.266963i	−0.991051	−	0.133482i	$-0.957384\pi$
$449$		−	5.65685i		−	0.266963i	0.991051	−	0.133482i	$-0.0426157\pi$
$450$		0			0
$451$			42.4264i			1.99778i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	10.0000			0.467780			0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000			0.467780			0.233890	+	0.972263i	$0.424854\pi$
$458$		0			0
$459$	−2.82843	+	10.0000i	−0.132020	+	0.466760i
$460$		0			0
$461$		−	34.0000i		−	1.58354i	−0.610821	−	0.791769i	$-0.709160\pi$
$461$		−	34.0000i		−	1.58354i	0.610821	−	0.791769i	$-0.290840\pi$
$462$		0			0
$463$		−	24.0000i		−	1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$		−	24.0000i		−	1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$		0			0
$465$	16.9706	−	24.0000i	0.786991	−	1.11297i
$466$		0			0
$467$	12.0000			0.555294			0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000			0.555294			0.277647	+	0.960683i	$0.410445\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−18.3848	+	26.0000i	−0.847126	+	1.19802i
$472$		0			0
$473$		−	50.9117i		−	2.34092i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−16.0000	+	5.65685i	−0.732590	+	0.259010i
$478$		0			0
$479$	−18.0000			−0.822441			−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000			−0.822441			−0.411220	+	0.911536i	$0.634897\pi$
$480$		0			0
$481$	8.48528			0.386896
$482$		0			0
$483$		0			0
$484$		0			0
$485$			14.1421i			0.642161i
$486$		0			0
$487$			24.0000i			1.08754i	0.839233	+	0.543772i	$0.183004\pi$
$487$			24.0000i			1.08754i	−0.839233	+	0.543772i	$0.816996\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−21.2132			−0.957338			−0.478669	−	0.877995i	$-0.658881\pi$
$491$	−21.2132			−0.957338			−0.478669	+	0.877995i	$0.658881\pi$
$492$		0			0
$493$	−16.9706			−0.764316
$494$		0			0
$495$	−24.0000	+	8.48528i	−1.07872	+	0.381385i
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		0			0
$501$	−18.0000	+	25.4558i	−0.804181	+	1.13728i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	4.00000			0.177998
$506$		0			0
$507$	−11.0000	+	15.5563i	−0.488527	+	0.690882i
$508$		0			0
$509$			10.0000i			0.443242i	0.975133	+	0.221621i	$0.0711348\pi$
$509$			10.0000i			0.443242i	−0.975133	+	0.221621i	$0.928865\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	16.9706			0.747812
$516$		0			0
$517$	25.4558			1.11955
$518$		0			0
$519$	−31.1127	−	22.0000i	−1.36570	−	0.965693i
$520$		0			0
$521$			14.0000i			0.613351i	0.951814	+	0.306676i	$0.0992167\pi$
$521$			14.0000i			0.613351i	−0.951814	+	0.306676i	$0.900783\pi$
$522$		0			0
$523$			25.4558i			1.11311i	0.830812	+	0.556553i	$0.187876\pi$
$523$			25.4558i			1.11311i	−0.830812	+	0.556553i	$0.812124\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−16.9706			−0.739249
$528$		0			0
$529$	−5.00000			−0.217391
$530$		0			0
$531$	12.0000	+	33.9411i	0.520756	+	1.47292i
$532$		0			0
$533$		−	14.1421i		−	0.612564i
$534$		0			0
$535$			8.48528i			0.366851i
$536$		0			0
$537$	−4.24264	+	6.00000i	−0.183083	+	0.258919i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	16.0000			0.687894			0.343947	−	0.938989i	$-0.388236\pi$
