LMFDB - Embedded newform 5472.2.k.d.2431.39

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5472,2,Mod(2431,5472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5472, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5472.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5472 = 2^{5} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5472.k (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:	$43.6941399860$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.39
Character	$\chi$	$=$	5472.2431
Dual form			5472.2.k.d.2431.38

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.07207 q^{5} -1.30098i q^{7} +O(q^{10})$	$q+4.07207 q^{5} -1.30098i q^{7} +4.01973i q^{11} +4.82142i q^{13} -2.18793 q^{17} +(-3.91777 + 1.91078i) q^{19} +3.65642i q^{23} +11.5818 q^{25} -9.24023i q^{29} +4.74740 q^{31} -5.29769i q^{35} +0.423562i q^{37} +3.88478i q^{41} +4.97333i q^{43} +4.57825i q^{47} +5.30745 q^{49} -1.30871i q^{53} +16.3686i q^{55} -10.7599 q^{59} +8.76020 q^{61} +19.6332i q^{65} -8.77896 q^{67} +14.2287 q^{71} -5.08785 q^{73} +5.22959 q^{77} -13.6744 q^{79} +12.3591i q^{83} -8.90941 q^{85} -2.16001i q^{89} +6.27257 q^{91} +(-15.9534 + 7.78083i) q^{95} +12.4610i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q+O(q^{10}) $ 40 * q	$ 40 q + 40 q^{25} - 24 q^{49} + 48 q^{61} + 16 q^{73} - 16 q^{85}+O(q^{100}) $ 40 * q + 40 * q^25 - 24 * q^49 + 48 * q^61 + 16 * q^73 - 16 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5472\mathbb{Z}\right)^\times$.

$n$	$1217$	$2053$	$3745$	$4447$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$5$	4.07207			1.82109			0.910543	−	0.413415i	$-0.135664\pi$
$5$	4.07207			1.82109			0.910543	+	0.413415i	$0.135664\pi$
$6$		0			0
$7$		−	1.30098i		−	0.491725i	−0.969305	−	0.245862i	$-0.920929\pi$
$7$		−	1.30098i		−	0.491725i	0.969305	−	0.245862i	$-0.0790711\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			4.01973i			1.21199i	0.795467	+	0.605997i	$0.207226\pi$
$11$			4.01973i			1.21199i	−0.795467	+	0.605997i	$0.792774\pi$
$12$		0			0
$13$			4.82142i			1.33722i	0.743613	+	0.668611i	$0.233111\pi$
$13$			4.82142i			1.33722i	−0.743613	+	0.668611i	$0.766889\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.18793			−0.530651			−0.265326	−	0.964159i	$-0.585479\pi$
$17$	−2.18793			−0.530651			−0.265326	+	0.964159i	$0.585479\pi$
$18$		0			0
$19$	−3.91777	+	1.91078i	−0.898798	+	0.438363i
$19$	−3.91777	+	1.91078i	−0.898798	+	0.438363i
$20$		0			0
$21$		0			0
$22$		0			0
$23$			3.65642i			0.762417i	0.924489	+	0.381208i	$0.124492\pi$
$23$			3.65642i			0.762417i	−0.924489	+	0.381208i	$0.875508\pi$
$24$		0			0
$25$	11.5818			2.31635
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	9.24023i		−	1.71587i	−0.513761	−	0.857933i	$-0.671748\pi$
$29$		−	9.24023i		−	1.71587i	0.513761	−	0.857933i	$-0.328252\pi$
$30$		0			0
$31$	4.74740			0.852658			0.426329	−	0.904568i	$-0.359807\pi$
$31$	4.74740			0.852658			0.426329	+	0.904568i	$0.359807\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	5.29769i		−	0.895472i
$36$		0			0
$37$			0.423562i			0.0696331i	0.999394	+	0.0348166i	$0.0110847\pi$
$37$			0.423562i			0.0696331i	−0.999394	+	0.0348166i	$0.988915\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			3.88478i			0.606701i	0.952879	+	0.303351i	$0.0981054\pi$
$41$			3.88478i			0.606701i	−0.952879	+	0.303351i	$0.901895\pi$
$42$		0			0
$43$			4.97333i			0.758426i	0.925309	+	0.379213i	$0.123805\pi$
$43$			4.97333i			0.758426i	−0.925309	+	0.379213i	$0.876195\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			4.57825i			0.667806i	0.942607	+	0.333903i	$0.108366\pi$
$47$			4.57825i			0.667806i	−0.942607	+	0.333903i	$0.891634\pi$
$48$		0			0
$49$	5.30745			0.758207
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	1.30871i		−	0.179765i	−0.995952	−	0.0898825i	$-0.971351\pi$
$53$		−	1.30871i		−	0.179765i	0.995952	−	0.0898825i	$-0.0286491\pi$
$54$		0			0
$55$			16.3686i			2.20715i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−10.7599			−1.40083			−0.700413	−	0.713738i	$-0.747001\pi$
$59$	−10.7599			−1.40083			−0.700413	+	0.713738i	$0.747001\pi$
$60$		0			0
$61$	8.76020			1.12163			0.560815	−	0.827941i	$-0.310488\pi$
$61$	8.76020			1.12163			0.560815	+	0.827941i	$0.310488\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			19.6332i			2.43519i
$66$		0			0
$67$	−8.77896			−1.07252			−0.536260	−	0.844053i	$-0.680163\pi$
$67$	−8.77896			−1.07252			−0.536260	+	0.844053i	$0.680163\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	14.2287			1.68863			0.844316	−	0.535846i	$-0.180007\pi$
$71$	14.2287			1.68863			0.844316	+	0.535846i	$0.180007\pi$
$72$		0			0
$73$	−5.08785			−0.595488			−0.297744	−	0.954646i	$-0.596234\pi$
$73$	−5.08785			−0.595488			−0.297744	+	0.954646i	$0.596234\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.22959			0.595967
$78$		0			0
$79$	−13.6744			−1.53849			−0.769245	−	0.638954i	$-0.779367\pi$
