LMFDB - Embedded newform 2240.2.h.f.671.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(671,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2240.671");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.h (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:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			671.8
Character	$\chi$	$=$	2240.671
Dual form			2240.2.h.f.671.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.91632i q^{3} +1.00000 q^{5} +(-2.36081 - 1.19440i) q^{7} -5.50491 q^{9} +O(q^{10})$	$q+2.91632i q^{3} +1.00000 q^{5} +(-2.36081 - 1.19440i) q^{7} -5.50491 q^{9} -0.723256 q^{11} +4.05292 q^{13} +2.91632i q^{15} -6.74988i q^{17} -8.55782i q^{19} +(3.48324 - 6.88487i) q^{21} +2.79744i q^{23} +1.00000 q^{25} -7.30511i q^{27} +6.62619i q^{29} -4.51695 q^{31} -2.10924i q^{33} +(-2.36081 - 1.19440i) q^{35} -10.4961i q^{37} +11.8196i q^{39} -9.91197i q^{41} -5.50449 q^{43} -5.50491 q^{45} +7.93120 q^{47} +(4.14684 + 5.63948i) q^{49} +19.6848 q^{51} -5.97913i q^{53} -0.723256 q^{55} +24.9573 q^{57} -6.56687i q^{59} +13.0189 q^{61} +(12.9960 + 6.57504i) q^{63} +4.05292 q^{65} -5.58402 q^{67} -8.15822 q^{69} +3.78929i q^{71} -7.79060i q^{73} +2.91632i q^{75} +(1.70747 + 0.863853i) q^{77} +3.02048i q^{79} +4.78929 q^{81} +4.62103i q^{83} -6.74988i q^{85} -19.3241 q^{87} +15.4067i q^{89} +(-9.56816 - 4.84078i) q^{91} -13.1729i q^{93} -8.55782i q^{95} -1.26819i q^{97} +3.98146 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{5} - 36 q^{9}+O(q^{10}) $ 24 * q + 24 * q^5 - 36 * q^9	$ 24 q + 24 q^{5} - 36 q^{9} + 4 q^{13} + 8 q^{21} + 24 q^{25} - 36 q^{45} + 24 q^{57} + 56 q^{61} + 4 q^{65} + 96 q^{69} - 12 q^{77} + 24 q^{81}+O(q^{100}) $ 24 * q + 24 * q^5 - 36 * q^9 + 4 * q^13 + 8 * q^21 + 24 * q^25 - 36 * q^45 + 24 * q^57 + 56 * q^61 + 4 * q^65 + 96 * q^69 - 12 * q^77 + 24 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$3$			2.91632i			1.68374i	0.539683	+	0.841868i	$0.318544\pi$
$3$			2.91632i			1.68374i	−0.539683	+	0.841868i	$0.681456\pi$
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	−2.36081	−	1.19440i	−0.892302	−	0.451439i
$7$	−2.36081	−	1.19440i	−0.892302	−	0.451439i
$8$		0			0
$9$	−5.50491			−1.83497
$10$		0			0
$11$	−0.723256			−0.218070			−0.109035	−	0.994038i	$-0.534776\pi$
$11$	−0.723256			−0.218070			−0.109035	+	0.994038i	$0.534776\pi$
$12$		0			0
$13$	4.05292			1.12408			0.562038	−	0.827111i	$-0.310017\pi$
$13$	4.05292			1.12408			0.562038	+	0.827111i	$0.310017\pi$
$14$		0			0
$15$			2.91632i			0.752990i
$16$		0			0
$17$		−	6.74988i		−	1.63709i	−0.574445	−	0.818543i	$-0.694782\pi$
$17$		−	6.74988i		−	1.63709i	0.574445	−	0.818543i	$-0.305218\pi$
$18$		0			0
$19$		−	8.55782i		−	1.96330i	−0.190693	−	0.981650i	$-0.561073\pi$
$19$		−	8.55782i		−	1.96330i	0.190693	−	0.981650i	$-0.438927\pi$
$20$		0			0
$21$	3.48324	−	6.88487i	0.760105	−	1.50240i
$22$		0			0
$23$			2.79744i			0.583307i	0.956524	+	0.291653i	$0.0942053\pi$
$23$			2.79744i			0.583307i	−0.956524	+	0.291653i	$0.905795\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		−	7.30511i		−	1.40587i
$28$		0			0
$29$			6.62619i			1.23045i	0.788350	+	0.615226i	$0.210935\pi$
$29$			6.62619i			1.23045i	−0.788350	+	0.615226i	$0.789065\pi$
$30$		0			0
$31$	−4.51695			−0.811268			−0.405634	−	0.914036i	$-0.632949\pi$
$31$	−4.51695			−0.811268			−0.405634	+	0.914036i	$0.632949\pi$
$32$		0			0
$33$		−	2.10924i		−	0.367172i
$34$		0			0
$35$	−2.36081	−	1.19440i	−0.399050	−	0.201890i
$36$		0			0
$37$		−	10.4961i		−	1.72555i	−0.505592	−	0.862773i	$-0.668726\pi$
$37$		−	10.4961i		−	1.72555i	0.505592	−	0.862773i	$-0.331274\pi$
$38$		0			0
$39$			11.8196i			1.89265i
$40$		0			0
$41$		−	9.91197i		−	1.54799i	−0.633192	−	0.773995i	$-0.718255\pi$
$41$		−	9.91197i		−	1.54799i	0.633192	−	0.773995i	$-0.281745\pi$
$42$		0			0
$43$	−5.50449			−0.839427			−0.419714	−	0.907657i	$-0.637870\pi$
$43$	−5.50449			−0.839427			−0.419714	+	0.907657i	$0.637870\pi$
$44$		0			0
$45$	−5.50491			−0.820623
$46$		0			0
$47$	7.93120			1.15688			0.578442	−	0.815723i	$-0.303661\pi$
$47$	7.93120			1.15688			0.578442	+	0.815723i	$0.303661\pi$
$48$		0			0
$49$	4.14684	+	5.63948i	0.592406	+	0.805640i
$50$		0			0
$51$	19.6848			2.75642
$52$		0			0
$53$		−	5.97913i		−	0.821297i	−0.911794	−	0.410649i	$-0.865302\pi$
$53$		−	5.97913i		−	0.821297i	0.911794	−	0.410649i	$-0.134698\pi$
$54$		0			0
$55$	−0.723256			−0.0975238
$56$		0			0
$57$	24.9573			3.30568
$58$		0			0
$59$		−	6.56687i		−	0.854934i	−0.904031	−	0.427467i	$-0.859406\pi$
$59$		−	6.56687i		−	0.854934i	0.904031	−	0.427467i	$-0.140594\pi$
$60$		0			0
$61$	13.0189			1.66690			0.833448	−	0.552599i	$-0.186364\pi$
$61$	13.0189			1.66690			0.833448	+	0.552599i	$0.186364\pi$
$62$		0			0
$63$	12.9960	+	6.57504i	1.63735	+	0.828377i
$64$		0			0
$65$	4.05292			0.502702
$66$		0			0
$67$	−5.58402			−0.682196			−0.341098	−	0.940028i	$-0.610799\pi$
$67$	−5.58402			−0.682196			−0.341098	+	0.940028i	$0.610799\pi$
$68$		0			0
$69$	−8.15822			−0.982135
$70$		0			0
$71$			3.78929i			0.449706i	0.974393	+	0.224853i	$0.0721902\pi$
$71$			3.78929i			0.449706i	−0.974393	+	0.224853i	$0.927810\pi$
$72$		0			0
$73$		−	7.79060i		−	0.911821i	−0.890026	−	0.455911i	$-0.849314\pi$
$73$		−	7.79060i		−	0.911821i	0.890026	−	0.455911i	$-0.150686\pi$
$74$		0			0
$75$			2.91632i			0.336747i
$76$		0			0
$77$	1.70747	+	0.863853i	0.194584	+	0.0984452i
