LMFDB - Embedded newform 2300.3.d.c.1149.11

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			1149.11
Character	$\chi$	$=$	2300.1149
Dual form			2300.3.d.c.1149.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.285226i q^{3} -5.85973 q^{7} +8.91865 q^{9} +O(q^{10})$	$q-0.285226i q^{3} -5.85973 q^{7} +8.91865 q^{9} +4.75053i q^{11} -11.4610i q^{13} +4.16740 q^{17} +21.4417i q^{19} +1.67135i q^{21} +(-13.1441 - 18.8741i) q^{23} -5.11087i q^{27} -4.76202 q^{29} -2.69584 q^{31} +1.35498 q^{33} +49.8033 q^{37} -3.26899 q^{39} +18.6032 q^{41} +15.1515 q^{43} -18.2985i q^{47} -14.6636 q^{49} -1.18865i q^{51} +29.2038 q^{53} +6.11572 q^{57} -38.0270 q^{59} +77.5848i q^{61} -52.2608 q^{63} -48.4591 q^{67} +(-5.38339 + 3.74905i) q^{69} +13.9756 q^{71} -15.2098i q^{73} -27.8368i q^{77} -111.294i q^{79} +78.8101 q^{81} +52.6827 q^{83} +1.35825i q^{87} -52.6530i q^{89} +67.1586i q^{91} +0.768923i q^{93} +37.6780 q^{97} +42.3683i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	0.285226i		−	0.0950754i	−0.998869	−	0.0475377i	$-0.984863\pi$
$3$		−	0.285226i		−	0.0950754i	0.998869	−	0.0475377i	$-0.0151374\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−5.85973			−0.837104			−0.418552	−	0.908193i	$-0.637462\pi$
$7$	−5.85973			−0.837104			−0.418552	+	0.908193i	$0.637462\pi$
$8$		0			0
$9$	8.91865			0.990961
$10$		0			0
$11$			4.75053i			0.431867i	0.976408	+	0.215933i	$0.0692794\pi$
$11$			4.75053i			0.431867i	−0.976408	+	0.215933i	$0.930721\pi$
$12$		0			0
$13$		−	11.4610i		−	0.881619i	−0.897601	−	0.440810i	$-0.854691\pi$
$13$		−	11.4610i		−	0.881619i	0.897601	−	0.440810i	$-0.145309\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.16740			0.245141			0.122571	−	0.992460i	$-0.460886\pi$
$17$	4.16740			0.245141			0.122571	+	0.992460i	$0.460886\pi$
$18$		0			0
$19$			21.4417i			1.12851i	0.825601	+	0.564254i	$0.190836\pi$
$19$			21.4417i			1.12851i	−0.825601	+	0.564254i	$0.809164\pi$
$20$		0			0
$21$			1.67135i			0.0795880i
$22$		0			0
$23$	−13.1441	−	18.8741i	−0.571483	−	0.820614i
$23$	−13.1441	−	18.8741i	−0.571483	−	0.820614i
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	5.11087i		−	0.189291i
$28$		0			0
$29$	−4.76202			−0.164208			−0.0821038	−	0.996624i	$-0.526164\pi$
$29$	−4.76202			−0.164208			−0.0821038	+	0.996624i	$0.526164\pi$
$30$		0			0
$31$	−2.69584			−0.0869624			−0.0434812	−	0.999054i	$-0.513845\pi$
$31$	−2.69584			−0.0869624			−0.0434812	+	0.999054i	$0.513845\pi$
$32$		0			0
$33$	1.35498			0.0410599
$34$		0			0
$35$		0			0
$36$		0			0
$37$	49.8033			1.34604			0.673018	−	0.739626i	$-0.264998\pi$
$37$	49.8033			1.34604			0.673018	+	0.739626i	$0.264998\pi$
$38$		0			0
$39$	−3.26899			−0.0838203
$40$		0			0
$41$	18.6032			0.453736			0.226868	−	0.973926i	$-0.427151\pi$
$41$	18.6032			0.453736			0.226868	+	0.973926i	$0.427151\pi$
$42$		0			0
$43$	15.1515			0.352361			0.176181	−	0.984358i	$-0.443626\pi$
$43$	15.1515			0.352361			0.176181	+	0.984358i	$0.443626\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	18.2985i		−	0.389330i	−0.980870	−	0.194665i	$-0.937638\pi$
$47$		−	18.2985i		−	0.389330i	0.980870	−	0.194665i	$-0.0623619\pi$
$48$		0			0
$49$	−14.6636			−0.299257
$50$		0			0
$51$		−	1.18865i		−	0.0233069i
$52$		0			0
$53$	29.2038			0.551015			0.275508	−	0.961299i	$-0.411154\pi$
$53$	29.2038			0.551015			0.275508	+	0.961299i	$0.411154\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	6.11572			0.107293
$58$		0			0
$59$	−38.0270			−0.644526			−0.322263	−	0.946650i	$-0.604444\pi$
$59$	−38.0270			−0.644526			−0.322263	+	0.946650i	$0.604444\pi$
$60$		0			0
$61$			77.5848i			1.27188i	0.771737	+	0.635941i	$0.219388\pi$
$61$			77.5848i			1.27188i	−0.771737	+	0.635941i	$0.780612\pi$
$62$		0			0
$63$	−52.2608			−0.829537
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−48.4591			−0.723270			−0.361635	−	0.932320i	$-0.617781\pi$
$67$	−48.4591			−0.723270			−0.361635	+	0.932320i	$0.617781\pi$
$68$		0			0
$69$	−5.38339	+	3.74905i	−0.0780202	+	0.0543340i
$70$		0			0
$71$	13.9756			0.196840			0.0984200	−	0.995145i	$-0.468621\pi$
$71$	13.9756			0.196840			0.0984200	+	0.995145i	$0.468621\pi$
$72$		0			0
$73$		−	15.2098i		−	0.208353i	−0.994559	−	0.104176i	$-0.966779\pi$
$73$		−	15.2098i		−	0.208353i	0.994559	−	0.104176i	$-0.0332207\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	27.8368i		−	0.361517i
$78$		0			0
$79$		−	111.294i		−	1.40878i	−0.709813	−	0.704390i	$-0.751220\pi$
$79$		−	111.294i		−	1.40878i	0.709813	−	0.704390i	$-0.248780\pi$
$80$		0			0
$81$	78.8101			0.972964
$82$		0			0
$83$	52.6827			0.634732			0.317366	−	0.948303i	$-0.397202\pi$
$83$	52.6827			0.634732			0.317366	+	0.948303i	$0.397202\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			1.35825i			0.0156121i
$88$		0			0
$89$		−	52.6530i		−	0.591606i	−0.955249	−	0.295803i	$-0.904413\pi$
$89$		−	52.6530i		−	0.591606i	0.955249	−	0.295803i	$-0.0955872\pi$
$90$		0			0
$91$			67.1586i			0.738007i
$92$		0			0
