LMFDB - Embedded newform 3420.3.o.a.721.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3420,3,Mod(721,3420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3420, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("3420.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3420.o (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$93.1882504112$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 62x^{10} + 1445x^{8} + 15924x^{6} + 83244x^{4} + 170640x^{2} + 55600 $ x^12 + 62x^10 + 1445x^8 + 15924x^6 + 83244x^4 + 170640*x^2 + 55600
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{9}\cdot 5 $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.3
Root			$-4.83157i$ of defining polynomial
Character	$\chi$	$=$	3420.721
Dual form			3420.3.o.a.721.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607 q^{5} -3.72856 q^{7} +O(q^{10})$	$q-2.23607 q^{5} -3.72856 q^{7} +13.8938 q^{11} -4.10699i q^{13} +15.3087 q^{17} +(-16.4588 + 9.49250i) q^{19} -17.4452 q^{23} +5.00000 q^{25} -16.3924i q^{29} +15.9121i q^{31} +8.33731 q^{35} -5.14304i q^{37} +42.2785i q^{41} +1.56664 q^{43} +54.5821 q^{47} -35.0979 q^{49} +55.9822i q^{53} -31.0676 q^{55} -101.822i q^{59} +80.5719 q^{61} +9.18350i q^{65} -24.8838i q^{67} -121.245i q^{71} -33.9798 q^{73} -51.8040 q^{77} +102.623i q^{79} -80.3324 q^{83} -34.2312 q^{85} +12.0884i q^{89} +15.3131i q^{91} +(36.8030 - 21.2259i) q^{95} +161.837i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{7}+O(q^{10}) $ 12 * q - 12 * q^7	$ 12 q - 12 q^{7} + 32 q^{11} + 12 q^{17} + 24 q^{19} - 4 q^{23} + 60 q^{25} - 40 q^{35} - 176 q^{43} + 72 q^{47} - 24 q^{49} + 152 q^{61} - 148 q^{73} - 376 q^{77} + 208 q^{83}+O(q^{100}) $ 12 * q - 12 * q^7 + 32 * q^11 + 12 * q^17 + 24 * q^19 - 4 * q^23 + 60 * q^25 - 40 * q^35 - 176 * q^43 + 72 * q^47 - 24 * q^49 + 152 * q^61 - 148 * q^73 - 376 * q^77 + 208 * q^83

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times$.

$n$	$1711$	$1901$	$2737$	$3061$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.23607			−0.447214
$5$	−2.23607			−0.447214
$6$		0			0
$7$	−3.72856			−0.532651			−0.266326	−	0.963883i	$-0.585810\pi$
$7$	−3.72856			−0.532651			−0.266326	+	0.963883i	$0.585810\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	13.8938			1.26308			0.631539	−	0.775344i	$-0.282424\pi$
$11$	13.8938			1.26308			0.631539	+	0.775344i	$0.282424\pi$
$12$		0			0
$13$		−	4.10699i		−	0.315922i	−0.987445	−	0.157961i	$-0.949508\pi$
$13$		−	4.10699i		−	0.315922i	0.987445	−	0.157961i	$-0.0504920\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	15.3087			0.900510			0.450255	−	0.892900i	$-0.351333\pi$
$17$	15.3087			0.900510			0.450255	+	0.892900i	$0.351333\pi$
$18$		0			0
$19$	−16.4588	+	9.49250i	−0.866253	+	0.499605i
$19$	−16.4588	+	9.49250i	−0.866253	+	0.499605i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−17.4452			−0.758486			−0.379243	−	0.925297i	$-0.623816\pi$
$23$	−17.4452			−0.758486			−0.379243	+	0.925297i	$0.623816\pi$
$24$		0			0
$25$	5.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	16.3924i		−	0.565254i	−0.959230	−	0.282627i	$-0.908794\pi$
$29$		−	16.3924i		−	0.565254i	0.959230	−	0.282627i	$-0.0912059\pi$
$30$		0			0
$31$			15.9121i			0.513292i	0.966505	+	0.256646i	$0.0826175\pi$
$31$			15.9121i			0.513292i	−0.966505	+	0.256646i	$0.917383\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	8.33731			0.238209
$36$		0			0
$37$		−	5.14304i		−	0.139001i	−0.997582	−	0.0695005i	$-0.977859\pi$
$37$		−	5.14304i		−	0.139001i	0.997582	−	0.0695005i	$-0.0221405\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			42.2785i			1.03118i	0.856835	+	0.515591i	$0.172428\pi$
$41$			42.2785i			1.03118i	−0.856835	+	0.515591i	$0.827572\pi$
$42$		0			0
$43$	1.56664			0.0364334			0.0182167	−	0.999834i	$-0.494201\pi$
$43$	1.56664			0.0364334			0.0182167	+	0.999834i	$0.494201\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	54.5821			1.16132			0.580660	−	0.814146i	$-0.302794\pi$
$47$	54.5821			1.16132			0.580660	+	0.814146i	$0.302794\pi$
$48$		0			0
$49$	−35.0979			−0.716283
$50$		0			0
$51$		0			0
$52$		0			0
$53$			55.9822i			1.05627i	0.849161	+	0.528134i	$0.177108\pi$
$53$			55.9822i			1.05627i	−0.849161	+	0.528134i	$0.822892\pi$
$54$		0			0
$55$	−31.0676			−0.564865
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	101.822i		−	1.72580i	−0.505376	−	0.862899i	$-0.668646\pi$
$59$		−	101.822i		−	1.72580i	0.505376	−	0.862899i	$-0.331354\pi$
$60$		0			0
$61$	80.5719			1.32085			0.660426	−	0.750892i	$-0.270376\pi$
$61$	80.5719			1.32085			0.660426	+	0.750892i	$0.270376\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			9.18350i			0.141285i
$66$		0			0
$67$		−	24.8838i		−	0.371400i	−0.982606	−	0.185700i	$-0.940545\pi$
$67$		−	24.8838i		−	0.371400i	0.982606	−	0.185700i	$-0.0594552\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	121.245i		−	1.70767i	−0.520545	−	0.853835i	$-0.674271\pi$
$71$		−	121.245i		−	1.70767i	0.520545	−	0.853835i	$-0.325729\pi$
$72$		0			0
$73$	−33.9798			−0.465476			−0.232738	−	0.972539i	$-0.574769\pi$
