LMFDB - Embedded newform 380.3.e.a.341.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [380,3,Mod(341,380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("380.341");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 380 = 2^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	380.e (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:	$10.3542500457$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 62x^{10} + 1445x^{8} + 15924x^{6} + 83244x^{4} + 170640x^{2} + 55600 $ x^12 + 62x^10 + 1445x^8 + 15924x^6 + 83244x^4 + 170640*x^2 + 55600
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{9}\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			341.3
Root			$-3.42504i$ of defining polynomial
Character	$\chi$	$=$	380.341
Dual form			380.3.e.a.341.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.42504i q^{3} +2.23607 q^{5} +9.18599 q^{7} -2.73093 q^{9} +O(q^{10})$	$q-3.42504i q^{3} +2.23607 q^{5} +9.18599 q^{7} -2.73093 q^{9} +11.4382 q^{11} +10.1235i q^{13} -7.65863i q^{15} -4.19261 q^{17} +(10.8104 + 15.6248i) q^{19} -31.4624i q^{21} -0.952351 q^{23} +5.00000 q^{25} -21.4719i q^{27} +9.43887i q^{29} -8.52508i q^{31} -39.1762i q^{33} +20.5405 q^{35} -20.0029i q^{37} +34.6736 q^{39} -61.2499i q^{41} -57.9875 q^{43} -6.10654 q^{45} -35.3773 q^{47} +35.3824 q^{49} +14.3599i q^{51} -57.6953i q^{53} +25.5765 q^{55} +(53.5158 - 37.0260i) q^{57} +16.7603i q^{59} -32.2192 q^{61} -25.0863 q^{63} +22.6369i q^{65} +6.07252i q^{67} +3.26184i q^{69} +113.671i q^{71} -27.9670 q^{73} -17.1252i q^{75} +105.071 q^{77} +65.7698i q^{79} -98.1204 q^{81} +60.2358 q^{83} -9.37497 q^{85} +32.3286 q^{87} +97.9852i q^{89} +92.9948i q^{91} -29.1988 q^{93} +(24.1727 + 34.9382i) q^{95} -92.0255i q^{97} -31.2368 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{7} - 16 q^{9}+O(q^{10}) $ 12 * q - 12 * q^7 - 16 * q^9	$ 12 q - 12 q^{7} - 16 q^{9} - 32 q^{11} - 12 q^{17} + 24 q^{19} + 4 q^{23} + 60 q^{25} + 40 q^{35} + 124 q^{39} - 176 q^{43} - 40 q^{45} - 72 q^{47} - 24 q^{49} + 140 q^{57} + 152 q^{61} + 48 q^{63} - 148 q^{73} + 376 q^{77} - 468 q^{81} - 208 q^{83} - 84 q^{87} - 184 q^{93} + 392 q^{99}+O(q^{100}) $ 12 * q - 12 * q^7 - 16 * q^9 - 32 * q^11 - 12 * q^17 + 24 * q^19 + 4 * q^23 + 60 * q^25 + 40 * q^35 + 124 * q^39 - 176 * q^43 - 40 * q^45 - 72 * q^47 - 24 * q^49 + 140 * q^57 + 152 * q^61 + 48 * q^63 - 148 * q^73 + 376 * q^77 - 468 * q^81 - 208 * q^83 - 84 * q^87 - 184 * q^93 + 392 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/380\mathbb{Z}\right)^\times$.

$n$	$21$	$77$	$191$
$\chi(n)$	$-1$	$1$	$1$

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
$3$		−	3.42504i		−	1.14168i	−0.821061	−	0.570841i	$-0.806617\pi$
$3$		−	3.42504i		−	1.14168i	0.821061	−	0.570841i	$-0.193383\pi$
$4$		0			0
$5$	2.23607			0.447214
$5$	2.23607			0.447214
$6$		0			0
$7$	9.18599			1.31228			0.656142	−	0.754637i	$-0.272187\pi$
$7$	9.18599			1.31228			0.656142	+	0.754637i	$0.272187\pi$
$8$		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
$13$			10.1235i			0.778734i	0.921083	+	0.389367i	$0.127306\pi$
$13$			10.1235i			0.778734i	−0.921083	+	0.389367i	$0.872694\pi$
$14$		0			0
$15$		−	7.65863i		−	0.510575i
$16$		0			0
$17$	−4.19261			−0.246624			−0.123312	−	0.992368i	$-0.539352\pi$
$17$	−4.19261			−0.246624			−0.123312	+	0.992368i	$0.539352\pi$
$18$		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
$23$	−0.952351			−0.0414066			−0.0207033	−	0.999786i	$-0.506591\pi$
$23$	−0.952351			−0.0414066			−0.0207033	+	0.999786i	$0.506591\pi$
$24$		0			0
$25$	5.00000			0.200000
$26$		0			0
$27$		−	21.4719i		−	0.795254i
$28$		0			0
$29$			9.43887i			0.325478i	0.986669	+	0.162739i	$0.0520329\pi$
$29$			9.43887i			0.325478i	−0.986669	+	0.162739i	$0.947967\pi$
$30$		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
$33$		−	39.1762i		−	1.18716i
$34$		0			0
$35$	20.5405			0.586871
$36$		0			0
$37$		−	20.0029i		−	0.540618i	−0.962774	−	0.270309i	$-0.912874\pi$
$37$		−	20.0029i		−	0.540618i	0.962774	−	0.270309i	$-0.0871258\pi$
$38$		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
$43$	−57.9875			−1.34855			−0.674273	−	0.738482i	$-0.735543\pi$
$43$	−57.9875			−1.34855			−0.674273	+	0.738482i	$0.735543\pi$
$44$		0			0
$45$	−6.10654			−0.135701
$46$		0			0
$47$	−35.3773			−0.752709			−0.376355	−	0.926476i	$-0.622823\pi$
$47$	−35.3773			−0.752709			−0.376355	+	0.926476i	$0.622823\pi$
$48$		0			0
$49$	35.3824			0.722091
$50$		0			0
$51$			14.3599i			0.281566i
$52$		0			0
$53$		−	57.6953i		−	1.08859i	−0.838893	−	0.544296i	$-0.816797\pi$
$53$		−	57.6953i		−	1.08859i	0.838893	−	0.544296i	$-0.183203\pi$
$54$		0			0
$55$	25.5765			0.465027
$56$		0			0
$57$	53.5158	−	37.0260i	0.938873	−	0.649579i
$58$		0			0