$93$			0.768923i			0.00826799i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	37.6780			0.388433			0.194216	−	0.980959i	$-0.437784\pi$
$97$	37.6780			0.388433			0.194216	+	0.980959i	$0.437784\pi$
$98$		0			0
$99$			42.3683i			0.427963i
$100$		0			0
$101$	138.110			1.36742			0.683711	−	0.729753i	$-0.260365\pi$
$101$	138.110			1.36742			0.683711	+	0.729753i	$0.260365\pi$
$102$		0			0
$103$	94.0019			0.912639			0.456320	−	0.889816i	$-0.349167\pi$
$103$	94.0019			0.912639			0.456320	+	0.889816i	$0.349167\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.6541			−0.118263			−0.0591315	−	0.998250i	$-0.518833\pi$
$107$	−12.6541			−0.118263			−0.0591315	+	0.998250i	$0.518833\pi$
$108$		0			0
$109$		−	87.4492i		−	0.802286i	−0.916015	−	0.401143i	$-0.868613\pi$
$109$		−	87.4492i		−	0.802286i	0.916015	−	0.401143i	$-0.131387\pi$
$110$		0			0
$111$		−	14.2052i		−	0.127975i
$112$		0			0
$113$	67.4596			0.596987			0.298494	−	0.954412i	$-0.403516\pi$
$113$	67.4596			0.596987			0.298494	+	0.954412i	$0.403516\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	102.217i		−	0.873650i
$118$		0			0
$119$	−24.4198			−0.205209
$120$		0			0
$121$	98.4324			0.813491
$122$		0			0
$123$		−	5.30611i		−	0.0431391i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	4.06787i		−	0.0320305i	−0.999872	−	0.0160153i	$-0.994902\pi$
$127$		−	4.06787i		−	0.0320305i	0.999872	−	0.0160153i	$-0.00509803\pi$
$128$		0			0
$129$		−	4.32161i		−	0.0335009i
$130$		0			0
$131$	60.5029			0.461854			0.230927	−	0.972971i	$-0.425824\pi$
$131$	60.5029			0.461854			0.230927	+	0.972971i	$0.425824\pi$
$132$		0			0
$133$		−	125.642i		−	0.944679i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−23.9899			−0.175109			−0.0875545	−	0.996160i	$-0.527905\pi$
$137$	−23.9899			−0.175109			−0.0875545	+	0.996160i	$0.527905\pi$
$138$		0			0
$139$	−63.5042			−0.456864			−0.228432	−	0.973560i	$-0.573360\pi$
$139$	−63.5042			−0.456864			−0.228432	+	0.973560i	$0.573360\pi$
$140$		0			0
$141$	−5.21921			−0.0370157
$142$		0			0
$143$	54.4461			0.380742
$144$		0			0
$145$		0			0
$146$		0			0
$147$			4.18244i			0.0284520i
$148$		0			0
$149$			194.217i			1.30347i	0.758446	+	0.651736i	$0.225959\pi$
$149$			194.217i			1.30347i	−0.758446	+	0.651736i	$0.774041\pi$
$150$		0			0
$151$	127.240			0.842646			0.421323	−	0.906911i	$-0.361566\pi$
$151$	127.240			0.842646			0.421323	+	0.906911i	$0.361566\pi$
$152$		0			0
$153$	37.1676			0.242925
$154$		0			0
$155$		0			0
$156$		0			0
$157$	290.123			1.84792			0.923960	−	0.382490i	$-0.124933\pi$
$157$	290.123			1.84792			0.923960	+	0.382490i	$0.124933\pi$
$158$		0			0
$159$		−	8.32969i		−	0.0523880i
$160$		0			0
$161$	77.0210	+	110.597i	0.478391	+	0.686939i
$162$		0			0
$163$		−	53.0475i		−	0.325444i	−0.986672	−	0.162722i	$-0.947973\pi$
$163$		−	53.0475i		−	0.325444i	0.986672	−	0.162722i	$-0.0520274\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	186.413i		−	1.11625i	−0.829758	−	0.558124i	$-0.811521\pi$
$167$		−	186.413i		−	1.11625i	0.829758	−	0.558124i	$-0.188479\pi$
$168$		0			0
$169$	37.6444			0.222748
$170$		0			0
$171$			191.231i			1.11831i
$172$		0			0
$173$		−	173.894i		−	1.00517i	−0.864529	−	0.502584i	$-0.832383\pi$
$173$		−	173.894i		−	1.00517i	0.864529	−	0.502584i	$-0.167617\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			10.8463i			0.0612786i
$178$		0			0
$179$	−30.3651			−0.169637			−0.0848187	−	0.996396i	$-0.527031\pi$
$179$	−30.3651			−0.169637			−0.0848187	+	0.996396i	$0.527031\pi$
$180$		0			0
$181$		−	103.618i		−	0.572477i	−0.958158	−	0.286239i	$-0.907595\pi$
$181$		−	103.618i		−	0.572477i	0.958158	−	0.286239i	$-0.0924050\pi$
$182$		0			0
$183$	22.1292			0.120925
$184$		0			0
$185$		0			0
$186$		0			0
$187$			19.7974i			0.105868i
$188$		0			0
$189$			29.9483i			0.158457i
$190$		0			0
$191$			194.288i			1.01721i	0.860999	+	0.508606i	$0.169839\pi$
$191$			194.288i			1.01721i	−0.860999	+	0.508606i	$0.830161\pi$
$192$		0			0
$193$		−	243.726i		−	1.26283i	−0.775446	−	0.631414i	$-0.782475\pi$
$193$		−	243.726i		−	1.26283i	0.775446	−	0.631414i	$-0.217525\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	306.178i		−	1.55420i	−0.629377	−	0.777101i	$-0.716690\pi$
$197$		−	306.178i		−	1.55420i	0.629377	−	0.777101i	$-0.283310\pi$
$198$		0			0
$199$		−	105.975i		−	0.532539i	−0.963899	−	0.266269i	$-0.914209\pi$
$199$		−	105.975i		−	0.532539i	0.963899	−	0.266269i	$-0.0857911\pi$
$200$		0			0
$201$			13.8218i			0.0687652i
$202$		0			0
$203$	27.9041			0.137459
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−117.228	−	168.332i	−0.566318	−	0.813196i
$208$		0			0
$209$	−101.859			−0.487365
$210$		0			0
$211$	160.352			0.759961			0.379981	−	0.924994i	$-0.375931\pi$
$211$	160.352			0.759961			0.379981	+	0.924994i	$0.375931\pi$
$212$		0			0
$213$		−	3.98622i		−	0.0187146i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	15.7969			0.0727966
$218$		0			0
$219$	−4.33822			−0.0198092
$220$		0			0
$221$		−	47.7628i		−	0.216121i
$222$		0			0
