LMFDB - Embedded newform 3800.2.d.n.3649.6

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			3649.6
Root			$-0.671462 + 1.24464i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.n.3649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.34292i q^{3} +1.19656i q^{7} -2.48929 q^{9} +O(q^{10})$	$q+2.34292i q^{3} +1.19656i q^{7} -2.48929 q^{9} +4.97858 q^{11} +6.63565i q^{13} +1.48929i q^{17} -1.00000 q^{19} -2.80344 q^{21} +0.510711i q^{23} +1.19656i q^{27} +7.88240 q^{29} -2.97858 q^{31} +11.6644i q^{33} -7.14637i q^{37} -15.5468 q^{39} +1.66442 q^{41} +6.39312i q^{43} -9.95715i q^{47} +5.56825 q^{49} -3.48929 q^{51} +11.4219i q^{53} -2.34292i q^{57} +11.8396 q^{59} +3.66442 q^{61} -2.97858i q^{63} +7.61423i q^{67} -1.19656 q^{69} -13.8396i q^{73} +5.95715i q^{77} -12.6858 q^{79} -10.2713 q^{81} +8.68585i q^{83} +18.4679i q^{87} +4.87819 q^{89} -7.93994 q^{91} -6.97858i q^{93} -6.81079i q^{97} -12.3931 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.34292i	1.35269i	0.736586	+	0.676344i	$0.236437\pi$
$3$	2.34292i	1.35269i	−0.736586	+	0.676344i	$0.763563\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.19656i	0.452256i	0.974098	+	0.226128i	$0.0726068\pi$
$7$	1.19656i	0.452256i	−0.974098	+	0.226128i	$0.927393\pi$
$8$	0	0
$9$	−2.48929	−0.829763
$10$	0	0
$11$	4.97858	1.50110	0.750549	−	0.660815i	$-0.229789\pi$
$11$	4.97858	1.50110	0.750549	+	0.660815i	$0.229789\pi$
$12$	0	0
$13$	6.63565i	1.84040i	0.391449	+	0.920200i	$0.371974\pi$
$13$	6.63565i	1.84040i	−0.391449	+	0.920200i	$0.628026\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.48929i	0.361206i	0.983556	+	0.180603i	$0.0578048\pi$
$17$	1.48929i	0.361206i	−0.983556	+	0.180603i	$0.942195\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−2.80344	−0.611761
$22$	0	0
$23$	0.510711i	0.106491i	0.998581	+	0.0532453i	$0.0169565\pi$
$23$	0.510711i	0.106491i	−0.998581	+	0.0532453i	$0.983043\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.19656i	0.230278i
$28$	0	0
$29$	7.88240	1.46373	0.731863	−	0.681452i	$-0.238651\pi$
$29$	7.88240	1.46373	0.731863	+	0.681452i	$0.238651\pi$
$30$	0	0
$31$	−2.97858	−0.534968	−0.267484	−	0.963562i	$-0.586192\pi$
$31$	−2.97858	−0.534968	−0.267484	+	0.963562i	$0.586192\pi$
$32$	0	0
$33$	11.6644i	2.03052i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 7.14637i	− 1.17486i	−0.809277	−	0.587428i	$-0.800141\pi$
$37$	− 7.14637i	− 1.17486i	0.809277	−	0.587428i	$-0.199859\pi$
$38$	0	0
$39$	−15.5468	−2.48948
$40$	0	0
$41$	1.66442	0.259939	0.129970	−	0.991518i	$-0.458512\pi$
$41$	1.66442	0.259939	0.129970	+	0.991518i	$0.458512\pi$
$42$	0	0
$43$	6.39312i	0.974941i	0.873139	+	0.487470i	$0.162080\pi$
$43$	6.39312i	0.974941i	−0.873139	+	0.487470i	$0.837920\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 9.95715i	− 1.45240i	−0.687483	−	0.726200i	$-0.741285\pi$
$47$	− 9.95715i	− 1.45240i	0.687483	−	0.726200i	$-0.258715\pi$
$48$	0	0
$49$	5.56825	0.795464
$50$	0	0
$51$	−3.48929	−0.488598
$52$	0	0
$53$	11.4219i	1.56892i	0.620182	+	0.784458i	$0.287059\pi$
$53$	11.4219i	1.56892i	−0.620182	+	0.784458i	$0.712941\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.34292i	− 0.310328i
$58$	0	0
$59$	11.8396	1.54138	0.770690	−	0.637211i	$-0.219912\pi$
$59$	11.8396	1.54138	0.770690	+	0.637211i	$0.219912\pi$
$60$	0	0
$61$	3.66442	0.469181	0.234591	−	0.972094i	$-0.424625\pi$
$61$	3.66442	0.469181	0.234591	+	0.972094i	$0.424625\pi$
$62$	0	0
$63$	− 2.97858i	− 0.375265i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.61423i	0.930226i	0.885251	+	0.465113i	$0.153986\pi$
$67$	7.61423i	0.930226i	−0.885251	+	0.465113i	$0.846014\pi$
$68$	0	0
$69$	−1.19656	−0.144049
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	− 13.8396i	− 1.61980i	−0.586570	−	0.809899i	$-0.699522\pi$
$73$	− 13.8396i	− 1.61980i	0.586570	−	0.809899i	$-0.300478\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.95715i	0.678881i
$78$	0	0
$79$	−12.6858	−1.42727	−0.713635	−	0.700518i	$-0.752952\pi$
$79$	−12.6858	−1.42727	−0.713635	+	0.700518i	$0.752952\pi$
$80$	0	0
$81$	−10.2713	−1.14126
$82$	0	0
$83$	8.68585i	0.953395i	0.879067	+	0.476698i	$0.158166\pi$
$83$	8.68585i	0.953395i	−0.879067	+	0.476698i	$0.841834\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	18.4679i	1.97996i
$88$	0	0
$89$	4.87819	0.517087	0.258544	−	0.966000i	$-0.416757\pi$
$89$	4.87819	0.517087	0.258544	+	0.966000i	$0.416757\pi$
$90$	0	0
$91$	−7.93994	−0.832332
$92$	0	0
$93$	− 6.97858i	− 0.723645i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 6.81079i	− 0.691531i	−0.938321	−	0.345765i	$-0.887619\pi$
