LMFDB - Embedded newform 700.6.e.f.449.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [700,6,Mod(449,700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(700, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("700.449");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 700 = 2^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	700.e (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$112.268673869$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{1009})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 505x^{2} + 63504 $ x^4 + 505*x^2 + 63504
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.4
Root			$16.3824i$ of defining polynomial
Character	$\chi$	$=$	700.449
Dual form			700.6.e.f.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+24.3824i q^{3} -49.0000i q^{7} -351.500 q^{9} +O(q^{10})$	$q+24.3824i q^{3} -49.0000i q^{7} -351.500 q^{9} +59.4414 q^{11} -873.678i q^{13} +414.795i q^{17} +1397.59 q^{19} +1194.74 q^{21} -839.708i q^{23} -2645.50i q^{27} +7593.04 q^{29} -7018.83 q^{31} +1449.32i q^{33} +2763.65i q^{37} +21302.3 q^{39} -14373.8 q^{41} +22368.9i q^{43} +12046.6i q^{47} -2401.00 q^{49} -10113.7 q^{51} +14422.9i q^{53} +34076.6i q^{57} -16743.3 q^{59} +33828.2 q^{61} +17223.5i q^{63} +12135.2i q^{67} +20474.1 q^{69} -43847.6 q^{71} -33432.7i q^{73} -2912.63i q^{77} +36877.6 q^{79} -20911.0 q^{81} +114590. i q^{83} +185136. i q^{87} +58796.4 q^{89} -42810.2 q^{91} -171136. i q^{93} +34474.9i q^{97} -20893.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 326 q^{9}+O(q^{10}) $ 4 * q - 326 * q^9	$ 4 q - 326 q^{9} - 334 q^{11} + 2668 q^{19} + 1666 q^{21} + 13918 q^{29} - 20960 q^{31} + 48934 q^{39} - 48220 q^{41} - 9604 q^{49} - 24890 q^{51} - 3952 q^{59} + 45228 q^{61} + 57628 q^{69} - 93056 q^{71} + 24962 q^{79} + 2756 q^{81} - 67596 q^{89} - 43610 q^{91} - 127156 q^{99}+O(q^{100}) $ 4 * q - 326 * q^9 - 334 * q^11 + 2668 * q^19 + 1666 * q^21 + 13918 * q^29 - 20960 * q^31 + 48934 * q^39 - 48220 * q^41 - 9604 * q^49 - 24890 * q^51 - 3952 * q^59 + 45228 * q^61 + 57628 * q^69 - 93056 * q^71 + 24962 * q^79 + 2756 * q^81 - 67596 * q^89 - 43610 * q^91 - 127156 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/700\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$477$
$\chi(n)$	$1$	$1$	$-1$

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	0
$2$	0	0
$3$	24.3824i	1.56413i	0.623197	+	0.782065i	$0.285834\pi$
$3$	24.3824i	1.56413i	−0.623197	+	0.782065i	$0.714166\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 49.0000i	− 0.377964i
$7$	− 49.0000i	− 0.377964i
$8$	0	0
$9$	−351.500	−1.44650
$10$	0	0
$11$	59.4414	0.148118	0.0740589	−	0.997254i	$-0.476405\pi$
$11$	59.4414	0.148118	0.0740589	+	0.997254i	$0.476405\pi$
$12$	0	0
$13$	− 873.678i	− 1.43381i	−0.697169	−	0.716907i	$-0.745557\pi$
$13$	− 873.678i	− 1.43381i	0.697169	−	0.716907i	$-0.254443\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	414.795i	0.348106i	0.984736	+	0.174053i	$0.0556863\pi$
$17$	414.795i	0.348106i	−0.984736	+	0.174053i	$0.944314\pi$
$18$	0	0
$19$	1397.59	0.888169	0.444085	−	0.895985i	$-0.353529\pi$
$19$	1397.59	0.888169	0.444085	+	0.895985i	$0.353529\pi$
$20$	0	0
$21$	1194.74	0.591186
$22$	0	0
$23$	− 839.708i	− 0.330985i	−0.986211	−	0.165493i	$-0.947079\pi$
$23$	− 839.708i	− 0.330985i	0.986211	−	0.165493i	$-0.0529214\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 2645.50i	− 0.698390i
$28$	0	0
$29$	7593.04	1.67657	0.838283	−	0.545236i	$-0.183560\pi$
$29$	7593.04	1.67657	0.838283	+	0.545236i	$0.183560\pi$
$30$	0	0
$31$	−7018.83	−1.31178	−0.655889	−	0.754857i	$-0.727706\pi$
$31$	−7018.83	−1.31178	−0.655889	+	0.754857i	$0.727706\pi$
$32$	0	0
$33$	1449.32i	0.231676i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2763.65i	0.331879i	0.986136	+	0.165939i	$0.0530656\pi$
$37$	2763.65i	0.331879i	−0.986136	+	0.165939i	$0.946934\pi$
$38$	0	0
$39$	21302.3	2.24267
$40$	0	0
$41$	−14373.8	−1.33540	−0.667702	−	0.744429i	$-0.732722\pi$
$41$	−14373.8	−1.33540	−0.667702	+	0.744429i	$0.732722\pi$
$42$	0	0
$43$	22368.9i	1.84490i	0.386113	+	0.922452i	$0.373818\pi$
$43$	22368.9i	1.84490i	−0.386113	+	0.922452i	$0.626182\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12046.6i	0.795461i	0.917502	+	0.397731i	$0.130202\pi$
$47$	12046.6i	0.795461i	−0.917502	+	0.397731i	$0.869798\pi$
$48$	0	0
$49$	−2401.00	−0.142857
$50$	0	0
$51$	−10113.7	−0.544482
$52$	0	0
$53$	14422.9i	0.705282i	0.935759	+	0.352641i	$0.114716\pi$
$53$	14422.9i	0.705282i	−0.935759	+	0.352641i	$0.885284\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	34076.6i	1.38921i
$58$	0	0
$59$	−16743.3	−0.626198	−0.313099	−	0.949721i	$-0.601367\pi$
$59$	−16743.3	−0.626198	−0.313099	+	0.949721i	$0.601367\pi$
$60$	0	0
$61$	33828.2	1.16400	0.582002	−	0.813187i	$-0.302270\pi$
$61$	33828.2	1.16400	0.582002	+	0.813187i	$0.302270\pi$
$62$	0	0
