LMFDB - Embedded newform 7056.2.b.z.1567.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7056,2,Mod(1567,7056)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7056.1567"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7056, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7056.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(53)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$56.3424436662$
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} + 114x^{12} - 336x^{10} + 755x^{8} - 336x^{6} + 114x^{4} - 12x^{2} + 1 $ x^16 - 12x^14 + 114x^12 - 336x^10 + 755x^8 - 336x^6 + 114x^4 - 12*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1567.7
Root			$1.45341 + 0.839125i$ of defining polynomial
Character	$\chi$	$=$	7056.1567
Dual form			7056.2.b.z.1567.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7056\mathbb{Z}\right)^\times$.

$n$	$785$	$1765$	$4609$	$6175$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 1.60804i	− 0.719136i	−0.933119	−	0.359568i	$-0.882924\pi$
$5$	− 1.60804i	− 0.719136i	0.933119	−	0.359568i	$-0.117076\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.14639i	1.55169i	0.630921	+	0.775847i	$0.282677\pi$
$11$	5.14639i	1.55169i	−0.630921	+	0.775847i	$0.717323\pi$
$12$	0	0
$13$	− 1.84776i	− 0.512476i	−0.966614	−	0.256238i	$-0.917517\pi$
$13$	− 1.84776i	− 0.512476i	0.966614	−	0.256238i	$-0.0824831\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.15626i	− 1.49311i	−0.665323	−	0.746556i	$-0.731706\pi$
$17$	− 6.15626i	− 1.49311i	0.665323	−	0.746556i	$-0.268294\pi$
$18$	0	0
$19$	4.52607	1.03835	0.519175	−	0.854668i	$-0.326239\pi$
$19$	4.52607	1.03835	0.519175	+	0.854668i	$0.326239\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.13170i	0.444491i	0.974991	+	0.222245i	$0.0713386\pi$
$23$	2.13170i	0.444491i	−0.974991	+	0.222245i	$0.928661\pi$
$24$	0	0
$25$	2.41421	0.482843
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.17327	1.33204	0.666022	−	0.745932i	$-0.267996\pi$
$29$	7.17327	1.33204	0.666022	+	0.745932i	$0.267996\pi$
$30$	0	0
$31$	−1.87476	−0.336717	−0.168358	−	0.985726i	$-0.553847\pi$
$31$	−1.87476	−0.336717	−0.168358	+	0.985726i	$0.553847\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.41421	−0.232495	−0.116248	−	0.993220i	$-0.537087\pi$
$37$	−1.41421	−0.232495	−0.116248	+	0.993220i	$0.537087\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 0.666071i	− 0.104023i	−0.998646	−	0.0520114i	$-0.983437\pi$
$41$	− 0.666071i	− 0.104023i	0.998646	−	0.0520114i	$-0.0165632\pi$
$42$	0	0
$43$	− 1.43488i	− 0.218817i	−0.993997	−	0.109408i	$-0.965104\pi$
$43$	− 1.43488i	− 0.218817i	0.993997	−	0.109408i	$-0.0348956\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.50928	−1.38707	−0.693536	−	0.720422i	$-0.743948\pi$
$47$	−9.50928	−1.38707	−0.693536	+	0.720422i	$0.743948\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.46148	−0.338110	−0.169055	−	0.985607i	$-0.554072\pi$
$53$	−2.46148	−0.338110	−0.169055	+	0.985607i	$0.554072\pi$
$54$	0	0
$55$	8.27558	1.11588
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.63153	−0.212408	−0.106204	−	0.994344i	$-0.533870\pi$
$59$	−1.63153	−0.212408	−0.106204	+	0.994344i	$0.533870\pi$
$60$	0	0
$61$	− 8.60474i	− 1.10172i	−0.834596	−	0.550862i	$-0.814299\pi$
$61$	− 8.60474i	− 1.10172i	0.834596	−	0.550862i	$-0.185701\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.97127	−0.368540
$66$	0	0
$67$	11.8272i	1.44492i	0.691413	+	0.722460i	$0.256989\pi$
$67$	11.8272i	1.44492i	−0.691413	+	0.722460i	$0.743011\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.40979i	1.11674i	0.829593	+	0.558368i	$0.188572\pi$
$71$	9.40979i	1.11674i	−0.829593	+	0.558368i	$0.811428\pi$
$72$	0	0
$73$	2.48181i	0.290474i	0.989397	+	0.145237i	$0.0463944\pi$
$73$	2.48181i	0.290474i	−0.989397	+	0.145237i	$0.953606\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 13.2621i	− 1.49210i	−0.665891	−	0.746049i	$-0.731948\pi$
$79$	− 13.2621i	− 1.49210i	0.665891	−	0.746049i	$-0.268052\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13.4482	1.47613	0.738063	−	0.674732i	$-0.235741\pi$
$83$	13.4482	1.47613	0.738063	+	0.674732i	$0.235741\pi$
$84$	0	0
$85$	−9.89949	−1.07375
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.04019i	0.852258i	0.904662	+	0.426129i	$0.140123\pi$