$78$		0			0
$79$			3.02048i			0.339831i	0.985459	+	0.169915i	$0.0543494\pi$
$79$			3.02048i			0.339831i	−0.985459	+	0.169915i	$0.945651\pi$
$80$		0			0
$81$	4.78929			0.532144
$82$		0			0
$83$			4.62103i			0.507223i	0.967306	+	0.253612i	$0.0816186\pi$
$83$			4.62103i			0.507223i	−0.967306	+	0.253612i	$0.918381\pi$
$84$		0			0
$85$		−	6.74988i		−	0.732128i
$86$		0			0
$87$	−19.3241			−2.07176
$88$		0			0
$89$			15.4067i			1.63311i	0.577270	+	0.816553i	$0.304118\pi$
$89$			15.4067i			1.63311i	−0.577270	+	0.816553i	$0.695882\pi$
$90$		0			0
$91$	−9.56816	−	4.84078i	−1.00302	−	0.507452i
$92$		0			0
$93$		−	13.1729i		−	1.36596i
$94$		0			0
$95$		−	8.55782i		−	0.878014i
$96$		0			0
$97$		−	1.26819i		−	0.128765i	−0.997925	−	0.0643826i	$-0.979492\pi$
$97$		−	1.26819i		−	0.128765i	0.997925	−	0.0643826i	$-0.0205078\pi$
$98$		0			0
$99$	3.98146			0.400151
$100$		0			0
$101$	−3.39694			−0.338008			−0.169004	−	0.985615i	$-0.554055\pi$
$101$	−3.39694			−0.338008			−0.169004	+	0.985615i	$0.554055\pi$
$102$		0			0
$103$	4.78124			0.471109			0.235555	−	0.971861i	$-0.424309\pi$
$103$	4.78124			0.471109			0.235555	+	0.971861i	$0.424309\pi$
$104$		0			0
$105$	3.48324	−	6.88487i	0.339929	−	0.671894i
$106$		0			0
$107$	5.96491			0.576650			0.288325	−	0.957533i	$-0.406902\pi$
$107$	5.96491			0.576650			0.288325	+	0.957533i	$0.406902\pi$
$108$		0			0
$109$			5.69703i			0.545677i	0.962060	+	0.272838i	$0.0879625\pi$
$109$			5.69703i			0.545677i	−0.962060	+	0.272838i	$0.912038\pi$
$110$		0			0
$111$	30.6099			2.90536
$112$		0			0
$113$	−10.6012			−0.997274			−0.498637	−	0.866811i	$-0.666166\pi$
$113$	−10.6012			−0.997274			−0.498637	+	0.866811i	$0.666166\pi$
$114$		0			0
$115$			2.79744i			0.260863i
$116$		0			0
$117$	−22.3109			−2.06265
$118$		0			0
$119$	−8.06203	+	15.9352i	−0.739045	+	1.46078i
$120$		0			0
$121$	−10.4769			−0.952446
$122$		0			0
$123$	28.9065			2.60641
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$		−	8.89422i		−	0.789235i	−0.918846	−	0.394617i	$-0.870877\pi$
$127$		−	8.89422i		−	0.789235i	0.918846	−	0.394617i	$-0.129123\pi$
$128$		0			0
$129$		−	16.0528i		−	1.41337i
$130$		0			0
$131$		−	3.62917i		−	0.317082i	−0.987352	−	0.158541i	$-0.949321\pi$
$131$		−	3.62917i		−	0.317082i	0.987352	−	0.158541i	$-0.0506791\pi$
$132$		0			0
$133$	−10.2214	+	20.2034i	−0.886310	+	1.75186i
$134$		0			0
$135$		−	7.30511i		−	0.628724i
$136$		0			0
$137$	6.95400			0.594120			0.297060	−	0.954859i	$-0.403994\pi$
$137$	6.95400			0.594120			0.297060	+	0.954859i	$0.403994\pi$
$138$		0			0
$139$		−	4.86064i		−	0.412274i	−0.978523	−	0.206137i	$-0.933911\pi$
$139$		−	4.86064i		−	0.412274i	0.978523	−	0.206137i	$-0.0660893\pi$
$140$		0			0
$141$			23.1299i			1.94789i
$142$		0			0
$143$	−2.93129			−0.245127
$144$		0			0
$145$			6.62619i			0.550275i
$146$		0			0
$147$	−16.4465	+	12.0935i	−1.35649	+	0.997455i
$148$		0			0
$149$		−	12.7041i		−	1.04076i	−0.853934	−	0.520381i	$-0.825790\pi$
$149$		−	12.7041i		−	1.04076i	0.853934	−	0.520381i	$-0.174210\pi$
$150$		0			0
$151$			3.25272i			0.264702i	0.991203	+	0.132351i	$0.0422526\pi$
$151$			3.25272i			0.264702i	−0.991203	+	0.132351i	$0.957747\pi$
$152$		0			0
$153$			37.1575i			3.00400i
$154$		0			0
$155$	−4.51695			−0.362810
$156$		0			0
$157$	−11.1609			−0.890736			−0.445368	−	0.895348i	$-0.646927\pi$
$157$	−11.1609			−0.890736			−0.445368	+	0.895348i	$0.646927\pi$
$158$		0			0
$159$	17.4371			1.38285
$160$		0			0
$161$	3.34125	−	6.60422i	0.263327	−	0.520486i
$162$		0			0
$163$	−2.40980			−0.188750			−0.0943751	−	0.995537i	$-0.530085\pi$
$163$	−2.40980			−0.188750			−0.0943751	+	0.995537i	$0.530085\pi$
$164$		0			0
$165$		−	2.10924i		−	0.164204i
$166$		0			0
$167$	−8.74278			−0.676537			−0.338268	−	0.941050i	$-0.609841\pi$
$167$	−8.74278			−0.676537			−0.338268	+	0.941050i	$0.609841\pi$
$168$		0			0
$169$	3.42612			0.263548
$170$		0			0
$171$			47.1100i			3.60259i
$172$		0			0
$173$	−9.88555			−0.751585			−0.375792	−	0.926704i	$-0.622629\pi$
$173$	−9.88555			−0.751585			−0.375792	+	0.926704i	$0.622629\pi$
$174$		0			0
$175$	−2.36081	−	1.19440i	−0.178460	−	0.0902878i
$176$		0			0
$177$	19.1511			1.43948
$178$		0			0
$179$	15.2830			1.14231			0.571153	−	0.820844i	$-0.306496\pi$
$179$	15.2830			1.14231			0.571153	+	0.820844i	$0.306496\pi$
$180$		0			0
$181$	6.61211			0.491474			0.245737	−	0.969337i	$-0.420970\pi$
$181$	6.61211			0.491474			0.245737	+	0.969337i	$0.420970\pi$
$182$		0			0
$183$			37.9671i			2.80661i
$184$		0			0
$185$		−	10.4961i		−	0.771687i
$186$		0			0
$187$			4.88189i			0.356999i
$188$		0			0
$189$	−8.72519	+	17.2460i	−0.634664	+	1.25446i
$190$		0			0
$191$		−	1.75710i		−	0.127139i	−0.997977	−	0.0635697i	$-0.979751\pi$
$191$		−	1.75710i		−	0.127139i	0.997977	−	0.0635697i	$-0.0202485\pi$
$192$		0			0
$193$	5.82359			0.419191			0.209595	−	0.977788i	$-0.432785\pi$
$193$	5.82359			0.419191			0.209595	+	0.977788i	$0.432785\pi$
$194$		0			0
$195$			11.8196i			0.846418i
$196$		0			0
$197$			6.83708i			0.487122i	0.969886	+	0.243561i	$0.0783156\pi$
$197$			6.83708i			0.487122i	−0.969886	+	0.243561i	$0.921684\pi$
$198$		0			0
$199$	14.7231			1.04370			0.521848	−	0.853038i	$-0.325243\pi$
$199$	14.7231			1.04370			0.521848	+	0.853038i	$0.325243\pi$
$200$		0			0
$201$		−	16.2848i		−	1.14864i
$202$		0			0
$203$	7.91429	−	15.6432i	0.555475	−	1.09794i
$204$		0			0
$205$		−	9.91197i		−	0.692282i
$206$		0			0
