LMFDB - Embedded newform 3800.2.d.n.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:	$6$
Coefficient field:	6.0.399424.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 $ x^6 - 2x^5 + 3x^4 - 6x^3 + 6x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 760)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.2
Root			$1.40680 + 0.144584i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.n.3649.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.81361i q^{3} -4.91638i q^{7} -0.289169 q^{9} +O(q^{10})$	$q-1.81361i q^{3} -4.91638i q^{7} -0.289169 q^{9} +0.578337 q^{11} +6.39194i q^{13} -0.710831i q^{17} -1.00000 q^{19} -8.91638 q^{21} +2.71083i q^{23} -4.91638i q^{27} -6.54359 q^{29} +1.42166 q^{31} -1.04888i q^{33} -9.10278i q^{37} +11.5925 q^{39} -11.0489 q^{41} -5.83276i q^{43} -1.15667i q^{47} -17.1708 q^{49} -1.28917 q^{51} -13.2736i q^{53} +1.81361i q^{57} -11.3869 q^{59} -9.04888 q^{61} +1.42166i q^{63} +2.97028i q^{67} +4.91638 q^{69} +9.38692i q^{73} -2.84333i q^{77} -4.37279 q^{79} -9.78389 q^{81} +0.372787i q^{83} +11.8675i q^{87} +16.6167 q^{89} +31.4252 q^{91} -2.57834i q^{93} +3.94610i q^{97} -0.167237 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q - 6 q^{19} - 26 q^{21} + 14 q^{29} + 12 q^{31} - 6 q^{39} - 44 q^{41} - 24 q^{49} - 6 q^{51} - 22 q^{59} - 32 q^{61} + 2 q^{69} - 52 q^{79} - 26 q^{81} + 12 q^{89} + 58 q^{91} - 56 q^{99}+O(q^{100}) $ 6 * q - 6 * q^19 - 26 * q^21 + 14 * q^29 + 12 * q^31 - 6 * q^39 - 44 * q^41 - 24 * q^49 - 6 * q^51 - 22 * q^59 - 32 * q^61 + 2 * q^69 - 52 * q^79 - 26 * q^81 + 12 * q^89 + 58 * q^91 - 56 * 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
$3$	− 1.81361i	− 1.04709i	−0.851999	−	0.523543i	$-0.824610\pi$
$3$	− 1.81361i	− 1.04709i	0.851999	−	0.523543i	$-0.175390\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.91638i	− 1.85822i	−0.369807	−	0.929109i	$-0.620576\pi$
$7$	− 4.91638i	− 1.85822i	0.369807	−	0.929109i	$-0.379424\pi$
$8$	0	0
$9$	−0.289169	−0.0963895
$10$	0	0
$11$	0.578337	0.174375	0.0871876	−	0.996192i	$-0.472212\pi$
$11$	0.578337	0.174375	0.0871876	+	0.996192i	$0.472212\pi$
$12$	0	0
$13$	6.39194i	1.77281i	0.462914	+	0.886403i	$0.346804\pi$
$13$	6.39194i	1.77281i	−0.462914	+	0.886403i	$0.653196\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.710831i	− 0.172402i	−0.996278	−	0.0862010i	$-0.972527\pi$
$17$	− 0.710831i	− 0.172402i	0.996278	−	0.0862010i	$-0.0274727\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−8.91638	−1.94571
$22$	0	0
$23$	2.71083i	0.565247i	0.959231	+	0.282624i	$0.0912048\pi$
$23$	2.71083i	0.565247i	−0.959231	+	0.282624i	$0.908795\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4.91638i	− 0.946158i
$28$	0	0
$29$	−6.54359	−1.21512	−0.607558	−	0.794276i	$-0.707851\pi$
$29$	−6.54359	−1.21512	−0.607558	+	0.794276i	$0.707851\pi$
$30$	0	0
$31$	1.42166	0.255338	0.127669	−	0.991817i	$-0.459250\pi$
$31$	1.42166	0.255338	0.127669	+	0.991817i	$0.459250\pi$
$32$	0	0
$33$	− 1.04888i	− 0.182586i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 9.10278i	− 1.49649i	−0.663424	−	0.748243i	$-0.730897\pi$
$37$	− 9.10278i	− 1.49649i	0.663424	−	0.748243i	$-0.269103\pi$
$38$	0	0
$39$	11.5925	1.85628
$40$	0	0
$41$	−11.0489	−1.72554	−0.862772	−	0.505593i	$-0.831274\pi$
$41$	−11.0489	−1.72554	−0.862772	+	0.505593i	$0.831274\pi$
$42$	0	0
$43$	− 5.83276i	− 0.889488i	−0.895658	−	0.444744i	$-0.853295\pi$
$43$	− 5.83276i	− 0.889488i	0.895658	−	0.444744i	$-0.146705\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 1.15667i	− 0.168718i	−0.996435	−	0.0843591i	$-0.973116\pi$
$47$	− 1.15667i	− 0.168718i	0.996435	−	0.0843591i	$-0.0268843\pi$
$48$	0	0
$49$	−17.1708	−2.45297
$50$	0	0
$51$	−1.28917	−0.180520
$52$	0	0
$53$	− 13.2736i	− 1.82327i	−0.411004	−	0.911633i	$-0.634822\pi$
$53$	− 13.2736i	− 1.82327i	0.411004	−	0.911633i	$-0.365178\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.81361i	0.240218i
$58$	0	0
$59$	−11.3869	−1.48245	−0.741225	−	0.671256i	$-0.765755\pi$
$59$	−11.3869	−1.48245	−0.741225	+	0.671256i	$0.765755\pi$
$60$	0	0
$61$	−9.04888	−1.15859	−0.579295	−	0.815118i	$-0.696672\pi$
$61$	−9.04888	−1.15859	−0.579295	+	0.815118i	$0.696672\pi$
$62$	0	0
$63$	1.42166i	0.179113i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.97028i	0.362878i	0.983402	+	0.181439i	$0.0580754\pi$
$67$	2.97028i	0.362878i	−0.983402	+	0.181439i	$0.941925\pi$
$68$	0	0
$69$	4.91638	0.591863
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	9.38692i	1.09866i	0.835607	+	0.549328i	$0.185116\pi$
$73$	9.38692i	1.09866i	−0.835607	+	0.549328i	$0.814884\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.84333i	− 0.324027i
$78$	0	0
