LMFDB - Embedded newform 1900.3.g.c.949.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,3,Mod(949,1900)]

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, 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("1900.949");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1900.g (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:	$51.7712502285$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			949.14
Character	$\chi$	$=$	1900.949
Dual form			1900.3.g.c.949.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.630185 q^{3} -3.98530i q^{7} -8.60287 q^{9} +O(q^{10})$	$q+0.630185 q^{3} -3.98530i q^{7} -8.60287 q^{9} -5.54431 q^{11} +23.1242 q^{13} +16.5013i q^{17} +(-11.6484 + 15.0105i) q^{19} -2.51147i q^{21} -6.86785i q^{23} -11.0931 q^{27} -37.0068i q^{29} +11.1127i q^{31} -3.49394 q^{33} -66.6951 q^{37} +14.5725 q^{39} -14.8753i q^{41} -34.7815i q^{43} +63.0151i q^{47} +33.1174 q^{49} +10.3989i q^{51} -36.8706 q^{53} +(-7.34066 + 9.45937i) q^{57} +19.0546i q^{59} +34.3686 q^{61} +34.2850i q^{63} -27.2417 q^{67} -4.32801i q^{69} +90.8927i q^{71} +19.7746i q^{73} +22.0957i q^{77} +111.986i q^{79} +70.4351 q^{81} +129.958i q^{83} -23.3211i q^{87} +66.5710i q^{89} -92.1568i q^{91} +7.00305i q^{93} -19.0605 q^{97} +47.6970 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 32 q^{9}+O(q^{10}) $ 24 * q + 32 * q^9	$ 24 q + 32 q^{9} - 64 q^{11} - 48 q^{19} - 248 q^{39} + 48 q^{49} + 304 q^{61} - 936 q^{81} - 784 q^{99}+O(q^{100}) $ 24 * q + 32 * q^9 - 64 * q^11 - 48 * q^19 - 248 * q^39 + 48 * q^49 + 304 * q^61 - 936 * q^81 - 784 * 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
$3$	0.630185			0.210062			0.105031	−	0.994469i	$-0.466506\pi$
$3$	0.630185			0.210062			0.105031	+	0.994469i	$0.466506\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	3.98530i		−	0.569328i	−0.958627	−	0.284664i	$-0.908118\pi$
$7$		−	3.98530i		−	0.569328i	0.958627	−	0.284664i	$-0.0918821\pi$
$8$		0			0
$9$	−8.60287			−0.955874
$10$		0			0
$11$	−5.54431			−0.504028			−0.252014	−	0.967724i	$-0.581093\pi$
$11$	−5.54431			−0.504028			−0.252014	+	0.967724i	$0.581093\pi$
$12$		0			0
$13$	23.1242			1.77878			0.889392	−	0.457145i	$-0.151128\pi$
$13$	23.1242			1.77878			0.889392	+	0.457145i	$0.151128\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			16.5013i			0.970664i	0.874330	+	0.485332i	$0.161301\pi$
$17$			16.5013i			0.970664i	−0.874330	+	0.485332i	$0.838699\pi$
$18$		0			0
$19$	−11.6484	+	15.0105i	−0.613075	+	0.790024i
$19$	−11.6484	+	15.0105i	−0.613075	+	0.790024i
$20$		0			0
$21$		−	2.51147i		−	0.119594i
$22$		0			0
$23$		−	6.86785i		−	0.298602i	−0.988792	−	0.149301i	$-0.952298\pi$
$23$		−	6.86785i		−	0.298602i	0.988792	−	0.149301i	$-0.0477023\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−11.0931			−0.410854
$28$		0			0
$29$		−	37.0068i		−	1.27610i	−0.769996	−	0.638049i	$-0.779742\pi$
$29$		−	37.0068i		−	1.27610i	0.769996	−	0.638049i	$-0.220258\pi$
$30$		0			0
$31$			11.1127i			0.358474i	0.983806	+	0.179237i	$0.0573629\pi$
$31$			11.1127i			0.358474i	−0.983806	+	0.179237i	$0.942637\pi$
$32$		0			0
$33$	−3.49394			−0.105877
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−66.6951			−1.80257			−0.901285	−	0.433227i	$-0.857375\pi$
$37$	−66.6951			−1.80257			−0.901285	+	0.433227i	$0.857375\pi$
$38$		0			0
$39$	14.5725			0.373654
$40$		0			0
$41$		−	14.8753i		−	0.362811i	−0.983408	−	0.181406i	$-0.941935\pi$
$41$		−	14.8753i		−	0.362811i	0.983408	−	0.181406i	$-0.0580647\pi$
$42$		0			0
$43$		−	34.7815i		−	0.808872i	−0.914566	−	0.404436i	$-0.867468\pi$
$43$		−	34.7815i		−	0.808872i	0.914566	−	0.404436i	$-0.132532\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			63.0151i			1.34075i	0.742024	+	0.670374i	$0.233866\pi$
$47$			63.0151i			1.34075i	−0.742024	+	0.670374i	$0.766134\pi$
$48$		0			0
$49$	33.1174			0.675865
$50$		0			0
$51$			10.3989i			0.203899i
$52$		0			0
$53$	−36.8706			−0.695672			−0.347836	−	0.937555i	$-0.613083\pi$
$53$	−36.8706			−0.695672			−0.347836	+	0.937555i	$0.613083\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−7.34066	+	9.45937i	−0.128784	+	0.165954i
$58$		0			0
$59$			19.0546i			0.322960i	0.986876	+	0.161480i	$0.0516267\pi$
$59$			19.0546i			0.322960i	−0.986876	+	0.161480i	$0.948373\pi$
$60$		0			0
$61$	34.3686			0.563420			0.281710	−	0.959500i	$-0.409098\pi$
$61$	34.3686			0.563420			0.281710	+	0.959500i	$0.409098\pi$
$62$		0			0
$63$			34.2850i			0.544206i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−27.2417			−0.406593			−0.203297	−	0.979117i	$-0.565166\pi$
$67$	−27.2417			−0.406593			−0.203297	+	0.979117i	$0.565166\pi$
$68$		0			0
$69$		−	4.32801i		−	0.0627248i
$70$		0			0
$71$			90.8927i			1.28018i	0.768300	+	0.640090i	$0.221103\pi$
$71$			90.8927i			1.28018i	−0.768300	+	0.640090i	$0.778897\pi$
$72$		0			0
$73$			19.7746i			0.270885i	0.990785	+	0.135443i	$0.0432457\pi$
$73$			19.7746i			0.270885i	−0.990785	+	0.135443i	$0.956754\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			22.0957i			0.286958i
$78$		0			0
$79$			111.986i			1.41754i	0.705439	+	0.708771i	$0.250750\pi$
