LMFDB - Embedded newform 1520.3.h.a.721.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,3,Mod(721,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1520.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1520.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$41.4170001828$
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} + 30x^{10} + 329x^{8} + 1620x^{6} + 3479x^{4} + 2470x^{2} + 55 $ x^12 + 30x^10 + 329x^8 + 1620x^6 + 3479x^4 + 2470*x^2 + 55
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.4
Root			$2.36559i$ of defining polynomial
Character	$\chi$	$=$	1520.721
Dual form			1520.3.h.a.721.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.55563i q^{3} -2.23607 q^{5} -11.0785 q^{7} -3.64250 q^{9} +O(q^{10})$	$q-3.55563i q^{3} -2.23607 q^{5} -11.0785 q^{7} -3.64250 q^{9} -16.7704 q^{11} +14.3035i q^{13} +7.95063i q^{15} -20.2160 q^{17} +(16.4480 + 9.51127i) q^{19} +39.3909i q^{21} +13.5882 q^{23} +5.00000 q^{25} -19.0493i q^{27} -30.3972i q^{29} +16.1256i q^{31} +59.6292i q^{33} +24.7722 q^{35} +51.2678i q^{37} +50.8580 q^{39} -74.6652i q^{41} -6.23197 q^{43} +8.14488 q^{45} +44.0252 q^{47} +73.7321 q^{49} +71.8806i q^{51} -30.8058i q^{53} +37.4997 q^{55} +(33.8185 - 58.4829i) q^{57} -73.0098i q^{59} -17.7828 q^{61} +40.3533 q^{63} -31.9837i q^{65} +20.5719i q^{67} -48.3147i q^{69} +7.48928i q^{71} -29.2302 q^{73} -17.7781i q^{75} +185.790 q^{77} -8.78894i q^{79} -100.515 q^{81} +63.7886 q^{83} +45.2043 q^{85} -108.081 q^{87} +162.908i q^{89} -158.461i q^{91} +57.3367 q^{93} +(-36.7788 - 21.2678i) q^{95} +75.5974i q^{97} +61.0861 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 20 q^{7} - 48 q^{9}+O(q^{10}) $ 12 * q - 20 * q^7 - 48 * q^9	$ 12 q - 20 q^{7} - 48 q^{9} - 32 q^{11} - 44 q^{17} - 8 q^{19} - 36 q^{23} + 60 q^{25} + 40 q^{35} - 76 q^{39} - 320 q^{43} - 40 q^{45} + 56 q^{47} + 72 q^{49} + 60 q^{57} - 296 q^{61} + 96 q^{63} - 244 q^{73} - 200 q^{77} - 372 q^{81} + 160 q^{83} + 160 q^{85} - 444 q^{87} + 296 q^{93} + 80 q^{95} + 312 q^{99}+O(q^{100}) $ 12 * q - 20 * q^7 - 48 * q^9 - 32 * q^11 - 44 * q^17 - 8 * q^19 - 36 * q^23 + 60 * q^25 + 40 * q^35 - 76 * q^39 - 320 * q^43 - 40 * q^45 + 56 * q^47 + 72 * q^49 + 60 * q^57 - 296 * q^61 + 96 * q^63 - 244 * q^73 - 200 * q^77 - 372 * q^81 + 160 * q^83 + 160 * q^85 - 444 * q^87 + 296 * q^93 + 80 * q^95 + 312 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$		−	3.55563i		−	1.18521i	−0.805493	−	0.592605i	$-0.798100\pi$
$3$		−	3.55563i		−	1.18521i	0.805493	−	0.592605i	$-0.201900\pi$
$4$		0			0
$5$	−2.23607			−0.447214
$5$	−2.23607			−0.447214
$6$		0			0
$7$	−11.0785			−1.58264			−0.791318	−	0.611405i	$-0.790605\pi$
$7$	−11.0785			−1.58264			−0.791318	+	0.611405i	$0.790605\pi$
$8$		0			0
$9$	−3.64250			−0.404723
$10$		0			0
$11$	−16.7704			−1.52458			−0.762289	−	0.647237i	$-0.775924\pi$
$11$	−16.7704			−1.52458			−0.762289	+	0.647237i	$0.775924\pi$
$12$		0			0
$13$			14.3035i			1.10027i	0.835075	+	0.550136i	$0.185424\pi$
$13$			14.3035i			1.10027i	−0.835075	+	0.550136i	$0.814576\pi$
$14$		0			0
$15$			7.95063i			0.530042i
$16$		0			0
$17$	−20.2160			−1.18918			−0.594588	−	0.804031i	$-0.702685\pi$
$17$	−20.2160			−1.18918			−0.594588	+	0.804031i	$0.702685\pi$
$18$		0			0
$19$	16.4480	+	9.51127i	0.865683	+	0.500593i
$19$	16.4480	+	9.51127i	0.865683	+	0.500593i
$20$		0			0
$21$			39.3909i			1.87576i
$22$		0			0
$23$	13.5882			0.590792			0.295396	−	0.955375i	$-0.404548\pi$
$23$	13.5882			0.590792			0.295396	+	0.955375i	$0.404548\pi$
$24$		0			0
$25$	5.00000			0.200000
$26$		0			0
$27$		−	19.0493i		−	0.705529i
$28$		0			0
$29$		−	30.3972i		−	1.04818i	−0.851663	−	0.524090i	$-0.824406\pi$
$29$		−	30.3972i		−	1.04818i	0.851663	−	0.524090i	$-0.175594\pi$
$30$		0			0
$31$			16.1256i			0.520181i	0.965584	+	0.260091i	$0.0837525\pi$
$31$			16.1256i			0.520181i	−0.965584	+	0.260091i	$0.916248\pi$
$32$		0			0
$33$			59.6292i			1.80694i
$34$		0			0
$35$	24.7722			0.707776
$36$		0			0
$37$			51.2678i			1.38562i	0.721122	+	0.692808i	$0.243627\pi$
$37$			51.2678i			1.38562i	−0.721122	+	0.692808i	$0.756373\pi$
$38$		0			0
$39$	50.8580			1.30405
$40$		0			0
$41$		−	74.6652i		−	1.82110i	−0.413395	−	0.910552i	$-0.635657\pi$
$41$		−	74.6652i		−	1.82110i	0.413395	−	0.910552i	$-0.364343\pi$
$42$		0			0
$43$	−6.23197			−0.144929			−0.0724647	−	0.997371i	$-0.523086\pi$
$43$	−6.23197			−0.144929			−0.0724647	+	0.997371i	$0.523086\pi$
$44$		0			0
$45$	8.14488			0.180997
$46$		0			0
$47$	44.0252			0.936706			0.468353	−	0.883541i	$-0.344848\pi$
$47$	44.0252			0.936706			0.468353	+	0.883541i	$0.344848\pi$
$48$		0			0
$49$	73.7321			1.50474
$50$		0			0
$51$			71.8806i			1.40942i
$52$		0			0
$53$		−	30.8058i		−	0.581241i	−0.956838	−	0.290620i	$-0.906138\pi$
$53$		−	30.8058i		−	0.581241i	0.956838	−	0.290620i	$-0.0938617\pi$
$54$		0			0
$55$	37.4997			0.681812
$56$		0			0
$57$	33.8185	−	58.4829i	0.593308	−	1.02602i
$58$		0			0
$59$		−	73.0098i		−	1.23745i	−0.785606	−	0.618727i	$-0.787649\pi$
$59$		−	73.0098i		−	1.23745i	0.785606	−	0.618727i	$-0.212351\pi$
$60$		0			0
$61$	−17.7828			−0.291522			−0.145761	−	0.989320i	$-0.546563\pi$
$61$	−17.7828			−0.291522			−0.145761	+	0.989320i	$0.546563\pi$
$62$		0			0
$63$	40.3533			0.640528
$64$		0			0
$65$		−	31.9837i		−	0.492056i
$66$		0			0
$67$			20.5719i			0.307044i	0.988145	+	0.153522i	$0.0490616\pi$
$67$			20.5719i			0.307044i	−0.988145	+	0.153522i	$0.950938\pi$
