LMFDB - Embedded newform 3024.2.k.l.1889.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3024,2,Mod(1889,3024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3024, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3024.1889");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3024 = 2^{4} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3024.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$24.1467615712$
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} - 6 x^{15} + 18 x^{14} - 36 x^{13} - 45 x^{12} + 306 x^{11} - 378 x^{10} + 1704 x^{9} + \cdots + 256 $ x^16 - 6x^15 + 18x^14 - 36x^13 - 45x^12 + 306x^11 - 378x^10 + 1704x^9 + 917x^8 - 12522x^7 + 27090x^6 - 35292x^5 + 30948x^4 - 18000x^3 + 7200x^2 - 1920*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.12
Root			$0.247793 - 0.924776i$ of defining polynomial
Character	$\chi$	$=$	3024.1889
Dual form			3024.2.k.l.1889.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.22223 q^{5} +(1.48321 + 2.19091i) q^{7} +O(q^{10})$	$q+1.22223 q^{5} +(1.48321 + 2.19091i) q^{7} +3.31156i q^{11} -1.60032i q^{13} -6.11387 q^{17} -1.35397i q^{19} +1.11697i q^{23} -3.50615 q^{25} +9.39495i q^{29} +7.00177i q^{31} +(1.81283 + 2.67780i) q^{35} -11.7065 q^{37} +5.86752 q^{41} +8.58954 q^{43} -3.15959 q^{47} +(-2.60018 + 6.49916i) q^{49} +0.539730i q^{53} +4.04749i q^{55} -6.87004 q^{59} -7.40968i q^{61} -1.95596i q^{65} +5.74008 q^{67} +4.81587i q^{71} -14.5107i q^{73} +(-7.25533 + 4.91174i) q^{77} -1.15055 q^{79} -7.27794 q^{83} -7.47257 q^{85} -11.4545 q^{89} +(3.50615 - 2.37360i) q^{91} -1.65486i q^{95} -4.04749i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{7}+O(q^{10}) $ 16 * q + 2 * q^7	$ 16 q + 2 q^{7} + 36 q^{25} - 8 q^{37} - 20 q^{43} + 2 q^{49} - 44 q^{67} - 40 q^{79} + 16 q^{85} - 36 q^{91}+O(q^{100}) $ 16 * q + 2 * q^7 + 36 * q^25 - 8 * q^37 - 20 * q^43 + 2 * q^49 - 44 * q^67 - 40 * q^79 + 16 * q^85 - 36 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1135$	$2593$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$5$	1.22223			0.546599			0.273299	−	0.961929i	$-0.411885\pi$
$5$	1.22223			0.546599			0.273299	+	0.961929i	$0.411885\pi$
$6$		0			0
$7$	1.48321	+	2.19091i	0.560601	+	0.828086i
$7$	1.48321	+	2.19091i	0.560601	+	0.828086i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			3.31156i			0.998472i	0.866466	+	0.499236i	$0.166386\pi$
$11$			3.31156i			0.998472i	−0.866466	+	0.499236i	$0.833614\pi$
$12$		0			0
$13$		−	1.60032i		−	0.443848i	−0.975064	−	0.221924i	$-0.928766\pi$
$13$		−	1.60032i		−	0.443848i	0.975064	−	0.221924i	$-0.0712337\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.11387			−1.48283			−0.741416	−	0.671046i	$-0.765845\pi$
$17$	−6.11387			−1.48283			−0.741416	+	0.671046i	$0.765845\pi$
$18$		0			0
$19$		−	1.35397i		−	0.310621i	−0.987866	−	0.155311i	$-0.950362\pi$
$19$		−	1.35397i		−	0.310621i	0.987866	−	0.155311i	$-0.0496378\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.11697i			0.232904i	0.993196	+	0.116452i	$0.0371521\pi$
$23$			1.11697i			0.232904i	−0.993196	+	0.116452i	$0.962848\pi$
$24$		0			0
$25$	−3.50615			−0.701230
$26$		0			0
$27$		0			0
$28$		0			0
$29$			9.39495i			1.74460i	0.488973	+	0.872299i	$0.337372\pi$
$29$			9.39495i			1.74460i	−0.488973	+	0.872299i	$0.662628\pi$
$30$		0			0
$31$			7.00177i			1.25756i	0.777585	+	0.628778i	$0.216445\pi$
$31$			7.00177i			1.25756i	−0.777585	+	0.628778i	$0.783555\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.81283	+	2.67780i	0.306424	+	0.452631i
$36$		0			0
$37$	−11.7065			−1.92454			−0.962269	−	0.272101i	$-0.912282\pi$
$37$	−11.7065			−1.92454			−0.962269	+	0.272101i	$0.912282\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.86752			0.916353			0.458177	−	0.888861i	$-0.348503\pi$
$41$	5.86752			0.916353			0.458177	+	0.888861i	$0.348503\pi$
$42$		0			0
$43$	8.58954			1.30989			0.654946	−	0.755676i	$-0.272691\pi$
$43$	8.58954			1.30989			0.654946	+	0.755676i	$0.272691\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.15959			−0.460873			−0.230437	−	0.973087i	$-0.574015\pi$
$47$	−3.15959			−0.460873			−0.230437	+	0.973087i	$0.574015\pi$
$48$		0			0
$49$	−2.60018	+	6.49916i	−0.371454	+	0.928451i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			0.539730i			0.0741376i	0.999313	+	0.0370688i	$0.0118021\pi$
$53$			0.539730i			0.0741376i	−0.999313	+	0.0370688i	$0.988198\pi$
$54$		0			0
$55$			4.04749i			0.545764i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.87004			−0.894403			−0.447202	−	0.894433i	$-0.647579\pi$
$59$	−6.87004			−0.894403			−0.447202	+	0.894433i	$0.647579\pi$
$60$		0			0
$61$		−	7.40968i		−	0.948712i	−0.880333	−	0.474356i	$-0.842681\pi$
$61$		−	7.40968i		−	0.948712i	0.880333	−	0.474356i	$-0.157319\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	1.95596i		−	0.242607i
$66$		0			0
$67$	5.74008			0.701263			0.350631	−	0.936514i	$-0.385967\pi$
$67$	5.74008			0.701263			0.350631	+	0.936514i	$0.385967\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			4.81587i			0.571539i	0.958298	+	0.285769i	$0.0922491\pi$
