LMFDB - Embedded newform 2300.3.f.c.1701.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,3,Mod(1701,2300)]

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, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2300.1701");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2300.f (of order $2$, degree $1$, 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:	$62.6704608029$
Analytic rank:	$0$
Dimension:	$16$
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} + 487 x^{14} + 91703 x^{12} + 8599549 x^{10} + 437649516 x^{8} + 12136718132 x^{6} + \cdots + 1845424439296 $ x^16 + 487x^14 + 91703x^12 + 8599549x^10 + 437649516x^8 + 12136718132x^6 + 169813416064x^4 + 973986747392*x^2 + 1845424439296
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1701.15
Root			$-6.97776i$ of defining polynomial
Character	$\chi$	$=$	2300.1701
Dual form			2300.3.f.c.1701.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.96261 q^{3} -6.97776i q^{7} +15.6275 q^{9} +O(q^{10})$	$q+4.96261 q^{3} -6.97776i q^{7} +15.6275 q^{9} +10.2862i q^{11} +10.1063 q^{13} +27.0186i q^{17} +19.5479i q^{19} -34.6279i q^{21} +(-19.4433 + 12.2865i) q^{23} +32.8896 q^{27} +44.8783 q^{29} +5.79870 q^{31} +51.0462i q^{33} -12.8954i q^{37} +50.1535 q^{39} -41.8490 q^{41} +63.2604i q^{43} +68.4156 q^{47} +0.310827 q^{49} +134.083i q^{51} -21.5403i q^{53} +97.0085i q^{57} +29.6932 q^{59} -69.8596i q^{61} -109.045i q^{63} +85.0306i q^{67} +(-96.4894 + 60.9732i) q^{69} -14.4925 q^{71} +1.89849 q^{73} +71.7744 q^{77} +24.6591i q^{79} +22.5708 q^{81} -157.827i q^{83} +222.713 q^{87} -109.733i q^{89} -70.5193i q^{91} +28.7767 q^{93} +174.332i q^{97} +160.747i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 64 q^{9}+O(q^{10}) $ 16 * q + 64 * q^9	$ 16 q + 64 q^{9} - 6 q^{13} + 8 q^{23} - 66 q^{27} - 6 q^{29} + 28 q^{31} + 74 q^{39} - 90 q^{41} - 40 q^{47} - 190 q^{49} - 174 q^{59} + 50 q^{69} + 116 q^{71} - 110 q^{73} - 198 q^{77} + 56 q^{81} + 362 q^{87} + 50 q^{93}+O(q^{100}) $ 16 * q + 64 * q^9 - 6 * q^13 + 8 * q^23 - 66 * q^27 - 6 * q^29 + 28 * q^31 + 74 * q^39 - 90 * q^41 - 40 * q^47 - 190 * q^49 - 174 * q^59 + 50 * q^69 + 116 * q^71 - 110 * q^73 - 198 * q^77 + 56 * q^81 + 362 * q^87 + 50 * q^93

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
$3$	4.96261			1.65420			0.827101	−	0.562053i	$-0.189988\pi$
$3$	4.96261			1.65420			0.827101	+	0.562053i	$0.189988\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	6.97776i		−	0.996823i	−0.866941	−	0.498412i	$-0.833917\pi$
$7$		−	6.97776i		−	0.996823i	0.866941	−	0.498412i	$-0.166083\pi$
$8$		0			0
$9$	15.6275			1.73639
$10$		0			0
$11$			10.2862i			0.935105i	0.883965	+	0.467553i	$0.154864\pi$
$11$			10.2862i			0.935105i	−0.883965	+	0.467553i	$0.845136\pi$
$12$		0			0
$13$	10.1063			0.777406			0.388703	−	0.921363i	$-0.372923\pi$
$13$	10.1063			0.777406			0.388703	+	0.921363i	$0.372923\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			27.0186i			1.58933i	0.607050	+	0.794664i	$0.292353\pi$
$17$			27.0186i			1.58933i	−0.607050	+	0.794664i	$0.707647\pi$
$18$		0			0
$19$			19.5479i			1.02884i	0.857540	+	0.514418i	$0.171992\pi$
$19$			19.5479i			1.02884i	−0.857540	+	0.514418i	$0.828008\pi$
$20$		0			0
$21$		−	34.6279i		−	1.64895i
$22$		0			0
$23$	−19.4433	+	12.2865i	−0.845360	+	0.534197i
$23$	−19.4433	+	12.2865i	−0.845360	+	0.534197i
$24$		0			0
$25$		0			0
$26$		0			0
$27$	32.8896			1.21813
$28$		0			0
$29$	44.8783			1.54753			0.773763	−	0.633475i	$-0.218372\pi$
$29$	44.8783			1.54753			0.773763	+	0.633475i	$0.218372\pi$
$30$		0			0
$31$	5.79870			0.187055			0.0935274	−	0.995617i	$-0.470186\pi$
$31$	5.79870			0.187055			0.0935274	+	0.995617i	$0.470186\pi$
$32$		0			0
$33$			51.0462i			1.54685i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	12.8954i		−	0.348525i	−0.984699	−	0.174262i	$-0.944246\pi$
$37$		−	12.8954i		−	0.348525i	0.984699	−	0.174262i	$-0.0557541\pi$
$38$		0			0
$39$	50.1535			1.28599
$40$		0			0
$41$	−41.8490			−1.02071			−0.510353	−	0.859965i	$-0.670485\pi$
$41$	−41.8490			−1.02071			−0.510353	+	0.859965i	$0.670485\pi$
$42$		0			0
$43$			63.2604i			1.47117i	0.677432	+	0.735586i	$0.263093\pi$
$43$			63.2604i			1.47117i	−0.677432	+	0.735586i	$0.736907\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	68.4156			1.45565			0.727826	−	0.685762i	$-0.240531\pi$
$47$	68.4156			1.45565			0.727826	+	0.685762i	$0.240531\pi$
$48$		0			0
$49$	0.310827			0.00634341
$50$		0			0
$51$			134.083i			2.62907i
$52$		0			0
$53$		−	21.5403i		−	0.406421i	−0.979135	−	0.203211i	$-0.934862\pi$
$53$		−	21.5403i		−	0.406421i	0.979135	−	0.203211i	$-0.0651376\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			97.0085i			1.70190i
$58$		0			0
$59$	29.6932			0.503275			0.251637	−	0.967822i	$-0.419031\pi$
$59$	29.6932			0.503275			0.251637	+	0.967822i	$0.419031\pi$
$60$		0			0
$61$		−	69.8596i		−	1.14524i	−0.819821	−	0.572620i	$-0.805927\pi$
$61$		−	69.8596i		−	1.14524i	0.819821	−	0.572620i	$-0.194073\pi$
$62$		0			0
$63$		−	109.045i		−	1.73087i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			85.0306i			1.26911i	0.772876	+	0.634557i	$0.218817\pi$
$67$			85.0306i			1.26911i	−0.772876	+	0.634557i	$0.781183\pi$
