LMFDB - Embedded newform 1800.3.v.n.1657.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,3,Mod(793,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))

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

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

chi := DirichletCharacter("1800.793");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1800.v (of order $4$, degree $2$, 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:	$49.0464475849$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1657.2
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1800.1657
Dual form			1800.3.v.n.793.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.44949 + 5.44949i) q^{7} +O(q^{10})$	$q+(5.44949 + 5.44949i) q^{7} +6.44949 q^{11} +(-14.4495 + 14.4495i) q^{13} +(-23.1464 - 23.1464i) q^{17} +16.6969i q^{19} +(-6.65153 + 6.65153i) q^{23} +0.0454077i q^{29} +4.49490 q^{31} +(-35.3485 - 35.3485i) q^{37} -20.2929 q^{41} +(-32.2929 + 32.2929i) q^{43} +(-50.5403 - 50.5403i) q^{47} +10.3939i q^{49} +(-5.50510 + 5.50510i) q^{53} -55.4393i q^{59} +47.8888 q^{61} +(85.2827 + 85.2827i) q^{67} -48.4041 q^{71} +(21.9898 - 21.9898i) q^{73} +(35.1464 + 35.1464i) q^{77} -126.697i q^{79} +(94.9444 - 94.9444i) q^{83} +71.7980i q^{89} -157.485 q^{91} +(37.0000 + 37.0000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{7}+O(q^{10}) $ 4 * q + 12 * q^7	$ 4 q + 12 q^{7} + 16 q^{11} - 48 q^{13} - 24 q^{17} - 56 q^{23} - 80 q^{31} - 112 q^{37} + 56 q^{41} + 8 q^{43} - 16 q^{47} - 120 q^{53} - 24 q^{61} + 8 q^{67} - 272 q^{71} - 108 q^{73} + 72 q^{77} + 272 q^{83} - 336 q^{91} + 148 q^{97}+O(q^{100}) $ 4 * q + 12 * q^7 + 16 * q^11 - 48 * q^13 - 24 * q^17 - 56 * q^23 - 80 * q^31 - 112 * q^37 + 56 * q^41 + 8 * q^43 - 16 * q^47 - 120 * q^53 - 24 * q^61 + 8 * q^67 - 272 * q^71 - 108 * q^73 + 72 * q^77 + 272 * q^83 - 336 * q^91 + 148 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	5.44949	+	5.44949i	0.778499	+	0.778499i	0.979575	−	0.201077i	$-0.0644441\pi$
$7$	5.44949	+	5.44949i	0.778499	+	0.778499i	−0.201077	+	0.979575i	$0.564444\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	6.44949			0.586317			0.293159	−	0.956064i	$-0.405294\pi$
$11$	6.44949			0.586317			0.293159	+	0.956064i	$0.405294\pi$
$12$		0			0
$13$	−14.4495	+	14.4495i	−1.11150	+	1.11150i	−0.118551	+	0.992948i	$0.537825\pi$
$13$	−14.4495	+	14.4495i	−1.11150	+	1.11150i	−0.992948	+	0.118551i	$0.962175\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−23.1464	−	23.1464i	−1.36155	−	1.36155i	−0.871938	−	0.489617i	$-0.837137\pi$
$17$	−23.1464	−	23.1464i	−1.36155	−	1.36155i	−0.489617	−	0.871938i	$-0.662863\pi$
$18$		0			0
$19$			16.6969i			0.878786i	0.898295	+	0.439393i	$0.144806\pi$
$19$			16.6969i			0.878786i	−0.898295	+	0.439393i	$0.855194\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.65153	+	6.65153i	−0.289197	+	0.289197i	−0.836763	−	0.547566i	$-0.815555\pi$
$23$	−6.65153	+	6.65153i	−0.289197	+	0.289197i	0.547566	+	0.836763i	$0.315555\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$			0.0454077i			0.00156578i	1.00000		0.000782891i	$0.000249202\pi$
$29$			0.0454077i			0.00156578i	−1.00000		0.000782891i	$0.999751\pi$
$30$		0			0
$31$	4.49490			0.144997			0.0724983	−	0.997369i	$-0.476903\pi$
$31$	4.49490			0.144997			0.0724983	+	0.997369i	$0.476903\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−35.3485	−	35.3485i	−0.955364	−	0.955364i	0.0436815	−	0.999046i	$-0.486091\pi$
$37$	−35.3485	−	35.3485i	−0.955364	−	0.955364i	−0.999046	+	0.0436815i	$0.986091\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−20.2929			−0.494948			−0.247474	−	0.968895i	$-0.579600\pi$
$41$	−20.2929			−0.494948			−0.247474	+	0.968895i	$0.579600\pi$
$42$		0			0
$43$	−32.2929	+	32.2929i	−0.750997	+	0.750997i	−0.974665	−	0.223669i	$-0.928197\pi$
$43$	−32.2929	+	32.2929i	−0.750997	+	0.750997i	0.223669	+	0.974665i	$0.428197\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−50.5403	−	50.5403i	−1.07533	−	1.07533i	−0.996922	−	0.0784040i	$-0.975018\pi$
$47$	−50.5403	−	50.5403i	−1.07533	−	1.07533i	−0.0784040	−	0.996922i	$-0.524982\pi$
$48$		0			0
$49$			10.3939i			0.212120i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.50510	+	5.50510i	−0.103870	+	0.103870i	−0.757132	−	0.653262i	$-0.773400\pi$
$53$	−5.50510	+	5.50510i	−0.103870	+	0.103870i	0.653262	+	0.757132i	$0.273400\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	55.4393i		−	0.939649i	−0.882760	−	0.469824i	$-0.844317\pi$
$59$		−	55.4393i		−	0.939649i	0.882760	−	0.469824i	$-0.155683\pi$
$60$		0			0
$61$	47.8888			0.785062			0.392531	−	0.919739i	$-0.371600\pi$
$61$	47.8888			0.785062			0.392531	+	0.919739i	$0.371600\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	85.2827	+	85.2827i	1.27288	+	1.27288i	0.944573	+	0.328303i	$0.106477\pi$
$67$	85.2827	+	85.2827i	1.27288	+	1.27288i	0.328303	+	0.944573i	$0.393523\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−48.4041			−0.681748			−0.340874	−	0.940109i	$-0.610723\pi$
$71$	−48.4041			−0.681748			−0.340874	+	0.940109i	$0.610723\pi$
$72$		0			0
$73$	21.9898	−	21.9898i	0.301230	−	0.301230i	−0.540265	−	0.841495i	$-0.681676\pi$
$73$	21.9898	−	21.9898i	0.301230	−	0.301230i	0.841495	+	0.540265i	$0.181676\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	35.1464	+	35.1464i	0.456447	+	0.456447i
