LMFDB - Embedded newform 2400.2.k.e.1201.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,2,Mod(1201,2400)]

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, 0]))

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

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

chi := DirichletCharacter("2400.1201");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.k (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:	$\Z[a_1, \ldots, a_{11}]$
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			1201.2
Root			$1.23291 + 0.692769i$ of defining polynomial
Character	$\chi$	$=$	2400.1201
Dual form			2400.2.k.e.1201.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -0.0802864 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -0.0802864 q^{7} -1.00000 q^{9} +2.41649i q^{11} +5.26785i q^{13} -0.255918 q^{17} -6.95864i q^{19} +0.0802864i q^{21} +1.64542 q^{23} +1.00000i q^{27} +4.51516i q^{29} -8.29484 q^{31} +2.41649 q^{33} +2.67241i q^{37} +5.26785 q^{39} -8.11921 q^{41} +4.08890i q^{43} -5.70272 q^{47} -6.99355 q^{49} +0.255918i q^{51} +11.5627i q^{53} -6.95864 q^{57} -12.6963i q^{59} +11.9403i q^{61} +0.0802864 q^{63} +7.27979i q^{67} -1.64542i q^{69} +11.3481 q^{71} -12.0779 q^{73} -0.194011i q^{77} -5.50539 q^{79} +1.00000 q^{81} +9.20811i q^{83} +4.51516 q^{87} +11.9173 q^{89} -0.422937i q^{91} +8.29484i q^{93} +8.50539 q^{97} -2.41649i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{7} - 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^7 - 8 * q^9	$ 8 q + 8 q^{7} - 8 q^{9} + 8 q^{23} - 8 q^{31} - 8 q^{57} - 8 q^{63} + 40 q^{71} + 16 q^{73} + 16 q^{79} + 8 q^{81} + 24 q^{87} + 8 q^{97}+O(q^{100}) $ 8 * q + 8 * q^7 - 8 * q^9 + 8 * q^23 - 8 * q^31 - 8 * q^57 - 8 * q^63 + 40 * q^71 + 16 * q^73 + 16 * q^79 + 8 * q^81 + 24 * q^87 + 8 * q^97

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.0802864	−0.0303454	−0.0151727	−	0.999885i	$-0.504830\pi$
$7$	−0.0802864	−0.0303454	−0.0151727	+	0.999885i	$0.504830\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.41649i	0.728599i	0.931282	+	0.364300i	$0.118692\pi$
$11$	2.41649i	0.728599i	−0.931282	+	0.364300i	$0.881308\pi$
$12$	0	0
$13$	5.26785i	1.46104i	0.682892	+	0.730520i	$0.260722\pi$
$13$	5.26785i	1.46104i	−0.682892	+	0.730520i	$0.739278\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.255918	−0.0620692	−0.0310346	−	0.999518i	$-0.509880\pi$
$17$	−0.255918	−0.0620692	−0.0310346	+	0.999518i	$0.509880\pi$
$18$	0	0
$19$	− 6.95864i	− 1.59642i	−0.602378	−	0.798211i	$-0.705780\pi$
$19$	− 6.95864i	− 1.59642i	0.602378	−	0.798211i	$-0.294220\pi$
$20$	0	0
$21$	0.0802864i	0.0175199i
$22$	0	0
$23$	1.64542	0.343093	0.171546	−	0.985176i	$-0.445124\pi$
$23$	1.64542	0.343093	0.171546	+	0.985176i	$0.445124\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	4.51516i	0.838444i	0.907884	+	0.419222i	$0.137697\pi$
$29$	4.51516i	0.838444i	−0.907884	+	0.419222i	$0.862303\pi$
$30$	0	0
$31$	−8.29484	−1.48980	−0.744899	−	0.667177i	$-0.767502\pi$
$31$	−8.29484	−1.48980	−0.744899	+	0.667177i	$0.767502\pi$
$32$	0	0
$33$	2.41649	0.420657
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.67241i	0.439341i	0.975574	+	0.219671i	$0.0704983\pi$
$37$	2.67241i	0.439341i	−0.975574	+	0.219671i	$0.929502\pi$
$38$	0	0
$39$	5.26785	0.843531
$40$	0	0
$41$	−8.11921	−1.26801	−0.634004	−	0.773330i	$-0.718590\pi$
$41$	−8.11921	−1.26801	−0.634004	+	0.773330i	$0.718590\pi$
$42$	0	0
$43$	4.08890i	0.623551i	0.950156	+	0.311776i	$0.100924\pi$
$43$	4.08890i	0.623551i	−0.950156	+	0.311776i	$0.899076\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.70272	−0.831827	−0.415914	−	0.909404i	$-0.636538\pi$
$47$	−5.70272	−0.831827	−0.415914	+	0.909404i	$0.636538\pi$
$48$	0	0
$49$	−6.99355	−0.999079
$50$	0	0
$51$	0.255918i	0.0358357i
$52$	0	0
$53$	11.5627i	1.58826i	0.607749	+	0.794129i	$0.292073\pi$
$53$	11.5627i	1.58826i	−0.607749	+	0.794129i	$0.707927\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−6.95864	−0.921694
$58$	0	0
$59$	− 12.6963i	− 1.65291i	−0.563000	−	0.826457i	$-0.690353\pi$
$59$	− 12.6963i	− 1.65291i	0.563000	−	0.826457i	$-0.309647\pi$
$60$	0	0
$61$	11.9403i	1.52879i	0.644746	+	0.764397i	$0.276963\pi$
$61$	11.9403i	1.52879i	−0.644746	+	0.764397i	$0.723037\pi$
$62$	0	0
$63$	0.0802864	0.0101151
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.27979i	0.889367i	0.895688	+	0.444684i	$0.146684\pi$
$67$	7.27979i	0.889367i	−0.895688	+	0.444684i	$0.853316\pi$
$68$	0	0
$69$	− 1.64542i	− 0.198085i
$70$	0	0
$71$	11.3481	1.34678	0.673388	−	0.739289i	$-0.264838\pi$
$71$	11.3481	1.34678	0.673388	+	0.739289i	$0.264838\pi$
$72$	0	0
$73$	−12.0779	−1.41361	−0.706803	−	0.707411i	$-0.749863\pi$
$73$	−12.0779	−1.41361	−0.706803	+	0.707411i	$0.749863\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.194011i	− 0.0221096i
$78$	0	0
$79$	−5.50539	−0.619405	−0.309702	−	0.950834i	$-0.600229\pi$