$89$	8.04019i	0.852258i	−0.904662	+	0.426129i	$0.859877\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 7.27809i	− 0.746716i
$96$	0	0
$97$	− 2.48181i	− 0.251990i	−0.992031	−	0.125995i	$-0.959788\pi$
$97$	− 2.48181i	− 0.251990i	0.992031	−	0.125995i	$-0.0402123\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	14.8625i	1.47888i	0.673225	+	0.739438i	$0.264909\pi$
$101$	14.8625i	1.47888i	−0.673225	+	0.739438i	$0.735091\pi$
$102$	0	0
$103$	16.2295	1.59914	0.799571	−	0.600572i	$-0.205060\pi$
$103$	16.2295	1.59914	0.799571	+	0.600572i	$0.205060\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.4245i	− 1.20112i	−0.799580	−	0.600560i	$-0.794945\pi$
$107$	− 12.4245i	− 1.20112i	0.799580	−	0.600560i	$-0.205055\pi$
$108$	0	0
$109$	4.58579	0.439239	0.219619	−	0.975586i	$-0.429518\pi$
$109$	4.58579	0.439239	0.219619	+	0.975586i	$0.429518\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−7.68306	−0.722762	−0.361381	−	0.932418i	$-0.617695\pi$
$113$	−7.68306	−0.722762	−0.361381	+	0.932418i	$0.617695\pi$
$114$	0	0
$115$	3.42786	0.319649
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−15.4853	−1.40775
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.9223i	− 1.06637i
$126$	0	0
$127$	− 7.76874i	− 0.689364i	−0.938720	−	0.344682i	$-0.887987\pi$
$127$	− 7.76874i	− 0.689364i	0.938720	−	0.344682i	$-0.112013\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.4482	1.17497	0.587485	−	0.809235i	$-0.300118\pi$
$131$	13.4482	1.17497	0.587485	+	0.809235i	$0.300118\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.0388	1.54116	0.770578	−	0.637346i	$-0.219968\pi$
$137$	18.0388	1.54116	0.770578	+	0.637346i	$0.219968\pi$
$138$	0	0
$139$	13.8999	1.17897	0.589485	−	0.807779i	$-0.299331\pi$
$139$	13.8999	1.17897	0.589485	+	0.807779i	$0.299331\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9.50928	0.795206
$144$	0	0
$145$	− 11.5349i	− 0.957921i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.92296	0.403304	0.201652	−	0.979457i	$-0.435369\pi$
$149$	4.92296	0.403304	0.201652	+	0.979457i	$0.435369\pi$
$150$	0	0
$151$	15.2913i	1.24439i	0.782863	+	0.622194i	$0.213758\pi$
$151$	15.2913i	1.24439i	−0.782863	+	0.622194i	$0.786242\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.01468i	0.242145i
$156$	0	0
$157$	− 5.99162i	− 0.478183i	−0.970997	−	0.239092i	$-0.923150\pi$
$157$	− 5.99162i	− 0.478183i	0.970997	−	0.239092i	$-0.0768496\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 21.6251i	− 1.69381i	−0.531743	−	0.846906i	$-0.678463\pi$
$163$	− 21.6251i	− 1.69381i	0.531743	−	0.846906i	$-0.321537\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.0797	−1.16690	−0.583451	−	0.812149i	$-0.698298\pi$
$167$	−15.0797	−1.16690	−0.583451	+	0.812149i	$0.698298\pi$
$168$	0	0
$169$	9.58579	0.737368
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.48840i	0.569332i	0.958627	+	0.284666i	$0.0918828\pi$
$173$	7.48840i	0.569332i	−0.958627	+	0.284666i	$0.908117\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.14639i	0.384659i	0.981330	+	0.192329i	$0.0616042\pi$
$179$	5.14639i	0.384659i	−0.981330	+	0.192329i	$0.938396\pi$
$180$	0	0
$181$	− 22.9385i	− 1.70500i	−0.522724	−	0.852502i	$-0.675084\pi$
$181$	− 22.9385i	− 1.70500i	0.522724	−	0.852502i	$-0.324916\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.27411i	0.167196i
$186$	0	0
$187$	31.6825	2.31685
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.4392i	1.11714i	0.829458	+	0.558569i	$0.188649\pi$
$191$	15.4392i	1.11714i	−0.829458	+	0.558569i	$0.811351\pi$
$192$	0	0
$193$	22.9706	1.65346	0.826729	−	0.562601i	$-0.190199\pi$
$193$	22.9706	1.65346	0.826729	+	0.562601i	$0.190199\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.0675	−1.07351	−0.536757	−	0.843737i	$-0.680351\pi$
$197$	−15.0675	−1.07351	−0.536757	+	0.843737i	$0.680351\pi$
$198$	0	0
$199$	−4.84772	−0.343646	−0.171823	−	0.985128i	$-0.554966\pi$
$199$	−4.84772	−0.343646	−0.171823	+	0.985128i	$0.554966\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.07107	−0.0748066
$206$	0	0
$207$	0	0
$208$	0	0
$209$	23.2929i	1.61120i
$210$	0	0
$211$	− 12.6677i	− 0.872081i	−0.899927	−	0.436041i	$-0.856380\pi$
$211$	− 12.6677i	− 0.872081i	0.899927	−	0.436041i	$-0.143620\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.30734	−0.157359
