LMFDB - Embedded newform 4925.2.a.l.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4925,2,Mod(1,4925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4925, 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("4925.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4925 = 5^{2} \cdot 197 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4925.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:	$39.3263229955$
Analytic rank:	$1$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 6 x^{16} - 8 x^{15} + 106 x^{14} - 60 x^{13} - 698 x^{12} + 877 x^{11} + 2076 x^{10} - 3556 x^{9} + \cdots + 2 $ x^17 - 6x^16 - 8x^15 + 106x^14 - 60x^13 - 698x^12 + 877x^11 + 2076x^10 - 3556x^9 - 2631x^8 + 5998x^7 + 1031x^6 - 4058x^5 - 119x^4 + 1073x^3 + 15x^2 - 75x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 985)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.22883$ of defining polynomial
Character	$\chi$	$=$	4925.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.22883 q^{2} -3.02604 q^{3} +2.96767 q^{4} +6.74452 q^{6} -1.28446 q^{7} -2.15678 q^{8} +6.15690 q^{9} -4.23897 q^{11} -8.98029 q^{12} +0.0326948 q^{13} +2.86284 q^{14} -1.12826 q^{16} -3.18747 q^{17} -13.7227 q^{18} -5.69971 q^{19} +3.88682 q^{21} +9.44795 q^{22} +6.17953 q^{23} +6.52650 q^{24} -0.0728710 q^{26} -9.55291 q^{27} -3.81186 q^{28} +7.19361 q^{29} +2.81208 q^{31} +6.82825 q^{32} +12.8273 q^{33} +7.10433 q^{34} +18.2717 q^{36} +11.8869 q^{37} +12.7037 q^{38} -0.0989356 q^{39} -3.01031 q^{41} -8.66306 q^{42} -11.6220 q^{43} -12.5799 q^{44} -13.7731 q^{46} -9.37541 q^{47} +3.41415 q^{48} -5.35016 q^{49} +9.64541 q^{51} +0.0970274 q^{52} -4.47527 q^{53} +21.2918 q^{54} +2.77030 q^{56} +17.2475 q^{57} -16.0333 q^{58} -6.52124 q^{59} +3.93428 q^{61} -6.26764 q^{62} -7.90829 q^{63} -12.9625 q^{64} -28.5898 q^{66} -3.36275 q^{67} -9.45938 q^{68} -18.6995 q^{69} +3.94676 q^{71} -13.2791 q^{72} +0.394938 q^{73} -26.4938 q^{74} -16.9149 q^{76} +5.44479 q^{77} +0.220510 q^{78} +5.02637 q^{79} +10.4367 q^{81} +6.70945 q^{82} -6.36108 q^{83} +11.5348 q^{84} +25.9035 q^{86} -21.7681 q^{87} +9.14253 q^{88} +8.91236 q^{89} -0.0419951 q^{91} +18.3388 q^{92} -8.50946 q^{93} +20.8962 q^{94} -20.6625 q^{96} +15.6077 q^{97} +11.9246 q^{98} -26.0990 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q - 6 q^{2} - 5 q^{3} + 18 q^{4} + 3 q^{6} - 7 q^{7} - 18 q^{8} + 22 q^{9} + 7 q^{11} - 20 q^{12} - 3 q^{13} + 17 q^{14} + 28 q^{16} - 6 q^{17} - 13 q^{18} - 23 q^{19} - q^{21} - 14 q^{22} - 49 q^{23}+ \cdots - 46 q^{99}+O(q^{100}) $ 17 * q - 6 * q^2 - 5 * q^3 + 18 * q^4 + 3 * q^6 - 7 * q^7 - 18 * q^8 + 22 * q^9 + 7 * q^11 - 20 * q^12 - 3 * q^13 + 17 * q^14 + 28 * q^16 - 6 * q^17 - 13 * q^18 - 23 * q^19 - q^21 - 14 * q^22 - 49 * q^23 + q^24 + 3 * q^26 - 29 * q^27 - 9 * q^28 + 19 * q^29 - 18 * q^31 - 42 * q^32 - 14 * q^33 - 19 * q^34 + 31 * q^36 - 10 * q^37 - 27 * q^38 + 21 * q^39 + 14 * q^41 - 31 * q^42 - 22 * q^43 + 27 * q^44 - 16 * q^46 - 51 * q^47 - 38 * q^48 + 8 * q^49 + 7 * q^51 - 17 * q^52 - 37 * q^53 - 36 * q^54 + 47 * q^56 + 11 * q^57 - 20 * q^58 + 23 * q^59 - 15 * q^61 + 6 * q^62 - 37 * q^63 - 2 * q^64 - 4 * q^66 - 24 * q^67 - 13 * q^68 + 6 * q^69 + 68 * q^71 + 33 * q^72 + 10 * q^73 - 9 * q^74 - 57 * q^76 + 16 * q^77 + 24 * q^78 + 3 * q^79 + 33 * q^81 + 40 * q^82 - 29 * q^83 - 83 * q^84 + 37 * q^86 - 23 * q^87 - 5 * q^88 - 11 * q^89 - 82 * q^91 - 15 * q^93 - 39 * q^94 + 2 * q^96 - 6 * q^97 - 15 * q^98 - 46 * 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 $

$2$	−2.22883	−1.57602	−0.788010	−	0.615663i	$-0.788888\pi$
$2$	−2.22883	−1.57602	−0.788010	+	0.615663i	$0.788888\pi$
$3$	−3.02604	−1.74708	−0.873542	−	0.486749i	$-0.838183\pi$
$3$	−3.02604	−1.74708	−0.873542	+	0.486749i	$0.838183\pi$
$4$	2.96767	1.48384
$5$	0	0
$5$	0	0
$6$	6.74452	2.75344
$7$	−1.28446	−0.485480	−0.242740	−	0.970091i	$-0.578046\pi$
$7$	−1.28446	−0.485480	−0.242740	+	0.970091i	$0.578046\pi$
$8$	−2.15678	−0.762537
$9$	6.15690	2.05230
$10$	0	0
$11$	−4.23897	−1.27810	−0.639050	−	0.769166i	$-0.720672\pi$
$11$	−4.23897	−1.27810	−0.639050	+	0.769166i	$0.720672\pi$
$12$	−8.98029	−2.59239
$13$	0.0326948	0.00906790	0.00453395	−	0.999990i	$-0.498557\pi$
$13$	0.0326948	0.00906790	0.00453395	+	0.999990i	$0.498557\pi$
$14$	2.86284	0.765126
$15$	0	0
$16$	−1.12826	−0.282065
$17$	−3.18747	−0.773076	−0.386538	−	0.922274i	$-0.626329\pi$
$17$	−3.18747	−0.773076	−0.386538	+	0.922274i	$0.626329\pi$
$18$	−13.7227	−3.23447
$19$	−5.69971	−1.30760	−0.653801	−	0.756666i	$-0.726827\pi$
$19$	−5.69971	−1.30760	−0.653801	+	0.756666i	$0.726827\pi$
$20$	0	0
$21$	3.88682	0.848174
$22$	9.44795	2.01431
$23$	6.17953	1.28852	0.644260	−	0.764806i	$-0.277165\pi$
$23$	6.17953	1.28852	0.644260	+	0.764806i	$0.277165\pi$
$24$	6.52650	1.33222
$25$	0	0
$26$	−0.0728710	−0.0142912
$27$	−9.55291	−1.83846
$28$	−3.81186	−0.720373
$29$	7.19361	1.33582	0.667910	−	0.744242i	$-0.267189\pi$
$29$	7.19361	1.33582	0.667910	+	0.744242i	$0.267189\pi$
$30$	0	0
$31$	2.81208	0.505064	0.252532	−	0.967588i	$-0.418737\pi$
$31$	2.81208	0.505064	0.252532	+	0.967588i	$0.418737\pi$
$32$	6.82825	1.20708
$33$	12.8273	2.23295
$34$	7.10433	1.21838
$35$	0	0
$36$	18.2717	3.04528
$37$	11.8869	1.95419	0.977096	−	0.212800i	$-0.0682584\pi$
$37$	11.8869	1.95419	0.977096	+	0.212800i	$0.0682584\pi$
$38$	12.7037	2.06081
$39$	−0.0989356	−0.0158424
$40$	0	0
$41$	−3.01031	−0.470131	−0.235065	−	0.971980i	$-0.575530\pi$
$41$	−3.01031	−0.470131	−0.235065	+	0.971980i	$0.575530\pi$
