LMFDB - Embedded newform 3872.2.c.c.1937.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3872,2,Mod(1937,3872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3872, 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("3872.1937");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3872 = 2^{5} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3872.c (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:	$30.9180756626$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1212153856.10
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 10x^{4} - 16x^{2} + 16 $ x^8 - 4x^6 + 10x^4 - 16*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 968)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1937.4
Root			$-1.36145 + 0.382683i$ of defining polynomial
Character	$\chi$	$=$	3872.1937
Dual form			3872.2.c.c.1937.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.47363i q^{3} +3.55765i q^{5} +3.85077 q^{7} +0.828427 q^{9} +4.77791i q^{13} +5.24264 q^{15} +1.59504 q^{17} -4.14386i q^{19} -5.67459i q^{21} +3.24264 q^{23} -7.65685 q^{25} -5.64167i q^{27} -7.39104i q^{29} +3.58579 q^{31} +13.6997i q^{35} -0.610396i q^{37} +7.04085 q^{39} +9.29658 q^{41} -3.06147i q^{43} +2.94725i q^{45} +4.82843 q^{47} +7.82843 q^{49} -2.35049i q^{51} -4.16804i q^{53} -6.10650 q^{57} +5.64167i q^{59} +2.16478i q^{61} +3.19008 q^{63} -16.9981 q^{65} -1.47363i q^{67} -4.77844i q^{69} -10.4142 q^{71} +9.29658 q^{73} +11.2833i q^{75} +3.19008 q^{79} -5.82843 q^{81} +12.6173i q^{83} +5.67459i q^{85} -10.8916 q^{87} -10.6569 q^{89} +18.3986i q^{91} -5.28411i q^{93} +14.7424 q^{95} -17.1421 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{9} + 8 q^{15} - 8 q^{23} - 16 q^{25} + 40 q^{31} + 16 q^{47} + 40 q^{49} - 72 q^{71} - 24 q^{81} - 40 q^{89} - 24 q^{97}+O(q^{100}) $ 8 * q - 16 * q^9 + 8 * q^15 - 8 * q^23 - 16 * q^25 + 40 * q^31 + 16 * q^47 + 40 * q^49 - 72 * q^71 - 24 * q^81 - 40 * q^89 - 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3872\mathbb{Z}\right)^\times$.

$n$	$485$	$1695$	$2785$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 1.47363i	− 0.850798i	−0.905006	−	0.425399i	$-0.860134\pi$
$3$	− 1.47363i	− 0.850798i	0.905006	−	0.425399i	$-0.139866\pi$
$4$	0	0
$5$	3.55765i	1.59103i	0.605935	+	0.795514i	$0.292799\pi$
$5$	3.55765i	1.59103i	−0.605935	+	0.795514i	$0.707201\pi$
$6$	0	0
$7$	3.85077	1.45545	0.727727	−	0.685867i	$-0.240577\pi$
$7$	3.85077	1.45545	0.727727	+	0.685867i	$0.240577\pi$
$8$	0	0
$9$	0.828427	0.276142
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.77791i	1.32515i	0.748994	+	0.662577i	$0.230537\pi$
$13$	4.77791i	1.32515i	−0.748994	+	0.662577i	$0.769463\pi$
$14$	0	0
$15$	5.24264	1.35364
$16$	0	0
$17$	1.59504	0.386854	0.193427	−	0.981115i	$-0.438040\pi$
$17$	1.59504	0.386854	0.193427	+	0.981115i	$0.438040\pi$
$18$	0	0
$19$	− 4.14386i	− 0.950667i	−0.879806	−	0.475333i	$-0.842327\pi$
$19$	− 4.14386i	− 0.950667i	0.879806	−	0.475333i	$-0.157673\pi$
$20$	0	0
$21$	− 5.67459i	− 1.23830i
$22$	0	0
$23$	3.24264	0.676137	0.338069	−	0.941121i	$-0.390226\pi$
$23$	3.24264	0.676137	0.338069	+	0.941121i	$0.390226\pi$
$24$	0	0
$25$	−7.65685	−1.53137
$26$	0	0
$27$	− 5.64167i	− 1.08574i
$28$	0	0
$29$	− 7.39104i	− 1.37248i	−0.727375	−	0.686240i	$-0.759260\pi$
$29$	− 7.39104i	− 1.37248i	0.727375	−	0.686240i	$-0.240740\pi$
$30$	0	0
$31$	3.58579	0.644026	0.322013	−	0.946735i	$-0.395640\pi$
$31$	3.58579	0.644026	0.322013	+	0.946735i	$0.395640\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	13.6997i	2.31567i
$36$	0	0
$37$	− 0.610396i	− 0.100348i	−0.998740	−	0.0501742i	$-0.984022\pi$
$37$	− 0.610396i	− 0.100348i	0.998740	−	0.0501742i	$-0.0159777\pi$
$38$	0	0
$39$	7.04085	1.12744
$40$	0	0
$41$	9.29658	1.45188	0.725941	−	0.687757i	$-0.241405\pi$
$41$	9.29658	1.45188	0.725941	+	0.687757i	$0.241405\pi$
$42$	0	0
$43$	− 3.06147i	− 0.466869i	−0.972372	−	0.233435i	$-0.925003\pi$
$43$	− 3.06147i	− 0.466869i	0.972372	−	0.233435i	$-0.0749965\pi$
$44$	0	0
$45$	2.94725i	0.439350i
$46$	0	0
$47$	4.82843	0.704298	0.352149	−	0.935944i	$-0.385451\pi$
$47$	4.82843	0.704298	0.352149	+	0.935944i	$0.385451\pi$
$48$	0	0
$49$	7.82843	1.11835
$50$	0	0
$51$	− 2.35049i	− 0.329135i
$52$	0	0
$53$	− 4.16804i	− 0.572525i	−0.958151	−	0.286262i	$-0.907587\pi$
$53$	− 4.16804i	− 0.572525i	0.958151	−	0.286262i	$-0.0924129\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−6.10650	−0.808825
$58$	0	0
$59$	5.64167i	0.734483i	0.930126	+	0.367241i	$0.119698\pi$
$59$	5.64167i	0.734483i	−0.930126	+	0.367241i	$0.880302\pi$
$60$	0	0
$61$	2.16478i	0.277172i	0.990350	+	0.138586i	$0.0442558\pi$
$61$	2.16478i	0.277172i	−0.990350	+	0.138586i	$0.955744\pi$
$62$	0	0
$63$	3.19008	0.401913
$64$	0	0
$65$	−16.9981	−2.10836
$66$	0	0
$67$	− 1.47363i	− 0.180032i	−0.995940	−	0.0900160i	$-0.971308\pi$
$67$	− 1.47363i	− 0.180032i	0.995940	−	0.0900160i	$-0.0286918\pi$
$68$	0	0