$207$		−	15.3997i		−	1.07035i
$208$		0			0
$209$			6.18949i			0.428136i
$210$		0			0
$211$	21.4295			1.47527			0.737634	−	0.675200i	$-0.235943\pi$
$211$	21.4295			1.47527			0.737634	+	0.675200i	$0.235943\pi$
$212$		0			0
$213$	−11.0508			−0.757187
$214$		0			0
$215$	−5.50449			−0.375403
$216$		0			0
$217$	10.6637	+	5.39502i	0.723896	+	0.366238i
$218$		0			0
$219$	22.7199			1.53527
$220$		0			0
$221$		−	27.3567i		−	1.84021i
$222$		0			0
$223$	−5.62373			−0.376593			−0.188296	−	0.982112i	$-0.560297\pi$
$223$	−5.62373			−0.376593			−0.188296	+	0.982112i	$0.560297\pi$
$224$		0			0
$225$	−5.50491			−0.366994
$226$		0			0
$227$		−	23.1214i		−	1.53462i	−0.641275	−	0.767311i	$-0.721594\pi$
$227$		−	23.1214i		−	1.53462i	0.641275	−	0.767311i	$-0.278406\pi$
$228$		0			0
$229$	13.7187			0.906558			0.453279	−	0.891369i	$-0.350254\pi$
$229$	13.7187			0.906558			0.453279	+	0.891369i	$0.350254\pi$
$230$		0			0
$231$	−2.51927	+	4.97952i	−0.165756	+	0.327628i
$232$		0			0
$233$	−6.23223			−0.408287			−0.204144	−	0.978941i	$-0.565441\pi$
$233$	−6.23223			−0.408287			−0.204144	+	0.978941i	$0.565441\pi$
$234$		0			0
$235$	7.93120			0.517375
$236$		0			0
$237$	−8.80868			−0.572185
$238$		0			0
$239$			23.8014i			1.53958i	0.638295	+	0.769792i	$0.279640\pi$
$239$			23.8014i			1.53958i	−0.638295	+	0.769792i	$0.720360\pi$
$240$		0			0
$241$			18.5892i			1.19744i	0.800959	+	0.598719i	$0.204323\pi$
$241$			18.5892i			1.19744i	−0.800959	+	0.598719i	$0.795677\pi$
$242$		0			0
$243$		−	7.94823i		−	0.509879i
$244$		0			0
$245$	4.14684	+	5.63948i	0.264932	+	0.360293i
$246$		0			0
$247$		−	34.6841i		−	2.20690i
$248$		0			0
$249$	−13.4764			−0.854031
$250$		0			0
$251$		−	7.89884i		−	0.498570i	−0.968430	−	0.249285i	$-0.919804\pi$
$251$		−	7.89884i		−	0.498570i	0.968430	−	0.249285i	$-0.0801957\pi$
$252$		0			0
$253$		−	2.02326i		−	0.127202i
$254$		0			0
$255$	19.6848			1.23271
$256$		0			0
$257$		−	12.4113i		−	0.774198i	−0.922038	−	0.387099i	$-0.873477\pi$
$257$		−	12.4113i		−	0.774198i	0.922038	−	0.387099i	$-0.126523\pi$
$258$		0			0
$259$	−12.5365	+	24.7793i	−0.778979	+	1.53971i
$260$		0			0
$261$		−	36.4766i		−	2.25784i
$262$		0			0
$263$		−	17.1254i		−	1.05600i	−0.849245	−	0.527999i	$-0.822942\pi$
$263$		−	17.1254i		−	1.05600i	0.849245	−	0.527999i	$-0.177058\pi$
$264$		0			0
$265$		−	5.97913i		−	0.367295i
$266$		0			0
$267$	−44.9308			−2.74972
$268$		0			0
$269$	−3.20508			−0.195417			−0.0977085	−	0.995215i	$-0.531151\pi$
$269$	−3.20508			−0.195417			−0.0977085	+	0.995215i	$0.531151\pi$
$270$		0			0
$271$	−2.68954			−0.163378			−0.0816891	−	0.996658i	$-0.526031\pi$
$271$	−2.68954			−0.163378			−0.0816891	+	0.996658i	$0.526031\pi$
$272$		0			0
$273$	14.1173	−	27.9038i	0.854416	−	1.68881i
$274$		0			0
$275$	−0.723256			−0.0436139
$276$		0			0
$277$		−	10.4166i		−	0.625870i	−0.949774	−	0.312935i	$-0.898688\pi$
$277$		−	10.4166i		−	0.625870i	0.949774	−	0.312935i	$-0.101312\pi$
$278$		0			0
$279$	24.8654			1.48865
$280$		0			0
$281$	28.2350			1.68436			0.842181	−	0.539195i	$-0.181271\pi$
$281$	28.2350			1.68436			0.842181	+	0.539195i	$0.181271\pi$
$282$		0			0
$283$		−	7.79421i		−	0.463318i	−0.972797	−	0.231659i	$-0.925585\pi$
$283$		−	7.79421i		−	0.463318i	0.972797	−	0.231659i	$-0.0744153\pi$
$284$		0			0
$285$	24.9573			1.47834
$286$		0			0
$287$	−11.8388	+	23.4003i	−0.698823	+	1.38127i
$288$		0			0
$289$	−28.5609			−1.68005
$290$		0			0
$291$	3.69845			0.216807
$292$		0			0
$293$	30.8954			1.80493			0.902463	−	0.430767i	$-0.141757\pi$
$293$	30.8954			1.80493			0.902463	+	0.430767i	$0.141757\pi$
$294$		0			0
$295$		−	6.56687i		−	0.382338i
$296$		0			0
$297$			5.28346i			0.306577i
$298$		0			0
$299$			11.3378i			0.655681i
$300$		0			0
$301$	12.9951	+	6.57454i	0.749023	+	0.378950i
$302$		0			0
$303$		−	9.90655i		−	0.569117i
$304$		0			0
$305$	13.0189			0.745458
$306$		0			0
$307$		−	18.9355i		−	1.08070i	−0.841439	−	0.540352i	$-0.818291\pi$
$307$		−	18.9355i		−	1.08070i	0.841439	−	0.540352i	$-0.181709\pi$
$308$		0			0
$309$			13.9436i			0.793224i
$310$		0			0
$311$	−2.16387			−0.122702			−0.0613510	−	0.998116i	$-0.519541\pi$
$311$	−2.16387			−0.122702			−0.0613510	+	0.998116i	$0.519541\pi$
$312$		0			0
$313$		−	13.8329i		−	0.781884i	−0.920415	−	0.390942i	$-0.872149\pi$
$313$		−	13.8329i		−	0.781884i	0.920415	−	0.390942i	$-0.127851\pi$
$314$		0			0
$315$	12.9960	+	6.57504i	0.732244	+	0.370461i
$316$		0			0
$317$			16.5616i			0.930193i	0.885260	+	0.465097i	$0.153980\pi$
$317$			16.5616i			0.930193i	−0.885260	+	0.465097i	$0.846020\pi$
$318$		0			0
$319$		−	4.79243i		−	0.268325i
$320$		0			0
$321$			17.3956i			0.970927i
$322$		0			0
$323$	−57.7643			−3.21409
$324$		0			0
$325$	4.05292			0.224815
$326$		0			0
$327$	−16.6144			−0.918776
$328$		0			0
$329$	−18.7241	−	9.47299i	−1.03229	−	0.522263i
$330$		0			0
$331$	−10.8147			−0.594428			−0.297214	−	0.954811i	$-0.596058\pi$
$331$	−10.8147			−0.594428			−0.297214	+	0.954811i	$0.596058\pi$
$332$		0			0
$333$			57.7800i			3.16632i
$334$		0			0
$335$	−5.58402			−0.305088
$336$		0			0
$337$	−29.3497			−1.59878			−0.799390	−	0.600813i	$-0.794844\pi$
$337$	−29.3497			−1.59878			−0.799390	+	0.600813i	$0.794844\pi$
$338$		0			0
$339$		−	30.9164i		−	1.67915i
$340$		0			0
$341$	3.26691			0.176913
$342$		0			0
$343$	−3.05413	−	18.2667i	−0.164907	−	0.986309i
$344$		0			0
$345$	−8.15822			−0.439224
$346$		0			0
