LMFDB - Embedded newform 6025.2.a.o.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6025,2,Mod(1,6025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6025 = 5^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6025.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:	$48.1098672178$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.33622 q^{2} +1.40541 q^{3} -0.214528 q^{4} -1.87793 q^{6} -0.850787 q^{7} +2.95909 q^{8} -1.02483 q^{9} +O(q^{10})$	\(q-1.33622 q^{2} +1.40541 q^{3} -0.214528 q^{4} -1.87793 q^{6} -0.850787 q^{7} +2.95909 q^{8} -1.02483 q^{9} -2.89105 q^{11} -0.301499 q^{12} -6.34069 q^{13} +1.13683 q^{14} -3.52492 q^{16} -4.06182 q^{17} +1.36940 q^{18} -4.69040 q^{19} -1.19570 q^{21} +3.86306 q^{22} -0.545361 q^{23} +4.15872 q^{24} +8.47253 q^{26} -5.65653 q^{27} +0.182518 q^{28} -9.52313 q^{29} +7.52351 q^{31} -1.20812 q^{32} -4.06310 q^{33} +5.42747 q^{34} +0.219855 q^{36} -8.64925 q^{37} +6.26738 q^{38} -8.91124 q^{39} +8.00829 q^{41} +1.59771 q^{42} -1.81253 q^{43} +0.620211 q^{44} +0.728719 q^{46} +10.4381 q^{47} -4.95395 q^{48} -6.27616 q^{49} -5.70851 q^{51} +1.36026 q^{52} +3.18773 q^{53} +7.55834 q^{54} -2.51755 q^{56} -6.59191 q^{57} +12.7250 q^{58} -9.30900 q^{59} +7.87599 q^{61} -10.0530 q^{62} +0.871914 q^{63} +8.66415 q^{64} +5.42918 q^{66} +4.22760 q^{67} +0.871375 q^{68} -0.766454 q^{69} -12.0642 q^{71} -3.03257 q^{72} +0.993013 q^{73} +11.5573 q^{74} +1.00622 q^{76} +2.45967 q^{77} +11.9073 q^{78} +12.9465 q^{79} -4.87522 q^{81} -10.7008 q^{82} +5.66443 q^{83} +0.256512 q^{84} +2.42193 q^{86} -13.3839 q^{87} -8.55486 q^{88} +9.01558 q^{89} +5.39457 q^{91} +0.116995 q^{92} +10.5736 q^{93} -13.9475 q^{94} -1.69790 q^{96} +15.5978 q^{97} +8.38630 q^{98} +2.96284 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9}+O(q^{10}) $ 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9	$ 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9} + q^{11} + 14 q^{12} + 9 q^{13} - q^{14} + 43 q^{16} + 12 q^{17} + 42 q^{18} + 2 q^{21} + 5 q^{22} + 77 q^{23} - 2 q^{24} + 2 q^{26} + 38 q^{27} + 42 q^{28} + 2 q^{29} + q^{31} + 72 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 28 q^{37} + 23 q^{38} - 2 q^{39} - 2 q^{41} + 37 q^{42} + 31 q^{43} + 3 q^{44} + 14 q^{46} + 96 q^{47} + 13 q^{48} + 40 q^{49} - 10 q^{51} + 42 q^{52} + 54 q^{53} + 4 q^{54} - 15 q^{56} + 37 q^{57} + 27 q^{58} + q^{59} + 5 q^{61} + 39 q^{62} + 70 q^{63} + 65 q^{64} - 52 q^{66} + 34 q^{67} + 52 q^{68} + 21 q^{69} - 9 q^{71} + 70 q^{72} + 25 q^{73} + 22 q^{74} - 47 q^{76} + 54 q^{77} + 58 q^{78} + 13 q^{79} + 12 q^{81} - 5 q^{82} + 63 q^{83} + 95 q^{84} - 18 q^{86} + 47 q^{87} + 13 q^{88} + 19 q^{89} - 31 q^{91} + 137 q^{92} + 52 q^{93} + 120 q^{94} - 49 q^{96} + 36 q^{97} + 64 q^{98} + 16 q^{99}+O(q^{100}) $ 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9 + q^11 + 14 * q^12 + 9 * q^13 - q^14 + 43 * q^16 + 12 * q^17 + 42 * q^18 + 2 * q^21 + 5 * q^22 + 77 * q^23 - 2 * q^24 + 2 * q^26 + 38 * q^27 + 42 * q^28 + 2 * q^29 + q^31 + 72 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 28 * q^37 + 23 * q^38 - 2 * q^39 - 2 * q^41 + 37 * q^42 + 31 * q^43 + 3 * q^44 + 14 * q^46 + 96 * q^47 + 13 * q^48 + 40 * q^49 - 10 * q^51 + 42 * q^52 + 54 * q^53 + 4 * q^54 - 15 * q^56 + 37 * q^57 + 27 * q^58 + q^59 + 5 * q^61 + 39 * q^62 + 70 * q^63 + 65 * q^64 - 52 * q^66 + 34 * q^67 + 52 * q^68 + 21 * q^69 - 9 * q^71 + 70 * q^72 + 25 * q^73 + 22 * q^74 - 47 * q^76 + 54 * q^77 + 58 * q^78 + 13 * q^79 + 12 * q^81 - 5 * q^82 + 63 * q^83 + 95 * q^84 - 18 * q^86 + 47 * q^87 + 13 * q^88 + 19 * q^89 - 31 * q^91 + 137 * q^92 + 52 * q^93 + 120 * q^94 - 49 * q^96 + 36 * q^97 + 64 * q^98 + 16 * q^99

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 $

$2$	−1.33622	−0.944847	−0.472424	−	0.881372i	$-0.656621\pi$
$2$	−1.33622	−0.944847	−0.472424	+	0.881372i	$0.656621\pi$
$3$	1.40541	0.811412	0.405706	−	0.914004i	$-0.367026\pi$
$3$	1.40541	0.811412	0.405706	+	0.914004i	$0.367026\pi$
$4$	−0.214528	−0.107264
$5$	0	0
$5$	0	0
$6$	−1.87793	−0.766660
$7$	−0.850787	−0.321567	−0.160784	−	0.986990i	$-0.551402\pi$
$7$	−0.850787	−0.321567	−0.160784	+	0.986990i	$0.551402\pi$
$8$	2.95909	1.04620
$9$	−1.02483	−0.341611
$10$	0	0
$11$	−2.89105	−0.871684	−0.435842	−	0.900023i	$-0.643549\pi$
$11$	−2.89105	−0.871684	−0.435842	+	0.900023i	$0.643549\pi$
$12$	−0.301499	−0.0870353
$13$	−6.34069	−1.75859	−0.879295	−	0.476277i	$-0.841986\pi$
$13$	−6.34069	−1.75859	−0.879295	+	0.476277i	$0.841986\pi$
$14$	1.13683	0.303832
$15$	0	0
$16$	−3.52492	−0.881230
$17$	−4.06182	−0.985136	−0.492568	−	0.870274i	$-0.663942\pi$
$17$	−4.06182	−0.985136	−0.492568	+	0.870274i	$0.663942\pi$
$18$	1.36940	0.322770
$19$	−4.69040	−1.07605	−0.538025	−	0.842929i	$-0.680829\pi$
$19$	−4.69040	−1.07605	−0.538025	+	0.842929i	$0.680829\pi$
$20$	0	0
$21$	−1.19570	−0.260923
$22$	3.86306	0.823608
$23$	−0.545361	−0.113716	−0.0568578	−	0.998382i	$-0.518108\pi$
$23$	−0.545361	−0.113716	−0.0568578	+	0.998382i	$0.518108\pi$
$24$	4.15872	0.848895
$25$	0	0
$26$	8.47253	1.66160
$27$	−5.65653	−1.08860
$28$	0.182518	0.0344926
$29$	−9.52313	−1.76840	−0.884200	−	0.467108i	$-0.845296\pi$
$29$	−9.52313	−1.76840	−0.884200	+	0.467108i	$0.845296\pi$
$30$	0	0
$31$	7.52351	1.35126	0.675631	−	0.737240i	$-0.263871\pi$
$31$	7.52351	1.35126	0.675631	+	0.737240i	$0.263871\pi$
$32$	−1.20812	−0.213567
$33$	−4.06310	−0.707295
$34$	5.42747	0.930803
$35$	0	0
$36$	0.219855	0.0366426
$37$	−8.64925	−1.42193	−0.710964	−	0.703229i	$-0.751741\pi$
$37$	−8.64925	−1.42193	−0.710964	+	0.703229i	$0.751741\pi$
$38$	6.26738	1.01670
