LMFDB - Embedded newform 784.4.f.h.783.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,Mod(783,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("784.783");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	784.f (of order $2$, degree $1$, 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:	$46.2574974445$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.12258833328.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 29x^{4} - 20x^{3} + 808x^{2} - 672x + 576 $ x^6 - x^5 + 29x^4 - 20x^3 + 808x^2 - 672x + 576
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3 $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			783.4
Root			$2.68858 + 4.65676i$ of defining polynomial
Character	$\chi$	$=$	784.783
Dual form			784.4.f.h.783.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.76834 q^{3} +16.8950i q^{5} -4.26295 q^{9} +O(q^{10})$	$q+4.76834 q^{3} +16.8950i q^{5} -4.26295 q^{9} -40.7424i q^{11} -56.7321i q^{13} +80.5609i q^{15} -122.834i q^{17} -75.4946 q^{19} +106.165i q^{23} -160.440 q^{25} -149.072 q^{27} -146.703 q^{29} -42.5912 q^{31} -194.274i q^{33} +80.9142 q^{37} -270.518i q^{39} +53.8052i q^{41} -341.166i q^{43} -72.0225i q^{45} -4.12785 q^{47} -585.714i q^{51} +279.697 q^{53} +688.342 q^{55} -359.984 q^{57} -174.831 q^{59} -467.087i q^{61} +958.488 q^{65} +753.251i q^{67} +506.229i q^{69} -669.583i q^{71} -835.774i q^{73} -765.033 q^{75} -1106.21i q^{79} -595.728 q^{81} +552.905 q^{83} +2075.28 q^{85} -699.530 q^{87} +122.469i q^{89} -203.089 q^{93} -1275.48i q^{95} +291.203i q^{97} +173.683i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 14 q^{3} + 156 q^{9}+O(q^{10}) $ 6 * q + 14 * q^3 + 156 * q^9	$ 6 q + 14 q^{3} + 156 q^{9} - 286 q^{19} - 612 q^{25} + 362 q^{27} - 348 q^{29} - 410 q^{31} + 498 q^{37} - 150 q^{47} + 1290 q^{53} + 918 q^{55} - 6 q^{57} - 642 q^{59} + 1224 q^{65} - 8276 q^{75} - 450 q^{81} + 24 q^{83} + 3786 q^{85} - 7284 q^{87} + 5982 q^{93}+O(q^{100}) $ 6 * q + 14 * q^3 + 156 * q^9 - 286 * q^19 - 612 * q^25 + 362 * q^27 - 348 * q^29 - 410 * q^31 + 498 * q^37 - 150 * q^47 + 1290 * q^53 + 918 * q^55 - 6 * q^57 - 642 * q^59 + 1224 * q^65 - 8276 * q^75 - 450 * q^81 + 24 * q^83 + 3786 * q^85 - 7284 * q^87 + 5982 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	4.76834	0.917667	0.458834	−	0.888522i	$-0.348267\pi$
$3$	4.76834	0.917667	0.458834	+	0.888522i	$0.348267\pi$
$4$	0	0
$5$	16.8950i	1.51113i	0.655072	+	0.755566i	$0.272638\pi$
$5$	16.8950i	1.51113i	−0.655072	+	0.755566i	$0.727362\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−4.26295	−0.157887
$10$	0	0
$11$	− 40.7424i	− 1.11676i	−0.829587	−	0.558378i	$-0.811424\pi$
$11$	− 40.7424i	− 1.11676i	0.829587	−	0.558378i	$-0.188576\pi$
$12$	0	0
$13$	− 56.7321i	− 1.21036i	−0.796089	−	0.605179i	$-0.793101\pi$
$13$	− 56.7321i	− 1.21036i	0.796089	−	0.605179i	$-0.206899\pi$
$14$	0	0
$15$	80.5609i	1.38672i
$16$	0	0
$17$	− 122.834i	− 1.75245i	−0.481903	−	0.876225i	$-0.660054\pi$
$17$	− 122.834i	− 1.75245i	0.481903	−	0.876225i	$-0.339946\pi$
$18$	0	0
$19$	−75.4946	−0.911561	−0.455780	−	0.890092i	$-0.650640\pi$
$19$	−75.4946	−0.911561	−0.455780	+	0.890092i	$0.650640\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	106.165i	0.962473i	0.876591	+	0.481236i	$0.159812\pi$
$23$	106.165i	0.962473i	−0.876591	+	0.481236i	$0.840188\pi$
$24$	0	0
$25$	−160.440	−1.28352
$26$	0	0
$27$	−149.072	−1.06255
$28$	0	0
$29$	−146.703	−0.939382	−0.469691	−	0.882831i	$-0.655635\pi$
$29$	−146.703	−0.939382	−0.469691	+	0.882831i	$0.655635\pi$
$30$	0	0
$31$	−42.5912	−0.246761	−0.123381	−	0.992359i	$-0.539374\pi$
$31$	−42.5912	−0.246761	−0.123381	+	0.992359i	$0.539374\pi$
$32$	0	0
$33$	− 194.274i	− 1.02481i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	80.9142	0.359519	0.179760	−	0.983711i	$-0.442468\pi$
$37$	80.9142	0.359519	0.179760	+	0.983711i	$0.442468\pi$
$38$	0	0
$39$	− 270.518i	− 1.11071i
$40$	0	0
$41$	53.8052i	0.204950i	0.994736	+	0.102475i	$0.0326762\pi$
$41$	53.8052i	0.204950i	−0.994736	+	0.102475i	$0.967324\pi$
$42$	0	0
$43$	− 341.166i	− 1.20994i	−0.796249	−	0.604970i	$-0.793185\pi$
$43$	− 341.166i	− 1.20994i	0.796249	−	0.604970i	$-0.206815\pi$
$44$	0	0
$45$	− 72.0225i	− 0.238588i
$46$	0	0
$47$	−4.12785	−0.0128108	−0.00640542	−	0.999979i	$-0.502039\pi$
$47$	−4.12785	−0.0128108	−0.00640542	+	0.999979i	$0.502039\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 585.714i	− 1.60817i
$52$	0	0
$53$	279.697	0.724894	0.362447	−	0.932004i	$-0.381941\pi$
$53$	279.697	0.724894	0.362447	+	0.932004i	$0.381941\pi$
$54$	0	0
$55$	688.342	1.68756
$56$	0	0
$57$	−359.984	−0.836509
$58$	0	0
$59$	−174.831	−0.385780	−0.192890	−	0.981220i	$-0.561786\pi$
$59$	−174.831	−0.385780	−0.192890	+	0.981220i	$0.561786\pi$
$60$	0	0
$61$	− 467.087i	− 0.980399i	−0.871610	−	0.490199i	$-0.836924\pi$
