LMFDB - Embedded newform 6800.2.a.be.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 170)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ 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)

$f(q)$	$=$	$q+1.56155 q^{3} -3.12311 q^{7} -0.561553 q^{9} +O(q^{10})$	$q+1.56155 q^{3} -3.12311 q^{7} -0.561553 q^{9} +4.00000 q^{11} -0.438447 q^{13} -1.00000 q^{17} -1.56155 q^{19} -4.87689 q^{21} +3.12311 q^{23} -5.56155 q^{27} +6.68466 q^{29} +2.43845 q^{31} +6.24621 q^{33} -1.12311 q^{37} -0.684658 q^{39} -12.2462 q^{41} +7.12311 q^{43} +2.43845 q^{47} +2.75379 q^{49} -1.56155 q^{51} -3.56155 q^{53} -2.43845 q^{57} +12.6847 q^{59} +11.5616 q^{61} +1.75379 q^{63} -0.876894 q^{67} +4.87689 q^{69} +16.6847 q^{71} -13.8078 q^{73} -12.4924 q^{77} -7.00000 q^{81} +10.2462 q^{83} +10.4384 q^{87} +2.68466 q^{89} +1.36932 q^{91} +3.80776 q^{93} -10.6847 q^{97} -2.24621 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q - q^3 + 2 * q^7 + 3 * q^9	$ 2 q - q^{3} + 2 q^{7} + 3 q^{9} + 8 q^{11} - 5 q^{13} - 2 q^{17} + q^{19} - 18 q^{21} - 2 q^{23} - 7 q^{27} + q^{29} + 9 q^{31} - 4 q^{33} + 6 q^{37} + 11 q^{39} - 8 q^{41} + 6 q^{43} + 9 q^{47} + 22 q^{49} + q^{51} - 3 q^{53} - 9 q^{57} + 13 q^{59} + 19 q^{61} + 20 q^{63} - 10 q^{67} + 18 q^{69} + 21 q^{71} - 7 q^{73} + 8 q^{77} - 14 q^{81} + 4 q^{83} + 25 q^{87} - 7 q^{89} - 22 q^{91} - 13 q^{93} - 9 q^{97} + 12 q^{99}+O(q^{100}) $ 2 * q - q^3 + 2 * q^7 + 3 * q^9 + 8 * q^11 - 5 * q^13 - 2 * q^17 + q^19 - 18 * q^21 - 2 * q^23 - 7 * q^27 + q^29 + 9 * q^31 - 4 * q^33 + 6 * q^37 + 11 * q^39 - 8 * q^41 + 6 * q^43 + 9 * q^47 + 22 * q^49 + q^51 - 3 * q^53 - 9 * q^57 + 13 * q^59 + 19 * q^61 + 20 * q^63 - 10 * q^67 + 18 * q^69 + 21 * q^71 - 7 * q^73 + 8 * q^77 - 14 * q^81 + 4 * q^83 + 25 * q^87 - 7 * q^89 - 22 * q^91 - 13 * q^93 - 9 * q^97 + 12 * 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
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.12311	−1.18042	−0.590211	−	0.807249i	$-0.700956\pi$
$7$	−3.12311	−1.18042	−0.590211	+	0.807249i	$0.700956\pi$
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	−0.438447	−0.121603	−0.0608017	−	0.998150i	$-0.519366\pi$
$13$	−0.438447	−0.121603	−0.0608017	+	0.998150i	$0.519366\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−1.56155	−0.358245	−0.179122	−	0.983827i	$-0.557326\pi$
$19$	−1.56155	−0.358245	−0.179122	+	0.983827i	$0.557326\pi$
$20$	0	0
$21$	−4.87689	−1.06423
$22$	0	0
$23$	3.12311	0.651213	0.325606	−	0.945505i	$-0.394432\pi$
$23$	3.12311	0.651213	0.325606	+	0.945505i	$0.394432\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.56155	−1.07032
$28$	0	0
$29$	6.68466	1.24131	0.620655	−	0.784084i	$-0.286867\pi$
$29$	6.68466	1.24131	0.620655	+	0.784084i	$0.286867\pi$
$30$	0	0
$31$	2.43845	0.437958	0.218979	−	0.975730i	$-0.429727\pi$
$31$	2.43845	0.437958	0.218979	+	0.975730i	$0.429727\pi$
$32$	0	0
$33$	6.24621	1.08733
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.12311	−0.184637	−0.0923187	−	0.995730i	$-0.529428\pi$
$37$	−1.12311	−0.184637	−0.0923187	+	0.995730i	$0.529428\pi$
$38$	0	0
$39$	−0.684658	−0.109633
$40$	0	0
$41$	−12.2462	−1.91254	−0.956268	−	0.292490i	$-0.905516\pi$
$41$	−12.2462	−1.91254	−0.956268	+	0.292490i	$0.905516\pi$
$42$	0	0
$43$	7.12311	1.08626	0.543132	−	0.839648i	$-0.317238\pi$
$43$	7.12311	1.08626	0.543132	+	0.839648i	$0.317238\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.43845	0.355684	0.177842	−	0.984059i	$-0.443088\pi$
$47$	2.43845	0.355684	0.177842	+	0.984059i	$0.443088\pi$
$48$	0	0
$49$	2.75379	0.393398
$50$	0	0
$51$	−1.56155	−0.218661
$52$	0	0
$53$	−3.56155	−0.489217	−0.244608	−	0.969622i	$-0.578659\pi$
$53$	−3.56155	−0.489217	−0.244608	+	0.969622i	$0.578659\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.43845	−0.322980
$58$	0	0
$59$	12.6847	1.65140	0.825701	−	0.564108i	$-0.190780\pi$
$59$	12.6847	1.65140	0.825701	+	0.564108i	$0.190780\pi$
$60$	0	0
$61$	11.5616	1.48031	0.740153	−	0.672439i	$-0.234753\pi$
$61$	11.5616	1.48031	0.740153	+	0.672439i	$0.234753\pi$
$62$	0	0
$63$	1.75379	0.220957
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−0.876894	−0.107130	−0.0535648	−	0.998564i	$-0.517058\pi$
$67$	−0.876894	−0.107130	−0.0535648	+	0.998564i	$0.517058\pi$
$68$	0	0
$69$	4.87689	0.587109
$70$	0	0
$71$	16.6847	1.98010	0.990052	−	0.140700i	$-0.0449352\pi$
$71$	16.6847	1.98010	0.990052	+	0.140700i	$0.0449352\pi$
$72$	0	0
$73$	−13.8078	−1.61608	−0.808038	−	0.589130i	$-0.799471\pi$