$63$	17223.5i	0.546727i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12135.2i	0.330263i	0.986272	+	0.165131i	$0.0528048\pi$
$67$	12135.2i	0.330263i	−0.986272	+	0.165131i	$0.947195\pi$
$68$	0	0
$69$	20474.1	0.517704
$70$	0	0
$71$	−43847.6	−1.03228	−0.516142	−	0.856503i	$-0.672633\pi$
$71$	−43847.6	−1.03228	−0.516142	+	0.856503i	$0.672633\pi$
$72$	0	0
$73$	− 33432.7i	− 0.734285i	−0.930165	−	0.367143i	$-0.880336\pi$
$73$	− 33432.7i	− 0.734285i	0.930165	−	0.367143i	$-0.119664\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2912.63i	− 0.0559833i
$78$	0	0
$79$	36877.6	0.664806	0.332403	−	0.943137i	$-0.392141\pi$
$79$	36877.6	0.664806	0.332403	+	0.943137i	$0.392141\pi$
$80$	0	0
$81$	−20911.0	−0.354130
$82$	0	0
$83$	114590.i	1.82579i	0.408191	+	0.912896i	$0.366160\pi$
$83$	114590.i	1.82579i	−0.408191	+	0.912896i	$0.633840\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	185136.i	2.62237i
$88$	0	0
$89$	58796.4	0.786821	0.393410	−	0.919363i	$-0.371295\pi$
$89$	58796.4	0.786821	0.393410	+	0.919363i	$0.371295\pi$
$90$	0	0
$91$	−42810.2	−0.541931
$92$	0	0
$93$	− 171136.i	− 2.05179i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	34474.9i	0.372026i	0.982547	+	0.186013i	$0.0595566\pi$
$97$	34474.9i	0.372026i	−0.982547	+	0.186013i	$0.940443\pi$
$98$	0	0
$99$	−20893.7	−0.214253
$100$	0	0
$101$	101322.	0.988330	0.494165	−	0.869368i	$-0.335474\pi$
$101$	101322.	0.988330	0.494165	+	0.869368i	$0.335474\pi$
$102$	0	0
$103$	28647.8i	0.266071i	0.991111	+	0.133036i	$0.0424725\pi$
$103$	28647.8i	0.266071i	−0.991111	+	0.133036i	$0.957528\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	89387.2i	0.754772i	0.926056	+	0.377386i	$0.123177\pi$
$107$	89387.2i	0.754772i	−0.926056	+	0.377386i	$0.876823\pi$
$108$	0	0
$109$	143199.	1.15444	0.577221	−	0.816588i	$-0.304137\pi$
$109$	143199.	1.15444	0.577221	+	0.816588i	$0.304137\pi$
$110$	0	0
$111$	−67384.4	−0.519101
$112$	0	0
$113$	130142.i	0.958784i	0.877601	+	0.479392i	$0.159143\pi$
$113$	130142.i	0.958784i	−0.877601	+	0.479392i	$0.840857\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	307098.i	2.07402i
$118$	0	0
$119$	20324.9	0.131572
$120$	0	0
$121$	−157518.	−0.978061
$122$	0	0
$123$	− 350468.i	− 2.08875i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 163710.i	− 0.900671i	−0.892859	−	0.450336i	$-0.851304\pi$
$127$	− 163710.i	− 0.900671i	0.892859	−	0.450336i	$-0.148696\pi$
$128$	0	0
$129$	−545407.	−2.88567
$130$	0	0
$131$	306287.	1.55937	0.779686	−	0.626171i	$-0.215379\pi$
$131$	306287.	1.55937	0.779686	+	0.626171i	$0.215379\pi$
$132$	0	0
$133$	− 68481.9i	− 0.335696i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 27852.7i	− 0.126784i	−0.997989	−	0.0633922i	$-0.979808\pi$
$137$	− 27852.7i	− 0.126784i	0.997989	−	0.0633922i	$-0.0201919\pi$
$138$	0	0
$139$	−176535.	−0.774986	−0.387493	−	0.921873i	$-0.626659\pi$
$139$	−176535.	−0.774986	−0.387493	+	0.921873i	$0.626659\pi$
$140$	0	0
$141$	−293724.	−1.24421
$142$	0	0
$143$	− 51932.6i	− 0.212373i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 58542.1i	− 0.223447i
$148$	0	0
$149$	47427.0	0.175009	0.0875045	−	0.996164i	$-0.472111\pi$
$149$	47427.0	0.175009	0.0875045	+	0.996164i	$0.472111\pi$
$150$	0	0
$151$	−509583.	−1.81875	−0.909374	−	0.415979i	$-0.863439\pi$
$151$	−509583.	−1.81875	−0.909374	+	0.415979i	$0.863439\pi$
$152$	0	0
$153$	− 145801.i	− 0.503536i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 285482.i	− 0.924334i	−0.886793	−	0.462167i	$-0.847072\pi$
$157$	− 285482.i	− 0.924334i	0.886793	−	0.462167i	$-0.152928\pi$
$158$	0	0
$159$	−351664.	−1.10315
$160$	0	0
$161$	−41145.7	−0.125101
$162$	0	0
$163$	312305.i	0.920682i	0.887742	+	0.460341i	$0.152273\pi$
$163$	312305.i	0.920682i	−0.887742	+	0.460341i	$0.847727\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 247746.i	− 0.687408i	−0.939078	−	0.343704i	$-0.888318\pi$
$167$	− 247746.i	− 0.687408i	0.939078	−	0.343704i	$-0.111682\pi$
$168$	0	0
$169$	−392020.	−1.05582
$170$	0	0
$171$	−491253.	−1.28474
$172$	0	0
$173$	301061.i	0.764784i	0.924000	+	0.382392i	$0.124900\pi$
$173$	301061.i	0.764784i	−0.924000	+	0.382392i	$0.875100\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 408242.i	− 0.979455i
$178$	0	0
$179$	−134306.	−0.313301	−0.156651	−	0.987654i	$-0.550070\pi$
$179$	−134306.	−0.313301	−0.156651	+	0.987654i	$0.550070\pi$
$180$	0	0
$181$	−384828.	−0.873113	−0.436556	−	0.899677i	$-0.643802\pi$
$181$	−384828.	−0.873113	−0.436556	+	0.899677i	$0.643802\pi$
$182$	0	0
$183$	824812.i	1.82065i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	24656.0i	0.0515607i
$188$	0	0
$189$	−129629.	−0.263967
$190$	0	0
$191$	−689905.	−1.36838	−0.684189	−	0.729305i	$-0.739843\pi$
$191$	−689905.	−1.36838	−0.684189	+	0.729305i	$0.739843\pi$
$192$	0	0
$193$	− 702104.i	− 1.35678i	−0.734704	−	0.678388i	$-0.762679\pi$
