LMFDB - Embedded newform 1900.3.g.c.949.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,3,Mod(949,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1900.949");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1900.g (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:	$51.7712502285$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			949.20
Character	$\chi$	$=$	1900.949
Dual form			1900.3.g.c.949.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.42504 q^{3} -9.18599i q^{7} +2.73093 q^{9} +O(q^{10})$	$q+3.42504 q^{3} -9.18599i q^{7} +2.73093 q^{9} +11.4382 q^{11} -10.1235 q^{13} +4.19261i q^{17} +(-10.8104 - 15.6248i) q^{19} -31.4624i q^{21} -0.952351i q^{23} -21.4719 q^{27} -9.43887i q^{29} -8.52508i q^{31} +39.1762 q^{33} -20.0029 q^{37} -34.6736 q^{39} -61.2499i q^{41} -57.9875i q^{43} +35.3773i q^{47} -35.3824 q^{49} +14.3599i q^{51} +57.6953 q^{53} +(-37.0260 - 53.5158i) q^{57} -16.7603i q^{59} -32.2192 q^{61} -25.0863i q^{63} +6.07252 q^{67} -3.26184i q^{69} +113.671i q^{71} -27.9670i q^{73} -105.071i q^{77} -65.7698i q^{79} -98.1204 q^{81} +60.2358i q^{83} -32.3286i q^{87} -97.9852i q^{89} +92.9948i q^{91} -29.1988i q^{93} -92.0255 q^{97} +31.2368 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 32 q^{9}+O(q^{10}) $ 24 * q + 32 * q^9	$ 24 q + 32 q^{9} - 64 q^{11} - 48 q^{19} - 248 q^{39} + 48 q^{49} + 304 q^{61} - 936 q^{81} - 784 q^{99}+O(q^{100}) $ 24 * q + 32 * q^9 - 64 * q^11 - 48 * q^19 - 248 * q^39 + 48 * q^49 + 304 * q^61 - 936 * q^81 - 784 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	3.42504			1.14168			0.570841	−	0.821061i	$-0.306617\pi$
$3$	3.42504			1.14168			0.570841	+	0.821061i	$0.306617\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	9.18599i		−	1.31228i	−0.754637	−	0.656142i	$-0.772187\pi$
$7$		−	9.18599i		−	1.31228i	0.754637	−	0.656142i	$-0.227813\pi$
$8$		0			0
$9$	2.73093			0.303436
$10$		0			0
$11$	11.4382			1.03983			0.519916	−	0.854217i	$-0.325963\pi$
$11$	11.4382			1.03983			0.519916	+	0.854217i	$0.325963\pi$
$12$		0			0
$13$	−10.1235			−0.778734			−0.389367	−	0.921083i	$-0.627306\pi$
$13$	−10.1235			−0.778734			−0.389367	+	0.921083i	$0.627306\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			4.19261i			0.246624i	0.992368	+	0.123312i	$0.0393517\pi$
$17$			4.19261i			0.246624i	−0.992368	+	0.123312i	$0.960648\pi$
$18$		0			0
$19$	−10.8104	−	15.6248i	−0.568967	−	0.822360i
$19$	−10.8104	−	15.6248i	−0.568967	−	0.822360i
$20$		0			0
$21$		−	31.4624i		−	1.49821i
$22$		0			0
$23$		−	0.952351i		−	0.0414066i	−0.999786	−	0.0207033i	$-0.993409\pi$
$23$		−	0.952351i		−	0.0414066i	0.999786	−	0.0207033i	$-0.00659053\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−21.4719			−0.795254
$28$		0			0
$29$		−	9.43887i		−	0.325478i	−0.986669	−	0.162739i	$-0.947967\pi$
$29$		−	9.43887i		−	0.325478i	0.986669	−	0.162739i	$-0.0520329\pi$
$30$		0			0
$31$		−	8.52508i		−	0.275002i	−0.990502	−	0.137501i	$-0.956093\pi$
$31$		−	8.52508i		−	0.275002i	0.990502	−	0.137501i	$-0.0439071\pi$
$32$		0			0
$33$	39.1762			1.18716
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−20.0029			−0.540618			−0.270309	−	0.962774i	$-0.587126\pi$
$37$	−20.0029			−0.540618			−0.270309	+	0.962774i	$0.587126\pi$
$38$		0			0
$39$	−34.6736			−0.889066
$40$		0			0
$41$		−	61.2499i		−	1.49390i	−0.664881	−	0.746950i	$-0.731518\pi$
$41$		−	61.2499i		−	1.49390i	0.664881	−	0.746950i	$-0.268482\pi$
$42$		0			0
$43$		−	57.9875i		−	1.34855i	−0.738482	−	0.674273i	$-0.764457\pi$
$43$		−	57.9875i		−	1.34855i	0.738482	−	0.674273i	$-0.235543\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			35.3773i			0.752709i	0.926476	+	0.376355i	$0.122823\pi$
$47$			35.3773i			0.752709i	−0.926476	+	0.376355i	$0.877177\pi$
$48$		0			0
$49$	−35.3824			−0.722091
$50$		0			0
$51$			14.3599i			0.281566i
$52$		0			0
$53$	57.6953			1.08859			0.544296	−	0.838893i	$-0.316797\pi$
$53$	57.6953			1.08859			0.544296	+	0.838893i	$0.316797\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−37.0260	−	53.5158i	−0.649579	−	0.938873i
$58$		0			0
$59$		−	16.7603i		−	0.284073i	−0.989861	−	0.142037i	$-0.954635\pi$
$59$		−	16.7603i		−	0.284073i	0.989861	−	0.142037i	$-0.0453651\pi$
$60$		0			0
$61$	−32.2192			−0.528183			−0.264092	−	0.964498i	$-0.585072\pi$
$61$	−32.2192			−0.528183			−0.264092	+	0.964498i	$0.585072\pi$
$62$		0			0
$63$		−	25.0863i		−	0.398195i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	6.07252			0.0906346			0.0453173	−	0.998973i	$-0.485570\pi$
$67$	6.07252			0.0906346			0.0453173	+	0.998973i	$0.485570\pi$
$68$		0			0
$69$		−	3.26184i		−	0.0472731i
$70$		0			0
$71$			113.671i			1.60101i	0.599328	+	0.800503i	$0.295434\pi$
$71$			113.671i			1.60101i	−0.599328	+	0.800503i	$0.704566\pi$
$72$		0			0
$73$		−	27.9670i		−	0.383109i	−0.981482	−	0.191554i	$-0.938647\pi$
$73$		−	27.9670i		−	0.383109i	0.981482	−	0.191554i	$-0.0613529\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	105.071i		−	1.36456i
$78$		0			0
$79$		−	65.7698i		−	0.832529i	−0.909244	−	0.416264i	$-0.863339\pi$
$79$		−	65.7698i		−	0.832529i	0.909244	−	0.416264i	$-0.136661\pi$
