LMFDB - Embedded newform 6012.2.h.a.3005.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6012,2,Mod(3005,6012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6012.3005");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6012 = 2^{2} \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6012.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$48.0060616952$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3005.5
Character	$\chi$	$=$	6012.3005
Dual form			6012.2.h.a.3005.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.47368 q^{5} +3.93299 q^{7} +O(q^{10})$	$q-3.47368 q^{5} +3.93299 q^{7} -4.34291i q^{11} +0.788280i q^{13} -5.32815 q^{17} +4.26219 q^{19} -6.69278 q^{23} +7.06648 q^{25} +5.99865i q^{29} -6.79664 q^{31} -13.6619 q^{35} +4.66119i q^{37} +4.62083 q^{41} -11.6022i q^{43} +12.6481i q^{47} +8.46837 q^{49} +0.406978 q^{53} +15.0859i q^{55} +6.56137 q^{59} -4.94490 q^{61} -2.73823i q^{65} +3.40874i q^{67} -0.507955 q^{71} -2.74265i q^{73} -17.0806i q^{77} +7.44663i q^{79} +6.51617 q^{83} +18.5083 q^{85} -15.1276i q^{89} +3.10029i q^{91} -14.8055 q^{95} -3.32370 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q+O(q^{10}) $ 56 * q	$ 56 q + 8 q^{19} + 64 q^{25} - 8 q^{31} + 56 q^{49} - 8 q^{61} + 32 q^{85} - 48 q^{97}+O(q^{100}) $ 56 * q + 8 * q^19 + 64 * q^25 - 8 * q^31 + 56 * q^49 - 8 * q^61 + 32 * q^85 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6012\mathbb{Z}\right)^\times$.

$n$	$3007$	$3341$	$4681$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	−3.47368			−1.55348			−0.776739	−	0.629822i	$-0.783128\pi$
$5$	−3.47368			−1.55348			−0.776739	+	0.629822i	$0.783128\pi$
$6$		0			0
$7$	3.93299			1.48653			0.743264	−	0.668998i	$-0.233276\pi$
$7$	3.93299			1.48653			0.743264	+	0.668998i	$0.233276\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	4.34291i		−	1.30944i	−0.755873	−	0.654718i	$-0.772787\pi$
$11$		−	4.34291i		−	1.30944i	0.755873	−	0.654718i	$-0.227213\pi$
$12$		0			0
$13$			0.788280i			0.218629i	0.994007	+	0.109315i	$0.0348656\pi$
$13$			0.788280i			0.218629i	−0.994007	+	0.109315i	$0.965134\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.32815			−1.29227			−0.646133	−	0.763225i	$-0.723615\pi$
$17$	−5.32815			−1.29227			−0.646133	+	0.763225i	$0.723615\pi$
$18$		0			0
$19$	4.26219			0.977812			0.488906	−	0.872336i	$-0.337396\pi$
$19$	4.26219			0.977812			0.488906	+	0.872336i	$0.337396\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.69278			−1.39554			−0.697770	−	0.716321i	$-0.745824\pi$
$23$	−6.69278			−1.39554			−0.697770	+	0.716321i	$0.745824\pi$
$24$		0			0
$25$	7.06648			1.41330
$26$		0			0
$27$		0			0
$28$		0			0
$29$			5.99865i			1.11392i	0.830539	+	0.556961i	$0.188033\pi$
$29$			5.99865i			1.11392i	−0.830539	+	0.556961i	$0.811967\pi$
$30$		0			0
$31$	−6.79664			−1.22071			−0.610356	−	0.792127i	$-0.708974\pi$
$31$	−6.79664			−1.22071			−0.610356	+	0.792127i	$0.708974\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−13.6619			−2.30929
$36$		0			0
$37$			4.66119i			0.766296i	0.923687	+	0.383148i	$0.125160\pi$
$37$			4.66119i			0.766296i	−0.923687	+	0.383148i	$0.874840\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.62083			0.721652			0.360826	−	0.932633i	$-0.382495\pi$
$41$	4.62083			0.721652			0.360826	+	0.932633i	$0.382495\pi$
$42$		0			0
$43$		−	11.6022i		−	1.76931i	−0.466243	−	0.884657i	$-0.654393\pi$
$43$		−	11.6022i		−	1.76931i	0.466243	−	0.884657i	$-0.345607\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			12.6481i			1.84491i	0.386106	+	0.922454i	$0.373820\pi$
$47$			12.6481i			1.84491i	−0.386106	+	0.922454i	$0.626180\pi$
$48$		0			0
$49$	8.46837			1.20977
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.406978			0.0559028			0.0279514	−	0.999609i	$-0.491102\pi$
$53$	0.406978			0.0559028			0.0279514	+	0.999609i	$0.491102\pi$
$54$		0			0
$55$			15.0859i			2.03418i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.56137			0.854218			0.427109	−	0.904200i	$-0.359532\pi$
$59$	6.56137			0.854218			0.427109	+	0.904200i	$0.359532\pi$
$60$		0			0
$61$	−4.94490			−0.633129			−0.316565	−	0.948571i	$-0.602529\pi$
$61$	−4.94490			−0.633129			−0.316565	+	0.948571i	$0.602529\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	2.73823i		−	0.339636i
$66$		0			0
$67$			3.40874i			0.416444i	0.978082	+	0.208222i	$0.0667677\pi$
$67$			3.40874i			0.416444i	−0.978082	+	0.208222i	$0.933232\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−0.507955			−0.0602832			−0.0301416	−	0.999546i	$-0.509596\pi$
$71$	−0.507955			−0.0602832			−0.0301416	+	0.999546i	$0.509596\pi$
$72$		0			0
$73$		−	2.74265i		−	0.321003i	−0.987036	−	0.160501i	$-0.948689\pi$
$73$		−	2.74265i		−	0.321003i	0.987036	−	0.160501i	$-0.0513111\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	17.0806i		−	1.94652i
$78$		0			0
$79$			7.44663i			0.837812i	0.908030	+	0.418906i	$0.137586\pi$