$193$	− 702104.i	− 1.35678i	0.734704	−	0.678388i	$-0.237321\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	544207.i	0.999077i	0.866292	+	0.499538i	$0.166497\pi$
$197$	544207.i	0.999077i	−0.866292	+	0.499538i	$0.833503\pi$
$198$	0	0
$199$	312684.	0.559723	0.279862	−	0.960040i	$-0.409711\pi$
$199$	312684.	0.559723	0.279862	+	0.960040i	$0.409711\pi$
$200$	0	0
$201$	−295885.	−0.516574
$202$	0	0
$203$	− 372059.i	− 0.633682i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	295158.i	0.478771i
$208$	0	0
$209$	83074.7	0.131554
$210$	0	0
$211$	552813.	0.854815	0.427407	−	0.904059i	$-0.359427\pi$
$211$	552813.	0.854815	0.427407	+	0.904059i	$0.359427\pi$
$212$	0	0
$213$	− 1.06911e6i	− 1.61463i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	343923.i	0.495805i
$218$	0	0
$219$	815170.	1.14852
$220$	0	0
$221$	362397.	0.499119
$222$	0	0
$223$	717458.i	0.966127i	0.875585	+	0.483064i	$0.160476\pi$
$223$	717458.i	0.966127i	−0.875585	+	0.483064i	$0.839524\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	680024.i	0.875911i	0.898997	+	0.437955i	$0.144297\pi$
$227$	680024.i	0.875911i	−0.898997	+	0.437955i	$0.855703\pi$
$228$	0	0
$229$	238359.	0.300361	0.150180	−	0.988659i	$-0.452015\pi$
$229$	238359.	0.300361	0.150180	+	0.988659i	$0.452015\pi$
$230$	0	0
$231$	71016.8	0.0875652
$232$	0	0
$233$	− 442042.i	− 0.533425i	−0.963776	−	0.266712i	$-0.914063\pi$
$233$	− 442042.i	− 0.533425i	0.963776	−	0.266712i	$-0.0859374\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	899164.i	1.03984i
$238$	0	0
$239$	−1.09492e6	−1.23990	−0.619952	−	0.784639i	$-0.712848\pi$
$239$	−1.09492e6	−1.23990	−0.619952	+	0.784639i	$0.712848\pi$
$240$	0	0
$241$	−1.47128e6	−1.63175	−0.815875	−	0.578228i	$-0.803744\pi$
$241$	−1.47128e6	−1.63175	−0.815875	+	0.578228i	$0.803744\pi$
$242$	0	0
$243$	− 1.15272e6i	− 1.25230i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.22104e6i	− 1.27347i
$248$	0	0
$249$	−2.79398e6	−2.85578
$250$	0	0
$251$	579697.	0.580787	0.290394	−	0.956907i	$-0.406214\pi$
$251$	579697.	0.580787	0.290394	+	0.956907i	$0.406214\pi$
$252$	0	0
$253$	− 49913.4i	− 0.0490248i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.05839e6i	0.999573i	0.866149	+	0.499786i	$0.166588\pi$
$257$	1.05839e6i	0.999573i	−0.866149	+	0.499786i	$0.833412\pi$
$258$	0	0
$259$	135419.	0.125438
$260$	0	0
$261$	−2.66896e6	−2.42516
$262$	0	0
$263$	1.76047e6i	1.56942i	0.619863	+	0.784710i	$0.287188\pi$
$263$	1.76047e6i	1.56942i	−0.619863	+	0.784710i	$0.712812\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.43360e6i	1.23069i
$268$	0	0
$269$	−385475.	−0.324799	−0.162400	−	0.986725i	$-0.551923\pi$
$269$	−385475.	−0.324799	−0.162400	+	0.986725i	$0.551923\pi$
$270$	0	0
$271$	1.02368e6	0.846723	0.423362	−	0.905961i	$-0.360850\pi$
$271$	1.02368e6	0.846723	0.423362	+	0.905961i	$0.360850\pi$
$272$	0	0
$273$	− 1.04381e6i	− 0.847650i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.29828e6i	1.01664i	0.861167	+	0.508322i	$0.169734\pi$
$277$	1.29828e6i	1.01664i	−0.861167	+	0.508322i	$0.830266\pi$
$278$	0	0
$279$	2.46712e6	1.89749
$280$	0	0
$281$	−1.98354e6	−1.49856	−0.749282	−	0.662251i	$-0.769601\pi$
$281$	−1.98354e6	−1.49856	−0.749282	+	0.662251i	$0.769601\pi$
$282$	0	0
$283$	1.84574e6i	1.36995i	0.728566	+	0.684975i	$0.240187\pi$
$283$	1.84574e6i	1.36995i	−0.728566	+	0.684975i	$0.759813\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	704318.i	0.504735i
$288$	0	0
$289$	1.24780e6	0.878823
$290$	0	0
$291$	−840579.	−0.581897
$292$	0	0
$293$	− 2.01175e6i	− 1.36901i	−0.729010	−	0.684504i	$-0.760019\pi$
$293$	− 2.01175e6i	− 1.36901i	0.729010	−	0.684504i	$-0.239981\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 157252.i	− 0.103444i
$298$	0	0
$299$	−733634.	−0.474571
$300$	0	0
$301$	1.09608e6	0.697308
$302$	0	0
$303$	2.47048e6i	1.54588i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.09488e6i	1.26857i	0.773100	+	0.634284i	$0.218705\pi$
$307$	2.09488e6i	1.26857i	−0.773100	+	0.634284i	$0.781295\pi$
$308$	0	0
$309$	−698501.	−0.416170
$310$	0	0
$311$	2.02927e6	1.18970	0.594852	−	0.803835i	$-0.297211\pi$
$311$	2.02927e6	1.18970	0.594852	+	0.803835i	$0.297211\pi$
$312$	0	0
$313$	− 233443.i	− 0.134685i	−0.997730	−	0.0673427i	$-0.978548\pi$
$313$	− 233443.i	− 0.134685i	0.997730	−	0.0673427i	$-0.0214521\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.88793e6i	1.61413i	0.590462	+	0.807065i	$0.298945\pi$
$317$	2.88793e6i	1.61413i	−0.590462	+	0.807065i	$0.701055\pi$
$318$	0	0
$319$	451341.	0.248329
$320$	0	0
$321$	−2.17947e6	−1.18056
$322$	0	0
$323$	579713.i	0.309177i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3.49152e6i	1.80570i
$328$	0	0
$329$	590282.	0.300656
$330$	0	0
$331$	1.33037e6	0.667423	0.333712	−	0.942675i	$-0.391699\pi$
$331$	1.33037e6	0.667423	0.333712	+	0.942675i	$0.391699\pi$
$332$	0	0
$333$	− 971425.i	− 0.480064i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 3.47180e6i	− 1.66525i	−0.553835	−	0.832626i	$-0.686836\pi$
