LMFDB - Embedded newform 1764.4.k.a.1549.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(361,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("1764.361");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.k (of order $3$, degree $2$, 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:	$104.079369250$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 196)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1549.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1764.1549
Dual form			1764.4.k.a.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-10.0000 + 17.3205i) q^{5} +O(q^{10})$	$q+(-10.0000 + 17.3205i) q^{5} +(22.0000 + 38.1051i) q^{11} -44.0000 q^{13} +(36.0000 + 62.3538i) q^{17} +(-50.0000 + 86.6025i) q^{19} +(-60.0000 + 103.923i) q^{23} +(-137.500 - 238.157i) q^{25} -218.000 q^{29} +(140.000 + 242.487i) q^{31} +(15.0000 - 25.9808i) q^{37} -120.000 q^{41} +220.000 q^{43} +(44.0000 - 76.2102i) q^{47} +(55.0000 + 95.2628i) q^{53} -880.000 q^{55} +(290.000 + 502.295i) q^{59} +(-190.000 + 329.090i) q^{61} +(440.000 - 762.102i) q^{65} +(490.000 + 848.705i) q^{67} +112.000 q^{71} +(320.000 + 554.256i) q^{73} +(244.000 - 422.620i) q^{79} -660.000 q^{83} -1440.00 q^{85} +(160.000 - 277.128i) q^{89} +(-1000.00 - 1732.05i) q^{95} +248.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 20 q^{5}+O(q^{10}) $ 2 * q - 20 * q^5	$ 2 q - 20 q^{5} + 44 q^{11} - 88 q^{13} + 72 q^{17} - 100 q^{19} - 120 q^{23} - 275 q^{25} - 436 q^{29} + 280 q^{31} + 30 q^{37} - 240 q^{41} + 440 q^{43} + 88 q^{47} + 110 q^{53} - 1760 q^{55} + 580 q^{59} - 380 q^{61} + 880 q^{65} + 980 q^{67} + 224 q^{71} + 640 q^{73} + 488 q^{79} - 1320 q^{83} - 2880 q^{85} + 320 q^{89} - 2000 q^{95} + 496 q^{97}+O(q^{100}) $ 2 * q - 20 * q^5 + 44 * q^11 - 88 * q^13 + 72 * q^17 - 100 * q^19 - 120 * q^23 - 275 * q^25 - 436 * q^29 + 280 * q^31 + 30 * q^37 - 240 * q^41 + 440 * q^43 + 88 * q^47 + 110 * q^53 - 1760 * q^55 + 580 * q^59 - 380 * q^61 + 880 * q^65 + 980 * q^67 + 224 * q^71 + 640 * q^73 + 488 * q^79 - 1320 * q^83 - 2880 * q^85 + 320 * q^89 - 2000 * q^95 + 496 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−10.0000	+	17.3205i	−0.894427	+	1.54919i	−0.0599153	+	0.998203i	$0.519083\pi$
$5$	−10.0000	+	17.3205i	−0.894427	+	1.54919i	−0.834512	+	0.550990i	$0.814250\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	22.0000	+	38.1051i	0.603023	+	1.04447i	0.992361	+	0.123371i	$0.0393705\pi$
$11$	22.0000	+	38.1051i	0.603023	+	1.04447i	−0.389338	+	0.921095i	$0.627296\pi$
$12$		0			0
$13$	−44.0000			−0.938723			−0.469362	−	0.883006i	$-0.655516\pi$
$13$	−44.0000			−0.938723			−0.469362	+	0.883006i	$0.655516\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	36.0000	+	62.3538i	0.513605	+	0.889590i	0.999875	+	0.0157814i	$0.00502359\pi$
$17$	36.0000	+	62.3538i	0.513605	+	0.889590i	−0.486271	+	0.873808i	$0.661643\pi$
$18$		0			0
$19$	−50.0000	+	86.6025i	−0.603726	+	1.04568i	0.388526	+	0.921438i	$0.372984\pi$
$19$	−50.0000	+	86.6025i	−0.603726	+	1.04568i	−0.992251	+	0.124246i	$0.960349\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−60.0000	+	103.923i	−0.543951	+	0.942150i	0.454721	+	0.890634i	$0.349739\pi$
$23$	−60.0000	+	103.923i	−0.543951	+	0.942150i	−0.998672	+	0.0515165i	$0.983595\pi$
$24$		0			0
$25$	−137.500	−	238.157i	−1.10000	−	1.90526i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−218.000			−1.39592			−0.697958	−	0.716138i	$-0.745908\pi$
$29$	−218.000			−1.39592			−0.697958	+	0.716138i	$0.745908\pi$
$30$		0			0
$31$	140.000	+	242.487i	0.811121	+	1.40490i	0.912080	+	0.410012i	$0.134475\pi$
$31$	140.000	+	242.487i	0.811121	+	1.40490i	−0.100960	+	0.994891i	$0.532191\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	15.0000	−	25.9808i	0.0666482	−	0.115438i	−0.830776	−	0.556607i	$-0.812103\pi$
$37$	15.0000	−	25.9808i	0.0666482	−	0.115438i	0.897424	+	0.441169i	$0.145436\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−120.000			−0.457094			−0.228547	−	0.973533i	$-0.573397\pi$
$41$	−120.000			−0.457094			−0.228547	+	0.973533i	$0.573397\pi$
$42$		0			0
$43$	220.000			0.780225			0.390113	−	0.920767i	$-0.372436\pi$
$43$	220.000			0.780225			0.390113	+	0.920767i	$0.372436\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	44.0000	−	76.2102i	0.136554	−	0.236519i	−0.789636	−	0.613576i	$-0.789730\pi$
$47$	44.0000	−	76.2102i	0.136554	−	0.236519i	0.926190	+	0.377057i	$0.123064\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	55.0000	+	95.2628i	0.142544	+	0.246893i	0.928454	−	0.371448i	$-0.121138\pi$
$53$	55.0000	+	95.2628i	0.142544	+	0.246893i	−0.785910	+	0.618341i	$0.787805\pi$
$54$		0			0
$55$	−880.000			−2.15744
$56$		0			0
$57$		0			0
$58$		0			0
$59$	290.000	+	502.295i	0.639912	+	1.10836i	0.985452	+	0.169955i	$0.0543624\pi$
$59$	290.000	+	502.295i	0.639912	+	1.10836i	−0.345540	+	0.938404i	$0.612304\pi$
$60$		0			0
$61$	−190.000	+	329.090i	−0.398803	+	0.690748i	−0.993579	−	0.113144i	$-0.963908\pi$
$61$	−190.000	+	329.090i	−0.398803	+	0.690748i	0.594775	+	0.803892i	$0.297241\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	440.000	−	762.102i	0.839620	−	1.45426i
$66$		0			0
$67$	490.000	+	848.705i	0.893478	+	1.54755i	0.835677	+	0.549221i	$0.185076\pi$
$67$	490.000	+	848.705i	0.893478	+	1.54755i	0.0578010	+	0.998328i	$0.481591\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	112.000			0.187211			0.0936053	−	0.995609i	$-0.470161\pi$
$71$	112.000			0.187211			0.0936053	+	0.995609i	$0.470161\pi$
$72$		0			0
