LMFDB - Embedded newform 40.4.c.a.9.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [40,4,Mod(9,40)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(40, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("40.9");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 40 = 2^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	40.c (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:	$2.36007640023$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			9.4
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	40.9
Dual form			40.4.c.a.9.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.89898i q^{3} +(-10.7980 + 2.89898i) q^{5} +12.6969i q^{7} -20.5959 q^{9} +59.1918 q^{11} +42.2020i q^{13} +(-20.0000 - 74.4949i) q^{15} -126.384i q^{17} +19.1918 q^{19} -87.5959 q^{21} +78.3133i q^{23} +(108.192 - 62.6061i) q^{25} +44.1816i q^{27} +148.384 q^{29} -139.151 q^{31} +408.363i q^{33} +(-36.8082 - 137.101i) q^{35} +66.5653i q^{37} -291.151 q^{39} -203.212 q^{41} +288.879i q^{43} +(222.394 - 59.7071i) q^{45} -360.434i q^{47} +181.788 q^{49} +871.918 q^{51} -686.888i q^{53} +(-639.151 + 171.596i) q^{55} +132.404i q^{57} +83.1102 q^{59} -208.829 q^{61} -261.505i q^{63} +(-122.343 - 455.696i) q^{65} -192.293i q^{67} -540.282 q^{69} +500.767 q^{71} +122.706i q^{73} +(431.918 + 746.413i) q^{75} +751.555i q^{77} +289.616 q^{79} -860.898 q^{81} -573.950i q^{83} +(366.384 + 1364.69i) q^{85} +1023.70i q^{87} +565.151 q^{89} -535.837 q^{91} -960.000i q^{93} +(-207.233 + 55.6367i) q^{95} -643.959i q^{97} -1219.11 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} - 4 q^{9} + 80 q^{11} - 80 q^{15} - 80 q^{19} - 272 q^{21} + 276 q^{25} + 280 q^{29} + 384 q^{31} - 304 q^{35} - 224 q^{39} - 1048 q^{41} + 772 q^{45} + 492 q^{49} + 1920 q^{51} - 1616 q^{55}+ \cdots - 3152 q^{99}+O(q^{100}) $ 4 * q - 4 * q^5 - 4 * q^9 + 80 * q^11 - 80 * q^15 - 80 * q^19 - 272 * q^21 + 276 * q^25 + 280 * q^29 + 384 * q^31 - 304 * q^35 - 224 * q^39 - 1048 * q^41 + 772 * q^45 + 492 * q^49 + 1920 * q^51 - 1616 * q^55 - 1392 * q^59 - 1384 * q^61 + 608 * q^65 + 112 * q^69 + 1376 * q^71 + 160 * q^75 + 1472 * q^79 - 1484 * q^81 + 1152 * q^85 + 1320 * q^89 + 992 * q^91 - 1456 * q^95 - 3152 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/40\mathbb{Z}\right)^\times$.

$n$	$17$	$21$	$31$
$\chi(n)$	$-1$	$1$	$1$

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
$3$			6.89898i			1.32771i	0.747862	+	0.663855i	$0.231081\pi$
$3$			6.89898i			1.32771i	−0.747862	+	0.663855i	$0.768919\pi$
$4$		0			0
$5$	−10.7980	+	2.89898i	−0.965799	+	0.259293i
$5$	−10.7980	+	2.89898i	−0.965799	+	0.259293i
$6$		0			0
$7$			12.6969i			0.685570i	0.939414	+	0.342785i	$0.111370\pi$
$7$			12.6969i			0.685570i	−0.939414	+	0.342785i	$0.888630\pi$
$8$		0			0
$9$	−20.5959			−0.762812
$10$		0			0
$11$	59.1918			1.62246			0.811228	−	0.584730i	$-0.198800\pi$
$11$	59.1918			1.62246			0.811228	+	0.584730i	$0.198800\pi$
$12$		0			0
$13$			42.2020i			0.900365i	0.892937	+	0.450182i	$0.148641\pi$
$13$			42.2020i			0.900365i	−0.892937	+	0.450182i	$0.851359\pi$
$14$		0			0
$15$	−20.0000	−	74.4949i	−0.344265	−	1.28230i
$16$		0			0
$17$		−	126.384i		−	1.80309i	−0.432685	−	0.901545i	$-0.642434\pi$
$17$		−	126.384i		−	1.80309i	0.432685	−	0.901545i	$-0.357566\pi$
$18$		0			0
$19$	19.1918			0.231732			0.115866	−	0.993265i	$-0.463036\pi$
$19$	19.1918			0.231732			0.115866	+	0.993265i	$0.463036\pi$
$20$		0			0
$21$	−87.5959			−0.910238
$22$		0			0
$23$			78.3133i			0.709976i	0.934871	+	0.354988i	$0.115515\pi$
$23$			78.3133i			0.709976i	−0.934871	+	0.354988i	$0.884485\pi$
$24$		0			0
$25$	108.192	−	62.6061i	0.865535	−	0.500849i
$26$		0			0
$27$			44.1816i			0.314917i
$28$		0			0
$29$	148.384			0.950143			0.475072	−	0.879947i	$-0.342422\pi$
$29$	148.384			0.950143			0.475072	+	0.879947i	$0.342422\pi$
$30$		0			0
$31$	−139.151			−0.806202			−0.403101	−	0.915156i	$-0.632068\pi$
$31$	−139.151			−0.806202			−0.403101	+	0.915156i	$0.632068\pi$
$32$		0			0
$33$			408.363i			2.15415i
$34$		0			0
$35$	−36.8082	−	137.101i	−0.177763	−	0.662123i
$36$		0			0
$37$			66.5653i			0.295764i	0.989005	+	0.147882i	$0.0472456\pi$
$37$			66.5653i			0.295764i	−0.989005	+	0.147882i	$0.952754\pi$
$38$		0			0
$39$	−291.151			−1.19542
$40$		0			0
$41$	−203.212			−0.774059			−0.387030	−	0.922067i	$-0.626499\pi$
$41$	−203.212			−0.774059			−0.387030	+	0.922067i	$0.626499\pi$
$42$		0			0
$43$			288.879i			1.02450i	0.858836	+	0.512251i	$0.171188\pi$
$43$			288.879i			1.02450i	−0.858836	+	0.512251i	$0.828812\pi$
$44$		0			0
$45$	222.394	−	59.7071i	0.736723	−	0.197791i
$46$		0			0
$47$		−	360.434i		−	1.11861i	−0.828962	−	0.559305i	$-0.811068\pi$
$47$		−	360.434i		−	1.11861i	0.828962	−	0.559305i	$-0.188932\pi$
$48$		0			0
$49$	181.788			0.529993
$50$		0			0
$51$	871.918			2.39398
$52$		0			0
$53$		−	686.888i		−	1.78021i	−0.455753	−	0.890106i	$-0.650630\pi$
$53$		−	686.888i		−	1.78021i	0.455753	−	0.890106i	$-0.349370\pi$
$54$		0			0
$55$	−639.151	+	171.596i	−1.56697	+	0.420691i
$56$		0			0
$57$			132.404i			0.307673i
$58$		0			0
$59$	83.1102			0.183390			0.0916951	−	0.995787i	$-0.470771\pi$
$59$	83.1102			0.183390			0.0916951	+	0.995787i	$0.470771\pi$
$60$		0			0
$61$	−208.829			−0.438324			−0.219162	−	0.975688i	$-0.570332\pi$
$61$	−208.829			−0.438324			−0.219162	+	0.975688i	$0.570332\pi$
$62$		0			0
$63$		−	261.505i		−	0.522961i
$64$		0			0
$65$	−122.343	−	455.696i	−0.233458	−	0.869571i
$66$		0			0
$67$		−	192.293i		−	0.350632i	−0.984512	−	0.175316i	$-0.943905\pi$
$67$		−	192.293i		−	0.350632i	0.984512	−	0.175316i	$-0.0560946\pi$
$68$		0			0
$69$	−540.282			−0.942642
$70$		0			0
