LMFDB - Embedded newform 1216.3.e.n.1025.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,3,Mod(1025,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, 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("1216.1025");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1216.e (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:	$33.1336001462$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 34x^{6} + 345x^{4} + 1064x^{2} + 256 $ x^8 + 34x^6 + 345x^4 + 1064*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.6
Root			$2.27869i$ of defining polynomial
Character	$\chi$	$=$	1216.1025
Dual form			1216.3.e.n.1025.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.27869i q^{3} -6.29008 q^{5} +1.03740 q^{7} +3.80758 q^{9} +O(q^{10})$	$q+2.27869i q^{3} -6.29008 q^{5} +1.03740 q^{7} +3.80758 q^{9} -4.67493 q^{11} -13.2282i q^{13} -14.3331i q^{15} +9.09766 q^{17} +(14.7726 + 11.9487i) q^{19} +2.36391i q^{21} +16.2925 q^{23} +14.5651 q^{25} +29.1845i q^{27} +12.4187i q^{29} +31.6747i q^{31} -10.6527i q^{33} -6.52533 q^{35} +57.9305i q^{37} +30.1429 q^{39} -33.8737i q^{41} -31.3008 q^{43} -23.9500 q^{45} -14.7654 q^{47} -47.9238 q^{49} +20.7307i q^{51} -2.57547i q^{53} +29.4057 q^{55} +(-27.2273 + 33.6621i) q^{57} -43.3981i q^{59} -50.5698 q^{61} +3.94998 q^{63} +83.2063i q^{65} -83.1046i q^{67} +37.1255i q^{69} -8.54578i q^{71} -145.369 q^{73} +33.1894i q^{75} -4.84977 q^{77} +100.604i q^{79} -32.2341 q^{81} +139.547 q^{83} -57.2250 q^{85} -28.2983 q^{87} +109.901i q^{89} -13.7229i q^{91} -72.1769 q^{93} +(-92.9208 - 75.1581i) q^{95} +81.1313i q^{97} -17.8001 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 14 q^{5} + 6 q^{7} + 4 q^{9}+O(q^{10}) $ 8 * q + 14 * q^5 + 6 * q^7 + 4 * q^9	$ 8 q + 14 q^{5} + 6 q^{7} + 4 q^{9} - 26 q^{11} - 18 q^{17} + 16 q^{19} - 12 q^{23} + 34 q^{25} - 50 q^{35} + 108 q^{39} + 62 q^{43} - 162 q^{45} - 22 q^{47} + 22 q^{49} - 174 q^{55} + 4 q^{57} + 158 q^{61} + 2 q^{63} - 170 q^{73} - 82 q^{77} - 256 q^{81} - 64 q^{83} - 410 q^{85} + 332 q^{87} + 344 q^{93} - 222 q^{95} + 526 q^{99}+O(q^{100}) $ 8 * q + 14 * q^5 + 6 * q^7 + 4 * q^9 - 26 * q^11 - 18 * q^17 + 16 * q^19 - 12 * q^23 + 34 * q^25 - 50 * q^35 + 108 * q^39 + 62 * q^43 - 162 * q^45 - 22 * q^47 + 22 * q^49 - 174 * q^55 + 4 * q^57 + 158 * q^61 + 2 * q^63 - 170 * q^73 - 82 * q^77 - 256 * q^81 - 64 * q^83 - 410 * q^85 + 332 * q^87 + 344 * q^93 - 222 * q^95 + 526 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$1$	$-1$	$1$

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
$3$			2.27869i			0.759563i	0.925076	+	0.379781i	$0.124001\pi$
$3$			2.27869i			0.759563i	−0.925076	+	0.379781i	$0.875999\pi$
$4$		0			0
$5$	−6.29008			−1.25802			−0.629008	−	0.777399i	$-0.716539\pi$
$5$	−6.29008			−1.25802			−0.629008	+	0.777399i	$0.716539\pi$
$6$		0			0
$7$	1.03740			0.148200			0.0741000	−	0.997251i	$-0.476392\pi$
$7$	1.03740			0.148200			0.0741000	+	0.997251i	$0.476392\pi$
$8$		0			0
$9$	3.80758			0.423064
$10$		0			0
$11$	−4.67493			−0.424993			−0.212497	−	0.977162i	$-0.568159\pi$
$11$	−4.67493			−0.424993			−0.212497	+	0.977162i	$0.568159\pi$
$12$		0			0
$13$		−	13.2282i		−	1.01755i	−0.860899	−	0.508776i	$-0.830098\pi$
$13$		−	13.2282i		−	1.01755i	0.860899	−	0.508776i	$-0.169902\pi$
$14$		0			0
$15$		−	14.3331i		−	0.955543i
$16$		0			0
$17$	9.09766			0.535156			0.267578	−	0.963536i	$-0.413777\pi$
$17$	9.09766			0.535156			0.267578	+	0.963536i	$0.413777\pi$
$18$		0			0
$19$	14.7726	+	11.9487i	0.777505	+	0.628877i
$19$	14.7726	+	11.9487i	0.777505	+	0.628877i
$20$		0			0
$21$			2.36391i			0.112567i
$22$		0			0
$23$	16.2925			0.708369			0.354185	−	0.935176i	$-0.384758\pi$
$23$	16.2925			0.708369			0.354185	+	0.935176i	$0.384758\pi$
$24$		0			0
$25$	14.5651			0.582605
$26$		0			0
$27$			29.1845i			1.08091i
$28$		0			0
$29$			12.4187i			0.428230i	0.976808	+	0.214115i	$0.0686867\pi$
$29$			12.4187i			0.428230i	−0.976808	+	0.214115i	$0.931313\pi$
$30$		0			0
$31$			31.6747i			1.02177i	0.859650	+	0.510883i	$0.170681\pi$
$31$			31.6747i			1.02177i	−0.859650	+	0.510883i	$0.829319\pi$
$32$		0			0
$33$		−	10.6527i		−	0.322809i
$34$		0			0
$35$	−6.52533			−0.186438
$36$		0			0
$37$			57.9305i			1.56569i	0.622217	+	0.782845i	$0.286232\pi$
$37$			57.9305i			1.56569i	−0.622217	+	0.782845i	$0.713768\pi$
$38$		0			0
$39$	30.1429			0.772895
$40$		0			0
$41$		−	33.8737i		−	0.826188i	−0.910688	−	0.413094i	$-0.864448\pi$
$41$		−	33.8737i		−	0.826188i	0.910688	−	0.413094i	$-0.135552\pi$
$42$		0			0
$43$	−31.3008			−0.727926			−0.363963	−	0.931413i	$-0.618577\pi$
$43$	−31.3008			−0.727926			−0.363963	+	0.931413i	$0.618577\pi$
$44$		0			0
$45$	−23.9500			−0.532222
$46$		0			0
$47$	−14.7654			−0.314157			−0.157079	−	0.987586i	$-0.550208\pi$
$47$	−14.7654			−0.314157			−0.157079	+	0.987586i	$0.550208\pi$
$48$		0			0
$49$	−47.9238			−0.978037
$50$		0			0
$51$			20.7307i			0.406485i
$52$		0			0
$53$		−	2.57547i		−	0.0485938i	−0.999705	−	0.0242969i	$-0.992265\pi$
$53$		−	2.57547i		−	0.0485938i	0.999705	−	0.0242969i	$-0.00773471\pi$
$54$		0			0
$55$	29.4057			0.534649
$56$		0			0
$57$	−27.2273	+	33.6621i	−0.477672	+	0.590564i
$58$		0			0
$59$		−	43.3981i		−	0.735561i	−0.929913	−	0.367780i	$-0.880118\pi$
$59$		−	43.3981i		−	0.735561i	0.929913	−	0.367780i	$-0.119882\pi$
$60$		0			0
$61$	−50.5698			−0.829014			−0.414507	−	0.910046i	$-0.636046\pi$
$61$	−50.5698			−0.829014			−0.414507	+	0.910046i	$0.636046\pi$
$62$		0			0
