LMFDB - Embedded newform 3800.2.d.j.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:	6.0.59105344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 21x^{4} + 116x^{2} + 64 $ x^6 + 21x^4 + 116x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.3
Root			$-3.29707i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.j.3649.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.786802i q^{3} -2.08387i q^{7} +2.38094 q^{9} +O(q^{10})$	$q-0.786802i q^{3} -2.08387i q^{7} +2.38094 q^{9} +1.29707 q^{11} -1.21320i q^{13} +4.08387i q^{17} +1.00000 q^{19} -1.63959 q^{21} +8.95455i q^{23} -4.23374i q^{27} +9.38094 q^{29} -1.02054i q^{33} -2.00000i q^{37} -0.954547 q^{39} +3.57360 q^{41} -7.72347i q^{43} +9.46482i q^{47} +2.65748 q^{49} +3.21320 q^{51} +11.9751i q^{53} -0.786802i q^{57} +7.21320 q^{59} +4.87067 q^{61} -4.96158i q^{63} +11.3809i q^{67} +7.04545 q^{69} -9.02054 q^{71} -5.65748i q^{73} -2.70293i q^{77} -9.57360 q^{79} +3.81172 q^{81} -10.7619i q^{83} -7.38094i q^{87} -11.0205 q^{89} -2.52815 q^{91} -8.59414i q^{97} +3.08825 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 24 q^{9}+O(q^{10}) $ 6 * q - 24 * q^9	$ 6 q - 24 q^{9} - 10 q^{11} + 6 q^{19} - 18 q^{21} + 18 q^{29} + 38 q^{39} + 16 q^{41} - 10 q^{49} + 22 q^{51} + 46 q^{59} + 6 q^{61} + 86 q^{69} - 24 q^{71} - 52 q^{79} + 94 q^{81} - 36 q^{89} + 34 q^{91} + 102 q^{99}+O(q^{100}) $ 6 * q - 24 * q^9 - 10 * q^11 + 6 * q^19 - 18 * q^21 + 18 * q^29 + 38 * q^39 + 16 * q^41 - 10 * q^49 + 22 * q^51 + 46 * q^59 + 6 * q^61 + 86 * q^69 - 24 * q^71 - 52 * q^79 + 94 * q^81 - 36 * q^89 + 34 * q^91 + 102 * 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
$3$	− 0.786802i	− 0.454260i	−0.973864	−	0.227130i	$-0.927066\pi$
$3$	− 0.786802i	− 0.454260i	0.973864	−	0.227130i	$-0.0729343\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.08387i	− 0.787630i	−0.919190	−	0.393815i	$-0.871155\pi$
$7$	− 2.08387i	− 0.787630i	0.919190	−	0.393815i	$-0.128845\pi$
$8$	0	0
$9$	2.38094	0.793648
$10$	0	0
$11$	1.29707	0.391081	0.195541	−	0.980696i	$-0.437354\pi$
$11$	1.29707	0.391081	0.195541	+	0.980696i	$0.437354\pi$
$12$	0	0
$13$	− 1.21320i	− 0.336481i	−0.985746	−	0.168240i	$-0.946192\pi$
$13$	− 1.21320i	− 0.336481i	0.985746	−	0.168240i	$-0.0538085\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.08387i	0.990485i	0.868755	+	0.495242i	$0.164921\pi$
$17$	4.08387i	0.990485i	−0.868755	+	0.495242i	$0.835079\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.63959	−0.357789
$22$	0	0
$23$	8.95455i	1.86715i	0.358379	+	0.933576i	$0.383329\pi$
$23$	8.95455i	1.86715i	−0.358379	+	0.933576i	$0.616671\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4.23374i	− 0.814783i
$28$	0	0
$29$	9.38094	1.74200	0.870999	−	0.491285i	$-0.163473\pi$
$29$	9.38094	1.74200	0.870999	+	0.491285i	$0.163473\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	− 1.02054i	− 0.177653i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	−0.954547	−0.152850
$40$	0	0
$41$	3.57360	0.558103	0.279052	−	0.960276i	$-0.409980\pi$
$41$	3.57360	0.558103	0.279052	+	0.960276i	$0.409980\pi$
$42$	0	0
$43$	− 7.72347i	− 1.17782i	−0.808199	−	0.588909i	$-0.799558\pi$
$43$	− 7.72347i	− 1.17782i	0.808199	−	0.588909i	$-0.200442\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.46482i	1.38059i	0.723530	+	0.690293i	$0.242518\pi$
$47$	9.46482i	1.38059i	−0.723530	+	0.690293i	$0.757482\pi$
$48$	0	0
$49$	2.65748	0.379639
$50$	0	0
$51$	3.21320	0.449938
$52$	0	0
$53$	11.9751i	1.64490i	0.568834	+	0.822452i	$0.307395\pi$
$53$	11.9751i	1.64490i	−0.568834	+	0.822452i	$0.692605\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 0.786802i	− 0.104214i
$58$	0	0
$59$	7.21320	0.939078	0.469539	−	0.882912i	$-0.344420\pi$
$59$	7.21320	0.939078	0.469539	+	0.882912i	$0.344420\pi$
$60$	0	0
$61$	4.87067	0.623626	0.311813	−	0.950144i	$-0.399064\pi$
$61$	4.87067	0.623626	0.311813	+	0.950144i	$0.399064\pi$
$62$	0	0
$63$	− 4.96158i	− 0.625100i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	11.3809i	1.39040i	0.718815	+	0.695202i	$0.244685\pi$
$67$	11.3809i	1.39040i	−0.718815	+	0.695202i	$0.755315\pi$
$68$	0	0
$69$	7.04545	0.848173
$70$	0	0
$71$	−9.02054	−1.07054	−0.535270	−	0.844681i	$-0.679790\pi$
$71$	−9.02054	−1.07054	−0.535270	+	0.844681i	$0.679790\pi$
$72$	0	0
$73$	− 5.65748i	− 0.662157i	−0.943603	−	0.331079i	$-0.892587\pi$
$73$	− 5.65748i	− 0.662157i	0.943603	−	0.331079i	$-0.107413\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.70293i	− 0.308027i
$78$	0	0
$79$	−9.57360	−1.07711	−0.538557	−	0.842589i	$-0.681030\pi$
$79$	−9.57360	−1.07711	−0.538557	+	0.842589i	$0.681030\pi$
$80$	0	0
$81$	3.81172	0.423524
$82$	0	0
