LMFDB - Embedded newform 6900.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6900,2,Mod(1,6900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6900, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6900.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6900.a (trivial)

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:	yes
Analytic conductor:	$55.0967773947$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 276)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.16228$ of defining polynomial
Character	$\chi$	$=$	6900.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.16228 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.16228 q^{7} +1.00000 q^{9} -4.00000 q^{13} -0.837722 q^{17} +5.16228 q^{19} +1.16228 q^{21} +1.00000 q^{23} +1.00000 q^{27} -4.32456 q^{29} +6.32456 q^{31} -4.32456 q^{37} -4.00000 q^{39} -2.00000 q^{41} +5.16228 q^{43} +12.3246 q^{47} -5.64911 q^{49} -0.837722 q^{51} +13.4868 q^{53} +5.16228 q^{57} +4.32456 q^{59} -12.3246 q^{61} +1.16228 q^{63} +5.16228 q^{67} +1.00000 q^{69} -10.6491 q^{73} +6.83772 q^{79} +1.00000 q^{81} +4.00000 q^{83} -4.32456 q^{87} +9.48683 q^{89} -4.64911 q^{91} +6.32456 q^{93} +16.3246 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 4 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 4 q^{7} + 2 q^{9} - 8 q^{13} - 8 q^{17} + 4 q^{19} - 4 q^{21} + 2 q^{23} + 2 q^{27} + 4 q^{29} + 4 q^{37} - 8 q^{39} - 4 q^{41} + 4 q^{43} + 12 q^{47} + 14 q^{49} - 8 q^{51} + 8 q^{53} + 4 q^{57} - 4 q^{59} - 12 q^{61} - 4 q^{63} + 4 q^{67} + 2 q^{69} + 4 q^{73} + 20 q^{79} + 2 q^{81} + 8 q^{83} + 4 q^{87} + 16 q^{91} + 20 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^7 + 2 * q^9 - 8 * q^13 - 8 * q^17 + 4 * q^19 - 4 * q^21 + 2 * q^23 + 2 * q^27 + 4 * q^29 + 4 * q^37 - 8 * q^39 - 4 * q^41 + 4 * q^43 + 12 * q^47 + 14 * q^49 - 8 * q^51 + 8 * q^53 + 4 * q^57 - 4 * q^59 - 12 * q^61 - 4 * q^63 + 4 * q^67 + 2 * q^69 + 4 * q^73 + 20 * q^79 + 2 * q^81 + 8 * q^83 + 4 * q^87 + 16 * q^91 + 20 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.16228	0.439300	0.219650	−	0.975579i	$-0.429509\pi$
$7$	1.16228	0.439300	0.219650	+	0.975579i	$0.429509\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.837722	−0.203178	−0.101589	−	0.994826i	$-0.532393\pi$
$17$	−0.837722	−0.203178	−0.101589	+	0.994826i	$0.532393\pi$
$18$	0	0
$19$	5.16228	1.18431	0.592154	−	0.805825i	$-0.298278\pi$
$19$	5.16228	1.18431	0.592154	+	0.805825i	$0.298278\pi$
$20$	0	0
$21$	1.16228	0.253630
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.32456	−0.803050	−0.401525	−	0.915848i	$-0.631520\pi$
$29$	−4.32456	−0.803050	−0.401525	+	0.915848i	$0.631520\pi$
$30$	0	0
$31$	6.32456	1.13592	0.567962	−	0.823055i	$-0.307732\pi$
$31$	6.32456	1.13592	0.567962	+	0.823055i	$0.307732\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.32456	−0.710953	−0.355476	−	0.934685i	$-0.615681\pi$
$37$	−4.32456	−0.710953	−0.355476	+	0.934685i	$0.615681\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	5.16228	0.787240	0.393620	−	0.919273i	$-0.371223\pi$
$43$	5.16228	0.787240	0.393620	+	0.919273i	$0.371223\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.3246	1.79772	0.898861	−	0.438235i	$-0.144396\pi$
$47$	12.3246	1.79772	0.898861	+	0.438235i	$0.144396\pi$
$48$	0	0
$49$	−5.64911	−0.807016
$50$	0	0
$51$	−0.837722	−0.117305
$52$	0	0
$53$	13.4868	1.85256	0.926279	−	0.376837i	$-0.122988\pi$
$53$	13.4868	1.85256	0.926279	+	0.376837i	$0.122988\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.16228	0.683760
$58$	0	0
$59$	4.32456	0.563009	0.281505	−	0.959560i	$-0.409167\pi$
$59$	4.32456	0.563009	0.281505	+	0.959560i	$0.409167\pi$
$60$	0	0
$61$	−12.3246	−1.57800	−0.788999	−	0.614395i	$-0.789400\pi$
$61$	−12.3246	−1.57800	−0.788999	+	0.614395i	$0.789400\pi$
$62$	0	0
$63$	1.16228	0.146433
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.16228	0.630673	0.315336	−	0.948980i	$-0.397883\pi$
$67$	5.16228	0.630673	0.315336	+	0.948980i	$0.397883\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−10.6491	−1.24638	−0.623192	−	0.782069i	$-0.714165\pi$
$73$	−10.6491	−1.24638	−0.623192	+	0.782069i	$0.714165\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.83772	0.769304	0.384652	−	0.923062i	$-0.374321\pi$
