LMFDB - Embedded newform 2300.2.i.f.1793.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,2,Mod(1057,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1, 2]))

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

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

chi := DirichletCharacter("2300.1057");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2300.i (of order $4$, degree $2$, 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:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 74x^{12} + 1357x^{8} + 3177x^{4} + 1296 $ x^16 + 74x^12 + 1357x^8 + 3177*x^4 + 1296
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1793.6
Root			$0.854787 + 0.854787i$ of defining polynomial
Character	$\chi$	$=$	2300.1793
Dual form			2300.2.i.f.1057.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.854787 + 0.854787i) q^{3} +(-3.24270 - 3.24270i) q^{7} -1.53868i q^{9} +O(q^{10})$	$q+(0.854787 + 0.854787i) q^{3} +(-3.24270 - 3.24270i) q^{7} -1.53868i q^{9} +4.86561i q^{11} +(4.90930 + 4.90930i) q^{13} +(-3.33320 - 3.33320i) q^{17} +1.13439 q^{19} -5.54364i q^{21} +(-1.49169 - 4.55795i) q^{23} +(3.87960 - 3.87960i) q^{27} +0.233144i q^{29} +6.17611 q^{31} +(-4.15906 + 4.15906i) q^{33} +(5.16592 + 5.16592i) q^{37} +8.39282i q^{39} +0.771821 q^{41} +(5.83599 - 5.83599i) q^{43} +(6.70938 - 6.70938i) q^{47} +14.0303i q^{49} -5.69835i q^{51} +(0.739915 - 0.739915i) q^{53} +(0.969659 + 0.969659i) q^{57} -5.83274i q^{59} +5.08729i q^{61} +(-4.98948 + 4.98948i) q^{63} +(1.97652 + 1.97652i) q^{67} +(2.62100 - 5.17115i) q^{69} +9.81355 q^{71} +(1.99512 + 1.99512i) q^{73} +(15.7777 - 15.7777i) q^{77} -9.05320 q^{79} +2.01644 q^{81} +(7.84970 - 7.84970i) q^{83} +(-0.199288 + 0.199288i) q^{87} +17.6984 q^{89} -31.8388i q^{91} +(5.27926 + 5.27926i) q^{93} +(-10.8255 - 10.8255i) q^{97} +7.48661 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 68 q^{19} - 4 q^{41} - 56 q^{69} - 8 q^{71} - 28 q^{79} + 64 q^{81} + 120 q^{89} - 28 q^{99}+O(q^{100}) $ 16 * q + 68 * q^19 - 4 * q^41 - 56 * q^69 - 8 * q^71 - 28 * q^79 + 64 * q^81 + 120 * q^89 - 28 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$277$	$1151$	$1201$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$-1$

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
$3$	0.854787	+	0.854787i	0.493512	+	0.493512i	0.909411	−	0.415899i	$-0.136533\pi$
$3$	0.854787	+	0.854787i	0.493512	+	0.493512i	−0.415899	+	0.909411i	$0.636533\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−3.24270	−	3.24270i	−1.22563	−	1.22563i	−0.965602	−	0.260025i	$-0.916269\pi$
$7$	−3.24270	−	3.24270i	−1.22563	−	1.22563i	−0.260025	−	0.965602i	$-0.583731\pi$
$8$		0			0
$9$		−	1.53868i		−	0.512893i
$10$		0			0
$11$			4.86561i			1.46704i	0.679669	+	0.733519i	$0.262123\pi$
$11$			4.86561i			1.46704i	−0.679669	+	0.733519i	$0.737877\pi$
$12$		0			0
$13$	4.90930	+	4.90930i	1.36160	+	1.36160i	0.871885	+	0.489711i	$0.162898\pi$
$13$	4.90930	+	4.90930i	1.36160	+	1.36160i	0.489711	+	0.871885i	$0.337102\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.33320	−	3.33320i	−0.808420	−	0.808420i	0.175975	−	0.984395i	$-0.443692\pi$
$17$	−3.33320	−	3.33320i	−0.808420	−	0.808420i	−0.984395	+	0.175975i	$0.943692\pi$
$18$		0			0
$19$	1.13439			0.260246			0.130123	−	0.991498i	$-0.458463\pi$
$19$	1.13439			0.260246			0.130123	+	0.991498i	$0.458463\pi$
$20$		0			0
$21$		−	5.54364i		−	1.20972i
$22$		0			0
$23$	−1.49169	−	4.55795i	−0.311038	−	0.950397i
$23$	−1.49169	−	4.55795i	−0.311038	−	0.950397i
$24$		0			0
$25$		0			0
$26$		0			0
$27$	3.87960	−	3.87960i	0.746630	−	0.746630i
$28$		0			0
$29$			0.233144i			0.0432937i	0.999766	+	0.0216468i	$0.00689094\pi$
$29$			0.233144i			0.0432937i	−0.999766	+	0.0216468i	$0.993109\pi$
$30$		0			0
$31$	6.17611			1.10926			0.554631	−	0.832096i	$-0.312859\pi$
$31$	6.17611			1.10926			0.554631	+	0.832096i	$0.312859\pi$
$32$		0			0
$33$	−4.15906	+	4.15906i	−0.724000	+	0.724000i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.16592	+	5.16592i	0.849272	+	0.849272i	0.990042	−	0.140770i	$-0.0449578\pi$
$37$	5.16592	+	5.16592i	0.849272	+	0.849272i	−0.140770	+	0.990042i	$0.544958\pi$
$38$		0			0
$39$			8.39282i			1.34393i
$40$		0			0
$41$	0.771821			0.120538			0.0602691	−	0.998182i	$-0.480804\pi$
$41$	0.771821			0.120538			0.0602691	+	0.998182i	$0.480804\pi$
$42$		0			0
$43$	5.83599	−	5.83599i	0.889980	−	0.889980i	−0.104541	−	0.994521i	$-0.533337\pi$
$43$	5.83599	−	5.83599i	0.889980	−	0.889980i	0.994521	+	0.104541i	$0.0333372\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.70938	−	6.70938i	0.978663	−	0.978663i	−0.0211140	−	0.999777i	$-0.506721\pi$
$47$	6.70938	−	6.70938i	0.978663	−	0.978663i	0.999777	+	0.0211140i	$0.00672131\pi$
$48$		0			0
$49$			14.0303i			2.00432i
$50$		0			0
$51$		−	5.69835i		−	0.797929i
$52$		0			0
$53$	0.739915	−	0.739915i	0.101635	−	0.101635i	−0.654461	−	0.756096i	$-0.727104\pi$
$53$	0.739915	−	0.739915i	0.101635	−	0.101635i	0.756096	+	0.654461i	$0.227104\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.969659	+	0.969659i	0.128434	+	0.128434i
$58$		0			0
$59$		−	5.83274i		−	0.759358i	−0.925118	−	0.379679i	$-0.876034\pi$
$59$		−	5.83274i		−	0.759358i	0.925118	−	0.379679i	$-0.123966\pi$
$60$		0			0
$61$			5.08729i			0.651360i	0.945480	+	0.325680i	$0.105593\pi$
$61$			5.08729i			0.651360i	−0.945480	+	0.325680i	$0.894407\pi$
$62$		0			0
$63$	−4.98948	+	4.98948i	−0.628615	+	0.628615i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	1.97652	+	1.97652i	0.241470	+	0.241470i	0.817458	−	0.575988i	$-0.195383\pi$
$67$	1.97652	+	1.97652i	0.241470	+	0.241470i	−0.575988	+	0.817458i	$0.695383\pi$
$68$		0			0
$69$	2.62100	−	5.17115i	0.315531	−	0.622533i
$70$		0			0
$71$	9.81355			1.16465			0.582327	−	0.812955i	$-0.302142\pi$
$71$	9.81355			1.16465			0.582327	+	0.812955i	$0.302142\pi$
$72$		0			0