$69$	− 4.77844i	− 0.575256i
$70$	0	0
$71$	−10.4142	−1.23594	−0.617970	−	0.786202i	$-0.712045\pi$
$71$	−10.4142	−1.23594	−0.617970	+	0.786202i	$0.712045\pi$
$72$	0	0
$73$	9.29658	1.08808	0.544041	−	0.839058i	$-0.316894\pi$
$73$	9.29658	1.08808	0.544041	+	0.839058i	$0.316894\pi$
$74$	0	0
$75$	11.2833i	1.30289i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	3.19008	0.358912	0.179456	−	0.983766i	$-0.442566\pi$
$79$	3.19008	0.358912	0.179456	+	0.983766i	$0.442566\pi$
$80$	0	0
$81$	−5.82843	−0.647603
$82$	0	0
$83$	12.6173i	1.38493i	0.721453	+	0.692464i	$0.243475\pi$
$83$	12.6173i	1.38493i	−0.721453	+	0.692464i	$0.756525\pi$
$84$	0	0
$85$	5.67459i	0.615496i
$86$	0	0
$87$	−10.8916	−1.16770
$88$	0	0
$89$	−10.6569	−1.12962	−0.564812	−	0.825220i	$-0.691051\pi$
$89$	−10.6569	−1.12962	−0.564812	+	0.825220i	$0.691051\pi$
$90$	0	0
$91$	18.3986i	1.92870i
$92$	0	0
$93$	− 5.28411i	− 0.547936i
$94$	0	0
$95$	14.7424	1.51254
$96$	0	0
$97$	−17.1421	−1.74052	−0.870260	−	0.492593i	$-0.836049\pi$
$97$	−17.1421	−1.74052	−0.870260	+	0.492593i	$0.836049\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.77791i	0.475420i	0.971336	+	0.237710i	$0.0763968\pi$
$101$	4.77791i	0.475420i	−0.971336	+	0.237710i	$0.923603\pi$
$102$	0	0
$103$	−9.31371	−0.917707	−0.458853	−	0.888512i	$-0.651740\pi$
$103$	−9.31371	−0.917707	−0.458853	+	0.888512i	$0.651740\pi$
$104$	0	0
$105$	20.1882	1.97017
$106$	0	0
$107$	7.20533i	0.696565i	0.937390	+	0.348283i	$0.113235\pi$
$107$	7.20533i	0.696565i	−0.937390	+	0.348283i	$0.886765\pi$
$108$	0	0
$109$	4.32957i	0.414697i	0.978267	+	0.207349i	$0.0664835\pi$
$109$	4.32957i	0.414697i	−0.978267	+	0.207349i	$0.933517\pi$
$110$	0	0
$111$	−0.899495	−0.0853763
$112$	0	0
$113$	3.48528	0.327868	0.163934	−	0.986471i	$-0.447582\pi$
$113$	3.48528	0.327868	0.163934	+	0.986471i	$0.447582\pi$
$114$	0	0
$115$	11.5362i	1.07575i
$116$	0	0
$117$	3.95815i	0.365931i
$118$	0	0
$119$	6.14214	0.563049
$120$	0	0
$121$	0	0
$122$	0	0
$123$	− 13.6997i	− 1.23526i
$124$	0	0
$125$	− 9.45215i	− 0.845426i
$126$	0	0
$127$	14.7424	1.30818	0.654088	−	0.756419i	$-0.273053\pi$
$127$	14.7424	1.30818	0.654088	+	0.756419i	$0.273053\pi$
$128$	0	0
$129$	−4.51146	−0.397212
$130$	0	0
$131$	10.6382i	0.929465i	0.885451	+	0.464732i	$0.153850\pi$
$131$	10.6382i	0.929465i	−0.885451	+	0.464732i	$0.846150\pi$
$132$	0	0
$133$	− 15.9570i	− 1.38365i
$134$	0	0
$135$	20.0711	1.72744
$136$	0	0
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	0	0
$139$	8.47343i	0.718707i	0.933202	+	0.359353i	$0.117003\pi$
$139$	8.47343i	0.718707i	−0.933202	+	0.359353i	$0.882997\pi$
$140$	0	0
$141$	− 7.11529i	− 0.599216i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	26.2947	2.18366
$146$	0	0
$147$	− 11.5362i	− 0.951487i
$148$	0	0
$149$	19.1886i	1.57199i	0.618234	+	0.785994i	$0.287848\pi$
$149$	19.1886i	1.57199i	−0.618234	+	0.785994i	$0.712152\pi$
$150$	0	0
$151$	7.70154	0.626742	0.313371	−	0.949631i	$-0.398542\pi$
$151$	7.70154	0.626742	0.313371	+	0.949631i	$0.398542\pi$
$152$	0	0
$153$	1.32138	0.106827
$154$	0	0
$155$	12.7570i	1.02466i
$156$	0	0
$157$	− 13.6202i	− 1.08701i	−0.839406	−	0.543505i	$-0.817097\pi$
$157$	− 13.6202i	− 1.08701i	0.839406	−	0.543505i	$-0.182903\pi$
$158$	0	0
$159$	−6.14214	−0.487103
$160$	0	0
$161$	12.4867	0.984087
$162$	0	0
$163$	− 20.1251i	− 1.57632i	−0.615471	−	0.788159i	$-0.711034\pi$
$163$	− 20.1251i	− 1.57632i	0.615471	−	0.788159i	$-0.288966\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.85077	0.297982	0.148991	−	0.988839i	$-0.452398\pi$
$167$	3.85077	0.297982	0.148991	+	0.988839i	$0.452398\pi$
$168$	0	0
$169$	−9.82843	−0.756033
$170$	0	0
$171$	− 3.43289i	− 0.262519i
$172$	0	0
$173$	5.22625i	0.397345i	0.980066	+	0.198672i	$0.0636629\pi$
$173$	5.22625i	0.397345i	−0.980066	+	0.198672i	$0.936337\pi$
$174$	0	0
$175$	−29.4848	−2.22884
$176$	0	0
$177$	8.31371	0.624897
$178$	0	0
$179$	− 19.8723i	− 1.48532i	−0.669667	−	0.742661i	$-0.733563\pi$
$179$	− 19.8723i	− 1.48532i	0.669667	−	0.742661i	$-0.266437\pi$
$180$	0	0
$181$	15.3467i	1.14071i	0.821399	+	0.570354i	$0.193194\pi$
$181$	15.3467i	1.14071i	−0.821399	+	0.570354i	$0.806806\pi$
$182$	0	0
$183$	3.19008	0.235818
$184$	0	0
$185$	2.17157	0.159657
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 21.7248i	− 1.58024i
$190$	0	0
$191$	12.0711	0.873432	0.436716	−	0.899599i	$-0.356141\pi$
$191$	12.0711	0.873432	0.436716	+	0.899599i	$0.356141\pi$
$192$	0	0
$193$	3.19008	0.229627	0.114814	−	0.993387i	$-0.463373\pi$
$193$	3.19008	0.229627	0.114814	+	0.993387i	$0.463373\pi$
$194$	0	0
$195$	25.0489i	1.79379i
$196$	0	0
$197$	15.6788i	1.11707i	0.829483	+	0.558533i	$0.188635\pi$
$197$	15.6788i	1.11707i	−0.829483	+	0.558533i	$0.811365\pi$
$198$	0	0
$199$	−16.8284	−1.19294	−0.596468	−	0.802637i	$-0.703430\pi$
$199$	−16.8284	−1.19294	−0.596468	+	0.802637i	$0.703430\pi$
$200$	0	0
$201$	−2.17157	−0.153171
$202$	0	0