$68$		0			0
$69$		−	48.3147i		−	0.700212i
$70$		0			0
$71$			7.48928i			0.105483i	0.998608	+	0.0527414i	$0.0167959\pi$
$71$			7.48928i			0.105483i	−0.998608	+	0.0527414i	$0.983204\pi$
$72$		0			0
$73$	−29.2302			−0.400414			−0.200207	−	0.979754i	$-0.564162\pi$
$73$	−29.2302			−0.400414			−0.200207	+	0.979754i	$0.564162\pi$
$74$		0			0
$75$		−	17.7781i		−	0.237042i
$76$		0			0
$77$	185.790			2.41285
$78$		0			0
$79$		−	8.78894i		−	0.111252i	−0.998452	−	0.0556262i	$-0.982284\pi$
$79$		−	8.78894i		−	0.111252i	0.998452	−	0.0556262i	$-0.0177155\pi$
$80$		0			0
$81$	−100.515			−1.24092
$82$		0			0
$83$	63.7886			0.768537			0.384269	−	0.923221i	$-0.374454\pi$
$83$	63.7886			0.768537			0.384269	+	0.923221i	$0.374454\pi$
$84$		0			0
$85$	45.2043			0.531815
$86$		0			0
$87$	−108.081			−1.24231
$88$		0			0
$89$			162.908i			1.83043i	0.402965	+	0.915215i	$0.367980\pi$
$89$			162.908i			1.83043i	−0.402965	+	0.915215i	$0.632020\pi$
$90$		0			0
$91$		−	158.461i		−	1.74133i
$92$		0			0
$93$	57.3367			0.616524
$94$		0			0
$95$	−36.7788	−	21.2678i	−0.387145	−	0.223872i
$96$		0			0
$97$			75.5974i			0.779355i	0.920951	+	0.389677i	$0.127413\pi$
$97$			75.5974i			0.779355i	−0.920951	+	0.389677i	$0.872587\pi$
$98$		0			0
$99$	61.0861			0.617031
$100$		0			0
$101$	15.7692			0.156131			0.0780656	−	0.996948i	$-0.475126\pi$
$101$	15.7692			0.156131			0.0780656	+	0.996948i	$0.475126\pi$
$102$		0			0
$103$			7.15141i			0.0694312i	0.999397	+	0.0347156i	$0.0110525\pi$
$103$			7.15141i			0.0694312i	−0.999397	+	0.0347156i	$0.988947\pi$
$104$		0			0
$105$		−	88.0807i		−	0.838864i
$106$		0			0
$107$			85.9872i			0.803619i	0.915723	+	0.401809i	$0.131619\pi$
$107$			85.9872i			0.803619i	−0.915723	+	0.401809i	$0.868381\pi$
$108$		0			0
$109$		−	95.0893i		−	0.872379i	−0.899855	−	0.436190i	$-0.856328\pi$
$109$		−	95.0893i		−	0.872379i	0.899855	−	0.436190i	$-0.143672\pi$
$110$		0			0
$111$	182.289			1.64225
$112$		0			0
$113$		−	19.0355i		−	0.168456i	−0.996447	−	0.0842281i	$-0.973158\pi$
$113$		−	19.0355i		−	0.168456i	0.996447	−	0.0842281i	$-0.0268424\pi$
$114$		0			0
$115$	−30.3842			−0.264210
$116$		0			0
$117$		−	52.1006i		−	0.445304i
$118$		0			0
$119$	223.962			1.88203
$120$		0			0
$121$	160.245			1.32434
$122$		0			0
$123$	−265.482			−2.15839
$124$		0			0
$125$	−11.1803			−0.0894427
$126$		0			0
$127$			93.4107i			0.735518i	0.929921	+	0.367759i	$0.119875\pi$
$127$			93.4107i			0.735518i	−0.929921	+	0.367759i	$0.880125\pi$
$128$		0			0
$129$			22.1586i			0.171772i
$130$		0			0
$131$	−126.496			−0.965619			−0.482810	−	0.875725i	$-0.660384\pi$
$131$	−126.496			−0.965619			−0.482810	+	0.875725i	$0.660384\pi$
$132$		0			0
$133$	−182.218	−	105.370i	−1.37006	−	0.792257i
$134$		0			0
$135$			42.5955i			0.315522i
$136$		0			0
$137$	93.8813			0.685265			0.342632	−	0.939470i	$-0.388681\pi$
$137$	93.8813			0.685265			0.342632	+	0.939470i	$0.388681\pi$
$138$		0			0
$139$	−41.0986			−0.295673			−0.147837	−	0.989012i	$-0.547231\pi$
$139$	−41.0986			−0.295673			−0.147837	+	0.989012i	$0.547231\pi$
$140$		0			0
$141$		−	156.537i		−	1.11019i
$142$		0			0
$143$		−	239.875i		−	1.67745i
$144$		0			0
$145$			67.9703i			0.468760i
$146$		0			0
$147$		−	262.164i		−	1.78343i
$148$		0			0
$149$	92.2911			0.619404			0.309702	−	0.950834i	$-0.399771\pi$
$149$	92.2911			0.619404			0.309702	+	0.950834i	$0.399771\pi$
$150$		0			0
$151$			7.26853i			0.0481359i	0.999710	+	0.0240680i	$0.00766181\pi$
$151$			7.26853i			0.0481359i	−0.999710	+	0.0240680i	$0.992338\pi$
$152$		0			0
$153$	73.6368			0.481286
$154$		0			0
$155$		−	36.0580i		−	0.232632i
$156$		0			0
$157$	229.214			1.45996			0.729981	−	0.683468i	$-0.239529\pi$
$157$	229.214			1.45996			0.729981	+	0.683468i	$0.239529\pi$
$158$		0			0
$159$	−109.534			−0.688892
$160$		0			0
$161$	−150.536			−0.935009
$162$		0			0
$163$	−34.0426			−0.208850			−0.104425	−	0.994533i	$-0.533300\pi$
$163$	−34.0426			−0.208850			−0.104425	+	0.994533i	$0.533300\pi$
$164$		0			0
$165$		−	133.335i		−	0.808090i
$166$		0			0
$167$			162.274i			0.971702i	0.874042	+	0.485851i	$0.161490\pi$
$167$			162.274i			0.971702i	−0.874042	+	0.485851i	$0.838510\pi$
$168$		0			0
$169$	−35.5908			−0.210597
$170$		0			0
$171$	−59.9118	−	34.6448i	−0.350361	−	0.202601i
$172$		0			0
$173$		−	29.7956i		−	0.172229i	−0.996285	−	0.0861145i	$-0.972555\pi$
$173$		−	29.7956i		−	0.172229i	0.996285	−	0.0861145i	$-0.0274451\pi$
$174$		0			0
$175$	−55.3923			−0.316527
$176$		0			0
$177$	−259.596			−1.46664
$178$		0			0
$179$		−	59.2752i		−	0.331146i	−0.986197	−	0.165573i	$-0.947053\pi$
$179$		−	59.2752i		−	0.331146i	0.986197	−	0.165573i	$-0.0529474\pi$
$180$		0			0
$181$		−	205.700i		−	1.13647i	−0.822868	−	0.568233i	$-0.807627\pi$
$181$		−	205.700i		−	1.13647i	0.822868	−	0.568233i	$-0.192373\pi$
$182$		0			0
$183$			63.2291i			0.345514i
$184$		0			0
$185$		−	114.638i		−	0.619667i
$186$		0			0
$187$	339.029			1.81299
$188$		0			0
$189$			211.037i			1.11660i
$190$		0			0
$191$	253.682			1.32818			0.664089	−	0.747654i	$-0.268820\pi$
$191$	253.682			1.32818			0.664089	+	0.747654i	$0.268820\pi$
$192$		0			0
$193$			289.646i			1.50076i	0.661009	+	0.750378i	$0.270128\pi$
$193$			289.646i			1.50076i	−0.661009	+	0.750378i	$0.729872\pi$
$194$		0			0
$195$	−113.722			−0.583190
$196$		0			0
$197$	178.190			0.904515			0.452258	−	0.891887i	$-0.350619\pi$
$197$	178.190			0.904515			0.452258	+	0.891887i	$0.350619\pi$
