LMFDB - Embedded newform 2300.2.c.j.1749.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,2,Mod(1749,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, 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("2300.1749");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2300.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:	$18.3655924649$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 53x^{4} + 29x^{2} + 4 $ x^8 + 14x^6 + 53x^4 + 29*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1749.7
Root			$2.50653i$ of defining polynomial
Character	$\chi$	$=$	2300.1749
Dual form			2300.2.c.j.1749.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.50653i q^{3} -0.797915i q^{7} -3.28271 q^{9} +O(q^{10})$	$q+2.50653i q^{3} -0.797915i q^{7} -3.28271 q^{9} +4.21515 q^{11} -1.48479i q^{13} -3.01307i q^{17} +2.21515 q^{19} +2.00000 q^{21} +1.00000i q^{23} -0.708618i q^{27} +2.57409 q^{29} +8.78924 q^{31} +10.5654i q^{33} -9.57848i q^{37} +3.72168 q^{39} -1.26097 q^{41} -2.21515i q^{43} +2.86547i q^{47} +6.36333 q^{49} +7.55235 q^{51} +5.41724i q^{53} +5.55235i q^{57} +1.21515 q^{59} +2.58276 q^{61} +2.61932i q^{63} +8.56542i q^{67} -2.50653 q^{69} +6.73036 q^{71} +3.02174i q^{73} -3.36333i q^{77} -0.189019 q^{79} -8.07195 q^{81} -7.22822i q^{83} +6.45204i q^{87} -6.38682 q^{89} -1.18474 q^{91} +22.0305i q^{93} +13.3999i q^{97} -13.8371 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{9}+O(q^{10}) $ 8 * q - 4 * q^9	$ 8 q - 4 q^{9} + 2 q^{11} - 14 q^{19} + 16 q^{21} + 10 q^{29} + 28 q^{31} - 22 q^{39} + 6 q^{41} - 2 q^{49} + 56 q^{51} - 22 q^{59} + 44 q^{61} + 36 q^{71} - 50 q^{79} + 50 q^{91} + 18 q^{99}+O(q^{100}) $ 8 * q - 4 * q^9 + 2 * q^11 - 14 * q^19 + 16 * q^21 + 10 * q^29 + 28 * q^31 - 22 * q^39 + 6 * q^41 - 2 * q^49 + 56 * q^51 - 22 * q^59 + 44 * q^61 + 36 * q^71 - 50 * q^79 + 50 * q^91 + 18 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.50653i	1.44715i	0.690247	+	0.723574i	$0.257502\pi$
$3$	2.50653i	1.44715i	−0.690247	+	0.723574i	$0.742498\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.797915i	− 0.301583i	−0.988566	−	0.150792i	$-0.951818\pi$
$7$	− 0.797915i	− 0.301583i	0.988566	−	0.150792i	$-0.0481823\pi$
$8$	0	0
$9$	−3.28271	−1.09424
$10$	0	0
$11$	4.21515	1.27092	0.635458	−	0.772135i	$-0.280811\pi$
$11$	4.21515	1.27092	0.635458	+	0.772135i	$0.280811\pi$
$12$	0	0
$13$	− 1.48479i	− 0.411808i	−0.978572	−	0.205904i	$-0.933987\pi$
$13$	− 1.48479i	− 0.411808i	0.978572	−	0.205904i	$-0.0660134\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.01307i	− 0.730776i	−0.930855	−	0.365388i	$-0.880936\pi$
$17$	− 3.01307i	− 0.730776i	0.930855	−	0.365388i	$-0.119064\pi$
$18$	0	0
$19$	2.21515	0.508191	0.254095	−	0.967179i	$-0.418222\pi$
$19$	2.21515	0.508191	0.254095	+	0.967179i	$0.418222\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 0.708618i	− 0.136374i
$28$	0	0
$29$	2.57409	0.477997	0.238998	−	0.971020i	$-0.423181\pi$
$29$	2.57409	0.477997	0.238998	+	0.971020i	$0.423181\pi$
$30$	0	0
$31$	8.78924	1.57859	0.789297	−	0.614011i	$-0.210445\pi$
$31$	8.78924	1.57859	0.789297	+	0.614011i	$0.210445\pi$
$32$	0	0
$33$	10.5654i	1.83920i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 9.57848i	− 1.57469i	−0.616511	−	0.787346i	$-0.711454\pi$
$37$	− 9.57848i	− 1.57469i	0.616511	−	0.787346i	$-0.288546\pi$
$38$	0	0
$39$	3.72168	0.595946
$40$	0	0
$41$	−1.26097	−0.196930	−0.0984651	−	0.995141i	$-0.531393\pi$
$41$	−1.26097	−0.196930	−0.0984651	+	0.995141i	$0.531393\pi$
$42$	0	0
$43$	− 2.21515i	− 0.337807i	−0.985633	−	0.168904i	$-0.945977\pi$
$43$	− 2.21515i	− 0.337807i	0.985633	−	0.168904i	$-0.0540227\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.86547i	0.417972i	0.977919	+	0.208986i	$0.0670163\pi$
$47$	2.86547i	0.417972i	−0.977919	+	0.208986i	$0.932984\pi$
$48$	0	0
$49$	6.36333	0.909047
$50$	0	0
$51$	7.55235	1.05754
$52$	0	0
$53$	5.41724i	0.744115i	0.928210	+	0.372057i	$0.121348\pi$
$53$	5.41724i	0.744115i	−0.928210	+	0.372057i	$0.878652\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.55235i	0.735427i
$58$	0	0
$59$	1.21515	0.158199	0.0790996	−	0.996867i	$-0.474795\pi$
$59$	1.21515	0.158199	0.0790996	+	0.996867i	$0.474795\pi$
$60$	0	0
$61$	2.58276	0.330689	0.165344	−	0.986236i	$-0.447126\pi$
$61$	2.58276	0.330689	0.165344	+	0.986236i	$0.447126\pi$
$62$	0	0
$63$	2.61932i	0.330004i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.56542i	1.04643i	0.852200	+	0.523216i	$0.175268\pi$
$67$	8.56542i	1.04643i	−0.852200	+	0.523216i	$0.824732\pi$
$68$	0	0
$69$	−2.50653	−0.301751
$70$	0	0
$71$	6.73036	0.798747	0.399373	−	0.916788i	$-0.369228\pi$
$71$	6.73036	0.798747	0.399373	+	0.916788i	$0.369228\pi$
$72$	0	0
$73$	3.02174i	0.353668i	0.984241	+	0.176834i	$0.0565856\pi$