$97$	− 6.81079i	− 0.691531i	0.938321	−	0.345765i	$-0.112381\pi$
$98$	0	0
$99$	−12.3931	−1.24555
$100$	0	0
$101$	−2.29273	−0.228135	−0.114068	−	0.993473i	$-0.536388\pi$
$101$	−2.29273	−0.228135	−0.114068	+	0.993473i	$0.536388\pi$
$102$	0	0
$103$	6.51806i	0.642243i	0.947038	+	0.321122i	$0.104060\pi$
$103$	6.51806i	0.642243i	−0.947038	+	0.321122i	$0.895940\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.71462i	0.745800i	0.927872	+	0.372900i	$0.121637\pi$
$107$	7.71462i	0.745800i	−0.927872	+	0.372900i	$0.878363\pi$
$108$	0	0
$109$	−15.5468	−1.48912	−0.744558	−	0.667558i	$-0.767340\pi$
$109$	−15.5468	−1.48912	−0.744558	+	0.667558i	$0.767340\pi$
$110$	0	0
$111$	16.7434	1.58921
$112$	0	0
$113$	0.753250i	0.0708598i	0.999372	+	0.0354299i	$0.0112801\pi$
$113$	0.753250i	0.0708598i	−0.999372	+	0.0354299i	$0.988720\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 16.5181i	− 1.52709i
$118$	0	0
$119$	−1.78202	−0.163357
$120$	0	0
$121$	13.7862	1.25329
$122$	0	0
$123$	3.89962i	0.351617i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	10.4177i	0.924419i	0.886771	+	0.462210i	$0.152943\pi$
$127$	10.4177i	0.924419i	−0.886771	+	0.462210i	$0.847057\pi$
$128$	0	0
$129$	−14.9786	−1.31879
$130$	0	0
$131$	−17.9572	−1.56892	−0.784462	−	0.620177i	$-0.787061\pi$
$131$	−17.9572	−1.56892	−0.784462	+	0.620177i	$0.787061\pi$
$132$	0	0
$133$	− 1.19656i	− 0.103755i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 9.83956i	− 0.840650i	−0.907374	−	0.420325i	$-0.861916\pi$
$137$	− 9.83956i	− 0.840650i	0.907374	−	0.420325i	$-0.138084\pi$
$138$	0	0
$139$	0.978577	0.0830018	0.0415009	−	0.999138i	$-0.486786\pi$
$139$	0.978577	0.0830018	0.0415009	+	0.999138i	$0.486786\pi$
$140$	0	0
$141$	23.3288	1.96464
$142$	0	0
$143$	33.0361i	2.76262i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	13.0460i	1.07601i
$148$	0	0
$149$	8.24989	0.675857	0.337928	−	0.941172i	$-0.390274\pi$
$149$	8.24989	0.675857	0.337928	+	0.941172i	$0.390274\pi$
$150$	0	0
$151$	13.8568	1.12765	0.563824	−	0.825895i	$-0.309330\pi$
$151$	13.8568	1.12765	0.563824	+	0.825895i	$0.309330\pi$
$152$	0	0
$153$	− 3.70727i	− 0.299715i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 15.7073i	− 1.25358i	−0.779190	−	0.626788i	$-0.784369\pi$
$157$	− 15.7073i	− 1.25358i	0.779190	−	0.626788i	$-0.215631\pi$
$158$	0	0
$159$	−26.7606	−2.12225
$160$	0	0
$161$	−0.611096	−0.0481611
$162$	0	0
$163$	− 5.80765i	− 0.454891i	−0.973791	−	0.227445i	$-0.926963\pi$
$163$	− 5.80765i	− 0.454891i	0.973791	−	0.227445i	$-0.0730373\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 13.8322i	− 1.07037i	−0.844735	−	0.535184i	$-0.820242\pi$
$167$	− 13.8322i	− 1.07037i	0.844735	−	0.535184i	$-0.179758\pi$
$168$	0	0
$169$	−31.0319	−2.38707
$170$	0	0
$171$	2.48929	0.190361
$172$	0	0
$173$	− 14.8108i	− 1.12604i	−0.826442	−	0.563022i	$-0.809639\pi$
$173$	− 14.8108i	− 1.12604i	0.826442	−	0.563022i	$-0.190361\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	27.7392i	2.08500i
$178$	0	0
$179$	−15.6644	−1.17081	−0.585407	−	0.810740i	$-0.699065\pi$
$179$	−15.6644	−1.17081	−0.585407	+	0.810740i	$0.699065\pi$
$180$	0	0
$181$	−14.7862	−1.09905	−0.549526	−	0.835477i	$-0.685192\pi$
$181$	−14.7862	−1.09905	−0.549526	+	0.835477i	$0.685192\pi$
$182$	0	0
$183$	8.58546i	0.634656i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	7.41454i	0.542205i
$188$	0	0
$189$	−1.43175	−0.104144
$190$	0	0
$191$	−6.36748	−0.460735	−0.230367	−	0.973104i	$-0.573993\pi$
$191$	−6.36748	−0.460735	−0.230367	+	0.973104i	$0.573993\pi$
$192$	0	0
$193$	19.5970i	1.41062i	0.708897	+	0.705312i	$0.249193\pi$
$193$	19.5970i	1.41062i	−0.708897	+	0.705312i	$0.750807\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.7862i	0.768487i	0.923232	+	0.384244i	$0.125538\pi$
$197$	10.7862i	0.768487i	−0.923232	+	0.384244i	$0.874462\pi$
$198$	0	0
$199$	−2.80344	−0.198731	−0.0993654	−	0.995051i	$-0.531681\pi$
$199$	−2.80344	−0.198731	−0.0993654	+	0.995051i	$0.531681\pi$
$200$	0	0
$201$	−17.8396	−1.25831
$202$	0	0
$203$	9.43175i	0.661979i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.27131i	− 0.0883620i
$208$	0	0
$209$	−4.97858	−0.344375
$210$	0	0
$211$	−11.2541	−0.774764	−0.387382	−	0.921919i	$-0.626621\pi$
$211$	−11.2541	−0.774764	−0.387382	+	0.921919i	$0.626621\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 3.56404i	− 0.241943i
$218$	0	0
$219$	32.4250	2.19108
$220$	0	0
$221$	−9.88240	−0.664762
$222$	0	0
$223$	− 17.4966i	− 1.17166i	−0.810433	−	0.585831i	$-0.800768\pi$