$203$	− 28.4612i	− 1.99758i
$204$	0	0
$205$	33.0740i	2.30999i
$206$	0	0
$207$	2.68629	0.186710
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 23.0698i	− 1.58819i	−0.607794	−	0.794095i	$-0.707945\pi$
$211$	− 23.0698i	− 1.58819i	0.607794	−	0.794095i	$-0.292055\pi$
$212$	0	0
$213$	15.3467i	1.05154i
$214$	0	0
$215$	10.8916	0.742802
$216$	0	0
$217$	13.8080	0.937351
$218$	0	0
$219$	− 13.6997i	− 0.925739i
$220$	0	0
$221$	7.62096i	0.512641i
$222$	0	0
$223$	−25.7279	−1.72287	−0.861435	−	0.507869i	$-0.830434\pi$
$223$	−25.7279	−1.72287	−0.861435	+	0.507869i	$0.830434\pi$
$224$	0	0
$225$	−6.34315	−0.422876
$226$	0	0
$227$	− 2.87576i	− 0.190871i	−0.995436	−	0.0954354i	$-0.969576\pi$
$227$	− 2.87576i	− 0.190871i	0.995436	−	0.0954354i	$-0.0304243\pi$
$228$	0	0
$229$	20.7355i	1.37024i	0.728430	+	0.685120i	$0.240250\pi$
$229$	20.7355i	1.37024i	−0.728430	+	0.685120i	$0.759750\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.8916	−0.713534	−0.356767	−	0.934193i	$-0.616121\pi$
$233$	−10.8916	−0.713534	−0.356767	+	0.934193i	$0.616121\pi$
$234$	0	0
$235$	17.1778i	1.12056i
$236$	0	0
$237$	− 4.70099i	− 0.305362i
$238$	0	0
$239$	23.1046	1.49451	0.747257	−	0.664535i	$-0.231371\pi$
$239$	23.1046	1.49451	0.747257	+	0.664535i	$0.231371\pi$
$240$	0	0
$241$	−29.4848	−1.89928	−0.949641	−	0.313340i	$-0.898552\pi$
$241$	−29.4848	−1.89928	−0.949641	+	0.313340i	$0.898552\pi$
$242$	0	0
$243$	− 8.33609i	− 0.534760i
$244$	0	0
$245$	27.8508i	1.77932i
$246$	0	0
$247$	19.7990	1.25978
$248$	0	0
$249$	18.5932	1.17829
$250$	0	0
$251$	14.4834i	0.914186i	0.889419	+	0.457093i	$0.151109\pi$
$251$	14.4834i	0.914186i	−0.889419	+	0.457093i	$0.848891\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	8.36223	0.523663
$256$	0	0
$257$	18.4853	1.15308	0.576540	−	0.817069i	$-0.304402\pi$
$257$	18.4853	1.15308	0.576540	+	0.817069i	$0.304402\pi$
$258$	0	0
$259$	− 2.35049i	− 0.146053i
$260$	0	0
$261$	− 6.12293i	− 0.379000i
$262$	0	0
$263$	−7.04085	−0.434158	−0.217079	−	0.976154i	$-0.569653\pi$
$263$	−7.04085	−0.434158	−0.217079	+	0.976154i	$0.569653\pi$
$264$	0	0
$265$	14.8284	0.910903
$266$	0	0
$267$	15.7042i	0.961082i
$268$	0	0
$269$	− 7.62096i	− 0.464658i	−0.972637	−	0.232329i	$-0.925365\pi$
$269$	− 7.62096i	− 0.464658i	0.972637	−	0.232329i	$-0.0746347\pi$
$270$	0	0
$271$	−16.0638	−0.975804	−0.487902	−	0.872898i	$-0.662238\pi$
$271$	−16.0638	−0.975804	−0.487902	+	0.872898i	$0.662238\pi$
$272$	0	0
$273$	27.1127	1.64094
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 5.59767i	− 0.336331i	−0.985759	−	0.168166i	$-0.946216\pi$
$277$	− 5.59767i	− 0.336331i	0.985759	−	0.168166i	$-0.0537844\pi$
$278$	0	0
$279$	2.97056	0.177843
$280$	0	0
$281$	23.1046	1.37831	0.689153	−	0.724616i	$-0.257983\pi$
$281$	23.1046	1.37831	0.689153	+	0.724616i	$0.257983\pi$
$282$	0	0
$283$	− 17.8435i	− 1.06069i	−0.847782	−	0.530344i	$-0.822063\pi$
$283$	− 17.8435i	− 1.06069i	0.847782	−	0.530344i	$-0.177937\pi$
$284$	0	0
$285$	− 21.7248i	− 1.28686i
$286$	0	0
$287$	35.7990	2.11315
$288$	0	0
$289$	−14.4558	−0.850344
$290$	0	0
$291$	25.2611i	1.48083i
$292$	0	0
$293$	2.98454i	0.174359i	0.996193	+	0.0871795i	$0.0277854\pi$
$293$	2.98454i	0.174359i	−0.996193	+	0.0871795i	$0.972215\pi$
$294$	0	0
$295$	−20.0711	−1.16858
$296$	0	0
$297$	0	0
$298$	0	0
$299$	15.4930i	0.895986i
$300$	0	0
$301$	− 11.7890i	− 0.679507i
$302$	0	0
$303$	7.04085	0.404486
$304$	0	0
$305$	−7.70154	−0.440989
$306$	0	0
$307$	− 28.1103i	− 1.60434i	−0.597095	−	0.802171i	$-0.703678\pi$
$307$	− 28.1103i	− 1.60434i	0.597095	−	0.802171i	$-0.296322\pi$
$308$	0	0
$309$	13.7249i	0.780783i
$310$	0	0
$311$	−10.4853	−0.594566	−0.297283	−	0.954789i	$-0.596080\pi$
$311$	−10.4853	−0.594566	−0.297283	+	0.954789i	$0.596080\pi$
$312$	0	0
$313$	−1.82843	−0.103349	−0.0516744	−	0.998664i	$-0.516456\pi$
$313$	−1.82843	−0.103349	−0.0516744	+	0.998664i	$0.516456\pi$
$314$	0	0
$315$	11.3492i	0.639454i
$316$	0	0
$317$	1.83119i	0.102850i	0.998677	+	0.0514249i	$0.0163763\pi$
$317$	1.83119i	0.102850i	−0.998677	+	0.0514249i	$0.983624\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	10.6180	0.592636
$322$	0	0
$323$	− 6.60963i	− 0.367769i
$324$	0	0
$325$	− 36.5838i	− 2.02930i
$326$	0	0
$327$	6.38016	0.352824
$328$	0	0
$329$	18.5932	1.02507
$330$	0	0
$331$	− 12.7570i	− 0.701186i	−0.936528	−	0.350593i	$-0.885980\pi$
$331$	− 12.7570i	− 0.701186i	0.936528	−	0.350593i	$-0.114020\pi$
$332$	0	0
$333$	− 0.505668i	− 0.0277105i
$334$	0	0
$335$	5.24264	0.286436
$336$	0	0
$337$	−6.10650	−0.332642	−0.166321	−	0.986072i	$-0.553189\pi$
$337$	−6.10650	−0.332642	−0.166321	+	0.986072i	$0.553189\pi$
$338$	0	0
$339$	− 5.13600i	− 0.278949i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	3.19008	0.172248
$344$	0	0
$345$	17.0000	0.915249
$346$	0	0
$347$	− 1.08239i	− 0.0581059i	−0.999578	−	0.0290529i	$-0.990751\pi$
$347$	− 1.08239i	− 0.0581059i	0.999578	−	0.0290529i	$-0.00924914\pi$
$348$	0	0