$79$			111.986i			1.41754i	−0.705439	+	0.708771i	$0.749250\pi$
$80$		0			0
$81$	70.4351			0.869569
$82$		0			0
$83$			129.958i			1.56576i	0.622171	+	0.782881i	$0.286251\pi$
$83$			129.958i			1.56576i	−0.622171	+	0.782881i	$0.713749\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		−	23.3211i		−	0.268059i
$88$		0			0
$89$			66.5710i			0.747989i	0.927431	+	0.373994i	$0.122012\pi$
$89$			66.5710i			0.747989i	−0.927431	+	0.373994i	$0.877988\pi$
$90$		0			0
$91$		−	92.1568i		−	1.01271i
$92$		0			0
$93$			7.00305i			0.0753016i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−19.0605			−0.196500			−0.0982498	−	0.995162i	$-0.531324\pi$
$97$	−19.0605			−0.196500			−0.0982498	+	0.995162i	$0.531324\pi$
$98$		0			0
$99$	47.6970			0.481788
$100$		0			0
$101$	−106.469			−1.05415			−0.527073	−	0.849820i	$-0.676711\pi$
$101$	−106.469			−1.05415			−0.527073	+	0.849820i	$0.676711\pi$
$102$		0			0
$103$	−123.262			−1.19672			−0.598359	−	0.801228i	$-0.704180\pi$
$103$	−123.262			−1.19672			−0.598359	+	0.801228i	$0.704180\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	102.138			0.954560			0.477280	−	0.878751i	$-0.341623\pi$
$107$	102.138			0.954560			0.477280	+	0.878751i	$0.341623\pi$
$108$		0			0
$109$			189.408i			1.73769i	0.495083	+	0.868846i	$0.335138\pi$
$109$			189.408i			1.73769i	−0.495083	+	0.868846i	$0.664862\pi$
$110$		0			0
$111$	−42.0302			−0.378651
$112$		0			0
$113$	−1.96880			−0.0174230			−0.00871150	−	0.999962i	$-0.502773\pi$
$113$	−1.96880			−0.0174230			−0.00871150	+	0.999962i	$0.502773\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−198.934			−1.70029
$118$		0			0
$119$	65.7625			0.552626
$120$		0			0
$121$	−90.2606			−0.745956
$122$		0			0
$123$		−	9.37417i		−	0.0762127i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	2.18645			0.0172162			0.00860809	−	0.999963i	$-0.497260\pi$
$127$	2.18645			0.0172162			0.00860809	+	0.999963i	$0.497260\pi$
$128$		0			0
$129$		−	21.9188i		−	0.169913i
$130$		0			0
$131$	−37.7731			−0.288344			−0.144172	−	0.989553i	$-0.546052\pi$
$131$	−37.7731			−0.288344			−0.144172	+	0.989553i	$0.546052\pi$
$132$		0			0
$133$	59.8212	+	46.4225i	0.449783	+	0.349041i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			117.952i			0.860964i	0.902599	+	0.430482i	$0.141656\pi$
$137$			117.952i			0.860964i	−0.902599	+	0.430482i	$0.858344\pi$
$138$		0			0
$139$	165.428			1.19013			0.595063	−	0.803679i	$-0.297127\pi$
$139$	165.428			1.19013			0.595063	+	0.803679i	$0.297127\pi$
$140$		0			0
$141$			39.7112i			0.281640i
$142$		0			0
$143$	−128.208			−0.896558
$144$		0			0
$145$		0			0
$146$		0			0
$147$	20.8701			0.141973
$148$		0			0
$149$	−121.995			−0.818757			−0.409378	−	0.912365i	$-0.634254\pi$
$149$	−121.995			−0.818757			−0.409378	+	0.912365i	$0.634254\pi$
$150$		0			0
$151$		−	261.820i		−	1.73391i	−0.498387	−	0.866955i	$-0.666074\pi$
$151$		−	261.820i		−	1.73391i	0.498387	−	0.866955i	$-0.333926\pi$
$152$		0			0
$153$		−	141.958i		−	0.927833i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			73.0950i			0.465573i	0.972528	+	0.232787i	$0.0747844\pi$
$157$			73.0950i			0.465573i	−0.972528	+	0.232787i	$0.925216\pi$
$158$		0			0
$159$	−23.2353			−0.146134
$160$		0			0
$161$	−27.3704			−0.170003
$162$		0			0
$163$			92.3779i			0.566735i	0.959011	+	0.283368i	$0.0914517\pi$
$163$			92.3779i			0.566735i	−0.959011	+	0.283368i	$0.908548\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−300.141			−1.79725			−0.898626	−	0.438716i	$-0.855433\pi$
$167$	−300.141			−1.79725			−0.898626	+	0.438716i	$0.855433\pi$
$168$		0			0
$169$	365.729			2.16407
$170$		0			0
$171$	100.210	−	129.133i	0.586023	−	0.755164i
$172$		0			0
$173$	−25.8449			−0.149392			−0.0746962	−	0.997206i	$-0.523799\pi$
$173$	−25.8449			−0.149392			−0.0746962	+	0.997206i	$0.523799\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			12.0079i			0.0678415i
$178$		0			0
$179$			198.054i			1.10644i	0.833034	+	0.553222i	$0.186602\pi$
$179$			198.054i			1.10644i	−0.833034	+	0.553222i	$0.813398\pi$
$180$		0			0
$181$		−	202.353i		−	1.11797i	−0.829177	−	0.558986i	$-0.811190\pi$
$181$		−	202.353i		−	1.11797i	0.829177	−	0.558986i	$-0.188810\pi$
$182$		0			0
$183$	21.6586			0.118353
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	91.4883i		−	0.489242i
$188$		0			0
$189$			44.2091i			0.233911i
$190$		0			0
$191$	5.50260			0.0288094			0.0144047	−	0.999896i	$-0.495415\pi$
$191$	5.50260			0.0288094			0.0144047	+	0.999896i	$0.495415\pi$
$192$		0			0
$193$	7.71141			0.0399555			0.0199777	−	0.999800i	$-0.493640\pi$
$193$	7.71141			0.0399555			0.0199777	+	0.999800i	$0.493640\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			366.507i			1.86044i	0.366997	+	0.930222i	$0.380386\pi$
$197$			366.507i			1.86044i	−0.366997	+	0.930222i	$0.619614\pi$
$198$		0			0
$199$	317.466			1.59531			0.797653	−	0.603117i	$-0.206075\pi$
$199$	317.466			1.59531			0.797653	+	0.603117i	$0.206075\pi$
$200$		0			0
$201$	−17.1673			−0.0854096
$202$		0			0
$203$	−147.483			−0.726518
$204$		0			0
$205$		0			0
$206$		0			0
$207$			59.0832i			0.285426i
$208$		0			0
$209$	64.5825	−	83.2227i	0.309007	−	0.398195i
