LMFDB - Embedded newform 6800.2.a.ch.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6800,2,Mod(1,6800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6800 = 2^{4} \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6800.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:	$54.2982733745$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.358395264.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 9x^{4} + 14x^{3} + 24x^{2} - 22x - 18 $ x^6 - 2x^5 - 9x^4 + 14x^3 + 24x^2 - 22*x - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 680)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.605009$ of defining polynomial
Character	$\chi$	$=$	6800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.02896 q^{3} +0.0907276 q^{7} +6.17457 q^{9} +O(q^{10})$	$q-3.02896 q^{3} +0.0907276 q^{7} +6.17457 q^{9} +6.07152 q^{11} +1.83254 q^{13} +1.00000 q^{17} -2.00000 q^{19} -0.274810 q^{21} -8.30786 q^{23} -9.61565 q^{27} +1.38049 q^{29} -3.92327 q^{31} -18.3904 q^{33} +1.38049 q^{37} -5.55070 q^{39} +6.71588 q^{41} -6.82686 q^{43} -1.14415 q^{47} -6.99177 q^{49} -3.02896 q^{51} -2.00000 q^{53} +6.05791 q^{57} -4.81719 q^{59} -9.86599 q^{61} +0.560204 q^{63} -9.09522 q^{67} +25.1642 q^{69} -10.6910 q^{71} +13.6955 q^{73} +0.550854 q^{77} +16.6925 q^{79} +10.6016 q^{81} -6.99069 q^{83} -4.18146 q^{87} +4.88334 q^{89} +0.166262 q^{91} +11.8834 q^{93} -0.824232 q^{97} +37.4890 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} + 2 q^{7} + 14 q^{9}+O(q^{10}) $ 6 * q - 4 * q^3 + 2 * q^7 + 14 * q^9	$ 6 q - 4 q^{3} + 2 q^{7} + 14 q^{9} - 10 q^{11} - 10 q^{13} + 6 q^{17} - 12 q^{19} + 12 q^{21} + 4 q^{23} - 22 q^{27} - 4 q^{31} - 10 q^{33} + 2 q^{39} + 14 q^{41} - 14 q^{43} - 6 q^{47} + 6 q^{49} - 4 q^{51} - 12 q^{53} + 8 q^{57} - 4 q^{59} + 2 q^{61} - 24 q^{63} - 30 q^{67} - 4 q^{69} - 26 q^{71} + 18 q^{73} - 30 q^{77} + 12 q^{79} + 46 q^{81} - 2 q^{83} - 28 q^{87} + 24 q^{89} + 2 q^{91} - 6 q^{93} - 8 q^{97} - 20 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 + 2 * q^7 + 14 * q^9 - 10 * q^11 - 10 * q^13 + 6 * q^17 - 12 * q^19 + 12 * q^21 + 4 * q^23 - 22 * q^27 - 4 * q^31 - 10 * q^33 + 2 * q^39 + 14 * q^41 - 14 * q^43 - 6 * q^47 + 6 * q^49 - 4 * q^51 - 12 * q^53 + 8 * q^57 - 4 * q^59 + 2 * q^61 - 24 * q^63 - 30 * q^67 - 4 * q^69 - 26 * q^71 + 18 * q^73 - 30 * q^77 + 12 * q^79 + 46 * q^81 - 2 * q^83 - 28 * q^87 + 24 * q^89 + 2 * q^91 - 6 * q^93 - 8 * q^97 - 20 * q^99

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
$3$	−3.02896	−1.74877	−0.874384	−	0.485234i	$-0.838734\pi$
$3$	−3.02896	−1.74877	−0.874384	+	0.485234i	$0.838734\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.0907276	0.0342918	0.0171459	−	0.999853i	$-0.494542\pi$
$7$	0.0907276	0.0342918	0.0171459	+	0.999853i	$0.494542\pi$
$8$	0	0
$9$	6.17457	2.05819
$10$	0	0
$11$	6.07152	1.83063	0.915316	−	0.402737i	$-0.131941\pi$
$11$	6.07152	1.83063	0.915316	+	0.402737i	$0.131941\pi$
$12$	0	0
$13$	1.83254	0.508256	0.254128	−	0.967171i	$-0.418211\pi$
$13$	1.83254	0.508256	0.254128	+	0.967171i	$0.418211\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−0.274810	−0.0599684
$22$	0	0
$23$	−8.30786	−1.73231	−0.866155	−	0.499776i	$-0.833416\pi$
$23$	−8.30786	−1.73231	−0.866155	+	0.499776i	$0.833416\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−9.61565	−1.85053
$28$	0	0
$29$	1.38049	0.256351	0.128176	−	0.991751i	$-0.459088\pi$
$29$	1.38049	0.256351	0.128176	+	0.991751i	$0.459088\pi$
$30$	0	0
$31$	−3.92327	−0.704640	−0.352320	−	0.935880i	$-0.614607\pi$
$31$	−3.92327	−0.704640	−0.352320	+	0.935880i	$0.614607\pi$
$32$	0	0
$33$	−18.3904	−3.20135
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.38049	0.226952	0.113476	−	0.993541i	$-0.463802\pi$
$37$	1.38049	0.226952	0.113476	+	0.993541i	$0.463802\pi$
$38$	0	0
$39$	−5.55070	−0.888823
$40$	0	0
$41$	6.71588	1.04884	0.524422	−	0.851458i	$-0.324281\pi$
$41$	6.71588	1.04884	0.524422	+	0.851458i	$0.324281\pi$
$42$	0	0
$43$	−6.82686	−1.04109	−0.520543	−	0.853835i	$-0.674270\pi$
$43$	−6.82686	−1.04109	−0.520543	+	0.853835i	$0.674270\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.14415	−0.166891	−0.0834456	−	0.996512i	$-0.526592\pi$
$47$	−1.14415	−0.166891	−0.0834456	+	0.996512i	$0.526592\pi$
$48$	0	0
$49$	−6.99177	−0.998824
$50$	0	0
$51$	−3.02896	−0.424139
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.05791	0.802390
$58$	0	0
$59$	−4.81719	−0.627145	−0.313573	−	0.949564i	$-0.601526\pi$
$59$	−4.81719	−0.627145	−0.313573	+	0.949564i	$0.601526\pi$
$60$	0	0
$61$	−9.86599	−1.26321	−0.631605	−	0.775290i	$-0.717604\pi$
$61$	−9.86599	−1.26321	−0.631605	+	0.775290i	$0.717604\pi$
$62$	0	0
$63$	0.560204	0.0705791
