LMFDB - Embedded newform 1900.3.e.g.1101.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,3,Mod(1101,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1900.1101");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1900.e (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$51.7712502285$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 89x^{12} + 3026x^{10} + 49092x^{8} + 390297x^{6} + 1440495x^{4} + 1994425x^{2} + 151875 $ x^14 + 89x^12 + 3026x^10 + 49092x^8 + 390297x^6 + 1440495x^4 + 1994425x^2 + 151875
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1101.13
Root			$4.84257i$ of defining polynomial
Character	$\chi$	$=$	1900.1101
Dual form			1900.3.e.g.1101.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.84257i q^{3} -6.85668 q^{7} -14.4504 q^{9} +O(q^{10})$	$q+4.84257i q^{3} -6.85668 q^{7} -14.4504 q^{9} -9.59871 q^{11} -10.0805i q^{13} -27.8791 q^{17} +(17.5630 - 7.24850i) q^{19} -33.2039i q^{21} +8.14968 q^{23} -26.3941i q^{27} -17.8174i q^{29} -20.3798i q^{31} -46.4824i q^{33} +65.1090i q^{37} +48.8154 q^{39} +16.0406i q^{41} +60.7072 q^{43} +9.60366 q^{47} -1.98597 q^{49} -135.006i q^{51} +14.7792i q^{53} +(35.1013 + 85.0500i) q^{57} +0.442058i q^{59} +65.2615 q^{61} +99.0820 q^{63} +103.423i q^{67} +39.4653i q^{69} -90.5453i q^{71} +88.8290 q^{73} +65.8153 q^{77} -86.7598i q^{79} -2.23865 q^{81} -47.4163 q^{83} +86.2817 q^{87} -58.0384i q^{89} +69.1187i q^{91} +98.6904 q^{93} -165.286i q^{97} +138.706 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 4 q^{7} - 52 q^{9}+O(q^{10}) $ 14 * q - 4 * q^7 - 52 * q^9	$ 14 q - 4 q^{7} - 52 q^{9} + 4 q^{11} + 6 q^{17} - 29 q^{19} + 28 q^{23} - 56 q^{39} - 22 q^{43} - 84 q^{47} + 138 q^{49} + 5 q^{57} - 50 q^{61} - 234 q^{63} + 204 q^{73} - 68 q^{77} + 66 q^{81} - 256 q^{83} - 18 q^{87} - 118 q^{93} + 92 q^{99}+O(q^{100}) $ 14 * q - 4 * q^7 - 52 * q^9 + 4 * q^11 + 6 * q^17 - 29 * q^19 + 28 * q^23 - 56 * q^39 - 22 * q^43 - 84 * q^47 + 138 * q^49 + 5 * q^57 - 50 * q^61 - 234 * q^63 + 204 * q^73 - 68 * q^77 + 66 * q^81 - 256 * q^83 - 18 * q^87 - 118 * q^93 + 92 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$1$	$-1$	$1$

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
$3$			4.84257i			1.61419i	0.590423	+	0.807094i	$0.298961\pi$
$3$			4.84257i			1.61419i	−0.590423	+	0.807094i	$0.701039\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−6.85668			−0.979525			−0.489763	−	0.871856i	$-0.662917\pi$
$7$	−6.85668			−0.979525			−0.489763	+	0.871856i	$0.662917\pi$
$8$		0			0
$9$	−14.4504			−1.60560
$10$		0			0
$11$	−9.59871			−0.872610			−0.436305	−	0.899799i	$-0.643713\pi$
$11$	−9.59871			−0.872610			−0.436305	+	0.899799i	$0.643713\pi$
$12$		0			0
$13$		−	10.0805i		−	0.775422i	−0.921781	−	0.387711i	$-0.873266\pi$
$13$		−	10.0805i		−	0.775422i	0.921781	−	0.387711i	$-0.126734\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−27.8791			−1.63995			−0.819973	−	0.572402i	$-0.806012\pi$
$17$	−27.8791			−1.63995			−0.819973	+	0.572402i	$0.806012\pi$
$18$		0			0
$19$	17.5630	−	7.24850i	0.924369	−	0.381500i
$19$	17.5630	−	7.24850i	0.924369	−	0.381500i
$20$		0			0
$21$		−	33.2039i		−	1.58114i
$22$		0			0
$23$	8.14968			0.354334			0.177167	−	0.984181i	$-0.443307\pi$
$23$	8.14968			0.354334			0.177167	+	0.984181i	$0.443307\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	26.3941i		−	0.977561i
$28$		0			0
$29$		−	17.8174i		−	0.614391i	−0.951646	−	0.307196i	$-0.900609\pi$
$29$		−	17.8174i		−	0.614391i	0.951646	−	0.307196i	$-0.0993906\pi$
$30$		0			0
$31$		−	20.3798i		−	0.657412i	−0.944432	−	0.328706i	$-0.893388\pi$
$31$		−	20.3798i		−	0.657412i	0.944432	−	0.328706i	$-0.106612\pi$
$32$		0			0
$33$		−	46.4824i		−	1.40856i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			65.1090i			1.75970i	0.475250	+	0.879851i	$0.342358\pi$
$37$			65.1090i			1.75970i	−0.475250	+	0.879851i	$0.657642\pi$
$38$		0			0
$39$	48.8154			1.25168
$40$		0			0
$41$			16.0406i			0.391235i	0.980680	+	0.195618i	$0.0626712\pi$
$41$			16.0406i			0.391235i	−0.980680	+	0.195618i	$0.937329\pi$
$42$		0			0
$43$	60.7072			1.41179			0.705897	−	0.708314i	$-0.250544\pi$
$43$	60.7072			1.41179			0.705897	+	0.708314i	$0.250544\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	9.60366			0.204333			0.102167	−	0.994767i	$-0.467423\pi$
$47$	9.60366			0.204333			0.102167	+	0.994767i	$0.467423\pi$
$48$		0			0
$49$	−1.98597			−0.0405299
$50$		0			0
$51$		−	135.006i		−	2.64718i
$52$		0			0
$53$			14.7792i			0.278854i	0.990232	+	0.139427i	$0.0445260\pi$
$53$			14.7792i			0.278854i	−0.990232	+	0.139427i	$0.955474\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	35.1013	+	85.0500i	0.615813	+	1.49211i
$58$		0			0
$59$			0.442058i			0.00749252i	0.999993	+	0.00374626i	$0.00119247\pi$
$59$			0.442058i			0.00749252i	−0.999993	+	0.00374626i	$0.998808\pi$
$60$		0			0
$61$	65.2615			1.06986			0.534930	−	0.844896i	$-0.320338\pi$
$61$	65.2615			1.06986			0.534930	+	0.844896i	$0.320338\pi$
$62$		0			0
$63$	99.0820			1.57273
$64$		0			0
$65$		0			0
$66$		0			0
$67$			103.423i			1.54363i	0.635845	+	0.771817i	$0.280652\pi$
$67$			103.423i			1.54363i	−0.635845	+	0.771817i	$0.719348\pi$
$68$		0			0