$223$	− 17.4966i	− 1.17166i	0.810433	−	0.585831i	$-0.199232\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3.84942i	− 0.255495i	−0.991807	−	0.127748i	$-0.959225\pi$
$227$	− 3.84942i	− 0.255495i	0.991807	−	0.127748i	$-0.0407747\pi$
$228$	0	0
$229$	14.6430	0.967637	0.483818	−	0.875168i	$-0.339250\pi$
$229$	14.6430	0.967637	0.483818	+	0.875168i	$0.339250\pi$
$230$	0	0
$231$	−13.9572	−0.918313
$232$	0	0
$233$	− 11.9572i	− 0.783339i	−0.920106	−	0.391670i	$-0.871898\pi$
$233$	− 11.9572i	− 0.783339i	0.920106	−	0.391670i	$-0.128102\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 29.7220i	− 1.93065i
$238$	0	0
$239$	−15.5897	−1.00841	−0.504206	−	0.863583i	$-0.668215\pi$
$239$	−15.5897	−1.00841	−0.504206	+	0.863583i	$0.668215\pi$
$240$	0	0
$241$	−16.0575	−1.03436	−0.517178	−	0.855878i	$-0.673018\pi$
$241$	−16.0575	−1.03436	−0.517178	+	0.855878i	$0.673018\pi$
$242$	0	0
$243$	− 20.4752i	− 1.31349i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 6.63565i	− 0.422217i
$248$	0	0
$249$	−20.3503	−1.28965
$250$	0	0
$251$	23.9143	1.50946	0.754729	−	0.656037i	$-0.227768\pi$
$251$	23.9143	1.50946	0.754729	+	0.656037i	$0.227768\pi$
$252$	0	0
$253$	2.54262i	0.159853i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 24.4324i	− 1.52405i	−0.647548	−	0.762025i	$-0.724206\pi$
$257$	− 24.4324i	− 1.52405i	0.647548	−	0.762025i	$-0.275794\pi$
$258$	0	0
$259$	8.55104	0.531336
$260$	0	0
$261$	−19.6216	−1.21455
$262$	0	0
$263$	14.4935i	0.893707i	0.894607	+	0.446854i	$0.147456\pi$
$263$	14.4935i	0.893707i	−0.894607	+	0.446854i	$0.852544\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	11.4292i	0.699458i
$268$	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
$271$	−6.76060	−0.410677	−0.205339	−	0.978691i	$-0.565830\pi$
$271$	−6.76060	−0.410677	−0.205339	+	0.978691i	$0.565830\pi$
$272$	0	0
$273$	− 18.6027i	− 1.12589i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 25.4292i	− 1.52789i	−0.645279	−	0.763947i	$-0.723259\pi$
$277$	− 25.4292i	− 1.52789i	0.645279	−	0.763947i	$-0.276741\pi$
$278$	0	0
$279$	7.41454	0.443897
$280$	0	0
$281$	9.07896	0.541605	0.270803	−	0.962635i	$-0.412711\pi$
$281$	9.07896	0.541605	0.270803	+	0.962635i	$0.412711\pi$
$282$	0	0
$283$	− 12.0147i	− 0.714199i	−0.934066	−	0.357100i	$-0.883766\pi$
$283$	− 12.0147i	− 0.714199i	0.934066	−	0.357100i	$-0.116234\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.99158i	0.117559i
$288$	0	0
$289$	14.7820	0.869531
$290$	0	0
$291$	15.9572	0.935425
$292$	0	0
$293$	− 23.6718i	− 1.38292i	−0.722415	−	0.691460i	$-0.756968\pi$
$293$	− 23.6718i	− 1.38292i	0.722415	−	0.691460i	$-0.243032\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.95715i	0.345669i
$298$	0	0
$299$	−3.38890	−0.195985
$300$	0	0
$301$	−7.64973	−0.440923
$302$	0	0
$303$	− 5.37169i	− 0.308596i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 16.0246i	− 0.914570i	−0.889320	−	0.457285i	$-0.848822\pi$
$307$	− 16.0246i	− 0.914570i	0.889320	−	0.457285i	$-0.151178\pi$
$308$	0	0
$309$	−15.2713	−0.868754
$310$	0	0
$311$	−27.5468	−1.56204	−0.781019	−	0.624508i	$-0.785300\pi$
$311$	−27.5468	−1.56204	−0.781019	+	0.624508i	$0.785300\pi$
$312$	0	0
$313$	− 10.1751i	− 0.575133i	−0.957761	−	0.287566i	$-0.907154\pi$
$313$	− 10.1751i	− 0.575133i	0.957761	−	0.287566i	$-0.0928462\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	29.6289i	1.66413i	0.554681	+	0.832063i	$0.312840\pi$
$317$	29.6289i	1.66413i	−0.554681	+	0.832063i	$0.687160\pi$
$318$	0	0
$319$	39.2432	2.19719
$320$	0	0
$321$	−18.0748	−1.00883
$322$	0	0
$323$	− 1.48929i	− 0.0828662i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 36.4250i	− 2.01431i
$328$	0	0
$329$	11.9143	0.656857
$330$	0	0
$331$	30.4679	1.67467	0.837333	−	0.546694i	$-0.184114\pi$
$331$	30.4679	1.67467	0.837333	+	0.546694i	$0.184114\pi$
$332$	0	0
$333$	17.7894i	0.974851i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	26.1396i	1.42392i	0.702222	+	0.711958i	$0.252192\pi$
$337$	26.1396i	1.42392i	−0.702222	+	0.711958i	$0.747808\pi$
$338$	0	0
$339$	−1.76481	−0.0958512
$340$	0	0
$341$	−14.8291	−0.803039
$342$	0	0
$343$	15.0386i	0.812010i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 7.31415i	− 0.392644i	−0.980539	−	0.196322i	$-0.937100\pi$
$347$	− 7.31415i	− 0.392644i	0.980539	−	0.196322i	$-0.0628998\pi$
$348$	0	0
$349$	−0.628308	−0.0336325	−0.0168163	−	0.999859i	$-0.505353\pi$
$349$	−0.628308	−0.0336325	−0.0168163	+	0.999859i	$0.505353\pi$
$350$	0	0
$351$	−7.93994	−0.423803
$352$	0	0
$353$	8.23267i	0.438181i	0.975705	+	0.219090i	$0.0703090\pi$