$210$		0			0
$211$			285.703i			1.35404i	0.735964	+	0.677020i	$0.236729\pi$
$211$			285.703i			1.35404i	−0.735964	+	0.677020i	$0.763271\pi$
$212$		0			0
$213$			57.2792i			0.268916i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	44.2874			0.204089
$218$		0			0
$219$			12.4617i			0.0569026i
$220$		0			0
$221$			381.579i			1.72660i
$222$		0			0
$223$	−152.576			−0.684198			−0.342099	−	0.939664i	$-0.611138\pi$
$223$	−152.576			−0.684198			−0.342099	+	0.939664i	$0.611138\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−405.597			−1.78677			−0.893386	−	0.449289i	$-0.851677\pi$
$227$	−405.597			−1.78677			−0.893386	+	0.449289i	$0.851677\pi$
$228$		0			0
$229$	−339.392			−1.48206			−0.741030	−	0.671472i	$-0.765662\pi$
$229$	−339.392			−1.48206			−0.741030	+	0.671472i	$0.765662\pi$
$230$		0			0
$231$			13.9244i			0.0602787i
$232$		0			0
$233$		−	245.708i		−	1.05454i	−0.849697	−	0.527271i	$-0.823215\pi$
$233$		−	245.708i		−	1.05454i	0.849697	−	0.527271i	$-0.176785\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			70.5717i			0.297771i
$238$		0			0
$239$	−273.105			−1.14270			−0.571349	−	0.820707i	$-0.693580\pi$
$239$	−273.105			−1.14270			−0.571349	+	0.820707i	$0.693580\pi$
$240$		0			0
$241$		−	114.540i		−	0.475271i	−0.971354	−	0.237635i	$-0.923628\pi$
$241$		−	114.540i		−	0.475271i	0.971354	−	0.237635i	$-0.0763723\pi$
$242$		0			0
$243$	144.225			0.593517
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−269.361	+	347.105i	−1.09053	+	1.40528i
$248$		0			0
$249$			81.8978i			0.328907i
$250$		0			0
$251$	287.738			1.14637			0.573183	−	0.819427i	$-0.305708\pi$
$251$	287.738			1.14637			0.573183	+	0.819427i	$0.305708\pi$
$252$		0			0
$253$			38.0775i			0.150504i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−238.684			−0.928730			−0.464365	−	0.885644i	$-0.653717\pi$
$257$	−238.684			−0.928730			−0.464365	+	0.885644i	$0.653717\pi$
$258$		0			0
$259$			265.800i			1.02625i
$260$		0			0
$261$			318.365i			1.21979i
$262$		0			0
$263$			450.262i			1.71202i	0.516958	+	0.856011i	$0.327064\pi$
$263$			450.262i			1.71202i	−0.516958	+	0.856011i	$0.672936\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			41.9520i			0.157124i
$268$		0			0
$269$		−	202.954i		−	0.754474i	−0.926117	−	0.377237i	$-0.876874\pi$
$269$		−	202.954i		−	0.754474i	0.926117	−	0.377237i	$-0.123126\pi$
$270$		0			0
$271$	−105.551			−0.389485			−0.194743	−	0.980854i	$-0.562387\pi$
$271$	−105.551			−0.389485			−0.194743	+	0.980854i	$0.562387\pi$
$272$		0			0
$273$		−	58.0758i		−	0.212732i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	253.935i		−	0.916734i	−0.888763	−	0.458367i	$-0.848435\pi$
$277$		−	253.935i		−	0.916734i	0.888763	−	0.458367i	$-0.151565\pi$
$278$		0			0
$279$		−	95.6010i		−	0.342656i
$280$		0			0
$281$		−	265.663i		−	0.945418i	−0.881219	−	0.472709i	$-0.843276\pi$
$281$		−	265.663i		−	0.945418i	0.881219	−	0.472709i	$-0.156724\pi$
$282$		0			0
$283$			62.3633i			0.220365i	0.993911	+	0.110183i	$0.0351435\pi$
$283$			62.3633i			0.220365i	−0.993911	+	0.110183i	$0.964856\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−59.2824			−0.206559
$288$		0			0
$289$	16.7075			0.0578114
$290$		0			0
$291$	−12.0116			−0.0412770
$292$		0			0
$293$	219.282			0.748401			0.374201	−	0.927348i	$-0.377917\pi$
$293$	219.282			0.748401			0.374201	+	0.927348i	$0.377917\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	61.5034			0.207082
$298$		0			0
$299$		−	158.813i		−	0.531149i
$300$		0			0
$301$	−138.615			−0.460514
$302$		0			0
$303$	−67.0950			−0.221436
$304$		0			0
$305$		0			0
$306$		0			0
$307$	217.314			0.707865			0.353932	−	0.935271i	$-0.384844\pi$
$307$	217.314			0.707865			0.353932	+	0.935271i	$0.384844\pi$
$308$		0			0
$309$	−77.6778			−0.251384
$310$		0			0
$311$	334.681			1.07615			0.538073	−	0.842899i	$-0.319153\pi$
$311$	334.681			1.07615			0.538073	+	0.842899i	$0.319153\pi$
$312$		0			0
$313$			50.1493i			0.160222i	0.996786	+	0.0801108i	$0.0255274\pi$
$313$			50.1493i			0.160222i	−0.996786	+	0.0801108i	$0.974473\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−147.124			−0.464113			−0.232056	−	0.972702i	$-0.574545\pi$
$317$	−147.124			−0.464113			−0.232056	+	0.972702i	$0.574545\pi$
$318$		0			0
$319$			205.177i			0.643189i
$320$		0			0
$321$	64.3657			0.200516
$322$		0			0
$323$	−247.692	−	192.214i	−0.766848	−	0.595090i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			119.362i			0.365022i
$328$		0			0
$329$	251.134			0.763325
$330$		0			0
$331$			371.342i			1.12188i	0.827857	+	0.560940i	$0.189560\pi$
$331$			371.342i			1.12188i	−0.827857	+	0.560940i	$0.810440\pi$
$332$		0			0
$333$	573.769			1.72303
$334$		0			0
$335$		0			0
$336$		0			0
$337$	588.911			1.74751			0.873755	−	0.486367i	$-0.161678\pi$
$337$	588.911			1.74751			0.873755	+	0.486367i	$0.161678\pi$
$338$		0			0
$339$	−1.24071			−0.00365990
$340$		0			0
$341$		−	61.6122i		−	0.180681i
$342$		0			0
$343$		−	327.262i		−	0.954117i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			116.937i			0.336993i	0.985702	+	0.168496i	$0.0538912\pi$
$347$			116.937i			0.336993i	−0.985702	+	0.168496i	$0.946109\pi$