$71$			4.81587i			0.571539i	−0.958298	+	0.285769i	$0.907751\pi$
$72$		0			0
$73$		−	14.5107i		−	1.69834i	−0.528116	−	0.849172i	$-0.677102\pi$
$73$		−	14.5107i		−	1.69834i	0.528116	−	0.849172i	$-0.322898\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−7.25533	+	4.91174i	−0.826821	+	0.559744i
$78$		0			0
$79$	−1.15055			−0.129447			−0.0647234	−	0.997903i	$-0.520617\pi$
$79$	−1.15055			−0.129447			−0.0647234	+	0.997903i	$0.520617\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.27794			−0.798858			−0.399429	−	0.916764i	$-0.630792\pi$
$83$	−7.27794			−0.798858			−0.399429	+	0.916764i	$0.630792\pi$
$84$		0			0
$85$	−7.47257			−0.810514
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−11.4545			−1.21417			−0.607085	−	0.794637i	$-0.707661\pi$
$89$	−11.4545			−1.21417			−0.607085	+	0.794637i	$0.707661\pi$
$90$		0			0
$91$	3.50615	−	2.37360i	0.367544	−	0.248821i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	1.65486i		−	0.169785i
$96$		0			0
$97$		−	4.04749i		−	0.410961i	−0.978661	−	0.205480i	$-0.934124\pi$
$97$		−	4.04749i		−	0.410961i	0.978661	−	0.205480i	$-0.0658756\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.62565			0.360766			0.180383	−	0.983596i	$-0.442266\pi$
$101$	3.62565			0.360766			0.180383	+	0.983596i	$0.442266\pi$
$102$		0			0
$103$		−	6.68185i		−	0.658383i	−0.944263	−	0.329191i	$-0.893224\pi$
$103$		−	6.68185i		−	0.658383i	0.944263	−	0.329191i	$-0.106776\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			15.7563i			1.52322i	0.648036	+	0.761610i	$0.275591\pi$
$107$			15.7563i			1.52322i	−0.648036	+	0.761610i	$0.724409\pi$
$108$		0			0
$109$	7.77367			0.744582			0.372291	−	0.928116i	$-0.378572\pi$
$109$	7.77367			0.744582			0.372291	+	0.928116i	$0.378572\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	0.111204i		−	0.0104612i	−0.999986	−	0.00523058i	$-0.998335\pi$
$113$		−	0.111204i		−	0.0104612i	0.999986	−	0.00523058i	$-0.00166495\pi$
$114$		0			0
$115$			1.36519i			0.127305i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−9.06815	−	13.3949i	−0.831276	−	1.22791i
$120$		0			0
$121$	0.0335804			0.00305277
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−10.3965			−0.929890
$126$		0			0
$127$	2.11697			0.187851			0.0939253	−	0.995579i	$-0.470059\pi$
$127$	2.11697			0.187851			0.0939253	+	0.995579i	$0.470059\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.38434			−0.470432			−0.235216	−	0.971943i	$-0.575580\pi$
$131$	−5.38434			−0.470432			−0.235216	+	0.971943i	$0.575580\pi$
$132$		0			0
$133$	2.96642	−	2.00822i	0.257221	−	0.174134i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			10.2108i			0.872369i	0.899857	+	0.436184i	$0.143670\pi$
$137$			10.2108i			0.872369i	−0.899857	+	0.436184i	$0.856330\pi$
$138$		0			0
$139$			20.9151i			1.77400i	0.461774	+	0.886998i	$0.347213\pi$
$139$			20.9151i			1.77400i	−0.461774	+	0.886998i	$0.652787\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	5.29954			0.443170
$144$		0			0
$145$			11.4828i			0.953595i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			12.1274i			0.993518i	0.867889	+	0.496759i	$0.165477\pi$
$149$			12.1274i			0.993518i	−0.867889	+	0.496759i	$0.834523\pi$
$150$		0			0
$151$	14.2837			1.16239			0.581197	−	0.813763i	$-0.302584\pi$
$151$	14.2837			1.16239			0.581197	+	0.813763i	$0.302584\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			8.55779i			0.687378i
$156$		0			0
$157$		−	7.36716i		−	0.587964i	−0.955811	−	0.293982i	$-0.905019\pi$
$157$		−	7.36716i		−	0.587964i	0.955811	−	0.293982i	$-0.0949805\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.44718	+	1.65670i	−0.192865	+	0.130566i
$162$		0			0
$163$	−3.57724			−0.280191			−0.140095	−	0.990138i	$-0.544741\pi$
$163$	−3.57724			−0.280191			−0.140095	+	0.990138i	$0.544741\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	15.9964			1.23784			0.618918	−	0.785456i	$-0.287571\pi$
$167$	15.9964			1.23784			0.618918	+	0.785456i	$0.287571\pi$
$168$		0			0
$169$	10.4390			0.802999
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.07012			0.461502			0.230751	−	0.973013i	$-0.425882\pi$
$173$	6.07012			0.461502			0.230751	+	0.973013i	$0.425882\pi$
$174$		0			0
$175$	−5.20035	−	7.68166i	−0.393110	−	0.580679i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			17.8004i			1.33046i	0.746638	+	0.665230i	$0.231667\pi$
$179$			17.8004i			1.33046i	−0.746638	+	0.665230i	$0.768333\pi$
$180$		0			0
$181$			10.6983i			0.795197i	0.917559	+	0.397599i	$0.130156\pi$
$181$			10.6983i			0.795197i	−0.917559	+	0.397599i	$0.869844\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−14.3081			−1.05195
$186$		0			0
$187$		−	20.2464i		−	1.48057i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			12.6903i			0.918236i	0.888375	+	0.459118i	$0.151835\pi$
$191$			12.6903i			0.918236i	−0.888375	+	0.459118i	$0.848165\pi$
$192$		0			0
$193$	−8.69890			−0.626161			−0.313080	−	0.949727i	$-0.601361\pi$
$193$	−8.69890			−0.626161			−0.313080	+	0.949727i	$0.601361\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	6.62128i		−	0.471747i	−0.971784	−	0.235873i	$-0.924205\pi$