$347$	−6.57011			−0.352702			−0.176351	−	0.984327i	$-0.556429\pi$
$347$	−6.57011			−0.352702			−0.176351	+	0.984327i	$0.556429\pi$
$348$		0			0
$349$	32.6174			1.74597			0.872984	−	0.487749i	$-0.162182\pi$
$349$	32.6174			1.74597			0.872984	+	0.487749i	$0.162182\pi$
$350$		0			0
$351$		−	29.6070i		−	1.58030i
$352$		0			0
$353$			11.2425i			0.598376i	0.954194	+	0.299188i	$0.0967158\pi$
$353$			11.2425i			0.598376i	−0.954194	+	0.299188i	$0.903284\pi$
$354$		0			0
$355$			3.78929i			0.201115i
$356$		0			0
$357$	−46.4721	−	23.5114i	−2.45956	−	1.24436i
$358$		0			0
$359$			8.30998i			0.438584i	0.975659	+	0.219292i	$0.0703747\pi$
$359$			8.30998i			0.438584i	−0.975659	+	0.219292i	$0.929625\pi$
$360$		0			0
$361$	−54.2363			−2.85454
$362$		0			0
$363$		−	30.5540i		−	1.60367i
$364$		0			0
$365$		−	7.79060i		−	0.407779i
$366$		0			0
$367$	−27.8619			−1.45438			−0.727191	−	0.686435i	$-0.759174\pi$
$367$	−27.8619			−1.45438			−0.727191	+	0.686435i	$0.759174\pi$
$368$		0			0
$369$			54.5645i			2.84051i
$370$		0			0
$371$	−7.14145	+	14.1156i	−0.370766	+	0.732845i
$372$		0			0
$373$		−	20.3173i		−	1.05199i	−0.850488	−	0.525995i	$-0.823693\pi$
$373$		−	20.3173i		−	1.05199i	0.850488	−	0.525995i	$-0.176307\pi$
$374$		0			0
$375$			2.91632i			0.150598i
$376$		0			0
$377$			26.8554i			1.38312i
$378$		0			0
$379$	1.21904			0.0626180			0.0313090	−	0.999510i	$-0.490032\pi$
$379$	1.21904			0.0626180			0.0313090	+	0.999510i	$0.490032\pi$
$380$		0			0
$381$	25.9384			1.32886
$382$		0			0
$383$	−19.2870			−0.985520			−0.492760	−	0.870165i	$-0.664012\pi$
$383$	−19.2870			−0.985520			−0.492760	+	0.870165i	$0.664012\pi$
$384$		0			0
$385$	1.70747	+	0.863853i	0.0870206	+	0.0440260i
$386$		0			0
$387$	30.3017			1.54032
$388$		0			0
$389$		−	8.97095i		−	0.454845i	−0.973796	−	0.227423i	$-0.926970\pi$
$389$		−	8.97095i		−	0.454845i	0.973796	−	0.227423i	$-0.0730299\pi$
$390$		0			0
$391$	18.8824			0.954924
$392$		0			0
$393$	10.5838			0.533883
$394$		0			0
$395$			3.02048i			0.151977i
$396$		0			0
$397$	−4.47489			−0.224588			−0.112294	−	0.993675i	$-0.535820\pi$
$397$	−4.47489			−0.224588			−0.112294	+	0.993675i	$0.535820\pi$
$398$		0			0
$399$	−58.9195	−	29.8089i	−2.94966	−	1.49231i
$400$		0			0
$401$	−23.1710			−1.15711			−0.578553	−	0.815644i	$-0.696383\pi$
$401$	−23.1710			−1.15711			−0.578553	+	0.815644i	$0.696383\pi$
$402$		0			0
$403$	−18.3068			−0.911927
$404$		0			0
$405$	4.78929			0.237982
$406$		0			0
$407$			7.59135i			0.376289i
$408$		0			0
$409$		−	23.9748i		−	1.18548i	−0.805395	−	0.592738i	$-0.798047\pi$
$409$		−	23.9748i		−	1.18548i	0.805395	−	0.592738i	$-0.201953\pi$
$410$		0			0
$411$			20.2801i			1.00034i
$412$		0			0
$413$	−7.84344	+	15.5031i	−0.385951	+	0.762859i
$414$		0			0
$415$			4.62103i			0.226837i
$416$		0			0
$417$	14.1752			0.694161
$418$		0			0
$419$		−	29.3282i		−	1.43277i	−0.697703	−	0.716387i	$-0.745794\pi$
$419$		−	29.3282i		−	1.43277i	0.697703	−	0.716387i	$-0.254206\pi$
$420$		0			0
$421$			10.2651i			0.500289i	0.968209	+	0.250144i	$0.0804781\pi$
$421$			10.2651i			0.500289i	−0.968209	+	0.250144i	$0.919522\pi$
$422$		0			0
$423$	−43.6605			−2.12285
$424$		0			0
$425$		−	6.74988i		−	0.327417i
$426$		0			0
$427$	−30.7351	−	15.5497i	−1.48737	−	0.752502i
$428$		0			0
$429$		−	8.54858i		−	0.412729i
$430$		0			0
$431$			5.21175i			0.251041i	0.992091	+	0.125521i	$0.0400601\pi$
$431$			5.21175i			0.251041i	−0.992091	+	0.125521i	$0.959940\pi$
$432$		0			0
$433$			24.4190i			1.17350i	0.809768	+	0.586751i	$0.199593\pi$
$433$			24.4190i			1.17350i	−0.809768	+	0.586751i	$0.800407\pi$
$434$		0			0
$435$	−19.3241			−0.926519
$436$		0			0
$437$	23.9400			1.14521
$438$		0			0
$439$	−37.3831			−1.78420			−0.892100	−	0.451838i	$-0.850769\pi$
$439$	−37.3831			−1.78420			−0.892100	+	0.451838i	$0.850769\pi$
$440$		0			0
$441$	−22.8280	−	31.0448i	−1.08705	−	1.47832i
$442$		0			0
$443$	23.7627			1.12900			0.564500	−	0.825433i	$-0.309069\pi$
$443$	23.7627			1.12900			0.564500	+	0.825433i	$0.309069\pi$
$444$		0			0
$445$			15.4067i			0.730347i
$446$		0			0
$447$	37.0493			1.75237
$448$		0			0
$449$	0.835362			0.0394232			0.0197116	−	0.999806i	$-0.493725\pi$
$449$	0.835362			0.0394232			0.0197116	+	0.999806i	$0.493725\pi$
$450$		0			0
$451$			7.16889i			0.337570i
$452$		0			0
$453$	−9.48595			−0.445689
$454$		0			0
$455$	−9.56816	−	4.84078i	−0.448562	−	0.226939i
$456$		0			0
$457$	−12.3597			−0.578162			−0.289081	−	0.957305i	$-0.593350\pi$
$457$	−12.3597			−0.578162			−0.289081	+	0.957305i	$0.593350\pi$
$458$		0			0
$459$	−49.3086			−2.30153
$460$		0			0
$461$	17.4251			0.811569			0.405785	−	0.913969i	$-0.366998\pi$
$461$	17.4251			0.811569			0.405785	+	0.913969i	$0.366998\pi$
$462$		0			0
$463$		−	12.3167i		−	0.572404i	−0.958169	−	0.286202i	$-0.907607\pi$
$463$		−	12.3167i		−	0.572404i	0.958169	−	0.286202i	$-0.0923929\pi$
$464$		0			0
$465$		−	13.1729i		−	0.610877i
$466$		0			0
$467$		−	36.6204i		−	1.69459i	−0.531124	−	0.847294i	$-0.678230\pi$
$467$		−	36.6204i		−	1.69459i	0.531124	−	0.847294i	$-0.321770\pi$
$468$		0			0
$469$	13.1828	+	6.66953i	0.608725	+	0.307970i
$470$		0			0
$471$		−	32.5487i		−	1.49976i
$472$		0			0
$473$	3.98115			0.183054
$474$		0			0
$475$		−	8.55782i		−	0.392660i
$476$		0			0
$477$			32.9146i			1.50706i
$478$		0			0
$479$	20.9678			0.958043			0.479022	−	0.877803i	$-0.340992\pi$
$479$	20.9678			0.958043			0.479022	+	0.877803i	$0.340992\pi$
$480$		0			0