$59$			16.7603i			0.284073i	0.989861	+	0.142037i	$0.0453651\pi$
$59$			16.7603i			0.284073i	−0.989861	+	0.142037i	$0.954635\pi$
$60$		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
$63$	−25.0863			−0.398195
$64$		0			0
$65$			22.6369i			0.348261i
$66$		0			0
$67$			6.07252i			0.0906346i	0.998973	+	0.0453173i	$0.0144299\pi$
$67$			6.07252i			0.0906346i	−0.998973	+	0.0453173i	$0.985570\pi$
$68$		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
$73$	−27.9670			−0.383109			−0.191554	−	0.981482i	$-0.561353\pi$
$73$	−27.9670			−0.383109			−0.191554	+	0.981482i	$0.561353\pi$
$74$		0			0
$75$		−	17.1252i		−	0.228336i
$76$		0			0
$77$	105.071			1.36456
$78$		0			0
$79$			65.7698i			0.832529i	0.909244	+	0.416264i	$0.136661\pi$
$79$			65.7698i			0.832529i	−0.909244	+	0.416264i	$0.863339\pi$
$80$		0			0
$81$	−98.1204			−1.21136
$82$		0			0
$83$	60.2358			0.725733			0.362866	−	0.931841i	$-0.381798\pi$
$83$	60.2358			0.725733			0.362866	+	0.931841i	$0.381798\pi$
$84$		0			0
$85$	−9.37497			−0.110294
$86$		0			0
$87$	32.3286			0.371593
$88$		0			0
$89$			97.9852i			1.10096i	0.834849	+	0.550478i	$0.185555\pi$
$89$			97.9852i			1.10096i	−0.834849	+	0.550478i	$0.814445\pi$
$90$		0			0
$91$			92.9948i			1.02192i
$92$		0			0
$93$	−29.1988			−0.313965
$94$		0			0
$95$	24.1727	+	34.9382i	0.254450	+	0.367771i
$96$		0			0
$97$		−	92.0255i		−	0.948717i	−0.880332	−	0.474358i	$-0.842680\pi$
$97$		−	92.0255i		−	0.948717i	0.880332	−	0.474358i	$-0.157320\pi$
$98$		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
$103$			204.226i			1.98278i	0.130940	+	0.991390i	$0.458200\pi$
$103$			204.226i			1.98278i	−0.130940	+	0.991390i	$0.541800\pi$
$104$		0			0
$105$		−	70.3521i		−	0.670020i
$106$		0			0
$107$		−	203.574i		−	1.90256i	−0.308324	−	0.951281i	$-0.599768\pi$
$107$		−	203.574i		−	1.90256i	0.308324	−	0.951281i	$-0.400232\pi$
$108$		0			0
$109$			63.6932i			0.584341i	0.956366	+	0.292171i	$0.0943775\pi$
$109$			63.6932i			0.584341i	−0.956366	+	0.292171i	$0.905622\pi$
$110$		0			0
$111$	−68.5107			−0.617213
$112$		0			0
$113$		−	2.01143i		−	0.0178003i	−0.999960	−	0.00890013i	$-0.997167\pi$
$113$		−	2.01143i		−	0.0178003i	0.999960	−	0.00890013i	$-0.00283304\pi$
$114$		0			0
$115$	−2.12952			−0.0185176
$116$		0			0
$117$		−	27.6467i		−	0.236296i
$118$		0			0
$119$	−38.5133			−0.323641
$120$		0			0
$121$	9.83149			0.0812520
$122$		0			0
$123$	−209.783			−1.70556
$124$		0			0
$125$	11.1803			0.0894427
$126$		0			0
$127$			41.7788i			0.328967i	0.986380	+	0.164484i	$0.0525958\pi$
$127$			41.7788i			0.328967i	−0.986380	+	0.164484i	$0.947404\pi$
$128$		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
$133$	99.3041	+	143.530i	0.746647	+	1.07917i
$134$		0			0
$135$		−	48.0125i		−	0.355648i
$136$		0			0
$137$	30.0467			0.219319			0.109659	−	0.993969i	$-0.465024\pi$
$137$	30.0467			0.219319			0.109659	+	0.993969i	$0.465024\pi$
$138$		0			0
$139$	−210.679			−1.51567			−0.757837	−	0.652444i	$-0.773744\pi$
$139$	−210.679			−1.51567			−0.757837	+	0.652444i	$0.773744\pi$
$140$		0			0
$141$			121.169i			0.859354i
$142$		0			0
$143$			115.795i			0.809753i
$144$		0			0
$145$			21.1060i			0.145558i
$146$		0			0
$147$		−	121.186i		−	0.824397i
$148$		0			0
$149$	−263.989			−1.77174			−0.885868	−	0.463938i	$-0.846436\pi$
$149$	−263.989			−1.77174			−0.885868	+	0.463938i	$0.846436\pi$
$150$		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
$153$	11.4497			0.0748348
$154$		0			0
$155$		−	19.0626i		−	0.122985i
$156$		0			0
$157$	108.051			0.688222			0.344111	−	0.938929i	$-0.388180\pi$
$157$	108.051			0.688222			0.344111	+	0.938929i	$0.388180\pi$
$158$		0			0
$159$	−197.609			−1.24282
$160$		0			0
$161$	−8.74829			−0.0543372
$162$		0			0
$163$	−127.846			−0.784333			−0.392167	−	0.919894i	$-0.628274\pi$
$163$	−127.846			−0.784333			−0.392167	+	0.919894i	$0.628274\pi$
$164$		0			0
$165$		−	87.6006i		−	0.530913i
$166$		0			0
$167$			143.920i			0.861798i	0.902400	+	0.430899i	$0.141803\pi$
$167$			143.920i			0.861798i	−0.902400	+	0.430899i	$0.858197\pi$
$168$		0			0
$169$	66.5138			0.393573
$170$		0			0
$171$	−29.5224	−	42.6703i	−0.172645	−	0.249534i
$172$		0			0
$173$		−	291.098i		−	1.68265i	−0.540532	−	0.841323i	$-0.681777\pi$
$173$		−	291.098i		−	1.68265i	0.540532	−	0.841323i	$-0.318223\pi$
$174$		0			0
$175$	45.9300			0.262457
$176$		0			0
$177$	57.4048			0.324321
$178$		0			0
$179$			91.3328i			0.510239i	0.966910	+	0.255120i	$0.0821148\pi$
$179$			91.3328i			0.510239i	−0.966910	+	0.255120i	$0.917885\pi$
$180$		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
$183$			110.352i			0.603017i
$184$		0			0
$185$		−	44.7277i		−	0.241772i
$186$		0			0
$187$	−47.9558			−0.256448
$188$		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
