LMFDB - Embedded newform 2288.2.j.k.1585.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2288,2,Mod(1585,2288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2288, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("2288.1585");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2288 = 2^{4} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2288.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$18.2697719825$
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} + 19x^{10} + 133x^{8} + 423x^{6} + 601x^{4} + 312x^{2} + 36 $ x^12 + 19x^10 + 133x^8 + 423x^6 + 601x^4 + 312*x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 143)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1585.11
Root			$-0.871160i$ of defining polynomial
Character	$\chi$	$=$	2288.1585
Dual form			2288.2.j.k.1585.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.06458 q^{3} -3.32707i q^{5} +2.06458i q^{7} +6.39165 q^{9} +O(q^{10})$	$q+3.06458 q^{3} -3.32707i q^{5} +2.06458i q^{7} +6.39165 q^{9} +1.00000i q^{11} +(2.24778 - 2.81912i) q^{13} -10.1961i q^{15} -1.15609 q^{17} +4.63592i q^{19} +6.32707i q^{21} +6.38917 q^{23} -6.06939 q^{25} +10.3940 q^{27} +3.75573 q^{29} -7.80923i q^{31} +3.06458i q^{33} +6.86900 q^{35} +0.727430i q^{37} +(6.88851 - 8.63943i) q^{39} +0.908490i q^{41} -11.6272 q^{43} -21.2655i q^{45} -0.353001i q^{47} +2.73751 q^{49} -3.54293 q^{51} -2.59483 q^{53} +3.32707 q^{55} +14.2071i q^{57} -7.06443i q^{59} +1.03861 q^{61} +13.1961i q^{63} +(-9.37942 - 7.47853i) q^{65} -11.3705i q^{67} +19.5801 q^{69} +8.01241i q^{71} +2.09051i q^{73} -18.6001 q^{75} -2.06458 q^{77} +1.62657 q^{79} +12.6782 q^{81} +8.35697i q^{83} +3.84639i q^{85} +11.5097 q^{87} +11.5737i q^{89} +(5.82031 + 4.64073i) q^{91} -23.9320i q^{93} +15.4240 q^{95} +1.80923i q^{97} +6.39165i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{3} + 12 q^{9}+O(q^{10}) $ 12 * q + 4 * q^3 + 12 * q^9	$ 12 q + 4 q^{3} + 12 q^{9} - 8 q^{13} - 12 q^{17} + 4 q^{23} - 16 q^{25} + 28 q^{27} + 8 q^{29} - 16 q^{35} - 4 q^{39} - 12 q^{43} + 32 q^{49} - 20 q^{53} + 8 q^{55} - 12 q^{61} - 20 q^{65} + 52 q^{69} + 8 q^{75} + 8 q^{77} + 48 q^{79} - 36 q^{81} - 12 q^{87} + 4 q^{95}+O(q^{100}) $ 12 * q + 4 * q^3 + 12 * q^9 - 8 * q^13 - 12 * q^17 + 4 * q^23 - 16 * q^25 + 28 * q^27 + 8 * q^29 - 16 * q^35 - 4 * q^39 - 12 * q^43 + 32 * q^49 - 20 * q^53 + 8 * q^55 - 12 * q^61 - 20 * q^65 + 52 * q^69 + 8 * q^75 + 8 * q^77 + 48 * q^79 - 36 * q^81 - 12 * q^87 + 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2288\mathbb{Z}\right)^\times$.

$n$	$287$	$353$	$1717$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	3.06458			1.76934			0.884668	−	0.466222i	$-0.154385\pi$
$3$	3.06458			1.76934			0.884668	+	0.466222i	$0.154385\pi$
$4$		0			0
$5$		−	3.32707i		−	1.48791i	−0.668229	−	0.743955i	$-0.732948\pi$
$5$		−	3.32707i		−	1.48791i	0.668229	−	0.743955i	$-0.267052\pi$
$6$		0			0
$7$			2.06458i			0.780338i	0.920743	+	0.390169i	$0.127583\pi$
$7$			2.06458i			0.780338i	−0.920743	+	0.390169i	$0.872417\pi$
$8$		0			0
$9$	6.39165			2.13055
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$	2.24778	−	2.81912i	0.623423	−	0.781885i
$13$	2.24778	−	2.81912i	0.623423	−	0.781885i
$14$		0			0
$15$		−	10.1961i		−	2.63261i
$16$		0			0
$17$	−1.15609			−0.280393			−0.140196	−	0.990124i	$-0.544773\pi$
$17$	−1.15609			−0.280393			−0.140196	+	0.990124i	$0.544773\pi$
$18$		0			0
$19$			4.63592i			1.06355i	0.846885	+	0.531777i	$0.178475\pi$
$19$			4.63592i			1.06355i	−0.846885	+	0.531777i	$0.821525\pi$
$20$		0			0
$21$			6.32707i			1.38068i
$22$		0			0
$23$	6.38917			1.33223			0.666117	−	0.745847i	$-0.267955\pi$
$23$	6.38917			1.33223			0.666117	+	0.745847i	$0.267955\pi$
$24$		0			0
$25$	−6.06939			−1.21388
$26$		0			0
$27$	10.3940			2.00032
$28$		0			0
$29$	3.75573			0.697421			0.348711	−	0.937230i	$-0.386620\pi$
$29$	3.75573			0.697421			0.348711	+	0.937230i	$0.386620\pi$
$30$		0			0
$31$		−	7.80923i		−	1.40258i	−0.712877	−	0.701290i	$-0.752608\pi$
$31$		−	7.80923i		−	1.40258i	0.712877	−	0.701290i	$-0.247392\pi$
$32$		0			0
$33$			3.06458i			0.533475i
$34$		0			0
$35$	6.86900			1.16107
$36$		0			0
$37$			0.727430i			0.119589i	0.998211	+	0.0597944i	$0.0190445\pi$
$37$			0.727430i			0.119589i	−0.998211	+	0.0597944i	$0.980955\pi$
$38$		0			0
$39$	6.88851	−	8.63943i	1.10305	−	1.38342i
$40$		0			0
$41$			0.908490i			0.141882i	0.997481	+	0.0709412i	$0.0226003\pi$
$41$			0.908490i			0.141882i	−0.997481	+	0.0709412i	$0.977400\pi$
$42$		0			0
$43$	−11.6272			−1.77313			−0.886566	−	0.462602i	$-0.846916\pi$
$43$	−11.6272			−1.77313			−0.886566	+	0.462602i	$0.846916\pi$
$44$		0			0
$45$		−	21.2655i		−	3.17007i
$46$		0			0
$47$		−	0.353001i		−	0.0514905i	−0.999669	−	0.0257452i	$-0.991804\pi$
$47$		−	0.353001i		−	0.0514905i	0.999669	−	0.0257452i	$-0.00819587\pi$
$48$		0			0
$49$	2.73751			0.391073
$50$		0			0
$51$	−3.54293			−0.496109
$52$		0			0
$53$	−2.59483			−0.356427			−0.178214	−	0.983992i	$-0.557032\pi$
$53$	−2.59483			−0.356427			−0.178214	+	0.983992i	$0.557032\pi$
$54$		0			0
$55$	3.32707			0.448622
$56$		0			0
$57$			14.2071i			1.88178i
$58$		0			0
$59$		−	7.06443i		−	0.919710i	−0.887994	−	0.459855i	$-0.847901\pi$
$59$		−	7.06443i		−	0.919710i	0.887994	−	0.459855i	$-0.152099\pi$
$60$		0			0
$61$	1.03861			0.132981			0.0664903	−	0.997787i	$-0.478820\pi$
$61$	1.03861			0.132981			0.0664903	+	0.997787i	$0.478820\pi$
$62$		0			0