$73$	3.02174i	0.353668i	−0.984241	+	0.176834i	$0.943414\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.36333i	− 0.383287i
$78$	0	0
$79$	−0.189019	−0.0212663	−0.0106331	−	0.999943i	$-0.503385\pi$
$79$	−0.189019	−0.0212663	−0.0106331	+	0.999943i	$0.503385\pi$
$80$	0	0
$81$	−8.07195	−0.896883
$82$	0	0
$83$	− 7.22822i	− 0.793400i	−0.917948	−	0.396700i	$-0.870155\pi$
$83$	− 7.22822i	− 0.793400i	0.917948	−	0.396700i	$-0.129845\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.45204i	0.691732i
$88$	0	0
$89$	−6.38682	−0.677002	−0.338501	−	0.940966i	$-0.609920\pi$
$89$	−6.38682	−0.677002	−0.338501	+	0.940966i	$0.609920\pi$
$90$	0	0
$91$	−1.18474	−0.124194
$92$	0	0
$93$	22.0305i	2.28446i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	13.3999i	1.36055i	0.732956	+	0.680276i	$0.238140\pi$
$97$	13.3999i	1.36055i	−0.732956	+	0.680276i	$0.761860\pi$
$98$	0	0
$99$	−13.8371	−1.39068
$100$	0	0
$101$	17.1978	1.71125	0.855623	−	0.517600i	$-0.173174\pi$
$101$	17.1978	1.71125	0.855623	+	0.517600i	$0.173174\pi$
$102$	0	0
$103$	− 0.393745i	− 0.0387968i	−0.999812	−	0.0193984i	$-0.993825\pi$
$103$	− 0.393745i	− 0.0387968i	0.999812	−	0.0193984i	$-0.00617509\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 0.969587i	− 0.0937336i	−0.998901	−	0.0468668i	$-0.985076\pi$
$107$	− 0.969587i	− 0.0937336i	0.998901	−	0.0468668i	$-0.0149236\pi$
$108$	0	0
$109$	−10.1612	−0.973271	−0.486635	−	0.873605i	$-0.661776\pi$
$109$	−10.1612	−0.973271	−0.486635	+	0.873605i	$0.661776\pi$
$110$	0	0
$111$	24.0088	2.27881
$112$	0	0
$113$	− 6.99572i	− 0.658102i	−0.944312	−	0.329051i	$-0.893271\pi$
$113$	− 6.99572i	− 0.658102i	0.944312	−	0.329051i	$-0.106729\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.87414i	0.450615i
$118$	0	0
$119$	−2.40417	−0.220390
$120$	0	0
$121$	6.76750	0.615227
$122$	0	0
$123$	− 3.16066i	− 0.284987i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.78924i	0.424976i	0.977164	+	0.212488i	$0.0681567\pi$
$127$	4.78924i	0.424976i	−0.977164	+	0.212488i	$0.931843\pi$
$128$	0	0
$129$	5.55235	0.488857
$130$	0	0
$131$	−5.11543	−0.446937	−0.223469	−	0.974711i	$-0.571738\pi$
$131$	−5.11543	−0.446937	−0.223469	+	0.974711i	$0.571738\pi$
$132$	0	0
$133$	− 1.76750i	− 0.153262i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.0261i	1.02746i	0.857951	+	0.513731i	$0.171737\pi$
$137$	12.0261i	1.02746i	−0.857951	+	0.513731i	$0.828263\pi$
$138$	0	0
$139$	−12.1041	−1.02666	−0.513329	−	0.858192i	$-0.671588\pi$
$139$	−12.1041	−1.02666	−0.513329	+	0.858192i	$0.671588\pi$
$140$	0	0
$141$	−7.18240	−0.604867
$142$	0	0
$143$	− 6.25863i	− 0.523373i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	15.9499i	1.31553i
$148$	0	0
$149$	−14.7701	−1.21002	−0.605009	−	0.796219i	$-0.706830\pi$
$149$	−14.7701	−1.21002	−0.605009	+	0.796219i	$0.706830\pi$
$150$	0	0
$151$	−6.40048	−0.520863	−0.260432	−	0.965492i	$-0.583865\pi$
$151$	−6.40048	−0.520863	−0.260432	+	0.965492i	$0.583865\pi$
$152$	0	0
$153$	9.89102i	0.799641i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.8780i	0.868155i	0.900876	+	0.434078i	$0.142926\pi$
$157$	10.8780i	0.868155i	−0.900876	+	0.434078i	$0.857074\pi$
$158$	0	0
$159$	−13.5785	−1.07684
$160$	0	0
$161$	0.797915	0.0628845
$162$	0	0
$163$	− 15.6237i	− 1.22374i	−0.790957	−	0.611872i	$-0.790417\pi$
$163$	− 15.6237i	− 1.22374i	0.790957	−	0.611872i	$-0.209583\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 12.7502i	− 0.986637i	−0.869849	−	0.493318i	$-0.835784\pi$
$167$	− 12.7502i	− 0.986637i	0.869849	−	0.493318i	$-0.164216\pi$
$168$	0	0
$169$	10.7954	0.830414
$170$	0	0
$171$	−7.27170	−0.556081
$172$	0	0
$173$	0.569697i	0.0433133i	0.999765	+	0.0216566i	$0.00689406\pi$
$173$	0.569697i	0.0433133i	−0.999765	+	0.0216566i	$0.993106\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.04582i	0.228938i
$178$	0	0
$179$	−3.99825	−0.298843	−0.149422	−	0.988774i	$-0.547741\pi$
$179$	−3.99825	−0.298843	−0.149422	+	0.988774i	$0.547741\pi$
$180$	0	0
$181$	−11.2178	−0.833812	−0.416906	−	0.908950i	$-0.636886\pi$
$181$	−11.2178	−0.833812	−0.416906	+	0.908950i	$0.636886\pi$
$182$	0	0
$183$	6.47378i	0.478556i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 12.7005i	− 0.928755i
$188$	0	0
$189$	−0.565417	−0.0411280
$190$	0	0
$191$	−0.0365585	−0.00264528	−0.00132264	−	0.999999i	$-0.500421\pi$
$191$	−0.0365585	−0.00264528	−0.00132264	+	0.999999i	$0.500421\pi$
$192$	0	0
$193$	10.2201i	0.735661i	0.929893	+	0.367831i	$0.119899\pi$
$193$	10.2201i	0.735661i	−0.929893	+	0.367831i	$0.880101\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.4657i	1.24438i	0.782867	+	0.622190i	$0.213757\pi$
$197$	17.4657i	1.24438i	−0.782867	+	0.622190i	$0.786243\pi$
$198$	0	0
$199$	4.37640	0.310235	0.155117	−	0.987896i	$-0.450424\pi$
