LMFDB - Embedded newform 5175.2.a.bv.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5175,2,Mod(1,5175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5175 = 3^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5175.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:	$41.3225830460$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.5744.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} - 2x + 1 $ x^4 - 5x^2 - 2x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 115)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.291367$ of defining polynomial
Character	$\chi$	$=$	5175.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.291367 q^{2} -1.91511 q^{4} +1.20647 q^{7} -1.14073 q^{8} +O(q^{10})$	$q+0.291367 q^{2} -1.91511 q^{4} +1.20647 q^{7} -1.14073 q^{8} -3.65773 q^{11} -0.859268 q^{13} +0.351526 q^{14} +3.49784 q^{16} +6.72347 q^{17} -1.51699 q^{19} -1.06574 q^{22} -1.00000 q^{23} -0.250362 q^{26} -2.31052 q^{28} -0.548747 q^{29} -5.99568 q^{31} +3.30062 q^{32} +1.95900 q^{34} -2.04100 q^{37} -0.442002 q^{38} +7.14998 q^{41} +10.0799 q^{43} +7.00493 q^{44} -0.291367 q^{46} -9.17040 q^{47} -5.54442 q^{49} +1.64559 q^{52} +5.37896 q^{53} -1.37626 q^{56} -0.159887 q^{58} +0.582734 q^{59} -8.83244 q^{61} -1.74694 q^{62} -6.03399 q^{64} +3.20647 q^{67} -12.8761 q^{68} +12.5784 q^{71} -8.62597 q^{73} -0.594681 q^{74} +2.90520 q^{76} -4.41294 q^{77} +0.0700619 q^{79} +2.08327 q^{82} +6.74197 q^{83} +2.93696 q^{86} +4.17248 q^{88} -4.96393 q^{89} -1.03668 q^{91} +1.91511 q^{92} -2.67195 q^{94} -11.3380 q^{97} -1.61546 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 6 q^{7} + 6 q^{8}+O(q^{10}) $ 4 * q + 2 * q^4 - 6 * q^7 + 6 * q^8	$ 4 q + 2 q^{4} - 6 q^{7} + 6 q^{8} - 2 q^{11} - 14 q^{13} + 4 q^{14} + 2 q^{16} + 14 q^{17} - 4 q^{19} - 4 q^{22} - 4 q^{23} - 6 q^{26} - 18 q^{28} - 4 q^{29} + 2 q^{32} + 14 q^{34} - 2 q^{37} - 10 q^{38} + 8 q^{41} - 4 q^{43} - 6 q^{44} + 2 q^{47} - 18 q^{52} + 4 q^{53} - 14 q^{56} - 8 q^{61} - 28 q^{62} - 20 q^{64} + 2 q^{67} + 8 q^{68} + 24 q^{71} - 18 q^{73} + 36 q^{74} - 18 q^{76} + 4 q^{77} + 24 q^{79} - 32 q^{82} - 6 q^{83} - 14 q^{86} + 10 q^{88} + 8 q^{89} + 26 q^{91} - 2 q^{92} - 42 q^{94} - 34 q^{97} - 28 q^{98}+O(q^{100}) $ 4 * q + 2 * q^4 - 6 * q^7 + 6 * q^8 - 2 * q^11 - 14 * q^13 + 4 * q^14 + 2 * q^16 + 14 * q^17 - 4 * q^19 - 4 * q^22 - 4 * q^23 - 6 * q^26 - 18 * q^28 - 4 * q^29 + 2 * q^32 + 14 * q^34 - 2 * q^37 - 10 * q^38 + 8 * q^41 - 4 * q^43 - 6 * q^44 + 2 * q^47 - 18 * q^52 + 4 * q^53 - 14 * q^56 - 8 * q^61 - 28 * q^62 - 20 * q^64 + 2 * q^67 + 8 * q^68 + 24 * q^71 - 18 * q^73 + 36 * q^74 - 18 * q^76 + 4 * q^77 + 24 * q^79 - 32 * q^82 - 6 * q^83 - 14 * q^86 + 10 * q^88 + 8 * q^89 + 26 * q^91 - 2 * q^92 - 42 * q^94 - 34 * q^97 - 28 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	0.291367	0.206028	0.103014	−	0.994680i	$-0.467151\pi$
$2$	0.291367	0.206028	0.103014	+	0.994680i	$0.467151\pi$
$3$	0	0
$3$	0	0
$4$	−1.91511	−0.957553
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.20647	0.456004	0.228002	−	0.973661i	$-0.426781\pi$
$7$	1.20647	0.456004	0.228002	+	0.973661i	$0.426781\pi$
$8$	−1.14073	−0.403310
$9$	0	0
$10$	0	0
$11$	−3.65773	−1.10285	−0.551423	−	0.834226i	$-0.685915\pi$
$11$	−3.65773	−1.10285	−0.551423	+	0.834226i	$0.685915\pi$
$12$	0	0
$13$	−0.859268	−0.238318	−0.119159	−	0.992875i	$-0.538020\pi$
$13$	−0.859268	−0.238318	−0.119159	+	0.992875i	$0.538020\pi$
$14$	0.351526	0.0939493
$15$	0	0
$16$	3.49784	0.874460
$17$	6.72347	1.63068	0.815340	−	0.578982i	$-0.196550\pi$
$17$	6.72347	1.63068	0.815340	+	0.578982i	$0.196550\pi$
$18$	0	0
$19$	−1.51699	−0.348022	−0.174011	−	0.984744i	$-0.555673\pi$
$19$	−1.51699	−0.348022	−0.174011	+	0.984744i	$0.555673\pi$
$20$	0	0
$21$	0	0
$22$	−1.06574	−0.227217
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	−0.250362	−0.0491001
$27$	0	0
$28$	−2.31052	−0.436648
$29$	−0.548747	−0.101900	−0.0509498	−	0.998701i	$-0.516225\pi$
$29$	−0.548747	−0.101900	−0.0509498	+	0.998701i	$0.516225\pi$
$30$	0	0
$31$	−5.99568	−1.07686	−0.538428	−	0.842672i	$-0.680982\pi$
$31$	−5.99568	−1.07686	−0.538428	+	0.842672i	$0.680982\pi$
$32$	3.30062	0.583472
$33$	0	0
$34$	1.95900	0.335965
$35$	0	0
$36$	0	0
$37$	−2.04100	−0.335539	−0.167770	−	0.985826i	$-0.553656\pi$
$37$	−2.04100	−0.335539	−0.167770	+	0.985826i	$0.553656\pi$
$38$	−0.442002	−0.0717021
$39$	0	0
$40$	0	0
$41$	7.14998	1.11664	0.558320	−	0.829626i	$-0.311446\pi$
$41$	7.14998	1.11664	0.558320	+	0.829626i	$0.311446\pi$
$42$	0	0
$43$	10.0799	1.53717	0.768587	−	0.639745i	$-0.220960\pi$
$43$	10.0799	1.53717	0.768587	+	0.639745i	$0.220960\pi$
$44$	7.00493	1.05603
$45$	0	0
$46$	−0.291367	−0.0429597
$47$	−9.17040	−1.33764	−0.668820	−	0.743424i	$-0.733200\pi$
$47$	−9.17040	−1.33764	−0.668820	+	0.743424i	$0.733200\pi$
$48$	0	0
$49$	−5.54442	−0.792061
$50$	0	0
$51$	0	0
$52$	1.64559	0.228202
$53$	5.37896	0.738857	0.369428	−	0.929259i	$-0.379554\pi$
