LMFDB - Embedded newform 3420.2.f.a.1369.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3420,2,Mod(1369,3420)]

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, 1, 0]))

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

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

chi := DirichletCharacter("3420.1369");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3420.f (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:	$27.3088374913$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 6x^{2} + 4 $ x^4 + 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1369.3
Root			$-2.28825i$ of defining polynomial
Character	$\chi$	$=$	3420.1369
Dual form			3420.2.f.a.1369.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{5} -2.82843i q^{7} +O(q^{10})$	$q+2.23607 q^{5} -2.82843i q^{7} +5.23607 q^{11} +4.03631i q^{13} -1.08036i q^{17} +1.00000 q^{19} +7.40492i q^{23} +5.00000 q^{25} +4.47214 q^{29} -4.00000 q^{31} -6.32456i q^{35} -6.86474i q^{37} +6.00000 q^{41} +8.48528i q^{43} +8.48528i q^{47} -1.00000 q^{49} +6.86474i q^{53} +11.7082 q^{55} -10.4721 q^{59} +1.70820 q^{61} +9.02546i q^{65} -1.62054i q^{67} +1.52786 q^{71} -13.7295i q^{73} -14.8098i q^{77} +11.4164 q^{79} +5.24419i q^{83} -2.41577i q^{85} -14.9443 q^{89} +11.4164 q^{91} +2.23607 q^{95} -4.44897i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 12 q^{11} + 4 q^{19} + 20 q^{25} - 16 q^{31} + 24 q^{41} - 4 q^{49} + 20 q^{55} - 24 q^{59} - 20 q^{61} + 24 q^{71} - 8 q^{79} - 24 q^{89} - 8 q^{91}+O(q^{100}) $ 4 * q + 12 * q^11 + 4 * q^19 + 20 * q^25 - 16 * q^31 + 24 * q^41 - 4 * q^49 + 20 * q^55 - 24 * q^59 - 20 * q^61 + 24 * q^71 - 8 * q^79 - 24 * q^89 - 8 * q^91

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	1.00000
$5$	2.23607	1.00000
$6$	0	0
$7$	− 2.82843i	− 1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	− 2.82843i	− 1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.23607	1.57873	0.789367	−	0.613922i	$-0.210409\pi$
$11$	5.23607	1.57873	0.789367	+	0.613922i	$0.210409\pi$
$12$	0	0
$13$	4.03631i	1.11947i	0.828671	+	0.559735i	$0.189097\pi$
$13$	4.03631i	1.11947i	−0.828671	+	0.559735i	$0.810903\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.08036i	− 0.262027i	−0.991381	−	0.131013i	$-0.958177\pi$
$17$	− 1.08036i	− 0.262027i	0.991381	−	0.131013i	$-0.0418230\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.40492i	1.54403i	0.635603	+	0.772016i	$0.280752\pi$
$23$	7.40492i	1.54403i	−0.635603	+	0.772016i	$0.719248\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 6.32456i	− 1.06904i
$36$	0	0
$37$	− 6.86474i	− 1.12856i	−0.825585	−	0.564278i	$-0.809155\pi$
$37$	− 6.86474i	− 1.12856i	0.825585	−	0.564278i	$-0.190845\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	8.48528i	1.29399i	0.762493	+	0.646997i	$0.223975\pi$
$43$	8.48528i	1.29399i	−0.762493	+	0.646997i	$0.776025\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.48528i	1.23771i	0.785507	+	0.618853i	$0.212402\pi$
$47$	8.48528i	1.23771i	−0.785507	+	0.618853i	$0.787598\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.86474i	0.942944i	0.881881	+	0.471472i	$0.156277\pi$
$53$	6.86474i	0.942944i	−0.881881	+	0.471472i	$0.843723\pi$
$54$	0	0
$55$	11.7082	1.57873
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.4721	−1.36336	−0.681678	−	0.731652i	$-0.738749\pi$
$59$	−10.4721	−1.36336	−0.681678	+	0.731652i	$0.738749\pi$
$60$	0	0
$61$	1.70820	0.218713	0.109357	−	0.994003i	$-0.465121\pi$
$61$	1.70820	0.218713	0.109357	+	0.994003i	$0.465121\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	9.02546i	1.11947i
$66$	0	0
$67$	− 1.62054i	− 0.197981i	−0.995088	−	0.0989905i	$-0.968439\pi$
$67$	− 1.62054i	− 0.197981i	0.995088	−	0.0989905i	$-0.0315613\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.52786	0.181324	0.0906621	−	0.995882i	$-0.471102\pi$
$71$	1.52786	0.181324	0.0906621	+	0.995882i	$0.471102\pi$
$72$	0	0
$73$	− 13.7295i	− 1.60691i	−0.595363	−	0.803457i	$-0.702992\pi$
$73$	− 13.7295i	− 1.60691i	0.595363	−	0.803457i	$-0.297008\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 14.8098i	− 1.68774i
$78$	0	0
$79$	11.4164	1.28445	0.642223	−	0.766518i	$-0.278012\pi$
$79$	11.4164	1.28445	0.642223	+	0.766518i	$0.278012\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.24419i	0.575625i	0.957687	+	0.287812i	$0.0929280\pi$
$83$	5.24419i	0.575625i	−0.957687	+	0.287812i	$0.907072\pi$
