LMFDB - Embedded newform 6012.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6012,2,Mod(1,6012)]

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, 0, 0]))

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

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

chi := DirichletCharacter("6012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6012 = 2^{2} \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6012.a (trivial)

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:	yes
Analytic conductor:	$48.0060616952$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.826865.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 6x - 1 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 6x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 668)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.147687$ of defining polynomial
Character	$\chi$	$=$	6012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{5} -0.516539 q^{7} +O(q^{10})$	$q-2.00000 q^{5} -0.516539 q^{7} -0.538350 q^{11} -0.295373 q^{13} +6.99507 q^{17} -4.06647 q^{19} -0.781327 q^{23} -1.00000 q^{25} -0.811912 q^{29} +3.07629 q^{31} +1.03308 q^{35} -3.95638 q^{37} -0.743318 q^{41} +0.590746 q^{43} +3.47572 q^{47} -6.73319 q^{49} -6.69970 q^{53} +1.07670 q^{55} +8.91427 q^{59} +4.63992 q^{61} +0.590746 q^{65} +6.55274 q^{67} -12.4655 q^{71} +7.77640 q^{73} +0.278079 q^{77} -4.88529 q^{79} +1.11388 q^{83} -13.9901 q^{85} +1.79742 q^{89} +0.152572 q^{91} +8.13294 q^{95} -17.1192 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 10 q^{5} + 9 q^{7}+O(q^{10}) $ 5 * q - 10 * q^5 + 9 * q^7	$ 5 q - 10 q^{5} + 9 q^{7} - 5 q^{11} - 4 q^{13} + 2 q^{17} + 5 q^{19} - 6 q^{23} - 5 q^{25} + 5 q^{29} + 9 q^{31} - 18 q^{35} + 8 q^{37} + 4 q^{41} + 8 q^{43} - 13 q^{47} + 14 q^{49} + 2 q^{53} + 10 q^{55} - 4 q^{59} + 11 q^{61} + 8 q^{65} + 28 q^{67} - 2 q^{71} + 8 q^{73} + 12 q^{77} - 10 q^{79} - 2 q^{83} - 4 q^{85} + 17 q^{89} - 12 q^{91} - 10 q^{95} - 27 q^{97}+O(q^{100}) $ 5 * q - 10 * q^5 + 9 * q^7 - 5 * q^11 - 4 * q^13 + 2 * q^17 + 5 * q^19 - 6 * q^23 - 5 * q^25 + 5 * q^29 + 9 * q^31 - 18 * q^35 + 8 * q^37 + 4 * q^41 + 8 * q^43 - 13 * q^47 + 14 * q^49 + 2 * q^53 + 10 * q^55 - 4 * q^59 + 11 * q^61 + 8 * q^65 + 28 * q^67 - 2 * q^71 + 8 * q^73 + 12 * q^77 - 10 * q^79 - 2 * q^83 - 4 * q^85 + 17 * q^89 - 12 * q^91 - 10 * q^95 - 27 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	−0.516539	−0.195233	−0.0976167	−	0.995224i	$-0.531122\pi$
$7$	−0.516539	−0.195233	−0.0976167	+	0.995224i	$0.531122\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.538350	−0.162319	−0.0811593	−	0.996701i	$-0.525862\pi$
$11$	−0.538350	−0.162319	−0.0811593	+	0.996701i	$0.525862\pi$
$12$	0	0
$13$	−0.295373	−0.0819218	−0.0409609	−	0.999161i	$-0.513042\pi$
$13$	−0.295373	−0.0819218	−0.0409609	+	0.999161i	$0.513042\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.99507	1.69655	0.848277	−	0.529553i	$-0.177640\pi$
$17$	6.99507	1.69655	0.848277	+	0.529553i	$0.177640\pi$
$18$	0	0
$19$	−4.06647	−0.932912	−0.466456	−	0.884544i	$-0.654469\pi$
$19$	−4.06647	−0.932912	−0.466456	+	0.884544i	$0.654469\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.781327	−0.162918	−0.0814590	−	0.996677i	$-0.525958\pi$
$23$	−0.781327	−0.162918	−0.0814590	+	0.996677i	$0.525958\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.811912	−0.150768	−0.0753841	−	0.997155i	$-0.524018\pi$
$29$	−0.811912	−0.150768	−0.0753841	+	0.997155i	$0.524018\pi$
$30$	0	0
$31$	3.07629	0.552517	0.276259	−	0.961083i	$-0.410905\pi$
$31$	3.07629	0.552517	0.276259	+	0.961083i	$0.410905\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.03308	0.174622
$36$	0	0
$37$	−3.95638	−0.650424	−0.325212	−	0.945641i	$-0.605436\pi$
$37$	−3.95638	−0.650424	−0.325212	+	0.945641i	$0.605436\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.743318	−0.116087	−0.0580434	−	0.998314i	$-0.518486\pi$
$41$	−0.743318	−0.116087	−0.0580434	+	0.998314i	$0.518486\pi$
$42$	0	0
$43$	0.590746	0.0900880	0.0450440	−	0.998985i	$-0.485657\pi$
$43$	0.590746	0.0900880	0.0450440	+	0.998985i	$0.485657\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.47572	0.506986	0.253493	−	0.967337i	$-0.418420\pi$
$47$	3.47572	0.506986	0.253493	+	0.967337i	$0.418420\pi$
$48$	0	0
$49$	−6.73319	−0.961884
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.69970	−0.920274	−0.460137	−	0.887848i	$-0.652200\pi$
$53$	−6.69970	−0.920274	−0.460137	+	0.887848i	$0.652200\pi$
$54$	0	0
$55$	1.07670	0.145182
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.91427	1.16054	0.580269	−	0.814425i	$-0.302947\pi$
