LMFDB - Embedded newform 2352.2.h.p.2255.14

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 12x^{14} + 97x^{12} - 432x^{10} + 1392x^{8} - 2502x^{6} + 3181x^{4} - 1650x^{2} + 625 $ x^16 - 12x^14 + 97x^12 - 432x^10 + 1392x^8 - 2502x^6 + 3181x^4 - 1650*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2255.14
Root			$0.667172 + 0.385192i$ of defining polynomial
Character	$\chi$	$=$	2352.2255
Dual form			2352.2.h.p.2255.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.70166 - 0.323042i) q^{3} +1.55481i q^{5} +(2.79129 - 1.09941i) q^{9} +O(q^{10})$	$q+(1.70166 - 0.323042i) q^{3} +1.55481i q^{5} +(2.79129 - 1.09941i) q^{9} -2.53326 q^{13} +(0.502268 + 2.64575i) q^{15} +4.33991i q^{17} +6.83723i q^{19} -3.80848 q^{23} +2.58258 q^{25} +(4.39466 - 2.77253i) q^{27} +8.33639i q^{29} -4.89898i q^{31} +1.58258 q^{37} +(-4.31075 + 0.818350i) q^{39} +7.44953i q^{41} +6.20520i q^{43} +(1.70938 + 4.33991i) q^{45} -5.38601 q^{47} +(1.40197 + 7.38505i) q^{51} -10.5352i q^{53} +(2.20871 + 11.6346i) q^{57} +10.2100 q^{59} +8.19012 q^{61} -3.93874i q^{65} -3.46410i q^{67} +(-6.48074 + 1.23030i) q^{69} +14.4391 q^{71} -11.5473 q^{73} +(4.39466 - 0.834280i) q^{75} -6.92820i q^{79} +(6.58258 - 6.13756i) q^{81} +4.82395 q^{83} -6.74773 q^{85} +(2.69300 + 14.1857i) q^{87} +4.33991i q^{89} +(-1.58258 - 8.33639i) q^{93} -10.6306 q^{95} +9.89949 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{9}+O(q^{10}) $ 16 * q + 8 * q^9	$ 16 q + 8 q^{9} - 32 q^{25} - 48 q^{37} + 72 q^{57} + 32 q^{81} + 112 q^{85} + 48 q^{93}+O(q^{100}) $ 16 * q + 8 * q^9 - 32 * q^25 - 48 * q^37 + 72 * q^57 + 32 * q^81 + 112 * q^85 + 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
$3$	1.70166	−	0.323042i	0.982453	−	0.186508i
$3$	1.70166	−	0.323042i	0.982453	−	0.186508i
$4$		0			0
$5$			1.55481i			0.695331i	0.937619	+	0.347665i	$0.113025\pi$
$5$			1.55481i			0.695331i	−0.937619	+	0.347665i	$0.886975\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	2.79129	−	1.09941i	0.930429	−	0.366471i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$	−2.53326			−0.702601			−0.351300	−	0.936263i	$-0.614260\pi$
$13$	−2.53326			−0.702601			−0.351300	+	0.936263i	$0.614260\pi$
$14$		0			0
$15$	0.502268	+	2.64575i	0.129685	+	0.683130i
$16$		0			0
$17$			4.33991i			1.05258i	0.850304	+	0.526292i	$0.176418\pi$
$17$			4.33991i			1.05258i	−0.850304	+	0.526292i	$0.823582\pi$
$18$		0			0
$19$			6.83723i			1.56857i	0.620402	+	0.784284i	$0.286970\pi$
$19$			6.83723i			1.56857i	−0.620402	+	0.784284i	$0.713030\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.80848			−0.794124			−0.397062	−	0.917792i	$-0.629970\pi$
$23$	−3.80848			−0.794124			−0.397062	+	0.917792i	$0.629970\pi$
$24$		0			0
$25$	2.58258			0.516515
$26$		0			0
$27$	4.39466	−	2.77253i	0.845753	−	0.533574i
$28$		0			0
$29$			8.33639i			1.54803i	0.633168	+	0.774015i	$0.281754\pi$
$29$			8.33639i			1.54803i	−0.633168	+	0.774015i	$0.718246\pi$
$30$		0			0
$31$		−	4.89898i		−	0.879883i	−0.898027	−	0.439941i	$-0.854999\pi$
$31$		−	4.89898i		−	0.879883i	0.898027	−	0.439941i	$-0.145001\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.58258			0.260174			0.130087	−	0.991503i	$-0.458474\pi$
$37$	1.58258			0.260174			0.130087	+	0.991503i	$0.458474\pi$
$38$		0			0
$39$	−4.31075	+	0.818350i	−0.690273	+	0.131041i
$40$		0			0
$41$			7.44953i			1.16342i	0.813396	+	0.581710i	$0.197616\pi$
$41$			7.44953i			1.16342i	−0.813396	+	0.581710i	$0.802384\pi$
$42$		0			0
$43$			6.20520i			0.946285i	0.880986	+	0.473142i	$0.156880\pi$
$43$			6.20520i			0.946285i	−0.880986	+	0.473142i	$0.843120\pi$
$44$		0			0
$45$	1.70938	+	4.33991i	0.254819	+	0.646956i
$46$		0			0
$47$	−5.38601			−0.785630			−0.392815	−	0.919617i	$-0.628499\pi$
$47$	−5.38601			−0.785630			−0.392815	+	0.919617i	$0.628499\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	1.40197	+	7.38505i	0.196316	+	1.03411i
$52$		0			0
$53$		−	10.5352i		−	1.44712i	−0.690259	−	0.723562i	$-0.742504\pi$
$53$		−	10.5352i		−	1.44712i	0.690259	−	0.723562i	$-0.257496\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	2.20871	+	11.6346i	0.292551	+	1.54105i
$58$		0			0
$59$	10.2100			1.32922			0.664611	−	0.747189i	$-0.268597\pi$
$59$	10.2100			1.32922			0.664611	+	0.747189i	$0.268597\pi$
$60$		0			0
$61$	8.19012			1.04864			0.524319	−	0.851522i	$-0.324320\pi$
$61$	8.19012			1.04864			0.524319	+	0.851522i	$0.324320\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	3.93874i		−	0.488540i
$66$		0			0
$67$		−	3.46410i		−	0.423207i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$		−	3.46410i		−	0.423207i	0.977356	−	0.211604i	$-0.0678686\pi$
$68$		0			0
$69$	−6.48074	+	1.23030i	−0.780189	+	0.148111i
$70$		0			0
$71$	14.4391			1.71360			0.856800	−	0.515648i	$-0.172449\pi$
$71$	14.4391			1.71360			0.856800	+	0.515648i	$0.172449\pi$
$72$		0			0
$73$	−11.5473			−1.35151			−0.675753	−	0.737128i	$-0.736181\pi$
$73$	−11.5473			−1.35151			−0.675753	+	0.737128i	$0.736181\pi$
$74$		0			0
$75$	4.39466	−	0.834280i	0.507452	−	0.0963344i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	6.92820i		−	0.779484i	−0.920924	−	0.389742i	$-0.872564\pi$
$79$		−	6.92820i		−	0.779484i	0.920924	−	0.389742i	$-0.127436\pi$
$80$		0			0
$81$	6.58258	−	6.13756i	0.731397	−	0.681952i
