LMFDB - Embedded newform 1127.2.c.b.1126.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1127,2,Mod(1126,1127)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1127, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1127.1126");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1127 = 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1127.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:	$8.99914030780$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1126.12
Character	$\chi$	$=$	1127.1126
Dual form			1127.2.c.b.1126.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.213257 q^{2} -3.34535i q^{3} -1.95452 q^{4} +0.713419i q^{6} +0.843330 q^{8} -8.19134 q^{9} +6.53855i q^{12} +7.10425i q^{13} +3.72920 q^{16} +1.74686 q^{18} +4.79583 q^{23} -2.82123i q^{24} -5.00000 q^{25} -1.51503i q^{26} +17.3668i q^{27} -3.94950 q^{29} +6.53053i q^{31} -2.48194 q^{32} +16.0102 q^{36} +23.7662 q^{39} -7.23958i q^{41} -1.02275 q^{46} +12.7941i q^{47} -12.4755i q^{48} +1.06629 q^{50} -13.8854i q^{52} -3.70361i q^{54} +0.842259 q^{58} -14.7571i q^{59} -1.39268i q^{62} -6.92910 q^{64} -16.0437i q^{69} -1.04102 q^{71} -6.90801 q^{72} +9.54442i q^{73} +16.7267i q^{75} -5.06831 q^{78} +33.5241 q^{81} +1.54389i q^{82} +13.2124i q^{87} -9.37356 q^{92} +21.8469 q^{93} -2.72843i q^{94} +8.30294i q^{96} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 48 q^{4} - 72 q^{9} + 96 q^{16} - 120 q^{25} - 144 q^{36} + 192 q^{64} + 216 q^{81}+O(q^{100}) $ 24 * q + 48 * q^4 - 72 * q^9 + 96 * q^16 - 120 * q^25 - 144 * q^36 + 192 * q^64 + 216 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1127\mathbb{Z}\right)^\times$.

$n$	$346$	$442$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−0.213257	−0.150796	−0.0753978	−	0.997154i	$-0.524023\pi$
$2$	−0.213257	−0.150796	−0.0753978	+	0.997154i	$0.524023\pi$
$3$	− 3.34535i	− 1.93144i	−0.259592	−	0.965718i	$-0.583588\pi$
$3$	− 3.34535i	− 1.93144i	0.259592	−	0.965718i	$-0.416412\pi$
$4$	−1.95452	−0.977261
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0.713419i	0.291252i
$7$	0	0
$7$	0	0
$8$	0.843330	0.298162
$9$	−8.19134	−2.73045
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	6.53855i	1.88752i
$13$	7.10425i	1.97036i	0.171515	+	0.985182i	$0.445134\pi$
$13$	7.10425i	1.97036i	−0.171515	+	0.985182i	$0.554866\pi$
$14$	0	0
$15$	0	0
$16$	3.72920	0.932299
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	1.74686	0.411740
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.79583	1.00000
$23$	4.79583	1.00000
$24$	− 2.82123i	− 0.575882i
$25$	−5.00000	−1.00000
$26$	− 1.51503i	− 0.297122i
$27$	17.3668i	3.34225i
$28$	0	0
$29$	−3.94950	−0.733404	−0.366702	−	0.930339i	$-0.619513\pi$
$29$	−3.94950	−0.733404	−0.366702	+	0.930339i	$0.619513\pi$
$30$	0	0
$31$	6.53053i	1.17292i	0.809979	+	0.586459i	$0.199479\pi$
$31$	6.53053i	1.17292i	−0.809979	+	0.586459i	$0.800521\pi$
$32$	−2.48194	−0.438749
$33$	0	0
$34$	0	0
$35$	0	0
$36$	16.0102	2.66836
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	23.7662	3.80563
$40$	0	0
$41$	− 7.23958i	− 1.13063i	−0.824874	−	0.565316i	$-0.808754\pi$
$41$	− 7.23958i	− 1.13063i	0.824874	−	0.565316i	$-0.191246\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	−1.02275	−0.150796
$47$	12.7941i	1.86621i	0.359608	+	0.933103i	$0.382910\pi$
$47$	12.7941i	1.86621i	−0.359608	+	0.933103i	$0.617090\pi$
$48$	− 12.4755i	− 1.80068i
$49$	0	0
$50$	1.06629	0.150796
$51$	0	0
$52$	− 13.8854i	− 1.92556i
