LMFDB - Embedded newform 2175.2.d.j.376.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,2,Mod(376,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2175.376");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2175 = 3 \cdot 5^{2} \cdot 29$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2175.d (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:	$17.3674624396$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 435)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			376.13
Character	$\chi$	$=$	2175.376
Dual form			2175.2.d.j.376.14

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.334522i q^{2} -1.00000i q^{3} +1.88809 q^{4} -0.334522 q^{6} -0.275019 q^{7} -1.30066i q^{8} -1.00000 q^{9} -0.541705i q^{11} -1.88809i q^{12} -5.03697 q^{13} +0.0920000i q^{14} +3.34109 q^{16} +3.32386i q^{17} +0.334522i q^{18} -6.31271i q^{19} +0.275019i q^{21} -0.181212 q^{22} -6.93673 q^{23} -1.30066 q^{24} +1.68498i q^{26} +1.00000i q^{27} -0.519262 q^{28} +(-2.25292 - 4.89125i) q^{29} -10.1057i q^{31} -3.71898i q^{32} -0.541705 q^{33} +1.11191 q^{34} -1.88809 q^{36} +4.07094i q^{37} -2.11174 q^{38} +5.03697i q^{39} +8.49843i q^{41} +0.0920000 q^{42} -5.97627i q^{43} -1.02279i q^{44} +2.32049i q^{46} -12.4310i q^{47} -3.34109i q^{48} -6.92436 q^{49} +3.32386 q^{51} -9.51028 q^{52} +3.98569 q^{53} +0.334522 q^{54} +0.357705i q^{56} -6.31271 q^{57} +(-1.63623 + 0.753651i) q^{58} +8.01554 q^{59} -4.84022i q^{61} -3.38058 q^{62} +0.275019 q^{63} +5.43810 q^{64} +0.181212i q^{66} -12.1664 q^{67} +6.27576i q^{68} +6.93673i q^{69} -7.20182 q^{71} +1.30066i q^{72} -1.23869i q^{73} +1.36182 q^{74} -11.9190i q^{76} +0.148979i q^{77} +1.68498 q^{78} +2.36972i q^{79} +1.00000 q^{81} +2.84292 q^{82} +10.3724 q^{83} +0.519262i q^{84} -1.99920 q^{86} +(-4.89125 + 2.25292i) q^{87} -0.704571 q^{88} -2.50630i q^{89} +1.38526 q^{91} -13.0972 q^{92} -10.1057 q^{93} -4.15845 q^{94} -3.71898 q^{96} -17.8627i q^{97} +2.31636i q^{98} +0.541705i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$24 q - 32 q^{4} - 24 q^{9} + 64 q^{16} - 16 q^{29} + 104 q^{34} + 32 q^{36} + 8 q^{49} - 16 q^{51} + 56 q^{59} + 8 q^{64} - 32 q^{71} + 56 q^{74} + 24 q^{81} - 32 q^{86} + 48 q^{91} + 56 q^{94} - 80 q^{96}+O(q^{100})$ 24 * q - 32 * q^4 - 24 * q^9 + 64 * q^16 - 16 * q^29 + 104 * q^34 + 32 * q^36 + 8 * q^49 - 16 * q^51 + 56 * q^59 + 8 * q^64 - 32 * q^71 + 56 * q^74 + 24 * q^81 - 32 * q^86 + 48 * q^91 + 56 * q^94 - 80 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times$ .

$n$	$901$	$1451$	$2002$
$\chi(n)$	$-1$	$1$	$1$

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

)

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

$2$		−	0.334522i		−	0.236543i	−0.992981	−	0.118272i	$-0.962265\pi$
$2$		−	0.334522i		−	0.236543i	0.992981	−	0.118272i	$-0.0377353\pi$
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$	1.88809			0.944047
$5$		0			0
$5$		0			0
$6$	−0.334522			−0.136568
$7$	−0.275019			−0.103947			−0.0519737	−	0.998648i	$-0.516551\pi$
$7$	−0.275019			−0.103947			−0.0519737	+	0.998648i	$0.516551\pi$
$8$		−	1.30066i		−	0.459851i
$9$	−1.00000			−0.333333
$10$		0			0
$11$		−	0.541705i		−	0.163330i	−0.996660	−	0.0816650i	$-0.973976\pi$
$11$		−	0.541705i		−	0.163330i	0.996660	−	0.0816650i	$-0.0260238\pi$
$12$		−	1.88809i		−	0.545046i
$13$	−5.03697			−1.39700			−0.698502	−	0.715608i	$-0.746150\pi$
$13$	−5.03697			−1.39700			−0.698502	+	0.715608i	$0.746150\pi$
$14$			0.0920000i			0.0245880i
$15$		0			0
$16$	3.34109			0.835273
$17$			3.32386i			0.806154i	0.915166	+	0.403077i	$0.132059\pi$
$17$			3.32386i			0.806154i	−0.915166	+	0.403077i	$0.867941\pi$
$18$			0.334522i			0.0788477i
$19$		−	6.31271i		−	1.44824i	−0.689676	−	0.724118i	$-0.742247\pi$
$19$		−	6.31271i		−	1.44824i	0.689676	−	0.724118i	$-0.257753\pi$
$20$		0			0
$21$			0.275019i			0.0600140i
$22$	−0.181212			−0.0386346
$23$	−6.93673			−1.44641			−0.723204	−	0.690635i	$-0.757331\pi$
$23$	−6.93673			−1.44641			−0.723204	+	0.690635i	$0.757331\pi$
$24$	−1.30066			−0.265495
$25$		0			0
$26$			1.68498i			0.330452i
$27$			1.00000i			0.192450i
$28$	−0.519262			−0.0981312
$29$	−2.25292	−	4.89125i	−0.418356	−	0.908283i
$29$	−2.25292	−	4.89125i	−0.418356	−	0.908283i
$30$		0			0
$31$		−	10.1057i		−	1.81504i	−0.420012	−	0.907518i	$-0.637974\pi$
$31$		−	10.1057i		−	1.81504i	0.420012	−	0.907518i	$-0.362026\pi$
$32$		−	3.71898i		−	0.657429i
$33$	−0.541705			−0.0942987
$34$	1.11191			0.190690
$35$		0			0
$36$	−1.88809			−0.314682
$37$			4.07094i			0.669259i	0.942350	+	0.334629i	$0.108611\pi$
$37$			4.07094i			0.669259i	−0.942350	+	0.334629i	$0.891389\pi$
$38$	−2.11174			−0.342570
$39$			5.03697i			0.806561i
$40$		0			0
$41$			8.49843i			1.32723i	0.748073	+	0.663616i	$0.230979\pi$
$41$			8.49843i			1.32723i	−0.748073	+	0.663616i	$0.769021\pi$
$42$	0.0920000			0.0141959
$43$		−	5.97627i		−	0.911372i	−0.890140	−	0.455686i	$-0.849394\pi$
$43$		−	5.97627i		−	0.911372i	0.890140	−	0.455686i	$-0.150606\pi$
$44$		−	1.02279i		−	0.154191i
$45$		0			0
$46$			2.32049i			0.342138i
$47$		−	12.4310i		−	1.81325i	−0.421939	−	0.906624i	$-0.638651\pi$
$47$		−	12.4310i		−	1.81325i	0.421939	−	0.906624i	$-0.361349\pi$
$48$		−	3.34109i		−	0.482245i
$49$	−6.92436			−0.989195
$50$		0			0
$51$	3.32386			0.465433
$52$	−9.51028			−1.31884
$53$	3.98569			0.547477			0.273739	−	0.961804i	$-0.411740\pi$
$53$	3.98569			0.547477			0.273739	+	0.961804i	$0.411740\pi$
$54$	0.334522			0.0455227
$55$		0			0
$56$			0.357705i			0.0478003i
$57$	−6.31271			−0.836139
$58$	−1.63623	+	0.753651i	−0.214848	+	0.0989593i
$59$	8.01554			1.04353			0.521767	−	0.853088i	$-0.325273\pi$
$59$	8.01554			1.04353			0.521767	+	0.853088i	$0.325273\pi$
$60$		0			0
$61$		−	4.84022i		−	0.619726i	−0.950781	−	0.309863i	$-0.899717\pi$
$61$		−	4.84022i		−	0.619726i	0.950781	−	0.309863i	$-0.100283\pi$
$62$	−3.38058			−0.429334
