LMFDB - Embedded newform 2300.3.d.c.1149.14

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			1149.14
Character	$\chi$	$=$	2300.1149
Dual form			2300.3.d.c.1149.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.38686i q^{3} -2.45527 q^{7} -10.2446 q^{9} +O(q^{10})$	$q+4.38686i q^{3} -2.45527 q^{7} -10.2446 q^{9} -3.07704i q^{11} -15.5683i q^{13} +14.1617 q^{17} +35.3379i q^{19} -10.7709i q^{21} +(20.3851 + 10.6512i) q^{23} -5.45975i q^{27} +46.6836 q^{29} -12.3639 q^{31} +13.4985 q^{33} -65.0040 q^{37} +68.2958 q^{39} +36.3294 q^{41} -68.8424 q^{43} +72.0807i q^{47} -42.9717 q^{49} +62.1252i q^{51} +79.0502 q^{53} -155.023 q^{57} +15.6136 q^{59} -0.819358i q^{61} +25.1531 q^{63} -86.0061 q^{67} +(-46.7252 + 89.4267i) q^{69} +20.1891 q^{71} +34.6295i q^{73} +7.55495i q^{77} +113.703i q^{79} -68.2499 q^{81} -88.9041 q^{83} +204.795i q^{87} -50.7631i q^{89} +38.2242i q^{91} -54.2388i q^{93} +29.7851 q^{97} +31.5229i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$			4.38686i			1.46229i	0.682223	+	0.731144i	$0.261013\pi$
$3$			4.38686i			1.46229i	−0.682223	+	0.731144i	$0.738987\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.45527			−0.350752			−0.175376	−	0.984501i	$-0.556114\pi$
$7$	−2.45527			−0.350752			−0.175376	+	0.984501i	$0.556114\pi$
$8$		0			0
$9$	−10.2446			−1.13829
$10$		0			0
$11$		−	3.07704i		−	0.279731i	−0.990171	−	0.139865i	$-0.955333\pi$
$11$		−	3.07704i		−	0.279731i	0.990171	−	0.139865i	$-0.0446670\pi$
$12$		0			0
$13$		−	15.5683i		−	1.19756i	−0.800914	−	0.598779i	$-0.795653\pi$
$13$		−	15.5683i		−	1.19756i	0.800914	−	0.598779i	$-0.204347\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	14.1617			0.833039			0.416519	−	0.909127i	$-0.363250\pi$
$17$	14.1617			0.833039			0.416519	+	0.909127i	$0.363250\pi$
$18$		0			0
$19$			35.3379i			1.85989i	0.367698	+	0.929945i	$0.380146\pi$
$19$			35.3379i			1.85989i	−0.367698	+	0.929945i	$0.619854\pi$
$20$		0			0
$21$		−	10.7709i		−	0.512901i
$22$		0			0
$23$	20.3851	+	10.6512i	0.886309	+	0.463094i
$23$	20.3851	+	10.6512i	0.886309	+	0.463094i
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	5.45975i		−	0.202213i
$28$		0			0
$29$	46.6836			1.60978			0.804890	−	0.593425i	$-0.202224\pi$
$29$	46.6836			1.60978			0.804890	+	0.593425i	$0.202224\pi$
$30$		0			0
$31$	−12.3639			−0.398836			−0.199418	−	0.979915i	$-0.563905\pi$
$31$	−12.3639			−0.398836			−0.199418	+	0.979915i	$0.563905\pi$
$32$		0			0
$33$	13.4985			0.409047
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−65.0040			−1.75687			−0.878433	−	0.477866i	$-0.841410\pi$
$37$	−65.0040			−1.75687			−0.878433	+	0.477866i	$0.841410\pi$
$38$		0			0
$39$	68.2958			1.75118
$40$		0			0
$41$	36.3294			0.886083			0.443042	−	0.896501i	$-0.353899\pi$
$41$	36.3294			0.886083			0.443042	+	0.896501i	$0.353899\pi$
$42$		0			0
$43$	−68.8424			−1.60099			−0.800493	−	0.599342i	$-0.795429\pi$
$43$	−68.8424			−1.60099			−0.800493	+	0.599342i	$0.795429\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			72.0807i			1.53363i	0.641867	+	0.766816i	$0.278160\pi$
$47$			72.0807i			1.53363i	−0.641867	+	0.766816i	$0.721840\pi$
$48$		0			0
$49$	−42.9717			−0.876973
$50$		0			0
$51$			62.1252i			1.21814i
$52$		0			0
$53$	79.0502			1.49151			0.745756	−	0.666219i	$-0.232088\pi$
$53$	79.0502			1.49151			0.745756	+	0.666219i	$0.232088\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−155.023			−2.71969
$58$		0			0
$59$	15.6136			0.264638			0.132319	−	0.991207i	$-0.457758\pi$
$59$	15.6136			0.264638			0.132319	+	0.991207i	$0.457758\pi$
$60$		0			0
$61$		−	0.819358i		−	0.0134321i	−0.999977	−	0.00671605i	$-0.997862\pi$
$61$		−	0.819358i		−	0.0134321i	0.999977	−	0.00671605i	$-0.00213780\pi$
$62$		0			0
$63$	25.1531			0.399256
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−86.0061			−1.28367			−0.641837	−	0.766841i	$-0.721827\pi$
$67$	−86.0061			−1.28367			−0.641837	+	0.766841i	$0.721827\pi$
$68$		0			0
$69$	−46.7252	+	89.4267i	−0.677177	+	1.29604i
$70$		0			0
$71$	20.1891			0.284353			0.142176	−	0.989841i	$-0.454590\pi$
$71$	20.1891			0.284353			0.142176	+	0.989841i	$0.454590\pi$
$72$		0			0
$73$			34.6295i			0.474377i	0.971464	+	0.237188i	$0.0762259\pi$
$73$			34.6295i			0.474377i	−0.971464	+	0.237188i	$0.923774\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			7.55495i			0.0981162i
$78$		0			0
$79$			113.703i			1.43928i	0.694348	+	0.719639i	$0.255693\pi$
$79$			113.703i			1.43928i	−0.694348	+	0.719639i	$0.744307\pi$
$80$		0			0
$81$	−68.2499			−0.842592
$82$		0			0
$83$	−88.9041			−1.07113			−0.535567	−	0.844493i	$-0.679902\pi$
$83$	−88.9041			−1.07113			−0.535567	+	0.844493i	$0.679902\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			204.795i			2.35396i
$88$		0			0
$89$		−	50.7631i		−	0.570372i	−0.958472	−	0.285186i	$-0.907945\pi$
$89$		−	50.7631i		−	0.570372i	0.958472	−	0.285186i	$-0.0920553\pi$
$90$		0			0
$91$			38.2242i			0.420047i
$92$		0			0
