LMFDB - Embedded newform 380.3.e.a.341.5

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			341.5
Root			$-2.03369i$ of defining polynomial
Character	$\chi$	$=$	380.341
Dual form			380.3.e.a.341.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.03369i q^{3} -2.23607 q^{5} +4.27843 q^{7} +4.86411 q^{9} +O(q^{10})$	$q-2.03369i q^{3} -2.23607 q^{5} +4.27843 q^{7} +4.86411 q^{9} +5.47217 q^{11} -0.634081i q^{13} +4.54746i q^{15} -1.69271 q^{17} +(-6.46218 - 17.8673i) q^{19} -8.70099i q^{21} +25.9867 q^{23} +5.00000 q^{25} -28.1953i q^{27} -48.8696i q^{29} -4.47053i q^{31} -11.1287i q^{33} -9.56686 q^{35} +6.12833i q^{37} -1.28952 q^{39} -16.9988i q^{41} +0.690441 q^{43} -10.8765 q^{45} +23.0801 q^{47} -30.6950 q^{49} +3.44245i q^{51} -54.3934i q^{53} -12.2361 q^{55} +(-36.3365 + 13.1420i) q^{57} +0.251179i q^{59} +39.5570 q^{61} +20.8108 q^{63} +1.41785i q^{65} +96.3354i q^{67} -52.8489i q^{69} +16.0568i q^{71} +70.0291 q^{73} -10.1684i q^{75} +23.4123 q^{77} +74.6211i q^{79} -13.5634 q^{81} +1.29672 q^{83} +3.78502 q^{85} -99.3856 q^{87} -141.348i q^{89} -2.71287i q^{91} -9.09167 q^{93} +(14.4499 + 39.9525i) q^{95} +125.110i q^{97} +26.6173 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	2.03369i		−	0.677896i	−0.940805	−	0.338948i	$-0.889929\pi$
$3$		−	2.03369i		−	0.677896i	0.940805	−	0.338948i	$-0.110071\pi$
$4$		0			0
$5$	−2.23607			−0.447214
$5$	−2.23607			−0.447214
$6$		0			0
$7$	4.27843			0.611204			0.305602	−	0.952159i	$-0.401142\pi$
$7$	4.27843			0.611204			0.305602	+	0.952159i	$0.401142\pi$
$8$		0			0
$9$	4.86411			0.540457
$10$		0			0
$11$	5.47217			0.497470			0.248735	−	0.968572i	$-0.419985\pi$
$11$	5.47217			0.497470			0.248735	+	0.968572i	$0.419985\pi$
$12$		0			0
$13$		−	0.634081i		−	0.0487754i	−0.999703	−	0.0243877i	$-0.992236\pi$
$13$		−	0.634081i		−	0.0487754i	0.999703	−	0.0243877i	$-0.00776362\pi$
$14$		0			0
$15$			4.54746i			0.303164i
$16$		0			0
$17$	−1.69271			−0.0995712			−0.0497856	−	0.998760i	$-0.515854\pi$
$17$	−1.69271			−0.0995712			−0.0497856	+	0.998760i	$0.515854\pi$
$18$		0			0
$19$	−6.46218	−	17.8673i	−0.340115	−	0.940384i
$19$	−6.46218	−	17.8673i	−0.340115	−	0.940384i
$20$		0			0
$21$		−	8.70099i		−	0.414333i
$22$		0			0
$23$	25.9867			1.12986			0.564929	−	0.825139i	$-0.308904\pi$
$23$	25.9867			1.12986			0.564929	+	0.825139i	$0.308904\pi$
$24$		0			0
$25$	5.00000			0.200000
$26$		0			0
$27$		−	28.1953i		−	1.04427i
$28$		0			0
$29$		−	48.8696i		−	1.68516i	−0.538571	−	0.842580i	$-0.681036\pi$
$29$		−	48.8696i		−	1.68516i	0.538571	−	0.842580i	$-0.318964\pi$
$30$		0			0
$31$		−	4.47053i		−	0.144211i	−0.997397	−	0.0721054i	$-0.977028\pi$
$31$		−	4.47053i		−	0.144211i	0.997397	−	0.0721054i	$-0.0229718\pi$
$32$		0			0
$33$		−	11.1287i		−	0.337233i
$34$		0			0
$35$	−9.56686			−0.273339
$36$		0			0
$37$			6.12833i			0.165630i	0.996565	+	0.0828152i	$0.0263911\pi$
$37$			6.12833i			0.165630i	−0.996565	+	0.0828152i	$0.973609\pi$
$38$		0			0
$39$	−1.28952			−0.0330647
$40$		0			0
$41$		−	16.9988i		−	0.414605i	−0.978277	−	0.207303i	$-0.933532\pi$
$41$		−	16.9988i		−	0.414605i	0.978277	−	0.207303i	$-0.0664685\pi$
$42$		0			0
$43$	0.690441			0.0160568			0.00802838	−	0.999968i	$-0.497444\pi$
$43$	0.690441			0.0160568			0.00802838	+	0.999968i	$0.497444\pi$
$44$		0			0
$45$	−10.8765			−0.241700
$46$		0			0
$47$	23.0801			0.491066			0.245533	−	0.969388i	$-0.421037\pi$
$47$	23.0801			0.491066			0.245533	+	0.969388i	$0.421037\pi$
$48$		0			0
$49$	−30.6950			−0.626429
$50$		0			0
$51$			3.44245i			0.0674989i
$52$		0			0
$53$		−	54.3934i		−	1.02629i	−0.858302	−	0.513145i	$-0.828480\pi$
$53$		−	54.3934i		−	1.02629i	0.858302	−	0.513145i	$-0.171520\pi$
$54$		0			0
$55$	−12.2361			−0.222475
$56$		0			0
$57$	−36.3365	+	13.1420i	−0.637482	+	0.230562i
$58$		0			0
$59$			0.251179i			0.00425727i	0.999998	+	0.00212863i	$0.000677566\pi$
$59$			0.251179i			0.00425727i	−0.999998	+	0.00212863i	$0.999322\pi$
$60$		0			0
$61$	39.5570			0.648475			0.324237	−	0.945976i	$-0.394892\pi$
$61$	39.5570			0.648475			0.324237	+	0.945976i	$0.394892\pi$
$62$		0			0
$63$	20.8108			0.330330
$64$		0			0
$65$			1.41785i			0.0218130i
$66$		0			0
$67$			96.3354i			1.43784i	0.695092	+	0.718921i	$0.255363\pi$
$67$			96.3354i			1.43784i	−0.695092	+	0.718921i	$0.744637\pi$
$68$		0			0
$69$		−	52.8489i		−	0.765926i
$70$		0			0
$71$			16.0568i			0.226152i	0.993586	+	0.113076i	$0.0360704\pi$
$71$			16.0568i			0.226152i	−0.993586	+	0.113076i	$0.963930\pi$
$72$		0			0
$73$	70.0291			0.959302			0.479651	−	0.877459i	$-0.340763\pi$
$73$	70.0291			0.959302			0.479651	+	0.877459i	$0.340763\pi$
$74$		0			0
$75$		−	10.1684i		−	0.135579i
$76$		0			0
$77$	23.4123			0.304056
$78$		0			0
$79$			74.6211i			0.944570i	0.881446	+	0.472285i	$0.156571\pi$
$79$			74.6211i			0.944570i	−0.881446	+	0.472285i	$0.843429\pi$
$80$		0			0
$81$	−13.5634			−0.167449
$82$		0			0
$83$	1.29672			0.0156232			0.00781159	−	0.999969i	$-0.497513\pi$
$83$	1.29672			0.0156232			0.00781159	+	0.999969i	$0.497513\pi$
$84$		0			0
$85$	3.78502			0.0445296
$86$		0			0
$87$	−99.3856			−1.14236
$88$		0			0
$89$		−	141.348i		−	1.58818i	−0.607800	−	0.794090i	$-0.707948\pi$
$89$		−	141.348i		−	1.58818i	0.607800	−	0.794090i	$-0.292052\pi$