$63$	3.94998			0.0626981
$64$		0			0
$65$			83.2063i			1.28010i
$66$		0			0
$67$		−	83.1046i		−	1.24037i	−0.784457	−	0.620184i	$-0.787058\pi$
$67$		−	83.1046i		−	1.24037i	0.784457	−	0.620184i	$-0.212942\pi$
$68$		0			0
$69$			37.1255i			0.538051i
$70$		0			0
$71$		−	8.54578i		−	0.120363i	−0.998187	−	0.0601815i	$-0.980832\pi$
$71$		−	8.54578i		−	0.120363i	0.998187	−	0.0601815i	$-0.0191680\pi$
$72$		0			0
$73$	−145.369			−1.99135			−0.995677	−	0.0928852i	$-0.970391\pi$
$73$	−145.369			−1.99135			−0.995677	+	0.0928852i	$0.970391\pi$
$74$		0			0
$75$			33.1894i			0.442525i
$76$		0			0
$77$	−4.84977			−0.0629840
$78$		0			0
$79$			100.604i			1.27347i	0.771081	+	0.636737i	$0.219716\pi$
$79$			100.604i			1.27347i	−0.771081	+	0.636737i	$0.780284\pi$
$80$		0			0
$81$	−32.2341			−0.397952
$82$		0			0
$83$	139.547			1.68129			0.840646	−	0.541585i	$-0.182176\pi$
$83$	139.547			1.68129			0.840646	+	0.541585i	$0.182176\pi$
$84$		0			0
$85$	−57.2250			−0.673236
$86$		0			0
$87$	−28.2983			−0.325268
$88$		0			0
$89$			109.901i			1.23484i	0.786635	+	0.617419i	$0.211822\pi$
$89$			109.901i			1.23484i	−0.786635	+	0.617419i	$0.788178\pi$
$90$		0			0
$91$		−	13.7229i		−	0.150801i
$92$		0			0
$93$	−72.1769			−0.776095
$94$		0			0
$95$	−92.9208	−	75.1581i	−0.978114	−	0.791138i
$96$		0			0
$97$			81.1313i			0.836405i	0.908354	+	0.418202i	$0.137340\pi$
$97$			81.1313i			0.836405i	−0.908354	+	0.418202i	$0.862660\pi$
$98$		0			0
$99$	−17.8001			−0.179799
$100$		0			0
$101$	−64.4920			−0.638534			−0.319267	−	0.947665i	$-0.603437\pi$
$101$	−64.4920			−0.638534			−0.319267	+	0.947665i	$0.603437\pi$
$102$		0			0
$103$		−	78.1486i		−	0.758724i	−0.925248	−	0.379362i	$-0.876143\pi$
$103$		−	78.1486i		−	0.758724i	0.925248	−	0.379362i	$-0.123857\pi$
$104$		0			0
$105$		−	14.8692i		−	0.141611i
$106$		0			0
$107$			82.2978i			0.769138i	0.923096	+	0.384569i	$0.125650\pi$
$107$			82.2978i			0.769138i	−0.923096	+	0.384569i	$0.874350\pi$
$108$		0			0
$109$			136.311i			1.25056i	0.780399	+	0.625282i	$0.215016\pi$
$109$			136.311i			1.25056i	−0.780399	+	0.625282i	$0.784984\pi$
$110$		0			0
$111$	−132.006			−1.18924
$112$		0			0
$113$			175.141i			1.54992i	0.632010	+	0.774960i	$0.282230\pi$
$113$			175.141i			1.54992i	−0.632010	+	0.774960i	$0.717770\pi$
$114$		0			0
$115$	−102.481			−0.891140
$116$		0			0
$117$		−	50.3673i		−	0.430490i
$118$		0			0
$119$	9.43791			0.0793102
$120$		0			0
$121$	−99.1451			−0.819381
$122$		0			0
$123$	77.1876			0.627541
$124$		0			0
$125$	65.6362			0.525089
$126$		0			0
$127$			108.592i			0.855056i	0.904002	+	0.427528i	$0.140616\pi$
$127$			108.592i			0.855056i	−0.904002	+	0.427528i	$0.859384\pi$
$128$		0			0
$129$		−	71.3248i		−	0.552905i
$130$		0			0
$131$	9.84988			0.0751900			0.0375950	−	0.999293i	$-0.488030\pi$
$131$	9.84988			0.0751900			0.0375950	+	0.999293i	$0.488030\pi$
$132$		0			0
$133$	15.3251	+	12.3955i	0.115226	+	0.0931996i
$134$		0			0
$135$		−	183.573i		−	1.35980i
$136$		0			0
$137$	−89.7964			−0.655448			−0.327724	−	0.944773i	$-0.606282\pi$
$137$	−89.7964			−0.655448			−0.327724	+	0.944773i	$0.606282\pi$
$138$		0			0
$139$	−23.1860			−0.166806			−0.0834030	−	0.996516i	$-0.526579\pi$
$139$	−23.1860			−0.166806			−0.0834030	+	0.996516i	$0.526579\pi$
$140$		0			0
$141$		−	33.6457i		−	0.238622i
$142$		0			0
$143$			61.8407i			0.432453i
$144$		0			0
$145$		−	78.1145i		−	0.538721i
$146$		0			0
$147$		−	109.203i		−	0.742880i
$148$		0			0
$149$	110.326			0.740443			0.370221	−	0.928944i	$-0.379282\pi$
$149$	110.326			0.740443			0.370221	+	0.928944i	$0.379282\pi$
$150$		0			0
$151$		−	115.413i		−	0.764326i	−0.924095	−	0.382163i	$-0.875179\pi$
$151$		−	115.413i		−	0.764326i	0.924095	−	0.382163i	$-0.124821\pi$
$152$		0			0
$153$	34.6401			0.226406
$154$		0			0
$155$		−	199.237i		−	1.28540i
$156$		0			0
$157$	−171.069			−1.08961			−0.544805	−	0.838563i	$-0.683396\pi$
$157$	−171.069			−1.08961			−0.544805	+	0.838563i	$0.683396\pi$
$158$		0			0
$159$	5.86870			0.0369101
$160$		0			0
$161$	16.9018			0.104980
$162$		0			0
$163$	−196.012			−1.20253			−0.601263	−	0.799051i	$-0.705335\pi$
$163$	−196.012			−1.20253			−0.601263	+	0.799051i	$0.705335\pi$
$164$		0			0
$165$			67.0064i			0.406099i
$166$		0			0
$167$			235.541i			1.41043i	0.708996	+	0.705213i	$0.249149\pi$
$167$			235.541i			1.41043i	−0.708996	+	0.705213i	$0.750851\pi$
$168$		0			0
$169$	−5.98462			−0.0354119
$170$		0			0
$171$	56.2478	+	45.4955i	0.328934	+	0.266055i
$172$		0			0
$173$		−	336.786i		−	1.94674i	−0.229237	−	0.973371i	$-0.573623\pi$
$173$		−	336.786i		−	1.94674i	0.229237	−	0.973371i	$-0.426377\pi$
$174$		0			0
$175$	15.1099			0.0863421
$176$		0			0
$177$	98.8907			0.558705
$178$		0			0
$179$		−	39.3051i		−	0.219582i	−0.993955	−	0.109791i	$-0.964982\pi$
$179$		−	39.3051i		−	0.219582i	0.993955	−	0.109791i	$-0.0350181\pi$
$180$		0			0
$181$			331.772i			1.83300i	0.400038	+	0.916498i	$0.368997\pi$
$181$			331.772i			1.83300i	−0.400038	+	0.916498i	$0.631003\pi$
$182$		0			0
$183$		−	115.233i		−	0.629688i
$184$		0			0
$185$		−	364.388i		−	1.96966i
$186$		0			0
$187$	−42.5309			−0.227438
$188$		0			0
$189$			30.2760i			0.160190i
$190$		0			0
$191$	−22.1093			−0.115755			−0.0578777	−	0.998324i	$-0.518433\pi$
$191$	−22.1093			−0.115755			−0.0578777	+	0.998324i	$0.518433\pi$
$192$		0			0
$193$			50.1326i			0.259754i	0.991530	+	0.129877i	$0.0414583\pi$
$193$			50.1326i			0.259754i	−0.991530	+	0.129877i	$0.958542\pi$
$194$		0			0
$195$	−189.601			−0.972314