$42$	−8.66306	−1.33674
$43$	−11.6220	−1.77234	−0.886172	−	0.463357i	$-0.846645\pi$
$43$	−11.6220	−1.77234	−0.886172	+	0.463357i	$0.846645\pi$
$44$	−12.5799	−1.89649
$45$	0	0
$46$	−13.7731	−2.03073
$47$	−9.37541	−1.36754	−0.683772	−	0.729695i	$-0.739662\pi$
$47$	−9.37541	−1.36754	−0.683772	+	0.729695i	$0.739662\pi$
$48$	3.41415	0.492790
$49$	−5.35016	−0.764309
$50$	0	0
$51$	9.64541	1.35063
$52$	0.0970274	0.0134553
$53$	−4.47527	−0.614726	−0.307363	−	0.951592i	$-0.599447\pi$
$53$	−4.47527	−0.614726	−0.307363	+	0.951592i	$0.599447\pi$
$54$	21.2918	2.89745
$55$	0	0
$56$	2.77030	0.370196
$57$	17.2475	2.28449
$58$	−16.0333	−2.10528
$59$	−6.52124	−0.848993	−0.424497	−	0.905429i	$-0.639549\pi$
$59$	−6.52124	−0.848993	−0.424497	+	0.905429i	$0.639549\pi$
$60$	0	0
$61$	3.93428	0.503733	0.251867	−	0.967762i	$-0.418956\pi$
$61$	3.93428	0.503733	0.251867	+	0.967762i	$0.418956\pi$
$62$	−6.26764	−0.795991
$63$	−7.90829	−0.996351
$64$	−12.9625	−1.62031
$65$	0	0
$66$	−28.5898	−3.51917
$67$	−3.36275	−0.410825	−0.205412	−	0.978675i	$-0.565854\pi$
$67$	−3.36275	−0.410825	−0.205412	+	0.978675i	$0.565854\pi$
$68$	−9.45938	−1.14712
$69$	−18.6995	−2.25115
$70$	0	0
$71$	3.94676	0.468394	0.234197	−	0.972189i	$-0.424754\pi$
$71$	3.94676	0.468394	0.234197	+	0.972189i	$0.424754\pi$
$72$	−13.2791	−1.56495
$73$	0.394938	0.0462240	0.0231120	−	0.999733i	$-0.492643\pi$
$73$	0.394938	0.0462240	0.0231120	+	0.999733i	$0.492643\pi$
$74$	−26.4938	−3.07984
$75$	0	0
$76$	−16.9149	−1.94027
$77$	5.44479	0.620492
$78$	0.220510	0.0249679
$79$	5.02637	0.565510	0.282755	−	0.959192i	$-0.408752\pi$
$79$	5.02637	0.565510	0.282755	+	0.959192i	$0.408752\pi$
$80$	0	0
$81$	10.4367	1.15964
$82$	6.70945	0.740935
$83$	−6.36108	−0.698219	−0.349110	−	0.937082i	$-0.613516\pi$
$83$	−6.36108	−0.698219	−0.349110	+	0.937082i	$0.613516\pi$
$84$	11.5348	1.25855
$85$	0	0
$86$	25.9035	2.79325
$87$	−21.7681	−2.33379
$88$	9.14253	0.974597
$89$	8.91236	0.944708	0.472354	−	0.881409i	$-0.343404\pi$
$89$	8.91236	0.944708	0.472354	+	0.881409i	$0.343404\pi$
$90$	0	0
$91$	−0.0419951	−0.00440228
$92$	18.3388	1.91195
$93$	−8.50946	−0.882390
$94$	20.8962	2.15528
$95$	0	0
$96$	−20.6625	−2.10886
$97$	15.6077	1.58472	0.792361	−	0.610052i	$-0.208852\pi$
$97$	15.6077	1.58472	0.792361	+	0.610052i	$0.208852\pi$
$98$	11.9246	1.20457
$99$	−26.0990	−2.62304
$100$	0	0
$101$	16.7632	1.66800	0.834002	−	0.551762i	$-0.186044\pi$
$101$	16.7632	1.66800	0.834002	+	0.551762i	$0.186044\pi$
$102$	−21.4980	−2.12862
$103$	2.86345	0.282144	0.141072	−	0.989999i	$-0.454945\pi$
$103$	2.86345	0.282144	0.141072	+	0.989999i	$0.454945\pi$
$104$	−0.0705154	−0.00691460
$105$	0	0
$106$	9.97461	0.968820
$107$	1.02937	0.0995130	0.0497565	−	0.998761i	$-0.484155\pi$
$107$	1.02937	0.0995130	0.0497565	+	0.998761i	$0.484155\pi$
$108$	−28.3499	−2.72797
$109$	−10.2929	−0.985883	−0.492941	−	0.870063i	$-0.664078\pi$
$109$	−10.2929	−0.985883	−0.492941	+	0.870063i	$0.664078\pi$
$110$	0	0
$111$	−35.9702	−3.41414
$112$	1.44920	0.136937
$113$	12.8305	1.20699	0.603495	−	0.797367i	$-0.293774\pi$
$113$	12.8305	1.20699	0.603495	+	0.797367i	$0.293774\pi$
$114$	−38.4418	−3.60040
$115$	0	0
$116$	21.3483	1.98214
$117$	0.201298	0.0186101
$118$	14.5347	1.33803
$119$	4.09418	0.375313
$120$	0	0
$121$	6.96891	0.633537
$122$	−8.76884	−0.793893
$123$	9.10930	0.821358
$124$	8.34534	0.749433
$125$	0	0
$126$	17.6262	1.57027
$127$	13.2366	1.17456	0.587279	−	0.809385i	$-0.300199\pi$
$127$	13.2366	1.17456	0.587279	+	0.809385i	$0.300199\pi$
$128$	15.2346	1.34656
$129$	35.1687	3.09643
$130$	0	0
$131$	10.5920	0.925425	0.462713	−	0.886508i	$-0.346876\pi$
$131$	10.5920	0.925425	0.462713	+	0.886508i	$0.346876\pi$
$132$	38.0672	3.31333
$133$	7.32104	0.634815
$134$	7.49499	0.647468
$135$	0	0
$136$	6.87468	0.589499
$137$	3.82013	0.326376	0.163188	−	0.986595i	$-0.447822\pi$
$137$	3.82013	0.326376	0.163188	+	0.986595i	$0.447822\pi$
$138$	41.6779	3.54786
$139$	3.20940	0.272218	0.136109	−	0.990694i	$-0.456540\pi$
$139$	3.20940	0.272218	0.136109	+	0.990694i	$0.456540\pi$
$140$	0	0
$141$	28.3704	2.38921
$142$	−8.79664	−0.738198
$143$	−0.138592	−0.0115897
$144$	−6.94658	−0.578881
$145$	0	0
$146$	−0.880249	−0.0728499
$147$	16.1898	1.33531
$148$	35.2764	2.89970
$149$	11.8165	0.968042	0.484021	−	0.875056i	$-0.339176\pi$
$149$	11.8165	0.968042	0.484021	+	0.875056i	$0.339176\pi$
$150$	0	0
$151$	−8.86121	−0.721115	−0.360558	−	0.932737i	$-0.617414\pi$
$151$	−8.86121	−0.721115	−0.360558	+	0.932737i	$0.617414\pi$
$152$	12.2930	0.997095
$153$	−19.6250	−1.58658
$154$	−12.1355	−0.977907
$155$	0	0
$156$	−0.293609	−0.0235075
$157$	0.0926825	0.00739687	0.00369843	−	0.999993i	$-0.498823\pi$
$157$	0.0926825	0.00739687	0.00369843	+	0.999993i	$0.498823\pi$
$158$	−11.2029	−0.891255
$159$	13.5423	1.07398
$160$	0	0
$161$	−7.93735	−0.625551
$162$	−23.2617	−1.82761
$163$	−5.54121	−0.434021	−0.217010	−	0.976169i	$-0.569631\pi$
$163$	−5.54121	−0.434021	−0.217010	+	0.976169i	$0.569631\pi$
$164$	−8.93361	−0.697598
$165$	0	0
$166$	14.1778	1.10041
$167$	22.8426	1.76761	0.883806	−	0.467854i	$-0.154973\pi$
$167$	22.8426	1.76761	0.883806	+	0.467854i	$0.154973\pi$
$168$	−8.38302	−0.646764
$169$	−12.9989	−0.999918
$170$	0	0
$171$	−35.0925	−2.68359
$172$	−34.4904	−2.62987
$173$	−19.4621	−1.47968	−0.739840	−	0.672783i	$-0.765099\pi$
$173$	−19.4621	−1.47968	−0.739840	+	0.672783i	$0.765099\pi$
$174$	48.5174	3.67810
$175$	0	0
$176$	4.78266	0.360506
$177$	19.7335	1.48326
$178$	−19.8641	−1.48888