$73$	−13.8078	−1.61608	−0.808038	+	0.589130i	$0.799471\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−12.4924	−1.42364
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	10.2462	1.12467	0.562334	−	0.826910i	$-0.309904\pi$
$83$	10.2462	1.12467	0.562334	+	0.826910i	$0.309904\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	10.4384	1.11912
$88$	0	0
$89$	2.68466	0.284573	0.142287	−	0.989825i	$-0.454555\pi$
$89$	2.68466	0.284573	0.142287	+	0.989825i	$0.454555\pi$
$90$	0	0
$91$	1.36932	0.143543
$92$	0	0
$93$	3.80776	0.394847
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.6847	−1.08486	−0.542431	−	0.840100i	$-0.682496\pi$
$97$	−10.6847	−1.08486	−0.542431	+	0.840100i	$0.682496\pi$
$98$	0	0
$99$	−2.24621	−0.225753
$100$	0	0
$101$	18.4924	1.84006	0.920032	−	0.391842i	$-0.128162\pi$
$101$	18.4924	1.84006	0.920032	+	0.391842i	$0.128162\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.4924	−1.59438	−0.797191	−	0.603727i	$-0.793682\pi$
$107$	−16.4924	−1.59438	−0.797191	+	0.603727i	$0.793682\pi$
$108$	0	0
$109$	−4.43845	−0.425126	−0.212563	−	0.977147i	$-0.568181\pi$
$109$	−4.43845	−0.425126	−0.212563	+	0.977147i	$0.568181\pi$
$110$	0	0
$111$	−1.75379	−0.166462
$112$	0	0
$113$	17.8078	1.67521	0.837607	−	0.546274i	$-0.183954\pi$
$113$	17.8078	1.67521	0.837607	+	0.546274i	$0.183954\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.246211	0.0227622
$118$	0	0
$119$	3.12311	0.286295
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−19.1231	−1.72427
$124$	0	0
$125$	0	0
$126$	0	0
$127$	13.5616	1.20339	0.601697	−	0.798725i	$-0.294492\pi$
$127$	13.5616	1.20339	0.601697	+	0.798725i	$0.294492\pi$
$128$	0	0
$129$	11.1231	0.979335
$130$	0	0
$131$	7.12311	0.622349	0.311174	−	0.950353i	$-0.399278\pi$
$131$	7.12311	0.622349	0.311174	+	0.950353i	$0.399278\pi$
$132$	0	0
$133$	4.87689	0.422880
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	16.4924	1.39887	0.699435	−	0.714697i	$-0.253435\pi$
$139$	16.4924	1.39887	0.699435	+	0.714697i	$0.253435\pi$
$140$	0	0
$141$	3.80776	0.320672
$142$	0	0
$143$	−1.75379	−0.146659
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.30019	0.354673
$148$	0	0
$149$	−5.12311	−0.419701	−0.209851	−	0.977733i	$-0.567298\pi$
$149$	−5.12311	−0.419701	−0.209851	+	0.977733i	$0.567298\pi$
$150$	0	0
$151$	−9.36932	−0.762464	−0.381232	−	0.924479i	$-0.624500\pi$
$151$	−9.36932	−0.762464	−0.381232	+	0.924479i	$0.624500\pi$
$152$	0	0
$153$	0.561553	0.0453989
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.4924	1.15662	0.578311	−	0.815817i	$-0.303712\pi$
$157$	14.4924	1.15662	0.578311	+	0.815817i	$0.303712\pi$
$158$	0	0
$159$	−5.56155	−0.441060
$160$	0	0
$161$	−9.75379	−0.768706
$162$	0	0
$163$	−2.24621	−0.175937	−0.0879684	−	0.996123i	$-0.528037\pi$
$163$	−2.24621	−0.175937	−0.0879684	+	0.996123i	$0.528037\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.12311	−0.241673	−0.120837	−	0.992672i	$-0.538558\pi$
$167$	−3.12311	−0.241673	−0.120837	+	0.992672i	$0.538558\pi$
$168$	0	0
$169$	−12.8078	−0.985213
$170$	0	0
$171$	0.876894	0.0670578
$172$	0	0
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	19.8078	1.48884
$178$	0	0
$179$	24.4924	1.83065	0.915325	−	0.402716i	$-0.131934\pi$
$179$	24.4924	1.83065	0.915325	+	0.402716i	$0.131934\pi$
$180$	0	0
$181$	−0.246211	−0.0183007	−0.00915037	−	0.999958i	$-0.502913\pi$
$181$	−0.246211	−0.0183007	−0.00915037	+	0.999958i	$0.502913\pi$
$182$	0	0
$183$	18.0540	1.33459
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	17.3693	1.26343
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	7.75379	0.558130	0.279065	−	0.960272i	$-0.409976\pi$
$193$	7.75379	0.558130	0.279065	+	0.960272i	$0.409976\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.3693	0.810030	0.405015	−	0.914310i	$-0.367266\pi$
$197$	11.3693	0.810030	0.405015	+	0.914310i	$0.367266\pi$
$198$	0	0
$199$	−10.4384	−0.739962	−0.369981	−	0.929039i	$-0.620636\pi$
$199$	−10.4384	−0.739962	−0.369981	+	0.929039i	$0.620636\pi$
$200$	0	0
$201$	−1.36932	−0.0965842
$202$	0	0
$203$	−20.8769	−1.46527
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.75379	−0.121897
$208$	0	0
$209$	−6.24621	−0.432059
$210$	0	0
$211$	−21.3693	−1.47112	−0.735562	−	0.677457i	$-0.763082\pi$
$211$	−21.3693	−1.47112	−0.735562	+	0.677457i	$0.763082\pi$
$212$	0	0
$213$	26.0540	1.78519
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−7.61553	−0.516976
$218$	0	0
$219$	−21.5616	−1.45699