$82$		0			0
$83$	4.82395			0.529497			0.264748	−	0.964318i	$-0.414711\pi$
$83$	4.82395			0.529497			0.264748	+	0.964318i	$0.414711\pi$
$84$		0			0
$85$	−6.74773			−0.731894
$86$		0			0
$87$	2.69300	+	14.1857i	0.288720	+	1.52087i
$88$		0			0
$89$			4.33991i			0.460030i	0.973187	+	0.230015i	$0.0738775\pi$
$89$			4.33991i			0.460030i	−0.973187	+	0.230015i	$0.926122\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−1.58258	−	8.33639i	−0.164105	−	0.864444i
$94$		0			0
$95$	−10.6306			−1.09067
$96$		0			0
$97$	9.89949			1.00514			0.502571	−	0.864536i	$-0.332388\pi$
$97$	9.89949			1.00514			0.502571	+	0.864536i	$0.332388\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			19.5635i			1.94664i	0.229455	+	0.973319i	$0.426306\pi$
$101$			19.5635i			1.94664i	−0.229455	+	0.973319i	$0.573694\pi$
$102$		0			0
$103$		−	3.87650i		−	0.381963i	−0.981594	−	0.190982i	$-0.938833\pi$
$103$		−	3.87650i		−	0.381963i	0.981594	−	0.190982i	$-0.0611671\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−14.4391			−1.39588			−0.697938	−	0.716158i	$-0.745899\pi$
$107$	−14.4391			−1.39588			−0.697938	+	0.716158i	$0.745899\pi$
$108$		0			0
$109$	−13.5826			−1.30097			−0.650487	−	0.759517i	$-0.725435\pi$
$109$	−13.5826			−1.30097			−0.650487	+	0.759517i	$0.725435\pi$
$110$		0			0
$111$	2.69300	−	0.511238i	0.255609	−	0.0485246i
$112$		0			0
$113$			14.4740i			1.36160i	0.732471	+	0.680798i	$0.238367\pi$
$113$			14.4740i			1.36160i	−0.732471	+	0.680798i	$0.761633\pi$
$114$		0			0
$115$		−	5.92146i		−	0.552179i
$116$		0			0
$117$	−7.07107	+	2.78511i	−0.653720	+	0.257483i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$	2.40651	+	12.6766i	0.216988	+	1.14301i
$124$		0			0
$125$			11.7894i			1.05448i
$126$		0			0
$127$		−	10.3923i		−	0.922168i	−0.887357	−	0.461084i	$-0.847461\pi$
$127$		−	10.3923i		−	0.922168i	0.887357	−	0.461084i	$-0.152539\pi$
$128$		0			0
$129$	2.00454	+	10.5591i	0.176490	+	0.929681i
$130$		0			0
$131$	15.5960			1.36263			0.681313	−	0.731992i	$-0.261409\pi$
$131$	15.5960			1.36263			0.681313	+	0.731992i	$0.261409\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	4.31075	+	6.83285i	0.371010	+	0.588078i
$136$		0			0
$137$		−	1.73991i		−	0.148650i	−0.997234	−	0.0743251i	$-0.976320\pi$
$137$		−	1.73991i		−	0.148650i	0.997234	−	0.0743251i	$-0.0236803\pi$
$138$		0			0
$139$			11.7362i			0.995452i	0.867334	+	0.497726i	$0.165832\pi$
$139$			11.7362i			0.995452i	−0.867334	+	0.497726i	$0.834168\pi$
$140$		0			0
$141$	−9.16515	+	1.73991i	−0.771845	+	0.146527i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−12.9615			−1.07639
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	10.5352i		−	0.863079i	−0.902094	−	0.431540i	$-0.857971\pi$
$149$		−	10.5352i		−	0.863079i	0.902094	−	0.431540i	$-0.142029\pi$
$150$		0			0
$151$		−	15.8745i		−	1.29185i	−0.763401	−	0.645925i	$-0.776472\pi$
$151$		−	15.8745i		−	1.29185i	0.763401	−	0.645925i	$-0.223528\pi$
$152$		0			0
$153$	4.77136	+	12.1139i	0.385742	+	0.979355i
$154$		0			0
$155$	7.61697			0.611809
$156$		0			0
$157$	−8.19012			−0.653643			−0.326821	−	0.945086i	$-0.605978\pi$
$157$	−8.19012			−0.653643			−0.326821	+	0.945086i	$0.605978\pi$
$158$		0			0
$159$	−3.40332	−	17.9274i	−0.269901	−	1.42173i
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	20.0616i		−	1.57135i	−0.618642	−	0.785673i	$-0.712317\pi$
$163$		−	20.0616i		−	1.57135i	0.618642	−	0.785673i	$-0.287683\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−6.58258			−0.506352
$170$		0			0
$171$	7.51695	+	19.0847i	0.574836	+	1.45944i
$172$		0			0
$173$			1.55481i			0.118210i	0.998252	+	0.0591049i	$0.0188246\pi$
$173$			1.55481i			0.118210i	−0.998252	+	0.0591049i	$0.981175\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	17.3739	−	3.29824i	1.30590	−	0.247911i
$178$		0			0
$179$	14.4391			1.07923			0.539613	−	0.841913i	$-0.318571\pi$
$179$	14.4391			1.07923			0.539613	+	0.841913i	$0.318571\pi$
$180$		0			0
$181$	−10.4282			−0.775123			−0.387562	−	0.921844i	$-0.626683\pi$
$181$	−10.4282			−0.775123			−0.387562	+	0.921844i	$0.626683\pi$
$182$		0			0
$183$	13.9368	−	2.64575i	1.03024	−	0.195580i
$184$		0			0
$185$			2.46060i			0.180907i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	22.0560			1.59592			0.797959	−	0.602712i	$-0.205913\pi$
$191$	22.0560			1.59592			0.797959	+	0.602712i	$0.205913\pi$
$192$		0			0
$193$	8.74773			0.629675			0.314838	−	0.949146i	$-0.398050\pi$
$193$	8.74773			0.629675			0.314838	+	0.949146i	$0.398050\pi$
$194$		0			0
$195$	−1.27238	−	6.70239i	−0.0911168	−	0.479968i
$196$		0			0
$197$			10.5352i			0.750604i	0.926903	+	0.375302i	$0.122461\pi$
$197$			10.5352i			0.750604i	−0.926903	+	0.375302i	$0.877539\pi$
$198$		0			0
$199$		−	9.79796i		−	0.694559i	−0.937762	−	0.347279i	$-0.887106\pi$
$199$		−	9.79796i		−	0.694559i	0.937762	−	0.347279i	$-0.112894\pi$
$200$		0			0
$201$	−1.11905	−	5.89472i	−0.0789317	−	0.415782i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−11.5826			−0.808962
$206$		0			0
$207$	−10.6306	+	4.18710i	−0.738876	+	0.291024i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	15.8745i		−	1.09285i	−0.837509	−	0.546423i	$-0.815989\pi$
$211$		−	15.8745i		−	1.09285i	0.837509	−	0.546423i	$-0.184011\pi$
$212$		0			0
$213$	24.5704	−	4.66442i	1.68353	−	0.319601i
