LMFDB - Embedded newform 7600.2.a.cc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.a (trivial)

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:	yes
Analytic conductor:	$60.6863055362$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.169.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x - 1 $ x^3 - x^2 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 475)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.273891$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.27389 q^{3} +0.726109 q^{7} -1.37720 q^{9} +0.273891 q^{11} +5.95328 q^{13} -5.27389 q^{17} -1.00000 q^{19} +0.924984 q^{21} -3.67939 q^{23} -5.57608 q^{27} -2.27389 q^{29} -3.19887 q^{31} +0.348907 q^{33} -8.12386 q^{37} +7.58383 q^{39} -9.43380 q^{41} +9.81100 q^{43} +12.1599 q^{47} -6.47277 q^{49} -6.71836 q^{51} -5.69781 q^{53} -1.27389 q^{57} +4.20662 q^{59} -0.103312 q^{61} -1.00000 q^{63} -11.7827 q^{67} -4.68714 q^{69} -5.75441 q^{71} -6.67939 q^{73} +0.198875 q^{77} -3.87826 q^{79} -2.97170 q^{81} -0.488265 q^{83} -2.89669 q^{87} -16.4338 q^{89} +4.32273 q^{91} -4.07502 q^{93} +4.44447 q^{97} -0.377203 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 4 q^{7} + q^{9} - q^{11} - 3 q^{13} - 14 q^{17} - 3 q^{19} - 6 q^{21} + 8 q^{23} - q^{27} - 5 q^{29} + q^{31} + 8 q^{33} - 5 q^{37} + 11 q^{39} + q^{41} - 5 q^{43} + 9 q^{47} - 7 q^{49}+ \cdots + 4 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 + 4 * q^7 + q^9 - q^11 - 3 * q^13 - 14 * q^17 - 3 * q^19 - 6 * q^21 + 8 * q^23 - q^27 - 5 * q^29 + q^31 + 8 * q^33 - 5 * q^37 + 11 * q^39 + q^41 - 5 * q^43 + 9 * q^47 - 7 * q^49 - 18 * q^51 - 31 * q^53 - 2 * q^57 + 6 * q^59 + 3 * q^61 - 3 * q^63 - 13 * q^67 + q^69 - 7 * q^71 - q^73 - 10 * q^77 + 18 * q^79 - 21 * q^81 + 3 * q^83 - 12 * q^87 - 20 * q^89 - 17 * q^91 - 21 * q^93 + 13 * q^97 + 4 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.27389	0.735481	0.367741	−	0.929928i	$-0.380131\pi$
$3$	1.27389	0.735481	0.367741	+	0.929928i	$0.380131\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.726109	0.274444	0.137222	−	0.990540i	$-0.456183\pi$
$7$	0.726109	0.274444	0.137222	+	0.990540i	$0.456183\pi$
$8$	0	0
$9$	−1.37720	−0.459068
$10$	0	0
$11$	0.273891	0.0825811	0.0412906	−	0.999147i	$-0.486853\pi$
$11$	0.273891	0.0825811	0.0412906	+	0.999147i	$0.486853\pi$
$12$	0	0
$13$	5.95328	1.65114	0.825571	−	0.564298i	$-0.190853\pi$
$13$	5.95328	1.65114	0.825571	+	0.564298i	$0.190853\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.27389	−1.27911	−0.639553	−	0.768747i	$-0.720881\pi$
$17$	−5.27389	−1.27911	−0.639553	+	0.768747i	$0.720881\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.924984	0.201848
$22$	0	0
$23$	−3.67939	−0.767206	−0.383603	−	0.923498i	$-0.625317\pi$
$23$	−3.67939	−0.767206	−0.383603	+	0.923498i	$0.625317\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.57608	−1.07312
$28$	0	0
$29$	−2.27389	−0.422251	−0.211125	−	0.977459i	$-0.567713\pi$
$29$	−2.27389	−0.422251	−0.211125	+	0.977459i	$0.567713\pi$
$30$	0	0
$31$	−3.19887	−0.574535	−0.287267	−	0.957850i	$-0.592747\pi$
$31$	−3.19887	−0.574535	−0.287267	+	0.957850i	$0.592747\pi$
$32$	0	0
$33$	0.348907	0.0607368
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.12386	−1.33555	−0.667777	−	0.744361i	$-0.732754\pi$
$37$	−8.12386	−1.33555	−0.667777	+	0.744361i	$0.732754\pi$
$38$	0	0
$39$	7.58383	1.21438
$40$	0	0
$41$	−9.43380	−1.47331	−0.736656	−	0.676268i	$-0.763596\pi$
$41$	−9.43380	−1.47331	−0.736656	+	0.676268i	$0.763596\pi$
$42$	0	0
$43$	9.81100	1.49616	0.748082	−	0.663607i	$-0.230975\pi$
$43$	9.81100	1.49616	0.748082	+	0.663607i	$0.230975\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.1599	1.77370	0.886852	−	0.462053i	$-0.152887\pi$
$47$	12.1599	1.77370	0.886852	+	0.462053i	$0.152887\pi$
$48$	0	0
$49$	−6.47277	−0.924681
$50$	0	0
$51$	−6.71836	−0.940758
$52$	0	0
$53$	−5.69781	−0.782655	−0.391327	−	0.920252i	$-0.627984\pi$
$53$	−5.69781	−0.782655	−0.391327	+	0.920252i	$0.627984\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.27389	−0.168731
$58$	0	0
$59$	4.20662	0.547656	0.273828	−	0.961779i	$-0.411710\pi$
$59$	4.20662	0.547656	0.273828	+	0.961779i	$0.411710\pi$
$60$	0	0
$61$	−0.103312	−0.0132278	−0.00661389	−	0.999978i	$-0.502105\pi$
$61$	−0.103312	−0.0132278	−0.00661389	+	0.999978i	$0.502105\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−11.7827	−1.43949	−0.719743	−	0.694241i	$-0.755740\pi$
$67$	−11.7827	−1.43949	−0.719743	+	0.694241i	$0.755740\pi$
$68$	0	0
$69$	−4.68714	−0.564265
$70$	0	0
$71$	−5.75441	−0.682922	−0.341461	−	0.939896i	$-0.610922\pi$
$71$	−5.75441	−0.682922	−0.341461	+	0.939896i	$0.610922\pi$