$78$		0			0
$79$		−	126.697i		−	1.60376i	−0.597486	−	0.801879i	$-0.703834\pi$
$79$		−	126.697i		−	1.60376i	0.597486	−	0.801879i	$-0.296166\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	94.9444	−	94.9444i	1.14391	−	1.14391i	0.156180	−	0.987729i	$-0.450082\pi$
$83$	94.9444	−	94.9444i	1.14391	−	1.14391i	0.987729	−	0.156180i	$-0.0499179\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			71.7980i			0.806719i	0.915042	+	0.403359i	$0.132158\pi$
$89$			71.7980i			0.806719i	−0.915042	+	0.403359i	$0.867842\pi$
$90$		0			0
$91$	−157.485			−1.73060
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	37.0000	+	37.0000i	0.381443	+	0.381443i	0.871622	−	0.490179i	$-0.163068\pi$
$97$	37.0000	+	37.0000i	0.381443	+	0.381443i	−0.490179	+	0.871622i	$0.663068\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−94.3383			−0.934042			−0.467021	−	0.884246i	$-0.654673\pi$
$101$	−94.3383			−0.934042			−0.467021	+	0.884246i	$0.654673\pi$
$102$		0			0
$103$	−72.6413	+	72.6413i	−0.705256	+	0.705256i	−0.965534	−	0.260278i	$-0.916186\pi$
$103$	−72.6413	+	72.6413i	−0.705256	+	0.705256i	0.260278	+	0.965534i	$0.416186\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−13.0556	−	13.0556i	−0.122015	−	0.122015i	0.643463	−	0.765478i	$-0.277497\pi$
$107$	−13.0556	−	13.0556i	−0.122015	−	0.122015i	−0.765478	+	0.643463i	$0.777497\pi$
$108$		0			0
$109$		−	69.2827i		−	0.635621i	−0.948154	−	0.317810i	$-0.897052\pi$
$109$		−	69.2827i		−	0.635621i	0.948154	−	0.317810i	$-0.102948\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−102.136	+	102.136i	−0.903860	+	0.903860i	−0.995768	−	0.0919072i	$-0.970704\pi$
$113$	−102.136	+	102.136i	−0.903860	+	0.903860i	0.0919072	+	0.995768i	$0.470704\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	252.272i		−	2.11994i
$120$		0			0
$121$	−79.4041			−0.656232
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−119.944	−	119.944i	−0.944444	−	0.944444i	0.0540920	−	0.998536i	$-0.482774\pi$
$127$	−119.944	−	119.944i	−0.944444	−	0.944444i	−0.998536	+	0.0540920i	$0.982774\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−121.146			−0.924782			−0.462391	−	0.886676i	$-0.653008\pi$
$131$	−121.146			−0.924782			−0.462391	+	0.886676i	$0.653008\pi$
$132$		0			0
$133$	−90.9898	+	90.9898i	−0.684134	+	0.684134i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	119.530	+	119.530i	0.872482	+	0.872482i	0.992742	−	0.120260i	$-0.0383728\pi$
$137$	119.530	+	119.530i	0.872482	+	0.872482i	−0.120260	+	0.992742i	$0.538373\pi$
$138$		0			0
$139$			140.788i			1.01286i	0.862281	+	0.506431i	$0.169035\pi$
$139$			140.788i			1.01286i	−0.862281	+	0.506431i	$0.830965\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−93.1918	+	93.1918i	−0.651691	+	0.651691i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$			53.5301i			0.359262i	0.983734	+	0.179631i	$0.0574904\pi$
$149$			53.5301i			0.359262i	−0.983734	+	0.179631i	$0.942510\pi$
$150$		0			0
$151$	−232.606			−1.54044			−0.770219	−	0.637780i	$-0.779853\pi$
$151$	−232.606			−1.54044			−0.770219	+	0.637780i	$0.779853\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−128.631	−	128.631i	−0.819306	−	0.819306i	0.166701	−	0.986007i	$-0.446689\pi$
$157$	−128.631	−	128.631i	−0.819306	−	0.819306i	−0.986007	+	0.166701i	$0.946689\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−72.4949			−0.450279
$162$		0			0
$163$	117.576	−	117.576i	0.721322	−	0.721322i	−0.247552	−	0.968875i	$-0.579626\pi$
$163$	117.576	−	117.576i	0.721322	−	0.721322i	0.968875	+	0.247552i	$0.0796262\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−33.6617	−	33.6617i	−0.201567	−	0.201567i	0.599104	−	0.800671i	$-0.295524\pi$
$167$	−33.6617	−	33.6617i	−0.201567	−	0.201567i	−0.800671	+	0.599104i	$0.795524\pi$
$168$		0			0
$169$		−	248.576i		−	1.47086i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	28.2474	−	28.2474i	0.163280	−	0.163280i	−0.620738	−	0.784018i	$-0.713167\pi$
$173$	28.2474	−	28.2474i	0.163280	−	0.163280i	0.784018	+	0.620738i	$0.213167\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$			45.2372i			0.252722i	0.991984	+	0.126361i	$0.0403298\pi$
$179$			45.2372i			0.252722i	−0.991984	+	0.126361i	$0.959670\pi$
$180$		0			0
$181$	−260.656			−1.44009			−0.720045	−	0.693928i	$-0.755879\pi$
$181$	−260.656			−1.44009			−0.720045	+	0.693928i	$0.755879\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−149.283	−	149.283i	−0.798303	−	0.798303i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	51.8684			0.271562			0.135781	−	0.990739i	$-0.456646\pi$
$191$	51.8684			0.271562			0.135781	+	0.990739i	$0.456646\pi$
$192$		0			0
$193$	−16.6163	+	16.6163i	−0.0860950	+	0.0860950i	−0.748843	−	0.662748i	$-0.769390\pi$
$193$	−16.6163	+	16.6163i	−0.0860950	+	0.0860950i	0.662748	+	0.748843i	$0.269390\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	86.0908	+	86.0908i	0.437009	+	0.437009i	0.891004	−	0.453995i	$-0.150002\pi$
$197$	86.0908	+	86.0908i	0.437009	+	0.437009i	−0.453995	+	0.891004i	$0.650002\pi$
$198$		0			0
$199$			28.5653i			0.143544i	0.997421	+	0.0717721i	$0.0228654\pi$