$59$	8.91427	1.16054	0.580269	+	0.814425i	$0.302947\pi$
$60$	0	0
$61$	4.63992	0.594081	0.297041	−	0.954865i	$-0.404000\pi$
$61$	4.63992	0.594081	0.297041	+	0.954865i	$0.404000\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.590746	0.0732731
$66$	0	0
$67$	6.55274	0.800544	0.400272	−	0.916396i	$-0.368916\pi$
$67$	6.55274	0.800544	0.400272	+	0.916396i	$0.368916\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.4655	−1.47938	−0.739691	−	0.672947i	$-0.765028\pi$
$71$	−12.4655	−1.47938	−0.739691	+	0.672947i	$0.765028\pi$
$72$	0	0
$73$	7.77640	0.910158	0.455079	−	0.890451i	$-0.349611\pi$
$73$	7.77640	0.910158	0.455079	+	0.890451i	$0.349611\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.278079	0.0316900
$78$	0	0
$79$	−4.88529	−0.549638	−0.274819	−	0.961496i	$-0.588618\pi$
$79$	−4.88529	−0.549638	−0.274819	+	0.961496i	$0.588618\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.11388	0.122264	0.0611321	−	0.998130i	$-0.480529\pi$
$83$	1.11388	0.122264	0.0611321	+	0.998130i	$0.480529\pi$
$84$	0	0
$85$	−13.9901	−1.51744
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.79742	0.190527	0.0952633	−	0.995452i	$-0.469631\pi$
$89$	1.79742	0.190527	0.0952633	+	0.995452i	$0.469631\pi$
$90$	0	0
$91$	0.152572	0.0159939
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.13294	0.834422
$96$	0	0
$97$	−17.1192	−1.73819	−0.869097	−	0.494641i	$-0.835300\pi$
$97$	−17.1192	−1.73819	−0.869097	+	0.494641i	$0.835300\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	14.0281	1.39585	0.697926	−	0.716170i	$-0.254106\pi$
$101$	14.0281	1.39585	0.697926	+	0.716170i	$0.254106\pi$
$102$	0	0
$103$	16.7553	1.65094	0.825472	−	0.564443i	$-0.190909\pi$
$103$	16.7553	1.65094	0.825472	+	0.564443i	$0.190909\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.47360	−0.915847	−0.457924	−	0.888992i	$-0.651407\pi$
$107$	−9.47360	−0.915847	−0.457924	+	0.888992i	$0.651407\pi$
$108$	0	0
$109$	20.3467	1.94886	0.974429	−	0.224695i	$-0.0721384\pi$
$109$	20.3467	1.94886	0.974429	+	0.224695i	$0.0721384\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−16.8270	−1.58295	−0.791476	−	0.611200i	$-0.790687\pi$
$113$	−16.8270	−1.58295	−0.791476	+	0.611200i	$0.790687\pi$
$114$	0	0
$115$	1.56265	0.145718
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.61322	−0.331224
$120$	0	0
$121$	−10.7102	−0.973653
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	9.52781	0.845456	0.422728	−	0.906257i	$-0.361073\pi$
$127$	9.52781	0.845456	0.422728	+	0.906257i	$0.361073\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	16.5809	1.44868	0.724339	−	0.689444i	$-0.242145\pi$
$131$	16.5809	1.44868	0.724339	+	0.689444i	$0.242145\pi$
$132$	0	0
$133$	2.10049	0.182136
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.53903	−0.302360	−0.151180	−	0.988506i	$-0.548307\pi$
$137$	−3.53903	−0.302360	−0.151180	+	0.988506i	$0.548307\pi$
$138$	0	0
$139$	−9.40022	−0.797316	−0.398658	−	0.917100i	$-0.630524\pi$
$139$	−9.40022	−0.797316	−0.398658	+	0.917100i	$0.630524\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.159014	0.0132974
$144$	0	0
$145$	1.62382	0.134851
$146$	0	0
$147$	0	0
$148$	0	0
$149$	18.5809	1.52221	0.761103	−	0.648631i	$-0.224658\pi$
$149$	18.5809	1.52221	0.761103	+	0.648631i	$0.224658\pi$
$150$	0	0
$151$	17.2644	1.40495	0.702477	−	0.711706i	$-0.252077\pi$
$151$	17.2644	1.40495	0.702477	+	0.711706i	$0.252077\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.15257	−0.494186
$156$	0	0
$157$	−7.19863	−0.574513	−0.287256	−	0.957854i	$-0.592743\pi$
$157$	−7.19863	−0.574513	−0.287256	+	0.957854i	$0.592743\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.403586	0.0318070
$162$	0	0
$163$	23.6948	1.85592	0.927959	−	0.372683i	$-0.121562\pi$
$163$	23.6948	1.85592	0.927959	+	0.372683i	$0.121562\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−12.9128	−0.993289
$170$	0	0
$171$	0	0
$172$	0	0
$173$	19.5278	1.48467	0.742336	−	0.670028i	$-0.233718\pi$
$173$	19.5278	1.48467	0.742336	+	0.670028i	$0.233718\pi$
$174$	0	0
$175$	0.516539	0.0390467
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.5038	0.785092	0.392546	−	0.919732i	$-0.371594\pi$
$179$	10.5038	0.785092	0.392546	+	0.919732i	$0.371594\pi$
$180$	0	0
$181$	8.36184	0.621531	0.310765	−	0.950487i	$-0.399415\pi$
$181$	8.36184	0.621531	0.310765	+	0.950487i	$0.399415\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.91275	0.581757
$186$	0	0
$187$	−3.76580	−0.275382
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−9.43844	−0.682942	−0.341471	−	0.939892i	$-0.610925\pi$
$191$	−9.43844	−0.682942	−0.341471	+	0.939892i	$0.610925\pi$
$192$	0	0
