LMFDB - Embedded newform 2400.2.d.g.49.6

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:	$8$
Coefficient field:	8.0.214798336.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 $ x^8 - 2x^7 - 2x^5 + 9x^4 - 4x^3 - 16*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.6
Root			$-1.08003 - 0.912978i$ of defining polynomial
Character	$\chi$	$=$	2400.49
Dual form			2400.2.d.g.49.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.33411i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.33411i q^{7} +1.00000 q^{9} +2.94418i q^{11} -2.04184 q^{13} -3.61241i q^{17} -5.35964i q^{19} -1.33411i q^{21} -8.59609i q^{23} -1.00000 q^{27} +5.26432i q^{29} +2.08134 q^{31} -2.94418i q^{33} -6.55659 q^{37} +2.04184 q^{39} +7.02786 q^{41} +8.50078 q^{43} +9.97204i q^{47} +5.22015 q^{49} +3.61241i q^{51} -6.12318 q^{53} +5.35964i q^{57} -4.75190i q^{59} +8.51476i q^{61} +1.33411i q^{63} +10.6961 q^{67} +8.59609i q^{69} +2.62405 q^{71} -15.3875i q^{73} -3.92787 q^{77} +10.4450 q^{79} +1.00000 q^{81} -1.52708 q^{83} -5.26432i q^{87} +12.7193 q^{89} -2.72404i q^{91} -2.08134 q^{93} +13.4450i q^{97} +2.94418i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 + 8 * q^9	$ 8 q - 8 q^{3} + 8 q^{9} - 8 q^{27} - 8 q^{31} - 8 q^{43} - 8 q^{53} + 24 q^{67} + 40 q^{71} + 24 q^{77} - 16 q^{79} + 8 q^{81} - 32 q^{83} + 8 q^{93}+O(q^{100}) $ 8 * q - 8 * q^3 + 8 * q^9 - 8 * q^27 - 8 * q^31 - 8 * q^43 - 8 * q^53 + 24 * q^67 + 40 * q^71 + 24 * q^77 - 16 * q^79 + 8 * q^81 - 32 * q^83 + 8 * 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
$7$	1.33411i	0.504247i	0.967695	+	0.252123i	$0.0811289\pi$
$7$	1.33411i	0.504247i	−0.967695	+	0.252123i	$0.918871\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.94418i	0.887705i	0.896100	+	0.443853i	$0.146389\pi$
$11$	2.94418i	0.887705i	−0.896100	+	0.443853i	$0.853611\pi$
$12$	0	0
$13$	−2.04184	−0.566304	−0.283152	−	0.959075i	$-0.591380\pi$
$13$	−2.04184	−0.566304	−0.283152	+	0.959075i	$0.591380\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.61241i	− 0.876138i	−0.898941	−	0.438069i	$-0.855663\pi$
$17$	− 3.61241i	− 0.876138i	0.898941	−	0.438069i	$-0.144337\pi$
$18$	0	0
$19$	− 5.35964i	− 1.22958i	−0.788689	−	0.614792i	$-0.789240\pi$
$19$	− 5.35964i	− 1.22958i	0.788689	−	0.614792i	$-0.210760\pi$
$20$	0	0
$21$	− 1.33411i	− 0.291127i
$22$	0	0
$23$	− 8.59609i	− 1.79241i	−0.443641	−	0.896205i	$-0.646313\pi$
$23$	− 8.59609i	− 1.79241i	0.443641	−	0.896205i	$-0.353687\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.26432i	0.977559i	0.872407	+	0.488780i	$0.162558\pi$
$29$	5.26432i	0.977559i	−0.872407	+	0.488780i	$0.837442\pi$
$30$	0	0
$31$	2.08134	0.373820	0.186910	−	0.982377i	$-0.440153\pi$
$31$	2.08134	0.373820	0.186910	+	0.982377i	$0.440153\pi$
$32$	0	0
$33$	− 2.94418i	− 0.512517i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.55659	−1.07790	−0.538949	−	0.842339i	$-0.681178\pi$
$37$	−6.55659	−1.07790	−0.538949	+	0.842339i	$0.681178\pi$
$38$	0	0
$39$	2.04184	0.326956
$40$	0	0
$41$	7.02786	1.09757	0.548784	−	0.835964i	$-0.315091\pi$
$41$	7.02786	1.09757	0.548784	+	0.835964i	$0.315091\pi$
$42$	0	0
$43$	8.50078	1.29636	0.648178	−	0.761489i	$-0.275531\pi$
$43$	8.50078	1.29636	0.648178	+	0.761489i	$0.275531\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.97204i	1.45457i	0.686334	+	0.727286i	$0.259219\pi$
$47$	9.97204i	1.45457i	−0.686334	+	0.727286i	$0.740781\pi$
$48$	0	0
$49$	5.22015	0.745735
$50$	0	0
$51$	3.61241i	0.505838i
$52$	0	0
$53$	−6.12318	−0.841083	−0.420541	−	0.907273i	$-0.638160\pi$
$53$	−6.12318	−0.841083	−0.420541	+	0.907273i	$0.638160\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.35964i	0.709901i
$58$	0	0
$59$	− 4.75190i	− 0.618644i	−0.950957	−	0.309322i	$-0.899898\pi$
$59$	− 4.75190i	− 0.618644i	0.950957	−	0.309322i	$-0.100102\pi$
$60$	0	0
$61$	8.51476i	1.09020i	0.838370	+	0.545101i	$0.183509\pi$
$61$	8.51476i	1.09020i	−0.838370	+	0.545101i	$0.816491\pi$
$62$	0	0
$63$	1.33411i	0.168082i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.6961	1.30673	0.653367	−	0.757041i	$-0.273356\pi$
$67$	10.6961	1.30673	0.653367	+	0.757041i	$0.273356\pi$
$68$	0	0
$69$	8.59609i	1.03485i
$70$	0	0
$71$	2.62405	0.311418	0.155709	−	0.987803i	$-0.450234\pi$
$71$	2.62405	0.311418	0.155709	+	0.987803i	$0.450234\pi$
$72$	0	0
$73$	− 15.3875i	− 1.80097i	−0.434887	−	0.900485i	$-0.643212\pi$
$73$	− 15.3875i	− 1.80097i	0.434887	−	0.900485i	$-0.356788\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.92787	−0.447622
$78$	0	0
$79$	10.4450	1.17515	0.587575	−	0.809170i	$-0.300083\pi$
$79$	10.4450	1.17515	0.587575	+	0.809170i	$0.300083\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.52708	−0.167619	−0.0838095	−	0.996482i	$-0.526709\pi$
$83$	−1.52708	−0.167619	−0.0838095	+	0.996482i	$0.526709\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 5.26432i	− 0.564394i
