LMFDB - Embedded newform 2240.2.h.e.671.12

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			671.12
Character	$\chi$	$=$	2240.671
Dual form			2240.2.h.e.671.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.421861i q^{3} -1.00000 q^{5} +(-1.02538 - 2.43897i) q^{7} +2.82203 q^{9} +O(q^{10})$	$q+0.421861i q^{3} -1.00000 q^{5} +(-1.02538 - 2.43897i) q^{7} +2.82203 q^{9} +1.89956 q^{11} -4.65608 q^{13} -0.421861i q^{15} -7.19944i q^{17} -0.834050i q^{19} +(1.02891 - 0.432568i) q^{21} +5.76981i q^{23} +1.00000 q^{25} +2.45609i q^{27} +4.84493i q^{29} -4.04358 q^{31} +0.801350i q^{33} +(1.02538 + 2.43897i) q^{35} -4.68101i q^{37} -1.96422i q^{39} -5.81861i q^{41} -12.6338 q^{43} -2.82203 q^{45} -6.30483 q^{47} +(-4.89718 + 5.00176i) q^{49} +3.03716 q^{51} -0.637428i q^{53} -1.89956 q^{55} +0.351853 q^{57} +5.32592i q^{59} +7.80404 q^{61} +(-2.89366 - 6.88287i) q^{63} +4.65608 q^{65} -4.64973 q^{67} -2.43406 q^{69} -6.42998i q^{71} +0.227709i q^{73} +0.421861i q^{75} +(-1.94778 - 4.63298i) q^{77} -7.17831i q^{79} +7.42998 q^{81} -4.76612i q^{83} +7.19944i q^{85} -2.04389 q^{87} -11.9442i q^{89} +(4.77427 + 11.3561i) q^{91} -1.70583i q^{93} +0.834050i q^{95} -10.3285i q^{97} +5.36063 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$			0.421861i			0.243561i	0.992557	+	0.121781i	$0.0388604\pi$
$3$			0.421861i			0.243561i	−0.992557	+	0.121781i	$0.961140\pi$
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$	−1.02538	−	2.43897i	−0.387558	−	0.921845i
$7$	−1.02538	−	2.43897i	−0.387558	−	0.921845i
$8$		0			0
$9$	2.82203			0.940678
$10$		0			0
$11$	1.89956			0.572739			0.286370	−	0.958119i	$-0.407551\pi$
$11$	1.89956			0.572739			0.286370	+	0.958119i	$0.407551\pi$
$12$		0			0
$13$	−4.65608			−1.29137			−0.645683	−	0.763606i	$-0.723427\pi$
$13$	−4.65608			−1.29137			−0.645683	+	0.763606i	$0.723427\pi$
$14$		0			0
$15$		−	0.421861i		−	0.108924i
$16$		0			0
$17$		−	7.19944i		−	1.74612i	−0.487613	−	0.873060i	$-0.662132\pi$
$17$		−	7.19944i		−	1.74612i	0.487613	−	0.873060i	$-0.337868\pi$
$18$		0			0
$19$		−	0.834050i		−	0.191344i	−0.995413	−	0.0956721i	$-0.969500\pi$
$19$		−	0.834050i		−	0.191344i	0.995413	−	0.0956721i	$-0.0305000\pi$
$20$		0			0
$21$	1.02891	−	0.432568i	0.224526	−	0.0943941i
$22$		0			0
$23$			5.76981i			1.20309i	0.798839	+	0.601544i	$0.205448\pi$
$23$			5.76981i			1.20309i	−0.798839	+	0.601544i	$0.794552\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$			2.45609i			0.472674i
$28$		0			0
$29$			4.84493i			0.899681i	0.893109	+	0.449841i	$0.148519\pi$
$29$			4.84493i			0.899681i	−0.893109	+	0.449841i	$0.851481\pi$
$30$		0			0
$31$	−4.04358			−0.726249			−0.363124	−	0.931741i	$-0.618290\pi$
$31$	−4.04358			−0.726249			−0.363124	+	0.931741i	$0.618290\pi$
$32$		0			0
$33$			0.801350i			0.139497i
$34$		0			0
$35$	1.02538	+	2.43897i	0.173321	+	0.412262i
$36$		0			0
$37$		−	4.68101i		−	0.769553i	−0.923010	−	0.384777i	$-0.874278\pi$
$37$		−	4.68101i		−	0.769553i	0.923010	−	0.384777i	$-0.125722\pi$
$38$		0			0
$39$		−	1.96422i		−	0.314527i
$40$		0			0
$41$		−	5.81861i		−	0.908714i	−0.890820	−	0.454357i	$-0.849869\pi$
$41$		−	5.81861i		−	0.908714i	0.890820	−	0.454357i	$-0.150131\pi$
$42$		0			0
$43$	−12.6338			−1.92663			−0.963316	−	0.268371i	$-0.913515\pi$
$43$	−12.6338			−1.92663			−0.963316	+	0.268371i	$0.913515\pi$
$44$		0			0
$45$	−2.82203			−0.420684
$46$		0			0
$47$	−6.30483			−0.919654			−0.459827	−	0.888008i	$-0.652089\pi$
$47$	−6.30483			−0.919654			−0.459827	+	0.888008i	$0.652089\pi$
$48$		0			0
$49$	−4.89718	+	5.00176i	−0.699597	+	0.714537i
$50$		0			0
$51$	3.03716			0.425287
$52$		0			0
$53$		−	0.637428i		−	0.0875574i	−0.999041	−	0.0437787i	$-0.986060\pi$
$53$		−	0.637428i		−	0.0875574i	0.999041	−	0.0437787i	$-0.0139397\pi$
$54$		0			0
$55$	−1.89956			−0.256137
$56$		0			0
$57$	0.351853			0.0466040
$58$		0			0
$59$			5.32592i			0.693376i	0.937981	+	0.346688i	$0.112694\pi$
$59$			5.32592i			0.693376i	−0.937981	+	0.346688i	$0.887306\pi$
$60$		0			0
$61$	7.80404			0.999204			0.499602	−	0.866255i	$-0.333480\pi$
$61$	7.80404			0.999204			0.499602	+	0.866255i	$0.333480\pi$
$62$		0			0
$63$	−2.89366	−	6.88287i	−0.364567	−	0.867159i
$64$		0			0
$65$	4.65608			0.577516
$66$		0			0
$67$	−4.64973			−0.568055			−0.284027	−	0.958816i	$-0.591671\pi$
$67$	−4.64973			−0.568055			−0.284027	+	0.958816i	$0.591671\pi$
$68$		0			0
$69$	−2.43406			−0.293026
$70$		0			0
$71$		−	6.42998i		−	0.763098i	−0.924349	−	0.381549i	$-0.875391\pi$
$71$		−	6.42998i		−	0.763098i	0.924349	−	0.381549i	$-0.124609\pi$
$72$		0			0
$73$			0.227709i			0.0266513i	0.999911	+	0.0133257i	$0.00424182\pi$
$73$			0.227709i			0.0266513i	−0.999911	+	0.0133257i	$0.995758\pi$
$74$		0			0
$75$			0.421861i			0.0487123i
$76$		0			0
$77$	−1.94778	−	4.63298i	−0.221970	−	0.527977i
$78$		0			0
$79$		−	7.17831i		−	0.807623i	−0.914842	−	0.403811i	$-0.867685\pi$
$79$		−	7.17831i		−	0.807623i	0.914842	−	0.403811i	$-0.132315\pi$
$80$		0			0
$81$	7.42998			0.825553
$82$		0			0
$83$		−	4.76612i		−	0.523150i	−0.965183	−	0.261575i	$-0.915758\pi$
$83$		−	4.76612i		−	0.523150i	0.965183	−	0.261575i	$-0.0842418\pi$
$84$		0			0
$85$			7.19944i			0.780889i
$86$		0			0
$87$	−2.04389			−0.219128
$88$		0			0
$89$		−	11.9442i		−	1.26609i	−0.774117	−	0.633043i	$-0.781806\pi$