$71$	500.767			0.837044			0.418522	−	0.908207i	$-0.362548\pi$
$71$	500.767			0.837044			0.418522	+	0.908207i	$0.362548\pi$
$72$		0			0
$73$			122.706i			0.196735i	0.995150	+	0.0983676i	$0.0313621\pi$
$73$			122.706i			0.196735i	−0.995150	+	0.0983676i	$0.968638\pi$
$74$		0			0
$75$	431.918	+	746.413i	0.664982	+	1.14918i
$76$		0			0
$77$			751.555i			1.11231i
$78$		0			0
$79$	289.616			0.412461			0.206230	−	0.978503i	$-0.433880\pi$
$79$	289.616			0.412461			0.206230	+	0.978503i	$0.433880\pi$
$80$		0			0
$81$	−860.898			−1.18093
$82$		0			0
$83$		−	573.950i		−	0.759026i	−0.925186	−	0.379513i	$-0.876091\pi$
$83$		−	573.950i		−	0.759026i	0.925186	−	0.379513i	$-0.123909\pi$
$84$		0			0
$85$	366.384	+	1364.69i	0.467528	+	1.74142i
$86$		0			0
$87$			1023.70i			1.26151i
$88$		0			0
$89$	565.151			0.673100			0.336550	−	0.941666i	$-0.390740\pi$
$89$	565.151			0.673100			0.336550	+	0.941666i	$0.390740\pi$
$90$		0			0
$91$	−535.837			−0.617263
$92$		0			0
$93$		−	960.000i		−	1.07040i
$94$		0			0
$95$	−207.233	+	55.6367i	−0.223807	+	0.0600864i
$96$		0			0
$97$		−	643.959i		−	0.674063i	−0.941493	−	0.337032i	$-0.890577\pi$
$97$		−	643.959i		−	0.674063i	0.941493	−	0.337032i	$-0.109423\pi$
$98$		0			0
$99$	−1219.11			−1.23763
$100$		0			0
$101$	−745.918			−0.734868			−0.367434	−	0.930050i	$-0.619764\pi$
$101$	−745.918			−0.734868			−0.367434	+	0.930050i	$0.619764\pi$
$102$		0			0
$103$		−	738.413i		−	0.706389i	−0.935550	−	0.353194i	$-0.885095\pi$
$103$		−	738.413i		−	0.706389i	0.935550	−	0.353194i	$-0.114905\pi$
$104$		0			0
$105$	945.857	−	253.939i	0.879107	−	0.236018i
$106$		0			0
$107$			558.536i			0.504633i	0.967645	+	0.252316i	$0.0811923\pi$
$107$			558.536i			0.504633i	−0.967645	+	0.252316i	$0.918808\pi$
$108$		0			0
$109$	1523.69			1.33893			0.669465	−	0.742843i	$-0.266523\pi$
$109$	1523.69			1.33893			0.669465	+	0.742843i	$0.266523\pi$
$110$		0			0
$111$	−459.233			−0.392689
$112$		0			0
$113$		−	58.4245i		−	0.0486382i	−0.999704	−	0.0243191i	$-0.992258\pi$
$113$		−	58.4245i		−	0.0486382i	0.999704	−	0.0243191i	$-0.00774177\pi$
$114$		0			0
$115$	−227.029	−	845.623i	−0.184091	−	0.685694i
$116$		0			0
$117$		−	869.190i		−	0.686809i
$118$		0			0
$119$	1604.69			1.23615
$120$		0			0
$121$	2172.67			1.63236
$122$		0			0
$123$		−	1401.96i		−	1.02773i
$124$		0			0
$125$	−986.757	+	989.664i	−0.706066	+	0.708146i
$126$		0			0
$127$		−	941.283i		−	0.657680i	−0.944386	−	0.328840i	$-0.893342\pi$
$127$		−	941.283i		−	0.657680i	0.944386	−	0.328840i	$-0.106658\pi$
$128$		0			0
$129$	−1992.97			−1.36024
$130$		0			0
$131$	−214.343			−0.142956			−0.0714779	−	0.997442i	$-0.522772\pi$
$131$	−214.343			−0.142956			−0.0714779	+	0.997442i	$0.522772\pi$
$132$		0			0
$133$			243.678i			0.158869i
$134$		0			0
$135$	−128.082	−	477.071i	−0.0816556	−	0.304146i
$136$		0			0
$137$			499.514i			0.311506i	0.987796	+	0.155753i	$0.0497805\pi$
$137$			499.514i			0.311506i	−0.987796	+	0.155753i	$0.950220\pi$
$138$		0			0
$139$	−2660.48			−1.62345			−0.811723	−	0.584042i	$-0.801470\pi$
$139$	−2660.48			−1.62345			−0.811723	+	0.584042i	$0.801470\pi$
$140$		0			0
$141$	2486.62			1.48519
$142$		0			0
$143$			2498.02i			1.46080i
$144$		0			0
$145$	−1602.24	+	430.161i	−0.917647	+	0.246365i
$146$		0			0
$147$			1254.15i			0.703677i
$148$		0			0
$149$	−1334.89			−0.733947			−0.366973	−	0.930231i	$-0.619606\pi$
$149$	−1334.89			−0.733947			−0.366973	+	0.930231i	$0.619606\pi$
$150$		0			0
$151$	−259.478			−0.139841			−0.0699205	−	0.997553i	$-0.522275\pi$
$151$	−259.478			−0.139841			−0.0699205	+	0.997553i	$0.522275\pi$
$152$		0			0
$153$			2602.99i			1.37542i
$154$		0			0
$155$	1502.55	−	403.396i	0.778629	−	0.209042i
$156$		0			0
$157$		−	1063.58i		−	0.540654i	−0.962769	−	0.270327i	$-0.912868\pi$
$157$		−	1063.58i		−	0.540654i	0.962769	−	0.270327i	$-0.0871319\pi$
$158$		0			0
$159$	4738.82			2.36361
$160$		0			0
$161$	−994.339			−0.486738
$162$		0			0
$163$			3139.67i			1.50870i	0.656474	+	0.754349i	$0.272047\pi$
$163$			3139.67i			1.50870i	−0.656474	+	0.754349i	$0.727953\pi$
$164$		0			0
$165$	−1183.84	−	4409.49i	−0.558555	−	2.08047i
$166$		0			0
$167$			936.411i			0.433902i	0.976182	+	0.216951i	$0.0696112\pi$
$167$			936.411i			0.433902i	−0.976182	+	0.216951i	$0.930389\pi$
$168$		0			0
$169$	415.988			0.189344
$170$		0			0
$171$	−395.273			−0.176768
$172$		0			0
$173$			1439.78i			0.632741i	0.948636	+	0.316371i	$0.102464\pi$
$173$			1439.78i			0.632741i	−0.948636	+	0.316371i	$0.897536\pi$
$174$		0			0
$175$	794.906	+	1373.71i	0.343367	+	0.593385i
$176$		0			0
$177$			573.376i			0.243489i
$178$		0			0
$179$	−2494.42			−1.04158			−0.520788	−	0.853686i	$-0.674362\pi$
$179$	−2494.42			−1.04158			−0.520788	+	0.853686i	$0.674362\pi$
$180$		0			0
$181$	−1590.36			−0.653096			−0.326548	−	0.945181i	$-0.605886\pi$
$181$	−1590.36			−0.653096			−0.326548	+	0.945181i	$0.605886\pi$
$182$		0			0
$183$		−	1440.70i		−	0.581967i
$184$		0			0
$185$	−192.971	−	718.769i	−0.0766894	−	0.285649i
$186$		0			0
$187$		−	7480.88i		−	2.92543i
$188$		0			0
$189$	−560.971			−0.215898
$190$		0			0
$191$	−2652.77			−1.00496			−0.502480	−	0.864589i	$-0.667579\pi$
$191$	−2652.77			−1.00496			−0.502480	+	0.864589i	$0.667579\pi$
$192$		0			0
$193$		−	3001.61i		−	1.11948i	−0.828667	−	0.559742i	$-0.810900\pi$
$193$		−	3001.61i		−	1.11948i	0.828667	−	0.559742i	$-0.189100\pi$
$194$		0			0
$195$	3143.84	−	844.041i	1.15454	−	0.309964i
$196$		0			0
$197$			2176.06i			0.786994i	0.919326	+	0.393497i	$0.128735\pi$
$197$			2176.06i			0.786994i	−0.919326	+	0.393497i	$0.871265\pi$
$198$		0			0
$199$	4270.30			1.52117			0.760587	−	0.649236i	$-0.224911\pi$
$199$	4270.30			1.52117			0.760587	+	0.649236i	$0.224911\pi$