$337$	− 3.47180e6i	− 1.66525i	0.553835	−	0.832626i	$-0.313164\pi$
$338$	0	0
$339$	−3.17317e6	−1.49966
$340$	0	0
$341$	−417209.	−0.194298
$342$	0	0
$343$	117649.i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.02779e6i	1.34990i	0.737862	+	0.674951i	$0.235835\pi$
$347$	3.02779e6i	1.34990i	−0.737862	+	0.674951i	$0.764165\pi$
$348$	0	0
$349$	3.98788e6	1.75258	0.876292	−	0.481781i	$-0.160010\pi$
$349$	3.98788e6	1.75258	0.876292	+	0.481781i	$0.160010\pi$
$350$	0	0
$351$	−2.31131e6	−1.00136
$352$	0	0
$353$	1.53257e6i	0.654611i	0.944919	+	0.327305i	$0.106141\pi$
$353$	1.53257e6i	0.654611i	−0.944919	+	0.327305i	$0.893859\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	495570.i	0.205795i
$358$	0	0
$359$	3.34312e6	1.36904	0.684519	−	0.728995i	$-0.260012\pi$
$359$	3.34312e6	1.36904	0.684519	+	0.728995i	$0.260012\pi$
$360$	0	0
$361$	−522843.	−0.211156
$362$	0	0
$363$	− 3.84066e6i	− 1.52982i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.80285e6i	1.47382i	0.675991	+	0.736910i	$0.263716\pi$
$367$	3.80285e6i	1.47382i	−0.675991	+	0.736910i	$0.736284\pi$
$368$	0	0
$369$	5.05241e6	1.93167
$370$	0	0
$371$	706722.	0.266571
$372$	0	0
$373$	1.26664e6i	0.471391i	0.971827	+	0.235695i	$0.0757368\pi$
$373$	1.26664e6i	0.471391i	−0.971827	+	0.235695i	$0.924263\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 6.63387e6i	− 2.40388i
$378$	0	0
$379$	−3.00151e6	−1.07335	−0.536675	−	0.843789i	$-0.680320\pi$
$379$	−3.00151e6	−1.07335	−0.536675	+	0.843789i	$0.680320\pi$
$380$	0	0
$381$	3.99164e6	1.40877
$382$	0	0
$383$	− 5.55564e6i	− 1.93525i	−0.252393	−	0.967625i	$-0.581218\pi$
$383$	− 5.55564e6i	− 1.93525i	0.252393	−	0.967625i	$-0.418782\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 7.86268e6i	− 2.66866i
$388$	0	0
$389$	−1.20865e6	−0.404972	−0.202486	−	0.979285i	$-0.564902\pi$
$389$	−1.20865e6	−0.404972	−0.202486	+	0.979285i	$0.564902\pi$
$390$	0	0
$391$	348306.	0.115218
$392$	0	0
$393$	7.46800e6i	2.43906i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 5.78510e6i	− 1.84219i	−0.389338	−	0.921095i	$-0.627296\pi$
$397$	− 5.78510e6i	− 1.84219i	0.389338	−	0.921095i	$-0.372704\pi$
$398$	0	0
$399$	1.66975e6	0.525073
$400$	0	0
$401$	202477.	0.0628802	0.0314401	−	0.999506i	$-0.489991\pi$
$401$	202477.	0.0628802	0.0314401	+	0.999506i	$0.489991\pi$
$402$	0	0
$403$	6.13219e6i	1.88085i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	164275.i	0.0491572i
$408$	0	0
$409$	−5.20097e6	−1.53736	−0.768681	−	0.639633i	$-0.779086\pi$
$409$	−5.20097e6	−1.53736	−0.768681	+	0.639633i	$0.779086\pi$
$410$	0	0
$411$	679115.	0.198307
$412$	0	0
$413$	820423.i	0.236681i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 4.30434e6i	− 1.21218i
$418$	0	0
$419$	3.77891e6	1.05155	0.525777	−	0.850622i	$-0.323775\pi$
$419$	3.77891e6	1.05155	0.525777	+	0.850622i	$0.323775\pi$
$420$	0	0
$421$	−5.87618e6	−1.61581	−0.807904	−	0.589314i	$-0.799398\pi$
$421$	−5.87618e6	−1.61581	−0.807904	+	0.589314i	$0.799398\pi$
$422$	0	0
$423$	− 4.23438e6i	− 1.15064i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1.65758e6i	− 0.439952i
$428$	0	0
$429$	1.26624e6	0.332180
$430$	0	0
$431$	4.43938e6	1.15114	0.575571	−	0.817751i	$-0.304780\pi$
$431$	4.43938e6	1.15114	0.575571	+	0.817751i	$0.304780\pi$
$432$	0	0
$433$	5.99703e6i	1.53715i	0.639759	+	0.768576i	$0.279034\pi$
$433$	5.99703e6i	1.53715i	−0.639759	+	0.768576i	$0.720966\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.17357e6i	− 0.293971i
$438$	0	0
$439$	−2.22003e6	−0.549791	−0.274895	−	0.961474i	$-0.588643\pi$
$439$	−2.22003e6	−0.549791	−0.274895	+	0.961474i	$0.588643\pi$
$440$	0	0
$441$	843953.	0.206643
$442$	0	0
$443$	− 4.93550e6i	− 1.19487i	−0.801916	−	0.597437i	$-0.796186\pi$
$443$	− 4.93550e6i	− 1.19487i	0.801916	−	0.597437i	$-0.203814\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.15638e6i	0.273737i
$448$	0	0
$449$	−6.01484e6	−1.40802	−0.704009	−	0.710191i	$-0.748609\pi$
$449$	−6.01484e6	−1.40802	−0.704009	+	0.710191i	$0.748609\pi$
$450$	0	0
$451$	−854401.	−0.197797
$452$	0	0
$453$	− 1.24248e7i	− 2.84476i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.74872e6i	0.615660i	0.951441	+	0.307830i	$0.0996028\pi$
$457$	2.74872e6i	0.615660i	−0.951441	+	0.307830i	$0.900397\pi$
$458$	0	0
$459$	1.09734e6	0.243114
$460$	0	0
$461$	3.53770e6	0.775297	0.387649	−	0.921807i	$-0.373287\pi$
$461$	3.53770e6	0.775297	0.387649	+	0.921807i	$0.373287\pi$
$462$	0	0
$463$	4.30850e6i	0.934058i	0.884242	+	0.467029i	$0.154676\pi$
$463$	4.30850e6i	0.934058i	−0.884242	+	0.467029i	$0.845324\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 2.69910e6i	− 0.572700i	−0.958125	−	0.286350i	$-0.907558\pi$
$467$	− 2.69910e6i	− 0.572700i	0.958125	−	0.286350i	$-0.0924421\pi$
$468$	0	0
$469$	594624.	0.124828
$470$	0	0
$471$	6.96072e6	1.44578
$472$	0	0
$473$	1.32964e6i	0.273263i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 5.06965e6i	− 1.02019i
