LMFDB - Embedded newform 9200.2.a.cl.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, 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("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9200 = 2^{4} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9200.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:	$73.4623698596$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.15317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 2 $ x^4 - 2x^3 - 4x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 115)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.32973$ of defining polynomial
Character	$\chi$	$=$	9200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155 q^{3} +4.06562 q^{7} -0.561553 q^{9} +O(q^{10})$	$q+1.56155 q^{3} +4.06562 q^{7} -0.561553 q^{9} -2.65945 q^{11} +5.91023 q^{13} +3.40617 q^{17} +2.65945 q^{19} +6.34868 q^{21} -1.00000 q^{23} -5.56155 q^{27} +5.84461 q^{29} +9.31640 q^{31} -4.15288 q^{33} +4.18059 q^{37} +9.22914 q^{39} +2.15539 q^{41} +2.34868 q^{43} +0.242644 q^{47} +9.52927 q^{49} +5.31891 q^{51} -9.03585 q^{53} +4.15288 q^{57} -14.4143 q^{59} +2.46365 q^{61} -2.28306 q^{63} -7.75485 q^{67} -1.56155 q^{69} -12.4395 q^{71} +2.08977 q^{73} -10.8123 q^{77} +4.15288 q^{79} -7.00000 q^{81} -12.5293 q^{83} +9.12667 q^{87} -9.35682 q^{89} +24.0288 q^{91} +14.5481 q^{93} -3.47179 q^{97} +1.49342 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 3 q^{7} + 6 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 - 3 * q^7 + 6 * q^9	$ 4 q - 2 q^{3} - 3 q^{7} + 6 q^{9} - 4 q^{11} + q^{17} + 4 q^{19} + 10 q^{21} - 4 q^{23} - 14 q^{27} + 19 q^{29} + q^{31} + 2 q^{33} + 3 q^{37} + 13 q^{41} - 6 q^{43} + 6 q^{47} + 9 q^{49} + 8 q^{51} - 19 q^{53} - 2 q^{57} - 23 q^{59} - 13 q^{63} - 3 q^{67} + 2 q^{69} + 3 q^{71} + 32 q^{73} - 18 q^{77} - 2 q^{79} - 28 q^{81} - 21 q^{83} - 18 q^{87} + 40 q^{91} + 8 q^{93} + 18 q^{97} - 6 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 3 * q^7 + 6 * q^9 - 4 * q^11 + q^17 + 4 * q^19 + 10 * q^21 - 4 * q^23 - 14 * q^27 + 19 * q^29 + q^31 + 2 * q^33 + 3 * q^37 + 13 * q^41 - 6 * q^43 + 6 * q^47 + 9 * q^49 + 8 * q^51 - 19 * q^53 - 2 * q^57 - 23 * q^59 - 13 * q^63 - 3 * q^67 + 2 * q^69 + 3 * q^71 + 32 * q^73 - 18 * q^77 - 2 * q^79 - 28 * q^81 - 21 * q^83 - 18 * q^87 + 40 * q^91 + 8 * q^93 + 18 * q^97 - 6 * 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
$7$	4.06562	1.53666	0.768330	−	0.640054i	$-0.221088\pi$
$7$	4.06562	1.53666	0.768330	+	0.640054i	$0.221088\pi$
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	−2.65945	−0.801856	−0.400928	−	0.916110i	$-0.631312\pi$
$11$	−2.65945	−0.801856	−0.400928	+	0.916110i	$0.631312\pi$
$12$	0	0
$13$	5.91023	1.63920	0.819602	−	0.572933i	$-0.194195\pi$
$13$	5.91023	1.63920	0.819602	+	0.572933i	$0.194195\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.40617	0.826117	0.413058	−	0.910705i	$-0.364461\pi$
$17$	3.40617	0.826117	0.413058	+	0.910705i	$0.364461\pi$
$18$	0	0
$19$	2.65945	0.610121	0.305060	−	0.952333i	$-0.401323\pi$
$19$	2.65945	0.610121	0.305060	+	0.952333i	$0.401323\pi$
$20$	0	0
$21$	6.34868	1.38540
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.56155	−1.07032
$28$	0	0
$29$	5.84461	1.08532	0.542659	−	0.839953i	$-0.317418\pi$
$29$	5.84461	1.08532	0.542659	+	0.839953i	$0.317418\pi$
$30$	0	0
$31$	9.31640	1.67327	0.836637	−	0.547757i	$-0.184518\pi$
$31$	9.31640	1.67327	0.836637	+	0.547757i	$0.184518\pi$
$32$	0	0
$33$	−4.15288	−0.722923
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.18059	0.687285	0.343642	−	0.939101i	$-0.388339\pi$
$37$	4.18059	0.687285	0.343642	+	0.939101i	$0.388339\pi$
$38$	0	0
$39$	9.22914	1.47785
$40$	0	0
$41$	2.15539	0.336615	0.168307	−	0.985735i	$-0.446170\pi$
$41$	2.15539	0.336615	0.168307	+	0.985735i	$0.446170\pi$
$42$	0	0
$43$	2.34868	0.358171	0.179085	−	0.983834i	$-0.442686\pi$
$43$	2.34868	0.358171	0.179085	+	0.983834i	$0.442686\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.242644	0.0353933	0.0176966	−	0.999843i	$-0.494367\pi$
$47$	0.242644	0.0353933	0.0176966	+	0.999843i	$0.494367\pi$
$48$	0	0
$49$	9.52927	1.36132
$50$	0	0
$51$	5.31891	0.744796
$52$	0	0
$53$	−9.03585	−1.24117	−0.620585	−	0.784140i	$-0.713105\pi$
$53$	−9.03585	−1.24117	−0.620585	+	0.784140i	$0.713105\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.15288	0.550062
$58$	0	0
$59$	−14.4143	−1.87658	−0.938291	−	0.345846i	$-0.887592\pi$
$59$	−14.4143	−1.87658	−0.938291	+	0.345846i	$0.887592\pi$
$60$	0	0
$61$	2.46365	0.315438	0.157719	−	0.987484i	$-0.449586\pi$
$61$	2.46365	0.315438	0.157719	+	0.987484i	$0.449586\pi$
$62$	0	0
$63$	−2.28306	−0.287639
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.75485	−0.947405	−0.473703	−	0.880685i	$-0.657083\pi$