$84$	0	0
$85$	− 2.41577i	− 0.262027i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−14.9443	−1.58409	−0.792045	−	0.610463i	$-0.790983\pi$
$89$	−14.9443	−1.58409	−0.792045	+	0.610463i	$0.790983\pi$
$90$	0	0
$91$	11.4164	1.19676
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.23607	0.229416
$96$	0	0
$97$	− 4.44897i	− 0.451725i	−0.974159	−	0.225862i	$-0.927480\pi$
$97$	− 4.44897i	− 0.451725i	0.974159	−	0.225862i	$-0.0725199\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−0.763932	−0.0760141	−0.0380070	−	0.999277i	$-0.512101\pi$
$101$	−0.763932	−0.0760141	−0.0380070	+	0.999277i	$0.512101\pi$
$102$	0	0
$103$	− 15.3500i	− 1.51248i	−0.654293	−	0.756241i	$-0.727034\pi$
$103$	− 15.3500i	− 1.51248i	0.654293	−	0.756241i	$-0.272966\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 16.4304i	− 1.58838i	−0.607666	−	0.794192i	$-0.707894\pi$
$107$	− 16.4304i	− 1.58838i	0.607666	−	0.794192i	$-0.292106\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 12.1089i	− 1.13911i	−0.821952	−	0.569556i	$-0.807115\pi$
$113$	− 12.1089i	− 1.13911i	0.821952	−	0.569556i	$-0.192885\pi$
$114$	0	0
$115$	16.5579i	1.54403i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.05573	−0.280118
$120$	0	0
$121$	16.4164	1.49240
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.1803	1.00000
$126$	0	0
$127$	10.1058i	0.896747i	0.893846	+	0.448374i	$0.147997\pi$
$127$	10.1058i	0.896747i	−0.893846	+	0.448374i	$0.852003\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	− 2.82843i	− 0.245256i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1.08036i	− 0.0923016i	−0.998934	−	0.0461508i	$-0.985305\pi$
$137$	− 1.08036i	− 0.0923016i	0.998934	−	0.0461508i	$-0.0146955\pi$
$138$	0	0
$139$	−15.7082	−1.33235	−0.666176	−	0.745794i	$-0.732070\pi$
$139$	−15.7082	−1.33235	−0.666176	+	0.745794i	$0.732070\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	21.1344i	1.76735i
$144$	0	0
$145$	10.0000	0.830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	18.6525	1.52807	0.764035	−	0.645175i	$-0.223215\pi$
$149$	18.6525	1.52807	0.764035	+	0.645175i	$0.223215\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−8.94427	−0.718421
$156$	0	0
$157$	19.3863i	1.54720i	0.633676	+	0.773599i	$0.281545\pi$
$157$	19.3863i	1.54720i	−0.633676	+	0.773599i	$0.718455\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	20.9443	1.65064
$162$	0	0
$163$	− 8.48528i	− 0.664619i	−0.943170	−	0.332309i	$-0.892172\pi$
$163$	− 8.48528i	− 0.664619i	0.943170	−	0.332309i	$-0.107828\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 4.70401i	− 0.364007i	−0.983298	−	0.182004i	$-0.941742\pi$
$167$	− 4.70401i	− 0.364007i	0.983298	−	0.182004i	$-0.0582583\pi$
$168$	0	0
$169$	−3.29180	−0.253215
$170$	0	0
$171$	0	0
$172$	0	0
$173$	13.1893i	1.00276i	0.865226	+	0.501382i	$0.167175\pi$
$173$	13.1893i	1.00276i	−0.865226	+	0.501382i	$0.832825\pi$
$174$	0	0
$175$	− 14.1421i	− 1.06904i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−13.5279	−1.01112	−0.505560	−	0.862791i	$-0.668714\pi$
$179$	−13.5279	−1.01112	−0.505560	+	0.862791i	$0.668714\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 15.3500i	− 1.12856i
$186$	0	0
$187$	− 5.65685i	− 0.413670i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.9443	1.51547	0.757737	−	0.652560i	$-0.226305\pi$
$191$	20.9443	1.51547	0.757737	+	0.652560i	$0.226305\pi$
$192$	0	0
$193$	4.44897i	0.320244i	0.987097	+	0.160122i	$0.0511888\pi$
$193$	4.44897i	0.320244i	−0.987097	+	0.160122i	$0.948811\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 12.6491i	− 0.901212i	−0.892723	−	0.450606i	$-0.851208\pi$
$197$	− 12.6491i	− 0.901212i	0.892723	−	0.450606i	$-0.148792\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 12.6491i	− 0.887794i
$204$	0	0
$205$	13.4164	0.937043
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.23607	0.362186
$210$	0	0
$211$	−11.4164	−0.785938	−0.392969	−	0.919552i	$-0.628552\pi$
$211$	−11.4164	−0.785938	−0.392969	+	0.919552i	$0.628552\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	18.9737i	1.29399i
$216$	0	0
$217$	11.3137i	0.768025i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.36068	0.293331
$222$	0	0
$223$	− 28.6668i	− 1.91967i	−0.280560	−	0.959836i	$-0.590520\pi$
$223$	− 28.6668i	− 1.91967i	0.280560	−	0.959836i	$-0.409480\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.3500i	1.01882i	0.860525	+	0.509408i	$0.170136\pi$
