LMFDB - Embedded newform 2240.2.b.g.1121.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1121,2240)]

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([0, 1, 0, 0]))

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

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

chi := DirichletCharacter("2240.1121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.b (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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} + 35 x^{10} - 120 x^{9} + 362 x^{8} - 794 x^{7} + 1463 x^{6} - 2048 x^{5} + 2258 x^{4} + \cdots + 61 $ x^12 - 6x^11 + 35x^10 - 120x^9 + 362x^8 - 794x^7 + 1463x^6 - 2048x^5 + 2258x^4 - 1822x^3 + 1035x^2 - 364*x + 61
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.4
Root			$0.500000 + 2.40029i$ of defining polynomial
Character	$\chi$	$=$	2240.1121
Dual form			2240.2.b.g.1121.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.47713i q^{3} -1.00000i q^{5} -1.00000 q^{7} +0.818075 q^{9} +O(q^{10})$	$q-1.47713i q^{3} -1.00000i q^{5} -1.00000 q^{7} +0.818075 q^{9} +5.66883i q^{11} +0.510061i q^{13} -1.47713 q^{15} +0.408601 q^{17} -6.68872i q^{19} +1.47713i q^{21} +0.912445 q^{23} -1.00000 q^{25} -5.63981i q^{27} -8.21427i q^{29} -3.62019 q^{31} +8.37362 q^{33} +1.00000i q^{35} -7.33081i q^{37} +0.753429 q^{39} +11.2958 q^{41} +8.39935i q^{43} -0.818075i q^{45} +11.4778 q^{47} +1.00000 q^{49} -0.603559i q^{51} -12.8932i q^{53} +5.66883 q^{55} -9.88014 q^{57} +8.85343i q^{59} -2.35375i q^{61} -0.818075 q^{63} +0.510061 q^{65} -14.9994i q^{67} -1.34780i q^{69} -13.3726 q^{71} +0.470044 q^{73} +1.47713i q^{75} -5.66883i q^{77} -10.1000 q^{79} -5.87653 q^{81} -7.14301i q^{83} -0.408601i q^{85} -12.1336 q^{87} +2.91244 q^{89} -0.510061i q^{91} +5.34751i q^{93} -6.68872 q^{95} +10.5196 q^{97} +4.63753i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{7} - 20 q^{9}+O(q^{10}) $ 12 * q - 12 * q^7 - 20 * q^9	$ 12 q - 12 q^{7} - 20 q^{9} + 16 q^{17} - 8 q^{23} - 12 q^{25} + 8 q^{31} + 72 q^{33} + 32 q^{39} + 32 q^{47} + 12 q^{49} - 8 q^{55} + 8 q^{57} + 20 q^{63} + 8 q^{65} - 48 q^{71} + 8 q^{73} + 16 q^{79} + 92 q^{81} + 32 q^{87} + 16 q^{89} + 32 q^{97}+O(q^{100}) $ 12 * q - 12 * q^7 - 20 * q^9 + 16 * q^17 - 8 * q^23 - 12 * q^25 + 8 * q^31 + 72 * q^33 + 32 * q^39 + 32 * q^47 + 12 * q^49 - 8 * q^55 + 8 * q^57 + 20 * q^63 + 8 * q^65 - 48 * q^71 + 8 * q^73 + 16 * q^79 + 92 * q^81 + 32 * q^87 + 16 * q^89 + 32 * q^97

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
$3$	− 1.47713i	− 0.852824i	−0.904529	−	0.426412i	$-0.859777\pi$
$3$	− 1.47713i	− 0.852824i	0.904529	−	0.426412i	$-0.140223\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0.818075	0.272692
$10$	0	0
$11$	5.66883i	1.70922i	0.519274	+	0.854608i	$0.326202\pi$
$11$	5.66883i	1.70922i	−0.519274	+	0.854608i	$0.673798\pi$
$12$	0	0
$13$	0.510061i	0.141466i	0.997495	+	0.0707328i	$0.0225338\pi$
$13$	0.510061i	0.141466i	−0.997495	+	0.0707328i	$0.977466\pi$
$14$	0	0
$15$	−1.47713	−0.381394
$16$	0	0
$17$	0.408601	0.0991004	0.0495502	−	0.998772i	$-0.484221\pi$
$17$	0.408601	0.0991004	0.0495502	+	0.998772i	$0.484221\pi$
$18$	0	0
$19$	− 6.68872i	− 1.53450i	−0.641349	−	0.767249i	$-0.721625\pi$
$19$	− 6.68872i	− 1.53450i	0.641349	−	0.767249i	$-0.278375\pi$
$20$	0	0
$21$	1.47713i	0.322337i
$22$	0	0
$23$	0.912445	0.190258	0.0951289	−	0.995465i	$-0.469674\pi$
$23$	0.912445	0.190258	0.0951289	+	0.995465i	$0.469674\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	− 5.63981i	− 1.08538i
$28$	0	0
$29$	− 8.21427i	− 1.52535i	−0.646781	−	0.762676i	$-0.723885\pi$
$29$	− 8.21427i	− 1.52535i	0.646781	−	0.762676i	$-0.276115\pi$
$30$	0	0
$31$	−3.62019	−0.650205	−0.325103	−	0.945679i	$-0.605399\pi$
$31$	−3.62019	−0.650205	−0.325103	+	0.945679i	$0.605399\pi$
$32$	0	0
$33$	8.37362	1.45766
$34$	0	0
$35$	1.00000i	0.169031i
$36$	0	0
$37$	− 7.33081i	− 1.20518i	−0.798052	−	0.602589i	$-0.794136\pi$
$37$	− 7.33081i	− 1.20518i	0.798052	−	0.602589i	$-0.205864\pi$
$38$	0	0
$39$	0.753429	0.120645
$40$	0	0
$41$	11.2958	1.76411	0.882056	−	0.471144i	$-0.156159\pi$
$41$	11.2958	1.76411	0.882056	+	0.471144i	$0.156159\pi$
$42$	0	0
$43$	8.39935i	1.28089i	0.768005	+	0.640444i	$0.221250\pi$
$43$	8.39935i	1.28089i	−0.768005	+	0.640444i	$0.778750\pi$
$44$	0	0
$45$	− 0.818075i	− 0.121951i
$46$	0	0
$47$	11.4778	1.67420	0.837101	−	0.547048i	$-0.184248\pi$
$47$	11.4778	1.67420	0.837101	+	0.547048i	$0.184248\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 0.603559i	− 0.0845152i
$52$	0	0
$53$	− 12.8932i	− 1.77102i	−0.464620	−	0.885510i	$-0.653809\pi$
$53$	− 12.8932i	− 1.77102i	0.464620	−	0.885510i	$-0.346191\pi$
$54$	0	0
$55$	5.66883	0.764385
$56$	0	0
$57$	−9.88014	−1.30866
$58$	0	0
$59$	8.85343i	1.15262i	0.817232	+	0.576309i	$0.195508\pi$
$59$	8.85343i	1.15262i	−0.817232	+	0.576309i	$0.804492\pi$
$60$	0	0