$69$			39.4653i			0.571962i
$70$		0			0
$71$		−	90.5453i		−	1.27529i	−0.770332	−	0.637643i	$-0.779909\pi$
$71$		−	90.5453i		−	1.27529i	0.770332	−	0.637643i	$-0.220091\pi$
$72$		0			0
$73$	88.8290			1.21684			0.608418	−	0.793617i	$-0.291804\pi$
$73$	88.8290			1.21684			0.608418	+	0.793617i	$0.291804\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	65.8153			0.854744
$78$		0			0
$79$		−	86.7598i		−	1.09823i	−0.835748	−	0.549113i	$-0.814966\pi$
$79$		−	86.7598i		−	1.09823i	0.835748	−	0.549113i	$-0.185034\pi$
$80$		0			0
$81$	−2.23865			−0.0276376
$82$		0			0
$83$	−47.4163			−0.571281			−0.285640	−	0.958337i	$-0.592206\pi$
$83$	−47.4163			−0.571281			−0.285640	+	0.958337i	$0.592206\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	86.2817			0.991744
$88$		0			0
$89$		−	58.0384i		−	0.652117i	−0.945350	−	0.326058i	$-0.894279\pi$
$89$		−	58.0384i		−	0.652117i	0.945350	−	0.326058i	$-0.105721\pi$
$90$		0			0
$91$			69.1187i			0.759546i
$92$		0			0
$93$	98.6904			1.06119
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	165.286i		−	1.70397i	−0.523562	−	0.851987i	$-0.675397\pi$
$97$		−	165.286i		−	1.70397i	0.523562	−	0.851987i	$-0.324603\pi$
$98$		0			0
$99$	138.706			1.40107
$100$		0			0
$101$	81.0614			0.802589			0.401294	−	0.915949i	$-0.368560\pi$
$101$	81.0614			0.802589			0.401294	+	0.915949i	$0.368560\pi$
$102$		0			0
$103$			76.4480i			0.742213i	0.928590	+	0.371107i	$0.121022\pi$
$103$			76.4480i			0.742213i	−0.928590	+	0.371107i	$0.878978\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			19.6386i			0.183538i	0.995780	+	0.0917691i	$0.0292522\pi$
$107$			19.6386i			0.183538i	−0.995780	+	0.0917691i	$0.970748\pi$
$108$		0			0
$109$			101.569i			0.931830i	0.884830	+	0.465915i	$0.154275\pi$
$109$			101.569i			0.931830i	−0.884830	+	0.465915i	$0.845725\pi$
$110$		0			0
$111$	−315.295			−2.84049
$112$		0			0
$113$		−	126.863i		−	1.12268i	−0.827585	−	0.561341i	$-0.810286\pi$
$113$		−	126.863i		−	1.12268i	0.827585	−	0.561341i	$-0.189714\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			145.668i			1.24502i
$118$		0			0
$119$	191.158			1.60637
$120$		0			0
$121$	−28.8647			−0.238552
$122$		0			0
$123$	−77.6779			−0.631528
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	62.1773i		−	0.489585i	−0.969575	−	0.244793i	$-0.921280\pi$
$127$		−	62.1773i		−	0.489585i	0.969575	−	0.244793i	$-0.0787199\pi$
$128$		0			0
$129$			293.978i			2.27890i
$130$		0			0
$131$	3.66734			0.0279949			0.0139975	−	0.999902i	$-0.495544\pi$
$131$	3.66734			0.0279949			0.0139975	+	0.999902i	$0.495544\pi$
$132$		0			0
$133$	−120.424	+	49.7006i	−0.905443	+	0.373689i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−223.349			−1.63028			−0.815142	−	0.579261i	$-0.803341\pi$
$137$	−223.349			−1.63028			−0.815142	+	0.579261i	$0.803341\pi$
$138$		0			0
$139$	30.4744			0.219240			0.109620	−	0.993974i	$-0.465037\pi$
$139$	30.4744			0.219240			0.109620	+	0.993974i	$0.465037\pi$
$140$		0			0
$141$			46.5063i			0.329832i
$142$		0			0
$143$			96.7597i			0.676642i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	9.61717i		−	0.0654229i
$148$		0			0
$149$	120.341			0.807655			0.403827	−	0.914835i	$-0.367680\pi$
$149$	120.341			0.807655			0.403827	+	0.914835i	$0.367680\pi$
$150$		0			0
$151$		−	111.480i		−	0.738275i	−0.929375	−	0.369138i	$-0.879653\pi$
$151$		−	111.480i		−	0.738275i	0.929375	−	0.369138i	$-0.120347\pi$
$152$		0			0
$153$	402.865			2.63311
$154$		0			0
$155$		0			0
$156$		0			0
$157$	139.390			0.887832			0.443916	−	0.896068i	$-0.353589\pi$
$157$	139.390			0.887832			0.443916	+	0.896068i	$0.353589\pi$
$158$		0			0
$159$	−71.5695			−0.450123
$160$		0			0
$161$	−55.8797			−0.347079
$162$		0			0
$163$	163.849			1.00521			0.502605	−	0.864516i	$-0.332375\pi$
$163$	163.849			1.00521			0.502605	+	0.864516i	$0.332375\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	282.206i		−	1.68986i	−0.534878	−	0.844929i	$-0.679643\pi$
$167$		−	282.206i		−	1.68986i	0.534878	−	0.844929i	$-0.320357\pi$
$168$		0			0
$169$	67.3837			0.398720
$170$		0			0
$171$	−253.793	+	104.744i	−1.48417	+	0.612538i
$172$		0			0
$173$			244.448i			1.41300i	0.707715	+	0.706498i	$0.249726\pi$
$173$			244.448i			1.41300i	−0.707715	+	0.706498i	$0.750274\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−2.14070			−0.0120943
$178$		0			0
$179$		−	319.507i		−	1.78496i	−0.451090	−	0.892478i	$-0.648965\pi$
$179$		−	319.507i		−	1.78496i	0.451090	−	0.892478i	$-0.351035\pi$
$180$		0			0
$181$			77.1074i			0.426008i	0.977051	+	0.213004i	$0.0683247\pi$
$181$			77.1074i			0.426008i	−0.977051	+	0.213004i	$0.931675\pi$
$182$		0			0
$183$			316.033i			1.72696i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	267.603			1.43103
$188$		0			0
$189$			180.976i			0.957546i
$190$		0			0
$191$	−73.2221			−0.383362			−0.191681	−	0.981457i	$-0.561394\pi$
$191$	−73.2221			−0.383362			−0.191681	+	0.981457i	$0.561394\pi$
$192$		0			0
$193$			118.144i			0.612144i	0.952008	+	0.306072i	$0.0990148\pi$
$193$			118.144i			0.612144i	−0.952008	+	0.306072i	$0.900985\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−16.1635			−0.0820485			−0.0410242	−	0.999158i	$-0.513062\pi$
$197$	−16.1635			−0.0820485			−0.0410242	+	0.999158i	$0.513062\pi$