$68$		0			0
$69$	−96.4894	+	60.9732i	−1.39840	+	0.883669i
$70$		0			0
$71$	−14.4925			−0.204120			−0.102060	−	0.994778i	$-0.532543\pi$
$71$	−14.4925			−0.204120			−0.102060	+	0.994778i	$0.532543\pi$
$72$		0			0
$73$	1.89849			0.0260067			0.0130034	−	0.999915i	$-0.495861\pi$
$73$	1.89849			0.0260067			0.0130034	+	0.999915i	$0.495861\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	71.7744			0.932135
$78$		0			0
$79$			24.6591i			0.312140i	0.987746	+	0.156070i	$0.0498825\pi$
$79$			24.6591i			0.312140i	−0.987746	+	0.156070i	$0.950117\pi$
$80$		0			0
$81$	22.5708			0.278652
$82$		0			0
$83$		−	157.827i		−	1.90153i	−0.309913	−	0.950765i	$-0.600300\pi$
$83$		−	157.827i		−	1.90153i	0.309913	−	0.950765i	$-0.399700\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	222.713			2.55992
$88$		0			0
$89$		−	109.733i		−	1.23296i	−0.787370	−	0.616480i	$-0.788558\pi$
$89$		−	109.733i		−	1.23296i	0.787370	−	0.616480i	$-0.211442\pi$
$90$		0			0
$91$		−	70.5193i		−	0.774937i
$92$		0			0
$93$	28.7767			0.309426
$94$		0			0
$95$		0			0
$96$		0			0
$97$			174.332i			1.79723i	0.438733	+	0.898617i	$0.355427\pi$
$97$			174.332i			1.79723i	−0.438733	+	0.898617i	$0.644573\pi$
$98$		0			0
$99$			160.747i			1.62370i
$100$		0			0
$101$	−18.4647			−0.182819			−0.0914095	−	0.995813i	$-0.529137\pi$
$101$	−18.4647			−0.182819			−0.0914095	+	0.995813i	$0.529137\pi$
$102$		0			0
$103$			67.4469i			0.654825i	0.944882	+	0.327412i	$0.106177\pi$
$103$			67.4469i			0.654825i	−0.944882	+	0.327412i	$0.893823\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	86.1864i		−	0.805480i	−0.915314	−	0.402740i	$-0.868058\pi$
$107$		−	86.1864i		−	0.805480i	0.915314	−	0.402740i	$-0.131942\pi$
$108$		0			0
$109$		−	70.4008i		−	0.645879i	−0.946420	−	0.322940i	$-0.895329\pi$
$109$		−	70.4008i		−	0.645879i	0.946420	−	0.322940i	$-0.104671\pi$
$110$		0			0
$111$		−	63.9949i		−	0.576531i
$112$		0			0
$113$		−	193.237i		−	1.71006i	−0.518580	−	0.855029i	$-0.673539\pi$
$113$		−	193.237i		−	1.71006i	0.518580	−	0.855029i	$-0.326461\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	157.936			1.34988
$118$		0			0
$119$	188.529			1.58428
$120$		0			0
$121$	15.1950			0.125578
$122$		0			0
$123$	−207.680			−1.68846
$124$		0			0
$125$		0			0
$126$		0			0
$127$	111.426			0.877372			0.438686	−	0.898640i	$-0.355444\pi$
$127$	111.426			0.877372			0.438686	+	0.898640i	$0.355444\pi$
$128$		0			0
$129$			313.936i			2.43362i
$130$		0			0
$131$	22.3359			0.170503			0.0852514	−	0.996359i	$-0.472831\pi$
$131$	22.3359			0.170503			0.0852514	+	0.996359i	$0.472831\pi$
$132$		0			0
$133$	136.400			1.02557
$134$		0			0
$135$		0			0
$136$		0			0
$137$			0.294939i			0.00215284i	0.999999	+	0.00107642i	$0.000342635\pi$
$137$			0.294939i			0.00215284i	−0.999999	+	0.00107642i	$0.999657\pi$
$138$		0			0
$139$	121.314			0.872765			0.436383	−	0.899761i	$-0.356259\pi$
$139$	121.314			0.872765			0.436383	+	0.899761i	$0.356259\pi$
$140$		0			0
$141$	339.520			2.40794
$142$		0			0
$143$			103.955i			0.726957i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	1.54251			0.0104933
$148$		0			0
$149$		−	80.8688i		−	0.542743i	−0.962475	−	0.271372i	$-0.912523\pi$
$149$		−	80.8688i		−	0.542743i	0.962475	−	0.271372i	$-0.0874772\pi$
$150$		0			0
$151$	230.813			1.52856			0.764281	−	0.644883i	$-0.223094\pi$
$151$	230.813			1.52856			0.764281	+	0.644883i	$0.223094\pi$
$152$		0			0
$153$			422.232i			2.75969i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	136.384i		−	0.868687i	−0.900747	−	0.434343i	$-0.856980\pi$
$157$		−	136.384i		−	0.868687i	0.900747	−	0.434343i	$-0.143020\pi$
$158$		0			0
$159$		−	106.896i		−	0.672303i
$160$		0			0
$161$	85.7324	+	135.671i	0.532500	+	0.842675i
$162$		0			0
$163$	−260.321			−1.59706			−0.798530	−	0.601954i	$-0.794389\pi$
$163$	−260.321			−1.59706			−0.798530	+	0.601954i	$0.794389\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−179.807			−1.07669			−0.538345	−	0.842725i	$-0.680950\pi$
$167$	−179.807			−1.07669			−0.538345	+	0.842725i	$0.680950\pi$
$168$		0			0
$169$	−66.8630			−0.395639
$170$		0			0
$171$			305.484i			1.78646i
$172$		0			0
$173$	−5.23637			−0.0302680			−0.0151340	−	0.999885i	$-0.504817\pi$
$173$	−5.23637			−0.0302680			−0.0151340	+	0.999885i	$0.504817\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	147.356			0.832518
$178$		0			0
$179$	227.005			1.26819			0.634093	−	0.773257i	$-0.281374\pi$
$179$	227.005			1.26819			0.634093	+	0.773257i	$0.281374\pi$
$180$		0			0
$181$			117.068i			0.646786i	0.946265	+	0.323393i	$0.104824\pi$
$181$			117.068i			0.646786i	−0.946265	+	0.323393i	$0.895176\pi$
$182$		0			0
$183$		−	346.686i		−	1.89446i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−277.917			−1.48619
$188$		0			0
$189$		−	229.496i		−	1.21426i
$190$		0			0
$191$		−	79.1718i		−	0.414512i	−0.978287	−	0.207256i	$-0.933547\pi$
$191$		−	79.1718i		−	0.414512i	0.978287	−	0.207256i	$-0.0664533\pi$
$192$		0			0
$193$	−29.3684			−0.152168			−0.0760840	−	0.997101i	$-0.524242\pi$
$193$	−29.3684			−0.152168			−0.0760840	+	0.997101i	$0.524242\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−237.076			−1.20343			−0.601715	−	0.798711i	$-0.705516\pi$