$61$	− 2.35375i	− 0.301366i	−0.988582	−	0.150683i	$-0.951853\pi$
$61$	− 2.35375i	− 0.301366i	0.988582	−	0.150683i	$-0.0481473\pi$
$62$	0	0
$63$	−0.818075	−0.103068
$64$	0	0
$65$	0.510061	0.0632653
$66$	0	0
$67$	− 14.9994i	− 1.83247i	−0.400641	−	0.916235i	$-0.631213\pi$
$67$	− 14.9994i	− 1.83247i	0.400641	−	0.916235i	$-0.368787\pi$
$68$	0	0
$69$	− 1.34780i	− 0.162256i
$70$	0	0
$71$	−13.3726	−1.58704	−0.793520	−	0.608544i	$-0.791754\pi$
$71$	−13.3726	−1.58704	−0.793520	+	0.608544i	$0.791754\pi$
$72$	0	0
$73$	0.470044	0.0550145	0.0275072	−	0.999622i	$-0.491243\pi$
$73$	0.470044	0.0550145	0.0275072	+	0.999622i	$0.491243\pi$
$74$	0	0
$75$	1.47713i	0.170565i
$76$	0	0
$77$	− 5.66883i	− 0.646023i
$78$	0	0
$79$	−10.1000	−1.13634	−0.568170	−	0.822912i	$-0.692348\pi$
$79$	−10.1000	−1.13634	−0.568170	+	0.822912i	$0.692348\pi$
$80$	0	0
$81$	−5.87653	−0.652948
$82$	0	0
$83$	− 7.14301i	− 0.784047i	−0.919955	−	0.392023i	$-0.871775\pi$
$83$	− 7.14301i	− 0.784047i	0.919955	−	0.392023i	$-0.128225\pi$
$84$	0	0
$85$	− 0.408601i	− 0.0443190i
$86$	0	0
$87$	−12.1336	−1.30086
$88$	0	0
$89$	2.91244	0.308719	0.154359	−	0.988015i	$-0.450669\pi$
$89$	2.91244	0.308719	0.154359	+	0.988015i	$0.450669\pi$
$90$	0	0
$91$	− 0.510061i	− 0.0534690i
$92$	0	0
$93$	5.34751i	0.554511i
$94$	0	0
$95$	−6.68872	−0.686248
$96$	0	0
$97$	10.5196	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$97$	10.5196	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$98$	0	0
$99$	4.63753i	0.466089i
$100$	0	0
$101$	− 8.62613i	− 0.858332i	−0.903226	−	0.429166i	$-0.858807\pi$
$101$	− 8.62613i	− 0.858332i	0.903226	−	0.429166i	$-0.141193\pi$
$102$	0	0
$103$	−11.4778	−1.13094	−0.565468	−	0.824770i	$-0.691305\pi$
$103$	−11.4778	−1.13094	−0.565468	+	0.824770i	$0.691305\pi$
$104$	0	0
$105$	1.47713	0.144154
$106$	0	0
$107$	− 13.4291i	− 1.29824i	−0.760685	−	0.649121i	$-0.775137\pi$
$107$	− 13.4291i	− 1.29824i	0.760685	−	0.649121i	$-0.224863\pi$
$108$	0	0
$109$	17.3056i	1.65758i	0.559563	+	0.828788i	$0.310969\pi$
$109$	17.3056i	1.65758i	−0.559563	+	0.828788i	$0.689031\pi$
$110$	0	0
$111$	−10.8286	−1.02780
$112$	0	0
$113$	0.899017	0.0845724	0.0422862	−	0.999106i	$-0.486536\pi$
$113$	0.899017	0.0845724	0.0422862	+	0.999106i	$0.486536\pi$
$114$	0	0
$115$	− 0.912445i	− 0.0850859i
$116$	0	0
$117$	0.417268i	0.0385765i
$118$	0	0
$119$	−0.408601	−0.0374564
$120$	0	0
$121$	−21.1356	−1.92142
$122$	0	0
$123$	− 16.6855i	− 1.50448i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	6.59042	0.584805	0.292403	−	0.956295i	$-0.405545\pi$
$127$	6.59042	0.584805	0.292403	+	0.956295i	$0.405545\pi$
$128$	0	0
$129$	12.4070	1.09237
$130$	0	0
$131$	6.71433i	0.586634i	0.956015	+	0.293317i	$0.0947591\pi$
$131$	6.71433i	0.586634i	−0.956015	+	0.293317i	$0.905241\pi$
$132$	0	0
$133$	6.68872i	0.579986i
$134$	0	0
$135$	−5.63981	−0.485397
$136$	0	0
$137$	−2.35375	−0.201094	−0.100547	−	0.994932i	$-0.532059\pi$
$137$	−2.35375	−0.201094	−0.100547	+	0.994932i	$0.532059\pi$
$138$	0	0
$139$	− 1.89916i	− 0.161085i	−0.996751	−	0.0805424i	$-0.974335\pi$
$139$	− 1.89916i	− 0.161085i	0.996751	−	0.0805424i	$-0.0256653\pi$
$140$	0	0
$141$	− 16.9542i	− 1.42780i
$142$	0	0
$143$	−2.89145	−0.241795
$144$	0	0
$145$	−8.21427	−0.682158
$146$	0	0
$147$	− 1.47713i	− 0.121832i
$148$	0	0
$149$	0.312176i	0.0255745i	0.999918	+	0.0127872i	$0.00407042\pi$
$149$	0.312176i	0.0255745i	−0.999918	+	0.0127872i	$0.995930\pi$
$150$	0	0
$151$	3.27744	0.266715	0.133357	−	0.991068i	$-0.457424\pi$
$151$	3.27744	0.266715	0.133357	+	0.991068i	$0.457424\pi$
$152$	0	0
$153$	0.334267	0.0270238
$154$	0	0
$155$	3.62019i	0.290781i
$156$	0	0
$157$	0.207031i	0.0165228i	0.999966	+	0.00826142i	$0.00262972\pi$
$157$	0.207031i	0.0165228i	−0.999966	+	0.00826142i	$0.997370\pi$
$158$	0	0
$159$	−19.0450	−1.51037
$160$	0	0
$161$	−0.912445	−0.0719107
$162$	0	0
$163$	10.5721i	0.828073i	0.910260	+	0.414036i	$0.135881\pi$
$163$	10.5721i	0.828073i	−0.910260	+	0.414036i	$0.864119\pi$
$164$	0	0
$165$	− 8.37362i	− 0.651885i
$166$	0	0
$167$	−1.24342	−0.0962184	−0.0481092	−	0.998842i	$-0.515320\pi$
$167$	−1.24342	−0.0962184	−0.0481092	+	0.998842i	$0.515320\pi$
$168$	0	0
$169$	12.7398	0.979987
$170$	0	0
$171$	− 5.47188i	− 0.418445i
$172$	0	0
$173$	− 21.9843i	− 1.67144i	−0.549158	−	0.835719i	$-0.685051\pi$
$173$	− 21.9843i	− 1.67144i	0.549158	−	0.835719i	$-0.314949\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	13.0777	0.982981
$178$	0	0
$179$	− 8.79114i	− 0.657080i	−0.944490	−	0.328540i	$-0.893443\pi$
$179$	− 8.79114i	− 0.657080i	0.944490	−	0.328540i	$-0.106557\pi$
$180$	0	0
$181$	6.55063i	0.486904i	0.969913	+	0.243452i	$0.0782799\pi$
$181$	6.55063i	0.486904i	−0.969913	+	0.243452i	$0.921720\pi$
$182$	0	0
$183$	−3.47680	−0.257012
$184$	0	0
$185$	−7.33081	−0.538972
$186$	0	0
$187$	2.31629i	0.169384i
$188$	0	0
$189$	5.63981i	0.410236i
$190$	0	0
$191$	3.91904	0.283572	0.141786	−	0.989897i	$-0.454716\pi$