$73$	320.000	+	554.256i	0.513057	+	0.888641i	0.999885	+	0.0151432i	$0.00482042\pi$
$73$	320.000	+	554.256i	0.513057	+	0.888641i	−0.486828	+	0.873498i	$0.661846\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	244.000	−	422.620i	0.347496	−	0.601880i	−0.638308	−	0.769781i	$-0.720366\pi$
$79$	244.000	−	422.620i	0.347496	−	0.601880i	0.985804	+	0.167901i	$0.0536989\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−660.000			−0.872824			−0.436412	−	0.899747i	$-0.643751\pi$
$83$	−660.000			−0.872824			−0.436412	+	0.899747i	$0.643751\pi$
$84$		0			0
$85$	−1440.00			−1.83753
$86$		0			0
$87$		0			0
$88$		0			0
$89$	160.000	−	277.128i	0.190561	−	0.330062i	−0.754875	−	0.655869i	$-0.772303\pi$
$89$	160.000	−	277.128i	0.190561	−	0.330062i	0.945436	+	0.325807i	$0.105636\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1000.00	−	1732.05i	−1.07998	−	1.87058i
$96$		0			0
$97$	248.000			0.259594			0.129797	−	0.991541i	$-0.458567\pi$
$97$	248.000			0.259594			0.129797	+	0.991541i	$0.458567\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	110.000	+	190.526i	0.108370	+	0.187703i	0.915110	−	0.403204i	$-0.132103\pi$
$101$	110.000	+	190.526i	0.108370	+	0.187703i	−0.806740	+	0.590907i	$0.798770\pi$
$102$		0			0
$103$	−668.000	+	1157.01i	−0.639029	+	1.10683i	0.346617	+	0.938007i	$0.387330\pi$
$103$	−668.000	+	1157.01i	−0.639029	+	1.10683i	−0.985646	+	0.168824i	$0.946003\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	50.0000	−	86.6025i	0.0451746	−	0.0782447i	−0.842554	−	0.538612i	$-0.818949\pi$
$107$	50.0000	−	86.6025i	0.0451746	−	0.0782447i	0.887729	+	0.460367i	$0.152282\pi$
$108$		0			0
$109$	−341.000	−	590.629i	−0.299650	−	0.519009i	0.676406	−	0.736529i	$-0.263537\pi$
$109$	−341.000	−	590.629i	−0.299650	−	0.519009i	−0.976056	+	0.217520i	$0.930203\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−370.000			−0.308024			−0.154012	−	0.988069i	$-0.549219\pi$
$113$	−370.000			−0.308024			−0.154012	+	0.988069i	$0.549219\pi$
$114$		0			0
$115$	−1200.00	−	2078.46i	−0.973048	−	1.68537i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−302.500	+	523.945i	−0.227273	+	0.393648i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3000.00			2.14663
$126$		0			0
$127$	−160.000			−0.111793			−0.0558965	−	0.998437i	$-0.517802\pi$
$127$	−160.000			−0.111793			−0.0558965	+	0.998437i	$0.517802\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−150.000	+	259.808i	−0.100042	+	0.173279i	−0.911702	−	0.410852i	$-0.865231\pi$
$131$	−150.000	+	259.808i	−0.100042	+	0.173279i	0.811659	+	0.584131i	$0.198565\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	595.000	+	1030.57i	0.371053	+	0.642683i	0.989728	−	0.142964i	$-0.0456634\pi$
$137$	595.000	+	1030.57i	0.371053	+	0.642683i	−0.618675	+	0.785647i	$0.712330\pi$
$138$		0			0
$139$	2220.00			1.35466			0.677331	−	0.735679i	$-0.263137\pi$
$139$	2220.00			1.35466			0.677331	+	0.735679i	$0.263137\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−968.000	−	1676.63i	−0.566072	−	0.980465i
$144$		0			0
$145$	2180.00	−	3775.87i	1.24855	−	2.16254i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1207.00	−	2090.59i	0.663633	−	1.14945i	−0.316021	−	0.948752i	$-0.602347\pi$
$149$	1207.00	−	2090.59i	0.663633	−	1.14945i	0.979654	−	0.200694i	$-0.0643196\pi$
$150$		0			0
$151$	−1028.00	−	1780.55i	−0.554023	−	0.959596i	−0.997979	−	0.0635472i	$-0.979759\pi$
$151$	−1028.00	−	1780.55i	−0.554023	−	0.959596i	0.443956	−	0.896049i	$-0.353575\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5600.00			−2.90195
$156$		0			0
$157$	−530.000	−	917.987i	−0.269418	−	0.466645i	0.699294	−	0.714834i	$-0.253498\pi$
$157$	−530.000	−	917.987i	−0.269418	−	0.466645i	−0.968712	+	0.248189i	$0.920165\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	990.000	−	1714.73i	0.475723	−	0.823976i	−0.523891	−	0.851786i	$-0.675520\pi$
$163$	990.000	−	1714.73i	0.475723	−	0.823976i	0.999613	+	0.0278098i	$0.00885328\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−488.000			−0.226123			−0.113062	−	0.993588i	$-0.536066\pi$
$167$	−488.000			−0.226123			−0.113062	+	0.993588i	$0.536066\pi$
$168$		0			0
$169$	−261.000			−0.118798
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1738.00	−	3010.30i	0.763802	−	1.32294i	−0.177076	−	0.984197i	$-0.556664\pi$
$173$	1738.00	−	3010.30i	0.763802	−	1.32294i	0.940878	−	0.338746i	$-0.110003\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−42.0000	−	72.7461i	−0.0175376	−	0.0303760i	0.857123	−	0.515111i	$-0.172249\pi$
$179$	−42.0000	−	72.7461i	−0.0175376	−	0.0303760i	−0.874661	+	0.484735i	$0.838916\pi$
$180$		0			0
$181$	2180.00			0.895238			0.447619	−	0.894224i	$-0.352272\pi$
$181$	2180.00			0.895238			0.447619	+	0.894224i	$0.352272\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	300.000	+	519.615i	0.119224	+	0.206502i
$186$		0			0
$187$	−1584.00	+	2743.57i	−0.619431	+	1.07289i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	348.000	−	602.754i	0.131835	−	0.228344i	−0.792549	−	0.609808i	$-0.791247\pi$
$191$	348.000	−	602.754i	0.131835	−	0.228344i	0.924384	+	0.381464i	$0.124580\pi$
$192$		0			0
$193$	−1345.00	−	2329.61i	−0.501633	−	0.868854i	−0.999998	−	0.00188695i	$-0.999399\pi$
$193$	−1345.00	−	2329.61i	−0.501633	−	0.868854i	0.498365	−	0.866967i	$-0.333934\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−5310.00			−1.92042			−0.960208	−	0.279287i	$-0.909902\pi$
$197$	−5310.00			−1.92042			−0.960208	+	0.279287i	$0.909902\pi$
$198$		0			0
$199$	−1540.00	−	2667.36i	−0.548581	−	0.950171i	−0.998372	−	0.0570369i	$-0.981835\pi$