$53$	5.37896	0.738857	0.369428	+	0.929259i	$0.379554\pi$
$54$	0	0
$55$	0	0
$56$	−1.37626	−0.183911
$57$	0	0
$58$	−0.159887	−0.0209941
$59$	0.582734	0.0758655	0.0379327	−	0.999280i	$-0.487923\pi$
$59$	0.582734	0.0758655	0.0379327	+	0.999280i	$0.487923\pi$
$60$	0	0
$61$	−8.83244	−1.13088	−0.565439	−	0.824790i	$-0.691293\pi$
$61$	−8.83244	−1.13088	−0.565439	+	0.824790i	$0.691293\pi$
$62$	−1.74694	−0.221862
$63$	0	0
$64$	−6.03399	−0.754248
$65$	0	0
$66$	0	0
$67$	3.20647	0.391733	0.195866	−	0.980631i	$-0.437248\pi$
$67$	3.20647	0.391733	0.195866	+	0.980631i	$0.437248\pi$
$68$	−12.8761	−1.56146
$69$	0	0
$70$	0	0
$71$	12.5784	1.49278	0.746391	−	0.665507i	$-0.231785\pi$
$71$	12.5784	1.49278	0.746391	+	0.665507i	$0.231785\pi$
$72$	0	0
$73$	−8.62597	−1.00959	−0.504797	−	0.863238i	$-0.668433\pi$
$73$	−8.62597	−1.00959	−0.504797	+	0.863238i	$0.668433\pi$
$74$	−0.594681	−0.0691303
$75$	0	0
$76$	2.90520	0.333250
$77$	−4.41294	−0.502902
$78$	0	0
$79$	0.0700619	0.00788258	0.00394129	−	0.999992i	$-0.498745\pi$
$79$	0.0700619	0.00788258	0.00394129	+	0.999992i	$0.498745\pi$
$80$	0	0
$81$	0	0
$82$	2.08327	0.230059
$83$	6.74197	0.740027	0.370014	−	0.929026i	$-0.379353\pi$
$83$	6.74197	0.740027	0.370014	+	0.929026i	$0.379353\pi$
$84$	0	0
$85$	0	0
$86$	2.93696	0.316700
$87$	0	0
$88$	4.17248	0.444788
$89$	−4.96393	−0.526175	−0.263088	−	0.964772i	$-0.584741\pi$
$89$	−4.96393	−0.526175	−0.263088	+	0.964772i	$0.584741\pi$
$90$	0	0
$91$	−1.03668	−0.108674
$92$	1.91511	0.199664
$93$	0	0
$94$	−2.67195	−0.275591
$95$	0	0
$96$	0	0
$97$	−11.3380	−1.15119	−0.575597	−	0.817733i	$-0.695230\pi$
$97$	−11.3380	−1.15119	−0.575597	+	0.817733i	$0.695230\pi$
$98$	−1.61546	−0.163186
$99$	0	0
$100$	0	0
$101$	−3.44693	−0.342983	−0.171491	−	0.985186i	$-0.554859\pi$
$101$	−3.44693	−0.342983	−0.171491	+	0.985186i	$0.554859\pi$
$102$	0	0
$103$	−9.13641	−0.900237	−0.450119	−	0.892969i	$-0.648618\pi$
$103$	−9.13641	−0.900237	−0.450119	+	0.892969i	$0.648618\pi$
$104$	0.980194	0.0961160
$105$	0	0
$106$	1.56725	0.152225
$107$	9.24539	0.893786	0.446893	−	0.894588i	$-0.352531\pi$
$107$	9.24539	0.893786	0.446893	+	0.894588i	$0.352531\pi$
$108$	0	0
$109$	1.64847	0.157895	0.0789476	−	0.996879i	$-0.474844\pi$
$109$	1.64847	0.157895	0.0789476	+	0.996879i	$0.474844\pi$
$110$	0	0
$111$	0	0
$112$	4.22005	0.398757
$113$	1.20647	0.113495	0.0567477	−	0.998389i	$-0.481927\pi$
$113$	1.20647	0.113495	0.0567477	+	0.998389i	$0.481927\pi$
$114$	0	0
$115$	0	0
$116$	1.05091	0.0975743
$117$	0	0
$118$	0.169789	0.0156304
$119$	8.11167	0.743596
$120$	0	0
$121$	2.37896	0.216269
$122$	−2.57348	−0.232992
$123$	0	0
$124$	11.4824	1.03115
$125$	0	0
$126$	0	0
$127$	−15.3542	−1.36247	−0.681233	−	0.732066i	$-0.738556\pi$
$127$	−15.3542	−1.36247	−0.681233	+	0.732066i	$0.738556\pi$
$128$	−8.35934	−0.738868
$129$	0	0
$130$	0	0
$131$	−13.0734	−1.14223	−0.571113	−	0.820872i	$-0.693488\pi$
$131$	−13.0734	−1.14223	−0.571113	+	0.820872i	$0.693488\pi$
$132$	0	0
$133$	−1.83021	−0.158699
$134$	0.934260	0.0807078
$135$	0	0
$136$	−7.66967	−0.657669
$137$	−13.1840	−1.12638	−0.563191	−	0.826327i	$-0.690427\pi$
$137$	−13.1840	−1.12638	−0.563191	+	0.826327i	$0.690427\pi$
$138$	0	0
$139$	−14.1160	−1.19730	−0.598652	−	0.801010i	$-0.704297\pi$
$139$	−14.1160	−1.19730	−0.598652	+	0.801010i	$0.704297\pi$
$140$	0	0
$141$	0	0
$142$	3.66493	0.307554
$143$	3.14297	0.262828
$144$	0	0
$145$	0	0
$146$	−2.51332	−0.208004
$147$	0	0
$148$	3.90874	0.321296
$149$	6.54442	0.536140	0.268070	−	0.963399i	$-0.413614\pi$
$149$	6.54442	0.536140	0.268070	+	0.963399i	$0.413614\pi$
$150$	0	0
$151$	2.61672	0.212946	0.106473	−	0.994316i	$-0.466044\pi$
$151$	2.61672	0.212946	0.106473	+	0.994316i	$0.466044\pi$
$152$	1.73048	0.140361
$153$	0	0
$154$	−1.28579	−0.103612
$155$	0	0
$156$	0	0
$157$	−8.81394	−0.703429	−0.351715	−	0.936107i	$-0.614401\pi$
$157$	−8.81394	−0.703429	−0.351715	+	0.936107i	$0.614401\pi$
$158$	0.0204137	0.00162403
$159$	0	0
$160$	0	0
$161$	−1.20647	−0.0950833
$162$	0	0
$163$	10.1747	0.796946	0.398473	−	0.917180i	$-0.369540\pi$
$163$	10.1747	0.796946	0.398473	+	0.917180i	$0.369540\pi$
$164$	−13.6930	−1.06924
$165$	0	0
$166$	1.96439	0.152466
$167$	−12.6647	−0.980027	−0.490014	−	0.871715i	$-0.663008\pi$
$167$	−12.6647	−0.980027	−0.490014	+	0.871715i	$0.663008\pi$
$168$	0	0
$169$	−12.2617	−0.943205
$170$	0	0
$171$	0	0
$172$	−19.3041	−1.47192
$173$	−15.6901	−1.19290	−0.596448	−	0.802652i	$-0.703422\pi$
$173$	−15.6901	−1.19290	−0.596448	+	0.802652i	$0.703422\pi$
$174$	0	0
$175$	0	0
$176$	−12.7941	−0.964394
$177$	0	0
$178$	−1.44632	−0.108407
$179$	2.39876	0.179292	0.0896460	−	0.995974i	$-0.471426\pi$
$179$	2.39876	0.179292	0.0896460	+	0.995974i	$0.471426\pi$
$180$	0	0
$181$	−9.87122	−0.733722	−0.366861	−	0.930276i	$-0.619567\pi$
$181$	−9.87122	−0.733722	−0.366861	+	0.930276i	$0.619567\pi$
$182$	−0.302055	−0.0223898
$183$	0	0
$184$	1.14073	0.0840959
$185$	0	0
$186$	0	0
$187$	−24.5926	−1.79839
$188$	17.5623	1.28086
$189$	0	0
$190$	0	0