$227$	15.3500i	1.01882i	−0.860525	+	0.509408i	$0.829864\pi$
$228$	0	0
$229$	5.70820	0.377209	0.188604	−	0.982053i	$-0.439604\pi$
$229$	5.70820	0.377209	0.188604	+	0.982053i	$0.439604\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 12.6491i	− 0.828671i	−0.910124	−	0.414335i	$-0.864014\pi$
$233$	− 12.6491i	− 0.828671i	0.910124	−	0.414335i	$-0.135986\pi$
$234$	0	0
$235$	18.9737i	1.23771i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5.88854	0.380898	0.190449	−	0.981697i	$-0.439006\pi$
$239$	5.88854	0.380898	0.190449	+	0.981697i	$0.439006\pi$
$240$	0	0
$241$	24.8328	1.59962	0.799811	−	0.600252i	$-0.204933\pi$
$241$	24.8328	1.59962	0.799811	+	0.600252i	$0.204933\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.23607	−0.142857
$246$	0	0
$247$	4.03631i	0.256824i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−29.8885	−1.88655	−0.943274	−	0.332015i	$-0.892272\pi$
$251$	−29.8885	−1.88655	−0.943274	+	0.332015i	$0.892272\pi$
$252$	0	0
$253$	38.7727i	2.43762i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.70401i	0.293428i	0.989179	+	0.146714i	$0.0468697\pi$
$257$	4.70401i	0.293428i	−0.989179	+	0.146714i	$0.953130\pi$
$258$	0	0
$259$	−19.4164	−1.20648
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 9.56564i	− 0.589843i	−0.955522	−	0.294921i	$-0.904707\pi$
$263$	− 9.56564i	− 0.589843i	0.955522	−	0.294921i	$-0.0952935\pi$
$264$	0	0
$265$	15.3500i	0.942944i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2.94427	−0.179515	−0.0897577	−	0.995964i	$-0.528609\pi$
$269$	−2.94427	−0.179515	−0.0897577	+	0.995964i	$0.528609\pi$
$270$	0	0
$271$	−19.1246	−1.16174	−0.580869	−	0.813997i	$-0.697287\pi$
$271$	−19.1246	−1.16174	−0.580869	+	0.813997i	$0.697287\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	26.1803	1.57873
$276$	0	0
$277$	3.24109i	0.194738i	0.995248	+	0.0973691i	$0.0310427\pi$
$277$	3.24109i	0.194738i	−0.995248	+	0.0973691i	$0.968957\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.41641	−0.0844958	−0.0422479	−	0.999107i	$-0.513452\pi$
$281$	−1.41641	−0.0844958	−0.0422479	+	0.999107i	$0.513452\pi$
$282$	0	0
$283$	16.5579i	0.984265i	0.870520	+	0.492133i	$0.163782\pi$
$283$	16.5579i	0.984265i	−0.870520	+	0.492133i	$0.836218\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 16.9706i	− 1.00174i
$288$	0	0
$289$	15.8328	0.931342
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.86474i	0.401042i	0.979689	+	0.200521i	$0.0642635\pi$
$293$	6.86474i	0.401042i	−0.979689	+	0.200521i	$0.935736\pi$
$294$	0	0
$295$	−23.4164	−1.36336
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−29.8885	−1.72850
$300$	0	0
$301$	24.0000	1.38334
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.81966	0.218713
$306$	0	0
$307$	− 4.03631i	− 0.230364i	−0.993344	−	0.115182i	$-0.963255\pi$
$307$	− 4.03631i	− 0.230364i	0.993344	−	0.115182i	$-0.0367452\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−3.70820	−0.210273	−0.105136	−	0.994458i	$-0.533528\pi$
$311$	−3.70820	−0.210273	−0.105136	+	0.994458i	$0.533528\pi$
$312$	0	0
$313$	27.4589i	1.55207i	0.630689	+	0.776036i	$0.282772\pi$
$313$	27.4589i	1.55207i	−0.630689	+	0.776036i	$0.717228\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.1893i	0.740784i	0.928875	+	0.370392i	$0.120777\pi$
$317$	13.1893i	0.740784i	−0.928875	+	0.370392i	$0.879223\pi$
$318$	0	0
$319$	23.4164	1.31107
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 1.08036i	− 0.0601130i
$324$	0	0
$325$	20.1815i	1.11947i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	−19.4164	−1.06722	−0.533611	−	0.845730i	$-0.679165\pi$
$331$	−19.4164	−1.06722	−0.533611	+	0.845730i	$0.679165\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 3.62365i	− 0.197981i
$336$	0	0
$337$	− 7.27740i	− 0.396425i	−0.980159	−	0.198213i	$-0.936486\pi$
$337$	− 7.27740i	− 0.396425i	0.980159	−	0.198213i	$-0.0635136\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−20.9443	−1.13420
$342$	0	0
$343$	− 16.9706i	− 0.916324i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	34.8639i	1.87159i	0.352544	+	0.935795i	$0.385317\pi$
$347$	34.8639i	1.87159i	−0.352544	+	0.935795i	$0.614683\pi$
$348$	0	0
$349$	1.41641	0.0758186	0.0379093	−	0.999281i	$-0.487930\pi$
$349$	1.41641	0.0758186	0.0379093	+	0.999281i	$0.487930\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 15.8902i	− 0.845750i	−0.906188	−	0.422875i	$-0.861021\pi$