$73$	1.99512	+	1.99512i	0.233511	+	0.233511i	0.814156	−	0.580646i	$-0.197200\pi$
$73$	1.99512	+	1.99512i	0.233511	+	0.233511i	−0.580646	+	0.814156i	$0.697200\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	15.7777	−	15.7777i	1.79804	−	1.79804i
$78$		0			0
$79$	−9.05320			−1.01856			−0.509282	−	0.860600i	$-0.670089\pi$
$79$	−9.05320			−1.01856			−0.509282	+	0.860600i	$0.670089\pi$
$80$		0			0
$81$	2.01644			0.224048
$82$		0			0
$83$	7.84970	−	7.84970i	0.861617	−	0.861617i	−0.129909	−	0.991526i	$-0.541469\pi$
$83$	7.84970	−	7.84970i	0.861617	−	0.861617i	0.991526	+	0.129909i	$0.0414685\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−0.199288	+	0.199288i	−0.0213659	+	0.0213659i
$88$		0			0
$89$	17.6984			1.87602			0.938011	−	0.346606i	$-0.112666\pi$
$89$	17.6984			1.87602			0.938011	+	0.346606i	$0.112666\pi$
$90$		0			0
$91$		−	31.8388i		−	3.33762i
$92$		0			0
$93$	5.27926	+	5.27926i	0.547434	+	0.547434i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−10.8255	−	10.8255i	−1.09916	−	1.09916i	−0.994509	−	0.104650i	$-0.966628\pi$
$97$	−10.8255	−	10.8255i	−1.09916	−	1.09916i	−0.104650	−	0.994509i	$-0.533372\pi$
$98$		0			0
$99$	7.48661			0.752433
$100$		0			0
$101$	−8.94297			−0.889859			−0.444929	−	0.895566i	$-0.646771\pi$
$101$	−8.94297			−0.889859			−0.444929	+	0.895566i	$0.646771\pi$
$102$		0			0
$103$	9.58432	−	9.58432i	0.944371	−	0.944371i	−0.0541615	−	0.998532i	$-0.517249\pi$
$103$	9.58432	−	9.58432i	0.944371	−	0.944371i	0.998532	+	0.0541615i	$0.0172486\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−10.6817	−	10.6817i	−1.03264	−	1.03264i	−0.999449	−	0.0331869i	$-0.989434\pi$
$107$	−10.6817	−	10.6817i	−1.03264	−	1.03264i	−0.0331869	−	0.999449i	$-0.510566\pi$
$108$		0			0
$109$	−0.653872			−0.0626295			−0.0313148	−	0.999510i	$-0.509969\pi$
$109$	−0.653872			−0.0626295			−0.0313148	+	0.999510i	$0.509969\pi$
$110$		0			0
$111$			8.83153i			0.838251i
$112$		0			0
$113$	3.00841	−	3.00841i	0.283007	−	0.283007i	−0.551300	−	0.834307i	$-0.685868\pi$
$113$	3.00841	−	3.00841i	0.283007	−	0.283007i	0.834307	+	0.551300i	$0.185868\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	7.55384	−	7.55384i	0.698353	−	0.698353i
$118$		0			0
$119$			21.6172i			1.98164i
$120$		0			0
$121$	−12.6742			−1.15220
$122$		0			0
$123$	0.659743	+	0.659743i	0.0594870	+	0.0594870i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−6.76334	+	6.76334i	−0.600149	+	0.600149i	−0.940352	−	0.340203i	$-0.889504\pi$
$127$	−6.76334	+	6.76334i	−0.600149	+	0.600149i	0.340203	+	0.940352i	$0.389504\pi$
$128$		0			0
$129$	9.97706			0.878431
$130$		0			0
$131$	19.8021			1.73012			0.865058	−	0.501672i	$-0.167282\pi$
$131$	19.8021			1.73012			0.865058	+	0.501672i	$0.167282\pi$
$132$		0			0
$133$	−3.67848	−	3.67848i	−0.318965	−	0.318965i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	11.4825	+	11.4825i	0.981016	+	0.981016i	0.999823	−	0.0188073i	$-0.00598690\pi$
$137$	11.4825	+	11.4825i	0.981016	+	0.981016i	−0.0188073	+	0.999823i	$0.505987\pi$
$138$		0			0
$139$			13.8135i			1.17165i	0.810438	+	0.585825i	$0.199229\pi$
$139$			13.8135i			1.17165i	−0.810438	+	0.585825i	$0.800771\pi$
$140$		0			0
$141$	11.4702			0.965963
$142$		0			0
$143$	−23.8868	+	23.8868i	−1.99751	+	1.99751i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−11.9929	+	11.9929i	−0.989156	+	0.989156i
$148$		0			0
$149$	9.42958			0.772501			0.386251	−	0.922394i	$-0.373770\pi$
$149$	9.42958			0.772501			0.386251	+	0.922394i	$0.373770\pi$
$150$		0			0
$151$	5.64891			0.459702			0.229851	−	0.973226i	$-0.426176\pi$
$151$	5.64891			0.459702			0.229851	+	0.973226i	$0.426176\pi$
$152$		0			0
$153$	−5.12872	+	5.12872i	−0.414633	+	0.414633i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−6.01243	−	6.01243i	−0.479844	−	0.479844i	0.425237	−	0.905082i	$-0.360191\pi$
$157$	−6.01243	−	6.01243i	−0.479844	−	0.479844i	−0.905082	+	0.425237i	$0.860191\pi$
$158$		0			0
$159$	1.26494			0.100316
$160$		0			0
$161$	−9.94297	+	19.6172i	−0.783616	+	1.54605i
$162$		0			0
$163$	−2.64453	−	2.64453i	−0.207136	−	0.207136i	0.595913	−	0.803049i	$-0.296790\pi$
$163$	−2.64453	−	2.64453i	−0.207136	−	0.207136i	−0.803049	+	0.595913i	$0.796790\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.84382	+	2.84382i	−0.220062	+	0.220062i	−0.808524	−	0.588463i	$-0.799733\pi$
$167$	−2.84382	+	2.84382i	−0.220062	+	0.220062i	0.588463	+	0.808524i	$0.299733\pi$
$168$		0			0
$169$			35.2025i			2.70789i
$170$		0			0
$171$		−	1.74546i		−	0.133478i
$172$		0			0
$173$	−9.91335	−	9.91335i	−0.753698	−	0.753698i	0.221469	−	0.975167i	$-0.428915\pi$
$173$	−9.91335	−	9.91335i	−0.753698	−	0.753698i	−0.975167	+	0.221469i	$0.928915\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	4.98575	−	4.98575i	0.374752	−	0.374752i
$178$		0			0
$179$		−	2.66534i		−	0.199217i	−0.995027	−	0.0996085i	$-0.968241\pi$
$179$		−	2.66534i		−	0.199217i	0.995027	−	0.0996085i	$-0.0317591\pi$
$180$		0			0
$181$			22.9404i			1.70514i	0.522611	+	0.852571i	$0.324958\pi$
$181$			22.9404i			1.70514i	−0.522611	+	0.852571i	$0.675042\pi$
$182$		0			0
$183$	−4.34855	+	4.34855i	−0.321454	+	0.321454i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	16.2181	−	16.2181i	1.18598	−	1.18598i
$188$		0			0
$189$	−25.1608			−1.83018
$190$		0			0
$191$			9.70707i			0.702379i	0.936304	+	0.351189i	$0.114223\pi$
$191$			9.70707i			0.702379i	−0.936304	+	0.351189i	$0.885777\pi$
$192$		0			0
$193$	−9.13265	−	9.13265i	−0.657383	−	0.657383i	0.297377	−	0.954760i	$-0.403888\pi$
$193$	−9.13265	−	9.13265i	−0.657383	−	0.657383i	−0.954760	+	0.297377i	$0.903888\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−10.5341	+	10.5341i	−0.750527	+	0.750527i	−0.974578	−	0.224050i	$-0.928072\pi$
$197$	−10.5341	+	10.5341i	−0.750527	+	0.750527i	0.224050	+	0.974578i	$0.428072\pi$
$198$		0			0
$199$	2.84267			0.201512			0.100756	−	0.994911i	$-0.467874\pi$
