LMFDB - Embedded newform 700.6.e.d.449.2

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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	700.449
Dual form			700.6.e.d.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{3} +49.0000i q^{7} +239.000 q^{9} +O(q^{10})$	$q+2.00000i q^{3} +49.0000i q^{7} +239.000 q^{9} -720.000 q^{11} -572.000i q^{13} +1254.00i q^{17} +94.0000 q^{19} -98.0000 q^{21} -96.0000i q^{23} +964.000i q^{27} +4374.00 q^{29} -6244.00 q^{31} -1440.00i q^{33} -10798.0i q^{37} +1144.00 q^{39} +12006.0 q^{41} +9160.00i q^{43} -25836.0i q^{47} -2401.00 q^{49} -2508.00 q^{51} -1014.00i q^{53} +188.000i q^{57} -1242.00 q^{59} +7592.00 q^{61} +11711.0i q^{63} +41132.0i q^{67} +192.000 q^{69} -37632.0 q^{71} +13438.0i q^{73} -35280.0i q^{77} -6248.00 q^{79} +56149.0 q^{81} +25254.0i q^{83} +8748.00i q^{87} +45126.0 q^{89} +28028.0 q^{91} -12488.0i q^{93} +107222. i q^{97} -172080. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 478 q^{9}+O(q^{10}) $ 2 * q + 478 * q^9	$ 2 q + 478 q^{9} - 1440 q^{11} + 188 q^{19} - 196 q^{21} + 8748 q^{29} - 12488 q^{31} + 2288 q^{39} + 24012 q^{41} - 4802 q^{49} - 5016 q^{51} - 2484 q^{59} + 15184 q^{61} + 384 q^{69} - 75264 q^{71} - 12496 q^{79} + 112298 q^{81} + 90252 q^{89} + 56056 q^{91} - 344160 q^{99}+O(q^{100}) $ 2 * q + 478 * q^9 - 1440 * q^11 + 188 * q^19 - 196 * q^21 + 8748 * q^29 - 12488 * q^31 + 2288 * q^39 + 24012 * q^41 - 4802 * q^49 - 5016 * q^51 - 2484 * q^59 + 15184 * q^61 + 384 * q^69 - 75264 * q^71 - 12496 * q^79 + 112298 * q^81 + 90252 * q^89 + 56056 * q^91 - 344160 * 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
$3$	2.00000i	0.128300i	0.997940	+	0.0641500i	$0.0204336\pi$
$3$	2.00000i	0.128300i	−0.997940	+	0.0641500i	$0.979566\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	49.0000i	0.377964i
$7$	49.0000i	0.377964i
$8$	0	0
$9$	239.000	0.983539
$10$	0	0
$11$	−720.000	−1.79412	−0.897059	−	0.441912i	$-0.854300\pi$
$11$	−720.000	−1.79412	−0.897059	+	0.441912i	$0.854300\pi$
$12$	0	0
$13$	− 572.000i	− 0.938723i	−0.883006	−	0.469362i	$-0.844484\pi$
$13$	− 572.000i	− 0.938723i	0.883006	−	0.469362i	$-0.155516\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1254.00i	1.05239i	0.850365	+	0.526193i	$0.176381\pi$
$17$	1254.00i	1.05239i	−0.850365	+	0.526193i	$0.823619\pi$
$18$	0	0
$19$	94.0000	0.0597371	0.0298685	−	0.999554i	$-0.490491\pi$
$19$	94.0000	0.0597371	0.0298685	+	0.999554i	$0.490491\pi$
$20$	0	0
$21$	−98.0000	−0.0484929
$22$	0	0
$23$	− 96.0000i	− 0.0378400i	−0.999821	−	0.0189200i	$-0.993977\pi$
$23$	− 96.0000i	− 0.0378400i	0.999821	−	0.0189200i	$-0.00602279\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	964.000i	0.254488i
$28$	0	0
$29$	4374.00	0.965792	0.482896	−	0.875678i	$-0.339585\pi$
$29$	4374.00	0.965792	0.482896	+	0.875678i	$0.339585\pi$
$30$	0	0
$31$	−6244.00	−1.16697	−0.583484	−	0.812125i	$-0.698311\pi$
$31$	−6244.00	−1.16697	−0.583484	+	0.812125i	$0.698311\pi$
$32$	0	0
$33$	− 1440.00i	− 0.230185i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 10798.0i	− 1.29670i	−0.761343	−	0.648349i	$-0.775460\pi$
$37$	− 10798.0i	− 1.29670i	0.761343	−	0.648349i	$-0.224540\pi$
$38$	0	0
$39$	1144.00	0.120438
$40$	0	0
$41$	12006.0	1.11542	0.557710	−	0.830036i	$-0.311680\pi$
$41$	12006.0	1.11542	0.557710	+	0.830036i	$0.311680\pi$
$42$	0	0
$43$	9160.00i	0.755482i	0.925911	+	0.377741i	$0.123299\pi$
$43$	9160.00i	0.755482i	−0.925911	+	0.377741i	$0.876701\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 25836.0i	− 1.70601i	−0.521906	−	0.853003i	$-0.674779\pi$
$47$	− 25836.0i	− 1.70601i	0.521906	−	0.853003i	$-0.325221\pi$
$48$	0	0
$49$	−2401.00	−0.142857
$50$	0	0
$51$	−2508.00	−0.135021
$52$	0	0
$53$	− 1014.00i	− 0.0495848i	−0.999693	−	0.0247924i	$-0.992108\pi$
$53$	− 1014.00i	− 0.0495848i	0.999693	−	0.0247924i	$-0.00789247\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	188.000i	0.00766427i
$58$	0	0
$59$	−1242.00	−0.0464506	−0.0232253	−	0.999730i	$-0.507394\pi$
$59$	−1242.00	−0.0464506	−0.0232253	+	0.999730i	$0.507394\pi$
$60$	0	0
$61$	7592.00	0.261235	0.130618	−	0.991433i	$-0.458304\pi$
$61$	7592.00	0.261235	0.130618	+	0.991433i	$0.458304\pi$
$62$	0	0
$63$	11711.0i	0.371743i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	41132.0i	1.11942i	0.828689	+	0.559710i	$0.189087\pi$
$67$	41132.0i	1.11942i	−0.828689	+	0.559710i	$0.810913\pi$
$68$	0	0
$69$	192.000	0.00485488
$70$	0	0
$71$	−37632.0	−0.885955	−0.442977	−	0.896533i	$-0.646078\pi$
$71$	−37632.0	−0.885955	−0.442977	+	0.896533i	$0.646078\pi$
$72$	0	0
$73$	13438.0i	0.295140i	0.989052	+	0.147570i	$0.0471451\pi$
$73$	13438.0i	0.295140i	−0.989052	+	0.147570i	$0.952855\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 35280.0i	− 0.678113i
$78$	0	0
$79$	−6248.00	−0.112635	−0.0563175	−	0.998413i	$-0.517936\pi$
$79$	−6248.00	−0.112635	−0.0563175	+	0.998413i	$0.517936\pi$