$79$	−5.50539	−0.619405	−0.309702	+	0.950834i	$0.600229\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.20811i	1.01072i	0.862908	+	0.505361i	$0.168641\pi$
$83$	9.20811i	1.01072i	−0.862908	+	0.505361i	$0.831359\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.51516	0.484076
$88$	0	0
$89$	11.9173	1.26323	0.631615	−	0.775283i	$-0.282393\pi$
$89$	11.9173	1.26323	0.631615	+	0.775283i	$0.282393\pi$
$90$	0	0
$91$	− 0.422937i	− 0.0443358i
$92$	0	0
$93$	8.29484i	0.860135i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.50539	0.863592	0.431796	−	0.901971i	$-0.357880\pi$
$97$	8.50539	0.863592	0.431796	+	0.901971i	$0.357880\pi$
$98$	0	0
$99$	− 2.41649i	− 0.242866i
$100$	0	0
$101$	7.56270i	0.752516i	0.926515	+	0.376258i	$0.122789\pi$
$101$	7.56270i	0.752516i	−0.926515	+	0.376258i	$0.877211\pi$
$102$	0	0
$103$	1.78544	0.175925	0.0879624	−	0.996124i	$-0.471964\pi$
$103$	1.78544	0.175925	0.0879624	+	0.996124i	$0.471964\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.4705i	1.01222i	0.862469	+	0.506110i	$0.168917\pi$
$107$	10.4705i	1.01222i	−0.862469	+	0.506110i	$0.831083\pi$
$108$	0	0
$109$	− 3.64298i	− 0.348934i	−0.984663	−	0.174467i	$-0.944180\pi$
$109$	− 3.64298i	− 0.348934i	0.984663	−	0.174467i	$-0.0558203\pi$
$110$	0	0
$111$	2.67241	0.253654
$112$	0	0
$113$	8.83298	0.830937	0.415468	−	0.909608i	$-0.363618\pi$
$113$	8.83298	0.830937	0.415468	+	0.909608i	$0.363618\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 5.26785i	− 0.487013i
$118$	0	0
$119$	0.0205467	0.00188351
$120$	0	0
$121$	5.16057	0.469143
$122$	0	0
$123$	8.11921i	0.732085i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	8.69628	0.771670	0.385835	−	0.922568i	$-0.373913\pi$
$127$	8.69628	0.771670	0.385835	+	0.922568i	$0.373913\pi$
$128$	0	0
$129$	4.08890	0.360008
$130$	0	0
$131$	10.7916i	0.942868i	0.881901	+	0.471434i	$0.156264\pi$
$131$	10.7916i	0.942868i	−0.881901	+	0.471434i	$0.843736\pi$
$132$	0	0
$133$	0.558684i	0.0484440i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.5421	0.986112	0.493056	−	0.869997i	$-0.335880\pi$
$137$	11.5421	0.986112	0.493056	+	0.869997i	$0.335880\pi$
$138$	0	0
$139$	− 0.214558i	− 0.0181986i	−0.999959	−	0.00909928i	$-0.997104\pi$
$139$	− 0.214558i	− 0.0181986i	0.999959	−	0.00909928i	$-0.00289643\pi$
$140$	0	0
$141$	5.70272i	0.480256i
$142$	0	0
$143$	−12.7297	−1.06451
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.99355i	0.576819i
$148$	0	0
$149$	− 23.0475i	− 1.88813i	−0.329762	−	0.944064i	$-0.606969\pi$
$149$	− 23.0475i	− 1.88813i	0.329762	−	0.944064i	$-0.393031\pi$
$150$	0	0
$151$	−9.48573	−0.771938	−0.385969	−	0.922512i	$-0.626133\pi$
$151$	−9.48573	−0.771938	−0.385969	+	0.922512i	$0.626133\pi$
$152$	0	0
$153$	0.255918	0.0206897
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 6.34413i	− 0.506316i	−0.967425	−	0.253158i	$-0.918531\pi$
$157$	− 6.34413i	− 0.506316i	0.967425	−	0.253158i	$-0.0814693\pi$
$158$	0	0
$159$	11.5627	0.916981
$160$	0	0
$161$	−0.132104	−0.0104113
$162$	0	0
$163$	12.4100i	0.972030i	0.873951	+	0.486015i	$0.161550\pi$
$163$	12.4100i	0.972030i	−0.873951	+	0.486015i	$0.838450\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−23.2654	−1.80033	−0.900166	−	0.435547i	$-0.856555\pi$
$167$	−23.2654	−1.80033	−0.900166	+	0.435547i	$0.856555\pi$
$168$	0	0
$169$	−14.7503	−1.13464
$170$	0	0
$171$	6.95864i	0.532140i
$172$	0	0
$173$	8.63897i	0.656809i	0.944537	+	0.328404i	$0.106511\pi$
$173$	8.63897i	0.656809i	−0.944537	+	0.328404i	$0.893489\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−12.6963	−0.954311
$178$	0	0
$179$	9.40544i	0.702996i	0.936189	+	0.351498i	$0.114328\pi$
$179$	9.40544i	0.702996i	−0.936189	+	0.351498i	$0.885672\pi$
$180$	0	0
$181$	− 6.43487i	− 0.478300i	−0.970983	−	0.239150i	$-0.923131\pi$
$181$	− 6.43487i	− 0.478300i	0.970983	−	0.239150i	$-0.0768688\pi$
$182$	0	0
$183$	11.9403	0.882649
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 0.618423i	− 0.0452236i
$188$	0	0
$189$	− 0.0802864i	− 0.00583997i
$190$	0	0
$191$	5.56270	0.402503	0.201251	−	0.979540i	$-0.435499\pi$
$191$	5.56270	0.402503	0.201251	+	0.979540i	$0.435499\pi$
$192$	0	0
$193$	−18.4227	−1.32609	−0.663046	−	0.748578i	$-0.730737\pi$
$193$	−18.4227	−1.32609	−0.663046	+	0.748578i	$0.730737\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.0239i	1.28415i	0.766643	+	0.642074i	$0.221926\pi$
$197$	18.0239i	1.28415i	−0.766643	+	0.642074i	$0.778074\pi$
$198$	0	0
$199$	−20.1214	−1.42637	−0.713183	−	0.700977i	$-0.752747\pi$
$199$	−20.1214	−1.42637	−0.713183	+	0.700977i	$0.752747\pi$
$200$	0	0
$201$	7.27979	0.513476
$202$	0	0
$203$	− 0.362505i	− 0.0254429i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.64542	−0.114364
$208$	0	0
$209$	16.8155	1.16315
$210$	0	0
$211$	3.25592i	0.224147i	0.993700	+	0.112073i	$0.0357492\pi$
$211$	3.25592i	0.224147i	−0.993700	+	0.112073i	$0.964251\pi$
$212$	0	0
$213$	− 11.3481i	− 0.777562i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.665963	0.0452085