$72$	0	0
$73$	−6.67939	−0.781763	−0.390882	−	0.920441i	$-0.627830\pi$
$73$	−6.67939	−0.781763	−0.390882	+	0.920441i	$0.627830\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.198875	0.0226639
$78$	0	0
$79$	−3.87826	−0.436339	−0.218169	−	0.975911i	$-0.570009\pi$
$79$	−3.87826	−0.436339	−0.218169	+	0.975911i	$0.570009\pi$
$80$	0	0
$81$	−2.97170	−0.330189
$82$	0	0
$83$	−0.488265	−0.0535941	−0.0267970	−	0.999641i	$-0.508531\pi$
$83$	−0.488265	−0.0535941	−0.0267970	+	0.999641i	$0.508531\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.89669	−0.310558
$88$	0	0
$89$	−16.4338	−1.74198	−0.870989	−	0.491302i	$-0.836521\pi$
$89$	−16.4338	−1.74198	−0.870989	+	0.491302i	$0.836521\pi$
$90$	0	0
$91$	4.32273	0.453146
$92$	0	0
$93$	−4.07502	−0.422559
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.44447	0.451267	0.225634	−	0.974212i	$-0.427555\pi$
$97$	4.44447	0.451267	0.225634	+	0.974212i	$0.427555\pi$
$98$	0	0
$99$	−0.377203	−0.0379103
$100$	0	0
$101$	4.38495	0.436319	0.218160	−	0.975913i	$-0.429995\pi$
$101$	4.38495	0.436319	0.218160	+	0.975913i	$0.429995\pi$
$102$	0	0
$103$	3.33048	0.328162	0.164081	−	0.986447i	$-0.447534\pi$
$103$	3.33048	0.328162	0.164081	+	0.986447i	$0.447534\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.4904	1.59419	0.797093	−	0.603857i	$-0.206370\pi$
$107$	16.4904	1.59419	0.797093	+	0.603857i	$0.206370\pi$
$108$	0	0
$109$	7.79045	0.746190	0.373095	−	0.927793i	$-0.378297\pi$
$109$	7.79045	0.746190	0.373095	+	0.927793i	$0.378297\pi$
$110$	0	0
$111$	−10.3489	−0.982275
$112$	0	0
$113$	0.142282	0.0133848	0.00669238	−	0.999978i	$-0.497870\pi$
$113$	0.142282	0.0133848	0.00669238	+	0.999978i	$0.497870\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−8.19887	−0.757986
$118$	0	0
$119$	−3.82942	−0.351043
$120$	0	0
$121$	−10.9250	−0.993180
$122$	0	0
$123$	−12.0176	−1.08359
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−15.1316	−1.34271	−0.671357	−	0.741135i	$-0.734288\pi$
$127$	−15.1316	−1.34271	−0.671357	+	0.741135i	$0.734288\pi$
$128$	0	0
$129$	12.4981	1.10040
$130$	0	0
$131$	−5.58383	−0.487861	−0.243931	−	0.969793i	$-0.578437\pi$
$131$	−5.58383	−0.487861	−0.243931	+	0.969793i	$0.578437\pi$
$132$	0	0
$133$	−0.726109	−0.0629617
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.8294	−1.09609	−0.548046	−	0.836448i	$-0.684628\pi$
$137$	−12.8294	−1.09609	−0.548046	+	0.836448i	$0.684628\pi$
$138$	0	0
$139$	15.2477	1.29329	0.646647	−	0.762789i	$-0.276171\pi$
$139$	15.2477	1.29329	0.646647	+	0.762789i	$0.276171\pi$
$140$	0	0
$141$	15.4904	1.30453
$142$	0	0
$143$	1.63055	0.136353
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−8.24559	−0.680085
$148$	0	0
$149$	13.8315	1.13312	0.566562	−	0.824019i	$-0.308273\pi$
$149$	13.8315	1.13312	0.566562	+	0.824019i	$0.308273\pi$
$150$	0	0
$151$	11.7077	0.952758	0.476379	−	0.879240i	$-0.341949\pi$
$151$	11.7077	0.952758	0.476379	+	0.879240i	$0.341949\pi$
$152$	0	0
$153$	7.26322	0.587196
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.79045	0.382320	0.191160	−	0.981559i	$-0.438775\pi$
$157$	4.79045	0.382320	0.191160	+	0.981559i	$0.438775\pi$
$158$	0	0
$159$	−7.25839	−0.575628
$160$	0	0
$161$	−2.67164	−0.210555
$162$	0	0
$163$	12.8011	1.00266	0.501331	−	0.865256i	$-0.332844\pi$
$163$	12.8011	1.00266	0.501331	+	0.865256i	$0.332844\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−20.9426	−1.62059	−0.810294	−	0.586024i	$-0.800692\pi$
$167$	−20.9426	−1.62059	−0.810294	+	0.586024i	$0.800692\pi$
$168$	0	0
$169$	22.4415	1.72627
$170$	0	0
$171$	1.37720	0.105317
$172$	0	0
$173$	−15.7282	−1.19580	−0.597898	−	0.801572i	$-0.703997\pi$
$173$	−15.7282	−1.19580	−0.597898	+	0.801572i	$0.703997\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.35878	0.402791
$178$	0	0
$179$	−3.41325	−0.255118	−0.127559	−	0.991831i	$-0.540714\pi$
$179$	−3.41325	−0.255118	−0.127559	+	0.991831i	$0.540714\pi$
$180$	0	0
$181$	23.5109	1.74755	0.873777	−	0.486327i	$-0.161664\pi$
$181$	23.5109	1.74755	0.873777	+	0.486327i	$0.161664\pi$
$182$	0	0
$183$	−0.131609	−0.00972878
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.44447	−0.105630
$188$	0	0
$189$	−4.04884	−0.294510
$190$	0	0
$191$	−12.4650	−0.901937	−0.450968	−	0.892540i	$-0.648921\pi$
$191$	−12.4650	−0.901937	−0.450968	+	0.892540i	$0.648921\pi$
$192$	0	0
$193$	−19.2993	−1.38919	−0.694596	−	0.719400i	$-0.744417\pi$
$193$	−19.2993	−1.38919	−0.694596	+	0.719400i	$0.744417\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−6.63055	−0.472407	−0.236203	−	0.971704i	$-0.575903\pi$
$197$	−6.63055	−0.472407	−0.236203	+	0.971704i	$0.575903\pi$
$198$	0	0
$199$	23.0849	1.63644	0.818222	−	0.574902i	$-0.194960\pi$