$541$	16.0000			0.687894			0.343947	+	0.938989i	$0.388236\pi$
$542$		0			0
$543$	−1.41421	+	2.00000i	−0.0606897	+	0.0858282i
$544$		0			0
$545$			24.0000i			1.02805i
$546$		0			0
$547$		−	12.0000i		−	0.513083i	−0.966533	−	0.256541i	$-0.917417\pi$
$547$		−	12.0000i		−	0.513083i	0.966533	−	0.256541i	$-0.0825830\pi$
$548$		0			0
$549$	−1.41421	−	4.00000i	−0.0603572	−	0.170716i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	16.9706	+	12.0000i	0.720360	+	0.509372i
$556$		0			0
$557$			5.65685i			0.239689i	0.992793	+	0.119844i	$0.0382395\pi$
$557$			5.65685i			0.239689i	−0.992793	+	0.119844i	$0.961760\pi$
$558$		0			0
$559$			16.9706i			0.717778i
$560$		0			0
$561$	12.0000	+	8.48528i	0.506640	+	0.358249i
$562$		0			0
$563$	36.0000			1.51722			0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000			1.51722			0.758610	+	0.651546i	$0.225879\pi$
$564$		0			0
$565$	−22.6274			−0.951943
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	8.48528i		−	0.355722i	−0.984056	−	0.177861i	$-0.943082\pi$
$569$		−	8.48528i		−	0.355722i	0.984056	−	0.177861i	$-0.0569177\pi$
$570$		0			0
$571$		−	12.0000i		−	0.502184i	−0.967963	−	0.251092i	$-0.919210\pi$
$571$		−	12.0000i		−	0.502184i	0.967963	−	0.251092i	$-0.0807897\pi$
$572$		0			0
$573$	−12.7279	+	18.0000i	−0.531717	+	0.751961i
$574$		0			0
$575$	−4.24264			−0.176930
$576$		0			0
$577$	9.89949			0.412121			0.206061	−	0.978539i	$-0.433936\pi$
$577$	9.89949			0.412121			0.206061	+	0.978539i	$0.433936\pi$
$578$		0			0
$579$	4.00000	−	5.65685i	0.166234	−	0.235091i
$580$		0			0
$581$		0			0
$582$		0			0
$583$			24.0000i			0.993978i
$584$		0			0
$585$	8.00000	−	2.82843i	0.330759	−	0.116941i
$586$		0			0
$587$	−12.0000			−0.495293			−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000			−0.495293			−0.247647	+	0.968850i	$0.579657\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	8.00000	+	5.65685i	0.329076	+	0.232692i
$592$		0			0
$593$		−	26.0000i		−	1.06769i	−0.845582	−	0.533846i	$-0.820746\pi$
$593$		−	26.0000i		−	1.06769i	0.845582	−	0.533846i	$-0.179254\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−24.0000	−	16.9706i	−0.982255	−	0.694559i
$598$		0			0
$599$	46.6690			1.90685			0.953423	−	0.301637i	$-0.0975331\pi$
$599$	46.6690			1.90685			0.953423	+	0.301637i	$0.0975331\pi$
$600$		0			0
$601$	15.5563			0.634557			0.317278	−	0.948332i	$-0.397231\pi$
$601$	15.5563			0.634557			0.317278	+	0.948332i	$0.397231\pi$
$602$		0			0
$603$	−33.9411	+	12.0000i	−1.38219	+	0.488678i
$604$		0			0
$605$			14.0000i			0.569181i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8.48528			−0.343278
$612$		0			0
$613$	36.0000			1.45403			0.727013	−	0.686624i	$-0.240908\pi$
$613$	36.0000			1.45403			0.727013	+	0.686624i	$0.240908\pi$
$614$		0			0
$615$	20.0000	−	28.2843i	0.806478	−	1.14053i
$616$		0			0
$617$			8.48528i			0.341605i	0.985305	+	0.170802i	$0.0546359\pi$
$617$			8.48528i			0.341605i	−0.985305	+	0.170802i	$0.945364\pi$
$618$		0			0
$619$			8.48528i			0.341052i	0.985353	+	0.170526i	$0.0545467\pi$
$619$			8.48528i			0.341052i	−0.985353	+	0.170526i	$0.945453\pi$
$620$		0			0
