LMFDB - Embedded newform 3800.2.d.k.3649.1

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			3649.1
Root			$1.32001 + 1.32001i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.k.3649.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.12489i q^{3} +1.51514i q^{7} -6.76491 q^{9} +O(q^{10})$	$q-3.12489i q^{3} +1.51514i q^{7} -6.76491 q^{9} +4.24977 q^{11} +4.15516i q^{13} +3.51514i q^{17} +1.00000 q^{19} +4.73463 q^{21} -8.73463i q^{23} +11.7649i q^{27} -1.45459 q^{29} -4.96972 q^{31} -13.2800i q^{33} -7.60975i q^{37} +12.9844 q^{39} -9.21949 q^{41} -8.31032i q^{43} -5.28005i q^{47} +4.70436 q^{49} +10.9844 q^{51} +0.155162i q^{53} -3.12489i q^{57} +2.48486 q^{59} -4.49954 q^{61} -10.2498i q^{63} -7.43521i q^{67} -27.2947 q^{69} +8.49954 q^{71} -15.0450i q^{73} +6.43899i q^{77} +0.310323 q^{79} +16.4693 q^{81} -8.96972i q^{83} +4.54541i q^{87} -0.719953 q^{89} -6.29564 q^{91} +15.5298i q^{93} -17.3893i q^{97} -28.7493 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 3.12489i	− 1.80415i	−0.431576	−	0.902077i	$-0.642042\pi$
$3$	− 3.12489i	− 1.80415i	0.431576	−	0.902077i	$-0.357958\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.51514i	0.572668i	0.958130	+	0.286334i	$0.0924368\pi$
$7$	1.51514i	0.572668i	−0.958130	+	0.286334i	$0.907563\pi$
$8$	0	0
$9$	−6.76491	−2.25497
$10$	0	0
$11$	4.24977	1.28135	0.640677	−	0.767810i	$-0.278654\pi$
$11$	4.24977	1.28135	0.640677	+	0.767810i	$0.278654\pi$
$12$	0	0
$13$	4.15516i	1.15243i	0.817297	+	0.576217i	$0.195472\pi$
$13$	4.15516i	1.15243i	−0.817297	+	0.576217i	$0.804528\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.51514i	0.852546i	0.904595	+	0.426273i	$0.140174\pi$
$17$	3.51514i	0.852546i	−0.904595	+	0.426273i	$0.859826\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	4.73463	1.03318
$22$	0	0
$23$	− 8.73463i	− 1.82130i	−0.413182	−	0.910648i	$-0.635583\pi$
$23$	− 8.73463i	− 1.82130i	0.413182	−	0.910648i	$-0.364417\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	11.7649i	2.26416i
$28$	0	0
$29$	−1.45459	−0.270110	−0.135055	−	0.990838i	$-0.543121\pi$
$29$	−1.45459	−0.270110	−0.135055	+	0.990838i	$0.543121\pi$
$30$	0	0
$31$	−4.96972	−0.892589	−0.446294	−	0.894886i	$-0.647257\pi$
$31$	−4.96972	−0.892589	−0.446294	+	0.894886i	$0.647257\pi$
$32$	0	0
$33$	− 13.2800i	− 2.31176i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 7.60975i	− 1.25103i	−0.780210	−	0.625517i	$-0.784888\pi$
$37$	− 7.60975i	− 1.25103i	0.780210	−	0.625517i	$-0.215112\pi$
$38$	0	0
$39$	12.9844	2.07917
$40$	0	0
$41$	−9.21949	−1.43984	−0.719922	−	0.694055i	$-0.755822\pi$
$41$	−9.21949	−1.43984	−0.719922	+	0.694055i	$0.755822\pi$
$42$	0	0
$43$	− 8.31032i	− 1.26731i	−0.773615	−	0.633656i	$-0.781553\pi$
$43$	− 8.31032i	− 1.26731i	0.773615	−	0.633656i	$-0.218447\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 5.28005i	− 0.770174i	−0.922880	−	0.385087i	$-0.874171\pi$
$47$	− 5.28005i	− 0.770174i	0.922880	−	0.385087i	$-0.125829\pi$
$48$	0	0
$49$	4.70436	0.672051
$50$	0	0
$51$	10.9844	1.53812
$52$	0	0
$53$	0.155162i	0.0213131i	0.999943	+	0.0106565i	$0.00339215\pi$
$53$	0.155162i	0.0213131i	−0.999943	+	0.0106565i	$0.996608\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 3.12489i	− 0.413901i
$58$	0	0
$59$	2.48486	0.323501	0.161751	−	0.986832i	$-0.448286\pi$
$59$	2.48486	0.323501	0.161751	+	0.986832i	$0.448286\pi$
$60$	0	0
$61$	−4.49954	−0.576107	−0.288054	−	0.957614i	$-0.593008\pi$
$61$	−4.49954	−0.576107	−0.288054	+	0.957614i	$0.593008\pi$
$62$	0	0
$63$	− 10.2498i	− 1.29135i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 7.43521i	− 0.908355i	−0.890911	−	0.454178i	$-0.849933\pi$
$67$	− 7.43521i	− 0.908355i	0.890911	−	0.454178i	$-0.150067\pi$
$68$	0	0
$69$	−27.2947	−3.28590
$70$	0	0
$71$	8.49954	1.00871	0.504355	−	0.863496i	$-0.331730\pi$
$71$	8.49954	1.00871	0.504355	+	0.863496i	$0.331730\pi$
$72$	0	0
$73$	− 15.0450i	− 1.76088i	−0.474159	−	0.880439i	$-0.657248\pi$
$73$	− 15.0450i	− 1.76088i	0.474159	−	0.880439i	$-0.342752\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.43899i	0.733791i
$78$	0	0
$79$	0.310323	0.0349141	0.0174570	−	0.999848i	$-0.494443\pi$
$79$	0.310323	0.0349141	0.0174570	+	0.999848i	$0.494443\pi$
$80$	0	0
$81$	16.4693	1.82992
$82$	0	0
$83$	− 8.96972i	− 0.984555i	−0.870438	−	0.492278i	$-0.836165\pi$
$83$	− 8.96972i	− 0.984555i	0.870438	−	0.492278i	$-0.163835\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.54541i	0.487320i
$88$	0	0
$89$	−0.719953	−0.0763149	−0.0381574	−	0.999272i	$-0.512149\pi$