$79$	−4.37279	−0.491977	−0.245988	−	0.969273i	$-0.579113\pi$
$79$	−4.37279	−0.491977	−0.245988	+	0.969273i	$0.579113\pi$
$80$	0	0
$81$	−9.78389	−1.08710
$82$	0	0
$83$	0.372787i	0.0409187i	0.999791	+	0.0204593i	$0.00651287\pi$
$83$	0.372787i	0.0409187i	−0.999791	+	0.0204593i	$0.993487\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	11.8675i	1.27233i
$88$	0	0
$89$	16.6167	1.76136	0.880681	−	0.473710i	$-0.157086\pi$
$89$	16.6167	1.76136	0.880681	+	0.473710i	$0.157086\pi$
$90$	0	0
$91$	31.4252	3.29426
$92$	0	0
$93$	− 2.57834i	− 0.267361i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.94610i	0.400666i	0.979728	+	0.200333i	$0.0642024\pi$
$97$	3.94610i	0.400666i	−0.979728	+	0.200333i	$0.935798\pi$
$98$	0	0
$99$	−0.167237	−0.0168079
$100$	0	0
$101$	−6.20555	−0.617475	−0.308738	−	0.951147i	$-0.599907\pi$
$101$	−6.20555	−0.617475	−0.308738	+	0.951147i	$0.599907\pi$
$102$	0	0
$103$	− 8.15165i	− 0.803206i	−0.915814	−	0.401603i	$-0.868453\pi$
$103$	− 8.15165i	− 0.803206i	0.915814	−	0.401603i	$-0.131547\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 13.0680i	− 1.26333i	−0.775240	−	0.631667i	$-0.782371\pi$
$107$	− 13.0680i	− 1.26333i	0.775240	−	0.631667i	$-0.217629\pi$
$108$	0	0
$109$	11.5925	1.11036	0.555179	−	0.831731i	$-0.312650\pi$
$109$	11.5925	1.11036	0.555179	+	0.831731i	$0.312650\pi$
$110$	0	0
$111$	−16.5089	−1.56695
$112$	0	0
$113$	14.9355i	1.40502i	0.711675	+	0.702509i	$0.247937\pi$
$113$	14.9355i	1.40502i	−0.711675	+	0.702509i	$0.752063\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 1.84835i	− 0.170880i
$118$	0	0
$119$	−3.49472	−0.320360
$120$	0	0
$121$	−10.6655	−0.969593
$122$	0	0
$123$	20.0383i	1.80679i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.8867i	1.05477i	0.849626	+	0.527385i	$0.176828\pi$
$127$	11.8867i	1.05477i	−0.849626	+	0.527385i	$0.823172\pi$
$128$	0	0
$129$	−10.5783	−0.931371
$130$	0	0
$131$	−9.15667	−0.800022	−0.400011	−	0.916510i	$-0.630994\pi$
$131$	−9.15667	−0.800022	−0.400011	+	0.916510i	$0.630994\pi$
$132$	0	0
$133$	4.91638i	0.426304i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13.3869i	1.14372i	0.820351	+	0.571861i	$0.193778\pi$
$137$	13.3869i	1.14372i	−0.820351	+	0.571861i	$0.806222\pi$
$138$	0	0
$139$	−3.42166	−0.290222	−0.145111	−	0.989415i	$-0.546354\pi$
$139$	−3.42166	−0.290222	−0.145111	+	0.989415i	$0.546354\pi$
$140$	0	0
$141$	−2.09775	−0.176663
$142$	0	0
$143$	3.69670i	0.309133i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	31.1411i	2.56847i
$148$	0	0
$149$	3.36222	0.275444	0.137722	−	0.990471i	$-0.456022\pi$
$149$	3.36222	0.275444	0.137722	+	0.990471i	$0.456022\pi$
$150$	0	0
$151$	21.1950	1.72482	0.862412	−	0.506207i	$-0.168953\pi$
$151$	21.1950	1.72482	0.862412	+	0.506207i	$0.168953\pi$
$152$	0	0
$153$	0.205550i	0.0166177i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 11.7944i	− 0.941300i	−0.882320	−	0.470650i	$-0.844020\pi$
$157$	− 11.7944i	− 0.941300i	0.882320	−	0.470650i	$-0.155980\pi$
$158$	0	0
$159$	−24.0731	−1.90912
$160$	0	0
$161$	13.3275	1.05035
$162$	0	0
$163$	14.2439i	1.11567i	0.829953	+	0.557833i	$0.188367\pi$
$163$	14.2439i	1.11567i	−0.829953	+	0.557833i	$0.811633\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 7.47556i	− 0.578476i	−0.957257	−	0.289238i	$-0.906598\pi$
$167$	− 7.47556i	− 0.578476i	0.957257	−	0.289238i	$-0.0934020\pi$
$168$	0	0
$169$	−27.8569	−2.14284
$170$	0	0
$171$	0.289169	0.0221133
$172$	0	0
$173$	− 4.05390i	− 0.308212i	−0.988054	−	0.154106i	$-0.950750\pi$
$173$	− 4.05390i	− 0.308212i	0.988054	−	0.154106i	$-0.0492498\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	20.6514i	1.55225i
$178$	0	0
$179$	−2.95112	−0.220577	−0.110289	−	0.993900i	$-0.535178\pi$
$179$	−2.95112	−0.220577	−0.110289	+	0.993900i	$0.535178\pi$
$180$	0	0
$181$	9.66553	0.718433	0.359216	−	0.933254i	$-0.383044\pi$
$181$	9.66553	0.718433	0.359216	+	0.933254i	$0.383044\pi$
$182$	0	0
$183$	16.4111i	1.21314i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 0.411100i	− 0.0300626i
$188$	0	0
$189$	−24.1708	−1.75817
$190$	0	0
$191$	−15.9058	−1.15090	−0.575452	−	0.817835i	$-0.695174\pi$
$191$	−15.9058	−1.15090	−0.575452	+	0.817835i	$0.695174\pi$
$192$	0	0
$193$	− 15.6116i	− 1.12375i	−0.827222	−	0.561875i	$-0.810080\pi$
$193$	− 15.6116i	− 1.12375i	0.827222	−	0.561875i	$-0.189920\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 13.6655i	− 0.973628i	−0.873506	−	0.486814i	$-0.838159\pi$
$197$	− 13.6655i	− 0.973628i	0.873506	−	0.486814i	$-0.161841\pi$
$198$	0	0
$199$	−8.91638	−0.632066	−0.316033	−	0.948748i	$-0.602351\pi$
$199$	−8.91638	−0.632066	−0.316033	+	0.948748i	$0.602351\pi$
$200$	0	0
$201$	5.38692	0.379964
$202$	0	0