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.09522	−1.11116	−0.555579	−	0.831464i	$-0.687503\pi$
$67$	−9.09522	−1.11116	−0.555579	+	0.831464i	$0.687503\pi$
$68$	0	0
$69$	25.1642	3.02941
$70$	0	0
$71$	−10.6910	−1.26879	−0.634396	−	0.773009i	$-0.718751\pi$
$71$	−10.6910	−1.26879	−0.634396	+	0.773009i	$0.718751\pi$
$72$	0	0
$73$	13.6955	1.60294	0.801469	−	0.598036i	$-0.204052\pi$
$73$	13.6955	1.60294	0.801469	+	0.598036i	$0.204052\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.550854	0.0627756
$78$	0	0
$79$	16.6925	1.87805	0.939024	−	0.343852i	$-0.111732\pi$
$79$	16.6925	1.87805	0.939024	+	0.343852i	$0.111732\pi$
$80$	0	0
$81$	10.6016	1.17796
$82$	0	0
$83$	−6.99069	−0.767328	−0.383664	−	0.923473i	$-0.625338\pi$
$83$	−6.99069	−0.767328	−0.383664	+	0.923473i	$0.625338\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−4.18146	−0.448299
$88$	0	0
$89$	4.88334	0.517633	0.258816	−	0.965927i	$-0.416668\pi$
$89$	4.88334	0.517633	0.258816	+	0.965927i	$0.416668\pi$
$90$	0	0
$91$	0.166262	0.0174290
$92$	0	0
$93$	11.8834	1.23225
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.824232	−0.0836881	−0.0418440	−	0.999124i	$-0.513323\pi$
$97$	−0.824232	−0.0836881	−0.0418440	+	0.999124i	$0.513323\pi$
$98$	0	0
$99$	37.4890	3.76779
$100$	0	0
$101$	5.93386	0.590441	0.295220	−	0.955429i	$-0.404607\pi$
$101$	5.93386	0.590441	0.295220	+	0.955429i	$0.404607\pi$
$102$	0	0
$103$	16.2396	1.60014	0.800068	−	0.599909i	$-0.204797\pi$
$103$	16.2396	1.60014	0.800068	+	0.599909i	$0.204797\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.84750	−0.661973	−0.330986	−	0.943636i	$-0.607381\pi$
$107$	−6.84750	−0.661973	−0.330986	+	0.943636i	$0.607381\pi$
$108$	0	0
$109$	−12.9891	−1.24413	−0.622065	−	0.782965i	$-0.713706\pi$
$109$	−12.9891	−1.24413	−0.622065	+	0.782965i	$0.713706\pi$
$110$	0	0
$111$	−4.18146	−0.396886
$112$	0	0
$113$	−16.2750	−1.53103	−0.765514	−	0.643419i	$-0.777515\pi$
$113$	−16.2750	−1.53103	−0.765514	+	0.643419i	$0.777515\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	11.3152	1.04609
$118$	0	0
$119$	0.0907276	0.00831698
$120$	0	0
$121$	25.8633	2.35121
$122$	0	0
$123$	−20.3421	−1.83419
$124$	0	0
$125$	0	0
$126$	0	0
$127$	17.2103	1.52716	0.763582	−	0.645711i	$-0.223439\pi$
$127$	17.2103	1.52716	0.763582	+	0.645711i	$0.223439\pi$
$128$	0	0
$129$	20.6782	1.82062
$130$	0	0
$131$	−15.3038	−1.33710	−0.668548	−	0.743669i	$-0.733084\pi$
$131$	−15.3038	−1.33710	−0.668548	+	0.743669i	$0.733084\pi$
$132$	0	0
$133$	−0.181455	−0.0157342
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−21.7599	−1.85907	−0.929536	−	0.368732i	$-0.879792\pi$
$137$	−21.7599	−1.85907	−0.929536	+	0.368732i	$0.879792\pi$
$138$	0	0
$139$	6.23102	0.528508	0.264254	−	0.964453i	$-0.414874\pi$
$139$	6.23102	0.528508	0.264254	+	0.964453i	$0.414874\pi$
$140$	0	0
$141$	3.46557	0.291854
$142$	0	0
$143$	11.1263	0.930430
$144$	0	0
$145$	0	0
$146$	0	0
$147$	21.1778	1.74671
$148$	0	0
$149$	19.2243	1.57491	0.787456	−	0.616371i	$-0.211398\pi$
$149$	19.2243	1.57491	0.787456	+	0.616371i	$0.211398\pi$
$150$	0	0
$151$	10.9857	0.894002	0.447001	−	0.894534i	$-0.352492\pi$
$151$	10.9857	0.894002	0.447001	+	0.894534i	$0.352492\pi$
$152$	0	0
$153$	6.17457	0.499185
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.934802	0.0746053	0.0373027	−	0.999304i	$-0.488123\pi$
$157$	0.934802	0.0746053	0.0373027	+	0.999304i	$0.488123\pi$
$158$	0	0
$159$	6.05791	0.480424
$160$	0	0
$161$	−0.753752	−0.0594040
$162$	0	0
$163$	−0.863690	−0.0676494	−0.0338247	−	0.999428i	$-0.510769\pi$
$163$	−0.863690	−0.0676494	−0.0338247	+	0.999428i	$0.510769\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.2066	0.944571	0.472286	−	0.881446i	$-0.343429\pi$
$167$	12.2066	0.944571	0.472286	+	0.881446i	$0.343429\pi$
$168$	0	0
$169$	−9.64178	−0.741675
$170$	0	0
$171$	−12.3491	−0.944363
$172$	0	0
$173$	−3.46036	−0.263087	−0.131543	−	0.991310i	$-0.541993\pi$
$173$	−3.46036	−0.263087	−0.131543	+	0.991310i	$0.541993\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	14.5911	1.09673
$178$	0	0
$179$	−13.7596	−1.02844	−0.514222	−	0.857657i	$-0.671919\pi$
$179$	−13.7596	−1.02844	−0.514222	+	0.857657i	$0.671919\pi$
$180$	0	0
$181$	−13.0019	−0.966424	−0.483212	−	0.875503i	$-0.660530\pi$
$181$	−13.0019	−0.966424	−0.483212	+	0.875503i	$0.660530\pi$
$182$	0	0
$183$	29.8837	2.20906
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.07152	0.443993
$188$	0	0
$189$	−0.872404	−0.0634581
$190$	0	0
$191$	17.2523	1.24833	0.624165	−	0.781293i	$-0.285439\pi$
$191$	17.2523	1.24833	0.624165	+	0.781293i	$0.285439\pi$
$192$	0	0
$193$	−11.6436	−0.838128	−0.419064	−	0.907957i	$-0.637642\pi$
$193$	−11.6436	−0.838128	−0.419064	+	0.907957i	$0.637642\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.6255	−0.757039	−0.378519	−	0.925593i	$-0.623567\pi$