$214$		0			0
$215$	−9.64789			−0.657981
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−19.6495	+	3.73025i	−1.32779	+	0.252067i
$220$		0			0
$221$		−	10.9941i		−	0.739546i
$222$		0			0
$223$			23.4724i			1.57183i	0.618335	+	0.785915i	$0.287808\pi$
$223$			23.4724i			1.57183i	−0.618335	+	0.785915i	$0.712192\pi$
$224$		0			0
$225$	7.20871	−	2.83932i	0.480581	−	0.189288i
$226$		0			0
$227$	4.82395			0.320177			0.160088	−	0.987103i	$-0.448822\pi$
$227$	4.82395			0.320177			0.160088	+	0.987103i	$0.448822\pi$
$228$		0			0
$229$	8.78044			0.580228			0.290114	−	0.956992i	$-0.406307\pi$
$229$	8.78044			0.580228			0.290114	+	0.956992i	$0.406307\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			1.73991i			0.113985i	0.998375	+	0.0569925i	$0.0181511\pi$
$233$			1.73991i			0.113985i	−0.998375	+	0.0569925i	$0.981849\pi$
$234$		0			0
$235$		−	8.37420i		−	0.546273i
$236$		0			0
$237$	−2.23810	−	11.7894i	−0.145380	−	0.765806i
$238$		0			0
$239$	−3.80848			−0.246350			−0.123175	−	0.992385i	$-0.539308\pi$
$239$	−3.80848			−0.246350			−0.123175	+	0.992385i	$0.539308\pi$
$240$		0			0
$241$	3.06199			0.197240			0.0986199	−	0.995125i	$-0.468557\pi$
$241$	3.06199			0.197240			0.0986199	+	0.995125i	$0.468557\pi$
$242$		0			0
$243$	9.21861	−	12.5705i	0.591374	−	0.806397i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	17.3205i		−	1.10208i
$248$		0			0
$249$	8.20871	−	1.55834i	0.520206	−	0.0987556i
$250$		0			0
$251$	0.562063			0.0354771			0.0177385	−	0.999843i	$-0.494353\pi$
$251$	0.562063			0.0354771			0.0177385	+	0.999843i	$0.494353\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−11.4823	+	2.17980i	−0.719052	+	0.136504i
$256$		0			0
$257$		−	13.6688i		−	0.852633i	−0.904574	−	0.426317i	$-0.859811\pi$
$257$		−	13.6688i		−	0.852633i	0.904574	−	0.426317i	$-0.140189\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	9.16515	+	23.2693i	0.567309	+	1.44033i
$262$		0			0
$263$	−29.6730			−1.82971			−0.914857	−	0.403777i	$-0.867697\pi$
$263$	−29.6730			−1.82971			−0.914857	+	0.403777i	$0.867697\pi$
$264$		0			0
$265$	16.3802			1.00623
$266$		0			0
$267$	1.40197	+	7.38505i	0.0857994	+	0.451958i
$268$		0			0
$269$		−	1.55481i		−	0.0947982i	−0.998876	−	0.0473991i	$-0.984907\pi$
$269$		−	1.55481i		−	0.0947982i	0.998876	−	0.0473991i	$-0.0150933\pi$
$270$		0			0
$271$			19.5959i			1.19037i	0.803590	+	0.595184i	$0.202921\pi$
$271$			19.5959i			1.19037i	−0.803590	+	0.595184i	$0.797079\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	2.00000			0.120168			0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000			0.120168			0.0600842	+	0.998193i	$0.480863\pi$
$278$		0			0
$279$	−5.38601	−	13.6745i	−0.322452	−	0.818669i
$280$		0			0
$281$		−	14.9329i		−	0.890821i	−0.895326	−	0.445410i	$-0.853058\pi$
$281$		−	14.9329i		−	0.890821i	0.895326	−	0.445410i	$-0.146942\pi$
$282$		0			0
$283$		−	26.4331i		−	1.57129i	−0.618679	−	0.785644i	$-0.712332\pi$
$283$		−	26.4331i		−	1.57129i	0.618679	−	0.785644i	$-0.287668\pi$
$284$		0			0
$285$	−18.0896	+	3.43412i	−1.07154	+	0.203420i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.83485			−0.107932
$290$		0			0
$291$	16.8456	−	3.19795i	0.987505	−	0.187467i
$292$		0			0
$293$		−	13.3442i		−	0.779579i	−0.920904	−	0.389790i	$-0.872548\pi$
$293$		−	13.3442i		−	0.779579i	0.920904	−	0.389790i	$-0.127452\pi$
$294$		0			0
$295$			15.8745i			0.924250i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	9.64789			0.557952
$300$		0			0
$301$		0			0
$302$		0			0
$303$	6.31982	+	33.2904i	0.363064	+	1.91248i
$304$		0			0
$305$			12.7341i			0.729150i
$306$		0			0
$307$		−	11.7362i		−	0.669821i	−0.942250	−	0.334910i	$-0.891294\pi$
$307$		−	11.7362i		−	0.669821i	0.942250	−	0.334910i	$-0.108706\pi$
$308$		0			0
$309$	−1.25227	−	6.59649i	−0.0712393	−	0.375261i
$310$		0			0
$311$	−15.0339			−0.852494			−0.426247	−	0.904607i	$-0.640165\pi$
$311$	−15.0339			−0.852494			−0.426247	+	0.904607i	$0.640165\pi$
$312$		0			0
$313$	5.89041			0.332946			0.166473	−	0.986046i	$-0.446762\pi$
$313$	5.89041			0.332946			0.166473	+	0.986046i	$0.446762\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	2.65775i		−	0.149274i	−0.997211	−	0.0746371i	$-0.976220\pi$
$317$		−	2.65775i		−	0.149274i	0.997211	−	0.0746371i	$-0.0237798\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−24.5704	+	4.66442i	−1.37138	+	0.260343i
$322$		0			0
$323$	−29.6730			−1.65105
$324$		0			0
$325$	−6.54234			−0.362904
$326$		0			0
$327$	−23.1129	+	4.38774i	−1.27815	+	0.242643i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	13.1334i		−	0.721877i	−0.932590	−	0.360939i	$-0.882456\pi$
$331$		−	13.1334i		−	0.721877i	0.932590	−	0.360939i	$-0.117544\pi$
$332$		0			0
$333$	4.41742	−	1.73991i	0.242073	−	0.0953463i
$334$		0			0
$335$	5.38601			0.294269
$336$		0			0
$337$	−1.25227			−0.0682157			−0.0341078	−	0.999418i	$-0.510859\pi$
$337$	−1.25227			−0.0682157			−0.0341078	+	0.999418i	$0.510859\pi$
$338$		0			0
$339$	4.67569	+	24.6297i	0.253949	+	1.33770i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−1.91288	−	10.0763i	−0.102986	−	0.542490i
$346$		0			0
$347$	36.4951			1.95916			0.979579	−	0.201058i	$-0.0644381\pi$
$347$	36.4951			1.95916			0.979579	+	0.201058i	$0.0644381\pi$
$348$		0			0
$349$	4.77136			0.255405			0.127703	−	0.991813i	$-0.459240\pi$
