LMFDB - Embedded newform 2432.2.c.e.1217.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2432,2,Mod(1217,2432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2432, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2432.1217");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2432 = 2^{7} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2432.c (of order $2$, degree $1$, 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:	$19.4196177716$
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^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1217.2
Root			$0.264658 - 1.38923i$ of defining polynomial
Character	$\chi$	$=$	2432.1217
Dual form			2432.2.c.e.1217.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.24914i q^{3} +1.47068i q^{5} -1.47068 q^{7} -2.05863 q^{9} +O(q^{10})$	$q-2.24914i q^{3} +1.47068i q^{5} -1.47068 q^{7} -2.05863 q^{9} +4.77846i q^{11} -4.49828i q^{13} +3.30777 q^{15} +1.71982 q^{17} -1.00000i q^{19} +3.30777i q^{21} -4.00000 q^{23} +2.83709 q^{25} -2.11727i q^{27} -7.80605i q^{29} -4.13187 q^{31} +10.7474 q^{33} -2.16291i q^{35} +3.30777i q^{37} -10.1173 q^{39} +9.80605 q^{41} -1.83709i q^{43} -3.02760i q^{45} -5.14486 q^{47} -4.83709 q^{49} -3.86813i q^{51} -12.8647i q^{53} -7.02760 q^{55} -2.24914 q^{57} -9.05863i q^{59} -8.91033i q^{61} +3.02760 q^{63} +6.61555 q^{65} +5.92332i q^{67} +8.99656i q^{69} +6.05520 q^{71} +0.335371 q^{73} -6.38101i q^{75} -7.02760i q^{77} -0.366407 q^{79} -10.9379 q^{81} -18.0552i q^{83} +2.52932i q^{85} -17.5569 q^{87} -2.36641 q^{89} +6.61555i q^{91} +9.29317i q^{93} +1.47068 q^{95} -7.05863 q^{97} -9.83709i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 8 q^{7} - 14 q^{9}+O(q^{10}) $ 6 * q - 8 * q^7 - 14 * q^9	$ 6 q - 8 q^{7} - 14 q^{9} + 4 q^{15} - 8 q^{17} - 24 q^{23} + 2 q^{25} - 4 q^{31} + 12 q^{33} - 64 q^{39} + 8 q^{41} - 14 q^{49} - 8 q^{55} + 4 q^{57} - 16 q^{63} + 8 q^{65} - 32 q^{71} - 48 q^{73} + 12 q^{79} + 6 q^{81} - 72 q^{87} + 8 q^{95} - 44 q^{97}+O(q^{100}) $ 6 * q - 8 * q^7 - 14 * q^9 + 4 * q^15 - 8 * q^17 - 24 * q^23 + 2 * q^25 - 4 * q^31 + 12 * q^33 - 64 * q^39 + 8 * q^41 - 14 * q^49 - 8 * q^55 + 4 * q^57 - 16 * q^63 + 8 * q^65 - 32 * q^71 - 48 * q^73 + 12 * q^79 + 6 * q^81 - 72 * q^87 + 8 * q^95 - 44 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2432\mathbb{Z}\right)^\times$.

$n$	$1407$	$1921$	$2053$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 2.24914i	− 1.29854i	−0.760557	−	0.649271i	$-0.775074\pi$
$3$	− 2.24914i	− 1.29854i	0.760557	−	0.649271i	$-0.224926\pi$
$4$	0	0
$5$	1.47068i	0.657710i	0.944380	+	0.328855i	$0.106663\pi$
$5$	1.47068i	0.657710i	−0.944380	+	0.328855i	$0.893337\pi$
$6$	0	0
$7$	−1.47068	−0.555866	−0.277933	−	0.960600i	$-0.589649\pi$
$7$	−1.47068	−0.555866	−0.277933	+	0.960600i	$0.589649\pi$
$8$	0	0
$9$	−2.05863	−0.686211
$10$	0	0
$11$	4.77846i	1.44076i	0.693580	+	0.720380i	$0.256032\pi$
$11$	4.77846i	1.44076i	−0.693580	+	0.720380i	$0.743968\pi$
$12$	0	0
$13$	− 4.49828i	− 1.24760i	−0.781585	−	0.623799i	$-0.785588\pi$
$13$	− 4.49828i	− 1.24760i	0.781585	−	0.623799i	$-0.214412\pi$
$14$	0	0
$15$	3.30777	0.854063
$16$	0	0
$17$	1.71982	0.417119	0.208559	−	0.978010i	$-0.433123\pi$
$17$	1.71982	0.417119	0.208559	+	0.978010i	$0.433123\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	3.30777i	0.721815i
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	2.83709	0.567418
$26$	0	0
$27$	− 2.11727i	− 0.407468i
$28$	0	0
$29$	− 7.80605i	− 1.44955i	−0.688987	−	0.724774i	$-0.741944\pi$
$29$	− 7.80605i	− 1.44955i	0.688987	−	0.724774i	$-0.258056\pi$
$30$	0	0
$31$	−4.13187	−0.742107	−0.371053	−	0.928612i	$-0.621003\pi$
$31$	−4.13187	−0.742107	−0.371053	+	0.928612i	$0.621003\pi$
$32$	0	0
$33$	10.7474	1.87089
$34$	0	0
$35$	− 2.16291i	− 0.365598i
$36$	0	0
$37$	3.30777i	0.543795i	0.962326	+	0.271897i	$0.0876511\pi$
$37$	3.30777i	0.543795i	−0.962326	+	0.271897i	$0.912349\pi$
$38$	0	0
$39$	−10.1173	−1.62006
$40$	0	0
$41$	9.80605	1.53145	0.765724	−	0.643169i	$-0.222381\pi$
$41$	9.80605	1.53145	0.765724	+	0.643169i	$0.222381\pi$
$42$	0	0
$43$	− 1.83709i	− 0.280154i	−0.990141	−	0.140077i	$-0.955265\pi$
$43$	− 1.83709i	− 0.280154i	0.990141	−	0.140077i	$-0.0447350\pi$
$44$	0	0
$45$	− 3.02760i	− 0.451328i
$46$	0	0
$47$	−5.14486	−0.750456	−0.375228	−	0.926933i	$-0.622436\pi$
$47$	−5.14486	−0.750456	−0.375228	+	0.926933i	$0.622436\pi$
$48$	0	0
$49$	−4.83709	−0.691013
$50$	0	0
$51$	− 3.86813i	− 0.541646i
$52$	0	0
$53$	− 12.8647i	− 1.76710i	−0.468336	−	0.883550i	$-0.655146\pi$
$53$	− 12.8647i	− 1.76710i	0.468336	−	0.883550i	$-0.344854\pi$
$54$	0	0
$55$	−7.02760	−0.947601
$56$	0	0
$57$	−2.24914	−0.297906
$58$	0	0
$59$	− 9.05863i	− 1.17933i	−0.807647	−	0.589667i	$-0.799259\pi$
$59$	− 9.05863i	− 1.17933i	0.807647	−	0.589667i	$-0.200741\pi$
$60$	0	0
$61$	− 8.91033i	− 1.14085i	−0.821349	−	0.570426i	$-0.806778\pi$