$39$	−8.91124	−1.42694
$40$	0	0
$41$	8.00829	1.25069	0.625343	−	0.780350i	$-0.284959\pi$
$41$	8.00829	1.25069	0.625343	+	0.780350i	$0.284959\pi$
$42$	1.59771	0.246533
$43$	−1.81253	−0.276408	−0.138204	−	0.990404i	$-0.544133\pi$
$43$	−1.81253	−0.276408	−0.138204	+	0.990404i	$0.544133\pi$
$44$	0.620211	0.0935004
$45$	0	0
$46$	0.728719	0.107444
$47$	10.4381	1.52255	0.761274	−	0.648431i	$-0.224574\pi$
$47$	10.4381	1.52255	0.761274	+	0.648431i	$0.224574\pi$
$48$	−4.95395	−0.715041
$49$	−6.27616	−0.896595
$50$	0	0
$51$	−5.70851	−0.799351
$52$	1.36026	0.188634
$53$	3.18773	0.437869	0.218935	−	0.975740i	$-0.429742\pi$
$53$	3.18773	0.437869	0.218935	+	0.975740i	$0.429742\pi$
$54$	7.55834	1.02856
$55$	0	0
$56$	−2.51755	−0.336422
$57$	−6.59191	−0.873120
$58$	12.7250	1.67087
$59$	−9.30900	−1.21193	−0.605964	−	0.795492i	$-0.707212\pi$
$59$	−9.30900	−1.21193	−0.605964	+	0.795492i	$0.707212\pi$
$60$	0	0
$61$	7.87599	1.00842	0.504209	−	0.863582i	$-0.331784\pi$
$61$	7.87599	1.00842	0.504209	+	0.863582i	$0.331784\pi$
$62$	−10.0530	−1.27674
$63$	0.871914	0.109851
$64$	8.66415	1.08302
$65$	0	0
$66$	5.42918	0.668285
$67$	4.22760	0.516484	0.258242	−	0.966080i	$-0.416857\pi$
$67$	4.22760	0.516484	0.258242	+	0.966080i	$0.416857\pi$
$68$	0.871375	0.105670
$69$	−0.766454	−0.0922702
$70$	0	0
$71$	−12.0642	−1.43176	−0.715879	−	0.698225i	$-0.753974\pi$
$71$	−12.0642	−1.43176	−0.715879	+	0.698225i	$0.753974\pi$
$72$	−3.03257	−0.357392
$73$	0.993013	0.116223	0.0581117	−	0.998310i	$-0.481492\pi$
$73$	0.993013	0.116223	0.0581117	+	0.998310i	$0.481492\pi$
$74$	11.5573	1.34350
$75$	0	0
$76$	1.00622	0.115422
$77$	2.45967	0.280305
$78$	11.9073	1.34824
$79$	12.9465	1.45660	0.728298	−	0.685261i	$-0.240312\pi$
$79$	12.9465	1.45660	0.728298	+	0.685261i	$0.240312\pi$
$80$	0	0
$81$	−4.87522	−0.541691
$82$	−10.7008	−1.18171
$83$	5.66443	0.621752	0.310876	−	0.950451i	$-0.399378\pi$
$83$	5.66443	0.621752	0.310876	+	0.950451i	$0.399378\pi$
$84$	0.256512	0.0279877
$85$	0	0
$86$	2.42193	0.261164
$87$	−13.3839	−1.43490
$88$	−8.55486	−0.911952
$89$	9.01558	0.955650	0.477825	−	0.878455i	$-0.341425\pi$
$89$	9.01558	0.955650	0.477825	+	0.878455i	$0.341425\pi$
$90$	0	0
$91$	5.39457	0.565505
$92$	0.116995	0.0121976
$93$	10.5736	1.09643
$94$	−13.9475	−1.43857
$95$	0	0
$96$	−1.69790	−0.173291
$97$	15.5978	1.58372	0.791858	−	0.610705i	$-0.209114\pi$
$97$	15.5978	1.58372	0.791858	+	0.610705i	$0.209114\pi$
$98$	8.38630	0.847145
$99$	2.96284	0.297777
$100$	0	0
$101$	−8.87740	−0.883335	−0.441667	−	0.897179i	$-0.645613\pi$
$101$	−8.87740	−0.883335	−0.441667	+	0.897179i	$0.645613\pi$
$102$	7.62780	0.755264
$103$	12.8125	1.26245	0.631226	−	0.775599i	$-0.282552\pi$
$103$	12.8125	1.26245	0.631226	+	0.775599i	$0.282552\pi$
$104$	−18.7626	−1.83983
$105$	0	0
$106$	−4.25950	−0.413719
$107$	1.62875	0.157457	0.0787285	−	0.996896i	$-0.474914\pi$
$107$	1.62875	0.157457	0.0787285	+	0.996896i	$0.474914\pi$
$108$	1.21348	0.116768
$109$	11.7960	1.12986	0.564928	−	0.825140i	$-0.308904\pi$
$109$	11.7960	1.12986	0.564928	+	0.825140i	$0.308904\pi$
$110$	0	0
$111$	−12.1557	−1.15377
$112$	2.99896	0.283375
$113$	18.1921	1.71137	0.855685	−	0.517497i	$-0.173136\pi$
$113$	18.1921	1.71137	0.855685	+	0.517497i	$0.173136\pi$
$114$	8.80821	0.824965
$115$	0	0
$116$	2.04298	0.189686
$117$	6.49814	0.600754
$118$	12.4388	1.14509
$119$	3.45574	0.316787
$120$	0	0
$121$	−2.64184	−0.240167
$122$	−10.5240	−0.952801
$123$	11.2549	1.01482
$124$	−1.61400	−0.144942
$125$	0	0
$126$	−1.16506	−0.103792
$127$	−6.68796	−0.593460	−0.296730	−	0.954961i	$-0.595896\pi$
$127$	−6.68796	−0.593460	−0.296730	+	0.954961i	$0.595896\pi$
$128$	−9.16093	−0.809720
$129$	−2.54734	−0.224281
$130$	0	0
$131$	2.09303	0.182869	0.0914343	−	0.995811i	$-0.470855\pi$
$131$	2.09303	0.182869	0.0914343	+	0.995811i	$0.470855\pi$
$132$	0.871649	0.0758673
$133$	3.99053	0.346022
$134$	−5.64899	−0.487998
$135$	0	0
$136$	−12.0193	−1.03064
$137$	−8.92382	−0.762413	−0.381207	−	0.924490i	$-0.624491\pi$
$137$	−8.92382	−0.762413	−0.381207	+	0.924490i	$0.624491\pi$
$138$	1.02415	0.0871812
$139$	−9.59317	−0.813682	−0.406841	−	0.913499i	$-0.633370\pi$
$139$	−9.59317	−0.813682	−0.406841	+	0.913499i	$0.633370\pi$
$140$	0	0
$141$	14.6697	1.23541
$142$	16.1204	1.35279
$143$	18.3312	1.53294
$144$	3.61245	0.301038
$145$	0	0
$146$	−1.32688	−0.109813
$147$	−8.82056	−0.727507
$148$	1.85551	0.152522
$149$	−5.34071	−0.437528	−0.218764	−	0.975778i	$-0.570203\pi$
$149$	−5.34071	−0.437528	−0.218764	+	0.975778i	$0.570203\pi$
$150$	0	0
$151$	−4.11132	−0.334575	−0.167287	−	0.985908i	$-0.553501\pi$
$151$	−4.11132	−0.334575	−0.167287	+	0.985908i	$0.553501\pi$
$152$	−13.8793	−1.12576
$153$	4.16269	0.336533
$154$	−3.28664	−0.264845
$155$	0	0
$156$	1.91171	0.153060
$157$	15.4815	1.23556	0.617779	−	0.786352i	$-0.288033\pi$
$157$	15.4815	1.23556	0.617779	+	0.786352i	$0.288033\pi$
$158$	−17.2993	−1.37626
$159$	4.48006	0.355292
$160$	0	0
$161$	0.463986	0.0365672
$162$	6.51435	0.511815
$163$	−12.1294	−0.950047	−0.475023	−	0.879973i	$-0.657560\pi$
$163$	−12.1294	−0.950047	−0.475023	+	0.879973i	$0.657560\pi$
$164$	−1.71800	−0.134154
$165$	0	0
$166$	−7.56890	−0.587460
$167$	9.21206	0.712850	0.356425	−	0.934324i	$-0.383995\pi$
$167$	9.21206	0.712850	0.356425	+	0.934324i	$0.383995\pi$
$168$	−3.53818	−0.272977
$169$	27.2043	2.09264
$170$	0	0
$171$	4.80687	0.367590
$172$	0.388839	0.0296487
$173$	10.6489	0.809623	0.404812	−	0.914400i	$-0.367337\pi$
$173$	10.6489	0.809623	0.404812	+	0.914400i	$0.367337\pi$
$174$	17.8837	1.35576
$175$	0	0
$176$	10.1907	0.768154
$177$	−13.0829	−0.983373
$178$	−12.0468	−0.902943