$197$		−	6.62128i		−	0.471747i	0.971784	−	0.235873i	$-0.0757951\pi$
$198$		0			0
$199$			19.6802i			1.39509i	0.716541	+	0.697545i	$0.245724\pi$
$199$			19.6802i			1.39509i	−0.716541	+	0.697545i	$0.754276\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−20.5835	+	13.9347i	−1.44468	+	0.978022i
$204$		0			0
$205$	7.17147			0.500877
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.48374			0.310147
$210$		0			0
$211$	−11.4390			−0.787492			−0.393746	−	0.919219i	$-0.628821\pi$
$211$	−11.4390			−0.787492			−0.393746	+	0.919219i	$0.628821\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	10.4984			0.715985
$216$		0			0
$217$	−15.3403	+	10.3851i	−1.04136	+	0.704987i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			9.78413i			0.658152i
$222$		0			0
$223$		−	17.8616i		−	1.19610i	−0.801458	−	0.598051i	$-0.795942\pi$
$223$		−	17.8616i		−	1.19610i	0.801458	−	0.598051i	$-0.204058\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−25.7215			−1.70719			−0.853597	−	0.520934i	$-0.825584\pi$
$227$	−25.7215			−1.70719			−0.853597	+	0.520934i	$0.825584\pi$
$228$		0			0
$229$			15.0232i			0.992760i	0.868105	+	0.496380i	$0.165338\pi$
$229$			15.0232i			0.992760i	−0.868105	+	0.496380i	$0.834662\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			3.03542i			0.198857i	0.995045	+	0.0994284i	$0.0317014\pi$
$233$			3.03542i			0.198857i	−0.995045	+	0.0994284i	$0.968299\pi$
$234$		0			0
$235$	−3.86175			−0.251913
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	6.90110i		−	0.446395i	−0.974773	−	0.223197i	$-0.928351\pi$
$239$		−	6.90110i		−	0.446395i	0.974773	−	0.223197i	$-0.0716494\pi$
$240$		0			0
$241$		−	26.6819i		−	1.71874i	−0.511358	−	0.859368i	$-0.670857\pi$
$241$		−	26.6819i		−	1.71874i	0.511358	−	0.859368i	$-0.329143\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.17802	+	7.94348i	−0.203036	+	0.507490i
$246$		0			0
$247$	−2.16677			−0.137869
$248$		0			0
$249$		0			0
$250$		0			0
$251$	19.3729			1.22281			0.611405	−	0.791318i	$-0.290605\pi$
$251$	19.3729			1.22281			0.611405	+	0.791318i	$0.290605\pi$
$252$		0			0
$253$	−3.69890			−0.232548
$254$		0			0
$255$		0			0
$256$		0			0
$257$	18.1363			1.13131			0.565656	−	0.824641i	$-0.308623\pi$
$257$	18.1363			1.13131			0.565656	+	0.824641i	$0.308623\pi$
$258$		0			0
$259$	−17.3632	−	25.6479i	−1.07890	−	1.59368i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	12.4072i		−	0.765063i	−0.923942	−	0.382532i	$-0.875052\pi$
$263$		−	12.4072i		−	0.765063i	0.923942	−	0.382532i	$-0.124948\pi$
$264$		0			0
$265$			0.659675i			0.0405235i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8.80468			−0.536831			−0.268416	−	0.963303i	$-0.586500\pi$
$269$	−8.80468			−0.536831			−0.268416	+	0.963303i	$0.586500\pi$
$270$		0			0
$271$		−	3.81237i		−	0.231585i	−0.993273	−	0.115792i	$-0.963059\pi$
$271$		−	3.81237i		−	0.231585i	0.993273	−	0.115792i	$-0.0369408\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	11.6108i		−	0.700159i
$276$		0			0
$277$	25.5722			1.53648			0.768242	−	0.640160i	$-0.221132\pi$
$277$	25.5722			1.53648			0.768242	+	0.640160i	$0.221132\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			31.5683i			1.88320i	0.336728	+	0.941602i	$0.390680\pi$
$281$			31.5683i			1.88320i	−0.336728	+	0.941602i	$0.609320\pi$
$282$		0			0
$283$			10.3214i			0.613546i	0.951783	+	0.306773i	$0.0992493\pi$
$283$			10.3214i			0.613546i	−0.951783	+	0.306773i	$0.900751\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8.70277	+	12.8552i	0.513708	+	0.758819i
$288$		0			0
$289$	20.3794			1.19879
$290$		0			0
$291$		0			0
$292$		0			0
$293$	17.7722			1.03826			0.519130	−	0.854695i	$-0.326256\pi$
$293$	17.7722			1.03826			0.519130	+	0.854695i	$0.326256\pi$
$294$		0			0
$295$	−8.39678			−0.488880
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.78750			0.103374
$300$		0			0
$301$	12.7401	+	18.8189i	0.734326	+	1.08470i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	9.05634i		−	0.518565i
$306$		0			0
$307$		−	5.38163i		−	0.307146i	−0.988137	−	0.153573i	$-0.950922\pi$
$307$		−	5.38163i		−	0.307146i	0.988137	−	0.153573i	$-0.0490780\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−26.7118			−1.51469			−0.757343	−	0.653018i	$-0.773503\pi$
$311$	−26.7118			−1.51469			−0.757343	+	0.653018i	$0.773503\pi$
$312$		0			0
$313$			19.2579i			1.08852i	0.838917	+	0.544259i	$0.183189\pi$
$313$			19.2579i			1.08852i	−0.838917	+	0.544259i	$0.816811\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	6.56571i		−	0.368767i	−0.982854	−	0.184383i	$-0.940971\pi$
$317$		−	6.56571i		−	0.368767i	0.982854	−	0.184383i	$-0.0590288\pi$
$318$		0			0
$319$	−31.1119			−1.74193
$320$		0			0
$321$		0			0
$322$		0			0
$323$			8.27798i			0.460599i
$324$		0			0
$325$			5.61095i			0.311239i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.68633	−	6.92238i	−0.258366	−	0.381643i
$330$		0			0
$331$	7.27614			0.399933			0.199966	−	0.979803i	$-0.435917\pi$
$331$	7.27614			0.399933			0.199966	+	0.979803i	$0.435917\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	7.01571			0.383309
$336$		0			0