$61$	− 467.087i	− 0.980399i	0.871610	−	0.490199i	$-0.163076\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	958.488	1.82901
$66$	0	0
$67$	753.251i	1.37350i	0.726895	+	0.686748i	$0.240963\pi$
$67$	753.251i	1.37350i	−0.726895	+	0.686748i	$0.759037\pi$
$68$	0	0
$69$	506.229i	0.883230i
$70$	0	0
$71$	− 669.583i	− 1.11922i	−0.828755	−	0.559611i	$-0.810950\pi$
$71$	− 669.583i	− 1.11922i	0.828755	−	0.559611i	$-0.189050\pi$
$72$	0	0
$73$	− 835.774i	− 1.34000i	−0.742361	−	0.670000i	$-0.766294\pi$
$73$	− 835.774i	− 1.34000i	0.742361	−	0.670000i	$-0.233706\pi$
$74$	0	0
$75$	−765.033	−1.17784
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 1106.21i	− 1.57543i	−0.616042	−	0.787713i	$-0.711265\pi$
$79$	− 1106.21i	− 1.57543i	0.616042	−	0.787713i	$-0.288735\pi$
$80$	0	0
$81$	−595.728	−0.817185
$82$	0	0
$83$	552.905	0.731195	0.365597	−	0.930773i	$-0.380865\pi$
$83$	552.905	0.731195	0.365597	+	0.930773i	$0.380865\pi$
$84$	0	0
$85$	2075.28	2.64818
$86$	0	0
$87$	−699.530	−0.862040
$88$	0	0
$89$	122.469i	0.145862i	0.997337	+	0.0729310i	$0.0232353\pi$
$89$	122.469i	0.145862i	−0.997337	+	0.0729310i	$0.976765\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−203.089	−0.226445
$94$	0	0
$95$	− 1275.48i	− 1.37749i
$96$	0	0
$97$	291.203i	0.304816i	0.988318	+	0.152408i	$0.0487028\pi$
$97$	291.203i	0.304816i	−0.988318	+	0.152408i	$0.951297\pi$
$98$	0	0
$99$	173.683i	0.176321i
$100$	0	0
$101$	1349.04i	1.32905i	0.747264	+	0.664527i	$0.231367\pi$
$101$	1349.04i	1.32905i	−0.747264	+	0.664527i	$0.768633\pi$
$102$	0	0
$103$	−1978.07	−1.89228	−0.946141	−	0.323756i	$-0.895054\pi$
$103$	−1978.07	−1.89228	−0.946141	+	0.323756i	$0.895054\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 846.316i	− 0.764640i	−0.924030	−	0.382320i	$-0.875125\pi$
$107$	− 846.316i	− 0.764640i	0.924030	−	0.382320i	$-0.124875\pi$
$108$	0	0
$109$	927.077	0.814659	0.407330	−	0.913281i	$-0.366460\pi$
$109$	927.077	0.814659	0.407330	+	0.913281i	$0.366460\pi$
$110$	0	0
$111$	385.826	0.329919
$112$	0	0
$113$	−599.053	−0.498709	−0.249355	−	0.968412i	$-0.580218\pi$
$113$	−599.053	−0.498709	−0.249355	+	0.968412i	$0.580218\pi$
$114$	0	0
$115$	−1793.65	−1.45442
$116$	0	0
$117$	241.846i	0.191100i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−328.946	−0.247142
$122$	0	0
$123$	256.561i	0.188076i
$124$	0	0
$125$	− 598.760i	− 0.428438i
$126$	0	0
$127$	1833.90i	1.28136i	0.767810	+	0.640678i	$0.221347\pi$
$127$	1833.90i	1.28136i	−0.767810	+	0.640678i	$0.778653\pi$
$128$	0	0
$129$	− 1626.80i	− 1.11032i
$130$	0	0
$131$	−1379.15	−0.919822	−0.459911	−	0.887965i	$-0.652119\pi$
$131$	−1379.15	−0.919822	−0.459911	+	0.887965i	$0.652119\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 2518.57i	− 1.60566i
$136$	0	0
$137$	1473.43	0.918857	0.459429	−	0.888215i	$-0.348054\pi$
$137$	1473.43	0.918857	0.459429	+	0.888215i	$0.348054\pi$
$138$	0	0
$139$	−1929.87	−1.17762	−0.588811	−	0.808271i	$-0.700404\pi$
$139$	−1929.87	−1.17762	−0.588811	+	0.808271i	$0.700404\pi$
$140$	0	0
$141$	−19.6830	−0.0117561
$142$	0	0
$143$	−2311.41	−1.35167
$144$	0	0
$145$	− 2478.54i	− 1.41953i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2306.84	1.26834	0.634172	−	0.773192i	$-0.281341\pi$
$149$	2306.84	1.26834	0.634172	+	0.773192i	$0.281341\pi$
$150$	0	0
$151$	1025.27i	0.552549i	0.961079	+	0.276275i	$0.0891000\pi$
$151$	1025.27i	0.552549i	−0.961079	+	0.276275i	$0.910900\pi$
$152$	0	0
$153$	523.636i	0.276689i
$154$	0	0
$155$	− 719.577i	− 0.372889i
$156$	0	0
$157$	− 77.0147i	− 0.0391493i	−0.999808	−	0.0195746i	$-0.993769\pi$
$157$	− 77.0147i	− 0.0391493i	0.999808	−	0.0195746i	$-0.00623120\pi$
$158$	0	0
$159$	1333.69	0.665212
$160$	0	0
$161$	0	0
$162$	0	0
$163$	3887.54i	1.86807i	0.357180	+	0.934036i	$0.383738\pi$
$163$	3887.54i	1.86807i	−0.357180	+	0.934036i	$0.616262\pi$
$164$	0	0
$165$	3282.25	1.54862
$166$	0	0
$167$	117.863	0.0546141	0.0273070	−	0.999627i	$-0.491307\pi$
$167$	117.863	0.0546141	0.0273070	+	0.999627i	$0.491307\pi$
$168$	0	0
$169$	−1021.54	−0.464969
$170$	0	0
$171$	321.830	0.143924
$172$	0	0
$173$	− 2864.21i	− 1.25874i	−0.777105	−	0.629370i	$-0.783313\pi$
$173$	− 2864.21i	− 1.25874i	0.777105	−	0.629370i	$-0.216687\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−833.653	−0.354018
$178$	0	0
$179$	− 1380.94i	− 0.576628i	−0.957536	−	0.288314i	$-0.906905\pi$
$179$	− 1380.94i	− 0.576628i	0.957536	−	0.288314i	$-0.0930947\pi$
$180$	0	0
$181$	204.817i	0.0841103i	0.999115	+	0.0420551i	$0.0133905\pi$
$181$	204.817i	0.0841103i	−0.999115	+	0.0420551i	$0.986609\pi$
$182$	0	0
$183$	− 2227.23i	− 0.899680i
$184$	0	0
$185$	1367.04i	0.543281i
$186$	0	0
$187$	−5004.56	−1.95706
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 4167.75i	− 1.57889i	−0.613821	−	0.789445i	$-0.710369\pi$
$191$	− 4167.75i	− 1.57889i	0.613821	−	0.789445i	$-0.289631\pi$
$192$	0	0
$193$	−4785.93	−1.78497	−0.892485	−	0.451078i	$-0.851040\pi$