$199$	4.37640	0.310235	0.155117	+	0.987896i	$0.450424\pi$
$200$	0	0
$201$	−21.4695	−1.51434
$202$	0	0
$203$	− 2.05390i	− 0.144156i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 3.28271i	− 0.228164i
$208$	0	0
$209$	9.33720	0.645868
$210$	0	0
$211$	11.6455	0.801706	0.400853	−	0.916142i	$-0.368714\pi$
$211$	11.6455	0.801706	0.400853	+	0.916142i	$0.368714\pi$
$212$	0	0
$213$	16.8699i	1.15590i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 7.01307i	− 0.476078i
$218$	0	0
$219$	−7.57409	−0.511810
$220$	0	0
$221$	−4.47378	−0.300939
$222$	0	0
$223$	− 18.4068i	− 1.23261i	−0.787507	−	0.616306i	$-0.788629\pi$
$223$	− 18.4068i	− 1.23261i	0.787507	−	0.616306i	$-0.211371\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	27.2747i	1.81029i	0.425105	+	0.905144i	$0.360237\pi$
$227$	27.2747i	1.81029i	−0.425105	+	0.905144i	$0.639763\pi$
$228$	0	0
$229$	−12.8606	−0.849853	−0.424926	−	0.905228i	$-0.639700\pi$
$229$	−12.8606	−0.849853	−0.424926	+	0.905228i	$0.639700\pi$
$230$	0	0
$231$	8.43030	0.554673
$232$	0	0
$233$	24.8847i	1.63025i	0.579285	+	0.815125i	$0.303332\pi$
$233$	24.8847i	1.63025i	−0.579285	+	0.815125i	$0.696668\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 0.473782i	− 0.0307754i
$238$	0	0
$239$	17.8160	1.15242	0.576209	−	0.817302i	$-0.304531\pi$
$239$	17.8160	1.15242	0.576209	+	0.817302i	$0.304531\pi$
$240$	0	0
$241$	13.2004	0.850315	0.425158	−	0.905119i	$-0.360219\pi$
$241$	13.2004	0.850315	0.425158	+	0.905119i	$0.360219\pi$
$242$	0	0
$243$	− 22.3585i	− 1.43430i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 3.28904i	− 0.209277i
$248$	0	0
$249$	18.1178	1.14817
$250$	0	0
$251$	−28.1266	−1.77533	−0.887666	−	0.460488i	$-0.847675\pi$
$251$	−28.1266	−1.77533	−0.887666	+	0.460488i	$0.847675\pi$
$252$	0	0
$253$	4.21515i	0.265004i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	17.1416i	1.06926i	0.845086	+	0.534631i	$0.179549\pi$
$257$	17.1416i	1.06926i	−0.845086	+	0.534631i	$0.820451\pi$
$258$	0	0
$259$	−7.64281	−0.474901
$260$	0	0
$261$	−8.44999	−0.523041
$262$	0	0
$263$	− 22.7093i	− 1.40032i	−0.713988	−	0.700158i	$-0.753113\pi$
$263$	− 22.7093i	− 1.40032i	0.713988	−	0.700158i	$-0.246887\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 16.0088i	− 0.979722i
$268$	0	0
$269$	19.3758	1.18136	0.590682	−	0.806904i	$-0.298859\pi$
$269$	19.3758	1.18136	0.590682	+	0.806904i	$0.298859\pi$
$270$	0	0
$271$	32.0549	1.94720	0.973598	−	0.228268i	$-0.0733061\pi$
$271$	32.0549	1.94720	0.973598	+	0.228268i	$0.0733061\pi$
$272$	0	0
$273$	− 2.96959i	− 0.179728i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6.36099i	0.382195i	0.981571	+	0.191098i	$0.0612047\pi$
$277$	6.36099i	0.382195i	−0.981571	+	0.191098i	$0.938795\pi$
$278$	0	0
$279$	−28.8525	−1.72736
$280$	0	0
$281$	−14.7440	−0.879554	−0.439777	−	0.898107i	$-0.644943\pi$
$281$	−14.7440	−0.879554	−0.439777	+	0.898107i	$0.644943\pi$
$282$	0	0
$283$	− 8.76136i	− 0.520809i	−0.965500	−	0.260404i	$-0.916144\pi$
$283$	− 8.76136i	− 0.520809i	0.965500	−	0.260404i	$-0.0838558\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.00615i	0.0593909i
$288$	0	0
$289$	7.92143	0.465967
$290$	0	0
$291$	−33.5873	−1.96892
$292$	0	0
$293$	− 10.2047i	− 0.596166i	−0.954540	−	0.298083i	$-0.903653\pi$
$293$	− 10.2047i	− 0.596166i	0.954540	−	0.298083i	$-0.0963473\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2.98693i	− 0.173319i
$298$	0	0
$299$	1.48479	0.0858678
$300$	0	0
$301$	−1.76750	−0.101877
$302$	0	0
$303$	43.1069i	2.47642i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 26.7267i	− 1.52537i	−0.646769	−	0.762686i	$-0.723880\pi$
$307$	− 26.7267i	− 1.52537i	0.646769	−	0.762686i	$-0.276120\pi$
$308$	0	0
$309$	0.986934	0.0561447
$310$	0	0
$311$	26.7807	1.51859	0.759297	−	0.650745i	$-0.225543\pi$
$311$	26.7807	1.51859	0.759297	+	0.650745i	$0.225543\pi$
$312$	0	0
$313$	− 12.9214i	− 0.730362i	−0.930936	−	0.365181i	$-0.881007\pi$
$313$	− 12.9214i	− 0.730362i	0.930936	−	0.365181i	$-0.118993\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.1717i	0.795960i	0.917394	+	0.397980i	$0.130289\pi$
$317$	14.1717i	0.795960i	−0.917394	+	0.397980i	$0.869711\pi$
$318$	0	0
$319$	10.8502	0.607493
$320$	0	0
$321$	2.43030	0.135646
$322$	0	0
$323$	− 6.67440i	− 0.371373i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 25.4695i	− 1.40847i
$328$	0	0
$329$	2.28640	0.126053
$330$	0	0
$331$	30.6766	1.68614	0.843068	−	0.537807i	$-0.180747\pi$
$331$	30.6766	1.68614	0.843068	+	0.537807i	$0.180747\pi$
$332$	0	0
$333$	31.4434i	1.72309i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 29.4260i	− 1.60294i	−0.598037	−	0.801469i	$-0.704052\pi$
$337$	− 29.4260i	− 1.60294i	0.598037	−	0.801469i	$-0.295948\pi$
$338$	0	0
$339$	17.5350	0.952371