$199$	2.84267			0.201512			0.100756	+	0.994911i	$0.467874\pi$
$200$		0			0
$201$			3.37900i			0.238336i
$202$		0			0
$203$	0.756015	−	0.756015i	0.0530619	−	0.0530619i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−7.01321	+	2.29523i	−0.487452	+	0.159529i
$208$		0			0
$209$			5.51949i			0.381791i
$210$		0			0
$211$	6.05320			0.416719			0.208360	−	0.978052i	$-0.433188\pi$
$211$	6.05320			0.416719			0.208360	+	0.978052i	$0.433188\pi$
$212$		0			0
$213$	8.38849	+	8.38849i	0.574770	+	0.574770i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−20.0273	−	20.0273i	−1.35954	−	1.35954i
$218$		0			0
$219$			3.41080i			0.230480i
$220$		0			0
$221$		−	32.7274i		−	2.20148i
$222$		0			0
$223$	4.50465	+	4.50465i	0.301653	+	0.301653i	0.841661	−	0.540007i	$-0.181578\pi$
$223$	4.50465	+	4.50465i	0.301653	+	0.301653i	−0.540007	+	0.841661i	$0.681578\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	12.4772	+	12.4772i	0.828140	+	0.828140i	0.987259	−	0.159119i	$-0.0508654\pi$
$227$	12.4772	+	12.4772i	0.828140	+	0.828140i	−0.159119	+	0.987259i	$0.550865\pi$
$228$		0			0
$229$	−10.3423			−0.683438			−0.341719	−	0.939802i	$-0.611009\pi$
$229$	−10.3423			−0.683438			−0.341719	+	0.939802i	$0.611009\pi$
$230$		0			0
$231$	26.9732			1.77471
$232$		0			0
$233$	11.3088	+	11.3088i	0.740862	+	0.740862i	0.972744	−	0.231882i	$-0.0744884\pi$
$233$	11.3088	+	11.3088i	0.740862	+	0.740862i	−0.231882	+	0.972744i	$0.574488\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−7.73856	−	7.73856i	−0.502673	−	0.502673i
$238$		0			0
$239$		−	12.2913i		−	0.795059i	−0.917589	−	0.397529i	$-0.869868\pi$
$239$		−	12.2913i		−	0.795059i	0.917589	−	0.397529i	$-0.130132\pi$
$240$		0			0
$241$			17.8531i			1.15002i	0.818147	+	0.575009i	$0.195001\pi$
$241$			17.8531i			1.15002i	−0.818147	+	0.575009i	$0.804999\pi$
$242$		0			0
$243$	−9.91519	−	9.91519i	−0.636060	−	0.636060i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	5.56905	+	5.56905i	0.354350	+	0.354350i
$248$		0			0
$249$	13.4197			0.850436
$250$		0			0
$251$		−	11.8959i		−	0.750861i	−0.926851	−	0.375430i	$-0.877495\pi$
$251$		−	11.8959i		−	0.750861i	0.926851	−	0.375430i	$-0.122505\pi$
$252$		0			0
$253$	22.1772	−	7.25797i	1.39427	−	0.456305i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.1426	−	10.1426i	0.632676	−	0.632676i	−0.316062	−	0.948738i	$-0.602361\pi$
$257$	10.1426	−	10.1426i	0.632676	−	0.632676i	0.948738	+	0.316062i	$0.102361\pi$
$258$		0			0
$259$		−	33.5031i		−	2.08178i
$260$		0			0
$261$	0.358733			0.0222050
$262$		0			0
$263$	−17.7466	+	17.7466i	−1.09431	+	1.09431i	−0.0992419	+	0.995063i	$0.531642\pi$
$263$	−17.7466	+	17.7466i	−1.09431	+	1.09431i	−0.995063	+	0.0992419i	$0.968358\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	15.1283	+	15.1283i	0.925838	+	0.925838i
$268$		0			0
$269$		−	3.62475i		−	0.221005i	−0.993876	−	0.110502i	$-0.964754\pi$
$269$		−	3.62475i		−	0.221005i	0.993876	−	0.110502i	$-0.0352460\pi$
$270$		0			0
$271$	−29.9391			−1.81867			−0.909337	−	0.416061i	$-0.863410\pi$
$271$	−29.9391			−1.81867			−0.909337	+	0.416061i	$0.863410\pi$
$272$		0			0
$273$	27.2154	−	27.2154i	1.64715	−	1.64715i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−4.52478	+	4.52478i	−0.271868	+	0.271868i	−0.829852	−	0.557984i	$-0.811575\pi$
$277$	−4.52478	+	4.52478i	−0.271868	+	0.271868i	0.557984	+	0.829852i	$0.311575\pi$
$278$		0			0
$279$		−	9.50305i		−	0.568933i
$280$		0			0
$281$		−	27.8859i		−	1.66354i	−0.555124	−	0.831768i	$-0.687329\pi$
$281$		−	27.8859i		−	1.66354i	0.555124	−	0.831768i	$-0.312671\pi$
$282$		0			0
$283$	−12.9251	+	12.9251i	−0.768319	+	0.768319i	−0.977810	−	0.209491i	$-0.932819\pi$
$283$	−12.9251	+	12.9251i	−0.768319	+	0.768319i	0.209491	+	0.977810i	$0.432819\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.50279	−	2.50279i	−0.147735	−	0.147735i
$288$		0			0
$289$			5.22046i			0.307086i
$290$		0			0
$291$		−	18.5069i		−	1.08490i
$292$		0			0
$293$	11.6513	−	11.6513i	0.680678	−	0.680678i	−0.279475	−	0.960153i	$-0.590160\pi$
$293$	11.6513	−	11.6513i	0.680678	−	0.680678i	0.960153	+	0.279475i	$0.0901604\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	18.8767	+	18.8767i	1.09533	+	1.09533i
$298$		0			0
$299$	15.0532	−	29.6995i	0.870549	−	1.71757i
$300$		0			0
$301$	−37.8488			−2.18157
$302$		0			0
$303$	−7.64434	−	7.64434i	−0.439156	−	0.439156i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	10.5866	−	10.5866i	0.604211	−	0.604211i	−0.337217	−	0.941427i	$-0.609485\pi$
$307$	10.5866	−	10.5866i	0.604211	−	0.604211i	0.941427	+	0.337217i	$0.109485\pi$
$308$		0			0
$309$	16.3851			0.932116
$310$		0			0
$311$	11.5551			0.655230			0.327615	−	0.944811i	$-0.393755\pi$
$311$	11.5551			0.655230			0.327615	+	0.944811i	$0.393755\pi$
$312$		0			0
$313$	−6.87550	+	6.87550i	−0.388626	+	0.388626i	−0.874197	−	0.485571i	$-0.838612\pi$
$313$	−6.87550	+	6.87550i	−0.388626	+	0.388626i	0.485571	+	0.874197i	$0.338612\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.39893	+	9.39893i	−0.527897	+	0.527897i	−0.919945	−	0.392048i	$-0.871767\pi$
$317$	−9.39893	+	9.39893i	−0.527897	+	0.527897i	0.392048	+	0.919945i	$0.371767\pi$
$318$		0			0
$319$	−1.13439			−0.0635134
$320$		0			0
$321$		−	18.2611i		−	1.01924i
$322$		0			0
$323$	−3.78114	−	3.78114i	−0.210388	−	0.210388i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−0.558921	−	0.558921i	−0.0309084	−	0.0309084i
$328$		0			0
$329$	−43.5130			−2.39895
$330$		0			0
$331$	−25.9568			−1.42671			−0.713357	−	0.700801i	$-0.752826\pi$
$331$	−25.9568			−1.42671			−0.713357	+	0.700801i	$0.752826\pi$
$332$		0			0
$333$	7.94869	−	7.94869i	0.435585	−	0.435585i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	13.8711	+	13.8711i	0.755606	+	0.755606i	0.975519	−	0.219914i	$-0.0705776\pi$
$337$	13.8711	+	13.8711i	0.755606	+	0.755606i	−0.219914	+	0.975519i	$0.570578\pi$