$83$	− 10.7619i	− 1.18127i	−0.806939	−	0.590635i	$-0.798877\pi$
$83$	− 10.7619i	− 1.18127i	0.806939	−	0.590635i	$-0.201123\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 7.38094i	− 0.791320i
$88$	0	0
$89$	−11.0205	−1.16817	−0.584087	−	0.811691i	$-0.698547\pi$
$89$	−11.0205	−1.16817	−0.584087	+	0.811691i	$0.698547\pi$
$90$	0	0
$91$	−2.52815	−0.265022
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 8.59414i	− 0.872603i	−0.899801	−	0.436301i	$-0.856288\pi$
$97$	− 8.59414i	− 0.872603i	0.899801	−	0.436301i	$-0.143712\pi$
$98$	0	0
$99$	3.08825	0.310381
$100$	0	0
$101$	9.14721	0.910181	0.455091	−	0.890445i	$-0.349607\pi$
$101$	9.14721	0.910181	0.455091	+	0.890445i	$0.349607\pi$
$102$	0	0
$103$	− 17.3560i	− 1.71014i	−0.518512	−	0.855070i	$-0.673514\pi$
$103$	− 17.3560i	− 1.71014i	0.518512	−	0.855070i	$-0.326486\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 0.786802i	− 0.0760630i	−0.999277	−	0.0380315i	$-0.987891\pi$
$107$	− 0.786802i	− 0.0760630i	0.999277	−	0.0380315i	$-0.0121087\pi$
$108$	0	0
$109$	13.5487	1.29773	0.648864	−	0.760904i	$-0.275244\pi$
$109$	13.5487	1.29773	0.648864	+	0.760904i	$0.275244\pi$
$110$	0	0
$111$	−1.57360	−0.149360
$112$	0	0
$113$	1.14721i	0.107920i	0.998543	+	0.0539601i	$0.0171844\pi$
$113$	1.14721i	0.107920i	−0.998543	+	0.0539601i	$0.982816\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2.88856i	− 0.267047i
$118$	0	0
$119$	8.51027	0.780135
$120$	0	0
$121$	−9.31761	−0.847055
$122$	0	0
$123$	− 2.81172i	− 0.253524i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1.02054i	− 0.0905581i	−0.998974	−	0.0452790i	$-0.985582\pi$
$127$	− 1.02054i	− 0.0905581i	0.998974	−	0.0452790i	$-0.0144177\pi$
$128$	0	0
$129$	−6.07684	−0.535036
$130$	0	0
$131$	−7.72347	−0.674802	−0.337401	−	0.941361i	$-0.609548\pi$
$131$	−7.72347	−0.674802	−0.337401	+	0.941361i	$0.609548\pi$
$132$	0	0
$133$	− 2.08387i	− 0.180695i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.2311i	1.30128i	0.759387	+	0.650639i	$0.225499\pi$
$137$	15.2311i	1.30128i	−0.759387	+	0.650639i	$0.774501\pi$
$138$	0	0
$139$	−16.0590	−1.36210	−0.681051	−	0.732236i	$-0.738477\pi$
$139$	−16.0590	−1.36210	−0.681051	+	0.732236i	$0.738477\pi$
$140$	0	0
$141$	7.44693	0.627145
$142$	0	0
$143$	− 1.57360i	− 0.131591i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2.09091i	− 0.172455i
$148$	0	0
$149$	2.10879	0.172759	0.0863793	−	0.996262i	$-0.472470\pi$
$149$	2.10879	0.172759	0.0863793	+	0.996262i	$0.472470\pi$
$150$	0	0
$151$	17.3560	1.41241	0.706207	−	0.708006i	$-0.250405\pi$
$151$	17.3560	1.41241	0.706207	+	0.708006i	$0.250405\pi$
$152$	0	0
$153$	9.72347i	0.786096i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	0	0
$159$	9.42202	0.747215
$160$	0	0
$161$	18.6601	1.47062
$162$	0	0
$163$	0.852793i	0.0667959i	0.999442	+	0.0333979i	$0.0106329\pi$
$163$	0.852793i	0.0667959i	−0.999442	+	0.0333979i	$0.989367\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 18.5941i	− 1.43886i	−0.694566	−	0.719429i	$-0.744404\pi$
$167$	− 18.5941i	− 1.43886i	0.694566	−	0.719429i	$-0.255596\pi$
$168$	0	0
$169$	11.5282	0.886781
$170$	0	0
$171$	2.38094	0.182075
$172$	0	0
$173$	− 2.85279i	− 0.216894i	−0.994102	−	0.108447i	$-0.965412\pi$
$173$	− 2.85279i	− 0.216894i	0.994102	−	0.108447i	$-0.0345877\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 5.67536i	− 0.426586i
$178$	0	0
$179$	26.0768	1.94907	0.974537	−	0.224226i	$-0.0719854\pi$
$179$	26.0768	1.94907	0.974537	+	0.224226i	$0.0719854\pi$
$180$	0	0
$181$	3.18828	0.236983	0.118492	−	0.992955i	$-0.462194\pi$
$181$	3.18828	0.236983	0.118492	+	0.992955i	$0.462194\pi$
$182$	0	0
$183$	− 3.83226i	− 0.283288i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.29707i	0.387360i
$188$	0	0
$189$	−8.82256	−0.641747
$190$	0	0
$191$	7.95720	0.575763	0.287881	−	0.957666i	$-0.407049\pi$
$191$	7.95720	0.575763	0.287881	+	0.957666i	$0.407049\pi$
$192$	0	0
$193$	− 20.9296i	− 1.50655i	−0.657707	−	0.753274i	$-0.728473\pi$
$193$	− 20.9296i	− 1.50655i	0.657707	−	0.753274i	$-0.271527\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3.57360i	− 0.254609i	−0.991864	−	0.127304i	$-0.959368\pi$
$197$	− 3.57360i	− 0.254609i	0.991864	−	0.127304i	$-0.0406325\pi$
$198$	0	0
$199$	26.4194	1.87282	0.936409	−	0.350909i	$-0.114127\pi$
$199$	26.4194	1.87282	0.936409	+	0.350909i	$0.114127\pi$
$200$	0	0
$201$	8.95455	0.631605
$202$	0	0
$203$	− 19.5487i	− 1.37205i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	21.3203i	1.48186i
$208$	0	0
$209$	1.29707	0.0897202
$210$	0	0
$211$	−23.5487	−1.62116	−0.810579	−	0.585629i	$-0.800848\pi$
$211$	−23.5487	−1.62116	−0.810579	+	0.585629i	$0.800848\pi$
$212$	0	0
