LMFDB - Embedded newform 2400.2.d.f.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,2,Mod(49,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2400.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$19.1640964851$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.399424.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 $ x^6 - 2x^5 + 3x^4 - 6x^3 + 6x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$-0.671462 - 1.24464i$ of defining polynomial
Character	$\chi$	$=$	2400.49
Dual form			2400.2.d.f.49.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -4.68585i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -4.68585i q^{7} +1.00000 q^{9} -2.29273i q^{11} -4.97858 q^{13} -2.97858i q^{17} +2.68585i q^{19} -4.68585i q^{21} +2.68585i q^{23} +1.00000 q^{27} -2.00000i q^{29} +6.97858 q^{31} -2.29273i q^{33} -4.39312 q^{37} -4.97858 q^{39} -11.3717 q^{41} -9.37169 q^{43} -7.27131i q^{47} -14.9572 q^{49} -2.97858i q^{51} -2.00000 q^{53} +2.68585i q^{57} +1.70727i q^{59} +4.58546i q^{61} -4.68585i q^{63} +4.00000 q^{67} +2.68585i q^{69} -0.585462 q^{71} +6.00000i q^{73} -10.7434 q^{77} +1.02142 q^{79} +1.00000 q^{81} +13.3717 q^{83} -2.00000i q^{87} -3.37169 q^{89} +23.3288i q^{91} +6.97858 q^{93} -3.95715i q^{97} -2.29273i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 + 6 * q^9	$ 6 q + 6 q^{3} + 6 q^{9} + 6 q^{27} + 12 q^{31} - 8 q^{37} - 20 q^{41} - 8 q^{43} - 30 q^{49} - 12 q^{53} + 24 q^{67} + 8 q^{71} + 32 q^{77} + 36 q^{79} + 6 q^{81} + 32 q^{83} + 28 q^{89} + 12 q^{93}+O(q^{100}) $ 6 * q + 6 * q^3 + 6 * q^9 + 6 * q^27 + 12 * q^31 - 8 * q^37 - 20 * q^41 - 8 * q^43 - 30 * q^49 - 12 * q^53 + 24 * q^67 + 8 * q^71 + 32 * q^77 + 36 * q^79 + 6 * q^81 + 32 * q^83 + 28 * q^89 + 12 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$-1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.68585i	− 1.77108i	−0.464560	−	0.885542i	$-0.653787\pi$
$7$	− 4.68585i	− 1.77108i	0.464560	−	0.885542i	$-0.346213\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	− 2.29273i	− 0.691284i	−0.938366	−	0.345642i	$-0.887661\pi$
$11$	− 2.29273i	− 0.691284i	0.938366	−	0.345642i	$-0.112339\pi$
$12$	0	0
$13$	−4.97858	−1.38081	−0.690404	−	0.723424i	$-0.742567\pi$
$13$	−4.97858	−1.38081	−0.690404	+	0.723424i	$0.742567\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.97858i	− 0.722411i	−0.932486	−	0.361206i	$-0.882365\pi$
$17$	− 2.97858i	− 0.722411i	0.932486	−	0.361206i	$-0.117635\pi$
$18$	0	0
$19$	2.68585i	0.616175i	0.951358	+	0.308088i	$0.0996890\pi$
$19$	2.68585i	0.616175i	−0.951358	+	0.308088i	$0.900311\pi$
$20$	0	0
$21$	− 4.68585i	− 1.02254i
$22$	0	0
$23$	2.68585i	0.560038i	0.959995	+	0.280019i	$0.0903407\pi$
$23$	2.68585i	0.560038i	−0.959995	+	0.280019i	$0.909659\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	− 2.00000i	− 0.371391i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$	− 2.00000i	− 0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$	0	0
$31$	6.97858	1.25339	0.626695	−	0.779265i	$-0.284407\pi$
$31$	6.97858	1.25339	0.626695	+	0.779265i	$0.284407\pi$
$32$	0	0
$33$	− 2.29273i	− 0.399113i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.39312	−0.722224	−0.361112	−	0.932523i	$-0.617603\pi$
$37$	−4.39312	−0.722224	−0.361112	+	0.932523i	$0.617603\pi$
$38$	0	0
$39$	−4.97858	−0.797210
$40$	0	0
$41$	−11.3717	−1.77596	−0.887980	−	0.459882i	$-0.847892\pi$
$41$	−11.3717	−1.77596	−0.887980	+	0.459882i	$0.847892\pi$
$42$	0	0
$43$	−9.37169	−1.42917	−0.714585	−	0.699549i	$-0.753384\pi$
$43$	−9.37169	−1.42917	−0.714585	+	0.699549i	$0.753384\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 7.27131i	− 1.06063i	−0.847801	−	0.530315i	$-0.822074\pi$
$47$	− 7.27131i	− 1.06063i	0.847801	−	0.530315i	$-0.177926\pi$
$48$	0	0
$49$	−14.9572	−2.13674
$50$	0	0
$51$	− 2.97858i	− 0.417084i
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.68585i	0.355749i
$58$	0	0
$59$	1.70727i	0.222267i	0.993805	+	0.111134i	$0.0354482\pi$
$59$	1.70727i	0.222267i	−0.993805	+	0.111134i	$0.964552\pi$
$60$	0	0
$61$	4.58546i	0.587108i	0.955942	+	0.293554i	$0.0948381\pi$
$61$	4.58546i	0.587108i	−0.955942	+	0.293554i	$0.905162\pi$
$62$	0	0
$63$	− 4.68585i	− 0.590361i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	2.68585i	0.323338i
$70$	0	0
$71$	−0.585462	−0.0694816	−0.0347408	−	0.999396i	$-0.511061\pi$
$71$	−0.585462	−0.0694816	−0.0347408	+	0.999396i	$0.511061\pi$
$72$	0	0
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−10.7434	−1.22432
$78$	0	0
$79$	1.02142	0.114919	0.0574595	−	0.998348i	$-0.481700\pi$
$79$	1.02142	0.114919	0.0574595	+	0.998348i	$0.481700\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.3717	1.46773	0.733867	−	0.679293i	$-0.237714\pi$
$83$	13.3717	1.46773	0.733867	+	0.679293i	$0.237714\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 2.00000i	− 0.214423i