$353$	− 15.8902i	− 0.845750i	0.906188	−	0.422875i	$-0.138979\pi$
$354$	0	0
$355$	3.41641	0.181324
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.1246	−0.587135	−0.293567	−	0.955938i	$-0.594842\pi$
$359$	−11.1246	−0.587135	−0.293567	+	0.955938i	$0.594842\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 30.7000i	− 1.60691i
$366$	0	0
$367$	− 8.48528i	− 0.442928i	−0.975169	−	0.221464i	$-0.928916\pi$
$367$	− 8.48528i	− 0.442928i	0.975169	−	0.221464i	$-0.0710835\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	19.4164	1.00805
$372$	0	0
$373$	− 34.3237i	− 1.77721i	−0.458670	−	0.888606i	$-0.651674\pi$
$373$	− 34.3237i	− 1.77721i	0.458670	−	0.888606i	$-0.348326\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	18.0509i	0.929670i
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 23.6777i	− 1.20987i	−0.796274	−	0.604936i	$-0.793199\pi$
$383$	− 23.6777i	− 1.20987i	0.796274	−	0.604936i	$-0.206801\pi$
$384$	0	0
$385$	− 33.1158i	− 1.68774i
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.94427	0.149281	0.0746403	−	0.997211i	$-0.476219\pi$
$389$	2.94427	0.149281	0.0746403	+	0.997211i	$0.476219\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	0	0
$394$	0	0
$395$	25.5279	1.28445
$396$	0	0
$397$	− 30.7000i	− 1.54079i	−0.637566	−	0.770395i	$-0.720059\pi$
$397$	− 30.7000i	− 1.54079i	0.637566	−	0.770395i	$-0.279941\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.47214	−0.223328	−0.111664	−	0.993746i	$-0.535618\pi$
$401$	−4.47214	−0.223328	−0.111664	+	0.993746i	$0.535618\pi$
$402$	0	0
$403$	− 16.1452i	− 0.804252i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 35.9442i	− 1.78169i
$408$	0	0
$409$	9.41641	0.465611	0.232806	−	0.972523i	$-0.425209\pi$
$409$	9.41641	0.465611	0.232806	+	0.972523i	$0.425209\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	29.6197i	1.45749i
$414$	0	0
$415$	11.7264i	0.575625i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3.05573	−0.149282	−0.0746410	−	0.997210i	$-0.523781\pi$
$419$	−3.05573	−0.149282	−0.0746410	+	0.997210i	$0.523781\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 5.40182i	− 0.262027i
$426$	0	0
$427$	− 4.83153i	− 0.233814i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10.4721	−0.504425	−0.252213	−	0.967672i	$-0.581158\pi$
$431$	−10.4721	−0.504425	−0.252213	+	0.967672i	$0.581158\pi$
$432$	0	0
$433$	− 37.9774i	− 1.82508i	−0.408989	−	0.912540i	$-0.634118\pi$
$433$	− 37.9774i	− 1.82508i	0.408989	−	0.912540i	$-0.365882\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.40492i	0.354225i
$438$	0	0
$439$	−34.8328	−1.66248	−0.831240	−	0.555914i	$-0.812368\pi$
$439$	−34.8328	−1.66248	−0.831240	+	0.555914i	$0.812368\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 5.24419i	− 0.249159i	−0.992210	−	0.124580i	$-0.960242\pi$
$443$	− 5.24419i	− 0.249159i	0.992210	−	0.124580i	$-0.0397582\pi$
$444$	0	0
$445$	−33.4164	−1.58409
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−7.52786	−0.355262	−0.177631	−	0.984097i	$-0.556843\pi$
$449$	−7.52786	−0.355262	−0.177631	+	0.984097i	$0.556843\pi$
$450$	0	0
$451$	31.4164	1.47934
$452$	0	0
$453$	0	0
$454$	0	0
$455$	25.5279	1.19676
$456$	0	0
$457$	37.9473i	1.77510i	0.460710	+	0.887551i	$0.347595\pi$
$457$	37.9473i	1.77510i	−0.460710	+	0.887551i	$0.652405\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	11.8885	0.553705	0.276852	−	0.960912i	$-0.410709\pi$
$461$	11.8885	0.553705	0.276852	+	0.960912i	$0.410709\pi$
$462$	0	0
$463$	33.5285i	1.55820i	0.626900	+	0.779100i	$0.284324\pi$
$463$	33.5285i	1.55820i	−0.626900	+	0.779100i	$0.715676\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11.7264i	0.542632i	0.962490	+	0.271316i	$0.0874588\pi$
$467$	11.7264i	0.542632i	−0.962490	+	0.271316i	$0.912541\pi$
$468$	0	0
$469$	−4.58359	−0.211651
$470$	0	0
$471$	0	0
$472$	0	0
$473$	44.4295i	2.04287i
$474$	0	0
$475$	5.00000	0.229416
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.76393	0.309052	0.154526	−	0.987989i	$-0.450615\pi$
$479$	6.76393	0.309052	0.154526	+	0.987989i	$0.450615\pi$
$480$	0	0
$481$	27.7082	1.26339
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 9.94820i	− 0.451725i
$486$	0	0
$487$	− 17.7658i	− 0.805044i	−0.915410	−	0.402522i	$-0.868134\pi$
$487$	− 17.7658i	− 0.805044i	0.915410	−	0.402522i	$-0.131866\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	29.8885	1.34885	0.674426	−	0.738343i	$-0.264391\pi$