$198$		0			0
$199$	−246.985			−1.24113			−0.620566	−	0.784154i	$-0.713097\pi$
$199$	−246.985			−1.24113			−0.620566	+	0.784154i	$0.713097\pi$
$200$		0			0
$201$	−500.835			−2.49172
$202$		0			0
$203$			122.168i			0.601812i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−117.766			−0.568920
$208$		0			0
$209$	−168.582	+	69.5763i	−0.806614	+	0.332901i
$210$		0			0
$211$		−	86.3447i		−	0.409217i	−0.978844	−	0.204608i	$-0.934408\pi$
$211$		−	86.3447i		−	0.409217i	0.978844	−	0.204608i	$-0.0655921\pi$
$212$		0			0
$213$	438.472			2.05855
$214$		0			0
$215$		0			0
$216$		0			0
$217$			139.737i			0.643952i
$218$		0			0
$219$			430.160i			1.96420i
$220$		0			0
$221$			281.035i			1.27165i
$222$		0			0
$223$			111.083i			0.498131i	0.968487	+	0.249065i	$0.0801234\pi$
$223$			111.083i			0.498131i	−0.968487	+	0.249065i	$0.919877\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	212.064i		−	0.934203i	−0.884204	−	0.467101i	$-0.845298\pi$
$227$		−	212.064i		−	0.934203i	0.884204	−	0.467101i	$-0.154702\pi$
$228$		0			0
$229$	−121.793			−0.531846			−0.265923	−	0.963994i	$-0.585677\pi$
$229$	−121.793			−0.531846			−0.265923	+	0.963994i	$0.585677\pi$
$230$		0			0
$231$			318.715i			1.37972i
$232$		0			0
$233$	334.305			1.43478			0.717392	−	0.696670i	$-0.245336\pi$
$233$	334.305			1.43478			0.717392	+	0.696670i	$0.245336\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	420.140			1.77274
$238$		0			0
$239$	453.267			1.89652			0.948258	−	0.317500i	$-0.102843\pi$
$239$	453.267			1.89652			0.948258	+	0.317500i	$0.102843\pi$
$240$		0			0
$241$			176.552i			0.732582i	0.930500	+	0.366291i	$0.119373\pi$
$241$			176.552i			0.732582i	−0.930500	+	0.366291i	$0.880627\pi$
$242$		0			0
$243$		−	248.388i		−	1.02217i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−73.0685	−	177.044i	−0.295824	−	0.716776i
$248$		0			0
$249$		−	229.616i		−	0.922154i
$250$		0			0
$251$	54.6634			0.217782			0.108891	−	0.994054i	$-0.465270\pi$
$251$	54.6634			0.217782			0.108891	+	0.994054i	$0.465270\pi$
$252$		0			0
$253$	−78.2264			−0.309195
$254$		0			0
$255$		0			0
$256$		0			0
$257$			84.2554i			0.327842i	0.986473	+	0.163921i	$0.0524142\pi$
$257$			84.2554i			0.327842i	−0.986473	+	0.163921i	$0.947586\pi$
$258$		0			0
$259$		−	446.431i		−	1.72367i
$260$		0			0
$261$			257.469i			0.986470i
$262$		0			0
$263$	198.029			0.752962			0.376481	−	0.926424i	$-0.377134\pi$
$263$	198.029			0.752962			0.376481	+	0.926424i	$0.377134\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	281.055			1.05264
$268$		0			0
$269$			477.641i			1.77562i	0.460213	+	0.887809i	$0.347773\pi$
$269$			477.641i			1.77562i	−0.460213	+	0.887809i	$0.652227\pi$
$270$		0			0
$271$	−129.912			−0.479379			−0.239689	−	0.970850i	$-0.577046\pi$
$271$	−129.912			−0.479379			−0.239689	+	0.970850i	$0.577046\pi$
$272$		0			0
$273$	−334.712			−1.22605
$274$		0			0
$275$		0			0
$276$		0			0
$277$	468.295			1.69059			0.845297	−	0.534297i	$-0.179423\pi$
$277$	468.295			1.69059			0.845297	+	0.534297i	$0.179423\pi$
$278$		0			0
$279$			294.497i			1.05554i
$280$		0			0
$281$		−	386.974i		−	1.37713i	−0.725175	−	0.688565i	$-0.758241\pi$
$281$		−	386.974i		−	1.37713i	0.725175	−	0.688565i	$-0.241759\pi$
$282$		0			0
$283$	−210.748			−0.744694			−0.372347	−	0.928094i	$-0.621447\pi$
$283$	−210.748			−0.744694			−0.372347	+	0.928094i	$0.621447\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	109.986i		−	0.383225i
$288$		0			0
$289$	488.243			1.68942
$290$		0			0
$291$	800.406			2.75054
$292$		0			0
$293$			385.803i			1.31674i	0.752697	+	0.658368i	$0.228753\pi$
$293$			385.803i			1.31674i	−0.752697	+	0.658368i	$0.771247\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			253.350i			0.853029i
$298$		0			0
$299$		−	82.1527i		−	0.274758i
$300$		0			0
$301$	−416.249			−1.38289
$302$		0			0
$303$			392.545i			1.29553i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			60.4268i			0.196830i	0.995145	+	0.0984150i	$0.0313773\pi$
$307$			60.4268i			0.196830i	−0.995145	+	0.0984150i	$0.968623\pi$
$308$		0			0
$309$	−370.204			−1.19807
$310$		0			0
$311$	244.412			0.785890			0.392945	−	0.919562i	$-0.371456\pi$
$311$	244.412			0.785890			0.392945	+	0.919562i	$0.371456\pi$
$312$		0			0
$313$	−479.471			−1.53186			−0.765929	−	0.642925i	$-0.777721\pi$
$313$	−479.471			−1.53186			−0.765929	+	0.642925i	$0.777721\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	87.4438i		−	0.275848i	−0.990443	−	0.137924i	$-0.955957\pi$
$317$		−	87.4438i		−	0.275848i	0.990443	−	0.137924i	$-0.0440430\pi$
$318$		0			0
$319$			171.024i			0.536124i
$320$		0			0
$321$	−95.1012			−0.296265
$322$		0			0
$323$	−489.641	+	202.082i	−1.51592	+	0.625640i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−491.857			−1.50415
$328$		0			0
$329$	−65.8492			−0.200149
$330$		0			0
$331$		−	398.587i		−	1.20419i	−0.798425	−	0.602095i	$-0.794333\pi$
$331$		−	398.587i		−	1.20419i	0.798425	−	0.602095i	$-0.205667\pi$
$332$		0			0
$333$		−	940.854i		−	2.82539i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	114.633i		−	0.340158i	−0.985430	−	0.170079i	$-0.945598\pi$
$337$		−	114.633i		−	0.340158i	0.985430	−	0.170079i	$-0.0544023\pi$