$218$	0	0
$219$	12.0779i	0.816146i
$220$	0	0
$221$	− 1.34814i	− 0.0906856i
$222$	0	0
$223$	−26.9911	−1.80746	−0.903730	−	0.428104i	$-0.859182\pi$
$223$	−26.9911	−1.80746	−0.903730	+	0.428104i	$0.859182\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 19.8219i	− 1.31563i	−0.753180	−	0.657814i	$-0.771481\pi$
$227$	− 19.8219i	− 1.31563i	0.753180	−	0.657814i	$-0.228519\pi$
$228$	0	0
$229$	− 21.6797i	− 1.43264i	−0.697773	−	0.716319i	$-0.745826\pi$
$229$	− 21.6797i	− 1.43264i	0.697773	−	0.716319i	$-0.254174\pi$
$230$	0	0
$231$	−0.194011	−0.0127650
$232$	0	0
$233$	17.2733	1.13161	0.565807	−	0.824538i	$-0.308565\pi$
$233$	17.2733	1.13161	0.565807	+	0.824538i	$0.308565\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	5.50539i	0.357614i
$238$	0	0
$239$	16.3718	1.05900	0.529502	−	0.848309i	$-0.322379\pi$
$239$	16.3718	1.05900	0.529502	+	0.848309i	$0.322379\pi$
$240$	0	0
$241$	−6.82654	−0.439736	−0.219868	−	0.975530i	$-0.570563\pi$
$241$	−6.82654	−0.439736	−0.219868	+	0.975530i	$0.570563\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	36.6571	2.33243
$248$	0	0
$249$	9.20811	0.583540
$250$	0	0
$251$	− 2.96969i	− 0.187445i	−0.995598	−	0.0937225i	$-0.970123\pi$
$251$	− 2.96969i	− 0.187445i	0.995598	−	0.0937225i	$-0.0298766\pi$
$252$	0	0
$253$	3.97613i	0.249977i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.03031	0.313782	0.156891	−	0.987616i	$-0.449853\pi$
$257$	5.03031	0.313782	0.156891	+	0.987616i	$0.449853\pi$
$258$	0	0
$259$	− 0.214558i	− 0.0133320i
$260$	0	0
$261$	− 4.51516i	− 0.279481i
$262$	0	0
$263$	2.70585	0.166850	0.0834248	−	0.996514i	$-0.473414\pi$
$263$	2.70585	0.166850	0.0834248	+	0.996514i	$0.473414\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 11.9173i	− 0.729326i
$268$	0	0
$269$	22.3718i	1.36403i	0.731337	+	0.682017i	$0.238897\pi$
$269$	22.3718i	1.36403i	−0.731337	+	0.682017i	$0.761103\pi$
$270$	0	0
$271$	0.869741	0.0528330	0.0264165	−	0.999651i	$-0.491590\pi$
$271$	0.869741	0.0528330	0.0264165	+	0.999651i	$0.491590\pi$
$272$	0	0
$273$	−0.422937	−0.0255973
$274$	0	0
$275$	0	0
$276$	0	0
$277$	28.6733i	1.72281i	0.507918	+	0.861406i	$0.330415\pi$
$277$	28.6733i	1.72281i	−0.507918	+	0.861406i	$0.669585\pi$
$278$	0	0
$279$	8.29484	0.496599
$280$	0	0
$281$	15.1429	0.903349	0.451674	−	0.892183i	$-0.350827\pi$
$281$	15.1429	0.903349	0.451674	+	0.892183i	$0.350827\pi$
$282$	0	0
$283$	6.23225i	0.370469i	0.982694	+	0.185234i	$0.0593044\pi$
$283$	6.23225i	0.370469i	−0.982694	+	0.185234i	$0.940696\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.651862	0.0384782
$288$	0	0
$289$	−16.9345	−0.996147
$290$	0	0
$291$	− 8.50539i	− 0.498595i
$292$	0	0
$293$	− 21.5054i	− 1.25636i	−0.778069	−	0.628179i	$-0.783800\pi$
$293$	− 21.5054i	− 1.25636i	0.778069	−	0.628179i	$-0.216200\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.41649	−0.140219
$298$	0	0
$299$	8.66781i	0.501272i
$300$	0	0
$301$	− 0.328283i	− 0.0189219i
$302$	0	0
$303$	7.56270	0.434466
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 3.57706i	− 0.204154i	−0.994777	−	0.102077i	$-0.967451\pi$
$307$	− 3.57706i	− 0.204154i	0.994777	−	0.102077i	$-0.0325488\pi$
$308$	0	0
$309$	− 1.78544i	− 0.101570i
$310$	0	0
$311$	2.49461	0.141456	0.0707282	−	0.997496i	$-0.477468\pi$
$311$	2.49461	0.141456	0.0707282	+	0.997496i	$0.477468\pi$
$312$	0	0
$313$	9.57246	0.541068	0.270534	−	0.962710i	$-0.412800\pi$
$313$	9.57246	0.541068	0.270534	+	0.962710i	$0.412800\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 3.16702i	− 0.177877i	−0.996037	−	0.0889387i	$-0.971652\pi$
$317$	− 3.16702i	− 0.177877i	0.996037	−	0.0889387i	$-0.0283475\pi$
$318$	0	0
$319$	−10.9108	−0.610889
$320$	0	0
$321$	10.4705	0.584405
$322$	0	0
$323$	1.78084i	0.0990887i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3.64298	−0.201457
$328$	0	0
$329$	0.457851	0.0252421
$330$	0	0
$331$	16.5118i	0.907573i	0.891111	+	0.453786i	$0.149927\pi$
$331$	16.5118i	0.907573i	−0.891111	+	0.453786i	$0.850073\pi$
$332$	0	0
$333$	− 2.67241i	− 0.146447i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−11.8330	−0.644584	−0.322292	−	0.946640i	$-0.604453\pi$
$337$	−11.8330	−0.644584	−0.322292	+	0.946640i	$0.604453\pi$
$338$	0	0
$339$	− 8.83298i	− 0.479742i
$340$	0	0
$341$	− 20.0444i	− 1.08547i
$342$	0	0
$343$	1.12349	0.0606628
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 23.9713i	− 1.28684i	−0.765511	−	0.643422i	$-0.777514\pi$
$347$	− 23.9713i	− 1.28684i	0.765511	−	0.643422i	$-0.222486\pi$
$348$	0	0
$349$	8.91570i	0.477247i	0.971112	+	0.238623i	$0.0766961\pi$
$349$	8.91570i	0.477247i	−0.971112	+	0.238623i	$0.923304\pi$
$350$	0	0
$351$	−5.26785	−0.281177
$352$	0	0
$353$	−7.35606	−0.391524	−0.195762	−	0.980651i	$-0.562718\pi$
$353$	−7.35606	−0.391524	−0.195762	+	0.980651i	$0.562718\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 0.0205467i	− 0.00108745i
$358$	0	0
$359$	−25.2114	−1.33061	−0.665304	−	0.746572i	$-0.731698\pi$