$179$	−7.78126	−0.581599	−0.290799	−	0.956784i	$-0.593921\pi$
$179$	−7.78126	−0.581599	−0.290799	+	0.956784i	$0.593921\pi$
$180$	0	0
$181$	6.89784	0.512713	0.256356	−	0.966582i	$-0.417478\pi$
$181$	6.89784	0.512713	0.256356	+	0.966582i	$0.417478\pi$
$182$	−7.20831	−0.534316
$183$	11.0690	0.818242
$184$	−1.61377	−0.118969
$185$	0	0
$186$	−14.1286	−1.03596
$187$	11.7429	0.858727
$188$	−2.23926	−0.163315
$189$	4.81250	0.350058
$190$	0	0
$191$	−23.4221	−1.69476	−0.847381	−	0.530985i	$-0.821822\pi$
$191$	−23.4221	−1.69476	−0.847381	+	0.530985i	$0.821822\pi$
$192$	12.1767	0.878774
$193$	−13.5773	−0.977313	−0.488657	−	0.872476i	$-0.662513\pi$
$193$	−13.5773	−0.977313	−0.488657	+	0.872476i	$0.662513\pi$
$194$	−20.8420	−1.49637
$195$	0	0
$196$	1.34641	0.0961724
$197$	−7.68983	−0.547877	−0.273939	−	0.961747i	$-0.588327\pi$
$197$	−7.68983	−0.547877	−0.273939	+	0.961747i	$0.588327\pi$
$198$	−3.95899	−0.281353
$199$	−10.8797	−0.771240	−0.385620	−	0.922658i	$-0.626012\pi$
$199$	−10.8797	−0.771240	−0.385620	+	0.922658i	$0.626012\pi$
$200$	0	0
$201$	5.94150	0.419081
$202$	11.8621	0.834616
$203$	8.10215	0.568660
$204$	1.22464	0.0857417
$205$	0	0
$206$	−17.1202	−1.19282
$207$	0.558903	0.0388465
$208$	22.3504	1.54972
$209$	13.5602	0.937976
$210$	0	0
$211$	−14.1357	−0.973139	−0.486569	−	0.873642i	$-0.661752\pi$
$211$	−14.1357	−0.973139	−0.486569	+	0.873642i	$0.661752\pi$
$212$	−0.683859	−0.0469676
$213$	−16.9551	−1.16174
$214$	−2.17636	−0.148773
$215$	0	0
$216$	−16.7382	−1.13889
$217$	−6.40090	−0.434521
$218$	−15.7621	−1.06754
$219$	1.39559	0.0943050
$220$	0	0
$221$	25.7547	1.73245
$222$	16.2426	1.09013
$223$	−9.26951	−0.620732	−0.310366	−	0.950617i	$-0.600452\pi$
$223$	−9.26951	−0.620732	−0.310366	+	0.950617i	$0.600452\pi$
$224$	1.02785	0.0686762
$225$	0	0
$226$	−24.3086	−1.61698
$227$	−13.3084	−0.883309	−0.441655	−	0.897185i	$-0.645608\pi$
$227$	−13.3084	−0.883309	−0.441655	+	0.897185i	$0.645608\pi$
$228$	1.41415	0.0936544
$229$	−26.8955	−1.77730	−0.888652	−	0.458583i	$-0.848357\pi$
$229$	−26.8955	−1.77730	−0.888652	+	0.458583i	$0.848357\pi$
$230$	0	0
$231$	3.45683	0.227443
$232$	−28.1798	−1.85009
$233$	28.6809	1.87895	0.939474	−	0.342621i	$-0.111315\pi$
$233$	28.6809	1.87895	0.939474	+	0.342621i	$0.111315\pi$
$234$	−8.68292	−0.567620
$235$	0	0
$236$	1.99704	0.129996
$237$	18.1951	1.18190
$238$	−4.61762	−0.299316
$239$	−13.1115	−0.848115	−0.424058	−	0.905635i	$-0.639395\pi$
$239$	−13.1115	−0.848115	−0.424058	+	0.905635i	$0.639395\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	3.53006	0.226921
$243$	10.1179	0.649064
$244$	−1.68962	−0.108167
$245$	0	0
$246$	−15.0390	−0.958850
$247$	29.7403	1.89233
$248$	22.2627	1.41368
$249$	7.96082	0.504497
$250$	0	0
$251$	13.1577	0.830504	0.415252	−	0.909706i	$-0.363693\pi$
$251$	13.1577	0.830504	0.415252	+	0.909706i	$0.363693\pi$
$252$	−0.187050	−0.0117830
$253$	1.57666	0.0991240
$254$	8.93655	0.560729
$255$	0	0
$256$	−5.08732	−0.317957
$257$	13.7576	0.858174	0.429087	−	0.903263i	$-0.358835\pi$
$257$	13.7576	0.858174	0.429087	+	0.903263i	$0.358835\pi$
$258$	3.40380	0.211911
$259$	7.35866	0.457245
$260$	0	0
$261$	9.75961	0.604105
$262$	−2.79674	−0.172783
$263$	−4.02854	−0.248410	−0.124205	−	0.992257i	$-0.539638\pi$
$263$	−4.02854	−0.248410	−0.124205	+	0.992257i	$0.539638\pi$
$264$	−12.0231	−0.739968
$265$	0	0
$266$	−5.33220	−0.326938
$267$	12.6706	0.775425
$268$	−0.906940	−0.0554002
$269$	−13.1288	−0.800476	−0.400238	−	0.916411i	$-0.631073\pi$
$269$	−13.1288	−0.800476	−0.400238	+	0.916411i	$0.631073\pi$
$270$	0	0
$271$	−18.8838	−1.14711	−0.573555	−	0.819167i	$-0.694436\pi$
$271$	−18.8838	−1.14711	−0.573555	+	0.819167i	$0.694436\pi$
$272$	14.3176	0.868132
$273$	7.58157	0.458857
$274$	11.9241	0.720364
$275$	0	0
$276$	0.164426	0.00989727
$277$	−23.5278	−1.41365	−0.706824	−	0.707390i	$-0.749873\pi$
$277$	−23.5278	−1.41365	−0.706824	+	0.707390i	$0.749873\pi$
$278$	12.8185	0.768805
$279$	−7.71033	−0.461606
$280$	0	0
$281$	3.26027	0.194492	0.0972458	−	0.995260i	$-0.468997\pi$
$281$	3.26027	0.194492	0.0972458	+	0.995260i	$0.468997\pi$
$282$	−19.6019	−1.16728
$283$	3.83263	0.227826	0.113913	−	0.993491i	$-0.463661\pi$
$283$	3.83263	0.227826	0.113913	+	0.993491i	$0.463661\pi$
$284$	2.58811	0.153576
$285$	0	0
$286$	−24.4945	−1.44839
$287$	−6.81335	−0.402179
$288$	1.23812	0.0729569
$289$	−0.501619	−0.0295070
$290$	0	0
$291$	21.9212	1.28505
$292$	−0.213029	−0.0124666
$293$	−9.47160	−0.553337	−0.276668	−	0.960965i	$-0.589230\pi$
$293$	−9.47160	−0.553337	−0.276668	+	0.960965i	$0.589230\pi$
$294$	11.7862	0.687383
$295$	0	0
$296$	−25.5939	−1.48761
$297$	16.3533	0.948914
$298$	7.13634	0.413397
$299$	3.45796	0.199979
$300$	0	0
$301$	1.54208	0.0888838
$302$	5.49361	0.316122
$303$	−12.4764	−0.716748
$304$	16.5333	0.948248
$305$	0	0
$306$	−5.56224	−0.317972
$307$	23.1309	1.32015	0.660074	−	0.751201i	$-0.270525\pi$
$307$	23.1309	1.32015	0.660074	+	0.751201i	$0.270525\pi$
$308$	−0.527668	−0.0300666
$309$	18.0067	1.02437
$310$	0	0
$311$	−13.9494	−0.790999	−0.395500	−	0.918466i	$-0.629429\pi$
$311$	−13.9494	−0.790999	−0.395500	+	0.918466i	$0.629429\pi$
$312$	−26.3691	−1.49286
$313$	13.1562	0.743630	0.371815	−	0.928307i	$-0.378736\pi$
$313$	13.1562	0.743630	0.371815	+	0.928307i	$0.378736\pi$
$314$	−20.6866	−1.16741
$315$	0	0
$316$	−2.77739	−0.156240
$317$	−22.5936	−1.26898	−0.634492	−	0.772929i	$-0.718791\pi$
$317$	−22.5936	−1.26898	−0.634492	+	0.772929i	$0.718791\pi$
$318$	−5.98633	−0.335697
$319$	27.5318	1.54149
$320$	0	0
$321$	2.28905	0.127762
$322$	−0.619985	−0.0345504
$323$	19.0515	1.06006
$324$	1.04587	0.0581040
$325$	0	0
$326$	16.2075	0.897649
