LMFDB - Embedded newform 6475.2.a.u.1.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6475,2,Mod(1,6475)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6475.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6475 = 5^{2} \cdot 7 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6475.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,1,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$51.7031353088$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 26 x^{13} + 24 x^{12} + 266 x^{11} - 222 x^{10} - 1368 x^{9} + 998 x^{8} + 3770 x^{7} + \cdots - 158 $ x^15 - x^14 - 26x^13 + 24x^12 + 266x^11 - 222x^10 - 1368x^9 + 998x^8 + 3770x^7 - 2257x^6 - 5537x^5 + 2394x^4 + 3982x^3 - 824x^2 - 1058*x - 158
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1295)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$2.26763$ of defining polynomial
Character	$\chi$	$=$	6475.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.26763 q^{2} +2.33889 q^{3} +3.14213 q^{4} +5.30373 q^{6} +1.00000 q^{7} +2.58991 q^{8} +2.47041 q^{9} +2.79940 q^{11} +7.34909 q^{12} -6.33892 q^{13} +2.26763 q^{14} -0.411300 q^{16} +5.73746 q^{17} +5.60196 q^{18} +5.46034 q^{19} +2.33889 q^{21} +6.34799 q^{22} +6.62426 q^{23} +6.05752 q^{24} -14.3743 q^{26} -1.23866 q^{27} +3.14213 q^{28} +7.31892 q^{29} -5.72336 q^{31} -6.11250 q^{32} +6.54749 q^{33} +13.0104 q^{34} +7.76233 q^{36} -1.00000 q^{37} +12.3820 q^{38} -14.8260 q^{39} -7.32058 q^{41} +5.30373 q^{42} +8.56324 q^{43} +8.79606 q^{44} +15.0213 q^{46} +1.83555 q^{47} -0.961985 q^{48} +1.00000 q^{49} +13.4193 q^{51} -19.9177 q^{52} -2.39441 q^{53} -2.80882 q^{54} +2.58991 q^{56} +12.7711 q^{57} +16.5966 q^{58} -5.54706 q^{59} +6.78124 q^{61} -12.9784 q^{62} +2.47041 q^{63} -13.0383 q^{64} +14.8472 q^{66} -1.71727 q^{67} +18.0278 q^{68} +15.4934 q^{69} +5.11520 q^{71} +6.39814 q^{72} +5.64803 q^{73} -2.26763 q^{74} +17.1571 q^{76} +2.79940 q^{77} -33.6199 q^{78} +2.53359 q^{79} -10.3083 q^{81} -16.6003 q^{82} +1.76178 q^{83} +7.34909 q^{84} +19.4182 q^{86} +17.1182 q^{87} +7.25020 q^{88} -13.0580 q^{89} -6.33892 q^{91} +20.8143 q^{92} -13.3863 q^{93} +4.16235 q^{94} -14.2965 q^{96} -13.1335 q^{97} +2.26763 q^{98} +6.91566 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + q^{2} + q^{3} + 23 q^{4} + 6 q^{6} + 15 q^{7} + 3 q^{8} + 30 q^{9} + 17 q^{11} - 8 q^{12} + 5 q^{13} + q^{14} + 39 q^{16} + 7 q^{17} - 12 q^{18} + 6 q^{19} + q^{21} - 18 q^{22} + 2 q^{23} + 4 q^{24}+ \cdots + 90 q^{99}+O(q^{100}) $ 15 * q + q^2 + q^3 + 23 * q^4 + 6 * q^6 + 15 * q^7 + 3 * q^8 + 30 * q^9 + 17 * q^11 - 8 * q^12 + 5 * q^13 + q^14 + 39 * q^16 + 7 * q^17 - 12 * q^18 + 6 * q^19 + q^21 - 18 * q^22 + 2 * q^23 + 4 * q^24 + 4 * q^26 - 5 * q^27 + 23 * q^28 + 13 * q^29 - 4 * q^31 + 7 * q^32 - 7 * q^33 + 38 * q^34 + 49 * q^36 - 15 * q^37 + 26 * q^38 + 33 * q^39 + 22 * q^41 + 6 * q^42 - 10 * q^43 + 58 * q^44 + 32 * q^46 + 9 * q^47 - 43 * q^48 + 15 * q^49 + 39 * q^51 - 6 * q^52 + 16 * q^53 + 30 * q^54 + 3 * q^56 - 16 * q^57 - 44 * q^58 + 6 * q^59 + 11 * q^62 + 30 * q^63 + 119 * q^64 + 3 * q^66 - 4 * q^67 + 70 * q^68 + 14 * q^69 + 48 * q^71 + 25 * q^72 - 38 * q^73 - q^74 - 3 * q^76 + 17 * q^77 + 40 * q^78 + 7 * q^79 + 111 * q^81 - 6 * q^82 + 6 * q^83 - 8 * q^84 - 40 * q^86 + 47 * q^87 - 23 * q^88 - 2 * q^89 + 5 * q^91 - 7 * q^92 - 26 * q^94 - 29 * q^96 - 79 * q^97 + q^98 + 90 * 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 $

$2$	2.26763	1.60345	0.801727	−	0.597691i	$-0.203915\pi$
$2$	2.26763	1.60345	0.801727	+	0.597691i	$0.203915\pi$
$3$	2.33889	1.35036	0.675179	−	0.737654i	$-0.264066\pi$
$3$	2.33889	1.35036	0.675179	+	0.737654i	$0.264066\pi$
$4$	3.14213	1.57106
$5$	0	0
$5$	0	0
$6$	5.30373	2.16524
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.58991	0.915672
$9$	2.47041	0.823469
$10$	0	0
$11$	2.79940	0.844051	0.422025	−	0.906584i	$-0.361319\pi$
$11$	2.79940	0.844051	0.422025	+	0.906584i	$0.361319\pi$
$12$	7.34909	2.12150
$13$	−6.33892	−1.75810	−0.879050	−	0.476730i	$-0.841822\pi$
$13$	−6.33892	−1.75810	−0.879050	+	0.476730i	$0.841822\pi$
$14$	2.26763	0.606048
$15$	0	0
$16$	−0.411300	−0.102825
$17$	5.73746	1.39154	0.695769	−	0.718266i	$-0.255064\pi$
$17$	5.73746	1.39154	0.695769	+	0.718266i	$0.255064\pi$
$18$	5.60196	1.32039
$19$	5.46034	1.25269	0.626344	−	0.779547i	$-0.284551\pi$
$19$	5.46034	1.25269	0.626344	+	0.779547i	$0.284551\pi$
$20$	0	0
$21$	2.33889	0.510388
$22$	6.34799	1.35340
$23$	6.62426	1.38125	0.690627	−	0.723211i	$-0.257335\pi$
$23$	6.62426	1.38125	0.690627	+	0.723211i	$0.257335\pi$
$24$	6.05752	1.23649
$25$	0	0
$26$	−14.3743	−2.81903
$27$	−1.23866	−0.238380
$28$	3.14213	0.593806
$29$	7.31892	1.35909	0.679545	−	0.733634i	$-0.262177\pi$
$29$	7.31892	1.35909	0.679545	+	0.733634i	$0.262177\pi$
$30$	0	0
$31$	−5.72336	−1.02795	−0.513973	−	0.857806i	$-0.671827\pi$
$31$	−5.72336	−1.02795	−0.513973	+	0.857806i	$0.671827\pi$
$32$	−6.11250	−1.08055
$33$	6.54749	1.13977
$34$	13.0104	2.23127
$35$	0	0
$36$	7.76233	1.29372
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	12.3820	2.00863
$39$	−14.8260	−2.37407
$40$	0	0
$41$	−7.32058	−1.14328	−0.571641	−	0.820504i	$-0.693693\pi$
$41$	−7.32058	−1.14328	−0.571641	+	0.820504i	$0.693693\pi$
$42$	5.30373	0.818383
$43$	8.56324	1.30588	0.652941	−	0.757409i	$-0.273535\pi$
$43$	8.56324	1.30588	0.652941	+	0.757409i	$0.273535\pi$
$44$	8.79606	1.32606
$45$	0	0
$46$	15.0213	2.21478