$61$	− 8.91033i	− 1.14085i	0.821349	−	0.570426i	$-0.193222\pi$
$62$	0	0
$63$	3.02760	0.381441
$64$	0	0
$65$	6.61555	0.820558
$66$	0	0
$67$	5.92332i	0.723649i	0.932246	+	0.361824i	$0.117846\pi$
$67$	5.92332i	0.723649i	−0.932246	+	0.361824i	$0.882154\pi$
$68$	0	0
$69$	8.99656i	1.08306i
$70$	0	0
$71$	6.05520	0.718619	0.359310	−	0.933218i	$-0.383012\pi$
$71$	6.05520	0.718619	0.359310	+	0.933218i	$0.383012\pi$
$72$	0	0
$73$	0.335371	0.0392522	0.0196261	−	0.999807i	$-0.493752\pi$
$73$	0.335371	0.0392522	0.0196261	+	0.999807i	$0.493752\pi$
$74$	0	0
$75$	− 6.38101i	− 0.736816i
$76$	0	0
$77$	− 7.02760i	− 0.800869i
$78$	0	0
$79$	−0.366407	−0.0412240	−0.0206120	−	0.999788i	$-0.506561\pi$
$79$	−0.366407	−0.0412240	−0.0206120	+	0.999788i	$0.506561\pi$
$80$	0	0
$81$	−10.9379	−1.21533
$82$	0	0
$83$	− 18.0552i	− 1.98182i	−0.134544	−	0.990908i	$-0.542957\pi$
$83$	− 18.0552i	− 1.98182i	0.134544	−	0.990908i	$-0.457043\pi$
$84$	0	0
$85$	2.52932i	0.274343i
$86$	0	0
$87$	−17.5569	−1.88230
$88$	0	0
$89$	−2.36641	−0.250839	−0.125419	−	0.992104i	$-0.540028\pi$
$89$	−2.36641	−0.250839	−0.125419	+	0.992104i	$0.540028\pi$
$90$	0	0
$91$	6.61555i	0.693498i
$92$	0	0
$93$	9.29317i	0.963656i
$94$	0	0
$95$	1.47068	0.150889
$96$	0	0
$97$	−7.05863	−0.716696	−0.358348	−	0.933588i	$-0.616660\pi$
$97$	−7.05863	−0.716696	−0.358348	+	0.933588i	$0.616660\pi$
$98$	0	0
$99$	− 9.83709i	− 0.988665i
$100$	0	0
$101$	− 6.94137i	− 0.690692i	−0.938475	−	0.345346i	$-0.887762\pi$
$101$	− 6.94137i	− 0.690692i	0.938475	−	0.345346i	$-0.112238\pi$
$102$	0	0
$103$	−0.824101	−0.0812010	−0.0406005	−	0.999175i	$-0.512927\pi$
$103$	−0.824101	−0.0812010	−0.0406005	+	0.999175i	$0.512927\pi$
$104$	0	0
$105$	−4.86469	−0.474745
$106$	0	0
$107$	− 11.4396i	− 1.10591i	−0.833210	−	0.552956i	$-0.813500\pi$
$107$	− 11.4396i	− 1.10591i	0.833210	−	0.552956i	$-0.186500\pi$
$108$	0	0
$109$	4.13187i	0.395762i	0.980226	+	0.197881i	$0.0634059\pi$
$109$	4.13187i	0.395762i	−0.980226	+	0.197881i	$0.936594\pi$
$110$	0	0
$111$	7.43965	0.706140
$112$	0	0
$113$	−7.92332	−0.745363	−0.372682	−	0.927959i	$-0.621562\pi$
$113$	−7.92332	−0.745363	−0.372682	+	0.927959i	$0.621562\pi$
$114$	0	0
$115$	− 5.88273i	− 0.548568i
$116$	0	0
$117$	9.26031i	0.856116i
$118$	0	0
$119$	−2.52932	−0.231862
$120$	0	0
$121$	−11.8337	−1.07579
$122$	0	0
$123$	− 22.0552i	− 1.98865i
$124$	0	0
$125$	11.5259i	1.03091i
$126$	0	0
$127$	1.19051	0.105640	0.0528202	−	0.998604i	$-0.483179\pi$
$127$	1.19051	0.105640	0.0528202	+	0.998604i	$0.483179\pi$
$128$	0	0
$129$	−4.13187	−0.363791
$130$	0	0
$131$	2.39744i	0.209466i	0.994500	+	0.104733i	$0.0333987\pi$
$131$	2.39744i	0.209466i	−0.994500	+	0.104733i	$0.966601\pi$
$132$	0	0
$133$	1.47068i	0.127524i
$134$	0	0
$135$	3.11383	0.267996
$136$	0	0
$137$	9.39400	0.802584	0.401292	−	0.915950i	$-0.368561\pi$
$137$	9.39400	0.802584	0.401292	+	0.915950i	$0.368561\pi$
$138$	0	0
$139$	12.2181i	1.03633i	0.855282	+	0.518163i	$0.173384\pi$
$139$	12.2181i	1.03633i	−0.855282	+	0.518163i	$0.826616\pi$
$140$	0	0
$141$	11.5715i	0.974498i
$142$	0	0
$143$	21.4948	1.79749
$144$	0	0
$145$	11.4802	0.953382
$146$	0	0
$147$	10.8793i	0.897309i
$148$	0	0
$149$	16.8190i	1.37787i	0.724823	+	0.688935i	$0.241921\pi$
$149$	16.8190i	1.37787i	−0.724823	+	0.688935i	$0.758079\pi$
$150$	0	0
$151$	7.17590	0.583966	0.291983	−	0.956423i	$-0.405685\pi$
$151$	7.17590	0.583966	0.291983	+	0.956423i	$0.405685\pi$
$152$	0	0
$153$	−3.54049	−0.286231
$154$	0	0
$155$	− 6.07668i	− 0.488091i
$156$	0	0
$157$	− 15.6742i	− 1.25094i	−0.780250	−	0.625468i	$-0.784908\pi$
$157$	− 15.6742i	− 1.25094i	0.780250	−	0.625468i	$-0.215092\pi$
$158$	0	0
$159$	−28.9345	−2.29465
$160$	0	0
$161$	5.88273	0.463624
$162$	0	0
$163$	9.32238i	0.730185i	0.930971	+	0.365093i	$0.118963\pi$
$163$	9.32238i	0.730185i	−0.930971	+	0.365093i	$0.881037\pi$
$164$	0	0
$165$	15.8061i	1.23050i
$166$	0	0
$167$	−20.3043	−1.57120	−0.785598	−	0.618737i	$-0.787645\pi$
$167$	−20.3043	−1.57120	−0.785598	+	0.618737i	$0.787645\pi$
$168$	0	0
$169$	−7.23453	−0.556503
$170$	0	0
$171$	2.05863i	0.157428i
$172$	0	0
$173$	0.560352i	0.0426028i	0.999773	+	0.0213014i	$0.00678096\pi$
$173$	0.560352i	0.0426028i	−0.999773	+	0.0213014i	$0.993219\pi$
$174$	0	0
$175$	−4.17246	−0.315408
$176$	0	0
$177$	−20.3741	−1.53141
$178$	0	0
$179$	4.36641i	0.326361i	0.986596	+	0.163180i	$0.0521752\pi$
$179$	4.36641i	0.326361i	−0.986596	+	0.163180i	$0.947825\pi$
$180$	0	0
$181$	5.05863i	0.376005i	0.982169	+	0.188003i	$0.0602013\pi$
$181$	5.05863i	0.376005i	−0.982169	+	0.188003i	$0.939799\pi$
$182$	0	0
$183$	−20.0406	−1.48144
$184$	0	0
$185$	−4.86469	−0.357659
$186$	0	0
$187$	8.21811i	0.600967i
$188$	0	0
$189$	3.11383i	0.226498i
$190$	0	0
$191$	−2.46725	−0.178524	−0.0892618	−	0.996008i	$-0.528451\pi$
$191$	−2.46725	−0.178524	−0.0892618	+	0.996008i	$0.528451\pi$
$192$	0	0
$193$	12.2491	0.881712	0.440856	−	0.897578i	$-0.354675\pi$