$213$	7.09738i	0.486304i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−4.45131	−0.300792
$220$	0	0
$221$	4.95455	0.333279
$222$	0	0
$223$	− 1.02054i	− 0.0683402i	−0.999416	−	0.0341701i	$-0.989121\pi$
$223$	− 1.02054i	− 0.0683402i	0.999416	−	0.0341701i	$-0.0108788\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.04545i	0.202134i	0.994880	+	0.101067i	$0.0322256\pi$
$227$	3.04545i	0.202134i	−0.994880	+	0.101067i	$0.967774\pi$
$228$	0	0
$229$	4.01788	0.265509	0.132755	−	0.991149i	$-0.457618\pi$
$229$	4.01788	0.265509	0.132755	+	0.991149i	$0.457618\pi$
$230$	0	0
$231$	−2.12667	−0.139925
$232$	0	0
$233$	− 25.2062i	− 1.65131i	−0.564175	−	0.825655i	$-0.690806\pi$
$233$	− 25.2062i	− 1.65131i	0.564175	−	0.825655i	$-0.309194\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	7.53253i	0.489290i
$238$	0	0
$239$	6.93667	0.448696	0.224348	−	0.974509i	$-0.427975\pi$
$239$	6.93667	0.448696	0.224348	+	0.974509i	$0.427975\pi$
$240$	0	0
$241$	0.761886	0.0490774	0.0245387	−	0.999699i	$-0.492188\pi$
$241$	0.761886	0.0490774	0.0245387	+	0.999699i	$0.492188\pi$
$242$	0	0
$243$	− 15.7003i	− 1.00717i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.21320i	− 0.0771940i
$248$	0	0
$249$	−8.46747	−0.536604
$250$	0	0
$251$	−6.14986	−0.388176	−0.194088	−	0.980984i	$-0.562175\pi$
$251$	−6.14986	−0.388176	−0.194088	+	0.980984i	$0.562175\pi$
$252$	0	0
$253$	11.6147i	0.730209i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 23.3560i	− 1.45691i	−0.685094	−	0.728454i	$-0.740239\pi$
$257$	− 23.3560i	− 1.45691i	0.685094	−	0.728454i	$-0.259761\pi$
$258$	0	0
$259$	−4.16774	−0.258971
$260$	0	0
$261$	22.3355	1.38253
$262$	0	0
$263$	− 12.2765i	− 0.757003i	−0.925601	−	0.378502i	$-0.876439\pi$
$263$	− 12.2765i	− 0.757003i	0.925601	−	0.378502i	$-0.123561\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	8.67098i	0.530655i
$268$	0	0
$269$	7.74135	0.471998	0.235999	−	0.971753i	$-0.424164\pi$
$269$	7.74135	0.471998	0.235999	+	0.971753i	$0.424164\pi$
$270$	0	0
$271$	−0.954547	−0.0579846	−0.0289923	−	0.999580i	$-0.509230\pi$
$271$	−0.954547	−0.0579846	−0.0289923	+	0.999580i	$0.509230\pi$
$272$	0	0
$273$	1.98915i	0.120389i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	17.0384i	1.02374i	0.859063	+	0.511870i	$0.171047\pi$
$277$	17.0384i	1.02374i	−0.859063	+	0.511870i	$0.828953\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−13.0974	−0.781324	−0.390662	−	0.920534i	$-0.627754\pi$
$281$	−13.0974	−0.781324	−0.390662	+	0.920534i	$0.627754\pi$
$282$	0	0
$283$	− 1.29707i	− 0.0771028i	−0.999257	−	0.0385514i	$-0.987726\pi$
$283$	− 1.29707i	− 0.0771028i	0.999257	−	0.0385514i	$-0.0122743\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 7.44693i	− 0.439579i
$288$	0	0
$289$	0.321987	0.0189404
$290$	0	0
$291$	−6.76189	−0.396389
$292$	0	0
$293$	21.5487i	1.25889i	0.777046	+	0.629444i	$0.216717\pi$
$293$	21.5487i	1.25889i	−0.777046	+	0.629444i	$0.783283\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 5.49145i	− 0.318646i
$298$	0	0
$299$	10.8636	0.628260
$300$	0	0
$301$	−16.0947	−0.927684
$302$	0	0
$303$	− 7.19704i	− 0.413459i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 9.18828i	− 0.524403i	−0.965013	−	0.262201i	$-0.915551\pi$
$307$	− 9.18828i	− 0.524403i	0.965013	−	0.262201i	$-0.0844485\pi$
$308$	0	0
$309$	−13.6558	−0.776849
$310$	0	0
$311$	−7.48973	−0.424704	−0.212352	−	0.977193i	$-0.568112\pi$
$311$	−7.48973	−0.424704	−0.212352	+	0.977193i	$0.568112\pi$
$312$	0	0
$313$	9.76626i	0.552022i	0.961154	+	0.276011i	$0.0890126\pi$
$313$	9.76626i	0.552022i	−0.961154	+	0.276011i	$0.910987\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.7164i	0.770392i	0.922835	+	0.385196i	$0.125866\pi$
$317$	13.7164i	0.770392i	−0.922835	+	0.385196i	$0.874134\pi$
$318$	0	0
$319$	12.1677	0.681263
$320$	0	0
$321$	−0.619057	−0.0345524
$322$	0	0
$323$	4.08387i	0.227233i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 10.6601i	− 0.589507i
$328$	0	0
$329$	19.7235	1.08739
$330$	0	0
$331$	−14.5282	−0.798539	−0.399270	−	0.916834i	$-0.630736\pi$
$331$	−14.5282	−0.798539	−0.399270	+	0.916834i	$0.630736\pi$
$332$	0	0
$333$	− 4.76189i	− 0.260950i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9.31495i	0.507418i	0.967281	+	0.253709i	$0.0816505\pi$
$337$	9.31495i	0.507418i	−0.967281	+	0.253709i	$0.918349\pi$
$338$	0	0
$339$	0.902625	0.0490238
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 20.1249i	− 1.08665i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 9.29707i	− 0.499093i	−0.968363	−	0.249546i	$-0.919718\pi$
$347$	− 9.29707i	− 0.499093i	0.968363	−	0.249546i	$-0.0802815\pi$
$348$	0	0
$349$	14.0590	0.752559	0.376279	−	0.926506i	$-0.377203\pi$