$90$		0			0
$91$		−	2.71287i		−	0.0298117i
$92$		0			0
$93$	−9.09167			−0.0977598
$94$		0			0
$95$	14.4499	+	39.9525i	0.152104	+	0.420553i
$96$		0			0
$97$			125.110i			1.28980i	0.764268	+	0.644898i	$0.223100\pi$
$97$			125.110i			1.28980i	−0.764268	+	0.644898i	$0.776900\pi$
$98$		0			0
$99$	26.6173			0.268861
$100$		0			0
$101$	102.871			1.01853			0.509263	−	0.860611i	$-0.329918\pi$
$101$	102.871			1.01853			0.509263	+	0.860611i	$0.329918\pi$
$102$		0			0
$103$			11.6950i			0.113544i	0.998387	+	0.0567720i	$0.0180808\pi$
$103$			11.6950i			0.113544i	−0.998387	+	0.0567720i	$0.981919\pi$
$104$		0			0
$105$			19.4560i			0.185295i
$106$		0			0
$107$			189.935i			1.77509i	0.460722	+	0.887545i	$0.347591\pi$
$107$			189.935i			1.77509i	−0.460722	+	0.887545i	$0.652409\pi$
$108$		0			0
$109$		−	43.6838i		−	0.400769i	−0.979717	−	0.200384i	$-0.935781\pi$
$109$		−	43.6838i		−	0.400769i	0.979717	−	0.200384i	$-0.0642191\pi$
$110$		0			0
$111$	12.4631			0.112280
$112$		0			0
$113$			100.896i			0.892884i	0.894813	+	0.446442i	$0.147309\pi$
$113$			100.896i			0.892884i	−0.894813	+	0.446442i	$0.852691\pi$
$114$		0			0
$115$	−58.1081			−0.505288
$116$		0			0
$117$		−	3.08424i		−	0.0263610i
$118$		0			0
$119$	−7.24214			−0.0608584
$120$		0			0
$121$	−91.0554			−0.752524
$122$		0			0
$123$	−34.5703			−0.281059
$124$		0			0
$125$	−11.1803			−0.0894427
$126$		0			0
$127$			140.225i			1.10414i	0.833799	+	0.552068i	$0.186161\pi$
$127$			140.225i			1.10414i	−0.833799	+	0.552068i	$0.813839\pi$
$128$		0			0
$129$		−	1.40414i		−	0.0108848i
$130$		0			0
$131$	−158.825			−1.21241			−0.606203	−	0.795310i	$-0.707308\pi$
$131$	−158.825			−1.21241			−0.606203	+	0.795310i	$0.707308\pi$
$132$		0			0
$133$	−27.6480	−	76.4440i	−0.207879	−	0.574767i
$134$		0			0
$135$			63.0466i			0.467012i
$136$		0			0
$137$	−74.8377			−0.546261			−0.273130	−	0.961977i	$-0.588059\pi$
$137$	−74.8377			−0.546261			−0.273130	+	0.961977i	$0.588059\pi$
$138$		0			0
$139$	−17.4925			−0.125846			−0.0629228	−	0.998018i	$-0.520042\pi$
$139$	−17.4925			−0.125846			−0.0629228	+	0.998018i	$0.520042\pi$
$140$		0			0
$141$		−	46.9377i		−	0.332892i
$142$		0			0
$143$		−	3.46980i		−	0.0242643i
$144$		0			0
$145$			109.276i			0.753626i
$146$		0			0
$147$			62.4241i			0.424654i
$148$		0			0
$149$	163.077			1.09448			0.547239	−	0.836976i	$-0.315679\pi$
$149$	163.077			1.09448			0.547239	+	0.836976i	$0.315679\pi$
$150$		0			0
$151$			147.367i			0.975940i	0.872861	+	0.487970i	$0.162262\pi$
$151$			147.367i			0.975940i	−0.872861	+	0.487970i	$0.837738\pi$
$152$		0			0
$153$	−8.23354			−0.0538140
$154$		0			0
$155$			9.99641i			0.0644930i
$156$		0			0
$157$	−208.993			−1.33117			−0.665583	−	0.746324i	$-0.731817\pi$
$157$	−208.993			−1.33117			−0.665583	+	0.746324i	$0.731817\pi$
$158$		0			0
$159$	−110.619			−0.695718
$160$		0			0
$161$	111.182			0.690574
$162$		0			0
$163$	−87.2475			−0.535261			−0.267630	−	0.963522i	$-0.586241\pi$
$163$	−87.2475			−0.535261			−0.267630	+	0.963522i	$0.586241\pi$
$164$		0			0
$165$			24.8845i			0.150815i
$166$		0			0
$167$			97.0793i			0.581313i	0.956827	+	0.290657i	$0.0938738\pi$
$167$			97.0793i			0.581313i	−0.956827	+	0.290657i	$0.906126\pi$
$168$		0			0
$169$	168.598			0.997621
$170$		0			0
$171$	−31.4328	−	86.9086i	−0.183817	−	0.508237i
$172$		0			0
$173$		−	70.7451i		−	0.408931i	−0.978874	−	0.204466i	$-0.934454\pi$
$173$		−	70.7451i		−	0.408931i	0.978874	−	0.204466i	$-0.0655456\pi$
$174$		0			0
$175$	21.3921			0.122241
$176$		0			0
$177$	0.510819			0.00288598
$178$		0			0
$179$		−	176.104i		−	0.983822i	−0.870646	−	0.491911i	$-0.836299\pi$
$179$		−	176.104i		−	0.983822i	0.870646	−	0.491911i	$-0.163701\pi$
$180$		0			0
$181$		−	57.5577i		−	0.317998i	−0.987279	−	0.158999i	$-0.949173\pi$
$181$		−	57.5577i		−	0.317998i	0.987279	−	0.158999i	$-0.0508267\pi$
$182$		0			0
$183$		−	80.4465i		−	0.439598i
$184$		0			0
$185$		−	13.7034i		−	0.0740722i
$186$		0			0
$187$	−9.26280			−0.0495337
$188$		0			0
$189$		−	120.632i		−	0.638262i
$190$		0			0
$191$	−145.682			−0.762731			−0.381365	−	0.924424i	$-0.624546\pi$
$191$	−145.682			−0.762731			−0.381365	+	0.924424i	$0.624546\pi$
$192$		0			0
$193$			64.1316i			0.332288i	0.986101	+	0.166144i	$0.0531317\pi$
$193$			64.1316i			0.332288i	−0.986101	+	0.166144i	$0.946868\pi$
$194$		0			0
$195$	2.88346			0.0147870
$196$		0			0
$197$	17.9219			0.0909739			0.0454870	−	0.998965i	$-0.485516\pi$
$197$	17.9219			0.0909739			0.0454870	+	0.998965i	$0.485516\pi$
$198$		0			0
$199$	209.349			1.05200			0.526002	−	0.850483i	$-0.323690\pi$
$199$	209.349			1.05200			0.526002	+	0.850483i	$0.323690\pi$
$200$		0			0
$201$	195.916			0.974707
$202$		0			0
$203$		−	209.085i		−	1.02998i
$204$		0			0
$205$			38.0105i			0.185417i
$206$		0			0
$207$	126.402			0.610640
$208$		0			0
$209$	−35.3621	−	97.7729i	−0.169197	−	0.467813i
$210$		0			0
$211$			232.975i			1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$211$			232.975i			1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$212$		0			0
$213$	32.6546			0.153308
$214$		0			0
$215$	−1.54387			−0.00718081
$216$		0			0
$217$		−	19.1269i		−	0.0881422i
$218$		0			0
$219$		−	142.417i		−	0.650307i
$220$		0			0