$63$	0.275019			0.0346491
$64$	5.43810			0.679762
$65$		0			0
$66$			0.181212i			0.0223057i
$67$	−12.1664			−1.48636			−0.743182	−	0.669090i	$-0.766684\pi$
$67$	−12.1664			−1.48636			−0.743182	+	0.669090i	$0.766684\pi$
$68$			6.27576i			0.761048i
$69$			6.93673i			0.835084i
$70$		0			0
$71$	−7.20182			−0.854698			−0.427349	−	0.904087i	$-0.640553\pi$
$71$	−7.20182			−0.854698			−0.427349	+	0.904087i	$0.640553\pi$
$72$			1.30066i			0.153284i
$73$		−	1.23869i		−	0.144977i	−0.997369	−	0.0724886i	$-0.976906\pi$
$73$		−	1.23869i		−	0.144977i	0.997369	−	0.0724886i	$-0.0230941\pi$
$74$	1.36182			0.158309
$75$		0			0
$76$		−	11.9190i		−	1.36720i
$77$			0.148979i			0.0169777i
$78$	1.68498			0.190786
$79$			2.36972i			0.266614i	0.991075	+	0.133307i	$0.0425597\pi$
$79$			2.36972i			0.266614i	−0.991075	+	0.133307i	$0.957440\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$	2.84292			0.313948
$83$	10.3724			1.13852			0.569258	−	0.822159i	$-0.307230\pi$
$83$	10.3724			1.13852			0.569258	+	0.822159i	$0.307230\pi$
$84$			0.519262i			0.0566561i
$85$		0			0
$86$	−1.99920			−0.215579
$87$	−4.89125	+	2.25292i	−0.524397	+	0.241538i
$88$	−0.704571			−0.0751075
$89$		−	2.50630i		−	0.265667i	−0.991138	−	0.132833i	$-0.957592\pi$
$89$		−	2.50630i		−	0.265667i	0.991138	−	0.132833i	$-0.0424076\pi$
$90$		0			0
$91$	1.38526			0.145215
$92$	−13.0972			−1.36548
$93$	−10.1057			−1.04791
$94$	−4.15845			−0.428911
$95$		0			0
$96$	−3.71898			−0.379567
$97$		−	17.8627i		−	1.81368i	−0.421476	−	0.906840i	$-0.638488\pi$
$97$		−	17.8627i		−	1.81368i	0.421476	−	0.906840i	$-0.361512\pi$
$98$			2.31636i			0.233987i
$99$			0.541705i			0.0544434i
$100$		0			0
$101$			13.6767i			1.36088i	0.732803	+	0.680441i	$0.238212\pi$
$101$			13.6767i			1.36088i	−0.732803	+	0.680441i	$0.761788\pi$
$102$		−	1.11191i		−	0.110095i
$103$	10.6870			1.05302			0.526511	−	0.850168i	$-0.323500\pi$
$103$	10.6870			1.05302			0.526511	+	0.850168i	$0.323500\pi$
$104$			6.55136i			0.642414i
$105$		0			0
$106$		−	1.33330i		−	0.129502i
$107$	4.59224			0.443948			0.221974	−	0.975053i	$-0.428750\pi$
$107$	4.59224			0.443948			0.221974	+	0.975053i	$0.428750\pi$
$108$			1.88809i			0.181682i
$109$	13.1473			1.25928			0.629639	−	0.776888i	$-0.283203\pi$
$109$	13.1473			1.25928			0.629639	+	0.776888i	$0.283203\pi$
$110$		0			0
$111$	4.07094			0.386397
$112$	−0.918863			−0.0868244
$113$			12.4441i			1.17065i	0.810800	+	0.585323i	$0.199032\pi$
$113$			12.4441i			1.17065i	−0.810800	+	0.585323i	$0.800968\pi$
$114$			2.11174i			0.197783i
$115$		0			0
$116$	−4.25372	−	9.23515i	−0.394948	−	0.857462i
$117$	5.03697			0.465668
$118$		−	2.68138i		−	0.246841i
$119$		−	0.914124i		−	0.0837976i
$120$		0			0
$121$	10.7066			0.973323
$122$	−1.61916			−0.146592
$123$	8.49843			0.766278
$124$		−	19.0805i		−	1.71348i
$125$		0			0
$126$		−	0.0920000i		−	0.00819601i
$127$		−	10.3846i		−	0.921487i	−0.887533	−	0.460744i	$-0.847583\pi$
$127$		−	10.3846i		−	0.921487i	0.887533	−	0.460744i	$-0.152417\pi$
$128$		−	9.25713i		−	0.818222i
$129$	−5.97627			−0.526181
$130$		0			0
$131$			3.80784i			0.332693i	0.986067	+	0.166346i	$0.0531970\pi$
$131$			3.80784i			0.332693i	−0.986067	+	0.166346i	$0.946803\pi$
$132$	−1.02279			−0.0890224
$133$			1.73611i			0.150540i
$134$			4.06994i			0.351589i
$135$		0			0
$136$	4.32319			0.370711
$137$			9.05968i			0.774021i	0.922075	+	0.387010i	$0.126492\pi$
$137$			9.05968i			0.774021i	−0.922075	+	0.387010i	$0.873508\pi$
$138$	2.32049			0.197533
$139$	−1.41853			−0.120318			−0.0601591	−	0.998189i	$-0.519161\pi$
$139$	−1.41853			−0.120318			−0.0601591	+	0.998189i	$0.519161\pi$
$140$		0			0
$141$	−12.4310			−1.04688
$142$			2.40917i			0.202173i
$143$			2.72855i			0.228173i
$144$	−3.34109			−0.278424
$145$		0			0
$146$	−0.414368			−0.0342933
$147$			6.92436i			0.571112i
$148$			7.68632i			0.631812i
$149$	−4.70591			−0.385523			−0.192762	−	0.981246i	$-0.561744\pi$
$149$	−4.70591			−0.385523			−0.192762	+	0.981246i	$0.561744\pi$
$150$		0			0
$151$	−5.99181			−0.487606			−0.243803	−	0.969825i	$-0.578395\pi$
$151$	−5.99181			−0.487606			−0.243803	+	0.969825i	$0.578395\pi$
$152$	−8.21066			−0.665973
$153$		−	3.32386i		−	0.268718i
$154$	0.0498368			0.00401596
$155$		0			0
$156$			9.51028i			0.761432i
$157$			11.2728i			0.899664i	0.893113	+	0.449832i	$0.148516\pi$
$157$			11.2728i			0.899664i	−0.893113	+	0.449832i	$0.851484\pi$
$158$	0.792725			0.0630658
$159$		−	3.98569i		−	0.316086i
$160$		0			0
$161$	1.90773			0.150350
$162$		−	0.334522i		−	0.0262826i
$163$			11.2859i			0.883980i	0.897020	+	0.441990i	$0.145727\pi$
$163$			11.2859i			0.883980i	−0.897020	+	0.441990i	$0.854273\pi$
$164$			16.0458i			1.25297i
$165$		0			0
$166$		−	3.46980i		−	0.269308i
$167$	−9.11150			−0.705069			−0.352535	−	0.935799i	$-0.614680\pi$
$167$	−9.11150			−0.705069			−0.352535	+	0.935799i	$0.614680\pi$
$168$	0.357705			0.0275975
$169$	12.3711			0.951621
$170$		0			0
$171$			6.31271i			0.482745i
$172$		−	11.2838i		−	0.860379i
$173$	−6.68099			−0.507946			−0.253973	−	0.967211i	$-0.581737\pi$
$173$	−6.68099			−0.507946			−0.253973	+	0.967211i	$0.581737\pi$
$174$	0.753651	+	1.63623i	0.0571342	+	0.124043i
$175$		0			0
$176$		−	1.80988i		−	0.136425i
$177$		−	8.01554i		−	0.602485i
$178$	−0.838413			−0.0628417
$179$	−17.6127			−1.31644			−0.658218	−	0.752827i	$-0.728690\pi$
$179$	−17.6127			−1.31644			−0.658218	+	0.752827i	$0.728690\pi$
$180$		0			0
$181$	0.799278			0.0594099			0.0297049	−	0.999559i	$-0.490543\pi$
$181$	0.799278			0.0594099			0.0297049	+	0.999559i	$0.490543\pi$
$182$		−	0.463401i		−	0.0343496i
$183$	−4.84022			−0.357799
$184$			9.02229i			0.665132i
$185$		0			0
$186$			3.38058i			0.247876i
$187$	1.80055			0.131669
$188$		−	23.4709i		−	1.71179i
$189$		−	0.275019i		−	0.0200047i
$190$		0			0
$191$			14.0344i			1.01549i	0.861506	+	0.507747i	$0.169522\pi$
$191$			14.0344i			1.01549i	−0.861506	+	0.507747i	$0.830478\pi$