$481$		−	42.5397i		−	1.93965i
$482$		0			0
$483$	19.2600	+	9.74415i	0.876361	+	0.443374i
$484$		0			0
$485$		−	1.26819i		−	0.0575856i
$486$		0			0
$487$			25.6074i			1.16038i	0.814480	+	0.580192i	$0.197022\pi$
$487$			25.6074i			1.16038i	−0.814480	+	0.580192i	$0.802978\pi$
$488$		0			0
$489$		−	7.02775i		−	0.317806i
$490$		0			0
$491$	13.7140			0.618903			0.309452	−	0.950915i	$-0.399854\pi$
$491$	13.7140			0.618903			0.309452	+	0.950915i	$0.399854\pi$
$492$		0			0
$493$	44.7260			2.01436
$494$		0			0
$495$	3.98146			0.178953
$496$		0			0
$497$	4.52591	−	8.94579i	0.203015	−	0.401274i
$498$		0			0
$499$	−29.6879			−1.32901			−0.664506	−	0.747283i	$-0.731358\pi$
$499$	−29.6879			−1.32901			−0.664506	+	0.747283i	$0.731358\pi$
$500$		0			0
$501$		−	25.4967i		−	1.13911i
$502$		0			0
$503$	22.3276			0.995536			0.497768	−	0.867310i	$-0.334153\pi$
$503$	22.3276			0.995536			0.497768	+	0.867310i	$0.334153\pi$
$504$		0			0
$505$	−3.39694			−0.151162
$506$		0			0
$507$			9.99167i			0.443745i
$508$		0			0
$509$	−5.88960			−0.261052			−0.130526	−	0.991445i	$-0.541667\pi$
$509$	−5.88960			−0.261052			−0.130526	+	0.991445i	$0.541667\pi$
$510$		0			0
$511$	−9.30506	+	18.3921i	−0.411632	+	0.813620i
$512$		0			0
$513$	−62.5158			−2.76014
$514$		0			0
$515$	4.78124			0.210686
$516$		0			0
$517$	−5.73629			−0.252282
$518$		0			0
$519$		−	28.8294i		−	1.26547i
$520$		0			0
$521$		−	13.2767i		−	0.581665i	−0.956774	−	0.290832i	$-0.906068\pi$
$521$		−	13.2767i		−	0.581665i	0.956774	−	0.290832i	$-0.0939322\pi$
$522$		0			0
$523$			1.69362i			0.0740569i	0.999314	+	0.0370284i	$0.0117892\pi$
$523$			1.69362i			0.0740569i	−0.999314	+	0.0370284i	$0.988211\pi$
$524$		0			0
$525$	3.48324	−	6.88487i	0.152021	−	0.300480i
$526$		0			0
$527$			30.4889i			1.32812i
$528$		0			0
$529$	15.1743			0.659753
$530$		0			0
$531$			36.1500i			1.56878i
$532$		0			0
$533$		−	40.1724i		−	1.74006i
$534$		0			0
$535$	5.96491			0.257886
$536$		0			0
$537$			44.5701i			1.92334i
$538$		0			0
$539$	−2.99922	−	4.07878i	−0.129186	−	0.175686i
$540$		0			0
$541$			26.1409i			1.12389i	0.827176	+	0.561943i	$0.189946\pi$
$541$			26.1409i			1.12389i	−0.827176	+	0.561943i	$0.810054\pi$
$542$		0			0
$543$			19.2830i			0.827513i
$544$		0			0
$545$			5.69703i			0.244034i
$546$		0			0
$547$	38.8638			1.66170			0.830849	−	0.556498i	$-0.187855\pi$
$547$	38.8638			1.66170			0.830849	+	0.556498i	$0.187855\pi$
$548$		0			0
$549$	−71.6677			−3.05870
$550$		0			0
$551$	56.7058			2.41575
$552$		0			0
$553$	3.60765	−	7.13078i	0.153413	−	0.303232i
$554$		0			0
$555$	30.6099			1.29932
$556$		0			0
$557$			7.92997i			0.336004i	0.985787	+	0.168002i	$0.0537314\pi$
$557$			7.92997i			0.336004i	−0.985787	+	0.168002i	$0.946269\pi$
$558$		0			0
$559$	−22.3092			−0.943581
$560$		0			0
$561$	−14.2371			−0.601093
$562$		0			0
$563$			43.3530i			1.82711i	0.406714	+	0.913555i	$0.366674\pi$
$563$			43.3530i			1.82711i	−0.406714	+	0.913555i	$0.633326\pi$
$564$		0			0
$565$	−10.6012			−0.445994
$566$		0			0
$567$	−11.3066	−	5.72031i	−0.474833	−	0.240230i
$568$		0			0
$569$	26.5702			1.11388			0.556941	−	0.830552i	$-0.311975\pi$
$569$	26.5702			1.11388			0.556941	+	0.830552i	$0.311975\pi$
$570$		0			0
$571$	−19.9436			−0.834612			−0.417306	−	0.908766i	$-0.637026\pi$
$571$	−19.9436			−0.834612			−0.417306	+	0.908766i	$0.637026\pi$
$572$		0			0
$573$	5.12427			0.214069
$574$		0			0
$575$			2.79744i			0.116661i
$576$		0			0
$577$			8.90827i			0.370856i	0.982658	+	0.185428i	$0.0593672\pi$
$577$			8.90827i			0.370856i	−0.982658	+	0.185428i	$0.940633\pi$
$578$		0			0
$579$			16.9834i			0.705807i
$580$		0			0
$581$	5.51933	−	10.9094i	0.228980	−	0.452597i
$582$		0			0
$583$			4.32444i			0.179100i
$584$		0			0
$585$	−22.3109			−0.922443
$586$		0			0
$587$			0.350119i			0.0144509i	0.999974	+	0.00722547i	$0.00229996\pi$
$587$			0.350119i			0.0144509i	−0.999974	+	0.00722547i	$0.997700\pi$
$588$		0			0
$589$			38.6553i			1.59276i
$590$		0			0
$591$	−19.9391			−0.820185
$592$		0			0
$593$		−	41.4280i		−	1.70125i	−0.525777	−	0.850623i	$-0.676225\pi$
$593$		−	41.4280i		−	1.70125i	0.525777	−	0.850623i	$-0.323775\pi$
$594$		0			0
$595$	−8.06203	+	15.9352i	−0.330511	+	0.653279i
$596$		0			0
$597$			42.9374i			1.75731i
$598$		0			0
$599$		−	30.8672i		−	1.26120i	−0.776108	−	0.630600i	$-0.782809\pi$
$599$		−	30.8672i		−	1.26120i	0.776108	−	0.630600i	$-0.217191\pi$
$600$		0			0
$601$			42.8294i			1.74705i	0.486783	+	0.873523i	$0.338170\pi$
$601$			42.8294i			1.74705i	−0.486783	+	0.873523i	$0.661830\pi$
$602$		0			0
$603$	30.7395			1.25181
$604$		0			0
$605$	−10.4769			−0.425947
$606$		0			0
$607$	−23.1344			−0.938997			−0.469499	−	0.882933i	$-0.655565\pi$
$607$	−23.1344			−0.938997			−0.469499	+	0.882933i	$0.655565\pi$
$608$		0			0
$609$	45.6205	+	23.0806i	1.84863	+	0.935273i
$610$		0			0
$611$	32.1445			1.30043
$612$		0			0
$613$		−	12.6600i		−	0.511331i	−0.966765	−	0.255665i	$-0.917705\pi$
$613$		−	12.6600i		−	0.511331i	0.966765	−	0.255665i	$-0.0822945\pi$
$614$		0			0
$615$	28.9065			1.16562
$616$		0			0
$617$	6.00229			0.241643			0.120822	−	0.992674i	$-0.461447\pi$
$617$	6.00229			0.241643			0.120822	+	0.992674i	$0.461447\pi$
$618$		0			0
$619$		−	2.57096i		−	0.103335i	−0.998664	−	0.0516677i	$-0.983546\pi$
$619$		−	2.57096i		−	0.103335i	0.998664	−	0.0516677i	$-0.0164537\pi$
$620$		0			0
$621$	20.4356			0.820053
$622$		0			0