$221$			1.07332i			0.00485663i
$222$		0			0
$223$		−	68.2129i		−	0.305887i	−0.988235	−	0.152944i	$-0.951125\pi$
$223$		−	68.2129i		−	0.305887i	0.988235	−	0.152944i	$-0.0488753\pi$
$224$		0			0
$225$	24.3206			0.108091
$226$		0			0
$227$			143.128i			0.630520i	0.949005	+	0.315260i	$0.102092\pi$
$227$			143.128i			0.630520i	−0.949005	+	0.315260i	$0.897908\pi$
$228$		0			0
$229$	135.889			0.593403			0.296701	−	0.954970i	$-0.404113\pi$
$229$	135.889			0.593403			0.296701	+	0.954970i	$0.404113\pi$
$230$		0			0
$231$		−	47.6133i		−	0.206118i
$232$		0			0
$233$	−270.764			−1.16208			−0.581039	−	0.813876i	$-0.697354\pi$
$233$	−270.764			−1.16208			−0.581039	+	0.813876i	$0.697354\pi$
$234$		0			0
$235$	−51.6087			−0.219611
$236$		0			0
$237$	151.756			0.640320
$238$		0			0
$239$	300.071			1.25553			0.627764	−	0.778404i	$-0.283970\pi$
$239$	300.071			1.25553			0.627764	+	0.778404i	$0.283970\pi$
$240$		0			0
$241$		−	177.442i		−	0.736275i	−0.929771	−	0.368138i	$-0.879996\pi$
$241$		−	177.442i		−	0.736275i	0.929771	−	0.368138i	$-0.120004\pi$
$242$		0			0
$243$		−	226.174i		−	0.930757i
$244$		0			0
$245$	68.6362			0.280148
$246$		0			0
$247$	−11.3293	+	4.09754i	−0.0458676	+	0.0165892i
$248$		0			0
$249$		−	2.63713i		−	0.0105909i
$250$		0			0
$251$	−80.0537			−0.318939			−0.159469	−	0.987203i	$-0.550978\pi$
$251$	−80.0537			−0.318939			−0.159469	+	0.987203i	$0.550978\pi$
$252$		0			0
$253$	142.204			0.562070
$254$		0			0
$255$		−	7.69754i		−	0.0301864i
$256$		0			0
$257$			147.621i			0.574402i	0.957870	+	0.287201i	$0.0927248\pi$
$257$			147.621i			0.574402i	−0.957870	+	0.287201i	$0.907275\pi$
$258$		0			0
$259$			26.2196i			0.101234i
$260$		0			0
$261$		−	237.707i		−	0.910757i
$262$		0			0
$263$	344.458			1.30973			0.654864	−	0.755747i	$-0.272726\pi$
$263$	344.458			1.30973			0.654864	+	0.755747i	$0.272726\pi$
$264$		0			0
$265$			121.627i			0.458971i
$266$		0			0
$267$	−287.458			−1.07662
$268$		0			0
$269$		−	190.965i		−	0.709908i	−0.934884	−	0.354954i	$-0.884497\pi$
$269$		−	190.965i		−	0.709908i	0.934884	−	0.354954i	$-0.115503\pi$
$270$		0			0
$271$	−265.131			−0.978342			−0.489171	−	0.872188i	$-0.662701\pi$
$271$	−265.131			−0.978342			−0.489171	+	0.872188i	$0.662701\pi$
$272$		0			0
$273$	−5.51713			−0.0202093
$274$		0			0
$275$	27.3608			0.0994940
$276$		0			0
$277$	−87.0682			−0.314326			−0.157163	−	0.987573i	$-0.550235\pi$
$277$	−87.0682			−0.314326			−0.157163	+	0.987573i	$0.550235\pi$
$278$		0			0
$279$		−	21.7452i		−	0.0779397i
$280$		0			0
$281$		−	233.988i		−	0.832697i	−0.909205	−	0.416348i	$-0.863310\pi$
$281$		−	233.988i		−	0.832697i	0.909205	−	0.416348i	$-0.136690\pi$
$282$		0			0
$283$	190.662			0.673718			0.336859	−	0.941555i	$-0.390635\pi$
$283$	190.662			0.673718			0.336859	+	0.941555i	$0.390635\pi$
$284$		0			0
$285$	81.2509	−	29.3865i	0.285091	−	0.103111i
$286$		0			0
$287$		−	72.7283i		−	0.253409i
$288$		0			0
$289$	−286.135			−0.990086
$290$		0			0
$291$	254.435			0.874348
$292$		0			0
$293$		−	230.400i		−	0.786348i	−0.919464	−	0.393174i	$-0.871377\pi$
$293$		−	230.400i		−	0.786348i	0.919464	−	0.393174i	$-0.128623\pi$
$294$		0			0
$295$		−	0.561653i		−	0.00190391i
$296$		0			0
$297$		−	154.289i		−	0.519493i
$298$		0			0
$299$		−	16.4777i		−	0.0551093i
$300$		0			0
$301$	2.95400			0.00981396
$302$		0			0
$303$		−	209.208i		−	0.690454i
$304$		0			0
$305$	−88.4521			−0.290007
$306$		0			0
$307$			370.642i			1.20730i	0.797248	+	0.603652i	$0.206288\pi$
$307$			370.642i			1.20730i	−0.797248	+	0.603652i	$0.793712\pi$
$308$		0			0
$309$	23.7840			0.0769710
$310$		0			0
$311$	−12.2047			−0.0392433			−0.0196216	−	0.999807i	$-0.506246\pi$
$311$	−12.2047			−0.0392433			−0.0196216	+	0.999807i	$0.506246\pi$
$312$		0			0
$313$	−263.418			−0.841591			−0.420795	−	0.907156i	$-0.638249\pi$
$313$	−263.418			−0.841591			−0.420795	+	0.907156i	$0.638249\pi$
$314$		0			0
$315$	−46.5343			−0.147728
$316$		0			0
$317$		−	328.524i		−	1.03635i	−0.855274	−	0.518177i	$-0.826611\pi$
$317$		−	328.524i		−	1.03635i	0.855274	−	0.518177i	$-0.173389\pi$
$318$		0			0
$319$		−	267.423i		−	0.838316i
$320$		0			0
$321$	386.268			1.20333
$322$		0			0
$323$	10.9386	+	30.2442i	0.0338656	+	0.0936352i
$324$		0			0
$325$		−	3.17040i		−	0.00975509i
$326$		0			0
$327$	−88.8392			−0.271679
$328$		0			0
$329$	98.7466			0.300142
$330$		0			0
$331$		−	29.2287i		−	0.0883043i	−0.999025	−	0.0441522i	$-0.985941\pi$
$331$		−	29.2287i		−	0.0883043i	0.999025	−	0.0441522i	$-0.0140586\pi$
$332$		0			0
$333$			29.8089i			0.0895162i
$334$		0			0
$335$		−	215.412i		−	0.643022i
$336$		0			0
$337$			283.044i			0.839894i	0.907549	+	0.419947i	$0.137951\pi$
$337$			283.044i			0.839894i	−0.907549	+	0.419947i	$0.862049\pi$
$338$		0			0
$339$	205.191			0.605282
$340$		0			0
$341$		−	24.4635i		−	0.0717405i
$342$		0			0
$343$	−340.970			−0.994080
$344$		0			0
$345$			118.174i			0.342533i
$346$		0			0
$347$	−214.566			−0.618347			−0.309174	−	0.951006i	$-0.600052\pi$
$347$	−214.566			−0.618347			−0.309174	+	0.951006i	$0.600052\pi$
$348$		0			0
$349$	−627.491			−1.79797			−0.898984	−	0.437981i	$-0.855694\pi$
$349$	−627.491			−1.79797			−0.898984	+	0.437981i	$0.855694\pi$
$350$		0			0
$351$	−17.8781			−0.0509347
$352$		0			0
