LMFDB - Embedded newform 2300.3.d.c.1149.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,3,Mod(1149,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2300.1149");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2300.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$62.6704608029$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1149.5
Character	$\chi$	$=$	2300.1149
Dual form			2300.3.d.c.1149.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.96261i q^{3} -6.97776 q^{7} -15.6275 q^{9} +O(q^{10})$	$q-4.96261i q^{3} -6.97776 q^{7} -15.6275 q^{9} -10.2862i q^{11} -10.1063i q^{13} +27.0186 q^{17} +19.5479i q^{19} +34.6279i q^{21} +(-12.2865 + 19.4433i) q^{23} +32.8896i q^{27} -44.8783 q^{29} +5.79870 q^{31} -51.0462 q^{33} -12.8954 q^{37} -50.1535 q^{39} -41.8490 q^{41} -63.2604 q^{43} +68.4156i q^{47} -0.310827 q^{49} -134.083i q^{51} +21.5403 q^{53} +97.0085 q^{57} -29.6932 q^{59} +69.8596i q^{61} +109.045 q^{63} +85.0306 q^{67} +(96.4894 + 60.9732i) q^{69} -14.4925 q^{71} -1.89849i q^{73} +71.7744i q^{77} +24.6591i q^{79} +22.5708 q^{81} +157.827 q^{83} +222.713i q^{87} -109.733i q^{89} +70.5193i q^{91} -28.7767i q^{93} +174.332 q^{97} +160.747i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 128 q^{9}+O(q^{10}) $ 32 * q - 128 * q^9	$ 32 q - 128 q^{9} + 12 q^{29} + 56 q^{31} - 148 q^{39} - 180 q^{41} + 380 q^{49} + 348 q^{59} - 100 q^{69} + 232 q^{71} + 112 q^{81}+O(q^{100}) $ 32 * q - 128 * q^9 + 12 * q^29 + 56 * q^31 - 148 * q^39 - 180 * q^41 + 380 * q^49 + 348 * q^59 - 100 * q^69 + 232 * q^71 + 112 * q^81

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
$3$		−	4.96261i		−	1.65420i	−0.562053	−	0.827101i	$-0.689988\pi$
$3$		−	4.96261i		−	1.65420i	0.562053	−	0.827101i	$-0.310012\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−6.97776			−0.996823			−0.498412	−	0.866941i	$-0.666083\pi$
$7$	−6.97776			−0.996823			−0.498412	+	0.866941i	$0.666083\pi$
$8$		0			0
$9$	−15.6275			−1.73639
$10$		0			0
$11$		−	10.2862i		−	0.935105i	−0.883965	−	0.467553i	$-0.845136\pi$
$11$		−	10.2862i		−	0.935105i	0.883965	−	0.467553i	$-0.154864\pi$
$12$		0			0
$13$		−	10.1063i		−	0.777406i	−0.921363	−	0.388703i	$-0.872923\pi$
$13$		−	10.1063i		−	0.777406i	0.921363	−	0.388703i	$-0.127077\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	27.0186			1.58933			0.794664	−	0.607050i	$-0.207647\pi$
$17$	27.0186			1.58933			0.794664	+	0.607050i	$0.207647\pi$
$18$		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
$23$	−12.2865	+	19.4433i	−0.534197	+	0.845360i
$23$	−12.2865	+	19.4433i	−0.534197	+	0.845360i
$24$		0			0
$25$		0			0
$26$		0			0
$27$			32.8896i			1.21813i
$28$		0			0
$29$	−44.8783			−1.54753			−0.773763	−	0.633475i	$-0.781628\pi$
$29$	−44.8783			−1.54753			−0.773763	+	0.633475i	$0.781628\pi$
$30$		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
$33$	−51.0462			−1.54685
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−12.8954			−0.348525			−0.174262	−	0.984699i	$-0.555754\pi$
$37$	−12.8954			−0.348525			−0.174262	+	0.984699i	$0.555754\pi$
$38$		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
$43$	−63.2604			−1.47117			−0.735586	−	0.677432i	$-0.763093\pi$
$43$	−63.2604			−1.47117			−0.735586	+	0.677432i	$0.763093\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			68.4156i			1.45565i	0.685762	+	0.727826i	$0.259469\pi$
$47$			68.4156i			1.45565i	−0.685762	+	0.727826i	$0.740531\pi$
$48$		0			0
$49$	−0.310827			−0.00634341
$50$		0			0
$51$		−	134.083i		−	2.62907i
$52$		0			0
$53$	21.5403			0.406421			0.203211	−	0.979135i	$-0.434862\pi$
$53$	21.5403			0.406421			0.203211	+	0.979135i	$0.434862\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	97.0085			1.70190
$58$		0			0
$59$	−29.6932			−0.503275			−0.251637	−	0.967822i	$-0.580969\pi$
$59$	−29.6932			−0.503275			−0.251637	+	0.967822i	$0.580969\pi$
$60$		0			0
$61$			69.8596i			1.14524i	0.819821	+	0.572620i	$0.194073\pi$
$61$			69.8596i			1.14524i	−0.819821	+	0.572620i	$0.805927\pi$
$62$		0			0
$63$	109.045			1.73087
$64$		0			0
$65$		0			0
$66$		0			0
$67$	85.0306			1.26911			0.634557	−	0.772876i	$-0.281183\pi$
$67$	85.0306			1.26911			0.634557	+	0.772876i	$0.281183\pi$
$68$		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
$73$		−	1.89849i		−	0.0260067i	−0.999915	−	0.0130034i	$-0.995861\pi$