$349$	− 18.6633i	− 0.999024i	−0.866307	−	0.499512i	$-0.833513\pi$
$349$	− 18.6633i	− 0.999024i	0.866307	−	0.499512i	$-0.166487\pi$
$350$	0	0
$351$	26.9554	1.43877
$352$	0	0
$353$	−16.7990	−0.894120	−0.447060	−	0.894504i	$-0.647529\pi$
$353$	−16.7990	−0.894120	−0.447060	+	0.894504i	$0.647529\pi$
$354$	0	0
$355$	− 37.0501i	− 1.96642i
$356$	0	0
$357$	− 9.05121i	− 0.479041i
$358$	0	0
$359$	17.9325	0.946440	0.473220	−	0.880944i	$-0.343092\pi$
$359$	17.9325	0.946440	0.473220	+	0.880944i	$0.343092\pi$
$360$	0	0
$361$	1.82843	0.0962330
$362$	0	0
$363$	0	0
$364$	0	0
$365$	33.0740i	1.73117i
$366$	0	0
$367$	5.92893	0.309488	0.154744	−	0.987955i	$-0.450545\pi$
$367$	5.92893	0.309488	0.154744	+	0.987955i	$0.450545\pi$
$368$	0	0
$369$	7.70154	0.400926
$370$	0	0
$371$	− 16.0502i	− 0.833284i
$372$	0	0
$373$	− 28.7444i	− 1.48833i	−0.667997	−	0.744164i	$-0.732848\pi$
$373$	− 28.7444i	− 1.48833i	0.667997	−	0.744164i	$-0.267152\pi$
$374$	0	0
$375$	−13.9289	−0.719287
$376$	0	0
$377$	35.3137	1.81875
$378$	0	0
$379$	2.69442i	0.138403i	0.997603	+	0.0692015i	$0.0220451\pi$
$379$	2.69442i	0.138403i	−0.997603	+	0.0692015i	$0.977955\pi$
$380$	0	0
$381$	− 21.7248i	− 1.11299i
$382$	0	0
$383$	−1.24264	−0.0634960	−0.0317480	−	0.999496i	$-0.510107\pi$
$383$	−1.24264	−0.0634960	−0.0317480	+	0.999496i	$0.510107\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 2.53620i	− 0.128922i
$388$	0	0
$389$	16.5674i	0.840003i	0.907523	+	0.420001i	$0.137970\pi$
$389$	16.5674i	0.840003i	−0.907523	+	0.420001i	$0.862030\pi$
$390$	0	0
$391$	5.17214	0.261567
$392$	0	0
$393$	15.6767	0.790787
$394$	0	0
$395$	11.3492i	0.571040i
$396$	0	0
$397$	36.0821i	1.81091i	0.424441	+	0.905455i	$0.360470\pi$
$397$	36.0821i	1.81091i	−0.424441	+	0.905455i	$0.639530\pi$
$398$	0	0
$399$	−23.5147	−1.17721
$400$	0	0
$401$	−11.4558	−0.572078	−0.286039	−	0.958218i	$-0.592339\pi$
$401$	−11.4558	−0.572078	−0.286039	+	0.958218i	$0.592339\pi$
$402$	0	0
$403$	17.1326i	0.853434i
$404$	0	0
$405$	− 20.7355i	− 1.03035i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	6.38016	0.315479	0.157739	−	0.987481i	$-0.449579\pi$
$409$	6.38016	0.315479	0.157739	+	0.987481i	$0.449579\pi$
$410$	0	0
$411$	− 4.42088i	− 0.218066i
$412$	0	0
$413$	21.7248i	1.06901i
$414$	0	0
$415$	−44.8879	−2.20346
$416$	0	0
$417$	12.4867	0.611474
$418$	0	0
$419$	− 11.7890i	− 0.575931i	−0.957641	−	0.287965i	$-0.907021\pi$
$419$	− 11.7890i	− 0.575931i	0.957641	−	0.287965i	$-0.0929788\pi$
$420$	0	0
$421$	− 4.16804i	− 0.203138i	−0.994828	−	0.101569i	$-0.967614\pi$
$421$	− 4.16804i	− 0.203138i	0.994828	−	0.101569i	$-0.0323863\pi$
$422$	0	0
$423$	4.00000	0.194487
$424$	0	0
$425$	−12.2130	−0.592417
$426$	0	0
$427$	8.33609i	0.403411i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−25.6340	−1.23475	−0.617373	−	0.786670i	$-0.711803\pi$
$431$	−25.6340	−1.23475	−0.617373	+	0.786670i	$0.711803\pi$
$432$	0	0
$433$	−3.34315	−0.160661	−0.0803307	−	0.996768i	$-0.525598\pi$
$433$	−3.34315	−0.160661	−0.0803307	+	0.996768i	$0.525598\pi$
$434$	0	0
$435$	− 38.7485i	− 1.85785i
$436$	0	0
$437$	− 13.4370i	− 0.642781i
$438$	0	0
$439$	−4.51146	−0.215320	−0.107660	−	0.994188i	$-0.534336\pi$
$439$	−4.51146	−0.215320	−0.107660	+	0.994188i	$0.534336\pi$
$440$	0	0
$441$	6.48528	0.308823
$442$	0	0
$443$	6.14734i	0.292069i	0.989280	+	0.146034i	$0.0466510\pi$
$443$	6.14734i	0.292069i	−0.989280	+	0.146034i	$0.953349\pi$
$444$	0	0
$445$	− 37.9133i	− 1.79726i
$446$	0	0
$447$	28.2768	1.33744
$448$	0	0
$449$	17.3431	0.818474	0.409237	−	0.912428i	$-0.365795\pi$
$449$	17.3431	0.818474	0.409237	+	0.912428i	$0.365795\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 11.3492i	− 0.533231i
$454$	0	0
$455$	−65.4558	−3.06862
$456$	0	0
$457$	−26.2947	−1.23001	−0.615007	−	0.788522i	$-0.710847\pi$
$457$	−26.2947	−1.23001	−0.615007	+	0.788522i	$0.710847\pi$
$458$	0	0
$459$	− 8.99869i	− 0.420023i
$460$	0	0
$461$	− 4.85483i	− 0.226112i	−0.993589	−	0.113056i	$-0.963936\pi$
$461$	− 4.85483i	− 0.226112i	0.993589	−	0.113056i	$-0.0360640\pi$
$462$	0	0
$463$	−7.72792	−0.359147	−0.179573	−	0.983745i	$-0.557472\pi$
$463$	−7.72792	−0.359147	−0.179573	+	0.983745i	$0.557472\pi$
$464$	0	0
$465$	18.7990	0.871782
$466$	0	0
$467$	28.7140i	1.32873i	0.747410	+	0.664363i	$0.231297\pi$
$467$	28.7140i	1.32873i	−0.747410	+	0.664363i	$0.768703\pi$
$468$	0	0
$469$	− 5.67459i	− 0.262028i
$470$	0	0
$471$	−20.0711	−0.924826
$472$	0	0
$473$	0	0
$474$	0	0
$475$	31.7289i	1.45582i
$476$	0	0
$477$	− 3.45292i	− 0.158098i
$478$	0	0
$479$	−3.85077	−0.175946	−0.0879731	−	0.996123i	$-0.528039\pi$
$479$	−3.85077	−0.175946	−0.0879731	+	0.996123i	$0.528039\pi$
$480$	0	0
$481$	2.91642	0.132977
$482$	0	0
$483$	− 18.4007i	− 0.837259i
$484$	0	0
$485$	− 60.9857i	− 2.76922i
$486$	0	0
$487$	34.8995	1.58145	0.790724	−	0.612173i	$-0.209705\pi$
$487$	34.8995	1.58145	0.790724	+	0.612173i	$0.209705\pi$
$488$	0	0
$489$	−29.6569	−1.34113