$179$	1.50622	0.112580	0.0562902	−	0.998414i	$-0.482073\pi$
$179$	1.50622	0.112580	0.0562902	+	0.998414i	$0.482073\pi$
$180$	0	0
$181$	−13.5177	−1.00476	−0.502380	−	0.864647i	$-0.667542\pi$
$181$	−13.5177	−1.00476	−0.502380	+	0.864647i	$0.667542\pi$
$182$	0.0935998	0.00693808
$183$	−11.9053	−0.880064
$184$	−13.3279	−0.982544
$185$	0	0
$186$	18.9661	1.39066
$187$	13.5116	0.988067
$188$	−27.8232	−2.02921
$189$	12.2703	0.892535
$190$	0	0
$191$	26.7463	1.93529	0.967646	−	0.252310i	$-0.0811904\pi$
$191$	26.7463	1.93529	0.967646	+	0.252310i	$0.0811904\pi$
$192$	39.2250	2.83082
$193$	−19.6630	−1.41538	−0.707688	−	0.706525i	$-0.750262\pi$
$193$	−19.6630	−1.41538	−0.707688	+	0.706525i	$0.750262\pi$
$194$	−34.7869	−2.49755
$195$	0	0
$196$	−15.8775	−1.13411
$197$	−1.00000	−0.0712470
$197$	−1.00000	−0.0712470
$198$	58.1701	4.13397
$199$	19.4206	1.37669	0.688345	−	0.725383i	$-0.258338\pi$
$199$	19.4206	1.37669	0.688345	+	0.725383i	$0.258338\pi$
$200$	0	0
$201$	10.1758	0.717746
$202$	−37.3624	−2.62881
$203$	−9.23990	−0.648514
$204$	28.6244	2.00411
$205$	0	0
$206$	−6.38214	−0.444665
$207$	38.0468	2.64443
$208$	−0.0368881	−0.00255773
$209$	24.1609	1.67124
$210$	0	0
$211$	9.62672	0.662731	0.331365	−	0.943502i	$-0.392491\pi$
$211$	9.62672	0.662731	0.331365	+	0.943502i	$0.392491\pi$
$212$	−13.2812	−0.912153
$213$	−11.9430	−0.818323
$214$	−2.29429	−0.156834
$215$	0	0
$216$	20.6035	1.40189
$217$	−3.61200	−0.245199
$218$	22.9411	1.55377
$219$	−1.19510	−0.0807572
$220$	0	0
$221$	−0.104214	−0.00701017
$222$	80.1713	5.38074
$223$	−27.1858	−1.82050	−0.910249	−	0.414061i	$-0.864110\pi$
$223$	−27.1858	−1.82050	−0.910249	+	0.414061i	$0.864110\pi$
$224$	−8.77061	−0.586011
$225$	0	0
$226$	−28.5969	−1.90224
$227$	20.7178	1.37509	0.687543	−	0.726144i	$-0.258689\pi$
$227$	20.7178	1.37509	0.687543	+	0.726144i	$0.258689\pi$
$228$	51.1850	3.38981
$229$	−16.6170	−1.09808	−0.549042	−	0.835795i	$-0.685007\pi$
$229$	−16.6170	−1.09808	−0.549042	+	0.835795i	$0.685007\pi$
$230$	0	0
$231$	−16.4761	−1.08405
$232$	−15.5150	−1.01861
$233$	0.693861	0.0454563	0.0227282	−	0.999742i	$-0.492765\pi$
$233$	0.693861	0.0454563	0.0227282	+	0.999742i	$0.492765\pi$
$234$	−0.448660	−0.0293298
$235$	0	0
$236$	−19.3529	−1.25977
$237$	−15.2100	−0.987994
$238$	−9.12522	−0.591500
$239$	29.2959	1.89500	0.947498	−	0.319763i	$-0.103603\pi$
$239$	29.2959	1.89500	0.947498	+	0.319763i	$0.103603\pi$
$240$	0	0
$241$	−2.16835	−0.139676	−0.0698378	−	0.997558i	$-0.522248\pi$
$241$	−2.16835	−0.139676	−0.0698378	+	0.997558i	$0.522248\pi$
$242$	−15.5325	−0.998467
$243$	−2.92326	−0.187528
$244$	11.6757	0.747458
$245$	0	0
$246$	−20.3031	−1.29448
$247$	−0.186351	−0.0118572
$248$	−6.06504	−0.385130
$249$	19.2489	1.21985
$250$	0	0
$251$	−14.5046	−0.915521	−0.457760	−	0.889076i	$-0.651348\pi$
$251$	−14.5046	−0.915521	−0.457760	+	0.889076i	$0.651348\pi$
$252$	−23.4692	−1.47842
$253$	−26.1949	−1.64686
$254$	−29.5021	−1.85113
$255$	0	0
$256$	−8.03043	−0.501902
$257$	−9.76616	−0.609196	−0.304598	−	0.952481i	$-0.598522\pi$
$257$	−9.76616	−0.609196	−0.304598	+	0.952481i	$0.598522\pi$
$258$	−78.3850	−4.88004
$259$	−15.2682	−0.948721
$260$	0	0
$261$	44.2903	2.74150
$262$	−23.6077	−1.45849
$263$	−6.64141	−0.409527	−0.204764	−	0.978811i	$-0.565643\pi$
$263$	−6.64141	−0.409527	−0.204764	+	0.978811i	$0.565643\pi$
$264$	−27.6656	−1.70270
$265$	0	0
$266$	−16.3173	−1.00048
$267$	−26.9691	−1.65048
$268$	−9.97954	−0.609597
$269$	5.15503	0.314308	0.157154	−	0.987574i	$-0.449768\pi$
$269$	5.15503	0.314308	0.157154	+	0.987574i	$0.449768\pi$
$270$	0	0
$271$	−1.63478	−0.0993061	−0.0496530	−	0.998767i	$-0.515812\pi$
$271$	−1.63478	−0.0993061	−0.0496530	+	0.998767i	$0.515812\pi$
$272$	3.59629	0.218057
$273$	0.127079	0.00769115
$274$	−8.51442	−0.514375
$275$	0	0
$276$	−55.4940	−3.34034
$277$	10.2128	0.613629	0.306814	−	0.951769i	$-0.400737\pi$
$277$	10.2128	0.613629	0.306814	+	0.951769i	$0.400737\pi$
$278$	−7.15320	−0.429020
$279$	17.3137	1.03654
$280$	0	0
$281$	−15.4518	−0.921779	−0.460890	−	0.887457i	$-0.652470\pi$
$281$	−15.4518	−0.921779	−0.460890	+	0.887457i	$0.652470\pi$
$282$	−63.2326	−3.76545
$283$	12.8317	0.762765	0.381383	−	0.924417i	$-0.375448\pi$
$283$	12.8317	0.762765	0.381383	+	0.924417i	$0.375448\pi$
$284$	11.7127	0.695020
$285$	0	0
$286$	0.308898	0.0182655
$287$	3.86662	0.228239
$288$	42.0409	2.47728
$289$	−6.84001	−0.402354
$290$	0	0
$291$	−47.2295	−2.76864
$292$	1.17205	0.0685889
$293$	13.1558	0.768571	0.384285	−	0.923214i	$-0.374448\pi$
$293$	13.1558	0.768571	0.384285	+	0.923214i	$0.374448\pi$
$294$	−36.0843	−2.10448
$295$	0	0
$296$	−25.6374	−1.49014
$297$	40.4945	2.34973
$298$	−26.3369	−1.52565
$299$	0.202038	0.0116842
$300$	0	0
$301$	14.9280	0.860437
$302$	19.7501	1.13649
$303$	−50.7262	−2.91414
$304$	6.43074	0.368828
$305$	0	0
$306$	43.7407	2.50049
$307$	5.07199	0.289474	0.144737	−	0.989470i	$-0.453766\pi$
$307$	5.07199	0.289474	0.144737	+	0.989470i	$0.453766\pi$
$308$	16.1584	0.920708
$309$	−8.66491	−0.492929
$310$	0	0
$311$	5.96977	0.338515	0.169257	−	0.985572i	$-0.445863\pi$
$311$	5.96977	0.338515	0.169257	+	0.985572i	$0.445863\pi$
$312$	0.213382	0.0120804
$313$	14.9712	0.846224	0.423112	−	0.906077i	$-0.360938\pi$
$313$	14.9712	0.846224	0.423112	+	0.906077i	$0.360938\pi$
$314$	−0.206573	−0.0116576
$315$	0	0
$316$	14.9166	0.839125
$317$	10.8677	0.610388	0.305194	−	0.952290i	$-0.401279\pi$
$317$	10.8677	0.610388	0.305194	+	0.952290i	$0.401279\pi$
$318$	−30.1836	−1.69261
$319$	−30.4935	−1.70731
$320$	0	0
$321$	−3.11491	−0.173857
