LMFDB - Embedded newform 350.6.c.l.99.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,6,Mod(99,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("350.99");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$350 = 2 \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	350.c (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:	$56.1343369345$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{79})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 39x^{2} + 400$ x^4 - 39*x^2 + 400
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{4}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.1
Root			$-4.44410 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	350.99
Dual form			350.6.c.l.99.4

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{2} -13.7764i q^{3} -16.0000 q^{4} -55.1056 q^{6} +49.0000i q^{7} +64.0000i q^{8} +53.2111 q^{9} +50.3292 q^{11} +220.422i q^{12} +79.3542i q^{13} +196.000 q^{14} +256.000 q^{16} -145.342i q^{17} -212.844i q^{18} +993.925 q^{19} +675.043 q^{21} -201.317i q^{22} -508.229i q^{23} +881.689 q^{24} +317.417 q^{26} -4080.72i q^{27} -784.000i q^{28} +3481.28 q^{29} +1548.70 q^{31} -1024.00i q^{32} -693.354i q^{33} -581.367 q^{34} -851.378 q^{36} -12177.2i q^{37} -3975.70i q^{38} +1093.21 q^{39} +1724.46 q^{41} -2700.17i q^{42} -266.279i q^{43} -805.267 q^{44} -2032.92 q^{46} -19049.1i q^{47} -3526.76i q^{48} -2401.00 q^{49} -2002.28 q^{51} -1269.67i q^{52} -4646.35i q^{53} -16322.9 q^{54} -3136.00 q^{56} -13692.7i q^{57} -13925.1i q^{58} -20004.6 q^{59} +8544.05 q^{61} -6194.78i q^{62} +2607.34i q^{63} -4096.00 q^{64} -2773.42 q^{66} -14263.8i q^{67} +2325.47i q^{68} -7001.56 q^{69} +14674.3 q^{71} +3405.51i q^{72} +16595.8i q^{73} -48708.6 q^{74} -15902.8 q^{76} +2466.13i q^{77} -4372.86i q^{78} -4456.04 q^{79} -43287.3 q^{81} -6897.85i q^{82} +60013.4i q^{83} -10800.7 q^{84} -1065.12 q^{86} -47959.5i q^{87} +3221.07i q^{88} -54263.7 q^{89} -3888.35 q^{91} +8131.67i q^{92} -21335.4i q^{93} -76196.4 q^{94} -14107.0 q^{96} -114107. i q^{97} +9604.00i q^{98} +2678.07 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 64 q^{4} + 64 q^{6} - 356 q^{9} - 12 q^{11} + 784 q^{14} + 1024 q^{16} + 136 q^{19} - 784 q^{21} - 1024 q^{24} + 5536 q^{26} - 5700 q^{29} + 7688 q^{31} - 4032 q^{34} + 5696 q^{36} - 24496 q^{39}+ \cdots + 31404 q^{99}+O(q^{100})$ 4 * q - 64 * q^4 + 64 * q^6 - 356 * q^9 - 12 * q^11 + 784 * q^14 + 1024 * q^16 + 136 * q^19 - 784 * q^21 - 1024 * q^24 + 5536 * q^26 - 5700 * q^29 + 7688 * q^31 - 4032 * q^34 + 5696 * q^36 - 24496 * q^39 + 30576 * q^41 + 192 * q^44 + 13200 * q^46 - 9604 * q^49 + 11616 * q^51 - 30592 * q^54 - 12544 * q^56 - 76392 * q^59 - 31312 * q^61 - 16384 * q^64 - 15360 * q^66 - 108000 * q^69 - 80172 * q^71 - 90736 * q^74 - 2176 * q^76 - 70940 * q^79 - 210124 * q^81 + 12544 * q^84 + 6832 * q^86 + 38712 * q^89 - 67816 * q^91 - 120480 * q^94 + 16384 * q^96 + 31404 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$ .

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	− 4.00000i	− 0.707107i
$2$	− 4.00000i	− 0.707107i
$3$	− 13.7764i	− 0.883756i	−0.897075	−	0.441878i	$-0.854312\pi$
$3$	− 13.7764i	− 0.883756i	0.897075	−	0.441878i	$-0.145688\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	−55.1056	−0.624910
$7$	49.0000i	0.377964i
$7$	49.0000i	0.377964i
$8$	64.0000i	0.353553i
$9$	53.2111	0.218976
$10$	0	0
$11$	50.3292	0.125412	0.0627058	−	0.998032i	$-0.480027\pi$
$11$	50.3292	0.125412	0.0627058	+	0.998032i	$0.480027\pi$
$12$	220.422i	0.441878i
$13$	79.3542i	0.130230i	0.997878	+	0.0651150i	$0.0207414\pi$
$13$	79.3542i	0.130230i	−0.997878	+	0.0651150i	$0.979259\pi$
$14$	196.000	0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	− 145.342i	− 0.121974i	−0.998139	−	0.0609871i	$-0.980575\pi$