$193$	12.2491	0.881712	0.440856	+	0.897578i	$0.354675\pi$
$194$	0	0
$195$	− 14.8793i	− 1.06553i
$196$	0	0
$197$	− 1.23109i	− 0.0877119i	−0.999038	−	0.0438559i	$-0.986036\pi$
$197$	− 1.23109i	− 0.0877119i	0.999038	−	0.0438559i	$-0.0139643\pi$
$198$	0	0
$199$	−27.0276	−1.91594	−0.957968	−	0.286876i	$-0.907383\pi$
$199$	−27.0276	−1.91594	−0.957968	+	0.286876i	$0.907383\pi$
$200$	0	0
$201$	13.3224	0.939688
$202$	0	0
$203$	11.4802i	0.805755i
$204$	0	0
$205$	14.4216i	1.00725i
$206$	0	0
$207$	8.23453	0.572340
$208$	0	0
$209$	4.77846	0.330533
$210$	0	0
$211$	0.325819i	0.0224303i	0.999937	+	0.0112152i	$0.00356997\pi$
$211$	0.325819i	0.0224303i	−0.999937	+	0.0112152i	$0.996430\pi$
$212$	0	0
$213$	− 13.6190i	− 0.933157i
$214$	0	0
$215$	2.70178	0.184260
$216$	0	0
$217$	6.07668	0.412512
$218$	0	0
$219$	− 0.754297i	− 0.0509707i
$220$	0	0
$221$	− 7.73625i	− 0.520397i
$222$	0	0
$223$	23.4182	1.56820	0.784098	−	0.620637i	$-0.213126\pi$
$223$	23.4182	1.56820	0.784098	+	0.620637i	$0.213126\pi$
$224$	0	0
$225$	−5.84053	−0.389369
$226$	0	0
$227$	− 23.8466i	− 1.58276i	−0.611326	−	0.791379i	$-0.709364\pi$
$227$	− 23.8466i	− 1.58276i	0.611326	−	0.791379i	$-0.290636\pi$
$228$	0	0
$229$	− 13.9690i	− 0.923095i	−0.887115	−	0.461548i	$-0.847294\pi$
$229$	− 13.9690i	− 0.923095i	0.887115	−	0.461548i	$-0.152706\pi$
$230$	0	0
$231$	−15.8061	−1.03996
$232$	0	0
$233$	12.8337	0.840761	0.420380	−	0.907348i	$-0.361897\pi$
$233$	12.8337	0.840761	0.420380	+	0.907348i	$0.361897\pi$
$234$	0	0
$235$	− 7.56647i	− 0.493582i
$236$	0	0
$237$	0.824101i	0.0535311i
$238$	0	0
$239$	24.2587	1.56916	0.784582	−	0.620025i	$-0.212877\pi$
$239$	24.2587	1.56916	0.784582	+	0.620025i	$0.212877\pi$
$240$	0	0
$241$	−26.0698	−1.67930	−0.839652	−	0.543125i	$-0.817241\pi$
$241$	−26.0698	−1.67930	−0.839652	+	0.543125i	$0.817241\pi$
$242$	0	0
$243$	18.2491i	1.17068i
$244$	0	0
$245$	− 7.11383i	− 0.454486i
$246$	0	0
$247$	−4.49828	−0.286219
$248$	0	0
$249$	−40.6087	−2.57347
$250$	0	0
$251$	− 23.3319i	− 1.47270i	−0.676602	−	0.736349i	$-0.736548\pi$
$251$	− 23.3319i	− 1.47270i	0.676602	−	0.736349i	$-0.263452\pi$
$252$	0	0
$253$	− 19.1138i	− 1.20168i
$254$	0	0
$255$	5.68879	0.356246
$256$	0	0
$257$	2.82410	0.176163	0.0880813	−	0.996113i	$-0.471926\pi$
$257$	2.82410	0.176163	0.0880813	+	0.996113i	$0.471926\pi$
$258$	0	0
$259$	− 4.86469i	− 0.302277i
$260$	0	0
$261$	16.0698i	0.994696i
$262$	0	0
$263$	−7.08967	−0.437168	−0.218584	−	0.975818i	$-0.570144\pi$
$263$	−7.08967	−0.437168	−0.218584	+	0.975818i	$0.570144\pi$
$264$	0	0
$265$	18.9199	1.16224
$266$	0	0
$267$	5.32238i	0.325724i
$268$	0	0
$269$	20.8647i	1.27214i	0.771630	+	0.636071i	$0.219442\pi$
$269$	20.8647i	1.27214i	−0.771630	+	0.636071i	$0.780558\pi$
$270$	0	0
$271$	0.0620710	0.00377055	0.00188527	−	0.999998i	$-0.499400\pi$
$271$	0.0620710	0.00377055	0.00188527	+	0.999998i	$0.499400\pi$
$272$	0	0
$273$	14.8793	0.900536
$274$	0	0
$275$	13.5569i	0.817513i
$276$	0	0
$277$	7.82248i	0.470007i	0.971994	+	0.235004i	$0.0755102\pi$
$277$	7.82248i	0.470007i	−0.971994	+	0.235004i	$0.924490\pi$
$278$	0	0
$279$	8.50601	0.509242
$280$	0	0
$281$	−20.6448	−1.23156	−0.615782	−	0.787917i	$-0.711160\pi$
$281$	−20.6448	−1.23156	−0.615782	+	0.787917i	$0.711160\pi$
$282$	0	0
$283$	− 21.3871i	− 1.27133i	−0.771964	−	0.635666i	$-0.780725\pi$
$283$	− 21.3871i	− 1.27133i	0.771964	−	0.635666i	$-0.219275\pi$
$284$	0	0
$285$	− 3.30777i	− 0.195936i
$286$	0	0
$287$	−14.4216	−0.851280
$288$	0	0
$289$	−14.0422	−0.826012
$290$	0	0
$291$	15.8759i	0.930659i
$292$	0	0
$293$	− 24.6087i	− 1.43765i	−0.695189	−	0.718827i	$-0.744679\pi$
$293$	− 24.6087i	− 1.43765i	0.695189	−	0.718827i	$-0.255321\pi$
$294$	0	0
$295$	13.3224	0.775659
$296$	0	0
$297$	10.1173	0.587063
$298$	0	0
$299$	17.9931i	1.04057i
$300$	0	0
$301$	2.70178i	0.155728i
$302$	0	0
$303$	−15.6121	−0.896892
$304$	0	0
$305$	13.1043	0.750349
$306$	0	0
$307$	− 10.4216i	− 0.594792i	−0.954754	−	0.297396i	$-0.903882\pi$
$307$	− 10.4216i	− 0.594792i	0.954754	−	0.297396i	$-0.0961182\pi$
$308$	0	0
$309$	1.85352i	0.105443i
$310$	0	0
$311$	−7.91377	−0.448749	−0.224374	−	0.974503i	$-0.572034\pi$
$311$	−7.91377	−0.448749	−0.224374	+	0.974503i	$0.572034\pi$
$312$	0	0
$313$	17.7846	1.00524	0.502622	−	0.864506i	$-0.332369\pi$
$313$	17.7846	1.00524	0.502622	+	0.864506i	$0.332369\pi$
$314$	0	0
$315$	4.45264i	0.250878i
$316$	0	0
$317$	30.0552i	1.68807i	0.536290	+	0.844034i	$0.319825\pi$
$317$	30.0552i	1.68807i	−0.536290	+	0.844034i	$0.680175\pi$
$318$	0	0
$319$	37.3009	2.08845
$320$	0	0
$321$	−25.7294	−1.43607
$322$	0	0
$323$	− 1.71982i	− 0.0956936i
$324$	0	0
$325$	− 12.7620i	− 0.707910i
$326$	0	0
$327$	9.29317	0.513913
$328$	0	0
$329$	7.56647	0.417153
$330$	0	0
$331$	18.8095i	1.03386i	0.856027	+	0.516932i	$0.172926\pi$
$331$	18.8095i	1.03386i	−0.856027	+	0.516932i	$0.827074\pi$
$332$	0	0
$333$	− 6.80949i	− 0.373158i
$334$	0	0
