LMFDB - Embedded newform 1400.2.x.c.993.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1400,2,Mod(657,1400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1400, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))

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

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

chi := DirichletCharacter("1400.657");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1400 = 2^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1400.x (of order $4$, degree $2$, 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:	$11.1790562830$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			993.1
Character	$\chi$	$=$	1400.993
Dual form			1400.2.x.c.657.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.09284 + 2.09284i) q^{3} +(0.510946 + 2.59595i) q^{7} -5.75996i q^{9} +O(q^{10})$	$q+(-2.09284 + 2.09284i) q^{3} +(0.510946 + 2.59595i) q^{7} -5.75996i q^{9} -3.94864 q^{11} +(1.69798 - 1.69798i) q^{13} +(2.66927 + 2.66927i) q^{17} -5.36102 q^{19} +(-6.50223 - 4.36357i) q^{21} +(-3.55527 - 3.55527i) q^{23} +(5.77616 + 5.77616i) q^{27} -7.24449i q^{29} +0.174160i q^{31} +(8.26386 - 8.26386i) q^{33} +(-4.85737 + 4.85737i) q^{37} +7.10719i q^{39} +0.732583i q^{41} +(-8.20187 - 8.20187i) q^{43} +(2.31716 + 2.31716i) q^{47} +(-6.47787 + 2.65278i) q^{49} -11.1727 q^{51} +(4.53449 + 4.53449i) q^{53} +(11.2198 - 11.2198i) q^{57} +13.1904 q^{59} -13.3556i q^{61} +(14.9525 - 2.94303i) q^{63} +(6.55658 - 6.55658i) q^{67} +14.8812 q^{69} +16.3312 q^{71} +(-7.02549 + 7.02549i) q^{73} +(-2.01754 - 10.2504i) q^{77} -6.63234i q^{79} -6.89727 q^{81} +(-10.4642 + 10.4642i) q^{83} +(15.1616 + 15.1616i) q^{87} -3.43554 q^{89} +(5.27543 + 3.54028i) q^{91} +(-0.364490 - 0.364490i) q^{93} +(-1.40892 - 1.40892i) q^{97} +22.7440i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q - 16 q^{11} - 40 q^{21} + 32 q^{51} + 128 q^{71}+O(q^{100}) $ 32 * q - 16 * q^11 - 40 * q^21 + 32 * q^51 + 128 * q^71

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$.

$n$	$351$	$701$	$801$	$1177$
$\chi(n)$	$1$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−2.09284	+	2.09284i	−1.20830	+	1.20830i	−0.236725	+	0.971577i	$0.576074\pi$
$3$	−2.09284	+	2.09284i	−1.20830	+	1.20830i	−0.971577	+	0.236725i	$0.923926\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.510946	+	2.59595i	0.193119	+	0.981175i
$7$	0.510946	+	2.59595i	0.193119	+	0.981175i
$8$		0			0
$9$		−	5.75996i		−	1.91999i
$10$		0			0
$11$	−3.94864			−1.19056			−0.595279	−	0.803519i	$-0.702959\pi$
$11$	−3.94864			−1.19056			−0.595279	+	0.803519i	$0.702959\pi$
$12$		0			0
$13$	1.69798	−	1.69798i	0.470934	−	0.470934i	−0.431283	−	0.902217i	$-0.641939\pi$
$13$	1.69798	−	1.69798i	0.470934	−	0.470934i	0.902217	+	0.431283i	$0.141939\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.66927	+	2.66927i	0.647392	+	0.647392i	0.952362	−	0.304970i	$-0.0986464\pi$
$17$	2.66927	+	2.66927i	0.647392	+	0.647392i	−0.304970	+	0.952362i	$0.598646\pi$
$18$		0			0
$19$	−5.36102			−1.22990			−0.614952	−	0.788565i	$-0.710824\pi$
$19$	−5.36102			−1.22990			−0.614952	+	0.788565i	$0.710824\pi$
$20$		0			0
$21$	−6.50223	−	4.36357i	−1.41890	−	0.952209i
$22$		0			0
$23$	−3.55527	−	3.55527i	−0.741325	−	0.741325i	0.231508	−	0.972833i	$-0.425634\pi$
$23$	−3.55527	−	3.55527i	−0.741325	−	0.741325i	−0.972833	+	0.231508i	$0.925634\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	5.77616	+	5.77616i	1.11162	+	1.11162i
$28$		0			0
$29$		−	7.24449i		−	1.34527i	−0.739975	−	0.672634i	$-0.765163\pi$
$29$		−	7.24449i		−	1.34527i	0.739975	−	0.672634i	$-0.234837\pi$
$30$		0			0
$31$			0.174160i			0.0312801i	0.999878	+	0.0156401i	$0.00497859\pi$
$31$			0.174160i			0.0312801i	−0.999878	+	0.0156401i	$0.995021\pi$
$32$		0			0
$33$	8.26386	−	8.26386i	1.43855	−	1.43855i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−4.85737	+	4.85737i	−0.798547	+	0.798547i	−0.982866	−	0.184320i	$-0.940992\pi$
$37$	−4.85737	+	4.85737i	−0.798547	+	0.798547i	0.184320	+	0.982866i	$0.440992\pi$
$38$		0			0
$39$			7.10719i			1.13806i
$40$		0			0
$41$			0.732583i			0.114410i	0.998362	+	0.0572051i	$0.0182189\pi$
$41$			0.732583i			0.114410i	−0.998362	+	0.0572051i	$0.981781\pi$
$42$		0			0
$43$	−8.20187	−	8.20187i	−1.25077	−	1.25077i	−0.955374	−	0.295400i	$-0.904547\pi$
$43$	−8.20187	−	8.20187i	−1.25077	−	1.25077i	−0.295400	−	0.955374i	$-0.595453\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.31716	+	2.31716i	0.337993	+	0.337993i	0.855612	−	0.517619i	$-0.173181\pi$
$47$	2.31716	+	2.31716i	0.337993	+	0.337993i	−0.517619	+	0.855612i	$0.673181\pi$
$48$		0			0
$49$	−6.47787	+	2.65278i	−0.925410	+	0.378968i
$50$		0			0
$51$	−11.1727			−1.56449
$52$		0			0
$53$	4.53449	+	4.53449i	0.622860	+	0.622860i	0.946262	−	0.323402i	$-0.104827\pi$
$53$	4.53449	+	4.53449i	0.622860	+	0.622860i	−0.323402	+	0.946262i	$0.604827\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	11.2198	−	11.2198i	1.48609	−	1.48609i
$58$		0			0
$59$	13.1904			1.71724			0.858620	−	0.512613i	$-0.171322\pi$
$59$	13.1904			1.71724			0.858620	+	0.512613i	$0.171322\pi$
$60$		0			0
$61$		−	13.3556i		−	1.71001i	−0.518622	−	0.855004i	$-0.673555\pi$
$61$		−	13.3556i		−	1.71001i	0.518622	−	0.855004i	$-0.326445\pi$
$62$		0			0
$63$	14.9525	−	2.94303i	1.88384	−	0.370787i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	6.55658	−	6.55658i	0.801014	−	0.801014i	−0.182240	−	0.983254i	$-0.558335\pi$
$67$	6.55658	−	6.55658i	0.801014	−	0.801014i	0.983254	+	0.182240i	$0.0583349\pi$
$68$		0			0
$69$	14.8812			1.79149
$70$		0			0
$71$	16.3312			1.93816			0.969081	−	0.246742i	$-0.0793599\pi$
$71$	16.3312			1.93816			0.969081	+	0.246742i	$0.0793599\pi$
$72$		0			0
$73$	−7.02549	+	7.02549i	−0.822271	+	0.822271i	−0.986433	−	0.164162i	$-0.947508\pi$
$73$	−7.02549	+	7.02549i	−0.822271	+	0.822271i	0.164162	+	0.986433i	$0.447508\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.01754	−	10.2504i	−0.229920	−	1.16815i