$89$		−	11.9442i		−	1.26609i	0.774117	−	0.633043i	$-0.218194\pi$
$90$		0			0
$91$	4.77427	+	11.3561i	0.500479	+	1.19044i
$92$		0			0
$93$		−	1.70583i		−	0.176886i
$94$		0			0
$95$			0.834050i			0.0855717i
$96$		0			0
$97$		−	10.3285i		−	1.04870i	−0.851502	−	0.524351i	$-0.824308\pi$
$97$		−	10.3285i		−	1.04870i	0.851502	−	0.524351i	$-0.175692\pi$
$98$		0			0
$99$	5.36063			0.538763
$100$		0			0
$101$	−15.0777			−1.50029			−0.750145	−	0.661273i	$-0.770016\pi$
$101$	−15.0777			−1.50029			−0.750145	+	0.661273i	$0.770016\pi$
$102$		0			0
$103$	−14.5333			−1.43201			−0.716006	−	0.698095i	$-0.754032\pi$
$103$	−14.5333			−1.43201			−0.716006	+	0.698095i	$0.754032\pi$
$104$		0			0
$105$	−1.02891	+	0.432568i	−0.100411	+	0.0422143i
$106$		0			0
$107$	−9.91022			−0.958057			−0.479029	−	0.877799i	$-0.659011\pi$
$107$	−9.91022			−0.958057			−0.479029	+	0.877799i	$0.659011\pi$
$108$		0			0
$109$		−	7.77892i		−	0.745085i	−0.928015	−	0.372543i	$-0.878486\pi$
$109$		−	7.77892i		−	0.745085i	0.928015	−	0.372543i	$-0.121514\pi$
$110$		0			0
$111$	1.97473			0.187433
$112$		0			0
$113$	4.75220			0.447050			0.223525	−	0.974698i	$-0.428244\pi$
$113$	4.75220			0.447050			0.223525	+	0.974698i	$0.428244\pi$
$114$		0			0
$115$		−	5.76981i		−	0.538038i
$116$		0			0
$117$	−13.1396			−1.21476
$118$		0			0
$119$	−17.5592	+	7.38218i	−1.60965	+	0.676723i
$120$		0			0
$121$	−7.39167			−0.671970
$122$		0			0
$123$	2.45464			0.221327
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$			5.70233i			0.506000i	0.967466	+	0.253000i	$0.0814172\pi$
$127$			5.70233i			0.506000i	−0.967466	+	0.253000i	$0.918583\pi$
$128$		0			0
$129$		−	5.32969i		−	0.469253i
$130$		0			0
$131$			16.9659i			1.48232i	0.671329	+	0.741159i	$0.265724\pi$
$131$			16.9659i			1.48232i	−0.671329	+	0.741159i	$0.734276\pi$
$132$		0			0
$133$	−2.03423	+	0.855220i	−0.176390	+	0.0741570i
$134$		0			0
$135$		−	2.45609i		−	0.211386i
$136$		0			0
$137$	−6.75958			−0.577510			−0.288755	−	0.957403i	$-0.593241\pi$
$137$	−6.75958			−0.577510			−0.288755	+	0.957403i	$0.593241\pi$
$138$		0			0
$139$			5.36998i			0.455476i	0.973722	+	0.227738i	$0.0731329\pi$
$139$			5.36998i			0.455476i	−0.973722	+	0.227738i	$0.926867\pi$
$140$		0			0
$141$		−	2.65976i		−	0.223992i
$142$		0			0
$143$	−8.84452			−0.739616
$144$		0			0
$145$		−	4.84493i		−	0.402350i
$146$		0			0
$147$	−2.11005	−	2.06593i	−0.174034	−	0.170395i
$148$		0			0
$149$		−	11.6260i		−	0.952436i	−0.879327	−	0.476218i	$-0.842007\pi$
$149$		−	11.6260i		−	0.952436i	0.879327	−	0.476218i	$-0.157993\pi$
$150$		0			0
$151$		−	5.29014i		−	0.430505i	−0.976558	−	0.215253i	$-0.930943\pi$
$151$		−	5.29014i		−	0.430505i	0.976558	−	0.215253i	$-0.0690575\pi$
$152$		0			0
$153$		−	20.3171i		−	1.64254i
$154$		0			0
$155$	4.04358			0.324788
$156$		0			0
$157$	21.9118			1.74875			0.874375	−	0.485250i	$-0.161272\pi$
$157$	21.9118			1.74875			0.874375	+	0.485250i	$0.161272\pi$
$158$		0			0
$159$	0.268906			0.0213256
$160$		0			0
$161$	14.0724	−	5.91626i	1.10906	−	0.466267i
$162$		0			0
$163$	0.817106			0.0640007			0.0320003	−	0.999488i	$-0.489812\pi$
$163$	0.817106			0.0640007			0.0320003	+	0.999488i	$0.489812\pi$
$164$		0			0
$165$		−	0.801350i		−	0.0623850i
$166$		0			0
$167$	13.5609			1.04937			0.524686	−	0.851296i	$-0.324183\pi$
$167$	13.5609			1.04937			0.524686	+	0.851296i	$0.324183\pi$
$168$		0			0
$169$	8.67912			0.667624
$170$		0			0
$171$		−	2.35372i		−	0.179993i
$172$		0			0
$173$	5.49980			0.418142			0.209071	−	0.977900i	$-0.432956\pi$
$173$	5.49980			0.418142			0.209071	+	0.977900i	$0.432956\pi$
$174$		0			0
$175$	−1.02538	−	2.43897i	−0.0775116	−	0.184369i
$176$		0			0
$177$	−2.24679			−0.168879
$178$		0			0
$179$	−23.9886			−1.79299			−0.896496	−	0.443052i	$-0.853896\pi$
$179$	−23.9886			−1.79299			−0.896496	+	0.443052i	$0.853896\pi$
$180$		0			0
$181$	−20.9178			−1.55481			−0.777403	−	0.629003i	$-0.783463\pi$
$181$	−20.9178			−1.55481			−0.777403	+	0.629003i	$0.783463\pi$
$182$		0			0
$183$			3.29221i			0.243368i
$184$		0			0
$185$			4.68101i			0.344155i
$186$		0			0
$187$		−	13.6758i		−	1.00007i
$188$		0			0
$189$	5.99033	−	2.51843i	0.435732	−	0.183189i
$190$		0			0
$191$		−	16.9342i		−	1.22532i	−0.790348	−	0.612658i	$-0.790100\pi$
$191$		−	16.9342i		−	1.22532i	0.790348	−	0.612658i	$-0.209900\pi$
$192$		0			0
$193$	5.00369			0.360173			0.180087	−	0.983651i	$-0.442362\pi$
$193$	5.00369			0.360173			0.180087	+	0.983651i	$0.442362\pi$
$194$		0			0
$195$			1.96422i			0.140661i
$196$		0			0
$197$		−	27.1869i		−	1.93699i	−0.249043	−	0.968493i	$-0.580116\pi$
$197$		−	27.1869i		−	1.93699i	0.249043	−	0.968493i	$-0.419884\pi$
$198$		0			0
$199$	−0.206346			−0.0146275			−0.00731373	−	0.999973i	$-0.502328\pi$
$199$	−0.206346			−0.0146275			−0.00731373	+	0.999973i	$0.502328\pi$
$200$		0			0
$201$		−	1.96154i		−	0.138356i
$202$		0			0
$203$	11.8167	−	4.96791i	0.829367	−	0.348679i
$204$		0			0
$205$			5.81861i			0.406389i
$206$		0			0
$207$			16.2826i			1.13172i
$208$		0			0
$209$		−	1.58433i		−	0.109590i
$210$		0			0
$211$	22.7712			1.56764			0.783818	−	0.620991i	$-0.213270\pi$
$211$	22.7712			1.56764			0.783818	+	0.620991i	$0.213270\pi$
$212$		0			0
$213$	2.71255			0.185861
$214$		0			0
$215$	12.6338			0.861616
$216$		0			0
$217$	4.14622	+	9.86219i	0.281464	+	0.669489i
$218$		0			0