$47$	1.83555	0.267743	0.133871	−	0.990999i	$-0.457259\pi$
$47$	1.83555	0.267743	0.133871	+	0.990999i	$0.457259\pi$
$48$	−0.961985	−0.138851
$49$	1.00000	0.142857
$50$	0	0
$51$	13.4193	1.87908
$52$	−19.9177	−2.76208
$53$	−2.39441	−0.328898	−0.164449	−	0.986386i	$-0.552585\pi$
$53$	−2.39441	−0.328898	−0.164449	+	0.986386i	$0.552585\pi$
$54$	−2.80882	−0.382231
$55$	0	0
$56$	2.58991	0.346092
$57$	12.7711	1.69158
$58$	16.5966	2.17924
$59$	−5.54706	−0.722165	−0.361083	−	0.932534i	$-0.617593\pi$
$59$	−5.54706	−0.722165	−0.361083	+	0.932534i	$0.617593\pi$
$60$	0	0
$61$	6.78124	0.868249	0.434125	−	0.900853i	$-0.357058\pi$
$61$	6.78124	0.868249	0.434125	+	0.900853i	$0.357058\pi$
$62$	−12.9784	−1.64826
$63$	2.47041	0.311242
$64$	−13.0383	−1.62978
$65$	0	0
$66$	14.8472	1.82757
$67$	−1.71727	−0.209798	−0.104899	−	0.994483i	$-0.533452\pi$
$67$	−1.71727	−0.209798	−0.104899	+	0.994483i	$0.533452\pi$
$68$	18.0278	2.18619
$69$	15.4934	1.86519
$70$	0	0
$71$	5.11520	0.607063	0.303531	−	0.952821i	$-0.401834\pi$
$71$	5.11520	0.607063	0.303531	+	0.952821i	$0.401834\pi$
$72$	6.39814	0.754028
$73$	5.64803	0.661052	0.330526	−	0.943797i	$-0.392774\pi$
$73$	5.64803	0.661052	0.330526	+	0.943797i	$0.392774\pi$
$74$	−2.26763	−0.263606
$75$	0	0
$76$	17.1571	1.96805
$77$	2.79940	0.319021
$78$	−33.6199	−3.80670
$79$	2.53359	0.285052	0.142526	−	0.989791i	$-0.454478\pi$
$79$	2.53359	0.285052	0.142526	+	0.989791i	$0.454478\pi$
$80$	0	0
$81$	−10.3083	−1.14537
$82$	−16.6003	−1.83320
$83$	1.76178	0.193381	0.0966903	−	0.995315i	$-0.469174\pi$
$83$	1.76178	0.193381	0.0966903	+	0.995315i	$0.469174\pi$
$84$	7.34909	0.801851
$85$	0	0
$86$	19.4182	2.09392
$87$	17.1182	1.83526
$88$	7.25020	0.772874
$89$	−13.0580	−1.38415	−0.692075	−	0.721826i	$-0.743303\pi$
$89$	−13.0580	−1.38415	−0.692075	+	0.721826i	$0.743303\pi$
$90$	0	0
$91$	−6.33892	−0.664499
$92$	20.8143	2.17004
$93$	−13.3863	−1.38810
$94$	4.16235	0.429313
$95$	0	0
$96$	−14.2965	−1.45913
$97$	−13.1335	−1.33351	−0.666753	−	0.745279i	$-0.732316\pi$
$97$	−13.1335	−1.33351	−0.666753	+	0.745279i	$0.732316\pi$
$98$	2.26763	0.229065
$99$	6.91566	0.695050
$100$	0	0
$101$	1.86164	0.185240	0.0926202	−	0.995702i	$-0.470476\pi$
$101$	1.86164	0.185240	0.0926202	+	0.995702i	$0.470476\pi$
$102$	30.4299	3.01301
$103$	0.996582	0.0981961	0.0490981	−	0.998794i	$-0.484365\pi$
$103$	0.996582	0.0981961	0.0490981	+	0.998794i	$0.484365\pi$
$104$	−16.4172	−1.60984
$105$	0	0
$106$	−5.42964	−0.527373
$107$	−12.2285	−1.18218	−0.591088	−	0.806607i	$-0.701301\pi$
$107$	−12.2285	−1.18218	−0.591088	+	0.806607i	$0.701301\pi$
$108$	−3.89202	−0.374510
$109$	17.7562	1.70074	0.850369	−	0.526186i	$-0.176379\pi$
$109$	17.7562	1.70074	0.850369	+	0.526186i	$0.176379\pi$
$110$	0	0
$111$	−2.33889	−0.221998
$112$	−0.411300	−0.0388642
$113$	−12.1725	−1.14509	−0.572547	−	0.819872i	$-0.694044\pi$
$113$	−12.1725	−1.14509	−0.572547	+	0.819872i	$0.694044\pi$
$114$	28.9601	2.71237
$115$	0	0
$116$	22.9970	2.13522
$117$	−15.6597	−1.44774
$118$	−12.5786	−1.15796
$119$	5.73746	0.525952
$120$	0	0
$121$	−3.16336	−0.287579
$122$	15.3773	1.39220
$123$	−17.1220	−1.54384
$124$	−17.9835	−1.61497
$125$	0	0
$126$	5.60196	0.499062
$127$	−18.8092	−1.66905	−0.834524	−	0.550972i	$-0.814257\pi$
$127$	−18.8092	−1.66905	−0.834524	+	0.550972i	$0.814257\pi$
$128$	−17.3409	−1.53273
$129$	20.0285	1.76341
$130$	0	0
$131$	12.3299	1.07727	0.538635	−	0.842539i	$-0.318940\pi$
$131$	12.3299	1.07727	0.538635	+	0.842539i	$0.318940\pi$
$132$	20.5730	1.79065
$133$	5.46034	0.473471
$134$	−3.89413	−0.336402
$135$	0	0
$136$	14.8595	1.27419
$137$	−0.877770	−0.0749929	−0.0374965	−	0.999297i	$-0.511938\pi$
$137$	−0.877770	−0.0749929	−0.0374965	+	0.999297i	$0.511938\pi$
$138$	35.1333	2.99074
$139$	−0.549892	−0.0466412	−0.0233206	−	0.999728i	$-0.507424\pi$
$139$	−0.549892	−0.0466412	−0.0233206	+	0.999728i	$0.507424\pi$
$140$	0	0
$141$	4.29316	0.361549
$142$	11.5994	0.973397
$143$	−17.7452	−1.48393
$144$	−1.01608	−0.0846731
$145$	0	0
$146$	12.8076	1.05997
$147$	2.33889	0.192908
$148$	−3.14213	−0.258281
$149$	3.38365	0.277200	0.138600	−	0.990348i	$-0.455740\pi$
$149$	3.38365	0.277200	0.138600	+	0.990348i	$0.455740\pi$
$150$	0	0
$151$	10.2026	0.830277	0.415138	−	0.909758i	$-0.363733\pi$
$151$	10.2026	0.830277	0.415138	+	0.909758i	$0.363733\pi$
$152$	14.1418	1.14705
$153$	14.1739	1.14589
$154$	6.34799	0.511536
$155$	0	0
$156$	−46.5853	−3.72981
$157$	−22.7953	−1.81926	−0.909632	−	0.415415i	$-0.863637\pi$
$157$	−22.7953	−1.81926	−0.909632	+	0.415415i	$0.863637\pi$
$158$	5.74524	0.457067
$159$	−5.60027	−0.444131
$160$	0	0
$161$	6.62426	0.522065
$162$	−23.3754	−1.83654
$163$	6.32276	0.495237	0.247618	−	0.968858i	$-0.420352\pi$
$163$	6.32276	0.495237	0.247618	+	0.968858i	$0.420352\pi$
$164$	−23.0022	−1.79617
$165$	0	0
$166$	3.99506	0.310077
$167$	6.36601	0.492617	0.246308	−	0.969192i	$-0.420782\pi$
$167$	6.36601	0.492617	0.246308	+	0.969192i	$0.420782\pi$
$168$	6.05752	0.467348
$169$	27.1819	2.09091
$170$	0	0
$171$	13.4893	1.03155
$172$	26.9068	2.05162
$173$	12.8684	0.978370	0.489185	−	0.872180i	$-0.337294\pi$
$173$	12.8684	0.978370	0.489185	+	0.872180i	$0.337294\pi$
$174$	38.8176	2.94275
$175$	0	0
$176$	−1.15139	−0.0867894
$177$	−12.9740	−0.975182
$178$	−29.6107	−2.21942
$179$	−23.1405	−1.72960	−0.864800	−	0.502117i	$-0.832555\pi$
$179$	−23.1405	−1.72960	−0.864800	+	0.502117i	$0.832555\pi$
$180$	0	0
$181$	−6.33145	−0.470613	−0.235306	−	0.971921i	$-0.575609\pi$