$79$	−13.6744			−1.53849			−0.769245	+	0.638954i	$0.779367\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			12.3591i			1.35659i	0.734792	+	0.678293i	$0.237280\pi$
$83$			12.3591i			1.35659i	−0.734792	+	0.678293i	$0.762720\pi$
$84$		0			0
$85$	−8.90941			−0.966361
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	2.16001i		−	0.228961i	−0.993426	−	0.114480i	$-0.963480\pi$
$89$		−	2.16001i		−	0.228961i	0.993426	−	0.114480i	$-0.0365203\pi$
$90$		0			0
$91$	6.27257			0.657545
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−15.9534	+	7.78083i	−1.63679	+	0.798297i
$96$		0			0
$97$			12.4610i			1.26522i	0.774470	+	0.632610i	$0.218017\pi$
$97$			12.4610i			1.26522i	−0.774470	+	0.632610i	$0.781983\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.33537			−0.431386			−0.215693	−	0.976461i	$-0.569201\pi$
$101$	−4.33537			−0.431386			−0.215693	+	0.976461i	$0.569201\pi$
$102$		0			0
$103$	−2.24102			−0.220814			−0.110407	−	0.993886i	$-0.535215\pi$
$103$	−2.24102			−0.220814			−0.110407	+	0.993886i	$0.535215\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−14.2777			−1.38028			−0.690140	−	0.723676i	$-0.742451\pi$
$107$	−14.2777			−1.38028			−0.690140	+	0.723676i	$0.742451\pi$
$108$		0			0
$109$			0.507733i			0.0486320i	0.999704	+	0.0243160i	$0.00774079\pi$
$109$			0.507733i			0.0486320i	−0.999704	+	0.0243160i	$0.992259\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			17.4732i			1.64374i	0.569678	+	0.821868i	$0.307068\pi$
$113$			17.4732i			1.64374i	−0.569678	+	0.821868i	$0.692932\pi$
$114$		0			0
$115$			14.8892i			1.38843i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			2.84646i			0.260934i
$120$		0			0
$121$	−5.15824			−0.468931
$122$		0			0
$123$		0			0
$124$		0			0
$125$	26.8014			2.39719
$126$		0			0
$127$	2.22426			0.197371			0.0986855	−	0.995119i	$-0.468536\pi$
$127$	2.22426			0.197371			0.0986855	+	0.995119i	$0.468536\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			7.06139i			0.616957i	0.951231	+	0.308478i	$0.0998198\pi$
$131$			7.06139i			0.616957i	−0.951231	+	0.308478i	$0.900180\pi$
$132$		0			0
$133$	2.48589	+	5.09694i	0.215554	+	0.441961i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	19.3839			1.65608			0.828039	−	0.560671i	$-0.189457\pi$
$137$	19.3839			1.65608			0.828039	+	0.560671i	$0.189457\pi$
$138$		0			0
$139$		−	19.8625i		−	1.68472i	−0.538916	−	0.842360i	$-0.681166\pi$
$139$		−	19.8625i		−	1.68472i	0.538916	−	0.842360i	$-0.318834\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−19.3808			−1.62071
$144$		0			0
$145$		−	37.6269i		−	3.12474i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	13.3332			1.09230			0.546151	−	0.837687i	$-0.316092\pi$
$149$	13.3332			1.09230			0.546151	+	0.837687i	$0.316092\pi$
$150$		0			0
$151$	12.8273			1.04387			0.521934	−	0.852986i	$-0.325211\pi$
$151$	12.8273			1.04387			0.521934	+	0.852986i	$0.325211\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	19.3318			1.55276
$156$		0			0
$157$	−7.49391			−0.598079			−0.299040	−	0.954241i	$-0.596666\pi$
$157$	−7.49391			−0.598079			−0.299040	+	0.954241i	$0.596666\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	4.75694			0.374899
$162$		0			0
$163$		−	11.0677i		−	0.866885i	−0.901181	−	0.433443i	$-0.857299\pi$
$163$		−	11.0677i		−	0.866885i	0.901181	−	0.433443i	$-0.142701\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.21235			0.0938145			0.0469073	−	0.998899i	$-0.485063\pi$
$167$	1.21235			0.0938145			0.0469073	+	0.998899i	$0.485063\pi$
$168$		0			0
$169$	−10.2461			−0.788161
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	1.30871i		−	0.0994993i	−0.998762	−	0.0497496i	$-0.984158\pi$
$173$		−	1.30871i		−	0.0994993i	0.998762	−	0.0497496i	$-0.0158423\pi$
$174$		0			0
$175$		−	15.0677i		−	1.13901i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	6.86920			0.513428			0.256714	−	0.966487i	$-0.417360\pi$
$179$	6.86920			0.513428			0.256714	+	0.966487i	$0.417360\pi$
$180$		0			0
$181$			15.0995i			1.12234i	0.827702	+	0.561168i	$0.189648\pi$
$181$			15.0995i			1.12234i	−0.827702	+	0.561168i	$0.810352\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			1.72477i			0.126808i
$186$		0			0
$187$		−	8.79490i		−	0.643146i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	5.29051i		−	0.382808i	−0.981511	−	0.191404i	$-0.938696\pi$
$191$		−	5.29051i		−	0.382808i	0.981511	−	0.191404i	$-0.0613040\pi$
$192$		0			0
$193$		−	18.6373i		−	1.34154i	−0.741666	−	0.670769i	$-0.765964\pi$
$193$		−	18.6373i		−	1.34154i	0.741666	−	0.670769i	$-0.234036\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−0.776449			−0.0553197			−0.0276598	−	0.999617i	$-0.508806\pi$
$197$	−0.776449			−0.0553197			−0.0276598	+	0.999617i	$0.508806\pi$
$198$		0			0
$199$			1.30098i			0.0922241i	0.998936	+	0.0461120i	$0.0146831\pi$
$199$			1.30098i			0.0922241i	−0.998936	+	0.0461120i	$0.985317\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−12.0214			−0.843734
$204$		0			0