$219$	−0.0960615			−0.00649123
$220$		0			0
$221$			33.5212i			2.25488i
$222$		0			0
$223$	14.9007			0.997828			0.498914	−	0.866652i	$-0.333732\pi$
$223$	14.9007			0.997828			0.498914	+	0.866652i	$0.333732\pi$
$224$		0			0
$225$	2.82203			0.188136
$226$		0			0
$227$			16.0299i			1.06394i	0.846762	+	0.531972i	$0.178549\pi$
$227$			16.0299i			1.06394i	−0.846762	+	0.531972i	$0.821451\pi$
$228$		0			0
$229$	−16.7458			−1.10660			−0.553298	−	0.832983i	$-0.686631\pi$
$229$	−16.7458			−1.10660			−0.553298	+	0.832983i	$0.686631\pi$
$230$		0			0
$231$	1.95447	−	0.821690i	0.128595	−	0.0540632i
$232$		0			0
$233$	−4.11183			−0.269375			−0.134687	−	0.990888i	$-0.543003\pi$
$233$	−4.11183			−0.269375			−0.134687	+	0.990888i	$0.543003\pi$
$234$		0			0
$235$	6.30483			0.411282
$236$		0			0
$237$	3.02825			0.196706
$238$		0			0
$239$		−	2.53385i		−	0.163901i	−0.996636	−	0.0819506i	$-0.973885\pi$
$239$		−	2.53385i		−	0.163901i	0.996636	−	0.0819506i	$-0.0261150\pi$
$240$		0			0
$241$			25.9867i			1.67395i	0.547240	+	0.836976i	$0.315679\pi$
$241$			25.9867i			1.67395i	−0.547240	+	0.836976i	$0.684321\pi$
$242$		0			0
$243$			10.5027i			0.673747i
$244$		0			0
$245$	4.89718	−	5.00176i	0.312869	−	0.319551i
$246$		0			0
$247$			3.88341i			0.247095i
$248$		0			0
$249$	2.01064			0.127419
$250$		0			0
$251$		−	11.9340i		−	0.753269i	−0.926362	−	0.376634i	$-0.877081\pi$
$251$		−	11.9340i		−	0.753269i	0.926362	−	0.376634i	$-0.122919\pi$
$252$		0			0
$253$			10.9601i			0.689056i
$254$		0			0
$255$	−3.03716			−0.190194
$256$		0			0
$257$		−	1.44000i		−	0.0898250i	−0.998991	−	0.0449125i	$-0.985699\pi$
$257$		−	1.44000i		−	0.0898250i	0.998991	−	0.0449125i	$-0.0143009\pi$
$258$		0			0
$259$	−11.4169	+	4.79982i	−0.709409	+	0.298247i
$260$		0			0
$261$			13.6726i			0.846310i
$262$		0			0
$263$		−	12.9905i		−	0.801026i	−0.916291	−	0.400513i	$-0.868832\pi$
$263$		−	12.9905i		−	0.801026i	0.916291	−	0.400513i	$-0.131168\pi$
$264$		0			0
$265$			0.637428i			0.0391569i
$266$		0			0
$267$	5.03880			0.308370
$268$		0			0
$269$	15.3509			0.935958			0.467979	−	0.883740i	$-0.344982\pi$
$269$	15.3509			0.935958			0.467979	+	0.883740i	$0.344982\pi$
$270$		0			0
$271$	14.3155			0.869604			0.434802	−	0.900526i	$-0.356818\pi$
$271$	14.3155			0.869604			0.434802	+	0.900526i	$0.356818\pi$
$272$		0			0
$273$	−4.79068	+	2.01407i	−0.289945	+	0.121897i
$274$		0			0
$275$	1.89956			0.114548
$276$		0			0
$277$			3.30303i			0.198460i	0.995065	+	0.0992298i	$0.0316379\pi$
$277$			3.30303i			0.198460i	−0.995065	+	0.0992298i	$0.968362\pi$
$278$		0			0
$279$	−11.4111			−0.683166
$280$		0			0
$281$	2.82096			0.168284			0.0841421	−	0.996454i	$-0.473185\pi$
$281$	2.82096			0.168284			0.0841421	+	0.996454i	$0.473185\pi$
$282$		0			0
$283$			22.2934i			1.32520i	0.748972	+	0.662602i	$0.230548\pi$
$283$			22.2934i			1.32520i	−0.748972	+	0.662602i	$0.769452\pi$
$284$		0			0
$285$	−0.351853			−0.0208420
$286$		0			0
$287$	−14.1914	+	5.96630i	−0.837693	+	0.352179i
$288$		0			0
$289$	−34.8319			−2.04893
$290$		0			0
$291$	4.35719			0.255423
$292$		0			0
$293$	−9.85574			−0.575778			−0.287889	−	0.957664i	$-0.592953\pi$
$293$	−9.85574			−0.575778			−0.287889	+	0.957664i	$0.592953\pi$
$294$		0			0
$295$		−	5.32592i		−	0.310087i
$296$		0			0
$297$			4.66549i			0.270719i
$298$		0			0
$299$		−	26.8647i		−	1.55363i
$300$		0			0
$301$	12.9544	+	30.8134i	0.746681	+	1.77606i
$302$		0			0
$303$		−	6.36070i		−	0.365413i
$304$		0			0
$305$	−7.80404			−0.446858
$306$		0			0
$307$		−	32.9065i		−	1.87808i	−0.343814	−	0.939038i	$-0.611719\pi$
$307$		−	32.9065i		−	1.87808i	0.343814	−	0.939038i	$-0.388281\pi$
$308$		0			0
$309$		−	6.13104i		−	0.348782i
$310$		0			0
$311$	−26.9734			−1.52952			−0.764759	−	0.644316i	$-0.777142\pi$
$311$	−26.9734			−1.52952			−0.764759	+	0.644316i	$0.777142\pi$
$312$		0			0
$313$			20.5471i			1.16139i	0.814121	+	0.580695i	$0.197219\pi$
$313$			20.5471i			1.16139i	−0.814121	+	0.580695i	$0.802781\pi$
$314$		0			0
$315$	2.89366	+	6.88287i	0.163039	+	0.387806i
$316$		0			0
$317$		−	7.66436i		−	0.430473i	−0.976562	−	0.215237i	$-0.930948\pi$
$317$		−	7.66436i		−	0.430473i	0.976562	−	0.215237i	$-0.0690523\pi$
$318$		0			0
$319$			9.20324i			0.515283i
$320$		0			0
$321$		−	4.18073i		−	0.233346i
$322$		0			0
$323$	−6.00469			−0.334110
$324$		0			0
$325$	−4.65608			−0.258273
$326$		0			0
$327$	3.28162			0.181474
$328$		0			0
$329$	6.46486	+	15.3773i	0.356419	+	0.847779i
$330$		0			0
$331$	21.0404			1.15649			0.578243	−	0.815864i	$-0.303738\pi$
$331$	21.0404			1.15649			0.578243	+	0.815864i	$0.303738\pi$
$332$		0			0
$333$		−	13.2100i		−	0.723902i
$334$		0			0
$335$	4.64973			0.254042
$336$		0			0
$337$	−25.8596			−1.40866			−0.704330	−	0.709873i	$-0.748752\pi$
$337$	−25.8596			−1.40866			−0.704330	+	0.709873i	$0.748752\pi$
$338$		0			0
$339$			2.00477i			0.108884i
$340$		0			0
$341$	−7.68103			−0.415951
$342$		0			0
$343$	17.2206	+	6.81538i	0.929827	+	0.367996i
$344$		0			0
$345$	2.43406			0.131045
$346$		0			0
$347$	−23.3946			−1.25589			−0.627944	−	0.778259i	$-0.716103\pi$
$347$	−23.3946			−1.25589			−0.627944	+	0.778259i	$0.716103\pi$
$348$		0			0
$349$	24.9021			1.33298			0.666489	−	0.745515i	$-0.267796\pi$
$349$	24.9021			1.33298			0.666489	+	0.745515i	$0.267796\pi$
$350$		0			0
$351$		−	11.4357i		−	0.610395i
