LMFDB - Embedded newform 6025.2.a.j.1.13

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:	$1$
Dimension:	$25$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
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)

$f(q)$	$=$	$q-0.362712 q^{2} -3.41577 q^{3} -1.86844 q^{4} +1.23894 q^{6} -3.34046 q^{7} +1.40313 q^{8} +8.66752 q^{9} +O(q^{10})$	\(q-0.362712 q^{2} -3.41577 q^{3} -1.86844 q^{4} +1.23894 q^{6} -3.34046 q^{7} +1.40313 q^{8} +8.66752 q^{9} -2.67832 q^{11} +6.38217 q^{12} -0.591068 q^{13} +1.21163 q^{14} +3.22795 q^{16} +1.07744 q^{17} -3.14382 q^{18} -5.88556 q^{19} +11.4103 q^{21} +0.971461 q^{22} -2.98266 q^{23} -4.79278 q^{24} +0.214388 q^{26} -19.3590 q^{27} +6.24145 q^{28} +3.96926 q^{29} -4.71138 q^{31} -3.97708 q^{32} +9.14854 q^{33} -0.390800 q^{34} -16.1947 q^{36} -8.86039 q^{37} +2.13477 q^{38} +2.01895 q^{39} -9.79063 q^{41} -4.13864 q^{42} +2.42881 q^{43} +5.00428 q^{44} +1.08185 q^{46} +9.02079 q^{47} -11.0259 q^{48} +4.15867 q^{49} -3.68028 q^{51} +1.10437 q^{52} +14.0701 q^{53} +7.02174 q^{54} -4.68710 q^{56} +20.1037 q^{57} -1.43970 q^{58} +9.68407 q^{59} -0.830604 q^{61} +1.70887 q^{62} -28.9535 q^{63} -5.01336 q^{64} -3.31829 q^{66} -10.9586 q^{67} -2.01313 q^{68} +10.1881 q^{69} +0.250176 q^{71} +12.1617 q^{72} +10.3124 q^{73} +3.21377 q^{74} +10.9968 q^{76} +8.94682 q^{77} -0.732300 q^{78} +10.1480 q^{79} +40.1233 q^{81} +3.55119 q^{82} +7.04557 q^{83} -21.3194 q^{84} -0.880961 q^{86} -13.5581 q^{87} -3.75804 q^{88} +6.66265 q^{89} +1.97444 q^{91} +5.57292 q^{92} +16.0930 q^{93} -3.27195 q^{94} +13.5848 q^{96} -0.0770398 q^{97} -1.50840 q^{98} -23.2144 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9}+O(q^{10}) $ 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9	$ 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9} + 2 q^{11} - 20 q^{12} - 14 q^{13} - 5 q^{14} + 38 q^{16} - 7 q^{17} - 9 q^{18} + 30 q^{19} + q^{21} - q^{22} - 43 q^{23} - 6 q^{24} - 22 q^{26} - 42 q^{27} - 32 q^{28} - 4 q^{29} + 14 q^{31} - 26 q^{32} - 4 q^{33} + 7 q^{34} + 15 q^{36} - 16 q^{37} - 14 q^{38} - 21 q^{39} - q^{41} + 25 q^{42} - 35 q^{43} - 52 q^{44} - 27 q^{46} - 50 q^{47} - 26 q^{48} + 46 q^{49} - 7 q^{51} - 3 q^{52} - 4 q^{53} - 31 q^{54} - 51 q^{56} - 2 q^{58} + 6 q^{59} + 19 q^{61} - 28 q^{63} + 49 q^{64} - 27 q^{66} - 65 q^{67} + 25 q^{68} + 2 q^{69} - 34 q^{71} + 10 q^{72} - 8 q^{73} - 42 q^{74} + 71 q^{76} - q^{77} + 59 q^{78} - 12 q^{79} + 29 q^{81} - 11 q^{82} - 41 q^{83} - 10 q^{84} - 13 q^{86} - 40 q^{87} + 52 q^{88} - 24 q^{89} + 46 q^{91} - 85 q^{92} + 30 q^{93} + 14 q^{94} - 30 q^{96} - 9 q^{97} + 64 q^{98} + 2 q^{99}+O(q^{100}) $ 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9 + 2 * q^11 - 20 * q^12 - 14 * q^13 - 5 * q^14 + 38 * q^16 - 7 * q^17 - 9 * q^18 + 30 * q^19 + q^21 - q^22 - 43 * q^23 - 6 * q^24 - 22 * q^26 - 42 * q^27 - 32 * q^28 - 4 * q^29 + 14 * q^31 - 26 * q^32 - 4 * q^33 + 7 * q^34 + 15 * q^36 - 16 * q^37 - 14 * q^38 - 21 * q^39 - q^41 + 25 * q^42 - 35 * q^43 - 52 * q^44 - 27 * q^46 - 50 * q^47 - 26 * q^48 + 46 * q^49 - 7 * q^51 - 3 * q^52 - 4 * q^53 - 31 * q^54 - 51 * q^56 - 2 * q^58 + 6 * q^59 + 19 * q^61 - 28 * q^63 + 49 * q^64 - 27 * q^66 - 65 * q^67 + 25 * q^68 + 2 * q^69 - 34 * q^71 + 10 * q^72 - 8 * q^73 - 42 * q^74 + 71 * q^76 - q^77 + 59 * q^78 - 12 * q^79 + 29 * q^81 - 11 * q^82 - 41 * q^83 - 10 * q^84 - 13 * q^86 - 40 * q^87 + 52 * q^88 - 24 * q^89 + 46 * q^91 - 85 * q^92 + 30 * q^93 + 14 * q^94 - 30 * q^96 - 9 * q^97 + 64 * q^98 + 2 * 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 $

$2$	−0.362712	−0.256476	−0.128238	−	0.991743i	$-0.540932\pi$
$2$	−0.362712	−0.256476	−0.128238	+	0.991743i	$0.540932\pi$
$3$	−3.41577	−1.97210	−0.986049	−	0.166454i	$-0.946768\pi$
$3$	−3.41577	−1.97210	−0.986049	+	0.166454i	$0.946768\pi$
$4$	−1.86844	−0.934220
$5$	0	0
$5$	0	0
$6$	1.23894	0.505797
$7$	−3.34046	−1.26257	−0.631287	−	0.775549i	$-0.717473\pi$
$7$	−3.34046	−1.26257	−0.631287	+	0.775549i	$0.717473\pi$
$8$	1.40313	0.496082
$9$	8.66752	2.88917
$10$	0	0
$11$	−2.67832	−0.807544	−0.403772	−	0.914860i	$-0.632301\pi$
$11$	−2.67832	−0.807544	−0.403772	+	0.914860i	$0.632301\pi$
$12$	6.38217	1.84237
$13$	−0.591068	−0.163933	−0.0819663	−	0.996635i	$-0.526120\pi$
$13$	−0.591068	−0.163933	−0.0819663	+	0.996635i	$0.526120\pi$
$14$	1.21163	0.323821
$15$	0	0
$16$	3.22795	0.806987
$17$	1.07744	0.261317	0.130658	−	0.991427i	$-0.458291\pi$
$17$	1.07744	0.261317	0.130658	+	0.991427i	$0.458291\pi$
$18$	−3.14382	−0.741005
$19$	−5.88556	−1.35024	−0.675120	−	0.737708i	$-0.735908\pi$
$19$	−5.88556	−1.35024	−0.675120	+	0.737708i	$0.735908\pi$
$20$	0	0
$21$	11.4103	2.48992
$22$	0.971461	0.207116
$23$	−2.98266	−0.621928	−0.310964	−	0.950422i	$-0.600652\pi$
$23$	−2.98266	−0.621928	−0.310964	+	0.950422i	$0.600652\pi$
$24$	−4.79278	−0.978322
$25$	0	0
$26$	0.214388	0.0420449
$27$	−19.3590	−3.72563
$28$	6.24145	1.17952
$29$	3.96926	0.737073	0.368536	−	0.929613i	$-0.379859\pi$
$29$	3.96926	0.737073	0.368536	+	0.929613i	$0.379859\pi$
$30$	0	0
$31$	−4.71138	−0.846188	−0.423094	−	0.906086i	$-0.639056\pi$
$31$	−4.71138	−0.846188	−0.423094	+	0.906086i	$0.639056\pi$
$32$	−3.97708	−0.703055
$33$	9.14854	1.59256
$34$	−0.390800	−0.0670216
$35$	0	0
$36$	−16.1947	−2.69912
$37$	−8.86039	−1.45664	−0.728319	−	0.685238i	$-0.759698\pi$
$37$	−8.86039	−1.45664	−0.728319	+	0.685238i	$0.759698\pi$
$38$	2.13477	0.346305
$39$	2.01895	0.323291
$40$	0	0
$41$	−9.79063	−1.52904	−0.764520	−	0.644600i	$-0.777024\pi$
$41$	−9.79063	−1.52904	−0.764520	+	0.644600i	$0.777024\pi$