$79$			7.44663i			0.837812i	−0.908030	+	0.418906i	$0.862414\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.51617			0.715242			0.357621	−	0.933867i	$-0.383588\pi$
$83$	6.51617			0.715242			0.357621	+	0.933867i	$0.383588\pi$
$84$		0			0
$85$	18.5083			2.00751
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	15.1276i		−	1.60352i	−0.597648	−	0.801759i	$-0.703898\pi$
$89$		−	15.1276i		−	1.60352i	0.597648	−	0.801759i	$-0.296102\pi$
$90$		0			0
$91$			3.10029i			0.324999i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−14.8055			−1.51901
$96$		0			0
$97$	−3.32370			−0.337471			−0.168735	−	0.985661i	$-0.553968\pi$
$97$	−3.32370			−0.337471			−0.168735	+	0.985661i	$0.553968\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−9.28457			−0.923849			−0.461925	−	0.886919i	$-0.652841\pi$
$101$	−9.28457			−0.923849			−0.461925	+	0.886919i	$0.652841\pi$
$102$		0			0
$103$		−	8.46976i		−	0.834551i	−0.908780	−	0.417275i	$-0.862985\pi$
$103$		−	8.46976i		−	0.834551i	0.908780	−	0.417275i	$-0.137015\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			15.4198i			1.49069i	0.666680	+	0.745344i	$0.267715\pi$
$107$			15.4198i			1.49069i	−0.666680	+	0.745344i	$0.732285\pi$
$108$		0			0
$109$			9.11253i			0.872822i	0.899747	+	0.436411i	$0.143751\pi$
$109$			9.11253i			0.872822i	−0.899747	+	0.436411i	$0.856249\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	18.7429			1.76318			0.881591	−	0.472013i	$-0.156473\pi$
$113$	18.7429			1.76318			0.881591	+	0.472013i	$0.156473\pi$
$114$		0			0
$115$	23.2486			2.16794
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−20.9555			−1.92099
$120$		0			0
$121$	−7.86087			−0.714625
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−7.17830			−0.642047
$126$		0			0
$127$	−18.9978			−1.68579			−0.842893	−	0.538081i	$-0.819149\pi$
$127$	−18.9978			−1.68579			−0.842893	+	0.538081i	$0.819149\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	12.1680			1.06312			0.531562	−	0.847019i	$-0.321605\pi$
$131$	12.1680			1.06312			0.531562	+	0.847019i	$0.321605\pi$
$132$		0			0
$133$	16.7631			1.45355
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	4.83479i		−	0.413064i	−0.978440	−	0.206532i	$-0.933782\pi$
$137$		−	4.83479i		−	0.413064i	0.978440	−	0.206532i	$-0.0662178\pi$
$138$		0			0
$139$			14.2167i			1.20585i	0.797799	+	0.602923i	$0.205997\pi$
$139$			14.2167i			1.20585i	−0.797799	+	0.602923i	$0.794003\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.42343			0.286281
$144$		0			0
$145$		−	20.8374i		−	1.73045i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.40998			0.197433			0.0987165	−	0.995116i	$-0.468526\pi$
$149$	2.40998			0.197433			0.0987165	+	0.995116i	$0.468526\pi$
$150$		0			0
$151$			4.37124i			0.355726i	0.984055	+	0.177863i	$0.0569184\pi$
$151$			4.37124i			0.355726i	−0.984055	+	0.177863i	$0.943082\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	23.6094			1.89635
$156$		0			0
$157$	1.86364			0.148735			0.0743673	−	0.997231i	$-0.476306\pi$
$157$	1.86364			0.148735			0.0743673	+	0.997231i	$0.476306\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−26.3226			−2.07451
$162$		0			0
$163$			22.5643i			1.76737i	0.468079	+	0.883686i	$0.344946\pi$
$163$			22.5643i			1.76737i	−0.468079	+	0.883686i	$0.655054\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.99239	+	8.19464i	0.773234	+	0.634121i
$167$	9.99239	+	8.19464i	0.773234	+	0.634121i
$168$		0			0
$169$	12.3786			0.952201
$170$		0			0
$171$		0			0
$172$		0			0
$173$			10.0210i			0.761885i	0.924599	+	0.380943i	$0.124400\pi$
$173$			10.0210i			0.761885i	−0.924599	+	0.380943i	$0.875600\pi$
$174$		0			0
$175$	27.7924			2.10091
$176$		0			0
$177$		0			0
$178$		0			0
$179$			18.7422i			1.40086i	0.713723	+	0.700428i	$0.247008\pi$
$179$			18.7422i			1.40086i	−0.713723	+	0.700428i	$0.752992\pi$
$180$		0			0
$181$	−10.2459			−0.761572			−0.380786	−	0.924663i	$-0.624347\pi$
$181$	−10.2459			−0.761572			−0.380786	+	0.924663i	$0.624347\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	16.1915i		−	1.19042i
$186$		0			0
$187$			23.1397i			1.69214i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	19.2270i		−	1.39122i	−0.718421	−	0.695608i	$-0.755135\pi$
$191$		−	19.2270i		−	1.39122i	0.718421	−	0.695608i	$-0.244865\pi$
$192$		0			0
$193$			5.61976i			0.404519i	0.979332	+	0.202259i	$0.0648284\pi$
$193$			5.61976i			0.404519i	−0.979332	+	0.202259i	$0.935172\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−15.1539			−1.07967			−0.539836	−	0.841770i	$-0.681514\pi$
$197$	−15.1539			−1.07967			−0.539836	+	0.841770i	$0.681514\pi$
$198$		0			0
$199$	−9.38899			−0.665568			−0.332784	−	0.943003i	$-0.607988\pi$
$199$	−9.38899			−0.665568			−0.332784	+	0.943003i	$0.607988\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			23.5926i			1.65588i
$204$		0			0
$205$	−16.0513			−1.12107