$340$	0	0
$341$	37.0480	2.00626
$342$	0	0
$343$	− 10.6628i	− 0.575737i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 13.9157i	− 0.747033i	−0.927624	−	0.373516i	$-0.878152\pi$
$347$	− 13.9157i	− 0.747033i	0.927624	−	0.373516i	$-0.121848\pi$
$348$	0	0
$349$	−10.2931	−0.550979	−0.275489	−	0.961304i	$-0.588840\pi$
$349$	−10.2931	−0.550979	−0.275489	+	0.961304i	$0.588840\pi$
$350$	0	0
$351$	−1.05215	−0.0561597
$352$	0	0
$353$	− 13.8960i	− 0.739609i	−0.929110	−	0.369805i	$-0.879425\pi$
$353$	− 13.8960i	− 0.739609i	0.929110	−	0.369805i	$-0.120575\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 6.02613i	− 0.318937i
$358$	0	0
$359$	32.8328	1.73285	0.866425	−	0.499307i	$-0.166412\pi$
$359$	32.8328	1.73285	0.866425	+	0.499307i	$0.166412\pi$
$360$	0	0
$361$	−14.0931	−0.741742
$362$	0	0
$363$	16.9630i	0.890325i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 3.47114i	− 0.181192i	−0.995888	−	0.0905961i	$-0.971123\pi$
$367$	− 3.47114i	− 0.181192i	0.995888	−	0.0905961i	$-0.0288772\pi$
$368$	0	0
$369$	4.13939	0.215488
$370$	0	0
$371$	4.32249	0.224413
$372$	0	0
$373$	− 24.7440i	− 1.28120i	−0.767876	−	0.640598i	$-0.778686\pi$
$373$	− 24.7440i	− 1.28120i	0.767876	−	0.640598i	$-0.221314\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3.82199i	− 0.196843i
$378$	0	0
$379$	−37.7832	−1.94079	−0.970397	−	0.241517i	$-0.922355\pi$
$379$	−37.7832	−1.94079	−0.970397	+	0.241517i	$0.922355\pi$
$380$	0	0
$381$	−12.0044	−0.615004
$382$	0	0
$383$	8.51151i	0.434918i	0.976069	+	0.217459i	$0.0697768\pi$
$383$	8.51151i	0.434918i	−0.976069	+	0.217459i	$0.930223\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.27170i	0.369641i
$388$	0	0
$389$	28.2309	1.43136	0.715681	−	0.698428i	$-0.246117\pi$
$389$	28.2309	1.43136	0.715681	+	0.698428i	$0.246117\pi$
$390$	0	0
$391$	3.01307	0.152377
$392$	0	0
$393$	− 12.8220i	− 0.646784i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 12.0714i	− 0.605844i	−0.953015	−	0.302922i	$-0.902038\pi$
$397$	− 12.0714i	− 0.605844i	0.953015	−	0.302922i	$-0.0979622\pi$
$398$	0	0
$399$	4.43030	0.221793
$400$	0	0
$401$	−32.1612	−1.60606	−0.803028	−	0.595941i	$-0.796779\pi$
$401$	−32.1612	−1.60606	−0.803028	+	0.595941i	$0.796779\pi$
$402$	0	0
$403$	− 13.0502i	− 0.650077i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 40.3748i	− 2.00130i
$408$	0	0
$409$	−17.6704	−0.873746	−0.436873	−	0.899523i	$-0.643914\pi$
$409$	−17.6704	−0.873746	−0.436873	+	0.899523i	$0.643914\pi$
$410$	0	0
$411$	−30.1439	−1.48689
$412$	0	0
$413$	− 0.969587i	− 0.0477103i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 30.3394i	− 1.48573i
$418$	0	0
$419$	16.8710	0.824204	0.412102	−	0.911138i	$-0.364795\pi$
$419$	16.8710	0.824204	0.412102	+	0.911138i	$0.364795\pi$
$420$	0	0
$421$	8.29519	0.404283	0.202141	−	0.979356i	$-0.435210\pi$
$421$	8.29519	0.404283	0.202141	+	0.979356i	$0.435210\pi$
$422$	0	0
$423$	− 9.40651i	− 0.457360i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 2.06083i	− 0.0997303i
$428$	0	0
$429$	15.6875	0.757398
$430$	0	0
$431$	−18.3213	−0.882507	−0.441254	−	0.897382i	$-0.645466\pi$
$431$	−18.3213	−0.882507	−0.441254	+	0.897382i	$0.645466\pi$
$432$	0	0
$433$	− 2.39538i	− 0.115115i	−0.998342	−	0.0575574i	$-0.981669\pi$
$433$	− 2.39538i	− 0.115115i	0.998342	−	0.0575574i	$-0.0183312\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.21515i	0.105965i
$438$	0	0
$439$	−3.00234	−0.143294	−0.0716469	−	0.997430i	$-0.522825\pi$
$439$	−3.00234	−0.143294	−0.0716469	+	0.997430i	$0.522825\pi$
$440$	0	0
$441$	−20.8890	−0.994713
$442$	0	0
$443$	17.0790i	0.811447i	0.913996	+	0.405724i	$0.132980\pi$
$443$	17.0790i	0.811447i	−0.913996	+	0.405724i	$0.867020\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 37.0219i	− 1.75107i
$448$	0	0
$449$	8.01999	0.378487	0.189243	−	0.981930i	$-0.439397\pi$
$449$	8.01999	0.378487	0.189243	+	0.981930i	$0.439397\pi$
$450$	0	0
$451$	−5.31518	−0.250282
$452$	0	0
$453$	− 16.0430i	− 0.753766i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 26.7081i	− 1.24935i	−0.780883	−	0.624677i	$-0.785231\pi$
$457$	− 26.7081i	− 1.24935i	0.780883	−	0.624677i	$-0.214769\pi$
$458$	0	0
$459$	−2.13511	−0.0996586
$460$	0	0
$461$	14.7895	0.688818	0.344409	−	0.938820i	$-0.388079\pi$
$461$	14.7895	0.688818	0.344409	+	0.938820i	$0.388079\pi$
$462$	0	0
$463$	− 1.64545i	− 0.0764708i	−0.999269	−	0.0382354i	$-0.987826\pi$
$463$	− 1.64545i	− 0.0764708i	0.999269	−	0.0382354i	$-0.0121737\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	22.3591i	1.03465i	0.855788	+	0.517327i	$0.173073\pi$
$467$	22.3591i	1.03465i	−0.855788	+	0.517327i	$0.826927\pi$
$468$	0	0
$469$	6.83447	0.315587
$470$	0	0
$471$	−27.2659	−1.25635
$472$	0	0