$199$	−1540.00	−	2667.36i	−0.548581	−	0.950171i	0.449791	−	0.893134i	$-0.351499\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	1200.00	−	2078.46i	0.408837	−	0.708127i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4400.00			−1.45624
$210$		0			0
$211$	−4.00000			−0.00130508			−0.000652539	−	1.00000i	$-0.500208\pi$
$211$	−4.00000			−0.00130508			−0.000652539		1.00000i	$0.500208\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2200.00	+	3810.51i	−0.697855	+	1.20872i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1584.00	−	2743.57i	−0.482133	−	0.835079i
$222$		0			0
$223$	−2416.00			−0.725504			−0.362752	−	0.931886i	$-0.618163\pi$
$223$	−2416.00			−0.725504			−0.362752	+	0.931886i	$0.618163\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2486.00	−	4305.88i	−0.726879	−	1.25899i	−0.958196	−	0.286113i	$-0.907637\pi$
$227$	−2486.00	−	4305.88i	−0.726879	−	1.25899i	0.231316	−	0.972879i	$-0.425697\pi$
$228$		0			0
$229$	2230.00	−	3862.47i	0.643505	−	1.11458i	−0.341140	−	0.940012i	$-0.610813\pi$
$229$	2230.00	−	3862.47i	0.643505	−	1.11458i	0.984645	−	0.174570i	$-0.0558536\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2275.00	−	3940.42i	0.639658	−	1.10792i	−0.345850	−	0.938290i	$-0.612410\pi$
$233$	2275.00	−	3940.42i	0.639658	−	1.10792i	0.985508	−	0.169630i	$-0.0542572\pi$
$234$		0			0
$235$	880.000	+	1524.20i	0.244276	+	0.423099i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2112.00			0.571606			0.285803	−	0.958288i	$-0.407740\pi$
$239$	2112.00			0.571606			0.285803	+	0.958288i	$0.407740\pi$
$240$		0			0
$241$	2420.00	+	4191.56i	0.646829	+	1.12034i	0.983876	+	0.178853i	$0.0572388\pi$
$241$	2420.00	+	4191.56i	0.646829	+	1.12034i	−0.337046	+	0.941488i	$0.609428\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	2200.00	−	3810.51i	0.566731	−	0.981608i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4340.00			1.09139			0.545694	−	0.837985i	$-0.316266\pi$
$251$	4340.00			1.09139			0.545694	+	0.837985i	$0.316266\pi$
$252$		0			0
$253$	−5280.00			−1.31206
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1760.00	+	3048.41i	−0.427182	+	0.739901i	−0.996621	−	0.0821323i	$-0.973827\pi$
$257$	−1760.00	+	3048.41i	−0.427182	+	0.739901i	0.569439	+	0.822033i	$0.307160\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1280.00	−	2217.03i	−0.300107	−	0.519801i	0.676053	−	0.736853i	$-0.263689\pi$
$263$	−1280.00	−	2217.03i	−0.300107	−	0.519801i	−0.976160	+	0.217052i	$0.930356\pi$
$264$		0			0
$265$	−2200.00			−0.509981
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3410.00	+	5906.29i	0.772905	+	1.33871i	0.935965	+	0.352094i	$0.114530\pi$
$269$	3410.00	+	5906.29i	0.772905	+	1.33871i	−0.163060	+	0.986616i	$0.552136\pi$
$270$		0			0
$271$	−1760.00	+	3048.41i	−0.394511	+	0.683312i	−0.993039	−	0.117789i	$-0.962419\pi$
$271$	−1760.00	+	3048.41i	−0.394511	+	0.683312i	0.598528	+	0.801102i	$0.295753\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	6050.00	−	10478.9i	1.32665	−	2.29783i
$276$		0			0
$277$	4345.00	+	7525.76i	0.942476	+	1.63242i	0.760728	+	0.649070i	$0.224842\pi$
$277$	4345.00	+	7525.76i	0.942476	+	1.63242i	0.181747	+	0.983345i	$0.441825\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3894.00			−0.826678			−0.413339	−	0.910577i	$-0.635638\pi$
$281$	−3894.00			−0.826678			−0.413339	+	0.910577i	$0.635638\pi$
$282$		0			0
$283$	3278.00	+	5677.66i	0.688540	+	1.19259i	0.972310	+	0.233694i	$0.0750815\pi$
$283$	3278.00	+	5677.66i	0.688540	+	1.19259i	−0.283770	+	0.958892i	$0.591585\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−135.500	+	234.693i	−0.0275799	+	0.0477698i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2484.00			0.495279			0.247640	−	0.968852i	$-0.420345\pi$
$293$	2484.00			0.495279			0.247640	+	0.968852i	$0.420345\pi$
$294$		0			0
$295$	−11600.0			−2.28942
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2640.00	−	4572.61i	0.510619	−	0.884418i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3800.00	−	6581.79i	−0.713401	−	1.23565i
$306$		0			0
$307$	308.000			0.0572589			0.0286295	−	0.999590i	$-0.490886\pi$
$307$	308.000			0.0572589			0.0286295	+	0.999590i	$0.490886\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	880.000	+	1524.20i	0.160451	+	0.277909i	0.935030	−	0.354567i	$-0.115372\pi$
$311$	880.000	+	1524.20i	0.160451	+	0.277909i	−0.774580	+	0.632476i	$0.782039\pi$
$312$		0			0
$313$	−12.0000	+	20.7846i	−0.00216703	+	0.00375340i	−0.867107	−	0.498122i	$-0.834023\pi$
$313$	−12.0000	+	20.7846i	−0.00216703	+	0.00375340i	0.864940	+	0.501876i	$0.167356\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2445.00	+	4234.86i	−0.433202	+	0.750327i	−0.997147	−	0.0754850i	$-0.975949\pi$
$317$	−2445.00	+	4234.86i	−0.433202	+	0.750327i	0.563945	+	0.825812i	$0.309283\pi$
$318$		0			0
$319$	−4796.00	−	8306.92i	−0.841769	−	1.45799i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−7200.00			−1.24031
$324$		0			0
$325$	6050.00	+	10478.9i	1.03260	+	1.78851i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−4054.00	+	7021.73i	−0.673196	+	1.16601i	0.303796	+	0.952737i	$0.401746\pi$
$331$	−4054.00	+	7021.73i	−0.673196	+	1.16601i	−0.976993	+	0.213273i	$0.931588\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−19600.0			−3.19660
$336$		0			0
$337$	2990.00			0.483311			0.241655	−	0.970362i	$-0.422310\pi$
$337$	2990.00			0.483311			0.241655	+	0.970362i	$0.422310\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6160.00	+	10669.4i	−0.978248	+	1.69438i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3930.00	−	6806.96i	−0.607993	−	1.05307i	−0.991571	−	0.129566i	$-0.958642\pi$