$199$			28.5653i			0.143544i	−0.997421	+	0.0717721i	$0.977135\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.247449	+	0.247449i	−0.00121896	+	0.00121896i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$			107.687i			0.515248i
$210$		0			0
$211$	−197.151			−0.934365			−0.467183	−	0.884161i	$-0.654731\pi$
$211$	−197.151			−0.934365			−0.467183	+	0.884161i	$0.654731\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	24.4949	+	24.4949i	0.112880	+	0.112880i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	668.908			3.02673
$222$		0			0
$223$	−287.338	+	287.338i	−1.28851	+	1.28851i	−0.352822	+	0.935690i	$0.614778\pi$
$223$	−287.338	+	287.338i	−1.28851	+	1.28851i	−0.935690	+	0.352822i	$0.885222\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−234.384	−	234.384i	−1.03253	−	1.03253i	−0.999453	−	0.0330743i	$-0.989470\pi$
$227$	−234.384	−	234.384i	−1.03253	−	1.03253i	−0.0330743	−	0.999453i	$-0.510530\pi$
$228$		0			0
$229$		−	284.969i		−	1.24441i	−0.782855	−	0.622204i	$-0.786237\pi$
$229$		−	284.969i		−	1.24441i	0.782855	−	0.622204i	$-0.213763\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−241.530	+	241.530i	−1.03661	+	1.03661i	−0.0373060	+	0.999304i	$0.511878\pi$
$233$	−241.530	+	241.530i	−1.03661	+	1.03661i	−0.999304	+	0.0373060i	$0.988122\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			327.737i			1.37128i	0.727939	+	0.685642i	$0.240478\pi$
$239$			327.737i			1.37128i	−0.727939	+	0.685642i	$0.759522\pi$
$240$		0			0
$241$	269.131			1.11672			0.558362	−	0.829597i	$-0.311430\pi$
$241$	269.131			1.11672			0.558362	+	0.829597i	$0.311430\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−241.262	−	241.262i	−0.976770	−	0.976770i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−186.136			−0.741579			−0.370789	−	0.928717i	$-0.620913\pi$
$251$	−186.136			−0.741579			−0.370789	+	0.928717i	$0.620913\pi$
$252$		0			0
$253$	−42.8990	+	42.8990i	−0.169561	+	0.169561i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	138.268	+	138.268i	0.538007	+	0.538007i	0.922943	−	0.384936i	$-0.125776\pi$
$257$	138.268	+	138.268i	0.538007	+	0.538007i	−0.384936	+	0.922943i	$0.625776\pi$
$258$		0			0
$259$		−	385.262i		−	1.48750i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−163.146	+	163.146i	−0.620329	+	0.620329i	−0.945615	−	0.325287i	$-0.894539\pi$
$263$	−163.146	+	163.146i	−0.620329	+	0.620329i	0.325287	+	0.945615i	$0.394539\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	108.227i		−	0.402331i	−0.979557	−	0.201165i	$-0.935527\pi$
$269$		−	108.227i		−	0.402331i	0.979557	−	0.201165i	$-0.0644729\pi$
$270$		0			0
$271$	324.384			1.19699			0.598494	−	0.801127i	$-0.295766\pi$
$271$	324.384			1.19699			0.598494	+	0.801127i	$0.295766\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	165.864	+	165.864i	0.598786	+	0.598786i	0.939990	−	0.341203i	$-0.110834\pi$
$277$	165.864	+	165.864i	0.598786	+	0.598786i	−0.341203	+	0.939990i	$0.610834\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−300.434			−1.06916			−0.534579	−	0.845118i	$-0.679530\pi$
$281$	−300.434			−1.06916			−0.534579	+	0.845118i	$0.679530\pi$
$282$		0			0
$283$	352.161	−	352.161i	1.24439	−	1.24439i	0.286223	−	0.958163i	$-0.407600\pi$
$283$	352.161	−	352.161i	1.24439	−	1.24439i	0.958163	−	0.286223i	$-0.0923998\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−110.586	−	110.586i	−0.385316	−	0.385316i
$288$		0			0
$289$			782.514i			2.70766i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−109.414	+	109.414i	−0.373428	+	0.373428i	−0.868724	−	0.495296i	$-0.835059\pi$
$293$	−109.414	+	109.414i	−0.373428	+	0.373428i	0.495296	+	0.868724i	$0.335059\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	192.222i		−	0.642884i
$300$		0			0
$301$	−351.959			−1.16930
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−110.627	−	110.627i	−0.360347	−	0.360347i	0.503594	−	0.863941i	$-0.332011\pi$
$307$	−110.627	−	110.627i	−0.360347	−	0.360347i	−0.863941	+	0.503594i	$0.832011\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.38367			−0.00766454			−0.00383227	−	0.999993i	$-0.501220\pi$
$311$	−2.38367			−0.00766454			−0.00383227	+	0.999993i	$0.501220\pi$
$312$		0			0
$313$	−132.959	+	132.959i	−0.424790	+	0.424790i	−0.886849	−	0.462059i	$-0.847111\pi$
$313$	−132.959	+	132.959i	−0.424790	+	0.424790i	0.462059	+	0.886849i	$0.347111\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−37.1464	−	37.1464i	−0.117181	−	0.117181i	0.646085	−	0.763266i	$-0.276405\pi$
$317$	−37.1464	−	37.1464i	−0.117181	−	0.117181i	−0.763266	+	0.646085i	$0.776405\pi$
$318$		0			0
$319$			0.292856i			0.000918045i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	386.474	−	386.474i	1.19652	−	1.19652i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	550.838i		−	1.67428i
$330$		0			0
$331$	21.6459			0.0653955			0.0326978	−	0.999465i	$-0.489590\pi$
$331$	21.6459			0.0653955			0.0326978	+	0.999465i	$0.489590\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	136.757	+	136.757i	0.405808	+	0.405808i	0.880274	−	0.474466i	$-0.157359\pi$
$337$	136.757	+	136.757i	0.405808	+	0.405808i	−0.474466	+	0.880274i	$0.657359\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	28.9898			0.0850141
$342$		0			0
$343$	210.384	−	210.384i	0.613363	−	0.613363i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	33.8480	+	33.8480i	0.0975445	+	0.0975445i	0.754195	−	0.656650i	$-0.228027\pi$
