LMFDB - Embedded newform 3800.2.d.k.3649.3

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:	$6$
Coefficient field:	6.0.5161984.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 4x^{3} + 25x^{2} - 20x + 8 $ x^6 - 4x^3 + 25x^2 - 20*x + 8
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			3649.3
Root			$-1.75233 + 1.75233i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.k.3649.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.363328i q^{3} +1.14134i q^{7} +2.86799 q^{9} +O(q^{10})$	$q-0.363328i q^{3} +1.14134i q^{7} +2.86799 q^{9} -2.72666 q^{11} +4.64600i q^{13} -0.858664i q^{17} +1.00000 q^{19} +0.414680 q^{21} +4.41468i q^{23} -2.13201i q^{27} -9.42401 q^{29} -10.2827 q^{31} +0.990671i q^{33} +6.77801i q^{37} +1.68802 q^{39} -7.55602 q^{41} -9.29200i q^{43} -7.00933i q^{47} +5.69735 q^{49} -0.311977 q^{51} +8.64600i q^{53} -0.363328i q^{57} +5.14134 q^{59} +9.45331 q^{61} +3.27334i q^{63} -13.6553i q^{67} +1.60398 q^{69} -5.45331 q^{71} -6.87732i q^{73} -3.11203i q^{77} -17.2920 q^{79} +7.82936 q^{81} +14.2827i q^{83} +3.42401i q^{87} -13.0093 q^{89} -5.30265 q^{91} +3.73599i q^{93} -9.68463i q^{97} -7.82003 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 8 q^{9}+O(q^{10}) $ 6 * q - 8 * q^9	$ 6 q - 8 q^{9} - 8 q^{11} + 6 q^{19} - 6 q^{21} - 6 q^{29} - 28 q^{31} + 10 q^{39} - 20 q^{41} - 8 q^{49} - 2 q^{51} + 14 q^{59} + 40 q^{61} - 66 q^{69} - 16 q^{71} - 28 q^{79} + 30 q^{81} - 36 q^{89} - 74 q^{91} - 72 q^{99}+O(q^{100}) $ 6 * q - 8 * q^9 - 8 * q^11 + 6 * q^19 - 6 * q^21 - 6 * q^29 - 28 * q^31 + 10 * q^39 - 20 * q^41 - 8 * q^49 - 2 * q^51 + 14 * q^59 + 40 * q^61 - 66 * q^69 - 16 * q^71 - 28 * q^79 + 30 * q^81 - 36 * q^89 - 74 * q^91 - 72 * 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
$3$	− 0.363328i	− 0.209768i	−0.994484	−	0.104884i	$-0.966553\pi$
$3$	− 0.363328i	− 0.209768i	0.994484	−	0.104884i	$-0.0334471\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.14134i	0.431385i	0.976461	+	0.215692i	$0.0692008\pi$
$7$	1.14134i	0.431385i	−0.976461	+	0.215692i	$0.930799\pi$
$8$	0	0
$9$	2.86799	0.955998
$10$	0	0
$11$	−2.72666	−0.822118	−0.411059	−	0.911609i	$-0.634841\pi$
$11$	−2.72666	−0.822118	−0.411059	+	0.911609i	$0.634841\pi$
$12$	0	0
$13$	4.64600i	1.28857i	0.764786	+	0.644284i	$0.222845\pi$
$13$	4.64600i	1.28857i	−0.764786	+	0.644284i	$0.777155\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.858664i	− 0.208257i	−0.994564	−	0.104128i	$-0.966795\pi$
$17$	− 0.858664i	− 0.208257i	0.994564	−	0.104128i	$-0.0332053\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0.414680	0.0904905
$22$	0	0
$23$	4.41468i	0.920524i	0.887783	+	0.460262i	$0.152245\pi$
$23$	4.41468i	0.920524i	−0.887783	+	0.460262i	$0.847755\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 2.13201i	− 0.410305i
$28$	0	0
$29$	−9.42401	−1.74999	−0.874997	−	0.484128i	$-0.839137\pi$
$29$	−9.42401	−1.74999	−0.874997	+	0.484128i	$0.839137\pi$
$30$	0	0
$31$	−10.2827	−1.84682	−0.923411	−	0.383812i	$-0.874611\pi$
$31$	−10.2827	−1.84682	−0.923411	+	0.383812i	$0.874611\pi$
$32$	0	0
$33$	0.990671i	0.172454i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.77801i	1.11430i	0.830413	+	0.557149i	$0.188105\pi$
$37$	6.77801i	1.11430i	−0.830413	+	0.557149i	$0.811895\pi$
$38$	0	0
$39$	1.68802	0.270300
$40$	0	0
$41$	−7.55602	−1.18005	−0.590026	−	0.807384i	$-0.700882\pi$
$41$	−7.55602	−1.18005	−0.590026	+	0.807384i	$0.700882\pi$
$42$	0	0
$43$	− 9.29200i	− 1.41702i	−0.705702	−	0.708508i	$-0.749368\pi$
$43$	− 9.29200i	− 1.41702i	0.705702	−	0.708508i	$-0.250632\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 7.00933i	− 1.02242i	−0.859457	−	0.511208i	$-0.829198\pi$
$47$	− 7.00933i	− 1.02242i	0.859457	−	0.511208i	$-0.170802\pi$
$48$	0	0
$49$	5.69735	0.813907
$50$	0	0
$51$	−0.311977	−0.0436855
$52$	0	0
$53$	8.64600i	1.18762i	0.804605	+	0.593810i	$0.202377\pi$
$53$	8.64600i	1.18762i	−0.804605	+	0.593810i	$0.797623\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 0.363328i	− 0.0481240i
$58$	0	0
$59$	5.14134	0.669345	0.334672	−	0.942335i	$-0.391374\pi$
$59$	5.14134	0.669345	0.334672	+	0.942335i	$0.391374\pi$
$60$	0	0
$61$	9.45331	1.21037	0.605186	−	0.796084i	$-0.293099\pi$
$61$	9.45331	1.21037	0.605186	+	0.796084i	$0.293099\pi$
$62$	0	0
$63$	3.27334i	0.412403i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 13.6553i	− 1.66826i	−0.551565	−	0.834132i	$-0.685969\pi$
$67$	− 13.6553i	− 1.66826i	0.551565	−	0.834132i	$-0.314031\pi$
$68$	0	0
$69$	1.60398	0.193096
$70$	0	0
$71$	−5.45331	−0.647189	−0.323595	−	0.946196i	$-0.604891\pi$
$71$	−5.45331	−0.647189	−0.323595	+	0.946196i	$0.604891\pi$
$72$	0	0
$73$	− 6.87732i	− 0.804930i	−0.915435	−	0.402465i	$-0.868154\pi$