$79$	6.83772	0.769304	0.384652	+	0.923062i	$0.374321\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−4.32456	−0.463641
$88$	0	0
$89$	9.48683	1.00560	0.502801	−	0.864402i	$-0.332303\pi$
$89$	9.48683	1.00560	0.502801	+	0.864402i	$0.332303\pi$
$90$	0	0
$91$	−4.64911	−0.487359
$92$	0	0
$93$	6.32456	0.655826
$94$	0	0
$95$	0	0
$96$	0	0
$97$	16.3246	1.65751	0.828754	−	0.559613i	$-0.189050\pi$
$97$	16.3246	1.65751	0.828754	+	0.559613i	$0.189050\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−11.4868	−1.13183	−0.565916	−	0.824463i	$-0.691477\pi$
$103$	−11.4868	−1.13183	−0.565916	+	0.824463i	$0.691477\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.0000	1.54678	0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000	1.54678	0.773389	+	0.633932i	$0.218560\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	−4.32456	−0.410469
$112$	0	0
$113$	−4.83772	−0.455095	−0.227547	−	0.973767i	$-0.573071\pi$
$113$	−4.83772	−0.455095	−0.227547	+	0.973767i	$0.573071\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−4.00000	−0.369800
$118$	0	0
$119$	−0.973666	−0.0892558
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	0	0
$126$	0	0
$127$	10.3246	0.916156	0.458078	−	0.888912i	$-0.348538\pi$
$127$	10.3246	0.916156	0.458078	+	0.888912i	$0.348538\pi$
$128$	0	0
$129$	5.16228	0.454513
$130$	0	0
$131$	−16.6491	−1.45464	−0.727320	−	0.686299i	$-0.759234\pi$
$131$	−16.6491	−1.45464	−0.727320	+	0.686299i	$0.759234\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.1623	−0.953658	−0.476829	−	0.878996i	$-0.658214\pi$
$137$	−11.1623	−0.953658	−0.476829	+	0.878996i	$0.658214\pi$
$138$	0	0
$139$	8.64911	0.733608	0.366804	−	0.930298i	$-0.380452\pi$
$139$	8.64911	0.733608	0.366804	+	0.930298i	$0.380452\pi$
$140$	0	0
$141$	12.3246	1.03791
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−5.64911	−0.465931
$148$	0	0
$149$	12.8377	1.05171	0.525854	−	0.850575i	$-0.323746\pi$
$149$	12.8377	1.05171	0.525854	+	0.850575i	$0.323746\pi$
$150$	0	0
$151$	20.6491	1.68040	0.840200	−	0.542276i	$-0.182437\pi$
$151$	20.6491	1.68040	0.840200	+	0.542276i	$0.182437\pi$
$152$	0	0
$153$	−0.837722	−0.0677258
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.324555	0.0259023	0.0129512	−	0.999916i	$-0.495877\pi$
$157$	0.324555	0.0259023	0.0129512	+	0.999916i	$0.495877\pi$
$158$	0	0
$159$	13.4868	1.06958
$160$	0	0
$161$	1.16228	0.0916003
$162$	0	0
$163$	−10.3246	−0.808682	−0.404341	−	0.914608i	$-0.632499\pi$
$163$	−10.3246	−0.808682	−0.404341	+	0.914608i	$0.632499\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.64911	−0.359759	−0.179879	−	0.983689i	$-0.557571\pi$
$167$	−4.64911	−0.359759	−0.179879	+	0.983689i	$0.557571\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	5.16228	0.394769
$172$	0	0
$173$	7.67544	0.583553	0.291777	−	0.956486i	$-0.405754\pi$
$173$	7.67544	0.583553	0.291777	+	0.956486i	$0.405754\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.32456	0.325053
$178$	0	0
$179$	−20.3246	−1.51913	−0.759564	−	0.650432i	$-0.774588\pi$
$179$	−20.3246	−1.51913	−0.759564	+	0.650432i	$0.774588\pi$
$180$	0	0
$181$	−18.6491	−1.38618	−0.693089	−	0.720852i	$-0.743751\pi$
$181$	−18.6491	−1.38618	−0.693089	+	0.720852i	$0.743751\pi$
$182$	0	0
$183$	−12.3246	−0.911057
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1.16228	0.0845433
$190$	0	0
$191$	18.9737	1.37289	0.686443	−	0.727183i	$-0.259171\pi$
$191$	18.9737	1.37289	0.686443	+	0.727183i	$0.259171\pi$
$192$	0	0
$193$	24.6491	1.77428	0.887141	−	0.461499i	$-0.152688\pi$
$193$	24.6491	1.77428	0.887141	+	0.461499i	$0.152688\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.3246	1.16308	0.581538	−	0.813519i	$-0.302451\pi$
$197$	16.3246	1.16308	0.581538	+	0.813519i	$0.302451\pi$
$198$	0	0
$199$	21.8114	1.54617	0.773084	−	0.634303i	$-0.218713\pi$
$199$	21.8114	1.54617	0.773084	+	0.634303i	$0.218713\pi$
$200$	0	0
$201$	5.16228	0.364119
$202$	0	0
$203$	−5.02633	−0.352779
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−2.32456	−0.160029	−0.0800145	−	0.996794i	$-0.525497\pi$
$211$	−2.32456	−0.160029	−0.0800145	+	0.996794i	$0.525497\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.35089	0.499011
$218$	0	0
$219$	−10.6491	−0.719600
$220$	0	0
$221$	3.35089	0.225405
$222$	0	0
$223$	12.0000	0.803579	0.401790	−	0.915732i	$-0.368388\pi$