$93$		−	54.2388i		−	0.583212i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	29.7851			0.307063			0.153532	−	0.988144i	$-0.450935\pi$
$97$	29.7851			0.307063			0.153532	+	0.988144i	$0.450935\pi$
$98$		0			0
$99$			31.5229i			0.318413i
$100$		0			0
$101$	−0.0442629			−0.000438246			−0.000219123	−	1.00000i	$-0.500070\pi$
$101$	−0.0442629			−0.000438246			−0.000219123		1.00000i	$0.500070\pi$
$102$		0			0
$103$	−166.740			−1.61883			−0.809415	−	0.587237i	$-0.800216\pi$
$103$	−166.740			−1.61883			−0.809415	+	0.587237i	$0.800216\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−142.522			−1.33198			−0.665992	−	0.745959i	$-0.731991\pi$
$107$	−142.522			−1.33198			−0.665992	+	0.745959i	$0.731991\pi$
$108$		0			0
$109$			46.2366i			0.424189i	0.977249	+	0.212095i	$0.0680285\pi$
$109$			46.2366i			0.424189i	−0.977249	+	0.212095i	$0.931972\pi$
$110$		0			0
$111$		−	285.164i		−	2.56904i
$112$		0			0
$113$	52.9534			0.468614			0.234307	−	0.972163i	$-0.424718\pi$
$113$	52.9534			0.468614			0.234307	+	0.972163i	$0.424718\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			159.490i			1.36316i
$118$		0			0
$119$	−34.7706			−0.292190
$120$		0			0
$121$	111.532			0.921751
$122$		0			0
$123$			159.372i			1.29571i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	245.218i		−	1.93085i	−0.260681	−	0.965425i	$-0.583947\pi$
$127$		−	245.218i		−	1.93085i	0.260681	−	0.965425i	$-0.416053\pi$
$128$		0			0
$129$		−	302.002i		−	2.34110i
$130$		0			0
$131$	−150.451			−1.14848			−0.574242	−	0.818686i	$-0.694703\pi$
$131$	−150.451			−1.14848			−0.574242	+	0.818686i	$0.694703\pi$
$132$		0			0
$133$		−	86.7640i		−	0.652361i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−54.1978			−0.395604			−0.197802	−	0.980242i	$-0.563380\pi$
$137$	−54.1978			−0.395604			−0.197802	+	0.980242i	$0.563380\pi$
$138$		0			0
$139$	243.095			1.74889			0.874444	−	0.485126i	$-0.161226\pi$
$139$	243.095			1.74889			0.874444	+	0.485126i	$0.161226\pi$
$140$		0			0
$141$	−316.208			−2.24261
$142$		0			0
$143$	−47.9041			−0.334994
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	188.511i		−	1.28239i
$148$		0			0
$149$			121.366i			0.814538i	0.913308	+	0.407269i	$0.133519\pi$
$149$			121.366i			0.814538i	−0.913308	+	0.407269i	$0.866481\pi$
$150$		0			0
$151$	109.104			0.722545			0.361272	−	0.932460i	$-0.382342\pi$
$151$	109.104			0.722545			0.361272	+	0.932460i	$0.382342\pi$
$152$		0			0
$153$	−145.080			−0.948236
$154$		0			0
$155$		0			0
$156$		0			0
$157$	69.6792			0.443816			0.221908	−	0.975068i	$-0.428771\pi$
$157$	69.6792			0.443816			0.221908	+	0.975068i	$0.428771\pi$
$158$		0			0
$159$			346.782i			2.18102i
$160$		0			0
$161$	−50.0509	−	26.1514i	−0.310875	−	0.162431i
$162$		0			0
$163$		−	8.54169i		−	0.0524030i	−0.999657	−	0.0262015i	$-0.991659\pi$
$163$		−	8.54169i		−	0.0524030i	0.999657	−	0.0262015i	$-0.00834116\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			199.039i			1.19185i	0.803040	+	0.595925i	$0.203215\pi$
$167$			199.039i			1.19185i	−0.803040	+	0.595925i	$0.796785\pi$
$168$		0			0
$169$	−73.3709			−0.434147
$170$		0			0
$171$		−	362.022i		−	2.11709i
$172$		0			0
$173$		−	36.4646i		−	0.210778i	−0.994431	−	0.105389i	$-0.966391\pi$
$173$		−	36.4646i		−	0.210778i	0.994431	−	0.105389i	$-0.0336088\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			68.4948i			0.386976i
$178$		0			0
$179$	108.394			0.605555			0.302777	−	0.953061i	$-0.402086\pi$
$179$	108.394			0.605555			0.302777	+	0.953061i	$0.402086\pi$
$180$		0			0
$181$		−	191.406i		−	1.05749i	−0.848779	−	0.528747i	$-0.822662\pi$
$181$		−	191.406i		−	1.05749i	0.848779	−	0.528747i	$-0.177338\pi$
$182$		0			0
$183$	3.59441			0.0196416
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	43.5759i		−	0.233026i
$188$		0			0
$189$			13.4051i			0.0709267i
$190$		0			0
$191$			100.395i			0.525629i	0.964846	+	0.262814i	$0.0846506\pi$
$191$			100.395i			0.525629i	−0.964846	+	0.262814i	$0.915349\pi$
$192$		0			0
$193$			173.428i			0.898592i	0.893383	+	0.449296i	$0.148325\pi$
$193$			173.428i			0.898592i	−0.893383	+	0.449296i	$0.851675\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			19.4500i			0.0987307i	0.998781	+	0.0493654i	$0.0157199\pi$
$197$			19.4500i			0.0987307i	−0.998781	+	0.0493654i	$0.984280\pi$
$198$		0			0
$199$			23.0162i			0.115659i	0.998326	+	0.0578297i	$0.0184181\pi$
$199$			23.0162i			0.115659i	−0.998326	+	0.0578297i	$0.981582\pi$
$200$		0			0
$201$		−	377.297i		−	1.87710i
$202$		0			0
$203$	−114.621			−0.564634
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−208.837	−	109.117i	−1.00887	−	0.527133i
$208$		0			0
$209$	108.736			0.520268
$210$		0			0
$211$	−88.3704			−0.418817			−0.209409	−	0.977828i	$-0.567154\pi$
$211$	−88.3704			−0.418817			−0.209409	+	0.977828i	$0.567154\pi$
$212$		0			0
$213$			88.5666i			0.415806i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	30.3567			0.139893
$218$		0			0
$219$	−151.915			−0.693675
$220$		0			0