$73$	− 6.87732i	− 0.804930i	0.915435	−	0.402465i	$-0.131846\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.11203i	− 0.354649i
$78$	0	0
$79$	−17.2920	−1.94550	−0.972751	−	0.231852i	$-0.925521\pi$
$79$	−17.2920	−1.94550	−0.972751	+	0.231852i	$0.925521\pi$
$80$	0	0
$81$	7.82936	0.869929
$82$	0	0
$83$	14.2827i	1.56773i	0.620933	+	0.783863i	$0.286754\pi$
$83$	14.2827i	1.56773i	−0.620933	+	0.783863i	$0.713246\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.42401i	0.367092i
$88$	0	0
$89$	−13.0093	−1.37899	−0.689493	−	0.724292i	$-0.742167\pi$
$89$	−13.0093	−1.37899	−0.689493	+	0.724292i	$0.742167\pi$
$90$	0	0
$91$	−5.30265	−0.555869
$92$	0	0
$93$	3.73599i	0.387404i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 9.68463i	− 0.983326i	−0.870786	−	0.491663i	$-0.836389\pi$
$97$	− 9.68463i	− 0.983326i	0.870786	−	0.491663i	$-0.163611\pi$
$98$	0	0
$99$	−7.82003	−0.785943
$100$	0	0
$101$	1.45331	0.144610	0.0723050	−	0.997383i	$-0.476964\pi$
$101$	1.45331	0.144610	0.0723050	+	0.997383i	$0.476964\pi$
$102$	0	0
$103$	− 2.23132i	− 0.219859i	−0.993939	−	0.109929i	$-0.964938\pi$
$103$	− 2.23132i	− 0.219859i	0.993939	−	0.109929i	$-0.0350624\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.6553i	1.32011i	0.751217	+	0.660055i	$0.229467\pi$
$107$	13.6553i	1.32011i	−0.751217	+	0.660055i	$0.770533\pi$
$108$	0	0
$109$	−20.1693	−1.93187	−0.965935	−	0.258784i	$-0.916678\pi$
$109$	−20.1693	−1.93187	−0.965935	+	0.258784i	$0.916678\pi$
$110$	0	0
$111$	2.46264	0.233744
$112$	0	0
$113$	10.0514i	0.945552i	0.881183	+	0.472776i	$0.156748\pi$
$113$	10.0514i	0.945552i	−0.881183	+	0.472776i	$0.843252\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	13.3247i	1.23187i
$118$	0	0
$119$	0.980024	0.0898387
$120$	0	0
$121$	−3.56534	−0.324122
$122$	0	0
$123$	2.74531i	0.247537i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 14.0700i	− 1.24851i	−0.781220	−	0.624256i	$-0.785402\pi$
$127$	− 14.0700i	− 1.24851i	0.781220	−	0.624256i	$-0.214598\pi$
$128$	0	0
$129$	−3.37605	−0.297244
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	1.14134i	0.0989664i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 4.85866i	− 0.415104i	−0.978224	−	0.207552i	$-0.933450\pi$
$137$	− 4.85866i	− 0.415104i	0.978224	−	0.207552i	$-0.0665496\pi$
$138$	0	0
$139$	4.17997	0.354540	0.177270	−	0.984162i	$-0.443273\pi$
$139$	4.17997	0.354540	0.177270	+	0.984162i	$0.443273\pi$
$140$	0	0
$141$	−2.54669	−0.214470
$142$	0	0
$143$	− 12.6680i	− 1.05936i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2.07001i	− 0.170731i
$148$	0	0
$149$	−1.45331	−0.119060	−0.0595300	−	0.998227i	$-0.518960\pi$
$149$	−1.45331	−0.119060	−0.0595300	+	0.998227i	$0.518960\pi$
$150$	0	0
$151$	7.27334	0.591896	0.295948	−	0.955204i	$-0.404364\pi$
$151$	7.27334	0.591896	0.295948	+	0.955204i	$0.404364\pi$
$152$	0	0
$153$	− 2.46264i	− 0.199093i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	17.0093i	1.35749i	0.734373	+	0.678746i	$0.237476\pi$
$157$	17.0093i	1.35749i	−0.734373	+	0.678746i	$0.762524\pi$
$158$	0	0
$159$	3.14134	0.249124
$160$	0	0
$161$	−5.03863	−0.397100
$162$	0	0
$163$	10.1800i	0.797357i	0.917091	+	0.398678i	$0.130531\pi$
$163$	10.1800i	0.797357i	−0.917091	+	0.398678i	$0.869469\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.07001i	0.469711i	0.972030	+	0.234856i	$0.0754618\pi$
$167$	6.07001i	0.469711i	−0.972030	+	0.234856i	$0.924538\pi$
$168$	0	0
$169$	−8.58532	−0.660409
$170$	0	0
$171$	2.86799	0.219321
$172$	0	0
$173$	5.22199i	0.397021i	0.980099	+	0.198510i	$0.0636103\pi$
$173$	5.22199i	0.397021i	−0.980099	+	0.198510i	$0.936390\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1.86799i	− 0.140407i
$178$	0	0
$179$	1.55602	0.116302	0.0581510	−	0.998308i	$-0.481480\pi$
$179$	1.55602	0.116302	0.0581510	+	0.998308i	$0.481480\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	− 3.43466i	− 0.253897i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.34128i	0.171211i
$188$	0	0
$189$	2.43334	0.176999
$190$	0	0
$191$	−5.03863	−0.364583	−0.182291	−	0.983245i	$-0.558351\pi$
$191$	−5.03863	−0.364583	−0.182291	+	0.983245i	$0.558351\pi$
$192$	0	0
$193$	9.68463i	0.697115i	0.937287	+	0.348558i	$0.113328\pi$
$193$	9.68463i	0.697115i	−0.937287	+	0.348558i	$0.886672\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2.00000i	− 0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	− 2.00000i	− 0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$	0	0
$199$	−19.8867	−1.40973	−0.704864	−	0.709343i	$-0.748992\pi$
$199$	−19.8867	−1.40973	−0.704864	+	0.709343i	$0.748992\pi$
$200$	0	0
$201$	−4.96137	−0.349948
$202$	0	0
$203$	− 10.7560i	− 0.754920i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	12.6613i	0.880019i
$208$	0	0