$88$	0	0
$89$	−3.37169	−0.357399	−0.178699	−	0.983904i	$-0.557189\pi$
$89$	−3.37169	−0.357399	−0.178699	+	0.983904i	$0.557189\pi$
$90$	0	0
$91$	23.3288i	2.44553i
$92$	0	0
$93$	6.97858	0.723645
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 3.95715i	− 0.401788i	−0.979613	−	0.200894i	$-0.935615\pi$
$97$	− 3.95715i	− 0.401788i	0.979613	−	0.200894i	$-0.0643847\pi$
$98$	0	0
$99$	− 2.29273i	− 0.230428i
$100$	0	0
$101$	− 2.00000i	− 0.199007i	−0.995037	−	0.0995037i	$-0.968274\pi$
$101$	− 2.00000i	− 0.199007i	0.995037	−	0.0995037i	$-0.0317255\pi$
$102$	0	0
$103$	− 14.6430i	− 1.44282i	−0.692509	−	0.721409i	$-0.743495\pi$
$103$	− 14.6430i	− 1.44282i	0.692509	−	0.721409i	$-0.256505\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.3288	−1.09520	−0.547600	−	0.836740i	$-0.684459\pi$
$107$	−11.3288	−1.09520	−0.547600	+	0.836740i	$0.684459\pi$
$108$	0	0
$109$	9.37169i	0.897645i	0.893621	+	0.448823i	$0.148157\pi$
$109$	9.37169i	0.897645i	−0.893621	+	0.448823i	$0.851843\pi$
$110$	0	0
$111$	−4.39312	−0.416976
$112$	0	0
$113$	− 19.7648i	− 1.85932i	−0.368423	−	0.929658i	$-0.620102\pi$
$113$	− 19.7648i	− 1.85932i	0.368423	−	0.929658i	$-0.379898\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−4.97858	−0.460270
$118$	0	0
$119$	−13.9572	−1.27945
$120$	0	0
$121$	5.74338	0.522126
$122$	0	0
$123$	−11.3717	−1.02535
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 6.64300i	− 0.589471i	−0.955579	−	0.294735i	$-0.904768\pi$
$127$	− 6.64300i	− 0.589471i	0.955579	−	0.294735i	$-0.0952315\pi$
$128$	0	0
$129$	−9.37169	−0.825132
$130$	0	0
$131$	− 7.07896i	− 0.618492i	−0.950982	−	0.309246i	$-0.899923\pi$
$131$	− 7.07896i	− 0.618492i	0.950982	−	0.309246i	$-0.100077\pi$
$132$	0	0
$133$	12.5855	1.09130
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 14.9786i	− 1.27971i	−0.768497	−	0.639853i	$-0.778995\pi$
$137$	− 14.9786i	− 1.27971i	0.768497	−	0.639853i	$-0.221005\pi$
$138$	0	0
$139$	4.64300i	0.393814i	0.980422	+	0.196907i	$0.0630897\pi$
$139$	4.64300i	0.393814i	−0.980422	+	0.196907i	$0.936910\pi$
$140$	0	0
$141$	− 7.27131i	− 0.612355i
$142$	0	0
$143$	11.4145i	0.954532i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−14.9572	−1.23365
$148$	0	0
$149$	− 2.00000i	− 0.163846i	−0.996639	−	0.0819232i	$-0.973894\pi$
$149$	− 2.00000i	− 0.163846i	0.996639	−	0.0819232i	$-0.0261062\pi$
$150$	0	0
$151$	8.35027	0.679535	0.339768	−	0.940509i	$-0.389652\pi$
$151$	8.35027	0.679535	0.339768	+	0.940509i	$0.389652\pi$
$152$	0	0
$153$	− 2.97858i	− 0.240804i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	22.3503	1.78375	0.891873	−	0.452286i	$-0.149391\pi$
$157$	22.3503	1.78375	0.891873	+	0.452286i	$0.149391\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	12.5855	0.991873
$162$	0	0
$163$	1.37169	0.107439	0.0537196	−	0.998556i	$-0.482892\pi$
$163$	1.37169	0.107439	0.0537196	+	0.998556i	$0.482892\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 11.2713i	− 0.872200i	−0.899898	−	0.436100i	$-0.856359\pi$
$167$	− 11.2713i	− 0.872200i	0.899898	−	0.436100i	$-0.143641\pi$
$168$	0	0
$169$	11.7862	0.906633
$170$	0	0
$171$	2.68585i	0.205392i
$172$	0	0
$173$	−10.7862	−0.820062	−0.410031	−	0.912072i	$-0.634482\pi$
$173$	−10.7862	−0.820062	−0.410031	+	0.912072i	$0.634482\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.70727i	0.128326i
$178$	0	0
$179$	3.66442i	0.273892i	0.990579	+	0.136946i	$0.0437287\pi$
$179$	3.66442i	0.273892i	−0.990579	+	0.136946i	$0.956271\pi$
$180$	0	0
$181$	− 6.62831i	− 0.492678i	−0.969184	−	0.246339i	$-0.920772\pi$
$181$	− 6.62831i	− 0.492678i	0.969184	−	0.246339i	$-0.0792277\pi$
$182$	0	0
$183$	4.58546i	0.338967i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−6.82908	−0.499392
$188$	0	0
$189$	− 4.68585i	− 0.340845i
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	− 1.21377i	− 0.0873690i	−0.999045	−	0.0436845i	$-0.986090\pi$
$193$	− 1.21377i	− 0.0873690i	0.999045	−	0.0436845i	$-0.0139096\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−23.9572	−1.70688	−0.853438	−	0.521194i	$-0.825487\pi$
$197$	−23.9572	−1.70688	−0.853438	+	0.521194i	$0.825487\pi$
$198$	0	0
$199$	0.350269	0.0248299	0.0124150	−	0.999923i	$-0.496048\pi$
$199$	0.350269	0.0248299	0.0124150	+	0.999923i	$0.496048\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	−9.37169	−0.657764
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.68585i	0.186679i
$208$	0	0
$209$	6.15792	0.425952
$210$	0	0
$211$	− 14.1004i	− 0.970710i	−0.874317	−	0.485355i	$-0.838690\pi$
$211$	− 14.1004i	− 0.970710i	0.874317	−	0.485355i	$-0.161310\pi$
$212$	0	0
$213$	−0.585462	−0.0401152
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 32.7005i	− 2.21986i
$218$	0	0
$219$	6.00000i	0.405442i
$220$	0	0
$221$	14.8291i	0.997512i
$222$	0	0
$223$	− 6.72869i	− 0.450587i	−0.974291	−	0.225293i	$-0.927666\pi$