$621$	21.2132	+	6.00000i	0.851257	+	0.240772i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	12.0000i		−	0.478471i
$630$		0			0
$631$		−	24.0000i		−	0.955425i	−0.878516	−	0.477712i	$-0.841466\pi$
$631$		−	24.0000i		−	0.955425i	0.878516	−	0.477712i	$-0.158534\pi$
$632$		0			0
$633$	16.9706	+	12.0000i	0.674519	+	0.476957i
$634$		0			0
$635$	−24.0000			−0.952411
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−12.7279	−	36.0000i	−0.503509	−	1.42414i
$640$		0			0
$641$			42.4264i			1.67574i	0.545868	+	0.837871i	$0.316200\pi$
$641$			42.4264i			1.67574i	−0.545868	+	0.837871i	$0.683800\pi$
$642$		0			0
$643$			33.9411i			1.33851i	0.743034	+	0.669254i	$0.233386\pi$
$643$			33.9411i			1.33851i	−0.743034	+	0.669254i	$0.766614\pi$
$644$		0			0
$645$	−24.0000	+	33.9411i	−0.944999	+	1.33643i
$646$		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
$649$	50.9117			1.99846
$650$		0			0
$651$		0			0
$652$		0			0
$653$			25.4558i			0.996164i	0.867130	+	0.498082i	$0.165962\pi$
$653$			25.4558i			0.996164i	−0.867130	+	0.498082i	$0.834038\pi$
$654$		0			0
$655$			36.0000i			1.40664i
$656$		0			0
$657$	9.89949	+	28.0000i	0.386216	+	1.09238i
$658$		0			0
$659$	4.24264			0.165270			0.0826349	−	0.996580i	$-0.473666\pi$
$659$	4.24264			0.165270			0.0826349	+	0.996580i	$0.473666\pi$
$660$		0			0
$661$	−1.41421			−0.0550065			−0.0275033	−	0.999622i	$-0.508756\pi$
$661$	−1.41421			−0.0550065			−0.0275033	+	0.999622i	$0.508756\pi$
$662$		0			0
$663$	−4.00000	−	2.82843i	−0.155347	−	0.109847i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			36.0000i			1.39393i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−6.00000			−0.231627
$672$		0			0
$673$	−30.0000			−1.15642			−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000			−1.15642			−0.578208	+	0.815890i	$0.696248\pi$
$674$		0			0
$675$	−5.00000	−	1.41421i	−0.192450	−	0.0544331i
$676$		0			0
$677$		−	34.0000i		−	1.30673i	−0.757045	−	0.653363i	$-0.773358\pi$
$677$		−	34.0000i		−	1.30673i	0.757045	−	0.653363i	$-0.226642\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	12.0000	−	16.9706i	0.459841	−	0.650313i
$682$		0			0
$683$	−29.6985			−1.13638			−0.568190	−	0.822897i	$-0.692356\pi$
$683$	−29.6985			−1.13638			−0.568190	+	0.822897i	$0.692356\pi$
$684$		0			0
$685$	−16.9706			−0.648412
$686$		0			0
$687$	7.07107	−	10.0000i	0.269778	−	0.381524i
$688$		0			0
$689$		−	8.00000i		−	0.304776i
$690$		0			0
$691$			25.4558i			0.968386i	0.874961	+	0.484193i	$0.160887\pi$
$691$			25.4558i			0.968386i	−0.874961	+	0.484193i	$0.839113\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	16.9706			0.643730
$696$		0			0
$697$	−20.0000			−0.757554
$698$		0			0
$699$	−12.0000	−	8.48528i	−0.453882	−	0.320943i
$700$		0			0
$701$			42.4264i			1.60242i	0.598381	+	0.801212i	$0.295811\pi$
$701$			42.4264i			1.60242i	−0.598381	+	0.801212i	$0.704189\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−16.9706	−	12.0000i	−0.639148	−	0.451946i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−12.0000			−0.450669			−0.225335	−	0.974281i	$-0.572348\pi$
$709$	−12.0000			−0.450669			−0.225335	+	0.974281i	$0.572348\pi$
$710$		0			0