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	18.5103i		−	1.28038i
$210$		0			0
$211$	−5.78342			−0.398147			−0.199074	−	0.979985i	$-0.563793\pi$
$211$	−5.78342			−0.398147			−0.199074	+	0.979985i	$0.563793\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			40.3023i			2.74859i
$216$		0			0
$217$	−26.7311			−1.81462
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	4.20007i		−	0.282528i
$222$		0			0
$223$	−19.3088			−1.29301			−0.646506	−	0.762909i	$-0.723770\pi$
$223$	−19.3088			−1.29301			−0.646506	+	0.762909i	$0.723770\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.98586			0.530040			0.265020	−	0.964243i	$-0.414621\pi$
$227$	7.98586			0.530040			0.265020	+	0.964243i	$0.414621\pi$
$228$		0			0
$229$	24.0329			1.58814			0.794070	−	0.607826i	$-0.207958\pi$
$229$	24.0329			1.58814			0.794070	+	0.607826i	$0.207958\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			14.9229i			0.977630i	0.872388	+	0.488815i	$0.162571\pi$
$233$			14.9229i			0.977630i	−0.872388	+	0.488815i	$0.837429\pi$
$234$		0			0
$235$		−	43.9353i		−	2.86603i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	1.14553i		−	0.0740981i	−0.999313	−	0.0370491i	$-0.988204\pi$
$239$		−	1.14553i		−	0.0740981i	0.999313	−	0.0370491i	$-0.0117958\pi$
$240$		0			0
$241$		−	9.14226i		−	0.588905i	−0.955666	−	0.294452i	$-0.904863\pi$
$241$		−	9.14226i		−	0.588905i	0.955666	−	0.294452i	$-0.0951372\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−29.4164			−1.87935
$246$		0			0
$247$			3.35979i			0.213779i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			5.57929i			0.352162i	0.984376	+	0.176081i	$0.0563420\pi$
$251$			5.57929i			0.352162i	−0.984376	+	0.176081i	$0.943658\pi$
$252$		0			0
$253$			29.0661i			1.82737i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	22.5972			1.40957			0.704787	−	0.709419i	$-0.251043\pi$
$257$	22.5972			1.40957			0.704787	+	0.709419i	$0.251043\pi$
$258$		0			0
$259$			18.3324i			1.13912i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			13.8743i			0.855524i	0.903891	+	0.427762i	$0.140698\pi$
$263$			13.8743i			0.855524i	−0.903891	+	0.427762i	$0.859302\pi$
$264$		0			0
$265$	−1.41371			−0.0868438
$266$		0			0
$267$		0			0
$268$		0			0
$269$	12.5090			0.762690			0.381345	−	0.924433i	$-0.375461\pi$
$269$	12.5090			0.762690			0.381345	+	0.924433i	$0.375461\pi$
$270$		0			0
$271$		−	12.7830i		−	0.776512i	−0.921552	−	0.388256i	$-0.873078\pi$
$271$		−	12.7830i		−	0.776512i	0.921552	−	0.388256i	$-0.126922\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	30.6891i		−	1.85062i
$276$		0			0
$277$			1.74046i			0.104574i	0.998632	+	0.0522869i	$0.0166510\pi$
$277$			1.74046i			0.104574i	−0.998632	+	0.0522869i	$0.983349\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			23.8524i			1.42292i	0.702728	+	0.711458i	$0.251965\pi$
$281$			23.8524i			1.42292i	−0.702728	+	0.711458i	$0.748035\pi$
$282$		0			0
$283$	−21.3744			−1.27057			−0.635287	−	0.772276i	$-0.719118\pi$
$283$	−21.3744			−1.27057			−0.635287	+	0.772276i	$0.719118\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	18.1736			1.07276
$288$		0			0
$289$	11.3892			0.669953
$290$		0			0
$291$		0			0
$292$		0			0
$293$			14.0998i			0.823719i	0.911247	+	0.411859i	$0.135120\pi$
$293$			14.0998i			0.823719i	−0.911247	+	0.411859i	$0.864880\pi$
$294$		0			0
$295$	−22.7921			−1.32701
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	5.27578i		−	0.305106i
$300$		0			0
$301$		−	45.6311i		−	2.63013i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	17.1770			0.983553
$306$		0			0
$307$			9.02299i			0.514969i	0.966282	+	0.257485i	$0.0828937\pi$
$307$			9.02299i			0.514969i	−0.966282	+	0.257485i	$0.917106\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	10.2184i		−	0.579432i	−0.957113	−	0.289716i	$-0.906439\pi$
$311$		−	10.2184i		−	0.579432i	0.957113	−	0.289716i	$-0.0935608\pi$
$312$		0			0
$313$			16.6078i			0.938726i	0.883005	+	0.469363i	$0.155516\pi$
$313$			16.6078i			0.938726i	−0.883005	+	0.469363i	$0.844484\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			25.5050i			1.43250i	0.697842	+	0.716252i	$0.254144\pi$
$317$			25.5050i			1.43250i	−0.697842	+	0.716252i	$0.745856\pi$
$318$		0			0
$319$	26.0516			1.45861
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−22.7096			−1.26359
$324$		0			0
$325$			5.57036i			0.308988i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			49.7446i			2.74251i
$330$		0			0
$331$			27.8955i			1.53328i	0.642080	+	0.766638i	$0.278072\pi$
$331$			27.8955i			1.53328i	−0.642080	+	0.766638i	$0.721928\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	11.8409i		−	0.646937i
$336$		0			0
$337$	25.8957			1.41063			0.705314	−	0.708895i	$-0.250806\pi$
$337$	25.8957			1.41063			0.705314	+	0.708895i	$0.250806\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			29.5172i			1.59845i
$342$		0			0
$343$	5.77508			0.311825