$63$			13.1961i			1.66255i
$64$		0			0
$65$	−9.37942	−	7.47853i	−1.16337	−	0.927598i
$66$		0			0
$67$		−	11.3705i		−	1.38913i	−0.719429	−	0.694565i	$-0.755597\pi$
$67$		−	11.3705i		−	1.38913i	0.719429	−	0.694565i	$-0.244403\pi$
$68$		0			0
$69$	19.5801			2.35717
$70$		0			0
$71$			8.01241i			0.950898i	0.879743	+	0.475449i	$0.157714\pi$
$71$			8.01241i			0.950898i	−0.879743	+	0.475449i	$0.842286\pi$
$72$		0			0
$73$			2.09051i			0.244676i	0.992489	+	0.122338i	$0.0390392\pi$
$73$			2.09051i			0.244676i	−0.992489	+	0.122338i	$0.960961\pi$
$74$		0			0
$75$	−18.6001			−2.14776
$76$		0			0
$77$	−2.06458			−0.235281
$78$		0			0
$79$	1.62657			0.183003			0.0915017	−	0.995805i	$-0.470833\pi$
$79$	1.62657			0.183003			0.0915017	+	0.995805i	$0.470833\pi$
$80$		0			0
$81$	12.6782			1.40869
$82$		0			0
$83$			8.35697i			0.917296i	0.888618	+	0.458648i	$0.151666\pi$
$83$			8.35697i			0.917296i	−0.888618	+	0.458648i	$0.848334\pi$
$84$		0			0
$85$			3.84639i			0.417200i
$86$		0			0
$87$	11.5097			1.23397
$88$		0			0
$89$			11.5737i			1.22681i	0.789768	+	0.613405i	$0.210201\pi$
$89$			11.5737i			1.22681i	−0.789768	+	0.613405i	$0.789799\pi$
$90$		0			0
$91$	5.82031	+	4.64073i	0.610134	+	0.486481i
$92$		0			0
$93$		−	23.9320i		−	2.48163i
$94$		0			0
$95$	15.4240			1.58247
$96$		0			0
$97$			1.80923i			0.183699i	0.995773	+	0.0918497i	$0.0292779\pi$
$97$			1.80923i			0.183699i	−0.995773	+	0.0918497i	$0.970722\pi$
$98$		0			0
$99$			6.39165i			0.642385i
$100$		0			0
$101$	−5.41614			−0.538926			−0.269463	−	0.963011i	$-0.586846\pi$
$101$	−5.41614			−0.538926			−0.269463	+	0.963011i	$0.586846\pi$
$102$		0			0
$103$	−14.3159			−1.41059			−0.705293	−	0.708916i	$-0.749184\pi$
$103$	−14.3159			−1.41059			−0.705293	+	0.708916i	$0.749184\pi$
$104$		0			0
$105$	21.0506			2.05433
$106$		0			0
$107$	−13.8945			−1.34323			−0.671616	−	0.740899i	$-0.734400\pi$
$107$	−13.8945			−1.34323			−0.671616	+	0.740899i	$0.734400\pi$
$108$		0			0
$109$		−	14.7470i		−	1.41251i	−0.707959	−	0.706254i	$-0.750384\pi$
$109$		−	14.7470i		−	1.41251i	0.707959	−	0.706254i	$-0.249616\pi$
$110$		0			0
$111$			2.22927i			0.211593i
$112$		0			0
$113$	8.30874			0.781620			0.390810	−	0.920471i	$-0.372195\pi$
$113$	8.30874			0.781620			0.390810	+	0.920471i	$0.372195\pi$
$114$		0			0
$115$		−	21.2572i		−	1.98224i
$116$		0			0
$117$	14.3670	−	18.0189i	1.32823	−	1.66584i
$118$		0			0
$119$		−	2.38684i		−	0.218801i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$			2.78414i			0.251037i
$124$		0			0
$125$			3.55793i			0.318231i
$126$		0			0
$127$	17.8333			1.58245			0.791226	−	0.611524i	$-0.209443\pi$
$127$	17.8333			1.58245			0.791226	+	0.611524i	$0.209443\pi$
$128$		0			0
$129$	−35.6325			−3.13727
$130$		0			0
$131$	−12.2085			−1.06666			−0.533330	−	0.845907i	$-0.679060\pi$
$131$	−12.2085			−1.06666			−0.533330	+	0.845907i	$0.679060\pi$
$132$		0			0
$133$	−9.57123			−0.829931
$134$		0			0
$135$		−	34.5815i		−	2.97630i
$136$		0			0
$137$			10.7281i			0.916561i	0.888808	+	0.458280i	$0.151534\pi$
$137$			10.7281i			0.916561i	−0.888808	+	0.458280i	$0.848466\pi$
$138$		0			0
$139$	9.35116			0.793155			0.396578	−	0.918001i	$-0.370198\pi$
$139$	9.35116			0.793155			0.396578	+	0.918001i	$0.370198\pi$
$140$		0			0
$141$		−	1.08180i		−	0.0911039i
$142$		0			0
$143$	2.81912	+	2.24778i	0.235747	+	0.187969i
$144$		0			0
$145$		−	12.4956i		−	1.03770i
$146$		0			0
$147$	8.38932			0.691939
$148$		0			0
$149$			4.24618i			0.347860i	0.984758	+	0.173930i	$0.0556467\pi$
$149$			4.24618i			0.347860i	−0.984758	+	0.173930i	$0.944353\pi$
$150$		0			0
$151$			1.10861i			0.0902176i	0.998982	+	0.0451088i	$0.0143634\pi$
$151$			1.10861i			0.0902176i	−0.998982	+	0.0451088i	$0.985637\pi$
$152$		0			0
$153$	−7.38932			−0.597391
$154$		0			0
$155$	−25.9818			−2.08691
$156$		0			0
$157$	−9.83299			−0.784758			−0.392379	−	0.919804i	$-0.628348\pi$
$157$	−9.83299			−0.784758			−0.392379	+	0.919804i	$0.628348\pi$
$158$		0			0
$159$	−7.95206			−0.630639
$160$		0			0
$161$			13.1909i			1.03959i
$162$		0			0
$163$			4.87396i			0.381758i	0.981614	+	0.190879i	$0.0611338\pi$
$163$			4.87396i			0.381758i	−0.981614	+	0.190879i	$0.938866\pi$
$164$		0			0
$165$	10.1961			0.793763
$166$		0			0
$167$			2.37997i			0.184167i	0.995751	+	0.0920837i	$0.0293527\pi$
$167$			2.37997i			0.184167i	−0.995751	+	0.0920837i	$0.970647\pi$
$168$		0			0
$169$	−2.89493	−	12.6736i	−0.222687	−	0.974890i
$170$		0			0
$171$			29.6312i			2.26595i
$172$		0			0
$173$	−5.10719			−0.388292			−0.194146	−	0.980973i	$-0.562194\pi$
$173$	−5.10719			−0.388292			−0.194146	+	0.980973i	$0.562194\pi$
$174$		0			0
$175$		−	12.5307i		−	0.947235i
$176$		0			0
$177$		−	21.6495i		−	1.62728i
$178$		0			0
$179$	−1.38157			−0.103263			−0.0516316	−	0.998666i	$-0.516442\pi$
$179$	−1.38157			−0.103263			−0.0516316	+	0.998666i	$0.516442\pi$
$180$		0			0
$181$	−10.4475			−0.776558			−0.388279	−	0.921542i	$-0.626930\pi$
$181$	−10.4475			−0.776558			−0.388279	+	0.921542i	$0.626930\pi$
$182$		0			0
$183$	3.18291			0.235287
$184$		0			0
$185$	2.42021			0.177937
$186$		0			0
$187$		−	1.15609i		−	0.0845417i
$188$		0			0
$189$			21.4592i			1.56093i
$190$		0			0
$191$	−5.69160			−0.411830			−0.205915	−	0.978570i	$-0.566017\pi$
$191$	−5.69160			−0.411830			−0.205915	+	0.978570i	$0.566017\pi$
$192$		0			0
$193$			8.63038i			0.621228i	0.950536	+	0.310614i	$0.100535\pi$
$193$			8.63038i			0.621228i	−0.950536	+	0.310614i	$0.899465\pi$