$196$		0			0
$197$	43.8312			0.222493			0.111247	−	0.993793i	$-0.464516\pi$
$197$	43.8312			0.222493			0.111247	+	0.993793i	$0.464516\pi$
$198$		0			0
$199$	376.691			1.89292			0.946461	−	0.322818i	$-0.104630\pi$
$199$	376.691			1.89292			0.946461	+	0.322818i	$0.104630\pi$
$200$		0			0
$201$	189.370			0.942137
$202$		0			0
$203$			12.8831i			0.0634637i
$204$		0			0
$205$			213.068i			1.03936i
$206$		0			0
$207$	62.0349			0.299686
$208$		0			0
$209$	−69.0608	−	55.8591i	−0.330434	−	0.267269i
$210$		0			0
$211$			299.111i			1.41759i	0.705416	+	0.708794i	$0.250760\pi$
$211$			299.111i			1.41759i	−0.705416	+	0.708794i	$0.749240\pi$
$212$		0			0
$213$	19.4732			0.0914233
$214$		0			0
$215$	196.885			0.915743
$216$		0			0
$217$			32.8594i			0.151426i
$218$		0			0
$219$		−	331.250i		−	1.51256i
$220$		0			0
$221$		−	120.345i		−	0.544550i
$222$		0			0
$223$			96.3080i			0.431874i	0.976407	+	0.215937i	$0.0692807\pi$
$223$			96.3080i			0.431874i	−0.976407	+	0.215937i	$0.930719\pi$
$224$		0			0
$225$	55.4579			0.246480
$226$		0			0
$227$			425.697i			1.87532i	0.347556	+	0.937659i	$0.387012\pi$
$227$			425.697i			1.87532i	−0.347556	+	0.937659i	$0.612988\pi$
$228$		0			0
$229$	−210.329			−0.918466			−0.459233	−	0.888316i	$-0.651876\pi$
$229$	−210.329			−0.918466			−0.459233	+	0.888316i	$0.651876\pi$
$230$		0			0
$231$		−	11.0511i		−	0.0478403i
$232$		0			0
$233$	−23.2820			−0.0999226			−0.0499613	−	0.998751i	$-0.515910\pi$
$233$	−23.2820			−0.0999226			−0.0499613	+	0.998751i	$0.515910\pi$
$234$		0			0
$235$	92.8755			0.395215
$236$		0			0
$237$	−229.246			−0.967283
$238$		0			0
$239$	−182.968			−0.765558			−0.382779	−	0.923840i	$-0.625033\pi$
$239$	−182.968			−0.765558			−0.382779	+	0.923840i	$0.625033\pi$
$240$		0			0
$241$			64.3761i			0.267121i	0.991041	+	0.133560i	$0.0426410\pi$
$241$			64.3761i			0.267121i	−0.991041	+	0.133560i	$0.957359\pi$
$242$		0			0
$243$			189.209i			0.778637i
$244$		0			0
$245$	301.445			1.23039
$246$		0			0
$247$	158.059	−	195.414i	0.639915	−	0.791151i
$248$		0			0
$249$			317.985i			1.27705i
$250$		0			0
$251$	396.215			1.57855			0.789273	−	0.614043i	$-0.210458\pi$
$251$	396.215			1.57855			0.789273	+	0.614043i	$0.210458\pi$
$252$		0			0
$253$	−76.1662			−0.301052
$254$		0			0
$255$		−	130.398i		−	0.511365i
$256$		0			0
$257$			53.3251i			0.207491i	0.994604	+	0.103745i	$0.0330827\pi$
$257$			53.3251i			0.207491i	−0.994604	+	0.103745i	$0.966917\pi$
$258$		0			0
$259$			60.0971i			0.232035i
$260$		0			0
$261$			47.2851i			0.181169i
$262$		0			0
$263$	−457.541			−1.73970			−0.869850	−	0.493316i	$-0.835785\pi$
$263$	−457.541			−1.73970			−0.869850	+	0.493316i	$0.835785\pi$
$264$		0			0
$265$			16.1999i			0.0611319i
$266$		0			0
$267$	−250.429			−0.937937
$268$		0			0
$269$			354.156i			1.31657i	0.752770	+	0.658283i	$0.228717\pi$
$269$			354.156i			1.31657i	−0.752770	+	0.658283i	$0.771283\pi$
$270$		0			0
$271$	−178.839			−0.659922			−0.329961	−	0.943994i	$-0.607036\pi$
$271$	−178.839			−0.659922			−0.329961	+	0.943994i	$0.607036\pi$
$272$		0			0
$273$	31.2702			0.114543
$274$		0			0
$275$	−68.0909			−0.247603
$276$		0			0
$277$	−249.791			−0.901772			−0.450886	−	0.892581i	$-0.648892\pi$
$277$	−249.791			−0.901772			−0.450886	+	0.892581i	$0.648892\pi$
$278$		0			0
$279$			120.604i			0.432272i
$280$		0			0
$281$		−	62.6513i		−	0.222958i	−0.993767	−	0.111479i	$-0.964441\pi$
$281$		−	62.6513i		−	0.222958i	0.993767	−	0.111479i	$-0.0355588\pi$
$282$		0			0
$283$	517.713			1.82938			0.914688	−	0.404160i	$-0.132436\pi$
$283$	517.713			1.82938			0.914688	+	0.404160i	$0.132436\pi$
$284$		0			0
$285$	171.262	−	211.738i	0.600919	−	0.742939i
$286$		0			0
$287$		−	35.1405i		−	0.122441i
$288$		0			0
$289$	−206.233			−0.713608
$290$		0			0
$291$	−184.873			−0.635302
$292$		0			0
$293$		−	416.666i		−	1.42207i	−0.703158	−	0.711034i	$-0.748227\pi$
$293$		−	416.666i		−	1.42207i	0.703158	−	0.711034i	$-0.251773\pi$
$294$		0			0
$295$			272.978i			0.925348i
$296$		0			0
$297$		−	136.435i		−	0.459378i
$298$		0			0
$299$		−	215.520i		−	0.720802i
$300$		0			0
$301$	−32.4714			−0.107879
$302$		0			0
$303$		−	146.957i		−	0.485007i
$304$		0			0
$305$	318.088			1.04291
$306$		0			0
$307$		−	163.173i		−	0.531508i	−0.964041	−	0.265754i	$-0.914379\pi$
$307$		−	163.173i		−	0.531508i	0.964041	−	0.265754i	$-0.0856209\pi$
$308$		0			0
$309$	178.076			0.576299
$310$		0			0
$311$	439.856			1.41433			0.707163	−	0.707050i	$-0.249975\pi$
$311$	439.856			1.41433			0.707163	+	0.707050i	$0.249975\pi$
$312$		0			0
$313$	98.4282			0.314467			0.157234	−	0.987561i	$-0.449742\pi$
$313$	98.4282			0.314467			0.157234	+	0.987561i	$0.449742\pi$
$314$		0			0
$315$	−24.8457			−0.0788752
$316$		0			0
$317$		−	392.584i		−	1.23843i	−0.785220	−	0.619217i	$-0.787450\pi$
$317$		−	392.584i		−	1.23843i	0.785220	−	0.619217i	$-0.212550\pi$
$318$		0			0
$319$		−	58.0564i		−	0.181995i
$320$		0			0
$321$	−187.531			−0.584209
$322$		0			0
$323$	134.396	+	108.705i	0.416087	+	0.336548i
$324$		0			0
$325$		−	192.670i		−	0.592831i
$326$		0			0
$327$	−310.611			−0.949882
$328$		0			0
$329$	−15.3176			−0.0465581
$330$		0			0
$331$			482.498i			1.45770i	0.684675	+	0.728849i	$0.259944\pi$
$331$			482.498i			1.45770i	−0.684675	+	0.728849i	$0.740056\pi$
$332$		0			0
$333$			220.575i			0.662387i
$334$		0			0
$335$			522.735i			1.56040i
$336$		0			0
$337$		−	158.934i		−	0.471614i	−0.971800	−	0.235807i	$-0.924227\pi$