$17$	− 145.342i	− 0.121974i	0.998139	−	0.0609871i	$-0.0194248\pi$
$18$	− 212.844i	− 0.154839i
$19$	993.925	0.631640	0.315820	−	0.948819i	$-0.397720\pi$
$19$	993.925	0.631640	0.315820	+	0.948819i	$0.397720\pi$
$20$	0	0
$21$	675.043	0.334028
$22$	− 201.317i	− 0.0886795i
$23$	− 508.229i	− 0.200327i	−0.994971	−	0.100164i	$-0.968063\pi$
$23$	− 508.229i	− 0.200327i	0.994971	−	0.100164i	$-0.0319366\pi$
$24$	881.689	0.312455
$25$	0	0
$26$	317.417	0.0920866
$27$	− 4080.72i	− 1.07728i
$28$	− 784.000i	− 0.188982i
$29$	3481.28	0.768678	0.384339	−	0.923192i	$-0.374429\pi$
$29$	3481.28	0.768678	0.384339	+	0.923192i	$0.374429\pi$
$30$	0	0
$31$	1548.70	0.289442	0.144721	−	0.989472i	$-0.453772\pi$
$31$	1548.70	0.289442	0.144721	+	0.989472i	$0.453772\pi$
$32$	− 1024.00i	− 0.176777i
$33$	− 693.354i	− 0.110833i
$34$	−581.367	−0.0862488
$35$	0	0
$36$	−851.378	−0.109488
$37$	− 12177.2i	− 1.46232i	−0.682207	−	0.731159i	$-0.738980\pi$
$37$	− 12177.2i	− 1.46232i	0.682207	−	0.731159i	$-0.261020\pi$
$38$	− 3975.70i	− 0.446637i
$39$	1093.21	0.115092
$40$	0	0
$41$	1724.46	0.160212	0.0801058	−	0.996786i	$-0.474474\pi$
$41$	1724.46	0.160212	0.0801058	+	0.996786i	$0.474474\pi$
$42$	− 2700.17i	− 0.236194i
$43$	− 266.279i	− 0.0219617i	−0.999940	−	0.0109809i	$-0.996505\pi$
$43$	− 266.279i	− 0.0219617i	0.999940	−	0.0109809i	$-0.00349538\pi$
$44$	−805.267	−0.0627058
$45$	0	0
$46$	−2032.92	−0.141653
$47$	− 19049.1i	− 1.25785i	−0.777465	−	0.628926i	$-0.783495\pi$
$47$	− 19049.1i	− 1.25785i	0.777465	−	0.628926i	$-0.216505\pi$
$48$	− 3526.76i	− 0.220939i
$49$	−2401.00	−0.142857
$50$	0	0
$51$	−2002.28	−0.107795
$52$	− 1269.67i	− 0.0651150i
$53$	− 4646.35i	− 0.227207i	−0.993526	−	0.113604i	$-0.963761\pi$
$53$	− 4646.35i	− 0.227207i	0.993526	−	0.113604i	$-0.0362394\pi$
$54$	−16322.9	−0.761750
$55$	0	0
$56$	−3136.00	−0.133631
$57$	− 13692.7i	− 0.558215i
$58$	− 13925.1i	− 0.543537i
$59$	−20004.6	−0.748169	−0.374085	−	0.927395i	$-0.622043\pi$
$59$	−20004.6	−0.748169	−0.374085	+	0.927395i	$0.622043\pi$
$60$	0	0
$61$	8544.05	0.293995	0.146997	−	0.989137i	$-0.453039\pi$
$61$	8544.05	0.293995	0.146997	+	0.989137i	$0.453039\pi$
$62$	− 6194.78i	− 0.204667i
$63$	2607.34i	0.0827651i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	−2773.42	−0.0783710
$67$	− 14263.8i	− 0.388192i	−0.980983	−	0.194096i	$-0.937823\pi$
$67$	− 14263.8i	− 0.388192i	0.980983	−	0.194096i	$-0.0621774\pi$
$68$	2325.47i	0.0609871i
$69$	−7001.56	−0.177040
$70$	0	0
$71$	14674.3	0.345471	0.172735	−	0.984968i	$-0.444739\pi$
$71$	14674.3	0.345471	0.172735	+	0.984968i	$0.444739\pi$
$72$	3405.51i	0.0774196i
$73$	16595.8i	0.364495i	0.983253	+	0.182248i	$0.0583373\pi$
$73$	16595.8i	0.364495i	−0.983253	+	0.182248i	$0.941663\pi$
$74$	−48708.6	−1.03401
$75$	0	0
$76$	−15902.8	−0.315820
$77$	2466.13i	0.0474012i
$78$	− 4372.86i	− 0.0813820i
$79$	−4456.04	−0.0803306	−0.0401653	−	0.999193i	$-0.512788\pi$
$79$	−4456.04	−0.0803306	−0.0401653	+	0.999193i	$0.512788\pi$
$80$	0	0
$81$	−43287.3	−0.733074
$82$	− 6897.85i	− 0.113287i
$83$	60013.4i	0.956209i	0.878303	+	0.478105i	$0.158676\pi$
$83$	60013.4i	0.956209i	−0.878303	+	0.478105i	$0.841324\pi$
$84$	−10800.7	−0.167014
$85$	0	0
$86$	−1065.12	−0.0155293
$87$	− 47959.5i	− 0.679323i
$88$	3221.07i	0.0443397i
$89$	−54263.7	−0.726163	−0.363082	−	0.931757i	$-0.618275\pi$
$89$	−54263.7	−0.726163	−0.363082	+	0.931757i	$0.618275\pi$
$90$	0	0
$91$	−3888.35	−0.0492223
$92$	8131.67i	0.100164i
$93$	− 21335.4i	− 0.255796i
$94$	−76196.4	−0.889436
$95$	0	0
$96$	−14107.0	−0.156227
$97$	− 114107.i	− 1.23136i	−0.787997	−	0.615679i	$-0.788882\pi$
$97$	− 114107.i	− 1.23136i	0.787997	−	0.615679i	$-0.211118\pi$
$98$	9604.00i	0.101015i
$99$	2678.07	0.0274621
$100$	0	0
$101$	42085.4	0.410514	0.205257	−	0.978708i	$-0.434197\pi$