$73$		−	1.89849i		−	0.0260067i	0.999915	−	0.0130034i	$-0.00413921\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			71.7744i			0.932135i
$78$		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
$83$	157.827			1.90153			0.950765	−	0.309913i	$-0.100300\pi$
$83$	157.827			1.90153			0.950765	+	0.309913i	$0.100300\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			222.713i			2.55992i
$88$		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
$93$		−	28.7767i		−	0.309426i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	174.332			1.79723			0.898617	−	0.438733i	$-0.144573\pi$
$97$	174.332			1.79723			0.898617	+	0.438733i	$0.144573\pi$
$98$		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
$103$	−67.4469			−0.654825			−0.327412	−	0.944882i	$-0.606177\pi$
$103$	−67.4469			−0.654825			−0.327412	+	0.944882i	$0.606177\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−86.1864			−0.805480			−0.402740	−	0.915314i	$-0.631942\pi$
$107$	−86.1864			−0.805480			−0.402740	+	0.915314i	$0.631942\pi$
$108$		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
$113$	193.237			1.71006			0.855029	−	0.518580i	$-0.173539\pi$
$113$	193.237			1.71006			0.855029	+	0.518580i	$0.173539\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			157.936i			1.34988i
$118$		0			0
$119$	−188.529			−1.58428
$120$		0			0
$121$	15.1950			0.125578
$122$		0			0
$123$			207.680i			1.68846i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			111.426i			0.877372i	0.898640	+	0.438686i	$0.144556\pi$
$127$			111.426i			0.877372i	−0.898640	+	0.438686i	$0.855444\pi$
$128$		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
$133$		−	136.400i		−	1.02557i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.294939			0.00215284			0.00107642	−	0.999999i	$-0.499657\pi$
$137$	0.294939			0.00215284			0.00107642	+	0.999999i	$0.499657\pi$
$138$		0			0
$139$	−121.314			−0.872765			−0.436383	−	0.899761i	$-0.643741\pi$
$139$	−121.314			−0.872765			−0.436383	+	0.899761i	$0.643741\pi$
$140$		0			0
$141$	339.520			2.40794
$142$		0			0
$143$	−103.955			−0.726957
$144$		0			0
$145$		0			0
$146$		0			0
$147$			1.54251i			0.0104933i
$148$		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
$153$	−422.232			−2.75969
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−136.384			−0.868687			−0.434343	−	0.900747i	$-0.643020\pi$
$157$	−136.384			−0.868687			−0.434343	+	0.900747i	$0.643020\pi$
$158$		0			0
$159$		−	106.896i		−	0.672303i
$160$		0			0
$161$	85.7324	−	135.671i	0.532500	−	0.842675i
$162$		0			0
$163$			260.321i			1.59706i	0.601954	+	0.798530i	$0.294389\pi$
$163$			260.321i			1.59706i	−0.601954	+	0.798530i	$0.705611\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	179.807i		−	1.07669i	−0.842725	−	0.538345i	$-0.819050\pi$
$167$		−	179.807i		−	1.07669i	0.842725	−	0.538345i	$-0.180950\pi$
$168$		0			0
$169$	66.8630			0.395639
$170$		0			0
$171$		−	305.484i		−	1.78646i
$172$		0			0
$173$			5.23637i			0.0302680i	0.999885	+	0.0151340i	$0.00481749\pi$
$173$			5.23637i			0.0302680i	−0.999885	+	0.0151340i	$0.995183\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			147.356i			0.832518i
$178$		0			0
$179$	−227.005			−1.26819			−0.634093	−	0.773257i	$-0.718626\pi$
$179$	−227.005			−1.26819			−0.634093	+	0.773257i	$0.718626\pi$
$180$		0			0
$181$		−	117.068i		−	0.646786i	−0.946265	−	0.323393i	$-0.895176\pi$
$181$		−	117.068i		−	0.646786i	0.946265	−	0.323393i	$-0.104824\pi$
$182$		0			0
$183$	346.686			1.89446
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	277.917i		−	1.48619i
$188$		0			0
$189$		−	229.496i		−	1.21426i
$190$		0			0
$191$			79.1718i			0.414512i	0.978287	+	0.207256i	$0.0664533\pi$
$191$			79.1718i			0.414512i	−0.978287	+	0.207256i	$0.933547\pi$
$192$		0			0
$193$			29.3684i			0.152168i	0.997101	+	0.0760840i	$0.0242417\pi$
$193$			29.3684i			0.152168i	−0.997101	+	0.0760840i	$0.975758\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	237.076i		−	1.20343i	−0.798711	−	0.601715i	$-0.794484\pi$
$197$		−	237.076i		−	1.20343i	0.798711	−	0.601715i	$-0.205516\pi$
$198$		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
$203$	313.150			1.54261
$204$		0			0
$205$		0			0
$206$		0			0
$207$	192.007	−	303.850i	0.927572	−	1.46787i
$208$		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
$213$			71.9205i			0.337655i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−40.4619			−0.186461
$218$		0			0