$322$	17.6910	0.985880
$323$	18.1677	1.01088
$324$	30.9729	1.72071
$325$	0	0
$326$	12.3504	0.684025
$327$	31.1468	1.72242
$328$	6.49257	0.358492
$329$	12.0423	0.663916
$330$	0	0
$331$	25.5182	1.40261	0.701304	−	0.712862i	$-0.252602\pi$
$331$	25.5182	1.40261	0.701304	+	0.712862i	$0.252602\pi$
$332$	−18.8776	−1.03604
$333$	73.1864	4.01059
$334$	−50.9122	−2.78579
$335$	0	0
$336$	−4.38534	−0.239240
$337$	26.3413	1.43490	0.717450	−	0.696610i	$-0.245309\pi$
$337$	26.3413	1.43490	0.717450	+	0.696610i	$0.245309\pi$
$338$	28.9724	1.57589
$339$	−38.8255	−2.10871
$340$	0	0
$341$	−11.9203	−0.645522
$342$	78.2152	4.22939
$343$	15.8633	0.856537
$344$	25.0662	1.35148
$345$	0	0
$346$	43.3778	2.33200
$347$	11.8578	0.636562	0.318281	−	0.947996i	$-0.396894\pi$
$347$	11.8578	0.636562	0.318281	+	0.947996i	$0.396894\pi$
$348$	−64.6007	−3.46296
$349$	−8.47396	−0.453601	−0.226800	−	0.973941i	$-0.572826\pi$
$349$	−8.47396	−0.453601	−0.226800	+	0.973941i	$0.572826\pi$
$350$	0	0
$351$	−0.312330	−0.0166709
$352$	−28.9448	−1.54276
$353$	−17.4148	−0.926894	−0.463447	−	0.886125i	$-0.653388\pi$
$353$	−17.4148	−0.926894	−0.463447	+	0.886125i	$0.653388\pi$
$354$	−43.9826	−2.33765
$355$	0	0
$356$	26.4490	1.40179
$357$	−12.3891	−0.655703
$358$	−3.35711	−0.177429
$359$	10.3720	0.547413	0.273706	−	0.961813i	$-0.411750\pi$
$359$	10.3720	0.547413	0.273706	+	0.961813i	$0.411750\pi$
$360$	0	0
$361$	13.4866	0.709823
$362$	30.1286	1.58352
$363$	−21.0882	−1.10684
$364$	−0.124628	−0.00653227
$365$	0	0
$366$	26.5348	1.38700
$367$	−33.3869	−1.74278	−0.871392	−	0.490588i	$-0.836782\pi$
$367$	−33.3869	−1.74278	−0.871392	+	0.490588i	$0.836782\pi$
$368$	−6.97210	−0.363446
$369$	−18.5342	−0.964850
$370$	0	0
$371$	5.74831	0.298437
$372$	−25.2533	−1.30932
$373$	22.5229	1.16619	0.583096	−	0.812403i	$-0.301841\pi$
$373$	22.5229	1.16619	0.583096	+	0.812403i	$0.301841\pi$
$374$	−30.1151	−1.55721
$375$	0	0
$376$	20.2207	1.04280
$377$	0.235193	0.0121131
$378$	−27.3484	−1.40665
$379$	−9.58707	−0.492455	−0.246227	−	0.969212i	$-0.579191\pi$
$379$	−9.58707	−0.492455	−0.246227	+	0.969212i	$0.579191\pi$
$380$	0	0
$381$	−40.0544	−2.05205
$382$	−59.6129	−3.05006
$383$	3.52599	0.180170	0.0900848	−	0.995934i	$-0.471286\pi$
$383$	3.52599	0.180170	0.0900848	+	0.995934i	$0.471286\pi$
$384$	−46.1006	−2.35256
$385$	0	0
$386$	43.8255	2.23066
$387$	−71.5557	−3.63738
$388$	46.3186	2.35147
$389$	−9.44773	−0.479019	−0.239509	−	0.970894i	$-0.576987\pi$
$389$	−9.44773	−0.479019	−0.239509	+	0.970894i	$0.576987\pi$
$390$	0	0
$391$	−19.6971	−0.996124
$392$	11.5391	0.582814
$393$	−32.0517	−1.61680
$394$	2.22883	0.112287
$395$	0	0
$396$	−77.4532	−3.89217
$397$	−21.3370	−1.07087	−0.535437	−	0.844575i	$-0.679853\pi$
$397$	−21.3370	−1.07087	−0.535437	+	0.844575i	$0.679853\pi$
$398$	−43.2852	−2.16969
$399$	−22.1537	−1.10907
$400$	0	0
$401$	35.9093	1.79322	0.896611	−	0.442819i	$-0.146021\pi$
$401$	35.9093	1.79322	0.896611	+	0.442819i	$0.146021\pi$
$402$	−22.6801	−1.13118
$403$	0.0919403	0.00457987
$404$	49.7478	2.47505
$405$	0	0
$406$	20.5941	1.02207
$407$	−50.3882	−2.49765
$408$	−20.8030	−1.02990
$409$	−8.75900	−0.433105	−0.216552	−	0.976271i	$-0.569481\pi$
$409$	−8.75900	−0.433105	−0.216552	+	0.976271i	$0.569481\pi$
$410$	0	0
$411$	−11.5599	−0.570206
$412$	8.49779	0.418656
$413$	8.37627	0.412169
$414$	−84.7997	−4.16768
$415$	0	0
$416$	0.223248	0.0109456
$417$	−9.71176	−0.475587
$418$	−53.8505	−2.63391
$419$	−39.8981	−1.94915	−0.974575	−	0.224063i	$-0.928068\pi$
$419$	−39.8981	−1.94915	−0.974575	+	0.224063i	$0.928068\pi$
$420$	0	0
$421$	−1.39340	−0.0679103	−0.0339552	−	0.999423i	$-0.510810\pi$
$421$	−1.39340	−0.0679103	−0.0339552	+	0.999423i	$0.510810\pi$
$422$	−21.4563	−1.04448
$423$	−57.7235	−2.80661
$424$	9.65218	0.468751
$425$	0	0
$426$	26.6190	1.28969
$427$	−5.05342	−0.244552
$428$	3.05484	0.147661
$429$	0.419385	0.0202481
$430$	0	0
$431$	28.4874	1.37219	0.686094	−	0.727513i	$-0.259324\pi$
$431$	28.4874	1.37219	0.686094	+	0.727513i	$0.259324\pi$
$432$	10.7781	0.518564
$433$	−4.19554	−0.201625	−0.100812	−	0.994905i	$-0.532144\pi$
$433$	−4.19554	−0.201625	−0.100812	+	0.994905i	$0.532144\pi$
$434$	8.05053	0.386438
$435$	0	0
$436$	−30.5460	−1.46289
$437$	−35.2215	−1.68487
$438$	2.66367	0.127275
$439$	−0.288756	−0.0137816	−0.00689079	−	0.999976i	$-0.502193\pi$
$439$	−0.288756	−0.0137816	−0.00689079	+	0.999976i	$0.502193\pi$
$440$	0	0
$441$	−32.9404	−1.56859
$442$	0.232274	0.0110482
$443$	−2.51971	−0.119715	−0.0598576	−	0.998207i	$-0.519065\pi$
$443$	−2.51971	−0.119715	−0.0598576	+	0.998207i	$0.519065\pi$
$444$	−106.748	−5.06602
$445$	0	0
$446$	60.5926	2.86914
$447$	−35.7571	−1.69125
$448$	16.6498	0.786628
$449$	11.0944	0.523579	0.261790	−	0.965125i	$-0.415687\pi$
$449$	11.0944	0.523579	0.261790	+	0.965125i	$0.415687\pi$
$450$	0	0
$451$	12.7606	0.600874
$452$	38.0767	1.79098
$453$	26.8144	1.25985
$454$	−46.1763	−2.16716
$455$	0	0
$456$	−37.1991	−1.74201
$457$	−11.7408	−0.549211	−0.274606	−	0.961557i	$-0.588547\pi$
$457$	−11.7408	−0.549211	−0.274606	+	0.961557i	$0.588547\pi$
$458$	37.0365	1.73060
$459$	30.4496	1.42127
$460$	0	0
$461$	−22.3702	−1.04188	−0.520941	−	0.853593i	$-0.674419\pi$
$461$	−22.3702	−1.04188	−0.520941	+	0.853593i	$0.674419\pi$
$462$	36.7225	1.70848
$463$	−33.4077	−1.55259	−0.776294	−	0.630371i	$-0.782903\pi$
$463$	−33.4077	−1.55259	−0.776294	+	0.630371i	$0.782903\pi$
$464$	−8.11625	−0.376787
$465$	0	0
$466$	−1.54650	−0.0716400
$467$	19.6617	0.909837	0.454919	−	0.890533i	$-0.349668\pi$