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	− 3.70361i	− 0.503997i
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0.842259	0.110594
$59$	− 14.7571i	− 1.92121i	−0.277910	−	0.960607i	$-0.589642\pi$
$59$	− 14.7571i	− 1.92121i	0.277910	−	0.960607i	$-0.410358\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	− 1.39268i	− 0.176871i
$63$	0	0
$64$	−6.92910	−0.866138
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	− 16.0437i	− 1.93144i
$70$	0	0
$71$	−1.04102	−0.123546	−0.0617729	−	0.998090i	$-0.519675\pi$
$71$	−1.04102	−0.123546	−0.0617729	+	0.998090i	$0.519675\pi$
$72$	−6.90801	−0.814117
$73$	9.54442i	1.11709i	0.829474	+	0.558545i	$0.188640\pi$
$73$	9.54442i	1.11709i	−0.829474	+	0.558545i	$0.811360\pi$
$74$	0	0
$75$	16.7267i	1.93144i
$76$	0	0
$77$	0	0
$78$	−5.06831	−0.573873
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	33.5241	3.72490
$82$	1.54389i	0.170494i
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	13.2124i	1.41652i
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	−9.37356	−0.977261
$93$	21.8469	2.26542
$94$	− 2.72843i	− 0.281416i
$95$	0	0
$96$	8.30294i	0.847416i
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	0	0
$100$	9.77261	0.977261
$101$	20.0192i	1.99198i	0.0894471	+	0.995992i	$0.471490\pi$
$101$	20.0192i	1.99198i	−0.0894471	+	0.995992i	$0.528510\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	5.99122i	0.587488i
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	− 33.9439i	− 3.26625i
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	7.71938	0.716727
$117$	− 58.1933i	− 5.37997i
$118$	3.14706i	0.289711i
$119$	0	0
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	−24.2189	−2.18375
$124$	− 12.7641i	− 1.14625i
$125$	0	0
$126$	0	0
$127$	−13.5285	−1.20046	−0.600232	−	0.799826i	$-0.704925\pi$
$127$	−13.5285	−1.20046	−0.600232	+	0.799826i	$0.704925\pi$
$128$	6.44156	0.569359
$129$	0	0
$130$	0	0
$131$	16.3094i	1.42496i	0.701692	+	0.712481i	$0.252428\pi$
$131$	16.3094i	1.42496i	−0.701692	+	0.712481i	$0.747572\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	3.42144i	0.291252i
$139$	19.2934i	1.63644i	0.574904	+	0.818221i	$0.305040\pi$
$139$	19.2934i	1.63644i	−0.574904	+	0.818221i	$0.694960\pi$
$140$	0	0
$141$	42.8006	3.60446
$142$	0.222004	0.0186302
$143$	0	0
$144$	−30.5471	−2.54559
$145$	0	0
$146$	− 2.03542i	− 0.168452i
$147$	0	0
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	− 3.56710i	− 0.291252i
$151$	23.1908	1.88724	0.943618	−	0.331035i	$-0.107398\pi$
$151$	23.1908	1.88724	0.943618	+	0.331035i	$0.107398\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−46.4515	−3.71909
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	−7.14925	−0.561698
$163$	3.42798	0.268500	0.134250	−	0.990947i	$-0.457137\pi$
$163$	3.42798	0.268500	0.134250	+	0.990947i	$0.457137\pi$
$164$	14.1499i	1.10492i
$165$	0	0
$166$	0	0
$167$	18.0459i	1.39644i	0.715885	+	0.698219i	$0.246024\pi$
$167$	18.0459i	1.39644i	−0.715885	+	0.698219i	$0.753976\pi$
$168$	0	0
$169$	−37.4703	−2.88233
$170$	0	0
$171$	0	0
$172$	0	0
$173$	9.28869i	0.706206i	0.935584	+	0.353103i	$0.114873\pi$
$173$	9.28869i	0.706206i	−0.935584	+	0.353103i	$0.885127\pi$
$174$	− 2.81765i	− 0.213605i
$175$	0	0
$176$	0	0
$177$	−49.3677	−3.71070
$178$	0	0
$179$	−23.9112	−1.78721	−0.893605	−	0.448855i	$-0.851832\pi$
$179$	−23.9112	−1.78721	−0.893605	+	0.448855i	$0.851832\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	4.04447	0.298162
$185$	0	0
