LMFDB - Embedded newform 400.9.h.b.399.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,9,Mod(399,400)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("400.399");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	400.h (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:	$162.951444024$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{35})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 17x^{2} + 81 $ x^4 - 17*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			399.3
Root			$-2.95804 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	400.399
Dual form			400.9.h.b.399.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+141.986 q^{3} +2555.75 q^{7} +13599.0 q^{9} -19168.1i q^{11} -27710.0i q^{13} -50370.0i q^{17} +108619. i q^{19} +362880. q^{21} +176347. q^{23} +999297. q^{27} -54978.0 q^{29} -1.17564e6i q^{31} -2.72160e6i q^{33} -793730. i q^{37} -3.93443e6i q^{39} -75582.0 q^{41} +499648. q^{43} -2.86755e6 q^{47} +767039. q^{49} -7.15183e6i q^{51} +1.11662e7i q^{53} +1.54224e7i q^{57} -2.18325e7i q^{59} -2.38266e7 q^{61} +3.47556e7 q^{63} -7.49473e6 q^{67} +2.50387e7 q^{69} +1.00824e7i q^{71} +6.51661e6i q^{73} -4.89888e7i q^{77} -4.87892e7i q^{79} +5.26630e7 q^{81} +7.34483e7 q^{83} -7.80610e6 q^{87} -8.67958e7 q^{89} -7.08197e7i q^{91} -1.66925e8i q^{93} +4.66703e7i q^{97} -2.60667e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 54396 q^{9} + 1451520 q^{21} - 219912 q^{29} - 302328 q^{41} + 3068156 q^{49} - 95306488 q^{61} + 100154880 q^{69} + 210652164 q^{81} - 347183112 q^{89}+O(q^{100}) $ 4 * q + 54396 * q^9 + 1451520 * q^21 - 219912 * q^29 - 302328 * q^41 + 3068156 * q^49 - 95306488 * q^61 + 100154880 * q^69 + 210652164 * q^81 - 347183112 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	141.986	1.75291	0.876456	−	0.481481i	$-0.159901\pi$
$3$	141.986	1.75291	0.876456	+	0.481481i	$0.159901\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2555.75	1.06445	0.532225	−	0.846603i	$-0.321356\pi$
$7$	2555.75	1.06445	0.532225	+	0.846603i	$0.321356\pi$
$8$	0	0
$9$	13599.0	2.07270
$10$	0	0
$11$	− 19168.1i	− 1.30921i	−0.755972	−	0.654603i	$-0.772836\pi$
$11$	− 19168.1i	− 1.30921i	0.755972	−	0.654603i	$-0.227164\pi$
$12$	0	0
$13$	− 27710.0i	− 0.970204i	−0.874458	−	0.485102i	$-0.838782\pi$
$13$	− 27710.0i	− 0.970204i	0.874458	−	0.485102i	$-0.161218\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 50370.0i	− 0.603082i	−0.953453	−	0.301541i	$-0.902499\pi$
$17$	− 50370.0i	− 0.603082i	0.953453	−	0.301541i	$-0.0975010\pi$
$18$	0	0
$19$	108619.i	0.833474i	0.909027	+	0.416737i	$0.136826\pi$
$19$	108619.i	0.833474i	−0.909027	+	0.416737i	$0.863174\pi$
$20$	0	0
$21$	362880.	1.86589
$22$	0	0
$23$	176347.	0.630167	0.315083	−	0.949064i	$-0.397968\pi$
$23$	176347.	0.630167	0.315083	+	0.949064i	$0.397968\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	999297.	1.88035
$28$	0	0
$29$	−54978.0	−0.0777315	−0.0388657	−	0.999244i	$-0.512374\pi$
$29$	−54978.0	−0.0777315	−0.0388657	+	0.999244i	$0.512374\pi$
$30$	0	0
$31$	− 1.17564e6i	− 1.27300i	−0.771276	−	0.636501i	$-0.780381\pi$
$31$	− 1.17564e6i	− 1.27300i	0.771276	−	0.636501i	$-0.219619\pi$
$32$	0	0
$33$	− 2.72160e6i	− 2.29493i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 793730.i	− 0.423512i	−0.977323	−	0.211756i	$-0.932082\pi$
$37$	− 793730.i	− 0.423512i	0.977323	−	0.211756i	$-0.0679182\pi$
$38$	0	0
$39$	− 3.93443e6i	− 1.70068i
$40$	0	0
$41$	−75582.0	−0.0267475	−0.0133737	−	0.999911i	$-0.504257\pi$
$41$	−75582.0	−0.0267475	−0.0133737	+	0.999911i	$0.504257\pi$
$42$	0	0
$43$	499648.	0.146147	0.0730736	−	0.997327i	$-0.476719\pi$
$43$	499648.	0.146147	0.0730736	+	0.997327i	$0.476719\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.86755e6	−0.587651	−0.293825	−	0.955859i	$-0.594928\pi$
$47$	−2.86755e6	−0.587651	−0.293825	+	0.955859i	$0.594928\pi$
$48$	0	0
$49$	767039.	0.133056
$50$	0	0
$51$	− 7.15183e6i	− 1.05715i
$52$	0	0
$53$	1.11662e7i	1.41515i	0.706639	+	0.707575i	$0.250211\pi$
$53$	1.11662e7i	1.41515i	−0.706639	+	0.707575i	$0.749789\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.54224e7i	1.46101i
$58$	0	0
$59$	− 2.18325e7i	− 1.80175i	−0.434078	−	0.900875i	$-0.642926\pi$
$59$	− 2.18325e7i	− 1.80175i	0.434078	−	0.900875i	$-0.357074\pi$
$60$	0	0
$61$	−2.38266e7	−1.72085	−0.860425	−	0.509577i	$-0.829802\pi$
$61$	−2.38266e7	−1.72085	−0.860425	+	0.509577i	$0.829802\pi$
$62$	0	0
$63$	3.47556e7	2.20629
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.49473e6	−0.371926	−0.185963	−	0.982557i	$-0.559540\pi$
$67$	−7.49473e6	−0.371926	−0.185963	+	0.982557i	$0.559540\pi$
$68$	0	0
$69$	2.50387e7	1.10463
$70$	0	0
$71$	1.00824e7i	0.396763i	0.980125	+	0.198382i	$0.0635685\pi$
$71$	1.00824e7i	0.396763i	−0.980125	+	0.198382i	$0.936431\pi$
$72$	0	0
$73$	6.51661e6i	0.229472i	0.993396	+	0.114736i	$0.0366023\pi$
$73$	6.51661e6i	0.229472i	−0.993396	+	0.114736i	$0.963398\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.89888e7i	− 1.39359i