$205$			15.8191i			1.10486i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−7.68082	−	15.7484i	−0.531294	−	1.08934i
$210$		0			0
$211$	19.4170			1.33672			0.668359	−	0.743838i	$-0.266997\pi$
$211$	19.4170			1.33672			0.668359	+	0.743838i	$0.266997\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			20.2518i			1.38116i
$216$		0			0
$217$		−	6.17628i		−	0.419273i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	10.5489i		−	0.709598i
$222$		0			0
$223$	22.3738			1.49826			0.749131	−	0.662422i	$-0.230471\pi$
$223$	22.3738			1.49826			0.749131	+	0.662422i	$0.230471\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	19.8855			1.31985			0.659924	−	0.751332i	$-0.270588\pi$
$227$	19.8855			1.31985			0.659924	+	0.751332i	$0.270588\pi$
$228$		0			0
$229$	−26.8690			−1.77555			−0.887777	−	0.460273i	$-0.847751\pi$
$229$	−26.8690			−1.77555			−0.887777	+	0.460273i	$0.847751\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	0.127114			0.00832749			0.00416375	−	0.999991i	$-0.498675\pi$
$233$	0.127114			0.00832749			0.00416375	+	0.999991i	$0.498675\pi$
$234$		0			0
$235$			18.6429i			1.21613i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	14.2659i		−	0.922787i	−0.887195	−	0.461394i	$-0.847350\pi$
$239$		−	14.2659i		−	0.922787i	0.887195	−	0.461394i	$-0.152650\pi$
$240$		0			0
$241$		−	15.5230i		−	0.999927i	−0.866047	−	0.499963i	$-0.833347\pi$
$241$		−	15.5230i		−	0.999927i	0.866047	−	0.499963i	$-0.166653\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	21.6123			1.38076
$246$		0			0
$247$	−9.21267	−	18.8892i	−0.586188	−	1.20189i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			18.8691i			1.19101i	0.803352	+	0.595505i	$0.203048\pi$
$251$			18.8691i			1.19101i	−0.803352	+	0.595505i	$0.796952\pi$
$252$		0			0
$253$	−14.6978			−0.924045
$254$		0			0
$255$		0			0
$256$		0			0
$257$			10.8563i			0.677198i	0.940931	+	0.338599i	$0.109953\pi$
$257$			10.8563i			0.677198i	−0.940931	+	0.338599i	$0.890047\pi$
$258$		0			0
$259$	0.551046			0.0342403
$260$		0			0
$261$		0			0
$262$		0			0
$263$			5.38585i			0.332106i	0.986117	+	0.166053i	$0.0531022\pi$
$263$			5.38585i			0.332106i	−0.986117	+	0.166053i	$0.946898\pi$
$264$		0			0
$265$		−	5.32915i		−	0.327367i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			24.9413i			1.52070i	0.649514	+	0.760349i	$0.274972\pi$
$269$			24.9413i			1.52070i	−0.649514	+	0.760349i	$0.725028\pi$
$270$		0			0
$271$		−	27.6163i		−	1.67757i	−0.544463	−	0.838785i	$-0.683267\pi$
$271$		−	27.6163i		−	1.67757i	0.544463	−	0.838785i	$-0.316733\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			46.5556i			2.80741i
$276$		0			0
$277$	11.8149			0.709890			0.354945	−	0.934887i	$-0.384500\pi$
$277$	11.8149			0.709890			0.354945	+	0.934887i	$0.384500\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			5.58739i			0.333316i	0.986015	+	0.166658i	$0.0532975\pi$
$281$			5.58739i			0.333316i	−0.986015	+	0.166658i	$0.946702\pi$
$282$		0			0
$283$		−	30.4645i		−	1.81093i	−0.424424	−	0.905464i	$-0.639523\pi$
$283$		−	30.4645i		−	1.81093i	0.424424	−	0.905464i	$-0.360477\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	5.05403			0.298330
$288$		0			0
$289$	−12.2130			−0.718409
$290$		0			0
$291$		0			0
$292$		0			0
$293$			19.4772i			1.13787i	0.822383	+	0.568934i	$0.192644\pi$
$293$			19.4772i			1.13787i	−0.822383	+	0.568934i	$0.807356\pi$
$294$		0			0
$295$	−43.8153			−2.55102
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−17.6291			−1.01952
$300$		0			0
$301$	6.47021			0.372937
$302$		0			0
$303$		0			0
$304$		0			0
$305$	35.6722			2.04258
$306$		0			0
$307$	−14.3357			−0.818181			−0.409091	−	0.912494i	$-0.634154\pi$
$307$	−14.3357			−0.818181			−0.409091	+	0.912494i	$0.634154\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			25.5281i			1.44757i	0.690027	+	0.723784i	$0.257599\pi$
$311$			25.5281i			1.44757i	−0.690027	+	0.723784i	$0.742401\pi$
$312$		0			0
$313$	24.2423			1.37026			0.685129	−	0.728422i	$-0.259746\pi$
$313$	24.2423			1.37026			0.685129	+	0.728422i	$0.259746\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	10.9428i		−	0.614611i	−0.951611	−	0.307305i	$-0.900573\pi$
$317$		−	10.9428i		−	0.614611i	0.951611	−	0.307305i	$-0.0994273\pi$
$318$		0			0
$319$	37.1432			2.07962
$320$		0			0
$321$		0			0
$322$		0			0
$323$	8.57181	−	4.18066i	0.476948	−	0.232618i
$324$		0			0
$325$			55.8406i			3.09748i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	5.95621			0.328377
$330$		0			0
$331$	−12.4489			−0.684251			−0.342125	−	0.939654i	$-0.611147\pi$
$331$	−12.4489			−0.684251			−0.342125	+	0.939654i	$0.611147\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−35.7486			−1.95315
$336$		0			0
$337$			7.30017i			0.397665i	0.980033	+	0.198833i	$0.0637151\pi$
$337$			7.30017i			0.397665i	−0.980033	+	0.198833i	$0.936285\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			19.0833i			1.03342i
$342$		0			0
$343$		−	16.0118i		−	0.864553i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	22.8639i		−	1.22740i	−0.789539	−	0.613700i	$-0.789680\pi$