$203$	32.1708i	2.25795i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 0.783887i	− 0.0544839i
$208$	0	0
$209$	−0.578337	−0.0400044
$210$	0	0
$211$	19.7980	1.36295	0.681476	−	0.731840i	$-0.261338\pi$
$211$	19.7980	1.36295	0.681476	+	0.731840i	$0.261338\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 6.98944i	− 0.474474i
$218$	0	0
$219$	17.0242	1.15039
$220$	0	0
$221$	4.54359	0.305635
$222$	0	0
$223$	1.57331i	0.105357i	0.998612	+	0.0526784i	$0.0167758\pi$
$223$	1.57331i	0.105357i	−0.998612	+	0.0526784i	$0.983224\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 28.0575i	− 1.86224i	−0.364713	−	0.931120i	$-0.618833\pi$
$227$	− 28.0575i	− 1.86224i	0.364713	−	0.931120i	$-0.381167\pi$
$228$	0	0
$229$	−2.47054	−0.163258	−0.0816289	−	0.996663i	$-0.526012\pi$
$229$	−2.47054	−0.163258	−0.0816289	+	0.996663i	$0.526012\pi$
$230$	0	0
$231$	−5.15667	−0.339284
$232$	0	0
$233$	− 3.15667i	− 0.206801i	−0.994640	−	0.103400i	$-0.967028\pi$
$233$	− 3.15667i	− 0.206801i	0.994640	−	0.103400i	$-0.0329723\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	7.93051i	0.515142i
$238$	0	0
$239$	2.74914	0.177827	0.0889137	−	0.996039i	$-0.471660\pi$
$239$	2.74914	0.177827	0.0889137	+	0.996039i	$0.471660\pi$
$240$	0	0
$241$	8.88164	0.572117	0.286058	−	0.958212i	$-0.407655\pi$
$241$	8.88164	0.572117	0.286058	+	0.958212i	$0.407655\pi$
$242$	0	0
$243$	2.99498i	0.192128i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 6.39194i	− 0.406710i
$248$	0	0
$249$	0.676089	0.0428454
$250$	0	0
$251$	6.31335	0.398495	0.199248	−	0.979949i	$-0.436150\pi$
$251$	6.31335	0.398495	0.199248	+	0.979949i	$0.436150\pi$
$252$	0	0
$253$	1.56777i	0.0985651i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.83830i	0.488940i	0.969657	+	0.244470i	$0.0786140\pi$
$257$	7.83830i	0.488940i	−0.969657	+	0.244470i	$0.921386\pi$
$258$	0	0
$259$	−44.7527	−2.78080
$260$	0	0
$261$	1.89220	0.117124
$262$	0	0
$263$	− 13.8711i	− 0.855327i	−0.903938	−	0.427664i	$-0.859337\pi$
$263$	− 13.8711i	− 0.855327i	0.903938	−	0.427664i	$-0.140663\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 30.1361i	− 1.84430i
$268$	0	0
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	−4.07306	−0.247421	−0.123710	−	0.992318i	$-0.539479\pi$
$271$	−4.07306	−0.247421	−0.123710	+	0.992318i	$0.539479\pi$
$272$	0	0
$273$	− 56.9930i	− 3.44937i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	16.1361i	0.969522i	0.874647	+	0.484761i	$0.161093\pi$
$277$	16.1361i	0.969522i	−0.874647	+	0.484761i	$0.838907\pi$
$278$	0	0
$279$	−0.411100	−0.0246119
$280$	0	0
$281$	−11.4600	−0.683645	−0.341822	−	0.939765i	$-0.611044\pi$
$281$	−11.4600	−0.683645	−0.341822	+	0.939765i	$0.611044\pi$
$282$	0	0
$283$	21.7250i	1.29142i	0.763585	+	0.645708i	$0.223437\pi$
$283$	21.7250i	1.29142i	−0.763585	+	0.645708i	$0.776563\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	54.3205i	3.20644i
$288$	0	0
$289$	16.4947	0.970278
$290$	0	0
$291$	7.15667	0.419532
$292$	0	0
$293$	5.91136i	0.345345i	0.984979	+	0.172673i	$0.0552403\pi$
$293$	5.91136i	0.345345i	−0.984979	+	0.172673i	$0.944760\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2.84333i	− 0.164986i
$298$	0	0
$299$	−17.3275	−1.00207
$300$	0	0
$301$	−28.6761	−1.65286
$302$	0	0
$303$	11.2544i	0.646550i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 29.7194i	− 1.69618i	−0.529854	−	0.848089i	$-0.677753\pi$
$307$	− 29.7194i	− 1.69618i	0.529854	−	0.848089i	$-0.322247\pi$
$308$	0	0
$309$	−14.7839	−0.841026
$310$	0	0
$311$	−0.407530	−0.0231089	−0.0115544	−	0.999933i	$-0.503678\pi$
$311$	−0.407530	−0.0231089	−0.0115544	+	0.999933i	$0.503678\pi$
$312$	0	0
$313$	0.338044i	0.0191074i	0.999954	+	0.00955370i	$0.00304108\pi$
$313$	0.338044i	0.0191074i	−0.999954	+	0.00955370i	$0.996959\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 8.75468i	− 0.491712i	−0.969306	−	0.245856i	$-0.920931\pi$
$317$	− 8.75468i	− 0.491712i	0.969306	−	0.245856i	$-0.0790690\pi$
$318$	0	0
$319$	−3.78440	−0.211886
$320$	0	0
$321$	−23.7003	−1.32282
$322$	0	0
$323$	0.710831i	0.0395517i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 21.0242i	− 1.16264i
$328$	0	0
$329$	−5.68665	−0.313515
$330$	0	0
$331$	23.8675	1.31188	0.655938	−	0.754814i	$-0.272273\pi$
$331$	23.8675	1.31188	0.655938	+	0.754814i	$0.272273\pi$
$332$	0	0
$333$	2.63224i	0.144246i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 10.0439i	− 0.547124i	−0.961854	−	0.273562i	$-0.911798\pi$
$337$	− 10.0439i	− 0.547124i	0.961854	−	0.273562i	$-0.0882018\pi$
$338$	0	0
$339$	27.0872	1.47117
$340$	0	0
$341$	0.822200	0.0445246
$342$	0	0
$343$	50.0036i	2.69994i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 15.6272i	− 0.838913i	−0.907775	−	0.419456i	$-0.862221\pi$