$223$	− 6.72869i	− 0.450587i	0.974291	−	0.225293i	$-0.0723340\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.95715	0.660880	0.330440	−	0.943827i	$-0.392803\pi$
$227$	9.95715	0.660880	0.330440	+	0.943827i	$0.392803\pi$
$228$	0	0
$229$	11.3288i	0.748631i	0.927301	+	0.374316i	$0.122122\pi$
$229$	11.3288i	0.748631i	−0.927301	+	0.374316i	$0.877878\pi$
$230$	0	0
$231$	−10.7434	−0.706863
$232$	0	0
$233$	18.9786i	1.24333i	0.783284	+	0.621664i	$0.213543\pi$
$233$	18.9786i	1.24333i	−0.783284	+	0.621664i	$0.786457\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.02142	0.0663485
$238$	0	0
$239$	2.62831	0.170011	0.0850055	−	0.996380i	$-0.472909\pi$
$239$	2.62831	0.170011	0.0850055	+	0.996380i	$0.472909\pi$
$240$	0	0
$241$	10.7862	0.694802	0.347401	−	0.937717i	$-0.387064\pi$
$241$	10.7862	0.694802	0.347401	+	0.937717i	$0.387064\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 13.3717i	− 0.850820i
$248$	0	0
$249$	13.3717	0.847397
$250$	0	0
$251$	− 30.9933i	− 1.95628i	−0.207952	−	0.978139i	$-0.566680\pi$
$251$	− 30.9933i	− 1.95628i	0.207952	−	0.978139i	$-0.433320\pi$
$252$	0	0
$253$	6.15792	0.387145
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20.9357i	1.30594i	0.757386	+	0.652968i	$0.226476\pi$
$257$	20.9357i	1.30594i	−0.757386	+	0.652968i	$0.773524\pi$
$258$	0	0
$259$	20.5855i	1.27912i
$260$	0	0
$261$	− 2.00000i	− 0.123797i
$262$	0	0
$263$	− 19.2713i	− 1.18832i	−0.804347	−	0.594160i	$-0.797485\pi$
$263$	− 19.2713i	− 1.18832i	0.804347	−	0.594160i	$-0.202515\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3.37169	−0.206344
$268$	0	0
$269$	24.7434i	1.50863i	0.656512	+	0.754315i	$0.272031\pi$
$269$	24.7434i	1.50863i	−0.656512	+	0.754315i	$0.727969\pi$
$270$	0	0
$271$	−27.5640	−1.67440	−0.837198	−	0.546900i	$-0.815808\pi$
$271$	−27.5640	−1.67440	−0.837198	+	0.546900i	$0.815808\pi$
$272$	0	0
$273$	23.3288i	1.41193i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	20.3074	1.22015	0.610077	−	0.792342i	$-0.291138\pi$
$277$	20.3074	1.22015	0.610077	+	0.792342i	$0.291138\pi$
$278$	0	0
$279$	6.97858	0.417796
$280$	0	0
$281$	−10.7862	−0.643453	−0.321726	−	0.946833i	$-0.604263\pi$
$281$	−10.7862	−0.643453	−0.321726	+	0.946833i	$0.604263\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	53.2860i	3.14537i
$288$	0	0
$289$	8.12808	0.478122
$290$	0	0
$291$	− 3.95715i	− 0.231972i
$292$	0	0
$293$	−21.9143	−1.28025	−0.640124	−	0.768272i	$-0.721117\pi$
$293$	−21.9143	−1.28025	−0.640124	+	0.768272i	$0.721117\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2.29273i	− 0.133038i
$298$	0	0
$299$	− 13.3717i	− 0.773305i
$300$	0	0
$301$	43.9143i	2.53118i
$302$	0	0
$303$	− 2.00000i	− 0.114897i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	26.5426	1.51487	0.757434	−	0.652912i	$-0.226453\pi$
$307$	26.5426	1.51487	0.757434	+	0.652912i	$0.226453\pi$
$308$	0	0
$309$	− 14.6430i	− 0.833011i
$310$	0	0
$311$	−12.2008	−0.691842	−0.345921	−	0.938264i	$-0.612434\pi$
$311$	−12.2008	−0.691842	−0.345921	+	0.938264i	$0.612434\pi$
$312$	0	0
$313$	− 15.9572i	− 0.901952i	−0.892536	−	0.450976i	$-0.851076\pi$
$313$	− 15.9572i	− 0.901952i	0.892536	−	0.450976i	$-0.148924\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	33.5296	1.88321	0.941605	−	0.336718i	$-0.109317\pi$
$317$	33.5296	1.88321	0.941605	+	0.336718i	$0.109317\pi$
$318$	0	0
$319$	−4.58546	−0.256737
$320$	0	0
$321$	−11.3288	−0.632315
$322$	0	0
$323$	8.00000	0.445132
$324$	0	0
$325$	0	0
$326$	0	0
$327$	9.37169i	0.518256i
$328$	0	0
$329$	−34.0722	−1.87846
$330$	0	0
$331$	19.8568i	1.09143i	0.837972	+	0.545713i	$0.183741\pi$
$331$	19.8568i	1.09143i	−0.837972	+	0.545713i	$0.816259\pi$
$332$	0	0
$333$	−4.39312	−0.240741
$334$	0	0
$335$	0	0
$336$	0	0
$337$	7.17092i	0.390625i	0.980741	+	0.195313i	$0.0625721\pi$
$337$	7.17092i	0.390625i	−0.980741	+	0.195313i	$0.937428\pi$
$338$	0	0
$339$	− 19.7648i	− 1.07348i
$340$	0	0
$341$	− 16.0000i	− 0.866449i
$342$	0	0
$343$	37.2860i	2.01325i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.786230	0.0422071	0.0211035	−	0.999777i	$-0.493282\pi$
$347$	0.786230	0.0422071	0.0211035	+	0.999777i	$0.493282\pi$
$348$	0	0
$349$	− 6.15792i	− 0.329626i	−0.986325	−	0.164813i	$-0.947298\pi$
$349$	− 6.15792i	− 0.329626i	0.986325	−	0.164813i	$-0.0527021\pi$
$350$	0	0
$351$	−4.97858	−0.265737
$352$	0	0
$353$	− 21.7220i	− 1.15614i	−0.815986	−	0.578072i	$-0.803805\pi$
$353$	− 21.7220i	− 1.15614i	0.815986	−	0.578072i	$-0.196195\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−13.9572	−0.738691
$358$	0	0
$359$	0.585462	0.0308995	0.0154498	−	0.999881i	$-0.495082\pi$
$359$	0.585462	0.0308995	0.0154498	+	0.999881i	$0.495082\pi$
$360$	0	0
$361$	11.7862	0.620328
$362$	0	0
$363$	5.74338	0.301450
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0.485078i	0.0253209i	0.999920	+	0.0126604i	$0.00403005\pi$
$367$	0.485078i	0.0253209i	−0.999920	+	0.0126604i	$0.995970\pi$