$78$	0	0
$79$	− 4.87892e7i	− 1.25261i	−0.779579	−	0.626304i	$-0.784567\pi$
$79$	− 4.87892e7i	− 1.25261i	0.779579	−	0.626304i	$-0.215433\pi$
$80$	0	0
$81$	5.26630e7	1.22339
$82$	0	0
$83$	7.34483e7	1.54764	0.773819	−	0.633407i	$-0.218344\pi$
$83$	7.34483e7	1.54764	0.773819	+	0.633407i	$0.218344\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−7.80610e6	−0.136256
$88$	0	0
$89$	−8.67958e7	−1.38337	−0.691685	−	0.722199i	$-0.743131\pi$
$89$	−8.67958e7	−1.38337	−0.691685	+	0.722199i	$0.743131\pi$
$90$	0	0
$91$	− 7.08197e7i	− 1.03273i
$92$	0	0
$93$	− 1.66925e8i	− 2.23146i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.66703e7i	0.527173i	0.964636	+	0.263587i	$0.0849055\pi$
$97$	4.66703e7i	0.527173i	−0.964636	+	0.263587i	$0.915095\pi$
$98$	0	0
$99$	− 2.60667e8i	− 2.71360i
$100$	0	0
$101$	6.59910e7	0.634161	0.317080	−	0.948399i	$-0.397297\pi$
$101$	6.59910e7	0.634161	0.317080	+	0.948399i	$0.397297\pi$
$102$	0	0
$103$	1.64884e8	1.46497	0.732486	−	0.680782i	$-0.238360\pi$
$103$	1.64884e8	1.46497	0.732486	+	0.680782i	$0.238360\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.27326e8	−0.971364	−0.485682	−	0.874136i	$-0.661429\pi$
$107$	−1.27326e8	−0.971364	−0.485682	+	0.874136i	$0.661429\pi$
$108$	0	0
$109$	1.56119e8	1.10598	0.552992	−	0.833186i	$-0.313486\pi$
$109$	1.56119e8	1.10598	0.552992	+	0.833186i	$0.313486\pi$
$110$	0	0
$111$	− 1.12698e8i	− 0.742380i
$112$	0	0
$113$	2.36346e8i	1.44955i	0.688984	+	0.724776i	$0.258057\pi$
$113$	2.36346e8i	1.44955i	−0.688984	+	0.724776i	$0.741943\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 3.76828e8i	− 2.01094i
$118$	0	0
$119$	− 1.28733e8i	− 0.641951i
$120$	0	0
$121$	−1.53057e8	−0.714023
$122$	0	0
$123$	−1.07316e7	−0.0468860
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.67741e8	1.41360	0.706802	−	0.707412i	$-0.250137\pi$
$127$	3.67741e8	1.41360	0.706802	+	0.707412i	$0.250137\pi$
$128$	0	0
$129$	7.09430e7	0.256183
$130$	0	0
$131$	− 2.16350e8i	− 0.734636i	−0.930095	−	0.367318i	$-0.880276\pi$
$131$	− 2.16350e8i	− 0.734636i	0.930095	−	0.367318i	$-0.119724\pi$
$132$	0	0
$133$	2.77603e8i	0.887193i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.86442e8i	1.09699i	0.836155	+	0.548494i	$0.184799\pi$
$137$	3.86442e8i	1.09699i	−0.836155	+	0.548494i	$0.815201\pi$
$138$	0	0
$139$	3.51077e8i	0.940465i	0.882543	+	0.470232i	$0.155830\pi$
$139$	3.51077e8i	0.940465i	−0.882543	+	0.470232i	$0.844170\pi$
$140$	0	0
$141$	−4.07151e8	−1.03010
$142$	0	0
$143$	−5.31148e8	−1.27020
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.08909e8	0.233235
$148$	0	0
$149$	4.54099e8	0.921308	0.460654	−	0.887580i	$-0.347615\pi$
$149$	4.54099e8	0.921308	0.460654	+	0.887580i	$0.347615\pi$
$150$	0	0
$151$	6.60188e8i	1.26987i	0.772565	+	0.634936i	$0.218973\pi$
$151$	6.60188e8i	1.26987i	−0.772565	+	0.634936i	$0.781027\pi$
$152$	0	0
$153$	− 6.84982e8i	− 1.25001i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 4.35318e8i	− 0.716486i	−0.933628	−	0.358243i	$-0.883376\pi$
$157$	− 4.35318e8i	− 0.716486i	0.933628	−	0.358243i	$-0.116624\pi$
$158$	0	0
$159$	1.58544e9i	2.48063i
$160$	0	0
$161$	4.50697e8	0.670782
$162$	0	0
$163$	2.44065e8	0.345744	0.172872	−	0.984944i	$-0.444695\pi$
$163$	2.44065e8	0.345744	0.172872	+	0.984944i	$0.444695\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.71351e8	−0.863145	−0.431573	−	0.902078i	$-0.642041\pi$
$167$	−6.71351e8	−0.863145	−0.431573	+	0.902078i	$0.642041\pi$
$168$	0	0
$169$	4.78866e7	0.0587040
$170$	0	0
$171$	1.47711e9i	1.72754i
$172$	0	0
$173$	− 1.76764e9i	− 1.97337i	−0.162644	−	0.986685i	$-0.552002\pi$
$173$	− 1.76764e9i	− 1.97337i	0.162644	−	0.986685i	$-0.447998\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 3.09990e9i	− 3.15831i
$178$	0	0
$179$	7.56967e8i	0.737335i	0.929561	+	0.368668i	$0.120186\pi$
$179$	7.56967e8i	0.737335i	−0.929561	+	0.368668i	$0.879814\pi$
$180$	0	0
$181$	6.27094e8	0.584277	0.292138	−	0.956376i	$-0.405633\pi$
$181$	6.27094e8	0.584277	0.292138	+	0.956376i	$0.405633\pi$
$182$	0	0
$183$	−3.38304e9	−3.01650
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−9.65497e8	−0.789559
$188$	0	0
$189$	2.55395e9	2.00154
$190$	0	0
$191$	− 1.07924e9i	− 0.810933i	−0.914110	−	0.405466i	$-0.867109\pi$
$191$	− 1.07924e9i	− 0.810933i	0.914110	−	0.405466i	$-0.132891\pi$
$192$	0	0
$193$	− 2.96757e7i	− 0.0213881i	−0.999943	−	0.0106940i	$-0.996596\pi$
$193$	− 2.96757e7i	− 0.0213881i	0.999943	−	0.0106940i	$-0.00340408\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.12484e9i	0.746837i	0.927663	+	0.373419i	$0.121814\pi$
$197$	1.12484e9i	0.746837i	−0.927663	+	0.373419i	$0.878186\pi$
$198$	0	0
$199$	1.04718e9i	0.667742i	0.942619	+	0.333871i	$0.108355\pi$
$199$	1.04718e9i	0.667742i	−0.942619	+	0.333871i	$0.891645\pi$
$200$	0	0
$201$	−1.06415e9	−0.651954
$202$	0	0
$203$	−1.40510e8	−0.0827413
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.39814e9	1.30615
$208$	0	0
$209$	2.08202e9	1.09119
$210$	0	0