$221$		−	220.472i		−	0.997613i
$222$		0			0
$223$			332.396i			1.49057i	0.666748	+	0.745283i	$0.267686\pi$
$223$			332.396i			1.49057i	−0.666748	+	0.745283i	$0.732314\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	390.309			1.71942			0.859712	−	0.510779i	$-0.170643\pi$
$227$	390.309			1.71942			0.859712	+	0.510779i	$0.170643\pi$
$228$		0			0
$229$			156.228i			0.682217i	0.940024	+	0.341109i	$0.110802\pi$
$229$			156.228i			0.682217i	−0.940024	+	0.341109i	$0.889198\pi$
$230$		0			0
$231$	−33.1425			−0.143474
$232$		0			0
$233$			290.131i			1.24520i	0.782541	+	0.622599i	$0.213923\pi$
$233$			290.131i			1.24520i	−0.782541	+	0.622599i	$0.786077\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−498.800			−2.10464
$238$		0			0
$239$	−386.670			−1.61787			−0.808934	−	0.587900i	$-0.799955\pi$
$239$	−386.670			−1.61787			−0.808934	+	0.587900i	$0.799955\pi$
$240$		0			0
$241$			78.3248i			0.324999i	0.986709	+	0.162500i	$0.0519556\pi$
$241$			78.3248i			0.324999i	−0.986709	+	0.162500i	$0.948044\pi$
$242$		0			0
$243$		−	348.541i		−	1.43432i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	550.150			2.22733
$248$		0			0
$249$		−	390.010i		−	1.56631i
$250$		0			0
$251$		−	167.063i		−	0.665591i	−0.942999	−	0.332795i	$-0.892008\pi$
$251$		−	167.063i		−	0.665591i	0.942999	−	0.332795i	$-0.107992\pi$
$252$		0			0
$253$	32.7740	−	62.7257i	0.129542	−	0.247928i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			2.84344i			0.0110640i	0.999985	+	0.00553198i	$0.00176089\pi$
$257$			2.84344i			0.0110640i	−0.999985	+	0.00553198i	$0.998239\pi$
$258$		0			0
$259$	159.602			0.616225
$260$		0			0
$261$	−478.253			−1.83239
$262$		0			0
$263$	−407.811			−1.55061			−0.775306	−	0.631586i	$-0.782404\pi$
$263$	−407.811			−1.55061			−0.775306	+	0.631586i	$0.782404\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	222.691			0.834048
$268$		0			0
$269$	−72.2061			−0.268424			−0.134212	−	0.990953i	$-0.542850\pi$
$269$	−72.2061			−0.268424			−0.134212	+	0.990953i	$0.542850\pi$
$270$		0			0
$271$	−357.105			−1.31773			−0.658865	−	0.752261i	$-0.728963\pi$
$271$	−357.105			−1.31773			−0.658865	+	0.752261i	$0.728963\pi$
$272$		0			0
$273$	−167.685			−0.614229
$274$		0			0
$275$		0			0
$276$		0			0
$277$			295.538i			1.06692i	0.845824	+	0.533462i	$0.179109\pi$
$277$			295.538i			1.06692i	−0.845824	+	0.533462i	$0.820891\pi$
$278$		0			0
$279$	126.663			0.453989
$280$		0			0
$281$		−	487.814i		−	1.73599i	−0.496570	−	0.867997i	$-0.665407\pi$
$281$		−	487.814i		−	1.73599i	0.496570	−	0.867997i	$-0.334593\pi$
$282$		0			0
$283$	−207.745			−0.734081			−0.367041	−	0.930205i	$-0.619629\pi$
$283$	−207.745			−0.734081			−0.367041	+	0.930205i	$0.619629\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−89.1984			−0.310796
$288$		0			0
$289$	−88.4475			−0.306047
$290$		0			0
$291$			130.663i			0.449015i
$292$		0			0
$293$	−461.647			−1.57559			−0.787793	−	0.615939i	$-0.788777\pi$
$293$	−461.647			−1.57559			−0.787793	+	0.615939i	$0.788777\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−16.7998			−0.0565652
$298$		0			0
$299$	165.820	−	317.361i	0.554582	−	1.06141i
$300$		0			0
$301$	169.026			0.561549
$302$		0			0
$303$		−	0.194175i		−	0.000640842i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	125.828i		−	0.409865i	−0.978776	−	0.204932i	$-0.934303\pi$
$307$		−	125.828i		−	0.409865i	0.978776	−	0.204932i	$-0.0656974\pi$
$308$		0			0
$309$		−	731.464i		−	2.36720i
$310$		0			0
$311$	360.853			1.16030			0.580150	−	0.814510i	$-0.302994\pi$
$311$	360.853			1.16030			0.580150	+	0.814510i	$0.302994\pi$
$312$		0			0
$313$	−422.870			−1.35102			−0.675511	−	0.737350i	$-0.736077\pi$
$313$	−422.870			−1.35102			−0.675511	+	0.737350i	$0.736077\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	515.381i		−	1.62581i	−0.582399	−	0.812903i	$-0.697886\pi$
$317$		−	515.381i		−	1.62581i	0.582399	−	0.812903i	$-0.302114\pi$
$318$		0			0
$319$		−	143.647i		−	0.450305i
$320$		0			0
$321$		−	625.225i		−	1.94774i
$322$		0			0
$323$			500.443i			1.54936i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−202.834			−0.620286
$328$		0			0
$329$		−	176.977i		−	0.537925i
$330$		0			0
$331$	−200.055			−0.604396			−0.302198	−	0.953245i	$-0.597720\pi$
$331$	−200.055			−0.604396			−0.302198	+	0.953245i	$0.597720\pi$
$332$		0			0
$333$	665.938			1.99981
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−256.390			−0.760801			−0.380400	−	0.924822i	$-0.624214\pi$
$337$	−256.390			−0.760801			−0.380400	+	0.924822i	$0.624214\pi$
$338$		0			0
$339$			232.299i			0.685248i
$340$		0			0
$341$			38.0442i			0.111567i
$342$		0			0
$343$	225.815			0.658353
$344$		0			0
$345$		0			0
$346$		0			0
$347$			210.605i			0.606932i	0.952842	+	0.303466i	$0.0981439\pi$
$347$			210.605i			0.606932i	−0.952842	+	0.303466i	$0.901856\pi$
$348$		0			0
$349$	35.2906			0.101119			0.0505595	−	0.998721i	$-0.483900\pi$
$349$	35.2906			0.101119			0.0505595	+	0.998721i	$0.483900\pi$
$350$		0			0