$335$	−8.71133	−0.475951
$336$	0	0
$337$	30.2017	1.64519	0.822595	−	0.568628i	$-0.192525\pi$
$337$	30.2017	1.64519	0.822595	+	0.568628i	$0.192525\pi$
$338$	0	0
$339$	17.8207i	0.967886i
$340$	0	0
$341$	− 19.7440i	− 1.06920i
$342$	0	0
$343$	17.4086	0.939977
$344$	0	0
$345$	−13.2311	−0.712338
$346$	0	0
$347$	− 9.36802i	− 0.502902i	−0.967870	−	0.251451i	$-0.919092\pi$
$347$	− 9.36802i	− 0.502902i	0.967870	−	0.251451i	$-0.0809077\pi$
$348$	0	0
$349$	21.3173i	1.14109i	0.821266	+	0.570545i	$0.193268\pi$
$349$	21.3173i	1.14109i	−0.821266	+	0.570545i	$0.806732\pi$
$350$	0	0
$351$	−9.52406	−0.508357
$352$	0	0
$353$	18.4362	0.981260	0.490630	−	0.871368i	$-0.336767\pi$
$353$	18.4362	0.981260	0.490630	+	0.871368i	$0.336767\pi$
$354$	0	0
$355$	8.90528i	0.472643i
$356$	0	0
$357$	5.68879i	0.301083i
$358$	0	0
$359$	−37.1932	−1.96298	−0.981491	−	0.191510i	$-0.938661\pi$
$359$	−37.1932	−1.96298	−0.981491	+	0.191510i	$0.938661\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	26.6155i	1.39695i
$364$	0	0
$365$	0.493225i	0.0258166i
$366$	0	0
$367$	11.6121	0.606147	0.303074	−	0.952967i	$-0.401987\pi$
$367$	11.6121	0.606147	0.303074	+	0.952967i	$0.401987\pi$
$368$	0	0
$369$	−20.1871	−1.05090
$370$	0	0
$371$	18.9199i	0.982271i
$372$	0	0
$373$	− 24.6087i	− 1.27419i	−0.770786	−	0.637094i	$-0.780136\pi$
$373$	− 24.6087i	− 1.27419i	0.770786	−	0.637094i	$-0.219864\pi$
$374$	0	0
$375$	25.9233	1.33867
$376$	0	0
$377$	−35.1138	−1.80845
$378$	0	0
$379$	1.05863i	0.0543783i	0.999630	+	0.0271892i	$0.00865565\pi$
$379$	1.05863i	0.0543783i	−0.999630	+	0.0271892i	$0.991344\pi$
$380$	0	0
$381$	− 2.67762i	− 0.137179i
$382$	0	0
$383$	18.4837	0.944472	0.472236	−	0.881472i	$-0.343447\pi$
$383$	18.4837	0.944472	0.472236	+	0.881472i	$0.343447\pi$
$384$	0	0
$385$	10.3354	0.526739
$386$	0	0
$387$	3.78189i	0.192245i
$388$	0	0
$389$	− 2.46725i	− 0.125094i	−0.998042	−	0.0625472i	$-0.980078\pi$
$389$	− 2.46725i	− 0.125094i	0.998042	−	0.0625472i	$-0.0199224\pi$
$390$	0	0
$391$	−6.87930	−0.347901
$392$	0	0
$393$	5.39218	0.272000
$394$	0	0
$395$	− 0.538868i	− 0.0271134i
$396$	0	0
$397$	6.91721i	0.347165i	0.984819	+	0.173582i	$0.0555343\pi$
$397$	6.91721i	0.347165i	−0.984819	+	0.173582i	$0.944466\pi$
$398$	0	0
$399$	3.30777	0.165596
$400$	0	0
$401$	20.8578	1.04159	0.520795	−	0.853682i	$-0.325636\pi$
$401$	20.8578	1.04159	0.520795	+	0.853682i	$0.325636\pi$
$402$	0	0
$403$	18.5863i	0.925851i
$404$	0	0
$405$	− 16.0862i	− 0.799331i
$406$	0	0
$407$	−15.8061	−0.783477
$408$	0	0
$409$	24.9820	1.23528	0.617639	−	0.786462i	$-0.288089\pi$
$409$	24.9820	1.23528	0.617639	+	0.786462i	$0.288089\pi$
$410$	0	0
$411$	− 21.1284i	− 1.04219i
$412$	0	0
$413$	13.3224i	0.655552i
$414$	0	0
$415$	26.5535	1.30346
$416$	0	0
$417$	27.4802	1.34571
$418$	0	0
$419$	− 31.6742i	− 1.54738i	−0.633561	−	0.773692i	$-0.718408\pi$
$419$	− 31.6742i	− 1.54738i	0.633561	−	0.773692i	$-0.281592\pi$
$420$	0	0
$421$	− 38.5941i	− 1.88096i	−0.339850	−	0.940480i	$-0.610376\pi$
$421$	− 38.5941i	− 1.88096i	0.339850	−	0.940480i	$-0.389624\pi$
$422$	0	0
$423$	10.5914	0.514971
$424$	0	0
$425$	4.87930	0.236681
$426$	0	0
$427$	13.1043i	0.634160i
$428$	0	0
$429$	− 48.3449i	− 2.33411i
$430$	0	0
$431$	37.4948	1.80606	0.903032	−	0.429574i	$-0.141336\pi$
$431$	37.4948	1.80606	0.903032	+	0.429574i	$0.141336\pi$
$432$	0	0
$433$	13.4182	0.644836	0.322418	−	0.946597i	$-0.395504\pi$
$433$	13.4182	0.644836	0.322418	+	0.946597i	$0.395504\pi$
$434$	0	0
$435$	− 25.8207i	− 1.23801i
$436$	0	0
$437$	4.00000i	0.191346i
$438$	0	0
$439$	36.3043	1.73271	0.866356	−	0.499427i	$-0.166456\pi$
$439$	36.3043	1.73271	0.866356	+	0.499427i	$0.166456\pi$
$440$	0	0
$441$	9.95779	0.474181
$442$	0	0
$443$	27.9215i	1.32659i	0.748358	+	0.663295i	$0.230843\pi$
$443$	27.9215i	1.32659i	−0.748358	+	0.663295i	$0.769157\pi$
$444$	0	0
$445$	− 3.48024i	− 0.164979i
$446$	0	0
$447$	37.8284	1.78922
$448$	0	0
$449$	−12.2086	−0.576157	−0.288079	−	0.957607i	$-0.593016\pi$
$449$	−12.2086	−0.576157	−0.288079	+	0.957607i	$0.593016\pi$
$450$	0	0
$451$	46.8578i	2.20645i
$452$	0	0
$453$	− 16.1396i	− 0.758305i
$454$	0	0
$455$	−9.72938	−0.456120
$456$	0	0
$457$	4.42666	0.207070	0.103535	−	0.994626i	$-0.466985\pi$
$457$	4.42666	0.207070	0.103535	+	0.994626i	$0.466985\pi$
$458$	0	0
$459$	− 3.64133i	− 0.169963i
$460$	0	0
$461$	6.96553i	0.324417i	0.986757	+	0.162208i	$0.0518617\pi$
$461$	6.96553i	0.324417i	−0.986757	+	0.162208i	$0.948138\pi$
$462$	0	0
$463$	−30.7501	−1.42908	−0.714539	−	0.699595i	$-0.753364\pi$
$463$	−30.7501	−1.42908	−0.714539	+	0.699595i	$0.753364\pi$
$464$	0	0
$465$	−13.6673	−0.633806
$466$	0	0
$467$	5.16635i	0.239070i	0.992830	+	0.119535i	$0.0381404\pi$
$467$	5.16635i	0.239070i	−0.992830	+	0.119535i	$0.961860\pi$
$468$	0	0
$469$	− 8.71133i	− 0.402252i
$470$	0	0
$471$	−35.2534	−1.62439
$472$	0	0
$473$	8.77846	0.403634
$474$	0	0
$475$	− 2.83709i	− 0.130175i
$476$	0	0