$327$	16.5782	0.916779
$328$	23.6972	1.30846
$329$	−8.88056	−0.489601
$330$	0	0
$331$	−20.1063	−1.10514	−0.552571	−	0.833466i	$-0.686353\pi$
$331$	−20.1063	−1.10514	−0.552571	+	0.833466i	$0.686353\pi$
$332$	−1.21518	−0.0666916
$333$	8.86403	0.485746
$334$	−12.3093	−0.673534
$335$	0	0
$336$	4.21475	0.229934
$337$	−2.51939	−0.137240	−0.0686199	−	0.997643i	$-0.521860\pi$
$337$	−2.51939	−0.137240	−0.0686199	+	0.997643i	$0.521860\pi$
$338$	−36.3508	−1.97722
$339$	25.5673	1.38863
$340$	0	0
$341$	−21.7508	−1.17787
$342$	−6.42301	−0.347317
$343$	11.2952	0.609882
$344$	−5.36344	−0.289177
$345$	0	0
$346$	−14.2293	−0.764970
$347$	6.19345	0.332482	0.166241	−	0.986085i	$-0.446837\pi$
$347$	6.19345	0.332482	0.166241	+	0.986085i	$0.446837\pi$
$348$	2.87122	0.153913
$349$	22.8230	1.22168	0.610842	−	0.791752i	$-0.290831\pi$
$349$	22.8230	1.22168	0.610842	+	0.791752i	$0.290831\pi$
$350$	0	0
$351$	35.8663	1.91440
$352$	3.49273	0.186163
$353$	7.52317	0.400418	0.200209	−	0.979753i	$-0.435838\pi$
$353$	7.52317	0.400418	0.200209	+	0.979753i	$0.435838\pi$
$354$	17.4816	0.929137
$355$	0	0
$356$	−1.93410	−0.102507
$357$	4.85672	0.257045
$358$	10.3974	0.549522
$359$	−24.2048	−1.27748	−0.638739	−	0.769423i	$-0.720544\pi$
$359$	−24.2048	−1.27748	−0.638739	+	0.769423i	$0.720544\pi$
$360$	0	0
$361$	2.99981	0.157885
$362$	−9.21701	−0.484435
$363$	−3.71286	−0.194874
$364$	−1.15729	−0.0606584
$365$	0	0
$366$	−14.7905	−0.773114
$367$	18.1874	0.949377	0.474689	−	0.880154i	$-0.342561\pi$
$367$	18.1874	0.949377	0.474689	+	0.880154i	$0.342561\pi$
$368$	1.92235	0.100210
$369$	−8.20716	−0.427248
$370$	0	0
$371$	−2.71208	−0.140804
$372$	−2.26833	−0.117608
$373$	−1.80623	−0.0935230	−0.0467615	−	0.998906i	$-0.514890\pi$
$373$	−1.80623	−0.0935230	−0.0467615	+	0.998906i	$0.514890\pi$
$374$	−15.6911	−0.811366
$375$	0	0
$376$	30.8871	1.59288
$377$	60.3832	3.10989
$378$	−6.43053	−0.330751
$379$	9.53751	0.489909	0.244955	−	0.969535i	$-0.421227\pi$
$379$	9.53751	0.489909	0.244955	+	0.969535i	$0.421227\pi$
$380$	0	0
$381$	−9.39930	−0.481541
$382$	31.2969	1.60129
$383$	11.8229	0.604121	0.302060	−	0.953289i	$-0.402326\pi$
$383$	11.8229	0.604121	0.302060	+	0.953289i	$0.402326\pi$
$384$	−12.8748	−0.657016
$385$	0	0
$386$	18.1422	0.923411
$387$	1.85754	0.0944241
$388$	−3.34617	−0.169876
$389$	13.0090	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$389$	13.0090	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$390$	0	0
$391$	2.21516	0.112025
$392$	−18.5717	−0.938013
$393$	2.94155	0.148382
$394$	10.2753	0.517660
$395$	0	0
$396$	−0.635613	−0.0319407
$397$	−37.9030	−1.90230	−0.951148	−	0.308736i	$-0.900094\pi$
$397$	−37.9030	−1.90230	−0.951148	+	0.308736i	$0.900094\pi$
$398$	14.5376	0.728704
$399$	5.60831	0.280767
$400$	0	0
$401$	28.9694	1.44666	0.723331	−	0.690501i	$-0.242610\pi$
$401$	28.9694	1.44666	0.723331	+	0.690501i	$0.242610\pi$
$402$	−7.93912	−0.395967
$403$	−47.7042	−2.37632
$404$	1.90445	0.0947501
$405$	0	0
$406$	−10.8262	−0.537296
$407$	25.0054	1.23947
$408$	−16.8920	−0.836277
$409$	14.5186	0.717899	0.358950	−	0.933357i	$-0.383135\pi$
$409$	14.5186	0.717899	0.358950	+	0.933357i	$0.383135\pi$
$410$	0	0
$411$	−12.5416	−0.618631
$412$	−2.74864	−0.135416
$413$	7.91997	0.389716
$414$	−0.746815	−0.0367040
$415$	0	0
$416$	7.66031	0.375577
$417$	−13.4823	−0.660231
$418$	−18.1193	−0.886244
$419$	29.5366	1.44296	0.721479	−	0.692436i	$-0.243462\pi$
$419$	29.5366	1.44296	0.721479	+	0.692436i	$0.243462\pi$
$420$	0	0
$421$	4.39009	0.213960	0.106980	−	0.994261i	$-0.465882\pi$
$421$	4.39009	0.213960	0.106980	+	0.994261i	$0.465882\pi$
$422$	18.8883	0.919467
$423$	−10.6973	−0.520119
$424$	9.43278	0.458096
$425$	0	0
$426$	22.6557	1.09767
$427$	−6.70079	−0.324274
$428$	−0.349412	−0.0168895
$429$	25.7628	1.24384
$430$	0	0
$431$	37.0789	1.78603	0.893015	−	0.450027i	$-0.148586\pi$
$431$	37.0789	1.78603	0.893015	+	0.450027i	$0.148586\pi$
$432$	19.9388	0.959306
$433$	31.4769	1.51268	0.756342	−	0.654177i	$-0.226985\pi$
$433$	31.4769	1.51268	0.756342	+	0.654177i	$0.226985\pi$
$434$	8.55298	0.410556
$435$	0	0
$436$	−2.53058	−0.121193
$437$	2.55796	0.122364
$438$	−1.86481	−0.0891038
$439$	11.7064	0.558716	0.279358	−	0.960187i	$-0.409878\pi$
$439$	11.7064	0.558716	0.279358	+	0.960187i	$0.409878\pi$
$440$	0	0
$441$	6.43201	0.306286
$442$	−34.4139	−1.63690
$443$	−3.69263	−0.175442	−0.0877212	−	0.996145i	$-0.527958\pi$
$443$	−3.69263	−0.175442	−0.0877212	+	0.996145i	$0.527958\pi$
$444$	2.60774	0.123758
$445$	0	0
$446$	12.3861	0.586497
$447$	−7.50587	−0.355015
$448$	−7.37134	−0.348263
$449$	−31.4241	−1.48300	−0.741498	−	0.670955i	$-0.765884\pi$
$449$	−31.4241	−1.48300	−0.741498	+	0.670955i	$0.765884\pi$
$450$	0	0
$451$	−23.1524	−1.09020
$452$	−3.90272	−0.183569
$453$	−5.77808	−0.271478
$454$	17.7829	0.834592
$455$	0	0
$456$	−19.5060	−0.913454
$457$	−6.35800	−0.297415	−0.148707	−	0.988881i	$-0.547511\pi$
$457$	−6.35800	−0.297415	−0.148707	+	0.988881i	$0.547511\pi$
$458$	35.9382	1.67928
$459$	22.9758	1.07242
$460$	0	0
$461$	−26.4491	−1.23186	−0.615929	−	0.787801i	$-0.711219\pi$
$461$	−26.4491	−1.23186	−0.615929	+	0.787801i	$0.711219\pi$
$462$	−4.61907	−0.214899
$463$	−35.1906	−1.63545	−0.817724	−	0.575611i	$-0.804764\pi$
$463$	−35.1906	−1.63545	−0.817724	+	0.575611i	$0.804764\pi$
$464$	33.5683	1.55837
$465$	0	0
$466$	−38.3239	−1.77532
$467$	−33.4422	−1.54752	−0.773759	−	0.633480i	$-0.781626\pi$
$467$	−33.4422	−1.54752	−0.773759	+	0.633480i	$0.781626\pi$
$468$	−1.39403	−0.0644393
$469$	−3.59679	−0.166084
$470$	0	0
$471$	21.7578	1.00255
$472$	−27.5461	−1.26791
$473$	5.24011	0.240941
$474$	−24.3126	−1.11671