$351$	−84.9988			−0.242162
$352$		0			0
$353$		−	255.855i		−	0.724801i	−0.932023	−	0.362400i	$-0.881957\pi$
$353$		−	255.855i		−	0.724801i	0.932023	−	0.362400i	$-0.118043\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		−	152.534i		−	0.427266i
$358$		0			0
$359$			533.701i			1.48663i	0.668940	+	0.743317i	$0.266748\pi$
$359$			533.701i			1.48663i	−0.668940	+	0.743317i	$0.733252\pi$
$360$		0			0
$361$	−887.769			−2.45919
$362$		0			0
$363$			489.275i			1.34786i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−162.646			−0.443176			−0.221588	−	0.975140i	$-0.571124\pi$
$367$	−162.646			−0.443176			−0.221588	+	0.975140i	$0.571124\pi$
$368$		0			0
$369$	−372.179			−1.00862
$370$		0			0
$371$	−194.089			−0.523152
$372$		0			0
$373$	−172.545			−0.462587			−0.231293	−	0.972884i	$-0.574296\pi$
$373$	−172.545			−0.462587			−0.231293	+	0.972884i	$0.574296\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	726.783i		−	1.92781i
$378$		0			0
$379$			670.313i			1.76864i	0.466886	+	0.884318i	$0.345376\pi$
$379$			670.313i			1.76864i	−0.466886	+	0.884318i	$0.654624\pi$
$380$		0			0
$381$	1075.74			2.82346
$382$		0			0
$383$	−49.5674			−0.129419			−0.0647094	−	0.997904i	$-0.520612\pi$
$383$	−49.5674			−0.129419			−0.0647094	+	0.997904i	$0.520612\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	705.261			1.82238
$388$		0			0
$389$			365.506i			0.939603i	0.882772	+	0.469802i	$0.155675\pi$
$389$			365.506i			0.939603i	−0.882772	+	0.469802i	$0.844325\pi$
$390$		0			0
$391$	288.687	+	150.838i	0.738330	+	0.385775i
$392$		0			0
$393$		−	660.010i		−	1.67941i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			309.764i			0.780262i	0.920759	+	0.390131i	$0.127570\pi$
$397$			309.764i			0.780262i	−0.920759	+	0.390131i	$0.872430\pi$
$398$		0			0
$399$	380.622			0.953939
$400$		0			0
$401$			774.795i			1.93216i	0.258251	+	0.966078i	$0.416854\pi$
$401$			774.795i			1.93216i	−0.258251	+	0.966078i	$0.583146\pi$
$402$		0			0
$403$			192.485i			0.477629i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			200.020i			0.491449i
$408$		0			0
$409$	−488.600			−1.19462			−0.597310	−	0.802010i	$-0.703764\pi$
$409$	−488.600			−1.19462			−0.597310	+	0.802010i	$0.703764\pi$
$410$		0			0
$411$		−	237.758i		−	0.578487i
$412$		0			0
$413$	−38.3356			−0.0928223
$414$		0			0
$415$		0			0
$416$		0			0
$417$			1066.43i			2.55738i
$418$		0			0
$419$			236.298i			0.563956i	0.959421	+	0.281978i	$0.0909905\pi$
$419$			236.298i			0.563956i	−0.959421	+	0.281978i	$0.909009\pi$
$420$		0			0
$421$		−	395.008i		−	0.938261i	−0.883129	−	0.469131i	$-0.844567\pi$
$421$		−	395.008i		−	0.938261i	0.883129	−	0.469131i	$-0.155433\pi$
$422$		0			0
$423$		−	738.435i		−	1.74571i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			2.01174i			0.00471134i
$428$		0			0
$429$		−	210.149i		−	0.489857i
$430$		0			0
$431$			307.817i			0.714194i	0.934067	+	0.357097i	$0.116233\pi$
$431$			307.817i			0.714194i	−0.934067	+	0.357097i	$0.883767\pi$
$432$		0			0
$433$	531.655			1.22784			0.613921	−	0.789368i	$-0.289592\pi$
$433$	531.655			1.22784			0.613921	+	0.789368i	$0.289592\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−376.390	+	720.367i	−0.861304	+	1.64844i
$438$		0			0
$439$	477.183			1.08698			0.543488	−	0.839417i	$-0.317103\pi$
$439$	477.183			1.08698			0.543488	+	0.839417i	$0.317103\pi$
$440$		0			0
$441$	440.226			0.998245
$442$		0			0
$443$		−	526.430i		−	1.18833i	−0.804344	−	0.594164i	$-0.797483\pi$
$443$		−	526.430i		−	1.18833i	0.804344	−	0.594164i	$-0.202517\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−532.417			−1.19109
$448$		0			0
$449$	−423.713			−0.943681			−0.471840	−	0.881684i	$-0.656410\pi$
$449$	−423.713			−0.943681			−0.471840	+	0.881684i	$0.656410\pi$
$450$		0			0
$451$		−	111.787i		−	0.247865i
$452$		0			0
$453$			478.625i			1.05657i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−107.913			−0.236134			−0.118067	−	0.993006i	$-0.537670\pi$
$457$	−107.913			−0.236134			−0.118067	+	0.993006i	$0.537670\pi$
$458$		0			0
$459$		−	77.3191i		−	0.168451i
$460$		0			0
$461$	−239.642			−0.519830			−0.259915	−	0.965632i	$-0.583695\pi$
$461$	−239.642			−0.519830			−0.259915	+	0.965632i	$0.583695\pi$
$462$		0			0
$463$			168.503i			0.363938i	0.983304	+	0.181969i	$0.0582471\pi$
$463$			168.503i			0.363938i	−0.983304	+	0.181969i	$0.941753\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	181.802			0.389297			0.194648	−	0.980873i	$-0.437643\pi$
$467$	181.802			0.389297			0.194648	+	0.980873i	$0.437643\pi$
$468$		0			0
$469$	211.168			0.450251
$470$		0			0
$471$			305.673i			0.648987i
$472$		0			0
$473$			211.831i			0.447845i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−809.835			−1.69777
$478$		0			0
$479$			607.374i			1.26801i	0.773331	+	0.634003i	$0.218589\pi$
$479$			607.374i			1.26801i	−0.773331	+	0.634003i	$0.781411\pi$
$480$		0			0
$481$			1012.00i			2.10395i
$482$		0			0