$349$	4.77136			0.255405			0.127703	+	0.991813i	$0.459240\pi$
$350$		0			0
$351$	−11.1328	+	7.02355i	−0.594227	+	0.374890i
$352$		0			0
$353$		−	27.9188i		−	1.48597i	−0.669309	−	0.742984i	$-0.733410\pi$
$353$		−	27.9188i		−	1.48597i	0.669309	−	0.742984i	$-0.266590\pi$
$354$		0			0
$355$			22.4499i			1.19152i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3.80848			0.201004			0.100502	−	0.994937i	$-0.467955\pi$
$359$	3.80848			0.201004			0.100502	+	0.994937i	$0.467955\pi$
$360$		0			0
$361$	−27.7477			−1.46041
$362$		0			0
$363$	−18.7183	+	3.55346i	−0.982453	+	0.186508i
$364$		0			0
$365$		−	17.9538i		−	0.939743i
$366$		0			0
$367$		−	23.4724i		−	1.22525i	−0.790374	−	0.612625i	$-0.790114\pi$
$367$		−	23.4724i		−	1.22525i	0.790374	−	0.612625i	$-0.209886\pi$
$368$		0			0
$369$	8.19012	+	20.7938i	0.426361	+	1.08248i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	10.0000			0.517780			0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000			0.517780			0.258890	+	0.965907i	$0.416643\pi$
$374$		0			0
$375$	3.80848	+	20.0616i	0.196669	+	1.03598i
$376$		0			0
$377$		−	21.1183i		−	1.08765i
$378$		0			0
$379$		−	6.20520i		−	0.318740i	−0.987219	−	0.159370i	$-0.949054\pi$
$379$		−	6.20520i		−	0.318740i	0.987219	−	0.159370i	$-0.0509463\pi$
$380$		0			0
$381$	−3.35715	−	17.6842i	−0.171992	−	0.905987i
$382$		0			0
$383$	−15.0339			−0.768196			−0.384098	−	0.923292i	$-0.625488\pi$
$383$	−15.0339			−0.768196			−0.384098	+	0.923292i	$0.625488\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	6.82209	+	17.3205i	0.346786	+	0.880451i
$388$		0			0
$389$			38.2022i			1.93693i	0.249157	+	0.968463i	$0.419846\pi$
$389$			38.2022i			1.93693i	−0.249157	+	0.968463i	$0.580154\pi$
$390$		0			0
$391$		−	16.5285i		−	0.835882i
$392$		0			0
$393$	26.5390	−	5.03815i	1.33872	−	0.254141i
$394$		0			0
$395$	10.7720			0.541999
$396$		0			0
$397$	38.1222			1.91330			0.956648	−	0.291246i	$-0.0940698\pi$
$397$	38.1222			1.91330			0.956648	+	0.291246i	$0.0940698\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			21.5294i			1.07513i	0.843224	+	0.537563i	$0.180655\pi$
$401$			21.5294i			1.07513i	−0.843224	+	0.537563i	$0.819345\pi$
$402$		0			0
$403$			12.4104i			0.618206i
$404$		0			0
$405$	9.54273	+	10.2346i	0.474182	+	0.508563i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−17.7944			−0.879879			−0.439939	−	0.898027i	$-0.645000\pi$
$409$	−17.7944			−0.879879			−0.439939	+	0.898027i	$0.645000\pi$
$410$		0			0
$411$	−0.562063	−	2.96073i	−0.0277245	−	0.146042i
$412$		0			0
$413$		0			0
$414$		0			0
$415$			7.50030i			0.368175i
$416$		0			0
$417$	3.79129	+	19.9710i	0.185660	+	0.977986i
$418$		0			0
$419$	15.5960			0.761913			0.380956	−	0.924593i	$-0.375595\pi$
$419$	15.5960			0.761913			0.380956	+	0.924593i	$0.375595\pi$
$420$		0			0
$421$	35.4955			1.72994			0.864971	−	0.501821i	$-0.167337\pi$
$421$	35.4955			1.72994			0.864971	+	0.501821i	$0.167337\pi$
$422$		0			0
$423$	−15.0339	+	5.92146i	−0.730973	+	0.287911i
$424$		0			0
$425$			11.2082i			0.543675i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−17.4527			−0.840665			−0.420333	−	0.907370i	$-0.638087\pi$
$431$	−17.4527			−0.840665			−0.420333	+	0.907370i	$0.638087\pi$
$432$		0			0
$433$	0.823886			0.0395935			0.0197967	−	0.999804i	$-0.493698\pi$
$433$	0.823886			0.0395935			0.0197967	+	0.999804i	$0.493698\pi$
$434$		0			0
$435$	−22.0560	+	4.18710i	−1.05751	+	0.200756i
$436$		0			0
$437$		−	26.0395i		−	1.24564i
$438$		0			0
$439$			14.6969i			0.701447i	0.936479	+	0.350723i	$0.114064\pi$
$439$			14.6969i			0.701447i	−0.936479	+	0.350723i	$0.885936\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6.82209			−0.324127			−0.162064	−	0.986780i	$-0.551815\pi$
$443$	−6.82209			−0.324127			−0.162064	+	0.986780i	$0.551815\pi$
$444$		0			0
$445$	−6.74773			−0.319873
$446$		0			0
$447$	−3.40332	−	17.9274i	−0.160971	−	0.847935i
$448$		0			0
$449$		−	28.9479i		−	1.36614i	−0.730355	−	0.683068i	$-0.760645\pi$
$449$		−	28.9479i		−	1.36614i	0.730355	−	0.683068i	$-0.239355\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−5.12813	−	27.0130i	−0.240941	−	1.26918i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−24.7477			−1.15765			−0.578825	−	0.815452i	$-0.696489\pi$
$457$	−24.7477			−1.15765			−0.578825	+	0.815452i	$0.696489\pi$
$458$		0			0
$459$	12.0325	+	19.0725i	0.561631	+	0.890226i
$460$		0			0
$461$		−	10.2346i		−	0.476674i	−0.971182	−	0.238337i	$-0.923398\pi$
$461$		−	10.2346i		−	0.476674i	0.971182	−	0.238337i	$-0.0766023\pi$
$462$		0			0
$463$			26.2668i			1.22072i	0.792123	+	0.610361i	$0.208976\pi$
$463$			26.2668i			1.22072i	−0.792123	+	0.610361i	$0.791024\pi$
$464$		0			0
$465$	12.9615	−	2.46060i	0.601074	−	0.114108i
$466$		0			0
$467$	25.2439			1.16815			0.584073	−	0.811701i	$-0.301458\pi$
$467$	25.2439			1.16815			0.584073	+	0.811701i	$0.301458\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−13.9368	+	2.64575i	−0.642173	+	0.121910i
$472$		0			0
$473$		0			0
$474$		0			0
$475$			17.6577i			0.810189i
$476$		0			0
$477$	−11.5826	−	29.4068i	−0.530330	−	1.34645i
$478$		0			0
$479$	−31.1919			−1.42520			−0.712598	−	0.701573i	$-0.752482\pi$
$479$	−31.1919			−1.42520			−0.712598	+	0.701573i	$0.752482\pi$
$480$		0			0
$481$	−4.00908			−0.182798