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−27.8704			−1.49616			−0.748081	−	0.663608i	$-0.769024\pi$
$347$	−27.8704			−1.49616			−0.748081	+	0.663608i	$0.769024\pi$
$348$		0			0
$349$			28.6541i			1.53382i	0.641757	+	0.766908i	$0.278206\pi$
$349$			28.6541i			1.53382i	−0.641757	+	0.766908i	$0.721794\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	4.19480i		−	0.223267i	−0.993749	−	0.111633i	$-0.964392\pi$
$353$		−	4.19480i		−	0.223267i	0.993749	−	0.111633i	$-0.0356083\pi$
$354$		0			0
$355$	1.76447			0.0936486
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	8.52346i		−	0.449851i	−0.974376	−	0.224925i	$-0.927786\pi$
$359$		−	8.52346i		−	0.449851i	0.974376	−	0.224925i	$-0.0722138\pi$
$360$		0			0
$361$	−0.833772			−0.0438827
$362$		0			0
$363$		0			0
$364$		0			0
$365$			9.52710i			0.498671i
$366$		0			0
$367$	−0.593513			−0.0309812			−0.0154906	−	0.999880i	$-0.504931\pi$
$367$	−0.593513			−0.0309812			−0.0154906	+	0.999880i	$0.504931\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.60064			0.0831011
$372$		0			0
$373$			24.8382i			1.28607i	0.765836	+	0.643036i	$0.222325\pi$
$373$			24.8382i			1.28607i	−0.765836	+	0.643036i	$0.777675\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.72861			−0.243536
$378$		0			0
$379$			30.1994i			1.55124i	0.631200	+	0.775620i	$0.282563\pi$
$379$			30.1994i			1.55124i	−0.631200	+	0.775620i	$0.717437\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			33.7964i			1.72692i	0.504420	+	0.863458i	$0.331706\pi$
$383$			33.7964i			1.72692i	−0.504420	+	0.863458i	$0.668294\pi$
$384$		0			0
$385$			59.3326i			3.02387i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−20.1882			−1.02358			−0.511791	−	0.859110i	$-0.671018\pi$
$389$	−20.1882			−1.02358			−0.511791	+	0.859110i	$0.671018\pi$
$390$		0			0
$391$	35.6601			1.80341
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	25.8673i		−	1.30152i
$396$		0			0
$397$	−18.5320			−0.930093			−0.465046	−	0.885286i	$-0.653962\pi$
$397$	−18.5320			−0.930093			−0.465046	+	0.885286i	$0.653962\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−16.0402			−0.801007			−0.400504	−	0.916295i	$-0.631165\pi$
$401$	−16.0402			−0.801007			−0.400504	+	0.916295i	$0.631165\pi$
$402$		0			0
$403$		−	5.35765i		−	0.266884i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	20.2432			1.00342
$408$		0			0
$409$	28.4443			1.40648			0.703239	−	0.710953i	$-0.251736\pi$
$409$	28.4443			1.40648			0.703239	+	0.710953i	$0.251736\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	25.8058			1.26982
$414$		0			0
$415$	−22.6351			−1.11111
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	9.24873i		−	0.451830i	−0.974147	−	0.225915i	$-0.927463\pi$
$419$		−	9.24873i		−	0.451830i	0.974147	−	0.225915i	$-0.0725372\pi$
$420$		0			0
$421$	35.8984			1.74958			0.874791	−	0.484501i	$-0.160999\pi$
$421$	35.8984			1.74958			0.874791	+	0.484501i	$0.160999\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−37.6513			−1.82636
$426$		0			0
$427$	−19.4482			−0.941165
$428$		0			0
$429$		0			0
$430$		0			0
$431$			14.0918i			0.678779i	0.940646	+	0.339389i	$0.110220\pi$
$431$			14.0918i			0.678779i	−0.940646	+	0.339389i	$0.889780\pi$
$432$		0			0
$433$	−0.284485			−0.0136715			−0.00683574	−	0.999977i	$-0.502176\pi$
$433$	−0.284485			−0.0136715			−0.00683574	+	0.999977i	$0.502176\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−28.5259			−1.36458
$438$		0			0
$439$			0.454366i			0.0216857i	0.999941	+	0.0108428i	$0.00345145\pi$
$439$			0.454366i			0.0216857i	−0.999941	+	0.0108428i	$0.996549\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−33.4825			−1.59080			−0.795400	−	0.606085i	$-0.792739\pi$
$443$	−33.4825			−1.59080			−0.795400	+	0.606085i	$0.792739\pi$
$444$		0			0
$445$			52.5483i			2.49103i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	18.3532i		−	0.866143i	−0.901360	−	0.433072i	$-0.857430\pi$
$449$		−	18.3532i		−	0.866143i	0.901360	−	0.433072i	$-0.142570\pi$
$450$		0			0
$451$		−	20.0678i		−	0.944957i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	10.7694i		−	0.504879i
$456$		0			0
$457$		−	12.3285i		−	0.576704i	−0.957525	−	0.288352i	$-0.906893\pi$
$457$		−	12.3285i		−	0.576704i	0.957525	−	0.288352i	$-0.0931073\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	11.7458i		−	0.547058i	−0.961864	−	0.273529i	$-0.911809\pi$
$461$		−	11.7458i		−	0.547058i	0.961864	−	0.273529i	$-0.0881910\pi$
$462$		0			0
$463$		−	19.8903i		−	0.924380i	−0.886781	−	0.462190i	$-0.847064\pi$
$463$		−	19.8903i		−	0.924380i	0.886781	−	0.462190i	$-0.152936\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	28.2627i		−	1.30784i	−0.756563	−	0.653921i	$-0.773123\pi$
$467$		−	28.2627i		−	1.30784i	0.756563	−	0.653921i	$-0.226877\pi$
$468$		0			0
$469$			13.4065i			0.619056i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−50.3872			−2.31680