$338$		0			0
$339$	5.14310			0.279335
$340$		0			0
$341$			30.0506i			1.62733i
$342$		0			0
$343$	22.7970	−	22.7970i	1.23092	−	1.23092i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	15.2808	−	15.2808i	0.820314	−	0.820314i	−0.165839	−	0.986153i	$-0.553033\pi$
$347$	15.2808	−	15.2808i	0.820314	−	0.820314i	0.986153	+	0.165839i	$0.0530332\pi$
$348$		0			0
$349$			1.61482i			0.0864393i	0.999066	+	0.0432196i	$0.0137615\pi$
$349$			1.61482i			0.0864393i	−0.999066	+	0.0432196i	$0.986238\pi$
$350$		0			0
$351$	38.0923			2.03322
$352$		0			0
$353$	−5.04344	−	5.04344i	−0.268435	−	0.268435i	0.560034	−	0.828469i	$-0.310788\pi$
$353$	−5.04344	−	5.04344i	−0.268435	−	0.268435i	−0.828469	+	0.560034i	$0.810788\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−18.4781	+	18.4781i	−0.977963	+	0.977963i
$358$		0			0
$359$	−21.8971			−1.15568			−0.577842	−	0.816149i	$-0.696105\pi$
$359$	−21.8971			−1.15568			−0.577842	+	0.816149i	$0.696105\pi$
$360$		0			0
$361$	−17.7132			−0.932272
$362$		0			0
$363$	−10.8337	−	10.8337i	−0.568624	−	0.568624i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−7.02384	−	7.02384i	−0.366642	−	0.366642i	0.499609	−	0.866251i	$-0.333477\pi$
$367$	−7.02384	−	7.02384i	−0.366642	−	0.366642i	−0.866251	+	0.499609i	$0.833477\pi$
$368$		0			0
$369$		−	1.18758i		−	0.0618232i
$370$		0			0
$371$	−4.79865			−0.249134
$372$		0			0
$373$	−9.72356	+	9.72356i	−0.503467	+	0.503467i	−0.912514	−	0.409047i	$-0.865861\pi$
$373$	−9.72356	+	9.72356i	−0.503467	+	0.503467i	0.409047	+	0.912514i	$0.365861\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.14457	+	1.14457i	−0.0589485	+	0.0589485i
$378$		0			0
$379$	18.3652			0.943359			0.471679	−	0.881770i	$-0.343648\pi$
$379$	18.3652			0.943359			0.471679	+	0.881770i	$0.343648\pi$
$380$		0			0
$381$	−11.5624			−0.592361
$382$		0			0
$383$	−8.22457	+	8.22457i	−0.420256	+	0.420256i	−0.885292	−	0.465036i	$-0.846041\pi$
$383$	−8.22457	+	8.22457i	−0.420256	+	0.420256i	0.465036	+	0.885292i	$0.346041\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−8.97971	−	8.97971i	−0.456464	−	0.456464i
$388$		0			0
$389$	24.1975			1.22686			0.613431	−	0.789748i	$-0.289789\pi$
$389$	24.1975			1.22686			0.613431	+	0.789748i	$0.289789\pi$
$390$		0			0
$391$	−10.2205	+	20.1646i	−0.516871	+	1.01977i
$392$		0			0
$393$	16.9266	+	16.9266i	0.853832	+	0.853832i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−25.1822	+	25.1822i	−1.26386	+	1.26386i	−0.314654	+	0.949206i	$0.601888\pi$
$397$	−25.1822	+	25.1822i	−1.26386	+	1.26386i	−0.949206	+	0.314654i	$0.898112\pi$
$398$		0			0
$399$		−	6.28863i		−	0.314825i
$400$		0			0
$401$		−	16.2382i		−	0.810895i	−0.914118	−	0.405448i	$-0.867116\pi$
$401$		−	16.2382i		−	0.810895i	0.914118	−	0.405448i	$-0.132884\pi$
$402$		0			0
$403$	30.3204	+	30.3204i	1.51037	+	1.51037i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−25.1354	+	25.1354i	−1.24591	+	1.24591i
$408$		0			0
$409$		−	19.9785i		−	0.987874i	−0.869498	−	0.493937i	$-0.835557\pi$
$409$		−	19.9785i		−	0.987874i	0.869498	−	0.493937i	$-0.164443\pi$
$410$		0			0
$411$			19.6302i			0.968285i
$412$		0			0
$413$	−18.9139	+	18.9139i	−0.930690	+	0.930690i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−11.8076	+	11.8076i	−0.578223	+	0.578223i
$418$		0			0
$419$	−12.9875			−0.634479			−0.317239	−	0.948346i	$-0.602756\pi$
$419$	−12.9875			−0.634479			−0.317239	+	0.948346i	$0.602756\pi$
$420$		0			0
$421$			18.4005i			0.896788i	0.893836	+	0.448394i	$0.148004\pi$
$421$			18.4005i			0.896788i	−0.893836	+	0.448394i	$0.851996\pi$
$422$		0			0
$423$	−10.3236	−	10.3236i	−0.501949	−	0.501949i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	16.4966	−	16.4966i	0.798325	−	0.798325i
$428$		0			0
$429$	−40.8362			−1.97159
$430$		0			0
$431$			10.2382i			0.493155i	0.969123	+	0.246578i	$0.0793060\pi$
$431$			10.2382i			0.493155i	−0.969123	+	0.246578i	$0.920694\pi$
$432$		0			0
$433$	3.62212	−	3.62212i	0.174068	−	0.174068i	−0.614696	−	0.788764i	$-0.710721\pi$
$433$	3.62212	−	3.62212i	0.174068	−	0.174068i	0.788764	+	0.614696i	$0.210721\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.69215	−	5.17047i	−0.0809465	−	0.247337i
$438$		0			0
$439$		−	21.9568i		−	1.04794i	−0.851737	−	0.523970i	$-0.824450\pi$
$439$		−	21.9568i		−	1.04794i	0.851737	−	0.523970i	$-0.175550\pi$
$440$		0			0
$441$	21.5880			1.02800
$442$		0			0
$443$	−11.0897	−	11.0897i	−0.526887	−	0.526887i	0.392756	−	0.919643i	$-0.371522\pi$
$443$	−11.0897	−	11.0897i	−0.526887	−	0.526887i	−0.919643	+	0.392756i	$0.871522\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	8.06029	+	8.06029i	0.381238	+	0.381238i
$448$		0			0
$449$		−	3.65509i		−	0.172494i	−0.996274	−	0.0862471i	$-0.972513\pi$
$449$		−	3.65509i		−	0.172494i	0.996274	−	0.0862471i	$-0.0274875\pi$
$450$		0			0
$451$			3.75538i			0.176834i
$452$		0			0
$453$	4.82861	+	4.82861i	0.226868	+	0.226868i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	14.2165	+	14.2165i	0.665021	+	0.665021i	0.956559	−	0.291539i	$-0.0941672\pi$
$457$	14.2165	+	14.2165i	0.665021	+	0.665021i	−0.291539	+	0.956559i	$0.594167\pi$
$458$		0			0
$459$	−25.8630			−1.20718
$460$		0			0
$461$	−8.65622			−0.403160			−0.201580	−	0.979472i	$-0.564608\pi$
$461$	−8.65622			−0.403160			−0.201580	+	0.979472i	$0.564608\pi$
$462$		0			0
$463$	−20.2125	−	20.2125i	−0.939357	−	0.939357i	0.0589069	−	0.998263i	$-0.481238\pi$
$463$	−20.2125	−	20.2125i	−0.939357	−	0.939357i	−0.998263	+	0.0589069i	$0.981238\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6.26766	+	6.26766i	0.290033	+	0.290033i	0.837093	−	0.547060i	$-0.184253\pi$
$467$	6.26766	+	6.26766i	0.290033	+	0.290033i	−0.547060	+	0.837093i	$0.684253\pi$
$468$		0			0
$469$		−	12.8185i		−	0.591904i
$470$		0			0
$471$		−	10.2787i		−	0.473618i
$472$		0			0
$473$	28.3957	+	28.3957i	1.30563	+	1.30563i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.13849	−	1.13849i	−0.0521280	−	0.0521280i