$73$	−33.9798			−0.465476			−0.232738	+	0.972539i	$0.574769\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−51.8040			−0.672779
$78$		0			0
$79$			102.623i			1.29902i	0.760351	+	0.649512i	$0.225027\pi$
$79$			102.623i			1.29902i	−0.760351	+	0.649512i	$0.774973\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−80.3324			−0.967860			−0.483930	−	0.875107i	$-0.660791\pi$
$83$	−80.3324			−0.967860			−0.483930	+	0.875107i	$0.660791\pi$
$84$		0			0
$85$	−34.2312			−0.402720
$86$		0			0
$87$		0			0
$88$		0			0
$89$			12.0884i			0.135824i	0.997691	+	0.0679121i	$0.0216338\pi$
$89$			12.0884i			0.135824i	−0.997691	+	0.0679121i	$0.978366\pi$
$90$		0			0
$91$			15.3131i			0.168276i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	36.8030	−	21.2259i	0.387400	−	0.223430i
$96$		0			0
$97$			161.837i			1.66842i	0.551447	+	0.834210i	$0.314076\pi$
$97$			161.837i			1.66842i	−0.551447	+	0.834210i	$0.685924\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.07921			0.0601901			0.0300951	−	0.999547i	$-0.490419\pi$
$101$	6.07921			0.0601901			0.0300951	+	0.999547i	$0.490419\pi$
$102$		0			0
$103$		−	27.8929i		−	0.270805i	−0.990791	−	0.135402i	$-0.956767\pi$
$103$		−	27.8929i		−	0.270805i	0.990791	−	0.135402i	$-0.0432327\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			61.4652i			0.574441i	0.957865	+	0.287220i	$0.0927312\pi$
$107$			61.4652i			0.574441i	−0.957865	+	0.287220i	$0.907269\pi$
$108$		0			0
$109$		−	30.2310i		−	0.277348i	−0.990338	−	0.138674i	$-0.955716\pi$
$109$		−	30.2310i		−	0.277348i	0.990338	−	0.138674i	$-0.0442841\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	178.939i		−	1.58353i	−0.610826	−	0.791765i	$-0.709163\pi$
$113$		−	178.939i		−	1.58353i	0.610826	−	0.791765i	$-0.290837\pi$
$114$		0			0
$115$	39.0086			0.339205
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−57.0793			−0.479658
$120$		0			0
$121$	72.0390			0.595364
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.1803			−0.0894427
$126$		0			0
$127$		−	53.4071i		−	0.420528i	−0.977645	−	0.210264i	$-0.932568\pi$
$127$		−	53.4071i		−	0.420528i	0.977645	−	0.210264i	$-0.0674324\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	109.035			0.832327			0.416163	−	0.909290i	$-0.363374\pi$
$131$	109.035			0.832327			0.416163	+	0.909290i	$0.363374\pi$
$132$		0			0
$133$	61.3676	−	35.3933i	0.461411	−	0.266115i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	24.9987			0.182472			0.0912362	−	0.995829i	$-0.470918\pi$
$137$	24.9987			0.182472			0.0912362	+	0.995829i	$0.470918\pi$
$138$		0			0
$139$	115.083			0.827936			0.413968	−	0.910291i	$-0.364143\pi$
$139$	115.083			0.827936			0.413968	+	0.910291i	$0.364143\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	57.0619i		−	0.399034i
$144$		0			0
$145$			36.6545i			0.252789i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	141.364			0.948750			0.474375	−	0.880323i	$-0.342674\pi$
$149$	141.364			0.948750			0.474375	+	0.880323i	$0.342674\pi$
$150$		0			0
$151$			250.107i			1.65634i	0.560478	+	0.828169i	$0.310617\pi$
$151$			250.107i			1.65634i	−0.560478	+	0.828169i	$0.689383\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	35.5804i		−	0.229551i
$156$		0			0
$157$	−14.5947			−0.0929601			−0.0464801	−	0.998919i	$-0.514800\pi$
$157$	−14.5947			−0.0929601			−0.0464801	+	0.998919i	$0.514800\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	65.0454			0.404009
$162$		0			0
$163$	−9.57924			−0.0587684			−0.0293842	−	0.999568i	$-0.509355\pi$
$163$	−9.57924			−0.0587684			−0.0293842	+	0.999568i	$0.509355\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			302.815i			1.81326i	0.421923	+	0.906631i	$0.361355\pi$
$167$			302.815i			1.81326i	−0.421923	+	0.906631i	$0.638645\pi$
$168$		0			0
$169$	152.133			0.900193
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	111.084i		−	0.642102i	−0.947062	−	0.321051i	$-0.895964\pi$
$173$		−	111.084i		−	0.642102i	0.947062	−	0.321051i	$-0.104036\pi$
$174$		0			0
$175$	−18.6428			−0.106530
$176$		0			0
$177$		0			0
$178$		0			0
$179$			247.724i			1.38393i	0.721929	+	0.691967i	$0.243256\pi$
$179$			247.724i			1.38393i	−0.721929	+	0.691967i	$0.756744\pi$
$180$		0			0
$181$			234.560i			1.29591i	0.761678	+	0.647956i	$0.224376\pi$
$181$			234.560i			1.29591i	−0.761678	+	0.647956i	$0.775624\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			11.5002i			0.0621631i
$186$		0			0
$187$	212.696			1.13741
$188$		0			0
$189$		0			0
$190$		0			0
$191$	250.384			1.31091			0.655455	−	0.755235i	$-0.272477\pi$
$191$	250.384			1.31091			0.655455	+	0.755235i	$0.272477\pi$
$192$		0			0
$193$			122.208i			0.633204i	0.948558	+	0.316602i	$0.102542\pi$
$193$			122.208i			0.633204i	−0.948558	+	0.316602i	$0.897458\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	343.177			1.74201			0.871007	−	0.491271i	$-0.163467\pi$
$197$	343.177			1.74201			0.871007	+	0.491271i	$0.163467\pi$
$198$		0			0
$199$	297.583			1.49539			0.747696	−	0.664042i	$-0.231160\pi$