$209$	−2.72666	−0.188607
$210$	0	0
$211$	4.31198	0.296849	0.148424	−	0.988924i	$-0.452580\pi$
$211$	4.31198	0.296849	0.148424	+	0.988924i	$0.452580\pi$
$212$	0	0
$213$	1.98134i	0.135759i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 11.7360i	− 0.796691i
$218$	0	0
$219$	−2.49873	−0.168848
$220$	0	0
$221$	3.98935	0.268353
$222$	0	0
$223$	18.5327i	1.24104i	0.784191	+	0.620519i	$0.213078\pi$
$223$	18.5327i	1.24104i	−0.784191	+	0.620519i	$0.786922\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.82597i	0.320311i	0.987092	+	0.160155i	$0.0511995\pi$
$227$	4.82597i	0.320311i	−0.987092	+	0.160155i	$0.948800\pi$
$228$	0	0
$229$	−23.7360	−1.56852	−0.784259	−	0.620434i	$-0.786957\pi$
$229$	−23.7360	−1.56852	−0.784259	+	0.620434i	$0.786957\pi$
$230$	0	0
$231$	−1.13069	−0.0743939
$232$	0	0
$233$	14.5653i	0.954207i	0.878847	+	0.477104i	$0.158313\pi$
$233$	14.5653i	0.954207i	−0.878847	+	0.477104i	$0.841687\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	6.28267i	0.408103i
$238$	0	0
$239$	−2.43334	−0.157399	−0.0786997	−	0.996898i	$-0.525077\pi$
$239$	−2.43334	−0.157399	−0.0786997	+	0.996898i	$0.525077\pi$
$240$	0	0
$241$	18.1986	1.17228	0.586138	−	0.810211i	$-0.300648\pi$
$241$	18.1986	1.17228	0.586138	+	0.810211i	$0.300648\pi$
$242$	0	0
$243$	− 9.24065i	− 0.592788i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.64600i	0.295618i
$248$	0	0
$249$	5.18930	0.328858
$250$	0	0
$251$	−15.1120	−0.953863	−0.476931	−	0.878940i	$-0.658251\pi$
$251$	−15.1120	−0.953863	−0.476931	+	0.878940i	$0.658251\pi$
$252$	0	0
$253$	− 12.0373i	− 0.756779i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.2313i	0.762969i	0.924375	+	0.381484i	$0.124587\pi$
$257$	12.2313i	0.762969i	−0.924375	+	0.381484i	$0.875413\pi$
$258$	0	0
$259$	−7.73599	−0.480691
$260$	0	0
$261$	−27.0280	−1.67299
$262$	0	0
$263$	− 22.5840i	− 1.39259i	−0.717756	−	0.696295i	$-0.754831\pi$
$263$	− 22.5840i	− 1.39259i	0.717756	−	0.696295i	$-0.245169\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4.72666i	0.289267i
$268$	0	0
$269$	13.1120	0.799455	0.399727	−	0.916634i	$-0.369105\pi$
$269$	13.1120	0.799455	0.399727	+	0.916634i	$0.369105\pi$
$270$	0	0
$271$	−20.8960	−1.26934	−0.634670	−	0.772783i	$-0.718864\pi$
$271$	−20.8960	−1.26934	−0.634670	+	0.772783i	$0.718864\pi$
$272$	0	0
$273$	1.92660i	0.116603i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 20.1800i	− 1.21250i	−0.795275	−	0.606248i	$-0.792674\pi$
$277$	− 20.1800i	− 1.21250i	0.795275	−	0.606248i	$-0.207326\pi$
$278$	0	0
$279$	−29.4906	−1.76556
$280$	0	0
$281$	−22.4626	−1.34001	−0.670004	−	0.742357i	$-0.733708\pi$
$281$	−22.4626	−1.34001	−0.670004	+	0.742357i	$0.733708\pi$
$282$	0	0
$283$	0.829359i	0.0493003i	0.999696	+	0.0246501i	$0.00784718\pi$
$283$	0.829359i	0.0493003i	−0.999696	+	0.0246501i	$0.992153\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 8.62395i	− 0.509056i
$288$	0	0
$289$	16.2627	0.956629
$290$	0	0
$291$	−3.51870	−0.206270
$292$	0	0
$293$	31.2886i	1.82790i	0.405827	+	0.913950i	$0.366984\pi$
$293$	31.2886i	1.82790i	−0.405827	+	0.913950i	$0.633016\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.81325i	0.337319i
$298$	0	0
$299$	−20.5106	−1.18616
$300$	0	0
$301$	10.6053	0.611279
$302$	0	0
$303$	− 0.528030i	− 0.0303345i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 3.11929i	− 0.178027i	−0.996030	−	0.0890136i	$-0.971629\pi$
$307$	− 3.11929i	− 0.178027i	0.996030	−	0.0890136i	$-0.0283715\pi$
$308$	0	0
$309$	−0.810702	−0.0461192
$310$	0	0
$311$	−21.4240	−1.21484	−0.607422	−	0.794379i	$-0.707796\pi$
$311$	−21.4240	−1.21484	−0.607422	+	0.794379i	$0.707796\pi$
$312$	0	0
$313$	12.1320i	0.685742i	0.939383	+	0.342871i	$0.111399\pi$
$313$	12.1320i	0.685742i	−0.939383	+	0.342871i	$0.888601\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.7487i	0.940701i	0.882480	+	0.470350i	$0.155872\pi$
$317$	16.7487i	0.940701i	−0.882480	+	0.470350i	$0.844128\pi$
$318$	0	0
$319$	25.6960	1.43870
$320$	0	0
$321$	4.96137	0.276916
$322$	0	0
$323$	− 0.858664i	− 0.0477773i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	7.32808i	0.405244i
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	−5.97070	−0.328179	−0.164090	−	0.986445i	$-0.552469\pi$
$331$	−5.97070	−0.328179	−0.164090	+	0.986445i	$0.552469\pi$
$332$	0	0
$333$	19.4393i	1.06527i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 20.3340i	− 1.10766i	−0.832628	−	0.553832i	$-0.813165\pi$
$337$	− 20.3340i	− 1.10766i	0.832628	−	0.553832i	$-0.186835\pi$
$338$	0	0
$339$	3.65194	0.198346
$340$	0	0
$341$	28.0373	1.51831
$342$	0	0
$343$	14.4919i	0.782492i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.8294i	1.11818i	0.829107	+	0.559089i	$0.188849\pi$
$347$	20.8294i	1.11818i	−0.829107	+	0.559089i	$0.811151\pi$