$349$	14.0590	0.752559	0.376279	+	0.926506i	$0.377203\pi$
$350$	0	0
$351$	−5.13636	−0.274159
$352$	0	0
$353$	20.4783i	1.08995i	0.838452	+	0.544975i	$0.183461\pi$
$353$	20.4783i	1.08995i	−0.838452	+	0.544975i	$0.816539\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 6.69589i	− 0.354384i
$358$	0	0
$359$	−18.9724	−1.00133	−0.500663	−	0.865642i	$-0.666910\pi$
$359$	−18.9724	−1.00133	−0.500663	+	0.865642i	$0.666910\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	7.33111i	0.384784i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 4.85279i	− 0.253314i	−0.991947	−	0.126657i	$-0.959575\pi$
$367$	− 4.85279i	− 0.253314i	0.991947	−	0.126657i	$-0.0404247\pi$
$368$	0	0
$369$	8.50855	0.442937
$370$	0	0
$371$	24.9545	1.29558
$372$	0	0
$373$	12.1071i	0.626880i	0.949608	+	0.313440i	$0.101481\pi$
$373$	12.1071i	0.626880i	−0.949608	+	0.313440i	$0.898519\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 11.3809i	− 0.586148i
$378$	0	0
$379$	2.02492	0.104013	0.0520065	−	0.998647i	$-0.483438\pi$
$379$	2.02492	0.104013	0.0520065	+	0.998647i	$0.483438\pi$
$380$	0	0
$381$	−0.802961	−0.0411369
$382$	0	0
$383$	1.23811i	0.0632647i	0.999500	+	0.0316323i	$0.0100706\pi$
$383$	1.23811i	0.0632647i	−0.999500	+	0.0316323i	$0.989929\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 18.3891i	− 0.934772i
$388$	0	0
$389$	−12.8707	−0.652569	−0.326285	−	0.945272i	$-0.605797\pi$
$389$	−12.8707	−0.652569	−0.326285	+	0.945272i	$0.605797\pi$
$390$	0	0
$391$	−36.5692	−1.84939
$392$	0	0
$393$	6.07684i	0.306536i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	17.3739i	0.871971i	0.899954	+	0.435986i	$0.143600\pi$
$397$	17.3739i	0.871971i	−0.899954	+	0.435986i	$0.856400\pi$
$398$	0	0
$399$	−1.63959	−0.0820824
$400$	0	0
$401$	19.0205	0.949840	0.474920	−	0.880029i	$-0.342477\pi$
$401$	19.0205	0.949840	0.474920	+	0.880029i	$0.342477\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 2.59414i	− 0.128587i
$408$	0	0
$409$	−20.4622	−1.01179	−0.505894	−	0.862595i	$-0.668837\pi$
$409$	−20.4622	−1.01179	−0.505894	+	0.862595i	$0.668837\pi$
$410$	0	0
$411$	11.9838	0.591119
$412$	0	0
$413$	− 15.0314i	− 0.739646i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	12.6352i	0.618749i
$418$	0	0
$419$	7.14721	0.349164	0.174582	−	0.984643i	$-0.444143\pi$
$419$	7.14721	0.349164	0.174582	+	0.984643i	$0.444143\pi$
$420$	0	0
$421$	12.3604	0.602409	0.301205	−	0.953560i	$-0.402611\pi$
$421$	12.3604	0.602409	0.301205	+	0.953560i	$0.402611\pi$
$422$	0	0
$423$	22.5352i	1.09570i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 10.1499i	− 0.491186i
$428$	0	0
$429$	−1.23811	−0.0597767
$430$	0	0
$431$	−1.90909	−0.0919578	−0.0459789	−	0.998942i	$-0.514641\pi$
$431$	−1.90909	−0.0919578	−0.0459789	+	0.998942i	$0.514641\pi$
$432$	0	0
$433$	− 4.04107i	− 0.194202i	−0.995275	−	0.0971008i	$-0.969043\pi$
$433$	− 4.04107i	− 0.194202i	0.995275	−	0.0971008i	$-0.0309569\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.95455i	0.428354i
$438$	0	0
$439$	−14.0768	−0.671851	−0.335925	−	0.941889i	$-0.609049\pi$
$439$	−14.0768	−0.671851	−0.335925	+	0.941889i	$0.609049\pi$
$440$	0	0
$441$	6.32730	0.301300
$442$	0	0
$443$	19.6736i	0.934723i	0.884066	+	0.467361i	$0.154795\pi$
$443$	19.6736i	0.934723i	−0.884066	+	0.467361i	$0.845205\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1.65920i	− 0.0784774i
$448$	0	0
$449$	−10.5531	−0.498030	−0.249015	−	0.968500i	$-0.580107\pi$
$449$	−10.5531	−0.498030	−0.249015	+	0.968500i	$0.580107\pi$
$450$	0	0
$451$	4.63522	0.218264
$452$	0	0
$453$	− 13.6558i	− 0.641603i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.72785i	0.127603i	0.997963	+	0.0638016i	$0.0203225\pi$
$457$	2.72785i	0.127603i	−0.997963	+	0.0638016i	$0.979678\pi$
$458$	0	0
$459$	17.2900	0.807030
$460$	0	0
$461$	−35.1153	−1.63548	−0.817740	−	0.575587i	$-0.804773\pi$
$461$	−35.1153	−1.63548	−0.817740	+	0.575587i	$0.804773\pi$
$462$	0	0
$463$	− 17.1293i	− 0.796067i	−0.917371	−	0.398034i	$-0.869693\pi$
$463$	− 17.1293i	− 0.796067i	0.917371	−	0.398034i	$-0.130307\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14.7029i	0.680370i	0.940358	+	0.340185i	$0.110490\pi$
$467$	14.7029i	0.680370i	−0.940358	+	0.340185i	$0.889510\pi$
$468$	0	0
$469$	23.7164	1.09512
$470$	0	0
$471$	11.0152	0.507555
$472$	0	0
$473$	− 10.0179i	− 0.460623i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	28.5120i	1.30547i
$478$	0	0
$479$	−10.0411	−0.458788	−0.229394	−	0.973334i	$-0.573674\pi$
$479$	−10.0411	−0.458788	−0.229394	+	0.973334i	$0.573674\pi$
$480$	0	0
$481$	−2.42640	−0.110634
$482$	0	0
$483$	− 14.6818i	− 0.668046i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 23.6504i	− 1.07170i	−0.844312	−	0.535852i	$-0.819991\pi$