$191$	3.91904	0.283572	0.141786	+	0.989897i	$0.454716\pi$
$192$	0	0
$193$	10.6279	0.765011	0.382506	−	0.923953i	$-0.375061\pi$
$193$	10.6279	0.765011	0.382506	+	0.923953i	$0.375061\pi$
$194$	0	0
$195$	− 0.753429i	− 0.0539542i
$196$	0	0
$197$	13.1498i	0.936888i	0.883493	+	0.468444i	$0.155185\pi$
$197$	13.1498i	0.936888i	−0.883493	+	0.468444i	$0.844815\pi$
$198$	0	0
$199$	12.7924	0.906828	0.453414	−	0.891300i	$-0.350206\pi$
$199$	12.7924	0.906828	0.453414	+	0.891300i	$0.350206\pi$
$200$	0	0
$201$	−22.1561	−1.56277
$202$	0	0
$203$	8.21427i	0.576528i
$204$	0	0
$205$	− 11.2958i	− 0.788935i
$206$	0	0
$207$	0.746448	0.0518817
$208$	0	0
$209$	37.9172	2.62279
$210$	0	0
$211$	6.14779i	0.423231i	0.977353	+	0.211616i	$0.0678725\pi$
$211$	6.14779i	0.423231i	−0.977353	+	0.211616i	$0.932128\pi$
$212$	0	0
$213$	19.7532i	1.35347i
$214$	0	0
$215$	8.39935	0.572831
$216$	0	0
$217$	3.62019	0.245754
$218$	0	0
$219$	− 0.694318i	− 0.0469177i
$220$	0	0
$221$	0.208412i	0.0140193i
$222$	0	0
$223$	13.7497	0.920747	0.460374	−	0.887725i	$-0.347715\pi$
$223$	13.7497	0.920747	0.460374	+	0.887725i	$0.347715\pi$
$224$	0	0
$225$	−0.818075	−0.0545383
$226$	0	0
$227$	− 12.8404i	− 0.852247i	−0.904665	−	0.426124i	$-0.859879\pi$
$227$	− 12.8404i	− 0.852247i	0.904665	−	0.426124i	$-0.140121\pi$
$228$	0	0
$229$	− 20.1667i	− 1.33265i	−0.745660	−	0.666327i	$-0.767866\pi$
$229$	− 20.1667i	− 1.33265i	0.745660	−	0.666327i	$-0.232134\pi$
$230$	0	0
$231$	−8.37362	−0.550944
$232$	0	0
$233$	−5.32397	−0.348785	−0.174393	−	0.984676i	$-0.555796\pi$
$233$	−5.32397	−0.348785	−0.174393	+	0.984676i	$0.555796\pi$
$234$	0	0
$235$	− 11.4778i	− 0.748726i
$236$	0	0
$237$	14.9191i	0.969097i
$238$	0	0
$239$	−17.6210	−1.13981	−0.569905	−	0.821710i	$-0.693020\pi$
$239$	−17.6210	−1.13981	−0.569905	+	0.821710i	$0.693020\pi$
$240$	0	0
$241$	12.4571	0.802434	0.401217	−	0.915983i	$-0.368587\pi$
$241$	12.4571	0.802434	0.401217	+	0.915983i	$0.368587\pi$
$242$	0	0
$243$	− 8.23900i	− 0.528532i
$244$	0	0
$245$	− 1.00000i	− 0.0638877i
$246$	0	0
$247$	3.41166	0.217079
$248$	0	0
$249$	−10.5512	−0.668654
$250$	0	0
$251$	17.5217i	1.10596i	0.833195	+	0.552979i	$0.186509\pi$
$251$	17.5217i	1.10596i	−0.833195	+	0.552979i	$0.813491\pi$
$252$	0	0
$253$	5.17249i	0.325192i
$254$	0	0
$255$	−0.603559	−0.0377963
$256$	0	0
$257$	−19.3739	−1.20851	−0.604255	−	0.796791i	$-0.706529\pi$
$257$	−19.3739	−1.20851	−0.604255	+	0.796791i	$0.706529\pi$
$258$	0	0
$259$	7.33081i	0.455515i
$260$	0	0
$261$	− 6.71988i	− 0.415950i
$262$	0	0
$263$	3.48536	0.214916	0.107458	−	0.994210i	$-0.465729\pi$
$263$	3.48536	0.214916	0.107458	+	0.994210i	$0.465729\pi$
$264$	0	0
$265$	−12.8932	−0.792024
$266$	0	0
$267$	− 4.30207i	− 0.263282i
$268$	0	0
$269$	30.1168i	1.83625i	0.396286	+	0.918127i	$0.370299\pi$
$269$	30.1168i	1.83625i	−0.396286	+	0.918127i	$0.629701\pi$
$270$	0	0
$271$	13.5543	0.823367	0.411684	−	0.911327i	$-0.364941\pi$
$271$	13.5543	0.823367	0.411684	+	0.911327i	$0.364941\pi$
$272$	0	0
$273$	−0.753429	−0.0455996
$274$	0	0
$275$	− 5.66883i	− 0.341843i
$276$	0	0
$277$	− 8.88124i	− 0.533622i	−0.963749	−	0.266811i	$-0.914030\pi$
$277$	− 8.88124i	− 0.533622i	0.963749	−	0.266811i	$-0.0859699\pi$
$278$	0	0
$279$	−2.96159	−0.177306
$280$	0	0
$281$	−15.9469	−0.951310	−0.475655	−	0.879632i	$-0.657789\pi$
$281$	−15.9469	−0.951310	−0.475655	+	0.879632i	$0.657789\pi$
$282$	0	0
$283$	7.88077i	0.468463i	0.972181	+	0.234232i	$0.0752574\pi$
$283$	7.88077i	0.468463i	−0.972181	+	0.234232i	$0.924743\pi$
$284$	0	0
$285$	9.88014i	0.585249i
$286$	0	0
$287$	−11.2958	−0.666772
$288$	0	0
$289$	−16.8330	−0.990179
$290$	0	0
$291$	− 15.5389i	− 0.910904i
$292$	0	0
$293$	19.8477i	1.15952i	0.814789	+	0.579758i	$0.196853\pi$
$293$	19.8477i	1.15952i	−0.814789	+	0.579758i	$0.803147\pi$
$294$	0	0
$295$	8.85343	0.515467
$296$	0	0
$297$	31.9711	1.85515
$298$	0	0
$299$	0.465403i	0.0269149i
$300$	0	0
$301$	− 8.39935i	− 0.484130i
$302$	0	0
$303$	−12.7420	−0.732006
$304$	0	0
$305$	−2.35375	−0.134775
$306$	0	0
$307$	− 8.32169i	− 0.474944i	−0.971394	−	0.237472i	$-0.923681\pi$
$307$	− 8.32169i	− 0.474944i	0.971394	−	0.237472i	$-0.0763188\pi$
$308$	0	0
$309$	16.9542i	0.964490i
$310$	0	0
$311$	−21.3678	−1.21166	−0.605828	−	0.795595i	$-0.707158\pi$
$311$	−21.3678	−1.21166	−0.605828	+	0.795595i	$0.707158\pi$
$312$	0	0
$313$	−10.1879	−0.575854	−0.287927	−	0.957652i	$-0.592966\pi$
$313$	−10.1879	−0.575854	−0.287927	+	0.957652i	$0.592966\pi$
$314$	0	0
$315$	0.818075i	0.0460933i
$316$	0	0
$317$	− 8.94489i	− 0.502395i	−0.967936	−	0.251198i	$-0.919176\pi$
$317$	− 8.94489i	− 0.502395i	0.967936	−	0.251198i	$-0.0808244\pi$
$318$	0	0
$319$	46.5653	2.60715
$320$	0	0
$321$	−19.8366	−1.10717
$322$	0	0
$323$	− 2.73302i	− 0.152069i
$324$	0	0
$325$	− 0.510061i	− 0.0282931i
$326$	0	0
$327$	25.5627	1.41362
$328$	0	0
$329$	−11.4778	−0.632789
$330$	0	0
$331$	32.8636i	1.80635i	0.429276	+	0.903173i	$0.358769\pi$