$368$	0	0
$369$	−11.3717	−0.591987
$370$	0	0
$371$	9.37169i	0.486554i
$372$	0	0
$373$	−12.3931	−0.641691	−0.320846	−	0.947132i	$-0.603967\pi$
$373$	−12.3931	−0.641691	−0.320846	+	0.947132i	$0.603967\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.95715i	0.512820i
$378$	0	0
$379$	− 26.0147i	− 1.33629i	−0.744033	−	0.668143i	$-0.767090\pi$
$379$	− 26.0147i	− 1.33629i	0.744033	−	0.668143i	$-0.232910\pi$
$380$	0	0
$381$	− 6.64300i	− 0.340331i
$382$	0	0
$383$	6.68585i	0.341631i	0.985303	+	0.170815i	$0.0546402\pi$
$383$	6.68585i	0.341631i	−0.985303	+	0.170815i	$0.945360\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.37169	−0.476390
$388$	0	0
$389$	− 29.9143i	− 1.51672i	−0.651838	−	0.758358i	$-0.726002\pi$
$389$	− 29.9143i	− 1.51672i	0.651838	−	0.758358i	$-0.273998\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	− 7.07896i	− 0.357086i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−9.76481	−0.490082	−0.245041	−	0.969513i	$-0.578801\pi$
$397$	−9.76481	−0.490082	−0.245041	+	0.969513i	$0.578801\pi$
$398$	0	0
$399$	12.5855	0.630061
$400$	0	0
$401$	−6.58546	−0.328862	−0.164431	−	0.986389i	$-0.552579\pi$
$401$	−6.58546	−0.328862	−0.164431	+	0.986389i	$0.552579\pi$
$402$	0	0
$403$	−34.7434	−1.73069
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.0722i	0.499262i
$408$	0	0
$409$	25.9143	1.28138	0.640690	−	0.767800i	$-0.278648\pi$
$409$	25.9143	1.28138	0.640690	+	0.767800i	$0.278648\pi$
$410$	0	0
$411$	− 14.9786i	− 0.738839i
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.64300i	0.227369i
$418$	0	0
$419$	− 12.2499i	− 0.598446i	−0.954183	−	0.299223i	$-0.903273\pi$
$419$	− 12.2499i	− 0.598446i	0.954183	−	0.299223i	$-0.0967275\pi$
$420$	0	0
$421$	− 4.67115i	− 0.227658i	−0.993500	−	0.113829i	$-0.963688\pi$
$421$	− 4.67115i	− 0.227658i	0.993500	−	0.113829i	$-0.0363116\pi$
$422$	0	0
$423$	− 7.27131i	− 0.353543i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	21.4868	1.03982
$428$	0	0
$429$	11.4145i	0.551099i
$430$	0	0
$431$	−0.585462	−0.0282007	−0.0141004	−	0.999901i	$-0.504488\pi$
$431$	−0.585462	−0.0282007	−0.0141004	+	0.999901i	$0.504488\pi$
$432$	0	0
$433$	21.9143i	1.05313i	0.850133	+	0.526567i	$0.176521\pi$
$433$	21.9143i	1.05313i	−0.850133	+	0.526567i	$0.823479\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.21377	−0.345081
$438$	0	0
$439$	−2.39312	−0.114217	−0.0571086	−	0.998368i	$-0.518188\pi$
$439$	−2.39312	−0.114217	−0.0571086	+	0.998368i	$0.518188\pi$
$440$	0	0
$441$	−14.9572	−0.712245
$442$	0	0
$443$	20.7005	0.983512	0.491756	−	0.870733i	$-0.336355\pi$
$443$	20.7005	0.983512	0.491756	+	0.870733i	$0.336355\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 2.00000i	− 0.0945968i
$448$	0	0
$449$	37.9143	1.78929	0.894643	−	0.446781i	$-0.147430\pi$
$449$	37.9143	1.78929	0.894643	+	0.446781i	$0.147430\pi$
$450$	0	0
$451$	26.0722i	1.22769i
$452$	0	0
$453$	8.35027	0.392330
$454$	0	0
$455$	0	0
$456$	0	0
$457$	38.7005i	1.81033i	0.425055	+	0.905167i	$0.360255\pi$
$457$	38.7005i	1.81033i	−0.425055	+	0.905167i	$0.639745\pi$
$458$	0	0
$459$	− 2.97858i	− 0.139028i
$460$	0	0
$461$	− 4.74338i	− 0.220921i	−0.993880	−	0.110461i	$-0.964767\pi$
$461$	− 4.74338i	− 0.220921i	0.993880	−	0.110461i	$-0.0352326\pi$
$462$	0	0
$463$	15.3142i	0.711709i	0.934541	+	0.355855i	$0.115810\pi$
$463$	15.3142i	0.711709i	−0.934541	+	0.355855i	$0.884190\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−30.5426	−1.41334	−0.706672	−	0.707541i	$-0.749804\pi$
$467$	−30.5426	−1.41334	−0.706672	+	0.707541i	$0.749804\pi$
$468$	0	0
$469$	− 18.7434i	− 0.865489i
$470$	0	0
$471$	22.3503	1.02985
$472$	0	0
$473$	21.4868i	0.987963i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	3.32885	0.152099	0.0760494	−	0.997104i	$-0.475769\pi$
$479$	3.32885	0.152099	0.0760494	+	0.997104i	$0.475769\pi$
$480$	0	0
$481$	21.8715	0.997253
$482$	0	0
$483$	12.5855	0.572658
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12.1004i	0.548321i	0.961684	+	0.274160i	$0.0883999\pi$
$487$	12.1004i	0.548321i	−0.961684	+	0.274160i	$0.911600\pi$
$488$	0	0
$489$	1.37169	0.0620301
$490$	0	0
$491$	− 14.2927i	− 0.645022i	−0.946566	−	0.322511i	$-0.895473\pi$
$491$	− 14.2927i	− 0.645022i	0.946566	−	0.322511i	$-0.104527\pi$
$492$	0	0
$493$	−5.95715	−0.268297
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.74338i	0.123058i
$498$	0	0
$499$	− 9.22846i	− 0.413123i	−0.978434	−	0.206561i	$-0.933773\pi$
$499$	− 9.22846i	− 0.413123i	0.978434	−	0.206561i	$-0.0662273\pi$
$500$	0	0
$501$	− 11.2713i	− 0.503565i
$502$	0	0
$503$	− 14.1004i	− 0.628705i	−0.949306	−	0.314353i	$-0.898213\pi$
$503$	− 14.1004i	− 0.628705i	0.949306	−	0.314353i	$-0.101787\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	11.7862	0.523445
$508$	0	0
$509$	− 43.4011i	− 1.92372i	−0.273544	−	0.961859i	$-0.588196\pi$