$478$	0	0
$479$	−7.97101e6	−1.58736	−0.793678	−	0.608338i	$-0.791836\pi$
$479$	−7.97101e6	−1.58736	−0.793678	+	0.608338i	$0.791836\pi$
$480$	0	0
$481$	2.41454e6	0.475852
$482$	0	0
$483$	− 1.00323e6i	− 0.195674i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 3.99688e6i	− 0.763658i	−0.924233	−	0.381829i	$-0.875294\pi$
$487$	− 3.99688e6i	− 0.763658i	0.924233	−	0.381829i	$-0.124706\pi$
$488$	0	0
$489$	−7.61473e6	−1.44007
$490$	0	0
$491$	789283.	0.147751	0.0738753	−	0.997267i	$-0.476463\pi$
$491$	789283.	0.147751	0.0738753	+	0.997267i	$0.476463\pi$
$492$	0	0
$493$	3.14955e6i	0.583622i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.14853e6i	0.390167i
$498$	0	0
$499$	−171664.	−0.0308622	−0.0154311	−	0.999881i	$-0.504912\pi$
$499$	−171664.	−0.0308622	−0.0154311	+	0.999881i	$0.504912\pi$
$500$	0	0
$501$	6.04063e6	1.07520
$502$	0	0
$503$	− 1.12380e6i	− 0.198047i	−0.995085	−	0.0990235i	$-0.968428\pi$
$503$	− 1.12380e6i	− 0.198047i	0.995085	−	0.0990235i	$-0.0315719\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9.55837e6i	− 1.65144i
$508$	0	0
$509$	8.64586e6	1.47915	0.739577	−	0.673072i	$-0.235025\pi$
$509$	8.64586e6	1.47915	0.739577	+	0.673072i	$0.235025\pi$
$510$	0	0
$511$	−1.63820e6	−0.277534
$512$	0	0
$513$	− 3.69732e6i	− 0.620289i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	716066.i	0.117822i
$518$	0	0
$519$	−7.34058e6	−1.19622
$520$	0	0
$521$	2.82649e6	0.456198	0.228099	−	0.973638i	$-0.426749\pi$
$521$	2.82649e6	0.456198	0.228099	+	0.973638i	$0.426749\pi$
$522$	0	0
$523$	− 9.25594e6i	− 1.47967i	−0.672786	−	0.739837i	$-0.734903\pi$
$523$	− 9.25594e6i	− 1.47967i	0.672786	−	0.739837i	$-0.265097\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 2.91137e6i	− 0.456637i
$528$	0	0
$529$	5.73123e6	0.890449
$530$	0	0
$531$	5.88529e6	0.905798
$532$	0	0
$533$	1.25581e7i	1.91472i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 3.27470e6i	− 0.490044i
$538$	0	0
$539$	−142719.	−0.0211597
$540$	0	0
$541$	6.96253e6	1.02276	0.511380	−	0.859355i	$-0.329134\pi$
$541$	6.96253e6	1.02276	0.511380	+	0.859355i	$0.329134\pi$
$542$	0	0
$543$	− 9.38303e6i	− 1.36566i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 4.71143e6i	− 0.673263i	−0.941637	−	0.336631i	$-0.890712\pi$
$547$	− 4.71143e6i	− 0.673263i	0.941637	−	0.336631i	$-0.109288\pi$
$548$	0	0
$549$	−1.18906e7	−1.68374
$550$	0	0
$551$	1.06119e7	1.48907
$552$	0	0
$553$	− 1.80700e6i	− 0.251273i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.69349e6i	0.641000i	0.947249	+	0.320500i	$0.103851\pi$
$557$	4.69349e6i	0.641000i	−0.947249	+	0.320500i	$0.896149\pi$
$558$	0	0
$559$	1.95432e7	2.64525
$560$	0	0
$561$	−601172.	−0.0806476
$562$	0	0
$563$	9.21973e6i	1.22588i	0.790130	+	0.612939i	$0.210013\pi$
$563$	9.21973e6i	1.22588i	−0.790130	+	0.612939i	$0.789987\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.02464e6i	0.133849i
$568$	0	0
$569$	162896.	0.0210926	0.0105463	−	0.999944i	$-0.496643\pi$
$569$	162896.	0.0210926	0.0105463	+	0.999944i	$0.496643\pi$
$570$	0	0
$571$	−733222.	−0.0941120	−0.0470560	−	0.998892i	$-0.514984\pi$
$571$	−733222.	−0.0941120	−0.0470560	+	0.998892i	$0.514984\pi$
$572$	0	0
$573$	− 1.68215e7i	− 2.14032i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1.25596e7i	− 1.57049i	−0.619185	−	0.785245i	$-0.712537\pi$
$577$	− 1.25596e7i	− 1.57049i	0.619185	−	0.785245i	$-0.287463\pi$
$578$	0	0
$579$	1.71190e7	2.12217
$580$	0	0
$581$	5.61491e6	0.690085
$582$	0	0
$583$	857317.i	0.104465i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.99237e6i	− 0.238658i	−0.992855	−	0.119329i	$-0.961926\pi$
$587$	− 1.99237e6i	− 0.238658i	0.992855	−	0.119329i	$-0.0380743\pi$
$588$	0	0
$589$	−9.80944e6	−1.16508
$590$	0	0
$591$	−1.32691e7	−1.56269
$592$	0	0
$593$	1.60192e7i	1.87070i	0.353725	+	0.935350i	$0.384915\pi$
$593$	1.60192e7i	1.87070i	−0.353725	+	0.935350i	$0.615085\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7.62399e6i	0.875480i
$598$	0	0
$599$	6.72762e6	0.766116	0.383058	−	0.923724i	$-0.374871\pi$
$599$	6.72762e6	0.766116	0.383058	+	0.923724i	$0.374871\pi$
$600$	0	0
$601$	1.53688e6	0.173561	0.0867806	−	0.996227i	$-0.472342\pi$
$601$	1.53688e6	0.173561	0.0867806	+	0.996227i	$0.472342\pi$
$602$	0	0
$603$	− 4.26553e6i	− 0.477727i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	8.43829e6i	0.929571i	0.885423	+	0.464786i	$0.153869\pi$
$607$	8.43829e6i	0.929571i	−0.885423	+	0.464786i	$0.846131\pi$
$608$	0	0
$609$	9.07168e6	0.991162
$610$	0	0
$611$	1.05248e7	1.14054
$612$	0	0
$613$	9.55026e6i	1.02651i	0.858236	+	0.513256i	$0.171561\pi$
$613$	9.55026e6i	1.02651i	−0.858236	+	0.513256i	$0.828439\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.46544e7i	1.54972i	0.632131	+	0.774862i	$0.282181\pi$
$617$	1.46544e7i	1.54972i	−0.632131	+	0.774862i	$0.717819\pi$
$618$	0	0
$619$	1.28694e7	1.34999	0.674997	−	0.737820i	$-0.264145\pi$