$331$	32.8636i	1.80635i	−0.429276	+	0.903173i	$0.641231\pi$
$332$	0	0
$333$	− 5.99715i	− 0.328642i
$334$	0	0
$335$	−14.9994	−0.819506
$336$	0	0
$337$	32.2398	1.75621	0.878107	−	0.478465i	$-0.158807\pi$
$337$	32.2398	1.75621	0.878107	+	0.478465i	$0.158807\pi$
$338$	0	0
$339$	− 1.32797i	− 0.0721254i
$340$	0	0
$341$	− 20.5222i	− 1.11134i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.34780	−0.0725633
$346$	0	0
$347$	− 20.4565i	− 1.09816i	−0.835769	−	0.549081i	$-0.814978\pi$
$347$	− 20.4565i	− 1.09816i	0.835769	−	0.549081i	$-0.185022\pi$
$348$	0	0
$349$	− 16.6986i	− 0.893855i	−0.894570	−	0.446927i	$-0.852518\pi$
$349$	− 16.6986i	− 0.893855i	0.894570	−	0.446927i	$-0.147482\pi$
$350$	0	0
$351$	2.87665	0.153544
$352$	0	0
$353$	11.6750	0.621400	0.310700	−	0.950508i	$-0.399437\pi$
$353$	11.6750	0.621400	0.310700	+	0.950508i	$0.399437\pi$
$354$	0	0
$355$	13.3726i	0.709746i
$356$	0	0
$357$	0.603559i	0.0319437i
$358$	0	0
$359$	13.1603	0.694571	0.347286	−	0.937759i	$-0.387103\pi$
$359$	13.1603	0.694571	0.347286	+	0.937759i	$0.387103\pi$
$360$	0	0
$361$	−25.7390	−1.35468
$362$	0	0
$363$	31.2201i	1.63863i
$364$	0	0
$365$	− 0.470044i	− 0.0246032i
$366$	0	0
$367$	26.2226	1.36881	0.684404	−	0.729103i	$-0.260063\pi$
$367$	26.2226	1.36881	0.684404	+	0.729103i	$0.260063\pi$
$368$	0	0
$369$	9.24084	0.481059
$370$	0	0
$371$	12.8932i	0.669383i
$372$	0	0
$373$	9.42182i	0.487843i	0.969795	+	0.243922i	$0.0784340\pi$
$373$	9.42182i	0.487843i	−0.969795	+	0.243922i	$0.921566\pi$
$374$	0	0
$375$	1.47713	0.0762789
$376$	0	0
$377$	4.18978	0.215785
$378$	0	0
$379$	− 11.2545i	− 0.578103i	−0.957314	−	0.289051i	$-0.906660\pi$
$379$	− 11.2545i	− 0.578103i	0.957314	−	0.289051i	$-0.0933399\pi$
$380$	0	0
$381$	− 9.73493i	− 0.498736i
$382$	0	0
$383$	−12.4546	−0.636400	−0.318200	−	0.948024i	$-0.603078\pi$
$383$	−12.4546	−0.636400	−0.318200	+	0.948024i	$0.603078\pi$
$384$	0	0
$385$	−5.66883	−0.288910
$386$	0	0
$387$	6.87129i	0.349288i
$388$	0	0
$389$	− 13.9543i	− 0.707512i	−0.935338	−	0.353756i	$-0.884904\pi$
$389$	− 13.9543i	− 0.707512i	0.935338	−	0.353756i	$-0.115096\pi$
$390$	0	0
$391$	0.372826	0.0188546
$392$	0	0
$393$	9.91797	0.500295
$394$	0	0
$395$	10.1000i	0.508186i
$396$	0	0
$397$	33.2703i	1.66979i	0.550410	+	0.834894i	$0.314471\pi$
$397$	33.2703i	1.66979i	−0.550410	+	0.834894i	$0.685529\pi$
$398$	0	0
$399$	9.88014	0.494626
$400$	0	0
$401$	3.48806	0.174186	0.0870928	−	0.996200i	$-0.472242\pi$
$401$	3.48806	0.174186	0.0870928	+	0.996200i	$0.472242\pi$
$402$	0	0
$403$	− 1.84652i	− 0.0919817i
$404$	0	0
$405$	5.87653i	0.292007i
$406$	0	0
$407$	41.5571	2.05991
$408$	0	0
$409$	−12.5270	−0.619419	−0.309710	−	0.950831i	$-0.600232\pi$
$409$	−12.5270	−0.619419	−0.309710	+	0.950831i	$0.600232\pi$
$410$	0	0
$411$	3.47680i	0.171498i
$412$	0	0
$413$	− 8.85343i	− 0.435649i
$414$	0	0
$415$	−7.14301	−0.350636
$416$	0	0
$417$	−2.80532	−0.137377
$418$	0	0
$419$	− 1.73823i	− 0.0849182i	−0.999098	−	0.0424591i	$-0.986481\pi$
$419$	− 1.73823i	− 0.0849182i	0.999098	−	0.0424591i	$-0.0135192\pi$
$420$	0	0
$421$	20.0433i	0.976852i	0.872605	+	0.488426i	$0.162429\pi$
$421$	20.0433i	0.976852i	−0.872605	+	0.488426i	$0.837571\pi$
$422$	0	0
$423$	9.38966	0.456541
$424$	0	0
$425$	−0.408601	−0.0198201
$426$	0	0
$427$	2.35375i	0.113906i
$428$	0	0
$429$	4.27106i	0.206209i
$430$	0	0
$431$	19.6703	0.947485	0.473742	−	0.880663i	$-0.342903\pi$
$431$	19.6703	0.947485	0.473742	+	0.880663i	$0.342903\pi$
$432$	0	0
$433$	38.3857	1.84470	0.922350	−	0.386355i	$-0.126266\pi$
$433$	38.3857	1.84470	0.922350	+	0.386355i	$0.126266\pi$
$434$	0	0
$435$	12.1336i	0.581760i
$436$	0	0
$437$	− 6.10309i	− 0.291950i
$438$	0	0
$439$	20.2319	0.965617	0.482809	−	0.875726i	$-0.339617\pi$
$439$	20.2319	0.965617	0.482809	+	0.875726i	$0.339617\pi$
$440$	0	0
$441$	0.818075	0.0389559
$442$	0	0
$443$	− 5.23855i	− 0.248891i	−0.992226	−	0.124445i	$-0.960285\pi$
$443$	− 5.23855i	− 0.248891i	0.992226	−	0.124445i	$-0.0397152\pi$
$444$	0	0
$445$	− 2.91244i	− 0.138063i
$446$	0	0
$447$	0.461126	0.0218105
$448$	0	0
$449$	13.5589	0.639885	0.319942	−	0.947437i	$-0.396336\pi$
$449$	13.5589	0.639885	0.319942	+	0.947437i	$0.396336\pi$
$450$	0	0
$451$	64.0341i	3.01525i
$452$	0	0
$453$	− 4.84122i	− 0.227460i
$454$	0	0
$455$	−0.510061	−0.0239120
$456$	0	0
$457$	−31.6356	−1.47985	−0.739926	−	0.672688i	$-0.765139\pi$
$457$	−31.6356	−1.47985	−0.739926	+	0.672688i	$0.765139\pi$
$458$	0	0
$459$	− 2.30443i	− 0.107562i
$460$	0	0
$461$	− 11.8569i	− 0.552229i	−0.961125	−	0.276115i	$-0.910953\pi$
$461$	− 11.8569i	− 0.552229i	0.961125	−	0.276115i	$-0.0890469\pi$
$462$	0	0
$463$	3.04905	0.141702	0.0708508	−	0.997487i	$-0.477429\pi$
$463$	3.04905	0.141702	0.0708508	+	0.997487i	$0.477429\pi$
$464$	0	0
$465$	5.34751	0.247985
$466$	0	0
$467$	− 5.39961i	− 0.249864i	−0.992165	−	0.124932i	$-0.960129\pi$
$467$	− 5.39961i	− 0.249864i	0.992165	−	0.124932i	$-0.0398713\pi$