$223$			30.6252i			0.137333i	0.997640	+	0.0686664i	$0.0218744\pi$
$223$			30.6252i			0.137333i	−0.997640	+	0.0686664i	$0.978126\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	389.626			1.71641			0.858207	−	0.513304i	$-0.171579\pi$
$227$	389.626			1.71641			0.858207	+	0.513304i	$0.171579\pi$
$228$		0			0
$229$		−	287.627i		−	1.25602i	−0.778207	−	0.628008i	$-0.783871\pi$
$229$		−	287.627i		−	1.25602i	0.778207	−	0.628008i	$-0.216129\pi$
$230$		0			0
$231$	−7.93980			−0.0343714
$232$		0			0
$233$		−	35.8448i		−	0.153840i	−0.997037	−	0.0769201i	$-0.975491\pi$
$233$		−	35.8448i		−	0.153840i	0.997037	−	0.0769201i	$-0.0245086\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−31.7439			−0.133940
$238$		0			0
$239$	−76.9035			−0.321772			−0.160886	−	0.986973i	$-0.551435\pi$
$239$	−76.9035			−0.321772			−0.160886	+	0.986973i	$0.551435\pi$
$240$		0			0
$241$			144.244i			0.598521i	0.954171	+	0.299261i	$0.0967400\pi$
$241$			144.244i			0.598521i	−0.954171	+	0.299261i	$0.903260\pi$
$242$		0			0
$243$		−	68.4765i		−	0.281796i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	245.744			0.994915
$248$		0			0
$249$		−	15.0265i		−	0.0603474i
$250$		0			0
$251$			272.682i			1.08638i	0.839609	+	0.543192i	$0.182784\pi$
$251$			272.682i			1.08638i	−0.839609	+	0.543192i	$0.817216\pi$
$252$		0			0
$253$	89.6621	−	62.4416i	0.354396	−	0.246805i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			84.8951i			0.330331i	0.986266	+	0.165165i	$0.0528158\pi$
$257$			84.8951i			0.330331i	−0.986266	+	0.165165i	$0.947184\pi$
$258$		0			0
$259$	−291.834			−1.12677
$260$		0			0
$261$	−42.4708			−0.162723
$262$		0			0
$263$	−279.641			−1.06327			−0.531637	−	0.846973i	$-0.678423\pi$
$263$	−279.641			−1.06327			−0.531637	+	0.846973i	$0.678423\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−15.0180			−0.0562472
$268$		0			0
$269$	184.412			0.685546			0.342773	−	0.939418i	$-0.388634\pi$
$269$	184.412			0.685546			0.342773	+	0.939418i	$0.388634\pi$
$270$		0			0
$271$	324.505			1.19743			0.598717	−	0.800960i	$-0.295677\pi$
$271$	324.505			1.19743			0.598717	+	0.800960i	$0.295677\pi$
$272$		0			0
$273$	19.1554			0.0701663
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	138.742i		−	0.500872i	−0.968133	−	0.250436i	$-0.919426\pi$
$277$		−	138.742i		−	0.500872i	0.968133	−	0.250436i	$-0.0805740\pi$
$278$		0			0
$279$	−24.0432			−0.0861764
$280$		0			0
$281$			219.290i			0.780393i	0.920732	+	0.390196i	$0.127593\pi$
$281$			219.290i			0.780393i	−0.920732	+	0.390196i	$0.872407\pi$
$282$		0			0
$283$	−122.495			−0.432845			−0.216422	−	0.976300i	$-0.569439\pi$
$283$	−122.495			−0.432845			−0.216422	+	0.976300i	$0.569439\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−109.009			−0.379824
$288$		0			0
$289$	−271.633			−0.939906
$290$		0			0
$291$		−	10.7467i		−	0.0369304i
$292$		0			0
$293$	329.831			1.12570			0.562851	−	0.826558i	$-0.309704\pi$
$293$	329.831			1.12570			0.562851	+	0.826558i	$0.309704\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	24.2793			0.0817486
$298$		0			0
$299$	−216.317	+	150.645i	−0.723469	+	0.503831i
$300$		0			0
$301$	−88.7838			−0.294963
$302$		0			0
$303$		−	39.3925i		−	0.130008i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			308.395i			1.00454i	0.864709	+	0.502272i	$0.167503\pi$
$307$			308.395i			1.00454i	−0.864709	+	0.502272i	$0.832497\pi$
$308$		0			0
$309$		−	26.8118i		−	0.0867695i
$310$		0			0
$311$	498.454			1.60275			0.801373	−	0.598165i	$-0.204103\pi$
$311$	498.454			1.60275			0.801373	+	0.598165i	$0.204103\pi$
$312$		0			0
$313$	−107.729			−0.344181			−0.172091	−	0.985081i	$-0.555052\pi$
$313$	−107.729			−0.344181			−0.172091	+	0.985081i	$0.555052\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			46.8123i			0.147673i	0.997270	+	0.0738365i	$0.0235243\pi$
$317$			46.8123i			0.147673i	−0.997270	+	0.0738365i	$0.976476\pi$
$318$		0			0
$319$		−	22.6221i		−	0.0709158i
$320$		0			0
$321$			3.60929i			0.0112439i
$322$		0			0
$323$			89.3560i			0.276644i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−24.9428			−0.0762776
$328$		0			0
$329$			107.224i			0.325910i
$330$		0			0
$331$	525.397			1.58730			0.793651	−	0.608374i	$-0.208178\pi$
$331$	525.397			1.58730			0.793651	+	0.608374i	$0.208178\pi$
$332$		0			0
$333$	444.178			1.33387
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−220.104			−0.653128			−0.326564	−	0.945175i	$-0.605891\pi$
$337$	−220.104			−0.653128			−0.326564	+	0.945175i	$0.605891\pi$
$338$		0			0
$339$		−	19.2412i		−	0.0567588i
$340$		0			0
$341$		−	12.8067i		−	0.0375562i
$342$		0			0
$343$	373.051			1.08761
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	178.154i		−	0.513411i	−0.966490	−	0.256705i	$-0.917363\pi$
$347$		−	178.154i		−	0.513411i	0.966490	−	0.256705i	$-0.0826370\pi$
$348$		0			0
$349$	224.147			0.642255			0.321128	−	0.947036i	$-0.395938\pi$
$349$	224.147			0.642255			0.321128	+	0.947036i	$0.395938\pi$
$350$		0			0
$351$	−58.5759			−0.166883
$352$		0			0