$67$	−7.75485	−0.947405	−0.473703	+	0.880685i	$0.657083\pi$
$68$	0	0
$69$	−1.56155	−0.187989
$70$	0	0
$71$	−12.4395	−1.47630	−0.738149	−	0.674638i	$-0.764300\pi$
$71$	−12.4395	−1.47630	−0.738149	+	0.674638i	$0.764300\pi$
$72$	0	0
$73$	2.08977	0.244589	0.122294	−	0.992494i	$-0.460975\pi$
$73$	2.08977	0.244589	0.122294	+	0.992494i	$0.460975\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−10.8123	−1.23218
$78$	0	0
$79$	4.15288	0.467235	0.233618	−	0.972329i	$-0.424944\pi$
$79$	4.15288	0.467235	0.233618	+	0.972329i	$0.424944\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−12.5293	−1.37527	−0.687633	−	0.726058i	$-0.741350\pi$
$83$	−12.5293	−1.37527	−0.687633	+	0.726058i	$0.741350\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	9.12667	0.978482
$88$	0	0
$89$	−9.35682	−0.991821	−0.495910	−	0.868374i	$-0.665166\pi$
$89$	−9.35682	−0.991821	−0.495910	+	0.868374i	$0.665166\pi$
$90$	0	0
$91$	24.0288	2.51890
$92$	0	0
$93$	14.5481	1.50856
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.47179	−0.352507	−0.176253	−	0.984345i	$-0.556398\pi$
$97$	−3.47179	−0.352507	−0.176253	+	0.984345i	$0.556398\pi$
$98$	0	0
$99$	1.49342	0.150095
$100$	0	0
$101$	9.21036	0.916465	0.458233	−	0.888832i	$-0.348483\pi$
$101$	9.21036	0.916465	0.458233	+	0.888832i	$0.348483\pi$
$102$	0	0
$103$	4.69736	0.462845	0.231422	−	0.972853i	$-0.425662\pi$
$103$	4.69736	0.462845	0.231422	+	0.972853i	$0.425662\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.376394	−0.0363873	−0.0181937	−	0.999834i	$-0.505792\pi$
$107$	−0.376394	−0.0363873	−0.0181937	+	0.999834i	$0.505792\pi$
$108$	0	0
$109$	19.0369	1.82340	0.911702	−	0.410851i	$-0.134768\pi$
$109$	19.0369	1.82340	0.911702	+	0.410851i	$0.134768\pi$
$110$	0	0
$111$	6.52821	0.619631
$112$	0	0
$113$	−2.94252	−0.276809	−0.138404	−	0.990376i	$-0.544197\pi$
$113$	−2.94252	−0.276809	−0.138404	+	0.990376i	$0.544197\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3.31891	−0.306833
$118$	0	0
$119$	13.8482	1.26946
$120$	0	0
$121$	−3.92730	−0.357028
$122$	0	0
$123$	3.36575	0.303479
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.00357	0.532730	0.266365	−	0.963872i	$-0.414177\pi$
$127$	6.00357	0.532730	0.266365	+	0.963872i	$0.414177\pi$
$128$	0	0
$129$	3.66759	0.322913
$130$	0	0
$131$	19.9606	1.74397	0.871985	−	0.489533i	$-0.162833\pi$
$131$	19.9606	1.74397	0.871985	+	0.489533i	$0.162833\pi$
$132$	0	0
$133$	10.8123	0.937548
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.7609	1.00480	0.502402	−	0.864634i	$-0.332450\pi$
$137$	11.7609	1.00480	0.502402	+	0.864634i	$0.332450\pi$
$138$	0	0
$139$	−12.3891	−1.05083	−0.525415	−	0.850846i	$-0.676090\pi$
$139$	−12.3891	−1.05083	−0.525415	+	0.850846i	$0.676090\pi$
$140$	0	0
$141$	0.378902	0.0319093
$142$	0	0
$143$	−15.7180	−1.31441
$144$	0	0
$145$	0	0
$146$	0	0
$147$	14.8805	1.22732
$148$	0	0
$149$	−2.13124	−0.174598	−0.0872991	−	0.996182i	$-0.527824\pi$
$149$	−2.13124	−0.174598	−0.0872991	+	0.996182i	$0.527824\pi$
$150$	0	0
$151$	0.787129	0.0640556	0.0320278	−	0.999487i	$-0.489803\pi$
$151$	0.787129	0.0640556	0.0320278	+	0.999487i	$0.489803\pi$
$152$	0	0
$153$	−1.91274	−0.154636
$154$	0	0
$155$	0	0
$156$	0	0
$157$	9.18873	0.733340	0.366670	−	0.930351i	$-0.380498\pi$
$157$	9.18873	0.733340	0.366670	+	0.930351i	$0.380498\pi$
$158$	0	0
$159$	−14.1100	−1.11899
$160$	0	0
$161$	−4.06562	−0.320416
$162$	0	0
$163$	−17.6928	−1.38581	−0.692903	−	0.721031i	$-0.743669\pi$
$163$	−17.6928	−1.38581	−0.692903	+	0.721031i	$0.743669\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.2462	1.10240	0.551202	−	0.834372i	$-0.314169\pi$
$167$	14.2462	1.10240	0.551202	+	0.834372i	$0.314169\pi$
$168$	0	0
$169$	21.9309	1.68699
$170$	0	0
$171$	−1.49342	−0.114205
$172$	0	0
$173$	1.20394	0.0915338	0.0457669	−	0.998952i	$-0.485427\pi$
$173$	1.20394	0.0915338	0.0457669	+	0.998952i	$0.485427\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−22.5087	−1.69186
$178$	0	0
$179$	−7.42495	−0.554967	−0.277483	−	0.960730i	$-0.589500\pi$
$179$	−7.42495	−0.554967	−0.277483	+	0.960730i	$0.589500\pi$
$180$	0	0
$181$	24.0450	1.78725	0.893627	−	0.448810i	$-0.148152\pi$
$181$	24.0450	1.78725	0.893627	+	0.448810i	$0.148152\pi$
$182$	0	0
$183$	3.84712	0.284387
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−9.05854	−0.662426
$188$	0	0
$189$	−22.6112	−1.64472
$190$	0	0
$191$	4.87689	0.352880	0.176440	−	0.984311i	$-0.443542\pi$
$191$	4.87689	0.352880	0.176440	+	0.984311i	$0.443542\pi$
$192$	0	0
$193$	−10.2714	−0.739353	−0.369676	−	0.929161i	$-0.620531\pi$
$193$	−10.2714	−0.739353	−0.369676	+	0.929161i	$0.620531\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.28557	−0.305334	−0.152667	−	0.988278i	$-0.548786\pi$