$509$	− 43.4011i	− 1.92372i	0.273544	−	0.961859i	$-0.411804\pi$
$510$	0	0
$511$	28.1151	1.24374
$512$	0	0
$513$	2.68585i	0.118583i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−16.6712	−0.733196
$518$	0	0
$519$	−10.7862	−0.473463
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	13.5725	0.593482	0.296741	−	0.954958i	$-0.404100\pi$
$523$	13.5725	0.593482	0.296741	+	0.954958i	$0.404100\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 20.7862i	− 0.905462i
$528$	0	0
$529$	15.7862	0.686358
$530$	0	0
$531$	1.70727i	0.0740892i
$532$	0	0
$533$	56.6148	2.45226
$534$	0	0
$535$	0	0
$536$	0	0
$537$	3.66442i	0.158132i
$538$	0	0
$539$	34.2927i	1.47709i
$540$	0	0
$541$	37.2860i	1.60305i	0.597961	+	0.801525i	$0.295978\pi$
$541$	37.2860i	1.60305i	−0.597961	+	0.801525i	$0.704022\pi$
$542$	0	0
$543$	− 6.62831i	− 0.284448i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0.200768	0.00858424	0.00429212	−	0.999991i	$-0.498634\pi$
$547$	0.200768	0.00858424	0.00429212	+	0.999991i	$0.498634\pi$
$548$	0	0
$549$	4.58546i	0.195703i
$550$	0	0
$551$	5.37169	0.228842
$552$	0	0
$553$	− 4.78623i	− 0.203531i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.21377	0.390400	0.195200	−	0.980763i	$-0.437464\pi$
$557$	9.21377	0.390400	0.195200	+	0.980763i	$0.437464\pi$
$558$	0	0
$559$	46.6577	1.97341
$560$	0	0
$561$	−6.82908	−0.288324
$562$	0	0
$563$	−36.7005	−1.54674	−0.773372	−	0.633953i	$-0.781431\pi$
$563$	−36.7005	−1.54674	−0.773372	+	0.633953i	$0.781431\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 4.68585i	− 0.196787i
$568$	0	0
$569$	−13.4145	−0.562367	−0.281183	−	0.959654i	$-0.590727\pi$
$569$	−13.4145	−0.562367	−0.281183	+	0.959654i	$0.590727\pi$
$570$	0	0
$571$	18.6858i	0.781978i	0.920395	+	0.390989i	$0.127867\pi$
$571$	18.6858i	0.781978i	−0.920395	+	0.390989i	$0.872133\pi$
$572$	0	0
$573$	8.00000	0.334205
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 2.78623i	− 0.115992i	−0.998317	−	0.0579961i	$-0.981529\pi$
$577$	− 2.78623i	− 0.115992i	0.998317	−	0.0579961i	$-0.0184711\pi$
$578$	0	0
$579$	− 1.21377i	− 0.0504425i
$580$	0	0
$581$	− 62.6577i	− 2.59948i
$582$	0	0
$583$	4.58546i	0.189910i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27.3288	1.12798	0.563991	−	0.825781i	$-0.309265\pi$
$587$	27.3288	1.12798	0.563991	+	0.825781i	$0.309265\pi$
$588$	0	0
$589$	18.7434i	0.772308i
$590$	0	0
$591$	−23.9572	−0.985466
$592$	0	0
$593$	6.97858i	0.286576i	0.989681	+	0.143288i	$0.0457675\pi$
$593$	6.97858i	0.286576i	−0.989681	+	0.143288i	$0.954233\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.350269	0.0143356
$598$	0	0
$599$	36.4998	1.49134	0.745670	−	0.666315i	$-0.232130\pi$
$599$	36.4998	1.49134	0.745670	+	0.666315i	$0.232130\pi$
$600$	0	0
$601$	−15.5725	−0.635214	−0.317607	−	0.948222i	$-0.602879\pi$
$601$	−15.5725	−0.635214	−0.317607	+	0.948222i	$0.602879\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	0	0
$606$	0	0
$607$	31.2285i	1.26752i	0.773528	+	0.633762i	$0.218490\pi$
$607$	31.2285i	1.26752i	−0.773528	+	0.633762i	$0.781510\pi$
$608$	0	0
$609$	−9.37169	−0.379760
$610$	0	0
$611$	36.2008i	1.46453i
$612$	0	0
$613$	0.978577	0.0395244	0.0197622	−	0.999805i	$-0.493709\pi$
$613$	0.978577	0.0395244	0.0197622	+	0.999805i	$0.493709\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	32.9357i	1.32594i	0.748645	+	0.662971i	$0.230705\pi$
$617$	32.9357i	1.32594i	−0.748645	+	0.662971i	$0.769295\pi$
$618$	0	0
$619$	− 3.35700i	− 0.134929i	−0.997722	−	0.0674646i	$-0.978509\pi$
$619$	− 3.35700i	− 0.134929i	0.997722	−	0.0674646i	$-0.0214910\pi$
$620$	0	0
$621$	2.68585i	0.107779i
$622$	0	0
$623$	15.7992i	0.632983i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6.15792	0.245924
$628$	0	0
$629$	13.0852i	0.521742i
$630$	0	0
$631$	27.7648	1.10530	0.552650	−	0.833414i	$-0.313617\pi$
$631$	27.7648	1.10530	0.552650	+	0.833414i	$0.313617\pi$
$632$	0	0
$633$	− 14.1004i	− 0.560440i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	74.4653	2.95042
$638$	0	0
$639$	−0.585462	−0.0231605
$640$	0	0
$641$	−21.1281	−0.834509	−0.417254	−	0.908790i	$-0.637008\pi$
$641$	−21.1281	−0.834509	−0.417254	+	0.908790i	$0.637008\pi$
$642$	0	0
$643$	−29.2860	−1.15493	−0.577464	−	0.816416i	$-0.695957\pi$
$643$	−29.2860	−1.15493	−0.577464	+	0.816416i	$0.695957\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 15.6728i	− 0.616163i	−0.951360	−	0.308082i	$-0.900313\pi$
$647$	− 15.6728i	− 0.616163i	0.951360	−	0.308082i	$-0.0996870\pi$
$648$	0	0
$649$	3.91431	0.153650
$650$	0	0
$651$	− 32.7005i	− 1.28164i
$652$	0	0
$653$	17.5296	0.685987	0.342993	−	0.939338i	$-0.388559\pi$
$653$	17.5296	0.685987	0.342993	+	0.939338i	$0.388559\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.00000i	0.234082i
$658$	0	0
$659$	− 23.8652i	− 0.929656i	−0.885401	−	0.464828i	$-0.846116\pi$
$659$	− 23.8652i	− 0.929656i	0.885401	−	0.464828i	$-0.153884\pi$