$482$		0			0
$483$		0			0
$484$		0			0
$485$			15.3918i			0.698906i
$486$		0			0
$487$		−	36.6591i		−	1.66118i	−0.556882	−	0.830592i	$-0.688002\pi$
$487$		−	36.6591i		−	1.66118i	0.556882	−	0.830592i	$-0.311998\pi$
$488$		0			0
$489$	−6.48074	−	34.1380i	−0.293069	−	1.54377i
$490$		0			0
$491$	−22.0560			−0.995374			−0.497687	−	0.867357i	$-0.665817\pi$
$491$	−22.0560			−0.995374			−0.497687	+	0.867357i	$0.665817\pi$
$492$		0			0
$493$	−36.1792			−1.62943
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$			8.94630i			0.400492i	0.979746	+	0.200246i	$0.0641741\pi$
$499$			8.94630i			0.400492i	−0.979746	+	0.200246i	$0.935826\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−31.1919			−1.39078			−0.695390	−	0.718633i	$-0.744768\pi$
$503$	−31.1919			−1.39078			−0.695390	+	0.718633i	$0.744768\pi$
$504$		0			0
$505$	−30.4174			−1.35356
$506$		0			0
$507$	−11.2013	+	2.12645i	−0.497467	+	0.0944389i
$508$		0			0
$509$			0.905793i			0.0401486i	0.999798	+	0.0200743i	$0.00639027\pi$
$509$			0.905793i			0.0401486i	−0.999798	+	0.0200743i	$0.993610\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	18.9564	+	30.0473i	0.836947	+	1.32662i
$514$		0			0
$515$	6.02721			0.265591
$516$		0			0
$517$		0			0
$518$		0			0
$519$	0.502268	+	2.64575i	0.0220471	+	0.116136i
$520$		0			0
$521$		−	27.9188i		−	1.22314i	−0.791189	−	0.611572i	$-0.790537\pi$
$521$		−	27.9188i		−	1.22314i	0.791189	−	0.611572i	$-0.209463\pi$
$522$		0			0
$523$		−	0.915775i		−	0.0400440i	−0.999800	−	0.0200220i	$-0.993626\pi$
$523$		−	0.915775i		−	0.0400440i	0.999800	−	0.0200220i	$-0.00637363\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	21.2611			0.926150
$528$		0			0
$529$	−8.49545			−0.369368
$530$		0			0
$531$	28.4989	−	11.2250i	1.23675	−	0.487122i
$532$		0			0
$533$		−	18.8716i		−	0.817420i
$534$		0			0
$535$		−	22.4499i		−	0.970596i
$536$		0			0
$537$	24.5704	−	4.66442i	1.06029	−	0.201285i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−23.4955			−1.01015			−0.505074	−	0.863076i	$-0.668535\pi$
$541$	−23.4955			−1.01015			−0.505074	+	0.863076i	$0.668535\pi$
$542$		0			0
$543$	−17.7453	+	3.36875i	−0.761523	+	0.144567i
$544$		0			0
$545$		−	21.1183i		−	0.904608i
$546$		0			0
$547$		−	11.6874i		−	0.499717i	−0.968282	−	0.249859i	$-0.919616\pi$
$547$		−	11.6874i		−	0.499717i	0.968282	−	0.249859i	$-0.0803842\pi$
$548$		0			0
$549$	22.8610	−	9.00433i	0.975683	−	0.384296i
$550$		0			0
$551$	−56.9978			−2.42819
$552$		0			0
$553$		0			0
$554$		0			0
$555$	0.794877	+	4.18710i	0.0337406	+	0.177733i
$556$		0			0
$557$		−	2.65775i		−	0.112613i	−0.998414	−	0.0563063i	$-0.982068\pi$
$557$		−	2.65775i		−	0.112613i	0.998414	−	0.0563063i	$-0.0179323\pi$
$558$		0			0
$559$		−	15.7194i		−	0.664860i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	0.562063			0.0236881			0.0118441	−	0.999930i	$-0.496230\pi$
$563$	0.562063			0.0236881			0.0118441	+	0.999930i	$0.496230\pi$
$564$		0			0
$565$	−22.5042			−0.946759
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	4.85658i		−	0.203598i	−0.994805	−	0.101799i	$-0.967540\pi$
$569$		−	4.85658i		−	0.203598i	0.994805	−	0.101799i	$-0.0324599\pi$
$570$		0			0
$571$		−	0.723000i		−	0.0302566i	−0.999886	−	0.0151283i	$-0.995184\pi$
$571$		−	0.723000i		−	0.0302566i	0.999886	−	0.0151283i	$-0.00481567\pi$
$572$		0			0
$573$	37.5318	−	7.12502i	1.56791	−	0.297652i
$574$		0			0
$575$	−9.83570			−0.410177
$576$		0			0
$577$	43.7174			1.81998			0.909990	−	0.414631i	$-0.136089\pi$
$577$	43.7174			1.81998			0.909990	+	0.414631i	$0.136089\pi$
$578$		0			0
$579$	14.8856	−	2.82588i	0.618627	−	0.117440i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−4.33030	−	10.9941i	−0.179036	−	0.454552i
$586$		0			0
$587$	−36.0159			−1.48653			−0.743267	−	0.668995i	$-0.766725\pi$
$587$	−36.0159			−1.48653			−0.743267	+	0.668995i	$0.766725\pi$
$588$		0			0
$589$	33.4955			1.38016
$590$		0			0
$591$	3.40332	+	17.9274i	0.139994	+	0.737433i
$592$		0			0
$593$		−	28.5678i		−	1.17314i	−0.809899	−	0.586570i	$-0.800478\pi$
$593$		−	28.5678i		−	1.17314i	0.809899	−	0.586570i	$-0.199522\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−3.16515	−	16.6728i	−0.129541	−	0.682372i
$598$		0			0
$599$	29.6730			1.21241			0.606203	−	0.795310i	$-0.292692\pi$
$599$	29.6730			1.21241			0.606203	+	0.795310i	$0.292692\pi$
$600$		0			0
$601$	−0.823886			−0.0336070			−0.0168035	−	0.999859i	$-0.505349\pi$
$601$	−0.823886			−0.0336070			−0.0168035	+	0.999859i	$0.505349\pi$
$602$		0			0
$603$	−3.80848	−	9.66930i	−0.155093	−	0.393765i
$604$		0			0
$605$		−	17.1029i		−	0.695331i
$606$		0			0
$607$		−	18.5734i		−	0.753873i	−0.926239	−	0.376936i	$-0.876978\pi$
$607$		−	18.5734i		−	0.753873i	0.926239	−	0.376936i	$-0.123022\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	13.6442			0.551984
$612$		0			0
$613$	38.2432			1.54463			0.772314	−	0.635241i	$-0.219099\pi$
$613$	38.2432			1.54463			0.772314	+	0.635241i	$0.219099\pi$
$614$		0			0
$615$	−19.7096	+	3.74166i	−0.794768	+	0.150878i
$616$		0			0
$617$		−	21.5294i		−	0.866740i	−0.901216	−	0.433370i	$-0.857324\pi$
$617$		−	21.5294i		−	0.866740i	0.901216	−	0.433370i	$-0.142676\pi$
$618$		0			0
$619$		−	7.85971i		−	0.315908i	−0.987446	−	0.157954i	$-0.949510\pi$
$619$		−	7.85971i		−	0.315908i	0.987446	−	0.157954i	$-0.0504898\pi$