$223$	12.0000	0.803579	0.401790	+	0.915732i	$0.368388\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.67544	0.376692	0.188346	−	0.982103i	$-0.439687\pi$
$227$	5.67544	0.376692	0.188346	+	0.982103i	$0.439687\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.6491	−1.22174	−0.610872	−	0.791729i	$-0.709181\pi$
$233$	−18.6491	−1.22174	−0.610872	+	0.791729i	$0.709181\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	6.83772	0.444158
$238$	0	0
$239$	28.6491	1.85316	0.926578	−	0.376102i	$-0.122736\pi$
$239$	28.6491	1.85316	0.926578	+	0.376102i	$0.122736\pi$
$240$	0	0
$241$	−7.67544	−0.494419	−0.247209	−	0.968962i	$-0.579514\pi$
$241$	−7.67544	−0.494419	−0.247209	+	0.968962i	$0.579514\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−20.6491	−1.31387
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	−1.67544	−0.105753	−0.0528766	−	0.998601i	$-0.516839\pi$
$251$	−1.67544	−0.105753	−0.0528766	+	0.998601i	$0.516839\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	−5.02633	−0.312321
$260$	0	0
$261$	−4.32456	−0.267683
$262$	0	0
$263$	−17.6754	−1.08991	−0.544957	−	0.838464i	$-0.683454\pi$
$263$	−17.6754	−1.08991	−0.544957	+	0.838464i	$0.683454\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	9.48683	0.580585
$268$	0	0
$269$	−16.3246	−0.995326	−0.497663	−	0.867371i	$-0.665808\pi$
$269$	−16.3246	−0.995326	−0.497663	+	0.867371i	$0.665808\pi$
$270$	0	0
$271$	−18.3246	−1.11314	−0.556569	−	0.830802i	$-0.687882\pi$
$271$	−18.3246	−1.11314	−0.556569	+	0.830802i	$0.687882\pi$
$272$	0	0
$273$	−4.64911	−0.281377
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−31.2982	−1.88053	−0.940264	−	0.340446i	$-0.889422\pi$
$277$	−31.2982	−1.88053	−0.940264	+	0.340446i	$0.889422\pi$
$278$	0	0
$279$	6.32456	0.378641
$280$	0	0
$281$	3.16228	0.188646	0.0943228	−	0.995542i	$-0.469931\pi$
$281$	3.16228	0.188646	0.0943228	+	0.995542i	$0.469931\pi$
$282$	0	0
$283$	−15.4868	−0.920597	−0.460298	−	0.887764i	$-0.652258\pi$
$283$	−15.4868	−0.920597	−0.460298	+	0.887764i	$0.652258\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.32456	−0.137214
$288$	0	0
$289$	−16.2982	−0.958719
$290$	0	0
$291$	16.3246	0.956962
$292$	0	0
$293$	−21.4868	−1.25527	−0.627637	−	0.778506i	$-0.715978\pi$
$293$	−21.4868	−1.25527	−0.627637	+	0.778506i	$0.715978\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.00000	−0.231326
$300$	0	0
$301$	6.00000	0.345834
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6.32456	0.360961	0.180481	−	0.983579i	$-0.442235\pi$
$307$	6.32456	0.360961	0.180481	+	0.983579i	$0.442235\pi$
$308$	0	0
$309$	−11.4868	−0.653463
$310$	0	0
$311$	−8.97367	−0.508850	−0.254425	−	0.967093i	$-0.581886\pi$
$311$	−8.97367	−0.508850	−0.254425	+	0.967093i	$0.581886\pi$
$312$	0	0
$313$	−18.6491	−1.05411	−0.527055	−	0.849831i	$-0.676704\pi$
$313$	−18.6491	−1.05411	−0.527055	+	0.849831i	$0.676704\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.3246	1.36620	0.683102	−	0.730323i	$-0.260631\pi$
$317$	24.3246	1.36620	0.683102	+	0.730323i	$0.260631\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	16.0000	0.893033
$322$	0	0
$323$	−4.32456	−0.240625
$324$	0	0
$325$	0	0
$326$	0	0
$327$	14.0000	0.774202
$328$	0	0
$329$	14.3246	0.789738
$330$	0	0
$331$	−6.32456	−0.347629	−0.173814	−	0.984778i	$-0.555609\pi$
$331$	−6.32456	−0.347629	−0.173814	+	0.984778i	$0.555609\pi$
$332$	0	0
$333$	−4.32456	−0.236984
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8.32456	0.453467	0.226734	−	0.973957i	$-0.427195\pi$
$337$	8.32456	0.453467	0.226734	+	0.973957i	$0.427195\pi$
$338$	0	0
$339$	−4.83772	−0.262749
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−14.7018	−0.793821
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.6754	0.841502	0.420751	−	0.907176i	$-0.361767\pi$
$347$	15.6754	0.841502	0.420751	+	0.907176i	$0.361767\pi$
$348$	0	0
$349$	16.6491	0.891206	0.445603	−	0.895231i	$-0.352989\pi$
$349$	16.6491	0.891206	0.445603	+	0.895231i	$0.352989\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	20.9737	1.11632	0.558158	−	0.829735i	$-0.311508\pi$
$353$	20.9737	1.11632	0.558158	+	0.829735i	$0.311508\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.973666	−0.0515319
$358$	0	0
$359$	18.9737	1.00139	0.500696	−	0.865623i	$-0.333077\pi$
$359$	18.9737	1.00139	0.500696	+	0.865623i	$0.333077\pi$
$360$	0	0
$361$	7.64911	0.402585