$473$	− 9.33720i	− 0.429325i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 17.7832i	− 0.814237i
$478$	0	0
$479$	−14.9245	−0.681916	−0.340958	−	0.940078i	$-0.610751\pi$
$479$	−14.9245	−0.681916	−0.340958	+	0.940078i	$0.610751\pi$
$480$	0	0
$481$	−14.2221	−0.648471
$482$	0	0
$483$	2.00000i	0.0910032i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 20.9994i	− 0.951574i	−0.879560	−	0.475787i	$-0.842163\pi$
$487$	− 20.9994i	− 0.951574i	0.879560	−	0.475787i	$-0.157837\pi$
$488$	0	0
$489$	39.1614	1.77094
$490$	0	0
$491$	−37.9743	−1.71376	−0.856878	−	0.515520i	$-0.827599\pi$
$491$	−37.9743	−1.71376	−0.856878	+	0.515520i	$0.827599\pi$
$492$	0	0
$493$	− 7.75590i	− 0.349308i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 5.37025i	− 0.240889i
$498$	0	0
$499$	−20.9630	−0.938431	−0.469216	−	0.883084i	$-0.655463\pi$
$499$	−20.9630	−0.938431	−0.469216	+	0.883084i	$0.655463\pi$
$500$	0	0
$501$	31.9587	1.42781
$502$	0	0
$503$	10.9157i	0.486706i	0.969938	+	0.243353i	$0.0782474\pi$
$503$	10.9157i	0.486706i	−0.969938	+	0.243353i	$0.921753\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	27.0590i	1.20173i
$508$	0	0
$509$	36.0068	1.59598	0.797988	−	0.602674i	$-0.205898\pi$
$509$	36.0068	1.59598	0.797988	+	0.602674i	$0.205898\pi$
$510$	0	0
$511$	2.41109	0.106660
$512$	0	0
$513$	− 1.56970i	− 0.0693038i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	12.0784i	0.531207i
$518$	0	0
$519$	−1.42796	−0.0626807
$520$	0	0
$521$	9.73973	0.426705	0.213353	−	0.976975i	$-0.431562\pi$
$521$	9.73973	0.426705	0.213353	+	0.976975i	$0.431562\pi$
$522$	0	0
$523$	− 22.8414i	− 0.998784i	−0.866376	−	0.499392i	$-0.833557\pi$
$523$	− 22.8414i	− 0.998784i	0.866376	−	0.499392i	$-0.166443\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 26.4826i	− 1.15360i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−3.98899	−0.173107
$532$	0	0
$533$	1.87228i	0.0810974i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 10.0217i	− 0.432470i
$538$	0	0
$539$	26.8224	1.15532
$540$	0	0
$541$	−43.1869	−1.85675	−0.928375	−	0.371645i	$-0.878794\pi$
$541$	−43.1869	−1.85675	−0.928375	+	0.371645i	$0.878794\pi$
$542$	0	0
$543$	− 28.1178i	− 1.20665i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	28.3438i	1.21189i	0.795506	+	0.605946i	$0.207205\pi$
$547$	28.3438i	1.21189i	−0.795506	+	0.605946i	$0.792795\pi$
$548$	0	0
$549$	−8.47846	−0.361852
$550$	0	0
$551$	5.70200	0.242913
$552$	0	0
$553$	0.150821i	0.00641356i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19.8214i	0.839860i	0.907557	+	0.419930i	$0.137945\pi$
$557$	19.8214i	0.839860i	−0.907557	+	0.419930i	$0.862055\pi$
$558$	0	0
$559$	−3.28904	−0.139112
$560$	0	0
$561$	31.8343	1.34405
$562$	0	0
$563$	35.2109i	1.48396i	0.670421	+	0.741981i	$0.266113\pi$
$563$	35.2109i	1.48396i	−0.670421	+	0.741981i	$0.733887\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	6.44073i	0.270485i
$568$	0	0
$569$	9.83875	0.412462	0.206231	−	0.978503i	$-0.433880\pi$
$569$	9.83875	0.412462	0.206231	+	0.978503i	$0.433880\pi$
$570$	0	0
$571$	32.5434	1.36190	0.680949	−	0.732331i	$-0.261567\pi$
$571$	32.5434	1.36190	0.680949	+	0.732331i	$0.261567\pi$
$572$	0	0
$573$	− 0.0916351i	− 0.00382811i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	16.7305i	0.696499i	0.937402	+	0.348249i	$0.113224\pi$
$577$	16.7305i	0.696499i	−0.937402	+	0.348249i	$0.886776\pi$
$578$	0	0
$579$	−25.6171	−1.06461
$580$	0	0
$581$	−5.76750	−0.239276
$582$	0	0
$583$	22.8345i	0.945707i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 5.51257i	− 0.227528i	−0.993508	−	0.113764i	$-0.963709\pi$
$587$	− 5.51257i	− 0.227528i	0.993508	−	0.113764i	$-0.0362908\pi$
$588$	0	0
$589$	19.4695	0.802227
$590$	0	0
$591$	−43.7783	−1.80080
$592$	0	0
$593$	− 2.61582i	− 0.107419i	−0.998557	−	0.0537094i	$-0.982896\pi$
$593$	− 2.61582i	− 0.107419i	0.998557	−	0.0537094i	$-0.0171044\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10.9696i	0.448955i
$598$	0	0
$599$	6.72402	0.274736	0.137368	−	0.990520i	$-0.456136\pi$
$599$	6.72402	0.274736	0.137368	+	0.990520i	$0.456136\pi$
$600$	0	0
$601$	−38.7249	−1.57962	−0.789811	−	0.613350i	$-0.789821\pi$
$601$	−38.7249	−1.57962	−0.789811	+	0.613350i	$0.789821\pi$
$602$	0	0
$603$	− 28.1178i	− 1.14504i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	24.4095i	0.990749i	0.868679	+	0.495375i	$0.164969\pi$
$607$	24.4095i	0.990749i	−0.868679	+	0.495375i	$0.835031\pi$
$608$	0	0
$609$	5.14818	0.208615
$610$	0	0
$611$	4.25463	0.172124
$612$	0	0
$613$	− 42.8224i	− 1.72958i	−0.502133	−	0.864790i	$-0.667451\pi$
$613$	− 42.8224i	− 1.72958i	0.502133	−	0.864790i	$-0.332549\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	40.6177i	1.63521i	0.575782	+	0.817603i	$0.304698\pi$
$617$	40.6177i	1.63521i	−0.575782	+	0.817603i	$0.695302\pi$
$618$	0	0