$88$	0	0
$89$	12.7193	1.34824	0.674120	−	0.738622i	$-0.264523\pi$
$89$	12.7193	1.34824	0.674120	+	0.738622i	$0.264523\pi$
$90$	0	0
$91$	− 2.72404i	− 0.285557i
$92$	0	0
$93$	−2.08134	−0.215825
$94$	0	0
$95$	0	0
$96$	0	0
$97$	13.4450i	1.36513i	0.730825	+	0.682565i	$0.239135\pi$
$97$	13.4450i	1.36513i	−0.730825	+	0.682565i	$0.760865\pi$
$98$	0	0
$99$	2.94418i	0.295902i
$100$	0	0
$101$	− 10.1232i	− 1.00729i	−0.863910	−	0.503647i	$-0.831991\pi$
$101$	− 10.1232i	− 1.00729i	0.863910	−	0.503647i	$-0.168009\pi$
$102$	0	0
$103$	− 10.7472i	− 1.05896i	−0.848324	−	0.529478i	$-0.822388\pi$
$103$	− 10.7472i	− 1.05896i	0.848324	−	0.529478i	$-0.177612\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.86518	−0.470335	−0.235167	−	0.971955i	$-0.575564\pi$
$107$	−4.86518	−0.470335	−0.235167	+	0.971955i	$0.575564\pi$
$108$	0	0
$109$	− 15.4573i	− 1.48054i	−0.672310	−	0.740270i	$-0.734698\pi$
$109$	− 15.4573i	− 1.48054i	0.672310	−	0.740270i	$-0.265302\pi$
$110$	0	0
$111$	6.55659	0.622324
$112$	0	0
$113$	− 9.88837i	− 0.930220i	−0.885253	−	0.465110i	$-0.846015\pi$
$113$	− 9.88837i	− 0.930220i	0.885253	−	0.465110i	$-0.153985\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.04184	−0.188768
$118$	0	0
$119$	4.81936	0.441790
$120$	0	0
$121$	2.33178	0.211980
$122$	0	0
$123$	−7.02786	−0.633681
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 8.75190i	− 0.776605i	−0.921532	−	0.388303i	$-0.873062\pi$
$127$	− 8.75190i	− 0.776605i	0.921532	−	0.388303i	$-0.126938\pi$
$128$	0	0
$129$	−8.50078	−0.748452
$130$	0	0
$131$	− 0.471266i	− 0.0411747i	−0.999788	−	0.0205874i	$-0.993446\pi$
$131$	− 0.471266i	− 0.0411747i	0.999788	−	0.0205874i	$-0.00655362\pi$
$132$	0	0
$133$	7.15035	0.620014
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1.30382i	− 0.111393i	−0.998448	−	0.0556964i	$-0.982262\pi$
$137$	− 1.30382i	− 0.111393i	0.998448	−	0.0556964i	$-0.0177379\pi$
$138$	0	0
$139$	− 8.74723i	− 0.741930i	−0.928647	−	0.370965i	$-0.879027\pi$
$139$	− 8.74723i	− 0.741930i	0.928647	−	0.370965i	$-0.120973\pi$
$140$	0	0
$141$	− 9.97204i	− 0.839798i
$142$	0	0
$143$	− 6.01155i	− 0.502711i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−5.22015	−0.430550
$148$	0	0
$149$	15.1411i	1.24041i	0.784439	+	0.620205i	$0.212951\pi$
$149$	15.1411i	1.24041i	−0.784439	+	0.620205i	$0.787049\pi$
$150$	0	0
$151$	23.2782	1.89435	0.947176	−	0.320713i	$-0.103922\pi$
$151$	23.2782	1.89435	0.947176	+	0.320713i	$0.103922\pi$
$152$	0	0
$153$	− 3.61241i	− 0.292046i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	21.8976	1.74762	0.873809	−	0.486270i	$-0.161643\pi$
$157$	21.8976	1.74762	0.873809	+	0.486270i	$0.161643\pi$
$158$	0	0
$159$	6.12318	0.485599
$160$	0	0
$161$	11.4682	0.903817
$162$	0	0
$163$	11.1643	0.874458	0.437229	−	0.899350i	$-0.355960\pi$
$163$	11.1643	0.874458	0.437229	+	0.899350i	$0.355960\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.0952i	0.781192i	0.920562	+	0.390596i	$0.127731\pi$
$167$	10.0952i	0.781192i	−0.920562	+	0.390596i	$0.872269\pi$
$168$	0	0
$169$	−8.83090	−0.679300
$170$	0	0
$171$	− 5.35964i	− 0.409862i
$172$	0	0
$173$	13.8162	1.05043	0.525215	−	0.850970i	$-0.323985\pi$
$173$	13.8162	1.05043	0.525215	+	0.850970i	$0.323985\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.75190i	0.357174i
$178$	0	0
$179$	21.9441i	1.64018i	0.572236	+	0.820089i	$0.306076\pi$
$179$	21.9441i	1.64018i	−0.572236	+	0.820089i	$0.693924\pi$
$180$	0	0
$181$	1.93021i	0.143471i	0.997424	+	0.0717356i	$0.0228538\pi$
$181$	1.93021i	0.143471i	−0.997424	+	0.0717356i	$0.977146\pi$
$182$	0	0
$183$	− 8.51476i	− 0.629429i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	10.6356	0.777752
$188$	0	0
$189$	− 1.33411i	− 0.0970423i
$190$	0	0
$191$	−12.1232	−0.877202	−0.438601	−	0.898682i	$-0.644526\pi$
$191$	−12.1232	−0.877202	−0.438601	+	0.898682i	$0.644526\pi$
$192$	0	0
$193$	− 1.27431i	− 0.0917267i	−0.998948	−	0.0458634i	$-0.985396\pi$
$193$	− 1.27431i	− 0.0917267i	0.998948	−	0.0458634i	$-0.0146039\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.30849	0.235720	0.117860	−	0.993030i	$-0.462397\pi$
$197$	3.30849	0.235720	0.117860	+	0.993030i	$0.462397\pi$
$198$	0	0
$199$	9.02718	0.639920	0.319960	−	0.947431i	$-0.396331\pi$
$199$	9.02718	0.639920	0.319960	+	0.947431i	$0.396331\pi$
$200$	0	0
$201$	−10.6961	−0.754443
$202$	0	0
$203$	−7.02319	−0.492931
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 8.59609i	− 0.597470i
$208$	0	0
$209$	15.7798	1.09151
$210$	0	0
$211$	6.61241i	0.455217i	0.973753	+	0.227608i	$0.0730906\pi$
$211$	6.61241i	0.455217i	−0.973753	+	0.227608i	$0.926909\pi$
$212$	0	0
$213$	−2.62405	−0.179797
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.77674i	0.188497i
$218$	0	0
$219$	15.3875i	1.03979i
$220$	0	0
$221$	7.37595i	0.496160i
$222$	0	0
$223$	− 0.833237i	− 0.0557976i	−0.999611	−	0.0278988i	$-0.991118\pi$