$348$	0	0
$349$	20.0187	1.07157	0.535787	−	0.844353i	$-0.320015\pi$
$349$	20.0187	1.07157	0.535787	+	0.844353i	$0.320015\pi$
$350$	0	0
$351$	9.90531	0.528706
$352$	0	0
$353$	27.1413i	1.44459i	0.691586	+	0.722294i	$0.256912\pi$
$353$	27.1413i	1.44459i	−0.691586	+	0.722294i	$0.743088\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 0.356070i	− 0.0188452i
$358$	0	0
$359$	4.59465	0.242496	0.121248	−	0.992622i	$-0.461310\pi$
$359$	4.59465	0.242496	0.121248	+	0.992622i	$0.461310\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	1.29539i	0.0679904i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 34.1214i	− 1.78112i	−0.454865	−	0.890560i	$-0.650313\pi$
$367$	− 34.1214i	− 1.78112i	0.454865	−	0.890560i	$-0.349687\pi$
$368$	0	0
$369$	−21.6706	−1.12813
$370$	0	0
$371$	−9.86799	−0.512321
$372$	0	0
$373$	7.81664i	0.404730i	0.979310	+	0.202365i	$0.0648628\pi$
$373$	7.81664i	0.404730i	−0.979310	+	0.202365i	$0.935137\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 43.7839i	− 2.25499i
$378$	0	0
$379$	23.0573	1.18437	0.592187	−	0.805801i	$-0.298265\pi$
$379$	23.0573	1.18437	0.592187	+	0.805801i	$0.298265\pi$
$380$	0	0
$381$	−5.11203	−0.261897
$382$	0	0
$383$	− 16.8807i	− 0.862564i	−0.902217	−	0.431282i	$-0.858061\pi$
$383$	− 16.8807i	− 0.862564i	0.902217	−	0.431282i	$-0.141939\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 26.6494i	− 1.35466i
$388$	0	0
$389$	18.3013	0.927914	0.463957	−	0.885858i	$-0.346429\pi$
$389$	18.3013	0.927914	0.463957	+	0.885858i	$0.346429\pi$
$390$	0	0
$391$	3.79073	0.191705
$392$	0	0
$393$	1.45331i	0.0733099i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	12.1800i	0.611295i	0.952145	+	0.305648i	$0.0988729\pi$
$397$	12.1800i	0.611295i	−0.952145	+	0.305648i	$0.901127\pi$
$398$	0	0
$399$	0.414680	0.0207599
$400$	0	0
$401$	16.5840	0.828166	0.414083	−	0.910239i	$-0.364102\pi$
$401$	16.5840	0.828166	0.414083	+	0.910239i	$0.364102\pi$
$402$	0	0
$403$	− 47.7733i	− 2.37976i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 18.4813i	− 0.916084i
$408$	0	0
$409$	−18.5653	−0.917997	−0.458999	−	0.888437i	$-0.651792\pi$
$409$	−18.5653	−0.917997	−0.458999	+	0.888437i	$0.651792\pi$
$410$	0	0
$411$	−1.76529	−0.0870753
$412$	0	0
$413$	5.86799i	0.288745i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 1.51870i	− 0.0743711i
$418$	0	0
$419$	−2.84802	−0.139135	−0.0695674	−	0.997577i	$-0.522162\pi$
$419$	−2.84802	−0.139135	−0.0695674	+	0.997577i	$0.522162\pi$
$420$	0	0
$421$	24.5360	1.19581	0.597907	−	0.801566i	$-0.295999\pi$
$421$	24.5360	1.19581	0.597907	+	0.801566i	$0.295999\pi$
$422$	0	0
$423$	− 20.1027i	− 0.977427i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	10.7894i	0.522136i
$428$	0	0
$429$	−4.60266	−0.222218
$430$	0	0
$431$	−7.53736	−0.363062	−0.181531	−	0.983385i	$-0.558105\pi$
$431$	−7.53736	−0.363062	−0.181531	+	0.983385i	$0.558105\pi$
$432$	0	0
$433$	0.392633i	0.0188687i	0.999955	+	0.00943437i	$0.00300310\pi$
$433$	0.392633i	0.0188687i	−0.999955	+	0.00943437i	$0.996997\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.41468i	0.211183i
$438$	0	0
$439$	−5.29200	−0.252573	−0.126287	−	0.991994i	$-0.540306\pi$
$439$	−5.29200	−0.252573	−0.126287	+	0.991994i	$0.540306\pi$
$440$	0	0
$441$	16.3400	0.778093
$442$	0	0
$443$	22.0187i	1.04614i	0.852290	+	0.523069i	$0.175213\pi$
$443$	22.0187i	1.04614i	−0.852290	+	0.523069i	$0.824787\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0.528030i	0.0249749i
$448$	0	0
$449$	4.90663	0.231558	0.115779	−	0.993275i	$-0.463064\pi$
$449$	4.90663	0.231558	0.115779	+	0.993275i	$0.463064\pi$
$450$	0	0
$451$	20.6027	0.970141
$452$	0	0
$453$	− 2.64261i	− 0.124161i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.9800i	1.26207i	0.775753	+	0.631036i	$0.217370\pi$
$457$	26.9800i	1.26207i	−0.775753	+	0.631036i	$0.782630\pi$
$458$	0	0
$459$	−1.83068	−0.0854487
$460$	0	0
$461$	−27.1307	−1.26360	−0.631801	−	0.775131i	$-0.717684\pi$
$461$	−27.1307	−1.26360	−0.631801	+	0.775131i	$0.717684\pi$
$462$	0	0
$463$	16.9253i	0.786585i	0.919413	+	0.393292i	$0.128664\pi$
$463$	16.9253i	0.786585i	−0.919413	+	0.393292i	$0.871336\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 42.6213i	− 1.97228i	−0.165917	−	0.986140i	$-0.553058\pi$
$467$	− 42.6213i	− 1.97228i	0.165917	−	0.986140i	$-0.446942\pi$
$468$	0	0
$469$	15.5853	0.719663
$470$	0	0
$471$	6.17997	0.284758
$472$	0	0
$473$	25.3361i	1.16495i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	24.7967i	1.13536i
$478$	0	0
$479$	25.3107	1.15647	0.578237	−	0.815869i	$-0.303741\pi$
$479$	25.3107	1.15647	0.578237	+	0.815869i	$0.303741\pi$
$480$	0	0
$481$	−31.4906	−1.43585
$482$	0	0