$186$	−4.65901	−0.341615
$187$	0	0
$188$	− 25.0063i	− 1.82377i
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	23.1802i	1.67289i
$193$	−18.7045	−1.34638	−0.673188	−	0.739471i	$-0.735076\pi$
$193$	−18.7045	−1.34638	−0.673188	+	0.739471i	$0.735076\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	24.7681	1.76466	0.882329	−	0.470634i	$-0.155975\pi$
$197$	24.7681	1.76466	0.882329	+	0.470634i	$0.155975\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	−4.21665	−0.298162
$201$	0	0
$202$	− 4.26924i	− 0.300382i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−39.2843	−2.73045
$208$	26.4931i	1.83697i
$209$	0	0
$210$	0	0
$211$	−17.5186	−1.20603	−0.603014	−	0.797731i	$-0.706034\pi$
$211$	−17.5186	−1.20603	−0.603014	+	0.797731i	$0.706034\pi$
$212$	0	0
$213$	3.48256i	0.238621i
$214$	0	0
$215$	0	0
$216$	14.6460i	0.996533i
$217$	0	0
$218$	0	0
$219$	31.9294	2.15759
$220$	0	0
$221$	0	0
$222$	0	0
$223$	18.4027i	1.23234i	0.787614	+	0.616170i	$0.211316\pi$
$223$	18.4027i	1.23234i	−0.787614	+	0.616170i	$0.788684\pi$
$224$	0	0
$225$	40.9567	2.73045
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	−3.33073	−0.218673
$233$	−22.6861	−1.48622	−0.743108	−	0.669171i	$-0.766649\pi$
$233$	−22.6861	−1.48622	−0.743108	+	0.669171i	$0.766649\pi$
$234$	12.4101i	0.811277i
$235$	0	0
$236$	28.8431i	1.87753i
$237$	0	0
$238$	0	0
$239$	−7.23684	−0.468112	−0.234056	−	0.972223i	$-0.575200\pi$
$239$	−7.23684	−0.468112	−0.234056	+	0.972223i	$0.575200\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	−2.34583	−0.150796
$243$	− 60.0492i	− 3.85216i
$244$	0	0
$245$	0	0
$246$	5.16486	0.329299
$247$	0	0
$248$	5.50740i	0.349720i
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	2.88506	0.181025
$255$	0	0
$256$	12.4845	0.780281
$257$	− 22.5430i	− 1.40619i	−0.711093	−	0.703097i	$-0.751800\pi$
$257$	− 22.5430i	− 1.40619i	0.711093	−	0.703097i	$-0.248200\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	32.3517	2.00252
$262$	− 3.47810i	− 0.214878i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 3.04512i	− 0.185664i	−0.995682	−	0.0928321i	$-0.970408\pi$
$269$	− 3.04512i	− 0.185664i	0.995682	−	0.0928321i	$-0.0295920\pi$
$270$	0	0
$271$	− 32.7076i	− 1.98684i	−0.114517	−	0.993421i	$-0.536532\pi$
$271$	− 32.7076i	− 1.98684i	0.114517	−	0.993421i	$-0.463468\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	31.3578i	1.88752i
$277$	−21.9112	−1.31652	−0.658259	−	0.752792i	$-0.728707\pi$
$277$	−21.9112	−1.31652	−0.658259	+	0.752792i	$0.728707\pi$
$278$	− 4.11445i	− 0.246768i
$279$	− 53.4939i	− 3.20259i
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	−9.12754	−0.543537
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	2.03469	0.120737
$285$	0	0
$286$	0	0
$287$	0	0
$288$	20.3304	1.19798
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	− 18.6548i	− 1.09169i
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	34.0708i	1.97036i
$300$	− 32.6928i	− 1.88752i
$301$	0	0
$302$	−4.94560	−0.284587
$303$	66.9711	3.84739
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 7.46588i	− 0.426100i	−0.977041	−	0.213050i	$-0.931660\pi$
$307$	− 7.46588i	− 0.426100i	0.977041	−	0.213050i	$-0.0683398\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	26.3160i	1.49225i	0.665808	+	0.746123i	$0.268087\pi$
$311$	26.3160i	1.49225i	−0.665808	+	0.746123i	$0.731913\pi$
$312$	20.0427	1.13470
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	34.7779	1.95332	0.976660	−	0.214792i	$-0.0689075\pi$