$194$		0			0
$195$	−28.7440	−	22.9186i	−2.05840	−	1.64123i
$196$		0			0
$197$			20.1705i			1.43709i	0.695482	+	0.718543i	$0.255191\pi$
$197$			20.1705i			1.43709i	−0.695482	+	0.718543i	$0.744809\pi$
$198$		0			0
$199$	−6.46631			−0.458384			−0.229192	−	0.973381i	$-0.573608\pi$
$199$	−6.46631			−0.458384			−0.229192	+	0.973381i	$0.573608\pi$
$200$		0			0
$201$		−	34.8459i		−	2.45784i
$202$		0			0
$203$			7.75400i			0.544224i
$204$		0			0
$205$	3.02261			0.211108
$206$		0			0
$207$	40.8373			2.83839
$208$		0			0
$209$	−4.63592			−0.320673
$210$		0			0
$211$	4.32386			0.297667			0.148833	−	0.988862i	$-0.452448\pi$
$211$	4.32386			0.297667			0.148833	+	0.988862i	$0.452448\pi$
$212$		0			0
$213$			24.5547i			1.68246i
$214$		0			0
$215$			38.6845i			2.63826i
$216$		0			0
$217$	16.1228			1.09449
$218$		0			0
$219$			6.40654i			0.432914i
$220$		0			0
$221$	−2.59864	+	3.25916i	−0.174803	+	0.219235i
$222$		0			0
$223$			3.89925i			0.261113i	0.991441	+	0.130557i	$0.0416764\pi$
$223$			3.89925i			0.261113i	−0.991441	+	0.130557i	$0.958324\pi$
$224$		0			0
$225$	−38.7934			−2.58623
$226$		0			0
$227$			26.5934i			1.76506i	0.470252	+	0.882532i	$0.344163\pi$
$227$			26.5934i			1.76506i	−0.470252	+	0.882532i	$0.655837\pi$
$228$		0			0
$229$			20.2894i			1.34076i	0.742018	+	0.670380i	$0.233869\pi$
$229$			20.2894i			1.34076i	−0.742018	+	0.670380i	$0.766131\pi$
$230$		0			0
$231$	−6.32707			−0.416291
$232$		0			0
$233$	27.5550			1.80519			0.902593	−	0.430494i	$-0.141661\pi$
$233$	27.5550			1.80519			0.902593	+	0.430494i	$0.141661\pi$
$234$		0			0
$235$	−1.17446			−0.0766132
$236$		0			0
$237$	4.98475			0.323794
$238$		0			0
$239$			5.47375i			0.354067i	0.984205	+	0.177034i	$0.0566502\pi$
$239$			5.47375i			0.354067i	−0.984205	+	0.177034i	$0.943350\pi$
$240$		0			0
$241$			22.0086i			1.41770i	0.705359	+	0.708850i	$0.250786\pi$
$241$			22.0086i			1.41770i	−0.705359	+	0.708850i	$0.749214\pi$
$242$		0			0
$243$	7.67151			0.492127
$244$		0			0
$245$		−	9.10789i		−	0.581882i
$246$		0			0
$247$	13.0692	+	10.4205i	0.831576	+	0.663044i
$248$		0			0
$249$			25.6106i			1.62301i
$250$		0			0
$251$	25.9587			1.63850			0.819250	−	0.573436i	$-0.194390\pi$
$251$	25.9587			1.63850			0.819250	+	0.573436i	$0.194390\pi$
$252$		0			0
$253$			6.38917i			0.401684i
$254$		0			0
$255$			11.7876i			0.738166i
$256$		0			0
$257$	5.38772			0.336077			0.168038	−	0.985780i	$-0.446257\pi$
$257$	5.38772			0.336077			0.168038	+	0.985780i	$0.446257\pi$
$258$		0			0
$259$	−1.50184			−0.0933197
$260$		0			0
$261$	24.0053			1.48589
$262$		0			0
$263$	−28.3609			−1.74881			−0.874405	−	0.485197i	$-0.838748\pi$
$263$	−28.3609			−1.74881			−0.874405	+	0.485197i	$0.838748\pi$
$264$		0			0
$265$			8.63318i			0.530332i
$266$		0			0
$267$			35.4685i			2.17064i
$268$		0			0
$269$	−4.51490			−0.275278			−0.137639	−	0.990482i	$-0.543951\pi$
$269$	−4.51490			−0.275278			−0.137639	+	0.990482i	$0.543951\pi$
$270$		0			0
$271$		−	22.3308i		−	1.35650i	−0.734832	−	0.678249i	$-0.762739\pi$
$271$		−	22.3308i		−	1.35650i	0.734832	−	0.678249i	$-0.237261\pi$
$272$		0			0
$273$	17.8368	+	14.2219i	1.07953	+	0.860748i
$274$		0			0
$275$		−	6.06939i		−	0.365998i
$276$		0			0
$277$	5.35812			0.321938			0.160969	−	0.986959i	$-0.448538\pi$
$277$	5.35812			0.321938			0.160969	+	0.986959i	$0.448538\pi$
$278$		0			0
$279$		−	49.9139i		−	2.98826i
$280$		0			0
$281$			10.5001i			0.626386i	0.949689	+	0.313193i	$0.101399\pi$
$281$			10.5001i			0.626386i	−0.949689	+	0.313193i	$0.898601\pi$
$282$		0			0
$283$	−12.0575			−0.716744			−0.358372	−	0.933579i	$-0.616668\pi$
$283$	−12.0575			−0.716744			−0.358372	+	0.933579i	$0.616668\pi$
$284$		0			0
$285$	47.2682			2.79992
$286$		0			0
$287$	−1.87565			−0.110716
$288$		0			0
$289$	−15.6635			−0.921380
$290$		0			0
$291$			5.54453i			0.325026i
$292$		0			0
$293$		−	18.8107i		−	1.09893i	−0.835516	−	0.549467i	$-0.814831\pi$
$293$		−	18.8107i		−	1.09893i	0.835516	−	0.549467i	$-0.185169\pi$
$294$		0			0
$295$	−23.5038			−1.36845
$296$		0			0
$297$			10.3940i			0.603120i
$298$		0			0
$299$	14.3615	−	18.0119i	0.830545	−	1.04165i
$300$		0			0
$301$		−	24.0053i		−	1.38364i
$302$		0			0
$303$	−16.5982			−0.953541
$304$		0			0
$305$		−	3.45553i		−	0.197863i
$306$		0			0
$307$		−	10.2726i		−	0.586288i	−0.956068	−	0.293144i	$-0.905298\pi$
$307$		−	10.2726i		−	0.586288i	0.956068	−	0.293144i	$-0.0947016\pi$
$308$		0			0
$309$	−43.8721			−2.49580
$310$		0			0
$311$	−2.03659			−0.115484			−0.0577421	−	0.998332i	$-0.518390\pi$
$311$	−2.03659			−0.115484			−0.0577421	+	0.998332i	$0.518390\pi$
$312$		0			0
$313$	−11.5631			−0.653582			−0.326791	−	0.945097i	$-0.605967\pi$
$313$	−11.5631			−0.653582			−0.326791	+	0.945097i	$0.605967\pi$
$314$		0			0
$315$	43.9042			2.47372
$316$		0			0
$317$		−	25.3143i		−	1.42179i	−0.703298	−	0.710895i	$-0.748290\pi$
$317$		−	25.3143i		−	1.42179i	0.703298	−	0.710895i	$-0.251710\pi$
$318$		0			0
$319$			3.75573i			0.210280i
$320$		0			0
$321$	−42.5808			−2.37663
$322$		0			0
$323$		−	5.35954i		−	0.298213i
$324$		0			0
$325$	−13.6427	+	17.1104i	−0.756760	+	0.949112i
$326$		0			0
$327$		−	45.1934i		−	2.49920i
$328$		0			0
$329$	0.728798			0.0401800
$330$		0			0
$331$			2.92453i			0.160747i	0.996765	+	0.0803733i	$0.0256112\pi$
$331$			2.92453i			0.160747i	−0.996765	+	0.0803733i	$0.974389\pi$
$332$		0			0
$333$			4.64948i			0.254790i
$334$		0			0
$335$	−37.8305			−2.06690
$336$		0			0