$353$			8.00237i			0.0226696i	0.999936	+	0.0113348i	$0.00360806\pi$
$353$			8.00237i			0.0226696i	−0.999936	+	0.0113348i	$0.996392\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			6.96518i			0.0195103i
$358$		0			0
$359$		−	603.858i		−	1.68206i	−0.540991	−	0.841028i	$-0.681951\pi$
$359$		−	603.858i		−	1.68206i	0.540991	−	0.841028i	$-0.318049\pi$
$360$		0			0
$361$	−98.7451			−0.273532
$362$		0			0
$363$		−	28.0755i		−	0.0773430i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−313.421			−0.854009			−0.427004	−	0.904250i	$-0.640431\pi$
$367$	−313.421			−0.854009			−0.427004	+	0.904250i	$0.640431\pi$
$368$		0			0
$369$	165.915			0.449634
$370$		0			0
$371$	−171.126			−0.461257
$372$		0			0
$373$	−385.887			−1.03455			−0.517274	−	0.855820i	$-0.673053\pi$
$373$	−385.887			−1.03455			−0.517274	+	0.855820i	$0.673053\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			54.5777i			0.144769i
$378$		0			0
$379$			38.3057i			0.101070i	0.998722	+	0.0505352i	$0.0160927\pi$
$379$			38.3057i			0.101070i	−0.998722	+	0.0505352i	$0.983907\pi$
$380$		0			0
$381$	−1.16026			−0.00304531
$382$		0			0
$383$	−141.743			−0.370086			−0.185043	−	0.982730i	$-0.559242\pi$
$383$	−141.743			−0.370086			−0.185043	+	0.982730i	$0.559242\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	135.131			0.349176
$388$		0			0
$389$			573.078i			1.47321i	0.676324	+	0.736605i	$0.263572\pi$
$389$			573.078i			1.47321i	−0.676324	+	0.736605i	$0.736428\pi$
$390$		0			0
$391$	−54.7768	−	78.6560i	−0.140094	−	0.201166i
$392$		0			0
$393$		−	17.2570i		−	0.0439109i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	488.751i		−	1.23111i	−0.788094	−	0.615555i	$-0.788932\pi$
$397$		−	488.751i		−	1.23111i	0.788094	−	0.615555i	$-0.211068\pi$
$398$		0			0
$399$	−35.8365			−0.0898157
$400$		0			0
$401$		−	294.910i		−	0.735437i	−0.929937	−	0.367718i	$-0.880139\pi$
$401$		−	294.910i		−	0.735437i	0.929937	−	0.367718i	$-0.119861\pi$
$402$		0			0
$403$			30.8971i			0.0766677i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			236.592i			0.581308i
$408$		0			0
$409$	166.122			0.406166			0.203083	−	0.979161i	$-0.434904\pi$
$409$	166.122			0.406166			0.203083	+	0.979161i	$0.434904\pi$
$410$		0			0
$411$			6.84256i			0.0166486i
$412$		0			0
$413$	222.828			0.539535
$414$		0			0
$415$		0			0
$416$		0			0
$417$			18.1130i			0.0434366i
$418$		0			0
$419$		−	649.591i		−	1.55034i	−0.631755	−	0.775168i	$-0.717665\pi$
$419$		−	649.591i		−	1.55034i	0.631755	−	0.775168i	$-0.282335\pi$
$420$		0			0
$421$			117.703i			0.279579i	0.990181	+	0.139789i	$0.0446426\pi$
$421$			117.703i			0.279579i	−0.990181	+	0.139789i	$0.955357\pi$
$422$		0			0
$423$		−	163.198i		−	0.385811i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	454.626i		−	1.06470i
$428$		0			0
$429$		−	15.5295i		−	0.0361992i
$430$		0			0
$431$		−	323.440i		−	0.750442i	−0.926935	−	0.375221i	$-0.877567\pi$
$431$		−	323.440i		−	0.750442i	0.926935	−	0.375221i	$-0.122433\pi$
$432$		0			0
$433$	−172.304			−0.397930			−0.198965	−	0.980007i	$-0.563758\pi$
$433$	−172.304			−0.397930			−0.198965	+	0.980007i	$0.563758\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	404.692	−	281.832i	0.926070	−	0.644924i
$438$		0			0
$439$	−352.510			−0.802984			−0.401492	−	0.915862i	$-0.631508\pi$
$439$	−352.510			−0.802984			−0.401492	+	0.915862i	$0.631508\pi$
$440$		0			0
$441$	−130.779			−0.296552
$442$		0			0
$443$		−	145.266i		−	0.327914i	−0.986467	−	0.163957i	$-0.947574\pi$
$443$		−	145.266i		−	0.327914i	0.986467	−	0.163957i	$-0.0524259\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	55.3958			0.123928
$448$		0			0
$449$	554.518			1.23501			0.617504	−	0.786568i	$-0.288144\pi$
$449$	554.518			1.23501			0.617504	+	0.786568i	$0.288144\pi$
$450$		0			0
$451$			88.3749i			0.195953i
$452$		0			0
$453$		−	36.2920i		−	0.0801149i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−36.8658			−0.0806693			−0.0403346	−	0.999186i	$-0.512842\pi$
$457$	−36.8658			−0.0806693			−0.0403346	+	0.999186i	$0.512842\pi$
$458$		0			0
$459$		−	21.2990i		−	0.0464031i
$460$		0			0
$461$	−257.831			−0.559286			−0.279643	−	0.960104i	$-0.590216\pi$
$461$	−257.831			−0.559286			−0.279643	+	0.960104i	$0.590216\pi$
$462$		0			0
$463$			586.092i			1.26586i	0.774210	+	0.632929i	$0.218147\pi$
$463$			586.092i			1.26586i	−0.774210	+	0.632929i	$0.781853\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−126.252			−0.270346			−0.135173	−	0.990822i	$-0.543159\pi$
$467$	−126.252			−0.270346			−0.135173	+	0.990822i	$0.543159\pi$
$468$		0			0
$469$	283.957			0.605453
$470$		0			0
$471$		−	82.7508i		−	0.175692i
$472$		0			0
$473$			71.9779i			0.152173i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	260.458			0.546034
$478$		0			0
$479$			389.718i			0.813608i	0.913516	+	0.406804i	$0.133357\pi$
$479$			389.718i			0.813608i	−0.913516	+	0.406804i	$0.866643\pi$
$480$		0			0
$481$		−	570.798i		−	1.18669i
$482$		0			0