$181$	−6.33145	−0.470613	−0.235306	+	0.971921i	$0.575609\pi$
$182$	−14.3743	−1.06549
$183$	15.8606	1.17245
$184$	17.1563	1.26478
$185$	0	0
$186$	−30.3551	−2.22575
$187$	16.0614	1.17453
$188$	5.76754	0.420641
$189$	−1.23866	−0.0900992
$190$	0	0
$191$	0.0775449	0.00561095	0.00280547	−	0.999996i	$-0.499107\pi$
$191$	0.0775449	0.00561095	0.00280547	+	0.999996i	$0.499107\pi$
$192$	−30.4950	−2.20079
$193$	−22.6338	−1.62922	−0.814610	−	0.580010i	$-0.803049\pi$
$193$	−22.6338	−1.62922	−0.814610	+	0.580010i	$0.803049\pi$
$194$	−29.7819	−2.13822
$195$	0	0
$196$	3.14213	0.224438
$197$	−0.320922	−0.0228647	−0.0114324	−	0.999935i	$-0.503639\pi$
$197$	−0.320922	−0.0228647	−0.0114324	+	0.999935i	$0.503639\pi$
$198$	15.6821	1.11448
$199$	−12.2904	−0.871243	−0.435622	−	0.900130i	$-0.643471\pi$
$199$	−12.2904	−0.871243	−0.435622	+	0.900130i	$0.643471\pi$
$200$	0	0
$201$	−4.01651	−0.283303
$202$	4.22151	0.297024
$203$	7.31892	0.513688
$204$	42.1651	2.95215
$205$	0	0
$206$	2.25987	0.157453
$207$	16.3646	1.13742
$208$	2.60720	0.180776
$209$	15.2857	1.05733
$210$	0	0
$211$	17.3974	1.19769	0.598844	−	0.800865i	$-0.295627\pi$
$211$	17.3974	1.19769	0.598844	+	0.800865i	$0.295627\pi$
$212$	−7.52355	−0.516720
$213$	11.9639	0.819753
$214$	−27.7297	−1.89556
$215$	0	0
$216$	−3.20802	−0.218278
$217$	−5.72336	−0.388527
$218$	40.2645	2.72705
$219$	13.2101	0.892658
$220$	0	0
$221$	−36.3693	−2.44646
$222$	−5.30373	−0.355963
$223$	24.1348	1.61618	0.808092	−	0.589056i	$-0.200500\pi$
$223$	24.1348	1.61618	0.808092	+	0.589056i	$0.200500\pi$
$224$	−6.11250	−0.408408
$225$	0	0
$226$	−27.6027	−1.83610
$227$	−0.703144	−0.0466693	−0.0233346	−	0.999728i	$-0.507428\pi$
$227$	−0.703144	−0.0466693	−0.0233346	+	0.999728i	$0.507428\pi$
$228$	40.1285	2.65757
$229$	−5.35876	−0.354117	−0.177058	−	0.984200i	$-0.556658\pi$
$229$	−5.35876	−0.354117	−0.177058	+	0.984200i	$0.556658\pi$
$230$	0	0
$231$	6.54749	0.430793
$232$	18.9554	1.24448
$233$	6.51843	0.427036	0.213518	−	0.976939i	$-0.431508\pi$
$233$	6.51843	0.427036	0.213518	+	0.976939i	$0.431508\pi$
$234$	−35.5104	−2.32138
$235$	0	0
$236$	−17.4295	−1.13457
$237$	5.92580	0.384922
$238$	13.0104	0.843339
$239$	26.8208	1.73489	0.867447	−	0.497530i	$-0.165759\pi$
$239$	26.8208	1.73489	0.867447	+	0.497530i	$0.165759\pi$
$240$	0	0
$241$	15.1452	0.975587	0.487794	−	0.872959i	$-0.337802\pi$
$241$	15.1452	0.975587	0.487794	+	0.872959i	$0.337802\pi$
$242$	−7.17333	−0.461119
$243$	−20.3940	−1.30828
$244$	21.3075	1.36407
$245$	0	0
$246$	−38.8264	−2.47548
$247$	−34.6126	−2.20235
$248$	−14.8230	−0.941261
$249$	4.12061	0.261133
$250$	0	0
$251$	−27.1608	−1.71437	−0.857186	−	0.515007i	$-0.827789\pi$
$251$	−27.1608	−1.71437	−0.857186	+	0.515007i	$0.827789\pi$
$252$	7.76233	0.488981
$253$	18.5440	1.16585
$254$	−42.6522	−2.67624
$255$	0	0
$256$	−13.2461	−0.827883
$257$	−17.0377	−1.06278	−0.531391	−	0.847126i	$-0.678331\pi$
$257$	−17.0377	−1.06278	−0.531391	+	0.847126i	$0.678331\pi$
$258$	45.4171	2.82754
$259$	−1.00000	−0.0621370
$260$	0	0
$261$	18.0807	1.11917
$262$	27.9597	1.72735
$263$	10.2405	0.631457	0.315728	−	0.948850i	$-0.397751\pi$
$263$	10.2405	0.631457	0.315728	+	0.948850i	$0.397751\pi$
$264$	16.9574	1.04366
$265$	0	0
$266$	12.3820	0.759189
$267$	−30.5413	−1.86910
$268$	−5.39588	−0.329606
$269$	−0.495295	−0.0301987	−0.0150993	−	0.999886i	$-0.504806\pi$
$269$	−0.495295	−0.0301987	−0.0150993	+	0.999886i	$0.504806\pi$
$270$	0	0
$271$	−8.96689	−0.544700	−0.272350	−	0.962198i	$-0.587801\pi$
$271$	−8.96689	−0.544700	−0.272350	+	0.962198i	$0.587801\pi$
$272$	−2.35981	−0.143085
$273$	−14.8260	−0.897312
$274$	−1.99045	−0.120248
$275$	0	0
$276$	48.6823	2.93033
$277$	5.66110	0.340143	0.170071	−	0.985432i	$-0.445600\pi$
$277$	5.66110	0.340143	0.170071	+	0.985432i	$0.445600\pi$
$278$	−1.24695	−0.0747870
$279$	−14.1390	−0.846481
$280$	0	0
$281$	5.72083	0.341276	0.170638	−	0.985334i	$-0.445417\pi$
$281$	5.72083	0.341276	0.170638	+	0.985334i	$0.445417\pi$
$282$	9.73527	0.579727
$283$	−25.0451	−1.48878	−0.744388	−	0.667748i	$-0.767258\pi$
$283$	−25.0451	−1.48878	−0.744388	+	0.667748i	$0.767258\pi$
$284$	16.0726	0.953734
$285$	0	0
$286$	−40.2394	−2.37940
$287$	−7.32058	−0.432120
$288$	−15.1004	−0.889797
$289$	15.9184	0.936378
$290$	0	0
$291$	−30.7178	−1.80071
$292$	17.7468	1.03855
$293$	−21.0069	−1.22724	−0.613619	−	0.789602i	$-0.710287\pi$
$293$	−21.0069	−1.22724	−0.613619	+	0.789602i	$0.710287\pi$
$294$	5.30373	0.309320
$295$	0	0
$296$	−2.58991	−0.150536
$297$	−3.46750	−0.201205
$298$	7.67286	0.444477
$299$	−41.9907	−2.42838
$300$	0	0
$301$	8.56324	0.493577
$302$	23.1357	1.33131
$303$	4.35418	0.250141
$304$	−2.24584	−0.128808
$305$	0	0
$306$	32.1410	1.83738
$307$	19.2470	1.09848	0.549241	−	0.835664i	$-0.314917\pi$
$307$	19.2470	1.09848	0.549241	+	0.835664i	$0.314917\pi$
$308$	8.79606	0.501202
$309$	2.33090	0.132600
$310$	0	0
$311$	−27.5686	−1.56327	−0.781635	−	0.623737i	$-0.785614\pi$
$311$	−27.5686	−1.56327	−0.781635	+	0.623737i	$0.785614\pi$
$312$	−38.3981	−2.17387
$313$	4.99170	0.282148	0.141074	−	0.989999i	$-0.454944\pi$
$313$	4.99170	0.282148	0.141074	+	0.989999i	$0.454944\pi$
$314$	−51.6912	−2.91711
$315$	0	0
$316$	7.96087	0.447834
$317$	−32.9576	−1.85108	−0.925542	−	0.378646i	$-0.876390\pi$
$317$	−32.9576	−1.85108	−0.925542	+	0.378646i	$0.876390\pi$
$318$	−12.6993	−0.712143
$319$	20.4886	1.14714
$320$	0	0
$321$	−28.6012	−1.59636
$322$	15.0213	0.837107
$323$	31.3285	1.74316
$324$	−32.3900	−1.79944
$325$	0	0