$467$	19.6617	0.909837	0.454919	+	0.890533i	$0.349668\pi$
$468$	0.597388	0.0276143
$469$	4.31931	0.199447
$470$	0	0
$471$	−0.280461	−0.0129229
$472$	14.0649	0.647389
$473$	49.2655	2.26523
$474$	33.9004	1.55710
$475$	0	0
$476$	12.1502	0.556903
$477$	−27.5538	−1.26160
$478$	−65.2956	−2.98655
$479$	−39.2770	−1.79461	−0.897306	−	0.441410i	$-0.854479\pi$
$479$	−39.2770	−1.79461	−0.897306	+	0.441410i	$0.854479\pi$
$480$	0	0
$481$	0.388639	0.0177204
$482$	4.83288	0.220131
$483$	24.0187	1.09289
$484$	20.6814	0.940066
$485$	0	0
$486$	6.51545	0.295547
$487$	−7.76387	−0.351815	−0.175907	−	0.984407i	$-0.556286\pi$
$487$	−7.76387	−0.351815	−0.175907	+	0.984407i	$0.556286\pi$
$488$	−8.48538	−0.384115
$489$	16.7679	0.758271
$490$	0	0
$491$	−35.9660	−1.62312	−0.811562	−	0.584267i	$-0.801382\pi$
$491$	−35.9660	−1.62312	−0.811562	+	0.584267i	$0.801382\pi$
$492$	27.0334	1.21876
$493$	−22.9294	−1.03269
$494$	0.415343	0.0186872
$495$	0	0
$496$	−3.17275	−0.142461
$497$	−5.06945	−0.227396
$498$	−42.9024	−1.92250
$499$	−12.6195	−0.564926	−0.282463	−	0.959278i	$-0.591151\pi$
$499$	−12.6195	−0.564926	−0.282463	+	0.959278i	$0.591151\pi$
$500$	0	0
$501$	−69.1225	−3.08816
$502$	32.3282	1.44288
$503$	−41.7449	−1.86131	−0.930656	−	0.365895i	$-0.880763\pi$
$503$	−41.7449	−1.86131	−0.930656	+	0.365895i	$0.880763\pi$
$504$	17.0564	0.759754
$505$	0	0
$506$	58.3838	2.59548
$507$	39.3353	1.74694
$508$	39.2819	1.74285
$509$	−9.54868	−0.423238	−0.211619	−	0.977352i	$-0.567874\pi$
$509$	−9.54868	−0.423238	−0.211619	+	0.977352i	$0.567874\pi$
$510$	0	0
$511$	−0.507282	−0.0224408
$512$	−12.5708	−0.555558
$513$	54.4488	2.40397
$514$	21.7671	0.960105
$515$	0	0
$516$	104.369	4.59460
$517$	39.7421	1.74786
$518$	34.0302	1.49520
$519$	58.8932	2.58512
$520$	0	0
$521$	9.48197	0.415412	0.207706	−	0.978191i	$-0.433400\pi$
$521$	9.48197	0.415412	0.207706	+	0.978191i	$0.433400\pi$
$522$	−98.7156	−4.32066
$523$	−21.5237	−0.941164	−0.470582	−	0.882356i	$-0.655956\pi$
$523$	−21.5237	−0.941164	−0.470582	+	0.882356i	$0.655956\pi$
$524$	31.4335	1.37318
$525$	0	0
$526$	14.8026	0.645423
$527$	−8.96343	−0.390453
$528$	−14.4725	−0.629835
$529$	15.1866	0.660285
$530$	0	0
$531$	−40.1507	−1.74239
$532$	21.7265	0.941962
$533$	−0.0984212	−0.00426310
$534$	60.1096	2.60120
$535$	0	0
$536$	7.25270	0.313269
$537$	−4.55788	−0.196687
$538$	−11.4897	−0.495355
$539$	22.6792	0.976863
$540$	0	0
$541$	5.18867	0.223078	0.111539	−	0.993760i	$-0.464422\pi$
$541$	5.18867	0.223078	0.111539	+	0.993760i	$0.464422\pi$
$542$	3.64365	0.156508
$543$	40.9050	1.75540
$544$	−21.7649	−0.933161
$545$	0	0
$546$	−0.283237	−0.0121214
$547$	−5.82831	−0.249201	−0.124600	−	0.992207i	$-0.539765\pi$
$547$	−5.82831	−0.249201	−0.124600	+	0.992207i	$0.539765\pi$
$548$	11.3369	0.484289
$549$	24.2230	1.03381
$550$	0	0
$551$	−41.0014	−1.74672
$552$	40.3307	1.71659
$553$	−6.45616	−0.274544
$554$	−22.7626	−0.967090
$555$	0	0
$556$	9.52445	0.403927
$557$	7.89296	0.334435	0.167218	−	0.985920i	$-0.446522\pi$
$557$	7.89296	0.334435	0.167218	+	0.985920i	$0.446522\pi$
$558$	−38.5893	−1.63361
$559$	−0.379980	−0.0160714
$560$	0	0
$561$	−40.8867	−1.72624
$562$	34.4395	1.45274
$563$	30.1329	1.26995	0.634974	−	0.772533i	$-0.281011\pi$
$563$	30.1329	1.26995	0.634974	+	0.772533i	$0.281011\pi$
$564$	84.1940	3.54521
$565$	0	0
$566$	−28.5997	−1.20213
$567$	−13.4056	−0.562981
$568$	−8.51228	−0.357167
$569$	−0.652674	−0.0273615	−0.0136808	−	0.999906i	$-0.504355\pi$
$569$	−0.652674	−0.0273615	−0.0136808	+	0.999906i	$0.504355\pi$
$570$	0	0
$571$	8.64196	0.361655	0.180827	−	0.983515i	$-0.442122\pi$
$571$	8.64196	0.361655	0.180827	+	0.983515i	$0.442122\pi$
$572$	−0.411297	−0.0171972
$573$	−80.9352	−3.38112
$574$	−8.61802	−0.359709
$575$	0	0
$576$	−79.8088	−3.32536
$577$	−39.1576	−1.63015	−0.815076	−	0.579354i	$-0.803305\pi$
$577$	−39.1576	−1.63015	−0.815076	+	0.579354i	$0.803305\pi$
$578$	15.2452	0.634117
$579$	59.5011	2.47278
$580$	0	0
$581$	8.17055	0.338972
$582$	105.266	4.36343
$583$	18.9706	0.785681
$584$	−0.851794	−0.0352475
$585$	0	0
$586$	−29.3220	−1.21128
$587$	19.2485	0.794472	0.397236	−	0.917716i	$-0.369969\pi$
$587$	19.2485	0.794472	0.397236	+	0.917716i	$0.369969\pi$
$588$	48.0460	1.98139
$589$	−16.0280	−0.660423
$590$	0	0
$591$	3.02604	0.124475
$592$	−13.4115	−0.551208
$593$	44.9942	1.84769	0.923845	−	0.382768i	$-0.125029\pi$
$593$	44.9942	1.84769	0.923845	+	0.382768i	$0.125029\pi$
$594$	−90.2554	−3.70322
$595$	0	0
$596$	35.0674	1.43642
$597$	−58.7675	−2.40519
$598$	−0.450308	−0.0184145
$599$	36.4837	1.49068	0.745342	−	0.666682i	$-0.232286\pi$
$599$	36.4837	1.49068	0.745342	+	0.666682i	$0.232286\pi$
$600$	0	0
$601$	36.7049	1.49722	0.748612	−	0.663008i	$-0.230720\pi$
$601$	36.7049	1.49722	0.748612	+	0.663008i	$0.230720\pi$
$602$	−33.2720	−1.35607
$603$	−20.7041	−0.843137
$604$	−26.2972	−1.07002
$605$	0	0
$606$	113.060	4.59274
$607$	−5.22622	−0.212126	−0.106063	−	0.994359i	$-0.533824\pi$
$607$	−5.22622	−0.212126	−0.106063	+	0.994359i	$0.533824\pi$
$608$	−38.9190	−1.57837
$609$	27.9603	1.13301
$610$	0	0
$611$	−0.306527	−0.0124008
$612$	−58.2405	−2.35423
$613$	−2.54201	−0.102671	−0.0513353	−	0.998681i	$-0.516348\pi$
$613$	−2.54201	−0.102671	−0.0513353	+	0.998681i	$0.516348\pi$
$614$	−11.3046	−0.456216
$615$	0	0
$616$	−11.7432	−0.473148
$617$	22.4800	0.905011	0.452506	−	0.891762i	$-0.350530\pi$
$617$	22.4800	0.905011	0.452506	+	0.891762i	$0.350530\pi$
$618$	19.3126	0.776866
$619$	13.4275	0.539696	0.269848	−	0.962903i	$-0.413026\pi$