$337$	3.04588			0.165920			0.0829598	−	0.996553i	$-0.473563\pi$
$337$	3.04588			0.165920			0.0829598	+	0.996553i	$0.473563\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−23.1868			−1.25563
$342$		0			0
$343$	−18.0957	+	3.94286i	−0.977075	+	0.212894i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			11.8675i			0.637082i	0.947909	+	0.318541i	$0.103193\pi$
$347$			11.8675i			0.637082i	−0.947909	+	0.318541i	$0.896807\pi$
$348$		0			0
$349$		−	22.6968i		−	1.21493i	−0.794346	−	0.607466i	$-0.792186\pi$
$349$		−	22.6968i		−	1.21493i	0.794346	−	0.607466i	$-0.207814\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−10.7139			−0.570245			−0.285123	−	0.958491i	$-0.592034\pi$
$353$	−10.7139			−0.570245			−0.285123	+	0.958491i	$0.592034\pi$
$354$		0			0
$355$			5.88611i			0.312402i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	37.1531i		−	1.96087i	−0.196855	−	0.980433i	$-0.563073\pi$
$359$		−	37.1531i		−	1.96087i	0.196855	−	0.980433i	$-0.436927\pi$
$360$		0			0
$361$	17.1668			0.903514
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	17.7354i		−	0.928313i
$366$		0			0
$367$			21.0454i			1.09856i	0.835639	+	0.549280i	$0.185098\pi$
$367$			21.0454i			1.09856i	−0.835639	+	0.549280i	$0.814902\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.18250	+	0.800532i	−0.0613923	+	0.0415616i
$372$		0			0
$373$	−0.928141			−0.0480573			−0.0240287	−	0.999711i	$-0.507649\pi$
$373$	−0.928141			−0.0480573			−0.0240287	+	0.999711i	$0.507649\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	15.0349			0.774336
$378$		0			0
$379$	−18.2625			−0.938080			−0.469040	−	0.883177i	$-0.655400\pi$
$379$	−18.2625			−0.938080			−0.469040	+	0.883177i	$0.655400\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	19.4262			0.992633			0.496317	−	0.868142i	$-0.334685\pi$
$383$	19.4262			0.992633			0.496317	+	0.868142i	$0.334685\pi$
$384$		0			0
$385$	−8.86769	+	6.00328i	−0.451939	+	0.305955i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	37.1137i		−	1.88174i	−0.338765	−	0.940871i	$-0.610009\pi$
$389$		−	37.1137i		−	1.88174i	0.338765	−	0.940871i	$-0.389991\pi$
$390$		0			0
$391$		−	6.82900i		−	0.345357i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.40624			−0.0707554
$396$		0			0
$397$			36.2726i			1.82047i	0.414092	+	0.910235i	$0.364099\pi$
$397$			36.2726i			1.82047i	−0.414092	+	0.910235i	$0.635901\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			5.67798i			0.283545i	0.989899	+	0.141772i	$0.0452801\pi$
$401$			5.67798i			0.283545i	−0.989899	+	0.141772i	$0.954720\pi$
$402$		0			0
$403$	11.2051			0.558163
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	38.7668i		−	1.92160i
$408$		0			0
$409$			25.9255i			1.28194i	0.767568	+	0.640968i	$0.221467\pi$
$409$			25.9255i			1.28194i	−0.767568	+	0.640968i	$0.778533\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−10.1897	−	15.0516i	−0.501403	−	0.740643i
$414$		0			0
$415$	−8.89533			−0.436655
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8.09103			−0.395273			−0.197636	−	0.980275i	$-0.563327\pi$
$419$	−8.09103			−0.395273			−0.197636	+	0.980275i	$0.563327\pi$
$420$		0			0
$421$	7.08732			0.345415			0.172707	−	0.984973i	$-0.444749\pi$
$421$	7.08732			0.345415			0.172707	+	0.984973i	$0.444749\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	21.4361			1.03981
$426$		0			0
$427$	16.2339	−	10.9901i	0.785615	−	0.531848i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	15.1791i		−	0.731150i	−0.930782	−	0.365575i	$-0.880872\pi$
$431$		−	15.1791i		−	0.731150i	0.930782	−	0.365575i	$-0.119128\pi$
$432$		0			0
$433$		−	17.5583i		−	0.843800i	−0.906642	−	0.421900i	$-0.861363\pi$
$433$		−	17.5583i		−	0.843800i	0.906642	−	0.421900i	$-0.138637\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.51234			0.0723449
$438$		0			0
$439$		−	40.2321i		−	1.92017i	−0.279700	−	0.960087i	$-0.590235\pi$
$439$		−	40.2321i		−	1.92017i	0.279700	−	0.960087i	$-0.409765\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			22.0440i			1.04734i	0.851920	+	0.523672i	$0.175438\pi$
$443$			22.0440i			1.04734i	−0.851920	+	0.523672i	$0.824562\pi$
$444$		0			0
$445$	−14.0000			−0.663664
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	33.6910i		−	1.58998i	−0.606625	−	0.794988i	$-0.707477\pi$
$449$		−	33.6910i		−	1.58998i	0.606625	−	0.794988i	$-0.292523\pi$
$450$		0			0
$451$			19.4306i			0.914953i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.28533	−	2.90110i	0.200899	−	0.136005i
$456$		0			0
$457$	3.92814			0.183751			0.0918753	−	0.995771i	$-0.470714\pi$
$457$	3.92814			0.183751			0.0918753	+	0.995771i	$0.470714\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	34.1525			1.59064			0.795320	−	0.606190i	$-0.207303\pi$
$461$	34.1525			1.59064			0.795320	+	0.606190i	$0.207303\pi$
$462$		0			0
$463$	3.43036			0.159422			0.0797112	−	0.996818i	$-0.474600\pi$
$463$	3.43036			0.159422			0.0797112	+	0.996818i	$0.474600\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−27.1436			−1.25606			−0.628028	−	0.778191i	$-0.716138\pi$
$467$	−27.1436			−1.25606			−0.628028	+	0.778191i	$0.716138\pi$
$468$		0			0