$80$	0	0
$81$	56149.0	0.950888
$82$	0	0
$83$	25254.0i	0.402379i	0.979552	+	0.201189i	$0.0644806\pi$
$83$	25254.0i	0.402379i	−0.979552	+	0.201189i	$0.935519\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8748.00i	0.123911i
$88$	0	0
$89$	45126.0	0.603882	0.301941	−	0.953327i	$-0.402365\pi$
$89$	45126.0	0.603882	0.301941	+	0.953327i	$0.402365\pi$
$90$	0	0
$91$	28028.0	0.354804
$92$	0	0
$93$	− 12488.0i	− 0.149722i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	107222.i	1.15706i	0.815662	+	0.578528i	$0.196373\pi$
$97$	107222.i	1.15706i	−0.815662	+	0.578528i	$0.803627\pi$
$98$	0	0
$99$	−172080.	−1.76458
$100$	0	0
$101$	47136.0	0.459779	0.229890	−	0.973217i	$-0.426163\pi$
$101$	47136.0	0.459779	0.229890	+	0.973217i	$0.426163\pi$
$102$	0	0
$103$	− 122204.i	− 1.13499i	−0.823377	−	0.567495i	$-0.807912\pi$
$103$	− 122204.i	− 1.13499i	0.823377	−	0.567495i	$-0.192088\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 129636.i	− 1.09463i	−0.836928	−	0.547314i	$-0.815651\pi$
$107$	− 129636.i	− 1.09463i	0.836928	−	0.547314i	$-0.184349\pi$
$108$	0	0
$109$	220558.	1.77810	0.889051	−	0.457809i	$-0.151365\pi$
$109$	220558.	1.77810	0.889051	+	0.457809i	$0.151365\pi$
$110$	0	0
$111$	21596.0	0.166366
$112$	0	0
$113$	− 170694.i	− 1.25754i	−0.777591	−	0.628770i	$-0.783559\pi$
$113$	− 170694.i	− 1.25754i	0.777591	−	0.628770i	$-0.216441\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 136708.i	− 0.923271i
$118$	0	0
$119$	−61446.0	−0.397765
$120$	0	0
$121$	357349.	2.21886
$122$	0	0
$123$	24012.0i	0.143109i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 249808.i	− 1.37435i	−0.726492	−	0.687175i	$-0.758851\pi$
$127$	− 249808.i	− 1.37435i	0.726492	−	0.687175i	$-0.241149\pi$
$128$	0	0
$129$	−18320.0	−0.0969284
$130$	0	0
$131$	12210.0	0.0621638	0.0310819	−	0.999517i	$-0.490105\pi$
$131$	12210.0	0.0621638	0.0310819	+	0.999517i	$0.490105\pi$
$132$	0	0
$133$	4606.00i	0.0225785i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 13902.0i	− 0.0632814i	−0.999499	−	0.0316407i	$-0.989927\pi$
$137$	− 13902.0i	− 0.0632814i	0.999499	−	0.0316407i	$-0.0100732\pi$
$138$	0	0
$139$	431794.	1.89557	0.947785	−	0.318911i	$-0.103317\pi$
$139$	431794.	1.89557	0.947785	+	0.318911i	$0.103317\pi$
$140$	0	0
$141$	51672.0	0.218881
$142$	0	0
$143$	411840.i	1.68418i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 4802.00i	− 0.0183286i
$148$	0	0
$149$	−326814.	−1.20597	−0.602983	−	0.797754i	$-0.706021\pi$
$149$	−326814.	−1.20597	−0.602983	+	0.797754i	$0.706021\pi$
$150$	0	0
$151$	173480.	0.619166	0.309583	−	0.950872i	$-0.399811\pi$
$151$	173480.	0.619166	0.309583	+	0.950872i	$0.399811\pi$
$152$	0	0
$153$	299706.i	1.03506i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 54532.0i	− 0.176564i	−0.996096	−	0.0882820i	$-0.971862\pi$
$157$	− 54532.0i	− 0.176564i	0.996096	−	0.0882820i	$-0.0281377\pi$
$158$	0	0
$159$	2028.00	0.00636173
$160$	0	0
$161$	4704.00	0.0143022
$162$	0	0
$163$	− 104960.i	− 0.309425i	−0.987960	−	0.154712i	$-0.950555\pi$
$163$	− 104960.i	− 0.309425i	0.987960	−	0.154712i	$-0.0494451\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	160788.i	0.446131i	0.974803	+	0.223066i	$0.0716064\pi$
$167$	160788.i	0.446131i	−0.974803	+	0.223066i	$0.928394\pi$
$168$	0	0
$169$	44109.0	0.118798
$170$	0	0
$171$	22466.0	0.0587537
$172$	0	0
$173$	− 360564.i	− 0.915940i	−0.888968	−	0.457970i	$-0.848577\pi$
$173$	− 360564.i	− 0.915940i	0.888968	−	0.457970i	$-0.151423\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 2484.00i	− 0.00595962i
$178$	0	0
$179$	312732.	0.729524	0.364762	−	0.931101i	$-0.381150\pi$
$179$	312732.	0.729524	0.364762	+	0.931101i	$0.381150\pi$
$180$	0	0
$181$	−123820.	−0.280928	−0.140464	−	0.990086i	$-0.544859\pi$
$181$	−123820.	−0.280928	−0.140464	+	0.990086i	$0.544859\pi$
$182$	0	0
$183$	15184.0i	0.0335165i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 902880.i	− 1.88810i
$188$	0	0
$189$	−47236.0	−0.0961875
$190$	0	0
$191$	323448.	0.641536	0.320768	−	0.947158i	$-0.396059\pi$
$191$	323448.	0.641536	0.320768	+	0.947158i	$0.396059\pi$
$192$	0	0
$193$	619954.i	1.19803i	0.800739	+	0.599013i	$0.204440\pi$
$193$	619954.i	1.19803i	−0.800739	+	0.599013i	$0.795560\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 499362.i	− 0.916748i	−0.888759	−	0.458374i	$-0.848432\pi$
$197$	− 499362.i	− 0.916748i	0.888759	−	0.458374i	$-0.151568\pi$
$198$	0	0
$199$	785932.	1.40686	0.703432	−	0.710762i	$-0.251650\pi$
$199$	785932.	1.40686	0.703432	+	0.710762i	$0.251650\pi$
$200$	0	0
$201$	−82264.0	−0.143622
$202$	0	0
$203$	214326.i	0.365035i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 22944.0i	− 0.0372172i
$208$	0	0
$209$	−67680.0	−0.107175
$210$	0	0
$211$	1.06276e6	1.64335	0.821676	−	0.569955i	$-0.193039\pi$
$211$	1.06276e6	1.64335	0.821676	+	0.569955i	$0.193039\pi$
$212$	0	0