$337$		−	158.934i		−	0.471614i	0.971800	−	0.235807i	$-0.0757733\pi$
$338$		0			0
$339$	−399.092			−1.17726
$340$		0			0
$341$		−	148.077i		−	0.434244i
$342$		0			0
$343$	−100.549			−0.293145
$344$		0			0
$345$		−	233.523i		−	0.676877i
$346$		0			0
$347$	421.620			1.21504			0.607522	−	0.794303i	$-0.292164\pi$
$347$	421.620			1.21504			0.607522	+	0.794303i	$0.292164\pi$
$348$		0			0
$349$	276.480			0.792206			0.396103	−	0.918206i	$-0.370362\pi$
$349$	276.480			0.792206			0.396103	+	0.918206i	$0.370362\pi$
$350$		0			0
$351$	386.057			1.09988
$352$		0			0
$353$	116.083			0.328846			0.164423	−	0.986390i	$-0.447424\pi$
$353$	116.083			0.328846			0.164423	+	0.986390i	$0.447424\pi$
$354$		0			0
$355$			53.7537i			0.151419i
$356$		0			0
$357$			21.5061i			0.0602410i
$358$		0			0
$359$	−102.430			−0.285319			−0.142660	−	0.989772i	$-0.545565\pi$
$359$	−102.430			−0.285319			−0.142660	+	0.989772i	$0.545565\pi$
$360$		0			0
$361$	75.4586	+	353.025i	0.209027	+	0.977910i
$362$		0			0
$363$		−	225.921i		−	0.622371i
$364$		0			0
$365$	914.382			2.50516
$366$		0			0
$367$	244.921			0.667360			0.333680	−	0.942686i	$-0.391710\pi$
$367$	244.921			0.667360			0.333680	+	0.942686i	$0.391710\pi$
$368$		0			0
$369$		−	128.977i		−	0.349530i
$370$		0			0
$371$		−	2.67180i		−	0.00720160i
$372$		0			0
$373$		−	205.089i		−	0.549837i	−0.961468	−	0.274918i	$-0.911349\pi$
$373$		−	205.089i		−	0.549837i	0.961468	−	0.274918i	$-0.0886509\pi$
$374$		0			0
$375$			149.564i			0.398838i
$376$		0			0
$377$	164.276			0.435746
$378$		0			0
$379$		−	488.205i		−	1.28814i	−0.764966	−	0.644070i	$-0.777244\pi$
$379$		−	488.205i		−	1.28814i	0.764966	−	0.644070i	$-0.222756\pi$
$380$		0			0
$381$	−247.448			−0.649469
$382$		0			0
$383$			539.079i			1.40752i	0.710439	+	0.703759i	$0.248496\pi$
$383$			539.079i			1.40752i	−0.710439	+	0.703759i	$0.751504\pi$
$384$		0			0
$385$	30.5054			0.0792349
$386$		0			0
$387$	−119.180			−0.307959
$388$		0			0
$389$	195.648			0.502951			0.251475	−	0.967864i	$-0.419084\pi$
$389$	195.648			0.502951			0.251475	+	0.967864i	$0.419084\pi$
$390$		0			0
$391$	148.224			0.379088
$392$		0			0
$393$			22.4448i			0.0571115i
$394$		0			0
$395$		−	632.810i		−	1.60205i
$396$		0			0
$397$	575.496			1.44961			0.724806	−	0.688953i	$-0.241929\pi$
$397$	575.496			1.44961			0.724806	+	0.688953i	$0.241929\pi$
$398$		0			0
$399$	−28.2456	+	34.9211i	−0.0707909	+	0.0875215i
$400$		0			0
$401$		−	548.087i		−	1.36680i	−0.730044	−	0.683400i	$-0.760501\pi$
$401$		−	548.087i		−	1.36680i	0.730044	−	0.683400i	$-0.239499\pi$
$402$		0			0
$403$	418.999			1.03970
$404$		0			0
$405$	202.755			0.500631
$406$		0			0
$407$		−	270.821i		−	0.665407i
$408$		0			0
$409$		−	480.592i		−	1.17504i	−0.809209	−	0.587520i	$-0.800104\pi$
$409$		−	480.592i		−	1.17504i	0.809209	−	0.587520i	$-0.199896\pi$
$410$		0			0
$411$		−	204.618i		−	0.497854i
$412$		0			0
$413$		−	45.0212i		−	0.109010i
$414$		0			0
$415$	−877.763			−2.11509
$416$		0			0
$417$		−	52.8338i		−	0.126700i
$418$		0			0
$419$	−636.344			−1.51872			−0.759360	−	0.650670i	$-0.774488\pi$
$419$	−636.344			−1.51872			−0.759360	+	0.650670i	$0.774488\pi$
$420$		0			0
$421$			39.9171i			0.0948150i	0.998876	+	0.0474075i	$0.0150959\pi$
$421$			39.9171i			0.0948150i	−0.998876	+	0.0474075i	$0.984904\pi$
$422$		0			0
$423$	−56.2204			−0.132909
$424$		0			0
$425$	132.509			0.311785
$426$		0			0
$427$	−52.4611			−0.122860
$428$		0			0
$429$	−140.916			−0.328475
$430$		0			0
$431$		−	763.895i		−	1.77238i	−0.463323	−	0.886189i	$-0.653343\pi$
$431$		−	763.895i		−	1.77238i	0.463323	−	0.886189i	$-0.346657\pi$
$432$		0			0
$433$			470.042i			1.08555i	0.839879	+	0.542774i	$0.182626\pi$
$433$			470.042i			1.08555i	−0.839879	+	0.542774i	$0.817374\pi$
$434$		0			0
$435$	177.999			0.409192
$436$		0			0
$437$	240.682	+	194.674i	0.550760	+	0.445477i
$438$		0			0
$439$		−	173.834i		−	0.395977i	−0.980204	−	0.197988i	$-0.936559\pi$
$439$		−	173.834i		−	0.395977i	0.980204	−	0.197988i	$-0.0634408\pi$
$440$		0			0
$441$	−182.474			−0.413772
$442$		0			0
$443$	494.990			1.11736			0.558680	−	0.829384i	$-0.311308\pi$
$443$	494.990			1.11736			0.558680	+	0.829384i	$0.311308\pi$
$444$		0			0
$445$		−	691.283i		−	1.55345i
$446$		0			0
$447$			251.398i			0.562413i
$448$		0			0
$449$		−	517.523i		−	1.15261i	−0.817234	−	0.576306i	$-0.804494\pi$
$449$		−	517.523i		−	1.15261i	0.817234	−	0.576306i	$-0.195506\pi$
$450$		0			0
$451$			158.357i			0.351124i
$452$		0			0
$453$	262.991			0.580554
$454$		0			0
$455$			86.3182i			0.189710i
$456$		0			0
$457$	354.686			0.776119			0.388059	−	0.921634i	$-0.373146\pi$
$457$	354.686			0.776119			0.388059	+	0.921634i	$0.373146\pi$
$458$		0			0
$459$			265.511i			0.578454i
$460$		0			0
$461$	176.578			0.383033			0.191516	−	0.981489i	$-0.438659\pi$
$461$	176.578			0.383033			0.191516	+	0.981489i	$0.438659\pi$
$462$		0			0
$463$	−36.7054			−0.0792773			−0.0396387	−	0.999214i	$-0.512621\pi$
$463$	−36.7054			−0.0792773			−0.0396387	+	0.999214i	$0.512621\pi$
$464$		0			0
$465$	453.998			0.976341
$466$		0			0
$467$	−781.984			−1.67448			−0.837242	−	0.546832i	$-0.815834\pi$
$467$	−781.984			−1.67448			−0.837242	+	0.546832i	$0.815834\pi$
$468$		0			0
$469$		−	86.2127i		−	0.183822i
$470$		0			0
$471$		−	389.812i		−	0.827627i
$472$		0			0
$473$	146.329			0.309364
$474$		0			0
$475$	215.165	+	174.034i	0.452978	+	0.366387i
$476$		0			0
$477$		−	9.80632i		−	0.0205583i
$478$		0			0
$479$	−357.010			−0.745324			−0.372662	−	0.927967i	$-0.621555\pi$