$474$		0			0
$475$	30.1187			1.38194
$476$		0			0
$477$		0			0
$478$		0			0
$479$	33.7430			1.54176			0.770879	−	0.636981i	$-0.219817\pi$
$479$	33.7430			1.54176			0.770879	+	0.636981i	$0.219817\pi$
$480$		0			0
$481$	−3.67432			−0.167535
$482$		0			0
$483$		0			0
$484$		0			0
$485$	11.5455			0.524254
$486$		0			0
$487$			23.3213i			1.05679i	0.848999	+	0.528395i	$0.177206\pi$
$487$			23.3213i			1.05679i	−0.848999	+	0.528395i	$0.822794\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	3.22659i		−	0.145614i	−0.997346	−	0.0728070i	$-0.976804\pi$
$491$		−	3.22659i		−	0.145614i	0.997346	−	0.0728070i	$-0.0231957\pi$
$492$		0			0
$493$		−	31.9617i		−	1.43948i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.99778			−0.0896126
$498$		0			0
$499$		−	35.1020i		−	1.57138i	−0.618621	−	0.785690i	$-0.712308\pi$
$499$		−	35.1020i		−	1.57138i	0.618621	−	0.785690i	$-0.287692\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			2.35604i			0.105051i	0.998620	+	0.0525253i	$0.0167270\pi$
$503$			2.35604i			0.105051i	−0.998620	+	0.0525253i	$0.983273\pi$
$504$		0			0
$505$	32.2517			1.43518
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	4.04422i		−	0.179257i	−0.995975	−	0.0896284i	$-0.971432\pi$
$509$		−	4.04422i		−	0.179257i	0.995975	−	0.0896284i	$-0.0285679\pi$
$510$		0			0
$511$		−	10.7868i		−	0.477180i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			29.4213i			1.29646i
$516$		0			0
$517$	54.9294			2.41579
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−34.7536			−1.52258			−0.761292	−	0.648410i	$-0.775434\pi$
$521$	−34.7536			−1.52258			−0.761292	+	0.648410i	$0.775434\pi$
$522$		0			0
$523$	6.98394			0.305386			0.152693	−	0.988274i	$-0.451205\pi$
$523$	6.98394			0.305386			0.152693	+	0.988274i	$0.451205\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	36.2135			1.57749
$528$		0			0
$529$	21.7933			0.947534
$530$		0			0
$531$		0			0
$532$		0			0
$533$			3.64250i			0.157774i
$534$		0			0
$535$		−	53.5635i		−	2.31575i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	36.7774i		−	1.58411i
$540$		0			0
$541$			13.1092i			0.563606i	0.959472	+	0.281803i	$0.0909325\pi$
$541$			13.1092i			0.563606i	−0.959472	+	0.281803i	$0.909067\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	31.6541i		−	1.35591i
$546$		0			0
$547$		−	23.7068i		−	1.01363i	−0.862055	−	0.506815i	$-0.830823\pi$
$547$		−	23.7068i		−	1.01363i	0.862055	−	0.506815i	$-0.169177\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			25.5674i			1.08921i
$552$		0			0
$553$			29.2875i			1.24543i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			13.9063i			0.589230i	0.955616	+	0.294615i	$0.0951914\pi$
$557$			13.9063i			0.589230i	−0.955616	+	0.294615i	$0.904809\pi$
$558$		0			0
$559$	9.14575			0.386824
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	4.83747i		−	0.203875i	−0.994791	−	0.101938i	$-0.967496\pi$
$563$		−	4.83747i		−	0.203875i	0.994791	−	0.101938i	$-0.0325042\pi$
$564$		0			0
$565$	−65.1069			−2.73907
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2.14692			−0.0900037			−0.0450019	−	0.998987i	$-0.514329\pi$
$569$	−2.14692			−0.0900037			−0.0450019	+	0.998987i	$0.514329\pi$
$570$		0			0
$571$			37.4846i			1.56868i	0.620329	+	0.784342i	$0.286999\pi$
$571$			37.4846i			1.56868i	−0.620329	+	0.784342i	$0.713001\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−47.2944			−1.97231
$576$		0			0
$577$	12.9723			0.540042			0.270021	−	0.962854i	$-0.412969\pi$
$577$	12.9723			0.540042			0.270021	+	0.962854i	$0.412969\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	25.6280			1.06323
$582$		0			0
$583$		−	1.76747i		−	0.0732012i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	28.5888			1.17999			0.589994	−	0.807408i	$-0.299130\pi$
$587$	28.5888			1.17999			0.589994	+	0.807408i	$0.299130\pi$
$588$		0			0
$589$	−28.9685			−1.19363
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−4.67910			−0.192148			−0.0960739	−	0.995374i	$-0.530629\pi$
$593$	−4.67910			−0.192148			−0.0960739	+	0.995374i	$0.530629\pi$
$594$		0			0
$595$	72.7929			2.98422
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	26.9092i		−	1.09948i	−0.835336	−	0.549740i	$-0.814727\pi$
$599$		−	26.9092i		−	1.09948i	0.835336	−	0.549740i	$-0.185273\pi$
$600$		0			0
$601$	15.5403			0.633902			0.316951	−	0.948442i	$-0.397341\pi$
$601$	15.5403			0.633902			0.316951	+	0.948442i	$0.397341\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	27.3062			1.11015
$606$		0			0
$607$		−	4.06302i		−	0.164913i	−0.996595	−	0.0824564i	$-0.973723\pi$
$607$		−	4.06302i		−	0.164913i	0.996595	−	0.0824564i	$-0.0262765\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−9.97020			−0.403351
$612$		0			0
$613$	23.8370			0.962769			0.481384	−	0.876510i	$-0.340134\pi$
$613$	23.8370			0.962769			0.481384	+	0.876510i	$0.340134\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1.21006i		−	0.0487152i	−0.999703	−	0.0243576i	$-0.992246\pi$