$220$	0	0
$221$	0.438447	0.0294931
$222$	0	0
$223$	−8.68466	−0.581568	−0.290784	−	0.956789i	$-0.593916\pi$
$223$	−8.68466	−0.581568	−0.290784	+	0.956789i	$0.593916\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	28.6847	1.90387	0.951934	−	0.306304i	$-0.0990923\pi$
$227$	28.6847	1.90387	0.951934	+	0.306304i	$0.0990923\pi$
$228$	0	0
$229$	18.4924	1.22201	0.611007	−	0.791625i	$-0.290765\pi$
$229$	18.4924	1.22201	0.611007	+	0.791625i	$0.290765\pi$
$230$	0	0
$231$	−19.5076	−1.28350
$232$	0	0
$233$	−18.6847	−1.22407	−0.612036	−	0.790830i	$-0.709649\pi$
$233$	−18.6847	−1.22407	−0.612036	+	0.790830i	$0.709649\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.36932	−0.0885737	−0.0442869	−	0.999019i	$-0.514102\pi$
$239$	−1.36932	−0.0885737	−0.0442869	+	0.999019i	$0.514102\pi$
$240$	0	0
$241$	−4.24621	−0.273523	−0.136761	−	0.990604i	$-0.543669\pi$
$241$	−4.24621	−0.273523	−0.136761	+	0.990604i	$0.543669\pi$
$242$	0	0
$243$	5.75379	0.369106
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.684658	0.0435638
$248$	0	0
$249$	16.0000	1.01396
$250$	0	0
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	12.4924	0.785392
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	3.50758	0.217950
$260$	0	0
$261$	−3.75379	−0.232354
$262$	0	0
$263$	4.19224	0.258504	0.129252	−	0.991612i	$-0.458742\pi$
$263$	4.19224	0.258504	0.129252	+	0.991612i	$0.458742\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4.19224	0.256561
$268$	0	0
$269$	17.8078	1.08576	0.542879	−	0.839811i	$-0.317334\pi$
$269$	17.8078	1.08576	0.542879	+	0.839811i	$0.317334\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	2.13826	0.129413
$274$	0	0
$275$	0	0
$276$	0	0
$277$	32.2462	1.93749	0.968744	−	0.248064i	$-0.0797945\pi$
$277$	32.2462	1.93749	0.968744	+	0.248064i	$0.0797945\pi$
$278$	0	0
$279$	−1.36932	−0.0819789
$280$	0	0
$281$	12.4384	0.742016	0.371008	−	0.928630i	$-0.379012\pi$
$281$	12.4384	0.742016	0.371008	+	0.928630i	$0.379012\pi$
$282$	0	0
$283$	20.6847	1.22958	0.614788	−	0.788693i	$-0.289242\pi$
$283$	20.6847	1.22958	0.614788	+	0.788693i	$0.289242\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	38.2462	2.25760
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−16.6847	−0.978072
$292$	0	0
$293$	12.4384	0.726662	0.363331	−	0.931660i	$-0.381639\pi$
$293$	12.4384	0.726662	0.363331	+	0.931660i	$0.381639\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−22.2462	−1.29086
$298$	0	0
$299$	−1.36932	−0.0791896
$300$	0	0
$301$	−22.2462	−1.28225
$302$	0	0
$303$	28.8769	1.65893
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−34.2462	−1.95453	−0.977267	−	0.212011i	$-0.931999\pi$
$307$	−34.2462	−1.95453	−0.977267	+	0.212011i	$0.931999\pi$
$308$	0	0
$309$	12.4924	0.710669
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.61553	0.540062	0.270031	−	0.962852i	$-0.412966\pi$
$317$	9.61553	0.540062	0.270031	+	0.962852i	$0.412966\pi$
$318$	0	0
$319$	26.7386	1.49708
$320$	0	0
$321$	−25.7538	−1.43744
$322$	0	0
$323$	1.56155	0.0868871
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−6.93087	−0.383278
$328$	0	0
$329$	−7.61553	−0.419858
$330$	0	0
$331$	14.0540	0.772476	0.386238	−	0.922399i	$-0.373774\pi$
$331$	14.0540	0.772476	0.386238	+	0.922399i	$0.373774\pi$
$332$	0	0
$333$	0.630683	0.0345612
$334$	0	0
$335$	0	0
$336$	0	0
$337$	22.6847	1.23571	0.617856	−	0.786291i	$-0.288001\pi$
$337$	22.6847	1.23571	0.617856	+	0.786291i	$0.288001\pi$
$338$	0	0
$339$	27.8078	1.51031
$340$	0	0
$341$	9.75379	0.528197
$342$	0	0
$343$	13.2614	0.716046
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.43845	−0.345634	−0.172817	−	0.984954i	$-0.555287\pi$
$347$	−6.43845	−0.345634	−0.172817	+	0.984954i	$0.555287\pi$
$348$	0	0
$349$	−6.87689	−0.368112	−0.184056	−	0.982916i	$-0.558923\pi$
$349$	−6.87689	−0.368112	−0.184056	+	0.982916i	$0.558923\pi$
$350$	0	0
$351$	2.43845	0.130155
$352$	0	0
$353$	10.4924	0.558455	0.279228	−	0.960225i	$-0.409922\pi$
$353$	10.4924	0.558455	0.279228	+	0.960225i	$0.409922\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.87689	0.258113
$358$	0	0
$359$	19.1231	1.00928	0.504639	−	0.863330i	$-0.331625\pi$
$359$	19.1231	1.00928	0.504639	+	0.863330i	$0.331625\pi$
$360$	0	0
$361$	−16.5616	−0.871661
$362$	0	0
$363$	7.80776	0.409801
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7.61553	0.397527	0.198764	−	0.980047i	$-0.436307\pi$
$367$	7.61553	0.397527	0.198764	+	0.980047i	$0.436307\pi$
$368$	0	0
$369$	6.87689	0.357997
$370$	0	0