$42$	−4.13864	−0.638606
$43$	2.42881	0.370391	0.185195	−	0.982702i	$-0.440708\pi$
$43$	2.42881	0.370391	0.185195	+	0.982702i	$0.440708\pi$
$44$	5.00428	0.754424
$45$	0	0
$46$	1.08185	0.159510
$47$	9.02079	1.31582	0.657909	−	0.753098i	$-0.271441\pi$
$47$	9.02079	1.31582	0.657909	+	0.753098i	$0.271441\pi$
$48$	−11.0259	−1.59146
$49$	4.15867	0.594095
$50$	0	0
$51$	−3.68028	−0.515342
$52$	1.10437	0.153149
$53$	14.0701	1.93268	0.966340	−	0.257270i	$-0.0828228\pi$
$53$	14.0701	1.93268	0.966340	+	0.257270i	$0.0828228\pi$
$54$	7.02174	0.955537
$55$	0	0
$56$	−4.68710	−0.626341
$57$	20.1037	2.66281
$58$	−1.43970	−0.189042
$59$	9.68407	1.26076	0.630379	−	0.776287i	$-0.282899\pi$
$59$	9.68407	1.26076	0.630379	+	0.776287i	$0.282899\pi$
$60$	0	0
$61$	−0.830604	−0.106348	−0.0531740	−	0.998585i	$-0.516934\pi$
$61$	−0.830604	−0.106348	−0.0531740	+	0.998585i	$0.516934\pi$
$62$	1.70887	0.217027
$63$	−28.9535	−3.64780
$64$	−5.01336	−0.626669
$65$	0	0
$66$	−3.31829	−0.408453
$67$	−10.9586	−1.33881	−0.669403	−	0.742899i	$-0.733450\pi$
$67$	−10.9586	−1.33881	−0.669403	+	0.742899i	$0.733450\pi$
$68$	−2.01313	−0.244127
$69$	10.1881	1.22650
$70$	0	0
$71$	0.250176	0.0296904	0.0148452	−	0.999890i	$-0.495274\pi$
$71$	0.250176	0.0296904	0.0148452	+	0.999890i	$0.495274\pi$
$72$	12.1617	1.43327
$73$	10.3124	1.20697	0.603486	−	0.797373i	$-0.293778\pi$
$73$	10.3124	1.20697	0.603486	+	0.797373i	$0.293778\pi$
$74$	3.21377	0.373594
$75$	0	0
$76$	10.9968	1.26142
$77$	8.94682	1.01959
$78$	−0.732300	−0.0829166
$79$	10.1480	1.14174	0.570871	−	0.821040i	$-0.306606\pi$
$79$	10.1480	1.14174	0.570871	+	0.821040i	$0.306606\pi$
$80$	0	0
$81$	40.1233	4.45814
$82$	3.55119	0.392163
$83$	7.04557	0.773351	0.386676	−	0.922216i	$-0.373623\pi$
$83$	7.04557	0.773351	0.386676	+	0.922216i	$0.373623\pi$
$84$	−21.3194	−2.32613
$85$	0	0
$86$	−0.880961	−0.0949965
$87$	−13.5581	−1.45358
$88$	−3.75804	−0.400608
$89$	6.66265	0.706240	0.353120	−	0.935578i	$-0.385121\pi$
$89$	6.66265	0.706240	0.353120	+	0.935578i	$0.385121\pi$
$90$	0	0
$91$	1.97444	0.206977
$92$	5.57292	0.581017
$93$	16.0930	1.66877
$94$	−3.27195	−0.337476
$95$	0	0
$96$	13.5848	1.38649
$97$	−0.0770398	−0.00782221	−0.00391110	−	0.999992i	$-0.501245\pi$
$97$	−0.0770398	−0.00782221	−0.00391110	+	0.999992i	$0.501245\pi$
$98$	−1.50840	−0.152372
$99$	−23.2144	−2.33313
$100$	0	0
$101$	4.48262	0.446038	0.223019	−	0.974814i	$-0.428409\pi$
$101$	4.48262	0.446038	0.223019	+	0.974814i	$0.428409\pi$
$102$	1.33488	0.132173
$103$	3.98186	0.392345	0.196172	−	0.980569i	$-0.437149\pi$
$103$	3.98186	0.392345	0.196172	+	0.980569i	$0.437149\pi$
$104$	−0.829345	−0.0813240
$105$	0	0
$106$	−5.10341	−0.495687
$107$	10.2751	0.993335	0.496668	−	0.867941i	$-0.334557\pi$
$107$	10.2751	0.993335	0.496668	+	0.867941i	$0.334557\pi$
$108$	36.1710	3.48056
$109$	−3.79466	−0.363463	−0.181731	−	0.983348i	$-0.558170\pi$
$109$	−3.79466	−0.363463	−0.181731	+	0.983348i	$0.558170\pi$
$110$	0	0
$111$	30.2651	2.87263
$112$	−10.7828	−1.01888
$113$	−10.3798	−0.976448	−0.488224	−	0.872718i	$-0.662355\pi$
$113$	−10.3798	−0.976448	−0.488224	+	0.872718i	$0.662355\pi$
$114$	−7.29188	−0.682947
$115$	0	0
$116$	−7.41632	−0.688588
$117$	−5.12309	−0.473630
$118$	−3.51253	−0.323355
$119$	−3.59913	−0.329932
$120$	0	0
$121$	−3.82659	−0.347872
$122$	0.301270	0.0272757
$123$	33.4426	3.01542
$124$	8.80292	0.790526
$125$	0	0
$126$	10.5018	0.935574
$127$	−8.55650	−0.759267	−0.379633	−	0.925137i	$-0.623950\pi$
$127$	−8.55650	−0.759267	−0.379633	+	0.925137i	$0.623950\pi$
$128$	9.77257	0.863781
$129$	−8.29628	−0.730447
$130$	0	0
$131$	−0.367465	−0.0321055	−0.0160528	−	0.999871i	$-0.505110\pi$
$131$	−0.367465	−0.0321055	−0.0160528	+	0.999871i	$0.505110\pi$
$132$	−17.0935	−1.48780
$133$	19.6605	1.70478
$134$	3.97482	0.343372
$135$	0	0
$136$	1.51179	0.129635
$137$	8.68523	0.742029	0.371015	−	0.928627i	$-0.379010\pi$
$137$	8.68523	0.742029	0.371015	+	0.928627i	$0.379010\pi$
$138$	−3.69535	−0.314569
$139$	13.2597	1.12468	0.562338	−	0.826907i	$-0.309902\pi$
$139$	13.2597	1.12468	0.562338	+	0.826907i	$0.309902\pi$
$140$	0	0
$141$	−30.8130	−2.59492
$142$	−0.0907418	−0.00761489
$143$	1.58307	0.132383
$144$	27.9783	2.33152
$145$	0	0
$146$	−3.74043	−0.309560
$147$	−14.2051	−1.17161
$148$	16.5551	1.36082
$149$	−13.5526	−1.11028	−0.555138	−	0.831759i	$-0.687334\pi$
$149$	−13.5526	−1.11028	−0.555138	+	0.831759i	$0.687334\pi$
$150$	0	0
$151$	−5.83929	−0.475195	−0.237597	−	0.971364i	$-0.576360\pi$
$151$	−5.83929	−0.475195	−0.237597	+	0.971364i	$0.576360\pi$
$152$	−8.25821	−0.669830
$153$	9.33870	0.754989
$154$	−3.24512	−0.261500
$155$	0	0
$156$	−3.77229	−0.302025
$157$	13.2138	1.05457	0.527286	−	0.849688i	$-0.323210\pi$
$157$	13.2138	1.05457	0.527286	+	0.849688i	$0.323210\pi$
$158$	−3.68082	−0.292830
$159$	−48.0604	−3.81143
$160$	0	0
$161$	9.96346	0.785230
$162$	−14.5532	−1.14341
$163$	24.5109	1.91984	0.959921	−	0.280272i	$-0.0904247\pi$
$163$	24.5109	1.91984	0.959921	+	0.280272i	$0.0904247\pi$
$164$	18.2932	1.42846
$165$	0	0
$166$	−2.55551	−0.198346
$167$	−16.9393	−1.31080	−0.655400	−	0.755282i	$-0.727500\pi$
$167$	−16.9393	−1.31080	−0.655400	+	0.755282i	$0.727500\pi$
$168$	16.0101	1.23521
$169$	−12.6506	−0.973126
$170$	0	0
$171$	−51.0132	−3.90108
$172$	−4.53809	−0.346026
$173$	22.5016	1.71077	0.855383	−	0.517996i	$-0.173322\pi$
$173$	22.5016	1.71077	0.855383	+	0.517996i	$0.173322\pi$
$174$	4.91769	0.372809
$175$	0	0
$176$	−8.64548	−0.651677
$177$	−33.0786	−2.48634
$178$	−2.41663	−0.181134
$179$	−3.45024	−0.257883	−0.128941	−	0.991652i	$-0.541158\pi$
$179$	−3.45024	−0.257883	−0.128941	+	0.991652i	$0.541158\pi$