$347$		−	22.8639i		−	1.22740i	0.789539	−	0.613700i	$-0.210320\pi$
$348$		0			0
$349$	−19.7565			−1.05754			−0.528769	−	0.848766i	$-0.677346\pi$
$349$	−19.7565			−1.05754			−0.528769	+	0.848766i	$0.677346\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.32639			0.0705966			0.0352983	−	0.999377i	$-0.488762\pi$
$353$	1.32639			0.0705966			0.0352983	+	0.999377i	$0.488762\pi$
$354$		0			0
$355$	57.9401			3.07514
$356$		0			0
$357$		0			0
$358$		0			0
$359$			24.4111i			1.28837i	0.764870	+	0.644184i	$0.222803\pi$
$359$			24.4111i			1.28837i	−0.764870	+	0.644184i	$0.777197\pi$
$360$		0			0
$361$	11.6978	−	14.9720i	0.615676	−	0.788000i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−20.7181			−1.08443
$366$		0			0
$367$			27.2594i			1.42293i	0.702722	+	0.711465i	$0.251968\pi$
$367$			27.2594i			1.42293i	−0.702722	+	0.711465i	$0.748032\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.70260			−0.0883948
$372$		0			0
$373$		−	21.0318i		−	1.08899i	−0.838765	−	0.544493i	$-0.816722\pi$
$373$		−	21.0318i		−	1.08899i	0.838765	−	0.544493i	$-0.183278\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	44.5510			2.29449
$378$		0			0
$379$	−11.2226			−0.576464			−0.288232	−	0.957561i	$-0.593067\pi$
$379$	−11.2226			−0.576464			−0.288232	+	0.957561i	$0.593067\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−19.8173			−1.01262			−0.506308	−	0.862353i	$-0.668990\pi$
$383$	−19.8173			−1.01262			−0.506308	+	0.862353i	$0.668990\pi$
$384$		0			0
$385$	21.2953			1.08531
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−18.6978			−0.948014			−0.474007	−	0.880521i	$-0.657193\pi$
$389$	−18.6978			−0.948014			−0.474007	+	0.880521i	$0.657193\pi$
$390$		0			0
$391$		−	8.00000i		−	0.404577i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−55.6831			−2.80172
$396$		0			0
$397$	−14.5525			−0.730371			−0.365185	−	0.930935i	$-0.618994\pi$
$397$	−14.5525			−0.730371			−0.365185	+	0.930935i	$0.618994\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	12.3210i		−	0.615282i	−0.951503	−	0.307641i	$-0.900460\pi$
$401$		−	12.3210i		−	0.615282i	0.951503	−	0.307641i	$-0.0995395\pi$
$402$		0			0
$403$			22.8892i			1.14019i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.70260			−0.0843950
$408$		0			0
$409$		−	20.1328i		−	0.995503i	−0.867320	−	0.497752i	$-0.834159\pi$
$409$		−	20.1328i		−	0.995503i	0.867320	−	0.497752i	$-0.165841\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			13.9985i			0.688820i
$414$		0			0
$415$			50.3271i			2.47046i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			1.14958i			0.0561606i	0.999606	+	0.0280803i	$0.00893941\pi$
$419$			1.14958i			0.0561606i	−0.999606	+	0.0280803i	$0.991061\pi$
$420$		0			0
$421$		−	34.1412i		−	1.66394i	−0.554819	−	0.831971i	$-0.687213\pi$
$421$		−	34.1412i		−	1.66394i	0.554819	−	0.831971i	$-0.312787\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−25.3401			−1.22918
$426$		0			0
$427$		−	11.3969i		−	0.551533i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	27.2450			1.31234			0.656172	−	0.754611i	$-0.272175\pi$
$431$	27.2450			1.31234			0.656172	+	0.754611i	$0.272175\pi$
$432$		0			0
$433$			12.7571i			0.613065i	0.951860	+	0.306532i	$0.0991688\pi$
$433$			12.7571i			0.613065i	−0.951860	+	0.306532i	$0.900831\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.98662	−	14.3250i	−0.334215	−	0.685259i
$438$		0			0
$439$	1.90079			0.0907199			0.0453600	−	0.998971i	$-0.485557\pi$
$439$	1.90079			0.0907199			0.0453600	+	0.998971i	$0.485557\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	3.12642i		−	0.148541i	−0.997238	−	0.0742703i	$-0.976337\pi$
$443$		−	3.12642i		−	0.148541i	0.997238	−	0.0742703i	$-0.0236627\pi$
$444$		0			0
$445$		−	8.79572i		−	0.416957i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	18.9379i		−	0.893733i	−0.894601	−	0.446866i	$-0.852540\pi$
$449$		−	18.9379i		−	0.893733i	0.894601	−	0.446866i	$-0.147460\pi$
$450$		0			0
$451$	−15.6158			−0.735319
$452$		0			0
$453$		0			0
$454$		0			0
$455$	25.5424			1.19744
$456$		0			0
$457$	12.2406			0.572589			0.286295	−	0.958142i	$-0.407576\pi$
$457$	12.2406			0.572589			0.286295	+	0.958142i	$0.407576\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	27.0971			1.26204			0.631020	−	0.775767i	$-0.282637\pi$
$461$	27.0971			1.26204			0.631020	+	0.775767i	$0.282637\pi$
$462$		0			0
$463$			1.95627i			0.0909157i	0.998966	+	0.0454579i	$0.0144747\pi$
$463$			1.95627i			0.0909157i	−0.998966	+	0.0454579i	$0.985525\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			16.8444i			0.779467i	0.920928	+	0.389734i	$0.127433\pi$
$467$			16.8444i			0.779467i	−0.920928	+	0.389734i	$0.872567\pi$
$468$		0			0
$469$			11.4213i			0.527385i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−19.9915			−0.919209
$474$		0			0
$475$	−45.3747	+	22.1302i	−2.08193	+	1.01540i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	27.4813i		−	1.25565i	−0.778354	−	0.627826i	$-0.783945\pi$