$200$		0			0
$201$	1326.62			0.465537
$202$		0			0
$203$			1884.02i			0.651390i
$204$		0			0
$205$	2194.28	−	589.108i	0.747585	−	0.200708i
$206$		0			0
$207$		−	1612.93i		−	0.541578i
$208$		0			0
$209$	1136.00			0.375975
$210$		0			0
$211$	−2978.78			−0.971886			−0.485943	−	0.873991i	$-0.661524\pi$
$211$	−2978.78			−0.971886			−0.485943	+	0.873991i	$0.661524\pi$
$212$		0			0
$213$			3454.78i			1.11135i
$214$		0			0
$215$	−837.453	−	3119.30i	−0.265646	−	0.989462i
$216$		0			0
$217$		−	1766.79i		−	0.552708i
$218$		0			0
$219$	−846.547			−0.261207
$220$		0			0
$221$	5333.65			1.62344
$222$		0			0
$223$			3183.16i			0.955875i	0.878394	+	0.477937i	$0.158615\pi$
$223$			3183.16i			0.955875i	−0.878394	+	0.477937i	$0.841385\pi$
$224$		0			0
$225$	−2228.31	+	1289.43i	−0.660240	+	0.382054i
$226$		0			0
$227$			5910.41i			1.72814i	0.503372	+	0.864070i	$0.332092\pi$
$227$			5910.41i			1.72814i	−0.503372	+	0.864070i	$0.667908\pi$
$228$		0			0
$229$	−3465.59			−1.00006			−0.500028	−	0.866009i	$-0.666677\pi$
$229$	−3465.59			−1.00006			−0.500028	+	0.866009i	$0.666677\pi$
$230$		0			0
$231$	−5184.96			−1.47682
$232$		0			0
$233$		−	4611.51i		−	1.29661i	−0.761380	−	0.648305i	$-0.775478\pi$
$233$		−	4611.51i		−	1.29661i	0.761380	−	0.648305i	$-0.224522\pi$
$234$		0			0
$235$	1044.89	+	3891.95i	0.290047	+	1.08035i
$236$		0			0
$237$			1998.06i			0.547628i
$238$		0			0
$239$	−4816.47			−1.30356			−0.651782	−	0.758406i	$-0.725978\pi$
$239$	−4816.47			−1.30356			−0.651782	+	0.758406i	$0.725978\pi$
$240$		0			0
$241$	3057.47			0.817217			0.408608	−	0.912710i	$-0.366014\pi$
$241$	3057.47			0.817217			0.408608	+	0.912710i	$0.366014\pi$
$242$		0			0
$243$		−	4746.41i		−	1.25301i
$244$		0			0
$245$	−1962.94	+	526.999i	−0.511867	+	0.137423i
$246$		0			0
$247$			809.935i			0.208643i
$248$		0			0
$249$	3959.67			1.00777
$250$		0			0
$251$	7498.54			1.88567			0.942836	−	0.333258i	$-0.108148\pi$
$251$	7498.54			1.88567			0.942836	+	0.333258i	$0.108148\pi$
$252$		0			0
$253$			4635.51i			1.15190i
$254$		0			0
$255$	−9414.94	+	2527.67i	−2.31210	+	0.620741i
$256$		0			0
$257$			2272.33i			0.551532i	0.961225	+	0.275766i	$0.0889316\pi$
$257$			2272.33i			0.551532i	−0.961225	+	0.275766i	$0.911068\pi$
$258$		0			0
$259$	−845.176			−0.202767
$260$		0			0
$261$	−3056.10			−0.724781
$262$		0			0
$263$		−	5271.34i		−	1.23591i	−0.786213	−	0.617955i	$-0.787961\pi$
$263$		−	5271.34i		−	1.23591i	0.786213	−	0.617955i	$-0.212039\pi$
$264$		0			0
$265$	1991.27	+	7416.99i	0.461596	+	1.71933i
$266$		0			0
$267$			3898.97i			0.893681i
$268$		0			0
$269$	4315.61			0.978169			0.489085	−	0.872236i	$-0.337331\pi$
$269$	4315.61			0.978169			0.489085	+	0.872236i	$0.337331\pi$
$270$		0			0
$271$	4471.27			1.00225			0.501125	−	0.865375i	$-0.332919\pi$
$271$	4471.27			1.00225			0.501125	+	0.865375i	$0.332919\pi$
$272$		0			0
$273$		−	3696.73i		−	0.819546i
$274$		0			0
$275$	6404.07	−	3705.77i	1.40429	−	0.812605i
$276$		0			0
$277$		−	8112.09i		−	1.75960i	−0.475348	−	0.879798i	$-0.657678\pi$
$277$		−	8112.09i		−	1.75960i	0.475348	−	0.879798i	$-0.342322\pi$
$278$		0			0
$279$	2865.94			0.614980
$280$		0			0
$281$	−3811.46			−0.809155			−0.404577	−	0.914504i	$-0.632581\pi$
$281$	−3811.46			−0.809155			−0.404577	+	0.914504i	$0.632581\pi$
$282$		0			0
$283$		−	2362.60i		−	0.496261i	−0.968727	−	0.248131i	$-0.920184\pi$
$283$		−	2362.60i		−	0.496261i	0.968727	−	0.248131i	$-0.0798162\pi$
$284$		0			0
$285$	−383.837	−	1429.69i	−0.0797773	−	0.297150i
$286$		0			0
$287$		−	2580.17i		−	0.530672i
$288$		0			0
$289$	−11059.8			−2.25114
$290$		0			0
$291$	4442.66			0.894960
$292$		0			0
$293$		−	6337.79i		−	1.26368i	−0.775099	−	0.631839i	$-0.782300\pi$
$293$		−	6337.79i		−	1.26368i	0.775099	−	0.631839i	$-0.217700\pi$
$294$		0			0
$295$	−897.420	+	240.935i	−0.177118	+	0.0475517i
$296$		0			0
$297$			2615.19i			0.510939i
$298$		0			0
$299$	−3304.98			−0.639237
$300$		0			0
$301$	−3667.87			−0.702368
$302$		0			0
$303$		−	5146.08i		−	0.975691i
$304$		0			0
$305$	2254.92	−	605.390i	0.423333	−	0.113654i
$306$		0			0
$307$			4170.44i			0.775308i	0.921805	+	0.387654i	$0.126715\pi$
$307$			4170.44i			0.775308i	−0.921805	+	0.387654i	$0.873285\pi$
$308$		0			0
$309$	5094.30			0.937879
$310$		0			0
$311$	−5083.71			−0.926915			−0.463457	−	0.886119i	$-0.653391\pi$
$311$	−5083.71			−0.926915			−0.463457	+	0.886119i	$0.653391\pi$
$312$		0			0
$313$		−	1522.86i		−	0.275007i	−0.990501	−	0.137504i	$-0.956092\pi$
$313$		−	1522.86i		−	0.275007i	0.990501	−	0.137504i	$-0.0439078\pi$
$314$		0			0
$315$	758.098	+	2823.72i	0.135600	+	0.505075i
$316$		0			0
$317$		−	4996.09i		−	0.885200i	−0.896719	−	0.442600i	$-0.854056\pi$
$317$		−	4996.09i		−	0.885200i	0.896719	−	0.442600i	$-0.145944\pi$
$318$		0			0
$319$	8783.10			1.54157
$320$		0			0
$321$	−3853.33			−0.670005
$322$		0			0
$323$		−	2425.53i		−	0.417834i
$324$		0			0
$325$	2642.11	+	4565.92i	0.450947	+	0.779297i
$326$		0			0
$327$			10511.9i			1.77771i
$328$		0			0
$329$	4576.40			0.766885
$330$		0			0
$331$	2832.04			0.470281			0.235141	−	0.971961i	$-0.424445\pi$
$331$	2832.04			0.470281			0.235141	+	0.971961i	$0.424445\pi$
$332$		0			0
$333$		−	1370.97i		−	0.225612i
$334$		0			0
$335$	557.453	+	2076.37i	0.0909162	+	0.338639i
$336$		0			0
$337$			1684.43i			0.272276i	0.990690	+	0.136138i	$0.0434690\pi$
$337$			1684.43i			0.272276i	−0.990690	+	0.136138i	$0.956531\pi$
$338$		0			0
$339$	403.069			0.0645774
$340$		0			0
$341$	−8236.60			−1.30803
$342$		0			0
$343$			6663.20i			1.04892i
$344$		0			0
$345$	5833.94	−	1566.27i	0.910402	−	0.244420i
$346$		0			0
$347$		−	9350.30i		−	1.44654i	−0.690564	−	0.723271i	$-0.742638\pi$