$371$	11.1231	0.577483
$372$	0	0
$373$	−24.7386	−1.28092	−0.640459	−	0.767992i	$-0.721256\pi$
$373$	−24.7386	−1.28092	−0.640459	+	0.767992i	$0.721256\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.93087	−0.150947
$378$	0	0
$379$	−2.24621	−0.115380	−0.0576901	−	0.998335i	$-0.518374\pi$
$379$	−2.24621	−0.115380	−0.0576901	+	0.998335i	$0.518374\pi$
$380$	0	0
$381$	21.1771	1.08493
$382$	0	0
$383$	3.80776	0.194568	0.0972838	−	0.995257i	$-0.468985\pi$
$383$	3.80776	0.194568	0.0972838	+	0.995257i	$0.468985\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	−11.3693	−0.576447	−0.288224	−	0.957563i	$-0.593065\pi$
$389$	−11.3693	−0.576447	−0.288224	+	0.957563i	$0.593065\pi$
$390$	0	0
$391$	−3.12311	−0.157942
$392$	0	0
$393$	11.1231	0.561086
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3.36932	0.169101	0.0845506	−	0.996419i	$-0.473055\pi$
$397$	3.36932	0.169101	0.0845506	+	0.996419i	$0.473055\pi$
$398$	0	0
$399$	7.61553	0.381253
$400$	0	0
$401$	35.3693	1.76626	0.883130	−	0.469129i	$-0.155432\pi$
$401$	35.3693	1.76626	0.883130	+	0.469129i	$0.155432\pi$
$402$	0	0
$403$	−1.06913	−0.0532572
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.49242	−0.222681
$408$	0	0
$409$	−14.6847	−0.726110	−0.363055	−	0.931768i	$-0.618266\pi$
$409$	−14.6847	−0.726110	−0.363055	+	0.931768i	$0.618266\pi$
$410$	0	0
$411$	−15.6155	−0.770257
$412$	0	0
$413$	−39.6155	−1.94935
$414$	0	0
$415$	0	0
$416$	0	0
$417$	25.7538	1.26117
$418$	0	0
$419$	16.8769	0.824490	0.412245	−	0.911073i	$-0.364745\pi$
$419$	16.8769	0.824490	0.412245	+	0.911073i	$0.364745\pi$
$420$	0	0
$421$	−33.6155	−1.63832	−0.819160	−	0.573565i	$-0.805560\pi$
$421$	−33.6155	−1.63832	−0.819160	+	0.573565i	$0.805560\pi$
$422$	0	0
$423$	−1.36932	−0.0665785
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−36.1080	−1.74739
$428$	0	0
$429$	−2.73863	−0.132222
$430$	0	0
$431$	−12.4924	−0.601739	−0.300869	−	0.953665i	$-0.597277\pi$
$431$	−12.4924	−0.601739	−0.300869	+	0.953665i	$0.597277\pi$
$432$	0	0
$433$	−30.4924	−1.46537	−0.732686	−	0.680567i	$-0.761734\pi$
$433$	−30.4924	−1.46537	−0.732686	+	0.680567i	$0.761734\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.87689	−0.233293
$438$	0	0
$439$	−32.9848	−1.57428	−0.787140	−	0.616774i	$-0.788439\pi$
$439$	−32.9848	−1.57428	−0.787140	+	0.616774i	$0.788439\pi$
$440$	0	0
$441$	−1.54640	−0.0736380
$442$	0	0
$443$	28.0000	1.33032	0.665160	−	0.746701i	$-0.268363\pi$
$443$	28.0000	1.33032	0.665160	+	0.746701i	$0.268363\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−8.00000	−0.378387
$448$	0	0
$449$	−25.1231	−1.18563	−0.592816	−	0.805338i	$-0.701984\pi$
$449$	−25.1231	−1.18563	−0.592816	+	0.805338i	$0.701984\pi$
$450$	0	0
$451$	−48.9848	−2.30661
$452$	0	0
$453$	−14.6307	−0.687409
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−0.246211	−0.0115173	−0.00575864	−	0.999983i	$-0.501833\pi$
$457$	−0.246211	−0.0115173	−0.00575864	+	0.999983i	$0.501833\pi$
$458$	0	0
$459$	5.56155	0.259591
$460$	0	0
$461$	−30.4924	−1.42017	−0.710087	−	0.704114i	$-0.751344\pi$
$461$	−30.4924	−1.42017	−0.710087	+	0.704114i	$0.751344\pi$
$462$	0	0
$463$	14.9309	0.693896	0.346948	−	0.937884i	$-0.387218\pi$
$463$	14.9309	0.693896	0.346948	+	0.937884i	$0.387218\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0.492423	0.0227866	0.0113933	−	0.999935i	$-0.496373\pi$
$467$	0.492423	0.0227866	0.0113933	+	0.999935i	$0.496373\pi$
$468$	0	0
$469$	2.73863	0.126458
$470$	0	0
$471$	22.6307	1.04277
$472$	0	0
$473$	28.4924	1.31008
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	−34.4384	−1.57353	−0.786766	−	0.617251i	$-0.788246\pi$
$479$	−34.4384	−1.57353	−0.786766	+	0.617251i	$0.788246\pi$
$480$	0	0
$481$	0.492423	0.0224525
$482$	0	0
$483$	−15.2311	−0.693037
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−15.6155	−0.707607	−0.353804	−	0.935320i	$-0.615112\pi$
$487$	−15.6155	−0.707607	−0.353804	+	0.935320i	$0.615112\pi$
$488$	0	0
$489$	−3.50758	−0.158618
$490$	0	0
$491$	1.56155	0.0704719	0.0352359	−	0.999379i	$-0.488782\pi$
$491$	1.56155	0.0704719	0.0352359	+	0.999379i	$0.488782\pi$
$492$	0	0
$493$	−6.68466	−0.301062
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−52.1080	−2.33736
$498$	0	0
$499$	0.876894	0.0392552	0.0196276	−	0.999807i	$-0.493752\pi$
$499$	0.876894	0.0392552	0.0196276	+	0.999807i	$0.493752\pi$
$500$	0	0
$501$	−4.87689	−0.217884
$502$	0	0
$503$	−26.7386	−1.19222	−0.596108	−	0.802904i	$-0.703287\pi$