$478$		0			0
$479$	−21.0104			−0.959989			−0.479995	−	0.877271i	$-0.659361\pi$
$479$	−21.0104			−0.959989			−0.479995	+	0.877271i	$0.659361\pi$
$480$		0			0
$481$			50.7222i			2.31273i
$482$		0			0
$483$	−25.2676	+	8.26938i	−1.14972	+	0.376270i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	3.54807	−	3.54807i	0.160778	−	0.160778i	−0.622133	−	0.782911i	$-0.713734\pi$
$487$	3.54807	−	3.54807i	0.160778	−	0.160778i	0.782911	+	0.622133i	$0.213734\pi$
$488$		0			0
$489$		−	4.52103i		−	0.204448i
$490$		0			0
$491$	−3.54251			−0.159871			−0.0799356	−	0.996800i	$-0.525471\pi$
$491$	−3.54251			−0.159871			−0.0799356	+	0.996800i	$0.525471\pi$
$492$		0			0
$493$	0.777114	−	0.777114i	0.0349995	−	0.0349995i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−31.8224	−	31.8224i	−1.42743	−	1.42743i
$498$		0			0
$499$		−	13.7668i		−	0.616285i	−0.951340	−	0.308143i	$-0.900293\pi$
$499$		−	13.7668i		−	0.616285i	0.951340	−	0.308143i	$-0.0997074\pi$
$500$		0			0
$501$	−4.86172			−0.217206
$502$		0			0
$503$	0.238681	−	0.238681i	0.0106422	−	0.0106422i	−0.701766	−	0.712408i	$-0.747605\pi$
$503$	0.238681	−	0.238681i	0.0106422	−	0.0106422i	0.712408	+	0.701766i	$0.247605\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−30.0907	+	30.0907i	−1.33637	+	1.33637i
$508$		0			0
$509$		−	24.3977i		−	1.08141i	−0.841213	−	0.540705i	$-0.818158\pi$
$509$		−	24.3977i		−	1.08141i	0.841213	−	0.540705i	$-0.181842\pi$
$510$		0			0
$511$		−	12.9391i		−	0.572394i
$512$		0			0
$513$	4.40097	−	4.40097i	0.194308	−	0.194308i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	32.6452	+	32.6452i	1.43574	+	1.43574i
$518$		0			0
$519$		−	16.9476i		−	0.743917i
$520$		0			0
$521$		−	8.00993i		−	0.350921i	−0.984486	−	0.175461i	$-0.943859\pi$
$521$		−	8.00993i		−	0.350921i	0.984486	−	0.175461i	$-0.0561415\pi$
$522$		0			0
$523$	7.47617	−	7.47617i	0.326910	−	0.326910i	−0.524500	−	0.851410i	$-0.675748\pi$
$523$	7.47617	−	7.47617i	0.326910	−	0.326910i	0.851410	+	0.524500i	$0.175748\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−20.5862	−	20.5862i	−0.896750	−	0.896750i
$528$		0			0
$529$	−18.5497	+	13.5981i	−0.806510	+	0.591220i
$530$		0			0
$531$	−8.97471			−0.389469
$532$		0			0
$533$	3.78911	+	3.78911i	0.164124	+	0.164124i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	2.27830	−	2.27830i	0.0983159	−	0.0983159i
$538$		0			0
$539$	−68.2658			−2.94042
$540$		0			0
$541$	11.2083			0.481883			0.240942	−	0.970540i	$-0.422544\pi$
$541$	11.2083			0.481883			0.240942	+	0.970540i	$0.422544\pi$
$542$		0			0
$543$	−19.6091	+	19.6091i	−0.841508	+	0.841508i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−4.52989	+	4.52989i	−0.193684	+	0.193684i	−0.797286	−	0.603602i	$-0.793732\pi$
$547$	−4.52989	+	4.52989i	−0.193684	+	0.193684i	0.603602	+	0.797286i	$0.293732\pi$
$548$		0			0
$549$	7.82769			0.334078
$550$		0			0
$551$			0.264475i			0.0112670i
$552$		0			0
$553$	29.3568	+	29.3568i	1.24838	+	1.24838i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	3.92626	+	3.92626i	0.166361	+	0.166361i	0.785378	−	0.619017i	$-0.212469\pi$
$557$	3.92626	+	3.92626i	0.166361	+	0.166361i	−0.619017	+	0.785378i	$0.712469\pi$
$558$		0			0
$559$	57.3013			2.42359
$560$		0			0
$561$	27.7260			1.17059
$562$		0			0
$563$	29.9395	−	29.9395i	1.26180	−	1.26180i	0.311578	−	0.950221i	$-0.399143\pi$
$563$	29.9395	−	29.9395i	1.26180	−	1.26180i	0.950221	−	0.311578i	$-0.100857\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−6.53871	−	6.53871i	−0.274600	−	0.274600i
$568$		0			0
$569$	18.4028			0.771486			0.385743	−	0.922606i	$-0.373945\pi$
$569$	18.4028			0.771486			0.385743	+	0.922606i	$0.373945\pi$
$570$		0			0
$571$			41.7359i			1.74659i	0.487188	+	0.873297i	$0.338023\pi$
$571$			41.7359i			1.74659i	−0.487188	+	0.873297i	$0.661977\pi$
$572$		0			0
$573$	−8.29748	+	8.29748i	−0.346632	+	0.346632i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	15.8513	−	15.8513i	0.659899	−	0.659899i	−0.295457	−	0.955356i	$-0.595472\pi$
$577$	15.8513	−	15.8513i	0.659899	−	0.659899i	0.955356	+	0.295457i	$0.0954719\pi$
$578$		0			0
$579$		−	15.6130i		−	0.648852i
$580$		0			0
$581$	−50.9085			−2.11204
$582$		0			0
$583$	3.60014	+	3.60014i	0.149103	+	0.149103i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.14927	+	1.14927i	−0.0474353	+	0.0474353i	−0.730427	−	0.682991i	$-0.760679\pi$
$587$	−1.14927	+	1.14927i	−0.0474353	+	0.0474353i	0.682991	+	0.730427i	$0.260679\pi$
$588$		0			0
$589$	7.00610			0.288681
$590$		0			0
$591$	−18.0089			−0.740788
$592$		0			0
$593$	−26.2162	−	26.2162i	−1.07657	−	1.07657i	−0.996814	−	0.0797552i	$-0.974586\pi$
$593$	−26.2162	−	26.2162i	−1.07657	−	1.07657i	−0.0797552	−	0.996814i	$-0.525414\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	2.42988	+	2.42988i	0.0994483	+	0.0994483i
$598$		0			0
$599$		−	18.0073i		−	0.735759i	−0.929874	−	0.367879i	$-0.880084\pi$
$599$		−	18.0073i		−	0.735759i	0.929874	−	0.367879i	$-0.119916\pi$
$600$		0			0
$601$	−20.8170			−0.849142			−0.424571	−	0.905395i	$-0.639575\pi$
$601$	−20.8170			−0.849142			−0.424571	+	0.905395i	$0.639575\pi$
$602$		0			0
$603$	3.04122	−	3.04122i	0.123848	−	0.123848i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	16.3663	−	16.3663i	0.664286	−	0.664286i	−0.292102	−	0.956387i	$-0.594355\pi$
$607$	16.3663	−	16.3663i	0.664286	−	0.664286i	0.956387	+	0.292102i	$0.0943545\pi$
$608$		0			0
$609$	1.29246			0.0523733
$610$		0			0
$611$	65.8767			2.66509
$612$		0			0
$613$	−22.7644	+	22.7644i	−0.919445	+	0.919445i	−0.996989	−	0.0775440i	$-0.975292\pi$
$613$	−22.7644	+	22.7644i	−0.919445	+	0.919445i	0.0775440	+	0.996989i	$0.475292\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	7.26968	+	7.26968i	0.292666	+	0.292666i	0.838133	−	0.545466i	$-0.183647\pi$