$347$	33.8480	+	33.8480i	0.0975445	+	0.0975445i	−0.656650	+	0.754195i	$0.728027\pi$
$348$		0			0
$349$		−	241.283i		−	0.691354i	−0.938354	−	0.345677i	$-0.887649\pi$
$349$		−	241.283i		−	0.691354i	0.938354	−	0.345677i	$-0.112351\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	126.833	−	126.833i	0.359301	−	0.359301i	−0.504254	−	0.863555i	$-0.668233\pi$
$353$	126.833	−	126.833i	0.359301	−	0.359301i	0.863555	+	0.504254i	$0.168233\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			22.1112i			0.0615912i	0.999526	+	0.0307956i	$0.00980409\pi$
$359$			22.1112i			0.0615912i	−0.999526	+	0.0307956i	$0.990196\pi$
$360$		0			0
$361$	82.2122			0.227735
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	213.944	+	213.944i	0.582955	+	0.582955i	0.935714	−	0.352759i	$-0.114757\pi$
$367$	213.944	+	213.944i	0.582955	+	0.582955i	−0.352759	+	0.935714i	$0.614757\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−60.0000			−0.161725
$372$		0			0
$373$	−210.025	+	210.025i	−0.563070	+	0.563070i	−0.930178	−	0.367108i	$-0.880348\pi$
$373$	−210.025	+	210.025i	−0.563070	+	0.563070i	0.367108	+	0.930178i	$0.380348\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.656118	−	0.656118i	−0.00174037	−	0.00174037i
$378$		0			0
$379$		−	124.343i		−	0.328081i	−0.986454	−	0.164041i	$-0.947547\pi$
$379$		−	124.343i		−	0.328081i	0.986454	−	0.164041i	$-0.0524528\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	418.540	−	418.540i	1.09279	−	1.09279i	0.0975654	−	0.995229i	$-0.468894\pi$
$383$	418.540	−	418.540i	1.09279	−	1.09279i	0.995229	−	0.0975654i	$-0.0311055\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			369.884i			0.950859i	0.879754	+	0.475430i	$0.157707\pi$
$389$			369.884i			0.950859i	−0.879754	+	0.475430i	$0.842293\pi$
$390$		0			0
$391$	307.918			0.787515
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	373.984	+	373.984i	0.942026	+	0.942026i	0.998409	−	0.0563835i	$-0.0179570\pi$
$397$	373.984	+	373.984i	0.942026	+	0.942026i	−0.0563835	+	0.998409i	$0.517957\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−113.151			−0.282172			−0.141086	−	0.989997i	$-0.545059\pi$
$401$	−113.151			−0.282172			−0.141086	+	0.989997i	$0.545059\pi$
$402$		0			0
$403$	−64.9490	+	64.9490i	−0.161164	+	0.161164i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−227.980	−	227.980i	−0.560146	−	0.560146i
$408$		0			0
$409$			90.3837i			0.220987i	0.993877	+	0.110493i	$0.0352431\pi$
$409$			90.3837i			0.220987i	−0.993877	+	0.110493i	$0.964757\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	302.116	−	302.116i	0.731515	−	0.731515i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	334.772i		−	0.798978i	−0.916738	−	0.399489i	$-0.869188\pi$
$419$		−	334.772i		−	0.798978i	0.916738	−	0.399489i	$-0.130812\pi$
$420$		0			0
$421$	57.2735			0.136042			0.0680208	−	0.997684i	$-0.478332\pi$
$421$	57.2735			0.136042			0.0680208	+	0.997684i	$0.478332\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	260.969	+	260.969i	0.611170	+	0.611170i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−442.656			−1.02704			−0.513522	−	0.858076i	$-0.671660\pi$
$431$	−442.656			−1.02704			−0.513522	+	0.858076i	$0.671660\pi$
$432$		0			0
$433$	14.8684	−	14.8684i	0.0343380	−	0.0343380i	−0.689729	−	0.724067i	$-0.742270\pi$
$433$	14.8684	−	14.8684i	0.0343380	−	0.0343380i	0.724067	+	0.689729i	$0.242270\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−111.060	−	111.060i	−0.254142	−	0.254142i
$438$		0			0
$439$			233.818i			0.532616i	0.963888	+	0.266308i	$0.0858038\pi$
$439$			233.818i			0.532616i	−0.963888	+	0.266308i	$0.914196\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−246.747	+	246.747i	−0.556991	+	0.556991i	−0.928449	−	0.371459i	$-0.878858\pi$
$443$	−246.747	+	246.747i	−0.556991	+	0.556991i	0.371459	+	0.928449i	$0.378858\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			282.758i			0.629751i	0.949133	+	0.314875i	$0.101963\pi$
$449$			282.758i			0.629751i	−0.949133	+	0.314875i	$0.898037\pi$
$450$		0			0
$451$	−130.879			−0.290196
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	298.171	+	298.171i	0.652454	+	0.652454i	0.953583	−	0.301129i	$-0.0973636\pi$
$457$	298.171	+	298.171i	0.652454	+	0.652454i	−0.301129	+	0.953583i	$0.597364\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	264.318			0.573358			0.286679	−	0.958027i	$-0.407449\pi$
$461$	264.318			0.573358			0.286679	+	0.958027i	$0.407449\pi$
$462$		0			0
$463$	−135.116	+	135.116i	−0.291827	+	0.291827i	−0.837802	−	0.545975i	$-0.816159\pi$
$463$	−135.116	+	135.116i	−0.291827	+	0.291827i	0.545975	+	0.837802i	$0.316159\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	558.631	+	558.631i	1.19621	+	1.19621i	0.975292	+	0.220920i	$0.0709061\pi$
$467$	558.631	+	558.631i	1.19621	+	1.19621i	0.220920	+	0.975292i	$0.429094\pi$
$468$		0			0
$469$			929.494i			1.98186i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−208.272	+	208.272i	−0.440322	+	0.440322i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$			456.141i			0.952277i	0.879370	+	0.476139i	$0.157964\pi$
$479$			456.141i			0.952277i	−0.879370	+	0.476139i	$0.842036\pi$
$480$		0			0
$481$	1021.53			2.12377