$483$	31.5452	−	21.9684i	0.0653110	−	0.0454832i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			587.014i			1.20537i	0.797980	+	0.602684i	$0.205902\pi$
$487$			587.014i			1.20537i	−0.797980	+	0.602684i	$0.794098\pi$
$488$		0			0
$489$	−15.1305			−0.0309418
$490$		0			0
$491$	−548.753			−1.11762			−0.558811	−	0.829295i	$-0.688742\pi$
$491$	−548.753			−1.11762			−0.558811	+	0.829295i	$0.688742\pi$
$492$		0			0
$493$	−19.8452			−0.0402540
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−81.8934			−0.164775
$498$		0			0
$499$	−47.2740			−0.0947374			−0.0473687	−	0.998877i	$-0.515084\pi$
$499$	−47.2740			−0.0947374			−0.0473687	+	0.998877i	$0.515084\pi$
$500$		0			0
$501$	−53.1700			−0.106128
$502$		0			0
$503$	620.845			1.23428			0.617142	−	0.786852i	$-0.288290\pi$
$503$	620.845			1.23428			0.617142	+	0.786852i	$0.288290\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	10.7372i		−	0.0211778i
$508$		0			0
$509$	290.221			0.570178			0.285089	−	0.958501i	$-0.407977\pi$
$509$	290.221			0.570178			0.285089	+	0.958501i	$0.407977\pi$
$510$		0			0
$511$			89.1251i			0.174413i
$512$		0			0
$513$	109.585			0.213617
$514$		0			0
$515$		0			0
$516$		0			0
$517$	86.9277			0.168139
$518$		0			0
$519$	−49.5991			−0.0955667
$520$		0			0
$521$			469.153i			0.900486i	0.892906	+	0.450243i	$0.148663\pi$
$521$			469.153i			0.900486i	−0.892906	+	0.450243i	$0.851337\pi$
$522$		0			0
$523$	−399.172			−0.763236			−0.381618	−	0.924320i	$-0.624633\pi$
$523$	−399.172			−0.763236			−0.381618	+	0.924320i	$0.624633\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−11.2346			−0.0213181
$528$		0			0
$529$	−183.464	+	496.167i	−0.346813	+	0.937934i
$530$		0			0
$531$	−339.150			−0.638700
$532$		0			0
$533$		−	213.212i		−	0.400022i
$534$		0			0
$535$		0			0
$536$		0			0
$537$			8.66091i			0.0161283i
$538$		0			0
$539$		−	69.6599i		−	0.129239i
$540$		0			0
$541$	185.967			0.343747			0.171873	−	0.985119i	$-0.445018\pi$
$541$	185.967			0.343747			0.171873	+	0.985119i	$0.445018\pi$
$542$		0			0
$543$	−29.5547			−0.0544285
$544$		0			0
$545$		0			0
$546$		0			0
$547$			778.387i			1.42301i	0.702681	+	0.711505i	$0.251986\pi$
$547$			778.387i			1.42301i	−0.702681	+	0.711505i	$0.748014\pi$
$548$		0			0
$549$			691.952i			1.26039i
$550$		0			0
$551$		−	102.106i		−	0.185310i
$552$		0			0
$553$			652.151i			1.17930i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−582.870			−1.04644			−0.523222	−	0.852196i	$-0.675270\pi$
$557$	−582.870			−1.04644			−0.523222	+	0.852196i	$0.675270\pi$
$558$		0			0
$559$		−	173.652i		−	0.310648i
$560$		0			0
$561$	5.64673			0.0100655
$562$		0			0
$563$	−725.427			−1.28850			−0.644251	−	0.764814i	$-0.722831\pi$
$563$	−725.427			−1.28850			−0.644251	+	0.764814i	$0.722831\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−461.806			−0.814472
$568$		0			0
$569$			647.932i			1.13872i	0.822088	+	0.569360i	$0.192809\pi$
$569$			647.932i			1.13872i	−0.822088	+	0.569360i	$0.807191\pi$
$570$		0			0
$571$		−	65.4969i		−	0.114706i	−0.998354	−	0.0573528i	$-0.981734\pi$
$571$		−	65.4969i		−	0.114706i	0.998354	−	0.0573528i	$-0.0182660\pi$
$572$		0			0
$573$	55.4159			0.0967119
$574$		0			0
$575$		0			0
$576$		0			0
$577$			77.4243i			0.134184i	0.997747	+	0.0670921i	$0.0213721\pi$
$577$			77.4243i			0.134184i	−0.997747	+	0.0670921i	$0.978628\pi$
$578$		0			0
$579$	−69.5170			−0.120064
$580$		0			0
$581$	−308.707			−0.531337
$582$		0			0
$583$			138.734i			0.237965i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			691.496i			1.17802i	0.808127	+	0.589009i	$0.200482\pi$
$587$			691.496i			1.17802i	−0.808127	+	0.589009i	$0.799518\pi$
$588$		0			0
$589$		−	57.8032i		−	0.0981379i
$590$		0			0
$591$	−87.3299			−0.147766
$592$		0			0
$593$		−	522.286i		−	0.880751i	−0.897814	−	0.440376i	$-0.854845\pi$
$593$		−	522.286i		−	0.880751i	0.897814	−	0.440376i	$-0.145155\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−30.2269			−0.0506313
$598$		0			0
$599$	−106.985			−0.178605			−0.0893027	−	0.996005i	$-0.528464\pi$
$599$	−106.985			−0.178605			−0.0893027	+	0.996005i	$0.528464\pi$
$600$		0			0
$601$	−549.886			−0.914951			−0.457476	−	0.889222i	$-0.651246\pi$
$601$	−549.886			−0.914951			−0.457476	+	0.889222i	$0.651246\pi$
$602$		0			0
$603$	−432.190			−0.716733
$604$		0			0
$605$		0			0
$606$		0			0
$607$			943.423i			1.55424i	0.629353	+	0.777120i	$0.283320\pi$
$607$			943.423i			1.55424i	−0.629353	+	0.777120i	$0.716680\pi$
$608$		0			0
$609$		−	7.95899i		−	0.0130690i
$610$		0			0
$611$	−209.720			−0.343241
$612$		0			0
$613$	−514.014			−0.838521			−0.419261	−	0.907866i	$-0.637711\pi$
$613$	−514.014			−0.838521			−0.419261	+	0.907866i	$0.637711\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−29.1372			−0.0472239			−0.0236120	−	0.999721i	$-0.507517\pi$
$617$	−29.1372			−0.0472239			−0.0236120	+	0.999721i	$0.507517\pi$
$618$		0			0
$619$		−	378.918i		−	0.612145i	−0.952008	−	0.306073i	$-0.900985\pi$
$619$		−	378.918i		−	0.612145i	0.952008	−	0.306073i	$-0.0990151\pi$