$362$	0	0
$363$	−11.0000	−0.577350
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−22.8377	−1.19212	−0.596060	−	0.802940i	$-0.703268\pi$
$367$	−22.8377	−1.19212	−0.596060	+	0.802940i	$0.703268\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	15.6754	0.813829
$372$	0	0
$373$	3.67544	0.190307	0.0951537	−	0.995463i	$-0.469666\pi$
$373$	3.67544	0.190307	0.0951537	+	0.995463i	$0.469666\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	17.2982	0.890904
$378$	0	0
$379$	18.8377	0.967629	0.483814	−	0.875171i	$-0.339251\pi$
$379$	18.8377	0.967629	0.483814	+	0.875171i	$0.339251\pi$
$380$	0	0
$381$	10.3246	0.528943
$382$	0	0
$383$	14.9737	0.765119	0.382559	−	0.923931i	$-0.375043\pi$
$383$	14.9737	0.765119	0.382559	+	0.923931i	$0.375043\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.16228	0.262413
$388$	0	0
$389$	0.837722	0.0424742	0.0212371	−	0.999774i	$-0.493240\pi$
$389$	0.837722	0.0424742	0.0212371	+	0.999774i	$0.493240\pi$
$390$	0	0
$391$	−0.837722	−0.0423654
$392$	0	0
$393$	−16.6491	−0.839837
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−6.00000	−0.301131	−0.150566	−	0.988600i	$-0.548110\pi$
$397$	−6.00000	−0.301131	−0.150566	+	0.988600i	$0.548110\pi$
$398$	0	0
$399$	6.00000	0.300376
$400$	0	0
$401$	13.4868	0.673500	0.336750	−	0.941594i	$-0.390672\pi$
$401$	13.4868	0.673500	0.336750	+	0.941594i	$0.390672\pi$
$402$	0	0
$403$	−25.2982	−1.26019
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	0	0
$411$	−11.1623	−0.550595
$412$	0	0
$413$	5.02633	0.247330
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8.64911	0.423549
$418$	0	0
$419$	−16.6491	−0.813362	−0.406681	−	0.913570i	$-0.633314\pi$
$419$	−16.6491	−0.813362	−0.406681	+	0.913570i	$0.633314\pi$
$420$	0	0
$421$	−10.6491	−0.519006	−0.259503	−	0.965742i	$-0.583559\pi$
$421$	−10.6491	−0.519006	−0.259503	+	0.965742i	$0.583559\pi$
$422$	0	0
$423$	12.3246	0.599240
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−14.3246	−0.693214
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	−6.64911	−0.319536	−0.159768	−	0.987155i	$-0.551075\pi$
$433$	−6.64911	−0.319536	−0.159768	+	0.987155i	$0.551075\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.16228	0.246945
$438$	0	0
$439$	−12.6491	−0.603709	−0.301855	−	0.953354i	$-0.597606\pi$
$439$	−12.6491	−0.603709	−0.301855	+	0.953354i	$0.597606\pi$
$440$	0	0
$441$	−5.64911	−0.269005
$442$	0	0
$443$	−8.64911	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$443$	−8.64911	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	12.8377	0.607203
$448$	0	0
$449$	−1.35089	−0.0637524	−0.0318762	−	0.999492i	$-0.510148\pi$
$449$	−1.35089	−0.0637524	−0.0318762	+	0.999492i	$0.510148\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	20.6491	0.970180
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.67544	−0.171930	−0.0859650	−	0.996298i	$-0.527397\pi$
$457$	−3.67544	−0.171930	−0.0859650	+	0.996298i	$0.527397\pi$
$458$	0	0
$459$	−0.837722	−0.0391015
$460$	0	0
$461$	36.9737	1.72204	0.861018	−	0.508575i	$-0.169828\pi$
$461$	36.9737	1.72204	0.861018	+	0.508575i	$0.169828\pi$
$462$	0	0
$463$	−16.6491	−0.773750	−0.386875	−	0.922132i	$-0.626445\pi$
$463$	−16.6491	−0.773750	−0.386875	+	0.922132i	$0.626445\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.9737	0.877997	0.438998	−	0.898488i	$-0.355333\pi$
$467$	18.9737	0.877997	0.438998	+	0.898488i	$0.355333\pi$
$468$	0	0
$469$	6.00000	0.277054
$470$	0	0
$471$	0.324555	0.0149547
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	13.4868	0.617520
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	17.2982	0.788731
$482$	0	0
$483$	1.16228	0.0528855
$484$	0	0
$485$	0	0
$486$	0	0
$487$	24.6491	1.11696	0.558479	−	0.829519i	$-0.311385\pi$
$487$	24.6491	1.11696	0.558479	+	0.829519i	$0.311385\pi$
$488$	0	0
$489$	−10.3246	−0.466893
$490$	0	0
$491$	16.9737	0.766011	0.383005	−	0.923746i	$-0.374889\pi$
$491$	16.9737	0.766011	0.383005	+	0.923746i	$0.374889\pi$
$492$	0	0
$493$	3.62278	0.163162
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−38.9737	−1.74470	−0.872350	−	0.488881i	$-0.837405\pi$
$499$	−38.9737	−1.74470	−0.872350	+	0.488881i	$0.837405\pi$
$500$	0	0
$501$	−4.64911	−0.207707
$502$	0	0
$503$	29.2982	1.30634	0.653172	−	0.757210i	$-0.273438\pi$