$198$		0			0
$199$	154.602			0.776893			0.388446	−	0.921471i	$-0.373012\pi$
$199$	154.602			0.776893			0.388446	+	0.921471i	$0.373012\pi$
$200$		0			0
$201$	73.1462			0.363912
$202$		0			0
$203$			336.754i			1.65889i
$204$		0			0
$205$			166.957i			0.814422i
$206$		0			0
$207$	−49.4951			−0.239107
$208$		0			0
$209$	−275.838	−	159.507i	−1.31980	−	0.763193i
$210$		0			0
$211$		−	273.846i		−	1.29785i	−0.760854	−	0.648923i	$-0.775220\pi$
$211$		−	273.846i		−	1.29785i	0.760854	−	0.648923i	$-0.224780\pi$
$212$		0			0
$213$	26.6291			0.125019
$214$		0			0
$215$	13.9351			0.0648144
$216$		0			0
$217$		−	178.647i		−	0.823258i
$218$		0			0
$219$			103.932i			0.474575i
$220$		0			0
$221$		−	289.160i		−	1.30842i
$222$		0			0
$223$			229.796i			1.03048i	0.857047	+	0.515238i	$0.172296\pi$
$223$			229.796i			1.03048i	−0.857047	+	0.515238i	$0.827704\pi$
$224$		0			0
$225$	−18.2125			−0.0809445
$226$		0			0
$227$		−	366.166i		−	1.61306i	−0.591190	−	0.806532i	$-0.701342\pi$
$227$		−	366.166i		−	1.61306i	0.591190	−	0.806532i	$-0.298658\pi$
$228$		0			0
$229$	334.098			1.45894			0.729471	−	0.684012i	$-0.239766\pi$
$229$	334.098			1.45894			0.729471	+	0.684012i	$0.239766\pi$
$230$		0			0
$231$		−	660.599i		−	2.85974i
$232$		0			0
$233$	83.9900			0.360472			0.180236	−	0.983623i	$-0.442314\pi$
$233$	83.9900			0.360472			0.180236	+	0.983623i	$0.442314\pi$
$234$		0			0
$235$	−98.4433			−0.418908
$236$		0			0
$237$	−31.2502			−0.131857
$238$		0			0
$239$	110.020			0.460335			0.230168	−	0.973151i	$-0.426073\pi$
$239$	110.020			0.460335			0.230168	+	0.973151i	$0.426073\pi$
$240$		0			0
$241$		−	321.641i		−	1.33461i	−0.744785	−	0.667305i	$-0.767448\pi$
$241$		−	321.641i		−	1.33461i	0.744785	−	0.667305i	$-0.232552\pi$
$242$		0			0
$243$			185.950i			0.765225i
$244$		0			0
$245$	−164.870			−0.672939
$246$		0			0
$247$	−136.045	+	235.264i	−0.550788	+	0.952486i
$248$		0			0
$249$		−	226.809i		−	0.910878i
$250$		0			0
$251$	−143.410			−0.571353			−0.285676	−	0.958326i	$-0.592218\pi$
$251$	−143.410			−0.571353			−0.285676	+	0.958326i	$0.592218\pi$
$252$		0			0
$253$	−227.879			−0.900708
$254$		0			0
$255$		−	160.730i		−	0.630313i
$256$		0			0
$257$			171.202i			0.666156i	0.942899	+	0.333078i	$0.108087\pi$
$257$			171.202i			0.666156i	−0.942899	+	0.333078i	$0.891913\pi$
$258$		0			0
$259$		−	567.968i		−	2.19293i
$260$		0			0
$261$			110.722i			0.424222i
$262$		0			0
$263$	184.993			0.703395			0.351697	−	0.936114i	$-0.385605\pi$
$263$	184.993			0.703395			0.351697	+	0.936114i	$0.385605\pi$
$264$		0			0
$265$			68.8838i			0.259939i
$266$		0			0
$267$	579.242			2.16944
$268$		0			0
$269$			453.194i			1.68473i	0.538904	+	0.842367i	$0.318839\pi$
$269$			453.194i			1.68473i	−0.538904	+	0.842367i	$0.681161\pi$
$270$		0			0
$271$	−336.795			−1.24279			−0.621393	−	0.783499i	$-0.713433\pi$
$271$	−336.795			−1.24279			−0.621393	+	0.783499i	$0.713433\pi$
$272$		0			0
$273$	−563.428			−2.06384
$274$		0			0
$275$	−83.8518			−0.304916
$276$		0			0
$277$	270.449			0.976351			0.488175	−	0.872746i	$-0.337663\pi$
$277$	270.449			0.976351			0.488175	+	0.872746i	$0.337663\pi$
$278$		0			0
$279$		−	58.7376i		−	0.210529i
$280$		0			0
$281$			151.335i			0.538558i	0.963062	+	0.269279i	$0.0867853\pi$
$281$			151.335i			0.538558i	−0.963062	+	0.269279i	$0.913215\pi$
$282$		0			0
$283$	437.891			1.54732			0.773658	−	0.633603i	$-0.218425\pi$
$283$	437.891			1.54732			0.773658	+	0.633603i	$0.218425\pi$
$284$		0			0
$285$	−75.6206	+	130.772i	−0.265335	+	0.458848i
$286$		0			0
$287$			827.175i			2.88214i
$288$		0			0
$289$	119.686			0.414139
$290$		0			0
$291$	268.796			0.923699
$292$		0			0
$293$			265.247i			0.905279i	0.891694	+	0.452640i	$0.149518\pi$
$293$			265.247i			0.905279i	−0.891694	+	0.452640i	$0.850482\pi$
$294$		0			0
$295$			163.255i			0.553407i
$296$		0			0
$297$			319.463i			1.07563i
$298$		0			0
$299$			194.359i			0.650031i
$300$		0			0
$301$	69.0406			0.229371
$302$		0			0
$303$		−	56.0696i		−	0.185048i
$304$		0			0
$305$	39.7636			0.130372
$306$		0			0
$307$			405.065i			1.31943i	0.751517	+	0.659714i	$0.229323\pi$
$307$			405.065i			1.31943i	−0.751517	+	0.659714i	$0.770677\pi$
$308$		0			0
$309$	25.4278			0.0822906
$310$		0			0
$311$	313.906			1.00934			0.504672	−	0.863311i	$-0.331613\pi$
$311$	313.906			1.00934			0.504672	+	0.863311i	$0.331613\pi$
$312$		0			0
$313$	513.442			1.64039			0.820194	−	0.572085i	$-0.193865\pi$
$313$	513.442			1.64039			0.820194	+	0.572085i	$0.193865\pi$
$314$		0			0
$315$	−90.2327			−0.286453
$316$		0			0
$317$			25.8051i			0.0814042i	0.999171	+	0.0407021i	$0.0129595\pi$
$317$			25.8051i			0.0814042i	−0.999171	+	0.0407021i	$0.987041\pi$
$318$		0			0
$319$			509.772i			1.59803i
$320$		0			0
$321$	305.739			0.952457
$322$		0			0
$323$	−332.512	−	192.280i	−1.02945	−	0.595293i
$324$		0			0
$325$			71.5176i			0.220054i
$326$		0			0
$327$	−338.102			−1.03395
$328$		0			0
$329$	−487.731			−1.48247
$330$		0			0
$331$		−	171.635i		−	0.518534i	−0.965806	−	0.259267i	$-0.916519\pi$
$331$		−	171.635i		−	0.518534i	0.965806	−	0.259267i	$-0.0834810\pi$
$332$		0			0
$333$		−	186.743i		−	0.560790i
$334$		0			0
$335$		−	46.0003i		−	0.137314i
$336$		0			0
$337$			206.114i			0.611613i	0.952094	+	0.305807i	$0.0989261\pi$
$337$			206.114i			0.611613i	−0.952094	+	0.305807i	$0.901074\pi$
$338$		0			0
$339$	−67.6833			−0.199656
$340$		0			0
$341$		−	270.432i		−	0.793057i