$80$		0			0
$81$	−98.1204			−1.21136
$82$		0			0
$83$			60.2358i			0.725733i	0.931841	+	0.362866i	$0.118202\pi$
$83$			60.2358i			0.725733i	−0.931841	+	0.362866i	$0.881798\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		−	32.3286i		−	0.371593i
$88$		0			0
$89$		−	97.9852i		−	1.10096i	−0.834849	−	0.550478i	$-0.814445\pi$
$89$		−	97.9852i		−	1.10096i	0.834849	−	0.550478i	$-0.185555\pi$
$90$		0			0
$91$			92.9948i			1.02192i
$92$		0			0
$93$		−	29.1988i		−	0.313965i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−92.0255			−0.948717			−0.474358	−	0.880332i	$-0.657320\pi$
$97$	−92.0255			−0.948717			−0.474358	+	0.880332i	$0.657320\pi$
$98$		0			0
$99$	31.2368			0.315523
$100$		0			0
$101$	64.7709			0.641296			0.320648	−	0.947198i	$-0.396099\pi$
$101$	64.7709			0.641296			0.320648	+	0.947198i	$0.396099\pi$
$102$		0			0
$103$	−204.226			−1.98278			−0.991390	−	0.130940i	$-0.958200\pi$
$103$	−204.226			−1.98278			−0.991390	+	0.130940i	$0.958200\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−203.574			−1.90256			−0.951281	−	0.308324i	$-0.900232\pi$
$107$	−203.574			−1.90256			−0.951281	+	0.308324i	$0.900232\pi$
$108$		0			0
$109$		−	63.6932i		−	0.584341i	−0.956366	−	0.292171i	$-0.905622\pi$
$109$		−	63.6932i		−	0.584341i	0.956366	−	0.292171i	$-0.0943775\pi$
$110$		0			0
$111$	−68.5107			−0.617213
$112$		0			0
$113$	2.01143			0.0178003			0.00890013	−	0.999960i	$-0.497167\pi$
$113$	2.01143			0.0178003			0.00890013	+	0.999960i	$0.497167\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−27.6467			−0.236296
$118$		0			0
$119$	38.5133			0.323641
$120$		0			0
$121$	9.83149			0.0812520
$122$		0			0
$123$		−	209.783i		−	1.70556i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	41.7788			0.328967			0.164484	−	0.986380i	$-0.447404\pi$
$127$	41.7788			0.328967			0.164484	+	0.986380i	$0.447404\pi$
$128$		0			0
$129$		−	198.610i		−	1.53961i
$130$		0			0
$131$	−50.6875			−0.386928			−0.193464	−	0.981107i	$-0.561972\pi$
$131$	−50.6875			−0.386928			−0.193464	+	0.981107i	$0.561972\pi$
$132$		0			0
$133$	−143.530	+	99.3041i	−1.07917	+	0.746647i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	30.0467i		−	0.219319i	−0.993969	−	0.109659i	$-0.965024\pi$
$137$		−	30.0467i		−	0.219319i	0.993969	−	0.109659i	$-0.0349760\pi$
$138$		0			0
$139$	210.679			1.51567			0.757837	−	0.652444i	$-0.226256\pi$
$139$	210.679			1.51567			0.757837	+	0.652444i	$0.226256\pi$
$140$		0			0
$141$			121.169i			0.859354i
$142$		0			0
$143$	−115.795			−0.809753
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−121.186			−0.824397
$148$		0			0
$149$	263.989			1.77174			0.885868	−	0.463938i	$-0.153564\pi$
$149$	263.989			1.77174			0.885868	+	0.463938i	$0.153564\pi$
$150$		0			0
$151$		−	90.3173i		−	0.598128i	−0.954233	−	0.299064i	$-0.903326\pi$
$151$		−	90.3173i		−	0.598128i	0.954233	−	0.299064i	$-0.0966743\pi$
$152$		0			0
$153$			11.4497i			0.0748348i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	108.051i		−	0.688222i	−0.938929	−	0.344111i	$-0.888180\pi$
$157$		−	108.051i		−	0.688222i	0.938929	−	0.344111i	$-0.111820\pi$
$158$		0			0
$159$	197.609			1.24282
$160$		0			0
$161$	−8.74829			−0.0543372
$162$		0			0
$163$		−	127.846i		−	0.784333i	−0.919894	−	0.392167i	$-0.871726\pi$
$163$		−	127.846i		−	0.784333i	0.919894	−	0.392167i	$-0.128274\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	143.920			0.861798			0.430899	−	0.902400i	$-0.358197\pi$
$167$	143.920			0.861798			0.430899	+	0.902400i	$0.358197\pi$
$168$		0			0
$169$	−66.5138			−0.393573
$170$		0			0
$171$	−29.5224	−	42.6703i	−0.172645	−	0.249534i
$172$		0			0
$173$	291.098			1.68265			0.841323	−	0.540532i	$-0.181777\pi$
$173$	291.098			1.68265			0.841323	+	0.540532i	$0.181777\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	57.4048i		−	0.324321i
$178$		0			0
$179$		−	91.3328i		−	0.510239i	−0.966910	−	0.255120i	$-0.917885\pi$
$179$		−	91.3328i		−	0.510239i	0.966910	−	0.255120i	$-0.0821148\pi$
$180$		0			0
$181$		−	182.671i		−	1.00923i	−0.863344	−	0.504617i	$-0.831634\pi$
$181$		−	182.671i		−	1.00923i	0.863344	−	0.504617i	$-0.168366\pi$
$182$		0			0
$183$	−110.352			−0.603017
$184$		0			0
$185$		0			0
$186$		0			0
$187$			47.9558i			0.256448i
$188$		0			0
$189$			197.240i			1.04360i
$190$		0			0
$191$	358.406			1.87647			0.938235	−	0.345998i	$-0.112460\pi$
$191$	358.406			1.87647			0.938235	+	0.345998i	$0.112460\pi$
$192$		0			0
$193$	−131.778			−0.682786			−0.341393	−	0.939921i	$-0.610899\pi$
$193$	−131.778			−0.682786			−0.341393	+	0.939921i	$0.610899\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	35.9154i		−	0.182312i	−0.995837	−	0.0911558i	$-0.970944\pi$
$197$		−	35.9154i		−	0.182312i	0.995837	−	0.0911558i	$-0.0290561\pi$
$198$		0			0
$199$	−121.742			−0.611769			−0.305885	−	0.952069i	$-0.598952\pi$
$199$	−121.742			−0.611769			−0.305885	+	0.952069i	$0.598952\pi$
$200$		0			0
$201$	20.7986			0.103476
$202$		0			0
$203$	−86.7054			−0.427120
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	2.60080i		−	0.0125643i
$208$		0			0
$209$	−123.651	−	178.719i	−0.591631	−	0.855117i
$210$		0			0