$192$		−	5.43810i		−	0.392461i
$193$		−	22.3693i		−	1.61018i	−0.593154	−	0.805089i	$-0.702117\pi$
$193$		−	22.3693i		−	1.61018i	0.593154	−	0.805089i	$-0.297883\pi$
$194$	−5.97546			−0.429013
$195$		0			0
$196$	−13.0739			−0.933847
$197$	−4.66274			−0.332207			−0.166103	−	0.986108i	$-0.553119\pi$
$197$	−4.66274			−0.332207			−0.166103	+	0.986108i	$0.553119\pi$
$198$	0.181212			0.0128782
$199$	−14.9769			−1.06169			−0.530843	−	0.847470i	$-0.678124\pi$
$199$	−14.9769			−1.06169			−0.530843	+	0.847470i	$0.678124\pi$
$200$		0			0
$201$			12.1664i			0.858152i
$202$	4.57516			0.321907
$203$	0.619595	+	1.34519i	0.0434870	+	0.0944136i
$204$	6.27576			0.439391
$205$		0			0
$206$		−	3.57504i		−	0.249085i
$207$	6.93673			0.482136
$208$	−16.8290			−1.16688
$209$	−3.41963			−0.236540
$210$		0			0
$211$		−	15.8960i		−	1.09432i	−0.837027	−	0.547162i	$-0.815708\pi$
$211$		−	15.8960i		−	1.09432i	0.837027	−	0.547162i	$-0.184292\pi$
$212$	7.52537			0.516844
$213$			7.20182i			0.493460i
$214$		−	1.53621i		−	0.105013i
$215$		0			0
$216$	1.30066			0.0884984
$217$			2.77926i			0.188668i
$218$		−	4.39805i		−	0.297874i
$219$	−1.23869			−0.0837026
$220$		0			0
$221$		−	16.7422i		−	1.12620i
$222$		−	1.36182i		−	0.0913995i
$223$	7.51910			0.503516			0.251758	−	0.967790i	$-0.418991\pi$
$223$	7.51910			0.503516			0.251758	+	0.967790i	$0.418991\pi$
$224$			1.02279i			0.0683380i
$225$		0			0
$226$	4.16284			0.276908
$227$	14.9639			0.993187			0.496594	−	0.867983i	$-0.334584\pi$
$227$	14.9639			0.993187			0.496594	+	0.867983i	$0.334584\pi$
$228$	−11.9190			−0.789355
$229$			12.3067i			0.813247i	0.913596	+	0.406623i	$0.133294\pi$
$229$			12.3067i			0.813247i	−0.913596	+	0.406623i	$0.866706\pi$
$230$		0			0
$231$	0.148979			0.00980210
$232$	−6.36183	+	2.93027i	−0.417675	+	0.192382i
$233$	−10.7500			−0.704258			−0.352129	−	0.935951i	$-0.614542\pi$
$233$	−10.7500			−0.704258			−0.352129	+	0.935951i	$0.614542\pi$
$234$		−	1.68498i		−	0.110151i
$235$		0			0
$236$	15.1341			0.985146
$237$	2.36972			0.153930
$238$	−0.305795			−0.0198217
$239$	6.04481			0.391006			0.195503	−	0.980703i	$-0.437366\pi$
$239$	6.04481			0.391006			0.195503	+	0.980703i	$0.437366\pi$
$240$		0			0
$241$	24.1998			1.55885			0.779423	−	0.626498i	$-0.215512\pi$
$241$	24.1998			1.55885			0.779423	+	0.626498i	$0.215512\pi$
$242$		−	3.58158i		−	0.230233i
$243$		−	1.00000i		−	0.0641500i
$244$		−	9.13879i		−	0.585051i
$245$		0			0
$246$		−	2.84292i		−	0.181258i
$247$			31.7969i			2.02319i
$248$	−13.1440			−0.834647
$249$		−	10.3724i		−	0.657323i
$250$		0			0
$251$			26.0151i			1.64206i	0.570886	+	0.821030i	$0.306600\pi$
$251$			26.0151i			1.64206i	−0.570886	+	0.821030i	$0.693400\pi$
$252$	0.519262			0.0327104
$253$			3.75766i			0.236242i
$254$	−3.47389			−0.217971
$255$		0			0
$256$	7.77948			0.486218
$257$	3.58709			0.223757			0.111878	−	0.993722i	$-0.464313\pi$
$257$	3.58709			0.223757			0.111878	+	0.993722i	$0.464313\pi$
$258$			1.99920i			0.124465i
$259$		−	1.11959i		−	0.0695677i
$260$		0			0
$261$	2.25292	+	4.89125i	0.139452	+	0.302761i
$262$	1.27381			0.0786962
$263$		−	3.92407i		−	0.241969i	−0.992654	−	0.120984i	$-0.961395\pi$
$263$		−	3.92407i		−	0.241969i	0.992654	−	0.120984i	$-0.0386051\pi$
$264$			0.704571i			0.0433633i
$265$		0			0
$266$	0.580769			0.0356093
$267$	−2.50630			−0.153383
$268$	−22.9713			−1.40320
$269$			13.8826i			0.846437i	0.906028	+	0.423218i	$0.139100\pi$
$269$			13.8826i			0.846437i	−0.906028	+	0.423218i	$0.860900\pi$
$270$		0			0
$271$		−	13.3880i		−	0.813265i	−0.913592	−	0.406632i	$-0.866703\pi$
$271$		−	13.3880i		−	0.813265i	0.913592	−	0.406632i	$-0.133297\pi$
$272$			11.1053i			0.673358i
$273$		−	1.38526i		−	0.0838398i
$274$	3.03067			0.183089
$275$		0			0
$276$			13.0972i			0.788359i
$277$	9.91356			0.595648			0.297824	−	0.954621i	$-0.403739\pi$
$277$	9.91356			0.595648			0.297824	+	0.954621i	$0.403739\pi$
$278$			0.474530i			0.0284604i
$279$			10.1057i			0.605012i
$280$		0			0
$281$	−18.7167			−1.11654			−0.558271	−	0.829658i	$-0.688535\pi$
$281$	−18.7167			−1.11654			−0.558271	+	0.829658i	$0.688535\pi$
$282$			4.15845i			0.247632i
$283$	−18.9462			−1.12624			−0.563118	−	0.826377i	$-0.690398\pi$
$283$	−18.9462			−1.12624			−0.563118	+	0.826377i	$0.690398\pi$
$284$	−13.5977			−0.806876
$285$		0			0
$286$	0.912761			0.0539727
$287$		−	2.33723i		−	0.137962i
$288$			3.71898i			0.219143i
$289$	5.95197			0.350116
$290$		0			0
$291$	−17.8627			−1.04713
$292$		−	2.33875i		−	0.136865i
$293$			7.13932i			0.417083i	0.978013	+	0.208542i	$0.0668717\pi$
$293$			7.13932i			0.417083i	−0.978013	+	0.208542i	$0.933128\pi$
$294$	2.31636			0.135093
$295$		0			0
$296$	5.29489			0.307759
$297$	0.541705			0.0314329
$298$			1.57423i			0.0911929i
$299$	34.9401			2.02064
$300$		0			0
$301$			1.64359i			0.0947347i
$302$			2.00439i			0.115340i
$303$	13.6767			0.785706
$304$		−	21.0913i		−	1.20967i
$305$		0			0
$306$	−1.11191			−0.0635634
$307$			10.0637i			0.574368i	0.957875	+	0.287184i	$0.0927192\pi$
$307$			10.0637i			0.574368i	−0.957875	+	0.287184i	$0.907281\pi$
$308$			0.281286i			0.0160278i
$309$		−	10.6870i		−	0.607963i
$310$		0			0
$311$		−	16.2493i		−	0.921412i	−0.887553	−	0.460706i	$-0.847596\pi$
$311$		−	16.2493i		−	0.921412i	0.887553	−	0.460706i	$-0.152404\pi$
$312$	6.55136			0.370898
$313$	21.4838			1.21434			0.607168	−	0.794573i	$-0.292305\pi$
$313$	21.4838			1.21434			0.607168	+	0.794573i	$0.292305\pi$
$314$	3.77099			0.212809
$315$		0			0
$316$			4.47426i			0.251697i
$317$		−	24.4204i		−	1.37159i	−0.727796	−	0.685794i	$-0.759455\pi$
$317$		−	24.4204i		−	1.37159i	0.727796	−	0.685794i	$-0.240545\pi$
$318$	−1.33330			−0.0747680
$319$	−2.64961	+	1.22042i	−0.148350	+	0.0683302i
$320$		0			0
$321$		−	4.59224i		−	0.256314i
$322$		−	0.638179i		−	0.0355643i
$323$	20.9826			1.16750
$324$	1.88809			0.104894
$325$		0			0