$479$	−357.010			−0.745324			−0.372662	+	0.927967i	$0.621555\pi$
$480$		0			0
$481$	766.315			1.59317
$482$		0			0
$483$			38.5140i			0.0797391i
$484$		0			0
$485$		−	510.322i		−	1.05221i
$486$		0			0
$487$		−	752.081i		−	1.54431i	−0.635432	−	0.772157i	$-0.719178\pi$
$487$		−	752.081i		−	1.54431i	0.635432	−	0.772157i	$-0.280822\pi$
$488$		0			0
$489$		−	446.650i		−	0.913394i
$490$		0			0
$491$	−481.529			−0.980711			−0.490356	−	0.871523i	$-0.663133\pi$
$491$	−481.529			−0.980711			−0.490356	+	0.871523i	$0.663133\pi$
$492$		0			0
$493$			112.981i			0.229170i
$494$		0			0
$495$	111.964			0.226191
$496$		0			0
$497$		−	8.86539i		−	0.0178378i
$498$		0			0
$499$	−322.939			−0.647172			−0.323586	−	0.946199i	$-0.604888\pi$
$499$	−322.939			−0.647172			−0.323586	+	0.946199i	$0.604888\pi$
$500$		0			0
$501$	−536.725			−1.07131
$502$		0			0
$503$	614.923			1.22251			0.611256	−	0.791433i	$-0.290665\pi$
$503$	614.923			1.22251			0.611256	+	0.791433i	$0.290665\pi$
$504$		0			0
$505$	405.660			0.803287
$506$		0			0
$507$		−	13.6371i		−	0.0268976i
$508$		0			0
$509$			408.948i			0.803434i	0.915764	+	0.401717i	$0.131586\pi$
$509$			408.948i			0.803434i	−0.915764	+	0.401717i	$0.868414\pi$
$510$		0			0
$511$	−150.806			−0.295118
$512$		0			0
$513$	−348.716	+	431.130i	−0.679758	+	0.840410i
$514$		0			0
$515$			491.561i			0.954487i
$516$		0			0
$517$	69.0271			0.133515
$518$		0			0
$519$	767.431			1.47867
$520$		0			0
$521$		−	17.1101i		−	0.0328408i	−0.999865	−	0.0164204i	$-0.994773\pi$
$521$		−	17.1101i		−	0.0328408i	0.999865	−	0.0164204i	$-0.00522701\pi$
$522$		0			0
$523$			306.964i			0.586929i	0.955970	+	0.293465i	$0.0948083\pi$
$523$			306.964i			0.586929i	−0.955970	+	0.293465i	$0.905192\pi$
$524$		0			0
$525$			34.4307i			0.0655822i
$526$		0			0
$527$			288.166i			0.546805i
$528$		0			0
$529$	−263.555			−0.498213
$530$		0			0
$531$		−	165.242i		−	0.311189i
$532$		0			0
$533$	−448.087			−0.840689
$534$		0			0
$535$		−	517.660i		−	0.967588i
$536$		0			0
$537$	89.5641			0.166786
$538$		0			0
$539$	224.040			0.415659
$540$		0			0
$541$	−727.145			−1.34408			−0.672038	−	0.740517i	$-0.734581\pi$
$541$	−727.145			−1.34408			−0.672038	+	0.740517i	$0.734581\pi$
$542$		0			0
$543$	−756.006			−1.39228
$544$		0			0
$545$		−	857.410i		−	1.57323i
$546$		0			0
$547$			153.277i			0.280213i	0.990136	+	0.140107i	$0.0447445\pi$
$547$			153.277i			0.280213i	−0.990136	+	0.140107i	$0.955255\pi$
$548$		0			0
$549$	−192.549			−0.350726
$550$		0			0
$551$	−148.387	+	183.456i	−0.269304	+	0.332951i
$552$		0			0
$553$			104.367i			0.188729i
$554$		0			0
$555$	830.326			1.49608
$556$		0			0
$557$	−294.840			−0.529335			−0.264668	−	0.964340i	$-0.585262\pi$
$557$	−294.840			−0.529335			−0.264668	+	0.964340i	$0.585262\pi$
$558$		0			0
$559$			414.053i			0.740702i
$560$		0			0
$561$		−	96.9147i		−	0.172753i
$562$		0			0
$563$		−	181.727i		−	0.322783i	−0.986890	−	0.161391i	$-0.948402\pi$
$563$		−	181.727i		−	0.322783i	0.986890	−	0.161391i	$-0.0515982\pi$
$564$		0			0
$565$		−	1101.65i		−	1.94983i
$566$		0			0
$567$	−33.4397			−0.0589765
$568$		0			0
$569$			940.429i			1.65278i	0.563102	+	0.826388i	$0.309608\pi$
$569$			940.429i			1.65278i	−0.563102	+	0.826388i	$0.690392\pi$
$570$		0			0
$571$	−546.519			−0.957125			−0.478563	−	0.878053i	$-0.658842\pi$
$571$	−546.519			−0.957125			−0.478563	+	0.878053i	$0.658842\pi$
$572$		0			0
$573$		−	50.3802i		−	0.0879235i
$574$		0			0
$575$	237.302			0.412700
$576$		0			0
$577$	13.6269			0.0236168			0.0118084	−	0.999930i	$-0.496241\pi$
$577$	13.6269			0.0236168			0.0118084	+	0.999930i	$0.496241\pi$
$578$		0			0
$579$	−114.237			−0.197300
$580$		0			0
$581$	144.766			0.249167
$582$		0			0
$583$			12.0402i			0.0206521i
$584$		0			0
$585$			316.815i			0.541563i
$586$		0			0
$587$	147.276			0.250896			0.125448	−	0.992100i	$-0.459963\pi$
$587$	147.276			0.250896			0.125448	+	0.992100i	$0.459963\pi$
$588$		0			0
$589$	−378.471	+	467.918i	−0.642565	+	0.794427i
$590$		0			0
$591$			99.8776i			0.168998i
$592$		0			0
$593$	−218.789			−0.368953			−0.184477	−	0.982837i	$-0.559059\pi$
$593$	−218.789			−0.368953			−0.184477	+	0.982837i	$0.559059\pi$
$594$		0			0
$595$	−59.3652			−0.0997735
$596$		0			0
$597$			858.363i			1.43779i
$598$		0			0
$599$		−	547.699i		−	0.914356i	−0.889375	−	0.457178i	$-0.848860\pi$
$599$		−	547.699i		−	0.914356i	0.889375	−	0.457178i	$-0.151140\pi$
$600$		0			0
$601$			245.418i			0.408349i	0.978934	+	0.204175i	$0.0654511\pi$
$601$			245.418i			0.408349i	−0.978934	+	0.204175i	$0.934549\pi$
$602$		0			0
$603$		−	316.427i		−	0.524755i
$604$		0			0
$605$	623.631			1.03079
$606$		0			0
$607$			333.015i			0.548625i	0.961641	+	0.274312i	$0.0884502\pi$
$607$			333.015i			0.548625i	−0.961641	+	0.274312i	$0.911550\pi$
$608$		0			0
$609$	−29.3566			−0.0482046
$610$		0			0
$611$			195.319i			0.319671i
$612$		0			0
$613$	−131.044			−0.213776			−0.106888	−	0.994271i	$-0.534089\pi$
$613$	−131.044			−0.213776			−0.106888	+	0.994271i	$0.534089\pi$
$614$		0			0
$615$	−485.516			−0.789457
$616$		0			0
$617$	1029.88			1.66917			0.834583	−	0.550882i	$-0.185709\pi$
$617$	1029.88			1.66917			0.834583	+	0.550882i	$0.185709\pi$
$618$		0			0
$619$	−841.181			−1.35893			−0.679467	−	0.733706i	$-0.737789\pi$
$619$	−841.181			−1.35893			−0.679467	+	0.733706i	$0.737789\pi$
$620$		0			0
$621$			475.488i			0.765681i
$622$		0			0
$623$			114.011i			0.183003i
$624$		0			0
$625$	−776.985			−1.24318
$626$		0			0