$475$	0	0
$476$	−0.741354	−0.0339799
$477$	−3.26689	−0.149581
$478$	17.5198	0.801339
$479$	−7.54273	−0.344636	−0.172318	−	0.985041i	$-0.555126\pi$
$479$	−7.54273	−0.344636	−0.172318	+	0.985041i	$0.555126\pi$
$480$	0	0
$481$	54.8422	2.50059
$482$	1.33622	0.0608629
$483$	0.652088	0.0296711
$484$	0.566749	0.0257613
$485$	0	0
$486$	−13.5197	−0.613266
$487$	28.8765	1.30852	0.654260	−	0.756270i	$-0.272980\pi$
$487$	28.8765	1.30852	0.654260	+	0.756270i	$0.272980\pi$
$488$	23.3058	1.05500
$489$	−17.0467	−0.770879
$490$	0	0
$491$	−35.2291	−1.58987	−0.794933	−	0.606697i	$-0.792494\pi$
$491$	−35.2291	−1.58987	−0.794933	+	0.606697i	$0.792494\pi$
$492$	−2.41449	−0.108854
$493$	38.6812	1.74212
$494$	−39.7395	−1.78796
$495$	0	0
$496$	−26.5198	−1.19077
$497$	10.2641	0.460406
$498$	−10.6374	−0.476672
$499$	−36.4205	−1.63041	−0.815203	−	0.579176i	$-0.803374\pi$
$499$	−36.4205	−1.63041	−0.815203	+	0.579176i	$0.803374\pi$
$500$	0	0
$501$	12.9467	0.578415
$502$	−17.5815	−0.784699
$503$	−12.9799	−0.578745	−0.289373	−	0.957217i	$-0.593447\pi$
$503$	−12.9799	−0.578745	−0.289373	+	0.957217i	$0.593447\pi$
$504$	2.58007	0.114925
$505$	0	0
$506$	−2.10676	−0.0936571
$507$	38.2331	1.69799
$508$	1.43476	0.0636570
$509$	22.1426	0.981455	0.490728	−	0.871313i	$-0.336731\pi$
$509$	22.1426	0.981455	0.490728	+	0.871313i	$0.336731\pi$
$510$	0	0
$511$	−0.844842	−0.0373736
$512$	25.1196	1.11014
$513$	26.5313	1.17139
$514$	−18.3831	−0.810843
$515$	0	0
$516$	0.546477	0.0240573
$517$	−30.1769	−1.32718
$518$	−9.83276	−0.432027
$519$	14.9661	0.656938
$520$	0	0
$521$	14.9523	0.655072	0.327536	−	0.944839i	$-0.393782\pi$
$521$	14.9523	0.655072	0.327536	+	0.944839i	$0.393782\pi$
$522$	−13.0409	−0.570787
$523$	−17.5348	−0.766744	−0.383372	−	0.923594i	$-0.625237\pi$
$523$	−17.5348	−0.766744	−0.383372	+	0.923594i	$0.625237\pi$
$524$	−0.449013	−0.0196152
$525$	0	0
$526$	5.38300	0.234710
$527$	−30.5591	−1.33118
$528$	14.3221	0.623290
$529$	−22.7026	−0.987069
$530$	0	0
$531$	9.54016	0.414008
$532$	−0.856080	−0.0371158
$533$	−50.7781	−2.19944
$534$	−16.9306	−0.732658
$535$	0	0
$536$	12.5098	0.540343
$537$	−10.9358	−0.471916
$538$	17.5429	0.756327
$539$	18.1447	0.781547
$540$	0	0
$541$	−20.3708	−0.875809	−0.437905	−	0.899021i	$-0.644279\pi$
$541$	−20.3708	−0.875809	−0.437905	+	0.899021i	$0.644279\pi$
$542$	25.2328	1.08384
$543$	9.69428	0.416021
$544$	4.90716	0.210393
$545$	0	0
$546$	−10.1306	−0.433550
$547$	7.72639	0.330357	0.165178	−	0.986264i	$-0.447180\pi$
$547$	7.72639	0.330357	0.165178	+	0.986264i	$0.447180\pi$
$548$	1.91441	0.0817795
$549$	−8.07158	−0.344486
$550$	0	0
$551$	44.6672	1.90289
$552$	−2.26800	−0.0965326
$553$	−11.0147	−0.468393
$554$	31.4382	1.33568
$555$	0	0
$556$	2.05800	0.0872788
$557$	37.0045	1.56793	0.783966	−	0.620804i	$-0.213194\pi$
$557$	37.0045	1.56793	0.783966	+	0.620804i	$0.213194\pi$
$558$	10.3027	0.436147
$559$	11.4927	0.486089
$560$	0	0
$561$	16.5036	0.696781
$562$	−4.35643	−0.183765
$563$	12.5828	0.530303	0.265151	−	0.964207i	$-0.414578\pi$
$563$	12.5828	0.530303	0.265151	+	0.964207i	$0.414578\pi$
$564$	−3.14707	−0.132515
$565$	0	0
$566$	−5.12122	−0.215261
$567$	4.14777	0.174190
$568$	−35.6990	−1.49790
$569$	−40.8208	−1.71130	−0.855648	−	0.517558i	$-0.826841\pi$
$569$	−40.8208	−1.71130	−0.855648	+	0.517558i	$0.826841\pi$
$570$	0	0
$571$	−11.1622	−0.467123	−0.233562	−	0.972342i	$-0.575038\pi$
$571$	−11.1622	−0.467123	−0.233562	+	0.972342i	$0.575038\pi$
$572$	−3.93257	−0.164429
$573$	−32.9175	−1.37515
$574$	9.10410	0.379998
$575$	0	0
$576$	−8.87930	−0.369971
$577$	−11.7563	−0.489421	−0.244711	−	0.969596i	$-0.578693\pi$
$577$	−11.7563	−0.489421	−0.244711	+	0.969596i	$0.578693\pi$
$578$	0.670271	0.0278796
$579$	−19.0816	−0.793003
$580$	0	0
$581$	−4.81922	−0.199935
$582$	−29.2915	−1.21417
$583$	−9.21590	−0.381683
$584$	2.93841	0.121592
$585$	0	0
$586$	12.6561	0.522818
$587$	−13.8302	−0.570835	−0.285418	−	0.958403i	$-0.592132\pi$
$587$	−13.8302	−0.570835	−0.285418	+	0.958403i	$0.592132\pi$
$588$	1.89226	0.0780354
$589$	−35.2882	−1.45403
$590$	0	0
$591$	−10.8073	−0.444554
$592$	30.4879	1.25305
$593$	−3.00332	−0.123332	−0.0616658	−	0.998097i	$-0.519641\pi$
$593$	−3.00332	−0.123332	−0.0616658	+	0.998097i	$0.519641\pi$
$594$	−21.8515	−0.896579
$595$	0	0
$596$	1.14573	0.0469310
$597$	−15.2904	−0.625793
$598$	−4.62058	−0.188950
$599$	41.1395	1.68091	0.840457	−	0.541878i	$-0.182287\pi$
$599$	41.1395	1.68091	0.840457	+	0.541878i	$0.182287\pi$
$600$	0	0
$601$	−3.21921	−0.131314	−0.0656572	−	0.997842i	$-0.520914\pi$
$601$	−3.21921	−0.131314	−0.0656572	+	0.997842i	$0.520914\pi$
$602$	−2.06055	−0.0839816
$603$	−4.33258	−0.176436
$604$	0.881995	0.0358879
$605$	0	0
$606$	16.6711	0.677217
$607$	18.0699	0.733434	0.366717	−	0.930333i	$-0.380482\pi$
$607$	18.0699	0.733434	0.366717	+	0.930333i	$0.380482\pi$
$608$	5.66656	0.229809
$609$	11.3868	0.461417
$610$	0	0
$611$	−66.1845	−2.67754
$612$	−0.893013	−0.0360979
$613$	−24.3324	−0.982776	−0.491388	−	0.870941i	$-0.663510\pi$
$613$	−24.3324	−0.982776	−0.491388	+	0.870941i	$0.663510\pi$
$614$	−30.9078	−1.24734
$615$	0	0
$616$	7.27836	0.293254
$617$	8.01500	0.322672	0.161336	−	0.986900i	$-0.448420\pi$
$617$	8.01500	0.322672	0.161336	+	0.986900i	$0.448420\pi$
$618$	−24.0609	−0.967871
$619$	8.47603	0.340680	0.170340	−	0.985385i	$-0.445513\pi$
$619$	8.47603	0.340680	0.170340	+	0.985385i	$0.445513\pi$
$620$	0	0
$621$	3.08485	0.123791
$622$	18.6394	0.747373
$623$	−7.67034	−0.307306
$624$	31.4114	1.25746
$625$	0	0
$626$	−17.5795	−0.702617
$627$	19.0575	0.761085
$628$	−3.32122	−0.132531
$629$	35.1317	1.40079
$630$	0	0