$620$		0			0
$621$	−16.7370	+	10.5591i	−0.671633	+	0.423724i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−5.41742			−0.216697
$626$		0			0
$627$		0			0
$628$		0			0
$629$			6.86824i			0.273855i
$630$		0			0
$631$		−	40.1232i		−	1.59728i	−0.601809	−	0.798640i	$-0.705553\pi$
$631$		−	40.1232i		−	1.59728i	0.601809	−	0.798640i	$-0.294447\pi$
$632$		0			0
$633$	−5.12813	−	27.0130i	−0.203825	−	1.07367i
$634$		0			0
$635$	16.1580			0.641212
$636$		0			0
$637$		0			0
$638$		0			0
$639$	40.3036	−	15.8745i	1.59438	−	0.627986i
$640$		0			0
$641$		−	4.85658i		−	0.191823i	−0.995390	−	0.0959117i	$-0.969423\pi$
$641$		−	4.85658i		−	0.191823i	0.995390	−	0.0959117i	$-0.0305766\pi$
$642$		0			0
$643$			39.0851i			1.54137i	0.637218	+	0.770684i	$0.280085\pi$
$643$			39.0851i			1.54137i	−0.637218	+	0.770684i	$0.719915\pi$
$644$		0			0
$645$	−16.4174	+	3.11667i	−0.646435	+	0.122719i
$646$		0			0
$647$	−25.8059			−1.01454			−0.507268	−	0.861789i	$-0.669344\pi$
$647$	−25.8059			−1.01454			−0.507268	+	0.861789i	$0.669344\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$			25.0092i			0.978685i	0.872092	+	0.489342i	$0.162763\pi$
$653$			25.0092i			0.978685i	−0.872092	+	0.489342i	$0.837237\pi$
$654$		0			0
$655$			24.2487i			0.947476i
$656$		0			0
$657$	−32.2317	+	12.6952i	−1.25748	+	0.495288i
$658$		0			0
$659$	−13.6442			−0.531502			−0.265751	−	0.964042i	$-0.585620\pi$
$659$	−13.6442			−0.531502			−0.265751	+	0.964042i	$0.585620\pi$
$660$		0			0
$661$	9.24756			0.359689			0.179844	−	0.983695i	$-0.442441\pi$
$661$	9.24756			0.359689			0.179844	+	0.983695i	$0.442441\pi$
$662$		0			0
$663$	−3.55157	−	18.7083i	−0.137932	−	0.726570i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	31.7490i		−	1.22933i
$668$		0			0
$669$	7.58258	+	39.9421i	0.293159	+	1.54425i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−15.1652			−0.584574			−0.292287	−	0.956331i	$-0.594416\pi$
$673$	−15.1652			−0.584574			−0.292287	+	0.956331i	$0.594416\pi$
$674$		0			0
$675$	11.3496	−	7.16027i	0.436844	−	0.275599i
$676$		0			0
$677$		−	0.905793i		−	0.0348124i	−0.999849	−	0.0174062i	$-0.994459\pi$
$677$		−	0.905793i		−	0.0348124i	0.999849	−	0.0174062i	$-0.00554085\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	8.20871	−	1.55834i	0.314559	−	0.0597156i
$682$		0			0
$683$	22.0560			0.843950			0.421975	−	0.906607i	$-0.361337\pi$
$683$	22.0560			0.843950			0.421975	+	0.906607i	$0.361337\pi$
$684$		0			0
$685$	2.70522			0.103361
$686$		0			0
$687$	14.9413	−	2.83645i	0.570047	−	0.108217i
$688$		0			0
$689$			26.6885i			1.01675i
$690$		0			0
$691$		−	12.7587i		−	0.485363i	−0.970106	−	0.242682i	$-0.921973\pi$
$691$		−	12.7587i		−	0.485363i	0.970106	−	0.242682i	$-0.0780270\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−18.2475			−0.692169
$696$		0			0
$697$	−32.3303			−1.22460
$698$		0			0
$699$	0.562063	+	2.96073i	0.0212592	+	0.111985i
$700$		0			0
$701$		−	4.85658i		−	0.183431i	−0.995785	−	0.0917153i	$-0.970765\pi$
$701$		−	4.85658i		−	0.183431i	0.995785	−	0.0917153i	$-0.0292350\pi$
$702$		0			0
$703$			10.8204i			0.408100i
$704$		0			0
$705$	−2.70522	−	14.2500i	−0.101884	−	0.536688i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	47.0780			1.76805			0.884026	−	0.467437i	$-0.154823\pi$
$709$	47.0780			1.76805			0.884026	+	0.467437i	$0.154823\pi$
$710$		0			0
$711$	−7.61697	−	19.3386i	−0.285659	−	0.725255i
$712$		0			0
$713$			18.6577i			0.698736i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−6.48074	+	1.23030i	−0.242028	+	0.0459464i
$718$		0			0
$719$	25.8059			0.962398			0.481199	−	0.876611i	$-0.340201\pi$
$719$	25.8059			0.962398			0.481199	+	0.876611i	$0.340201\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	5.21046	−	0.989150i	0.193779	−	0.0367869i
$724$		0			0
$725$			21.5294i			0.799581i
$726$		0			0
$727$			20.6184i			0.764694i	0.924019	+	0.382347i	$0.124884\pi$
$727$			20.6184i			0.764694i	−0.924019	+	0.382347i	$0.875116\pi$
$728$		0			0
$729$	11.6261	−	24.3687i	0.430598	−	0.902544i
$730$		0			0
$731$	−26.9300			−0.996044
$732$		0			0
$733$	−1.94294			−0.0717640			−0.0358820	−	0.999356i	$-0.511424\pi$
$733$	−1.94294			−0.0717640			−0.0358820	+	0.999356i	$0.511424\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	40.8462i		−	1.50255i	−0.659988	−	0.751276i	$-0.729439\pi$
$739$		−	40.8462i		−	1.50255i	0.659988	−	0.751276i	$-0.270561\pi$
$740$		0			0
$741$	−5.59525	−	29.4736i	−0.205547	−	1.08274i
$742$		0			0
$743$	25.0696			0.919716			0.459858	−	0.887993i	$-0.347900\pi$
$743$	25.0696			0.919716			0.459858	+	0.887993i	$0.347900\pi$
$744$		0			0
$745$	16.3802			0.600125
$746$		0			0
$747$	13.4650	−	5.30352i	0.492659	−	0.194046i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	22.8027i		−	0.832083i	−0.909346	−	0.416041i	$-0.863417\pi$
$751$		−	22.8027i		−	0.832083i	0.909346	−	0.416041i	$-0.136583\pi$
$752$		0			0
$753$	0.956439	−	0.181570i	0.0348546	−	0.00661677i
$754$		0			0
$755$	24.6818			0.898262
$756$		0			0
$757$	−4.74773			−0.172559			−0.0862795	−	0.996271i	$-0.527498\pi$
$757$	−4.74773			−0.172559			−0.0862795	+	0.996271i	$0.527498\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			13.6688i			0.495492i	0.968825	+	0.247746i	$0.0796898\pi$