$197$	−4.28557	−0.305334	−0.152667	+	0.988278i	$0.548786\pi$
$198$	0	0
$199$	−4.02164	−0.285086	−0.142543	−	0.989789i	$-0.545528\pi$
$199$	−4.02164	−0.285086	−0.142543	+	0.989789i	$0.545528\pi$
$200$	0	0
$201$	−12.1096	−0.854146
$202$	0	0
$203$	23.7620	1.66776
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.561553	0.0390306
$208$	0	0
$209$	−7.07270	−0.489229
$210$	0	0
$211$	−15.3416	−1.05616	−0.528080	−	0.849195i	$-0.677088\pi$
$211$	−15.3416	−1.05616	−0.528080	+	0.849195i	$0.677088\pi$
$212$	0	0
$213$	−19.4249	−1.33098
$214$	0	0
$215$	0	0
$216$	0	0
$217$	37.8770	2.57126
$218$	0	0
$219$	3.26328	0.220512
$220$	0	0
$221$	20.1312	1.35417
$222$	0	0
$223$	−17.6801	−1.18395	−0.591973	−	0.805958i	$-0.701651\pi$
$223$	−17.6801	−1.18395	−0.591973	+	0.805958i	$0.701651\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.63817	0.573335	0.286668	−	0.958030i	$-0.407452\pi$
$227$	8.63817	0.573335	0.286668	+	0.958030i	$0.407452\pi$
$228$	0	0
$229$	−3.34055	−0.220749	−0.110375	−	0.993890i	$-0.535205\pi$
$229$	−3.34055	−0.220749	−0.110375	+	0.993890i	$0.535205\pi$
$230$	0	0
$231$	−16.8840	−1.11089
$232$	0	0
$233$	−8.23728	−0.539642	−0.269821	−	0.962910i	$-0.586965\pi$
$233$	−8.23728	−0.539642	−0.269821	+	0.962910i	$0.586965\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	6.48494	0.421242
$238$	0	0
$239$	2.86525	0.185338	0.0926688	−	0.995697i	$-0.470460\pi$
$239$	2.86525	0.185338	0.0926688	+	0.995697i	$0.470460\pi$
$240$	0	0
$241$	4.68109	0.301536	0.150768	−	0.988569i	$-0.451825\pi$
$241$	4.68109	0.301536	0.150768	+	0.988569i	$0.451825\pi$
$242$	0	0
$243$	5.75379	0.369106
$244$	0	0
$245$	0	0
$246$	0	0
$247$	15.7180	1.00011
$248$	0	0
$249$	−19.5651	−1.23989
$250$	0	0
$251$	18.5824	1.17291	0.586455	−	0.809982i	$-0.300523\pi$
$251$	18.5824	1.17291	0.586455	+	0.809982i	$0.300523\pi$
$252$	0	0
$253$	2.65945	0.167198
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.7226	1.79166	0.895832	−	0.444392i	$-0.146580\pi$
$257$	28.7226	1.79166	0.895832	+	0.444392i	$0.146580\pi$
$258$	0	0
$259$	16.9967	1.05612
$260$	0	0
$261$	−3.28206	−0.203154
$262$	0	0
$263$	−9.20500	−0.567604	−0.283802	−	0.958883i	$-0.591596\pi$
$263$	−9.20500	−0.567604	−0.283802	+	0.958883i	$0.591596\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−14.6112	−0.894189
$268$	0	0
$269$	16.9194	1.03160	0.515798	−	0.856710i	$-0.327496\pi$
$269$	16.9194	1.03160	0.515798	+	0.856710i	$0.327496\pi$
$270$	0	0
$271$	−1.15082	−0.0699072	−0.0349536	−	0.999389i	$-0.511128\pi$
$271$	−1.15082	−0.0699072	−0.0349536	+	0.999389i	$0.511128\pi$
$272$	0	0
$273$	37.5222	2.27095
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2.57505	−0.154720	−0.0773600	−	0.997003i	$-0.524649\pi$
$277$	−2.57505	−0.154720	−0.0773600	+	0.997003i	$0.524649\pi$
$278$	0	0
$279$	−5.23165	−0.313211
$280$	0	0
$281$	27.4718	1.63883	0.819415	−	0.573201i	$-0.194299\pi$
$281$	27.4718	1.63883	0.819415	+	0.573201i	$0.194299\pi$
$282$	0	0
$283$	−2.36389	−0.140519	−0.0702595	−	0.997529i	$-0.522383\pi$
$283$	−2.36389	−0.140519	−0.0702595	+	0.997529i	$0.522383\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.76298	0.517263
$288$	0	0
$289$	−5.39803	−0.317531
$290$	0	0
$291$	−5.42138	−0.317807
$292$	0	0
$293$	−34.1198	−1.99330	−0.996650	−	0.0817846i	$-0.973938\pi$
$293$	−34.1198	−1.99330	−0.996650	+	0.0817846i	$0.973938\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	14.7907	0.858243
$298$	0	0
$299$	−5.91023	−0.341798
$300$	0	0
$301$	9.54885	0.550386
$302$	0	0
$303$	14.3825	0.826251
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.49342	0.0852342	0.0426171	−	0.999091i	$-0.486430\pi$
$307$	1.49342	0.0852342	0.0426171	+	0.999091i	$0.486430\pi$
$308$	0	0
$309$	7.33518	0.417284
$310$	0	0
$311$	14.1060	0.799880	0.399940	−	0.916541i	$-0.369031\pi$
$311$	14.1060	0.799880	0.399940	+	0.916541i	$0.369031\pi$
$312$	0	0
$313$	32.8563	1.85715	0.928574	−	0.371146i	$-0.121035\pi$
$313$	32.8563	1.85715	0.928574	+	0.371146i	$0.121035\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−12.5824	−0.706698	−0.353349	−	0.935492i	$-0.614957\pi$
$317$	−12.5824	−0.706698	−0.353349	+	0.935492i	$0.614957\pi$
$318$	0	0
$319$	−15.5435	−0.870268
$320$	0	0
$321$	−0.587758	−0.0328055
$322$	0	0
$323$	9.05854	0.504031
$324$	0	0
$325$	0	0
$326$	0	0
$327$	29.7271	1.64391
$328$	0	0
$329$	0.986499	0.0543874
$330$	0	0
$331$	−1.57677	−0.0866669	−0.0433334	−	0.999061i	$-0.513798\pi$
$331$	−1.57677	−0.0866669	−0.0433334	+	0.999061i	$0.513798\pi$
$332$	0	0
$333$	−2.34762	−0.128649
$334$	0	0
$335$	0	0
$336$	0	0
$337$	15.4718	0.842802	0.421401	−	0.906874i	$-0.361539\pi$
$337$	15.4718	0.842802	0.421401	+	0.906874i	$0.361539\pi$