$479$		−	27.4813i		−	1.25565i	0.778354	−	0.627826i	$-0.216055\pi$
$480$		0			0
$481$	−2.04217			−0.0931149
$482$		0			0
$483$		0			0
$484$		0			0
$485$			50.7420i			2.30408i
$486$		0			0
$487$	−2.52034			−0.114207			−0.0571037	−	0.998368i	$-0.518187\pi$
$487$	−2.52034			−0.114207			−0.0571037	+	0.998368i	$0.518187\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	33.8402i		−	1.52719i	−0.645698	−	0.763593i	$-0.723434\pi$
$491$		−	33.8402i		−	1.52719i	0.645698	−	0.763593i	$-0.276566\pi$
$492$		0			0
$493$			20.2170i			0.910527i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	18.5112i		−	0.830342i
$498$		0			0
$499$			38.0090i			1.70152i	0.525557	+	0.850759i	$0.323857\pi$
$499$			38.0090i			1.70152i	−0.525557	+	0.850759i	$0.676143\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	26.3814i		−	1.17629i	−0.808756	−	0.588144i	$-0.799859\pi$
$503$		−	26.3814i		−	1.17629i	0.808756	−	0.588144i	$-0.200141\pi$
$504$		0			0
$505$	−17.6539			−0.785590
$506$		0			0
$507$		0			0
$508$		0			0
$509$			23.0324i			1.02089i	0.859909	+	0.510447i	$0.170520\pi$
$509$			23.0324i			1.02089i	−0.859909	+	0.510447i	$0.829480\pi$
$510$		0			0
$511$			6.61920i			0.292816i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−9.12557			−0.402121
$516$		0			0
$517$	−18.4033			−0.809377
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	31.5995i		−	1.38440i	−0.721706	−	0.692200i	$-0.756642\pi$
$521$		−	31.5995i		−	1.38440i	0.721706	−	0.692200i	$-0.243358\pi$
$522$		0			0
$523$	−11.4369			−0.500102			−0.250051	−	0.968233i	$-0.580447\pi$
$523$	−11.4369			−0.500102			−0.250051	+	0.968233i	$0.580447\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.3870			−0.452464
$528$		0			0
$529$	9.63057			0.418721
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−18.7302			−0.811294
$534$		0			0
$535$	−58.1399			−2.51361
$536$		0			0
$537$		0			0
$538$		0			0
$539$			21.3345i			0.918943i
$540$		0			0
$541$	8.52806			0.366650			0.183325	−	0.983052i	$-0.441314\pi$
$541$	8.52806			0.366650			0.183325	+	0.983052i	$0.441314\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			2.06753i			0.0885631i
$546$		0			0
$547$	44.8307			1.91682			0.958412	−	0.285389i	$-0.0921228\pi$
$547$	44.8307			1.91682			0.958412	+	0.285389i	$0.0921228\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	17.6560	+	36.2011i	0.752173	+	1.54222i
$552$		0			0
$553$			17.7901i			0.756513i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−8.65729			−0.366821			−0.183411	−	0.983036i	$-0.558714\pi$
$557$	−8.65729			−0.366821			−0.183411	+	0.983036i	$0.558714\pi$
$558$		0			0
$559$	−23.9785			−1.01418
$560$		0			0
$561$		0			0
$562$		0			0
$563$	29.7187			1.25250			0.626248	−	0.779624i	$-0.284590\pi$
$563$	29.7187			1.25250			0.626248	+	0.779624i	$0.284590\pi$
$564$		0			0
$565$			71.1519i			2.99338i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	28.0673i		−	1.17664i	−0.808628	−	0.588321i	$-0.799789\pi$
$569$		−	28.0673i		−	1.17664i	0.808628	−	0.588321i	$-0.200211\pi$
$570$		0			0
$571$			0.727070i			0.0304269i	0.999884	+	0.0152135i	$0.00484278\pi$
$571$			0.727070i			0.0304269i	−0.999884	+	0.0152135i	$0.995157\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			42.3478i			1.76603i
$576$		0			0
$577$	45.3727			1.88889			0.944446	−	0.328667i	$-0.106599\pi$
$577$	45.3727			1.88889			0.944446	+	0.328667i	$0.106599\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	16.0789			0.667066
$582$		0			0
$583$	5.26066			0.217874
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	4.49113i		−	0.185369i	−0.995696	−	0.0926843i	$-0.970455\pi$
$587$		−	4.49113i		−	0.185369i	0.995696	−	0.0926843i	$-0.0295447\pi$
$588$		0			0
$589$	−18.5992	+	9.07124i	−0.766368	+	0.373774i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	29.7969			1.22361			0.611807	−	0.791007i	$-0.290443\pi$
$593$	29.7969			1.22361			0.611807	+	0.791007i	$0.290443\pi$
$594$		0			0
$595$			11.5910i			0.475184i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	28.9919			1.18458			0.592289	−	0.805726i	$-0.298224\pi$
$599$	28.9919			1.18458			0.592289	+	0.805726i	$0.298224\pi$
$600$		0			0
$601$		−	29.8466i		−	1.21747i	−0.793374	−	0.608734i	$-0.791678\pi$
$601$		−	29.8466i		−	1.21747i	0.793374	−	0.608734i	$-0.208322\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−21.0047			−0.853964
$606$		0			0
$607$	14.2939			0.580173			0.290087	−	0.957000i	$-0.406316\pi$
$607$	14.2939			0.580173			0.290087	+	0.957000i	$0.406316\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−22.0737			−0.893004
$612$		0			0
$613$	21.3088			0.860656			0.430328	−	0.902673i	$-0.358398\pi$
$613$	21.3088			0.860656			0.430328	+	0.902673i	$0.358398\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4.62982			0.186390			0.0931948	−	0.995648i	$-0.470292\pi$
$617$	4.62982			0.186390			0.0931948	+	0.995648i	$0.470292\pi$
$618$		0			0
