LMFDB - Embedded newform 3800.2.d.i.3649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(3649,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, 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("3800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.d (of order $2$, degree $1$, not 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:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 760)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.2
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.i.3649.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{3} +2.82843i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.41421i q^{3} +2.82843i q^{7} +1.00000 q^{9} +4.82843 q^{11} -0.585786i q^{13} -0.828427i q^{17} +1.00000 q^{19} +4.00000 q^{21} -4.00000i q^{23} -5.65685i q^{27} -4.82843 q^{29} -6.82843i q^{33} +1.75736i q^{37} -0.828427 q^{39} +4.82843 q^{41} +2.82843i q^{43} +8.48528i q^{47} -1.00000 q^{49} -1.17157 q^{51} +1.07107i q^{53} -1.41421i q^{57} +2.82843 q^{59} +9.65685 q^{61} +2.82843i q^{63} -6.58579i q^{67} -5.65685 q^{69} -7.31371 q^{71} +16.1421i q^{73} +13.6569i q^{77} +14.8284 q^{79} -5.00000 q^{81} -8.00000i q^{83} +6.82843i q^{87} +8.82843 q^{89} +1.65685 q^{91} +6.24264i q^{97} +4.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^9	$ 4 q + 4 q^{9} + 8 q^{11} + 4 q^{19} + 16 q^{21} - 8 q^{29} + 8 q^{39} + 8 q^{41} - 4 q^{49} - 16 q^{51} + 16 q^{61} + 16 q^{71} + 48 q^{79} - 20 q^{81} + 24 q^{89} - 16 q^{91} + 8 q^{99}+O(q^{100}) $ 4 * q + 4 * q^9 + 8 * q^11 + 4 * q^19 + 16 * q^21 - 8 * q^29 + 8 * q^39 + 8 * q^41 - 4 * q^49 - 16 * q^51 + 16 * q^61 + 16 * q^71 + 48 * q^79 - 20 * q^81 + 24 * q^89 - 16 * q^91 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$.

$n$	$401$	$951$	$1901$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 1.41421i	− 0.816497i	−0.912871	−	0.408248i	$-0.866140\pi$
$3$	− 1.41421i	− 0.816497i	0.912871	−	0.408248i	$-0.133860\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.82843i	1.06904i	0.845154	+	0.534522i	$0.179509\pi$
$7$	2.82843i	1.06904i	−0.845154	+	0.534522i	$0.820491\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.82843	1.45583	0.727913	−	0.685670i	$-0.240491\pi$
$11$	4.82843	1.45583	0.727913	+	0.685670i	$0.240491\pi$
$12$	0	0
$13$	− 0.585786i	− 0.162468i	−0.996695	−	0.0812340i	$-0.974114\pi$
$13$	− 0.585786i	− 0.162468i	0.996695	−	0.0812340i	$-0.0258861\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.828427i	− 0.200923i	−0.994941	−	0.100462i	$-0.967968\pi$
$17$	− 0.828427i	− 0.200923i	0.994941	−	0.100462i	$-0.0320319\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	4.00000	0.872872
$22$	0	0
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.65685i	− 1.08866i
$28$	0	0
$29$	−4.82843	−0.896616	−0.448308	−	0.893879i	$-0.647973\pi$
$29$	−4.82843	−0.896616	−0.448308	+	0.893879i	$0.647973\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	− 6.82843i	− 1.18868i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.75736i	0.288908i	0.989512	+	0.144454i	$0.0461426\pi$
$37$	1.75736i	0.288908i	−0.989512	+	0.144454i	$0.953857\pi$
$38$	0	0
$39$	−0.828427	−0.132655
$40$	0	0
$41$	4.82843	0.754074	0.377037	−	0.926198i	$-0.376943\pi$
$41$	4.82843	0.754074	0.377037	+	0.926198i	$0.376943\pi$
$42$	0	0
$43$	2.82843i	0.431331i	0.976467	+	0.215666i	$0.0691921\pi$
$43$	2.82843i	0.431331i	−0.976467	+	0.215666i	$0.930808\pi$
$44$	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
$51$	−1.17157	−0.164053
$52$	0	0
$53$	1.07107i	0.147122i	0.997291	+	0.0735612i	$0.0234364\pi$
$53$	1.07107i	0.147122i	−0.997291	+	0.0735612i	$0.976564\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.41421i	− 0.187317i
$58$	0	0
$59$	2.82843	0.368230	0.184115	−	0.982905i	$-0.441058\pi$
$59$	2.82843	0.368230	0.184115	+	0.982905i	$0.441058\pi$
$60$	0	0
$61$	9.65685	1.23643	0.618217	−	0.786008i	$-0.287855\pi$
$61$	9.65685	1.23643	0.618217	+	0.786008i	$0.287855\pi$
$62$	0	0
$63$	2.82843i	0.356348i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 6.58579i	− 0.804582i	−0.915512	−	0.402291i	$-0.868214\pi$
$67$	− 6.58579i	− 0.804582i	0.915512	−	0.402291i	$-0.131786\pi$
$68$	0	0
$69$	−5.65685	−0.681005
$70$	0	0
$71$	−7.31371	−0.867978	−0.433989	−	0.900918i	$-0.642894\pi$
$71$	−7.31371	−0.867978	−0.433989	+	0.900918i	$0.642894\pi$
$72$	0	0
$73$	16.1421i	1.88929i	0.328088	+	0.944647i	$0.393596\pi$
$73$	16.1421i	1.88929i	−0.328088	+	0.944647i	$0.606404\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	13.6569i	1.55634i
$78$	0	0
$79$	14.8284	1.66833	0.834164	−	0.551516i	$-0.185951\pi$