$338$	0	0
$339$	−4.59489	−0.249560
$340$	0	0
$341$	−24.7765	−1.34172
$342$	0	0
$343$	10.2831	0.555233
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9.00312	−0.483313	−0.241656	−	0.970362i	$-0.577691\pi$
$347$	−9.00312	−0.483313	−0.241656	+	0.970362i	$0.577691\pi$
$348$	0	0
$349$	1.10398	0.0590945	0.0295473	−	0.999563i	$-0.490593\pi$
$349$	1.10398	0.0590945	0.0295473	+	0.999563i	$0.490593\pi$
$350$	0	0
$351$	−32.8701	−1.75448
$352$	0	0
$353$	−1.96566	−0.104621	−0.0523107	−	0.998631i	$-0.516659\pi$
$353$	−1.96566	−0.104621	−0.0523107	+	0.998631i	$0.516659\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	21.6247	1.14450
$358$	0	0
$359$	2.60403	0.137435	0.0687177	−	0.997636i	$-0.478109\pi$
$359$	2.60403	0.137435	0.0687177	+	0.997636i	$0.478109\pi$
$360$	0	0
$361$	−11.9273	−0.627753
$362$	0	0
$363$	−6.13269	−0.321883
$364$	0	0
$365$	0	0
$366$	0	0
$367$	6.98042	0.364375	0.182188	−	0.983264i	$-0.441682\pi$
$367$	6.98042	0.364375	0.182188	+	0.983264i	$0.441682\pi$
$368$	0	0
$369$	−1.21036	−0.0630090
$370$	0	0
$371$	−36.7363	−1.90726
$372$	0	0
$373$	−24.3220	−1.25935	−0.629673	−	0.776860i	$-0.716811\pi$
$373$	−24.3220	−1.25935	−0.629673	+	0.776860i	$0.716811\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	34.5430	1.77906
$378$	0	0
$379$	24.6541	1.26640	0.633198	−	0.773990i	$-0.281742\pi$
$379$	24.6541	1.26640	0.633198	+	0.773990i	$0.281742\pi$
$380$	0	0
$381$	9.37489	0.480290
$382$	0	0
$383$	−4.99292	−0.255126	−0.127563	−	0.991830i	$-0.540716\pi$
$383$	−4.99292	−0.255126	−0.127563	+	0.991830i	$0.540716\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.31891	−0.0670439
$388$	0	0
$389$	−32.0234	−1.62365	−0.811826	−	0.583900i	$-0.801526\pi$
$389$	−32.0234	−1.62365	−0.811826	+	0.583900i	$0.801526\pi$
$390$	0	0
$391$	−3.40617	−0.172257
$392$	0	0
$393$	31.1696	1.57230
$394$	0	0
$395$	0	0
$396$	0	0
$397$	17.3441	0.870476	0.435238	−	0.900315i	$-0.356664\pi$
$397$	17.3441	0.870476	0.435238	+	0.900315i	$0.356664\pi$
$398$	0	0
$399$	16.8840	0.845259
$400$	0	0
$401$	−21.3352	−1.06543	−0.532714	−	0.846295i	$-0.678828\pi$
$401$	−21.3352	−1.06543	−0.532714	+	0.846295i	$0.678828\pi$
$402$	0	0
$403$	55.0621	2.74284
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−11.1181	−0.551103
$408$	0	0
$409$	−12.5328	−0.619709	−0.309855	−	0.950784i	$-0.600280\pi$
$409$	−12.5328	−0.619709	−0.309855	+	0.950784i	$0.600280\pi$
$410$	0	0
$411$	18.3653	0.905894
$412$	0	0
$413$	−58.6031	−2.88367
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−19.3462	−0.947389
$418$	0	0
$419$	−12.0288	−0.587644	−0.293822	−	0.955860i	$-0.594927\pi$
$419$	−12.0288	−0.587644	−0.293822	+	0.955860i	$0.594927\pi$
$420$	0	0
$421$	−38.3721	−1.87014	−0.935071	−	0.354462i	$-0.884664\pi$
$421$	−38.3721	−1.87014	−0.935071	+	0.354462i	$0.884664\pi$
$422$	0	0
$423$	−0.136257	−0.00662507
$424$	0	0
$425$	0	0
$426$	0	0
$427$	10.0163	0.484721
$428$	0	0
$429$	−24.5445	−1.18502
$430$	0	0
$431$	29.4789	1.41995	0.709975	−	0.704227i	$-0.248706\pi$
$431$	29.4789	1.41995	0.709975	+	0.704227i	$0.248706\pi$
$432$	0	0
$433$	12.0531	0.579236	0.289618	−	0.957142i	$-0.406472\pi$
$433$	12.0531	0.579236	0.289618	+	0.957142i	$0.406472\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.65945	−0.127219
$438$	0	0
$439$	−16.1656	−0.771541	−0.385771	−	0.922595i	$-0.626064\pi$
$439$	−16.1656	−0.771541	−0.385771	+	0.922595i	$0.626064\pi$
$440$	0	0
$441$	−5.35119	−0.254819
$442$	0	0
$443$	19.3729	0.920433	0.460217	−	0.887807i	$-0.347772\pi$
$443$	19.3729	0.920433	0.460217	+	0.887807i	$0.347772\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−3.32805	−0.157411
$448$	0	0
$449$	21.4728	1.01337	0.506683	−	0.862132i	$-0.330871\pi$
$449$	21.4728	1.01337	0.506683	+	0.862132i	$0.330871\pi$
$450$	0	0
$451$	−5.73215	−0.269916
$452$	0	0
$453$	1.22914	0.0577502
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.95602	0.278611	0.139305	−	0.990249i	$-0.455513\pi$
$457$	5.95602	0.278611	0.139305	+	0.990249i	$0.455513\pi$
$458$	0	0
$459$	−18.9436	−0.884210
$460$	0	0
$461$	34.6918	1.61576	0.807879	−	0.589348i	$-0.200615\pi$
$461$	34.6918	1.61576	0.807879	+	0.589348i	$0.200615\pi$
$462$	0	0
$463$	−30.6399	−1.42396	−0.711979	−	0.702200i	$-0.752201\pi$
$463$	−30.6399	−1.42396	−0.711979	+	0.702200i	$0.752201\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8.89547	−0.411633	−0.205817	−	0.978591i	$-0.565985\pi$
$467$	−8.89547	−0.411633	−0.205817	+	0.978591i	$0.565985\pi$
$468$	0	0
$469$	−31.5283	−1.45584
$470$	0	0
$471$	14.3487	0.661152
$472$	0	0
$473$	−6.24621	−0.287201
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.07411	0.232327
$478$	0	0
$479$	9.13224	0.417263	0.208631	−	0.977994i	$-0.433099\pi$