$761$			13.6688i			0.495492i	−0.968825	+	0.247746i	$0.920310\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−18.8348	+	7.41855i	−0.680975	+	0.268218i
$766$		0			0
$767$	−25.8645			−0.933913
$768$		0			0
$769$	11.0801			0.399560			0.199780	−	0.979841i	$-0.435977\pi$
$769$	11.0801			0.399560			0.199780	+	0.979841i	$0.435977\pi$
$770$		0			0
$771$	−4.41558	−	23.2596i	−0.159023	−	0.837673i
$772$		0			0
$773$			1.55481i			0.0559225i	0.999609	+	0.0279613i	$0.00890150\pi$
$773$			1.55481i			0.0559225i	−0.999609	+	0.0279613i	$0.991098\pi$
$774$		0			0
$775$		−	12.6520i		−	0.454473i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−50.9341			−1.82490
$780$		0			0
$781$		0			0
$782$		0			0
$783$	23.1129	+	36.6356i	0.825988	+	1.30925i
$784$		0			0
$785$		−	12.7341i		−	0.454498i
$786$		0			0
$787$			29.2872i			1.04398i	0.852953	+	0.521988i	$0.174809\pi$
$787$			29.2872i			1.04398i	−0.852953	+	0.521988i	$0.825191\pi$
$788$		0			0
$789$	−50.4933	+	9.58562i	−1.79761	+	0.341257i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−20.7477			−0.736773
$794$		0			0
$795$	27.8736	−	5.29150i	0.988574	−	0.187670i
$796$		0			0
$797$			7.77403i			0.275370i	0.990476	+	0.137685i	$0.0439662\pi$
$797$			7.77403i			0.275370i	−0.990476	+	0.137685i	$0.956034\pi$
$798$		0			0
$799$		−	23.3748i		−	0.826941i
$800$		0			0
$801$	4.77136	+	12.1139i	0.168588	+	0.428025i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−0.502268	−	2.64575i	−0.0176807	−	0.0931349i
$808$		0			0
$809$			22.3514i			0.785834i	0.919574	+	0.392917i	$0.128534\pi$
$809$			22.3514i			0.785834i	−0.919574	+	0.392917i	$0.871466\pi$
$810$		0			0
$811$		−	54.8045i		−	1.92445i	−0.272259	−	0.962224i	$-0.587771\pi$
$811$		−	54.8045i		−	1.92445i	0.272259	−	0.962224i	$-0.412229\pi$
$812$		0			0
$813$	6.33030	+	33.3456i	0.222013	+	1.16948i
$814$		0			0
$815$	31.1919			1.09261
$816$		0			0
$817$	−42.4264			−1.48431
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	2.65775i		−	0.0927562i	−0.998924	−	0.0463781i	$-0.985232\pi$
$821$		−	2.65775i		−	0.0927562i	0.998924	−	0.0463781i	$-0.0147679\pi$
$822$		0			0
$823$			40.1232i			1.39861i	0.714825	+	0.699304i	$0.246507\pi$
$823$			40.1232i			1.39861i	−0.714825	+	0.699304i	$0.753493\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	22.0560			0.766963			0.383482	−	0.923549i	$-0.374725\pi$
$827$	22.0560			0.766963			0.383482	+	0.923549i	$0.374725\pi$
$828$		0			0
$829$	3.71392			0.128990			0.0644948	−	0.997918i	$-0.479456\pi$
$829$	3.71392			0.128990			0.0644948	+	0.997918i	$0.479456\pi$
$830$		0			0
$831$	3.40332	−	0.646084i	0.118060	−	0.0224124i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−13.5826	−	21.5294i	−0.469482	−	0.744164i
$838$		0			0
$839$	−25.8059			−0.890919			−0.445460	−	0.895302i	$-0.646960\pi$
$839$	−25.8059			−0.890919			−0.445460	+	0.895302i	$0.646960\pi$
$840$		0			0
$841$	−40.4955			−1.39639
$842$		0			0
$843$	−4.82395	−	25.4107i	−0.166146	−	0.875190i
$844$		0			0
$845$		−	10.2346i		−	0.352082i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−8.53901	−	44.9802i	−0.293058	−	1.54372i
$850$		0			0
$851$	−6.02721			−0.206610
$852$		0			0
$853$	30.2272			1.03496			0.517480	−	0.855695i	$-0.326870\pi$
$853$	30.2272			1.03496			0.517480	+	0.855695i	$0.326870\pi$
$854$		0			0
$855$	−29.6730	+	11.6874i	−1.01479	+	0.399701i
$856$		0			0
$857$			22.9976i			0.785583i	0.919628	+	0.392791i	$0.128491\pi$
$857$			22.9976i			0.785583i	−0.919628	+	0.392791i	$0.871509\pi$
$858$		0			0
$859$			5.81475i			0.198397i	0.995068	+	0.0991984i	$0.0316279\pi$
$859$			5.81475i			0.198397i	−0.995068	+	0.0991984i	$0.968372\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	35.7002			1.21525			0.607625	−	0.794224i	$-0.292123\pi$
$863$	35.7002			1.21525			0.607625	+	0.794224i	$0.292123\pi$
$864$		0			0
$865$	−2.41742			−0.0821949
$866$		0			0
$867$	−3.12229	+	0.592733i	−0.106038	+	0.0201303i
$868$		0			0
$869$		0			0
$870$		0			0
$871$			8.77548i			0.297346i
$872$		0			0
$873$	27.6323	−	10.8836i	0.935213	−	0.368356i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	10.4174			0.351771			0.175886	−	0.984411i	$-0.443721\pi$
$877$	10.4174			0.351771			0.175886	+	0.984411i	$0.443721\pi$
$878$		0			0
$879$	−4.31075	−	22.7074i	−0.145398	−	0.765900i
$880$		0			0
$881$		−	16.7784i		−	0.565278i	−0.959226	−	0.282639i	$-0.908790\pi$
$881$		−	16.7784i		−	0.565278i	0.959226	−	0.282639i	$-0.0912098\pi$
$882$		0			0
$883$			21.3567i			0.718711i	0.933201	+	0.359355i	$0.117003\pi$
$883$			21.3567i			0.718711i	−0.933201	+	0.359355i	$0.882997\pi$
$884$		0			0
$885$	5.12813	+	27.0130i	0.172380	+	0.908032i
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	36.8254i		−	1.23231i
$894$		0			0
$895$			22.4499i			0.750419i
$896$		0			0
$897$	16.4174	−	3.11667i	0.548162	−	0.104063i
$898$		0			0
$899$	40.8398			1.36208
$900$		0			0
$901$	45.7220			1.52322
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	16.2139i		−	0.538967i
$906$		0			0
$907$			43.5873i			1.44729i	0.690171	+	0.723647i	$0.257535\pi$
$907$			43.5873i			1.44729i	−0.690171	+	0.723647i	$0.742465\pi$
$908$		0			0
$909$	21.5084	+	54.6073i	0.713388	+	1.81121i
$910$		0			0
$911$	25.0696			0.830594			0.415297	−	0.909686i	$-0.363678\pi$