$191$	9.60617	0.695078	0.347539	−	0.937666i	$-0.387017\pi$
$191$	9.60617	0.695078	0.347539	+	0.937666i	$0.387017\pi$
$192$	0	0
$193$	−24.9709	−1.79745	−0.898724	−	0.438515i	$-0.855505\pi$
$193$	−24.9709	−1.79745	−0.898724	+	0.438515i	$0.855505\pi$
$194$	−3.30350	−0.237178
$195$	0	0
$196$	10.6182	0.758440
$197$	−9.92292	−0.706979	−0.353489	−	0.935439i	$-0.615005\pi$
$197$	−9.92292	−0.706979	−0.353489	+	0.935439i	$0.615005\pi$
$198$	0	0
$199$	7.01818	0.497506	0.248753	−	0.968567i	$-0.419979\pi$
$199$	7.01818	0.497506	0.248753	+	0.968567i	$0.419979\pi$
$200$	0	0
$201$	0	0
$202$	−1.00432	−0.0706638
$203$	−0.662047	−0.0464666
$204$	0	0
$205$	0	0
$206$	−2.66205	−0.185474
$207$	0	0
$208$	−3.00558	−0.208400
$209$	5.54875	0.383815
$210$	0	0
$211$	7.44693	0.512668	0.256334	−	0.966588i	$-0.417485\pi$
$211$	7.44693	0.512668	0.256334	+	0.966588i	$0.417485\pi$
$212$	−10.3013	−0.707494
$213$	0	0
$214$	2.69380	0.184144
$215$	0	0
$216$	0	0
$217$	−7.23362	−0.491050
$218$	0.480311	0.0325307
$219$	0	0
$220$	0	0
$221$	−5.77726	−0.388620
$222$	0	0
$223$	10.2180	0.684245	0.342123	−	0.939655i	$-0.388854\pi$
$223$	10.2180	0.684245	0.342123	+	0.939655i	$0.388854\pi$
$224$	3.98210	0.266066
$225$	0	0
$226$	0.351526	0.0233832
$227$	−5.83291	−0.387144	−0.193572	−	0.981086i	$-0.562007\pi$
$227$	−5.83291	−0.387144	−0.193572	+	0.981086i	$0.562007\pi$
$228$	0	0
$229$	5.87061	0.387941	0.193970	−	0.981007i	$-0.437863\pi$
$229$	5.87061	0.387941	0.193970	+	0.981007i	$0.437863\pi$
$230$	0	0
$231$	0	0
$232$	0.625973	0.0410971
$233$	−0.839462	−0.0549950	−0.0274975	−	0.999622i	$-0.508754\pi$
$233$	−0.839462	−0.0549950	−0.0274975	+	0.999622i	$0.508754\pi$
$234$	0	0
$235$	0	0
$236$	−1.11600	−0.0726452
$237$	0	0
$238$	2.36347	0.153201
$239$	21.3296	1.37970	0.689850	−	0.723953i	$-0.257677\pi$
$239$	21.3296	1.37970	0.689850	+	0.723953i	$0.257677\pi$
$240$	0	0
$241$	−20.7037	−1.33364	−0.666820	−	0.745219i	$-0.732345\pi$
$241$	−20.7037	−1.33364	−0.666820	+	0.745219i	$0.732345\pi$
$242$	0.693149	0.0445573
$243$	0	0
$244$	16.9151	1.08288
$245$	0	0
$246$	0	0
$247$	1.30350	0.0829400
$248$	6.83946	0.434306
$249$	0	0
$250$	0	0
$251$	−23.7290	−1.49776	−0.748881	−	0.662705i	$-0.769408\pi$
$251$	−23.7290	−1.49776	−0.748881	+	0.662705i	$0.769408\pi$
$252$	0	0
$253$	3.65773	0.229959
$254$	−4.47371	−0.280706
$255$	0	0
$256$	9.63234	0.602021
$257$	−18.2481	−1.13828	−0.569142	−	0.822239i	$-0.692725\pi$
$257$	−18.2481	−1.13828	−0.569142	+	0.822239i	$0.692725\pi$
$258$	0	0
$259$	−2.46242	−0.153007
$260$	0	0
$261$	0	0
$262$	−3.80915	−0.235330
$263$	25.8718	1.59532	0.797662	−	0.603104i	$-0.206070\pi$
$263$	25.8718	1.59532	0.797662	+	0.603104i	$0.206070\pi$
$264$	0	0
$265$	0	0
$266$	−0.533263	−0.0326964
$267$	0	0
$268$	−6.14073	−0.375105
$269$	−7.34497	−0.447831	−0.223915	−	0.974609i	$-0.571884\pi$
$269$	−7.34497	−0.447831	−0.223915	+	0.974609i	$0.571884\pi$
$270$	0	0
$271$	30.2278	1.83621	0.918105	−	0.396338i	$-0.129719\pi$
$271$	30.2278	1.83621	0.918105	+	0.396338i	$0.129719\pi$
$272$	23.5176	1.42596
$273$	0	0
$274$	−3.84137	−0.232066
$275$	0	0
$276$	0	0
$277$	−2.50983	−0.150801	−0.0754005	−	0.997153i	$-0.524024\pi$
$277$	−2.50983	−0.150801	−0.0754005	+	0.997153i	$0.524024\pi$
$278$	−4.11293	−0.246677
$279$	0	0
$280$	0	0
$281$	10.4836	0.625400	0.312700	−	0.949852i	$-0.398767\pi$
$281$	10.4836	0.625400	0.312700	+	0.949852i	$0.398767\pi$
$282$	0	0
$283$	−3.83946	−0.228232	−0.114116	−	0.993467i	$-0.536404\pi$
$283$	−3.83946	−0.228232	−0.114116	+	0.993467i	$0.536404\pi$
$284$	−24.0890	−1.42942
$285$	0	0
$286$	0.915756	0.0541498
$287$	8.62626	0.509192
$288$	0	0
$289$	28.2050	1.65912
$290$	0	0
$291$	0	0
$292$	16.5196	0.966739
$293$	0.544425	0.0318056	0.0159028	−	0.999874i	$-0.494938\pi$
$293$	0.544425	0.0318056	0.0159028	+	0.999874i	$0.494938\pi$
$294$	0	0
$295$	0	0
$296$	2.32824	0.135326
$297$	0	0
$298$	1.90683	0.110460
$299$	0.859268	0.0496927
$300$	0	0
$301$	12.1611	0.700957
$302$	0.762426	0.0438727
$303$	0	0
$304$	−5.30620	−0.304331
$305$	0	0
$306$	0	0
$307$	−16.1850	−0.923729	−0.461865	−	0.886950i	$-0.652819\pi$
$307$	−16.1850	−0.923729	−0.461865	+	0.886950i	$0.652819\pi$
$308$	8.45125	0.481555
$309$	0	0
$310$	0	0
$311$	7.51923	0.426376	0.213188	−	0.977011i	$-0.431615\pi$
$311$	7.51923	0.426376	0.213188	+	0.977011i	$0.431615\pi$
$312$	0	0
$313$	−25.2475	−1.42707	−0.713536	−	0.700619i	$-0.752907\pi$
$313$	−25.2475	−1.42707	−0.713536	+	0.700619i	$0.752907\pi$
$314$	−2.56809	−0.144926
$315$	0	0
$316$	−0.134176	−0.00754799
$317$	−13.4173	−0.753589	−0.376794	−	0.926297i	$-0.622974\pi$
$317$	−13.4173	−0.753589	−0.376794	+	0.926297i	$0.622974\pi$
$318$	0	0
$319$	2.00716	0.112380
$320$	0	0
$321$	0	0
$322$	−0.351526	−0.0195898
$323$	−10.1995	−0.567513
$324$	0	0
$325$	0	0
$326$	2.96458	0.164193
$327$	0	0
$328$	−8.15622	−0.450352
$329$	−11.0638	−0.609969
$330$	0	0
$331$	−24.3748	−1.33976	−0.669880	−	0.742470i	$-0.733654\pi$
$331$	−24.3748	−1.33976	−0.669880	+	0.742470i	$0.733654\pi$
$332$	−12.9116	−0.708615
$333$	0	0