$347$	−3930.00	−	6806.96i	−0.607993	−	1.05307i	0.383578	−	0.923508i	$-0.374692\pi$
$348$		0			0
$349$	−6060.00			−0.929468			−0.464734	−	0.885450i	$-0.653850\pi$
$349$	−6060.00			−0.929468			−0.464734	+	0.885450i	$0.653850\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1512.00	−	2618.86i	−0.227976	−	0.394867i	0.729232	−	0.684267i	$-0.239878\pi$
$353$	−1512.00	−	2618.86i	−0.227976	−	0.394867i	−0.957208	+	0.289400i	$0.906544\pi$
$354$		0			0
$355$	−1120.00	+	1939.90i	−0.167446	+	0.290025i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2948.00	−	5106.09i	0.433397	−	0.750665i	−0.563766	−	0.825934i	$-0.690648\pi$
$359$	2948.00	−	5106.09i	0.433397	−	0.750665i	0.997163	+	0.0752688i	$0.0239815\pi$
$360$		0			0
$361$	−1570.50	−	2720.19i	−0.228969	−	0.396586i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−12800.0			−1.83557
$366$		0			0
$367$	−1096.00	−	1898.33i	−0.155888	−	0.270005i	0.777494	−	0.628890i	$-0.216490\pi$
$367$	−1096.00	−	1898.33i	−0.155888	−	0.270005i	−0.933382	+	0.358885i	$0.883157\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1705.00	−	2953.15i	0.236680	−	0.409941i	−0.723080	−	0.690765i	$-0.757274\pi$
$373$	1705.00	−	2953.15i	0.236680	−	0.409941i	0.959760	+	0.280823i	$0.0906074\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	9592.00			1.31038
$378$		0			0
$379$	5916.00			0.801806			0.400903	−	0.916120i	$-0.368696\pi$
$379$	5916.00			0.801806			0.400903	+	0.916120i	$0.368696\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1020.00	−	1766.69i	0.136082	−	0.235702i	−0.789928	−	0.613200i	$-0.789882\pi$
$383$	1020.00	−	1766.69i	0.136082	−	0.235702i	0.926010	+	0.377498i	$0.123215\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5889.00	+	10200.0i	0.767569	+	1.32947i	0.938878	+	0.344250i	$0.111867\pi$
$389$	5889.00	+	10200.0i	0.767569	+	1.32947i	−0.171309	+	0.985217i	$0.554800\pi$
$390$		0			0
$391$	−8640.00			−1.11750
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4880.00	+	8452.41i	0.621619	+	1.07668i
$396$		0			0
$397$	−2354.00	+	4077.25i	−0.297592	+	0.515444i	−0.975584	−	0.219625i	$-0.929517\pi$
$397$	−2354.00	+	4077.25i	−0.297592	+	0.515444i	0.677993	+	0.735069i	$0.262850\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2937.00	+	5087.03i	−0.365753	+	0.633502i	−0.988897	−	0.148605i	$-0.952522\pi$
$401$	−2937.00	+	5087.03i	−0.365753	+	0.633502i	0.623144	+	0.782107i	$0.285855\pi$
$402$		0			0
$403$	−6160.00	−	10669.4i	−0.761418	−	1.31881i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1320.00			0.160762
$408$		0			0
$409$	−7020.00	−	12159.0i	−0.848696	−	1.46998i	−0.882372	−	0.470552i	$-0.844055\pi$
$409$	−7020.00	−	12159.0i	−0.848696	−	1.46998i	0.0336764	−	0.999433i	$-0.489278\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	6600.00	−	11431.5i	0.780678	−	1.35217i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7940.00			0.925762			0.462881	−	0.886420i	$-0.346816\pi$
$419$	7940.00			0.925762			0.462881	+	0.886420i	$0.346816\pi$
$420$		0			0
$421$	5214.00			0.603598			0.301799	−	0.953372i	$-0.402413\pi$
$421$	5214.00			0.603598			0.301799	+	0.953372i	$0.402413\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	9900.00	−	17147.3i	1.12993	−	1.95710i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	5192.00	+	8992.81i	0.580255	+	1.00503i	0.995449	+	0.0952980i	$0.0303804\pi$
$431$	5192.00	+	8992.81i	0.580255	+	1.00503i	−0.415194	+	0.909733i	$0.636286\pi$
$432$		0			0
$433$	6520.00			0.723629			0.361814	−	0.932250i	$-0.382157\pi$
$433$	6520.00			0.723629			0.361814	+	0.932250i	$0.382157\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6000.00	−	10392.3i	−0.656794	−	1.13760i
$438$		0			0
$439$	5760.00	−	9976.61i	0.626218	−	1.08464i	−0.362086	−	0.932145i	$-0.617935\pi$
$439$	5760.00	−	9976.61i	0.626218	−	1.08464i	0.988304	−	0.152497i	$-0.0487314\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6690.00	−	11587.4i	0.717498	−	1.24274i	−0.244491	−	0.969652i	$-0.578621\pi$
$443$	6690.00	−	11587.4i	0.717498	−	1.24274i	0.961988	−	0.273091i	$-0.0880460\pi$
$444$		0			0
$445$	3200.00	+	5542.56i	0.340887	+	0.590433i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−4098.00			−0.430727			−0.215364	−	0.976534i	$-0.569094\pi$
$449$	−4098.00			−0.430727			−0.215364	+	0.976534i	$0.569094\pi$
$450$		0			0
$451$	−2640.00	−	4572.61i	−0.275638	−	0.477419i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	8195.00	−	14194.2i	0.838831	−	1.45290i	−0.0520411	−	0.998645i	$-0.516573\pi$
$457$	8195.00	−	14194.2i	0.838831	−	1.45290i	0.890872	−	0.454254i	$-0.150094\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−9540.00			−0.963822			−0.481911	−	0.876220i	$-0.660057\pi$
$461$	−9540.00			−0.963822			−0.481911	+	0.876220i	$0.660057\pi$
$462$		0			0
$463$	−8920.00			−0.895351			−0.447676	−	0.894196i	$-0.647748\pi$
$463$	−8920.00			−0.895351			−0.447676	+	0.894196i	$0.647748\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4214.00	+	7298.86i	−0.417560	+	0.723236i	−0.995693	−	0.0927068i	$-0.970448\pi$
$467$	−4214.00	+	7298.86i	−0.417560	+	0.723236i	0.578133	+	0.815942i	$0.303781\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4840.00	+	8383.13i	0.470494	+	0.814919i
$474$		0			0
$475$	27500.0			2.65639
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−7260.00	−	12574.7i	−0.692522	−	1.19948i	−0.971009	−	0.239043i	$-0.923166\pi$
$479$	−7260.00	−	12574.7i	−0.692522	−	1.19948i	0.278487	−	0.960440i	$-0.410167\pi$
$480$		0			0