$317$	34.7779	1.95332	0.976660	+	0.214792i	$0.0689075\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−65.5236	−3.64020
$325$	− 35.5212i	− 1.97036i
$326$	−0.731042	−0.0404886
$327$	0	0
$328$	− 6.10536i	− 0.337112i
$329$	0	0
$330$	0	0
$331$	−25.7498	−1.41534	−0.707670	−	0.706543i	$-0.750253\pi$
$331$	−25.7498	−1.41534	−0.707670	+	0.706543i	$0.750253\pi$
$332$	0	0
$333$	0	0
$334$	− 3.84843i	− 0.210577i
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	7.99081	0.434643
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	− 1.98088i	− 0.106493i
$347$	32.2382	1.73064	0.865318	−	0.501223i	$-0.167117\pi$
$347$	32.2382	1.73064	0.865318	+	0.501223i	$0.167117\pi$
$348$	− 25.8240i	− 1.38431i
$349$	34.7798i	1.86172i	0.365375	+	0.930860i	$0.380941\pi$
$349$	34.7798i	1.86172i	−0.365375	+	0.930860i	$0.619059\pi$
$350$	0	0
$351$	−123.378	−6.58545
$352$	0	0
$353$	− 11.3061i	− 0.601765i	−0.953661	−	0.300882i	$-0.902719\pi$
$353$	− 11.3061i	− 0.601765i	0.953661	−	0.300882i	$-0.0972811\pi$
$354$	10.5280	0.559558
$355$	0	0
$356$	0	0
$357$	0	0
$358$	5.09924	0.269503
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	− 36.7988i	− 1.93144i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	17.8846	0.932299
$369$	59.3019i	3.08713i
$370$	0	0
$371$	0	0
$372$	−42.7002	−2.21390
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	10.7896i	0.556432i
$377$	− 28.0582i	− 1.44507i
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	45.2577i	2.31862i
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	− 21.5492i	− 1.09968i
$385$	0	0
$386$	3.98886	0.203028
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	54.5607	2.75222
$394$	−5.28199	−0.266103
$395$	0	0
$396$	0	0
$397$	− 1.14196i	− 0.0573133i	−0.999589	−	0.0286566i	$-0.990877\pi$
$397$	− 1.14196i	− 0.0573133i	0.999589	−	0.0286566i	$-0.00912294\pi$
$398$	0	0
$399$	0	0
$400$	−18.6460	−0.932299
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	−46.3945	−2.31108
$404$	− 39.1279i	− 1.94669i
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	− 21.2140i	− 1.04897i	−0.851421	−	0.524483i	$-0.824258\pi$
$409$	− 21.2140i	− 1.04897i	0.851421	−	0.524483i	$-0.175742\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	8.37766	0.411740
$415$	0	0
$416$	− 17.6323i	− 0.864495i
$417$	64.5430	3.16068
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	3.73596	0.181864
$423$	− 104.801i	− 5.09558i
$424$	0	0
$425$	0	0
$426$	− 0.742681i	− 0.0359830i
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	64.7644i	3.11598i
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	−6.80917	−0.325355
$439$	0.204531i	0.00976175i	0.999988	+	0.00488088i	$0.00155364\pi$
$439$	0.204531i	0.00976175i	−0.999988	+	0.00488088i	$0.998446\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	41.8799	1.98978	0.994888	−	0.100985i	$-0.0321996\pi$
$443$	41.8799	1.98978	0.994888	+	0.100985i	$0.0321996\pi$
$444$	0	0
$445$	0	0
$446$	− 3.92452i	− 0.185831i
$447$	0	0
$448$	0	0
$449$	39.8572	1.88098	0.940490	−	0.339822i	$-0.110367\pi$
$449$	39.8572	1.88098	0.940490	+	0.339822i	$0.110367\pi$
$450$	−8.73432	−0.411740
$451$	0	0
$452$	0	0
$453$	− 77.5811i	− 3.64508i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	27.9047i	1.29965i	0.760083	+	0.649826i	$0.225158\pi$
$461$	27.9047i	1.29965i	−0.760083	+	0.649826i	$0.774842\pi$
$462$	0	0
$463$	2.28043	0.105980	0.0529902	−	0.998595i	$-0.483125\pi$
$463$	2.28043	0.105980	0.0529902	+	0.998595i	$0.483125\pi$