$89$	−0.719953	−0.0763149	−0.0381574	+	0.999272i	$0.512149\pi$
$90$	0	0
$91$	−6.29564	−0.659963
$92$	0	0
$93$	15.5298i	1.61037i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 17.3893i	− 1.76562i	−0.469731	−	0.882810i	$-0.655649\pi$
$97$	− 17.3893i	− 1.76562i	0.469731	−	0.882810i	$-0.344351\pi$
$98$	0	0
$99$	−28.7493	−2.88941
$100$	0	0
$101$	−12.4995	−1.24375	−0.621875	−	0.783116i	$-0.713629\pi$
$101$	−12.4995	−1.24375	−0.621875	+	0.783116i	$0.713629\pi$
$102$	0	0
$103$	− 10.8898i	− 1.07300i	−0.843899	−	0.536502i	$-0.819746\pi$
$103$	− 10.8898i	− 1.07300i	0.843899	−	0.536502i	$-0.180254\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.43521i	0.718789i	0.933186	+	0.359394i	$0.117017\pi$
$107$	7.43521i	0.718789i	−0.933186	+	0.359394i	$0.882983\pi$
$108$	0	0
$109$	19.3553	1.85390	0.926950	−	0.375186i	$-0.122421\pi$
$109$	19.3553	1.85390	0.926950	+	0.375186i	$0.122421\pi$
$110$	0	0
$111$	−23.7796	−2.25706
$112$	0	0
$113$	− 17.8595i	− 1.68008i	−0.542523	−	0.840041i	$-0.682531\pi$
$113$	− 17.8595i	− 1.68008i	0.542523	−	0.840041i	$-0.317469\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 28.1093i	− 2.59870i
$118$	0	0
$119$	−5.32592	−0.488226
$120$	0	0
$121$	7.06055	0.641868
$122$	0	0
$123$	28.8099i	2.59770i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2.70058i	− 0.239637i	−0.992796	−	0.119819i	$-0.961769\pi$
$127$	− 2.70058i	− 0.239637i	0.992796	−	0.119819i	$-0.0382313\pi$
$128$	0	0
$129$	−25.9688	−2.28643
$130$	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
$133$	1.51514i	0.131379i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.51514i	0.642062i	0.947069	+	0.321031i	$0.104029\pi$
$137$	7.51514i	0.642062i	−0.947069	+	0.321031i	$0.895971\pi$
$138$	0	0
$139$	−16.7493	−1.42066	−0.710329	−	0.703870i	$-0.751454\pi$
$139$	−16.7493	−1.42066	−0.710329	+	0.703870i	$0.751454\pi$
$140$	0	0
$141$	−16.4995	−1.38951
$142$	0	0
$143$	17.6585i	1.47668i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 14.7006i	− 1.21248i
$148$	0	0
$149$	12.4995	1.02400	0.512001	−	0.858985i	$-0.328904\pi$
$149$	12.4995	1.02400	0.512001	+	0.858985i	$0.328904\pi$
$150$	0	0
$151$	14.2498	1.15963	0.579815	−	0.814748i	$-0.303125\pi$
$151$	14.2498	1.15963	0.579815	+	0.814748i	$0.303125\pi$
$152$	0	0
$153$	− 23.7796i	− 1.92247i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 4.71995i	− 0.376693i	−0.982103	−	0.188347i	$-0.939687\pi$
$157$	− 4.71995i	− 0.376693i	0.982103	−	0.188347i	$-0.0603128\pi$
$158$	0	0
$159$	0.484862	0.0384521
$160$	0	0
$161$	13.2342	1.04300
$162$	0	0
$163$	10.7493i	0.841951i	0.907072	+	0.420976i	$0.138312\pi$
$163$	10.7493i	0.841951i	−0.907072	+	0.420976i	$0.861688\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.7006i	0.828035i	0.910269	+	0.414018i	$0.135875\pi$
$167$	10.7006i	0.828035i	−0.910269	+	0.414018i	$0.864125\pi$
$168$	0	0
$169$	−4.26537	−0.328105
$170$	0	0
$171$	−6.76491	−0.517326
$172$	0	0
$173$	− 4.39025i	− 0.333785i	−0.985975	−	0.166892i	$-0.946627\pi$
$173$	− 4.39025i	− 0.333785i	0.985975	−	0.166892i	$-0.0533732\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 7.76491i	− 0.583646i
$178$	0	0
$179$	3.21949	0.240636	0.120318	−	0.992735i	$-0.461609\pi$
$179$	3.21949	0.240636	0.120318	+	0.992735i	$0.461609\pi$
$180$	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
$183$	14.0606i	1.03939i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	14.9385i	1.09241i
$188$	0	0
$189$	−17.8255	−1.29661
$190$	0	0
$191$	13.2342	0.957591	0.478796	−	0.877926i	$-0.341073\pi$
$191$	13.2342	0.957591	0.478796	+	0.877926i	$0.341073\pi$
$192$	0	0
$193$	17.3893i	1.25171i	0.779939	+	0.625856i	$0.215250\pi$
$193$	17.3893i	1.25171i	−0.779939	+	0.625856i	$0.784750\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	0	0
$199$	14.3250	1.01547	0.507736	−	0.861513i	$-0.330482\pi$
$199$	14.3250	1.01547	0.507736	+	0.861513i	$0.330482\pi$
$200$	0	0
$201$	−23.2342	−1.63881
$202$	0	0
$203$	− 2.20390i	− 0.154683i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	59.0890i	4.10697i
$208$	0	0
$209$	4.24977	0.293963
$210$	0	0
$211$	−6.98440	−0.480826	−0.240413	−	0.970671i	$-0.577283\pi$
$211$	−6.98440	−0.480826	−0.240413	+	0.970671i	$0.577283\pi$
$212$	0	0
$213$	− 26.5601i	− 1.81987i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 7.52982i	− 0.511157i
$218$	0	0
$219$	−47.0138	−3.17690
$220$	0	0
$221$	−14.6060	−0.982504
$222$	0	0
$223$	24.4802i	1.63931i	0.572855	+	0.819657i	$0.305836\pi$
$223$	24.4802i	1.63931i	−0.572855	+	0.819657i	$0.694164\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.9045i	1.65297i	0.562960	+	0.826484i	$0.309662\pi$