$211$	2.77676e9i	1.40090i	0.713699	+	0.700452i	$0.247018\pi$
$211$	2.77676e9i	1.40090i	−0.713699	+	0.700452i	$0.752982\pi$
$212$	0	0
$213$	1.43156e9i	0.695491i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 3.00465e9i	− 1.35505i
$218$	0	0
$219$	9.25267e8i	0.402245i
$220$	0	0
$221$	−1.39575e9	−0.585113
$222$	0	0
$223$	2.27822e9	0.921248	0.460624	−	0.887595i	$-0.347626\pi$
$223$	2.27822e9	0.921248	0.460624	+	0.887595i	$0.347626\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.48863e9	0.560639	0.280319	−	0.959907i	$-0.409560\pi$
$227$	1.48863e9	0.560639	0.280319	+	0.959907i	$0.409560\pi$
$228$	0	0
$229$	−1.69447e9	−0.616157	−0.308079	−	0.951361i	$-0.599686\pi$
$229$	−1.69447e9	−0.616157	−0.308079	+	0.951361i	$0.599686\pi$
$230$	0	0
$231$	− 6.95572e9i	− 2.44284i
$232$	0	0
$233$	5.21423e8i	0.176916i	0.996080	+	0.0884580i	$0.0281939\pi$
$233$	5.21423e8i	0.176916i	−0.996080	+	0.0884580i	$0.971806\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 6.92738e9i	− 2.19571i
$238$	0	0
$239$	− 4.40690e9i	− 1.35065i	−0.737522	−	0.675323i	$-0.764004\pi$
$239$	− 4.40690e9i	− 1.35065i	0.737522	−	0.675323i	$-0.235996\pi$
$240$	0	0
$241$	−1.62148e9	−0.480666	−0.240333	−	0.970691i	$-0.577257\pi$
$241$	−1.62148e9	−0.480666	−0.240333	+	0.970691i	$0.577257\pi$
$242$	0	0
$243$	9.21023e8	0.264147
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.00984e9	0.808640
$248$	0	0
$249$	1.04286e10	2.71287
$250$	0	0
$251$	1.31321e8i	0.0330855i	0.999863	+	0.0165428i	$0.00526597\pi$
$251$	1.31321e8i	0.0330855i	−0.999863	+	0.0165428i	$0.994734\pi$
$252$	0	0
$253$	− 3.38023e9i	− 0.825019i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 5.27789e9i	− 1.20984i	−0.796287	−	0.604920i	$-0.793205\pi$
$257$	− 5.27789e9i	− 1.20984i	0.796287	−	0.604920i	$-0.206795\pi$
$258$	0	0
$259$	− 2.02857e9i	− 0.450808i
$260$	0	0
$261$	−7.47646e8	−0.161114
$262$	0	0
$263$	7.38745e9	1.54409	0.772044	−	0.635570i	$-0.219235\pi$
$263$	7.38745e9	1.54409	0.772044	+	0.635570i	$0.219235\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1.23238e10	−2.42493
$268$	0	0
$269$	−3.46450e8	−0.0661655	−0.0330828	−	0.999453i	$-0.510532\pi$
$269$	−3.46450e8	−0.0661655	−0.0330828	+	0.999453i	$0.510532\pi$
$270$	0	0
$271$	− 6.51715e6i	− 0.00120832i	−1.00000		0.000604158i	$-0.999808\pi$
$271$	− 6.51715e6i	− 0.00120832i	1.00000		0.000604158i	$-0.000192310\pi$
$272$	0	0
$273$	− 1.00554e10i	− 1.81029i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 2.15061e9i	− 0.365293i	−0.983179	−	0.182647i	$-0.941534\pi$
$277$	− 2.15061e9i	− 0.365293i	0.983179	−	0.182647i	$-0.0584664\pi$
$278$	0	0
$279$	− 1.59876e10i	− 2.63855i
$280$	0	0
$281$	1.04256e10	1.67215	0.836074	−	0.548616i	$-0.184845\pi$
$281$	1.04256e10	1.67215	0.836074	+	0.548616i	$0.184845\pi$
$282$	0	0
$283$	1.28042e9	0.199622	0.0998108	−	0.995006i	$-0.468176\pi$
$283$	1.28042e9	0.199622	0.0998108	+	0.995006i	$0.468176\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.93168e8	−0.0284714
$288$	0	0
$289$	4.43862e9	0.636292
$290$	0	0
$291$	6.62652e9i	0.924089i
$292$	0	0
$293$	− 2.13786e9i	− 0.290074i	−0.989426	−	0.145037i	$-0.953670\pi$
$293$	− 2.13786e9i	− 0.290074i	0.989426	−	0.145037i	$-0.0463301\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1.91546e10i	− 2.46177i
$298$	0	0
$299$	− 4.88656e9i	− 0.611390i
$300$	0	0
$301$	1.27697e9	0.155567
$302$	0	0
$303$	9.36980e9	1.11163
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−5.45140e9	−0.613698	−0.306849	−	0.951758i	$-0.599275\pi$
$307$	−5.45140e9	−0.613698	−0.306849	+	0.951758i	$0.599275\pi$
$308$	0	0
$309$	2.34112e10	2.56797
$310$	0	0
$311$	− 1.07550e10i	− 1.14965i	−0.818275	−	0.574827i	$-0.805069\pi$
$311$	− 1.07550e10i	− 1.14965i	0.818275	−	0.574827i	$-0.194931\pi$
$312$	0	0
$313$	− 2.99804e8i	− 0.0312364i	−0.999878	−	0.0156182i	$-0.995028\pi$
$313$	− 2.99804e8i	− 0.0312364i	0.999878	−	0.0156182i	$-0.00497163\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 5.31172e9i	− 0.526015i	−0.964794	−	0.263007i	$-0.915286\pi$
$317$	− 5.31172e9i	− 0.526015i	0.964794	−	0.263007i	$-0.0847144\pi$
$318$	0	0
$319$	1.05382e9i	0.101767i
$320$	0	0
$321$	−1.80785e10	−1.70272
$322$	0	0
$323$	5.47115e9	0.502653
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.21667e10	1.93869
$328$	0	0
$329$	−7.32872e9	−0.625525
$330$	0	0
$331$	1.01004e10i	0.841446i	0.907189	+	0.420723i	$0.138224\pi$
$331$	1.01004e10i	0.841446i	−0.907189	+	0.420723i	$0.861776\pi$
$332$	0	0
$333$	− 1.07939e10i	− 0.877815i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.84359e10i	1.42937i	0.699448	+	0.714684i	$0.253429\pi$
$337$	1.84359e10i	1.42937i	−0.699448	+	0.714684i	$0.746571\pi$
$338$	0	0
$339$	3.35578e10i	2.54094i
$340$	0	0
$341$	−2.25348e10	−1.66662
$342$	0	0
$343$	−1.27730e10	−0.922820
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.27822e10	−0.881636	−0.440818	−	0.897597i	$-0.645312\pi$
$347$	−1.27822e10	−0.881636	−0.440818	+	0.897597i	$0.645312\pi$
$348$	0	0