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−11.3753	−0.765184
$222$	0	0
$223$	16.5512	1.10835	0.554174	−	0.832401i	$-0.313034\pi$
$223$	16.5512	1.10835	0.554174	+	0.832401i	$0.313034\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.20194	0.478010	0.239005	−	0.971018i	$-0.423179\pi$
$227$	7.20194	0.478010	0.239005	+	0.971018i	$0.423179\pi$
$228$	0	0
$229$	− 9.94977i	− 0.657499i	−0.944417	−	0.328750i	$-0.893373\pi$
$229$	− 9.94977i	− 0.657499i	0.944417	−	0.328750i	$-0.106627\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.1354	0.926040	0.463020	−	0.886348i	$-0.346766\pi$
$233$	14.1354	0.926040	0.463020	+	0.886348i	$0.346766\pi$
$234$	0	0
$235$	15.2913i	0.997493i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 16.6879i	− 1.07945i	−0.841842	−	0.539725i	$-0.818528\pi$
$239$	− 16.6879i	− 1.07945i	0.841842	−	0.539725i	$-0.181472\pi$
$240$	0	0
$241$	11.5893i	0.746532i	0.927724	+	0.373266i	$0.121762\pi$
$241$	11.5893i	0.746532i	−0.927724	+	0.373266i	$0.878238\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 8.36308i	− 0.532130i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−19.6944	−1.24310	−0.621549	−	0.783376i	$-0.713496\pi$
$251$	−19.6944	−1.24310	−0.621549	+	0.783376i	$0.713496\pi$
$252$	0	0
$253$	−10.9706	−0.689713
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 23.9590i	− 1.49452i	−0.664533	−	0.747259i	$-0.731369\pi$
$257$	− 23.9590i	− 1.49452i	0.664533	−	0.747259i	$-0.268631\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 0.882980i	− 0.0544469i	−0.999629	−	0.0272234i	$-0.991333\pi$
$263$	− 0.882980i	− 0.0544469i	0.999629	−	0.0272234i	$-0.00866656\pi$
$264$	0	0
$265$	3.95815i	0.243147i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 19.9625i	− 1.21714i	−0.793501	−	0.608568i	$-0.791744\pi$
$269$	− 19.9625i	− 1.21714i	0.793501	−	0.608568i	$-0.208256\pi$
$270$	0	0
$271$	31.6825	1.92457	0.962286	−	0.272038i	$-0.0876977\pi$
$271$	31.6825	1.92457	0.962286	+	0.272038i	$0.0876977\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	12.4245i	0.749224i
$276$	0	0
$277$	3.51472	0.211179	0.105589	−	0.994410i	$-0.466327\pi$
$277$	3.51472	0.211179	0.105589	+	0.994410i	$0.466327\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	21.5198	1.28376	0.641882	−	0.766803i	$-0.278154\pi$
$281$	21.5198	1.28376	0.641882	+	0.766803i	$0.278154\pi$
$282$	0	0
$283$	−18.4259	−1.09531	−0.547654	−	0.836705i	$-0.684479\pi$
$283$	−18.4259	−1.09531	−0.547654	+	0.836705i	$0.684479\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−20.8995	−1.22938
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.21786i	0.0711483i	0.999367	+	0.0355741i	$0.0113260\pi$
$293$	1.21786i	0.0711483i	−0.999367	+	0.0355741i	$0.988674\pi$
$294$	0	0
$295$	2.62357i	0.152750i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.93887	0.227791
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−13.8368	−0.792290
$306$	0	0
$307$	27.4781	1.56826	0.784128	−	0.620599i	$-0.213111\pi$
$307$	27.4781	1.56826	0.784128	+	0.620599i	$0.213111\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−30.8352	−1.74850	−0.874251	−	0.485474i	$-0.838647\pi$
$311$	−30.8352	−1.74850	−0.874251	+	0.485474i	$0.838647\pi$
$312$	0	0
$313$	− 31.4119i	− 1.77551i	−0.460320	−	0.887753i	$-0.652265\pi$
$313$	− 31.4119i	− 1.77551i	0.460320	−	0.887753i	$-0.347735\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−35.3566	−1.98582	−0.992912	−	0.118855i	$-0.962077\pi$
$317$	−35.3566	−1.98582	−0.992912	+	0.118855i	$0.962077\pi$
$318$	0	0
$319$	36.9164i	2.06692i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 27.8636i	− 1.55037i
$324$	0	0
$325$	− 4.46088i	− 0.247445i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 19.5959i	− 1.07709i	−0.842597	−	0.538545i	$-0.818974\pi$
$331$	− 19.5959i	− 1.07709i	0.842597	−	0.538545i	$-0.181026\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	19.0186	1.03909
$336$	0	0
$337$	−7.07107	−0.385186	−0.192593	−	0.981279i	$-0.561690\pi$
$337$	−7.07107	−0.385186	−0.192593	+	0.981279i	$0.561690\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 9.64823i	− 0.522481i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 14.1904i	− 0.761782i	−0.924620	−	0.380891i	$-0.875617\pi$
$347$	− 14.1904i	− 0.761782i	0.924620	−	0.380891i	$-0.124383\pi$
$348$	0	0