$338$		0			0
$339$	614.343			1.81222
$340$		0			0
$341$			195.619i			0.573664i
$342$		0			0
$343$	349.594			1.01923
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−234.225			−0.674999			−0.337499	−	0.941326i	$-0.609581\pi$
$347$	−234.225			−0.674999			−0.337499	+	0.941326i	$0.609581\pi$
$348$		0			0
$349$	−163.364			−0.468093			−0.234046	−	0.972225i	$-0.575197\pi$
$349$	−163.364			−0.468093			−0.234046	+	0.972225i	$0.575197\pi$
$350$		0			0
$351$	−266.066			−0.758023
$352$		0			0
$353$	671.997			1.90367			0.951837	−	0.306605i	$-0.0991931\pi$
$353$	671.997			1.90367			0.951837	+	0.306605i	$0.0991931\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			925.695i			2.59298i
$358$		0			0
$359$	160.300			0.446517			0.223259	−	0.974759i	$-0.428331\pi$
$359$	160.300			0.446517			0.223259	+	0.974759i	$0.428331\pi$
$360$		0			0
$361$	255.918	−	254.611i	0.708915	−	0.705294i
$362$		0			0
$363$		−	139.779i		−	0.385067i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	34.2401			0.0932973			0.0466487	−	0.998911i	$-0.485146\pi$
$367$	34.2401			0.0932973			0.0466487	+	0.998911i	$0.485146\pi$
$368$		0			0
$369$		−	231.794i		−	0.628169i
$370$		0			0
$371$		−	101.337i		−	0.273144i
$372$		0			0
$373$		−	415.310i		−	1.11343i	−0.830703	−	0.556716i	$-0.812061\pi$
$373$		−	415.310i		−	1.11343i	0.830703	−	0.556716i	$-0.187939\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−179.608			−0.476413
$378$		0			0
$379$		−	514.295i		−	1.35698i	−0.734610	−	0.678489i	$-0.762635\pi$
$379$		−	514.295i		−	1.35698i	0.734610	−	0.678489i	$-0.237365\pi$
$380$		0			0
$381$	301.098			0.790283
$382$		0			0
$383$		−	658.940i		−	1.72047i	−0.509897	−	0.860235i	$-0.670317\pi$
$383$		−	658.940i		−	1.72047i	0.509897	−	0.860235i	$-0.329683\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−877.245			−2.26678
$388$		0			0
$389$	−284.073			−0.730265			−0.365133	−	0.930955i	$-0.618976\pi$
$389$	−284.073			−0.730265			−0.365133	+	0.930955i	$0.618976\pi$
$390$		0			0
$391$	−227.206			−0.581088
$392$		0			0
$393$			17.7593i			0.0451891i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	184.434			0.464570			0.232285	−	0.972648i	$-0.425380\pi$
$397$	184.434			0.464570			0.232285	+	0.972648i	$0.425380\pi$
$398$		0			0
$399$	−240.679	−	583.161i	−0.603205	−	1.46156i
$400$		0			0
$401$		−	378.243i		−	0.943250i	−0.881799	−	0.471625i	$-0.843668\pi$
$401$		−	378.243i		−	0.943250i	0.881799	−	0.471625i	$-0.156332\pi$
$402$		0			0
$403$	−205.438			−0.509772
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	624.962i		−	1.53553i
$408$		0			0
$409$			418.384i			1.02294i	0.859300	+	0.511471i	$0.170899\pi$
$409$			418.384i			1.02294i	−0.859300	+	0.511471i	$0.829101\pi$
$410$		0			0
$411$		−	1081.58i		−	2.63159i
$412$		0			0
$413$		−	3.03105i		−	0.00733911i
$414$		0			0
$415$		0			0
$416$		0			0
$417$			147.574i			0.353896i
$418$		0			0
$419$	695.935			1.66094			0.830471	−	0.557061i	$-0.188071\pi$
$419$	695.935			1.66094			0.830471	+	0.557061i	$0.188071\pi$
$420$		0			0
$421$		−	402.866i		−	0.956927i	−0.878108	−	0.478463i	$-0.841194\pi$
$421$		−	402.866i		−	0.956927i	0.878108	−	0.478463i	$-0.158806\pi$
$422$		0			0
$423$	−138.777			−0.328078
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−447.477			−1.04796
$428$		0			0
$429$	−468.565			−1.09223
$430$		0			0
$431$			245.727i			0.570132i	0.958508	+	0.285066i	$0.0920156\pi$
$431$			245.727i			0.570132i	−0.958508	+	0.285066i	$0.907984\pi$
$432$		0			0
$433$		−	230.070i		−	0.531339i	−0.964064	−	0.265670i	$-0.914407\pi$
$433$		−	230.070i		−	0.531339i	0.964064	−	0.265670i	$-0.0855930\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	143.133	−	59.0729i	0.327535	−	0.135178i
$438$		0			0
$439$		−	543.857i		−	1.23885i	−0.785054	−	0.619427i	$-0.787365\pi$
$439$		−	543.857i		−	1.23885i	0.785054	−	0.619427i	$-0.212635\pi$
$440$		0			0
$441$	28.6981			0.0650750
$442$		0			0
$443$	−38.6987			−0.0873560			−0.0436780	−	0.999046i	$-0.513908\pi$
$443$	−38.6987			−0.0873560			−0.0436780	+	0.999046i	$0.513908\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			582.757i			1.30371i
$448$		0			0
$449$			103.840i			0.231270i	0.993292	+	0.115635i	$0.0368902\pi$
$449$			103.840i			0.231270i	−0.993292	+	0.115635i	$0.963110\pi$
$450$		0			0
$451$		−	153.970i		−	0.341396i
$452$		0			0
$453$	539.847			1.19172
$454$		0			0
$455$		0			0
$456$		0			0
$457$	93.1389			0.203805			0.101902	−	0.994794i	$-0.467507\pi$
$457$	93.1389			0.203805			0.101902	+	0.994794i	$0.467507\pi$
$458$		0			0
$459$			735.844i			1.60315i
$460$		0			0
$461$	−117.789			−0.255507			−0.127753	−	0.991806i	$-0.540777\pi$
$461$	−117.789			−0.255507			−0.127753	+	0.991806i	$0.540777\pi$
$462$		0			0
$463$	93.5692			0.202093			0.101047	−	0.994882i	$-0.467781\pi$
$463$	93.5692			0.202093			0.101047	+	0.994882i	$0.467781\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−351.018			−0.751645			−0.375823	−	0.926692i	$-0.622640\pi$
$467$	−351.018			−0.751645			−0.375823	+	0.926692i	$0.622640\pi$
$468$		0			0
$469$		−	709.141i		−	1.51203i
$470$		0			0
$471$			675.004i			1.43313i
$472$		0			0
$473$	−582.710			−1.23195
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	213.567i		−	0.447729i