$78$		0			0
$79$		−	6.63234i		−	0.746197i	−0.927792	−	0.373098i	$-0.878295\pi$
$79$		−	6.63234i		−	0.746197i	0.927792	−	0.373098i	$-0.121705\pi$
$80$		0			0
$81$	−6.89727			−0.766363
$82$		0			0
$83$	−10.4642	+	10.4642i	−1.14860	+	1.14860i	−0.161766	+	0.986829i	$0.551719\pi$
$83$	−10.4642	+	10.4642i	−1.14860	+	1.14860i	−0.986829	+	0.161766i	$0.948281\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	15.1616	+	15.1616i	1.62549	+	1.62549i
$88$		0			0
$89$	−3.43554			−0.364166			−0.182083	−	0.983283i	$-0.558284\pi$
$89$	−3.43554			−0.364166			−0.182083	+	0.983283i	$0.558284\pi$
$90$		0			0
$91$	5.27543	+	3.54028i	0.553015	+	0.371122i
$92$		0			0
$93$	−0.364490	−	0.364490i	−0.0377958	−	0.0377958i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.40892	−	1.40892i	−0.143054	−	0.143054i	0.631953	−	0.775007i	$-0.282254\pi$
$97$	−1.40892	−	1.40892i	−0.143054	−	0.143054i	−0.775007	+	0.631953i	$0.782254\pi$
$98$		0			0
$99$			22.7440i			2.28586i
$100$		0			0
$101$		−	12.1783i		−	1.21178i	−0.795547	−	0.605891i	$-0.792817\pi$
$101$		−	12.1783i		−	1.21178i	0.795547	−	0.605891i	$-0.207183\pi$
$102$		0			0
$103$	9.52421	−	9.52421i	0.938449	−	0.938449i	−0.0597638	−	0.998213i	$-0.519035\pi$
$103$	9.52421	−	9.52421i	0.938449	−	0.938449i	0.998213	+	0.0597638i	$0.0190348\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8.98949	−	8.98949i	0.869046	−	0.869046i	−0.123321	−	0.992367i	$-0.539354\pi$
$107$	8.98949	−	8.98949i	0.869046	−	0.869046i	0.992367	+	0.123321i	$0.0393543\pi$
$108$		0			0
$109$		−	3.86269i		−	0.369979i	−0.982741	−	0.184989i	$-0.940775\pi$
$109$		−	3.86269i		−	0.369979i	0.982741	−	0.184989i	$-0.0592251\pi$
$110$		0			0
$111$		−	20.3314i		−	1.92977i
$112$		0			0
$113$	−2.94525	−	2.94525i	−0.277066	−	0.277066i	0.554870	−	0.831937i	$-0.312768\pi$
$113$	−2.94525	−	2.94525i	−0.277066	−	0.277066i	−0.831937	+	0.554870i	$0.812768\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−9.78028	−	9.78028i	−0.904187	−	0.904187i
$118$		0			0
$119$	−5.56542	+	8.29312i	−0.510181	+	0.760229i
$120$		0			0
$121$	4.59172			0.417429
$122$		0			0
$123$	−1.53318	−	1.53318i	−0.138242	−	0.138242i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	1.73709	−	1.73709i	0.154142	−	0.154142i	−0.625823	−	0.779965i	$-0.715237\pi$
$127$	1.73709	−	1.73709i	0.154142	−	0.154142i	0.779965	+	0.625823i	$0.215237\pi$
$128$		0			0
$129$	34.3304			3.02262
$130$		0			0
$131$		−	2.70296i		−	0.236158i	−0.993004	−	0.118079i	$-0.962326\pi$
$131$		−	2.70296i		−	0.236158i	0.993004	−	0.118079i	$-0.0376737\pi$
$132$		0			0
$133$	−2.73919	−	13.9169i	−0.237518	−	1.20675i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.26814	+	3.26814i	−0.279216	+	0.279216i	−0.832796	−	0.553580i	$-0.813261\pi$
$137$	−3.26814	+	3.26814i	−0.279216	+	0.279216i	0.553580	+	0.832796i	$0.313261\pi$
$138$		0			0
$139$	−21.4638			−1.82054			−0.910269	−	0.414017i	$-0.864126\pi$
$139$	−21.4638			−1.82054			−0.910269	+	0.414017i	$0.864126\pi$
$140$		0			0
$141$	−9.69891			−0.816795
$142$		0			0
$143$	−6.70469	+	6.70469i	−0.560674	+	0.560674i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	8.00531	−	19.1090i	0.660267	−	1.57608i
$148$		0			0
$149$		−	2.09645i		−	0.171747i	−0.996306	−	0.0858737i	$-0.972632\pi$
$149$		−	2.09645i		−	0.171747i	0.996306	−	0.0858737i	$-0.0273682\pi$
$150$		0			0
$151$	1.78039			0.144886			0.0724432	−	0.997373i	$-0.476920\pi$
$151$	1.78039			0.144886			0.0724432	+	0.997373i	$0.476920\pi$
$152$		0			0
$153$	15.3749	−	15.3749i	1.24298	−	1.24298i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	4.64528	+	4.64528i	0.370734	+	0.370734i	0.867744	−	0.497011i	$-0.165569\pi$
$157$	4.64528	+	4.64528i	0.370734	+	0.370734i	−0.497011	+	0.867744i	$0.665569\pi$
$158$		0			0
$159$	−18.9799			−1.50521
$160$		0			0
$161$	7.41274	−	11.0458i	0.584205	−	0.870534i
$162$		0			0
$163$	−1.67419	−	1.67419i	−0.131132	−	0.131132i	0.638494	−	0.769627i	$-0.279558\pi$
$163$	−1.67419	−	1.67419i	−0.131132	−	0.131132i	−0.769627	+	0.638494i	$0.779558\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.65522	−	1.65522i	−0.128084	−	0.128084i	0.640158	−	0.768243i	$-0.278869\pi$
$167$	−1.65522	−	1.65522i	−0.128084	−	0.128084i	−0.768243	+	0.640158i	$0.778869\pi$
$168$		0			0
$169$			7.23375i			0.556443i
$170$		0			0
$171$			30.8793i			2.36140i
$172$		0			0
$173$	−0.949313	+	0.949313i	−0.0721750	+	0.0721750i	−0.742273	−	0.670098i	$-0.766252\pi$
$173$	−0.949313	+	0.949313i	−0.0721750	+	0.0721750i	0.670098	+	0.742273i	$0.266252\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−27.6053	+	27.6053i	−2.07494	+	2.07494i
$178$		0			0
$179$		−	16.6926i		−	1.24767i	−0.781558	−	0.623833i	$-0.785575\pi$
$179$		−	16.6926i		−	1.24767i	0.781558	−	0.623833i	$-0.214425\pi$
$180$		0			0
$181$			18.6406i			1.38554i	0.721158	+	0.692771i	$0.243610\pi$
$181$			18.6406i			1.38554i	−0.721158	+	0.692771i	$0.756390\pi$
$182$		0			0
$183$	27.9511	+	27.9511i	2.06621	+	2.06621i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−10.5400	−	10.5400i	−0.770758	−	0.770758i
$188$		0			0
$189$	−12.0433	+	17.9459i	−0.876020	+	1.30537i
$190$		0			0
$191$	−4.33648			−0.313777			−0.156888	−	0.987616i	$-0.550146\pi$
$191$	−4.33648			−0.313777			−0.156888	+	0.987616i	$0.550146\pi$
$192$		0			0
$193$	−11.8884	−	11.8884i	−0.855747	−	0.855747i	0.135087	−	0.990834i	$-0.456869\pi$
$193$	−11.8884	−	11.8884i	−0.855747	−	0.855747i	−0.990834	+	0.135087i	$0.956869\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.85131	+	2.85131i	−0.203148	+	0.203148i	−0.801347	−	0.598200i	$-0.795883\pi$
$197$	−2.85131	+	2.85131i	−0.203148	+	0.203148i	0.598200	+	0.801347i	$0.295883\pi$
$198$		0			0
$199$	4.34228			0.307816			0.153908	−	0.988085i	$-0.450814\pi$
$199$	4.34228			0.307816			0.153908	+	0.988085i	$0.450814\pi$
$200$		0			0
$201$			27.4438i			1.93573i