$219$	−9.42147			−0.0430204
$220$		0			0
$221$		−	273.057i		−	1.23555i
$222$		0			0
$223$			227.702i			1.02109i	0.859852	+	0.510543i	$0.170556\pi$
$223$			227.702i			1.02109i	−0.859852	+	0.510543i	$0.829444\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−283.476			−1.24879			−0.624396	−	0.781108i	$-0.714655\pi$
$227$	−283.476			−1.24879			−0.624396	+	0.781108i	$0.714655\pi$
$228$		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
$233$			87.4912i			0.375499i	0.982217	+	0.187749i	$0.0601193\pi$
$233$			87.4912i			0.375499i	−0.982217	+	0.187749i	$0.939881\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	122.373			0.516343
$238$		0			0
$239$	−155.688			−0.651414			−0.325707	−	0.945471i	$-0.605602\pi$
$239$	−155.688			−0.651414			−0.325707	+	0.945471i	$0.605602\pi$
$240$		0			0
$241$		−	130.278i		−	0.540573i	−0.962780	−	0.270286i	$-0.912882\pi$
$241$		−	130.278i		−	0.540573i	0.962780	−	0.270286i	$-0.0871184\pi$
$242$		0			0
$243$			183.996i			0.757186i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	197.556			0.799824
$248$		0			0
$249$		−	783.233i		−	3.14552i
$250$		0			0
$251$			339.659i			1.35322i	0.736340	+	0.676612i	$0.236553\pi$
$251$			339.659i			1.35322i	−0.736340	+	0.676612i	$0.763447\pi$
$252$		0			0
$253$	199.997	+	126.381i	0.790501	+	0.499530i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			421.707i			1.64088i	0.571730	+	0.820442i	$0.306272\pi$
$257$			421.707i			1.64088i	−0.571730	+	0.820442i	$0.693728\pi$
$258$		0			0
$259$	89.9812			0.347418
$260$		0			0
$261$	701.334			2.68710
$262$		0			0
$263$	50.8196			0.193231			0.0966153	−	0.995322i	$-0.469198\pi$
$263$	50.8196			0.193231			0.0966153	+	0.995322i	$0.469198\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−544.564			−2.03957
$268$		0			0
$269$	337.599			1.25501			0.627507	−	0.778611i	$-0.284075\pi$
$269$	337.599			1.25501			0.627507	+	0.778611i	$0.284075\pi$
$270$		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
$273$	349.959			1.28190
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	98.0305i		−	0.353901i	−0.984220	−	0.176950i	$-0.943377\pi$
$277$		−	98.0305i		−	0.353901i	0.984220	−	0.176950i	$-0.0566232\pi$
$278$		0			0
$279$	−90.6190			−0.324799
$280$		0			0
$281$			113.715i			0.404678i	0.979316	+	0.202339i	$0.0648543\pi$
$281$			113.715i			0.404678i	−0.979316	+	0.202339i	$0.935146\pi$
$282$		0			0
$283$	−231.462			−0.817887			−0.408944	−	0.912560i	$-0.634103\pi$
$283$	−231.462			−0.817887			−0.408944	+	0.912560i	$0.634103\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	292.012			1.01746
$288$		0			0
$289$	441.003			1.52596
$290$		0			0
$291$		−	865.140i		−	2.97299i
$292$		0			0
$293$	−474.148			−1.61825			−0.809126	−	0.587635i	$-0.800059\pi$
$293$	−474.148			−1.61825			−0.809126	+	0.587635i	$0.800059\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	338.307			1.13908
$298$		0			0
$299$	196.499	+	124.171i	0.657189	+	0.415288i
$300$		0			0
$301$	441.416			1.46650
$302$		0			0
$303$			91.6332i			0.302420i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	25.3248i		−	0.0824914i	−0.999149	−	0.0412457i	$-0.986867\pi$
$307$		−	25.3248i		−	0.0824914i	0.999149	−	0.0412457i	$-0.0131326\pi$
$308$		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
$313$	97.6690			0.312042			0.156021	−	0.987754i	$-0.450133\pi$
$313$	97.6690			0.312042			0.156021	+	0.987754i	$0.450133\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			28.6106i			0.0902542i	0.998981	+	0.0451271i	$0.0143693\pi$
$317$			28.6106i			0.0902542i	−0.998981	+	0.0451271i	$0.985631\pi$
$318$		0			0
$319$			461.625i			1.44710i
$320$		0			0
$321$			427.709i			1.33243i
$322$		0			0
$323$			528.156i			1.63516i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−349.372			−1.06842
$328$		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
$333$	201.523			0.605174
$334$		0			0
$335$		0			0
$336$		0			0
$337$	175.571			0.520981			0.260491	−	0.965476i	$-0.416116\pi$
$337$	175.571			0.520981			0.260491	+	0.965476i	$0.416116\pi$
$338$		0			0
$339$		−	958.957i		−	2.82878i
$340$		0			0
$341$		−	59.6463i		−	0.174916i
$342$		0			0
$343$	344.079			1.00315
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	555.150i		−	1.59986i	−0.600096	−	0.799928i	$-0.704871\pi$
$347$		−	555.150i		−	1.59986i	0.600096	−	0.799928i	$-0.295129\pi$
$348$		0			0
$349$	−509.836			−1.46085			−0.730424	−	0.682994i	$-0.760678\pi$