$197$	−10.6255	−0.757039	−0.378519	+	0.925593i	$0.623567\pi$
$198$	0	0
$199$	−5.43235	−0.385089	−0.192545	−	0.981288i	$-0.561674\pi$
$199$	−5.43235	−0.385089	−0.192545	+	0.981288i	$0.561674\pi$
$200$	0	0
$201$	27.5490	1.94316
$202$	0	0
$203$	0.125249	0.00879075
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−51.2975	−3.56542
$208$	0	0
$209$	−12.1430	−0.839951
$210$	0	0
$211$	−27.0297	−1.86080	−0.930401	−	0.366543i	$-0.880541\pi$
$211$	−27.0297	−1.86080	−0.930401	+	0.366543i	$0.880541\pi$
$212$	0	0
$213$	32.3826	2.21882
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−0.355949	−0.0241634
$218$	0	0
$219$	−41.4831	−2.80317
$220$	0	0
$221$	1.83254	0.123270
$222$	0	0
$223$	16.4211	1.09964	0.549818	−	0.835285i	$-0.314697\pi$
$223$	16.4211	1.09964	0.549818	+	0.835285i	$0.314697\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.5560	−0.833370	−0.416685	−	0.909051i	$-0.636808\pi$
$227$	−12.5560	−0.833370	−0.416685	+	0.909051i	$0.636808\pi$
$228$	0	0
$229$	−21.1027	−1.39451	−0.697253	−	0.716826i	$-0.745594\pi$
$229$	−21.1027	−1.39451	−0.697253	+	0.716826i	$0.745594\pi$
$230$	0	0
$231$	−1.66851	−0.109780
$232$	0	0
$233$	14.7203	0.964356	0.482178	−	0.876073i	$-0.339846\pi$
$233$	14.7203	0.964356	0.482178	+	0.876073i	$0.339846\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−50.5607	−3.28427
$238$	0	0
$239$	−11.9291	−0.771630	−0.385815	−	0.922576i	$-0.626080\pi$
$239$	−11.9291	−0.771630	−0.385815	+	0.922576i	$0.626080\pi$
$240$	0	0
$241$	7.44091	0.479311	0.239656	−	0.970858i	$-0.422965\pi$
$241$	7.44091	0.479311	0.239656	+	0.970858i	$0.422965\pi$
$242$	0	0
$243$	−3.26498	−0.209449
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.66509	−0.233204
$248$	0	0
$249$	21.1745	1.34188
$250$	0	0
$251$	18.7980	1.18652	0.593259	−	0.805012i	$-0.297841\pi$
$251$	18.7980	1.18652	0.593259	+	0.805012i	$0.297841\pi$
$252$	0	0
$253$	−50.4413	−3.17122
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.7599	−0.858317	−0.429159	−	0.903229i	$-0.641190\pi$
$257$	−13.7599	−0.858317	−0.429159	+	0.903229i	$0.641190\pi$
$258$	0	0
$259$	0.125249	0.00778258
$260$	0	0
$261$	8.52396	0.527620
$262$	0	0
$263$	14.0046	0.863559	0.431780	−	0.901979i	$-0.357886\pi$
$263$	14.0046	0.863559	0.431780	+	0.901979i	$0.357886\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−14.7914	−0.905220
$268$	0	0
$269$	6.27100	0.382350	0.191175	−	0.981556i	$-0.438770\pi$
$269$	6.27100	0.382350	0.191175	+	0.981556i	$0.438770\pi$
$270$	0	0
$271$	−18.2718	−1.10993	−0.554967	−	0.831872i	$-0.687269\pi$
$271$	−18.2718	−1.10993	−0.554967	+	0.831872i	$0.687269\pi$
$272$	0	0
$273$	−0.503601	−0.0304793
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−10.0018	−0.600952	−0.300476	−	0.953789i	$-0.597145\pi$
$277$	−10.0018	−0.600952	−0.300476	+	0.953789i	$0.597145\pi$
$278$	0	0
$279$	−24.2245	−1.45028
$280$	0	0
$281$	1.43979	0.0858909	0.0429454	−	0.999077i	$-0.486326\pi$
$281$	1.43979	0.0858909	0.0429454	+	0.999077i	$0.486326\pi$
$282$	0	0
$283$	−2.91070	−0.173023	−0.0865117	−	0.996251i	$-0.527572\pi$
$283$	−2.91070	−0.173023	−0.0865117	+	0.996251i	$0.527572\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.609316	0.0359668
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	2.49656	0.146351
$292$	0	0
$293$	13.7713	0.804529	0.402265	−	0.915523i	$-0.368223\pi$
$293$	13.7713	0.804529	0.402265	+	0.915523i	$0.368223\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−58.3816	−3.38764
$298$	0	0
$299$	−15.2245	−0.880457
$300$	0	0
$301$	−0.619384	−0.0357007
$302$	0	0
$303$	−17.9734	−1.03254
$304$	0	0
$305$	0	0
$306$	0	0
$307$	25.9495	1.48102	0.740509	−	0.672047i	$-0.234585\pi$
$307$	25.9495	1.48102	0.740509	+	0.672047i	$0.234585\pi$
$308$	0	0
$309$	−49.1891	−2.79827
$310$	0	0
$311$	28.1204	1.59456	0.797281	−	0.603609i	$-0.206271\pi$
$311$	28.1204	1.59456	0.797281	+	0.603609i	$0.206271\pi$
$312$	0	0
$313$	−22.7178	−1.28409	−0.642044	−	0.766668i	$-0.721913\pi$
$313$	−22.7178	−1.28409	−0.642044	+	0.766668i	$0.721913\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−18.5211	−1.04025	−0.520123	−	0.854091i	$-0.674114\pi$
$317$	−18.5211	−1.04025	−0.520123	+	0.854091i	$0.674114\pi$
$318$	0	0
$319$	8.38169	0.469285
$320$	0	0
$321$	20.7408	1.15764
$322$	0	0
$323$	−2.00000	−0.111283
$324$	0	0
$325$	0	0
$326$	0	0
$327$	39.3434	2.17570
$328$	0	0
$329$	−0.103806	−0.00572300
$330$	0	0
$331$	−17.3129	−0.951601	−0.475801	−	0.879553i	$-0.657842\pi$
$331$	−17.3129	−0.951601	−0.475801	+	0.879553i	$0.657842\pi$
$332$	0	0
$333$	8.52396	0.467110
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−10.6314	−0.579129	−0.289565	−	0.957158i	$-0.593511\pi$
$337$	−10.6314	−0.579129	−0.289565	+	0.957158i	$0.593511\pi$