$193$	−16.3291	−1.17540	−0.587698	−	0.809080i	$-0.699966\pi$
$193$	−16.3291	−1.17540	−0.587698	+	0.809080i	$0.699966\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.3460	1.23585	0.617926	−	0.786237i	$-0.287973\pi$
$197$	17.3460	1.23585	0.617926	+	0.786237i	$0.287973\pi$
$198$	0	0
$199$	6.46097	0.458006	0.229003	−	0.973426i	$-0.426453\pi$
$199$	6.46097	0.458006	0.229003	+	0.973426i	$0.426453\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.419384	0.0294350
$204$	0	0
$205$	1.48664	0.103831
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.18918	0.151429
$210$	0	0
$211$	−2.41035	−0.165935	−0.0829677	−	0.996552i	$-0.526440\pi$
$211$	−2.41035	−0.165935	−0.0829677	+	0.996552i	$0.526440\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.18149	−0.0805771
$216$	0	0
$217$	−1.58902	−0.107870
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.06616	−0.138985
$222$	0	0
$223$	−0.798106	−0.0534452	−0.0267226	−	0.999643i	$-0.508507\pi$
$223$	−0.798106	−0.0534452	−0.0267226	+	0.999643i	$0.508507\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.4979	0.696769	0.348385	−	0.937352i	$-0.386730\pi$
$227$	10.4979	0.696769	0.348385	+	0.937352i	$0.386730\pi$
$228$	0	0
$229$	11.6916	0.772602	0.386301	−	0.922373i	$-0.373753\pi$
$229$	11.6916	0.772602	0.386301	+	0.922373i	$0.373753\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	23.8858	1.56481	0.782404	−	0.622771i	$-0.213993\pi$
$233$	23.8858	1.56481	0.782404	+	0.622771i	$0.213993\pi$
$234$	0	0
$235$	−6.95145	−0.453462
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−21.8051	−1.41046	−0.705228	−	0.708981i	$-0.749155\pi$
$239$	−21.8051	−1.41046	−0.705228	+	0.708981i	$0.749155\pi$
$240$	0	0
$241$	17.4951	1.12696	0.563481	−	0.826129i	$-0.309462\pi$
$241$	17.4951	1.12696	0.563481	+	0.826129i	$0.309462\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	13.4664	0.860335
$246$	0	0
$247$	1.20113	0.0764258
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.26889	0.0800917	0.0400458	−	0.999198i	$-0.487250\pi$
$251$	1.26889	0.0800917	0.0400458	+	0.999198i	$0.487250\pi$
$252$	0	0
$253$	0.420628	0.0264446
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−26.5168	−1.65407	−0.827036	−	0.562149i	$-0.809975\pi$
$257$	−26.5168	−1.65407	−0.827036	+	0.562149i	$0.809975\pi$
$258$	0	0
$259$	2.04362	0.126985
$260$	0	0
$261$	0	0
$262$	0	0
$263$	14.2975	0.881622	0.440811	−	0.897600i	$-0.354691\pi$
$263$	14.2975	0.881622	0.440811	+	0.897600i	$0.354691\pi$
$264$	0	0
$265$	13.3994	0.823118
$266$	0	0
$267$	0	0
$268$	0	0
$269$	18.0183	1.09859	0.549297	−	0.835627i	$-0.314896\pi$
$269$	18.0183	1.09859	0.549297	+	0.835627i	$0.314896\pi$
$270$	0	0
$271$	5.40925	0.328589	0.164294	−	0.986411i	$-0.447465\pi$
$271$	5.40925	0.328589	0.164294	+	0.986411i	$0.447465\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.538350	0.0324637
$276$	0	0
$277$	0.475355	0.0285613	0.0142806	−	0.999898i	$-0.495454\pi$
$277$	0.475355	0.0285613	0.0142806	+	0.999898i	$0.495454\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	9.23067	0.550655	0.275328	−	0.961350i	$-0.411214\pi$
$281$	9.23067	0.550655	0.275328	+	0.961350i	$0.411214\pi$
$282$	0	0
$283$	22.6333	1.34541	0.672704	−	0.739911i	$-0.265133\pi$
$283$	22.6333	1.34541	0.672704	+	0.739911i	$0.265133\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.383953	0.0226640
$288$	0	0
$289$	31.9310	1.87829
$290$	0	0
$291$	0	0
$292$	0	0
$293$	10.0171	0.585207	0.292604	−	0.956234i	$-0.405478\pi$
$293$	10.0171	0.585207	0.292604	+	0.956234i	$0.405478\pi$
$294$	0	0
$295$	−17.8285	−1.03802
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.230783	0.0133465
$300$	0	0
$301$	−0.305143	−0.0175882
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−9.27984	−0.531362
$306$	0	0
$307$	16.0119	0.913849	0.456925	−	0.889505i	$-0.348951\pi$
$307$	16.0119	0.913849	0.456925	+	0.889505i	$0.348951\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	10.3613	0.587537	0.293768	−	0.955877i	$-0.405091\pi$
$311$	10.3613	0.587537	0.293768	+	0.955877i	$0.405091\pi$
$312$	0	0
$313$	9.36002	0.529059	0.264530	−	0.964378i	$-0.414783\pi$
$313$	9.36002	0.529059	0.264530	+	0.964378i	$0.414783\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−12.8595	−0.722260	−0.361130	−	0.932515i	$-0.617609\pi$
$317$	−12.8595	−0.722260	−0.361130	+	0.932515i	$0.617609\pi$
$318$	0	0
$319$	0.437093	0.0244725
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−28.4452	−1.58273
$324$	0	0
$325$	0.295373	0.0163844
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.79535	−0.0989806
$330$	0	0
$331$	−2.81289	−0.154611	−0.0773053	−	0.997007i	$-0.524632\pi$