$79$	14.8284	1.66833	0.834164	+	0.551516i	$0.185951\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	− 8.00000i	− 0.878114i	−0.898459	−	0.439057i	$-0.855313\pi$
$83$	− 8.00000i	− 0.878114i	0.898459	−	0.439057i	$-0.144687\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.82843i	0.732084i
$88$	0	0
$89$	8.82843	0.935811	0.467906	−	0.883778i	$-0.345009\pi$
$89$	8.82843	0.935811	0.467906	+	0.883778i	$0.345009\pi$
$90$	0	0
$91$	1.65685	0.173686
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.24264i	0.633844i	0.948452	+	0.316922i	$0.102649\pi$
$97$	6.24264i	0.633844i	−0.948452	+	0.316922i	$0.897351\pi$
$98$	0	0
$99$	4.82843	0.485275
$100$	0	0
$101$	6.34315	0.631167	0.315583	−	0.948898i	$-0.397800\pi$
$101$	6.34315	0.631167	0.315583	+	0.948898i	$0.397800\pi$
$102$	0	0
$103$	0.242641i	0.0239081i	0.999929	+	0.0119540i	$0.00380518\pi$
$103$	0.242641i	0.0239081i	−0.999929	+	0.0119540i	$0.996195\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.41421i	− 0.136717i	−0.997661	−	0.0683586i	$-0.978224\pi$
$107$	− 1.41421i	− 0.136717i	0.997661	−	0.0683586i	$-0.0217762\pi$
$108$	0	0
$109$	7.65685	0.733394	0.366697	−	0.930341i	$-0.380489\pi$
$109$	7.65685	0.733394	0.366697	+	0.930341i	$0.380489\pi$
$110$	0	0
$111$	2.48528	0.235892
$112$	0	0
$113$	− 10.7279i	− 1.00920i	−0.863354	−	0.504599i	$-0.831640\pi$
$113$	− 10.7279i	− 1.00920i	0.863354	−	0.504599i	$-0.168360\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 0.585786i	− 0.0541560i
$118$	0	0
$119$	2.34315	0.214796
$120$	0	0
$121$	12.3137	1.11943
$122$	0	0
$123$	− 6.82843i	− 0.615699i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 10.5858i	− 0.939337i	−0.882843	−	0.469668i	$-0.844374\pi$
$127$	− 10.5858i	− 0.939337i	0.882843	−	0.469668i	$-0.155626\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	−15.3137	−1.33796	−0.668982	−	0.743278i	$-0.733270\pi$
$131$	−15.3137	−1.33796	−0.668982	+	0.743278i	$0.733270\pi$
$132$	0	0
$133$	2.82843i	0.245256i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 12.8284i	− 1.09601i	−0.836476	−	0.548003i	$-0.815388\pi$
$137$	− 12.8284i	− 1.09601i	0.836476	−	0.548003i	$-0.184612\pi$
$138$	0	0
$139$	8.82843	0.748817	0.374409	−	0.927264i	$-0.377846\pi$
$139$	8.82843	0.748817	0.374409	+	0.927264i	$0.377846\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	0	0
$143$	− 2.82843i	− 0.236525i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.41421i	0.116642i
$148$	0	0
$149$	9.65685	0.791120	0.395560	−	0.918440i	$-0.370550\pi$
$149$	9.65685	0.791120	0.395560	+	0.918440i	$0.370550\pi$
$150$	0	0
$151$	−22.1421	−1.80190	−0.900951	−	0.433921i	$-0.857130\pi$
$151$	−22.1421	−1.80190	−0.900951	+	0.433921i	$0.857130\pi$
$152$	0	0
$153$	− 0.828427i	− 0.0669744i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.48528i	0.517582i	0.965933	+	0.258791i	$0.0833241\pi$
$157$	6.48528i	0.517582i	−0.965933	+	0.258791i	$0.916676\pi$
$158$	0	0
$159$	1.51472	0.120125
$160$	0	0
$161$	11.3137	0.891645
$162$	0	0
$163$	− 9.17157i	− 0.718373i	−0.933266	−	0.359187i	$-0.883054\pi$
$163$	− 9.17157i	− 0.718373i	0.933266	−	0.359187i	$-0.116946\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.8995i	1.69463i	0.531088	+	0.847317i	$0.321783\pi$
$167$	21.8995i	1.69463i	−0.531088	+	0.847317i	$0.678217\pi$
$168$	0	0
$169$	12.6569	0.973604
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	− 14.7279i	− 1.11974i	−0.828579	−	0.559872i	$-0.810850\pi$
$173$	− 14.7279i	− 1.11974i	0.828579	−	0.559872i	$-0.189150\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 4.00000i	− 0.300658i
$178$	0	0
$179$	8.48528	0.634220	0.317110	−	0.948389i	$-0.397288\pi$
$179$	8.48528	0.634220	0.317110	+	0.948389i	$0.397288\pi$
$180$	0	0
$181$	17.3137	1.28692	0.643459	−	0.765481i	$-0.277499\pi$
$181$	17.3137	1.28692	0.643459	+	0.765481i	$0.277499\pi$
$182$	0	0
$183$	− 13.6569i	− 1.00954i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 4.00000i	− 0.292509i
$188$	0	0
$189$	16.0000	1.16383
$190$	0	0
$191$	−19.3137	−1.39749	−0.698745	−	0.715370i	$-0.746258\pi$
$191$	−19.3137	−1.39749	−0.698745	+	0.715370i	$0.746258\pi$
$192$	0	0
$193$	21.0711i	1.51673i	0.651831	+	0.758364i	$0.274001\pi$
$193$	21.0711i	1.51673i	−0.651831	+	0.758364i	$0.725999\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.68629i	0.191390i	0.995411	+	0.0956952i	$0.0305074\pi$
$197$	2.68629i	0.191390i	−0.995411	+	0.0956952i	$0.969493\pi$
$198$	0	0
$199$	−5.65685	−0.401004	−0.200502	−	0.979693i	$-0.564257\pi$
$199$	−5.65685	−0.401004	−0.200502	+	0.979693i	$0.564257\pi$
$200$	0	0
$201$	−9.31371	−0.656938
$202$	0	0
$203$	− 13.6569i	− 0.958523i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 4.00000i	− 0.278019i
$208$	0	0