$334$	−3.69009	−0.201913
$335$	0	0
$336$	0	0
$337$	−19.2454	−1.04836	−0.524182	−	0.851607i	$-0.675629\pi$
$337$	−19.2454	−1.04836	−0.524182	+	0.851607i	$0.675629\pi$
$338$	−3.57264	−0.194326
$339$	0	0
$340$	0	0
$341$	21.9305	1.18761
$342$	0	0
$343$	−15.1345	−0.817186
$344$	−11.4985	−0.619957
$345$	0	0
$346$	−4.57157	−0.245769
$347$	−7.51044	−0.403181	−0.201591	−	0.979470i	$-0.564611\pi$
$347$	−7.51044	−0.403181	−0.201591	+	0.979470i	$0.564611\pi$
$348$	0	0
$349$	12.3208	0.659520	0.329760	−	0.944065i	$-0.393032\pi$
$349$	12.3208	0.659520	0.329760	+	0.944065i	$0.393032\pi$
$350$	0	0
$351$	0	0
$352$	−12.0728	−0.643480
$353$	−11.7333	−0.624502	−0.312251	−	0.950000i	$-0.601083\pi$
$353$	−11.7333	−0.624502	−0.312251	+	0.950000i	$0.601083\pi$
$354$	0	0
$355$	0	0
$356$	9.50644	0.503840
$357$	0	0
$358$	0.698920	0.0369391
$359$	18.6879	0.986307	0.493154	−	0.869942i	$-0.335844\pi$
$359$	18.6879	0.986307	0.493154	+	0.869942i	$0.335844\pi$
$360$	0	0
$361$	−16.6987	−0.878881
$362$	−2.87615	−0.151167
$363$	0	0
$364$	1.98536	0.104061
$365$	0	0
$366$	0	0
$367$	23.0103	1.20113	0.600564	−	0.799576i	$-0.294943\pi$
$367$	23.0103	1.20113	0.600564	+	0.799576i	$0.294943\pi$
$368$	−3.49784	−0.182337
$369$	0	0
$370$	0	0
$371$	6.48956	0.336921
$372$	0	0
$373$	−35.7402	−1.85056	−0.925278	−	0.379290i	$-0.876168\pi$
$373$	−35.7402	−1.85056	−0.925278	+	0.379290i	$0.876168\pi$
$374$	−7.16547	−0.370518
$375$	0	0
$376$	10.4610	0.539483
$377$	0.471520	0.0242845
$378$	0	0
$379$	9.65178	0.495779	0.247889	−	0.968788i	$-0.420263\pi$
$379$	9.65178	0.495779	0.247889	+	0.968788i	$0.420263\pi$
$380$	0	0
$381$	0	0
$382$	2.79892	0.143205
$383$	−25.0954	−1.28232	−0.641158	−	0.767409i	$-0.721546\pi$
$383$	−25.0954	−1.28232	−0.641158	+	0.767409i	$0.721546\pi$
$384$	0	0
$385$	0	0
$386$	−7.27571	−0.370324
$387$	0	0
$388$	21.7134	1.10233
$389$	−16.8259	−0.853106	−0.426553	−	0.904462i	$-0.640272\pi$
$389$	−16.8259	−0.853106	−0.426553	+	0.904462i	$0.640272\pi$
$390$	0	0
$391$	−6.72347	−0.340020
$392$	6.32470	0.319446
$393$	0	0
$394$	−2.89121	−0.145657
$395$	0	0
$396$	0	0
$397$	−27.8080	−1.39564	−0.697822	−	0.716272i	$-0.745847\pi$
$397$	−27.8080	−1.39564	−0.697822	+	0.716272i	$0.745847\pi$
$398$	2.04487	0.102500
$399$	0	0
$400$	0	0
$401$	−1.46483	−0.0731500	−0.0365750	−	0.999331i	$-0.511645\pi$
$401$	−1.46483	−0.0731500	−0.0365750	+	0.999331i	$0.511645\pi$
$402$	0	0
$403$	5.15189	0.256634
$404$	6.60124	0.328424
$405$	0	0
$406$	−0.192899	−0.00957340
$407$	7.46544	0.370048
$408$	0	0
$409$	0.514906	0.0254605	0.0127302	−	0.999919i	$-0.495948\pi$
$409$	0.514906	0.0254605	0.0127302	+	0.999919i	$0.495948\pi$
$410$	0	0
$411$	0	0
$412$	17.4972	0.862025
$413$	0.703052	0.0345949
$414$	0	0
$415$	0	0
$416$	−2.83612	−0.139052
$417$	0	0
$418$	1.61672	0.0790764
$419$	9.65387	0.471622	0.235811	−	0.971799i	$-0.424225\pi$
$419$	9.65387	0.471622	0.235811	+	0.971799i	$0.424225\pi$
$420$	0	0
$421$	14.2788	0.695905	0.347952	−	0.937512i	$-0.386877\pi$
$421$	14.2788	0.695905	0.347952	+	0.937512i	$0.386877\pi$
$422$	2.16979	0.105624
$423$	0	0
$424$	−6.13595	−0.297988
$425$	0	0
$426$	0	0
$427$	−10.6561	−0.515685
$428$	−17.7059	−0.855847
$429$	0	0
$430$	0	0
$431$	−5.71198	−0.275136	−0.137568	−	0.990492i	$-0.543929\pi$
$431$	−5.71198	−0.275136	−0.137568	+	0.990492i	$0.543929\pi$
$432$	0	0
$433$	30.3302	1.45758	0.728789	−	0.684738i	$-0.240083\pi$
$433$	30.3302	1.45758	0.728789	+	0.684738i	$0.240083\pi$
$434$	−2.10764	−0.101170
$435$	0	0
$436$	−3.15700	−0.151193
$437$	1.51699	0.0725676
$438$	0	0
$439$	−19.7579	−0.942994	−0.471497	−	0.881868i	$-0.656286\pi$
$439$	−19.7579	−0.942994	−0.471497	+	0.881868i	$0.656286\pi$
$440$	0	0
$441$	0	0
$442$	−1.68330	−0.0800665
$443$	16.0643	0.763236	0.381618	−	0.924320i	$-0.375367\pi$
$443$	16.0643	0.763236	0.381618	+	0.924320i	$0.375367\pi$
$444$	0	0
$445$	0	0
$446$	2.97717	0.140973
$447$	0	0
$448$	−7.27984	−0.343940
$449$	−28.5881	−1.34916	−0.674579	−	0.738203i	$-0.735675\pi$
$449$	−28.5881	−1.34916	−0.674579	+	0.738203i	$0.735675\pi$
$450$	0	0
$451$	−26.1527	−1.23148
$452$	−2.31052	−0.108678
$453$	0	0
$454$	−1.69952	−0.0797622
$455$	0	0
$456$	0	0
$457$	−10.1209	−0.473437	−0.236718	−	0.971578i	$-0.576072\pi$
$457$	−10.1209	−0.473437	−0.236718	+	0.971578i	$0.576072\pi$
$458$	1.71050	0.0799264
$459$	0	0
$460$	0	0
$461$	−15.6931	−0.730901	−0.365450	−	0.930831i	$-0.619085\pi$
$461$	−15.6931	−0.730901	−0.365450	+	0.930831i	$0.619085\pi$
$462$	0	0
$463$	−12.9124	−0.600089	−0.300044	−	0.953925i	$-0.597001\pi$
$463$	−12.9124	−0.600089	−0.300044	+	0.953925i	$0.597001\pi$
$464$	−1.91943	−0.0891072
$465$	0	0
$466$	−0.244592	−0.0113305
$467$	−7.10196	−0.328640	−0.164320	−	0.986407i	$-0.552543\pi$
$467$	−7.10196	−0.328640	−0.164320	+	0.986407i	$0.552543\pi$
$468$	0	0
$469$	3.86852	0.178632
$470$	0	0
$471$	0	0
$472$	−0.664743	−0.0305973
$473$	−36.8696	−1.69527
$474$	0	0
$475$	0	0
$476$	−15.5347	−0.712032
$477$	0	0
$478$	6.21475	0.284256
$479$	−16.7758	−0.766506	−0.383253	−	0.923643i	$-0.625196\pi$