$223$	− 0.833237i	− 0.0557976i	0.999611	−	0.0278988i	$-0.00888162\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.9999	−0.730089	−0.365045	−	0.930990i	$-0.618946\pi$
$227$	−10.9999	−0.730089	−0.365045	+	0.930990i	$0.618946\pi$
$228$	0	0
$229$	− 15.2061i	− 1.00485i	−0.864622	−	0.502423i	$-0.832442\pi$
$229$	− 15.2061i	− 1.00485i	0.864622	−	0.502423i	$-0.167558\pi$
$230$	0	0
$231$	3.92787	0.258435
$232$	0	0
$233$	2.47594i	0.162204i	0.996706	+	0.0811020i	$0.0258439\pi$
$233$	2.47594i	0.162204i	−0.996706	+	0.0811020i	$0.974156\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−10.4450	−0.678473
$238$	0	0
$239$	−21.0737	−1.36314	−0.681572	−	0.731751i	$-0.738703\pi$
$239$	−21.0737	−1.36314	−0.681572	+	0.731751i	$0.738703\pi$
$240$	0	0
$241$	−6.10852	−0.393484	−0.196742	−	0.980455i	$-0.563036\pi$
$241$	−6.10852	−0.393484	−0.196742	+	0.980455i	$0.563036\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	10.9435i	0.696318i
$248$	0	0
$249$	1.52708	0.0967748
$250$	0	0
$251$	− 22.5286i	− 1.42199i	−0.703195	−	0.710997i	$-0.748244\pi$
$251$	− 22.5286i	− 1.42199i	0.703195	−	0.710997i	$-0.251756\pi$
$252$	0	0
$253$	25.3085	1.59113
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 14.5286i	− 0.906271i	−0.891442	−	0.453136i	$-0.850305\pi$
$257$	− 14.5286i	− 0.906271i	0.891442	−	0.453136i	$-0.149695\pi$
$258$	0	0
$259$	− 8.74723i	− 0.543526i
$260$	0	0
$261$	5.26432i	0.325853i
$262$	0	0
$263$	− 5.29694i	− 0.326624i	−0.986575	−	0.163312i	$-0.947782\pi$
$263$	− 5.29694i	− 0.326624i	0.986575	−	0.163312i	$-0.0522177\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−12.7193	−0.778407
$268$	0	0
$269$	− 27.0737i	− 1.65071i	−0.564613	−	0.825356i	$-0.690975\pi$
$269$	− 27.0737i	− 1.65071i	0.564613	−	0.825356i	$-0.309025\pi$
$270$	0	0
$271$	−15.8604	−0.963451	−0.481726	−	0.876322i	$-0.659990\pi$
$271$	−15.8604	−0.963451	−0.481726	+	0.876322i	$0.659990\pi$
$272$	0	0
$273$	2.72404i	0.164866i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	9.98592	0.599996	0.299998	−	0.953940i	$-0.403014\pi$
$277$	9.98592	0.599996	0.299998	+	0.953940i	$0.403014\pi$
$278$	0	0
$279$	2.08134	0.124607
$280$	0	0
$281$	13.4218	0.800676	0.400338	−	0.916368i	$-0.368893\pi$
$281$	13.4218	0.800676	0.400338	+	0.916368i	$0.368893\pi$
$282$	0	0
$283$	−3.83722	−0.228099	−0.114050	−	0.993475i	$-0.536382\pi$
$283$	−3.83722	−0.228099	−0.114050	+	0.993475i	$0.536382\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.37595i	0.553445i
$288$	0	0
$289$	3.95051	0.232383
$290$	0	0
$291$	− 13.4450i	− 0.788158i
$292$	0	0
$293$	−26.4450	−1.54493	−0.772466	−	0.635057i	$-0.780977\pi$
$293$	−26.4450	−1.54493	−0.772466	+	0.635057i	$0.780977\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2.94418i	− 0.170839i
$298$	0	0
$299$	17.5518i	1.01505i
$300$	0	0
$301$	11.3410i	0.653684i
$302$	0	0
$303$	10.1232i	0.581561i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.27596	0.0728230	0.0364115	−	0.999337i	$-0.488407\pi$
$307$	1.27596	0.0728230	0.0364115	+	0.999337i	$0.488407\pi$
$308$	0	0
$309$	10.7472i	0.611388i
$310$	0	0
$311$	−2.44496	−0.138641	−0.0693205	−	0.997594i	$-0.522083\pi$
$311$	−2.44496	−0.138641	−0.0693205	+	0.997594i	$0.522083\pi$
$312$	0	0
$313$	22.8325i	1.29057i	0.763943	+	0.645283i	$0.223261\pi$
$313$	22.8325i	1.29057i	−0.763943	+	0.645283i	$0.776739\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.11163	0.118601	0.0593005	−	0.998240i	$-0.481113\pi$
$317$	2.11163	0.118601	0.0593005	+	0.998240i	$0.481113\pi$
$318$	0	0
$319$	−15.4991	−0.867784
$320$	0	0
$321$	4.86518	0.271548
$322$	0	0
$323$	−19.3612	−1.07729
$324$	0	0
$325$	0	0
$326$	0	0
$327$	15.4573i	0.854790i
$328$	0	0
$329$	−13.3038	−0.733463
$330$	0	0
$331$	23.2248i	1.27655i	0.769808	+	0.638276i	$0.220352\pi$
$331$	23.2248i	1.27655i	−0.769808	+	0.638276i	$0.779648\pi$
$332$	0	0
$333$	−6.55659	−0.359299
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 12.8884i	− 0.702074i	−0.936362	−	0.351037i	$-0.885829\pi$
$337$	− 12.8884i	− 0.702074i	0.936362	−	0.351037i	$-0.114171\pi$
$338$	0	0
$339$	9.88837i	0.537063i
$340$	0	0
$341$	6.12785i	0.331841i
$342$	0	0
$343$	16.3030i	0.880281i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.79827	−0.364951	−0.182475	−	0.983210i	$-0.558411\pi$
$347$	−6.79827	−0.364951	−0.182475	+	0.983210i	$0.558411\pi$
$348$	0	0
$349$	− 34.6076i	− 1.85250i	−0.376904	−	0.926252i	$-0.623011\pi$
$349$	− 34.6076i	− 1.85250i	0.376904	−	0.926252i	$-0.376989\pi$
$350$	0	0
$351$	2.04184	0.108985
$352$	0	0
$353$	12.2433i	0.651647i	0.945431	+	0.325823i	$0.105642\pi$
$353$	12.2433i	0.651647i	−0.945431	+	0.325823i	$0.894358\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4.81936	−0.255067
$358$	0	0
$359$	−2.01622	−0.106412	−0.0532059	−	0.998584i	$-0.516944\pi$
$359$	−2.01622	−0.106412	−0.0532059	+	0.998584i	$0.516944\pi$
$360$	0	0
$361$	−9.72569	−0.511878
$362$	0	0
$363$	−2.33178	−0.122387