$503$	29.2982	1.30634	0.653172	+	0.757210i	$0.273438\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.00000	0.133235
$508$	0	0
$509$	15.6754	0.694802	0.347401	−	0.937717i	$-0.387064\pi$
$509$	15.6754	0.694802	0.347401	+	0.937717i	$0.387064\pi$
$510$	0	0
$511$	−12.3772	−0.547536
$512$	0	0
$513$	5.16228	0.227920
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	7.67544	0.336915
$520$	0	0
$521$	−39.8114	−1.74417	−0.872084	−	0.489356i	$-0.837232\pi$
$521$	−39.8114	−1.74417	−0.872084	+	0.489356i	$0.837232\pi$
$522$	0	0
$523$	17.8114	0.778838	0.389419	−	0.921061i	$-0.372676\pi$
$523$	17.8114	0.778838	0.389419	+	0.921061i	$0.372676\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.29822	−0.230794
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	4.32456	0.187670
$532$	0	0
$533$	8.00000	0.346518
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−20.3246	−0.877069
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	0	0
$543$	−18.6491	−0.800310
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14.9737	0.640228	0.320114	−	0.947379i	$-0.396279\pi$
$547$	14.9737	0.640228	0.320114	+	0.947379i	$0.396279\pi$
$548$	0	0
$549$	−12.3246	−0.525999
$550$	0	0
$551$	−22.3246	−0.951058
$552$	0	0
$553$	7.94733	0.337955
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−38.7851	−1.64338	−0.821688	−	0.569938i	$-0.806967\pi$
$557$	−38.7851	−1.64338	−0.821688	+	0.569938i	$0.806967\pi$
$558$	0	0
$559$	−20.6491	−0.873364
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−21.6754	−0.913511	−0.456756	−	0.889592i	$-0.650989\pi$
$563$	−21.6754	−0.913511	−0.456756	+	0.889592i	$0.650989\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.16228	0.0488111
$568$	0	0
$569$	−5.48683	−0.230020	−0.115010	−	0.993364i	$-0.536690\pi$
$569$	−5.48683	−0.230020	−0.115010	+	0.993364i	$0.536690\pi$
$570$	0	0
$571$	14.4605	0.605153	0.302577	−	0.953125i	$-0.402153\pi$
$571$	14.4605	0.605153	0.302577	+	0.953125i	$0.402153\pi$
$572$	0	0
$573$	18.9737	0.792636
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−40.6491	−1.69224	−0.846122	−	0.532989i	$-0.821069\pi$
$577$	−40.6491	−1.69224	−0.846122	+	0.532989i	$0.821069\pi$
$578$	0	0
$579$	24.6491	1.02438
$580$	0	0
$581$	4.64911	0.192878
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.6491	0.687182	0.343591	−	0.939119i	$-0.388357\pi$
$587$	16.6491	0.687182	0.343591	+	0.939119i	$0.388357\pi$
$588$	0	0
$589$	32.6491	1.34528
$590$	0	0
$591$	16.3246	0.671502
$592$	0	0
$593$	9.35089	0.383995	0.191998	−	0.981395i	$-0.438503\pi$
$593$	9.35089	0.383995	0.191998	+	0.981395i	$0.438503\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	21.8114	0.892681
$598$	0	0
$599$	−28.6491	−1.17057	−0.585285	−	0.810827i	$-0.699018\pi$
$599$	−28.6491	−1.17057	−0.585285	+	0.810827i	$0.699018\pi$
$600$	0	0
$601$	−16.6491	−0.679131	−0.339566	−	0.940582i	$-0.610280\pi$
$601$	−16.6491	−0.679131	−0.339566	+	0.940582i	$0.610280\pi$
$602$	0	0
$603$	5.16228	0.210224
$604$	0	0
$605$	0	0
$606$	0	0
$607$	19.3509	0.785428	0.392714	−	0.919661i	$-0.371536\pi$
$607$	19.3509	0.785428	0.392714	+	0.919661i	$0.371536\pi$
$608$	0	0
$609$	−5.02633	−0.203677
$610$	0	0
$611$	−49.2982	−1.99439
$612$	0	0
$613$	25.6228	1.03489	0.517447	−	0.855715i	$-0.326882\pi$
$613$	25.6228	1.03489	0.517447	+	0.855715i	$0.326882\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−28.8377	−1.16096	−0.580481	−	0.814273i	$-0.697136\pi$
$617$	−28.8377	−1.16096	−0.580481	+	0.814273i	$0.697136\pi$
$618$	0	0
$619$	25.8114	1.03745	0.518724	−	0.854942i	$-0.326407\pi$
$619$	25.8114	1.03745	0.518724	+	0.854942i	$0.326407\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	11.0263	0.441761
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.62278	0.144450
$630$	0	0
$631$	5.81139	0.231348	0.115674	−	0.993287i	$-0.463097\pi$
$631$	5.81139	0.231348	0.115674	+	0.993287i	$0.463097\pi$
$632$	0	0
$633$	−2.32456	−0.0923928
$634$	0	0
$635$	0	0
$636$	0	0
$637$	22.5964	0.895304
$638$	0	0
$639$	0	0
$640$	0	0
$641$	40.8377	1.61299	0.806497	−	0.591239i	$-0.201361\pi$
$641$	40.8377	1.61299	0.806497	+	0.591239i	$0.201361\pi$
$642$	0	0
$643$	−2.83772	−0.111909	−0.0559544	−	0.998433i	$-0.517820\pi$
$643$	−2.83772	−0.111909	−0.0559544	+	0.998433i	$0.517820\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	46.2719	1.81914	0.909568	−	0.415556i	$-0.136413\pi$