$353$	8.23267i	0.438181i	−0.975705	+	0.219090i	$0.929691\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 4.17513i	− 0.220972i
$358$	0	0
$359$	−12.1323	−0.640318	−0.320159	−	0.947364i	$-0.603736\pi$
$359$	−12.1323	−0.640318	−0.320159	+	0.947364i	$0.603736\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	32.3001i	1.69531i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	17.9572i	0.937356i	0.883369	+	0.468678i	$0.155270\pi$
$367$	17.9572i	0.937356i	−0.883369	+	0.468678i	$0.844730\pi$
$368$	0	0
$369$	−4.14323	−0.215688
$370$	0	0
$371$	−13.6669	−0.709552
$372$	0	0
$373$	− 29.6289i	− 1.53413i	−0.641571	−	0.767064i	$-0.721717\pi$
$373$	− 29.6289i	− 1.53413i	0.641571	−	0.767064i	$-0.278283\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	52.3049i	2.69384i
$378$	0	0
$379$	6.80344	0.349469	0.174735	−	0.984616i	$-0.444093\pi$
$379$	6.80344	0.349469	0.174735	+	0.984616i	$0.444093\pi$
$380$	0	0
$381$	−24.4078	−1.25045
$382$	0	0
$383$	20.2253i	1.03347i	0.856147	+	0.516733i	$0.172852\pi$
$383$	20.2253i	1.03347i	−0.856147	+	0.516733i	$0.827148\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 15.9143i	− 0.808970i
$388$	0	0
$389$	−0.393115	−0.0199317	−0.00996587	−	0.999950i	$-0.503172\pi$
$389$	−0.393115	−0.0199317	−0.00996587	+	0.999950i	$0.503172\pi$
$390$	0	0
$391$	−0.760597	−0.0384650
$392$	0	0
$393$	− 42.0722i	− 2.12226i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	17.4292i	0.874748i	0.899280	+	0.437374i	$0.144091\pi$
$397$	17.4292i	0.874748i	−0.899280	+	0.437374i	$0.855909\pi$
$398$	0	0
$399$	2.80344	0.140348
$400$	0	0
$401$	13.1281	0.655585	0.327792	−	0.944750i	$-0.393695\pi$
$401$	13.1281	0.655585	0.327792	+	0.944750i	$0.393695\pi$
$402$	0	0
$403$	− 19.7648i	− 0.984555i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 35.5787i	− 1.76357i
$408$	0	0
$409$	18.7862	0.928919	0.464460	−	0.885594i	$-0.346249\pi$
$409$	18.7862	0.928919	0.464460	+	0.885594i	$0.346249\pi$
$410$	0	0
$411$	23.0533	1.13714
$412$	0	0
$413$	14.1667i	0.697098i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.29273i	0.112276i
$418$	0	0
$419$	5.22219	0.255121	0.127560	−	0.991831i	$-0.459285\pi$
$419$	5.22219	0.255121	0.127560	+	0.991831i	$0.459285\pi$
$420$	0	0
$421$	−7.68164	−0.374380	−0.187190	−	0.982324i	$-0.559938\pi$
$421$	−7.68164	−0.374380	−0.187190	+	0.982324i	$0.559938\pi$
$422$	0	0
$423$	24.7862i	1.20515i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.38469i	0.212190i
$428$	0	0
$429$	−77.4011	−3.73696
$430$	0	0
$431$	−17.7073	−0.852929	−0.426465	−	0.904504i	$-0.640241\pi$
$431$	−17.7073	−0.852929	−0.426465	+	0.904504i	$0.640241\pi$
$432$	0	0
$433$	− 17.9901i	− 0.864551i	−0.901742	−	0.432275i	$-0.857711\pi$
$433$	− 17.9901i	− 0.864551i	0.901742	−	0.432275i	$-0.142289\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 0.510711i	− 0.0244306i
$438$	0	0
$439$	−14.8438	−0.708454	−0.354227	−	0.935159i	$-0.615256\pi$
$439$	−14.8438	−0.708454	−0.354227	+	0.935159i	$0.615256\pi$
$440$	0	0
$441$	−13.8610	−0.660047
$442$	0	0
$443$	− 22.4078i	− 1.06463i	−0.846547	−	0.532314i	$-0.821323\pi$
$443$	− 22.4078i	− 1.06463i	0.846547	−	0.532314i	$-0.178677\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	19.3288i	0.914223i
$448$	0	0
$449$	40.7434	1.92280	0.961400	−	0.275156i	$-0.0887295\pi$
$449$	40.7434	1.92280	0.961400	+	0.275156i	$0.0887295\pi$
$450$	0	0
$451$	8.28646	0.390194
$452$	0	0
$453$	32.4653i	1.52536i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.01721i	0.375029i	0.982262	+	0.187515i	$0.0600432\pi$
$457$	8.01721i	0.375029i	−0.982262	+	0.187515i	$0.939957\pi$
$458$	0	0
$459$	−1.78202	−0.0831775
$460$	0	0
$461$	−25.1281	−1.17033	−0.585166	−	0.810914i	$-0.698971\pi$
$461$	−25.1281	−1.17033	−0.585166	+	0.810914i	$0.698971\pi$
$462$	0	0
$463$	31.7795i	1.47692i	0.674298	+	0.738459i	$0.264446\pi$
$463$	31.7795i	1.47692i	−0.674298	+	0.738459i	$0.735554\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.9210i	0.968110i	0.875038	+	0.484055i	$0.160837\pi$
$467$	20.9210i	0.968110i	−0.875038	+	0.484055i	$0.839163\pi$
$468$	0	0
$469$	−9.11087	−0.420701
$470$	0	0
$471$	36.8009	1.69570
$472$	0	0
$473$	31.8286i	1.46348i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 28.4324i	− 1.30183i
$478$	0	0
$479$	−27.7220	−1.26665	−0.633324	−	0.773886i	$-0.718310\pi$
$479$	−27.7220	−1.26665	−0.633324	+	0.773886i	$0.718310\pi$
$480$	0	0
$481$	47.4208	2.16220
$482$	0	0
$483$	− 1.43175i	− 0.0651469i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3.20390i	0.145183i	0.997362	+	0.0725914i	$0.0231269\pi$