$619$			34.3137i			1.37919i	0.724197	+	0.689593i	$0.242211\pi$
$619$			34.3137i			1.37919i	−0.724197	+	0.689593i	$0.757789\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2.81013			−0.112586
$624$		0			0
$625$	51.2284			2.04914
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	0.926724i		−	0.0369509i
$630$		0			0
$631$			13.1912i			0.525134i	0.964914	+	0.262567i	$0.0845690\pi$
$631$			13.1912i			0.525134i	−0.964914	+	0.262567i	$0.915431\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	9.05734			0.359430
$636$		0			0
$637$			25.5894i			1.01389i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	2.42008i		−	0.0955875i	−0.998857	−	0.0477937i	$-0.984781\pi$
$641$		−	2.42008i		−	0.0955875i	0.998857	−	0.0477937i	$-0.0152190\pi$
$642$		0			0
$643$		−	45.6120i		−	1.79876i	−0.437167	−	0.899381i	$-0.644018\pi$
$643$		−	45.6120i		−	1.79876i	0.437167	−	0.899381i	$-0.355982\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			1.50508i			0.0591707i	0.999562	+	0.0295854i	$0.00941869\pi$
$647$			1.50508i			0.0591707i	−0.999562	+	0.0295854i	$0.990581\pi$
$648$		0			0
$649$		−	43.2521i		−	1.69779i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	19.0340			0.744857			0.372428	−	0.928061i	$-0.378525\pi$
$653$	19.0340			0.744857			0.372428	+	0.928061i	$0.378525\pi$
$654$		0			0
$655$			28.7545i			1.12353i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	29.4822			1.14846			0.574231	−	0.818693i	$-0.305301\pi$
$659$	29.4822			1.14846			0.574231	+	0.818693i	$0.305301\pi$
$660$		0			0
$661$			36.0038i			1.40039i	0.713953	+	0.700193i	$0.246903\pi$
$661$			36.0038i			1.40039i	−0.713953	+	0.700193i	$0.753097\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	10.1227	+	20.7551i	0.392542	+	0.804849i
$666$		0			0
$667$	33.7862			1.30821
$668$		0			0
$669$		0			0
$670$		0			0
$671$			35.2137i			1.35941i
$672$		0			0
$673$		−	42.4050i		−	1.63459i	−0.576219	−	0.817296i	$-0.695472\pi$
$673$		−	42.4050i		−	1.63459i	0.576219	−	0.817296i	$-0.304528\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			32.0407i			1.23142i	0.787971	+	0.615712i	$0.211132\pi$
$677$			32.0407i			1.23142i	−0.787971	+	0.615712i	$0.788868\pi$
$678$		0			0
$679$	16.2115			0.622140
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−29.1650			−1.11597			−0.557983	−	0.829852i	$-0.688425\pi$
$683$	−29.1650			−1.11597			−0.557983	+	0.829852i	$0.688425\pi$
$684$		0			0
$685$	78.9326			3.01586
$686$		0			0
$687$		0			0
$688$		0			0
$689$	6.30983			0.240385
$690$		0			0
$691$		−	43.2714i		−	1.64612i	−0.567953	−	0.823061i	$-0.692265\pi$
$691$		−	43.2714i		−	1.64612i	0.567953	−	0.823061i	$-0.307735\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	80.8817i		−	3.06802i
$696$		0			0
$697$		−	8.49964i		−	0.321947i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−3.42770			−0.129462			−0.0647312	−	0.997903i	$-0.520619\pi$
$701$	−3.42770			−0.129462			−0.0647312	+	0.997903i	$0.520619\pi$
$702$		0			0
$703$	−0.809333	−	1.65942i	−0.0305246	−	0.0625861i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			5.64023i			0.212123i
$708$		0			0
$709$	0.207672			0.00779928			0.00389964	−	0.999992i	$-0.498759\pi$
$709$	0.207672			0.00779928			0.00389964	+	0.999992i	$0.498759\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			17.3585i			0.650081i
$714$		0			0
$715$	−78.9201			−2.95144
$716$		0			0
$717$		0			0
$718$		0			0
$719$			20.3808i			0.760075i	0.924971	+	0.380037i	$0.124089\pi$
$719$			20.3808i			0.760075i	−0.924971	+	0.380037i	$0.875911\pi$
$720$		0			0
$721$			2.91552i			0.108580i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	107.018i		−	3.97455i
$726$		0			0
$727$		−	1.43663i		−	0.0532817i	−0.999645	−	0.0266408i	$-0.991519\pi$
$727$		−	1.43663i		−	0.0532817i	0.999645	−	0.0266408i	$-0.00848105\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	10.8813i		−	0.402460i
$732$		0			0
$733$	−31.7700			−1.17345			−0.586726	−	0.809786i	$-0.699583\pi$
$733$	−31.7700			−1.17345			−0.586726	+	0.809786i	$0.699583\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	35.2891i		−	1.29989i
$738$		0			0
$739$			19.6684i			0.723515i	0.932272	+	0.361757i	$0.117823\pi$
$739$			19.6684i			0.723515i	−0.932272	+	0.361757i	$0.882177\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−15.1554			−0.555997			−0.277999	−	0.960581i	$-0.589671\pi$
$743$	−15.1554			−0.555997			−0.277999	+	0.960581i	$0.589671\pi$
$744$		0			0
$745$	54.2939			1.98918
$746$		0			0
$747$		0			0
$748$		0			0
$749$			18.5750i			0.678717i
$750$		0			0
$751$	17.5957			0.642077			0.321038	−	0.947066i	$-0.395968\pi$
$751$	17.5957			0.642077			0.321038	+	0.947066i	$0.395968\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	52.2336			1.90097
$756$		0			0
$757$	−21.6651			−0.787431			−0.393716	−	0.919232i	$-0.628811\pi$
$757$	−21.6651			−0.787431			−0.393716	+	0.919232i	$0.628811\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	8.92470			0.323520			0.161760	−	0.986830i	$-0.448283\pi$