$481$	−660.000	+	1143.15i	−0.0625643	+	0.108364i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2480.00	+	4295.49i	−0.232188	+	0.402161i
$486$		0			0
$487$	5260.00	+	9110.59i	0.489432	+	0.847721i	0.999926	−	0.0121603i	$-0.00387083\pi$
$487$	5260.00	+	9110.59i	0.489432	+	0.847721i	−0.510494	+	0.859881i	$0.670537\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	5436.00			0.499640			0.249820	−	0.968292i	$-0.419629\pi$
$491$	5436.00			0.499640			0.249820	+	0.968292i	$0.419629\pi$
$492$		0			0
$493$	−7848.00	−	13593.1i	−0.716950	−	1.24179i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−9174.00	+	15889.8i	−0.823015	+	1.42550i	0.0804114	+	0.996762i	$0.474377\pi$
$499$	−9174.00	+	15889.8i	−0.823015	+	1.42550i	−0.903427	+	0.428743i	$0.858957\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	20944.0			1.85655			0.928277	−	0.371889i	$-0.121290\pi$
$503$	20944.0			1.85655			0.928277	+	0.371889i	$0.121290\pi$
$504$		0			0
$505$	−4400.00			−0.387718
$506$		0			0
$507$		0			0
$508$		0			0
$509$	130.000	−	225.167i	0.0113205	−	0.0196077i	−0.860310	−	0.509772i	$-0.829730\pi$
$509$	130.000	−	225.167i	0.0113205	−	0.0196077i	0.871630	+	0.490164i	$0.163063\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−13360.0	−	23140.2i	−1.14313	−	1.97996i
$516$		0			0
$517$	3872.00			0.329382
$518$		0			0
$519$		0			0
$520$		0			0
$521$	7300.00	+	12644.0i	0.613856	+	1.06323i	0.990584	+	0.136905i	$0.0437157\pi$
$521$	7300.00	+	12644.0i	0.613856	+	1.06323i	−0.376728	+	0.926324i	$0.622951\pi$
$522$		0			0
$523$	−3410.00	+	5906.29i	−0.285103	+	0.493813i	−0.972634	−	0.232342i	$-0.925361\pi$
$523$	−3410.00	+	5906.29i	−0.285103	+	0.493813i	0.687531	+	0.726155i	$0.258694\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10080.0	+	17459.1i	−0.833191	+	1.44313i
$528$		0			0
$529$	−1116.50	−	1933.83i	−0.0917646	−	0.158941i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	5280.00			0.429085
$534$		0			0
$535$	1000.00	+	1732.05i	0.0808108	+	0.139968i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−6083.00	+	10536.1i	−0.483417	+	0.837303i	−0.999819	−	0.0190437i	$-0.993938\pi$
$541$	−6083.00	+	10536.1i	−0.483417	+	0.837303i	0.516402	+	0.856347i	$0.327271\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	13640.0			1.07206
$546$		0			0
$547$	22660.0			1.77125			0.885623	−	0.464405i	$-0.153732\pi$
$547$	22660.0			1.77125			0.885623	+	0.464405i	$0.153732\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	10900.0	−	18879.4i	0.842751	−	1.45969i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4245.00	−	7352.56i	−0.322920	−	0.559314i	0.658169	−	0.752870i	$-0.271331\pi$
$557$	−4245.00	−	7352.56i	−0.322920	−	0.559314i	−0.981089	+	0.193556i	$0.937998\pi$
$558$		0			0
$559$	−9680.00			−0.732416
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7150.00	+	12384.2i	0.535234	+	0.927052i	0.999152	+	0.0411740i	$0.0131098\pi$
$563$	7150.00	+	12384.2i	0.535234	+	0.927052i	−0.463918	+	0.885878i	$0.653557\pi$
$564$		0			0
$565$	3700.00	−	6408.59i	0.275505	−	0.477188i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	9147.00	−	15843.1i	0.673923	−	1.16727i	−0.302859	−	0.953035i	$-0.597941\pi$
$569$	9147.00	−	15843.1i	0.673923	−	1.16727i	0.976782	−	0.214234i	$-0.0687254\pi$
$570$		0			0
$571$	7194.00	+	12460.4i	0.527250	+	0.913223i	0.999496	+	0.0317563i	$0.0101101\pi$
$571$	7194.00	+	12460.4i	0.527250	+	0.913223i	−0.472246	+	0.881467i	$0.656557\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	33000.0			2.39338
$576$		0			0
$577$	−12616.0	−	21851.6i	−0.910244	−	1.57659i	−0.813719	−	0.581259i	$-0.802560\pi$
$577$	−12616.0	−	21851.6i	−0.910244	−	1.57659i	−0.0965256	−	0.995331i	$-0.530773\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−2420.00	+	4191.56i	−0.171915	+	0.297765i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−16060.0			−1.12925			−0.564623	−	0.825349i	$-0.690978\pi$
$587$	−16060.0			−1.12925			−0.564623	+	0.825349i	$0.690978\pi$
$588$		0			0
$589$	−28000.0			−1.95878
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−11928.0	+	20659.9i	−0.826011	+	1.43069i	0.0751340	+	0.997173i	$0.476062\pi$
$593$	−11928.0	+	20659.9i	−0.826011	+	1.43069i	−0.901145	+	0.433519i	$0.857272\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1584.00	+	2743.57i	0.108048	+	0.187144i	0.914979	−	0.403501i	$-0.132207\pi$
$599$	1584.00	+	2743.57i	0.108048	+	0.187144i	−0.806932	+	0.590645i	$0.798873\pi$
$600$		0			0
$601$	−12320.0			−0.836179			−0.418089	−	0.908406i	$-0.637300\pi$
$601$	−12320.0			−0.836179			−0.418089	+	0.908406i	$0.637300\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6050.00	−	10478.9i	−0.406558	−	0.704179i
$606$		0			0
$607$	4464.00	−	7731.87i	0.298498	−	0.517013i	−0.677295	−	0.735712i	$-0.736848\pi$
$607$	4464.00	−	7731.87i	0.298498	−	0.517013i	0.975792	+	0.218698i	$0.0701811\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1936.00	+	3353.25i	−0.128187	+	0.222026i
$612$		0			0
$613$	2255.00	+	3905.77i	0.148578	+	0.257345i	0.930702	−	0.365778i	$-0.119197\pi$
$613$	2255.00	+	3905.77i	0.148578	+	0.257345i	−0.782124	+	0.623123i	$0.785864\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−5830.00			−0.380400			−0.190200	−	0.981745i	$-0.560914\pi$
$617$	−5830.00			−0.380400			−0.190200	+	0.981745i	$0.560914\pi$
$618$		0			0
$619$	12230.0	+	21183.0i	0.794128	+	1.37547i	0.923391	+	0.383860i	$0.125405\pi$
$619$	12230.0	+	21183.0i	0.794128	+	1.37547i	−0.129263	+	0.991610i	$0.541261\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12812.5	+	22191.9i	−0.820000	+	1.42028i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2160.00			0.136923