$197$	−237.076			−1.20343			−0.601715	+	0.798711i	$0.705516\pi$
$198$		0			0
$199$			192.923i			0.969462i	0.874663	+	0.484731i	$0.161082\pi$
$199$			192.923i			0.969462i	−0.874663	+	0.484731i	$0.838918\pi$
$200$		0			0
$201$			421.974i			2.09937i
$202$		0			0
$203$		−	313.150i		−	1.54261i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−303.850	+	192.007i	−1.46787	+	0.927572i
$208$		0			0
$209$	−201.073			−0.962070
$210$		0			0
$211$	250.021			1.18493			0.592467	−	0.805595i	$-0.298154\pi$
$211$	250.021			1.18493			0.592467	+	0.805595i	$0.298154\pi$
$212$		0			0
$213$	−71.9205			−0.337655
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	40.4619i		−	0.186461i
$218$		0			0
$219$	9.42147			0.0430204
$220$		0			0
$221$			273.057i			1.23555i
$222$		0			0
$223$	−227.702			−1.02109			−0.510543	−	0.859852i	$-0.670556\pi$
$223$	−227.702			−1.02109			−0.510543	+	0.859852i	$0.670556\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	283.476i		−	1.24879i	−0.781108	−	0.624396i	$-0.785345\pi$
$227$		−	283.476i		−	1.24879i	0.781108	−	0.624396i	$-0.214655\pi$
$228$		0			0
$229$			341.991i			1.49341i	0.665156	+	0.746704i	$0.268365\pi$
$229$			341.991i			1.49341i	−0.665156	+	0.746704i	$0.731635\pi$
$230$		0			0
$231$	356.188			1.54194
$232$		0			0
$233$	−87.4912			−0.375499			−0.187749	−	0.982217i	$-0.560119\pi$
$233$	−87.4912			−0.375499			−0.187749	+	0.982217i	$0.560119\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			122.373i			0.516343i
$238$		0			0
$239$	155.688			0.651414			0.325707	−	0.945471i	$-0.394398\pi$
$239$	155.688			0.651414			0.325707	+	0.945471i	$0.394398\pi$
$240$		0			0
$241$			130.278i			0.540573i	0.962780	+	0.270286i	$0.0871184\pi$
$241$			130.278i			0.540573i	−0.962780	+	0.270286i	$0.912882\pi$
$242$		0			0
$243$	−183.996			−0.757186
$244$		0			0
$245$		0			0
$246$		0			0
$247$			197.556i			0.799824i
$248$		0			0
$249$		−	783.233i		−	3.14552i
$250$		0			0
$251$		−	339.659i		−	1.35322i	−0.736340	−	0.676612i	$-0.763447\pi$
$251$		−	339.659i		−	1.35322i	0.736340	−	0.676612i	$-0.236553\pi$
$252$		0			0
$253$	−126.381	−	199.997i	−0.499530	−	0.790501i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	421.707			1.64088			0.820442	−	0.571730i	$-0.193728\pi$
$257$	421.707			1.64088			0.820442	+	0.571730i	$0.193728\pi$
$258$		0			0
$259$	−89.9812			−0.347418
$260$		0			0
$261$	701.334			2.68710
$262$		0			0
$263$		−	50.8196i		−	0.193231i	−0.995322	−	0.0966153i	$-0.969198\pi$
$263$		−	50.8196i		−	0.193231i	0.995322	−	0.0966153i	$-0.0308016\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	544.564i		−	2.03957i
$268$		0			0
$269$	−337.599			−1.25501			−0.627507	−	0.778611i	$-0.715925\pi$
$269$	−337.599			−1.25501			−0.627507	+	0.778611i	$0.715925\pi$
$270$		0			0
$271$	−315.074			−1.16263			−0.581317	−	0.813677i	$-0.697462\pi$
$271$	−315.074			−1.16263			−0.581317	+	0.813677i	$0.697462\pi$
$272$		0			0
$273$		−	349.959i		−	1.28190i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−98.0305			−0.353901			−0.176950	−	0.984220i	$-0.556623\pi$
$277$	−98.0305			−0.353901			−0.176950	+	0.984220i	$0.556623\pi$
$278$		0			0
$279$	90.6190			0.324799
$280$		0			0
$281$		−	113.715i		−	0.404678i	−0.979316	−	0.202339i	$-0.935146\pi$
$281$		−	113.715i		−	0.404678i	0.979316	−	0.202339i	$-0.0648543\pi$
$282$		0			0
$283$			231.462i			0.817887i	0.912560	+	0.408944i	$0.134103\pi$
$283$			231.462i			0.817887i	−0.912560	+	0.408944i	$0.865897\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			292.012i			1.01746i
$288$		0			0
$289$	−441.003			−1.52596
$290$		0			0
$291$			865.140i			2.97299i
$292$		0			0
$293$			474.148i			1.61825i	0.587635	+	0.809126i	$0.300059\pi$
$293$			474.148i			1.61825i	−0.587635	+	0.809126i	$0.699941\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			338.307i			1.13908i
$298$		0			0
$299$	−196.499	+	124.171i	−0.657189	+	0.415288i
$300$		0			0
$301$	441.416			1.46650
$302$		0			0
$303$	−91.6332			−0.302420
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−25.3248			−0.0824914			−0.0412457	−	0.999149i	$-0.513133\pi$
$307$	−25.3248			−0.0824914			−0.0412457	+	0.999149i	$0.513133\pi$
$308$		0			0
$309$			334.713i			1.08321i
$310$		0			0
$311$	−194.820			−0.626432			−0.313216	−	0.949682i	$-0.601406\pi$
$311$	−194.820			−0.626432			−0.313216	+	0.949682i	$0.601406\pi$
$312$		0			0
$313$		−	97.6690i		−	0.312042i	−0.987754	−	0.156021i	$-0.950133\pi$
$313$		−	97.6690i		−	0.312042i	0.987754	−	0.156021i	$-0.0498667\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	28.6106			0.0902542			0.0451271	−	0.998981i	$-0.485631\pi$
$317$	28.6106			0.0902542			0.0451271	+	0.998981i	$0.485631\pi$
$318$		0			0
$319$			461.625i			1.44710i
$320$		0			0
$321$		−	427.709i		−	1.33243i
$322$		0			0
$323$	−528.156			−1.63516
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	349.372i		−	1.06842i
$328$		0			0
$329$		−	477.388i		−	1.45103i
$330$		0			0
$331$	−388.452			−1.17357			−0.586786	−	0.809742i	$-0.699607\pi$
$331$	−388.452			−1.17357			−0.586786	+	0.809742i	$0.699607\pi$
$332$		0			0
$333$		−	201.523i		−	0.605174i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			175.571i			0.520981i	0.965476	+	0.260491i	$0.0838843\pi$