$487$	3.20390i	0.145183i	−0.997362	+	0.0725914i	$0.976873\pi$
$488$	0	0
$489$	13.6069	0.615325
$490$	0	0
$491$	−21.5212	−0.971238	−0.485619	−	0.874171i	$-0.661406\pi$
$491$	−21.5212	−0.971238	−0.485619	+	0.874171i	$0.661406\pi$
$492$	0	0
$493$	11.7392i	0.528706i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	23.1709	1.03727	0.518637	−	0.854995i	$-0.326440\pi$
$499$	23.1709	1.03727	0.518637	+	0.854995i	$0.326440\pi$
$500$	0	0
$501$	32.4078	1.44787
$502$	0	0
$503$	1.88240i	0.0839322i	0.999119	+	0.0419661i	$0.0133622\pi$
$503$	1.88240i	0.0839322i	−0.999119	+	0.0419661i	$0.986638\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 72.7054i	− 3.22896i
$508$	0	0
$509$	−5.46365	−0.242172	−0.121086	−	0.992642i	$-0.538638\pi$
$509$	−5.46365	−0.242172	−0.121086	+	0.992642i	$0.538638\pi$
$510$	0	0
$511$	16.5598	0.732564
$512$	0	0
$513$	− 1.19656i	− 0.0528293i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 49.5725i	− 2.18019i
$518$	0	0
$519$	34.7005	1.52318
$520$	0	0
$521$	11.2222	0.491653	0.245827	−	0.969314i	$-0.420941\pi$
$521$	11.2222	0.491653	0.245827	+	0.969314i	$0.420941\pi$
$522$	0	0
$523$	29.1867i	1.27624i	0.769935	+	0.638122i	$0.220289\pi$
$523$	29.1867i	1.27624i	−0.769935	+	0.638122i	$0.779711\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 4.43596i	− 0.193233i
$528$	0	0
$529$	22.7392	0.988660
$530$	0	0
$531$	−29.4721	−1.27898
$532$	0	0
$533$	11.0445i	0.478392i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 36.7005i	− 1.58375i
$538$	0	0
$539$	27.7220	1.19407
$540$	0	0
$541$	20.2499	0.870611	0.435305	−	0.900283i	$-0.356640\pi$
$541$	20.2499	0.870611	0.435305	+	0.900283i	$0.356640\pi$
$542$	0	0
$543$	− 34.6430i	− 1.48667i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	42.8830i	1.83355i	0.399409	+	0.916773i	$0.369215\pi$
$547$	42.8830i	1.83355i	−0.399409	+	0.916773i	$0.630785\pi$
$548$	0	0
$549$	−9.12181	−0.389309
$550$	0	0
$551$	−7.88240	−0.335802
$552$	0	0
$553$	− 15.1793i	− 0.645491i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	28.2583i	1.19734i	0.800995	+	0.598671i	$0.204304\pi$
$557$	28.2583i	1.19734i	−0.800995	+	0.598671i	$0.795696\pi$
$558$	0	0
$559$	−42.4225	−1.79428
$560$	0	0
$561$	−17.3717	−0.733433
$562$	0	0
$563$	20.2253i	0.852396i	0.904630	+	0.426198i	$0.140147\pi$
$563$	20.2253i	0.852396i	−0.904630	+	0.426198i	$0.859853\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 12.2902i	− 0.516140i
$568$	0	0
$569$	31.1365	1.30531	0.652655	−	0.757655i	$-0.273655\pi$
$569$	31.1365	1.30531	0.652655	+	0.757655i	$0.273655\pi$
$570$	0	0
$571$	27.4868	1.15029	0.575143	−	0.818053i	$-0.304947\pi$
$571$	27.4868	1.15029	0.575143	+	0.818053i	$0.304947\pi$
$572$	0	0
$573$	− 14.9185i	− 0.623230i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	23.7820i	0.990058i	0.868876	+	0.495029i	$0.164843\pi$
$577$	23.7820i	0.990058i	−0.868876	+	0.495029i	$0.835157\pi$
$578$	0	0
$579$	−45.9143	−1.90813
$580$	0	0
$581$	−10.3931	−0.431179
$582$	0	0
$583$	56.8647i	2.35510i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	24.0147i	0.991192i	0.868553	+	0.495596i	$0.165050\pi$
$587$	24.0147i	0.991192i	−0.868553	+	0.495596i	$0.834950\pi$
$588$	0	0
$589$	2.97858	0.122730
$590$	0	0
$591$	−25.2713	−1.03952
$592$	0	0
$593$	− 46.6148i	− 1.91424i	−0.289688	−	0.957121i	$-0.593551\pi$
$593$	− 46.6148i	− 1.91424i	0.289688	−	0.957121i	$-0.406449\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 6.56825i	− 0.268821i
$598$	0	0
$599$	44.6002	1.82231	0.911156	−	0.412061i	$-0.135191\pi$
$599$	44.6002	1.82231	0.911156	+	0.412061i	$0.135191\pi$
$600$	0	0
$601$	−17.5787	−0.717051	−0.358526	−	0.933520i	$-0.616720\pi$
$601$	−17.5787	−0.717051	−0.358526	+	0.933520i	$0.616720\pi$
$602$	0	0
$603$	− 18.9540i	− 0.771867i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	37.2467i	1.51180i	0.654688	+	0.755899i	$0.272800\pi$
$607$	37.2467i	1.51180i	−0.654688	+	0.755899i	$0.727200\pi$
$608$	0	0
$609$	−22.0979	−0.895451
$610$	0	0
$611$	66.0722	2.67300
$612$	0	0
$613$	14.8866i	0.601265i	0.953740	+	0.300632i	$0.0971977\pi$
$613$	14.8866i	0.601265i	−0.953740	+	0.300632i	$0.902802\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.70727i	0.149249i	0.997212	+	0.0746245i	$0.0237758\pi$
$617$	3.70727i	0.149249i	−0.997212	+	0.0746245i	$0.976224\pi$
$618$	0	0
$619$	−17.7648	−0.714028	−0.357014	−	0.934099i	$-0.616205\pi$
$619$	−17.7648	−0.714028	−0.357014	+	0.934099i	$0.616205\pi$
$620$	0	0
$621$	−0.611096	−0.0245224
$622$	0	0