$349$	6.39381e8	0.0430981	0.0215490	−	0.999768i	$-0.493140\pi$
$349$	6.39381e8	0.0430981	0.0215490	+	0.999768i	$0.493140\pi$
$350$	0	0
$351$	− 2.76905e10i	− 1.82433i
$352$	0	0
$353$	2.59837e10i	1.67341i	0.547653	+	0.836705i	$0.315521\pi$
$353$	2.59837e10i	1.67341i	−0.547653	+	0.836705i	$0.684479\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1.82783e10i	− 1.12528i
$358$	0	0
$359$	− 2.22541e10i	− 1.33978i	−0.742461	−	0.669889i	$-0.766342\pi$
$359$	− 2.22541e10i	− 1.33978i	0.742461	−	0.669889i	$-0.233658\pi$
$360$	0	0
$361$	5.18543e9	0.305320
$362$	0	0
$363$	−2.17320e10	−1.25162
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.86229e10	1.02656	0.513280	−	0.858221i	$-0.328430\pi$
$367$	1.86229e10	1.02656	0.513280	+	0.858221i	$0.328430\pi$
$368$	0	0
$369$	−1.02784e9	−0.0554396
$370$	0	0
$371$	2.85380e10i	1.50636i
$372$	0	0
$373$	1.19680e10i	0.618283i	0.951016	+	0.309141i	$0.100042\pi$
$373$	1.19680e10i	0.618283i	−0.951016	+	0.309141i	$0.899958\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.52344e9i	0.0754154i
$378$	0	0
$379$	2.30787e10i	1.11855i	0.828982	+	0.559275i	$0.188920\pi$
$379$	2.30787e10i	1.11855i	−0.828982	+	0.559275i	$0.811080\pi$
$380$	0	0
$381$	5.22141e10	2.47792
$382$	0	0
$383$	−1.43419e10	−0.666518	−0.333259	−	0.942835i	$-0.608148\pi$
$383$	−1.43419e10	−0.666518	−0.333259	+	0.942835i	$0.608148\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.79472e9	0.302920
$388$	0	0
$389$	2.73457e10	1.19424	0.597119	−	0.802152i	$-0.296312\pi$
$389$	2.73457e10	1.19424	0.597119	+	0.802152i	$0.296312\pi$
$390$	0	0
$391$	− 8.88257e9i	− 0.380042i
$392$	0	0
$393$	− 3.07187e10i	− 1.28775i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 3.99456e10i	− 1.60808i	−0.594576	−	0.804039i	$-0.702680\pi$
$397$	− 3.99456e10i	− 1.60808i	0.594576	−	0.804039i	$-0.297320\pi$
$398$	0	0
$399$	3.94157e10i	1.55517i
$400$	0	0
$401$	−2.12767e10	−0.822863	−0.411431	−	0.911441i	$-0.634971\pi$
$401$	−2.12767e10	−0.822863	−0.411431	+	0.911441i	$0.634971\pi$
$402$	0	0
$403$	−3.25771e10	−1.23507
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.52143e10	−0.554465
$408$	0	0
$409$	−1.14283e10	−0.408404	−0.204202	−	0.978929i	$-0.565460\pi$
$409$	−1.14283e10	−0.408404	−0.204202	+	0.978929i	$0.565460\pi$
$410$	0	0
$411$	5.48693e10i	1.92292i
$412$	0	0
$413$	− 5.57982e10i	− 1.91788i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.98479e10i	1.64855i
$418$	0	0
$419$	1.10009e10i	0.356922i	0.983947	+	0.178461i	$0.0571119\pi$
$419$	1.10009e10i	0.356922i	−0.983947	+	0.178461i	$0.942888\pi$
$420$	0	0
$421$	2.28766e10	0.728220	0.364110	−	0.931356i	$-0.381373\pi$
$421$	2.28766e10	0.728220	0.364110	+	0.931356i	$0.381373\pi$
$422$	0	0
$423$	−3.89958e10	−1.21802
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.08948e10	−1.83176
$428$	0	0
$429$	−7.54155e10	−2.22655
$430$	0	0
$431$	9.55108e8i	0.0276786i	0.999904	+	0.0138393i	$0.00440532\pi$
$431$	9.55108e8i	0.0276786i	−0.999904	+	0.0138393i	$0.995595\pi$
$432$	0	0
$433$	3.82225e10i	1.08735i	0.839297	+	0.543673i	$0.182967\pi$
$433$	3.82225e10i	1.08735i	−0.839297	+	0.543673i	$0.817033\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.91546e10i	0.525228i
$438$	0	0
$439$	6.40288e10i	1.72392i	0.506976	+	0.861960i	$0.330763\pi$
$439$	6.40288e10i	1.72392i	−0.506976	+	0.861960i	$0.669237\pi$
$440$	0	0
$441$	1.04310e10	0.275785
$442$	0	0
$443$	7.47659e10	1.94128	0.970641	−	0.240533i	$-0.0773222\pi$
$443$	7.47659e10	1.94128	0.970641	+	0.240533i	$0.0773222\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.44756e10	1.61497
$448$	0	0
$449$	−2.51987e10	−0.620001	−0.310000	−	0.950736i	$-0.600329\pi$
$449$	−2.51987e10	−0.620001	−0.310000	+	0.950736i	$0.600329\pi$
$450$	0	0
$451$	1.44876e9i	0.0350180i
$452$	0	0
$453$	9.37373e10i	2.22597i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 4.66828e9i	− 0.107027i	−0.998567	−	0.0535133i	$-0.982958\pi$
$457$	− 4.66828e9i	− 0.107027i	0.998567	−	0.0535133i	$-0.0170420\pi$
$458$	0	0
$459$	− 5.03346e10i	− 1.13401i
$460$	0	0
$461$	−3.88096e10	−0.859281	−0.429641	−	0.903000i	$-0.641360\pi$
$461$	−3.88096e10	−0.859281	−0.429641	+	0.903000i	$0.641360\pi$
$462$	0	0
$463$	−3.23432e10	−0.703817	−0.351908	−	0.936034i	$-0.614467\pi$
$463$	−3.23432e10	−0.703817	−0.351908	+	0.936034i	$0.614467\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.31902e10	0.487570	0.243785	−	0.969829i	$-0.421611\pi$
$467$	2.31902e10	0.487570	0.243785	+	0.969829i	$0.421611\pi$
$468$	0	0
$469$	−1.91546e10	−0.395897
$470$	0	0
$471$	− 6.18090e10i	− 1.25594i
$472$	0	0
$473$	− 9.57731e9i	− 0.191337i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.51849e11i	2.93318i
$478$	0	0
$479$	− 4.83542e9i	− 0.0918528i	−0.998945	−	0.0459264i	$-0.985376\pi$
$479$	− 4.83542e9i	− 0.0918528i	0.998945	−	0.0459264i	$-0.0146240\pi$
$480$	0	0
$481$	−2.19943e10	−0.410893
$482$	0	0
$483$	6.39926e10	1.17582
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−3.03878e10	−0.540236	−0.270118	−	0.962827i	$-0.587063\pi$