$348$		0			0
$349$	−236.153			−0.676657			−0.338329	−	0.941028i	$-0.609862\pi$
$349$	−236.153			−0.676657			−0.338329	+	0.941028i	$0.609862\pi$
$350$		0			0
$351$	−256.518			−0.730821
$352$		0			0
$353$		−	176.061i		−	0.498755i	−0.968406	−	0.249378i	$-0.919774\pi$
$353$		−	176.061i		−	0.498755i	0.968406	−	0.249378i	$-0.0802260\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	41.4426			0.116086
$358$		0			0
$359$	277.373			0.772626			0.386313	−	0.922368i	$-0.373749\pi$
$359$	277.373			0.772626			0.386313	+	0.922368i	$0.373749\pi$
$360$		0			0
$361$	−89.6281	−	349.697i	−0.248277	−	0.968689i
$362$		0			0
$363$	−56.8809			−0.156697
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	214.948i		−	0.585689i	−0.956160	−	0.292845i	$-0.905398\pi$
$367$		−	214.948i		−	0.585689i	0.956160	−	0.292845i	$-0.0946019\pi$
$368$		0			0
$369$			127.970i			0.346802i
$370$		0			0
$371$			146.940i			0.396066i
$372$		0			0
$373$	−217.265			−0.582479			−0.291239	−	0.956650i	$-0.594068\pi$
$373$	−217.265			−0.582479			−0.291239	+	0.956650i	$0.594068\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	855.753i		−	2.26990i
$378$		0			0
$379$		−	648.851i		−	1.71201i	−0.516970	−	0.856003i	$-0.672940\pi$
$379$		−	648.851i		−	1.71201i	0.516970	−	0.856003i	$-0.327060\pi$
$380$		0			0
$381$	1.37787			0.00361646
$382$		0			0
$383$	−74.9159			−0.195603			−0.0978014	−	0.995206i	$-0.531181\pi$
$383$	−74.9159			−0.195603			−0.0978014	+	0.995206i	$0.531181\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			299.221i			0.773180i
$388$		0			0
$389$	539.065			1.38577			0.692886	−	0.721048i	$-0.256339\pi$
$389$	539.065			1.38577			0.692886	+	0.721048i	$0.256339\pi$
$390$		0			0
$391$	113.328			0.289842
$392$		0			0
$393$	−23.8040			−0.0605700
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	71.2291i		−	0.179418i	−0.995968	−	0.0897092i	$-0.971406\pi$
$397$		−	71.2291i		−	0.179418i	0.995968	−	0.0897092i	$-0.0285938\pi$
$398$		0			0
$399$	37.6984	+	29.2547i	0.0944822	+	0.0733201i
$400$		0			0
$401$		−	471.620i		−	1.17611i	−0.808821	−	0.588054i	$-0.799894\pi$
$401$		−	471.620i		−	1.17611i	0.808821	−	0.588054i	$-0.200106\pi$
$402$		0			0
$403$			256.972i			0.637648i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	369.778			0.908546
$408$		0			0
$409$			456.208i			1.11542i	0.830035	+	0.557712i	$0.188320\pi$
$409$			456.208i			1.11542i	−0.830035	+	0.557712i	$0.811680\pi$
$410$		0			0
$411$			74.3316i			0.180855i
$412$		0			0
$413$	75.9384			0.183870
$414$		0			0
$415$		0			0
$416$		0			0
$417$	104.250			0.250000
$418$		0			0
$419$	−367.066			−0.876052			−0.438026	−	0.898962i	$-0.644322\pi$
$419$	−367.066			−0.876052			−0.438026	+	0.898962i	$0.644322\pi$
$420$		0			0
$421$		−	484.369i		−	1.15052i	−0.817971	−	0.575260i	$-0.804901\pi$
$421$		−	484.369i		−	1.15052i	0.817971	−	0.575260i	$-0.195099\pi$
$422$		0			0
$423$		−	542.111i		−	1.28159i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	136.969i		−	0.320771i
$428$		0			0
$429$	−80.7946			−0.188332
$430$		0			0
$431$			227.149i			0.527028i	0.964656	+	0.263514i	$0.0848815\pi$
$431$			227.149i			0.527028i	−0.964656	+	0.263514i	$0.915119\pi$
$432$		0			0
$433$	−243.982			−0.563468			−0.281734	−	0.959493i	$-0.590910\pi$
$433$	−243.982			−0.563468			−0.281734	+	0.959493i	$0.590910\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	103.090	+	79.9996i	0.235903	+	0.183066i
$438$		0			0
$439$			106.971i			0.243671i	0.992550	+	0.121835i	$0.0388780\pi$
$439$			106.971i			0.243671i	−0.992550	+	0.121835i	$0.961122\pi$
$440$		0			0
$441$	−284.905			−0.646042
$442$		0			0
$443$			508.879i			1.14871i	0.818606	+	0.574356i	$0.194747\pi$
$443$			508.879i			1.14871i	−0.818606	+	0.574356i	$0.805253\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−76.8793			−0.171989
$448$		0			0
$449$			158.585i			0.353195i	0.984283	+	0.176598i	$0.0565091\pi$
$449$			158.585i			0.353195i	−0.984283	+	0.176598i	$0.943491\pi$
$450$		0			0
$451$			82.4731i			0.182867i
$452$		0			0
$453$		−	164.995i		−	0.364228i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	204.194i		−	0.446814i	−0.974725	−	0.223407i	$-0.928282\pi$
$457$		−	204.194i		−	0.446814i	0.974725	−	0.223407i	$-0.0717179\pi$
$458$		0			0
$459$		−	183.050i		−	0.398801i
$460$		0			0
$461$	842.357			1.82724			0.913619	−	0.406571i	$-0.133276\pi$
$461$	842.357			1.82724			0.913619	+	0.406571i	$0.133276\pi$
$462$		0			0
$463$			677.168i			1.46257i	0.682075	+	0.731283i	$0.261078\pi$
$463$			677.168i			1.46257i	−0.682075	+	0.731283i	$0.738922\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			15.6256i			0.0334596i	0.999860	+	0.0167298i	$0.00532550\pi$
$467$			15.6256i			0.0334596i	−0.999860	+	0.0167298i	$0.994674\pi$
$468$		0			0
$469$			108.566i			0.231485i
$470$		0			0
$471$			46.0634i			0.0977991i
$472$		0			0
$473$			192.840i			0.407695i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	317.193			0.664975
$478$		0			0
$479$	−835.461			−1.74418			−0.872088	−	0.489349i	$-0.837235\pi$
$479$	−835.461			−1.74418			−0.872088	+	0.489349i	$0.837235\pi$
$480$		0			0
$481$	−1542.27			−3.20638