$349$	−509.836			−1.46085			−0.730424	+	0.682994i	$0.760678\pi$
$350$		0			0
$351$	332.392			0.946984
$352$		0			0
$353$			325.703i			0.922670i	0.887226	+	0.461335i	$0.152629\pi$
$353$			325.703i			0.922670i	−0.887226	+	0.461335i	$0.847371\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			935.596i			2.62072i
$358$		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
$363$		−	75.4066i		−	0.207732i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−727.546			−1.98241			−0.991207	−	0.132320i	$-0.957757\pi$
$367$	−727.546			−1.98241			−0.991207	+	0.132320i	$0.957757\pi$
$368$		0			0
$369$	653.994			1.77234
$370$		0			0
$371$	−150.303			−0.405130
$372$		0			0
$373$	175.303			0.469980			0.234990	−	0.971998i	$-0.424494\pi$
$373$	175.303			0.469980			0.234990	+	0.971998i	$0.424494\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			453.552i			1.20306i
$378$		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
$383$	−300.247			−0.783933			−0.391967	−	0.919979i	$-0.628205\pi$
$383$	−300.247			−0.783933			−0.391967	+	0.919979i	$0.628205\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	988.600			2.55452
$388$		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
$393$		−	110.844i		−	0.282046i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	691.800i		−	1.74257i	−0.490777	−	0.871285i	$-0.663287\pi$
$397$		−	691.800i		−	1.74257i	0.490777	−	0.871285i	$-0.336713\pi$
$398$		0			0
$399$	−676.902			−1.69650
$400$		0			0
$401$		−	262.510i		−	0.654639i	−0.944914	−	0.327320i	$-0.893855\pi$
$401$		−	262.510i		−	0.654639i	0.944914	−	0.327320i	$-0.106145\pi$
$402$		0			0
$403$		−	58.6033i		−	0.145418i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			132.644i			0.325907i
$408$		0			0
$409$	−63.4146			−0.155048			−0.0775240	−	0.996990i	$-0.524701\pi$
$409$	−63.4146			−0.155048			−0.0775240	+	0.996990i	$0.524701\pi$
$410$		0			0
$411$		−	1.46367i		−	0.00356124i
$412$		0			0
$413$	207.192			0.501676
$414$		0			0
$415$		0			0
$416$		0			0
$417$			602.036i			1.44373i
$418$		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	$-0.796991\pi$
$421$		−	501.333i		−	1.19081i	0.803424	−	0.595407i	$-0.203009\pi$
$422$		0			0
$423$		−	1069.16i		−	2.52757i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	487.464i		−	1.14160i
$428$		0			0
$429$			515.887i			1.20253i
$430$		0			0
$431$			534.282i			1.23963i	0.784747	+	0.619816i	$0.212793\pi$
$431$			534.282i			1.23963i	−0.784747	+	0.619816i	$0.787207\pi$
$432$		0			0
$433$	346.798			0.800920			0.400460	−	0.916314i	$-0.368850\pi$
$433$	346.798			0.800920			0.400460	+	0.916314i	$0.368850\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−380.075	−	240.175i	−0.869737	−	0.549601i
$438$		0			0
$439$	−552.167			−1.25778			−0.628892	−	0.777492i	$-0.716491\pi$
$439$	−552.167			−1.25778			−0.628892	+	0.777492i	$0.716491\pi$
$440$		0			0
$441$	4.85745			0.0110146
$442$		0			0
$443$		−	550.487i		−	1.24263i	−0.783559	−	0.621317i	$-0.786598\pi$
$443$		−	550.487i		−	1.24263i	0.783559	−	0.621317i	$-0.213402\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−401.320			−0.897807
$448$		0			0
$449$	−504.192			−1.12292			−0.561462	−	0.827503i	$-0.689761\pi$
$449$	−504.192			−1.12292			−0.561462	+	0.827503i	$0.689761\pi$
$450$		0			0
$451$			430.465i			0.954468i
$452$		0			0
$453$		−	1145.43i		−	2.52855i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	19.5978			0.0428836			0.0214418	−	0.999770i	$-0.493174\pi$
$457$	19.5978			0.0428836			0.0214418	+	0.999770i	$0.493174\pi$
$458$		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
$463$			308.904i			0.667180i	0.942718	+	0.333590i	$0.108260\pi$
$463$			308.904i			0.667180i	−0.942718	+	0.333590i	$0.891740\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	417.558			0.894128			0.447064	−	0.894502i	$-0.352470\pi$
$467$	417.558			0.894128			0.447064	+	0.894502i	$0.352470\pi$
$468$		0			0
$469$	−593.324			−1.26508
$470$		0			0
$471$			676.819i			1.43698i
$472$		0			0
$473$			650.706i			1.37570i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−336.621			−0.705704
$478$		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
$483$	−673.280	−	425.456i	−1.39395	−	0.880862i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			834.672i			1.71391i	0.515395	+	0.856953i	$0.327645\pi$
$487$			834.672i			1.71391i	−0.515395	+	0.856953i	$0.672355\pi$