$193$			131.778i			0.682786i	0.939921	+	0.341393i	$0.110899\pi$
$193$			131.778i			0.682786i	−0.939921	+	0.341393i	$0.889101\pi$
$194$		0			0
$195$	77.5325			0.397603
$196$		0			0
$197$	35.9154			0.182312			0.0911558	−	0.995837i	$-0.470944\pi$
$197$	35.9154			0.182312			0.0911558	+	0.995837i	$0.470944\pi$
$198$		0			0
$199$	121.742			0.611769			0.305885	−	0.952069i	$-0.401048\pi$
$199$	121.742			0.611769			0.305885	+	0.952069i	$0.401048\pi$
$200$		0			0
$201$	20.7986			0.103476
$202$		0			0
$203$			86.7054i			0.427120i
$204$		0			0
$205$		−	136.959i		−	0.668092i
$206$		0			0
$207$	2.60080			0.0125643
$208$		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
$213$	389.330			1.82784
$214$		0			0
$215$	−129.664			−0.603088
$216$		0			0
$217$		−	78.3113i		−	0.360881i
$218$		0			0
$219$			95.7881i			0.437388i
$220$		0			0
$221$		−	42.4441i		−	0.192055i
$222$		0			0
$223$			81.1828i			0.364048i	0.983294	+	0.182024i	$0.0582649\pi$
$223$			81.1828i			0.364048i	−0.983294	+	0.182024i	$0.941735\pi$
$224$		0			0
$225$	−13.6546			−0.0606873
$226$		0			0
$227$		−	337.268i		−	1.48576i	−0.669423	−	0.742882i	$-0.733459\pi$
$227$		−	337.268i		−	1.48576i	0.669423	−	0.742882i	$-0.266541\pi$
$228$		0			0
$229$	−240.819			−1.05161			−0.525806	−	0.850605i	$-0.676236\pi$
$229$	−240.819			−1.05161			−0.525806	+	0.850605i	$0.676236\pi$
$230$		0			0
$231$		−	359.872i		−	1.55789i
$232$		0			0
$233$	136.526			0.585948			0.292974	−	0.956120i	$-0.405355\pi$
$233$	136.526			0.585948			0.292974	+	0.956120i	$0.405355\pi$
$234$		0			0
$235$	−79.1061			−0.336622
$236$		0			0
$237$	225.264			0.950483
$238$		0			0
$239$	−320.734			−1.34198			−0.670991	−	0.741466i	$-0.734131\pi$
$239$	−320.734			−1.34198			−0.670991	+	0.741466i	$0.734131\pi$
$240$		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
$243$			142.820i			0.587736i
$244$		0			0
$245$	79.1175			0.322929
$246$		0			0
$247$	−158.179	+	109.439i	−0.640400	+	0.443074i
$248$		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
$253$	−10.8931			−0.0430559
$254$		0			0
$255$			32.1097i			0.125920i
$256$		0			0
$257$		−	56.5607i		−	0.220080i	−0.993927	−	0.110040i	$-0.964902\pi$
$257$		−	56.5607i		−	0.220080i	0.993927	−	0.110040i	$-0.0350980\pi$
$258$		0			0
$259$		−	183.746i		−	0.709444i
$260$		0			0
$261$		−	25.7769i		−	0.0987620i
$262$		0			0
$263$	280.514			1.06659			0.533296	−	0.845929i	$-0.320953\pi$
$263$	280.514			1.06659			0.533296	+	0.845929i	$0.320953\pi$
$264$		0			0
$265$		−	129.011i		−	0.486833i
$266$		0			0
$267$	335.603			1.25694
$268$		0			0
$269$		−	102.156i		−	0.379763i	−0.981807	−	0.189881i	$-0.939190\pi$
$269$		−	102.156i		−	0.379763i	0.981807	−	0.189881i	$-0.0608103\pi$
$270$		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
$273$	318.511			1.16671
$274$		0			0
$275$	57.1908			0.207967
$276$		0			0
$277$	264.070			0.953323			0.476661	−	0.879087i	$-0.341847\pi$
$277$	264.070			0.953323			0.476661	+	0.879087i	$0.341847\pi$
$278$		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
$283$	−271.524			−0.959450			−0.479725	−	0.877419i	$-0.659264\pi$
$283$	−271.524			−0.959450			−0.479725	+	0.877419i	$0.659264\pi$
$284$		0			0
$285$	119.665	−	82.7927i	0.419877	−	0.290501i
$286$		0			0
$287$		−	562.641i		−	1.96042i
$288$		0			0
$289$	−271.422			−0.939176
$290$		0			0
$291$	−315.191			−1.08313
$292$		0			0
$293$		−	188.495i		−	0.643328i	−0.946854	−	0.321664i	$-0.895758\pi$
$293$		−	188.495i		−	0.643328i	0.946854	−	0.321664i	$-0.104242\pi$
$294$		0			0
$295$			37.4772i			0.127041i
$296$		0			0
$297$		−	245.598i		−	0.826931i
$298$		0			0
$299$		−	9.64117i		−	0.0322447i
$300$		0			0
$301$	−532.672			−1.76968
$302$		0			0
$303$		−	221.843i		−	0.732156i
$304$		0			0
$305$	−72.0443			−0.236211
$306$		0			0
$307$		−	262.646i		−	0.855524i	−0.903891	−	0.427762i	$-0.859302\pi$
$307$		−	262.646i		−	0.855524i	0.903891	−	0.427762i	$-0.140698\pi$
$308$		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
$313$	−483.965			−1.54621			−0.773106	−	0.634277i	$-0.781298\pi$
$313$	−483.965			−1.54621			−0.773106	+	0.634277i	$0.781298\pi$
$314$		0			0
$315$	−56.0946			−0.178078
$316$		0			0
$317$		−	80.3004i		−	0.253313i	−0.991947	−	0.126657i	$-0.959575\pi$
$317$		−	80.3004i		−	0.253313i	0.991947	−	0.126657i	$-0.0404247\pi$
$318$		0			0
$319$			107.963i			0.338443i
$320$		0			0
$321$	−697.251			−2.17212
$322$		0			0
$323$	−45.3238	−	65.5089i	−0.140321	−	0.202814i
$324$		0			0
$325$			50.6177i			0.155747i
$326$		0			0
$327$	218.152			0.667132
$328$		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
$333$			54.6263i			0.164043i
$334$		0			0
$335$			13.5786i			0.0405330i
$336$		0			0
$337$			413.317i			1.22646i	0.789904	+	0.613230i	$0.210130\pi$
$337$			413.317i			1.22646i	−0.789904	+	0.613230i	$0.789870\pi$