$482$		0			0
$483$	−17.2484			−0.0357110
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−171.272			−0.351688			−0.175844	−	0.984418i	$-0.556266\pi$
$487$	−171.272			−0.351688			−0.175844	+	0.984418i	$0.556266\pi$
$488$		0			0
$489$			58.2151i			0.119049i
$490$		0			0
$491$	−401.468			−0.817653			−0.408826	−	0.912612i	$-0.634062\pi$
$491$	−401.468			−0.817653			−0.408826	+	0.912612i	$0.634062\pi$
$492$		0			0
$493$	610.660			1.23866
$494$		0			0
$495$		0			0
$496$		0			0
$497$	362.235			0.728842
$498$		0			0
$499$	−711.651			−1.42615			−0.713077	−	0.701086i	$-0.752699\pi$
$499$	−711.651			−1.42615			−0.713077	+	0.701086i	$0.752699\pi$
$500$		0			0
$501$	−189.144			−0.377534
$502$		0			0
$503$			333.140i			0.662306i	0.943577	+	0.331153i	$0.107438\pi$
$503$			333.140i			0.662306i	−0.943577	+	0.331153i	$0.892562\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	230.477			0.454589
$508$		0			0
$509$		−	571.857i		−	1.12349i	−0.827310	−	0.561746i	$-0.810130\pi$
$509$		−	571.857i		−	1.12349i	0.827310	−	0.561746i	$-0.189870\pi$
$510$		0			0
$511$	78.8078			0.154223
$512$		0			0
$513$	129.217	−	166.512i	0.251884	−	0.324585i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	349.375i		−	0.675775i
$518$		0			0
$519$	−16.2870			−0.0313816
$520$		0			0
$521$		−	193.824i		−	0.372022i	−0.982548	−	0.186011i	$-0.940444\pi$
$521$		−	193.824i		−	0.372022i	0.982548	−	0.186011i	$-0.0595561\pi$
$522$		0			0
$523$	−749.167			−1.43244			−0.716221	−	0.697873i	$-0.754130\pi$
$523$	−749.167			−1.43244			−0.716221	+	0.697873i	$0.754130\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−183.374			−0.347958
$528$		0			0
$529$	481.833			0.910837
$530$		0			0
$531$		−	163.925i		−	0.308709i
$532$		0			0
$533$		−	343.979i		−	0.645363i
$534$		0			0
$535$		0			0
$536$		0			0
$537$			124.810i			0.232421i
$538$		0			0
$539$	−183.613			−0.340655
$540$		0			0
$541$	156.375			0.289048			0.144524	−	0.989501i	$-0.453835\pi$
$541$	156.375			0.289048			0.144524	+	0.989501i	$0.453835\pi$
$542$		0			0
$543$		−	127.520i		−	0.234843i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	453.990			0.829964			0.414982	−	0.909830i	$-0.363788\pi$
$547$	453.990			0.829964			0.414982	+	0.909830i	$0.363788\pi$
$548$		0			0
$549$	−295.669			−0.538559
$550$		0			0
$551$	555.489	+	431.071i	1.00815	+	0.782344i
$552$		0			0
$553$	446.297			0.807046
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	481.620i		−	0.864668i	−0.901714	−	0.432334i	$-0.857690\pi$
$557$		−	481.620i		−	0.864668i	0.901714	−	0.432334i	$-0.142310\pi$
$558$		0			0
$559$		−	804.295i		−	1.43881i
$560$		0			0
$561$		−	57.6545i		−	0.102771i
$562$		0			0
$563$	182.504			0.324163			0.162081	−	0.986777i	$-0.448179\pi$
$563$	182.504			0.324163			0.162081	+	0.986777i	$0.448179\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	280.705i		−	0.495070i
$568$		0			0
$569$			675.222i			1.18668i	0.804951	+	0.593341i	$0.202191\pi$
$569$			675.222i			1.18668i	−0.804951	+	0.593341i	$0.797809\pi$
$570$		0			0
$571$	633.002			1.10858			0.554292	−	0.832322i	$-0.312989\pi$
$571$	633.002			1.10858			0.554292	+	0.832322i	$0.312989\pi$
$572$		0			0
$573$	3.46765			0.00605175
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	66.1452i		−	0.114636i	−0.998356	−	0.0573182i	$-0.981745\pi$
$577$		−	66.1452i		−	0.114636i	0.998356	−	0.0573182i	$-0.0182550\pi$
$578$		0			0
$579$	4.85961			0.00839311
$580$		0			0
$581$	517.923			0.891433
$582$		0			0
$583$	204.422			0.350638
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	730.051i		−	1.24370i	−0.783137	−	0.621850i	$-0.786382\pi$
$587$		−	730.051i		−	1.24370i	0.783137	−	0.621850i	$-0.213618\pi$
$588$		0			0
$589$	−166.807	−	129.445i	−0.283203	−	0.219771i
$590$		0			0
$591$			230.967i			0.390808i
$592$		0			0
$593$			591.105i			0.996805i	0.866946	+	0.498403i	$0.166080\pi$
$593$			591.105i			0.996805i	−0.866946	+	0.498403i	$0.833920\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	200.062			0.335112
$598$		0			0
$599$		−	885.437i		−	1.47819i	−0.673600	−	0.739096i	$-0.735253\pi$
$599$		−	885.437i		−	1.47819i	0.673600	−	0.739096i	$-0.264747\pi$
$600$		0			0
$601$		−	591.878i		−	0.984823i	−0.870363	−	0.492411i	$-0.836116\pi$
$601$		−	591.878i		−	0.984823i	0.870363	−	0.492411i	$-0.163884\pi$
$602$		0			0
$603$	234.357			0.388652
$604$		0			0
$605$		0			0
$606$		0			0
$607$	747.061			1.23074			0.615371	−	0.788237i	$-0.289006\pi$
$607$	747.061			1.23074			0.615371	+	0.788237i	$0.289006\pi$
$608$		0			0
$609$	−92.9416			−0.152614
$610$		0			0
$611$			1457.17i			2.38490i
$612$		0			0
$613$		−	1164.45i		−	1.89960i	−0.312859	−	0.949800i	$-0.601287\pi$
$613$		−	1164.45i		−	1.89960i	0.312859	−	0.949800i	$-0.398713\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			352.505i			0.571321i	0.958331	+	0.285660i	$0.0922129\pi$
$617$			352.505i			0.571321i	−0.958331	+	0.285660i	$0.907787\pi$
$618$		0			0
$619$	240.601			0.388693			0.194347	−	0.980933i	$-0.437741\pi$
$619$	240.601			0.388693			0.194347	+	0.980933i	$0.437741\pi$
$620$		0			0
$621$			76.1854i			0.122682i
$622$		0			0
$623$	265.305			0.425851
$624$		0			0