$352$		0			0
$353$			17.9466i			0.955201i	0.878577	+	0.477600i	$0.158493\pi$
$353$			17.9466i			0.955201i	−0.878577	+	0.477600i	$0.841507\pi$
$354$		0			0
$355$			6.42998i			0.341268i
$356$		0			0
$357$	−3.11425	−	7.40755i	−0.164823	−	0.392049i
$358$		0			0
$359$			32.1939i			1.69913i	0.527483	+	0.849565i	$0.323136\pi$
$359$			32.1939i			1.69913i	−0.527483	+	0.849565i	$0.676864\pi$
$360$		0			0
$361$	18.3044			0.963387
$362$		0			0
$363$		−	3.11825i		−	0.163666i
$364$		0			0
$365$		−	0.227709i		−	0.0119188i
$366$		0			0
$367$	−8.96776			−0.468113			−0.234057	−	0.972223i	$-0.575200\pi$
$367$	−8.96776			−0.468113			−0.234057	+	0.972223i	$0.575200\pi$
$368$		0			0
$369$		−	16.4203i		−	0.854807i
$370$		0			0
$371$	−1.55467	+	0.653607i	−0.0807144	+	0.0339336i
$372$		0			0
$373$		−	19.2075i		−	0.994524i	−0.867600	−	0.497262i	$-0.834339\pi$
$373$		−	19.2075i		−	0.994524i	0.867600	−	0.497262i	$-0.165661\pi$
$374$		0			0
$375$		−	0.421861i		−	0.0217848i
$376$		0			0
$377$		−	22.5584i		−	1.16182i
$378$		0			0
$379$	−21.5548			−1.10719			−0.553597	−	0.832784i	$-0.686745\pi$
$379$	−21.5548			−1.10719			−0.553597	+	0.832784i	$0.686745\pi$
$380$		0			0
$381$	−2.40559			−0.123242
$382$		0			0
$383$	23.1089			1.18081			0.590406	−	0.807107i	$-0.298968\pi$
$383$	23.1089			1.18081			0.590406	+	0.807107i	$0.298968\pi$
$384$		0			0
$385$	1.94778	+	4.63298i	0.0992679	+	0.236118i
$386$		0			0
$387$	−35.6529			−1.81234
$388$		0			0
$389$		−	14.3793i		−	0.729058i	−0.931192	−	0.364529i	$-0.881230\pi$
$389$		−	14.3793i		−	0.729058i	0.931192	−	0.364529i	$-0.118770\pi$
$390$		0			0
$391$	41.5394			2.10074
$392$		0			0
$393$	−7.15725			−0.361035
$394$		0			0
$395$			7.17831i			0.361180i
$396$		0			0
$397$	17.4913			0.877864			0.438932	−	0.898520i	$-0.355357\pi$
$397$	17.4913			0.877864			0.438932	+	0.898520i	$0.355357\pi$
$398$		0			0
$399$	−0.360784	−	0.858160i	−0.0180618	−	0.0429617i
$400$		0			0
$401$	34.9119			1.74342			0.871709	−	0.490024i	$-0.163012\pi$
$401$	34.9119			1.74342			0.871709	+	0.490024i	$0.163012\pi$
$402$		0			0
$403$	18.8273			0.937852
$404$		0			0
$405$	−7.42998			−0.369198
$406$		0			0
$407$		−	8.89186i		−	0.440753i
$408$		0			0
$409$			34.0866i			1.68547i	0.538327	+	0.842736i	$0.319057\pi$
$409$			34.0866i			1.68547i	−0.538327	+	0.842736i	$0.680943\pi$
$410$		0			0
$411$		−	2.85160i		−	0.140659i
$412$		0			0
$413$	12.9898	−	5.46110i	0.639185	−	0.268723i
$414$		0			0
$415$			4.76612i			0.233960i
$416$		0			0
$417$	−2.26538			−0.110936
$418$		0			0
$419$		−	25.3178i		−	1.23685i	−0.785842	−	0.618427i	$-0.787770\pi$
$419$		−	25.3178i		−	1.23685i	0.785842	−	0.618427i	$-0.212230\pi$
$420$		0			0
$421$			22.7943i			1.11093i	0.831541	+	0.555463i	$0.187459\pi$
$421$			22.7943i			1.11093i	−0.831541	+	0.555463i	$0.812541\pi$
$422$		0			0
$423$	−17.7924			−0.865098
$424$		0			0
$425$		−	7.19944i		−	0.349224i
$426$		0			0
$427$	−8.00212	−	19.0338i	−0.387250	−	0.921112i
$428$		0			0
$429$		−	3.73115i		−	0.180142i
$430$		0			0
$431$			1.06648i			0.0513707i	0.999670	+	0.0256854i	$0.00817680\pi$
$431$			1.06648i			0.0513707i	−0.999670	+	0.0256854i	$0.991823\pi$
$432$		0			0
$433$		−	11.8877i		−	0.571286i	−0.958336	−	0.285643i	$-0.907793\pi$
$433$		−	11.8877i		−	0.571286i	0.958336	−	0.285643i	$-0.0922072\pi$
$434$		0			0
$435$	2.04389			0.0979968
$436$		0			0
$437$	4.81231			0.230204
$438$		0			0
$439$	40.4015			1.92826			0.964128	−	0.265437i	$-0.0855162\pi$
$439$	40.4015			1.92826			0.964128	+	0.265437i	$0.0855162\pi$
$440$		0			0
$441$	−13.8200	+	14.1151i	−0.658096	+	0.672149i
$442$		0			0
$443$	−5.09808			−0.242217			−0.121109	−	0.992639i	$-0.538645\pi$
$443$	−5.09808			−0.242217			−0.121109	+	0.992639i	$0.538645\pi$
$444$		0			0
$445$			11.9442i			0.566211i
$446$		0			0
$447$	4.90454			0.231977
$448$		0			0
$449$	−15.3965			−0.726608			−0.363304	−	0.931671i	$-0.618351\pi$
$449$	−15.3965			−0.726608			−0.363304	+	0.931671i	$0.618351\pi$
$450$		0			0
$451$		−	11.0528i		−	0.520456i
$452$		0			0
$453$	2.23170			0.104854
$454$		0			0
$455$	−4.77427	−	11.3561i	−0.223821	−	0.532380i
$456$		0			0
$457$	15.3274			0.716987			0.358493	−	0.933532i	$-0.383291\pi$
$457$	15.3274			0.716987			0.358493	+	0.933532i	$0.383291\pi$
$458$		0			0
$459$	17.6824			0.825346
$460$		0			0
$461$	−11.2470			−0.523824			−0.261912	−	0.965092i	$-0.584353\pi$
$461$	−11.2470			−0.523824			−0.261912	+	0.965092i	$0.584353\pi$
$462$		0			0
$463$			30.4061i			1.41309i	0.707667	+	0.706546i	$0.249748\pi$
$463$			30.4061i			1.41309i	−0.707667	+	0.706546i	$0.750252\pi$
$464$		0			0
$465$			1.70583i			0.0791059i
$466$		0			0
$467$		−	7.30584i		−	0.338074i	−0.985610	−	0.169037i	$-0.945934\pi$
$467$		−	7.30584i		−	0.338074i	0.985610	−	0.169037i	$-0.0540658\pi$
$468$		0			0
$469$	4.76775	+	11.3406i	0.220154	+	0.523659i
$470$		0			0
$471$			9.24372i			0.425928i
$472$		0			0
$473$	−23.9986			−1.10346
$474$		0			0
$475$		−	0.834050i		−	0.0382688i
$476$		0			0
$477$		−	1.79884i		−	0.0823633i
$478$		0			0
$479$	−5.42872			−0.248045			−0.124022	−	0.992279i	$-0.539579\pi$
$479$	−5.42872			−0.248045			−0.124022	+	0.992279i	$0.539579\pi$
$480$		0			0
$481$			21.7952i			0.993774i
$482$		0			0
$483$	2.49584	+	5.93660i	0.113565	+	0.270124i
$484$		0			0
$485$			10.3285i			0.468994i
$486$		0			0