$711$	−33.9411	+	12.0000i	−1.27289	+	0.450035i
$712$		0			0
$713$			36.0000i			1.34821i
$714$		0			0
$715$		−	12.0000i		−	0.448775i
$716$		0			0
$717$	−21.2132	+	30.0000i	−0.792222	+	1.12037i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−24.0416	+	34.0000i	−0.894118	+	1.26447i
$724$		0			0
$725$		−	8.48528i		−	0.315135i
$726$		0			0
$727$		−	8.48528i		−	0.314702i	−0.987543	−	0.157351i	$-0.949705\pi$
$727$		−	8.48528i		−	0.314702i	0.987543	−	0.157351i	$-0.0502953\pi$
$728$		0			0
$729$	23.0000	+	14.1421i	0.851852	+	0.523783i
$730$		0			0
$731$	24.0000			0.887672
$732$		0			0
$733$	−9.89949			−0.365646			−0.182823	−	0.983146i	$-0.558524\pi$
$733$	−9.89949			−0.365646			−0.182823	+	0.983146i	$0.558524\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			50.9117i			1.87536i
$738$		0			0
$739$			24.0000i			0.882854i	0.897297	+	0.441427i	$0.145528\pi$
$739$			24.0000i			0.882854i	−0.897297	+	0.441427i	$0.854472\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	12.7279			0.466942			0.233471	−	0.972364i	$-0.424992\pi$
$743$	12.7279			0.466942			0.233471	+	0.972364i	$0.424992\pi$
$744$		0			0
$745$	−22.6274			−0.829004
$746$		0			0
$747$	6.00000	+	16.9706i	0.219529	+	0.620920i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	12.0000i		−	0.437886i	−0.975738	−	0.218943i	$-0.929739\pi$
$751$		−	12.0000i		−	0.437886i	0.975738	−	0.218943i	$-0.0702609\pi$
$752$		0			0
$753$	12.0000	−	16.9706i	0.437304	−	0.618442i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	36.0000			1.30844			0.654221	−	0.756303i	$-0.272997\pi$
$757$	36.0000			1.30844			0.654221	+	0.756303i	$0.272997\pi$
$758$		0			0
$759$	18.0000	−	25.4558i	0.653359	−	0.923989i
$760$		0			0
$761$		−	14.0000i		−	0.507500i	−0.967270	−	0.253750i	$-0.918336\pi$
$761$		−	14.0000i		−	0.507500i	0.967270	−	0.253750i	$-0.0816640\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−4.00000	−	11.3137i	−0.144620	−	0.409048i
$766$		0			0
$767$	−16.9706			−0.612772
$768$		0			0
$769$	−24.0416			−0.866963			−0.433482	−	0.901162i	$-0.642715\pi$
$769$	−24.0416			−0.866963			−0.433482	+	0.901162i	$0.642715\pi$
$770$		0			0
$771$	19.7990	+	14.0000i	0.713043	+	0.504198i
$772$		0			0
$773$		−	10.0000i		−	0.359675i	−0.983696	−	0.179838i	$-0.942443\pi$
$773$		−	10.0000i		−	0.359675i	0.983696	−	0.179838i	$-0.0575572\pi$
$774$		0			0
$775$		−	8.48528i		−	0.304800i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−54.0000			−1.93227
$782$		0			0
$783$	−12.0000	+	42.4264i	−0.428845	+	1.51620i
$784$		0			0
$785$		−	36.7696i		−	1.31236i
$786$		0			0
$787$		−	8.48528i		−	0.302468i	−0.988498	−	0.151234i	$-0.951675\pi$
$787$		−	8.48528i		−	0.302468i	0.988498	−	0.151234i	$-0.0483246\pi$
$788$		0			0
$789$	4.24264	−	6.00000i	0.151042	−	0.213606i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	2.00000			0.0710221
$794$		0			0
$795$	11.3137	−	16.0000i	0.401256	−	0.567462i
$796$		0			0
$797$		−	38.0000i		−	1.34603i	−0.739629	−	0.673015i	$-0.764999\pi$
$797$		−	38.0000i		−	1.34603i	0.739629	−	0.673015i	$-0.235001\pi$
$798$		0			0
$799$			12.0000i			0.424529i
$800$		0			0
$801$	−5.65685	+	2.00000i	−0.199875	+	0.0706665i