$338$		0			0
$339$	−6.88924			−0.0203222
$340$		0			0
$341$		−	97.5112i		−	0.285957i
$342$		0			0
$343$	−125.091			−0.364696
$344$		0			0
$345$			7.29370i			0.0211412i
$346$		0			0
$347$	207.384			0.597648			0.298824	−	0.954308i	$-0.403406\pi$
$347$	207.384			0.597648			0.298824	+	0.954308i	$0.403406\pi$
$348$		0			0
$349$	−176.833			−0.506684			−0.253342	−	0.967377i	$-0.581530\pi$
$349$	−176.833			−0.506684			−0.253342	+	0.967377i	$0.581530\pi$
$350$		0			0
$351$	217.371			0.619291
$352$		0			0
$353$	−398.294			−1.12831			−0.564156	−	0.825668i	$-0.690798\pi$
$353$	−398.294			−1.12831			−0.564156	+	0.825668i	$0.690798\pi$
$354$		0			0
$355$			254.177i			0.715992i
$356$		0			0
$357$			131.910i			0.369495i
$358$		0			0
$359$	89.1125			0.248224			0.124112	−	0.992268i	$-0.460392\pi$
$359$	89.1125			0.248224			0.124112	+	0.992268i	$0.460392\pi$
$360$		0			0
$361$	−127.271	+	337.821i	−0.352552	+	0.935792i
$362$		0			0
$363$		−	33.6733i		−	0.0927639i
$364$		0			0
$365$	−62.5360			−0.171332
$366$		0			0
$367$	−450.345			−1.22710			−0.613550	−	0.789656i	$-0.710259\pi$
$367$	−450.345			−1.22710			−0.613550	+	0.789656i	$0.710259\pi$
$368$		0			0
$369$			167.269i			0.453303i
$370$		0			0
$371$		−	529.989i		−	1.42854i
$372$		0			0
$373$		−	442.338i		−	1.18589i	−0.805242	−	0.592946i	$-0.797965\pi$
$373$		−	442.338i		−	1.18589i	0.805242	−	0.592946i	$-0.202035\pi$
$374$		0			0
$375$		−	38.2932i		−	0.102115i
$376$		0			0
$377$	−95.5549			−0.253461
$378$		0			0
$379$			573.993i			1.51449i	0.653128	+	0.757247i	$0.273456\pi$
$379$			573.993i			1.51449i	−0.653128	+	0.757247i	$0.726544\pi$
$380$		0			0
$381$	143.094			0.375576
$382$		0			0
$383$			159.706i			0.416987i	0.978024	+	0.208494i	$0.0668560\pi$
$383$			159.706i			0.416987i	−0.978024	+	0.208494i	$0.933144\pi$
$384$		0			0
$385$	234.946			0.610248
$386$		0			0
$387$	158.360			0.409198
$388$		0			0
$389$	620.531			1.59519			0.797597	−	0.603190i	$-0.206104\pi$
$389$	620.531			1.59519			0.797597	+	0.603190i	$0.206104\pi$
$390$		0			0
$391$	3.99284			0.0102119
$392$		0			0
$393$			173.607i			0.441748i
$394$		0			0
$395$			147.066i			0.372318i
$396$		0			0
$397$	−765.284			−1.92767			−0.963833	−	0.266506i	$-0.914131\pi$
$397$	−765.284			−1.92767			−0.963833	+	0.266506i	$0.914131\pi$
$398$		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
$403$	86.3040			0.214154
$404$		0			0
$405$	−219.404			−0.541738
$406$		0			0
$407$		−	228.796i		−	0.562152i
$408$		0			0
$409$			225.432i			0.551179i	0.961275	+	0.275589i	$0.0888730\pi$
$409$			225.432i			0.551179i	−0.961275	+	0.275589i	$0.911127\pi$
$410$		0			0
$411$		−	102.911i		−	0.250392i
$412$		0			0
$413$			153.960i			0.372785i
$414$		0			0
$415$	134.691			0.324558
$416$		0			0
$417$			721.584i			1.73042i
$418$		0			0
$419$	314.462			0.750506			0.375253	−	0.926922i	$-0.377556\pi$
$419$	314.462			0.750506			0.375253	+	0.926922i	$0.377556\pi$
$420$		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
$423$	96.6129			0.228399
$424$		0			0
$425$	−20.9631			−0.0493249
$426$		0			0
$427$	−295.965			−0.693127
$428$		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
$433$			37.2858i			0.0861104i	0.999073	+	0.0430552i	$0.0137091\pi$
$433$			37.2858i			0.0861104i	−0.999073	+	0.0430552i	$0.986291\pi$
$434$		0			0
$435$	72.2888			0.166181
$436$		0			0
$437$	−10.2953	−	14.8803i	−0.0235590	−	0.0340511i
$438$		0			0
$439$		−	365.662i		−	0.832944i	−0.909149	−	0.416472i	$-0.863266\pi$
$439$		−	365.662i		−	0.832944i	0.909149	−	0.416472i	$-0.136734\pi$
$440$		0			0
$441$	−96.6269			−0.219109
$442$		0			0
$443$	−177.806			−0.401367			−0.200684	−	0.979656i	$-0.564316\pi$
$443$	−177.806			−0.401367			−0.200684	+	0.979656i	$0.564316\pi$
$444$		0			0
$445$			219.101i			0.492363i
$446$		0			0
$447$			904.172i			2.02276i
$448$		0			0
$449$			204.290i			0.454988i	0.973780	+	0.227494i	$0.0730532\pi$
$449$			204.290i			0.454988i	−0.973780	+	0.227494i	$0.926947\pi$
$450$		0			0
$451$		−	700.586i		−	1.55341i
$452$		0			0
$453$	−309.341			−0.682871
$454$		0			0
$455$			207.943i			0.457017i
$456$		0			0
$457$	−229.796			−0.502837			−0.251418	−	0.967879i	$-0.580897\pi$
$457$	−229.796			−0.502837			−0.251418	+	0.967879i	$0.580897\pi$
$458$		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
$463$	907.359			1.95974			0.979869	−	0.199639i	$-0.0639771\pi$
$463$	907.359			1.95974			0.979869	+	0.199639i	$0.0639771\pi$
$464$		0			0
$465$	−65.2904			−0.140409
$466$		0			0
$467$	−270.399			−0.579013			−0.289507	−	0.957176i	$-0.593491\pi$
$467$	−270.399			−0.579013			−0.289507	+	0.957176i	$0.593491\pi$
$468$		0			0
$469$			55.7821i			0.118938i
$470$		0			0
$471$		−	370.079i		−	0.785730i
$472$		0			0
$473$	−663.270			−1.40226
$474$		0			0