$349$	− 25.3659i	− 1.35781i	−0.734228	−	0.678903i	$-0.762456\pi$
$349$	− 25.3659i	− 1.35781i	0.734228	−	0.678903i	$-0.237544\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 8.43037i	− 0.448703i	−0.974508	−	0.224352i	$-0.927974\pi$
$353$	− 8.43037i	− 0.448703i	0.974508	−	0.224352i	$-0.0720264\pi$
$354$	0	0
$355$	15.1313	0.803086
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 31.2441i	− 1.64900i	−0.565863	−	0.824499i	$-0.691457\pi$
$359$	− 31.2441i	− 1.64900i	0.565863	−	0.824499i	$-0.308543\pi$
$360$	0	0
$361$	1.48528	0.0781727
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.99084	0.208890
$366$	0	0
$367$	17.0061	0.887709	0.443855	−	0.896099i	$-0.353611\pi$
$367$	17.0061	0.887709	0.443855	+	0.896099i	$0.353611\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	8.48528	0.439351	0.219676	−	0.975573i	$-0.429500\pi$
$373$	8.48528	0.439351	0.219676	+	0.975573i	$0.429500\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 13.2545i	− 0.682640i
$378$	0	0
$379$	4.05845i	0.208468i	0.994553	+	0.104234i	$0.0332392\pi$
$379$	4.05845i	0.208468i	−0.994553	+	0.104234i	$0.966761\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	15.7555	0.805068	0.402534	−	0.915405i	$-0.368130\pi$
$383$	15.7555	0.805068	0.402534	+	0.915405i	$0.368130\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−28.4819	−1.44409	−0.722046	−	0.691845i	$-0.756798\pi$
$389$	−28.4819	−1.44409	−0.722046	+	0.691845i	$0.756798\pi$
$390$	0	0
$391$	13.1233	0.663674
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−21.3259	−1.07302
$396$	0	0
$397$	− 16.2584i	− 0.815986i	−0.912985	−	0.407993i	$-0.866229\pi$
$397$	− 16.2584i	− 0.815986i	0.912985	−	0.407993i	$-0.133771\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.1354	0.705887	0.352944	−	0.935645i	$-0.385181\pi$
$401$	14.1354	0.705887	0.352944	+	0.935645i	$0.385181\pi$
$402$	0	0
$403$	3.46410i	0.172559i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 7.27809i	− 0.360761i
$408$	0	0
$409$	21.5935i	1.06773i	0.845571	+	0.533864i	$0.179260\pi$
$409$	21.5935i	1.06773i	−0.845571	+	0.533864i	$0.820740\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 21.6251i	− 1.06154i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	6.24621	0.305147	0.152574	−	0.988292i	$-0.451244\pi$
$419$	6.24621	0.305147	0.152574	+	0.988292i	$0.451244\pi$
$420$	0	0
$421$	−16.9706	−0.827095	−0.413547	−	0.910483i	$-0.635710\pi$
$421$	−16.9706	−0.827095	−0.413547	+	0.910483i	$0.635710\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 14.8625i	− 0.720938i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0.882980i	0.0425317i	0.999774	+	0.0212658i	$0.00676963\pi$
$431$	0.882980i	0.0425317i	−0.999774	+	0.0212658i	$0.993230\pi$
$432$	0	0
$433$	24.0978i	1.15807i	0.815304	+	0.579033i	$0.196570\pi$
$433$	24.0978i	1.15807i	−0.815304	+	0.579033i	$0.803430\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.64823i	0.461537i
$438$	0	0
$439$	28.2546	1.34852	0.674259	−	0.738495i	$-0.264463\pi$
$439$	28.2546	1.34852	0.674259	+	0.738495i	$0.264463\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5.14639i	0.244512i	0.992499	+	0.122256i	$0.0390129\pi$
$443$	5.14639i	0.244512i	−0.992499	+	0.122256i	$0.960987\pi$
$444$	0	0
$445$	12.9289	0.612890
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−20.5877	−0.971594	−0.485797	−	0.874072i	$-0.661471\pi$
$449$	−20.5877	−0.971594	−0.485797	+	0.874072i	$0.661471\pi$
$450$	0	0
$451$	3.42786	0.161412
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−33.4558	−1.56500	−0.782499	−	0.622652i	$-0.786055\pi$
$457$	−33.4558	−1.56500	−0.782499	+	0.622652i	$0.786055\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	7.48840i	0.348770i	0.984678	+	0.174385i	$0.0557936\pi$
$461$	7.48840i	0.348770i	−0.984678	+	0.174385i	$0.944206\pi$
$462$	0	0
$463$	− 26.5241i	− 1.23268i	−0.787480	−	0.616340i	$-0.788615\pi$
$463$	− 26.5241i	− 1.23268i	0.787480	−	0.616340i	$-0.211385\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−36.4056	−1.68465	−0.842325	−	0.538970i	$-0.818814\pi$
$467$	−36.4056	−1.68465	−0.842325	+	0.538970i	$0.818814\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.38443	0.339537
$474$	0	0
$475$	10.9269	0.501360
$476$	0	0
$477$	0	0
$478$	0	0
$479$	17.3870	0.794433	0.397217	−	0.917725i	$-0.369976\pi$