$331$	−2.81289	−0.154611	−0.0773053	+	0.997007i	$0.524632\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−13.1055	−0.716029
$336$	0	0
$337$	19.3021	1.05145	0.525725	−	0.850654i	$-0.323794\pi$
$337$	19.3021	1.05145	0.525725	+	0.850654i	$0.323794\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.65612	−0.0896839
$342$	0	0
$343$	7.09373	0.383025
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−28.4120	−1.52524	−0.762618	−	0.646849i	$-0.776086\pi$
$347$	−28.4120	−1.52524	−0.762618	+	0.646849i	$0.776086\pi$
$348$	0	0
$349$	7.74264	0.414454	0.207227	−	0.978293i	$-0.433556\pi$
$349$	7.74264	0.414454	0.207227	+	0.978293i	$0.433556\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−19.3204	−1.02832	−0.514161	−	0.857694i	$-0.671896\pi$
$353$	−19.3204	−1.02832	−0.514161	+	0.857694i	$0.671896\pi$
$354$	0	0
$355$	24.9310	1.32320
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.57278	0.135786	0.0678932	−	0.997693i	$-0.478372\pi$
$359$	2.57278	0.135786	0.0678932	+	0.997693i	$0.478372\pi$
$360$	0	0
$361$	−2.46383	−0.129675
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−15.5528	−0.814070
$366$	0	0
$367$	−23.1615	−1.20902	−0.604509	−	0.796598i	$-0.706631\pi$
$367$	−23.1615	−1.20902	−0.604509	+	0.796598i	$0.706631\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.46065	0.179668
$372$	0	0
$373$	20.9233	1.08337	0.541684	−	0.840582i	$-0.317787\pi$
$373$	20.9233	1.08337	0.541684	+	0.840582i	$0.317787\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.239817	0.0123512
$378$	0	0
$379$	30.4945	1.56640	0.783198	−	0.621773i	$-0.213587\pi$
$379$	30.4945	1.56640	0.783198	+	0.621773i	$0.213587\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−27.8997	−1.42561	−0.712804	−	0.701363i	$-0.752575\pi$
$383$	−27.8997	−1.42561	−0.712804	+	0.701363i	$0.752575\pi$
$384$	0	0
$385$	−0.556158	−0.0283444
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−18.9087	−0.958707	−0.479353	−	0.877622i	$-0.659129\pi$
$389$	−18.9087	−0.958707	−0.479353	+	0.877622i	$0.659129\pi$
$390$	0	0
$391$	−5.46544	−0.276399
$392$	0	0
$393$	0	0
$394$	0	0
$395$	9.77058	0.491611
$396$	0	0
$397$	−23.2771	−1.16824	−0.584122	−	0.811666i	$-0.698561\pi$
$397$	−23.2771	−1.16824	−0.584122	+	0.811666i	$0.698561\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	23.3692	1.16700	0.583500	−	0.812113i	$-0.301683\pi$
$401$	23.3692	1.16700	0.583500	+	0.812113i	$0.301683\pi$
$402$	0	0
$403$	−0.908652	−0.0452632
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.12992	0.105576
$408$	0	0
$409$	−13.5940	−0.672178	−0.336089	−	0.941830i	$-0.609104\pi$
$409$	−13.5940	−0.672178	−0.336089	+	0.941830i	$0.609104\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4.60456	−0.226576
$414$	0	0
$415$	−2.22776	−0.109356
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−20.6740	−1.00999	−0.504996	−	0.863121i	$-0.668506\pi$
$419$	−20.6740	−1.00999	−0.504996	+	0.863121i	$0.668506\pi$
$420$	0	0
$421$	33.9625	1.65523	0.827614	−	0.561297i	$-0.189698\pi$
$421$	33.9625	1.65523	0.827614	+	0.561297i	$0.189698\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.99507	−0.339311
$426$	0	0
$427$	−2.39670	−0.115984
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.35940	0.209985	0.104992	−	0.994473i	$-0.466518\pi$
$431$	4.35940	0.209985	0.104992	+	0.994473i	$0.466518\pi$
$432$	0	0
$433$	−8.95196	−0.430204	−0.215102	−	0.976592i	$-0.569008\pi$
$433$	−8.95196	−0.430204	−0.215102	+	0.976592i	$0.569008\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.17724	0.151988
$438$	0	0
$439$	30.0473	1.43408	0.717039	−	0.697033i	$-0.245497\pi$
$439$	30.0473	1.43408	0.717039	+	0.697033i	$0.245497\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14.2252	0.675858	0.337929	−	0.941172i	$-0.390274\pi$
$443$	14.2252	0.675858	0.337929	+	0.941172i	$0.390274\pi$
$444$	0	0
$445$	−3.59485	−0.170412
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5.54334	0.261606	0.130803	−	0.991408i	$-0.458244\pi$
$449$	5.54334	0.261606	0.130803	+	0.991408i	$0.458244\pi$
$450$	0	0
$451$	0.400165	0.0188431
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.305143	−0.0143053
$456$	0	0
$457$	−11.2195	−0.524826	−0.262413	−	0.964956i	$-0.584518\pi$
$457$	−11.2195	−0.524826	−0.262413	+	0.964956i	$0.584518\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.92450	0.0896328	0.0448164	−	0.998995i	$-0.485730\pi$
$461$	1.92450	0.0896328	0.0448164	+	0.998995i	$0.485730\pi$
$462$	0	0
$463$	26.4339	1.22849	0.614244	−	0.789116i	$-0.289461\pi$
$463$	26.4339	1.22849	0.614244	+	0.789116i	$0.289461\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	31.5897	1.46180	0.730898	−	0.682486i	$-0.239101\pi$