$617$	7.26968	+	7.26968i	0.292666	+	0.292666i	−0.545466	+	0.838133i	$0.683647\pi$
$618$		0			0
$619$	−12.8514			−0.516541			−0.258270	−	0.966073i	$-0.583153\pi$
$619$	−12.8514			−0.516541			−0.258270	+	0.966073i	$0.583153\pi$
$620$		0			0
$621$	−23.4702	−	11.8959i	−0.941826	−	0.477365i
$622$		0			0
$623$	−57.3905	−	57.3905i	−2.29930	−	2.29930i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−4.71799	+	4.71799i	−0.188418	+	0.188418i
$628$		0			0
$629$		−	34.4381i		−	1.37314i
$630$		0			0
$631$		−	29.5195i		−	1.17515i	−0.809169	−	0.587576i	$-0.800082\pi$
$631$		−	29.5195i		−	1.17515i	0.809169	−	0.587576i	$-0.199918\pi$
$632$		0			0
$633$	5.17420	+	5.17420i	0.205656	+	0.205656i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−68.8788	+	68.8788i	−2.72908	+	2.72908i
$638$		0			0
$639$		−	15.0999i		−	0.597342i
$640$		0			0
$641$		−	30.0506i		−	1.18693i	−0.804861	−	0.593463i	$-0.797760\pi$
$641$		−	30.0506i		−	1.18693i	0.804861	−	0.593463i	$-0.202240\pi$
$642$		0			0
$643$	−5.07542	+	5.07542i	−0.200155	+	0.200155i	−0.800067	−	0.599911i	$-0.795203\pi$
$643$	−5.07542	+	5.07542i	−0.200155	+	0.200155i	0.599911	+	0.800067i	$0.295203\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−32.9012	+	32.9012i	−1.29348	+	1.29348i	−0.360858	+	0.932621i	$0.617516\pi$
$647$	−32.9012	+	32.9012i	−1.29348	+	1.29348i	−0.932621	+	0.360858i	$0.882484\pi$
$648$		0			0
$649$	28.3799			1.11401
$650$		0			0
$651$		−	34.2382i		−	1.34190i
$652$		0			0
$653$	−4.77378	−	4.77378i	−0.186813	−	0.186813i	0.607504	−	0.794317i	$-0.292171\pi$
$653$	−4.77378	−	4.77378i	−0.186813	−	0.186813i	−0.794317	+	0.607504i	$0.792171\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	3.06984	−	3.06984i	0.119766	−	0.119766i
$658$		0			0
$659$	5.62362			0.219065			0.109532	−	0.993983i	$-0.465065\pi$
$659$	5.62362			0.219065			0.109532	+	0.993983i	$0.465065\pi$
$660$		0			0
$661$		−	18.6056i		−	0.723672i	−0.932242	−	0.361836i	$-0.882150\pi$
$661$		−	18.6056i		−	0.723672i	0.932242	−	0.361836i	$-0.117850\pi$
$662$		0			0
$663$	27.9750	−	27.9750i	1.08646	−	1.08646i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1.06266	−	0.347777i	0.0411462	−	0.0134660i
$668$		0			0
$669$			7.70103i			0.297739i
$670$		0			0
$671$	−24.7528			−0.955570
$672$		0			0
$673$	18.1526	+	18.1526i	0.699732	+	0.699732i	0.964353	−	0.264620i	$-0.0852467\pi$
$673$	18.1526	+	18.1526i	0.699732	+	0.699732i	−0.264620	+	0.964353i	$0.585247\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	19.7469	+	19.7469i	0.758934	+	0.758934i	0.976128	−	0.217194i	$-0.0696905\pi$
$677$	19.7469	+	19.7469i	0.758934	+	0.758934i	−0.217194	+	0.976128i	$0.569691\pi$
$678$		0			0
$679$			70.2076i			2.69432i
$680$		0			0
$681$			21.3307i			0.817394i
$682$		0			0
$683$	−11.2131	−	11.2131i	−0.429056	−	0.429056i	0.459251	−	0.888307i	$-0.348118\pi$
$683$	−11.2131	−	11.2131i	−0.429056	−	0.429056i	−0.888307	+	0.459251i	$0.848118\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−8.84046	−	8.84046i	−0.337285	−	0.337285i
$688$		0			0
$689$	7.26494			0.276772
$690$		0			0
$691$	2.21557			0.0842844			0.0421422	−	0.999112i	$-0.486582\pi$
$691$	2.21557			0.0842844			0.0421422	+	0.999112i	$0.486582\pi$
$692$		0			0
$693$	−24.2769	−	24.2769i	−0.922202	−	0.922202i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−2.57264	−	2.57264i	−0.0974455	−	0.0974455i
$698$		0			0
$699$			19.3332i			0.731248i
$700$		0			0
$701$			37.7213i			1.42471i	0.701817	+	0.712357i	$0.252372\pi$
$701$			37.7213i			1.42471i	−0.701817	+	0.712357i	$0.747628\pi$
$702$		0			0
$703$	5.86015	+	5.86015i	0.221020	+	0.221020i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	28.9994	+	28.9994i	1.09063	+	1.09063i
$708$		0			0
$709$	38.1799			1.43388			0.716939	−	0.697136i	$-0.245543\pi$
$709$	38.1799			1.43388			0.716939	+	0.697136i	$0.245543\pi$
$710$		0			0
$711$			13.9300i			0.522414i
$712$		0			0
$713$	−9.21283	−	28.1504i	−0.345023	−	1.05424i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	10.5065	−	10.5065i	0.392371	−	0.392371i
$718$		0			0
$719$			19.4450i			0.725177i	0.931949	+	0.362589i	$0.118107\pi$
$719$			19.4450i			0.725177i	−0.931949	+	0.362589i	$0.881893\pi$
$720$		0			0
$721$	−62.1582			−2.31489
$722$		0			0
$723$	−15.2606	+	15.2606i	−0.567547	+	0.567547i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	4.70144	+	4.70144i	0.174367	+	0.174367i	0.788895	−	0.614528i	$-0.210654\pi$
$727$	4.70144	+	4.70144i	0.174367	+	0.174367i	−0.614528	+	0.788895i	$0.710654\pi$
$728$		0			0
$729$		−	23.0001i		−	0.851854i
$730$		0			0
$731$	−38.9051			−1.43896
$732$		0			0
$733$	−9.35780	+	9.35780i	−0.345638	+	0.345638i	−0.858482	−	0.512844i	$-0.828592\pi$
$733$	−9.35780	+	9.35780i	−0.345638	+	0.345638i	0.512844	+	0.858482i	$0.328592\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−9.61696	+	9.61696i	−0.354245	+	0.354245i
$738$		0			0
$739$		−	25.5958i		−	0.941555i	−0.882252	−	0.470778i	$-0.843973\pi$
$739$		−	25.5958i		−	0.941555i	0.882252	−	0.470778i	$-0.156027\pi$
$740$		0			0
$741$			9.52070i			0.349752i
$742$		0			0
$743$	−14.1632	+	14.1632i	−0.519598	+	0.519598i	−0.917450	−	0.397852i	$-0.869756\pi$
$743$	−14.1632	+	14.1632i	−0.519598	+	0.519598i	0.397852	+	0.917450i	$0.369756\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−12.0782	−	12.0782i	−0.441917	−	0.441917i
$748$		0			0
$749$			69.2750i			2.53125i
$750$		0			0
$751$		−	48.3656i		−	1.76489i	−0.470418	−	0.882444i	$-0.655897\pi$
$751$		−	48.3656i		−	1.76489i	0.470418	−	0.882444i	$-0.344103\pi$
$752$		0			0
$753$	10.1684	−	10.1684i	0.370558	−	0.370558i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	22.4988	+	22.4988i	0.817732	+	0.817732i	0.985779	−	0.168047i	$-0.0537461\pi$
$757$	22.4988	+	22.4988i	0.817732	+	0.817732i	−0.168047	+	0.985779i	$0.553746\pi$
$758$		0			0
$759$	25.1608	+	12.7528i	0.913280	+	0.462896i
$760$		0			0
$761$	10.6688			0.386745			0.193372	−	0.981125i	$-0.438057\pi$