$468$	0	0
$469$	14.9994i	0.692609i
$470$	0	0
$471$	0.305812	0.0140911
$472$	0	0
$473$	−47.6145	−2.18931
$474$	0	0
$475$	6.68872i	0.306900i
$476$	0	0
$477$	− 10.5476i	− 0.482942i
$478$	0	0
$479$	8.87240	0.405390	0.202695	−	0.979242i	$-0.435030\pi$
$479$	8.87240	0.405390	0.202695	+	0.979242i	$0.435030\pi$
$480$	0	0
$481$	3.73917	0.170491
$482$	0	0
$483$	1.34780i	0.0613272i
$484$	0	0
$485$	− 10.5196i	− 0.477671i
$486$	0	0
$487$	−2.13752	−0.0968603	−0.0484302	−	0.998827i	$-0.515422\pi$
$487$	−2.13752	−0.0968603	−0.0484302	+	0.998827i	$0.515422\pi$
$488$	0	0
$489$	15.6164	0.706200
$490$	0	0
$491$	27.7538i	1.25251i	0.779618	+	0.626255i	$0.215413\pi$
$491$	27.7538i	1.25251i	−0.779618	+	0.626255i	$0.784587\pi$
$492$	0	0
$493$	− 3.35636i	− 0.151163i
$494$	0	0
$495$	4.63753	0.208441
$496$	0	0
$497$	13.3726	0.599845
$498$	0	0
$499$	− 8.59703i	− 0.384856i	−0.981311	−	0.192428i	$-0.938364\pi$
$499$	− 8.59703i	− 0.384856i	0.981311	−	0.192428i	$-0.0616362\pi$
$500$	0	0
$501$	1.83669i	0.0820573i
$502$	0	0
$503$	23.7738	1.06002	0.530010	−	0.847991i	$-0.322188\pi$
$503$	23.7738	1.06002	0.530010	+	0.847991i	$0.322188\pi$
$504$	0	0
$505$	−8.62613	−0.383858
$506$	0	0
$507$	− 18.8184i	− 0.835757i
$508$	0	0
$509$	22.2061i	0.984267i	0.870520	+	0.492134i	$0.163783\pi$
$509$	22.2061i	0.984267i	−0.870520	+	0.492134i	$0.836217\pi$
$510$	0	0
$511$	−0.470044	−0.0207935
$512$	0	0
$513$	−37.7231	−1.66552
$514$	0	0
$515$	11.4778i	0.505770i
$516$	0	0
$517$	65.0654i	2.86157i
$518$	0	0
$519$	−32.4738	−1.42544
$520$	0	0
$521$	−18.1789	−0.796432	−0.398216	−	0.917292i	$-0.630370\pi$
$521$	−18.1789	−0.796432	−0.398216	+	0.917292i	$0.630370\pi$
$522$	0	0
$523$	− 41.4996i	− 1.81465i	−0.420428	−	0.907326i	$-0.638120\pi$
$523$	− 41.4996i	− 1.81465i	0.420428	−	0.907326i	$-0.361880\pi$
$524$	0	0
$525$	− 1.47713i	− 0.0644674i
$526$	0	0
$527$	−1.47921	−0.0644356
$528$	0	0
$529$	−22.1674	−0.963802
$530$	0	0
$531$	7.24277i	0.314309i
$532$	0	0
$533$	5.76157i	0.249561i
$534$	0	0
$535$	−13.4291	−0.580591
$536$	0	0
$537$	−12.9857	−0.560374
$538$	0	0
$539$	5.66883i	0.244174i
$540$	0	0
$541$	19.9092i	0.855965i	0.903787	+	0.427983i	$0.140776\pi$
$541$	19.9092i	0.855965i	−0.903787	+	0.427983i	$0.859224\pi$
$542$	0	0
$543$	9.67616	0.415244
$544$	0	0
$545$	17.3056	0.741290
$546$	0	0
$547$	39.7370i	1.69903i	0.527563	+	0.849516i	$0.323106\pi$
$547$	39.7370i	1.69903i	−0.527563	+	0.849516i	$0.676894\pi$
$548$	0	0
$549$	− 1.92554i	− 0.0821800i
$550$	0	0
$551$	−54.9429	−2.34065
$552$	0	0
$553$	10.1000	0.429496
$554$	0	0
$555$	10.8286i	0.459648i
$556$	0	0
$557$	− 7.80234i	− 0.330596i	−0.986244	−	0.165298i	$-0.947141\pi$
$557$	− 7.80234i	− 0.330596i	0.986244	−	0.165298i	$-0.0528586\pi$
$558$	0	0
$559$	−4.28418	−0.181202
$560$	0	0
$561$	3.42147	0.144455
$562$	0	0
$563$	− 5.89074i	− 0.248265i	−0.992266	−	0.124133i	$-0.960385\pi$
$563$	− 5.89074i	− 0.248265i	0.992266	−	0.124133i	$-0.0396148\pi$
$564$	0	0
$565$	− 0.899017i	− 0.0378219i
$566$	0	0
$567$	5.87653	0.246791
$568$	0	0
$569$	26.6634	1.11779	0.558894	−	0.829239i	$-0.311226\pi$
$569$	26.6634	1.11779	0.558894	+	0.829239i	$0.311226\pi$
$570$	0	0
$571$	− 8.92039i	− 0.373307i	−0.982426	−	0.186653i	$-0.940236\pi$
$571$	− 8.92039i	− 0.373307i	0.982426	−	0.186653i	$-0.0597641\pi$
$572$	0	0
$573$	− 5.78894i	− 0.241837i
$574$	0	0
$575$	−0.912445	−0.0380516
$576$	0	0
$577$	−2.95557	−0.123042	−0.0615211	−	0.998106i	$-0.519595\pi$
$577$	−2.95557	−0.123042	−0.0615211	+	0.998106i	$0.519595\pi$
$578$	0	0
$579$	− 15.6988i	− 0.652420i
$580$	0	0
$581$	7.14301i	0.296342i
$582$	0	0
$583$	73.0894	3.02706
$584$	0	0
$585$	0.417268	0.0172519
$586$	0	0
$587$	− 11.8829i	− 0.490461i	−0.969465	−	0.245230i	$-0.921136\pi$
$587$	− 11.8829i	− 0.490461i	0.969465	−	0.245230i	$-0.0788636\pi$
$588$	0	0
$589$	24.2144i	0.997739i
$590$	0	0
$591$	19.4241	0.799000
$592$	0	0
$593$	−10.8318	−0.444808	−0.222404	−	0.974955i	$-0.571390\pi$
$593$	−10.8318	−0.444808	−0.222404	+	0.974955i	$0.571390\pi$
$594$	0	0
$595$	0.408601i	0.0167510i
$596$	0	0
$597$	− 18.8961i	− 0.773364i
$598$	0	0
$599$	20.1393	0.822871	0.411435	−	0.911439i	$-0.365028\pi$
$599$	20.1393	0.822871	0.411435	+	0.911439i	$0.365028\pi$
$600$	0	0
$601$	16.0902	0.656335	0.328167	−	0.944620i	$-0.393569\pi$
$601$	16.0902	0.656335	0.328167	+	0.944620i	$0.393569\pi$
$602$	0	0
$603$	− 12.2706i	− 0.499699i
$604$	0	0
$605$	21.1356i	0.859285i
$606$	0	0
$607$	46.8337	1.90092	0.950461	−	0.310845i	$-0.100612\pi$
$607$	46.8337	1.90092	0.950461	+	0.310845i	$0.100612\pi$
$608$	0	0
$609$	12.1336	0.491677
$610$	0	0
$611$	5.85436i	0.236842i
$612$	0	0
$613$	32.0439i	1.29424i	0.762388	+	0.647120i	$0.224027\pi$
$613$	32.0439i	1.29424i	−0.762388	+	0.647120i	$0.775973\pi$
$614$	0	0
$615$	−16.6855	−0.672823
$616$	0	0
$617$	11.7188	0.471781	0.235890	−	0.971780i	$-0.424199\pi$
$617$	11.7188	0.471781	0.235890	+	0.971780i	$0.424199\pi$