$483$	1.83068i	0.0832987i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	25.4020i	1.15107i	0.817776	+	0.575536i	$0.195207\pi$
$487$	25.4020i	1.15107i	−0.817776	+	0.575536i	$0.804793\pi$
$488$	0	0
$489$	3.69867	0.167260
$490$	0	0
$491$	20.6240	0.930746	0.465373	−	0.885115i	$-0.345920\pi$
$491$	20.6240	0.930746	0.465373	+	0.885115i	$0.345920\pi$
$492$	0	0
$493$	8.09206i	0.364448i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 6.22406i	− 0.279187i
$498$	0	0
$499$	−17.3693	−0.777555	−0.388778	−	0.921332i	$-0.627103\pi$
$499$	−17.3693	−0.777555	−0.388778	+	0.921332i	$0.627103\pi$
$500$	0	0
$501$	2.20541	0.0985303
$502$	0	0
$503$	14.9987i	0.668758i	0.942439	+	0.334379i	$0.108527\pi$
$503$	14.9987i	0.668758i	−0.942439	+	0.334379i	$0.891473\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.11929i	0.138533i
$508$	0	0
$509$	19.9160	0.882759	0.441380	−	0.897320i	$-0.354489\pi$
$509$	19.9160	0.882759	0.441380	+	0.897320i	$0.354489\pi$
$510$	0	0
$511$	7.84934	0.347234
$512$	0	0
$513$	− 2.13201i	− 0.0941304i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	19.1120i	0.840546i
$518$	0	0
$519$	1.89730	0.0832821
$520$	0	0
$521$	−14.8294	−0.649686	−0.324843	−	0.945768i	$-0.605311\pi$
$521$	−14.8294	−0.649686	−0.324843	+	0.945768i	$0.605311\pi$
$522$	0	0
$523$	− 11.9966i	− 0.524575i	−0.964990	−	0.262288i	$-0.915523\pi$
$523$	− 11.9966i	− 0.524575i	0.964990	−	0.262288i	$-0.0844769\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.82936i	0.384613i
$528$	0	0
$529$	3.51060	0.152635
$530$	0	0
$531$	14.7453	0.639892
$532$	0	0
$533$	− 35.1053i	− 1.52058i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 0.565344i	− 0.0243964i
$538$	0	0
$539$	−15.5347	−0.669128
$540$	0	0
$541$	−2.01866	−0.0867889	−0.0433944	−	0.999058i	$-0.513817\pi$
$541$	−2.01866	−0.0867889	−0.0433944	+	0.999058i	$0.513817\pi$
$542$	0	0
$543$	− 7.99322i	− 0.343022i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 18.6940i	− 0.799296i	−0.916669	−	0.399648i	$-0.869132\pi$
$547$	− 18.6940i	− 0.799296i	0.916669	−	0.399648i	$-0.130868\pi$
$548$	0	0
$549$	27.1120	1.15711
$550$	0	0
$551$	−9.42401	−0.401476
$552$	0	0
$553$	− 19.7360i	− 0.839259i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11.0866i	0.469754i	0.972025	+	0.234877i	$0.0754688\pi$
$557$	11.0866i	0.469754i	−0.972025	+	0.234877i	$0.924531\pi$
$558$	0	0
$559$	43.1706	1.82592
$560$	0	0
$561$	0.850654	0.0359146
$562$	0	0
$563$	− 2.69396i	− 0.113537i	−0.998387	−	0.0567685i	$-0.981920\pi$
$563$	− 2.69396i	− 0.113537i	0.998387	−	0.0567685i	$-0.0180797\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	8.93593i	0.375274i
$568$	0	0
$569$	−29.4134	−1.23307	−0.616536	−	0.787327i	$-0.711465\pi$
$569$	−29.4134	−1.23307	−0.616536	+	0.787327i	$0.711465\pi$
$570$	0	0
$571$	−33.0466	−1.38296	−0.691479	−	0.722396i	$-0.743041\pi$
$571$	−33.0466	−1.38296	−0.691479	+	0.722396i	$0.743041\pi$
$572$	0	0
$573$	1.83068i	0.0764777i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 0.396022i	− 0.0164866i	−0.999966	−	0.00824331i	$-0.997376\pi$
$577$	− 0.396022i	− 0.0164866i	0.999966	−	0.00824331i	$-0.00262396\pi$
$578$	0	0
$579$	3.51870	0.146232
$580$	0	0
$581$	−16.3013	−0.676293
$582$	0	0
$583$	− 23.5747i	− 0.976363i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 14.0773i	− 0.581031i	−0.956870	−	0.290515i	$-0.906173\pi$
$587$	− 14.0773i	− 0.581031i	0.956870	−	0.290515i	$-0.0938268\pi$
$588$	0	0
$589$	−10.2827	−0.423690
$590$	0	0
$591$	−0.726656	−0.0298907
$592$	0	0
$593$	− 5.43466i	− 0.223175i	−0.993755	−	0.111587i	$-0.964407\pi$
$593$	− 5.43466i	− 0.223175i	0.993755	−	0.111587i	$-0.0355935\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7.22538i	0.295715i
$598$	0	0
$599$	−11.6333	−0.475323	−0.237662	−	0.971348i	$-0.576381\pi$
$599$	−11.6333	−0.475323	−0.237662	+	0.971348i	$0.576381\pi$
$600$	0	0
$601$	8.12136	0.331277	0.165639	−	0.986187i	$-0.447031\pi$
$601$	8.12136	0.331277	0.165639	+	0.986187i	$0.447031\pi$
$602$	0	0
$603$	− 39.1634i	− 1.59486i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	10.5327i	0.427507i	0.976888	+	0.213754i	$0.0685689\pi$
$607$	10.5327i	0.427507i	−0.976888	+	0.213754i	$0.931431\pi$
$608$	0	0
$609$	−3.90794	−0.158358
$610$	0	0
$611$	32.5653	1.31745
$612$	0	0
$613$	− 47.3293i	− 1.91161i	−0.293996	−	0.955807i	$-0.594985\pi$
$613$	− 47.3293i	− 1.91161i	0.293996	−	0.955807i	$-0.405015\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 33.3693i	− 1.34340i	−0.740825	−	0.671698i	$-0.765565\pi$
$617$	− 33.3693i	− 1.34340i	0.740825	−	0.671698i	$-0.234435\pi$
$618$	0	0
$619$	33.8760	1.36159	0.680796	−	0.732473i	$-0.261634\pi$
$619$	33.8760	1.36159	0.680796	+	0.732473i	$0.261634\pi$
$620$	0	0
$621$	9.41213	0.377696
$622$	0	0