$660$	0	0
$661$	− 30.1579i	− 1.17301i	−0.809947	−	0.586504i	$-0.800504\pi$
$661$	− 30.1579i	− 1.17301i	0.809947	−	0.586504i	$-0.199496\pi$
$662$	0	0
$663$	14.8291i	0.575914i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.37169	0.207993
$668$	0	0
$669$	− 6.72869i	− 0.260146i
$670$	0	0
$671$	10.5132	0.405859
$672$	0	0
$673$	18.0000i	0.693849i	0.937893	+	0.346925i	$0.112774\pi$
$673$	18.0000i	0.693849i	−0.937893	+	0.346925i	$0.887226\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−9.61531	−0.369546	−0.184773	−	0.982781i	$-0.559155\pi$
$677$	−9.61531	−0.369546	−0.184773	+	0.982781i	$0.559155\pi$
$678$	0	0
$679$	−18.5426	−0.711600
$680$	0	0
$681$	9.95715	0.381559
$682$	0	0
$683$	−18.6283	−0.712792	−0.356396	−	0.934335i	$-0.615995\pi$
$683$	−18.6283	−0.712792	−0.356396	+	0.934335i	$0.615995\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	11.3288i	0.432222i
$688$	0	0
$689$	9.95715	0.379337
$690$	0	0
$691$	13.4292i	0.510872i	0.966826	+	0.255436i	$0.0822190\pi$
$691$	13.4292i	0.510872i	−0.966826	+	0.255436i	$0.917781\pi$
$692$	0	0
$693$	−10.7434	−0.408107
$694$	0	0
$695$	0	0
$696$	0	0
$697$	33.8715i	1.28297i
$698$	0	0
$699$	18.9786i	0.717836i
$700$	0	0
$701$	19.1709i	0.724076i	0.932163	+	0.362038i	$0.117919\pi$
$701$	19.1709i	0.724076i	−0.932163	+	0.362038i	$0.882081\pi$
$702$	0	0
$703$	− 11.7992i	− 0.445016i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.37169	−0.352459
$708$	0	0
$709$	− 15.4145i	− 0.578905i	−0.957192	−	0.289453i	$-0.906527\pi$
$709$	− 15.4145i	− 0.578905i	0.957192	−	0.289453i	$-0.0934732\pi$
$710$	0	0
$711$	1.02142	0.0383063
$712$	0	0
$713$	18.7434i	0.701945i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.62831	0.0981559
$718$	0	0
$719$	20.7862	0.775196	0.387598	−	0.921829i	$-0.373305\pi$
$719$	20.7862	0.775196	0.387598	+	0.921829i	$0.373305\pi$
$720$	0	0
$721$	−68.6148	−2.55535
$722$	0	0
$723$	10.7862	0.401144
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 12.3012i	− 0.456224i	−0.973635	−	0.228112i	$-0.926745\pi$
$727$	− 12.3012i	− 0.456224i	0.973635	−	0.228112i	$-0.0732553\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	27.9143i	1.03245i
$732$	0	0
$733$	−35.9227	−1.32684	−0.663418	−	0.748249i	$-0.730895\pi$
$733$	−35.9227	−1.32684	−0.663418	+	0.748249i	$0.730895\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 9.17092i	− 0.337815i
$738$	0	0
$739$	29.0277i	1.06780i	0.845547	+	0.533900i	$0.179274\pi$
$739$	29.0277i	1.06780i	−0.845547	+	0.533900i	$0.820726\pi$
$740$	0	0
$741$	− 13.3717i	− 0.491221i
$742$	0	0
$743$	2.60015i	0.0953904i	0.998862	+	0.0476952i	$0.0151876\pi$
$743$	2.60015i	0.0953904i	−0.998862	+	0.0476952i	$0.984812\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	13.3717	0.489245
$748$	0	0
$749$	53.0852i	1.93969i
$750$	0	0
$751$	10.8929	0.397487	0.198744	−	0.980052i	$-0.436314\pi$
$751$	10.8929	0.397487	0.198744	+	0.980052i	$0.436314\pi$
$752$	0	0
$753$	− 30.9933i	− 1.12946i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−34.3503	−1.24848	−0.624241	−	0.781232i	$-0.714592\pi$
$757$	−34.3503	−1.24848	−0.624241	+	0.781232i	$0.714592\pi$
$758$	0	0
$759$	6.15792	0.223518
$760$	0	0
$761$	−19.0852	−0.691839	−0.345920	−	0.938264i	$-0.612433\pi$
$761$	−19.0852	−0.691839	−0.345920	+	0.938264i	$0.612433\pi$
$762$	0	0
$763$	43.9143	1.58980
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 8.49977i	− 0.306909i
$768$	0	0
$769$	−31.8715	−1.14931	−0.574657	−	0.818394i	$-0.694865\pi$
$769$	−31.8715	−1.14931	−0.574657	+	0.818394i	$0.694865\pi$
$770$	0	0
$771$	20.9357i	0.753982i
$772$	0	0
$773$	−11.9572	−0.430069	−0.215034	−	0.976606i	$-0.568986\pi$
$773$	−11.9572	−0.430069	−0.215034	+	0.976606i	$0.568986\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	20.5855i	0.738499i
$778$	0	0
$779$	− 30.5426i	− 1.09430i
$780$	0	0
$781$	1.34231i	0.0480315i
$782$	0	0
$783$	− 2.00000i	− 0.0714742i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	33.0852	1.17936	0.589681	−	0.807637i	$-0.299254\pi$
$787$	33.0852	1.17936	0.589681	+	0.807637i	$0.299254\pi$
$788$	0	0
$789$	− 19.2713i	− 0.686077i
$790$	0	0
$791$	−92.6148	−3.29300
$792$	0	0
$793$	− 22.8291i	− 0.810684i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	10.0000	0.354218	0.177109	−	0.984191i	$-0.443325\pi$
$797$	10.0000	0.354218	0.177109	+	0.984191i	$0.443325\pi$
$798$	0	0
$799$	−21.6582	−0.766210
$800$	0	0
$801$	−3.37169	−0.119133
$802$	0	0
$803$	13.7564	0.485452
$804$	0	0
$805$	0	0
$806$	0	0
$807$	24.7434i	0.871008i
$808$	0	0
$809$	30.6148	1.07636	0.538180	−	0.842830i	$-0.319112\pi$
$809$	30.6148	1.07636	0.538180	+	0.842830i	$0.319112\pi$
$810$	0	0
$811$	− 53.9290i	− 1.89370i	−0.321670	−	0.946852i	$-0.604244\pi$
$811$	− 53.9290i	− 1.89370i	0.321670	−	0.946852i	$-0.395756\pi$
$812$	0	0
$813$	−27.5640	−0.966713
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 25.1709i	− 0.880619i