$487$	− 23.6504i	− 1.07170i	0.844312	−	0.535852i	$-0.180009\pi$
$488$	0	0
$489$	0.670979	0.0303427
$490$	0	0
$491$	−38.3766	−1.73191	−0.865955	−	0.500122i	$-0.833289\pi$
$491$	−38.3766	−1.73191	−0.865955	+	0.500122i	$0.833289\pi$
$492$	0	0
$493$	38.3106i	1.72542i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	18.7976i	0.843190i
$498$	0	0
$499$	−10.3176	−0.461880	−0.230940	−	0.972968i	$-0.574180\pi$
$499$	−10.3176	−0.461880	−0.230940	+	0.972968i	$0.574180\pi$
$500$	0	0
$501$	−14.6299	−0.653616
$502$	0	0
$503$	20.4372i	0.911252i	0.890171	+	0.455626i	$0.150584\pi$
$503$	20.4372i	0.911252i	−0.890171	+	0.455626i	$0.849416\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9.07037i	− 0.402829i
$508$	0	0
$509$	15.4059	0.682853	0.341426	−	0.939909i	$-0.389090\pi$
$509$	15.4059	0.682853	0.341426	+	0.939909i	$0.389090\pi$
$510$	0	0
$511$	−11.7895	−0.521535
$512$	0	0
$513$	− 4.23374i	− 0.186924i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	12.2765i	0.539921i
$518$	0	0
$519$	−2.24458	−0.0985262
$520$	0	0
$521$	36.0270	1.57837	0.789186	−	0.614154i	$-0.210503\pi$
$521$	36.0270	1.57837	0.789186	+	0.614154i	$0.210503\pi$
$522$	0	0
$523$	− 44.7370i	− 1.95621i	−0.208109	−	0.978106i	$-0.566731\pi$
$523$	− 44.7370i	− 1.95621i	0.208109	−	0.978106i	$-0.433269\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−57.1839	−2.48626
$530$	0	0
$531$	17.1742	0.745297
$532$	0	0
$533$	− 4.33549i	− 0.187791i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 20.5173i	− 0.885387i
$538$	0	0
$539$	3.44693	0.148470
$540$	0	0
$541$	−37.5059	−1.61250	−0.806252	−	0.591572i	$-0.798507\pi$
$541$	−37.5059	−1.61250	−0.806252	+	0.591572i	$0.798507\pi$
$542$	0	0
$543$	− 2.50855i	− 0.107652i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	12.0000i	0.513083i	0.966533	+	0.256541i	$0.0825830\pi$
$547$	12.0000i	0.513083i	−0.966533	+	0.256541i	$0.917417\pi$
$548$	0	0
$549$	11.5968	0.494939
$550$	0	0
$551$	9.38094	0.399642
$552$	0	0
$553$	19.9502i	0.848367i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 33.3381i	− 1.41258i	−0.707921	−	0.706291i	$-0.750367\pi$
$557$	− 33.3381i	− 1.41258i	0.707921	−	0.706291i	$-0.249633\pi$
$558$	0	0
$559$	−9.37010	−0.396313
$560$	0	0
$561$	4.16774	0.175962
$562$	0	0
$563$	− 3.44693i	− 0.145271i	−0.997359	−	0.0726355i	$-0.976859\pi$
$563$	− 3.44693i	− 0.145271i	0.997359	−	0.0726355i	$-0.0231410\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 7.94313i	− 0.333580i
$568$	0	0
$569$	−27.0205	−1.13276	−0.566380	−	0.824144i	$-0.691657\pi$
$569$	−27.0205	−1.13276	−0.566380	+	0.824144i	$0.691657\pi$
$570$	0	0
$571$	−5.70559	−0.238771	−0.119386	−	0.992848i	$-0.538092\pi$
$571$	−5.70559	−0.238771	−0.119386	+	0.992848i	$0.538092\pi$
$572$	0	0
$573$	− 6.26074i	− 0.261546i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	6.34252i	0.264043i	0.991247	+	0.132021i	$0.0421467\pi$
$577$	6.34252i	0.264043i	−0.991247	+	0.132021i	$0.957853\pi$
$578$	0	0
$579$	−16.4675	−0.684365
$580$	0	0
$581$	−22.4264	−0.930404
$582$	0	0
$583$	15.5325i	0.643292i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 37.5827i	− 1.55121i	−0.631222	−	0.775603i	$-0.717446\pi$
$587$	− 37.5827i	− 1.55121i	0.631222	−	0.775603i	$-0.282554\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−2.81172	−0.115659
$592$	0	0
$593$	9.48270i	0.389408i	0.980862	+	0.194704i	$0.0623746\pi$
$593$	9.48270i	0.389408i	−0.980862	+	0.194704i	$0.937625\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 20.7868i	− 0.850747i
$598$	0	0
$599$	−24.6710	−1.00803	−0.504014	−	0.863695i	$-0.668144\pi$
$599$	−24.6710	−1.00803	−0.504014	+	0.863695i	$0.668144\pi$
$600$	0	0
$601$	41.4329	1.69008	0.845041	−	0.534702i	$-0.179576\pi$
$601$	41.4329	1.69008	0.845041	+	0.534702i	$0.179576\pi$
$602$	0	0
$603$	27.0974i	1.10349i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5.32026i	0.215943i	0.994154	+	0.107971i	$0.0344355\pi$
$607$	5.32026i	0.215943i	−0.994154	+	0.107971i	$0.965564\pi$
$608$	0	0
$609$	−15.3809	−0.623267
$610$	0	0
$611$	11.4827	0.464540
$612$	0	0
$613$	27.2472i	1.10051i	0.834998	+	0.550253i	$0.185469\pi$
$613$	27.2472i	1.10051i	−0.834998	+	0.550253i	$0.814531\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 30.9117i	− 1.24446i	−0.782834	−	0.622230i	$-0.786227\pi$
$617$	− 30.9117i	− 1.24446i	0.782834	−	0.622230i	$-0.213773\pi$
$618$	0	0
$619$	24.2857	0.976123	0.488061	−	0.872809i	$-0.337704\pi$
$619$	24.2857	0.976123	0.488061	+	0.872809i	$0.337704\pi$
$620$	0	0
$621$	37.9112	1.52132
$622$	0	0
$623$	22.9654i	0.920089i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1.02054i	− 0.0407563i