$483$	114.723	−	219.566i	0.237521	−	0.454589i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	531.999i		−	1.09240i	−0.837655	−	0.546200i	$-0.816074\pi$
$487$		−	531.999i		−	1.09240i	0.837655	−	0.546200i	$-0.183926\pi$
$488$		0			0
$489$	37.4712			0.0766283
$490$		0			0
$491$	564.887			1.15048			0.575241	−	0.817984i	$-0.304908\pi$
$491$	564.887			1.15048			0.575241	+	0.817984i	$0.304908\pi$
$492$		0			0
$493$	661.117			1.34101
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−49.5695			−0.0997374
$498$		0			0
$499$	291.197			0.583560			0.291780	−	0.956485i	$-0.405752\pi$
$499$	291.197			0.583560			0.291780	+	0.956485i	$0.405752\pi$
$500$		0			0
$501$	−873.157			−1.74283
$502$		0			0
$503$	22.0340			0.0438053			0.0219026	−	0.999760i	$-0.493028\pi$
$503$	22.0340			0.0438053			0.0219026	+	0.999760i	$0.493028\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	321.868i		−	0.634848i
$508$		0			0
$509$	795.664			1.56319			0.781596	−	0.623786i	$-0.214406\pi$
$509$	795.664			1.56319			0.781596	+	0.623786i	$0.214406\pi$
$510$		0			0
$511$		−	85.0246i		−	0.166389i
$512$		0			0
$513$	192.936			0.376094
$514$		0			0
$515$		0			0
$516$		0			0
$517$	221.795			0.429004
$518$		0			0
$519$	159.965			0.308218
$520$		0			0
$521$			312.095i			0.599030i	0.954092	+	0.299515i	$0.0968249\pi$
$521$			312.095i			0.599030i	−0.954092	+	0.299515i	$0.903175\pi$
$522$		0			0
$523$	347.028			0.663533			0.331766	−	0.943362i	$-0.392355\pi$
$523$	347.028			0.663533			0.331766	+	0.943362i	$0.392355\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−175.093			−0.332245
$528$		0			0
$529$	302.106	+	434.250i	0.571088	+	0.820889i
$530$		0			0
$531$	−159.955			−0.301233
$532$		0			0
$533$		−	565.586i		−	1.06114i
$534$		0			0
$535$		0			0
$536$		0			0
$537$			475.511i			0.885495i
$538$		0			0
$539$			132.225i			0.245316i
$540$		0			0
$541$	649.815			1.20114			0.600568	−	0.799574i	$-0.294941\pi$
$541$	649.815			1.20114			0.600568	+	0.799574i	$0.294941\pi$
$542$		0			0
$543$	839.674			1.54636
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	647.571i		−	1.18386i	−0.805990	−	0.591929i	$-0.798366\pi$
$547$		−	647.571i		−	1.18386i	0.805990	−	0.591929i	$-0.201634\pi$
$548$		0			0
$549$			8.39397i			0.0152896i
$550$		0			0
$551$			1649.70i			2.99401i
$552$		0			0
$553$		−	279.171i		−	0.504830i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	151.319			0.271668			0.135834	−	0.990732i	$-0.456629\pi$
$557$	151.319			0.271668			0.135834	+	0.990732i	$0.456629\pi$
$558$		0			0
$559$			1071.76i			1.91727i
$560$		0			0
$561$	191.162			0.340752
$562$		0			0
$563$	−395.021			−0.701637			−0.350818	−	0.936444i	$-0.614097\pi$
$563$	−395.021			−0.701637			−0.350818	+	0.936444i	$0.614097\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	167.572			0.295541
$568$		0			0
$569$		−	1091.70i		−	1.91864i	−0.282325	−	0.959319i	$-0.591106\pi$
$569$		−	1091.70i		−	1.91864i	0.282325	−	0.959319i	$-0.408894\pi$
$570$		0			0
$571$		−	204.628i		−	0.358368i	−0.983816	−	0.179184i	$-0.942654\pi$
$571$		−	204.628i		−	0.358368i	0.983816	−	0.179184i	$-0.0573457\pi$
$572$		0			0
$573$	−440.419			−0.768620
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	70.2706i		−	0.121786i	−0.998144	−	0.0608931i	$-0.980605\pi$
$577$		−	70.2706i		−	0.121786i	0.998144	−	0.0608931i	$-0.0193949\pi$
$578$		0			0
$579$	−760.806			−1.31400
$580$		0			0
$581$	218.283			0.375703
$582$		0			0
$583$		−	243.240i		−	0.417222i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	782.912i		−	1.33375i	−0.745169	−	0.666876i	$-0.767631\pi$
$587$		−	782.912i		−	1.33375i	0.745169	−	0.666876i	$-0.232369\pi$
$588$		0			0
$589$		−	436.915i		−	0.741791i
$590$		0			0
$591$	−85.3243			−0.144373
$592$		0			0
$593$			703.518i			1.18637i	0.805066	+	0.593185i	$0.202130\pi$
$593$			703.518i			1.18637i	−0.805066	+	0.593185i	$0.797870\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−100.969			−0.169127
$598$		0			0
$599$	733.329			1.22426			0.612128	−	0.790759i	$-0.290314\pi$
$599$	733.329			1.22426			0.612128	+	0.790759i	$0.290314\pi$
$600$		0			0
$601$	503.715			0.838129			0.419064	−	0.907956i	$-0.362358\pi$
$601$	503.715			0.838129			0.419064	+	0.907956i	$0.362358\pi$
$602$		0			0
$603$	881.095			1.46119
$604$		0			0
$605$		0			0
$606$		0			0
$607$			330.458i			0.544412i	0.962239	+	0.272206i	$0.0877532\pi$
$607$			330.458i			0.544412i	−0.962239	+	0.272206i	$0.912247\pi$
$608$		0			0
$609$		−	502.825i		−	0.825657i
$610$		0			0
$611$	1122.17			1.83661
$612$		0			0
$613$	−198.653			−0.324068			−0.162034	−	0.986785i	$-0.551805\pi$
$613$	−198.653			−0.324068			−0.162034	+	0.986785i	$0.551805\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−615.338			−0.997306			−0.498653	−	0.866802i	$-0.666172\pi$
$617$	−615.338			−0.997306			−0.498653	+	0.866802i	$0.666172\pi$
$618$		0			0
$619$		−	923.689i		−	1.49223i	−0.665818	−	0.746114i	$-0.731917\pi$
$619$		−	923.689i		−	1.49223i	0.665818	−	0.746114i	$-0.268083\pi$
$620$		0			0