$338$	0	0
$339$	49.2964	2.67741
$340$	0	0
$341$	−23.8202	−1.28994
$342$	0	0
$343$	−1.26944	−0.0685433
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−21.2206	−1.13918	−0.569592	−	0.821928i	$-0.692899\pi$
$347$	−21.2206	−1.13918	−0.569592	+	0.821928i	$0.692899\pi$
$348$	0	0
$349$	−6.56724	−0.351536	−0.175768	−	0.984432i	$-0.556241\pi$
$349$	−6.56724	−0.351536	−0.175768	+	0.984432i	$0.556241\pi$
$350$	0	0
$351$	−17.6211	−0.940545
$352$	0	0
$353$	−11.5320	−0.613784	−0.306892	−	0.951744i	$-0.599289\pi$
$353$	−11.5320	−0.613784	−0.306892	+	0.951744i	$0.599289\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.274810	−0.0145445
$358$	0	0
$359$	−20.9347	−1.10489	−0.552446	−	0.833549i	$-0.686306\pi$
$359$	−20.9347	−1.10489	−0.552446	+	0.833549i	$0.686306\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−78.3389	−4.11172
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−29.2281	−1.52569	−0.762847	−	0.646580i	$-0.776199\pi$
$367$	−29.2281	−1.52569	−0.762847	+	0.646580i	$0.776199\pi$
$368$	0	0
$369$	41.4677	2.15872
$370$	0	0
$371$	−0.181455	−0.00942068
$372$	0	0
$373$	−24.5271	−1.26997	−0.634983	−	0.772526i	$-0.718993\pi$
$373$	−24.5271	−1.26997	−0.634983	+	0.772526i	$0.718993\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.52982	0.130292
$378$	0	0
$379$	19.1832	0.985373	0.492686	−	0.870207i	$-0.336015\pi$
$379$	19.1832	0.985373	0.492686	+	0.870207i	$0.336015\pi$
$380$	0	0
$381$	−52.1291	−2.67066
$382$	0	0
$383$	7.54656	0.385611	0.192806	−	0.981237i	$-0.438241\pi$
$383$	7.54656	0.385611	0.192806	+	0.981237i	$0.438241\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−42.1529	−2.14275
$388$	0	0
$389$	13.6610	0.692642	0.346321	−	0.938116i	$-0.387431\pi$
$389$	13.6610	0.692642	0.346321	+	0.938116i	$0.387431\pi$
$390$	0	0
$391$	−8.30786	−0.420147
$392$	0	0
$393$	46.3544	2.33827
$394$	0	0
$395$	0	0
$396$	0	0
$397$	28.5876	1.43477	0.717386	−	0.696676i	$-0.245338\pi$
$397$	28.5876	1.43477	0.717386	+	0.696676i	$0.245338\pi$
$398$	0	0
$399$	0.549620	0.0275154
$400$	0	0
$401$	6.88740	0.343940	0.171970	−	0.985102i	$-0.444987\pi$
$401$	6.88740	0.343940	0.171970	+	0.985102i	$0.444987\pi$
$402$	0	0
$403$	−7.18957	−0.358138
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.38169	0.415465
$408$	0	0
$409$	−10.4580	−0.517115	−0.258558	−	0.965996i	$-0.583247\pi$
$409$	−10.4580	−0.517115	−0.258558	+	0.965996i	$0.583247\pi$
$410$	0	0
$411$	65.9097	3.25109
$412$	0	0
$413$	−0.437052	−0.0215059
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−18.8735	−0.924238
$418$	0	0
$419$	−17.0816	−0.834490	−0.417245	−	0.908794i	$-0.637004\pi$
$419$	−17.0816	−0.834490	−0.417245	+	0.908794i	$0.637004\pi$
$420$	0	0
$421$	3.53591	0.172330	0.0861648	−	0.996281i	$-0.472539\pi$
$421$	3.53591	0.172330	0.0861648	+	0.996281i	$0.472539\pi$
$422$	0	0
$423$	−7.06463	−0.343494
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−0.895117	−0.0433178
$428$	0	0
$429$	−33.7012	−1.62711
$430$	0	0
$431$	−14.7872	−0.712274	−0.356137	−	0.934434i	$-0.615906\pi$
$431$	−14.7872	−0.712274	−0.356137	+	0.934434i	$0.615906\pi$
$432$	0	0
$433$	17.8221	0.856477	0.428238	−	0.903666i	$-0.359134\pi$
$433$	17.8221	0.856477	0.428238	+	0.903666i	$0.359134\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	16.6157	0.794838
$438$	0	0
$439$	−14.3781	−0.686228	−0.343114	−	0.939294i	$-0.611482\pi$
$439$	−14.3781	−0.686228	−0.343114	+	0.939294i	$0.611482\pi$
$440$	0	0
$441$	−43.1712	−2.05577
$442$	0	0
$443$	36.5921	1.73854	0.869271	−	0.494335i	$-0.164588\pi$
$443$	36.5921	1.73854	0.869271	+	0.494335i	$0.164588\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−58.2294	−2.75416
$448$	0	0
$449$	−17.3396	−0.818305	−0.409152	−	0.912466i	$-0.634176\pi$
$449$	−17.3396	−0.818305	−0.409152	+	0.912466i	$0.634176\pi$
$450$	0	0
$451$	40.7756	1.92005
$452$	0	0
$453$	−33.2751	−1.56340
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.37361	−0.157811	−0.0789055	−	0.996882i	$-0.525143\pi$
$457$	−3.37361	−0.157811	−0.0789055	+	0.996882i	$0.525143\pi$
$458$	0	0
$459$	−9.61565	−0.448820
$460$	0	0
$461$	3.52670	0.164255	0.0821274	−	0.996622i	$-0.473829\pi$
$461$	3.52670	0.164255	0.0821274	+	0.996622i	$0.473829\pi$
$462$	0	0
$463$	−2.87873	−0.133786	−0.0668929	−	0.997760i	$-0.521309\pi$
$463$	−2.87873	−0.133786	−0.0668929	+	0.997760i	$0.521309\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−16.2115	−0.750178	−0.375089	−	0.926989i	$-0.622388\pi$
$467$	−16.2115	−0.750178	−0.375089	+	0.926989i	$0.622388\pi$
$468$	0	0
$469$	−0.825187	−0.0381036
$470$	0	0
$471$	−2.83147	−0.130467
$472$	0	0
$473$	−41.4494	−1.90584
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−12.3491	−0.565429
$478$	0	0