$213$	− 75264.0i	− 0.113668i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 305956.i	− 0.441072i
$218$	0	0
$219$	−26876.0	−0.0378664
$220$	0	0
$221$	717288.	0.987900
$222$	0	0
$223$	− 707720.i	− 0.953014i	−0.879171	−	0.476507i	$-0.841903\pi$
$223$	− 707720.i	− 0.953014i	0.879171	−	0.476507i	$-0.158097\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.04437e6i	− 1.34520i	−0.740005	−	0.672602i	$-0.765177\pi$
$227$	− 1.04437e6i	− 1.34520i	0.740005	−	0.672602i	$-0.234823\pi$
$228$	0	0
$229$	539716.	0.680106	0.340053	−	0.940406i	$-0.389555\pi$
$229$	539716.	0.680106	0.340053	+	0.940406i	$0.389555\pi$
$230$	0	0
$231$	70560.0	0.0870019
$232$	0	0
$233$	− 177114.i	− 0.213729i	−0.994274	−	0.106864i	$-0.965919\pi$
$233$	− 177114.i	− 0.213729i	0.994274	−	0.106864i	$-0.0340811\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 12496.0i	− 0.0144511i
$238$	0	0
$239$	655464.	0.742257	0.371128	−	0.928582i	$-0.378971\pi$
$239$	655464.	0.742257	0.371128	+	0.928582i	$0.378971\pi$
$240$	0	0
$241$	1.38709e6	1.53838	0.769189	−	0.639021i	$-0.220660\pi$
$241$	1.38709e6	1.53838	0.769189	+	0.639021i	$0.220660\pi$
$242$	0	0
$243$	346550.i	0.376487i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 53768.0i	− 0.0560766i
$248$	0	0
$249$	−50508.0	−0.0516252
$250$	0	0
$251$	1.88811e6	1.89166	0.945830	−	0.324663i	$-0.105251\pi$
$251$	1.88811e6	1.89166	0.945830	+	0.324663i	$0.105251\pi$
$252$	0	0
$253$	69120.0i	0.0678895i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	346194.i	0.326954i	0.986547	+	0.163477i	$0.0522710\pi$
$257$	346194.i	0.326954i	−0.986547	+	0.163477i	$0.947729\pi$
$258$	0	0
$259$	529102.	0.490106
$260$	0	0
$261$	1.04539e6	0.949895
$262$	0	0
$263$	929088.i	0.828262i	0.910217	+	0.414131i	$0.135914\pi$
$263$	929088.i	0.828262i	−0.910217	+	0.414131i	$0.864086\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	90252.0i	0.0774780i
$268$	0	0
$269$	−382068.	−0.321929	−0.160964	−	0.986960i	$-0.551460\pi$
$269$	−382068.	−0.321929	−0.160964	+	0.986960i	$0.551460\pi$
$270$	0	0
$271$	−1.58056e6	−1.30734	−0.653669	−	0.756781i	$-0.726771\pi$
$271$	−1.58056e6	−1.30734	−0.653669	+	0.756781i	$0.726771\pi$
$272$	0	0
$273$	56056.0i	0.0455214i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1.36911e6i	− 1.07211i	−0.844182	−	0.536056i	$-0.819914\pi$
$277$	− 1.36911e6i	− 1.07211i	0.844182	−	0.536056i	$-0.180086\pi$
$278$	0	0
$279$	−1.49232e6	−1.14776
$280$	0	0
$281$	−394854.	−0.298312	−0.149156	−	0.988814i	$-0.547656\pi$
$281$	−394854.	−0.298312	−0.149156	+	0.988814i	$0.547656\pi$
$282$	0	0
$283$	− 673034.i	− 0.499541i	−0.968305	−	0.249770i	$-0.919645\pi$
$283$	− 673034.i	− 0.499541i	0.968305	−	0.249770i	$-0.0803551\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	588294.i	0.421589i
$288$	0	0
$289$	−152659.	−0.107517
$290$	0	0
$291$	−214444.	−0.148450
$292$	0	0
$293$	− 1.83468e6i	− 1.24851i	−0.781222	−	0.624254i	$-0.785403\pi$
$293$	− 1.83468e6i	− 1.24851i	0.781222	−	0.624254i	$-0.214597\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 694080.i	− 0.456582i
$298$	0	0
$299$	−54912.0	−0.0355213
$300$	0	0
$301$	−448840.	−0.285545
$302$	0	0
$303$	94272.0i	0.0589897i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.51056e6i	− 0.914727i	−0.889280	−	0.457363i	$-0.848794\pi$
$307$	− 1.51056e6i	− 0.914727i	0.889280	−	0.457363i	$-0.151206\pi$
$308$	0	0
$309$	244408.	0.145619
$310$	0	0
$311$	−1.87529e6	−1.09943	−0.549714	−	0.835353i	$-0.685263\pi$
$311$	−1.87529e6	−1.09943	−0.549714	+	0.835353i	$0.685263\pi$
$312$	0	0
$313$	1.51076e6i	0.871636i	0.900035	+	0.435818i	$0.143541\pi$
$313$	1.51076e6i	0.871636i	−0.900035	+	0.435818i	$0.856459\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.02709e6i	1.13299i	0.824065	+	0.566495i	$0.191701\pi$
$317$	2.02709e6i	1.13299i	−0.824065	+	0.566495i	$0.808299\pi$
$318$	0	0
$319$	−3.14928e6	−1.73274
$320$	0	0
$321$	259272.	0.140441
$322$	0	0
$323$	117876.i	0.0628665i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	441116.i	0.228131i
$328$	0	0
$329$	1.26596e6	0.644810
$330$	0	0
$331$	1.54009e6	0.772637	0.386319	−	0.922365i	$-0.373747\pi$
$331$	1.54009e6	0.772637	0.386319	+	0.922365i	$0.373747\pi$
$332$	0	0
$333$	− 2.58072e6i	− 1.27535i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.01166e6i	0.485245i	0.970121	+	0.242622i	$0.0780076\pi$
$337$	1.01166e6i	0.485245i	−0.970121	+	0.242622i	$0.921992\pi$
$338$	0	0
$339$	341388.	0.161343
$340$	0	0
$341$	4.49568e6	2.09368
$342$	0	0
$343$	− 117649.i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.15748e6i	− 0.961885i	−0.876752	−	0.480942i	$-0.840295\pi$
$347$	− 2.15748e6i	− 0.961885i	0.876752	−	0.480942i	$-0.159705\pi$
$348$	0	0
$349$	1.15798e6	0.508906	0.254453	−	0.967085i	$-0.418105\pi$
$349$	1.15798e6	0.508906	0.254453	+	0.967085i	$0.418105\pi$
$350$	0	0
$351$	551408.	0.238894
$352$	0	0
$353$	3.17566e6i	1.35643i	0.734863	+	0.678215i	$0.237246\pi$