$487$			34.7819i			1.57612i	0.615599	+	0.788060i	$0.288914\pi$
$487$			34.7819i			1.57612i	−0.615599	+	0.788060i	$0.711086\pi$
$488$		0			0
$489$			0.344705i			0.0155881i
$490$		0			0
$491$	36.1387			1.63092			0.815458	−	0.578816i	$-0.196485\pi$
$491$	36.1387			1.63092			0.815458	+	0.578816i	$0.196485\pi$
$492$		0			0
$493$	34.8808			1.57095
$494$		0			0
$495$	−5.36063			−0.240942
$496$		0			0
$497$	−15.6825	+	6.59318i	−0.703458	+	0.295745i
$498$		0			0
$499$	25.2593			1.13076			0.565381	−	0.824830i	$-0.308729\pi$
$499$	25.2593			1.13076			0.565381	+	0.824830i	$0.308729\pi$
$500$		0			0
$501$			5.72080i			0.255586i
$502$		0			0
$503$	38.0796			1.69789			0.848943	−	0.528485i	$-0.177240\pi$
$503$	38.0796			1.69789			0.848943	+	0.528485i	$0.177240\pi$
$504$		0			0
$505$	15.0777			0.670950
$506$		0			0
$507$			3.66138i			0.162607i
$508$		0			0
$509$	−4.52937			−0.200761			−0.100380	−	0.994949i	$-0.532006\pi$
$509$	−4.52937			−0.200761			−0.100380	+	0.994949i	$0.532006\pi$
$510$		0			0
$511$	0.555376	−	0.233489i	0.0245684	−	0.0103289i
$512$		0			0
$513$	2.04850			0.0904434
$514$		0			0
$515$	14.5333			0.640415
$516$		0			0
$517$	−11.9764			−0.526722
$518$		0			0
$519$			2.32015i			0.101843i
$520$		0			0
$521$			24.4806i			1.07251i	0.844055	+	0.536257i	$0.180162\pi$
$521$			24.4806i			1.07251i	−0.844055	+	0.536257i	$0.819838\pi$
$522$		0			0
$523$		−	24.1816i		−	1.05739i	−0.848812	−	0.528694i	$-0.822682\pi$
$523$		−	24.1816i		−	1.05739i	0.848812	−	0.528694i	$-0.177318\pi$
$524$		0			0
$525$	1.02891	−	0.432568i	0.0449052	−	0.0188788i
$526$		0			0
$527$			29.1115i			1.26812i
$528$		0			0
$529$	−10.2907			−0.447422
$530$		0			0
$531$			15.0299i			0.652243i
$532$		0			0
$533$			27.0919i			1.17348i
$534$		0			0
$535$	9.91022			0.428456
$536$		0			0
$537$		−	10.1198i		−	0.436704i
$538$		0			0
$539$	−9.30250	+	9.50115i	−0.400687	+	0.409243i
$540$		0			0
$541$			3.51920i			0.151302i	0.997134	+	0.0756512i	$0.0241035\pi$
$541$			3.51920i			0.151302i	−0.997134	+	0.0756512i	$0.975896\pi$
$542$		0			0
$543$		−	8.82438i		−	0.378690i
$544$		0			0
$545$			7.77892i			0.333212i
$546$		0			0
$547$	−15.3167			−0.654893			−0.327446	−	0.944870i	$-0.606188\pi$
$547$	−15.3167			−0.654893			−0.327446	+	0.944870i	$0.606188\pi$
$548$		0			0
$549$	22.0233			0.939929
$550$		0			0
$551$	4.04092			0.172149
$552$		0			0
$553$	−17.5077	+	7.36051i	−0.744503	+	0.313001i
$554$		0			0
$555$	−1.97473			−0.0838228
$556$		0			0
$557$		−	19.0219i		−	0.805985i	−0.915203	−	0.402993i	$-0.867970\pi$
$557$		−	19.0219i		−	0.805985i	0.915203	−	0.402993i	$-0.132030\pi$
$558$		0			0
$559$	58.8239			2.48798
$560$		0			0
$561$	5.76927			0.243579
$562$		0			0
$563$		−	17.8461i		−	0.752125i	−0.926594	−	0.376063i	$-0.877278\pi$
$563$		−	17.8461i		−	0.752125i	0.926594	−	0.376063i	$-0.122722\pi$
$564$		0			0
$565$	−4.75220			−0.199927
$566$		0			0
$567$	−7.61857	−	18.1215i	−0.319950	−	0.761032i
$568$		0			0
$569$	3.78552			0.158697			0.0793486	−	0.996847i	$-0.474716\pi$
$569$	3.78552			0.158697			0.0793486	+	0.996847i	$0.474716\pi$
$570$		0			0
$571$	9.65474			0.404038			0.202019	−	0.979382i	$-0.435250\pi$
$571$	9.65474			0.404038			0.202019	+	0.979382i	$0.435250\pi$
$572$		0			0
$573$	7.14387			0.298440
$574$		0			0
$575$			5.76981i			0.240618i
$576$		0			0
$577$			38.2814i			1.59368i	0.604194	+	0.796838i	$0.293495\pi$
$577$			38.2814i			1.59368i	−0.604194	+	0.796838i	$0.706505\pi$
$578$		0			0
$579$			2.11086i			0.0877243i
$580$		0			0
$581$	−11.6244	+	4.88710i	−0.482263	+	0.202751i
$582$		0			0
$583$		−	1.21083i		−	0.0501476i
$584$		0			0
$585$	13.1396			0.543257
$586$		0			0
$587$			5.95450i			0.245768i	0.992421	+	0.122884i	$0.0392144\pi$
$587$			5.95450i			0.245768i	−0.992421	+	0.122884i	$0.960786\pi$
$588$		0			0
$589$			3.37255i			0.138963i
$590$		0			0
$591$	11.4691			0.471775
$592$		0			0
$593$		−	5.82852i		−	0.239349i	−0.992813	−	0.119674i	$-0.961815\pi$
$593$		−	5.82852i		−	0.239349i	0.992813	−	0.119674i	$-0.0381850\pi$
$594$		0			0
$595$	17.5592	−	7.38218i	0.719858	−	0.302640i
$596$		0			0
$597$		−	0.0870491i		−	0.00356268i
$598$		0			0
$599$		−	17.5376i		−	0.716568i	−0.933613	−	0.358284i	$-0.883362\pi$
$599$		−	17.5376i		−	0.716568i	0.933613	−	0.358284i	$-0.116638\pi$
$600$		0			0
$601$		−	4.75204i		−	0.193840i	−0.995292	−	0.0969199i	$-0.969101\pi$
$601$		−	4.75204i		−	0.193840i	0.995292	−	0.0969199i	$-0.0308991\pi$
$602$		0			0
$603$	−13.1217			−0.534357
$604$		0			0
$605$	7.39167			0.300514
$606$		0			0
$607$	16.2743			0.660555			0.330277	−	0.943884i	$-0.392858\pi$
$607$	16.2743			0.660555			0.330277	+	0.943884i	$0.392858\pi$
$608$		0			0
$609$	2.09576	+	4.98498i	0.0849246	+	0.202002i
$610$		0			0
$611$	29.3558			1.18761
$612$		0			0
$613$		−	31.6544i		−	1.27851i	−0.768996	−	0.639254i	$-0.779243\pi$
$613$		−	31.6544i		−	1.27851i	0.768996	−	0.639254i	$-0.220757\pi$
$614$		0			0
$615$	−2.45464			−0.0989807
$616$		0			0
$617$	−29.6932			−1.19540			−0.597701	−	0.801719i	$-0.703919\pi$
$617$	−29.6932			−1.19540			−0.597701	+	0.801719i	$0.703919\pi$
$618$		0			0
$619$		−	16.1041i		−	0.647277i	−0.946181	−	0.323638i	$-0.895094\pi$
$619$		−	16.1041i		−	0.647277i	0.946181	−	0.323638i	$-0.104906\pi$
$620$		0			0
$621$	−14.1712			−0.568669
$622$		0			0