$326$	3.77539			0.209099
$327$		−	13.1473i		−	0.727045i
$328$	11.0535			0.610329
$329$			3.41876i			0.188482i
$330$		0			0
$331$			0.627675i			0.0345001i	0.999851	+	0.0172501i	$0.00549114\pi$
$331$			0.627675i			0.0345001i	−0.999851	+	0.0172501i	$0.994509\pi$
$332$	19.5840			1.07481
$333$		−	4.07094i		−	0.223086i
$334$			3.04800i			0.166779i
$335$		0			0
$336$			0.918863i			0.0501281i
$337$		−	27.2068i		−	1.48205i	−0.671479	−	0.741024i	$-0.734341\pi$
$337$		−	27.2068i		−	1.48205i	0.671479	−	0.741024i	$-0.265659\pi$
$338$		−	4.13840i		−	0.225099i
$339$	12.4441			0.675873
$340$		0			0
$341$	−5.47430			−0.296450
$342$	2.11174			0.114190
$343$	3.82946			0.206772
$344$	−7.77306			−0.419096
$345$		0			0
$346$			2.23494i			0.120151i
$347$	−13.0161			−0.698739			−0.349369	−	0.936985i	$-0.613604\pi$
$347$	−13.0161			−0.698739			−0.349369	+	0.936985i	$0.613604\pi$
$348$	−9.23515	+	4.25372i	−0.495056	+	0.228023i
$349$	−23.2480			−1.24444			−0.622218	−	0.782844i	$-0.713768\pi$
$349$	−23.2480			−1.24444			−0.622218	+	0.782844i	$0.713768\pi$
$350$		0			0
$351$		−	5.03697i		−	0.268854i
$352$	−2.01459			−0.107378
$353$	33.0268			1.75784			0.878920	−	0.476969i	$-0.158265\pi$
$353$	33.0268			1.75784			0.878920	+	0.476969i	$0.158265\pi$
$354$	−2.68138			−0.142514
$355$		0			0
$356$		−	4.73213i		−	0.250802i
$357$	−0.914124			−0.0483805
$358$			5.89185i			0.311394i
$359$		−	30.0802i		−	1.58757i	−0.608195	−	0.793787i	$-0.708106\pi$
$359$		−	30.0802i		−	1.58757i	0.608195	−	0.793787i	$-0.291894\pi$
$360$		0			0
$361$	−20.8503			−1.09739
$362$		−	0.267376i		−	0.0140530i
$363$		−	10.7066i		−	0.561948i
$364$	2.61551			0.137090
$365$		0			0
$366$			1.61916i			0.0846349i
$367$		−	3.89299i		−	0.203213i	−0.994825	−	0.101606i	$-0.967602\pi$
$367$		−	3.89299i		−	0.203213i	0.994825	−	0.101606i	$-0.0323982\pi$
$368$	−23.1762			−1.20814
$369$		−	8.49843i		−	0.442411i
$370$		0			0
$371$	−1.09614			−0.0569088
$372$	−19.0805			−0.989279
$373$	−1.78697			−0.0925260			−0.0462630	−	0.998929i	$-0.514731\pi$
$373$	−1.78697			−0.0925260			−0.0462630	+	0.998929i	$0.514731\pi$
$374$		−	0.602324i		−	0.0311454i
$375$		0			0
$376$	−16.1684			−0.833824
$377$	11.3479	+	24.6371i	0.584445	+	1.26888i
$378$	−0.0920000			−0.00473197
$379$		−	18.4165i		−	0.945993i	−0.881064	−	0.472996i	$-0.843172\pi$
$379$		−	18.4165i		−	0.945993i	0.881064	−	0.472996i	$-0.156828\pi$
$380$		0			0
$381$	−10.3846			−0.532021
$382$	4.69482			0.240208
$383$	28.3971			1.45102			0.725512	−	0.688210i	$-0.241603\pi$
$383$	28.3971			1.45102			0.725512	+	0.688210i	$0.241603\pi$
$384$	−9.25713			−0.472401
$385$		0			0
$386$	−7.48304			−0.380877
$387$			5.97627i			0.303791i
$388$		−	33.7264i		−	1.71220i
$389$		−	8.49677i		−	0.430803i	−0.976526	−	0.215402i	$-0.930894\pi$
$389$		−	8.49677i		−	0.430803i	0.976526	−	0.215402i	$-0.0691061\pi$
$390$		0			0
$391$		−	23.0567i		−	1.16603i
$392$			9.00621i			0.454882i
$393$	3.80784			0.192080
$394$			1.55979i			0.0785812i
$395$		0			0
$396$			1.02279i			0.0513971i
$397$	−1.07708			−0.0540571			−0.0270286	−	0.999635i	$-0.508605\pi$
$397$	−1.07708			−0.0540571			−0.0270286	+	0.999635i	$0.508605\pi$
$398$			5.01011i			0.251134i
$399$	1.73611			0.0869144
$400$		0			0
$401$	18.2009			0.908910			0.454455	−	0.890770i	$-0.349834\pi$
$401$	18.2009			0.908910			0.454455	+	0.890770i	$0.349834\pi$
$402$	4.06994			0.202990
$403$			50.9021i			2.53561i
$404$			25.8229i			1.28474i
$405$		0			0
$406$	0.449995	−	0.207268i	0.0223329	−	0.0102866i
$407$	2.20525			0.109310
$408$		−	4.32319i		−	0.214030i
$409$			29.4035i			1.45391i	0.686684	+	0.726956i	$0.259066\pi$
$409$			29.4035i			1.45391i	−0.686684	+	0.726956i	$0.740934\pi$
$410$		0			0
$411$	9.05968			0.446881
$412$	20.1781			0.994103
$413$	−2.20442			−0.108473
$414$		−	2.32049i		−	0.114046i
$415$		0			0
$416$			18.7324i			0.918431i
$417$			1.41853i			0.0694657i
$418$			1.14394i			0.0559520i
$419$	7.33375			0.358277			0.179139	−	0.983824i	$-0.442669\pi$
$419$	7.33375			0.358277			0.179139	+	0.983824i	$0.442669\pi$
$420$		0			0
$421$			21.0761i			1.02718i	0.858034	+	0.513592i	$0.171686\pi$
$421$			21.0761i			1.02718i	−0.858034	+	0.513592i	$0.828314\pi$
$422$	−5.31756			−0.258855
$423$			12.4310i			0.604416i
$424$		−	5.18401i		−	0.251758i
$425$		0			0
$426$	2.40917			0.116725
$427$			1.33115i			0.0644189i
$428$	8.67058			0.419108
$429$	2.72855			0.131736
$430$		0			0
$431$	−10.6334			−0.512195			−0.256098	−	0.966651i	$-0.582437\pi$
$431$	−10.6334			−0.512195			−0.256098	+	0.966651i	$0.582437\pi$
$432$			3.34109i			0.160748i
$433$			12.7281i			0.611674i	0.952084	+	0.305837i	$0.0989363\pi$
$433$			12.7281i			0.611674i	−0.952084	+	0.305837i	$0.901064\pi$
$434$	0.929724			0.0446282
$435$		0			0
$436$	24.8233			1.18882
$437$			43.7896i			2.09474i
$438$			0.414368i			0.0197993i
$439$	4.48366			0.213994			0.106997	−	0.994259i	$-0.465877\pi$
$439$	4.48366			0.213994			0.106997	+	0.994259i	$0.465877\pi$
$440$		0			0
$441$	6.92436			0.329732
$442$	−5.60063			−0.266395
$443$			12.8049i			0.608379i	0.952612	+	0.304189i	$0.0983855\pi$
$443$			12.8049i			0.608379i	−0.952612	+	0.304189i	$0.901614\pi$
$444$	7.68632			0.364777
$445$		0			0
$446$		−	2.51531i		−	0.119103i
$447$			4.70591i			0.222582i
$448$	−1.49558			−0.0706595
$449$			22.1787i			1.04668i	0.852124	+	0.523340i	$0.175314\pi$
$449$			22.1787i			1.04668i	−0.852124	+	0.523340i	$0.824686\pi$
$450$		0			0
$451$	4.60364			0.216777
$452$			23.4957i			1.10515i
$453$			5.99181i			0.281520i
$454$		−	5.00575i		−	0.234932i
$455$		0			0
$456$			8.21066i			0.384499i
$457$	12.2239			0.571808			0.285904	−	0.958258i	$-0.407706\pi$
$457$	12.2239			0.571808			0.285904	+	0.958258i	$0.407706\pi$
$458$	4.11685			0.192368
$459$	−3.32386			−0.155144
$460$		0			0
$461$		−	26.1495i		−	1.21790i	−0.793207	−	0.608952i	$-0.791590\pi$
$461$		−	26.1495i		−	1.21790i	0.793207	−	0.608952i	$-0.208410\pi$