$619$	1.28694e7	1.34999	0.674997	+	0.737820i	$0.264145\pi$
$620$	0	0
$621$	−2.22145e6	−0.231157
$622$	0	0
$623$	− 2.88102e6i	− 0.297390i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.02556e6i	0.205767i
$628$	0	0
$629$	−1.14635e6	−0.115529
$630$	0	0
$631$	1.95977e7	1.95944	0.979719	−	0.200378i	$-0.0642171\pi$
$631$	1.95977e7	1.95944	0.979719	+	0.200378i	$0.0642171\pi$
$632$	0	0
$633$	1.34789e7i	1.33704i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.09770e6i	0.204831i
$638$	0	0
$639$	1.54124e7	1.49320
$640$	0	0
$641$	−1.23136e7	−1.18369	−0.591847	−	0.806050i	$-0.701601\pi$
$641$	−1.23136e7	−1.18369	−0.591847	+	0.806050i	$0.701601\pi$
$642$	0	0
$643$	− 423026.i	− 0.0403496i	−0.999796	−	0.0201748i	$-0.993578\pi$
$643$	− 423026.i	− 0.0403496i	0.999796	−	0.0201748i	$-0.00642227\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 3.53941e6i	− 0.332407i	−0.986091	−	0.166203i	$-0.946849\pi$
$647$	− 3.53941e6i	− 0.332407i	0.986091	−	0.166203i	$-0.0531509\pi$
$648$	0	0
$649$	−995247.	−0.0927511
$650$	0	0
$651$	−8.38565e6	−0.775504
$652$	0	0
$653$	− 544725.i	− 0.0499913i	−0.999688	−	0.0249956i	$-0.992043\pi$
$653$	− 544725.i	− 0.0499913i	0.999688	−	0.0249956i	$-0.00795718\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.17516e7i	1.06215i
$658$	0	0
$659$	−1.50778e7	−1.35246	−0.676229	−	0.736691i	$-0.736387\pi$
$659$	−1.50778e7	−1.35246	−0.676229	+	0.736691i	$0.736387\pi$
$660$	0	0
$661$	2.87160e6	0.255635	0.127817	−	0.991798i	$-0.459203\pi$
$661$	2.87160e6	0.255635	0.127817	+	0.991798i	$0.459203\pi$
$662$	0	0
$663$	8.83610e6i	0.780687i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 6.37593e6i	− 0.554918i
$668$	0	0
$669$	−1.74933e7	−1.51115
$670$	0	0
$671$	2.01080e6	0.172410
$672$	0	0
$673$	2.11706e7i	1.80175i	0.434078	+	0.900875i	$0.357074\pi$
$673$	2.11706e7i	1.80175i	−0.434078	+	0.900875i	$0.642926\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.86653e6i	0.491938i	0.969278	+	0.245969i	$0.0791061\pi$
$677$	5.86653e6i	0.491938i	−0.969278	+	0.245969i	$0.920894\pi$
$678$	0	0
$679$	1.68927e6	0.140613
$680$	0	0
$681$	−1.65806e7	−1.37004
$682$	0	0
$683$	− 441447.i	− 0.0362099i	−0.999836	−	0.0181049i	$-0.994237\pi$
$683$	− 441447.i	− 0.0362099i	0.999836	−	0.0181049i	$-0.00576329\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5.81176e6i	0.469803i
$688$	0	0
$689$	1.26010e7	1.01124
$690$	0	0
$691$	−5.89504e6	−0.469669	−0.234834	−	0.972035i	$-0.575455\pi$
$691$	−5.89504e6	−0.469669	−0.234834	+	0.972035i	$0.575455\pi$
$692$	0	0
$693$	1.02379e6i	0.0809801i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 5.96219e6i	− 0.464862i
$698$	0	0
$699$	1.07780e7	0.834346
$700$	0	0
$701$	−1.25764e7	−0.966635	−0.483317	−	0.875445i	$-0.660568\pi$
$701$	−1.25764e7	−0.966635	−0.483317	+	0.875445i	$0.660568\pi$
$702$	0	0
$703$	3.86245e6i	0.294764i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.96480e6i	− 0.373554i
$708$	0	0
$709$	−1.41839e7	−1.05969	−0.529847	−	0.848093i	$-0.677751\pi$
$709$	−1.41839e7	−1.05969	−0.529847	+	0.848093i	$0.677751\pi$
$710$	0	0
$711$	−1.29625e7	−0.961645
$712$	0	0
$713$	5.89376e6i	0.434179i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 2.66968e7i	− 1.93937i
$718$	0	0
$719$	2.16964e7	1.56518	0.782591	−	0.622537i	$-0.213898\pi$
$719$	2.16964e7	1.56518	0.782591	+	0.622537i	$0.213898\pi$
$720$	0	0
$721$	1.40374e6	0.100565
$722$	0	0
$723$	− 3.58734e7i	− 2.55227i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.51847e7i	1.76726i	0.468187	+	0.883629i	$0.344907\pi$
$727$	2.51847e7i	1.76726i	−0.468187	+	0.883629i	$0.655093\pi$
$728$	0	0
$729$	2.30246e7	1.60462
$730$	0	0
$731$	−9.27850e6	−0.642221
$732$	0	0
$733$	1.92772e7i	1.32521i	0.748971	+	0.662603i	$0.230548\pi$
$733$	1.92772e7i	1.32521i	−0.748971	+	0.662603i	$0.769452\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	721333.i	0.0489178i
$738$	0	0
$739$	−4.95610e6	−0.333833	−0.166916	−	0.985971i	$-0.553381\pi$
$739$	−4.95610e6	−0.333833	−0.166916	+	0.985971i	$0.553381\pi$
$740$	0	0
$741$	2.97719e7	1.99187
$742$	0	0
$743$	− 1.03298e7i	− 0.686465i	−0.939250	−	0.343233i	$-0.888478\pi$
$743$	− 1.03298e7i	− 0.686465i	0.939250	−	0.343233i	$-0.111522\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 4.02784e7i	− 2.64102i
$748$	0	0
$749$	4.37997e6	0.285277
$750$	0	0
$751$	−1.23752e7	−0.800665	−0.400333	−	0.916370i	$-0.631105\pi$
$751$	−1.23752e7	−0.800665	−0.400333	+	0.916370i	$0.631105\pi$
$752$	0	0
$753$	1.41344e7i	0.908427i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.53583e7i	1.60835i	0.594392	+	0.804175i	$0.297393\pi$
$757$	2.53583e7i	1.60835i	−0.594392	+	0.804175i	$0.702607\pi$
$758$	0	0
$759$	1.21701e6	0.0766812
$760$	0	0
$761$	2.94609e7	1.84410	0.922048	−	0.387075i	$-0.126514\pi$
$761$	2.94609e7	1.84410	0.922048	+	0.387075i	$0.126514\pi$
$762$	0	0
$763$	− 7.01673e6i	− 0.436338i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.46283e7i	0.897851i