$209$	4.82843	0.333989
$210$	0	0
$211$	0.485281	0.0334081	0.0167041	−	0.999860i	$-0.494683\pi$
$211$	0.485281	0.0334081	0.0167041	+	0.999860i	$0.494683\pi$
$212$	0	0
$213$	10.3431i	0.708701i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	22.8284	1.54260
$220$	0	0
$221$	−0.485281	−0.0326436
$222$	0	0
$223$	− 10.5858i	− 0.708877i	−0.935079	−	0.354438i	$-0.884672\pi$
$223$	− 10.5858i	− 0.708877i	0.935079	−	0.354438i	$-0.115328\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 24.0416i	− 1.59570i	−0.602857	−	0.797850i	$-0.705971\pi$
$227$	− 24.0416i	− 1.59570i	0.602857	−	0.797850i	$-0.294029\pi$
$228$	0	0
$229$	11.3137	0.747631	0.373815	−	0.927503i	$-0.378049\pi$
$229$	11.3137	0.747631	0.373815	+	0.927503i	$0.378049\pi$
$230$	0	0
$231$	19.3137	1.27075
$232$	0	0
$233$	10.0000i	0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$	10.0000i	0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 20.9706i	− 1.36218i
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	−20.1421	−1.29747	−0.648735	−	0.761015i	$-0.724701\pi$
$241$	−20.1421	−1.29747	−0.648735	+	0.761015i	$0.724701\pi$
$242$	0	0
$243$	− 9.89949i	− 0.635053i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 0.585786i	− 0.0372727i
$248$	0	0
$249$	−11.3137	−0.716977
$250$	0	0
$251$	−4.97056	−0.313739	−0.156870	−	0.987619i	$-0.550140\pi$
$251$	−4.97056	−0.313739	−0.156870	+	0.987619i	$0.550140\pi$
$252$	0	0
$253$	− 19.3137i	− 1.21424i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 23.8995i	− 1.49081i	−0.666612	−	0.745405i	$-0.732256\pi$
$257$	− 23.8995i	− 1.49081i	0.666612	−	0.745405i	$-0.267744\pi$
$258$	0	0
$259$	−4.97056	−0.308856
$260$	0	0
$261$	−4.82843	−0.298872
$262$	0	0
$263$	− 4.68629i	− 0.288969i	−0.989507	−	0.144485i	$-0.953848\pi$
$263$	− 4.68629i	− 0.288969i	0.989507	−	0.144485i	$-0.0461524\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 12.4853i	− 0.764087i
$268$	0	0
$269$	6.68629	0.407670	0.203835	−	0.979005i	$-0.434659\pi$
$269$	6.68629	0.407670	0.203835	+	0.979005i	$0.434659\pi$
$270$	0	0
$271$	−26.4853	−1.60887	−0.804433	−	0.594043i	$-0.797531\pi$
$271$	−26.4853	−1.60887	−0.804433	+	0.594043i	$0.797531\pi$
$272$	0	0
$273$	− 2.34315i	− 0.141814i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	24.8284i	1.49180i	0.666060	+	0.745898i	$0.267979\pi$
$277$	24.8284i	1.49180i	−0.666060	+	0.745898i	$0.732021\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.8284	1.00390	0.501950	−	0.864897i	$-0.332616\pi$
$281$	16.8284	1.00390	0.501950	+	0.864897i	$0.332616\pi$
$282$	0	0
$283$	− 17.6569i	− 1.04959i	−0.851228	−	0.524796i	$-0.824142\pi$
$283$	− 17.6569i	− 1.04959i	0.851228	−	0.524796i	$-0.175858\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	13.6569i	0.806139i
$288$	0	0
$289$	16.3137	0.959630
$290$	0	0
$291$	8.82843	0.517532
$292$	0	0
$293$	4.38478i	0.256161i	0.991764	+	0.128081i	$0.0408816\pi$
$293$	4.38478i	0.256161i	−0.991764	+	0.128081i	$0.959118\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 27.3137i	− 1.58490i
$298$	0	0
$299$	−2.34315	−0.135508
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	− 8.97056i	− 0.515345i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.41421i	− 0.0807134i	−0.999185	−	0.0403567i	$-0.987151\pi$
$307$	− 1.41421i	− 0.0807134i	0.999185	−	0.0403567i	$-0.0128494\pi$
$308$	0	0
$309$	0.343146	0.0195209
$310$	0	0
$311$	24.8284	1.40789	0.703945	−	0.710254i	$-0.251420\pi$
$311$	24.8284	1.40789	0.703945	+	0.710254i	$0.251420\pi$
$312$	0	0
$313$	− 17.3137i	− 0.978629i	−0.872107	−	0.489314i	$-0.837247\pi$
$313$	− 17.3137i	− 0.978629i	0.872107	−	0.489314i	$-0.162753\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	19.8995i	1.11767i	0.829280	+	0.558833i	$0.188751\pi$
$317$	19.8995i	1.11767i	−0.829280	+	0.558833i	$0.811249\pi$
$318$	0	0
$319$	−23.3137	−1.30532
$320$	0	0
$321$	−2.00000	−0.111629
$322$	0	0
$323$	− 0.828427i	− 0.0460949i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 10.8284i	− 0.598813i
$328$	0	0
$329$	−24.0000	−1.32316
$330$	0	0
$331$	33.4558	1.83890	0.919450	−	0.393208i	$-0.128635\pi$
$331$	33.4558	1.83890	0.919450	+	0.393208i	$0.128635\pi$
$332$	0	0
$333$	1.75736i	0.0963027i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	16.3848i	0.892536i	0.894899	+	0.446268i	$0.147247\pi$
$337$	16.3848i	0.892536i	−0.894899	+	0.446268i	$0.852753\pi$
$338$	0	0
$339$	−15.1716	−0.824007
$340$	0	0
$341$	0	0
$342$	0	0
$343$	16.9706i	0.916324i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	36.2843i	1.94784i	0.226888	+	0.973921i	$0.427145\pi$
$347$	36.2843i	1.94784i	−0.226888	+	0.973921i	$0.572855\pi$
$348$	0	0