$479$	−16.7758	−0.766506	−0.383253	+	0.923643i	$0.625196\pi$
$480$	0	0
$481$	1.75377	0.0799650
$482$	−6.03236	−0.274767
$483$	0	0
$484$	−4.55595	−0.207089
$485$	0	0
$486$	0	0
$487$	−7.02488	−0.318328	−0.159164	−	0.987252i	$-0.550880\pi$
$487$	−7.02488	−0.318328	−0.159164	+	0.987252i	$0.550880\pi$
$488$	10.0755	0.456094
$489$	0	0
$490$	0	0
$491$	−11.1500	−0.503192	−0.251596	−	0.967832i	$-0.580955\pi$
$491$	−11.1500	−0.503192	−0.251596	+	0.967832i	$0.580955\pi$
$492$	0	0
$493$	−3.68948	−0.166166
$494$	0.379798	0.0170879
$495$	0	0
$496$	−20.9719	−0.941667
$497$	15.1755	0.680714
$498$	0	0
$499$	14.8347	0.664091	0.332046	−	0.943263i	$-0.392261\pi$
$499$	14.8347	0.664091	0.332046	+	0.943263i	$0.392261\pi$
$500$	0	0
$501$	0	0
$502$	−6.91385	−0.308580
$503$	−3.10196	−0.138310	−0.0691548	−	0.997606i	$-0.522030\pi$
$503$	−3.10196	−0.138310	−0.0691548	+	0.997606i	$0.522030\pi$
$504$	0	0
$505$	0	0
$506$	1.06574	0.0473779
$507$	0	0
$508$	29.4050	1.30463
$509$	22.3353	0.989993	0.494996	−	0.868895i	$-0.335169\pi$
$509$	22.3353	0.989993	0.494996	+	0.868895i	$0.335169\pi$
$510$	0	0
$511$	−10.4070	−0.460378
$512$	19.5252	0.862901
$513$	0	0
$514$	−5.31689	−0.234518
$515$	0	0
$516$	0	0
$517$	33.5428	1.47521
$518$	−0.717466	−0.0315237
$519$	0	0
$520$	0	0
$521$	14.2147	0.622758	0.311379	−	0.950286i	$-0.399209\pi$
$521$	14.2147	0.622758	0.311379	+	0.950286i	$0.399209\pi$
$522$	0	0
$523$	−4.09494	−0.179059	−0.0895297	−	0.995984i	$-0.528536\pi$
$523$	−4.09494	−0.179059	−0.0895297	+	0.995984i	$0.528536\pi$
$524$	25.0369	1.09374
$525$	0	0
$526$	7.53819	0.328681
$527$	−40.3117	−1.75601
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	3.50505	0.151963
$533$	−6.14375	−0.266115
$534$	0	0
$535$	0	0
$536$	−3.65773	−0.157990
$537$	0	0
$538$	−2.14008	−0.0922654
$539$	20.2800	0.873521
$540$	0	0
$541$	33.6902	1.44846	0.724228	−	0.689560i	$-0.242196\pi$
$541$	33.6902	1.44846	0.724228	+	0.689560i	$0.242196\pi$
$542$	8.80739	0.378310
$543$	0	0
$544$	22.1916	0.951457
$545$	0	0
$546$	0	0
$547$	32.9358	1.40823	0.704117	−	0.710084i	$-0.251343\pi$
$547$	32.9358	1.40823	0.704117	+	0.710084i	$0.251343\pi$
$548$	25.2487	1.07857
$549$	0	0
$550$	0	0
$551$	0.832445	0.0354633
$552$	0	0
$553$	0.0845278	0.00359449
$554$	−0.731281	−0.0310692
$555$	0	0
$556$	27.0336	1.14648
$557$	20.8337	0.882751	0.441375	−	0.897323i	$-0.354491\pi$
$557$	20.8337	0.882751	0.441375	+	0.897323i	$0.354491\pi$
$558$	0	0
$559$	−8.66135	−0.366336
$560$	0	0
$561$	0	0
$562$	3.05458	0.128850
$563$	−30.4368	−1.28276	−0.641380	−	0.767223i	$-0.721638\pi$
$563$	−30.4368	−1.28276	−0.641380	+	0.767223i	$0.721638\pi$
$564$	0	0
$565$	0	0
$566$	−1.11869	−0.0470221
$567$	0	0
$568$	−14.3486	−0.602054
$569$	−39.4801	−1.65509	−0.827545	−	0.561399i	$-0.810263\pi$
$569$	−39.4801	−1.65509	−0.827545	+	0.561399i	$0.810263\pi$
$570$	0	0
$571$	27.0301	1.13118	0.565588	−	0.824688i	$-0.308649\pi$
$571$	27.0301	1.13118	0.565588	+	0.824688i	$0.308649\pi$
$572$	−6.01911	−0.251672
$573$	0	0
$574$	2.51341	0.104908
$575$	0	0
$576$	0	0
$577$	4.68818	0.195171	0.0975857	−	0.995227i	$-0.468888\pi$
$577$	4.68818	0.195171	0.0975857	+	0.995227i	$0.468888\pi$
$578$	8.21800	0.341824
$579$	0	0
$580$	0	0
$581$	8.13400	0.337455
$582$	0	0
$583$	−19.6747	−0.814845
$584$	9.83992	0.407179
$585$	0	0
$586$	0.158627	0.00655284
$587$	−9.61718	−0.396944	−0.198472	−	0.980107i	$-0.563598\pi$
$587$	−9.61718	−0.396944	−0.198472	+	0.980107i	$0.563598\pi$
$588$	0	0
$589$	9.09541	0.374770
$590$	0	0
$591$	0	0
$592$	−7.13911	−0.293415
$593$	−8.68902	−0.356815	−0.178408	−	0.983957i	$-0.557095\pi$
$593$	−8.68902	−0.356815	−0.178408	+	0.983957i	$0.557095\pi$
$594$	0	0
$595$	0	0
$596$	−12.5333	−0.513382
$597$	0	0
$598$	0.250362	0.0102381
$599$	−11.7581	−0.480421	−0.240211	−	0.970721i	$-0.577217\pi$
$599$	−11.7581	−0.480421	−0.240211	+	0.970721i	$0.577217\pi$
$600$	0	0
$601$	37.9388	1.54756	0.773778	−	0.633457i	$-0.218365\pi$
$601$	37.9388	1.54756	0.773778	+	0.633457i	$0.218365\pi$
$602$	3.54336	0.144416
$603$	0	0
$604$	−5.01130	−0.203907
$605$	0	0
$606$	0	0
$607$	9.01418	0.365874	0.182937	−	0.983125i	$-0.441440\pi$
$607$	9.01418	0.365874	0.182937	+	0.983125i	$0.441440\pi$
$608$	−5.00702	−0.203061
$609$	0	0
$610$	0	0
$611$	7.87983	0.318784
$612$	0	0
$613$	42.5846	1.71997	0.859987	−	0.510316i	$-0.170472\pi$
$613$	42.5846	1.71997	0.859987	+	0.510316i	$0.170472\pi$
$614$	−4.71578	−0.190314
$615$	0	0
$616$	5.03399	0.202825
$617$	12.6190	0.508020	0.254010	−	0.967202i	$-0.418250\pi$
$617$	12.6190	0.508020	0.254010	+	0.967202i	$0.418250\pi$
$618$	0	0
$619$	−26.0638	−1.04759	−0.523796	−	0.851844i	$-0.675485\pi$
$619$	−26.0638	−1.04759	−0.523796	+	0.851844i	$0.675485\pi$
$620$	0	0
$621$	0	0
$622$	2.19085	0.0878452
$623$	−5.98884	−0.239938
$624$	0	0
$625$	0	0
$626$	−7.35628	−0.294016
$627$	0	0
$628$	16.8796	0.673570
$629$	−13.7226	−0.547157
$630$	0	0
$631$	−46.1083	−1.83554	−0.917771	−	0.397110i	$-0.870013\pi$
$631$	−46.1083	−1.83554	−0.917771	+	0.397110i	$0.870013\pi$