$347$	− 15.6272i	− 0.838913i	0.907775	−	0.419456i	$-0.137779\pi$
$348$	0	0
$349$	−17.2544	−0.923608	−0.461804	−	0.886982i	$-0.652798\pi$
$349$	−17.2544	−0.923608	−0.461804	+	0.886982i	$0.652798\pi$
$350$	0	0
$351$	31.4252	1.67735
$352$	0	0
$353$	− 27.2197i	− 1.44876i	−0.689402	−	0.724379i	$-0.742127\pi$
$353$	− 27.2197i	− 1.44876i	0.689402	−	0.724379i	$-0.257873\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6.33804i	0.335445i
$358$	0	0
$359$	7.18137	0.379018	0.189509	−	0.981879i	$-0.439310\pi$
$359$	7.18137	0.379018	0.189509	+	0.981879i	$0.439310\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	19.3431i	1.01525i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	9.15667i	0.477975i	0.971023	+	0.238987i	$0.0768154\pi$
$367$	9.15667i	0.477975i	−0.971023	+	0.238987i	$0.923185\pi$
$368$	0	0
$369$	3.19499	0.166324
$370$	0	0
$371$	−65.2580	−3.38803
$372$	0	0
$373$	8.75468i	0.453300i	0.973976	+	0.226650i	$0.0727774\pi$
$373$	8.75468i	0.453300i	−0.973976	+	0.226650i	$0.927223\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 41.8263i	− 2.15416i
$378$	0	0
$379$	12.9164	0.663470	0.331735	−	0.943373i	$-0.392366\pi$
$379$	12.9164	0.663470	0.331735	+	0.943373i	$0.392366\pi$
$380$	0	0
$381$	21.5577	1.10444
$382$	0	0
$383$	1.64280i	0.0839431i	0.999119	+	0.0419716i	$0.0133639\pi$
$383$	1.64280i	0.0839431i	−0.999119	+	0.0419716i	$0.986636\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.68665i	0.0857373i
$388$	0	0
$389$	11.8328	0.599945	0.299972	−	0.953948i	$-0.403023\pi$
$389$	11.8328	0.599945	0.299972	+	0.953948i	$0.403023\pi$
$390$	0	0
$391$	1.92694	0.0974498
$392$	0	0
$393$	16.6066i	0.837692i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 24.1361i	− 1.21135i	−0.795710	−	0.605677i	$-0.792902\pi$
$397$	− 24.1361i	− 1.21135i	0.795710	−	0.605677i	$-0.207098\pi$
$398$	0	0
$399$	8.91638	0.446377
$400$	0	0
$401$	19.9789	0.997697	0.498849	−	0.866689i	$-0.333756\pi$
$401$	19.9789	0.997697	0.498849	+	0.866689i	$0.333756\pi$
$402$	0	0
$403$	9.08719i	0.452665i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 5.26447i	− 0.260950i
$408$	0	0
$409$	−5.66553	−0.280142	−0.140071	−	0.990141i	$-0.544733\pi$
$409$	−5.66553	−0.280142	−0.140071	+	0.990141i	$0.544733\pi$
$410$	0	0
$411$	24.2786	1.19758
$412$	0	0
$413$	55.9824i	2.75472i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	6.20555i	0.303887i
$418$	0	0
$419$	−22.6550	−1.10677	−0.553384	−	0.832926i	$-0.686664\pi$
$419$	−22.6550	−1.10677	−0.553384	+	0.832926i	$0.686664\pi$
$420$	0	0
$421$	−25.5330	−1.24440	−0.622202	−	0.782857i	$-0.713762\pi$
$421$	−25.5330	−1.24440	−0.622202	+	0.782857i	$0.713762\pi$
$422$	0	0
$423$	0.334474i	0.0162627i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	44.4877i	2.15291i
$428$	0	0
$429$	6.70436	0.323689
$430$	0	0
$431$	−13.7944	−0.664455	−0.332228	−	0.943199i	$-0.607800\pi$
$431$	−13.7944	−0.664455	−0.332228	+	0.943199i	$0.607800\pi$
$432$	0	0
$433$	29.4444i	1.41501i	0.706710	+	0.707504i	$0.250179\pi$
$433$	29.4444i	1.41501i	−0.706710	+	0.707504i	$0.749821\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.71083i	− 0.129677i
$438$	0	0
$439$	34.5472	1.64885	0.824423	−	0.565974i	$-0.191500\pi$
$439$	34.5472	1.64885	0.824423	+	0.565974i	$0.191500\pi$
$440$	0	0
$441$	4.96526	0.236441
$442$	0	0
$443$	23.5577i	1.11926i	0.828742	+	0.559631i	$0.189057\pi$
$443$	23.5577i	1.11926i	−0.828742	+	0.559631i	$0.810943\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 6.09775i	− 0.288414i
$448$	0	0
$449$	7.49115	0.353529	0.176765	−	0.984253i	$-0.443437\pi$
$449$	7.49115	0.353529	0.176765	+	0.984253i	$0.443437\pi$
$450$	0	0
$451$	−6.38997	−0.300892
$452$	0	0
$453$	− 38.4394i	− 1.80604i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	38.5819i	1.80479i	0.430914	+	0.902393i	$0.358191\pi$
$457$	38.5819i	1.80479i	−0.430914	+	0.902393i	$0.641809\pi$
$458$	0	0
$459$	−3.49472	−0.163119
$460$	0	0
$461$	−31.9789	−1.48940	−0.744702	−	0.667397i	$-0.767409\pi$
$461$	−31.9789	−1.48940	−0.744702	+	0.667397i	$0.767409\pi$
$462$	0	0
$463$	− 30.8122i	− 1.43196i	−0.698120	−	0.715981i	$-0.745980\pi$
$463$	− 30.8122i	− 1.43196i	0.698120	−	0.715981i	$-0.254020\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	41.4600i	1.91854i	0.282493	+	0.959269i	$0.408839\pi$
$467$	41.4600i	1.91854i	−0.282493	+	0.959269i	$0.591161\pi$
$468$	0	0
$469$	14.6030	0.674305
$470$	0	0
$471$	−21.3905	−0.985622
$472$	0	0
$473$	− 3.37330i	− 0.155105i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.83830i	0.175744i
$478$	0	0
$479$	9.93051	0.453737	0.226868	−	0.973925i	$-0.427151\pi$
$479$	9.93051	0.453737	0.226868	+	0.973925i	$0.427151\pi$
$480$	0	0
$481$	58.1844	2.65298