$101$	42085.4	0.410514	0.205257	+	0.978708i	$0.434197\pi$
$102$	8009.13i	0.0762228i
$103$	108760.i	1.01013i	0.863081	+	0.505065i	$0.168531\pi$
$103$	108760.i	1.01013i	−0.863081	+	0.505065i	$0.831469\pi$
$104$	−5078.67	−0.0460433
$105$	0	0
$106$	−18585.4	−0.160660
$107$	− 154379.i	− 1.30356i	−0.758410	−	0.651778i	$-0.774023\pi$
$107$	− 154379.i	− 1.30356i	0.758410	−	0.651778i	$-0.225977\pi$
$108$	65291.5i	0.538638i
$109$	−81080.6	−0.653658	−0.326829	−	0.945083i	$-0.605980\pi$
$109$	−81080.6	−0.653658	−0.326829	+	0.945083i	$0.605980\pi$
$110$	0	0
$111$	−167757.	−1.29233
$112$	12544.0i	0.0944911i
$113$	102542.i	0.755448i	0.925918	+	0.377724i	$0.123293\pi$
$113$	102542.i	0.755448i	−0.925918	+	0.377724i	$0.876707\pi$
$114$	−54770.8	−0.394718
$115$	0	0
$116$	−55700.5	−0.384339
$117$	4222.52i	0.0285172i
$118$	80018.4i	0.529035i
$119$	7121.74	0.0461019
$120$	0	0
$121$	−158518.	−0.984272
$122$	− 34176.2i	− 0.207886i
$123$	− 23756.9i	− 0.141588i
$124$	−24779.1	−0.144721
$125$	0	0
$126$	10429.4	0.0585237
$127$	− 208233.i	− 1.14562i	−0.819689	−	0.572809i	$-0.805854\pi$
$127$	− 208233.i	− 1.14562i	0.819689	−	0.572809i	$-0.194146\pi$
$128$	16384.0i	0.0883883i
$129$	−3668.37	−0.0194088
$130$	0	0
$131$	−96652.5	−0.492079	−0.246040	−	0.969260i	$-0.579129\pi$
$131$	−96652.5	−0.492079	−0.246040	+	0.969260i	$0.579129\pi$
$132$	11093.7i	0.0554167i
$133$	48702.3i	0.238737i
$134$	−57055.0	−0.274493
$135$	0	0
$136$	9301.87	0.0431244
$137$	18521.3i	0.0843084i	0.999111	+	0.0421542i	$0.0134221\pi$
$137$	18521.3i	0.0843084i	−0.999111	+	0.0421542i	$0.986578\pi$
$138$	28006.3i	0.125186i
$139$	188850.	0.829047	0.414524	−	0.910039i	$-0.363948\pi$
$139$	188850.	0.829047	0.414524	+	0.910039i	$0.363948\pi$
$140$	0	0
$141$	−262428.	−1.11163
$142$	− 58697.1i	− 0.244285i
$143$	3993.83i	0.0163324i
$144$	13622.0	0.0547439
$145$	0	0
$146$	66383.3	0.257737
$147$	33077.1i	0.126251i
$148$	194835.i	0.731159i
$149$	294645.	1.08726	0.543630	−	0.839325i	$-0.317050\pi$
$149$	294645.	1.08726	0.543630	+	0.839325i	$0.317050\pi$
$150$	0	0
$151$	−489919.	−1.74857	−0.874283	−	0.485416i	$-0.838668\pi$
$151$	−489919.	−1.74857	−0.874283	+	0.485416i	$0.838668\pi$
$152$	63611.2i	0.223318i
$153$	− 7733.79i	− 0.0267094i
$154$	9864.52	0.0335177
$155$	0	0
$156$	−17491.4	−0.0575458
$157$	− 323421.i	− 1.04717i	−0.851972	−	0.523587i	$-0.824594\pi$
$157$	− 323421.i	− 1.04717i	0.851972	−	0.523587i	$-0.175406\pi$
$158$	17824.2i	0.0568023i
$159$	−64009.9	−0.200796
$160$	0	0
$161$	24903.2	0.0757166
$162$	173149.i	0.518361i
$163$	− 493738.i	− 1.45555i	−0.685816	−	0.727775i	$-0.740554\pi$
$163$	− 493738.i	− 1.45555i	0.685816	−	0.727775i	$-0.259446\pi$
$164$	−27591.4	−0.0801058
$165$	0	0
$166$	240054.	0.676142
$167$	− 310107.i	− 0.860440i	−0.902724	−	0.430220i	$-0.858436\pi$
$167$	− 310107.i	− 0.860440i	0.902724	−	0.430220i	$-0.141564\pi$
$168$	43202.8i	0.118097i
$169$	364996.	0.983040
$170$	0	0
$171$	52887.9	0.138314
$172$	4260.47i	0.0109809i
$173$	− 513313.i	− 1.30397i	−0.758232	−	0.651984i	$-0.773937\pi$
$173$	− 513313.i	− 1.30397i	0.758232	−	0.651984i	$-0.226063\pi$
$174$	−191838.	−0.480354
$175$	0	0
$176$	12884.3	0.0313529
$177$	275591.i	0.661199i
$178$	217055.i	0.513475i
$179$	−263095.	−0.613734	−0.306867	−	0.951752i	$-0.599281\pi$
$179$	−263095.	−0.613734	−0.306867	+	0.951752i	$0.599281\pi$
$180$	0	0
$181$	−26410.8	−0.0599219	−0.0299610	−	0.999551i	$-0.509538\pi$
$181$	−26410.8	−0.0599219	−0.0299610	+	0.999551i	$0.509538\pi$
$182$	15553.4i	0.0348055i
$183$	− 117706.i	− 0.259819i
$184$	32526.7	0.0708264
$185$	0	0
$186$	−85341.7	−0.180875
$187$	− 7314.92i	− 0.0152970i
$188$	304786.i	0.628926i
$189$	199955.	0.407172
$190$	0	0
$191$	330442.	0.655408	0.327704	−	0.944780i	$-0.393725\pi$