$464$	−14.7285	−0.683752
$465$	0	0
$466$	4.83798	0.224115
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	113.740i	5.25764i
$469$	0	0
$470$	0	0
$471$	0	0
$472$	− 12.4451i	− 0.572834i
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	1.54331	0.0705892
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−21.4997	−0.977261
$485$	0	0
$486$	12.8059i	0.580888i
$487$	−27.1250	−1.22915	−0.614575	−	0.788858i	$-0.710672\pi$
$487$	−27.1250	−1.22915	−0.614575	+	0.788858i	$0.710672\pi$
$488$	0	0
$489$	− 11.4678i	− 0.518591i
$490$	0	0
$491$	26.0819	1.17706	0.588531	−	0.808475i	$-0.299707\pi$
$491$	26.0819	1.17706	0.588531	+	0.808475i	$0.299707\pi$
$492$	47.3364	2.13409
$493$	0	0
$494$	0	0
$495$	0	0
$496$	24.3536i	1.09351i
$497$	0	0
$498$	0	0
$499$	−44.6766	−2.00000	−1.00000		0.000508342i	$-0.999838\pi$
$499$	−44.6766	−2.00000	−1.00000		0.000508342i	$0.999838\pi$
$500$	0	0
$501$	60.3699	2.69713
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	125.351i	5.56704i
$508$	26.4418	1.17317
$509$	− 12.2394i	− 0.542503i	−0.962508	−	0.271252i	$-0.912562\pi$
$509$	− 12.2394i	− 0.542503i	0.962508	−	0.271252i	$-0.0874375\pi$
$510$	0	0
$511$	0	0
$512$	−15.5455	−0.687022
$513$	0	0
$514$	4.80746i	0.212048i
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	31.0739	1.36399
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	−6.89924	−0.301971
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	− 31.8771i	− 1.39256i
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	120.881i	5.24578i
$532$	0	0
$533$	51.4318	2.22776
$534$	0	0
$535$	0	0
$536$	0	0
$537$	79.9914i	3.45188i
$538$	0.649394i	0.0279973i
$539$	0	0
$540$	0	0
$541$	−11.4831	−0.493695	−0.246847	−	0.969054i	$-0.579395\pi$
$541$	−11.4831	−0.493695	−0.246847	+	0.969054i	$0.579395\pi$
$542$	6.97512i	0.299607i
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	20.6317	0.882146	0.441073	−	0.897471i	$-0.354598\pi$
$547$	20.6317	0.882146	0.441073	+	0.897471i	$0.354598\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	− 13.5302i	− 0.575882i
$553$	0	0
$554$	4.67272	0.198525
$555$	0	0
$556$	− 37.7093i	− 1.59923i
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	11.4080i	0.482937i
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	−83.6547	−3.52250
$565$	0	0
$566$	0	0
$567$	0	0
$568$	−0.877920	−0.0368367
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−23.9792	−1.00000
$576$	56.7587	2.36494
$577$	− 47.8409i	− 1.99164i	−0.0913277	−	0.995821i	$-0.529111\pi$
$577$	− 47.8409i	− 1.99164i	0.0913277	−	0.995821i	$-0.470889\pi$
$578$	3.62537	0.150796
$579$	62.5729i	2.60044i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	8.04910i	0.333074i
$585$	0	0
$586$	0	0
$587$	5.11244i	0.211013i	0.994419	+	0.105507i	$0.0336464\pi$
$587$	5.11244i	0.211013i	−0.994419	+	0.105507i	$0.966354\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	− 82.8580i	− 3.40832i
$592$	0	0
$593$	23.9657i	0.984152i	0.870552	+	0.492076i	$0.163762\pi$
$593$	23.9657i	0.984152i	−0.870552	+	0.492076i	$0.836238\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	− 7.26584i	− 0.297122i
$599$	−27.1588	−1.10968	−0.554839	−	0.831958i	$-0.687220\pi$
$599$	−27.1588	−1.10968	−0.554839	+	0.831958i	$0.687220\pi$
$600$	14.1062i	0.575882i
$601$	− 18.9301i	− 0.772176i	−0.922462	−	0.386088i	$-0.873826\pi$
$601$	− 18.9301i	− 0.772176i	0.922462	−	0.386088i	$-0.126174\pi$
$602$	0	0
$603$	0	0
$604$	−45.3268	−1.84432
$605$	0	0
$606$	−14.2821	−0.580170