$619$	−13.6132	−0.547160	−0.273580	−	0.961849i	$-0.588208\pi$
$619$	−13.6132	−0.547160	−0.273580	+	0.961849i	$0.588208\pi$
$620$	0	0
$621$	0.708618	0.0284359
$622$	0	0
$623$	5.09614i	0.204173i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	23.4040i	0.934666i
$628$	0	0
$629$	−28.8606	−1.15075
$630$	0	0
$631$	−37.7108	−1.50124	−0.750621	−	0.660733i	$-0.770246\pi$
$631$	−37.7108	−1.50124	−0.750621	+	0.660733i	$0.770246\pi$
$632$	0	0
$633$	29.1897i	1.16019i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 9.44824i	− 0.374353i
$638$	0	0
$639$	−22.0938	−0.874017
$640$	0	0
$641$	−30.0261	−1.18596	−0.592980	−	0.805217i	$-0.702049\pi$
$641$	−30.0261	−1.18596	−0.592980	+	0.805217i	$0.702049\pi$
$642$	0	0
$643$	24.5030i	0.966302i	0.875537	+	0.483151i	$0.160508\pi$
$643$	24.5030i	0.966302i	−0.875537	+	0.483151i	$0.839492\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 19.3535i	− 0.760864i	−0.924809	−	0.380432i	$-0.875775\pi$
$647$	− 19.3535i	− 0.760864i	0.924809	−	0.380432i	$-0.124225\pi$
$648$	0	0
$649$	5.12205	0.201058
$650$	0	0
$651$	17.5785	0.688955
$652$	0	0
$653$	36.2953i	1.42034i	0.704028	+	0.710172i	$0.251383\pi$
$653$	36.2953i	1.42034i	−0.704028	+	0.710172i	$0.748617\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 9.91949i	− 0.386996i
$658$	0	0
$659$	−30.0553	−1.17079	−0.585394	−	0.810749i	$-0.699060\pi$
$659$	−30.0553	−1.17079	−0.585394	+	0.810749i	$0.699060\pi$
$660$	0	0
$661$	1.29947	0.0505435	0.0252717	−	0.999681i	$-0.491955\pi$
$661$	1.29947	0.0505435	0.0252717	+	0.999681i	$0.491955\pi$
$662$	0	0
$663$	− 11.2137i	− 0.435503i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.57409i	0.0996692i
$668$	0	0
$669$	46.1373	1.78377
$670$	0	0
$671$	10.8867	0.420278
$672$	0	0
$673$	− 50.4441i	− 1.94448i	−0.233994	−	0.972238i	$-0.575180\pi$
$673$	− 50.4441i	− 1.94448i	0.233994	−	0.972238i	$-0.424820\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.84754i	− 0.0710067i	−0.999370	−	0.0355034i	$-0.988697\pi$
$677$	− 1.84754i	− 0.0710067i	0.999370	−	0.0355034i	$-0.0113034\pi$
$678$	0	0
$679$	10.6920	0.410320
$680$	0	0
$681$	−68.3650	−2.61975
$682$	0	0
$683$	− 32.0332i	− 1.22572i	−0.790193	−	0.612858i	$-0.790020\pi$
$683$	− 32.0332i	− 1.22572i	0.790193	−	0.612858i	$-0.209980\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 32.2355i	− 1.22986i
$688$	0	0
$689$	8.04348	0.306432
$690$	0	0
$691$	25.5585	0.972291	0.486146	−	0.873878i	$-0.338402\pi$
$691$	25.5585	0.972291	0.486146	+	0.873878i	$0.338402\pi$
$692$	0	0
$693$	11.0408i	0.419407i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.79938i	0.143912i
$698$	0	0
$699$	−62.3743	−2.35921
$700$	0	0
$701$	22.5707	0.852484	0.426242	−	0.904609i	$-0.359837\pi$
$701$	22.5707	0.852484	0.426242	+	0.904609i	$0.359837\pi$
$702$	0	0
$703$	− 21.2178i	− 0.800244i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 13.7224i	− 0.516083i
$708$	0	0
$709$	−13.6916	−0.514198	−0.257099	−	0.966385i	$-0.582767\pi$
$709$	−13.6916	−0.514198	−0.257099	+	0.966385i	$0.582767\pi$
$710$	0	0
$711$	0.620494	0.0232703
$712$	0	0
$713$	8.78924i	0.329160i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	44.6563i	1.66772i
$718$	0	0
$719$	−19.5003	−0.727239	−0.363619	−	0.931548i	$-0.618459\pi$
$719$	−19.5003	−0.727239	−0.363619	+	0.931548i	$0.618459\pi$
$720$	0	0
$721$	−0.314175	−0.0117005
$722$	0	0
$723$	33.0874i	1.23053i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 50.2432i	− 1.86342i	−0.363209	−	0.931708i	$-0.618319\pi$
$727$	− 50.2432i	− 1.86342i	0.363209	−	0.931708i	$-0.381681\pi$
$728$	0	0
$729$	31.8264	1.17876
$730$	0	0
$731$	−6.67440	−0.246862
$732$	0	0
$733$	− 20.7355i	− 0.765881i	−0.923773	−	0.382941i	$-0.874911\pi$
$733$	− 20.7355i	− 0.765881i	0.923773	−	0.382941i	$-0.125089\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	36.1045i	1.32993i
$738$	0	0
$739$	−33.6672	−1.23847	−0.619234	−	0.785207i	$-0.712557\pi$
$739$	−33.6672	−1.23847	−0.619234	+	0.785207i	$0.712557\pi$
$740$	0	0
$741$	8.24410	0.302854
$742$	0	0
$743$	− 36.5412i	− 1.34056i	−0.742106	−	0.670282i	$-0.766173\pi$
$743$	− 36.5412i	− 1.34056i	0.742106	−	0.670282i	$-0.233827\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	23.7281i	0.868167i
$748$	0	0
$749$	−0.773648	−0.0282685
$750$	0	0
$751$	25.3495	0.925016	0.462508	−	0.886615i	$-0.346950\pi$
$751$	25.3495	0.925016	0.462508	+	0.886615i	$0.346950\pi$
$752$	0	0
$753$	− 70.5001i	− 2.56917i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	9.14818i	0.332496i	0.986084	+	0.166248i	$0.0531653\pi$
$757$	9.14818i	0.332496i	−0.986084	+	0.166248i	$0.946835\pi$
$758$	0	0
$759$	−10.5654	−0.383500
$760$	0	0
$761$	20.9609	0.759833	0.379916	−	0.925021i	$-0.375953\pi$
$761$	20.9609	0.759833	0.379916	+	0.925021i	$0.375953\pi$