$193$	−4785.93	−1.78497	−0.892485	+	0.451078i	$0.851040\pi$
$194$	0	0
$195$	4570.39	1.67842
$196$	0	0
$197$	−377.368	−0.136479	−0.0682395	−	0.997669i	$-0.521738\pi$
$197$	−377.368	−0.136479	−0.0682395	+	0.997669i	$0.521738\pi$
$198$	0	0
$199$	1619.53	0.576911	0.288455	−	0.957493i	$-0.406858\pi$
$199$	1619.53	0.576911	0.288455	+	0.957493i	$0.406858\pi$
$200$	0	0
$201$	3591.76i	1.26041i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−909.037	−0.309707
$206$	0	0
$207$	− 452.575i	− 0.151962i
$208$	0	0
$209$	3075.83i	1.01799i
$210$	0	0
$211$	− 3010.54i	− 0.982247i	−0.871090	−	0.491123i	$-0.836586\pi$
$211$	− 3010.54i	− 0.982247i	0.871090	−	0.491123i	$-0.163414\pi$
$212$	0	0
$213$	− 3192.80i	− 1.02707i
$214$	0	0
$215$	5764.00	1.82838
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 3985.26i	− 1.22967i
$220$	0	0
$221$	−6968.64	−2.12109
$222$	0	0
$223$	−879.692	−0.264164	−0.132082	−	0.991239i	$-0.542166\pi$
$223$	−879.692	−0.264164	−0.132082	+	0.991239i	$0.542166\pi$
$224$	0	0
$225$	683.948	0.202651
$226$	0	0
$227$	−1424.63	−0.416547	−0.208273	−	0.978071i	$-0.566784\pi$
$227$	−1424.63	−0.416547	−0.208273	+	0.978071i	$0.566784\pi$
$228$	0	0
$229$	238.730i	0.0688896i	0.999407	+	0.0344448i	$0.0109663\pi$
$229$	238.730i	0.0688896i	−0.999407	+	0.0344448i	$0.989034\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5841.72	1.64251	0.821253	−	0.570565i	$-0.193276\pi$
$233$	5841.72	1.64251	0.821253	+	0.570565i	$0.193276\pi$
$234$	0	0
$235$	− 69.7399i	− 0.0193589i
$236$	0	0
$237$	− 5274.80i	− 1.44572i
$238$	0	0
$239$	2101.91i	0.568876i	0.958694	+	0.284438i	$0.0918070\pi$
$239$	2101.91i	0.568876i	−0.958694	+	0.284438i	$0.908193\pi$
$240$	0	0
$241$	− 2175.18i	− 0.581393i	−0.956815	−	0.290697i	$-0.906113\pi$
$241$	− 2175.18i	− 0.581393i	0.956815	−	0.290697i	$-0.0938871\pi$
$242$	0	0
$243$	1184.32	0.312652
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4282.97i	1.10332i
$248$	0	0
$249$	2636.44	0.670993
$250$	0	0
$251$	2069.98	0.520542	0.260271	−	0.965536i	$-0.416188\pi$
$251$	2069.98	0.520542	0.260271	+	0.965536i	$0.416188\pi$
$252$	0	0
$253$	4325.41	1.07485
$254$	0	0
$255$	9895.63	2.43015
$256$	0	0
$257$	5194.72i	1.26085i	0.776252	+	0.630423i	$0.217119\pi$
$257$	5194.72i	1.26085i	−0.776252	+	0.630423i	$0.782881\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	625.388	0.148316
$262$	0	0
$263$	− 1824.70i	− 0.427818i	−0.976854	−	0.213909i	$-0.931380\pi$
$263$	− 1824.70i	− 0.427818i	0.976854	−	0.213909i	$-0.0686196\pi$
$264$	0	0
$265$	4725.48i	1.09541i
$266$	0	0
$267$	583.975i	0.133853i
$268$	0	0
$269$	− 2497.04i	− 0.565974i	−0.959124	−	0.282987i	$-0.908675\pi$
$269$	− 2497.04i	− 0.565974i	0.959124	−	0.282987i	$-0.0913253\pi$
$270$	0	0
$271$	−1922.58	−0.430954	−0.215477	−	0.976509i	$-0.569131\pi$
$271$	−1922.58	−0.430954	−0.215477	+	0.976509i	$0.569131\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6536.72i	1.43338i
$276$	0	0
$277$	−1885.43	−0.408970	−0.204485	−	0.978870i	$-0.565552\pi$
$277$	−1885.43	−0.408970	−0.204485	+	0.978870i	$0.565552\pi$
$278$	0	0
$279$	181.564	0.0389604
$280$	0	0
$281$	−2666.13	−0.566006	−0.283003	−	0.959119i	$-0.591331\pi$
$281$	−2666.13	−0.566006	−0.283003	+	0.959119i	$0.591331\pi$
$282$	0	0
$283$	3747.87	0.787236	0.393618	−	0.919274i	$-0.371223\pi$
$283$	3747.87	0.787236	0.393618	+	0.919274i	$0.371223\pi$
$284$	0	0
$285$	− 6081.92i	− 1.26408i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−10175.2	−2.07108
$290$	0	0
$291$	1388.55i	0.279720i
$292$	0	0
$293$	− 6486.41i	− 1.29331i	−0.762782	−	0.646655i	$-0.776167\pi$
$293$	− 6486.41i	− 1.29331i	0.762782	−	0.646655i	$-0.223833\pi$
$294$	0	0
$295$	− 2953.76i	− 0.582965i
$296$	0	0
$297$	6073.57i	1.18661i
$298$	0	0
$299$	6022.95	1.16494
$300$	0	0
$301$	0	0
$302$	0	0
$303$	6432.68i	1.21963i
$304$	0	0
$305$	7891.42	1.48151
$306$	0	0
$307$	−8884.08	−1.65160	−0.825800	−	0.563963i	$-0.809276\pi$
$307$	−8884.08	−1.65160	−0.825800	+	0.563963i	$0.809276\pi$
$308$	0	0
$309$	−9432.11	−1.73648
$310$	0	0
$311$	−7001.77	−1.27664	−0.638318	−	0.769773i	$-0.720370\pi$
$311$	−7001.77	−1.27664	−0.638318	+	0.769773i	$0.720370\pi$
$312$	0	0
$313$	4781.08i	0.863395i	0.902018	+	0.431697i	$0.142085\pi$
$313$	4781.08i	0.863395i	−0.902018	+	0.431697i	$0.857915\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3603.43	−0.638451	−0.319226	−	0.947679i	$-0.603423\pi$
$317$	−3603.43	−0.638451	−0.319226	+	0.947679i	$0.603423\pi$
$318$	0	0
$319$	5977.04i	1.04906i
$320$	0	0
$321$	− 4035.52i	− 0.701685i
$322$	0	0
$323$	9273.31i	1.59746i
$324$	0	0
$325$	9102.11i	1.55352i
$326$	0	0
$327$	4420.62	0.747586
$328$	0	0
$329$	0	0
$330$	0	0
$331$	3304.08i	0.548667i	0.961635	+	0.274333i	$0.0884571\pi$
$331$	3304.08i	0.548667i	−0.961635	+	0.274333i	$0.911543\pi$
$332$	0	0
$333$	−344.933	−0.0567635
$334$	0	0