$619$	13.4275	0.539696	0.269848	+	0.962903i	$0.413026\pi$
$620$	0	0
$621$	−59.0325	−2.36889
$622$	−13.3056	−0.533506
$623$	−11.4476	−0.458637
$624$	0.111625	0.00446857
$625$	0	0
$626$	−33.3683	−1.33367
$627$	−73.1118	−2.91980
$628$	0.275051	0.0109757
$629$	−37.8891	−1.51074
$630$	0	0
$631$	−32.2619	−1.28433	−0.642163	−	0.766568i	$-0.721963\pi$
$631$	−32.2619	−1.28433	−0.642163	+	0.766568i	$0.721963\pi$
$632$	−10.8408	−0.431222
$633$	−29.1308	−1.15785
$634$	−24.2221	−0.961984
$635$	0	0
$636$	40.1893	1.59361
$637$	−0.174922	−0.00693068
$638$	67.9648	2.69075
$639$	24.2998	0.961285
$640$	0	0
$641$	9.71139	0.383577	0.191788	−	0.981436i	$-0.438571\pi$
$641$	9.71139	0.383577	0.191788	+	0.981436i	$0.438571\pi$
$642$	6.94260	0.274003
$643$	11.3068	0.445897	0.222949	−	0.974830i	$-0.428432\pi$
$643$	11.3068	0.445897	0.222949	+	0.974830i	$0.428432\pi$
$644$	−23.5555	−0.928216
$645$	0	0
$646$	−40.4926	−1.59316
$647$	−33.4004	−1.31311	−0.656553	−	0.754280i	$-0.727986\pi$
$647$	−33.4004	−1.31311	−0.656553	+	0.754280i	$0.727986\pi$
$648$	−22.5098	−0.884267
$649$	27.6434	1.08510
$650$	0	0
$651$	10.9301	0.428383
$652$	−16.4445	−0.644016
$653$	−39.2091	−1.53437	−0.767185	−	0.641426i	$-0.778343\pi$
$653$	−39.2091	−1.53437	−0.767185	+	0.641426i	$0.778343\pi$
$654$	−69.4208	−2.71457
$655$	0	0
$656$	3.39640	0.132607
$657$	2.43159	0.0948655
$658$	−26.8403	−1.04634
$659$	13.5095	0.526254	0.263127	−	0.964761i	$-0.415246\pi$
$659$	13.5095	0.526254	0.263127	+	0.964761i	$0.415246\pi$
$660$	0	0
$661$	−33.2439	−1.29304	−0.646519	−	0.762898i	$-0.723776\pi$
$661$	−33.2439	−1.29304	−0.646519	+	0.762898i	$0.723776\pi$
$662$	−56.8757	−2.21054
$663$	0.315355	0.0122474
$664$	13.7194	0.532418
$665$	0	0
$666$	−163.120	−6.32077
$667$	44.4531	1.72123
$668$	67.7893	2.62285
$669$	82.2654	3.18056
$670$	0	0
$671$	−16.6773	−0.643821
$672$	26.5402	1.02381
$673$	−17.3913	−0.670386	−0.335193	−	0.942149i	$-0.608802\pi$
$673$	−17.3913	−0.670386	−0.335193	+	0.942149i	$0.608802\pi$
$674$	−58.7102	−2.26143
$675$	0	0
$676$	−38.5766	−1.48372
$677$	1.49868	0.0575991	0.0287995	−	0.999585i	$-0.490832\pi$
$677$	1.49868	0.0575991	0.0287995	+	0.999585i	$0.490832\pi$
$678$	86.5354	3.32337
$679$	−20.0475	−0.769351
$680$	0	0
$681$	−62.6927	−2.40239
$682$	26.5684	1.01736
$683$	−38.8528	−1.48666	−0.743331	−	0.668924i	$-0.766755\pi$
$683$	−38.8528	−1.48666	−0.743331	+	0.668924i	$0.766755\pi$
$684$	−104.143	−3.98202
$685$	0	0
$686$	−35.3565	−1.34992
$687$	50.2837	1.91844
$688$	13.1127	0.499915
$689$	−0.146318	−0.00557427
$690$	0	0
$691$	−31.4387	−1.19598	−0.597992	−	0.801502i	$-0.704035\pi$
$691$	−31.4387	−1.19598	−0.597992	+	0.801502i	$0.704035\pi$
$692$	−57.7573	−2.19560
$693$	33.5231	1.27344
$694$	−26.4291	−1.00323
$695$	0	0
$696$	46.9490	1.77960
$697$	9.59527	0.363447
$698$	18.8870	0.714883
$699$	−2.09965	−0.0794160
$700$	0	0
$701$	−14.6371	−0.552835	−0.276418	−	0.961038i	$-0.589147\pi$
$701$	−14.6371	−0.552835	−0.276418	+	0.961038i	$0.589147\pi$
$702$	0.696130	0.0262737
$703$	−67.7517	−2.55530
$704$	54.9476	2.07092
$705$	0	0
$706$	38.8145	1.46080
$707$	−21.5317	−0.809783
$708$	58.5627	2.20092
$709$	3.84602	0.144440	0.0722201	−	0.997389i	$-0.476992\pi$
$709$	3.84602	0.144440	0.0722201	+	0.997389i	$0.476992\pi$
$710$	0	0
$711$	30.9468	1.16060
$712$	−19.2220	−0.720375
$713$	17.3773	0.650786
$714$	27.6133	1.03340
$715$	0	0
$716$	4.46998	0.167051
$717$	−88.6505	−3.31072
$718$	−23.1174	−0.862733
$719$	−43.8516	−1.63539	−0.817693	−	0.575654i	$-0.804748\pi$
$719$	−43.8516	−1.63539	−0.817693	+	0.575654i	$0.804748\pi$
$720$	0	0
$721$	−3.67798	−0.136975
$722$	−30.0594	−1.11870
$723$	6.56150	0.244025
$724$	−40.1160	−1.49090
$725$	0	0
$726$	47.0019	1.74440
$727$	35.0618	1.30037	0.650186	−	0.759775i	$-0.274691\pi$
$727$	35.0618	1.30037	0.650186	+	0.759775i	$0.274691\pi$
$728$	0.0905741	0.00335690
$729$	−22.4643	−0.832012
$730$	0	0
$731$	37.0449	1.37016
$732$	−35.3310	−1.30587
$733$	17.3916	0.642375	0.321187	−	0.947016i	$-0.395918\pi$
$733$	17.3916	0.642375	0.321187	+	0.947016i	$0.395918\pi$
$734$	74.4137	2.74666
$735$	0	0
$736$	42.1954	1.55534
$737$	14.2546	0.525075
$738$	41.3095	1.52062
$739$	−35.9634	−1.32293	−0.661467	−	0.749974i	$-0.730066\pi$
$739$	−35.9634	−1.32293	−0.661467	+	0.749974i	$0.730066\pi$
$740$	0	0
$741$	0.563904	0.0207155
$742$	−12.8120	−0.470343
$743$	−7.32782	−0.268832	−0.134416	−	0.990925i	$-0.542916\pi$
$743$	−7.32782	−0.268832	−0.134416	+	0.990925i	$0.542916\pi$
$744$	18.3530	0.672855
$745$	0	0
$746$	−50.1997	−1.83794
$747$	−39.1646	−1.43296
$748$	40.0981	1.46613
$749$	−1.32218	−0.0483116
$750$	0	0
$751$	−4.38020	−0.159836	−0.0799179	−	0.996801i	$-0.525466\pi$
$751$	−4.38020	−0.159836	−0.0799179	+	0.996801i	$0.525466\pi$
$752$	10.5779	0.385736
$753$	43.8914	1.59949
$754$	−0.524205	−0.0190904
$755$	0	0
$756$	36.4143	1.32438
$757$	−14.6109	−0.531042	−0.265521	−	0.964105i	$-0.585544\pi$
$757$	−14.6109	−0.531042	−0.265521	+	0.964105i	$0.585544\pi$
$758$	21.3679	0.776118
$759$	79.2666	2.87720
$760$	0	0
$761$	25.1421	0.911401	0.455700	−	0.890133i	$-0.349389\pi$
$761$	25.1421	0.911401	0.455700	+	0.890133i	$0.349389\pi$
$762$	89.2745	3.23407
$763$	13.2208	0.478626
$764$	79.3742	2.87166
$765$	0	0
$766$	−7.85883	−0.283951
$767$	−0.213210	−0.00769858
$768$	24.3004	0.876864
$769$	−8.07722	−0.291272	−0.145636	−	0.989338i	$-0.546523\pi$
$769$	−8.07722	−0.291272	−0.145636	+	0.989338i	$0.546523\pi$
$770$	0	0
$771$	29.5528	1.06432
$772$	−58.3535	−2.10019
$773$	−35.4654	−1.27560	−0.637802	−	0.770201i	$-0.720156\pi$