$479$	9.13224	0.417263	0.208631	+	0.977994i	$0.433099\pi$
$480$	0	0
$481$	24.7083	1.12660
$482$	0	0
$483$	−6.34868	−0.288875
$484$	0	0
$485$	0	0
$486$	0	0
$487$	14.2085	0.643849	0.321924	−	0.946765i	$-0.395670\pi$
$487$	14.2085	0.643849	0.321924	+	0.946765i	$0.395670\pi$
$488$	0	0
$489$	−27.6282	−1.24939
$490$	0	0
$491$	23.1760	1.04592	0.522960	−	0.852357i	$-0.324828\pi$
$491$	23.1760	1.04592	0.522960	+	0.852357i	$0.324828\pi$
$492$	0	0
$493$	19.9077	0.896599
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−50.5743	−2.26857
$498$	0	0
$499$	2.19831	0.0984099	0.0492050	−	0.998789i	$-0.484331\pi$
$499$	2.19831	0.0984099	0.0492050	+	0.998789i	$0.484331\pi$
$500$	0	0
$501$	22.2462	0.993887
$502$	0	0
$503$	−44.0461	−1.96392	−0.981959	−	0.189092i	$-0.939445\pi$
$503$	−44.0461	−1.96392	−0.981959	+	0.189092i	$0.939445\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	34.2462	1.52093
$508$	0	0
$509$	−9.09254	−0.403020	−0.201510	−	0.979486i	$-0.564585\pi$
$509$	−9.09254	−0.403020	−0.201510	+	0.979486i	$0.564585\pi$
$510$	0	0
$511$	8.49619	0.375850
$512$	0	0
$513$	−14.7907	−0.653025
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−0.645301	−0.0283803
$518$	0	0
$519$	1.88001	0.0825235
$520$	0	0
$521$	−17.1231	−0.750177	−0.375088	−	0.926989i	$-0.622388\pi$
$521$	−17.1231	−0.750177	−0.375088	+	0.926989i	$0.622388\pi$
$522$	0	0
$523$	−26.6112	−1.16362	−0.581812	−	0.813323i	$-0.697656\pi$
$523$	−26.6112	−1.16362	−0.581812	+	0.813323i	$0.697656\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	31.7332	1.38232
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	8.09439	0.351267
$532$	0	0
$533$	12.7388	0.551780
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−11.5944	−0.500337
$538$	0	0
$539$	−25.3427	−1.09159
$540$	0	0
$541$	31.5941	1.35834	0.679168	−	0.733983i	$-0.262341\pi$
$541$	31.5941	1.35834	0.679168	+	0.733983i	$0.262341\pi$
$542$	0	0
$543$	37.5476	1.61132
$544$	0	0
$545$	0	0
$546$	0	0
$547$	22.7009	0.970622	0.485311	−	0.874342i	$-0.338706\pi$
$547$	22.7009	0.970622	0.485311	+	0.874342i	$0.338706\pi$
$548$	0	0
$549$	−1.38347	−0.0590451
$550$	0	0
$551$	15.5435	0.662175
$552$	0	0
$553$	16.8840	0.717982
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−35.1292	−1.48847	−0.744236	−	0.667917i	$-0.767186\pi$
$557$	−35.1292	−1.48847	−0.744236	+	0.667917i	$0.767186\pi$
$558$	0	0
$559$	13.8813	0.587115
$560$	0	0
$561$	−14.1454	−0.597219
$562$	0	0
$563$	32.0511	1.35079	0.675397	−	0.737455i	$-0.263972\pi$
$563$	32.0511	1.35079	0.675397	+	0.737455i	$0.263972\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−28.4593	−1.19518
$568$	0	0
$569$	−22.0575	−0.924700	−0.462350	−	0.886697i	$-0.652994\pi$
$569$	−22.0575	−0.924700	−0.462350	+	0.886697i	$0.652994\pi$
$570$	0	0
$571$	−8.49242	−0.355397	−0.177698	−	0.984085i	$-0.556865\pi$
$571$	−8.49242	−0.355397	−0.177698	+	0.984085i	$0.556865\pi$
$572$	0	0
$573$	7.61553	0.318143
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−27.1554	−1.13050	−0.565248	−	0.824921i	$-0.691220\pi$
$577$	−27.1554	−1.13050	−0.565248	+	0.824921i	$0.691220\pi$
$578$	0	0
$579$	−16.0394	−0.666573
$580$	0	0
$581$	−50.9393	−2.11332
$582$	0	0
$583$	24.0304	0.995239
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−30.8322	−1.27258	−0.636290	−	0.771450i	$-0.719532\pi$
$587$	−30.8322	−1.27258	−0.636290	+	0.771450i	$0.719532\pi$
$588$	0	0
$589$	24.7765	1.02090
$590$	0	0
$591$	−6.69214	−0.275278
$592$	0	0
$593$	27.7559	1.13980	0.569899	−	0.821715i	$-0.306982\pi$
$593$	27.7559	1.13980	0.569899	+	0.821715i	$0.306982\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.28000	−0.257023
$598$	0	0
$599$	30.7712	1.25728	0.628638	−	0.777698i	$-0.283613\pi$
$599$	30.7712	1.25728	0.628638	+	0.777698i	$0.283613\pi$
$600$	0	0
$601$	−17.9830	−0.733541	−0.366771	−	0.930311i	$-0.619537\pi$
$601$	−17.9830	−0.733541	−0.366771	+	0.930311i	$0.619537\pi$
$602$	0	0
$603$	4.35476	0.177339
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−18.3220	−0.743668	−0.371834	−	0.928299i	$-0.621271\pi$
$607$	−18.3220	−0.743668	−0.371834	+	0.928299i	$0.621271\pi$
$608$	0	0
$609$	37.1056	1.50359
$610$	0	0
$611$	1.43408	0.0580168
$612$	0	0
$613$	−35.4451	−1.43162	−0.715808	−	0.698297i	$-0.753941\pi$
$613$	−35.4451	−1.43162	−0.715808	+	0.698297i	$0.753941\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−43.1567	−1.73742	−0.868712	−	0.495318i	$-0.835052\pi$
$617$	−43.1567	−1.73742	−0.868712	+	0.495318i	$0.835052\pi$
$618$	0	0
$619$	−5.81133	−0.233577	−0.116789	−	0.993157i	$-0.537260\pi$
$619$	−5.81133	−0.233577	−0.116789	+	0.993157i	$0.537260\pi$
$620$	0	0
$621$	5.56155	0.223177
$622$	0	0
$623$	−38.0413	−1.52409
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−11.0444	−0.441070