$621$	58.1527	−	111.298i	0.0936436	−	0.179223i
$622$		0			0
$623$			124.637i			0.200059i
$624$		0			0
$625$		0			0
$626$		0			0
$627$			477.010i			0.760782i
$628$		0			0
$629$	−920.564			−1.46354
$630$		0			0
$631$		−	44.8428i		−	0.0710662i	−0.999368	−	0.0355331i	$-0.988687\pi$
$631$		−	44.8428i		−	0.0710662i	0.999368	−	0.0355331i	$-0.0113129\pi$
$632$		0			0
$633$		−	387.669i		−	0.612431i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			668.994i			1.05023i
$638$		0			0
$639$	−206.828			−0.323675
$640$		0			0
$641$			489.747i			0.764036i	0.924155	+	0.382018i	$0.124771\pi$
$641$			489.747i			0.764036i	−0.924155	+	0.382018i	$0.875229\pi$
$642$		0			0
$643$	−598.613			−0.930969			−0.465485	−	0.885056i	$-0.654120\pi$
$643$	−598.613			−0.930969			−0.465485	+	0.885056i	$0.654120\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			239.957i			0.370876i	0.982656	+	0.185438i	$0.0593704\pi$
$647$			239.957i			0.370876i	−0.982656	+	0.185438i	$0.940630\pi$
$648$		0			0
$649$		−	48.0437i		−	0.0740273i
$650$		0			0
$651$			133.171i			0.204563i
$652$		0			0
$653$			656.132i			1.00480i	0.864636	+	0.502398i	$0.167549\pi$
$653$			656.132i			1.00480i	−0.864636	+	0.502398i	$0.832451\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	354.764i		−	0.539976i
$658$		0			0
$659$		−	475.996i		−	0.722301i	−0.932508	−	0.361150i	$-0.882384\pi$
$659$		−	475.996i		−	0.722301i	0.932508	−	0.361150i	$-0.117616\pi$
$660$		0			0
$661$			385.603i			0.583363i	0.956516	+	0.291681i	$0.0942147\pi$
$661$			385.603i			0.583363i	−0.956516	+	0.291681i	$0.905785\pi$
$662$		0			0
$663$	967.182			1.45880
$664$		0			0
$665$		0			0
$666$		0			0
$667$	951.650	+	497.235i	1.42676	+	0.745479i
$668$		0			0
$669$	−1458.18			−2.17964
$670$		0			0
$671$	−2.52120			−0.00375737
$672$		0			0
$673$			764.494i			1.13595i	0.823046	+	0.567975i	$0.192273\pi$
$673$			764.494i			1.13595i	−0.823046	+	0.567975i	$0.807727\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	663.808			0.980515			0.490257	−	0.871578i	$-0.336903\pi$
$677$	663.808			0.980515			0.490257	+	0.871578i	$0.336903\pi$
$678$		0			0
$679$	−73.1304			−0.107703
$680$		0			0
$681$			1712.23i			2.51429i
$682$		0			0
$683$			837.423i			1.22609i	0.790046	+	0.613047i	$0.210057\pi$
$683$			837.423i			1.22609i	−0.790046	+	0.613047i	$0.789943\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−685.350			−0.997598
$688$		0			0
$689$		−	1230.67i		−	1.78617i
$690$		0			0
$691$	665.022			0.962406			0.481203	−	0.876609i	$-0.340200\pi$
$691$	665.022			0.962406			0.481203	+	0.876609i	$0.340200\pi$
$692$		0			0
$693$		−	77.3971i		−	0.111684i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	514.485			0.738142
$698$		0			0
$699$	−1272.77			−1.82084
$700$		0			0
$701$		−	337.288i		−	0.481152i	−0.970630	−	0.240576i	$-0.922664\pi$
$701$		−	337.288i		−	0.481152i	0.970630	−	0.240576i	$-0.0773364\pi$
$702$		0			0
$703$		−	2297.11i		−	3.26758i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.108677			0.000153716
$708$		0			0
$709$			1192.17i			1.68148i	0.541440	+	0.840739i	$0.317879\pi$
$709$			1192.17i			1.68148i	−0.541440	+	0.840739i	$0.682121\pi$
$710$		0			0
$711$		−	1164.84i		−	1.63831i
$712$		0			0
$713$	−252.040	−	131.690i	−0.353492	−	0.184698i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	1696.27i		−	2.36579i
$718$		0			0
$719$	694.141			0.965426			0.482713	−	0.875779i	$-0.339651\pi$
$719$	694.141			0.965426			0.482713	+	0.875779i	$0.339651\pi$
$720$		0			0
$721$	409.390			0.567809
$722$		0			0
$723$	−343.600			−0.475242
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−284.314			−0.391078			−0.195539	−	0.980696i	$-0.562646\pi$
$727$	−284.314			−0.391078			−0.195539	+	0.980696i	$0.562646\pi$
$728$		0			0
$729$	914.752			1.25480
$730$		0			0
$731$	−974.922			−1.33368
$732$		0			0
$733$	151.817			0.207118			0.103559	−	0.994623i	$-0.466977\pi$
$733$	151.817			0.207118			0.103559	+	0.994623i	$0.466977\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			264.644i			0.359083i
$738$		0			0
$739$	621.688			0.841256			0.420628	−	0.907233i	$-0.361810\pi$
$739$	621.688			0.841256			0.420628	+	0.907233i	$0.361810\pi$
$740$		0			0
$741$			2413.43i			3.25699i
$742$		0			0
$743$	1014.19			1.36500			0.682499	−	0.730886i	$-0.260893\pi$
$743$	1014.19			1.36500			0.682499	+	0.730886i	$0.260893\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	910.784			1.21926
$748$		0			0
$749$	349.930			0.467196
$750$		0			0
$751$		−	565.970i		−	0.753622i	−0.926290	−	0.376811i	$-0.877021\pi$
$751$		−	565.970i		−	0.753622i	0.926290	−	0.376811i	$-0.122979\pi$
$752$		0			0
$753$	732.884			0.973285
$754$		0			0
$755$		0			0
$756$		0			0
$757$	897.599			1.18573			0.592866	−	0.805301i	$-0.297996\pi$
$757$	897.599			1.18573			0.592866	+	0.805301i	$0.297996\pi$
$758$		0			0
$759$	275.169	+	143.775i	0.362542	+	0.189427i
$760$		0			0
$761$	1250.72			1.64352			0.821758	−	0.569836i	$-0.192993\pi$