$180$	0	0
$181$	15.2794	1.13571	0.567853	−	0.823130i	$-0.307774\pi$
$181$	15.2794	1.13571	0.567853	+	0.823130i	$0.307774\pi$
$182$	−0.716153	−0.0530848
$183$	2.83716	0.209729
$184$	−4.18506	−0.308527
$185$	0	0
$186$	−5.83713	−0.427999
$187$	−2.88572	−0.211025
$188$	−16.8548	−1.22926
$189$	64.6678	4.70389
$190$	0	0
$191$	3.66386	0.265107	0.132554	−	0.991176i	$-0.457682\pi$
$191$	3.66386	0.265107	0.132554	+	0.991176i	$0.457682\pi$
$192$	17.1245	1.23585
$193$	4.66451	0.335759	0.167880	−	0.985808i	$-0.446308\pi$
$193$	4.66451	0.335759	0.167880	+	0.985808i	$0.446308\pi$
$194$	0.0279433	0.00200621
$195$	0	0
$196$	−7.77022	−0.555016
$197$	−15.4498	−1.10075	−0.550377	−	0.834916i	$-0.685516\pi$
$197$	−15.4498	−1.10075	−0.550377	+	0.834916i	$0.685516\pi$
$198$	8.42015	0.598394
$199$	−5.40435	−0.383104	−0.191552	−	0.981482i	$-0.561352\pi$
$199$	−5.40435	−0.383104	−0.191552	+	0.981482i	$0.561352\pi$
$200$	0	0
$201$	37.4321	2.64026
$202$	−1.62590	−0.114398
$203$	−13.2591	−0.930609
$204$	6.87638	0.481443
$205$	0	0
$206$	−1.44427	−0.100627
$207$	−25.8523	−1.79686
$208$	−1.90793	−0.132291
$209$	15.7634	1.09038
$210$	0	0
$211$	1.59042	0.109489	0.0547446	−	0.998500i	$-0.482566\pi$
$211$	1.59042	0.109489	0.0547446	+	0.998500i	$0.482566\pi$
$212$	−26.2892	−1.80555
$213$	−0.854544	−0.0585524
$214$	−3.72692	−0.254767
$215$	0	0
$216$	−27.1632	−1.84822
$217$	15.7382	1.06838
$218$	1.37637	0.0932197
$219$	−35.2248	−2.38027
$220$	0	0
$221$	−0.636838	−0.0428384
$222$	−10.9775	−0.736763
$223$	−22.8974	−1.53333	−0.766663	−	0.642049i	$-0.778084\pi$
$223$	−22.8974	−1.53333	−0.766663	+	0.642049i	$0.778084\pi$
$224$	13.2853	0.887660
$225$	0	0
$226$	3.76488	0.250436
$227$	20.7584	1.37779	0.688893	−	0.724863i	$-0.258097\pi$
$227$	20.7584	1.37779	0.688893	+	0.724863i	$0.258097\pi$
$228$	−37.5626	−2.48765
$229$	−2.70281	−0.178607	−0.0893033	−	0.996004i	$-0.528464\pi$
$229$	−2.70281	−0.178607	−0.0893033	+	0.996004i	$0.528464\pi$
$230$	0	0
$231$	−30.5603	−2.01072
$232$	5.56939	0.365648
$233$	4.42131	0.289650	0.144825	−	0.989457i	$-0.453738\pi$
$233$	4.42131	0.289650	0.144825	+	0.989457i	$0.453738\pi$
$234$	1.85821	0.121475
$235$	0	0
$236$	−18.0941	−1.17783
$237$	−34.6634	−2.25163
$238$	1.30545	0.0846198
$239$	−15.5478	−1.00570	−0.502851	−	0.864373i	$-0.667715\pi$
$239$	−15.5478	−1.00570	−0.502851	+	0.864373i	$0.667715\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	1.38795	0.0892210
$243$	−78.9752	−5.06626
$244$	1.55193	0.0993523
$245$	0	0
$246$	−12.1300	−0.773384
$247$	3.47876	0.221348
$248$	−6.61068	−0.419778
$249$	−24.0661	−1.52512
$250$	0	0
$251$	25.2828	1.59584	0.797920	−	0.602764i	$-0.205934\pi$
$251$	25.2828	1.59584	0.797920	+	0.602764i	$0.205934\pi$
$252$	54.0978	3.40784
$253$	7.98852	0.502234
$254$	3.10355	0.194734
$255$	0	0
$256$	6.48208	0.405130
$257$	18.6057	1.16059	0.580296	−	0.814405i	$-0.302937\pi$
$257$	18.6057	1.16059	0.580296	+	0.814405i	$0.302937\pi$
$258$	3.00917	0.187342
$259$	29.5978	1.83912
$260$	0	0
$261$	34.4036	2.12953
$262$	0.133284	0.00823431
$263$	7.45593	0.459753	0.229876	−	0.973220i	$-0.426168\pi$
$263$	7.45593	0.459753	0.229876	+	0.973220i	$0.426168\pi$
$264$	12.8366	0.790039
$265$	0	0
$266$	−7.13110	−0.437236
$267$	−22.7581	−1.39277
$268$	20.4755	1.25074
$269$	31.0344	1.89220	0.946101	−	0.323871i	$-0.104984\pi$
$269$	31.0344	1.89220	0.946101	+	0.323871i	$0.104984\pi$
$270$	0	0
$271$	−13.0676	−0.793802	−0.396901	−	0.917861i	$-0.629914\pi$
$271$	−13.0676	−0.793802	−0.396901	+	0.917861i	$0.629914\pi$
$272$	3.47791	0.210879
$273$	−6.74423	−0.408179
$274$	−3.15024	−0.190313
$275$	0	0
$276$	−19.0358	−1.14582
$277$	16.4559	0.988740	0.494370	−	0.869252i	$-0.335399\pi$
$277$	16.4559	0.988740	0.494370	+	0.869252i	$0.335399\pi$
$278$	−4.80947	−0.288453
$279$	−40.8359	−2.44478
$280$	0	0
$281$	15.2677	0.910793	0.455397	−	0.890289i	$-0.349497\pi$
$281$	15.2677	0.910793	0.455397	+	0.890289i	$0.349497\pi$
$282$	11.1763	0.665536
$283$	−15.0853	−0.896726	−0.448363	−	0.893852i	$-0.647993\pi$
$283$	−15.0853	−0.896726	−0.448363	+	0.893852i	$0.647993\pi$
$284$	−0.467438	−0.0277374
$285$	0	0
$286$	−0.574199	−0.0339531
$287$	32.7052	1.93053
$288$	−34.4714	−2.03125
$289$	−15.8391	−0.931714
$290$	0	0
$291$	0.263151	0.0154262
$292$	−19.2681	−1.12758
$293$	26.1109	1.52541	0.762706	−	0.646745i	$-0.223870\pi$
$293$	26.1109	1.52541	0.762706	+	0.646745i	$0.223870\pi$
$294$	5.15236	0.300492
$295$	0	0
$296$	−12.4323	−0.722612
$297$	51.8495	3.00861
$298$	4.91571	0.284759
$299$	1.76295	0.101954
$300$	0	0
$301$	−8.11336	−0.467646
$302$	2.11798	0.121876
$303$	−15.3116	−0.879630
$304$	−18.9983	−1.08963
$305$	0	0
$306$	−3.38726	−0.193637
$307$	−0.0551646	−0.00314841	−0.00157421	−	0.999999i	$-0.500501\pi$
$307$	−0.0551646	−0.00314841	−0.00157421	+	0.999999i	$0.500501\pi$
$308$	−16.7166	−0.952517
$309$	−13.6011	−0.773742
$310$	0	0
$311$	−20.8676	−1.18330	−0.591648	−	0.806197i	$-0.701522\pi$
$311$	−20.8676	−1.18330	−0.591648	+	0.806197i	$0.701522\pi$
$312$	2.83286	0.160379
$313$	−25.7693	−1.45657	−0.728283	−	0.685276i	$-0.759682\pi$
$313$	−25.7693	−1.45657	−0.728283	+	0.685276i	$0.759682\pi$
$314$	−4.79279	−0.270473
$315$	0	0
$316$	−18.9610	−1.06664
$317$	−2.63788	−0.148158	−0.0740792	−	0.997252i	$-0.523602\pi$
$317$	−2.63788	−0.148158	−0.0740792	+	0.997252i	$0.523602\pi$
$318$	17.4321	0.977543
$319$	−10.6309	−0.595219
$320$	0	0
$321$	−35.0976	−1.95895
$322$	−3.61387	−0.201393
$323$	−6.34132	−0.352840
$324$	−74.9679	−4.16489
$325$	0	0
$326$	−8.89041	−0.492394
$327$	12.9617	0.716785
$328$	−13.7375	−0.758529
$329$	−30.1336	−1.66132