$482$	0	0
$483$	− 24.1708i	− 1.09981i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 19.7789i	− 0.896266i	−0.893967	−	0.448133i	$-0.852089\pi$
$487$	− 19.7789i	− 0.896266i	0.893967	−	0.448133i	$-0.147911\pi$
$488$	0	0
$489$	25.8328	1.16820
$490$	0	0
$491$	−16.1461	−0.728664	−0.364332	−	0.931269i	$-0.618703\pi$
$491$	−16.1461	−0.728664	−0.364332	+	0.931269i	$0.618703\pi$
$492$	0	0
$493$	4.65139i	0.209488i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	38.8222	1.73792	0.868960	−	0.494882i	$-0.164789\pi$
$499$	38.8222	1.73792	0.868960	+	0.494882i	$0.164789\pi$
$500$	0	0
$501$	−13.5577	−0.605715
$502$	0	0
$503$	− 12.5436i	− 0.559291i	−0.960103	−	0.279646i	$-0.909783\pi$
$503$	− 12.5436i	− 0.559291i	0.960103	−	0.279646i	$-0.0902170\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	50.5215i	2.24374i
$508$	0	0
$509$	−25.0278	−1.10934	−0.554668	−	0.832072i	$-0.687155\pi$
$509$	−25.0278	−1.10934	−0.554668	+	0.832072i	$0.687155\pi$
$510$	0	0
$511$	46.1497	2.04154
$512$	0	0
$513$	4.91638i	0.217064i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 0.668948i	− 0.0294203i
$518$	0	0
$519$	−7.35218	−0.322725
$520$	0	0
$521$	−16.6550	−0.729667	−0.364834	−	0.931073i	$-0.618874\pi$
$521$	−16.6550	−0.729667	−0.364834	+	0.931073i	$0.618874\pi$
$522$	0	0
$523$	− 24.3608i	− 1.06522i	−0.846360	−	0.532611i	$-0.821211\pi$
$523$	− 24.3608i	− 1.06522i	0.846360	−	0.532611i	$-0.178789\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.01056i	− 0.0440208i
$528$	0	0
$529$	15.6514	0.680495
$530$	0	0
$531$	3.29274	0.142893
$532$	0	0
$533$	− 70.6238i	− 3.05906i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	5.35218i	0.230964i
$538$	0	0
$539$	−9.93051	−0.427738
$540$	0	0
$541$	15.3622	0.660474	0.330237	−	0.943898i	$-0.392871\pi$
$541$	15.3622	0.660474	0.330237	+	0.943898i	$0.392871\pi$
$542$	0	0
$543$	− 17.5295i	− 0.752261i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 26.5527i	− 1.13531i	−0.823266	−	0.567656i	$-0.807850\pi$
$547$	− 26.5527i	− 1.13531i	0.823266	−	0.567656i	$-0.192150\pi$
$548$	0	0
$549$	2.61665	0.111676
$550$	0	0
$551$	6.54359	0.278767
$552$	0	0
$553$	21.4983i	0.914200i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 28.9583i	− 1.22700i	−0.789694	−	0.613501i	$-0.789761\pi$
$557$	− 28.9583i	− 1.22700i	0.789694	−	0.613501i	$-0.210239\pi$
$558$	0	0
$559$	37.2827	1.57689
$560$	0	0
$561$	−0.745574	−0.0314782
$562$	0	0
$563$	1.64280i	0.0692357i	0.999401	+	0.0346179i	$0.0110214\pi$
$563$	1.64280i	0.0692357i	−0.999401	+	0.0346179i	$0.988979\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	48.1013i	2.02007i
$568$	0	0
$569$	−14.3416	−0.601232	−0.300616	−	0.953745i	$-0.597192\pi$
$569$	−14.3416	−0.601232	−0.300616	+	0.953745i	$0.597192\pi$
$570$	0	0
$571$	−39.0177	−1.63284	−0.816420	−	0.577458i	$-0.804045\pi$
$571$	−39.0177	−1.63284	−0.816420	+	0.577458i	$0.804045\pi$
$572$	0	0
$573$	28.8469i	1.20510i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	25.4947i	1.06136i	0.847573	+	0.530680i	$0.178063\pi$
$577$	25.4947i	1.06136i	−0.847573	+	0.530680i	$0.821937\pi$
$578$	0	0
$579$	−28.3133	−1.17666
$580$	0	0
$581$	1.83276	0.0760358
$582$	0	0
$583$	− 7.67661i	− 0.317933i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 9.72496i	− 0.401392i	−0.979654	−	0.200696i	$-0.935680\pi$
$587$	− 9.72496i	− 0.401392i	0.979654	−	0.200696i	$-0.0643204\pi$
$588$	0	0
$589$	−1.42166	−0.0585786
$590$	0	0
$591$	−24.7839	−1.01947
$592$	0	0
$593$	13.0388i	0.535441i	0.963497	+	0.267720i	$0.0862703\pi$
$593$	13.0388i	0.535441i	−0.963497	+	0.267720i	$0.913730\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.1708i	0.661827i
$598$	0	0
$599$	18.6861	0.763495	0.381747	−	0.924267i	$-0.375322\pi$
$599$	18.6861	0.763495	0.381747	+	0.924267i	$0.375322\pi$
$600$	0	0
$601$	12.7355	0.519493	0.259747	−	0.965677i	$-0.416361\pi$
$601$	12.7355	0.519493	0.259747	+	0.965677i	$0.416361\pi$
$602$	0	0
$603$	− 0.858912i	− 0.0349776i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	23.0645i	0.936158i	0.883687	+	0.468079i	$0.155054\pi$
$607$	23.0645i	0.936158i	−0.883687	+	0.468079i	$0.844946\pi$
$608$	0	0
$609$	58.3452	2.36427
$610$	0	0
$611$	7.39340	0.299105
$612$	0	0
$613$	− 25.7038i	− 1.03817i	−0.854723	−	0.519084i	$-0.826273\pi$
$613$	− 25.7038i	− 1.03817i	0.854723	−	0.519084i	$-0.173727\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 0.205550i	− 0.00827514i	−0.999991	−	0.00413757i	$-0.998683\pi$
$617$	− 0.205550i	− 0.00827514i	0.999991	−	0.00413757i	$-0.00131703\pi$
$618$	0	0
$619$	11.0872	0.445632	0.222816	−	0.974861i	$-0.428475\pi$
$619$	11.0872	0.445632	0.222816	+	0.974861i	$0.428475\pi$
$620$	0	0
$621$	13.3275	0.534813
$622$	0	0