$326$	14.3376	0.794089
$327$	41.5299	2.29661
$328$	−18.9597	−1.04687
$329$	1.83555	0.101197
$330$	0	0
$331$	−31.3772	−1.72465	−0.862324	−	0.506357i	$-0.830992\pi$
$331$	−31.3772	−1.72465	−0.862324	+	0.506357i	$0.830992\pi$
$332$	5.53574	0.303813
$333$	−2.47041	−0.135377
$334$	14.4357	0.789888
$335$	0	0
$336$	−0.961985	−0.0524806
$337$	−24.9408	−1.35861	−0.679306	−	0.733856i	$-0.737719\pi$
$337$	−24.9408	−1.35861	−0.679306	+	0.733856i	$0.737719\pi$
$338$	61.6383	3.35268
$339$	−28.4702	−1.54629
$340$	0	0
$341$	−16.0220	−0.867638
$342$	30.5886	1.65404
$343$	1.00000	0.0539949
$344$	22.1780	1.19576
$345$	0	0
$346$	29.1808	1.56877
$347$	−24.3317	−1.30619	−0.653096	−	0.757275i	$-0.726530\pi$
$347$	−24.3317	−1.30619	−0.653096	+	0.757275i	$0.726530\pi$
$348$	53.7874	2.88331
$349$	17.9676	0.961783	0.480892	−	0.876780i	$-0.340313\pi$
$349$	17.9676	0.961783	0.480892	+	0.876780i	$0.340313\pi$
$350$	0	0
$351$	7.85176	0.419096
$352$	−17.1113	−0.912037
$353$	−25.6470	−1.36505	−0.682526	−	0.730861i	$-0.739119\pi$
$353$	−25.6470	−1.36505	−0.682526	+	0.730861i	$0.739119\pi$
$354$	−29.4201	−1.56366
$355$	0	0
$356$	−41.0300	−2.17459
$357$	13.4193	0.710224
$358$	−52.4739	−2.77333
$359$	12.7052	0.670556	0.335278	−	0.942119i	$-0.391170\pi$
$359$	12.7052	0.670556	0.335278	+	0.942119i	$0.391170\pi$
$360$	0	0
$361$	10.8153	0.569226
$362$	−14.3573	−0.754605
$363$	−7.39876	−0.388334
$364$	−19.9177	−1.04397
$365$	0	0
$366$	35.9659	1.87997
$367$	13.6395	0.711976	0.355988	−	0.934491i	$-0.384144\pi$
$367$	13.6395	0.711976	0.355988	+	0.934491i	$0.384144\pi$
$368$	−2.72456	−0.142027
$369$	−18.0848	−0.941458
$370$	0	0
$371$	−2.39441	−0.124312
$372$	−42.0615	−2.18078
$373$	−31.5785	−1.63507	−0.817537	−	0.575876i	$-0.804661\pi$
$373$	−31.5785	−1.63507	−0.817537	+	0.575876i	$0.804661\pi$
$374$	36.4213	1.88330
$375$	0	0
$376$	4.75392	0.245165
$377$	−46.3941	−2.38942
$378$	−2.80882	−0.144470
$379$	3.33671	0.171395	0.0856977	−	0.996321i	$-0.472688\pi$
$379$	3.33671	0.171395	0.0856977	+	0.996321i	$0.472688\pi$
$380$	0	0
$381$	−43.9927	−2.25381
$382$	0.175843	0.00899690
$383$	19.7865	1.01104	0.505521	−	0.862815i	$-0.331300\pi$
$383$	19.7865	1.01104	0.505521	+	0.862815i	$0.331300\pi$
$384$	−40.5584	−2.06974
$385$	0	0
$386$	−51.3251	−2.61238
$387$	21.1547	1.07535
$388$	−41.2671	−2.09502
$389$	37.7879	1.91592	0.957962	−	0.286894i	$-0.0926229\pi$
$389$	37.7879	1.91592	0.957962	+	0.286894i	$0.0926229\pi$
$390$	0	0
$391$	38.0064	1.92207
$392$	2.58991	0.130810
$393$	28.8383	1.45470
$394$	−0.727730	−0.0366625
$395$	0	0
$396$	21.7299	1.09197
$397$	−26.2198	−1.31594	−0.657968	−	0.753046i	$-0.728584\pi$
$397$	−26.2198	−1.31594	−0.657968	+	0.753046i	$0.728584\pi$
$398$	−27.8700	−1.39700
$399$	12.7711	0.639356
$400$	0	0
$401$	−26.1126	−1.30400	−0.652000	−	0.758219i	$-0.726070\pi$
$401$	−26.1126	−1.30400	−0.652000	+	0.758219i	$0.726070\pi$
$402$	−9.10794	−0.454263
$403$	36.2799	1.80723
$404$	5.84952	0.291024
$405$	0	0
$406$	16.5966	0.823674
$407$	−2.79940	−0.138761
$408$	34.7548	1.72062
$409$	−6.79254	−0.335869	−0.167935	−	0.985798i	$-0.553710\pi$
$409$	−6.79254	−0.335869	−0.167935	+	0.985798i	$0.553710\pi$
$410$	0	0
$411$	−2.05301	−0.101267
$412$	3.13138	0.154272
$413$	−5.54706	−0.272953
$414$	37.1088	1.82380
$415$	0	0
$416$	38.7466	1.89971
$417$	−1.28614	−0.0629824
$418$	34.6622	1.69538
$419$	−2.49664	−0.121969	−0.0609844	−	0.998139i	$-0.519424\pi$
$419$	−2.49664	−0.121969	−0.0609844	+	0.998139i	$0.519424\pi$
$420$	0	0
$421$	−30.5515	−1.48899	−0.744494	−	0.667629i	$-0.767309\pi$
$421$	−30.5515	−1.48899	−0.744494	+	0.667629i	$0.767309\pi$
$422$	39.4509	1.92044
$423$	4.53456	0.220478
$424$	−6.20132	−0.301163
$425$	0	0
$426$	27.1296	1.31444
$427$	6.78124	0.328167
$428$	−38.4236	−1.85727
$429$	−41.5040	−2.00383
$430$	0	0
$431$	−0.189462	−0.00912608	−0.00456304	−	0.999990i	$-0.501452\pi$
$431$	−0.189462	−0.00912608	−0.00456304	+	0.999990i	$0.501452\pi$
$432$	0.509460	0.0245114
$433$	−15.3070	−0.735610	−0.367805	−	0.929903i	$-0.619891\pi$
$433$	−15.3070	−0.735610	−0.367805	+	0.929903i	$0.619891\pi$
$434$	−12.9784	−0.622985
$435$	0	0
$436$	55.7923	2.67197
$437$	36.1707	1.73028
$438$	29.9556	1.43134
$439$	35.1270	1.67652	0.838261	−	0.545270i	$-0.183573\pi$
$439$	35.1270	1.67652	0.838261	+	0.545270i	$0.183573\pi$
$440$	0	0
$441$	2.47041	0.117638
$442$	−82.4719	−3.92279
$443$	−40.2651	−1.91305	−0.956526	−	0.291647i	$-0.905797\pi$
$443$	−40.2651	−1.91305	−0.956526	+	0.291647i	$0.905797\pi$
$444$	−7.34909	−0.348772
$445$	0	0
$446$	54.7286	2.59148
$447$	7.91399	0.374319
$448$	−13.0383	−0.616000
$449$	−9.30052	−0.438919	−0.219459	−	0.975622i	$-0.570429\pi$
$449$	−9.30052	−0.438919	−0.219459	+	0.975622i	$0.570429\pi$
$450$	0	0
$451$	−20.4932	−0.964988
$452$	−38.2475	−1.79901
$453$	23.8628	1.12117
$454$	−1.59447	−0.0748320
$455$	0	0
$456$	33.0761	1.54893
$457$	33.6074	1.57209	0.786045	−	0.618169i	$-0.212125\pi$
$457$	33.6074	1.57209	0.786045	+	0.618169i	$0.212125\pi$
$458$	−12.1516	−0.567809
$459$	−7.10676	−0.331715
$460$	0	0
$461$	15.2698	0.711187	0.355594	−	0.934641i	$-0.384279\pi$
$461$	15.2698	0.711187	0.355594	+	0.934641i	$0.384279\pi$
$462$	14.8472	0.690757
$463$	19.5853	0.910206	0.455103	−	0.890439i	$-0.349602\pi$
$463$	19.5853	0.910206	0.455103	+	0.890439i	$0.349602\pi$
$464$	−3.01027	−0.139748
$465$	0	0
$466$	14.7814	0.684733
$467$	−16.3744	−0.757717	−0.378859	−	0.925455i	$-0.623683\pi$
$467$	−16.3744	−0.757717	−0.378859	+	0.925455i	$0.623683\pi$