$630$		0			0
$631$	15216.0			0.959967			0.479984	−	0.877277i	$-0.340643\pi$
$631$	15216.0			0.959967			0.479984	+	0.877277i	$0.340643\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1600.00	−	2771.28i	0.0999907	−	0.173189i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6239.00	+	10806.3i	0.384439	+	0.665869i	0.991691	−	0.128641i	$-0.0410614\pi$
$641$	6239.00	+	10806.3i	0.384439	+	0.665869i	−0.607252	+	0.794509i	$0.707728\pi$
$642$		0			0
$643$	15996.0			0.981059			0.490529	−	0.871425i	$-0.336803\pi$
$643$	15996.0			0.981059			0.490529	+	0.871425i	$0.336803\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−2940.00	−	5092.23i	−0.178645	−	0.309422i	0.762772	−	0.646668i	$-0.223838\pi$
$647$	−2940.00	−	5092.23i	−0.178645	−	0.309422i	−0.941417	+	0.337246i	$0.890505\pi$
$648$		0			0
$649$	−12760.0	+	22101.0i	−0.771762	+	1.33673i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3755.00	+	6503.85i	−0.225030	+	0.389763i	−0.956328	−	0.292294i	$-0.905581\pi$
$653$	−3755.00	+	6503.85i	−0.225030	+	0.389763i	0.731299	+	0.682058i	$0.238915\pi$
$654$		0			0
$655$	−3000.00	−	5196.15i	−0.178961	−	0.309970i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	16508.0			0.975812			0.487906	−	0.872896i	$-0.337761\pi$
$659$	16508.0			0.975812			0.487906	+	0.872896i	$0.337761\pi$
$660$		0			0
$661$	610.000	+	1056.55i	0.0358945	+	0.0621711i	0.883415	−	0.468592i	$-0.155239\pi$
$661$	610.000	+	1056.55i	0.0358945	+	0.0621711i	−0.847520	+	0.530763i	$0.821905\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	13080.0	−	22655.2i	0.759310	−	1.31516i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−16720.0			−0.961950
$672$		0			0
$673$	−13090.0			−0.749751			−0.374875	−	0.927075i	$-0.622315\pi$
$673$	−13090.0			−0.749751			−0.374875	+	0.927075i	$0.622315\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7554.00	−	13083.9i	0.428839	−	0.742770i	−0.567932	−	0.823076i	$-0.692256\pi$
$677$	7554.00	−	13083.9i	0.428839	−	0.742770i	0.996770	+	0.0803053i	$0.0255895\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	770.000	+	1333.68i	0.0431380	+	0.0747171i	0.886788	−	0.462176i	$-0.152931\pi$
$683$	770.000	+	1333.68i	0.0431380	+	0.0747171i	−0.843650	+	0.536893i	$0.819598\pi$
$684$		0			0
$685$	−23800.0			−1.32752
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−2420.00	−	4191.56i	−0.133809	−	0.231765i
$690$		0			0
$691$	−2890.00	+	5005.63i	−0.159104	+	0.275576i	−0.934546	−	0.355843i	$-0.884194\pi$
$691$	−2890.00	+	5005.63i	−0.159104	+	0.275576i	0.775442	+	0.631419i	$0.217527\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−22200.0	+	38451.5i	−1.21165	+	2.09863i
$696$		0			0
$697$	−4320.00	−	7482.46i	−0.234766	−	0.406626i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	10406.0			0.560669			0.280335	−	0.959902i	$-0.409555\pi$
$701$	10406.0			0.560669			0.280335	+	0.959902i	$0.409555\pi$
$702$		0			0
$703$	1500.00	+	2598.08i	0.0804745	+	0.139386i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−14321.0	+	24804.7i	−0.758585	+	1.31391i	0.184988	+	0.982741i	$0.440775\pi$
$709$	−14321.0	+	24804.7i	−0.758585	+	1.31391i	−0.943572	+	0.331166i	$0.892558\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−33600.0			−1.76484
$714$		0			0
$715$	38720.0			2.02524
$716$		0			0
$717$		0			0
$718$		0			0
$719$	3780.00	−	6547.15i	0.196064	−	0.339593i	−0.751185	−	0.660092i	$-0.770517\pi$
$719$	3780.00	−	6547.15i	0.196064	−	0.339593i	0.947249	+	0.320499i	$0.103851\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	29975.0	+	51918.2i	1.53551	+	2.65958i
$726$		0			0
$727$	−20360.0			−1.03867			−0.519333	−	0.854572i	$-0.673820\pi$
$727$	−20360.0			−1.03867			−0.519333	+	0.854572i	$0.673820\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	7920.00	+	13717.8i	0.400727	+	0.694080i
$732$		0			0
$733$	−11638.0	+	20157.6i	−0.586438	+	1.01574i	0.408256	+	0.912867i	$0.366137\pi$
$733$	−11638.0	+	20157.6i	−0.586438	+	1.01574i	−0.994694	+	0.102874i	$0.967196\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−21560.0	+	37343.0i	−1.07758	+	1.86641i
$738$		0			0
$739$	−1586.00	−	2747.03i	−0.0789472	−	0.136740i	0.823849	−	0.566810i	$-0.191823\pi$
$739$	−1586.00	−	2747.03i	−0.0789472	−	0.136740i	−0.902796	+	0.430069i	$0.858489\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	6600.00			0.325882			0.162941	−	0.986636i	$-0.447902\pi$
$743$	6600.00			0.325882			0.162941	+	0.986636i	$0.447902\pi$
$744$		0			0
$745$	24140.0	+	41811.7i	1.18714	+	2.05619i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	8208.00	−	14216.7i	0.398820	−	0.690777i	−0.594760	−	0.803903i	$-0.702753\pi$
$751$	8208.00	−	14216.7i	0.398820	−	0.690777i	0.993581	+	0.113126i	$0.0360863\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	41120.0			1.98213
$756$		0			0
$757$	36850.0			1.76927			0.884634	−	0.466286i	$-0.154408\pi$
$757$	36850.0			1.76927			0.884634	+	0.466286i	$0.154408\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−18700.0	+	32389.4i	−0.890768	+	1.54286i	−0.0518116	+	0.998657i	$0.516500\pi$
$761$	−18700.0	+	32389.4i	−0.890768	+	1.54286i	−0.838956	+	0.544199i	$0.816834\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−12760.0	−	22101.0i	−0.600700	−	1.04044i
$768$		0			0
$769$	−5720.00			−0.268229			−0.134115	−	0.990966i	$-0.542819\pi$
$769$	−5720.00			−0.268229			−0.134115	+	0.990966i	$0.542819\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	3678.00	+	6370.48i	0.171136	+	0.296417i	0.938817	−	0.344415i	$-0.111923\pi$