$469$	8.51375	+	12.5760i	0.393128	+	0.580706i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			28.4448i			1.30789i
$474$		0			0
$475$			4.74721i			0.217817i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	32.3877			1.47983			0.739915	−	0.672700i	$-0.234866\pi$
$479$	32.3877			1.47983			0.739915	+	0.672700i	$0.234866\pi$
$480$		0			0
$481$			18.7341i			0.854202i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	4.94697i		−	0.224631i
$486$		0			0
$487$	24.3257			1.10230			0.551151	−	0.834405i	$-0.314189\pi$
$487$	24.3257			1.10230			0.551151	+	0.834405i	$0.314189\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			34.4906i			1.55654i	0.627930	+	0.778270i	$0.283903\pi$
$491$			34.4906i			1.55654i	−0.627930	+	0.778270i	$0.716097\pi$
$492$		0			0
$493$		−	57.4395i		−	2.58694i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−10.5511	+	7.14295i	−0.473283	+	0.320405i
$498$		0			0
$499$	−14.0285			−0.628003			−0.314002	−	0.949422i	$-0.601670\pi$
$499$	−14.0285			−0.628003			−0.314002	+	0.949422i	$0.601670\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	37.1573			1.65676			0.828382	−	0.560164i	$-0.189262\pi$
$503$	37.1573			1.65676			0.828382	+	0.560164i	$0.189262\pi$
$504$		0			0
$505$	4.43139			0.197194
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−16.9141			−0.749703			−0.374852	−	0.927085i	$-0.622306\pi$
$509$	−16.9141			−0.749703			−0.374852	+	0.927085i	$0.622306\pi$
$510$		0			0
$511$	31.7915	−	21.5223i	1.40638	−	0.952093i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	8.16677i		−	0.359871i
$516$		0			0
$517$		−	10.4632i		−	0.460169i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−21.5989			−0.946267			−0.473133	−	0.880991i	$-0.656877\pi$
$521$	−21.5989			−0.946267			−0.473133	+	0.880991i	$0.656877\pi$
$522$		0			0
$523$		−	3.24610i		−	0.141942i	−0.997478	−	0.0709710i	$-0.977390\pi$
$523$		−	3.24610i		−	0.141942i	0.997478	−	0.0709710i	$-0.0226098\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	42.8080i		−	1.86474i
$528$		0			0
$529$	21.7524			0.945756
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	9.38989i		−	0.406721i
$534$		0			0
$535$			19.2579i			0.832590i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−21.5223	−	8.61064i	−0.927033	−	0.370887i
$540$		0			0
$541$	−32.7267			−1.40703			−0.703514	−	0.710681i	$-0.748387\pi$
$541$	−32.7267			−1.40703			−0.703514	+	0.710681i	$0.748387\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	9.50122			0.406988
$546$		0			0
$547$	19.1830			0.820206			0.410103	−	0.912039i	$-0.365493\pi$
$547$	19.1830			0.820206			0.410103	+	0.912039i	$0.365493\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	12.7204			0.541909
$552$		0			0
$553$	−1.70650	−	2.52075i	−0.0725679	−	0.107193i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			26.1751i			1.10908i	0.832158	+	0.554538i	$0.187105\pi$
$557$			26.1751i			1.10908i	−0.832158	+	0.554538i	$0.812895\pi$
$558$		0			0
$559$		−	13.7460i		−	0.581393i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	27.7130			1.16797			0.583983	−	0.811766i	$-0.301494\pi$
$563$	27.7130			1.16797			0.583983	+	0.811766i	$0.301494\pi$
$564$		0			0
$565$		−	0.135917i		−	0.00571805i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	25.7661i		−	1.08017i	−0.841611	−	0.540085i	$-0.818392\pi$
$569$		−	25.7661i		−	1.08017i	0.841611	−	0.540085i	$-0.181608\pi$
$570$		0			0
$571$	−30.5895			−1.28013			−0.640066	−	0.768320i	$-0.721093\pi$
$571$	−30.5895			−1.28013			−0.640066	+	0.768320i	$0.721093\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	3.91626i		−	0.163319i
$576$		0			0
$577$		−	32.4033i		−	1.34897i	−0.738289	−	0.674484i	$-0.764366\pi$
$577$		−	32.4033i		−	1.34897i	0.738289	−	0.674484i	$-0.235634\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−10.7947	−	15.9453i	−0.447840	−	0.661523i
$582$		0			0
$583$	−1.78735			−0.0740243
$584$		0			0
$585$		0			0
$586$		0			0
$587$	21.1925			0.874709			0.437354	−	0.899289i	$-0.355916\pi$
$587$	21.1925			0.874709			0.437354	+	0.899289i	$0.355916\pi$
$588$		0			0
$589$	9.48017			0.390624
$590$		0			0
$591$		0			0
$592$		0			0
$593$	46.5951			1.91343			0.956715	−	0.291027i	$-0.0939968\pi$
$593$	46.5951			1.91343			0.956715	+	0.291027i	$0.0939968\pi$
$594$		0			0
$595$	−11.0834	−	16.3717i	−0.454375	−	0.671175i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	14.6412i		−	0.598222i	−0.954218	−	0.299111i	$-0.903310\pi$
$599$		−	14.6412i		−	0.598222i	0.954218	−	0.299111i	$-0.0966902\pi$
$600$		0			0
$601$			11.1517i			0.454885i	0.973791	+	0.227443i	$0.0730364\pi$
$601$			11.1517i			0.454885i	−0.973791	+	0.227443i	$0.926964\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.0410431			0.00166864
$606$		0			0
$607$			14.7259i			0.597708i	0.954299	+	0.298854i	$0.0966043\pi$
$607$			14.7259i			0.597708i	−0.954299	+	0.298854i	$0.903396\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			5.05634i			0.204558i
$612$		0			0
$613$	−40.7589			−1.64623			−0.823117	−	0.567871i	$-0.807767\pi$