$199$	23.0849	1.63644	0.818222	+	0.574902i	$0.194960\pi$
$200$	0	0
$201$	−15.0099	−1.05871
$202$	0	0
$203$	−1.65109	−0.115884
$204$	0	0
$205$	0	0
$206$	0	0
$207$	5.06727	0.352199
$208$	0	0
$209$	−0.273891	−0.0189454
$210$	0	0
$211$	7.54778	0.519611	0.259805	−	0.965661i	$-0.416342\pi$
$211$	7.54778	0.519611	0.259805	+	0.965661i	$0.416342\pi$
$212$	0	0
$213$	−7.33048	−0.502276
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2.32273	−0.157677
$218$	0	0
$219$	−8.50881	−0.574972
$220$	0	0
$221$	−31.3969	−2.11199
$222$	0	0
$223$	−1.09344	−0.0732221	−0.0366111	−	0.999330i	$-0.511656\pi$
$223$	−1.09344	−0.0732221	−0.0366111	+	0.999330i	$0.511656\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.1316	−1.33618	−0.668091	−	0.744080i	$-0.732888\pi$
$227$	−20.1316	−1.33618	−0.668091	+	0.744080i	$0.732888\pi$
$228$	0	0
$229$	−5.51656	−0.364545	−0.182272	−	0.983248i	$-0.558345\pi$
$229$	−5.51656	−0.364545	−0.182272	+	0.983248i	$0.558345\pi$
$230$	0	0
$231$	0.253344	0.0166688
$232$	0	0
$233$	18.1805	1.19104	0.595520	−	0.803340i	$-0.296946\pi$
$233$	18.1805	1.19104	0.595520	+	0.803340i	$0.296946\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−4.94048	−0.320919
$238$	0	0
$239$	−21.9164	−1.41766	−0.708828	−	0.705381i	$-0.750776\pi$
$239$	−21.9164	−1.41766	−0.708828	+	0.705381i	$0.750776\pi$
$240$	0	0
$241$	−28.1882	−1.81576	−0.907881	−	0.419228i	$-0.862301\pi$
$241$	−28.1882	−1.81576	−0.907881	+	0.419228i	$0.862301\pi$
$242$	0	0
$243$	12.9426	0.830269
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.95328	−0.378798
$248$	0	0
$249$	−0.621996	−0.0394174
$250$	0	0
$251$	−9.00987	−0.568698	−0.284349	−	0.958721i	$-0.591777\pi$
$251$	−9.00987	−0.568698	−0.284349	+	0.958721i	$0.591777\pi$
$252$	0	0
$253$	−1.00775	−0.0633567
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.86064	−0.427955	−0.213978	−	0.976839i	$-0.568642\pi$
$257$	−6.86064	−0.427955	−0.213978	+	0.976839i	$0.568642\pi$
$258$	0	0
$259$	−5.89881	−0.366534
$260$	0	0
$261$	3.13161	0.193842
$262$	0	0
$263$	−9.25547	−0.570717	−0.285358	−	0.958421i	$-0.592113\pi$
$263$	−9.25547	−0.570717	−0.285358	+	0.958421i	$0.592113\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−20.9349	−1.28119
$268$	0	0
$269$	0.498939	0.0304208	0.0152104	−	0.999884i	$-0.495158\pi$
$269$	0.498939	0.0304208	0.0152104	+	0.999884i	$0.495158\pi$
$270$	0	0
$271$	−3.71061	−0.225403	−0.112702	−	0.993629i	$-0.535950\pi$
$271$	−3.71061	−0.225403	−0.112702	+	0.993629i	$0.535950\pi$
$272$	0	0
$273$	5.50669	0.333280
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−4.58675	−0.275591	−0.137796	−	0.990461i	$-0.544002\pi$
$277$	−4.58675	−0.275591	−0.137796	+	0.990461i	$0.544002\pi$
$278$	0	0
$279$	4.40550	0.263750
$280$	0	0
$281$	27.2653	1.62651	0.813257	−	0.581905i	$-0.197692\pi$
$281$	27.2653	1.62651	0.813257	+	0.581905i	$0.197692\pi$
$282$	0	0
$283$	−10.2661	−0.610259	−0.305129	−	0.952311i	$-0.598700\pi$
$283$	−10.2661	−0.610259	−0.305129	+	0.952311i	$0.598700\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.84997	−0.404341
$288$	0	0
$289$	10.8139	0.636113
$290$	0	0
$291$	5.66177	0.331899
$292$	0	0
$293$	1.87051	0.109277	0.0546383	−	0.998506i	$-0.482599\pi$
$293$	1.87051	0.109277	0.0546383	+	0.998506i	$0.482599\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.52723	−0.0886192
$298$	0	0
$299$	−21.9044	−1.26677
$300$	0	0
$301$	7.12386	0.410612
$302$	0	0
$303$	5.58595	0.320904
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0.227171	0.0129653	0.00648266	−	0.999979i	$-0.497936\pi$
$307$	0.227171	0.0129653	0.00648266	+	0.999979i	$0.497936\pi$
$308$	0	0
$309$	4.24267	0.241357
$310$	0	0
$311$	−20.9554	−1.18827	−0.594136	−	0.804365i	$-0.702506\pi$
$311$	−20.9554	−1.18827	−0.594136	+	0.804365i	$0.702506\pi$
$312$	0	0
$313$	11.2349	0.635035	0.317518	−	0.948252i	$-0.397151\pi$
$313$	11.2349	0.635035	0.317518	+	0.948252i	$0.397151\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−18.6228	−1.04596	−0.522980	−	0.852345i	$-0.675180\pi$
$317$	−18.6228	−1.04596	−0.522980	+	0.852345i	$0.675180\pi$
$318$	0	0
$319$	−0.622797	−0.0348699
$320$	0	0
$321$	21.0069	1.17249
$322$	0	0
$323$	5.27389	0.293447
$324$	0	0
$325$	0	0
$326$	0	0
$327$	9.92418	0.548809
$328$	0	0
$329$	8.82942	0.486782
$330$	0	0
$331$	14.1054	0.775305	0.387652	−	0.921806i	$-0.373286\pi$
$331$	14.1054	0.775305	0.387652	+	0.921806i	$0.373286\pi$
$332$	0	0
$333$	11.1882	0.613110
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−22.9709	−1.25130	−0.625652	−	0.780102i	$-0.715167\pi$
$337$	−22.9709	−1.25130	−0.625652	+	0.780102i	$0.715167\pi$
$338$	0	0
$339$	0.181252	0.00984424
$340$	0	0