$353$	3.17566e6i	1.35643i	−0.734863	+	0.678215i	$0.762754\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 122892.i	− 0.0510332i
$358$	0	0
$359$	74616.0	0.0305560	0.0152780	−	0.999883i	$-0.495137\pi$
$359$	74616.0	0.0305560	0.0152780	+	0.999883i	$0.495137\pi$
$360$	0	0
$361$	−2.46726e6	−0.996431
$362$	0	0
$363$	714698.i	0.284679i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1.79807e6i	− 0.696854i	−0.937336	−	0.348427i	$-0.886716\pi$
$367$	− 1.79807e6i	− 0.696854i	0.937336	−	0.348427i	$-0.113284\pi$
$368$	0	0
$369$	2.86943e6	1.09706
$370$	0	0
$371$	49686.0	0.0187413
$372$	0	0
$373$	− 2.20461e6i	− 0.820463i	−0.911981	−	0.410231i	$-0.865448\pi$
$373$	− 2.20461e6i	− 0.820463i	0.911981	−	0.410231i	$-0.134552\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.50193e6i	− 0.906612i
$378$	0	0
$379$	177568.	0.0634990	0.0317495	−	0.999496i	$-0.489892\pi$
$379$	177568.	0.0634990	0.0317495	+	0.999496i	$0.489892\pi$
$380$	0	0
$381$	499616.	0.176329
$382$	0	0
$383$	2.87468e6i	1.00137i	0.865630	+	0.500683i	$0.166918\pi$
$383$	2.87468e6i	1.00137i	−0.865630	+	0.500683i	$0.833082\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.18924e6i	0.743046i
$388$	0	0
$389$	4.79965e6	1.60818	0.804091	−	0.594506i	$-0.202652\pi$
$389$	4.79965e6	1.60818	0.804091	+	0.594506i	$0.202652\pi$
$390$	0	0
$391$	120384.	0.0398223
$392$	0	0
$393$	24420.0i	0.00797562i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 2.81643e6i	− 0.896855i	−0.893819	−	0.448428i	$-0.851984\pi$
$397$	− 2.81643e6i	− 0.896855i	0.893819	−	0.448428i	$-0.148016\pi$
$398$	0	0
$399$	−9212.00	−0.00289682
$400$	0	0
$401$	−2.83797e6	−0.881347	−0.440673	−	0.897667i	$-0.645260\pi$
$401$	−2.83797e6	−0.881347	−0.440673	+	0.897667i	$0.645260\pi$
$402$	0	0
$403$	3.57157e6i	1.09546i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.77456e6i	2.32643i
$408$	0	0
$409$	−154286.	−0.0456056	−0.0228028	−	0.999740i	$-0.507259\pi$
$409$	−154286.	−0.0456056	−0.0228028	+	0.999740i	$0.507259\pi$
$410$	0	0
$411$	27804.0	0.00811900
$412$	0	0
$413$	− 60858.0i	− 0.0175567i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	863588.i	0.243202i
$418$	0	0
$419$	−3.72865e6	−1.03757	−0.518783	−	0.854906i	$-0.673615\pi$
$419$	−3.72865e6	−1.03757	−0.518783	+	0.854906i	$0.673615\pi$
$420$	0	0
$421$	−2.32623e6	−0.639658	−0.319829	−	0.947475i	$-0.603626\pi$
$421$	−2.32623e6	−0.639658	−0.319829	+	0.947475i	$0.603626\pi$
$422$	0	0
$423$	− 6.17480e6i	− 1.67792i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	372008.i	0.0987376i
$428$	0	0
$429$	−823680.	−0.216080
$430$	0	0
$431$	−2.61482e6	−0.678031	−0.339015	−	0.940781i	$-0.610094\pi$
$431$	−2.61482e6	−0.678031	−0.339015	+	0.940781i	$0.610094\pi$
$432$	0	0
$433$	1.19226e6i	0.305598i	0.988257	+	0.152799i	$0.0488287\pi$
$433$	1.19226e6i	0.305598i	−0.988257	+	0.152799i	$0.951171\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 9024.00i	− 0.00226045i
$438$	0	0
$439$	−1.05793e6	−0.261996	−0.130998	−	0.991383i	$-0.541818\pi$
$439$	−1.05793e6	−0.261996	−0.130998	+	0.991383i	$0.541818\pi$
$440$	0	0
$441$	−573839.	−0.140506
$442$	0	0
$443$	− 4.12756e6i	− 0.999272i	−0.866235	−	0.499636i	$-0.833467\pi$
$443$	− 4.12756e6i	− 0.999272i	0.866235	−	0.499636i	$-0.166533\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 653628.i	− 0.154725i
$448$	0	0
$449$	−3.75823e6	−0.879766	−0.439883	−	0.898055i	$-0.644980\pi$
$449$	−3.75823e6	−0.879766	−0.439883	+	0.898055i	$0.644980\pi$
$450$	0	0
$451$	−8.64432e6	−2.00120
$452$	0	0
$453$	346960.i	0.0794390i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 451114.i	− 0.101041i	−0.998723	−	0.0505203i	$-0.983912\pi$
$457$	− 451114.i	− 0.101041i	0.998723	−	0.0505203i	$-0.0160880\pi$
$458$	0	0
$459$	−1.20886e6	−0.267820
$460$	0	0
$461$	−1.95186e6	−0.427756	−0.213878	−	0.976860i	$-0.568610\pi$
$461$	−1.95186e6	−0.427756	−0.213878	+	0.976860i	$0.568610\pi$
$462$	0	0
$463$	7.20218e6i	1.56139i	0.624913	+	0.780695i	$0.285135\pi$
$463$	7.20218e6i	1.56139i	−0.624913	+	0.780695i	$0.714865\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.17801e6i	1.52304i	0.648140	+	0.761521i	$0.275547\pi$
$467$	7.17801e6i	1.52304i	−0.648140	+	0.761521i	$0.724453\pi$
$468$	0	0
$469$	−2.01547e6	−0.423101
$470$	0	0
$471$	109064.	0.0226532
$472$	0	0
$473$	− 6.59520e6i	− 1.35542i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 242346.i	− 0.0487686i
$478$	0	0
$479$	−6.17632e6	−1.22996	−0.614980	−	0.788543i	$-0.710836\pi$
$479$	−6.17632e6	−1.22996	−0.614980	+	0.788543i	$0.710836\pi$
$480$	0	0
$481$	−6.17646e6	−1.21724
$482$	0	0
$483$	9408.00i	0.00183497i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7.59330e6i	1.45080i	0.688327	+	0.725401i	$0.258345\pi$
$487$	7.59330e6i	1.45080i	−0.688327	+	0.725401i	$0.741655\pi$
$488$	0	0
$489$	209920.	0.0396992
$490$	0	0
$491$	−1.51878e6	−0.284309	−0.142155	−	0.989844i	$-0.545403\pi$
$491$	−1.51878e6	−0.284309	−0.142155	+	0.989844i	$0.545403\pi$