$490$	0	0
$491$	− 23.4412i	− 1.05789i	−0.848657	−	0.528944i	$-0.822588\pi$
$491$	− 23.4412i	− 1.05789i	0.848657	−	0.528944i	$-0.177412\pi$
$492$	0	0
$493$	− 11.7890i	− 0.530950i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−40.1027	−1.79885
$498$	0	0
$499$	− 31.9141i	− 1.42867i	−0.699803	−	0.714336i	$-0.746729\pi$
$499$	− 31.9141i	− 1.42867i	0.699803	−	0.714336i	$-0.253271\pi$
$500$	0	0
$501$	− 5.67459i	− 0.253522i
$502$	0	0
$503$	10.8916	0.485633	0.242817	−	0.970072i	$-0.421929\pi$
$503$	10.8916	0.485633	0.242817	+	0.970072i	$0.421929\pi$
$504$	0	0
$505$	−16.9981	−0.756406
$506$	0	0
$507$	14.4834i	0.643231i
$508$	0	0
$509$	3.05198i	0.135277i	0.997710	+	0.0676383i	$0.0215464\pi$
$509$	3.05198i	0.135277i	−0.997710	+	0.0676383i	$0.978454\pi$
$510$	0	0
$511$	35.7990	1.58365
$512$	0	0
$513$	−23.3783	−1.03218
$514$	0	0
$515$	− 33.1349i	− 1.46010i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	7.70154	0.338060
$520$	0	0
$521$	1.48528	0.0650714	0.0325357	−	0.999471i	$-0.489642\pi$
$521$	1.48528	0.0650714	0.0325357	+	0.999471i	$0.489642\pi$
$522$	0	0
$523$	28.8532i	1.26166i	0.775921	+	0.630831i	$0.217286\pi$
$523$	28.8532i	1.26166i	−0.775921	+	0.630831i	$0.782714\pi$
$524$	0	0
$525$	43.4495i	1.89629i
$526$	0	0
$527$	5.71948	0.249144
$528$	0	0
$529$	−12.4853	−0.542838
$530$	0	0
$531$	4.67371i	0.202822i
$532$	0	0
$533$	44.4182i	1.92397i
$534$	0	0
$535$	−25.6340	−1.10826
$536$	0	0
$537$	−29.2843	−1.26371
$538$	0	0
$539$	0	0
$540$	0	0
$541$	4.77791i	0.205418i	0.994711	+	0.102709i	$0.0327511\pi$
$541$	4.77791i	0.205418i	−0.994711	+	0.102709i	$0.967249\pi$
$542$	0	0
$543$	22.6152	0.970512
$544$	0	0
$545$	−15.4031	−0.659795
$546$	0	0
$547$	− 40.5419i	− 1.73345i	−0.498789	−	0.866724i	$-0.666222\pi$
$547$	− 40.5419i	− 1.73345i	0.498789	−	0.866724i	$-0.333778\pi$
$548$	0	0
$549$	1.79337i	0.0765390i
$550$	0	0
$551$	−30.6274	−1.30477
$552$	0	0
$553$	12.2843	0.522380
$554$	0	0
$555$	− 3.20009i	− 0.135836i
$556$	0	0
$557$	− 24.4148i	− 1.03449i	−0.855838	−	0.517244i	$-0.826958\pi$
$557$	− 24.4148i	− 1.03449i	0.855838	−	0.517244i	$-0.173042\pi$
$558$	0	0
$559$	14.6274	0.618674
$560$	0	0
$561$	0	0
$562$	0	0
$563$	18.2150i	0.767669i	0.923402	+	0.383834i	$0.125397\pi$
$563$	18.2150i	0.767669i	−0.923402	+	0.383834i	$0.874603\pi$
$564$	0	0
$565$	12.3994i	0.521647i
$566$	0	0
$567$	−22.4439	−0.942556
$568$	0	0
$569$	−14.0817	−0.590336	−0.295168	−	0.955445i	$-0.595376\pi$
$569$	−14.0817	−0.590336	−0.295168	+	0.955445i	$0.595376\pi$
$570$	0	0
$571$	40.3881i	1.69019i	0.534618	+	0.845094i	$0.320456\pi$
$571$	40.3881i	1.69019i	−0.534618	+	0.845094i	$0.679544\pi$
$572$	0	0
$573$	− 17.7882i	− 0.743114i
$574$	0	0
$575$	−24.8284	−1.03542
$576$	0	0
$577$	−25.3431	−1.05505	−0.527524	−	0.849540i	$-0.676880\pi$
$577$	−25.3431	−1.05505	−0.527524	+	0.849540i	$0.676880\pi$
$578$	0	0
$579$	− 4.70099i	− 0.195366i
$580$	0	0
$581$	48.5863i	2.01570i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−14.0817	−0.582207
$586$	0	0
$587$	− 6.60963i	− 0.272808i	−0.990653	−	0.136404i	$-0.956445\pi$
$587$	− 6.60963i	− 0.272808i	0.990653	−	0.136404i	$-0.0435546\pi$
$588$	0	0
$589$	− 14.8590i	− 0.612254i
$590$	0	0
$591$	23.1046	0.950397
$592$	0	0
$593$	21.7832	0.894531	0.447265	−	0.894401i	$-0.352398\pi$
$593$	21.7832	0.894531	0.447265	+	0.894401i	$0.352398\pi$
$594$	0	0
$595$	21.8516i	0.895826i
$596$	0	0
$597$	24.7988i	1.01495i
$598$	0	0
$599$	−42.7696	−1.74752	−0.873758	−	0.486360i	$-0.838324\pi$
$599$	−42.7696	−1.74752	−0.873758	+	0.486360i	$0.838324\pi$
$600$	0	0
$601$	−26.5684	−1.08375	−0.541873	−	0.840460i	$-0.682285\pi$
$601$	−26.5684	−1.08375	−0.541873	+	0.840460i	$0.682285\pi$
$602$	0	0
$603$	− 1.22079i	− 0.0497145i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	32.6749	1.32623	0.663116	−	0.748517i	$-0.269234\pi$
$607$	32.6749	1.32623	0.663116	+	0.748517i	$0.269234\pi$
$608$	0	0
$609$	−41.9411	−1.69954
$610$	0	0
$611$	23.0698i	0.933304i
$612$	0	0
$613$	19.1116i	0.771912i	0.922517	+	0.385956i	$0.126128\pi$
$613$	19.1116i	0.771912i	−0.922517	+	0.385956i	$0.873872\pi$
$614$	0	0
$615$	48.7386	1.96533
$616$	0	0
$617$	13.7990	0.555526	0.277763	−	0.960650i	$-0.410407\pi$
$617$	13.7990	0.555526	0.277763	+	0.960650i	$0.410407\pi$
$618$	0	0
$619$	2.69442i	0.108298i	0.998533	+	0.0541489i	$0.0172446\pi$
$619$	2.69442i	0.108298i	−0.998533	+	0.0541489i	$0.982755\pi$
$620$	0	0
$621$	− 18.2939i	− 0.734109i
$622$	0	0
$623$	−41.0371	−1.64412
$624$	0	0
$625$	−4.65685	−0.186274
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 0.973606i	− 0.0388202i
$630$	0	0
$631$	−6.61522	−0.263348	−0.131674	−	0.991293i	$-0.542035\pi$
$631$	−6.61522	−0.263348	−0.131674	+	0.991293i	$0.542035\pi$
$632$	0	0
$633$	−33.9962	−1.35123
$634$	0	0
$635$	52.4482i	2.08134i
$636$	0	0
$637$	37.4035i	1.48198i
$638$	0	0
$639$	−8.62742	−0.341295
$640$	0	0
$641$	2.17157	0.0857720	0.0428860	−	0.999080i	$-0.486345\pi$