$337$	−7.19897			−0.392153			−0.196077	−	0.980589i	$-0.562820\pi$
$337$	−7.19897			−0.392153			−0.196077	+	0.980589i	$0.562820\pi$
$338$		0			0
$339$	25.4628			1.38295
$340$		0			0
$341$	7.80923			0.422893
$342$		0			0
$343$			20.1039i			1.08551i
$344$		0			0
$345$		−	65.1444i		−	3.50726i
$346$		0			0
$347$	35.8182			1.92282			0.961410	−	0.275120i	$-0.0887176\pi$
$347$	35.8182			1.92282			0.961410	+	0.275120i	$0.0887176\pi$
$348$		0			0
$349$		−	3.82095i		−	0.204531i	−0.994757	−	0.102265i	$-0.967391\pi$
$349$		−	3.82095i		−	0.204531i	0.994757	−	0.102265i	$-0.0326091\pi$
$350$		0			0
$351$	23.3634	−	29.3019i	1.24705	−	1.56402i
$352$		0			0
$353$			9.19855i			0.489589i	0.969575	+	0.244795i	$0.0787206\pi$
$353$			9.19855i			0.489589i	−0.969575	+	0.244795i	$0.921279\pi$
$354$		0			0
$355$	26.6578			1.41485
$356$		0			0
$357$		−	7.31466i		−	0.387133i
$358$		0			0
$359$			4.37439i			0.230872i	0.993315	+	0.115436i	$0.0368265\pi$
$359$			4.37439i			0.230872i	−0.993315	+	0.115436i	$0.963174\pi$
$360$		0			0
$361$	−2.49176			−0.131145
$362$		0			0
$363$	−3.06458			−0.160849
$364$		0			0
$365$	6.95528			0.364056
$366$		0			0
$367$	24.9387			1.30179			0.650895	−	0.759168i	$-0.274394\pi$
$367$	24.9387			1.30179			0.650895	+	0.759168i	$0.274394\pi$
$368$		0			0
$369$			5.80675i			0.302287i
$370$		0			0
$371$		−	5.35723i		−	0.278134i
$372$		0			0
$373$	−21.0672			−1.09082			−0.545410	−	0.838169i	$-0.683626\pi$
$373$	−21.0672			−1.09082			−0.545410	+	0.838169i	$0.683626\pi$
$374$		0			0
$375$			10.9036i			0.563058i
$376$		0			0
$377$	8.44207	−	10.5879i	0.434789	−	0.545303i
$378$		0			0
$379$			11.0128i			0.565689i	0.959166	+	0.282845i	$0.0912781\pi$
$379$			11.0128i			0.565689i	−0.959166	+	0.282845i	$0.908722\pi$
$380$		0			0
$381$	54.6516			2.79989
$382$		0			0
$383$			1.11191i			0.0568158i	0.999596	+	0.0284079i	$0.00904373\pi$
$383$			1.11191i			0.0568158i	−0.999596	+	0.0284079i	$0.990956\pi$
$384$		0			0
$385$			6.86900i			0.350077i
$386$		0			0
$387$	−74.3170			−3.77775
$388$		0			0
$389$	−18.1420			−0.919836			−0.459918	−	0.887962i	$-0.652121\pi$
$389$	−18.1420			−0.919836			−0.459918	+	0.887962i	$0.652121\pi$
$390$		0			0
$391$	−7.38645			−0.373549
$392$		0			0
$393$	−37.4139			−1.88728
$394$		0			0
$395$		−	5.41171i		−	0.272293i
$396$		0			0
$397$			31.5241i			1.58215i	0.611720	+	0.791074i	$0.290478\pi$
$397$			31.5241i			1.58215i	−0.611720	+	0.791074i	$0.709522\pi$
$398$		0			0
$399$	−29.3318			−1.46843
$400$		0			0
$401$			30.3679i			1.51650i	0.651963	+	0.758251i	$0.273946\pi$
$401$			30.3679i			1.51650i	−0.651963	+	0.758251i	$0.726054\pi$
$402$		0			0
$403$	−22.0152	−	17.5535i	−1.09665	−	0.874400i
$404$		0			0
$405$		−	42.1813i		−	2.09601i
$406$		0			0
$407$	−0.727430			−0.0360574
$408$		0			0
$409$		−	32.6574i		−	1.61481i	−0.590001	−	0.807403i	$-0.700873\pi$
$409$		−	32.6574i		−	1.61481i	0.590001	−	0.807403i	$-0.299127\pi$
$410$		0			0
$411$			32.8770i			1.62170i
$412$		0			0
$413$	14.5851			0.717685
$414$		0			0
$415$	27.8042			1.36486
$416$		0			0
$417$	28.6574			1.40336
$418$		0			0
$419$	12.1917			0.595602			0.297801	−	0.954628i	$-0.403747\pi$
$419$	12.1917			0.595602			0.297801	+	0.954628i	$0.403747\pi$
$420$		0			0
$421$		−	31.7855i		−	1.54913i	−0.632495	−	0.774565i	$-0.717969\pi$
$421$		−	31.7855i		−	1.54913i	0.632495	−	0.774565i	$-0.282031\pi$
$422$		0			0
$423$		−	2.25626i		−	0.109703i
$424$		0			0
$425$	7.01676			0.340363
$426$		0			0
$427$			2.14430i			0.103770i
$428$		0			0
$429$	8.63943	+	6.88851i	0.417116	+	0.332581i
$430$		0			0
$431$			12.1690i			0.586160i	0.956088	+	0.293080i	$0.0946802\pi$
$431$			12.1690i			0.586160i	−0.956088	+	0.293080i	$0.905320\pi$
$432$		0			0
$433$	3.78376			0.181836			0.0909179	−	0.995858i	$-0.471020\pi$
$433$	3.78376			0.181836			0.0909179	+	0.995858i	$0.471020\pi$
$434$		0			0
$435$		−	38.2937i		−	1.83604i
$436$		0			0
$437$			29.6197i			1.41690i
$438$		0			0
$439$	35.1463			1.67744			0.838721	−	0.544561i	$-0.183304\pi$
$439$	35.1463			1.67744			0.838721	+	0.544561i	$0.183304\pi$
$440$		0			0
$441$	17.4972			0.833200
$442$		0			0
$443$	−20.6838			−0.982719			−0.491360	−	0.870957i	$-0.663500\pi$
$443$	−20.6838			−0.982719			−0.491360	+	0.870957i	$0.663500\pi$
$444$		0			0
$445$	38.5065			1.82538
$446$		0			0
$447$			13.0127i			0.615482i
$448$		0			0
$449$			3.16448i			0.149341i	0.997208	+	0.0746706i	$0.0237905\pi$
$449$			3.16448i			0.149341i	−0.997208	+	0.0746706i	$0.976209\pi$
$450$		0			0
$451$	−0.908490			−0.0427791
$452$		0			0
$453$			3.39743i			0.159625i
$454$		0			0
$455$	15.4400	−	19.3646i	0.723840	−	0.907825i
$456$		0			0
$457$			9.72817i			0.455064i	0.973771	+	0.227532i	$0.0730657\pi$
$457$			9.72817i			0.455064i	−0.973771	+	0.227532i	$0.926934\pi$
$458$		0			0
$459$	−12.0164			−0.560876
$460$		0			0
$461$		−	22.5640i		−	1.05091i	−0.850821	−	0.525455i	$-0.823895\pi$
$461$		−	22.5640i		−	1.05091i	0.850821	−	0.525455i	$-0.176105\pi$
$462$		0			0
$463$			31.0928i			1.44500i	0.691369	+	0.722502i	$0.257008\pi$
$463$			31.0928i			1.44500i	−0.691369	+	0.722502i	$0.742992\pi$
$464$		0			0
$465$	−79.6234			−3.69245
$466$		0			0
$467$	13.0692			0.604769			0.302384	−	0.953186i	$-0.402217\pi$
$467$	13.0692			0.604769			0.302384	+	0.953186i	$0.402217\pi$
$468$		0			0
$469$	23.4754			1.08399
$470$		0			0
$471$	−30.1340			−1.38850
$472$		0			0
$473$		−	11.6272i		−	0.534620i
$474$		0			0
$475$		−	28.1372i		−	1.29102i
$476$		0			0