$349$	−23.6569	−1.26632	−0.633161	−	0.774020i	$-0.718243\pi$
$349$	−23.6569	−1.26632	−0.633161	+	0.774020i	$0.718243\pi$
$350$	0	0
$351$	−3.31371	−0.176873
$352$	0	0
$353$	− 16.1421i	− 0.859159i	−0.903029	−	0.429580i	$-0.858662\pi$
$353$	− 16.1421i	− 0.859159i	0.903029	−	0.429580i	$-0.141338\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 3.31371i	− 0.175380i
$358$	0	0
$359$	−31.4558	−1.66018	−0.830088	−	0.557632i	$-0.811710\pi$
$359$	−31.4558	−1.66018	−0.830088	+	0.557632i	$0.811710\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 17.4142i	− 0.914009i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 31.1127i	− 1.62407i	−0.583609	−	0.812035i	$-0.698360\pi$
$367$	− 31.1127i	− 1.62407i	0.583609	−	0.812035i	$-0.301640\pi$
$368$	0	0
$369$	4.82843	0.251358
$370$	0	0
$371$	−3.02944	−0.157281
$372$	0	0
$373$	2.44365i	0.126527i	0.997997	+	0.0632637i	$0.0201509\pi$
$373$	2.44365i	0.126527i	−0.997997	+	0.0632637i	$0.979849\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.82843i	0.145671i
$378$	0	0
$379$	1.65685	0.0851069	0.0425534	−	0.999094i	$-0.486451\pi$
$379$	1.65685	0.0851069	0.0425534	+	0.999094i	$0.486451\pi$
$380$	0	0
$381$	−14.9706	−0.766965
$382$	0	0
$383$	− 7.27208i	− 0.371586i	−0.982589	−	0.185793i	$-0.940515\pi$
$383$	− 7.27208i	− 0.371586i	0.982589	−	0.185793i	$-0.0594854\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.82843i	0.143777i
$388$	0	0
$389$	13.3137	0.675032	0.337516	−	0.941320i	$-0.390413\pi$
$389$	13.3137	0.675032	0.337516	+	0.941320i	$0.390413\pi$
$390$	0	0
$391$	−3.31371	−0.167581
$392$	0	0
$393$	21.6569i	1.09244i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	31.1716i	1.56446i	0.622992	+	0.782228i	$0.285917\pi$
$397$	31.1716i	1.56446i	−0.622992	+	0.782228i	$0.714083\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	0	0
$401$	−18.9706	−0.947345	−0.473672	−	0.880701i	$-0.657072\pi$
$401$	−18.9706	−0.947345	−0.473672	+	0.880701i	$0.657072\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.48528i	0.420600i
$408$	0	0
$409$	−26.2843	−1.29967	−0.649837	−	0.760074i	$-0.725163\pi$
$409$	−26.2843	−1.29967	−0.649837	+	0.760074i	$0.725163\pi$
$410$	0	0
$411$	−18.1421	−0.894886
$412$	0	0
$413$	8.00000i	0.393654i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 12.4853i	− 0.611407i
$418$	0	0
$419$	9.65685	0.471768	0.235884	−	0.971781i	$-0.424201\pi$
$419$	9.65685	0.471768	0.235884	+	0.971781i	$0.424201\pi$
$420$	0	0
$421$	−15.1716	−0.739417	−0.369709	−	0.929148i	$-0.620542\pi$
$421$	−15.1716	−0.739417	−0.369709	+	0.929148i	$0.620542\pi$
$422$	0	0
$423$	8.48528i	0.412568i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	27.3137i	1.32180i
$428$	0	0
$429$	−4.00000	−0.193122
$430$	0	0
$431$	−9.17157	−0.441779	−0.220890	−	0.975299i	$-0.570896\pi$
$431$	−9.17157	−0.441779	−0.220890	+	0.975299i	$0.570896\pi$
$432$	0	0
$433$	− 30.0416i	− 1.44371i	−0.692045	−	0.721854i	$-0.743290\pi$
$433$	− 30.0416i	− 1.44371i	0.692045	−	0.721854i	$-0.256710\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4.00000i	− 0.191346i
$438$	0	0
$439$	14.1421	0.674967	0.337484	−	0.941331i	$-0.390424\pi$
$439$	14.1421	0.674967	0.337484	+	0.941331i	$0.390424\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	− 23.3137i	− 1.10767i	−0.832627	−	0.553834i	$-0.813164\pi$
$443$	− 23.3137i	− 1.10767i	0.832627	−	0.553834i	$-0.186836\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 13.6569i	− 0.645947i
$448$	0	0
$449$	30.2843	1.42920	0.714602	−	0.699532i	$-0.246608\pi$
$449$	30.2843	1.42920	0.714602	+	0.699532i	$0.246608\pi$
$450$	0	0
$451$	23.3137	1.09780
$452$	0	0
$453$	31.3137i	1.47125i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	22.2843i	1.04241i	0.853430	+	0.521207i	$0.174518\pi$
$457$	22.2843i	1.04241i	−0.853430	+	0.521207i	$0.825482\pi$
$458$	0	0
$459$	−4.68629	−0.218737
$460$	0	0
$461$	36.6274	1.70591	0.852954	−	0.521985i	$-0.174808\pi$
$461$	36.6274	1.70591	0.852954	+	0.521985i	$0.174808\pi$
$462$	0	0
$463$	18.6274i	0.865689i	0.901468	+	0.432845i	$0.142490\pi$
$463$	18.6274i	0.865689i	−0.901468	+	0.432845i	$0.857510\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	24.0000i	1.11059i	0.831654	+	0.555294i	$0.187394\pi$
$467$	24.0000i	1.11059i	−0.831654	+	0.555294i	$0.812606\pi$
$468$	0	0
$469$	18.6274	0.860134
$470$	0	0
$471$	9.17157	0.422604
$472$	0	0
$473$	13.6569i	0.627943i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.07107i	0.0490408i
$478$	0	0
$479$	27.4558	1.25449	0.627245	−	0.778822i	$-0.284183\pi$
$479$	27.4558	1.25449	0.627245	+	0.778822i	$0.284183\pi$
$480$	0	0
$481$	1.02944	0.0469383
$482$	0	0