$632$	−0.0799219	−0.00317912
$633$	0	0
$634$	−3.90935	−0.155260
$635$	0	0
$636$	0	0
$637$	4.76415	0.188762
$638$	0.584821	0.0231533
$639$	0	0
$640$	0	0
$641$	−19.3319	−0.763563	−0.381781	−	0.924253i	$-0.624689\pi$
$641$	−19.3319	−0.763563	−0.381781	+	0.924253i	$0.624689\pi$
$642$	0	0
$643$	−2.27891	−0.0898713	−0.0449356	−	0.998990i	$-0.514308\pi$
$643$	−2.27891	−0.0898713	−0.0449356	+	0.998990i	$0.514308\pi$
$644$	2.31052	0.0910473
$645$	0	0
$646$	−2.97178	−0.116923
$647$	47.5054	1.86763	0.933815	−	0.357755i	$-0.116458\pi$
$647$	47.5054	1.86763	0.933815	+	0.357755i	$0.116458\pi$
$648$	0	0
$649$	−2.13148	−0.0836679
$650$	0	0
$651$	0	0
$652$	−19.4857	−0.763117
$653$	2.73226	0.106921	0.0534607	−	0.998570i	$-0.482975\pi$
$653$	2.73226	0.106921	0.0534607	+	0.998570i	$0.482975\pi$
$654$	0	0
$655$	0	0
$656$	25.0095	0.976457
$657$	0	0
$658$	−3.22363	−0.125670
$659$	12.1375	0.472809	0.236405	−	0.971655i	$-0.424031\pi$
$659$	12.1375	0.472809	0.236405	+	0.971655i	$0.424031\pi$
$660$	0	0
$661$	41.1992	1.60246	0.801232	−	0.598354i	$-0.204178\pi$
$661$	41.1992	1.60246	0.801232	+	0.598354i	$0.204178\pi$
$662$	−7.10200	−0.276027
$663$	0	0
$664$	−7.69078	−0.298460
$665$	0	0
$666$	0	0
$667$	0.548747	0.0212476
$668$	24.2543	0.938428
$669$	0	0
$670$	0	0
$671$	32.3067	1.24718
$672$	0	0
$673$	−24.2175	−0.933516	−0.466758	−	0.884385i	$-0.654578\pi$
$673$	−24.2175	−0.933516	−0.466758	+	0.884385i	$0.654578\pi$
$674$	−5.60747	−0.215992
$675$	0	0
$676$	23.4824	0.903168
$677$	48.1512	1.85060	0.925300	−	0.379235i	$-0.123813\pi$
$677$	48.1512	1.85060	0.925300	+	0.379235i	$0.123813\pi$
$678$	0	0
$679$	−13.6789	−0.524949
$680$	0	0
$681$	0	0
$682$	6.38984	0.244679
$683$	12.1351	0.464337	0.232169	−	0.972676i	$-0.425418\pi$
$683$	12.1351	0.464337	0.232169	+	0.972676i	$0.425418\pi$
$684$	0	0
$685$	0	0
$686$	−4.40969	−0.168363
$687$	0	0
$688$	35.2579	1.34420
$689$	−4.62197	−0.176083
$690$	0	0
$691$	−13.8289	−0.526076	−0.263038	−	0.964785i	$-0.584725\pi$
$691$	−13.8289	−0.526076	−0.263038	+	0.964785i	$0.584725\pi$
$692$	30.0482	1.14226
$693$	0	0
$694$	−2.18829	−0.0830665
$695$	0	0
$696$	0	0
$697$	48.0727	1.82088
$698$	3.58989	0.135879
$699$	0	0
$700$	0	0
$701$	15.2263	0.575089	0.287544	−	0.957767i	$-0.407161\pi$
$701$	15.2263	0.575089	0.287544	+	0.957767i	$0.407161\pi$
$702$	0	0
$703$	3.09619	0.116775
$704$	22.0707	0.831820
$705$	0	0
$706$	−3.41870	−0.128665
$707$	−4.15863	−0.156401
$708$	0	0
$709$	15.2317	0.572037	0.286019	−	0.958224i	$-0.407668\pi$
$709$	15.2317	0.572037	0.286019	+	0.958224i	$0.407668\pi$
$710$	0	0
$711$	0	0
$712$	5.66251	0.212211
$713$	5.99568	0.224540
$714$	0	0
$715$	0	0
$716$	−4.59388	−0.171681
$717$	0	0
$718$	5.44502	0.203206
$719$	−4.58943	−0.171157	−0.0855784	−	0.996331i	$-0.527274\pi$
$719$	−4.58943	−0.171157	−0.0855784	+	0.996331i	$0.527274\pi$
$720$	0	0
$721$	−11.0228	−0.410511
$722$	−4.86546	−0.181074
$723$	0	0
$724$	18.9044	0.702577
$725$	0	0
$726$	0	0
$727$	−21.2414	−0.787800	−0.393900	−	0.919153i	$-0.628874\pi$
$727$	−21.2414	−0.787800	−0.393900	+	0.919153i	$0.628874\pi$
$728$	1.18258	0.0438292
$729$	0	0
$730$	0	0
$731$	67.7720	2.50664
$732$	0	0
$733$	−40.5297	−1.49700	−0.748499	−	0.663136i	$-0.769225\pi$
$733$	−40.5297	−1.49700	−0.748499	+	0.663136i	$0.769225\pi$
$734$	6.70445	0.247466
$735$	0	0
$736$	−3.30062	−0.121662
$737$	−11.7284	−0.432021
$738$	0	0
$739$	30.6476	1.12739	0.563695	−	0.825983i	$-0.309379\pi$
$739$	30.6476	1.12739	0.563695	+	0.825983i	$0.309379\pi$
$740$	0	0
$741$	0	0
$742$	1.89084	0.0694151
$743$	1.01357	0.0371844	0.0185922	−	0.999827i	$-0.494082\pi$
$743$	1.01357	0.0371844	0.0185922	+	0.999827i	$0.494082\pi$
$744$	0	0
$745$	0	0
$746$	−10.4135	−0.381265
$747$	0	0
$748$	47.0974	1.72205
$749$	11.1543	0.407569
$750$	0	0
$751$	−21.0644	−0.768652	−0.384326	−	0.923197i	$-0.625566\pi$
$751$	−21.0644	−0.768652	−0.384326	+	0.923197i	$0.625566\pi$
$752$	−32.0766	−1.16971
$753$	0	0
$754$	0.137385	0.00500328
$755$	0	0
$756$	0	0
$757$	13.2109	0.480160	0.240080	−	0.970753i	$-0.422826\pi$
$757$	13.2109	0.480160	0.240080	+	0.970753i	$0.422826\pi$
$758$	2.81221	0.102144
$759$	0	0
$760$	0	0
$761$	11.9759	0.434125	0.217063	−	0.976158i	$-0.430352\pi$
$761$	11.9759	0.434125	0.217063	+	0.976158i	$0.430352\pi$
$762$	0	0
$763$	1.98884	0.0720008
$764$	−18.3968	−0.665574
$765$	0	0
$766$	−7.31197	−0.264192
$767$	−0.500724	−0.0180801
$768$	0	0
$769$	39.6569	1.43006	0.715031	−	0.699092i	$-0.246412\pi$
$769$	39.6569	1.43006	0.715031	+	0.699092i	$0.246412\pi$
$770$	0	0
$771$	0	0
$772$	47.8220	1.72115
$773$	16.0305	0.576575	0.288288	−	0.957544i	$-0.406914\pi$
$773$	16.0305	0.576575	0.288288	+	0.957544i	$0.406914\pi$
$774$	0	0
$775$	0	0
$776$	12.9336	0.464288
$777$	0	0
$778$	−4.90251	−0.175763
$779$	−10.8465	−0.388615
$780$	0	0
$781$	−46.0084	−1.64631
$782$	−1.95900	−0.0700535
$783$	0	0
$784$	−19.3935	−0.692625
$785$	0	0
$786$	0	0
$787$	39.0352	1.39145	0.695727	−	0.718306i	$-0.255082\pi$