$359$	−25.2114	−1.33061	−0.665304	+	0.746572i	$0.731698\pi$
$360$	0	0
$361$	−29.4227	−1.54856
$362$	0	0
$363$	− 5.16057i	− 0.270860i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−5.86573	−0.306189	−0.153094	−	0.988212i	$-0.548924\pi$
$367$	−5.86573	−0.306189	−0.153094	+	0.988212i	$0.548924\pi$
$368$	0	0
$369$	8.11921	0.422669
$370$	0	0
$371$	− 0.928327i	− 0.0481963i
$372$	0	0
$373$	− 27.5063i	− 1.42422i	−0.702067	−	0.712111i	$-0.747739\pi$
$373$	− 27.5063i	− 1.42422i	0.702067	−	0.712111i	$-0.252261\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−23.7852	−1.22500
$378$	0	0
$379$	11.7549i	0.603807i	0.953339	+	0.301903i	$0.0976220\pi$
$379$	11.7549i	0.603807i	−0.953339	+	0.301903i	$0.902378\pi$
$380$	0	0
$381$	− 8.69628i	− 0.445524i
$382$	0	0
$383$	34.3335	1.75436	0.877180	−	0.480162i	$-0.159422\pi$
$383$	34.3335	1.75436	0.877180	+	0.480162i	$0.159422\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 4.08890i	− 0.207850i
$388$	0	0
$389$	− 2.89515i	− 0.146790i	−0.997303	−	0.0733951i	$-0.976617\pi$
$389$	− 2.89515i	− 0.146790i	0.997303	−	0.0733951i	$-0.0233834\pi$
$390$	0	0
$391$	−0.421092	−0.0212955
$392$	0	0
$393$	10.7916	0.544365
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 22.9099i	− 1.14982i	−0.818218	−	0.574909i	$-0.805038\pi$
$397$	− 22.9099i	− 1.14982i	0.818218	−	0.574909i	$-0.194962\pi$
$398$	0	0
$399$	0.558684	0.0279692
$400$	0	0
$401$	−12.4337	−0.620910	−0.310455	−	0.950588i	$-0.600481\pi$
$401$	−12.4337	−0.620910	−0.310455	+	0.950588i	$0.600481\pi$
$402$	0	0
$403$	− 43.6960i	− 2.17665i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.45785	−0.320104
$408$	0	0
$409$	−32.0886	−1.58668	−0.793340	−	0.608778i	$-0.791660\pi$
$409$	−32.0886	−1.58668	−0.793340	+	0.608778i	$0.791660\pi$
$410$	0	0
$411$	− 11.5421i	− 0.569332i
$412$	0	0
$413$	1.01934i	0.0501583i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−0.214558	−0.0105069
$418$	0	0
$419$	16.1364i	0.788317i	0.919043	+	0.394158i	$0.128964\pi$
$419$	16.1364i	0.788317i	−0.919043	+	0.394158i	$0.871036\pi$
$420$	0	0
$421$	− 28.7675i	− 1.40204i	−0.713141	−	0.701021i	$-0.752728\pi$
$421$	− 28.7675i	− 1.40204i	0.713141	−	0.701021i	$-0.247272\pi$
$422$	0	0
$423$	5.70272	0.277276
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 0.958640i	− 0.0463918i
$428$	0	0
$429$	12.7297i	0.614596i
$430$	0	0
$431$	−24.7297	−1.19119	−0.595594	−	0.803285i	$-0.703083\pi$
$431$	−24.7297	−1.19119	−0.595594	+	0.803285i	$0.703083\pi$
$432$	0	0
$433$	−4.48816	−0.215687	−0.107844	−	0.994168i	$-0.534395\pi$
$433$	−4.48816	−0.215687	−0.107844	+	0.994168i	$0.534395\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 11.4499i	− 0.547721i
$438$	0	0
$439$	−5.96081	−0.284494	−0.142247	−	0.989831i	$-0.545433\pi$
$439$	−5.96081	−0.284494	−0.142247	+	0.989831i	$0.545433\pi$
$440$	0	0
$441$	6.99355	0.333026
$442$	0	0
$443$	− 14.2924i	− 0.679053i	−0.940597	−	0.339526i	$-0.889733\pi$
$443$	− 14.2924i	− 0.679053i	0.940597	−	0.339526i	$-0.110267\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−23.0475	−1.09011
$448$	0	0
$449$	24.5529	1.15872	0.579362	−	0.815070i	$-0.303302\pi$
$449$	24.5529	1.15872	0.579362	+	0.815070i	$0.303302\pi$
$450$	0	0
$451$	− 19.6200i	− 0.923870i
$452$	0	0
$453$	9.48573i	0.445678i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.9108	1.35239	0.676196	−	0.736722i	$-0.263627\pi$
$457$	28.9108	1.35239	0.676196	+	0.736722i	$0.263627\pi$
$458$	0	0
$459$	− 0.255918i	− 0.0119452i
$460$	0	0
$461$	4.35458i	0.202813i	0.994845	+	0.101407i	$0.0323343\pi$
$461$	4.35458i	0.202813i	−0.994845	+	0.101407i	$0.967666\pi$
$462$	0	0
$463$	−11.1303	−0.517267	−0.258634	−	0.965976i	$-0.583272\pi$
$463$	−11.1303	−0.517267	−0.258634	+	0.965976i	$0.583272\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.1257i	0.885030i	0.896761	+	0.442515i	$0.145914\pi$
$467$	19.1257i	0.885030i	−0.896761	+	0.442515i	$0.854086\pi$
$468$	0	0
$469$	− 0.584467i	− 0.0269882i
$470$	0	0
$471$	−6.34413	−0.292322
$472$	0	0
$473$	−9.88079	−0.454319
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 11.5627i	− 0.529419i
$478$	0	0
$479$	25.6358	1.17133	0.585666	−	0.810553i	$-0.300833\pi$
$479$	25.6358	1.17133	0.585666	+	0.810553i	$0.300833\pi$
$480$	0	0
$481$	−14.0779	−0.641895
$482$	0	0
$483$	0.132104i	0.00601096i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12.8434	0.581992	0.290996	−	0.956724i	$-0.406013\pi$
$487$	12.8434	0.581992	0.290996	+	0.956724i	$0.406013\pi$
$488$	0	0
$489$	12.4100	0.561202
$490$	0	0
$491$	− 16.9887i	− 0.766689i	−0.923605	−	0.383344i	$-0.874772\pi$
$491$	− 16.9887i	− 0.766689i	0.923605	−	0.383344i	$-0.125228\pi$
$492$	0	0
$493$	− 1.15551i	− 0.0520415i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.911101	−0.0408684
$498$	0	0
$499$	14.0521i	0.629060i	0.949248	+	0.314530i	$0.101847\pi$
$499$	14.0521i	0.629060i	−0.949248	+	0.314530i	$0.898153\pi$
$500$	0	0