$618$	0	0
$619$	16.4804i	0.662405i	0.943560	+	0.331203i	$0.107454\pi$
$619$	16.4804i	0.662405i	−0.943560	+	0.331203i	$0.892546\pi$
$620$	0	0
$621$	− 5.14601i	− 0.206502i
$622$	0	0
$623$	−2.91244	−0.116685
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	− 56.0088i	− 2.23678i
$628$	0	0
$629$	− 2.99538i	− 0.119434i
$630$	0	0
$631$	−16.4917	−0.656523	−0.328261	−	0.944587i	$-0.606463\pi$
$631$	−16.4917	−0.656523	−0.328261	+	0.944587i	$0.606463\pi$
$632$	0	0
$633$	9.08111	0.360942
$634$	0	0
$635$	− 6.59042i	− 0.261533i
$636$	0	0
$637$	0.510061i	0.0202094i
$638$	0	0
$639$	−10.9398	−0.432773
$640$	0	0
$641$	−43.7638	−1.72857	−0.864283	−	0.503007i	$-0.832227\pi$
$641$	−43.7638	−1.72857	−0.864283	+	0.503007i	$0.832227\pi$
$642$	0	0
$643$	29.5227i	1.16426i	0.813095	+	0.582132i	$0.197781\pi$
$643$	29.5227i	1.16426i	−0.813095	+	0.582132i	$0.802219\pi$
$644$	0	0
$645$	− 12.4070i	− 0.488524i
$646$	0	0
$647$	−42.2249	−1.66003	−0.830016	−	0.557740i	$-0.811669\pi$
$647$	−42.2249	−1.66003	−0.830016	+	0.557740i	$0.811669\pi$
$648$	0	0
$649$	−50.1886	−1.97007
$650$	0	0
$651$	− 5.34751i	− 0.209585i
$652$	0	0
$653$	38.4027i	1.50281i	0.659839	+	0.751407i	$0.270624\pi$
$653$	38.4027i	1.50281i	−0.659839	+	0.751407i	$0.729376\pi$
$654$	0	0
$655$	6.71433	0.262351
$656$	0	0
$657$	0.384531	0.0150020
$658$	0	0
$659$	− 22.8036i	− 0.888302i	−0.895952	−	0.444151i	$-0.853505\pi$
$659$	− 22.8036i	− 0.888302i	0.895952	−	0.444151i	$-0.146495\pi$
$660$	0	0
$661$	24.2618i	0.943674i	0.881686	+	0.471837i	$0.156409\pi$
$661$	24.2618i	0.943674i	−0.881686	+	0.471837i	$0.843591\pi$
$662$	0	0
$663$	0.307852	0.0119560
$664$	0	0
$665$	6.68872	0.259378
$666$	0	0
$667$	− 7.49506i	− 0.290210i
$668$	0	0
$669$	− 20.3101i	− 0.785235i
$670$	0	0
$671$	13.3430	0.515100
$672$	0	0
$673$	−33.3553	−1.28575	−0.642876	−	0.765970i	$-0.722259\pi$
$673$	−33.3553	−1.28575	−0.642876	+	0.765970i	$0.722259\pi$
$674$	0	0
$675$	5.63981i	0.217076i
$676$	0	0
$677$	19.0404i	0.731783i	0.930657	+	0.365892i	$0.119236\pi$
$677$	19.0404i	0.731783i	−0.930657	+	0.365892i	$0.880764\pi$
$678$	0	0
$679$	−10.5196	−0.403705
$680$	0	0
$681$	−18.9670	−0.726817
$682$	0	0
$683$	− 15.3312i	− 0.586631i	−0.956016	−	0.293315i	$-0.905241\pi$
$683$	− 15.3312i	− 0.586631i	0.956016	−	0.293315i	$-0.0947586\pi$
$684$	0	0
$685$	2.35375i	0.0899320i
$686$	0	0
$687$	−29.7889	−1.13652
$688$	0	0
$689$	6.57633	0.250538
$690$	0	0
$691$	− 6.24280i	− 0.237487i	−0.992925	−	0.118744i	$-0.962113\pi$
$691$	− 6.24280i	− 0.237487i	0.992925	−	0.118744i	$-0.0378867\pi$
$692$	0	0
$693$	− 4.63753i	− 0.176165i
$694$	0	0
$695$	−1.89916	−0.0720394
$696$	0	0
$697$	4.61549	0.174824
$698$	0	0
$699$	7.86422i	0.297452i
$700$	0	0
$701$	14.9020i	0.562839i	0.959585	+	0.281420i	$0.0908053\pi$
$701$	14.9020i	0.562839i	−0.959585	+	0.281420i	$0.909195\pi$
$702$	0	0
$703$	−49.0338	−1.84934
$704$	0	0
$705$	−16.9542	−0.638532
$706$	0	0
$707$	8.62613i	0.324419i
$708$	0	0
$709$	− 25.1684i	− 0.945221i	−0.881272	−	0.472610i	$-0.843312\pi$
$709$	− 25.1684i	− 0.945221i	0.881272	−	0.472610i	$-0.156688\pi$
$710$	0	0
$711$	−8.26256	−0.309870
$712$	0	0
$713$	−3.30322	−0.123707
$714$	0	0
$715$	2.89145i	0.108134i
$716$	0	0
$717$	26.0286i	0.972057i
$718$	0	0
$719$	34.3157	1.27976	0.639879	−	0.768476i	$-0.278984\pi$
$719$	34.3157	1.27976	0.639879	+	0.768476i	$0.278984\pi$
$720$	0	0
$721$	11.4778	0.427454
$722$	0	0
$723$	− 18.4009i	− 0.684335i
$724$	0	0
$725$	8.21427i	0.305070i
$726$	0	0
$727$	−16.3929	−0.607977	−0.303989	−	0.952676i	$-0.598318\pi$
$727$	−16.3929	−0.607977	−0.303989	+	0.952676i	$0.598318\pi$
$728$	0	0
$729$	−29.7997	−1.10369
$730$	0	0
$731$	3.43199i	0.126937i
$732$	0	0
$733$	11.8792i	0.438770i	0.975638	+	0.219385i	$0.0704050\pi$
$733$	11.8792i	0.438770i	−0.975638	+	0.219385i	$0.929595\pi$
$734$	0	0
$735$	−1.47713	−0.0544849
$736$	0	0
$737$	85.0291	3.13209
$738$	0	0
$739$	9.83652i	0.361842i	0.983498	+	0.180921i	$0.0579079\pi$
$739$	9.83652i	0.361842i	−0.983498	+	0.180921i	$0.942092\pi$
$740$	0	0
$741$	− 5.03948i	− 0.185130i
$742$	0	0
$743$	2.51084	0.0921137	0.0460569	−	0.998939i	$-0.485334\pi$
$743$	2.51084	0.0921137	0.0460569	+	0.998939i	$0.485334\pi$
$744$	0	0
$745$	0.312176	0.0114373
$746$	0	0
$747$	− 5.84351i	− 0.213803i
$748$	0	0
$749$	13.4291i	0.490689i
$750$	0	0
$751$	−18.2715	−0.666735	−0.333367	−	0.942797i	$-0.608185\pi$
$751$	−18.2715	−0.666735	−0.333367	+	0.942797i	$0.608185\pi$
$752$	0	0
$753$	25.8819	0.943188
$754$	0	0
$755$	− 3.27744i	− 0.119278i
$756$	0	0
$757$	13.0029i	0.472598i	0.971680	+	0.236299i	$0.0759344\pi$
$757$	13.0029i	0.472598i	−0.971680	+	0.236299i	$0.924066\pi$
$758$	0	0
$759$	7.64046	0.277331
$760$	0	0
$761$	−3.91848	−0.142045	−0.0710224	−	0.997475i	$-0.522626\pi$
$761$	−3.91848	−0.142045	−0.0710224	+	0.997475i	$0.522626\pi$
$762$	0	0
$763$	− 17.3056i	− 0.626505i
$764$	0	0
$765$	− 0.334267i	− 0.0120854i
$766$	0	0