$628$	0	0
$629$	14.2398	0.567777
$630$	0	0
$631$	27.4626	1.09327	0.546635	−	0.837371i	$-0.315908\pi$
$631$	27.4626	1.09327	0.546635	+	0.837371i	$0.315908\pi$
$632$	0	0
$633$	−23.9567	−0.952194
$634$	0	0
$635$	0	0
$636$	0	0
$637$	56.3202	2.23149
$638$	0	0
$639$	6.98544	0.276340
$640$	0	0
$641$	37.7938	1.49277	0.746383	−	0.665517i	$-0.231789\pi$
$641$	37.7938	1.49277	0.746383	+	0.665517i	$0.231789\pi$
$642$	0	0
$643$	12.5343	0.494304	0.247152	−	0.968977i	$-0.420505\pi$
$643$	12.5343	0.494304	0.247152	+	0.968977i	$0.420505\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−21.5133	−0.845774	−0.422887	−	0.906183i	$-0.638983\pi$
$647$	−21.5133	−0.845774	−0.422887	+	0.906183i	$0.638983\pi$
$648$	0	0
$649$	38.3342	1.50475
$650$	0	0
$651$	59.1469	2.31815
$652$	0	0
$653$	−19.9011	−0.778790	−0.389395	−	0.921071i	$-0.627316\pi$
$653$	−19.9011	−0.778790	−0.389395	+	0.921071i	$0.627316\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.17351	−0.0457831
$658$	0	0
$659$	4.95773	0.193126	0.0965628	−	0.995327i	$-0.469215\pi$
$659$	4.95773	0.193126	0.0965628	+	0.995327i	$0.469215\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	0	0
$663$	31.4360	1.22087
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.84461	−0.226304
$668$	0	0
$669$	−27.6084	−1.06740
$670$	0	0
$671$	−6.55197	−0.252936
$672$	0	0
$673$	28.5176	1.09927	0.549637	−	0.835404i	$-0.314766\pi$
$673$	28.5176	1.09927	0.549637	+	0.835404i	$0.314766\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−28.6214	−1.10001	−0.550004	−	0.835162i	$-0.685374\pi$
$677$	−28.6214	−1.10001	−0.550004	+	0.835162i	$0.685374\pi$
$678$	0	0
$679$	−14.1150	−0.541683
$680$	0	0
$681$	13.4890	0.516898
$682$	0	0
$683$	11.3983	0.436144	0.218072	−	0.975933i	$-0.430023\pi$
$683$	11.3983	0.436144	0.218072	+	0.975933i	$0.430023\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−5.21644	−0.199020
$688$	0	0
$689$	−53.4040	−2.03453
$690$	0	0
$691$	−45.1898	−1.71910	−0.859550	−	0.511051i	$-0.829256\pi$
$691$	−45.1898	−1.71910	−0.859550	+	0.511051i	$0.829256\pi$
$692$	0	0
$693$	6.07170	0.230645
$694$	0	0
$695$	0	0
$696$	0	0
$697$	7.34160	0.278083
$698$	0	0
$699$	−12.8629	−0.486521
$700$	0	0
$701$	40.4871	1.52918	0.764588	−	0.644520i	$-0.222943\pi$
$701$	40.4871	1.52918	0.764588	+	0.644520i	$0.222943\pi$
$702$	0	0
$703$	11.1181	0.419327
$704$	0	0
$705$	0	0
$706$	0	0
$707$	37.4458	1.40830
$708$	0	0
$709$	−41.6443	−1.56398	−0.781992	−	0.623288i	$-0.785796\pi$
$709$	−41.6443	−1.56398	−0.781992	+	0.623288i	$0.785796\pi$
$710$	0	0
$711$	−2.33206	−0.0874591
$712$	0	0
$713$	−9.31640	−0.348902
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.47424	0.167093
$718$	0	0
$719$	−49.2288	−1.83592	−0.917961	−	0.396670i	$-0.870166\pi$
$719$	−49.2288	−1.83592	−0.917961	+	0.396670i	$0.870166\pi$
$720$	0	0
$721$	19.0977	0.711235
$722$	0	0
$723$	7.30977	0.271853
$724$	0	0
$725$	0	0
$726$	0	0
$727$	27.9581	1.03691	0.518455	−	0.855105i	$-0.326507\pi$
$727$	27.9581	1.03691	0.518455	+	0.855105i	$0.326507\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	18.6655	0.689427	0.344714	−	0.938708i	$-0.387976\pi$
$733$	18.6655	0.689427	0.344714	+	0.938708i	$0.387976\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	20.6237	0.759682
$738$	0	0
$739$	−2.50407	−0.0921136	−0.0460568	−	0.998939i	$-0.514666\pi$
$739$	−2.50407	−0.0921136	−0.0460568	+	0.998939i	$0.514666\pi$
$740$	0	0
$741$	24.5445	0.901664
$742$	0	0
$743$	4.69736	0.172330	0.0861648	−	0.996281i	$-0.472539\pi$
$743$	4.69736	0.172330	0.0861648	+	0.996281i	$0.472539\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	7.03585	0.257428
$748$	0	0
$749$	−1.53027	−0.0559150
$750$	0	0
$751$	16.0720	0.586477	0.293239	−	0.956039i	$-0.405267\pi$
$751$	16.0720	0.586477	0.293239	+	0.956039i	$0.405267\pi$
$752$	0	0
$753$	29.0174	1.05745
$754$	0	0
$755$	0	0
$756$	0	0
$757$	25.8357	0.939014	0.469507	−	0.882929i	$-0.344432\pi$
$757$	25.8357	0.939014	0.469507	+	0.882929i	$0.344432\pi$
$758$	0	0
$759$	4.15288	0.150740
$760$	0	0
$761$	15.6005	0.565518	0.282759	−	0.959191i	$-0.408750\pi$
$761$	15.6005	0.565518	0.282759	+	0.959191i	$0.408750\pi$
$762$	0	0
$763$	77.3968	2.80195
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−85.1919	−3.07610
$768$	0	0
$769$	−37.7722	−1.36210	−0.681050	−	0.732237i	$-0.738476\pi$
$769$	−37.7722	−1.36210	−0.681050	+	0.732237i	$0.738476\pi$
$770$	0	0
$771$	44.8518	1.61530
$772$	0	0
$773$	42.9078	1.54329	0.771643	−	0.636056i	$-0.219435\pi$
$773$	42.9078	1.54329	0.771643	+	0.636056i	$0.219435\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	26.5412	0.952162
$778$	0	0
$779$	5.73215	0.205376
$780$	0	0
$781$	33.0823	1.18378
$782$	0	0
$783$	−32.5051	−1.16164