$477$	26.4837i	1.21260i
$478$	0	0
$479$	−23.3776	−1.06815	−0.534074	−	0.845437i	$-0.679340\pi$
$479$	−23.3776	−1.06815	−0.534074	+	0.845437i	$0.679340\pi$
$480$	0	0
$481$	14.8793	0.678437
$482$	0	0
$483$	− 13.2311i	− 0.602036i
$484$	0	0
$485$	− 10.3810i	− 0.471378i
$486$	0	0
$487$	41.9018	1.89875	0.949377	−	0.314140i	$-0.101716\pi$
$487$	41.9018	1.89875	0.949377	+	0.314140i	$0.101716\pi$
$488$	0	0
$489$	20.9673	0.948176
$490$	0	0
$491$	4.90528i	0.221372i	0.993855	+	0.110686i	$0.0353048\pi$
$491$	4.90528i	0.221372i	−0.993855	+	0.110686i	$0.964695\pi$
$492$	0	0
$493$	− 13.4250i	− 0.604633i
$494$	0	0
$495$	14.4672	0.650254
$496$	0	0
$497$	−8.90528	−0.399456
$498$	0	0
$499$	− 5.77502i	− 0.258525i	−0.991610	−	0.129263i	$-0.958739\pi$
$499$	− 5.77502i	− 0.258525i	0.991610	−	0.129263i	$-0.0412610\pi$
$500$	0	0
$501$	45.6673i	2.04026i
$502$	0	0
$503$	31.3776	1.39906	0.699529	−	0.714605i	$-0.253393\pi$
$503$	31.3776	1.39906	0.699529	+	0.714605i	$0.253393\pi$
$504$	0	0
$505$	10.2086	0.454275
$506$	0	0
$507$	16.2715i	0.722642i
$508$	0	0
$509$	− 17.2603i	− 0.765050i	−0.923945	−	0.382525i	$-0.875055\pi$
$509$	− 17.2603i	− 0.765050i	0.923945	−	0.382525i	$-0.124945\pi$
$510$	0	0
$511$	−0.493225	−0.0218190
$512$	0	0
$513$	−2.11727	−0.0934796
$514$	0	0
$515$	− 1.21199i	− 0.0534067i
$516$	0	0
$517$	− 24.5845i	− 1.08123i
$518$	0	0
$519$	1.26031	0.0553215
$520$	0	0
$521$	18.8939	0.827757	0.413878	−	0.910332i	$-0.364174\pi$
$521$	18.8939	0.827757	0.413878	+	0.910332i	$0.364174\pi$
$522$	0	0
$523$	40.2017i	1.75790i	0.476917	+	0.878948i	$0.341754\pi$
$523$	40.2017i	1.75790i	−0.476917	+	0.878948i	$0.658246\pi$
$524$	0	0
$525$	9.38445i	0.409571i
$526$	0	0
$527$	−7.10610	−0.309546
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	18.6484i	0.809272i
$532$	0	0
$533$	− 44.1104i	− 1.91063i
$534$	0	0
$535$	16.8241	0.727369
$536$	0	0
$537$	9.82066	0.423793
$538$	0	0
$539$	− 23.1138i	− 0.995583i
$540$	0	0
$541$	22.4052i	0.963274i	0.876371	+	0.481637i	$0.159958\pi$
$541$	22.4052i	0.963274i	−0.876371	+	0.481637i	$0.840042\pi$
$542$	0	0
$543$	11.3776	0.488259
$544$	0	0
$545$	−6.07668	−0.260296
$546$	0	0
$547$	− 37.9931i	− 1.62447i	−0.583331	−	0.812234i	$-0.698251\pi$
$547$	− 37.9931i	− 1.62447i	0.583331	−	0.812234i	$-0.301749\pi$
$548$	0	0
$549$	18.3431i	0.782865i
$550$	0	0
$551$	−7.80605	−0.332549
$552$	0	0
$553$	0.538868	0.0229150
$554$	0	0
$555$	10.9414i	0.464435i
$556$	0	0
$557$	− 16.1775i	− 0.685463i	−0.939433	−	0.342732i	$-0.888648\pi$
$557$	− 16.1775i	− 0.685463i	0.939433	−	0.342732i	$-0.111352\pi$
$558$	0	0
$559$	−8.26375	−0.349519
$560$	0	0
$561$	18.4837	0.780381
$562$	0	0
$563$	− 9.59750i	− 0.404486i	−0.979335	−	0.202243i	$-0.935177\pi$
$563$	− 9.59750i	− 0.404486i	0.979335	−	0.202243i	$-0.0648232\pi$
$564$	0	0
$565$	− 11.6527i	− 0.490233i
$566$	0	0
$567$	16.0862	0.675558
$568$	0	0
$569$	−23.9018	−1.00202	−0.501009	−	0.865442i	$-0.667037\pi$
$569$	−23.9018	−1.00202	−0.501009	+	0.865442i	$0.667037\pi$
$570$	0	0
$571$	36.1725i	1.51377i	0.653548	+	0.756885i	$0.273280\pi$
$571$	36.1725i	1.51377i	−0.653548	+	0.756885i	$0.726720\pi$
$572$	0	0
$573$	5.54918i	0.231820i
$574$	0	0
$575$	−11.3484	−0.473259
$576$	0	0
$577$	−12.5699	−0.523292	−0.261646	−	0.965164i	$-0.584265\pi$
$577$	−12.5699	−0.523292	−0.261646	+	0.965164i	$0.584265\pi$
$578$	0	0
$579$	− 27.5500i	− 1.14494i
$580$	0	0
$581$	26.5535i	1.10162i
$582$	0	0
$583$	61.4734	2.54597
$584$	0	0
$585$	−13.6190	−0.563076
$586$	0	0
$587$	0.107714i	0.00444585i	0.999998	+	0.00222292i	$0.000707579\pi$
$587$	0.107714i	0.00444585i	−0.999998	+	0.00222292i	$0.999292\pi$
$588$	0	0
$589$	4.13187i	0.170251i
$590$	0	0
$591$	−2.76891	−0.113898
$592$	0	0
$593$	25.7846	1.05885	0.529423	−	0.848358i	$-0.322409\pi$
$593$	25.7846	1.05885	0.529423	+	0.848358i	$0.322409\pi$
$594$	0	0
$595$	− 3.71982i	− 0.152498i
$596$	0	0
$597$	60.7889i	2.48792i
$598$	0	0
$599$	−23.6336	−0.965642	−0.482821	−	0.875719i	$-0.660388\pi$
$599$	−23.6336	−0.965642	−0.482821	+	0.875719i	$0.660388\pi$
$600$	0	0
$601$	−35.3415	−1.44161	−0.720805	−	0.693138i	$-0.756228\pi$
$601$	−35.3415	−1.44161	−0.720805	+	0.693138i	$0.756228\pi$
$602$	0	0
$603$	− 12.1939i	− 0.496576i
$604$	0	0
$605$	− 17.4036i	− 0.707555i
$606$	0	0
$607$	−28.8432	−1.17071	−0.585355	−	0.810777i	$-0.699045\pi$
$607$	−28.8432	−1.17071	−0.585355	+	0.810777i	$0.699045\pi$
$608$	0	0
$609$	25.8207	1.04631
$610$	0	0
$611$	23.1430i	0.936267i
$612$	0	0
$613$	27.7604i	1.12123i	0.828076	+	0.560616i	$0.189436\pi$
$613$	27.7604i	1.12123i	−0.828076	+	0.560616i	$0.810564\pi$
$614$	0	0
$615$	32.4362	1.30795
$616$	0	0
$617$	−32.5370	−1.30989	−0.654946	−	0.755676i	$-0.727309\pi$
$617$	−32.5370	−1.30989	−0.654946	+	0.755676i	$0.727309\pi$
$618$	0	0
$619$	6.20855i	0.249543i	0.992186	+	0.124771i	$0.0398198\pi$
$619$	6.20855i	0.249543i	−0.992186	+	0.124771i	$0.960180\pi$
$620$	0	0
$621$	8.46907i	0.339852i