$647$	46.2719	1.81914	0.909568	+	0.415556i	$0.136413\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	7.35089	0.288104
$652$	0	0
$653$	−39.9473	−1.56326	−0.781630	−	0.623742i	$-0.785611\pi$
$653$	−39.9473	−1.56326	−0.781630	+	0.623742i	$0.785611\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−10.6491	−0.415461
$658$	0	0
$659$	−15.6228	−0.608577	−0.304288	−	0.952580i	$-0.598419\pi$
$659$	−15.6228	−0.608577	−0.304288	+	0.952580i	$0.598419\pi$
$660$	0	0
$661$	−32.3246	−1.25728	−0.628640	−	0.777697i	$-0.716388\pi$
$661$	−32.3246	−1.25728	−0.628640	+	0.777697i	$0.716388\pi$
$662$	0	0
$663$	3.35089	0.130138
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.32456	−0.167447
$668$	0	0
$669$	12.0000	0.463947
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−4.64911	−0.179210	−0.0896050	−	0.995977i	$-0.528560\pi$
$673$	−4.64911	−0.179210	−0.0896050	+	0.995977i	$0.528560\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	35.1623	1.35140	0.675698	−	0.737178i	$-0.263842\pi$
$677$	35.1623	1.35140	0.675698	+	0.737178i	$0.263842\pi$
$678$	0	0
$679$	18.9737	0.728142
$680$	0	0
$681$	5.67544	0.217484
$682$	0	0
$683$	−29.2982	−1.12107	−0.560533	−	0.828132i	$-0.689404\pi$
$683$	−29.2982	−1.12107	−0.560533	+	0.828132i	$0.689404\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	6.00000	0.228914
$688$	0	0
$689$	−53.9473	−2.05523
$690$	0	0
$691$	−46.9737	−1.78696	−0.893481	−	0.449101i	$-0.851745\pi$
$691$	−46.9737	−1.78696	−0.893481	+	0.449101i	$0.851745\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.67544	0.0634620
$698$	0	0
$699$	−18.6491	−0.705374
$700$	0	0
$701$	43.1623	1.63022	0.815108	−	0.579309i	$-0.196677\pi$
$701$	43.1623	1.63022	0.815108	+	0.579309i	$0.196677\pi$
$702$	0	0
$703$	−22.3246	−0.841987
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.97367	0.262272
$708$	0	0
$709$	−39.9473	−1.50025	−0.750127	−	0.661294i	$-0.770008\pi$
$709$	−39.9473	−1.50025	−0.750127	+	0.661294i	$0.770008\pi$
$710$	0	0
$711$	6.83772	0.256435
$712$	0	0
$713$	6.32456	0.236856
$714$	0	0
$715$	0	0
$716$	0	0
$717$	28.6491	1.06992
$718$	0	0
$719$	11.0263	0.411213	0.205606	−	0.978635i	$-0.434083\pi$
$719$	11.0263	0.411213	0.205606	+	0.978635i	$0.434083\pi$
$720$	0	0
$721$	−13.3509	−0.497213
$722$	0	0
$723$	−7.67544	−0.285453
$724$	0	0
$725$	0	0
$726$	0	0
$727$	27.4868	1.01943	0.509715	−	0.860343i	$-0.329751\pi$
$727$	27.4868	1.01943	0.509715	+	0.860343i	$0.329751\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.32456	−0.159949
$732$	0	0
$733$	−18.0000	−0.664845	−0.332423	−	0.943131i	$-0.607866\pi$
$733$	−18.0000	−0.664845	−0.332423	+	0.943131i	$0.607866\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	−20.6491	−0.758564
$742$	0	0
$743$	−40.2719	−1.47743	−0.738716	−	0.674017i	$-0.764568\pi$
$743$	−40.2719	−1.47743	−0.738716	+	0.674017i	$0.764568\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4.00000	0.146352
$748$	0	0
$749$	18.5964	0.679499
$750$	0	0
$751$	−4.51317	−0.164688	−0.0823439	−	0.996604i	$-0.526241\pi$
$751$	−4.51317	−0.164688	−0.0823439	+	0.996604i	$0.526241\pi$
$752$	0	0
$753$	−1.67544	−0.0610566
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−47.9473	−1.74268	−0.871338	−	0.490684i	$-0.836747\pi$
$757$	−47.9473	−1.74268	−0.871338	+	0.490684i	$0.836747\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	48.9737	1.77529	0.887647	−	0.460524i	$-0.152339\pi$
$761$	48.9737	1.77529	0.887647	+	0.460524i	$0.152339\pi$
$762$	0	0
$763$	16.2719	0.589082
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−17.2982	−0.624603
$768$	0	0
$769$	−5.35089	−0.192958	−0.0964790	−	0.995335i	$-0.530758\pi$
$769$	−5.35089	−0.192958	−0.0964790	+	0.995335i	$0.530758\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	0	0
$773$	−6.13594	−0.220695	−0.110347	−	0.993893i	$-0.535196\pi$
$773$	−6.13594	−0.220695	−0.110347	+	0.993893i	$0.535196\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−5.02633	−0.180319
$778$	0	0
$779$	−10.3246	−0.369916
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−4.32456	−0.154547
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−26.8377	−0.956661	−0.478331	−	0.878180i	$-0.658758\pi$
$787$	−26.8377	−0.956661	−0.478331	+	0.878180i	$0.658758\pi$
$788$	0	0
$789$	−17.6754	−0.629262
$790$	0	0
$791$	−5.62278	−0.199923
$792$	0	0
$793$	49.2982	1.75063