$761$	10.6688			0.386745			0.193372	+	0.981125i	$0.438057\pi$
$762$		0			0
$763$	2.12031	+	2.12031i	0.0767604	+	0.0767604i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	28.6347	−	28.6347i	1.03394	−	1.03394i
$768$		0			0
$769$	18.4511			0.665364			0.332682	−	0.943039i	$-0.392046\pi$
$769$	18.4511			0.665364			0.332682	+	0.943039i	$0.392046\pi$
$770$		0			0
$771$	17.3395			0.624466
$772$		0			0
$773$	3.87147	−	3.87147i	0.139247	−	0.139247i	−0.634047	−	0.773294i	$-0.718608\pi$
$773$	3.87147	−	3.87147i	0.139247	−	0.139247i	0.773294	+	0.634047i	$0.218608\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	28.6380	−	28.6380i	1.02738	−	1.02738i
$778$		0			0
$779$	0.875544			0.0313696
$780$		0			0
$781$			47.7489i			1.70859i
$782$		0			0
$783$	0.904504	+	0.904504i	0.0323243	+	0.0323243i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	21.9536	+	21.9536i	0.782561	+	0.782561i	0.980262	−	0.197701i	$-0.0633477\pi$
$787$	21.9536	+	21.9536i	0.782561	+	0.782561i	−0.197701	+	0.980262i	$0.563348\pi$
$788$		0			0
$789$	−30.3392			−1.08010
$790$		0			0
$791$	−19.5108			−0.693723
$792$		0			0
$793$	−24.9750	+	24.9750i	−0.886889	+	0.886889i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	29.0407	+	29.0407i	1.02867	+	1.02867i	0.999577	+	0.0290978i	$0.00926343\pi$
$797$	29.0407	+	29.0407i	1.02867	+	1.02867i	0.0290978	+	0.999577i	$0.490737\pi$
$798$		0			0
$799$	−44.7274			−1.58234
$800$		0			0
$801$		−	27.2321i		−	0.962198i
$802$		0			0
$803$	−9.70746	+	9.70746i	−0.342569	+	0.342569i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	3.09839	−	3.09839i	0.109068	−	0.109068i
$808$		0			0
$809$			30.3001i			1.06529i	0.846337	+	0.532647i	$0.178803\pi$
$809$			30.3001i			1.06529i	−0.846337	+	0.532647i	$0.821197\pi$
$810$		0			0
$811$	23.7379			0.833549			0.416775	−	0.909010i	$-0.363160\pi$
$811$	23.7379			0.833549			0.416775	+	0.909010i	$0.363160\pi$
$812$		0			0
$813$	−25.5916	−	25.5916i	−0.897536	−	0.897536i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	6.62027	−	6.62027i	0.231614	−	0.231614i
$818$		0			0
$819$	−48.9897			−1.71184
$820$		0			0
$821$	50.3508			1.75726			0.878628	−	0.477507i	$-0.158459\pi$
$821$	50.3508			1.75726			0.878628	+	0.477507i	$0.158459\pi$
$822$		0			0
$823$	21.4945	+	21.4945i	0.749250	+	0.749250i	0.974338	−	0.225088i	$-0.0722670\pi$
$823$	21.4945	+	21.4945i	0.749250	+	0.749250i	−0.225088	+	0.974338i	$0.572267\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−38.6531	−	38.6531i	−1.34410	−	1.34410i	−0.891933	−	0.452167i	$-0.850651\pi$
$827$	−38.6531	−	38.6531i	−1.34410	−	1.34410i	−0.452167	−	0.891933i	$-0.649349\pi$
$828$		0			0
$829$		−	8.24200i		−	0.286256i	−0.989704	−	0.143128i	$-0.954284\pi$
$829$		−	8.24200i		−	0.286256i	0.989704	−	0.143128i	$-0.0457161\pi$
$830$		0			0
$831$	−7.73544			−0.268340
$832$		0			0
$833$	46.7657	−	46.7657i	1.62033	−	1.62033i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	23.9609	−	23.9609i	0.828209	−	0.828209i
$838$		0			0
$839$	−13.3747			−0.461746			−0.230873	−	0.972984i	$-0.574158\pi$
$839$	−13.3747			−0.461746			−0.230873	+	0.972984i	$0.574158\pi$
$840$		0			0
$841$	28.9456			0.998126
$842$		0			0
$843$	23.8365	−	23.8365i	0.820974	−	0.820974i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	41.0987	+	41.0987i	1.41217	+	1.41217i
$848$		0			0
$849$	−22.0965			−0.758349
$850$		0			0
$851$	15.8401	−	31.2519i	0.542990	−	1.07130i
$852$		0			0
$853$	26.5782	+	26.5782i	0.910019	+	0.910019i	0.996273	−	0.0862539i	$-0.0274896\pi$
$853$	26.5782	+	26.5782i	0.910019	+	0.910019i	−0.0862539	+	0.996273i	$0.527490\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	5.82649	−	5.82649i	0.199029	−	0.199029i	−0.600555	−	0.799584i	$-0.705054\pi$
$857$	5.82649	−	5.82649i	0.199029	−	0.199029i	0.799584	+	0.600555i	$0.205054\pi$
$858$		0			0
$859$		−	26.5339i		−	0.905323i	−0.891682	−	0.452662i	$-0.850475\pi$
$859$		−	26.5339i		−	0.905323i	0.891682	−	0.452662i	$-0.149525\pi$
$860$		0			0
$861$		−	4.27870i		−	0.145818i
$862$		0			0
$863$	−3.51808	−	3.51808i	−0.119757	−	0.119757i	0.644688	−	0.764445i	$-0.276987\pi$
$863$	−3.51808	−	3.51808i	−0.119757	−	0.119757i	−0.764445	+	0.644688i	$0.776987\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−4.46238	+	4.46238i	−0.151550	+	0.151550i
$868$		0			0
$869$		−	44.0494i		−	1.49427i
$870$		0			0
$871$			19.4066i			0.657569i
$872$		0			0
$873$	−16.6569	+	16.6569i	−0.563751	+	0.563751i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−33.8282	+	33.8282i	−1.14230	+	1.14230i	−0.154269	+	0.988029i	$0.549302\pi$
$877$	−33.8282	+	33.8282i	−1.14230	+	1.14230i	−0.988029	+	0.154269i	$0.950698\pi$
$878$		0			0
$879$	19.9188			0.671845
$880$		0			0
$881$		−	1.81242i		−	0.0610618i	−0.999534	−	0.0305309i	$-0.990280\pi$
$881$		−	1.81242i		−	0.0610618i	0.999534	−	0.0305309i	$-0.00971980\pi$
$882$		0			0
$883$	21.6924	+	21.6924i	0.730007	+	0.730007i	0.970621	−	0.240614i	$-0.0773489\pi$
$883$	21.6924	+	21.6924i	0.730007	+	0.730007i	−0.240614	+	0.970621i	$0.577349\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	9.81340	−	9.81340i	0.329502	−	0.329502i	−0.522895	−	0.852397i	$-0.675148\pi$
$887$	9.81340	−	9.81340i	0.329502	−	0.329502i	0.852397	+	0.522895i	$0.175148\pi$
$888$		0			0
$889$	43.8630			1.47112
$890$		0			0
$891$			9.81120i			0.328688i
$892$		0			0
$893$	7.61102	−	7.61102i	0.254693	−	0.254693i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	38.2540	−	12.5195i	1.27726	−	0.418013i
$898$		0			0
$899$			1.43992i			0.0480240i
$900$		0			0
$901$	−4.93257			−0.164328
$902$		0			0
$903$	−32.3526	−	32.3526i	−1.07663	−	1.07663i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−9.79812	−	9.79812i	−0.325341	−	0.325341i	0.525471	−	0.850812i	$-0.323889\pi$
$907$	−9.79812	−	9.79812i	−0.325341	−	0.325341i	−0.850812	+	0.525471i	$0.823889\pi$
$908$		0			0