$641$	2.17157	0.0857720	0.0428860	+	0.999080i	$0.486345\pi$
$642$	0	0
$643$	− 40.5030i	− 1.59728i	−0.601807	−	0.798642i	$-0.705552\pi$
$643$	− 40.5030i	− 1.59728i	0.601807	−	0.798642i	$-0.294448\pi$
$644$	0	0
$645$	− 16.0502i	− 0.631975i
$646$	0	0
$647$	−27.3848	−1.07661	−0.538303	−	0.842751i	$-0.680935\pi$
$647$	−27.3848	−1.07661	−0.538303	+	0.842751i	$0.680935\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	− 20.3479i	− 0.797496i
$652$	0	0
$653$	3.05198i	0.119433i	0.998215	+	0.0597166i	$0.0190197\pi$
$653$	3.05198i	0.119433i	−0.998215	+	0.0597166i	$0.980980\pi$
$654$	0	0
$655$	−37.8470	−1.47881
$656$	0	0
$657$	7.70154	0.300466
$658$	0	0
$659$	49.2011i	1.91660i	0.285763	+	0.958300i	$0.407753\pi$
$659$	49.2011i	1.91660i	−0.285763	+	0.958300i	$0.592247\pi$
$660$	0	0
$661$	28.5659i	1.11108i	0.831488	+	0.555542i	$0.187489\pi$
$661$	28.5659i	1.11108i	−0.831488	+	0.555542i	$0.812511\pi$
$662$	0	0
$663$	11.2304	0.436154
$664$	0	0
$665$	56.7696	2.20143
$666$	0	0
$667$	− 23.9665i	− 0.927986i
$668$	0	0
$669$	37.9133i	1.46581i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	9.57025	0.368906	0.184453	−	0.982841i	$-0.440949\pi$
$673$	9.57025	0.368906	0.184453	+	0.982841i	$0.440949\pi$
$674$	0	0
$675$	43.1974i	1.66267i
$676$	0	0
$677$	33.0740i	1.27114i	0.772045	+	0.635568i	$0.219234\pi$
$677$	33.0740i	1.27114i	−0.772045	+	0.635568i	$0.780766\pi$
$678$	0	0
$679$	−66.0104	−2.53325
$680$	0	0
$681$	−4.23779	−0.162393
$682$	0	0
$683$	16.6722i	0.637943i	0.947764	+	0.318971i	$0.103337\pi$
$683$	16.6722i	0.637943i	−0.947764	+	0.318971i	$0.896663\pi$
$684$	0	0
$685$	10.6729i	0.407792i
$686$	0	0
$687$	30.5563	1.16580
$688$	0	0
$689$	19.9145	0.758683
$690$	0	0
$691$	23.3252i	0.887332i	0.896192	+	0.443666i	$0.146322\pi$
$691$	23.3252i	0.887332i	−0.896192	+	0.443666i	$0.853678\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−30.1455	−1.14348
$696$	0	0
$697$	14.8284	0.561667
$698$	0	0
$699$	16.0502i	0.607073i
$700$	0	0
$701$	− 27.8477i	− 1.05179i	−0.850549	−	0.525897i	$-0.823730\pi$
$701$	− 27.8477i	− 1.05179i	0.850549	−	0.525897i	$-0.176270\pi$
$702$	0	0
$703$	−2.52939	−0.0953979
$704$	0	0
$705$	25.3137	0.953369
$706$	0	0
$707$	18.3986i	0.691952i
$708$	0	0
$709$	− 7.72569i	− 0.290144i	−0.989421	−	0.145072i	$-0.953659\pi$
$709$	− 7.72569i	− 0.290144i	0.989421	−	0.145072i	$-0.0463414\pi$
$710$	0	0
$711$	2.64275	0.0991109
$712$	0	0
$713$	11.6274	0.435450
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 34.0476i	− 1.27153i
$718$	0	0
$719$	35.5858	1.32713	0.663563	−	0.748120i	$-0.269043\pi$
$719$	35.5858	1.32713	0.663563	+	0.748120i	$0.269043\pi$
$720$	0	0
$721$	−35.8649	−1.33568
$722$	0	0
$723$	43.4495i	1.61591i
$724$	0	0
$725$	56.5921i	2.10178i
$726$	0	0
$727$	8.07107	0.299339	0.149670	−	0.988736i	$-0.452179\pi$
$727$	8.07107	0.299339	0.149670	+	0.988736i	$0.452179\pi$
$728$	0	0
$729$	−29.7696	−1.10258
$730$	0	0
$731$	− 4.88317i	− 0.180610i
$732$	0	0
$733$	6.49435i	0.239874i	0.992781	+	0.119937i	$0.0382693\pi$
$733$	6.49435i	0.239874i	−0.992781	+	0.119937i	$0.961731\pi$
$734$	0	0
$735$	41.0416	1.51384
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 23.0698i	− 0.848636i	−0.905513	−	0.424318i	$-0.860514\pi$
$739$	− 23.0698i	− 0.848636i	0.905513	−	0.424318i	$-0.139486\pi$
$740$	0	0
$741$	− 29.1763i	− 1.07182i
$742$	0	0
$743$	−38.5077	−1.41271	−0.706355	−	0.707858i	$-0.749662\pi$
$743$	−38.5077	−1.41271	−0.706355	+	0.707858i	$0.749662\pi$
$744$	0	0
$745$	−68.2661	−2.50108
$746$	0	0
$747$	10.4525i	0.382437i
$748$	0	0
$749$	27.7461i	1.01382i
$750$	0	0
$751$	−14.7574	−0.538504	−0.269252	−	0.963070i	$-0.586776\pi$
$751$	−14.7574	−0.538504	−0.269252	+	0.963070i	$0.586776\pi$
$752$	0	0
$753$	21.3431	0.777787
$754$	0	0
$755$	27.3994i	0.997165i
$756$	0	0
$757$	− 49.3014i	− 1.79189i	−0.444165	−	0.895945i	$-0.646500\pi$
$757$	− 49.3014i	− 1.79189i	0.444165	−	0.895945i	$-0.353500\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	29.2111	1.05890	0.529451	−	0.848340i	$-0.322398\pi$
$761$	29.2111	1.05890	0.529451	+	0.848340i	$0.322398\pi$
$762$	0	0
$763$	16.6722i	0.603573i
$764$	0	0
$765$	4.70099i	0.169965i
$766$	0	0
$767$	−26.9554	−0.973303
$768$	0	0
$769$	40.1027	1.44614	0.723071	−	0.690774i	$-0.242730\pi$
$769$	40.1027	1.44614	0.723071	+	0.690774i	$0.242730\pi$
$770$	0	0
$771$	− 27.2404i	− 0.981039i
$772$	0	0
$773$	− 48.5863i	− 1.74753i	−0.486351	−	0.873763i	$-0.661672\pi$
$773$	− 48.5863i	− 1.74753i	0.486351	−	0.873763i	$-0.338328\pi$
$774$	0	0
$775$	−27.4558	−0.986243
$776$	0	0
$777$	−3.46375	−0.124261
$778$	0	0
$779$	− 38.5237i	− 1.38026i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−41.6978	−1.49016
$784$	0	0
$785$	48.4558	1.72946
$786$	0	0
$787$	− 0.896683i	− 0.0319633i	−0.999872	−	0.0159816i	$-0.994913\pi$
$787$	− 0.896683i	− 0.0319633i	0.999872	−	0.0159816i	$-0.00508733\pi$
$788$	0	0
$789$	10.3756i	0.369380i
$790$	0	0