$483$	− 16.0000i	− 0.728025i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 3.55635i	− 0.161154i	−0.996748	−	0.0805768i	$-0.974324\pi$
$487$	− 3.55635i	− 0.161154i	0.996748	−	0.0805768i	$-0.0256762\pi$
$488$	0	0
$489$	−12.9706	−0.586549
$490$	0	0
$491$	20.9706	0.946388	0.473194	−	0.880958i	$-0.343101\pi$
$491$	20.9706	0.946388	0.473194	+	0.880958i	$0.343101\pi$
$492$	0	0
$493$	4.00000i	0.180151i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 20.6863i	− 0.927907i
$498$	0	0
$499$	24.1421	1.08075	0.540375	−	0.841424i	$-0.318282\pi$
$499$	24.1421	1.08075	0.540375	+	0.841424i	$0.318282\pi$
$500$	0	0
$501$	30.9706	1.38366
$502$	0	0
$503$	− 9.65685i	− 0.430578i	−0.976550	−	0.215289i	$-0.930931\pi$
$503$	− 9.65685i	− 0.430578i	0.976550	−	0.215289i	$-0.0690693\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 17.8995i	− 0.794944i
$508$	0	0
$509$	−35.4558	−1.57155	−0.785776	−	0.618511i	$-0.787736\pi$
$509$	−35.4558	−1.57155	−0.785776	+	0.618511i	$0.787736\pi$
$510$	0	0
$511$	−45.6569	−2.01974
$512$	0	0
$513$	− 5.65685i	− 0.249756i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	40.9706i	1.80188i
$518$	0	0
$519$	−20.8284	−0.914266
$520$	0	0
$521$	37.3137	1.63474	0.817372	−	0.576111i	$-0.195430\pi$
$521$	37.3137	1.63474	0.817372	+	0.576111i	$0.195430\pi$
$522$	0	0
$523$	27.7574i	1.21374i	0.794799	+	0.606872i	$0.207576\pi$
$523$	27.7574i	1.21374i	−0.794799	+	0.606872i	$0.792424\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	2.82843	0.122743
$532$	0	0
$533$	− 2.82843i	− 0.122513i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 12.0000i	− 0.517838i
$538$	0	0
$539$	−4.82843	−0.207975
$540$	0	0
$541$	−20.2843	−0.872089	−0.436044	−	0.899925i	$-0.643621\pi$
$541$	−20.2843	−0.872089	−0.436044	+	0.899925i	$0.643621\pi$
$542$	0	0
$543$	− 24.4853i	− 1.05076i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 24.9289i	− 1.06588i	−0.846152	−	0.532942i	$-0.821086\pi$
$547$	− 24.9289i	− 1.06588i	0.846152	−	0.532942i	$-0.178914\pi$
$548$	0	0
$549$	9.65685	0.412144
$550$	0	0
$551$	−4.82843	−0.205698
$552$	0	0
$553$	41.9411i	1.78352i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	33.7990i	1.43211i	0.698044	+	0.716055i	$0.254054\pi$
$557$	33.7990i	1.43211i	−0.698044	+	0.716055i	$0.745946\pi$
$558$	0	0
$559$	1.65685	0.0700775
$560$	0	0
$561$	−5.65685	−0.238833
$562$	0	0
$563$	− 4.24264i	− 0.178806i	−0.995996	−	0.0894030i	$-0.971504\pi$
$563$	− 4.24264i	− 0.178806i	0.995996	−	0.0894030i	$-0.0284959\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 14.1421i	− 0.593914i
$568$	0	0
$569$	−35.9411	−1.50673	−0.753365	−	0.657602i	$-0.771571\pi$
$569$	−35.9411	−1.50673	−0.753365	+	0.657602i	$0.771571\pi$
$570$	0	0
$571$	−13.5147	−0.565573	−0.282787	−	0.959183i	$-0.591259\pi$
$571$	−13.5147	−0.565573	−0.282787	+	0.959183i	$0.591259\pi$
$572$	0	0
$573$	27.3137i	1.14105i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 0.627417i	− 0.0261197i	−0.999915	−	0.0130599i	$-0.995843\pi$
$577$	− 0.627417i	− 0.0261197i	0.999915	−	0.0130599i	$-0.00415720\pi$
$578$	0	0
$579$	29.7990	1.23840
$580$	0	0
$581$	22.6274	0.938743
$582$	0	0
$583$	5.17157i	0.214185i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.65685i	0.233483i	0.993162	+	0.116742i	$0.0372450\pi$
$587$	5.65685i	0.233483i	−0.993162	+	0.116742i	$0.962755\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	3.79899	0.156270
$592$	0	0
$593$	28.6274i	1.17559i	0.809011	+	0.587794i	$0.200003\pi$
$593$	28.6274i	1.17559i	−0.809011	+	0.587794i	$0.799997\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.00000i	0.327418i
$598$	0	0
$599$	22.1421	0.904703	0.452352	−	0.891840i	$-0.350585\pi$
$599$	22.1421	0.904703	0.452352	+	0.891840i	$0.350585\pi$
$600$	0	0
$601$	2.48528	0.101377	0.0506884	−	0.998715i	$-0.483858\pi$
$601$	2.48528	0.101377	0.0506884	+	0.998715i	$0.483858\pi$
$602$	0	0
$603$	− 6.58579i	− 0.268194i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 16.2426i	− 0.659268i	−0.944109	−	0.329634i	$-0.893075\pi$
$607$	− 16.2426i	− 0.659268i	0.944109	−	0.329634i	$-0.106925\pi$
$608$	0	0
$609$	−19.3137	−0.782631
$610$	0	0
$611$	4.97056	0.201087
$612$	0	0
$613$	21.7990i	0.880453i	0.897887	+	0.440226i	$0.145102\pi$
$613$	21.7990i	0.880453i	−0.897887	+	0.440226i	$0.854898\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.79899i	0.0724246i	0.999344	+	0.0362123i	$0.0115293\pi$
$617$	1.79899i	0.0724246i	−0.999344	+	0.0362123i	$0.988471\pi$
$618$	0	0
$619$	−11.8579	−0.476608	−0.238304	−	0.971191i	$-0.576591\pi$
$619$	−11.8579	−0.476608	−0.238304	+	0.971191i	$0.576591\pi$
$620$	0	0
$621$	−22.6274	−0.908007
$622$	0	0
$623$	24.9706i	1.00042i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 6.82843i	− 0.272701i