$622$	0	0
$623$	3.48024	0.139433
$624$	0	0
$625$	−2.76547	−0.110619
$626$	0	0
$627$	− 10.7474i	− 0.429211i
$628$	0	0
$629$	5.68879i	0.226827i
$630$	0	0
$631$	−2.67256	−0.106393	−0.0531965	−	0.998584i	$-0.516941\pi$
$631$	−2.67256	−0.106393	−0.0531965	+	0.998584i	$0.516941\pi$
$632$	0	0
$633$	0.732814	0.0291267
$634$	0	0
$635$	1.75086i	0.0694807i
$636$	0	0
$637$	21.7586i	0.862107i
$638$	0	0
$639$	−12.4654	−0.493125
$640$	0	0
$641$	−24.6639	−0.974164	−0.487082	−	0.873356i	$-0.661939\pi$
$641$	−24.6639	−0.974164	−0.487082	+	0.873356i	$0.661939\pi$
$642$	0	0
$643$	25.5405i	1.00722i	0.863932	+	0.503609i	$0.167995\pi$
$643$	25.5405i	1.00722i	−0.863932	+	0.503609i	$0.832005\pi$
$644$	0	0
$645$	− 6.07668i	− 0.239269i
$646$	0	0
$647$	21.7535	0.855220	0.427610	−	0.903963i	$-0.359356\pi$
$647$	21.7535	0.855220	0.427610	+	0.903963i	$0.359356\pi$
$648$	0	0
$649$	43.2863	1.69914
$650$	0	0
$651$	− 13.6673i	− 0.535664i
$652$	0	0
$653$	− 9.37940i	− 0.367044i	−0.983016	−	0.183522i	$-0.941250\pi$
$653$	− 9.37940i	− 0.367044i	0.983016	−	0.183522i	$-0.0587499\pi$
$654$	0	0
$655$	−3.52588	−0.137767
$656$	0	0
$657$	−0.690407	−0.0269353
$658$	0	0
$659$	39.8138i	1.55092i	0.631394	+	0.775462i	$0.282483\pi$
$659$	39.8138i	1.55092i	−0.631394	+	0.775462i	$0.717517\pi$
$660$	0	0
$661$	19.1138i	0.743442i	0.928345	+	0.371721i	$0.121232\pi$
$661$	19.1138i	0.743442i	−0.928345	+	0.371721i	$0.878768\pi$
$662$	0	0
$663$	−17.3999	−0.675757
$664$	0	0
$665$	−2.16291	−0.0838740
$666$	0	0
$667$	31.2242i	1.20901i
$668$	0	0
$669$	− 52.6707i	− 2.03637i
$670$	0	0
$671$	42.5776	1.64369
$672$	0	0
$673$	−21.1138	−0.813878	−0.406939	−	0.913455i	$-0.633404\pi$
$673$	−21.1138	−0.813878	−0.406939	+	0.913455i	$0.633404\pi$
$674$	0	0
$675$	− 6.00688i	− 0.231205i
$676$	0	0
$677$	36.3855i	1.39841i	0.714922	+	0.699204i	$0.246462\pi$
$677$	36.3855i	1.39841i	−0.714922	+	0.699204i	$0.753538\pi$
$678$	0	0
$679$	10.3810	0.398387
$680$	0	0
$681$	−53.6344	−2.05528
$682$	0	0
$683$	− 12.6087i	− 0.482457i	−0.970468	−	0.241229i	$-0.922450\pi$
$683$	− 12.6087i	− 0.482457i	0.970468	−	0.241229i	$-0.0775504\pi$
$684$	0	0
$685$	13.8156i	0.527867i
$686$	0	0
$687$	−31.4182	−1.19868
$688$	0	0
$689$	−57.8690	−2.20463
$690$	0	0
$691$	16.8405i	0.640644i	0.947309	+	0.320322i	$0.103791\pi$
$691$	16.8405i	0.640644i	−0.947309	+	0.320322i	$0.896209\pi$
$692$	0	0
$693$	14.4672i	0.549565i
$694$	0	0
$695$	−17.9690	−0.681602
$696$	0	0
$697$	16.8647	0.638796
$698$	0	0
$699$	− 28.8647i	− 1.09176i
$700$	0	0
$701$	33.9311i	1.28156i	0.767725	+	0.640779i	$0.221389\pi$
$701$	33.9311i	1.28156i	−0.767725	+	0.640779i	$0.778611\pi$
$702$	0	0
$703$	3.30777	0.124755
$704$	0	0
$705$	−17.0180	−0.640937
$706$	0	0
$707$	10.2086i	0.383932i
$708$	0	0
$709$	30.8984i	1.16041i	0.814469	+	0.580207i	$0.197028\pi$
$709$	30.8984i	1.16041i	−0.814469	+	0.580207i	$0.802972\pi$
$710$	0	0
$711$	0.754297	0.0282884
$712$	0	0
$713$	16.5275	0.618960
$714$	0	0
$715$	31.6121i	1.18223i
$716$	0	0
$717$	− 54.5612i	− 2.03763i
$718$	0	0
$719$	26.8122	0.999925	0.499963	−	0.866047i	$-0.333347\pi$
$719$	26.8122	0.999925	0.499963	+	0.866047i	$0.333347\pi$
$720$	0	0
$721$	1.21199	0.0451369
$722$	0	0
$723$	58.6347i	2.18065i
$724$	0	0
$725$	− 22.1465i	− 0.822500i
$726$	0	0
$727$	−30.1897	−1.11968	−0.559838	−	0.828602i	$-0.689137\pi$
$727$	−30.1897	−1.11968	−0.559838	+	0.828602i	$0.689137\pi$
$728$	0	0
$729$	8.23109	0.304855
$730$	0	0
$731$	− 3.15947i	− 0.116857i
$732$	0	0
$733$	− 37.8207i	− 1.39694i	−0.715640	−	0.698469i	$-0.753865\pi$
$733$	− 37.8207i	− 1.39694i	0.715640	−	0.698469i	$-0.246135\pi$
$734$	0	0
$735$	−16.0000	−0.590169
$736$	0	0
$737$	−28.3043	−1.04260
$738$	0	0
$739$	− 39.3803i	− 1.44863i	−0.689471	−	0.724313i	$-0.742157\pi$
$739$	− 39.3803i	− 1.44863i	0.689471	−	0.724313i	$-0.257843\pi$
$740$	0	0
$741$	10.1173i	0.371667i
$742$	0	0
$743$	−13.9854	−0.513074	−0.256537	−	0.966534i	$-0.582582\pi$
$743$	−13.9854	−0.513074	−0.256537	+	0.966534i	$0.582582\pi$
$744$	0	0
$745$	−24.7355	−0.906238
$746$	0	0
$747$	37.1690i	1.35994i
$748$	0	0
$749$	16.8241i	0.614739i
$750$	0	0
$751$	25.0777	0.915100	0.457550	−	0.889184i	$-0.348727\pi$
$751$	25.0777	0.915100	0.457550	+	0.889184i	$0.348727\pi$
$752$	0	0
$753$	−52.4768	−1.91236
$754$	0	0
$755$	10.5535i	0.384080i
$756$	0	0
$757$	24.6137i	0.894601i	0.894384	+	0.447301i	$0.147615\pi$
$757$	24.6137i	0.894601i	−0.894384	+	0.447301i	$0.852385\pi$
$758$	0	0
$759$	−42.9897	−1.56043
$760$	0	0
$761$	37.8010	1.37029	0.685143	−	0.728409i	$-0.259740\pi$
$761$	37.8010	1.37029	0.685143	+	0.728409i	$0.259740\pi$
$762$	0	0
$763$	− 6.07668i	− 0.219991i
$764$	0	0
$765$	− 5.20693i	− 0.188257i
$766$	0	0
$767$	−40.7483	−1.47134
$768$	0	0
$769$	9.98357	0.360017	0.180008	−	0.983665i	$-0.442388\pi$
$769$	9.98357	0.360017	0.180008	+	0.983665i	$0.442388\pi$
$770$	0	0
$771$	− 6.35180i	− 0.228754i
$772$	0	0