$613$	−40.7589			−1.64623			−0.823117	+	0.567871i	$0.807767\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	28.2549i		−	1.13750i	−0.822511	−	0.568749i	$-0.807428\pi$
$617$		−	28.2549i		−	1.13750i	0.822511	−	0.568749i	$-0.192572\pi$
$618$		0			0
$619$			0.538166i			0.0216307i	0.999942	+	0.0108154i	$0.00344270\pi$
$619$			0.538166i			0.0216307i	−0.999942	+	0.0108154i	$0.996557\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−16.9894	−	25.0957i	−0.680664	−	1.00544i
$624$		0			0
$625$	4.82383			0.192953
$626$		0			0
$627$		0			0
$628$		0			0
$629$	71.5721			2.85377
$630$		0			0
$631$	16.1416			0.642586			0.321293	−	0.946980i	$-0.395883\pi$
$631$	16.1416			0.642586			0.321293	+	0.946980i	$0.395883\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.58743			0.102679
$636$		0			0
$637$	10.4007	+	4.16111i	0.412091	+	0.164869i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	13.5205i		−	0.534029i	−0.963692	−	0.267015i	$-0.913963\pi$
$641$		−	13.5205i		−	0.534029i	0.963692	−	0.267015i	$-0.0860372\pi$
$642$		0			0
$643$			29.1685i			1.15029i	0.818051	+	0.575146i	$0.195055\pi$
$643$			29.1685i			1.15029i	−0.818051	+	0.575146i	$0.804945\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−29.3787			−1.15499			−0.577497	−	0.816393i	$-0.695970\pi$
$647$	−29.3787			−1.15499			−0.577497	+	0.816393i	$0.695970\pi$
$648$		0			0
$649$		−	22.7505i		−	0.893037i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			15.8726i			0.621142i	0.950550	+	0.310571i	$0.100520\pi$
$653$			15.8726i			0.621142i	−0.950550	+	0.310571i	$0.899480\pi$
$654$		0			0
$655$	−6.58091			−0.257137
$656$		0			0
$657$		0			0
$658$		0			0
$659$			3.06603i			0.119436i	0.998215	+	0.0597178i	$0.0190201\pi$
$659$			3.06603i			0.119436i	−0.998215	+	0.0597178i	$0.980980\pi$
$660$		0			0
$661$		−	43.3592i		−	1.68648i	−0.537540	−	0.843239i	$-0.680646\pi$
$661$		−	43.3592i		−	1.68648i	0.537540	−	0.843239i	$-0.319354\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3.62565	−	2.45451i	0.140597	−	0.0951817i
$666$		0			0
$667$	−10.4939			−0.406324
$668$		0			0
$669$		0			0
$670$		0			0
$671$	24.5376			0.947263
$672$		0			0
$673$	−16.8733			−0.650417			−0.325209	−	0.945642i	$-0.605435\pi$
$673$	−16.8733			−0.650417			−0.325209	+	0.945642i	$0.605435\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	4.77260			0.183426			0.0917130	−	0.995785i	$-0.470766\pi$
$677$	4.77260			0.183426			0.0917130	+	0.995785i	$0.470766\pi$
$678$		0			0
$679$	8.86769	−	6.00328i	0.340311	−	0.230385i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	13.9097i		−	0.532241i	−0.963940	−	0.266120i	$-0.914258\pi$
$683$		−	13.9097i		−	0.532241i	0.963940	−	0.266120i	$-0.0857419\pi$
$684$		0			0
$685$			12.4800i			0.476836i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.863738			0.0329058
$690$		0			0
$691$			17.0116i			0.647151i	0.946202	+	0.323575i	$0.104885\pi$
$691$			17.0116i			0.647151i	−0.946202	+	0.323575i	$0.895115\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			25.5631i			0.969664i
$696$		0			0
$697$	−35.8733			−1.35880
$698$		0			0
$699$		0			0
$700$		0			0
$701$			21.5779i			0.814988i	0.913208	+	0.407494i	$0.133597\pi$
$701$			21.5779i			0.814988i	−0.913208	+	0.407494i	$0.866403\pi$
$702$		0			0
$703$			15.8502i			0.597802i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	5.37760	+	7.94348i	0.202246	+	0.298745i
$708$		0			0
$709$	−11.1542			−0.418906			−0.209453	−	0.977819i	$-0.567168\pi$
$709$	−11.1542			−0.418906			−0.209453	+	0.977819i	$0.567168\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−7.82076			−0.292890
$714$		0			0
$715$	6.47727			0.242236
$716$		0			0
$717$		0			0
$718$		0			0
$719$	17.5141			0.653166			0.326583	−	0.945168i	$-0.394103\pi$
$719$	17.5141			0.653166			0.326583	+	0.945168i	$0.394103\pi$
$720$		0			0
$721$	14.6393	−	9.91059i	0.545198	−	0.369090i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	32.9401i		−	1.22336i
$726$		0			0
$727$			18.2920i			0.678411i	0.940712	+	0.339206i	$0.110158\pi$
$727$			18.2920i			0.678411i	−0.940712	+	0.339206i	$0.889842\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−52.5153			−1.94235
$732$		0			0
$733$			23.9149i			0.883315i	0.897184	+	0.441658i	$0.145609\pi$
$733$			23.9149i			0.883315i	−0.897184	+	0.441658i	$0.854391\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			19.0086i			0.700192i
$738$		0			0
$739$	17.6989			0.651065			0.325532	−	0.945531i	$-0.394456\pi$
$739$	17.6989			0.651065			0.325532	+	0.945531i	$0.394456\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	21.1300i		−	0.775184i	−0.921831	−	0.387592i	$-0.873307\pi$
$743$		−	21.1300i		−	0.775184i	0.921831	−	0.387592i	$-0.126693\pi$
$744$		0			0
$745$			14.8225i			0.543056i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−34.5207	+	23.3699i	−1.26136	+	0.853918i
$750$		0			0
$751$	51.6928			1.88630			0.943149	−	0.332370i	$-0.107848\pi$
$751$	51.6928			1.88630			0.943149	+	0.332370i	$0.107848\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	17.4580			0.635363
$756$		0			0
$757$	−0.0248524			−0.000903274			−0.000451637	−	1.00000i	$-0.500144\pi$