$191$	330442.	0.655408	0.327704	+	0.944780i	$0.393725\pi$
$192$	56428.1i	0.110469i
$193$	− 661283.i	− 1.27789i	−0.769252	−	0.638946i	$-0.779371\pi$
$193$	− 661283.i	− 1.27789i	0.769252	−	0.638946i	$-0.220629\pi$
$194$	−456429.	−0.870701
$195$	0	0
$196$	38416.0	0.0714286
$197$	449944.i	0.826025i	0.910725	+	0.413012i	$0.135523\pi$
$197$	449944.i	0.826025i	−0.910725	+	0.413012i	$0.864477\pi$
$198$	− 10712.3i	− 0.0194187i
$199$	39456.2	0.0706289	0.0353145	−	0.999376i	$-0.488757\pi$
$199$	39456.2	0.0706289	0.0353145	+	0.999376i	$0.488757\pi$
$200$	0	0
$201$	−196503.	−0.343067
$202$	− 168342.i	− 0.290277i
$203$	170583.i	0.290533i
$204$	32036.5	0.0538977
$205$	0	0
$206$	435041.	0.714269
$207$	− 27043.4i	− 0.0438668i
$208$	20314.7i	0.0325575i
$209$	50023.4	0.0792150
$210$	0	0
$211$	559638.	0.865369	0.432685	−	0.901545i	$-0.357566\pi$
$211$	559638.	0.865369	0.432685	+	0.901545i	$0.357566\pi$
$212$	74341.6i	0.113604i
$213$	− 202159.i	− 0.305312i
$214$	−617517.	−0.921753
$215$	0	0
$216$	261166.	0.380875
$217$	75886.1i	0.109399i
$218$	324322.i	0.462206i
$219$	228631.	0.322125
$220$	0	0
$221$	11533.5	0.0158847
$222$	671029.i	0.913816i
$223$	723973.i	0.974900i	0.873151	+	0.487450i	$0.162073\pi$
$223$	723973.i	0.974900i	−0.873151	+	0.487450i	$0.837927\pi$
$224$	50176.0	0.0668153
$225$	0	0
$226$	410167.	0.534182
$227$	− 596271.i	− 0.768032i	−0.923327	−	0.384016i	$-0.874541\pi$
$227$	− 596271.i	− 0.768032i	0.923327	−	0.384016i	$-0.125459\pi$
$228$	219083.i	0.279108i
$229$	−91835.9	−0.115724	−0.0578620	−	0.998325i	$-0.518428\pi$
$229$	−91835.9	−0.115724	−0.0578620	+	0.998325i	$0.518428\pi$
$230$	0	0
$231$	33974.4	0.0418911
$232$	222802.i	0.271769i
$233$	− 285704.i	− 0.344767i	−0.985030	−	0.172384i	$-0.944853\pi$
$233$	− 285704.i	− 0.344767i	0.985030	−	0.172384i	$-0.0551469\pi$
$234$	16890.1	0.0201647
$235$	0	0
$236$	320074.	0.374085
$237$	61388.1i	0.0709926i
$238$	− 28487.0i	− 0.0325990i
$239$	110997.	0.125694	0.0628472	−	0.998023i	$-0.479982\pi$
$239$	110997.	0.125694	0.0628472	+	0.998023i	$0.479982\pi$
$240$	0	0
$241$	1.44863e6	1.60663	0.803315	−	0.595554i	$-0.203068\pi$
$241$	1.44863e6	1.60663	0.803315	+	0.595554i	$0.203068\pi$
$242$	634072.i	0.695985i
$243$	− 395272.i	− 0.429419i
$244$	−136705.	−0.146997
$245$	0	0
$246$	−95027.5	−0.100118
$247$	78872.1i	0.0822585i
$248$	99116.5i	0.102333i
$249$	826768.	0.845055
$250$	0	0
$251$	347891.	0.348545	0.174272	−	0.984697i	$-0.444243\pi$
$251$	347891.	0.348545	0.174272	+	0.984697i	$0.444243\pi$
$252$	− 41717.5i	− 0.0413825i
$253$	− 25578.8i	− 0.0251234i
$254$	−832930.	−0.810074
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.47105e6i	1.38930i	0.719348	+	0.694650i	$0.244441\pi$
$257$	1.47105e6i	1.38930i	−0.719348	+	0.694650i	$0.755559\pi$
$258$	14673.5i	0.0137241i
$259$	596681.	0.552704
$260$	0	0
$261$	185243.	0.168322
$262$	386610.i	0.347952i
$263$	− 1.73199e6i	− 1.54403i	−0.635602	−	0.772017i	$-0.719248\pi$
$263$	− 1.73199e6i	− 1.54403i	0.635602	−	0.772017i	$-0.280752\pi$
$264$	44374.7	0.0391855
$265$	0	0
$266$	194809.	0.168813
$267$	747557.i	0.641751i
$268$	228220.i	0.194096i
$269$	1.50181e6	1.26542	0.632708	−	0.774391i	$-0.281943\pi$
$269$	1.50181e6	1.26542	0.632708	+	0.774391i	$0.281943\pi$
$270$	0	0
$271$	226531.	0.187372	0.0936861	−	0.995602i	$-0.470135\pi$
$271$	226531.	0.187372	0.0936861	+	0.995602i	$0.470135\pi$
$272$	− 37207.5i	− 0.0304935i
$273$	53567.5i	0.0435005i
$274$	74085.4	0.0596151
$275$	0	0
$276$	112025.	0.0885202
$277$	− 3802.20i	− 0.00297739i	−0.999999	−	0.00148870i	$-0.999526\pi$
$277$	− 3802.20i	− 0.00297739i	0.999999	−	0.00148870i	$-0.000473867\pi$
$278$	− 755399.i	− 0.586225i
$279$	82407.8	0.0633808
$280$	0	0
$281$	2.17365e6	1.64219	0.821097	−	0.570788i	$-0.193362\pi$
$281$	2.17365e6	1.64219	0.821097	+	0.570788i	$0.193362\pi$