$627$	127.286	−	157.368i	0.203007	−	0.250986i
$628$		0			0
$629$			527.032i			0.837889i
$630$		0			0
$631$	999.068			1.58331			0.791655	−	0.610969i	$-0.209220\pi$
$631$	999.068			1.58331			0.791655	+	0.610969i	$0.209220\pi$
$632$		0			0
$633$	−681.581			−1.07675
$634$		0			0
$635$		−	683.054i		−	1.07567i
$636$		0			0
$637$			633.944i			0.995203i
$638$		0			0
$639$		−	32.5387i		−	0.0509213i
$640$		0			0
$641$			176.109i			0.274741i	0.990520	+	0.137371i	$0.0438651\pi$
$641$			176.109i			0.274741i	−0.990520	+	0.137371i	$0.956135\pi$
$642$		0			0
$643$	155.981			0.242583			0.121292	−	0.992617i	$-0.461296\pi$
$643$	155.981			0.242583			0.121292	+	0.992617i	$0.461296\pi$
$644$		0			0
$645$			448.639i			0.695564i
$646$		0			0
$647$	595.470			0.920356			0.460178	−	0.887827i	$-0.347786\pi$
$647$	595.470			0.920356			0.460178	+	0.887827i	$0.347786\pi$
$648$		0			0
$649$			202.883i			0.312608i
$650$		0			0
$651$	−74.8762			−0.115017
$652$		0			0
$653$	608.312			0.931565			0.465782	−	0.884899i	$-0.345773\pi$
$653$	608.312			0.931565			0.465782	+	0.884899i	$0.345773\pi$
$654$		0			0
$655$	−61.9566			−0.0945902
$656$		0			0
$657$	−553.503			−0.842471
$658$		0			0
$659$		−	210.781i		−	0.319849i	−0.987129	−	0.159925i	$-0.948875\pi$
$659$		−	210.781i		−	0.319849i	0.987129	−	0.159925i	$-0.0511251\pi$
$660$		0			0
$661$			475.867i			0.719920i	0.932968	+	0.359960i	$0.117210\pi$
$661$			475.867i			0.719920i	−0.932968	+	0.359960i	$0.882790\pi$
$662$		0			0
$663$	274.230			0.413620
$664$		0			0
$665$	−96.3960	−	77.9690i	−0.144956	−	0.117247i
$666$		0			0
$667$			202.331i			0.303345i
$668$		0			0
$669$	−219.456			−0.328036
$670$		0			0
$671$	236.410			0.352325
$672$		0			0
$673$		−	860.374i		−	1.27842i	−0.769034	−	0.639208i	$-0.779262\pi$
$673$		−	860.374i		−	1.27842i	0.769034	−	0.639208i	$-0.220738\pi$
$674$		0			0
$675$			425.076i			0.629742i
$676$		0			0
$677$		−	805.478i		−	1.18978i	−0.803809	−	0.594888i	$-0.797197\pi$
$677$		−	805.478i		−	1.18978i	0.803809	−	0.594888i	$-0.202803\pi$
$678$		0			0
$679$			84.1655i			0.123955i
$680$		0			0
$681$	−970.032			−1.42442
$682$		0			0
$683$			678.667i			0.993656i	0.867849	+	0.496828i	$0.165502\pi$
$683$			678.667i			0.993656i	−0.867849	+	0.496828i	$0.834498\pi$
$684$		0			0
$685$	564.827			0.824564
$686$		0			0
$687$		−	479.274i		−	0.697633i
$688$		0			0
$689$	−34.0688			−0.0494468
$690$		0			0
$691$	636.921			0.921738			0.460869	−	0.887468i	$-0.347538\pi$
$691$	636.921			0.921738			0.460869	+	0.887468i	$0.347538\pi$
$692$		0			0
$693$	−18.4659			−0.0266463
$694$		0			0
$695$	145.842			0.209845
$696$		0			0
$697$		−	308.171i		−	0.442140i
$698$		0			0
$699$		−	53.0524i		−	0.0758975i
$700$		0			0
$701$	−1039.31			−1.48261			−0.741306	−	0.671167i	$-0.765793\pi$
$701$	−1039.31			−1.48261			−0.741306	+	0.671167i	$0.765793\pi$
$702$		0			0
$703$	−692.192	+	855.783i	−0.984626	+	1.21733i
$704$		0			0
$705$			211.634i			0.300191i
$706$		0			0
$707$	−66.9039			−0.0946307
$708$		0			0
$709$	−1068.61			−1.50721			−0.753606	−	0.657326i	$-0.771687\pi$
$709$	−1068.61			−1.50721			−0.753606	+	0.657326i	$0.771687\pi$
$710$		0			0
$711$			383.059i			0.538761i
$712$		0			0
$713$			516.060i			0.723787i
$714$		0			0
$715$		−	388.983i		−	0.544033i
$716$		0			0
$717$		−	416.928i		−	0.581489i
$718$		0			0
$719$	−1046.98			−1.45616			−0.728078	−	0.685494i	$-0.759586\pi$
$719$	−1046.98			−1.45616			−0.728078	+	0.685494i	$0.759586\pi$
$720$		0			0
$721$		−	81.0713i		−	0.112443i
$722$		0			0
$723$	−146.693			−0.202895
$724$		0			0
$725$			180.880i			0.249489i
$726$		0			0
$727$	610.242			0.839398			0.419699	−	0.907663i	$-0.362136\pi$
$727$	610.242			0.839398			0.419699	+	0.907663i	$0.362136\pi$
$728$		0			0
$729$	−721.255			−0.989376
$730$		0			0
$731$	−284.764			−0.389554
$732$		0			0
$733$	111.114			0.151588			0.0757941	−	0.997123i	$-0.475851\pi$
$733$	111.114			0.151588			0.0757941	+	0.997123i	$0.475851\pi$
$734$		0			0
$735$			686.899i			0.934556i
$736$		0			0
$737$			388.508i			0.527148i
$738$		0			0
$739$	1056.20			1.42923			0.714613	−	0.699520i	$-0.246603\pi$
$739$	1056.20			1.42923			0.714613	+	0.699520i	$0.246603\pi$
$740$		0			0
$741$	445.288	+	360.167i	0.600929	+	0.486056i
$742$		0			0
$743$			760.688i			1.02381i	0.859043	+	0.511903i	$0.171059\pi$
$743$			760.688i			1.02381i	−0.859043	+	0.511903i	$0.828941\pi$
$744$		0			0
$745$	−693.959			−0.931489
$746$		0			0
$747$	531.337			0.711294
$748$		0			0
$749$			85.3757i			0.113986i
$750$		0			0
$751$			10.7326i			0.0142910i	0.999974	+	0.00714551i	$0.00227451\pi$
$751$			10.7326i			0.0142910i	−0.999974	+	0.00714551i	$0.997725\pi$
$752$		0			0
$753$			902.850i			1.19900i
$754$		0			0
$755$			725.959i			0.961535i
$756$		0			0
$757$	592.194			0.782291			0.391146	−	0.920329i	$-0.372079\pi$
$757$	592.194			0.782291			0.391146	+	0.920329i	$0.372079\pi$
$758$		0			0
$759$		−	173.559i		−	0.228668i
$760$		0			0
$761$	488.932			0.642486			0.321243	−	0.946997i	$-0.395899\pi$
$761$	488.932			0.642486			0.321243	+	0.946997i	$0.395899\pi$
$762$		0			0
$763$			141.409i			0.185333i
$764$		0			0
$765$	−217.889			−0.284822
$766$		0			0
$767$	−574.078			−0.748471
$768$		0			0
$769$	−785.922			−1.02201			−0.511003	−	0.859579i	$-0.670726\pi$
$769$	−785.922			−1.02201			−0.511003	+	0.859579i	$0.670726\pi$
$770$		0			0
$771$	−121.511			−0.157602
$772$		0			0
$773$			225.009i			0.291085i	0.989352	+	0.145543i	$0.0464928\pi$
$773$			225.009i			0.291085i	−0.989352	+	0.145543i	$0.953507\pi$