$211$			272.967i			1.29368i	0.762626	+	0.646840i	$0.223910\pi$
$211$			272.967i			1.29368i	−0.762626	+	0.646840i	$0.776090\pi$
$212$		0			0
$213$			389.330i			1.82784i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−78.3113			−0.360881
$218$		0			0
$219$		−	95.7881i		−	0.437388i
$220$		0			0
$221$		−	42.4441i		−	0.192055i
$222$		0			0
$223$	−81.1828			−0.364048			−0.182024	−	0.983294i	$-0.558265\pi$
$223$	−81.1828			−0.364048			−0.182024	+	0.983294i	$0.558265\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−337.268			−1.48576			−0.742882	−	0.669423i	$-0.766541\pi$
$227$	−337.268			−1.48576			−0.742882	+	0.669423i	$0.766541\pi$
$228$		0			0
$229$	240.819			1.05161			0.525806	−	0.850605i	$-0.323764\pi$
$229$	240.819			1.05161			0.525806	+	0.850605i	$0.323764\pi$
$230$		0			0
$231$		−	359.872i		−	1.55789i
$232$		0			0
$233$			136.526i			0.585948i	0.956120	+	0.292974i	$0.0946449\pi$
$233$			136.526i			0.585948i	−0.956120	+	0.292974i	$0.905355\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	225.264i		−	0.950483i
$238$		0			0
$239$	320.734			1.34198			0.670991	−	0.741466i	$-0.265869\pi$
$239$	320.734			1.34198			0.670991	+	0.741466i	$0.265869\pi$
$240$		0			0
$241$			359.039i			1.48979i	0.667183	+	0.744894i	$0.267500\pi$
$241$			359.039i			1.48979i	−0.667183	+	0.744894i	$0.732500\pi$
$242$		0			0
$243$	−142.820			−0.587736
$244$		0			0
$245$		0			0
$246$		0			0
$247$	109.439	+	158.179i	0.443074	+	0.640400i
$248$		0			0
$249$			206.310i			0.828556i
$250$		0			0
$251$	103.786			0.413490			0.206745	−	0.978395i	$-0.433713\pi$
$251$	103.786			0.413490			0.206745	+	0.978395i	$0.433713\pi$
$252$		0			0
$253$		−	10.8931i		−	0.0430559i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−56.5607			−0.220080			−0.110040	−	0.993927i	$-0.535098\pi$
$257$	−56.5607			−0.220080			−0.110040	+	0.993927i	$0.535098\pi$
$258$		0			0
$259$			183.746i			0.709444i
$260$		0			0
$261$		−	25.7769i		−	0.0987620i
$262$		0			0
$263$			280.514i			1.06659i	0.845929	+	0.533296i	$0.179047\pi$
$263$			280.514i			1.06659i	−0.845929	+	0.533296i	$0.820953\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	335.603i		−	1.25694i
$268$		0			0
$269$			102.156i			0.379763i	0.981807	+	0.189881i	$0.0608103\pi$
$269$			102.156i			0.379763i	−0.981807	+	0.189881i	$0.939190\pi$
$270$		0			0
$271$	476.115			1.75688			0.878441	−	0.477851i	$-0.158584\pi$
$271$	476.115			1.75688			0.878441	+	0.477851i	$0.158584\pi$
$272$		0			0
$273$			318.511i			1.16671i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	264.070i		−	0.953323i	−0.879087	−	0.476661i	$-0.841847\pi$
$277$		−	264.070i		−	0.953323i	0.879087	−	0.476661i	$-0.158153\pi$
$278$		0			0
$279$		−	23.2814i		−	0.0834457i
$280$		0			0
$281$		−	242.447i		−	0.862800i	−0.902161	−	0.431400i	$-0.858020\pi$
$281$		−	242.447i		−	0.862800i	0.902161	−	0.431400i	$-0.141980\pi$
$282$		0			0
$283$		−	271.524i		−	0.959450i	−0.877419	−	0.479725i	$-0.840736\pi$
$283$		−	271.524i		−	0.959450i	0.877419	−	0.479725i	$-0.159264\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−562.641			−1.96042
$288$		0			0
$289$	271.422			0.939176
$290$		0			0
$291$	−315.191			−1.08313
$292$		0			0
$293$	188.495			0.643328			0.321664	−	0.946854i	$-0.395758\pi$
$293$	188.495			0.643328			0.321664	+	0.946854i	$0.395758\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−245.598			−0.826931
$298$		0			0
$299$			9.64117i			0.0322447i
$300$		0			0
$301$	−532.672			−1.76968
$302$		0			0
$303$	221.843			0.732156
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−262.646			−0.855524			−0.427762	−	0.903891i	$-0.640698\pi$
$307$	−262.646			−0.855524			−0.427762	+	0.903891i	$0.640698\pi$
$308$		0			0
$309$	−699.484			−2.26370
$310$		0			0
$311$	352.334			1.13291			0.566454	−	0.824093i	$-0.308315\pi$
$311$	352.334			1.13291			0.566454	+	0.824093i	$0.308315\pi$
$312$		0			0
$313$		−	483.965i		−	1.54621i	−0.634277	−	0.773106i	$-0.718702\pi$
$313$		−	483.965i		−	1.54621i	0.634277	−	0.773106i	$-0.281298\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−80.3004			−0.253313			−0.126657	−	0.991947i	$-0.540425\pi$
$317$	−80.3004			−0.253313			−0.126657	+	0.991947i	$0.540425\pi$
$318$		0			0
$319$		−	107.963i		−	0.338443i
$320$		0			0
$321$	−697.251			−2.17212
$322$		0			0
$323$	65.5089	−	45.3238i	0.202814	−	0.140321i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	218.152i		−	0.667132i
$328$		0			0
$329$	324.976			0.987769
$330$		0			0
$331$			583.642i			1.76327i	0.471933	+	0.881634i	$0.343556\pi$
$331$			583.642i			1.76327i	−0.471933	+	0.881634i	$0.656444\pi$
$332$		0			0
$333$	−54.6263			−0.164043
$334$		0			0
$335$		0			0
$336$		0			0
$337$	413.317			1.22646			0.613230	−	0.789904i	$-0.289870\pi$
$337$	413.317			1.22646			0.613230	+	0.789904i	$0.289870\pi$
$338$		0			0
$339$	6.88924			0.0203222
$340$		0			0
$341$		−	97.5112i		−	0.285957i
$342$		0			0
$343$		−	125.091i		−	0.364696i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	207.384i		−	0.597648i	−0.954308	−	0.298824i	$-0.903406\pi$
$347$		−	207.384i		−	0.597648i	0.954308	−	0.298824i	$-0.0965944\pi$