$623$	− 81.6938i	− 3.27299i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.04888i	0.0418881i
$628$	0	0
$629$	−6.47054	−0.257997
$630$	0	0
$631$	10.4806	0.417226	0.208613	−	0.977998i	$-0.433105\pi$
$631$	10.4806	0.417226	0.208613	+	0.977998i	$0.433105\pi$
$632$	0	0
$633$	− 35.9058i	− 1.42713i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 109.755i	− 4.34864i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	37.3694	1.47600	0.738001	−	0.674800i	$-0.235770\pi$
$641$	37.3694	1.47600	0.738001	+	0.674800i	$0.235770\pi$
$642$	0	0
$643$	− 42.0172i	− 1.65700i	−0.559992	−	0.828498i	$-0.689196\pi$
$643$	− 42.0172i	− 1.65700i	0.559992	−	0.828498i	$-0.310804\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	23.4635i	0.922447i	0.887284	+	0.461224i	$0.152589\pi$
$647$	23.4635i	0.922447i	−0.887284	+	0.461224i	$0.847411\pi$
$648$	0	0
$649$	−6.58548	−0.258503
$650$	0	0
$651$	−12.6761	−0.496815
$652$	0	0
$653$	− 33.6272i	− 1.31593i	−0.753047	−	0.657967i	$-0.771417\pi$
$653$	− 33.6272i	− 1.31593i	0.753047	−	0.657967i	$-0.228583\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.71440i	− 0.105899i
$658$	0	0
$659$	−17.9653	−0.699827	−0.349914	−	0.936782i	$-0.613789\pi$
$659$	−17.9653	−0.699827	−0.349914	+	0.936782i	$0.613789\pi$
$660$	0	0
$661$	15.5542	0.604987	0.302493	−	0.953152i	$-0.402181\pi$
$661$	15.5542	0.604987	0.302493	+	0.953152i	$0.402181\pi$
$662$	0	0
$663$	− 8.24029i	− 0.320026i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 17.7386i	− 0.686841i
$668$	0	0
$669$	2.85337	0.110318
$670$	0	0
$671$	−5.23330	−0.202029
$672$	0	0
$673$	− 12.2111i	− 0.470703i	−0.971910	−	0.235351i	$-0.924376\pi$
$673$	− 12.2111i	− 0.470703i	0.971910	−	0.235351i	$-0.0756241\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 43.0680i	− 1.65524i	−0.561290	−	0.827619i	$-0.689695\pi$
$677$	− 43.0680i	− 1.65524i	0.561290	−	0.827619i	$-0.310305\pi$
$678$	0	0
$679$	19.4005	0.744524
$680$	0	0
$681$	−50.8852	−1.94993
$682$	0	0
$683$	45.3850i	1.73661i	0.496033	+	0.868303i	$0.334789\pi$
$683$	45.3850i	1.73661i	−0.496033	+	0.868303i	$0.665211\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	4.48059i	0.170945i
$688$	0	0
$689$	84.8440	3.23230
$690$	0	0
$691$	−0.508852	−0.0193576	−0.00967882	−	0.999953i	$-0.503081\pi$
$691$	−0.508852	−0.0193576	−0.00967882	+	0.999953i	$0.503081\pi$
$692$	0	0
$693$	0.822200i	0.0312328i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	7.85389i	0.297487i
$698$	0	0
$699$	−5.72496	−0.216538
$700$	0	0
$701$	−40.1844	−1.51774	−0.758872	−	0.651239i	$-0.774249\pi$
$701$	−40.1844	−1.51774	−0.758872	+	0.651239i	$0.774249\pi$
$702$	0	0
$703$	9.10278i	0.343318i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	30.5089i	1.14740i
$708$	0	0
$709$	−20.0978	−0.754787	−0.377393	−	0.926053i	$-0.623180\pi$
$709$	−20.0978	−0.754787	−0.377393	+	0.926053i	$0.623180\pi$
$710$	0	0
$711$	1.26447	0.0474214
$712$	0	0
$713$	3.85389i	0.144329i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 4.98587i	− 0.186201i
$718$	0	0
$719$	−35.5925	−1.32738	−0.663688	−	0.748010i	$-0.731010\pi$
$719$	−35.5925	−1.32738	−0.663688	+	0.748010i	$0.731010\pi$
$720$	0	0
$721$	−40.0766	−1.49253
$722$	0	0
$723$	− 16.1078i	− 0.599055i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 20.1708i	− 0.748094i	−0.927410	−	0.374047i	$-0.877970\pi$
$727$	− 20.1708i	− 0.748094i	0.927410	−	0.374047i	$-0.122030\pi$
$728$	0	0
$729$	−23.9200	−0.885924
$730$	0	0
$731$	−4.14611	−0.153349
$732$	0	0
$733$	− 30.1461i	− 1.11347i	−0.830689	−	0.556736i	$-0.812053\pi$
$733$	− 30.1461i	− 1.11347i	0.830689	−	0.556736i	$-0.187947\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.71782i	0.0632768i
$738$	0	0
$739$	−26.0283	−0.957465	−0.478733	−	0.877961i	$-0.658904\pi$
$739$	−26.0283	−0.957465	−0.478733	+	0.877961i	$0.658904\pi$
$740$	0	0
$741$	−11.5925	−0.425860
$742$	0	0
$743$	− 0.221136i	− 0.00811269i	−0.999992	−	0.00405635i	$-0.998709\pi$
$743$	− 0.221136i	− 0.00811269i	0.999992	−	0.00405635i	$-0.00129118\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 0.107798i	− 0.00394413i
$748$	0	0
$749$	−64.2474	−2.34755
$750$	0	0
$751$	41.9406	1.53043	0.765216	−	0.643773i	$-0.222632\pi$
$751$	41.9406	1.53043	0.765216	+	0.643773i	$0.222632\pi$
$752$	0	0
$753$	− 11.4499i	− 0.417259i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 26.7456i	− 0.972084i	−0.873935	−	0.486042i	$-0.838440\pi$
$757$	− 26.7456i	− 0.972084i	0.873935	−	0.486042i	$-0.161560\pi$
$758$	0	0
$759$	2.84333	0.103206
$760$	0	0
$761$	−11.6519	−0.422381	−0.211191	−	0.977445i	$-0.567734\pi$
$761$	−11.6519	−0.422381	−0.211191	+	0.977445i	$0.567734\pi$
$762$	0	0