$347$		−	9350.30i		−	1.44654i	0.690564	−	0.723271i	$-0.257362\pi$
$348$		0			0
$349$	−4174.20			−0.640228			−0.320114	−	0.947379i	$-0.603721\pi$
$349$	−4174.20			−0.640228			−0.320114	+	0.947379i	$0.603721\pi$
$350$		0			0
$351$	−1864.55			−0.283540
$352$		0			0
$353$			6217.04i			0.937394i	0.883359	+	0.468697i	$0.155276\pi$
$353$			6217.04i			0.937394i	−0.883359	+	0.468697i	$0.844724\pi$
$354$		0			0
$355$	−5407.27	+	1451.71i	−0.808416	+	0.217039i
$356$		0			0
$357$			11070.7i			1.64124i
$358$		0			0
$359$	−8149.73			−1.19812			−0.599062	−	0.800703i	$-0.704460\pi$
$359$	−8149.73			−1.19812			−0.599062	+	0.800703i	$0.704460\pi$
$360$		0			0
$361$	−6490.67			−0.946300
$362$		0			0
$363$			14989.2i			2.16730i
$364$		0			0
$365$	−355.723	−	1324.98i	−0.0510120	−	0.190007i
$366$		0			0
$367$		−	11868.2i		−	1.68806i	−0.536299	−	0.844028i	$-0.680178\pi$
$367$		−	11868.2i		−	1.68806i	0.536299	−	0.844028i	$-0.319822\pi$
$368$		0			0
$369$	4185.34			0.590461
$370$		0			0
$371$	8721.37			1.22046
$372$		0			0
$373$			3286.85i			0.456265i	0.973630	+	0.228132i	$0.0732619\pi$
$373$			3286.85i			0.456265i	−0.973630	+	0.228132i	$0.926738\pi$
$374$		0			0
$375$	−6827.67	−	6807.62i	−0.940212	−	0.937450i
$376$		0			0
$377$			6262.09i			0.855475i
$378$		0			0
$379$	−8326.24			−1.12847			−0.564235	−	0.825614i	$-0.690829\pi$
$379$	−8326.24			−1.12847			−0.564235	+	0.825614i	$0.690829\pi$
$380$		0			0
$381$	6493.89			0.873207
$382$		0			0
$383$		−	4249.23i		−	0.566908i	−0.958986	−	0.283454i	$-0.908520\pi$
$383$		−	4249.23i		−	0.566908i	0.958986	−	0.283454i	$-0.0914803\pi$
$384$		0			0
$385$	−2178.74	−	8115.26i	−0.288413	−	1.07426i
$386$		0			0
$387$		−	5949.72i		−	0.781502i
$388$		0			0
$389$	7841.83			1.02210			0.511050	−	0.859551i	$-0.329257\pi$
$389$	7841.83			1.02210			0.511050	+	0.859551i	$0.329257\pi$
$390$		0			0
$391$	9897.52			1.28015
$392$		0			0
$393$		−	1478.75i		−	0.189804i
$394$		0			0
$395$	−3127.27	+	839.592i	−0.398354	+	0.106948i
$396$		0			0
$397$			401.794i			0.0507946i	0.999677	+	0.0253973i	$0.00808508\pi$
$397$			401.794i			0.0507946i	−0.999677	+	0.0253973i	$0.991915\pi$
$398$		0			0
$399$	−1681.13			−0.210931
$400$		0			0
$401$	9732.89			1.21206			0.606032	−	0.795441i	$-0.292761\pi$
$401$	9732.89			1.21206			0.606032	+	0.795441i	$0.292761\pi$
$402$		0			0
$403$		−	5872.46i		−	0.725876i
$404$		0			0
$405$	9295.94	−	2495.73i	1.14054	−	0.306206i
$406$		0			0
$407$			3940.12i			0.479864i
$408$		0			0
$409$	5629.67			0.680609			0.340305	−	0.940315i	$-0.389470\pi$
$409$	5629.67			0.680609			0.340305	+	0.940315i	$0.389470\pi$
$410$		0			0
$411$	−3446.14			−0.413590
$412$		0			0
$413$			1055.25i			0.125727i
$414$		0			0
$415$	1663.87	+	6197.49i	0.196810	+	0.733067i
$416$		0			0
$417$		−	18354.6i		−	2.15547i
$418$		0			0
$419$	−9773.77			−1.13957			−0.569785	−	0.821794i	$-0.692974\pi$
$419$	−9773.77			−1.13957			−0.569785	+	0.821794i	$0.692974\pi$
$420$		0			0
$421$	3037.95			0.351688			0.175844	−	0.984418i	$-0.443735\pi$
$421$	3037.95			0.351688			0.175844	+	0.984418i	$0.443735\pi$
$422$		0			0
$423$			7423.46i			0.853289i
$424$		0			0
$425$	−7912.39	−	13673.7i	−0.903076	−	1.56064i
$426$		0			0
$427$		−	2651.48i		−	0.300502i
$428$		0			0
$429$	−17233.8			−1.93952
$430$		0			0
$431$	1873.45			0.209376			0.104688	−	0.994505i	$-0.466616\pi$
$431$	1873.45			0.209376			0.104688	+	0.994505i	$0.466616\pi$
$432$		0			0
$433$			11855.8i			1.31583i	0.753092	+	0.657915i	$0.228561\pi$
$433$			11855.8i			1.31583i	−0.753092	+	0.657915i	$0.771439\pi$
$434$		0			0
$435$	−2967.67	−	11053.8i	−0.327101	−	1.21837i
$436$		0			0
$437$			1502.98i			0.164524i
$438$		0			0
$439$	16243.8			1.76600			0.882999	−	0.469375i	$-0.155521\pi$
$439$	16243.8			1.76600			0.882999	+	0.469375i	$0.155521\pi$
$440$		0			0
$441$	−3744.09			−0.404285
$442$		0			0
$443$			3905.52i			0.418864i	0.977823	+	0.209432i	$0.0671614\pi$
$443$			3905.52i			0.418864i	−0.977823	+	0.209432i	$0.932839\pi$
$444$		0			0
$445$	−6102.48	+	1638.36i	−0.650079	+	0.174530i
$446$		0			0
$447$		−	9209.35i		−	0.974468i
$448$		0			0
$449$	6852.71			0.720265			0.360133	−	0.932901i	$-0.382731\pi$
$449$	6852.71			0.720265			0.360133	+	0.932901i	$0.382731\pi$
$450$		0			0
$451$	−12028.5			−1.25588
$452$		0			0
$453$		−	1790.13i		−	0.185668i
$454$		0			0
$455$	5785.94	−	1553.38i	0.596152	−	0.160052i
$456$		0			0
$457$		−	5800.75i		−	0.593758i	−0.954915	−	0.296879i	$-0.904054\pi$
$457$		−	5800.75i		−	0.593758i	0.954915	−	0.296879i	$-0.0959458\pi$
$458$		0			0
$459$	5583.84			0.567824
$460$		0			0
$461$	−5263.63			−0.531783			−0.265891	−	0.964003i	$-0.585666\pi$
$461$	−5263.63			−0.531783			−0.265891	+	0.964003i	$0.585666\pi$
$462$		0			0
$463$			13636.6i			1.36879i	0.729113	+	0.684394i	$0.239933\pi$
$463$			13636.6i			1.36879i	−0.729113	+	0.684394i	$0.760067\pi$
$464$		0			0
$465$	2783.02	+	10366.0i	0.277547	+	1.03379i
$466$		0			0
$467$			13158.3i			1.30384i	0.758287	+	0.651920i	$0.226036\pi$
$467$			13158.3i			1.30384i	−0.758287	+	0.651920i	$0.773964\pi$
$468$		0			0
$469$	2441.53			0.240383
$470$		0			0
$471$	7337.60			0.717832
$472$		0			0
$473$			17099.3i			1.66221i
$474$		0			0
$475$	2076.40	−	1201.53i	0.200572	−	0.116063i
$476$		0			0
$477$			14147.1i			1.35797i
$478$		0			0
$479$	6775.90			0.646344			0.323172	−	0.946340i	$-0.395251\pi$
$479$	6775.90			0.646344			0.323172	+	0.946340i	$0.395251\pi$
$480$		0			0
$481$	−2809.19			−0.266295
$482$		0			0
$483$		−	6859.92i		−	0.646247i
$484$		0			0
$485$	1866.82	+	6953.44i	0.174780	+	0.651010i
$486$		0			0
$487$			5933.64i			0.552113i	0.961141	+	0.276057i	$0.0890277\pi$
$487$			5933.64i			0.552113i	−0.961141	+	0.276057i	$0.910972\pi$