$348$		0			0
$349$	176.833			0.506684			0.253342	−	0.967377i	$-0.418470\pi$
$349$	176.833			0.506684			0.253342	+	0.967377i	$0.418470\pi$
$350$		0			0
$351$	217.371			0.619291
$352$		0			0
$353$		−	398.294i		−	1.12831i	−0.825668	−	0.564156i	$-0.809202\pi$
$353$		−	398.294i		−	1.12831i	0.825668	−	0.564156i	$-0.190798\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	131.910			0.369495
$358$		0			0
$359$	−89.1125			−0.248224			−0.124112	−	0.992268i	$-0.539608\pi$
$359$	−89.1125			−0.248224			−0.124112	+	0.992268i	$0.539608\pi$
$360$		0			0
$361$	−127.271	+	337.821i	−0.352552	+	0.935792i
$362$		0			0
$363$	33.6733			0.0927639
$364$		0			0
$365$		0			0
$366$		0			0
$367$			450.345i			1.22710i	0.789656	+	0.613550i	$0.210259\pi$
$367$			450.345i			1.22710i	−0.789656	+	0.613550i	$0.789741\pi$
$368$		0			0
$369$		−	167.269i		−	0.453303i
$370$		0			0
$371$		−	529.989i		−	1.42854i
$372$		0			0
$373$	442.338			1.18589			0.592946	−	0.805242i	$-0.297965\pi$
$373$	442.338			1.18589			0.592946	+	0.805242i	$0.297965\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			95.5549i			0.253461i
$378$		0			0
$379$		−	573.993i		−	1.51449i	−0.653128	−	0.757247i	$-0.726544\pi$
$379$		−	573.993i		−	1.51449i	0.653128	−	0.757247i	$-0.273456\pi$
$380$		0			0
$381$	143.094			0.375576
$382$		0			0
$383$	−159.706			−0.416987			−0.208494	−	0.978024i	$-0.566856\pi$
$383$	−159.706			−0.416987			−0.208494	+	0.978024i	$0.566856\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	158.360i		−	0.409198i
$388$		0			0
$389$	−620.531			−1.59519			−0.797597	−	0.603190i	$-0.793896\pi$
$389$	−620.531			−1.59519			−0.797597	+	0.603190i	$0.793896\pi$
$390$		0			0
$391$	3.99284			0.0102119
$392$		0			0
$393$	−173.607			−0.441748
$394$		0			0
$395$		0			0
$396$		0			0
$397$			765.284i			1.92767i	0.266506	+	0.963833i	$0.414131\pi$
$397$			765.284i			1.92767i	−0.266506	+	0.963833i	$0.585869\pi$
$398$		0			0
$399$	−491.595	+	340.121i	−1.23207	+	0.852433i
$400$		0			0
$401$			255.996i			0.638393i	0.947688	+	0.319197i	$0.103413\pi$
$401$			255.996i			0.638393i	−0.947688	+	0.319197i	$0.896587\pi$
$402$		0			0
$403$			86.3040i			0.214154i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−228.796			−0.562152
$408$		0			0
$409$		−	225.432i		−	0.551179i	−0.961275	−	0.275589i	$-0.911127\pi$
$409$		−	225.432i		−	0.551179i	0.961275	−	0.275589i	$-0.0888730\pi$
$410$		0			0
$411$		−	102.911i		−	0.250392i
$412$		0			0
$413$	−153.960			−0.372785
$414$		0			0
$415$		0			0
$416$		0			0
$417$	721.584			1.73042
$418$		0			0
$419$	−314.462			−0.750506			−0.375253	−	0.926922i	$-0.622444\pi$
$419$	−314.462			−0.750506			−0.375253	+	0.926922i	$0.622444\pi$
$420$		0			0
$421$			354.990i			0.843206i	0.906781	+	0.421603i	$0.138532\pi$
$421$			354.990i			0.843206i	−0.906781	+	0.421603i	$0.861468\pi$
$422$		0			0
$423$			96.6129i			0.228399i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			295.965i			0.693127i
$428$		0			0
$429$	−396.602			−0.924480
$430$		0			0
$431$		−	178.559i		−	0.414290i	−0.978310	−	0.207145i	$-0.933583\pi$
$431$		−	178.559i		−	0.414290i	0.978310	−	0.207145i	$-0.0664172\pi$
$432$		0			0
$433$	−37.2858			−0.0861104			−0.0430552	−	0.999073i	$-0.513709\pi$
$433$	−37.2858			−0.0861104			−0.0430552	+	0.999073i	$0.513709\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−14.8803	+	10.2953i	−0.0340511	+	0.0235590i
$438$		0			0
$439$			365.662i			0.832944i	0.909149	+	0.416472i	$0.136734\pi$
$439$			365.662i			0.832944i	−0.909149	+	0.416472i	$0.863266\pi$
$440$		0			0
$441$	−96.6269			−0.219109
$442$		0			0
$443$		−	177.806i		−	0.401367i	−0.979656	−	0.200684i	$-0.935684\pi$
$443$		−	177.806i		−	0.401367i	0.979656	−	0.200684i	$-0.0643163\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	904.172			2.02276
$448$		0			0
$449$		−	204.290i		−	0.454988i	−0.973780	−	0.227494i	$-0.926947\pi$
$449$		−	204.290i		−	0.454988i	0.973780	−	0.227494i	$-0.0730532\pi$
$450$		0			0
$451$		−	700.586i		−	1.55341i
$452$		0			0
$453$		−	309.341i		−	0.682871i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			229.796i			0.502837i	0.967879	+	0.251418i	$0.0808970\pi$
$457$			229.796i			0.502837i	−0.967879	+	0.251418i	$0.919103\pi$
$458$		0			0
$459$		−	90.0232i		−	0.196129i
$460$		0			0
$461$	−249.439			−0.541082			−0.270541	−	0.962708i	$-0.587203\pi$
$461$	−249.439			−0.541082			−0.270541	+	0.962708i	$0.587203\pi$
$462$		0			0
$463$			907.359i			1.95974i	0.199639	+	0.979869i	$0.436023\pi$
$463$			907.359i			1.95974i	−0.199639	+	0.979869i	$0.563977\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			270.399i			0.579013i	0.957176	+	0.289507i	$0.0934912\pi$
$467$			270.399i			0.579013i	−0.957176	+	0.289507i	$0.906509\pi$
$468$		0			0
$469$		−	55.7821i		−	0.118938i
$470$		0			0
$471$		−	370.079i		−	0.785730i
$472$		0			0
$473$		−	663.270i		−	1.40226i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	157.562			0.330318
$478$		0			0
$479$	221.920			0.463299			0.231649	−	0.972799i	$-0.425588\pi$
$479$	221.920			0.463299			0.231649	+	0.972799i	$0.425588\pi$
$480$		0			0
$481$	202.500			0.420997
$482$		0			0