$628$	0	0
$629$	8.16774	0.325669
$630$	0	0
$631$	24.9117	0.991721	0.495861	−	0.868402i	$-0.334853\pi$
$631$	24.9117	0.991721	0.495861	+	0.868402i	$0.334853\pi$
$632$	0	0
$633$	18.5282i	0.736428i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 3.22405i	− 0.127741i
$638$	0	0
$639$	−21.4774	−0.849632
$640$	0	0
$641$	4.25865	0.168207	0.0841033	−	0.996457i	$-0.473197\pi$
$641$	4.25865	0.168207	0.0841033	+	0.996457i	$0.473197\pi$
$642$	0	0
$643$	13.9323i	0.549436i	0.961525	+	0.274718i	$0.0885845\pi$
$643$	13.9323i	0.549436i	−0.961525	+	0.274718i	$0.911416\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.58064i	0.219398i	0.993965	+	0.109699i	$0.0349886\pi$
$647$	5.58064i	0.219398i	−0.993965	+	0.109699i	$0.965011\pi$
$648$	0	0
$649$	9.35603	0.367256
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.77330i	0.0693945i	0.999398	+	0.0346973i	$0.0110467\pi$
$653$	1.77330i	0.0693945i	−0.999398	+	0.0346973i	$0.988953\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 13.4701i	− 0.525520i
$658$	0	0
$659$	1.12229	0.0437183	0.0218591	−	0.999761i	$-0.493041\pi$
$659$	1.12229	0.0437183	0.0218591	+	0.999761i	$0.493041\pi$
$660$	0	0
$661$	21.9340	0.853134	0.426567	−	0.904456i	$-0.359723\pi$
$661$	21.9340	0.853134	0.426567	+	0.904456i	$0.359723\pi$
$662$	0	0
$663$	− 3.89825i	− 0.151395i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	84.0021i	3.25257i
$668$	0	0
$669$	−0.802961	−0.0310443
$670$	0	0
$671$	6.31761	0.243889
$672$	0	0
$673$	8.24458i	0.317805i	0.987294	+	0.158903i	$0.0507956\pi$
$673$	8.24458i	0.317805i	−0.987294	+	0.158903i	$0.949204\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 5.76626i	− 0.221616i	−0.993842	−	0.110808i	$-0.964656\pi$
$677$	− 5.76626i	− 0.221616i	0.993842	−	0.110808i	$-0.0353438\pi$
$678$	0	0
$679$	−17.9091	−0.687288
$680$	0	0
$681$	2.39617	0.0918214
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 3.16128i	− 0.120610i
$688$	0	0
$689$	14.5282	0.553478
$690$	0	0
$691$	23.0384	0.876423	0.438211	−	0.898872i	$-0.355612\pi$
$691$	23.0384	0.876423	0.438211	+	0.898872i	$0.355612\pi$
$692$	0	0
$693$	− 6.43552i	− 0.244465i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	14.5941i	0.552793i
$698$	0	0
$699$	−19.8323	−0.750125
$700$	0	0
$701$	−3.90909	−0.147644	−0.0738222	−	0.997271i	$-0.523520\pi$
$701$	−3.90909	−0.147644	−0.0738222	+	0.997271i	$0.523520\pi$
$702$	0	0
$703$	− 2.00000i	− 0.0754314i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 19.0616i	− 0.716886i
$708$	0	0
$709$	12.0411	0.452212	0.226106	−	0.974103i	$-0.427400\pi$
$709$	12.0411	0.452212	0.226106	+	0.974103i	$0.427400\pi$
$710$	0	0
$711$	−22.7942	−0.854849
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 5.45778i	− 0.203825i
$718$	0	0
$719$	−32.2927	−1.20431	−0.602157	−	0.798378i	$-0.705692\pi$
$719$	−32.2927	−1.20431	−0.602157	+	0.798378i	$0.705692\pi$
$720$	0	0
$721$	−36.1677	−1.34696
$722$	0	0
$723$	− 0.599453i	− 0.0222939i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	41.1812i	1.52733i	0.645614	+	0.763664i	$0.276602\pi$
$727$	41.1812i	1.52733i	−0.645614	+	0.763664i	$0.723398\pi$
$728$	0	0
$729$	−0.917850	−0.0339945
$730$	0	0
$731$	31.5417	1.16661
$732$	0	0
$733$	− 1.27919i	− 0.0472479i	−0.999721	−	0.0236240i	$-0.992480\pi$
$733$	− 1.27919i	− 0.0472479i	0.999721	−	0.0236240i	$-0.00752044\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14.7619i	0.543761i
$738$	0	0
$739$	−30.3534	−1.11657	−0.558283	−	0.829650i	$-0.688540\pi$
$739$	−30.3534	−1.11657	−0.558283	+	0.829650i	$0.688540\pi$
$740$	0	0
$741$	−0.954547	−0.0350661
$742$	0	0
$743$	19.4827i	0.714751i	0.933961	+	0.357375i	$0.116328\pi$
$743$	19.4827i	0.714751i	−0.933961	+	0.357375i	$0.883672\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 25.6234i	− 0.937512i
$748$	0	0
$749$	−1.63959	−0.0599095
$750$	0	0
$751$	−48.6710	−1.77603	−0.888015	−	0.459815i	$-0.847916\pi$
$751$	−48.6710	−1.77603	−0.888015	+	0.459815i	$0.847916\pi$
$752$	0	0
$753$	4.83872i	0.176333i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	41.7235i	1.51647i	0.651984	+	0.758233i	$0.273937\pi$
$757$	41.7235i	1.51647i	−0.651984	+	0.758233i	$0.726063\pi$
$758$	0	0
$759$	9.13845	0.331705
$760$	0	0
$761$	−9.10441	−0.330035	−0.165017	−	0.986291i	$-0.552768\pi$
$761$	−9.10441	−0.330035	−0.165017	+	0.986291i	$0.552768\pi$
$762$	0	0
$763$	− 28.2337i	− 1.02213i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 8.75104i	− 0.315982i
$768$	0	0
$769$	−7.14548	−0.257673	−0.128836	−	0.991666i	$-0.541124\pi$
$769$	−7.14548	−0.257673	−0.128836	+	0.991666i	$0.541124\pi$
$770$	0	0
$771$	−18.3766	−0.661816
$772$	0	0
$773$	28.9956i	1.04290i	0.853282	+	0.521450i	$0.174609\pi$