$773$	3678.00	+	6370.48i	0.171136	+	0.296417i	−0.767681	+	0.640832i	$0.778589\pi$
$774$		0			0
$775$	38500.0	−	66684.0i	1.78447	−	3.09079i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	6000.00	−	10392.3i	0.275959	−	0.477976i
$780$		0			0
$781$	2464.00	+	4267.77i	0.112892	+	0.195535i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	21200.0			0.963899
$786$		0			0
$787$	19286.0	+	33404.3i	0.873535	+	1.51301i	0.858316	+	0.513122i	$0.171511\pi$
$787$	19286.0	+	33404.3i	0.873535	+	1.51301i	0.0152188	+	0.999884i	$0.495156\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	8360.00	−	14479.9i	0.374366	−	0.648421i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−7812.00			−0.347196			−0.173598	−	0.984817i	$-0.555539\pi$
$797$	−7812.00			−0.347196			−0.173598	+	0.984817i	$0.555539\pi$
$798$		0			0
$799$	6336.00			0.280540
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−14080.0	+	24387.3i	−0.618770	+	1.07174i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	22509.0	+	38986.7i	0.978213	+	1.69431i	0.668897	+	0.743355i	$0.266767\pi$
$809$	22509.0	+	38986.7i	0.978213	+	1.69431i	0.309316	+	0.950959i	$0.399900\pi$
$810$		0			0
$811$	2740.00			0.118637			0.0593184	−	0.998239i	$-0.481107\pi$
$811$	2740.00			0.118637			0.0593184	+	0.998239i	$0.481107\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	19800.0	+	34294.6i	0.850998	+	1.47397i
$816$		0			0
$817$	−11000.0	+	19052.6i	−0.471042	+	0.815869i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	727.000	−	1259.20i	0.0309044	−	0.0535279i	−0.850160	−	0.526525i	$-0.823495\pi$
$821$	727.000	−	1259.20i	0.0309044	−	0.0535279i	0.881064	+	0.472997i	$0.156828\pi$
$822$		0			0
$823$	−18560.0	−	32146.9i	−0.786101	−	1.36157i	−0.928339	−	0.371734i	$-0.878763\pi$
$823$	−18560.0	−	32146.9i	−0.786101	−	1.36157i	0.142239	−	0.989832i	$-0.454570\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	7260.00			0.305266			0.152633	−	0.988283i	$-0.451225\pi$
$827$	7260.00			0.305266			0.152633	+	0.988283i	$0.451225\pi$
$828$		0			0
$829$	5570.00	+	9647.52i	0.233358	+	0.404189i	0.958794	−	0.284101i	$-0.0916951\pi$
$829$	5570.00	+	9647.52i	0.233358	+	0.404189i	−0.725436	+	0.688290i	$0.758362\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	4880.00	−	8452.41i	0.202251	−	0.350309i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−8760.00			−0.360463			−0.180232	−	0.983624i	$-0.557685\pi$
$839$	−8760.00			−0.360463			−0.180232	+	0.983624i	$0.557685\pi$
$840$		0			0
$841$	23135.0			0.948583
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2610.00	−	4520.65i	0.106256	−	0.184042i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1800.00	+	3117.69i	0.0725067	+	0.125585i
$852$		0			0
$853$	820.000			0.0329147			0.0164574	−	0.999865i	$-0.494761\pi$
$853$	820.000			0.0329147			0.0164574	+	0.999865i	$0.494761\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−660.000	−	1143.15i	−0.0263071	−	0.0455652i	0.852572	−	0.522610i	$-0.175041\pi$
$857$	−660.000	−	1143.15i	−0.0263071	−	0.0455652i	−0.878879	+	0.477044i	$0.841708\pi$
$858$		0			0
$859$	−21390.0	+	37048.6i	−0.849613	+	1.47157i	0.0319414	+	0.999490i	$0.489831\pi$
$859$	−21390.0	+	37048.6i	−0.849613	+	1.47157i	−0.881554	+	0.472083i	$0.843502\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	14940.0	−	25876.8i	0.589297	−	1.02069i	−0.405027	−	0.914305i	$-0.632738\pi$
$863$	14940.0	−	25876.8i	0.589297	−	1.02069i	0.994325	−	0.106388i	$-0.0339286\pi$
$864$		0			0
$865$	34760.0	+	60206.1i	1.36633	+	2.36655i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	21472.0			0.838191
$870$		0			0
$871$	−21560.0	−	37343.0i	−0.838729	−	1.45272i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−14085.0	+	24395.9i	−0.542322	+	0.939330i	0.456448	+	0.889750i	$0.349121\pi$
$877$	−14085.0	+	24395.9i	−0.542322	+	0.939330i	−0.998770	+	0.0495796i	$0.984212\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−20800.0			−0.795425			−0.397713	−	0.917510i	$-0.630196\pi$
$881$	−20800.0			−0.795425			−0.397713	+	0.917510i	$0.630196\pi$
$882$		0			0
$883$	−20900.0			−0.796536			−0.398268	−	0.917269i	$-0.630389\pi$
$883$	−20900.0			−0.796536			−0.398268	+	0.917269i	$0.630389\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	5820.00	−	10080.5i	0.220312	−	0.381591i	−0.734591	−	0.678510i	$-0.762626\pi$
$887$	5820.00	−	10080.5i	0.220312	−	0.381591i	0.954903	+	0.296919i	$0.0959592\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4400.00	+	7621.02i	0.164883	+	0.285585i
$894$		0			0
$895$	1680.00			0.0627444
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−30520.0	−	52862.2i	−1.13226	−	1.96113i
$900$		0			0
$901$	−3960.00	+	6858.92i	−0.146423	+	0.253611i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−21800.0	+	37758.7i	−0.800725	+	1.38690i
$906$		0			0
$907$	18790.0	+	32545.2i	0.687885	+	1.19145i	0.972521	+	0.232816i	$0.0747940\pi$
$907$	18790.0	+	32545.2i	0.687885	+	1.19145i	−0.284636	+	0.958636i	$0.591873\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	40832.0			1.48499			0.742494	−	0.669852i	$-0.233643\pi$
$911$	40832.0			1.48499			0.742494	+	0.669852i	$0.233643\pi$
$912$		0			0
$913$	−14520.0	−	25149.4i	−0.526333	−	0.911635i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−21384.0	+	37038.2i	−0.767566	+	1.32946i	0.171313	+	0.985217i	$0.445199\pi$
$919$	−21384.0	+	37038.2i	−0.767566	+	1.32946i	−0.938879	+	0.344247i	$0.888134\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4928.00			−0.175739
$924$		0			0
$925$	−8250.00			−0.293252