$483$	−29.9633			−0.0620358
$484$		0			0
$485$		0			0
$486$		0			0
$487$	881.561			1.81019			0.905094	−	0.425212i	$-0.139801\pi$
$487$	881.561			1.81019			0.905094	+	0.425212i	$0.139801\pi$
$488$		0			0
$489$		−	437.879i		−	0.895458i
$490$		0			0
$491$	91.7890			0.186943			0.0934715	−	0.995622i	$-0.470204\pi$
$491$	91.7890			0.186943			0.0934715	+	0.995622i	$0.470204\pi$
$492$		0			0
$493$	39.5736			0.0802709
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1044.19			2.10098
$498$		0			0
$499$	−471.959			−0.945809			−0.472905	−	0.881114i	$-0.656795\pi$
$499$	−471.959			−0.945809			−0.472905	+	0.881114i	$0.656795\pi$
$500$		0			0
$501$	492.933			0.983899
$502$		0			0
$503$		−	888.718i		−	1.76684i	−0.468586	−	0.883418i	$-0.655237\pi$
$503$		−	888.718i		−	1.76684i	0.468586	−	0.883418i	$-0.344763\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−227.813			−0.449335
$508$		0			0
$509$			418.738i			0.822669i	0.911485	+	0.411334i	$0.134937\pi$
$509$			418.738i			0.822669i	−0.911485	+	0.411334i	$0.865063\pi$
$510$		0			0
$511$	−256.904			−0.502748
$512$		0			0
$513$	232.119	+	335.494i	0.452473	+	0.653985i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			404.652i			0.782692i
$518$		0			0
$519$	997.023			1.92105
$520$		0			0
$521$			186.146i			0.357285i	0.983914	+	0.178643i	$0.0571706\pi$
$521$			186.146i			0.357285i	−0.983914	+	0.178643i	$0.942829\pi$
$522$		0			0
$523$	674.426			1.28953			0.644766	−	0.764380i	$-0.276955\pi$
$523$	674.426			1.28953			0.644766	+	0.764380i	$0.276955\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	35.7424			0.0678223
$528$		0			0
$529$	528.093			0.998285
$530$		0			0
$531$		−	45.7712i		−	0.0861981i
$532$		0			0
$533$			620.066i			1.16335i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		−	312.819i		−	0.582531i
$538$		0			0
$539$	−404.710			−0.750853
$540$		0			0
$541$	−641.545			−1.18585			−0.592925	−	0.805258i	$-0.702027\pi$
$541$	−641.545			−1.18585			−0.592925	+	0.805258i	$0.702027\pi$
$542$		0			0
$543$		−	625.657i		−	1.15222i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	183.171			0.334865			0.167432	−	0.985884i	$-0.446452\pi$
$547$	183.171			0.334865			0.167432	+	0.985884i	$0.446452\pi$
$548$		0			0
$549$	−87.9882			−0.160270
$550$		0			0
$551$	−147.481	+	102.038i	−0.267660	+	0.185187i
$552$		0			0
$553$	−604.161			−1.09251
$554$		0			0
$555$		0			0
$556$		0			0
$557$			713.778i			1.28147i	0.767763	+	0.640734i	$0.221370\pi$
$557$			713.778i			1.28147i	−0.767763	+	0.640734i	$0.778630\pi$
$558$		0			0
$559$			587.039i			1.05016i
$560$		0			0
$561$			164.251i			0.292782i
$562$		0			0
$563$	−316.069			−0.561401			−0.280700	−	0.959795i	$-0.590567\pi$
$563$	−316.069			−0.561401			−0.280700	+	0.959795i	$0.590567\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$			901.333i			1.58965i
$568$		0			0
$569$			883.884i			1.55340i	0.629871	+	0.776699i	$0.283108\pi$
$569$			883.884i			1.55340i	−0.629871	+	0.776699i	$0.716892\pi$
$570$		0			0
$571$	168.849			0.295708			0.147854	−	0.989009i	$-0.452763\pi$
$571$	168.849			0.295708			0.147854	+	0.989009i	$0.452763\pi$
$572$		0			0
$573$	1227.56			2.14233
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	380.352i		−	0.659188i	−0.944123	−	0.329594i	$-0.893088\pi$
$577$		−	380.352i		−	0.659188i	0.944123	−	0.329594i	$-0.106912\pi$
$578$		0			0
$579$	−451.344			−0.779524
$580$		0			0
$581$	553.326			0.952368
$582$		0			0
$583$	659.929			1.13195
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	123.630i		−	0.210613i	−0.994440	−	0.105306i	$-0.966418\pi$
$587$		−	123.630i		−	0.210613i	0.994440	−	0.105306i	$-0.0335823\pi$
$588$		0			0
$589$	−133.203	+	92.1593i	−0.226151	+	0.156467i
$590$		0			0
$591$		−	123.012i		−	0.208142i
$592$		0			0
$593$			191.622i			0.323141i	0.986861	+	0.161570i	$0.0516559\pi$
$593$			191.622i			0.323141i	−0.986861	+	0.161570i	$0.948344\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−416.972			−0.698446
$598$		0			0
$599$		−	277.360i		−	0.463038i	−0.972830	−	0.231519i	$-0.925630\pi$
$599$		−	277.360i		−	0.463038i	0.972830	−	0.231519i	$-0.0743695\pi$
$600$		0			0
$601$			603.631i			1.00438i	0.864758	+	0.502188i	$0.167472\pi$
$601$			603.631i			1.00438i	−0.864758	+	0.502188i	$0.832528\pi$
$602$		0			0
$603$	16.5836			0.0275018
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−1035.05			−1.70519			−0.852597	−	0.522568i	$-0.824974\pi$
$607$	−1035.05			−1.70519			−0.852597	+	0.522568i	$0.824974\pi$
$608$		0			0
$609$	−296.970			−0.487635
$610$		0			0
$611$		−	358.144i		−	0.586160i
$612$		0			0
$613$			965.052i			1.57431i	0.616756	+	0.787155i	$0.288447\pi$
$613$			965.052i			1.57431i	−0.616756	+	0.787155i	$0.711553\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			967.261i			1.56768i	0.620960	+	0.783842i	$0.286743\pi$
$617$			967.261i			1.56768i	−0.620960	+	0.783842i	$0.713257\pi$
$618$		0			0
$619$	−995.539			−1.60830			−0.804151	−	0.594425i	$-0.797380\pi$
$619$	−995.539			−1.60830			−0.804151	+	0.594425i	$0.797380\pi$
$620$		0			0
$621$			20.4487i			0.0329287i
$622$		0			0
$623$	−900.091			−1.44477