$787$	39.0352	1.39145	0.695727	+	0.718306i	$0.255082\pi$
$788$	19.0034	0.676969
$789$	0	0
$790$	0	0
$791$	1.45558	0.0517543
$792$	0	0
$793$	7.58944	0.269509
$794$	−8.10233	−0.287541
$795$	0	0
$796$	−13.4406	−0.476388
$797$	−3.37687	−0.119615	−0.0598074	−	0.998210i	$-0.519049\pi$
$797$	−3.37687	−0.119615	−0.0598074	+	0.998210i	$0.519049\pi$
$798$	0	0
$799$	−61.6569	−2.18126
$800$	0	0
$801$	0	0
$802$	−0.426802	−0.0150709
$803$	31.5514	1.11343
$804$	0	0
$805$	0	0
$806$	1.50109	0.0528737
$807$	0	0
$808$	3.93203	0.138328
$809$	49.3406	1.73472	0.867361	−	0.497680i	$-0.165814\pi$
$809$	49.3406	1.73472	0.867361	+	0.497680i	$0.165814\pi$
$810$	0	0
$811$	18.3582	0.644645	0.322322	−	0.946630i	$-0.395537\pi$
$811$	18.3582	0.644645	0.322322	+	0.946630i	$0.395537\pi$
$812$	1.26789	0.0444942
$813$	0	0
$814$	2.17518	0.0762400
$815$	0	0
$816$	0	0
$817$	−15.2912	−0.534971
$818$	0.150027	0.00524556
$819$	0	0
$820$	0	0
$821$	−19.4610	−0.679192	−0.339596	−	0.940571i	$-0.610290\pi$
$821$	−19.4610	−0.679192	−0.339596	+	0.940571i	$0.610290\pi$
$822$	0	0
$823$	47.2973	1.64868	0.824341	−	0.566094i	$-0.191546\pi$
$823$	47.2973	1.64868	0.824341	+	0.566094i	$0.191546\pi$
$824$	10.4222	0.363074
$825$	0	0
$826$	0.204846	0.00712751
$827$	26.2901	0.914197	0.457098	−	0.889416i	$-0.348889\pi$
$827$	26.2901	0.914197	0.457098	+	0.889416i	$0.348889\pi$
$828$	0	0
$829$	−20.0140	−0.695116	−0.347558	−	0.937658i	$-0.612989\pi$
$829$	−20.0140	−0.695116	−0.347558	+	0.937658i	$0.612989\pi$
$830$	0	0
$831$	0	0
$832$	5.18481	0.179751
$833$	−37.2778	−1.29160
$834$	0	0
$835$	0	0
$836$	−10.6264	−0.367523
$837$	0	0
$838$	2.81282	0.0971671
$839$	40.7210	1.40585	0.702923	−	0.711266i	$-0.251878\pi$
$839$	40.7210	1.40585	0.702923	+	0.711266i	$0.251878\pi$
$840$	0	0
$841$	−28.6989	−0.989616
$842$	4.16036	0.143375
$843$	0	0
$844$	−14.2617	−0.490907
$845$	0	0
$846$	0	0
$847$	2.87015	0.0986194
$848$	18.8147	0.646100
$849$	0	0
$850$	0	0
$851$	2.04100	0.0699647
$852$	0	0
$853$	−5.34390	−0.182972	−0.0914858	−	0.995806i	$-0.529162\pi$
$853$	−5.34390	−0.182972	−0.0914858	+	0.995806i	$0.529162\pi$
$854$	−3.10483	−0.106245
$855$	0	0
$856$	−10.5465	−0.360472
$857$	48.4603	1.65537	0.827686	−	0.561192i	$-0.189657\pi$
$857$	48.4603	1.65537	0.827686	+	0.561192i	$0.189657\pi$
$858$	0	0
$859$	−51.3123	−1.75075	−0.875377	−	0.483440i	$-0.839387\pi$
$859$	−51.3123	−1.75075	−0.875377	+	0.483440i	$0.839387\pi$
$860$	0	0
$861$	0	0
$862$	−1.66428	−0.0566857
$863$	24.5507	0.835714	0.417857	−	0.908513i	$-0.362781\pi$
$863$	24.5507	0.835714	0.417857	+	0.908513i	$0.362781\pi$
$864$	0	0
$865$	0	0
$866$	8.83723	0.300301
$867$	0	0
$868$	13.8531	0.470206
$869$	−0.256267	−0.00869327
$870$	0	0
$871$	−2.75522	−0.0933570
$872$	−1.88047	−0.0636807
$873$	0	0
$874$	0.442002	0.0149509
$875$	0	0
$876$	0	0
$877$	−30.1400	−1.01776	−0.508878	−	0.860838i	$-0.669940\pi$
$877$	−30.1400	−1.01776	−0.508878	+	0.860838i	$0.669940\pi$
$878$	−5.75680	−0.194283
$879$	0	0
$880$	0	0
$881$	−36.4729	−1.22880	−0.614401	−	0.788994i	$-0.710602\pi$
$881$	−36.4729	−1.22880	−0.614401	+	0.788994i	$0.710602\pi$
$882$	0	0
$883$	2.17426	0.0731696	0.0365848	−	0.999331i	$-0.488352\pi$
$883$	2.17426	0.0731696	0.0365848	+	0.999331i	$0.488352\pi$
$884$	11.0641	0.372125
$885$	0	0
$886$	4.68059	0.157248
$887$	−16.1463	−0.542139	−0.271069	−	0.962560i	$-0.587377\pi$
$887$	−16.1463	−0.542139	−0.271069	+	0.962560i	$0.587377\pi$
$888$	0	0
$889$	−18.5244	−0.621290
$890$	0	0
$891$	0	0
$892$	−19.5685	−0.655201
$893$	13.9114	0.465528
$894$	0	0
$895$	0	0
$896$	−10.0853	−0.336927
$897$	0	0
$898$	−8.32963	−0.277963
$899$	3.29011	0.109731
$900$	0	0
$901$	36.1652	1.20484
$902$	−7.62002	−0.253719
$903$	0	0
$904$	−1.37626	−0.0457738
$905$	0	0
$906$	0	0
$907$	4.74475	0.157547	0.0787734	−	0.996893i	$-0.474900\pi$
$907$	4.74475	0.157547	0.0787734	+	0.996893i	$0.474900\pi$
$908$	11.1706	0.370710
$909$	0	0
$910$	0	0
$911$	−38.8650	−1.28765	−0.643827	−	0.765171i	$-0.722654\pi$
$911$	−38.8650	−1.28765	−0.643827	+	0.765171i	$0.722654\pi$
$912$	0	0
$913$	−24.6603	−0.816136
$914$	−2.94890	−0.0975410
$915$	0	0
$916$	−11.2428	−0.371474
$917$	−15.7727	−0.520859
$918$	0	0
$919$	40.5516	1.33767	0.668837	−	0.743409i	$-0.266793\pi$
$919$	40.5516	1.33767	0.668837	+	0.743409i	$0.266793\pi$
$920$	0	0
$921$	0	0
$922$	−4.57245	−0.150586
$923$	−10.8082	−0.355757
$924$	0	0
$925$	0	0
$926$	−3.76224	−0.123635
$927$	0	0
$928$	−1.81120	−0.0594557
$929$	−19.4994	−0.639755	−0.319878	−	0.947459i	$-0.603642\pi$
$929$	−19.4994	−0.639755	−0.319878	+	0.947459i	$0.603642\pi$
$930$	0	0
$931$	8.41086	0.275655
$932$	1.60766	0.0526606
$933$	0	0
$934$	−2.06928	−0.0677088
$935$	0	0
$936$	0	0
$937$	24.9361	0.814626	0.407313	−	0.913289i	$-0.366466\pi$
$937$	24.9361	0.814626	0.407313	+	0.913289i	$0.366466\pi$
$938$	1.12716	0.0368030
$939$	0	0
$940$	0	0
$941$	41.9160	1.36642	0.683211	−	0.730221i	$-0.260583\pi$
$941$	41.9160	1.36642	0.683211	+	0.730221i	$0.260583\pi$
$942$	0	0
$943$	−7.14998	−0.232836
$944$	2.03831	0.0663413