$478$		0			0
$479$	−70.1471			−0.146445			−0.0732224	−	0.997316i	$-0.523328\pi$
$479$	−70.1471			−0.146445			−0.0732224	+	0.997316i	$0.523328\pi$
$480$		0			0
$481$	656.331			1.36451
$482$		0			0
$483$		−	270.601i		−	0.560251i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	765.631i		−	1.57214i	−0.618140	−	0.786068i	$-0.712113\pi$
$487$		−	765.631i		−	1.57214i	0.618140	−	0.786068i	$-0.287887\pi$
$488$		0			0
$489$			793.451i			1.62260i
$490$		0			0
$491$	244.304			0.497564			0.248782	−	0.968560i	$-0.419970\pi$
$491$	244.304			0.497564			0.248782	+	0.968560i	$0.419970\pi$
$492$		0			0
$493$			496.731i			1.00757i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			620.840i			1.24918i
$498$		0			0
$499$	73.0277			0.146348			0.0731741	−	0.997319i	$-0.476687\pi$
$499$	73.0277			0.146348			0.0731741	+	0.997319i	$0.476687\pi$
$500$		0			0
$501$	1366.60			2.72775
$502$		0			0
$503$	−710.302			−1.41213			−0.706065	−	0.708147i	$-0.749532\pi$
$503$	−710.302			−1.41213			−0.706065	+	0.708147i	$0.749532\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			326.310i			0.643609i
$508$		0			0
$509$		−	936.262i		−	1.83942i	−0.392604	−	0.919708i	$-0.628426\pi$
$509$		−	936.262i		−	1.83942i	0.392604	−	0.919708i	$-0.371574\pi$
$510$		0			0
$511$	−609.072			−1.19192
$512$		0			0
$513$	−191.318	−	463.560i	−0.372939	−	0.903627i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−92.1827			−0.178303
$518$		0			0
$519$	−1183.76			−2.28084
$520$		0			0
$521$		−	689.775i		−	1.32394i	−0.749528	−	0.661972i	$-0.769720\pi$
$521$		−	689.775i		−	1.32394i	0.749528	−	0.661972i	$-0.230280\pi$
$522$		0			0
$523$			126.270i			0.241434i	0.992687	+	0.120717i	$0.0385194\pi$
$523$			126.270i			0.241434i	−0.992687	+	0.120717i	$0.961481\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			568.169i			1.07812i
$528$		0			0
$529$	−462.583			−0.874448
$530$		0			0
$531$		−	6.38794i		−	0.0120300i
$532$		0			0
$533$	161.698			0.303373
$534$		0			0
$535$		0			0
$536$		0			0
$537$	1547.24			2.88126
$538$		0			0
$539$	19.0627			0.0353668
$540$		0			0
$541$	952.485			1.76060			0.880300	−	0.474417i	$-0.157341\pi$
$541$	952.485			1.76060			0.880300	+	0.474417i	$0.157341\pi$
$542$		0			0
$543$	−373.398			−0.687657
$544$		0			0
$545$		0			0
$546$		0			0
$547$			764.558i			1.39773i	0.715254	+	0.698865i	$0.246311\pi$
$547$			764.558i			1.39773i	−0.715254	+	0.698865i	$0.753689\pi$
$548$		0			0
$549$	−943.057			−1.71777
$550$		0			0
$551$	−129.149	−	312.926i	−0.234390	−	0.567924i
$552$		0			0
$553$			594.884i			1.07574i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−465.851			−0.836358			−0.418179	−	0.908365i	$-0.637331\pi$
$557$	−465.851			−0.836358			−0.418179	+	0.908365i	$0.637331\pi$
$558$		0			0
$559$		−	611.958i		−	1.09474i
$560$		0			0
$561$			1295.89i			2.30996i
$562$		0			0
$563$		−	442.560i		−	0.786075i	−0.919522	−	0.393038i	$-0.871424\pi$
$563$		−	442.560i		−	0.786075i	0.919522	−	0.393038i	$-0.128576\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	15.3497			0.0270718
$568$		0			0
$569$		−	663.147i		−	1.16546i	−0.812666	−	0.582730i	$-0.801984\pi$
$569$		−	663.147i		−	1.16546i	0.812666	−	0.582730i	$-0.198016\pi$
$570$		0			0
$571$	161.428			0.282710			0.141355	−	0.989959i	$-0.454854\pi$
$571$	161.428			0.282710			0.141355	+	0.989959i	$0.454854\pi$
$572$		0			0
$573$		−	354.583i		−	0.618818i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−517.669			−0.897174			−0.448587	−	0.893739i	$-0.648073\pi$
$577$	−517.669			−0.897174			−0.448587	+	0.893739i	$0.648073\pi$
$578$		0			0
$579$	−572.119			−0.988115
$580$		0			0
$581$	325.118			0.559584
$582$		0			0
$583$		−	141.862i		−	0.243331i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−389.141			−0.662931			−0.331466	−	0.943467i	$-0.607543\pi$
$587$	−389.141			−0.662931			−0.331466	+	0.943467i	$0.607543\pi$
$588$		0			0
$589$	−147.723	−	357.930i	−0.250803	−	0.607691i
$590$		0			0
$591$		−	78.2730i		−	0.132442i
$592$		0			0
$593$	−809.753			−1.36552			−0.682760	−	0.730643i	$-0.739221\pi$
$593$	−809.753			−1.36552			−0.682760	+	0.730643i	$0.739221\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	1196.04i		−	2.00342i
$598$		0			0
$599$			1074.40i			1.79366i	0.442374	+	0.896831i	$0.354136\pi$
$599$			1074.40i			1.79366i	−0.442374	+	0.896831i	$0.645864\pi$
$600$		0			0
$601$			1073.76i			1.78662i	0.449439	+	0.893311i	$0.351624\pi$
$601$			1073.76i			1.78662i	−0.449439	+	0.893311i	$0.648376\pi$
$602$		0			0
$603$		−	1494.51i		−	2.47847i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	381.395i		−	0.628327i	−0.949369	−	0.314164i	$-0.898276\pi$
$607$		−	381.395i		−	0.628327i	0.949369	−	0.314164i	$-0.101724\pi$
$608$		0			0
$609$	−591.606			−0.971438
$610$		0			0
$611$		−	96.8096i		−	0.158444i
$612$		0			0
$613$	−544.647			−0.888495			−0.444247	−	0.895904i	$-0.646529\pi$
$613$	−544.647			−0.888495			−0.444247	+	0.895904i	$0.646529\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	413.071			0.669482			0.334741	−	0.942310i	$-0.391351\pi$
$617$	413.071			0.669482			0.334741	+	0.942310i	$0.391351\pi$
$618$		0			0
$619$	1108.99			1.79158			0.895792	−	0.444473i	$-0.146609\pi$