$768$	0	0
$769$	−1.05271e7	−0.641936	−0.320968	−	0.947090i	$-0.604008\pi$
$769$	−1.05271e7	−0.641936	−0.320968	+	0.947090i	$0.604008\pi$
$770$	0	0
$771$	−2.58062e7	−1.56346
$772$	0	0
$773$	− 1.11836e7i	− 0.673180i	−0.941651	−	0.336590i	$-0.890726\pi$
$773$	− 1.11836e7i	− 0.673180i	0.941651	−	0.336590i	$-0.109274\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.30184e6i	0.196202i
$778$	0	0
$779$	−2.00887e7	−1.18606
$780$	0	0
$781$	−2.60636e6	−0.152900
$782$	0	0
$783$	− 2.00874e7i	− 1.17090i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	3.14414e6i	0.180953i	0.995899	+	0.0904764i	$0.0288390\pi$
$787$	3.14414e6i	0.180953i	−0.995899	+	0.0904764i	$0.971161\pi$
$788$	0	0
$789$	−4.29244e7	−2.45478
$790$	0	0
$791$	6.37695e6	0.362386
$792$	0	0
$793$	− 2.95550e7i	− 1.66897i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.10953e7i	− 0.618720i	−0.950945	−	0.309360i	$-0.899885\pi$
$797$	− 1.10953e7i	− 0.618720i	0.950945	−	0.309360i	$-0.100115\pi$
$798$	0	0
$799$	−4.99686e6	−0.276905
$800$	0	0
$801$	−2.06670e7	−1.13814
$802$	0	0
$803$	− 1.98729e6i	− 0.108761i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 9.39879e6i	− 0.508029i
$808$	0	0
$809$	3.05083e6	0.163888	0.0819439	−	0.996637i	$-0.473887\pi$
$809$	3.05083e6	0.163888	0.0819439	+	0.996637i	$0.473887\pi$
$810$	0	0
$811$	−5.20426e6	−0.277848	−0.138924	−	0.990303i	$-0.544364\pi$
$811$	−5.20426e6	−0.277848	−0.138924	+	0.990303i	$0.544364\pi$
$812$	0	0
$813$	2.49598e7i	1.32439i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.12625e7i	1.63859i
$818$	0	0
$819$	1.50478e7	0.783905
$820$	0	0
$821$	2.92071e7	1.51227	0.756137	−	0.654413i	$-0.227084\pi$
$821$	2.92071e7	1.51227	0.756137	+	0.654413i	$0.227084\pi$
$822$	0	0
$823$	5.08765e6i	0.261829i	0.991394	+	0.130915i	$0.0417914\pi$
$823$	5.08765e6i	0.261829i	−0.991394	+	0.130915i	$0.958209\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.43585e6i	0.327222i	0.986525	+	0.163611i	$0.0523142\pi$
$827$	6.43585e6i	0.327222i	−0.986525	+	0.163611i	$0.947686\pi$
$828$	0	0
$829$	−3.80871e7	−1.92483	−0.962414	−	0.271586i	$-0.912452\pi$
$829$	−3.80871e7	−1.92483	−0.962414	+	0.271586i	$0.912452\pi$
$830$	0	0
$831$	−3.16552e7	−1.59016
$832$	0	0
$833$	− 995922.i	− 0.0497294i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.85683e7i	0.916133i
$838$	0	0
$839$	1.84812e7	0.906412	0.453206	−	0.891406i	$-0.350280\pi$
$839$	1.84812e7	0.906412	0.453206	+	0.891406i	$0.350280\pi$
$840$	0	0
$841$	3.71431e7	1.81087
$842$	0	0
$843$	− 4.83635e7i	− 2.34395i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	7.71837e6i	0.369672i
$848$	0	0
$849$	−4.50036e7	−2.14278
$850$	0	0
$851$	2.32066e6	0.109847
$852$	0	0
$853$	− 1.45787e7i	− 0.686033i	−0.939329	−	0.343017i	$-0.888551\pi$
$853$	− 1.45787e7i	− 0.686033i	0.939329	−	0.343017i	$-0.111449\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1.71760e7i	− 0.798858i	−0.916764	−	0.399429i	$-0.869208\pi$
$857$	− 1.71760e7i	− 0.798858i	0.916764	−	0.399429i	$-0.130792\pi$
$858$	0	0
$859$	−1.06813e7	−0.493904	−0.246952	−	0.969028i	$-0.579429\pi$
$859$	−1.06813e7	−0.493904	−0.246952	+	0.969028i	$0.579429\pi$
$860$	0	0
$861$	−1.71729e7	−0.789472
$862$	0	0
$863$	− 7.01227e6i	− 0.320503i	−0.987076	−	0.160251i	$-0.948770\pi$
$863$	− 7.01227e6i	− 0.320503i	0.987076	−	0.160251i	$-0.0512305\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.04244e7i	1.37459i
$868$	0	0
$869$	2.19206e6	0.0984697
$870$	0	0
$871$	1.06022e7	0.473535
$872$	0	0
$873$	− 1.21179e7i	− 0.538137i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.38378e7i	1.04657i	0.852158	+	0.523284i	$0.175293\pi$
$877$	2.38378e7i	1.04657i	−0.852158	+	0.523284i	$0.824707\pi$
$878$	0	0
$879$	4.90513e7	2.14131
$880$	0	0
$881$	−1.43085e7	−0.621088	−0.310544	−	0.950559i	$-0.600511\pi$
$881$	−1.43085e7	−0.621088	−0.310544	+	0.950559i	$0.600511\pi$
$882$	0	0
$883$	− 1.61976e7i	− 0.699114i	−0.936915	−	0.349557i	$-0.886332\pi$
$883$	− 1.61976e7i	− 0.699114i	0.936915	−	0.349557i	$-0.113668\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.76151e7i	− 0.751755i	−0.926669	−	0.375878i	$-0.877341\pi$
$887$	− 1.76151e7i	− 0.751755i	0.926669	−	0.375878i	$-0.122659\pi$
$888$	0	0
$889$	−8.02180e6	−0.340422
$890$	0	0
$891$	−1.24298e6	−0.0524530
$892$	0	0
$893$	1.68362e7i	0.706504i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.78877e7i	− 0.742291i
$898$	0	0
$899$	−5.32942e7	−2.19928
$900$	0	0
$901$	−5.98254e6	−0.245512
$902$	0	0
$903$	2.67249e7i	1.09068i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 3.08532e7i	− 1.24532i	−0.782491	−	0.622662i	$-0.786051\pi$
$907$	− 3.08532e7i	− 1.24532i	0.782491	−	0.622662i	$-0.213949\pi$
$908$	0	0
$909$	−3.56149e7	−1.42962
$910$	0	0
$911$	−1.09518e7	−0.437210	−0.218605	−	0.975813i	$-0.570151\pi$
$911$	−1.09518e7	−0.437210	−0.218605	+	0.975813i	$0.570151\pi$
$912$	0	0