$945$	0	0
$946$	−10.7426	−0.349271
$947$	−29.3006	−0.952141	−0.476070	−	0.879407i	$-0.657939\pi$
$947$	−29.3006	−0.952141	−0.476070	+	0.879407i	$0.657939\pi$
$948$	0	0
$949$	7.41202	0.240604
$950$	0	0
$951$	0	0
$952$	−9.25325	−0.299899
$953$	−24.9759	−0.809048	−0.404524	−	0.914527i	$-0.632563\pi$
$953$	−24.9759	−0.809048	−0.404524	+	0.914527i	$0.632563\pi$
$954$	0	0
$955$	0	0
$956$	−40.8485	−1.32113
$957$	0	0
$958$	−4.88792	−0.157921
$959$	−15.9061	−0.513635
$960$	0	0
$961$	4.94816	0.159618
$962$	0.510990	0.0164750
$963$	0	0
$964$	39.6497	1.27703
$965$	0	0
$966$	0	0
$967$	−0.405142	−0.0130285	−0.00651424	−	0.999979i	$-0.502074\pi$
$967$	−0.405142	−0.0130285	−0.00651424	+	0.999979i	$0.502074\pi$
$968$	−2.71375	−0.0872233
$969$	0	0
$970$	0	0
$971$	−38.8032	−1.24525	−0.622627	−	0.782519i	$-0.713935\pi$
$971$	−38.8032	−1.24525	−0.622627	+	0.782519i	$0.713935\pi$
$972$	0	0
$973$	−17.0306	−0.545975
$974$	−2.04682	−0.0655843
$975$	0	0
$976$	−30.8945	−0.988908
$977$	−39.9940	−1.27952	−0.639760	−	0.768575i	$-0.720966\pi$
$977$	−39.9940	−1.27952	−0.639760	+	0.768575i	$0.720966\pi$
$978$	0	0
$979$	18.1567	0.580290
$980$	0	0
$981$	0	0
$982$	−3.24874	−0.103671
$983$	21.3563	0.681160	0.340580	−	0.940216i	$-0.389377\pi$
$983$	21.3563	0.681160	0.340580	+	0.940216i	$0.389377\pi$
$984$	0	0
$985$	0	0
$986$	−1.07499	−0.0342347
$987$	0	0
$988$	−2.49635	−0.0794194
$989$	−10.0799	−0.320523
$990$	0	0
$991$	4.39444	0.139594	0.0697970	−	0.997561i	$-0.477765\pi$
$991$	4.39444	0.139594	0.0697970	+	0.997561i	$0.477765\pi$
$992$	−19.7894	−0.628316
$993$	0	0
$994$	4.42164	0.140246
$995$	0	0
$996$	0	0
$997$	−28.9420	−0.916603	−0.458302	−	0.888797i	$-0.651542\pi$
$997$	−28.9420	−0.916603	−0.458302	+	0.888797i	$0.651542\pi$
$998$	4.32233	0.136821
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5175.2.a.bv.1.3		4
3.2	odd	2		575.2.a.i.1.2		4
5.2	odd	4		1035.2.b.e.829.5		8
5.3	odd	4		1035.2.b.e.829.4		8
5.4	even	2		5175.2.a.bw.1.2		4
12.11	even	2		9200.2.a.cq.1.4		4
15.2	even	4		115.2.b.b.24.4	✓	8
15.8	even	4		115.2.b.b.24.5	yes	8
15.14	odd	2		575.2.a.j.1.3		4
60.23	odd	4		1840.2.e.d.369.8		8
60.47	odd	4		1840.2.e.d.369.1		8
60.59	even	2		9200.2.a.ck.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.b.b.24.4	✓	8	15.2	even	4
115.2.b.b.24.5	yes	8	15.8	even	4
575.2.a.i.1.2		4	3.2	odd	2
575.2.a.j.1.3		4	15.14	odd	2
1035.2.b.e.829.4		8	5.3	odd	4
1035.2.b.e.829.5		8	5.2	odd	4
1840.2.e.d.369.1		8	60.47	odd	4
1840.2.e.d.369.8		8	60.23	odd	4
5175.2.a.bv.1.3		4	1.1	even	1	trivial
5175.2.a.bw.1.2		4	5.4	even	2
9200.2.a.ck.1.1		4	60.59	even	2
9200.2.a.cq.1.4		4	12.11	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:	\( N \)	\(=\)	\( 5175 = 3^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5175.a (trivial)

\(f(q)\)	\(=\)	\(q+0.291367 q^{2} -1.91511 q^{4} +1.20647 q^{7} -1.14073 q^{8} +O(q^{10})\)	\(q+0.291367 q^{2} -1.91511 q^{4} +1.20647 q^{7} -1.14073 q^{8} -3.65773 q^{11} -0.859268 q^{13} +0.351526 q^{14} +3.49784 q^{16} +6.72347 q^{17} -1.51699 q^{19} -1.06574 q^{22} -1.00000 q^{23} -0.250362 q^{26} -2.31052 q^{28} -0.548747 q^{29} -5.99568 q^{31} +3.30062 q^{32} +1.95900 q^{34} -2.04100 q^{37} -0.442002 q^{38} +7.14998 q^{41} +10.0799 q^{43} +7.00493 q^{44} -0.291367 q^{46} -9.17040 q^{47} -5.54442 q^{49} +1.64559 q^{52} +5.37896 q^{53} -1.37626 q^{56} -0.159887 q^{58} +0.582734 q^{59} -8.83244 q^{61} -1.74694 q^{62} -6.03399 q^{64} +3.20647 q^{67} -12.8761 q^{68} +12.5784 q^{71} -8.62597 q^{73} -0.594681 q^{74} +2.90520 q^{76} -4.41294 q^{77} +0.0700619 q^{79} +2.08327 q^{82} +6.74197 q^{83} +2.93696 q^{86} +4.17248 q^{88} -4.96393 q^{89} -1.03668 q^{91} +1.91511 q^{92} -2.67195 q^{94} -11.3380 q^{97} -1.61546 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 6 q^{7} + 6 q^{8}+O(q^{10}) \) 4 * q + 2 * q^4 - 6 * q^7 + 6 * q^8	\( 4 q + 2 q^{4} - 6 q^{7} + 6 q^{8} - 2 q^{11} - 14 q^{13} + 4 q^{14} + 2 q^{16} + 14 q^{17} - 4 q^{19} - 4 q^{22} - 4 q^{23} - 6 q^{26} - 18 q^{28} - 4 q^{29} + 2 q^{32} + 14 q^{34} - 2 q^{37} - 10 q^{38} + 8 q^{41} - 4 q^{43} - 6 q^{44} + 2 q^{47} - 18 q^{52} + 4 q^{53} - 14 q^{56} - 8 q^{61} - 28 q^{62} - 20 q^{64} + 2 q^{67} + 8 q^{68} + 24 q^{71} - 18 q^{73} + 36 q^{74} - 18 q^{76} + 4 q^{77} + 24 q^{79} - 32 q^{82} - 6 q^{83} - 14 q^{86} + 10 q^{88} + 8 q^{89} + 26 q^{91} - 2 q^{92} - 42 q^{94} - 34 q^{97} - 28 q^{98}+O(q^{100}) \) 4 * q + 2 * q^4 - 6 * q^7 + 6 * q^8 - 2 * q^11 - 14 * q^13 + 4 * q^14 + 2 * q^16 + 14 * q^17 - 4 * q^19 - 4 * q^22 - 4 * q^23 - 6 * q^26 - 18 * q^28 - 4 * q^29 + 2 * q^32 + 14 * q^34 - 2 * q^37 - 10 * q^38 + 8 * q^41 - 4 * q^43 - 6 * q^44 + 2 * q^47 - 18 * q^52 + 4 * q^53 - 14 * q^56 - 8 * q^61 - 28 * q^62 - 20 * q^64 + 2 * q^67 + 8 * q^68 + 24 * q^71 - 18 * q^73 + 36 * q^74 - 18 * q^76 + 4 * q^77 + 24 * q^79 - 32 * q^82 - 6 * q^83 - 14 * q^86 + 10 * q^88 + 8 * q^89 + 26 * q^91 - 2 * q^92 - 42 * q^94 - 34 * q^97 - 28 * q^98

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