$617$		−	1.21006i		−	0.0487152i	0.999703	−	0.0243576i	$-0.00775403\pi$
$618$		0			0
$619$			18.5597i			0.745977i	0.927836	+	0.372988i	$0.121667\pi$
$619$			18.5597i			0.745977i	−0.927836	+	0.372988i	$0.878333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	59.4965i		−	2.38367i
$624$		0			0
$625$	−10.3973			−0.415890
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	24.8356i		−	0.990258i
$630$		0			0
$631$	−41.6100			−1.65647			−0.828234	−	0.560382i	$-0.810654\pi$
$631$	−41.6100			−1.65647			−0.828234	+	0.560382i	$0.810654\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	65.9925			2.61883
$636$		0			0
$637$			6.67544i			0.264491i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−9.20133			−0.363431			−0.181715	−	0.983351i	$-0.558165\pi$
$641$	−9.20133			−0.363431			−0.181715	+	0.983351i	$0.558165\pi$
$642$		0			0
$643$		−	5.18930i		−	0.204646i	−0.994751	−	0.102323i	$-0.967372\pi$
$643$		−	5.18930i		−	0.204646i	0.994751	−	0.102323i	$-0.0326275\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	31.7966			1.25005			0.625027	−	0.780603i	$-0.285088\pi$
$647$	31.7966			1.25005			0.625027	+	0.780603i	$0.285088\pi$
$648$		0			0
$649$		−	28.4954i		−	1.11854i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	41.2069i		−	1.61255i	−0.591542	−	0.806275i	$-0.701480\pi$
$653$		−	41.2069i		−	1.61255i	0.591542	−	0.806275i	$-0.298520\pi$
$654$		0			0
$655$	−42.2678			−1.65154
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−34.0275			−1.32552			−0.662761	−	0.748831i	$-0.730616\pi$
$659$	−34.0275			−1.32552			−0.662761	+	0.748831i	$0.730616\pi$
$660$		0			0
$661$		−	8.80722i		−	0.342561i	−0.985222	−	0.171281i	$-0.945210\pi$
$661$		−	8.80722i		−	0.342561i	0.985222	−	0.171281i	$-0.0547905\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−58.2298			−2.25805
$666$		0			0
$667$		−	40.1476i		−	1.55452i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			21.4752i			0.829043i
$672$		0			0
$673$		−	8.37773i		−	0.322938i	−0.986878	−	0.161469i	$-0.948377\pi$
$673$		−	8.37773i		−	0.322938i	0.986878	−	0.161469i	$-0.0516231\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			5.16558i			0.198529i	0.995061	+	0.0992647i	$0.0316490\pi$
$677$			5.16558i			0.198529i	−0.995061	+	0.0992647i	$0.968351\pi$
$678$		0			0
$679$	−13.0721			−0.501660
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−17.6447			−0.675157			−0.337579	−	0.941297i	$-0.609608\pi$
$683$	−17.6447			−0.675157			−0.337579	+	0.941297i	$0.609608\pi$
$684$		0			0
$685$			16.7945i			0.641686i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			0.320813i			0.0122220i
$690$		0			0
$691$			44.5617i			1.69521i	0.530629	+	0.847604i	$0.321956\pi$
$691$			44.5617i			1.69521i	−0.530629	+	0.847604i	$0.678044\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	49.3844i		−	1.87326i
$696$		0			0
$697$	−24.6205			−0.932567
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	6.36642i		−	0.240456i	−0.992746	−	0.120228i	$-0.961637\pi$
$701$		−	6.36642i		−	0.240456i	0.992746	−	0.120228i	$-0.0383626\pi$
$702$		0			0
$703$			19.8669i			0.749293i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−36.5161			−1.37333
$708$		0			0
$709$		−	12.1344i		−	0.455718i	−0.973694	−	0.227859i	$-0.926828\pi$
$709$		−	12.1344i		−	0.455718i	0.973694	−	0.227859i	$-0.0731725\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	45.4884			1.70355
$714$		0			0
$715$	−11.8919			−0.444732
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.35686			0.162483			0.0812417	−	0.996694i	$-0.474111\pi$
$719$	4.35686			0.162483			0.0812417	+	0.996694i	$0.474111\pi$
$720$		0			0
$721$		−	33.3115i		−	1.24058i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			42.3893i			1.57430i
$726$		0			0
$727$		−	47.6676i		−	1.76789i	−0.467587	−	0.883947i	$-0.654876\pi$
$727$		−	47.6676i		−	1.76789i	0.467587	−	0.883947i	$-0.345124\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			61.8181i			2.28642i
$732$		0			0
$733$	−17.4188			−0.643377			−0.321688	−	0.946846i	$-0.604250\pi$
$733$	−17.4188			−0.643377			−0.321688	+	0.946846i	$0.604250\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	14.8039			0.545307
$738$		0			0
$739$			10.8508i			0.399153i	0.979882	+	0.199576i	$0.0639566\pi$
$739$			10.8508i			0.399153i	−0.979882	+	0.199576i	$0.936043\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			13.2937i			0.487699i	0.969813	+	0.243850i	$0.0784103\pi$
$743$			13.2937i			0.487699i	−0.969813	+	0.243850i	$0.921590\pi$
$744$		0			0
$745$	−8.37150			−0.306708
$746$		0			0
$747$		0			0
$748$		0			0
$749$			60.6458i			2.21595i
$750$		0			0
$751$		−	7.66618i		−	0.279743i	−0.990170	−	0.139871i	$-0.955331\pi$
$751$		−	7.66618i		−	0.279743i	0.990170	−	0.139871i	$-0.0446689\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	15.1843i		−	0.552613i