$337$			175.571i			0.520981i	−0.965476	+	0.260491i	$0.916116\pi$
$338$		0			0
$339$		−	958.957i		−	2.82878i
$340$		0			0
$341$			59.6463i			0.174916i
$342$		0			0
$343$		−	344.079i		−	1.00315i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−555.150			−1.59986			−0.799928	−	0.600096i	$-0.795129\pi$
$347$	−555.150			−1.59986			−0.799928	+	0.600096i	$0.795129\pi$
$348$		0			0
$349$	509.836			1.46085			0.730424	−	0.682994i	$-0.239322\pi$
$349$	509.836			1.46085			0.730424	+	0.682994i	$0.239322\pi$
$350$		0			0
$351$	332.392			0.946984
$352$		0			0
$353$	−325.703			−0.922670			−0.461335	−	0.887226i	$-0.652629\pi$
$353$	−325.703			−0.922670			−0.461335	+	0.887226i	$0.652629\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	935.596			2.62072
$358$		0			0
$359$			373.666i			1.04085i	0.853907	+	0.520426i	$0.174227\pi$
$359$			373.666i			1.04085i	−0.853907	+	0.520426i	$0.825773\pi$
$360$		0			0
$361$	−21.1196			−0.0585031
$362$		0			0
$363$	75.4066			0.207732
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	727.546i		−	1.98241i	−0.132320	−	0.991207i	$-0.542243\pi$
$367$		−	727.546i		−	1.98241i	0.132320	−	0.991207i	$-0.457757\pi$
$368$		0			0
$369$	−653.994			−1.77234
$370$		0			0
$371$	−150.303			−0.405130
$372$		0			0
$373$		−	175.303i		−	0.469980i	−0.971998	−	0.234990i	$-0.924494\pi$
$373$		−	175.303i		−	0.469980i	0.971998	−	0.234990i	$-0.0755057\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	453.552			1.20306
$378$		0			0
$379$			170.384i			0.449562i	0.974409	+	0.224781i	$0.0721666\pi$
$379$			170.384i			0.449562i	−0.974409	+	0.224781i	$0.927833\pi$
$380$		0			0
$381$	552.965			1.45135
$382$		0			0
$383$			300.247i			0.783933i	0.919979	+	0.391967i	$0.128205\pi$
$383$			300.247i			0.783933i	−0.919979	+	0.391967i	$0.871795\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			988.600i			2.55452i
$388$		0			0
$389$		−	25.7716i		−	0.0662509i	−0.999451	−	0.0331254i	$-0.989454\pi$
$389$		−	25.7716i		−	0.0662509i	0.999451	−	0.0331254i	$-0.0105461\pi$
$390$		0			0
$391$	−331.964	−	525.330i	−0.849013	−	1.34355i
$392$		0			0
$393$	110.844			0.282046
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−691.800			−1.74257			−0.871285	−	0.490777i	$-0.836713\pi$
$397$	−691.800			−1.74257			−0.871285	+	0.490777i	$0.836713\pi$
$398$		0			0
$399$	676.902			1.69650
$400$		0			0
$401$			262.510i			0.654639i	0.944914	+	0.327320i	$0.106145\pi$
$401$			262.510i			0.654639i	−0.944914	+	0.327320i	$0.893855\pi$
$402$		0			0
$403$	58.6033			0.145418
$404$		0			0
$405$		0			0
$406$		0			0
$407$	132.644			0.325907
$408$		0			0
$409$	63.4146			0.155048			0.0775240	−	0.996990i	$-0.475299\pi$
$409$	63.4146			0.155048			0.0775240	+	0.996990i	$0.475299\pi$
$410$		0			0
$411$			1.46367i			0.00356124i
$412$		0			0
$413$		−	207.192i		−	0.501676i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	602.036			1.44373
$418$		0			0
$419$			187.856i			0.448343i	0.974550	+	0.224172i	$0.0719676\pi$
$419$			187.856i			0.448343i	−0.974550	+	0.224172i	$0.928032\pi$
$420$		0			0
$421$			501.333i			1.19081i	0.803424	+	0.595407i	$0.203009\pi$
$421$			501.333i			1.19081i	−0.803424	+	0.595407i	$0.796991\pi$
$422$		0			0
$423$	1069.16			2.52757
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−487.464			−1.14160
$428$		0			0
$429$			515.887i			1.20253i
$430$		0			0
$431$		−	534.282i		−	1.23963i	−0.784747	−	0.619816i	$-0.787207\pi$
$431$		−	534.282i		−	1.23963i	0.784747	−	0.619816i	$-0.212793\pi$
$432$		0			0
$433$		−	346.798i		−	0.800920i	−0.916314	−	0.400460i	$-0.868850\pi$
$433$		−	346.798i		−	0.800920i	0.916314	−	0.400460i	$-0.131150\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−240.175	−	380.075i	−0.549601	−	0.869737i
$438$		0			0
$439$	552.167			1.25778			0.628892	−	0.777492i	$-0.283509\pi$
$439$	552.167			1.25778			0.628892	+	0.777492i	$0.283509\pi$
$440$		0			0
$441$	4.85745			0.0110146
$442$		0			0
$443$	550.487			1.24263			0.621317	−	0.783559i	$-0.286598\pi$
$443$	550.487			1.24263			0.621317	+	0.783559i	$0.286598\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	401.320i		−	0.897807i
$448$		0			0
$449$	504.192			1.12292			0.561462	−	0.827503i	$-0.310239\pi$
$449$	504.192			1.12292			0.561462	+	0.827503i	$0.310239\pi$
$450$		0			0
$451$		−	430.465i		−	0.954468i
$452$		0			0
$453$	1145.43			2.52855
$454$		0			0
$455$		0			0
$456$		0			0
$457$			19.5978i			0.0428836i	0.999770	+	0.0214418i	$0.00682565\pi$
$457$			19.5978i			0.0428836i	−0.999770	+	0.0214418i	$0.993174\pi$
$458$		0			0
$459$			888.629i			1.93601i
$460$		0			0
$461$	216.475			0.469578			0.234789	−	0.972046i	$-0.424560\pi$
$461$	216.475			0.469578			0.234789	+	0.972046i	$0.424560\pi$
$462$		0			0
$463$	−308.904			−0.667180			−0.333590	−	0.942718i	$-0.608260\pi$
$463$	−308.904			−0.667180			−0.333590	+	0.942718i	$0.608260\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			417.558i			0.894128i	0.894502	+	0.447064i	$0.147530\pi$
$467$			417.558i			0.894128i	−0.894502	+	0.447064i	$0.852470\pi$
$468$		0			0
$469$	593.324			1.26508
$470$		0			0
$471$		−	676.819i		−	1.43698i
$472$		0			0
$473$	−650.706			−1.37570