$501$	23.2654i	1.03942i
$502$	0	0
$503$	9.53258	0.425037	0.212518	−	0.977157i	$-0.431833\pi$
$503$	9.53258	0.425037	0.212518	+	0.977157i	$0.431833\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	14.7503i	0.655082i
$508$	0	0
$509$	30.3450i	1.34502i	0.740088	+	0.672510i	$0.234784\pi$
$509$	30.3450i	1.34502i	−0.740088	+	0.672510i	$0.765216\pi$
$510$	0	0
$511$	0.969687	0.0428964
$512$	0	0
$513$	6.95864	0.307231
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 13.7806i	− 0.606069i
$518$	0	0
$519$	8.63897	0.379209
$520$	0	0
$521$	14.4245	0.631949	0.315975	−	0.948768i	$-0.397669\pi$
$521$	14.4245	0.631949	0.315975	+	0.948768i	$0.397669\pi$
$522$	0	0
$523$	− 28.2207i	− 1.23401i	−0.786961	−	0.617003i	$-0.788346\pi$
$523$	− 28.2207i	− 1.23401i	0.786961	−	0.617003i	$-0.211654\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.12280	0.0924706
$528$	0	0
$529$	−20.2926	−0.882287
$530$	0	0
$531$	12.6963i	0.550971i
$532$	0	0
$533$	− 42.7708i	− 1.85261i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	9.40544	0.405875
$538$	0	0
$539$	− 16.8999i	− 0.727928i
$540$	0	0
$541$	− 13.4695i	− 0.579100i	−0.957163	−	0.289550i	$-0.906494\pi$
$541$	− 13.4695i	− 0.579100i	0.957163	−	0.289550i	$-0.0935056\pi$
$542$	0	0
$543$	−6.43487	−0.276147
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 4.42773i	− 0.189316i	−0.995510	−	0.0946581i	$-0.969824\pi$
$547$	− 4.42773i	− 0.189316i	0.995510	−	0.0946581i	$-0.0301758\pi$
$548$	0	0
$549$	− 11.9403i	− 0.509598i
$550$	0	0
$551$	31.4193	1.33851
$552$	0	0
$553$	0.442008	0.0187961
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 40.2017i	− 1.70340i	−0.524030	−	0.851700i	$-0.675572\pi$
$557$	− 40.2017i	− 1.70340i	0.524030	−	0.851700i	$-0.324428\pi$
$558$	0	0
$559$	−21.5397	−0.911033
$560$	0	0
$561$	−0.618423	−0.0261099
$562$	0	0
$563$	− 13.1128i	− 0.552637i	−0.961066	−	0.276319i	$-0.910885\pi$
$563$	− 13.1128i	− 0.552637i	0.961066	−	0.276319i	$-0.0891145\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−0.0802864	−0.00337171
$568$	0	0
$569$	−11.0257	−0.462222	−0.231111	−	0.972927i	$-0.574236\pi$
$569$	−11.0257	−0.462222	−0.231111	+	0.972927i	$0.574236\pi$
$570$	0	0
$571$	− 45.6960i	− 1.91232i	−0.292847	−	0.956159i	$-0.594603\pi$
$571$	− 45.6960i	− 1.91232i	0.292847	−	0.956159i	$-0.405397\pi$
$572$	0	0
$573$	− 5.56270i	− 0.232385i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	17.2685	0.718899	0.359449	−	0.933165i	$-0.382965\pi$
$577$	17.2685	0.718899	0.359449	+	0.933165i	$0.382965\pi$
$578$	0	0
$579$	18.4227i	0.765620i
$580$	0	0
$581$	− 0.739286i	− 0.0306707i
$582$	0	0
$583$	−27.9411	−1.15720
$584$	0	0
$585$	0	0
$586$	0	0
$587$	34.7155i	1.43286i	0.697657	+	0.716432i	$0.254226\pi$
$587$	34.7155i	1.43286i	−0.697657	+	0.716432i	$0.745774\pi$
$588$	0	0
$589$	57.7208i	2.37835i
$590$	0	0
$591$	18.0239	0.741403
$592$	0	0
$593$	9.34022	0.383557	0.191778	−	0.981438i	$-0.438575\pi$
$593$	9.34022	0.383557	0.191778	+	0.981438i	$0.438575\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	20.1214i	0.823513i
$598$	0	0
$599$	13.9110	0.568389	0.284195	−	0.958767i	$-0.408274\pi$
$599$	13.9110	0.568389	0.284195	+	0.958767i	$0.408274\pi$
$600$	0	0
$601$	11.7330	0.478600	0.239300	−	0.970946i	$-0.423082\pi$
$601$	11.7330	0.478600	0.239300	+	0.970946i	$0.423082\pi$
$602$	0	0
$603$	− 7.27979i	− 0.296456i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−10.8158	−0.438998	−0.219499	−	0.975613i	$-0.570442\pi$
$607$	−10.8158	−0.438998	−0.219499	+	0.975613i	$0.570442\pi$
$608$	0	0
$609$	−0.362505	−0.0146895
$610$	0	0
$611$	− 30.0411i	− 1.21533i
$612$	0	0
$613$	− 17.9632i	− 0.725528i	−0.931881	−	0.362764i	$-0.881833\pi$
$613$	− 17.9632i	− 0.725528i	0.931881	−	0.362764i	$-0.118167\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12.3576	−0.497500	−0.248750	−	0.968568i	$-0.580020\pi$
$617$	−12.3576	−0.497500	−0.248750	+	0.968568i	$0.580020\pi$
$618$	0	0
$619$	− 4.99540i	− 0.200782i	−0.994948	−	0.100391i	$-0.967991\pi$
$619$	− 4.99540i	− 0.200782i	0.994948	−	0.100391i	$-0.0320094\pi$
$620$	0	0
$621$	1.64542i	0.0660283i
$622$	0	0
$623$	−0.956795	−0.0383332
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 16.8155i	− 0.671546i
$628$	0	0
$629$	− 0.683917i	− 0.0272696i
$630$	0	0
$631$	17.9674	0.715273	0.357636	−	0.933861i	$-0.383583\pi$
$631$	17.9674	0.715273	0.357636	+	0.933861i	$0.383583\pi$
$632$	0	0
$633$	3.25592	0.129411
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 36.8410i	− 1.45969i
$638$	0	0
$639$	−11.3481	−0.448925
$640$	0	0
$641$	27.3638	1.08081	0.540403	−	0.841406i	$-0.318272\pi$
$641$	27.3638	1.08081	0.540403	+	0.841406i	$0.318272\pi$
$642$	0	0
$643$	− 2.27518i	− 0.0897245i	−0.998993	−	0.0448623i	$-0.985715\pi$
$643$	− 2.27518i	− 0.0897245i	0.998993	−	0.0448623i	$-0.0142849\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.4769	0.490516	0.245258	−	0.969458i	$-0.421127\pi$
$647$	12.4769	0.490516	0.245258	+	0.969458i	$0.421127\pi$