$468$	−49.2048	−2.27449
$469$	−1.71727	−0.0792963
$470$	0	0
$471$	−53.3157	−2.45666
$472$	−14.3664	−0.661267
$473$	23.9719	1.10223
$474$	13.4375	0.617205
$475$	0	0
$476$	18.0278	0.826303
$477$	−5.91518	−0.270837
$478$	60.8195	2.78182
$479$	0.805367	0.0367982	0.0183991	−	0.999831i	$-0.494143\pi$
$479$	0.805367	0.0367982	0.0183991	+	0.999831i	$0.494143\pi$
$480$	0	0
$481$	6.33892	0.289030
$482$	34.3436	1.56431
$483$	15.4934	0.704975
$484$	−9.93969	−0.451804
$485$	0	0
$486$	−46.2460	−2.09776
$487$	7.55283	0.342252	0.171126	−	0.985249i	$-0.445260\pi$
$487$	7.55283	0.342252	0.171126	+	0.985249i	$0.445260\pi$
$488$	17.5628	0.795032
$489$	14.7882	0.668747
$490$	0	0
$491$	30.6550	1.38344	0.691720	−	0.722166i	$-0.256853\pi$
$491$	30.6550	1.38344	0.691720	+	0.722166i	$0.256853\pi$
$492$	−53.7996	−2.42547
$493$	41.9920	1.89123
$494$	−78.4885	−3.53137
$495$	0	0
$496$	2.35402	0.105698
$497$	5.11520	0.229448
$498$	9.34400	0.418715
$499$	24.4146	1.09295	0.546475	−	0.837475i	$-0.315969\pi$
$499$	24.4146	1.09295	0.546475	+	0.837475i	$0.315969\pi$
$500$	0	0
$501$	14.8894	0.665209
$502$	−61.5904	−2.74892
$503$	1.19590	0.0533227	0.0266613	−	0.999645i	$-0.491512\pi$
$503$	1.19590	0.0533227	0.0266613	+	0.999645i	$0.491512\pi$
$504$	6.39814	0.284996
$505$	0	0
$506$	42.0507	1.86938
$507$	63.5755	2.82349
$508$	−59.1009	−2.62218
$509$	10.6289	0.471117	0.235558	−	0.971860i	$-0.424308\pi$
$509$	10.6289	0.471117	0.235558	+	0.971860i	$0.424308\pi$
$510$	0	0
$511$	5.64803	0.249854
$512$	4.64453	0.205261
$513$	−6.76350	−0.298616
$514$	−38.6351	−1.70412
$515$	0	0
$516$	62.9320	2.77043
$517$	5.13844	0.225989
$518$	−2.26763	−0.0996337
$519$	30.0979	1.32115
$520$	0	0
$521$	28.5436	1.25052	0.625258	−	0.780418i	$-0.284994\pi$
$521$	28.5436	1.25052	0.625258	+	0.780418i	$0.284994\pi$
$522$	41.0003	1.79453
$523$	16.5675	0.724447	0.362224	−	0.932091i	$-0.382018\pi$
$523$	16.5675	0.724447	0.362224	+	0.932091i	$0.382018\pi$
$524$	38.7422	1.69246
$525$	0	0
$526$	23.2216	1.01251
$527$	−32.8375	−1.43043
$528$	−2.69298	−0.117197
$529$	20.8808	0.907862
$530$	0	0
$531$	−13.7035	−0.594681
$532$	17.1571	0.743853
$533$	46.4046	2.01000
$534$	−69.2563	−2.99701
$535$	0	0
$536$	−4.44758	−0.192106
$537$	−54.1230	−2.33558
$538$	−1.12314	−0.0484222
$539$	2.79940	0.120579
$540$	0	0
$541$	14.8297	0.637580	0.318790	−	0.947825i	$-0.396723\pi$
$541$	14.8297	0.637580	0.318790	+	0.947825i	$0.396723\pi$
$542$	−20.3336	−0.873401
$543$	−14.8086	−0.635496
$544$	−35.0702	−1.50362
$545$	0	0
$546$	−33.6199	−1.43880
$547$	28.3226	1.21099	0.605493	−	0.795851i	$-0.292976\pi$
$547$	28.3226	1.21099	0.605493	+	0.795851i	$0.292976\pi$
$548$	−2.75806	−0.117819
$549$	16.7524	0.714976
$550$	0	0
$551$	39.9638	1.70252
$552$	40.1266	1.70790
$553$	2.53359	0.107739
$554$	12.8373	0.545403
$555$	0	0
$556$	−1.72783	−0.0732763
$557$	−21.2463	−0.900236	−0.450118	−	0.892969i	$-0.648618\pi$
$557$	−21.2463	−0.900236	−0.450118	+	0.892969i	$0.648618\pi$
$558$	−32.0620	−1.35729
$559$	−54.2817	−2.29587
$560$	0	0
$561$	37.5659	1.58603
$562$	12.9727	0.547221
$563$	−19.3839	−0.816932	−0.408466	−	0.912774i	$-0.633936\pi$
$563$	−19.3839	−0.816932	−0.408466	+	0.912774i	$0.633936\pi$
$564$	13.4896	0.568016
$565$	0	0
$566$	−56.7928	−2.38718
$567$	−10.3083	−0.432908
$568$	13.2479	0.555871
$569$	0.848892	0.0355874	0.0177937	−	0.999842i	$-0.494336\pi$
$569$	0.848892	0.0355874	0.0177937	+	0.999842i	$0.494336\pi$
$570$	0	0
$571$	27.8653	1.16613	0.583063	−	0.812427i	$-0.301854\pi$
$571$	27.8653	1.16613	0.583063	+	0.812427i	$0.301854\pi$
$572$	−55.7575	−2.33134
$573$	0.181369	0.00757679
$574$	−16.6003	−0.692884
$575$	0	0
$576$	−32.2098	−1.34208
$577$	−6.82820	−0.284262	−0.142131	−	0.989848i	$-0.545395\pi$
$577$	−6.82820	−0.284262	−0.142131	+	0.989848i	$0.545395\pi$
$578$	36.0970	1.50144
$579$	−52.9381	−2.20003
$580$	0	0
$581$	1.76178	0.0730910
$582$	−69.6566	−2.88736
$583$	−6.70292	−0.277607
$584$	14.6279	0.605307
$585$	0	0
$586$	−47.6359	−1.96782
$587$	−19.8166	−0.817918	−0.408959	−	0.912553i	$-0.634108\pi$
$587$	−19.8166	−0.817918	−0.408959	+	0.912553i	$0.634108\pi$
$588$	7.34909	0.303071
$589$	−31.2515	−1.28769
$590$	0	0
$591$	−0.750600	−0.0308756
$592$	0.411300	0.0169043
$593$	−43.2139	−1.77458	−0.887290	−	0.461212i	$-0.847415\pi$
$593$	−43.2139	−1.77458	−0.887290	+	0.461212i	$0.847415\pi$
$594$	−7.86300	−0.322623
$595$	0	0
$596$	10.6319	0.435498
$597$	−28.7459	−1.17649
$598$	−95.2191	−3.89380
$599$	22.1144	0.903570	0.451785	−	0.892127i	$-0.350787\pi$
$599$	22.1144	0.903570	0.451785	+	0.892127i	$0.350787\pi$
$600$	0	0
$601$	0.0269834	0.00110068	0.000550339	−	1.00000i	$-0.499825\pi$
$601$	0.0269834	0.00110068	0.000550339		1.00000i	$0.499825\pi$
$602$	19.4182	0.791428
$603$	−4.24236	−0.172762
$604$	32.0579	1.30442
$605$	0	0
$606$	9.87365	0.401090
$607$	31.2344	1.26777	0.633883	−	0.773429i	$-0.281460\pi$
$607$	31.2344	1.26777	0.633883	+	0.773429i	$0.281460\pi$
$608$	−33.3763	−1.35359
$609$	17.1182	0.693663
$610$	0	0
$611$	−11.6354	−0.470719
$612$	44.5360	1.80026
$613$	33.1657	1.33955	0.669774	−	0.742565i	$-0.266391\pi$
$613$	33.1657	1.33955	0.669774	+	0.742565i	$0.266391\pi$
$614$	43.6449	1.76136
$615$	0	0
$616$	7.25020	0.292119
$617$	16.9802	0.683599	0.341799	−	0.939773i	$-0.388964\pi$
$617$	16.9802	0.683599	0.341799	+	0.939773i	$0.388964\pi$
$618$	5.28560	0.212618
$619$	35.6539	1.43305	0.716525	−	0.697561i	$-0.245731\pi$
$619$	35.6539	1.43305	0.716525	+	0.697561i	$0.245731\pi$