$625$		0			0
$626$		0			0
$627$	40.6989	−	52.4457i	0.0649106	−	0.0836454i
$628$		0			0
$629$		−	1100.55i		−	1.74969i
$630$		0			0
$631$	582.735			0.923510			0.461755	−	0.887007i	$-0.347220\pi$
$631$	582.735			0.923510			0.461755	+	0.887007i	$0.347220\pi$
$632$		0			0
$633$			180.045i			0.284432i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	765.813			1.20222
$638$		0			0
$639$		−	781.938i		−	1.22369i
$640$		0			0
$641$		−	314.886i		−	0.491242i	−0.969366	−	0.245621i	$-0.921008\pi$
$641$		−	314.886i		−	0.491242i	0.969366	−	0.245621i	$-0.0789919\pi$
$642$		0			0
$643$		−	650.686i		−	1.01195i	−0.862547	−	0.505977i	$-0.831132\pi$
$643$		−	650.686i		−	1.01195i	0.862547	−	0.505977i	$-0.168868\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			2.19128i			0.00338683i	0.999999	+	0.00169341i	$0.000539030\pi$
$647$			2.19128i			0.00338683i	−0.999999	+	0.00169341i	$0.999461\pi$
$648$		0			0
$649$		−	105.645i		−	0.162781i
$650$		0			0
$651$	27.9092			0.0428713
$652$		0			0
$653$		−	955.448i		−	1.46317i	−0.681752	−	0.731584i	$-0.738782\pi$
$653$		−	955.448i		−	1.46317i	0.681752	−	0.731584i	$-0.261218\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	170.119i		−	0.258932i
$658$		0			0
$659$		−	76.0625i		−	0.115421i	−0.998333	−	0.0577106i	$-0.981620\pi$
$659$		−	76.0625i		−	0.115421i	0.998333	−	0.0577106i	$-0.0183801\pi$
$660$		0			0
$661$			662.480i			1.00224i	0.865378	+	0.501119i	$0.167078\pi$
$661$			662.480i			1.00224i	−0.865378	+	0.501119i	$0.832922\pi$
$662$		0			0
$663$			240.465i			0.362693i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−254.157			−0.381045
$668$		0			0
$669$	−96.1512			−0.143724
$670$		0			0
$671$	−190.550			−0.283980
$672$		0			0
$673$	150.670			0.223879			0.111939	−	0.993715i	$-0.464294\pi$
$673$	150.670			0.223879			0.111939	+	0.993715i	$0.464294\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	392.546			0.579832			0.289916	−	0.957052i	$-0.406373\pi$
$677$	392.546			0.579832			0.289916	+	0.957052i	$0.406373\pi$
$678$		0			0
$679$			75.9616i			0.111873i
$680$		0			0
$681$	−255.601			−0.375332
$682$		0			0
$683$	−737.192			−1.07934			−0.539672	−	0.841875i	$-0.681452\pi$
$683$	−737.192			−1.07934			−0.539672	+	0.841875i	$0.681452\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−213.879			−0.311324
$688$		0			0
$689$	−852.603			−1.23745
$690$		0			0
$691$	828.415			1.19886			0.599432	−	0.800426i	$-0.295393\pi$
$691$	828.415			1.19886			0.599432	+	0.800426i	$0.295393\pi$
$692$		0			0
$693$		−	190.087i		−	0.274295i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	245.461			0.352168
$698$		0			0
$699$		−	154.842i		−	0.221519i
$700$		0			0
$701$	−1040.93			−1.48493			−0.742463	−	0.669887i	$-0.766343\pi$
$701$	−1040.93			−1.48493			−0.742463	+	0.669887i	$0.766343\pi$
$702$		0			0
$703$	776.893	−	1001.12i	1.10511	−	1.42407i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			424.310i			0.600155i
$708$		0			0
$709$	−29.3704			−0.0414251			−0.0207126	−	0.999785i	$-0.506593\pi$
$709$	−29.3704			−0.0414251			−0.0207126	+	0.999785i	$0.506593\pi$
$710$		0			0
$711$		−	963.399i		−	1.35499i
$712$		0			0
$713$	76.3202			0.107041
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−172.106			−0.240037
$718$		0			0
$719$	241.324			0.335638			0.167819	−	0.985818i	$-0.446328\pi$
$719$	241.324			0.335638			0.167819	+	0.985818i	$0.446328\pi$
$720$		0			0
$721$			491.235i			0.681325i
$722$		0			0
$723$		−	72.1815i		−	0.0998361i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			142.920i			0.196589i	0.995157	+	0.0982943i	$0.0313387\pi$
$727$			142.920i			0.196589i	−0.995157	+	0.0982943i	$0.968661\pi$
$728$		0			0
$729$	−543.028			−0.744894
$730$		0			0
$731$	573.940			0.785143
$732$		0			0
$733$			316.452i			0.431721i	0.976424	+	0.215861i	$0.0692557\pi$
$733$			316.452i			0.431721i	−0.976424	+	0.215861i	$0.930744\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	151.037			0.204934
$738$		0			0
$739$	497.784			0.673592			0.336796	−	0.941578i	$-0.390657\pi$
$739$	497.784			0.673592			0.336796	+	0.941578i	$0.390657\pi$
$740$		0			0
$741$	−169.747	+	218.740i	−0.229078	+	0.295196i
$742$		0			0
$743$	−554.732			−0.746611			−0.373306	−	0.927708i	$-0.621776\pi$
$743$	−554.732			−0.746611			−0.373306	+	0.927708i	$0.621776\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	1118.01i		−	1.49667i
$748$		0			0
$749$		−	407.050i		−	0.543458i
$750$		0			0
$751$			694.808i			0.925177i	0.886573	+	0.462589i	$0.153079\pi$
$751$			694.808i			0.925177i	−0.886573	+	0.462589i	$0.846921\pi$
$752$		0			0
$753$	181.328			0.240808
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	927.589i		−	1.22535i	−0.790335	−	0.612675i	$-0.790094\pi$
$757$		−	927.589i		−	1.22535i	0.790335	−	0.612675i	$-0.209906\pi$
$758$		0			0
$759$			23.9958i			0.0316151i
$760$		0			0
$761$	83.5130			0.109741			0.0548706	−	0.998493i	$-0.482525\pi$
$761$	83.5130			0.109741			0.0548706	+	0.998493i	$0.482525\pi$
$762$		0			0
$763$	754.849			0.989317
$764$		0			0
$765$		0			0
$766$		0			0
$767$			440.623i			0.574476i
$768$		0			0
$769$	−91.0434			−0.118392			−0.0591960	−	0.998246i	$-0.518854\pi$