$330$	0	0
$331$	−2.00015	−0.109938	−0.0549691	−	0.998488i	$-0.517506\pi$
$331$	−2.00015	−0.109938	−0.0549691	+	0.998488i	$0.517506\pi$
$332$	−13.1642	−0.722480
$333$	−76.7975	−4.20848
$334$	6.14408	0.336189
$335$	0	0
$336$	36.8317	2.00933
$337$	−31.5650	−1.71945	−0.859727	−	0.510754i	$-0.829366\pi$
$337$	−31.5650	−1.71945	−0.859727	+	0.510754i	$0.829366\pi$
$338$	4.58854	0.249584
$339$	35.4550	1.92565
$340$	0	0
$341$	12.6186	0.683334
$342$	18.5031	1.00053
$343$	9.49135	0.512485
$344$	3.40795	0.183744
$345$	0	0
$346$	−8.16162	−0.438771
$347$	11.1162	0.596747	0.298373	−	0.954449i	$-0.403556\pi$
$347$	11.1162	0.596747	0.298373	+	0.954449i	$0.403556\pi$
$348$	25.3325	1.35796
$349$	2.19652	0.117577	0.0587884	−	0.998270i	$-0.481276\pi$
$349$	2.19652	0.117577	0.0587884	+	0.998270i	$0.481276\pi$
$350$	0	0
$351$	11.4425	0.610753
$352$	10.6519	0.567748
$353$	1.90559	0.101424	0.0507122	−	0.998713i	$-0.483851\pi$
$353$	1.90559	0.101424	0.0507122	+	0.998713i	$0.483851\pi$
$354$	11.9980	0.637688
$355$	0	0
$356$	−12.4488	−0.659783
$357$	12.2938	0.650659
$358$	1.25144	0.0661409
$359$	−8.45511	−0.446244	−0.223122	−	0.974791i	$-0.571625\pi$
$359$	−8.45511	−0.446244	−0.223122	+	0.974791i	$0.571625\pi$
$360$	0	0
$361$	15.6398	0.823149
$362$	−5.54201	−0.291282
$363$	13.0708	0.686038
$364$	−3.68912	−0.193362
$365$	0	0
$366$	−1.02907	−0.0537904
$367$	−30.7371	−1.60446	−0.802232	−	0.597013i	$-0.796354\pi$
$367$	−30.7371	−1.60446	−0.802232	+	0.597013i	$0.796354\pi$
$368$	−9.62787	−0.501887
$369$	−84.8605	−4.41766
$370$	0	0
$371$	−47.0007	−2.44015
$372$	−30.0688	−1.55899
$373$	−7.23017	−0.374364	−0.187182	−	0.982325i	$-0.559935\pi$
$373$	−7.23017	−0.374364	−0.187182	+	0.982325i	$0.559935\pi$
$374$	1.04669	0.0541229
$375$	0	0
$376$	12.6574	0.652753
$377$	−2.34610	−0.120830
$378$	−23.4558	−1.20644
$379$	−16.3620	−0.840459	−0.420229	−	0.907418i	$-0.638050\pi$
$379$	−16.3620	−0.840459	−0.420229	+	0.907418i	$0.638050\pi$
$380$	0	0
$381$	29.2271	1.49735
$382$	−1.32893	−0.0679938
$383$	17.0940	0.873462	0.436731	−	0.899592i	$-0.356136\pi$
$383$	17.0940	0.873462	0.436731	+	0.899592i	$0.356136\pi$
$384$	−33.3809	−1.70346
$385$	0	0
$386$	−1.69188	−0.0861143
$387$	21.0518	1.07012
$388$	0.143944	0.00730766
$389$	−18.4363	−0.934760	−0.467380	−	0.884057i	$-0.654802\pi$
$389$	−18.4363	−0.934760	−0.467380	+	0.884057i	$0.654802\pi$
$390$	0	0
$391$	−3.21363	−0.162520
$392$	5.83516	0.294720
$393$	1.25518	0.0633153
$394$	5.60384	0.282317
$395$	0	0
$396$	43.3747	2.17966
$397$	−34.6231	−1.73769	−0.868843	−	0.495088i	$-0.835136\pi$
$397$	−34.6231	−1.73769	−0.868843	+	0.495088i	$0.835136\pi$
$398$	1.96023	0.0982573
$399$	−67.1558	−3.36199
$400$	0	0
$401$	18.7453	0.936094	0.468047	−	0.883703i	$-0.344958\pi$
$401$	18.7453	0.936094	0.468047	+	0.883703i	$0.344958\pi$
$402$	−13.5771	−0.677164
$403$	2.78474	0.138718
$404$	−8.37551	−0.416697
$405$	0	0
$406$	4.80926	0.238679
$407$	23.7310	1.17630
$408$	−5.16392	−0.255652
$409$	36.1825	1.78911	0.894554	−	0.446960i	$-0.147493\pi$
$409$	36.1825	1.78911	0.894554	+	0.446960i	$0.147493\pi$
$410$	0	0
$411$	−29.6668	−1.46335
$412$	−7.43987	−0.366536
$413$	−32.3492	−1.59180
$414$	9.37694	0.460851
$415$	0	0
$416$	2.35072	0.115254
$417$	−45.2923	−2.21797
$418$	−5.71759	−0.279657
$419$	27.8441	1.36027	0.680137	−	0.733085i	$-0.261920\pi$
$419$	27.8441	1.36027	0.680137	+	0.733085i	$0.261920\pi$
$420$	0	0
$421$	−31.4544	−1.53299	−0.766497	−	0.642248i	$-0.778002\pi$
$421$	−31.4544	−1.53299	−0.766497	+	0.642248i	$0.778002\pi$
$422$	−0.576866	−0.0280814
$423$	78.1879	3.80162
$424$	19.7422	0.958767
$425$	0	0
$426$	0.309954	0.0150173
$427$	2.77460	0.134272
$428$	−19.1985	−0.927993
$429$	−5.40741	−0.261072
$430$	0	0
$431$	−31.3748	−1.51127	−0.755635	−	0.654993i	$-0.772672\pi$
$431$	−31.3748	−1.51127	−0.755635	+	0.654993i	$0.772672\pi$
$432$	−62.4897	−3.00654
$433$	−27.6739	−1.32992	−0.664962	−	0.746877i	$-0.731552\pi$
$433$	−27.6739	−1.32992	−0.664962	+	0.746877i	$0.731552\pi$
$434$	−5.70843	−0.274013
$435$	0	0
$436$	7.09010	0.339554
$437$	17.5546	0.839752
$438$	12.7765	0.610483
$439$	35.9019	1.71350	0.856752	−	0.515729i	$-0.172479\pi$
$439$	35.9019	1.71350	0.856752	+	0.515729i	$0.172479\pi$
$440$	0	0
$441$	36.0453	1.71644
$442$	0.230989	0.0109870
$443$	−6.70280	−0.318460	−0.159230	−	0.987242i	$-0.550901\pi$
$443$	−6.70280	−0.318460	−0.159230	+	0.987242i	$0.550901\pi$
$444$	−56.5485	−2.68367
$445$	0	0
$446$	8.30519	0.393262
$447$	46.2927	2.18957
$448$	16.7469	0.791217
$449$	4.17189	0.196884	0.0984418	−	0.995143i	$-0.468614\pi$
$449$	4.17189	0.196884	0.0984418	+	0.995143i	$0.468614\pi$
$450$	0	0
$451$	26.2225	1.23477
$452$	19.3940	0.912217
$453$	19.9457	0.937131
$454$	−7.52935	−0.353370
$455$	0	0
$456$	28.2082	1.32097
$457$	−13.4988	−0.631448	−0.315724	−	0.948851i	$-0.602247\pi$
$457$	−13.4988	−0.631448	−0.315724	+	0.948851i	$0.602247\pi$
$458$	0.980342	0.0458084
$459$	−20.8581	−0.973571
$460$	0	0
$461$	−36.6435	−1.70666	−0.853328	−	0.521374i	$-0.825420\pi$
$461$	−36.6435	−1.70666	−0.853328	+	0.521374i	$0.825420\pi$
$462$	11.0846	0.515703
$463$	−2.14127	−0.0995130	−0.0497565	−	0.998761i	$-0.515845\pi$
$463$	−2.14127	−0.0995130	−0.0497565	+	0.998761i	$0.515845\pi$
$464$	12.8125	0.594808
$465$	0	0
$466$	−1.60367	−0.0742884
$467$	0.514805	0.0238223	0.0119112	−	0.999929i	$-0.496208\pi$
$467$	0.514805	0.0238223	0.0119112	+	0.999929i	$0.496208\pi$
$468$	9.57218	0.442474
$469$	36.6068	1.69034
$470$	0	0
$471$	−45.1352	−2.07972
$472$	13.5880	0.625439
$473$	−6.50515	−0.299107
$474$	12.5728	0.577490
$475$	0	0
$476$	6.72476	0.308229