$341$	−0.876142	−0.0474457
$342$	0	0
$343$	−9.78270	−0.528216
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.93273	−0.211120	−0.105560	−	0.994413i	$-0.533664\pi$
$347$	−3.93273	−0.211120	−0.105560	+	0.994413i	$0.533664\pi$
$348$	0	0
$349$	−34.4252	−1.84274	−0.921371	−	0.388685i	$-0.872929\pi$
$349$	−34.4252	−1.84274	−0.921371	+	0.388685i	$0.872929\pi$
$350$	0	0
$351$	−33.1960	−1.77187
$352$	0	0
$353$	4.25547	0.226496	0.113248	−	0.993567i	$-0.463875\pi$
$353$	4.25547	0.226496	0.113248	+	0.993567i	$0.463875\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4.87826	−0.258185
$358$	0	0
$359$	20.2944	1.07110	0.535550	−	0.844504i	$-0.320104\pi$
$359$	20.2944	1.07110	0.535550	+	0.844504i	$0.320104\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−13.9172	−0.730465
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.85289	0.201119	0.100560	−	0.994931i	$-0.467937\pi$
$367$	3.85289	0.201119	0.100560	+	0.994931i	$0.467937\pi$
$368$	0	0
$369$	12.9922	0.676350
$370$	0	0
$371$	−4.13724	−0.214795
$372$	0	0
$373$	−14.6356	−0.757802	−0.378901	−	0.925437i	$-0.623698\pi$
$373$	−14.6356	−0.757802	−0.378901	+	0.925437i	$0.623698\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−13.5371	−0.697197
$378$	0	0
$379$	22.0099	1.13057	0.565286	−	0.824895i	$-0.308766\pi$
$379$	22.0099	1.13057	0.565286	+	0.824895i	$0.308766\pi$
$380$	0	0
$381$	−19.2760	−0.987540
$382$	0	0
$383$	−3.08569	−0.157671	−0.0788357	−	0.996888i	$-0.525120\pi$
$383$	−3.08569	−0.157671	−0.0788357	+	0.996888i	$0.525120\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−13.5117	−0.686840
$388$	0	0
$389$	8.77203	0.444760	0.222380	−	0.974960i	$-0.428618\pi$
$389$	8.77203	0.444760	0.222380	+	0.974960i	$0.428618\pi$
$390$	0	0
$391$	19.4047	0.981338
$392$	0	0
$393$	−7.11319	−0.358813
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1.59450	0.0800257	0.0400129	−	0.999199i	$-0.487260\pi$
$397$	1.59450	0.0800257	0.0400129	+	0.999199i	$0.487260\pi$
$398$	0	0
$399$	−0.924984	−0.0463071
$400$	0	0
$401$	−17.5526	−0.876535	−0.438268	−	0.898844i	$-0.644408\pi$
$401$	−17.5526	−0.876535	−0.438268	+	0.898844i	$0.644408\pi$
$402$	0	0
$403$	−19.0438	−0.948639
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.22505	−0.110292
$408$	0	0
$409$	36.6815	1.81378	0.906892	−	0.421363i	$-0.138448\pi$
$409$	36.6815	1.81378	0.906892	+	0.421363i	$0.138448\pi$
$410$	0	0
$411$	−16.3433	−0.806155
$412$	0	0
$413$	3.05447	0.150301
$414$	0	0
$415$	0	0
$416$	0	0
$417$	19.4239	0.951194
$418$	0	0
$419$	18.8187	0.919356	0.459678	−	0.888086i	$-0.347965\pi$
$419$	18.8187	0.919356	0.459678	+	0.888086i	$0.347965\pi$
$420$	0	0
$421$	−33.7819	−1.64643	−0.823215	−	0.567730i	$-0.807822\pi$
$421$	−33.7819	−1.64643	−0.823215	+	0.567730i	$0.807822\pi$
$422$	0	0
$423$	−16.7467	−0.814250
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−0.0750160	−0.00363028
$428$	0	0
$429$	2.07714	0.100285
$430$	0	0
$431$	12.7651	0.614872	0.307436	−	0.951569i	$-0.400529\pi$
$431$	12.7651	0.614872	0.307436	+	0.951569i	$0.400529\pi$
$432$	0	0
$433$	16.0771	0.772618	0.386309	−	0.922369i	$-0.373750\pi$
$433$	16.0771	0.772618	0.386309	+	0.922369i	$0.373750\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.67939	0.176009
$438$	0	0
$439$	−1.36945	−0.0653604	−0.0326802	−	0.999466i	$-0.510404\pi$
$439$	−1.36945	−0.0653604	−0.0326802	+	0.999466i	$0.510404\pi$
$440$	0	0
$441$	8.91431	0.424491
$442$	0	0
$443$	4.62280	0.219636	0.109818	−	0.993952i	$-0.464973\pi$
$443$	4.62280	0.219636	0.109818	+	0.993952i	$0.464973\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	17.6199	0.833391
$448$	0	0
$449$	23.2555	1.09749	0.548747	−	0.835989i	$-0.315105\pi$
$449$	23.2555	1.09749	0.548747	+	0.835989i	$0.315105\pi$
$450$	0	0
$451$	−2.58383	−0.121668
$452$	0	0
$453$	14.9143	0.700735
$454$	0	0
$455$	0	0
$456$	0	0
$457$	35.8443	1.67673	0.838364	−	0.545111i	$-0.183513\pi$
$457$	35.8443	1.67673	0.838364	+	0.545111i	$0.183513\pi$
$458$	0	0
$459$	29.4076	1.37263
$460$	0	0
$461$	−14.8812	−0.693086	−0.346543	−	0.938034i	$-0.612645\pi$
$461$	−14.8812	−0.693086	−0.346543	+	0.938034i	$0.612645\pi$
$462$	0	0
$463$	−29.9554	−1.39215	−0.696073	−	0.717971i	$-0.745071\pi$
$463$	−29.9554	−1.39215	−0.696073	+	0.717971i	$0.745071\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.73598	−0.311704	−0.155852	−	0.987780i	$-0.549812\pi$
$467$	−6.73598	−0.311704	−0.155852	+	0.987780i	$0.549812\pi$
$468$	0	0
$469$	−8.55553	−0.395058
$470$	0	0
$471$	6.10251	0.281189
$472$	0	0
$473$	2.68714	0.123555
$474$	0	0
$475$	0	0
$476$	0	0
$477$	7.84704	0.359291
$478$	0	0
$479$	−16.6978	−0.762943	−0.381471	−	0.924381i	$-0.624582\pi$