$773$	12.5896i	0.452815i	0.974033	+	0.226408i	$0.0726981\pi$
$773$	12.5896i	0.452815i	−0.974033	+	0.226408i	$0.927302\pi$
$774$	0	0
$775$	−11.7225	−0.421085
$776$	0	0
$777$	−10.9414	−0.392519
$778$	0	0
$779$	− 9.80605i	− 0.351338i
$780$	0	0
$781$	28.9345i	1.03536i
$782$	0	0
$783$	−16.5275	−0.590645
$784$	0	0
$785$	23.0518	0.822753
$786$	0	0
$787$	29.2603i	1.04302i	0.853246	+	0.521509i	$0.174631\pi$
$787$	29.2603i	1.04302i	−0.853246	+	0.521509i	$0.825369\pi$
$788$	0	0
$789$	15.9457i	0.567681i
$790$	0	0
$791$	11.6527	0.414322
$792$	0	0
$793$	−40.0812	−1.42332
$794$	0	0
$795$	− 42.5535i	− 1.50922i
$796$	0	0
$797$	− 46.8578i	− 1.65979i	−0.557920	−	0.829894i	$-0.688401\pi$
$797$	− 46.8578i	− 1.65979i	0.557920	−	0.829894i	$-0.311599\pi$
$798$	0	0
$799$	−8.84826	−0.313029
$800$	0	0
$801$	4.87156	0.172128
$802$	0	0
$803$	1.60256i	0.0565530i
$804$	0	0
$805$	8.65164i	0.304930i
$806$	0	0
$807$	46.9276	1.65193
$808$	0	0
$809$	22.6681	0.796967	0.398483	−	0.917176i	$-0.369537\pi$
$809$	22.6681	0.796967	0.398483	+	0.917176i	$0.369537\pi$
$810$	0	0
$811$	32.7620i	1.15043i	0.818002	+	0.575215i	$0.195082\pi$
$811$	32.7620i	1.15043i	−0.818002	+	0.575215i	$0.804918\pi$
$812$	0	0
$813$	− 0.139606i	− 0.00489621i
$814$	0	0
$815$	−13.7103	−0.480250
$816$	0	0
$817$	−1.83709	−0.0642717
$818$	0	0
$819$	− 13.6190i	− 0.475886i
$820$	0	0
$821$	3.20006i	0.111683i	0.998440	+	0.0558414i	$0.0177841\pi$
$821$	3.20006i	0.111683i	−0.998440	+	0.0558414i	$0.982216\pi$
$822$	0	0
$823$	24.4741	0.853114	0.426557	−	0.904461i	$-0.359726\pi$
$823$	24.4741	0.853114	0.426557	+	0.904461i	$0.359726\pi$
$824$	0	0
$825$	30.4914	1.06157
$826$	0	0
$827$	− 24.5681i	− 0.854316i	−0.904177	−	0.427158i	$-0.859515\pi$
$827$	− 24.5681i	− 0.854316i	0.904177	−	0.427158i	$-0.140485\pi$
$828$	0	0
$829$	11.2380i	0.390311i	0.980772	+	0.195155i	$0.0625211\pi$
$829$	11.2380i	0.390311i	−0.980772	+	0.195155i	$0.937479\pi$
$830$	0	0
$831$	17.5939	0.610324
$832$	0	0
$833$	−8.31894	−0.288234
$834$	0	0
$835$	− 29.8613i	− 1.03339i
$836$	0	0
$837$	8.74828i	0.302385i
$838$	0	0
$839$	−1.94480	−0.0671421	−0.0335711	−	0.999436i	$-0.510688\pi$
$839$	−1.94480	−0.0671421	−0.0335711	+	0.999436i	$0.510688\pi$
$840$	0	0
$841$	−31.9345	−1.10119
$842$	0	0
$843$	46.4330i	1.59924i
$844$	0	0
$845$	− 10.6397i	− 0.366017i
$846$	0	0
$847$	17.4036	0.597993
$848$	0	0
$849$	−48.1027	−1.65088
$850$	0	0
$851$	− 13.2311i	− 0.453556i
$852$	0	0
$853$	− 29.8207i	− 1.02104i	−0.859866	−	0.510520i	$-0.829453\pi$
$853$	− 29.8207i	− 1.02104i	0.859866	−	0.510520i	$-0.170547\pi$
$854$	0	0
$855$	−3.02760	−0.103542
$856$	0	0
$857$	6.49828	0.221977	0.110989	−	0.993822i	$-0.464598\pi$
$857$	6.49828	0.221977	0.110989	+	0.993822i	$0.464598\pi$
$858$	0	0
$859$	− 21.0422i	− 0.717951i	−0.933347	−	0.358975i	$-0.883126\pi$
$859$	− 21.0422i	− 0.717951i	0.933347	−	0.358975i	$-0.116874\pi$
$860$	0	0
$861$	32.4362i	1.10542i
$862$	0	0
$863$	7.80605	0.265721	0.132861	−	0.991135i	$-0.457584\pi$
$863$	7.80605	0.265721	0.132861	+	0.991135i	$0.457584\pi$
$864$	0	0
$865$	−0.824101	−0.0280203
$866$	0	0
$867$	31.5829i	1.07261i
$868$	0	0
$869$	− 1.75086i	− 0.0593938i
$870$	0	0
$871$	26.6448	0.902823
$872$	0	0
$873$	14.5311	0.491804
$874$	0	0
$875$	− 16.9509i	− 0.573046i
$876$	0	0
$877$	4.32582i	0.146073i	0.997329	+	0.0730363i	$0.0232689\pi$
$877$	4.32582i	0.146073i	−0.997329	+	0.0730363i	$0.976731\pi$
$878$	0	0
$879$	−55.3484	−1.86685
$880$	0	0
$881$	−5.65775	−0.190615	−0.0953073	−	0.995448i	$-0.530383\pi$
$881$	−5.65775	−0.190615	−0.0953073	+	0.995448i	$0.530383\pi$
$882$	0	0
$883$	19.6578i	0.661536i	0.943712	+	0.330768i	$0.107308\pi$
$883$	19.6578i	0.661536i	−0.943712	+	0.330768i	$0.892692\pi$
$884$	0	0
$885$	− 29.9639i	− 1.00723i
$886$	0	0
$887$	40.7811	1.36930	0.684648	−	0.728874i	$-0.259956\pi$
$887$	40.7811	1.36930	0.684648	+	0.728874i	$0.259956\pi$
$888$	0	0
$889$	−1.75086	−0.0587219
$890$	0	0
$891$	− 52.2664i	− 1.75099i
$892$	0	0
$893$	5.14486i	0.172166i
$894$	0	0
$895$	−6.42160	−0.214650
$896$	0	0
$897$	40.4691	1.35122
$898$	0	0
$899$	32.2536i	1.07572i
$900$	0	0
$901$	− 22.1250i	− 0.737091i
$902$	0	0
$903$	6.07668	0.202219
$904$	0	0
$905$	−7.43965	−0.247302
$906$	0	0
$907$	37.1475i	1.23346i	0.787173	+	0.616732i	$0.211544\pi$
$907$	37.1475i	1.23346i	−0.787173	+	0.616732i	$0.788456\pi$
$908$	0	0
$909$	14.2897i	0.473960i
$910$	0	0
$911$	20.4070	0.676114	0.338057	−	0.941126i	$-0.390230\pi$
$911$	20.4070	0.676114	0.338057	+	0.941126i	$0.390230\pi$
$912$	0	0
$913$	86.2760	2.85532
$914$	0	0
$915$	− 29.4734i	− 0.974359i
$916$	0	0
$917$	− 3.52588i	− 0.116435i
$918$	0	0
$919$	52.9536	1.74678	0.873389	−	0.487023i	$-0.161917\pi$
$919$	52.9536	1.74678	0.873389	+	0.487023i	$0.161917\pi$
$920$	0	0
$921$	−23.4396	−0.772363
$922$	0	0
$923$	− 27.2380i	− 0.896549i
$924$	0	0
$925$	9.38445i	0.308559i
$926$	0	0
$927$	1.69652	0.0557210
$928$	0	0
$929$	56.8984	1.86678	0.933388	−	0.358869i	$-0.116837\pi$