$227$	24.9045i	1.65297i	−0.562960	+	0.826484i	$0.690338\pi$
$228$	0	0
$229$	−4.47018	−0.295398	−0.147699	−	0.989032i	$-0.547187\pi$
$229$	−4.47018	−0.295398	−0.147699	+	0.989032i	$0.547187\pi$
$230$	0	0
$231$	20.1211	1.32387
$232$	0	0
$233$	− 3.93945i	− 0.258082i	−0.991639	−	0.129041i	$-0.958810\pi$
$233$	− 3.93945i	− 0.258082i	0.991639	−	0.129041i	$-0.0411899\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 0.969724i	− 0.0629903i
$238$	0	0
$239$	17.8255	1.15303	0.576517	−	0.817085i	$-0.304412\pi$
$239$	17.8255	1.15303	0.576517	+	0.817085i	$0.304412\pi$
$240$	0	0
$241$	−27.3094	−1.75915	−0.879577	−	0.475757i	$-0.842174\pi$
$241$	−27.3094	−1.75915	−0.879577	+	0.475757i	$0.842174\pi$
$242$	0	0
$243$	− 16.1698i	− 1.03730i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.15516i	0.264387i
$248$	0	0
$249$	−28.0294	−1.77629
$250$	0	0
$251$	−18.4390	−1.16386	−0.581929	−	0.813239i	$-0.697702\pi$
$251$	−18.4390	−1.16386	−0.581929	+	0.813239i	$0.697702\pi$
$252$	0	0
$253$	− 37.1202i	− 2.33373i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.889794i	0.0555038i	0.999615	+	0.0277519i	$0.00883485\pi$
$257$	0.889794i	0.0555038i	−0.999615	+	0.0277519i	$0.991165\pi$
$258$	0	0
$259$	11.5298	0.716428
$260$	0	0
$261$	9.84014	0.609089
$262$	0	0
$263$	− 12.6206i	− 0.778222i	−0.921191	−	0.389111i	$-0.872782\pi$
$263$	− 12.6206i	− 0.778222i	0.921191	−	0.389111i	$-0.127218\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.24977i	0.137684i
$268$	0	0
$269$	16.4390	1.00230	0.501151	−	0.865360i	$-0.332910\pi$
$269$	16.4390	1.00230	0.501151	+	0.865360i	$0.332910\pi$
$270$	0	0
$271$	25.6050	1.55540	0.777698	−	0.628638i	$-0.216387\pi$
$271$	25.6050	1.55540	0.777698	+	0.628638i	$0.216387\pi$
$272$	0	0
$273$	19.6732i	1.19067i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 0.749313i	− 0.0450218i	−0.999747	−	0.0225109i	$-0.992834\pi$
$277$	− 0.749313i	− 0.0450218i	0.999747	−	0.0225109i	$-0.00716605\pi$
$278$	0	0
$279$	33.6197	2.01276
$280$	0	0
$281$	3.77959	0.225471	0.112736	−	0.993625i	$-0.464039\pi$
$281$	3.77959	0.225471	0.112736	+	0.993625i	$0.464039\pi$
$282$	0	0
$283$	− 9.46927i	− 0.562889i	−0.959577	−	0.281445i	$-0.909186\pi$
$283$	− 9.46927i	− 0.562889i	0.959577	−	0.281445i	$-0.0908136\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 13.9688i	− 0.824553i
$288$	0	0
$289$	4.64380	0.273165
$290$	0	0
$291$	−54.3397	−3.18545
$292$	0	0
$293$	24.6841i	1.44206i	0.692905	+	0.721029i	$0.256331\pi$
$293$	24.6841i	1.44206i	−0.692905	+	0.721029i	$0.743669\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	49.9982i	2.90119i
$298$	0	0
$299$	36.2938	2.09893
$300$	0	0
$301$	12.5913	0.725750
$302$	0	0
$303$	39.0596i	2.24392i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 13.3288i	− 0.760714i	−0.924840	−	0.380357i	$-0.875801\pi$
$307$	− 13.3288i	− 0.760714i	0.924840	−	0.380357i	$-0.124199\pi$
$308$	0	0
$309$	−34.0294	−1.93586
$310$	0	0
$311$	−13.4546	−0.762940	−0.381470	−	0.924381i	$-0.624582\pi$
$311$	−13.4546	−0.762940	−0.381470	+	0.924381i	$0.624582\pi$
$312$	0	0
$313$	− 21.7649i	− 1.23023i	−0.788439	−	0.615113i	$-0.789111\pi$
$313$	− 21.7649i	− 1.23023i	0.788439	−	0.615113i	$-0.210889\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 23.5639i	− 1.32348i	−0.749734	−	0.661740i	$-0.769818\pi$
$317$	− 23.5639i	− 1.32348i	0.749734	−	0.661740i	$-0.230182\pi$
$318$	0	0
$319$	−6.18166	−0.346106
$320$	0	0
$321$	23.2342	1.29681
$322$	0	0
$323$	3.51514i	0.195588i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 60.4830i	− 3.34472i
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	−11.9541	−0.657058	−0.328529	−	0.944494i	$-0.606553\pi$
$331$	−11.9541	−0.657058	−0.328529	+	0.944494i	$0.606553\pi$
$332$	0	0
$333$	51.4792i	2.82105i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	22.8292i	1.24359i	0.783181	+	0.621794i	$0.213596\pi$
$337$	22.8292i	1.24359i	−0.783181	+	0.621794i	$0.786404\pi$
$338$	0	0
$339$	−55.8089	−3.03113
$340$	0	0
$341$	−21.1202	−1.14372
$342$	0	0
$343$	17.7337i	0.957531i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 29.4693i	− 1.58199i	−0.611821	−	0.790997i	$-0.709563\pi$
$347$	− 29.4693i	− 1.58199i	0.611821	−	0.790997i	$-0.290437\pi$
$348$	0	0
$349$	−4.56009	−0.244096	−0.122048	−	0.992524i	$-0.538946\pi$
$349$	−4.56009	−0.244096	−0.122048	+	0.992524i	$0.538946\pi$
$350$	0	0
$351$	−48.8851	−2.60929
$352$	0	0
$353$	− 24.4849i	− 1.30320i	−0.758564	−	0.651599i	$-0.774099\pi$
$353$	− 24.4849i	− 1.30320i	0.758564	−	0.651599i	$-0.225901\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	16.6429i	0.880835i