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	336.621i		−	0.705704i
$478$		0			0
$479$			191.149i			0.399057i	0.979892	+	0.199529i	$0.0639411\pi$
$479$			191.149i			0.399057i	−0.979892	+	0.199529i	$0.936059\pi$
$480$		0			0
$481$		−	130.325i		−	0.270945i
$482$		0			0
$483$	425.456	+	673.280i	0.880862	+	1.39395i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	834.672			1.71391			0.856953	−	0.515395i	$-0.172355\pi$
$487$	834.672			1.71391			0.856953	+	0.515395i	$0.172355\pi$
$488$		0			0
$489$	−1291.87			−2.64186
$490$		0			0
$491$	−178.865			−0.364287			−0.182143	−	0.983272i	$-0.558303\pi$
$491$	−178.865			−0.364287			−0.182143	+	0.983272i	$0.558303\pi$
$492$		0			0
$493$			1212.55i			2.45953i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			101.125i			0.203471i
$498$		0			0
$499$	718.540			1.43996			0.719980	−	0.693994i	$-0.244151\pi$
$499$	718.540			1.43996			0.719980	+	0.693994i	$0.244151\pi$
$500$		0			0
$501$	−892.312			−1.78106
$502$		0			0
$503$		−	490.987i		−	0.976117i	−0.872811	−	0.488059i	$-0.837705\pi$
$503$		−	490.987i		−	0.976117i	0.872811	−	0.488059i	$-0.162295\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−331.815			−0.654467
$508$		0			0
$509$	−114.915			−0.225765			−0.112883	−	0.993608i	$-0.536008\pi$
$509$	−114.915			−0.225765			−0.112883	+	0.993608i	$0.536008\pi$
$510$		0			0
$511$		−	13.2472i		−	0.0259241i
$512$		0			0
$513$			642.922i			1.25326i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			703.734i			1.36119i
$518$		0			0
$519$	−25.9861			−0.0500695
$520$		0			0
$521$		−	479.276i		−	0.919916i	−0.887941	−	0.459958i	$-0.847865\pi$
$521$		−	479.276i		−	0.919916i	0.887941	−	0.459958i	$-0.152135\pi$
$522$		0			0
$523$		−	413.778i		−	0.791162i	−0.918431	−	0.395581i	$-0.870543\pi$
$523$		−	413.778i		−	0.791162i	0.918431	−	0.395581i	$-0.129457\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			156.672i			0.297291i
$528$		0			0
$529$	227.083	−	477.781i	0.429268	−	0.903177i
$530$		0			0
$531$	464.030			0.873879
$532$		0			0
$533$	−422.937			−0.793504
$534$		0			0
$535$		0			0
$536$		0			0
$537$	1126.54			2.09784
$538$		0			0
$539$			3.19722i			0.00593176i
$540$		0			0
$541$	−762.358			−1.40916			−0.704582	−	0.709622i	$-0.748866\pi$
$541$	−762.358			−1.40916			−0.704582	+	0.709622i	$0.748866\pi$
$542$		0			0
$543$			580.964i			1.06992i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−464.746			−0.849627			−0.424813	−	0.905281i	$-0.639660\pi$
$547$	−464.746			−0.849627			−0.424813	+	0.905281i	$0.639660\pi$
$548$		0			0
$549$		−	1091.73i		−	1.98858i
$550$		0			0
$551$			877.275i			1.59215i
$552$		0			0
$553$	172.065			0.311148
$554$		0			0
$555$		0			0
$556$		0			0
$557$			383.306i			0.688161i	0.938940	+	0.344080i	$0.111809\pi$
$557$			383.306i			0.688161i	−0.938940	+	0.344080i	$0.888191\pi$
$558$		0			0
$559$			639.327i			1.14370i
$560$		0			0
$561$	−1379.19			−2.45846
$562$		0			0
$563$		−	441.290i		−	0.783819i	−0.920004	−	0.391910i	$-0.871815\pi$
$563$		−	441.290i		−	0.783819i	0.920004	−	0.391910i	$-0.128185\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	157.494i		−	0.277767i
$568$		0			0
$569$			848.231i			1.49074i	0.666651	+	0.745370i	$0.267727\pi$
$569$			848.231i			1.49074i	−0.666651	+	0.745370i	$0.732273\pi$
$570$		0			0
$571$			960.641i			1.68238i	0.540737	+	0.841192i	$0.318145\pi$
$571$			960.641i			1.68238i	−0.540737	+	0.841192i	$0.681855\pi$
$572$		0			0
$573$		−	392.898i		−	0.685687i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−539.287			−0.934639			−0.467319	−	0.884089i	$-0.654780\pi$
$577$	−539.287			−0.934639			−0.467319	+	0.884089i	$0.654780\pi$
$578$		0			0
$579$	−145.744			−0.251717
$580$		0			0
$581$	−1101.28			−1.89549
$582$		0			0
$583$	221.567			0.380046
$584$		0			0
$585$		0			0
$586$		0			0
$587$	830.035			1.41403			0.707014	−	0.707199i	$-0.250042\pi$
$587$	830.035			1.41403			0.707014	+	0.707199i	$0.250042\pi$
$588$		0			0
$589$			113.352i			0.192449i
$590$		0			0
$591$	−1176.51			−1.99072
$592$		0			0
$593$	251.455			0.424038			0.212019	−	0.977266i	$-0.431996\pi$
$593$	251.455			0.424038			0.212019	+	0.977266i	$0.431996\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			957.401i			1.60369i
$598$		0			0
$599$	149.525			0.249624			0.124812	−	0.992180i	$-0.460167\pi$
$599$	149.525			0.249624			0.124812	+	0.992180i	$0.460167\pi$
$600$		0			0
$601$	−1057.76			−1.76000			−0.880000	−	0.474975i	$-0.842457\pi$
$601$	−1057.76			−1.76000			−0.880000	+	0.474975i	$0.842457\pi$
$602$		0			0
$603$			1328.81i			2.20367i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	1117.60			1.84119			0.920593	−	0.390524i	$-0.127706\pi$
$607$	1117.60			1.84119			0.920593	+	0.390524i	$0.127706\pi$
$608$		0			0
$609$		−	1554.04i		−	2.55179i
$610$		0			0
$611$	691.428			1.13163
$612$		0			0
$613$		−	593.931i		−	0.968892i	−0.874821	−	0.484446i	$-0.839021\pi$
$613$		−	593.931i		−	0.968892i	0.874821	−	0.484446i	$-0.160979\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	415.912i		−	0.674087i	−0.941489	−	0.337043i	$-0.890573\pi$
$617$		−	415.912i		−	0.674087i	0.941489	−	0.337043i	$-0.109427\pi$
$618$		0			0