$353$	61.3501			0.173796			0.0868982	−	0.996217i	$-0.472305\pi$
$353$	61.3501			0.173796			0.0868982	+	0.996217i	$0.472305\pi$
$354$		0			0
$355$		−	35.9041i		−	0.101138i
$356$		0			0
$357$			14.7283i			0.0412556i
$358$		0			0
$359$	303.982			0.846748			0.423374	−	0.905955i	$-0.360846\pi$
$359$	303.982			0.846748			0.423374	+	0.905955i	$0.360846\pi$
$360$		0			0
$361$	−277.481	+	230.923i	−0.768644	+	0.639677i
$362$		0			0
$363$			185.178i			0.510133i
$364$		0			0
$365$	−156.590			−0.429013
$366$		0			0
$367$	542.618			1.47852			0.739262	−	0.673418i	$-0.235175\pi$
$367$	542.618			1.47852			0.739262	+	0.673418i	$0.235175\pi$
$368$		0			0
$369$		−	82.6842i		−	0.224076i
$370$		0			0
$371$		−	232.718i		−	0.627273i
$372$		0			0
$373$			568.106i			1.52307i	0.648123	+	0.761536i	$0.275554\pi$
$373$			568.106i			1.52307i	−0.648123	+	0.761536i	$0.724446\pi$
$374$		0			0
$375$			22.7373i			0.0606329i
$376$		0			0
$377$	−30.9873			−0.0821944
$378$		0			0
$379$			322.531i			0.851005i	0.904957	+	0.425502i	$0.139903\pi$
$379$			322.531i			0.851005i	−0.904957	+	0.425502i	$0.860097\pi$
$380$		0			0
$381$	285.175			0.748490
$382$		0			0
$383$		−	134.256i		−	0.350539i	−0.984521	−	0.175269i	$-0.943920\pi$
$383$		−	134.256i		−	0.350539i	0.984521	−	0.175269i	$-0.0560796\pi$
$384$		0			0
$385$	−52.3515			−0.135978
$386$		0			0
$387$	3.35838			0.00867800
$388$		0			0
$389$	−47.9667			−0.123308			−0.0616539	−	0.998098i	$-0.519638\pi$
$389$	−47.9667			−0.123308			−0.0616539	+	0.998098i	$0.519638\pi$
$390$		0			0
$391$	−43.9880			−0.112501
$392$		0			0
$393$			323.001i			0.821885i
$394$		0			0
$395$		−	166.858i		−	0.422425i
$396$		0			0
$397$	161.948			0.407928			0.203964	−	0.978978i	$-0.434617\pi$
$397$	161.948			0.407928			0.203964	+	0.978978i	$0.434617\pi$
$398$		0			0
$399$	−155.463	+	56.2273i	−0.389632	+	0.140921i
$400$		0			0
$401$			203.228i			0.506802i	0.967361	+	0.253401i	$0.0815492\pi$
$401$			203.228i			0.506802i	−0.967361	+	0.253401i	$0.918451\pi$
$402$		0			0
$403$	−2.83468			−0.00703394
$404$		0			0
$405$	30.3286			0.0748854
$406$		0			0
$407$			33.5352i			0.0823962i
$408$		0			0
$409$			651.844i			1.59375i	0.604144	+	0.796875i	$0.293515\pi$
$409$			651.844i			1.59375i	−0.604144	+	0.796875i	$0.706485\pi$
$410$		0			0
$411$			152.197i			0.370308i
$412$		0			0
$413$			1.07465i			0.00260206i
$414$		0			0
$415$	−2.89956			−0.00698690
$416$		0			0
$417$			35.5744i			0.0853102i
$418$		0			0
$419$	−787.718			−1.88000			−0.939998	−	0.341180i	$-0.889173\pi$
$419$	−787.718			−1.88000			−0.939998	+	0.341180i	$0.889173\pi$
$420$		0			0
$421$		−	93.6475i		−	0.222441i	−0.993796	−	0.111220i	$-0.964524\pi$
$421$		−	93.6475i		−	0.222441i	0.993796	−	0.111220i	$-0.0354759\pi$
$422$		0			0
$423$	112.264			0.265400
$424$		0			0
$425$	−8.46356			−0.0199142
$426$		0			0
$427$	169.242			0.396351
$428$		0			0
$429$	−7.05648			−0.0164487
$430$		0			0
$431$			546.360i			1.26766i	0.773474	+	0.633828i	$0.218517\pi$
$431$			546.360i			1.26766i	−0.773474	+	0.633828i	$0.781483\pi$
$432$		0			0
$433$			843.569i			1.94820i	0.226128	+	0.974098i	$0.427393\pi$
$433$			843.569i			1.94820i	−0.226128	+	0.974098i	$0.572607\pi$
$434$		0			0
$435$	222.233			0.510880
$436$		0			0
$437$	−167.931	−	464.313i	−0.384281	−	1.06250i
$438$		0			0
$439$		−	118.320i		−	0.269521i	−0.990878	−	0.134761i	$-0.956973\pi$
$439$		−	118.320i		−	0.269521i	0.990878	−	0.134761i	$-0.0430266\pi$
$440$		0			0
$441$	−149.304			−0.338558
$442$		0			0
$443$	883.490			1.99433			0.997167	−	0.0752227i	$-0.0239668\pi$
$443$	883.490			1.99433			0.997167	+	0.0752227i	$0.0239668\pi$
$444$		0			0
$445$			316.064i			0.710256i
$446$		0			0
$447$		−	331.648i		−	0.741943i
$448$		0			0
$449$			358.093i			0.797534i	0.917052	+	0.398767i	$0.130562\pi$
$449$			358.093i			0.797534i	−0.917052	+	0.398767i	$0.869438\pi$
$450$		0			0
$451$		−	93.0204i		−	0.206254i
$452$		0			0
$453$	299.698			0.661585
$454$		0			0
$455$			6.06616i			0.0133322i
$456$		0			0
$457$	−329.340			−0.720657			−0.360328	−	0.932825i	$-0.617335\pi$
$457$	−329.340			−0.720657			−0.360328	+	0.932825i	$0.617335\pi$
$458$		0			0
$459$			47.7265i			0.103979i
$460$		0			0
$461$	−613.352			−1.33048			−0.665240	−	0.746629i	$-0.731671\pi$
$461$	−613.352			−1.33048			−0.665240	+	0.746629i	$0.731671\pi$
$462$		0			0
$463$	−142.260			−0.307258			−0.153629	−	0.988129i	$-0.549096\pi$
$463$	−142.260			−0.307258			−0.153629	+	0.988129i	$0.549096\pi$
$464$		0			0
$465$	20.3296			0.0437195
$466$		0			0
$467$	435.333			0.932191			0.466096	−	0.884734i	$-0.345660\pi$
$467$	435.333			0.932191			0.466096	+	0.884734i	$0.345660\pi$
$468$		0			0
$469$			412.164i			0.878814i
$470$		0			0
$471$			425.027i			0.902392i
$472$		0			0
$473$	3.77821			0.00798776
$474$		0			0
$475$	−32.3109	−	89.3365i	−0.0680229	−	0.188077i
$476$		0			0
$477$		−	264.575i		−	0.554666i
$478$		0			0
$479$	−80.5389			−0.168140			−0.0840698	−	0.996460i	$-0.526792\pi$
$479$	−80.5389			−0.168140			−0.0840698	+	0.996460i	$0.526792\pi$
$480$		0			0
$481$	3.88585			0.00807870
$482$		0			0
$483$		−	226.110i		−	0.468137i
$484$		0			0
$485$		−	279.755i		−	0.576815i
$486$		0			0
$487$		−	168.483i		−	0.345960i	−0.984925	−	0.172980i	$-0.944660\pi$