$335$	−12726.2	−2.07554
$336$	0	0
$337$	−6650.35	−1.07498	−0.537489	−	0.843271i	$-0.680627\pi$
$337$	−6650.35	−1.07498	−0.537489	+	0.843271i	$0.680627\pi$
$338$	0	0
$339$	−2856.49	−0.457649
$340$	0	0
$341$	1735.27i	0.275572i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−8552.73	−1.33468
$346$	0	0
$347$	235.365i	0.0364123i	0.999834	+	0.0182061i	$0.00579551\pi$
$347$	235.365i	0.0364123i	−0.999834	+	0.0182061i	$0.994204\pi$
$348$	0	0
$349$	2984.02i	0.457681i	0.973464	+	0.228841i	$0.0734935\pi$
$349$	2984.02i	0.457681i	−0.973464	+	0.228841i	$0.926507\pi$
$350$	0	0
$351$	8457.19i	1.28607i
$352$	0	0
$353$	2446.28i	0.368845i	0.982847	+	0.184422i	$0.0590414\pi$
$353$	2446.28i	0.368845i	−0.982847	+	0.184422i	$0.940959\pi$
$354$	0	0
$355$	11312.6	1.69129
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5694.08i	0.837108i	0.908192	+	0.418554i	$0.137463\pi$
$359$	5694.08i	0.837108i	−0.908192	+	0.418554i	$0.862537\pi$
$360$	0	0
$361$	−1159.56	−0.169057
$362$	0	0
$363$	−1568.53	−0.226794
$364$	0	0
$365$	14120.4	2.02492
$366$	0	0
$367$	336.321	0.0478361	0.0239180	−	0.999714i	$-0.492386\pi$
$367$	336.321	0.0478361	0.0239180	+	0.999714i	$0.492386\pi$
$368$	0	0
$369$	− 229.369i	− 0.0323590i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	9816.84	1.36273	0.681363	−	0.731945i	$-0.261387\pi$
$373$	9816.84	1.36273	0.681363	+	0.731945i	$0.261387\pi$
$374$	0	0
$375$	− 2855.09i	− 0.393163i
$376$	0	0
$377$	8322.78i	1.13699i
$378$	0	0
$379$	3978.71i	0.539242i	0.962967	+	0.269621i	$0.0868984\pi$
$379$	3978.71i	0.539242i	−0.962967	+	0.269621i	$0.913102\pi$
$380$	0	0
$381$	8744.65i	1.17586i
$382$	0	0
$383$	−3725.99	−0.497100	−0.248550	−	0.968619i	$-0.579954\pi$
$383$	−3725.99	−0.497100	−0.248550	+	0.968619i	$0.579954\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1454.38i	0.191034i
$388$	0	0
$389$	12693.8	1.65450	0.827250	−	0.561834i	$-0.189904\pi$
$389$	12693.8	1.65450	0.827250	+	0.561834i	$0.189904\pi$
$390$	0	0
$391$	13040.6	1.68669
$392$	0	0
$393$	−6576.24	−0.844091
$394$	0	0
$395$	18689.4	2.38068
$396$	0	0
$397$	2072.26i	0.261974i	0.991384	+	0.130987i	$0.0418147\pi$
$397$	2072.26i	0.261974i	−0.991384	+	0.130987i	$0.958185\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9857.39	−1.22757	−0.613784	−	0.789474i	$-0.710353\pi$
$401$	−9857.39	−1.22757	−0.613784	+	0.789474i	$0.710353\pi$
$402$	0	0
$403$	2416.29i	0.298670i
$404$	0	0
$405$	− 10064.8i	− 1.23487i
$406$	0	0
$407$	− 3296.64i	− 0.401495i
$408$	0	0
$409$	− 1374.47i	− 0.166169i	−0.996542	−	0.0830847i	$-0.973523\pi$
$409$	− 1374.47i	− 0.166169i	0.996542	−	0.0830847i	$-0.0264772\pi$
$410$	0	0
$411$	7025.80	0.843205
$412$	0	0
$413$	0	0
$414$	0	0
$415$	9341.31i	1.10493i
$416$	0	0
$417$	−9202.27	−1.08066
$418$	0	0
$419$	−11066.7	−1.29032	−0.645158	−	0.764049i	$-0.723209\pi$
$419$	−11066.7	−1.29032	−0.645158	+	0.764049i	$0.723209\pi$
$420$	0	0
$421$	15288.1	1.76982	0.884910	−	0.465761i	$-0.154219\pi$
$421$	15288.1	1.76982	0.884910	+	0.465761i	$0.154219\pi$
$422$	0	0
$423$	17.5968	0.00202266
$424$	0	0
$425$	19707.5i	2.24931i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−11021.6	−1.24039
$430$	0	0
$431$	− 2615.48i	− 0.292305i	−0.989262	−	0.146152i	$-0.953311\pi$
$431$	− 2615.48i	− 0.292305i	0.989262	−	0.146152i	$-0.0466890\pi$
$432$	0	0
$433$	− 3152.43i	− 0.349875i	−0.984580	−	0.174938i	$-0.944028\pi$
$433$	− 3152.43i	− 0.349875i	0.984580	−	0.174938i	$-0.0559723\pi$
$434$	0	0
$435$	− 11818.5i	− 1.30266i
$436$	0	0
$437$	− 8014.87i	− 0.877352i
$438$	0	0
$439$	−6940.47	−0.754557	−0.377278	−	0.926100i	$-0.623140\pi$
$439$	−6940.47	−0.754557	−0.377278	+	0.926100i	$0.623140\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 12533.9i	− 1.34425i	−0.740439	−	0.672123i	$-0.765382\pi$
$443$	− 12533.9i	− 1.34425i	0.740439	−	0.672123i	$-0.234618\pi$
$444$	0	0
$445$	−2069.12	−0.220417
$446$	0	0
$447$	10999.8	1.16392
$448$	0	0
$449$	8637.25	0.907833	0.453917	−	0.891044i	$-0.350026\pi$
$449$	8637.25	0.907833	0.453917	+	0.891044i	$0.350026\pi$
$450$	0	0
$451$	2192.15	0.228879
$452$	0	0
$453$	4888.82i	0.507056i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	16496.7	1.68858	0.844291	−	0.535885i	$-0.180022\pi$
$457$	16496.7	1.68858	0.844291	+	0.535885i	$0.180022\pi$
$458$	0	0
$459$	18311.2i	1.86207i
$460$	0	0
$461$	− 6423.89i	− 0.649003i	−0.945885	−	0.324501i	$-0.894803\pi$
$461$	− 6423.89i	− 0.649003i	0.945885	−	0.324501i	$-0.105197\pi$
$462$	0	0
$463$	3390.22i	0.340295i	0.985419	+	0.170148i	$0.0544245\pi$
$463$	3390.22i	0.340295i	−0.985419	+	0.170148i	$0.945576\pi$
$464$	0	0
$465$	− 3431.19i	− 0.342188i
$466$	0	0
$467$	−1160.52	−0.114995	−0.0574975	−	0.998346i	$-0.518312\pi$
$467$	−1160.52	−0.114995	−0.0574975	+	0.998346i	$0.518312\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 367.232i	− 0.0359260i
$472$	0	0
$473$	−13899.9	−1.35121
$474$	0	0