$358$	0	0
$359$	−12.0147	−0.634111	−0.317055	−	0.948407i	$-0.602694\pi$
$359$	−12.0147	−0.634111	−0.317055	+	0.948407i	$0.602694\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 22.0634i	− 1.15803i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	25.1589i	1.31329i	0.754201	+	0.656643i	$0.228024\pi$
$367$	25.1589i	1.31329i	−0.754201	+	0.656643i	$0.771976\pi$
$368$	0	0
$369$	62.3690	3.24680
$370$	0	0
$371$	−0.235091	−0.0122053
$372$	0	0
$373$	9.62443i	0.498334i	0.968461	+	0.249167i	$0.0801568\pi$
$373$	9.62443i	0.498334i	−0.968461	+	0.249167i	$0.919843\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 6.04404i	− 0.311284i
$378$	0	0
$379$	−19.7943	−1.01676	−0.508382	−	0.861132i	$-0.669756\pi$
$379$	−19.7943	−1.01676	−0.508382	+	0.861132i	$0.669756\pi$
$380$	0	0
$381$	−8.43899	−0.432343
$382$	0	0
$383$	33.3288i	1.70302i	0.524337	+	0.851511i	$0.324313\pi$
$383$	33.3288i	1.70302i	−0.524337	+	0.851511i	$0.675687\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	56.2186i	2.85775i
$388$	0	0
$389$	−11.5904	−0.587655	−0.293828	−	0.955858i	$-0.594929\pi$
$389$	−11.5904	−0.587655	−0.293828	+	0.955858i	$0.594929\pi$
$390$	0	0
$391$	30.7034	1.55274
$392$	0	0
$393$	12.4995i	0.630518i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8.74931i	0.439115i	0.975600	+	0.219558i	$0.0704614\pi$
$397$	8.74931i	0.439115i	−0.975600	+	0.219558i	$0.929539\pi$
$398$	0	0
$399$	4.73463	0.237028
$400$	0	0
$401$	−18.6206	−0.929871	−0.464935	−	0.885345i	$-0.653922\pi$
$401$	−18.6206	−0.929871	−0.464935	+	0.885345i	$0.653922\pi$
$402$	0	0
$403$	− 20.6500i	− 1.02865i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 32.3397i	− 1.60302i
$408$	0	0
$409$	−7.93945	−0.392580	−0.196290	−	0.980546i	$-0.562889\pi$
$409$	−7.93945	−0.392580	−0.196290	+	0.980546i	$0.562889\pi$
$410$	0	0
$411$	23.4839	1.15838
$412$	0	0
$413$	3.76491i	0.185259i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	52.3397i	2.56309i
$418$	0	0
$419$	13.0908	0.639529	0.319764	−	0.947497i	$-0.396396\pi$
$419$	13.0908	0.639529	0.319764	+	0.947497i	$0.396396\pi$
$420$	0	0
$421$	19.8936	0.969554	0.484777	−	0.874638i	$-0.338901\pi$
$421$	19.8936	0.969554	0.484777	+	0.874638i	$0.338901\pi$
$422$	0	0
$423$	35.7190i	1.73672i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 6.81743i	− 0.329918i
$428$	0	0
$429$	55.1807	2.66415
$430$	0	0
$431$	−33.7796	−1.62711	−0.813553	−	0.581491i	$-0.802470\pi$
$431$	−33.7796	−1.62711	−0.813553	+	0.581491i	$0.802470\pi$
$432$	0	0
$433$	9.07901i	0.436310i	0.975914	+	0.218155i	$0.0700038\pi$
$433$	9.07901i	0.436310i	−0.975914	+	0.218155i	$0.929996\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 8.73463i	− 0.417834i
$438$	0	0
$439$	12.3103	0.587540	0.293770	−	0.955876i	$-0.405090\pi$
$439$	12.3103	0.587540	0.293770	+	0.955876i	$0.405090\pi$
$440$	0	0
$441$	−31.8245	−1.51545
$442$	0	0
$443$	2.56009i	0.121634i	0.998149	+	0.0608169i	$0.0193706\pi$
$443$	2.56009i	0.121634i	−0.998149	+	0.0608169i	$0.980629\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 39.0596i	− 1.84746i
$448$	0	0
$449$	−22.9991	−1.08539	−0.542697	−	0.839929i	$-0.682597\pi$
$449$	−22.9991	−1.08539	−0.542697	+	0.839929i	$0.682597\pi$
$450$	0	0
$451$	−39.1807	−1.84495
$452$	0	0
$453$	− 44.5289i	− 2.09215i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 20.6741i	− 0.967093i	−0.875319	−	0.483546i	$-0.839348\pi$
$457$	− 20.6741i	− 0.967093i	0.875319	−	0.483546i	$-0.160652\pi$
$458$	0	0
$459$	−41.3553	−1.93030
$460$	0	0
$461$	−5.87890	−0.273807	−0.136904	−	0.990584i	$-0.543715\pi$
$461$	−5.87890	−0.273807	−0.136904	+	0.990584i	$0.543715\pi$
$462$	0	0
$463$	35.5592i	1.65258i	0.563249	+	0.826288i	$0.309551\pi$
$463$	35.5592i	1.65258i	−0.563249	+	0.826288i	$0.690449\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 41.7408i	− 1.93154i	−0.259408	−	0.965768i	$-0.583528\pi$
$467$	− 41.7408i	− 1.93154i	0.259408	−	0.965768i	$-0.416472\pi$
$468$	0	0
$469$	11.2654	0.520186
$470$	0	0
$471$	−14.7493	−0.679612
$472$	0	0
$473$	− 35.3170i	− 1.62388i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1.04965i	− 0.0480603i
$478$	0	0
$479$	−16.8704	−0.770829	−0.385415	−	0.922744i	$-0.625942\pi$
$479$	−16.8704	−0.770829	−0.385415	+	0.922744i	$0.625942\pi$
$480$	0	0
$481$	31.6197	1.44174
$482$	0	0
$483$	− 41.3553i	− 1.88173i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 3.64094i	− 0.164987i	−0.996592	−	0.0824934i	$-0.973712\pi$
$487$	− 3.64094i	− 0.164987i	0.996592	−	0.0824934i	$-0.0262883\pi$
$488$	0	0