$491$	29.8885	1.34885	0.674426	+	0.738343i	$0.264391\pi$
$492$	0	0
$493$	− 4.83153i	− 0.217601i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 4.32145i	− 0.193844i
$498$	0	0
$499$	−23.1246	−1.03520	−0.517600	−	0.855623i	$-0.673174\pi$
$499$	−23.1246	−1.03520	−0.517600	+	0.855623i	$0.673174\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 39.1853i	− 1.74719i	−0.486656	−	0.873593i	$-0.661784\pi$
$503$	− 39.1853i	− 1.74719i	0.486656	−	0.873593i	$-0.338216\pi$
$504$	0	0
$505$	−1.70820	−0.0760141
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−37.4164	−1.65845	−0.829227	−	0.558913i	$-0.811219\pi$
$509$	−37.4164	−1.65845	−0.829227	+	0.558913i	$0.811219\pi$
$510$	0	0
$511$	−38.8328	−1.71786
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 34.3237i	− 1.51248i
$516$	0	0
$517$	44.4295i	1.95401i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−35.8885	−1.57231	−0.786153	−	0.618032i	$-0.787930\pi$
$521$	−35.8885	−1.57231	−0.786153	+	0.618032i	$0.787930\pi$
$522$	0	0
$523$	3.62365i	0.158451i	0.996857	+	0.0792255i	$0.0252447\pi$
$523$	3.62365i	0.158451i	−0.996857	+	0.0792255i	$0.974755\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.32145i	0.188245i
$528$	0	0
$529$	−31.8328	−1.38404
$530$	0	0
$531$	0	0
$532$	0	0
$533$	24.2179i	1.04899i
$534$	0	0
$535$	− 36.7394i	− 1.58838i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.23607	−0.225533
$540$	0	0
$541$	−1.70820	−0.0734414	−0.0367207	−	0.999326i	$-0.511691\pi$
$541$	−1.70820	−0.0734414	−0.0367207	+	0.999326i	$0.511691\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−4.47214	−0.191565
$546$	0	0
$547$	− 1.20788i	− 0.0516453i	−0.999667	−	0.0258227i	$-0.991779\pi$
$547$	− 1.20788i	− 0.0516453i	0.999667	−	0.0258227i	$-0.00822052\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.47214	0.190519
$552$	0	0
$553$	− 32.2905i	− 1.37313i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 11.5687i	− 0.490184i	−0.969500	−	0.245092i	$-0.921182\pi$
$557$	− 11.5687i	− 0.490184i	0.969500	−	0.245092i	$-0.0788181\pi$
$558$	0	0
$559$	−34.2492	−1.44859
$560$	0	0
$561$	0	0
$562$	0	0
$563$	17.3531i	0.731347i	0.930743	+	0.365673i	$0.119161\pi$
$563$	17.3531i	0.731347i	−0.930743	+	0.365673i	$0.880839\pi$
$564$	0	0
$565$	− 27.0764i	− 1.13911i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−21.0557	−0.882702	−0.441351	−	0.897335i	$-0.645501\pi$
$569$	−21.0557	−0.882702	−0.441351	+	0.897335i	$0.645501\pi$
$570$	0	0
$571$	−15.1246	−0.632945	−0.316473	−	0.948602i	$-0.602499\pi$
$571$	−15.1246	−0.632945	−0.316473	+	0.948602i	$0.602499\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	37.0246i	1.54403i
$576$	0	0
$577$	5.65685i	0.235498i	0.993043	+	0.117749i	$0.0375678\pi$
$577$	5.65685i	0.235498i	−0.993043	+	0.117749i	$0.962432\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	14.8328	0.615369
$582$	0	0
$583$	35.9442i	1.48866i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	37.0246i	1.52817i	0.645117	+	0.764084i	$0.276809\pi$
$587$	37.0246i	1.52817i	−0.645117	+	0.764084i	$0.723191\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	0	0
$592$	0	0
$593$	14.8098i	0.608167i	0.952645	+	0.304084i	$0.0983502\pi$
$593$	14.8098i	0.608167i	−0.952645	+	0.304084i	$0.901650\pi$
$594$	0	0
$595$	−6.83282	−0.280118
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	−36.8328	−1.50244	−0.751221	−	0.660051i	$-0.770535\pi$
$601$	−36.8328	−1.50244	−0.751221	+	0.660051i	$0.770535\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	36.7082	1.49240
$606$	0	0
$607$	− 12.9343i	− 0.524985i	−0.964934	−	0.262493i	$-0.915455\pi$
$607$	− 12.9343i	− 0.524985i	0.964934	−	0.262493i	$-0.0845445\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−34.2492	−1.38558
$612$	0	0
$613$	− 20.2117i	− 0.816341i	−0.912906	−	0.408170i	$-0.866167\pi$
$613$	− 20.2117i	− 0.816341i	0.912906	−	0.408170i	$-0.133833\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 28.5393i	− 1.14895i	−0.818522	−	0.574475i	$-0.805206\pi$
$617$	− 28.5393i	− 1.14895i	0.818522	−	0.574475i	$-0.194794\pi$
$618$	0	0
$619$	−0.875388	−0.0351848	−0.0175924	−	0.999845i	$-0.505600\pi$
$619$	−0.875388	−0.0351848	−0.0175924	+	0.999845i	$0.505600\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	42.2688i	1.69346i
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7.41641	−0.295712
$630$	0	0
$631$	11.7082	0.466096	0.233048	−	0.972465i	$-0.425130\pi$