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−11.9444	−	11.9444i	−0.0245265	−	0.0245265i	0.694737	−	0.719264i	$-0.255521\pi$
$487$	−11.9444	−	11.9444i	−0.0245265	−	0.0245265i	−0.719264	+	0.694737i	$0.755521\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−822.468			−1.67509			−0.837544	−	0.546370i	$-0.816009\pi$
$491$	−822.468			−1.67509			−0.837544	+	0.546370i	$0.816009\pi$
$492$		0			0
$493$	1.05103	−	1.05103i	0.00213190	−	0.00213190i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−263.778	−	263.778i	−0.530740	−	0.530740i
$498$		0			0
$499$			312.474i			0.626201i	0.949720	+	0.313101i	$0.101368\pi$
$499$			312.474i			0.626201i	−0.949720	+	0.313101i	$0.898632\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−371.530	+	371.530i	−0.738628	+	0.738628i	−0.972313	−	0.233684i	$-0.924922\pi$
$503$	−371.530	+	371.530i	−0.738628	+	0.738628i	0.233684	+	0.972313i	$0.424922\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	171.228i		−	0.336401i	−0.985753	−	0.168200i	$-0.946204\pi$
$509$		−	171.228i		−	0.336401i	0.985753	−	0.168200i	$-0.0537956\pi$
$510$		0			0
$511$	239.666			0.469014
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−325.959	−	325.959i	−0.630482	−	0.630482i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	206.313			0.395995			0.197997	−	0.980203i	$-0.436556\pi$
$521$	206.313			0.395995			0.197997	+	0.980203i	$0.436556\pi$
$522$		0			0
$523$	135.526	−	135.526i	0.259131	−	0.259131i	−0.565570	−	0.824701i	$-0.691344\pi$
$523$	135.526	−	135.526i	0.259131	−	0.259131i	0.824701	+	0.565570i	$0.191344\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−104.041	−	104.041i	−0.197421	−	0.197421i
$528$		0			0
$529$			440.514i			0.832730i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	293.221	−	293.221i	0.550134	−	0.550134i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$			67.0352i			0.124370i
$540$		0			0
$541$	303.485			0.560970			0.280485	−	0.959858i	$-0.409505\pi$
$541$	303.485			0.560970			0.280485	+	0.959858i	$0.409505\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	71.3939	+	71.3939i	0.130519	+	0.130519i	0.769348	−	0.638829i	$-0.220581\pi$
$547$	71.3939	+	71.3939i	0.130519	+	0.130519i	−0.638829	+	0.769348i	$0.720581\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−0.758169			−0.00137599
$552$		0			0
$553$	690.434	−	690.434i	1.24852	−	1.24852i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	753.019	+	753.019i	1.35192	+	1.35192i	0.883513	+	0.468407i	$0.155172\pi$
$557$	753.019	+	753.019i	1.35192	+	1.35192i	0.468407	+	0.883513i	$0.344828\pi$
$558$		0			0
$559$		−	933.231i		−	1.66946i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	703.464	−	703.464i	1.24949	−	1.24949i	0.293548	−	0.955944i	$-0.405164\pi$
$563$	703.464	−	703.464i	1.24949	−	1.24949i	0.955944	−	0.293548i	$-0.0948361\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	434.504i		−	0.763628i	−0.924239	−	0.381814i	$-0.875300\pi$
$569$		−	434.504i		−	0.763628i	0.924239	−	0.381814i	$-0.124700\pi$
$570$		0			0
$571$	−131.040			−0.229492			−0.114746	−	0.993395i	$-0.536605\pi$
$571$	−131.040			−0.229492			−0.114746	+	0.993395i	$0.536605\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−290.444	−	290.444i	−0.503369	−	0.503369i	0.409114	−	0.912483i	$-0.365838\pi$
$577$	−290.444	−	290.444i	−0.503369	−	0.503369i	−0.912483	+	0.409114i	$0.865838\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1034.80			1.78106
$582$		0			0
$583$	−35.5051	+	35.5051i	−0.0609007	+	0.0609007i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	180.288	+	180.288i	0.307135	+	0.307135i	0.843797	−	0.536662i	$-0.180315\pi$
$587$	180.288	+	180.288i	0.307135	+	0.307135i	−0.536662	+	0.843797i	$0.680315\pi$
$588$		0			0
$589$			75.0510i			0.127421i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−25.0556	+	25.0556i	−0.0422523	+	0.0422523i	−0.727917	−	0.685665i	$-0.759512\pi$
$593$	−25.0556	+	25.0556i	−0.0422523	+	0.0422523i	0.685665	+	0.727917i	$0.259512\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	509.807i		−	0.851097i	−0.904936	−	0.425549i	$-0.860081\pi$
$599$		−	509.807i		−	0.851097i	0.904936	−	0.425549i	$-0.139919\pi$
$600$		0			0
$601$	179.757			0.299097			0.149548	−	0.988754i	$-0.452218\pi$
$601$	179.757			0.299097			0.149548	+	0.988754i	$0.452218\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−33.1566	−	33.1566i	−0.0546238	−	0.0546238i	0.679267	−	0.733891i	$-0.262298\pi$
$607$	−33.1566	−	33.1566i	−0.0546238	−	0.0546238i	−0.733891	+	0.679267i	$0.762298\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1460.56			2.39045
$612$		0			0
$613$	261.712	−	261.712i	0.426936	−	0.426936i	−0.460647	−	0.887583i	$-0.652383\pi$
$613$	261.712	−	261.712i	0.426936	−	0.426936i	0.887583	+	0.460647i	$0.152383\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−762.075	−	762.075i	−1.23513	−	1.23513i	−0.961968	−	0.273161i	$-0.911931\pi$
$617$	−762.075	−	762.075i	−1.23513	−	1.23513i	−0.273161	−	0.961968i	$-0.588069\pi$
$618$		0			0
$619$		−	81.6367i		−	0.131885i	−0.997823	−	0.0659424i	$-0.978995\pi$
$619$		−	81.6367i		−	0.131885i	0.997823	−	0.0659424i	$-0.0210054\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−391.262	+	391.262i	−0.628029	+	0.628029i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$			1636.38i			2.60156i