$489$	33.5904	1.51901
$490$	0	0
$491$	−1.96881	−0.0888510	−0.0444255	−	0.999013i	$-0.514146\pi$
$491$	−1.96881	−0.0888510	−0.0444255	+	0.999013i	$0.514146\pi$
$492$	0	0
$493$	− 5.11307i	− 0.230281i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.8780i	0.577656i
$498$	0	0
$499$	36.7787	1.64644	0.823220	−	0.567723i	$-0.192175\pi$
$499$	36.7787	1.64644	0.823220	+	0.567723i	$0.192175\pi$
$500$	0	0
$501$	33.4381	1.49390
$502$	0	0
$503$	15.8860i	0.708322i	0.935184	+	0.354161i	$0.115234\pi$
$503$	15.8860i	0.708322i	−0.935184	+	0.354161i	$0.884766\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	13.3288i	0.591952i
$508$	0	0
$509$	−20.2791	−0.898857	−0.449428	−	0.893316i	$-0.648372\pi$
$509$	−20.2791	−0.898857	−0.449428	+	0.893316i	$0.648372\pi$
$510$	0	0
$511$	22.7952	1.00840
$512$	0	0
$513$	11.7649i	0.519433i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 22.4390i	− 0.986866i
$518$	0	0
$519$	−13.7190	−0.602199
$520$	0	0
$521$	−23.4693	−1.02821	−0.514104	−	0.857728i	$-0.671875\pi$
$521$	−23.4693	−1.02821	−0.514104	+	0.857728i	$0.671875\pi$
$522$	0	0
$523$	− 26.3737i	− 1.15324i	−0.817011	−	0.576622i	$-0.804371\pi$
$523$	− 26.3737i	− 1.15324i	0.817011	−	0.576622i	$-0.195629\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 17.4693i	− 0.760973i
$528$	0	0
$529$	−53.2938	−2.31712
$530$	0	0
$531$	−16.8099	−0.729486
$532$	0	0
$533$	− 38.3085i	− 1.65932i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 10.0606i	− 0.434145i
$538$	0	0
$539$	19.9924	0.861135
$540$	0	0
$541$	22.5601	0.969934	0.484967	−	0.874532i	$-0.338832\pi$
$541$	22.5601	0.969934	0.484967	+	0.874532i	$0.338832\pi$
$542$	0	0
$543$	− 68.7475i	− 2.95024i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 20.6694i	− 0.883759i	−0.897074	−	0.441879i	$-0.854312\pi$
$547$	− 20.6694i	− 0.883759i	0.897074	−	0.441879i	$-0.145688\pi$
$548$	0	0
$549$	30.4390	1.29910
$550$	0	0
$551$	−1.45459	−0.0619674
$552$	0	0
$553$	0.470182i	0.0199942i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	37.7484i	1.59945i	0.600366	+	0.799725i	$0.295022\pi$
$557$	37.7484i	1.59945i	−0.600366	+	0.799725i	$0.704978\pi$
$558$	0	0
$559$	34.5307	1.46049
$560$	0	0
$561$	46.6812	1.97088
$562$	0	0
$563$	− 36.6694i	− 1.54543i	−0.634753	−	0.772715i	$-0.718898\pi$
$563$	− 36.6694i	− 1.54543i	0.634753	−	0.772715i	$-0.281102\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	24.9532i	1.04794i
$568$	0	0
$569$	−2.84862	−0.119420	−0.0597102	−	0.998216i	$-0.519018\pi$
$569$	−2.84862	−0.119420	−0.0597102	+	0.998216i	$0.519018\pi$
$570$	0	0
$571$	28.4002	1.18851	0.594256	−	0.804276i	$-0.297446\pi$
$571$	28.4002	1.18851	0.594256	+	0.804276i	$0.297446\pi$
$572$	0	0
$573$	− 41.3553i	− 1.72764i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	29.2947i	1.21956i	0.792572	+	0.609778i	$0.208741\pi$
$577$	29.2947i	1.21956i	−0.792572	+	0.609778i	$0.791259\pi$
$578$	0	0
$579$	54.3397	2.25828
$580$	0	0
$581$	13.5904	0.563824
$582$	0	0
$583$	0.659401i	0.0273096i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 22.4683i	− 0.927368i	−0.886001	−	0.463684i	$-0.846527\pi$
$587$	− 22.4683i	− 0.927368i	0.886001	−	0.463684i	$-0.153473\pi$
$588$	0	0
$589$	−4.96972	−0.204774
$590$	0	0
$591$	6.24977	0.257081
$592$	0	0
$593$	16.0606i	0.659528i	0.944063	+	0.329764i	$0.106969\pi$
$593$	16.0606i	0.659528i	−0.944063	+	0.329764i	$0.893031\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 44.7640i	− 1.83207i
$598$	0	0
$599$	23.2489	0.949922	0.474961	−	0.880007i	$-0.342462\pi$
$599$	23.2489	0.949922	0.474961	+	0.880007i	$0.342462\pi$
$600$	0	0
$601$	−0.841057	−0.0343074	−0.0171537	−	0.999853i	$-0.505460\pi$
$601$	−0.841057	−0.0343074	−0.0171537	+	0.999853i	$0.505460\pi$
$602$	0	0
$603$	50.2985i	2.04831i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	32.4802i	1.31833i	0.751999	+	0.659165i	$0.229090\pi$
$607$	32.4802i	1.31833i	−0.751999	+	0.659165i	$0.770910\pi$
$608$	0	0
$609$	−6.88693	−0.279072
$610$	0	0
$611$	21.9394	0.887575
$612$	0	0
$613$	− 19.4305i	− 0.784791i	−0.919797	−	0.392395i	$-0.871646\pi$
$613$	− 19.4305i	− 0.784791i	0.919797	−	0.392395i	$-0.128354\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 20.7787i	− 0.836518i	−0.908328	−	0.418259i	$-0.862640\pi$
$617$	− 20.7787i	− 0.836518i	0.908328	−	0.418259i	$-0.137360\pi$
$618$	0	0
$619$	−18.9310	−0.760900	−0.380450	−	0.924802i	$-0.624231\pi$
$619$	−18.9310	−0.760900	−0.380450	+	0.924802i	$0.624231\pi$
$620$	0	0
$621$	102.762	4.12370
$622$	0	0
$623$	− 1.09083i	− 0.0437031i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 13.2800i	− 0.530354i