$487$		−	168.483i		−	0.345960i	0.984925	−	0.172980i	$-0.0553396\pi$
$488$		0			0
$489$			177.434i			0.362851i
$490$		0			0
$491$	467.906			0.952965			0.476482	−	0.879184i	$-0.341912\pi$
$491$	467.906			0.952965			0.476482	+	0.879184i	$0.341912\pi$
$492$		0			0
$493$			82.7222i			0.167793i
$494$		0			0
$495$	−59.5180			−0.120238
$496$		0			0
$497$			68.6980i			0.138225i
$498$		0			0
$499$	−487.521			−0.976997			−0.488498	−	0.872565i	$-0.662455\pi$
$499$	−487.521			−0.976997			−0.488498	+	0.872565i	$0.662455\pi$
$500$		0			0
$501$	197.429			0.394070
$502$		0			0
$503$	−500.167			−0.994369			−0.497184	−	0.867645i	$-0.665633\pi$
$503$	−500.167			−0.994369			−0.497184	+	0.867645i	$0.665633\pi$
$504$		0			0
$505$	−230.027			−0.455498
$506$		0			0
$507$		−	342.876i		−	0.676283i
$508$		0			0
$509$			254.504i			0.500007i	0.968245	+	0.250004i	$0.0804318\pi$
$509$			254.504i			0.500007i	−0.968245	+	0.250004i	$0.919568\pi$
$510$		0			0
$511$	299.614			0.586329
$512$		0			0
$513$	−503.773	+	182.203i	−0.982014	+	0.355171i
$514$		0			0
$515$		−	26.1509i		−	0.0507784i
$516$		0			0
$517$	126.298			0.244291
$518$		0			0
$519$	−143.873			−0.277213
$520$		0			0
$521$			676.089i			1.29767i	0.760927	+	0.648837i	$0.224744\pi$
$521$			676.089i			1.29767i	−0.760927	+	0.648837i	$0.775256\pi$
$522$		0			0
$523$			34.9971i			0.0669162i	0.999440	+	0.0334581i	$0.0106520\pi$
$523$			34.9971i			0.0669162i	−0.999440	+	0.0334581i	$0.989348\pi$
$524$		0			0
$525$		−	43.5049i		−	0.0828666i
$526$		0			0
$527$			7.56732i			0.0143592i
$528$		0			0
$529$	146.311			0.276580
$530$		0			0
$531$			1.22176i			0.00230087i
$532$		0			0
$533$	−10.7786			−0.0202226
$534$		0			0
$535$		−	424.707i		−	0.793844i
$536$		0			0
$537$	−358.141			−0.666929
$538$		0			0
$539$	−167.968			−0.311630
$540$		0			0
$541$	411.260			0.760184			0.380092	−	0.924949i	$-0.375892\pi$
$541$	411.260			0.760184			0.380092	+	0.924949i	$0.375892\pi$
$542$		0			0
$543$	−117.054			−0.215570
$544$		0			0
$545$			97.6799i			0.179229i
$546$		0			0
$547$		−	355.828i		−	0.650508i	−0.945627	−	0.325254i	$-0.894550\pi$
$547$		−	355.828i		−	0.650508i	0.945627	−	0.325254i	$-0.105450\pi$
$548$		0			0
$549$	192.410			0.350473
$550$		0			0
$551$	−873.168	+	315.804i	−1.58470	+	0.573147i
$552$		0			0
$553$			319.261i			0.577325i
$554$		0			0
$555$	−27.8683			−0.0502132
$556$		0			0
$557$	823.586			1.47861			0.739305	−	0.673370i	$-0.235154\pi$
$557$	823.586			1.47861			0.739305	+	0.673370i	$0.235154\pi$
$558$		0			0
$559$		−	0.437795i		−	0.000783176i
$560$		0			0
$561$			18.8376i			0.0335787i
$562$		0			0
$563$		−	753.891i		−	1.33906i	−0.742785	−	0.669531i	$-0.766495\pi$
$563$		−	753.891i		−	1.33906i	0.742785	−	0.669531i	$-0.233505\pi$
$564$		0			0
$565$		−	225.610i		−	0.399310i
$566$		0			0
$567$	−58.0299			−0.102345
$568$		0			0
$569$		−	183.744i		−	0.322924i	−0.986879	−	0.161462i	$-0.948379\pi$
$569$		−	183.744i		−	0.322924i	0.986879	−	0.161462i	$-0.0516209\pi$
$570$		0			0
$571$	−544.748			−0.954024			−0.477012	−	0.878897i	$-0.658280\pi$
$571$	−544.748			−0.954024			−0.477012	+	0.878897i	$0.658280\pi$
$572$		0			0
$573$			296.271i			0.517052i
$574$		0			0
$575$	129.934			0.225972
$576$		0			0
$577$	−68.9024			−0.119415			−0.0597074	−	0.998216i	$-0.519017\pi$
$577$	−68.9024			−0.119415			−0.0597074	+	0.998216i	$0.519017\pi$
$578$		0			0
$579$	130.424			0.225257
$580$		0			0
$581$	5.54794			0.00954895
$582$		0			0
$583$		−	297.650i		−	0.510548i
$584$		0			0
$585$			6.89657i			0.0117890i
$586$		0			0
$587$	110.222			0.187771			0.0938857	−	0.995583i	$-0.470071\pi$
$587$	110.222			0.187771			0.0938857	+	0.995583i	$0.470071\pi$
$588$		0			0
$589$	−79.8763	+	28.8894i	−0.135613	+	0.0490482i
$590$		0			0
$591$		−	36.4475i		−	0.0616708i
$592$		0			0
$593$	−15.0895			−0.0254460			−0.0127230	−	0.999919i	$-0.504050\pi$
$593$	−15.0895			−0.0254460			−0.0127230	+	0.999919i	$0.504050\pi$
$594$		0			0
$595$	16.1939			0.0272167
$596$		0			0
$597$		−	425.750i		−	0.713150i
$598$		0			0
$599$			557.545i			0.930793i	0.885102	+	0.465396i	$0.154088\pi$
$599$			557.545i			0.930793i	−0.885102	+	0.465396i	$0.845912\pi$
$600$		0			0
$601$		−	409.791i		−	0.681849i	−0.940091	−	0.340924i	$-0.889260\pi$
$601$		−	409.791i		−	0.681849i	0.940091	−	0.340924i	$-0.110740\pi$
$602$		0			0
$603$			468.586i			0.777092i
$604$		0			0
$605$	203.606			0.336539
$606$		0			0
$607$		−	720.249i		−	1.18657i	−0.804992	−	0.593286i	$-0.797830\pi$
$607$		−	720.249i		−	1.18657i	0.804992	−	0.593286i	$-0.202170\pi$
$608$		0			0
$609$	−425.214			−0.698217
$610$		0			0
$611$		−	14.6346i		−	0.0239520i
$612$		0			0
$613$	−662.266			−1.08037			−0.540184	−	0.841547i	$-0.681646\pi$
$613$	−662.266			−1.08037			−0.540184	+	0.841547i	$0.681646\pi$
$614$		0			0
$615$	77.3015			0.125694
$616$		0			0
$617$	656.881			1.06464			0.532319	−	0.846544i	$-0.321321\pi$
$617$	656.881			1.06464			0.532319	+	0.846544i	$0.321321\pi$
$618$		0			0
$619$	454.752			0.734656			0.367328	−	0.930092i	$-0.380273\pi$
$619$	454.752			0.734656			0.367328	+	0.930092i	$0.380273\pi$
$620$		0			0
$621$		−	732.703i		−	1.17988i
$622$		0			0
$623$		−	604.748i		−	0.970703i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−198.839	+	71.9155i	−0.317128	+	0.114698i