$623$	18.4017	−	36.3723i	0.737248	−	1.45722i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−18.0505			−0.720869
$628$		0			0
$629$	−70.8473			−2.82487
$630$		0			0
$631$		−	34.2136i		−	1.36202i	−0.732274	−	0.681011i	$-0.761541\pi$
$631$		−	34.2136i		−	1.36202i	0.732274	−	0.681011i	$-0.238459\pi$
$632$		0			0
$633$			62.4953i			2.48396i
$634$		0			0
$635$		−	8.89422i		−	0.352956i
$636$		0			0
$637$	16.8068	+	22.8563i	0.665909	+	0.905601i
$638$		0			0
$639$		−	20.8597i		−	0.825197i
$640$		0			0
$641$	2.06020			0.0813730			0.0406865	−	0.999172i	$-0.487046\pi$
$641$	2.06020			0.0813730			0.0406865	+	0.999172i	$0.487046\pi$
$642$		0			0
$643$			13.0544i			0.514814i	0.966303	+	0.257407i	$0.0828681\pi$
$643$			13.0544i			0.514814i	−0.966303	+	0.257407i	$0.917132\pi$
$644$		0			0
$645$		−	16.0528i		−	0.632080i
$646$		0			0
$647$	11.8545			0.466047			0.233024	−	0.972471i	$-0.425138\pi$
$647$	11.8545			0.466047			0.233024	+	0.972471i	$0.425138\pi$
$648$		0			0
$649$			4.74953i			0.186435i
$650$		0			0
$651$	−15.7336	+	31.0986i	−0.616649	+	1.21885i
$652$		0			0
$653$			11.4383i			0.447614i	0.974633	+	0.223807i	$0.0718486\pi$
$653$			11.4383i			0.447614i	−0.974633	+	0.223807i	$0.928151\pi$
$654$		0			0
$655$		−	3.62917i		−	0.141804i
$656$		0			0
$657$			42.8866i			1.67316i
$658$		0			0
$659$	−13.4641			−0.524488			−0.262244	−	0.965002i	$-0.584463\pi$
$659$	−13.4641			−0.524488			−0.262244	+	0.965002i	$0.584463\pi$
$660$		0			0
$661$	7.84521			0.305143			0.152572	−	0.988292i	$-0.451244\pi$
$661$	7.84521			0.305143			0.152572	+	0.988292i	$0.451244\pi$
$662$		0			0
$663$	79.7808			3.09843
$664$		0			0
$665$	−10.2214	+	20.2034i	−0.396370	+	0.783454i
$666$		0			0
$667$	−18.5364			−0.717731
$668$		0			0
$669$		−	16.4006i		−	0.634083i
$670$		0			0
$671$	−9.41597			−0.363499
$672$		0			0
$673$	−34.9983			−1.34909			−0.674543	−	0.738236i	$-0.735659\pi$
$673$	−34.9983			−1.34909			−0.674543	+	0.738236i	$0.735659\pi$
$674$		0			0
$675$		−	7.30511i		−	0.281174i
$676$		0			0
$677$	8.42861			0.323938			0.161969	−	0.986796i	$-0.448216\pi$
$677$	8.42861			0.323938			0.161969	+	0.986796i	$0.448216\pi$
$678$		0			0
$679$	−1.51472	+	2.99396i	−0.0581297	+	0.114897i
$680$		0			0
$681$	67.4294			2.58390
$682$		0			0
$683$	17.8717			0.683840			0.341920	−	0.939729i	$-0.388923\pi$
$683$	17.8717			0.683840			0.341920	+	0.939729i	$0.388923\pi$
$684$		0			0
$685$	6.95400			0.265699
$686$		0			0
$687$			40.0081i			1.52640i
$688$		0			0
$689$		−	24.2329i		−	0.923201i
$690$		0			0
$691$			12.2698i			0.466764i	0.972385	+	0.233382i	$0.0749793\pi$
$691$			12.2698i			0.466764i	−0.972385	+	0.233382i	$0.925021\pi$
$692$		0			0
$693$	−9.39946	−	4.75543i	−0.357056	−	0.180644i
$694$		0			0
$695$		−	4.86064i		−	0.184375i
$696$		0			0
$697$	−66.9047			−2.53419
$698$		0			0
$699$		−	18.1752i		−	0.687448i
$700$		0			0
$701$		−	9.12819i		−	0.344767i	−0.985030	−	0.172384i	$-0.944853\pi$
$701$		−	9.12819i		−	0.344767i	0.985030	−	0.172384i	$-0.0551469\pi$
$702$		0			0
$703$	−89.8236			−3.38776
$704$		0			0
$705$			23.1299i			0.871122i
$706$		0			0
$707$	8.01953	+	4.05729i	0.301605	+	0.152590i
$708$		0			0
$709$			0.381427i			0.0143248i	0.999974	+	0.00716239i	$0.00227988\pi$
$709$			0.381427i			0.0143248i	−0.999974	+	0.00716239i	$0.997720\pi$
$710$		0			0
$711$		−	16.6275i		−	0.623579i
$712$		0			0
$713$		−	12.6359i		−	0.473218i
$714$		0			0
$715$	−2.93129			−0.109624
$716$		0			0
$717$	−69.4124			−2.59226
$718$		0			0
$719$	−43.2872			−1.61434			−0.807171	−	0.590318i	$-0.799002\pi$
$719$	−43.2872			−1.61434			−0.807171	+	0.590318i	$0.799002\pi$
$720$		0			0
$721$	−11.2876	−	5.71069i	−0.420372	−	0.212677i
$722$		0			0
$723$	−54.2121			−2.01617
$724$		0			0
$725$			6.62619i			0.246091i
$726$		0			0
$727$	23.1436			0.858348			0.429174	−	0.903222i	$-0.358805\pi$
$727$	23.1436			0.858348			0.429174	+	0.903222i	$0.358805\pi$
$728$		0			0
$729$	37.5474			1.39065
$730$		0			0
$731$			37.1547i			1.37422i
$732$		0			0
$733$	−32.5388			−1.20185			−0.600924	−	0.799306i	$-0.705201\pi$
$733$	−32.5388			−1.20185			−0.600924	+	0.799306i	$0.705201\pi$
$734$		0			0
$735$	−16.4465	+	12.0935i	−0.606639	+	0.446075i
$736$		0			0
$737$	4.03867			0.148766
$738$		0			0
$739$	16.7623			0.616611			0.308306	−	0.951287i	$-0.400238\pi$
$739$	16.7623			0.616611			0.308306	+	0.951287i	$0.400238\pi$
$740$		0			0
$741$	101.150			3.71584
$742$		0			0
$743$		−	42.8089i		−	1.57051i	−0.619175	−	0.785253i	$-0.712533\pi$
$743$		−	42.8089i		−	1.57051i	0.619175	−	0.785253i	$-0.287467\pi$
$744$		0			0
$745$		−	12.7041i		−	0.465443i
$746$		0			0
$747$		−	25.4383i		−	0.930740i
$748$		0			0
$749$	−14.0820	−	7.12447i	−0.514546	−	0.260322i
$750$		0			0
$751$		−	19.5692i		−	0.714089i	−0.934087	−	0.357044i	$-0.883784\pi$
$751$		−	19.5692i		−	0.714089i	0.934087	−	0.357044i	$-0.116216\pi$
$752$		0			0
$753$	23.0355			0.839461
$754$		0			0
$755$			3.25272i			0.118378i
$756$		0			0
$757$			2.72657i			0.0990987i	0.998772	+	0.0495494i	$0.0157785\pi$
$757$			2.72657i			0.0990987i	−0.998772	+	0.0495494i	$0.984221\pi$
$758$		0			0
$759$	5.90048			0.214174
$760$		0			0
$761$			31.7752i			1.15185i	0.817502	+	0.575926i	$0.195358\pi$
$761$			31.7752i			1.15185i	−0.817502	+	0.575926i	$0.804642\pi$
$762$		0			0
$763$	6.80451	−	13.4496i	0.246340	−	0.486909i
$764$		0			0
$765$			37.1575i			1.34343i
$766$		0			0
$767$		−	26.6150i		−	0.961011i
$768$		0			0