$492$	0	0
$493$	5.48500e6i	1.01639i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.84397e6i	− 0.334859i
$498$	0	0
$499$	−1.47576e6	−0.265316	−0.132658	−	0.991162i	$-0.542351\pi$
$499$	−1.47576e6	−0.265316	−0.132658	+	0.991162i	$0.542351\pi$
$500$	0	0
$501$	−321576.	−0.0572386
$502$	0	0
$503$	1.31309e6i	0.231406i	0.993284	+	0.115703i	$0.0369120\pi$
$503$	1.31309e6i	0.231406i	−0.993284	+	0.115703i	$0.963088\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	88218.0i	0.0152418i
$508$	0	0
$509$	−4.40932e6	−0.754357	−0.377178	−	0.926141i	$-0.623106\pi$
$509$	−4.40932e6	−0.754357	−0.377178	+	0.926141i	$0.623106\pi$
$510$	0	0
$511$	−658462.	−0.111552
$512$	0	0
$513$	90616.0i	0.0152024i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.86019e7i	3.06078i
$518$	0	0
$519$	721128.	0.117515
$520$	0	0
$521$	2.97629e6	0.480376	0.240188	−	0.970726i	$-0.422791\pi$
$521$	2.97629e6	0.480376	0.240188	+	0.970726i	$0.422791\pi$
$522$	0	0
$523$	− 6.34627e6i	− 1.01453i	−0.861790	−	0.507265i	$-0.830657\pi$
$523$	− 6.34627e6i	− 1.01453i	0.861790	−	0.507265i	$-0.169343\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 7.82998e6i	− 1.22810i
$528$	0	0
$529$	6.42713e6	0.998568
$530$	0	0
$531$	−296838.	−0.0456860
$532$	0	0
$533$	− 6.86743e6i	− 1.04707i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	625464.i	0.0935980i
$538$	0	0
$539$	1.72872e6	0.256302
$540$	0	0
$541$	−1.36667e6	−0.200756	−0.100378	−	0.994949i	$-0.532005\pi$
$541$	−1.36667e6	−0.200756	−0.100378	+	0.994949i	$0.532005\pi$
$542$	0	0
$543$	− 247640.i	− 0.0360430i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 9.55818e6i	− 1.36586i	−0.730483	−	0.682931i	$-0.760705\pi$
$547$	− 9.55818e6i	− 1.36586i	0.730483	−	0.682931i	$-0.239295\pi$
$548$	0	0
$549$	1.81449e6	0.256935
$550$	0	0
$551$	411156.	0.0576936
$552$	0	0
$553$	− 306152.i	− 0.0425720i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.94287e6i	0.948202i	0.880470	+	0.474101i	$0.157227\pi$
$557$	6.94287e6i	0.948202i	−0.880470	+	0.474101i	$0.842773\pi$
$558$	0	0
$559$	5.23952e6	0.709189
$560$	0	0
$561$	1.80576e6	0.242244
$562$	0	0
$563$	− 5.24662e6i	− 0.697604i	−0.937196	−	0.348802i	$-0.886589\pi$
$563$	− 5.24662e6i	− 0.697604i	0.937196	−	0.348802i	$-0.113411\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.75130e6i	0.359402i
$568$	0	0
$569$	−3.46551e6	−0.448731	−0.224366	−	0.974505i	$-0.572031\pi$
$569$	−3.46551e6	−0.448731	−0.224366	+	0.974505i	$0.572031\pi$
$570$	0	0
$571$	4.90069e6	0.629023	0.314512	−	0.949254i	$-0.398159\pi$
$571$	4.90069e6	0.629023	0.314512	+	0.949254i	$0.398159\pi$
$572$	0	0
$573$	646896.i	0.0823091i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.28346e6i	0.285531i	0.989757	+	0.142766i	$0.0455995\pi$
$577$	2.28346e6i	0.285531i	−0.989757	+	0.142766i	$0.954401\pi$
$578$	0	0
$579$	−1.23991e6	−0.153707
$580$	0	0
$581$	−1.23745e6	−0.152085
$582$	0	0
$583$	730080.i	0.0889609i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.03157e7i	1.23568i	0.786305	+	0.617838i	$0.211991\pi$
$587$	1.03157e7i	1.23568i	−0.786305	+	0.617838i	$0.788009\pi$
$588$	0	0
$589$	−586936.	−0.0697112
$590$	0	0
$591$	998724.	0.117619
$592$	0	0
$593$	− 3.52838e6i	− 0.412039i	−0.978548	−	0.206020i	$-0.933949\pi$
$593$	− 3.52838e6i	− 0.412039i	0.978548	−	0.206020i	$-0.0660510\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.57186e6i	0.180501i
$598$	0	0
$599$	−2.92260e6	−0.332815	−0.166407	−	0.986057i	$-0.553217\pi$
$599$	−2.92260e6	−0.332815	−0.166407	+	0.986057i	$0.553217\pi$
$600$	0	0
$601$	−1.17567e7	−1.32770	−0.663849	−	0.747866i	$-0.731078\pi$
$601$	−1.17567e7	−1.32770	−0.663849	+	0.747866i	$0.731078\pi$
$602$	0	0
$603$	9.83055e6i	1.10099i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 4.71491e6i	− 0.519400i	−0.965689	−	0.259700i	$-0.916376\pi$
$607$	− 4.71491e6i	− 0.519400i	0.965689	−	0.259700i	$-0.0836237\pi$
$608$	0	0
$609$	−428652.	−0.0468340
$610$	0	0
$611$	−1.47782e7	−1.60147
$612$	0	0
$613$	− 213842.i	− 0.0229849i	−0.999934	−	0.0114924i	$-0.996342\pi$
$613$	− 213842.i	− 0.0229849i	0.999934	−	0.0114924i	$-0.00365823\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 336666.i	− 0.0356030i	−0.999842	−	0.0178015i	$-0.994333\pi$
$617$	− 336666.i	− 0.0356030i	0.999842	−	0.0178015i	$-0.00566669\pi$
$618$	0	0
$619$	−1.42655e7	−1.49645	−0.748223	−	0.663447i	$-0.769093\pi$
$619$	−1.42655e7	−1.49645	−0.748223	+	0.663447i	$0.769093\pi$
$620$	0	0
$621$	92544.0	0.00962984
$622$	0	0
$623$	2.21117e6i	0.228246i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 135360.i	− 0.0137506i
$628$	0	0
$629$	1.35407e7	1.36463
$630$	0	0
$631$	−6.59637e6	−0.659525	−0.329763	−	0.944064i	$-0.606969\pi$
$631$	−6.59637e6	−0.659525	−0.329763	+	0.944064i	$0.606969\pi$
$632$	0	0
$633$	2.12553e6i	0.210842i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.37337e6i	0.134103i
$638$	0	0
$639$	−8.99405e6	−0.871371