$761$	8.92470			0.323520			0.161760	+	0.986830i	$0.448283\pi$
$762$		0			0
$763$	0.660552			0.0239136
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	51.8782i		−	1.87321i
$768$		0			0
$769$	−42.6341			−1.53743			−0.768713	−	0.639594i	$-0.779103\pi$
$769$	−42.6341			−1.53743			−0.768713	+	0.639594i	$0.779103\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			42.4839i			1.52804i	0.645192	+	0.764020i	$0.276777\pi$
$773$			42.4839i			1.52804i	−0.645192	+	0.764020i	$0.723223\pi$
$774$		0			0
$775$	54.9833			1.97506
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7.42297	−	15.2197i	−0.265955	−	0.545302i
$780$		0			0
$781$			57.1954i			2.04661i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−30.5158			−1.08915
$786$		0			0
$787$	−9.72238			−0.346565			−0.173283	−	0.984872i	$-0.555437\pi$
$787$	−9.72238			−0.346565			−0.173283	+	0.984872i	$0.555437\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	22.7322			0.808265
$792$		0			0
$793$			42.2366i			1.49987i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			0.694777i			0.0246103i	0.999924	+	0.0123051i	$0.00391694\pi$
$797$			0.694777i			0.0246103i	−0.999924	+	0.0123051i	$0.996083\pi$
$798$		0			0
$799$		−	10.0169i		−	0.354372i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	20.4518i		−	0.721728i
$804$		0			0
$805$	19.3706			0.682723
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−24.1408			−0.848746			−0.424373	−	0.905487i	$-0.639505\pi$
$809$	−24.1408			−0.848746			−0.424373	+	0.905487i	$0.639505\pi$
$810$		0			0
$811$	20.5252			0.720737			0.360368	−	0.932810i	$-0.382651\pi$
$811$	20.5252			0.720737			0.360368	+	0.932810i	$0.382651\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	45.0683i		−	1.57867i
$816$		0			0
$817$	−9.50295	−	19.4844i	−0.332466	−	0.681672i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−24.3004			−0.848088			−0.424044	−	0.905642i	$-0.639390\pi$
$821$	−24.3004			−0.848088			−0.424044	+	0.905642i	$0.639390\pi$
$822$		0			0
$823$		−	7.35743i		−	0.256464i	−0.991744	−	0.128232i	$-0.959070\pi$
$823$		−	7.35743i		−	0.256464i	0.991744	−	0.128232i	$-0.0409302\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	29.4966			1.02570			0.512849	−	0.858479i	$-0.328590\pi$
$827$	29.4966			1.02570			0.512849	+	0.858479i	$0.328590\pi$
$828$		0			0
$829$		−	49.1132i		−	1.70577i	−0.522096	−	0.852886i	$-0.674850\pi$
$829$		−	49.1132i		−	1.70577i	0.522096	−	0.852886i	$-0.325150\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−11.6123			−0.402343
$834$		0			0
$835$	4.93678			0.170844
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−16.5485			−0.571319			−0.285660	−	0.958331i	$-0.592213\pi$
$839$	−16.5485			−0.571319			−0.285660	+	0.958331i	$0.592213\pi$
$840$		0			0
$841$	−56.3818			−1.94420
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−41.7228			−1.43531
$846$		0			0
$847$			6.71077i			0.230585i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.54872			−0.0530895
$852$		0			0
$853$	11.5867			0.396720			0.198360	−	0.980129i	$-0.436438\pi$
$853$	11.5867			0.396720			0.198360	+	0.980129i	$0.436438\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			17.7630i			0.606772i	0.952868	+	0.303386i	$0.0981171\pi$
$857$			17.7630i			0.606772i	−0.952868	+	0.303386i	$0.901883\pi$
$858$		0			0
$859$		−	22.2031i		−	0.757561i	−0.925487	−	0.378781i	$-0.876344\pi$
$859$		−	22.2031i		−	0.757561i	0.925487	−	0.378781i	$-0.123656\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−56.8320			−1.93458			−0.967291	−	0.253668i	$-0.918363\pi$
$863$	−56.8320			−1.93458			−0.967291	+	0.253668i	$0.918363\pi$
$864$		0			0
$865$		−	5.32915i		−	0.181197i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	54.9674i		−	1.86464i
$870$		0			0
$871$		−	42.3271i		−	1.43420i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	34.8681i		−	1.17876i
$876$		0			0
$877$		−	36.8509i		−	1.24437i	−0.782871	−	0.622184i	$-0.786246\pi$
$877$		−	36.8509i		−	1.24437i	0.782871	−	0.622184i	$-0.213754\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−34.9534			−1.17761			−0.588804	−	0.808276i	$-0.700401\pi$
$881$	−34.9534			−1.17761			−0.588804	+	0.808276i	$0.700401\pi$
$882$		0			0
$883$		−	9.71297i		−	0.326867i	−0.986554	−	0.163434i	$-0.947743\pi$
$883$		−	9.71297i		−	0.326867i	0.986554	−	0.163434i	$-0.0522570\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	10.1014			0.339170			0.169585	−	0.985516i	$-0.445757\pi$
$887$	10.1014			0.339170			0.169585	+	0.985516i	$0.445757\pi$
$888$		0			0
$889$		−	2.89372i		−	0.0970522i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−8.74802	−	17.9365i	−0.292741	−	0.600223i
$894$		0			0
$895$	27.9719			0.934997
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	43.8671i		−	1.46305i
$900$		0			0
$901$			2.86336i			0.0953925i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			61.4861i			2.04387i
$906$		0			0
$907$	−2.05122			−0.0681098			−0.0340549	−	0.999420i	$-0.510842\pi$