$479$	17.3870	0.794433	0.397217	+	0.917725i	$0.369976\pi$
$480$	0	0
$481$	2.61313i	0.119148i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.99084	−0.181215
$486$	0	0
$487$	5.49333i	0.248926i	0.992224	+	0.124463i	$0.0397209\pi$
$487$	5.49333i	0.248926i	−0.992224	+	0.124463i	$0.960279\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	37.2734i	1.68213i	0.540937	+	0.841063i	$0.318070\pi$
$491$	37.2734i	1.68213i	−0.540937	+	0.841063i	$0.681930\pi$
$492$	0	0
$493$	− 44.1605i	− 1.98889i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 17.3205i	− 0.775372i	−0.921791	−	0.387686i	$-0.873274\pi$
$499$	− 17.3205i	− 0.775372i	0.921791	−	0.387686i	$-0.126726\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−10.1851	−0.454130	−0.227065	−	0.973880i	$-0.572913\pi$
$503$	−10.1851	−0.454130	−0.227065	+	0.973880i	$0.572913\pi$
$504$	0	0
$505$	23.8995	1.06351
$506$	0	0
$507$	0	0
$508$	0	0
$509$	33.6072i	1.48961i	0.667281	+	0.744806i	$0.267458\pi$
$509$	33.6072i	1.48961i	−0.667281	+	0.744806i	$0.732542\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 26.0977i	− 1.15000i
$516$	0	0
$517$	− 48.9384i	− 2.15231i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.43037i	0.369341i	0.982800	+	0.184671i	$0.0591218\pi$
$521$	8.43037i	0.369341i	−0.982800	+	0.184671i	$0.940878\pi$
$522$	0	0
$523$	15.4530	0.675711	0.337855	−	0.941198i	$-0.390299\pi$
$523$	15.4530	0.675711	0.337855	+	0.941198i	$0.390299\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11.5415i	0.502755i
$528$	0	0
$529$	18.4558	0.802428
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.23074	−0.0533092
$534$	0	0
$535$	−19.9790	−0.863769
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	5.51472	0.237096	0.118548	−	0.992948i	$-0.462176\pi$
$541$	5.51472	0.237096	0.118548	+	0.992948i	$0.462176\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 7.37412i	− 0.315873i
$546$	0	0
$547$	− 6.33386i	− 0.270816i	−0.990790	−	0.135408i	$-0.956765\pi$
$547$	− 6.33386i	− 0.270816i	0.990790	−	0.135408i	$-0.0432345\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	32.4667	1.38313
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	43.3383	1.83630	0.918151	−	0.396232i	$-0.129682\pi$
$557$	43.3383	1.83630	0.918151	+	0.396232i	$0.129682\pi$
$558$	0	0
$559$	−2.65131	−0.112138
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.8352	1.29955	0.649774	−	0.760128i	$-0.274864\pi$
$563$	30.8352	1.29955	0.649774	+	0.760128i	$0.274864\pi$
$564$	0	0
$565$	12.3547i	0.519764i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	25.4232	1.06580	0.532898	−	0.846180i	$-0.321103\pi$
$569$	25.4232	1.06580	0.532898	+	0.846180i	$0.321103\pi$
$570$	0	0
$571$	25.0892i	1.04995i	0.851117	+	0.524976i	$0.175926\pi$
$571$	25.0892i	1.04995i	−0.851117	+	0.524976i	$0.824074\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.14639i	0.214619i
$576$	0	0
$577$	21.4077i	0.891216i	0.895228	+	0.445608i	$0.147012\pi$
$577$	21.4077i	0.891216i	−0.895228	+	0.445608i	$0.852988\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 12.6677i	− 0.524643i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−31.7909	−1.31215	−0.656076	−	0.754695i	$-0.727785\pi$
$587$	−31.7909	−1.31215	−0.656076	+	0.754695i	$0.727785\pi$
$588$	0	0
$589$	−8.48528	−0.349630
$590$	0	0
$591$	0	0
$592$	0	0
$593$	39.0974i	1.60554i	0.596291	+	0.802768i	$0.296640\pi$
$593$	39.0974i	1.60554i	−0.596291	+	0.802768i	$0.703360\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	44.5515i	1.82033i	0.414251	+	0.910163i	$0.364044\pi$
$599$	44.5515i	1.82033i	−0.414251	+	0.910163i	$0.635956\pi$
$600$	0	0
$601$	24.9176i	1.01641i	0.861237	+	0.508204i	$0.169690\pi$
$601$	24.9176i	1.01641i	−0.861237	+	0.508204i	$0.830310\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	24.9009i	1.01237i
$606$	0	0
$607$	−17.6494	−0.716366	−0.358183	−	0.933651i	$-0.616604\pi$
$607$	−17.6494	−0.716366	−0.358183	+	0.933651i	$0.616604\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	17.5709i	0.710841i
$612$	0	0
$613$	8.10051	0.327176	0.163588	−	0.986529i	$-0.447693\pi$
$613$	8.10051	0.327176	0.163588	+	0.986529i	$0.447693\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.26989	0.131641	0.0658205	−	0.997831i	$-0.479034\pi$
$617$	3.26989	0.131641	0.0658205	+	0.997831i	$0.479034\pi$
$618$	0	0