$773$	28.9956i	1.04290i	−0.853282	+	0.521450i	$0.825391\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.27919i	0.117640i
$778$	0	0
$779$	3.57360	0.128038
$780$	0	0
$781$	−11.7003	−0.418669
$782$	0	0
$783$	− 39.7164i	− 1.41935i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 23.3311i	− 0.831664i	−0.909441	−	0.415832i	$-0.863490\pi$
$787$	− 23.3311i	− 0.831664i	0.909441	−	0.415832i	$-0.136510\pi$
$788$	0	0
$789$	−9.65920	−0.343877
$790$	0	0
$791$	2.39063	0.0850011
$792$	0	0
$793$	− 5.90909i	− 0.209838i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	19.2543i	0.682021i	0.940059	+	0.341011i	$0.110769\pi$
$797$	19.2543i	0.682021i	−0.940059	+	0.341011i	$0.889231\pi$
$798$	0	0
$799$	−38.6531	−1.36745
$800$	0	0
$801$	−26.2393	−0.927119
$802$	0	0
$803$	− 7.33815i	− 0.258958i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 6.09091i	− 0.214410i
$808$	0	0
$809$	−27.9983	−0.984367	−0.492184	−	0.870491i	$-0.663801\pi$
$809$	−27.9983	−0.984367	−0.492184	+	0.870491i	$0.663801\pi$
$810$	0	0
$811$	−47.4631	−1.66665	−0.833327	−	0.552780i	$-0.813567\pi$
$811$	−47.4631	−1.66665	−0.833327	+	0.552780i	$0.813567\pi$
$812$	0	0
$813$	0.751039i	0.0263401i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 7.72347i	− 0.270210i
$818$	0	0
$819$	−6.01938	−0.210334
$820$	0	0
$821$	2.14455	0.0748454	0.0374227	−	0.999300i	$-0.488085\pi$
$821$	2.14455	0.0748454	0.0374227	+	0.999300i	$0.488085\pi$
$822$	0	0
$823$	32.8458i	1.14493i	0.819929	+	0.572466i	$0.194013\pi$
$823$	32.8458i	1.14493i	−0.819929	+	0.572466i	$0.805987\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.6103i	0.925331i	0.886533	+	0.462665i	$0.153107\pi$
$827$	26.6103i	0.925331i	−0.886533	+	0.462665i	$0.846893\pi$
$828$	0	0
$829$	−50.6871	−1.76044	−0.880219	−	0.474569i	$-0.842604\pi$
$829$	−50.6871	−1.76044	−0.880219	+	0.474569i	$0.842604\pi$
$830$	0	0
$831$	13.4059	0.465044
$832$	0	0
$833$	10.8528i	0.376027i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	51.2651	1.76987	0.884934	−	0.465716i	$-0.154203\pi$
$839$	51.2651	1.76987	0.884934	+	0.465716i	$0.154203\pi$
$840$	0	0
$841$	59.0021	2.03455
$842$	0	0
$843$	10.3050i	0.354924i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	19.4167i	0.667166i
$848$	0	0
$849$	−1.02054	−0.0350248
$850$	0	0
$851$	17.9091	0.613916
$852$	0	0
$853$	7.95893i	0.272508i	0.990674	+	0.136254i	$0.0435064\pi$
$853$	7.95893i	0.272508i	−0.990674	+	0.136254i	$0.956494\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 4.09091i	− 0.139743i	−0.997556	−	0.0698714i	$-0.977741\pi$
$857$	− 4.09091i	− 0.139743i	0.997556	−	0.0698714i	$-0.0222589\pi$
$858$	0	0
$859$	13.3328	0.454910	0.227455	−	0.973789i	$-0.426959\pi$
$859$	13.3328	0.454910	0.227455	+	0.973789i	$0.426959\pi$
$860$	0	0
$861$	−5.85926	−0.199683
$862$	0	0
$863$	20.1677i	0.686518i	0.939241	+	0.343259i	$0.111531\pi$
$863$	20.1677i	0.686518i	−0.939241	+	0.343259i	$0.888469\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 0.253340i	− 0.00860386i
$868$	0	0
$869$	−12.4176	−0.421240
$870$	0	0
$871$	13.8073	0.467844
$872$	0	0
$873$	− 20.4622i	− 0.692539i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	23.8572i	0.805599i	0.915288	+	0.402800i	$0.131963\pi$
$877$	23.8572i	0.805599i	−0.915288	+	0.402800i	$0.868037\pi$
$878$	0	0
$879$	16.9545	0.571863
$880$	0	0
$881$	−47.9182	−1.61441	−0.807203	−	0.590274i	$-0.799020\pi$
$881$	−47.9182	−1.61441	−0.807203	+	0.590274i	$0.799020\pi$
$882$	0	0
$883$	28.5622i	0.961194i	0.876942	+	0.480597i	$0.159580\pi$
$883$	28.5622i	0.961194i	−0.876942	+	0.480597i	$0.840420\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	47.3150i	1.58868i	0.607473	+	0.794340i	$0.292183\pi$
$887$	47.3150i	1.58868i	−0.607473	+	0.794340i	$0.707817\pi$
$888$	0	0
$889$	−2.12667	−0.0713262
$890$	0	0
$891$	4.94407	0.165632
$892$	0	0
$893$	9.46482i	0.316728i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 8.54753i	− 0.285394i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−48.9047	−1.62925
$902$	0	0
$903$	12.6634i	0.421410i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 15.8982i	− 0.527893i	−0.964537	−	0.263946i	$-0.914976\pi$
$907$	− 15.8982i	− 0.527893i	0.964537	−	0.263946i	$-0.0850242\pi$
$908$	0	0
$909$	21.7790	0.722363
$910$	0	0
$911$	−20.2997	−0.672560	−0.336280	−	0.941762i	$-0.609169\pi$
$911$	−20.2997	−0.672560	−0.336280	+	0.941762i	$0.609169\pi$
$912$	0	0
$913$	− 13.9589i	− 0.461973i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.0947i	0.531494i
$918$	0	0
$919$	−32.6191	−1.07600	−0.538002	−	0.842944i	$-0.680821\pi$
$919$	−32.6191	−1.07600	−0.538002	+	0.842944i	$0.680821\pi$
$920$	0	0
$921$	−7.22936	−0.238215
$922$	0	0