$929$	56.8984	1.86678	0.933388	+	0.358869i	$0.116837\pi$
$930$	0	0
$931$	4.83709i	0.158529i
$932$	0	0
$933$	17.7992i	0.582719i
$934$	0	0
$935$	−12.0862	−0.395262
$936$	0	0
$937$	−40.9148	−1.33663	−0.668315	−	0.743879i	$-0.732984\pi$
$937$	−40.9148	−1.33663	−0.668315	+	0.743879i	$0.732984\pi$
$938$	0	0
$939$	− 40.0000i	− 1.30535i
$940$	0	0
$941$	− 2.59644i	− 0.0846416i	−0.999104	−	0.0423208i	$-0.986525\pi$
$941$	− 2.59644i	− 0.0846416i	0.999104	−	0.0423208i	$-0.0134752\pi$
$942$	0	0
$943$	−39.2242	−1.27732
$944$	0	0
$945$	−4.57946	−0.148970
$946$	0	0
$947$	25.7586i	0.837042i	0.908207	+	0.418521i	$0.137451\pi$
$947$	25.7586i	0.837042i	−0.908207	+	0.418521i	$0.862549\pi$
$948$	0	0
$949$	− 1.50859i	− 0.0489711i
$950$	0	0
$951$	67.5984	2.19203
$952$	0	0
$953$	−2.25602	−0.0730795	−0.0365398	−	0.999332i	$-0.511634\pi$
$953$	−2.25602	−0.0730795	−0.0365398	+	0.999332i	$0.511634\pi$
$954$	0	0
$955$	− 3.62854i	− 0.117417i
$956$	0	0
$957$	− 83.8950i	− 2.71194i
$958$	0	0
$959$	−13.8156	−0.446129
$960$	0	0
$961$	−13.9276	−0.449278
$962$	0	0
$963$	23.5500i	0.758889i
$964$	0	0
$965$	18.0146i	0.579911i
$966$	0	0
$967$	−45.0449	−1.44855	−0.724273	−	0.689513i	$-0.757824\pi$
$967$	−45.0449	−1.44855	−0.724273	+	0.689513i	$0.757824\pi$
$968$	0	0
$969$	−3.86813	−0.124262
$970$	0	0
$971$	− 17.5760i	− 0.564041i	−0.959408	−	0.282021i	$-0.908995\pi$
$971$	− 17.5760i	− 0.564041i	0.959408	−	0.282021i	$-0.0910047\pi$
$972$	0	0
$973$	− 17.9690i	− 0.576059i
$974$	0	0
$975$	−28.7036	−0.919251
$976$	0	0
$977$	−15.9042	−0.508821	−0.254410	−	0.967096i	$-0.581881\pi$
$977$	−15.9042	−0.508821	−0.254410	+	0.967096i	$0.581881\pi$
$978$	0	0
$979$	− 11.3078i	− 0.361398i
$980$	0	0
$981$	− 8.50601i	− 0.271576i
$982$	0	0
$983$	−41.7294	−1.33096	−0.665480	−	0.746416i	$-0.731773\pi$
$983$	−41.7294	−1.33096	−0.665480	+	0.746416i	$0.731773\pi$
$984$	0	0
$985$	1.81055	0.0576889
$986$	0	0
$987$	− 17.0180i	− 0.541690i
$988$	0	0
$989$	7.34836i	0.233664i
$990$	0	0
$991$	−0.193945	−0.00616087	−0.00308044	−	0.999995i	$-0.500981\pi$
$991$	−0.193945	−0.00616087	−0.00308044	+	0.999995i	$0.500981\pi$
$992$	0	0
$993$	42.3052	1.34251
$994$	0	0
$995$	− 39.7490i	− 1.26013i
$996$	0	0
$997$	40.6949i	1.28882i	0.764680	+	0.644410i	$0.222897\pi$
$997$	40.6949i	1.28882i	−0.764680	+	0.644410i	$0.777103\pi$
$998$	0	0
$999$	7.00344	0.221579

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2432.2.c.e.1217.2	✓	6
4.3	odd	2		2432.2.c.h.1217.5	yes	6
8.3	odd	2		2432.2.c.h.1217.2	yes	6
8.5	even	2	inner	2432.2.c.e.1217.5	yes	6
16.3	odd	4		4864.2.a.bh.1.1		3
16.5	even	4		4864.2.a.bg.1.1		3
16.11	odd	4		4864.2.a.ba.1.3		3
16.13	even	4		4864.2.a.bb.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.e.1217.2	✓	6	1.1	even	1	trivial
2432.2.c.e.1217.5	yes	6	8.5	even	2	inner
2432.2.c.h.1217.2	yes	6	8.3	odd	2
2432.2.c.h.1217.5	yes	6	4.3	odd	2
4864.2.a.ba.1.3		3	16.11	odd	4
4864.2.a.bb.1.3		3	16.13	even	4
4864.2.a.bg.1.1		3	16.5	even	4
4864.2.a.bh.1.1		3	16.3	odd	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:	\( N \)	\(=\)	\( 2432 = 2^{7} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2432.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.24914i q^{3} +1.47068i q^{5} -1.47068 q^{7} -2.05863 q^{9} +O(q^{10})\)	\(q-2.24914i q^{3} +1.47068i q^{5} -1.47068 q^{7} -2.05863 q^{9} +4.77846i q^{11} -4.49828i q^{13} +3.30777 q^{15} +1.71982 q^{17} -1.00000i q^{19} +3.30777i q^{21} -4.00000 q^{23} +2.83709 q^{25} -2.11727i q^{27} -7.80605i q^{29} -4.13187 q^{31} +10.7474 q^{33} -2.16291i q^{35} +3.30777i q^{37} -10.1173 q^{39} +9.80605 q^{41} -1.83709i q^{43} -3.02760i q^{45} -5.14486 q^{47} -4.83709 q^{49} -3.86813i q^{51} -12.8647i q^{53} -7.02760 q^{55} -2.24914 q^{57} -9.05863i q^{59} -8.91033i q^{61} +3.02760 q^{63} +6.61555 q^{65} +5.92332i q^{67} +8.99656i q^{69} +6.05520 q^{71} +0.335371 q^{73} -6.38101i q^{75} -7.02760i q^{77} -0.366407 q^{79} -10.9379 q^{81} -18.0552i q^{83} +2.52932i q^{85} -17.5569 q^{87} -2.36641 q^{89} +6.61555i q^{91} +9.29317i q^{93} +1.47068 q^{95} -7.05863 q^{97} -9.83709i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{7} - 14 q^{9}+O(q^{10}) \) 6 * q - 8 * q^7 - 14 * q^9	\( 6 q - 8 q^{7} - 14 q^{9} + 4 q^{15} - 8 q^{17} - 24 q^{23} + 2 q^{25} - 4 q^{31} + 12 q^{33} - 64 q^{39} + 8 q^{41} - 14 q^{49} - 8 q^{55} + 4 q^{57} - 16 q^{63} + 8 q^{65} - 32 q^{71} - 48 q^{73} + 12 q^{79} + 6 q^{81} - 72 q^{87} + 8 q^{95} - 44 q^{97}+O(q^{100}) \) 6 * q - 8 * q^7 - 14 * q^9 + 4 * q^15 - 8 * q^17 - 24 * q^23 + 2 * q^25 - 4 * q^31 + 12 * q^33 - 64 * q^39 + 8 * q^41 - 14 * q^49 - 8 * q^55 + 4 * q^57 - 16 * q^63 + 8 * q^65 - 32 * q^71 - 48 * q^73 + 12 * q^79 + 6 * q^81 - 72 * q^87 + 8 * q^95 - 44 * q^97

Introduction

L-functions

Modular forms

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Database

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