$342$		0			0
$343$	−273.994			−0.798815
$344$		0			0
$345$			108.035i			0.313145i
$346$		0			0
$347$	7.91893			0.0228211			0.0114106	−	0.999935i	$-0.496368\pi$
$347$	7.91893			0.0228211			0.0114106	+	0.999935i	$0.496368\pi$
$348$		0			0
$349$	409.448			1.17320			0.586602	−	0.809876i	$-0.300465\pi$
$349$	409.448			1.17320			0.586602	+	0.809876i	$0.300465\pi$
$350$		0			0
$351$	272.472			0.776273
$352$		0			0
$353$	−637.919			−1.80714			−0.903568	−	0.428445i	$-0.859061\pi$
$353$	−637.919			−1.80714			−0.903568	+	0.428445i	$0.859061\pi$
$354$		0			0
$355$		−	16.7465i		−	0.0471733i
$356$		0			0
$357$		−	796.325i		−	2.23060i
$358$		0			0
$359$	237.272			0.660926			0.330463	−	0.943819i	$-0.392795\pi$
$359$	237.272			0.660926			0.330463	+	0.943819i	$0.392795\pi$
$360$		0			0
$361$	180.072	+	312.882i	0.498813	+	0.866710i
$362$		0			0
$363$		−	569.771i		−	1.56962i
$364$		0			0
$365$	65.3608			0.179071
$366$		0			0
$367$	−542.889			−1.47926			−0.739631	−	0.673013i	$-0.765000\pi$
$367$	−542.889			−1.47926			−0.739631	+	0.673013i	$0.765000\pi$
$368$		0			0
$369$			271.968i			0.737042i
$370$		0			0
$371$			341.280i			0.919892i
$372$		0			0
$373$		−	444.746i		−	1.19235i	−0.802855	−	0.596175i	$-0.796687\pi$
$373$		−	444.746i		−	1.19235i	0.802855	−	0.596175i	$-0.203313\pi$
$374$		0			0
$375$			39.7531i			0.106008i
$376$		0			0
$377$	434.787			1.15328
$378$		0			0
$379$		−	150.221i		−	0.396362i	−0.980165	−	0.198181i	$-0.936497\pi$
$379$		−	150.221i		−	0.396362i	0.980165	−	0.198181i	$-0.0635035\pi$
$380$		0			0
$381$	332.134			0.871743
$382$		0			0
$383$			85.8011i			0.224024i	0.993707	+	0.112012i	$0.0357295\pi$
$383$			85.8011i			0.224024i	−0.993707	+	0.112012i	$0.964271\pi$
$384$		0			0
$385$	−415.438			−1.07906
$386$		0			0
$387$	22.7000			0.0586562
$388$		0			0
$389$	−249.669			−0.641822			−0.320911	−	0.947109i	$-0.603989\pi$
$389$	−249.669			−0.641822			−0.320911	+	0.947109i	$0.603989\pi$
$390$		0			0
$391$	−274.699			−0.702555
$392$		0			0
$393$			449.773i			1.14446i
$394$		0			0
$395$			19.6527i			0.0497536i
$396$		0			0
$397$	−340.476			−0.857622			−0.428811	−	0.903394i	$-0.641067\pi$
$397$	−340.476			−0.857622			−0.428811	+	0.903394i	$0.641067\pi$
$398$		0			0
$399$	−374.657	+	647.900i	−0.938990	+	1.62381i
$400$		0			0
$401$			491.002i			1.22444i	0.790686	+	0.612221i	$0.209724\pi$
$401$			491.002i			1.22444i	−0.790686	+	0.612221i	$0.790276\pi$
$402$		0			0
$403$	−230.653			−0.572341
$404$		0			0
$405$	224.758			0.554957
$406$		0			0
$407$		−	859.779i		−	2.11248i
$408$		0			0
$409$			115.833i			0.283210i	0.989923	+	0.141605i	$0.0452263\pi$
$409$			115.833i			0.283210i	−0.989923	+	0.141605i	$0.954774\pi$
$410$		0			0
$411$		−	333.807i		−	0.812183i
$412$		0			0
$413$			808.836i			1.95844i
$414$		0			0
$415$	−142.636			−0.343700
$416$		0			0
$417$			146.131i			0.350435i
$418$		0			0
$419$	−81.4969			−0.194503			−0.0972516	−	0.995260i	$-0.531005\pi$
$419$	−81.4969			−0.194503			−0.0972516	+	0.995260i	$0.531005\pi$
$420$		0			0
$421$			221.652i			0.526490i	0.964729	+	0.263245i	$0.0847927\pi$
$421$			221.652i			0.526490i	−0.964729	+	0.263245i	$0.915207\pi$
$422$		0			0
$423$	−160.362			−0.379106
$424$		0			0
$425$	−101.080			−0.237835
$426$		0			0
$427$	197.006			0.461372
$428$		0			0
$429$	−852.907			−1.98813
$430$		0			0
$431$			211.834i			0.491494i	0.969334	+	0.245747i	$0.0790333\pi$
$431$			211.834i			0.491494i	−0.969334	+	0.245747i	$0.920967\pi$
$432$		0			0
$433$		−	389.534i		−	0.899616i	−0.893125	−	0.449808i	$-0.851492\pi$
$433$		−	389.534i		−	0.899616i	0.893125	−	0.449808i	$-0.148508\pi$
$434$		0			0
$435$	241.677			0.555579
$436$		0			0
$437$	223.499	+	129.241i	0.511438	+	0.295746i
$438$		0			0
$439$			465.691i			1.06080i	0.847748	+	0.530400i	$0.177958\pi$
$439$			465.691i			1.06080i	−0.847748	+	0.530400i	$0.822042\pi$
$440$		0			0
$441$	−268.569			−0.609001
$442$		0			0
$443$	382.479			0.863383			0.431692	−	0.902021i	$-0.357917\pi$
$443$	382.479			0.863383			0.431692	+	0.902021i	$0.357917\pi$
$444$		0			0
$445$		−	364.274i		−	0.818593i
$446$		0			0
$447$		−	328.153i		−	0.734123i
$448$		0			0
$449$		−	666.393i		−	1.48417i	−0.670305	−	0.742086i	$-0.733837\pi$
$449$		−	666.393i		−	1.48417i	0.670305	−	0.742086i	$-0.266163\pi$
$450$		0			0
$451$			1252.16i			2.77641i
$452$		0			0
$453$	25.8442			0.0570512
$454$		0			0
$455$			354.329i			0.778746i
$456$		0			0
$457$	−627.200			−1.37243			−0.686214	−	0.727399i	$-0.740729\pi$
$457$	−627.200			−1.37243			−0.686214	+	0.727399i	$0.740729\pi$
$458$		0			0
$459$			385.100i			0.838998i
$460$		0			0
$461$	798.338			1.73175			0.865876	−	0.500258i	$-0.166762\pi$
$461$	798.338			1.73175			0.865876	+	0.500258i	$0.166762\pi$
$462$		0			0
$463$	−791.786			−1.71012			−0.855061	−	0.518528i	$-0.826480\pi$
$463$	−791.786			−1.71012			−0.855061	+	0.518528i	$0.826480\pi$
$464$		0			0
$465$	−128.209			−0.275718
$466$		0			0
$467$	418.723			0.896622			0.448311	−	0.893878i	$-0.352026\pi$
$467$	418.723			0.896622			0.448311	+	0.893878i	$0.352026\pi$
$468$		0			0
$469$		−	227.905i		−	0.485939i
$470$		0			0
$471$		−	815.000i		−	1.73036i
$472$		0			0
$473$	104.512			0.220956
$474$		0			0
$475$	82.2399	+	47.5563i	0.173137	+	0.100119i
$476$		0			0
$477$			112.210i			0.235241i
$478$		0			0
$479$	−63.6160			−0.132810			−0.0664050	−	0.997793i	$-0.521153\pi$
$479$	−63.6160			−0.132810			−0.0664050	+	0.997793i	$0.521153\pi$
$480$		0			0