$630$		0			0
$631$	−16.4133			−0.0260115			−0.0130057	−	0.999915i	$-0.504140\pi$
$631$	−16.4133			−0.0260115			−0.0130057	+	0.999915i	$0.504140\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−150.186	−	150.186i	−0.235771	−	0.235771i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	546.041			0.851858			0.425929	−	0.904757i	$-0.359947\pi$
$641$	546.041			0.851858			0.425929	+	0.904757i	$0.359947\pi$
$642$		0			0
$643$	142.879	−	142.879i	0.222206	−	0.222206i	−0.587221	−	0.809427i	$-0.699778\pi$
$643$	142.879	−	142.879i	0.222206	−	0.222206i	0.809427	+	0.587221i	$0.199778\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−649.691	−	649.691i	−1.00416	−	1.00416i	−0.999991	−	0.00416838i	$-0.998673\pi$
$647$	−649.691	−	649.691i	−1.00416	−	1.00416i	−0.00416838	−	0.999991i	$-0.501327\pi$
$648$		0			0
$649$		−	357.555i		−	0.550932i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−150.904	+	150.904i	−0.231093	+	0.231093i	−0.813149	−	0.582056i	$-0.802248\pi$
$653$	−150.904	+	150.904i	−0.231093	+	0.231093i	0.582056	+	0.813149i	$0.302248\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$			268.802i			0.407893i	0.978982	+	0.203947i	$0.0653769\pi$
$659$			268.802i			0.407893i	−0.978982	+	0.203947i	$0.934623\pi$
$660$		0			0
$661$	311.162			0.470745			0.235372	−	0.971905i	$-0.424369\pi$
$661$	311.162			0.470745			0.235372	+	0.971905i	$0.424369\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−0.302031	−	0.302031i	−0.000452820	−	0.000452820i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	308.858			0.460295
$672$		0			0
$673$	547.756	−	547.756i	0.813902	−	0.813902i	−0.171314	−	0.985216i	$-0.554801\pi$
$673$	547.756	−	547.756i	0.813902	−	0.813902i	0.985216	+	0.171314i	$0.0548014\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	696.257	+	696.257i	1.02844	+	1.02844i	0.999583	+	0.0288606i	$0.00918789\pi$
$677$	696.257	+	696.257i	1.02844	+	1.02844i	0.0288606	+	0.999583i	$0.490812\pi$
$678$		0			0
$679$			403.262i			0.593906i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−84.0250	+	84.0250i	−0.123023	+	0.123023i	−0.765938	−	0.642915i	$-0.777725\pi$
$683$	−84.0250	+	84.0250i	−0.123023	+	0.123023i	0.642915	+	0.765938i	$0.277725\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	159.092i		−	0.230903i
$690$		0			0
$691$	1157.06			1.67447			0.837236	−	0.546842i	$-0.184170\pi$
$691$	1157.06			1.67447			0.837236	+	0.546842i	$0.184170\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	469.707	+	469.707i	0.673898	+	0.673898i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−843.691			−1.20355			−0.601777	−	0.798664i	$-0.705540\pi$
$701$	−843.691			−1.20355			−0.601777	+	0.798664i	$0.705540\pi$
$702$		0			0
$703$	590.211	−	590.211i	0.839561	−	0.839561i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−514.095	−	514.095i	−0.727150	−	0.727150i
$708$		0			0
$709$		−	170.686i		−	0.240741i	−0.992729	−	0.120371i	$-0.961592\pi$
$709$		−	170.686i		−	0.240741i	0.992729	−	0.120371i	$-0.0384083\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−29.8979	+	29.8979i	−0.0419326	+	0.0419326i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$			605.435i			0.842051i	0.907049	+	0.421026i	$0.138330\pi$
$719$			605.435i			0.842051i	−0.907049	+	0.421026i	$0.861670\pi$
$720$		0			0
$721$	−791.716			−1.09808
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	50.8025	+	50.8025i	0.0698797	+	0.0698797i	0.741183	−	0.671303i	$-0.234265\pi$
$727$	50.8025	+	50.8025i	0.0698797	+	0.0698797i	−0.671303	+	0.741183i	$0.734265\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1494.93			2.04505
$732$		0			0
$733$	−516.529	+	516.529i	−0.704678	+	0.704678i	−0.965411	−	0.260733i	$-0.916036\pi$
$733$	−516.529	+	516.529i	−0.704678	+	0.704678i	0.260733	+	0.965411i	$0.416036\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	550.030	+	550.030i	0.746309	+	0.746309i
$738$		0			0
$739$			650.109i			0.879715i	0.898068	+	0.439857i	$0.144971\pi$
$739$			650.109i			0.879715i	−0.898068	+	0.439857i	$0.855029\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	90.6811	−	90.6811i	0.122047	−	0.122047i	−0.643445	−	0.765492i	$-0.722496\pi$
$743$	90.6811	−	90.6811i	0.122047	−	0.122047i	0.765492	+	0.643445i	$0.222496\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	142.293i		−	0.189977i
$750$		0			0
$751$	−300.050			−0.399534			−0.199767	−	0.979843i	$-0.564019\pi$
$751$	−300.050			−0.399534			−0.199767	+	0.979843i	$0.564019\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−523.176	−	523.176i	−0.691118	−	0.691118i	0.271360	−	0.962478i	$-0.412527\pi$
$757$	−523.176	−	523.176i	−0.691118	−	0.691118i	−0.962478	+	0.271360i	$0.912527\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	724.130			0.951550			0.475775	−	0.879567i	$-0.342168\pi$
$761$	724.130			0.951550			0.475775	+	0.879567i	$0.342168\pi$
$762$		0			0
$763$	377.555	−	377.555i	0.494830	−	0.494830i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	801.069	+	801.069i	1.04442	+	1.04442i
$768$		0			0
$769$		−	13.7775i		−	0.0179162i	−0.999960	−	0.00895809i	$-0.997149\pi$
$769$		−	13.7775i		−	0.0179162i	0.999960	−	0.00895809i	$-0.00285149\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−564.207	+	564.207i	−0.729892	+	0.729892i	−0.970598	−	0.240706i	$-0.922621\pi$