$488$		0			0
$489$	−21660.5			−2.00311
$490$		0			0
$491$	−11934.5			−1.09694			−0.548468	−	0.836171i	$-0.684789\pi$
$491$	−11934.5			−1.09694			−0.548468	+	0.836171i	$0.684789\pi$
$492$		0			0
$493$		−	18753.3i		−	1.71319i
$494$		0			0
$495$	13163.9	−	3534.18i	1.19530	−	0.320908i
$496$		0			0
$497$			6358.21i			0.573853i
$498$		0			0
$499$	1567.65			0.140636			0.0703182	−	0.997525i	$-0.477599\pi$
$499$	1567.65			0.140636			0.0703182	+	0.997525i	$0.477599\pi$
$500$		0			0
$501$	−6460.28			−0.576096
$502$		0			0
$503$			450.509i			0.0399348i	0.999801	+	0.0199674i	$0.00635624\pi$
$503$			450.509i			0.0399348i	−0.999801	+	0.0199674i	$0.993644\pi$
$504$		0			0
$505$	8054.40	−	2162.40i	0.709734	−	0.190546i
$506$		0			0
$507$			2869.89i			0.251393i
$508$		0			0
$509$	7016.11			0.610969			0.305485	−	0.952197i	$-0.401182\pi$
$509$	7016.11			0.610969			0.305485	+	0.952197i	$0.401182\pi$
$510$		0			0
$511$	−1557.99			−0.134876
$512$		0			0
$513$			847.927i			0.0729764i
$514$		0			0
$515$	2140.64	+	7973.36i	0.183161	+	0.682229i
$516$		0			0
$517$		−	21334.7i		−	1.81489i
$518$		0			0
$519$	−9933.00			−0.840097
$520$		0			0
$521$	−16356.0			−1.37538			−0.687688	−	0.726006i	$-0.741374\pi$
$521$	−16356.0			−1.37538			−0.687688	+	0.726006i	$0.741374\pi$
$522$		0			0
$523$		−	13190.5i		−	1.10283i	−0.834232	−	0.551414i	$-0.814089\pi$
$523$		−	13190.5i		−	1.10283i	0.834232	−	0.551414i	$-0.185911\pi$
$524$		0			0
$525$	−9477.16	+	5484.04i	−0.787842	+	0.455892i
$526$		0			0
$527$			17586.4i			1.45366i
$528$		0			0
$529$	6034.03			0.495934
$530$		0			0
$531$	−1711.73			−0.139892
$532$		0			0
$533$		−	8575.97i		−	0.696935i
$534$		0			0
$535$	−1619.18	−	6031.05i	−0.130847	−	0.487374i
$536$		0			0
$537$		−	17209.0i		−	1.38291i
$538$		0			0
$539$	10760.4			0.859891
$540$		0			0
$541$	21997.6			1.74815			0.874076	−	0.485789i	$-0.161468\pi$
$541$	21997.6			1.74815			0.874076	+	0.485789i	$0.161468\pi$
$542$		0			0
$543$		−	10971.9i		−	0.867122i
$544$		0			0
$545$	−16452.8	+	4417.16i	−1.29314	+	0.347175i
$546$		0			0
$547$			212.663i			0.0166230i	0.999965	+	0.00831151i	$0.00264567\pi$
$547$			212.663i			0.0166230i	−0.999965	+	0.00831151i	$0.997354\pi$
$548$		0			0
$549$	4301.02			0.334359
$550$		0			0
$551$	2847.76			0.220179
$552$		0			0
$553$			3677.24i			0.282771i
$554$		0			0
$555$	4958.78	−	1331.31i	0.379258	−	0.101821i
$556$		0			0
$557$		−	6641.66i		−	0.505235i	−0.967566	−	0.252618i	$-0.918709\pi$
$557$		−	6641.66i		−	0.505235i	0.967566	−	0.252618i	$-0.0812915\pi$
$558$		0			0
$559$	−12191.3			−0.922425
$560$		0			0
$561$	51610.4			3.88413
$562$		0			0
$563$			7122.94i			0.533208i	0.963806	+	0.266604i	$0.0859016\pi$
$563$			7122.94i			0.533208i	−0.963806	+	0.266604i	$0.914098\pi$
$564$		0			0
$565$	169.371	+	630.865i	0.0126115	+	0.0469747i
$566$		0			0
$567$		−	10930.8i		−	0.809610i
$568$		0			0
$569$	−5951.39			−0.438480			−0.219240	−	0.975671i	$-0.570358\pi$
$569$	−5951.39			−0.438480			−0.219240	+	0.975671i	$0.570358\pi$
$570$		0			0
$571$	−23612.2			−1.73054			−0.865272	−	0.501303i	$-0.832854\pi$
$571$	−23612.2			−1.73054			−0.865272	+	0.501303i	$0.832854\pi$
$572$		0			0
$573$		−	18301.4i		−	1.33430i
$574$		0			0
$575$	4902.89	+	8472.86i	0.355591	+	0.614509i
$576$		0			0
$577$			25398.1i			1.83247i	0.400636	+	0.916237i	$0.368789\pi$
$577$			25398.1i			1.83247i	−0.400636	+	0.916237i	$0.631211\pi$
$578$		0			0
$579$	20708.0			1.48635
$580$		0			0
$581$	7287.41			0.520366
$582$		0			0
$583$		−	40658.1i		−	2.88832i
$584$		0			0
$585$	2519.76	+	9385.48i	0.178084	+	0.663319i
$586$		0			0
$587$		−	17994.6i		−	1.26528i	−0.774448	−	0.632638i	$-0.781972\pi$
$587$		−	17994.6i		−	1.26528i	0.774448	−	0.632638i	$-0.218028\pi$
$588$		0			0
$589$	−2670.56			−0.186823
$590$		0			0
$591$	−15012.6			−1.04490
$592$		0			0
$593$		−	14842.0i		−	1.02780i	−0.857850	−	0.513900i	$-0.828200\pi$
$593$		−	14842.0i		−	1.02780i	0.857850	−	0.513900i	$-0.171800\pi$
$594$		0			0
$595$	−17327.3	+	4651.95i	−1.19387	+	0.320523i
$596$		0			0
$597$			29460.7i			2.01968i
$598$		0			0
$599$	−25071.6			−1.71018			−0.855089	−	0.518481i	$-0.826498\pi$
$599$	−25071.6			−1.71018			−0.855089	+	0.518481i	$0.826498\pi$
$600$		0			0
$601$	−2772.27			−0.188159			−0.0940794	−	0.995565i	$-0.529991\pi$
$601$	−2772.27			−0.188159			−0.0940794	+	0.995565i	$0.529991\pi$
$602$		0			0
$603$			3960.45i			0.267466i
$604$		0			0
$605$	−23460.4	+	6298.54i	−1.57653	+	0.423259i
$606$		0			0
$607$			19585.8i			1.30966i	0.755776	+	0.654830i	$0.227260\pi$
$607$			19585.8i			1.30966i	−0.755776	+	0.654830i	$0.772740\pi$
$608$		0			0
$609$	−12997.8			−0.864856
$610$		0			0
$611$	15211.0			1.00716
$612$		0			0
$613$			24247.5i			1.59763i	0.601579	+	0.798814i	$0.294539\pi$
$613$			24247.5i			1.59763i	−0.601579	+	0.798814i	$0.705461\pi$
$614$		0			0
$615$	4064.24	+	15138.3i	0.266482	+	0.992576i
$616$		0			0
$617$		−	1225.83i		−	0.0799841i	−0.999200	−	0.0399920i	$-0.987267\pi$
$617$		−	1225.83i		−	0.0799841i	0.999200	−	0.0399920i	$-0.0127333\pi$
$618$		0			0
$619$	12430.1			0.807122			0.403561	−	0.914953i	$-0.367772\pi$
$619$	12430.1			0.807122			0.403561	+	0.914953i	$0.367772\pi$
$620$		0			0
$621$	−3460.01			−0.223583
$622$		0			0
$623$			7175.69i			0.461457i
$624$		0			0
$625$	7785.95	−	13546.9i	0.498301	−	0.867004i
$626$		0			0
$627$			7837.24i			0.499185i
$628$		0			0
$629$	8412.77			0.533289
$630$		0			0
$631$	−6215.00			−0.392100			−0.196050	−	0.980594i	$-0.562812\pi$
$631$	−6215.00			−0.392100			−0.196050	+	0.980594i	$0.562812\pi$
$632$		0			0
$633$		−	20550.6i		−	1.29038i
$634$		0			0
$635$	2728.76	+	10163.9i	0.170531	+	0.635186i
$636$		0			0
$637$			7671.81i			0.477187i