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−423.510	−	612.122i	−0.675454	−	0.976271i
$628$		0			0
$629$		−	83.8643i		−	0.133330i
$630$		0			0
$631$	689.387			1.09253			0.546266	−	0.837612i	$-0.316049\pi$
$631$	689.387			1.09253			0.546266	+	0.837612i	$0.316049\pi$
$632$		0			0
$633$			934.922i			1.47697i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	358.196			0.562317
$638$		0			0
$639$			310.429i			0.485804i
$640$		0			0
$641$		−	732.715i		−	1.14308i	−0.820574	−	0.571540i	$-0.806346\pi$
$641$		−	732.715i		−	1.14308i	0.820574	−	0.571540i	$-0.193654\pi$
$642$		0			0
$643$		−	509.087i		−	0.791738i	−0.918307	−	0.395869i	$-0.870443\pi$
$643$		−	509.087i		−	0.791738i	0.918307	−	0.395869i	$-0.129557\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1046.09i		−	1.61683i	−0.588611	−	0.808416i	$-0.700325\pi$
$647$		−	1046.09i		−	1.61683i	0.588611	−	0.808416i	$-0.299675\pi$
$648$		0			0
$649$		−	191.707i		−	0.295389i
$650$		0			0
$651$	−268.220			−0.412012
$652$		0			0
$653$			351.804i			0.538750i	0.963035	+	0.269375i	$0.0868171\pi$
$653$			351.804i			0.538750i	−0.963035	+	0.269375i	$0.913183\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	76.3757i		−	0.116249i
$658$		0			0
$659$			1238.23i			1.87895i	0.342621	+	0.939474i	$0.388685\pi$
$659$			1238.23i			1.87895i	−0.342621	+	0.939474i	$0.611315\pi$
$660$		0			0
$661$		−	414.630i		−	0.627277i	−0.949542	−	0.313638i	$-0.898452\pi$
$661$		−	414.630i		−	0.627277i	0.949542	−	0.313638i	$-0.101548\pi$
$662$		0			0
$663$		−	145.373i		−	0.219265i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−8.98912			−0.0134769
$668$		0			0
$669$	−278.055			−0.415627
$670$		0			0
$671$	−368.528			−0.549222
$672$		0			0
$673$	1163.57			1.72892			0.864462	−	0.502698i	$-0.167659\pi$
$673$	1163.57			1.72892			0.864462	+	0.502698i	$0.167659\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	71.7621			0.106000			0.0530001	−	0.998595i	$-0.483122\pi$
$677$	71.7621			0.106000			0.0530001	+	0.998595i	$0.483122\pi$
$678$		0			0
$679$			845.345i			1.24499i
$680$		0			0
$681$	−1155.16			−1.69627
$682$		0			0
$683$	−558.227			−0.817316			−0.408658	−	0.912688i	$-0.634003\pi$
$683$	−558.227			−0.817316			−0.408658	+	0.912688i	$0.634003\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	824.816			1.20061
$688$		0			0
$689$	−584.081			−0.847723
$690$		0			0
$691$	310.776			0.449748			0.224874	−	0.974388i	$-0.427803\pi$
$691$	310.776			0.449748			0.224874	+	0.974388i	$0.427803\pi$
$692$		0			0
$693$		−	286.941i		−	0.414056i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	256.797			0.368432
$698$		0			0
$699$			467.607i			0.668965i
$700$		0			0
$701$	940.642			1.34186			0.670929	−	0.741522i	$-0.265896\pi$
$701$	940.642			1.34186			0.670929	+	0.741522i	$0.265896\pi$
$702$		0			0
$703$	216.238	+	312.541i	0.307594	+	0.444582i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	594.985i		−	0.841563i
$708$		0			0
$709$	780.564			1.10094			0.550468	−	0.834856i	$-0.314449\pi$
$709$	780.564			1.10094			0.550468	+	0.834856i	$0.314449\pi$
$710$		0			0
$711$		−	179.612i		−	0.252620i
$712$		0			0
$713$	−8.11886			−0.0113869
$714$		0			0
$715$		0			0
$716$		0			0
$717$	1098.53			1.53212
$718$		0			0
$719$	606.339			0.843308			0.421654	−	0.906757i	$-0.361450\pi$
$719$	606.339			0.843308			0.421654	+	0.906757i	$0.361450\pi$
$720$		0			0
$721$			1876.02i			2.60197i
$722$		0			0
$723$			1229.72i			1.70086i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	266.418i		−	0.366463i	−0.983070	−	0.183231i	$-0.941344\pi$
$727$		−	266.418i		−	0.366463i	0.983070	−	0.183231i	$-0.0586557\pi$
$728$		0			0
$729$	393.919			0.540355
$730$		0			0
$731$	243.119			0.332584
$732$		0			0
$733$			951.526i			1.29813i	0.760735	+	0.649063i	$0.224839\pi$
$733$			951.526i			1.29813i	−0.760735	+	0.649063i	$0.775161\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	69.4584			0.0942448
$738$		0			0
$739$	−504.613			−0.682832			−0.341416	−	0.939912i	$-0.610906\pi$
$739$	−504.613			−0.682832			−0.341416	+	0.939912i	$0.610906\pi$
$740$		0			0
$741$	374.835	+	541.769i	0.505850	+	0.731133i
$742$		0			0
$743$	−1081.91			−1.45613			−0.728067	−	0.685506i	$-0.759581\pi$
$743$	−1081.91			−1.45613			−0.728067	+	0.685506i	$0.759581\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			164.500i			0.220214i
$748$		0			0
$749$			1870.03i			2.49670i
$750$		0			0
$751$		−	873.565i		−	1.16320i	−0.813474	−	0.581601i	$-0.802426\pi$
$751$		−	873.565i		−	1.16320i	0.813474	−	0.581601i	$-0.197574\pi$
$752$		0			0
$753$	355.472			0.472074
$754$		0			0
$755$		0			0
$756$		0			0
$757$			264.661i			0.349619i	0.984602	+	0.174809i	$0.0559309\pi$
$757$			264.661i			0.349619i	−0.984602	+	0.174809i	$0.944069\pi$
$758$		0			0
$759$		−	37.3095i		−	0.0491561i
$760$		0			0
$761$	681.093			0.894998			0.447499	−	0.894284i	$-0.352315\pi$
$761$	681.093			0.894998			0.447499	+	0.894284i	$0.352315\pi$
$762$		0			0
$763$	−585.085			−0.766822
$764$		0			0
$765$		0			0
$766$		0			0
$767$			169.674i			0.221218i
$768$		0			0
$769$	962.529			1.25166			0.625832	−	0.779958i	$-0.284760\pi$
$769$	962.529			1.25166			0.625832	+	0.779958i	$0.284760\pi$