$475$	12112.4	1.17001
$476$	0	0
$477$	−1192.34	−0.114451
$478$	0	0
$479$	−4084.93	−0.389656	−0.194828	−	0.980837i	$-0.562415\pi$
$479$	−4084.93	−0.389656	−0.194828	+	0.980837i	$0.562415\pi$
$480$	0	0
$481$	− 4590.44i	− 0.435147i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4919.86	−0.460617
$486$	0	0
$487$	7320.89i	0.681193i	0.940210	+	0.340597i	$0.110629\pi$
$487$	7320.89i	0.681193i	−0.940210	+	0.340597i	$0.889371\pi$
$488$	0	0
$489$	18537.1i	1.71427i
$490$	0	0
$491$	− 32.0507i	− 0.00294588i	−0.999999	−	0.00147294i	$-0.999531\pi$
$491$	− 32.0507i	− 0.00294588i	0.999999	−	0.00147294i	$-0.000468851\pi$
$492$	0	0
$493$	18020.1i	1.64622i
$494$	0	0
$495$	−2934.37	−0.266445
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 7521.66i	− 0.674781i	−0.941365	−	0.337391i	$-0.890456\pi$
$499$	− 7521.66i	− 0.674781i	0.941365	−	0.337391i	$-0.109544\pi$
$500$	0	0
$501$	562.013	0.0501175
$502$	0	0
$503$	−4599.64	−0.407729	−0.203865	−	0.978999i	$-0.565350\pi$
$503$	−4599.64	−0.407729	−0.203865	+	0.978999i	$0.565350\pi$
$504$	0	0
$505$	−22792.0	−2.00838
$506$	0	0
$507$	−4871.03	−0.426686
$508$	0	0
$509$	− 20791.8i	− 1.81057i	−0.424803	−	0.905286i	$-0.639657\pi$
$509$	− 20791.8i	− 1.81057i	0.424803	−	0.905286i	$-0.360343\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	11254.2	0.968583
$514$	0	0
$515$	− 33419.4i	− 2.85949i
$516$	0	0
$517$	168.179i	0.0143066i
$518$	0	0
$519$	− 13657.5i	− 1.15511i
$520$	0	0
$521$	− 19292.6i	− 1.62231i	−0.584829	−	0.811156i	$-0.698839\pi$
$521$	− 19292.6i	− 1.62231i	0.584829	−	0.811156i	$-0.301161\pi$
$522$	0	0
$523$	16688.0	1.39525	0.697626	−	0.716462i	$-0.254240\pi$
$523$	16688.0	1.39525	0.697626	+	0.716462i	$0.254240\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5231.65i	0.432437i
$528$	0	0
$529$	896.048	0.0736458
$530$	0	0
$531$	745.296	0.0609098
$532$	0	0
$533$	3052.48	0.248063
$534$	0	0
$535$	14298.5	1.15547
$536$	0	0
$537$	− 6584.80i	− 0.529153i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	3051.34	0.242491	0.121245	−	0.992623i	$-0.461311\pi$
$541$	3051.34	0.242491	0.121245	+	0.992623i	$0.461311\pi$
$542$	0	0
$543$	976.639i	0.0771853i
$544$	0	0
$545$	15662.9i	1.23106i
$546$	0	0
$547$	12022.3i	0.939740i	0.882736	+	0.469870i	$0.155699\pi$
$547$	12022.3i	0.939740i	−0.882736	+	0.469870i	$0.844301\pi$
$548$	0	0
$549$	1991.17i	0.154792i
$550$	0	0
$551$	11075.3	0.856304
$552$	0	0
$553$	0	0
$554$	0	0
$555$	6518.53i	0.498551i
$556$	0	0
$557$	11827.8	0.899747	0.449873	−	0.893092i	$-0.351469\pi$
$557$	11827.8	0.899747	0.449873	+	0.893092i	$0.351469\pi$
$558$	0	0
$559$	−19355.1	−1.46446
$560$	0	0
$561$	−23863.4	−1.79593
$562$	0	0
$563$	−13682.6	−1.02425	−0.512127	−	0.858910i	$-0.671142\pi$
$563$	−13682.6	−1.02425	−0.512127	+	0.858910i	$0.671142\pi$
$564$	0	0
$565$	− 10121.0i	− 0.753616i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1633.47	0.120349	0.0601747	−	0.998188i	$-0.480834\pi$
$569$	1633.47	0.120349	0.0601747	+	0.998188i	$0.480834\pi$
$570$	0	0
$571$	− 14967.3i	− 1.09696i	−0.836164	−	0.548480i	$-0.815207\pi$
$571$	− 14967.3i	− 1.09696i	0.836164	−	0.548480i	$-0.184793\pi$
$572$	0	0
$573$	− 19873.3i	− 1.44890i
$574$	0	0
$575$	− 17033.1i	− 1.23535i
$576$	0	0
$577$	− 12929.6i	− 0.932873i	−0.884555	−	0.466436i	$-0.845538\pi$
$577$	− 12929.6i	− 0.932873i	0.884555	−	0.466436i	$-0.154462\pi$
$578$	0	0
$579$	−22821.0	−1.63801
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 11395.6i	− 0.809530i
$584$	0	0
$585$	−4085.99	−0.288777
$586$	0	0
$587$	7486.36	0.526397	0.263199	−	0.964742i	$-0.415223\pi$
$587$	7486.36	0.526397	0.263199	+	0.964742i	$0.415223\pi$
$588$	0	0
$589$	3215.40	0.224938
$590$	0	0
$591$	−1799.42	−0.125242
$592$	0	0
$593$	2649.26i	0.183461i	0.995784	+	0.0917304i	$0.0292398\pi$
$593$	2649.26i	0.183461i	−0.995784	+	0.0917304i	$0.970760\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7722.46	0.529412
$598$	0	0
$599$	17510.6i	1.19443i	0.802082	+	0.597213i	$0.203725\pi$
$599$	17510.6i	1.19443i	−0.802082	+	0.597213i	$0.796275\pi$
$600$	0	0
$601$	− 12124.7i	− 0.822922i	−0.911427	−	0.411461i	$-0.865019\pi$
$601$	− 12124.7i	− 0.822922i	0.911427	−	0.411461i	$-0.134981\pi$
$602$	0	0
$603$	− 3211.07i	− 0.216857i
$604$	0	0
$605$	− 5557.53i	− 0.373464i
$606$	0	0
$607$	−11412.5	−0.763131	−0.381566	−	0.924342i	$-0.624615\pi$
$607$	−11412.5	−0.763131	−0.381566	+	0.924342i	$0.624615\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	234.182i	0.0155057i
$612$	0	0
$613$	6758.76	0.445324	0.222662	−	0.974896i	$-0.428525\pi$
$613$	6758.76	0.445324	0.222662	+	0.974896i	$0.428525\pi$
$614$	0	0
$615$	−4334.60	−0.284208
$616$	0	0
$617$	−9991.14	−0.651909	−0.325955	−	0.945385i	$-0.605686\pi$
$617$	−9991.14	−0.651909	−0.325955	+	0.945385i	$0.605686\pi$
$618$	0	0
$619$	−20664.1	−1.34178	−0.670889	−	0.741558i	$-0.734087\pi$