$462$		−	0.0498368i		−	0.00231862i
$463$	23.5519			1.09455			0.547275	−	0.836953i	$-0.315665\pi$
$463$	23.5519			1.09455			0.547275	+	0.836953i	$0.315665\pi$
$464$	−7.52720	−	16.3421i	−0.349442	−	0.758664i
$465$		0			0
$466$			3.59613i			0.166587i
$467$		−	12.7060i		−	0.587963i	−0.955811	−	0.293981i	$-0.905020\pi$
$467$		−	12.7060i		−	0.587963i	0.955811	−	0.293981i	$-0.0949804\pi$
$468$	9.51028			0.439613
$469$	3.34599			0.154503
$470$		0			0
$471$	11.2728			0.519421
$472$		−	10.4255i		−	0.479870i
$473$	−3.23737			−0.148855
$474$		−	0.792725i		−	0.0364111i
$475$		0			0
$476$		−	1.72595i		−	0.0791089i
$477$	−3.98569			−0.182492
$478$		−	2.02212i		−	0.0924898i
$479$		−	14.2867i		−	0.652775i	−0.945236	−	0.326388i	$-0.894169\pi$
$479$		−	14.2867i		−	0.652775i	0.945236	−	0.326388i	$-0.105831\pi$
$480$		0			0
$481$		−	20.5052i		−	0.934957i
$482$		−	8.09538i		−	0.368734i
$483$		−	1.90773i		−	0.0868047i
$484$	20.2150			0.918863
$485$		0			0
$486$	−0.334522			−0.0151742
$487$	34.4239			1.55990			0.779948	−	0.625844i	$-0.215246\pi$
$487$	34.4239			1.55990			0.779948	+	0.625844i	$0.215246\pi$
$488$	−6.29545			−0.284982
$489$	11.2859			0.510366
$490$		0			0
$491$		−	20.5793i		−	0.928733i	−0.885643	−	0.464366i	$-0.846282\pi$
$491$		−	20.5793i		−	0.928733i	0.885643	−	0.464366i	$-0.153718\pi$
$492$	16.0458			0.723403
$493$	16.2578	−	7.48838i	0.732216	−	0.337260i
$494$	10.6368			0.478572
$495$		0			0
$496$		−	33.7641i		−	1.51605i
$497$	1.98064			0.0888436
$498$	−3.46980			−0.155485
$499$	19.5569			0.875487			0.437743	−	0.899100i	$-0.355778\pi$
$499$	19.5569			0.875487			0.437743	+	0.899100i	$0.355778\pi$
$500$		0			0
$501$			9.11150i			0.407072i
$502$	8.70264			0.388418
$503$			38.5345i			1.71817i	0.511835	+	0.859084i	$0.328966\pi$
$503$			38.5345i			1.71817i	−0.511835	+	0.859084i	$0.671034\pi$
$504$		−	0.357705i		−	0.0159334i
$505$		0			0
$506$	1.25702			0.0558814
$507$		−	12.3711i		−	0.549418i
$508$		−	19.6072i		−	0.869928i
$509$	−9.66065			−0.428201			−0.214100	−	0.976812i	$-0.568682\pi$
$509$	−9.66065			−0.428201			−0.214100	+	0.976812i	$0.568682\pi$
$510$		0			0
$511$			0.340662i			0.0150700i
$512$		−	21.1167i		−	0.933234i
$513$	6.31271			0.278713
$514$		−	1.19996i		−	0.0529281i
$515$		0			0
$516$	−11.2838			−0.496740
$517$	−6.73393			−0.296158
$518$	−0.374527			−0.0164557
$519$			6.68099i			0.293263i
$520$		0			0
$521$	20.3242			0.890421			0.445211	−	0.895426i	$-0.353129\pi$
$521$	20.3242			0.890421			0.445211	+	0.895426i	$0.353129\pi$
$522$	1.63623	−	0.753651i	0.0716160	−	0.0329864i
$523$	40.6870			1.77912			0.889559	−	0.456820i	$-0.151012\pi$
$523$	40.6870			1.77912			0.889559	+	0.456820i	$0.151012\pi$
$524$			7.18957i			0.314078i
$525$		0			0
$526$	−1.31269			−0.0572360
$527$	33.5899			1.46320
$528$	−1.80988			−0.0787651
$529$	25.1182			1.09209
$530$		0			0
$531$	−8.01554			−0.347845
$532$			3.27795i			0.142117i
$533$		−	42.8064i		−	1.85415i
$534$			0.838413i			0.0362817i
$535$		0			0
$536$			15.8243i			0.683506i
$537$			17.6127i			0.760045i
$538$	4.64404			0.200219
$539$			3.75096i			0.161565i
$540$		0			0
$541$		−	5.02402i		−	0.216000i	−0.994151	−	0.108000i	$-0.965555\pi$
$541$		−	5.02402i		−	0.216000i	0.994151	−	0.108000i	$-0.0344446\pi$
$542$	−4.47860			−0.192372
$543$		−	0.799278i		−	0.0343003i
$544$	12.3614			0.529989
$545$		0			0
$546$	−0.463401			−0.0198317
$547$	−8.61344			−0.368284			−0.184142	−	0.982900i	$-0.558951\pi$
$547$	−8.61344			−0.368284			−0.184142	+	0.982900i	$0.558951\pi$
$548$			17.1055i			0.730712i
$549$			4.84022i			0.206575i
$550$		0			0
$551$	−30.8771	+	14.2220i	−1.31541	+	0.605878i
$552$	9.02229			0.384014
$553$		−	0.651718i		−	0.0277139i
$554$		−	3.31631i		−	0.140896i
$555$		0			0
$556$	−2.67832			−0.113586
$557$	0.455853			0.0193151			0.00965755	−	0.999953i	$-0.496926\pi$
$557$	0.455853			0.0193151			0.00965755	+	0.999953i	$0.496926\pi$
$558$	3.38058			0.143111
$559$			30.1023i			1.27319i
$560$		0			0
$561$		−	1.80055i		−	0.0760192i
$562$			6.26114i			0.264110i
$563$		−	0.120515i		−	0.00507911i	−0.999997	−	0.00253956i	$-0.999192\pi$
$563$		−	0.120515i		−	0.00507911i	0.999997	−	0.00253956i	$-0.000808366\pi$
$564$	−23.4709			−0.988304
$565$		0			0
$566$			6.33794i			0.266403i
$567$	−0.275019			−0.0115497
$568$			9.36708i			0.393034i
$569$		−	22.7315i		−	0.952954i	−0.879187	−	0.476477i	$-0.841914\pi$
$569$		−	22.7315i		−	0.952954i	0.879187	−	0.476477i	$-0.158086\pi$
$570$		0			0
$571$	26.7167			1.11806			0.559030	−	0.829147i	$-0.311174\pi$
$571$	26.7167			1.11806			0.559030	+	0.829147i	$0.311174\pi$
$572$			5.15176i			0.215406i
$573$	14.0344			0.586296
$574$	−0.781856			−0.0326340
$575$		0			0
$576$	−5.43810			−0.226587
$577$			15.3024i			0.637049i	0.947915	+	0.318524i	$0.103187\pi$
$577$			15.3024i			0.637049i	−0.947915	+	0.318524i	$0.896813\pi$
$578$		−	1.99107i		−	0.0828175i
$579$	−22.3693			−0.929637
$580$		0			0
$581$	−2.85260			−0.118346
$582$			5.97546i			0.247691i
$583$		−	2.15907i		−	0.0894195i
$584$	−1.61110			−0.0666679
$585$		0			0
$586$	2.38826			0.0986582
$587$	41.6229			1.71796			0.858980	−	0.512009i	$-0.171098\pi$
$587$	41.6229			1.71796			0.858980	+	0.512009i	$0.171098\pi$
$588$			13.0739i			0.539157i
$589$	−63.7944			−2.62860
$590$		0			0
$591$			4.66274i			0.191800i
$592$			13.6014i			0.559013i
$593$	11.7493			0.482487			0.241244	−	0.970465i	$-0.422445\pi$
$593$	11.7493			0.482487			0.241244	+	0.970465i	$0.422445\pi$
$594$		−	0.181212i		−	0.00743523i
$595$		0			0
$596$	−8.88521			−0.363952
$597$			14.9769i			0.612964i
$598$		−	11.6882i		−	0.477968i
$599$		−	8.86844i		−	0.362354i	−0.983450	−	0.181177i	$-0.942009\pi$
$599$		−	8.86844i		−	0.362354i	0.983450	−	0.181177i	$-0.0579908\pi$
$600$		0			0
$601$		−	0.339050i		−	0.0138301i	−0.999976	−	0.00691506i	$-0.997799\pi$
$601$		−	0.339050i		−	0.0138301i	0.999976	−	0.00691506i	$-0.00220115\pi$