$802$		0			0
$803$	42.0000			1.48215
$804$		0			0
$805$		0			0
$806$		0			0
$807$	14.1421	+	10.0000i	0.497827	+	0.352017i
$808$		0			0
$809$		−	45.2548i		−	1.59108i	−0.605904	−	0.795538i	$-0.707189\pi$
$809$		−	45.2548i		−	1.59108i	0.605904	−	0.795538i	$-0.292811\pi$
$810$		0			0
$811$		−	25.4558i		−	0.893876i	−0.894565	−	0.446938i	$-0.852515\pi$
$811$		−	25.4558i		−	0.893876i	0.894565	−	0.446938i	$-0.147485\pi$
$812$		0			0
$813$	−12.0000	−	8.48528i	−0.420858	−	0.297592i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$			22.6274i			0.789702i	0.918745	+	0.394851i	$0.129204\pi$
$821$			22.6274i			0.789702i	−0.918745	+	0.394851i	$0.870796\pi$
$822$		0			0
$823$		−	36.0000i		−	1.25488i	−0.778664	−	0.627441i	$-0.784103\pi$
$823$		−	36.0000i		−	1.25488i	0.778664	−	0.627441i	$-0.215897\pi$
$824$		0			0
$825$	−4.24264	+	6.00000i	−0.147710	+	0.208893i
$826$		0			0
$827$	−21.2132			−0.737655			−0.368828	−	0.929498i	$-0.620241\pi$
$827$	−21.2132			−0.737655			−0.368828	+	0.929498i	$0.620241\pi$
$828$		0			0
$829$	26.8701			0.933236			0.466618	−	0.884459i	$-0.345472\pi$
$829$	26.8701			0.933236			0.466618	+	0.884459i	$0.345472\pi$
$830$		0			0
$831$	8.00000	−	11.3137i	0.277517	−	0.392468i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		−	36.0000i		−	1.24583i
$836$		0			0
$837$	−12.0000	+	42.4264i	−0.414781	+	1.46647i
$838$		0			0
$839$	6.00000			0.207143			0.103572	−	0.994622i	$-0.466973\pi$
$839$	6.00000			0.207143			0.103572	+	0.994622i	$0.466973\pi$
$840$		0			0
$841$	−43.0000			−1.48276
$842$		0			0
$843$	12.0000	+	8.48528i	0.413302	+	0.292249i
$844$		0			0
$845$		−	22.0000i		−	0.756823i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	24.0000	+	16.9706i	0.823678	+	0.582428i
$850$		0			0
$851$	−25.4558			−0.872615
$852$		0			0
$853$	7.07107			0.242109			0.121054	−	0.992646i	$-0.461372\pi$
$853$	7.07107			0.242109			0.121054	+	0.992646i	$0.461372\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			10.0000i			0.341593i	0.985306	+	0.170797i	$0.0546341\pi$
$857$			10.0000i			0.341593i	−0.985306	+	0.170797i	$0.945366\pi$
$858$		0			0
$859$		−	50.9117i		−	1.73708i	−0.495615	−	0.868542i	$-0.665057\pi$
$859$		−	50.9117i		−	1.73708i	0.495615	−	0.868542i	$-0.334943\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−12.7279			−0.433264			−0.216632	−	0.976253i	$-0.569507\pi$
$863$	−12.7279			−0.433264			−0.216632	+	0.976253i	$0.569507\pi$
$864$		0			0
$865$	44.0000			1.49604
$866$		0			0
$867$	13.0000	−	18.3848i	0.441503	−	0.624380i
$868$		0			0
$869$			50.9117i			1.72706i
$870$		0			0
$871$		−	16.9706i		−	0.575026i
$872$		0			0
$873$	−7.07107	−	20.0000i	−0.239319	−	0.676897i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	12.0000			0.405211			0.202606	−	0.979260i	$-0.435059\pi$
$877$	12.0000			0.405211			0.202606	+	0.979260i	$0.435059\pi$
$878$		0			0
$879$	31.1127	+	22.0000i	1.04941	+	0.742042i
$880$		0			0
$881$		−	14.0000i		−	0.471672i	−0.971793	−	0.235836i	$-0.924217\pi$
$881$		−	14.0000i		−	0.471672i	0.971793	−	0.235836i	$-0.0757828\pi$
$882$		0			0
$883$			48.0000i			1.61533i	0.589643	+	0.807664i	$0.299269\pi$