$623$	− 14.8480i	− 0.594873i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.990671i	0.0395636i
$628$	0	0
$629$	5.82003	0.232060
$630$	0	0
$631$	27.8573	1.10898	0.554492	−	0.832189i	$-0.312913\pi$
$631$	27.8573	1.10898	0.554492	+	0.832189i	$0.312913\pi$
$632$	0	0
$633$	− 1.56666i	− 0.0622693i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	26.4699i	1.04878i
$638$	0	0
$639$	−15.6401	−0.618711
$640$	0	0
$641$	−4.74531	−0.187429	−0.0937143	−	0.995599i	$-0.529874\pi$
$641$	−4.74531	−0.187429	−0.0937143	+	0.995599i	$0.529874\pi$
$642$	0	0
$643$	6.28267i	0.247764i	0.992297	+	0.123882i	$0.0395345\pi$
$643$	6.28267i	0.247764i	−0.992297	+	0.123882i	$0.960466\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.1507i	0.634948i	0.948267	+	0.317474i	$0.102835\pi$
$647$	16.1507i	0.634948i	−0.948267	+	0.317474i	$0.897165\pi$
$648$	0	0
$649$	−14.0187	−0.550280
$650$	0	0
$651$	−4.26401	−0.167120
$652$	0	0
$653$	− 29.8760i	− 1.16914i	−0.811344	−	0.584569i	$-0.801264\pi$
$653$	− 29.8760i	− 1.16914i	0.811344	−	0.584569i	$-0.198736\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 19.7241i	− 0.769511i
$658$	0	0
$659$	17.4427	0.679470	0.339735	−	0.940521i	$-0.389663\pi$
$659$	17.4427	0.679470	0.339735	+	0.940521i	$0.389663\pi$
$660$	0	0
$661$	−2.87732	−0.111915	−0.0559574	−	0.998433i	$-0.517821\pi$
$661$	−2.87732	−0.111915	−0.0559574	+	0.998433i	$0.517821\pi$
$662$	0	0
$663$	− 1.44944i	− 0.0562918i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 41.6040i	− 1.61091i
$668$	0	0
$669$	6.73344	0.260330
$670$	0	0
$671$	−25.7759	−0.995069
$672$	0	0
$673$	− 10.9393i	− 0.421680i	−0.977521	−	0.210840i	$-0.932380\pi$
$673$	− 10.9393i	− 0.421680i	0.977521	−	0.210840i	$-0.0676199\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	19.0900i	0.733688i	0.930283	+	0.366844i	$0.119562\pi$
$677$	19.0900i	0.733688i	−0.930283	+	0.366844i	$0.880438\pi$
$678$	0	0
$679$	11.0534	0.424191
$680$	0	0
$681$	1.75341	0.0671909
$682$	0	0
$683$	− 25.7687i	− 0.986011i	−0.870026	−	0.493006i	$-0.835898\pi$
$683$	− 25.7687i	− 0.986011i	0.870026	−	0.493006i	$-0.164102\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	8.62395i	0.329024i
$688$	0	0
$689$	−40.1693	−1.53033
$690$	0	0
$691$	−27.2334	−1.03601	−0.518004	−	0.855378i	$-0.673325\pi$
$691$	−27.2334	−1.03601	−0.518004	+	0.855378i	$0.673325\pi$
$692$	0	0
$693$	− 8.92528i	− 0.339043i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6.48808i	0.245753i
$698$	0	0
$699$	5.29200	0.200162
$700$	0	0
$701$	49.6960	1.87699	0.938497	−	0.345288i	$-0.112219\pi$
$701$	49.6960	1.87699	0.938497	+	0.345288i	$0.112219\pi$
$702$	0	0
$703$	6.77801i	0.255637i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.65872i	0.0623825i
$708$	0	0
$709$	−5.64006	−0.211817	−0.105908	−	0.994376i	$-0.533775\pi$
$709$	−5.64006	−0.211817	−0.105908	+	0.994376i	$0.533775\pi$
$710$	0	0
$711$	−49.5933	−1.85990
$712$	0	0
$713$	− 45.3947i	− 1.70005i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.884100i	0.0330173i
$718$	0	0
$719$	33.4240	1.24651	0.623253	−	0.782021i	$-0.285811\pi$
$719$	33.4240	1.24651	0.623253	+	0.782021i	$0.285811\pi$
$720$	0	0
$721$	2.54669	0.0948436
$722$	0	0
$723$	− 6.61208i	− 0.245906i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 14.3306i	− 0.531494i	−0.964043	−	0.265747i	$-0.914381\pi$
$727$	− 14.3306i	− 0.531494i	0.964043	−	0.265747i	$-0.0856185\pi$
$728$	0	0
$729$	20.1307	0.745581
$730$	0	0
$731$	−7.97871	−0.295103
$732$	0	0
$733$	40.8853i	1.51013i	0.655648	+	0.755067i	$0.272396\pi$
$733$	40.8853i	1.51013i	−0.655648	+	0.755067i	$0.727604\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	37.2334i	1.37151i
$738$	0	0
$739$	28.8294	1.06051	0.530253	−	0.847840i	$-0.322097\pi$
$739$	28.8294	1.06051	0.530253	+	0.847840i	$0.322097\pi$
$740$	0	0
$741$	1.68802	0.0620111
$742$	0	0
$743$	8.05135i	0.295375i	0.989034	+	0.147688i	$0.0471831\pi$
$743$	8.05135i	0.295375i	−0.989034	+	0.147688i	$0.952817\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	40.9626i	1.49874i
$748$	0	0
$749$	−15.5853	−0.569475
$750$	0	0
$751$	47.3107	1.72639	0.863195	−	0.504870i	$-0.168460\pi$
$751$	47.3107	1.72639	0.863195	+	0.504870i	$0.168460\pi$
$752$	0	0
$753$	5.49063i	0.200090i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	6.20541i	0.225539i	0.993621	+	0.112770i	$0.0359722\pi$
$757$	6.20541i	0.225539i	−0.993621	+	0.112770i	$0.964028\pi$
$758$	0	0
$759$	−4.37350	−0.158748
$760$	0	0
$761$	43.8280	1.58877	0.794383	−	0.607418i	$-0.207795\pi$
$761$	43.8280	1.58877	0.794383	+	0.607418i	$0.207795\pi$
$762$	0	0
$763$	− 23.0200i	− 0.833379i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	23.8867i	0.862497i
$768$	0	0
$769$	3.22538	0.116310	0.0581551	−	0.998308i	$-0.481478\pi$
$769$	3.22538	0.116310	0.0581551	+	0.998308i	$0.481478\pi$