$619$	1108.99			1.79158			0.895792	+	0.444473i	$0.146609\pi$
$620$		0			0
$621$		−	215.104i		−	0.346383i
$622$		0			0
$623$			397.951i			0.638765i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−336.928	−	816.371i	−0.537365	−	1.30203i
$628$		0			0
$629$		−	1815.18i		−	2.88582i
$630$		0			0
$631$	−61.5377			−0.0975241			−0.0487621	−	0.998810i	$-0.515528\pi$
$631$	−61.5377			−0.0975241			−0.0487621	+	0.998810i	$0.515528\pi$
$632$		0			0
$633$	418.130			0.660553
$634$		0			0
$635$		0			0
$636$		0			0
$637$			20.0195i			0.0314278i
$638$		0			0
$639$			1308.42i			2.04761i
$640$		0			0
$641$			461.117i			0.719371i	0.933074	+	0.359686i	$0.117116\pi$
$641$			461.117i			0.719371i	−0.933074	+	0.359686i	$0.882884\pi$
$642$		0			0
$643$	446.784			0.694843			0.347422	−	0.937709i	$-0.387057\pi$
$643$	446.784			0.694843			0.347422	+	0.937709i	$0.387057\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	146.630			0.226631			0.113315	−	0.993559i	$-0.463853\pi$
$647$	146.630			0.226631			0.113315	+	0.993559i	$0.463853\pi$
$648$		0			0
$649$		−	4.24319i		−	0.00653805i
$650$		0			0
$651$	−676.688			−1.03946
$652$		0			0
$653$	−1257.57			−1.92583			−0.962917	−	0.269797i	$-0.913044\pi$
$653$	−1257.57			−1.92583			−0.962917	+	0.269797i	$0.913044\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−1283.62			−1.95376
$658$		0			0
$659$		−	302.508i		−	0.459041i	−0.973304	−	0.229521i	$-0.926284\pi$
$659$		−	302.508i		−	0.459041i	0.973304	−	0.229521i	$-0.0737158\pi$
$660$		0			0
$661$			649.086i			0.981976i	0.871166	+	0.490988i	$0.163364\pi$
$661$			649.086i			0.981976i	−0.871166	+	0.490988i	$0.836636\pi$
$662$		0			0
$663$	−1360.93			−2.05268
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	145.206i		−	0.217700i
$668$		0			0
$669$	−537.928			−0.804077
$670$		0			0
$671$	−626.426			−0.933571
$672$		0			0
$673$			1130.94i			1.68044i	0.542243	+	0.840222i	$0.317575\pi$
$673$			1130.94i			1.68044i	−0.542243	+	0.840222i	$0.682425\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	732.839i		−	1.08248i	−0.840868	−	0.541240i	$-0.817955\pi$
$677$		−	732.839i		−	1.08248i	0.840868	−	0.541240i	$-0.182045\pi$
$678$		0			0
$679$			1133.31i			1.66909i
$680$		0			0
$681$	1026.93			1.50798
$682$		0			0
$683$		−	1054.81i		−	1.54438i	−0.635390	−	0.772191i	$-0.719161\pi$
$683$		−	1054.81i		−	1.54438i	0.635390	−	0.772191i	$-0.280839\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	589.789i		−	0.858499i
$688$		0			0
$689$	148.982			0.216229
$690$		0			0
$691$	1062.69			1.53790			0.768951	−	0.639307i	$-0.220779\pi$
$691$	1062.69			1.53790			0.768951	+	0.639307i	$0.220779\pi$
$692$		0			0
$693$	−951.060			−1.37238
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	447.199i		−	0.641605i
$698$		0			0
$699$			1618.89i			2.31601i
$700$		0			0
$701$	−504.359			−0.719486			−0.359743	−	0.933051i	$-0.617136\pi$
$701$	−504.359			−0.719486			−0.359743	+	0.933051i	$0.617136\pi$
$702$		0			0
$703$	471.943	+	1143.51i	0.671327	+	1.62661i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−555.812			−0.786156
$708$		0			0
$709$	740.831			1.04490			0.522448	−	0.852671i	$-0.325019\pi$
$709$	740.831			1.04490			0.522448	+	0.852671i	$0.325019\pi$
$710$		0			0
$711$			1253.72i			1.76332i
$712$		0			0
$713$		−	166.088i		−	0.232943i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			2194.98i			3.06134i
$718$		0			0
$719$	380.067			0.528606			0.264303	−	0.964440i	$-0.414858\pi$
$719$	380.067			0.528606			0.264303	+	0.964440i	$0.414858\pi$
$720$		0			0
$721$		−	524.179i		−	0.727017i
$722$		0			0
$723$	−854.966			−1.18253
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−32.1100			−0.0441678			−0.0220839	−	0.999756i	$-0.507030\pi$
$727$	−32.1100			−0.0441678			−0.0220839	+	0.999756i	$0.507030\pi$
$728$		0			0
$729$	1182.69			1.62234
$730$		0			0
$731$	−1692.46			−2.31527
$732$		0			0
$733$	−716.127			−0.976980			−0.488490	−	0.872569i	$-0.662452\pi$
$733$	−716.127			−0.976980			−0.488490	+	0.872569i	$0.662452\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	992.732i		−	1.34699i
$738$		0			0
$739$	−487.621			−0.659839			−0.329919	−	0.944009i	$-0.607022\pi$
$739$	−487.621			−0.659839			−0.329919	+	0.944009i	$0.607022\pi$
$740$		0			0
$741$	857.346	−	353.839i	1.15701	−	0.477515i
$742$		0			0
$743$			1355.45i			1.82430i	0.409858	+	0.912149i	$0.365578\pi$
$743$			1355.45i			1.82430i	−0.409858	+	0.912149i	$0.634422\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	685.186			0.917251
$748$		0			0
$749$		−	134.655i		−	0.179780i
$750$		0			0
$751$			308.439i			0.410705i	0.978688	+	0.205352i	$0.0658340\pi$
$751$			308.439i			0.410705i	−0.978688	+	0.205352i	$0.934166\pi$
$752$		0			0
$753$			264.711i			0.351542i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	126.540			0.167160			0.0835800	−	0.996501i	$-0.473365\pi$
$757$	126.540			0.167160			0.0835800	+	0.996501i	$0.473365\pi$
$758$		0			0
$759$		−	378.816i		−	0.499099i
$760$		0			0
$761$	−116.745			−0.153411			−0.0767053	−	0.997054i	$-0.524440\pi$
$761$	−116.745			−0.153411			−0.0767053	+	0.997054i	$0.524440\pi$
$762$		0			0
$763$		−	696.429i		−	0.912751i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4.45617			0.00580987