$467$	31.5897	1.46180	0.730898	+	0.682486i	$0.239101\pi$
$468$	0	0
$469$	−3.38474	−0.156293
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−0.318028	−0.0146230
$474$	0	0
$475$	4.06647	0.186582
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−12.8045	−0.585052	−0.292526	−	0.956258i	$-0.594496\pi$
$479$	−12.8045	−0.585052	−0.292526	+	0.956258i	$0.594496\pi$
$480$	0	0
$481$	1.16861	0.0532839
$482$	0	0
$483$	0	0
$484$	0	0
$485$	34.2385	1.55469
$486$	0	0
$487$	−38.2482	−1.73319	−0.866596	−	0.499011i	$-0.833697\pi$
$487$	−38.2482	−1.73319	−0.866596	+	0.499011i	$0.833697\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	38.4176	1.73376	0.866881	−	0.498514i	$-0.166121\pi$
$491$	38.4176	1.73376	0.866881	+	0.498514i	$0.166121\pi$
$492$	0	0
$493$	−5.67938	−0.255786
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.43891	0.288825
$498$	0	0
$499$	−3.61884	−0.162001	−0.0810007	−	0.996714i	$-0.525812\pi$
$499$	−3.61884	−0.162001	−0.0810007	+	0.996714i	$0.525812\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	31.3565	1.39812	0.699059	−	0.715064i	$-0.253602\pi$
$503$	31.3565	1.39812	0.699059	+	0.715064i	$0.253602\pi$
$504$	0	0
$505$	−28.0563	−1.24849
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−22.9722	−1.01822	−0.509111	−	0.860701i	$-0.670026\pi$
$509$	−22.9722	−1.01822	−0.509111	+	0.860701i	$0.670026\pi$
$510$	0	0
$511$	−4.01681	−0.177693
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−33.5105	−1.47665
$516$	0	0
$517$	−1.87116	−0.0822934
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−12.1707	−0.533210	−0.266605	−	0.963806i	$-0.585902\pi$
$521$	−12.1707	−0.533210	−0.266605	+	0.963806i	$0.585902\pi$
$522$	0	0
$523$	6.62991	0.289906	0.144953	−	0.989439i	$-0.453697\pi$
$523$	6.62991	0.289906	0.144953	+	0.989439i	$0.453697\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	21.5188	0.937375
$528$	0	0
$529$	−22.3895	−0.973458
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0.219556	0.00951003
$534$	0	0
$535$	18.9472	0.819159
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.62481	0.156132
$540$	0	0
$541$	−14.6963	−0.631842	−0.315921	−	0.948785i	$-0.602313\pi$
$541$	−14.6963	−0.631842	−0.315921	+	0.948785i	$0.602313\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−40.6934	−1.74311
$546$	0	0
$547$	40.9360	1.75030	0.875148	−	0.483856i	$-0.160764\pi$
$547$	40.9360	1.75030	0.875148	+	0.483856i	$0.160764\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.30162	0.140654
$552$	0	0
$553$	2.52344	0.107308
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−41.2803	−1.74910	−0.874552	−	0.484932i	$-0.838844\pi$
$557$	−41.2803	−1.74910	−0.874552	+	0.484932i	$0.838844\pi$
$558$	0	0
$559$	−0.174491	−0.00738017
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−15.4103	−0.649467	−0.324733	−	0.945806i	$-0.605275\pi$
$563$	−15.4103	−0.649467	−0.324733	+	0.945806i	$0.605275\pi$
$564$	0	0
$565$	33.6540	1.41584
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−11.2870	−0.473177	−0.236588	−	0.971610i	$-0.576029\pi$
$569$	−11.2870	−0.473177	−0.236588	+	0.971610i	$0.576029\pi$
$570$	0	0
$571$	1.50980	0.0631830	0.0315915	−	0.999501i	$-0.489942\pi$
$571$	1.50980	0.0631830	0.0315915	+	0.999501i	$0.489942\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0.781327	0.0325836
$576$	0	0
$577$	24.9800	1.03993	0.519966	−	0.854187i	$-0.325945\pi$
$577$	24.9800	1.03993	0.519966	+	0.854187i	$0.325945\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−0.575363	−0.0238701
$582$	0	0
$583$	3.60678	0.149378
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.9400	−0.864287	−0.432144	−	0.901805i	$-0.642243\pi$
$587$	−20.9400	−0.864287	−0.432144	+	0.901805i	$0.642243\pi$
$588$	0	0
$589$	−12.5096	−0.515450
$590$	0	0
$591$	0	0
$592$	0	0
$593$	32.8904	1.35065	0.675323	−	0.737522i	$-0.264004\pi$
$593$	32.8904	1.35065	0.675323	+	0.737522i	$0.264004\pi$
$594$	0	0
$595$	7.22645	0.296256
$596$	0	0
$597$	0	0
$598$	0	0
$599$	18.3583	0.750098	0.375049	−	0.927005i	$-0.377626\pi$
$599$	18.3583	0.750098	0.375049	+	0.927005i	$0.377626\pi$
$600$	0	0
$601$	−45.8253	−1.86925	−0.934626	−	0.355633i	$-0.884265\pi$
$601$	−45.8253	−1.86925	−0.934626	+	0.355633i	$0.884265\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	21.4204	0.870861
$606$	0	0
$607$	17.6455	0.716210	0.358105	−	0.933681i	$-0.383423\pi$
$607$	17.6455	0.716210	0.358105	+	0.933681i	$0.383423\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.02664	−0.0415332
$612$	0	0
$613$	2.27761	0.0919919	0.0459960	−	0.998942i	$-0.485354\pi$
$613$	2.27761	0.0919919	0.0459960	+	0.998942i	$0.485354\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.33514	−0.0537507	−0.0268753	−	0.999639i	$-0.508556\pi$