$791$	13.4210	0.477196
$792$	0	0
$793$	−10.3431	−0.367296
$794$	0	0
$795$	− 21.8516i	− 0.774995i
$796$	0	0
$797$	5.99923i	0.212504i	0.994339	+	0.106252i	$0.0338850\pi$
$797$	5.99923i	0.212504i	−0.994339	+	0.106252i	$0.966115\pi$
$798$	0	0
$799$	7.70154	0.272461
$800$	0	0
$801$	−8.82843	−0.311937
$802$	0	0
$803$	0	0
$804$	0	0
$805$	44.4231i	1.56571i
$806$	0	0
$807$	−11.2304	−0.395330
$808$	0	0
$809$	−9.29658	−0.326850	−0.163425	−	0.986556i	$-0.552254\pi$
$809$	−9.29658	−0.326850	−0.163425	+	0.986556i	$0.552254\pi$
$810$	0	0
$811$	2.69005i	0.0944604i	0.998884	+	0.0472302i	$0.0150394\pi$
$811$	2.69005i	0.0944604i	−0.998884	+	0.0472302i	$0.984961\pi$
$812$	0	0
$813$	23.6720i	0.830213i
$814$	0	0
$815$	71.5980	2.50797
$816$	0	0
$817$	−12.6863	−0.443837
$818$	0	0
$819$	15.2419i	0.532596i
$820$	0	0
$821$	1.79337i	0.0625889i	0.999510	+	0.0312945i	$0.00996296\pi$
$821$	1.79337i	0.0625889i	−0.999510	+	0.0312945i	$0.990037\pi$
$822$	0	0
$823$	2.27208	0.0791997	0.0395998	−	0.999216i	$-0.487392\pi$
$823$	2.27208	0.0791997	0.0395998	+	0.999216i	$0.487392\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 22.8841i	− 0.795758i	−0.917438	−	0.397879i	$-0.869746\pi$
$827$	− 22.8841i	− 0.795758i	0.917438	−	0.397879i	$-0.130254\pi$
$828$	0	0
$829$	− 31.5132i	− 1.09450i	−0.836970	−	0.547249i	$-0.815675\pi$
$829$	− 31.5132i	− 1.09450i	0.836970	−	0.547249i	$-0.184325\pi$
$830$	0	0
$831$	−8.24887	−0.286150
$832$	0	0
$833$	12.4867	0.432637
$834$	0	0
$835$	13.6997i	0.474097i
$836$	0	0
$837$	− 20.2298i	− 0.699245i
$838$	0	0
$839$	5.24264	0.180996	0.0904980	−	0.995897i	$-0.471154\pi$
$839$	5.24264	0.180996	0.0904980	+	0.995897i	$0.471154\pi$
$840$	0	0
$841$	−25.6274	−0.883704
$842$	0	0
$843$	− 34.0476i	− 1.17266i
$844$	0	0
$845$	− 34.9661i	− 1.20287i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−26.2947	−0.902432
$850$	0	0
$851$	− 1.97929i	− 0.0678493i
$852$	0	0
$853$	32.1003i	1.09909i	0.835462	+	0.549547i	$0.185200\pi$
$853$	32.1003i	1.09909i	−0.835462	+	0.549547i	$0.814800\pi$
$854$	0	0
$855$	12.2130	0.417676
$856$	0	0
$857$	24.9733	0.853073	0.426536	−	0.904470i	$-0.359734\pi$
$857$	24.9733	0.853073	0.426536	+	0.904470i	$0.359734\pi$
$858$	0	0
$859$	− 49.5542i	− 1.69077i	−0.534159	−	0.845384i	$-0.679372\pi$
$859$	− 49.5542i	− 1.69077i	0.534159	−	0.845384i	$-0.320628\pi$
$860$	0	0
$861$	− 52.7543i	− 1.79786i
$862$	0	0
$863$	−33.5147	−1.14085	−0.570427	−	0.821348i	$-0.693222\pi$
$863$	−33.5147	−1.14085	−0.570427	+	0.821348i	$0.693222\pi$
$864$	0	0
$865$	−18.5932	−0.632186
$866$	0	0
$867$	21.3025i	0.723471i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	7.04085	0.238570
$872$	0	0
$873$	−14.2010	−0.480631
$874$	0	0
$875$	− 36.3981i	− 1.23048i
$876$	0	0
$877$	− 13.9623i	− 0.471474i	−0.971817	−	0.235737i	$-0.924250\pi$
$877$	− 13.9623i	− 0.471474i	0.971817	−	0.235737i	$-0.0757504\pi$
$878$	0	0
$879$	4.39810	0.148344
$880$	0	0
$881$	−24.1716	−0.814361	−0.407180	−	0.913348i	$-0.633488\pi$
$881$	−24.1716	−0.814361	−0.407180	+	0.913348i	$0.633488\pi$
$882$	0	0
$883$	26.7347i	0.899695i	0.893105	+	0.449847i	$0.148522\pi$
$883$	26.7347i	0.899695i	−0.893105	+	0.449847i	$0.851478\pi$
$884$	0	0
$885$	29.5772i	0.994228i
$886$	0	0
$887$	−41.6978	−1.40007	−0.700037	−	0.714106i	$-0.746833\pi$
$887$	−41.6978	−1.40007	−0.700037	+	0.714106i	$0.746833\pi$
$888$	0	0
$889$	56.7696	1.90399
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 20.0083i	− 0.669553i
$894$	0	0
$895$	70.6985	2.36319
$896$	0	0
$897$	22.8310	0.762303
$898$	0	0
$899$	− 26.5027i	− 0.883914i
$900$	0	0
$901$	− 6.64820i	− 0.221484i
$902$	0	0
$903$	−17.3726	−0.578123
$904$	0	0
$905$	−54.5980	−1.81490
$906$	0	0
$907$	45.1334i	1.49863i	0.662215	+	0.749314i	$0.269617\pi$
$907$	45.1334i	1.49863i	−0.662215	+	0.749314i	$0.730383\pi$
$908$	0	0
$909$	3.95815i	0.131284i
$910$	0	0
$911$	23.1716	0.767708	0.383854	−	0.923394i	$-0.374597\pi$
$911$	23.1716	0.767708	0.383854	+	0.923394i	$0.374597\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	11.3492i	0.375193i
$916$	0	0
$917$	40.9653i	1.35279i
$918$	0	0
$919$	−32.6749	−1.07784	−0.538922	−	0.842356i	$-0.681168\pi$
$919$	−32.6749	−1.07784	−0.538922	+	0.842356i	$0.681168\pi$
$920$	0	0
$921$	−41.4241	−1.36497
$922$	0	0
$923$	− 49.7582i	− 1.63781i
$924$	0	0
$925$	4.67371i	0.153671i
$926$	0	0
$927$	−7.71573	−0.253418
$928$	0	0
$929$	58.0833	1.90565	0.952825	−	0.303520i	$-0.0981620\pi$
$929$	58.0833	1.90565	0.952825	+	0.303520i	$0.0981620\pi$
$930$	0	0
$931$	− 32.4399i	− 1.06317i
$932$	0	0
$933$	15.4514i	0.505855i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	38.7814	1.26693	0.633466	−	0.773771i	$-0.281632\pi$
$937$	38.7814	1.26693	0.633466	+	0.773771i	$0.281632\pi$
$938$	0	0
$939$	2.69442i	0.0879290i
$940$	0	0
$941$	− 52.6339i	− 1.71582i	−0.513802	−	0.857909i	$-0.671763\pi$
$941$	− 52.6339i	− 1.71582i	0.513802	−	0.857909i	$-0.328237\pi$
$942$	0	0