$487$	−3.03878e10	−0.540236	−0.270118	+	0.962827i	$0.587063\pi$
$488$	0	0
$489$	3.46538e10	0.606059
$490$	0	0
$491$	5.56483e10i	0.957472i	0.877959	+	0.478736i	$0.158905\pi$
$491$	5.56483e10i	0.957472i	−0.877959	+	0.478736i	$0.841095\pi$
$492$	0	0
$493$	2.76924e9i	0.0468784i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.57681e10i	0.422335i
$498$	0	0
$499$	− 7.88458e10i	− 1.27168i	−0.771822	−	0.635838i	$-0.780655\pi$
$499$	− 7.88458e10i	− 1.27168i	0.771822	−	0.635838i	$-0.219345\pi$
$500$	0	0
$501$	−9.53224e10	−1.51302
$502$	0	0
$503$	4.41092e10	0.689061	0.344530	−	0.938775i	$-0.388038\pi$
$503$	4.41092e10	0.689061	0.344530	+	0.938775i	$0.388038\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.79923e9	0.102903
$508$	0	0
$509$	−1.05927e10	−0.157811	−0.0789055	−	0.996882i	$-0.525143\pi$
$509$	−1.05927e10	−0.157811	−0.0789055	+	0.996882i	$0.525143\pi$
$510$	0	0
$511$	1.66548e10i	0.244262i
$512$	0	0
$513$	1.08543e11i	1.56723i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.49654e10i	0.769356i
$518$	0	0
$519$	− 2.50979e11i	− 3.45914i
$520$	0	0
$521$	1.24958e11	1.69595	0.847973	−	0.530039i	$-0.177823\pi$
$521$	1.24958e11	1.69595	0.847973	+	0.530039i	$0.177823\pi$
$522$	0	0
$523$	2.80408e10	0.374786	0.187393	−	0.982285i	$-0.439996\pi$
$523$	2.80408e10	0.374786	0.187393	+	0.982285i	$0.439996\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.92172e10	−0.767724
$528$	0	0
$529$	−4.72129e10	−0.602890
$530$	0	0
$531$	− 2.96900e11i	− 3.73449i
$532$	0	0
$533$	2.09438e9i	0.0259505i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.07479e11i	1.29248i
$538$	0	0
$539$	− 1.47027e10i	− 0.174197i
$540$	0	0
$541$	1.44659e11	1.68871	0.844356	−	0.535782i	$-0.179983\pi$
$541$	1.44659e11	1.68871	0.844356	+	0.535782i	$0.179983\pi$
$542$	0	0
$543$	8.90386e10	1.02419
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−1.03774e10	−0.115915	−0.0579573	−	0.998319i	$-0.518459\pi$
$547$	−1.03774e10	−0.115915	−0.0579573	+	0.998319i	$0.518459\pi$
$548$	0	0
$549$	−3.24018e11	−3.56681
$550$	0	0
$551$	− 5.97167e9i	− 0.0647872i
$552$	0	0
$553$	− 1.24693e11i	− 1.33334i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.47312e9i	0.0568610i	0.999596	+	0.0284305i	$0.00905093\pi$
$557$	5.47312e9i	0.0568610i	−0.999596	+	0.0284305i	$0.990949\pi$
$558$	0	0
$559$	− 1.38453e10i	− 0.141793i
$560$	0	0
$561$	−1.37087e11	−1.38403
$562$	0	0
$563$	−4.36118e10	−0.434081	−0.217040	−	0.976163i	$-0.569640\pi$
$563$	−4.36118e10	−0.434081	−0.217040	+	0.976163i	$0.569640\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.34593e11	1.30224
$568$	0	0
$569$	−1.27822e10	−0.121943	−0.0609716	−	0.998140i	$-0.519420\pi$
$569$	−1.27822e10	−0.121943	−0.0609716	+	0.998140i	$0.519420\pi$
$570$	0	0
$571$	7.59455e10i	0.714427i	0.934023	+	0.357213i	$0.116273\pi$
$571$	7.59455e10i	0.714427i	−0.934023	+	0.357213i	$0.883727\pi$
$572$	0	0
$573$	− 1.53237e11i	− 1.42149i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 2.13827e10i	− 0.192912i	−0.995337	−	0.0964560i	$-0.969249\pi$
$577$	− 2.13827e10i	− 0.192912i	0.995337	−	0.0964560i	$-0.0307507\pi$
$578$	0	0
$579$	− 4.21353e9i	− 0.0374914i
$580$	0	0
$581$	1.87715e11	1.64738
$582$	0	0
$583$	2.14035e11	1.85272
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−6.07298e10	−0.511504	−0.255752	−	0.966742i	$-0.582323\pi$
$587$	−6.07298e10	−0.511504	−0.255752	+	0.966742i	$0.582323\pi$
$588$	0	0
$589$	1.27697e11	1.06101
$590$	0	0
$591$	1.59711e11i	1.30914i
$592$	0	0
$593$	1.15978e11i	0.937899i	0.883225	+	0.468949i	$0.155367\pi$
$593$	1.15978e11i	0.937899i	−0.883225	+	0.468949i	$0.844633\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.48685e11i	1.17049i
$598$	0	0
$599$	2.40647e11i	1.86927i	0.355607	+	0.934636i	$0.384274\pi$
$599$	2.40647e11i	1.86927i	−0.355607	+	0.934636i	$0.615726\pi$
$600$	0	0
$601$	−1.92942e11	−1.47887	−0.739434	−	0.673229i	$-0.764907\pi$
$601$	−1.92942e11	−1.47887	−0.739434	+	0.673229i	$0.764907\pi$
$602$	0	0
$603$	−1.01921e11	−0.770892
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.62042e11	−1.19364	−0.596819	−	0.802376i	$-0.703569\pi$
$607$	−1.62042e11	−1.19364	−0.596819	+	0.802376i	$0.703569\pi$
$608$	0	0
$609$	−1.99504e10	−0.145038
$610$	0	0
$611$	7.94597e10i	0.570141i
$612$	0	0
$613$	1.76424e11i	1.24944i	0.780847	+	0.624722i	$0.214788\pi$
$613$	1.76424e11i	1.24944i	−0.780847	+	0.624722i	$0.785212\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.84986e10i	0.679656i	0.940488	+	0.339828i	$0.110369\pi$
$617$	9.84986e10i	0.679656i	−0.940488	+	0.339828i	$0.889631\pi$
$618$	0	0
$619$	− 1.28596e10i	− 0.0875923i	−0.999040	−	0.0437961i	$-0.986055\pi$
$619$	− 1.28596e10i	− 0.0875923i	0.999040	−	0.0437961i	$-0.0139452\pi$
$620$	0	0
$621$	1.76223e11	1.18494
$622$	0	0
$623$	−2.21828e11	−1.47253
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.95618e11	1.91276
$628$	0	0
$629$	−3.99802e10	−0.255413
$630$	0	0
$631$	1.73463e11i	1.09418i	0.837073	+	0.547091i	$0.184265\pi$