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.18861	0.148368	0.0741841	−	0.997245i	$-0.476365\pi$
$797$	4.18861	0.148368	0.0741841	+	0.997245i	$0.476365\pi$
$798$	0	0
$799$	−10.3246	−0.365257
$800$	0	0
$801$	9.48683	0.335201
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−16.3246	−0.574652
$808$	0	0
$809$	28.9737	1.01866	0.509330	−	0.860571i	$-0.329893\pi$
$809$	28.9737	1.01866	0.509330	+	0.860571i	$0.329893\pi$
$810$	0	0
$811$	−18.3246	−0.643462	−0.321731	−	0.946831i	$-0.604265\pi$
$811$	−18.3246	−0.643462	−0.321731	+	0.946831i	$0.604265\pi$
$812$	0	0
$813$	−18.3246	−0.642670
$814$	0	0
$815$	0	0
$816$	0	0
$817$	26.6491	0.932334
$818$	0	0
$819$	−4.64911	−0.162453
$820$	0	0
$821$	−11.2982	−0.394311	−0.197155	−	0.980372i	$-0.563170\pi$
$821$	−11.2982	−0.394311	−0.197155	+	0.980372i	$0.563170\pi$
$822$	0	0
$823$	−27.6228	−0.962869	−0.481435	−	0.876482i	$-0.659884\pi$
$823$	−27.6228	−0.962869	−0.481435	+	0.876482i	$0.659884\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−42.9737	−1.49434	−0.747170	−	0.664633i	$-0.768588\pi$
$827$	−42.9737	−1.49434	−0.747170	+	0.664633i	$0.768588\pi$
$828$	0	0
$829$	−17.2982	−0.600792	−0.300396	−	0.953815i	$-0.597119\pi$
$829$	−17.2982	−0.600792	−0.300396	+	0.953815i	$0.597119\pi$
$830$	0	0
$831$	−31.2982	−1.08572
$832$	0	0
$833$	4.73239	0.163967
$834$	0	0
$835$	0	0
$836$	0	0
$837$	6.32456	0.218609
$838$	0	0
$839$	39.6228	1.36793	0.683965	−	0.729515i	$-0.260254\pi$
$839$	39.6228	1.36793	0.683965	+	0.729515i	$0.260254\pi$
$840$	0	0
$841$	−10.2982	−0.355111
$842$	0	0
$843$	3.16228	0.108915
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−12.7851	−0.439300
$848$	0	0
$849$	−15.4868	−0.531507
$850$	0	0
$851$	−4.32456	−0.148244
$852$	0	0
$853$	23.2982	0.797716	0.398858	−	0.917013i	$-0.369407\pi$
$853$	23.2982	0.797716	0.398858	+	0.917013i	$0.369407\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−34.6491	−1.18359	−0.591796	−	0.806088i	$-0.701581\pi$
$857$	−34.6491	−1.18359	−0.591796	+	0.806088i	$0.701581\pi$
$858$	0	0
$859$	−9.29822	−0.317251	−0.158626	−	0.987339i	$-0.550706\pi$
$859$	−9.29822	−0.317251	−0.158626	+	0.987339i	$0.550706\pi$
$860$	0	0
$861$	−2.32456	−0.0792206
$862$	0	0
$863$	−44.9737	−1.53092	−0.765461	−	0.643483i	$-0.777489\pi$
$863$	−44.9737	−1.53092	−0.765461	+	0.643483i	$0.777489\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−16.2982	−0.553517
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−20.6491	−0.699668
$872$	0	0
$873$	16.3246	0.552502
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−47.9473	−1.61907	−0.809533	−	0.587074i	$-0.800280\pi$
$877$	−47.9473	−1.61907	−0.809533	+	0.587074i	$0.800280\pi$
$878$	0	0
$879$	−21.4868	−0.724733
$880$	0	0
$881$	3.16228	0.106540	0.0532699	−	0.998580i	$-0.483036\pi$
$881$	3.16228	0.106540	0.0532699	+	0.998580i	$0.483036\pi$
$882$	0	0
$883$	−1.67544	−0.0563832	−0.0281916	−	0.999603i	$-0.508975\pi$
$883$	−1.67544	−0.0563832	−0.0281916	+	0.999603i	$0.508975\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.3246	1.08535	0.542676	−	0.839942i	$-0.317411\pi$
$887$	32.3246	1.08535	0.542676	+	0.839942i	$0.317411\pi$
$888$	0	0
$889$	12.0000	0.402467
$890$	0	0
$891$	0	0
$892$	0	0
$893$	63.6228	2.12906
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4.00000	−0.133556
$898$	0	0
$899$	−27.3509	−0.912203
$900$	0	0
$901$	−11.2982	−0.376398
$902$	0	0
$903$	6.00000	0.199667
$904$	0	0
$905$	0	0
$906$	0	0
$907$	13.1623	0.437046	0.218523	−	0.975832i	$-0.429876\pi$
$907$	13.1623	0.437046	0.218523	+	0.975832i	$0.429876\pi$
$908$	0	0
$909$	6.00000	0.199007
$910$	0	0
$911$	28.0000	0.927681	0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000	0.927681	0.463841	+	0.885919i	$0.346471\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−19.3509	−0.639023
$918$	0	0
$919$	26.4605	0.872851	0.436426	−	0.899740i	$-0.356244\pi$
$919$	26.4605	0.872851	0.436426	+	0.899740i	$0.356244\pi$
$920$	0	0
$921$	6.32456	0.208401
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−11.4868	−0.377277
$928$	0	0
$929$	2.64911	0.0869145	0.0434573	−	0.999055i	$-0.486163\pi$
$929$	2.64911	0.0869145	0.0434573	+	0.999055i	$0.486163\pi$
$930$	0	0
$931$	−29.1623	−0.955755
$932$	0	0
$933$	−8.97367	−0.293785
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	−18.6491	−0.608591