$619$		−	629.480i		−	1.01693i	−0.861082	−	0.508465i	$-0.830213\pi$
$619$		−	629.480i		−	1.01693i	0.861082	−	0.508465i	$-0.169787\pi$
$620$		0			0
$621$	−639.482	+	404.099i	−1.02976	+	0.650722i
$622$		0			0
$623$	−765.694			−1.22904
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−997.844			−1.59146
$628$		0			0
$629$	348.416			0.553920
$630$		0			0
$631$		−	851.765i		−	1.34986i	−0.737880	−	0.674932i	$-0.764173\pi$
$631$		−	851.765i		−	1.34986i	0.737880	−	0.674932i	$-0.235827\pi$
$632$		0			0
$633$	1240.76			1.96012
$634$		0			0
$635$		0			0
$636$		0			0
$637$	3.14131			0.00493141
$638$		0			0
$639$	−226.481			−0.354430
$640$		0			0
$641$		−	1062.56i		−	1.65765i	−0.559506	−	0.828826i	$-0.689009\pi$
$641$		−	1062.56i		−	1.65765i	0.559506	−	0.828826i	$-0.310991\pi$
$642$		0			0
$643$			225.914i			0.351343i	0.984449	+	0.175672i	$0.0562097\pi$
$643$			225.914i			0.351343i	−0.984449	+	0.175672i	$0.943790\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−169.983			−0.262725			−0.131363	−	0.991334i	$-0.541935\pi$
$647$	−169.983			−0.262725			−0.131363	+	0.991334i	$0.541935\pi$
$648$		0			0
$649$			305.429i			0.470615i
$650$		0			0
$651$		−	200.797i		−	0.308444i
$652$		0			0
$653$	−551.093			−0.843941			−0.421970	−	0.906610i	$-0.638661\pi$
$653$	−551.093			−0.843941			−0.421970	+	0.906610i	$0.638661\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	29.6686			0.0451577
$658$		0			0
$659$			363.865i			0.552147i	0.961137	+	0.276073i	$0.0890333\pi$
$659$			363.865i			0.552147i	−0.961137	+	0.276073i	$0.910967\pi$
$660$		0			0
$661$		−	920.556i		−	1.39267i	−0.717716	−	0.696336i	$-0.754813\pi$
$661$		−	920.556i		−	1.39267i	0.717716	−	0.696336i	$-0.245187\pi$
$662$		0			0
$663$			1355.08i			2.04386i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−872.581	+	551.398i	−1.30822	+	0.826683i
$668$		0			0
$669$	−1130.00			−1.68908
$670$		0			0
$671$	718.587			1.07092
$672$		0			0
$673$	−919.345			−1.36604			−0.683020	−	0.730400i	$-0.739334\pi$
$673$	−919.345			−1.36604			−0.683020	+	0.730400i	$0.739334\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	46.6213i		−	0.0688646i	−0.999407	−	0.0344323i	$-0.989038\pi$
$677$		−	46.6213i		−	0.0688646i	0.999407	−	0.0344323i	$-0.0109623\pi$
$678$		0			0
$679$	1216.45			1.79153
$680$		0			0
$681$		−	1406.78i		−	2.06576i
$682$		0			0
$683$	601.428			0.880568			0.440284	−	0.897859i	$-0.354878\pi$
$683$	601.428			0.880568			0.440284	+	0.897859i	$0.354878\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$			1697.16i			2.47040i
$688$		0			0
$689$		−	217.693i		−	0.315954i
$690$		0			0
$691$	510.417			0.738664			0.369332	−	0.929297i	$-0.379586\pi$
$691$	510.417			0.738664			0.369332	+	0.929297i	$0.379586\pi$
$692$		0			0
$693$	1121.65			1.61855
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	1130.70i		−	1.62224i
$698$		0			0
$699$	−434.185			−0.621151
$700$		0			0
$701$		−	686.332i		−	0.979075i	−0.871982	−	0.489538i	$-0.837166\pi$
$701$		−	686.332i		−	0.979075i	0.871982	−	0.489538i	$-0.162834\pi$
$702$		0			0
$703$	252.078			0.358575
$704$		0			0
$705$		0			0
$706$		0			0
$707$			128.842i			0.182238i
$708$		0			0
$709$		−	695.760i		−	0.981326i	−0.871350	−	0.490663i	$-0.836755\pi$
$709$		−	695.760i		−	0.981326i	0.871350	−	0.490663i	$-0.163245\pi$
$710$		0			0
$711$			385.359i			0.541996i
$712$		0			0
$713$	−112.746	+	71.2458i	−0.158129	+	0.0999240i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	772.618			1.07757
$718$		0			0
$719$	−842.750			−1.17211			−0.586057	−	0.810270i	$-0.699321\pi$
$719$	−842.750			−1.17211			−0.586057	+	0.810270i	$0.699321\pi$
$720$		0			0
$721$	470.629			0.652744
$722$		0			0
$723$			646.519i			0.894217i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	797.255i		−	1.09664i	−0.836270	−	0.548318i	$-0.815268\pi$
$727$		−	797.255i		−	1.09664i	0.836270	−	0.548318i	$-0.184732\pi$
$728$		0			0
$729$	−1116.24			−1.53119
$730$		0			0
$731$	−1709.20			−2.33817
$732$		0			0
$733$			238.571i			0.325473i	0.986670	+	0.162736i	$0.0520320\pi$
$733$			238.571i			0.325473i	−0.986670	+	0.162736i	$0.947968\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−874.638			−1.18676
$738$		0			0
$739$	−235.936			−0.319264			−0.159632	−	0.987177i	$-0.551031\pi$
$739$	−235.936			−0.319264			−0.159632	+	0.987177i	$0.551031\pi$
$740$		0			0
$741$			980.395i			1.32307i
$742$		0			0
$743$		−	448.670i		−	0.603863i	−0.953330	−	0.301931i	$-0.902369\pi$
$743$		−	448.670i		−	0.603863i	0.953330	−	0.301931i	$-0.0976314\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	2466.44i		−	3.30179i
$748$		0			0
$749$	−601.388			−0.802922
$750$		0			0
$751$		−	325.795i		−	0.433815i	−0.976192	−	0.216907i	$-0.930403\pi$
$751$		−	325.795i		−	0.433815i	0.976192	−	0.216907i	$-0.0695970\pi$
$752$		0			0
$753$		−	1685.60i		−	2.23851i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	554.633i		−	0.732673i	−0.930482	−	0.366336i	$-0.880612\pi$
$757$		−	554.633i		−	0.732673i	0.930482	−	0.366336i	$-0.119388\pi$
$758$		0			0
$759$	−627.180	−	992.505i	−0.826324	−	1.30765i
$760$		0			0
$761$	151.828			0.199511			0.0997553	−	0.995012i	$-0.468194\pi$
$761$	151.828			0.199511			0.0997553	+	0.995012i	$0.468194\pi$
$762$		0			0