$199$	297.583			1.49539			0.747696	+	0.664042i	$0.231160\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			61.1199i			0.301083i
$204$		0			0
$205$		−	94.5376i		−	0.461159i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−228.676	+	131.887i	−1.09414	+	0.631040i
$210$		0			0
$211$			237.792i			1.12698i	0.826124	+	0.563488i	$0.190541\pi$
$211$			237.792i			1.12698i	−0.826124	+	0.563488i	$0.809459\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.50311			−0.0162935
$216$		0			0
$217$		−	59.3290i		−	0.273406i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	62.8725i		−	0.284491i
$222$		0			0
$223$			152.390i			0.683365i	0.939815	+	0.341682i	$0.110997\pi$
$223$			152.390i			0.683365i	−0.939815	+	0.341682i	$0.889003\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	205.814i		−	0.906670i	−0.891340	−	0.453335i	$-0.850234\pi$
$227$		−	205.814i		−	0.906670i	0.891340	−	0.453335i	$-0.149766\pi$
$228$		0			0
$229$	135.060			0.589783			0.294892	−	0.955531i	$-0.404716\pi$
$229$	135.060			0.589783			0.294892	+	0.955531i	$0.404716\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−100.593			−0.431728			−0.215864	−	0.976423i	$-0.569257\pi$
$233$	−100.593			−0.431728			−0.215864	+	0.976423i	$0.569257\pi$
$234$		0			0
$235$	−122.049			−0.519358
$236$		0			0
$237$		0			0
$238$		0			0
$239$	328.565			1.37475			0.687373	−	0.726304i	$-0.258764\pi$
$239$	328.565			1.37475			0.687373	+	0.726304i	$0.258764\pi$
$240$		0			0
$241$		−	144.249i		−	0.598543i	−0.954168	−	0.299272i	$-0.903256\pi$
$241$		−	144.249i		−	0.598543i	0.954168	−	0.299272i	$-0.0967437\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	78.4812			0.320331
$246$		0			0
$247$	38.9856	+	67.5961i	0.157836	+	0.273669i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−66.0565			−0.263173			−0.131587	−	0.991305i	$-0.542007\pi$
$251$	−66.0565			−0.263173			−0.131587	+	0.991305i	$0.542007\pi$
$252$		0			0
$253$	−242.381			−0.958027
$254$		0			0
$255$		0			0
$256$		0			0
$257$			268.255i			1.04379i	0.853009	+	0.521896i	$0.174775\pi$
$257$			268.255i			1.04379i	−0.853009	+	0.521896i	$0.825225\pi$
$258$		0			0
$259$			19.1761i			0.0740390i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	241.131			0.916848			0.458424	−	0.888734i	$-0.348414\pi$
$263$	241.131			0.916848			0.458424	+	0.888734i	$0.348414\pi$
$264$		0			0
$265$		−	125.180i		−	0.472378i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			270.274i			1.00473i	0.864654	+	0.502367i	$0.167538\pi$
$269$			270.274i			1.00473i	−0.864654	+	0.502367i	$0.832462\pi$
$270$		0			0
$271$	159.872			0.589934			0.294967	−	0.955507i	$-0.404691\pi$
$271$	159.872			0.589934			0.294967	+	0.955507i	$0.404691\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	69.4692			0.252615
$276$		0			0
$277$	480.862			1.73596			0.867981	−	0.496597i	$-0.165417\pi$
$277$	480.862			1.73596			0.867981	+	0.496597i	$0.165417\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	125.506i		−	0.446639i	−0.974745	−	0.223320i	$-0.928311\pi$
$281$		−	125.506i		−	0.446639i	0.974745	−	0.223320i	$-0.0716894\pi$
$282$		0			0
$283$	−169.824			−0.600086			−0.300043	−	0.953926i	$-0.597001\pi$
$283$	−169.824			−0.600086			−0.300043	+	0.953926i	$0.597001\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	157.638i		−	0.549261i
$288$		0			0
$289$	−54.6445			−0.189081
$290$		0			0
$291$		0			0
$292$		0			0
$293$			103.213i			0.352263i	0.984367	+	0.176131i	$0.0563583\pi$
$293$			103.213i			0.352263i	−0.984367	+	0.176131i	$0.943642\pi$
$294$		0			0
$295$			227.681i			0.771801i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			71.6472i			0.239623i
$300$		0			0
$301$	−5.84130			−0.0194063
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−180.164			−0.590703
$306$		0			0
$307$		−	470.447i		−	1.53240i	−0.642602	−	0.766200i	$-0.722145\pi$
$307$		−	470.447i		−	1.53240i	0.642602	−	0.766200i	$-0.277855\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	104.966			0.337511			0.168756	−	0.985658i	$-0.446025\pi$
$311$	104.966			0.337511			0.168756	+	0.985658i	$0.446025\pi$
$312$		0			0
$313$	358.845			1.14647			0.573234	−	0.819392i	$-0.305689\pi$
$313$	358.845			1.14647			0.573234	+	0.819392i	$0.305689\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			367.285i			1.15863i	0.815104	+	0.579314i	$0.196680\pi$
$317$			367.285i			1.15863i	−0.815104	+	0.579314i	$0.803320\pi$
$318$		0			0
$319$		−	227.753i		−	0.713960i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−251.963	+	145.318i	−0.780070	+	0.449900i
$324$		0			0
$325$		−	20.5349i		−	0.0631844i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−203.512			−0.618579
$330$		0			0
$331$		−	496.428i		−	1.49978i	−0.661561	−	0.749892i	$-0.730106\pi$
$331$		−	496.428i		−	1.49978i	0.661561	−	0.749892i	$-0.269894\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			55.6419i			0.166095i
$336$		0			0
$337$		−	52.5036i		−	0.155797i	−0.996961	−	0.0778986i	$-0.975179\pi$
$337$		−	52.5036i		−	0.155797i	0.996961	−	0.0778986i	$-0.0248210\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			221.080i			0.648328i
$342$		0			0