$202$		0			0
$203$	18.8063	−	3.70154i	1.31994	−	0.259797i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−20.4782	+	20.4782i	−1.42333	+	1.42333i
$208$		0			0
$209$	21.1687			1.46427
$210$		0			0
$211$	−18.8362			−1.29674			−0.648369	−	0.761326i	$-0.724549\pi$
$211$	−18.8362			−1.29674			−0.648369	+	0.761326i	$0.724549\pi$
$212$		0			0
$213$	−34.1787	+	34.1787i	−2.34189	+	2.34189i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.452111	+	0.0889865i	−0.0306913	+	0.00604080i
$218$		0			0
$219$		−	29.4065i		−	1.98710i
$220$		0			0
$221$	9.06470			0.609758
$222$		0			0
$223$	−13.9232	+	13.9232i	−0.932366	+	0.932366i	−0.997853	−	0.0654872i	$-0.979140\pi$
$223$	−13.9232	+	13.9232i	−0.932366	+	0.932366i	0.0654872	+	0.997853i	$0.479140\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.91799	+	6.91799i	0.459163	+	0.459163i	0.898381	−	0.439218i	$-0.144744\pi$
$227$	6.91799	+	6.91799i	0.459163	+	0.459163i	−0.439218	+	0.898381i	$0.644744\pi$
$228$		0			0
$229$	−1.64658			−0.108809			−0.0544047	−	0.998519i	$-0.517326\pi$
$229$	−1.64658			−0.108809			−0.0544047	+	0.998519i	$0.517326\pi$
$230$		0			0
$231$	25.6749	+	17.2302i	1.68929	+	1.13366i
$232$		0			0
$233$	−4.73081	−	4.73081i	−0.309926	−	0.309926i	0.534955	−	0.844881i	$-0.320329\pi$
$233$	−4.73081	−	4.73081i	−0.309926	−	0.309926i	−0.844881	+	0.534955i	$0.820329\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	13.8804	+	13.8804i	0.901631	+	0.901631i
$238$		0			0
$239$		−	5.18344i		−	0.335289i	−0.985848	−	0.167644i	$-0.946384\pi$
$239$		−	5.18344i		−	0.335289i	0.985848	−	0.167644i	$-0.0536160\pi$
$240$		0			0
$241$		−	8.65109i		−	0.557265i	−0.960398	−	0.278633i	$-0.910119\pi$
$241$		−	8.65109i		−	0.557265i	0.960398	−	0.278633i	$-0.0898812\pi$
$242$		0			0
$243$	−2.89359	+	2.89359i	−0.185624	+	0.185624i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−9.10289	+	9.10289i	−0.579203	+	0.579203i
$248$		0			0
$249$		−	43.7998i		−	2.77570i
$250$		0			0
$251$			11.0974i			0.700459i	0.936664	+	0.350230i	$0.113896\pi$
$251$			11.0974i			0.700459i	−0.936664	+	0.350230i	$0.886104\pi$
$252$		0			0
$253$	14.0385	+	14.0385i	0.882591	+	0.882591i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.62391	+	1.62391i	0.101297	+	0.101297i	0.755939	−	0.654642i	$-0.227181\pi$
$257$	1.62391	+	1.62391i	0.101297	+	0.101297i	−0.654642	+	0.755939i	$0.727181\pi$
$258$		0			0
$259$	−15.0913	−	10.1276i	−0.937729	−	0.629300i
$260$		0			0
$261$	−41.7280			−2.58290
$262$		0			0
$263$	−20.7161	−	20.7161i	−1.27741	−	1.27741i	−0.942111	−	0.335301i	$-0.891162\pi$
$263$	−20.7161	−	20.7161i	−1.27741	−	1.27741i	−0.335301	−	0.942111i	$-0.608838\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	7.19003	−	7.19003i	0.440023	−	0.440023i
$268$		0			0
$269$	−31.0977			−1.89606			−0.948031	−	0.318179i	$-0.896929\pi$
$269$	−31.0977			−1.89606			−0.948031	+	0.318179i	$0.896929\pi$
$270$		0			0
$271$		−	9.24962i		−	0.561875i	−0.959726	−	0.280937i	$-0.909355\pi$
$271$		−	9.24962i		−	0.561875i	0.959726	−	0.280937i	$-0.0906453\pi$
$272$		0			0
$273$	−18.4499	+	3.63139i	−1.11664	+	0.219782i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−14.3476	+	14.3476i	−0.862066	+	0.862066i	−0.991578	−	0.129512i	$-0.958659\pi$
$277$	−14.3476	+	14.3476i	−0.862066	+	0.862066i	0.129512	+	0.991578i	$0.458659\pi$
$278$		0			0
$279$	1.00316			0.0600574
$280$		0			0
$281$	12.3108			0.734402			0.367201	−	0.930142i	$-0.380316\pi$
$281$	12.3108			0.734402			0.367201	+	0.930142i	$0.380316\pi$
$282$		0			0
$283$	7.14401	−	7.14401i	0.424667	−	0.424667i	−0.462140	−	0.886807i	$-0.652918\pi$
$283$	7.14401	−	7.14401i	0.424667	−	0.424667i	0.886807	+	0.462140i	$0.152918\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.90174	+	0.374310i	−0.112256	+	0.0220948i
$288$		0			0
$289$		−	2.75003i		−	0.161766i
$290$		0			0
$291$	5.89727			0.345704
$292$		0			0
$293$	−10.8037	+	10.8037i	−0.631156	+	0.631156i	−0.948358	−	0.317202i	$-0.897257\pi$
$293$	−10.8037	+	10.8037i	−0.631156	+	0.631156i	0.317202	+	0.948358i	$0.397257\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−22.8079	−	22.8079i	−1.32345	−	1.32345i
$298$		0			0
$299$	−12.0735			−0.698230
$300$		0			0
$301$	17.1009	−	25.4823i	0.985680	−	1.46878i
$302$		0			0
$303$	25.4872	+	25.4872i	1.46420	+	1.46420i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−21.5142	−	21.5142i	−1.22788	−	1.22788i	−0.964765	−	0.263113i	$-0.915251\pi$
$307$	−21.5142	−	21.5142i	−1.22788	−	1.22788i	−0.263113	−	0.964765i	$-0.584749\pi$
$308$		0			0
$309$			39.8653i			2.26786i
$310$		0			0
$311$			15.8682i			0.899802i	0.893078	+	0.449901i	$0.148541\pi$
$311$			15.8682i			0.899802i	−0.893078	+	0.449901i	$0.851459\pi$
$312$		0			0
$313$	16.6132	−	16.6132i	0.939036	−	0.939036i	−0.0592093	−	0.998246i	$-0.518858\pi$
$313$	16.6132	−	16.6132i	0.939036	−	0.939036i	0.998246	+	0.0592093i	$0.0188579\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.53714	+	9.53714i	−0.535659	+	0.535659i	−0.922251	−	0.386592i	$-0.873652\pi$
$317$	−9.53714	+	9.53714i	−0.535659	+	0.535659i	0.386592	+	0.922251i	$0.373652\pi$
$318$		0			0
$319$			28.6059i			1.60162i
$320$		0			0
$321$			37.6271i			2.10014i
$322$		0			0
$323$	−14.3100	−	14.3100i	−0.796230	−	0.796230i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	8.08400	+	8.08400i	0.447046	+	0.447046i
$328$		0			0
$329$	−4.83129	+	7.19917i	−0.266357	+	0.396903i
$330$		0			0
$331$	12.9177			0.710021			0.355011	−	0.934862i	$-0.384477\pi$
$331$	12.9177			0.710021			0.355011	+	0.934862i	$0.384477\pi$
$332$		0			0
$333$	27.9783	+	27.9783i	1.53320	+	1.53320i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−7.12871	+	7.12871i	−0.388326	+	0.388326i	−0.874090	−	0.485764i	$-0.838541\pi$
$337$	−7.12871	+	7.12871i	−0.388326	+	0.388326i	0.485764	+	0.874090i	$0.338541\pi$
$338$		0			0
$339$	12.3279			0.669559
$340$		0			0
$341$		−	0.687696i		−	0.0372408i