$477$	−16.5852			−0.759386
$478$		0			0
$479$			14.4440i			0.659963i	0.943987	+	0.329982i	$0.107043\pi$
$479$			14.4440i			0.659963i	−0.943987	+	0.329982i	$0.892957\pi$
$480$		0			0
$481$	2.05072	+	1.63511i	0.0935046	+	0.0745544i
$482$		0			0
$483$			40.4247i			1.83939i
$484$		0			0
$485$	6.01943			0.273328
$486$		0			0
$487$		−	42.2880i		−	1.91625i	−0.286346	−	0.958126i	$-0.592441\pi$
$487$		−	42.2880i		−	1.91625i	0.286346	−	0.958126i	$-0.407559\pi$
$488$		0			0
$489$			14.9366i			0.675458i
$490$		0			0
$491$	−1.96012			−0.0884589			−0.0442295	−	0.999021i	$-0.514083\pi$
$491$	−1.96012			−0.0884589			−0.0442295	+	0.999021i	$0.514083\pi$
$492$		0			0
$493$	−4.34196			−0.195552
$494$		0			0
$495$	21.2655			0.955811
$496$		0			0
$497$	−16.5423			−0.742022
$498$		0			0
$499$			1.01120i			0.0452674i	0.999744	+	0.0226337i	$0.00720515\pi$
$499$			1.01120i			0.0452674i	−0.999744	+	0.0226337i	$0.992795\pi$
$500$		0			0
$501$			7.29360i			0.325854i
$502$		0			0
$503$	−16.4128			−0.731811			−0.365905	−	0.930652i	$-0.619241\pi$
$503$	−16.4128			−0.731811			−0.365905	+	0.930652i	$0.619241\pi$
$504$		0			0
$505$			18.0199i			0.801873i
$506$		0			0
$507$	−8.87175	−	38.8392i	−0.394008	−	1.72491i
$508$		0			0
$509$			13.6343i			0.604329i	0.953256	+	0.302165i	$0.0977092\pi$
$509$			13.6343i			0.604329i	−0.953256	+	0.302165i	$0.902291\pi$
$510$		0			0
$511$	−4.31603			−0.190930
$512$		0			0
$513$			48.1857i			2.12745i
$514$		0			0
$515$			47.6299i			2.09882i
$516$		0			0
$517$	0.353001			0.0155250
$518$		0			0
$519$	−15.6514			−0.687020
$520$		0			0
$521$	−19.7115			−0.863574			−0.431787	−	0.901976i	$-0.642117\pi$
$521$	−19.7115			−0.863574			−0.431787	+	0.901976i	$0.642117\pi$
$522$		0			0
$523$	−13.6339			−0.596167			−0.298084	−	0.954540i	$-0.596347\pi$
$523$	−13.6339			−0.596167			−0.298084	+	0.954540i	$0.596347\pi$
$524$		0			0
$525$		−	38.4014i		−	1.67598i
$526$		0			0
$527$			9.02817i			0.393273i
$528$		0			0
$529$	17.8215			0.774847
$530$		0			0
$531$		−	45.1534i		−	1.95949i
$532$		0			0
$533$	2.56115	+	2.04209i	0.110936	+	0.0884527i
$534$		0			0
$535$			46.2280i			1.99861i
$536$		0			0
$537$	−4.23393			−0.182707
$538$		0			0
$539$			2.73751i			0.117913i
$540$		0			0
$541$			14.8788i			0.639689i	0.947470	+	0.319844i	$0.103631\pi$
$541$			14.8788i			0.639689i	−0.947470	+	0.319844i	$0.896369\pi$
$542$		0			0
$543$	−32.0172			−1.37399
$544$		0			0
$545$	−49.0643			−2.10169
$546$		0			0
$547$	−9.91020			−0.423730			−0.211865	−	0.977299i	$-0.567954\pi$
$547$	−9.91020			−0.423730			−0.211865	+	0.977299i	$0.567954\pi$
$548$		0			0
$549$	6.63844			0.283322
$550$		0			0
$551$			17.4113i			0.741745i
$552$		0			0
$553$			3.35818i			0.142804i
$554$		0			0
$555$	7.41693			0.314831
$556$		0			0
$557$		−	17.0997i		−	0.724539i	−0.932073	−	0.362270i	$-0.882002\pi$
$557$		−	17.0997i		−	0.724539i	0.932073	−	0.362270i	$-0.117998\pi$
$558$		0			0
$559$	−26.1355	+	32.7786i	−1.10541	+	1.38639i
$560$		0			0
$561$		−	3.54293i		−	0.149583i
$562$		0			0
$563$	2.32531			0.0980003			0.0490002	−	0.998799i	$-0.484397\pi$
$563$	2.32531			0.0980003			0.0490002	+	0.998799i	$0.484397\pi$
$564$		0			0
$565$		−	27.6437i		−	1.16298i
$566$		0			0
$567$			26.1752i			1.09926i
$568$		0			0
$569$	29.2314			1.22545			0.612723	−	0.790298i	$-0.290074\pi$
$569$	29.2314			1.22545			0.612723	+	0.790298i	$0.290074\pi$
$570$		0			0
$571$	4.07170			0.170395			0.0851976	−	0.996364i	$-0.472848\pi$
$571$	4.07170			0.170395			0.0851976	+	0.996364i	$0.472848\pi$
$572$		0			0
$573$	−17.4424			−0.728665
$574$		0			0
$575$	−38.7784			−1.61717
$576$		0			0
$577$		−	22.0559i		−	0.918200i	−0.888384	−	0.459100i	$-0.848172\pi$
$577$		−	22.0559i		−	0.918200i	0.888384	−	0.459100i	$-0.151828\pi$
$578$		0			0
$579$			26.4485i			1.09916i
$580$		0			0
$581$	−17.2536			−0.715801
$582$		0			0
$583$		−	2.59483i		−	0.107467i
$584$		0			0
$585$	−59.9500	−	47.8002i	−2.47863	−	1.97629i
$586$		0			0
$587$			8.17148i			0.337273i	0.985678	+	0.168636i	$0.0539364\pi$
$587$			8.17148i			0.337273i	−0.985678	+	0.168636i	$0.946064\pi$
$588$		0			0
$589$	36.2030			1.49172
$590$		0			0
$591$			61.8140i			2.54269i
$592$		0			0
$593$			14.4272i			0.592454i	0.955118	+	0.296227i	$0.0957285\pi$
$593$			14.4272i			0.592454i	−0.955118	+	0.296227i	$0.904271\pi$
$594$		0			0
$595$	−7.94118			−0.325557
$596$		0			0
$597$	−19.8165			−0.811036
$598$		0			0
$599$	5.29819			0.216478			0.108239	−	0.994125i	$-0.465479\pi$
$599$	5.29819			0.216478			0.108239	+	0.994125i	$0.465479\pi$
$600$		0			0
$601$	−32.3516			−1.31965			−0.659825	−	0.751419i	$-0.729370\pi$
$601$	−32.3516			−1.31965			−0.659825	+	0.751419i	$0.729370\pi$
$602$		0			0
$603$		−	72.6764i		−	2.95961i
$604$		0			0
$605$			3.32707i			0.135265i
$606$		0			0
$607$	−23.4371			−0.951284			−0.475642	−	0.879639i	$-0.657784\pi$
$607$	−23.4371			−0.951284			−0.475642	+	0.879639i	$0.657784\pi$
$608$		0			0
$609$			23.7628i			0.962915i
$610$		0			0
$611$	−0.995153	−	0.793470i	−0.0402596	−	0.0321003i
$612$		0			0
$613$		−	34.3459i		−	1.38722i	−0.720351	−	0.693610i	$-0.756019\pi$
$613$		−	34.3459i		−	1.38722i	0.720351	−	0.693610i	$-0.243981\pi$
$614$		0			0
$615$	9.26303			0.373521
$616$		0			0
$617$			13.7945i			0.555347i	0.960676	+	0.277673i	$0.0895633\pi$
$617$			13.7945i			0.555347i	−0.960676	+	0.277673i	$0.910437\pi$
$618$		0			0
$619$			5.73959i			0.230694i	0.993325	+	0.115347i	$0.0367980\pi$
$619$			5.73959i			0.230694i	−0.993325	+	0.115347i	$0.963202\pi$