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 13.4131i	− 0.700159i	−0.936720	−	0.350079i	$-0.886155\pi$
$367$	− 13.4131i	− 0.700159i	0.936720	−	0.350079i	$-0.113845\pi$
$368$	0	0
$369$	7.02786	0.365856
$370$	0	0
$371$	− 8.16900i	− 0.424113i
$372$	0	0
$373$	10.0976	0.522832	0.261416	−	0.965226i	$-0.415811\pi$
$373$	10.0976	0.522832	0.261416	+	0.965226i	$0.415811\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 10.7489i	− 0.553595i
$378$	0	0
$379$	− 18.2775i	− 0.938853i	−0.882972	−	0.469426i	$-0.844461\pi$
$379$	− 18.2775i	− 0.938853i	0.882972	−	0.469426i	$-0.155539\pi$
$380$	0	0
$381$	8.75190i	0.448373i
$382$	0	0
$383$	11.7734i	0.601594i	0.953688	+	0.300797i	$0.0972527\pi$
$383$	11.7734i	0.601594i	−0.953688	+	0.300797i	$0.902747\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.50078	0.432119
$388$	0	0
$389$	33.4270i	1.69482i	0.530942	+	0.847408i	$0.321838\pi$
$389$	33.4270i	1.69482i	−0.530942	+	0.847408i	$0.678162\pi$
$390$	0	0
$391$	−31.0526	−1.57040
$392$	0	0
$393$	0.471266i	0.0237722i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	39.0434	1.95953	0.979766	−	0.200147i	$-0.0641420\pi$
$397$	39.0434	1.95953	0.979766	+	0.200147i	$0.0641420\pi$
$398$	0	0
$399$	−7.15035	−0.357965
$400$	0	0
$401$	−24.6140	−1.22916	−0.614581	−	0.788853i	$-0.710675\pi$
$401$	−24.6140	−1.22916	−0.614581	+	0.788853i	$0.710675\pi$
$402$	0	0
$403$	−4.24976	−0.211695
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 19.3038i	− 0.956855i
$408$	0	0
$409$	14.5024	0.717099	0.358550	−	0.933511i	$-0.383271\pi$
$409$	14.5024	0.717099	0.358550	+	0.933511i	$0.383271\pi$
$410$	0	0
$411$	1.30382i	0.0643127i
$412$	0	0
$413$	6.33956	0.311949
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8.74723i	0.428354i
$418$	0	0
$419$	− 12.6419i	− 0.617598i	−0.951127	−	0.308799i	$-0.900073\pi$
$419$	− 12.6419i	− 0.617598i	0.951127	−	0.308799i	$-0.0999271\pi$
$420$	0	0
$421$	− 16.8389i	− 0.820677i	−0.911933	−	0.410338i	$-0.865411\pi$
$421$	− 16.8389i	− 0.820677i	0.911933	−	0.410338i	$-0.134589\pi$
$422$	0	0
$423$	9.97204i	0.484857i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−11.3596	−0.549731
$428$	0	0
$429$	6.01155i	0.290240i
$430$	0	0
$431$	−5.98845	−0.288454	−0.144227	−	0.989545i	$-0.546070\pi$
$431$	−5.98845	−0.288454	−0.144227	+	0.989545i	$0.546070\pi$
$432$	0	0
$433$	− 2.22482i	− 0.106918i	−0.998570	−	0.0534589i	$-0.982975\pi$
$433$	− 2.22482i	− 0.106918i	0.998570	−	0.0534589i	$-0.0170246\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−46.0719	−2.20392
$438$	0	0
$439$	−2.30460	−0.109993	−0.0549963	−	0.998487i	$-0.517515\pi$
$439$	−2.30460	−0.109993	−0.0549963	+	0.998487i	$0.517515\pi$
$440$	0	0
$441$	5.22015	0.248578
$442$	0	0
$443$	22.1347	1.05165	0.525826	−	0.850592i	$-0.323756\pi$
$443$	22.1347	1.05165	0.525826	+	0.850592i	$0.323756\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 15.1411i	− 0.716151i
$448$	0	0
$449$	−21.5861	−1.01871	−0.509356	−	0.860556i	$-0.670116\pi$
$449$	−21.5861	−1.01871	−0.509356	+	0.860556i	$0.670116\pi$
$450$	0	0
$451$	20.6913i	0.974316i
$452$	0	0
$453$	−23.2782	−1.09371
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.50088i	0.116986i	0.998288	+	0.0584930i	$0.0186295\pi$
$457$	2.50088i	0.116986i	−0.998288	+	0.0584930i	$0.981370\pi$
$458$	0	0
$459$	3.61241i	0.168613i
$460$	0	0
$461$	− 2.59609i	− 0.120912i	−0.998171	−	0.0604561i	$-0.980744\pi$
$461$	− 2.59609i	− 0.120912i	0.998171	−	0.0604561i	$-0.0192555\pi$
$462$	0	0
$463$	27.8604i	1.29478i	0.762158	+	0.647392i	$0.224140\pi$
$463$	27.8604i	1.29478i	−0.762158	+	0.647392i	$0.775860\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5.75200	−0.266171	−0.133085	−	0.991105i	$-0.542488\pi$
$467$	−5.75200	−0.266171	−0.133085	+	0.991105i	$0.542488\pi$
$468$	0	0
$469$	14.2698i	0.658917i
$470$	0	0
$471$	−21.8976	−1.00899
$472$	0	0
$473$	25.0279i	1.15078i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−6.12318	−0.280361
$478$	0	0
$479$	−12.5473	−0.573299	−0.286649	−	0.958036i	$-0.592541\pi$
$479$	−12.5473	−0.573299	−0.286649	+	0.958036i	$0.592541\pi$
$480$	0	0
$481$	13.3875	0.610417
$482$	0	0
$483$	−11.4682	−0.521819
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 8.60530i	− 0.389944i	−0.980809	−	0.194972i	$-0.937538\pi$
$487$	− 8.60530i	− 0.389944i	0.980809	−	0.194972i	$-0.0624615\pi$
$488$	0	0
$489$	−11.1643	−0.504868
$490$	0	0
$491$	36.8866i	1.66467i	0.554273	+	0.832335i	$0.312996\pi$
$491$	36.8866i	1.66467i	−0.554273	+	0.832335i	$0.687004\pi$
$492$	0	0
$493$	19.0169	0.856477
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.50078i	0.157031i
$498$	0	0
$499$	− 36.2496i	− 1.62275i	−0.584524	−	0.811377i	$-0.698719\pi$
$499$	− 36.2496i	− 1.62275i	0.584524	−	0.811377i	$-0.301281\pi$
$500$	0	0
$501$	− 10.0952i	− 0.451021i
$502$	0	0
$503$	23.3527i	1.04124i	0.853787	+	0.520622i	$0.174300\pi$