$479$	14.8905	0.680366	0.340183	−	0.940359i	$-0.389511\pi$
$479$	14.8905	0.680366	0.340183	+	0.940359i	$0.389511\pi$
$480$	0	0
$481$	2.52982	0.115350
$482$	0	0
$483$	2.28308	0.103884
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−24.8887	−1.12781	−0.563907	−	0.825838i	$-0.690702\pi$
$487$	−24.8887	−1.12781	−0.563907	+	0.825838i	$0.690702\pi$
$488$	0	0
$489$	2.61608	0.118303
$490$	0	0
$491$	−38.1206	−1.72036	−0.860180	−	0.509990i	$-0.829649\pi$
$491$	−38.1206	−1.72036	−0.860180	+	0.509990i	$0.829649\pi$
$492$	0	0
$493$	1.38049	0.0621743
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.969971	−0.0435091
$498$	0	0
$499$	−11.9299	−0.534056	−0.267028	−	0.963689i	$-0.586042\pi$
$499$	−11.9299	−0.534056	−0.267028	+	0.963689i	$0.586042\pi$
$500$	0	0
$501$	−36.9731	−1.65184
$502$	0	0
$503$	−2.59937	−0.115900	−0.0579502	−	0.998319i	$-0.518456\pi$
$503$	−2.59937	−0.115900	−0.0579502	+	0.998319i	$0.518456\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	29.2045	1.29702
$508$	0	0
$509$	0.332209	0.0147249	0.00736246	−	0.999973i	$-0.497656\pi$
$509$	0.332209	0.0147249	0.00736246	+	0.999973i	$0.497656\pi$
$510$	0	0
$511$	1.24256	0.0549677
$512$	0	0
$513$	19.2313	0.849082
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−6.94671	−0.305516
$518$	0	0
$519$	10.4813	0.460077
$520$	0	0
$521$	−1.64473	−0.0720570	−0.0360285	−	0.999351i	$-0.511471\pi$
$521$	−1.64473	−0.0720570	−0.0360285	+	0.999351i	$0.511471\pi$
$522$	0	0
$523$	8.08181	0.353393	0.176697	−	0.984265i	$-0.443459\pi$
$523$	8.08181	0.353393	0.176697	+	0.984265i	$0.443459\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.92327	−0.170900
$528$	0	0
$529$	46.0206	2.00090
$530$	0	0
$531$	−29.7441	−1.29079
$532$	0	0
$533$	12.3072	0.533082
$534$	0	0
$535$	0	0
$536$	0	0
$537$	41.6773	1.79851
$538$	0	0
$539$	−42.4506	−1.82848
$540$	0	0
$541$	2.09471	0.0900585	0.0450292	−	0.998986i	$-0.485662\pi$
$541$	2.09471	0.0900585	0.0450292	+	0.998986i	$0.485662\pi$
$542$	0	0
$543$	39.3822	1.69005
$544$	0	0
$545$	0	0
$546$	0	0
$547$	21.2082	0.906797	0.453398	−	0.891308i	$-0.350212\pi$
$547$	21.2082	0.906797	0.453398	+	0.891308i	$0.350212\pi$
$548$	0	0
$549$	−60.9183	−2.59993
$550$	0	0
$551$	−2.76099	−0.117622
$552$	0	0
$553$	1.51447	0.0644016
$554$	0	0
$555$	0	0
$556$	0	0
$557$	36.3525	1.54030	0.770152	−	0.637860i	$-0.220180\pi$
$557$	36.3525	1.54030	0.770152	+	0.637860i	$0.220180\pi$
$558$	0	0
$559$	−12.5105	−0.529139
$560$	0	0
$561$	−18.3904	−0.776442
$562$	0	0
$563$	7.75988	0.327040	0.163520	−	0.986540i	$-0.447715\pi$
$563$	7.75988	0.327040	0.163520	+	0.986540i	$0.447715\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.961862	0.0403944
$568$	0	0
$569$	−15.2759	−0.640400	−0.320200	−	0.947350i	$-0.603750\pi$
$569$	−15.2759	−0.640400	−0.320200	+	0.947350i	$0.603750\pi$
$570$	0	0
$571$	−31.5407	−1.31994	−0.659968	−	0.751294i	$-0.729430\pi$
$571$	−31.5407	−1.31994	−0.659968	+	0.751294i	$0.729430\pi$
$572$	0	0
$573$	−52.2563	−2.18304
$574$	0	0
$575$	0	0
$576$	0	0
$577$	41.4963	1.72751	0.863757	−	0.503908i	$-0.168105\pi$
$577$	41.4963	1.72751	0.863757	+	0.503908i	$0.168105\pi$
$578$	0	0
$579$	35.2681	1.46569
$580$	0	0
$581$	−0.634248	−0.0263131
$582$	0	0
$583$	−12.1430	−0.502913
$584$	0	0
$585$	0	0
$586$	0	0
$587$	20.3157	0.838521	0.419260	−	0.907866i	$-0.362289\pi$
$587$	20.3157	0.838521	0.419260	+	0.907866i	$0.362289\pi$
$588$	0	0
$589$	7.84654	0.323311
$590$	0	0
$591$	32.1843	1.32389
$592$	0	0
$593$	23.2628	0.955290	0.477645	−	0.878553i	$-0.341490\pi$
$593$	23.2628	0.955290	0.477645	+	0.878553i	$0.341490\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.4544	0.673432
$598$	0	0
$599$	28.7476	1.17459	0.587297	−	0.809372i	$-0.300192\pi$
$599$	28.7476	1.17459	0.587297	+	0.809372i	$0.300192\pi$
$600$	0	0
$601$	12.7378	0.519584	0.259792	−	0.965665i	$-0.416346\pi$
$601$	12.7378	0.519584	0.259792	+	0.965665i	$0.416346\pi$
$602$	0	0
$603$	−56.1591	−2.28698
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−11.9486	−0.484979	−0.242489	−	0.970154i	$-0.577964\pi$
$607$	−11.9486	−0.484979	−0.242489	+	0.970154i	$0.577964\pi$
$608$	0	0
$609$	−0.379373	−0.0153730
$610$	0	0
$611$	−2.09670	−0.0848235
$612$	0	0
$613$	20.4037	0.824097	0.412048	−	0.911162i	$-0.364813\pi$
$613$	20.4037	0.824097	0.412048	+	0.911162i	$0.364813\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13.6963	0.551392	0.275696	−	0.961245i	$-0.411092\pi$
$617$	13.6963	0.551392	0.275696	+	0.961245i	$0.411092\pi$
$618$	0	0
$619$	−7.25818	−0.291731	−0.145866	−	0.989304i	$-0.546597\pi$
$619$	−7.25818	−0.291731	−0.145866	+	0.989304i	$0.546597\pi$
$620$	0	0
$621$	79.8855	3.20569
$622$	0	0
$623$	0.443053	0.0177506
$624$	0	0
$625$	0	0
$626$	0	0
$627$	36.7807	1.46888
$628$	0	0
$629$	1.38049	0.0550439