$761$	1250.72			1.64352			0.821758	+	0.569836i	$0.192993\pi$
$762$		0			0
$763$		−	113.523i		−	0.148785i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	243.077i		−	0.316919i
$768$		0			0
$769$			232.071i			0.301783i	0.988550	+	0.150891i	$0.0482143\pi$
$769$			232.071i			0.301783i	−0.988550	+	0.150891i	$0.951786\pi$
$770$		0			0
$771$	−12.4738			−0.0161787
$772$		0			0
$773$	−563.021			−0.728359			−0.364179	−	0.931329i	$-0.618651\pi$
$773$	−563.021			−0.728359			−0.364179	+	0.931329i	$0.618651\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$			700.153i			0.901098i
$778$		0			0
$779$			1283.81i			1.64802i
$780$		0			0
$781$		−	62.1225i		−	0.0795422i
$782$		0			0
$783$		−	254.881i		−	0.325518i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	527.934			0.670819			0.335409	−	0.942072i	$-0.391125\pi$
$787$	527.934			0.670819			0.335409	+	0.942072i	$0.391125\pi$
$788$		0			0
$789$		−	1789.01i		−	2.26744i
$790$		0			0
$791$	−130.015			−0.164367
$792$		0			0
$793$	−12.7560			−0.0160857
$794$		0			0
$795$		0			0
$796$		0			0
$797$	233.898			0.293473			0.146736	−	0.989176i	$-0.453123\pi$
$797$	233.898			0.293473			0.146736	+	0.989176i	$0.453123\pi$
$798$		0			0
$799$			1020.78i			1.27757i
$800$		0			0
$801$			520.046i			0.649246i
$802$		0			0
$803$	106.556			0.132698
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	316.758i		−	0.392513i
$808$		0			0
$809$	−1236.80			−1.52880			−0.764400	−	0.644742i	$-0.776965\pi$
$809$	−1236.80			−1.52880			−0.764400	+	0.644742i	$0.776965\pi$
$810$		0			0
$811$	−872.659			−1.07603			−0.538014	−	0.842936i	$-0.680825\pi$
$811$	−872.659			−1.07603			−0.538014	+	0.842936i	$0.680825\pi$
$812$		0			0
$813$		−	1566.57i		−	1.92690i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	2432.75i		−	2.97766i
$818$		0			0
$819$		−	391.591i		−	0.478133i
$820$		0			0
$821$	−411.704			−0.501466			−0.250733	−	0.968056i	$-0.580672\pi$
$821$	−411.704			−0.501466			−0.250733	+	0.968056i	$0.580672\pi$
$822$		0			0
$823$		−	1023.52i		−	1.24364i	−0.783159	−	0.621821i	$-0.786393\pi$
$823$		−	1023.52i		−	1.24364i	0.783159	−	0.621821i	$-0.213607\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	929.910			1.12444			0.562219	−	0.826988i	$-0.309948\pi$
$827$	929.910			1.12444			0.562219	+	0.826988i	$0.309948\pi$
$828$		0			0
$829$	−67.6740			−0.0816332			−0.0408166	−	0.999167i	$-0.512996\pi$
$829$	−67.6740			−0.0816332			−0.0408166	+	0.999167i	$0.512996\pi$
$830$		0			0
$831$	−1296.48			−1.56015
$832$		0			0
$833$	−608.550			−0.730552
$834$		0			0
$835$		0			0
$836$		0			0
$837$			67.5038i			0.0806497i
$838$		0			0
$839$		−	581.263i		−	0.692805i	−0.938086	−	0.346402i	$-0.887403\pi$
$839$		−	581.263i		−	0.692805i	0.938086	−	0.346402i	$-0.112597\pi$
$840$		0			0
$841$	1338.36			1.59139
$842$		0			0
$843$	2139.97			2.53852
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−273.840			−0.323306
$848$		0			0
$849$		−	911.349i		−	1.07344i
$850$		0			0
$851$	−1325.11	−	692.368i	−1.55713	−	0.813594i
$852$		0			0
$853$			503.134i			0.589841i	0.955522	+	0.294920i	$0.0952931\pi$
$853$			503.134i			0.589841i	−0.955522	+	0.294920i	$0.904707\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			398.956i			0.465527i	0.972533	+	0.232763i	$0.0747767\pi$
$857$			398.956i			0.465527i	−0.972533	+	0.232763i	$0.925223\pi$
$858$		0			0
$859$	−478.946			−0.557562			−0.278781	−	0.960355i	$-0.589930\pi$
$859$	−478.946			−0.557562			−0.278781	+	0.960355i	$0.589930\pi$
$860$		0			0
$861$		−	391.301i		−	0.454473i
$862$		0			0
$863$			89.9267i			0.104202i	0.998642	+	0.0521012i	$0.0165918\pi$
$863$			89.9267i			0.104202i	−0.998642	+	0.0521012i	$0.983408\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	388.007i		−	0.447529i
$868$		0			0
$869$	349.868			0.402610
$870$		0			0
$871$			1338.97i			1.53727i
$872$		0			0
$873$	−305.136			−0.349526
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	812.731i		−	0.926718i	−0.886171	−	0.463359i	$-0.846644\pi$
$877$		−	812.731i		−	0.926718i	0.886171	−	0.463359i	$-0.153356\pi$
$878$		0			0
$879$		−	2025.18i		−	2.30396i
$880$		0			0
$881$			780.094i			0.885465i	0.896654	+	0.442732i	$0.145991\pi$
$881$			780.094i			0.885465i	−0.896654	+	0.442732i	$0.854009\pi$
$882$		0			0
$883$			510.424i			0.578057i	0.957321	+	0.289028i	$0.0933322\pi$
$883$			510.424i			0.578057i	−0.957321	+	0.289028i	$0.906668\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	420.588i		−	0.474169i	−0.971489	−	0.237085i	$-0.923808\pi$
$887$		−	420.588i		−	0.474169i	0.971489	−	0.237085i	$-0.0761918\pi$
$888$		0			0
$889$			602.075i			0.677250i
$890$		0			0
$891$			210.008i			0.235699i
$892$		0			0
$893$	−2547.18			−2.85239
$894$		0			0
$895$		0			0
$896$		0			0
$897$	1392.22	+	727.430i	1.55208	+	0.810959i
$898$		0			0
$899$	−577.192			−0.642037
$900$		0			0
$901$	1119.48			1.24249
$902$		0			0
$903$			741.496i			0.821147i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	551.424			0.607965			0.303982	−	0.952678i	$-0.401684\pi$