$763$	− 56.9930i	− 2.06329i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 72.7846i	− 2.62810i
$768$	0	0
$769$	9.28917	0.334976	0.167488	−	0.985874i	$-0.446434\pi$
$769$	9.28917	0.334976	0.167488	+	0.985874i	$0.446434\pi$
$770$	0	0
$771$	14.2156	0.511962
$772$	0	0
$773$	10.3225i	0.371273i	0.982618	+	0.185637i	$0.0594347\pi$
$773$	10.3225i	0.371273i	−0.982618	+	0.185637i	$0.940565\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	81.1638i	2.91174i
$778$	0	0
$779$	11.0489	0.395867
$780$	0	0
$781$	0	0
$782$	0	0
$783$	32.1708i	1.14969i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	49.8902i	1.77839i	0.457524	+	0.889197i	$0.348736\pi$
$787$	49.8902i	1.77839i	−0.457524	+	0.889197i	$0.651264\pi$
$788$	0	0
$789$	−25.1567	−0.895601
$790$	0	0
$791$	73.4288	2.61083
$792$	0	0
$793$	− 57.8399i	− 2.05396i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	28.7436i	1.01815i	0.860722	+	0.509075i	$0.170013\pi$
$797$	28.7436i	1.01815i	−0.860722	+	0.509075i	$0.829987\pi$
$798$	0	0
$799$	−0.822200	−0.0290874
$800$	0	0
$801$	−4.80501	−0.169777
$802$	0	0
$803$	5.42880i	0.191578i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 25.3905i	− 0.893788i
$808$	0	0
$809$	−38.5124	−1.35402	−0.677012	−	0.735972i	$-0.736726\pi$
$809$	−38.5124	−1.35402	−0.677012	+	0.735972i	$0.736726\pi$
$810$	0	0
$811$	−28.1013	−0.986771	−0.493385	−	0.869811i	$-0.664241\pi$
$811$	−28.1013	−0.986771	−0.493385	+	0.869811i	$0.664241\pi$
$812$	0	0
$813$	7.38692i	0.259071i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	5.83276i	0.204063i
$818$	0	0
$819$	−9.08719	−0.317532
$820$	0	0
$821$	22.2650	0.777053	0.388527	−	0.921437i	$-0.372984\pi$
$821$	22.2650	0.777053	0.388527	+	0.921437i	$0.372984\pi$
$822$	0	0
$823$	− 39.8363i	− 1.38861i	−0.719682	−	0.694304i	$-0.755712\pi$
$823$	− 39.8363i	− 1.38861i	0.719682	−	0.694304i	$-0.244288\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 48.5874i	− 1.68955i	−0.535121	−	0.844776i	$-0.679734\pi$
$827$	− 48.5874i	− 1.68955i	0.535121	−	0.844776i	$-0.320266\pi$
$828$	0	0
$829$	45.9058	1.59437	0.797187	−	0.603732i	$-0.206320\pi$
$829$	45.9058	1.59437	0.797187	+	0.603732i	$0.206320\pi$
$830$	0	0
$831$	29.2645	1.01517
$832$	0	0
$833$	12.2056i	0.422897i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 6.98944i	− 0.241590i
$838$	0	0
$839$	27.9688	0.965591	0.482796	−	0.875733i	$-0.339621\pi$
$839$	27.9688	0.965591	0.482796	+	0.875733i	$0.339621\pi$
$840$	0	0
$841$	13.8186	0.476504
$842$	0	0
$843$	20.7839i	0.715835i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	52.4358i	1.80172i
$848$	0	0
$849$	39.4005	1.35222
$850$	0	0
$851$	24.6761	0.845885
$852$	0	0
$853$	− 7.56777i	− 0.259116i	−0.991572	−	0.129558i	$-0.958644\pi$
$853$	− 7.56777i	− 0.259116i	0.991572	−	0.129558i	$-0.0413558\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21.4756i	0.733591i	0.930302	+	0.366796i	$0.119545\pi$
$857$	21.4756i	0.733591i	−0.930302	+	0.366796i	$0.880455\pi$
$858$	0	0
$859$	33.0177	1.12655	0.563275	−	0.826270i	$-0.309541\pi$
$859$	33.0177	1.12655	0.563275	+	0.826270i	$0.309541\pi$
$860$	0	0
$861$	98.5160	3.35742
$862$	0	0
$863$	− 18.6605i	− 0.635211i	−0.948223	−	0.317605i	$-0.897121\pi$
$863$	− 18.6605i	− 0.635211i	0.948223	−	0.317605i	$-0.102879\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 29.9149i	− 1.01596i
$868$	0	0
$869$	−2.52894	−0.0857886
$870$	0	0
$871$	−18.9859	−0.643312
$872$	0	0
$873$	− 1.14109i	− 0.0386200i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 57.7925i	− 1.95151i	−0.218858	−	0.975757i	$-0.570233\pi$
$877$	− 57.7925i	− 1.95151i	0.218858	−	0.975757i	$-0.429767\pi$
$878$	0	0
$879$	10.7209	0.361606
$880$	0	0
$881$	36.2338	1.22075	0.610374	−	0.792113i	$-0.291019\pi$
$881$	36.2338	1.22075	0.610374	+	0.792113i	$0.291019\pi$
$882$	0	0
$883$	2.98944i	0.100603i	0.998734	+	0.0503013i	$0.0160182\pi$
$883$	2.98944i	0.100603i	−0.998734	+	0.0503013i	$0.983982\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 42.2494i	− 1.41860i	−0.704909	−	0.709298i	$-0.749012\pi$
$887$	− 42.2494i	− 1.41860i	0.704909	−	0.709298i	$-0.250988\pi$
$888$	0	0
$889$	58.4394	1.95999
$890$	0	0
$891$	−5.65838	−0.189563
$892$	0	0
$893$	1.15667i	0.0387066i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	31.4252i	1.04926i
$898$	0	0
$899$	−9.30279	−0.310265
$900$	0	0
$901$	−9.43528	−0.314335
$902$	0	0
$903$	52.0071i	1.73069i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.97080i	0.0654392i	0.999465	+	0.0327196i	$0.0104168\pi$
$907$	1.97080i	0.0654392i	−0.999465	+	0.0327196i	$0.989583\pi$
$908$	0	0
$909$	1.79445	0.0595181
$910$	0	0
$911$	9.98995	0.330982	0.165491	−	0.986211i	$-0.447079\pi$
$911$	9.98995	0.330982	0.165491	+	0.986211i	$0.447079\pi$
$912$	0	0