$620$		0			0
$621$	−96.4631	+	67.1778i	−0.155335	+	0.108177i
$622$		0			0
$623$			308.532i			0.495236i
$624$		0			0
$625$		0			0
$626$		0			0
$627$			29.0530i			0.0463365i
$628$		0			0
$629$	207.550			0.329969
$630$		0			0
$631$			520.669i			0.825149i	0.910924	+	0.412574i	$0.135370\pi$
$631$			520.669i			0.825149i	−0.910924	+	0.412574i	$0.864630\pi$
$632$		0			0
$633$		−	45.7365i		−	0.0722536i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			168.060i			0.263831i
$638$		0			0
$639$	124.644			0.195061
$640$		0			0
$641$		−	850.408i		−	1.32669i	−0.748314	−	0.663345i	$-0.769136\pi$
$641$		−	850.408i		−	1.32669i	0.748314	−	0.663345i	$-0.230864\pi$
$642$		0			0
$643$	−883.735			−1.37439			−0.687197	−	0.726471i	$-0.741159\pi$
$643$	−883.735			−1.37439			−0.687197	+	0.726471i	$0.741159\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	118.054i		−	0.182464i	−0.995830	−	0.0912321i	$-0.970919\pi$
$647$		−	118.054i		−	0.182464i	0.995830	−	0.0912321i	$-0.0290805\pi$
$648$		0			0
$649$		−	180.649i		−	0.278349i
$650$		0			0
$651$		−	4.50568i		−	0.00692117i
$652$		0			0
$653$		−	670.046i		−	1.02610i	−0.858358	−	0.513052i	$-0.828515\pi$
$653$		−	670.046i		−	1.02610i	0.858358	−	0.513052i	$-0.171485\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	135.651i		−	0.206470i
$658$		0			0
$659$		−	552.664i		−	0.838641i	−0.907838	−	0.419320i	$-0.862268\pi$
$659$		−	552.664i		−	0.838641i	0.907838	−	0.419320i	$-0.137732\pi$
$660$		0			0
$661$			200.319i			0.303055i	0.988453	+	0.151527i	$0.0484191\pi$
$661$			200.319i			0.303055i	−0.988453	+	0.151527i	$0.951581\pi$
$662$		0			0
$663$	−13.6232			−0.0205478
$664$		0			0
$665$		0			0
$666$		0			0
$667$	62.5925	+	89.8789i	0.0938419	+	0.134751i
$668$		0			0
$669$	8.73511			0.0130570
$670$		0			0
$671$	−368.569			−0.549284
$672$		0			0
$673$			797.196i			1.18454i	0.805740	+	0.592270i	$0.201768\pi$
$673$			797.196i			1.18454i	−0.805740	+	0.592270i	$0.798232\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−464.170			−0.685627			−0.342814	−	0.939403i	$-0.611380\pi$
$677$	−464.170			−0.685627			−0.342814	+	0.939403i	$0.611380\pi$
$678$		0			0
$679$	−220.783			−0.325159
$680$		0			0
$681$		−	111.131i		−	0.163189i
$682$		0			0
$683$			903.065i			1.32220i	0.750296	+	0.661102i	$0.229911\pi$
$683$			903.065i			1.32220i	−0.750296	+	0.661102i	$0.770089\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−82.0389			−0.119416
$688$		0			0
$689$		−	334.706i		−	0.485785i
$690$		0			0
$691$	−325.464			−0.471005			−0.235502	−	0.971874i	$-0.575674\pi$
$691$	−325.464			−0.471005			−0.235502	+	0.971874i	$0.575674\pi$
$692$		0			0
$693$		−	248.267i		−	0.358250i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	77.5268			0.111229
$698$		0			0
$699$	−10.2239			−0.0146264
$700$		0			0
$701$		−	930.942i		−	1.32802i	−0.747723	−	0.664010i	$-0.768853\pi$
$701$		−	930.942i		−	1.32802i	0.747723	−	0.664010i	$-0.231147\pi$
$702$		0			0
$703$			1067.87i			1.51901i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−809.285			−1.14467
$708$		0			0
$709$			730.650i			1.03054i	0.857029	+	0.515268i	$0.172308\pi$
$709$			730.650i			1.03054i	−0.857029	+	0.515268i	$0.827692\pi$
$710$		0			0
$711$		−	992.589i		−	1.39605i
$712$		0			0
$713$	35.4344	+	50.8815i	0.0496976	+	0.0713626i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			21.9349i			0.0305926i
$718$		0			0
$719$	83.0314			0.115482			0.0577409	−	0.998332i	$-0.481610\pi$
$719$	83.0314			0.115482			0.0577409	+	0.998332i	$0.481610\pi$
$720$		0			0
$721$	−550.825			−0.763974
$722$		0			0
$723$	41.1420			0.0569046
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−895.949			−1.23239			−0.616196	−	0.787593i	$-0.711327\pi$
$727$	−895.949			−1.23239			−0.616196	+	0.787593i	$0.711327\pi$
$728$		0			0
$729$	689.759			0.946172
$730$		0			0
$731$	63.1425			0.0863782
$732$		0			0
$733$	59.5511			0.0812430			0.0406215	−	0.999175i	$-0.487066\pi$
$733$	59.5511			0.0812430			0.0406215	+	0.999175i	$0.487066\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	230.207i		−	0.312357i
$738$		0			0
$739$	−291.410			−0.394330			−0.197165	−	0.980370i	$-0.563173\pi$
$739$	−291.410			−0.394330			−0.197165	+	0.980370i	$0.563173\pi$
$740$		0			0
$741$		−	70.0926i		−	0.0945919i
$742$		0			0
$743$	−860.645			−1.15834			−0.579169	−	0.815208i	$-0.696623\pi$
$743$	−860.645			−1.15834			−0.579169	+	0.815208i	$0.696623\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	469.859			0.628994
$748$		0			0
$749$	74.1499			0.0989985
$750$		0			0
$751$		−	590.806i		−	0.786692i	−0.919391	−	0.393346i	$-0.871317\pi$
$751$		−	590.806i		−	0.786692i	0.919391	−	0.393346i	$-0.128683\pi$
$752$		0			0
$753$	77.7761			0.103288
$754$		0			0
$755$		0			0
$756$		0			0
$757$	416.621			0.550359			0.275179	−	0.961393i	$-0.411263\pi$
$757$	416.621			0.550359			0.275179	+	0.961393i	$0.411263\pi$
$758$		0			0
$759$	−17.8100	−	25.5740i	−0.0234651	−	0.0336943i
$760$		0			0
$761$	−920.381			−1.20944			−0.604718	−	0.796439i	$-0.706714\pi$