$481$	−733.310			−1.52455
$482$		0			0
$483$			535.252i			1.10818i
$484$		0			0
$485$		−	169.041i		−	0.348538i
$486$		0			0
$487$			556.997i			1.14373i	0.820347	+	0.571866i	$0.193780\pi$
$487$			556.997i			1.14373i	−0.820347	+	0.571866i	$0.806220\pi$
$488$		0			0
$489$			121.043i			0.247531i
$490$		0			0
$491$	−856.809			−1.74503			−0.872514	−	0.488589i	$-0.837512\pi$
$491$	−856.809			−1.74503			−0.872514	+	0.488589i	$0.837512\pi$
$492$		0			0
$493$			614.510i			1.24647i
$494$		0			0
$495$	−136.593			−0.275945
$496$		0			0
$497$		−	82.9696i		−	0.166941i
$498$		0			0
$499$	−761.632			−1.52632			−0.763158	−	0.646211i	$-0.776352\pi$
$499$	−761.632			−1.52632			−0.763158	+	0.646211i	$0.776352\pi$
$500$		0			0
$501$	576.987			1.15167
$502$		0			0
$503$	34.7192			0.0690242			0.0345121	−	0.999404i	$-0.489012\pi$
$503$	34.7192			0.0690242			0.0345121	+	0.999404i	$0.489012\pi$
$504$		0			0
$505$	−35.2611			−0.0698240
$506$		0			0
$507$			126.548i			0.249601i
$508$		0			0
$509$			9.31181i			0.0182943i	0.999958	+	0.00914716i	$0.00291167\pi$
$509$			9.31181i			0.0182943i	−0.999958	+	0.00914716i	$0.997088\pi$
$510$		0			0
$511$	323.826			0.633710
$512$		0			0
$513$	181.183	−	313.322i	0.353183	−	0.610764i
$514$		0			0
$515$		−	15.9910i		−	0.0310506i
$516$		0			0
$517$	−738.318			−1.42808
$518$		0			0
$519$	−105.942			−0.204128
$520$		0			0
$521$			88.0858i			0.169071i	0.996420	+	0.0845353i	$0.0269406\pi$
$521$			88.0858i			0.169071i	−0.996420	+	0.0845353i	$0.973059\pi$
$522$		0			0
$523$			377.616i			0.722019i	0.932562	+	0.361009i	$0.117568\pi$
$523$			377.616i			0.722019i	−0.932562	+	0.361009i	$0.882432\pi$
$524$		0			0
$525$			196.954i			0.375151i
$526$		0			0
$527$		−	325.995i		−	0.618587i
$528$		0			0
$529$	−344.360			−0.650965
$530$		0			0
$531$			265.938i			0.500826i
$532$		0			0
$533$	1067.98			2.00371
$534$		0			0
$535$		−	192.273i		−	0.359389i
$536$		0			0
$537$	−210.761			−0.392478
$538$		0			0
$539$	−1236.51			−2.29409
$540$		0			0
$541$	−225.756			−0.417293			−0.208647	−	0.977991i	$-0.566906\pi$
$541$	−225.756			−0.417293			−0.208647	+	0.977991i	$0.566906\pi$
$542$		0			0
$543$	−731.394			−1.34695
$544$		0			0
$545$			212.626i			0.390140i
$546$		0			0
$547$		−	1050.58i		−	1.92062i	−0.278938	−	0.960309i	$-0.589982\pi$
$547$		−	1050.58i		−	1.92062i	0.278938	−	0.960309i	$-0.410018\pi$
$548$		0			0
$549$	64.7739			0.117985
$550$		0			0
$551$	289.116	−	499.973i	0.524712	−	0.907391i
$552$		0			0
$553$			97.3678i			0.176072i
$554$		0			0
$555$	−407.611			−0.734435
$556$		0			0
$557$	580.999			1.04309			0.521543	−	0.853225i	$-0.325356\pi$
$557$	580.999			1.04309			0.521543	+	0.853225i	$0.325356\pi$
$558$		0			0
$559$		−	89.1391i		−	0.159462i
$560$		0			0
$561$		−	1205.46i		−	2.14877i
$562$		0			0
$563$			796.409i			1.41458i	0.706923	+	0.707291i	$0.250083\pi$
$563$			796.409i			1.41458i	−0.706923	+	0.707291i	$0.749917\pi$
$564$		0			0
$565$			42.5648i			0.0753359i
$566$		0			0
$567$	1113.55			1.96393
$568$		0			0
$569$			171.428i			0.301280i	0.988589	+	0.150640i	$0.0481334\pi$
$569$			171.428i			0.301280i	−0.988589	+	0.150640i	$0.951867\pi$
$570$		0			0
$571$	673.960			1.18031			0.590157	−	0.807288i	$-0.299066\pi$
$571$	673.960			1.18031			0.590157	+	0.807288i	$0.299066\pi$
$572$		0			0
$573$		−	901.999i		−	1.57417i
$574$		0			0
$575$	67.9411			0.118158
$576$		0			0
$577$	−795.916			−1.37940			−0.689702	−	0.724094i	$-0.742258\pi$
$577$	−795.916			−1.37940			−0.689702	+	0.724094i	$0.742258\pi$
$578$		0			0
$579$	1029.87			1.77871
$580$		0			0
$581$	−706.679			−1.21632
$582$		0			0
$583$			516.623i			0.886146i
$584$		0			0
$585$			116.501i			0.199146i
$586$		0			0
$587$	19.6929			0.0335484			0.0167742	−	0.999859i	$-0.494660\pi$
$587$	19.6929			0.0335484			0.0167742	+	0.999859i	$0.494660\pi$
$588$		0			0
$589$	−153.375	+	265.234i	−0.260399	+	0.450312i
$590$		0			0
$591$		−	633.576i		−	1.07204i
$592$		0			0
$593$	−636.058			−1.07261			−0.536306	−	0.844024i	$-0.680181\pi$
$593$	−636.058			−1.07261			−0.536306	+	0.844024i	$0.680181\pi$
$594$		0			0
$595$	−500.794			−0.841670
$596$		0			0
$597$		−	549.706i		−	0.920781i
$598$		0			0
$599$		−	936.525i		−	1.56348i	−0.623604	−	0.781741i	$-0.714332\pi$
$599$		−	936.525i		−	1.56348i	0.623604	−	0.781741i	$-0.285668\pi$
$600$		0			0
$601$		−	252.090i		−	0.419451i	−0.977760	−	0.209725i	$-0.932743\pi$
$601$		−	252.090i		−	0.419451i	0.977760	−	0.209725i	$-0.0672570\pi$
$602$		0			0
$603$		−	74.9334i		−	0.124268i
$604$		0			0
$605$	−358.318			−0.592261
$606$		0			0
$607$		−	498.526i		−	0.821295i	−0.911794	−	0.410648i	$-0.865303\pi$
$607$		−	498.526i		−	0.821295i	0.911794	−	0.410648i	$-0.134697\pi$
$608$		0			0
$609$	1197.37			1.96613
$610$		0			0
$611$			629.715i			1.03063i
$612$		0			0
$613$	−638.709			−1.04194			−0.520969	−	0.853575i	$-0.674429\pi$
$613$	−638.709			−1.04194			−0.520969	+	0.853575i	$0.674429\pi$
$614$		0			0
$615$	593.636			0.965261
$616$		0			0
$617$	702.709			1.13891			0.569457	−	0.822021i	$-0.307154\pi$
$617$	702.709			1.13891			0.569457	+	0.822021i	$0.307154\pi$
$618$		0			0
$619$	−195.693			−0.316144			−0.158072	−	0.987428i	$-0.550528\pi$
$619$	−195.693			−0.316144			−0.158072	+	0.987428i	$0.550528\pi$
$620$		0			0
$621$		−	258.846i		−	0.416821i
$622$		0			0
$623$		−	1804.77i		−	2.89691i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−567.149	+	980.779i	−0.904544	+	1.56424i
$628$		0			0