$503$	−26.7386	−1.19222	−0.596108	+	0.802904i	$0.703287\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−20.0000	−0.888231
$508$	0	0
$509$	25.1231	1.11356	0.556781	−	0.830659i	$-0.312036\pi$
$509$	25.1231	1.11356	0.556781	+	0.830659i	$0.312036\pi$
$510$	0	0
$511$	43.1231	1.90765
$512$	0	0
$513$	8.68466	0.383437
$514$	0	0
$515$	0	0
$516$	0	0
$517$	9.75379	0.428971
$518$	0	0
$519$	28.1080	1.23380
$520$	0	0
$521$	2.38447	0.104466	0.0522328	−	0.998635i	$-0.483366\pi$
$521$	2.38447	0.104466	0.0522328	+	0.998635i	$0.483366\pi$
$522$	0	0
$523$	−17.8617	−0.781039	−0.390520	−	0.920595i	$-0.627705\pi$
$523$	−17.8617	−0.781039	−0.390520	+	0.920595i	$0.627705\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.43845	−0.106220
$528$	0	0
$529$	−13.2462	−0.575922
$530$	0	0
$531$	−7.12311	−0.309116
$532$	0	0
$533$	5.36932	0.232571
$534$	0	0
$535$	0	0
$536$	0	0
$537$	38.2462	1.65045
$538$	0	0
$539$	11.0152	0.474456
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	0	0
$543$	−0.384472	−0.0164993
$544$	0	0
$545$	0	0
$546$	0	0
$547$	33.5616	1.43499	0.717494	−	0.696564i	$-0.245289\pi$
$547$	33.5616	1.43499	0.717494	+	0.696564i	$0.245289\pi$
$548$	0	0
$549$	−6.49242	−0.277090
$550$	0	0
$551$	−10.4384	−0.444693
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−32.4384	−1.37446	−0.687231	−	0.726439i	$-0.741174\pi$
$557$	−32.4384	−1.37446	−0.687231	+	0.726439i	$0.741174\pi$
$558$	0	0
$559$	−3.12311	−0.132093
$560$	0	0
$561$	−6.24621	−0.263715
$562$	0	0
$563$	−29.3693	−1.23777	−0.618885	−	0.785482i	$-0.712415\pi$
$563$	−29.3693	−1.23777	−0.618885	+	0.785482i	$0.712415\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	21.8617	0.918107
$568$	0	0
$569$	24.9309	1.04516	0.522578	−	0.852591i	$-0.324970\pi$
$569$	24.9309	1.04516	0.522578	+	0.852591i	$0.324970\pi$
$570$	0	0
$571$	−16.8769	−0.706276	−0.353138	−	0.935571i	$-0.614885\pi$
$571$	−16.8769	−0.706276	−0.353138	+	0.935571i	$0.614885\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−19.3693	−0.806355	−0.403178	−	0.915122i	$-0.632094\pi$
$577$	−19.3693	−0.806355	−0.403178	+	0.915122i	$0.632094\pi$
$578$	0	0
$579$	12.1080	0.503189
$580$	0	0
$581$	−32.0000	−1.32758
$582$	0	0
$583$	−14.2462	−0.590018
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−15.1231	−0.624197	−0.312099	−	0.950050i	$-0.601032\pi$
$587$	−15.1231	−0.624197	−0.312099	+	0.950050i	$0.601032\pi$
$588$	0	0
$589$	−3.80776	−0.156896
$590$	0	0
$591$	17.7538	0.730293
$592$	0	0
$593$	−8.24621	−0.338631	−0.169316	−	0.985562i	$-0.554156\pi$
$593$	−8.24621	−0.338631	−0.169316	+	0.985562i	$0.554156\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−16.3002	−0.667122
$598$	0	0
$599$	−31.6155	−1.29178	−0.645888	−	0.763432i	$-0.723513\pi$
$599$	−31.6155	−1.29178	−0.645888	+	0.763432i	$0.723513\pi$
$600$	0	0
$601$	33.6155	1.37121	0.685603	−	0.727976i	$-0.259539\pi$
$601$	33.6155	1.37121	0.685603	+	0.727976i	$0.259539\pi$
$602$	0	0
$603$	0.492423	0.0200530
$604$	0	0
$605$	0	0
$606$	0	0
$607$	13.8617	0.562631	0.281315	−	0.959615i	$-0.409229\pi$
$607$	13.8617	0.562631	0.281315	+	0.959615i	$0.409229\pi$
$608$	0	0
$609$	−32.6004	−1.32103
$610$	0	0
$611$	−1.06913	−0.0432524
$612$	0	0
$613$	−4.93087	−0.199156	−0.0995780	−	0.995030i	$-0.531749\pi$
$613$	−4.93087	−0.199156	−0.0995780	+	0.995030i	$0.531749\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−18.6847	−0.752216	−0.376108	−	0.926576i	$-0.622738\pi$
$617$	−18.6847	−0.752216	−0.376108	+	0.926576i	$0.622738\pi$
$618$	0	0
$619$	−0.876894	−0.0352454	−0.0176227	−	0.999845i	$-0.505610\pi$
$619$	−0.876894	−0.0352454	−0.0176227	+	0.999845i	$0.505610\pi$
$620$	0	0
$621$	−17.3693	−0.697007
$622$	0	0
$623$	−8.38447	−0.335917
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−9.75379	−0.389529
$628$	0	0
$629$	1.12311	0.0447812
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	0	0
$633$	−33.3693	−1.32631
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1.20739	−0.0478386
$638$	0	0
$639$	−9.36932	−0.370644
$640$	0	0
$641$	−7.75379	−0.306256	−0.153128	−	0.988206i	$-0.548935\pi$
$641$	−7.75379	−0.306256	−0.153128	+	0.988206i	$0.548935\pi$
$642$	0	0
$643$	7.50758	0.296070	0.148035	−	0.988982i	$-0.452705\pi$
$643$	7.50758	0.296070	0.148035	+	0.988982i	$0.452705\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−0.684658	−0.0269167	−0.0134584	−	0.999909i	$-0.504284\pi$
$647$	−0.684658	−0.0269167	−0.0134584	+	0.999909i	$0.504284\pi$
$648$	0	0
$649$	50.7386	1.99167
$650$	0	0