$623$	5.83704i	0.233856i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 11.6644i	− 0.465832i
$628$	0	0
$629$	10.6430	0.424364
$630$	0	0
$631$	40.3074	1.60461	0.802307	−	0.596912i	$-0.203606\pi$
$631$	40.3074	1.60461	0.802307	+	0.596912i	$0.203606\pi$
$632$	0	0
$633$	− 26.3675i	− 1.04801i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	36.9490i	1.46397i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−27.6728	−1.09301	−0.546506	−	0.837455i	$-0.684042\pi$
$641$	−27.6728	−1.09301	−0.546506	+	0.837455i	$0.684042\pi$
$642$	0	0
$643$	− 19.0277i	− 0.750379i	−0.926948	−	0.375190i	$-0.877578\pi$
$643$	− 19.0277i	− 0.750379i	0.926948	−	0.375190i	$-0.122422\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 32.0403i	− 1.25964i	−0.776743	−	0.629818i	$-0.783130\pi$
$647$	− 32.0403i	− 1.25964i	0.776743	−	0.629818i	$-0.216870\pi$
$648$	0	0
$649$	58.9442	2.31376
$650$	0	0
$651$	8.35027	0.327273
$652$	0	0
$653$	− 25.3142i	− 0.990619i	−0.868716	−	0.495310i	$-0.835055\pi$
$653$	− 25.3142i	− 0.990619i	0.868716	−	0.495310i	$-0.164945\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	34.4507i	1.34405i
$658$	0	0
$659$	0.860981	0.0335391	0.0167695	−	0.999859i	$-0.494662\pi$
$659$	0.860981	0.0335391	0.0167695	+	0.999859i	$0.494662\pi$
$660$	0	0
$661$	4.55356	0.177113	0.0885564	−	0.996071i	$-0.471775\pi$
$661$	4.55356	0.177113	0.0885564	+	0.996071i	$0.471775\pi$
$662$	0	0
$663$	− 23.1537i	− 0.899216i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.02563i	0.155873i
$668$	0	0
$669$	40.9933	1.58489
$670$	0	0
$671$	18.2436	0.704287
$672$	0	0
$673$	11.7465i	0.452795i	0.974035	+	0.226398i	$0.0726948\pi$
$673$	11.7465i	0.452795i	−0.974035	+	0.226398i	$0.927305\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 22.2854i	− 0.856497i	−0.903661	−	0.428248i	$-0.859131\pi$
$677$	− 22.2854i	− 0.856497i	0.903661	−	0.428248i	$-0.140869\pi$
$678$	0	0
$679$	8.14950	0.312749
$680$	0	0
$681$	9.01890	0.345605
$682$	0	0
$683$	7.23833i	0.276967i	0.990365	+	0.138483i	$0.0442228\pi$
$683$	7.23833i	0.276967i	−0.990365	+	0.138483i	$0.955777\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	34.3074i	1.30891i
$688$	0	0
$689$	−75.7917	−2.88743
$690$	0	0
$691$	32.7434	1.24562	0.622809	−	0.782374i	$-0.285992\pi$
$691$	32.7434	1.24562	0.622809	+	0.782374i	$0.285992\pi$
$692$	0	0
$693$	− 14.8291i	− 0.563310i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.47881i	0.0938915i
$698$	0	0
$699$	28.0147	1.05961
$700$	0	0
$701$	−29.4208	−1.11121	−0.555604	−	0.831447i	$-0.687513\pi$
$701$	−29.4208	−1.11121	−0.555604	+	0.831447i	$0.687513\pi$
$702$	0	0
$703$	7.14637i	0.269530i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 2.74338i	− 0.103176i
$708$	0	0
$709$	5.32885	0.200129	0.100065	−	0.994981i	$-0.468095\pi$
$709$	5.32885	0.200129	0.100065	+	0.994981i	$0.468095\pi$
$710$	0	0
$711$	31.5787	1.18429
$712$	0	0
$713$	− 1.52119i	− 0.0569691i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 36.5254i	− 1.36407i
$718$	0	0
$719$	−8.45317	−0.315250	−0.157625	−	0.987499i	$-0.550384\pi$
$719$	−8.45317	−0.315250	−0.157625	+	0.987499i	$0.550384\pi$
$720$	0	0
$721$	−7.79923	−0.290459
$722$	0	0
$723$	− 37.6216i	− 1.39916i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.56825i	0.0952511i	0.998865	+	0.0476256i	$0.0151654\pi$
$727$	2.56825i	0.0952511i	−0.998865	+	0.0476256i	$0.984835\pi$
$728$	0	0
$729$	17.1579	0.635479
$730$	0	0
$731$	−9.52119	−0.352154
$732$	0	0
$733$	− 35.5212i	− 1.31201i	−0.754759	−	0.656003i	$-0.772246\pi$
$733$	− 35.5212i	− 1.31201i	0.754759	−	0.656003i	$-0.227754\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	37.9080i	1.39636i
$738$	0	0
$739$	37.0508	1.36294	0.681468	−	0.731848i	$-0.261342\pi$
$739$	37.0508	1.36294	0.681468	+	0.731848i	$0.261342\pi$
$740$	0	0
$741$	15.5468	0.571127
$742$	0	0
$743$	− 23.2039i	− 0.851269i	−0.904895	−	0.425634i	$-0.860051\pi$
$743$	− 23.2039i	− 0.851269i	0.904895	−	0.425634i	$-0.139949\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 21.6216i	− 0.791092i
$748$	0	0
$749$	−9.23098	−0.337293
$750$	0	0
$751$	51.2285	1.86935	0.934677	−	0.355499i	$-0.115689\pi$
$751$	51.2285	1.86935	0.934677	+	0.355499i	$0.115689\pi$
$752$	0	0
$753$	56.0294i	2.04182i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 43.3717i	− 1.57637i	−0.615438	−	0.788185i	$-0.711021\pi$
$757$	− 43.3717i	− 1.57637i	0.615438	−	0.788185i	$-0.288979\pi$
$758$	0	0
$759$	−5.95715	−0.216231
$760$	0	0
$761$	24.7753	0.898104	0.449052	−	0.893506i	$-0.351762\pi$
$761$	24.7753	0.898104	0.449052	+	0.893506i	$0.351762\pi$