$488$		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
$493$	−1212.55			−2.45953
$494$		0			0
$495$		0			0
$496$		0			0
$497$	101.125			0.203471
$498$		0			0
$499$	−718.540			−1.43996			−0.719980	−	0.693994i	$-0.755849\pi$
$499$	−718.540			−1.43996			−0.719980	+	0.693994i	$0.755849\pi$
$500$		0			0
$501$	−892.312			−1.78106
$502$		0			0
$503$	490.987			0.976117			0.488059	−	0.872811i	$-0.337705\pi$
$503$	490.987			0.976117			0.488059	+	0.872811i	$0.337705\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	331.815i		−	0.654467i
$508$		0			0
$509$	114.915			0.225765			0.112883	−	0.993608i	$-0.463992\pi$
$509$	114.915			0.225765			0.112883	+	0.993608i	$0.463992\pi$
$510$		0			0
$511$			13.2472i			0.0259241i
$512$		0			0
$513$	−642.922			−1.25326
$514$		0			0
$515$		0			0
$516$		0			0
$517$	703.734			1.36119
$518$		0			0
$519$	25.9861			0.0500695
$520$		0			0
$521$			479.276i			0.919916i	0.887941	+	0.459958i	$0.152135\pi$
$521$			479.276i			0.919916i	−0.887941	+	0.459958i	$0.847865\pi$
$522$		0			0
$523$	413.778			0.791162			0.395581	−	0.918431i	$-0.370543\pi$
$523$	413.778			0.791162			0.395581	+	0.918431i	$0.370543\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	156.672			0.297291
$528$		0			0
$529$	−227.083	−	477.781i	−0.429268	−	0.903177i
$530$		0			0
$531$	464.030			0.873879
$532$		0			0
$533$			422.937i			0.793504i
$534$		0			0
$535$		0			0
$536$		0			0
$537$			1126.54i			2.09784i
$538$		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
$543$	−580.964			−1.06992
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	464.746i		−	0.849627i	−0.905281	−	0.424813i	$-0.860340\pi$
$547$		−	464.746i		−	0.849627i	0.905281	−	0.424813i	$-0.139660\pi$
$548$		0			0
$549$		−	1091.73i		−	1.98858i
$550$		0			0
$551$		−	877.275i		−	1.59215i
$552$		0			0
$553$		−	172.065i		−	0.311148i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	383.306			0.688161			0.344080	−	0.938940i	$-0.388191\pi$
$557$	383.306			0.688161			0.344080	+	0.938940i	$0.388191\pi$
$558$		0			0
$559$			639.327i			1.14370i
$560$		0			0
$561$	−1379.19			−2.45846
$562$		0			0
$563$	441.290			0.783819			0.391910	−	0.920004i	$-0.371815\pi$
$563$	441.290			0.783819			0.391910	+	0.920004i	$0.371815\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−157.494			−0.277767
$568$		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	$-0.681855\pi$
$571$		−	960.641i		−	1.68238i	0.540737	−	0.841192i	$-0.318145\pi$
$572$		0			0
$573$	392.898			0.685687
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	539.287i		−	0.934639i	−0.884089	−	0.467319i	$-0.845220\pi$
$577$		−	539.287i		−	0.934639i	0.884089	−	0.467319i	$-0.154780\pi$
$578$		0			0
$579$	145.744			0.251717
$580$		0			0
$581$	−1101.28			−1.89549
$582$		0			0
$583$		−	221.567i		−	0.380046i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			830.035i			1.41403i	0.707199	+	0.707014i	$0.249958\pi$
$587$			830.035i			1.41403i	−0.707199	+	0.707014i	$0.750042\pi$
$588$		0			0
$589$			113.352i			0.192449i
$590$		0			0
$591$	−1176.51			−1.99072
$592$		0			0
$593$		−	251.455i		−	0.424038i	−0.977266	−	0.212019i	$-0.931996\pi$
$593$		−	251.455i		−	0.424038i	0.977266	−	0.212019i	$-0.0680039\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	957.401			1.60369
$598$		0			0
$599$	−149.525			−0.249624			−0.124812	−	0.992180i	$-0.539833\pi$
$599$	−149.525			−0.249624			−0.124812	+	0.992180i	$0.539833\pi$
$600$		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
$603$	−1328.81			−2.20367
$604$		0			0
$605$		0			0
$606$		0			0
$607$			1117.60i			1.84119i	0.390524	+	0.920593i	$0.372294\pi$
$607$			1117.60i			1.84119i	−0.390524	+	0.920593i	$0.627706\pi$
$608$		0			0
$609$		−	1554.04i		−	2.55179i
$610$		0			0
$611$	691.428			1.13163
$612$		0			0
$613$	593.931			0.968892			0.484446	−	0.874821i	$-0.339021\pi$
$613$	593.931			0.968892			0.484446	+	0.874821i	$0.339021\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−415.912			−0.674087			−0.337043	−	0.941489i	$-0.609427\pi$
$617$	−415.912			−0.674087			−0.337043	+	0.941489i	$0.609427\pi$
$618$		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
$623$			765.694i			1.22904i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		−	997.844i		−	1.59146i
$628$		0			0
$629$	−348.416			−0.553920
$630$		0			0
$631$			851.765i			1.34986i	0.737880	+	0.674932i	$0.235827\pi$