$343$	313.564			0.914180
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−409.556			−1.18028			−0.590138	−	0.807302i	$-0.700927\pi$
$347$	−409.556			−1.18028			−0.590138	+	0.807302i	$0.700927\pi$
$348$		0			0
$349$	562.063			1.61050			0.805248	−	0.592938i	$-0.202032\pi$
$349$	562.063			1.61050			0.805248	+	0.592938i	$0.202032\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−252.453			−0.715165			−0.357583	−	0.933881i	$-0.616399\pi$
$353$	−252.453			−0.715165			−0.357583	+	0.933881i	$0.616399\pi$
$354$		0			0
$355$			271.111i			0.763693i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	119.344			0.332433			0.166217	−	0.986089i	$-0.446845\pi$
$359$	119.344			0.332433			0.166217	+	0.986089i	$0.446845\pi$
$360$		0			0
$361$	180.785	−	312.470i	0.500789	−	0.865569i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	75.9811			0.208167
$366$		0			0
$367$	−690.635			−1.88184			−0.940920	−	0.338629i	$-0.890037\pi$
$367$	−690.635			−1.88184			−0.940920	+	0.338629i	$0.890037\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	208.733i		−	0.562623i
$372$		0			0
$373$		−	491.052i		−	1.31649i	−0.752803	−	0.658246i	$-0.771299\pi$
$373$		−	491.052i		−	1.31649i	0.752803	−	0.658246i	$-0.228701\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−67.3233			−0.178576
$378$		0			0
$379$			691.775i			1.82526i	0.408782	+	0.912632i	$0.365954\pi$
$379$			691.775i			1.82526i	−0.408782	+	0.912632i	$0.634046\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	252.889i		−	0.660285i	−0.943931	−	0.330142i	$-0.892903\pi$
$383$		−	252.889i		−	0.660285i	0.943931	−	0.330142i	$-0.107097\pi$
$384$		0			0
$385$	115.837			0.300876
$386$		0			0
$387$		0			0
$388$		0			0
$389$	504.764			1.29759			0.648797	−	0.760961i	$-0.275272\pi$
$389$	504.764			1.29759			0.648797	+	0.760961i	$0.275272\pi$
$390$		0			0
$391$	−267.063			−0.683025
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	229.472i		−	0.580941i
$396$		0			0
$397$	313.581			0.789876			0.394938	−	0.918708i	$-0.370766\pi$
$397$	313.581			0.789876			0.394938	+	0.918708i	$0.370766\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			561.838i			1.40109i	0.713607	+	0.700546i	$0.247060\pi$
$401$			561.838i			1.40109i	−0.713607	+	0.700546i	$0.752940\pi$
$402$		0			0
$403$	65.3506			0.162160
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	71.4566i		−	0.175569i
$408$		0			0
$409$			165.613i			0.404921i	0.979290	+	0.202460i	$0.0648937\pi$
$409$			165.613i			0.404921i	−0.979290	+	0.202460i	$0.935106\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			379.650i			0.919249i
$414$		0			0
$415$	179.629			0.432840
$416$		0			0
$417$		0			0
$418$		0			0
$419$	440.399			1.05107			0.525536	−	0.850771i	$-0.323865\pi$
$419$	440.399			1.05107			0.525536	+	0.850771i	$0.323865\pi$
$420$		0			0
$421$			786.831i			1.86896i	0.356018	+	0.934479i	$0.384134\pi$
$421$			786.831i			1.86896i	−0.356018	+	0.934479i	$0.615866\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	76.5434			0.180102
$426$		0			0
$427$	−300.417			−0.703553
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	219.539i		−	0.509372i	−0.967024	−	0.254686i	$-0.918028\pi$
$431$		−	219.539i		−	0.509372i	0.967024	−	0.254686i	$-0.0819721\pi$
$432$		0			0
$433$			450.900i			1.04134i	0.853758	+	0.520670i	$0.174318\pi$
$433$			450.900i			1.04134i	−0.853758	+	0.520670i	$0.825682\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	287.127	−	165.598i	0.657041	−	0.378944i
$438$		0			0
$439$		−	319.069i		−	0.726808i	−0.931632	−	0.363404i	$-0.881615\pi$
$439$		−	319.069i		−	0.726808i	0.931632	−	0.363404i	$-0.118385\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−780.776			−1.76247			−0.881237	−	0.472674i	$-0.843289\pi$
$443$	−780.776			−1.76247			−0.881237	+	0.472674i	$0.843289\pi$
$444$		0			0
$445$		−	27.0304i		−	0.0607425i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	523.244i		−	1.16535i	−0.812704	−	0.582677i	$-0.802005\pi$
$449$		−	523.244i		−	1.16535i	0.812704	−	0.582677i	$-0.197995\pi$
$450$		0			0
$451$			587.411i			1.30246i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	34.2412i		−	0.0752554i
$456$		0			0
$457$	874.643			1.91388			0.956940	−	0.290285i	$-0.0937500\pi$
$457$	874.643			1.91388			0.956940	+	0.290285i	$0.0937500\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−120.368			−0.261102			−0.130551	−	0.991442i	$-0.541675\pi$
$461$	−120.368			−0.261102			−0.130551	+	0.991442i	$0.541675\pi$
$462$		0			0
$463$	256.911			0.554883			0.277441	−	0.960743i	$-0.410514\pi$
$463$	256.911			0.554883			0.277441	+	0.960743i	$0.410514\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−490.911			−1.05120			−0.525601	−	0.850731i	$-0.676160\pi$
$467$	−490.911			−1.05120			−0.525601	+	0.850731i	$0.676160\pi$
$468$		0			0
$469$			92.7807i			0.197827i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	21.7666			0.0460182
$474$		0			0
$475$	−82.2941	+	47.4625i	−0.173251	+	0.0999210i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−545.752			−1.13936			−0.569679	−	0.821868i	$-0.692932\pi$