$475$	54.0519	+	78.1242i	0.113793	+	0.164472i
$476$		0			0
$477$			157.562i			0.330318i
$478$		0			0
$479$	−221.920			−0.463299			−0.231649	−	0.972799i	$-0.574412\pi$
$479$	−221.920			−0.463299			−0.231649	+	0.972799i	$0.574412\pi$
$480$		0			0
$481$	202.500			0.420997
$482$		0			0
$483$			29.9633i			0.0620358i
$484$		0			0
$485$		−	205.775i		−	0.424279i
$486$		0			0
$487$			881.561i			1.81019i	0.425212	+	0.905094i	$0.360199\pi$
$487$			881.561i			1.81019i	−0.425212	+	0.905094i	$0.639801\pi$
$488$		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
$493$		−	39.5736i		−	0.0802709i
$494$		0			0
$495$	−69.8476			−0.141106
$496$		0			0
$497$			1044.19i			2.10098i
$498$		0			0
$499$	471.959			0.945809			0.472905	−	0.881114i	$-0.343205\pi$
$499$	471.959			0.945809			0.472905	+	0.881114i	$0.343205\pi$
$500$		0			0
$501$	492.933			0.983899
$502$		0			0
$503$	−888.718			−1.76684			−0.883418	−	0.468586i	$-0.844763\pi$
$503$	−888.718			−1.76684			−0.883418	+	0.468586i	$0.844763\pi$
$504$		0			0
$505$	144.832			0.286796
$506$		0			0
$507$		−	227.813i		−	0.449335i
$508$		0			0
$509$		−	418.738i		−	0.822669i	−0.911485	−	0.411334i	$-0.865063\pi$
$509$		−	418.738i		−	0.822669i	0.911485	−	0.411334i	$-0.134937\pi$
$510$		0			0
$511$	−256.904			−0.502748
$512$		0			0
$513$	335.494	−	232.119i	0.653985	−	0.452473i
$514$		0			0
$515$			456.664i			0.886726i
$516$		0			0
$517$	−404.652			−0.782692
$518$		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
$523$		−	674.426i		−	1.28953i	−0.764380	−	0.644766i	$-0.776955\pi$
$523$		−	674.426i		−	1.28953i	0.764380	−	0.644766i	$-0.223045\pi$
$524$		0			0
$525$		−	157.312i		−	0.299642i
$526$		0			0
$527$			35.7424i			0.0678223i
$528$		0			0
$529$	−528.093			−0.998285
$530$		0			0
$531$		−	45.7712i		−	0.0861981i
$532$		0			0
$533$	620.066			1.16335
$534$		0			0
$535$		−	455.206i		−	0.850852i
$536$		0			0
$537$	312.819			0.582531
$538$		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
$543$	−625.657			−1.15222
$544$		0			0
$545$			142.422i			0.261325i
$546$		0			0
$547$			183.171i			0.334865i	0.985884	+	0.167432i	$0.0535476\pi$
$547$			183.171i			0.334865i	−0.985884	+	0.167432i	$0.946452\pi$
$548$		0			0
$549$	87.9882			0.160270
$550$		0			0
$551$	−147.481	+	102.038i	−0.267660	+	0.185187i
$552$		0			0
$553$			604.161i			1.09251i
$554$		0			0
$555$	−153.194			−0.276026
$556$		0			0
$557$	−713.778			−1.28147			−0.640734	−	0.767763i	$-0.721370\pi$
$557$	−713.778			−1.28147			−0.640734	+	0.767763i	$0.721370\pi$
$558$		0			0
$559$		−	587.039i		−	1.05016i
$560$		0			0
$561$			164.251i			0.292782i
$562$		0			0
$563$			316.069i			0.561401i	0.959795	+	0.280700i	$0.0905667\pi$
$563$			316.069i			0.561401i	−0.959795	+	0.280700i	$0.909433\pi$
$564$		0			0
$565$		−	4.49770i		−	0.00796052i
$566$		0			0
$567$	−901.333			−1.58965
$568$		0			0
$569$		−	883.884i		−	1.55340i	−0.629871	−	0.776699i	$-0.716892\pi$
$569$		−	883.884i		−	1.55340i	0.629871	−	0.776699i	$-0.283108\pi$
$570$		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
$573$		−	1227.56i		−	2.14233i
$574$		0			0
$575$	−4.76175			−0.00828131
$576$		0			0
$577$	380.352			0.659188			0.329594	−	0.944123i	$-0.393088\pi$
$577$	380.352			0.659188			0.329594	+	0.944123i	$0.393088\pi$
$578$		0			0
$579$	451.344			0.779524
$580$		0			0
$581$	553.326			0.952368
$582$		0			0
$583$		−	659.929i		−	1.13195i
$584$		0			0
$585$		−	61.8198i		−	0.105675i
$586$		0			0
$587$	123.630			0.210613			0.105306	−	0.994440i	$-0.466418\pi$
$587$	123.630			0.210613			0.105306	+	0.994440i	$0.466418\pi$
$588$		0			0
$589$	133.203	−	92.1593i	0.226151	−	0.156467i
$590$		0			0
$591$		−	123.012i		−	0.208142i
$592$		0			0
$593$	191.622			0.323141			0.161570	−	0.986861i	$-0.448344\pi$
$593$	191.622			0.323141			0.161570	+	0.986861i	$0.448344\pi$
$594$		0			0
$595$	−86.1184			−0.144737
$596$		0			0
$597$		−	416.972i		−	0.698446i
$598$		0			0
$599$			277.360i			0.463038i	0.972830	+	0.231519i	$0.0743695\pi$
$599$			277.360i			0.463038i	−0.972830	+	0.231519i	$0.925630\pi$
$600$		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
$603$		−	16.5836i		−	0.0275018i
$604$		0			0
$605$	21.9839			0.0363370
$606$		0			0
$607$		−	1035.05i		−	1.70519i	−0.522568	−	0.852597i	$-0.675026\pi$
$607$		−	1035.05i		−	1.70519i	0.522568	−	0.852597i	$-0.324974\pi$
$608$		0			0
$609$	296.970			0.487635
$610$		0			0
$611$		−	358.144i		−	0.586160i
$612$		0			0
$613$	965.052			1.57431			0.787155	−	0.616756i	$-0.211553\pi$
$613$	965.052			1.57431			0.787155	+	0.616756i	$0.211553\pi$
$614$		0			0
$615$	−469.090			−0.762748
$616$		0			0
$617$	−967.261			−1.56768			−0.783842	−	0.620960i	$-0.786743\pi$
$617$	−967.261			−1.56768			−0.783842	+	0.620960i	$0.786743\pi$
$618$		0			0