$342$		0			0
$343$	−10.1963	−	15.4608i	−0.550549	−	0.834803i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.28294	+	1.28294i	−0.0688717	+	0.0688717i	−0.740704	−	0.671832i	$-0.765508\pi$
$347$	−1.28294	+	1.28294i	−0.0688717	+	0.0688717i	0.671832	+	0.740704i	$0.265508\pi$
$348$		0			0
$349$	−14.4635			−0.774213			−0.387106	−	0.922035i	$-0.626525\pi$
$349$	−14.4635			−0.774213			−0.387106	+	0.922035i	$0.626525\pi$
$350$		0			0
$351$	19.6156			1.04700
$352$		0			0
$353$	−1.19731	+	1.19731i	−0.0637263	+	0.0637263i	−0.738252	−	0.674525i	$-0.764348\pi$
$353$	−1.19731	+	1.19731i	−0.0637263	+	0.0637263i	0.674525	+	0.738252i	$0.264348\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−5.70864	−	29.0037i	−0.302133	−	1.53504i
$358$		0			0
$359$		−	27.1832i		−	1.43467i	−0.696726	−	0.717337i	$-0.745361\pi$
$359$		−	27.1832i		−	1.43467i	0.696726	−	0.717337i	$-0.254639\pi$
$360$		0			0
$361$	9.74058			0.512662
$362$		0			0
$363$	−9.60974	+	9.60974i	−0.504380	+	0.504380i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−18.0535	−	18.0535i	−0.942383	−	0.942383i	0.0560450	−	0.998428i	$-0.482151\pi$
$367$	−18.0535	−	18.0535i	−0.942383	−	0.942383i	−0.998428	+	0.0560450i	$0.982151\pi$
$368$		0			0
$369$	4.21965			0.219666
$370$		0			0
$371$	−9.45441	+	14.0882i	−0.490849	+	0.731421i
$372$		0			0
$373$	−23.4769	−	23.4769i	−1.21559	−	1.21559i	−0.969161	−	0.246430i	$-0.920743\pi$
$373$	−23.4769	−	23.4769i	−1.21559	−	1.21559i	−0.246430	−	0.969161i	$-0.579257\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−12.3010	−	12.3010i	−0.633533	−	0.633533i
$378$		0			0
$379$			32.9476i			1.69241i	0.532861	+	0.846203i	$0.321117\pi$
$379$			32.9476i			1.69241i	−0.532861	+	0.846203i	$0.678883\pi$
$380$		0			0
$381$			7.27092i			0.372501i
$382$		0			0
$383$	−12.6328	+	12.6328i	−0.645506	+	0.645506i	−0.951904	−	0.306398i	$-0.900876\pi$
$383$	−12.6328	+	12.6328i	−0.645506	+	0.645506i	0.306398	+	0.951904i	$0.400876\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−47.2425	+	47.2425i	−2.40147	+	2.40147i
$388$		0			0
$389$			3.77597i			0.191449i	0.995408	+	0.0957246i	$0.0305168\pi$
$389$			3.77597i			0.191449i	−0.995408	+	0.0957246i	$0.969483\pi$
$390$		0			0
$391$		−	18.9799i		−	0.959856i
$392$		0			0
$393$	5.65685	+	5.65685i	0.285351	+	0.285351i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−16.0271	−	16.0271i	−0.804375	−	0.804375i	0.179401	−	0.983776i	$-0.442584\pi$
$397$	−16.0271	−	16.0271i	−0.804375	−	0.804375i	−0.983776	+	0.179401i	$0.942584\pi$
$398$		0			0
$399$	34.8586	+	23.3932i	1.74511	+	1.17113i
$400$		0			0
$401$	3.52985			0.176272			0.0881362	−	0.996108i	$-0.471909\pi$
$401$	3.52985			0.176272			0.0881362	+	0.996108i	$0.471909\pi$
$402$		0			0
$403$	0.295720	+	0.295720i	0.0147309	+	0.0147309i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	19.1800	−	19.1800i	0.950717	−	0.950717i
$408$		0			0
$409$	−5.07318			−0.250852			−0.125426	−	0.992103i	$-0.540030\pi$
$409$	−5.07318			−0.250852			−0.125426	+	0.992103i	$0.540030\pi$
$410$		0			0
$411$		−	13.6794i		−	0.674754i
$412$		0			0
$413$	6.73957	+	34.2415i	0.331632	+	1.68491i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	44.9204	−	44.9204i	2.19976	−	2.19976i
$418$		0			0
$419$	9.44663			0.461498			0.230749	−	0.973013i	$-0.425882\pi$
$419$	9.44663			0.461498			0.230749	+	0.973013i	$0.425882\pi$
$420$		0			0
$421$	30.4827			1.48564			0.742818	−	0.669493i	$-0.233489\pi$
$421$	30.4827			1.48564			0.742818	+	0.669493i	$0.233489\pi$
$422$		0			0
$423$	13.3468	−	13.3468i	0.648942	−	0.648942i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	34.6704	−	6.82398i	1.67782	−	0.330236i
$428$		0			0
$429$		−	28.0637i		−	1.35493i
$430$		0			0
$431$	−11.4845			−0.553191			−0.276595	−	0.960987i	$-0.589206\pi$
$431$	−11.4845			−0.553191			−0.276595	+	0.960987i	$0.589206\pi$
$432$		0			0
$433$	26.8641	−	26.8641i	1.29101	−	1.29101i	0.356845	−	0.934164i	$-0.383852\pi$
$433$	26.8641	−	26.8641i	1.29101	−	1.29101i	0.934164	−	0.356845i	$-0.116148\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	19.0599	+	19.0599i	0.911758	+	0.911758i
$438$		0			0
$439$	13.5387			0.646166			0.323083	−	0.946371i	$-0.395281\pi$
$439$	13.5387			0.646166			0.323083	+	0.946371i	$0.395281\pi$
$440$		0			0
$441$	15.2799	+	37.3123i	0.727614	+	1.77677i
$442$		0			0
$443$	7.27325	+	7.27325i	0.345563	+	0.345563i	0.858454	−	0.512891i	$-0.171426\pi$
$443$	7.27325	+	7.27325i	0.345563	+	0.345563i	−0.512891	+	0.858454i	$0.671426\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	4.38752	+	4.38752i	0.207523	+	0.207523i
$448$		0			0
$449$			4.85641i			0.229188i	0.993412	+	0.114594i	$0.0365567\pi$
$449$			4.85641i			0.229188i	−0.993412	+	0.114594i	$0.963443\pi$
$450$		0			0
$451$		−	2.89270i		−	0.136212i
$452$		0			0
$453$	−3.72608	+	3.72608i	−0.175066	+	0.175066i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	14.5716	−	14.5716i	0.681629	−	0.681629i	−0.278738	−	0.960367i	$-0.589916\pi$
$457$	14.5716	−	14.5716i	0.681629	−	0.681629i	0.960367	+	0.278738i	$0.0899162\pi$
$458$		0			0
$459$			30.8362i			1.43931i
$460$		0			0
$461$		−	7.19256i		−	0.334991i	−0.985873	−	0.167495i	$-0.946432\pi$
$461$		−	7.19256i		−	0.334991i	0.985873	−	0.167495i	$-0.0535680\pi$
$462$		0			0
$463$	−3.74846	−	3.74846i	−0.174206	−	0.174206i	0.614619	−	0.788824i	$-0.289310\pi$
$463$	−3.74846	−	3.74846i	−0.174206	−	0.174206i	−0.788824	+	0.614619i	$0.789310\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−17.4030	−	17.4030i	−0.805314	−	0.805314i	0.178607	−	0.983921i	$-0.442841\pi$
$467$	−17.4030	−	17.4030i	−0.805314	−	0.805314i	−0.983921	+	0.178607i	$0.942841\pi$
$468$		0			0
$469$	20.3706	+	13.6705i	0.940626	+	0.631244i
$470$		0			0
$471$	−19.4437			−0.895917
$472$		0			0
$473$	32.3862	+	32.3862i	1.48912	+	1.48912i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	26.1185	−	26.1185i	1.19588	−	1.19588i