$818$	0	0
$819$	23.3288i	0.815176i
$820$	0	0
$821$	12.6577i	0.441757i	0.975301	+	0.220878i	$0.0708924\pi$
$821$	12.6577i	0.441757i	−0.975301	+	0.220878i	$0.929108\pi$
$822$	0	0
$823$	− 19.8139i	− 0.690670i	−0.938479	−	0.345335i	$-0.887765\pi$
$823$	− 19.8139i	− 0.690670i	0.938479	−	0.345335i	$-0.112235\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	0	0
$829$	41.3717i	1.43690i	0.695580	+	0.718449i	$0.255148\pi$
$829$	41.3717i	1.43690i	−0.695580	+	0.718449i	$0.744852\pi$
$830$	0	0
$831$	20.3074	0.704457
$832$	0	0
$833$	44.5510i	1.54360i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	6.97858	0.241215
$838$	0	0
$839$	20.9013	0.721593	0.360797	−	0.932645i	$-0.382505\pi$
$839$	20.9013	0.721593	0.360797	+	0.932645i	$0.382505\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	−10.7862	−0.371498
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 26.9126i	− 0.924728i
$848$	0	0
$849$	20.0000	0.686398
$850$	0	0
$851$	− 11.7992i	− 0.404472i
$852$	0	0
$853$	6.63673	0.227237	0.113619	−	0.993524i	$-0.463756\pi$
$853$	6.63673	0.227237	0.113619	+	0.993524i	$0.463756\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9.80765i	0.335023i	0.985870	+	0.167512i	$0.0535731\pi$
$857$	9.80765i	0.335023i	−0.985870	+	0.167512i	$0.946427\pi$
$858$	0	0
$859$	39.3864i	1.34385i	0.740621	+	0.671923i	$0.234531\pi$
$859$	39.3864i	1.34385i	−0.740621	+	0.671923i	$0.765469\pi$
$860$	0	0
$861$	53.2860i	1.81598i
$862$	0	0
$863$	7.07054i	0.240684i	0.992732	+	0.120342i	$0.0383991\pi$
$863$	7.07054i	0.240684i	−0.992732	+	0.120342i	$0.961601\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.12808	0.276044
$868$	0	0
$869$	− 2.34185i	− 0.0794417i
$870$	0	0
$871$	−19.9143	−0.674771
$872$	0	0
$873$	− 3.95715i	− 0.133929i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−23.1365	−0.781264	−0.390632	−	0.920547i	$-0.627744\pi$
$877$	−23.1365	−0.781264	−0.390632	+	0.920547i	$0.627744\pi$
$878$	0	0
$879$	−21.9143	−0.739151
$880$	0	0
$881$	28.4569	0.958738	0.479369	−	0.877613i	$-0.340866\pi$
$881$	28.4569	0.958738	0.479369	+	0.877613i	$0.340866\pi$
$882$	0	0
$883$	41.2003	1.38650	0.693250	−	0.720697i	$-0.256178\pi$
$883$	41.2003	1.38650	0.693250	+	0.720697i	$0.256178\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.55777i	0.119458i	0.998215	+	0.0597291i	$0.0190237\pi$
$887$	3.55777i	0.119458i	−0.998215	+	0.0597291i	$0.980976\pi$
$888$	0	0
$889$	−31.1281	−1.04400
$890$	0	0
$891$	− 2.29273i	− 0.0768094i
$892$	0	0
$893$	19.5296	0.653534
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 13.3717i	− 0.446468i
$898$	0	0
$899$	− 13.9572i	− 0.465497i
$900$	0	0
$901$	5.95715i	0.198462i
$902$	0	0
$903$	43.9143i	1.46138i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−50.6577	−1.68206	−0.841031	−	0.540988i	$-0.818051\pi$
$907$	−50.6577	−1.68206	−0.841031	+	0.540988i	$0.818051\pi$
$908$	0	0
$909$	− 2.00000i	− 0.0663358i
$910$	0	0
$911$	26.4569	0.876557	0.438279	−	0.898839i	$-0.355588\pi$
$911$	26.4569	0.876557	0.438279	+	0.898839i	$0.355588\pi$
$912$	0	0
$913$	− 30.6577i	− 1.01462i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−33.1709	−1.09540
$918$	0	0
$919$	29.8077	0.983264	0.491632	−	0.870803i	$-0.336401\pi$
$919$	29.8077	0.983264	0.491632	+	0.870803i	$0.336401\pi$
$920$	0	0
$921$	26.5426	0.874609
$922$	0	0
$923$	2.91477	0.0959407
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 14.6430i	− 0.480939i
$928$	0	0
$929$	8.82908	0.289673	0.144836	−	0.989456i	$-0.453734\pi$
$929$	8.82908	0.289673	0.144836	+	0.989456i	$0.453734\pi$
$930$	0	0
$931$	− 40.1726i	− 1.31660i
$932$	0	0
$933$	−12.2008	−0.399435
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 42.2302i	− 1.37960i	−0.724000	−	0.689799i	$-0.757699\pi$
$937$	− 42.2302i	− 1.37960i	0.724000	−	0.689799i	$-0.242301\pi$
$938$	0	0
$939$	− 15.9572i	− 0.520742i
$940$	0	0
$941$	32.7434i	1.06740i	0.845673	+	0.533702i	$0.179200\pi$
$941$	32.7434i	1.06740i	−0.845673	+	0.533702i	$0.820800\pi$
$942$	0	0
$943$	− 30.5426i	− 0.994604i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−15.9143	−0.517146	−0.258573	−	0.965992i	$-0.583252\pi$
$947$	−15.9143	−0.517146	−0.258573	+	0.965992i	$0.583252\pi$
$948$	0	0
$949$	− 29.8715i	− 0.969669i
$950$	0	0
$951$	33.5296	1.08727
$952$	0	0
$953$	− 55.6791i	− 1.80362i	−0.432129	−	0.901812i	$-0.642238\pi$
$953$	− 55.6791i	− 1.80362i	0.432129	−	0.901812i	$-0.357762\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−4.58546	−0.148227
$958$	0	0
$959$	−70.1873	−2.26647
$960$	0	0
$961$	17.7005	0.570985
$962$	0	0
$963$	−11.3288	−0.365067
$964$	0	0
$965$	0	0
$966$	0	0
$967$	54.7581i	1.76090i	0.474138	+	0.880451i	$0.342760\pi$
$967$	54.7581i	1.76090i	−0.474138	+	0.880451i	$0.657240\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	30.3221i	0.973083i	0.873657	+	0.486542i	$0.161742\pi$