$767$	−4.51579	−0.163056
$768$	0	0
$769$	−21.9151	−0.790280	−0.395140	−	0.918621i	$-0.629304\pi$
$769$	−21.9151	−0.790280	−0.395140	+	0.918621i	$0.629304\pi$
$770$	0	0
$771$	28.6178i	1.03065i
$772$	0	0
$773$	− 54.0731i	− 1.94487i	−0.233168	−	0.972437i	$-0.574909\pi$
$773$	− 54.0731i	− 1.94487i	0.233168	−	0.972437i	$-0.425091\pi$
$774$	0	0
$775$	3.62019	0.130041
$776$	0	0
$777$	10.8286	0.388474
$778$	0	0
$779$	− 75.5547i	− 2.70703i
$780$	0	0
$781$	− 75.8072i	− 2.71259i
$782$	0	0
$783$	−46.3269	−1.65559
$784$	0	0
$785$	0.207031	0.00738924
$786$	0	0
$787$	23.2548i	0.828942i	0.910062	+	0.414471i	$0.136033\pi$
$787$	23.2548i	0.828942i	−0.910062	+	0.414471i	$0.863967\pi$
$788$	0	0
$789$	− 5.14834i	− 0.183286i
$790$	0	0
$791$	−0.899017	−0.0319654
$792$	0	0
$793$	1.20055	0.0426329
$794$	0	0
$795$	19.0450i	0.675457i
$796$	0	0
$797$	26.8454i	0.950914i	0.879739	+	0.475457i	$0.157717\pi$
$797$	26.8454i	0.950914i	−0.879739	+	0.475457i	$0.842283\pi$
$798$	0	0
$799$	4.68983	0.165914
$800$	0	0
$801$	2.38260	0.0841849
$802$	0	0
$803$	2.66460i	0.0940316i
$804$	0	0
$805$	0.912445i	0.0321594i
$806$	0	0
$807$	44.4866	1.56600
$808$	0	0
$809$	1.86439	0.0655485	0.0327743	−	0.999463i	$-0.489566\pi$
$809$	1.86439	0.0655485	0.0327743	+	0.999463i	$0.489566\pi$
$810$	0	0
$811$	17.0803i	0.599769i	0.953976	+	0.299884i	$0.0969481\pi$
$811$	17.0803i	0.599769i	−0.953976	+	0.299884i	$0.903052\pi$
$812$	0	0
$813$	− 20.0216i	− 0.702187i
$814$	0	0
$815$	10.5721	0.370325
$816$	0	0
$817$	56.1809	1.96552
$818$	0	0
$819$	− 0.417268i	− 0.0145805i
$820$	0	0
$821$	− 13.0461i	− 0.455314i	−0.973741	−	0.227657i	$-0.926894\pi$
$821$	− 13.0461i	− 0.455314i	0.973741	−	0.227657i	$-0.0731064\pi$
$822$	0	0
$823$	42.3612	1.47662	0.738308	−	0.674463i	$-0.235625\pi$
$823$	42.3612	1.47662	0.738308	+	0.674463i	$0.235625\pi$
$824$	0	0
$825$	−8.37362	−0.291532
$826$	0	0
$827$	9.28794i	0.322973i	0.986875	+	0.161487i	$0.0516289\pi$
$827$	9.28794i	0.322973i	−0.986875	+	0.161487i	$0.948371\pi$
$828$	0	0
$829$	40.3923i	1.40288i	0.712728	+	0.701441i	$0.247459\pi$
$829$	40.3923i	1.40288i	−0.712728	+	0.701441i	$0.752541\pi$
$830$	0	0
$831$	−13.1188	−0.455086
$832$	0	0
$833$	0.408601	0.0141572
$834$	0	0
$835$	1.24342i	0.0430302i
$836$	0	0
$837$	20.4172i	0.705721i
$838$	0	0
$839$	21.0874	0.728016	0.364008	−	0.931396i	$-0.381408\pi$
$839$	21.0874	0.728016	0.364008	+	0.931396i	$0.381408\pi$
$840$	0	0
$841$	−38.4742	−1.32670
$842$	0	0
$843$	23.5557i	0.811300i
$844$	0	0
$845$	− 12.7398i	− 0.438264i
$846$	0	0
$847$	21.1356	0.726228
$848$	0	0
$849$	11.6410	0.399517
$850$	0	0
$851$	− 6.68896i	− 0.229295i
$852$	0	0
$853$	7.51271i	0.257230i	0.991695	+	0.128615i	$0.0410532\pi$
$853$	7.51271i	0.257230i	−0.991695	+	0.128615i	$0.958947\pi$
$854$	0	0
$855$	−5.47188	−0.187134
$856$	0	0
$857$	30.5590	1.04388	0.521939	−	0.852983i	$-0.325209\pi$
$857$	30.5590	1.04388	0.521939	+	0.852983i	$0.325209\pi$
$858$	0	0
$859$	− 41.1532i	− 1.40413i	−0.712114	−	0.702064i	$-0.752262\pi$
$859$	− 41.1532i	− 1.40413i	0.712114	−	0.702064i	$-0.247738\pi$
$860$	0	0
$861$	16.6855i	0.568639i
$862$	0	0
$863$	2.03063	0.0691235	0.0345618	−	0.999403i	$-0.488996\pi$
$863$	2.03063	0.0691235	0.0345618	+	0.999403i	$0.488996\pi$
$864$	0	0
$865$	−21.9843	−0.747490
$866$	0	0
$867$	24.8647i	0.844448i
$868$	0	0
$869$	− 57.2552i	− 1.94225i
$870$	0	0
$871$	7.65062	0.259231
$872$	0	0
$873$	8.60582	0.291263
$874$	0	0
$875$	− 1.00000i	− 0.0338062i
$876$	0	0
$877$	13.2790i	0.448401i	0.974543	+	0.224201i	$0.0719771\pi$
$877$	13.2790i	0.448401i	−0.974543	+	0.224201i	$0.928023\pi$
$878$	0	0
$879$	29.3177	0.988863
$880$	0	0
$881$	−41.1953	−1.38790	−0.693952	−	0.720021i	$-0.744132\pi$
$881$	−41.1953	−1.38790	−0.693952	+	0.720021i	$0.744132\pi$
$882$	0	0
$883$	34.5680i	1.16331i	0.813437	+	0.581653i	$0.197594\pi$
$883$	34.5680i	1.16331i	−0.813437	+	0.581653i	$0.802406\pi$
$884$	0	0
$885$	− 13.0777i	− 0.439602i
$886$	0	0
$887$	−34.9834	−1.17463	−0.587314	−	0.809359i	$-0.699815\pi$
$887$	−34.9834	−1.17463	−0.587314	+	0.809359i	$0.699815\pi$
$888$	0	0
$889$	−6.59042	−0.221036
$890$	0	0
$891$	− 33.3130i	− 1.11603i
$892$	0	0
$893$	− 76.7715i	− 2.56906i
$894$	0	0
$895$	−8.79114	−0.293855
$896$	0	0
$897$	0.687462	0.0229537
$898$	0	0
$899$	29.7372i	0.991791i
$900$	0	0
$901$	− 5.26819i	− 0.175509i
$902$	0	0
$903$	−12.4070	−0.412878
$904$	0	0
$905$	6.55063	0.217750
$906$	0	0
$907$	− 6.89907i	− 0.229080i	−0.993419	−	0.114540i	$-0.963461\pi$
$907$	− 6.89907i	− 0.229080i	0.993419	−	0.114540i	$-0.0365394\pi$
$908$	0	0
$909$	− 7.05682i	− 0.234060i
$910$	0	0
$911$	6.11442	0.202580	0.101290	−	0.994857i	$-0.467703\pi$
$911$	6.11442	0.202580	0.101290	+	0.994857i	$0.467703\pi$
$912$	0	0
$913$	40.4925	1.34011
$914$	0	0
$915$	3.47680i	0.114939i
$916$	0	0
$917$	− 6.71433i	− 0.221727i
$918$	0	0
$919$	−55.0100	−1.81461	−0.907306	−	0.420471i	$-0.861865\pi$
$919$	−55.0100	−1.81461	−0.907306	+	0.420471i	$0.861865\pi$