$757$	−0.0248524			−0.000903274			−0.000451637		1.00000i	$0.500144\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.930657			0.0337363			0.0168682	−	0.999858i	$-0.494630\pi$
$761$	0.930657			0.0337363			0.0168682	+	0.999858i	$0.494630\pi$
$762$		0			0
$763$	11.5300	+	17.0314i	0.417413	+	0.616578i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			10.9942i			0.396979i
$768$		0			0
$769$			13.6327i			0.491609i	0.969319	+	0.245805i	$0.0790521\pi$
$769$			13.6327i			0.491609i	−0.969319	+	0.245805i	$0.920948\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	11.8294			0.425472			0.212736	−	0.977110i	$-0.431762\pi$
$773$	11.8294			0.425472			0.212736	+	0.977110i	$0.431762\pi$
$774$		0			0
$775$		−	24.5493i		−	0.881836i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	7.94443i		−	0.284639i
$780$		0			0
$781$	−15.9480			−0.570666
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	9.00438i		−	0.321380i
$786$		0			0
$787$			27.6561i			0.985834i	0.870076	+	0.492917i	$0.164069\pi$
$787$			27.6561i			0.985834i	−0.870076	+	0.492917i	$0.835931\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	0.243637	−	0.164938i	0.00866274	−	0.00586453i
$792$		0			0
$793$	−11.8578			−0.421084
$794$		0			0
$795$		0			0
$796$		0			0
$797$	0.974413			0.0345155			0.0172577	−	0.999851i	$-0.494506\pi$
$797$	0.974413			0.0345155			0.0172577	+	0.999851i	$0.494506\pi$
$798$		0			0
$799$	19.3173			0.683398
$800$		0			0
$801$		0			0
$802$		0			0
$803$	48.0529			1.69575
$804$		0			0
$805$	−2.99102	+	2.02487i	−0.105419	+	0.0713672i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	20.4715i		−	0.719741i	−0.933002	−	0.359871i	$-0.882821\pi$
$809$		−	20.4715i		−	0.719741i	0.933002	−	0.359871i	$-0.117179\pi$
$810$		0			0
$811$			41.6484i			1.46247i	0.682123	+	0.731237i	$0.261057\pi$
$811$			41.6484i			1.46247i	−0.682123	+	0.731237i	$0.738943\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.37221			−0.153152
$816$		0			0
$817$		−	11.6299i		−	0.406880i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	43.2166i		−	1.50827i	−0.656720	−	0.754135i	$-0.728057\pi$
$821$		−	43.2166i		−	1.50827i	0.656720	−	0.754135i	$-0.271943\pi$
$822$		0			0
$823$	9.16172			0.319357			0.159679	−	0.987169i	$-0.448954\pi$
$823$	9.16172			0.319357			0.159679	+	0.987169i	$0.448954\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	12.7458i		−	0.443217i	−0.975136	−	0.221608i	$-0.928869\pi$
$827$		−	12.7458i		−	0.443217i	0.975136	−	0.221608i	$-0.0711306\pi$
$828$		0			0
$829$		−	3.92018i		−	0.136154i	−0.997680	−	0.0680768i	$-0.978314\pi$
$829$		−	3.92018i		−	0.136154i	0.997680	−	0.0680768i	$-0.0216863\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	15.8972	−	39.7350i	0.550804	−	1.37674i
$834$		0			0
$835$	19.5513			0.676599
$836$		0			0
$837$		0			0
$838$		0			0
$839$	53.2940			1.83991			0.919957	−	0.392019i	$-0.128223\pi$
$839$	53.2940			1.83991			0.919957	+	0.392019i	$0.128223\pi$
$840$		0			0
$841$	−59.2650			−2.04362
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12.7589			0.438918
$846$		0			0
$847$	0.0498068	+	0.0735717i	0.00171138	+	0.00252795i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	13.0758i		−	0.448232i
$852$		0			0
$853$		−	27.2338i		−	0.932468i	−0.884661	−	0.466234i	$-0.845610\pi$
$853$		−	27.2338i		−	0.932468i	0.884661	−	0.466234i	$-0.154390\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−46.0202			−1.57202			−0.786010	−	0.618214i	$-0.787857\pi$
$857$	−46.0202			−1.57202			−0.786010	+	0.618214i	$0.787857\pi$
$858$		0			0
$859$		−	41.6542i		−	1.42122i	−0.703584	−	0.710612i	$-0.748418\pi$
$859$		−	41.6542i		−	1.42122i	0.703584	−	0.710612i	$-0.251582\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	41.2375i		−	1.40374i	−0.712305	−	0.701871i	$-0.752349\pi$
$863$		−	41.2375i		−	1.40374i	0.712305	−	0.701871i	$-0.247651\pi$
$864$		0			0
$865$	7.41909			0.252257
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	3.81011i		−	0.129249i
$870$		0			0
$871$		−	9.18595i		−	0.311254i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−15.4202	−	22.7778i	−0.521297	−	0.770029i
$876$		0			0
$877$	13.1469			0.443938			0.221969	−	0.975054i	$-0.428752\pi$
$877$	13.1469			0.443938			0.221969	+	0.975054i	$0.428752\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	22.9485			0.773154			0.386577	−	0.922257i	$-0.373657\pi$
$881$	22.9485			0.773154			0.386577	+	0.922257i	$0.373657\pi$
$882$		0			0
$883$	46.1693			1.55372			0.776861	−	0.629673i	$-0.216811\pi$
$883$	46.1693			1.55372			0.776861	+	0.629673i	$0.216811\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	27.9382			0.938072			0.469036	−	0.883179i	$-0.344601\pi$
$887$	27.9382			0.938072			0.469036	+	0.883179i	$0.344601\pi$
$888$		0			0
$889$	3.13991	+	4.63809i	0.105309	+	0.155556i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			4.27798i			0.143157i
$894$		0			0
$895$			21.7562i			0.727228i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−65.7813			−2.19393
$900$		0			0
$901$		−	3.29984i		−	0.109934i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			13.0758i			0.434654i
$906$		0			0
$907$	4.91661			0.163253			0.0816267	−	0.996663i	$-0.473988\pi$