$623$	−29.1317	+	12.2474i	−1.16714	+	0.490682i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	0.668366			0.0266920
$628$		0			0
$629$	−33.7006			−1.34373
$630$		0			0
$631$		−	6.78949i		−	0.270285i	−0.990826	−	0.135143i	$-0.956851\pi$
$631$		−	6.78949i		−	0.270285i	0.990826	−	0.135143i	$-0.0431493\pi$
$632$		0			0
$633$			9.60628i			0.381815i
$634$		0			0
$635$		−	5.70233i		−	0.226290i
$636$		0			0
$637$	22.8017	−	23.2886i	0.903436	−	0.922728i
$638$		0			0
$639$		−	18.1456i		−	0.717829i
$640$		0			0
$641$	−10.2906			−0.406456			−0.203228	−	0.979131i	$-0.565143\pi$
$641$	−10.2906			−0.406456			−0.203228	+	0.979131i	$0.565143\pi$
$642$		0			0
$643$		−	44.1456i		−	1.74093i	−0.492230	−	0.870465i	$-0.663818\pi$
$643$		−	44.1456i		−	1.74093i	0.492230	−	0.870465i	$-0.336182\pi$
$644$		0			0
$645$			5.32969i			0.209856i
$646$		0			0
$647$	−6.57865			−0.258634			−0.129317	−	0.991603i	$-0.541278\pi$
$647$	−6.57865			−0.258634			−0.129317	+	0.991603i	$0.541278\pi$
$648$		0			0
$649$			10.1169i			0.397123i
$650$		0			0
$651$	−4.16047	+	1.74913i	−0.163062	+	0.0685536i
$652$		0			0
$653$			31.9386i			1.24985i	0.780683	+	0.624927i	$0.214871\pi$
$653$			31.9386i			1.24985i	−0.780683	+	0.624927i	$0.785129\pi$
$654$		0			0
$655$		−	16.9659i		−	0.662913i
$656$		0			0
$657$			0.642603i			0.0250703i
$658$		0			0
$659$	−37.4842			−1.46018			−0.730088	−	0.683353i	$-0.760521\pi$
$659$	−37.4842			−1.46018			−0.730088	+	0.683353i	$0.760521\pi$
$660$		0			0
$661$	19.7477			0.768098			0.384049	−	0.923313i	$-0.374529\pi$
$661$	19.7477			0.768098			0.384049	+	0.923313i	$0.374529\pi$
$662$		0			0
$663$	−14.1413			−0.549201
$664$		0			0
$665$	2.03423	−	0.855220i	0.0788839	−	0.0331640i
$666$		0			0
$667$	−27.9543			−1.08240
$668$		0			0
$669$			6.28604i			0.243032i
$670$		0			0
$671$	14.8242			0.572284
$672$		0			0
$673$	38.1738			1.47149			0.735745	−	0.677258i	$-0.236832\pi$
$673$	38.1738			1.47149			0.735745	+	0.677258i	$0.236832\pi$
$674$		0			0
$675$			2.45609i			0.0945348i
$676$		0			0
$677$	29.5570			1.13597			0.567984	−	0.823040i	$-0.307724\pi$
$677$	29.5570			1.13597			0.567984	+	0.823040i	$0.307724\pi$
$678$		0			0
$679$	−25.1910	+	10.5907i	−0.966741	+	0.406433i
$680$		0			0
$681$	−6.76240			−0.259136
$682$		0			0
$683$	−5.95463			−0.227848			−0.113924	−	0.993489i	$-0.536342\pi$
$683$	−5.95463			−0.227848			−0.113924	+	0.993489i	$0.536342\pi$
$684$		0			0
$685$	6.75958			0.258270
$686$		0			0
$687$		−	7.06441i		−	0.269524i
$688$		0			0
$689$			2.96792i			0.113069i
$690$		0			0
$691$			9.60671i			0.365456i	0.983163	+	0.182728i	$0.0584928\pi$
$691$			9.60671i			0.365456i	−0.983163	+	0.182728i	$0.941507\pi$
$692$		0			0
$693$	−5.49669	−	13.0744i	−0.208802	−	0.496656i
$694$		0			0
$695$		−	5.36998i		−	0.203695i
$696$		0			0
$697$	−41.8907			−1.58672
$698$		0			0
$699$		−	1.73462i		−	0.0656092i
$700$		0			0
$701$			12.6155i			0.476481i	0.971206	+	0.238240i	$0.0765706\pi$
$701$			12.6155i			0.476481i	−0.971206	+	0.238240i	$0.923429\pi$
$702$		0			0
$703$	−3.90420			−0.147250
$704$		0			0
$705$			2.65976i			0.100172i
$706$		0			0
$707$	15.4604	+	36.7742i	0.581450	+	1.38304i
$708$		0			0
$709$		−	18.4188i		−	0.691731i	−0.938284	−	0.345866i	$-0.887585\pi$
$709$		−	18.4188i		−	0.691731i	0.938284	−	0.345866i	$-0.112415\pi$
$710$		0			0
$711$		−	20.2574i		−	0.759713i
$712$		0			0
$713$		−	23.3307i		−	0.873742i
$714$		0			0
$715$	8.84452			0.330766
$716$		0			0
$717$	1.06893			0.0399200
$718$		0			0
$719$	−40.9006			−1.52534			−0.762668	−	0.646791i	$-0.776111\pi$
$719$	−40.9006			−1.52534			−0.762668	+	0.646791i	$0.776111\pi$
$720$		0			0
$721$	14.9022	+	35.4464i	0.554988	+	1.32009i
$722$		0			0
$723$	−10.9628			−0.407710
$724$		0			0
$725$			4.84493i			0.179936i
$726$		0			0
$727$	21.0462			0.780559			0.390279	−	0.920696i	$-0.372378\pi$
$727$	21.0462			0.780559			0.390279	+	0.920696i	$0.372378\pi$
$728$		0			0
$729$	17.8593			0.661454
$730$		0			0
$731$			90.9560i			3.36413i
$732$		0			0
$733$	9.67124			0.357215			0.178608	−	0.983920i	$-0.442841\pi$
$733$	9.67124			0.357215			0.178608	+	0.983920i	$0.442841\pi$
$734$		0			0
$735$	2.11005	+	2.06593i	0.0778302	+	0.0762029i
$736$		0			0
$737$	−8.83245			−0.325347
$738$		0			0
$739$	29.0213			1.06757			0.533783	−	0.845621i	$-0.320770\pi$
$739$	29.0213			1.06757			0.533783	+	0.845621i	$0.320770\pi$
$740$		0			0
$741$	−1.63826			−0.0601828
$742$		0			0
$743$		−	9.59397i		−	0.351969i	−0.984393	−	0.175984i	$-0.943689\pi$
$743$		−	9.59397i		−	0.351969i	0.984393	−	0.175984i	$-0.0563108\pi$
$744$		0			0
$745$			11.6260i			0.425942i
$746$		0			0
$747$		−	13.4502i		−	0.492115i
$748$		0			0
$749$	10.1618	+	24.1708i	0.371303	+	0.883180i
$750$		0			0
$751$			0.422026i			0.0153999i	0.999970	+	0.00769997i	$0.00245100\pi$
$751$			0.422026i			0.0153999i	−0.999970	+	0.00769997i	$0.997549\pi$
$752$		0			0
$753$	5.03449			0.183467
$754$		0			0
$755$			5.29014i			0.192528i
$756$		0			0
$757$		−	39.3511i		−	1.43024i	−0.699001	−	0.715121i	$-0.746372\pi$
$757$		−	39.3511i		−	1.43024i	0.699001	−	0.715121i	$-0.253628\pi$
$758$		0			0
$759$	−4.62364			−0.167827
$760$		0			0
$761$		−	18.8389i		−	0.682911i	−0.939898	−	0.341455i	$-0.889080\pi$
$761$		−	18.8389i		−	0.682911i	0.939898	−	0.341455i	$-0.110920\pi$
$762$		0			0
$763$	−18.9726	+	7.97636i	−0.686853	+	0.288764i
$764$		0			0