$628$	0	0
$629$	1.45584	0.0580483
$630$	0	0
$631$	−17.5147	−0.697250	−0.348625	−	0.937262i	$-0.613351\pi$
$631$	−17.5147	−0.697250	−0.348625	+	0.937262i	$0.613351\pi$
$632$	0	0
$633$	− 0.686292i	− 0.0272776i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0.585786i	0.0232097i
$638$	0	0
$639$	−7.31371	−0.289326
$640$	0	0
$641$	−23.4558	−0.926450	−0.463225	−	0.886241i	$-0.653308\pi$
$641$	−23.4558	−0.926450	−0.463225	+	0.886241i	$0.653308\pi$
$642$	0	0
$643$	12.6863i	0.500298i	0.968207	+	0.250149i	$0.0804797\pi$
$643$	12.6863i	0.500298i	−0.968207	+	0.250149i	$0.919520\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 39.3137i	− 1.54558i	−0.634661	−	0.772791i	$-0.718860\pi$
$647$	− 39.3137i	− 1.54558i	0.634661	−	0.772791i	$-0.281140\pi$
$648$	0	0
$649$	13.6569	0.536078
$650$	0	0
$651$	0	0
$652$	0	0
$653$	16.1421i	0.631691i	0.948811	+	0.315845i	$0.102288\pi$
$653$	16.1421i	0.631691i	−0.948811	+	0.315845i	$0.897712\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	16.1421i	0.629765i
$658$	0	0
$659$	11.5147	0.448550	0.224275	−	0.974526i	$-0.427999\pi$
$659$	11.5147	0.448550	0.224275	+	0.974526i	$0.427999\pi$
$660$	0	0
$661$	3.17157	0.123360	0.0616799	−	0.998096i	$-0.480354\pi$
$661$	3.17157	0.123360	0.0616799	+	0.998096i	$0.480354\pi$
$662$	0	0
$663$	0.686292i	0.0266534i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	19.3137i	0.747830i
$668$	0	0
$669$	−14.9706	−0.578795
$670$	0	0
$671$	46.6274	1.80003
$672$	0	0
$673$	28.3848i	1.09415i	0.837083	+	0.547076i	$0.184259\pi$
$673$	28.3848i	1.09415i	−0.837083	+	0.547076i	$0.815741\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 44.3848i	− 1.70585i	−0.522037	−	0.852923i	$-0.674828\pi$
$677$	− 44.3848i	− 1.70585i	0.522037	−	0.852923i	$-0.325172\pi$
$678$	0	0
$679$	−17.6569	−0.677608
$680$	0	0
$681$	−34.0000	−1.30288
$682$	0	0
$683$	− 9.89949i	− 0.378794i	−0.981901	−	0.189397i	$-0.939347\pi$
$683$	− 9.89949i	− 0.378794i	0.981901	−	0.189397i	$-0.0606533\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 16.0000i	− 0.610438i
$688$	0	0
$689$	0.627417	0.0239027
$690$	0	0
$691$	−40.8284	−1.55319	−0.776593	−	0.630002i	$-0.783054\pi$
$691$	−40.8284	−1.55319	−0.776593	+	0.630002i	$0.783054\pi$
$692$	0	0
$693$	13.6569i	0.518781i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 4.00000i	− 0.151511i
$698$	0	0
$699$	14.1421	0.534905
$700$	0	0
$701$	−24.0000	−0.906467	−0.453234	−	0.891392i	$-0.649730\pi$
$701$	−24.0000	−0.906467	−0.453234	+	0.891392i	$0.649730\pi$
$702$	0	0
$703$	1.75736i	0.0662801i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	17.9411i	0.674745i
$708$	0	0
$709$	−38.2843	−1.43780	−0.718898	−	0.695116i	$-0.755353\pi$
$709$	−38.2843	−1.43780	−0.718898	+	0.695116i	$0.755353\pi$
$710$	0	0
$711$	14.8284	0.556109
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	11.3137i	0.422518i
$718$	0	0
$719$	−19.4558	−0.725581	−0.362790	−	0.931871i	$-0.618176\pi$
$719$	−19.4558	−0.725581	−0.362790	+	0.931871i	$0.618176\pi$
$720$	0	0
$721$	−0.686292	−0.0255588
$722$	0	0
$723$	28.4853i	1.05938i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	9.45584i	0.350698i	0.984506	+	0.175349i	$0.0561054\pi$
$727$	9.45584i	0.350698i	−0.984506	+	0.175349i	$0.943895\pi$
$728$	0	0
$729$	−29.0000	−1.07407
$730$	0	0
$731$	2.34315	0.0866644
$732$	0	0
$733$	− 15.6569i	− 0.578299i	−0.957284	−	0.289150i	$-0.906628\pi$
$733$	− 15.6569i	− 0.578299i	0.957284	−	0.289150i	$-0.0933725\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 31.7990i	− 1.17133i
$738$	0	0
$739$	−23.3137	−0.857609	−0.428804	−	0.903397i	$-0.641065\pi$
$739$	−23.3137	−0.857609	−0.428804	+	0.903397i	$0.641065\pi$
$740$	0	0
$741$	−0.828427	−0.0304330
$742$	0	0
$743$	24.2426i	0.889376i	0.895685	+	0.444688i	$0.146685\pi$
$743$	24.2426i	0.889376i	−0.895685	+	0.444688i	$0.853315\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 8.00000i	− 0.292705i
$748$	0	0
$749$	4.00000	0.146157
$750$	0	0
$751$	−27.5147	−1.00403	−0.502013	−	0.864860i	$-0.667407\pi$
$751$	−27.5147	−1.00403	−0.502013	+	0.864860i	$0.667407\pi$
$752$	0	0
$753$	7.02944i	0.256167i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	34.2843i	1.24608i	0.782189	+	0.623042i	$0.214103\pi$
$757$	34.2843i	1.24608i	−0.782189	+	0.623042i	$0.785897\pi$
$758$	0	0
$759$	−27.3137	−0.991425
$760$	0	0
$761$	46.6274	1.69024	0.845121	−	0.534575i	$-0.179528\pi$
$761$	46.6274	1.69024	0.845121	+	0.534575i	$0.179528\pi$
$762$	0	0
$763$	21.6569i	0.784031i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.65685i	− 0.0598255i
$768$	0	0
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	−33.7990	−1.21724
$772$	0	0