$631$	11.7082	0.466096	0.233048	+	0.972465i	$0.425130\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	22.5973i	0.896747i
$636$	0	0
$637$	− 4.03631i	− 0.159924i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	47.8885	1.89148	0.945742	−	0.324919i	$-0.105337\pi$
$641$	47.8885	1.89148	0.945742	+	0.324919i	$0.105337\pi$
$642$	0	0
$643$	− 11.7264i	− 0.462443i	−0.972901	−	0.231221i	$-0.925728\pi$
$643$	− 11.7264i	− 0.462443i	0.972901	−	0.231221i	$-0.0742722\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	34.8639i	1.37064i	0.728242	+	0.685320i	$0.240338\pi$
$647$	34.8639i	1.37064i	−0.728242	+	0.685320i	$0.759662\pi$
$648$	0	0
$649$	−54.8328	−2.15238
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.24109i	0.126834i	0.997987	+	0.0634168i	$0.0201998\pi$
$653$	3.24109i	0.126834i	−0.997987	+	0.0634168i	$0.979800\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−27.0557	−1.05394	−0.526971	−	0.849883i	$-0.676672\pi$
$659$	−27.0557	−1.05394	−0.526971	+	0.849883i	$0.676672\pi$
$660$	0	0
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 6.32456i	− 0.245256i
$666$	0	0
$667$	33.1158i	1.28225i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8.94427	0.345290
$672$	0	0
$673$	42.8090i	1.65016i	0.565013	+	0.825082i	$0.308871\pi$
$673$	42.8090i	1.65016i	−0.565013	+	0.825082i	$0.691129\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 30.0022i	− 1.15308i	−0.817069	−	0.576540i	$-0.804403\pi$
$677$	− 30.0022i	− 1.15308i	0.817069	−	0.576540i	$-0.195597\pi$
$678$	0	0
$679$	−12.5836	−0.482914
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 10.1058i	− 0.386689i	−0.981131	−	0.193344i	$-0.938067\pi$
$683$	− 10.1058i	− 0.386689i	0.981131	−	0.193344i	$-0.0619334\pi$
$684$	0	0
$685$	− 2.41577i	− 0.0923016i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−27.7082	−1.05560
$690$	0	0
$691$	−15.1246	−0.575367	−0.287684	−	0.957725i	$-0.592885\pi$
$691$	−15.1246	−0.575367	−0.287684	+	0.957725i	$0.592885\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−35.1246	−1.33235
$696$	0	0
$697$	− 6.48218i	− 0.245530i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−41.1246	−1.55326	−0.776628	−	0.629960i	$-0.783071\pi$
$701$	−41.1246	−1.55326	−0.776628	+	0.629960i	$0.783071\pi$
$702$	0	0
$703$	− 6.86474i	− 0.258908i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.16073i	0.0812625i
$708$	0	0
$709$	−6.58359	−0.247252	−0.123626	−	0.992329i	$-0.539452\pi$
$709$	−6.58359	−0.247252	−0.123626	+	0.992329i	$0.539452\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 29.6197i	− 1.10927i
$714$	0	0
$715$	47.2579i	1.76735i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−21.5967	−0.805423	−0.402711	−	0.915327i	$-0.631932\pi$
$719$	−21.5967	−0.805423	−0.402711	+	0.915327i	$0.631932\pi$
$720$	0	0
$721$	−43.4164	−1.61691
$722$	0	0
$723$	0	0
$724$	0	0
$725$	22.3607	0.830455
$726$	0	0
$727$	− 8.48528i	− 0.314702i	−0.987543	−	0.157351i	$-0.949705\pi$
$727$	− 8.48528i	− 0.314702i	0.987543	−	0.157351i	$-0.0502953\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9.16718	0.339061
$732$	0	0
$733$	− 33.9411i	− 1.25364i	−0.779162	−	0.626822i	$-0.784355\pi$
$733$	− 33.9411i	− 1.25364i	0.779162	−	0.626822i	$-0.215645\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8.48528i	− 0.312559i
$738$	0	0
$739$	−34.8328	−1.28135	−0.640673	−	0.767814i	$-0.721345\pi$
$739$	−34.8328	−1.28135	−0.640673	+	0.767814i	$0.721345\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 6.86474i	− 0.251843i	−0.992040	−	0.125921i	$-0.959811\pi$
$743$	− 6.86474i	− 0.251843i	0.992040	−	0.125921i	$-0.0401887\pi$
$744$	0	0
$745$	41.7082	1.52807
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−46.4721	−1.69805
$750$	0	0
$751$	47.4164	1.73025	0.865125	−	0.501557i	$-0.167239\pi$
$751$	47.4164	1.73025	0.865125	+	0.501557i	$0.167239\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−17.8885	−0.651031
$756$	0	0
$757$	0.825324i	0.0299969i	0.999888	+	0.0149985i	$0.00477434\pi$
$757$	0.825324i	0.0299969i	−0.999888	+	0.0149985i	$0.995226\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−24.5410	−0.889611	−0.444806	−	0.895627i	$-0.646727\pi$
$761$	−24.5410	−0.889611	−0.444806	+	0.895627i	$0.646727\pi$
$762$	0	0
$763$	5.65685i	0.204792i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 42.2688i	− 1.52624i
$768$	0	0
$769$	−28.5410	−1.02922	−0.514608	−	0.857426i	$-0.672062\pi$