$762$	0	0
$763$	− 18.6027i	− 0.673462i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	78.5632i	2.83675i
$768$	0	0
$769$	11.4893	0.414314	0.207157	−	0.978308i	$-0.433579\pi$
$769$	11.4893	0.414314	0.207157	+	0.978308i	$0.433579\pi$
$770$	0	0
$771$	57.2432	2.06156
$772$	0	0
$773$	− 27.0863i	− 0.974227i	−0.873339	−	0.487113i	$-0.838050\pi$
$773$	− 27.0863i	− 0.974227i	0.873339	−	0.487113i	$-0.161950\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	20.0344i	0.718731i
$778$	0	0
$779$	−1.66442	−0.0596342
$780$	0	0
$781$	0	0
$782$	0	0
$783$	9.43175i	0.337063i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	13.4563i	0.479666i	0.970814	+	0.239833i	$0.0770926\pi$
$787$	13.4563i	0.479666i	−0.970814	+	0.239833i	$0.922907\pi$
$788$	0	0
$789$	−33.9572	−1.20891
$790$	0	0
$791$	−0.901307	−0.0320468
$792$	0	0
$793$	24.3158i	0.863481i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	30.4496i	1.07858i	0.842120	+	0.539290i	$0.181307\pi$
$797$	30.4496i	1.07858i	−0.842120	+	0.539290i	$0.818693\pi$
$798$	0	0
$799$	14.8291	0.524615
$800$	0	0
$801$	−12.1432	−0.429060
$802$	0	0
$803$	− 68.9013i	− 2.43147i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	32.8009i	1.15465i
$808$	0	0
$809$	29.7047	1.04436	0.522182	−	0.852834i	$-0.325118\pi$
$809$	29.7047	1.04436	0.522182	+	0.852834i	$0.325118\pi$
$810$	0	0
$811$	32.2902	1.13386	0.566931	−	0.823765i	$-0.308130\pi$
$811$	32.2902	1.13386	0.566931	+	0.823765i	$0.308130\pi$
$812$	0	0
$813$	− 15.8396i	− 0.555518i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 6.39312i	− 0.223667i
$818$	0	0
$819$	19.7648	0.690638
$820$	0	0
$821$	9.06427	0.316345	0.158173	−	0.987411i	$-0.449440\pi$
$821$	9.06427	0.316345	0.158173	+	0.987411i	$0.449440\pi$
$822$	0	0
$823$	7.35448i	0.256361i	0.991751	+	0.128181i	$0.0409137\pi$
$823$	7.35448i	0.256361i	−0.991751	+	0.128181i	$0.959086\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.02204i	0.0703132i	0.999382	+	0.0351566i	$0.0111930\pi$
$827$	2.02204i	0.0703132i	−0.999382	+	0.0351566i	$0.988807\pi$
$828$	0	0
$829$	36.3675	1.26309	0.631547	−	0.775337i	$-0.282420\pi$
$829$	36.3675	1.26309	0.631547	+	0.775337i	$0.282420\pi$
$830$	0	0
$831$	59.5787	2.06676
$832$	0	0
$833$	8.29273i	0.287326i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 3.56404i	− 0.123191i
$838$	0	0
$839$	−25.8223	−0.891486	−0.445743	−	0.895161i	$-0.647061\pi$
$839$	−25.8223	−0.891486	−0.445743	+	0.895161i	$0.647061\pi$
$840$	0	0
$841$	33.1323	1.14249
$842$	0	0
$843$	21.2713i	0.732623i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	16.4960i	0.566810i
$848$	0	0
$849$	28.1495	0.966088
$850$	0	0
$851$	3.64973	0.125111
$852$	0	0
$853$	− 8.54262i	− 0.292494i	−0.989248	−	0.146247i	$-0.953281\pi$
$853$	− 8.54262i	− 0.292494i	0.989248	−	0.146247i	$-0.0467194\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	27.8322i	0.950730i	0.879789	+	0.475365i	$0.157684\pi$
$857$	27.8322i	0.950730i	−0.879789	+	0.475365i	$0.842316\pi$
$858$	0	0
$859$	−33.4868	−1.14255	−0.571277	−	0.820757i	$-0.693552\pi$
$859$	−33.4868	−1.14255	−0.571277	+	0.820757i	$0.693552\pi$
$860$	0	0
$861$	−4.66611	−0.159021
$862$	0	0
$863$	29.2614i	0.996071i	0.867157	+	0.498036i	$0.165945\pi$
$863$	29.2614i	0.996071i	−0.867157	+	0.498036i	$0.834055\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	34.6331i	1.17620i
$868$	0	0
$869$	−63.1575	−2.14247
$870$	0	0
$871$	−50.5254	−1.71199
$872$	0	0
$873$	16.9540i	0.573807i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 46.7852i	− 1.57982i	−0.613221	−	0.789911i	$-0.710127\pi$
$877$	− 46.7852i	− 1.57982i	0.613221	−	0.789911i	$-0.289873\pi$
$878$	0	0
$879$	55.4611	1.87066
$880$	0	0
$881$	−30.7581	−1.03627	−0.518133	−	0.855300i	$-0.673373\pi$
$881$	−30.7581	−1.03627	−0.518133	+	0.855300i	$0.673373\pi$
$882$	0	0
$883$	− 0.435961i	− 0.0146713i	−0.999973	−	0.00733563i	$-0.997665\pi$
$883$	− 0.435961i	− 0.0146713i	0.999973	−	0.00733563i	$-0.00233502\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 2.15310i	− 0.0722939i	−0.999346	−	0.0361469i	$-0.988492\pi$
$887$	− 2.15310i	− 0.0722939i	0.999346	−	0.0361469i	$-0.0115084\pi$
$888$	0	0
$889$	−12.4653	−0.418074
$890$	0	0
$891$	−51.1365	−1.71314
$892$	0	0
$893$	9.95715i	0.333203i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 7.93994i	− 0.265107i
$898$	0	0
$899$	−23.4783	−0.783047
$900$	0	0
$901$	−17.0105	−0.566701
$902$	0	0
$903$	− 17.9227i	− 0.596431i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 36.9002i	− 1.22525i	−0.790373	−	0.612626i	$-0.790113\pi$