$774$		0			0
$775$			461.347i			0.595286i
$776$		0			0
$777$	−136.943			−0.176245
$778$		0			0
$779$	404.745	−	500.402i	0.519571	−	0.642365i
$780$		0			0
$781$			39.9509i			0.0511535i
$782$		0			0
$783$	−362.433			−0.462877
$784$		0			0
$785$	1076.04			1.37075
$786$		0			0
$787$		−	447.434i		−	0.568531i	−0.958746	−	0.284266i	$-0.908250\pi$
$787$		−	447.434i		−	0.568531i	0.958746	−	0.284266i	$-0.0917498\pi$
$788$		0			0
$789$		−	1042.59i		−	1.32141i
$790$		0			0
$791$			181.691i			0.229698i
$792$		0			0
$793$			668.947i			0.843564i
$794$		0			0
$795$	−36.9146			−0.0464335
$796$		0			0
$797$		−	81.7849i		−	0.102616i	−0.998683	−	0.0513080i	$-0.983661\pi$
$797$		−	81.7849i		−	0.102616i	0.998683	−	0.0513080i	$-0.0163390\pi$
$798$		0			0
$799$	−134.331			−0.168123
$800$		0			0
$801$			418.455i			0.522416i
$802$		0			0
$803$	679.588			0.846312
$804$		0			0
$805$	−106.314			−0.132067
$806$		0			0
$807$	−807.012			−1.00001
$808$		0			0
$809$	−277.637			−0.343185			−0.171593	−	0.985168i	$-0.554891\pi$
$809$	−277.637			−0.343185			−0.171593	+	0.985168i	$0.554891\pi$
$810$		0			0
$811$			21.8435i			0.0269340i	0.999909	+	0.0134670i	$0.00428681\pi$
$811$			21.8435i			0.0269340i	−0.999909	+	0.0134670i	$0.995713\pi$
$812$		0			0
$813$		−	407.518i		−	0.501253i
$814$		0			0
$815$	1232.93			1.51280
$816$		0			0
$817$	−462.394	−	374.003i	−0.565966	−	0.457776i
$818$		0			0
$819$		−	52.2510i		−	0.0637986i
$820$		0			0
$821$	−445.574			−0.542720			−0.271360	−	0.962478i	$-0.587473\pi$
$821$	−445.574			−0.542720			−0.271360	+	0.962478i	$0.587473\pi$
$822$		0			0
$823$	652.986			0.793422			0.396711	−	0.917944i	$-0.370152\pi$
$823$	652.986			0.793422			0.396711	+	0.917944i	$0.370152\pi$
$824$		0			0
$825$		−	155.158i		−	0.188070i
$826$		0			0
$827$			1085.77i			1.31290i	0.754370	+	0.656450i	$0.227943\pi$
$827$			1085.77i			1.31290i	−0.754370	+	0.656450i	$0.772057\pi$
$828$		0			0
$829$		−	578.238i		−	0.697512i	−0.937214	−	0.348756i	$-0.886604\pi$
$829$		−	578.238i		−	0.697512i	0.937214	−	0.348756i	$-0.113396\pi$
$830$		0			0
$831$		−	569.196i		−	0.684953i
$832$		0			0
$833$	−435.994			−0.523403
$834$		0			0
$835$		−	1481.57i		−	1.77434i
$836$		0			0
$837$	−924.411			−1.10443
$838$		0			0
$839$		−	1128.80i		−	1.34541i	−0.739911	−	0.672704i	$-0.765133\pi$
$839$		−	1128.80i		−	1.34541i	0.739911	−	0.672704i	$-0.234867\pi$
$840$		0			0
$841$	686.777			0.816619
$842$		0			0
$843$	142.763			0.169351
$844$		0			0
$845$	37.6437			0.0445488
$846$		0			0
$847$	−102.853			−0.121432
$848$		0			0
$849$			1179.71i			1.38953i
$850$		0			0
$851$			943.832i			1.10909i
$852$		0			0
$853$	838.329			0.982800			0.491400	−	0.870934i	$-0.336485\pi$
$853$	838.329			0.982800			0.491400	+	0.870934i	$0.336485\pi$
$854$		0			0
$855$	−353.803	−	286.170i	−0.413805	−	0.334702i
$856$		0			0
$857$		−	971.203i		−	1.13326i	−0.823973	−	0.566630i	$-0.808247\pi$
$857$		−	971.203i		−	1.13326i	0.823973	−	0.566630i	$-0.191753\pi$
$858$		0			0
$859$	154.488			0.179846			0.0899232	−	0.995949i	$-0.471338\pi$
$859$	154.488			0.179846			0.0899232	+	0.995949i	$0.471338\pi$
$860$		0			0
$861$	80.0744			0.0930016
$862$		0			0
$863$			198.110i			0.229559i	0.993391	+	0.114780i	$0.0366162\pi$
$863$			198.110i			0.229559i	−0.993391	+	0.114780i	$0.963384\pi$
$864$		0			0
$865$			2118.41i			2.44903i
$866$		0			0
$867$		−	469.940i		−	0.542030i
$868$		0			0
$869$		−	470.318i		−	0.541217i
$870$		0			0
$871$	−1099.32			−1.26214
$872$		0			0
$873$			308.914i			0.353853i
$874$		0			0
$875$	68.0909			0.0778182
$876$		0			0
$877$			173.558i			0.197900i	0.995092	+	0.0989498i	$0.0315483\pi$
$877$			173.558i			0.197900i	−0.995092	+	0.0989498i	$0.968452\pi$
$878$		0			0
$879$	949.451			1.08015
$880$		0			0
$881$	561.120			0.636913			0.318456	−	0.947938i	$-0.396836\pi$
$881$	561.120			0.636913			0.318456	+	0.947938i	$0.396836\pi$
$882$		0			0
$883$	−449.856			−0.509463			−0.254732	−	0.967012i	$-0.581987\pi$
$883$	−449.856			−0.509463			−0.254732	+	0.967012i	$0.581987\pi$
$884$		0			0
$885$	−622.031			−0.702860
$886$		0			0
$887$		−	532.156i		−	0.599950i	−0.953947	−	0.299975i	$-0.903022\pi$
$887$		−	532.156i		−	0.599950i	0.953947	−	0.299975i	$-0.0969784\pi$
$888$		0			0
$889$			112.653i			0.126719i
$890$		0			0
$891$	150.692			0.169127
$892$		0			0
$893$	−218.123	−	176.427i	−0.244259	−	0.197566i
$894$		0			0
$895$			247.232i			0.276237i
$896$		0			0
$897$	491.103			0.547495
$898$		0			0
$899$	−393.358			−0.437551
$900$		0			0
$901$		−	23.4308i		−	0.0260053i
$902$		0			0
$903$		−	73.9923i		−	0.0819406i
$904$		0			0
$905$		−	2086.88i		−	2.30594i
$906$		0			0
$907$			586.664i			0.646818i	0.946259	+	0.323409i	$0.104829\pi$
$907$			586.664i			0.646818i	−0.946259	+	0.323409i	$0.895171\pi$
$908$		0			0
$909$	−245.558			−0.270141
$910$		0			0
$911$			42.8590i			0.0470461i	0.999723	+	0.0235230i	$0.00748830\pi$
$911$			42.8590i			0.0470461i	−0.999723	+	0.0235230i	$0.992512\pi$
$912$		0			0
$913$	−652.373			−0.714538
$914$		0			0
$915$			724.824i			0.792158i
$916$		0			0
$917$	10.2183			0.0111431
$918$		0			0
$919$	−949.641			−1.03334			−0.516671	−	0.856184i	$-0.672829\pi$
$919$	−949.641			−1.03334			−0.516671	+	0.856184i	$0.672829\pi$
$920$		0			0
$921$	371.820			0.403714
$922$		0			0
$923$	−113.045			−0.122476
$924$		0			0
$925$			843.766i			0.912179i
$926$		0			0
$927$		−	297.557i		−	0.320989i
$928$		0			0
$929$	475.866			0.512235			0.256117	−	0.966646i	$-0.417557\pi$