$648$	0	0
$649$	30.6804	1.20431
$650$	0	0
$651$	− 0.665963i	− 0.0261011i
$652$	0	0
$653$	29.3055i	1.14681i	0.819271	+	0.573406i	$0.194378\pi$
$653$	29.3055i	1.14681i	−0.819271	+	0.573406i	$0.805622\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	12.0779	0.471202
$658$	0	0
$659$	− 18.6009i	− 0.724589i	−0.932064	−	0.362295i	$-0.881993\pi$
$659$	− 18.6009i	− 0.724589i	0.932064	−	0.362295i	$-0.118007\pi$
$660$	0	0
$661$	16.1318i	0.627456i	0.949513	+	0.313728i	$0.101578\pi$
$661$	16.1318i	0.627456i	−0.949513	+	0.313728i	$0.898422\pi$
$662$	0	0
$663$	−1.34814	−0.0523573
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.42931i	0.287664i
$668$	0	0
$669$	26.9911i	1.04354i
$670$	0	0
$671$	−28.8535	−1.11388
$672$	0	0
$673$	34.1385	1.31594	0.657971	−	0.753043i	$-0.271415\pi$
$673$	34.1385	1.31594	0.657971	+	0.753043i	$0.271415\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	12.1940i	0.468654i	0.972158	+	0.234327i	$0.0752886\pi$
$677$	12.1940i	0.468654i	−0.972158	+	0.234327i	$0.924711\pi$
$678$	0	0
$679$	−0.682867	−0.0262060
$680$	0	0
$681$	−19.8219	−0.759578
$682$	0	0
$683$	− 21.8567i	− 0.836322i	−0.908373	−	0.418161i	$-0.862675\pi$
$683$	− 21.8567i	− 0.836322i	0.908373	−	0.418161i	$-0.137325\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−21.6797	−0.827134
$688$	0	0
$689$	−60.9106	−2.32051
$690$	0	0
$691$	− 6.17780i	− 0.235015i	−0.993072	−	0.117507i	$-0.962510\pi$
$691$	− 6.17780i	− 0.235015i	0.993072	−	0.117507i	$-0.0374903\pi$
$692$	0	0
$693$	0.194011i	0.00736988i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.07785	0.0787043
$698$	0	0
$699$	− 17.2733i	− 0.653338i
$700$	0	0
$701$	33.9746i	1.28320i	0.767038	+	0.641601i	$0.221730\pi$
$701$	33.9746i	1.28320i	−0.767038	+	0.641601i	$0.778270\pi$
$702$	0	0
$703$	18.5963	0.701374
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 0.607181i	− 0.0228354i
$708$	0	0
$709$	− 22.3441i	− 0.839151i	−0.907720	−	0.419576i	$-0.862179\pi$
$709$	− 22.3441i	− 0.839151i	0.907720	−	0.419576i	$-0.137821\pi$
$710$	0	0
$711$	5.50539	0.206468
$712$	0	0
$713$	−13.6485	−0.511139
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 16.3718i	− 0.611416i
$718$	0	0
$719$	17.8427	0.665422	0.332711	−	0.943029i	$-0.392037\pi$
$719$	17.8427	0.665422	0.332711	+	0.943029i	$0.392037\pi$
$720$	0	0
$721$	−0.143347	−0.00533851
$722$	0	0
$723$	6.82654i	0.253882i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−23.9148	−0.886953	−0.443476	−	0.896286i	$-0.646255\pi$
$727$	−23.9148	−0.886953	−0.443476	+	0.896286i	$0.646255\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 1.04642i	− 0.0387034i
$732$	0	0
$733$	15.6789i	0.579112i	0.957161	+	0.289556i	$0.0935076\pi$
$733$	15.6789i	0.579112i	−0.957161	+	0.289556i	$0.906492\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−17.5915	−0.647992
$738$	0	0
$739$	− 22.3083i	− 0.820622i	−0.911946	−	0.410311i	$-0.865420\pi$
$739$	− 22.3083i	− 0.820622i	0.911946	−	0.410311i	$-0.134580\pi$
$740$	0	0
$741$	− 36.6571i	− 1.34663i
$742$	0	0
$743$	9.78057	0.358814	0.179407	−	0.983775i	$-0.442582\pi$
$743$	9.78057	0.358814	0.179407	+	0.983775i	$0.442582\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 9.20811i	− 0.336907i
$748$	0	0
$749$	− 0.840636i	− 0.0307162i
$750$	0	0
$751$	8.05399	0.293894	0.146947	−	0.989144i	$-0.453055\pi$
$751$	8.05399	0.293894	0.146947	+	0.989144i	$0.453055\pi$
$752$	0	0
$753$	−2.96969	−0.108221
$754$	0	0
$755$	0	0
$756$	0	0
$757$	28.4889i	1.03545i	0.855549	+	0.517723i	$0.173220\pi$
$757$	28.4889i	1.03545i	−0.855549	+	0.517723i	$0.826780\pi$
$758$	0	0
$759$	3.97613	0.144324
$760$	0	0
$761$	−21.5005	−0.779393	−0.389697	−	0.920943i	$-0.627420\pi$
$761$	−21.5005	−0.779393	−0.389697	+	0.920943i	$0.627420\pi$
$762$	0	0
$763$	0.292482i	0.0105886i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	66.8821	2.41497
$768$	0	0
$769$	23.5596	0.849580	0.424790	−	0.905292i	$-0.360348\pi$
$769$	23.5596	0.849580	0.424790	+	0.905292i	$0.360348\pi$
$770$	0	0
$771$	− 5.03031i	− 0.181162i
$772$	0	0
$773$	− 31.1655i	− 1.12094i	−0.828173	−	0.560472i	$-0.810620\pi$
$773$	− 31.1655i	− 1.12094i	0.828173	−	0.560472i	$-0.189380\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−0.214558	−0.00769722
$778$	0	0
$779$	56.4987i	2.02428i
$780$	0	0
$781$	27.4227i	0.981260i
$782$	0	0
$783$	−4.51516	−0.161359
$784$	0	0
$785$	0	0
$786$	0	0
$787$	5.07812i	0.181015i	0.995896	+	0.0905077i	$0.0288490\pi$
$787$	5.07812i	0.181015i	−0.995896	+	0.0905077i	$0.971151\pi$
$788$	0	0
$789$	− 2.70585i	− 0.0963307i
$790$	0	0
$791$	−0.709168	−0.0252151
$792$	0	0
$793$	−62.8995	−2.23363
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.43418i	− 0.0508012i	−0.999677	−	0.0254006i	$-0.991914\pi$
$797$	− 1.43418i	− 0.0508012i	0.999677	−	0.0254006i	$-0.00808613\pi$
$798$	0	0
$799$	1.45943	0.0516309
$800$	0	0
$801$	−11.9173	−0.421076
$802$	0	0
$803$	− 29.1860i	− 1.02995i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	22.3718	0.787525