$503$	23.3527i	1.04124i	−0.853787	+	0.520622i	$0.825700\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8.83090	0.392194
$508$	0	0
$509$	− 3.35506i	− 0.148711i	−0.997232	−	0.0743553i	$-0.976310\pi$
$509$	− 3.35506i	− 0.148711i	0.997232	−	0.0743553i	$-0.0236899\pi$
$510$	0	0
$511$	20.5286	0.908133
$512$	0	0
$513$	5.35964i	0.236634i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−29.3595	−1.29123
$518$	0	0
$519$	−13.8162	−0.606466
$520$	0	0
$521$	−33.6029	−1.47217	−0.736084	−	0.676890i	$-0.763327\pi$
$521$	−33.6029	−1.47217	−0.736084	+	0.676890i	$0.763327\pi$
$522$	0	0
$523$	0.965721	0.0422280	0.0211140	−	0.999777i	$-0.493279\pi$
$523$	0.965721	0.0422280	0.0211140	+	0.999777i	$0.493279\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 7.51865i	− 0.327517i
$528$	0	0
$529$	−50.8928	−2.21273
$530$	0	0
$531$	− 4.75190i	− 0.206215i
$532$	0	0
$533$	−14.3497	−0.621556
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 21.9441i	− 0.946957i
$538$	0	0
$539$	15.3691i	0.661993i
$540$	0	0
$541$	6.34877i	0.272955i	0.990643	+	0.136478i	$0.0435781\pi$
$541$	6.34877i	0.272955i	−0.990643	+	0.136478i	$0.956422\pi$
$542$	0	0
$543$	− 1.93021i	− 0.0828331i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.07433	0.0886921	0.0443460	−	0.999016i	$-0.485880\pi$
$547$	2.07433	0.0886921	0.0443460	+	0.999016i	$0.485880\pi$
$548$	0	0
$549$	8.51476i	0.363401i
$550$	0	0
$551$	28.2148	1.20199
$552$	0	0
$553$	13.9347i	0.592566i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	27.6931	1.17339	0.586696	−	0.809807i	$-0.300428\pi$
$557$	27.6931	1.17339	0.586696	+	0.809807i	$0.300428\pi$
$558$	0	0
$559$	−17.3572	−0.734131
$560$	0	0
$561$	−10.6356	−0.449035
$562$	0	0
$563$	3.80771	0.160476	0.0802380	−	0.996776i	$-0.474432\pi$
$563$	3.80771	0.160476	0.0802380	+	0.996776i	$0.474432\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.33411i	0.0560274i
$568$	0	0
$569$	−38.6371	−1.61975	−0.809875	−	0.586603i	$-0.800465\pi$
$569$	−38.6371	−1.61975	−0.809875	+	0.586603i	$0.800465\pi$
$570$	0	0
$571$	− 6.24976i	− 0.261544i	−0.991412	−	0.130772i	$-0.958254\pi$
$571$	− 6.24976i	− 0.261544i	0.991412	−	0.130772i	$-0.0417456\pi$
$572$	0	0
$573$	12.1232	0.506453
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.17377i	0.0904952i	0.998976	+	0.0452476i	$0.0144077\pi$
$577$	2.17377i	0.0904952i	−0.998976	+	0.0452476i	$0.985592\pi$
$578$	0	0
$579$	1.27431i	0.0529585i
$580$	0	0
$581$	− 2.03730i	− 0.0845213i
$582$	0	0
$583$	− 18.0278i	− 0.746634i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	34.1688	1.41030	0.705149	−	0.709059i	$-0.250880\pi$
$587$	34.1688	1.41030	0.705149	+	0.709059i	$0.250880\pi$
$588$	0	0
$589$	− 11.1552i	− 0.459643i
$590$	0	0
$591$	−3.30849	−0.136093
$592$	0	0
$593$	12.9952i	0.533650i	0.963745	+	0.266825i	$0.0859746\pi$
$593$	12.9952i	0.533650i	−0.963745	+	0.266825i	$0.914025\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−9.02718	−0.369458
$598$	0	0
$599$	47.2572	1.93088	0.965439	−	0.260628i	$-0.0839295\pi$
$599$	47.2572	1.93088	0.965439	+	0.260628i	$0.0839295\pi$
$600$	0	0
$601$	−23.5007	−0.958613	−0.479306	−	0.877648i	$-0.659112\pi$
$601$	−23.5007	−0.958613	−0.479306	+	0.877648i	$0.659112\pi$
$602$	0	0
$603$	10.6961	0.435578
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 0.218591i	− 0.00887233i	−0.999990	−	0.00443617i	$-0.998588\pi$
$607$	− 0.218591i	− 0.00887233i	0.999990	−	0.00443617i	$-0.00141208\pi$
$608$	0	0
$609$	7.02319	0.284594
$610$	0	0
$611$	− 20.3613i	− 0.823730i
$612$	0	0
$613$	−35.7488	−1.44388	−0.721940	−	0.691956i	$-0.756749\pi$
$613$	−35.7488	−1.44388	−0.721940	+	0.691956i	$0.756749\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	33.0836i	1.33189i	0.745999	+	0.665947i	$0.231972\pi$
$617$	33.0836i	1.33189i	−0.745999	+	0.665947i	$0.768028\pi$
$618$	0	0
$619$	− 25.1084i	− 1.00919i	−0.863355	−	0.504596i	$-0.831641\pi$
$619$	− 25.1084i	− 1.00919i	0.863355	−	0.504596i	$-0.168359\pi$
$620$	0	0
$621$	8.59609i	0.344949i
$622$	0	0
$623$	16.9689i	0.679846i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−15.7798	−0.630183
$628$	0	0
$629$	23.6851i	0.944386i
$630$	0	0
$631$	−23.2829	−0.926876	−0.463438	−	0.886129i	$-0.653384\pi$
$631$	−23.2829	−0.926876	−0.463438	+	0.886129i	$0.653384\pi$
$632$	0	0
$633$	− 6.61241i	− 0.262820i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−10.6587	−0.422313
$638$	0	0
$639$	2.62405	0.103806
$640$	0	0
$641$	−38.3021	−1.51284	−0.756420	−	0.654086i	$-0.773054\pi$
$641$	−38.3021	−1.51284	−0.756420	+	0.654086i	$0.773054\pi$
$642$	0	0
$643$	45.8045	1.80635	0.903177	−	0.429269i	$-0.141229\pi$
$643$	45.8045	1.80635	0.903177	+	0.429269i	$0.141229\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 48.1114i	− 1.89146i	−0.324960	−	0.945728i	$-0.605351\pi$
$647$	− 48.1114i	− 1.89146i	0.324960	−	0.945728i	$-0.394649\pi$
$648$	0	0
$649$	13.9905	0.549174
$650$	0	0
$651$	− 2.77674i	− 0.108829i