$607$	− 11.2773i	− 0.457731i	−0.973458	−	0.228865i	$-0.926498\pi$
$607$	− 11.2773i	− 0.457731i	0.973458	−	0.228865i	$-0.0735016\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−90.8922	−3.67710
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	1.59215i	0.0642540i
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	83.2885i	3.34225i
$622$	− 5.61209i	− 0.225024i
$623$	0	0
$624$	88.6287	3.54799
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	58.6057i	2.32937i
$634$	−7.41663	−0.294552
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	8.52732	0.337336
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	48.5499i	1.90869i	0.298699	+	0.954347i	$0.403447\pi$
$647$	48.5499i	1.90869i	−0.298699	+	0.954347i	$0.596553\pi$
$648$	28.2719	1.11062
$649$	0	0
$650$	7.57516i	0.297122i
$651$	0	0
$652$	−6.70006	−0.262395
$653$	1.10036	0.0430605	0.0215302	−	0.999768i	$-0.493146\pi$
$653$	1.10036	0.0430605	0.0215302	+	0.999768i	$0.493146\pi$
$654$	0	0
$655$	0	0
$656$	− 26.9978i	− 1.05409i
$657$	− 78.1816i	− 3.05016i
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	5.49134	0.213427
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−18.9411	−0.733404
$668$	− 35.2712i	− 1.36468i
$669$	61.5636	2.38019
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−4.78599	−0.184487	−0.0922433	−	0.995737i	$-0.529404\pi$
$673$	−4.78599	−0.184487	−0.0922433	+	0.995737i	$0.529404\pi$
$674$	0	0
$675$	− 86.8342i	− 3.34225i
$676$	73.2365	2.81679
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−46.4132	−1.77595	−0.887977	−	0.459889i	$-0.847889\pi$
$683$	−46.4132	−1.77595	−0.887977	+	0.459889i	$0.847889\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 43.4227i	− 1.65188i	−0.563760	−	0.825939i	$-0.690646\pi$
$691$	− 43.4227i	− 1.65188i	0.563760	−	0.825939i	$-0.309354\pi$
$692$	− 18.1549i	− 0.690147i
$693$	0	0
$694$	−6.87502	−0.260972
$695$	0	0
$696$	11.1425i	0.422354i
$697$	0	0
$698$	− 7.41705i	− 0.280739i
$699$	75.8929i	2.87053i
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	26.3113	0.993057
$703$	0	0
$704$	0	0
$705$	0	0
$706$	2.41111i	0.0907435i
$707$	0	0
$708$	96.4902	3.62632
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	31.3193i	1.17292i
$714$	0	0
$715$	0	0
$716$	46.7350	1.74657
$717$	24.2097i	0.904129i
$718$	0	0
$719$	− 3.82026i	− 0.142472i	−0.997460	−	0.0712358i	$-0.977306\pi$
$719$	− 3.82026i	− 0.142472i	0.997460	−	0.0712358i	$-0.0226943\pi$
$720$	0	0
$721$	0	0
$722$	4.05189	0.150796
$723$	0	0
$724$	0	0
$725$	19.7475	0.733404
$726$	7.84761i	0.291252i
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−100.313	−3.71530
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	−11.9030	−0.438749
$737$	0	0
$738$	− 12.6466i	− 0.465526i
$739$	12.6950	0.466994	0.233497	−	0.972357i	$-0.424983\pi$
$739$	12.6950	0.466994	0.233497	+	0.972357i	$0.424983\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	18.4242	0.675462
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	47.7116i	1.73986i
$753$	0	0
$754$	5.98362i	0.217910i
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 47.9768i	− 1.73916i	−0.493794	−	0.869579i	$-0.664390\pi$
$761$	− 47.9768i	− 1.73916i	0.493794	−	0.869579i	$-0.335610\pi$
$762$	− 9.65152i	− 0.349638i
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	104.838	3.78549
$768$	− 41.7650i	− 1.50706i
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	−75.4142	−2.71598
$772$	36.5583	1.31576