$631$			851.765i			1.34986i	−0.737880	+	0.674932i	$0.764173\pi$
$632$		0			0
$633$		−	1240.76i		−	1.96012i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			3.14131i			0.00493141i
$638$		0			0
$639$	226.481			0.354430
$640$		0			0
$641$			1062.56i			1.65765i	0.559506	+	0.828826i	$0.310991\pi$
$641$			1062.56i			1.65765i	−0.559506	+	0.828826i	$0.689009\pi$
$642$		0			0
$643$	−225.914			−0.351343			−0.175672	−	0.984449i	$-0.556210\pi$
$643$	−225.914			−0.351343			−0.175672	+	0.984449i	$0.556210\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	169.983i		−	0.262725i	−0.991334	−	0.131363i	$-0.958065\pi$
$647$		−	169.983i		−	0.262725i	0.991334	−	0.131363i	$-0.0419352\pi$
$648$		0			0
$649$			305.429i			0.470615i
$650$		0			0
$651$			200.797i			0.308444i
$652$		0			0
$653$			551.093i			0.843941i	0.906610	+	0.421970i	$0.138661\pi$
$653$			551.093i			0.843941i	−0.906610	+	0.421970i	$0.861339\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			29.6686i			0.0451577i
$658$		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	$0.245187\pi$
$661$			920.556i			1.39267i	−0.717716	+	0.696336i	$0.754813\pi$
$662$		0			0
$663$	−1355.08			−2.04386
$664$		0			0
$665$		0			0
$666$		0			0
$667$	551.398	−	872.581i	0.826683	−	1.30822i
$668$		0			0
$669$	1130.00			1.68908
$670$		0			0
$671$	718.587			1.07092
$672$		0			0
$673$			919.345i			1.36604i	0.730400	+	0.683020i	$0.239334\pi$
$673$			919.345i			1.36604i	−0.730400	+	0.683020i	$0.760666\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−46.6213			−0.0688646			−0.0344323	−	0.999407i	$-0.510962\pi$
$677$	−46.6213			−0.0688646			−0.0344323	+	0.999407i	$0.510962\pi$
$678$		0			0
$679$	−1216.45			−1.79153
$680$		0			0
$681$			1406.78i			2.06576i
$682$		0			0
$683$		−	601.428i		−	0.880568i	−0.897859	−	0.440284i	$-0.854878\pi$
$683$		−	601.428i		−	0.880568i	0.897859	−	0.440284i	$-0.145122\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1697.16			2.47040
$688$		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
$693$		−	1121.65i		−	1.61855i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1130.70			−1.62224
$698$		0			0
$699$	434.185			0.621151
$700$		0			0
$701$			686.332i			0.979075i	0.871982	+	0.489538i	$0.162834\pi$
$701$			686.332i			0.979075i	−0.871982	+	0.489538i	$0.837166\pi$
$702$		0			0
$703$		−	252.078i		−	0.358575i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	128.842			0.182238
$708$		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
$713$	−71.2458	+	112.746i	−0.0999240	+	0.158129i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			772.618i			1.07757i
$718$		0			0
$719$	842.750			1.17211			0.586057	−	0.810270i	$-0.300679\pi$
$719$	842.750			1.17211			0.586057	+	0.810270i	$0.300679\pi$
$720$		0			0
$721$	470.629			0.652744
$722$		0			0
$723$	−646.519			−0.894217
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−797.255			−1.09664			−0.548318	−	0.836270i	$-0.684732\pi$
$727$	−797.255			−1.09664			−0.548318	+	0.836270i	$0.684732\pi$
$728$		0			0
$729$	1116.24			1.53119
$730$		0			0
$731$	−1709.20			−2.33817
$732$		0			0
$733$	−238.571			−0.325473			−0.162736	−	0.986670i	$-0.552032\pi$
$733$	−238.571			−0.325473			−0.162736	+	0.986670i	$0.552032\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	874.638i		−	1.18676i
$738$		0			0
$739$	235.936			0.319264			0.159632	−	0.987177i	$-0.448969\pi$
$739$	235.936			0.319264			0.159632	+	0.987177i	$0.448969\pi$
$740$		0			0
$741$		−	980.395i		−	1.32307i
$742$		0			0
$743$	448.670			0.603863			0.301931	−	0.953330i	$-0.402369\pi$
$743$	448.670			0.603863			0.301931	+	0.953330i	$0.402369\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−2466.44			−3.30179
$748$		0			0
$749$	601.388			0.802922
$750$		0			0
$751$			325.795i			0.433815i	0.976192	+	0.216907i	$0.0695970\pi$
$751$			325.795i			0.433815i	−0.976192	+	0.216907i	$0.930403\pi$
$752$		0			0
$753$	1685.60			2.23851
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−554.633			−0.732673			−0.366336	−	0.930482i	$-0.619388\pi$
$757$	−554.633			−0.732673			−0.366336	+	0.930482i	$0.619388\pi$
$758$		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
$763$			491.240i			0.643828i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			300.088i			0.391249i
$768$		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
$773$	815.293			1.05471			0.527357	−	0.849644i	$-0.323183\pi$