$628$	0	0
$629$	26.7493	1.06656
$630$	0	0
$631$	−0.370875	−0.0147643	−0.00738215	−	0.999973i	$-0.502350\pi$
$631$	−0.370875	−0.0147643	−0.00738215	+	0.999973i	$0.502350\pi$
$632$	0	0
$633$	21.8255i	0.867484i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	19.5474i	0.774495i
$638$	0	0
$639$	−57.4986	−2.27461
$640$	0	0
$641$	26.8099	1.05893	0.529463	−	0.848333i	$-0.322393\pi$
$641$	26.8099	1.05893	0.529463	+	0.848333i	$0.322393\pi$
$642$	0	0
$643$	− 0.969724i	− 0.0382422i	−0.999817	−	0.0191211i	$-0.993913\pi$
$643$	− 0.969724i	− 0.0382422i	0.999817	−	0.0191211i	$-0.00608680\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.20482i	− 0.0473662i	−0.999720	−	0.0236831i	$-0.992461\pi$
$647$	− 1.20482i	− 0.0473662i	0.999720	−	0.0236831i	$-0.00753926\pi$
$648$	0	0
$649$	10.5601	0.414520
$650$	0	0
$651$	−23.5298	−0.922206
$652$	0	0
$653$	− 22.9310i	− 0.897358i	−0.893693	−	0.448679i	$-0.851895\pi$
$653$	− 22.9310i	− 0.897358i	0.893693	−	0.448679i	$-0.148105\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	101.778i	3.97073i
$658$	0	0
$659$	−15.1055	−0.588427	−0.294214	−	0.955740i	$-0.595058\pi$
$659$	−15.1055	−0.588427	−0.294214	+	0.955740i	$0.595058\pi$
$660$	0	0
$661$	19.0450	0.740763	0.370381	−	0.928880i	$-0.379227\pi$
$661$	19.0450	0.740763	0.370381	+	0.928880i	$0.379227\pi$
$662$	0	0
$663$	45.6420i	1.77259i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12.7053i	0.491950i
$668$	0	0
$669$	76.4977	2.95757
$670$	0	0
$671$	−19.1220	−0.738197
$672$	0	0
$673$	15.4205i	0.594418i	0.954812	+	0.297209i	$0.0960558\pi$
$673$	15.4205i	0.594418i	−0.954812	+	0.297209i	$0.903944\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 8.62534i	− 0.331499i	−0.986168	−	0.165749i	$-0.946996\pi$
$677$	− 8.62534i	− 0.331499i	0.986168	−	0.165749i	$-0.0530043\pi$
$678$	0	0
$679$	26.3472	1.01111
$680$	0	0
$681$	77.8236	2.98221
$682$	0	0
$683$	38.8898i	1.48808i	0.668137	+	0.744038i	$0.267092\pi$
$683$	38.8898i	1.48808i	−0.668137	+	0.744038i	$0.732908\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	13.9688i	0.532943i
$688$	0	0
$689$	−0.644721	−0.0245619
$690$	0	0
$691$	−21.5979	−0.821624	−0.410812	−	0.911720i	$-0.634755\pi$
$691$	−21.5979	−0.821624	−0.410812	+	0.911720i	$0.634755\pi$
$692$	0	0
$693$	− 43.5592i	− 1.65468i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 32.4078i	− 1.22753i
$698$	0	0
$699$	−12.3103	−0.465619
$700$	0	0
$701$	17.8183	0.672990	0.336495	−	0.941685i	$-0.390759\pi$
$701$	17.8183	0.672990	0.336495	+	0.941685i	$0.390759\pi$
$702$	0	0
$703$	− 7.60975i	− 0.287007i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 18.9385i	− 0.712257i
$708$	0	0
$709$	−47.4986	−1.78385	−0.891924	−	0.452185i	$-0.850645\pi$
$709$	−47.4986	−1.78385	−0.891924	+	0.452185i	$0.850645\pi$
$710$	0	0
$711$	−2.09931	−0.0787302
$712$	0	0
$713$	43.4087i	1.62567i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 55.7025i	− 2.08025i
$718$	0	0
$719$	25.4546	0.949296	0.474648	−	0.880176i	$-0.342575\pi$
$719$	25.4546	0.949296	0.474648	+	0.880176i	$0.342575\pi$
$720$	0	0
$721$	16.4995	0.614475
$722$	0	0
$723$	85.3388i	3.17378i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 21.5445i	− 0.799041i	−0.916724	−	0.399521i	$-0.869177\pi$
$727$	− 21.5445i	− 0.799041i	0.916724	−	0.399521i	$-0.130823\pi$
$728$	0	0
$729$	−1.12110	−0.0415224
$730$	0	0
$731$	29.2119	1.08044
$732$	0	0
$733$	24.2110i	0.894254i	0.894470	+	0.447127i	$0.147553\pi$
$733$	24.2110i	0.894254i	−0.894470	+	0.447127i	$0.852447\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 31.5979i	− 1.16392i
$738$	0	0
$739$	37.4693	1.37833	0.689165	−	0.724605i	$-0.257978\pi$
$739$	37.4693	1.37833	0.689165	+	0.724605i	$0.257978\pi$
$740$	0	0
$741$	12.9844	0.476994
$742$	0	0
$743$	− 15.8595i	− 0.581829i	−0.956749	−	0.290915i	$-0.906040\pi$
$743$	− 15.8595i	− 0.581829i	0.956749	−	0.290915i	$-0.0939596\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	60.6794i	2.22014i
$748$	0	0
$749$	−11.2654	−0.411628
$750$	0	0
$751$	5.12958	0.187181	0.0935906	−	0.995611i	$-0.470166\pi$
$751$	5.12958	0.187181	0.0935906	+	0.995611i	$0.470166\pi$
$752$	0	0
$753$	57.6197i	2.09978i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 37.4381i	− 1.36071i	−0.732883	−	0.680355i	$-0.761826\pi$
$757$	− 37.4381i	− 1.36071i	0.732883	−	0.680355i	$-0.238174\pi$
$758$	0	0
$759$	−115.996	−4.21040
$760$	0	0
$761$	21.5833	0.782392	0.391196	−	0.920307i	$-0.372061\pi$
$761$	21.5833	0.782392	0.391196	+	0.920307i	$0.372061\pi$
$762$	0	0