$652$	0	0
$653$	−38.3331	−1.50009	−0.750046	−	0.661386i	$-0.769969\pi$
$653$	−38.3331	−1.50009	−0.750046	+	0.661386i	$0.769969\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 15.3875i	− 0.600323i
$658$	0	0
$659$	− 5.03253i	− 0.196040i	−0.995184	−	0.0980198i	$-0.968749\pi$
$659$	− 5.03253i	− 0.196040i	0.995184	−	0.0980198i	$-0.0312508\pi$
$660$	0	0
$661$	− 17.4665i	− 0.679368i	−0.940540	−	0.339684i	$-0.889680\pi$
$661$	− 17.4665i	− 0.679368i	0.940540	−	0.339684i	$-0.110320\pi$
$662$	0	0
$663$	− 7.37595i	− 0.286458i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	45.2526	1.75219
$668$	0	0
$669$	0.833237i	0.0322148i
$670$	0	0
$671$	−25.0690	−0.967779
$672$	0	0
$673$	32.4448i	1.25065i	0.780363	+	0.625327i	$0.215034\pi$
$673$	32.4448i	1.25065i	−0.780363	+	0.625327i	$0.784966\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.07213	−0.310237	−0.155119	−	0.987896i	$-0.549576\pi$
$677$	−8.07213	−0.310237	−0.155119	+	0.987896i	$0.549576\pi$
$678$	0	0
$679$	−17.9371	−0.688362
$680$	0	0
$681$	10.9999	0.421517
$682$	0	0
$683$	−36.3380	−1.39043	−0.695217	−	0.718799i	$-0.744692\pi$
$683$	−36.3380	−1.39043	−0.695217	+	0.718799i	$0.744692\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	15.2061i	0.580148i
$688$	0	0
$689$	12.5025	0.476308
$690$	0	0
$691$	− 15.0016i	− 0.570686i	−0.958425	−	0.285343i	$-0.907892\pi$
$691$	− 15.0016i	− 0.570686i	0.958425	−	0.285343i	$-0.0921075\pi$
$692$	0	0
$693$	−3.92787	−0.149207
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 25.3875i	− 0.961620i
$698$	0	0
$699$	− 2.47594i	− 0.0936485i
$700$	0	0
$701$	− 13.2874i	− 0.501859i	−0.968005	−	0.250929i	$-0.919264\pi$
$701$	− 13.2874i	− 0.501859i	0.968005	−	0.250929i	$-0.0807362\pi$
$702$	0	0
$703$	35.1409i	1.32537i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.5054	0.507925
$708$	0	0
$709$	37.8976i	1.42327i	0.702548	+	0.711637i	$0.252046\pi$
$709$	37.8976i	1.42327i	−0.702548	+	0.711637i	$0.747954\pi$
$710$	0	0
$711$	10.4450	0.391717
$712$	0	0
$713$	− 17.8914i	− 0.670038i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	21.0737	0.787011
$718$	0	0
$719$	−4.17909	−0.155854	−0.0779269	−	0.996959i	$-0.524830\pi$
$719$	−4.17909	−0.155854	−0.0779269	+	0.996959i	$0.524830\pi$
$720$	0	0
$721$	14.3380	0.533975
$722$	0	0
$723$	6.10852	0.227178
$724$	0	0
$725$	0	0
$726$	0	0
$727$	26.7727i	0.992943i	0.868053	+	0.496471i	$0.165371\pi$
$727$	26.7727i	0.992943i	−0.868053	+	0.496471i	$0.834629\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	− 30.7083i	− 1.13579i
$732$	0	0
$733$	21.3364	0.788080	0.394040	−	0.919093i	$-0.371077\pi$
$733$	21.3364	0.788080	0.394040	+	0.919093i	$0.371077\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	31.4912i	1.15999i
$738$	0	0
$739$	13.1038i	0.482033i	0.970521	+	0.241016i	$0.0774807\pi$
$739$	13.1038i	0.482033i	−0.970521	+	0.241016i	$0.922519\pi$
$740$	0	0
$741$	− 10.9435i	− 0.402020i
$742$	0	0
$743$	33.3595i	1.22384i	0.790919	+	0.611921i	$0.209603\pi$
$743$	33.3595i	1.22384i	−0.790919	+	0.611921i	$0.790397\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1.52708	−0.0558730
$748$	0	0
$749$	− 6.49069i	− 0.237165i
$750$	0	0
$751$	1.92100	0.0700981	0.0350491	−	0.999386i	$-0.488841\pi$
$751$	1.92100	0.0700981	0.0350491	+	0.999386i	$0.488841\pi$
$752$	0	0
$753$	22.5286i	0.820989i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−13.9908	−0.508504	−0.254252	−	0.967138i	$-0.581829\pi$
$757$	−13.9908	−0.508504	−0.254252	+	0.967138i	$0.581829\pi$
$758$	0	0
$759$	−25.3085	−0.918640
$760$	0	0
$761$	25.6618	0.930240	0.465120	−	0.885248i	$-0.346011\pi$
$761$	25.6618	0.930240	0.465120	+	0.885248i	$0.346011\pi$
$762$	0	0
$763$	20.6217	0.746557
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.70260i	0.350341i
$768$	0	0
$769$	12.3922	0.446873	0.223436	−	0.974719i	$-0.428272\pi$
$769$	12.3922	0.446873	0.223436	+	0.974719i	$0.428272\pi$
$770$	0	0
$771$	14.5286i	0.523236i
$772$	0	0
$773$	38.4843	1.38418	0.692091	−	0.721810i	$-0.256689\pi$
$773$	38.4843	1.38418	0.692091	+	0.721810i	$0.256689\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.74723i	0.313805i
$778$	0	0
$779$	− 37.6668i	− 1.34955i
$780$	0	0
$781$	7.72569i	0.276447i
$782$	0	0
$783$	− 5.26432i	− 0.188131i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0.389147	0.0138716	0.00693579	−	0.999976i	$-0.497792\pi$
$787$	0.389147	0.0138716	0.00693579	+	0.999976i	$0.497792\pi$
$788$	0	0
$789$	5.29694i	0.188576i
$790$	0	0
$791$	13.1922	0.469060
$792$	0	0
$793$	− 17.3857i	− 0.617386i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0.854188	0.0302569	0.0151284	−	0.999886i	$-0.495184\pi$
$797$	0.854188	0.0302569	0.0151284	+	0.999886i	$0.495184\pi$
$798$	0	0
$799$	36.0231	1.27441
$800$	0	0
$801$	12.7193	0.449413
$802$	0	0
$803$	45.3036	1.59873
$804$	0	0
$805$	0	0
$806$	0	0
$807$	27.0737i	0.953039i
$808$	0	0