$631$	39.6020	1.57653	0.788266	−	0.615335i	$-0.210979\pi$
$631$	39.6020	1.57653	0.788266	+	0.615335i	$0.210979\pi$
$632$	38.3098	1.52388
$633$	−19.8663	−0.789616
$634$	30.1900	1.19900
$635$	0	0
$636$	−0.961100	−0.0381101
$637$	39.7952	1.57674
$638$	−36.7885	−1.45647
$639$	12.3638	0.489104
$640$	0	0
$641$	7.10164	0.280498	0.140249	−	0.990116i	$-0.455210\pi$
$641$	7.10164	0.280498	0.140249	+	0.990116i	$0.455210\pi$
$642$	−3.05867	−0.120716
$643$	−7.76572	−0.306250	−0.153125	−	0.988207i	$-0.548934\pi$
$643$	−7.76572	−0.306250	−0.153125	+	0.988207i	$0.548934\pi$
$644$	−0.0995380	−0.00392235
$645$	0	0
$646$	−25.4570	−1.00159
$647$	38.8734	1.52827	0.764136	−	0.645056i	$-0.223166\pi$
$647$	38.8734	1.52827	0.764136	+	0.645056i	$0.223166\pi$
$648$	−14.4262	−0.566715
$649$	26.9128	1.05642
$650$	0	0
$651$	−8.99586	−0.352576
$652$	2.60210	0.101906
$653$	−6.50525	−0.254570	−0.127285	−	0.991866i	$-0.540626\pi$
$653$	−6.50525	−0.254570	−0.127285	+	0.991866i	$0.540626\pi$
$654$	−22.1521	−0.866216
$655$	0	0
$656$	−28.2286	−1.10214
$657$	−1.01767	−0.0397032
$658$	11.8663	0.462598
$659$	−14.2807	−0.556299	−0.278149	−	0.960538i	$-0.589721\pi$
$659$	−14.2807	−0.556299	−0.278149	+	0.960538i	$0.589721\pi$
$660$	0	0
$661$	3.38853	0.131799	0.0658994	−	0.997826i	$-0.479008\pi$
$661$	3.38853	0.131799	0.0658994	+	0.997826i	$0.479008\pi$
$662$	26.8663	1.04419
$663$	36.1959	1.40573
$664$	16.7615	0.650474
$665$	0	0
$666$	−11.8443	−0.458955
$667$	5.19354	0.201095
$668$	−1.97625	−0.0764632
$669$	−13.0274	−0.503670
$670$	0	0
$671$	−22.7699	−0.879022
$672$	1.44455	0.0557247
$673$	−26.7238	−1.03013	−0.515064	−	0.857152i	$-0.672232\pi$
$673$	−26.7238	−1.03013	−0.515064	+	0.857152i	$0.672232\pi$
$674$	3.36645	0.129671
$675$	0	0
$676$	−5.83609	−0.224465
$677$	25.0910	0.964326	0.482163	−	0.876082i	$-0.339851\pi$
$677$	25.0910	0.964326	0.482163	+	0.876082i	$0.339851\pi$
$678$	−34.1634	−1.31204
$679$	−13.2704	−0.509271
$680$	0	0
$681$	−18.7037	−0.716728
$682$	29.0638	1.11291
$683$	9.23834	0.353495	0.176748	−	0.984256i	$-0.443442\pi$
$683$	9.23834	0.353495	0.176748	+	0.984256i	$0.443442\pi$
$684$	−1.03121	−0.0394293
$685$	0	0
$686$	−15.0928	−0.576246
$687$	−37.7991	−1.44212
$688$	6.38903	0.243579
$689$	−20.2124	−0.770032
$690$	0	0
$691$	44.5304	1.69402	0.847009	−	0.531579i	$-0.178401\pi$
$691$	44.5304	1.69402	0.847009	+	0.531579i	$0.178401\pi$
$692$	−2.28450	−0.0868435
$693$	−2.52075	−0.0957552
$694$	−8.27579	−0.314145
$695$	0	0
$696$	−39.6040	−1.50119
$697$	−32.5282	−1.23210
$698$	−30.4964	−1.15431
$699$	40.3083	1.52460
$700$	0	0
$701$	−5.47161	−0.206660	−0.103330	−	0.994647i	$-0.532950\pi$
$701$	−5.47161	−0.206660	−0.103330	+	0.994647i	$0.532950\pi$
$702$	−47.9251	−1.80881
$703$	40.5684	1.53007
$704$	−25.0485	−0.944050
$705$	0	0
$706$	−10.0526	−0.378334
$707$	7.55278	0.284051
$708$	2.80666	0.105481
$709$	22.0297	0.827343	0.413672	−	0.910426i	$-0.364246\pi$
$709$	22.0297	0.827343	0.413672	+	0.910426i	$0.364246\pi$
$710$	0	0
$711$	−13.2680	−0.497589
$712$	26.6779	0.999796
$713$	−4.10302	−0.153659
$714$	−6.48963	−0.242868
$715$	0	0
$716$	1.66930	0.0623846
$717$	−18.4270	−0.688171
$718$	32.3428	1.20702
$719$	18.9045	0.705020	0.352510	−	0.935808i	$-0.385328\pi$
$719$	18.9045	0.705020	0.352510	+	0.935808i	$0.385328\pi$
$720$	0	0
$721$	−10.9007	−0.405963
$722$	−4.00839	−0.149177
$723$	−1.40541	−0.0522676
$724$	−1.47978	−0.0549957
$725$	0	0
$726$	4.96117	0.184126
$727$	52.4245	1.94432	0.972159	−	0.234321i	$-0.0752867\pi$
$727$	52.4245	1.94432	0.972159	+	0.234321i	$0.0752867\pi$
$728$	15.9630	0.591628
$729$	28.8454	1.06835
$730$	0	0
$731$	7.36217	0.272300
$732$	−2.37461	−0.0877680
$733$	38.2731	1.41365	0.706825	−	0.707388i	$-0.250127\pi$
$733$	38.2731	1.41365	0.706825	+	0.707388i	$0.250127\pi$
$734$	−24.3024	−0.897016
$735$	0	0
$736$	0.658861	0.0242859
$737$	−12.2222	−0.450211
$738$	10.9665	0.403684
$739$	18.1631	0.668140	0.334070	−	0.942548i	$-0.391578\pi$
$739$	18.1631	0.668140	0.334070	+	0.942548i	$0.391578\pi$
$740$	0	0
$741$	41.7973	1.53546
$742$	3.62393	0.133039
$743$	−19.9933	−0.733484	−0.366742	−	0.930323i	$-0.619527\pi$
$743$	−19.9933	−0.733484	−0.366742	+	0.930323i	$0.619527\pi$
$744$	31.2882	1.14708
$745$	0	0
$746$	2.41351	0.0883649
$747$	−5.80509	−0.212397
$748$	−2.51919	−0.0921106
$749$	−1.38572	−0.0506330
$750$	0	0
$751$	−33.7675	−1.23219	−0.616097	−	0.787670i	$-0.711287\pi$
$751$	−33.7675	−1.23219	−0.616097	+	0.787670i	$0.711287\pi$
$752$	−36.7933	−1.34172
$753$	18.4919	0.673881
$754$	−80.6850	−2.93837
$755$	0	0
$756$	−1.03242	−0.0375486
$757$	18.8110	0.683699	0.341849	−	0.939755i	$-0.388947\pi$
$757$	18.8110	0.683699	0.341849	+	0.939755i	$0.388947\pi$
$758$	−12.7442	−0.462889
$759$	2.21585	0.0804304
$760$	0	0
$761$	31.2239	1.13186	0.565932	−	0.824452i	$-0.308516\pi$
$761$	31.2239	1.13186	0.565932	+	0.824452i	$0.308516\pi$
$762$	12.5595	0.454982
$763$	−10.0359	−0.363325
$764$	5.02470	0.181787
$765$	0	0
$766$	−15.7979	−0.570802
$767$	59.0254	2.13129
$768$	−7.14975	−0.257994
$769$	−32.1220	−1.15835	−0.579175	−	0.815203i	$-0.696625\pi$
$769$	−32.1220	−1.15835	−0.579175	+	0.815203i	$0.696625\pi$
$770$	0	0
$771$	19.3350	0.696332
$772$	2.91271	0.104831
$773$	31.0604	1.11716	0.558582	−	0.829450i	$-0.311346\pi$
$773$	31.0604	1.11716	0.558582	+	0.829450i	$0.311346\pi$
$774$	−2.48207	−0.0892163
$775$	0	0
$776$	46.1552	1.65688
$777$	10.3419	0.371014
$778$	−17.3828	−0.623202
$779$	−37.5621	−1.34580
$780$	0	0
$781$	34.8782	1.24804
$782$	−2.95993	−0.105847
$783$	53.8678	1.92508
$784$	22.1230	0.790106
$785$	0	0
$786$	−3.93055	−0.140198
$787$	−19.4577	−0.693592	−0.346796	−	0.937941i	$-0.612730\pi$