$640$	0	0
$641$	−1.02490e7	−0.985225	−0.492613	−	0.870249i	$-0.663958\pi$
$641$	−1.02490e7	−0.985225	−0.492613	+	0.870249i	$0.663958\pi$
$642$	0	0
$643$	4.16543e6i	0.397312i	0.980069	+	0.198656i	$0.0636577\pi$
$643$	4.16543e6i	0.397312i	−0.980069	+	0.198656i	$0.936342\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 3.35051e6i	− 0.314666i	−0.987546	−	0.157333i	$-0.949710\pi$
$647$	− 3.35051e6i	− 0.314666i	0.987546	−	0.157333i	$-0.0502896\pi$
$648$	0	0
$649$	894240.	0.0833379
$650$	0	0
$651$	611912.	0.0565896
$652$	0	0
$653$	9.05408e6i	0.830924i	0.909611	+	0.415462i	$0.136380\pi$
$653$	9.05408e6i	0.830924i	−0.909611	+	0.415462i	$0.863620\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.21168e6i	0.290281i
$658$	0	0
$659$	−6.45382e6	−0.578899	−0.289450	−	0.957193i	$-0.593472\pi$
$659$	−6.45382e6	−0.578899	−0.289450	+	0.957193i	$0.593472\pi$
$660$	0	0
$661$	1.43167e7	1.27450	0.637250	−	0.770657i	$-0.280072\pi$
$661$	1.43167e7	1.27450	0.637250	+	0.770657i	$0.280072\pi$
$662$	0	0
$663$	1.43458e6i	0.126748i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 419904.i	− 0.0365456i
$668$	0	0
$669$	1.41544e6	0.122272
$670$	0	0
$671$	−5.46624e6	−0.468686
$672$	0	0
$673$	− 2.27250e7i	− 1.93404i	−0.254701	−	0.967020i	$-0.581977\pi$
$673$	− 2.27250e7i	− 1.93404i	0.254701	−	0.967020i	$-0.418023\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 8.53249e6i	− 0.715491i	−0.933819	−	0.357746i	$-0.883546\pi$
$677$	− 8.53249e6i	− 0.715491i	0.933819	−	0.357746i	$-0.116454\pi$
$678$	0	0
$679$	−5.25388e6	−0.437326
$680$	0	0
$681$	2.08873e6	0.172590
$682$	0	0
$683$	2.24921e7i	1.84492i	0.386090	+	0.922461i	$0.373825\pi$
$683$	2.24921e7i	1.84492i	−0.386090	+	0.922461i	$0.626175\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.07943e6i	0.0872576i
$688$	0	0
$689$	−580008.	−0.0465464
$690$	0	0
$691$	−1.26894e7	−1.01099	−0.505495	−	0.862830i	$-0.668690\pi$
$691$	−1.26894e7	−1.01099	−0.505495	+	0.862830i	$0.668690\pi$
$692$	0	0
$693$	− 8.43192e6i	− 0.666950i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.50555e7i	1.17385i
$698$	0	0
$699$	354228.	0.0274214
$700$	0	0
$701$	−5.13939e6	−0.395018	−0.197509	−	0.980301i	$-0.563285\pi$
$701$	−5.13939e6	−0.395018	−0.197509	+	0.980301i	$0.563285\pi$
$702$	0	0
$703$	− 1.01501e6i	− 0.0774610i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.30966e6i	0.173780i
$708$	0	0
$709$	1.16065e7	0.867132	0.433566	−	0.901122i	$-0.357255\pi$
$709$	1.16065e7	0.867132	0.433566	+	0.901122i	$0.357255\pi$
$710$	0	0
$711$	−1.49327e6	−0.110781
$712$	0	0
$713$	599424.i	0.0441581i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.31093e6i	0.0952316i
$718$	0	0
$719$	−1.50998e7	−1.08930	−0.544650	−	0.838663i	$-0.683338\pi$
$719$	−1.50998e7	−1.08930	−0.544650	+	0.838663i	$0.683338\pi$
$720$	0	0
$721$	5.98800e6	0.428986
$722$	0	0
$723$	2.77419e6i	0.197374i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 2.32536e7i	− 1.63175i	−0.578229	−	0.815874i	$-0.696256\pi$
$727$	− 2.32536e7i	− 1.63175i	0.578229	−	0.815874i	$-0.303744\pi$
$728$	0	0
$729$	1.29511e7	0.902585
$730$	0	0
$731$	−1.14866e7	−0.795059
$732$	0	0
$733$	− 2.37814e7i	− 1.63485i	−0.576038	−	0.817423i	$-0.695402\pi$
$733$	− 2.37814e7i	− 1.63485i	0.576038	−	0.817423i	$-0.304598\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2.96150e7i	− 2.00837i
$738$	0	0
$739$	2.51392e6	0.169333	0.0846663	−	0.996409i	$-0.473018\pi$
$739$	2.51392e6	0.169333	0.0846663	+	0.996409i	$0.473018\pi$
$740$	0	0
$741$	107536.	0.00719463
$742$	0	0
$743$	2.22646e7i	1.47959i	0.672830	+	0.739797i	$0.265078\pi$
$743$	2.22646e7i	1.47959i	−0.672830	+	0.739797i	$0.734922\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.03571e6i	0.395755i
$748$	0	0
$749$	6.35216e6	0.413730
$750$	0	0
$751$	2.30108e7	1.48878	0.744392	−	0.667743i	$-0.232740\pi$
$751$	2.30108e7	1.48878	0.744392	+	0.667743i	$0.232740\pi$
$752$	0	0
$753$	3.77622e6i	0.242700i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.59335e7i	1.01058i	0.862950	+	0.505290i	$0.168614\pi$
$757$	1.59335e7i	1.01058i	−0.862950	+	0.505290i	$0.831386\pi$
$758$	0	0
$759$	−138240.	−0.00871022
$760$	0	0
$761$	1.68629e7	1.05553	0.527764	−	0.849391i	$-0.323031\pi$
$761$	1.68629e7	1.05553	0.527764	+	0.849391i	$0.323031\pi$
$762$	0	0
$763$	1.08073e7i	0.672059i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	710424.i	0.0436043i
$768$	0	0
$769$	2.75402e7	1.67939	0.839694	−	0.543060i	$-0.182735\pi$
$769$	2.75402e7	1.67939	0.839694	+	0.543060i	$0.182735\pi$
$770$	0	0
$771$	−692388.	−0.0419482
$772$	0	0
$773$	− 2.26820e7i	− 1.36532i	−0.730738	−	0.682658i	$-0.760824\pi$
$773$	− 2.26820e7i	− 1.36532i	0.730738	−	0.682658i	$-0.239176\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.05820e6i	0.0628806i
$778$	0	0
$779$	1.12856e6	0.0666320
$780$	0	0
$781$	2.70950e7	1.58951
$782$	0	0
