LMFDB - Embedded newform 1050.2.g.a.799.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,2,Mod(799,1050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, 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("1050.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1050.g (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:	$8.38429221223$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1050.799
Dual form			1050.2.g.a.799.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} +6.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} +1.00000 q^{21} +4.00000i q^{22} +8.00000i q^{23} +1.00000 q^{24} +6.00000 q^{26} +1.00000i q^{27} -1.00000i q^{28} +2.00000 q^{29} -1.00000i q^{32} +4.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} +10.0000i q^{37} -4.00000i q^{38} +6.00000 q^{39} -6.00000 q^{41} -1.00000i q^{42} -4.00000i q^{43} +4.00000 q^{44} +8.00000 q^{46} -1.00000i q^{48} -1.00000 q^{49} -2.00000 q^{51} -6.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} -1.00000 q^{56} -4.00000i q^{57} -2.00000i q^{58} -4.00000 q^{59} +6.00000 q^{61} -1.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} -4.00000i q^{67} +2.00000i q^{68} +8.00000 q^{69} +8.00000 q^{71} -1.00000i q^{72} +10.0000i q^{73} +10.0000 q^{74} -4.00000 q^{76} -4.00000i q^{77} -6.00000i q^{78} +1.00000 q^{81} +6.00000i q^{82} -4.00000i q^{83} -1.00000 q^{84} -4.00000 q^{86} -2.00000i q^{87} -4.00000i q^{88} +6.00000 q^{89} -6.00000 q^{91} -8.00000i q^{92} -1.00000 q^{96} +14.0000i q^{97} +1.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 8 q^{11} + 2 q^{14} + 2 q^{16} + 8 q^{19} + 2 q^{21} + 2 q^{24} + 12 q^{26} + 4 q^{29} - 4 q^{34} + 2 q^{36} + 12 q^{39} - 12 q^{41} + 8 q^{44} + 16 q^{46} - 2 q^{49} - 4 q^{51} + 2 q^{54} - 2 q^{56} - 8 q^{59} + 12 q^{61} - 2 q^{64} + 8 q^{66} + 16 q^{69} + 16 q^{71} + 20 q^{74} - 8 q^{76} + 2 q^{81} - 2 q^{84} - 8 q^{86} + 12 q^{89} - 12 q^{91} - 2 q^{96} + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 8 * q^11 + 2 * q^14 + 2 * q^16 + 8 * q^19 + 2 * q^21 + 2 * q^24 + 12 * q^26 + 4 * q^29 - 4 * q^34 + 2 * q^36 + 12 * q^39 - 12 * q^41 + 8 * q^44 + 16 * q^46 - 2 * q^49 - 4 * q^51 + 2 * q^54 - 2 * q^56 - 8 * q^59 + 12 * q^61 - 2 * q^64 + 8 * q^66 + 16 * q^69 + 16 * q^71 + 20 * q^74 - 8 * q^76 + 2 * q^81 - 2 * q^84 - 8 * q^86 + 12 * q^89 - 12 * q^91 - 2 * q^96 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$701$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	1.00000i	0.288675i
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	1.00000i	0.235702i
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	4.00000i	0.852803i
$23$	8.00000i	1.66812i	0.551677	+	0.834058i	$0.313988\pi$
$23$	8.00000i	1.66812i	−0.551677	+	0.834058i	$0.686012\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	6.00000	1.17670
$27$	1.00000i	0.192450i
$28$	− 1.00000i	− 0.188982i
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	− 1.00000i	− 0.176777i
$33$	4.00000i	0.696311i
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	10.0000i	1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$	10.0000i	1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$	− 4.00000i	− 0.648886i
$39$	6.00000	0.960769
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	− 1.00000i	− 0.154303i
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	8.00000	1.17954
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−2.00000	−0.280056
$52$	− 6.00000i	− 0.832050i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−1.00000	−0.133631
$57$	− 4.00000i	− 0.529813i
$58$	− 2.00000i	− 0.262613i
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	− 1.00000i	− 0.125988i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	4.00000	0.492366
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	2.00000i	0.242536i
$69$	8.00000	0.963087
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	− 1.00000i	− 0.117851i
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	−4.00000	−0.458831
$77$	− 4.00000i	− 0.455842i
$78$	− 6.00000i	− 0.679366i
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000i	0.662589i
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	−4.00000	−0.431331
$87$	− 2.00000i	− 0.214423i
$88$	− 4.00000i	− 0.426401i
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	− 8.00000i	− 0.834058i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	−1.00000	−0.102062
$97$	14.0000i	1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$	14.0000i	1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	1.00000i	0.101015i