$907$	4.91661			0.163253			0.0816267	+	0.996663i	$0.473988\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	32.1936i		−	1.06662i	−0.845920	−	0.533310i	$-0.820948\pi$
$911$		−	32.1936i		−	1.06662i	0.845920	−	0.533310i	$-0.179052\pi$
$912$		0			0
$913$		−	24.1013i		−	0.797637i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−7.98610	−	11.7966i	−0.263724	−	0.389558i
$918$		0			0
$919$	−33.2289			−1.09612			−0.548060	−	0.836439i	$-0.684633\pi$
$919$	−33.2289			−1.09612			−0.548060	+	0.836439i	$0.684633\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	7.70692			0.253676
$924$		0			0
$925$	41.0448			1.34954
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.26452			0.0414875			0.0207437	−	0.999785i	$-0.493397\pi$
$929$	1.26452			0.0414875			0.0207437	+	0.999785i	$0.493397\pi$
$930$		0			0
$931$	8.79965	+	3.52055i	0.288397	+	0.115381i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	24.7458i		−	0.809276i
$936$		0			0
$937$		−	26.1127i		−	0.853065i	−0.904472	−	0.426533i	$-0.859735\pi$
$937$		−	26.1127i		−	0.853065i	0.904472	−	0.426533i	$-0.140265\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.25105			0.0407830			0.0203915	−	0.999792i	$-0.493509\pi$
$941$	1.25105			0.0407830			0.0203915	+	0.999792i	$0.493509\pi$
$942$		0			0
$943$			6.55383i			0.213422i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			44.5809i			1.44868i	0.689440	+	0.724342i	$0.257857\pi$
$947$			44.5809i			1.44868i	−0.689440	+	0.724342i	$0.742143\pi$
$948$		0			0
$949$	−23.2216			−0.753806
$950$		0			0
$951$		0			0
$952$		0			0
$953$			35.2115i			1.14061i	0.821432	+	0.570307i	$0.193176\pi$
$953$			35.2115i			1.14061i	−0.821432	+	0.570307i	$0.806824\pi$
$954$		0			0
$955$			15.5105i			0.501907i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−22.3710	+	15.1448i	−0.722397	+	0.489051i
$960$		0			0
$961$	−18.0249			−0.581447
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−10.6321			−0.342259
$966$		0			0
$967$	−45.7600			−1.47154			−0.735771	−	0.677230i	$-0.763180\pi$
$967$	−45.7600			−1.47154			−0.735771	+	0.677230i	$0.763180\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−3.45116			−0.110753			−0.0553766	−	0.998466i	$-0.517636\pi$
$971$	−3.45116			−0.110753			−0.0553766	+	0.998466i	$0.517636\pi$
$972$		0			0
$973$	−45.8231	+	31.0215i	−1.46902	+	0.994503i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	51.2902i		−	1.64092i	−0.571704	−	0.820460i	$-0.693718\pi$
$977$		−	51.2902i		−	1.64092i	0.571704	−	0.820460i	$-0.306282\pi$
$978$		0			0
$979$		−	37.9321i		−	1.21232i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	30.3484			0.967964			0.483982	−	0.875078i	$-0.339190\pi$
$983$	30.3484			0.967964			0.483982	+	0.875078i	$0.339190\pi$
$984$		0			0
$985$		−	8.09274i		−	0.257856i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			9.59423i			0.305079i
$990$		0			0
$991$	7.11407			0.225986			0.112993	−	0.993596i	$-0.463956\pi$
$991$	7.11407			0.225986			0.112993	+	0.993596i	$0.463956\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			24.0537i			0.762555i
$996$		0			0
$997$			28.2710i			0.895353i	0.894196	+	0.447676i	$0.147748\pi$
$997$			28.2710i			0.895353i	−0.894196	+	0.447676i	$0.852252\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3024.2.k.l.1889.12		16
3.2	odd	2	inner	3024.2.k.l.1889.6		16
4.3	odd	2		1512.2.k.b.377.11	yes	16
7.6	odd	2	inner	3024.2.k.l.1889.5		16
12.11	even	2		1512.2.k.b.377.5	✓	16
21.20	even	2	inner	3024.2.k.l.1889.11		16
28.27	even	2		1512.2.k.b.377.6	yes	16
84.83	odd	2		1512.2.k.b.377.12	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.k.b.377.5	✓	16	12.11	even	2
1512.2.k.b.377.6	yes	16	28.27	even	2
1512.2.k.b.377.11	yes	16	4.3	odd	2
1512.2.k.b.377.12	yes	16	84.83	odd	2
3024.2.k.l.1889.5		16	7.6	odd	2	inner
3024.2.k.l.1889.6		16	3.2	odd	2	inner
3024.2.k.l.1889.11		16	21.20	even	2	inner
3024.2.k.l.1889.12		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3024.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.22223 q^{5} +(1.48321 + 2.19091i) q^{7} +O(q^{10})\)	\(q+1.22223 q^{5} +(1.48321 + 2.19091i) q^{7} +3.31156i q^{11} -1.60032i q^{13} -6.11387 q^{17} -1.35397i q^{19} +1.11697i q^{23} -3.50615 q^{25} +9.39495i q^{29} +7.00177i q^{31} +(1.81283 + 2.67780i) q^{35} -11.7065 q^{37} +5.86752 q^{41} +8.58954 q^{43} -3.15959 q^{47} +(-2.60018 + 6.49916i) q^{49} +0.539730i q^{53} +4.04749i q^{55} -6.87004 q^{59} -7.40968i q^{61} -1.95596i q^{65} +5.74008 q^{67} +4.81587i q^{71} -14.5107i q^{73} +(-7.25533 + 4.91174i) q^{77} -1.15055 q^{79} -7.27794 q^{83} -7.47257 q^{85} -11.4545 q^{89} +(3.50615 - 2.37360i) q^{91} -1.65486i q^{95} -4.04749i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{7}+O(q^{10}) \) 16 * q + 2 * q^7	\( 16 q + 2 q^{7} + 36 q^{25} - 8 q^{37} - 20 q^{43} + 2 q^{49} - 44 q^{67} - 40 q^{79} + 16 q^{85} - 36 q^{91}+O(q^{100}) \) 16 * q + 2 * q^7 + 36 * q^25 - 8 * q^37 - 20 * q^43 + 2 * q^49 - 44 * q^67 - 40 * q^79 + 16 * q^85 - 36 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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