$628$		0			0
$629$		−	10.3735i		−	0.0164920i
$630$		0			0
$631$	283.098			0.448650			0.224325	−	0.974514i	$-0.427982\pi$
$631$	283.098			0.448650			0.224325	+	0.974514i	$0.427982\pi$
$632$		0			0
$633$	473.799			0.748497
$634$		0			0
$635$		−	313.553i		−	0.493785i
$636$		0			0
$637$			19.4631i			0.0305544i
$638$		0			0
$639$			78.1022i			0.122226i
$640$		0			0
$641$		−	470.172i		−	0.733497i	−0.930320	−	0.366749i	$-0.880471\pi$
$641$		−	470.172i		−	0.733497i	0.930320	−	0.366749i	$-0.119529\pi$
$642$		0			0
$643$	916.438			1.42525			0.712627	−	0.701544i	$-0.247505\pi$
$643$	916.438			1.42525			0.712627	+	0.701544i	$0.247505\pi$
$644$		0			0
$645$			3.13976i			0.00486784i
$646$		0			0
$647$	−582.366			−0.900101			−0.450051	−	0.893003i	$-0.648594\pi$
$647$	−582.366			−0.900101			−0.450051	+	0.893003i	$0.648594\pi$
$648$		0			0
$649$			1.37449i			0.00211786i
$650$		0			0
$651$	−38.8980			−0.0597512
$652$		0			0
$653$	−228.094			−0.349301			−0.174651	−	0.984630i	$-0.555880\pi$
$653$	−228.094			−0.349301			−0.174651	+	0.984630i	$0.555880\pi$
$654$		0			0
$655$	355.144			0.542204
$656$		0			0
$657$	340.629			0.518462
$658$		0			0
$659$			250.379i			0.379938i	0.981790	+	0.189969i	$0.0608387\pi$
$659$			250.379i			0.379938i	−0.981790	+	0.189969i	$0.939161\pi$
$660$		0			0
$661$		−	830.133i		−	1.25587i	−0.778264	−	0.627937i	$-0.783900\pi$
$661$		−	830.133i		−	1.25587i	0.778264	−	0.627937i	$-0.216100\pi$
$662$		0			0
$663$	2.18279			0.00329229
$664$		0			0
$665$	61.8227	+	170.934i	0.0929665	+	0.257043i
$666$		0			0
$667$		−	1269.96i		−	1.90399i
$668$		0			0
$669$	−138.724			−0.207360
$670$		0			0
$671$	216.462			0.322597
$672$		0			0
$673$			1120.67i			1.66519i	0.553882	+	0.832595i	$0.313146\pi$
$673$			1120.67i			1.66519i	−0.553882	+	0.832595i	$0.686854\pi$
$674$		0			0
$675$		−	140.976i		−	0.208854i
$676$		0			0
$677$		−	500.474i		−	0.739253i	−0.929181	−	0.369626i	$-0.879486\pi$
$677$		−	500.474i		−	0.739253i	0.929181	−	0.369626i	$-0.120514\pi$
$678$		0			0
$679$			535.275i			0.788329i
$680$		0			0
$681$	291.078			0.427427
$682$		0			0
$683$		−	253.863i		−	0.371688i	−0.982579	−	0.185844i	$-0.940498\pi$
$683$		−	253.863i		−	0.371688i	0.982579	−	0.185844i	$-0.0595020\pi$
$684$		0			0
$685$	167.342			0.244295
$686$		0			0
$687$		−	276.356i		−	0.402265i
$688$		0			0
$689$	−34.4898			−0.0500577
$690$		0			0
$691$	210.478			0.304599			0.152300	−	0.988334i	$-0.451332\pi$
$691$	210.478			0.304599			0.152300	+	0.988334i	$0.451332\pi$
$692$		0			0
$693$	113.880			0.164329
$694$		0			0
$695$	39.1145			0.0562799
$696$		0			0
$697$			28.7741i			0.0412828i
$698$		0			0
$699$			550.650i			0.787768i
$700$		0			0
$701$	146.797			0.209410			0.104705	−	0.994503i	$-0.466610\pi$
$701$	146.797			0.209410			0.104705	+	0.994503i	$0.466610\pi$
$702$		0			0
$703$	109.497	−	39.6023i	0.155756	−	0.0563333i
$704$		0			0
$705$			104.956i			0.148874i
$706$		0			0
$707$	440.127			0.622527
$708$		0			0
$709$	803.083			1.13270			0.566349	−	0.824166i	$-0.308356\pi$
$709$	803.083			1.13270			0.566349	+	0.824166i	$0.308356\pi$
$710$		0			0
$711$			362.965i			0.510500i
$712$		0			0
$713$		−	116.175i		−	0.162938i
$714$		0			0
$715$			7.75870i			0.0108513i
$716$		0			0
$717$		−	610.251i		−	0.851117i
$718$		0			0
$719$	637.968			0.887299			0.443650	−	0.896200i	$-0.353683\pi$
$719$	637.968			0.887299			0.443650	+	0.896200i	$0.353683\pi$
$720$		0			0
$721$			50.0364i			0.0693986i
$722$		0			0
$723$	−360.862			−0.499118
$724$		0			0
$725$		−	244.348i		−	0.337032i
$726$		0			0
$727$	−319.305			−0.439210			−0.219605	−	0.975589i	$-0.570477\pi$
$727$	−319.305			−0.439210			−0.219605	+	0.975589i	$0.570477\pi$
$728$		0			0
$729$	−582.037			−0.798405
$730$		0			0
$731$	−1.16872			−0.00159879
$732$		0			0
$733$	−1185.22			−1.61694			−0.808470	−	0.588537i	$-0.799704\pi$
$733$	−1185.22			−1.61694			−0.808470	+	0.588537i	$0.799704\pi$
$734$		0			0
$735$		−	139.585i		−	0.189911i
$736$		0			0
$737$			527.163i			0.715283i
$738$		0			0
$739$	182.023			0.246311			0.123155	−	0.992387i	$-0.460699\pi$
$739$	182.023			0.246311			0.123155	+	0.992387i	$0.460699\pi$
$740$		0			0
$741$	8.33312	+	23.0403i	0.0112458	+	0.0310935i
$742$		0			0
$743$		−	827.162i		−	1.11327i	−0.830756	−	0.556637i	$-0.812092\pi$
$743$		−	827.162i		−	1.11327i	0.830756	−	0.556637i	$-0.187908\pi$
$744$		0			0
$745$	−364.652			−0.489466
$746$		0			0
$747$	6.30741			0.00844366
$748$		0			0
$749$			812.622i			1.08494i
$750$		0			0
$751$		−	1154.06i		−	1.53669i	−0.640034	−	0.768346i	$-0.721080\pi$
$751$		−	1154.06i		−	1.53669i	0.640034	−	0.768346i	$-0.278920\pi$
$752$		0			0
$753$			162.804i			0.216207i
$754$		0			0
$755$		−	329.522i		−	0.436453i
$756$		0			0
$757$	1361.46			1.79850			0.899250	−	0.437435i	$-0.144113\pi$
$757$	1361.46			1.79850			0.899250	+	0.437435i	$0.144113\pi$
$758$		0			0
$759$		−	289.198i		−	0.381025i
$760$		0			0
$761$	−646.107			−0.849023			−0.424512	−	0.905422i	$-0.639554\pi$
$761$	−646.107			−0.849023			−0.424512	+	0.905422i	$0.639554\pi$
$762$		0			0
$763$		−	186.898i		−	0.244952i
$764$		0			0
$765$	18.4108			0.0240663
$766$		0			0
$767$	0.159268			0.000207650
$768$		0			0
$769$	1226.28			1.59464			0.797318	−	0.603559i	$-0.206251\pi$