$99$	4.00000	0.402015
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	2.00000i	0.198030i
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	6.00000	0.582772
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	1.00000i	0.0944911i
$113$	− 14.0000i	− 1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	− 14.0000i	− 1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	−2.00000	−0.185695
$117$	− 6.00000i	− 0.554700i
$118$	4.00000i	0.368230i
$119$	2.00000	0.183340
$120$	0	0
$121$	5.00000	0.454545
$122$	− 6.00000i	− 0.543214i
$123$	6.00000i	0.541002i
$124$	0	0
$125$	0	0
$126$	−1.00000	−0.0890871
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.00000i	0.0883883i
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	− 4.00000i	− 0.348155i
$133$	4.00000i	0.346844i
$134$	−4.00000	−0.345547
$135$	0	0
$136$	2.00000	0.171499
$137$	− 10.0000i	− 0.854358i	−0.904167	−	0.427179i	$-0.859507\pi$
$137$	− 10.0000i	− 0.854358i	0.904167	−	0.427179i	$-0.140493\pi$
$138$	− 8.00000i	− 0.681005i
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	− 8.00000i	− 0.671345i
$143$	− 24.0000i	− 2.00698i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	10.0000	0.827606
$147$	1.00000i	0.0824786i
$148$	− 10.0000i	− 0.821995i
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	4.00000i	0.324443i
$153$	2.00000i	0.161690i
$154$	−4.00000	−0.322329
$155$	0	0
$156$	−6.00000	−0.480384
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	−8.00000	−0.630488
$162$	− 1.00000i	− 0.0785674i
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−4.00000	−0.310460
$167$	8.00000i	0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$	8.00000i	0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$	1.00000i	0.0771517i
$169$	−23.0000	−1.76923
$170$	0	0
$171$	−4.00000	−0.305888
$172$	4.00000i	0.304997i
$173$	22.0000i	1.67263i	0.548250	+	0.836315i	$0.315294\pi$
$173$	22.0000i	1.67263i	−0.548250	+	0.836315i	$0.684706\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	−4.00000	−0.301511
$177$	4.00000i	0.300658i
$178$	− 6.00000i	− 0.449719i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	6.00000i	0.444750i
$183$	− 6.00000i	− 0.443533i
$184$	−8.00000	−0.589768
$185$	0	0
$186$	0	0
$187$	8.00000i	0.585018i
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000i	0.0721688i
$193$	2.00000i	0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$	2.00000i	0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	1.00000	0.0714286
$197$	10.0000i	0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$	10.0000i	0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$	− 4.00000i	− 0.284268i
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	2.00000i	0.140720i
$203$	2.00000i	0.140372i
$204$	2.00000	0.140028
$205$	0	0
$206$	8.00000	0.557386
$207$	− 8.00000i	− 0.556038i
$208$	6.00000i	0.416025i
$209$	−16.0000	−1.10674
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	− 6.00000i	− 0.412082i
$213$	− 8.00000i	− 0.548151i
$214$	−12.0000	−0.820303
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	− 2.00000i	− 0.135457i
$219$	10.0000	0.675737
$220$	0	0
$221$	12.0000	0.807207
$222$	− 10.0000i	− 0.671156i
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−14.0000	−0.931266
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	4.00000i	0.264906i
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	2.00000i	0.131306i
$233$	− 22.0000i	− 1.44127i	−0.693316	−	0.720634i	$-0.743851\pi$
$233$	− 22.0000i	− 1.44127i	0.693316	−	0.720634i	$-0.256149\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	4.00000	0.260378
$237$	0	0
$238$	− 2.00000i	− 0.129641i
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	− 5.00000i	− 0.321412i
$243$	− 1.00000i	− 0.0641500i
$244$	−6.00000	−0.384111
$245$	0	0
$246$	6.00000	0.382546
$247$	24.0000i	1.52708i
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	1.00000i	0.0629941i
$253$	− 32.0000i	− 2.01182i
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	30.0000i	1.87135i	0.352865	+	0.935674i	$0.385208\pi$
$257$	30.0000i	1.87135i	−0.352865	+	0.935674i	$0.614792\pi$
$258$	4.00000i	0.249029i
$259$	−10.0000	−0.621370
$260$	0	0
$261$	−2.00000	−0.123797
$262$	20.0000i	1.23560i
$263$	− 24.0000i	− 1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$	− 24.0000i	− 1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$	−4.00000	−0.246183
$265$	0	0
$266$	4.00000	0.245256
$267$	− 6.00000i	− 0.367194i
$268$	4.00000i	0.244339i