$920$	0	0
$921$	−12.2923	−0.405043
$922$	0	0
$923$	− 6.82087i	− 0.224512i
$924$	0	0
$925$	7.33081i	0.241036i
$926$	0	0
$927$	−9.38966	−0.308397
$928$	0	0
$929$	−1.71389	−0.0562309	−0.0281154	−	0.999605i	$-0.508951\pi$
$929$	−1.71389	−0.0562309	−0.0281154	+	0.999605i	$0.508951\pi$
$930$	0	0
$931$	− 6.68872i	− 0.219214i
$932$	0	0
$933$	31.5631i	1.03333i
$934$	0	0
$935$	2.31629	0.0757508
$936$	0	0
$937$	−24.5891	−0.803290	−0.401645	−	0.915795i	$-0.631561\pi$
$937$	−24.5891	−0.803290	−0.401645	+	0.915795i	$0.631561\pi$
$938$	0	0
$939$	15.0489i	0.491102i
$940$	0	0
$941$	− 8.89981i	− 0.290125i	−0.989422	−	0.145063i	$-0.953662\pi$
$941$	− 8.89981i	− 0.290125i	0.989422	−	0.145063i	$-0.0463384\pi$
$942$	0	0
$943$	10.3068	0.335636
$944$	0	0
$945$	5.63981	0.183463
$946$	0	0
$947$	10.9396i	0.355488i	0.984077	+	0.177744i	$0.0568799\pi$
$947$	10.9396i	0.355488i	−0.984077	+	0.177744i	$0.943120\pi$
$948$	0	0
$949$	0.239751i	0.00778265i
$950$	0	0
$951$	−13.2128	−0.428455
$952$	0	0
$953$	53.3106	1.72690	0.863449	−	0.504436i	$-0.168299\pi$
$953$	53.3106	1.72690	0.863449	+	0.504436i	$0.168299\pi$
$954$	0	0
$955$	− 3.91904i	− 0.126817i
$956$	0	0
$957$	− 68.7831i	− 2.22344i
$958$	0	0
$959$	2.35375	0.0760064
$960$	0	0
$961$	−17.8942	−0.577233
$962$	0	0
$963$	− 10.9860i	− 0.354020i
$964$	0	0
$965$	− 10.6279i	− 0.342123i
$966$	0	0
$967$	−32.1122	−1.03266	−0.516329	−	0.856390i	$-0.672702\pi$
$967$	−32.1122	−1.03266	−0.516329	+	0.856390i	$0.672702\pi$
$968$	0	0
$969$	−4.03704	−0.129688
$970$	0	0
$971$	− 33.6762i	− 1.08072i	−0.841434	−	0.540360i	$-0.818288\pi$
$971$	− 33.6762i	− 1.08072i	0.841434	−	0.540360i	$-0.181712\pi$
$972$	0	0
$973$	1.89916i	0.0608844i
$974$	0	0
$975$	−0.753429	−0.0241290
$976$	0	0
$977$	−31.2493	−0.999754	−0.499877	−	0.866096i	$-0.666621\pi$
$977$	−31.2493	−0.999754	−0.499877	+	0.866096i	$0.666621\pi$
$978$	0	0
$979$	16.5101i	0.527667i
$980$	0	0
$981$	14.1573i	0.452007i
$982$	0	0
$983$	−33.4001	−1.06530	−0.532648	−	0.846337i	$-0.678803\pi$
$983$	−33.4001	−1.06530	−0.532648	+	0.846337i	$0.678803\pi$
$984$	0	0
$985$	13.1498	0.418989
$986$	0	0
$987$	16.9542i	0.539658i
$988$	0	0
$989$	7.66394i	0.243699i
$990$	0	0
$991$	−48.7791	−1.54952	−0.774760	−	0.632255i	$-0.782129\pi$
$991$	−48.7791	−1.54952	−0.774760	+	0.632255i	$0.782129\pi$
$992$	0	0
$993$	48.5439	1.54050
$994$	0	0
$995$	− 12.7924i	− 0.405546i
$996$	0	0
$997$	19.8477i	0.628583i	0.949326	+	0.314292i	$0.101767\pi$
$997$	19.8477i	0.628583i	−0.949326	+	0.314292i	$0.898233\pi$
$998$	0	0
$999$	−41.3444	−1.30808

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.b.g.1121.4	✓	12
4.3	odd	2		2240.2.b.h.1121.9	yes	12
8.3	odd	2		2240.2.b.h.1121.4	yes	12
8.5	even	2	inner	2240.2.b.g.1121.9	yes	12
16.3	odd	4		8960.2.a.cb.1.2		6
16.5	even	4		8960.2.a.ch.1.2		6
16.11	odd	4		8960.2.a.ce.1.5		6
16.13	even	4		8960.2.a.cc.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.b.g.1121.4	✓	12	1.1	even	1	trivial
2240.2.b.g.1121.9	yes	12	8.5	even	2	inner
2240.2.b.h.1121.4	yes	12	8.3	odd	2
2240.2.b.h.1121.9	yes	12	4.3	odd	2
8960.2.a.cb.1.2		6	16.3	odd	4
8960.2.a.cc.1.5		6	16.13	even	4
8960.2.a.ce.1.5		6	16.11	odd	4
8960.2.a.ch.1.2		6	16.5	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.47713i q^{3} -1.00000i q^{5} -1.00000 q^{7} +0.818075 q^{9} +O(q^{10})\)	\(q-1.47713i q^{3} -1.00000i q^{5} -1.00000 q^{7} +0.818075 q^{9} +5.66883i q^{11} +0.510061i q^{13} -1.47713 q^{15} +0.408601 q^{17} -6.68872i q^{19} +1.47713i q^{21} +0.912445 q^{23} -1.00000 q^{25} -5.63981i q^{27} -8.21427i q^{29} -3.62019 q^{31} +8.37362 q^{33} +1.00000i q^{35} -7.33081i q^{37} +0.753429 q^{39} +11.2958 q^{41} +8.39935i q^{43} -0.818075i q^{45} +11.4778 q^{47} +1.00000 q^{49} -0.603559i q^{51} -12.8932i q^{53} +5.66883 q^{55} -9.88014 q^{57} +8.85343i q^{59} -2.35375i q^{61} -0.818075 q^{63} +0.510061 q^{65} -14.9994i q^{67} -1.34780i q^{69} -13.3726 q^{71} +0.470044 q^{73} +1.47713i q^{75} -5.66883i q^{77} -10.1000 q^{79} -5.87653 q^{81} -7.14301i q^{83} -0.408601i q^{85} -12.1336 q^{87} +2.91244 q^{89} -0.510061i q^{91} +5.34751i q^{93} -6.68872 q^{95} +10.5196 q^{97} +4.63753i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{7} - 20 q^{9}+O(q^{10}) \) 12 * q - 12 * q^7 - 20 * q^9	\( 12 q - 12 q^{7} - 20 q^{9} + 16 q^{17} - 8 q^{23} - 12 q^{25} + 8 q^{31} + 72 q^{33} + 32 q^{39} + 32 q^{47} + 12 q^{49} - 8 q^{55} + 8 q^{57} + 20 q^{63} + 8 q^{65} - 48 q^{71} + 8 q^{73} + 16 q^{79} + 92 q^{81} + 32 q^{87} + 16 q^{89} + 32 q^{97}+O(q^{100}) \) 12 * q - 12 * q^7 - 20 * q^9 + 16 * q^17 - 8 * q^23 - 12 * q^25 + 8 * q^31 + 72 * q^33 + 32 * q^39 + 32 * q^47 + 12 * q^49 - 8 * q^55 + 8 * q^57 + 20 * q^63 + 8 * q^65 - 48 * q^71 + 8 * q^73 + 16 * q^79 + 92 * q^81 + 32 * q^87 + 16 * q^89 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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