$479$	−545.752			−1.13936			−0.569679	+	0.821868i	$0.692932\pi$
$480$		0			0
$481$	−21.1224			−0.0439135
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	361.878i		−	0.746140i
$486$		0			0
$487$			243.032i			0.499039i	0.968370	+	0.249519i	$0.0802726\pi$
$487$			243.032i			0.499039i	−0.968370	+	0.249519i	$0.919727\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	555.332			1.13102			0.565511	−	0.824740i	$-0.308679\pi$
$491$	555.332			1.13102			0.565511	+	0.824740i	$0.308679\pi$
$492$		0			0
$493$		−	250.946i		−	0.509017i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			452.067i			0.909592i
$498$		0			0
$499$	−231.235			−0.463397			−0.231698	−	0.972788i	$-0.574428\pi$
$499$	−231.235			−0.463397			−0.231698	+	0.972788i	$0.574428\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	206.048			0.409638			0.204819	−	0.978800i	$-0.434340\pi$
$503$	206.048			0.409638			0.204819	+	0.978800i	$0.434340\pi$
$504$		0			0
$505$	−13.5935			−0.0269179
$506$		0			0
$507$		0			0
$508$		0			0
$509$			398.832i			0.783560i	0.920059	+	0.391780i	$0.128141\pi$
$509$			398.832i			0.783560i	−0.920059	+	0.391780i	$0.871859\pi$
$510$		0			0
$511$	126.696			0.247936
$512$		0			0
$513$		0			0
$514$		0			0
$515$			62.3704i			0.121108i
$516$		0			0
$517$	758.355			1.46684
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	262.617i		−	0.504064i	−0.967719	−	0.252032i	$-0.918901\pi$
$521$		−	262.617i		−	0.504064i	0.967719	−	0.252032i	$-0.0810988\pi$
$522$		0			0
$523$		−	814.031i		−	1.55646i	−0.627977	−	0.778232i	$-0.716117\pi$
$523$		−	814.031i		−	1.55646i	0.627977	−	0.778232i	$-0.283883\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			243.592i			0.462225i
$528$		0			0
$529$	−224.666			−0.424699
$530$		0			0
$531$		0			0
$532$		0			0
$533$	173.637			0.325773
$534$		0			0
$535$		−	137.440i		−	0.256898i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−487.644			−0.904720
$540$		0			0
$541$	−196.753			−0.363684			−0.181842	−	0.983328i	$-0.558206\pi$
$541$	−196.753			−0.363684			−0.181842	+	0.983328i	$0.558206\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			67.5985i			0.124034i
$546$		0			0
$547$		−	759.052i		−	1.38766i	−0.720137	−	0.693832i	$-0.755921\pi$
$547$		−	759.052i		−	1.38766i	0.720137	−	0.693832i	$-0.244079\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	155.605	+	269.799i	0.282404	+	0.489653i
$552$		0			0
$553$		−	382.636i		−	0.691927i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−705.968			−1.26745			−0.633723	−	0.773560i	$-0.718474\pi$
$557$	−705.968			−1.26745			−0.633723	+	0.773560i	$0.718474\pi$
$558$		0			0
$559$		−	6.43416i		−	0.0115101i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			357.744i			0.635424i	0.948187	+	0.317712i	$0.102914\pi$
$563$			357.744i			0.635424i	−0.948187	+	0.317712i	$0.897086\pi$
$564$		0			0
$565$			400.119i			0.708176i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			448.411i			0.788069i	0.919096	+	0.394034i	$0.128921\pi$
$569$			448.411i			0.788069i	−0.919096	+	0.394034i	$0.871079\pi$
$570$		0			0
$571$	203.672			0.356694			0.178347	−	0.983968i	$-0.442925\pi$
$571$	203.672			0.356694			0.178347	+	0.983968i	$0.442925\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−87.2259			−0.151697
$576$		0			0
$577$	232.447			0.402854			0.201427	−	0.979504i	$-0.435442\pi$
$577$	232.447			0.402854			0.201427	+	0.979504i	$0.435442\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	299.524			0.515532
$582$		0			0
$583$			777.809i			1.33415i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1101.80			1.87701			0.938503	−	0.345271i	$-0.112213\pi$
$587$	1101.80			1.87701			0.938503	+	0.345271i	$0.112213\pi$
$588$		0			0
$589$	−151.045	−	261.894i	−0.256443	−	0.444641i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	387.409			0.653303			0.326651	−	0.945145i	$-0.394080\pi$
$593$	387.409			0.653303			0.326651	+	0.945145i	$0.394080\pi$
$594$		0			0
$595$	127.633			0.214509
$596$		0			0
$597$		0			0
$598$		0			0
$599$			649.288i			1.08395i	0.840393	+	0.541977i	$0.182324\pi$
$599$			649.288i			1.08395i	−0.840393	+	0.541977i	$0.817676\pi$
$600$		0			0
$601$			564.191i			0.938754i	0.882998	+	0.469377i	$0.155521\pi$
$601$			564.191i			0.938754i	−0.882998	+	0.469377i	$0.844479\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−161.084			−0.266255
$606$		0			0
$607$		−	431.505i		−	0.710882i	−0.934699	−	0.355441i	$-0.884331\pi$
$607$		−	431.505i		−	0.710882i	0.934699	−	0.355441i	$-0.115669\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	224.168i		−	0.366887i
$612$		0			0
$613$	360.161			0.587539			0.293769	−	0.955876i	$-0.405090\pi$
$613$	360.161			0.587539			0.293769	+	0.955876i	$0.405090\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	255.900			0.414749			0.207375	−	0.978262i	$-0.433508\pi$
$617$	255.900			0.414749			0.207375	+	0.978262i	$0.433508\pi$
$618$		0			0
$619$	−231.220			−0.373538			−0.186769	−	0.982404i	$-0.559802\pi$