$883$			48.0000i			1.61533i	−0.589643	+	0.807664i	$0.700731\pi$
$884$		0			0
$885$	−33.9411	−	24.0000i	−1.14092	−	0.806751i
$886$		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
$889$		0			0
$890$		0			0
$891$	29.6985	−	24.0000i	0.994937	−	0.804030i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		−	8.48528i		−	0.283632i
$896$		0			0
$897$	−6.00000	+	8.48528i	−0.200334	+	0.283315i
$898$		0			0
$899$	−72.0000			−2.40133
$900$		0			0
$901$	−11.3137			−0.376914
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	2.82843i		−	0.0940201i
$906$		0			0
$907$		−	24.0000i		−	0.796907i	−0.917189	−	0.398453i	$-0.869547\pi$
$907$		−	24.0000i		−	0.796907i	0.917189	−	0.398453i	$-0.130453\pi$
$908$		0			0
$909$	−5.65685	+	2.00000i	−0.187626	+	0.0663358i
$910$		0			0
$911$	4.24264			0.140565			0.0702825	−	0.997527i	$-0.477610\pi$
$911$	4.24264			0.140565			0.0702825	+	0.997527i	$0.477610\pi$
$912$		0			0
$913$	25.4558			0.842465
$914$		0			0
$915$	4.00000	+	2.82843i	0.132236	+	0.0935049i
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	12.0000i		−	0.395843i	−0.980218	−	0.197922i	$-0.936581\pi$
$919$		−	12.0000i		−	0.395843i	0.980218	−	0.197922i	$-0.0634192\pi$
$920$		0			0
$921$	24.0000	+	16.9706i	0.790827	+	0.559199i
$922$		0			0
$923$	18.0000			0.592477
$924$		0			0
$925$	6.00000			0.197279
$926$		0			0
$927$	−24.0000	+	8.48528i	−0.788263	+	0.278693i
$928$		0			0
$929$		−	14.0000i		−	0.459325i	−0.973270	−	0.229663i	$-0.926238\pi$
$929$		−	14.0000i		−	0.459325i	0.973270	−	0.229663i	$-0.0737623\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−6.00000	+	8.48528i	−0.196431	+	0.277796i
$934$		0			0
$935$	−16.9706			−0.554997
$936$		0			0
$937$	−35.3553			−1.15501			−0.577504	−	0.816388i	$-0.695973\pi$
$937$	−35.3553			−1.15501			−0.577504	+	0.816388i	$0.695973\pi$
$938$		0			0
$939$	15.5563	−	22.0000i	0.507662	−	0.717943i
$940$		0			0
$941$			22.0000i			0.717180i	0.933495	+	0.358590i	$0.116742\pi$
$941$			22.0000i			0.717180i	−0.933495	+	0.358590i	$0.883258\pi$
$942$		0			0
$943$			42.4264i			1.38159i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−46.6690			−1.51654			−0.758270	−	0.651940i	$-0.773955\pi$
$947$	−46.6690			−1.51654			−0.758270	+	0.651940i	$0.773955\pi$
$948$		0			0
$949$	−14.0000			−0.454459
$950$		0			0
$951$	−40.0000	−	28.2843i	−1.29709	−	0.917180i
$952$		0			0
$953$		−	45.2548i		−	1.46595i	−0.680257	−	0.732974i	$-0.738132\pi$
$953$		−	45.2548i		−	1.46595i	0.680257	−	0.732974i	$-0.261868\pi$
$954$		0			0
$955$		−	25.4558i		−	0.823732i
$956$		0			0
$957$	50.9117	+	36.0000i	1.64574	+	1.16371i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−41.0000			−1.32258
$962$		0			0
$963$	−4.24264	−	12.0000i	−0.136717	−	0.386695i
$964$		0			0
$965$			8.00000i			0.257529i
$966$		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
$969$		0			0
$970$		0			0
$971$	18.0000			0.577647			0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000			0.577647			0.288824	+	0.957382i	$0.406736\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	1.41421	−	2.00000i	0.0452911	−	0.0640513i
$976$		0			0