$765$			20.3171i			0.734565i
$766$		0			0
$767$		−	24.7979i		−	0.895401i
$768$		0			0
$769$		−	51.7564i		−	1.86638i	−0.359380	−	0.933191i	$-0.617012\pi$
$769$		−	51.7564i		−	1.86638i	0.359380	−	0.933191i	$-0.382988\pi$
$770$		0			0
$771$	0.607481			0.0218779
$772$		0			0
$773$	−33.1875			−1.19367			−0.596836	−	0.802363i	$-0.703576\pi$
$773$	−33.1875			−1.19367			−0.596836	+	0.802363i	$0.703576\pi$
$774$		0			0
$775$	−4.04358			−0.145250
$776$		0			0
$777$	−2.02486	−	4.81632i	−0.0726413	−	0.172785i
$778$		0			0
$779$	−4.85301			−0.173877
$780$		0			0
$781$		−	12.2141i		−	0.437056i
$782$		0			0
$783$	−11.8996			−0.425256
$784$		0			0
$785$	−21.9118			−0.782065
$786$		0			0
$787$		−	30.7062i		−	1.09456i	−0.836951	−	0.547278i	$-0.815664\pi$
$787$		−	30.7062i		−	1.09456i	0.836951	−	0.547278i	$-0.184336\pi$
$788$		0			0
$789$	5.48016			0.195099
$790$		0			0
$791$	−4.87283	−	11.5905i	−0.173258	−	0.412111i
$792$		0			0
$793$	−36.3362			−1.29034
$794$		0			0
$795$	−0.268906			−0.00953710
$796$		0			0
$797$	−1.18345			−0.0419198			−0.0209599	−	0.999780i	$-0.506672\pi$
$797$	−1.18345			−0.0419198			−0.0209599	+	0.999780i	$0.506672\pi$
$798$		0			0
$799$			45.3912i			1.60583i
$800$		0			0
$801$		−	33.7070i		−	1.19098i
$802$		0			0
$803$			0.432547i			0.0152643i
$804$		0			0
$805$	−14.0724	+	5.91626i	−0.495987	+	0.208521i
$806$		0			0
$807$			6.47592i			0.227963i
$808$		0			0
$809$	14.5452			0.511383			0.255691	−	0.966758i	$-0.417697\pi$
$809$	14.5452			0.511383			0.255691	+	0.966758i	$0.417697\pi$
$810$		0			0
$811$		−	38.4140i		−	1.34890i	−0.738321	−	0.674449i	$-0.764381\pi$
$811$		−	38.4140i		−	1.34890i	0.738321	−	0.674449i	$-0.235619\pi$
$812$		0			0
$813$			6.03914i			0.211802i
$814$		0			0
$815$	−0.817106			−0.0286220
$816$		0			0
$817$			10.5372i			0.368650i
$818$		0			0
$819$	13.4731	+	32.0472i	0.470790	+	1.11982i
$820$		0			0
$821$			25.0218i			0.873265i	0.899640	+	0.436633i	$0.143829\pi$
$821$			25.0218i			0.873265i	−0.899640	+	0.436633i	$0.856171\pi$
$822$		0			0
$823$			29.1029i			1.01446i	0.861810	+	0.507232i	$0.169331\pi$
$823$			29.1029i			1.01446i	−0.861810	+	0.507232i	$0.830669\pi$
$824$		0			0
$825$			0.801350i			0.0278994i
$826$		0			0
$827$	36.9359			1.28439			0.642194	−	0.766542i	$-0.278024\pi$
$827$	36.9359			1.28439			0.642194	+	0.766542i	$0.278024\pi$
$828$		0			0
$829$	−5.85695			−0.203420			−0.101710	−	0.994814i	$-0.532431\pi$
$829$	−5.85695			−0.203420			−0.101710	+	0.994814i	$0.532431\pi$
$830$		0			0
$831$	−1.39342			−0.0483371
$832$		0			0
$833$	36.0099	+	35.2570i	1.24767	+	1.22158i
$834$		0			0
$835$	−13.5609			−0.469294
$836$		0			0
$837$		−	9.93139i		−	0.343279i
$838$		0			0
$839$	−7.99477			−0.276010			−0.138005	−	0.990432i	$-0.544069\pi$
$839$	−7.99477			−0.276010			−0.138005	+	0.990432i	$0.544069\pi$
$840$		0			0
$841$	5.52664			0.190574
$842$		0			0
$843$			1.19005i			0.0409875i
$844$		0			0
$845$	−8.67912			−0.298571
$846$		0			0
$847$	7.57929	+	18.0281i	0.260427	+	0.619452i
$848$		0			0
$849$	−9.40470			−0.322768
$850$		0			0
$851$	27.0085			0.925841
$852$		0			0
$853$	−29.4916			−1.00977			−0.504886	−	0.863186i	$-0.668465\pi$
$853$	−29.4916			−1.00977			−0.504886	+	0.863186i	$0.668465\pi$
$854$		0			0
$855$			2.35372i			0.0804954i
$856$		0			0
$857$		−	4.42236i		−	0.151065i	−0.997143	−	0.0755325i	$-0.975934\pi$
$857$		−	4.42236i		−	0.151065i	0.997143	−	0.0755325i	$-0.0240657\pi$
$858$		0			0
$859$			29.7417i			1.01477i	0.861718	+	0.507387i	$0.169389\pi$
$859$			29.7417i			1.01477i	−0.861718	+	0.507387i	$0.830611\pi$
$860$		0			0
$861$	−2.51694	−	5.98680i	−0.0857773	−	0.204030i
$862$		0			0
$863$		−	11.3281i		−	0.385613i	−0.981237	−	0.192806i	$-0.938241\pi$
$863$		−	11.3281i		−	0.385613i	0.981237	−	0.192806i	$-0.0617589\pi$
$864$		0			0
$865$	−5.49980			−0.186999
$866$		0			0
$867$		−	14.6942i		−	0.499041i
$868$		0			0
$869$		−	13.6356i		−	0.462557i
$870$		0			0
$871$	21.6495			0.733566
$872$		0			0
$873$		−	29.1474i		−	0.986491i
$874$		0			0
$875$	1.02538	+	2.43897i	0.0346642	+	0.0824523i
$876$		0			0
$877$			42.1172i			1.42220i	0.703093	+	0.711098i	$0.251802\pi$
$877$			42.1172i			1.42220i	−0.703093	+	0.711098i	$0.748198\pi$
$878$		0			0
$879$		−	4.15775i		−	0.140237i
$880$		0			0
$881$		−	38.9740i		−	1.31307i	−0.754296	−	0.656534i	$-0.772022\pi$
$881$		−	38.9740i		−	1.31307i	0.754296	−	0.656534i	$-0.227978\pi$
$882$		0			0
$883$	−13.2610			−0.446268			−0.223134	−	0.974788i	$-0.571629\pi$
$883$	−13.2610			−0.446268			−0.223134	+	0.974788i	$0.571629\pi$
$884$		0			0
$885$	2.24679			0.0755252
$886$		0			0
$887$	−11.0616			−0.371411			−0.185706	−	0.982605i	$-0.559457\pi$
$887$	−11.0616			−0.371411			−0.185706	+	0.982605i	$0.559457\pi$
$888$		0			0
$889$	13.9078	−	5.84706i	0.466453	−	0.196104i
$890$		0			0
$891$	14.1137			0.472826
$892$		0			0
$893$			5.25854i			0.175970i
$894$		0			0
$895$	23.9886			0.801851
$896$		0			0
$897$	11.3332			0.378403
$898$		0			0
$899$		−	19.5909i		−	0.653392i
$900$		0			0
$901$	−4.58912			−0.152886
$902$		0			0
$903$	−12.9990	+	5.46497i	−0.432578	+	0.181863i
$904$		0			0
$905$	20.9178			0.695330
$906$		0			0
$907$	−18.0808			−0.600362			−0.300181	−	0.953882i	$-0.597047\pi$
$907$	−18.0808			−0.600362			−0.300181	+	0.953882i	$0.597047\pi$
$908$		0			0
$909$	−42.5499			−1.41129