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	− 32.6527i	− 1.17292i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 68.5904i	− 2.45122i
$784$	0	0
$785$	0	0
$786$	−11.6355	−0.415023
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	−48.4099	−1.72453
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0.243531i	0.00864259i
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	12.4097	0.438749
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	9.89396	0.348500
$807$	−10.1870	−0.358599
$808$	16.8828i	0.593934i
$809$	38.3667	1.34890	0.674450	−	0.738321i	$-0.264381\pi$
$809$	38.3667	1.34890	0.674450	+	0.738321i	$0.264381\pi$
$810$	0	0
$811$	20.5113i	0.720250i	0.932904	+	0.360125i	$0.117266\pi$
$811$	20.5113i	0.720250i	−0.932904	+	0.360125i	$0.882734\pi$
$812$	0	0
$813$	−109.418	−3.83746
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	4.52405i	0.158180i
$819$	0	0
$820$	0	0
$821$	−24.6191	−0.859213	−0.429606	−	0.903016i	$-0.641348\pi$
$821$	−24.6191	−0.859213	−0.429606	+	0.903016i	$0.641348\pi$
$822$	0	0
$823$	34.7592	1.21163	0.605815	−	0.795605i	$-0.292847\pi$
$823$	34.7592	1.21163	0.605815	+	0.795605i	$0.292847\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	76.7820	2.66836
$829$	23.8712i	0.829080i	0.910031	+	0.414540i	$0.136058\pi$
$829$	23.8712i	0.829080i	−0.910031	+	0.414540i	$0.863942\pi$
$830$	0	0
$831$	73.3006i	2.54277i
$832$	− 49.2260i	− 1.70661i
$833$	0	0
$834$	−13.7643	−0.476617
$835$	0	0
$836$	0	0
$837$	−113.415	−3.92019
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−13.4015	−0.462119
$842$	0	0
$843$	0	0
$844$	34.2404	1.17860
$845$	0	0
$846$	22.3495i	0.768391i
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	− 6.80674i	− 0.233195i
$853$	− 49.3424i	− 1.68945i	−0.535199	−	0.844726i	$-0.679763\pi$
$853$	− 49.3424i	− 1.68945i	0.535199	−	0.844726i	$-0.320237\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 32.3115i	− 1.10374i	−0.833930	−	0.551870i	$-0.813914\pi$
$857$	− 32.3115i	− 1.10374i	0.833930	−	0.551870i	$-0.186086\pi$
$858$	0	0
$859$	− 7.90566i	− 0.269738i	−0.990863	−	0.134869i	$-0.956939\pi$
$859$	− 7.90566i	− 0.269738i	0.990863	−	0.134869i	$-0.0430613\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−18.1829	−0.618954	−0.309477	−	0.950907i	$-0.600154\pi$
$863$	−18.1829	−0.618954	−0.309477	+	0.950907i	$0.600154\pi$
$864$	− 43.1034i	− 1.46641i
$865$	0	0
$866$	0	0
$867$	56.8709i	1.93144i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	−62.4067	−2.10853
$877$	−57.5500	−1.94332	−0.971662	−	0.236373i	$-0.924041\pi$
$877$	−57.5500	−1.94332	−0.971662	+	0.236373i	$0.924041\pi$
$878$	− 0.0436178i	− 0.00147203i
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	−57.1165	−1.92212	−0.961062	−	0.276332i	$-0.910881\pi$
$883$	−57.1165	−1.92212	−0.961062	+	0.276332i	$0.910881\pi$
$884$	0	0
$885$	0	0
$886$	−8.93120	−0.300049
$887$	− 58.9349i	− 1.97884i	−0.145078	−	0.989420i	$-0.546343\pi$
$887$	− 58.9349i	− 1.97884i	0.145078	−	0.989420i	$-0.453657\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	− 35.9686i	− 1.20432i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	113.979	3.80563
$898$	−8.49984	−0.283643
$899$	− 25.7923i	− 0.860223i
$900$	−80.0508	−2.66836
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	16.5447i	0.549662i
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	− 163.984i	− 5.43901i
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−24.9760	−0.822986
$922$	− 5.95089i	− 0.195982i
$923$	− 7.39563i	− 0.243430i