$769$	−91.0434			−0.118392			−0.0591960	+	0.998246i	$0.518854\pi$
$770$		0			0
$771$	−150.415			−0.195091
$772$		0			0
$773$	920.649			1.19101			0.595504	−	0.803352i	$-0.296952\pi$
$773$	920.649			1.19101			0.595504	+	0.803352i	$0.296952\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$			167.503i			0.215577i
$778$		0			0
$779$	223.285	+	173.274i	0.286630	+	0.222431i
$780$		0			0
$781$		−	503.937i		−	0.645246i
$782$		0			0
$783$			410.519i			0.524290i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−1386.63			−1.76192			−0.880962	−	0.473188i	$-0.843103\pi$
$787$	−1386.63			−1.76192			−0.880962	+	0.473188i	$0.843103\pi$
$788$		0			0
$789$			283.748i			0.359630i
$790$		0			0
$791$			7.84625i			0.00991941i
$792$		0			0
$793$	794.747			1.00220
$794$		0			0
$795$		0			0
$796$		0			0
$797$	327.713			0.411183			0.205592	−	0.978638i	$-0.434088\pi$
$797$	327.713			0.411183			0.205592	+	0.978638i	$0.434088\pi$
$798$		0			0
$799$	−1039.83			−1.30142
$800$		0			0
$801$		−	572.701i		−	0.714983i
$802$		0			0
$803$		−	109.637i		−	0.136534i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	127.898i		−	0.158486i
$808$		0			0
$809$	−130.207			−0.160948			−0.0804738	−	0.996757i	$-0.525643\pi$
$809$	−130.207			−0.160948			−0.0804738	+	0.996757i	$0.525643\pi$
$810$		0			0
$811$		−	732.273i		−	0.902926i	−0.892290	−	0.451463i	$-0.850902\pi$
$811$		−	732.273i		−	0.902926i	0.892290	−	0.451463i	$-0.149098\pi$
$812$		0			0
$813$	−66.5163			−0.0818159
$814$		0			0
$815$		0			0
$816$		0			0
$817$	522.087	+	405.150i	0.639029	+	0.495900i
$818$		0			0
$819$			792.813i			0.968025i
$820$		0			0
$821$	163.217			0.198803			0.0994014	−	0.995047i	$-0.468307\pi$
$821$	163.217			0.198803			0.0994014	+	0.995047i	$0.468307\pi$
$822$		0			0
$823$			533.268i			0.647957i	0.946065	+	0.323978i	$0.105020\pi$
$823$			533.268i			0.647957i	−0.946065	+	0.323978i	$0.894980\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1276.45			1.54347			0.771733	−	0.635947i	$-0.219390\pi$
$827$	1276.45			1.54347			0.771733	+	0.635947i	$0.219390\pi$
$828$		0			0
$829$		−	1194.31i		−	1.44067i	−0.693628	−	0.720333i	$-0.743989\pi$
$829$		−	1194.31i		−	1.44067i	0.693628	−	0.720333i	$-0.256011\pi$
$830$		0			0
$831$		−	160.026i		−	0.192571i
$832$		0			0
$833$			546.480i			0.656038i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	123.274i		−	0.147280i
$838$		0			0
$839$			1239.77i			1.47768i	0.673883	+	0.738839i	$0.264625\pi$
$839$			1239.77i			1.47768i	−0.673883	+	0.738839i	$0.735375\pi$
$840$		0			0
$841$	−528.504			−0.628424
$842$		0			0
$843$		−	167.416i		−	0.198596i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			359.715i			0.424694i
$848$		0			0
$849$			39.3004i			0.0462902i
$850$		0			0
$851$			458.052i			0.538251i
$852$		0			0
$853$			644.073i			0.755068i	0.925996	+	0.377534i	$0.123228\pi$
$853$			644.073i			0.755068i	−0.925996	+	0.377534i	$0.876772\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−263.355			−0.307299			−0.153650	−	0.988125i	$-0.549103\pi$
$857$	−263.355			−0.307299			−0.153650	+	0.988125i	$0.549103\pi$
$858$		0			0
$859$	−931.348			−1.08422			−0.542112	−	0.840307i	$-0.682375\pi$
$859$	−931.348			−1.08422			−0.542112	+	0.840307i	$0.682375\pi$
$860$		0			0
$861$	−37.3588			−0.0433901
$862$		0			0
$863$	−1043.40			−1.20903			−0.604517	−	0.796592i	$-0.706634\pi$
$863$	−1043.40			−1.20903			−0.604517	+	0.796592i	$0.706634\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	10.5288			0.0121439
$868$		0			0
$869$		−	620.884i		−	0.714481i
$870$		0			0
$871$	−629.943			−0.723241
$872$		0			0
$873$	163.975			0.187829
$874$		0			0
$875$		0			0
$876$		0			0
$877$	366.106			0.417453			0.208727	−	0.977974i	$-0.433068\pi$
$877$	366.106			0.417453			0.208727	+	0.977974i	$0.433068\pi$
$878$		0			0
$879$	138.188			0.157210
$880$		0			0
$881$	742.098			0.842336			0.421168	−	0.906983i	$-0.361620\pi$
$881$	742.098			0.842336			0.421168	+	0.906983i	$0.361620\pi$
$882$		0			0
$883$			197.913i			0.224137i	0.993701	+	0.112068i	$0.0357475\pi$
$883$			197.913i			0.224137i	−0.993701	+	0.112068i	$0.964252\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−356.570			−0.401995			−0.200998	−	0.979592i	$-0.564418\pi$
$887$	−356.570			−0.401995			−0.200998	+	0.979592i	$0.564418\pi$
$888$		0			0
$889$		−	8.71367i		−	0.00980165i
$890$		0			0
$891$	−390.514			−0.438288
$892$		0			0
$893$	−945.886	−	734.027i	−1.05922	−	0.821979i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		−	100.082i		−	0.111574i
$898$		0			0
$899$	411.245			0.457447
$900$		0			0
$901$		−	608.412i		−	0.675263i
$902$		0			0
$903$	−87.3529			−0.0967363
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−562.349			−0.620010			−0.310005	−	0.950735i	$-0.600331\pi$
$907$	−562.349			−0.620010			−0.310005	+	0.950735i	$0.600331\pi$
$908$		0			0
$909$	915.937			1.00763
$910$		0			0
$911$			1310.45i			1.43847i	0.694765	+	0.719237i	$0.255509\pi$
$911$			1310.45i			1.43847i	−0.694765	+	0.719237i	$0.744491\pi$
$912$		0			0
$913$		−	720.529i		−	0.789189i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			150.537i			0.164162i