$477$	121.953	5.58384
$478$	5.63937	0.257939
$479$	28.0468	1.28149	0.640745	−	0.767754i	$-0.278626\pi$
$479$	28.0468	1.28149	0.640745	+	0.767754i	$0.278626\pi$
$480$	0	0
$481$	5.23709	0.238791
$482$	0.362712	0.0165211
$483$	−34.0329	−1.54855
$484$	7.14976	0.324989
$485$	0	0
$486$	28.6453	1.29938
$487$	−22.7761	−1.03208	−0.516041	−	0.856564i	$-0.672595\pi$
$487$	−22.7761	−1.03208	−0.516041	+	0.856564i	$0.672595\pi$
$488$	−1.16545	−0.0527573
$489$	−83.7237	−3.78612
$490$	0	0
$491$	25.8901	1.16840	0.584201	−	0.811609i	$-0.301408\pi$
$491$	25.8901	1.16840	0.584201	+	0.811609i	$0.301408\pi$
$492$	−62.4855	−2.81706
$493$	4.27662	0.192609
$494$	−1.26179	−0.0567707
$495$	0	0
$496$	−15.2081	−0.682862
$497$	−0.835702	−0.0374863
$498$	8.72906	0.391159
$499$	−1.54718	−0.0692611	−0.0346306	−	0.999400i	$-0.511025\pi$
$499$	−1.54718	−0.0692611	−0.0346306	+	0.999400i	$0.511025\pi$
$500$	0	0
$501$	57.8607	2.58503
$502$	−9.17040	−0.409295
$503$	6.37809	0.284385	0.142192	−	0.989839i	$-0.454585\pi$
$503$	6.37809	0.284385	0.142192	+	0.989839i	$0.454585\pi$
$504$	−40.6255	−1.80961
$505$	0	0
$506$	−2.89754	−0.128811
$507$	43.2117	1.91910
$508$	15.9873	0.709322
$509$	26.8650	1.19077	0.595386	−	0.803440i	$-0.296999\pi$
$509$	26.8650	1.19077	0.595386	+	0.803440i	$0.296999\pi$
$510$	0	0
$511$	−34.4481	−1.52389
$512$	−21.8963	−0.967687
$513$	113.938	5.03050
$514$	−6.74853	−0.297665
$515$	0	0
$516$	15.5011	0.682398
$517$	−24.1606	−1.06258
$518$	−10.7355	−0.471690
$519$	−76.8604	−3.37380
$520$	0	0
$521$	7.51004	0.329021	0.164511	−	0.986375i	$-0.447396\pi$
$521$	7.51004	0.329021	0.164511	+	0.986375i	$0.447396\pi$
$522$	−12.4786	−0.546174
$523$	−33.2991	−1.45607	−0.728034	−	0.685542i	$-0.759565\pi$
$523$	−33.2991	−1.45607	−0.728034	+	0.685542i	$0.759565\pi$
$524$	0.686585	0.0299936
$525$	0	0
$526$	−2.70436	−0.117916
$527$	−5.07621	−0.221123
$528$	29.5310	1.28517
$529$	−14.1037	−0.613206
$530$	0	0
$531$	83.9368	3.64255
$532$	−36.7344	−1.59264
$533$	5.78693	0.250660
$534$	8.25465	0.357214
$535$	0	0
$536$	−15.3764	−0.664157
$537$	11.7852	0.508571
$538$	−11.2566	−0.485305
$539$	−11.1383	−0.479758
$540$	0	0
$541$	−21.8383	−0.938900	−0.469450	−	0.882959i	$-0.655548\pi$
$541$	−21.8383	−0.938900	−0.469450	+	0.882959i	$0.655548\pi$
$542$	4.73979	0.203592
$543$	−52.1908	−2.23972
$544$	−4.28505	−0.183720
$545$	0	0
$546$	2.44622	0.104688
$547$	14.9034	0.637223	0.318612	−	0.947885i	$-0.396783\pi$
$547$	14.9034	0.637223	0.318612	+	0.947885i	$0.396783\pi$
$548$	−16.2278	−0.693218
$549$	−7.19927	−0.307257
$550$	0	0
$551$	−23.3613	−0.995225
$552$	14.2952	0.608446
$553$	−33.8991	−1.44154
$554$	−5.96876	−0.253589
$555$	0	0
$556$	−24.7750	−1.05069
$557$	−37.2999	−1.58045	−0.790224	−	0.612819i	$-0.790036\pi$
$557$	−37.2999	−1.58045	−0.790224	+	0.612819i	$0.790036\pi$
$558$	14.8117	0.627029
$559$	−1.43559	−0.0607191
$560$	0	0
$561$	9.85698	0.416162
$562$	−5.53778	−0.233597
$563$	−15.4464	−0.650989	−0.325494	−	0.945544i	$-0.605531\pi$
$563$	−15.4464	−0.650989	−0.325494	+	0.945544i	$0.605531\pi$
$564$	57.5722	2.42423
$565$	0	0
$566$	5.47161	0.229989
$567$	−134.030	−5.62874
$568$	0.351029	0.0147289
$569$	29.8232	1.25025	0.625127	−	0.780523i	$-0.285047\pi$
$569$	29.8232	1.25025	0.625127	+	0.780523i	$0.285047\pi$
$570$	0	0
$571$	35.6202	1.49066	0.745329	−	0.666697i	$-0.232292\pi$
$571$	35.6202	1.49066	0.745329	+	0.666697i	$0.232292\pi$
$572$	−2.95787	−0.123675
$573$	−12.5149	−0.522818
$574$	−11.8626	−0.495135
$575$	0	0
$576$	−43.4533	−1.81056
$577$	21.9411	0.913419	0.456709	−	0.889616i	$-0.349028\pi$
$577$	21.9411	0.913419	0.456709	+	0.889616i	$0.349028\pi$
$578$	5.74505	0.238963
$579$	−15.9329	−0.662150
$580$	0	0
$581$	−23.5354	−0.976414
$582$	−0.0954480	−0.00395645
$583$	−37.6843	−1.56072
$584$	14.4696	0.598757
$585$	0	0
$586$	−9.47073	−0.391232
$587$	−7.00544	−0.289146	−0.144573	−	0.989494i	$-0.546181\pi$
$587$	−7.00544	−0.289146	−0.144573	+	0.989494i	$0.546181\pi$
$588$	26.5413	1.09455
$589$	27.7291	1.14256
$590$	0	0
$591$	52.7731	2.17079
$592$	−28.6009	−1.17549
$593$	−15.1748	−0.623155	−0.311577	−	0.950221i	$-0.600857\pi$
$593$	−15.1748	−0.623155	−0.311577	+	0.950221i	$0.600857\pi$
$594$	−18.8065	−0.771639
$595$	0	0
$596$	25.3223	1.03724
$597$	18.4601	0.755520
$598$	−0.639445	−0.0261489
$599$	−13.0172	−0.531870	−0.265935	−	0.963991i	$-0.585681\pi$
$599$	−13.0172	−0.531870	−0.265935	+	0.963991i	$0.585681\pi$
$600$	0	0
$601$	36.1757	1.47564	0.737818	−	0.674999i	$-0.235856\pi$
$601$	36.1757	1.47564	0.737818	+	0.674999i	$0.235856\pi$
$602$	2.94282	0.119940
$603$	−94.9838	−3.86804
$604$	10.9104	0.443936
$605$	0	0
$606$	5.55372	0.225604
$607$	4.90393	0.199044	0.0995221	−	0.995035i	$-0.468269\pi$
$607$	4.90393	0.199044	0.0995221	+	0.995035i	$0.468269\pi$
$608$	23.4073	0.949293
$609$	45.2902	1.83525
$610$	0	0
$611$	−5.33190	−0.215705
$612$	−17.4488	−0.705326
$613$	−5.66442	−0.228784	−0.114392	−	0.993436i	$-0.536492\pi$
$613$	−5.66442	−0.228784	−0.114392	+	0.993436i	$0.536492\pi$
$614$	0.0200089	0.000807494 0
$615$	0	0
$616$	12.5536	0.505798
$617$	−26.7780	−1.07804	−0.539020	−	0.842293i	$-0.681205\pi$
$617$	−26.7780	−1.07804	−0.539020	+	0.842293i	$0.681205\pi$
$618$	4.93331	0.198447
$619$	39.9899	1.60733	0.803665	−	0.595082i	$-0.202880\pi$
$619$	39.9899	1.60733	0.803665	+	0.595082i	$0.202880\pi$
$620$	0	0
$621$	57.7412	2.31707
$622$	7.56895	0.303487
$623$	−22.2563	−0.891680
$624$	6.51707	0.260892
$625$	0	0
$626$	9.34685	0.373575
$627$	−53.8443	−2.15033
$628$	−24.6891	−0.985202
$629$	−9.54651	−0.380644
$630$	0	0
$631$	32.9213	1.31058	0.655288	−	0.755379i	$-0.272547\pi$