$619$	995.539			1.60830			0.804151	−	0.594425i	$-0.202620\pi$
$619$	995.539			1.60830			0.804151	+	0.594425i	$0.202620\pi$
$620$		0			0
$621$			20.4487i			0.0329287i
$622$		0			0
$623$			900.091i			1.44477i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	612.122	−	423.510i	0.976271	−	0.675454i
$628$		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
$633$	934.922			1.47697
$634$		0			0
$635$			93.4203i			0.147119i
$636$		0			0
$637$			358.196i			0.562317i
$638$		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
$643$	−509.087			−0.791738			−0.395869	−	0.918307i	$-0.629557\pi$
$643$	−509.087			−0.791738			−0.395869	+	0.918307i	$0.629557\pi$
$644$		0			0
$645$			444.105i			0.688534i
$646$		0			0
$647$	1046.09			1.61683			0.808416	−	0.588611i	$-0.200325\pi$
$647$	1046.09			1.61683			0.808416	+	0.588611i	$0.200325\pi$
$648$		0			0
$649$			191.707i			0.295389i
$650$		0			0
$651$	−268.220			−0.412012
$652$		0			0
$653$	351.804			0.538750			0.269375	−	0.963035i	$-0.413183\pi$
$653$	351.804			0.538750			0.269375	+	0.963035i	$0.413183\pi$
$654$		0			0
$655$	−113.341			−0.173039
$656$		0			0
$657$	76.3757			0.116249
$658$		0			0
$659$		−	1238.23i		−	1.87895i	−0.342621	−	0.939474i	$-0.611315\pi$
$659$		−	1238.23i		−	1.87895i	0.342621	−	0.939474i	$-0.388685\pi$
$660$		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
$663$	−145.373			−0.219265
$664$		0			0
$665$	222.051	+	320.942i	0.333911	+	0.482620i
$666$		0			0
$667$		−	8.98912i		−	0.0134769i
$668$		0			0
$669$	278.055			0.415627
$670$		0			0
$671$	−368.528			−0.549222
$672$		0			0
$673$		−	1163.57i		−	1.72892i	−0.502698	−	0.864462i	$-0.667659\pi$
$673$		−	1163.57i		−	1.72892i	0.502698	−	0.864462i	$-0.332341\pi$
$674$		0			0
$675$		−	107.359i		−	0.159051i
$676$		0			0
$677$			71.7621i			0.106000i	0.998595	+	0.0530001i	$0.0168784\pi$
$677$			71.7621i			0.106000i	−0.998595	+	0.0530001i	$0.983122\pi$
$678$		0			0
$679$		−	845.345i		−	1.24499i
$680$		0			0
$681$	−1155.16			−1.69627
$682$		0			0
$683$			558.227i			0.817316i	0.912688	+	0.408658i	$0.134003\pi$
$683$			558.227i			0.817316i	−0.912688	+	0.408658i	$0.865997\pi$
$684$		0			0
$685$	67.1865			0.0980824
$686$		0			0
$687$			824.816i			1.20061i
$688$		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
$693$	−286.941			−0.414056
$694$		0			0
$695$	−471.092			−0.677830
$696$		0			0
$697$			256.797i			0.368432i
$698$		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
$703$	312.541	−	216.238i	0.444582	−	0.307594i
$704$		0			0
$705$			270.942i			0.384315i
$706$		0			0
$707$	594.985			0.841563
$708$		0			0
$709$	−780.564			−1.10094			−0.550468	−	0.834856i	$-0.685551\pi$
$709$	−780.564			−1.10094			−0.550468	+	0.834856i	$0.685551\pi$
$710$		0			0
$711$		−	179.612i		−	0.252620i
$712$		0			0
$713$			8.11886i			0.0113869i
$714$		0			0
$715$			258.925i			0.362133i
$716$		0			0
$717$			1098.53i			1.53212i
$718$		0			0
$719$	−606.339			−0.843308			−0.421654	−	0.906757i	$-0.638550\pi$
$719$	−606.339			−0.843308			−0.421654	+	0.906757i	$0.638550\pi$
$720$		0			0
$721$			1876.02i			2.60197i
$722$		0			0
$723$	1229.72			1.70086
$724$		0			0
$725$			47.1944i			0.0650957i
$726$		0			0
$727$	266.418			0.366463			0.183231	−	0.983070i	$-0.441344\pi$
$727$	266.418			0.366463			0.183231	+	0.983070i	$0.441344\pi$
$728$		0			0
$729$	−393.919			−0.540355
$730$		0			0
$731$	243.119			0.332584
$732$		0			0
$733$	951.526			1.29813			0.649063	−	0.760735i	$-0.275161\pi$
$733$	951.526			1.29813			0.649063	+	0.760735i	$0.275161\pi$
$734$		0			0
$735$		−	270.981i		−	0.368682i
$736$		0			0
$737$			69.4584i			0.0942448i
$738$		0			0
$739$	504.613			0.682832			0.341416	−	0.939912i	$-0.389094\pi$
$739$	504.613			0.682832			0.341416	+	0.939912i	$0.389094\pi$
$740$		0			0
$741$	374.835	+	541.769i	0.505850	+	0.731133i
$742$		0			0
$743$			1081.91i			1.45613i	0.685506	+	0.728067i	$0.259581\pi$
$743$			1081.91i			1.45613i	−0.685506	+	0.728067i	$0.740419\pi$
$744$		0			0
$745$	−590.296			−0.792344
$746$		0			0
$747$	−164.500			−0.220214
$748$		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
$753$		−	355.472i		−	0.472074i
$754$		0			0
$755$		−	201.956i		−	0.267491i
$756$		0			0
$757$	−264.661			−0.349619			−0.174809	−	0.984602i	$-0.555931\pi$
$757$	−264.661			−0.349619			−0.174809	+	0.984602i	$0.555931\pi$
$758$		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
$763$			585.085i			0.766822i
$764$		0			0
$765$	25.6024			0.0334671
$766$		0			0
$767$	−169.674			−0.221218
$768$		0			0
$769$	−962.529			−1.25166			−0.625832	−	0.779958i	$-0.715240\pi$
$769$	−962.529			−1.25166			−0.625832	+	0.779958i	$0.715240\pi$
$770$		0			0
$771$	−193.723			−0.251262
$772$		0			0
$773$		−	27.0726i		−	0.0350227i	−0.999847	−	0.0175114i	$-0.994426\pi$