$783$	4.21654e6i	0.245783i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 2.34266e7i	− 1.34826i	−0.738614	−	0.674129i	$-0.764519\pi$
$787$	− 2.34266e7i	− 1.34826i	0.738614	−	0.674129i	$-0.235481\pi$
$788$	0	0
$789$	−1.85818e6	−0.106266
$790$	0	0
$791$	8.36401e6	0.475306
$792$	0	0
$793$	− 4.34262e6i	− 0.245228i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.82051e7i	− 1.01519i	−0.861596	−	0.507594i	$-0.830535\pi$
$797$	− 1.82051e7i	− 1.01519i	0.861596	−	0.507594i	$-0.169465\pi$
$798$	0	0
$799$	3.23983e7	1.79538
$800$	0	0
$801$	1.07851e7	0.593941
$802$	0	0
$803$	− 9.67536e6i	− 0.529515i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 764136.i	− 0.0413035i
$808$	0	0
$809$	−1.47411e7	−0.791878	−0.395939	−	0.918277i	$-0.629581\pi$
$809$	−1.47411e7	−0.791878	−0.395939	+	0.918277i	$0.629581\pi$
$810$	0	0
$811$	1.69629e7	0.905625	0.452812	−	0.891606i	$-0.350421\pi$
$811$	1.69629e7	0.905625	0.452812	+	0.891606i	$0.350421\pi$
$812$	0	0
$813$	− 3.16112e6i	− 0.167731i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	861040.i	0.0451303i
$818$	0	0
$819$	6.69869e6	0.348964
$820$	0	0
$821$	8.03929e6	0.416255	0.208128	−	0.978102i	$-0.433263\pi$
$821$	8.03929e6	0.416255	0.208128	+	0.978102i	$0.433263\pi$
$822$	0	0
$823$	− 386648.i	− 0.0198983i	−0.999951	−	0.00994915i	$-0.996833\pi$
$823$	− 386648.i	− 0.0198983i	0.999951	−	0.00994915i	$-0.00316697\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.55021e7i	1.80505i	0.430635	+	0.902526i	$0.358290\pi$
$827$	3.55021e7i	1.80505i	−0.430635	+	0.902526i	$0.641710\pi$
$828$	0	0
$829$	−2.48814e7	−1.25745	−0.628723	−	0.777630i	$-0.716422\pi$
$829$	−2.48814e7	−1.25745	−0.628723	+	0.777630i	$0.716422\pi$
$830$	0	0
$831$	2.73823e6	0.137552
$832$	0	0
$833$	− 3.01085e6i	− 0.150341i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 6.01922e6i	− 0.296979i
$838$	0	0
$839$	−3.41458e7	−1.67468	−0.837340	−	0.546682i	$-0.815891\pi$
$839$	−3.41458e7	−1.67468	−0.837340	+	0.546682i	$0.815891\pi$
$840$	0	0
$841$	−1.37927e6	−0.0672450
$842$	0	0
$843$	− 789708.i	− 0.0382734i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.75101e7i	0.838649i
$848$	0	0
$849$	1.34607e6	0.0640911
$850$	0	0
$851$	−1.03661e6	−0.0490671
$852$	0	0
$853$	− 2.50701e7i	− 1.17973i	−0.807502	−	0.589865i	$-0.799181\pi$
$853$	− 2.50701e7i	− 1.17973i	0.807502	−	0.589865i	$-0.200819\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1.81938e7i	− 0.846199i	−0.906083	−	0.423099i	$-0.860942\pi$
$857$	− 1.81938e7i	− 0.846199i	0.906083	−	0.423099i	$-0.139058\pi$
$858$	0	0
$859$	−6.91797e6	−0.319886	−0.159943	−	0.987126i	$-0.551131\pi$
$859$	−6.91797e6	−0.319886	−0.159943	+	0.987126i	$0.551131\pi$
$860$	0	0
$861$	−1.17659e6	−0.0540899
$862$	0	0
$863$	2.78069e7i	1.27094i	0.772125	+	0.635471i	$0.219194\pi$
$863$	2.78069e7i	1.27094i	−0.772125	+	0.635471i	$0.780806\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 305318.i	− 0.0137945i
$868$	0	0
$869$	4.49856e6	0.202080
$870$	0	0
$871$	2.35275e7	1.05083
$872$	0	0
$873$	2.56261e7i	1.13801i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.79587e6i	0.166653i	0.996522	+	0.0833263i	$0.0265544\pi$
$877$	3.79587e6i	0.166653i	−0.996522	+	0.0833263i	$0.973446\pi$
$878$	0	0
$879$	3.66936e6	0.160184
$880$	0	0
$881$	2.48904e7	1.08042	0.540210	−	0.841530i	$-0.318345\pi$
$881$	2.48904e7	1.08042	0.540210	+	0.841530i	$0.318345\pi$
$882$	0	0
$883$	3.13568e6i	0.135341i	0.997708	+	0.0676705i	$0.0215567\pi$
$883$	3.13568e6i	0.135341i	−0.997708	+	0.0676705i	$0.978443\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 2.02437e7i	− 0.863933i	−0.901890	−	0.431966i	$-0.857820\pi$
$887$	− 2.02437e7i	− 0.863933i	0.901890	−	0.431966i	$-0.142180\pi$
$888$	0	0
$889$	1.22406e7	0.519455
$890$	0	0
$891$	−4.04273e7	−1.70600
$892$	0	0
$893$	− 2.42858e6i	− 0.101912i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 109824.i	− 0.00455739i
$898$	0	0
$899$	−2.73113e7	−1.12705
$900$	0	0
$901$	1.27156e6	0.0521823
$902$	0	0
$903$	− 897680.i	− 0.0366355i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 6.86324e6i	− 0.277020i	−0.990361	−	0.138510i	$-0.955769\pi$
$907$	− 6.86324e6i	− 0.277020i	0.990361	−	0.138510i	$-0.0442313\pi$
$908$	0	0
$909$	1.12655e7	0.452211
$910$	0	0
$911$	−9.40661e6	−0.375523	−0.187762	−	0.982215i	$-0.560123\pi$
$911$	−9.40661e6	−0.375523	−0.187762	+	0.982215i	$0.560123\pi$
$912$	0	0
$913$	− 1.81829e7i	− 0.721914i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	598290.i	0.0234957i
$918$	0	0
$919$	2.10280e7	0.821313	0.410656	−	0.911790i	$-0.365300\pi$
$919$	2.10280e7	0.821313	0.410656	+	0.911790i	$0.365300\pi$
$920$	0	0
$921$	3.02112e6	0.117360
$922$	0	0
$923$	2.15255e7i	0.831666i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 2.92068e7i	− 1.11631i
$928$	0	0
$929$	4.30928e7	1.63819	0.819096	−	0.573656i	$-0.194475\pi$
$929$	4.30928e7	1.63819	0.819096	+	0.573656i	$0.194475\pi$
$930$	0	0
$931$	−225694.	−0.00853387