$809$	10.4107	0.366020	0.183010	−	0.983111i	$-0.441416\pi$
$809$	10.4107	0.366020	0.183010	+	0.983111i	$0.441416\pi$
$810$	0	0
$811$	− 6.08825i	− 0.213787i	−0.994270	−	0.106894i	$-0.965910\pi$
$811$	− 6.08825i	− 0.213787i	0.994270	−	0.106894i	$-0.0340904\pi$
$812$	0	0
$813$	15.8604	0.556249
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 45.5611i	− 1.59398i
$818$	0	0
$819$	− 2.72404i	− 0.0951856i
$820$	0	0
$821$	− 35.3908i	− 1.23515i	−0.786513	−	0.617574i	$-0.788116\pi$
$821$	− 35.3908i	− 1.23515i	0.786513	−	0.617574i	$-0.211884\pi$
$822$	0	0
$823$	− 16.2846i	− 0.567646i	−0.958877	−	0.283823i	$-0.908397\pi$
$823$	− 16.2846i	− 0.567646i	0.958877	−	0.283823i	$-0.0916028\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−32.1362	−1.11748	−0.558742	−	0.829341i	$-0.688716\pi$
$827$	−32.1362	−1.11748	−0.558742	+	0.829341i	$0.688716\pi$
$828$	0	0
$829$	− 22.4682i	− 0.780355i	−0.920740	−	0.390177i	$-0.872414\pi$
$829$	− 22.4682i	− 0.780355i	0.920740	−	0.390177i	$-0.127586\pi$
$830$	0	0
$831$	−9.98592	−0.346408
$832$	0	0
$833$	− 18.8573i	− 0.653367i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.08134	−0.0719416
$838$	0	0
$839$	−16.1358	−0.557070	−0.278535	−	0.960426i	$-0.589849\pi$
$839$	−16.1358	−0.557070	−0.278535	+	0.960426i	$0.589849\pi$
$840$	0	0
$841$	1.28695	0.0443777
$842$	0	0
$843$	−13.4218	−0.462270
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.11085i	0.106890i
$848$	0	0
$849$	3.83722	0.131693
$850$	0	0
$851$	56.3611i	1.93203i
$852$	0	0
$853$	−44.6262	−1.52797	−0.763986	−	0.645233i	$-0.776760\pi$
$853$	−44.6262	−1.52797	−0.763986	+	0.645233i	$0.776760\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 4.52553i	− 0.154589i	−0.997008	−	0.0772945i	$-0.975372\pi$
$857$	− 4.52553i	− 0.154589i	0.997008	−	0.0772945i	$-0.0246282\pi$
$858$	0	0
$859$	42.7783i	1.45958i	0.683673	+	0.729788i	$0.260381\pi$
$859$	42.7783i	1.45958i	−0.683673	+	0.729788i	$0.739619\pi$
$860$	0	0
$861$	− 9.37595i	− 0.319531i
$862$	0	0
$863$	− 23.7734i	− 0.809257i	−0.914481	−	0.404629i	$-0.867401\pi$
$863$	− 23.7734i	− 0.809257i	0.914481	−	0.404629i	$-0.132599\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−3.95051	−0.134166
$868$	0	0
$869$	30.7519i	1.04319i
$870$	0	0
$871$	−21.8397	−0.740009
$872$	0	0
$873$	13.4450i	0.455043i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13.1470	0.443944	0.221972	−	0.975053i	$-0.428751\pi$
$877$	13.1470	0.443944	0.221972	+	0.975053i	$0.428751\pi$
$878$	0	0
$879$	26.4450	0.891966
$880$	0	0
$881$	38.9132	1.31102	0.655510	−	0.755187i	$-0.272454\pi$
$881$	38.9132	1.31102	0.655510	+	0.755187i	$0.272454\pi$
$882$	0	0
$883$	−44.5843	−1.50038	−0.750190	−	0.661223i	$-0.770038\pi$
$883$	−44.5843	−1.50038	−0.750190	+	0.661223i	$0.770038\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 32.3240i	− 1.08533i	−0.839948	−	0.542667i	$-0.817415\pi$
$887$	− 32.3240i	− 1.08533i	0.839948	−	0.542667i	$-0.182585\pi$
$888$	0	0
$889$	11.6760	0.391601
$890$	0	0
$891$	2.94418i	0.0986339i
$892$	0	0
$893$	53.4465	1.78852
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 17.5518i	− 0.586038i
$898$	0	0
$899$	10.9568i	0.365431i
$900$	0	0
$901$	22.1194i	0.736904i
$902$	0	0
$903$	− 11.3410i	− 0.377404i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−14.8309	−0.492452	−0.246226	−	0.969212i	$-0.579191\pi$
$907$	−14.8309	−0.492452	−0.246226	+	0.969212i	$0.579191\pi$
$908$	0	0
$909$	− 10.1232i	− 0.335765i
$910$	0	0
$911$	−11.6108	−0.384681	−0.192341	−	0.981328i	$-0.561608\pi$
$911$	−11.6108	−0.384681	−0.192341	+	0.981328i	$0.561608\pi$
$912$	0	0
$913$	− 4.49601i	− 0.148796i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.628722	0.0207622
$918$	0	0
$919$	−58.2518	−1.92155	−0.960775	−	0.277330i	$-0.910550\pi$
$919$	−58.2518	−1.92155	−0.960775	+	0.277330i	$0.910550\pi$
$920$	0	0
$921$	−1.27596	−0.0420444
$922$	0	0
$923$	−5.35789	−0.176357
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 10.7472i	− 0.352985i
$928$	0	0
$929$	−18.4433	−0.605105	−0.302553	−	0.953133i	$-0.597839\pi$
$929$	−18.4433	−0.605105	−0.302553	+	0.953133i	$0.597839\pi$
$930$	0	0
$931$	− 27.9781i	− 0.916944i
$932$	0	0
$933$	2.44496	0.0800445
$934$	0	0
$935$	0	0
$936$	0	0
$937$	16.1005i	0.525982i	0.964798	+	0.262991i	$0.0847089\pi$
$937$	16.1005i	0.525982i	−0.964798	+	0.262991i	$0.915291\pi$
$938$	0	0
$939$	− 22.8325i	− 0.745109i
$940$	0	0
$941$	32.0974i	1.04635i	0.852226	+	0.523173i	$0.175252\pi$
$941$	32.0974i	1.04635i	−0.852226	+	0.523173i	$0.824748\pi$
$942$	0	0
$943$	− 60.4121i	− 1.96729i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.10998	0.133556	0.0667782	−	0.997768i	$-0.478728\pi$
$947$	4.10998	0.133556	0.0667782	+	0.997768i	$0.478728\pi$
$948$	0	0
$949$	31.4188i	1.01990i
$950$	0	0
$951$	−2.11163	−0.0684743
$952$	0	0
$953$	− 31.7208i	− 1.02754i	−0.857928	−	0.513769i	$-0.828249\pi$