$769$	1226.28			1.59464			0.797318	+	0.603559i	$0.206251\pi$
$770$		0			0
$771$	300.216			0.389385
$772$		0			0
$773$			544.100i			0.703881i	0.936022	+	0.351940i	$0.114478\pi$
$773$			544.100i			0.703881i	−0.936022	+	0.351940i	$0.885522\pi$
$774$		0			0
$775$		−	22.3527i		−	0.0288421i
$776$		0			0
$777$	53.3225			0.0686261
$778$		0			0
$779$	−303.723	+	109.849i	−0.389888	+	0.141013i
$780$		0			0
$781$			87.8656i			0.112504i
$782$		0			0
$783$	−1377.89			−1.75976
$784$		0			0
$785$	467.323			0.595316
$786$		0			0
$787$		−	1035.63i		−	1.31592i	−0.753054	−	0.657958i	$-0.771420\pi$
$787$		−	1035.63i		−	1.31592i	0.753054	−	0.657958i	$-0.228580\pi$
$788$		0			0
$789$		−	700.521i		−	0.887859i
$790$		0			0
$791$			431.676i			0.545734i
$792$		0			0
$793$		−	25.0823i		−	0.0316296i
$794$		0			0
$795$	247.352			0.311134
$796$		0			0
$797$		−	1237.45i		−	1.55264i	−0.630341	−	0.776318i	$-0.717085\pi$
$797$		−	1237.45i		−	1.55264i	0.630341	−	0.776318i	$-0.282915\pi$
$798$		0			0
$799$	−39.0680			−0.0488961
$800$		0			0
$801$		−	687.533i		−	0.858344i
$802$		0			0
$803$	383.211			0.477224
$804$		0			0
$805$	−248.611			−0.308834
$806$		0			0
$807$	−388.363			−0.481243
$808$		0			0
$809$	−456.534			−0.564319			−0.282160	−	0.959367i	$-0.591051\pi$
$809$	−456.534			−0.564319			−0.282160	+	0.959367i	$0.591051\pi$
$810$		0			0
$811$			1238.50i			1.52712i	0.645735	+	0.763562i	$0.276551\pi$
$811$			1238.50i			1.52712i	−0.645735	+	0.763562i	$0.723449\pi$
$812$		0			0
$813$			539.193i			0.663214i
$814$		0			0
$815$	195.091			0.239376
$816$		0			0
$817$	−4.46175	−	12.3363i	−0.00546114	−	0.0150995i
$818$		0			0
$819$		−	13.1957i		−	0.0161120i
$820$		0			0
$821$	503.379			0.613130			0.306565	−	0.951850i	$-0.400820\pi$
$821$	503.379			0.613130			0.306565	+	0.951850i	$0.400820\pi$
$822$		0			0
$823$	864.599			1.05055			0.525273	−	0.850934i	$-0.323963\pi$
$823$	864.599			1.05055			0.525273	+	0.850934i	$0.323963\pi$
$824$		0			0
$825$		−	55.6434i		−	0.0674466i
$826$		0			0
$827$			1528.37i			1.84809i	0.382280	+	0.924047i	$0.375139\pi$
$827$			1528.37i			1.84809i	−0.382280	+	0.924047i	$0.624861\pi$
$828$		0			0
$829$		−	696.807i		−	0.840539i	−0.907399	−	0.420270i	$-0.861935\pi$
$829$		−	696.807i		−	0.840539i	0.907399	−	0.420270i	$-0.138065\pi$
$830$		0			0
$831$			177.070i			0.213080i
$832$		0			0
$833$	51.9578			0.0623744
$834$		0			0
$835$		−	217.076i		−	0.259971i
$836$		0			0
$837$	−126.048			−0.150595
$838$		0			0
$839$			1372.41i			1.63577i	0.575380	+	0.817886i	$0.304854\pi$
$839$			1372.41i			1.63577i	−0.575380	+	0.817886i	$0.695146\pi$
$840$		0			0
$841$	−1547.24			−1.83976
$842$		0			0
$843$	−475.858			−0.564482
$844$		0			0
$845$	−376.996			−0.446150
$846$		0			0
$847$	−389.574			−0.459946
$848$		0			0
$849$		−	387.748i		−	0.456711i
$850$		0			0
$851$			159.255i			0.187139i
$852$		0			0
$853$	−152.915			−0.179268			−0.0896339	−	0.995975i	$-0.528570\pi$
$853$	−152.915			−0.179268			−0.0896339	+	0.995975i	$0.528570\pi$
$854$		0			0
$855$	70.2858	+	194.333i	0.0822056	+	0.227291i
$856$		0			0
$857$		−	621.971i		−	0.725754i	−0.931837	−	0.362877i	$-0.881795\pi$
$857$		−	621.971i		−	0.725754i	0.931837	−	0.362877i	$-0.118205\pi$
$858$		0			0
$859$	−710.294			−0.826885			−0.413442	−	0.910530i	$-0.635674\pi$
$859$	−710.294			−0.826885			−0.413442	+	0.910530i	$0.635674\pi$
$860$		0			0
$861$	−147.907			−0.171785
$862$		0			0
$863$		−	450.250i		−	0.521726i	−0.965376	−	0.260863i	$-0.915993\pi$
$863$		−	450.250i		−	0.521726i	0.965376	−	0.260863i	$-0.0840072\pi$
$864$		0			0
$865$			158.191i			0.182880i
$866$		0			0
$867$			581.909i			0.671175i
$868$		0			0
$869$			408.339i			0.469895i
$870$		0			0
$871$	61.0844			0.0701313
$872$		0			0
$873$			608.551i			0.697080i
$874$		0			0
$875$	−47.8343			−0.0546678
$876$		0			0
$877$		−	687.264i		−	0.783654i	−0.920039	−	0.391827i	$-0.871843\pi$
$877$		−	687.264i		−	0.783654i	0.920039	−	0.391827i	$-0.128157\pi$
$878$		0			0
$879$	−468.561			−0.533062
$880$		0			0
$881$	−848.052			−0.962602			−0.481301	−	0.876555i	$-0.659836\pi$
$881$	−848.052			−0.962602			−0.481301	+	0.876555i	$0.659836\pi$
$882$		0			0
$883$	267.540			0.302990			0.151495	−	0.988458i	$-0.451591\pi$
$883$	267.540			0.302990			0.151495	+	0.988458i	$0.451591\pi$
$884$		0			0
$885$	−1.14223			−0.00129065
$886$		0			0
$887$			1189.74i			1.34130i	0.741773	+	0.670651i	$0.233985\pi$
$887$			1189.74i			1.34130i	−0.741773	+	0.670651i	$0.766015\pi$
$888$		0			0
$889$			599.944i			0.674853i
$890$		0			0
$891$	−74.2210			−0.0833008
$892$		0			0
$893$	−149.148	−	412.379i	−0.167019	−	0.461791i
$894$		0			0
$895$			393.781i			0.439978i
$896$		0			0
$897$	−33.5105			−0.0373584
$898$		0			0
$899$	−218.473			−0.243018
$900$		0			0
$901$			92.0722i			0.102189i
$902$		0			0
$903$		−	6.00752i		−	0.00665285i
$904$		0			0
$905$			128.703i			0.142213i
$906$		0			0
$907$			1069.54i			1.17921i	0.807692	+	0.589605i	$0.200717\pi$
$907$			1069.54i			1.17921i	−0.807692	+	0.589605i	$0.799283\pi$
$908$		0			0
$909$	500.377			0.550469
$910$		0			0
$911$			1455.41i			1.59760i	0.601597	+	0.798800i	$0.294531\pi$
$911$			1455.41i			1.59760i	−0.601597	+	0.798800i	$0.705469\pi$
$912$		0			0