$651$	−11.8920	−0.466086
$652$	0	0
$653$	5.50758	0.215528	0.107764	−	0.994176i	$-0.465631\pi$
$653$	5.50758	0.215528	0.107764	+	0.994176i	$0.465631\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	7.75379	0.302504
$658$	0	0
$659$	13.0691	0.509101	0.254551	−	0.967059i	$-0.418072\pi$
$659$	13.0691	0.509101	0.254551	+	0.967059i	$0.418072\pi$
$660$	0	0
$661$	−39.8617	−1.55044	−0.775221	−	0.631690i	$-0.782362\pi$
$661$	−39.8617	−1.55044	−0.775221	+	0.631690i	$0.782362\pi$
$662$	0	0
$663$	0.684658	0.0265899
$664$	0	0
$665$	0	0
$666$	0	0
$667$	20.8769	0.808357
$668$	0	0
$669$	−13.5616	−0.524320
$670$	0	0
$671$	46.2462	1.78532
$672$	0	0
$673$	41.4233	1.59675	0.798375	−	0.602160i	$-0.205693\pi$
$673$	41.4233	1.59675	0.798375	+	0.602160i	$0.205693\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.73863	−0.335853	−0.167926	−	0.985800i	$-0.553707\pi$
$677$	−8.73863	−0.335853	−0.167926	+	0.985800i	$0.553707\pi$
$678$	0	0
$679$	33.3693	1.28060
$680$	0	0
$681$	44.7926	1.71646
$682$	0	0
$683$	12.3002	0.470654	0.235327	−	0.971916i	$-0.424384\pi$
$683$	12.3002	0.470654	0.235327	+	0.971916i	$0.424384\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	28.8769	1.10172
$688$	0	0
$689$	1.56155	0.0594904
$690$	0	0
$691$	−26.2462	−0.998453	−0.499226	−	0.866472i	$-0.666382\pi$
$691$	−26.2462	−0.998453	−0.499226	+	0.866472i	$0.666382\pi$
$692$	0	0
$693$	7.01515	0.266484
$694$	0	0
$695$	0	0
$696$	0	0
$697$	12.2462	0.463858
$698$	0	0
$699$	−29.1771	−1.10358
$700$	0	0
$701$	−44.3542	−1.67523	−0.837617	−	0.546258i	$-0.816052\pi$
$701$	−44.3542	−1.67523	−0.837617	+	0.546258i	$0.816052\pi$
$702$	0	0
$703$	1.75379	0.0661454
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−57.7538	−2.17205
$708$	0	0
$709$	−32.5464	−1.22231	−0.611153	−	0.791513i	$-0.709294\pi$
$709$	−32.5464	−1.22231	−0.611153	+	0.791513i	$0.709294\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.61553	0.285204
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−2.13826	−0.0798548
$718$	0	0
$719$	−35.8078	−1.33540	−0.667702	−	0.744429i	$-0.732722\pi$
$719$	−35.8078	−1.33540	−0.667702	+	0.744429i	$0.732722\pi$
$720$	0	0
$721$	−24.9848	−0.930484
$722$	0	0
$723$	−6.63068	−0.246598
$724$	0	0
$725$	0	0
$726$	0	0
$727$	42.4384	1.57395	0.786977	−	0.616982i	$-0.211645\pi$
$727$	42.4384	1.57395	0.786977	+	0.616982i	$0.211645\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	−7.12311	−0.263458
$732$	0	0
$733$	−10.4924	−0.387546	−0.193773	−	0.981046i	$-0.562073\pi$
$733$	−10.4924	−0.387546	−0.193773	+	0.981046i	$0.562073\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.50758	−0.129203
$738$	0	0
$739$	−25.1771	−0.926154	−0.463077	−	0.886318i	$-0.653255\pi$
$739$	−25.1771	−0.926154	−0.463077	+	0.886318i	$0.653255\pi$
$740$	0	0
$741$	1.06913	0.0392755
$742$	0	0
$743$	−12.8769	−0.472407	−0.236204	−	0.971704i	$-0.575903\pi$
$743$	−12.8769	−0.472407	−0.236204	+	0.971704i	$0.575903\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−5.75379	−0.210520
$748$	0	0
$749$	51.5076	1.88205
$750$	0	0
$751$	−21.1771	−0.772763	−0.386381	−	0.922339i	$-0.626275\pi$
$751$	−21.1771	−0.772763	−0.386381	+	0.922339i	$0.626275\pi$
$752$	0	0
$753$	6.24621	0.227625
$754$	0	0
$755$	0	0
$756$	0	0
$757$	38.7926	1.40994	0.704971	−	0.709236i	$-0.250960\pi$
$757$	38.7926	1.40994	0.704971	+	0.709236i	$0.250960\pi$
$758$	0	0
$759$	19.5076	0.708080
$760$	0	0
$761$	28.7386	1.04177	0.520887	−	0.853625i	$-0.325601\pi$
$761$	28.7386	1.04177	0.520887	+	0.853625i	$0.325601\pi$
$762$	0	0
$763$	13.8617	0.501829
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.56155	−0.200816
$768$	0	0
$769$	15.5616	0.561164	0.280582	−	0.959830i	$-0.409473\pi$
$769$	15.5616	0.561164	0.280582	+	0.959830i	$0.409473\pi$
$770$	0	0
$771$	−3.12311	−0.112476
$772$	0	0
$773$	28.7386	1.03366	0.516828	−	0.856089i	$-0.327113\pi$
$773$	28.7386	1.03366	0.516828	+	0.856089i	$0.327113\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	5.47727	0.196496
$778$	0	0
$779$	19.1231	0.685156
$780$	0	0
$781$	66.7386	2.38810
$782$	0	0
$783$	−37.1771	−1.32860
$784$	0	0
$785$	0	0
$786$	0	0
$787$	42.5464	1.51662	0.758308	−	0.651897i	$-0.226027\pi$
$787$	42.5464	1.51662	0.758308	+	0.651897i	$0.226027\pi$
$788$	0	0
$789$	6.54640	0.233058
$790$	0	0
$791$	−55.6155	−1.97746
$792$	0	0
$793$	−5.06913	−0.180010
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.4924	0.513348	0.256674	−	0.966498i	$-0.417373\pi$
$797$	14.4924	0.513348	0.256674	+	0.966498i	$0.417373\pi$