$773$	−564.207	+	564.207i	−0.729892	+	0.729892i	0.240706	+	0.970598i	$0.422621\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	338.829i		−	0.434953i
$780$		0			0
$781$	−312.182			−0.399720
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	212.424	+	212.424i	0.269917	+	0.269917i	0.829067	−	0.559150i	$-0.188872\pi$
$787$	212.424	+	212.424i	0.269917	+	0.269917i	−0.559150	+	0.829067i	$0.688872\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1113.18			−1.40731
$792$		0			0
$793$	−691.968	+	691.968i	−0.872596	+	0.872596i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−15.6617	−	15.6617i	−0.0196509	−	0.0196509i	0.697213	−	0.716864i	$-0.254423\pi$
$797$	−15.6617	−	15.6617i	−0.0196509	−	0.0196509i	−0.716864	+	0.697213i	$0.754423\pi$
$798$		0			0
$799$			2339.66i			2.92823i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	141.823	−	141.823i	0.176616	−	0.176616i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			816.788i			1.00963i	0.863229	+	0.504813i	$0.168439\pi$
$809$			816.788i			1.00963i	−0.863229	+	0.504813i	$0.831561\pi$
$810$		0			0
$811$	830.504			1.02405			0.512025	−	0.858971i	$-0.328896\pi$
$811$	830.504			1.02405			0.512025	+	0.858971i	$0.328896\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−539.192	−	539.192i	−0.659966	−	0.659966i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−569.217			−0.693321			−0.346661	−	0.937991i	$-0.612684\pi$
$821$	−569.217			−0.693321			−0.346661	+	0.937991i	$0.612684\pi$
$822$		0			0
$823$	−497.095	+	497.095i	−0.604004	+	0.604004i	−0.941373	−	0.337369i	$-0.890463\pi$
$823$	−497.095	+	497.095i	−0.604004	+	0.604004i	0.337369	+	0.941373i	$0.390463\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	874.070	+	874.070i	1.05692	+	1.05692i	0.998279	+	0.0586377i	$0.0186757\pi$
$827$	874.070	+	874.070i	1.05692	+	1.05692i	0.0586377	+	0.998279i	$0.481324\pi$
$828$		0			0
$829$		−	1548.77i		−	1.86824i	−0.356954	−	0.934122i	$-0.616185\pi$
$829$		−	1548.77i		−	1.86824i	0.356954	−	0.934122i	$-0.383815\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	240.581	−	240.581i	0.288813	−	0.288813i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	731.523i		−	0.871899i	−0.899971	−	0.435950i	$-0.856413\pi$
$839$		−	731.523i		−	0.871899i	0.899971	−	0.435950i	$-0.143587\pi$
$840$		0			0
$841$	840.998			0.999998
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−432.712	−	432.712i	−0.510876	−	0.510876i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	470.243			0.552577
$852$		0			0
$853$	554.166	−	554.166i	0.649667	−	0.649667i	−0.303246	−	0.952912i	$-0.598070\pi$
$853$	554.166	−	554.166i	0.649667	−	0.649667i	0.952912	+	0.303246i	$0.0980703\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	687.206	+	687.206i	0.801874	+	0.801874i	0.983388	−	0.181515i	$-0.0581000\pi$
$857$	687.206	+	687.206i	0.801874	+	0.801874i	−0.181515	+	0.983388i	$0.558100\pi$
$858$		0			0
$859$		−	1285.29i		−	1.49627i	−0.663549	−	0.748133i	$-0.730951\pi$
$859$		−	1285.29i		−	1.49627i	0.663549	−	0.748133i	$-0.269049\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−712.985	+	712.985i	−0.826171	+	0.826171i	−0.986985	−	0.160814i	$-0.948588\pi$
$863$	−712.985	+	712.985i	−0.826171	+	0.826171i	0.160814	+	0.986985i	$0.448588\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	817.131i		−	0.940311i
$870$		0			0
$871$	−2464.58			−2.82960
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	474.297	+	474.297i	0.540818	+	0.540818i	0.923769	−	0.382951i	$-0.125092\pi$
$877$	474.297	+	474.297i	0.540818	+	0.540818i	−0.382951	+	0.923769i	$0.625092\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1652.71			−1.87594			−0.937972	−	0.346712i	$-0.887298\pi$
$881$	−1652.71			−1.87594			−0.937972	+	0.346712i	$0.887298\pi$
$882$		0			0
$883$	−18.9286	+	18.9286i	−0.0214367	+	0.0214367i	−0.717744	−	0.696307i	$-0.754825\pi$
$883$	−18.9286	+	18.9286i	−0.0214367	+	0.0214367i	0.696307	+	0.717744i	$0.254825\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−382.338	−	382.338i	−0.431047	−	0.431047i	0.457938	−	0.888984i	$-0.348588\pi$
$887$	−382.338	−	382.338i	−0.431047	−	0.431047i	−0.888984	+	0.457938i	$0.848588\pi$
$888$		0			0
$889$		−	1307.27i		−	1.47050i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	843.868	−	843.868i	0.944981	−	0.944981i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$			0.204103i			0.000227033i
$900$		0			0
$901$	254.847			0.282849
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−725.485	−	725.485i	−0.799873	−	0.799873i	0.183202	−	0.983075i	$-0.441354\pi$
$907$	−725.485	−	725.485i	−0.799873	−	0.799873i	−0.983075	+	0.183202i	$0.941354\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	154.474			0.169566			0.0847829	−	0.996399i	$-0.472980\pi$
$911$	154.474			0.169566			0.0847829	+	0.996399i	$0.472980\pi$
$912$		0			0
$913$	612.343	−	612.343i	0.670693	−	0.670693i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−660.186	−	660.186i	−0.719941	−	0.719941i
$918$		0			0
$919$			659.079i			0.717169i	0.933497	+	0.358585i	$0.116741\pi$
$919$			659.079i			0.717169i	−0.933497	+	0.358585i	$0.883259\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	699.414	−	699.414i	0.757762	−	0.757762i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	368.556i		−	0.396723i	−0.980129	−	0.198362i	$-0.936438\pi$