$602$	0.549817			0.0224089
$603$	12.1664			0.495454
$604$	−11.3131			−0.460324
$605$		0			0
$606$		−	4.57516i		−	0.185853i
$607$		−	8.27856i		−	0.336017i	−0.985786	−	0.168008i	$-0.946266\pi$
$607$		−	8.27856i		−	0.336017i	0.985786	−	0.168008i	$-0.0537335\pi$
$608$	−23.4769			−0.952112
$609$	1.34519	−	0.619595i	0.0545097	−	0.0251072i
$610$		0			0
$611$			62.6146i			2.53312i
$612$		−	6.27576i		−	0.253683i
$613$	−42.0873			−1.69989			−0.849946	−	0.526870i	$-0.823365\pi$
$613$	−42.0873			−1.69989			−0.849946	+	0.526870i	$0.823365\pi$
$614$	3.36655			0.135863
$615$		0			0
$616$	0.193770			0.00780723
$617$		−	28.7087i		−	1.15577i	−0.816119	−	0.577884i	$-0.803879\pi$
$617$		−	28.7087i		−	1.15577i	0.816119	−	0.577884i	$-0.196121\pi$
$618$	−3.57504			−0.143809
$619$		−	8.90716i		−	0.358009i	−0.983848	−	0.179005i	$-0.942712\pi$
$619$		−	8.90716i		−	0.358009i	0.983848	−	0.179005i	$-0.0572877\pi$
$620$		0			0
$621$		−	6.93673i		−	0.278361i
$622$	−5.43575			−0.217954
$623$			0.689279i			0.0276154i
$624$			16.8290i			0.673698i
$625$		0			0
$626$		−	7.18681i		−	0.287243i
$627$			3.41963i			0.136567i
$628$			21.2840i			0.849325i
$629$	−13.5312			−0.539525
$630$		0			0
$631$	17.9351			0.713986			0.356993	−	0.934107i	$-0.383802\pi$
$631$	17.9351			0.713986			0.356993	+	0.934107i	$0.383802\pi$
$632$	3.08219			0.122603
$633$	−15.8960			−0.631809
$634$	−8.16918			−0.324440
$635$		0			0
$636$		−	7.52537i		−	0.298400i
$637$	34.8778			1.38191
$638$	0.408256	+	0.886356i	0.0161630	+	0.0350912i
$639$	7.20182			0.284899
$640$		0			0
$641$			0.220741i			0.00871875i	0.999990	+	0.00435938i	$0.00138764\pi$
$641$			0.220741i			0.00871875i	−0.999990	+	0.00435938i	$0.998612\pi$
$642$	−1.53621			−0.0606292
$643$	6.25440			0.246650			0.123325	−	0.992366i	$-0.460644\pi$
$643$	6.25440			0.246650			0.123325	+	0.992366i	$0.460644\pi$
$644$	3.60198			0.141938
$645$		0			0
$646$		−	7.01914i		−	0.276164i
$647$	24.3697			0.958071			0.479035	−	0.877796i	$-0.340987\pi$
$647$	24.3697			0.958071			0.479035	+	0.877796i	$0.340987\pi$
$648$		−	1.30066i		−	0.0510946i
$649$		−	4.34205i		−	0.170441i
$650$		0			0
$651$	2.77926			0.108928
$652$			21.3088i			0.834519i
$653$		−	41.1834i		−	1.61163i	−0.592167	−	0.805815i	$-0.701727\pi$
$653$		−	41.1834i		−	1.61163i	0.592167	−	0.805815i	$-0.298273\pi$
$654$	−4.39805			−0.171977
$655$		0			0
$656$			28.3940i			1.10860i
$657$			1.23869i			0.0483257i
$658$	1.14365			0.0445842
$659$		−	4.18021i		−	0.162838i	−0.996680	−	0.0814190i	$-0.974055\pi$
$659$		−	4.18021i		−	0.162838i	0.996680	−	0.0814190i	$-0.0259452\pi$
$660$		0			0
$661$	−8.11965			−0.315818			−0.157909	−	0.987454i	$-0.550475\pi$
$661$	−8.11965			−0.315818			−0.157909	+	0.987454i	$0.550475\pi$
$662$	0.209971			0.00816077
$663$	−16.7422			−0.650212
$664$		−	13.4909i		−	0.523548i
$665$		0			0
$666$	−1.36182			−0.0527695
$667$	15.6279	+	33.9293i	0.605113	+	1.31375i
$668$	−17.2034			−0.665619
$669$		−	7.51910i		−	0.290705i
$670$		0			0
$671$	−2.62197			−0.101220
$672$	1.02279			0.0394550
$673$	−33.1749			−1.27880			−0.639399	−	0.768875i	$-0.720817\pi$
$673$	−33.1749			−1.27880			−0.639399	+	0.768875i	$0.720817\pi$
$674$	−9.10128			−0.350568
$675$		0			0
$676$	23.3577			0.898375
$677$		−	14.7844i		−	0.568212i	−0.958793	−	0.284106i	$-0.908303\pi$
$677$		−	14.7844i		−	0.568212i	0.958793	−	0.284106i	$-0.0916967\pi$
$678$		−	4.16284i		−	0.159873i
$679$			4.91257i			0.188527i
$680$		0			0
$681$		−	14.9639i		−	0.573417i
$682$			1.83128i			0.0701232i
$683$	−33.5573			−1.28404			−0.642018	−	0.766690i	$-0.721902\pi$
$683$	−33.5573			−1.28404			−0.642018	+	0.766690i	$0.721902\pi$
$684$			11.9190i			0.455734i
$685$		0			0
$686$		−	1.28104i		−	0.0489104i
$687$	12.3067			0.469528
$688$		−	19.9673i		−	0.761245i
$689$	−20.0758			−0.764828
$690$		0			0
$691$	−11.4767			−0.436596			−0.218298	−	0.975882i	$-0.570050\pi$
$691$	−11.4767			−0.436596			−0.218298	+	0.975882i	$0.570050\pi$
$692$	−12.6143			−0.479525
$693$		−	0.148979i		−	0.00565924i
$694$			4.35417i			0.165282i
$695$		0			0
$696$	2.93027	+	6.36183i	0.111072	+	0.241145i
$697$	−28.2476			−1.06995
$698$			7.77697i			0.294363i
$699$			10.7500i			0.406604i
$700$		0			0
$701$	11.0065			0.415711			0.207855	−	0.978160i	$-0.433352\pi$
$701$	11.0065			0.415711			0.207855	+	0.978160i	$0.433352\pi$
$702$	−1.68498			−0.0635955
$703$	25.6987			0.969244
$704$		−	2.94584i		−	0.111026i
$705$		0			0
$706$		−	11.0482i		−	0.415805i
$707$		−	3.76135i		−	0.141460i
$708$		−	15.1341i		−	0.568774i
$709$	−21.4248			−0.804624			−0.402312	−	0.915503i	$-0.631793\pi$
$709$	−21.4248			−0.804624			−0.402312	+	0.915503i	$0.631793\pi$
$710$		0			0
$711$		−	2.36972i		−	0.0888715i
$712$	−3.25983			−0.122167
$713$			70.1005i			2.62528i
$714$			0.305795i			0.0114441i
$715$		0			0
$716$	−33.2545			−1.24278
$717$		−	6.04481i		−	0.225747i
$718$	−10.0625			−0.375530
$719$	21.8797			0.815977			0.407988	−	0.912987i	$-0.366230\pi$
$719$	21.8797			0.815977			0.407988	+	0.912987i	$0.366230\pi$
$720$		0			0
$721$	−2.93913			−0.109459
$722$			6.97491i			0.259579i
$723$		−	24.1998i		−	0.900000i
$724$	1.50911			0.0560857
$725$		0			0
$726$	−3.58158			−0.132925
$727$			36.6219i			1.35823i	0.734032	+	0.679115i	$0.237636\pi$
$727$			36.6219i			1.35823i	−0.734032	+	0.679115i	$0.762364\pi$
$728$		−	1.80175i		−	0.0667772i
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	19.8643			0.734707
$732$	−9.13879			−0.337779
$733$			7.68719i			0.283933i	0.989871	+	0.141966i	$0.0453425\pi$
$733$			7.68719i			0.283933i	−0.989871	+	0.141966i	$0.954658\pi$
$734$	−1.30229			−0.0480686
$735$		0			0
$736$			25.7975i			0.950910i
$737$			6.59060i			0.242768i
$738$	−2.84292			−0.104649
$739$			26.2160i			0.964371i	0.876069	+	0.482186i	$0.160157\pi$
$739$			26.2160i			0.964371i	−0.876069	+	0.482186i	$0.839843\pi$
$740$		0			0
$741$	31.7969			1.16809
$742$			0.366684i			0.0134614i