$638$		0			0
$639$	−10313.8			−0.638507
$640$		0			0
$641$	5059.76			0.311776			0.155888	−	0.987775i	$-0.450176\pi$
$641$	5059.76			0.311776			0.155888	+	0.987775i	$0.450176\pi$
$642$		0			0
$643$			8676.69i			0.532154i	0.963952	+	0.266077i	$0.0857276\pi$
$643$			8676.69i			0.532154i	−0.963952	+	0.266077i	$0.914272\pi$
$644$		0			0
$645$	21520.0	−	5777.57i	1.31372	−	0.352700i
$646$		0			0
$647$		−	22787.4i		−	1.38465i	−0.721587	−	0.692323i	$-0.756587\pi$
$647$		−	22787.4i		−	1.38465i	0.721587	−	0.692323i	$-0.243413\pi$
$648$		0			0
$649$	4919.45			0.297543
$650$		0			0
$651$	12189.1			0.733836
$652$		0			0
$653$			26813.1i			1.60686i	0.595402	+	0.803428i	$0.296993\pi$
$653$			26813.1i			1.60686i	−0.595402	+	0.803428i	$0.703007\pi$
$654$		0			0
$655$	2314.47	−	621.376i	0.138067	−	0.0370674i
$656$		0			0
$657$		−	2527.25i		−	0.150072i
$658$		0			0
$659$	1935.78			0.114427			0.0572134	−	0.998362i	$-0.481778\pi$
$659$	1935.78			0.114427			0.0572134	+	0.998362i	$0.481778\pi$
$660$		0			0
$661$	3371.84			0.198411			0.0992053	−	0.995067i	$-0.468370\pi$
$661$	3371.84			0.198411			0.0992053	+	0.995067i	$0.468370\pi$
$662$		0			0
$663$			36796.7i			2.15546i
$664$		0			0
$665$	−706.416	−	2631.22i	−0.0411935	−	0.153435i
$666$		0			0
$667$			11620.4i			0.674579i
$668$		0			0
$669$	−21960.5			−1.26912
$670$		0			0
$671$	−12360.9			−0.711161
$672$		0			0
$673$		−	22061.3i		−	1.26360i	−0.775133	−	0.631798i	$-0.782317\pi$
$673$		−	22061.3i		−	1.26360i	0.775133	−	0.631798i	$-0.217683\pi$
$674$		0			0
$675$	2766.04	+	4780.09i	0.157726	+	0.272572i
$676$		0			0
$677$		−	12747.5i		−	0.723674i	−0.932241	−	0.361837i	$-0.882150\pi$
$677$		−	12747.5i		−	0.723674i	0.932241	−	0.361837i	$-0.117850\pi$
$678$		0			0
$679$	8176.31			0.462118
$680$		0			0
$681$	−40775.8			−2.29447
$682$		0			0
$683$		−	24562.0i		−	1.37605i	−0.725689	−	0.688023i	$-0.758479\pi$
$683$		−	24562.0i		−	1.37605i	0.725689	−	0.688023i	$-0.241521\pi$
$684$		0			0
$685$	−1448.08	−	5393.73i	−0.0807713	−	0.300853i
$686$		0			0
$687$		−	23909.0i		−	1.32778i
$688$		0			0
$689$	28988.1			1.60284
$690$		0			0
$691$	−15853.1			−0.872767			−0.436383	−	0.899761i	$-0.643741\pi$
$691$	−15853.1			−0.872767			−0.436383	+	0.899761i	$0.643741\pi$
$692$		0			0
$693$		−	15479.0i		−	0.848481i
$694$		0			0
$695$	28727.8	−	7712.68i	1.56792	−	0.420948i
$696$		0			0
$697$			25682.7i			1.39570i
$698$		0			0
$699$	31814.7			1.72152
$700$		0			0
$701$	−18288.0			−0.985347			−0.492673	−	0.870214i	$-0.663980\pi$
$701$	−18288.0			−0.985347			−0.492673	+	0.870214i	$0.663980\pi$
$702$		0			0
$703$			1277.51i			0.0685380i
$704$		0			0
$705$	−26850.5	+	7208.67i	−1.43439	+	0.385098i
$706$		0			0
$707$		−	9470.88i		−	0.503804i
$708$		0			0
$709$	−11985.3			−0.634864			−0.317432	−	0.948281i	$-0.602821\pi$
$709$	−11985.3			−0.634864			−0.317432	+	0.948281i	$0.602821\pi$
$710$		0			0
$711$	−5964.91			−0.314630
$712$		0			0
$713$		−	10897.4i		−	0.572384i
$714$		0			0
$715$	−7241.70	−	26973.5i	−0.378775	−	1.41084i
$716$		0			0
$717$		−	33228.8i		−	1.73075i
$718$		0			0
$719$	−17036.3			−0.883653			−0.441827	−	0.897101i	$-0.645669\pi$
$719$	−17036.3			−0.883653			−0.441827	+	0.897101i	$0.645669\pi$
$720$		0			0
$721$	9375.59			0.484279
$722$		0			0
$723$			21093.4i			1.08503i
$724$		0			0
$725$	16053.9	−	9289.73i	0.822382	−	0.475878i
$726$		0			0
$727$		−	1219.85i		−	0.0622307i	−0.999516	−	0.0311154i	$-0.990094\pi$
$727$		−	1219.85i		−	0.0622307i	0.999516	−	0.0311154i	$-0.00990592\pi$
$728$		0			0
$729$	9501.16			0.482709
$730$		0			0
$731$	36509.5			1.84727
$732$		0			0
$733$			17750.6i			0.894453i	0.894421	+	0.447226i	$0.147588\pi$
$733$			17750.6i			0.894453i	−0.894421	+	0.447226i	$0.852412\pi$
$734$		0			0
$735$	−3635.76	−	13542.3i	−0.182458	−	0.679611i
$736$		0			0
$737$		−	11382.2i		−	0.568884i
$738$		0			0
$739$	25825.4			1.28553			0.642763	−	0.766065i	$-0.277788\pi$
$739$	25825.4			1.28553			0.642763	+	0.766065i	$0.277788\pi$
$740$		0			0
$741$	−5587.72			−0.277018
$742$		0			0
$743$		−	11549.8i		−	0.570282i	−0.958486	−	0.285141i	$-0.907959\pi$
$743$		−	11549.8i		−	0.570282i	0.958486	−	0.285141i	$-0.0920405\pi$
$744$		0			0
$745$	14414.0	−	3869.81i	0.708845	−	0.190307i
$746$		0			0
$747$			11821.0i			0.578994i
$748$		0			0
$749$	−7091.69			−0.345961
$750$		0			0
$751$	29468.3			1.43184			0.715921	−	0.698182i	$-0.246007\pi$
$751$	29468.3			1.43184			0.715921	+	0.698182i	$0.246007\pi$
$752$		0			0
$753$			51732.3i			2.50362i
$754$		0			0
$755$	2801.83	−	752.220i	0.135058	−	0.0362597i
$756$		0			0
$757$			13820.2i			0.663545i	0.943359	+	0.331772i	$0.107647\pi$
$757$			13820.2i			0.663545i	−0.943359	+	0.331772i	$0.892353\pi$
$758$		0			0
$759$	−31980.3			−1.52939
$760$		0			0
$761$	−1188.09			−0.0565943			−0.0282971	−	0.999600i	$-0.509008\pi$
$761$	−1188.09			−0.0565943			−0.0282971	+	0.999600i	$0.509008\pi$
$762$		0			0
$763$			19346.2i			0.917931i
$764$		0			0
$765$	−7546.01	−	28107.0i	−0.356636	−	1.32838i
$766$		0			0
$767$			3507.42i			0.165118i
$768$		0			0
$769$	−29907.3			−1.40245			−0.701225	−	0.712940i	$-0.747363\pi$
$769$	−29907.3			−1.40245			−0.701225	+	0.712940i	$0.747363\pi$
$770$		0			0
$771$	−15676.7			−0.732275
$772$		0			0
$773$			27528.3i			1.28088i	0.768007	+	0.640442i	$0.221249\pi$
$773$			27528.3i			1.28088i	−0.768007	+	0.640442i	$0.778751\pi$
$774$		0			0
$775$	−15055.0	+	8711.71i	−0.697796	+	0.403785i
$776$		0			0
$777$		−	5830.85i		−	0.269216i
$778$		0			0
$779$	−3900.02			−0.179374
$780$		0			0
$781$	29641.3			1.35807
$782$		0			0
$783$			6555.83i			0.299216i
$784$		0			0
$785$	3083.29	+	11484.5i	0.140188	+	0.522163i
$786$		0			0
$787$		−	10038.2i		−	0.454666i	−0.973817	−	0.227333i	$-0.926999\pi$