$787$	−19.4577	−0.693592	−0.346796	+	0.937941i	$0.612730\pi$
$788$	1.64968	0.0587676
$789$	−5.66173	−0.201563
$790$	0	0
$791$	−15.4776	−0.550320
$792$	8.76730	0.311533
$793$	−49.9392	−1.77339
$794$	50.6465	1.79738
$795$	0	0
$796$	2.33400	0.0827264
$797$	38.3830	1.35959	0.679797	−	0.733400i	$-0.262068\pi$
$797$	38.3830	1.35959	0.679797	+	0.733400i	$0.262068\pi$
$798$	−7.49391	−0.265282
$799$	−42.3975	−1.49992
$800$	0	0
$801$	−9.23946	−0.326460
$802$	−38.7094	−1.36687
$803$	−2.87085	−0.101310
$804$	−1.27462	−0.0449523
$805$	0	0
$806$	63.7431	2.24525
$807$	−18.4513	−0.649516
$808$	−26.2690	−0.924140
$809$	7.27781	0.255874	0.127937	−	0.991782i	$-0.459164\pi$
$809$	7.27781	0.255874	0.127937	+	0.991782i	$0.459164\pi$
$810$	0	0
$811$	22.5263	0.791004	0.395502	−	0.918465i	$-0.370571\pi$
$811$	22.5263	0.791004	0.395502	+	0.918465i	$0.370571\pi$
$812$	−1.73814	−0.0609967
$813$	−26.5394	−0.930778
$814$	−33.4126	−1.17111
$815$	0	0
$816$	20.1220	0.704412
$817$	8.50149	0.297429
$818$	−19.4000	−0.678305
$819$	−5.52853	−0.193183
$820$	0	0
$821$	19.0471	0.664749	0.332374	−	0.943148i	$-0.392150\pi$
$821$	19.0471	0.664749	0.332374	+	0.943148i	$0.392150\pi$
$822$	16.7583	0.584512
$823$	−18.9246	−0.659670	−0.329835	−	0.944039i	$-0.606993\pi$
$823$	−18.9246	−0.659670	−0.329835	+	0.944039i	$0.606993\pi$
$824$	37.9132	1.32077
$825$	0	0
$826$	−10.5828	−0.368222
$827$	45.9742	1.59868	0.799339	−	0.600880i	$-0.205183\pi$
$827$	45.9742	1.59868	0.799339	+	0.600880i	$0.205183\pi$
$828$	−0.119901	−0.00416683
$829$	51.6174	1.79275	0.896373	−	0.443301i	$-0.146193\pi$
$829$	51.6174	1.79275	0.896373	+	0.443301i	$0.146193\pi$
$830$	0	0
$831$	−33.0661	−1.14705
$832$	−54.9367	−1.90459
$833$	25.4926	0.883268
$834$	18.0153	0.623817
$835$	0	0
$836$	−2.90904	−0.100611
$837$	−42.5569	−1.47098
$838$	−39.4673	−1.36338
$839$	56.7594	1.95955	0.979777	−	0.200095i	$-0.0641251\pi$
$839$	56.7594	1.95955	0.979777	+	0.200095i	$0.0641251\pi$
$840$	0	0
$841$	61.6900	2.12724
$842$	−5.86610	−0.202159
$843$	4.58201	0.157813
$844$	3.03250	0.104383
$845$	0	0
$846$	14.2938	0.491433
$847$	2.24764	0.0772298
$848$	−11.2365	−0.385863
$849$	5.38641	0.184861
$850$	0	0
$851$	4.71696	0.161695
$852$	3.63735	0.124614
$853$	−44.7742	−1.53304	−0.766520	−	0.642220i	$-0.778013\pi$
$853$	−44.7742	−1.53304	−0.766520	+	0.642220i	$0.778013\pi$
$854$	8.95370	0.306389
$855$	0	0
$856$	4.81961	0.164731
$857$	−1.95226	−0.0666879	−0.0333439	−	0.999444i	$-0.510616\pi$
$857$	−1.95226	−0.0666879	−0.0333439	+	0.999444i	$0.510616\pi$
$858$	−34.4247	−1.17524
$859$	−39.3917	−1.34403	−0.672014	−	0.740539i	$-0.734570\pi$
$859$	−39.3917	−1.34403	−0.672014	+	0.740539i	$0.734570\pi$
$860$	0	0
$861$	−9.57552	−0.326333
$862$	−49.5455	−1.68753
$863$	26.5098	0.902403	0.451202	−	0.892422i	$-0.350996\pi$
$863$	26.5098	0.902403	0.451202	+	0.892422i	$0.350996\pi$
$864$	6.83376	0.232489
$865$	0	0
$866$	−42.0599	−1.42925
$867$	−0.704978	−0.0239423
$868$	1.37317	0.0466085
$869$	−37.4290	−1.26969
$870$	0	0
$871$	−26.8059	−0.908283
$872$	34.9055	1.18205
$873$	−15.9851	−0.541014
$874$	−3.41798	−0.115615
$875$	0	0
$876$	−0.299393	−0.0101155
$877$	−26.8885	−0.907960	−0.453980	−	0.891012i	$-0.649996\pi$
$877$	−26.8885	−0.907960	−0.453980	+	0.891012i	$0.649996\pi$
$878$	−15.6423	−0.527901
$879$	−13.3114	−0.448984
$880$	0	0
$881$	38.6948	1.30366	0.651831	−	0.758364i	$-0.274001\pi$
$881$	38.6948	1.30366	0.651831	+	0.758364i	$0.274001\pi$
$882$	−8.59456	−0.289394
$883$	0.146816	0.00494075	0.00247037	−	0.999997i	$-0.499214\pi$
$883$	0.146816	0.00494075	0.00247037	+	0.999997i	$0.499214\pi$
$884$	−5.52512	−0.185830
$885$	0	0
$886$	4.93416	0.165766
$887$	0.829698	0.0278585	0.0139293	−	0.999903i	$-0.495566\pi$
$887$	0.829698	0.0278585	0.0139293	+	0.999903i	$0.495566\pi$
$888$	−35.9698	−1.20707
$889$	5.69002	0.190837
$890$	0	0
$891$	14.0945	0.472184
$892$	1.98857	0.0665823
$893$	−48.9586	−1.63834
$894$	10.0295	0.335435
$895$	0	0
$896$	7.79400	0.260379
$897$	4.85984	0.162265
$898$	41.9894	1.40120
$899$	−71.6473	−2.38957
$900$	0	0
$901$	−12.9480	−0.431361
$902$	30.9365	1.03007
$903$	2.16725	0.0721214
$904$	53.8320	1.79043
$905$	0	0
$906$	7.72076	0.256505
$907$	5.96847	0.198180	0.0990899	−	0.995078i	$-0.468407\pi$
$907$	5.96847	0.198180	0.0990899	+	0.995078i	$0.468407\pi$
$908$	2.85503	0.0947474
$909$	9.09785	0.301757
$910$	0	0
$911$	−39.1949	−1.29858	−0.649292	−	0.760539i	$-0.724935\pi$
$911$	−39.1949	−1.29858	−0.649292	+	0.760539i	$0.724935\pi$
$912$	23.2360	0.769420
$913$	−16.3761	−0.541971
$914$	8.49566	0.281011
$915$	0	0
$916$	5.76984	0.190641
$917$	−1.78072	−0.0588046
$918$	−30.7006	−1.01327
$919$	18.6944	0.616670	0.308335	−	0.951278i	$-0.400228\pi$
$919$	18.6944	0.616670	0.308335	+	0.951278i	$0.400228\pi$
$920$	0	0
$921$	32.5083	1.07118
$922$	35.3417	1.16392
$923$	76.4953	2.51788
$924$	−0.741587	−0.0243964
$925$	0	0
$926$	47.0223	1.54525
$927$	−13.1306	−0.431267
$928$	11.5051	0.377673
$929$	−22.3347	−0.732778	−0.366389	−	0.930462i	$-0.619406\pi$
$929$	−22.3347	−0.732778	−0.366389	+	0.930462i	$0.619406\pi$
$930$	0	0
$931$	29.4377	0.964781
$932$	−6.15286	−0.201544
$933$	−19.6046	−0.641826
$934$	44.6859	1.46217
$935$	0	0
$936$	19.2286	0.628505
$937$	18.9229	0.618183	0.309092	−	0.951032i	$-0.399975\pi$
$937$	18.9229	0.618183	0.309092	+	0.951032i	$0.399975\pi$
$938$	4.80608	0.156924
$939$	18.4898	0.603390
$940$	0	0
$941$	−15.3143	−0.499231	−0.249616	−	0.968345i	$-0.580304\pi$
$941$	−15.3143	−0.499231	−0.249616	+	0.968345i	$0.580304\pi$
$942$	−29.0731	−0.947252
$943$	−4.36741	−0.142222
$944$	32.8135	1.06799
$945$	0	0
$946$	−7.00192	−0.227652