$761$	−920.381			−1.20944			−0.604718	+	0.796439i	$0.706714\pi$
$762$		0			0
$763$			512.428i			0.671597i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			435.830i			0.568227i
$768$		0			0
$769$		−	520.185i		−	0.676444i	−0.941066	−	0.338222i	$-0.890175\pi$
$769$		−	520.185i		−	0.676444i	0.941066	−	0.338222i	$-0.109825\pi$
$770$		0			0
$771$	24.2143			0.0314063
$772$		0			0
$773$	277.589			0.359106			0.179553	−	0.983748i	$-0.442535\pi$
$773$	277.589			0.359106			0.179553	+	0.983748i	$0.442535\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$			83.2387i			0.107128i
$778$		0			0
$779$			398.883i			0.512045i
$780$		0			0
$781$			66.3917i			0.0850086i
$782$		0			0
$783$			24.3380i			0.0310831i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−516.191			−0.655897			−0.327948	−	0.944696i	$-0.606357\pi$
$787$	−516.191			−0.655897			−0.327948	+	0.944696i	$0.606357\pi$
$788$		0			0
$789$			79.7609i			0.101091i
$790$		0			0
$791$	−395.295			−0.499741
$792$		0			0
$793$	889.204			1.12132
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1344.17			1.68653			0.843266	−	0.537496i	$-0.180630\pi$
$797$	1344.17			1.68653			0.843266	+	0.537496i	$0.180630\pi$
$798$		0			0
$799$		−	76.2572i		−	0.0954408i
$800$		0			0
$801$		−	469.593i		−	0.586259i
$802$		0			0
$803$	72.2545			0.0899807
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	52.5990i		−	0.0651785i
$808$		0			0
$809$	1054.25			1.30315			0.651577	−	0.758582i	$-0.274108\pi$
$809$	1054.25			1.30315			0.651577	+	0.758582i	$0.274108\pi$
$810$		0			0
$811$	−202.244			−0.249376			−0.124688	−	0.992196i	$-0.539793\pi$
$811$	−202.244			−0.249376			−0.124688	+	0.992196i	$0.539793\pi$
$812$		0			0
$813$		−	92.5572i		−	0.113847i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			324.874i			0.397643i
$818$		0			0
$819$			598.964i			0.731336i
$820$		0			0
$821$	242.591			0.295483			0.147741	−	0.989026i	$-0.452800\pi$
$821$	242.591			0.295483			0.147741	+	0.989026i	$0.452800\pi$
$822$		0			0
$823$			254.478i			0.309208i	0.987977	+	0.154604i	$0.0494101\pi$
$823$			254.478i			0.309208i	−0.987977	+	0.154604i	$0.950590\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−366.951			−0.443713			−0.221856	−	0.975079i	$-0.571212\pi$
$827$	−366.951			−0.443713			−0.221856	+	0.975079i	$0.571212\pi$
$828$		0			0
$829$	133.052			0.160498			0.0802488	−	0.996775i	$-0.474429\pi$
$829$	133.052			0.160498			0.0802488	+	0.996775i	$0.474429\pi$
$830$		0			0
$831$	−39.5727			−0.0476206
$832$		0			0
$833$	−61.1090			−0.0733602
$834$		0			0
$835$		0			0
$836$		0			0
$837$			13.7781i			0.0164612i
$838$		0			0
$839$			1381.18i			1.64622i	0.567880	+	0.823112i	$0.307764\pi$
$839$			1381.18i			1.64622i	−0.567880	+	0.823112i	$0.692236\pi$
$840$		0			0
$841$	−818.323			−0.973036
$842$		0			0
$843$	62.5473			0.0741961
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−576.787			−0.680977
$848$		0			0
$849$			34.9388i			0.0411529i
$850$		0			0
$851$	−654.621	−	939.994i	−0.769237	−	1.10458i
$852$		0			0
$853$		−	741.672i		−	0.869486i	−0.900555	−	0.434743i	$-0.856839\pi$
$853$		−	741.672i		−	0.869486i	0.900555	−	0.434743i	$-0.143161\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	180.242i		−	0.210317i	−0.994455	−	0.105158i	$-0.966465\pi$
$857$		−	180.242i		−	0.210317i	0.994455	−	0.105158i	$-0.0335350\pi$
$858$		0			0
$859$	−849.613			−0.989072			−0.494536	−	0.869157i	$-0.664662\pi$
$859$	−849.613			−0.989072			−0.494536	+	0.869157i	$0.664662\pi$
$860$		0			0
$861$			31.0923i			0.0361119i
$862$		0			0
$863$			1174.16i			1.36056i	0.732952	+	0.680280i	$0.238142\pi$
$863$			1174.16i			1.36056i	−0.732952	+	0.680280i	$0.761858\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			77.4768i			0.0893619i
$868$		0			0
$869$	528.705			0.608406
$870$		0			0
$871$			555.392i			0.637649i
$872$		0			0
$873$	336.036			0.384922
$874$		0			0
$875$		0			0
$876$		0			0
$877$			267.454i			0.304965i	0.988306	+	0.152482i	$0.0487267\pi$
$877$			267.454i			0.304965i	−0.988306	+	0.152482i	$0.951273\pi$
$878$		0			0
$879$		−	94.0764i		−	0.107027i
$880$		0			0
$881$		−	1570.79i		−	1.78296i	−0.453056	−	0.891482i	$-0.649666\pi$
$881$		−	1570.79i		−	1.78296i	0.453056	−	0.891482i	$-0.350334\pi$
$882$		0			0
$883$			823.664i			0.932802i	0.884573	+	0.466401i	$0.154450\pi$
$883$			823.664i			0.932802i	−0.884573	+	0.466401i	$0.845550\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	389.703i		−	0.439350i	−0.975573	−	0.219675i	$-0.929500\pi$
$887$		−	389.703i		−	0.439350i	0.975573	−	0.219675i	$-0.0704996\pi$
$888$		0			0
$889$			23.8366i			0.0268129i
$890$		0			0
$891$			374.390i			0.420191i
$892$		0			0
$893$	392.350			0.439362
$894$		0			0
$895$		0			0
$896$		0			0
$897$	42.9680	+	61.6993i	0.0479019	+	0.0687841i
$898$		0			0
$899$	12.8376			0.0142799
$900$		0			0
$901$	121.704			0.135077
$902$		0			0
$903$			25.3235i			0.0280437i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−877.582			−0.967566			−0.483783	−	0.875188i	$-0.660738\pi$