$620$	0	0
$621$	−8.20520	−0.329264
$622$	−62.5151	−2.50663
$623$	−13.0580	−0.523159
$624$	6.09794	0.244113
$625$	0	0
$626$	11.3193	0.452411
$627$	35.7515	1.42778
$628$	−71.6257	−2.85818
$629$	−5.73746	−0.228767
$630$	0	0
$631$	−0.441024	−0.0175569	−0.00877844	−	0.999961i	$-0.502794\pi$
$631$	−0.441024	−0.0175569	−0.00877844	+	0.999961i	$0.502794\pi$
$632$	6.56179	0.261014
$633$	40.6907	1.61731
$634$	−74.7355	−2.96813
$635$	0	0
$636$	−17.5968	−0.697757
$637$	−6.33892	−0.251157
$638$	46.4604	1.83939
$639$	12.6366	0.499897
$640$	0	0
$641$	−13.2345	−0.522733	−0.261367	−	0.965240i	$-0.584173\pi$
$641$	−13.2345	−0.522733	−0.261367	+	0.965240i	$0.584173\pi$
$642$	−64.8567	−2.55969
$643$	−17.1263	−0.675396	−0.337698	−	0.941254i	$-0.609648\pi$
$643$	−17.1263	−0.675396	−0.337698	+	0.941254i	$0.609648\pi$
$644$	20.8143	0.820197
$645$	0	0
$646$	71.0412	2.79508
$647$	1.53143	0.0602067	0.0301033	−	0.999547i	$-0.490416\pi$
$647$	1.53143	0.0602067	0.0301033	+	0.999547i	$0.490416\pi$
$648$	−26.6976	−1.04878
$649$	−15.5284	−0.609544
$650$	0	0
$651$	−13.3863	−0.524651
$652$	19.8669	0.778048
$653$	−17.3691	−0.679704	−0.339852	−	0.940479i	$-0.610377\pi$
$653$	−17.3691	−0.679704	−0.339852	+	0.940479i	$0.610377\pi$
$654$	94.1742	3.68250
$655$	0	0
$656$	3.01095	0.117558
$657$	13.9529	0.544356
$658$	4.16235	0.162265
$659$	40.9466	1.59505	0.797526	−	0.603284i	$-0.206142\pi$
$659$	40.9466	1.59505	0.797526	+	0.603284i	$0.206142\pi$
$660$	0	0
$661$	−17.6456	−0.686336	−0.343168	−	0.939274i	$-0.611500\pi$
$661$	−17.6456	−0.686336	−0.343168	+	0.939274i	$0.611500\pi$
$662$	−71.1518	−2.76539
$663$	−85.0638	−3.30360
$664$	4.56286	0.177073
$665$	0	0
$666$	−5.60196	−0.217071
$667$	48.4825	1.87725
$668$	20.0028	0.773932
$669$	56.4486	2.18243
$670$	0	0
$671$	18.9834	0.732846
$672$	−14.2965	−0.551498
$673$	1.69907	0.0654945	0.0327472	−	0.999464i	$-0.489574\pi$
$673$	1.69907	0.0654945	0.0327472	+	0.999464i	$0.489574\pi$
$674$	−56.5564	−2.17847
$675$	0	0
$676$	85.4089	3.28496
$677$	−47.3233	−1.81878	−0.909390	−	0.415944i	$-0.863451\pi$
$677$	−47.3233	−1.81878	−0.909390	+	0.415944i	$0.863451\pi$
$678$	−64.5597	−2.47940
$679$	−13.1335	−0.504018
$680$	0	0
$681$	−1.64458	−0.0630203
$682$	−36.3318	−1.39122
$683$	11.1193	0.425468	0.212734	−	0.977110i	$-0.431763\pi$
$683$	11.1193	0.425468	0.212734	+	0.977110i	$0.431763\pi$
$684$	42.3849	1.62063
$685$	0	0
$686$	2.26763	0.0865783
$687$	−12.5335	−0.478184
$688$	−3.52206	−0.134277
$689$	15.1780	0.578236
$690$	0	0
$691$	6.09962	0.232041	0.116020	−	0.993247i	$-0.462986\pi$
$691$	6.09962	0.232041	0.116020	+	0.993247i	$0.462986\pi$
$692$	40.4343	1.53708
$693$	6.91566	0.262704
$694$	−55.1751	−2.09442
$695$	0	0
$696$	44.3345	1.68050
$697$	−42.0015	−1.59092
$698$	40.7438	1.54217
$699$	15.2459	0.576652
$700$	0	0
$701$	−34.3205	−1.29627	−0.648134	−	0.761526i	$-0.724451\pi$
$701$	−34.3205	−1.29627	−0.648134	+	0.761526i	$0.724451\pi$
$702$	17.8049	0.672001
$703$	−5.46034	−0.205941
$704$	−36.4993	−1.37562
$705$	0	0
$706$	−58.1578	−2.18880
$707$	1.86164	0.0700143
$708$	−40.7658	−1.53207
$709$	21.9943	0.826013	0.413006	−	0.910728i	$-0.364479\pi$
$709$	21.9943	0.826013	0.413006	+	0.910728i	$0.364479\pi$
$710$	0	0
$711$	6.25901	0.234731
$712$	−33.8192	−1.26743
$713$	−37.9130	−1.41985
$714$	30.4299	1.13881
$715$	0	0
$716$	−72.7102	−2.71731
$717$	62.7309	2.34273
$718$	28.8107	1.07520
$719$	33.1549	1.23647	0.618234	−	0.785994i	$-0.287848\pi$
$719$	33.1549	1.23647	0.618234	+	0.785994i	$0.287848\pi$
$720$	0	0
$721$	0.996582	0.0371146
$722$	24.5250	0.912728
$723$	35.4229	1.31739
$724$	−19.8942	−0.739362
$725$	0	0
$726$	−16.7776	−0.622676
$727$	18.2312	0.676158	0.338079	−	0.941118i	$-0.390223\pi$
$727$	18.2312	0.676158	0.338079	+	0.941118i	$0.390223\pi$
$728$	−16.4172	−0.608463
$729$	−16.7745	−0.621276
$730$	0	0
$731$	49.1312	1.81718
$732$	49.8359	1.84199
$733$	−6.20286	−0.229108	−0.114554	−	0.993417i	$-0.536544\pi$
$733$	−6.20286	−0.229108	−0.114554	+	0.993417i	$0.536544\pi$
$734$	30.9293	1.14162
$735$	0	0
$736$	−40.4908	−1.49251
$737$	−4.80733	−0.177080
$738$	−41.0096	−1.50958
$739$	−30.7511	−1.13120	−0.565598	−	0.824681i	$-0.691355\pi$
$739$	−30.7511	−1.13120	−0.565598	+	0.824681i	$0.691355\pi$
$740$	0	0
$741$	−80.9552	−2.97396
$742$	−5.42964	−0.199328
$743$	26.2672	0.963650	0.481825	−	0.876268i	$-0.339974\pi$
$743$	26.2672	0.963650	0.481825	+	0.876268i	$0.339974\pi$
$744$	−34.6694	−1.27104
$745$	0	0
$746$	−71.6083	−2.62177
$747$	4.35232	0.159243
$748$	50.4670	1.84526
$749$	−12.2285	−0.446821
$750$	0	0
$751$	28.6757	1.04639	0.523195	−	0.852213i	$-0.324740\pi$
$751$	28.6757	1.04639	0.523195	+	0.852213i	$0.324740\pi$
$752$	−0.754962	−0.0275306
$753$	−63.5260	−2.31502
$754$	−105.204	−3.83132
$755$	0	0
$756$	−3.89202	−0.141552
$757$	30.1764	1.09678	0.548389	−	0.836223i	$-0.315241\pi$
$757$	30.1764	1.09678	0.548389	+	0.836223i	$0.315241\pi$
$758$	7.56642	0.274825
$759$	43.3723	1.57431
$760$	0	0
$761$	−43.0936	−1.56214	−0.781071	−	0.624442i	$-0.785326\pi$
$761$	−43.0936	−1.56214	−0.781071	+	0.624442i	$0.785326\pi$
$762$	−99.7589	−3.61388
$763$	17.7562	0.642819
$764$	0.243656	0.00881515
$765$	0	0
$766$	44.8683	1.62116
$767$	35.1623	1.26964
$768$	−30.9812	−1.11794
$769$	−16.8603	−0.607997	−0.303999	−	0.952672i	$-0.598322\pi$
$769$	−16.8603	−0.607997	−0.303999	+	0.952672i	$0.598322\pi$
$770$	0	0
$771$	−39.8493	−1.43514
$772$	−71.1184	−2.55961
$773$	−34.4543	−1.23923	−0.619617	−	0.784904i	$-0.712712\pi$