$269$	−22.0000	−1.34136	−0.670682	−	0.741745i	$-0.733998\pi$
$269$	−22.0000	−1.34136	−0.670682	+	0.741745i	$0.733998\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	− 2.00000i	− 0.121268i
$273$	6.00000i	0.363137i
$274$	−10.0000	−0.604122
$275$	0	0
$276$	−8.00000	−0.481543
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	4.00000i	0.239904i
$279$	0	0
$280$	0	0
$281$	26.0000	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	−24.0000	−1.41915
$287$	− 6.00000i	− 0.354169i
$288$	1.00000i	0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	14.0000	0.820695
$292$	− 10.0000i	− 0.585206i
$293$	30.0000i	1.75262i	0.481749	+	0.876309i	$0.340002\pi$
$293$	30.0000i	1.75262i	−0.481749	+	0.876309i	$0.659998\pi$
$294$	1.00000	0.0583212
$295$	0	0
$296$	−10.0000	−0.581238
$297$	− 4.00000i	− 0.232104i
$298$	6.00000i	0.347571i
$299$	−48.0000	−2.77591
$300$	0	0
$301$	4.00000	0.230556
$302$	8.00000i	0.460348i
$303$	2.00000i	0.114897i
$304$	4.00000	0.229416
$305$	0	0
$306$	2.00000	0.114332
$307$	− 28.0000i	− 1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$	− 28.0000i	− 1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	4.00000i	0.227921i
$309$	8.00000	0.455104
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	6.00000i	0.339683i
$313$	10.0000i	0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$	10.0000i	0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	0	0
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	− 6.00000i	− 0.336463i
$319$	−8.00000	−0.447914
$320$	0	0
$321$	−12.0000	−0.669775
$322$	8.00000i	0.445823i
$323$	− 8.00000i	− 0.445132i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	20.0000	1.10770
$327$	− 2.00000i	− 0.110600i
$328$	− 6.00000i	− 0.331295i
$329$	0	0
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	4.00000i	0.219529i
$333$	− 10.0000i	− 0.547997i
$334$	8.00000	0.437741
$335$	0	0
$336$	1.00000	0.0545545
$337$	− 18.0000i	− 0.980522i	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	− 18.0000i	− 0.980522i	0.871576	−	0.490261i	$-0.163099\pi$
$338$	23.0000i	1.25104i
$339$	−14.0000	−0.760376
$340$	0	0
$341$	0	0
$342$	4.00000i	0.216295i
$343$	− 1.00000i	− 0.0539949i
$344$	4.00000	0.215666
$345$	0	0
$346$	22.0000	1.18273
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	2.00000i	0.107211i
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	0	0
$351$	−6.00000	−0.320256
$352$	4.00000i	0.213201i
$353$	− 30.0000i	− 1.59674i	−0.602168	−	0.798369i	$-0.705696\pi$
$353$	− 30.0000i	− 1.59674i	0.602168	−	0.798369i	$-0.294304\pi$
$354$	4.00000	0.212598
$355$	0	0
$356$	−6.00000	−0.317999
$357$	− 2.00000i	− 0.105851i
$358$	− 12.0000i	− 0.634220i
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	18.0000i	0.946059i
$363$	− 5.00000i	− 0.262432i
$364$	6.00000	0.314485
$365$	0	0
$366$	−6.00000	−0.313625
$367$	− 32.0000i	− 1.67039i	−0.549957	−	0.835193i	$-0.685356\pi$
$367$	− 32.0000i	− 1.67039i	0.549957	−	0.835193i	$-0.314644\pi$
$368$	8.00000i	0.417029i
$369$	6.00000	0.312348
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	22.0000i	1.13912i	0.821951	+	0.569558i	$0.192886\pi$
$373$	22.0000i	1.13912i	−0.821951	+	0.569558i	$0.807114\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	0	0
$377$	12.0000i	0.618031i
$378$	1.00000i	0.0514344i
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 16.0000i	− 0.817562i	−0.912633	−	0.408781i	$-0.865954\pi$
$383$	− 16.0000i	− 0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	2.00000	0.101797
$387$	4.00000i	0.203331i
$388$	− 14.0000i	− 0.710742i
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	− 1.00000i	− 0.0505076i
$393$	20.0000i	1.00887i
$394$	10.0000	0.503793
$395$	0	0
$396$	−4.00000	−0.201008
$397$	− 6.00000i	− 0.301131i	−0.988600	−	0.150566i	$-0.951890\pi$
$397$	− 6.00000i	− 0.301131i	0.988600	−	0.150566i	$-0.0481095\pi$
$398$	8.00000i	0.401004i
$399$	4.00000	0.200250
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	4.00000i	0.199502i
$403$	0	0
$404$	2.00000	0.0995037
$405$	0	0
$406$	2.00000	0.0992583
$407$	− 40.0000i	− 1.98273i
$408$	− 2.00000i	− 0.0990148i
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	−10.0000	−0.493264
$412$	− 8.00000i	− 0.394132i
$413$	− 4.00000i	− 0.196827i
$414$	−8.00000	−0.393179
$415$	0	0
$416$	6.00000	0.294174
$417$	4.00000i	0.195881i
$418$	16.0000i	0.782586i
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	− 20.0000i	− 0.973585i
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	−8.00000	−0.387601
$427$	6.00000i	0.290360i
$428$	12.0000i	0.580042i