$769$			23.4373i			0.845172i	0.906323	+	0.422586i	$0.138878\pi$
$769$			23.4373i			0.845172i	−0.906323	+	0.422586i	$0.861122\pi$
$770$		0			0
$771$	36.1954			1.30354
$772$		0			0
$773$	−18.1504			−0.652824			−0.326412	−	0.945228i	$-0.605840\pi$
$773$	−18.1504			−0.652824			−0.326412	+	0.945228i	$0.605840\pi$
$774$		0			0
$775$	−4.51695			−0.162254
$776$		0			0
$777$	−72.2642	−	36.5603i	−2.59246	−	1.31160i
$778$		0			0
$779$	−84.8249			−3.03917
$780$		0			0
$781$		−	2.74063i		−	0.0980673i
$782$		0			0
$783$	48.4051			1.72986
$784$		0			0
$785$	−11.1609			−0.398349
$786$		0			0
$787$			29.4416i			1.04948i	0.851263	+	0.524740i	$0.175837\pi$
$787$			29.4416i			1.04948i	−0.851263	+	0.524740i	$0.824163\pi$
$788$		0			0
$789$	49.9431			1.77802
$790$		0			0
$791$	25.0273	+	12.6620i	0.889870	+	0.450208i
$792$		0			0
$793$	52.7644			1.87372
$794$		0			0
$795$	17.4371			0.618429
$796$		0			0
$797$	35.7160			1.26513			0.632563	−	0.774509i	$-0.282003\pi$
$797$	35.7160			1.26513			0.632563	+	0.774509i	$0.282003\pi$
$798$		0			0
$799$		−	53.5347i		−	1.89392i
$800$		0			0
$801$		−	84.8125i		−	2.99670i
$802$		0			0
$803$			5.63460i			0.198841i
$804$		0			0
$805$	3.34125	−	6.60422i	0.117764	−	0.232768i
$806$		0			0
$807$		−	9.34702i		−	0.329031i
$808$		0			0
$809$	−0.930510			−0.0327150			−0.0163575	−	0.999866i	$-0.505207\pi$
$809$	−0.930510			−0.0327150			−0.0163575	+	0.999866i	$0.505207\pi$
$810$		0			0
$811$			19.6579i			0.690281i	0.938551	+	0.345140i	$0.112169\pi$
$811$			19.6579i			0.690281i	−0.938551	+	0.345140i	$0.887831\pi$
$812$		0			0
$813$		−	7.84356i		−	0.275086i
$814$		0			0
$815$	−2.40980			−0.0844116
$816$		0			0
$817$			47.1065i			1.64805i
$818$		0			0
$819$	52.6718	+	26.6481i	1.84050	+	0.931159i
$820$		0			0
$821$			39.0522i			1.36293i	0.731850	+	0.681465i	$0.238657\pi$
$821$			39.0522i			1.36293i	−0.731850	+	0.681465i	$0.761343\pi$
$822$		0			0
$823$		−	46.6302i		−	1.62543i	−0.582664	−	0.812713i	$-0.697990\pi$
$823$		−	46.6302i		−	1.62543i	0.582664	−	0.812713i	$-0.302010\pi$
$824$		0			0
$825$		−	2.10924i		−	0.0734344i
$826$		0			0
$827$	34.2405			1.19066			0.595330	−	0.803481i	$-0.297021\pi$
$827$	34.2405			1.19066			0.595330	+	0.803481i	$0.297021\pi$
$828$		0			0
$829$	−54.1765			−1.88163			−0.940814	−	0.338923i	$-0.889937\pi$
$829$	−54.1765			−1.88163			−0.940814	+	0.338923i	$0.889937\pi$
$830$		0			0
$831$	30.3780			1.05380
$832$		0			0
$833$	38.0658	−	27.9907i	1.31890	−	0.969819i
$834$		0			0
$835$	−8.74278			−0.302556
$836$		0			0
$837$			32.9968i			1.14054i
$838$		0			0
$839$	45.8131			1.58164			0.790822	−	0.612046i	$-0.209653\pi$
$839$	45.8131			1.58164			0.790822	+	0.612046i	$0.209653\pi$
$840$		0			0
$841$	−14.9064			−0.514015
$842$		0			0
$843$			82.3424i			2.83602i
$844$		0			0
$845$	3.42612			0.117862
$846$		0			0
$847$	24.7340	+	12.5136i	0.849869	+	0.429971i
$848$		0			0
$849$	22.7304			0.780105
$850$		0			0
$851$	29.3622			1.00652
$852$		0			0
$853$	−51.8458			−1.77517			−0.887583	−	0.460647i	$-0.847617\pi$
$853$	−51.8458			−1.77517			−0.887583	+	0.460647i	$0.847617\pi$
$854$		0			0
$855$			47.1100i			1.61113i
$856$		0			0
$857$		−	9.36591i		−	0.319934i	−0.987122	−	0.159967i	$-0.948861\pi$
$857$		−	9.36591i		−	0.319934i	0.987122	−	0.159967i	$-0.0511387\pi$
$858$		0			0
$859$		−	14.8372i		−	0.506240i	−0.967435	−	0.253120i	$-0.918543\pi$
$859$		−	14.8372i		−	0.506240i	0.967435	−	0.253120i	$-0.0814567\pi$
$860$		0			0
$861$	−68.2426	−	34.5257i	−2.32570	−	1.17663i
$862$		0			0
$863$			34.0888i			1.16040i	0.814475	+	0.580198i	$0.197025\pi$
$863$			34.0888i			1.16040i	−0.814475	+	0.580198i	$0.802975\pi$
$864$		0			0
$865$	−9.88555			−0.336119
$866$		0			0
$867$		−	83.2927i		−	2.82877i
$868$		0			0
$869$		−	2.18458i		−	0.0741068i
$870$		0			0
$871$	−22.6316			−0.766841
$872$		0			0
$873$			6.98127i			0.236280i
$874$		0			0
$875$	−2.36081	−	1.19440i	−0.0798099	−	0.0403779i
$876$		0			0
$877$			42.0210i			1.41895i	0.704732	+	0.709473i	$0.251067\pi$
$877$			42.0210i			1.41895i	−0.704732	+	0.709473i	$0.748933\pi$
$878$		0			0
$879$			90.1007i			3.03902i
$880$		0			0
$881$		−	10.9294i		−	0.368221i	−0.982906	−	0.184111i	$-0.941060\pi$
$881$		−	10.9294i		−	0.368221i	0.982906	−	0.184111i	$-0.0589405\pi$
$882$		0			0
$883$	−21.9353			−0.738181			−0.369091	−	0.929393i	$-0.620331\pi$
$883$	−21.9353			−0.738181			−0.369091	+	0.929393i	$0.620331\pi$
$884$		0			0
$885$	19.1511			0.643757
$886$		0			0
$887$	43.6481			1.46556			0.732779	−	0.680466i	$-0.238223\pi$
$887$	43.6481			1.46556			0.732779	+	0.680466i	$0.238223\pi$
$888$		0			0
$889$	−10.6232	+	20.9976i	−0.356291	+	0.704236i
$890$		0			0
$891$	−3.46388			−0.116044
$892$		0			0
$893$		−	67.8738i		−	2.27131i
$894$		0			0
$895$	15.2830			0.510855
$896$		0			0
$897$	−33.0646			−1.10399
$898$		0			0
$899$		−	29.9302i		−	0.998227i
$900$		0			0
$901$	−40.3585			−1.34454
$902$		0			0
$903$	−19.1735	+	37.8977i	−0.638052	+	1.26116i
$904$		0			0
$905$	6.61211			0.219794
$906$		0			0
$907$	4.18859			0.139080			0.0695399	−	0.997579i	$-0.477847\pi$
$907$	4.18859			0.139080			0.0695399	+	0.997579i	$0.477847\pi$
$908$		0			0
$909$	18.6998			0.620235
$910$		0			0
$911$			45.9339i			1.52186i	0.648835	+	0.760929i	$0.275256\pi$
$911$			45.9339i			1.52186i	−0.648835	+	0.760929i	$0.724744\pi$
$912$		0			0
$913$		−	3.34218i		−	0.110610i
$914$		0			0
$915$			37.9671i			1.25516i
$916$		0			0