$619$	17.0061	0.683531	0.341766	−	0.939785i	$-0.388975\pi$
$619$	17.0061	0.683531	0.341766	+	0.939785i	$0.388975\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−7.10051	−0.284020
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8.70626i	0.347141i
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−12.4924	−0.495747
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	31.9630	1.26246	0.631231	−	0.775595i	$-0.282550\pi$
$641$	31.9630	1.26246	0.631231	+	0.775595i	$0.282550\pi$
$642$	0	0
$643$	−6.07917	−0.239739	−0.119869	−	0.992790i	$-0.538248\pi$
$643$	−6.07917	−0.239739	−0.119869	+	0.992790i	$0.538248\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	41.0203	1.61267	0.806336	−	0.591457i	$-0.201447\pi$
$647$	41.0203	1.61267	0.806336	+	0.591457i	$0.201447\pi$
$648$	0	0
$649$	− 8.39651i	− 0.329592i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.0388	0.705911	0.352956	−	0.935640i	$-0.385177\pi$
$653$	18.0388	0.705911	0.352956	+	0.935640i	$0.385177\pi$
$654$	0	0
$655$	− 21.6251i	− 0.844964i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	9.40979i	0.366553i	0.983061	+	0.183277i	$0.0586704\pi$
$659$	9.40979i	0.366553i	−0.983061	+	0.183277i	$0.941330\pi$
$660$	0	0
$661$	19.7682i	0.768895i	0.923147	+	0.384447i	$0.125608\pi$
$661$	19.7682i	0.768895i	−0.923147	+	0.384447i	$0.874392\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	15.2913i	0.592081i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	44.2833	1.70954
$672$	0	0
$673$	15.8995	0.612880	0.306440	−	0.951890i	$-0.400862\pi$
$673$	15.8995	0.612880	0.306440	+	0.951890i	$0.400862\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 21.1331i	− 0.812209i	−0.913827	−	0.406105i	$-0.866887\pi$
$677$	− 21.1331i	− 0.812209i	0.913827	−	0.406105i	$-0.133113\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	21.4685i	0.821470i	0.911755	+	0.410735i	$0.134728\pi$
$683$	21.4685i	0.821470i	−0.911755	+	0.410735i	$0.865272\pi$
$684$	0	0
$685$	− 29.0070i	− 1.10830i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.54822i	0.173273i
$690$	0	0
$691$	20.3007	0.772274	0.386137	−	0.922441i	$-0.373809\pi$
$691$	20.3007	0.772274	0.386137	+	0.922441i	$0.373809\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 22.3515i	− 0.847841i
$696$	0	0
$697$	−4.10051	−0.155318
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−6.75095	−0.254980	−0.127490	−	0.991840i	$-0.540692\pi$
$701$	−6.75095	−0.254980	−0.127490	+	0.991840i	$0.540692\pi$
$702$	0	0
$703$	−6.40083	−0.241412
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	15.8995	0.597118	0.298559	−	0.954391i	$-0.403494\pi$
$709$	15.8995	0.597118	0.298559	+	0.954391i	$0.403494\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 3.99643i	− 0.149667i
$714$	0	0
$715$	− 15.2913i	− 0.571862i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	7.87775	0.293790	0.146895	−	0.989152i	$-0.453072\pi$
$719$	7.87775	0.293790	0.146895	+	0.989152i	$0.453072\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	17.3178	0.643167
$726$	0	0
$727$	17.7826	0.659520	0.329760	−	0.944065i	$-0.393032\pi$
$727$	17.7826	0.659520	0.329760	+	0.944065i	$0.393032\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−8.83348	−0.326718
$732$	0	0
$733$	− 9.94977i	− 0.367503i	−0.982973	−	0.183752i	$-0.941176\pi$
$733$	− 9.94977i	− 0.367503i	0.982973	−	0.183752i	$-0.0588242\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−60.8672	−2.24207
$738$	0	0
$739$	9.79796i	0.360424i	0.983628	+	0.180212i	$0.0576783\pi$
$739$	9.79796i	0.360424i	−0.983628	+	0.180212i	$0.942322\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	40.2881i	1.47803i	0.673691	+	0.739014i	$0.264708\pi$
$743$	40.2881i	1.47803i	−0.673691	+	0.739014i	$0.735292\pi$
$744$	0	0
$745$	− 7.91630i	− 0.290031i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	10.6385i	0.388204i	0.980981	+	0.194102i	$0.0621793\pi$
$751$	10.6385i	0.388204i	−0.980981	+	0.194102i	$0.937821\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	24.5890	0.894884
$756$	0	0
$757$	−42.0416	−1.52803	−0.764015	−	0.645199i	$-0.776774\pi$
$757$	−42.0416	−1.52803	−0.764015	+	0.645199i	$0.776774\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.60804i	0.0582913i	0.999575	+	0.0291457i	$0.00927867\pi$