$631$	1.73463e11i	1.09418i	−0.837073	+	0.547091i	$0.815735\pi$
$632$	0	0
$633$	3.94261e11i	2.45566i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 2.12547e10i	− 0.129091i
$638$	0	0
$639$	1.37111e11i	0.822372i
$640$	0	0
$641$	−1.13903e11	−0.674690	−0.337345	−	0.941381i	$-0.609529\pi$
$641$	−1.13903e11	−0.674690	−0.337345	+	0.941381i	$0.609529\pi$
$642$	0	0
$643$	−7.04067e10	−0.411879	−0.205940	−	0.978565i	$-0.566025\pi$
$643$	−7.04067e10	−0.411879	−0.205940	+	0.978565i	$0.566025\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.99175e11	−1.13663	−0.568314	−	0.822812i	$-0.692404\pi$
$647$	−1.99175e11	−1.13663	−0.568314	+	0.822812i	$0.692404\pi$
$648$	0	0
$649$	−4.18487e11	−2.35886
$650$	0	0
$651$	− 4.26617e11i	− 2.37528i
$652$	0	0
$653$	6.49972e9i	0.0357472i	0.999840	+	0.0178736i	$0.00568964\pi$
$653$	6.49972e9i	0.0357472i	−0.999840	+	0.0178736i	$0.994310\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8.86194e10i	0.475628i
$658$	0	0
$659$	2.20982e11i	1.17170i	0.810421	+	0.585848i	$0.199239\pi$
$659$	2.20982e11i	1.17170i	−0.810421	+	0.585848i	$0.800761\pi$
$660$	0	0
$661$	−2.69549e11	−1.41199	−0.705995	−	0.708217i	$-0.749500\pi$
$661$	−2.69549e11	−1.41199	−0.705995	+	0.708217i	$0.749500\pi$
$662$	0	0
$663$	−1.98177e11	−1.02565
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−9.69518e9	−0.0489838
$668$	0	0
$669$	3.23476e11	1.61487
$670$	0	0
$671$	4.56711e11i	2.25295i
$672$	0	0
$673$	9.44470e10i	0.460392i	0.973144	+	0.230196i	$0.0739368\pi$
$673$	9.44470e10i	0.460392i	−0.973144	+	0.230196i	$0.926063\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 8.02735e10i	− 0.382136i	−0.981577	−	0.191068i	$-0.938805\pi$
$677$	− 8.02735e10i	− 0.382136i	0.981577	−	0.191068i	$-0.0611950\pi$
$678$	0	0
$679$	1.19277e11i	0.561150i
$680$	0	0
$681$	2.11364e11	0.982751
$682$	0	0
$683$	−3.00783e11	−1.38220	−0.691099	−	0.722760i	$-0.742873\pi$
$683$	−3.00783e11	−1.38220	−0.691099	+	0.722760i	$0.742873\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2.40591e11	−1.08007
$688$	0	0
$689$	3.09416e11	1.37298
$690$	0	0
$691$	3.06208e11i	1.34309i	0.740964	+	0.671544i	$0.234369\pi$
$691$	3.06208e11i	1.34309i	−0.740964	+	0.671544i	$0.765631\pi$
$692$	0	0
$693$	− 6.66199e11i	− 2.88849i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.80707e9i	0.0161309i
$698$	0	0
$699$	7.40348e10i	0.310118i
$700$	0	0
$701$	−2.73603e11	−1.13305	−0.566524	−	0.824045i	$-0.691712\pi$
$701$	−2.73603e11	−1.13305	−0.566524	+	0.824045i	$0.691712\pi$
$702$	0	0
$703$	8.62143e10	0.352987
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.68656e11	0.675033
$708$	0	0
$709$	1.76662e11	0.699129	0.349564	−	0.936912i	$-0.386330\pi$
$709$	1.76662e11	0.699129	0.349564	+	0.936912i	$0.386330\pi$
$710$	0	0
$711$	− 6.63484e11i	− 2.59628i
$712$	0	0
$713$	− 2.07321e11i	− 0.802203i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 6.25718e11i	− 2.36756i
$718$	0	0
$719$	2.25510e11i	0.843821i	0.906637	+	0.421911i	$0.138640\pi$
$719$	2.25510e11i	0.843821i	−0.906637	+	0.421911i	$0.861360\pi$
$720$	0	0
$721$	4.21402e11	1.55939
$722$	0	0
$723$	−2.30227e11	−0.842565
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.87080e11	1.02770	0.513849	−	0.857881i	$-0.328219\pi$
$727$	2.87080e11	1.02770	0.513849	+	0.857881i	$0.328219\pi$
$728$	0	0
$729$	−2.14750e11	−0.760366
$730$	0	0
$731$	− 2.51673e10i	− 0.0881388i
$732$	0	0
$733$	− 2.94176e11i	− 1.01904i	−0.860459	−	0.509520i	$-0.829823\pi$
$733$	− 2.94176e11i	− 1.01904i	0.860459	−	0.509520i	$-0.170177\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.43660e11i	0.486928i
$738$	0	0
$739$	9.69888e10i	0.325195i	0.986692	+	0.162598i	$0.0519872\pi$
$739$	9.69888e10i	0.325195i	−0.986692	+	0.162598i	$0.948013\pi$
$740$	0	0
$741$	4.27355e11	1.41748
$742$	0	0
$743$	1.01567e11	0.333271	0.166635	−	0.986019i	$-0.446710\pi$
$743$	1.01567e11	0.333271	0.166635	+	0.986019i	$0.446710\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.98824e11	3.20779
$748$	0	0
$749$	−3.25413e11	−1.03397
$750$	0	0
$751$	4.17899e11i	1.31375i	0.754001	+	0.656873i	$0.228121\pi$
$751$	4.17899e11i	1.31375i	−0.754001	+	0.656873i	$0.771879\pi$
$752$	0	0
$753$	1.86457e10i	0.0579960i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 1.82006e11i	− 0.554244i	−0.960835	−	0.277122i	$-0.910619\pi$
$757$	− 1.82006e11i	− 0.554244i	0.960835	−	0.277122i	$-0.0893806\pi$
$758$	0	0
$759$	− 4.79945e11i	− 1.44619i
$760$	0	0
$761$	−4.27419e11	−1.27443	−0.637213	−	0.770687i	$-0.719913\pi$
$761$	−4.27419e11	−1.27443	−0.637213	+	0.770687i	$0.719913\pi$
$762$	0	0
$763$	3.99000e11	1.17727
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.04978e11	−1.74807
$768$	0	0
$769$	5.09969e11	1.45827	0.729136	−	0.684368i	$-0.239922\pi$
$769$	5.09969e11	1.45827	0.729136	+	0.684368i	$0.239922\pi$
$770$	0	0
$771$	− 7.49386e11i	− 2.12074i
$772$	0	0
$773$	− 1.49408e11i	− 0.418462i	−0.977866	−	0.209231i	$-0.932904\pi$