$630$	0	0
$631$	1.25902	0.0501209	0.0250605	−	0.999686i	$-0.492022\pi$
$631$	1.25902	0.0501209	0.0250605	+	0.999686i	$0.492022\pi$
$632$	0	0
$633$	81.8718	3.25411
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−12.8127	−0.507659
$638$	0	0
$639$	−66.0125	−2.61142
$640$	0	0
$641$	−18.4418	−0.728408	−0.364204	−	0.931319i	$-0.618659\pi$
$641$	−18.4418	−0.728408	−0.364204	+	0.931319i	$0.618659\pi$
$642$	0	0
$643$	−38.9825	−1.53732	−0.768660	−	0.639658i	$-0.779076\pi$
$643$	−38.9825	−1.53732	−0.768660	+	0.639658i	$0.779076\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−28.0873	−1.10423	−0.552113	−	0.833769i	$-0.686178\pi$
$647$	−28.0873	−1.10423	−0.552113	+	0.833769i	$0.686178\pi$
$648$	0	0
$649$	−29.2477	−1.14807
$650$	0	0
$651$	1.07815	0.0422562
$652$	0	0
$653$	12.2327	0.478704	0.239352	−	0.970933i	$-0.423065\pi$
$653$	12.2327	0.478704	0.239352	+	0.970933i	$0.423065\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	84.5640	3.29916
$658$	0	0
$659$	3.79819	0.147956	0.0739782	−	0.997260i	$-0.476430\pi$
$659$	3.79819	0.147956	0.0739782	+	0.997260i	$0.476430\pi$
$660$	0	0
$661$	−19.2907	−0.750320	−0.375160	−	0.926960i	$-0.622412\pi$
$661$	−19.2907	−0.750320	−0.375160	+	0.926960i	$0.622412\pi$
$662$	0	0
$663$	−5.55070	−0.215571
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−11.4690	−0.444080
$668$	0	0
$669$	−49.7387	−1.92301
$670$	0	0
$671$	−59.9015	−2.31247
$672$	0	0
$673$	−46.0506	−1.77512	−0.887561	−	0.460691i	$-0.847602\pi$
$673$	−46.0506	−1.77512	−0.887561	+	0.460691i	$0.847602\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−23.6106	−0.907429	−0.453715	−	0.891147i	$-0.649901\pi$
$677$	−23.6106	−0.907429	−0.453715	+	0.891147i	$0.649901\pi$
$678$	0	0
$679$	−0.0747806	−0.00286982
$680$	0	0
$681$	38.0315	1.45737
$682$	0	0
$683$	36.2324	1.38639	0.693197	−	0.720748i	$-0.256201\pi$
$683$	36.2324	1.38639	0.693197	+	0.720748i	$0.256201\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	63.9191	2.43867
$688$	0	0
$689$	−3.66509	−0.139629
$690$	0	0
$691$	24.5616	0.934368	0.467184	−	0.884160i	$-0.345269\pi$
$691$	24.5616	0.934368	0.467184	+	0.884160i	$0.345269\pi$
$692$	0	0
$693$	3.40129	0.129204
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6.71588	0.254382
$698$	0	0
$699$	−44.5870	−1.68644
$700$	0	0
$701$	−40.1420	−1.51614	−0.758071	−	0.652172i	$-0.773858\pi$
$701$	−40.1420	−1.51614	−0.758071	+	0.652172i	$0.773858\pi$
$702$	0	0
$703$	−2.76099	−0.104133
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.538364	0.0202473
$708$	0	0
$709$	21.7147	0.815512	0.407756	−	0.913091i	$-0.366311\pi$
$709$	21.7147	0.815512	0.407756	+	0.913091i	$0.366311\pi$
$710$	0	0
$711$	103.069	3.86538
$712$	0	0
$713$	32.5940	1.22066
$714$	0	0
$715$	0	0
$716$	0	0
$717$	36.1328	1.34940
$718$	0	0
$719$	−32.2545	−1.20289	−0.601445	−	0.798914i	$-0.705408\pi$
$719$	−32.2545	−1.20289	−0.601445	+	0.798914i	$0.705408\pi$
$720$	0	0
$721$	1.47338	0.0548715
$722$	0	0
$723$	−22.5382	−0.838205
$724$	0	0
$725$	0	0
$726$	0	0
$727$	23.7917	0.882387	0.441193	−	0.897412i	$-0.354555\pi$
$727$	23.7917	0.882387	0.441193	+	0.897412i	$0.354555\pi$
$728$	0	0
$729$	−21.9155	−0.811683
$730$	0	0
$731$	−6.82686	−0.252500
$732$	0	0
$733$	14.7042	0.543113	0.271557	−	0.962422i	$-0.412462\pi$
$733$	14.7042	0.543113	0.271557	+	0.962422i	$0.412462\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−55.2218	−2.03412
$738$	0	0
$739$	35.7687	1.31577	0.657887	−	0.753117i	$-0.271450\pi$
$739$	35.7687	1.31577	0.657887	+	0.753117i	$0.271450\pi$
$740$	0	0
$741$	11.1014	0.407820
$742$	0	0
$743$	−1.11269	−0.0408207	−0.0204103	−	0.999792i	$-0.506497\pi$
$743$	−1.11269	−0.0408207	−0.0204103	+	0.999792i	$0.506497\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−43.1645	−1.57931
$748$	0	0
$749$	−0.621257	−0.0227002
$750$	0	0
$751$	−4.19406	−0.153043	−0.0765217	−	0.997068i	$-0.524381\pi$
$751$	−4.19406	−0.153043	−0.0765217	+	0.997068i	$0.524381\pi$
$752$	0	0
$753$	−56.9382	−2.07494
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−21.2720	−0.773145	−0.386572	−	0.922259i	$-0.626341\pi$
$757$	−21.2720	−0.773145	−0.386572	+	0.922259i	$0.626341\pi$
$758$	0	0
$759$	152.785	5.54573
$760$	0	0
$761$	22.8189	0.827183	0.413592	−	0.910463i	$-0.364274\pi$
$761$	22.8189	0.827183	0.413592	+	0.910463i	$0.364274\pi$
$762$	0	0
$763$	−1.17847	−0.0426635
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.82772	−0.318751
$768$	0	0
$769$	−53.5261	−1.93020	−0.965100	−	0.261883i	$-0.915657\pi$
$769$	−53.5261	−1.93020	−0.965100	+	0.261883i	$0.915657\pi$
$770$	0	0
$771$	41.6781	1.50100
$772$	0	0
$773$	−39.8504	−1.43332	−0.716660	−	0.697422i	$-0.754330\pi$
$773$	−39.8504	−1.43332	−0.716660	+	0.697422i	$0.754330\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−0.379373	−0.0136099