$917$	−4.33467	+	8.56779i	−0.143143	+	0.282933i
$918$		0			0
$919$			24.0663i			0.793874i	0.917846	+	0.396937i	$0.129927\pi$
$919$			24.0663i			0.793874i	−0.917846	+	0.396937i	$0.870073\pi$
$920$		0			0
$921$	55.2218			1.81962
$922$		0			0
$923$			15.3577i			0.505504i
$924$		0			0
$925$		−	10.4961i		−	0.345109i
$926$		0			0
$927$	−26.3203			−0.864471
$928$		0			0
$929$			53.4609i			1.75399i	0.480495	+	0.876997i	$0.340457\pi$
$929$			53.4609i			1.75399i	−0.480495	+	0.876997i	$0.659543\pi$
$930$		0			0
$931$	48.2617	−	35.4879i	1.58171	−	1.16307i
$932$		0			0
$933$		−	6.31054i		−	0.206598i
$934$		0			0
$935$			4.88189i			0.159655i
$936$		0			0
$937$			33.8278i			1.10510i	0.833478	+	0.552552i	$0.186346\pi$
$937$			33.8278i			1.10510i	−0.833478	+	0.552552i	$0.813654\pi$
$938$		0			0
$939$	40.3413			1.31649
$940$		0			0
$941$	41.0372			1.33777			0.668887	−	0.743364i	$-0.266771\pi$
$941$	41.0372			1.33777			0.668887	+	0.743364i	$0.266771\pi$
$942$		0			0
$943$	27.7282			0.902953
$944$		0			0
$945$	−8.72519	+	17.2460i	−0.283830	+	0.561011i
$946$		0			0
$947$	29.1650			0.947734			0.473867	−	0.880596i	$-0.342858\pi$
$947$	29.1650			0.947734			0.473867	+	0.880596i	$0.342858\pi$
$948$		0			0
$949$		−	31.5747i		−	1.02496i
$950$		0			0
$951$	−48.2990			−1.56620
$952$		0			0
$953$	37.4864			1.21430			0.607152	−	0.794586i	$-0.292312\pi$
$953$	37.4864			1.21430			0.607152	+	0.794586i	$0.292312\pi$
$954$		0			0
$955$		−	1.75710i		−	0.0568585i
$956$		0			0
$957$	13.9762			0.451788
$958$		0			0
$959$	−16.4171	−	8.30582i	−0.530134	−	0.268209i
$960$		0			0
$961$	−10.5972			−0.341844
$962$		0			0
$963$	−32.8363			−1.05814
$964$		0			0
$965$	5.82359			0.187468
$966$		0			0
$967$			40.8218i			1.31274i	0.754438	+	0.656371i	$0.227909\pi$
$967$			40.8218i			1.31274i	−0.754438	+	0.656371i	$0.772091\pi$
$968$		0			0
$969$		−	168.459i		−	5.41168i
$970$		0			0
$971$		−	29.7061i		−	0.953315i	−0.879089	−	0.476658i	$-0.841848\pi$
$971$		−	29.7061i		−	0.953315i	0.879089	−	0.476658i	$-0.158152\pi$
$972$		0			0
$973$	−5.80553	+	11.4750i	−0.186117	+	0.367873i
$974$		0			0
$975$			11.8196i			0.378530i
$976$		0			0
$977$	1.20259			0.0384743			0.0192371	−	0.999815i	$-0.493876\pi$
$977$	1.20259			0.0384743			0.0192371	+	0.999815i	$0.493876\pi$
$978$		0			0
$979$		−	11.1430i		−	0.356131i
$980$		0			0
$981$		−	31.3617i		−	1.00130i
$982$		0			0
$983$	−47.1151			−1.50274			−0.751369	−	0.659882i	$-0.770606\pi$
$983$	−47.1151			−1.50274			−0.751369	+	0.659882i	$0.770606\pi$
$984$		0			0
$985$			6.83708i			0.217848i
$986$		0			0
$987$	27.6262	−	54.6053i	0.879353	−	1.73811i
$988$		0			0
$989$		−	15.3985i		−	0.489643i
$990$		0			0
$991$			0.625176i			0.0198594i	0.999951	+	0.00992969i	$0.00316077\pi$
$991$			0.625176i			0.0198594i	−0.999951	+	0.00992969i	$0.996839\pi$
$992$		0			0
$993$		−	31.5390i		−	1.00086i
$994$		0			0
$995$	14.7231			0.466755
$996$		0			0
$997$	−11.7960			−0.373584			−0.186792	−	0.982400i	$-0.559809\pi$
$997$	−11.7960			−0.373584			−0.186792	+	0.982400i	$0.559809\pi$
$998$		0			0
$999$	−76.6750			−2.42589

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.h.f.671.8	yes	24
4.3	odd	2	inner	2240.2.h.f.671.17	yes	24
7.6	odd	2		2240.2.h.e.671.17	yes	24
8.3	odd	2		2240.2.h.e.671.18	yes	24
8.5	even	2		2240.2.h.e.671.7	✓	24
28.27	even	2		2240.2.h.e.671.8	yes	24
56.13	odd	2	inner	2240.2.h.f.671.18	yes	24
56.27	even	2	inner	2240.2.h.f.671.7	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.h.e.671.7	✓	24	8.5	even	2
2240.2.h.e.671.8	yes	24	28.27	even	2
2240.2.h.e.671.17	yes	24	7.6	odd	2
2240.2.h.e.671.18	yes	24	8.3	odd	2
2240.2.h.f.671.7	yes	24	56.27	even	2	inner
2240.2.h.f.671.8	yes	24	1.1	even	1	trivial
2240.2.h.f.671.17	yes	24	4.3	odd	2	inner
2240.2.h.f.671.18	yes	24	56.13	odd	2	inner

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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.91632i q^{3} +1.00000 q^{5} +(-2.36081 - 1.19440i) q^{7} -5.50491 q^{9} +O(q^{10})\)	\(q+2.91632i q^{3} +1.00000 q^{5} +(-2.36081 - 1.19440i) q^{7} -5.50491 q^{9} -0.723256 q^{11} +4.05292 q^{13} +2.91632i q^{15} -6.74988i q^{17} -8.55782i q^{19} +(3.48324 - 6.88487i) q^{21} +2.79744i q^{23} +1.00000 q^{25} -7.30511i q^{27} +6.62619i q^{29} -4.51695 q^{31} -2.10924i q^{33} +(-2.36081 - 1.19440i) q^{35} -10.4961i q^{37} +11.8196i q^{39} -9.91197i q^{41} -5.50449 q^{43} -5.50491 q^{45} +7.93120 q^{47} +(4.14684 + 5.63948i) q^{49} +19.6848 q^{51} -5.97913i q^{53} -0.723256 q^{55} +24.9573 q^{57} -6.56687i q^{59} +13.0189 q^{61} +(12.9960 + 6.57504i) q^{63} +4.05292 q^{65} -5.58402 q^{67} -8.15822 q^{69} +3.78929i q^{71} -7.79060i q^{73} +2.91632i q^{75} +(1.70747 + 0.863853i) q^{77} +3.02048i q^{79} +4.78929 q^{81} +4.62103i q^{83} -6.74988i q^{85} -19.3241 q^{87} +15.4067i q^{89} +(-9.56816 - 4.84078i) q^{91} -13.1729i q^{93} -8.55782i q^{95} -1.26819i q^{97} +3.98146 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{5} - 36 q^{9}+O(q^{10}) \) 24 * q + 24 * q^5 - 36 * q^9	\( 24 q + 24 q^{5} - 36 q^{9} + 4 q^{13} + 8 q^{21} + 24 q^{25} - 36 q^{45} + 24 q^{57} + 56 q^{61} + 4 q^{65} + 96 q^{69} - 12 q^{77} + 24 q^{81}+O(q^{100}) \) 24 * q + 24 * q^5 - 36 * q^9 + 4 * q^13 + 8 * q^21 + 24 * q^25 - 36 * q^45 + 24 * q^57 + 56 * q^61 + 4 * q^65 + 96 * q^69 - 12 * q^77 + 24 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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