$808$	0	0
$809$	23.7153	0.833787	0.416894	−	0.908955i	$-0.363119\pi$
$809$	23.7153	0.833787	0.416894	+	0.908955i	$0.363119\pi$
$810$	0	0
$811$	47.8394i	1.67987i	0.542689	+	0.839933i	$0.317406\pi$
$811$	47.8394i	1.67987i	−0.542689	+	0.839933i	$0.682594\pi$
$812$	0	0
$813$	− 0.869741i	− 0.0305031i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	28.4532	0.995451
$818$	0	0
$819$	0.422937i	0.0147786i
$820$	0	0
$821$	− 51.9216i	− 1.81208i	−0.423195	−	0.906038i	$-0.639092\pi$
$821$	− 51.9216i	− 1.81208i	0.423195	−	0.906038i	$-0.360908\pi$
$822$	0	0
$823$	27.8542	0.970937	0.485469	−	0.874254i	$-0.338649\pi$
$823$	27.8542	0.970937	0.485469	+	0.874254i	$0.338649\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 43.9365i	− 1.52782i	−0.645321	−	0.763912i	$-0.723276\pi$
$827$	− 43.9365i	− 1.52782i	0.645321	−	0.763912i	$-0.276724\pi$
$828$	0	0
$829$	41.6898i	1.44795i	0.689827	+	0.723974i	$0.257686\pi$
$829$	41.6898i	1.44795i	−0.689827	+	0.723974i	$0.742314\pi$
$830$	0	0
$831$	28.6733	0.994666
$832$	0	0
$833$	1.78978	0.0620121
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 8.29484i	− 0.286712i
$838$	0	0
$839$	25.4733	0.879437	0.439719	−	0.898136i	$-0.355078\pi$
$839$	25.4733	0.879437	0.439719	+	0.898136i	$0.355078\pi$
$840$	0	0
$841$	8.61336	0.297012
$842$	0	0
$843$	− 15.1429i	− 0.521549i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−0.414324	−0.0142363
$848$	0	0
$849$	6.23225	0.213890
$850$	0	0
$851$	4.39722i	0.150735i
$852$	0	0
$853$	12.5366i	0.429245i	0.976697	+	0.214622i	$0.0688521\pi$
$853$	12.5366i	0.429245i	−0.976697	+	0.214622i	$0.931148\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.61409	−0.0892956	−0.0446478	−	0.999003i	$-0.514217\pi$
$857$	−2.61409	−0.0892956	−0.0446478	+	0.999003i	$0.514217\pi$
$858$	0	0
$859$	− 31.8438i	− 1.08650i	−0.839573	−	0.543248i	$-0.817195\pi$
$859$	− 31.8438i	− 1.08650i	0.839573	−	0.543248i	$-0.182805\pi$
$860$	0	0
$861$	− 0.651862i	− 0.0222154i
$862$	0	0
$863$	−22.3335	−0.760241	−0.380121	−	0.924937i	$-0.624118\pi$
$863$	−22.3335	−0.760241	−0.380121	+	0.924937i	$0.624118\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	16.9345i	0.575126i
$868$	0	0
$869$	− 13.3037i	− 0.451298i
$870$	0	0
$871$	−38.3488	−1.29940
$872$	0	0
$873$	−8.50539	−0.287864
$874$	0	0
$875$	0	0
$876$	0	0
$877$	37.4408i	1.26429i	0.774852	+	0.632143i	$0.217824\pi$
$877$	37.4408i	1.26429i	−0.774852	+	0.632143i	$0.782176\pi$
$878$	0	0
$879$	−21.5054	−0.725359
$880$	0	0
$881$	53.1952	1.79219	0.896096	−	0.443860i	$-0.146391\pi$
$881$	53.1952	1.79219	0.896096	+	0.443860i	$0.146391\pi$
$882$	0	0
$883$	36.0907i	1.21455i	0.794493	+	0.607274i	$0.207737\pi$
$883$	36.0907i	1.21455i	−0.794493	+	0.607274i	$0.792263\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−43.3018	−1.45393	−0.726966	−	0.686673i	$-0.759070\pi$
$887$	−43.3018	−1.45393	−0.726966	+	0.686673i	$0.759070\pi$
$888$	0	0
$889$	−0.698192	−0.0234166
$890$	0	0
$891$	2.41649i	0.0809555i
$892$	0	0
$893$	39.6832i	1.32795i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	8.66781	0.289410
$898$	0	0
$899$	− 37.4525i	− 1.24911i
$900$	0	0
$901$	− 2.95910i	− 0.0985820i
$902$	0	0
$903$	−0.328283	−0.0109246
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 8.75026i	− 0.290548i	−0.989391	−	0.145274i	$-0.953594\pi$
$907$	− 8.75026i	− 0.290548i	0.989391	−	0.145274i	$-0.0464063\pi$
$908$	0	0
$909$	− 7.56270i	− 0.250839i
$910$	0	0
$911$	13.7438	0.455353	0.227676	−	0.973737i	$-0.426887\pi$
$911$	13.7438	0.455353	0.227676	+	0.973737i	$0.426887\pi$
$912$	0	0
$913$	−22.2513	−0.736411
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 0.866420i	− 0.0286117i
$918$	0	0
$919$	0.989347	0.0326355	0.0163178	−	0.999867i	$-0.494806\pi$
$919$	0.989347	0.0326355	0.0163178	+	0.999867i	$0.494806\pi$
$920$	0	0
$921$	−3.57706	−0.117868
$922$	0	0
$923$	59.7803i	1.96769i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.78544	−0.0586416
$928$	0	0
$929$	−8.49434	−0.278690	−0.139345	−	0.990244i	$-0.544500\pi$
$929$	−8.49434	−0.278690	−0.139345	+	0.990244i	$0.544500\pi$
$930$	0	0
$931$	48.6656i	1.59495i
$932$	0	0
$933$	− 2.49461i	− 0.0816699i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−18.6912	−0.610615	−0.305308	−	0.952254i	$-0.598759\pi$
$937$	−18.6912	−0.610615	−0.305308	+	0.952254i	$0.598759\pi$
$938$	0	0
$939$	− 9.57246i	− 0.312386i
$940$	0	0
$941$	− 3.03170i	− 0.0988305i	−0.998778	−	0.0494152i	$-0.984264\pi$
$941$	− 3.03170i	− 0.0988305i	0.998778	−	0.0494152i	$-0.0157358\pi$
$942$	0	0
$943$	−13.3595	−0.435045
$944$	0	0
$945$	0	0
$946$	0	0
$947$	16.8327i	0.546990i	0.961873	+	0.273495i	$0.0881797\pi$
$947$	16.8327i	0.546990i	−0.961873	+	0.273495i	$0.911820\pi$
$948$	0	0
$949$	− 63.6243i	− 2.06533i
$950$	0	0
$951$	−3.16702	−0.102698
$952$	0	0
$953$	−1.73948	−0.0563473	−0.0281737	−	0.999603i	$-0.508969\pi$