$770$		0			0
$771$	−193.723			−0.251262
$772$		0			0
$773$	27.0726			0.0350227			0.0175114	−	0.999847i	$-0.494426\pi$
$773$	27.0726			0.0350227			0.0175114	+	0.999847i	$0.494426\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$			629.338i			0.809959i
$778$		0			0
$779$	−957.019	+	662.134i	−1.22852	+	0.849980i
$780$		0			0
$781$			1300.19i			1.66478i
$782$		0			0
$783$			202.670i			0.258838i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	49.7504			0.0632152			0.0316076	−	0.999500i	$-0.489937\pi$
$787$	49.7504			0.0632152			0.0316076	+	0.999500i	$0.489937\pi$
$788$		0			0
$789$			960.771i			1.21771i
$790$		0			0
$791$		−	18.4770i		−	0.0233590i
$792$		0			0
$793$	326.172			0.411314
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−221.385			−0.277773			−0.138886	−	0.990308i	$-0.544352\pi$
$797$	−221.385			−0.277773			−0.138886	+	0.990308i	$0.544352\pi$
$798$		0			0
$799$	−148.324			−0.185636
$800$		0			0
$801$		−	267.590i		−	0.334070i
$802$		0			0
$803$		−	319.890i		−	0.398369i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			349.889i			0.433568i
$808$		0			0
$809$	933.918			1.15441			0.577205	−	0.816599i	$-0.304143\pi$
$809$	933.918			1.15441			0.577205	+	0.816599i	$0.304143\pi$
$810$		0			0
$811$		−	1390.98i		−	1.71514i	−0.514364	−	0.857572i	$-0.671972\pi$
$811$		−	1390.98i		−	1.71514i	0.514364	−	0.857572i	$-0.328028\pi$
$812$		0			0
$813$	1630.71			2.00580
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−906.045	+	626.867i	−1.10899	+	0.767279i
$818$		0			0
$819$			253.962i			0.310088i
$820$		0			0
$821$	1066.11			1.29855			0.649275	−	0.760553i	$-0.275072\pi$
$821$	1066.11			1.29855			0.649275	+	0.760553i	$0.275072\pi$
$822$		0			0
$823$			495.548i			0.602124i	0.953605	+	0.301062i	$0.0973411\pi$
$823$			495.548i			0.602124i	−0.953605	+	0.301062i	$0.902659\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1036.35			1.25314			0.626571	−	0.779365i	$-0.284458\pi$
$827$	1036.35			1.25314			0.626571	+	0.779365i	$0.284458\pi$
$828$		0			0
$829$		−	355.400i		−	0.428710i	−0.976756	−	0.214355i	$-0.931235\pi$
$829$		−	355.400i		−	0.428710i	0.976756	−	0.214355i	$-0.0687649\pi$
$830$		0			0
$831$		−	904.453i		−	1.08839i
$832$		0			0
$833$		−	148.345i		−	0.178085i
$834$		0			0
$835$		0			0
$836$		0			0
$837$			183.049i			0.218697i
$838$		0			0
$839$		−	429.992i		−	0.512505i	−0.966610	−	0.256253i	$-0.917512\pi$
$839$		−	429.992i		−	0.512505i	0.966610	−	0.256253i	$-0.0824879\pi$
$840$		0			0
$841$	751.908			0.894064
$842$		0			0
$843$		−	830.391i		−	0.985043i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	90.3120i		−	0.106626i
$848$		0			0
$849$		−	929.983i		−	1.09539i
$850$		0			0
$851$			19.0497i			0.0223851i
$852$		0			0
$853$			557.873i			0.654013i	0.945022	+	0.327007i	$0.106040\pi$
$853$			557.873i			0.654013i	−0.945022	+	0.327007i	$0.893960\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−813.712			−0.949489			−0.474745	−	0.880124i	$-0.657460\pi$
$857$	−813.712			−0.949489			−0.474745	+	0.880124i	$0.657460\pi$
$858$		0			0
$859$	368.216			0.428657			0.214328	−	0.976762i	$-0.431244\pi$
$859$	368.216			0.428657			0.214328	+	0.976762i	$0.431244\pi$
$860$		0			0
$861$	−1927.07			−2.23818
$862$		0			0
$863$	−963.303			−1.11623			−0.558113	−	0.829765i	$-0.688474\pi$
$863$	−963.303			−1.11623			−0.558113	+	0.829765i	$0.688474\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	929.632			1.07224
$868$		0			0
$869$		−	752.285i		−	0.865691i
$870$		0			0
$871$	−61.4754			−0.0705802
$872$		0			0
$873$	−251.315			−0.287875
$874$		0			0
$875$		0			0
$876$		0			0
$877$	938.800			1.07047			0.535234	−	0.844704i	$-0.320223\pi$
$877$	938.800			1.07047			0.535234	+	0.844704i	$0.320223\pi$
$878$		0			0
$879$	645.604			0.734476
$880$		0			0
$881$	417.023			0.473352			0.236676	−	0.971589i	$-0.423942\pi$
$881$	417.023			0.473352			0.236676	+	0.971589i	$0.423942\pi$
$882$		0			0
$883$		−	326.997i		−	0.370325i	−0.982708	−	0.185163i	$-0.940719\pi$
$883$		−	326.997i		−	0.370325i	0.982708	−	0.185163i	$-0.0592812\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−366.885			−0.413624			−0.206812	−	0.978381i	$-0.566309\pi$
$887$	−366.885			−0.413624			−0.206812	+	0.978381i	$0.566309\pi$
$888$		0			0
$889$		−	383.780i		−	0.431699i
$890$		0			0
$891$	−1122.32			−1.25961
$892$		0			0
$893$	552.765	−	382.442i	0.618998	−	0.428267i
$894$		0			0
$895$		0			0
$896$		0			0
$897$			33.0214i			0.0368132i
$898$		0			0
$899$	−80.4671			−0.0895073
$900$		0			0
$901$			241.894i			0.268473i
$902$		0			0
$903$	−1824.43			−2.02041
$904$		0			0
$905$		0			0
$906$		0			0
$907$	948.571			1.04583			0.522917	−	0.852384i	$-0.324844\pi$
$907$	948.571			1.04583			0.522917	+	0.852384i	$0.324844\pi$
$908$		0			0
$909$	176.885			0.194592
$910$		0			0
$911$			1288.33i			1.41419i	0.707118	+	0.707096i	$0.249995\pi$
$911$			1288.33i			1.41419i	−0.707118	+	0.707096i	$0.750005\pi$
$912$		0			0
$913$			688.987i			0.754641i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			465.615i			0.507759i
$918$		0			0
$919$	20.3298			0.0221216			0.0110608	−	0.999939i	$-0.496479\pi$