$619$	−231.220			−0.373538			−0.186769	+	0.982404i	$0.559802\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	45.0722i		−	0.0723470i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	78.7331i		−	0.125172i
$630$		0			0
$631$	55.6688			0.0882232			0.0441116	−	0.999027i	$-0.485954\pi$
$631$	55.6688			0.0882232			0.0441116	+	0.999027i	$0.485954\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			119.422i			0.188066i
$636$		0			0
$637$			144.146i			0.226290i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	1086.58i		−	1.69513i	−0.530692	−	0.847565i	$-0.678068\pi$
$641$		−	1086.58i		−	1.69513i	0.530692	−	0.847565i	$-0.321932\pi$
$642$		0			0
$643$	132.021			0.205320			0.102660	−	0.994717i	$-0.467265\pi$
$643$	132.021			0.205320			0.102660	+	0.994717i	$0.467265\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−139.001			−0.214840			−0.107420	−	0.994214i	$-0.534259\pi$
$647$	−139.001			−0.214840			−0.107420	+	0.994214i	$0.534259\pi$
$648$		0			0
$649$		−	1414.70i		−	2.17982i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.61433			0.00400356			0.00200178	−	0.999998i	$-0.499363\pi$
$653$	2.61433			0.00400356			0.00200178	+	0.999998i	$0.499363\pi$
$654$		0			0
$655$	−243.809			−0.372228
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	1115.25i		−	1.69234i	−0.532911	−	0.846171i	$-0.678902\pi$
$659$		−	1115.25i		−	1.69234i	0.532911	−	0.846171i	$-0.321098\pi$
$660$		0			0
$661$			1032.56i			1.56212i	0.624453	+	0.781062i	$0.285322\pi$
$661$			1032.56i			1.56212i	−0.624453	+	0.781062i	$0.714678\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−137.222	+	79.1419i	−0.206349	+	0.119010i
$666$		0			0
$667$			285.968i			0.428738i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1119.45			1.66834
$672$		0			0
$673$			923.804i			1.37267i	0.727288	+	0.686333i	$0.240781\pi$
$673$			923.804i			1.37267i	−0.727288	+	0.686333i	$0.759219\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			1144.25i			1.69017i	0.534631	+	0.845086i	$0.320451\pi$
$677$			1144.25i			1.69017i	−0.534631	+	0.845086i	$0.679549\pi$
$678$		0			0
$679$		−	603.417i		−	0.888686i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	686.997i		−	1.00585i	−0.864330	−	0.502926i	$-0.832257\pi$
$683$		−	686.997i		−	1.00585i	0.864330	−	0.502926i	$-0.167743\pi$
$684$		0			0
$685$	−55.8988			−0.0816041
$686$		0			0
$687$		0			0
$688$		0			0
$689$	229.918			0.333699
$690$		0			0
$691$	−1216.77			−1.76088			−0.880442	−	0.474154i	$-0.842754\pi$
$691$	−1216.77			−1.76088			−0.880442	+	0.474154i	$0.842754\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−257.334			−0.370264
$696$		0			0
$697$			647.228i			0.928591i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1082.00			1.54351			0.771753	−	0.635922i	$-0.219380\pi$
$701$	1082.00			1.54351			0.771753	+	0.635922i	$0.219380\pi$
$702$		0			0
$703$	48.8203	+	84.6482i	0.0694456	+	0.120410i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−22.6667			−0.0320604
$708$		0			0
$709$	−639.509			−0.901987			−0.450994	−	0.892527i	$-0.648930\pi$
$709$	−639.509			−0.901987			−0.450994	+	0.892527i	$0.648930\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	277.589i		−	0.389325i
$714$		0			0
$715$			127.594i			0.178453i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	95.9878			0.133502			0.0667509	−	0.997770i	$-0.478737\pi$
$719$	95.9878			0.133502			0.0667509	+	0.997770i	$0.478737\pi$
$720$		0			0
$721$			104.000i			0.144245i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	81.9619i		−	0.113051i
$726$		0			0
$727$	−368.875			−0.507394			−0.253697	−	0.967284i	$-0.581647\pi$
$727$	−368.875			−0.507394			−0.253697	+	0.967284i	$0.581647\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	23.9831			0.0328087
$732$		0			0
$733$	−865.251			−1.18042			−0.590212	−	0.807248i	$-0.700956\pi$
$733$	−865.251			−1.18042			−0.590212	+	0.807248i	$0.700956\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	345.732i		−	0.469107i
$738$		0			0
$739$	−934.331			−1.26432			−0.632159	−	0.774839i	$-0.717831\pi$
$739$	−934.331			−1.26432			−0.632159	+	0.774839i	$0.717831\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	235.551i		−	0.317027i	−0.987357	−	0.158514i	$-0.949330\pi$
$743$		−	235.551i		−	0.317027i	0.987357	−	0.158514i	$-0.0506702\pi$
$744$		0			0
$745$	−316.099			−0.424294
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	229.176i		−	0.305976i
$750$		0			0
$751$			971.042i			1.29300i	0.762915	+	0.646499i	$0.223768\pi$
$751$			971.042i			1.29300i	−0.762915	+	0.646499i	$0.776232\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	559.256i		−	0.740737i
$756$		0			0
$757$	−177.074			−0.233916			−0.116958	−	0.993137i	$-0.537314\pi$
$757$	−177.074			−0.233916			−0.116958	+	0.993137i	$0.537314\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	159.521			0.209621			0.104810	−	0.994492i	$-0.466576\pi$
$761$	159.521			0.209621			0.104810	+	0.994492i	$0.466576\pi$
$762$		0			0
$763$			112.718i			0.147730i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−418.182			−0.545218
$768$		0			0
$769$	707.134			0.919551			0.459775	−	0.888035i	$-0.347930\pi$