$478$		0			0
$479$	29.8582			1.36426			0.682129	−	0.731232i	$-0.261054\pi$
$479$	29.8582			1.36426			0.682129	+	0.731232i	$0.261054\pi$
$480$		0			0
$481$			16.4954i			0.752125i
$482$		0			0
$483$	7.60350	+	38.6308i	0.345971	+	1.75776i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−18.7577	+	18.7577i	−0.849993	+	0.849993i	−0.990132	−	0.140139i	$-0.955245\pi$
$487$	−18.7577	+	18.7577i	−0.849993	+	0.849993i	0.140139	+	0.990132i	$0.455245\pi$
$488$		0			0
$489$	7.00761			0.316895
$490$		0			0
$491$	20.7440			0.936163			0.468081	−	0.883685i	$-0.344945\pi$
$491$	20.7440			0.936163			0.468081	+	0.883685i	$0.344945\pi$
$492$		0			0
$493$	19.3375	−	19.3375i	0.870917	−	0.870917i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	8.34438	+	42.3950i	0.374297	+	1.90168i
$498$		0			0
$499$		−	1.19553i		−	0.0535192i	−0.999642	−	0.0267596i	$-0.991481\pi$
$499$		−	1.19553i		−	0.0535192i	0.999642	−	0.0267596i	$-0.00851886\pi$
$500$		0			0
$501$	6.92820			0.309529
$502$		0			0
$503$	−12.8461	+	12.8461i	−0.572779	+	0.572779i	−0.932904	−	0.360125i	$-0.882734\pi$
$503$	−12.8461	+	12.8461i	−0.572779	+	0.572779i	0.360125	+	0.932904i	$0.382734\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−15.1391	−	15.1391i	−0.672351	−	0.672351i
$508$		0			0
$509$	18.4311			0.816943			0.408471	−	0.912771i	$-0.366062\pi$
$509$	18.4311			0.816943			0.408471	+	0.912771i	$0.366062\pi$
$510$		0			0
$511$	−21.8274	−	14.6481i	−0.965589	−	0.647996i
$512$		0			0
$513$	−30.9661	−	30.9661i	−1.36719	−	1.36719i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−9.14963	−	9.14963i	−0.402400	−	0.402400i
$518$		0			0
$519$		−	3.97352i		−	0.174418i
$520$		0			0
$521$		−	30.4435i		−	1.33375i	−0.745168	−	0.666877i	$-0.767631\pi$
$521$		−	30.4435i		−	1.33375i	0.745168	−	0.666877i	$-0.232369\pi$
$522$		0			0
$523$	−12.5460	+	12.5460i	−0.548599	+	0.548599i	−0.926036	−	0.377436i	$-0.876806\pi$
$523$	−12.5460	+	12.5460i	−0.548599	+	0.548599i	0.377436	+	0.926036i	$0.376806\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.464880	+	0.464880i	−0.0202505	+	0.0202505i
$528$		0			0
$529$			2.27988i			0.0991254i
$530$		0			0
$531$		−	75.9760i		−	3.29708i
$532$		0			0
$533$	1.24391	+	1.24391i	0.0538796	+	0.0538796i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	34.9350	+	34.9350i	1.50756	+	1.50756i
$538$		0			0
$539$	25.5787	−	10.4748i	1.10175	−	0.451183i
$540$		0			0
$541$	−20.0700			−0.862875			−0.431438	−	0.902143i	$-0.641994\pi$
$541$	−20.0700			−0.862875			−0.431438	+	0.902143i	$0.641994\pi$
$542$		0			0
$543$	−39.0117	−	39.0117i	−1.67415	−	1.67415i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	12.3298	−	12.3298i	0.527185	−	0.527185i	−0.392547	−	0.919732i	$-0.628406\pi$
$547$	12.3298	−	12.3298i	0.527185	−	0.527185i	0.919732	+	0.392547i	$0.128406\pi$
$548$		0			0
$549$	−76.9277			−3.28319
$550$		0			0
$551$			38.8379i			1.65455i
$552$		0			0
$553$	17.2172	−	3.38877i	0.732150	−	0.144105i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	10.8455	−	10.8455i	0.459541	−	0.459541i	−0.438964	−	0.898505i	$-0.644654\pi$
$557$	10.8455	−	10.8455i	0.459541	−	0.459541i	0.898505	+	0.438964i	$0.144654\pi$
$558$		0			0
$559$	−27.8532			−1.17806
$560$		0			0
$561$	44.1169			1.86262
$562$		0			0
$563$	29.8745	−	29.8745i	1.25906	−	1.25906i	0.307517	−	0.951542i	$-0.400502\pi$
$563$	29.8745	−	29.8745i	1.25906	−	1.25906i	0.951542	−	0.307517i	$-0.0994982\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−3.52413	−	17.9049i	−0.148000	−	0.751937i
$568$		0			0
$569$		−	16.2736i		−	0.682225i	−0.940023	−	0.341112i	$-0.889196\pi$
$569$		−	16.2736i		−	0.682225i	0.940023	−	0.341112i	$-0.110804\pi$
$570$		0			0
$571$	−24.0495			−1.00644			−0.503221	−	0.864158i	$-0.667852\pi$
$571$	−24.0495			−1.00644			−0.503221	+	0.864158i	$0.667852\pi$
$572$		0			0
$573$	9.07557	−	9.07557i	0.379137	−	0.379137i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−11.1701	−	11.1701i	−0.465017	−	0.465017i	0.435278	−	0.900296i	$-0.356650\pi$
$577$	−11.1701	−	11.1701i	−0.465017	−	0.465017i	−0.900296	+	0.435278i	$0.856650\pi$
$578$		0			0
$579$	49.7611			2.06800
$580$		0			0
$581$	−32.5111	−	21.8179i	−1.34879	−	0.905157i
$582$		0			0
$583$	−17.9050	−	17.9050i	−0.741551	−	0.741551i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	15.4617	+	15.4617i	0.638171	+	0.638171i	0.950104	−	0.311933i	$-0.100977\pi$
$587$	15.4617	+	15.4617i	0.638171	+	0.638171i	−0.311933	+	0.950104i	$0.600977\pi$
$588$		0			0
$589$		−	0.933678i		−	0.0384715i
$590$		0			0
$591$		−	11.9347i		−	0.490928i
$592$		0			0
$593$	−32.5438	+	32.5438i	−1.33641	+	1.33641i	−0.436904	+	0.899508i	$0.643925\pi$
$593$	−32.5438	+	32.5438i	−1.33641	+	1.33641i	−0.899508	+	0.436904i	$0.856075\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−9.08770	+	9.08770i	−0.371935	+	0.371935i
$598$		0			0
$599$			21.7804i			0.889923i	0.895550	+	0.444961i	$0.146783\pi$
$599$			21.7804i			0.889923i	−0.895550	+	0.444961i	$0.853217\pi$
$600$		0			0
$601$			19.8736i			0.810660i	0.914170	+	0.405330i	$0.132843\pi$
$601$			19.8736i			0.810660i	−0.914170	+	0.405330i	$0.867157\pi$
$602$		0			0
$603$	−37.7657	−	37.7657i	−1.53794	−	1.53794i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−6.80167	−	6.80167i	−0.276071	−	0.276071i	0.555467	−	0.831538i	$-0.312539\pi$
$607$	−6.80167	−	6.80167i	−0.276071	−	0.276071i	−0.831538	+	0.555467i	$0.812539\pi$
$608$		0			0
$609$	−31.6119	+	47.1054i	−1.28098	+	1.90881i
$610$		0			0
$611$	7.86898			0.318345
$612$		0			0
$613$	13.0751	+	13.0751i	0.528098	+	0.528098i	0.920005	−	0.391907i	$-0.128185\pi$
$613$	13.0751	+	13.0751i	0.528098	+	0.528098i	−0.391907	+	0.920005i	$0.628185\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	21.6256	−	21.6256i	0.870614	−	0.870614i	−0.121925	−	0.992539i	$-0.538907\pi$
$617$	21.6256	−	21.6256i	0.870614	−	0.870614i	0.992539	+	0.121925i	$0.0389068\pi$
$618$		0			0