$778$	0	0
$779$	−13.4318	−0.481243
$780$	0	0
$781$	−64.9107	−2.32269
$782$	0	0
$783$	−13.2743	−0.474386
$784$	0	0
$785$	0	0
$786$	0	0
$787$	17.7971	0.634396	0.317198	−	0.948359i	$-0.397258\pi$
$787$	17.7971	0.634396	0.317198	+	0.948359i	$0.397258\pi$
$788$	0	0
$789$	−42.4192	−1.51016
$790$	0	0
$791$	−1.47660	−0.0525017
$792$	0	0
$793$	−18.0799	−0.642035
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−14.0229	−0.496716	−0.248358	−	0.968668i	$-0.579891\pi$
$797$	−14.0229	−0.496716	−0.248358	+	0.968668i	$0.579891\pi$
$798$	0	0
$799$	−1.14415	−0.0404770
$800$	0	0
$801$	30.1525	1.06539
$802$	0	0
$803$	83.1526	2.93439
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−18.9946	−0.668641
$808$	0	0
$809$	−16.9735	−0.596758	−0.298379	−	0.954447i	$-0.596446\pi$
$809$	−16.9735	−0.596758	−0.298379	+	0.954447i	$0.596446\pi$
$810$	0	0
$811$	−44.9162	−1.57722	−0.788610	−	0.614893i	$-0.789199\pi$
$811$	−44.9162	−1.57722	−0.788610	+	0.614893i	$0.789199\pi$
$812$	0	0
$813$	55.3446	1.94102
$814$	0	0
$815$	0	0
$816$	0	0
$817$	13.6537	0.477683
$818$	0	0
$819$	1.02660	0.0358723
$820$	0	0
$821$	−26.0195	−0.908088	−0.454044	−	0.890979i	$-0.650019\pi$
$821$	−26.0195	−0.908088	−0.454044	+	0.890979i	$0.650019\pi$
$822$	0	0
$823$	23.6870	0.825676	0.412838	−	0.910805i	$-0.364538\pi$
$823$	23.6870	0.825676	0.412838	+	0.910805i	$0.364538\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−10.7176	−0.372687	−0.186344	−	0.982485i	$-0.559664\pi$
$827$	−10.7176	−0.372687	−0.186344	+	0.982485i	$0.559664\pi$
$828$	0	0
$829$	33.4319	1.16114	0.580569	−	0.814211i	$-0.302830\pi$
$829$	33.4319	1.16114	0.580569	+	0.814211i	$0.302830\pi$
$830$	0	0
$831$	30.2951	1.05093
$832$	0	0
$833$	−6.99177	−0.242250
$834$	0	0
$835$	0	0
$836$	0	0
$837$	37.7248	1.30396
$838$	0	0
$839$	46.7292	1.61327	0.806635	−	0.591049i	$-0.201286\pi$
$839$	46.7292	1.61327	0.806635	+	0.591049i	$0.201286\pi$
$840$	0	0
$841$	−27.0942	−0.934284
$842$	0	0
$843$	−4.36107	−0.150203
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.34652	0.0806273
$848$	0	0
$849$	8.81640	0.302578
$850$	0	0
$851$	−11.4690	−0.393151
$852$	0	0
$853$	6.88353	0.235688	0.117844	−	0.993032i	$-0.462402\pi$
$853$	6.88353	0.235688	0.117844	+	0.993032i	$0.462402\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.0263	0.376652	0.188326	−	0.982107i	$-0.439694\pi$
$857$	11.0263	0.376652	0.188326	+	0.982107i	$0.439694\pi$
$858$	0	0
$859$	−8.33213	−0.284289	−0.142144	−	0.989846i	$-0.545400\pi$
$859$	−8.33213	−0.284289	−0.142144	+	0.989846i	$0.545400\pi$
$860$	0	0
$861$	−1.84559	−0.0628975
$862$	0	0
$863$	−0.0159450	−0.000542775 0	−0.000271387	−	1.00000i	$-0.500086\pi$
$863$	−0.0159450	−0.000542775 0	−0.000271387		1.00000i	$0.500086\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−3.02896	−0.102869
$868$	0	0
$869$	101.349	3.43801
$870$	0	0
$871$	−16.6674	−0.564753
$872$	0	0
$873$	−5.08928	−0.172246
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−29.9506	−1.01136	−0.505680	−	0.862721i	$-0.668758\pi$
$877$	−29.9506	−1.01136	−0.505680	+	0.862721i	$0.668758\pi$
$878$	0	0
$879$	−41.7127	−1.40694
$880$	0	0
$881$	8.33396	0.280778	0.140389	−	0.990096i	$-0.455165\pi$
$881$	8.33396	0.280778	0.140389	+	0.990096i	$0.455165\pi$
$882$	0	0
$883$	−2.40317	−0.0808732	−0.0404366	−	0.999182i	$-0.512875\pi$
$883$	−2.40317	−0.0808732	−0.0404366	+	0.999182i	$0.512875\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14.5305	0.487887	0.243944	−	0.969789i	$-0.421559\pi$
$887$	14.5305	0.487887	0.243944	+	0.969789i	$0.421559\pi$
$888$	0	0
$889$	1.56144	0.0523692
$890$	0	0
$891$	64.3681	2.15641
$892$	0	0
$893$	2.28830	0.0765749
$894$	0	0
$895$	0	0
$896$	0	0
$897$	46.1144	1.53972
$898$	0	0
$899$	−5.41605	−0.180635
$900$	0	0
$901$	−2.00000	−0.0666297
$902$	0	0
$903$	1.87609	0.0624323
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−28.4182	−0.943611	−0.471806	−	0.881703i	$-0.656398\pi$
$907$	−28.4182	−0.943611	−0.471806	+	0.881703i	$0.656398\pi$
$908$	0	0
$909$	36.6390	1.21524
$910$	0	0
$911$	−7.84562	−0.259937	−0.129969	−	0.991518i	$-0.541488\pi$
$911$	−7.84562	−0.259937	−0.129969	+	0.991518i	$0.541488\pi$
$912$	0	0
$913$	−42.4441	−1.40469
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.38847	−0.0458514
$918$	0	0
$919$	15.1748	0.500570	0.250285	−	0.968172i	$-0.419476\pi$
$919$	15.1748	0.500570	0.250285	+	0.968172i	$0.419476\pi$
$920$	0	0
$921$	−78.6000	−2.58996
$922$	0	0
$923$	−19.5918	−0.644871
$924$	0	0
$925$	0	0
$926$	0	0
$927$	100.273	3.29339
$928$	0	0
$929$	−37.1215	−1.21792	−0.608959	−	0.793202i	$-0.708412\pi$
$929$	−37.1215	−1.21792	−0.608959	+	0.793202i	$0.708412\pi$
$930$	0	0
$931$	13.9835	0.458292
$932$	0	0