$911$	25.0696			0.830594			0.415297	+	0.909686i	$0.363678\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	4.11363	+	21.6690i	0.135992	+	0.716356i
$916$		0			0
$917$		0			0
$918$		0			0
$919$			15.3024i			0.504780i	0.967626	+	0.252390i	$0.0812166\pi$
$919$			15.3024i			0.504780i	−0.967626	+	0.252390i	$0.918783\pi$
$920$		0			0
$921$	−3.79129	−	19.9710i	−0.124927	−	0.658068i
$922$		0			0
$923$	−36.5779			−1.20398
$924$		0			0
$925$	4.08712			0.134384
$926$		0			0
$927$	−4.26188	−	10.8204i	−0.139979	−	0.355390i
$928$		0			0
$929$			51.4977i			1.68958i	0.535095	+	0.844792i	$0.320276\pi$
$929$			51.4977i			1.68958i	−0.535095	+	0.844792i	$0.679724\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−25.5826	+	4.85658i	−0.837536	+	0.158997i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−37.0031			−1.20884			−0.604419	−	0.796666i	$-0.706595\pi$
$937$	−37.0031			−1.20884			−0.604419	+	0.796666i	$0.706595\pi$
$938$		0			0
$939$	10.0235	−	1.90285i	0.327104	−	0.0620972i
$940$		0			0
$941$		−	22.6731i		−	0.739122i	−0.929207	−	0.369561i	$-0.879508\pi$
$941$		−	22.6731i		−	0.739122i	0.929207	−	0.369561i	$-0.120492\pi$
$942$		0			0
$943$		−	28.3714i		−	0.923900i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	22.8509			0.742554			0.371277	−	0.928522i	$-0.378920\pi$
$947$	22.8509			0.742554			0.371277	+	0.928522i	$0.378920\pi$
$948$		0			0
$949$	29.2523			0.949569
$950$		0			0
$951$	−0.858565	−	4.52259i	−0.0278409	−	0.146655i
$952$		0			0
$953$			55.3339i			1.79244i	0.443610	+	0.896220i	$0.353697\pi$
$953$			55.3339i			1.79244i	−0.443610	+	0.896220i	$0.646303\pi$
$954$		0			0
$955$			34.2929i			1.10969i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	7.00000			0.225806
$962$		0			0
$963$	−40.3036	+	15.8745i	−1.29876	+	0.511549i
$964$		0			0
$965$			13.6010i			0.437833i
$966$		0			0
$967$			2.01810i			0.0648978i	0.999473	+	0.0324489i	$0.0103306\pi$
$967$			2.01810i			0.0648978i	−0.999473	+	0.0324489i	$0.989669\pi$
$968$		0			0
$969$	−50.4933	+	9.58562i	−1.62208	+	0.307934i
$970$		0			0
$971$	−36.0159			−1.15580			−0.577902	−	0.816106i	$-0.696128\pi$
$971$	−36.0159			−1.15580			−0.577902	+	0.816106i	$0.696128\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−11.1328	+	2.11345i	−0.356536	+	0.0676846i
$976$		0			0
$977$		−	44.7986i		−	1.43324i	−0.697466	−	0.716618i	$-0.745689\pi$
$977$		−	44.7986i		−	1.43324i	0.697466	−	0.716618i	$-0.254311\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−37.9129	+	14.9329i	−1.21047	+	0.476770i
$982$		0			0
$983$	5.38601			0.171787			0.0858935	−	0.996304i	$-0.472626\pi$
$983$	5.38601			0.171787			0.0858935	+	0.996304i	$0.472626\pi$
$984$		0			0
$985$	−16.3802			−0.521918
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	23.6324i		−	0.751467i
$990$		0			0
$991$		−	59.4618i		−	1.88887i	−0.328702	−	0.944434i	$-0.606611\pi$
$991$		−	59.4618i		−	1.88887i	0.328702	−	0.944434i	$-0.393389\pi$
$992$		0			0
$993$	−4.24264	−	22.3486i	−0.134636	−	0.709211i
$994$		0			0
$995$	15.2339			0.482948
$996$		0			0
$997$	−38.5893			−1.22213			−0.611067	−	0.791579i	$-0.709260\pi$
$997$	−38.5893			−1.22213			−0.611067	+	0.791579i	$0.709260\pi$
$998$		0			0
$999$	6.95489	−	4.38774i	0.220043	−	0.138822i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.h.p.2255.14	yes	16
3.2	odd	2	inner	2352.2.h.p.2255.1	✓	16
4.3	odd	2	inner	2352.2.h.p.2255.4	yes	16
7.6	odd	2	inner	2352.2.h.p.2255.3	yes	16
12.11	even	2	inner	2352.2.h.p.2255.15	yes	16
21.20	even	2	inner	2352.2.h.p.2255.16	yes	16
28.27	even	2	inner	2352.2.h.p.2255.13	yes	16
84.83	odd	2	inner	2352.2.h.p.2255.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2352.2.h.p.2255.1	✓	16	3.2	odd	2	inner
2352.2.h.p.2255.2	yes	16	84.83	odd	2	inner
2352.2.h.p.2255.3	yes	16	7.6	odd	2	inner
2352.2.h.p.2255.4	yes	16	4.3	odd	2	inner
2352.2.h.p.2255.13	yes	16	28.27	even	2	inner
2352.2.h.p.2255.14	yes	16	1.1	even	1	trivial
2352.2.h.p.2255.15	yes	16	12.11	even	2	inner
2352.2.h.p.2255.16	yes	16	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)

\(f(q)\)	\(=\)	\(q+(1.70166 - 0.323042i) q^{3} +1.55481i q^{5} +(2.79129 - 1.09941i) q^{9} +O(q^{10})\)	\(q+(1.70166 - 0.323042i) q^{3} +1.55481i q^{5} +(2.79129 - 1.09941i) q^{9} -2.53326 q^{13} +(0.502268 + 2.64575i) q^{15} +4.33991i q^{17} +6.83723i q^{19} -3.80848 q^{23} +2.58258 q^{25} +(4.39466 - 2.77253i) q^{27} +8.33639i q^{29} -4.89898i q^{31} +1.58258 q^{37} +(-4.31075 + 0.818350i) q^{39} +7.44953i q^{41} +6.20520i q^{43} +(1.70938 + 4.33991i) q^{45} -5.38601 q^{47} +(1.40197 + 7.38505i) q^{51} -10.5352i q^{53} +(2.20871 + 11.6346i) q^{57} +10.2100 q^{59} +8.19012 q^{61} -3.93874i q^{65} -3.46410i q^{67} +(-6.48074 + 1.23030i) q^{69} +14.4391 q^{71} -11.5473 q^{73} +(4.39466 - 0.834280i) q^{75} -6.92820i q^{79} +(6.58258 - 6.13756i) q^{81} +4.82395 q^{83} -6.74773 q^{85} +(2.69300 + 14.1857i) q^{87} +4.33991i q^{89} +(-1.58258 - 8.33639i) q^{93} -10.6306 q^{95} +9.89949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{9}+O(q^{10}) \) 16 * q + 8 * q^9	\( 16 q + 8 q^{9} - 32 q^{25} - 48 q^{37} + 72 q^{57} + 32 q^{81} + 112 q^{85} + 48 q^{93}+O(q^{100}) \) 16 * q + 8 * q^9 - 32 * q^25 - 48 * q^37 + 72 * q^57 + 32 * q^81 + 112 * q^85 + 48 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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