$773$	−35.4654	−1.27560	−0.637802	+	0.770201i	$0.720156\pi$
$774$	159.485	5.73258
$775$	0	0
$776$	−33.6624	−1.20841
$777$	46.2022	1.65749
$778$	21.0574	0.754943
$779$	17.1579	0.614744
$780$	0	0
$781$	−16.7302	−0.598654
$782$	43.9014	1.56991
$783$	−68.7199	−2.45585
$784$	6.03637	0.215585
$785$	0	0
$786$	71.4378	2.54810
$787$	−36.2547	−1.29234	−0.646171	−	0.763193i	$-0.723631\pi$
$787$	−36.2547	−1.29234	−0.646171	+	0.763193i	$0.723631\pi$
$788$	−2.96767	−0.105719
$789$	20.0972	0.715478
$790$	0	0
$791$	−16.4802	−0.585969
$792$	56.2897	2.00017
$793$	0.128630	0.00456780
$794$	47.5566	1.68772
$795$	0	0
$796$	57.6340	2.04278
$797$	−37.6915	−1.33510	−0.667551	−	0.744565i	$-0.732657\pi$
$797$	−37.6915	−1.33510	−0.667551	+	0.744565i	$0.732657\pi$
$798$	49.3769	1.74792
$799$	29.8839	1.05722
$800$	0	0
$801$	54.8725	1.93883
$802$	−80.0355	−2.82615
$803$	−1.67413	−0.0590788
$804$	30.1985	1.06502
$805$	0	0
$806$	−0.204919	−0.00721797
$807$	−15.5993	−0.549122
$808$	−36.1546	−1.27191
$809$	−40.7120	−1.43136	−0.715678	−	0.698430i	$-0.753882\pi$
$809$	−40.7120	−1.43136	−0.715678	+	0.698430i	$0.753882\pi$
$810$	0	0
$811$	−50.3877	−1.76935	−0.884676	−	0.466207i	$-0.845620\pi$
$811$	−50.3877	−1.76935	−0.884676	+	0.466207i	$0.845620\pi$
$812$	−27.4210	−0.962289
$813$	4.94692	0.173496
$814$	112.307	3.93634
$815$	0	0
$816$	−10.8825	−0.380964
$817$	66.2421	2.31752
$818$	19.5223	0.682581
$819$	−0.258560	−0.00903481
$820$	0	0
$821$	27.4410	0.957697	0.478849	−	0.877897i	$-0.341054\pi$
$821$	27.4410	0.957697	0.478849	+	0.877897i	$0.341054\pi$
$822$	25.7650	0.898656
$823$	35.1356	1.22475	0.612375	−	0.790567i	$-0.290214\pi$
$823$	35.1356	1.22475	0.612375	+	0.790567i	$0.290214\pi$
$824$	−6.17583	−0.215145
$825$	0	0
$826$	−18.6693	−0.649587
$827$	−6.76979	−0.235409	−0.117704	−	0.993049i	$-0.537553\pi$
$827$	−6.76979	−0.235409	−0.117704	+	0.993049i	$0.537553\pi$
$828$	112.910	3.92391
$829$	−37.4923	−1.30216	−0.651080	−	0.759009i	$-0.725684\pi$
$829$	−37.4923	−1.30216	−0.651080	+	0.759009i	$0.725684\pi$
$830$	0	0
$831$	−30.9044	−1.07206
$832$	−0.423805	−0.0146928
$833$	17.0535	0.590869
$834$	21.6458	0.749534
$835$	0	0
$836$	71.7017	2.47986
$837$	−26.8635	−0.928540
$838$	88.9260	3.07190
$839$	39.5332	1.36484	0.682419	−	0.730961i	$-0.260928\pi$
$839$	39.5332	1.36484	0.682419	+	0.730961i	$0.260928\pi$
$840$	0	0
$841$	22.7480	0.784413
$842$	3.10566	0.107028
$843$	46.7578	1.61043
$844$	28.5690	0.983385
$845$	0	0
$846$	128.656	4.42328
$847$	−8.95128	−0.307570
$848$	5.04926	0.173392
$849$	−38.8292	−1.33261
$850$	0	0
$851$	73.4553	2.51802
$852$	−35.4430	−1.21426
$853$	48.5066	1.66084	0.830418	−	0.557141i	$-0.188102\pi$
$853$	48.5066	1.66084	0.830418	+	0.557141i	$0.188102\pi$
$854$	11.2632	0.385419
$855$	0	0
$856$	−2.22012	−0.0758823
$857$	23.5135	0.803205	0.401602	−	0.915814i	$-0.368453\pi$
$857$	23.5135	0.803205	0.401602	+	0.915814i	$0.368453\pi$
$858$	−0.934738	−0.0319114
$859$	−53.5959	−1.82867	−0.914335	−	0.404959i	$-0.867286\pi$
$859$	−53.5959	−1.82867	−0.914335	+	0.404959i	$0.867286\pi$
$860$	0	0
$861$	−11.7005	−0.398753
$862$	−63.4935	−2.16260
$863$	−21.5279	−0.732819	−0.366409	−	0.930454i	$-0.619413\pi$
$863$	−21.5279	−0.732819	−0.366409	+	0.930454i	$0.619413\pi$
$864$	−65.2297	−2.21916
$865$	0	0
$866$	9.35113	0.317764
$867$	20.6981	0.702946
$868$	−10.7192	−0.363835
$869$	−21.3066	−0.722778
$870$	0	0
$871$	−0.109944	−0.00372532
$872$	22.1996	0.751772
$873$	96.0951	3.25233
$874$	78.5026	2.65539
$875$	0	0
$876$	−3.54666	−0.119830
$877$	−7.08676	−0.239303	−0.119651	−	0.992816i	$-0.538178\pi$
$877$	−7.08676	−0.239303	−0.119651	+	0.992816i	$0.538178\pi$
$878$	0.643588	0.0217200
$879$	−39.8100	−1.34276
$880$	0	0
$881$	−38.2223	−1.28774	−0.643871	−	0.765134i	$-0.722673\pi$
$881$	−38.2223	−1.28774	−0.643871	+	0.765134i	$0.722673\pi$
$882$	73.4186	2.47213
$883$	−58.3397	−1.96329	−0.981644	−	0.190723i	$-0.938917\pi$
$883$	−58.3397	−1.96329	−0.981644	+	0.190723i	$0.938917\pi$
$884$	−0.309272	−0.0104020
$885$	0	0
$886$	5.61601	0.188673
$887$	−25.8682	−0.868570	−0.434285	−	0.900776i	$-0.642999\pi$
$887$	−25.8682	−0.868570	−0.434285	+	0.900776i	$0.642999\pi$
$888$	77.5797	2.60340
$889$	−17.0019	−0.570224
$890$	0	0
$891$	−44.2411	−1.48213
$892$	−80.6787	−2.70132
$893$	53.4371	1.78820
$894$	79.6963	2.66544
$895$	0	0
$896$	−19.5683	−0.653730
$897$	−0.611375	−0.0204132
$898$	−24.7276	−0.825171
$899$	20.2290	0.674675
$900$	0	0
$901$	14.2648	0.475230
$902$	−28.4412	−0.946989
$903$	−45.1728	−1.50326
$904$	−27.6725	−0.920374
$905$	0	0
$906$	−59.7646	−1.98555
$907$	34.0342	1.13009	0.565043	−	0.825061i	$-0.308860\pi$
$907$	34.0342	1.13009	0.565043	+	0.825061i	$0.308860\pi$
$908$	61.4835	2.04040
$909$	103.210	3.42325
$910$	0	0
$911$	−38.9474	−1.29039	−0.645193	−	0.764020i	$-0.723223\pi$
$911$	−38.9474	−1.29039	−0.645193	+	0.764020i	$0.723223\pi$
$912$	−19.4597	−0.644374
$913$	26.9645	0.892394
$914$	26.1682	0.865568
$915$	0	0
$916$	−49.3139	−1.62938
$917$	−13.6050	−0.449276
$918$	−67.8670	−2.23994
$919$	5.14400	0.169685	0.0848424	−	0.996394i	$-0.472961\pi$
$919$	5.14400	0.169685	0.0848424	+	0.996394i	$0.472961\pi$
$920$	0	0
$921$	−15.3480	−0.505735
$922$	49.8592	1.64203
$923$	0.129038	0.00424735
$924$	−48.8958	−1.60855
$925$	0	0
$926$	74.4601	2.44691
$927$	17.6300	0.579045
$928$	49.1198	1.61244
$929$	44.9253	1.47395	0.736976	−	0.675919i	$-0.236253\pi$
$929$	44.9253	1.47395	0.736976	+	0.675919i	$0.236253\pi$
$930$	0	0
$931$	30.4944	0.999412
$932$	2.05915	0.0674498