$429$	−24.0000	−1.15873
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	1.00000i	0.0481125i
$433$	2.00000i	0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$	2.00000i	0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$	0	0
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	32.0000i	1.53077i
$438$	− 10.0000i	− 0.477818i
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	− 12.0000i	− 0.570782i
$443$	− 4.00000i	− 0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$	− 4.00000i	− 0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	−10.0000	−0.474579
$445$	0	0
$446$	−16.0000	−0.757622
$447$	6.00000i	0.283790i
$448$	− 1.00000i	− 0.0472456i
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	0	0
$451$	24.0000	1.13012
$452$	14.0000i	0.658505i
$453$	8.00000i	0.375873i
$454$	−12.0000	−0.563188
$455$	0	0
$456$	4.00000	0.187317
$457$	− 10.0000i	− 0.467780i	−0.972263	−	0.233890i	$-0.924854\pi$
$457$	− 10.0000i	− 0.467780i	0.972263	−	0.233890i	$-0.0751456\pi$
$458$	− 2.00000i	− 0.0934539i
$459$	2.00000	0.0933520
$460$	0	0
$461$	22.0000	1.02464	0.512321	−	0.858794i	$-0.328786\pi$
$461$	22.0000	1.02464	0.512321	+	0.858794i	$0.328786\pi$
$462$	4.00000i	0.186097i
$463$	− 32.0000i	− 1.48717i	−0.668644	−	0.743583i	$-0.733125\pi$
$463$	− 32.0000i	− 1.48717i	0.668644	−	0.743583i	$-0.266875\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−22.0000	−1.01913
$467$	− 28.0000i	− 1.29569i	−0.761774	−	0.647843i	$-0.775671\pi$
$467$	− 28.0000i	− 1.29569i	0.761774	−	0.647843i	$-0.224329\pi$
$468$	6.00000i	0.277350i
$469$	4.00000	0.184703
$470$	0	0
$471$	10.0000	0.460776
$472$	− 4.00000i	− 0.184115i
$473$	16.0000i	0.735681i
$474$	0	0
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	− 6.00000i	− 0.274721i
$478$	0	0
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	−60.0000	−2.73576
$482$	− 2.00000i	− 0.0910975i
$483$	8.00000i	0.364013i
$484$	−5.00000	−0.227273
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	− 8.00000i	− 0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$	− 8.00000i	− 0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	6.00000i	0.271607i
$489$	20.0000	0.904431
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	− 6.00000i	− 0.270501i
$493$	− 4.00000i	− 0.180151i
$494$	24.0000	1.07981
$495$	0	0
$496$	0	0
$497$	8.00000i	0.358849i
$498$	4.00000i	0.179244i
$499$	44.0000	1.96971	0.984855	−	0.173379i	$-0.0554684\pi$
$499$	44.0000	1.96971	0.984855	+	0.173379i	$0.0554684\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	12.0000i	0.535586i
$503$	24.0000i	1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$	24.0000i	1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	−32.0000	−1.42257
$507$	23.0000i	1.02147i
$508$	0	0
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	− 1.00000i	− 0.0441942i
$513$	4.00000i	0.176604i
$514$	30.0000	1.32324
$515$	0	0
$516$	4.00000	0.176090
$517$	0	0
$518$	10.0000i	0.439375i
$519$	22.0000	0.965693
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	2.00000i	0.0875376i
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	20.0000	0.873704
$525$	0	0
$526$	−24.0000	−1.04645
$527$	0	0
$528$	4.00000i	0.174078i
$529$	−41.0000	−1.78261
$530$	0	0
$531$	4.00000	0.173585
$532$	− 4.00000i	− 0.173422i
$533$	− 36.0000i	− 1.55933i
$534$	−6.00000	−0.259645
$535$	0	0
$536$	4.00000	0.172774
$537$	− 12.0000i	− 0.517838i
$538$	22.0000i	0.948487i
$539$	4.00000	0.172292
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	0	0
$543$	18.0000i	0.772454i
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	6.00000	0.256776
$547$	12.0000i	0.513083i	0.966533	+	0.256541i	$0.0825830\pi$
$547$	12.0000i	0.513083i	−0.966533	+	0.256541i	$0.917417\pi$
$548$	10.0000i	0.427179i
$549$	−6.00000	−0.256074
$550$	0	0
$551$	8.00000	0.340811
$552$	8.00000i	0.340503i
$553$	0	0
$554$	10.0000	0.424859
$555$	0	0
$556$	4.00000	0.169638
$557$	2.00000i	0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$	2.00000i	0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	0	0
$561$	8.00000	0.337760
$562$	− 26.0000i	− 1.09674i
$563$	44.0000i	1.85438i	0.374593	+	0.927189i	$0.377783\pi$
$563$	44.0000i	1.85438i	−0.374593	+	0.927189i	$0.622217\pi$
$564$	0	0
$565$	0	0
$566$	4.00000	0.168133
$567$	1.00000i	0.0419961i
$568$	8.00000i	0.335673i
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	24.0000i	1.00349i
$573$	0	0
$574$	−6.00000	−0.250435
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 34.0000i	− 1.41544i	−0.706494	−	0.707719i	$-0.749724\pi$