$798$	0	0
$799$	−2.43845	−0.0862661
$800$	0	0
$801$	−1.50758	−0.0532676
$802$	0	0
$803$	−55.2311	−1.94906
$804$	0	0
$805$	0	0
$806$	0	0
$807$	27.8078	0.978880
$808$	0	0
$809$	−46.9848	−1.65190	−0.825950	−	0.563744i	$-0.809360\pi$
$809$	−46.9848	−1.65190	−0.825950	+	0.563744i	$0.809360\pi$
$810$	0	0
$811$	21.3693	0.750378	0.375189	−	0.926948i	$-0.377578\pi$
$811$	21.3693	0.750378	0.375189	+	0.926948i	$0.377578\pi$
$812$	0	0
$813$	−24.9848	−0.876257
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−11.1231	−0.389148
$818$	0	0
$819$	−0.768944	−0.0268691
$820$	0	0
$821$	38.3002	1.33669	0.668343	−	0.743853i	$-0.267004\pi$
$821$	38.3002	1.33669	0.668343	+	0.743853i	$0.267004\pi$
$822$	0	0
$823$	25.3693	0.884319	0.442159	−	0.896936i	$-0.354213\pi$
$823$	25.3693	0.884319	0.442159	+	0.896936i	$0.354213\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−32.4924	−1.12987	−0.564936	−	0.825135i	$-0.691099\pi$
$827$	−32.4924	−1.12987	−0.564936	+	0.825135i	$0.691099\pi$
$828$	0	0
$829$	15.3693	0.533798	0.266899	−	0.963724i	$-0.414001\pi$
$829$	15.3693	0.533798	0.266899	+	0.963724i	$0.414001\pi$
$830$	0	0
$831$	50.3542	1.74677
$832$	0	0
$833$	−2.75379	−0.0954131
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−13.5616	−0.468756
$838$	0	0
$839$	2.05398	0.0709111	0.0354556	−	0.999371i	$-0.488712\pi$
$839$	2.05398	0.0709111	0.0354556	+	0.999371i	$0.488712\pi$
$840$	0	0
$841$	15.6847	0.540850
$842$	0	0
$843$	19.4233	0.668974
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−15.6155	−0.536556
$848$	0	0
$849$	32.3002	1.10854
$850$	0	0
$851$	−3.50758	−0.120238
$852$	0	0
$853$	−31.7538	−1.08723	−0.543615	−	0.839335i	$-0.682945\pi$
$853$	−31.7538	−1.08723	−0.543615	+	0.839335i	$0.682945\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−47.1771	−1.61154	−0.805769	−	0.592230i	$-0.798248\pi$
$857$	−47.1771	−1.61154	−0.805769	+	0.592230i	$0.798248\pi$
$858$	0	0
$859$	−34.5464	−1.17871	−0.589354	−	0.807875i	$-0.700618\pi$
$859$	−34.5464	−1.17871	−0.589354	+	0.807875i	$0.700618\pi$
$860$	0	0
$861$	59.7235	2.03537
$862$	0	0
$863$	44.4924	1.51454	0.757270	−	0.653102i	$-0.226533\pi$
$863$	44.4924	1.51454	0.757270	+	0.653102i	$0.226533\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.56155	0.0530331
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0.384472	0.0130273
$872$	0	0
$873$	6.00000	0.203069
$874$	0	0
$875$	0	0
$876$	0	0
$877$	10.3845	0.350659	0.175329	−	0.984510i	$-0.443901\pi$
$877$	10.3845	0.350659	0.175329	+	0.984510i	$0.443901\pi$
$878$	0	0
$879$	19.4233	0.655131
$880$	0	0
$881$	20.7386	0.698702	0.349351	−	0.936992i	$-0.386402\pi$
$881$	20.7386	0.698702	0.349351	+	0.936992i	$0.386402\pi$
$882$	0	0
$883$	2.63068	0.0885295	0.0442648	−	0.999020i	$-0.485905\pi$
$883$	2.63068	0.0885295	0.0442648	+	0.999020i	$0.485905\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−56.6004	−1.90045	−0.950227	−	0.311558i	$-0.899149\pi$
$887$	−56.6004	−1.90045	−0.950227	+	0.311558i	$0.899149\pi$
$888$	0	0
$889$	−42.3542	−1.42051
$890$	0	0
$891$	−28.0000	−0.938035
$892$	0	0
$893$	−3.80776	−0.127422
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.13826	−0.0713944
$898$	0	0
$899$	16.3002	0.543642
$900$	0	0
$901$	3.56155	0.118653
$902$	0	0
$903$	−34.7386	−1.15603
$904$	0	0
$905$	0	0
$906$	0	0
$907$	14.4384	0.479421	0.239710	−	0.970844i	$-0.422948\pi$
$907$	14.4384	0.479421	0.239710	+	0.970844i	$0.422948\pi$
$908$	0	0
$909$	−10.3845	−0.344431
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	40.9848	1.35640
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−22.2462	−0.734635
$918$	0	0
$919$	12.8769	0.424770	0.212385	−	0.977186i	$-0.431877\pi$
$919$	12.8769	0.424770	0.212385	+	0.977186i	$0.431877\pi$
$920$	0	0
$921$	−53.4773	−1.76214
$922$	0	0
$923$	−7.31534	−0.240787
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−4.49242	−0.147551
$928$	0	0
$929$	39.4773	1.29521	0.647604	−	0.761977i	$-0.275771\pi$
$929$	39.4773	1.29521	0.647604	+	0.761977i	$0.275771\pi$
$930$	0	0
$931$	−4.30019	−0.140933
$932$	0	0
$933$	−12.4924	−0.408984
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−30.1080	−0.983584	−0.491792	−	0.870713i	$-0.663658\pi$
$937$	−30.1080	−0.983584	−0.491792	+	0.870713i	$0.663658\pi$
$938$	0	0
$939$	9.36932	0.305756
$940$	0	0
$941$	−31.5616	−1.02888	−0.514439	−	0.857527i	$-0.672000\pi$
$941$	−31.5616	−1.02888	−0.514439	+	0.857527i	$0.672000\pi$
$942$	0	0
$943$	−38.2462	−1.24547