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−33.5324	−1.19530	−0.597650	−	0.801757i	$-0.703899\pi$
$787$	−33.5324	−1.19530	−0.597650	+	0.801757i	$0.703899\pi$
$788$	0	0
$789$	−14.3741	−0.511731
$790$	0	0
$791$	−11.9632	−0.425361
$792$	0	0
$793$	14.5608	0.517068
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−32.7488	−1.16002	−0.580012	−	0.814608i	$-0.696952\pi$
$797$	−32.7488	−1.16002	−0.580012	+	0.814608i	$0.696952\pi$
$798$	0	0
$799$	0.826486	0.0292390
$800$	0	0
$801$	5.25435	0.185653
$802$	0	0
$803$	−5.55764	−0.196125
$804$	0	0
$805$	0	0
$806$	0	0
$807$	26.4206	0.930049
$808$	0	0
$809$	28.8513	1.01436	0.507179	−	0.861841i	$-0.330688\pi$
$809$	28.8513	1.01436	0.507179	+	0.861841i	$0.330688\pi$
$810$	0	0
$811$	−36.1791	−1.27042	−0.635211	−	0.772339i	$-0.719087\pi$
$811$	−36.1791	−1.27042	−0.635211	+	0.772339i	$0.719087\pi$
$812$	0	0
$813$	−1.79706	−0.0630257
$814$	0	0
$815$	0	0
$816$	0	0
$817$	6.24621	0.218527
$818$	0	0
$819$	−13.4934	−0.471498
$820$	0	0
$821$	5.73752	0.200241	0.100120	−	0.994975i	$-0.468077\pi$
$821$	5.73752	0.200241	0.100120	+	0.994975i	$0.468077\pi$
$822$	0	0
$823$	−6.87333	−0.239589	−0.119795	−	0.992799i	$-0.538224\pi$
$823$	−6.87333	−0.239589	−0.119795	+	0.992799i	$0.538224\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−19.7332	−0.686191	−0.343095	−	0.939301i	$-0.611475\pi$
$827$	−19.7332	−0.686191	−0.343095	+	0.939301i	$0.611475\pi$
$828$	0	0
$829$	−2.88833	−0.100316	−0.0501580	−	0.998741i	$-0.515972\pi$
$829$	−2.88833	−0.100316	−0.0501580	+	0.998741i	$0.515972\pi$
$830$	0	0
$831$	−4.02108	−0.139490
$832$	0	0
$833$	32.4583	1.12461
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−51.8137	−1.79094
$838$	0	0
$839$	−24.1904	−0.835147	−0.417573	−	0.908643i	$-0.637119\pi$
$839$	−24.1904	−0.835147	−0.417573	+	0.908643i	$0.637119\pi$
$840$	0	0
$841$	5.15951	0.177914
$842$	0	0
$843$	42.8986	1.47751
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−15.9669	−0.548630
$848$	0	0
$849$	−3.69135	−0.126687
$850$	0	0
$851$	−4.18059	−0.143309
$852$	0	0
$853$	−27.5381	−0.942887	−0.471444	−	0.881896i	$-0.656267\pi$
$853$	−27.5381	−0.942887	−0.471444	+	0.881896i	$0.656267\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	17.3995	0.594357	0.297178	−	0.954822i	$-0.403954\pi$
$857$	17.3995	0.594357	0.297178	+	0.954822i	$0.403954\pi$
$858$	0	0
$859$	−1.24560	−0.0424993	−0.0212497	−	0.999774i	$-0.506764\pi$
$859$	−1.24560	−0.0424993	−0.0212497	+	0.999774i	$0.506764\pi$
$860$	0	0
$861$	13.6839	0.466345
$862$	0	0
$863$	−4.60383	−0.156716	−0.0783580	−	0.996925i	$-0.524968\pi$
$863$	−4.60383	−0.156716	−0.0783580	+	0.996925i	$0.524968\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−8.42931	−0.286274
$868$	0	0
$869$	−11.0444	−0.374655
$870$	0	0
$871$	−45.8330	−1.55299
$872$	0	0
$873$	1.94959	0.0659837
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−33.8276	−1.14228	−0.571138	−	0.820854i	$-0.693498\pi$
$877$	−33.8276	−1.14228	−0.571138	+	0.820854i	$0.693498\pi$
$878$	0	0
$879$	−53.2799	−1.79709
$880$	0	0
$881$	26.1134	0.879782	0.439891	−	0.898051i	$-0.355017\pi$
$881$	26.1134	0.879782	0.439891	+	0.898051i	$0.355017\pi$
$882$	0	0
$883$	−54.5412	−1.83546	−0.917729	−	0.397206i	$-0.869980\pi$
$883$	−54.5412	−1.83546	−0.917729	+	0.397206i	$0.869980\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	22.6201	0.759509	0.379754	−	0.925087i	$-0.376009\pi$
$887$	22.6201	0.759509	0.379754	+	0.925087i	$0.376009\pi$
$888$	0	0
$889$	24.4082	0.818626
$890$	0	0
$891$	18.6162	0.623666
$892$	0	0
$893$	0.645301	0.0215942
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−9.22914	−0.308152
$898$	0	0
$899$	54.4508	1.81603
$900$	0	0
$901$	−30.7776	−1.02535
$902$	0	0
$903$	14.9110	0.496208
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−3.62149	−0.120250	−0.0601248	−	0.998191i	$-0.519150\pi$
$907$	−3.62149	−0.120250	−0.0601248	+	0.998191i	$0.519150\pi$
$908$	0	0
$909$	−5.17211	−0.171548
$910$	0	0
$911$	−26.9241	−0.892034	−0.446017	−	0.895025i	$-0.647158\pi$
$911$	−26.9241	−0.892034	−0.446017	+	0.895025i	$0.647158\pi$
$912$	0	0
$913$	33.3210	1.10277
$914$	0	0
$915$	0	0
$916$	0	0
$917$	81.1524	2.67989
$918$	0	0
$919$	19.5059	0.643441	0.321721	−	0.946835i	$-0.395739\pi$
$919$	19.5059	0.643441	0.321721	+	0.946835i	$0.395739\pi$
$920$	0	0
$921$	2.33206	0.0768440
$922$	0	0
$923$	−73.5204	−2.41995
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−2.63782	−0.0866373
$928$	0	0
$929$	27.2139	0.892860	0.446430	−	0.894819i	$-0.352695\pi$
$929$	27.2139	0.892860	0.446430	+	0.894819i	$0.352695\pi$
$930$	0	0
$931$	25.3427	0.830572
$932$	0	0
$933$	22.0273	0.721142
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−3.85626	−0.125978	−0.0629892	−	0.998014i	$-0.520063\pi$