$577$	− 34.0000i	− 1.41544i	0.706494	−	0.707719i	$-0.250276\pi$
$578$	− 13.0000i	− 0.540729i
$579$	2.00000	0.0831172
$580$	0	0
$581$	4.00000	0.165948
$582$	− 14.0000i	− 0.580319i
$583$	− 24.0000i	− 0.993978i
$584$	−10.0000	−0.413803
$585$	0	0
$586$	30.0000	1.23929
$587$	28.0000i	1.15568i	0.816149	+	0.577842i	$0.196105\pi$
$587$	28.0000i	1.15568i	−0.816149	+	0.577842i	$0.803895\pi$
$588$	− 1.00000i	− 0.0412393i
$589$	0	0
$590$	0	0
$591$	10.0000	0.411345
$592$	10.0000i	0.410997i
$593$	18.0000i	0.739171i	0.929197	+	0.369586i	$0.120500\pi$
$593$	18.0000i	0.739171i	−0.929197	+	0.369586i	$0.879500\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	6.00000	0.245770
$597$	8.00000i	0.327418i
$598$	48.0000i	1.96287i
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	− 4.00000i	− 0.163028i
$603$	4.00000i	0.162893i
$604$	8.00000	0.325515
$605$	0	0
$606$	2.00000	0.0812444
$607$	− 48.0000i	− 1.94826i	−0.225989	−	0.974130i	$-0.572561\pi$
$607$	− 48.0000i	− 1.94826i	0.225989	−	0.974130i	$-0.427439\pi$
$608$	− 4.00000i	− 0.162221i
$609$	2.00000	0.0810441
$610$	0	0
$611$	0	0
$612$	− 2.00000i	− 0.0808452i
$613$	− 42.0000i	− 1.69636i	−0.529705	−	0.848182i	$-0.677697\pi$
$613$	− 42.0000i	− 1.69636i	0.529705	−	0.848182i	$-0.322303\pi$
$614$	−28.0000	−1.12999
$615$	0	0
$616$	4.00000	0.161165
$617$	22.0000i	0.885687i	0.896599	+	0.442843i	$0.146030\pi$
$617$	22.0000i	0.885687i	−0.896599	+	0.442843i	$0.853970\pi$
$618$	− 8.00000i	− 0.321807i
$619$	44.0000	1.76851	0.884255	−	0.467005i	$-0.154667\pi$
$619$	44.0000	1.76851	0.884255	+	0.467005i	$0.154667\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	8.00000i	0.320771i
$623$	6.00000i	0.240385i
$624$	6.00000	0.240192
$625$	0	0
$626$	10.0000	0.399680
$627$	16.0000i	0.638978i
$628$	− 10.0000i	− 0.399043i
$629$	20.0000	0.797452
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	0	0
$633$	− 20.0000i	− 0.794929i
$634$	18.0000	0.714871
$635$	0	0
$636$	−6.00000	−0.237915
$637$	− 6.00000i	− 0.237729i
$638$	8.00000i	0.316723i
$639$	−8.00000	−0.316475
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	12.0000i	0.473602i
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	−8.00000	−0.314756
$647$	− 24.0000i	− 0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$647$	− 24.0000i	− 0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$	1.00000i	0.0392837i
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	− 20.0000i	− 0.783260i
$653$	− 18.0000i	− 0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$	− 18.0000i	− 0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$	−2.00000	−0.0782062
$655$	0	0
$656$	−6.00000	−0.234261
$657$	− 10.0000i	− 0.390137i
$658$	0	0
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	0	0
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	4.00000i	0.155464i
$663$	− 12.0000i	− 0.466041i
$664$	4.00000	0.155230
$665$	0	0
$666$	−10.0000	−0.387492
$667$	16.0000i	0.619522i
$668$	− 8.00000i	− 0.309529i
$669$	−16.0000	−0.618596
$670$	0	0
$671$	−24.0000	−0.926510
$672$	− 1.00000i	− 0.0385758i
$673$	2.00000i	0.0770943i	0.999257	+	0.0385472i	$0.0122730\pi$
$673$	2.00000i	0.0770943i	−0.999257	+	0.0385472i	$0.987727\pi$
$674$	−18.0000	−0.693334
$675$	0	0
$676$	23.0000	0.884615
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	14.0000i	0.537667i
$679$	−14.0000	−0.537271
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	− 2.00000i	− 0.0763048i
$688$	− 4.00000i	− 0.152499i
$689$	−36.0000	−1.37149
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	− 22.0000i	− 0.836315i
$693$	4.00000i	0.151947i
$694$	−12.0000	−0.455514
$695$	0	0
$696$	2.00000	0.0758098
$697$	12.0000i	0.454532i
$698$	22.0000i	0.832712i
$699$	−22.0000	−0.832116
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	6.00000i	0.226455i
$703$	40.0000i	1.50863i
$704$	4.00000	0.150756
$705$	0	0
$706$	−30.0000	−1.12906
$707$	− 2.00000i	− 0.0752177i
$708$	− 4.00000i	− 0.150329i
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	0	0
$712$	6.00000i	0.224860i
$713$	0	0
$714$	−2.00000	−0.0748481
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	− 8.00000i	− 0.298557i
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	3.00000i	0.111648i
$723$	− 2.00000i	− 0.0743808i
$724$	18.0000	0.668965
$725$	0	0
$726$	−5.00000	−0.185567
$727$	8.00000i	0.296704i	0.988935	+	0.148352i	$0.0473968\pi$
$727$	8.00000i	0.296704i	−0.988935	+	0.148352i	$0.952603\pi$
$728$	− 6.00000i	− 0.222375i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−8.00000	−0.295891
$732$	6.00000i	0.221766i