$629$		−	1036.43i		−	1.64774i
$630$		0			0
$631$	1063.16			1.68489			0.842444	−	0.538784i	$-0.181116\pi$
$631$	1063.16			1.68489			0.842444	+	0.538784i	$0.181116\pi$
$632$		0			0
$633$	−973.694			−1.53822
$634$		0			0
$635$		−	208.873i		−	0.328933i
$636$		0			0
$637$			1054.63i			1.65562i
$638$		0			0
$639$		−	27.2797i		−	0.0426913i
$640$		0			0
$641$			744.630i			1.16167i	0.814021	+	0.580835i	$0.197274\pi$
$641$			744.630i			1.16167i	−0.814021	+	0.580835i	$0.802726\pi$
$642$		0			0
$643$	−195.660			−0.304293			−0.152146	−	0.988358i	$-0.548618\pi$
$643$	−195.660			−0.304293			−0.152146	+	0.988358i	$0.548618\pi$
$644$		0			0
$645$		−	49.5481i		−	0.0768187i
$646$		0			0
$647$	128.045			0.197906			0.0989529	−	0.995092i	$-0.468451\pi$
$647$	128.045			0.197906			0.0989529	+	0.995092i	$0.468451\pi$
$648$		0			0
$649$			1224.40i			1.88660i
$650$		0			0
$651$	−635.202			−0.975733
$652$		0			0
$653$	−717.897			−1.09938			−0.549692	−	0.835368i	$-0.685255\pi$
$653$	−717.897			−1.09938			−0.549692	+	0.835368i	$0.685255\pi$
$654$		0			0
$655$	282.854			0.431838
$656$		0			0
$657$	106.471			0.162057
$658$		0			0
$659$		−	544.617i		−	0.826430i	−0.910633	−	0.413215i	$-0.864406\pi$
$659$		−	544.617i		−	0.826430i	0.910633	−	0.413215i	$-0.135594\pi$
$660$		0			0
$661$		−	435.420i		−	0.658729i	−0.944203	−	0.329365i	$-0.893165\pi$
$661$		−	435.420i		−	0.658729i	0.944203	−	0.329365i	$-0.106835\pi$
$662$		0			0
$663$	−1028.15			−1.55075
$664$		0			0
$665$	407.452	+	235.615i	0.612710	+	0.354308i
$666$		0			0
$667$		−	413.044i		−	0.619256i
$668$		0			0
$669$	817.070			1.22133
$670$		0			0
$671$	298.224			0.444447
$672$		0			0
$673$			397.296i			0.590336i	0.955445	+	0.295168i	$0.0953756\pi$
$673$			397.296i			0.590336i	−0.955445	+	0.295168i	$0.904624\pi$
$674$		0			0
$675$		−	95.2464i		−	0.141106i
$676$		0			0
$677$		−	456.859i		−	0.674829i	−0.941356	−	0.337415i	$-0.890448\pi$
$677$		−	456.859i		−	0.674829i	0.941356	−	0.337415i	$-0.109552\pi$
$678$		0			0
$679$		−	837.502i		−	1.23343i
$680$		0			0
$681$	−1301.95			−1.91182
$682$		0			0
$683$			880.553i			1.28924i	0.764502	+	0.644621i	$0.222985\pi$
$683$			880.553i			1.28924i	−0.764502	+	0.644621i	$0.777015\pi$
$684$		0			0
$685$	−209.925			−0.306460
$686$		0			0
$687$		−	1187.93i		−	1.72915i
$688$		0			0
$689$	440.631			0.639522
$690$		0			0
$691$	34.3920			0.0497713			0.0248857	−	0.999690i	$-0.492078\pi$
$691$	34.3920			0.0497713			0.0248857	+	0.999690i	$0.492078\pi$
$692$		0			0
$693$	−676.739			−0.976535
$694$		0			0
$695$	91.8992			0.132229
$696$		0			0
$697$			1509.43i			2.16561i
$698$		0			0
$699$		−	298.637i		−	0.427235i
$700$		0			0
$701$	419.321			0.598175			0.299088	−	0.954226i	$-0.403318\pi$
$701$	419.321			0.598175			0.299088	+	0.954226i	$0.403318\pi$
$702$		0			0
$703$	−487.622	+	843.251i	−0.693630	+	1.19950i
$704$		0			0
$705$			350.028i			0.496494i
$706$		0			0
$707$	−174.699			−0.247099
$708$		0			0
$709$	373.670			0.527038			0.263519	−	0.964654i	$-0.415117\pi$
$709$	373.670			0.527038			0.263519	+	0.964654i	$0.415117\pi$
$710$		0			0
$711$			32.0137i			0.0450263i
$712$		0			0
$713$			219.118i			0.307319i
$714$		0			0
$715$			536.377i			0.750178i
$716$		0			0
$717$		−	391.191i		−	0.545594i
$718$		0			0
$719$	328.089			0.456312			0.228156	−	0.973625i	$-0.426730\pi$
$719$	328.089			0.456312			0.228156	+	0.973625i	$0.426730\pi$
$720$		0			0
$721$		−	79.2266i		−	0.109884i
$722$		0			0
$723$	−1143.64			−1.58179
$724$		0			0
$725$		−	151.986i		−	0.209636i
$726$		0			0
$727$	−1131.05			−1.55578			−0.777889	−	0.628402i	$-0.783709\pi$
$727$	−1131.05			−1.55578			−0.777889	+	0.628402i	$0.783709\pi$
$728$		0			0
$729$	−243.465			−0.333971
$730$		0			0
$731$	125.985			0.172347
$732$		0			0
$733$	−450.895			−0.615137			−0.307568	−	0.951526i	$-0.599515\pi$
$733$	−450.895			−0.615137			−0.307568	+	0.951526i	$0.599515\pi$
$734$		0			0
$735$			586.217i			0.797574i
$736$		0			0
$737$		−	344.999i		−	0.468112i
$738$		0			0
$739$	195.478			0.264517			0.132258	−	0.991215i	$-0.457777\pi$
$739$	195.478			0.264517			0.132258	+	0.991215i	$0.457777\pi$
$740$		0			0
$741$	836.512	+	483.724i	1.12890	+	0.652800i
$742$		0			0
$743$			194.450i			0.261710i	0.991402	+	0.130855i	$0.0417722\pi$
$743$			194.450i			0.261710i	−0.991402	+	0.130855i	$0.958228\pi$
$744$		0			0
$745$	−206.369			−0.277006
$746$		0			0
$747$	−232.350			−0.311044
$748$		0			0
$749$		−	952.605i		−	1.27184i
$750$		0			0
$751$			943.646i			1.25652i	0.778004	+	0.628259i	$0.216232\pi$
$751$			943.646i			1.25652i	−0.778004	+	0.628259i	$0.783768\pi$
$752$		0			0
$753$			509.911i			0.677173i
$754$		0			0
$755$		−	16.2529i		−	0.0215270i
$756$		0			0
$757$	891.082			1.17712			0.588561	−	0.808453i	$-0.299695\pi$
$757$	891.082			1.17712			0.588561	+	0.808453i	$0.299695\pi$
$758$		0			0
$759$			810.254i			1.06753i
$760$		0			0
$761$	759.322			0.997795			0.498898	−	0.866661i	$-0.333738\pi$
$761$	759.322			0.997795			0.498898	+	0.866661i	$0.333738\pi$
$762$		0			0
$763$			1053.44i			1.38066i
$764$		0			0
$765$	−164.657			−0.215238
$766$		0			0
$767$	1044.30			1.36154
$768$		0			0
$769$	225.596			0.293363			0.146681	−	0.989184i	$-0.453141\pi$
$769$	225.596			0.293363			0.146681	+	0.989184i	$0.453141\pi$
$770$		0			0
$771$	608.731			0.789535
$772$		0			0
$773$			1189.39i			1.53867i	0.638843	+	0.769337i	$0.279413\pi$
$773$			1189.39i			1.53867i	−0.638843	+	0.769337i	$0.720587\pi$