$907$	−2.05122			−0.0681098			−0.0340549	+	0.999420i	$0.510842\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	7.18640			0.238096			0.119048	−	0.992888i	$-0.462016\pi$
$911$	7.18640			0.238096			0.119048	+	0.992888i	$0.462016\pi$
$912$		0			0
$913$	−49.6802			−1.64417
$914$		0			0
$915$		0			0
$916$		0			0
$917$	9.18674			0.303373
$918$		0			0
$919$		−	42.3484i		−	1.39695i	−0.715637	−	0.698473i	$-0.753863\pi$
$919$		−	42.3484i		−	1.39695i	0.715637	−	0.698473i	$-0.246137\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			68.6024i			2.25807i
$924$		0			0
$925$			4.90559i			0.161295i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−33.2577			−1.09115			−0.545575	−	0.838062i	$-0.683689\pi$
$929$	−33.2577			−1.09115			−0.545575	+	0.838062i	$0.683689\pi$
$930$		0			0
$931$	−20.7934	+	10.1414i	−0.681475	+	0.332370i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	35.8134i		−	1.17122i
$936$		0			0
$937$	−13.0794			−0.427286			−0.213643	−	0.976912i	$-0.568533\pi$
$937$	−13.0794			−0.427286			−0.213643	+	0.976912i	$0.568533\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			14.9278i			0.486634i	0.969947	+	0.243317i	$0.0782355\pi$
$941$			14.9278i			0.486634i	−0.969947	+	0.243317i	$0.921764\pi$
$942$		0			0
$943$	−14.2044			−0.462559
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	45.9387i		−	1.49281i	−0.665494	−	0.746403i	$-0.731779\pi$
$947$		−	45.9387i		−	1.49281i	0.665494	−	0.746403i	$-0.268221\pi$
$948$		0			0
$949$		−	24.5307i		−	0.796299i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	39.4977i		−	1.27946i	−0.768601	−	0.639729i	$-0.779047\pi$
$953$		−	39.4977i		−	1.27946i	0.768601	−	0.639729i	$-0.220953\pi$
$954$		0			0
$955$		−	21.5433i		−	0.697126i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	25.2181i		−	0.814334i
$960$		0			0
$961$	−8.46219			−0.272974
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	75.8922i		−	2.44306i
$966$		0			0
$967$		−	0.851558i		−	0.0273843i	−0.999906	−	0.0136921i	$-0.995642\pi$
$967$		−	0.851558i		−	0.0273843i	0.999906	−	0.0136921i	$-0.00435848\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−9.61110			−0.308435			−0.154218	−	0.988037i	$-0.549286\pi$
$971$	−9.61110			−0.308435			−0.154218	+	0.988037i	$0.549286\pi$
$972$		0			0
$973$	−25.8408			−0.828418
$974$		0			0
$975$		0			0
$976$		0			0
$977$			40.6164i			1.29943i	0.760177	+	0.649717i	$0.225112\pi$
$977$			40.6164i			1.29943i	−0.760177	+	0.649717i	$0.774888\pi$
$978$		0			0
$979$	8.68266			0.277499
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−3.35142			−0.106894			−0.0534469	−	0.998571i	$-0.517021\pi$
$983$	−3.35142			−0.106894			−0.0534469	+	0.998571i	$0.517021\pi$
$984$		0			0
$985$	−3.16175			−0.100742
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−18.1846			−0.578237
$990$		0			0
$991$	−34.2994			−1.08956			−0.544779	−	0.838580i	$-0.683386\pi$
$991$	−34.2994			−1.08956			−0.544779	+	0.838580i	$0.683386\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			5.29769i			0.167948i
$996$		0			0
$997$	−51.9747			−1.64606			−0.823028	−	0.568001i	$-0.807717\pi$
$997$	−51.9747			−1.64606			−0.823028	+	0.568001i	$0.807717\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5472.2.k.d.2431.39	yes	40
3.2	odd	2	inner	5472.2.k.d.2431.1	✓	40
4.3	odd	2	inner	5472.2.k.d.2431.37	yes	40
12.11	even	2	inner	5472.2.k.d.2431.3	yes	40
19.18	odd	2	inner	5472.2.k.d.2431.40	yes	40
57.56	even	2	inner	5472.2.k.d.2431.2	yes	40
76.75	even	2	inner	5472.2.k.d.2431.38	yes	40
228.227	odd	2	inner	5472.2.k.d.2431.4	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5472.2.k.d.2431.1	✓	40	3.2	odd	2	inner
5472.2.k.d.2431.2	yes	40	57.56	even	2	inner
5472.2.k.d.2431.3	yes	40	12.11	even	2	inner
5472.2.k.d.2431.4	yes	40	228.227	odd	2	inner
5472.2.k.d.2431.37	yes	40	4.3	odd	2	inner
5472.2.k.d.2431.38	yes	40	76.75	even	2	inner
5472.2.k.d.2431.39	yes	40	1.1	even	1	trivial
5472.2.k.d.2431.40	yes	40	19.18	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5472.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.07207 q^{5} -1.30098i q^{7} +O(q^{10})\)	\(q+4.07207 q^{5} -1.30098i q^{7} +4.01973i q^{11} +4.82142i q^{13} -2.18793 q^{17} +(-3.91777 + 1.91078i) q^{19} +3.65642i q^{23} +11.5818 q^{25} -9.24023i q^{29} +4.74740 q^{31} -5.29769i q^{35} +0.423562i q^{37} +3.88478i q^{41} +4.97333i q^{43} +4.57825i q^{47} +5.30745 q^{49} -1.30871i q^{53} +16.3686i q^{55} -10.7599 q^{59} +8.76020 q^{61} +19.6332i q^{65} -8.77896 q^{67} +14.2287 q^{71} -5.08785 q^{73} +5.22959 q^{77} -13.6744 q^{79} +12.3591i q^{83} -8.90941 q^{85} -2.16001i q^{89} +6.27257 q^{91} +(-15.9534 + 7.78083i) q^{95} +12.4610i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q+O(q^{10}) \) 40 * q	\( 40 q + 40 q^{25} - 24 q^{49} + 48 q^{61} + 16 q^{73} - 16 q^{85}+O(q^{100}) \) 40 * q + 40 * q^25 - 24 * q^49 + 48 * q^61 + 16 * q^73 - 16 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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