$907$	− 36.9002i	− 1.22525i	0.790373	−	0.612626i	$-0.209887\pi$
$908$	0	0
$909$	5.70727	0.189298
$910$	0	0
$911$	−36.9504	−1.22422	−0.612111	−	0.790772i	$-0.709679\pi$
$911$	−36.9504	−1.22422	−0.612111	+	0.790772i	$0.709679\pi$
$912$	0	0
$913$	43.2432i	1.43114i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 21.4868i	− 0.709556i
$918$	0	0
$919$	−19.1537	−0.631823	−0.315911	−	0.948789i	$-0.602310\pi$
$919$	−19.1537	−0.631823	−0.315911	+	0.948789i	$0.602310\pi$
$920$	0	0
$921$	37.5443	1.23713
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 16.2253i	− 0.532910i
$928$	0	0
$929$	56.9185	1.86744	0.933718	−	0.358009i	$-0.116544\pi$
$929$	56.9185	1.86744	0.933718	+	0.358009i	$0.116544\pi$
$930$	0	0
$931$	−5.56825	−0.182492
$932$	0	0
$933$	− 64.5401i	− 2.11295i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 24.1176i	− 0.787888i	−0.919135	−	0.393944i	$-0.871110\pi$
$937$	− 24.1176i	− 0.787888i	0.919135	−	0.393944i	$-0.128890\pi$
$938$	0	0
$939$	23.8396	0.777975
$940$	0	0
$941$	3.19656	0.104205	0.0521024	−	0.998642i	$-0.483408\pi$
$941$	3.19656	0.104205	0.0521024	+	0.998642i	$0.483408\pi$
$942$	0	0
$943$	0.850040i	0.0276811i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.63504i	0.118123i	0.998254	+	0.0590614i	$0.0188108\pi$
$947$	3.63504i	0.118123i	−0.998254	+	0.0590614i	$0.981189\pi$
$948$	0	0
$949$	91.8345	2.98107
$950$	0	0
$951$	−69.4183	−2.25104
$952$	0	0
$953$	48.0821i	1.55753i	0.627315	+	0.778766i	$0.284154\pi$
$953$	48.0821i	1.55753i	−0.627315	+	0.778766i	$0.715846\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	91.9437i	2.97212i
$958$	0	0
$959$	11.7736	0.380189
$960$	0	0
$961$	−22.1281	−0.713809
$962$	0	0
$963$	− 19.2039i	− 0.618837i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 23.4637i	− 0.754540i	−0.926103	−	0.377270i	$-0.876863\pi$
$967$	− 23.4637i	− 0.754540i	0.926103	−	0.377270i	$-0.123137\pi$
$968$	0	0
$969$	3.48929	0.112092
$970$	0	0
$971$	17.4868	0.561177	0.280589	−	0.959828i	$-0.409470\pi$
$971$	17.4868	0.561177	0.280589	+	0.959828i	$0.409470\pi$
$972$	0	0
$973$	1.17092i	0.0375381i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20.2316i	0.647266i	0.946183	+	0.323633i	$0.104904\pi$
$977$	20.2316i	0.647266i	−0.946183	+	0.323633i	$0.895096\pi$
$978$	0	0
$979$	24.2865	0.776199
$980$	0	0
$981$	38.7005	1.23561
$982$	0	0
$983$	− 28.1249i	− 0.897046i	−0.893771	−	0.448523i	$-0.851950\pi$
$983$	− 28.1249i	− 0.897046i	0.893771	−	0.448523i	$-0.148050\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	27.9143i	0.888522i
$988$	0	0
$989$	−3.26504	−0.103822
$990$	0	0
$991$	−33.9718	−1.07915	−0.539576	−	0.841937i	$-0.681415\pi$
$991$	−33.9718	−1.07915	−0.539576	+	0.841937i	$0.681415\pi$
$992$	0	0
$993$	71.3839i	2.26530i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 23.8223i	− 0.754461i	−0.926119	−	0.377231i	$-0.876876\pi$
$997$	− 23.8223i	− 0.754461i	0.926119	−	0.377231i	$-0.123124\pi$
$998$	0	0
$999$	8.55104	0.270543

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.n.3649.6		6
5.2	odd	4		3800.2.a.w.1.3		3
5.3	odd	4		760.2.a.i.1.1	✓	3
5.4	even	2	inner	3800.2.d.n.3649.1		6
15.8	even	4		6840.2.a.bm.1.2		3
20.3	even	4		1520.2.a.q.1.3		3
20.7	even	4		7600.2.a.bp.1.1		3
40.3	even	4		6080.2.a.br.1.1		3
40.13	odd	4		6080.2.a.bx.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.i.1.1	✓	3	5.3	odd	4
1520.2.a.q.1.3		3	20.3	even	4
3800.2.a.w.1.3		3	5.2	odd	4
3800.2.d.n.3649.1		6	5.4	even	2	inner
3800.2.d.n.3649.6		6	1.1	even	1	trivial
6080.2.a.br.1.1		3	40.3	even	4
6080.2.a.bx.1.3		3	40.13	odd	4
6840.2.a.bm.1.2		3	15.8	even	4
7600.2.a.bp.1.1		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)

\(f(q)\)	\(=\)	\(q+2.34292i q^{3} +1.19656i q^{7} -2.48929 q^{9} +O(q^{10})\)	\(q+2.34292i q^{3} +1.19656i q^{7} -2.48929 q^{9} +4.97858 q^{11} +6.63565i q^{13} +1.48929i q^{17} -1.00000 q^{19} -2.80344 q^{21} +0.510711i q^{23} +1.19656i q^{27} +7.88240 q^{29} -2.97858 q^{31} +11.6644i q^{33} -7.14637i q^{37} -15.5468 q^{39} +1.66442 q^{41} +6.39312i q^{43} -9.95715i q^{47} +5.56825 q^{49} -3.48929 q^{51} +11.4219i q^{53} -2.34292i q^{57} +11.8396 q^{59} +3.66442 q^{61} -2.97858i q^{63} +7.61423i q^{67} -1.19656 q^{69} -13.8396i q^{73} +5.95715i q^{77} -12.6858 q^{79} -10.2713 q^{81} +8.68585i q^{83} +18.4679i q^{87} +4.87819 q^{89} -7.93994 q^{91} -6.97858i q^{93} -6.81079i q^{97} -12.3931 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 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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