$773$	− 4.87006i	− 0.175164i	−0.996157	−	0.0875819i	$-0.972086\pi$
$773$	− 4.87006i	− 0.175164i	0.996157	−	0.0875819i	$-0.0279139\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	7.02944i	0.252180i
$778$	0	0
$779$	4.82843	0.172996
$780$	0	0
$781$	−35.3137	−1.26362
$782$	0	0
$783$	27.3137i	0.976112i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.41421i	0.0504113i	0.999682	+	0.0252056i	$0.00802405\pi$
$787$	1.41421i	0.0504113i	−0.999682	+	0.0252056i	$0.991976\pi$
$788$	0	0
$789$	−6.62742	−0.235942
$790$	0	0
$791$	30.3431	1.07888
$792$	0	0
$793$	− 5.65685i	− 0.200881i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 31.6985i	− 1.12282i	−0.827538	−	0.561409i	$-0.810259\pi$
$797$	− 31.6985i	− 1.12282i	0.827538	−	0.561409i	$-0.189741\pi$
$798$	0	0
$799$	7.02944	0.248684
$800$	0	0
$801$	8.82843	0.311937
$802$	0	0
$803$	77.9411i	2.75048i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 9.45584i	− 0.332861i
$808$	0	0
$809$	−38.0000	−1.33601	−0.668004	−	0.744157i	$-0.732851\pi$
$809$	−38.0000	−1.33601	−0.668004	+	0.744157i	$0.732851\pi$
$810$	0	0
$811$	−23.3137	−0.818655	−0.409328	−	0.912388i	$-0.634237\pi$
$811$	−23.3137	−0.818655	−0.409328	+	0.912388i	$0.634237\pi$
$812$	0	0
$813$	37.4558i	1.31363i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.82843i	0.0989541i
$818$	0	0
$819$	1.65685	0.0578952
$820$	0	0
$821$	29.5980	1.03298	0.516488	−	0.856294i	$-0.327239\pi$
$821$	29.5980	1.03298	0.516488	+	0.856294i	$0.327239\pi$
$822$	0	0
$823$	− 0.485281i	− 0.0169158i	−0.999964	−	0.00845792i	$-0.997308\pi$
$823$	− 0.485281i	− 0.0169158i	0.999964	−	0.00845792i	$-0.00269227\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 45.6985i	− 1.58909i	−0.607204	−	0.794546i	$-0.707709\pi$
$827$	− 45.6985i	− 1.58909i	0.607204	−	0.794546i	$-0.292291\pi$
$828$	0	0
$829$	−29.5980	−1.02798	−0.513990	−	0.857796i	$-0.671833\pi$
$829$	−29.5980	−1.02798	−0.513990	+	0.857796i	$0.671833\pi$
$830$	0	0
$831$	35.1127	1.21805
$832$	0	0
$833$	0.828427i	0.0287033i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−39.7990	−1.37401	−0.687007	−	0.726651i	$-0.741076\pi$
$839$	−39.7990	−1.37401	−0.687007	+	0.726651i	$0.741076\pi$
$840$	0	0
$841$	−5.68629	−0.196079
$842$	0	0
$843$	− 23.7990i	− 0.819681i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	34.8284i	1.19672i
$848$	0	0
$849$	−24.9706	−0.856987
$850$	0	0
$851$	7.02944	0.240966
$852$	0	0
$853$	− 39.9411i	− 1.36756i	−0.729689	−	0.683779i	$-0.760335\pi$
$853$	− 39.9411i	− 1.36756i	0.729689	−	0.683779i	$-0.239665\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 11.2132i	− 0.383036i	−0.981489	−	0.191518i	$-0.938659\pi$
$857$	− 11.2132i	− 0.383036i	0.981489	−	0.191518i	$-0.0613410\pi$
$858$	0	0
$859$	−17.6569	−0.602444	−0.301222	−	0.953554i	$-0.597395\pi$
$859$	−17.6569	−0.602444	−0.301222	+	0.953554i	$0.597395\pi$
$860$	0	0
$861$	19.3137	0.658209
$862$	0	0
$863$	− 4.04163i	− 0.137579i	−0.997631	−	0.0687894i	$-0.978086\pi$
$863$	− 4.04163i	− 0.137579i	0.997631	−	0.0687894i	$-0.0219136\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 23.0711i	− 0.783535i
$868$	0	0
$869$	71.5980	2.42880
$870$	0	0
$871$	−3.85786	−0.130719
$872$	0	0
$873$	6.24264i	0.211281i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 6.72792i	− 0.227186i	−0.993527	−	0.113593i	$-0.963764\pi$
$877$	− 6.72792i	− 0.227186i	0.993527	−	0.113593i	$-0.0362360\pi$
$878$	0	0
$879$	6.20101	0.209155
$880$	0	0
$881$	9.65685	0.325348	0.162674	−	0.986680i	$-0.447988\pi$
$881$	9.65685	0.325348	0.162674	+	0.986680i	$0.447988\pi$
$882$	0	0
$883$	5.17157i	0.174037i	0.996207	+	0.0870186i	$0.0277340\pi$
$883$	5.17157i	0.174037i	−0.996207	+	0.0870186i	$0.972266\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 40.7279i	− 1.36751i	−0.729712	−	0.683755i	$-0.760346\pi$
$887$	− 40.7279i	− 1.36751i	0.729712	−	0.683755i	$-0.239654\pi$
$888$	0	0
$889$	29.9411	1.00419
$890$	0	0
$891$	−24.1421	−0.808792
$892$	0	0
$893$	8.48528i	0.283949i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.31371i	0.110642i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0.887302	0.0295603
$902$	0	0
$903$	11.3137i	0.376497i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 0.0416306i	− 0.00138232i	−1.00000		0.000691160i	$-0.999780\pi$
$907$	− 0.0416306i	− 0.00138232i	1.00000		0.000691160i	$-0.000220003\pi$
$908$	0	0
$909$	6.34315	0.210389
$910$	0	0
$911$	−14.8284	−0.491288	−0.245644	−	0.969360i	$-0.578999\pi$
$911$	−14.8284	−0.491288	−0.245644	+	0.969360i	$0.578999\pi$
$912$	0	0
$913$	− 38.6274i	− 1.27838i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 43.3137i	− 1.43034i
$918$	0	0
$919$	−49.9411	−1.64741	−0.823703	−	0.567022i	$-0.808096\pi$