$768$		0			0
$769$	178.462			0.232071			0.116035	−	0.993245i	$-0.462981\pi$
$769$	178.462			0.232071			0.116035	+	0.993245i	$0.462981\pi$
$770$		0			0
$771$	−408.012			−0.529199
$772$		0			0
$773$			757.415i			0.979838i	0.871768	+	0.489919i	$0.162974\pi$
$773$			757.415i			0.979838i	−0.871768	+	0.489919i	$0.837026\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	2161.87			2.78233
$778$		0			0
$779$	116.271	+	281.722i	0.149256	+	0.361646i
$780$		0			0
$781$			869.119i			1.11283i
$782$		0			0
$783$	−470.274			−0.600605
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	394.797i		−	0.501649i	−0.968033	−	0.250824i	$-0.919298\pi$
$787$		−	394.797i		−	0.501649i	0.968033	−	0.250824i	$-0.0807016\pi$
$788$		0			0
$789$			958.969i			1.21542i
$790$		0			0
$791$			869.859i			1.09970i
$792$		0			0
$793$		−	657.868i		−	0.829594i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1395.92i		−	1.75147i	−0.482790	−	0.875736i	$-0.660377\pi$
$797$		−	1395.92i		−	1.75147i	0.482790	−	0.875736i	$-0.339623\pi$
$798$		0			0
$799$	−267.741			−0.335095
$800$		0			0
$801$			838.680i			1.04704i
$802$		0			0
$803$	−852.644			−1.06182
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−2313.01			−2.86618
$808$		0			0
$809$	−1037.98			−1.28304			−0.641520	−	0.767106i	$-0.721696\pi$
$809$	−1037.98			−1.28304			−0.641520	+	0.767106i	$0.721696\pi$
$810$		0			0
$811$		−	856.583i		−	1.05621i	−0.849180	−	0.528103i	$-0.822903\pi$
$811$		−	856.583i		−	1.05621i	0.849180	−	0.528103i	$-0.177097\pi$
$812$		0			0
$813$		−	629.106i		−	0.773808i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1066.20	−	440.036i	1.30502	−	0.538600i
$818$		0			0
$819$		−	998.796i		−	1.21953i
$820$		0			0
$821$	−1246.94			−1.51881			−0.759403	−	0.650620i	$-0.774509\pi$
$821$	−1246.94			−1.51881			−0.759403	+	0.650620i	$0.774509\pi$
$822$		0			0
$823$	−565.419			−0.687021			−0.343511	−	0.939149i	$-0.611616\pi$
$823$	−565.419			−0.687021			−0.343511	+	0.939149i	$0.611616\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			385.248i			0.465838i	0.972496	+	0.232919i	$0.0748278\pi$
$827$			385.248i			0.465838i	−0.972496	+	0.232919i	$0.925172\pi$
$828$		0			0
$829$		−	291.671i		−	0.351835i	−0.984405	−	0.175918i	$-0.943711\pi$
$829$		−	291.671i		−	0.351835i	0.984405	−	0.175918i	$-0.0562892\pi$
$830$		0			0
$831$			2267.75i			2.72894i
$832$		0			0
$833$	55.3669			0.0664669
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−537.906			−0.642660
$838$		0			0
$839$			681.986i			0.812855i	0.913683	+	0.406428i	$0.133226\pi$
$839$			681.986i			0.812855i	−0.913683	+	0.406428i	$0.866774\pi$
$840$		0			0
$841$	523.542			0.622523
$842$		0			0
$843$	1873.94			2.22295
$844$		0			0
$845$		0			0
$846$		0			0
$847$	197.916			0.233667
$848$		0			0
$849$		−	1020.56i		−	1.20208i
$850$		0			0
$851$			530.617i			0.623522i
$852$		0			0
$853$	1226.89			1.43833			0.719163	−	0.694842i	$-0.244526\pi$
$853$	1226.89			1.43833			0.719163	+	0.694842i	$0.244526\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1490.25i		−	1.73891i	−0.494011	−	0.869455i	$-0.664470\pi$
$857$		−	1490.25i		−	1.73891i	0.494011	−	0.869455i	$-0.335530\pi$
$858$		0			0
$859$	−353.772			−0.411842			−0.205921	−	0.978569i	$-0.566019\pi$
$859$	−353.772			−0.411842			−0.205921	+	0.978569i	$0.566019\pi$
$860$		0			0
$861$	532.612			0.618597
$862$		0			0
$863$			191.399i			0.221783i	0.993833	+	0.110892i	$0.0353707\pi$
$863$			191.399i			0.221783i	−0.993833	+	0.110892i	$0.964629\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			2364.35i			2.72705i
$868$		0			0
$869$			832.782i			0.958323i
$870$		0			0
$871$	1042.56			1.19697
$872$		0			0
$873$			2388.45i			2.73591i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	215.594i		−	0.245831i	−0.992417	−	0.122915i	$-0.960776\pi$
$877$		−	215.594i		−	0.245831i	0.992417	−	0.122915i	$-0.0392244\pi$
$878$		0			0
$879$	−1868.28			−2.12546
$880$		0			0
$881$	−738.673			−0.838448			−0.419224	−	0.907883i	$-0.637698\pi$
$881$	−738.673			−0.838448			−0.419224	+	0.907883i	$0.637698\pi$
$882$		0			0
$883$	676.569			0.766216			0.383108	−	0.923703i	$-0.374854\pi$
$883$	676.569			0.766216			0.383108	+	0.923703i	$0.374854\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	947.023i		−	1.06767i	−0.845589	−	0.533835i	$-0.820750\pi$
$887$		−	947.023i		−	1.06767i	0.845589	−	0.533835i	$-0.179250\pi$
$888$		0			0
$889$			426.330i			0.479561i
$890$		0			0
$891$	21.4881			0.0241169
$892$		0			0
$893$	168.669	−	69.6121i	0.188879	−	0.0779531i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	397.830			0.443512
$898$		0			0
$899$	−363.113			−0.403908
$900$		0			0
$901$		−	412.032i		−	0.457305i
$902$		0			0
$903$		−	2015.72i		−	2.23224i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			1224.31i			1.34985i	0.737886	+	0.674925i	$0.235824\pi$
$907$			1224.31i			1.34985i	−0.737886	+	0.674925i	$0.764176\pi$
$908$		0			0
$909$	−1171.37			−1.28864
$910$		0			0
$911$		−	1146.43i		−	1.25844i	−0.777229	−	0.629218i	$-0.783375\pi$
$911$		−	1146.43i		−	1.25844i	0.777229	−	0.629218i	$-0.216625\pi$
$912$		0			0
$913$	455.135			0.498505
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−25.1457			−0.0274217