$773$	− 1.49408e11i	− 0.418462i	0.977866	−	0.209231i	$-0.0670961\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 2.88029e11i	− 0.790227i
$778$	0	0
$779$	− 8.20966e9i	− 0.0222933i
$780$	0	0
$781$	1.93261e11	0.519445
$782$	0	0
$783$	−5.49393e10	−0.146163
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−7.33252e11	−1.91141	−0.955706	−	0.294323i	$-0.904906\pi$
$787$	−7.33252e11	−1.91141	−0.955706	+	0.294323i	$0.904906\pi$
$788$	0	0
$789$	1.04891e12	2.70665
$790$	0	0
$791$	6.04040e11i	1.54298i
$792$	0	0
$793$	6.60236e11i	1.66958i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 3.02703e11i	− 0.750212i	−0.926982	−	0.375106i	$-0.877606\pi$
$797$	− 3.02703e11i	− 0.750212i	0.926982	−	0.375106i	$-0.122394\pi$
$798$	0	0
$799$	1.44438e11i	0.354401i
$800$	0	0
$801$	−1.18034e12	−2.86732
$802$	0	0
$803$	1.24911e11	0.300427
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−4.91911e10	−0.115982
$808$	0	0
$809$	5.84316e11	1.36412	0.682062	−	0.731295i	$-0.261084\pi$
$809$	5.84316e11	1.36412	0.682062	+	0.731295i	$0.261084\pi$
$810$	0	0
$811$	1.21470e11i	0.280793i	0.990095	+	0.140396i	$0.0448377\pi$
$811$	1.21470e11i	0.280793i	−0.990095	+	0.140396i	$0.955162\pi$
$812$	0	0
$813$	− 9.25344e8i	− 0.00211807i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	5.42714e10i	0.121810i
$818$	0	0
$819$	− 9.63078e11i	− 2.14055i
$820$	0	0
$821$	4.52470e11	0.995903	0.497952	−	0.867205i	$-0.334086\pi$
$821$	4.52470e11	0.995903	0.497952	+	0.867205i	$0.334086\pi$
$822$	0	0
$823$	3.06704e11	0.668528	0.334264	−	0.942479i	$-0.391512\pi$
$823$	3.06704e11	0.668528	0.334264	+	0.942479i	$0.391512\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.93276e11	−0.626982	−0.313491	−	0.949591i	$-0.601499\pi$
$827$	−2.93276e11	−0.626982	−0.313491	+	0.949591i	$0.601499\pi$
$828$	0	0
$829$	−3.35532e11	−0.710421	−0.355210	−	0.934786i	$-0.615591\pi$
$829$	−3.35532e11	−0.710421	−0.355210	+	0.934786i	$0.615591\pi$
$830$	0	0
$831$	− 3.05356e11i	− 0.640327i
$832$	0	0
$833$	− 3.86358e10i	− 0.0802434i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1.17482e12i	− 2.39369i
$838$	0	0
$839$	− 3.42844e11i	− 0.691908i	−0.938252	−	0.345954i	$-0.887555\pi$
$839$	− 3.42844e11i	− 0.691908i	0.938252	−	0.345954i	$-0.112445\pi$
$840$	0	0
$841$	−4.97224e11	−0.993958
$842$	0	0
$843$	1.48029e12	2.93113
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−3.91175e11	−0.760042
$848$	0	0
$849$	1.81802e11	0.349919
$850$	0	0
$851$	− 1.39972e11i	− 0.266883i
$852$	0	0
$853$	− 5.08662e11i	− 0.960801i	−0.877049	−	0.480400i	$-0.840491\pi$
$853$	− 5.08662e11i	− 0.960801i	0.877049	−	0.480400i	$-0.159509\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 6.06764e11i	− 1.12486i	−0.826846	−	0.562428i	$-0.809867\pi$
$857$	− 6.06764e11i	− 1.12486i	0.826846	−	0.562428i	$-0.190133\pi$
$858$	0	0
$859$	− 9.49431e11i	− 1.74378i	−0.489705	−	0.871888i	$-0.662895\pi$
$859$	− 9.49431e11i	− 1.74378i	0.489705	−	0.871888i	$-0.337105\pi$
$860$	0	0
$861$	−2.74272e10	−0.0499078
$862$	0	0
$863$	−2.99836e10	−0.0540556	−0.0270278	−	0.999635i	$-0.508604\pi$
$863$	−2.99836e10	−0.0540556	−0.0270278	+	0.999635i	$0.508604\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6.30222e11	1.11536
$868$	0	0
$869$	−9.35196e11	−1.63992
$870$	0	0
$871$	2.07679e11i	0.360844i
$872$	0	0
$873$	6.34669e11i	1.09267i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.80195e11i	1.48792i	0.668223	+	0.743961i	$0.267055\pi$
$877$	8.80195e11i	1.48792i	−0.668223	+	0.743961i	$0.732945\pi$
$878$	0	0
$879$	− 3.03546e11i	− 0.508474i
$880$	0	0
$881$	1.04085e12	1.72776	0.863879	−	0.503699i	$-0.168028\pi$
$881$	1.04085e12	1.72776	0.863879	+	0.503699i	$0.168028\pi$
$882$	0	0
$883$	−7.60446e11	−1.25091	−0.625454	−	0.780261i	$-0.715086\pi$
$883$	−7.60446e11	−1.25091	−0.625454	+	0.780261i	$0.715086\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3.34097e11	−0.539732	−0.269866	−	0.962898i	$-0.586979\pi$
$887$	−3.34097e11	−0.539732	−0.269866	+	0.962898i	$0.586979\pi$
$888$	0	0
$889$	9.39853e11	1.50471
$890$	0	0
$891$	− 1.00945e12i	− 1.60167i
$892$	0	0
$893$	− 3.11471e11i	− 0.489792i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 6.93823e11i	− 1.07171i
$898$	0	0
$899$	6.46345e10i	0.0989523i
$900$	0	0
$901$	5.62442e11	0.853451
$902$	0	0
$903$	1.81312e11	0.272695
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−7.65213e11	−1.13071	−0.565357	−	0.824846i	$-0.691262\pi$
$907$	−7.65213e11	−1.13071	−0.565357	+	0.824846i	$0.691262\pi$
$908$	0	0
$909$	8.97412e11	1.31443
$910$	0	0
$911$	3.83541e11i	0.556851i	0.960458	+	0.278425i	$0.0898125\pi$
$911$	3.83541e11i	0.556851i	−0.960458	+	0.278425i	$0.910188\pi$
$912$	0	0
$913$	− 1.40786e12i	− 2.02618i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 5.52937e11i	− 0.781984i
$918$	0	0
$919$	6.82775e11i	0.957229i	0.878025	+	0.478615i	$0.158861\pi$
$919$	6.82775e11i	0.957229i	−0.878025	+	0.478615i	$0.841139\pi$
$920$	0	0
$921$	−7.74022e11	−1.07576
$922$	0	0
$923$	2.79384e11	0.384941
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.24226e12	3.03645