$923$	10.9437i	0.360216i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 41.3237i	− 1.35725i
$928$	0	0
$929$	−53.4631	−1.75407	−0.877034	−	0.480429i	$-0.840481\pi$
$929$	−53.4631	−1.75407	−0.877034	+	0.480429i	$0.840481\pi$
$930$	0	0
$931$	2.65748	0.0870953
$932$	0	0
$933$	5.89293i	0.192926i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 43.0135i	− 1.40519i	−0.711590	−	0.702595i	$-0.752025\pi$
$937$	− 43.0135i	− 1.40519i	0.711590	−	0.702595i	$-0.247975\pi$
$938$	0	0
$939$	7.68411	0.250762
$940$	0	0
$941$	25.3311	0.825771	0.412885	−	0.910783i	$-0.364521\pi$
$941$	25.3311	0.825771	0.412885	+	0.910783i	$0.364521\pi$
$942$	0	0
$943$	32.0000i	1.04206i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12.6710i	0.411751i	0.978578	+	0.205876i	$0.0660043\pi$
$947$	12.6710i	0.411751i	−0.978578	+	0.205876i	$0.933996\pi$
$948$	0	0
$949$	−6.86364	−0.222803
$950$	0	0
$951$	10.7921	0.349958
$952$	0	0
$953$	31.2240i	1.01145i	0.862696	+	0.505723i	$0.168774\pi$
$953$	31.2240i	1.01145i	−0.862696	+	0.505723i	$0.831226\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 9.57360i	− 0.309471i
$958$	0	0
$959$	31.7396	1.02493
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	− 1.87333i	− 0.0603672i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	6.29441i	0.202415i	0.994865	+	0.101207i	$0.0322706\pi$
$967$	6.29441i	0.202415i	−0.994865	+	0.101207i	$0.967729\pi$
$968$	0	0
$969$	3.21320	0.103223
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	0	0
$973$	33.4648i	1.07283i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 34.1677i	− 1.09312i	−0.837419	−	0.546561i	$-0.815936\pi$
$977$	− 34.1677i	− 1.09312i	0.837419	−	0.546561i	$-0.184064\pi$
$978$	0	0
$979$	−14.2944	−0.456851
$980$	0	0
$981$	32.2587	1.02994
$982$	0	0
$983$	− 30.4264i	− 0.970451i	−0.874389	−	0.485226i	$-0.838737\pi$
$983$	− 30.4264i	− 0.970451i	0.874389	−	0.485226i	$-0.161263\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 15.5185i	− 0.493958i
$988$	0	0
$989$	69.1601	2.19916
$990$	0	0
$991$	−24.1179	−0.766131	−0.383065	−	0.923721i	$-0.625132\pi$
$991$	−24.1179	−0.766131	−0.383065	+	0.923721i	$0.625132\pi$
$992$	0	0
$993$	11.4308i	0.362745i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	15.9323i	0.504581i	0.967652	+	0.252290i	$0.0811838\pi$
$997$	15.9323i	0.504581i	−0.967652	+	0.252290i	$0.918816\pi$
$998$	0	0
$999$	−8.46747	−0.267899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.j.3649.3		6
5.2	odd	4		3800.2.a.r.1.2		3
5.3	odd	4		152.2.a.c.1.2	✓	3
5.4	even	2	inner	3800.2.d.j.3649.4		6
15.8	even	4		1368.2.a.n.1.1		3
20.3	even	4		304.2.a.g.1.2		3
20.7	even	4		7600.2.a.bv.1.2		3
35.13	even	4		7448.2.a.bf.1.2		3
40.3	even	4		1216.2.a.v.1.2		3
40.13	odd	4		1216.2.a.u.1.2		3
60.23	odd	4		2736.2.a.bd.1.1		3
95.18	even	4		2888.2.a.o.1.2		3
380.303	odd	4		5776.2.a.bp.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.a.c.1.2	✓	3	5.3	odd	4
304.2.a.g.1.2		3	20.3	even	4
1216.2.a.u.1.2		3	40.13	odd	4
1216.2.a.v.1.2		3	40.3	even	4
1368.2.a.n.1.1		3	15.8	even	4
2736.2.a.bd.1.1		3	60.23	odd	4
2888.2.a.o.1.2		3	95.18	even	4
3800.2.a.r.1.2		3	5.2	odd	4
3800.2.d.j.3649.3		6	1.1	even	1	trivial
3800.2.d.j.3649.4		6	5.4	even	2	inner
5776.2.a.bp.1.2		3	380.303	odd	4
7448.2.a.bf.1.2		3	35.13	even	4
7600.2.a.bv.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)

\(f(q)\)	\(=\)	\(q-0.786802i q^{3} -2.08387i q^{7} +2.38094 q^{9} +O(q^{10})\)	\(q-0.786802i q^{3} -2.08387i q^{7} +2.38094 q^{9} +1.29707 q^{11} -1.21320i q^{13} +4.08387i q^{17} +1.00000 q^{19} -1.63959 q^{21} +8.95455i q^{23} -4.23374i q^{27} +9.38094 q^{29} -1.02054i q^{33} -2.00000i q^{37} -0.954547 q^{39} +3.57360 q^{41} -7.72347i q^{43} +9.46482i q^{47} +2.65748 q^{49} +3.21320 q^{51} +11.9751i q^{53} -0.786802i q^{57} +7.21320 q^{59} +4.87067 q^{61} -4.96158i q^{63} +11.3809i q^{67} +7.04545 q^{69} -9.02054 q^{71} -5.65748i q^{73} -2.70293i q^{77} -9.57360 q^{79} +3.81172 q^{81} -10.7619i q^{83} -7.38094i q^{87} -11.0205 q^{89} -2.52815 q^{91} -8.59414i q^{97} +3.08825 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 24 q^{9}+O(q^{10}) \) 6 * q - 24 * q^9	\( 6 q - 24 q^{9} - 10 q^{11} + 6 q^{19} - 18 q^{21} + 18 q^{29} + 38 q^{39} + 16 q^{41} - 10 q^{49} + 22 q^{51} + 46 q^{59} + 6 q^{61} + 86 q^{69} - 24 q^{71} - 52 q^{79} + 94 q^{81} - 36 q^{89} + 34 q^{91} + 102 q^{99}+O(q^{100}) \) 6 * q - 24 * q^9 - 10 * q^11 + 6 * q^19 - 18 * q^21 + 18 * q^29 + 38 * q^39 + 16 * q^41 - 10 * q^49 + 22 * q^51 + 46 * q^59 + 6 * q^61 + 86 * q^69 - 24 * q^71 - 52 * q^79 + 94 * q^81 - 36 * q^89 + 34 * q^91 + 102 * q^99

Introduction

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