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−4380.00	+	7586.38i	−0.154686	+	0.267924i	−0.932945	−	0.360020i	$-0.882770\pi$
$929$	−4380.00	+	7586.38i	−0.154686	+	0.267924i	0.778259	+	0.627944i	$0.216103\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−31680.0	−	54871.4i	−1.10807	−	1.91924i
$936$		0			0
$937$	49632.0			1.73042			0.865212	−	0.501407i	$-0.167184\pi$
$937$	49632.0			1.73042			0.865212	+	0.501407i	$0.167184\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−18450.0	−	31956.3i	−0.639163	−	1.10706i	−0.985617	−	0.168996i	$-0.945947\pi$
$941$	−18450.0	−	31956.3i	−0.639163	−	1.10706i	0.346453	−	0.938067i	$-0.387386\pi$
$942$		0			0
$943$	7200.00	−	12470.8i	0.248637	−	0.430651i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−16670.0	+	28873.3i	−0.572019	+	0.990766i	0.424339	+	0.905503i	$0.360506\pi$
$947$	−16670.0	+	28873.3i	−0.572019	+	0.990766i	−0.996358	+	0.0852631i	$0.972827\pi$
$948$		0			0
$949$	−14080.0	−	24387.3i	−0.481619	−	0.834188i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−5610.00			−0.190688			−0.0953440	−	0.995444i	$-0.530395\pi$
$953$	−5610.00			−0.190688			−0.0953440	+	0.995444i	$0.530395\pi$
$954$		0			0
$955$	6960.00	+	12055.1i	0.235833	+	0.408474i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−24304.5	+	42096.6i	−0.815834	+	1.41307i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	53800.0			1.79470
$966$		0			0
$967$	−25160.0			−0.836702			−0.418351	−	0.908285i	$-0.637392\pi$
$967$	−25160.0			−0.836702			−0.418351	+	0.908285i	$0.637392\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	530.000	−	917.987i	0.0175165	−	0.0303394i	−0.857134	−	0.515093i	$-0.827757\pi$
$971$	530.000	−	917.987i	0.0175165	−	0.0303394i	0.874651	+	0.484754i	$0.161091\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−19385.0	−	33575.8i	−0.634781	−	1.09947i	−0.986561	−	0.163390i	$-0.947757\pi$
$977$	−19385.0	−	33575.8i	−0.634781	−	1.09947i	0.351780	−	0.936083i	$-0.385576\pi$
$978$		0			0
$979$	14080.0			0.459651
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−12148.0	−	21041.0i	−0.394162	−	0.682708i	0.598832	−	0.800875i	$-0.295632\pi$
$983$	−12148.0	−	21041.0i	−0.394162	−	0.682708i	−0.992994	+	0.118166i	$0.962298\pi$
$984$		0			0
$985$	53100.0	−	91971.9i	1.71767	−	2.97509i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−13200.0	+	22863.1i	−0.424404	+	0.735089i
$990$		0			0
$991$	836.000	+	1447.99i	0.0267976	+	0.0464148i	0.879113	−	0.476613i	$-0.158136\pi$
$991$	836.000	+	1447.99i	0.0267976	+	0.0464148i	−0.852316	+	0.523028i	$0.824802\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	61600.0			1.96266
$996$		0			0
$997$	−22814.0	−	39515.0i	−0.724701	−	1.25522i	−0.959097	−	0.283077i	$-0.908645\pi$
$997$	−22814.0	−	39515.0i	−0.724701	−	1.25522i	0.234396	−	0.972141i	$-0.424689\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.k.a.1549.1		2
3.2	odd	2		196.4.e.e.177.1		2
7.2	even	3		1764.4.a.m.1.1		1
7.3	odd	6		1764.4.k.p.361.1		2
7.4	even	3	inner	1764.4.k.a.361.1		2
7.5	odd	6		1764.4.a.a.1.1		1
7.6	odd	2		1764.4.k.p.1549.1		2
21.2	odd	6		196.4.a.a.1.1	✓	1
21.5	even	6		196.4.a.c.1.1	yes	1
21.11	odd	6		196.4.e.e.165.1		2
21.17	even	6		196.4.e.b.165.1		2
21.20	even	2		196.4.e.b.177.1		2
84.23	even	6		784.4.a.m.1.1		1
84.47	odd	6		784.4.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.4.a.a.1.1	✓	1	21.2	odd	6
196.4.a.c.1.1	yes	1	21.5	even	6
196.4.e.b.165.1		2	21.17	even	6
196.4.e.b.177.1		2	21.20	even	2
196.4.e.e.165.1		2	21.11	odd	6
196.4.e.e.177.1		2	3.2	odd	2
784.4.a.f.1.1		1	84.47	odd	6
784.4.a.m.1.1		1	84.23	even	6
1764.4.a.a.1.1		1	7.5	odd	6
1764.4.a.m.1.1		1	7.2	even	3
1764.4.k.a.361.1		2	7.4	even	3	inner
1764.4.k.a.1549.1		2	1.1	even	1	trivial
1764.4.k.p.361.1		2	7.3	odd	6
1764.4.k.p.1549.1		2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.k (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-10.0000 + 17.3205i) q^{5} +O(q^{10})\)	\(q+(-10.0000 + 17.3205i) q^{5} +(22.0000 + 38.1051i) q^{11} -44.0000 q^{13} +(36.0000 + 62.3538i) q^{17} +(-50.0000 + 86.6025i) q^{19} +(-60.0000 + 103.923i) q^{23} +(-137.500 - 238.157i) q^{25} -218.000 q^{29} +(140.000 + 242.487i) q^{31} +(15.0000 - 25.9808i) q^{37} -120.000 q^{41} +220.000 q^{43} +(44.0000 - 76.2102i) q^{47} +(55.0000 + 95.2628i) q^{53} -880.000 q^{55} +(290.000 + 502.295i) q^{59} +(-190.000 + 329.090i) q^{61} +(440.000 - 762.102i) q^{65} +(490.000 + 848.705i) q^{67} +112.000 q^{71} +(320.000 + 554.256i) q^{73} +(244.000 - 422.620i) q^{79} -660.000 q^{83} -1440.00 q^{85} +(160.000 - 277.128i) q^{89} +(-1000.00 - 1732.05i) q^{95} +248.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 20 q^{5}+O(q^{10}) \) 2 * q - 20 * q^5	\( 2 q - 20 q^{5} + 44 q^{11} - 88 q^{13} + 72 q^{17} - 100 q^{19} - 120 q^{23} - 275 q^{25} - 436 q^{29} + 280 q^{31} + 30 q^{37} - 240 q^{41} + 440 q^{43} + 88 q^{47} + 110 q^{53} - 1760 q^{55} + 580 q^{59} - 380 q^{61} + 880 q^{65} + 980 q^{67} + 224 q^{71} + 640 q^{73} + 488 q^{79} - 1320 q^{83} - 2880 q^{85} + 320 q^{89} - 2000 q^{95} + 496 q^{97}+O(q^{100}) \) 2 * q - 20 * q^5 + 44 * q^11 - 88 * q^13 + 72 * q^17 - 100 * q^19 - 120 * q^23 - 275 * q^25 - 436 * q^29 + 280 * q^31 + 30 * q^37 - 240 * q^41 + 440 * q^43 + 88 * q^47 + 110 * q^53 - 1760 * q^55 + 580 * q^59 - 380 * q^61 + 880 * q^65 + 980 * q^67 + 224 * q^71 + 640 * q^73 + 488 * q^79 - 1320 * q^83 - 2880 * q^85 + 320 * q^89 - 2000 * q^95 + 496 * q^97

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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