$617$	−1.33514	−0.0537507	−0.0268753	+	0.999639i	$0.508556\pi$
$618$	0	0
$619$	16.6808	0.670458	0.335229	−	0.942137i	$-0.391186\pi$
$619$	16.6808	0.670458	0.335229	+	0.942137i	$0.391186\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−0.928440	−0.0371971
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−27.6751	−1.10348
$630$	0	0
$631$	−3.60704	−0.143594	−0.0717971	−	0.997419i	$-0.522873\pi$
$631$	−3.60704	−0.143594	−0.0717971	+	0.997419i	$0.522873\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−19.0556	−0.756199
$636$	0	0
$637$	1.98880	0.0787992
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−10.2139	−0.403426	−0.201713	−	0.979445i	$-0.564651\pi$
$641$	−10.2139	−0.403426	−0.201713	+	0.979445i	$0.564651\pi$
$642$	0	0
$643$	2.82338	0.111343	0.0556717	−	0.998449i	$-0.482270\pi$
$643$	2.82338	0.111343	0.0556717	+	0.998449i	$0.482270\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	41.4408	1.62921	0.814604	−	0.580018i	$-0.196954\pi$
$647$	41.4408	1.62921	0.814604	+	0.580018i	$0.196954\pi$
$648$	0	0
$649$	−4.79900	−0.188377
$650$	0	0
$651$	0	0
$652$	0	0
$653$	43.7584	1.71240	0.856200	−	0.516645i	$-0.172819\pi$
$653$	43.7584	1.71240	0.856200	+	0.516645i	$0.172819\pi$
$654$	0	0
$655$	−33.1618	−1.29574
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−7.47042	−0.291006	−0.145503	−	0.989358i	$-0.546480\pi$
$659$	−7.47042	−0.291006	−0.145503	+	0.989358i	$0.546480\pi$
$660$	0	0
$661$	28.3065	1.10099	0.550497	−	0.834837i	$-0.314438\pi$
$661$	28.3065	1.10099	0.550497	+	0.834837i	$0.314438\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4.20098	−0.162907
$666$	0	0
$667$	0.634369	0.0245629
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−2.49790	−0.0964305
$672$	0	0
$673$	−31.8789	−1.22884	−0.614421	−	0.788979i	$-0.710610\pi$
$673$	−31.8789	−1.22884	−0.614421	+	0.788979i	$0.710610\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−34.6470	−1.33159	−0.665797	−	0.746133i	$-0.731908\pi$
$677$	−34.6470	−1.33159	−0.665797	+	0.746133i	$0.731908\pi$
$678$	0	0
$679$	8.84275	0.339354
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−7.53159	−0.288188	−0.144094	−	0.989564i	$-0.546027\pi$
$683$	−7.53159	−0.288188	−0.144094	+	0.989564i	$0.546027\pi$
$684$	0	0
$685$	7.07806	0.270439
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.97891	0.0753905
$690$	0	0
$691$	−34.3094	−1.30519	−0.652596	−	0.757706i	$-0.726320\pi$
$691$	−34.3094	−1.30519	−0.652596	+	0.757706i	$0.726320\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	18.8004	0.713141
$696$	0	0
$697$	−5.19956	−0.196947
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−0.0251855	−0.000951242 0	−0.000475621	−	1.00000i	$-0.500151\pi$
$701$	−0.0251855	−0.000951242 0	−0.000475621		1.00000i	$0.500151\pi$
$702$	0	0
$703$	16.0885	0.606789
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−7.24608	−0.272517
$708$	0	0
$709$	−28.2131	−1.05957	−0.529783	−	0.848133i	$-0.677727\pi$
$709$	−28.2131	−1.05957	−0.529783	+	0.848133i	$0.677727\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.40359	−0.0900150
$714$	0	0
$715$	−0.318028	−0.0118936
$716$	0	0
$717$	0	0
$718$	0	0
$719$	24.5507	0.915585	0.457792	−	0.889059i	$-0.348640\pi$
$719$	24.5507	0.915585	0.457792	+	0.889059i	$0.348640\pi$
$720$	0	0
$721$	−8.65474	−0.322319
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0.811912	0.0301537
$726$	0	0
$727$	16.2103	0.601208	0.300604	−	0.953749i	$-0.402812\pi$
$727$	16.2103	0.601208	0.300604	+	0.953749i	$0.402812\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	4.13231	0.152839
$732$	0	0
$733$	−16.0087	−0.591296	−0.295648	−	0.955297i	$-0.595535\pi$
$733$	−16.0087	−0.591296	−0.295648	+	0.955297i	$0.595535\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.52767	−0.129943
$738$	0	0
$739$	−11.9536	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$739$	−11.9536	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−38.6259	−1.41705	−0.708524	−	0.705686i	$-0.750639\pi$
$743$	−38.6259	−1.41705	−0.708524	+	0.705686i	$0.750639\pi$
$744$	0	0
$745$	−37.1618	−1.36150
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.89348	0.178804
$750$	0	0
$751$	21.6589	0.790346	0.395173	−	0.918607i	$-0.370685\pi$
$751$	21.6589	0.790346	0.395173	+	0.918607i	$0.370685\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−34.5287	−1.25663
$756$	0	0
$757$	43.6016	1.58473	0.792364	−	0.610048i	$-0.208850\pi$
$757$	43.6016	1.58473	0.792364	+	0.610048i	$0.208850\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−32.5442	−1.17973	−0.589863	−	0.807504i	$-0.700818\pi$