$773$	815.293			1.05471			0.527357	+	0.849644i	$0.323183\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		−	446.541i		−	0.574699i
$778$		0			0
$779$		−	818.058i		−	1.05014i
$780$		0			0
$781$			149.072i			0.190873i
$782$		0			0
$783$		−	1476.03i		−	1.88509i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−851.333			−1.08175			−0.540873	−	0.841105i	$-0.681906\pi$
$787$	−851.333			−1.08175			−0.540873	+	0.841105i	$0.681906\pi$
$788$		0			0
$789$		−	252.198i		−	0.319642i
$790$		0			0
$791$	−1348.36			−1.70463
$792$		0			0
$793$	706.021			0.890317
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1341.91			−1.68371			−0.841853	−	0.539707i	$-0.818535\pi$
$797$	−1341.91			−1.68371			−0.841853	+	0.539707i	$0.818535\pi$
$798$		0			0
$799$			1848.49i			2.31351i
$800$		0			0
$801$			1714.86i			2.14090i
$802$		0			0
$803$	−19.5282			−0.0243190
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	1675.37i		−	2.07605i
$808$		0			0
$809$	94.3160			0.116583			0.0582917	−	0.998300i	$-0.481435\pi$
$809$	94.3160			0.116583			0.0582917	+	0.998300i	$0.481435\pi$
$810$		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
$813$			1563.59i			1.92323i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	1236.61i		−	1.51359i
$818$		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
$823$		−	1571.25i		−	1.90918i	−0.297928	−	0.954588i	$-0.596295\pi$
$823$		−	1571.25i		−	1.90918i	0.297928	−	0.954588i	$-0.403705\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1208.04			−1.46075			−0.730377	−	0.683044i	$-0.760656\pi$
$827$	−1208.04			−1.46075			−0.730377	+	0.683044i	$0.760656\pi$
$828$		0			0
$829$	−200.190			−0.241484			−0.120742	−	0.992684i	$-0.538527\pi$
$829$	−200.190			−0.241484			−0.120742	+	0.992684i	$0.538527\pi$
$830$		0			0
$831$	−486.487			−0.585423
$832$		0			0
$833$	−8.39811			−0.0100818
$834$		0			0
$835$		0			0
$836$		0			0
$837$			190.717i			0.227858i
$838$		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
$843$	564.321			0.669419
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−106.027			−0.125179
$848$		0			0
$849$			1148.66i			1.35295i
$850$		0			0
$851$	158.440	−	250.729i	0.186181	−	0.294629i
$852$		0			0
$853$			1363.07i			1.59797i	0.601349	+	0.798987i	$0.294630\pi$
$853$			1363.07i			1.59797i	−0.601349	+	0.798987i	$0.705370\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1402.94i		−	1.63704i	−0.574481	−	0.818518i	$-0.694796\pi$
$857$		−	1402.94i		−	1.63704i	0.574481	−	0.818518i	$-0.305204\pi$
$858$		0			0
$859$	−517.579			−0.602536			−0.301268	−	0.953539i	$-0.597410\pi$
$859$	−517.579			−0.602536			−0.301268	+	0.953539i	$0.597410\pi$
$860$		0			0
$861$		−	1449.14i		−	1.68309i
$862$		0			0
$863$		−	188.469i		−	0.218389i	−0.994020	−	0.109194i	$-0.965173\pi$
$863$		−	188.469i		−	0.218389i	0.994020	−	0.109194i	$-0.0348271\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	2188.52i		−	2.52425i
$868$		0			0
$869$	253.647			0.291884
$870$		0			0
$871$		−	859.344i		−	0.986617i
$872$		0			0
$873$	−2724.37			−3.12069
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	1581.90i		−	1.80376i	−0.431988	−	0.901879i	$-0.642188\pi$
$877$		−	1581.90i		−	1.80376i	0.431988	−	0.901879i	$-0.357812\pi$
$878$		0			0
$879$			2353.01i			2.67692i
$880$		0			0
$881$		−	570.406i		−	0.647453i	−0.946151	−	0.323727i	$-0.895064\pi$
$881$		−	570.406i		−	0.647453i	0.946151	−	0.323727i	$-0.104936\pi$
$882$		0			0
$883$		−	369.107i		−	0.418015i	−0.977914	−	0.209007i	$-0.932977\pi$
$883$		−	369.107i		−	0.418015i	0.977914	−	0.209007i	$-0.0670232\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			432.646i			0.487763i	0.969805	+	0.243882i	$0.0784208\pi$
$887$			432.646i			0.487763i	−0.969805	+	0.243882i	$0.921579\pi$
$888$		0			0
$889$		−	777.506i		−	0.874585i
$890$		0			0
$891$		−	232.167i		−	0.260569i
$892$		0			0
$893$	−1337.38			−1.49763
$894$		0			0
$895$		0			0
$896$		0			0
$897$	616.212	−	975.149i	0.686970	−	1.08712i
$898$		0			0
$899$	−260.235			−0.289472
$900$		0			0
$901$	581.988			0.645936
$902$		0			0
$903$		−	2190.57i		−	2.42589i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−406.615			−0.448307			−0.224154	−	0.974554i	$-0.571962\pi$
$907$	−406.615			−0.448307			−0.224154	+	0.974554i	$0.571962\pi$
$908$		0			0
$909$	288.557			0.317445
$910$		0			0
$911$		−	1544.38i		−	1.69525i	−0.530593	−	0.847627i	$-0.678031\pi$