$479$	−16.6978	−0.762943	−0.381471	+	0.924381i	$0.624582\pi$
$480$	0	0
$481$	−48.3636	−2.20519
$482$	0	0
$483$	−3.40338	−0.154859
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3.64042	0.164963	0.0824816	−	0.996593i	$-0.473715\pi$
$487$	3.64042	0.164963	0.0824816	+	0.996593i	$0.473715\pi$
$488$	0	0
$489$	16.3072	0.737439
$490$	0	0
$491$	33.3249	1.50393	0.751965	−	0.659203i	$-0.229106\pi$
$491$	33.3249	1.50393	0.751965	+	0.659203i	$0.229106\pi$
$492$	0	0
$493$	11.9922	0.540104
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.17833	−0.187424
$498$	0	0
$499$	37.9914	1.70073	0.850365	−	0.526193i	$-0.176381\pi$
$499$	37.9914	1.70073	0.850365	+	0.526193i	$0.176381\pi$
$500$	0	0
$501$	−26.6786	−1.19191
$502$	0	0
$503$	−42.1826	−1.88083	−0.940414	−	0.340032i	$-0.889562\pi$
$503$	−42.1826	−1.88083	−0.940414	+	0.340032i	$0.889562\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	28.5881	1.26964
$508$	0	0
$509$	−21.9971	−0.975003	−0.487502	−	0.873122i	$-0.662092\pi$
$509$	−21.9971	−0.975003	−0.487502	+	0.873122i	$0.662092\pi$
$510$	0	0
$511$	−4.84997	−0.214550
$512$	0	0
$513$	5.57608	0.246190
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3.33048	0.146474
$518$	0	0
$519$	−20.0360	−0.879485
$520$	0	0
$521$	20.0977	0.880496	0.440248	−	0.897876i	$-0.354891\pi$
$521$	20.0977	0.880496	0.440248	+	0.897876i	$0.354891\pi$
$522$	0	0
$523$	−4.64817	−0.203250	−0.101625	−	0.994823i	$-0.532404\pi$
$523$	−4.64817	−0.203250	−0.101625	+	0.994823i	$0.532404\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16.8705	0.734891
$528$	0	0
$529$	−9.46209	−0.411395
$530$	0	0
$531$	−5.79338	−0.251411
$532$	0	0
$533$	−56.1620	−2.43265
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−4.34811	−0.187635
$538$	0	0
$539$	−1.77283	−0.0763612
$540$	0	0
$541$	20.0673	0.862759	0.431380	−	0.902171i	$-0.358027\pi$
$541$	20.0673	0.862759	0.431380	+	0.902171i	$0.358027\pi$
$542$	0	0
$543$	29.9504	1.28529
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−37.2010	−1.59060	−0.795300	−	0.606216i	$-0.792687\pi$
$547$	−37.2010	−1.59060	−0.795300	+	0.606216i	$0.792687\pi$
$548$	0	0
$549$	0.142282	0.00607245
$550$	0	0
$551$	2.27389	0.0968710
$552$	0	0
$553$	−2.81604	−0.119750
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−44.8393	−1.89990	−0.949951	−	0.312399i	$-0.898867\pi$
$557$	−44.8393	−1.89990	−0.949951	+	0.312399i	$0.898867\pi$
$558$	0	0
$559$	58.4076	2.47038
$560$	0	0
$561$	−1.84010	−0.0776889
$562$	0	0
$563$	−21.9172	−0.923701	−0.461851	−	0.886958i	$-0.652814\pi$
$563$	−21.9172	−0.923701	−0.461851	+	0.886958i	$0.652814\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.15778	−0.0906183
$568$	0	0
$569$	9.90656	0.415305	0.207652	−	0.978203i	$-0.433418\pi$
$569$	9.90656	0.415305	0.207652	+	0.978203i	$0.433418\pi$
$570$	0	0
$571$	−17.6404	−0.738229	−0.369114	−	0.929384i	$-0.620339\pi$
$571$	−17.6404	−0.738229	−0.369114	+	0.929384i	$0.620339\pi$
$572$	0	0
$573$	−15.8791	−0.663357
$574$	0	0
$575$	0	0
$576$	0	0
$577$	12.7048	0.528906	0.264453	−	0.964399i	$-0.414809\pi$
$577$	12.7048	0.528906	0.264453	+	0.964399i	$0.414809\pi$
$578$	0	0
$579$	−24.5851	−1.02172
$580$	0	0
$581$	−0.354534	−0.0147085
$582$	0	0
$583$	−1.56058	−0.0646325
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15.0438	0.620924	0.310462	−	0.950586i	$-0.399516\pi$
$587$	15.0438	0.620924	0.310462	+	0.950586i	$0.399516\pi$
$588$	0	0
$589$	3.19887	0.131807
$590$	0	0
$591$	−8.44659	−0.347446
$592$	0	0
$593$	16.4231	0.674417	0.337208	−	0.941430i	$-0.390517\pi$
$593$	16.4231	0.674417	0.337208	+	0.941430i	$0.390517\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	29.4076	1.20357
$598$	0	0
$599$	19.1260	0.781466	0.390733	−	0.920504i	$-0.372222\pi$
$599$	19.1260	0.781466	0.390733	+	0.920504i	$0.372222\pi$
$600$	0	0
$601$	31.4124	1.28134	0.640670	−	0.767816i	$-0.278657\pi$
$601$	31.4124	1.28134	0.640670	+	0.767816i	$0.278657\pi$
$602$	0	0
$603$	16.2272	0.660821
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−41.5315	−1.68571	−0.842855	−	0.538140i	$-0.819127\pi$
$607$	−41.5315	−1.68571	−0.842855	+	0.538140i	$0.819127\pi$
$608$	0	0
$609$	−2.10331	−0.0852305
$610$	0	0
$611$	72.3913	2.92864
$612$	0	0
$613$	−21.7274	−0.877563	−0.438781	−	0.898594i	$-0.644590\pi$
$613$	−21.7274	−0.877563	−0.438781	+	0.898594i	$0.644590\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−33.6065	−1.35295	−0.676473	−	0.736467i	$-0.736493\pi$
$617$	−33.6065	−1.35295	−0.676473	+	0.736467i	$0.736493\pi$
$618$	0	0
$619$	27.6036	1.10948	0.554741	−	0.832023i	$-0.312817\pi$
$619$	27.6036	1.10948	0.554741	+	0.832023i	$0.312817\pi$
$620$	0	0
$621$	20.5166	0.823301
$622$	0	0
$623$	−11.9327	−0.478075