$762$	0	0
$763$	8.10781i	0.293522i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.80425i	− 0.0651477i
$768$	0	0
$769$	−36.5915	−1.31953	−0.659763	−	0.751474i	$-0.729343\pi$
$769$	−36.5915	−1.31953	−0.659763	+	0.751474i	$0.729343\pi$
$770$	0	0
$771$	−42.9659	−1.54738
$772$	0	0
$773$	− 38.1219i	− 1.37115i	−0.728003	−	0.685574i	$-0.759551\pi$
$773$	− 38.1219i	− 1.37115i	0.728003	−	0.685574i	$-0.240449\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 19.1570i	− 0.687252i
$778$	0	0
$779$	−2.79324	−0.100078
$780$	0	0
$781$	28.3695	1.01514
$782$	0	0
$783$	− 1.82405i	− 0.0651861i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 20.4495i	− 0.728946i	−0.931214	−	0.364473i	$-0.881249\pi$
$787$	− 20.4495i	− 0.728946i	0.931214	−	0.364473i	$-0.118751\pi$
$788$	0	0
$789$	56.9217	2.02646
$790$	0	0
$791$	−5.58199	−0.198473
$792$	0	0
$793$	− 3.83487i	− 0.136180i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27.5177i	0.974725i	0.873200	+	0.487363i	$0.162041\pi$
$797$	27.5177i	0.974725i	−0.873200	+	0.487363i	$0.837959\pi$
$798$	0	0
$799$	8.63386	0.305444
$800$	0	0
$801$	20.9661	0.740800
$802$	0	0
$803$	12.7371i	0.449482i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	48.5661i	1.70961i
$808$	0	0
$809$	−16.1917	−0.569268	−0.284634	−	0.958636i	$-0.591872\pi$
$809$	−16.1917	−0.569268	−0.284634	+	0.958636i	$0.591872\pi$
$810$	0	0
$811$	−17.2872	−0.607036	−0.303518	−	0.952826i	$-0.598161\pi$
$811$	−17.2872	−0.607036	−0.303518	+	0.952826i	$0.598161\pi$
$812$	0	0
$813$	80.3467i	2.81788i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 4.90690i	− 0.171671i
$818$	0	0
$819$	3.88915	0.135898
$820$	0	0
$821$	−32.6395	−1.13913	−0.569564	−	0.821947i	$-0.692888\pi$
$821$	−32.6395	−1.13913	−0.569564	+	0.821947i	$0.692888\pi$
$822$	0	0
$823$	39.7419i	1.38532i	0.721266	+	0.692658i	$0.243560\pi$
$823$	39.7419i	1.38532i	−0.721266	+	0.692658i	$0.756440\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.46764i	0.329222i	0.986359	+	0.164611i	$0.0526369\pi$
$827$	9.46764i	0.329222i	−0.986359	+	0.164611i	$0.947363\pi$
$828$	0	0
$829$	−43.6920	−1.51748	−0.758742	−	0.651391i	$-0.774186\pi$
$829$	−43.6920	−1.51748	−0.758742	+	0.651391i	$0.774186\pi$
$830$	0	0
$831$	−15.9440	−0.553093
$832$	0	0
$833$	− 19.1731i	− 0.664310i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 6.22822i	− 0.215279i
$838$	0	0
$839$	1.37212	0.0473708	0.0236854	−	0.999719i	$-0.492460\pi$
$839$	1.37212	0.0473708	0.0236854	+	0.999719i	$0.492460\pi$
$840$	0	0
$841$	−22.3741	−0.771519
$842$	0	0
$843$	− 36.9564i	− 1.27284i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 5.39989i	− 0.185542i
$848$	0	0
$849$	21.9606	0.753687
$850$	0	0
$851$	9.57848	0.328346
$852$	0	0
$853$	− 18.6367i	− 0.638107i	−0.947737	−	0.319054i	$-0.896635\pi$
$853$	− 18.6367i	− 0.638107i	0.947737	−	0.319054i	$-0.103365\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21.9201i	0.748776i	0.927272	+	0.374388i	$0.122147\pi$
$857$	21.9201i	0.748776i	−0.927272	+	0.374388i	$0.877853\pi$
$858$	0	0
$859$	−43.1587	−1.47256	−0.736278	−	0.676679i	$-0.763419\pi$
$859$	−43.1587	−1.47256	−0.736278	+	0.676679i	$0.763419\pi$
$860$	0	0
$861$	−2.52194	−0.0859474
$862$	0	0
$863$	38.0107i	1.29390i	0.762532	+	0.646950i	$0.223956\pi$
$863$	38.0107i	1.29390i	−0.762532	+	0.646950i	$0.776044\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	19.8553i	0.674322i
$868$	0	0
$869$	−0.796743	−0.0270277
$870$	0	0
$871$	12.7179	0.430929
$872$	0	0
$873$	− 43.9879i	− 1.48877i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	27.9976i	0.945411i	0.881220	+	0.472706i	$0.156723\pi$
$877$	27.9976i	0.945411i	−0.881220	+	0.472706i	$0.843277\pi$
$878$	0	0
$879$	25.5785	0.862741
$880$	0	0
$881$	−8.84326	−0.297937	−0.148968	−	0.988842i	$-0.547595\pi$
$881$	−8.84326	−0.297937	−0.148968	+	0.988842i	$0.547595\pi$
$882$	0	0
$883$	19.3282i	0.650447i	0.945637	+	0.325224i	$0.105440\pi$
$883$	19.3282i	0.650447i	−0.945637	+	0.325224i	$0.894560\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 22.4857i	− 0.754995i	−0.926011	−	0.377498i	$-0.876785\pi$
$887$	− 22.4857i	− 0.754995i	0.926011	−	0.377498i	$-0.123215\pi$
$888$	0	0
$889$	3.82141	0.128166
$890$	0	0
$891$	−34.0245	−1.13986
$892$	0	0
$893$	6.34745i	0.212409i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.72168i	0.124263i
$898$	0	0
$899$	22.6243	0.754563
$900$	0	0
$901$	16.3225	0.543781
$902$	0	0
$903$	− 4.43030i	− 0.147431i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	28.5934i	0.949429i	0.880140	+	0.474714i	$0.157449\pi$
$907$	28.5934i	0.949429i	−0.880140	+	0.474714i	$0.842551\pi$
$908$	0	0
$909$	−56.4554	−1.87251
$910$	0	0
$911$	−42.2774	−1.40071	−0.700356	−	0.713794i	$-0.746975\pi$