$773$		−	27.0726i		−	0.0350227i	0.999847	−	0.0175114i	$-0.00557433\pi$
$774$		0			0
$775$		−	42.6254i		−	0.0550005i
$776$		0			0
$777$	−629.338			−0.809959
$778$		0			0
$779$	957.019	−	662.134i	1.22852	−	0.849980i
$780$		0			0
$781$			1300.19i			1.66478i
$782$		0			0
$783$	202.670			0.258838
$784$		0			0
$785$	241.609			0.307782
$786$		0			0
$787$			49.7504i			0.0632152i	0.999500	+	0.0316076i	$0.0100627\pi$
$787$			49.7504i			0.0632152i	−0.999500	+	0.0316076i	$0.989937\pi$
$788$		0			0
$789$		−	960.771i		−	1.21771i
$790$		0			0
$791$		−	18.4770i		−	0.0233590i
$792$		0			0
$793$		−	326.172i		−	0.411314i
$794$		0			0
$795$	−441.867			−0.555808
$796$		0			0
$797$		−	221.385i		−	0.277773i	−0.990308	−	0.138886i	$-0.955648\pi$
$797$		−	221.385i		−	0.277773i	0.990308	−	0.138886i	$-0.0443523\pi$
$798$		0			0
$799$	148.324			0.185636
$800$		0			0
$801$		−	267.590i		−	0.334070i
$802$		0			0
$803$	−319.890			−0.398369
$804$		0			0
$805$	−19.5618			−0.0243003
$806$		0			0
$807$	−349.889			−0.433568
$808$		0			0
$809$	−933.918			−1.15441			−0.577205	−	0.816599i	$-0.695857\pi$
$809$	−933.918			−1.15441			−0.577205	+	0.816599i	$0.695857\pi$
$810$		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
$813$		−	1630.71i		−	2.00580i
$814$		0			0
$815$	−285.873			−0.350764
$816$		0			0
$817$	−626.867	−	906.045i	−0.767279	−	1.10899i
$818$		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
$823$	495.548			0.602124			0.301062	−	0.953605i	$-0.402659\pi$
$823$	495.548			0.602124			0.301062	+	0.953605i	$0.402659\pi$
$824$		0			0
$825$		−	195.881i		−	0.237432i
$826$		0			0
$827$			1036.35i			1.25314i	0.779365	+	0.626571i	$0.215542\pi$
$827$			1036.35i			1.25314i	−0.779365	+	0.626571i	$0.784458\pi$
$828$		0			0
$829$			355.400i			0.428710i	0.976756	+	0.214355i	$0.0687649\pi$
$829$			355.400i			0.428710i	−0.976756	+	0.214355i	$0.931235\pi$
$830$		0			0
$831$		−	904.453i		−	1.08839i
$832$		0			0
$833$	−148.345			−0.178085
$834$		0			0
$835$			321.815i			0.385408i
$836$		0			0
$837$	−183.049			−0.218697
$838$		0			0
$839$			429.992i			0.512505i	0.966610	+	0.256253i	$0.0824879\pi$
$839$			429.992i			0.512505i	−0.966610	+	0.256253i	$0.917512\pi$
$840$		0			0
$841$	751.908			0.894064
$842$		0			0
$843$	−830.391			−0.985043
$844$		0			0
$845$	148.729			0.176011
$846$		0			0
$847$	90.3120			0.106626
$848$		0			0
$849$			929.983i			1.09539i
$850$		0			0
$851$			19.0497i			0.0223851i
$852$		0			0
$853$	557.873			0.654013			0.327007	−	0.945022i	$-0.393960\pi$
$853$	557.873			0.654013			0.327007	+	0.945022i	$0.393960\pi$
$854$		0			0
$855$	−66.0140	−	95.4137i	−0.0772094	−	0.111595i
$856$		0			0
$857$		−	813.712i		−	0.949489i	−0.880124	−	0.474745i	$-0.842540\pi$
$857$		−	813.712i		−	0.949489i	0.880124	−	0.474745i	$-0.157460\pi$
$858$		0			0
$859$	−368.216			−0.428657			−0.214328	−	0.976762i	$-0.568756\pi$
$859$	−368.216			−0.428657			−0.214328	+	0.976762i	$0.568756\pi$
$860$		0			0
$861$	−1927.07			−2.23818
$862$		0			0
$863$			963.303i			1.11623i	0.829765	+	0.558113i	$0.188474\pi$
$863$			963.303i			1.11623i	−0.829765	+	0.558113i	$0.811526\pi$
$864$		0			0
$865$		−	650.915i		−	0.752502i
$866$		0			0
$867$			929.632i			1.07224i
$868$		0			0
$869$			752.285i			0.865691i
$870$		0			0
$871$	−61.4754			−0.0705802
$872$		0			0
$873$			251.315i			0.287875i
$874$		0			0
$875$	102.703			0.117374
$876$		0			0
$877$			938.800i			1.07047i	0.844704	+	0.535234i	$0.179777\pi$
$877$			938.800i			1.07047i	−0.844704	+	0.535234i	$0.820223\pi$
$878$		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
$883$	−326.997			−0.370325			−0.185163	−	0.982708i	$-0.559281\pi$
$883$	−326.997			−0.370325			−0.185163	+	0.982708i	$0.559281\pi$
$884$		0			0
$885$	128.361			0.145041
$886$		0			0
$887$		−	366.885i		−	0.413624i	−0.978381	−	0.206812i	$-0.933691\pi$
$887$		−	366.885i		−	0.413624i	0.978381	−	0.206812i	$-0.0663089\pi$
$888$		0			0
$889$			383.780i			0.431699i
$890$		0			0
$891$	−1122.32			−1.25961
$892$		0			0
$893$	−382.442	−	552.765i	−0.428267	−	0.618998i
$894$		0			0
$895$			204.226i			0.228186i
$896$		0			0
$897$	−33.0214			−0.0368132
$898$		0			0
$899$	80.4671			0.0895073
$900$		0			0
$901$			241.894i			0.268473i
$902$		0			0
$903$			1824.43i			2.02041i
$904$		0			0
$905$		−	408.465i		−	0.451343i
$906$		0			0
$907$			948.571i			1.04583i	0.852384	+	0.522917i	$0.175156\pi$
$907$			948.571i			1.04583i	−0.852384	+	0.522917i	$0.824844\pi$
$908$		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
$913$	688.987			0.754641
$914$		0			0
$915$			246.755i			0.269677i
$916$		0			0
$917$	−465.615			−0.507759
$918$		0			0
$919$	−20.3298			−0.0221216			−0.0110608	−	0.999939i	$-0.503521\pi$
$919$	−20.3298			−0.0221216			−0.0110608	+	0.999939i	$0.503521\pi$
$920$		0			0