$911$		−	1544.38i		−	1.69525i	0.530593	−	0.847627i	$-0.321969\pi$
$912$		0			0
$913$		−	1623.43i		−	1.77813i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−155.854			−0.169961
$918$		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
$923$			146.465i			0.158684i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	1054.03			1.13703
$928$		0			0
$929$	−376.630			−0.405414			−0.202707	−	0.979239i	$-0.564974\pi$
$929$	−376.630			−0.405414			−0.202707	+	0.979239i	$0.564974\pi$
$930$		0			0
$931$		−	6.07601i		−	0.00652633i
$932$		0			0
$933$			966.817i			1.03625i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−171.692			−0.183235			−0.0916177	−	0.995794i	$-0.529204\pi$
$937$	−171.692			−0.183235			−0.0916177	+	0.995794i	$0.529204\pi$
$938$		0			0
$939$		−	484.693i		−	0.516180i
$940$		0			0
$941$			1114.50i			1.18438i	0.805799	+	0.592190i	$0.201736\pi$
$941$			1114.50i			1.18438i	−0.805799	+	0.592190i	$0.798264\pi$
$942$		0			0
$943$	514.178	−	813.681i	0.545258	−	0.862865i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	193.105i		−	0.203912i	−0.994789	−	0.101956i	$-0.967490\pi$
$947$		−	193.105i		−	0.203912i	0.994789	−	0.101956i	$-0.0325101\pi$
$948$		0			0
$949$	−19.1867			−0.0202178
$950$		0			0
$951$	141.983			0.149299
$952$		0			0
$953$	1881.46			1.97425			0.987125	−	0.159951i	$-0.0511336\pi$
$953$	1881.46			1.97425			0.987125	+	0.159951i	$0.0511336\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	2290.86			2.39380
$958$		0			0
$959$	−2.05802			−0.00214600
$960$		0			0
$961$	−927.375			−0.965011
$962$		0			0
$963$	1346.88			1.39863
$964$		0			0
$965$		0			0
$966$		0			0
$967$			405.013i			0.418835i	0.977826	+	0.209417i	$0.0671567\pi$
$967$			405.013i			0.418835i	−0.977826	+	0.209417i	$0.932843\pi$
$968$		0			0
$969$	2621.03			2.70488
$970$		0			0
$971$		−	50.8734i		−	0.0523928i	−0.999657	−	0.0261964i	$-0.991660\pi$
$971$		−	50.8734i		−	0.0523928i	0.999657	−	0.0261964i	$-0.00833952\pi$
$972$		0			0
$973$	846.503			0.869993
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1663.88			1.70305			0.851527	−	0.524310i	$-0.175677\pi$
$977$	1663.88			1.70305			0.851527	+	0.524310i	$0.175677\pi$
$978$		0			0
$979$	−1128.74			−1.15295
$980$		0			0
$981$			1100.19i			1.12150i
$982$		0			0
$983$	−1079.48			−1.09814			−0.549072	−	0.835775i	$-0.685019\pi$
$983$	−1079.48			−1.09814			−0.549072	+	0.835775i	$0.685019\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−2369.09			−2.40029
$988$		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
$993$			1927.74i			1.94133i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	782.208i		−	0.784562i	−0.919845	−	0.392281i	$-0.871686\pi$
$997$		−	782.208i		−	0.784562i	0.919845	−	0.392281i	$-0.128314\pi$
$998$		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.d.c.1149.5		32
5.2	odd	4		2300.3.f.d.1701.1	yes	16
5.3	odd	4		2300.3.f.c.1701.16	yes	16
5.4	even	2	inner	2300.3.d.c.1149.28		32
23.22	odd	2	inner	2300.3.d.c.1149.27		32
115.22	even	4		2300.3.f.d.1701.2	yes	16
115.68	even	4		2300.3.f.c.1701.15	✓	16
115.114	odd	2	inner	2300.3.d.c.1149.6		32

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2300.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.96261i q^{3} -6.97776 q^{7} -15.6275 q^{9} +O(q^{10})\)	\(q-4.96261i q^{3} -6.97776 q^{7} -15.6275 q^{9} -10.2862i q^{11} -10.1063i q^{13} +27.0186 q^{17} +19.5479i q^{19} +34.6279i q^{21} +(-12.2865 + 19.4433i) q^{23} +32.8896i q^{27} -44.8783 q^{29} +5.79870 q^{31} -51.0462 q^{33} -12.8954 q^{37} -50.1535 q^{39} -41.8490 q^{41} -63.2604 q^{43} +68.4156i q^{47} -0.310827 q^{49} -134.083i q^{51} +21.5403 q^{53} +97.0085 q^{57} -29.6932 q^{59} +69.8596i q^{61} +109.045 q^{63} +85.0306 q^{67} +(96.4894 + 60.9732i) q^{69} -14.4925 q^{71} -1.89849i q^{73} +71.7744i q^{77} +24.6591i q^{79} +22.5708 q^{81} +157.827 q^{83} +222.713i q^{87} -109.733i q^{89} +70.5193i q^{91} -28.7767i q^{93} +174.332 q^{97} +160.747i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 128 q^{9}+O(q^{10}) \) 32 * q - 128 * q^9	\( 32 q - 128 q^{9} + 12 q^{29} + 56 q^{31} - 148 q^{39} - 180 q^{41} + 380 q^{49} + 348 q^{59} - 100 q^{69} + 232 q^{71} + 112 q^{81}+O(q^{100}) \) 32 * q - 128 * q^9 + 12 * q^29 + 56 * q^31 - 148 * q^39 - 180 * q^41 + 380 * q^49 + 348 * q^59 - 100 * q^69 + 232 * q^71 + 112 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more