$787$		−	10038.2i		−	0.454666i	0.973817	−	0.227333i	$-0.0730006\pi$
$788$		0			0
$789$	36366.8			1.64093
$790$		0			0
$791$	741.812			0.0333449
$792$		0			0
$793$		−	8812.99i		−	0.394651i
$794$		0			0
$795$	−51169.6	+	13737.8i	−2.28277	+	0.612865i
$796$		0			0
$797$		−	41075.3i		−	1.82555i	−0.408466	−	0.912773i	$-0.633936\pi$
$797$		−	41075.3i		−	1.82555i	0.408466	−	0.912773i	$-0.366064\pi$
$798$		0			0
$799$	−45552.9			−2.01695
$800$		0			0
$801$	−11639.8			−0.513449
$802$		0			0
$803$			7263.20i			0.319194i
$804$		0			0
$805$	10736.8	−	2882.57i	0.470091	−	0.126208i
$806$		0			0
$807$			29773.3i			1.29872i
$808$		0			0
$809$	2134.47			0.0927616			0.0463808	−	0.998924i	$-0.485231\pi$
$809$	2134.47			0.0927616			0.0463808	+	0.998924i	$0.485231\pi$
$810$		0			0
$811$	5866.38			0.254003			0.127001	−	0.991903i	$-0.459465\pi$
$811$	5866.38			0.254003			0.127001	+	0.991903i	$0.459465\pi$
$812$		0			0
$813$			30847.2i			1.33070i
$814$		0			0
$815$	−9101.83	−	33902.0i	−0.391194	−	1.45710i
$816$		0			0
$817$			5544.11i			0.237410i
$818$		0			0
$819$	11036.0			0.470856
$820$		0			0
$821$	−28572.6			−1.21460			−0.607301	−	0.794472i	$-0.707748\pi$
$821$	−28572.6			−1.21460			−0.607301	+	0.794472i	$0.707748\pi$
$822$		0			0
$823$			15900.5i			0.673460i	0.941601	+	0.336730i	$0.109321\pi$
$823$			15900.5i			0.673460i	−0.941601	+	0.336730i	$0.890679\pi$
$824$		0			0
$825$	25566.0	+	44181.6i	1.07890	+	1.86449i
$826$		0			0
$827$			34717.6i			1.45979i	0.683558	+	0.729896i	$0.260431\pi$
$827$			34717.6i			1.45979i	−0.683558	+	0.729896i	$0.739569\pi$
$828$		0			0
$829$	812.559			0.0340426			0.0170213	−	0.999855i	$-0.494582\pi$
$829$	812.559			0.0340426			0.0170213	+	0.999855i	$0.494582\pi$
$830$		0			0
$831$	55965.1			2.33623
$832$		0			0
$833$		−	22975.0i		−	0.955626i
$834$		0			0
$835$	−2714.64	−	10111.3i	−0.112508	−	0.419062i
$836$		0			0
$837$		−	6147.92i		−	0.253887i
$838$		0			0
$839$	852.588			0.0350829			0.0175415	−	0.999846i	$-0.494416\pi$
$839$	852.588			0.0350829			0.0175415	+	0.999846i	$0.494416\pi$
$840$		0			0
$841$	−2371.29			−0.0972277
$842$		0			0
$843$		−	26295.2i		−	1.07432i
$844$		0			0
$845$	−4491.82	+	1205.94i	−0.182868	+	0.0490954i
$846$		0			0
$847$			27586.3i			1.11910i
$848$		0			0
$849$	16299.5			0.658891
$850$		0			0
$851$	−5212.95			−0.209985
$852$		0			0
$853$		−	8851.54i		−	0.355300i	−0.984094	−	0.177650i	$-0.943150\pi$
$853$		−	8851.54i		−	0.355300i	0.984094	−	0.177650i	$-0.0568495\pi$
$854$		0			0
$855$	4268.15	−	1145.89i	0.170722	−	0.0458346i
$856$		0			0
$857$		−	1712.28i		−	0.0682502i	−0.999418	−	0.0341251i	$-0.989136\pi$
$857$		−	1712.28i		−	0.0682502i	0.999418	−	0.0341251i	$-0.0108645\pi$
$858$		0			0
$859$	−33276.7			−1.32175			−0.660877	−	0.750495i	$-0.729815\pi$
$859$	−33276.7			−1.32175			−0.660877	+	0.750495i	$0.729815\pi$
$860$		0			0
$861$	17800.6			0.704578
$862$		0			0
$863$		−	5913.49i		−	0.233253i	−0.993176	−	0.116627i	$-0.962792\pi$
$863$		−	5913.49i		−	0.233253i	0.993176	−	0.116627i	$-0.0372081\pi$
$864$		0			0
$865$	−4173.89	−	15546.7i	−0.164065	−	0.611101i
$866$		0			0
$867$		−	76301.6i		−	2.98885i
$868$		0			0
$869$	17142.9			0.669199
$870$		0			0
$871$	8115.15			0.315696
$872$		0			0
$873$			13262.9i			0.514183i
$874$		0			0
$875$	−12565.7	−	12528.8i	−0.485484	−	0.484058i
$876$		0			0
$877$			38661.1i			1.48859i	0.667851	+	0.744295i	$0.267214\pi$
$877$			38661.1i			1.48859i	−0.667851	+	0.744295i	$0.732786\pi$
$878$		0			0
$879$	43724.3			1.67780
$880$		0			0
$881$	20876.4			0.798348			0.399174	−	0.916875i	$-0.369297\pi$
$881$	20876.4			0.798348			0.399174	+	0.916875i	$0.369297\pi$
$882$		0			0
$883$		−	14713.8i		−	0.560770i	−0.959888	−	0.280385i	$-0.909538\pi$
$883$		−	14713.8i		−	0.560770i	0.959888	−	0.280385i	$-0.0904622\pi$
$884$		0			0
$885$	−1662.20	−	6191.29i	−0.0631349	−	0.235161i
$886$		0			0
$887$		−	38213.1i		−	1.44653i	−0.690571	−	0.723264i	$-0.742641\pi$
$887$		−	38213.1i		−	1.44653i	0.690571	−	0.723264i	$-0.257359\pi$
$888$		0			0
$889$	11951.4			0.450886
$890$		0			0
$891$	−50958.1			−1.91601
$892$		0			0
$893$		−	6917.38i		−	0.259218i
$894$		0			0
$895$	26934.7	−	7231.29i	1.00595	−	0.270073i
$896$		0			0
$897$		−	22801.0i		−	0.848721i
$898$		0			0
$899$	−20647.7			−0.766007
$900$		0			0
$901$	−86811.4			−3.20989
$902$		0			0
$903$		−	25304.6i		−	0.932540i
$904$		0			0
$905$	17172.6	−	4610.42i	0.630760	−	0.169343i
$906$		0			0
$907$			30912.4i			1.13168i	0.824516	+	0.565838i	$0.191447\pi$
$907$			30912.4i			1.13168i	−0.824516	+	0.565838i	$0.808553\pi$
$908$		0			0
$909$	15362.9			0.560566
$910$		0			0
$911$	−1390.58			−0.0505730			−0.0252865	−	0.999680i	$-0.508050\pi$
$911$	−1390.58			−0.0505730			−0.0252865	+	0.999680i	$0.508050\pi$
$912$		0			0
$913$		−	33973.2i		−	1.23149i
$914$		0			0
$915$	4176.57	+	15556.7i	0.150900	+	0.562063i
$916$		0			0
$917$		−	2721.50i		−	0.0980063i
$918$		0			0
$919$	14064.7			0.504843			0.252422	−	0.967617i	$-0.418773\pi$
$919$	14064.7			0.504843			0.252422	+	0.967617i	$0.418773\pi$
$920$		0			0
$921$	−28771.8			−1.02938
$922$		0			0
$923$			21133.4i			0.753645i
$924$		0			0
$925$	4167.40	+	7201.82i	0.148133	+	0.255994i
$926$		0			0
$927$			15208.3i			0.538841i
$928$		0			0
$929$	−51692.7			−1.82560			−0.912799	−	0.408408i	$-0.866084\pi$
$929$	−51692.7			−1.82560			−0.912799	+	0.408408i	$0.866084\pi$
$930$		0			0
$931$	3488.84			0.122816
$932$		0			0
$933$		−	35072.4i		−	1.23067i
$934$		0			0
$935$	21686.9	+	80778.3i	0.758544	+	2.82538i
$936$		0			0
$937$		−	12479.3i		−	0.435092i	−0.976050	−	0.217546i	$-0.930195\pi$
$937$		−	12479.3i		−	0.435092i	0.976050	−	0.217546i	$-0.0698052\pi$