$913$	6.81139e6i	0.270433i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.50080e7i	− 0.589387i
$918$	0	0
$919$	−1.71813e7	−0.671067	−0.335534	−	0.942028i	$-0.608917\pi$
$919$	−1.71813e7	−0.671067	−0.335534	+	0.942028i	$0.608917\pi$
$920$	0	0
$921$	−5.10782e7	−1.98421
$922$	0	0
$923$	3.83086e7i	1.48010i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 1.00697e7i	− 0.384873i
$928$	0	0
$929$	−4.19050e6	−0.159304	−0.0796520	−	0.996823i	$-0.525381\pi$
$929$	−4.19050e6	−0.159304	−0.0796520	+	0.996823i	$0.525381\pi$
$930$	0	0
$931$	−3.35561e6	−0.126881
$932$	0	0
$933$	4.94784e7i	1.86085i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 3.45468e7i	− 1.28546i	−0.766092	−	0.642731i	$-0.777801\pi$
$937$	− 3.45468e7i	− 1.28546i	0.766092	−	0.642731i	$-0.222199\pi$
$938$	0	0
$939$	5.69190e6	0.210666
$940$	0	0
$941$	−2.31574e7	−0.852542	−0.426271	−	0.904596i	$-0.640173\pi$
$941$	−2.31574e7	−0.852542	−0.426271	+	0.904596i	$0.640173\pi$
$942$	0	0
$943$	1.20698e7i	0.441999i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.71790e7i	0.622477i	0.950332	+	0.311238i	$0.100744\pi$
$947$	1.71790e7i	0.622477i	−0.950332	+	0.311238i	$0.899256\pi$
$948$	0	0
$949$	−2.92094e7	−1.05283
$950$	0	0
$951$	−7.04146e7	−2.52471
$952$	0	0
$953$	− 2.26025e7i	− 0.806164i	−0.915164	−	0.403082i	$-0.867939\pi$
$953$	− 2.26025e7i	− 0.806164i	0.915164	−	0.403082i	$-0.132061\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.10048e7i	0.388419i
$958$	0	0
$959$	−1.36478e6	−0.0479200
$960$	0	0
$961$	2.06348e7	0.720761
$962$	0	0
$963$	− 3.14196e7i	− 1.09178i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	3.56482e7i	1.22595i	0.790103	+	0.612974i	$0.210027\pi$
$967$	3.56482e7i	1.22595i	−0.790103	+	0.612974i	$0.789973\pi$
$968$	0	0
$969$	−1.41348e7	−0.483592
$970$	0	0
$971$	−8.70185e6	−0.296185	−0.148093	−	0.988973i	$-0.547313\pi$
$971$	−8.70185e6	−0.296185	−0.148093	+	0.988973i	$0.547313\pi$
$972$	0	0
$973$	8.65021e6i	0.292917i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 703138.i	− 0.0235670i	−0.999931	−	0.0117835i	$-0.996249\pi$
$977$	− 703138.i	− 0.0235670i	0.999931	−	0.0117835i	$-0.00375089\pi$
$978$	0	0
$979$	3.49494e6	0.116542
$980$	0	0
$981$	−5.03344e7	−1.66991
$982$	0	0
$983$	− 3.74622e7i	− 1.23654i	−0.785964	−	0.618272i	$-0.787833\pi$
$983$	− 3.74622e7i	− 1.23654i	0.785964	−	0.618272i	$-0.212167\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.43925e7i	0.470265i
$988$	0	0
$989$	1.87833e7	0.610635
$990$	0	0
$991$	2.39813e7	0.775689	0.387845	−	0.921725i	$-0.373220\pi$
$991$	2.39813e7	0.775689	0.387845	+	0.921725i	$0.373220\pi$
$992$	0	0
$993$	3.24375e7i	1.04394i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 8.35171e6i	− 0.266096i	−0.991110	−	0.133048i	$-0.957524\pi$
$997$	− 8.35171e6i	− 0.266096i	0.991110	−	0.133048i	$-0.0424764\pi$
$998$	0	0
$999$	7.31124e6	0.231781

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.6.e.f.449.4		4
5.2	odd	4		700.6.a.g.1.2		2
5.3	odd	4		140.6.a.b.1.1	✓	2
5.4	even	2	inner	700.6.e.f.449.1		4
20.3	even	4		560.6.a.o.1.2		2
35.13	even	4		980.6.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.6.a.b.1.1	✓	2	5.3	odd	4
560.6.a.o.1.2		2	20.3	even	4
700.6.a.g.1.2		2	5.2	odd	4
700.6.e.f.449.1		4	5.4	even	2	inner
700.6.e.f.449.4		4	1.1	even	1	trivial
980.6.a.f.1.2		2	35.13	even	4

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:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	700.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+24.3824i q^{3} -49.0000i q^{7} -351.500 q^{9} +O(q^{10})\)	\(q+24.3824i q^{3} -49.0000i q^{7} -351.500 q^{9} +59.4414 q^{11} -873.678i q^{13} +414.795i q^{17} +1397.59 q^{19} +1194.74 q^{21} -839.708i q^{23} -2645.50i q^{27} +7593.04 q^{29} -7018.83 q^{31} +1449.32i q^{33} +2763.65i q^{37} +21302.3 q^{39} -14373.8 q^{41} +22368.9i q^{43} +12046.6i q^{47} -2401.00 q^{49} -10113.7 q^{51} +14422.9i q^{53} +34076.6i q^{57} -16743.3 q^{59} +33828.2 q^{61} +17223.5i q^{63} +12135.2i q^{67} +20474.1 q^{69} -43847.6 q^{71} -33432.7i q^{73} -2912.63i q^{77} +36877.6 q^{79} -20911.0 q^{81} +114590. i q^{83} +185136. i q^{87} +58796.4 q^{89} -42810.2 q^{91} -171136. i q^{93} +34474.9i q^{97} -20893.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 326 q^{9}+O(q^{10}) \) 4 * q - 326 * q^9	\( 4 q - 326 q^{9} - 334 q^{11} + 2668 q^{19} + 1666 q^{21} + 13918 q^{29} - 20960 q^{31} + 48934 q^{39} - 48220 q^{41} - 9604 q^{49} - 24890 q^{51} - 3952 q^{59} + 45228 q^{61} + 57628 q^{69} - 93056 q^{71} + 24962 q^{79} + 2756 q^{81} - 67596 q^{89} - 43610 q^{91} - 127156 q^{99}+O(q^{100}) \) 4 * q - 326 * q^9 - 334 * q^11 + 2668 * q^19 + 1666 * q^21 + 13918 * q^29 - 20960 * q^31 + 48934 * q^39 - 48220 * q^41 - 9604 * q^49 - 24890 * q^51 - 3952 * q^59 + 45228 * q^61 + 57628 * q^69 - 93056 * q^71 + 24962 * q^79 + 2756 * q^81 - 67596 * q^89 - 43610 * q^91 - 127156 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more