$919$	−49.9411	−1.64741	−0.823703	+	0.567022i	$0.808096\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	0	0
$923$	4.28427i	0.141019i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0.242641i	0.00796937i
$928$	0	0
$929$	−18.6863	−0.613077	−0.306539	−	0.951858i	$-0.599171\pi$
$929$	−18.6863	−0.613077	−0.306539	+	0.951858i	$0.599171\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	− 35.1127i	− 1.14954i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.54416i	0.148451i	0.997241	+	0.0742256i	$0.0236485\pi$
$937$	4.54416i	0.148451i	−0.997241	+	0.0742256i	$0.976352\pi$
$938$	0	0
$939$	−24.4853	−0.799047
$940$	0	0
$941$	−57.5980	−1.87764	−0.938820	−	0.344408i	$-0.888080\pi$
$941$	−57.5980	−1.87764	−0.938820	+	0.344408i	$0.888080\pi$
$942$	0	0
$943$	− 19.3137i	− 0.628941i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	24.6863i	0.802197i	0.916035	+	0.401098i	$0.131371\pi$
$947$	24.6863i	0.802197i	−0.916035	+	0.401098i	$0.868629\pi$
$948$	0	0
$949$	9.45584	0.306950
$950$	0	0
$951$	28.1421	0.912571
$952$	0	0
$953$	9.27208i	0.300352i	0.988659	+	0.150176i	$0.0479840\pi$
$953$	9.27208i	0.300352i	−0.988659	+	0.150176i	$0.952016\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	32.9706i	1.06579i
$958$	0	0
$959$	36.2843	1.17168
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	− 1.41421i	− 0.0455724i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	26.3431i	0.847138i	0.905864	+	0.423569i	$0.139223\pi$
$967$	26.3431i	0.847138i	−0.905864	+	0.423569i	$0.860777\pi$
$968$	0	0
$969$	−1.17157	−0.0376363
$970$	0	0
$971$	−41.9411	−1.34595	−0.672977	−	0.739663i	$-0.734985\pi$
$971$	−41.9411	−1.34595	−0.672977	+	0.739663i	$0.734985\pi$
$972$	0	0
$973$	24.9706i	0.800519i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	33.8406i	1.08266i	0.840811	+	0.541329i	$0.182079\pi$
$977$	33.8406i	1.08266i	−0.840811	+	0.541329i	$0.817921\pi$
$978$	0	0
$979$	42.6274	1.36238
$980$	0	0
$981$	7.65685	0.244465
$982$	0	0
$983$	44.0416i	1.40471i	0.711827	+	0.702355i	$0.247868\pi$
$983$	44.0416i	1.40471i	−0.711827	+	0.702355i	$0.752132\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	33.9411i	1.08036i
$988$	0	0
$989$	11.3137	0.359755
$990$	0	0
$991$	−28.7696	−0.913895	−0.456947	−	0.889494i	$-0.651057\pi$
$991$	−28.7696	−0.913895	−0.456947	+	0.889494i	$0.651057\pi$
$992$	0	0
$993$	− 47.3137i	− 1.50146i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	7.17157i	0.227126i	0.993531	+	0.113563i	$0.0362264\pi$
$997$	7.17157i	0.227126i	−0.993531	+	0.113563i	$0.963774\pi$
$998$	0	0
$999$	9.94113	0.314523

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.i.3649.2		4
5.2	odd	4		3800.2.a.n.1.1		2
5.3	odd	4		760.2.a.f.1.2	✓	2
5.4	even	2	inner	3800.2.d.i.3649.4		4
15.8	even	4		6840.2.a.z.1.2		2
20.3	even	4		1520.2.a.m.1.1		2
20.7	even	4		7600.2.a.ba.1.2		2
40.3	even	4		6080.2.a.bg.1.2		2
40.13	odd	4		6080.2.a.bf.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.f.1.2	✓	2	5.3	odd	4
1520.2.a.m.1.1		2	20.3	even	4
3800.2.a.n.1.1		2	5.2	odd	4
3800.2.d.i.3649.2		4	1.1	even	1	trivial
3800.2.d.i.3649.4		4	5.4	even	2	inner
6080.2.a.bf.1.1		2	40.13	odd	4
6080.2.a.bg.1.2		2	40.3	even	4
6840.2.a.z.1.2		2	15.8	even	4
7600.2.a.ba.1.2		2	20.7	even	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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{3} +2.82843i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.41421i q^{3} +2.82843i q^{7} +1.00000 q^{9} +4.82843 q^{11} -0.585786i q^{13} -0.828427i q^{17} +1.00000 q^{19} +4.00000 q^{21} -4.00000i q^{23} -5.65685i q^{27} -4.82843 q^{29} -6.82843i q^{33} +1.75736i q^{37} -0.828427 q^{39} +4.82843 q^{41} +2.82843i q^{43} +8.48528i q^{47} -1.00000 q^{49} -1.17157 q^{51} +1.07107i q^{53} -1.41421i q^{57} +2.82843 q^{59} +9.65685 q^{61} +2.82843i q^{63} -6.58579i q^{67} -5.65685 q^{69} -7.31371 q^{71} +16.1421i q^{73} +13.6569i q^{77} +14.8284 q^{79} -5.00000 q^{81} -8.00000i q^{83} +6.82843i q^{87} +8.82843 q^{89} +1.65685 q^{91} +6.24264i q^{97} +4.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^9	\( 4 q + 4 q^{9} + 8 q^{11} + 4 q^{19} + 16 q^{21} - 8 q^{29} + 8 q^{39} + 8 q^{41} - 4 q^{49} - 16 q^{51} + 16 q^{61} + 16 q^{71} + 48 q^{79} - 20 q^{81} + 24 q^{89} - 16 q^{91} + 8 q^{99}+O(q^{100}) \) 4 * q + 4 * q^9 + 8 * q^11 + 4 * q^19 + 16 * q^21 - 8 * q^29 + 8 * q^39 + 8 * q^41 - 4 * q^49 - 16 * q^51 + 16 * q^61 + 16 * q^71 + 48 * q^79 - 20 * q^81 + 24 * q^89 - 16 * q^91 + 8 * q^99

Introduction

L-functions

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Database

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