$769$	707.134			0.919551			0.459775	+	0.888035i	$0.347930\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	799.572i		−	1.03438i	−0.855872	−	0.517188i	$-0.826979\pi$
$773$		−	799.572i		−	1.03438i	0.855872	−	0.517188i	$-0.173021\pi$
$774$		0			0
$775$			79.5603i			0.102658i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−401.329	−	695.854i	−0.515184	−	0.893265i
$780$		0			0
$781$		−	1684.55i		−	2.15692i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	32.6348			0.0415730
$786$		0			0
$787$			1366.97i			1.73693i	0.495749	+	0.868466i	$0.334893\pi$
$787$			1366.97i			1.73693i	−0.495749	+	0.868466i	$0.665107\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			667.184i			0.843469i
$792$		0			0
$793$		−	330.908i		−	0.417286i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	121.258i		−	0.152143i	−0.997102	−	0.0760717i	$-0.975762\pi$
$797$		−	121.258i		−	0.152143i	0.997102	−	0.0760717i	$-0.0242378\pi$
$798$		0			0
$799$	835.579			1.04578
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−472.110			−0.587933
$804$		0			0
$805$	−145.446			−0.180678
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−820.488			−1.01420			−0.507100	−	0.861887i	$-0.669283\pi$
$809$	−820.488			−1.01420			−0.507100	+	0.861887i	$0.669283\pi$
$810$		0			0
$811$		−	537.722i		−	0.663036i	−0.943449	−	0.331518i	$-0.892439\pi$
$811$		−	537.722i		−	0.663036i	0.943449	−	0.331518i	$-0.107561\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	21.4198			0.0262820
$816$		0			0
$817$	−25.7850	+	14.8713i	−0.0315606	+	0.0182023i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	54.0424			0.0658250			0.0329125	−	0.999458i	$-0.489522\pi$
$821$	54.0424			0.0658250			0.0329125	+	0.999458i	$0.489522\pi$
$822$		0			0
$823$	−599.052			−0.727889			−0.363944	−	0.931421i	$-0.618570\pi$
$823$	−599.052			−0.727889			−0.363944	+	0.931421i	$0.618570\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			620.339i			0.750108i	0.927003	+	0.375054i	$0.122376\pi$
$827$			620.339i			0.750108i	−0.927003	+	0.375054i	$0.877624\pi$
$828$		0			0
$829$		−	36.8107i		−	0.0444038i	−0.999754	−	0.0222019i	$-0.992932\pi$
$829$		−	36.8107i		−	0.0444038i	0.999754	−	0.0222019i	$-0.00706766\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−537.302			−0.645020
$834$		0			0
$835$		−	677.115i		−	0.810916i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			531.549i			0.633551i	0.948501	+	0.316776i	$0.102600\pi$
$839$			531.549i			0.633551i	−0.948501	+	0.316776i	$0.897400\pi$
$840$		0			0
$841$	572.290			0.680488
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−340.179			−0.402579
$846$		0			0
$847$	−268.602			−0.317121
$848$		0			0
$849$		0			0
$850$		0			0
$851$			89.7212i			0.105430i
$852$		0			0
$853$	16.6518			0.0195214			0.00976072	−	0.999952i	$-0.496893\pi$
$853$	16.6518			0.0195214			0.00976072	+	0.999952i	$0.496893\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			656.701i			0.766279i	0.923691	+	0.383139i	$0.125157\pi$
$857$			656.701i			0.766279i	−0.923691	+	0.383139i	$0.874843\pi$
$858$		0			0
$859$	−7.72122			−0.00898862			−0.00449431	−	0.999990i	$-0.501431\pi$
$859$	−7.72122			−0.00898862			−0.00449431	+	0.999990i	$0.501431\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1132.71i		−	1.31253i	−0.754530	−	0.656266i	$-0.772135\pi$
$863$		−	1132.71i		−	1.31253i	0.754530	−	0.656266i	$-0.227865\pi$
$864$		0			0
$865$			248.390i			0.287157i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1425.83i			1.64077i
$870$		0			0
$871$	−102.197			−0.117333
$872$		0			0
$873$		0			0
$874$		0			0
$875$	41.6865			0.0476418
$876$		0			0
$877$		−	826.040i		−	0.941893i	−0.882162	−	0.470947i	$-0.843912\pi$
$877$		−	826.040i		−	0.941893i	0.882162	−	0.470947i	$-0.156088\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−691.162			−0.784520			−0.392260	−	0.919854i	$-0.628307\pi$
$881$	−691.162			−0.784520			−0.392260	+	0.919854i	$0.628307\pi$
$882$		0			0
$883$	−1339.08			−1.51651			−0.758255	−	0.651958i	$-0.773948\pi$
$883$	−1339.08			−1.51651			−0.758255	+	0.651958i	$0.773948\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	1263.16i		−	1.42408i	−0.702139	−	0.712040i	$-0.747772\pi$
$887$		−	1263.16i		−	1.42408i	0.702139	−	0.712040i	$-0.252228\pi$
$888$		0			0
$889$			199.131i			0.223995i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−898.356	+	518.120i	−1.00600	+	0.580202i
$894$		0			0
$895$		−	553.928i		−	0.618914i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	260.836			0.290141
$900$		0			0
$901$			857.014i			0.951181i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	524.492i		−	0.579549i
$906$		0			0
$907$		−	531.490i		−	0.585987i	−0.956114	−	0.292994i	$-0.905349\pi$
$907$		−	531.490i		−	0.585987i	0.956114	−	0.292994i	$-0.0946515\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	214.421i		−	0.235368i	−0.993051	−	0.117684i	$-0.962453\pi$
$911$		−	214.421i		−	0.235368i	0.993051	−	0.117684i	$-0.0375471\pi$
$912$		0			0
$913$	−1116.13			−1.22248
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−406.542			−0.443340