$971$	30.3221i	0.973083i	−0.873657	+	0.486542i	$0.838258\pi$
$972$	0	0
$973$	21.7564	0.697478
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 29.7220i	− 0.950890i	−0.879746	−	0.475445i	$-0.842287\pi$
$977$	− 29.7220i	− 0.950890i	0.879746	−	0.475445i	$-0.157713\pi$
$978$	0	0
$979$	7.73038i	0.247064i
$980$	0	0
$981$	9.37169i	0.299215i
$982$	0	0
$983$	− 49.5443i	− 1.58022i	−0.612966	−	0.790109i	$-0.710024\pi$
$983$	− 49.5443i	− 1.58022i	0.612966	−	0.790109i	$-0.289976\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−34.0722	−1.08453
$988$	0	0
$989$	− 25.1709i	− 0.800389i
$990$	0	0
$991$	−19.0937	−0.606530	−0.303265	−	0.952906i	$-0.598077\pi$
$991$	−19.0937	−0.606530	−0.303265	+	0.952906i	$0.598077\pi$
$992$	0	0
$993$	19.8568i	0.630136i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−38.8500	−1.23039	−0.615197	−	0.788374i	$-0.710923\pi$
$997$	−38.8500	−1.23039	−0.615197	+	0.788374i	$0.710923\pi$
$998$	0	0
$999$	−4.39312	−0.138992

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.d.f.49.1		6
3.2	odd	2		7200.2.d.q.2449.1		6
4.3	odd	2		600.2.d.e.349.1		6
5.2	odd	4		2400.2.k.c.1201.3		6
5.3	odd	4		480.2.k.b.241.4		6
5.4	even	2		2400.2.d.e.49.6		6
8.3	odd	2		600.2.d.f.349.5		6
8.5	even	2		2400.2.d.e.49.1		6
12.11	even	2		1800.2.d.q.1549.6		6
15.2	even	4		7200.2.k.p.3601.6		6
15.8	even	4		1440.2.k.f.721.4		6
15.14	odd	2		7200.2.d.r.2449.6		6
20.3	even	4		120.2.k.b.61.2	yes	6
20.7	even	4		600.2.k.c.301.5		6
20.19	odd	2		600.2.d.f.349.6		6
24.5	odd	2		7200.2.d.r.2449.1		6
24.11	even	2		1800.2.d.r.1549.2		6
40.3	even	4		120.2.k.b.61.1	✓	6
40.13	odd	4		480.2.k.b.241.1		6
40.19	odd	2		600.2.d.e.349.2		6
40.27	even	4		600.2.k.c.301.6		6
40.29	even	2	inner	2400.2.d.f.49.6		6
40.37	odd	4		2400.2.k.c.1201.6		6
60.23	odd	4		360.2.k.f.181.5		6
60.47	odd	4		1800.2.k.p.901.2		6
60.59	even	2		1800.2.d.r.1549.1		6
80.3	even	4		3840.2.a.bq.1.1		3
80.13	odd	4		3840.2.a.bo.1.3		3
80.43	even	4		3840.2.a.bp.1.1		3
80.53	odd	4		3840.2.a.br.1.3		3
120.29	odd	2		7200.2.d.q.2449.6		6
120.53	even	4		1440.2.k.f.721.1		6
120.59	even	2		1800.2.d.q.1549.5		6
120.77	even	4		7200.2.k.p.3601.5		6
120.83	odd	4		360.2.k.f.181.6		6
120.107	odd	4		1800.2.k.p.901.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.k.b.61.1	✓	6	40.3	even	4
120.2.k.b.61.2	yes	6	20.3	even	4
360.2.k.f.181.5		6	60.23	odd	4
360.2.k.f.181.6		6	120.83	odd	4
480.2.k.b.241.1		6	40.13	odd	4
480.2.k.b.241.4		6	5.3	odd	4
600.2.d.e.349.1		6	4.3	odd	2
600.2.d.e.349.2		6	40.19	odd	2
600.2.d.f.349.5		6	8.3	odd	2
600.2.d.f.349.6		6	20.19	odd	2
600.2.k.c.301.5		6	20.7	even	4
600.2.k.c.301.6		6	40.27	even	4
1440.2.k.f.721.1		6	120.53	even	4
1440.2.k.f.721.4		6	15.8	even	4
1800.2.d.q.1549.5		6	120.59	even	2
1800.2.d.q.1549.6		6	12.11	even	2
1800.2.d.r.1549.1		6	60.59	even	2
1800.2.d.r.1549.2		6	24.11	even	2
1800.2.k.p.901.1		6	120.107	odd	4
1800.2.k.p.901.2		6	60.47	odd	4
2400.2.d.e.49.1		6	8.5	even	2
2400.2.d.e.49.6		6	5.4	even	2
2400.2.d.f.49.1		6	1.1	even	1	trivial
2400.2.d.f.49.6		6	40.29	even	2	inner
2400.2.k.c.1201.3		6	5.2	odd	4
2400.2.k.c.1201.6		6	40.37	odd	4
3840.2.a.bo.1.3		3	80.13	odd	4
3840.2.a.bp.1.1		3	80.43	even	4
3840.2.a.bq.1.1		3	80.3	even	4
3840.2.a.br.1.3		3	80.53	odd	4
7200.2.d.q.2449.1		6	3.2	odd	2
7200.2.d.q.2449.6		6	120.29	odd	2
7200.2.d.r.2449.1		6	24.5	odd	2
7200.2.d.r.2449.6		6	15.14	odd	2
7200.2.k.p.3601.5		6	120.77	even	4
7200.2.k.p.3601.6		6	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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2400.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -4.68585i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -4.68585i q^{7} +1.00000 q^{9} -2.29273i q^{11} -4.97858 q^{13} -2.97858i q^{17} +2.68585i q^{19} -4.68585i q^{21} +2.68585i q^{23} +1.00000 q^{27} -2.00000i q^{29} +6.97858 q^{31} -2.29273i q^{33} -4.39312 q^{37} -4.97858 q^{39} -11.3717 q^{41} -9.37169 q^{43} -7.27131i q^{47} -14.9572 q^{49} -2.97858i q^{51} -2.00000 q^{53} +2.68585i q^{57} +1.70727i q^{59} +4.58546i q^{61} -4.68585i q^{63} +4.00000 q^{67} +2.68585i q^{69} -0.585462 q^{71} +6.00000i q^{73} -10.7434 q^{77} +1.02142 q^{79} +1.00000 q^{81} +13.3717 q^{83} -2.00000i q^{87} -3.37169 q^{89} +23.3288i q^{91} +6.97858 q^{93} -3.95715i q^{97} -2.29273i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 + 6 * q^9	\( 6 q + 6 q^{3} + 6 q^{9} + 6 q^{27} + 12 q^{31} - 8 q^{37} - 20 q^{41} - 8 q^{43} - 30 q^{49} - 12 q^{53} + 24 q^{67} + 8 q^{71} + 32 q^{77} + 36 q^{79} + 6 q^{81} + 32 q^{83} + 28 q^{89} + 12 q^{93}+O(q^{100}) \) 6 * q + 6 * q^3 + 6 * q^9 + 6 * q^27 + 12 * q^31 - 8 * q^37 - 20 * q^41 - 8 * q^43 - 30 * q^49 - 12 * q^53 + 24 * q^67 + 8 * q^71 + 32 * q^77 + 36 * q^79 + 6 * q^81 + 32 * q^83 + 28 * q^89 + 12 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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