$953$	−1.73948	−0.0563473	−0.0281737	+	0.999603i	$0.508969\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	10.9108i	0.352697i
$958$	0	0
$959$	−0.926677	−0.0299240
$960$	0	0
$961$	37.8044	1.21950
$962$	0	0
$963$	− 10.4705i	− 0.337406i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−17.3399	−0.557615	−0.278808	−	0.960347i	$-0.589939\pi$
$967$	−17.3399	−0.557615	−0.278808	+	0.960347i	$0.589939\pi$
$968$	0	0
$969$	1.78084	0.0572089
$970$	0	0
$971$	45.4054i	1.45713i	0.684977	+	0.728565i	$0.259812\pi$
$971$	45.4054i	1.45713i	−0.684977	+	0.728565i	$0.740188\pi$
$972$	0	0
$973$	0.0172261i	0 0.000552243i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−43.4336	−1.38957	−0.694783	−	0.719220i	$-0.744499\pi$
$977$	−43.4336	−1.38957	−0.694783	+	0.719220i	$0.744499\pi$
$978$	0	0
$979$	28.7980i	0.920388i
$980$	0	0
$981$	3.64298i	0.116311i
$982$	0	0
$983$	−47.4465	−1.51331	−0.756655	−	0.653815i	$-0.773168\pi$
$983$	−47.4465	−1.51331	−0.756655	+	0.653815i	$0.773168\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 0.457851i	− 0.0145735i
$988$	0	0
$989$	6.72794i	0.213936i
$990$	0	0
$991$	28.8434	0.916242	0.458121	−	0.888890i	$-0.348523\pi$
$991$	28.8434	0.916242	0.458121	+	0.888890i	$0.348523\pi$
$992$	0	0
$993$	16.5118	0.523987
$994$	0	0
$995$	0	0
$996$	0	0
$997$	33.1449i	1.04971i	0.851192	+	0.524855i	$0.175881\pi$
$997$	33.1449i	1.04971i	−0.851192	+	0.524855i	$0.824119\pi$
$998$	0	0
$999$	−2.67241	−0.0845513

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.k.e.1201.2		8
3.2	odd	2		7200.2.k.s.3601.3		8
4.3	odd	2		600.2.k.e.301.2	yes	8
5.2	odd	4		2400.2.d.g.49.4		8
5.3	odd	4		2400.2.d.h.49.5		8
5.4	even	2		2400.2.k.d.1201.7		8
8.3	odd	2		600.2.k.e.301.1	yes	8
8.5	even	2	inner	2400.2.k.e.1201.6		8
12.11	even	2		1800.2.k.q.901.7		8
15.2	even	4		7200.2.d.s.2449.4		8
15.8	even	4		7200.2.d.t.2449.5		8
15.14	odd	2		7200.2.k.r.3601.5		8
20.3	even	4		600.2.d.g.349.6		8
20.7	even	4		600.2.d.h.349.3		8
20.19	odd	2		600.2.k.d.301.7	✓	8
24.5	odd	2		7200.2.k.s.3601.4		8
24.11	even	2		1800.2.k.q.901.8		8
40.3	even	4		600.2.d.h.349.4		8
40.13	odd	4		2400.2.d.g.49.5		8
40.19	odd	2		600.2.k.d.301.8	yes	8
40.27	even	4		600.2.d.g.349.5		8
40.29	even	2		2400.2.k.d.1201.3		8
40.37	odd	4		2400.2.d.h.49.4		8
60.23	odd	4		1800.2.d.t.1549.3		8
60.47	odd	4		1800.2.d.s.1549.6		8
60.59	even	2		1800.2.k.t.901.2		8
120.29	odd	2		7200.2.k.r.3601.6		8
120.53	even	4		7200.2.d.s.2449.5		8
120.59	even	2		1800.2.k.t.901.1		8
120.77	even	4		7200.2.d.t.2449.4		8
120.83	odd	4		1800.2.d.s.1549.5		8
120.107	odd	4		1800.2.d.t.1549.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.2.d.g.349.5		8	40.27	even	4
600.2.d.g.349.6		8	20.3	even	4
600.2.d.h.349.3		8	20.7	even	4
600.2.d.h.349.4		8	40.3	even	4
600.2.k.d.301.7	✓	8	20.19	odd	2
600.2.k.d.301.8	yes	8	40.19	odd	2
600.2.k.e.301.1	yes	8	8.3	odd	2
600.2.k.e.301.2	yes	8	4.3	odd	2
1800.2.d.s.1549.5		8	120.83	odd	4
1800.2.d.s.1549.6		8	60.47	odd	4
1800.2.d.t.1549.3		8	60.23	odd	4
1800.2.d.t.1549.4		8	120.107	odd	4
1800.2.k.q.901.7		8	12.11	even	2
1800.2.k.q.901.8		8	24.11	even	2
1800.2.k.t.901.1		8	120.59	even	2
1800.2.k.t.901.2		8	60.59	even	2
2400.2.d.g.49.4		8	5.2	odd	4
2400.2.d.g.49.5		8	40.13	odd	4
2400.2.d.h.49.4		8	40.37	odd	4
2400.2.d.h.49.5		8	5.3	odd	4
2400.2.k.d.1201.3		8	40.29	even	2
2400.2.k.d.1201.7		8	5.4	even	2
2400.2.k.e.1201.2		8	1.1	even	1	trivial
2400.2.k.e.1201.6		8	8.5	even	2	inner
7200.2.d.s.2449.4		8	15.2	even	4
7200.2.d.s.2449.5		8	120.53	even	4
7200.2.d.t.2449.4		8	120.77	even	4
7200.2.d.t.2449.5		8	15.8	even	4
7200.2.k.r.3601.5		8	15.14	odd	2
7200.2.k.r.3601.6		8	120.29	odd	2
7200.2.k.s.3601.3		8	3.2	odd	2
7200.2.k.s.3601.4		8	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -0.0802864 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -0.0802864 q^{7} -1.00000 q^{9} +2.41649i q^{11} +5.26785i q^{13} -0.255918 q^{17} -6.95864i q^{19} +0.0802864i q^{21} +1.64542 q^{23} +1.00000i q^{27} +4.51516i q^{29} -8.29484 q^{31} +2.41649 q^{33} +2.67241i q^{37} +5.26785 q^{39} -8.11921 q^{41} +4.08890i q^{43} -5.70272 q^{47} -6.99355 q^{49} +0.255918i q^{51} +11.5627i q^{53} -6.95864 q^{57} -12.6963i q^{59} +11.9403i q^{61} +0.0802864 q^{63} +7.27979i q^{67} -1.64542i q^{69} +11.3481 q^{71} -12.0779 q^{73} -0.194011i q^{77} -5.50539 q^{79} +1.00000 q^{81} +9.20811i q^{83} +4.51516 q^{87} +11.9173 q^{89} -0.422937i q^{91} +8.29484i q^{93} +8.50539 q^{97} -2.41649i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{7} - 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^7 - 8 * q^9	\( 8 q + 8 q^{7} - 8 q^{9} + 8 q^{23} - 8 q^{31} - 8 q^{57} - 8 q^{63} + 40 q^{71} + 16 q^{73} + 16 q^{79} + 8 q^{81} + 24 q^{87} + 8 q^{97}+O(q^{100}) \) 8 * q + 8 * q^7 - 8 * q^9 + 8 * q^23 - 8 * q^31 - 8 * q^57 - 8 * q^63 + 40 * q^71 + 16 * q^73 + 16 * q^79 + 8 * q^81 + 24 * q^87 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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