$761$	−32.5442	−1.17973	−0.589863	+	0.807504i	$0.700818\pi$
$762$	0	0
$763$	−10.5099	−0.380482
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.63303	−0.0950734
$768$	0	0
$769$	−11.9886	−0.432319	−0.216159	−	0.976358i	$-0.569353\pi$
$769$	−11.9886	−0.432319	−0.216159	+	0.976358i	$0.569353\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−31.3355	−1.12706	−0.563530	−	0.826096i	$-0.690557\pi$
$773$	−31.3355	−1.12706	−0.563530	+	0.826096i	$0.690557\pi$
$774$	0	0
$775$	−3.07629	−0.110503
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.02268	0.108299
$780$	0	0
$781$	6.71080	0.240131
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14.3973	0.513860
$786$	0	0
$787$	9.39669	0.334956	0.167478	−	0.985876i	$-0.446438\pi$
$787$	9.39669	0.334956	0.167478	+	0.985876i	$0.446438\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	8.69181	0.309045
$792$	0	0
$793$	−1.37051	−0.0486682
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−21.7756	−0.771330	−0.385665	−	0.922639i	$-0.626028\pi$
$797$	−21.7756	−0.771330	−0.385665	+	0.922639i	$0.626028\pi$
$798$	0	0
$799$	24.3129	0.860129
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−4.18642	−0.147736
$804$	0	0
$805$	−0.807172	−0.0284491
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.83916	0.0646614	0.0323307	−	0.999477i	$-0.489707\pi$
$809$	1.83916	0.0646614	0.0323307	+	0.999477i	$0.489707\pi$
$810$	0	0
$811$	7.27579	0.255488	0.127744	−	0.991807i	$-0.459226\pi$
$811$	7.27579	0.255488	0.127744	+	0.991807i	$0.459226\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−47.3895	−1.65998
$816$	0	0
$817$	−2.40225	−0.0840441
$818$	0	0
$819$	0	0
$820$	0	0
$821$	45.0675	1.57287	0.786434	−	0.617674i	$-0.211925\pi$
$821$	45.0675	1.57287	0.786434	+	0.617674i	$0.211925\pi$
$822$	0	0
$823$	−8.71724	−0.303864	−0.151932	−	0.988391i	$-0.548549\pi$
$823$	−8.71724	−0.303864	−0.151932	+	0.988391i	$0.548549\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.18072	0.284471	0.142236	−	0.989833i	$-0.454571\pi$
$827$	8.18072	0.284471	0.142236	+	0.989833i	$0.454571\pi$
$828$	0	0
$829$	−43.1349	−1.49814	−0.749068	−	0.662493i	$-0.769498\pi$
$829$	−43.1349	−1.49814	−0.749068	+	0.662493i	$0.769498\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−47.0991	−1.63189
$834$	0	0
$835$	−2.00000	−0.0692129
$836$	0	0
$837$	0	0
$838$	0	0
$839$	37.2048	1.28445	0.642226	−	0.766515i	$-0.278011\pi$
$839$	37.2048	1.28445	0.642226	+	0.766515i	$0.278011\pi$
$840$	0	0
$841$	−28.3408	−0.977269
$842$	0	0
$843$	0	0
$844$	0	0
$845$	25.8255	0.888425
$846$	0	0
$847$	5.53222	0.190089
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.09123	0.105966
$852$	0	0
$853$	11.9553	0.409341	0.204671	−	0.978831i	$-0.434388\pi$
$853$	11.9553	0.409341	0.204671	+	0.978831i	$0.434388\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.28383	0.0780142	0.0390071	−	0.999239i	$-0.487580\pi$
$857$	2.28383	0.0780142	0.0390071	+	0.999239i	$0.487580\pi$
$858$	0	0
$859$	−16.8387	−0.574529	−0.287265	−	0.957851i	$-0.592746\pi$
$859$	−16.8387	−0.574529	−0.287265	+	0.957851i	$0.592746\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	31.2591	1.06407	0.532036	−	0.846721i	$-0.321427\pi$
$863$	31.2591	1.06407	0.532036	+	0.846721i	$0.321427\pi$
$864$	0	0
$865$	−39.0556	−1.32793
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.63000	0.0892165
$870$	0	0
$871$	−1.93550	−0.0655820
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−6.19847	−0.209546
$876$	0	0
$877$	−6.86671	−0.231872	−0.115936	−	0.993257i	$-0.536987\pi$
$877$	−6.86671	−0.231872	−0.115936	+	0.993257i	$0.536987\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	9.06820	0.305515	0.152758	−	0.988264i	$-0.451185\pi$
$881$	9.06820	0.305515	0.152758	+	0.988264i	$0.451185\pi$
$882$	0	0
$883$	−2.28390	−0.0768594	−0.0384297	−	0.999261i	$-0.512236\pi$
$883$	−2.28390	−0.0768594	−0.0384297	+	0.999261i	$0.512236\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	31.8293	1.06872	0.534361	−	0.845256i	$-0.320552\pi$
$887$	31.8293	1.06872	0.534361	+	0.845256i	$0.320552\pi$
$888$	0	0
$889$	−4.92148	−0.165061
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−14.1339	−0.472974
$894$	0	0
$895$	−21.0076	−0.702208
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.49767	−0.0833021
$900$	0	0
$901$	−46.8648	−1.56129
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−16.7237	−0.555914
$906$	0	0
$907$	−54.0598	−1.79502	−0.897512	−	0.440989i	$-0.854628\pi$
$907$	−54.0598	−1.79502	−0.897512	+	0.440989i	$0.854628\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−6.19867	−0.205371	−0.102686	−	0.994714i	$-0.532744\pi$