$631$	32.9213	1.31058	0.655288	+	0.755379i	$0.272547\pi$
$632$	14.2390	0.566398
$633$	−5.43252	−0.215923
$634$	0.956793	0.0379991
$635$	0	0
$636$	89.7979	3.56072
$637$	−2.45805	−0.0973916
$638$	3.85598	0.152660
$639$	2.16840	0.0857806
$640$	0	0
$641$	−19.2412	−0.759983	−0.379991	−	0.924990i	$-0.624073\pi$
$641$	−19.2412	−0.759983	−0.379991	+	0.924990i	$0.624073\pi$
$642$	12.7303	0.502426
$643$	−37.4059	−1.47514	−0.737572	−	0.675268i	$-0.764028\pi$
$643$	−37.4059	−1.47514	−0.737572	+	0.675268i	$0.764028\pi$
$644$	−18.6161	−0.733578
$645$	0	0
$646$	2.30008	0.0904953
$647$	38.1467	1.49970	0.749851	−	0.661607i	$-0.230125\pi$
$647$	38.1467	1.49970	0.749851	+	0.661607i	$0.230125\pi$
$648$	56.2982	2.21160
$649$	−25.9371	−1.01812
$650$	0	0
$651$	−53.7580	−2.10694
$652$	−45.7971	−1.79355
$653$	26.3157	1.02981	0.514906	−	0.857247i	$-0.327827\pi$
$653$	26.3157	1.02981	0.514906	+	0.857247i	$0.327827\pi$
$654$	−4.70138	−0.183838
$655$	0	0
$656$	−31.6036	−1.23391
$657$	89.3827	3.48715
$658$	10.9298	0.426089
$659$	−25.5777	−0.996366	−0.498183	−	0.867072i	$-0.665999\pi$
$659$	−25.5777	−0.996366	−0.498183	+	0.867072i	$0.665999\pi$
$660$	0	0
$661$	16.8601	0.655781	0.327890	−	0.944716i	$-0.393662\pi$
$661$	16.8601	0.655781	0.327890	+	0.944716i	$0.393662\pi$
$662$	0.725479	0.0281966
$663$	2.17529	0.0844815
$664$	9.88585	0.383646
$665$	0	0
$666$	27.8554	1.07938
$667$	−11.8389	−0.458406
$668$	31.6500	1.22458
$669$	78.2125	3.02387
$670$	0	0
$671$	2.22462	0.0858807
$672$	−45.3795	−1.75055
$673$	−13.1203	−0.505751	−0.252875	−	0.967499i	$-0.581376\pi$
$673$	−13.1203	−0.505751	−0.252875	+	0.967499i	$0.581376\pi$
$674$	11.4490	0.440999
$675$	0	0
$676$	23.6370	0.909114
$677$	4.07063	0.156447	0.0782236	−	0.996936i	$-0.475075\pi$
$677$	4.07063	0.156447	0.0782236	+	0.996936i	$0.475075\pi$
$678$	−12.8600	−0.493884
$679$	0.257348	0.00987612
$680$	0	0
$681$	−70.9062	−2.71713
$682$	−4.57692	−0.175259
$683$	−24.7733	−0.947924	−0.473962	−	0.880545i	$-0.657177\pi$
$683$	−24.7733	−0.947924	−0.473962	+	0.880545i	$0.657177\pi$
$684$	95.3151	3.64446
$685$	0	0
$686$	−3.44263	−0.131440
$687$	9.23218	0.352230
$688$	7.84008	0.298900
$689$	−8.31639	−0.316829
$690$	0	0
$691$	−15.7267	−0.598270	−0.299135	−	0.954211i	$-0.596698\pi$
$691$	−15.7267	−0.598270	−0.299135	+	0.954211i	$0.596698\pi$
$692$	−42.0429	−1.59823
$693$	77.5467	2.94576
$694$	−4.03197	−0.153051
$695$	0	0
$696$	−19.0238	−0.721094
$697$	−10.5488	−0.399564
$698$	−0.796704	−0.0301557
$699$	−15.1022	−0.571218
$700$	0	0
$701$	23.0979	0.872397	0.436199	−	0.899850i	$-0.356325\pi$
$701$	23.0979	0.872397	0.436199	+	0.899850i	$0.356325\pi$
$702$	−4.15032	−0.156644
$703$	52.1483	1.96681
$704$	13.4274	0.506063
$705$	0	0
$706$	−0.691182	−0.0260130
$707$	−14.9740	−0.563156
$708$	61.8054	2.32279
$709$	1.65188	0.0620376	0.0310188	−	0.999519i	$-0.490125\pi$
$709$	1.65188	0.0620376	0.0310188	+	0.999519i	$0.490125\pi$
$710$	0	0
$711$	87.9582	3.29869
$712$	9.34858	0.350353
$713$	14.0524	0.526268
$714$	−4.45913	−0.166879
$715$	0	0
$716$	6.44656	0.240919
$717$	53.1077	1.98334
$718$	3.06677	0.114451
$719$	−30.6145	−1.14173	−0.570865	−	0.821044i	$-0.693392\pi$
$719$	−30.6145	−1.14173	−0.570865	+	0.821044i	$0.693392\pi$
$720$	0	0
$721$	−13.3013	−0.495365
$722$	−5.67276	−0.211118
$723$	3.41577	0.127034
$724$	−28.5485	−1.06100
$725$	0	0
$726$	−4.74094	−0.175953
$727$	20.1615	0.747750	0.373875	−	0.927479i	$-0.378029\pi$
$727$	20.1615	0.747750	0.373875	+	0.927479i	$0.378029\pi$
$728$	2.77039	0.102678
$729$	149.392	5.53303
$730$	0	0
$731$	2.61689	0.0967893
$732$	−5.30105	−0.195933
$733$	−16.5899	−0.612762	−0.306381	−	0.951909i	$-0.599118\pi$
$733$	−16.5899	−0.612762	−0.306381	+	0.951909i	$0.599118\pi$
$734$	11.1487	0.411507
$735$	0	0
$736$	11.8623	0.437249
$737$	29.3507	1.08115
$738$	30.7800	1.13303
$739$	44.8663	1.65043	0.825217	−	0.564816i	$-0.191053\pi$
$739$	44.8663	1.65043	0.825217	+	0.564816i	$0.191053\pi$
$740$	0	0
$741$	−11.8827	−0.436521
$742$	17.0477	0.625842
$743$	15.0186	0.550978	0.275489	−	0.961304i	$-0.411160\pi$
$743$	15.0186	0.550978	0.275489	+	0.961304i	$0.411160\pi$
$744$	22.5806	0.827844
$745$	0	0
$746$	2.62247	0.0960156
$747$	61.0676	2.23434
$748$	5.39180	0.197144
$749$	−34.3237	−1.25416
$750$	0	0
$751$	−15.0984	−0.550948	−0.275474	−	0.961309i	$-0.588835\pi$
$751$	−15.0984	−0.550948	−0.275474	+	0.961309i	$0.588835\pi$
$752$	29.1186	1.06185
$753$	−86.3605	−3.14715
$754$	0.850959	0.0309901
$755$	0	0
$756$	−120.828	−4.39447
$757$	12.1390	0.441198	0.220599	−	0.975365i	$-0.429199\pi$
$757$	12.1390	0.441198	0.220599	+	0.975365i	$0.429199\pi$
$758$	5.93470	0.215558
$759$	−27.2870	−0.990455
$760$	0	0
$761$	−26.6306	−0.965358	−0.482679	−	0.875797i	$-0.660336\pi$
$761$	−26.6306	−0.965358	−0.482679	+	0.875797i	$0.660336\pi$
$762$	−10.6010	−0.384035
$763$	12.6759	0.458899
$764$	−6.84570	−0.247669
$765$	0	0
$766$	−6.20020	−0.224022
$767$	−5.72394	−0.206679
$768$	−22.1413	−0.798956
$769$	−19.3398	−0.697411	−0.348705	−	0.937232i	$-0.613379\pi$
$769$	−19.3398	−0.697411	−0.348705	+	0.937232i	$0.613379\pi$
$770$	0	0
$771$	−63.5529	−2.28880
$772$	−8.71536	−0.313673
$773$	35.1241	1.26333	0.631663	−	0.775243i	$-0.282373\pi$
$773$	35.1241	1.26333	0.631663	+	0.775243i	$0.282373\pi$
$774$	−7.63575	−0.274461
$775$	0	0
$776$	−0.108097	−0.00388046
$777$	−101.099	−3.62692
$778$	6.68709	0.239744
$779$	57.6234	2.06457
$780$	0	0
$781$	−0.670051	−0.0239763
$782$	1.16562	0.0416826
$783$	−76.8407	−2.74606
$784$	13.4240	0.479427
$785$	0	0
$786$	−0.455268	−0.0162389
$787$	−33.6906	−1.20094	−0.600469	−	0.799648i	$-0.705020\pi$