$761$	1.60804i	0.0582913i	−0.999575	+	0.0291457i	$0.990721\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.01468i	0.108854i
$768$	0	0
$769$	44.3687i	1.59998i	0.600015	+	0.799989i	$0.295161\pi$
$769$	44.3687i	1.59998i	−0.600015	+	0.799989i	$0.704839\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 7.48840i	− 0.269339i	−0.990891	−	0.134669i	$-0.957003\pi$
$773$	− 7.48840i	− 0.269339i	0.990891	−	0.134669i	$-0.0429973\pi$
$774$	0	0
$775$	−4.52607	−0.162581
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 3.01468i	− 0.108012i
$780$	0	0
$781$	−48.4264	−1.73283
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−9.63475	−0.343879
$786$	0	0
$787$	−18.5592	−0.661563	−0.330781	−	0.943707i	$-0.607312\pi$
$787$	−18.5592	−0.661563	−0.330781	+	0.943707i	$0.607312\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−15.8995	−0.564608
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 24.5107i	− 0.868215i	−0.900861	−	0.434108i	$-0.857064\pi$
$797$	− 24.5107i	− 0.868215i	0.900861	−	0.434108i	$-0.142936\pi$
$798$	0	0
$799$	58.5416i	2.07105i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−12.7723	−0.450726
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−38.1167	−1.34011	−0.670056	−	0.742311i	$-0.733730\pi$
$809$	−38.1167	−1.34011	−0.670056	+	0.742311i	$0.733730\pi$
$810$	0	0
$811$	−16.5512	−0.581190	−0.290595	−	0.956846i	$-0.593853\pi$
$811$	−16.5512	−0.581190	−0.290595	+	0.956846i	$0.593853\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−34.7741	−1.21808
$816$	0	0
$817$	− 6.49435i	− 0.227209i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	45.7997	1.59842	0.799211	−	0.601051i	$-0.205251\pi$
$821$	45.7997	1.59842	0.799211	+	0.601051i	$0.205251\pi$
$822$	0	0
$823$	− 2.62357i	− 0.0914519i	−0.998954	−	0.0457259i	$-0.985440\pi$
$823$	− 2.62357i	− 0.0914519i	0.998954	−	0.0457259i	$-0.0145601\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 9.92703i	− 0.345197i	−0.984992	−	0.172598i	$-0.944784\pi$
$827$	− 9.92703i	− 0.345197i	0.984992	−	0.172598i	$-0.0552162\pi$
$828$	0	0
$829$	− 39.6996i	− 1.37882i	−0.724369	−	0.689412i	$-0.757869\pi$
$829$	− 39.6996i	− 1.37882i	0.724369	−	0.689412i	$-0.242131\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	24.2487i	0.839161i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3.93887	−0.135985	−0.0679925	−	0.997686i	$-0.521659\pi$
$839$	−3.93887	−0.135985	−0.0679925	+	0.997686i	$0.521659\pi$
$840$	0	0
$841$	22.4558	0.774339
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 15.4143i	− 0.530268i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 3.01468i	− 0.103342i
$852$	0	0
$853$	− 4.08947i	− 0.140021i	−0.997546	−	0.0700103i	$-0.977697\pi$
$853$	− 4.08947i	− 0.140021i	0.997546	−	0.0700103i	$-0.0223032\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 25.6813i	− 0.877256i	−0.898669	−	0.438628i	$-0.855465\pi$
$857$	− 25.6813i	− 0.877256i	0.898669	−	0.438628i	$-0.144535\pi$
$858$	0	0
$859$	−48.2336	−1.64571	−0.822855	−	0.568251i	$-0.807620\pi$
$859$	−48.2336	−1.64571	−0.822855	+	0.568251i	$0.807620\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 16.6879i	− 0.568062i	−0.958815	−	0.284031i	$-0.908328\pi$
$863$	− 16.6879i	− 0.568062i	0.958815	−	0.284031i	$-0.0916719\pi$
$864$	0	0
$865$	12.0416	0.409428
$866$	0	0
$867$	0	0
$868$	0	0
$869$	68.2517	2.31528
$870$	0	0
$871$	21.8538	0.740487
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	56.8701	1.92037	0.960183	−	0.279373i	$-0.0901265\pi$
$877$	56.8701	1.92037	0.960183	+	0.279373i	$0.0901265\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	7.48840i	0.252291i	0.992012	+	0.126145i	$0.0402606\pi$
$881$	7.48840i	0.252291i	−0.992012	+	0.126145i	$0.959739\pi$
$882$	0	0
$883$	23.0600i	0.776031i	0.921653	+	0.388016i	$0.126839\pi$
$883$	23.0600i	0.776031i	−0.921653	+	0.388016i	$0.873161\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−50.8095	−1.70602	−0.853008	−	0.521899i	$-0.825224\pi$
$887$	−50.8095	−1.70602	−0.853008	+	0.521899i	$0.825224\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−43.0396	−1.44027
$894$	0	0
$895$	8.27558	0.276622
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−13.4482	−0.448521
$900$	0	0
$901$	15.1535i	0.504836i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−36.8859	−1.22613
$906$	0	0
$907$	− 37.7570i	− 1.25370i	−0.779140	−	0.626850i	$-0.784344\pi$