$763$	29.3259i	1.06167i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	10.3250i	0.372814i
$768$	0	0
$769$	40.7640	1.46999	0.734994	−	0.678074i	$-0.237185\pi$
$769$	40.7640	1.46999	0.734994	+	0.678074i	$0.237185\pi$
$770$	0	0
$771$	2.78051	0.100137
$772$	0	0
$773$	48.1845i	1.73308i	0.499110	+	0.866538i	$0.333660\pi$
$773$	48.1845i	1.73308i	−0.499110	+	0.866538i	$0.666340\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 36.0294i	− 1.29255i
$778$	0	0
$779$	−9.21949	−0.330323
$780$	0	0
$781$	36.1211	1.29251
$782$	0	0
$783$	− 17.1131i	− 0.611571i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 27.6925i	− 0.987132i	−0.869708	−	0.493566i	$-0.835693\pi$
$787$	− 27.6925i	− 0.987132i	0.869708	−	0.493566i	$-0.164307\pi$
$788$	0	0
$789$	−39.4381	−1.40403
$790$	0	0
$791$	27.0596	0.962130
$792$	0	0
$793$	− 18.6963i	− 0.663926i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 10.3368i	− 0.366149i	−0.983099	−	0.183074i	$-0.941395\pi$
$797$	− 10.3368i	− 0.366149i	0.983099	−	0.183074i	$-0.0586049\pi$
$798$	0	0
$799$	18.5601	0.656609
$800$	0	0
$801$	4.87042	0.172088
$802$	0	0
$803$	− 63.9376i	− 2.25631i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 51.3700i	− 1.80831i
$808$	0	0
$809$	23.2342	0.816870	0.408435	−	0.912787i	$-0.366075\pi$
$809$	23.2342	0.816870	0.408435	+	0.912787i	$0.366075\pi$
$810$	0	0
$811$	32.1433	1.12871	0.564353	−	0.825534i	$-0.309126\pi$
$811$	32.1433	1.12871	0.564353	+	0.825534i	$0.309126\pi$
$812$	0	0
$813$	− 80.0128i	− 2.80617i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 8.31032i	− 0.290741i
$818$	0	0
$819$	42.5895	1.48820
$820$	0	0
$821$	−27.0908	−0.945476	−0.472738	−	0.881203i	$-0.656734\pi$
$821$	−27.0908	−0.945476	−0.472738	+	0.881203i	$0.656734\pi$
$822$	0	0
$823$	− 14.4849i	− 0.504911i	−0.967609	−	0.252455i	$-0.918762\pi$
$823$	− 14.4849i	− 0.504911i	0.967609	−	0.252455i	$-0.0812381\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	54.0247i	1.87862i	0.343067	+	0.939311i	$0.388534\pi$
$827$	54.0247i	1.87862i	−0.343067	+	0.939311i	$0.611466\pi$
$828$	0	0
$829$	−18.7640	−0.651700	−0.325850	−	0.945421i	$-0.605651\pi$
$829$	−18.7640	−0.651700	−0.325850	+	0.945421i	$0.605651\pi$
$830$	0	0
$831$	−2.34152	−0.0812263
$832$	0	0
$833$	16.5365i	0.572954i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 58.4683i	− 2.02096i
$838$	0	0
$839$	7.84014	0.270672	0.135336	−	0.990800i	$-0.456789\pi$
$839$	7.84014	0.270672	0.135336	+	0.990800i	$0.456789\pi$
$840$	0	0
$841$	−26.8842	−0.927041
$842$	0	0
$843$	− 11.8108i	− 0.406785i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.6977i	0.367578i
$848$	0	0
$849$	−29.5904	−1.01554
$850$	0	0
$851$	−66.4683	−2.27851
$852$	0	0
$853$	36.8174i	1.26060i	0.776350	+	0.630302i	$0.217069\pi$
$853$	36.8174i	1.26060i	−0.776350	+	0.630302i	$0.782931\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5.23887i	0.178956i	0.995989	+	0.0894782i	$0.0285199\pi$
$857$	5.23887i	0.178956i	−0.995989	+	0.0894782i	$0.971480\pi$
$858$	0	0
$859$	29.6197	1.01061	0.505306	−	0.862940i	$-0.331380\pi$
$859$	29.6197	1.01061	0.505306	+	0.862940i	$0.331380\pi$
$860$	0	0
$861$	−43.6509	−1.48762
$862$	0	0
$863$	− 35.8889i	− 1.22167i	−0.791757	−	0.610836i	$-0.790834\pi$
$863$	− 35.8889i	− 1.22167i	0.791757	−	0.610836i	$-0.209166\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 14.5114i	− 0.492832i
$868$	0	0
$869$	1.31880	0.0447373
$870$	0	0
$871$	30.8945	1.04682
$872$	0	0
$873$	117.637i	3.98142i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	29.8136i	1.00674i	0.864072	+	0.503368i	$0.167906\pi$
$877$	29.8136i	1.00674i	−0.864072	+	0.503368i	$0.832094\pi$
$878$	0	0
$879$	77.1349	2.60169
$880$	0	0
$881$	−38.9679	−1.31286	−0.656431	−	0.754386i	$-0.727935\pi$
$881$	−38.9679	−1.31286	−0.656431	+	0.754386i	$0.727935\pi$
$882$	0	0
$883$	− 28.5677i	− 0.961378i	−0.876891	−	0.480689i	$-0.840387\pi$
$883$	− 28.5677i	− 0.961378i	0.876891	−	0.480689i	$-0.159613\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 27.5104i	− 0.923710i	−0.886955	−	0.461855i	$-0.847184\pi$
$887$	− 27.5104i	− 0.923710i	0.886955	−	0.461855i	$-0.152816\pi$
$888$	0	0
$889$	4.09174	0.137233
$890$	0	0
$891$	69.9906	2.34477
$892$	0	0
$893$	− 5.28005i	− 0.176690i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 113.414i	− 3.78678i
$898$	0	0
$899$	7.22889	0.241097
$900$	0	0
$901$	−0.545414	−0.0181704
$902$	0	0
$903$	− 39.3463i	− 1.30936i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	55.4939i	1.84265i	0.388799	+	0.921323i	$0.372890\pi$
$907$	55.4939i	1.84265i	−0.388799	+	0.921323i	$0.627110\pi$
$908$	0	0