$743$		−	7.62063i		−	0.279574i	−0.990182	−	0.139787i	$-0.955358\pi$
$743$		−	7.62063i		−	0.279574i	0.990182	−	0.139787i	$-0.0446417\pi$
$744$			13.1440i			0.481883i
$745$		0			0
$746$			0.597783i			0.0218864i
$747$	−10.3724			−0.379506
$748$	3.39961			0.124302
$749$	−1.26295			−0.0461472
$750$		0			0
$751$		−	28.5397i		−	1.04143i	−0.853731	−	0.520714i	$-0.825666\pi$
$751$		−	28.5397i		−	1.04143i	0.853731	−	0.520714i	$-0.174334\pi$
$752$		−	41.5331i		−	1.51456i
$753$	26.0151			0.948043
$754$	8.24166	−	3.79612i	0.300144	−	0.138247i
$755$		0			0
$756$		−	0.519262i		−	0.0188854i
$757$		−	19.8761i		−	0.722408i	−0.932487	−	0.361204i	$-0.882366\pi$
$757$		−	19.8761i		−	0.722408i	0.932487	−	0.361204i	$-0.117634\pi$
$758$	−6.16074			−0.223768
$759$	3.75766			0.136394
$760$		0			0
$761$	33.8887			1.22847			0.614233	−	0.789125i	$-0.289465\pi$
$761$	33.8887			1.22847			0.614233	+	0.789125i	$0.289465\pi$
$762$			3.47389i			0.125846i
$763$	−3.61574			−0.130899
$764$			26.4983i			0.958674i
$765$		0			0
$766$		−	9.49947i		−	0.343230i
$767$	−40.3740			−1.45782
$768$		−	7.77948i		−	0.280718i
$769$		−	37.6604i		−	1.35807i	−0.734107	−	0.679034i	$-0.762399\pi$
$769$		−	37.6604i		−	1.35807i	0.734107	−	0.679034i	$-0.237601\pi$
$770$		0			0
$771$		−	3.58709i		−	0.129186i
$772$		−	42.2354i		−	1.52008i
$773$		−	23.1391i		−	0.832255i	−0.909306	−	0.416127i	$-0.863387\pi$
$773$		−	23.1391i		−	0.832255i	0.909306	−	0.416127i	$-0.136613\pi$
$774$	1.99920			0.0718596
$775$		0			0
$776$	−23.2332			−0.834022
$777$	−1.11959			−0.0401649
$778$	−2.84236			−0.101904
$779$	53.6482			1.92214
$780$		0			0
$781$			3.90126i			0.139598i
$782$	−7.71298			−0.275816
$783$	4.89125	−	2.25292i	0.174799	−	0.0805127i
$784$	−23.1349			−0.826248
$785$		0			0
$786$		−	1.27381i		−	0.0454353i
$787$	11.0598			0.394238			0.197119	−	0.980380i	$-0.436841\pi$
$787$	11.0598			0.394238			0.197119	+	0.980380i	$0.436841\pi$
$788$	−8.80370			−0.313619
$789$	−3.92407			−0.139701
$790$		0			0
$791$		−	3.42237i		−	0.121686i
$792$	0.704571			0.0250358
$793$			24.3800i			0.865760i
$794$			0.360308i			0.0127868i
$795$		0			0
$796$	−28.2778			−1.00228
$797$		−	11.5108i		−	0.407733i	−0.978999	−	0.203866i	$-0.934649\pi$
$797$		−	11.5108i		−	0.407733i	0.978999	−	0.203866i	$-0.0653508\pi$
$798$		−	0.580769i		−	0.0205590i
$799$	41.3189			1.46176
$800$		0			0
$801$			2.50630i			0.0885557i
$802$		−	6.08861i		−	0.214996i
$803$	−0.671001			−0.0236791
$804$			22.9713i			0.810136i
$805$		0			0
$806$	17.0279			0.599782
$807$	13.8826			0.488690
$808$	17.7887			0.625803
$809$			26.4897i			0.931330i	0.884961	+	0.465665i	$0.154185\pi$
$809$			26.4897i			0.931330i	−0.884961	+	0.465665i	$0.845815\pi$
$810$		0			0
$811$	−40.5686			−1.42455			−0.712277	−	0.701898i	$-0.752336\pi$
$811$	−40.5686			−1.42455			−0.712277	+	0.701898i	$0.752336\pi$
$812$	1.16985	+	2.53984i	0.0410538	+	0.0891309i
$813$	−13.3880			−0.469539
$814$		−	0.737705i		−	0.0258565i
$815$		0			0
$816$	11.1053			0.388764
$817$	−37.7265			−1.31988
$818$	9.83615			0.343913
$819$	−1.38526			−0.0484050
$820$		0			0
$821$	−34.6649			−1.20981			−0.604906	−	0.796297i	$-0.706789\pi$
$821$	−34.6649			−1.20981			−0.604906	+	0.796297i	$0.706789\pi$
$822$		−	3.03067i		−	0.105707i
$823$		−	37.6231i		−	1.31146i	−0.754996	−	0.655729i	$-0.772361\pi$
$823$		−	37.6231i		−	1.31146i	0.754996	−	0.655729i	$-0.227639\pi$
$824$		−	13.9001i		−	0.484233i
$825$		0			0
$826$			0.737429i			0.0256585i
$827$			45.0238i			1.56563i	0.622253	+	0.782816i	$0.286217\pi$
$827$			45.0238i			1.56563i	−0.622253	+	0.782816i	$0.713783\pi$
$828$	13.0972			0.455159
$829$		−	12.5148i		−	0.434656i	−0.976099	−	0.217328i	$-0.930266\pi$
$829$		−	12.5148i		−	0.434656i	0.976099	−	0.217328i	$-0.0697341\pi$
$830$		0			0
$831$		−	9.91356i		−	0.343898i
$832$	−27.3915			−0.949631
$833$		−	23.0156i		−	0.797443i
$834$	0.474530			0.0164316
$835$		0			0
$836$	−6.45658			−0.223305
$837$	10.1057			0.349304
$838$		−	2.45330i		−	0.0847480i
$839$		−	35.4253i		−	1.22302i	−0.791238	−	0.611508i	$-0.790563\pi$
$839$		−	35.4253i		−	1.22302i	0.791238	−	0.611508i	$-0.209437\pi$
$840$		0			0
$841$	−18.8487	+	22.0392i	−0.649956	+	0.759972i
$842$	7.05042			0.242973
$843$			18.7167i			0.644636i
$844$		−	30.0131i		−	1.03309i
$845$		0			0
$846$	4.15845			0.142970
$847$	−2.94450			−0.101174
$848$	13.3166			0.457293
$849$			18.9462i			0.650233i
$850$		0			0
$851$		−	28.2390i		−	0.968021i
$852$			13.5977i			0.465850i
$853$			4.63924i			0.158844i	0.996841	+	0.0794222i	$0.0253075\pi$
$853$			4.63924i			0.158844i	−0.996841	+	0.0794222i	$0.974692\pi$
$854$	0.445300			0.0152379
$855$		0			0
$856$		−	5.97292i		−	0.204150i
$857$	−52.9487			−1.80869			−0.904346	−	0.426800i	$-0.859641\pi$
$857$	−52.9487			−1.80869			−0.904346	+	0.426800i	$0.859641\pi$
$858$		−	0.912761i		−	0.0311612i
$859$			37.3538i			1.27450i	0.770658	+	0.637248i	$0.219927\pi$
$859$			37.3538i			1.27450i	−0.770658	+	0.637248i	$0.780073\pi$
$860$		0			0
$861$	−2.33723			−0.0796525
$862$			3.55713i			0.121156i
$863$	−10.2002			−0.347219			−0.173609	−	0.984815i	$-0.555543\pi$
$863$	−10.2002			−0.347219			−0.173609	+	0.984815i	$0.555543\pi$
$864$	3.71898			0.126522
$865$		0			0
$866$	4.25784			0.144687
$867$		−	5.95197i		−	0.202139i
$868$			5.24750i			0.178112i
$869$	1.28369			0.0435461
$870$		0			0
$871$	61.2818			2.07646
$872$		−	17.1000i		−	0.579081i
$873$			17.8627i			0.604560i
$874$	14.6486			0.495496
$875$		0			0
$876$	−2.33875			−0.0790192
$877$	8.25649			0.278802			0.139401	−	0.990236i	$-0.455482\pi$
$877$	8.25649			0.278802			0.139401	+	0.990236i	$0.455482\pi$
$878$		−	1.49989i		−	0.0506187i
$879$	7.13932			0.240803
$880$		0			0
$881$		−	4.94367i		−	0.166556i	−0.996526	−	0.0832782i	$-0.973461\pi$
$881$		−	4.94367i		−	0.166556i	0.996526	−	0.0832782i	$-0.0265390\pi$
$882$		−	2.31636i		−	0.0779958i
$883$	−35.6378			−1.19931			−0.599654	−	0.800259i	$-0.704695\pi$