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−0.348907	−0.0139340
$628$	0	0
$629$	42.8443	1.70832
$630$	0	0
$631$	−1.94048	−0.0772495	−0.0386247	−	0.999254i	$-0.512298\pi$
$631$	−1.94048	−0.0772495	−0.0386247	+	0.999254i	$0.512298\pi$
$632$	0	0
$633$	9.61505	0.382164
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−38.5342	−1.52678
$638$	0	0
$639$	7.92498	0.313508
$640$	0	0
$641$	1.01975	0.0402775	0.0201388	−	0.999797i	$-0.493589\pi$
$641$	1.01975	0.0402775	0.0201388	+	0.999797i	$0.493589\pi$
$642$	0	0
$643$	36.9866	1.45861	0.729305	−	0.684189i	$-0.239844\pi$
$643$	36.9866	1.45861	0.729305	+	0.684189i	$0.239844\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.1182	0.948186	0.474093	−	0.880475i	$-0.342776\pi$
$647$	24.1182	0.948186	0.474093	+	0.880475i	$0.342776\pi$
$648$	0	0
$649$	1.15215	0.0452260
$650$	0	0
$651$	−2.95891	−0.115969
$652$	0	0
$653$	−37.2603	−1.45811	−0.729054	−	0.684456i	$-0.760040\pi$
$653$	−37.2603	−1.45811	−0.729054	+	0.684456i	$0.760040\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	9.19887	0.358882
$658$	0	0
$659$	−21.4386	−0.835130	−0.417565	−	0.908647i	$-0.637116\pi$
$659$	−21.4386	−0.835130	−0.417565	+	0.908647i	$0.637116\pi$
$660$	0	0
$661$	0.783503	0.0304747	0.0152374	−	0.999884i	$-0.495150\pi$
$661$	0.783503	0.0304747	0.0152374	+	0.999884i	$0.495150\pi$
$662$	0	0
$663$	−39.9963	−1.55333
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.36653	0.323953
$668$	0	0
$669$	−1.39292	−0.0538535
$670$	0	0
$671$	−0.0282963	−0.00109237
$672$	0	0
$673$	50.1903	1.93469	0.967347	−	0.253454i	$-0.0815667\pi$
$673$	50.1903	1.93469	0.967347	+	0.253454i	$0.0815667\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29.8804	1.14840	0.574198	−	0.818716i	$-0.305314\pi$
$677$	29.8804	1.14840	0.574198	+	0.818716i	$0.305314\pi$
$678$	0	0
$679$	3.22717	0.123847
$680$	0	0
$681$	−25.6455	−0.982736
$682$	0	0
$683$	12.3326	0.471894	0.235947	−	0.971766i	$-0.424181\pi$
$683$	12.3326	0.471894	0.235947	+	0.971766i	$0.424181\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−7.02750	−0.268116
$688$	0	0
$689$	−33.9207	−1.29227
$690$	0	0
$691$	−3.62200	−0.137787	−0.0688936	−	0.997624i	$-0.521947\pi$
$691$	−3.62200	−0.137787	−0.0688936	+	0.997624i	$0.521947\pi$
$692$	0	0
$693$	−0.273891	−0.0104042
$694$	0	0
$695$	0	0
$696$	0	0
$697$	49.7528	1.88452
$698$	0	0
$699$	23.1599	0.875988
$700$	0	0
$701$	34.1209	1.28873	0.644365	−	0.764718i	$-0.277122\pi$
$701$	34.1209	1.28873	0.644365	+	0.764718i	$0.277122\pi$
$702$	0	0
$703$	8.12386	0.306397
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.18396	0.119745
$708$	0	0
$709$	−17.1209	−0.642990	−0.321495	−	0.946911i	$-0.604185\pi$
$709$	−17.1209	−0.642990	−0.321495	+	0.946911i	$0.604185\pi$
$710$	0	0
$711$	5.34116	0.200309
$712$	0	0
$713$	11.7699	0.440786
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−27.9191	−1.04266
$718$	0	0
$719$	7.02750	0.262081	0.131041	−	0.991377i	$-0.458168\pi$
$719$	7.02750	0.262081	0.131041	+	0.991377i	$0.458168\pi$
$720$	0	0
$721$	2.41830	0.0900620
$722$	0	0
$723$	−35.9087	−1.33546
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−11.8938	−0.441115	−0.220558	−	0.975374i	$-0.570788\pi$
$727$	−11.8938	−0.441115	−0.220558	+	0.975374i	$0.570788\pi$
$728$	0	0
$729$	25.4026	0.940836
$730$	0	0
$731$	−51.7421	−1.91375
$732$	0	0
$733$	−20.7154	−0.765142	−0.382571	−	0.923926i	$-0.624961\pi$
$733$	−20.7154	−0.765142	−0.382571	+	0.923926i	$0.624961\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.22717	−0.118874
$738$	0	0
$739$	−33.8620	−1.24563	−0.622816	−	0.782368i	$-0.714012\pi$
$739$	−33.8620	−1.24563	−0.622816	+	0.782368i	$0.714012\pi$
$740$	0	0
$741$	−7.58383	−0.278599
$742$	0	0
$743$	42.7381	1.56791	0.783955	−	0.620818i	$-0.213200\pi$
$743$	42.7381	1.56791	0.783955	+	0.620818i	$0.213200\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.672440	0.0246033
$748$	0	0
$749$	11.9738	0.437514
$750$	0	0
$751$	−11.9581	−0.436358	−0.218179	−	0.975909i	$-0.570012\pi$
$751$	−11.9581	−0.436358	−0.218179	+	0.975909i	$0.570012\pi$
$752$	0	0
$753$	−11.4776	−0.418267
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−29.1103	−1.05803	−0.529015	−	0.848612i	$-0.677439\pi$
$757$	−29.1103	−1.05803	−0.529015	+	0.848612i	$0.677439\pi$
$758$	0	0
$759$	−1.28376	−0.0465977
$760$	0	0
$761$	−22.3014	−0.808425	−0.404212	−	0.914665i	$-0.632454\pi$
$761$	−22.3014	−0.808425	−0.404212	+	0.914665i	$0.632454\pi$
$762$	0	0
$763$	5.65672	0.204787
$764$	0	0
$765$	0	0
$766$	0	0
$767$	25.0432	0.904258
$768$	0	0
$769$	3.95891	0.142762	0.0713809	−	0.997449i	$-0.477259\pi$
$769$	3.95891	0.142762	0.0713809	+	0.997449i	$0.477259\pi$