$907$	−877.582			−0.967566			−0.483783	+	0.875188i	$0.660738\pi$
$908$		0			0
$909$	1231.75			1.35506
$910$		0			0
$911$		−	242.747i		−	0.266463i	−0.991085	−	0.133231i	$-0.957465\pi$
$911$		−	242.747i		−	0.266463i	0.991085	−	0.133231i	$-0.0425353\pi$
$912$		0			0
$913$			250.271i			0.274120i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−354.530			−0.386620
$918$		0			0
$919$		−	1558.46i		−	1.69583i	−0.530135	−	0.847913i	$-0.677859\pi$
$919$		−	1558.46i		−	1.69583i	0.530135	−	0.847913i	$-0.322141\pi$
$920$		0			0
$921$	87.9624			0.0955075
$922$		0			0
$923$		−	160.175i		−	0.173538i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	838.369			0.904390
$928$		0			0
$929$	1029.50			1.10818			0.554088	−	0.832458i	$-0.313067\pi$
$929$	1029.50			1.10818			0.554088	+	0.832458i	$0.313067\pi$
$930$		0			0
$931$		−	314.412i		−	0.337714i
$932$		0			0
$933$		−	142.172i		−	0.152382i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−623.103			−0.664998			−0.332499	−	0.943104i	$-0.607892\pi$
$937$	−623.103			−0.664998			−0.332499	+	0.943104i	$0.607892\pi$
$938$		0			0
$939$			30.7271i			0.0327232i
$940$		0			0
$941$			1421.23i			1.51034i	0.655531	+	0.755169i	$0.272445\pi$
$941$			1421.23i			1.51034i	−0.655531	+	0.755169i	$0.727555\pi$
$942$		0			0
$943$	−244.522	−	351.118i	−0.259302	−	0.372342i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			288.402i			0.304542i	0.988339	+	0.152271i	$0.0486587\pi$
$947$			288.402i			0.304542i	−0.988339	+	0.152271i	$0.951341\pi$
$948$		0			0
$949$	−174.320			−0.183688
$950$		0			0
$951$	13.3521			0.0140401
$952$		0			0
$953$	−1270.93			−1.33361			−0.666805	−	0.745232i	$-0.732339\pi$
$953$	−1270.93			−1.33361			−0.666805	+	0.745232i	$0.732339\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−6.45243			−0.00674235
$958$		0			0
$959$	140.575			0.146584
$960$		0			0
$961$	−953.732			−0.992438
$962$		0			0
$963$	−112.858			−0.117194
$964$		0			0
$965$		0			0
$966$		0			0
$967$			379.110i			0.392047i	0.980599	+	0.196024i	$0.0628030\pi$
$967$			379.110i			0.392047i	−0.980599	+	0.196024i	$0.937197\pi$
$968$		0			0
$969$	25.4867			0.0263020
$970$		0			0
$971$			34.3648i			0.0353912i	0.999843	+	0.0176956i	$0.00563297\pi$
$971$			34.3648i			0.0353912i	−0.999843	+	0.0176956i	$0.994367\pi$
$972$		0			0
$973$	372.117			0.382443
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1371.73			1.40402			0.702010	−	0.712167i	$-0.252286\pi$
$977$	1371.73			1.40402			0.702010	+	0.712167i	$0.252286\pi$
$978$		0			0
$979$	250.130			0.255495
$980$		0			0
$981$		−	779.928i		−	0.795034i
$982$		0			0
$983$	499.903			0.508548			0.254274	−	0.967132i	$-0.418163\pi$
$983$	499.903			0.508548			0.254274	+	0.967132i	$0.418163\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	30.5832			0.0309860
$988$		0			0
$989$	−199.153	−	285.972i	−0.201368	−	0.289152i
$990$		0			0
$991$	−261.315			−0.263689			−0.131844	−	0.991270i	$-0.542090\pi$
$991$	−261.315			−0.263689			−0.131844	+	0.991270i	$0.542090\pi$
$992$		0			0
$993$		−	149.857i		−	0.150913i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	77.8322i		−	0.0780664i	−0.999238	−	0.0390332i	$-0.987572\pi$
$997$		−	77.8322i		−	0.0780664i	0.999238	−	0.0390332i	$-0.0124278\pi$
$998$		0			0
$999$		−	254.538i		−	0.254793i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.3.d.c.1149.11		32
5.2	odd	4		2300.3.f.c.1701.7	✓	16
5.3	odd	4		2300.3.f.d.1701.10	yes	16
5.4	even	2	inner	2300.3.d.c.1149.22		32
23.22	odd	2	inner	2300.3.d.c.1149.21		32
115.22	even	4		2300.3.f.c.1701.8	yes	16
115.68	even	4		2300.3.f.d.1701.9	yes	16
115.114	odd	2	inner	2300.3.d.c.1149.12		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2300.3.d.c.1149.11		32	1.1	even	1	trivial
2300.3.d.c.1149.12		32	115.114	odd	2	inner
2300.3.d.c.1149.21		32	23.22	odd	2	inner
2300.3.d.c.1149.22		32	5.4	even	2	inner
2300.3.f.c.1701.7	✓	16	5.2	odd	4
2300.3.f.c.1701.8	yes	16	115.22	even	4
2300.3.f.d.1701.9	yes	16	115.68	even	4
2300.3.f.d.1701.10	yes	16	5.3	odd	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)

\(f(q)\)	\(=\)	\(q-0.285226i q^{3} -5.85973 q^{7} +8.91865 q^{9} +O(q^{10})\)	\(q-0.285226i q^{3} -5.85973 q^{7} +8.91865 q^{9} +4.75053i q^{11} -11.4610i q^{13} +4.16740 q^{17} +21.4417i q^{19} +1.67135i q^{21} +(-13.1441 - 18.8741i) q^{23} -5.11087i q^{27} -4.76202 q^{29} -2.69584 q^{31} +1.35498 q^{33} +49.8033 q^{37} -3.26899 q^{39} +18.6032 q^{41} +15.1515 q^{43} -18.2985i q^{47} -14.6636 q^{49} -1.18865i q^{51} +29.2038 q^{53} +6.11572 q^{57} -38.0270 q^{59} +77.5848i q^{61} -52.2608 q^{63} -48.4591 q^{67} +(-5.38339 + 3.74905i) q^{69} +13.9756 q^{71} -15.2098i q^{73} -27.8368i q^{77} -111.294i q^{79} +78.8101 q^{81} +52.6827 q^{83} +1.35825i q^{87} -52.6530i q^{89} +67.1586i q^{91} +0.768923i q^{93} +37.6780 q^{97} +42.3683i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 128 q^{9}+O(q^{10}) \) 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