$910$		0			0
$911$		−	32.9387i		−	1.09131i	−0.838011	−	0.545654i	$-0.816281\pi$
$911$		−	32.9387i		−	1.09131i	0.838011	−	0.545654i	$-0.183719\pi$
$912$		0			0
$913$		−	9.05354i		−	0.299628i
$914$		0			0
$915$		−	3.29221i		−	0.108837i
$916$		0			0
$917$	41.3794	−	17.3965i	1.36647	−	0.574484i
$918$		0			0
$919$		−	49.4808i		−	1.63222i	−0.577896	−	0.816111i	$-0.696126\pi$
$919$		−	49.4808i		−	1.63222i	0.577896	−	0.816111i	$-0.303874\pi$
$920$		0			0
$921$	13.8820			0.457426
$922$		0			0
$923$			29.9385i			0.985438i
$924$		0			0
$925$		−	4.68101i		−	0.153911i
$926$		0			0
$927$	−41.0135			−1.34706
$928$		0			0
$929$		−	21.0077i		−	0.689241i	−0.938742	−	0.344621i	$-0.888008\pi$
$929$		−	21.0077i		−	0.689241i	0.938742	−	0.344621i	$-0.111992\pi$
$930$		0			0
$931$	4.17172	+	4.08449i	0.136723	+	0.133864i
$932$		0			0
$933$		−	11.3790i		−	0.372532i
$934$		0			0
$935$			13.6758i			0.447245i
$936$		0			0
$937$		−	9.45821i		−	0.308986i	−0.987994	−	0.154493i	$-0.950626\pi$
$937$		−	9.45821i		−	0.308986i	0.987994	−	0.154493i	$-0.0493744\pi$
$938$		0			0
$939$	−8.66801			−0.282870
$940$		0			0
$941$	29.2269			0.952771			0.476385	−	0.879237i	$-0.341947\pi$
$941$	29.2269			0.952771			0.476385	+	0.879237i	$0.341947\pi$
$942$		0			0
$943$	33.5723			1.09326
$944$		0			0
$945$	−5.99033	+	2.51843i	−0.194865	+	0.0819244i
$946$		0			0
$947$	−28.7292			−0.933573			−0.466787	−	0.884370i	$-0.654588\pi$
$947$	−28.7292			−0.933573			−0.466787	+	0.884370i	$0.654588\pi$
$948$		0			0
$949$		−	1.06023i		−	0.0344166i
$950$		0			0
$951$	3.23329			0.104847
$952$		0			0
$953$	22.8837			0.741276			0.370638	−	0.928777i	$-0.379139\pi$
$953$	22.8837			0.741276			0.370638	+	0.928777i	$0.379139\pi$
$954$		0			0
$955$			16.9342i			0.547978i
$956$		0			0
$957$	−3.88249			−0.125503
$958$		0			0
$959$	6.93116	+	16.4864i	0.223819	+	0.532375i
$960$		0			0
$961$	−14.6494			−0.472563
$962$		0			0
$963$	−27.9670			−0.901223
$964$		0			0
$965$	−5.00369			−0.161074
$966$		0			0
$967$			23.1050i			0.743006i	0.928432	+	0.371503i	$0.121157\pi$
$967$			23.1050i			0.743006i	−0.928432	+	0.371503i	$0.878843\pi$
$968$		0			0
$969$		−	2.53314i		−	0.0813762i
$970$		0			0
$971$		−	55.3373i		−	1.77586i	−0.459981	−	0.887929i	$-0.652144\pi$
$971$		−	55.3373i		−	1.77586i	0.459981	−	0.887929i	$-0.347856\pi$
$972$		0			0
$973$	13.0972	−	5.50628i	0.419878	−	0.176523i
$974$		0			0
$975$		−	1.96422i		−	0.0629053i
$976$		0			0
$977$	56.5869			1.81038			0.905188	−	0.425011i	$-0.139730\pi$
$977$	56.5869			1.81038			0.905188	+	0.425011i	$0.139730\pi$
$978$		0			0
$979$		−	22.6888i		−	0.725137i
$980$		0			0
$981$		−	21.9524i		−	0.700885i
$982$		0			0
$983$	−33.1612			−1.05768			−0.528839	−	0.848722i	$-0.677373\pi$
$983$	−33.1612			−1.05768			−0.528839	+	0.848722i	$0.677373\pi$
$984$		0			0
$985$			27.1869i			0.866246i
$986$		0			0
$987$	−6.48708	+	2.72727i	−0.206486	+	0.0868100i
$988$		0			0
$989$		−	72.8944i		−	2.31791i
$990$		0			0
$991$			0.564256i			0.0179242i	0.999960	+	0.00896210i	$0.00285276\pi$
$991$			0.564256i			0.0179242i	−0.999960	+	0.00896210i	$0.997147\pi$
$992$		0			0
$993$			8.87613i			0.281675i
$994$		0			0
$995$	0.206346			0.00654160
$996$		0			0
$997$	−16.2119			−0.513437			−0.256719	−	0.966486i	$-0.582641\pi$
$997$	−16.2119			−0.513437			−0.256719	+	0.966486i	$0.582641\pi$
$998$		0			0
$999$	11.4970			0.363748

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.h.e.671.12	yes	24
4.3	odd	2	inner	2240.2.h.e.671.13	yes	24
7.6	odd	2		2240.2.h.f.671.13	yes	24
8.3	odd	2		2240.2.h.f.671.14	yes	24
8.5	even	2		2240.2.h.f.671.11	yes	24
28.27	even	2		2240.2.h.f.671.12	yes	24
56.13	odd	2	inner	2240.2.h.e.671.14	yes	24
56.27	even	2	inner	2240.2.h.e.671.11	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.h.e.671.11	✓	24	56.27	even	2	inner
2240.2.h.e.671.12	yes	24	1.1	even	1	trivial
2240.2.h.e.671.13	yes	24	4.3	odd	2	inner
2240.2.h.e.671.14	yes	24	56.13	odd	2	inner
2240.2.h.f.671.11	yes	24	8.5	even	2
2240.2.h.f.671.12	yes	24	28.27	even	2
2240.2.h.f.671.13	yes	24	7.6	odd	2
2240.2.h.f.671.14	yes	24	8.3	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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.421861i q^{3} -1.00000 q^{5} +(-1.02538 - 2.43897i) q^{7} +2.82203 q^{9} +O(q^{10})\)	\(q+0.421861i q^{3} -1.00000 q^{5} +(-1.02538 - 2.43897i) q^{7} +2.82203 q^{9} +1.89956 q^{11} -4.65608 q^{13} -0.421861i q^{15} -7.19944i q^{17} -0.834050i q^{19} +(1.02891 - 0.432568i) q^{21} +5.76981i q^{23} +1.00000 q^{25} +2.45609i q^{27} +4.84493i q^{29} -4.04358 q^{31} +0.801350i q^{33} +(1.02538 + 2.43897i) q^{35} -4.68101i q^{37} -1.96422i q^{39} -5.81861i q^{41} -12.6338 q^{43} -2.82203 q^{45} -6.30483 q^{47} +(-4.89718 + 5.00176i) q^{49} +3.03716 q^{51} -0.637428i q^{53} -1.89956 q^{55} +0.351853 q^{57} +5.32592i q^{59} +7.80404 q^{61} +(-2.89366 - 6.88287i) q^{63} +4.65608 q^{65} -4.64973 q^{67} -2.43406 q^{69} -6.42998i q^{71} +0.227709i q^{73} +0.421861i q^{75} +(-1.94778 - 4.63298i) q^{77} -7.17831i q^{79} +7.42998 q^{81} -4.76612i q^{83} +7.19944i q^{85} -2.04389 q^{87} -11.9442i q^{89} +(4.77427 + 11.3561i) q^{91} -1.70583i q^{93} +0.834050i q^{95} -10.3285i q^{97} +5.36063 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{5} - 36 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more