$620$		0			0
$621$	66.4089			2.66490
$622$		0			0
$623$	−23.8948			−0.957327
$624$		0			0
$625$	−18.5095			−0.740378
$626$		0			0
$627$	−14.2071			−0.567379
$628$		0			0
$629$		−	0.840975i		−	0.0335319i
$630$		0			0
$631$		−	36.1004i		−	1.43714i	−0.695457	−	0.718568i	$-0.744798\pi$
$631$		−	36.1004i		−	1.43714i	0.695457	−	0.718568i	$-0.255202\pi$
$632$		0			0
$633$	13.2508			0.526673
$634$		0			0
$635$		−	59.3327i		−	2.35455i
$636$		0			0
$637$	6.15333	−	7.71738i	0.243804	−	0.305774i
$638$		0			0
$639$			51.2125i			2.02594i
$640$		0			0
$641$	3.03511			0.119880			0.0599398	−	0.998202i	$-0.480909\pi$
$641$	3.03511			0.119880			0.0599398	+	0.998202i	$0.480909\pi$
$642$		0			0
$643$		−	39.7851i		−	1.56897i	−0.620147	−	0.784485i	$-0.712927\pi$
$643$		−	39.7851i		−	1.56897i	0.620147	−	0.784485i	$-0.287073\pi$
$644$		0			0
$645$			118.552i			4.66797i
$646$		0			0
$647$	24.1158			0.948091			0.474045	−	0.880500i	$-0.342793\pi$
$647$	24.1158			0.948091			0.474045	+	0.880500i	$0.342793\pi$
$648$		0			0
$649$	7.06443			0.277303
$650$		0			0
$651$	49.4095			1.93651
$652$		0			0
$653$	−21.5840			−0.844648			−0.422324	−	0.906445i	$-0.638786\pi$
$653$	−21.5840			−0.844648			−0.422324	+	0.906445i	$0.638786\pi$
$654$		0			0
$655$			40.6185i			1.58709i
$656$		0			0
$657$			13.3618i			0.521294i
$658$		0			0
$659$	−50.8417			−1.98051			−0.990255	−	0.139265i	$-0.955526\pi$
$659$	−50.8417			−1.98051			−0.990255	+	0.139265i	$0.955526\pi$
$660$		0			0
$661$		−	1.54106i		−	0.0599405i	−0.999551	−	0.0299702i	$-0.990459\pi$
$661$		−	1.54106i		−	0.0599405i	0.999551	−	0.0299702i	$-0.00954125\pi$
$662$		0			0
$663$	−7.96374	+	9.98796i	−0.309286	+	0.387900i
$664$		0			0
$665$			31.8441i			1.23486i
$666$		0			0
$667$	23.9960			0.929128
$668$		0			0
$669$			11.9496i			0.461997i
$670$		0			0
$671$			1.03861i			0.0400951i
$672$		0			0
$673$	−14.3971			−0.554967			−0.277484	−	0.960730i	$-0.589500\pi$
$673$	−14.3971			−0.554967			−0.277484	+	0.960730i	$0.589500\pi$
$674$		0			0
$675$	−63.0851			−2.42815
$676$		0			0
$677$	1.50672			0.0579081			0.0289541	−	0.999581i	$-0.490782\pi$
$677$	1.50672			0.0579081			0.0289541	+	0.999581i	$0.490782\pi$
$678$		0			0
$679$	−3.73530			−0.143348
$680$		0			0
$681$			81.4975i			3.12299i
$682$		0			0
$683$			6.42054i			0.245675i	0.992427	+	0.122838i	$0.0391994\pi$
$683$			6.42054i			0.245675i	−0.992427	+	0.122838i	$0.960801\pi$
$684$		0			0
$685$	35.6930			1.36376
$686$		0			0
$687$			62.1784i			2.37225i
$688$		0			0
$689$	−5.83262	+	7.31515i	−0.222205	+	0.278685i
$690$		0			0
$691$			21.8206i			0.830097i	0.909799	+	0.415048i	$0.136235\pi$
$691$			21.8206i			0.830097i	−0.909799	+	0.415048i	$0.863765\pi$
$692$		0			0
$693$	−13.1961			−0.501277
$694$		0			0
$695$		−	31.1120i		−	1.18014i
$696$		0			0
$697$		−	1.05030i		−	0.0397828i
$698$		0			0
$699$	84.4444			3.19398
$700$		0			0
$701$	−24.9703			−0.943117			−0.471558	−	0.881835i	$-0.656308\pi$
$701$	−24.9703			−0.943117			−0.471558	+	0.881835i	$0.656308\pi$
$702$		0			0
$703$	−3.37231			−0.127189
$704$		0			0
$705$	−3.59922			−0.135554
$706$		0			0
$707$		−	11.1820i		−	0.420544i
$708$		0			0
$709$		−	19.7274i		−	0.740877i	−0.928857	−	0.370438i	$-0.879208\pi$
$709$		−	19.7274i		−	0.740877i	0.928857	−	0.370438i	$-0.120792\pi$
$710$		0			0
$711$	10.3965			0.389898
$712$		0			0
$713$		−	49.8945i		−	1.86856i
$714$		0			0
$715$	7.47853	−	9.37942i	0.279681	−	0.350771i
$716$		0			0
$717$			16.7747i			0.626464i
$718$		0			0
$719$	−50.1049			−1.86860			−0.934299	−	0.356491i	$-0.883973\pi$
$719$	−50.1049			−1.86860			−0.934299	+	0.356491i	$0.883973\pi$
$720$		0			0
$721$		−	29.5563i		−	1.10073i
$722$		0			0
$723$			67.4472i			2.50839i
$724$		0			0
$725$	−22.7950			−0.846584
$726$		0			0
$727$	−18.9646			−0.703359			−0.351680	−	0.936120i	$-0.614389\pi$
$727$	−18.9646			−0.703359			−0.351680	+	0.936120i	$0.614389\pi$
$728$		0			0
$729$	−14.5247			−0.537953
$730$		0			0
$731$	13.4421			0.497174
$732$		0			0
$733$			7.73257i			0.285609i	0.989751	+	0.142805i	$0.0456120\pi$
$733$			7.73257i			0.285609i	−0.989751	+	0.142805i	$0.954388\pi$
$734$		0			0
$735$		−	27.9118i		−	1.02954i
$736$		0			0
$737$	11.3705			0.418839
$738$		0			0
$739$			10.1797i			0.374466i	0.982316	+	0.187233i	$0.0599520\pi$
$739$			10.1797i			0.374466i	−0.982316	+	0.187233i	$0.940048\pi$
$740$		0			0
$741$	40.0517	+	31.9346i	1.47134	+	1.17315i
$742$		0			0
$743$		−	14.5273i		−	0.532957i	−0.963841	−	0.266478i	$-0.914140\pi$
$743$		−	14.5273i		−	0.532957i	0.963841	−	0.266478i	$-0.0858601\pi$
$744$		0			0
$745$	14.1273			0.517585
$746$		0			0
$747$			53.4148i			1.95435i
$748$		0			0
$749$		−	28.6863i		−	1.04818i
$750$		0			0
$751$	18.5630			0.677372			0.338686	−	0.940899i	$-0.390018\pi$
$751$	18.5630			0.677372			0.338686	+	0.940899i	$0.390018\pi$
$752$		0			0
$753$	79.5526			2.89906
$754$		0			0
$755$	3.68843			0.134236
$756$		0			0
$757$	40.8436			1.48448			0.742242	−	0.670132i	$-0.233762\pi$
$757$	40.8436			1.48448			0.742242	+	0.670132i	$0.233762\pi$
$758$		0			0
$759$			19.5801i			0.710713i
$760$		0			0
$761$		−	20.6214i		−	0.747524i	−0.927525	−	0.373762i	$-0.878068\pi$
$761$		−	20.6214i		−	0.747524i	0.927525	−	0.373762i	$-0.121932\pi$
$762$		0			0
$763$	30.4464			1.10223
$764$		0			0
$765$			24.5848i			0.888864i
$766$		0			0
$767$	−19.9155	−	15.8793i	−0.719107	−	0.573369i
$768$		0			0
$769$			41.1742i			1.48478i	0.669968	+	0.742390i	$0.266308\pi$