$924$	0	0
$925$	0	0
$926$	−0.486318	−0.0159814
$927$	0	0
$928$	9.80241	0.321780
$929$	53.9316i	1.76944i	0.466124	+	0.884719i	$0.345650\pi$
$929$	53.9316i	1.76944i	−0.466124	+	0.884719i	$0.654350\pi$
$930$	0	0
$931$	0	0
$932$	44.3405	1.45242
$933$	88.0363	2.88218
$934$	0	0
$935$	0	0
$936$	− 49.0762i	− 1.60411i
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	− 34.7198i	− 1.13063i
$944$	− 55.0322i	− 1.79115i
$945$	0	0
$946$	0	0
$947$	−38.3849	−1.24734	−0.623671	−	0.781687i	$-0.714359\pi$
$947$	−38.3849	−1.24734	−0.623671	+	0.781687i	$0.714359\pi$
$948$	0	0
$949$	−67.8059	−2.20107
$950$	0	0
$951$	− 116.344i	− 3.77271i
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	14.1446	0.457467
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−11.6479	−0.375738
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	28.5131	0.916920	0.458460	−	0.888715i	$-0.348401\pi$
$967$	28.5131	0.916920	0.458460	+	0.888715i	$0.348401\pi$
$968$	9.27663	0.298162
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	117.367i	3.76456i
$973$	0	0
$974$	5.78460	0.185350
$975$	−118.831	−3.80563
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	2.44559i	0.0782013i
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−5.56216	−0.177496
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	−20.4245	−0.651110
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−28.7750	−0.914068	−0.457034	−	0.889449i	$-0.651088\pi$
$991$	−28.7750	−0.914068	−0.457034	+	0.889449i	$0.651088\pi$
$992$	− 16.2084i	− 0.514617i
$993$	86.1422i	2.73364i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1.99744i	− 0.0632597i	−0.999500	−	0.0316299i	$-0.989930\pi$
$997$	− 1.99744i	− 0.0632597i	0.999500	−	0.0316299i	$-0.0100698\pi$
$998$	9.52761	0.301591
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1127.2.c.b.1126.12	yes	24
7.6	odd	2	inner	1127.2.c.b.1126.11	✓	24
23.22	odd	2	CM	1127.2.c.b.1126.12	yes	24
161.160	even	2	inner	1127.2.c.b.1126.11	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1127.2.c.b.1126.11	✓	24	7.6	odd	2	inner
1127.2.c.b.1126.11	✓	24	161.160	even	2	inner
1127.2.c.b.1126.12	yes	24	1.1	even	1	trivial
1127.2.c.b.1126.12	yes	24	23.22	odd	2	CM

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:	\( N \)	\(=\)	\( 1127 = 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1127.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.213257 q^{2} -3.34535i q^{3} -1.95452 q^{4} +0.713419i q^{6} +0.843330 q^{8} -8.19134 q^{9} +6.53855i q^{12} +7.10425i q^{13} +3.72920 q^{16} +1.74686 q^{18} +4.79583 q^{23} -2.82123i q^{24} -5.00000 q^{25} -1.51503i q^{26} +17.3668i q^{27} -3.94950 q^{29} +6.53053i q^{31} -2.48194 q^{32} +16.0102 q^{36} +23.7662 q^{39} -7.23958i q^{41} -1.02275 q^{46} +12.7941i q^{47} -12.4755i q^{48} +1.06629 q^{50} -13.8854i q^{52} -3.70361i q^{54} +0.842259 q^{58} -14.7571i q^{59} -1.39268i q^{62} -6.92910 q^{64} -16.0437i q^{69} -1.04102 q^{71} -6.90801 q^{72} +9.54442i q^{73} +16.7267i q^{75} -5.06831 q^{78} +33.5241 q^{81} +1.54389i q^{82} +13.2124i q^{87} -9.37356 q^{92} +21.8469 q^{93} -2.72843i q^{94} +8.30294i q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 48 q^{4} - 72 q^{9} + 96 q^{16} - 120 q^{25} - 144 q^{36} + 192 q^{64} + 216 q^{81}+O(q^{100}) \) 24 * q + 48 * q^4 - 72 * q^9 + 96 * q^16 - 120 * q^25 - 144 * q^36 + 192 * q^64 + 216 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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