$774$		0			0
$775$			80.6281i			0.104036i
$776$		0			0
$777$	−2019.48			−2.59908
$778$		0			0
$779$	710.161	−	1228.09i	0.911632	−	1.57650i
$780$		0			0
$781$		−	125.598i		−	0.160817i
$782$		0			0
$783$	−579.045			−0.739521
$784$		0			0
$785$	−512.538			−0.652915
$786$		0			0
$787$			254.533i			0.323422i	0.986838	+	0.161711i	$0.0517013\pi$
$787$			254.533i			0.323422i	−0.986838	+	0.161711i	$0.948299\pi$
$788$		0			0
$789$		−	657.766i		−	0.833670i
$790$		0			0
$791$			210.884i			0.266605i
$792$		0			0
$793$		−	254.357i		−	0.320753i
$794$		0			0
$795$	244.925			0.308082
$796$		0			0
$797$		−	719.295i		−	0.902503i	−0.892397	−	0.451251i	$-0.850978\pi$
$797$		−	719.295i		−	0.902503i	0.892397	−	0.451251i	$-0.149022\pi$
$798$		0			0
$799$	−890.013			−1.11391
$800$		0			0
$801$		−	593.394i		−	0.740816i
$802$		0			0
$803$	490.202			0.610463
$804$		0			0
$805$	336.610			0.418149
$806$		0			0
$807$	1611.39			1.99676
$808$		0			0
$809$	67.3018			0.0831914			0.0415957	−	0.999135i	$-0.486756\pi$
$809$	67.3018			0.0831914			0.0415957	+	0.999135i	$0.486756\pi$
$810$		0			0
$811$		−	401.142i		−	0.494626i	−0.968936	−	0.247313i	$-0.920452\pi$
$811$		−	401.142i		−	0.494626i	0.968936	−	0.247313i	$-0.0795477\pi$
$812$		0			0
$813$			1197.52i			1.47296i
$814$		0			0
$815$	76.1215			0.0934006
$816$		0			0
$817$	−102.503	−	59.2739i	−0.125463	−	0.0725507i
$818$		0			0
$819$			577.194i			0.704755i
$820$		0			0
$821$	−99.8212			−0.121585			−0.0607925	−	0.998150i	$-0.519363\pi$
$821$	−99.8212			−0.121585			−0.0607925	+	0.998150i	$0.519363\pi$
$822$		0			0
$823$	93.5359			0.113652			0.0568262	−	0.998384i	$-0.481902\pi$
$823$	93.5359			0.113652			0.0568262	+	0.998384i	$0.481902\pi$
$824$		0			0
$825$			298.146i			0.361389i
$826$		0			0
$827$		−	398.725i		−	0.482135i	−0.970508	−	0.241067i	$-0.922503\pi$
$827$		−	398.725i		−	0.482135i	0.970508	−	0.241067i	$-0.0774975\pi$
$828$		0			0
$829$			805.348i			0.971470i	0.874106	+	0.485735i	$0.161448\pi$
$829$			805.348i			0.971470i	−0.874106	+	0.485735i	$0.838552\pi$
$830$		0			0
$831$		−	961.617i		−	1.15718i
$832$		0			0
$833$	−1490.57			−1.78940
$834$		0			0
$835$		−	362.856i		−	0.434558i
$836$		0			0
$837$	307.181			0.367003
$838$		0			0
$839$			1547.08i			1.84396i	0.387235	+	0.921981i	$0.373430\pi$
$839$			1547.08i			1.84396i	−0.387235	+	0.921981i	$0.626570\pi$
$840$		0			0
$841$	−82.9912			−0.0986815
$842$		0			0
$843$	538.090			0.638304
$844$		0			0
$845$	79.5835			0.0941817
$846$		0			0
$847$	−1775.26			−2.09594
$848$		0			0
$849$		−	1556.98i		−	1.83390i
$850$		0			0
$851$			696.638i			0.818611i
$852$		0			0
$853$	−252.303			−0.295784			−0.147892	−	0.989004i	$-0.547249\pi$
$853$	−252.303			−0.295784			−0.147892	+	0.989004i	$0.547249\pi$
$854$		0			0
$855$	133.967	+	77.4682i	0.156686	+	0.0906061i
$856$		0			0
$857$		−	1397.35i		−	1.63051i	−0.579102	−	0.815255i	$-0.696597\pi$
$857$		−	1397.35i		−	1.63051i	0.579102	−	0.815255i	$-0.303403\pi$
$858$		0			0
$859$	1472.96			1.71473			0.857366	−	0.514706i	$-0.172099\pi$
$859$	1472.96			1.71473			0.857366	+	0.514706i	$0.172099\pi$
$860$		0			0
$861$	2941.13			3.41595
$862$		0			0
$863$			63.9533i			0.0741058i	0.999313	+	0.0370529i	$0.0117970\pi$
$863$			63.9533i			0.0741058i	−0.999313	+	0.0370529i	$0.988203\pi$
$864$		0			0
$865$			66.6250i			0.0770232i
$866$		0			0
$867$		−	425.559i		−	0.490841i
$868$		0			0
$869$			147.394i			0.169613i
$870$		0			0
$871$	−294.251			−0.337832
$872$		0			0
$873$		−	275.364i		−	0.315422i
$874$		0			0
$875$	123.861			0.141555
$876$		0			0
$877$		−	399.492i		−	0.455522i	−0.973717	−	0.227761i	$-0.926860\pi$
$877$		−	399.492i		−	0.455522i	0.973717	−	0.227761i	$-0.0731404\pi$
$878$		0			0
$879$	943.119			1.07295
$880$		0			0
$881$	514.180			0.583632			0.291816	−	0.956474i	$-0.405740\pi$
$881$	514.180			0.583632			0.291816	+	0.956474i	$0.405740\pi$
$882$		0			0
$883$	−323.869			−0.366783			−0.183392	−	0.983040i	$-0.558708\pi$
$883$	−323.869			−0.366783			−0.183392	+	0.983040i	$0.558708\pi$
$884$		0			0
$885$	580.474			0.655903
$886$		0			0
$887$		−	277.999i		−	0.313415i	−0.987645	−	0.156707i	$-0.949912\pi$
$887$		−	277.999i		−	0.313415i	0.987645	−	0.156707i	$-0.0500879\pi$
$888$		0			0
$889$		−	1034.85i		−	1.16406i
$890$		0			0
$891$	1685.67			1.89188
$892$		0			0
$893$	724.125	+	418.735i	0.810890	+	0.468909i
$894$		0			0
$895$			132.543i			0.148093i
$896$		0			0
$897$	691.070			0.770424
$898$		0			0
$899$	490.174			0.545244
$900$		0			0
$901$			622.769i			0.691197i
$902$		0			0
$903$		−	245.483i		−	0.271852i
$904$		0			0
$905$			459.960i			0.508243i
$906$		0			0
$907$			401.356i			0.442509i	0.975216	+	0.221255i	$0.0710152\pi$
$907$			401.356i			0.442509i	−0.975216	+	0.221255i	$0.928985\pi$
$908$		0			0
$909$	−57.4395			−0.0631898
$910$		0			0
$911$			378.750i			0.415752i	0.978155	+	0.207876i	$0.0666551\pi$
$911$			378.750i			0.415752i	−0.978155	+	0.207876i	$0.933345\pi$
$912$		0			0
$913$	−1069.76			−1.17170
$914$		0			0
$915$		−	141.385i		−	0.154519i
$916$		0			0
$917$	1401.38			1.52822
$918$		0			0
$919$	102.234			0.111244			0.0556222	−	0.998452i	$-0.482286\pi$
$919$	102.234			0.111244			0.0556222	+	0.998452i	$0.482286\pi$
$920$		0			0
$921$	1440.26			1.56380
$922$		0			0
$923$	−107.123			−0.116060
$924$		0			0
$925$			256.339i			0.277123i
$926$		0			0
$927$		−	26.0490i		−	0.0281004i
$928$		0			0
$929$	1794.45			1.93160			0.965798	−	0.259295i	$-0.0834903\pi$