$770$	0	0
$771$	4.44398	0.160046
$772$	0	0
$773$	− 6.16470i	− 0.221729i	−0.993836	−	0.110864i	$-0.964638\pi$
$773$	− 6.16470i	− 0.221729i	0.993836	−	0.110864i	$-0.0353619\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2.81070i	0.100833i
$778$	0	0
$779$	−7.55602	−0.270722
$780$	0	0
$781$	14.8693	0.532066
$782$	0	0
$783$	20.0921i	0.718031i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	35.1527i	1.25306i	0.779397	+	0.626530i	$0.215525\pi$
$787$	35.1527i	1.25306i	−0.779397	+	0.626530i	$0.784475\pi$
$788$	0	0
$789$	−8.20541	−0.292120
$790$	0	0
$791$	−11.4720	−0.407896
$792$	0	0
$793$	43.9201i	1.55965i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 30.3420i	− 1.07477i	−0.843337	−	0.537385i	$-0.819412\pi$
$797$	− 30.3420i	− 1.07477i	0.843337	−	0.537385i	$-0.180588\pi$
$798$	0	0
$799$	−6.01866	−0.212925
$800$	0	0
$801$	−37.3107	−1.31831
$802$	0	0
$803$	18.7521i	0.661747i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 4.76397i	− 0.167700i
$808$	0	0
$809$	4.96137	0.174432	0.0872162	−	0.996189i	$-0.472203\pi$
$809$	4.96137	0.174432	0.0872162	+	0.996189i	$0.472203\pi$
$810$	0	0
$811$	29.8094	1.04675	0.523375	−	0.852103i	$-0.324673\pi$
$811$	29.8094	1.04675	0.523375	+	0.852103i	$0.324673\pi$
$812$	0	0
$813$	7.59210i	0.266267i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 9.29200i	− 0.325086i
$818$	0	0
$819$	−15.2080	−0.531409
$820$	0	0
$821$	−11.1520	−0.389207	−0.194603	−	0.980882i	$-0.562342\pi$
$821$	−11.1520	−0.389207	−0.194603	+	0.980882i	$0.562342\pi$
$822$	0	0
$823$	17.1413i	0.597509i	0.954330	+	0.298755i	$0.0965712\pi$
$823$	17.1413i	0.597509i	−0.954330	+	0.298755i	$0.903429\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	24.8633i	0.864581i	0.901734	+	0.432291i	$0.142295\pi$
$827$	24.8633i	0.864581i	−0.901734	+	0.432291i	$0.857705\pi$
$828$	0	0
$829$	18.7746	0.652069	0.326035	−	0.945358i	$-0.394287\pi$
$829$	18.7746	0.652069	0.326035	+	0.945358i	$0.394287\pi$
$830$	0	0
$831$	−7.33195	−0.254343
$832$	0	0
$833$	− 4.89211i	− 0.169502i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	21.9227i	0.757761i
$838$	0	0
$839$	−29.0280	−1.00216	−0.501079	−	0.865402i	$-0.667063\pi$
$839$	−29.0280	−1.00216	−0.501079	+	0.865402i	$0.667063\pi$
$840$	0	0
$841$	59.8119	2.06248
$842$	0	0
$843$	8.16131i	0.281091i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 4.06926i	− 0.139821i
$848$	0	0
$849$	0.301330	0.0103416
$850$	0	0
$851$	−29.9227	−1.02574
$852$	0	0
$853$	− 40.7894i	− 1.39660i	−0.715804	−	0.698301i	$-0.753940\pi$
$853$	− 40.7894i	− 1.39660i	0.715804	−	0.698301i	$-0.246060\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 32.6354i	− 1.11480i	−0.830243	−	0.557401i	$-0.811799\pi$
$857$	− 32.6354i	− 1.11480i	0.830243	−	0.557401i	$-0.188201\pi$
$858$	0	0
$859$	−33.4906	−1.14269	−0.571343	−	0.820712i	$-0.693577\pi$
$859$	−33.4906	−1.14269	−0.571343	+	0.820712i	$0.693577\pi$
$860$	0	0
$861$	−3.13333	−0.106783
$862$	0	0
$863$	− 5.13795i	− 0.174898i	−0.996169	−	0.0874489i	$-0.972129\pi$
$863$	− 5.13795i	− 0.174898i	0.996169	−	0.0874489i	$-0.0278714\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 5.90870i	− 0.200670i
$868$	0	0
$869$	47.1493	1.59943
$870$	0	0
$871$	63.4427	2.14967
$872$	0	0
$873$	− 27.7755i	− 0.940057i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 16.0220i	− 0.541026i	−0.962716	−	0.270513i	$-0.912807\pi$
$877$	− 16.0220i	− 0.541026i	0.962716	−	0.270513i	$-0.0871933\pi$
$878$	0	0
$879$	11.3680	0.383434
$880$	0	0
$881$	11.5306	0.388475	0.194238	−	0.980955i	$-0.437777\pi$
$881$	11.5306	0.388475	0.194238	+	0.980955i	$0.437777\pi$
$882$	0	0
$883$	39.5161i	1.32982i	0.746923	+	0.664911i	$0.231530\pi$
$883$	39.5161i	1.32982i	−0.746923	+	0.664911i	$0.768470\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 20.8153i	− 0.698910i	−0.936953	−	0.349455i	$-0.886367\pi$
$887$	− 20.8153i	− 0.698910i	0.936953	−	0.349455i	$-0.113633\pi$
$888$	0	0
$889$	16.0586	0.538588
$890$	0	0
$891$	−21.3480	−0.715184
$892$	0	0
$893$	− 7.00933i	− 0.234558i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	7.45208i	0.248818i
$898$	0	0
$899$	96.9040	3.23193
$900$	0	0
$901$	7.42401	0.247330
$902$	0	0
$903$	− 3.85320i	− 0.128227i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	32.0339i	1.06367i	0.846848	+	0.531835i	$0.178497\pi$
$907$	32.0339i	1.06367i	−0.846848	+	0.531835i	$0.821503\pi$
$908$	0	0
$909$	4.16809	0.138247
$910$	0	0
$911$	−2.12136	−0.0702838	−0.0351419	−	0.999382i	$-0.511188\pi$
$911$	−2.12136	−0.0702838	−0.0351419	+	0.999382i	$0.511188\pi$
$912$	0	0
$913$	− 38.9439i	− 1.28886i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 4.56534i	− 0.150761i
$918$	0	0
$919$	26.4333	0.871955	0.435978	−	0.899957i	$-0.356403\pi$