$921$	−899.573			−0.976735
$922$		0			0
$923$	−1150.76			−1.24676
$924$		0			0
$925$		−	100.014i		−	0.108124i
$926$		0			0
$927$		−	557.727i		−	0.601648i
$928$		0			0
$929$	−356.003			−0.383210			−0.191605	−	0.981472i	$-0.561369\pi$
$929$	−356.003			−0.383210			−0.191605	+	0.981472i	$0.561369\pi$
$930$		0			0
$931$	382.498	+	552.845i	0.410846	+	0.593819i
$932$		0			0
$933$		−	1206.76i		−	1.29342i
$934$		0			0
$935$	−107.232			−0.114687
$936$		0			0
$937$	−822.306			−0.877595			−0.438797	−	0.898586i	$-0.644595\pi$
$937$	−822.306			−0.877595			−0.438797	+	0.898586i	$0.644595\pi$
$938$		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
$943$			58.3314i			0.0618572i
$944$		0			0
$945$		−	441.043i		−	0.466712i
$946$		0			0
$947$	−1226.54			−1.29518			−0.647591	−	0.761988i	$-0.724223\pi$
$947$	−1226.54			−1.29518			−0.647591	+	0.761988i	$0.724223\pi$
$948$		0			0
$949$		−	283.125i		−	0.298340i
$950$		0			0
$951$	−275.032			−0.289203
$952$		0			0
$953$			1329.78i			1.39537i	0.716406	+	0.697683i	$0.245786\pi$
$953$			1329.78i			1.39537i	−0.716406	+	0.697683i	$0.754214\pi$
$954$		0			0
$955$	801.420			0.839183
$956$		0			0
$957$	369.779			0.386394
$958$		0			0
$959$	276.009			0.287809
$960$		0			0
$961$	888.323			0.924374
$962$		0			0
$963$			555.946i			0.577307i
$964$		0			0
$965$			294.664i			0.305351i
$966$		0			0
$967$	1006.28			1.04062			0.520309	−	0.853978i	$-0.325817\pi$
$967$	1006.28			1.04062			0.520309	+	0.853978i	$0.325817\pi$
$968$		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
$973$	−1935.29			−1.98900
$974$		0			0
$975$	173.368			0.177813
$976$		0			0
$977$		−	750.278i		−	0.767940i	−0.923346	−	0.383970i	$-0.874557\pi$
$977$		−	750.278i		−	0.767940i	0.923346	−	0.383970i	$-0.125443\pi$
$978$		0			0
$979$			1120.77i			1.14481i
$980$		0			0
$981$		−	173.941i		−	0.177310i
$982$		0			0
$983$			603.980i			0.614425i	0.951641	+	0.307213i	$0.0993963\pi$
$983$			603.980i			0.614425i	−0.951641	+	0.307213i	$0.900604\pi$
$984$		0			0
$985$	80.3092			0.0815322
$986$		0			0
$987$			1113.06i			1.12772i
$988$		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
$993$	1999.00			2.01309
$994$		0			0
$995$	272.224			0.273592
$996$		0			0
$997$	1616.79			1.62165			0.810827	−	0.585285i	$-0.199018\pi$
$997$	1616.79			1.62165			0.810827	+	0.585285i	$0.199018\pi$
$998$		0			0
$999$	−429.498			−0.429928

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.e.a.341.3	✓	12	1.1	even	1	trivial
380.3.e.a.341.10	yes	12	19.18	odd	2	inner
1520.3.h.b.721.3		12	76.75	even	2
1520.3.h.b.721.10		12	4.3	odd	2
1900.3.e.f.1101.3		12	95.94	odd	2
1900.3.e.f.1101.10		12	5.4	even	2
1900.3.g.c.949.5		24	5.2	odd	4
1900.3.g.c.949.6		24	95.18	even	4
1900.3.g.c.949.19		24	95.37	even	4
1900.3.g.c.949.20		24	5.3	odd	4
3420.3.o.a.721.5		12	57.56	even	2
3420.3.o.a.721.6		12	3.2	odd	2

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:	\( N \)	\(=\)	\( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	380.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.42504i q^{3} +2.23607 q^{5} +9.18599 q^{7} -2.73093 q^{9} +O(q^{10})\)	\(q-3.42504i q^{3} +2.23607 q^{5} +9.18599 q^{7} -2.73093 q^{9} +11.4382 q^{11} +10.1235i q^{13} -7.65863i q^{15} -4.19261 q^{17} +(10.8104 + 15.6248i) q^{19} -31.4624i q^{21} -0.952351 q^{23} +5.00000 q^{25} -21.4719i q^{27} +9.43887i q^{29} -8.52508i q^{31} -39.1762i q^{33} +20.5405 q^{35} -20.0029i q^{37} +34.6736 q^{39} -61.2499i q^{41} -57.9875 q^{43} -6.10654 q^{45} -35.3773 q^{47} +35.3824 q^{49} +14.3599i q^{51} -57.6953i q^{53} +25.5765 q^{55} +(53.5158 - 37.0260i) q^{57} +16.7603i q^{59} -32.2192 q^{61} -25.0863 q^{63} +22.6369i q^{65} +6.07252i q^{67} +3.26184i q^{69} +113.671i q^{71} -27.9670 q^{73} -17.1252i q^{75} +105.071 q^{77} +65.7698i q^{79} -98.1204 q^{81} +60.2358 q^{83} -9.37497 q^{85} +32.3286 q^{87} +97.9852i q^{89} +92.9948i q^{91} -29.1988 q^{93} +(24.1727 + 34.9382i) q^{95} -92.0255i q^{97} -31.2368 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{7} - 16 q^{9}+O(q^{10}) \) 12 * q - 12 * q^7 - 16 * q^9	\( 12 q - 12 q^{7} - 16 q^{9} - 32 q^{11} - 12 q^{17} + 24 q^{19} + 4 q^{23} + 60 q^{25} + 40 q^{35} + 124 q^{39} - 176 q^{43} - 40 q^{45} - 72 q^{47} - 24 q^{49} + 140 q^{57} + 152 q^{61} + 48 q^{63} - 148 q^{73} + 376 q^{77} - 468 q^{81} - 208 q^{83} - 84 q^{87} - 184 q^{93} + 392 q^{99}+O(q^{100}) \) 12 * q - 12 * q^7 - 16 * q^9 - 32 * q^11 - 12 * q^17 + 24 * q^19 + 4 * q^23 + 60 * q^25 + 40 * q^35 + 124 * q^39 - 176 * q^43 - 40 * q^45 - 72 * q^47 - 24 * q^49 + 140 * q^57 + 152 * q^61 + 48 * q^63 - 148 * q^73 + 376 * q^77 - 468 * q^81 - 208 * q^83 - 84 * q^87 - 184 * q^93 + 392 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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