$919$	20.3298			0.0221216			0.0110608	+	0.999939i	$0.496479\pi$
$920$		0			0
$921$	−899.573			−0.976735
$922$		0			0
$923$		−	1150.76i		−	1.24676i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−557.727			−0.601648
$928$		0			0
$929$	356.003			0.383210			0.191605	−	0.981472i	$-0.438631\pi$
$929$	356.003			0.383210			0.191605	+	0.981472i	$0.438631\pi$
$930$		0			0
$931$	382.498	+	552.845i	0.410846	+	0.593819i
$932$		0			0
$933$	1206.76			1.29342
$934$		0			0
$935$		0			0
$936$		0			0
$937$			822.306i			0.877595i	0.898586	+	0.438797i	$0.144595\pi$
$937$			822.306i			0.877595i	−0.898586	+	0.438797i	$0.855405\pi$
$938$		0			0
$939$		−	1657.60i		−	1.76528i
$940$		0			0
$941$		−	1734.01i		−	1.84273i	−0.388699	−	0.921365i	$-0.627075\pi$
$941$		−	1734.01i		−	1.84273i	0.388699	−	0.921365i	$-0.372925\pi$
$942$		0			0
$943$	−58.3314			−0.0618572
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1226.54i			1.29518i	0.761988	+	0.647591i	$0.224223\pi$
$947$			1226.54i			1.29518i	−0.761988	+	0.647591i	$0.775777\pi$
$948$		0			0
$949$			283.125i			0.298340i
$950$		0			0
$951$	−275.032			−0.289203
$952$		0			0
$953$	−1329.78			−1.39537			−0.697683	−	0.716406i	$-0.745786\pi$
$953$	−1329.78			−1.39537			−0.697683	+	0.716406i	$0.745786\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	369.779i		−	0.386394i
$958$		0			0
$959$	−276.009			−0.287809
$960$		0			0
$961$	888.323			0.924374
$962$		0			0
$963$	−555.946			−0.577307
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	1006.28i		−	1.04062i	−0.853978	−	0.520309i	$-0.825817\pi$
$967$		−	1006.28i		−	1.04062i	0.853978	−	0.520309i	$-0.174183\pi$
$968$		0			0
$969$	224.371	−	155.236i	0.231549	−	0.160202i
$970$		0			0
$971$			1521.07i			1.56650i	0.621710	+	0.783248i	$0.286438\pi$
$971$			1521.07i			1.56650i	−0.621710	+	0.783248i	$0.713562\pi$
$972$		0			0
$973$		−	1935.29i		−	1.98900i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−750.278			−0.767940			−0.383970	−	0.923346i	$-0.625443\pi$
$977$	−750.278			−0.767940			−0.383970	+	0.923346i	$0.625443\pi$
$978$		0			0
$979$		−	1120.77i		−	1.14481i
$980$		0			0
$981$		−	173.941i		−	0.177310i
$982$		0			0
$983$	−603.980			−0.614425			−0.307213	−	0.951641i	$-0.599396\pi$
$983$	−603.980			−0.614425			−0.307213	+	0.951641i	$0.599396\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	1113.06			1.12772
$988$		0			0
$989$	−55.2244			−0.0558386
$990$		0			0
$991$		−	1522.67i		−	1.53650i	−0.640152	−	0.768248i	$-0.721129\pi$
$991$		−	1522.67i		−	1.53650i	0.640152	−	0.768248i	$-0.278871\pi$
$992$		0			0
$993$			1999.00i			2.01309i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	1616.79i		−	1.62165i	−0.585285	−	0.810827i	$-0.699018\pi$
$997$		−	1616.79i		−	1.62165i	0.585285	−	0.810827i	$-0.300982\pi$
$998$		0			0
$999$	429.498			0.429928

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.3.g.c.949.20		24
5.2	odd	4		380.3.e.a.341.3	✓	12
5.3	odd	4		1900.3.e.f.1101.10		12
5.4	even	2	inner	1900.3.g.c.949.5		24
15.2	even	4		3420.3.o.a.721.6		12
19.18	odd	2	inner	1900.3.g.c.949.6		24
20.7	even	4		1520.3.h.b.721.10		12
95.18	even	4		1900.3.e.f.1101.3		12
95.37	even	4		380.3.e.a.341.10	yes	12
95.94	odd	2	inner	1900.3.g.c.949.19		24
285.227	odd	4		3420.3.o.a.721.5		12
380.227	odd	4		1520.3.h.b.721.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.e.a.341.3	✓	12	5.2	odd	4
380.3.e.a.341.10	yes	12	95.37	even	4
1520.3.h.b.721.3		12	380.227	odd	4
1520.3.h.b.721.10		12	20.7	even	4
1900.3.e.f.1101.3		12	95.18	even	4
1900.3.e.f.1101.10		12	5.3	odd	4
1900.3.g.c.949.5		24	5.4	even	2	inner
1900.3.g.c.949.6		24	19.18	odd	2	inner
1900.3.g.c.949.19		24	95.94	odd	2	inner
1900.3.g.c.949.20		24	1.1	even	1	trivial
3420.3.o.a.721.5		12	285.227	odd	4
3420.3.o.a.721.6		12	15.2	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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1900.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.42504 q^{3} -9.18599i q^{7} +2.73093 q^{9} +O(q^{10})\)	\(q+3.42504 q^{3} -9.18599i q^{7} +2.73093 q^{9} +11.4382 q^{11} -10.1235 q^{13} +4.19261i q^{17} +(-10.8104 - 15.6248i) q^{19} -31.4624i q^{21} -0.952351i q^{23} -21.4719 q^{27} -9.43887i q^{29} -8.52508i q^{31} +39.1762 q^{33} -20.0029 q^{37} -34.6736 q^{39} -61.2499i q^{41} -57.9875i q^{43} +35.3773i q^{47} -35.3824 q^{49} +14.3599i q^{51} +57.6953 q^{53} +(-37.0260 - 53.5158i) q^{57} -16.7603i q^{59} -32.2192 q^{61} -25.0863i q^{63} +6.07252 q^{67} -3.26184i q^{69} +113.671i q^{71} -27.9670i q^{73} -105.071i q^{77} -65.7698i q^{79} -98.1204 q^{81} +60.2358i q^{83} -32.3286i q^{87} -97.9852i q^{89} +92.9948i q^{91} -29.1988i q^{93} -92.0255 q^{97} +31.2368 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 32 q^{9}+O(q^{10}) \) 24 * q + 32 * q^9	\( 24 q + 32 q^{9} - 64 q^{11} - 48 q^{19} - 248 q^{39} + 48 q^{49} + 304 q^{61} - 936 q^{81} - 784 q^{99}+O(q^{100}) \) 24 * q + 32 * q^9 - 64 * q^11 - 48 * q^19 - 248 * q^39 + 48 * q^49 + 304 * q^61 - 936 * q^81 - 784 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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