$918$		0			0
$919$	−1523.80			−1.65811			−0.829055	−	0.559167i	$-0.811121\pi$
$919$	−1523.80			−1.65811			−0.829055	+	0.559167i	$0.811121\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−497.950			−0.539490
$924$		0			0
$925$		−	25.7152i		−	0.0278002i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	269.533			0.290132			0.145066	−	0.989422i	$-0.453661\pi$
$929$	269.533			0.290132			0.145066	+	0.989422i	$0.453661\pi$
$930$		0			0
$931$	577.669	−	333.166i	0.620482	−	0.357859i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−475.604			−0.508667
$936$		0			0
$937$	−492.223			−0.525318			−0.262659	−	0.964889i	$-0.584599\pi$
$937$	−492.223			−0.525318			−0.262659	+	0.964889i	$0.584599\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1364.05i		−	1.44958i	−0.688972	−	0.724788i	$-0.741938\pi$
$941$		−	1364.05i		−	1.44958i	0.688972	−	0.724788i	$-0.258062\pi$
$942$		0			0
$943$		−	737.556i		−	0.782138i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1082.40			1.14298			0.571490	−	0.820609i	$-0.306366\pi$
$947$	1082.40			1.14298			0.571490	+	0.820609i	$0.306366\pi$
$948$		0			0
$949$			139.554i			0.147054i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			196.828i			0.206535i	0.994654	+	0.103268i	$0.0329298\pi$
$953$			196.828i			0.206535i	−0.994654	+	0.103268i	$0.967070\pi$
$954$		0			0
$955$	−559.875			−0.586256
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−93.2092			−0.0971941
$960$		0			0
$961$	707.806			0.736531
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	273.266i		−	0.283177i
$966$		0			0
$967$	1381.14			1.42828			0.714138	−	0.700005i	$-0.246819\pi$
$967$	1381.14			1.42828			0.714138	+	0.700005i	$0.246819\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1116.21i			1.14954i	0.818314	+	0.574771i	$0.194909\pi$
$971$			1116.21i			1.14954i	−0.818314	+	0.574771i	$0.805091\pi$
$972$		0			0
$973$	−429.094			−0.441001
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1784.43i		−	1.82644i	−0.407471	−	0.913218i	$-0.633589\pi$
$977$		−	1784.43i		−	1.82644i	0.407471	−	0.913218i	$-0.366411\pi$
$978$		0			0
$979$			167.954i			0.171557i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	600.203i		−	0.610583i	−0.952259	−	0.305291i	$-0.901246\pi$
$983$		−	600.203i		−	0.610583i	0.952259	−	0.305291i	$-0.0987538\pi$
$984$		0			0
$985$	−767.366			−0.779052
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−27.3303			−0.0276343
$990$		0			0
$991$		−	14.0179i		−	0.0141452i	−0.999975	−	0.00707259i	$-0.997749\pi$
$991$		−	14.0179i		−	0.0141452i	0.999975	−	0.00707259i	$-0.00225129\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−665.415			−0.668759
$996$		0			0
$997$	137.727			0.138142			0.0690708	−	0.997612i	$-0.477997\pi$
$997$	137.727			0.138142			0.0690708	+	0.997612i	$0.477997\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3420.3.o.a.721.3		12
3.2	odd	2		380.3.e.a.341.12	yes	12
12.11	even	2		1520.3.h.b.721.1		12
15.2	even	4		1900.3.g.c.949.24		24
15.8	even	4		1900.3.g.c.949.1		24
15.14	odd	2		1900.3.e.f.1101.1		12
19.18	odd	2	inner	3420.3.o.a.721.4		12
57.56	even	2		380.3.e.a.341.1	✓	12
228.227	odd	2		1520.3.h.b.721.12		12
285.113	odd	4		1900.3.g.c.949.23		24
285.227	odd	4		1900.3.g.c.949.2		24
285.284	even	2		1900.3.e.f.1101.12		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.e.a.341.1	✓	12	57.56	even	2
380.3.e.a.341.12	yes	12	3.2	odd	2
1520.3.h.b.721.1		12	12.11	even	2
1520.3.h.b.721.12		12	228.227	odd	2
1900.3.e.f.1101.1		12	15.14	odd	2
1900.3.e.f.1101.12		12	285.284	even	2
1900.3.g.c.949.1		24	15.8	even	4
1900.3.g.c.949.2		24	285.227	odd	4
1900.3.g.c.949.23		24	285.113	odd	4
1900.3.g.c.949.24		24	15.2	even	4
3420.3.o.a.721.3		12	1.1	even	1	trivial
3420.3.o.a.721.4		12	19.18	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3420.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607 q^{5} -3.72856 q^{7} +O(q^{10})\)	\(q-2.23607 q^{5} -3.72856 q^{7} +13.8938 q^{11} -4.10699i q^{13} +15.3087 q^{17} +(-16.4588 + 9.49250i) q^{19} -17.4452 q^{23} +5.00000 q^{25} -16.3924i q^{29} +15.9121i q^{31} +8.33731 q^{35} -5.14304i q^{37} +42.2785i q^{41} +1.56664 q^{43} +54.5821 q^{47} -35.0979 q^{49} +55.9822i q^{53} -31.0676 q^{55} -101.822i q^{59} +80.5719 q^{61} +9.18350i q^{65} -24.8838i q^{67} -121.245i q^{71} -33.9798 q^{73} -51.8040 q^{77} +102.623i q^{79} -80.3324 q^{83} -34.2312 q^{85} +12.0884i q^{89} +15.3131i q^{91} +(36.8030 - 21.2259i) q^{95} +161.837i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{7}+O(q^{10}) \) 12 * q - 12 * q^7	\( 12 q - 12 q^{7} + 32 q^{11} + 12 q^{17} + 24 q^{19} - 4 q^{23} + 60 q^{25} - 40 q^{35} - 176 q^{43} + 72 q^{47} - 24 q^{49} + 152 q^{61} - 148 q^{73} - 376 q^{77} + 208 q^{83}+O(q^{100}) \) 12 * q - 12 * q^7 + 32 * q^11 + 12 * q^17 + 24 * q^19 - 4 * q^23 + 60 * q^25 - 40 * q^35 - 176 * q^43 + 72 * q^47 - 24 * q^49 + 152 * q^61 - 148 * q^73 - 376 * q^77 + 208 * q^83

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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