$619$	−42.9528			−1.72642			−0.863210	−	0.504845i	$-0.831549\pi$
$619$	−42.9528			−1.72642			−0.863210	+	0.504845i	$0.831549\pi$
$620$		0			0
$621$		−	41.0716i		−	1.64815i
$622$		0			0
$623$	−1.75537	−	8.91847i	−0.0703276	−	0.357311i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−44.3028	+	44.3028i	−1.76928	+	1.76928i
$628$		0			0
$629$	−25.9312			−1.03395
$630$		0			0
$631$	−22.7175			−0.904370			−0.452185	−	0.891924i	$-0.649355\pi$
$631$	−22.7175			−0.904370			−0.452185	+	0.891924i	$0.649355\pi$
$632$		0			0
$633$	39.4212	−	39.4212i	1.56685	−	1.56685i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−6.49492	+	15.5036i	−0.257338	+	0.614276i
$638$		0			0
$639$		−	94.0674i		−	3.72125i
$640$		0			0
$641$	−26.0435			−1.02866			−0.514328	−	0.857594i	$-0.671959\pi$
$641$	−26.0435			−1.02866			−0.514328	+	0.857594i	$0.671959\pi$
$642$		0			0
$643$	12.7907	−	12.7907i	0.504416	−	0.504416i	−0.408391	−	0.912807i	$-0.633910\pi$
$643$	12.7907	−	12.7907i	0.504416	−	0.504416i	0.912807	+	0.408391i	$0.133910\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.81311	+	1.81311i	0.0712805	+	0.0712805i	0.741848	−	0.670568i	$-0.233949\pi$
$647$	1.81311	+	1.81311i	0.0712805	+	0.0712805i	−0.670568	+	0.741848i	$0.733949\pi$
$648$		0			0
$649$	−52.0840			−2.04447
$650$		0			0
$651$	0.759961	−	1.13243i	0.0297852	−	0.0443834i
$652$		0			0
$653$	21.4676	+	21.4676i	0.840092	+	0.840092i	0.988871	−	0.148778i	$-0.0475340\pi$
$653$	21.4676	+	21.4676i	0.840092	+	0.840092i	−0.148778	+	0.988871i	$0.547534\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	40.4665	+	40.4665i	1.57875	+	1.57875i
$658$		0			0
$659$		−	27.6423i		−	1.07679i	−0.842692	−	0.538396i	$-0.819031\pi$
$659$		−	27.6423i		−	1.07679i	0.842692	−	0.538396i	$-0.180969\pi$
$660$		0			0
$661$		−	7.38292i		−	0.287162i	−0.989639	−	0.143581i	$-0.954138\pi$
$661$		−	7.38292i		−	0.287162i	0.989639	−	0.143581i	$-0.0458618\pi$
$662$		0			0
$663$	−18.9710	+	18.9710i	−0.736772	+	0.736772i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−25.7561	+	25.7561i	−0.997281	+	0.997281i
$668$		0			0
$669$		−	58.2781i		−	2.25316i
$670$		0			0
$671$			52.7363i			2.03586i
$672$		0			0
$673$	−12.4963	−	12.4963i	−0.481697	−	0.481697i	0.423976	−	0.905673i	$-0.360634\pi$
$673$	−12.4963	−	12.4963i	−0.481697	−	0.481697i	−0.905673	+	0.423976i	$0.860634\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	27.0753	+	27.0753i	1.04059	+	1.04059i	0.999141	+	0.0414483i	$0.0131972\pi$
$677$	27.0753	+	27.0753i	1.04059	+	1.04059i	0.0414483	+	0.999141i	$0.486803\pi$
$678$		0			0
$679$	2.93759	−	4.37735i	0.112734	−	0.167987i
$680$		0			0
$681$	−28.9565			−1.10962
$682$		0			0
$683$	17.7113	+	17.7113i	0.677704	+	0.677704i	0.959480	−	0.281776i	$-0.0909235\pi$
$683$	17.7113	+	17.7113i	0.677704	+	0.677704i	−0.281776	+	0.959480i	$0.590924\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	3.44604	−	3.44604i	0.131474	−	0.131474i
$688$		0			0
$689$	15.3989			0.586652
$690$		0			0
$691$		−	2.75386i		−	0.104762i	−0.998627	−	0.0523810i	$-0.983319\pi$
$691$		−	2.75386i		−	0.104762i	0.998627	−	0.0523810i	$-0.0166810\pi$
$692$		0			0
$693$	−59.0422	+	11.6209i	−2.24283	+	0.441443i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1.95546	+	1.95546i	−0.0740683	+	0.0740683i
$698$		0			0
$699$	19.8017			0.748968
$700$		0			0
$701$	−1.66352			−0.0628301			−0.0314151	−	0.999506i	$-0.510001\pi$
$701$	−1.66352			−0.0628301			−0.0314151	+	0.999506i	$0.510001\pi$
$702$		0			0
$703$	26.0405	−	26.0405i	0.982135	−	0.982135i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	31.6141	−	6.22244i	1.18897	−	0.234019i
$708$		0			0
$709$			28.7442i			1.07951i	0.841822	+	0.539756i	$0.181484\pi$
$709$			28.7442i			1.07951i	−0.841822	+	0.539756i	$0.818516\pi$
$710$		0			0
$711$	−38.2020			−1.43269
$712$		0			0
$713$	0.619187	−	0.619187i	0.0231887	−	0.0231887i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	10.8481	+	10.8481i	0.405130	+	0.405130i
$718$		0			0
$719$	42.1059			1.57029			0.785143	−	0.619314i	$-0.212589\pi$
$719$	42.1059			1.57029			0.785143	+	0.619314i	$0.212589\pi$
$720$		0			0
$721$	29.5907	+	19.8580i	1.10202	+	0.739550i
$722$		0			0
$723$	18.1053	+	18.1053i	0.673345	+	0.673345i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−15.6639	−	15.6639i	−0.580943	−	0.580943i	0.354219	−	0.935162i	$-0.384747\pi$
$727$	−15.6639	−	15.6639i	−0.580943	−	0.580943i	−0.935162	+	0.354219i	$0.884747\pi$
$728$		0			0
$729$		−	32.8035i		−	1.21494i
$730$		0			0
$731$		−	43.7860i		−	1.61948i
$732$		0			0
$733$	−3.73703	+	3.73703i	−0.138031	+	0.138031i	−0.772746	−	0.634715i	$-0.781117\pi$
$733$	−3.73703	+	3.73703i	−0.138031	+	0.138031i	0.634715	+	0.772746i	$0.281117\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−25.8895	+	25.8895i	−0.953654	+	0.953654i
$738$		0			0
$739$		−	14.2445i		−	0.523993i	−0.965069	−	0.261997i	$-0.915619\pi$
$739$		−	14.2445i		−	0.523993i	0.965069	−	0.261997i	$-0.0843809\pi$
$740$		0			0
$741$		−	38.1018i		−	1.39970i
$742$		0			0
$743$	4.30083	+	4.30083i	0.157782	+	0.157782i	0.781583	−	0.623801i	$-0.214412\pi$
$743$	4.30083	+	4.30083i	0.157782	+	0.157782i	−0.623801	+	0.781583i	$0.714412\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	60.2734	+	60.2734i	2.20529	+	2.20529i
$748$		0			0
$749$	27.9294	+	18.7431i	1.02052	+	0.684857i
$750$		0			0
$751$	−28.2906			−1.03234			−0.516170	−	0.856486i	$-0.672643\pi$
$751$	−28.2906			−1.03234			−0.516170	+	0.856486i	$0.672643\pi$
$752$		0			0
$753$	−23.2250	−	23.2250i	−0.846366	−	0.846366i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−15.1132	+	15.1132i	−0.549299	+	0.549299i	−0.926238	−	0.376939i	$-0.876977\pi$
$757$	−15.1132	+	15.1132i	−0.549299	+	0.549299i	0.376939	+	0.926238i	$0.376977\pi$
$758$		0			0
$759$	−58.7605			−2.13287
$760$		0			0
$761$			23.7113i			0.859533i	0.902940	+	0.429766i	$0.141404\pi$
$761$			23.7113i			0.859533i	−0.902940	+	0.429766i	$0.858596\pi$