$619$	−20664.1	−1.34178	−0.670889	+	0.741558i	$0.734087\pi$
$620$	0	0
$621$	− 15826.2i	− 1.02268i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−9938.98	−0.636095
$626$	0	0
$627$	14666.6i	0.934176i
$628$	0	0
$629$	− 9939.02i	− 0.630039i
$630$	0	0
$631$	28295.8i	1.78516i	0.450887	+	0.892581i	$0.351108\pi$
$631$	28295.8i	1.78516i	−0.450887	+	0.892581i	$0.648892\pi$
$632$	0	0
$633$	− 14355.3i	− 0.901376i
$634$	0	0
$635$	−30983.7	−1.93630
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2854.40i	0.176711i
$640$	0	0
$641$	−16415.7	−1.01152	−0.505758	−	0.862676i	$-0.668787\pi$
$641$	−16415.7	−1.01152	−0.505758	+	0.862676i	$0.668787\pi$
$642$	0	0
$643$	14139.9	0.867220	0.433610	−	0.901101i	$-0.357240\pi$
$643$	14139.9	0.867220	0.433610	+	0.901101i	$0.357240\pi$
$644$	0	0
$645$	27484.7	1.67784
$646$	0	0
$647$	−21949.1	−1.33371	−0.666853	−	0.745189i	$-0.732359\pi$
$647$	−21949.1	−1.33371	−0.666853	+	0.745189i	$0.732359\pi$
$648$	0	0
$649$	7123.04i	0.430822i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5877.76	0.352243	0.176121	−	0.984368i	$-0.443645\pi$
$653$	5877.76	0.352243	0.176121	+	0.984368i	$0.443645\pi$
$654$	0	0
$655$	− 23300.7i	− 1.38997i
$656$	0	0
$657$	3562.87i	0.211569i
$658$	0	0
$659$	11002.0i	0.650344i	0.945655	+	0.325172i	$0.105422\pi$
$659$	11002.0i	0.650344i	−0.945655	+	0.325172i	$0.894578\pi$
$660$	0	0
$661$	− 3691.25i	− 0.217206i	−0.994085	−	0.108603i	$-0.965362\pi$
$661$	− 3691.25i	− 0.217206i	0.994085	−	0.108603i	$-0.0346377\pi$
$662$	0	0
$663$	−33228.8	−1.94646
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 15574.7i	− 0.904130i
$668$	0	0
$669$	−4194.67	−0.242414
$670$	0	0
$671$	−19030.2	−1.09487
$672$	0	0
$673$	6819.19	0.390580	0.195290	−	0.980746i	$-0.437435\pi$
$673$	6819.19	0.390580	0.195290	+	0.980746i	$0.437435\pi$
$674$	0	0
$675$	23917.2	1.36381
$676$	0	0
$677$	15800.2i	0.896973i	0.893790	+	0.448487i	$0.148037\pi$
$677$	15800.2i	0.896973i	−0.893790	+	0.448487i	$0.851963\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−6793.12	−0.382251
$682$	0	0
$683$	− 21534.7i	− 1.20644i	−0.797573	−	0.603222i	$-0.793883\pi$
$683$	− 21534.7i	− 1.20644i	0.797573	−	0.603222i	$-0.206117\pi$
$684$	0	0
$685$	24893.5i	1.38851i
$686$	0	0
$687$	1138.35i	0.0632177i
$688$	0	0
$689$	− 15867.8i	− 0.877382i
$690$	0	0
$691$	4038.54	0.222335	0.111167	−	0.993802i	$-0.464541\pi$
$691$	4038.54	0.222335	0.111167	+	0.993802i	$0.464541\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 32605.1i	− 1.77954i
$696$	0	0
$697$	6609.11	0.359165
$698$	0	0
$699$	27855.3	1.50727
$700$	0	0
$701$	11823.9	0.637066	0.318533	−	0.947912i	$-0.396810\pi$
$701$	11823.9	0.637066	0.318533	+	0.947912i	$0.396810\pi$
$702$	0	0
$703$	−6108.59	−0.327724
$704$	0	0
$705$	− 332.544i	− 0.0177650i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	26286.5	1.39240	0.696200	−	0.717848i	$-0.254873\pi$
$709$	26286.5	1.39240	0.696200	+	0.717848i	$0.254873\pi$
$710$	0	0
$711$	4715.73i	0.248739i
$712$	0	0
$713$	− 4521.68i	− 0.237501i
$714$	0	0
$715$	− 39051.1i	− 2.04256i
$716$	0	0
$717$	10022.6i	0.522039i
$718$	0	0
$719$	19229.9	0.997435	0.498717	−	0.866765i	$-0.333805\pi$
$719$	19229.9	0.997435	0.498717	+	0.866765i	$0.333805\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 10372.0i	− 0.533526i
$724$	0	0
$725$	23537.1	1.20572
$726$	0	0
$727$	33771.2	1.72284	0.861419	−	0.507895i	$-0.169576\pi$
$727$	33771.2	1.72284	0.861419	+	0.507895i	$0.169576\pi$
$728$	0	0
$729$	21731.9	1.10409
$730$	0	0
$731$	−41906.9	−2.12036
$732$	0	0
$733$	2119.95i	0.106824i	0.998573	+	0.0534121i	$0.0170097\pi$
$733$	2119.95i	0.106824i	−0.998573	+	0.0534121i	$0.982990\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	30689.3	1.53386
$738$	0	0
$739$	10667.5i	0.531004i	0.964110	+	0.265502i	$0.0855377\pi$
$739$	10667.5i	0.531004i	−0.964110	+	0.265502i	$0.914462\pi$
$740$	0	0
$741$	20422.7i	1.01248i
$742$	0	0
$743$	− 26684.2i	− 1.31756i	−0.752335	−	0.658781i	$-0.771072\pi$
$743$	− 26684.2i	− 1.31756i	0.752335	−	0.658781i	$-0.228928\pi$
$744$	0	0
$745$	38973.9i	1.91664i
$746$	0	0
$747$	−2357.01	−0.115446
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 35120.5i	− 1.70648i	−0.521519	−	0.853240i	$-0.674634\pi$
$751$	− 35120.5i	− 1.70648i	0.521519	−	0.853240i	$-0.325366\pi$
$752$	0	0
$753$	9870.37	0.477684
$754$	0	0
$755$	−17321.8	−0.834975
$756$	0	0
$757$	10060.1	0.483011	0.241506	−	0.970399i	$-0.422359\pi$
$757$	10060.1	0.483011	0.241506	+	0.970399i	$0.422359\pi$
$758$	0	0
$759$	20625.0	0.986351
$760$	0	0
$761$	− 36350.6i	− 1.73155i	−0.500435	−	0.865774i	$-0.666826\pi$
$761$	− 36350.6i	− 1.73155i	0.500435	−	0.865774i	$-0.333174\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−8846.81	−0.418114
$766$	0	0
$767$	9918.53i	0.466933i
$768$	0	0
$769$	17664.9i	0.828366i	0.910194	+	0.414183i	$0.135933\pi$
$769$	17664.9i	0.828366i	−0.910194	+	0.414183i	$0.864067\pi$