$953$	− 31.7208i	− 1.02754i	0.857928	−	0.513769i	$-0.171751\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	15.4991	0.501016
$958$	0	0
$959$	1.73944	0.0561695
$960$	0	0
$961$	−26.6680	−0.860259
$962$	0	0
$963$	−4.86518	−0.156778
$964$	0	0
$965$	0	0
$966$	0	0
$967$	26.9936i	0.868055i	0.900900	+	0.434027i	$0.142908\pi$
$967$	26.9936i	0.868055i	−0.900900	+	0.434027i	$0.857092\pi$
$968$	0	0
$969$	19.3612	0.621971
$970$	0	0
$971$	14.0559i	0.451076i	0.974234	+	0.225538i	$0.0724139\pi$
$971$	14.0559i	0.451076i	−0.974234	+	0.225538i	$0.927586\pi$
$972$	0	0
$973$	11.6698	0.374116
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.14251i	0.0365520i	0.999833	+	0.0182760i	$0.00581776\pi$
$977$	1.14251i	0.0365520i	−0.999833	+	0.0182760i	$0.994182\pi$
$978$	0	0
$979$	37.4479i	1.19684i
$980$	0	0
$981$	− 15.4573i	− 0.493513i
$982$	0	0
$983$	6.41720i	0.204677i	0.994750	+	0.102338i	$0.0326325\pi$
$983$	6.41720i	0.204677i	−0.994750	+	0.102338i	$0.967368\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	13.3038	0.423465
$988$	0	0
$989$	− 73.0735i	− 2.32360i
$990$	0	0
$991$	7.39470	0.234900	0.117450	−	0.993079i	$-0.462528\pi$
$991$	7.39470	0.234900	0.117450	+	0.993079i	$0.462528\pi$
$992$	0	0
$993$	− 23.2248i	− 0.737017i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	31.6649	1.00284	0.501419	−	0.865205i	$-0.332812\pi$
$997$	31.6649	1.00284	0.501419	+	0.865205i	$0.332812\pi$
$998$	0	0
$999$	6.55659	0.207441

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.d.g.49.6		8
3.2	odd	2		7200.2.d.s.2449.6		8
4.3	odd	2		600.2.d.h.349.7		8
5.2	odd	4		2400.2.k.d.1201.6		8
5.3	odd	4		2400.2.k.e.1201.3		8
5.4	even	2		2400.2.d.h.49.3		8
8.3	odd	2		600.2.d.g.349.1		8
8.5	even	2		2400.2.d.h.49.6		8
12.11	even	2		1800.2.d.s.1549.2		8
15.2	even	4		7200.2.k.r.3601.3		8
15.8	even	4		7200.2.k.s.3601.5		8
15.14	odd	2		7200.2.d.t.2449.3		8
20.3	even	4		600.2.k.e.301.3	yes	8
20.7	even	4		600.2.k.d.301.6	yes	8
20.19	odd	2		600.2.d.g.349.2		8
24.5	odd	2		7200.2.d.t.2449.6		8
24.11	even	2		1800.2.d.t.1549.8		8
40.3	even	4		600.2.k.e.301.4	yes	8
40.13	odd	4		2400.2.k.e.1201.7		8
40.19	odd	2		600.2.d.h.349.8		8
40.27	even	4		600.2.k.d.301.5	✓	8
40.29	even	2	inner	2400.2.d.g.49.3		8
40.37	odd	4		2400.2.k.d.1201.2		8
60.23	odd	4		1800.2.k.q.901.6		8
60.47	odd	4		1800.2.k.t.901.3		8
60.59	even	2		1800.2.d.t.1549.7		8
120.29	odd	2		7200.2.d.s.2449.3		8
120.53	even	4		7200.2.k.s.3601.6		8
120.59	even	2		1800.2.d.s.1549.1		8
120.77	even	4		7200.2.k.r.3601.4		8
120.83	odd	4		1800.2.k.q.901.5		8
120.107	odd	4		1800.2.k.t.901.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.2.d.g.349.1		8	8.3	odd	2
600.2.d.g.349.2		8	20.19	odd	2
600.2.d.h.349.7		8	4.3	odd	2
600.2.d.h.349.8		8	40.19	odd	2
600.2.k.d.301.5	✓	8	40.27	even	4
600.2.k.d.301.6	yes	8	20.7	even	4
600.2.k.e.301.3	yes	8	20.3	even	4
600.2.k.e.301.4	yes	8	40.3	even	4
1800.2.d.s.1549.1		8	120.59	even	2
1800.2.d.s.1549.2		8	12.11	even	2
1800.2.d.t.1549.7		8	60.59	even	2
1800.2.d.t.1549.8		8	24.11	even	2
1800.2.k.q.901.5		8	120.83	odd	4
1800.2.k.q.901.6		8	60.23	odd	4
1800.2.k.t.901.3		8	60.47	odd	4
1800.2.k.t.901.4		8	120.107	odd	4
2400.2.d.g.49.3		8	40.29	even	2	inner
2400.2.d.g.49.6		8	1.1	even	1	trivial
2400.2.d.h.49.3		8	5.4	even	2
2400.2.d.h.49.6		8	8.5	even	2
2400.2.k.d.1201.2		8	40.37	odd	4
2400.2.k.d.1201.6		8	5.2	odd	4
2400.2.k.e.1201.3		8	5.3	odd	4
2400.2.k.e.1201.7		8	40.13	odd	4
7200.2.d.s.2449.3		8	120.29	odd	2
7200.2.d.s.2449.6		8	3.2	odd	2
7200.2.d.t.2449.3		8	15.14	odd	2
7200.2.d.t.2449.6		8	24.5	odd	2
7200.2.k.r.3601.3		8	15.2	even	4
7200.2.k.r.3601.4		8	120.77	even	4
7200.2.k.s.3601.5		8	15.8	even	4
7200.2.k.s.3601.6		8	120.53	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.33411i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.33411i q^{7} +1.00000 q^{9} +2.94418i q^{11} -2.04184 q^{13} -3.61241i q^{17} -5.35964i q^{19} -1.33411i q^{21} -8.59609i q^{23} -1.00000 q^{27} +5.26432i q^{29} +2.08134 q^{31} -2.94418i q^{33} -6.55659 q^{37} +2.04184 q^{39} +7.02786 q^{41} +8.50078 q^{43} +9.97204i q^{47} +5.22015 q^{49} +3.61241i q^{51} -6.12318 q^{53} +5.35964i q^{57} -4.75190i q^{59} +8.51476i q^{61} +1.33411i q^{63} +10.6961 q^{67} +8.59609i q^{69} +2.62405 q^{71} -15.3875i q^{73} -3.92787 q^{77} +10.4450 q^{79} +1.00000 q^{81} -1.52708 q^{83} -5.26432i q^{87} +12.7193 q^{89} -2.72404i q^{91} -2.08134 q^{93} +13.4450i q^{97} +2.94418i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 + 8 * q^9	\( 8 q - 8 q^{3} + 8 q^{9} - 8 q^{27} - 8 q^{31} - 8 q^{43} - 8 q^{53} + 24 q^{67} + 40 q^{71} + 24 q^{77} - 16 q^{79} + 8 q^{81} - 32 q^{83} + 8 q^{93}+O(q^{100}) \) 8 * q - 8 * q^3 + 8 * q^9 - 8 * q^27 - 8 * q^31 - 8 * q^43 - 8 * q^53 + 24 * q^67 + 40 * q^71 + 24 * q^77 - 16 * q^79 + 8 * q^81 - 32 * q^83 + 8 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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