$933$	−85.1754	−2.78852
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.18385	0.0713431	0.0356716	−	0.999364i	$-0.488643\pi$
$937$	2.18385	0.0713431	0.0356716	+	0.999364i	$0.488643\pi$
$938$	0	0
$939$	68.8113	2.24557
$940$	0	0
$941$	−50.8979	−1.65923	−0.829613	−	0.558339i	$-0.811439\pi$
$941$	−50.8979	−1.65923	−0.829613	+	0.558339i	$0.811439\pi$
$942$	0	0
$943$	−55.7946	−1.81692
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.14716	0.199756	0.0998780	−	0.995000i	$-0.468155\pi$
$947$	6.14716	0.199756	0.0998780	+	0.995000i	$0.468155\pi$
$948$	0	0
$949$	25.0976	0.814704
$950$	0	0
$951$	56.0995	1.81915
$952$	0	0
$953$	−1.74919	−0.0566617	−0.0283308	−	0.999599i	$-0.509019\pi$
$953$	−1.74919	−0.0566617	−0.0283308	+	0.999599i	$0.509019\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−25.3878	−0.820670
$958$	0	0
$959$	−1.97422	−0.0637509
$960$	0	0
$961$	−15.6079	−0.503482
$962$	0	0
$963$	−42.2804	−1.36247
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−21.3980	−0.688113	−0.344057	−	0.938949i	$-0.611801\pi$
$967$	−21.3980	−0.688113	−0.344057	+	0.938949i	$0.611801\pi$
$968$	0	0
$969$	6.05791	0.194608
$970$	0	0
$971$	8.92971	0.286568	0.143284	−	0.989682i	$-0.454234\pi$
$971$	8.92971	0.286568	0.143284	+	0.989682i	$0.454234\pi$
$972$	0	0
$973$	0.565325	0.0181235
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−41.7751	−1.33651	−0.668253	−	0.743934i	$-0.732957\pi$
$977$	−41.7751	−1.33651	−0.668253	+	0.743934i	$0.732957\pi$
$978$	0	0
$979$	29.6493	0.947595
$980$	0	0
$981$	−80.2022	−2.56066
$982$	0	0
$983$	−3.24827	−0.103604	−0.0518018	−	0.998657i	$-0.516496\pi$
$983$	−3.24827	−0.103604	−0.0518018	+	0.998657i	$0.516496\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.314423	0.0100082
$988$	0	0
$989$	56.7166	1.80348
$990$	0	0
$991$	−9.10342	−0.289180	−0.144590	−	0.989492i	$-0.546186\pi$
$991$	−9.10342	−0.289180	−0.144590	+	0.989492i	$0.546186\pi$
$992$	0	0
$993$	52.4399	1.66413
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−26.4321	−0.837112	−0.418556	−	0.908191i	$-0.637464\pi$
$997$	−26.4321	−0.837112	−0.418556	+	0.908191i	$0.637464\pi$
$998$	0	0
$999$	−13.2743	−0.419982

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6800.2.a.ch.1.2		6
4.3	odd	2		3400.2.a.y.1.5		6
5.2	odd	4		1360.2.e.g.1089.11		12
5.3	odd	4		1360.2.e.g.1089.2		12
5.4	even	2		6800.2.a.ci.1.5		6
20.3	even	4		680.2.e.c.409.11	yes	12
20.7	even	4		680.2.e.c.409.2	✓	12
20.19	odd	2		3400.2.a.x.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
680.2.e.c.409.2	✓	12	20.7	even	4
680.2.e.c.409.11	yes	12	20.3	even	4
1360.2.e.g.1089.2		12	5.3	odd	4
1360.2.e.g.1089.11		12	5.2	odd	4
3400.2.a.x.1.2		6	20.19	odd	2
3400.2.a.y.1.5		6	4.3	odd	2
6800.2.a.ch.1.2		6	1.1	even	1	trivial
6800.2.a.ci.1.5		6	5.4	even	2

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:	\( N \)	\(=\)	\( 6800 = 2^{4} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6800.a (trivial)

\(f(q)\)	\(=\)	\(q-3.02896 q^{3} +0.0907276 q^{7} +6.17457 q^{9} +O(q^{10})\)	\(q-3.02896 q^{3} +0.0907276 q^{7} +6.17457 q^{9} +6.07152 q^{11} +1.83254 q^{13} +1.00000 q^{17} -2.00000 q^{19} -0.274810 q^{21} -8.30786 q^{23} -9.61565 q^{27} +1.38049 q^{29} -3.92327 q^{31} -18.3904 q^{33} +1.38049 q^{37} -5.55070 q^{39} +6.71588 q^{41} -6.82686 q^{43} -1.14415 q^{47} -6.99177 q^{49} -3.02896 q^{51} -2.00000 q^{53} +6.05791 q^{57} -4.81719 q^{59} -9.86599 q^{61} +0.560204 q^{63} -9.09522 q^{67} +25.1642 q^{69} -10.6910 q^{71} +13.6955 q^{73} +0.550854 q^{77} +16.6925 q^{79} +10.6016 q^{81} -6.99069 q^{83} -4.18146 q^{87} +4.88334 q^{89} +0.166262 q^{91} +11.8834 q^{93} -0.824232 q^{97} +37.4890 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} + 2 q^{7} + 14 q^{9}+O(q^{10}) \) 6 * q - 4 * q^3 + 2 * q^7 + 14 * q^9	\( 6 q - 4 q^{3} + 2 q^{7} + 14 q^{9} - 10 q^{11} - 10 q^{13} + 6 q^{17} - 12 q^{19} + 12 q^{21} + 4 q^{23} - 22 q^{27} - 4 q^{31} - 10 q^{33} + 2 q^{39} + 14 q^{41} - 14 q^{43} - 6 q^{47} + 6 q^{49} - 4 q^{51} - 12 q^{53} + 8 q^{57} - 4 q^{59} + 2 q^{61} - 24 q^{63} - 30 q^{67} - 4 q^{69} - 26 q^{71} + 18 q^{73} - 30 q^{77} + 12 q^{79} + 46 q^{81} - 2 q^{83} - 28 q^{87} + 24 q^{89} + 2 q^{91} - 6 q^{93} - 8 q^{97} - 20 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 + 2 * q^7 + 14 * q^9 - 10 * q^11 - 10 * q^13 + 6 * q^17 - 12 * q^19 + 12 * q^21 + 4 * q^23 - 22 * q^27 - 4 * q^31 - 10 * q^33 + 2 * q^39 + 14 * q^41 - 14 * q^43 - 6 * q^47 + 6 * q^49 - 4 * q^51 - 12 * q^53 + 8 * q^57 - 4 * q^59 + 2 * q^61 - 24 * q^63 - 30 * q^67 - 4 * q^69 - 26 * q^71 + 18 * q^73 - 30 * q^77 + 12 * q^79 + 46 * q^81 - 2 * q^83 - 28 * q^87 + 24 * q^89 + 2 * q^91 - 6 * q^93 - 8 * q^97 - 20 * q^99

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