$769$			41.1742i			1.48478i	−0.669968	+	0.742390i	$0.733692\pi$
$770$		0			0
$771$	16.5111			0.594633
$772$		0			0
$773$			36.3951i			1.30904i	0.756045	+	0.654520i	$0.227129\pi$
$773$			36.3951i			1.30904i	−0.756045	+	0.654520i	$0.772871\pi$
$774$		0			0
$775$			47.3973i			1.70256i
$776$		0			0
$777$	−4.60250			−0.165114
$778$		0			0
$779$	−4.21169			−0.150899
$780$		0			0
$781$	−8.01241			−0.286707
$782$		0			0
$783$	39.0370			1.39507
$784$		0			0
$785$			32.7150i			1.16765i
$786$		0			0
$787$		−	44.1270i		−	1.57296i	−0.617616	−	0.786480i	$-0.711902\pi$
$787$		−	44.1270i		−	1.57296i	0.617616	−	0.786480i	$-0.288098\pi$
$788$		0			0
$789$	−86.9143			−3.09423
$790$		0			0
$791$			17.1541i			0.609928i
$792$		0			0
$793$	2.33457	−	2.92797i	0.0829031	−	0.103975i
$794$		0			0
$795$			26.4571i			0.938335i
$796$		0			0
$797$	−29.9481			−1.06082			−0.530408	−	0.847743i	$-0.677961\pi$
$797$	−29.9481			−1.06082			−0.530408	+	0.847743i	$0.677961\pi$
$798$		0			0
$799$			0.408101i			0.0144376i
$800$		0			0
$801$			73.9751i			2.61378i
$802$		0			0
$803$	−2.09051			−0.0737725
$804$		0			0
$805$	43.8872			1.54682
$806$		0			0
$807$	−13.8363			−0.487060
$808$		0			0
$809$	−16.7839			−0.590089			−0.295045	−	0.955483i	$-0.595335\pi$
$809$	−16.7839			−0.590089			−0.295045	+	0.955483i	$0.595335\pi$
$810$		0			0
$811$		−	53.1025i		−	1.86468i	−0.361584	−	0.932340i	$-0.617764\pi$
$811$		−	53.1025i		−	1.86468i	0.361584	−	0.932340i	$-0.382236\pi$
$812$		0			0
$813$		−	68.4344i		−	2.40010i
$814$		0			0
$815$	16.2160			0.568022
$816$		0			0
$817$		−	53.9028i		−	1.88582i
$818$		0			0
$819$	37.2014	+	29.6619i	1.29992	+	1.03647i
$820$		0			0
$821$		−	9.55934i		−	0.333623i	−0.985989	−	0.166812i	$-0.946653\pi$
$821$		−	9.55934i		−	0.333623i	0.985989	−	0.166812i	$-0.0533472\pi$
$822$		0			0
$823$	−6.49297			−0.226331			−0.113165	−	0.993576i	$-0.536099\pi$
$823$	−6.49297			−0.226331			−0.113165	+	0.993576i	$0.536099\pi$
$824$		0			0
$825$		−	18.6001i		−	0.647573i
$826$		0			0
$827$		−	47.7011i		−	1.65873i	−0.558706	−	0.829365i	$-0.688702\pi$
$827$		−	47.7011i		−	1.65873i	0.558706	−	0.829365i	$-0.311298\pi$
$828$		0			0
$829$	42.3951			1.47244			0.736221	−	0.676741i	$-0.236608\pi$
$829$	42.3951			1.47244			0.736221	+	0.676741i	$0.236608\pi$
$830$		0			0
$831$	16.4204			0.569617
$832$		0			0
$833$	−3.16481			−0.109654
$834$		0			0
$835$	7.91832			0.274025
$836$		0			0
$837$		−	81.1690i		−	2.80561i
$838$		0			0
$839$		−	48.7919i		−	1.68448i	−0.539100	−	0.842242i	$-0.681236\pi$
$839$		−	48.7919i		−	1.68448i	0.539100	−	0.842242i	$-0.318764\pi$
$840$		0			0
$841$	−14.8945			−0.513604
$842$		0			0
$843$			32.1785i			1.10829i
$844$		0			0
$845$	−42.1658	+	9.63164i	−1.45055	+	0.331338i
$846$		0			0
$847$		−	2.06458i		−	0.0709398i
$848$		0			0
$849$	−36.9512			−1.26816
$850$		0			0
$851$			4.64768i			0.159320i
$852$		0			0
$853$		−	2.70440i		−	0.0925970i	−0.998928	−	0.0462985i	$-0.985257\pi$
$853$		−	2.70440i		−	0.0925970i	0.998928	−	0.0462985i	$-0.0147425\pi$
$854$		0			0
$855$	98.5850			3.37153
$856$		0			0
$857$	−21.0821			−0.720152			−0.360076	−	0.932923i	$-0.617249\pi$
$857$	−21.0821			−0.720152			−0.360076	+	0.932923i	$0.617249\pi$
$858$		0			0
$859$	21.1606			0.721990			0.360995	−	0.932568i	$-0.382437\pi$
$859$	21.1606			0.721990			0.360995	+	0.932568i	$0.382437\pi$
$860$		0			0
$861$	−5.74808			−0.195894
$862$		0			0
$863$		−	27.8038i		−	0.946454i	−0.880941	−	0.473227i	$-0.843089\pi$
$863$		−	27.8038i		−	0.946454i	0.880941	−	0.473227i	$-0.156911\pi$
$864$		0			0
$865$			16.9920i			0.577744i
$866$		0			0
$867$	−48.0019			−1.63023
$868$		0			0
$869$			1.62657i			0.0551776i
$870$		0			0
$871$	−32.0549	−	25.5585i	−1.08614	−	0.866016i
$872$		0			0
$873$			11.5640i			0.391381i
$874$		0			0
$875$	−7.34563			−0.248328
$876$		0			0
$877$			26.5016i			0.894895i	0.894310	+	0.447448i	$0.147667\pi$
$877$			26.5016i			0.894895i	−0.894310	+	0.447448i	$0.852333\pi$
$878$		0			0
$879$		−	57.6469i		−	1.94438i
$880$		0			0
$881$	−4.68828			−0.157952			−0.0789761	−	0.996877i	$-0.525165\pi$
$881$	−4.68828			−0.157952			−0.0789761	+	0.996877i	$0.525165\pi$
$882$		0			0
$883$	40.5074			1.36318			0.681591	−	0.731733i	$-0.261288\pi$
$883$	40.5074			1.36318			0.681591	+	0.731733i	$0.261288\pi$
$884$		0			0
$885$	−72.0294			−2.42124
$886$		0			0
$887$	6.76759			0.227233			0.113617	−	0.993525i	$-0.463756\pi$
$887$	6.76759			0.227233			0.113617	+	0.993525i	$0.463756\pi$
$888$		0			0
$889$			36.8183i			1.23485i
$890$		0			0
$891$			12.6782i			0.424737i
$892$		0			0
$893$	1.63648			0.0547628
$894$		0			0
$895$			4.59657i			0.153647i
$896$		0			0
$897$	44.0119	−	55.1988i	1.46951	−	1.84303i
$898$		0			0
$899$		−	29.3293i		−	0.978188i
$900$		0			0
$901$	2.99986			0.0999397
$902$		0			0
$903$		−	73.5662i		−	2.44813i
$904$		0			0
$905$			34.7596i			1.15545i
$906$		0			0
$907$	39.1121			1.29869			0.649347	−	0.760492i	$-0.275042\pi$
$907$	39.1121			1.29869			0.649347	+	0.760492i	$0.275042\pi$
$908$		0			0
$909$	−34.6180			−1.14821
$910$		0			0
$911$	−7.08198			−0.234636			−0.117318	−	0.993094i	$-0.537430\pi$
$911$	−7.08198			−0.234636			−0.117318	+	0.993094i	$0.537430\pi$
$912$		0			0
$913$	−8.35697			−0.276575
$914$		0			0
$915$		−	10.5898i		−	0.350086i
$916$		0			0
$917$		−	25.2054i		−	0.832355i
$918$		0			0
$919$	−5.09837			−0.168180			−0.0840899	−	0.996458i	$-0.526798\pi$
$919$	−5.09837			−0.168180			−0.0840899	+	0.996458i	$0.526798\pi$