$929$		−	368.556i		−	0.396723i	0.980129	−	0.198362i	$-0.0635621\pi$
$930$		0			0
$931$	−173.546			−0.186408
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	820.898	+	820.898i	0.876092	+	0.876092i	0.993128	−	0.117036i	$-0.0373393\pi$
$937$	820.898	+	820.898i	0.876092	+	0.876092i	−0.117036	+	0.993128i	$0.537339\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1758.50			−1.86875			−0.934377	−	0.356286i	$-0.884043\pi$
$941$	−1758.50			−1.86875			−0.934377	+	0.356286i	$0.884043\pi$
$942$		0			0
$943$	134.979	−	134.979i	0.143137	−	0.143137i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−118.080	−	118.080i	−0.124688	−	0.124688i	0.642009	−	0.766697i	$-0.278101\pi$
$947$	−118.080	−	118.080i	−0.124688	−	0.124688i	−0.766697	+	0.642009i	$0.778101\pi$
$948$		0			0
$949$			635.483i			0.669634i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−217.955	+	217.955i	−0.228704	+	0.228704i	−0.812151	−	0.583447i	$-0.801703\pi$
$953$	−217.955	+	217.955i	−0.228704	+	0.228704i	0.583447	+	0.812151i	$0.301703\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$			1302.76i			1.35845i
$960$		0			0
$961$	−940.796			−0.978976
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	323.983	+	323.983i	0.335039	+	0.335039i	0.854497	−	0.519457i	$-0.173866\pi$
$967$	323.983	+	323.983i	0.335039	+	0.335039i	−0.519457	+	0.854497i	$0.673866\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−743.743			−0.765956			−0.382978	−	0.923757i	$-0.625102\pi$
$971$	−743.743			−0.765956			−0.382978	+	0.923757i	$0.625102\pi$
$972$		0			0
$973$	−767.221	+	767.221i	−0.788511	+	0.788511i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1050.52	−	1050.52i	−1.07525	−	1.07525i	−0.996928	−	0.0783226i	$-0.975044\pi$
$977$	−1050.52	−	1050.52i	−1.07525	−	1.07525i	−0.0783226	−	0.996928i	$-0.524956\pi$
$978$		0			0
$979$			463.060i			0.472993i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	72.9648	−	72.9648i	0.0742267	−	0.0742267i	−0.669019	−	0.743245i	$-0.733285\pi$
$983$	72.9648	−	72.9648i	0.0742267	−	0.0742267i	0.743245	+	0.669019i	$0.233285\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	429.594i		−	0.434372i
$990$		0			0
$991$	444.624			0.448662			0.224331	−	0.974513i	$-0.427980\pi$
$991$	444.624			0.448662			0.224331	+	0.974513i	$0.427980\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−12.6424	−	12.6424i	−0.0126804	−	0.0126804i	0.700738	−	0.713419i	$-0.252854\pi$
$997$	−12.6424	−	12.6424i	−0.0126804	−	0.0126804i	−0.713419	+	0.700738i	$0.752854\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.3.v.n.1657.2		4
3.2	odd	2		600.3.u.e.457.2		4
5.2	odd	4		360.3.v.b.73.2		4
5.3	odd	4	inner	1800.3.v.n.793.2		4
5.4	even	2		360.3.v.b.217.2		4
12.11	even	2		1200.3.bg.e.1057.1		4
15.2	even	4		120.3.u.a.73.1	✓	4
15.8	even	4		600.3.u.e.193.2		4
15.14	odd	2		120.3.u.a.97.1	yes	4
20.7	even	4		720.3.bh.g.433.2		4
20.19	odd	2		720.3.bh.g.577.2		4
60.23	odd	4		1200.3.bg.e.193.1		4
60.47	odd	4		240.3.bg.c.193.2		4
60.59	even	2		240.3.bg.c.97.2		4
120.29	odd	2		960.3.bg.c.577.2		4
120.59	even	2		960.3.bg.d.577.1		4
120.77	even	4		960.3.bg.c.193.2		4
120.107	odd	4		960.3.bg.d.193.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.u.a.73.1	✓	4	15.2	even	4
120.3.u.a.97.1	yes	4	15.14	odd	2
240.3.bg.c.97.2		4	60.59	even	2
240.3.bg.c.193.2		4	60.47	odd	4
360.3.v.b.73.2		4	5.2	odd	4
360.3.v.b.217.2		4	5.4	even	2
600.3.u.e.193.2		4	15.8	even	4
600.3.u.e.457.2		4	3.2	odd	2
720.3.bh.g.433.2		4	20.7	even	4
720.3.bh.g.577.2		4	20.19	odd	2
960.3.bg.c.193.2		4	120.77	even	4
960.3.bg.c.577.2		4	120.29	odd	2
960.3.bg.d.193.1		4	120.107	odd	4
960.3.bg.d.577.1		4	120.59	even	2
1200.3.bg.e.193.1		4	60.23	odd	4
1200.3.bg.e.1057.1		4	12.11	even	2
1800.3.v.n.793.2		4	5.3	odd	4	inner
1800.3.v.n.1657.2		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1800.v (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(5.44949 + 5.44949i) q^{7} +O(q^{10})\)	\(q+(5.44949 + 5.44949i) q^{7} +6.44949 q^{11} +(-14.4495 + 14.4495i) q^{13} +(-23.1464 - 23.1464i) q^{17} +16.6969i q^{19} +(-6.65153 + 6.65153i) q^{23} +0.0454077i q^{29} +4.49490 q^{31} +(-35.3485 - 35.3485i) q^{37} -20.2929 q^{41} +(-32.2929 + 32.2929i) q^{43} +(-50.5403 - 50.5403i) q^{47} +10.3939i q^{49} +(-5.50510 + 5.50510i) q^{53} -55.4393i q^{59} +47.8888 q^{61} +(85.2827 + 85.2827i) q^{67} -48.4041 q^{71} +(21.9898 - 21.9898i) q^{73} +(35.1464 + 35.1464i) q^{77} -126.697i q^{79} +(94.9444 - 94.9444i) q^{83} +71.7980i q^{89} -157.485 q^{91} +(37.0000 + 37.0000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{7}+O(q^{10}) \) 4 * q + 12 * q^7	\( 4 q + 12 q^{7} + 16 q^{11} - 48 q^{13} - 24 q^{17} - 56 q^{23} - 80 q^{31} - 112 q^{37} + 56 q^{41} + 8 q^{43} - 16 q^{47} - 120 q^{53} - 24 q^{61} + 8 q^{67} - 272 q^{71} - 108 q^{73} + 72 q^{77} + 272 q^{83} - 336 q^{91} + 148 q^{97}+O(q^{100}) \) 4 * q + 12 * q^7 + 16 * q^11 - 48 * q^13 - 24 * q^17 - 56 * q^23 - 80 * q^31 - 112 * q^37 + 56 * q^41 + 8 * q^43 - 16 * q^47 - 120 * q^53 - 24 * q^61 + 8 * q^67 - 272 * q^71 - 108 * q^73 + 72 * q^77 + 272 * q^83 - 336 * q^91 + 148 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more