$977$			25.4558i			0.814405i	0.913338	+	0.407202i	$0.133496\pi$
$977$			25.4558i			0.814405i	−0.913338	+	0.407202i	$0.866504\pi$
$978$		0			0
$979$			8.48528i			0.271191i
$980$		0			0
$981$	−12.0000	−	33.9411i	−0.383131	−	1.08366i
$982$		0			0
$983$	−48.0000			−1.53096			−0.765481	−	0.643458i	$-0.777499\pi$
$983$	−48.0000			−1.53096			−0.765481	+	0.643458i	$0.777499\pi$
$984$		0			0
$985$	−11.3137			−0.360485
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	50.9117i		−	1.61890i
$990$		0			0
$991$		−	36.0000i		−	1.14358i	−0.820401	−	0.571789i	$-0.806250\pi$
$991$		−	36.0000i		−	1.14358i	0.820401	−	0.571789i	$-0.193750\pi$
$992$		0			0
$993$	33.9411	+	24.0000i	1.07709	+	0.761617i
$994$		0			0
$995$	33.9411			1.07601
$996$		0			0
$997$	24.0416			0.761406			0.380703	−	0.924697i	$-0.375682\pi$
$997$	24.0416			0.761406			0.380703	+	0.924697i	$0.375682\pi$
$998$		0			0
$999$	−30.0000	−	8.48528i	−0.949158	−	0.268462i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.h.k.2255.2	yes	4
3.2	odd	2		2352.2.h.f.2255.1	✓	4
4.3	odd	2		2352.2.h.f.2255.4	yes	4
7.6	odd	2		2352.2.h.f.2255.3	yes	4
12.11	even	2	inner	2352.2.h.k.2255.3	yes	4
21.20	even	2	inner	2352.2.h.k.2255.4	yes	4
28.27	even	2	inner	2352.2.h.k.2255.1	yes	4
84.83	odd	2		2352.2.h.f.2255.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2352.2.h.f.2255.1	✓	4	3.2	odd	2
2352.2.h.f.2255.2	yes	4	84.83	odd	2
2352.2.h.f.2255.3	yes	4	7.6	odd	2
2352.2.h.f.2255.4	yes	4	4.3	odd	2
2352.2.h.k.2255.1	yes	4	28.27	even	2	inner
2352.2.h.k.2255.2	yes	4	1.1	even	1	trivial
2352.2.h.k.2255.3	yes	4	12.11	even	2	inner
2352.2.h.k.2255.4	yes	4	21.20	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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.41421i) q^{3} +2.00000i q^{5} +(-1.00000 - 2.82843i) q^{9} +O(q^{10})\)	\(q+(1.00000 - 1.41421i) q^{3} +2.00000i q^{5} +(-1.00000 - 2.82843i) q^{9} -4.24264 q^{11} +1.41421 q^{13} +(2.82843 + 2.00000i) q^{15} -2.00000i q^{17} -4.24264 q^{23} +1.00000 q^{25} +(-5.00000 - 1.41421i) q^{27} -8.48528i q^{29} -8.48528i q^{31} +(-4.24264 + 6.00000i) q^{33} +6.00000 q^{37} +(1.41421 - 2.00000i) q^{39} -10.0000i q^{41} +12.0000i q^{43} +(5.65685 - 2.00000i) q^{45} -6.00000 q^{47} +(-2.82843 - 2.00000i) q^{51} -5.65685i q^{53} -8.48528i q^{55} -12.0000 q^{59} +1.41421 q^{61} +2.82843i q^{65} -12.0000i q^{67} +(-4.24264 + 6.00000i) q^{69} +12.7279 q^{71} -9.89949 q^{73} +(1.00000 - 1.41421i) q^{75} -12.0000i q^{79} +(-7.00000 + 5.65685i) q^{81} -6.00000 q^{83} +4.00000 q^{85} +(-12.0000 - 8.48528i) q^{87} -2.00000i q^{89} +(-12.0000 - 8.48528i) q^{93} +7.07107 q^{97} +(4.24264 + 12.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 4 * q^9	\( 4 q + 4 q^{3} - 4 q^{9} + 4 q^{25} - 20 q^{27} + 24 q^{37} - 24 q^{47} - 48 q^{59} + 4 q^{75} - 28 q^{81} - 24 q^{83} + 16 q^{85} - 48 q^{87} - 48 q^{93}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^9 + 4 * q^25 - 20 * q^27 + 24 * q^37 - 24 * q^47 - 48 * q^59 + 4 * q^75 - 28 * q^81 - 24 * q^83 + 16 * q^85 - 48 * q^87 - 48 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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