$929$	1794.45			1.93160			0.965798	+	0.259295i	$0.0834903\pi$
$930$		0			0
$931$	1212.74	+	701.286i	1.30262	+	0.753261i
$932$		0			0
$933$		−	1116.13i		−	1.19629i
$934$		0			0
$935$	−758.092			−0.810794
$936$		0			0
$937$	−823.194			−0.878542			−0.439271	−	0.898355i	$-0.644763\pi$
$937$	−823.194			−0.878542			−0.439271	+	0.898355i	$0.644763\pi$
$938$		0			0
$939$		−	1825.61i		−	1.94421i
$940$		0			0
$941$		−	270.097i		−	0.287032i	−0.989648	−	0.143516i	$-0.954159\pi$
$941$		−	270.097i		−	0.287032i	0.989648	−	0.143516i	$-0.0458409\pi$
$942$		0			0
$943$		−	1014.57i		−	1.07589i
$944$		0			0
$945$		−	471.892i		−	0.499357i
$946$		0			0
$947$	1412.14			1.49118			0.745588	−	0.666407i	$-0.232169\pi$
$947$	1412.14			1.49118			0.745588	+	0.666407i	$0.232169\pi$
$948$		0			0
$949$		−	418.096i		−	0.440564i
$950$		0			0
$951$	91.7535			0.0964811
$952$		0			0
$953$			432.598i			0.453933i	0.973903	+	0.226966i	$0.0728808\pi$
$953$			432.598i			0.453933i	−0.973903	+	0.226966i	$0.927119\pi$
$954$		0			0
$955$	−567.250			−0.593979
$956$		0			0
$957$	1812.56			1.89400
$958$		0			0
$959$	−1040.06			−1.08452
$960$		0			0
$961$	700.964			0.729411
$962$		0			0
$963$		−	313.209i		−	0.325243i
$964$		0			0
$965$		−	647.668i		−	0.671158i
$966$		0			0
$967$	−10.0222			−0.0103642			−0.00518210	−	0.999987i	$-0.501650\pi$
$967$	−10.0222			−0.0103642			−0.00518210	+	0.999987i	$0.501650\pi$
$968$		0			0
$969$	−683.675	+	1182.29i	−0.705547	+	1.22011i
$970$		0			0
$971$			339.417i			0.349554i	0.984608	+	0.174777i	$0.0559205\pi$
$971$			339.417i			0.349554i	−0.984608	+	0.174777i	$0.944080\pi$
$972$		0			0
$973$	455.309			0.467943
$974$		0			0
$975$	254.290			0.260810
$976$		0			0
$977$		−	673.762i		−	0.689623i	−0.938672	−	0.344812i	$-0.887943\pi$
$977$		−	673.762i		−	0.689623i	0.938672	−	0.344812i	$-0.112057\pi$
$978$		0			0
$979$		−	2732.03i		−	2.79063i
$980$		0			0
$981$			346.363i			0.353071i
$982$		0			0
$983$			428.230i			0.435636i	0.975989	+	0.217818i	$0.0698940\pi$
$983$			428.230i			0.435636i	−0.975989	+	0.217818i	$0.930106\pi$
$984$		0			0
$985$	−398.444			−0.404512
$986$		0			0
$987$			1734.19i			1.75703i
$988$		0			0
$989$	−84.6813			−0.0856232
$990$		0			0
$991$			1310.89i			1.32280i	0.750033	+	0.661400i	$0.230037\pi$
$991$			1310.89i			1.32280i	−0.750033	+	0.661400i	$0.769963\pi$
$992$		0			0
$993$	−610.270			−0.614572
$994$		0			0
$995$	−345.700			−0.347437
$996$		0			0
$997$	−690.240			−0.692316			−0.346158	−	0.938176i	$-0.612514\pi$
$997$	−690.240			−0.692316			−0.346158	+	0.938176i	$0.612514\pi$
$998$		0			0
$999$	976.615			0.977592

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.3.h.a.721.4		12
4.3	odd	2		95.3.c.a.56.3	✓	12
12.11	even	2		855.3.e.a.721.10		12
19.18	odd	2	inner	1520.3.h.a.721.9		12
20.3	even	4		475.3.d.c.474.6		24
20.7	even	4		475.3.d.c.474.19		24
20.19	odd	2		475.3.c.g.151.10		12
76.75	even	2		95.3.c.a.56.10	yes	12
228.227	odd	2		855.3.e.a.721.3		12
380.227	odd	4		475.3.d.c.474.5		24
380.303	odd	4		475.3.d.c.474.20		24
380.379	even	2		475.3.c.g.151.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.3.c.a.56.3	✓	12	4.3	odd	2
95.3.c.a.56.10	yes	12	76.75	even	2
475.3.c.g.151.3		12	380.379	even	2
475.3.c.g.151.10		12	20.19	odd	2
475.3.d.c.474.5		24	380.227	odd	4
475.3.d.c.474.6		24	20.3	even	4
475.3.d.c.474.19		24	20.7	even	4
475.3.d.c.474.20		24	380.303	odd	4
855.3.e.a.721.3		12	228.227	odd	2
855.3.e.a.721.10		12	12.11	even	2
1520.3.h.a.721.4		12	1.1	even	1	trivial
1520.3.h.a.721.9		12	19.18	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1520.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.55563i q^{3} -2.23607 q^{5} -11.0785 q^{7} -3.64250 q^{9} +O(q^{10})\)	\(q-3.55563i q^{3} -2.23607 q^{5} -11.0785 q^{7} -3.64250 q^{9} -16.7704 q^{11} +14.3035i q^{13} +7.95063i q^{15} -20.2160 q^{17} +(16.4480 + 9.51127i) q^{19} +39.3909i q^{21} +13.5882 q^{23} +5.00000 q^{25} -19.0493i q^{27} -30.3972i q^{29} +16.1256i q^{31} +59.6292i q^{33} +24.7722 q^{35} +51.2678i q^{37} +50.8580 q^{39} -74.6652i q^{41} -6.23197 q^{43} +8.14488 q^{45} +44.0252 q^{47} +73.7321 q^{49} +71.8806i q^{51} -30.8058i q^{53} +37.4997 q^{55} +(33.8185 - 58.4829i) q^{57} -73.0098i q^{59} -17.7828 q^{61} +40.3533 q^{63} -31.9837i q^{65} +20.5719i q^{67} -48.3147i q^{69} +7.48928i q^{71} -29.2302 q^{73} -17.7781i q^{75} +185.790 q^{77} -8.78894i q^{79} -100.515 q^{81} +63.7886 q^{83} +45.2043 q^{85} -108.081 q^{87} +162.908i q^{89} -158.461i q^{91} +57.3367 q^{93} +(-36.7788 - 21.2678i) q^{95} +75.5974i q^{97} +61.0861 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 20 q^{7} - 48 q^{9}+O(q^{10}) \) 12 * q - 20 * q^7 - 48 * q^9	\( 12 q - 20 q^{7} - 48 q^{9} - 32 q^{11} - 44 q^{17} - 8 q^{19} - 36 q^{23} + 60 q^{25} + 40 q^{35} - 76 q^{39} - 320 q^{43} - 40 q^{45} + 56 q^{47} + 72 q^{49} + 60 q^{57} - 296 q^{61} + 96 q^{63} - 244 q^{73} - 200 q^{77} - 372 q^{81} + 160 q^{83} + 160 q^{85} - 444 q^{87} + 296 q^{93} + 80 q^{95} + 312 q^{99}+O(q^{100}) \) 12 * q - 20 * q^7 - 48 * q^9 - 32 * q^11 - 44 * q^17 - 8 * q^19 - 36 * q^23 + 60 * q^25 + 40 * q^35 - 76 * q^39 - 320 * q^43 - 40 * q^45 + 56 * q^47 + 72 * q^49 + 60 * q^57 - 296 * q^61 + 96 * q^63 - 244 * q^73 - 200 * q^77 - 372 * q^81 + 160 * q^83 + 160 * q^85 - 444 * q^87 + 296 * q^93 + 80 * q^95 + 312 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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