$762$		0			0
$763$	10.0273	−	1.97363i	0.363014	−	0.0714501i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	22.3969	−	22.3969i	0.808706	−	0.808706i
$768$		0			0
$769$	27.3228			0.985286			0.492643	−	0.870232i	$-0.336031\pi$
$769$	27.3228			0.985286			0.492643	+	0.870232i	$0.336031\pi$
$770$		0			0
$771$	−6.79718			−0.244794
$772$		0			0
$773$	−7.79374	+	7.79374i	−0.280321	+	0.280321i	−0.833237	−	0.552916i	$-0.813515\pi$
$773$	−7.79374	+	7.79374i	−0.280321	+	0.280321i	0.552916	+	0.833237i	$0.313515\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	52.7792	−	10.3882i	1.89344	−	0.372676i
$778$		0			0
$779$		−	3.92739i		−	0.140713i
$780$		0			0
$781$	−64.4861			−2.30750
$782$		0			0
$783$	41.8453	−	41.8453i	1.49543	−	1.49543i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	6.99375	+	6.99375i	0.249300	+	0.249300i	0.820683	−	0.571383i	$-0.193593\pi$
$787$	6.99375	+	6.99375i	0.249300	+	0.249300i	−0.571383	+	0.820683i	$0.693593\pi$
$788$		0			0
$789$	86.7112			3.08700
$790$		0			0
$791$	6.14086	−	9.15059i	0.218344	−	0.325357i
$792$		0			0
$793$	−22.6775	−	22.6775i	−0.805300	−	0.805300i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−14.1365	−	14.1365i	−0.500740	−	0.500740i	0.410928	−	0.911668i	$-0.365205\pi$
$797$	−14.1365	−	14.1365i	−0.500740	−	0.500740i	−0.911668	+	0.410928i	$0.865205\pi$
$798$		0			0
$799$			12.3703i			0.437628i
$800$		0			0
$801$			19.7886i			0.699195i
$802$		0			0
$803$	27.7411	−	27.7411i	0.978962	−	0.978962i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	65.0826	−	65.0826i	2.29101	−	2.29101i
$808$		0			0
$809$		−	7.37370i		−	0.259246i	−0.991563	−	0.129623i	$-0.958623\pi$
$809$		−	7.37370i		−	0.259246i	0.991563	−	0.129623i	$-0.0413766\pi$
$810$		0			0
$811$			25.8396i			0.907351i	0.891167	+	0.453675i	$0.149887\pi$
$811$			25.8396i			0.907351i	−0.891167	+	0.453675i	$0.850113\pi$
$812$		0			0
$813$	19.3580	+	19.3580i	0.678914	+	0.678914i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	43.9704	+	43.9704i	1.53833	+	1.53833i
$818$		0			0
$819$	20.3919	−	30.3863i	0.712550	−	1.06178i
$820$		0			0
$821$	−19.6580			−0.686070			−0.343035	−	0.939323i	$-0.611455\pi$
$821$	−19.6580			−0.686070			−0.343035	+	0.939323i	$0.611455\pi$
$822$		0			0
$823$	2.50942	+	2.50942i	0.0874728	+	0.0874728i	0.749489	−	0.662016i	$-0.230299\pi$
$823$	2.50942	+	2.50942i	0.0874728	+	0.0874728i	−0.662016	+	0.749489i	$0.730299\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−24.3351	+	24.3351i	−0.846213	+	0.846213i	−0.989658	−	0.143445i	$-0.954182\pi$
$827$	−24.3351	+	24.3351i	−0.846213	+	0.846213i	0.143445	+	0.989658i	$0.454182\pi$
$828$		0			0
$829$	−32.2235			−1.11917			−0.559584	−	0.828773i	$-0.689039\pi$
$829$	−32.2235			−1.11917			−0.559584	+	0.828773i	$0.689039\pi$
$830$		0			0
$831$		−	60.0546i		−	2.08327i
$832$		0			0
$833$	−24.3721	−	10.2102i	−0.844444	−	0.353762i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1.00598	+	1.00598i	−0.0347717	+	0.0347717i
$838$		0			0
$839$	−18.6243			−0.642984			−0.321492	−	0.946912i	$-0.604184\pi$
$839$	−18.6243			−0.642984			−0.321492	+	0.946912i	$0.604184\pi$
$840$		0			0
$841$	−23.4827			−0.809748
$842$		0			0
$843$	−25.7646	+	25.7646i	−0.887379	+	0.887379i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	2.34612	+	11.9199i	0.0806136	+	0.409571i
$848$		0			0
$849$			29.9025i			1.02625i
$850$		0			0
$851$	34.5385			1.18397
$852$		0			0
$853$	−0.641601	+	0.641601i	−0.0219680	+	0.0219680i	−0.718005	−	0.696037i	$-0.754945\pi$
$853$	−0.641601	+	0.641601i	−0.0219680	+	0.0219680i	0.696037	+	0.718005i	$0.254945\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−22.0372	−	22.0372i	−0.752778	−	0.752778i	0.222219	−	0.974997i	$-0.428670\pi$
$857$	−22.0372	−	22.0372i	−0.752778	−	0.752778i	−0.974997	+	0.222219i	$0.928670\pi$
$858$		0			0
$859$	−0.584800			−0.0199531			−0.00997656	−	0.999950i	$-0.503176\pi$
$859$	−0.584800			−0.0199531			−0.00997656	+	0.999950i	$0.503176\pi$
$860$		0			0
$861$	3.19668	−	4.76342i	0.108942	−	0.162337i
$862$		0			0
$863$	−1.85864	−	1.85864i	−0.0632688	−	0.0632688i	0.674764	−	0.738033i	$-0.264245\pi$
$863$	−1.85864	−	1.85864i	−0.0632688	−	0.0632688i	−0.738033	+	0.674764i	$0.764245\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	5.75537	+	5.75537i	0.195463	+	0.195463i
$868$		0			0
$869$			26.1887i			0.888391i
$870$		0			0
$871$		−	22.2658i		−	0.754449i
$872$		0			0
$873$	−8.11530	+	8.11530i	−0.274661	+	0.274661i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−2.60420	+	2.60420i	−0.0879377	+	0.0879377i	−0.749707	−	0.661770i	$-0.769806\pi$
$877$	−2.60420	+	2.60420i	−0.0879377	+	0.0879377i	0.661770	+	0.749707i	$0.269806\pi$
$878$		0			0
$879$		−	45.2206i		−	1.52525i
$880$		0			0
$881$		−	9.41484i		−	0.317194i	−0.987343	−	0.158597i	$-0.949303\pi$
$881$		−	9.41484i		−	0.317194i	0.987343	−	0.158597i	$-0.0506971\pi$
$882$		0			0
$883$	21.5450	+	21.5450i	0.725046	+	0.725046i	0.969628	−	0.244583i	$-0.0786510\pi$
$883$	21.5450	+	21.5450i	0.725046	+	0.725046i	−0.244583	+	0.969628i	$0.578651\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−11.6835	−	11.6835i	−0.392293	−	0.392293i	0.483211	−	0.875504i	$-0.339470\pi$
$887$	−11.6835	−	11.6835i	−0.392293	−	0.392293i	−0.875504	+	0.483211i	$0.839470\pi$
$888$		0			0
$889$	5.39697	+	3.62184i	0.181008	+	0.121473i
$890$		0			0
$891$	27.2348			0.912400
$892$		0			0
$893$	−12.4224	−	12.4224i	−0.415699	−	0.415699i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	25.2680	−	25.2680i	0.843673	−	0.843673i
$898$		0			0
$899$	1.26170			0.0420802
$900$		0			0
$901$			24.2075i			0.806470i
$902$		0			0
$903$	17.5410	+	89.1199i	0.583727	+	2.96572i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−9.07325	+	9.07325i	−0.301272	+	0.301272i	−0.841512	−	0.540239i	$-0.818334\pi$
$907$	−9.07325	+	9.07325i	−0.301272	+	0.301272i	0.540239	+	0.841512i	$0.318334\pi$