$282$	1.04971e6i	0.786044i
$283$	− 135231.i	− 0.100372i	−0.998740	−	0.0501858i	$-0.984019\pi$
$283$	− 135231.i	− 0.100372i	0.998740	−	0.0501858i	$-0.0159813\pi$
$284$	−234789.	−0.172735
$285$	0	0
$286$	15975.3	0.0115487
$287$	84498.7i	0.0605543i
$288$	− 54488.2i	− 0.0387098i
$289$	1.39873e6	0.985122
$290$	0	0
$291$	−1.57199e6	−1.08822
$292$	− 265533.i	− 0.182248i
$293$	1.53017e6i	1.04129i	0.853774	+	0.520644i	$0.174308\pi$
$293$	1.53017e6i	1.04129i	−0.853774	+	0.520644i	$0.825692\pi$
$294$	132308.	0.0892728
$295$	0	0
$296$	779338.	0.517007
$297$	− 205379.i	− 0.135103i
$298$	− 1.17858e6i	− 0.768808i
$299$	40330.1	0.0260886
$300$	0	0
$301$	13047.7	0.00830074
$302$	1.95968e6i	1.23642i
$303$	− 579785.i	− 0.362794i
$304$	254445.	0.157910
$305$	0	0
$306$	−30935.2	−0.0188864
$307$	− 2.22690e6i	− 1.34851i	−0.738497	−	0.674257i	$-0.764464\pi$
$307$	− 2.22690e6i	− 1.34851i	0.738497	−	0.674257i	$-0.235536\pi$
$308$	− 39458.1i	− 0.0237006i
$309$	1.49832e6	0.892708
$310$	0	0
$311$	−497393.	−0.291608	−0.145804	−	0.989314i	$-0.546577\pi$
$311$	−497393.	−0.291608	−0.145804	+	0.989314i	$0.546577\pi$
$312$	69965.7i	0.0406910i
$313$	− 649722.i	− 0.374858i	−0.982278	−	0.187429i	$-0.939985\pi$
$313$	− 649722.i	− 0.374858i	0.982278	−	0.187429i	$-0.0600155\pi$
$314$	−1.29368e6	−0.740463
$315$	0	0
$316$	71296.6	0.0401653
$317$	2.61927e6i	1.46397i	0.681320	+	0.731986i	$0.261406\pi$
$317$	2.61927e6i	1.46397i	−0.681320	+	0.731986i	$0.738594\pi$
$318$	256040.i	0.141984i
$319$	175210.	0.0964012
$320$	0	0
$321$	−2.12679e6	−1.15203
$322$	− 99612.9i	− 0.0535397i
$323$	− 144459.i	− 0.0770438i
$324$	692596.	0.366537
$325$	0	0
$326$	−1.97495e6	−1.02923
$327$	1.11700e6i	0.577674i
$328$	110366.i	0.0566434i
$329$	933406.	0.475424
$330$	0	0
$331$	31686.5	0.0158966	0.00794831	−	0.999968i	$-0.497470\pi$
$331$	31686.5	0.0158966	0.00794831	+	0.999968i	$0.497470\pi$
$332$	− 960214.i	− 0.478105i
$333$	− 647960.i	− 0.320212i
$334$	−1.24043e6	−0.608423
$335$	0	0
$336$	172811.	0.0835071
$337$	73508.6i	0.0352585i	0.999845	+	0.0176292i	$0.00561185\pi$
$337$	73508.6i	0.0352585i	−0.999845	+	0.0176292i	$0.994388\pi$
$338$	− 1.45998e6i	− 0.695114i
$339$	1.41266e6	0.667632
$340$	0	0
$341$	77944.6	0.0362994
$342$	− 211551.i	− 0.0978027i
$343$	− 117649.i	− 0.0539949i
$344$	17041.9	0.00776464
$345$	0	0
$346$	−2.05325e6	−0.922045
$347$	− 1.79688e6i	− 0.801118i	−0.916271	−	0.400559i	$-0.868816\pi$
$347$	− 1.79688e6i	− 0.801118i	0.916271	−	0.400559i	$-0.131184\pi$
$348$	767352.i	0.339662i
$349$	268466.	0.117985	0.0589923	−	0.998258i	$-0.481211\pi$
$349$	268466.	0.117985	0.0589923	+	0.998258i	$0.481211\pi$
$350$	0	0
$351$	323822.	0.140294
$352$	− 51537.1i	− 0.0221699i
$353$	291901.i	0.124680i	0.998055	+	0.0623402i	$0.0198564\pi$
$353$	291901.i	0.124680i	−0.998055	+	0.0623402i	$0.980144\pi$
$354$	1.10236e6	0.467538
$355$	0	0
$356$	868219.	0.363082
$357$	− 98111.9i	− 0.0407428i
$358$	1.05238e6i	0.433976i
$359$	−573376.	−0.234803	−0.117402	−	0.993085i	$-0.537456\pi$
$359$	−573376.	−0.234803	−0.117402	+	0.993085i	$0.537456\pi$
$360$	0	0
$361$	−1.48821e6	−0.601031
$362$	105643.i	0.0423712i
$363$	2.18381e6i	0.869856i
$364$	62213.7	0.0246112
$365$	0	0
$366$	−470825.	−0.183720
$367$	− 1.73711e6i	− 0.673228i	−0.941643	−	0.336614i	$-0.890718\pi$
$367$	− 1.73711e6i	− 0.673228i	0.941643	−	0.336614i	$-0.109282\pi$
$368$	− 130107.i	− 0.0500818i
$369$	91760.6	0.0350825
$370$	0	0
$371$	227671.	0.0858763
$372$	341367.i	0.127898i
$373$	− 559504.i	− 0.208224i	−0.994566	−	0.104112i	$-0.966800\pi$
$373$	− 559504.i	− 0.208224i	0.994566	−	0.104112i	$-0.0332001\pi$
$374$	−29259.7	−0.0108166
$375$	0	0
$376$	1.21914e6	0.444718
$377$	276254.i	0.100105i
$378$	− 799821.i	− 0.287914i
$379$	−1.81746e6	−0.649929	−0.324964	−	0.945726i	$-0.605352\pi$