$733$	6.00000i	0.221615i	0.993842	+	0.110808i	$0.0353437\pi$
$733$	6.00000i	0.221615i	−0.993842	+	0.110808i	$0.964656\pi$
$734$	−32.0000	−1.18114
$735$	0	0
$736$	8.00000	0.294884
$737$	16.0000i	0.589368i
$738$	− 6.00000i	− 0.220863i
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	0	0
$741$	24.0000	0.881662
$742$	6.00000i	0.220267i
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	0	0
$745$	0	0
$746$	22.0000	0.805477
$747$	4.00000i	0.146352i
$748$	− 8.00000i	− 0.292509i
$749$	12.0000	0.438470
$750$	0	0
$751$	48.0000	1.75154	0.875772	−	0.482724i	$-0.160353\pi$
$751$	48.0000	1.75154	0.875772	+	0.482724i	$0.160353\pi$
$752$	0	0
$753$	12.0000i	0.437304i
$754$	12.0000	0.437014
$755$	0	0
$756$	1.00000	0.0363696
$757$	− 6.00000i	− 0.218074i	−0.994038	−	0.109037i	$-0.965223\pi$
$757$	− 6.00000i	− 0.218074i	0.994038	−	0.109037i	$-0.0347767\pi$
$758$	− 20.0000i	− 0.726433i
$759$	−32.0000	−1.16153
$760$	0	0
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	0	0
$763$	2.00000i	0.0724049i
$764$	0	0
$765$	0	0
$766$	−16.0000	−0.578103
$767$	− 24.0000i	− 0.866590i
$768$	− 1.00000i	− 0.0360844i
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	30.0000	1.08042
$772$	− 2.00000i	− 0.0719816i
$773$	− 2.00000i	− 0.0719350i	−0.999353	−	0.0359675i	$-0.988549\pi$
$773$	− 2.00000i	− 0.0719350i	0.999353	−	0.0359675i	$-0.0114513\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	−14.0000	−0.502571
$777$	10.0000i	0.358748i
$778$	− 26.0000i	− 0.932145i
$779$	−24.0000	−0.859889
$780$	0	0
$781$	−32.0000	−1.14505
$782$	− 16.0000i	− 0.572159i
$783$	2.00000i	0.0714742i
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	20.0000	0.713376
$787$	36.0000i	1.28326i	0.767014	+	0.641631i	$0.221742\pi$
$787$	36.0000i	1.28326i	−0.767014	+	0.641631i	$0.778258\pi$
$788$	− 10.0000i	− 0.356235i
$789$	−24.0000	−0.854423
$790$	0	0
$791$	14.0000	0.497783
$792$	4.00000i	0.142134i
$793$	36.0000i	1.27840i
$794$	−6.00000	−0.212932
$795$	0	0
$796$	8.00000	0.283552
$797$	− 6.00000i	− 0.212531i	−0.994338	−	0.106265i	$-0.966111\pi$
$797$	− 6.00000i	− 0.212531i	0.994338	−	0.106265i	$-0.0338893\pi$
$798$	− 4.00000i	− 0.141598i
$799$	0	0
$800$	0	0
$801$	−6.00000	−0.212000
$802$	− 18.0000i	− 0.635602i
$803$	− 40.0000i	− 1.41157i
$804$	4.00000	0.141069
$805$	0	0
$806$	0	0
$807$	22.0000i	0.774437i
$808$	− 2.00000i	− 0.0703598i
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	−44.0000	−1.54505	−0.772524	−	0.634985i	$-0.781006\pi$
$811$	−44.0000	−1.54505	−0.772524	+	0.634985i	$0.781006\pi$
$812$	− 2.00000i	− 0.0701862i
$813$	0	0
$814$	−40.0000	−1.40200
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	− 16.0000i	− 0.559769i
$818$	− 22.0000i	− 0.769212i
$819$	6.00000	0.209657
$820$	0	0
$821$	38.0000	1.32621	0.663105	−	0.748527i	$-0.269238\pi$
$821$	38.0000	1.32621	0.663105	+	0.748527i	$0.269238\pi$
$822$	10.0000i	0.348790i
$823$	− 56.0000i	− 1.95204i	−0.217687	−	0.976019i	$-0.569851\pi$
$823$	− 56.0000i	− 1.95204i	0.217687	−	0.976019i	$-0.430149\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	−4.00000	−0.139178
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	8.00000i	0.278019i
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	− 6.00000i	− 0.208013i
$833$	2.00000i	0.0692959i
$834$	4.00000	0.138509
$835$	0	0
$836$	16.0000	0.553372
$837$	0	0
$838$	− 36.0000i	− 1.24360i
$839$	−56.0000	−1.93333	−0.966667	−	0.256036i	$-0.917584\pi$
$839$	−56.0000	−1.93333	−0.966667	+	0.256036i	$0.917584\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	− 6.00000i	− 0.206774i
$843$	− 26.0000i	− 0.895488i
$844$	−20.0000	−0.688428
$845$	0	0
$846$	0	0
$847$	5.00000i	0.171802i
$848$	6.00000i	0.206041i
$849$	4.00000	0.137280
$850$	0	0
$851$	−80.0000	−2.74236
$852$	8.00000i	0.274075i
$853$	14.0000i	0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$	14.0000i	0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$	6.00000	0.205316
$855$	0	0
$856$	12.0000	0.410152
$857$	− 42.0000i	− 1.43469i	−0.696717	−	0.717346i	$-0.745357\pi$
$857$	− 42.0000i	− 1.43469i	0.696717	−	0.717346i	$-0.254643\pi$
$858$	24.0000i	0.819346i
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	0	0
$863$	− 32.0000i	− 1.08929i	−0.838666	−	0.544646i	$-0.816664\pi$
$863$	− 32.0000i	− 1.08929i	0.838666	−	0.544646i	$-0.183336\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	2.00000	0.0679628
$867$	− 13.0000i	− 0.441503i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	24.0000	0.813209
$872$	2.00000i	0.0677285i
$873$	− 14.0000i	− 0.473828i