$911$	−42.2774	−1.40071	−0.700356	+	0.713794i	$0.746975\pi$
$912$	0	0
$913$	− 30.4680i	− 1.00834i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.08168i	0.134789i
$918$	0	0
$919$	36.3783	1.20001	0.600004	−	0.799997i	$-0.295166\pi$
$919$	36.3783	1.20001	0.600004	+	0.799997i	$0.295166\pi$
$920$	0	0
$921$	66.9913	2.20744
$922$	0	0
$923$	− 9.99319i	− 0.328930i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.29255i	0.0424529i
$928$	0	0
$929$	−11.0567	−0.362757	−0.181379	−	0.983413i	$-0.558056\pi$
$929$	−11.0567	−0.362757	−0.181379	+	0.983413i	$0.558056\pi$
$930$	0	0
$931$	14.0957	0.461969
$932$	0	0
$933$	67.1267i	2.19763i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 44.7081i	− 1.46055i	−0.683153	−	0.730276i	$-0.739392\pi$
$937$	− 44.7081i	− 1.46055i	0.683153	−	0.730276i	$-0.260608\pi$
$938$	0	0
$939$	32.3880	1.05694
$940$	0	0
$941$	25.1343	0.819356	0.409678	−	0.912230i	$-0.365641\pi$
$941$	25.1343	0.819356	0.409678	+	0.912230i	$0.365641\pi$
$942$	0	0
$943$	− 1.26097i	− 0.0410628i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 50.2426i	− 1.63266i	−0.577583	−	0.816332i	$-0.696004\pi$
$947$	− 50.2426i	− 1.63266i	0.577583	−	0.816332i	$-0.303996\pi$
$948$	0	0
$949$	4.48666	0.145643
$950$	0	0
$951$	−35.5218	−1.15187
$952$	0	0
$953$	37.2648i	1.20712i	0.797316	+	0.603562i	$0.206253\pi$
$953$	37.2648i	1.20712i	−0.797316	+	0.603562i	$0.793747\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	27.1963i	0.879133i
$958$	0	0
$959$	9.59583	0.309866
$960$	0	0
$961$	46.2508	1.49196
$962$	0	0
$963$	3.18287i	0.102567i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 6.29267i	− 0.202359i	−0.994868	−	0.101179i	$-0.967738\pi$
$967$	− 6.29267i	− 0.202359i	0.994868	−	0.101179i	$-0.0322616\pi$
$968$	0	0
$969$	16.7296	0.537432
$970$	0	0
$971$	11.1451	0.357665	0.178832	−	0.983880i	$-0.442768\pi$
$971$	11.1451	0.357665	0.178832	+	0.983880i	$0.442768\pi$
$972$	0	0
$973$	9.65805i	0.309623i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 31.1605i	− 0.996912i	−0.866915	−	0.498456i	$-0.833901\pi$
$977$	− 31.1605i	− 0.996912i	0.866915	−	0.498456i	$-0.166099\pi$
$978$	0	0
$979$	−26.9214	−0.860413
$980$	0	0
$981$	33.3564	1.06499
$982$	0	0
$983$	− 34.4322i	− 1.09822i	−0.835752	−	0.549108i	$-0.814968\pi$
$983$	− 34.4322i	− 1.09822i	0.835752	−	0.549108i	$-0.185032\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	5.73094i	0.182418i
$988$	0	0
$989$	2.21515	0.0704377
$990$	0	0
$991$	5.54092	0.176013	0.0880066	−	0.996120i	$-0.471950\pi$
$991$	5.54092	0.176013	0.0880066	+	0.996120i	$0.471950\pi$
$992$	0	0
$993$	76.8918i	2.44009i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	12.0230i	0.380773i	0.981709	+	0.190387i	$0.0609741\pi$
$997$	12.0230i	0.380773i	−0.981709	+	0.190387i	$0.939026\pi$
$998$	0	0
$999$	−6.78749	−0.214747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.2.c.j.1749.7		8
5.2	odd	4		2300.2.a.l.1.4	✓	4
5.3	odd	4		2300.2.a.m.1.1	yes	4
5.4	even	2	inner	2300.2.c.j.1749.2		8
20.3	even	4		9200.2.a.cm.1.4		4
20.7	even	4		9200.2.a.co.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2300.2.a.l.1.4	✓	4	5.2	odd	4
2300.2.a.m.1.1	yes	4	5.3	odd	4
2300.2.c.j.1749.2		8	5.4	even	2	inner
2300.2.c.j.1749.7		8	1.1	even	1	trivial
9200.2.a.cm.1.4		4	20.3	even	4
9200.2.a.co.1.1		4	20.7	even	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:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2300.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.50653i q^{3} -0.797915i q^{7} -3.28271 q^{9} +O(q^{10})\)	\(q+2.50653i q^{3} -0.797915i q^{7} -3.28271 q^{9} +4.21515 q^{11} -1.48479i q^{13} -3.01307i q^{17} +2.21515 q^{19} +2.00000 q^{21} +1.00000i q^{23} -0.708618i q^{27} +2.57409 q^{29} +8.78924 q^{31} +10.5654i q^{33} -9.57848i q^{37} +3.72168 q^{39} -1.26097 q^{41} -2.21515i q^{43} +2.86547i q^{47} +6.36333 q^{49} +7.55235 q^{51} +5.41724i q^{53} +5.55235i q^{57} +1.21515 q^{59} +2.58276 q^{61} +2.61932i q^{63} +8.56542i q^{67} -2.50653 q^{69} +6.73036 q^{71} +3.02174i q^{73} -3.36333i q^{77} -0.189019 q^{79} -8.07195 q^{81} -7.22822i q^{83} +6.45204i q^{87} -6.38682 q^{89} -1.18474 q^{91} +22.0305i q^{93} +13.3999i q^{97} -13.8371 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{9}+O(q^{10}) \) 8 * q - 4 * q^9	\( 8 q - 4 q^{9} + 2 q^{11} - 14 q^{19} + 16 q^{21} + 10 q^{29} + 28 q^{31} - 22 q^{39} + 6 q^{41} - 2 q^{49} + 56 q^{51} - 22 q^{59} + 44 q^{61} + 36 q^{71} - 50 q^{79} + 50 q^{91} + 18 q^{99}+O(q^{100}) \) 8 * q - 4 * q^9 + 2 * q^11 - 14 * q^19 + 16 * q^21 + 10 * q^29 + 28 * q^31 - 22 * q^39 + 6 * q^41 - 2 * q^49 + 56 * q^51 - 22 * q^59 + 44 * q^61 + 36 * q^71 - 50 * q^79 + 50 * q^91 + 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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