$911$	−6.19867	−0.205371	−0.102686	+	0.994714i	$0.532744\pi$
$912$	0	0
$913$	−0.599658	−0.0198458
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−8.56467	−0.282830
$918$	0	0
$919$	6.53624	0.215611	0.107805	−	0.994172i	$-0.465618\pi$
$919$	6.53624	0.215611	0.107805	+	0.994172i	$0.465618\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3.68197	0.121194
$924$	0	0
$925$	3.95638	0.130085
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−21.4372	−0.703331	−0.351666	−	0.936126i	$-0.614385\pi$
$929$	−21.4372	−0.703331	−0.351666	+	0.936126i	$0.614385\pi$
$930$	0	0
$931$	27.3803	0.897353
$932$	0	0
$933$	0	0
$934$	0	0
$935$	7.53159	0.246309
$936$	0	0
$937$	−1.58083	−0.0516434	−0.0258217	−	0.999667i	$-0.508220\pi$
$937$	−1.58083	−0.0516434	−0.0258217	+	0.999667i	$0.508220\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−10.6995	−0.348794	−0.174397	−	0.984675i	$-0.555798\pi$
$941$	−10.6995	−0.348794	−0.174397	+	0.984675i	$0.555798\pi$
$942$	0	0
$943$	0.580775	0.0189126
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−29.2664	−0.951030	−0.475515	−	0.879708i	$-0.657738\pi$
$947$	−29.2664	−0.951030	−0.475515	+	0.879708i	$0.657738\pi$
$948$	0	0
$949$	−2.29694	−0.0745618
$950$	0	0
$951$	0	0
$952$	0	0
$953$	45.0155	1.45820	0.729098	−	0.684410i	$-0.239940\pi$
$953$	45.0155	1.45820	0.729098	+	0.684410i	$0.239940\pi$
$954$	0	0
$955$	18.8769	0.610842
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.82805	0.0590307
$960$	0	0
$961$	−21.5365	−0.694725
$962$	0	0
$963$	0	0
$964$	0	0
$965$	32.6583	1.05131
$966$	0	0
$967$	61.7031	1.98424	0.992118	−	0.125304i	$-0.0399907\pi$
$967$	61.7031	1.98424	0.992118	+	0.125304i	$0.0399907\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	11.4233	0.366590	0.183295	−	0.983058i	$-0.441324\pi$
$971$	11.4233	0.366590	0.183295	+	0.983058i	$0.441324\pi$
$972$	0	0
$973$	4.85558	0.155663
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.28909	0.105227	0.0526137	−	0.998615i	$-0.483245\pi$
$977$	3.28909	0.105227	0.0526137	+	0.998615i	$0.483245\pi$
$978$	0	0
$979$	−0.967644	−0.0309260
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−15.4572	−0.493009	−0.246505	−	0.969142i	$-0.579282\pi$
$983$	−15.4572	−0.493009	−0.246505	+	0.969142i	$0.579282\pi$
$984$	0	0
$985$	−34.6920	−1.10538
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.461566	−0.0146770
$990$	0	0
$991$	−21.3319	−0.677630	−0.338815	−	0.940853i	$-0.610026\pi$
$991$	−21.3319	−0.677630	−0.338815	+	0.940853i	$0.610026\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−12.9219	−0.409653
$996$	0	0
$997$	−36.9757	−1.17103	−0.585516	−	0.810661i	$-0.699108\pi$
$997$	−36.9757	−1.17103	−0.585516	+	0.810661i	$0.699108\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.d.1.2		5
3.2	odd	2		668.2.a.b.1.5	✓	5
12.11	even	2		2672.2.a.j.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.a.b.1.5	✓	5	3.2	odd	2
2672.2.a.j.1.1		5	12.11	even	2
6012.2.a.d.1.2		5	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:	\( N \)	\(=\)	\( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6012.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{5} -0.516539 q^{7} +O(q^{10})\)	\(q-2.00000 q^{5} -0.516539 q^{7} -0.538350 q^{11} -0.295373 q^{13} +6.99507 q^{17} -4.06647 q^{19} -0.781327 q^{23} -1.00000 q^{25} -0.811912 q^{29} +3.07629 q^{31} +1.03308 q^{35} -3.95638 q^{37} -0.743318 q^{41} +0.590746 q^{43} +3.47572 q^{47} -6.73319 q^{49} -6.69970 q^{53} +1.07670 q^{55} +8.91427 q^{59} +4.63992 q^{61} +0.590746 q^{65} +6.55274 q^{67} -12.4655 q^{71} +7.77640 q^{73} +0.278079 q^{77} -4.88529 q^{79} +1.11388 q^{83} -13.9901 q^{85} +1.79742 q^{89} +0.152572 q^{91} +8.13294 q^{95} -17.1192 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 10 q^{5} + 9 q^{7}+O(q^{10}) \) 5 * q - 10 * q^5 + 9 * q^7	\( 5 q - 10 q^{5} + 9 q^{7} - 5 q^{11} - 4 q^{13} + 2 q^{17} + 5 q^{19} - 6 q^{23} - 5 q^{25} + 5 q^{29} + 9 q^{31} - 18 q^{35} + 8 q^{37} + 4 q^{41} + 8 q^{43} - 13 q^{47} + 14 q^{49} + 2 q^{53} + 10 q^{55} - 4 q^{59} + 11 q^{61} + 8 q^{65} + 28 q^{67} - 2 q^{71} + 8 q^{73} + 12 q^{77} - 10 q^{79} - 2 q^{83} - 4 q^{85} + 17 q^{89} - 12 q^{91} - 10 q^{95} - 27 q^{97}+O(q^{100}) \) 5 * q - 10 * q^5 + 9 * q^7 - 5 * q^11 - 4 * q^13 + 2 * q^17 + 5 * q^19 - 6 * q^23 - 5 * q^25 + 5 * q^29 + 9 * q^31 - 18 * q^35 + 8 * q^37 + 4 * q^41 + 8 * q^43 - 13 * q^47 + 14 * q^49 + 2 * q^53 + 10 * q^55 - 4 * q^59 + 11 * q^61 + 8 * q^65 + 28 * q^67 - 2 * q^71 + 8 * q^73 + 12 * q^77 - 10 * q^79 - 2 * q^83 - 4 * q^85 + 17 * q^89 - 12 * q^91 - 10 * q^95 - 27 * q^97

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