$770$	0	0
$771$	−8.73971	−0.314753
$772$	0	0
$773$	−27.8139	−1.00040	−0.500199	−	0.865911i	$-0.666740\pi$
$773$	−27.8139	−1.00040	−0.500199	+	0.865911i	$0.666740\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−7.51444	−0.269579
$778$	0	0
$779$	9.43380	0.338001
$780$	0	0
$781$	−1.57608	−0.0563965
$782$	0	0
$783$	12.6794	0.453124
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.82460	−0.0650398	−0.0325199	−	0.999471i	$-0.510353\pi$
$787$	−1.82460	−0.0650398	−0.0325199	+	0.999471i	$0.510353\pi$
$788$	0	0
$789$	−11.7905	−0.419751
$790$	0	0
$791$	0.103312	0.00367336
$792$	0	0
$793$	−0.615047	−0.0218410
$794$	0	0
$795$	0	0
$796$	0	0
$797$	21.0360	0.745135	0.372567	−	0.928005i	$-0.378478\pi$
$797$	21.0360	0.745135	0.372567	+	0.928005i	$0.378478\pi$
$798$	0	0
$799$	−64.1300	−2.26876
$800$	0	0
$801$	22.6327	0.799686
$802$	0	0
$803$	−1.82942	−0.0645589
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0.635593	0.0223739
$808$	0	0
$809$	−0.0819654	−0.00288175	−0.00144088	−	0.999999i	$-0.500459\pi$
$809$	−0.0819654	−0.00288175	−0.00144088	+	0.999999i	$0.500459\pi$
$810$	0	0
$811$	6.72531	0.236158	0.118079	−	0.993004i	$-0.462326\pi$
$811$	6.72531	0.236158	0.118079	+	0.993004i	$0.462326\pi$
$812$	0	0
$813$	−4.72691	−0.165780
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−9.81100	−0.343243
$818$	0	0
$819$	−5.95328	−0.208024
$820$	0	0
$821$	30.9426	1.07990	0.539952	−	0.841696i	$-0.318442\pi$
$821$	30.9426	1.07990	0.539952	+	0.841696i	$0.318442\pi$
$822$	0	0
$823$	26.5908	0.926896	0.463448	−	0.886124i	$-0.346612\pi$
$823$	26.5908	0.926896	0.463448	+	0.886124i	$0.346612\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−5.57900	−0.194001	−0.0970004	−	0.995284i	$-0.530925\pi$
$827$	−5.57900	−0.194001	−0.0970004	+	0.995284i	$0.530925\pi$
$828$	0	0
$829$	18.9765	0.659082	0.329541	−	0.944141i	$-0.393106\pi$
$829$	18.9765	0.659082	0.329541	+	0.944141i	$0.393106\pi$
$830$	0	0
$831$	−5.84302	−0.202692
$832$	0	0
$833$	34.1367	1.18276
$834$	0	0
$835$	0	0
$836$	0	0
$837$	17.8372	0.616543
$838$	0	0
$839$	−12.9143	−0.445852	−0.222926	−	0.974835i	$-0.571561\pi$
$839$	−12.9143	−0.445852	−0.222926	+	0.974835i	$0.571561\pi$
$840$	0	0
$841$	−23.8294	−0.821704
$842$	0	0
$843$	34.7331	1.19627
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−7.93273	−0.272572
$848$	0	0
$849$	−13.0779	−0.448834
$850$	0	0
$851$	29.8908	1.02464
$852$	0	0
$853$	−6.46077	−0.221213	−0.110606	−	0.993864i	$-0.535279\pi$
$853$	−6.46077	−0.221213	−0.110606	+	0.993864i	$0.535279\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12.4055	0.423764	0.211882	−	0.977295i	$-0.432041\pi$
$857$	12.4055	0.423764	0.211882	+	0.977295i	$0.432041\pi$
$858$	0	0
$859$	−40.3425	−1.37647	−0.688234	−	0.725489i	$-0.741614\pi$
$859$	−40.3425	−1.37647	−0.688234	+	0.725489i	$0.741614\pi$
$860$	0	0
$861$	−8.72611	−0.297385
$862$	0	0
$863$	1.83235	0.0623738	0.0311869	−	0.999514i	$-0.490071\pi$
$863$	1.83235	0.0623738	0.0311869	+	0.999514i	$0.490071\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.7758	0.467849
$868$	0	0
$869$	−1.06222	−0.0360333
$870$	0	0
$871$	−70.1457	−2.37680
$872$	0	0
$873$	−6.12094	−0.207162
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.14419	0.275010	0.137505	−	0.990501i	$-0.456092\pi$
$877$	8.14419	0.275010	0.137505	+	0.990501i	$0.456092\pi$
$878$	0	0
$879$	2.38283	0.0803709
$880$	0	0
$881$	12.1706	0.410037	0.205019	−	0.978758i	$-0.434275\pi$
$881$	12.1706	0.410037	0.205019	+	0.978758i	$0.434275\pi$
$882$	0	0
$883$	46.7614	1.57364	0.786822	−	0.617179i	$-0.211725\pi$
$883$	46.7614	1.57364	0.786822	+	0.617179i	$0.211725\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−34.8804	−1.17117	−0.585584	−	0.810611i	$-0.699135\pi$
$887$	−34.8804	−1.17117	−0.585584	+	0.810611i	$0.699135\pi$
$888$	0	0
$889$	−10.9872	−0.368499
$890$	0	0
$891$	−0.813922	−0.0272674
$892$	0	0
$893$	−12.1599	−0.406916
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−27.9039	−0.931683
$898$	0	0
$899$	7.27389	0.242598
$900$	0	0
$901$	30.0496	1.00110
$902$	0	0
$903$	9.07502	0.301998
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−25.5080	−0.846980	−0.423490	−	0.905901i	$-0.639195\pi$
$907$	−25.5080	−0.846980	−0.423490	+	0.905901i	$0.639195\pi$
$908$	0	0
$909$	−6.03897	−0.200300
$910$	0	0
$911$	21.5032	0.712432	0.356216	−	0.934404i	$-0.384067\pi$
$911$	21.5032	0.712432	0.356216	+	0.934404i	$0.384067\pi$
$912$	0	0
$913$	−0.133731	−0.00442586
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.05447	−0.133890
$918$	0	0
$919$	−37.1386	−1.22509	−0.612544	−	0.790436i	$-0.709854\pi$
$919$	−37.1386	−1.22509	−0.612544	+	0.790436i	$0.709854\pi$
$920$	0	0