$909$			13.7603i			0.456402i
$910$		0			0
$911$			11.5753i			0.383507i	0.981443	+	0.191753i	$0.0614174\pi$
$911$			11.5753i			0.383507i	−0.981443	+	0.191753i	$0.938583\pi$
$912$		0			0
$913$	38.1936	+	38.1936i	1.26402	+	1.26402i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−64.2123	−	64.2123i	−2.12048	−	2.12048i
$918$		0			0
$919$	21.4649			0.708061			0.354031	−	0.935234i	$-0.384811\pi$
$919$	21.4649			0.708061			0.354031	+	0.935234i	$0.384811\pi$
$920$		0			0
$921$	18.0986			0.596370
$922$		0			0
$923$	48.1777	+	48.1777i	1.58579	+	1.58579i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−14.7472	−	14.7472i	−0.484361	−	0.484361i
$928$		0			0
$929$			41.8702i			1.37372i	0.726792	+	0.686858i	$0.241011\pi$
$929$			41.8702i			1.37372i	−0.726792	+	0.686858i	$0.758989\pi$
$930$		0			0
$931$			15.9157i			0.521617i
$932$		0			0
$933$	9.87716	+	9.87716i	0.323364	+	0.323364i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−41.5997	−	41.5997i	−1.35900	−	1.35900i	−0.875147	−	0.483857i	$-0.839235\pi$
$937$	−41.5997	−	41.5997i	−1.35900	−	1.35900i	−0.483857	−	0.875147i	$-0.660765\pi$
$938$		0			0
$939$	−11.7542			−0.383583
$940$		0			0
$941$		−	6.30942i		−	0.205681i	−0.994698	−	0.102841i	$-0.967207\pi$
$941$		−	6.30942i		−	0.205681i	0.994698	−	0.102841i	$-0.0327932\pi$
$942$		0			0
$943$	−1.15132	−	3.51792i	−0.0374920	−	0.114559i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−3.13509	+	3.13509i	−0.101877	+	0.101877i	−0.756208	−	0.654331i	$-0.772950\pi$
$947$	−3.13509	+	3.13509i	−0.101877	+	0.101877i	0.654331	+	0.756208i	$0.272950\pi$
$948$		0			0
$949$			19.5893i			0.635894i
$950$		0			0
$951$	−16.0682			−0.521046
$952$		0			0
$953$	−15.6666	+	15.6666i	−0.507491	+	0.507491i	−0.913755	−	0.406265i	$-0.866831\pi$
$953$	−15.6666	+	15.6666i	−0.507491	+	0.507491i	0.406265	+	0.913755i	$0.366831\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−0.969659	−	0.969659i	−0.0313446	−	0.0313446i
$958$		0			0
$959$		−	74.4687i		−	2.40472i
$960$		0			0
$961$	7.14437			0.230464
$962$		0			0
$963$	−16.4357	+	16.4357i	−0.529631	+	0.529631i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−16.8244	+	16.8244i	−0.541035	+	0.541035i	−0.923832	−	0.382798i	$-0.874961\pi$
$967$	−16.8244	+	16.8244i	−0.541035	+	0.541035i	0.382798	+	0.923832i	$0.374961\pi$
$968$		0			0
$969$		−	6.46414i		−	0.207658i
$970$		0			0
$971$			37.3419i			1.19836i	0.800614	+	0.599180i	$0.204507\pi$
$971$			37.3419i			1.19836i	−0.800614	+	0.599180i	$0.795493\pi$
$972$		0			0
$973$	44.7932	−	44.7932i	1.43601	−	1.43601i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−13.9259	−	13.9259i	−0.445528	−	0.445528i	0.448337	−	0.893865i	$-0.352016\pi$
$977$	−13.9259	−	13.9259i	−0.445528	−	0.445528i	−0.893865	+	0.448337i	$0.852016\pi$
$978$		0			0
$979$			86.1134i			2.75219i
$980$		0			0
$981$			1.00610i			0.0321222i
$982$		0			0
$983$	−33.6878	+	33.6878i	−1.07447	+	1.07447i	−0.0774800	+	0.996994i	$0.524687\pi$
$983$	−33.6878	+	33.6878i	−1.07447	+	1.07447i	−0.996994	+	0.0774800i	$0.975313\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−37.1944	−	37.1944i	−1.18391	−	1.18391i
$988$		0			0
$989$	−35.3056	−	17.8947i	−1.12265	−	0.569017i
$990$		0			0
$991$	−44.2103			−1.40439			−0.702194	−	0.711986i	$-0.747796\pi$
$991$	−44.2103			−1.40439			−0.702194	+	0.711986i	$0.747796\pi$
$992$		0			0
$993$	−22.1875	−	22.1875i	−0.704100	−	0.704100i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.98060	−	1.98060i	0.0627262	−	0.0627262i	−0.675048	−	0.737774i	$-0.735877\pi$
$997$	1.98060	−	1.98060i	0.0627262	−	0.0627262i	0.737774	+	0.675048i	$0.235877\pi$
$998$		0			0
$999$	40.0835			1.26818

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.2.i.f.1793.6	yes	16
5.2	odd	4		2300.2.i.e.1057.6	yes	16
5.3	odd	4		2300.2.i.e.1057.3	✓	16
5.4	even	2	inner	2300.2.i.f.1793.3	yes	16
23.22	odd	2		2300.2.i.e.1793.6	yes	16
115.22	even	4	inner	2300.2.i.f.1057.6	yes	16
115.68	even	4	inner	2300.2.i.f.1057.3	yes	16
115.114	odd	2		2300.2.i.e.1793.3	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2300.2.i.e.1057.3	✓	16	5.3	odd	4
2300.2.i.e.1057.6	yes	16	5.2	odd	4
2300.2.i.e.1793.3	yes	16	115.114	odd	2
2300.2.i.e.1793.6	yes	16	23.22	odd	2
2300.2.i.f.1057.3	yes	16	115.68	even	4	inner
2300.2.i.f.1057.6	yes	16	115.22	even	4	inner
2300.2.i.f.1793.3	yes	16	5.4	even	2	inner
2300.2.i.f.1793.6	yes	16	1.1	even	1	trivial

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.i (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.854787 + 0.854787i) q^{3} +(-3.24270 - 3.24270i) q^{7} -1.53868i q^{9} +O(q^{10})\)	\(q+(0.854787 + 0.854787i) q^{3} +(-3.24270 - 3.24270i) q^{7} -1.53868i q^{9} +4.86561i q^{11} +(4.90930 + 4.90930i) q^{13} +(-3.33320 - 3.33320i) q^{17} +1.13439 q^{19} -5.54364i q^{21} +(-1.49169 - 4.55795i) q^{23} +(3.87960 - 3.87960i) q^{27} +0.233144i q^{29} +6.17611 q^{31} +(-4.15906 + 4.15906i) q^{33} +(5.16592 + 5.16592i) q^{37} +8.39282i q^{39} +0.771821 q^{41} +(5.83599 - 5.83599i) q^{43} +(6.70938 - 6.70938i) q^{47} +14.0303i q^{49} -5.69835i q^{51} +(0.739915 - 0.739915i) q^{53} +(0.969659 + 0.969659i) q^{57} -5.83274i q^{59} +5.08729i q^{61} +(-4.98948 + 4.98948i) q^{63} +(1.97652 + 1.97652i) q^{67} +(2.62100 - 5.17115i) q^{69} +9.81355 q^{71} +(1.99512 + 1.99512i) q^{73} +(15.7777 - 15.7777i) q^{77} -9.05320 q^{79} +2.01644 q^{81} +(7.84970 - 7.84970i) q^{83} +(-0.199288 + 0.199288i) q^{87} +17.6984 q^{89} -31.8388i q^{91} +(5.27926 + 5.27926i) q^{93} +(-10.8255 - 10.8255i) q^{97} +7.48661 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 68 q^{19} - 4 q^{41} - 56 q^{69} - 8 q^{71} - 28 q^{79} + 64 q^{81} + 120 q^{89} - 28 q^{99}+O(q^{100}) \) 16 * q + 68 * q^19 - 4 * q^41 - 56 * q^69 - 8 * q^71 - 28 * q^79 + 64 * q^81 + 120 * q^89 - 28 * q^99

Introduction

L-functions

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