$773$	−34.4543	−1.23923	−0.619617	+	0.784904i	$0.712712\pi$
$774$	47.9709	1.72428
$775$	0	0
$776$	−34.0146	−1.22105
$777$	−2.33889	−0.0839072
$778$	85.6889	3.07210
$779$	−39.9728	−1.43218
$780$	0	0
$781$	14.3195	0.512392
$782$	86.1843	3.08195
$783$	−9.06565	−0.323980
$784$	−0.411300	−0.0146893
$785$	0	0
$786$	65.3946	2.33255
$787$	−3.43686	−0.122511	−0.0612554	−	0.998122i	$-0.519510\pi$
$787$	−3.43686	−0.122511	−0.0612554	+	0.998122i	$0.519510\pi$
$788$	−1.00838	−0.0359219
$789$	23.9514	0.852693
$790$	0	0
$791$	−12.1725	−0.432805
$792$	17.9109	0.636438
$793$	−42.9858	−1.52647
$794$	−59.4567	−2.11004
$795$	0	0
$796$	−38.6180	−1.36878
$797$	37.3130	1.32169	0.660847	−	0.750520i	$-0.270197\pi$
$797$	37.3130	1.32169	0.660847	+	0.750520i	$0.270197\pi$
$798$	28.9601	1.02518
$799$	10.5314	0.372574
$800$	0	0
$801$	−32.2587	−1.13980
$802$	−59.2135	−2.09090
$803$	15.8111	0.557962
$804$	−12.6204	−0.445087
$805$	0	0
$806$	82.2692	2.89781
$807$	−1.15844	−0.0407791
$808$	4.82149	0.169620
$809$	12.4049	0.436132	0.218066	−	0.975934i	$-0.430025\pi$
$809$	12.4049	0.436132	0.218066	+	0.975934i	$0.430025\pi$
$810$	0	0
$811$	−44.1420	−1.55004	−0.775018	−	0.631939i	$-0.782259\pi$
$811$	−44.1420	−1.55004	−0.775018	+	0.631939i	$0.782259\pi$
$812$	22.9970	0.807036
$813$	−20.9726	−0.735540
$814$	−6.34799	−0.222497
$815$	0	0
$816$	−5.51935	−0.193216
$817$	46.7582	1.63586
$818$	−15.4029	−0.538551
$819$	−15.6597	−0.547195
$820$	0	0
$821$	−6.07487	−0.212014	−0.106007	−	0.994365i	$-0.533807\pi$
$821$	−6.07487	−0.212014	−0.106007	+	0.994365i	$0.533807\pi$
$822$	−4.65545	−0.162377
$823$	35.5628	1.23964	0.619821	−	0.784743i	$-0.287205\pi$
$823$	35.5628	1.23964	0.619821	+	0.784743i	$0.287205\pi$
$824$	2.58106	0.0899155
$825$	0	0
$826$	−12.5786	−0.437667
$827$	−12.7340	−0.442805	−0.221403	−	0.975183i	$-0.571063\pi$
$827$	−12.7340	−0.442805	−0.221403	+	0.975183i	$0.571063\pi$
$828$	51.4197	1.78696
$829$	−18.1704	−0.631084	−0.315542	−	0.948912i	$-0.602186\pi$
$829$	−18.1704	−0.631084	−0.315542	+	0.948912i	$0.602186\pi$
$830$	0	0
$831$	13.2407	0.459314
$832$	82.6484	2.86532
$833$	5.73746	0.198791
$834$	−2.91648	−0.100989
$835$	0	0
$836$	48.0295	1.66113
$837$	7.08929	0.245042
$838$	−5.66144	−0.195571
$839$	25.2744	0.872570	0.436285	−	0.899809i	$-0.356294\pi$
$839$	25.2744	0.872570	0.436285	+	0.899809i	$0.356294\pi$
$840$	0	0
$841$	24.5666	0.847125
$842$	−69.2793	−2.38752
$843$	13.3804	0.460845
$844$	54.6649	1.88164
$845$	0	0
$846$	10.2827	0.353526
$847$	−3.16336	−0.108694
$848$	0.984822	0.0338189
$849$	−58.5777	−2.01038
$850$	0	0
$851$	−6.62426	−0.227077
$852$	37.5921	1.28788
$853$	−9.01117	−0.308537	−0.154268	−	0.988029i	$-0.549302\pi$
$853$	−9.01117	−0.308537	−0.154268	+	0.988029i	$0.549302\pi$
$854$	15.3773	0.526201
$855$	0	0
$856$	−31.6708	−1.08249
$857$	5.12017	0.174902	0.0874509	−	0.996169i	$-0.472128\pi$
$857$	5.12017	0.174902	0.0874509	+	0.996169i	$0.472128\pi$
$858$	−94.1155	−3.21305
$859$	23.7331	0.809765	0.404882	−	0.914369i	$-0.367313\pi$
$859$	23.7331	0.809765	0.404882	+	0.914369i	$0.367313\pi$
$860$	0	0
$861$	−17.1220	−0.583517
$862$	−0.429630	−0.0146332
$863$	30.7629	1.04718	0.523591	−	0.851970i	$-0.324592\pi$
$863$	30.7629	1.04718	0.523591	+	0.851970i	$0.324592\pi$
$864$	7.57130	0.257581
$865$	0	0
$866$	−34.7106	−1.17952
$867$	37.2314	1.26445
$868$	−17.9835	−0.610400
$869$	7.09254	0.240598
$870$	0	0
$871$	10.8857	0.368846
$872$	45.9871	1.55732
$873$	−32.4451	−1.09810
$874$	82.0216	2.77442
$875$	0	0
$876$	41.5079	1.40242
$877$	−10.2878	−0.347393	−0.173696	−	0.984799i	$-0.555571\pi$
$877$	−10.2878	−0.347393	−0.173696	+	0.984799i	$0.555571\pi$
$878$	79.6549	2.68822
$879$	−49.1329	−1.65721
$880$	0	0
$881$	−10.5360	−0.354967	−0.177483	−	0.984124i	$-0.556796\pi$
$881$	−10.5360	−0.354967	−0.177483	+	0.984124i	$0.556796\pi$
$882$	5.60196	0.188628
$883$	19.3672	0.651759	0.325879	−	0.945411i	$-0.394340\pi$
$883$	19.3672	0.651759	0.325879	+	0.945411i	$0.394340\pi$
$884$	−114.277	−3.84355
$885$	0	0
$886$	−91.3061	−3.06749
$887$	−8.62651	−0.289650	−0.144825	−	0.989457i	$-0.546262\pi$
$887$	−8.62651	−0.289650	−0.144825	+	0.989457i	$0.546262\pi$
$888$	−6.05752	−0.203277
$889$	−18.8092	−0.630841
$890$	0	0
$891$	−28.8571	−0.966748
$892$	75.8345	2.53913
$893$	10.0227	0.335398
$894$	17.9460	0.600203
$895$	0	0
$896$	−17.3409	−0.579318
$897$	−98.2115	−3.27919
$898$	−21.0901	−0.703786
$899$	−41.8888	−1.39707
$900$	0	0
$901$	−13.7379	−0.457674
$902$	−46.4710	−1.54731
$903$	20.0285	0.666506
$904$	−31.5257	−1.04853
$905$	0	0
$906$	54.1119	1.79775
$907$	−14.2661	−0.473698	−0.236849	−	0.971546i	$-0.576115\pi$
$907$	−14.2661	−0.473698	−0.236849	+	0.971546i	$0.576115\pi$
$908$	−2.20937	−0.0733204
$909$	4.59902	0.152540
$910$	0	0
$911$	13.4933	0.447054	0.223527	−	0.974698i	$-0.428243\pi$
$911$	13.4933	0.447054	0.223527	+	0.974698i	$0.428243\pi$
$912$	−5.25276	−0.173936
$913$	4.93193	0.163223
$914$	76.2091	2.52077
$915$	0	0
$916$	−16.8379	−0.556339
$917$	12.3299	0.407170
$918$	−16.1155	−0.531890
$919$	4.46309	0.147224	0.0736118	−	0.997287i	$-0.476547\pi$
$919$	4.46309	0.147224	0.0736118	+	0.997287i	$0.476547\pi$
$920$	0	0
$921$	45.0165	1.48335
$922$	34.6263	1.14036
$923$	−32.4249	−1.06728
$924$	20.5730	0.676803
$925$	0	0
$926$	44.4121	1.45947
$927$	2.46196	0.0808615
$928$	−44.7369	−1.46856
$929$	38.9438	1.27771	0.638853	−	0.769329i	$-0.279409\pi$
$929$	38.9438	1.27771	0.638853	+	0.769329i	$0.279409\pi$