$938$		0			0
$939$	10506.2			0.365129
$940$		0			0
$941$	41800.9			1.44811			0.724054	−	0.689743i	$-0.242277\pi$
$941$	41800.9			1.44811			0.724054	+	0.689743i	$0.242277\pi$
$942$		0			0
$943$		−	15914.2i		−	0.549563i
$944$		0			0
$945$	6057.35	−	1626.24i	0.208514	−	0.0559807i
$946$		0			0
$947$		−	6482.77i		−	0.222452i	−0.993795	−	0.111226i	$-0.964522\pi$
$947$		−	6482.77i		−	0.222452i	0.993795	−	0.111226i	$-0.0354777\pi$
$948$		0			0
$949$	−5178.45			−0.177133
$950$		0			0
$951$	34467.9			1.17529
$952$		0			0
$953$		−	28635.8i		−	0.973351i	−0.873583	−	0.486675i	$-0.838209\pi$
$953$		−	28635.8i		−	0.973351i	0.873583	−	0.486675i	$-0.161791\pi$
$954$		0			0
$955$	28644.5	−	7690.32i	0.970590	−	0.260579i
$956$		0			0
$957$			60594.4i			2.04675i
$958$		0			0
$959$	−6342.30			−0.213560
$960$		0			0
$961$	−10428.0			−0.350038
$962$		0			0
$963$		−	11503.6i		−	0.384940i
$964$		0			0
$965$	8701.60	+	32411.2i	0.290274	+	1.08120i
$966$		0			0
$967$		−	3086.04i		−	0.102627i	−0.998683	−	0.0513135i	$-0.983659\pi$
$967$		−	3086.04i		−	0.102627i	0.998683	−	0.0513135i	$-0.0163408\pi$
$968$		0			0
$969$	16733.7			0.554762
$970$		0			0
$971$	11625.0			0.384205			0.192102	−	0.981375i	$-0.438469\pi$
$971$	11625.0			0.384205			0.192102	+	0.981375i	$0.438469\pi$
$972$		0			0
$973$		−	33780.0i		−	1.11299i
$974$		0			0
$975$	−31500.2	+	18227.8i	−1.03468	+	0.598726i
$976$		0			0
$977$			38232.0i			1.25194i	0.779845	+	0.625972i	$0.215298\pi$
$977$			38232.0i			1.25194i	−0.779845	+	0.625972i	$0.784702\pi$
$978$		0			0
$979$	33452.3			1.09207
$980$		0			0
$981$	−31381.9			−1.02135
$982$		0			0
$983$			54290.3i			1.76154i	0.473546	+	0.880769i	$0.342974\pi$
$983$			54290.3i			1.76154i	−0.473546	+	0.880769i	$0.657026\pi$
$984$		0			0
$985$	−6308.35	−	23497.0i	−0.204062	−	0.760078i
$986$		0			0
$987$			31572.5i			1.01820i
$988$		0			0
$989$	−22623.0			−0.727371
$990$		0			0
$991$	13673.7			0.438303			0.219152	−	0.975691i	$-0.429671\pi$
$991$	13673.7			0.438303			0.219152	+	0.975691i	$0.429671\pi$
$992$		0			0
$993$			19538.2i			0.624397i
$994$		0			0
$995$	−46110.5	+	12379.5i	−1.46915	+	0.394429i
$996$		0			0
$997$		−	3392.56i		−	0.107767i	−0.998547	−	0.0538834i	$-0.982840\pi$
$997$		−	3392.56i		−	0.107767i	0.998547	−	0.0538834i	$-0.0171599\pi$
$998$		0			0
$999$	−2940.96			−0.0931411

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	40.4.c.a.9.4	yes	4
3.2	odd	2		360.4.f.e.289.3		4
4.3	odd	2		80.4.c.c.49.1		4
5.2	odd	4		200.4.a.l.1.2		2
5.3	odd	4		200.4.a.k.1.1		2
5.4	even	2	inner	40.4.c.a.9.1	✓	4
8.3	odd	2		320.4.c.h.129.4		4
8.5	even	2		320.4.c.g.129.1		4
12.11	even	2		720.4.f.m.289.3		4
15.2	even	4		1800.4.a.bp.1.1		2
15.8	even	4		1800.4.a.bk.1.2		2
15.14	odd	2		360.4.f.e.289.4		4
20.3	even	4		400.4.a.x.1.2		2
20.7	even	4		400.4.a.v.1.1		2
20.19	odd	2		80.4.c.c.49.4		4
40.3	even	4		1600.4.a.cf.1.1		2
40.13	odd	4		1600.4.a.cl.1.2		2
40.19	odd	2		320.4.c.h.129.1		4
40.27	even	4		1600.4.a.cm.1.2		2
40.29	even	2		320.4.c.g.129.4		4
40.37	odd	4		1600.4.a.ce.1.1		2
60.59	even	2		720.4.f.m.289.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.c.a.9.1	✓	4	5.4	even	2	inner
40.4.c.a.9.4	yes	4	1.1	even	1	trivial
80.4.c.c.49.1		4	4.3	odd	2
80.4.c.c.49.4		4	20.19	odd	2
200.4.a.k.1.1		2	5.3	odd	4
200.4.a.l.1.2		2	5.2	odd	4
320.4.c.g.129.1		4	8.5	even	2
320.4.c.g.129.4		4	40.29	even	2
320.4.c.h.129.1		4	40.19	odd	2
320.4.c.h.129.4		4	8.3	odd	2
360.4.f.e.289.3		4	3.2	odd	2
360.4.f.e.289.4		4	15.14	odd	2
400.4.a.v.1.1		2	20.7	even	4
400.4.a.x.1.2		2	20.3	even	4
720.4.f.m.289.3		4	12.11	even	2
720.4.f.m.289.4		4	60.59	even	2
1600.4.a.ce.1.1		2	40.37	odd	4
1600.4.a.cf.1.1		2	40.3	even	4
1600.4.a.cl.1.2		2	40.13	odd	4
1600.4.a.cm.1.2		2	40.27	even	4
1800.4.a.bk.1.2		2	15.8	even	4
1800.4.a.bp.1.1		2	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 40 = 2^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	40.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+6.89898i q^{3} +(-10.7980 + 2.89898i) q^{5} +12.6969i q^{7} -20.5959 q^{9} +59.1918 q^{11} +42.2020i q^{13} +(-20.0000 - 74.4949i) q^{15} -126.384i q^{17} +19.1918 q^{19} -87.5959 q^{21} +78.3133i q^{23} +(108.192 - 62.6061i) q^{25} +44.1816i q^{27} +148.384 q^{29} -139.151 q^{31} +408.363i q^{33} +(-36.8082 - 137.101i) q^{35} +66.5653i q^{37} -291.151 q^{39} -203.212 q^{41} +288.879i q^{43} +(222.394 - 59.7071i) q^{45} -360.434i q^{47} +181.788 q^{49} +871.918 q^{51} -686.888i q^{53} +(-639.151 + 171.596i) q^{55} +132.404i q^{57} +83.1102 q^{59} -208.829 q^{61} -261.505i q^{63} +(-122.343 - 455.696i) q^{65} -192.293i q^{67} -540.282 q^{69} +500.767 q^{71} +122.706i q^{73} +(431.918 + 746.413i) q^{75} +751.555i q^{77} +289.616 q^{79} -860.898 q^{81} -573.950i q^{83} +(366.384 + 1364.69i) q^{85} +1023.70i q^{87} +565.151 q^{89} -535.837 q^{91} -960.000i q^{93} +(-207.233 + 55.6367i) q^{95} -643.959i q^{97} -1219.11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 4 q^{9} + 80 q^{11} - 80 q^{15} - 80 q^{19} - 272 q^{21} + 276 q^{25} + 280 q^{29} + 384 q^{31} - 304 q^{35} - 224 q^{39} - 1048 q^{41} + 772 q^{45} + 492 q^{49} + 1920 q^{51} - 1616 q^{55}+ \cdots - 3152 q^{99}+O(q^{100}) \) 4 * q - 4 * q^5 - 4 * q^9 + 80 * q^11 - 80 * q^15 - 80 * q^19 - 272 * q^21 + 276 * q^25 + 280 * q^29 + 384 * q^31 - 304 * q^35 - 224 * q^39 - 1048 * q^41 + 772 * q^45 + 492 * q^49 + 1920 * q^51 - 1616 * q^55 - 1392 * q^59 - 1384 * q^61 + 608 * q^65 + 112 * q^69 + 1376 * q^71 + 160 * q^75 + 1472 * q^79 - 1484 * q^81 + 1152 * q^85 + 1320 * q^89 + 992 * q^91 - 1456 * q^95 - 3152 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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