$874$	32.0000	1.08242
$875$	0	0
$876$	−10.0000	−0.337869
$877$	2.00000i	0.0675352i	0.999430	+	0.0337676i	$0.0107506\pi$
$877$	2.00000i	0.0675352i	−0.999430	+	0.0337676i	$0.989249\pi$
$878$	− 24.0000i	− 0.809961i
$879$	30.0000	1.01187
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	− 1.00000i	− 0.0336718i
$883$	20.0000i	0.673054i	0.941674	+	0.336527i	$0.109252\pi$
$883$	20.0000i	0.673054i	−0.941674	+	0.336527i	$0.890748\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	−4.00000	−0.134383
$887$	24.0000i	0.805841i	0.915235	+	0.402921i	$0.132005\pi$
$887$	24.0000i	0.805841i	−0.915235	+	0.402921i	$0.867995\pi$
$888$	10.0000i	0.335578i
$889$	0	0
$890$	0	0
$891$	−4.00000	−0.134005
$892$	16.0000i	0.535720i
$893$	0	0
$894$	6.00000	0.200670
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	48.0000i	1.60267i
$898$	34.0000i	1.13459i
$899$	0	0
$900$	0	0
$901$	12.0000	0.399778
$902$	− 24.0000i	− 0.799113i
$903$	− 4.00000i	− 0.133112i
$904$	14.0000	0.465633
$905$	0	0
$906$	8.00000	0.265782
$907$	− 12.0000i	− 0.398453i	−0.979953	−	0.199227i	$-0.936157\pi$
$907$	− 12.0000i	− 0.398453i	0.979953	−	0.199227i	$-0.0638430\pi$
$908$	12.0000i	0.398234i
$909$	2.00000	0.0663358
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	− 4.00000i	− 0.132453i
$913$	16.0000i	0.529523i
$914$	−10.0000	−0.330771
$915$	0	0
$916$	−2.00000	−0.0660819
$917$	− 20.0000i	− 0.660458i
$918$	− 2.00000i	− 0.0660098i
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	−28.0000	−0.922631
$922$	− 22.0000i	− 0.724531i
$923$	48.0000i	1.57994i
$924$	4.00000	0.131590
$925$	0	0
$926$	−32.0000	−1.05159
$927$	− 8.00000i	− 0.262754i
$928$	− 2.00000i	− 0.0656532i
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	22.0000i	0.720634i
$933$	8.00000i	0.261908i
$934$	−28.0000	−0.916188
$935$	0	0
$936$	6.00000	0.196116
$937$	22.0000i	0.718709i	0.933201	+	0.359354i	$0.117003\pi$
$937$	22.0000i	0.718709i	−0.933201	+	0.359354i	$0.882997\pi$
$938$	− 4.00000i	− 0.130605i
$939$	10.0000	0.326338
$940$	0	0
$941$	−26.0000	−0.847576	−0.423788	−	0.905761i	$-0.639300\pi$
$941$	−26.0000	−0.847576	−0.423788	+	0.905761i	$0.639300\pi$
$942$	− 10.0000i	− 0.325818i
$943$	− 48.0000i	− 1.56310i
$944$	−4.00000	−0.130189
$945$	0	0
$946$	16.0000	0.520205
$947$	− 4.00000i	− 0.129983i	−0.997886	−	0.0649913i	$-0.979298\pi$
$947$	− 4.00000i	− 0.129983i	0.997886	−	0.0649913i	$-0.0207020\pi$
$948$	0	0
$949$	−60.0000	−1.94768
$950$	0	0
$951$	18.0000	0.583690
$952$	2.00000i	0.0648204i
$953$	26.0000i	0.842223i	0.907009	+	0.421111i	$0.138360\pi$
$953$	26.0000i	0.842223i	−0.907009	+	0.421111i	$0.861640\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	0	0
$957$	8.00000i	0.258603i
$958$	− 16.0000i	− 0.516937i
$959$	10.0000	0.322917
$960$	0	0
$961$	−31.0000	−1.00000
$962$	60.0000i	1.93448i
$963$	12.0000i	0.386695i
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	8.00000	0.257396
$967$	− 8.00000i	− 0.257263i	−0.991692	−	0.128631i	$-0.958942\pi$
$967$	− 8.00000i	− 0.257263i	0.991692	−	0.128631i	$-0.0410584\pi$
$968$	5.00000i	0.160706i
$969$	−8.00000	−0.256997
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	1.00000i	0.0320750i
$973$	− 4.00000i	− 0.128234i
$974$	−8.00000	−0.256337
$975$	0	0
$976$	6.00000	0.192055
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	− 20.0000i	− 0.639529i
$979$	−24.0000	−0.767043
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	− 12.0000i	− 0.382935i
$983$	− 24.0000i	− 0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$	− 24.0000i	− 0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	−4.00000	−0.127386
$987$	0	0
$988$	− 24.0000i	− 0.763542i
$989$	32.0000	1.01754
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	4.00000i	0.126936i
$994$	8.00000	0.253745
$995$	0	0
$996$	4.00000	0.126745
$997$	− 14.0000i	− 0.443384i	−0.975117	−	0.221692i	$-0.928842\pi$
$997$	− 14.0000i	− 0.443384i	0.975117	−	0.221692i	$-0.0711580\pi$
$998$	− 44.0000i	− 1.39280i
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.2.g.a.799.1		2
3.2	odd	2		3150.2.g.r.2899.2		2
5.2	odd	4		42.2.a.a.1.1	✓	1
5.3	odd	4		1050.2.a.i.1.1		1
5.4	even	2	inner	1050.2.g.a.799.2		2
15.2	even	4		126.2.a.a.1.1		1
15.8	even	4		3150.2.a.bo.1.1		1
15.14	odd	2		3150.2.g.r.2899.1		2
20.3	even	4		8400.2.a.k.1.1		1
20.7	even	4		336.2.a.d.1.1		1
35.2	odd	12		294.2.e.c.67.1		2
35.12	even	12		294.2.e.a.67.1		2
35.13	even	4		7350.2.a.f.1.1		1
35.17	even	12		294.2.e.a.79.1		2
35.27	even	4		294.2.a.g.1.1		1
35.32	odd	12		294.2.e.c.79.1		2
40.27	even	4		1344.2.a.i.1.1		1