$909$	84.5583	2.80462
$910$	0	0
$911$	6.84106	0.226654	0.113327	−	0.993558i	$-0.463849\pi$
$911$	6.84106	0.226654	0.113327	+	0.993558i	$0.463849\pi$
$912$	0	0
$913$	− 38.1193i	− 1.26156i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 6.06055i	− 0.200137i
$918$	0	0
$919$	6.17454	0.203679	0.101840	−	0.994801i	$-0.467527\pi$
$919$	6.17454	0.203679	0.101840	+	0.994801i	$0.467527\pi$
$920$	0	0
$921$	−41.6509	−1.37244
$922$	0	0
$923$	35.3170i	1.16247i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	73.6685i	2.41959i
$928$	0	0
$929$	−44.7034	−1.46667	−0.733336	−	0.679866i	$-0.762038\pi$
$929$	−44.7034	−1.46667	−0.733336	+	0.679866i	$0.762038\pi$
$930$	0	0
$931$	4.70436	0.154179
$932$	0	0
$933$	42.0440i	1.37646i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	12.1064i	0.395500i	0.980253	+	0.197750i	$0.0633634\pi$
$937$	12.1064i	0.395500i	−0.980253	+	0.197750i	$0.936637\pi$
$938$	0	0
$939$	−68.0128	−2.21952
$940$	0	0
$941$	−26.2645	−0.856197	−0.428098	−	0.903732i	$-0.640816\pi$
$941$	−26.2645	−0.856197	−0.428098	+	0.903732i	$0.640816\pi$
$942$	0	0
$943$	80.5289i	2.62238i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 7.37935i	− 0.239797i	−0.992786	−	0.119898i	$-0.961743\pi$
$947$	− 7.37935i	− 0.239797i	0.992786	−	0.119898i	$-0.0382569\pi$
$948$	0	0
$949$	62.5142	2.02930
$950$	0	0
$951$	−73.6344	−2.38776
$952$	0	0
$953$	− 21.5398i	− 0.697743i	−0.937171	−	0.348871i	$-0.886565\pi$
$953$	− 21.5398i	− 0.697743i	0.937171	−	0.348871i	$-0.113435\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	19.3170i	0.624429i
$958$	0	0
$959$	−11.3865	−0.367688
$960$	0	0
$961$	−6.30184	−0.203285
$962$	0	0
$963$	− 50.2985i	− 1.62085i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	39.8789i	1.28242i	0.767366	+	0.641209i	$0.221567\pi$
$967$	39.8789i	1.28242i	−0.767366	+	0.641209i	$0.778433\pi$
$968$	0	0
$969$	10.9844	0.352870
$970$	0	0
$971$	−29.7408	−0.954429	−0.477214	−	0.878787i	$-0.658353\pi$
$971$	−29.7408	−0.954429	−0.477214	+	0.878787i	$0.658353\pi$
$972$	0	0
$973$	− 25.3775i	− 0.813566i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	58.6694i	1.87700i	0.345279	+	0.938500i	$0.387784\pi$
$977$	58.6694i	1.87700i	−0.345279	+	0.938500i	$0.612216\pi$
$978$	0	0
$979$	−3.05964	−0.0977864
$980$	0	0
$981$	−130.937	−4.18049
$982$	0	0
$983$	11.7384i	0.374397i	0.982322	+	0.187199i	$0.0599408\pi$
$983$	11.7384i	0.374397i	−0.982322	+	0.187199i	$0.940059\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 24.9991i	− 0.795730i
$988$	0	0
$989$	−72.5876	−2.30815
$990$	0	0
$991$	−26.7493	−0.849720	−0.424860	−	0.905259i	$-0.639677\pi$
$991$	−26.7493	−0.849720	−0.424860	+	0.905259i	$0.639677\pi$
$992$	0	0
$993$	37.3553i	1.18543i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 45.8401i	− 1.45177i	−0.687815	−	0.725886i	$-0.741430\pi$
$997$	− 45.8401i	− 1.45177i	0.687815	−	0.725886i	$-0.258570\pi$
$998$	0	0
$999$	89.5280	2.83254

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.k.3649.1		6
5.2	odd	4		760.2.a.h.1.1	✓	3
5.3	odd	4		3800.2.a.y.1.3		3
5.4	even	2	inner	3800.2.d.k.3649.6		6
15.2	even	4		6840.2.a.bj.1.2		3
20.3	even	4		7600.2.a.bo.1.1		3
20.7	even	4		1520.2.a.r.1.3		3
40.27	even	4		6080.2.a.bs.1.1		3
40.37	odd	4		6080.2.a.bw.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.h.1.1	✓	3	5.2	odd	4
1520.2.a.r.1.3		3	20.7	even	4
3800.2.a.y.1.3		3	5.3	odd	4
3800.2.d.k.3649.1		6	1.1	even	1	trivial
3800.2.d.k.3649.6		6	5.4	even	2	inner
6080.2.a.bs.1.1		3	40.27	even	4
6080.2.a.bw.1.3		3	40.37	odd	4
6840.2.a.bj.1.2		3	15.2	even	4
7600.2.a.bo.1.1		3	20.3	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)

\(f(q)\)	\(=\)	\(q-3.12489i q^{3} +1.51514i q^{7} -6.76491 q^{9} +O(q^{10})\)	\(q-3.12489i q^{3} +1.51514i q^{7} -6.76491 q^{9} +4.24977 q^{11} +4.15516i q^{13} +3.51514i q^{17} +1.00000 q^{19} +4.73463 q^{21} -8.73463i q^{23} +11.7649i q^{27} -1.45459 q^{29} -4.96972 q^{31} -13.2800i q^{33} -7.60975i q^{37} +12.9844 q^{39} -9.21949 q^{41} -8.31032i q^{43} -5.28005i q^{47} +4.70436 q^{49} +10.9844 q^{51} +0.155162i q^{53} -3.12489i q^{57} +2.48486 q^{59} -4.49954 q^{61} -10.2498i q^{63} -7.43521i q^{67} -27.2947 q^{69} +8.49954 q^{71} -15.0450i q^{73} +6.43899i q^{77} +0.310323 q^{79} +16.4693 q^{81} -8.96972i q^{83} +4.54541i q^{87} -0.719953 q^{89} -6.29564 q^{91} +15.5298i q^{93} -17.3893i q^{97} -28.7493 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{9}+O(q^{10}) \) 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

Groups

Database

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