$379$	−1.81746e6	−0.649929	−0.324964	+	0.945726i	$0.605352\pi$
$380$	0	0
$381$	−2.86869e6	−1.01245
$382$	− 1.32177e6i	− 0.463443i
$383$	2.67383e6i	0.931402i	0.884942	+	0.465701i	$0.154198\pi$
$383$	2.67383e6i	0.931402i	−0.884942	+	0.465701i	$0.845802\pi$
$384$	225712.	0.0781137
$385$	0	0
$386$	−2.64513e6	−0.903606
$387$	− 14169.0i	− 0.00480908i
$388$	1.82572e6i	0.615679i
$389$	−2.11315e6	−0.708038	−0.354019	−	0.935238i	$-0.615185\pi$
$389$	−2.11315e6	−0.708038	−0.354019	+	0.935238i	$0.615185\pi$
$390$	0	0
$391$	−73866.9	−0.0244347
$392$	− 153664.i	− 0.0505076i
$393$	1.33152e6i	0.434878i
$394$	1.79978e6	0.584088
$395$	0	0
$396$	−42849.1	−0.0137311
$397$	3.62861e6i	1.15548i	0.816220	+	0.577741i	$0.196066\pi$
$397$	3.62861e6i	1.15548i	−0.816220	+	0.577741i	$0.803934\pi$
$398$	− 157825.i	− 0.0499422i
$399$	670942.	0.210986
$400$	0	0
$401$	−1.46779e6	−0.455829	−0.227915	−	0.973681i	$-0.573191\pi$
$401$	−1.46779e6	−0.455829	−0.227915	+	0.973681i	$0.573191\pi$
$402$	786012.i	0.242585i
$403$	122895.i	0.0376941i
$404$	−673367.	−0.205257
$405$	0	0
$406$	682332.	0.205438
$407$	− 612866.i	− 0.183392i
$408$	− 128146.i	− 0.0381114i
$409$	−405109.	−0.119747	−0.0598734	−	0.998206i	$-0.519070\pi$
$409$	−405109.	−0.119747	−0.0598734	+	0.998206i	$0.519070\pi$
$410$	0	0
$411$	255157.	0.0745081
$412$	− 1.74016e6i	− 0.505065i
$413$	− 980225.i	− 0.282781i
$414$	−108174.	−0.0310185
$415$	0	0
$416$	81258.7	0.0230216
$417$	− 2.60167e6i	− 0.732675i
$418$	− 200094.i	− 0.0560135i
$419$	−5.59025e6	−1.55559	−0.777797	−	0.628515i	$-0.783663\pi$
$419$	−5.59025e6	−1.55559	−0.777797	+	0.628515i	$0.783663\pi$
$420$	0	0
$421$	4.32984e6	1.19060	0.595302	−	0.803502i	$-0.297033\pi$
$421$	4.32984e6	1.19060	0.595302	+	0.803502i	$0.297033\pi$
$422$	− 2.23855e6i	− 0.611908i
$423$	− 1.01362e6i	− 0.275439i
$424$	297366.	0.0803299
$425$	0	0
$426$	−808635.	−0.215888
$427$	418659.i	0.111120i
$428$	2.47007e6i	0.651778i
$429$	55020.5	0.0144338
$430$	0	0
$431$	2.26685e6	0.587801	0.293901	−	0.955836i	$-0.405047\pi$
$431$	2.26685e6	0.587801	0.293901	+	0.955836i	$0.405047\pi$
$432$	− 1.04466e6i	− 0.269319i
$433$	3.77679e6i	0.968061i	0.875051	+	0.484030i	$0.160828\pi$
$433$	3.77679e6i	0.968061i	−0.875051	+	0.484030i	$0.839172\pi$
$434$	303544.	0.0773567
$435$	0	0
$436$	1.29729e6	0.326829
$437$	− 505142.i	− 0.126535i
$438$	− 914523.i	− 0.227777i
$439$	−5.66625e6	−1.40325	−0.701624	−	0.712548i	$-0.747541\pi$
$439$	−5.66625e6	−1.40325	−0.701624	+	0.712548i	$0.747541\pi$
$440$	0	0
$441$	−127760.	−0.0312823
$442$	− 46133.9i	− 0.0112322i
$443$	− 443096.i	− 0.107272i	−0.998561	−	0.0536362i	$-0.982919\pi$
$443$	− 443096.i	− 0.107272i	0.998561	−	0.0536362i	$-0.0170811\pi$
$444$	2.68412e6	0.646166
$445$	0	0
$446$	2.89589e6	0.689359
$447$	− 4.05914e6i	− 0.960872i
$448$	− 200704.i	− 0.0472456i
$449$	−6.00344e6	−1.40535	−0.702674	−	0.711512i	$-0.748011\pi$
$449$	−6.00344e6	−1.40535	−0.702674	+	0.711512i	$0.748011\pi$
$450$	0	0
$451$	86790.8	0.0200924
$452$	− 1.64067e6i	− 0.377724i
$453$	6.74932e6i	1.54531i
$454$	−2.38508e6	−0.543080
$455$	0	0
$456$	876333.	0.197359
$457$	− 1.87447e6i	− 0.419845i	−0.977718	−	0.209922i	$-0.932679\pi$
$457$	− 1.87447e6i	− 0.419845i	0.977718	−	0.209922i	$-0.0673211\pi$
$458$	367343.i	0.0818293i
$459$	−593099.	−0.131400
$460$	0	0
$461$	−3.52843e6	−0.773268	−0.386634	−	0.922233i	$-0.626362\pi$
$461$	−3.52843e6	−0.773268	−0.386634	+	0.922233i	$0.626362\pi$
$462$	− 135897.i	− 0.0296214i
$463$	− 3.23963e6i	− 0.702332i	−0.936313	−	0.351166i	$-0.885785\pi$
$463$	− 3.23963e6i	− 0.702332i	0.936313	−	0.351166i	$-0.114215\pi$
$464$	891209.	0.192169
$465$	0	0
$466$	−1.14281e6	−0.243787
$467$	4.69897e6i	0.997036i	0.866879	+	0.498518i	$0.166122\pi$
$467$	4.69897e6i	0.997036i	−0.866879	+	0.498518i	$0.833878\pi$