$763$	−491.240			−0.643828
$764$		0			0
$765$		0			0
$766$		0			0
$767$	300.088			0.391249
$768$		0			0
$769$			401.826i			0.522530i	0.965267	+	0.261265i	$0.0841397\pi$
$769$			401.826i			0.522530i	−0.965267	+	0.261265i	$0.915860\pi$
$770$		0			0
$771$	2092.77			2.71435
$772$		0			0
$773$		−	815.293i		−	1.05471i	−0.849644	−	0.527357i	$-0.823183\pi$
$773$		−	815.293i		−	1.05471i	0.849644	−	0.527357i	$-0.176817\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−446.541			−0.574699
$778$		0			0
$779$		−	818.058i		−	1.05014i
$780$		0			0
$781$		−	149.072i		−	0.190873i
$782$		0			0
$783$	1476.03			1.88509
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	851.333i		−	1.08175i	−0.841105	−	0.540873i	$-0.818094\pi$
$787$		−	851.333i		−	1.08175i	0.841105	−	0.540873i	$-0.181906\pi$
$788$		0			0
$789$		−	252.198i		−	0.319642i
$790$		0			0
$791$	−1348.36			−1.70463
$792$		0			0
$793$		−	706.021i		−	0.890317i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1341.91i		−	1.68371i	−0.539707	−	0.841853i	$-0.681465\pi$
$797$		−	1341.91i		−	1.68371i	0.539707	−	0.841853i	$-0.318535\pi$
$798$		0			0
$799$			1848.49i			2.31351i
$800$		0			0
$801$		−	1714.86i		−	2.14090i
$802$		0			0
$803$			19.5282i			0.0243190i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−1675.37			−2.07605
$808$		0			0
$809$	−94.3160			−0.116583			−0.0582917	−	0.998300i	$-0.518565\pi$
$809$	−94.3160			−0.116583			−0.0582917	+	0.998300i	$0.518565\pi$
$810$		0			0
$811$	709.408			0.874732			0.437366	−	0.899284i	$-0.355911\pi$
$811$	709.408			0.874732			0.437366	+	0.899284i	$0.355911\pi$
$812$		0			0
$813$	−1563.59			−1.92323
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1236.61			−1.51359
$818$		0			0
$819$		−	1102.04i		−	1.34559i
$820$		0			0
$821$	−1580.42			−1.92499			−0.962495	−	0.271300i	$-0.912546\pi$
$821$	−1580.42			−1.92499			−0.962495	+	0.271300i	$0.912546\pi$
$822$		0			0
$823$	1571.25			1.90918			0.954588	−	0.297928i	$-0.0962955\pi$
$823$	1571.25			1.90918			0.954588	+	0.297928i	$0.0962955\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1208.04i		−	1.46075i	−0.683044	−	0.730377i	$-0.739344\pi$
$827$		−	1208.04i		−	1.46075i	0.683044	−	0.730377i	$-0.260656\pi$
$828$		0			0
$829$	200.190			0.241484			0.120742	−	0.992684i	$-0.461473\pi$
$829$	200.190			0.241484			0.120742	+	0.992684i	$0.461473\pi$
$830$		0			0
$831$	−486.487			−0.585423
$832$		0			0
$833$			8.39811i			0.0100818i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	190.717			0.227858
$838$		0			0
$839$			193.500i			0.230632i	0.993329	+	0.115316i	$0.0367881\pi$
$839$			193.500i			0.230632i	−0.993329	+	0.115316i	$0.963212\pi$
$840$		0			0
$841$	1173.06			1.39484
$842$		0			0
$843$		−	564.321i		−	0.669419i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	106.027i		−	0.125179i
$848$		0			0
$849$			1148.66i			1.35295i
$850$		0			0
$851$	158.440	+	250.729i	0.186181	+	0.294629i
$852$		0			0
$853$	−1363.07			−1.59797			−0.798987	−	0.601349i	$-0.794630\pi$
$853$	−1363.07			−1.59797			−0.798987	+	0.601349i	$0.794630\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1402.94			−1.63704			−0.818518	−	0.574481i	$-0.805204\pi$
$857$	−1402.94			−1.63704			−0.818518	+	0.574481i	$0.805204\pi$
$858$		0			0
$859$	517.579			0.602536			0.301268	−	0.953539i	$-0.402590\pi$
$859$	517.579			0.602536			0.301268	+	0.953539i	$0.402590\pi$
$860$		0			0
$861$			1449.14i			1.68309i
$862$		0			0
$863$	188.469			0.218389			0.109194	−	0.994020i	$-0.465173\pi$
$863$	188.469			0.218389			0.109194	+	0.994020i	$0.465173\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−2188.52			−2.52425
$868$		0			0
$869$	−253.647			−0.291884
$870$		0			0
$871$			859.344i			0.986617i
$872$		0			0
$873$			2724.37i			3.12069i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1581.90			−1.80376			−0.901879	−	0.431988i	$-0.857812\pi$
$877$	−1581.90			−1.80376			−0.901879	+	0.431988i	$0.857812\pi$
$878$		0			0
$879$			2353.01i			2.67692i
$880$		0			0
$881$			570.406i			0.647453i	0.946151	+	0.323727i	$0.104936\pi$
$881$			570.406i			0.647453i	−0.946151	+	0.323727i	$0.895064\pi$
$882$		0			0
$883$	369.107			0.418015			0.209007	−	0.977914i	$-0.432977\pi$
$883$	369.107			0.418015			0.209007	+	0.977914i	$0.432977\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	432.646			0.487763			0.243882	−	0.969805i	$-0.421579\pi$
$887$	432.646			0.487763			0.243882	+	0.969805i	$0.421579\pi$
$888$		0			0
$889$		−	777.506i		−	0.874585i
$890$		0			0
$891$			232.167i			0.260569i
$892$		0			0
$893$			1337.38i			1.49763i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−975.149	+	616.212i	−1.08712	+	0.686970i
$898$		0			0
$899$	260.235			0.289472
$900$		0			0
$901$	581.988			0.645936
$902$		0			0
$903$	2190.57			2.42589
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	406.615i		−	0.448307i	−0.974554	−	0.224154i	$-0.928038\pi$
$907$		−	406.615i		−	0.448307i	0.974554	−	0.224154i	$-0.0719617\pi$
$908$		0			0
$909$	−288.557			−0.317445
$910$		0			0
$911$			1544.38i			1.69525i	0.530593	+	0.847627i	$0.321969\pi$
$911$			1544.38i			1.69525i	−0.530593	+	0.847627i	$0.678031\pi$
$912$		0			0
$913$	1623.43			1.77813