$944$	0	0
$945$	0	0
$946$	0	0
$947$	32.1922	1.04611	0.523054	−	0.852300i	$-0.324793\pi$
$947$	32.1922	1.04611	0.523054	+	0.852300i	$0.324793\pi$
$948$	0	0
$949$	6.05398	0.196520
$950$	0	0
$951$	15.0152	0.486900
$952$	0	0
$953$	−30.8769	−1.00020	−0.500100	−	0.865967i	$-0.666704\pi$
$953$	−30.8769	−1.00020	−0.500100	+	0.865967i	$0.666704\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	41.7538	1.34971
$958$	0	0
$959$	31.2311	1.00850
$960$	0	0
$961$	−25.0540	−0.808193
$962$	0	0
$963$	9.26137	0.298443
$964$	0	0
$965$	0	0
$966$	0	0
$967$	52.4924	1.68804	0.844021	−	0.536310i	$-0.180182\pi$
$967$	52.4924	1.68804	0.844021	+	0.536310i	$0.180182\pi$
$968$	0	0
$969$	2.43845	0.0783342
$970$	0	0
$971$	−3.31534	−0.106394	−0.0531972	−	0.998584i	$-0.516941\pi$
$971$	−3.31534	−0.106394	−0.0531972	+	0.998584i	$0.516941\pi$
$972$	0	0
$973$	−51.5076	−1.65126
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.8769	0.603925	0.301963	−	0.953320i	$-0.402358\pi$
$977$	18.8769	0.603925	0.301963	+	0.953320i	$0.402358\pi$
$978$	0	0
$979$	10.7386	0.343208
$980$	0	0
$981$	2.49242	0.0795769
$982$	0	0
$983$	−28.8769	−0.921030	−0.460515	−	0.887652i	$-0.652335\pi$
$983$	−28.8769	−0.921030	−0.460515	+	0.887652i	$0.652335\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−11.8920	−0.378528
$988$	0	0
$989$	22.2462	0.707388
$990$	0	0
$991$	27.4233	0.871130	0.435565	−	0.900157i	$-0.356549\pi$
$991$	27.4233	0.871130	0.435565	+	0.900157i	$0.356549\pi$
$992$	0	0
$993$	21.9460	0.696436
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−28.2462	−0.894566	−0.447283	−	0.894392i	$-0.647608\pi$
$997$	−28.2462	−0.894566	−0.447283	+	0.894392i	$0.647608\pi$
$998$	0	0
$999$	6.24621	0.197621

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6800.2.a.be.1.2		2
4.3	odd	2		850.2.a.n.1.1		2
5.4	even	2		1360.2.a.m.1.1		2
12.11	even	2		7650.2.a.de.1.2		2
20.3	even	4		850.2.c.i.749.3		4
20.7	even	4		850.2.c.i.749.2		4
20.19	odd	2		170.2.a.f.1.2	✓	2
40.19	odd	2		5440.2.a.bj.1.1		2
40.29	even	2		5440.2.a.bd.1.2		2
60.59	even	2		1530.2.a.r.1.1		2
140.139	even	2		8330.2.a.bq.1.1		2
340.259	odd	4		2890.2.b.i.2311.2		4
340.319	odd	4		2890.2.b.i.2311.3		4
340.339	odd	2		2890.2.a.u.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.2.a.f.1.2	✓	2	20.19	odd	2
850.2.a.n.1.1		2	4.3	odd	2
850.2.c.i.749.2		4	20.7	even	4
850.2.c.i.749.3		4	20.3	even	4
1360.2.a.m.1.1		2	5.4	even	2
1530.2.a.r.1.1		2	60.59	even	2
2890.2.a.u.1.1		2	340.339	odd	2
2890.2.b.i.2311.2		4	340.259	odd	4
2890.2.b.i.2311.3		4	340.319	odd	4
5440.2.a.bd.1.2		2	40.29	even	2
5440.2.a.bj.1.1		2	40.19	odd	2
6800.2.a.be.1.2		2	1.1	even	1	trivial
7650.2.a.de.1.2		2	12.11	even	2
8330.2.a.bq.1.1		2	140.139	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)

\(f(q)\)	\(=\)	\(q+1.56155 q^{3} -3.12311 q^{7} -0.561553 q^{9} +O(q^{10})\)	\(q+1.56155 q^{3} -3.12311 q^{7} -0.561553 q^{9} +4.00000 q^{11} -0.438447 q^{13} -1.00000 q^{17} -1.56155 q^{19} -4.87689 q^{21} +3.12311 q^{23} -5.56155 q^{27} +6.68466 q^{29} +2.43845 q^{31} +6.24621 q^{33} -1.12311 q^{37} -0.684658 q^{39} -12.2462 q^{41} +7.12311 q^{43} +2.43845 q^{47} +2.75379 q^{49} -1.56155 q^{51} -3.56155 q^{53} -2.43845 q^{57} +12.6847 q^{59} +11.5616 q^{61} +1.75379 q^{63} -0.876894 q^{67} +4.87689 q^{69} +16.6847 q^{71} -13.8078 q^{73} -12.4924 q^{77} -7.00000 q^{81} +10.2462 q^{83} +10.4384 q^{87} +2.68466 q^{89} +1.36932 q^{91} +3.80776 q^{93} -10.6847 q^{97} -2.24621 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q - q^3 + 2 * q^7 + 3 * q^9	\( 2 q - q^{3} + 2 q^{7} + 3 q^{9} + 8 q^{11} - 5 q^{13} - 2 q^{17} + q^{19} - 18 q^{21} - 2 q^{23} - 7 q^{27} + q^{29} + 9 q^{31} - 4 q^{33} + 6 q^{37} + 11 q^{39} - 8 q^{41} + 6 q^{43} + 9 q^{47} + 22 q^{49} + q^{51} - 3 q^{53} - 9 q^{57} + 13 q^{59} + 19 q^{61} + 20 q^{63} - 10 q^{67} + 18 q^{69} + 21 q^{71} - 7 q^{73} + 8 q^{77} - 14 q^{81} + 4 q^{83} + 25 q^{87} - 7 q^{89} - 22 q^{91} - 13 q^{93} - 9 q^{97} + 12 q^{99}+O(q^{100}) \) 2 * q - q^3 + 2 * q^7 + 3 * q^9 + 8 * q^11 - 5 * q^13 - 2 * q^17 + q^19 - 18 * q^21 - 2 * q^23 - 7 * q^27 + q^29 + 9 * q^31 - 4 * q^33 + 6 * q^37 + 11 * q^39 - 8 * q^41 + 6 * q^43 + 9 * q^47 + 22 * q^49 + q^51 - 3 * q^53 - 9 * q^57 + 13 * q^59 + 19 * q^61 + 20 * q^63 - 10 * q^67 + 18 * q^69 + 21 * q^71 - 7 * q^73 + 8 * q^77 - 14 * q^81 + 4 * q^83 + 25 * q^87 - 7 * q^89 - 22 * q^91 - 13 * q^93 - 9 * q^97 + 12 * q^99

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