$933$	−18.0647	−0.591413
$934$	−43.8227	−1.43392
$935$	0	0
$936$	−0.434156	−0.0141908
$937$	32.4004	1.05848	0.529238	−	0.848473i	$-0.322478\pi$
$937$	32.4004	1.05848	0.529238	+	0.848473i	$0.322478\pi$
$938$	−9.62700	−0.314333
$939$	−45.3035	−1.47842
$940$	0	0
$941$	−14.0427	−0.457777	−0.228889	−	0.973453i	$-0.573509\pi$
$941$	−14.0427	−0.457777	−0.228889	+	0.973453i	$0.573509\pi$
$942$	0.625098	0.0203668
$943$	−18.6023	−0.605773
$944$	7.35765	0.239471
$945$	0	0
$946$	−109.804	−3.57005
$947$	−15.7483	−0.511751	−0.255876	−	0.966710i	$-0.582364\pi$
$947$	−15.7483	−0.511751	−0.255876	+	0.966710i	$0.582364\pi$
$948$	−45.1382	−1.46602
$949$	0.0129124	0.000419154 0
$950$	0	0
$951$	−32.8859	−1.06640
$952$	−8.83024	−0.286190
$953$	−7.73667	−0.250615	−0.125308	−	0.992118i	$-0.539992\pi$
$953$	−7.73667	−0.250615	−0.125308	+	0.992118i	$0.539992\pi$
$954$	61.4127	1.98831
$955$	0	0
$956$	86.9407	2.81186
$957$	92.2745	2.98281
$958$	87.5417	2.82834
$959$	−4.90681	−0.158449
$960$	0	0
$961$	−23.0922	−0.744910
$962$	−0.866209	−0.0279277
$963$	6.33773	0.204231
$964$	−6.43495	−0.207256
$965$	0	0
$966$	−53.5336	−1.72242
$967$	31.6140	1.01664	0.508319	−	0.861169i	$-0.330267\pi$
$967$	31.6140	1.01664	0.508319	+	0.861169i	$0.330267\pi$
$968$	−15.0304	−0.483095
$969$	−54.9760	−1.76608
$970$	0	0
$971$	−35.4323	−1.13708	−0.568539	−	0.822657i	$-0.692491\pi$
$971$	−35.4323	−1.13708	−0.568539	+	0.822657i	$0.692491\pi$
$972$	−8.67530	−0.278260
$973$	−4.12234	−0.132156
$974$	17.3043	0.554467
$975$	0	0
$976$	−4.43889	−0.142085
$977$	−55.8056	−1.78538	−0.892690	−	0.450671i	$-0.851185\pi$
$977$	−55.8056	−1.78538	−0.892690	+	0.450671i	$0.851185\pi$
$978$	−37.3728	−1.19505
$979$	−37.7793	−1.20743
$980$	0	0
$981$	−63.3725	−2.02333
$982$	80.1620	2.55807
$983$	29.2455	0.932787	0.466393	−	0.884577i	$-0.345553\pi$
$983$	29.2455	0.932787	0.466393	+	0.884577i	$0.345553\pi$
$984$	−19.6467	−0.626315
$985$	0	0
$986$	51.1058	1.62754
$987$	−36.4406	−1.15992
$988$	−0.553028	−0.0175942
$989$	−71.8187	−2.28370
$990$	0	0
$991$	−15.9250	−0.505875	−0.252937	−	0.967483i	$-0.581397\pi$
$991$	−15.9250	−0.505875	−0.252937	+	0.967483i	$0.581397\pi$
$992$	19.2016	0.609651
$993$	−77.2191	−2.45047
$994$	11.2989	0.358380
$995$	0	0
$996$	57.1244	1.81006
$997$	−7.45247	−0.236022	−0.118011	−	0.993012i	$-0.537652\pi$
$997$	−7.45247	−0.236022	−0.118011	+	0.993012i	$0.537652\pi$
$998$	28.1267	0.890334
$999$	−113.554	−3.59270

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4925.2.a.l.1.4		17
5.4	even	2		985.2.a.g.1.14	✓	17
15.14	odd	2		8865.2.a.z.1.4		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
985.2.a.g.1.14	✓	17	5.4	even	2
4925.2.a.l.1.4		17	1.1	even	1	trivial
8865.2.a.z.1.4		17	15.14	odd	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:	\( N \)	\(=\)	\( 4925 = 5^{2} \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4925.a (trivial)

\(f(q)\)	\(=\)	\(q-2.22883 q^{2} -3.02604 q^{3} +2.96767 q^{4} +6.74452 q^{6} -1.28446 q^{7} -2.15678 q^{8} +6.15690 q^{9} -4.23897 q^{11} -8.98029 q^{12} +0.0326948 q^{13} +2.86284 q^{14} -1.12826 q^{16} -3.18747 q^{17} -13.7227 q^{18} -5.69971 q^{19} +3.88682 q^{21} +9.44795 q^{22} +6.17953 q^{23} +6.52650 q^{24} -0.0728710 q^{26} -9.55291 q^{27} -3.81186 q^{28} +7.19361 q^{29} +2.81208 q^{31} +6.82825 q^{32} +12.8273 q^{33} +7.10433 q^{34} +18.2717 q^{36} +11.8869 q^{37} +12.7037 q^{38} -0.0989356 q^{39} -3.01031 q^{41} -8.66306 q^{42} -11.6220 q^{43} -12.5799 q^{44} -13.7731 q^{46} -9.37541 q^{47} +3.41415 q^{48} -5.35016 q^{49} +9.64541 q^{51} +0.0970274 q^{52} -4.47527 q^{53} +21.2918 q^{54} +2.77030 q^{56} +17.2475 q^{57} -16.0333 q^{58} -6.52124 q^{59} +3.93428 q^{61} -6.26764 q^{62} -7.90829 q^{63} -12.9625 q^{64} -28.5898 q^{66} -3.36275 q^{67} -9.45938 q^{68} -18.6995 q^{69} +3.94676 q^{71} -13.2791 q^{72} +0.394938 q^{73} -26.4938 q^{74} -16.9149 q^{76} +5.44479 q^{77} +0.220510 q^{78} +5.02637 q^{79} +10.4367 q^{81} +6.70945 q^{82} -6.36108 q^{83} +11.5348 q^{84} +25.9035 q^{86} -21.7681 q^{87} +9.14253 q^{88} +8.91236 q^{89} -0.0419951 q^{91} +18.3388 q^{92} -8.50946 q^{93} +20.8962 q^{94} -20.6625 q^{96} +15.6077 q^{97} +11.9246 q^{98} -26.0990 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - 6 q^{2} - 5 q^{3} + 18 q^{4} + 3 q^{6} - 7 q^{7} - 18 q^{8} + 22 q^{9} + 7 q^{11} - 20 q^{12} - 3 q^{13} + 17 q^{14} + 28 q^{16} - 6 q^{17} - 13 q^{18} - 23 q^{19} - q^{21} - 14 q^{22} - 49 q^{23}+ \cdots - 46 q^{99}+O(q^{100}) \) 17 * q - 6 * q^2 - 5 * q^3 + 18 * q^4 + 3 * q^6 - 7 * q^7 - 18 * q^8 + 22 * q^9 + 7 * q^11 - 20 * q^12 - 3 * q^13 + 17 * q^14 + 28 * q^16 - 6 * q^17 - 13 * q^18 - 23 * q^19 - q^21 - 14 * q^22 - 49 * q^23 + q^24 + 3 * q^26 - 29 * q^27 - 9 * q^28 + 19 * q^29 - 18 * q^31 - 42 * q^32 - 14 * q^33 - 19 * q^34 + 31 * q^36 - 10 * q^37 - 27 * q^38 + 21 * q^39 + 14 * q^41 - 31 * q^42 - 22 * q^43 + 27 * q^44 - 16 * q^46 - 51 * q^47 - 38 * q^48 + 8 * q^49 + 7 * q^51 - 17 * q^52 - 37 * q^53 - 36 * q^54 + 47 * q^56 + 11 * q^57 - 20 * q^58 + 23 * q^59 - 15 * q^61 + 6 * q^62 - 37 * q^63 - 2 * q^64 - 4 * q^66 - 24 * q^67 - 13 * q^68 + 6 * q^69 + 68 * q^71 + 33 * q^72 + 10 * q^73 - 9 * q^74 - 57 * q^76 + 16 * q^77 + 24 * q^78 + 3 * q^79 + 33 * q^81 + 40 * q^82 - 29 * q^83 - 83 * q^84 + 37 * q^86 - 23 * q^87 - 5 * q^88 - 11 * q^89 - 82 * q^91 - 15 * q^93 - 39 * q^94 + 2 * q^96 - 6 * q^97 - 15 * q^98 - 46 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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