$468$	− 67560.4i	− 0.0142586i
$469$	698924.	0.146723
$470$	0	0
$471$	−4.45557e6	−0.925445
$472$	− 1.28029e6i	− 0.264518i
$473$	− 13401.6i	− 0.00275425i
$474$	245552.	0.0501994
$475$	0	0
$476$	−113948.	−0.0230509
$477$	− 247237.i	− 0.0497529i
$478$	− 443988.i	− 0.0888794i
$479$	−9.35047e6	−1.86206	−0.931032	−	0.364938i	$-0.881090\pi$
$479$	−9.35047e6	−1.86206	−0.931032	+	0.364938i	$0.881090\pi$
$480$	0	0
$481$	966308.	0.190438
$482$	− 5.79453e6i	− 1.13606i
$483$	− 343077.i	− 0.0669150i
$484$	2.53629e6	0.492136
$485$	0	0
$486$	−1.58109e6	−0.303645
$487$	− 9.31032e6i	− 1.77886i	−0.457071	−	0.889430i	$-0.651101\pi$
$487$	− 9.31032e6i	− 1.77886i	0.457071	−	0.889430i	$-0.348899\pi$
$488$	546819.i	0.103943i
$489$	−6.80192e6	−1.28635
$490$	0	0
$491$	8.23217e6	1.54103	0.770514	−	0.637423i	$-0.220000\pi$
$491$	8.23217e6	1.54103	0.770514	+	0.637423i	$0.220000\pi$
$492$	380110.i	0.0707940i
$493$	− 505976.i	− 0.0937588i
$494$	315488.	0.0581656
$495$	0	0
$496$	396466.	0.0723606
$497$	719040.i	0.130576i
$498$	− 3.30707e6i	− 0.597544i
$499$	1.05044e7	1.88851	0.944257	−	0.329209i	$-0.106782\pi$
$499$	1.05044e7	1.88851	0.944257	+	0.329209i	$0.106782\pi$
$500$	0	0
$501$	−4.27216e6	−0.760419
$502$	− 1.39156e6i	− 0.246458i
$503$	− 2.01109e6i	− 0.354415i	−0.984173	−	0.177208i	$-0.943294\pi$
$503$	− 2.01109e6i	− 0.354415i	0.984173	−	0.177208i	$-0.0567064\pi$
$504$	−166870.	−0.0292619
$505$	0	0
$506$	−102315.	−0.0177649
$507$	− 5.02833e6i	− 0.868767i
$508$	3.33172e6i	0.572809i
$509$	−1.02138e7	−1.74741	−0.873703	−	0.486460i	$-0.838288\pi$
$509$	−1.02138e7	−1.74741	−0.873703	+	0.486460i	$0.838288\pi$
$510$	0	0
$511$	−813196.	−0.137766
$512$	− 262144.i	− 0.0441942i
$513$	− 4.05593e6i	− 0.680451i
$514$	5.88422e6	0.982383
$515$	0	0
$516$	58693.8	0.00970439
$517$	− 958725.i	− 0.157749i
$518$	− 2.38672e6i	− 0.390821i
$519$	−7.07160e6	−1.15239
$520$	0	0
$521$	−8.16451e6	−1.31776	−0.658879	−	0.752249i	$-0.728969\pi$
$521$	−8.16451e6	−1.31776	−0.658879	+	0.752249i	$0.728969\pi$
$522$	− 740972.i	− 0.119022i
$523$	4.58812e6i	0.733467i	0.930326	+	0.366733i	$0.119524\pi$
$523$	4.58812e6i	0.733467i	−0.930326	+	0.366733i	$0.880476\pi$
$524$	1.54644e6	0.246040
$525$	0	0
$526$	−6.92797e6	−1.09180
$527$	− 225090.i	− 0.0353045i
$528$	− 177499.i	− 0.0277083i
$529$	6.17805e6	0.959869
$530$	0	0
$531$	−1.06447e6	−0.163831
$532$	− 779237.i	− 0.119369i
$533$	136843.i	0.0208644i
$534$	2.99023e6	0.453786
$535$	0	0
$536$	912880.	0.137247
$537$	3.62450e6i	0.542391i
$538$	− 6.00722e6i	− 0.894784i
$539$	−120840.	−0.0179160
$540$	0	0
$541$	−671740.	−0.0986753	−0.0493376	−	0.998782i	$-0.515711\pi$
$541$	−671740.	−0.0986753	−0.0493376	+	0.998782i	$0.515711\pi$
$542$	− 906126.i	− 0.132492i
$543$	363846.i	0.0529563i
$544$	−148830.	−0.0215622
$545$	0	0
$546$	214270.	0.0307595
$547$	1.60632e6i	0.229543i	0.993392	+	0.114772i	$0.0366136\pi$
$547$	1.60632e6i	0.229543i	−0.993392	+	0.114772i	$0.963386\pi$
$548$	− 296341.i	− 0.0421542i
$549$	454639.	0.0643777
$550$	0	0
$551$	3.46013e6	0.485528
$552$	− 448100.i	− 0.0625932i
$553$	− 218346.i	− 0.0303621i
$554$	−15208.8	−0.00210533
$555$	0	0
$556$	−3.02160e6	−0.414524
$557$	− 3.86497e6i	− 0.527847i	−0.964544	−	0.263923i	$-0.914983\pi$
$557$	− 3.86497e6i	− 0.527847i	0.964544	−	0.263923i	$-0.0850166\pi$
$558$	− 329631.i	− 0.0448170i
$559$	21130.4	0.00286007
$560$	0	0
$561$	−100773.	−0.0135188
$562$	− 8.69462e6i	− 1.16121i
$563$	− 6.43081e6i	− 0.855057i	−0.904002	−	0.427528i	$-0.859384\pi$
$563$	− 6.43081e6i	− 0.855057i	0.904002	−	0.427528i	$-0.140616\pi$
$564$	4.19884e6	0.555817
$565$	0	0
$566$	−540925.	−0.0709734
$567$	− 2.12108e6i	− 0.277076i
$568$	939154.i	0.122142i
$569$	6.69245e6	0.866572	0.433286	−	0.901256i	$-0.357354\pi$