$943$	30.1455	0.981672
$944$	0	0
$945$	77.2891	2.51421
$946$	0	0
$947$	29.9348i	0.972750i	0.873750	+	0.486375i	$0.161681\pi$
$947$	29.9348i	0.972750i	−0.873750	+	0.486375i	$0.838319\pi$
$948$	0	0
$949$	44.4182i	1.44188i
$950$	0	0
$951$	2.69848	0.0875044
$952$	0	0
$953$	−44.6142	−1.44520	−0.722598	−	0.691269i	$-0.757052\pi$
$953$	−44.6142	−1.44520	−0.722598	+	0.691269i	$0.757052\pi$
$954$	0	0
$955$	42.9446i	1.38965i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	11.5523	0.373044
$960$	0	0
$961$	−18.1421	−0.585230
$962$	0	0
$963$	5.96909i	0.192351i
$964$	0	0
$965$	11.3492i	0.365343i
$966$	0	0
$967$	−33.9962	−1.09325	−0.546623	−	0.837379i	$-0.684087\pi$
$967$	−33.9962	−1.09325	−0.546623	+	0.837379i	$0.684087\pi$
$968$	0	0
$969$	−9.74012	−0.312898
$970$	0	0
$971$	− 24.5460i	− 0.787718i	−0.919171	−	0.393859i	$-0.871140\pi$
$971$	− 24.5460i	− 0.787718i	0.919171	−	0.393859i	$-0.128860\pi$
$972$	0	0
$973$	32.6292i	1.04604i
$974$	0	0
$975$	−53.9108	−1.72653
$976$	0	0
$977$	−60.3137	−1.92961	−0.964803	−	0.262973i	$-0.915297\pi$
$977$	−60.3137	−1.92961	−0.964803	+	0.262973i	$0.915297\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	3.58673i	0.114516i
$982$	0	0
$983$	−43.7279	−1.39470	−0.697352	−	0.716729i	$-0.745639\pi$
$983$	−43.7279	−1.39470	−0.697352	+	0.716729i	$0.745639\pi$
$984$	0	0
$985$	−55.7795	−1.77728
$986$	0	0
$987$	− 27.3994i	− 0.872131i
$988$	0	0
$989$	− 9.92724i	− 0.315668i
$990$	0	0
$991$	39.4558	1.25336	0.626678	−	0.779278i	$-0.284414\pi$
$991$	39.4558	1.25336	0.626678	+	0.779278i	$0.284414\pi$
$992$	0	0
$993$	−18.7990	−0.596568
$994$	0	0
$995$	− 59.8696i	− 1.89799i
$996$	0	0
$997$	12.1689i	0.385394i	0.981258	+	0.192697i	$0.0617235\pi$
$997$	12.1689i	0.385394i	−0.981258	+	0.192697i	$0.938276\pi$
$998$	0	0
$999$	−3.44365	−0.108952

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3872.2.c.c.1937.4		8
4.3	odd	2		968.2.c.c.485.2	yes	8
8.3	odd	2		968.2.c.c.485.1	✓	8
8.5	even	2	inner	3872.2.c.c.1937.6		8
11.10	odd	2	inner	3872.2.c.c.1937.3		8
44.3	odd	10		968.2.o.c.493.4		32
44.7	even	10		968.2.o.c.269.7		32
44.15	odd	10		968.2.o.c.269.2		32
44.19	even	10		968.2.o.c.493.5		32
44.27	odd	10		968.2.o.c.245.6		32
44.31	odd	10		968.2.o.c.565.7		32
44.35	even	10		968.2.o.c.565.2		32
44.39	even	10		968.2.o.c.245.3		32
44.43	even	2		968.2.c.c.485.7	yes	8
88.3	odd	10		968.2.o.c.493.2		32
88.19	even	10		968.2.o.c.493.7		32
88.21	odd	2	inner	3872.2.c.c.1937.5		8
88.27	odd	10		968.2.o.c.245.7		32
88.35	even	10		968.2.o.c.565.3		32
88.43	even	2		968.2.c.c.485.8	yes	8
88.51	even	10		968.2.o.c.269.5		32
88.59	odd	10		968.2.o.c.269.4		32
88.75	odd	10		968.2.o.c.565.6		32
88.83	even	10		968.2.o.c.245.2		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
968.2.c.c.485.1	✓	8	8.3	odd	2
968.2.c.c.485.2	yes	8	4.3	odd	2
968.2.c.c.485.7	yes	8	44.43	even	2
968.2.c.c.485.8	yes	8	88.43	even	2
968.2.o.c.245.2		32	88.83	even	10
968.2.o.c.245.3		32	44.39	even	10
968.2.o.c.245.6		32	44.27	odd	10
968.2.o.c.245.7		32	88.27	odd	10
968.2.o.c.269.2		32	44.15	odd	10
968.2.o.c.269.4		32	88.59	odd	10
968.2.o.c.269.5		32	88.51	even	10
968.2.o.c.269.7		32	44.7	even	10
968.2.o.c.493.2		32	88.3	odd	10
968.2.o.c.493.4		32	44.3	odd	10
968.2.o.c.493.5		32	44.19	even	10
968.2.o.c.493.7		32	88.19	even	10
968.2.o.c.565.2		32	44.35	even	10
968.2.o.c.565.3		32	88.35	even	10
968.2.o.c.565.6		32	88.75	odd	10
968.2.o.c.565.7		32	44.31	odd	10
3872.2.c.c.1937.3		8	11.10	odd	2	inner
3872.2.c.c.1937.4		8	1.1	even	1	trivial
3872.2.c.c.1937.5		8	88.21	odd	2	inner
3872.2.c.c.1937.6		8	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 3872 = 2^{5} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3872.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.47363i q^{3} +3.55765i q^{5} +3.85077 q^{7} +0.828427 q^{9} +4.77791i q^{13} +5.24264 q^{15} +1.59504 q^{17} -4.14386i q^{19} -5.67459i q^{21} +3.24264 q^{23} -7.65685 q^{25} -5.64167i q^{27} -7.39104i q^{29} +3.58579 q^{31} +13.6997i q^{35} -0.610396i q^{37} +7.04085 q^{39} +9.29658 q^{41} -3.06147i q^{43} +2.94725i q^{45} +4.82843 q^{47} +7.82843 q^{49} -2.35049i q^{51} -4.16804i q^{53} -6.10650 q^{57} +5.64167i q^{59} +2.16478i q^{61} +3.19008 q^{63} -16.9981 q^{65} -1.47363i q^{67} -4.77844i q^{69} -10.4142 q^{71} +9.29658 q^{73} +11.2833i q^{75} +3.19008 q^{79} -5.82843 q^{81} +12.6173i q^{83} +5.67459i q^{85} -10.8916 q^{87} -10.6569 q^{89} +18.3986i q^{91} -5.28411i q^{93} +14.7424 q^{95} -17.1421 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{9} + 8 q^{15} - 8 q^{23} - 16 q^{25} + 40 q^{31} + 16 q^{47} + 40 q^{49} - 72 q^{71} - 24 q^{81} - 40 q^{89} - 24 q^{97}+O(q^{100}) \) 8 * q - 16 * q^9 + 8 * q^15 - 8 * q^23 - 16 * q^25 + 40 * q^31 + 16 * q^47 + 40 * q^49 - 72 * q^71 - 24 * q^81 - 40 * q^89 - 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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