$769$	−28.5410	−1.02922	−0.514608	+	0.857426i	$0.672062\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 15.3500i	− 0.552102i	−0.961143	−	0.276051i	$-0.910974\pi$
$773$	− 15.3500i	− 0.552102i	0.961143	−	0.276051i	$-0.0890258\pi$
$774$	0	0
$775$	−20.0000	−0.718421
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.00000	0.214972
$780$	0	0
$781$	8.00000	0.286263
$782$	0	0
$783$	0	0
$784$	0	0
$785$	43.3491i	1.54720i
$786$	0	0
$787$	3.62365i	0.129169i	0.997912	+	0.0645845i	$0.0205722\pi$
$787$	3.62365i	0.129169i	−0.997912	+	0.0645845i	$0.979428\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−34.2492	−1.21776
$792$	0	0
$793$	6.89484i	0.244843i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.1120i	0.499874i	0.968262	+	0.249937i	$0.0804099\pi$
$797$	14.1120i	0.499874i	−0.968262	+	0.249937i	$0.919590\pi$
$798$	0	0
$799$	9.16718	0.324312
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 71.8885i	− 2.53689i
$804$	0	0
$805$	46.8328	1.65064
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−16.4721	−0.579129	−0.289565	−	0.957158i	$-0.593511\pi$
$809$	−16.4721	−0.579129	−0.289565	+	0.957158i	$0.593511\pi$
$810$	0	0
$811$	8.00000	0.280918	0.140459	−	0.990086i	$-0.455142\pi$
$811$	8.00000	0.280918	0.140459	+	0.990086i	$0.455142\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 18.9737i	− 0.664619i
$816$	0	0
$817$	8.48528i	0.296862i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−21.0557	−0.734850	−0.367425	−	0.930053i	$-0.619761\pi$
$821$	−21.0557	−0.734850	−0.367425	+	0.930053i	$0.619761\pi$
$822$	0	0
$823$	2.00310i	0.0698238i	0.999390	+	0.0349119i	$0.0111151\pi$
$823$	2.00310i	0.0698238i	−0.999390	+	0.0349119i	$0.988885\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.5942i	0.716131i	0.933696	+	0.358065i	$0.116564\pi$
$827$	20.5942i	0.716131i	−0.933696	+	0.358065i	$0.883436\pi$
$828$	0	0
$829$	−18.0000	−0.625166	−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000	−0.625166	−0.312583	+	0.949890i	$0.601194\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.08036i	0.0374324i
$834$	0	0
$835$	− 10.5185i	− 0.364007i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−13.3050	−0.459338	−0.229669	−	0.973269i	$-0.573764\pi$
$839$	−13.3050	−0.459338	−0.229669	+	0.973269i	$0.573764\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−7.36068	−0.253215
$846$	0	0
$847$	− 46.4326i	− 1.59544i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	50.8328	1.74253
$852$	0	0
$853$	10.4884i	0.359115i	0.983747	+	0.179558i	$0.0574667\pi$
$853$	10.4884i	0.359115i	−0.983747	+	0.179558i	$0.942533\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	40.8059i	1.39390i	0.717119	+	0.696951i	$0.245460\pi$
$857$	40.8059i	1.39390i	−0.717119	+	0.696951i	$0.754540\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	20.5942i	0.701035i	0.936556	+	0.350518i	$0.113994\pi$
$863$	20.5942i	0.701035i	−0.936556	+	0.350518i	$0.886006\pi$
$864$	0	0
$865$	29.4922i	1.00276i
$866$	0	0
$867$	0	0
$868$	0	0
$869$	59.7771	2.02780
$870$	0	0
$871$	6.54102	0.221634
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 31.6228i	− 1.06904i
$876$	0	0
$877$	− 4.44897i	− 0.150231i	−0.997175	−	0.0751155i	$-0.976067\pi$
$877$	− 4.44897i	− 0.150231i	0.997175	−	0.0751155i	$-0.0239326\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	30.6525	1.03271	0.516354	−	0.856375i	$-0.327289\pi$
$881$	30.6525	1.03271	0.516354	+	0.856375i	$0.327289\pi$
$882$	0	0
$883$	− 14.9675i	− 0.503695i	−0.967767	−	0.251848i	$-0.918962\pi$
$883$	− 14.9675i	− 0.503695i	0.967767	−	0.251848i	$-0.0810382\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	36.3268i	1.21973i	0.792504	+	0.609867i	$0.208777\pi$
$887$	36.3268i	1.21973i	−0.792504	+	0.609867i	$0.791223\pi$
$888$	0	0
$889$	28.5836	0.958663
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.48528i	0.283949i
$894$	0	0
$895$	−30.2492	−1.01112
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−17.8885	−0.596616
$900$	0	0
$901$	7.41641	0.247076
$902$	0	0
$903$	0	0
$904$	0	0
$905$	13.4164	0.445976
$906$	0	0
$907$	− 8.86784i	− 0.294452i	−0.989103	−	0.147226i	$-0.952966\pi$
$907$	− 8.86784i	− 0.294452i	0.989103	−	0.147226i	$-0.0470344\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	37.5279	1.24335	0.621677	−	0.783274i	$-0.286452\pi$
$911$	37.5279	1.24335	0.621677	+	0.783274i	$0.286452\pi$
$912$	0	0
$913$	27.4589i	0.908759i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	26.8328	0.885133	0.442566	−	0.896736i	$-0.354068\pi$