$756$		0			0
$757$	1.28794			0.0468109			0.0234055	−	0.999726i	$-0.492549\pi$
$757$	1.28794			0.0468109			0.0234055	+	0.999726i	$0.492549\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	19.1314i		−	0.693511i	−0.937956	−	0.346756i	$-0.887283\pi$
$761$		−	19.1314i		−	0.693511i	0.937956	−	0.346756i	$-0.112717\pi$
$762$		0			0
$763$			35.8395i			1.29748i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			5.17219i			0.186757i
$768$		0			0
$769$			25.6056i			0.923360i	0.887047	+	0.461680i	$0.152753\pi$
$769$			25.6056i			0.923360i	−0.887047	+	0.461680i	$0.847247\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−42.6184			−1.53288			−0.766439	−	0.642317i	$-0.777973\pi$
$773$	−42.6184			−1.53288			−0.766439	+	0.642317i	$0.777973\pi$
$774$		0			0
$775$	−48.0283			−1.72523
$776$		0			0
$777$		0			0
$778$		0			0
$779$	19.6948			0.705640
$780$		0			0
$781$			2.20600i			0.0789370i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−6.47369			−0.231056
$786$		0			0
$787$		−	54.9021i		−	1.95705i	−0.206132	−	0.978524i	$-0.566088\pi$
$787$		−	54.9021i		−	1.95705i	0.206132	−	0.978524i	$-0.433912\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	73.7155			2.62102
$792$		0			0
$793$		−	3.89796i		−	0.138421i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−26.3457			−0.933211			−0.466606	−	0.884465i	$-0.654523\pi$
$797$	−26.3457			−0.933211			−0.466606	+	0.884465i	$0.654523\pi$
$798$		0			0
$799$		−	67.3908i		−	2.38411i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−11.9111			−0.420333
$804$		0			0
$805$	91.4364			3.22271
$806$		0			0
$807$		0			0
$808$		0			0
$809$			44.8080i			1.57537i	0.616081	+	0.787683i	$0.288719\pi$
$809$			44.8080i			1.57537i	−0.616081	+	0.787683i	$0.711281\pi$
$810$		0			0
$811$			8.29490i			0.291273i	0.989338	+	0.145637i	$0.0465230\pi$
$811$			8.29490i			0.291273i	−0.989338	+	0.145637i	$0.953477\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	78.3813i		−	2.74558i
$816$		0			0
$817$		−	49.4506i		−	1.73006i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5.71316			−0.199391			−0.0996954	−	0.995018i	$-0.531787\pi$
$821$	−5.71316			−0.199391			−0.0996954	+	0.995018i	$0.531787\pi$
$822$		0			0
$823$		−	56.0954i		−	1.95536i	−0.210094	−	0.977681i	$-0.567377\pi$
$823$		−	56.0954i		−	1.95536i	0.210094	−	0.977681i	$-0.432623\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	42.6624			1.48352			0.741759	−	0.670666i	$-0.233992\pi$
$827$	42.6624			1.48352			0.741759	+	0.670666i	$0.233992\pi$
$828$		0			0
$829$		−	15.1129i		−	0.524892i	−0.964947	−	0.262446i	$-0.915471\pi$
$829$		−	15.1129i		−	0.524892i	0.964947	−	0.262446i	$-0.0845291\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−45.1208			−1.56334
$834$		0			0
$835$	−34.7104	−	28.4656i	−1.20120	−	0.985093i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	5.04212i		−	0.174073i	−0.996205	−	0.0870367i	$-0.972260\pi$
$839$		−	5.04212i		−	0.174073i	0.996205	−	0.0870367i	$-0.0277397\pi$
$840$		0			0
$841$	−6.98379			−0.240820
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−42.9994			−1.47922
$846$		0			0
$847$	−30.9167			−1.06231
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	31.1963i		−	1.06940i
$852$		0			0
$853$	41.3744			1.41663			0.708317	−	0.705895i	$-0.249455\pi$
$853$	41.3744			1.41663			0.708317	+	0.705895i	$0.249455\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	55.3879i		−	1.89202i	−0.324145	−	0.946008i	$-0.605076\pi$
$857$		−	55.3879i		−	1.89202i	0.324145	−	0.946008i	$-0.394924\pi$
$858$		0			0
$859$	4.34936			0.148398			0.0741991	−	0.997243i	$-0.476360\pi$
$859$	4.34936			0.148398			0.0741991	+	0.997243i	$0.476360\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			48.5230i			1.65174i	0.563859	+	0.825871i	$0.309316\pi$
$863$			48.5230i			1.65174i	−0.563859	+	0.825871i	$0.690684\pi$
$864$		0			0
$865$		−	34.8099i		−	1.18357i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	32.3401			1.09706
$870$		0			0
$871$	−2.68704			−0.0910469
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−28.2322			−0.954421
$876$		0			0
$877$	−47.3468			−1.59879			−0.799393	−	0.600808i	$-0.794846\pi$
$877$	−47.3468			−1.59879			−0.799393	+	0.600808i	$0.794846\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−18.5404			−0.624643			−0.312322	−	0.949976i	$-0.601107\pi$
$881$	−18.5404			−0.624643			−0.312322	+	0.949976i	$0.601107\pi$
$882$		0			0
$883$	−17.3061			−0.582396			−0.291198	−	0.956663i	$-0.594054\pi$
$883$	−17.3061			−0.582396			−0.291198	+	0.956663i	$0.594054\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−8.15097			−0.273683			−0.136841	−	0.990593i	$-0.543695\pi$
$887$	−8.15097			−0.273683			−0.136841	+	0.990593i	$0.543695\pi$
$888$		0			0
$889$	−74.7182			−2.50597
$890$		0			0
$891$		0			0
$892$		0			0
$893$			53.9084i			1.80397i
$894$		0			0