$913$	7.09589			0.00777206
$914$		0			0
$915$			179.884i			0.196594i
$916$		0			0
$917$	−679.522			−0.741027
$918$		0			0
$919$	104.189			0.113372			0.0566862	−	0.998392i	$-0.481947\pi$
$919$	104.189			0.113372			0.0566862	+	0.998392i	$0.481947\pi$
$920$		0			0
$921$	753.771			0.818426
$922$		0			0
$923$	10.1813			0.0110307
$924$		0			0
$925$			30.6416i			0.0331261i
$926$		0			0
$927$			56.8860i			0.0613657i
$928$		0			0
$929$	59.7100			0.0642734			0.0321367	−	0.999483i	$-0.489769\pi$
$929$	59.7100			0.0642734			0.0321367	+	0.999483i	$0.489769\pi$
$930$		0			0
$931$	198.357	+	548.437i	0.213058	+	0.589084i
$932$		0			0
$933$			24.8205i			0.0266028i
$934$		0			0
$935$	20.7123			0.0221521
$936$		0			0
$937$	−907.749			−0.968783			−0.484391	−	0.874851i	$-0.660959\pi$
$937$	−907.749			−0.968783			−0.484391	+	0.874851i	$0.660959\pi$
$938$		0			0
$939$			535.710i			0.570511i
$940$		0			0
$941$		−	60.4652i		−	0.0642563i	−0.999484	−	0.0321282i	$-0.989772\pi$
$941$		−	60.4652i		−	0.0642563i	0.999484	−	0.0321282i	$-0.0102285\pi$
$942$		0			0
$943$		−	441.744i		−	0.468445i
$944$		0			0
$945$			269.740i			0.285439i
$946$		0			0
$947$	−96.4321			−0.101829			−0.0509145	−	0.998703i	$-0.516214\pi$
$947$	−96.4321			−0.101829			−0.0509145	+	0.998703i	$0.516214\pi$
$948$		0			0
$949$		−	44.4041i		−	0.0467904i
$950$		0			0
$951$	−668.115			−0.702539
$952$		0			0
$953$		−	334.185i		−	0.350667i	−0.984509	−	0.175333i	$-0.943900\pi$
$953$		−	334.185i		−	0.350667i	0.984509	−	0.175333i	$-0.0561003\pi$
$954$		0			0
$955$	325.754			0.341103
$956$		0			0
$957$	−543.855			−0.568291
$958$		0			0
$959$	−320.188			−0.333877
$960$		0			0
$961$	941.014			0.979203
$962$		0			0
$963$			923.863i			0.959360i
$964$		0			0
$965$		−	143.403i		−	0.148604i
$966$		0			0
$967$	1230.05			1.27203			0.636016	−	0.771676i	$-0.280581\pi$
$967$	1230.05			1.27203			0.636016	+	0.771676i	$0.280581\pi$
$968$		0			0
$969$	61.5072	−	22.2457i	0.0634749	−	0.0229574i
$970$		0			0
$971$			1215.04i			1.25133i	0.780091	+	0.625667i	$0.215173\pi$
$971$			1215.04i			1.25133i	−0.780091	+	0.625667i	$0.784827\pi$
$972$		0			0
$973$	−74.8406			−0.0769174
$974$		0			0
$975$	−6.44761			−0.00661293
$976$		0			0
$977$			337.236i			0.345175i	0.984994	+	0.172588i	$0.0552128\pi$
$977$			337.236i			0.345175i	−0.984994	+	0.172588i	$0.944787\pi$
$978$		0			0
$979$		−	773.481i		−	0.790072i
$980$		0			0
$981$		−	212.483i		−	0.216598i
$982$		0			0
$983$			73.3754i			0.0746443i	0.999303	+	0.0373222i	$0.0118828\pi$
$983$			73.3754i			0.0746443i	−0.999303	+	0.0373222i	$0.988117\pi$
$984$		0			0
$985$	−40.0745			−0.0406848
$986$		0			0
$987$		−	200.820i		−	0.203465i
$988$		0			0
$989$	17.9423			0.0181419
$990$		0			0
$991$		−	635.844i		−	0.641618i	−0.947144	−	0.320809i	$-0.896045\pi$
$991$		−	635.844i		−	0.641618i	0.947144	−	0.320809i	$-0.103955\pi$
$992$		0			0
$993$	−59.4421			−0.0598611
$994$		0			0
$995$	−468.119			−0.470471
$996$		0			0
$997$	1405.38			1.40960			0.704802	−	0.709404i	$-0.251036\pi$
$997$	1405.38			1.40960			0.704802	+	0.709404i	$0.251036\pi$
$998$		0			0
$999$	172.790			0.172963

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	380.3.e.a.341.5	✓	12
3.2	odd	2		3420.3.o.a.721.11		12
4.3	odd	2		1520.3.h.b.721.8		12
5.2	odd	4		1900.3.g.c.949.10		24
5.3	odd	4		1900.3.g.c.949.15		24
5.4	even	2		1900.3.e.f.1101.8		12
19.18	odd	2	inner	380.3.e.a.341.8	yes	12
57.56	even	2		3420.3.o.a.721.12		12
76.75	even	2		1520.3.h.b.721.5		12
95.18	even	4		1900.3.g.c.949.9		24
95.37	even	4		1900.3.g.c.949.16		24
95.94	odd	2		1900.3.e.f.1101.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.e.a.341.5	✓	12	1.1	even	1	trivial
380.3.e.a.341.8	yes	12	19.18	odd	2	inner
1520.3.h.b.721.5		12	76.75	even	2
1520.3.h.b.721.8		12	4.3	odd	2
1900.3.e.f.1101.5		12	95.94	odd	2
1900.3.e.f.1101.8		12	5.4	even	2
1900.3.g.c.949.9		24	95.18	even	4
1900.3.g.c.949.10		24	5.2	odd	4
1900.3.g.c.949.15		24	5.3	odd	4
1900.3.g.c.949.16		24	95.37	even	4
3420.3.o.a.721.11		12	3.2	odd	2
3420.3.o.a.721.12		12	57.56	even	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)

\(f(q)\)	\(=\)	\(q-2.03369i q^{3} -2.23607 q^{5} +4.27843 q^{7} +4.86411 q^{9} +O(q^{10})\)	\(q-2.03369i q^{3} -2.23607 q^{5} +4.27843 q^{7} +4.86411 q^{9} +5.47217 q^{11} -0.634081i q^{13} +4.54746i q^{15} -1.69271 q^{17} +(-6.46218 - 17.8673i) q^{19} -8.70099i q^{21} +25.9867 q^{23} +5.00000 q^{25} -28.1953i q^{27} -48.8696i q^{29} -4.47053i q^{31} -11.1287i q^{33} -9.56686 q^{35} +6.12833i q^{37} -1.28952 q^{39} -16.9988i q^{41} +0.690441 q^{43} -10.8765 q^{45} +23.0801 q^{47} -30.6950 q^{49} +3.44245i q^{51} -54.3934i q^{53} -12.2361 q^{55} +(-36.3365 + 13.1420i) q^{57} +0.251179i q^{59} +39.5570 q^{61} +20.8108 q^{63} +1.41785i q^{65} +96.3354i q^{67} -52.8489i q^{69} +16.0568i q^{71} +70.0291 q^{73} -10.1684i q^{75} +23.4123 q^{77} +74.6211i q^{79} -13.5634 q^{81} +1.29672 q^{83} +3.78502 q^{85} -99.3856 q^{87} -141.348i q^{89} -2.71287i q^{91} -9.09167 q^{93} +(14.4499 + 39.9525i) q^{95} +125.110i q^{97} +26.6173 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{7} - 16 q^{9}+O(q^{10}) \) 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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