$921$	0.289391	0.00953575
$922$	0	0
$923$	−34.2576	−1.12760
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−4.58675	−0.150649
$928$	0	0
$929$	−3.36170	−0.110294	−0.0551469	−	0.998478i	$-0.517563\pi$
$929$	−3.36170	−0.110294	−0.0551469	+	0.998478i	$0.517563\pi$
$930$	0	0
$931$	6.47277	0.212136
$932$	0	0
$933$	−26.6949	−0.873951
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−10.0694	−0.328953	−0.164476	−	0.986381i	$-0.552593\pi$
$937$	−10.0694	−0.328953	−0.164476	+	0.986381i	$0.552593\pi$
$938$	0	0
$939$	14.3121	0.467056
$940$	0	0
$941$	20.8139	0.678514	0.339257	−	0.940694i	$-0.389824\pi$
$941$	20.8139	0.678514	0.339257	+	0.940694i	$0.389824\pi$
$942$	0	0
$943$	34.7106	1.13033
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30.4904	−0.990804	−0.495402	−	0.868664i	$-0.664979\pi$
$947$	−30.4904	−0.990804	−0.495402	+	0.868664i	$0.664979\pi$
$948$	0	0
$949$	−39.7643	−1.29080
$950$	0	0
$951$	−23.7234	−0.769284
$952$	0	0
$953$	−7.58383	−0.245664	−0.122832	−	0.992427i	$-0.539198\pi$
$953$	−7.58383	−0.245664	−0.122832	+	0.992427i	$0.539198\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−0.793375	−0.0256462
$958$	0	0
$959$	−9.31556	−0.300815
$960$	0	0
$961$	−20.7672	−0.669910
$962$	0	0
$963$	−22.7106	−0.731839
$964$	0	0
$965$	0	0
$966$	0	0
$967$	54.6687	1.75803	0.879014	−	0.476797i	$-0.158202\pi$
$967$	54.6687	1.75803	0.879014	+	0.476797i	$0.158202\pi$
$968$	0	0
$969$	6.71836	0.215825
$970$	0	0
$971$	39.9632	1.28248	0.641239	−	0.767341i	$-0.278421\pi$
$971$	39.9632	1.28248	0.641239	+	0.767341i	$0.278421\pi$
$972$	0	0
$973$	11.0715	0.354936
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−15.2400	−0.487570	−0.243785	−	0.969829i	$-0.578389\pi$
$977$	−15.2400	−0.487570	−0.243785	+	0.969829i	$0.578389\pi$
$978$	0	0
$979$	−4.50106	−0.143855
$980$	0	0
$981$	−10.7290	−0.342552
$982$	0	0
$983$	−45.3609	−1.44679	−0.723394	−	0.690435i	$-0.757419\pi$
$983$	−45.3609	−1.44679	−0.723394	+	0.690435i	$0.757419\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	11.2477	0.358019
$988$	0	0
$989$	−36.0985	−1.14787
$990$	0	0
$991$	16.3537	0.519493	0.259747	−	0.965677i	$-0.416361\pi$
$991$	16.3537	0.519493	0.259747	+	0.965677i	$0.416361\pi$
$992$	0	0
$993$	17.9688	0.570222
$994$	0	0
$995$	0	0
$996$	0	0
$997$	33.2037	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$997$	33.2037	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$998$	0	0
$999$	45.2993	1.43321

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.cc.1.2		3
4.3	odd	2		475.2.a.e.1.1	✓	3
5.4	even	2		7600.2.a.bh.1.2		3
12.11	even	2		4275.2.a.bm.1.3		3
20.3	even	4		475.2.b.b.324.6		6
20.7	even	4		475.2.b.b.324.1		6
20.19	odd	2		475.2.a.g.1.3	yes	3
60.59	even	2		4275.2.a.ba.1.1		3
76.75	even	2		9025.2.a.bc.1.3		3
380.379	even	2		9025.2.a.y.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.a.e.1.1	✓	3	4.3	odd	2
475.2.a.g.1.3	yes	3	20.19	odd	2
475.2.b.b.324.1		6	20.7	even	4
475.2.b.b.324.6		6	20.3	even	4
4275.2.a.ba.1.1		3	60.59	even	2
4275.2.a.bm.1.3		3	12.11	even	2
7600.2.a.bh.1.2		3	5.4	even	2
7600.2.a.cc.1.2		3	1.1	even	1	trivial
9025.2.a.y.1.1		3	380.379	even	2
9025.2.a.bc.1.3		3	76.75	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q+1.27389 q^{3} +0.726109 q^{7} -1.37720 q^{9} +0.273891 q^{11} +5.95328 q^{13} -5.27389 q^{17} -1.00000 q^{19} +0.924984 q^{21} -3.67939 q^{23} -5.57608 q^{27} -2.27389 q^{29} -3.19887 q^{31} +0.348907 q^{33} -8.12386 q^{37} +7.58383 q^{39} -9.43380 q^{41} +9.81100 q^{43} +12.1599 q^{47} -6.47277 q^{49} -6.71836 q^{51} -5.69781 q^{53} -1.27389 q^{57} +4.20662 q^{59} -0.103312 q^{61} -1.00000 q^{63} -11.7827 q^{67} -4.68714 q^{69} -5.75441 q^{71} -6.67939 q^{73} +0.198875 q^{77} -3.87826 q^{79} -2.97170 q^{81} -0.488265 q^{83} -2.89669 q^{87} -16.4338 q^{89} +4.32273 q^{91} -4.07502 q^{93} +4.44447 q^{97} -0.377203 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 4 q^{7} + q^{9} - q^{11} - 3 q^{13} - 14 q^{17} - 3 q^{19} - 6 q^{21} + 8 q^{23} - q^{27} - 5 q^{29} + q^{31} + 8 q^{33} - 5 q^{37} + 11 q^{39} + q^{41} - 5 q^{43} + 9 q^{47} - 7 q^{49}+ \cdots + 4 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 + 4 * q^7 + q^9 - q^11 - 3 * q^13 - 14 * q^17 - 3 * q^19 - 6 * q^21 + 8 * q^23 - q^27 - 5 * q^29 + q^31 + 8 * q^33 - 5 * q^37 + 11 * q^39 + q^41 - 5 * q^43 + 9 * q^47 - 7 * q^49 - 18 * q^51 - 31 * q^53 - 2 * q^57 + 6 * q^59 + 3 * q^61 - 3 * q^63 - 13 * q^67 + q^69 - 7 * q^71 - q^73 - 10 * q^77 + 18 * q^79 - 21 * q^81 + 3 * q^83 - 12 * q^87 - 20 * q^89 - 17 * q^91 - 21 * q^93 + 13 * q^97 + 4 * q^99

Introduction

L-functions

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