$920$		0			0
$921$		−	31.4812i		−	1.03734i
$922$		0			0
$923$	22.5880	+	18.0102i	0.743493	+	0.592812i
$924$		0			0
$925$		−	4.41506i		−	0.145166i
$926$		0			0
$927$	−91.5021			−3.00532
$928$		0			0
$929$		−	32.8585i		−	1.07805i	−0.842289	−	0.539027i	$-0.818792\pi$
$929$		−	32.8585i		−	1.07805i	0.842289	−	0.539027i	$-0.181208\pi$
$930$		0			0
$931$			12.6909i			0.415927i
$932$		0			0
$933$	−6.24128			−0.204330
$934$		0			0
$935$	−3.84639			−0.125790
$936$		0			0
$937$	34.7493			1.13521			0.567605	−	0.823301i	$-0.307870\pi$
$937$	34.7493			1.13521			0.567605	+	0.823301i	$0.307870\pi$
$938$		0			0
$939$	−35.4359			−1.15641
$940$		0			0
$941$		−	9.67858i		−	0.315513i	−0.987478	−	0.157756i	$-0.949574\pi$
$941$		−	9.67858i		−	0.315513i	0.987478	−	0.157756i	$-0.0504261\pi$
$942$		0			0
$943$			5.80450i			0.189020i
$944$		0			0
$945$	71.3962			2.32252
$946$		0			0
$947$			33.2936i			1.08190i	0.841056	+	0.540948i	$0.181934\pi$
$947$			33.2936i			1.08190i	−0.841056	+	0.540948i	$0.818066\pi$
$948$		0			0
$949$	5.89341	+	4.69902i	0.191308	+	0.152537i
$950$		0			0
$951$		−	77.5776i		−	2.51563i
$952$		0			0
$953$	−6.57404			−0.212954			−0.106477	−	0.994315i	$-0.533957\pi$
$953$	−6.57404			−0.212954			−0.106477	+	0.994315i	$0.533957\pi$
$954$		0			0
$955$			18.9363i			0.612766i
$956$		0			0
$957$			11.5097i			0.372057i
$958$		0			0
$959$	−22.1490			−0.715227
$960$		0			0
$961$	−29.9841			−0.967228
$962$		0			0
$963$	−88.8088			−2.86182
$964$		0			0
$965$	28.7139			0.924332
$966$		0			0
$967$			58.8304i			1.89186i	0.324371	+	0.945930i	$0.394847\pi$
$967$			58.8304i			1.89186i	−0.324371	+	0.945930i	$0.605153\pi$
$968$		0			0
$969$		−	16.4247i		−	0.527639i
$970$		0			0
$971$	0.768602			0.0246656			0.0123328	−	0.999924i	$-0.496074\pi$
$971$	0.768602			0.0246656			0.0123328	+	0.999924i	$0.496074\pi$
$972$		0			0
$973$			19.3062i			0.618929i
$974$		0			0
$975$	−41.8091	+	52.4361i	−1.33896	+	1.67930i
$976$		0			0
$977$		−	6.32343i		−	0.202304i	−0.994871	−	0.101152i	$-0.967747\pi$
$977$		−	6.32343i		−	0.202304i	0.994871	−	0.101152i	$-0.0322529\pi$
$978$		0			0
$979$	−11.5737			−0.369897
$980$		0			0
$981$		−	94.2578i		−	3.00942i
$982$		0			0
$983$			10.4400i			0.332986i	0.986043	+	0.166493i	$0.0532443\pi$
$983$			10.4400i			0.332986i	−0.986043	+	0.166493i	$0.946756\pi$
$984$		0			0
$985$	67.1085			2.13826
$986$		0			0
$987$	2.23346			0.0710918
$988$		0			0
$989$	−74.2882			−2.36223
$990$		0			0
$991$	1.84883			0.0587301			0.0293651	−	0.999569i	$-0.490651\pi$
$991$	1.84883			0.0587301			0.0293651	+	0.999569i	$0.490651\pi$
$992$		0			0
$993$			8.96245i			0.284415i
$994$		0			0
$995$			21.5139i			0.682035i
$996$		0			0
$997$	−23.5629			−0.746244			−0.373122	−	0.927782i	$-0.621713\pi$
$997$	−23.5629			−0.746244			−0.373122	+	0.927782i	$0.621713\pi$
$998$		0			0
$999$			7.56090i			0.239216i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2288.2.j.k.1585.11		12
4.3	odd	2		143.2.b.a.12.8	yes	12
12.11	even	2		1287.2.b.b.298.5		12
13.12	even	2	inner	2288.2.j.k.1585.12		12
52.31	even	4		1859.2.a.j.1.4		6
52.47	even	4		1859.2.a.n.1.3		6
52.51	odd	2		143.2.b.a.12.5	✓	12
156.155	even	2		1287.2.b.b.298.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.2.b.a.12.5	✓	12	52.51	odd	2
143.2.b.a.12.8	yes	12	4.3	odd	2
1287.2.b.b.298.5		12	12.11	even	2
1287.2.b.b.298.8		12	156.155	even	2
1859.2.a.j.1.4		6	52.31	even	4
1859.2.a.n.1.3		6	52.47	even	4
2288.2.j.k.1585.11		12	1.1	even	1	trivial
2288.2.j.k.1585.12		12	13.12	even	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:	\( N \)	\(=\)	\( 2288 = 2^{4} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2288.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.06458 q^{3} -3.32707i q^{5} +2.06458i q^{7} +6.39165 q^{9} +O(q^{10})\)	\(q+3.06458 q^{3} -3.32707i q^{5} +2.06458i q^{7} +6.39165 q^{9} +1.00000i q^{11} +(2.24778 - 2.81912i) q^{13} -10.1961i q^{15} -1.15609 q^{17} +4.63592i q^{19} +6.32707i q^{21} +6.38917 q^{23} -6.06939 q^{25} +10.3940 q^{27} +3.75573 q^{29} -7.80923i q^{31} +3.06458i q^{33} +6.86900 q^{35} +0.727430i q^{37} +(6.88851 - 8.63943i) q^{39} +0.908490i q^{41} -11.6272 q^{43} -21.2655i q^{45} -0.353001i q^{47} +2.73751 q^{49} -3.54293 q^{51} -2.59483 q^{53} +3.32707 q^{55} +14.2071i q^{57} -7.06443i q^{59} +1.03861 q^{61} +13.1961i q^{63} +(-9.37942 - 7.47853i) q^{65} -11.3705i q^{67} +19.5801 q^{69} +8.01241i q^{71} +2.09051i q^{73} -18.6001 q^{75} -2.06458 q^{77} +1.62657 q^{79} +12.6782 q^{81} +8.35697i q^{83} +3.84639i q^{85} +11.5097 q^{87} +11.5737i q^{89} +(5.82031 + 4.64073i) q^{91} -23.9320i q^{93} +15.4240 q^{95} +1.80923i q^{97} +6.39165i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{3} + 12 q^{9}+O(q^{10}) \) 12 * q + 4 * q^3 + 12 * q^9	\( 12 q + 4 q^{3} + 12 q^{9} - 8 q^{13} - 12 q^{17} + 4 q^{23} - 16 q^{25} + 28 q^{27} + 8 q^{29} - 16 q^{35} - 4 q^{39} - 12 q^{43} + 32 q^{49} - 20 q^{53} + 8 q^{55} - 12 q^{61} - 20 q^{65} + 52 q^{69} + 8 q^{75} + 8 q^{77} + 48 q^{79} - 36 q^{81} - 12 q^{87} + 4 q^{95}+O(q^{100}) \) 12 * q + 4 * q^3 + 12 * q^9 - 8 * q^13 - 12 * q^17 + 4 * q^23 - 16 * q^25 + 28 * q^27 + 8 * q^29 - 16 * q^35 - 4 * q^39 - 12 * q^43 + 32 * q^49 - 20 * q^53 + 8 * q^55 - 12 * q^61 - 20 * q^65 + 52 * q^69 + 8 * q^75 + 8 * q^77 + 48 * q^79 - 36 * q^81 - 12 * q^87 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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