$928$	0	0
$929$	−2.94973e11	−0.396021	−0.198011	−	0.980200i	$-0.563448\pi$
$929$	−2.94973e11	−0.396021	−0.198011	+	0.980200i	$0.563448\pi$
$930$	0	0
$931$	8.33152e10i	0.110898i
$932$	0	0
$933$	− 1.52705e12i	− 2.01524i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1.03941e12i	− 1.34843i	−0.738533	−	0.674217i	$-0.764481\pi$
$937$	− 1.03941e12i	− 1.34843i	0.738533	−	0.674217i	$-0.235519\pi$
$938$	0	0
$939$	− 4.25680e10i	− 0.0547546i
$940$	0	0
$941$	1.26891e11	0.161836	0.0809178	−	0.996721i	$-0.474215\pi$
$941$	1.26891e11	0.161836	0.0809178	+	0.996721i	$0.474215\pi$
$942$	0	0
$943$	−1.33286e10	−0.0168554
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.13064e11	0.762265	0.381133	−	0.924520i	$-0.375534\pi$
$947$	6.13064e11	0.762265	0.381133	+	0.924520i	$0.375534\pi$
$948$	0	0
$949$	1.80575e11	0.222635
$950$	0	0
$951$	− 7.54189e11i	− 0.922058i
$952$	0	0
$953$	6.58227e11i	0.798002i	0.916951	+	0.399001i	$0.130643\pi$
$953$	6.58227e11i	0.798002i	−0.916951	+	0.399001i	$0.869357\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.49628e11i	0.178388i
$958$	0	0
$959$	9.87647e11i	1.16769i
$960$	0	0
$961$	−5.29246e11	−0.620532
$962$	0	0
$963$	−1.73151e12	−2.01335
$964$	0	0
$965$	0	0
$966$	0	0
$967$	5.41485e11	0.619271	0.309635	−	0.950855i	$-0.399793\pi$
$967$	5.41485e11	0.619271	0.309635	+	0.950855i	$0.399793\pi$
$968$	0	0
$969$	7.76826e11	0.881107
$970$	0	0
$971$	3.54981e11i	0.399327i	0.979865	+	0.199663i	$0.0639849\pi$
$971$	3.54981e11i	0.399327i	−0.979865	+	0.199663i	$0.936015\pi$
$972$	0	0
$973$	8.97263e11i	1.00108i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 6.02238e11i	− 0.660982i	−0.943809	−	0.330491i	$-0.892786\pi$
$977$	− 6.02238e11i	− 0.660982i	0.943809	−	0.330491i	$-0.107214\pi$
$978$	0	0
$979$	1.66371e12i	1.81112i
$980$	0	0
$981$	2.12306e12	2.29238
$982$	0	0
$983$	1.37465e12	1.47223	0.736117	−	0.676854i	$-0.236657\pi$
$983$	1.37465e12	1.47223	0.736117	+	0.676854i	$0.236657\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.04058e12	−1.09649
$988$	0	0
$989$	8.81113e10	0.0920972
$990$	0	0
$991$	− 1.01081e12i	− 1.04803i	−0.851709	−	0.524015i	$-0.824433\pi$
$991$	− 1.01081e12i	− 1.04803i	0.851709	−	0.524015i	$-0.175567\pi$
$992$	0	0
$993$	1.43411e12i	1.47498i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3.28556e11i	0.332528i	0.986081	+	0.166264i	$0.0531704\pi$
$997$	3.28556e11i	0.332528i	−0.986081	+	0.166264i	$0.946830\pi$
$998$	0	0
$999$	− 7.93172e11i	− 0.796353i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.9.h.b.399.3		4
4.3	odd	2	inner	400.9.h.b.399.2		4
5.2	odd	4		16.9.c.a.15.1	✓	2
5.3	odd	4		400.9.b.c.351.2		2
5.4	even	2	inner	400.9.h.b.399.1		4
15.2	even	4		144.9.g.g.127.2		2
20.3	even	4		400.9.b.c.351.1		2
20.7	even	4		16.9.c.a.15.2	yes	2
20.19	odd	2	inner	400.9.h.b.399.4		4
40.27	even	4		64.9.c.d.63.1		2
40.37	odd	4		64.9.c.d.63.2		2
60.47	odd	4		144.9.g.g.127.1		2
80.27	even	4		256.9.d.f.127.3		4
80.37	odd	4		256.9.d.f.127.1		4
80.67	even	4		256.9.d.f.127.2		4
80.77	odd	4		256.9.d.f.127.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.9.c.a.15.1	✓	2	5.2	odd	4
16.9.c.a.15.2	yes	2	20.7	even	4
64.9.c.d.63.1		2	40.27	even	4
64.9.c.d.63.2		2	40.37	odd	4
144.9.g.g.127.1		2	60.47	odd	4
144.9.g.g.127.2		2	15.2	even	4
256.9.d.f.127.1		4	80.37	odd	4
256.9.d.f.127.2		4	80.67	even	4
256.9.d.f.127.3		4	80.27	even	4
256.9.d.f.127.4		4	80.77	odd	4
400.9.b.c.351.1		2	20.3	even	4
400.9.b.c.351.2		2	5.3	odd	4
400.9.h.b.399.1		4	5.4	even	2	inner
400.9.h.b.399.2		4	4.3	odd	2	inner
400.9.h.b.399.3		4	1.1	even	1	trivial
400.9.h.b.399.4		4	20.19	odd	2	inner

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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	400.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+141.986 q^{3} +2555.75 q^{7} +13599.0 q^{9} -19168.1i q^{11} -27710.0i q^{13} -50370.0i q^{17} +108619. i q^{19} +362880. q^{21} +176347. q^{23} +999297. q^{27} -54978.0 q^{29} -1.17564e6i q^{31} -2.72160e6i q^{33} -793730. i q^{37} -3.93443e6i q^{39} -75582.0 q^{41} +499648. q^{43} -2.86755e6 q^{47} +767039. q^{49} -7.15183e6i q^{51} +1.11662e7i q^{53} +1.54224e7i q^{57} -2.18325e7i q^{59} -2.38266e7 q^{61} +3.47556e7 q^{63} -7.49473e6 q^{67} +2.50387e7 q^{69} +1.00824e7i q^{71} +6.51661e6i q^{73} -4.89888e7i q^{77} -4.87892e7i q^{79} +5.26630e7 q^{81} +7.34483e7 q^{83} -7.80610e6 q^{87} -8.67958e7 q^{89} -7.08197e7i q^{91} -1.66925e8i q^{93} +4.66703e7i q^{97} -2.60667e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 54396 q^{9} + 1451520 q^{21} - 219912 q^{29} - 302328 q^{41} + 3068156 q^{49} - 95306488 q^{61} + 100154880 q^{69} + 210652164 q^{81} - 347183112 q^{89}+O(q^{100}) \) 4 * q + 54396 * q^9 + 1451520 * q^21 - 219912 * q^29 - 302328 * q^41 + 3068156 * q^49 - 95306488 * q^61 + 100154880 * q^69 + 210652164 * q^81 - 347183112 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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