$770$	0	0
$771$	24770.2i	1.15704i
$772$	0	0
$773$	7908.42i	0.367977i	0.982928	+	0.183988i	$0.0589009\pi$
$773$	7908.42i	0.367977i	−0.982928	+	0.183988i	$0.941099\pi$
$774$	0	0
$775$	6833.33	0.316723
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 4062.00i	− 0.186825i
$780$	0	0
$781$	−27280.4	−1.24990
$782$	0	0
$783$	21869.4	0.998145
$784$	0	0
$785$	1301.16	0.0591598
$786$	0	0
$787$	11881.1	0.538140	0.269070	−	0.963121i	$-0.413284\pi$
$787$	11881.1	0.538140	0.269070	+	0.963121i	$0.413284\pi$
$788$	0	0
$789$	− 8700.81i	− 0.392594i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−26498.8	−1.18663
$794$	0	0
$795$	22532.7i	1.00522i
$796$	0	0
$797$	− 10203.4i	− 0.453480i	−0.973955	−	0.226740i	$-0.927193\pi$
$797$	− 10203.4i	− 0.453480i	0.973955	−	0.226740i	$-0.0728068\pi$
$798$	0	0
$799$	507.041i	0.0224503i
$800$	0	0
$801$	− 522.081i	− 0.0230297i
$802$	0	0
$803$	−34051.5	−1.49645
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 11906.7i	− 0.519376i
$808$	0	0
$809$	20942.1	0.910117	0.455059	−	0.890461i	$-0.349618\pi$
$809$	20942.1	0.910117	0.455059	+	0.890461i	$0.349618\pi$
$810$	0	0
$811$	5434.74	0.235314	0.117657	−	0.993054i	$-0.462462\pi$
$811$	5434.74	0.235314	0.117657	+	0.993054i	$0.462462\pi$
$812$	0	0
$813$	−9167.53	−0.395473
$814$	0	0
$815$	−65679.9	−2.82290
$816$	0	0
$817$	25756.2i	1.10293i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2132.86	−0.0906667	−0.0453334	−	0.998972i	$-0.514435\pi$
$821$	−2132.86	−0.0906667	−0.0453334	+	0.998972i	$0.514435\pi$
$822$	0	0
$823$	19157.9i	0.811424i	0.914001	+	0.405712i	$0.132976\pi$
$823$	19157.9i	0.811424i	−0.914001	+	0.405712i	$0.867024\pi$
$824$	0	0
$825$	31169.3i	1.31536i
$826$	0	0
$827$	− 27208.2i	− 1.14404i	−0.820239	−	0.572021i	$-0.806160\pi$
$827$	− 27208.2i	− 1.14404i	0.820239	−	0.572021i	$-0.193840\pi$
$828$	0	0
$829$	− 17206.0i	− 0.720857i	−0.932787	−	0.360429i	$-0.882630\pi$
$829$	− 17206.0i	− 0.720857i	0.932787	−	0.360429i	$-0.117370\pi$
$830$	0	0
$831$	−8990.39	−0.375299
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1991.30i	0.0825291i
$836$	0	0
$837$	6349.17	0.262197
$838$	0	0
$839$	9526.10	0.391988	0.195994	−	0.980605i	$-0.437207\pi$
$839$	9526.10	0.391988	0.195994	+	0.980605i	$0.437207\pi$
$840$	0	0
$841$	−2867.21	−0.117562
$842$	0	0
$843$	−12713.0	−0.519405
$844$	0	0
$845$	− 17258.8i	− 0.702629i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	17871.1	0.722421
$850$	0	0
$851$	8590.24i	0.346028i
$852$	0	0
$853$	− 8447.62i	− 0.339087i	−0.985523	−	0.169543i	$-0.945771\pi$
$853$	− 8447.62i	− 0.339087i	0.985523	−	0.169543i	$-0.0542293\pi$
$854$	0	0
$855$	5437.31i	0.217488i
$856$	0	0
$857$	25647.7i	1.02230i	0.859492	+	0.511150i	$0.170780\pi$
$857$	25647.7i	1.02230i	−0.859492	+	0.511150i	$0.829220\pi$
$858$	0	0
$859$	−7477.32	−0.297000	−0.148500	−	0.988912i	$-0.547444\pi$
$859$	−7477.32	−0.297000	−0.148500	+	0.988912i	$0.547444\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	23203.2i	0.915234i	0.889150	+	0.457617i	$0.151297\pi$
$863$	23203.2i	0.915234i	−0.889150	+	0.457617i	$0.848703\pi$
$864$	0	0
$865$	48390.8	1.90212
$866$	0	0
$867$	−48518.9	−1.90056
$868$	0	0
$869$	−45069.8	−1.75937
$870$	0	0
$871$	42733.6	1.66242
$872$	0	0
$873$	− 1241.38i	− 0.0481265i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	12283.2	0.472947	0.236474	−	0.971638i	$-0.424008\pi$
$877$	12283.2	0.472947	0.236474	+	0.971638i	$0.424008\pi$
$878$	0	0
$879$	− 30929.4i	− 1.18683i
$880$	0	0
$881$	− 44697.8i	− 1.70931i	−0.519193	−	0.854657i	$-0.673767\pi$
$881$	− 44697.8i	− 1.70931i	0.519193	−	0.854657i	$-0.326233\pi$
$882$	0	0
$883$	− 9474.88i	− 0.361104i	−0.983565	−	0.180552i	$-0.942212\pi$
$883$	− 9474.88i	− 0.361104i	0.983565	−	0.180552i	$-0.0577884\pi$
$884$	0	0
$885$	− 14084.5i	− 0.534968i
$886$	0	0
$887$	15004.6	0.567987	0.283993	−	0.958826i	$-0.408341\pi$
$887$	15004.6	0.567987	0.283993	+	0.958826i	$0.408341\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	24271.4i	0.912595i
$892$	0	0
$893$	311.631	0.0116778
$894$	0	0
$895$	23331.0	0.871361
$896$	0	0
$897$	28719.5	1.06902
$898$	0	0
$899$	6248.26	0.231803
$900$	0	0
$901$	− 34356.4i	− 1.27034i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3460.39	−0.127102
$906$	0	0
$907$	27994.5i	1.02485i	0.858731	+	0.512427i	$0.171253\pi$
$907$	27994.5i	1.02485i	−0.858731	+	0.512427i	$0.828747\pi$
$908$	0	0
$909$	− 5750.89i	− 0.209840i
$910$	0	0
$911$	27683.1i	1.00679i	0.864057	+	0.503393i	$0.167915\pi$
$911$	27683.1i	1.00679i	−0.864057	+	0.503393i	$0.832085\pi$
$912$	0	0
$913$	− 22526.7i	− 0.816566i
$914$	0	0
$915$	37628.9	1.35953
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 17.4351i	0 0.000625822i	−1.00000		0.000312911i	$-0.999900\pi$
$919$	− 17.4351i	0 0.000625822i	1.00000		0.000312911i	$-9.96027e-5\pi$
$920$	0	0
$921$	−42362.3	−1.51562
$922$	0	0