$918$		0			0
$919$	−1510.92			−1.64409			−0.822043	−	0.569425i	$-0.807166\pi$
$919$	−1510.92			−1.64409			−0.822043	+	0.569425i	$0.807166\pi$
$920$		0			0
$921$	−292.621			−0.317721
$922$		0			0
$923$	−912.742			−0.988886
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	1104.71i		−	1.19170i
$928$		0			0
$929$	350.718			0.377522			0.188761	−	0.982023i	$-0.439553\pi$
$929$	350.718			0.377522			0.188761	+	0.982023i	$0.439553\pi$
$930$		0			0
$931$	−34.8795	+	14.3953i	−0.0374646	+	0.0154622i
$932$		0			0
$933$			1183.58i			1.26858i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1152.96			1.23048			0.615242	−	0.788338i	$-0.289058\pi$
$937$	1152.96			1.23048			0.615242	+	0.788338i	$0.289058\pi$
$938$		0			0
$939$		−	2321.87i		−	2.47271i
$940$		0			0
$941$		−	765.775i		−	0.813788i	−0.913475	−	0.406894i	$-0.866612\pi$
$941$		−	765.775i		−	0.813788i	0.913475	−	0.406894i	$-0.133388\pi$
$942$		0			0
$943$			130.726i			0.138628i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−242.099			−0.255648			−0.127824	−	0.991797i	$-0.540799\pi$
$947$	−242.099			−0.255648			−0.127824	+	0.991797i	$0.540799\pi$
$948$		0			0
$949$		−	895.440i		−	0.943562i
$950$		0			0
$951$	423.453			0.445271
$952$		0			0
$953$		−	402.050i		−	0.421878i	−0.977499	−	0.210939i	$-0.932348\pi$
$953$		−	402.050i		−	0.421878i	0.977499	−	0.210939i	$-0.0676522\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−828.193			−0.865406
$958$		0			0
$959$	1531.43			1.59690
$960$		0			0
$961$	545.665			0.567810
$962$		0			0
$963$		−	283.786i		−	0.294690i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1022.23			−1.05711			−0.528557	−	0.848898i	$-0.677267\pi$
$967$	−1022.23			−1.05711			−0.528557	+	0.848898i	$0.677267\pi$
$968$		0			0
$969$	−978.593	−	2371.12i	−1.00990	−	2.44697i
$970$		0			0
$971$		−	1739.87i		−	1.79183i	−0.444226	−	0.895915i	$-0.646521\pi$
$971$		−	1739.87i		−	1.79183i	0.444226	−	0.895915i	$-0.353479\pi$
$972$		0			0
$973$	−208.953			−0.214752
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1454.92i			1.48917i	0.667528	+	0.744585i	$0.267352\pi$
$977$			1454.92i			1.48917i	−0.667528	+	0.744585i	$0.732648\pi$
$978$		0			0
$979$			557.094i			0.569044i
$980$		0			0
$981$		−	1467.72i		−	1.49615i
$982$		0			0
$983$		−	785.609i		−	0.799195i	−0.916691	−	0.399597i	$-0.869150\pi$
$983$		−	785.609i		−	0.799195i	0.916691	−	0.399597i	$-0.130850\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	318.879i		−	0.323079i
$988$		0			0
$989$	494.744			0.500246
$990$		0			0
$991$			1039.65i			1.04909i	0.851383	+	0.524544i	$0.175764\pi$
$991$			1039.65i			1.04909i	−0.851383	+	0.524544i	$0.824236\pi$
$992$		0			0
$993$	1930.18			1.94379
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1628.52			1.63342			0.816709	−	0.577050i	$-0.195796\pi$
$997$	1628.52			1.63342			0.816709	+	0.577050i	$0.195796\pi$
$998$		0			0
$999$	1718.50			1.72022

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.3.e.g.1101.13	yes	14
5.2	odd	4		1900.3.g.d.949.25		28
5.3	odd	4		1900.3.g.d.949.4		28
5.4	even	2		1900.3.e.h.1101.2	yes	14
19.18	odd	2	inner	1900.3.e.g.1101.2	✓	14
95.18	even	4		1900.3.g.d.949.26		28
95.37	even	4		1900.3.g.d.949.3		28
95.94	odd	2		1900.3.e.h.1101.13	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1900.3.e.g.1101.2	✓	14	19.18	odd	2	inner
1900.3.e.g.1101.13	yes	14	1.1	even	1	trivial
1900.3.e.h.1101.2	yes	14	5.4	even	2
1900.3.e.h.1101.13	yes	14	95.94	odd	2
1900.3.g.d.949.3		28	95.37	even	4
1900.3.g.d.949.4		28	5.3	odd	4
1900.3.g.d.949.25		28	5.2	odd	4
1900.3.g.d.949.26		28	95.18	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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1900.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.84257i q^{3} -6.85668 q^{7} -14.4504 q^{9} +O(q^{10})\)	\(q+4.84257i q^{3} -6.85668 q^{7} -14.4504 q^{9} -9.59871 q^{11} -10.0805i q^{13} -27.8791 q^{17} +(17.5630 - 7.24850i) q^{19} -33.2039i q^{21} +8.14968 q^{23} -26.3941i q^{27} -17.8174i q^{29} -20.3798i q^{31} -46.4824i q^{33} +65.1090i q^{37} +48.8154 q^{39} +16.0406i q^{41} +60.7072 q^{43} +9.60366 q^{47} -1.98597 q^{49} -135.006i q^{51} +14.7792i q^{53} +(35.1013 + 85.0500i) q^{57} +0.442058i q^{59} +65.2615 q^{61} +99.0820 q^{63} +103.423i q^{67} +39.4653i q^{69} -90.5453i q^{71} +88.8290 q^{73} +65.8153 q^{77} -86.7598i q^{79} -2.23865 q^{81} -47.4163 q^{83} +86.2817 q^{87} -58.0384i q^{89} +69.1187i q^{91} +98.6904 q^{93} -165.286i q^{97} +138.706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 4 q^{7} - 52 q^{9}+O(q^{10}) \) 14 * q - 4 * q^7 - 52 * q^9	\( 14 q - 4 q^{7} - 52 q^{9} + 4 q^{11} + 6 q^{17} - 29 q^{19} + 28 q^{23} - 56 q^{39} - 22 q^{43} - 84 q^{47} + 138 q^{49} + 5 q^{57} - 50 q^{61} - 234 q^{63} + 204 q^{73} - 68 q^{77} + 66 q^{81} - 256 q^{83} - 18 q^{87} - 118 q^{93} + 92 q^{99}+O(q^{100}) \) 14 * q - 4 * q^7 - 52 * q^9 + 4 * q^11 + 6 * q^17 - 29 * q^19 + 28 * q^23 - 56 * q^39 - 22 * q^43 - 84 * q^47 + 138 * q^49 + 5 * q^57 - 50 * q^61 - 234 * q^63 + 204 * q^73 - 68 * q^77 + 66 * q^81 - 256 * q^83 - 18 * q^87 - 118 * q^93 + 92 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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