$918$		0			0
$919$	768.042			0.835736			0.417868	−	0.908508i	$-0.362777\pi$
$919$	768.042			0.835736			0.417868	+	0.908508i	$0.362777\pi$
$920$		0			0
$921$	136.948			0.148695
$922$		0			0
$923$			2101.82i			2.27716i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	1060.41			1.14391
$928$		0			0
$929$	478.127			0.514669			0.257334	−	0.966322i	$-0.417156\pi$
$929$	478.127			0.514669			0.257334	+	0.966322i	$0.417156\pi$
$930$		0			0
$931$	−385.766	+	497.108i	−0.414356	+	0.533950i
$932$		0			0
$933$	210.911			0.226057
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	427.941i		−	0.456714i	−0.973577	−	0.228357i	$-0.926665\pi$
$937$		−	427.941i		−	0.456714i	0.973577	−	0.228357i	$-0.0733353\pi$
$938$		0			0
$939$			31.6033i			0.0336564i
$940$		0			0
$941$			1089.92i			1.15826i	0.815237	+	0.579128i	$0.196607\pi$
$941$			1089.92i			1.15826i	−0.815237	+	0.579128i	$0.803393\pi$
$942$		0			0
$943$	−102.161			−0.108336
$944$		0			0
$945$		0			0
$946$		0			0
$947$			950.576i			1.00378i	0.864933	+	0.501888i	$0.167361\pi$
$947$			950.576i			1.00378i	−0.864933	+	0.501888i	$0.832639\pi$
$948$		0			0
$949$			457.272i			0.481847i
$950$		0			0
$951$	−92.7151			−0.0974922
$952$		0			0
$953$	−1262.54			−1.32481			−0.662405	−	0.749146i	$-0.730464\pi$
$953$	−1262.54			−1.32481			−0.662405	+	0.749146i	$0.730464\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			129.300i			0.135109i
$958$		0			0
$959$	470.074			0.490171
$960$		0			0
$961$	837.508			0.871497
$962$		0			0
$963$	−878.679			−0.912439
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	1658.76i		−	1.71536i	−0.514180	−	0.857682i	$-0.671904\pi$
$967$		−	1658.76i		−	1.71536i	0.514180	−	0.857682i	$-0.328096\pi$
$968$		0			0
$969$	−156.092	−	121.130i	−0.161085	−	0.125006i
$970$		0			0
$971$		−	1083.30i		−	1.11565i	−0.829957	−	0.557827i	$-0.811635\pi$
$971$		−	1083.30i		−	1.11565i	0.829957	−	0.557827i	$-0.188365\pi$
$972$		0			0
$973$		−	659.278i		−	0.677573i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1248.74			−1.27813			−0.639066	−	0.769152i	$-0.720679\pi$
$977$	−1248.74			−1.27813			−0.639066	+	0.769152i	$0.720679\pi$
$978$		0			0
$979$		−	369.090i		−	0.377007i
$980$		0			0
$981$		−	1629.45i		−	1.66101i
$982$		0			0
$983$	−1806.81			−1.83806			−0.919028	−	0.394192i	$-0.871024\pi$
$983$	−1806.81			−1.83806			−0.919028	+	0.394192i	$0.871024\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	158.261			0.160345
$988$		0			0
$989$	−238.874			−0.241531
$990$		0			0
$991$		−	697.379i		−	0.703712i	−0.936054	−	0.351856i	$-0.885551\pi$
$991$		−	697.379i		−	0.703712i	0.936054	−	0.351856i	$-0.114449\pi$
$992$		0			0
$993$			234.014i			0.235664i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	1233.03i		−	1.23674i	−0.785887	−	0.618370i	$-0.787793\pi$
$997$		−	1233.03i		−	1.23674i	0.785887	−	0.618370i	$-0.212207\pi$
$998$		0			0
$999$	739.853			0.740593

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.3.g.c.949.14		24
5.2	odd	4		1900.3.e.f.1101.6		12
5.3	odd	4		380.3.e.a.341.7	yes	12
5.4	even	2	inner	1900.3.g.c.949.11		24
15.8	even	4		3420.3.o.a.721.2		12
19.18	odd	2	inner	1900.3.g.c.949.12		24
20.3	even	4		1520.3.h.b.721.6		12
95.18	even	4		380.3.e.a.341.6	✓	12
95.37	even	4		1900.3.e.f.1101.7		12
95.94	odd	2	inner	1900.3.g.c.949.13		24
285.113	odd	4		3420.3.o.a.721.1		12
380.303	odd	4		1520.3.h.b.721.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.e.a.341.6	✓	12	95.18	even	4
380.3.e.a.341.7	yes	12	5.3	odd	4
1520.3.h.b.721.6		12	20.3	even	4
1520.3.h.b.721.7		12	380.303	odd	4
1900.3.e.f.1101.6		12	5.2	odd	4
1900.3.e.f.1101.7		12	95.37	even	4
1900.3.g.c.949.11		24	5.4	even	2	inner
1900.3.g.c.949.12		24	19.18	odd	2	inner
1900.3.g.c.949.13		24	95.94	odd	2	inner
1900.3.g.c.949.14		24	1.1	even	1	trivial
3420.3.o.a.721.1		12	285.113	odd	4
3420.3.o.a.721.2		12	15.8	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.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.630185 q^{3} -3.98530i q^{7} -8.60287 q^{9} +O(q^{10})\)	\(q+0.630185 q^{3} -3.98530i q^{7} -8.60287 q^{9} -5.54431 q^{11} +23.1242 q^{13} +16.5013i q^{17} +(-11.6484 + 15.0105i) q^{19} -2.51147i q^{21} -6.86785i q^{23} -11.0931 q^{27} -37.0068i q^{29} +11.1127i q^{31} -3.49394 q^{33} -66.6951 q^{37} +14.5725 q^{39} -14.8753i q^{41} -34.7815i q^{43} +63.0151i q^{47} +33.1174 q^{49} +10.3989i q^{51} -36.8706 q^{53} +(-7.34066 + 9.45937i) q^{57} +19.0546i q^{59} +34.3686 q^{61} +34.2850i q^{63} -27.2417 q^{67} -4.32801i q^{69} +90.8927i q^{71} +19.7746i q^{73} +22.0957i q^{77} +111.986i q^{79} +70.4351 q^{81} +129.958i q^{83} -23.3211i q^{87} +66.5710i q^{89} -92.1568i q^{91} +7.00305i q^{93} -19.0605 q^{97} +47.6970 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 32 q^{9}+O(q^{10}) \) 24 * q + 32 * q^9	\( 24 q + 32 q^{9} - 64 q^{11} - 48 q^{19} - 248 q^{39} + 48 q^{49} + 304 q^{61} - 936 q^{81} - 784 q^{99}+O(q^{100}) \) 24 * q + 32 * q^9 - 64 * q^11 - 48 * q^19 - 248 * q^39 + 48 * q^49 + 304 * q^61 - 936 * q^81 - 784 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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