$923$	−37986.9	−1.35466
$924$	0	0
$925$	−12981.9	−0.461451
$926$	0	0
$927$	8432.42	0.298767
$928$	0	0
$929$	42595.2i	1.50431i	0.658986	+	0.752155i	$0.270985\pi$
$929$	42595.2i	1.50431i	−0.658986	+	0.752155i	$0.729015\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−33386.8	−1.17153
$934$	0	0
$935$	− 84551.9i	− 2.95737i
$936$	0	0
$937$	26316.4i	0.917522i	0.888560	+	0.458761i	$0.151707\pi$
$937$	26316.4i	0.917522i	−0.888560	+	0.458761i	$0.848293\pi$
$938$	0	0
$939$	22797.8i	0.792309i
$940$	0	0
$941$	− 29453.0i	− 1.02034i	−0.860074	−	0.510170i	$-0.829583\pi$
$941$	− 29453.0i	− 1.02034i	0.860074	−	0.510170i	$-0.170417\pi$
$942$	0	0
$943$	−5712.21	−0.197259
$944$	0	0
$945$	0	0
$946$	0	0
$947$	21005.0i	0.720770i	0.932804	+	0.360385i	$0.117355\pi$
$947$	21005.0i	0.720770i	−0.932804	+	0.360385i	$0.882645\pi$
$948$	0	0
$949$	−47415.3	−1.62188
$950$	0	0
$951$	−17182.4	−0.585885
$952$	0	0
$953$	10707.4	0.363952	0.181976	−	0.983303i	$-0.441751\pi$
$953$	10707.4	0.363952	0.181976	+	0.983303i	$0.441751\pi$
$954$	0	0
$955$	70414.1	2.38591
$956$	0	0
$957$	28500.5i	0.962687i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27977.0	−0.939109
$962$	0	0
$963$	3607.81i	0.120727i
$964$	0	0
$965$	− 80858.2i	− 2.69733i
$966$	0	0
$967$	20353.4i	0.676859i	0.940992	+	0.338429i	$0.109896\pi$
$967$	20353.4i	0.676859i	−0.940992	+	0.338429i	$0.890104\pi$
$968$	0	0
$969$	44218.3i	1.46594i
$970$	0	0
$971$	55329.3	1.82863	0.914316	−	0.405002i	$-0.132729\pi$
$971$	55329.3	1.82863	0.914316	+	0.405002i	$0.132729\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	43401.9i	1.42562i
$976$	0	0
$977$	29925.0	0.979925	0.489963	−	0.871743i	$-0.337010\pi$
$977$	29925.0	0.979925	0.489963	+	0.871743i	$0.337010\pi$
$978$	0	0
$979$	4989.70	0.162892
$980$	0	0
$981$	−3952.08	−0.128624
$982$	0	0
$983$	−4562.19	−0.148028	−0.0740139	−	0.997257i	$-0.523581\pi$
$983$	−4562.19	−0.148028	−0.0740139	+	0.997257i	$0.523581\pi$
$984$	0	0
$985$	− 6375.63i	− 0.206238i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	36219.8	1.16453
$990$	0	0
$991$	1224.37i	0.0392467i	0.999807	+	0.0196233i	$0.00624671\pi$
$991$	1224.37i	0.0392467i	−0.999807	+	0.0196233i	$0.993753\pi$
$992$	0	0
$993$	15755.0i	0.503493i
$994$	0	0
$995$	27361.9i	0.871789i
$996$	0	0
$997$	− 46098.0i	− 1.46433i	−0.681127	−	0.732165i	$-0.738510\pi$
$997$	− 46098.0i	− 1.46433i	0.681127	−	0.732165i	$-0.261490\pi$
$998$	0	0
$999$	−12062.1	−0.382009

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.f.h.783.4		6
4.3	odd	2		784.4.f.g.783.4		6
7.2	even	3		112.4.p.f.31.2	✓	6
7.3	odd	6		112.4.p.g.47.2	yes	6
7.6	odd	2		784.4.f.g.783.3		6
28.3	even	6		112.4.p.f.47.2	yes	6
28.23	odd	6		112.4.p.g.31.2	yes	6
28.27	even	2	inner	784.4.f.h.783.3		6
56.3	even	6		448.4.p.g.383.2		6
56.37	even	6		448.4.p.g.255.2		6
56.45	odd	6		448.4.p.f.383.2		6
56.51	odd	6		448.4.p.f.255.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.4.p.f.31.2	✓	6	7.2	even	3
112.4.p.f.47.2	yes	6	28.3	even	6
112.4.p.g.31.2	yes	6	28.23	odd	6
112.4.p.g.47.2	yes	6	7.3	odd	6
448.4.p.f.255.2		6	56.51	odd	6
448.4.p.f.383.2		6	56.45	odd	6
448.4.p.g.255.2		6	56.37	even	6
448.4.p.g.383.2		6	56.3	even	6
784.4.f.g.783.3		6	7.6	odd	2
784.4.f.g.783.4		6	4.3	odd	2
784.4.f.h.783.3		6	28.27	even	2	inner
784.4.f.h.783.4		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.76834 q^{3} +16.8950i q^{5} -4.26295 q^{9} +O(q^{10})\)	\(q+4.76834 q^{3} +16.8950i q^{5} -4.26295 q^{9} -40.7424i q^{11} -56.7321i q^{13} +80.5609i q^{15} -122.834i q^{17} -75.4946 q^{19} +106.165i q^{23} -160.440 q^{25} -149.072 q^{27} -146.703 q^{29} -42.5912 q^{31} -194.274i q^{33} +80.9142 q^{37} -270.518i q^{39} +53.8052i q^{41} -341.166i q^{43} -72.0225i q^{45} -4.12785 q^{47} -585.714i q^{51} +279.697 q^{53} +688.342 q^{55} -359.984 q^{57} -174.831 q^{59} -467.087i q^{61} +958.488 q^{65} +753.251i q^{67} +506.229i q^{69} -669.583i q^{71} -835.774i q^{73} -765.033 q^{75} -1106.21i q^{79} -595.728 q^{81} +552.905 q^{83} +2075.28 q^{85} -699.530 q^{87} +122.469i q^{89} -203.089 q^{93} -1275.48i q^{95} +291.203i q^{97} +173.683i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 14 q^{3} + 156 q^{9}+O(q^{10}) \) 6 * q + 14 * q^3 + 156 * q^9	\( 6 q + 14 q^{3} + 156 q^{9} - 286 q^{19} - 612 q^{25} + 362 q^{27} - 348 q^{29} - 410 q^{31} + 498 q^{37} - 150 q^{47} + 1290 q^{53} + 918 q^{55} - 6 q^{57} - 642 q^{59} + 1224 q^{65} - 8276 q^{75} - 450 q^{81} + 24 q^{83} + 3786 q^{85} - 7284 q^{87} + 5982 q^{93}+O(q^{100}) \) 6 * q + 14 * q^3 + 156 * q^9 - 286 * q^19 - 612 * q^25 + 362 * q^27 - 348 * q^29 - 410 * q^31 + 498 * q^37 - 150 * q^47 + 1290 * q^53 + 918 * q^55 - 6 * q^57 - 642 * q^59 + 1224 * q^65 - 8276 * q^75 - 450 * q^81 + 24 * q^83 + 3786 * q^85 - 7284 * q^87 + 5982 * q^93

Introduction

L-functions

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