$929$	475.866			0.512235			0.256117	+	0.966646i	$0.417557\pi$
$930$		0			0
$931$	−707.959	−	572.626i	−0.760428	−	0.615065i
$932$		0			0
$933$			1002.29i			1.07427i
$934$		0			0
$935$	267.523			0.286121
$936$		0			0
$937$	−398.277			−0.425056			−0.212528	−	0.977155i	$-0.568170\pi$
$937$	−398.277			−0.425056			−0.212528	+	0.977155i	$0.568170\pi$
$938$		0			0
$939$			224.287i			0.238857i
$940$		0			0
$941$			913.121i			0.970373i	0.874411	+	0.485187i	$0.161248\pi$
$941$			913.121i			0.970373i	−0.874411	+	0.485187i	$0.838752\pi$
$942$		0			0
$943$		−	551.887i		−	0.585246i
$944$		0			0
$945$		−	190.438i		−	0.201522i
$946$		0			0
$947$	−287.682			−0.303782			−0.151891	−	0.988397i	$-0.548536\pi$
$947$	−287.682			−0.303782			−0.151891	+	0.988397i	$0.548536\pi$
$948$		0			0
$949$			1922.96i			2.02631i
$950$		0			0
$951$	894.576			0.940669
$952$		0			0
$953$			821.664i			0.862187i	0.902307	+	0.431093i	$0.141872\pi$
$953$			821.664i			0.862187i	−0.902307	+	0.431093i	$0.858128\pi$
$954$		0			0
$955$	139.069			0.145622
$956$		0			0
$957$	132.292			0.138237
$958$		0			0
$959$	−93.1547			−0.0971374
$960$		0			0
$961$	−42.2888			−0.0440050
$962$		0			0
$963$			313.355i			0.325395i
$964$		0			0
$965$		−	315.338i		−	0.326775i
$966$		0			0
$967$	239.162			0.247324			0.123662	−	0.992324i	$-0.460536\pi$
$967$	239.162			0.247324			0.123662	+	0.992324i	$0.460536\pi$
$968$		0			0
$969$	−247.705	+	306.247i	−0.255629	+	0.316044i
$970$		0			0
$971$		−	1935.48i		−	1.99328i	−0.0819035	−	0.996640i	$-0.526100\pi$
$971$		−	1935.48i		−	1.99328i	0.0819035	−	0.996640i	$-0.473900\pi$
$972$		0			0
$973$	−24.0532			−0.0247206
$974$		0			0
$975$	439.035			0.450293
$976$		0			0
$977$		−	1095.11i		−	1.12089i	−0.828193	−	0.560444i	$-0.810631\pi$
$977$		−	1095.11i		−	1.12089i	0.828193	−	0.560444i	$-0.189369\pi$
$978$		0			0
$979$		−	513.777i		−	0.524798i
$980$		0			0
$981$			519.017i			0.529069i
$982$		0			0
$983$		−	219.814i		−	0.223615i	−0.993730	−	0.111807i	$-0.964336\pi$
$983$		−	219.814i		−	0.223615i	0.993730	−	0.111807i	$-0.0356640\pi$
$984$		0			0
$985$	−275.702			−0.279900
$986$		0			0
$987$		−	34.9041i		−	0.0353638i
$988$		0			0
$989$	−509.968			−0.515640
$990$		0			0
$991$			887.026i			0.895082i	0.894263	+	0.447541i	$0.147700\pi$
$991$			887.026i			0.895082i	−0.894263	+	0.447541i	$0.852300\pi$
$992$		0			0
$993$	−1099.46			−1.10721
$994$		0			0
$995$	−2369.42			−2.38133
$996$		0			0
$997$	227.450			0.228135			0.114067	−	0.993473i	$-0.463612\pi$
$997$	227.450			0.228135			0.114067	+	0.993473i	$0.463612\pi$
$998$		0			0
$999$	−1690.67			−1.69236

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.e.n.1025.6		8
4.3	odd	2		1216.3.e.m.1025.3		8
8.3	odd	2		152.3.e.b.113.6	yes	8
8.5	even	2		304.3.e.g.113.3		8
19.18	odd	2	inner	1216.3.e.n.1025.3		8
24.5	odd	2		2736.3.o.p.721.2		8
24.11	even	2		1368.3.o.b.721.2		8
76.75	even	2		1216.3.e.m.1025.6		8
152.37	odd	2		304.3.e.g.113.6		8
152.75	even	2		152.3.e.b.113.3	✓	8
456.227	odd	2		1368.3.o.b.721.1		8
456.341	even	2		2736.3.o.p.721.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.3.e.b.113.3	✓	8	152.75	even	2
152.3.e.b.113.6	yes	8	8.3	odd	2
304.3.e.g.113.3		8	8.5	even	2
304.3.e.g.113.6		8	152.37	odd	2
1216.3.e.m.1025.3		8	4.3	odd	2
1216.3.e.m.1025.6		8	76.75	even	2
1216.3.e.n.1025.3		8	19.18	odd	2	inner
1216.3.e.n.1025.6		8	1.1	even	1	trivial
1368.3.o.b.721.1		8	456.227	odd	2
1368.3.o.b.721.2		8	24.11	even	2
2736.3.o.p.721.1		8	456.341	even	2
2736.3.o.p.721.2		8	24.5	odd	2

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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.27869i q^{3} -6.29008 q^{5} +1.03740 q^{7} +3.80758 q^{9} +O(q^{10})\)	\(q+2.27869i q^{3} -6.29008 q^{5} +1.03740 q^{7} +3.80758 q^{9} -4.67493 q^{11} -13.2282i q^{13} -14.3331i q^{15} +9.09766 q^{17} +(14.7726 + 11.9487i) q^{19} +2.36391i q^{21} +16.2925 q^{23} +14.5651 q^{25} +29.1845i q^{27} +12.4187i q^{29} +31.6747i q^{31} -10.6527i q^{33} -6.52533 q^{35} +57.9305i q^{37} +30.1429 q^{39} -33.8737i q^{41} -31.3008 q^{43} -23.9500 q^{45} -14.7654 q^{47} -47.9238 q^{49} +20.7307i q^{51} -2.57547i q^{53} +29.4057 q^{55} +(-27.2273 + 33.6621i) q^{57} -43.3981i q^{59} -50.5698 q^{61} +3.94998 q^{63} +83.2063i q^{65} -83.1046i q^{67} +37.1255i q^{69} -8.54578i q^{71} -145.369 q^{73} +33.1894i q^{75} -4.84977 q^{77} +100.604i q^{79} -32.2341 q^{81} +139.547 q^{83} -57.2250 q^{85} -28.2983 q^{87} +109.901i q^{89} -13.7229i q^{91} -72.1769 q^{93} +(-92.9208 - 75.1581i) q^{95} +81.1313i q^{97} -17.8001 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 14 q^{5} + 6 q^{7} + 4 q^{9}+O(q^{10}) \) 8 * q + 14 * q^5 + 6 * q^7 + 4 * q^9	\( 8 q + 14 q^{5} + 6 q^{7} + 4 q^{9} - 26 q^{11} - 18 q^{17} + 16 q^{19} - 12 q^{23} + 34 q^{25} - 50 q^{35} + 108 q^{39} + 62 q^{43} - 162 q^{45} - 22 q^{47} + 22 q^{49} - 174 q^{55} + 4 q^{57} + 158 q^{61} + 2 q^{63} - 170 q^{73} - 82 q^{77} - 256 q^{81} - 64 q^{83} - 410 q^{85} + 332 q^{87} + 344 q^{93} - 222 q^{95} + 526 q^{99}+O(q^{100}) \) 8 * q + 14 * q^5 + 6 * q^7 + 4 * q^9 - 26 * q^11 - 18 * q^17 + 16 * q^19 - 12 * q^23 + 34 * q^25 - 50 * q^35 + 108 * q^39 + 62 * q^43 - 162 * q^45 - 22 * q^47 + 22 * q^49 - 174 * q^55 + 4 * q^57 + 158 * q^61 + 2 * q^63 - 170 * q^73 - 82 * q^77 - 256 * q^81 - 64 * q^83 - 410 * q^85 + 332 * q^87 + 344 * q^93 - 222 * q^95 + 526 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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