$932$	0	0
$933$	− 3.75058e6i	− 0.141057i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 3.85862e6i	− 0.143576i	−0.997420	−	0.0717882i	$-0.977129\pi$
$937$	− 3.85862e6i	− 0.143576i	0.997420	−	0.0717882i	$-0.0228706\pi$
$938$	0	0
$939$	−3.02152e6	−0.111831
$940$	0	0
$941$	1.59601e7	0.587572	0.293786	−	0.955871i	$-0.405085\pi$
$941$	1.59601e7	0.587572	0.293786	+	0.955871i	$0.405085\pi$
$942$	0	0
$943$	− 1.15258e6i	− 0.0422076i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.66831e6i	− 0.169155i	−0.996417	−	0.0845775i	$-0.973046\pi$
$947$	− 4.66831e6i	− 0.169155i	0.996417	−	0.0845775i	$-0.0269541\pi$
$948$	0	0
$949$	7.68654e6	0.277054
$950$	0	0
$951$	−4.05419e6	−0.145363
$952$	0	0
$953$	3.43457e7i	1.22501i	0.790466	+	0.612505i	$0.209838\pi$
$953$	3.43457e7i	1.22501i	−0.790466	+	0.612505i	$0.790162\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 6.29856e6i	− 0.222311i
$958$	0	0
$959$	681198.	0.0239181
$960$	0	0
$961$	1.03584e7	0.361813
$962$	0	0
$963$	− 3.09830e7i	− 1.07661i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 2.28181e7i	− 0.784718i	−0.919812	−	0.392359i	$-0.871659\pi$
$967$	− 2.28181e7i	− 0.784718i	0.919812	−	0.392359i	$-0.128341\pi$
$968$	0	0
$969$	−235752.	−0.00806577
$970$	0	0
$971$	4.94042e7	1.68157	0.840786	−	0.541367i	$-0.182093\pi$
$971$	4.94042e7	1.68157	0.840786	+	0.541367i	$0.182093\pi$
$972$	0	0
$973$	2.11579e7i	0.716458i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7.17542e6i	0.240498i	0.992744	+	0.120249i	$0.0383692\pi$
$977$	7.17542e6i	0.240498i	−0.992744	+	0.120249i	$0.961631\pi$
$978$	0	0
$979$	−3.24907e7	−1.08343
$980$	0	0
$981$	5.27134e7	1.74883
$982$	0	0
$983$	4.22279e6i	0.139385i	0.997569	+	0.0696924i	$0.0222018\pi$
$983$	4.22279e6i	0.139385i	−0.997569	+	0.0696924i	$0.977798\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.53193e6i	0.0827291i
$988$	0	0
$989$	879360.	0.0285875
$990$	0	0
$991$	1.65645e7	0.535789	0.267895	−	0.963448i	$-0.413672\pi$
$991$	1.65645e7	0.535789	0.267895	+	0.963448i	$0.413672\pi$
$992$	0	0
$993$	3.08018e6i	0.0991294i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 4.40973e7i	− 1.40499i	−0.711687	−	0.702496i	$-0.752069\pi$
$997$	− 4.40973e7i	− 1.40499i	0.711687	−	0.702496i	$-0.247931\pi$
$998$	0	0
$999$	1.04093e7	0.329994

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.6.e.d.449.2		2
5.2	odd	4		700.6.a.d.1.1		1
5.3	odd	4		28.6.a.a.1.1	✓	1
5.4	even	2	inner	700.6.e.d.449.1		2
15.8	even	4		252.6.a.d.1.1		1
20.3	even	4		112.6.a.e.1.1		1
35.3	even	12		196.6.e.e.177.1		2
35.13	even	4		196.6.a.d.1.1		1
35.18	odd	12		196.6.e.f.177.1		2
35.23	odd	12		196.6.e.f.165.1		2
35.33	even	12		196.6.e.e.165.1		2
40.3	even	4		448.6.a.h.1.1		1
40.13	odd	4		448.6.a.i.1.1		1
60.23	odd	4		1008.6.a.bb.1.1		1
140.83	odd	4		784.6.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.6.a.a.1.1	✓	1	5.3	odd	4
112.6.a.e.1.1		1	20.3	even	4
196.6.a.d.1.1		1	35.13	even	4
196.6.e.e.165.1		2	35.33	even	12
196.6.e.e.177.1		2	35.3	even	12
196.6.e.f.165.1		2	35.23	odd	12
196.6.e.f.177.1		2	35.18	odd	12
252.6.a.d.1.1		1	15.8	even	4
448.6.a.h.1.1		1	40.3	even	4
448.6.a.i.1.1		1	40.13	odd	4
700.6.a.d.1.1		1	5.2	odd	4
700.6.e.d.449.1		2	5.4	even	2	inner
700.6.e.d.449.2		2	1.1	even	1	trivial
784.6.a.f.1.1		1	140.83	odd	4
1008.6.a.bb.1.1		1	60.23	odd	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)

\(f(q)\)	\(=\)	\(q+2.00000i q^{3} +49.0000i q^{7} +239.000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{3} +49.0000i q^{7} +239.000 q^{9} -720.000 q^{11} -572.000i q^{13} +1254.00i q^{17} +94.0000 q^{19} -98.0000 q^{21} -96.0000i q^{23} +964.000i q^{27} +4374.00 q^{29} -6244.00 q^{31} -1440.00i q^{33} -10798.0i q^{37} +1144.00 q^{39} +12006.0 q^{41} +9160.00i q^{43} -25836.0i q^{47} -2401.00 q^{49} -2508.00 q^{51} -1014.00i q^{53} +188.000i q^{57} -1242.00 q^{59} +7592.00 q^{61} +11711.0i q^{63} +41132.0i q^{67} +192.000 q^{69} -37632.0 q^{71} +13438.0i q^{73} -35280.0i q^{77} -6248.00 q^{79} +56149.0 q^{81} +25254.0i q^{83} +8748.00i q^{87} +45126.0 q^{89} +28028.0 q^{91} -12488.0i q^{93} +107222. i q^{97} -172080. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 478 q^{9}+O(q^{10}) \) 2 * q + 478 * q^9	\( 2 q + 478 q^{9} - 1440 q^{11} + 188 q^{19} - 196 q^{21} + 8748 q^{29} - 12488 q^{31} + 2288 q^{39} + 24012 q^{41} - 4802 q^{49} - 5016 q^{51} - 2484 q^{59} + 15184 q^{61} + 384 q^{69} - 75264 q^{71} - 12496 q^{79} + 112298 q^{81} + 90252 q^{89} + 56056 q^{91} - 344160 q^{99}+O(q^{100}) \) 2 * q + 478 * q^9 - 1440 * q^11 + 188 * q^19 - 196 * q^21 + 8748 * q^29 - 12488 * q^31 + 2288 * q^39 + 24012 * q^41 - 4802 * q^49 - 5016 * q^51 - 2484 * q^59 + 15184 * q^61 + 384 * q^69 - 75264 * q^71 - 12496 * q^79 + 112298 * q^81 + 90252 * q^89 + 56056 * q^91 - 344160 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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