$787$	−33.6906	−1.20094	−0.600469	+	0.799648i	$0.705020\pi$
$788$	28.8671	1.02835
$789$	−25.4678	−0.906677
$790$	0	0
$791$	34.6733	1.23284
$792$	−32.5728	−1.15743
$793$	0.490943	0.0174339
$794$	12.5582	0.445675
$795$	0	0
$796$	10.0977	0.357904
$797$	44.2786	1.56843	0.784215	−	0.620489i	$-0.213066\pi$
$797$	44.2786	1.56843	0.784215	+	0.620489i	$0.213066\pi$
$798$	24.3582	0.862272
$799$	9.71933	0.343845
$800$	0	0
$801$	57.7486	2.04045
$802$	−6.79915	−0.240086
$803$	−27.6199	−0.974684
$804$	−69.9396	−2.46658
$805$	0	0
$806$	−1.01006	−0.0355779
$807$	−106.007	−3.73161
$808$	6.28971	0.221271
$809$	−40.3465	−1.41851	−0.709253	−	0.704954i	$-0.750968\pi$
$809$	−40.3465	−1.41851	−0.709253	+	0.704954i	$0.750968\pi$
$810$	0	0
$811$	−5.93409	−0.208374	−0.104187	−	0.994558i	$-0.533224\pi$
$811$	−5.93409	−0.208374	−0.104187	+	0.994558i	$0.533224\pi$
$812$	24.7739	0.869394
$813$	44.6361	1.56546
$814$	−8.60752	−0.301693
$815$	0	0
$816$	−11.8798	−0.415874
$817$	−14.2949	−0.500117
$818$	−13.1238	−0.458864
$819$	17.1135	0.597993
$820$	0	0
$821$	−34.3672	−1.19942	−0.599711	−	0.800217i	$-0.704718\pi$
$821$	−34.3672	−1.19942	−0.599711	+	0.800217i	$0.704718\pi$
$822$	10.7605	0.375316
$823$	36.1276	1.25933	0.629665	−	0.776867i	$-0.283192\pi$
$823$	36.1276	1.25933	0.629665	+	0.776867i	$0.283192\pi$
$824$	5.58708	0.194635
$825$	0	0
$826$	11.7335	0.408260
$827$	−27.2587	−0.947877	−0.473938	−	0.880558i	$-0.657168\pi$
$827$	−27.2587	−0.947877	−0.473938	+	0.880558i	$0.657168\pi$
$828$	48.3034	1.67866
$829$	31.7188	1.10164	0.550819	−	0.834625i	$-0.314315\pi$
$829$	31.7188	1.10164	0.550819	+	0.834625i	$0.314315\pi$
$830$	0	0
$831$	−56.2097	−1.94989
$832$	2.96323	0.102732
$833$	4.48070	0.155247
$834$	16.4281	0.568858
$835$	0	0
$836$	−29.4530	−1.01865
$837$	91.2073	3.15259
$838$	−10.0994	−0.348878
$839$	19.0221	0.656715	0.328357	−	0.944554i	$-0.393505\pi$
$839$	19.0221	0.656715	0.328357	+	0.944554i	$0.393505\pi$
$840$	0	0
$841$	−13.2450	−0.456724
$842$	11.4089	0.393177
$843$	−52.1510	−1.79617
$844$	−2.97161	−0.102287
$845$	0	0
$846$	−28.3597	−0.975027
$847$	12.7826	0.439215
$848$	45.4176	1.55965
$849$	51.5279	1.76843
$850$	0	0
$851$	26.4275	0.905924
$852$	1.59666	0.0547008
$853$	44.3027	1.51690	0.758448	−	0.651733i	$-0.225958\pi$
$853$	44.3027	1.51690	0.758448	+	0.651733i	$0.225958\pi$
$854$	−1.00638	−0.0344377
$855$	0	0
$856$	14.4174	0.492776
$857$	4.65097	0.158874	0.0794371	−	0.996840i	$-0.474688\pi$
$857$	4.65097	0.158874	0.0794371	+	0.996840i	$0.474688\pi$
$858$	1.96133	0.0669588
$859$	−19.3303	−0.659540	−0.329770	−	0.944061i	$-0.606971\pi$
$859$	−19.3303	−0.659540	−0.329770	+	0.944061i	$0.606971\pi$
$860$	0	0
$861$	−111.714	−3.80719
$862$	11.3800	0.387605
$863$	−22.7992	−0.776094	−0.388047	−	0.921640i	$-0.626850\pi$
$863$	−22.7992	−0.776094	−0.388047	+	0.921640i	$0.626850\pi$
$864$	76.9921	2.61932
$865$	0	0
$866$	10.0377	0.341094
$867$	54.1029	1.83743
$868$	−29.4058	−0.998098
$869$	−27.1797	−0.922007
$870$	0	0
$871$	6.47727	0.219474
$872$	−5.32441	−0.180307
$873$	−0.667744	−0.0225997
$874$	−6.36728	−0.215377
$875$	0	0
$876$	65.8154	2.22369
$877$	−52.0581	−1.75788	−0.878939	−	0.476934i	$-0.841748\pi$
$877$	−52.0581	−1.75788	−0.878939	+	0.476934i	$0.841748\pi$
$878$	−13.0221	−0.439473
$879$	−89.1888	−3.00826
$880$	0	0
$881$	−30.1695	−1.01644	−0.508218	−	0.861229i	$-0.669695\pi$
$881$	−30.1695	−1.01644	−0.508218	+	0.861229i	$0.669695\pi$
$882$	−13.0741	−0.440228
$883$	−33.7863	−1.13700	−0.568500	−	0.822683i	$-0.692476\pi$
$883$	−33.7863	−1.13700	−0.568500	+	0.822683i	$0.692476\pi$
$884$	1.18989	0.0400204
$885$	0	0
$886$	2.43119	0.0816774
$887$	−35.3082	−1.18553	−0.592767	−	0.805374i	$-0.701964\pi$
$887$	−35.3082	−1.18553	−0.592767	+	0.805374i	$0.701964\pi$
$888$	42.4659	1.42506
$889$	28.5827	0.958631
$890$	0	0
$891$	−107.463	−3.60015
$892$	42.7825	1.43246
$893$	−53.0924	−1.77667
$894$	−16.7910	−0.561574
$895$	0	0
$896$	−32.6449	−1.09059
$897$	−6.02185	−0.201064
$898$	−1.51320	−0.0504960
$899$	−18.7007	−0.623702
$900$	0	0
$901$	15.1597	0.505042
$902$	−9.51122	−0.316689
$903$	27.7134	0.922244
$904$	−14.5642	−0.484398
$905$	0	0
$906$	−7.23456	−0.240352
$907$	13.2957	0.441476	0.220738	−	0.975333i	$-0.429153\pi$
$907$	13.2957	0.441476	0.220738	+	0.975333i	$0.429153\pi$
$908$	−38.7859	−1.28716
$909$	38.8532	1.28868
$910$	0	0
$911$	32.8035	1.08683	0.543415	−	0.839464i	$-0.317131\pi$
$911$	32.8035	1.08683	0.543415	+	0.839464i	$0.317131\pi$
$912$	64.8938	2.14885
$913$	−18.8703	−0.624515
$914$	4.89619	0.161951
$915$	0	0
$916$	5.05003	0.166858
$917$	1.22750	0.0405356
$918$	7.56548	0.249698
$919$	4.52202	0.149168	0.0745838	−	0.997215i	$-0.476237\pi$
$919$	4.52202	0.149168	0.0745838	+	0.997215i	$0.476237\pi$
$920$	0	0
$921$	0.188430	0.00620898
$922$	13.2910	0.437717
$923$	−0.147871	−0.00486722
$924$	57.1001	1.87846
$925$	0	0
$926$	0.776664	0.0255227
$927$	34.5129	1.13355
$928$	−15.7860	−0.518202
$929$	40.2300	1.31990	0.659952	−	0.751308i	$-0.270577\pi$
$929$	40.2300	1.31990	0.659952	+	0.751308i	$0.270577\pi$
$930$	0	0
$931$	−24.4761	−0.802172
$932$	−8.26096	−0.270597
$933$	71.2792	2.33357
$934$	−0.186726	−0.00610987
$935$	0	0
$936$	−7.18836	−0.234959
$937$	−34.8986	−1.14009	−0.570044	−	0.821614i	$-0.693074\pi$
$937$	−34.8986	−1.14009	−0.570044	+	0.821614i	$0.693074\pi$
$938$	−13.2777	−0.433533
$939$	88.0221	2.87249
$940$	0	0
$941$	−19.4140	−0.632879	−0.316439	−	0.948613i	$-0.602487\pi$
$941$	−19.4140	−0.632879	−0.316439	+	0.948613i	$0.602487\pi$
$942$	16.3711	0.533399
$943$	29.2021	0.950953
$944$	31.2597	1.01742
$945$	0	0
$946$	2.35950	0.0767139