$907$	551.424			0.607965			0.303982	+	0.952678i	$0.401684\pi$
$908$		0			0
$909$	0.453454			0.000498849
$910$		0			0
$911$		−	749.871i		−	0.823129i	−0.911381	−	0.411565i	$-0.864982\pi$
$911$		−	749.871i		−	0.823129i	0.911381	−	0.411565i	$-0.135018\pi$
$912$		0			0
$913$			273.561i			0.299629i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	369.398			0.402833
$918$		0			0
$919$			1094.12i			1.19055i	0.803522	+	0.595275i	$0.202957\pi$
$919$			1094.12i			1.19055i	−0.803522	+	0.595275i	$0.797043\pi$
$920$		0			0
$921$	551.992			0.599340
$922$		0			0
$923$		−	314.309i		−	0.340529i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	1708.17			1.84269
$928$		0			0
$929$	1384.10			1.48988			0.744939	−	0.667132i	$-0.232478\pi$
$929$	1384.10			1.48988			0.744939	+	0.667132i	$0.232478\pi$
$930$		0			0
$931$		−	1518.53i		−	1.63107i
$932$		0			0
$933$			1583.01i			1.69669i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	756.867			0.807756			0.403878	−	0.914813i	$-0.367662\pi$
$937$	756.867			0.807756			0.403878	+	0.914813i	$0.367662\pi$
$938$		0			0
$939$		−	1855.07i		−	1.97558i
$940$		0			0
$941$		−	389.483i		−	0.413904i	−0.978351	−	0.206952i	$-0.933646\pi$
$941$		−	389.483i		−	0.413904i	0.978351	−	0.206952i	$-0.0663544\pi$
$942$		0			0
$943$	740.579	+	386.950i	0.785344	+	0.410340i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1214.13i		−	1.28208i	−0.767507	−	0.641041i	$-0.778503\pi$
$947$		−	1214.13i		−	1.28208i	0.767507	−	0.641041i	$-0.221497\pi$
$948$		0			0
$949$	539.121			0.568094
$950$		0			0
$951$	2260.90			2.37740
$952$		0			0
$953$	448.128			0.470229			0.235114	−	0.971968i	$-0.424453\pi$
$953$	448.128			0.470229			0.235114	+	0.971968i	$0.424453\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	630.160			0.658475
$958$		0			0
$959$	133.070			0.138759
$960$		0			0
$961$	−808.134			−0.840930
$962$		0			0
$963$	1460.08			1.51618
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	510.756i		−	0.528186i	−0.964497	−	0.264093i	$-0.914927\pi$
$967$		−	510.756i		−	0.528186i	0.964497	−	0.264093i	$-0.0850725\pi$
$968$		0			0
$969$	−2195.38			−2.26561
$970$		0			0
$971$		−	1188.95i		−	1.22446i	−0.790682	−	0.612228i	$-0.790274\pi$
$971$		−	1188.95i		−	1.22446i	0.790682	−	0.612228i	$-0.209726\pi$
$972$		0			0
$973$	−596.864			−0.613427
$974$		0			0
$975$		0			0
$976$		0			0
$977$	708.996			0.725687			0.362843	−	0.931850i	$-0.381806\pi$
$977$	708.996			0.725687			0.362843	+	0.931850i	$0.381806\pi$
$978$		0			0
$979$	−156.200			−0.159550
$980$		0			0
$981$		−	473.674i		−	0.482848i
$982$		0			0
$983$	1066.61			1.08506			0.542530	−	0.840036i	$-0.317466\pi$
$983$	1066.61			1.08506			0.542530	+	0.840036i	$0.317466\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	776.375			0.786601
$988$		0			0
$989$	−1403.36	−	733.251i	−1.41897	−	0.741407i
$990$		0			0
$991$	−229.226			−0.231308			−0.115654	−	0.993290i	$-0.536896\pi$
$991$	−229.226			−0.231308			−0.115654	+	0.993290i	$0.536896\pi$
$992$		0			0
$993$		−	877.615i		−	0.883801i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	1388.05i		−	1.39223i	−0.717932	−	0.696113i	$-0.754911\pi$
$997$		−	1388.05i		−	1.39223i	0.717932	−	0.696113i	$-0.245089\pi$
$998$		0			0
$999$			354.906i			0.355261i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.3.d.c.1149.14		32
5.2	odd	4		2300.3.f.c.1701.13	✓	16
5.3	odd	4		2300.3.f.d.1701.4	yes	16
5.4	even	2	inner	2300.3.d.c.1149.19		32
23.22	odd	2	inner	2300.3.d.c.1149.20		32
115.22	even	4		2300.3.f.c.1701.14	yes	16
115.68	even	4		2300.3.f.d.1701.3	yes	16
115.114	odd	2	inner	2300.3.d.c.1149.13		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2300.3.d.c.1149.13		32	115.114	odd	2	inner
2300.3.d.c.1149.14		32	1.1	even	1	trivial
2300.3.d.c.1149.19		32	5.4	even	2	inner
2300.3.d.c.1149.20		32	23.22	odd	2	inner
2300.3.f.c.1701.13	✓	16	5.2	odd	4
2300.3.f.c.1701.14	yes	16	115.22	even	4
2300.3.f.d.1701.3	yes	16	115.68	even	4
2300.3.f.d.1701.4	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)

\(f(q)\)	\(=\)	\(q+4.38686i q^{3} -2.45527 q^{7} -10.2446 q^{9} +O(q^{10})\)	\(q+4.38686i q^{3} -2.45527 q^{7} -10.2446 q^{9} -3.07704i q^{11} -15.5683i q^{13} +14.1617 q^{17} +35.3379i q^{19} -10.7709i q^{21} +(20.3851 + 10.6512i) q^{23} -5.45975i q^{27} +46.6836 q^{29} -12.3639 q^{31} +13.4985 q^{33} -65.0040 q^{37} +68.2958 q^{39} +36.3294 q^{41} -68.8424 q^{43} +72.0807i q^{47} -42.9717 q^{49} +62.1252i q^{51} +79.0502 q^{53} -155.023 q^{57} +15.6136 q^{59} -0.819358i q^{61} +25.1531 q^{63} -86.0061 q^{67} +(-46.7252 + 89.4267i) q^{69} +20.1891 q^{71} +34.6295i q^{73} +7.55495i q^{77} +113.703i q^{79} -68.2499 q^{81} -88.9041 q^{83} +204.795i q^{87} -50.7631i q^{89} +38.2242i q^{91} -54.2388i q^{93} +29.7851 q^{97} +31.5229i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 128 q^{9}+O(q^{10}) \) 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

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