$913$	0.215597i	0.00713520i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	45.0177i	1.48662i
$918$	0	0
$919$	−4.24029	−0.139874	−0.0699372	−	0.997551i	$-0.522280\pi$
$919$	−4.24029	−0.139874	−0.0699372	+	0.997551i	$0.522280\pi$
$920$	0	0
$921$	−53.8993	−1.77604
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.35720i	0.0774206i
$928$	0	0
$929$	13.1531	0.431539	0.215770	−	0.976444i	$-0.430774\pi$
$929$	13.1531	0.431539	0.215770	+	0.976444i	$0.430774\pi$
$930$	0	0
$931$	17.1708	0.562750
$932$	0	0
$933$	0.739098i	0.0241970i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 38.5436i	− 1.25916i	−0.776934	−	0.629582i	$-0.783226\pi$
$937$	− 38.5436i	− 1.25916i	0.776934	−	0.629582i	$-0.216774\pi$
$938$	0	0
$939$	0.613080	0.0200071
$940$	0	0
$941$	−2.91638	−0.0950713	−0.0475357	−	0.998870i	$-0.515137\pi$
$941$	−2.91638	−0.0950713	−0.0475357	+	0.998870i	$0.515137\pi$
$942$	0	0
$943$	− 29.9516i	− 0.975360i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	58.4011i	1.89778i	0.315608	+	0.948890i	$0.397792\pi$
$947$	58.4011i	1.89778i	−0.315608	+	0.948890i	$0.602208\pi$
$948$	0	0
$949$	−60.0007	−1.94770
$950$	0	0
$951$	−15.8776	−0.514865
$952$	0	0
$953$	36.8378i	1.19329i	0.802504	+	0.596646i	$0.203501\pi$
$953$	36.8378i	1.19329i	−0.802504	+	0.596646i	$0.796499\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.86342i	0.221863i
$958$	0	0
$959$	65.8152	2.12528
$960$	0	0
$961$	−28.9789	−0.934802
$962$	0	0
$963$	3.77886i	0.121772i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 43.0278i	− 1.38368i	−0.722051	−	0.691840i	$-0.756801\pi$
$967$	− 43.0278i	− 1.38368i	0.722051	−	0.691840i	$-0.243199\pi$
$968$	0	0
$969$	1.28917	0.0414141
$970$	0	0
$971$	−49.0177	−1.57305	−0.786526	−	0.617557i	$-0.788123\pi$
$971$	−49.0177	−1.57305	−0.786526	+	0.617557i	$0.788123\pi$
$972$	0	0
$973$	16.8222i	0.539295i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20.2383i	0.647481i	0.946146	+	0.323741i	$0.104941\pi$
$977$	20.2383i	0.647481i	−0.946146	+	0.323741i	$0.895059\pi$
$978$	0	0
$979$	9.61003	0.307138
$980$	0	0
$981$	−3.35218	−0.107027
$982$	0	0
$983$	− 25.6811i	− 0.819100i	−0.912288	−	0.409550i	$-0.865686\pi$
$983$	− 25.6811i	− 0.819100i	0.912288	−	0.409550i	$-0.134314\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	10.3133i	0.328277i
$988$	0	0
$989$	15.8116	0.502781
$990$	0	0
$991$	8.56829	0.272181	0.136090	−	0.990696i	$-0.456546\pi$
$991$	8.56829	0.272181	0.136090	+	0.990696i	$0.456546\pi$
$992$	0	0
$993$	− 43.2863i	− 1.37365i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	29.9688i	0.949122i	0.880223	+	0.474561i	$0.157393\pi$
$997$	29.9688i	0.949122i	−0.880223	+	0.474561i	$0.842607\pi$
$998$	0	0
$999$	−44.7527	−1.41591

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.n.3649.2		6
5.2	odd	4		3800.2.a.w.1.1		3
5.3	odd	4		760.2.a.i.1.3	✓	3
5.4	even	2	inner	3800.2.d.n.3649.5		6
15.8	even	4		6840.2.a.bm.1.1		3
20.3	even	4		1520.2.a.q.1.1		3
20.7	even	4		7600.2.a.bp.1.3		3
40.3	even	4		6080.2.a.br.1.3		3
40.13	odd	4		6080.2.a.bx.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.i.1.3	✓	3	5.3	odd	4
1520.2.a.q.1.1		3	20.3	even	4
3800.2.a.w.1.1		3	5.2	odd	4
3800.2.d.n.3649.2		6	1.1	even	1	trivial
3800.2.d.n.3649.5		6	5.4	even	2	inner
6080.2.a.br.1.3		3	40.3	even	4
6080.2.a.bx.1.1		3	40.13	odd	4
6840.2.a.bm.1.1		3	15.8	even	4
7600.2.a.bp.1.3		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)

\(f(q)\)	\(=\)	\(q-1.81361i q^{3} -4.91638i q^{7} -0.289169 q^{9} +O(q^{10})\)	\(q-1.81361i q^{3} -4.91638i q^{7} -0.289169 q^{9} +0.578337 q^{11} +6.39194i q^{13} -0.710831i q^{17} -1.00000 q^{19} -8.91638 q^{21} +2.71083i q^{23} -4.91638i q^{27} -6.54359 q^{29} +1.42166 q^{31} -1.04888i q^{33} -9.10278i q^{37} +11.5925 q^{39} -11.0489 q^{41} -5.83276i q^{43} -1.15667i q^{47} -17.1708 q^{49} -1.28917 q^{51} -13.2736i q^{53} +1.81361i q^{57} -11.3869 q^{59} -9.04888 q^{61} +1.42166i q^{63} +2.97028i q^{67} +4.91638 q^{69} +9.38692i q^{73} -2.84333i q^{77} -4.37279 q^{79} -9.78389 q^{81} +0.372787i q^{83} +11.8675i q^{87} +16.6167 q^{89} +31.4252 q^{91} -2.57834i q^{93} +3.94610i q^{97} -0.167237 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q - 6 q^{19} - 26 q^{21} + 14 q^{29} + 12 q^{31} - 6 q^{39} - 44 q^{41} - 24 q^{49} - 6 q^{51} - 22 q^{59} - 32 q^{61} + 2 q^{69} - 52 q^{79} - 26 q^{81} + 12 q^{89} + 58 q^{91} - 56 q^{99}+O(q^{100}) \) 6 * q - 6 * q^19 - 26 * q^21 + 14 * q^29 + 12 * q^31 - 6 * q^39 - 44 * q^41 - 24 * q^49 - 6 * q^51 - 22 * q^59 - 32 * q^61 + 2 * q^69 - 52 * q^79 - 26 * q^81 + 12 * q^89 + 58 * q^91 - 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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