$919$	26.8328	0.885133	0.442566	+	0.896736i	$0.354068\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	6.16693i	0.202987i
$924$	0	0
$925$	− 34.3237i	− 1.12856i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	16.4721	0.540433	0.270217	−	0.962800i	$-0.412905\pi$
$929$	16.4721	0.540433	0.270217	+	0.962800i	$0.412905\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 12.6491i	− 0.413670i
$936$	0	0
$937$	− 51.6768i	− 1.68821i	−0.536180	−	0.844104i	$-0.680133\pi$
$937$	− 51.6768i	− 1.68821i	0.536180	−	0.844104i	$-0.319867\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	21.0557	0.686397	0.343199	−	0.939263i	$-0.388490\pi$
$941$	21.0557	0.686397	0.343199	+	0.939263i	$0.388490\pi$
$942$	0	0
$943$	44.4295i	1.44682i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 28.6969i	− 0.932525i	−0.884646	−	0.466263i	$-0.845600\pi$
$947$	− 28.6969i	− 0.932525i	0.884646	−	0.466263i	$-0.154400\pi$
$948$	0	0
$949$	55.4164	1.79889
$950$	0	0
$951$	0	0
$952$	0	0
$953$	27.0764i	0.877090i	0.898709	+	0.438545i	$0.144506\pi$
$953$	27.0764i	0.877090i	−0.898709	+	0.438545i	$0.855494\pi$
$954$	0	0
$955$	46.8328	1.51547
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.05573	−0.0986746
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	0	0
$964$	0	0
$965$	9.94820i	0.320244i
$966$	0	0
$967$	− 16.5579i	− 0.532466i	−0.963909	−	0.266233i	$-0.914221\pi$
$967$	− 16.5579i	− 0.532466i	0.963909	−	0.266233i	$-0.0857791\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−4.36068	−0.139941	−0.0699704	−	0.997549i	$-0.522290\pi$
$971$	−4.36068	−0.139941	−0.0699704	+	0.997549i	$0.522290\pi$
$972$	0	0
$973$	44.4295i	1.42434i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	47.1304i	1.50784i	0.656969	+	0.753918i	$0.271838\pi$
$977$	47.1304i	1.50784i	−0.656969	+	0.753918i	$0.728162\pi$
$978$	0	0
$979$	−78.2492	−2.50086
$980$	0	0
$981$	0	0
$982$	0	0
$983$	33.4009i	1.06532i	0.846328	+	0.532662i	$0.178808\pi$
$983$	33.4009i	1.06532i	−0.846328	+	0.532662i	$0.821192\pi$
$984$	0	0
$985$	− 28.2843i	− 0.901212i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−62.8328	−1.99797
$990$	0	0
$991$	−14.8328	−0.471180	−0.235590	−	0.971853i	$-0.575702\pi$
$991$	−14.8328	−0.471180	−0.235590	+	0.971853i	$0.575702\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	35.7771	1.13421
$996$	0	0
$997$	30.7000i	0.972280i	0.873881	+	0.486140i	$0.161595\pi$
$997$	30.7000i	0.972280i	−0.873881	+	0.486140i	$0.838405\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3420.2.f.a.1369.3		4
3.2	odd	2		380.2.c.a.229.1	✓	4
5.4	even	2	inner	3420.2.f.a.1369.4		4
12.11	even	2		1520.2.d.f.609.4		4
15.2	even	4		1900.2.a.j.1.1		4
15.8	even	4		1900.2.a.j.1.4		4
15.14	odd	2		380.2.c.a.229.4	yes	4
60.23	odd	4		7600.2.a.ce.1.1		4
60.47	odd	4		7600.2.a.ce.1.4		4
60.59	even	2		1520.2.d.f.609.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.c.a.229.1	✓	4	3.2	odd	2
380.2.c.a.229.4	yes	4	15.14	odd	2
1520.2.d.f.609.1		4	60.59	even	2
1520.2.d.f.609.4		4	12.11	even	2
1900.2.a.j.1.1		4	15.2	even	4
1900.2.a.j.1.4		4	15.8	even	4
3420.2.f.a.1369.3		4	1.1	even	1	trivial
3420.2.f.a.1369.4		4	5.4	even	2	inner
7600.2.a.ce.1.1		4	60.23	odd	4
7600.2.a.ce.1.4		4	60.47	odd	4

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3420.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.23607 q^{5} -2.82843i q^{7} +O(q^{10})\)	\(q+2.23607 q^{5} -2.82843i q^{7} +5.23607 q^{11} +4.03631i q^{13} -1.08036i q^{17} +1.00000 q^{19} +7.40492i q^{23} +5.00000 q^{25} +4.47214 q^{29} -4.00000 q^{31} -6.32456i q^{35} -6.86474i q^{37} +6.00000 q^{41} +8.48528i q^{43} +8.48528i q^{47} -1.00000 q^{49} +6.86474i q^{53} +11.7082 q^{55} -10.4721 q^{59} +1.70820 q^{61} +9.02546i q^{65} -1.62054i q^{67} +1.52786 q^{71} -13.7295i q^{73} -14.8098i q^{77} +11.4164 q^{79} +5.24419i q^{83} -2.41577i q^{85} -14.9443 q^{89} +11.4164 q^{91} +2.23607 q^{95} -4.44897i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 12 q^{11} + 4 q^{19} + 20 q^{25} - 16 q^{31} + 24 q^{41} - 4 q^{49} + 20 q^{55} - 24 q^{59} - 20 q^{61} + 24 q^{71} - 8 q^{79} - 24 q^{89} - 8 q^{91}+O(q^{100}) \) 4 * q + 12 * q^11 + 4 * q^19 + 20 * q^25 - 16 * q^31 + 24 * q^41 - 4 * q^49 + 20 * q^55 - 24 * q^59 - 20 * q^61 + 24 * q^71 - 8 * q^79 - 24 * q^89 - 8 * q^91

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