$895$		−	65.1044i		−	2.17620i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	40.7706i		−	1.35978i
$900$		0			0
$901$	−2.16844			−0.0722413
$902$		0			0
$903$		0			0
$904$		0			0
$905$	35.5910			1.18309
$906$		0			0
$907$	48.3172			1.60435			0.802173	−	0.597092i	$-0.203677\pi$
$907$	48.3172			1.60435			0.802173	+	0.597092i	$0.203677\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	41.6278i		−	1.37919i	−0.724195	−	0.689595i	$-0.757788\pi$
$911$		−	41.6278i		−	1.37919i	0.724195	−	0.689595i	$-0.242212\pi$
$912$		0			0
$913$		−	28.2991i		−	0.936565i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	47.8566			1.58036
$918$		0			0
$919$	15.2093			0.501707			0.250853	−	0.968025i	$-0.419289\pi$
$919$	15.2093			0.501707			0.250853	+	0.968025i	$0.419289\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	0.400410i		−	0.0131797i
$924$		0			0
$925$			32.9382i			1.08300i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			14.7431i			0.483704i	0.970313	+	0.241852i	$0.0777548\pi$
$929$			14.7431i			0.483704i	−0.970313	+	0.241852i	$0.922245\pi$
$930$		0			0
$931$	36.0938			1.18293
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	80.3800i		−	2.62871i
$936$		0			0
$937$			57.8416i			1.88960i	0.327643	+	0.944801i	$0.393746\pi$
$937$			57.8416i			1.88960i	−0.327643	+	0.944801i	$0.606254\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	21.5591			0.702805			0.351403	−	0.936224i	$-0.385705\pi$
$941$	21.5591			0.702805			0.351403	+	0.936224i	$0.385705\pi$
$942$		0			0
$943$	−30.9262			−1.00709
$944$		0			0
$945$		0			0
$946$		0			0
$947$			54.2270i			1.76214i	0.472986	+	0.881070i	$0.343176\pi$
$947$			54.2270i			1.76214i	−0.472986	+	0.881070i	$0.656824\pi$
$948$		0			0
$949$	2.16198			0.0701807
$950$		0			0
$951$		0			0
$952$		0			0
$953$	37.7716			1.22354			0.611771	−	0.791035i	$-0.290457\pi$
$953$	37.7716			1.22354			0.611771	+	0.791035i	$0.290457\pi$
$954$		0			0
$955$			66.7885i			2.16122i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	19.0152i		−	0.614032i
$960$		0			0
$961$	15.1943			0.490138
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	19.5213i		−	0.628411i
$966$		0			0
$967$	−52.5711			−1.69057			−0.845287	−	0.534313i	$-0.820570\pi$
$967$	−52.5711			−1.69057			−0.845287	+	0.534313i	$0.820570\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−0.253042			−0.00812051			−0.00406026	−	0.999992i	$-0.501292\pi$
$971$	−0.253042			−0.00812051			−0.00406026	+	0.999992i	$0.501292\pi$
$972$		0			0
$973$			55.9141i			1.79252i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	52.9558			1.69421			0.847103	−	0.531429i	$-0.178345\pi$
$977$	52.9558			1.69421			0.847103	+	0.531429i	$0.178345\pi$
$978$		0			0
$979$	−65.6976			−2.09971
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−58.1838			−1.85577			−0.927887	−	0.372862i	$-0.878377\pi$
$983$	−58.1838			−1.85577			−0.927887	+	0.372862i	$0.878377\pi$
$984$		0			0
$985$	52.6400			1.67725
$986$		0			0
$987$		0			0
$988$		0			0
$989$			77.6507i			2.46915i
$990$		0			0
$991$			47.4082i			1.50597i	0.658036	+	0.752986i	$0.271387\pi$
$991$			47.4082i			1.50597i	−0.658036	+	0.752986i	$0.728613\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	32.6144			1.03395
$996$		0			0
$997$	−21.1546			−0.669973			−0.334986	−	0.942223i	$-0.608732\pi$
$997$	−21.1546			−0.669973			−0.334986	+	0.942223i	$0.608732\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.h.a.3005.5	✓	56
3.2	odd	2	inner	6012.2.h.a.3005.52	yes	56
167.166	odd	2	inner	6012.2.h.a.3005.51	yes	56
501.500	even	2	inner	6012.2.h.a.3005.6	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6012.2.h.a.3005.5	✓	56	1.1	even	1	trivial
6012.2.h.a.3005.6	yes	56	501.500	even	2	inner
6012.2.h.a.3005.51	yes	56	167.166	odd	2	inner
6012.2.h.a.3005.52	yes	56	3.2	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:	\( N \)	\(=\)	\( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6012.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.47368 q^{5} +3.93299 q^{7} +O(q^{10})\)	\(q-3.47368 q^{5} +3.93299 q^{7} -4.34291i q^{11} +0.788280i q^{13} -5.32815 q^{17} +4.26219 q^{19} -6.69278 q^{23} +7.06648 q^{25} +5.99865i q^{29} -6.79664 q^{31} -13.6619 q^{35} +4.66119i q^{37} +4.62083 q^{41} -11.6022i q^{43} +12.6481i q^{47} +8.46837 q^{49} +0.406978 q^{53} +15.0859i q^{55} +6.56137 q^{59} -4.94490 q^{61} -2.73823i q^{65} +3.40874i q^{67} -0.507955 q^{71} -2.74265i q^{73} -17.0806i q^{77} +7.44663i q^{79} +6.51617 q^{83} +18.5083 q^{85} -15.1276i q^{89} +3.10029i q^{91} -14.8055 q^{95} -3.32370 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q+O(q^{10}) \) 56 * q	\( 56 q + 8 q^{19} + 64 q^{25} - 8 q^{31} + 56 q^{49} - 8 q^{61} + 32 q^{85} - 48 q^{97}+O(q^{100}) \) 56 * q + 8 * q^19 + 64 * q^25 - 8 * q^31 + 56 * q^49 - 8 * q^61 + 32 * q^85 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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