$919$	26.4333	0.871955	0.435978	+	0.899957i	$0.356403\pi$
$920$	0	0
$921$	−1.13333	−0.0373444
$922$	0	0
$923$	− 25.3361i	− 0.833948i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 6.39941i	− 0.210184i
$928$	0	0
$929$	−17.7907	−0.583695	−0.291847	−	0.956465i	$-0.594270\pi$
$929$	−17.7907	−0.583695	−0.291847	+	0.956465i	$0.594270\pi$
$930$	0	0
$931$	5.69735	0.186723
$932$	0	0
$933$	7.78395i	0.254835i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 7.46396i	− 0.243837i	−0.992540	−	0.121918i	$-0.961095\pi$
$937$	− 7.46396i	− 0.243837i	0.992540	−	0.121918i	$-0.0389046\pi$
$938$	0	0
$939$	4.40790	0.143846
$940$	0	0
$941$	−2.67869	−0.0873229	−0.0436615	−	0.999046i	$-0.513902\pi$
$941$	−2.67869	−0.0873229	−0.0436615	+	0.999046i	$0.513902\pi$
$942$	0	0
$943$	− 33.3574i	− 1.08627i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	42.5840i	1.38379i	0.721996	+	0.691897i	$0.243225\pi$
$947$	42.5840i	1.38379i	−0.721996	+	0.691897i	$0.756775\pi$
$948$	0	0
$949$	31.9520	1.03721
$950$	0	0
$951$	6.08528	0.197329
$952$	0	0
$953$	− 60.0046i	− 1.94374i	−0.235517	−	0.971870i	$-0.575678\pi$
$953$	− 60.0046i	− 1.94374i	0.235517	−	0.971870i	$-0.424322\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 9.33609i	− 0.301793i
$958$	0	0
$959$	5.54537	0.179069
$960$	0	0
$961$	74.7333	2.41075
$962$	0	0
$963$	39.1634i	1.26202i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 61.1307i	− 1.96583i	−0.184059	−	0.982915i	$-0.558924\pi$
$967$	− 61.1307i	− 1.96583i	0.184059	−	0.982915i	$-0.441076\pi$
$968$	0	0
$969$	−0.311977	−0.0100221
$970$	0	0
$971$	54.6213	1.75288	0.876441	−	0.481510i	$-0.159911\pi$
$971$	54.6213	1.75288	0.876441	+	0.481510i	$0.159911\pi$
$972$	0	0
$973$	4.77075i	0.152943i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 19.3060i	− 0.617655i	−0.951118	−	0.308827i	$-0.900063\pi$
$977$	− 19.3060i	− 0.617655i	0.951118	−	0.308827i	$-0.0999366\pi$
$978$	0	0
$979$	35.4720	1.13369
$980$	0	0
$981$	−57.8455	−1.84686
$982$	0	0
$983$	− 25.1820i	− 0.803182i	−0.915819	−	0.401591i	$-0.868457\pi$
$983$	− 25.1820i	− 0.803182i	0.915819	−	0.401591i	$-0.131543\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 2.90663i	− 0.0925189i
$988$	0	0
$989$	41.0212	1.30440
$990$	0	0
$991$	−5.82003	−0.184879	−0.0924397	−	0.995718i	$-0.529467\pi$
$991$	−5.82003	−0.184879	−0.0924397	+	0.995718i	$0.529467\pi$
$992$	0	0
$993$	2.16932i	0.0688414i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	8.97201i	0.284147i	0.989856	+	0.142073i	$0.0453769\pi$
$997$	8.97201i	0.284147i	−0.989856	+	0.142073i	$0.954623\pi$
$998$	0	0
$999$	14.4508	0.457202

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.k.3649.3		6
5.2	odd	4		3800.2.a.y.1.2		3
5.3	odd	4		760.2.a.h.1.2	✓	3
5.4	even	2	inner	3800.2.d.k.3649.4		6
15.8	even	4		6840.2.a.bj.1.3		3
20.3	even	4		1520.2.a.r.1.2		3
20.7	even	4		7600.2.a.bo.1.2		3
40.3	even	4		6080.2.a.bs.1.2		3
40.13	odd	4		6080.2.a.bw.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.h.1.2	✓	3	5.3	odd	4
1520.2.a.r.1.2		3	20.3	even	4
3800.2.a.y.1.2		3	5.2	odd	4
3800.2.d.k.3649.3		6	1.1	even	1	trivial
3800.2.d.k.3649.4		6	5.4	even	2	inner
6080.2.a.bs.1.2		3	40.3	even	4
6080.2.a.bw.1.2		3	40.13	odd	4
6840.2.a.bj.1.3		3	15.8	even	4
7600.2.a.bo.1.2		3	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)

\(f(q)\)	\(=\)	\(q-0.363328i q^{3} +1.14134i q^{7} +2.86799 q^{9} +O(q^{10})\)	\(q-0.363328i q^{3} +1.14134i q^{7} +2.86799 q^{9} -2.72666 q^{11} +4.64600i q^{13} -0.858664i q^{17} +1.00000 q^{19} +0.414680 q^{21} +4.41468i q^{23} -2.13201i q^{27} -9.42401 q^{29} -10.2827 q^{31} +0.990671i q^{33} +6.77801i q^{37} +1.68802 q^{39} -7.55602 q^{41} -9.29200i q^{43} -7.00933i q^{47} +5.69735 q^{49} -0.311977 q^{51} +8.64600i q^{53} -0.363328i q^{57} +5.14134 q^{59} +9.45331 q^{61} +3.27334i q^{63} -13.6553i q^{67} +1.60398 q^{69} -5.45331 q^{71} -6.87732i q^{73} -3.11203i q^{77} -17.2920 q^{79} +7.82936 q^{81} +14.2827i q^{83} +3.42401i q^{87} -13.0093 q^{89} -5.30265 q^{91} +3.73599i q^{93} -9.68463i q^{97} -7.82003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{9}+O(q^{10}) \) 6 * q - 8 * q^9	\( 6 q - 8 q^{9} - 8 q^{11} + 6 q^{19} - 6 q^{21} - 6 q^{29} - 28 q^{31} + 10 q^{39} - 20 q^{41} - 8 q^{49} - 2 q^{51} + 14 q^{59} + 40 q^{61} - 66 q^{69} - 16 q^{71} - 28 q^{79} + 30 q^{81} - 36 q^{89} - 74 q^{91} - 72 q^{99}+O(q^{100}) \) 6 * q - 8 * q^9 - 8 * q^11 + 6 * q^19 - 6 * q^21 - 6 * q^29 - 28 * q^31 + 10 * q^39 - 20 * q^41 - 8 * q^49 - 2 * q^51 + 14 * q^59 + 40 * q^61 - 66 * q^69 - 16 * q^71 - 28 * q^79 + 30 * q^81 - 36 * q^89 - 74 * q^91 - 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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