$940$	0	0
$941$	−36.8377	−1.20088	−0.600438	−	0.799672i	$-0.705007\pi$
$941$	−36.8377	−1.20088	−0.600438	+	0.799672i	$0.705007\pi$
$942$	0	0
$943$	−2.00000	−0.0651290
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−20.9737	−0.681553	−0.340776	−	0.940144i	$-0.610690\pi$
$947$	−20.9737	−0.681553	−0.340776	+	0.940144i	$0.610690\pi$
$948$	0	0
$949$	42.5964	1.38274
$950$	0	0
$951$	24.3246	0.788778
$952$	0	0
$953$	18.1359	0.587481	0.293740	−	0.955885i	$-0.405100\pi$
$953$	18.1359	0.587481	0.293740	+	0.955885i	$0.405100\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12.9737	−0.418942
$960$	0	0
$961$	9.00000	0.290323
$962$	0	0
$963$	16.0000	0.515593
$964$	0	0
$965$	0	0
$966$	0	0
$967$	26.9737	0.867415	0.433707	−	0.901054i	$-0.357205\pi$
$967$	26.9737	0.867415	0.433707	+	0.901054i	$0.357205\pi$
$968$	0	0
$969$	−4.32456	−0.138925
$970$	0	0
$971$	59.6228	1.91339	0.956693	−	0.291099i	$-0.0940209\pi$
$971$	59.6228	1.91339	0.956693	+	0.291099i	$0.0940209\pi$
$972$	0	0
$973$	10.0527	0.322274
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−30.1359	−0.964134	−0.482067	−	0.876134i	$-0.660114\pi$
$977$	−30.1359	−0.964134	−0.482067	+	0.876134i	$0.660114\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	14.0000	0.446986
$982$	0	0
$983$	2.70178	0.0861734	0.0430867	−	0.999071i	$-0.486281\pi$
$983$	2.70178	0.0861734	0.0430867	+	0.999071i	$0.486281\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	14.3246	0.455956
$988$	0	0
$989$	5.16228	0.164151
$990$	0	0
$991$	−29.6754	−0.942672	−0.471336	−	0.881954i	$-0.656228\pi$
$991$	−29.6754	−0.942672	−0.471336	+	0.881954i	$0.656228\pi$
$992$	0	0
$993$	−6.32456	−0.200704
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−24.6491	−0.780645	−0.390323	−	0.920678i	$-0.627637\pi$
$997$	−24.6491	−0.780645	−0.390323	+	0.920678i	$0.627637\pi$
$998$	0	0
$999$	−4.32456	−0.136823

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6900.2.a.p.1.2		2
5.2	odd	4		6900.2.f.k.6349.2		4
5.3	odd	4		6900.2.f.k.6349.3		4
5.4	even	2		276.2.a.a.1.2	✓	2
15.14	odd	2		828.2.a.f.1.1		2
20.19	odd	2		1104.2.a.n.1.2		2
40.19	odd	2		4416.2.a.be.1.1		2
40.29	even	2		4416.2.a.bk.1.1		2
60.59	even	2		3312.2.a.y.1.1		2
115.114	odd	2		6348.2.a.e.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
276.2.a.a.1.2	✓	2	5.4	even	2
828.2.a.f.1.1		2	15.14	odd	2
1104.2.a.n.1.2		2	20.19	odd	2
3312.2.a.y.1.1		2	60.59	even	2
4416.2.a.be.1.1		2	40.19	odd	2
4416.2.a.bk.1.1		2	40.29	even	2
6348.2.a.e.1.1		2	115.114	odd	2
6900.2.a.p.1.2		2	1.1	even	1	trivial
6900.2.f.k.6349.2		4	5.2	odd	4
6900.2.f.k.6349.3		4	5.3	odd	4

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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:	\( N \)	\(=\)	\( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6900.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.16228 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.16228 q^{7} +1.00000 q^{9} -4.00000 q^{13} -0.837722 q^{17} +5.16228 q^{19} +1.16228 q^{21} +1.00000 q^{23} +1.00000 q^{27} -4.32456 q^{29} +6.32456 q^{31} -4.32456 q^{37} -4.00000 q^{39} -2.00000 q^{41} +5.16228 q^{43} +12.3246 q^{47} -5.64911 q^{49} -0.837722 q^{51} +13.4868 q^{53} +5.16228 q^{57} +4.32456 q^{59} -12.3246 q^{61} +1.16228 q^{63} +5.16228 q^{67} +1.00000 q^{69} -10.6491 q^{73} +6.83772 q^{79} +1.00000 q^{81} +4.00000 q^{83} -4.32456 q^{87} +9.48683 q^{89} -4.64911 q^{91} +6.32456 q^{93} +16.3246 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 4 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 4 q^{7} + 2 q^{9} - 8 q^{13} - 8 q^{17} + 4 q^{19} - 4 q^{21} + 2 q^{23} + 2 q^{27} + 4 q^{29} + 4 q^{37} - 8 q^{39} - 4 q^{41} + 4 q^{43} + 12 q^{47} + 14 q^{49} - 8 q^{51} + 8 q^{53} + 4 q^{57} - 4 q^{59} - 12 q^{61} - 4 q^{63} + 4 q^{67} + 2 q^{69} + 4 q^{73} + 20 q^{79} + 2 q^{81} + 8 q^{83} + 4 q^{87} + 16 q^{91} + 20 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^7 + 2 * q^9 - 8 * q^13 - 8 * q^17 + 4 * q^19 - 4 * q^21 + 2 * q^23 + 2 * q^27 + 4 * q^29 + 4 * q^37 - 8 * q^39 - 4 * q^41 + 4 * q^43 + 12 * q^47 + 14 * q^49 - 8 * q^51 + 8 * q^53 + 4 * q^57 - 4 * q^59 - 12 * q^61 - 4 * q^63 + 4 * q^67 + 2 * q^69 + 4 * q^73 + 20 * q^79 + 2 * q^81 + 8 * q^83 + 4 * q^87 + 16 * q^91 + 20 * q^97

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