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	155.854i		−	0.169961i
$918$		0			0
$919$		−	949.509i		−	1.03320i	−0.856228	−	0.516599i	$-0.827198\pi$
$919$		−	949.509i		−	1.03320i	0.856228	−	0.516599i	$-0.172802\pi$
$920$		0			0
$921$	−125.677			−0.136457
$922$		0			0
$923$	−146.465			−0.158684
$924$		0			0
$925$		0			0
$926$		0			0
$927$			1054.03i			1.13703i
$928$		0			0
$929$	376.630			0.405414			0.202707	−	0.979239i	$-0.435026\pi$
$929$	376.630			0.405414			0.202707	+	0.979239i	$0.435026\pi$
$930$		0			0
$931$			6.07601i			0.00652633i
$932$		0			0
$933$	−966.817			−1.03625
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	171.692i		−	0.183235i	−0.995794	−	0.0916177i	$-0.970796\pi$
$937$		−	171.692i		−	0.183235i	0.995794	−	0.0916177i	$-0.0292038\pi$
$938$		0			0
$939$		−	484.693i		−	0.516180i
$940$		0			0
$941$		−	1114.50i		−	1.18438i	−0.805799	−	0.592190i	$-0.798264\pi$
$941$		−	1114.50i		−	1.18438i	0.805799	−	0.592190i	$-0.201736\pi$
$942$		0			0
$943$	813.681	−	514.178i	0.862865	−	0.545258i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−193.105			−0.203912			−0.101956	−	0.994789i	$-0.532510\pi$
$947$	−193.105			−0.203912			−0.101956	+	0.994789i	$0.532510\pi$
$948$		0			0
$949$	19.1867			0.0202178
$950$		0			0
$951$	141.983			0.149299
$952$		0			0
$953$		−	1881.46i		−	1.97425i	−0.159951	−	0.987125i	$-0.551134\pi$
$953$		−	1881.46i		−	1.97425i	0.159951	−	0.987125i	$-0.448866\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			2290.86i			2.39380i
$958$		0			0
$959$	2.05802			0.00214600
$960$		0			0
$961$	−927.375			−0.965011
$962$		0			0
$963$		−	1346.88i		−	1.39863i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	405.013			0.418835			0.209417	−	0.977826i	$-0.432843\pi$
$967$	405.013			0.418835			0.209417	+	0.977826i	$0.432843\pi$
$968$		0			0
$969$	−2621.03			−2.70488
$970$		0			0
$971$			50.8734i			0.0523928i	0.999657	+	0.0261964i	$0.00833952\pi$
$971$			50.8734i			0.0523928i	−0.999657	+	0.0261964i	$0.991660\pi$
$972$		0			0
$973$		−	846.503i		−	0.869993i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1663.88i			1.70305i	0.524310	+	0.851527i	$0.324323\pi$
$977$			1663.88i			1.70305i	−0.524310	+	0.851527i	$0.675677\pi$
$978$		0			0
$979$	1128.74			1.15295
$980$		0			0
$981$		−	1100.19i		−	1.12150i
$982$		0			0
$983$			1079.48i			1.09814i	0.835775	+	0.549072i	$0.185019\pi$
$983$			1079.48i			1.09814i	−0.835775	+	0.549072i	$0.814981\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	2369.09i		−	2.40029i
$988$		0			0
$989$	−777.250	−	1229.99i	−0.785895	−	1.24367i
$990$		0			0
$991$	−415.008			−0.418777			−0.209388	−	0.977833i	$-0.567147\pi$
$991$	−415.008			−0.418777			−0.209388	+	0.977833i	$0.567147\pi$
$992$		0			0
$993$	−1927.74			−1.94133
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−782.208			−0.784562			−0.392281	−	0.919845i	$-0.628314\pi$
$997$	−782.208			−0.784562			−0.392281	+	0.919845i	$0.628314\pi$
$998$		0			0
$999$		−	424.125i		−	0.424550i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.3.f.c.1701.15	✓	16
5.2	odd	4		2300.3.d.c.1149.27		32
5.3	odd	4		2300.3.d.c.1149.6		32
5.4	even	2		2300.3.f.d.1701.2	yes	16
23.22	odd	2	inner	2300.3.f.c.1701.16	yes	16
115.22	even	4		2300.3.d.c.1149.5		32
115.68	even	4		2300.3.d.c.1149.28		32
115.114	odd	2		2300.3.f.d.1701.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2300.3.d.c.1149.5		32	115.22	even	4
2300.3.d.c.1149.6		32	5.3	odd	4
2300.3.d.c.1149.27		32	5.2	odd	4
2300.3.d.c.1149.28		32	115.68	even	4
2300.3.f.c.1701.15	✓	16	1.1	even	1	trivial
2300.3.f.c.1701.16	yes	16	23.22	odd	2	inner
2300.3.f.d.1701.1	yes	16	115.114	odd	2
2300.3.f.d.1701.2	yes	16	5.4	even	2

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2300.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.96261 q^{3} -6.97776i q^{7} +15.6275 q^{9} +O(q^{10})\)	\(q+4.96261 q^{3} -6.97776i q^{7} +15.6275 q^{9} +10.2862i q^{11} +10.1063 q^{13} +27.0186i q^{17} +19.5479i q^{19} -34.6279i q^{21} +(-19.4433 + 12.2865i) q^{23} +32.8896 q^{27} +44.8783 q^{29} +5.79870 q^{31} +51.0462i q^{33} -12.8954i q^{37} +50.1535 q^{39} -41.8490 q^{41} +63.2604i q^{43} +68.4156 q^{47} +0.310827 q^{49} +134.083i q^{51} -21.5403i q^{53} +97.0085i q^{57} +29.6932 q^{59} -69.8596i q^{61} -109.045i q^{63} +85.0306i q^{67} +(-96.4894 + 60.9732i) q^{69} -14.4925 q^{71} +1.89849 q^{73} +71.7744 q^{77} +24.6591i q^{79} +22.5708 q^{81} -157.827i q^{83} +222.713 q^{87} -109.733i q^{89} -70.5193i q^{91} +28.7767 q^{93} +174.332i q^{97} +160.747i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 64 q^{9}+O(q^{10}) \) 16 * q + 64 * q^9	\( 16 q + 64 q^{9} - 6 q^{13} + 8 q^{23} - 66 q^{27} - 6 q^{29} + 28 q^{31} + 74 q^{39} - 90 q^{41} - 40 q^{47} - 190 q^{49} - 174 q^{59} + 50 q^{69} + 116 q^{71} - 110 q^{73} - 198 q^{77} + 56 q^{81} + 362 q^{87} + 50 q^{93}+O(q^{100}) \) 16 * q + 64 * q^9 - 6 * q^13 + 8 * q^23 - 66 * q^27 - 6 * q^29 + 28 * q^31 + 74 * q^39 - 90 * q^41 - 40 * q^47 - 190 * q^49 - 174 * q^59 + 50 * q^69 + 116 * q^71 - 110 * q^73 - 198 * q^77 + 56 * q^81 + 362 * q^87 + 50 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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