$930$	0	0
$931$	5.46034	0.178955
$932$	20.4817	0.670901
$933$	−64.4798	−2.11097
$934$	−37.1310	−1.21496
$935$	0	0
$936$	−40.5573	−1.32566
$937$	50.8106	1.65991	0.829955	−	0.557830i	$-0.188366\pi$
$937$	50.8106	1.65991	0.829955	+	0.557830i	$0.188366\pi$
$938$	−3.89413	−0.127148
$939$	11.6750	0.381001
$940$	0	0
$941$	48.0236	1.56553	0.782763	−	0.622320i	$-0.213810\pi$
$941$	48.0236	1.56553	0.782763	+	0.622320i	$0.213810\pi$
$942$	−120.900	−3.93914
$943$	−48.4934	−1.57916
$944$	2.28150	0.0742566
$945$	0	0
$946$	54.3594	1.76738
$947$	10.6956	0.347562	0.173781	−	0.984784i	$-0.444402\pi$
$947$	10.6956	0.347562	0.173781	+	0.984784i	$0.444402\pi$
$948$	18.6196	0.604737
$949$	−35.8024	−1.16220
$950$	0	0
$951$	−77.0842	−2.49963
$952$	14.8595	0.481600
$953$	−5.38151	−0.174324	−0.0871621	−	0.996194i	$-0.527780\pi$
$953$	−5.38151	−0.174324	−0.0871621	+	0.996194i	$0.527780\pi$
$954$	−13.4134	−0.434275
$955$	0	0
$956$	84.2743	2.72563
$957$	47.9206	1.54905
$958$	1.82627	0.0590042
$959$	−0.877770	−0.0283447
$960$	0	0
$961$	1.75684	0.0566723
$962$	14.3743	0.463446
$963$	−30.2094	−0.973485
$964$	47.5881	1.53271
$965$	0	0
$966$	35.1333	1.13039
$967$	−16.5761	−0.533050	−0.266525	−	0.963828i	$-0.585876\pi$
$967$	−16.5761	−0.533050	−0.266525	+	0.963828i	$0.585876\pi$
$968$	−8.19284	−0.263328
$969$	73.2738	2.35389
$970$	0	0
$971$	43.3305	1.39054	0.695270	−	0.718748i	$-0.255285\pi$
$971$	43.3305	1.39054	0.695270	+	0.718748i	$0.255285\pi$
$972$	−64.0806	−2.05539
$973$	−0.549892	−0.0176287
$974$	17.1270	0.548784
$975$	0	0
$976$	−2.78912	−0.0892777
$977$	−0.629398	−0.0201362	−0.0100681	−	0.999949i	$-0.503205\pi$
$977$	−0.629398	−0.0201362	−0.0100681	+	0.999949i	$0.503205\pi$
$978$	33.5342	1.07230
$979$	−36.5547	−1.16829
$980$	0	0
$981$	43.8651	1.40051
$982$	69.5140	2.21828
$983$	59.1378	1.88620	0.943101	−	0.332507i	$-0.107895\pi$
$983$	59.1378	1.88620	0.943101	+	0.332507i	$0.107895\pi$
$984$	−44.3446	−1.41365
$985$	0	0
$986$	95.2222	3.03249
$987$	4.29316	0.136653
$988$	−108.757	−3.46003
$989$	56.7251	1.80375
$990$	0	0
$991$	26.7223	0.848861	0.424431	−	0.905460i	$-0.360474\pi$
$991$	26.7223	0.848861	0.424431	+	0.905460i	$0.360474\pi$
$992$	34.9840	1.11074
$993$	−73.3878	−2.32889
$994$	11.5994	0.367909
$995$	0	0
$996$	12.9475	0.410257
$997$	33.5537	1.06266	0.531329	−	0.847166i	$-0.321693\pi$
$997$	33.5537	1.06266	0.531329	+	0.847166i	$0.321693\pi$
$998$	55.3633	1.75249
$999$	1.23866	0.0391895

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6475.2.a.u.1.13		15
5.4	even	2		1295.2.a.j.1.3	✓	15
35.34	odd	2		9065.2.a.o.1.3		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1295.2.a.j.1.3	✓	15	5.4	even	2
6475.2.a.u.1.13		15	1.1	even	1	trivial
9065.2.a.o.1.3		15	35.34	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6475 = 5^{2} \cdot 7 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6475.a (trivial)

\(f(q)\)	\(=\)	\(q+2.26763 q^{2} +2.33889 q^{3} +3.14213 q^{4} +5.30373 q^{6} +1.00000 q^{7} +2.58991 q^{8} +2.47041 q^{9} +2.79940 q^{11} +7.34909 q^{12} -6.33892 q^{13} +2.26763 q^{14} -0.411300 q^{16} +5.73746 q^{17} +5.60196 q^{18} +5.46034 q^{19} +2.33889 q^{21} +6.34799 q^{22} +6.62426 q^{23} +6.05752 q^{24} -14.3743 q^{26} -1.23866 q^{27} +3.14213 q^{28} +7.31892 q^{29} -5.72336 q^{31} -6.11250 q^{32} +6.54749 q^{33} +13.0104 q^{34} +7.76233 q^{36} -1.00000 q^{37} +12.3820 q^{38} -14.8260 q^{39} -7.32058 q^{41} +5.30373 q^{42} +8.56324 q^{43} +8.79606 q^{44} +15.0213 q^{46} +1.83555 q^{47} -0.961985 q^{48} +1.00000 q^{49} +13.4193 q^{51} -19.9177 q^{52} -2.39441 q^{53} -2.80882 q^{54} +2.58991 q^{56} +12.7711 q^{57} +16.5966 q^{58} -5.54706 q^{59} +6.78124 q^{61} -12.9784 q^{62} +2.47041 q^{63} -13.0383 q^{64} +14.8472 q^{66} -1.71727 q^{67} +18.0278 q^{68} +15.4934 q^{69} +5.11520 q^{71} +6.39814 q^{72} +5.64803 q^{73} -2.26763 q^{74} +17.1571 q^{76} +2.79940 q^{77} -33.6199 q^{78} +2.53359 q^{79} -10.3083 q^{81} -16.6003 q^{82} +1.76178 q^{83} +7.34909 q^{84} +19.4182 q^{86} +17.1182 q^{87} +7.25020 q^{88} -13.0580 q^{89} -6.33892 q^{91} +20.8143 q^{92} -13.3863 q^{93} +4.16235 q^{94} -14.2965 q^{96} -13.1335 q^{97} +2.26763 q^{98} +6.91566 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + q^{2} + q^{3} + 23 q^{4} + 6 q^{6} + 15 q^{7} + 3 q^{8} + 30 q^{9} + 17 q^{11} - 8 q^{12} + 5 q^{13} + q^{14} + 39 q^{16} + 7 q^{17} - 12 q^{18} + 6 q^{19} + q^{21} - 18 q^{22} + 2 q^{23} + 4 q^{24}+ \cdots + 90 q^{99}+O(q^{100}) \) 15 * q + q^2 + q^3 + 23 * q^4 + 6 * q^6 + 15 * q^7 + 3 * q^8 + 30 * q^9 + 17 * q^11 - 8 * q^12 + 5 * q^13 + q^14 + 39 * q^16 + 7 * q^17 - 12 * q^18 + 6 * q^19 + q^21 - 18 * q^22 + 2 * q^23 + 4 * q^24 + 4 * q^26 - 5 * q^27 + 23 * q^28 + 13 * q^29 - 4 * q^31 + 7 * q^32 - 7 * q^33 + 38 * q^34 + 49 * q^36 - 15 * q^37 + 26 * q^38 + 33 * q^39 + 22 * q^41 + 6 * q^42 - 10 * q^43 + 58 * q^44 + 32 * q^46 + 9 * q^47 - 43 * q^48 + 15 * q^49 + 39 * q^51 - 6 * q^52 + 16 * q^53 + 30 * q^54 + 3 * q^56 - 16 * q^57 - 44 * q^58 + 6 * q^59 + 11 * q^62 + 30 * q^63 + 119 * q^64 + 3 * q^66 - 4 * q^67 + 70 * q^68 + 14 * q^69 + 48 * q^71 + 25 * q^72 - 38 * q^73 - q^74 - 3 * q^76 + 17 * q^77 + 40 * q^78 + 7 * q^79 + 111 * q^81 - 6 * q^82 + 6 * q^83 - 8 * q^84 - 40 * q^86 + 47 * q^87 - 23 * q^88 - 2 * q^89 + 5 * q^91 - 7 * q^92 - 26 * q^94 - 29 * q^96 - 79 * q^97 + q^98 + 90 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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