$947$	−35.5445	−1.15504	−0.577521	−	0.816376i	$-0.695980\pi$
$947$	−35.5445	−1.15504	−0.577521	+	0.816376i	$0.695980\pi$
$948$	−3.90336	−0.126775
$949$	−6.29639	−0.204389
$950$	0	0
$951$	−31.7532	−1.02967
$952$	10.2258	0.331421
$953$	41.5088	1.34460	0.672301	−	0.740278i	$-0.265306\pi$
$953$	41.5088	1.34460	0.672301	+	0.740278i	$0.265306\pi$
$954$	4.36527	0.141331
$955$	0	0
$956$	2.81279	0.0909723
$957$	38.6934	1.25078
$958$	10.0787	0.325628
$959$	7.59226	0.245167
$960$	0	0
$961$	25.6031	0.825908
$962$	−73.2810	−2.36267
$963$	−1.66919	−0.0537890
$964$	0.214528	0.00690949
$965$	0	0
$966$	−0.871331	−0.0280346
$967$	−40.8051	−1.31220	−0.656102	−	0.754672i	$-0.727796\pi$
$967$	−40.8051	−1.31220	−0.656102	+	0.754672i	$0.727796\pi$
$968$	−7.81743	−0.251262
$969$	26.7752	0.860142
$970$	0	0
$971$	48.7222	1.56357	0.781785	−	0.623548i	$-0.214309\pi$
$971$	48.7222	1.56357	0.781785	+	0.623548i	$0.214309\pi$
$972$	−2.17058	−0.0696213
$973$	8.16174	0.261653
$974$	−38.5853	−1.23635
$975$	0	0
$976$	−27.7623	−0.888648
$977$	−33.9600	−1.08648	−0.543239	−	0.839578i	$-0.682802\pi$
$977$	−33.9600	−1.08648	−0.543239	+	0.839578i	$0.682802\pi$
$978$	22.7781	0.728363
$979$	−26.0645	−0.833025
$980$	0	0
$981$	−12.0890	−0.385971
$982$	47.0736	1.50218
$983$	6.53175	0.208330	0.104165	−	0.994560i	$-0.466783\pi$
$983$	6.53175	0.208330	0.104165	+	0.994560i	$0.466783\pi$
$984$	33.3042	1.06170
$985$	0	0
$986$	−51.6865	−1.64603
$987$	−12.4808	−0.397268
$988$	−6.38014	−0.202979
$989$	0.988483	0.0314319
$990$	0	0
$991$	36.9169	1.17270	0.586352	−	0.810057i	$-0.300564\pi$
$991$	36.9169	1.17270	0.586352	+	0.810057i	$0.300564\pi$
$992$	−9.08929	−0.288585
$993$	−28.2575	−0.896725
$994$	−13.7150	−0.435013
$995$	0	0
$996$	−1.70782	−0.0541144
$997$	−18.4225	−0.583446	−0.291723	−	0.956503i	$-0.594229\pi$
$997$	−18.4225	−0.583446	−0.291723	+	0.956503i	$0.594229\pi$
$998$	48.6656	1.54048
$999$	48.9247	1.54791

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.o.1.11	yes	40
5.4	even	2		6025.2.a.l.1.30	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.l.1.30	✓	40	5.4	even	2
6025.2.a.o.1.11	yes	40	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q-1.33622 q^{2} +1.40541 q^{3} -0.214528 q^{4} -1.87793 q^{6} -0.850787 q^{7} +2.95909 q^{8} -1.02483 q^{9} +O(q^{10})\)	\(q-1.33622 q^{2} +1.40541 q^{3} -0.214528 q^{4} -1.87793 q^{6} -0.850787 q^{7} +2.95909 q^{8} -1.02483 q^{9} -2.89105 q^{11} -0.301499 q^{12} -6.34069 q^{13} +1.13683 q^{14} -3.52492 q^{16} -4.06182 q^{17} +1.36940 q^{18} -4.69040 q^{19} -1.19570 q^{21} +3.86306 q^{22} -0.545361 q^{23} +4.15872 q^{24} +8.47253 q^{26} -5.65653 q^{27} +0.182518 q^{28} -9.52313 q^{29} +7.52351 q^{31} -1.20812 q^{32} -4.06310 q^{33} +5.42747 q^{34} +0.219855 q^{36} -8.64925 q^{37} +6.26738 q^{38} -8.91124 q^{39} +8.00829 q^{41} +1.59771 q^{42} -1.81253 q^{43} +0.620211 q^{44} +0.728719 q^{46} +10.4381 q^{47} -4.95395 q^{48} -6.27616 q^{49} -5.70851 q^{51} +1.36026 q^{52} +3.18773 q^{53} +7.55834 q^{54} -2.51755 q^{56} -6.59191 q^{57} +12.7250 q^{58} -9.30900 q^{59} +7.87599 q^{61} -10.0530 q^{62} +0.871914 q^{63} +8.66415 q^{64} +5.42918 q^{66} +4.22760 q^{67} +0.871375 q^{68} -0.766454 q^{69} -12.0642 q^{71} -3.03257 q^{72} +0.993013 q^{73} +11.5573 q^{74} +1.00622 q^{76} +2.45967 q^{77} +11.9073 q^{78} +12.9465 q^{79} -4.87522 q^{81} -10.7008 q^{82} +5.66443 q^{83} +0.256512 q^{84} +2.42193 q^{86} -13.3839 q^{87} -8.55486 q^{88} +9.01558 q^{89} +5.39457 q^{91} +0.116995 q^{92} +10.5736 q^{93} -13.9475 q^{94} -1.69790 q^{96} +15.5978 q^{97} +8.38630 q^{98} +2.96284 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9}+O(q^{10}) \) 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9	\( 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9} + q^{11} + 14 q^{12} + 9 q^{13} - q^{14} + 43 q^{16} + 12 q^{17} + 42 q^{18} + 2 q^{21} + 5 q^{22} + 77 q^{23} - 2 q^{24} + 2 q^{26} + 38 q^{27} + 42 q^{28} + 2 q^{29} + q^{31} + 72 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 28 q^{37} + 23 q^{38} - 2 q^{39} - 2 q^{41} + 37 q^{42} + 31 q^{43} + 3 q^{44} + 14 q^{46} + 96 q^{47} + 13 q^{48} + 40 q^{49} - 10 q^{51} + 42 q^{52} + 54 q^{53} + 4 q^{54} - 15 q^{56} + 37 q^{57} + 27 q^{58} + q^{59} + 5 q^{61} + 39 q^{62} + 70 q^{63} + 65 q^{64} - 52 q^{66} + 34 q^{67} + 52 q^{68} + 21 q^{69} - 9 q^{71} + 70 q^{72} + 25 q^{73} + 22 q^{74} - 47 q^{76} + 54 q^{77} + 58 q^{78} + 13 q^{79} + 12 q^{81} - 5 q^{82} + 63 q^{83} + 95 q^{84} - 18 q^{86} + 47 q^{87} + 13 q^{88} + 19 q^{89} - 31 q^{91} + 137 q^{92} + 52 q^{93} + 120 q^{94} - 49 q^{96} + 36 q^{97} + 64 q^{98} + 16 q^{99}+O(q^{100}) \) 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9 + q^11 + 14 * q^12 + 9 * q^13 - q^14 + 43 * q^16 + 12 * q^17 + 42 * q^18 + 2 * q^21 + 5 * q^22 + 77 * q^23 - 2 * q^24 + 2 * q^26 + 38 * q^27 + 42 * q^28 + 2 * q^29 + q^31 + 72 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 28 * q^37 + 23 * q^38 - 2 * q^39 - 2 * q^41 + 37 * q^42 + 31 * q^43 + 3 * q^44 + 14 * q^46 + 96 * q^47 + 13 * q^48 + 40 * q^49 - 10 * q^51 + 42 * q^52 + 54 * q^53 + 4 * q^54 - 15 * q^56 + 37 * q^57 + 27 * q^58 + q^59 + 5 * q^61 + 39 * q^62 + 70 * q^63 + 65 * q^64 - 52 * q^66 + 34 * q^67 + 52 * q^68 + 21 * q^69 - 9 * q^71 + 70 * q^72 + 25 * q^73 + 22 * q^74 - 47 * q^76 + 54 * q^77 + 58 * q^78 + 13 * q^79 + 12 * q^81 - 5 * q^82 + 63 * q^83 + 95 * q^84 - 18 * q^86 + 47 * q^87 + 13 * q^88 + 19 * q^89 - 31 * q^91 + 137 * q^92 + 52 * q^93 + 120 * q^94 - 49 * q^96 + 36 * q^97 + 64 * q^98 + 16 * q^99

Introduction

L-functions

Modular forms

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