$947$	−19.3006	−0.627184	−0.313592	−	0.949558i	$-0.601532\pi$
$947$	−19.3006	−0.627184	−0.313592	+	0.949558i	$0.601532\pi$
$948$	64.7664	2.10352
$949$	−6.09531	−0.197862
$950$	0	0
$951$	9.01042	0.292183
$952$	−5.05006	−0.163673
$953$	−6.26171	−0.202837	−0.101418	−	0.994844i	$-0.532338\pi$
$953$	−6.26171	−0.202837	−0.101418	+	0.994844i	$0.532338\pi$
$954$	−44.2339	−1.43212
$955$	0	0
$956$	29.0501	0.939546
$957$	36.3129	1.17383
$958$	−10.1729	−0.328672
$959$	−29.0127	−0.936867
$960$	0	0
$961$	−8.80294	−0.283966
$962$	−1.89956	−0.0612442
$963$	89.0599	2.86992
$964$	1.86844	0.0601784
$965$	0	0
$966$	12.3442	0.397167
$967$	16.5403	0.531899	0.265950	−	0.963987i	$-0.414315\pi$
$967$	16.5403	0.531899	0.265950	+	0.963987i	$0.414315\pi$
$968$	−5.36921	−0.172573
$969$	21.6605	0.695836
$970$	0	0
$971$	12.6336	0.405433	0.202717	−	0.979237i	$-0.435023\pi$
$971$	12.6336	0.405433	0.202717	+	0.979237i	$0.435023\pi$
$972$	147.560	4.73300
$973$	−44.2936	−1.41999
$974$	8.26116	0.264705
$975$	0	0
$976$	−2.68114	−0.0858213
$977$	−0.0361421	−0.00115629	−0.000578145	−	1.00000i	$-0.500184\pi$
$977$	−0.0361421	−0.00115629	−0.000578145		1.00000i	$0.500184\pi$
$978$	30.3676	0.971050
$979$	−17.8447	−0.570320
$980$	0	0
$981$	−32.8903	−1.05011
$982$	−9.39065	−0.299668
$983$	−21.7663	−0.694236	−0.347118	−	0.937821i	$-0.612840\pi$
$983$	−21.7663	−0.694236	−0.347118	+	0.937821i	$0.612840\pi$
$984$	46.9244	1.49589
$985$	0	0
$986$	−1.55118	−0.0493998
$987$	102.930	3.27628
$988$	−6.49986	−0.206788
$989$	−7.24433	−0.230356
$990$	0	0
$991$	12.9908	0.412665	0.206332	−	0.978482i	$-0.433847\pi$
$991$	12.9908	0.412665	0.206332	+	0.978482i	$0.433847\pi$
$992$	18.7375	0.594917
$993$	6.83206	0.216809
$994$	0.303119	0.00961437
$995$	0	0
$996$	44.9660	1.42480
$997$	2.82932	0.0896055	0.0448028	−	0.998996i	$-0.485734\pi$
$997$	2.82932	0.0896055	0.0448028	+	0.998996i	$0.485734\pi$
$998$	0.561180	0.0177638
$999$	171.528	5.42690

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.j.1.13		25
5.4	even	2		1205.2.a.e.1.13	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.a.e.1.13	✓	25	5.4	even	2
6025.2.a.j.1.13		25	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)

\(f(q)\)	\(=\)	\(q-0.362712 q^{2} -3.41577 q^{3} -1.86844 q^{4} +1.23894 q^{6} -3.34046 q^{7} +1.40313 q^{8} +8.66752 q^{9} +O(q^{10})\)	\(q-0.362712 q^{2} -3.41577 q^{3} -1.86844 q^{4} +1.23894 q^{6} -3.34046 q^{7} +1.40313 q^{8} +8.66752 q^{9} -2.67832 q^{11} +6.38217 q^{12} -0.591068 q^{13} +1.21163 q^{14} +3.22795 q^{16} +1.07744 q^{17} -3.14382 q^{18} -5.88556 q^{19} +11.4103 q^{21} +0.971461 q^{22} -2.98266 q^{23} -4.79278 q^{24} +0.214388 q^{26} -19.3590 q^{27} +6.24145 q^{28} +3.96926 q^{29} -4.71138 q^{31} -3.97708 q^{32} +9.14854 q^{33} -0.390800 q^{34} -16.1947 q^{36} -8.86039 q^{37} +2.13477 q^{38} +2.01895 q^{39} -9.79063 q^{41} -4.13864 q^{42} +2.42881 q^{43} +5.00428 q^{44} +1.08185 q^{46} +9.02079 q^{47} -11.0259 q^{48} +4.15867 q^{49} -3.68028 q^{51} +1.10437 q^{52} +14.0701 q^{53} +7.02174 q^{54} -4.68710 q^{56} +20.1037 q^{57} -1.43970 q^{58} +9.68407 q^{59} -0.830604 q^{61} +1.70887 q^{62} -28.9535 q^{63} -5.01336 q^{64} -3.31829 q^{66} -10.9586 q^{67} -2.01313 q^{68} +10.1881 q^{69} +0.250176 q^{71} +12.1617 q^{72} +10.3124 q^{73} +3.21377 q^{74} +10.9968 q^{76} +8.94682 q^{77} -0.732300 q^{78} +10.1480 q^{79} +40.1233 q^{81} +3.55119 q^{82} +7.04557 q^{83} -21.3194 q^{84} -0.880961 q^{86} -13.5581 q^{87} -3.75804 q^{88} +6.66265 q^{89} +1.97444 q^{91} +5.57292 q^{92} +16.0930 q^{93} -3.27195 q^{94} +13.5848 q^{96} -0.0770398 q^{97} -1.50840 q^{98} -23.2144 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9}+O(q^{10}) \) 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9	\( 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9} + 2 q^{11} - 20 q^{12} - 14 q^{13} - 5 q^{14} + 38 q^{16} - 7 q^{17} - 9 q^{18} + 30 q^{19} + q^{21} - q^{22} - 43 q^{23} - 6 q^{24} - 22 q^{26} - 42 q^{27} - 32 q^{28} - 4 q^{29} + 14 q^{31} - 26 q^{32} - 4 q^{33} + 7 q^{34} + 15 q^{36} - 16 q^{37} - 14 q^{38} - 21 q^{39} - q^{41} + 25 q^{42} - 35 q^{43} - 52 q^{44} - 27 q^{46} - 50 q^{47} - 26 q^{48} + 46 q^{49} - 7 q^{51} - 3 q^{52} - 4 q^{53} - 31 q^{54} - 51 q^{56} - 2 q^{58} + 6 q^{59} + 19 q^{61} - 28 q^{63} + 49 q^{64} - 27 q^{66} - 65 q^{67} + 25 q^{68} + 2 q^{69} - 34 q^{71} + 10 q^{72} - 8 q^{73} - 42 q^{74} + 71 q^{76} - q^{77} + 59 q^{78} - 12 q^{79} + 29 q^{81} - 11 q^{82} - 41 q^{83} - 10 q^{84} - 13 q^{86} - 40 q^{87} + 52 q^{88} - 24 q^{89} + 46 q^{91} - 85 q^{92} + 30 q^{93} + 14 q^{94} - 30 q^{96} - 9 q^{97} + 64 q^{98} + 2 q^{99}+O(q^{100}) \) 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9 + 2 * q^11 - 20 * q^12 - 14 * q^13 - 5 * q^14 + 38 * q^16 - 7 * q^17 - 9 * q^18 + 30 * q^19 + q^21 - q^22 - 43 * q^23 - 6 * q^24 - 22 * q^26 - 42 * q^27 - 32 * q^28 - 4 * q^29 + 14 * q^31 - 26 * q^32 - 4 * q^33 + 7 * q^34 + 15 * q^36 - 16 * q^37 - 14 * q^38 - 21 * q^39 - q^41 + 25 * q^42 - 35 * q^43 - 52 * q^44 - 27 * q^46 - 50 * q^47 - 26 * q^48 + 46 * q^49 - 7 * q^51 - 3 * q^52 - 4 * q^53 - 31 * q^54 - 51 * q^56 - 2 * q^58 + 6 * q^59 + 19 * q^61 - 28 * q^63 + 49 * q^64 - 27 * q^66 - 65 * q^67 + 25 * q^68 + 2 * q^69 - 34 * q^71 + 10 * q^72 - 8 * q^73 - 42 * q^74 + 71 * q^76 - q^77 + 59 * q^78 - 12 * q^79 + 29 * q^81 - 11 * q^82 - 41 * q^83 - 10 * q^84 - 13 * q^86 - 40 * q^87 + 52 * q^88 - 24 * q^89 + 46 * q^91 - 85 * q^92 + 30 * q^93 + 14 * q^94 - 30 * q^96 - 9 * q^97 + 64 * q^98 + 2 * q^99

Introduction

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