$883$	−35.6378			−1.19931			−0.599654	+	0.800259i	$0.704695\pi$
$884$		−	31.6108i		−	1.06319i
$885$		0			0
$886$	4.28352			0.143908
$887$			2.14284i			0.0719497i	0.999353	+	0.0359748i	$0.0114536\pi$
$887$			2.14284i			0.0719497i	−0.999353	+	0.0359748i	$0.988546\pi$
$888$		−	5.29489i		−	0.177685i
$889$			2.85597i			0.0957862i
$890$		0			0
$891$		−	0.541705i		−	0.0181478i
$892$	14.1968			0.475343
$893$	−78.4733			−2.62601
$894$	1.57423			0.0526503
$895$		0			0
$896$			2.54588i			0.0850520i
$897$		−	34.9401i		−	1.16662i
$898$	7.41929			0.247585
$899$	−49.4295	+	22.7673i	−1.64857	+	0.759332i
$900$		0			0
$901$			13.2479i			0.441351i
$902$		−	1.54002i		−	0.0512771i
$903$	1.64359			0.0546951
$904$	16.1855			0.538323
$905$		0			0
$906$	2.00439			0.0665916
$907$		−	27.7834i		−	0.922533i	−0.887262	−	0.461266i	$-0.847395\pi$
$907$		−	27.7834i		−	0.922533i	0.887262	−	0.461266i	$-0.152605\pi$
$908$	28.2532			0.937616
$909$		−	13.6767i		−	0.453627i
$910$		0			0
$911$			7.92701i			0.262634i	0.991340	+	0.131317i	$0.0419205\pi$
$911$			7.92701i			0.262634i	−0.991340	+	0.131317i	$0.958080\pi$
$912$	−21.0913			−0.698404
$913$		−	5.61877i		−	0.185954i
$914$		−	4.08916i		−	0.135257i
$915$		0			0
$916$			23.2361i			0.767743i
$917$		−	1.04723i		−	0.0345825i
$918$			1.11191i			0.0366983i
$919$	−24.5480			−0.809764			−0.404882	−	0.914369i	$-0.632687\pi$
$919$	−24.5480			−0.809764			−0.404882	+	0.914369i	$0.632687\pi$
$920$		0			0
$921$	10.0637			0.331612
$922$	−8.74760			−0.288087
$923$	36.2753			1.19402
$924$	0.281286			0.00925364
$925$		0			0
$926$		−	7.87864i		−	0.258908i
$927$	−10.6870			−0.351007
$928$	−18.1905	+	8.37855i	−0.597132	+	0.275040i
$929$	−2.45572			−0.0805695			−0.0402847	−	0.999188i	$-0.512827\pi$
$929$	−2.45572			−0.0805695			−0.0402847	+	0.999188i	$0.512827\pi$
$930$		0			0
$931$			43.7115i			1.43259i
$932$	−20.2971			−0.664853
$933$	−16.2493			−0.531977
$934$	−4.25044			−0.139079
$935$		0			0
$936$		−	6.55136i		−	0.214138i
$937$	−39.7814			−1.29960			−0.649801	−	0.760104i	$-0.725148\pi$
$937$	−39.7814			−1.29960			−0.649801	+	0.760104i	$0.725148\pi$
$938$		−	1.11931i		−	0.0365467i
$939$		−	21.4838i		−	0.701097i
$940$		0			0
$941$	33.7621			1.10061			0.550306	−	0.834963i	$-0.314511\pi$
$941$	33.7621			1.10061			0.550306	+	0.834963i	$0.314511\pi$
$942$		−	3.77099i		−	0.122866i
$943$		−	58.9513i		−	1.91972i
$944$	26.7806			0.871636
$945$		0			0
$946$			1.08297i			0.0352105i
$947$		−	1.32780i		−	0.0431476i	−0.999767	−	0.0215738i	$-0.993132\pi$
$947$		−	1.32780i		−	0.0431476i	0.999767	−	0.0215738i	$-0.00686768\pi$
$948$	4.47426			0.145317
$949$			6.23922i			0.202534i
$950$		0			0
$951$	−24.4204			−0.791887
$952$	−1.18896			−0.0385344
$953$	21.2315			0.687755			0.343878	−	0.939014i	$-0.388259\pi$
$953$	21.2315			0.687755			0.343878	+	0.939014i	$0.388259\pi$
$954$			1.33330i			0.0431673i
$955$		0			0
$956$	11.4132			0.369128
$957$	1.22042	+	2.64961i	0.0394504	+	0.0856499i
$958$	−4.77922			−0.154409
$959$		−	2.49158i		−	0.0804574i
$960$		0			0
$961$	−71.1251			−2.29436
$962$	−6.85945			−0.221158
$963$	−4.59224			−0.147983
$964$	45.6915			1.47162
$965$		0			0
$966$	−0.638179			−0.0205331
$967$		−	8.54151i		−	0.274677i	−0.990524	−	0.137338i	$-0.956145\pi$
$967$		−	8.54151i		−	0.274677i	0.990524	−	0.137338i	$-0.0438547\pi$
$968$		−	13.9255i		−	0.447584i
$969$		−	20.9826i		−	0.674057i
$970$		0			0
$971$		−	18.0454i		−	0.579104i	−0.957162	−	0.289552i	$-0.906494\pi$
$971$		−	18.0454i		−	0.579104i	0.957162	−	0.289552i	$-0.0935063\pi$
$972$		−	1.88809i		−	0.0605607i
$973$	0.390123			0.0125068
$974$		−	11.5156i		−	0.368983i
$975$		0			0
$976$		−	16.1716i		−	0.517641i
$977$	−15.2948			−0.489324			−0.244662	−	0.969608i	$-0.578677\pi$
$977$	−15.2948			−0.489324			−0.244662	+	0.969608i	$0.578677\pi$
$978$		−	3.77539i		−	0.120724i
$979$	−1.35767			−0.0433914
$980$		0			0
$981$	−13.1473			−0.419760
$982$	−6.88425			−0.219685
$983$			38.2643i			1.22044i	0.792231	+	0.610221i	$0.208919\pi$
$983$			38.2643i			1.22044i	−0.792231	+	0.610221i	$0.791081\pi$
$984$		−	11.0535i		−	0.352374i
$985$		0			0
$986$	−2.50503	−	5.43861i	−0.0797764	−	0.173201i
$987$	3.41876			0.108820
$988$			60.0356i			1.90999i
$989$			41.4557i			1.31822i
$990$		0			0
$991$	22.5768			0.717174			0.358587	−	0.933496i	$-0.383259\pi$
$991$	22.5768			0.717174			0.358587	+	0.933496i	$0.383259\pi$
$992$	−37.5829			−1.19326
$993$	0.627675			0.0199187
$994$		−	0.662567i		−	0.0210153i
$995$		0			0
$996$		−	19.5840i		−	0.620544i
$997$		−	28.4370i		−	0.900608i	−0.892875	−	0.450304i	$-0.851316\pi$
$997$		−	28.4370i		−	0.900608i	0.892875	−	0.450304i	$-0.148684\pi$
$998$		−	6.54222i		−	0.207090i
$999$	−4.07094			−0.128799

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.d.j.376.13		24
5.2	odd	4		435.2.f.e.289.8	yes	12
5.3	odd	4		435.2.f.f.289.5	yes	12
5.4	even	2	inner	2175.2.d.j.376.12		24
15.2	even	4		1305.2.f.k.289.5		12
15.8	even	4		1305.2.f.l.289.8		12
29.28	even	2	inner	2175.2.d.j.376.14		24
145.28	odd	4		435.2.f.e.289.7	✓	12
145.57	odd	4		435.2.f.f.289.6	yes	12
145.144	even	2	inner	2175.2.d.j.376.11		24
435.173	even	4		1305.2.f.k.289.6		12
435.347	even	4		1305.2.f.l.289.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.f.e.289.7	✓	12	145.28	odd	4
435.2.f.e.289.8	yes	12	5.2	odd	4
435.2.f.f.289.5	yes	12	5.3	odd	4
435.2.f.f.289.6	yes	12	145.57	odd	4
1305.2.f.k.289.5		12	15.2	even	4
1305.2.f.k.289.6		12	435.173	even	4
1305.2.f.l.289.7		12	435.347	even	4
1305.2.f.l.289.8		12	15.8	even	4
2175.2.d.j.376.11		24	145.144	even	2	inner
2175.2.d.j.376.12		24	5.4	even	2	inner
2175.2.d.j.376.13		24	1.1	even	1	trivial
2175.2.d.j.376.14		24	29.28	even	2	inner

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2175.2.d.j.376.13

Level	$2175$
Weight	$2$
Character	2175.376
Analytic conductor	$17.367$
Analytic rank	$0$
Dimension	$24$
Inner twists	$4$