$937$	−3.85626	−0.125978	−0.0629892	+	0.998014i	$0.520063\pi$
$938$	0	0
$939$	51.3069	1.67434
$940$	0	0
$941$	−60.6471	−1.97704	−0.988519	−	0.151097i	$-0.951720\pi$
$941$	−60.6471	−1.97704	−0.988519	+	0.151097i	$0.951720\pi$
$942$	0	0
$943$	−2.15539	−0.0701890
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.11954	−0.0363801	−0.0181901	−	0.999835i	$-0.505790\pi$
$947$	−1.11954	−0.0363801	−0.0181901	+	0.999835i	$0.505790\pi$
$948$	0	0
$949$	12.3510	0.400931
$950$	0	0
$951$	−19.6481	−0.637132
$952$	0	0
$953$	−14.8590	−0.481331	−0.240666	−	0.970608i	$-0.577366\pi$
$953$	−14.8590	−0.481331	−0.240666	+	0.970608i	$0.577366\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−24.2720	−0.784601
$958$	0	0
$959$	47.8155	1.54404
$960$	0	0
$961$	55.7953	1.79985
$962$	0	0
$963$	0.211365	0.00681114
$964$	0	0
$965$	0	0
$966$	0	0
$967$	38.5718	1.24039	0.620193	−	0.784449i	$-0.287054\pi$
$967$	38.5718	1.24039	0.620193	+	0.784449i	$0.287054\pi$
$968$	0	0
$969$	14.1454	0.454416
$970$	0	0
$971$	25.4984	0.818284	0.409142	−	0.912471i	$-0.365828\pi$
$971$	25.4984	0.818284	0.409142	+	0.912471i	$0.365828\pi$
$972$	0	0
$973$	−50.3694	−1.61477
$974$	0	0
$975$	0	0
$976$	0	0
$977$	15.8952	0.508533	0.254267	−	0.967134i	$-0.418166\pi$
$977$	15.8952	0.508533	0.254267	+	0.967134i	$0.418166\pi$
$978$	0	0
$979$	24.8840	0.795297
$980$	0	0
$981$	−10.6902	−0.341313
$982$	0	0
$983$	−2.50587	−0.0799247	−0.0399624	−	0.999201i	$-0.512724\pi$
$983$	−2.50587	−0.0799247	−0.0399624	+	0.999201i	$0.512724\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.54047	0.0490337
$988$	0	0
$989$	−2.34868	−0.0746837
$990$	0	0
$991$	−12.5272	−0.397938	−0.198969	−	0.980006i	$-0.563759\pi$
$991$	−12.5272	−0.397938	−0.198969	+	0.980006i	$0.563759\pi$
$992$	0	0
$993$	−2.46220	−0.0781356
$994$	0	0
$995$	0	0
$996$	0	0
$997$	9.34803	0.296055	0.148028	−	0.988983i	$-0.452708\pi$
$997$	9.34803	0.296055	0.148028	+	0.988983i	$0.452708\pi$
$998$	0	0
$999$	−23.2506	−0.735616

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.cl.1.4		4
4.3	odd	2		575.2.a.h.1.3		4
5.4	even	2		1840.2.a.u.1.1		4
12.11	even	2		5175.2.a.bx.1.2		4
20.3	even	4		575.2.b.e.24.4		8
20.7	even	4		575.2.b.e.24.5		8
20.19	odd	2		115.2.a.c.1.2	✓	4
40.19	odd	2		7360.2.a.cj.1.2		4
40.29	even	2		7360.2.a.cg.1.3		4
60.59	even	2		1035.2.a.o.1.3		4
140.139	even	2		5635.2.a.v.1.2		4
460.459	even	2		2645.2.a.m.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.a.c.1.2	✓	4	20.19	odd	2
575.2.a.h.1.3		4	4.3	odd	2
575.2.b.e.24.4		8	20.3	even	4
575.2.b.e.24.5		8	20.7	even	4
1035.2.a.o.1.3		4	60.59	even	2
1840.2.a.u.1.1		4	5.4	even	2
2645.2.a.m.1.2		4	460.459	even	2
5175.2.a.bx.1.2		4	12.11	even	2
5635.2.a.v.1.2		4	140.139	even	2
7360.2.a.cg.1.3		4	40.29	even	2
7360.2.a.cj.1.2		4	40.19	odd	2
9200.2.a.cl.1.4		4	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:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56155 q^{3} +4.06562 q^{7} -0.561553 q^{9} +O(q^{10})\)	\(q+1.56155 q^{3} +4.06562 q^{7} -0.561553 q^{9} -2.65945 q^{11} +5.91023 q^{13} +3.40617 q^{17} +2.65945 q^{19} +6.34868 q^{21} -1.00000 q^{23} -5.56155 q^{27} +5.84461 q^{29} +9.31640 q^{31} -4.15288 q^{33} +4.18059 q^{37} +9.22914 q^{39} +2.15539 q^{41} +2.34868 q^{43} +0.242644 q^{47} +9.52927 q^{49} +5.31891 q^{51} -9.03585 q^{53} +4.15288 q^{57} -14.4143 q^{59} +2.46365 q^{61} -2.28306 q^{63} -7.75485 q^{67} -1.56155 q^{69} -12.4395 q^{71} +2.08977 q^{73} -10.8123 q^{77} +4.15288 q^{79} -7.00000 q^{81} -12.5293 q^{83} +9.12667 q^{87} -9.35682 q^{89} +24.0288 q^{91} +14.5481 q^{93} -3.47179 q^{97} +1.49342 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 3 q^{7} + 6 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 - 3 * q^7 + 6 * q^9	\( 4 q - 2 q^{3} - 3 q^{7} + 6 q^{9} - 4 q^{11} + q^{17} + 4 q^{19} + 10 q^{21} - 4 q^{23} - 14 q^{27} + 19 q^{29} + q^{31} + 2 q^{33} + 3 q^{37} + 13 q^{41} - 6 q^{43} + 6 q^{47} + 9 q^{49} + 8 q^{51} - 19 q^{53} - 2 q^{57} - 23 q^{59} - 13 q^{63} - 3 q^{67} + 2 q^{69} + 3 q^{71} + 32 q^{73} - 18 q^{77} - 2 q^{79} - 28 q^{81} - 21 q^{83} - 18 q^{87} + 40 q^{91} + 8 q^{93} + 18 q^{97} - 6 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 3 * q^7 + 6 * q^9 - 4 * q^11 + q^17 + 4 * q^19 + 10 * q^21 - 4 * q^23 - 14 * q^27 + 19 * q^29 + q^31 + 2 * q^33 + 3 * q^37 + 13 * q^41 - 6 * q^43 + 6 * q^47 + 9 * q^49 + 8 * q^51 - 19 * q^53 - 2 * q^57 - 23 * q^59 - 13 * q^63 - 3 * q^67 + 2 * q^69 + 3 * q^71 + 32 * q^73 - 18 * q^77 - 2 * q^79 - 28 * q^81 - 21 * q^83 - 18 * q^87 + 40 * q^91 + 8 * q^93 + 18 * q^97 - 6 * q^99

Introduction

L-functions

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