$908$		0			0
$909$	−70.1464			−2.32661
$910$		0			0
$911$	−6.95674			−0.230487			−0.115243	−	0.993337i	$-0.536765\pi$
$911$	−6.95674			−0.230487			−0.115243	+	0.993337i	$0.536765\pi$
$912$		0			0
$913$	41.3193	−	41.3193i	1.36747	−	1.36747i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7.01673	−	1.38106i	0.231713	−	0.0456067i
$918$		0			0
$919$			14.8965i			0.491390i	0.969347	+	0.245695i	$0.0790161\pi$
$919$			14.8965i			0.491390i	−0.969347	+	0.245695i	$0.920984\pi$
$920$		0			0
$921$	90.0514			2.96729
$922$		0			0
$923$	27.7301	−	27.7301i	0.912746	−	0.912746i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−54.8591	−	54.8591i	−1.80181	−	1.80181i
$928$		0			0
$929$	2.87543			0.0943397			0.0471699	−	0.998887i	$-0.484980\pi$
$929$	2.87543			0.0943397			0.0471699	+	0.998887i	$0.484980\pi$
$930$		0			0
$931$	34.7280	−	14.2216i	1.13816	−	0.466094i
$932$		0			0
$933$	−33.2096	−	33.2096i	−1.08723	−	1.08723i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−25.5847	−	25.5847i	−0.835816	−	0.835816i	0.152489	−	0.988305i	$-0.451271\pi$
$937$	−25.5847	−	25.5847i	−0.835816	−	0.835816i	−0.988305	+	0.152489i	$0.951271\pi$
$938$		0			0
$939$			69.5377i			2.26928i
$940$		0			0
$941$			37.0907i			1.20912i	0.796558	+	0.604562i	$0.206652\pi$
$941$			37.0907i			1.20912i	−0.796558	+	0.604562i	$0.793348\pi$
$942$		0			0
$943$	2.60453	−	2.60453i	0.0848151	−	0.0848151i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.321147	+	0.321147i	−0.0104359	+	0.0104359i	−0.712305	−	0.701870i	$-0.752349\pi$
$947$	−0.321147	+	0.321147i	−0.0104359	+	0.0104359i	0.701870	+	0.712305i	$0.252349\pi$
$948$		0			0
$949$			23.8582i			0.774471i
$950$		0			0
$951$		−	39.9194i		−	1.29448i
$952$		0			0
$953$	4.73528	+	4.73528i	0.153391	+	0.153391i	0.779630	−	0.626240i	$-0.215407\pi$
$953$	4.73528	+	4.73528i	0.153391	+	0.153391i	−0.626240	+	0.779630i	$0.715407\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−59.8675	−	59.8675i	−1.93524	−	1.93524i
$958$		0			0
$959$	−10.1537	−	6.81406i	−0.327882	−	0.220038i
$960$		0			0
$961$	30.9697			0.999022
$962$		0			0
$963$	−51.7791	−	51.7791i	−1.66856	−	1.66856i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−6.63821	+	6.63821i	−0.213470	+	0.213470i	−0.805740	−	0.592269i	$-0.798232\pi$
$967$	−6.63821	+	6.63821i	−0.213470	+	0.213470i	0.592269	+	0.805740i	$0.298232\pi$
$968$		0			0
$969$	59.8971			1.92417
$970$		0			0
$971$		−	15.9800i		−	0.512823i	−0.966568	−	0.256412i	$-0.917460\pi$
$971$		−	15.9800i		−	0.512823i	0.966568	−	0.256412i	$-0.0825402\pi$
$972$		0			0
$973$	−10.9669	−	55.7189i	−0.351581	−	1.78627i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−22.8732	+	22.8732i	−0.731780	+	0.731780i	−0.970972	−	0.239193i	$-0.923117\pi$
$977$	−22.8732	+	22.8732i	−0.731780	+	0.731780i	0.239193	+	0.970972i	$0.423117\pi$
$978$		0			0
$979$	13.5657			0.433561
$980$		0			0
$981$	−22.2490			−0.710354
$982$		0			0
$983$	12.0432	−	12.0432i	0.384119	−	0.384119i	−0.488465	−	0.872584i	$-0.662443\pi$
$983$	12.0432	−	12.0432i	0.384119	−	0.384119i	0.872584	+	0.488465i	$0.162443\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−4.95562	−	25.1778i	−0.157739	−	0.801419i
$988$		0			0
$989$			58.3197i			1.85446i
$990$		0			0
$991$	−22.0158			−0.699354			−0.349677	−	0.936870i	$-0.613709\pi$
$991$	−22.0158			−0.699354			−0.349677	+	0.936870i	$0.613709\pi$
$992$		0			0
$993$	−27.0347	+	27.0347i	−0.857920	+	0.857920i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	34.1156	+	34.1156i	1.08045	+	1.08045i	0.996467	+	0.0839845i	$0.0267646\pi$
$997$	34.1156	+	34.1156i	1.08045	+	1.08045i	0.0839845	+	0.996467i	$0.473235\pi$
$998$		0			0
$999$	−56.1139			−1.77536

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1400.2.x.c.993.1	yes	32
5.2	odd	4	inner	1400.2.x.c.657.15	yes	32
5.3	odd	4	inner	1400.2.x.c.657.2	yes	32
5.4	even	2	inner	1400.2.x.c.993.16	yes	32
7.6	odd	2	inner	1400.2.x.c.993.15	yes	32
35.13	even	4	inner	1400.2.x.c.657.16	yes	32
35.27	even	4	inner	1400.2.x.c.657.1	✓	32
35.34	odd	2	inner	1400.2.x.c.993.2	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1400.2.x.c.657.1	✓	32	35.27	even	4	inner
1400.2.x.c.657.2	yes	32	5.3	odd	4	inner
1400.2.x.c.657.15	yes	32	5.2	odd	4	inner
1400.2.x.c.657.16	yes	32	35.13	even	4	inner
1400.2.x.c.993.1	yes	32	1.1	even	1	trivial
1400.2.x.c.993.2	yes	32	35.34	odd	2	inner
1400.2.x.c.993.15	yes	32	7.6	odd	2	inner
1400.2.x.c.993.16	yes	32	5.4	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:	\( N \)	\(=\)	\( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1400.x (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.09284 + 2.09284i) q^{3} +(0.510946 + 2.59595i) q^{7} -5.75996i q^{9} +O(q^{10})\)	\(q+(-2.09284 + 2.09284i) q^{3} +(0.510946 + 2.59595i) q^{7} -5.75996i q^{9} -3.94864 q^{11} +(1.69798 - 1.69798i) q^{13} +(2.66927 + 2.66927i) q^{17} -5.36102 q^{19} +(-6.50223 - 4.36357i) q^{21} +(-3.55527 - 3.55527i) q^{23} +(5.77616 + 5.77616i) q^{27} -7.24449i q^{29} +0.174160i q^{31} +(8.26386 - 8.26386i) q^{33} +(-4.85737 + 4.85737i) q^{37} +7.10719i q^{39} +0.732583i q^{41} +(-8.20187 - 8.20187i) q^{43} +(2.31716 + 2.31716i) q^{47} +(-6.47787 + 2.65278i) q^{49} -11.1727 q^{51} +(4.53449 + 4.53449i) q^{53} +(11.2198 - 11.2198i) q^{57} +13.1904 q^{59} -13.3556i q^{61} +(14.9525 - 2.94303i) q^{63} +(6.55658 - 6.55658i) q^{67} +14.8812 q^{69} +16.3312 q^{71} +(-7.02549 + 7.02549i) q^{73} +(-2.01754 - 10.2504i) q^{77} -6.63234i q^{79} -6.89727 q^{81} +(-10.4642 + 10.4642i) q^{83} +(15.1616 + 15.1616i) q^{87} -3.43554 q^{89} +(5.27543 + 3.54028i) q^{91} +(-0.364490 - 0.364490i) q^{93} +(-1.40892 - 1.40892i) q^{97} +22.7440i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q - 16 q^{11} - 40 q^{21} + 32 q^{51} + 128 q^{71}+O(q^{100}) \) 32 * q - 16 * q^11 - 40 * q^21 + 32 * q^51 + 128 * q^71

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