$907$	− 37.7570i	− 1.25370i	0.779140	−	0.626850i	$-0.215656\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 21.4685i	− 0.711284i	−0.934622	−	0.355642i	$-0.884262\pi$
$911$	− 21.4685i	− 0.711284i	0.934622	−	0.355642i	$-0.115738\pi$
$912$	0	0
$913$	69.2094i	2.29050i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	8.36308i	0.275873i	0.990441	+	0.137936i	$0.0440469\pi$
$919$	8.36308i	0.275873i	−0.990441	+	0.137936i	$0.955953\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	17.3870	0.572301
$924$	0	0
$925$	−3.41421	−0.112259
$926$	0	0
$927$	0	0
$928$	0	0
$929$	28.8973i	0.948091i	0.880500	+	0.474046i	$0.157207\pi$
$929$	28.8973i	0.948091i	−0.880500	+	0.474046i	$0.842793\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 50.9466i	− 1.66613i
$936$	0	0
$937$	30.5152i	0.996889i	0.866922	+	0.498444i	$0.166095\pi$
$937$	30.5152i	0.996889i	−0.866922	+	0.498444i	$0.833905\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 37.6036i	− 1.22584i	−0.790144	−	0.612921i	$-0.789994\pi$
$941$	− 37.6036i	− 1.22584i	0.790144	−	0.612921i	$-0.210006\pi$
$942$	0	0
$943$	1.41987	0.0462372
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.13170i	0.0692710i	0.999400	+	0.0346355i	$0.0110270\pi$
$947$	2.13170i	0.0692710i	−0.999400	+	0.0346355i	$0.988973\pi$
$948$	0	0
$949$	4.58579	0.148861
$950$	0	0
$951$	0	0
$952$	0	0
$953$	38.1167	1.23472	0.617360	−	0.786681i	$-0.288202\pi$
$953$	38.1167	1.23472	0.617360	+	0.786681i	$0.288202\pi$
$954$	0	0
$955$	24.8268	0.803375
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27.4853	−0.886622
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 36.9375i	− 1.18906i
$966$	0	0
$967$	− 44.4390i	− 1.42906i	−0.699604	−	0.714531i	$-0.746640\pi$
$967$	− 44.4390i	− 1.42906i	0.699604	−	0.714531i	$-0.253360\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−8.83348	−0.283480	−0.141740	−	0.989904i	$-0.545270\pi$
$971$	−8.83348	−0.283480	−0.141740	+	0.989904i	$0.545270\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−25.0009	−0.799849	−0.399924	−	0.916548i	$-0.630964\pi$
$977$	−25.0009	−0.799849	−0.399924	+	0.916548i	$0.630964\pi$
$978$	0	0
$979$	−41.3779	−1.32244
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−57.0557	−1.81979	−0.909897	−	0.414835i	$-0.863839\pi$
$983$	−57.0557	−1.81979	−0.909897	+	0.414835i	$0.863839\pi$
$984$	0	0
$985$	24.2291i	0.772004i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.05873	0.0972620
$990$	0	0
$991$	− 27.1185i	− 0.861446i	−0.902484	−	0.430723i	$-0.858259\pi$
$991$	− 27.1185i	− 0.861446i	0.902484	−	0.430723i	$-0.141741\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	7.79533i	0.247128i
$996$	0	0
$997$	− 31.9690i	− 1.01247i	−0.862396	−	0.506235i	$-0.831037\pi$
$997$	− 31.9690i	− 1.01247i	0.862396	−	0.506235i	$-0.168963\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7056.2.b.z.1567.7	yes	16
3.2	odd	2	inner	7056.2.b.z.1567.9	yes	16
4.3	odd	2	inner	7056.2.b.z.1567.5	✓	16
7.6	odd	2	inner	7056.2.b.z.1567.12	yes	16
12.11	even	2	inner	7056.2.b.z.1567.11	yes	16
21.20	even	2	inner	7056.2.b.z.1567.6	yes	16
28.27	even	2	inner	7056.2.b.z.1567.10	yes	16
84.83	odd	2	inner	7056.2.b.z.1567.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7056.2.b.z.1567.5	✓	16	4.3	odd	2	inner
7056.2.b.z.1567.6	yes	16	21.20	even	2	inner
7056.2.b.z.1567.7	yes	16	1.1	even	1	trivial
7056.2.b.z.1567.8	yes	16	84.83	odd	2	inner
7056.2.b.z.1567.9	yes	16	3.2	odd	2	inner
7056.2.b.z.1567.10	yes	16	28.27	even	2	inner
7056.2.b.z.1567.11	yes	16	12.11	even	2	inner
7056.2.b.z.1567.12	yes	16	7.6	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7056.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.60804i q^{5} +5.14639i q^{11} -1.84776i q^{13} -6.15626i q^{17} +4.52607 q^{19} +2.13170i q^{23} +2.41421 q^{25} +7.17327 q^{29} -1.87476 q^{31} -1.41421 q^{37} -0.666071i q^{41} -1.43488i q^{43} -9.50928 q^{47} -2.46148 q^{53} +8.27558 q^{55} -1.63153 q^{59} -8.60474i q^{61} -2.97127 q^{65} +11.8272i q^{67} +9.40979i q^{71} +2.48181i q^{73} -13.2621i q^{79} +13.4482 q^{83} -9.89949 q^{85} +8.04019i q^{89} -7.27809i q^{95} -2.48181i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{25}+O(q^{100}) \) 16 * q + 16 * q^25

Introduction

L-functions

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