40.37	odd	4		1344.2.a.q.1.1		1
45.2	even	12		1134.2.f.j.757.1		2
45.7	odd	12		1134.2.f.g.757.1		2
45.22	odd	12		1134.2.f.g.379.1		2
45.32	even	12		1134.2.f.j.379.1		2
55.32	even	4		5082.2.a.d.1.1		1
60.47	odd	4		1008.2.a.j.1.1		1
65.12	odd	4		7098.2.a.f.1.1		1
80.27	even	4		5376.2.c.e.2689.1		2
80.37	odd	4		5376.2.c.bc.2689.2		2
80.67	even	4		5376.2.c.e.2689.2		2
80.77	odd	4		5376.2.c.bc.2689.1		2
105.2	even	12		882.2.g.h.361.1		2
105.17	odd	12		882.2.g.j.667.1		2
105.32	even	12		882.2.g.h.667.1		2
105.47	odd	12		882.2.g.j.361.1		2
105.62	odd	4		882.2.a.b.1.1		1
120.77	even	4		4032.2.a.e.1.1		1
120.107	odd	4		4032.2.a.m.1.1		1
140.27	odd	4		2352.2.a.l.1.1		1
140.47	odd	12		2352.2.q.n.1537.1		2
140.67	even	12		2352.2.q.i.961.1		2
140.87	odd	12		2352.2.q.n.961.1		2
140.107	even	12		2352.2.q.i.1537.1		2
280.27	odd	4		9408.2.a.bw.1.1		1
280.237	even	4		9408.2.a.n.1.1		1
420.167	even	4		7056.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.2.a.a.1.1	✓	1	5.2	odd	4
126.2.a.a.1.1		1	15.2	even	4
294.2.a.g.1.1		1	35.27	even	4
294.2.e.a.67.1		2	35.12	even	12
294.2.e.a.79.1		2	35.17	even	12
294.2.e.c.67.1		2	35.2	odd	12
294.2.e.c.79.1		2	35.32	odd	12
336.2.a.d.1.1		1	20.7	even	4
882.2.a.b.1.1		1	105.62	odd	4
882.2.g.h.361.1		2	105.2	even	12
882.2.g.h.667.1		2	105.32	even	12
882.2.g.j.361.1		2	105.47	odd	12
882.2.g.j.667.1		2	105.17	odd	12
1008.2.a.j.1.1		1	60.47	odd	4
1050.2.a.i.1.1		1	5.3	odd	4
1050.2.g.a.799.1		2	1.1	even	1	trivial
1050.2.g.a.799.2		2	5.4	even	2	inner
1134.2.f.g.379.1		2	45.22	odd	12
1134.2.f.g.757.1		2	45.7	odd	12
1134.2.f.j.379.1		2	45.32	even	12
1134.2.f.j.757.1		2	45.2	even	12
1344.2.a.i.1.1		1	40.27	even	4
1344.2.a.q.1.1		1	40.37	odd	4
2352.2.a.l.1.1		1	140.27	odd	4
2352.2.q.i.961.1		2	140.67	even	12
2352.2.q.i.1537.1		2	140.107	even	12
2352.2.q.n.961.1		2	140.87	odd	12
2352.2.q.n.1537.1		2	140.47	odd	12
3150.2.a.bo.1.1		1	15.8	even	4
3150.2.g.r.2899.1		2	15.14	odd	2
3150.2.g.r.2899.2		2	3.2	odd	2
4032.2.a.e.1.1		1	120.77	even	4
4032.2.a.m.1.1		1	120.107	odd	4
5082.2.a.d.1.1		1	55.32	even	4
5376.2.c.e.2689.1		2	80.27	even	4
5376.2.c.e.2689.2		2	80.67	even	4
5376.2.c.bc.2689.1		2	80.77	odd	4
5376.2.c.bc.2689.2		2	80.37	odd	4
7056.2.a.k.1.1		1	420.167	even	4
7098.2.a.f.1.1		1	65.12	odd	4
7350.2.a.f.1.1		1	35.13	even	4
8400.2.a.k.1.1		1	20.3	even	4
9408.2.a.n.1.1		1	280.237	even	4
9408.2.a.bw.1.1		1	280.27	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:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1050.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} +6.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} +1.00000 q^{21} +4.00000i q^{22} +8.00000i q^{23} +1.00000 q^{24} +6.00000 q^{26} +1.00000i q^{27} -1.00000i q^{28} +2.00000 q^{29} -1.00000i q^{32} +4.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} +10.0000i q^{37} -4.00000i q^{38} +6.00000 q^{39} -6.00000 q^{41} -1.00000i q^{42} -4.00000i q^{43} +4.00000 q^{44} +8.00000 q^{46} -1.00000i q^{48} -1.00000 q^{49} -2.00000 q^{51} -6.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} -1.00000 q^{56} -4.00000i q^{57} -2.00000i q^{58} -4.00000 q^{59} +6.00000 q^{61} -1.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} -4.00000i q^{67} +2.00000i q^{68} +8.00000 q^{69} +8.00000 q^{71} -1.00000i q^{72} +10.0000i q^{73} +10.0000 q^{74} -4.00000 q^{76} -4.00000i q^{77} -6.00000i q^{78} +1.00000 q^{81} +6.00000i q^{82} -4.00000i q^{83} -1.00000 q^{84} -4.00000 q^{86} -2.00000i q^{87} -4.00000i q^{88} +6.00000 q^{89} -6.00000 q^{91} -8.00000i q^{92} -1.00000 q^{96} +14.0000i q^{97} +1.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 8 q^{11} + 2 q^{14} + 2 q^{16} + 8 q^{19} + 2 q^{21} + 2 q^{24} + 12 q^{26} + 4 q^{29} - 4 q^{34} + 2 q^{36} + 12 q^{39} - 12 q^{41} + 8 q^{44} + 16 q^{46} - 2 q^{49} - 4 q^{51} + 2 q^{54} - 2 q^{56} - 8 q^{59} + 12 q^{61} - 2 q^{64} + 8 q^{66} + 16 q^{69} + 16 q^{71} + 20 q^{74} - 8 q^{76} + 2 q^{81} - 2 q^{84} - 8 q^{86} + 12 q^{89} - 12 q^{91} - 2 q^{96} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 8 * q^11 + 2 * q^14 + 2 * q^16 + 8 * q^19 + 2 * q^21 + 2 * q^24 + 12 * q^26 + 4 * q^29 - 4 * q^34 + 2 * q^36 + 12 * q^39 - 12 * q^41 + 8 * q^44 + 16 * q^46 - 2 * q^49 - 4 * q^51 + 2 * q^54 - 2 * q^56 - 8 * q^59 + 12 * q^61 - 2 * q^64 + 8 * q^66 + 16 * q^69 + 16 * q^71 + 20 * q^74 - 8 * q^76 + 2 * q^81 - 2 * q^84 - 8 * q^86 + 12 * q^89 - 12 * q^91 - 2 * q^96 + 8 * q^99

Introduction

L-functions

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