$569$	6.69245e6	0.866572	0.433286	+	0.901256i	$0.357354\pi$
$570$	0	0
$571$	1.18842e7	1.52539	0.762694	−	0.646760i	$-0.223876\pi$
$571$	1.18842e7	1.52539	0.762694	+	0.646760i	$0.223876\pi$
$572$	− 63901.3i	− 0.00816619i
$573$	− 4.55230e6i	− 0.579220i
$574$	337995.	0.0428184
$575$	0	0
$576$	−217953.	−0.0273720
$577$	5.90879e6i	0.738855i	0.929260	+	0.369427i	$0.120446\pi$
$577$	5.90879e6i	0.738855i	−0.929260	+	0.369427i	$0.879554\pi$
$578$	− 5.59493e6i	− 0.696587i
$579$	−9.11009e6	−1.12934
$580$	0	0
$581$	−2.94066e6	−0.361413
$582$	6.28795e6i	0.769487i
$583$	− 233847.i	− 0.0284944i
$584$	−1.06213e6	−0.128869
$585$	0	0
$586$	6.12069e6	0.736302
$587$	7.19723e6i	0.862125i	0.902322	+	0.431062i	$0.141861\pi$
$587$	7.19723e6i	0.862125i	−0.902322	+	0.431062i	$0.858139\pi$
$588$	− 529234.i	− 0.0631254i
$589$	1.53929e6	0.182823
$590$	0	0
$591$	6.19861e6	0.730004
$592$	− 3.11735e6i	− 0.365579i
$593$	1.46242e7i	1.70779i	0.520447	+	0.853894i	$0.325766\pi$
$593$	1.46242e7i	1.70779i	−0.520447	+	0.853894i	$0.674234\pi$
$594$	−821517.	−0.0955323
$595$	0	0
$596$	−4.71432e6	−0.543630
$597$	− 543564.i	− 0.0624187i
$598$	− 161320.i	− 0.0184475i
$599$	1.26967e7	1.44585	0.722924	−	0.690927i	$-0.242798\pi$
$599$	1.26967e7	1.44585	0.722924	+	0.690927i	$0.242798\pi$
$600$	0	0
$601$	−2.83830e6	−0.320533	−0.160266	−	0.987074i	$-0.551235\pi$
$601$	−2.83830e6	−0.320533	−0.160266	+	0.987074i	$0.551235\pi$
$602$	− 52190.7i	− 0.00586951i
$603$	− 758990.i	− 0.0850047i
$604$	7.83871e6	0.874283
$605$	0	0
$606$	−2.31914e6	−0.256534
$607$	8.78430e6i	0.967689i	0.875154	+	0.483844i	$0.160760\pi$
$607$	8.78430e6i	0.967689i	−0.875154	+	0.483844i	$0.839240\pi$
$608$	− 1.01778e6i	− 0.111659i
$609$	2.35002e6	0.256760
$610$	0	0
$611$	1.51163e6	0.163810
$612$	123741.i	0.0133547i
$613$	1.24669e7i	1.34001i	0.742355	+	0.670006i	$0.233709\pi$
$613$	1.24669e7i	1.34001i	−0.742355	+	0.670006i	$0.766291\pi$
$614$	−8.90761e6	−0.953543
$615$	0	0
$616$	−157832.	−0.0167588
$617$	1.04133e7i	1.10122i	0.834762	+	0.550611i	$0.185605\pi$
$617$	1.04133e7i	1.10122i	−0.834762	+	0.550611i	$0.814395\pi$
$618$	− 5.99329e6i	− 0.631240i
$619$	7.59748e6	0.796971	0.398486	−	0.917175i	$-0.369536\pi$
$619$	7.59748e6	0.796971	0.398486	+	0.917175i	$0.369536\pi$
$620$	0	0
$621$	−2.07394e6	−0.215808
$622$	1.98957e6i	0.206198i
$623$	− 2.65892e6i	− 0.274464i
$624$	279863.	0.0287729
$625$	0	0
$626$	−2.59889e6	−0.265065
$627$	− 689142.i	− 0.0700067i
$628$	5.17473e6i	0.523587i
$629$	−1.76985e6	−0.178365
$630$	0	0
$631$	9.92192e6	0.992024	0.496012	−	0.868316i	$-0.334797\pi$
$631$	9.92192e6	0.992024	0.496012	+	0.868316i	$0.334797\pi$
$632$	− 285186.i	− 0.0284012i
$633$	− 7.70980e6i	− 0.764775i
$634$	1.04771e7	1.03518
$635$	0	0
$636$	1.02416e6	0.100398
$637$	− 190529.i	− 0.0186043i
$638$	− 700840.i	− 0.0681659i
$639$	780835.	0.0756497
$640$	0	0
$641$	3.49761e6	0.336222	0.168111	−	0.985768i	$-0.446233\pi$
$641$	3.49761e6	0.336222	0.168111	+	0.985768i	$0.446233\pi$
$642$	8.50716e6i	0.814605i
$643$	1.02287e7i	0.975645i	0.872943	+	0.487823i	$0.162209\pi$
$643$	1.02287e7i	0.975645i	−0.872943	+	0.487823i	$0.837791\pi$
$644$	−398452.	−0.0378583
$645$	0	0
$646$	−577835.	−0.0544782
$647$	8.58075e6i	0.805869i	0.915229	+	0.402934i	$0.132010\pi$
$647$	8.58075e6i	0.805869i	−0.915229	+	0.402934i	$0.867990\pi$
$648$	− 2.77039e6i	− 0.259181i
$649$	−1.00681e6	−0.0938291
$650$	0	0
$651$	1.04544e6	0.0966819
$652$	7.89980e6i	0.727775i
$653$	7.47927e6i	0.686399i	0.939263	+	0.343199i	$0.111511\pi$
$653$	7.47927e6i	0.686399i	−0.939263	+	0.343199i	$0.888489\pi$
$654$	4.46799e6	0.408477
$655$	0	0
$656$	441462.	0.0400529
$657$	883083.i	0.0798157i
$658$	− 3.73362e6i	− 0.336175i
$659$	−5.63102e6	−0.505095	−0.252548	−	0.967584i	$-0.581268\pi$
$659$	−5.63102e6	−0.505095	−0.252548	+	0.967584i	$0.581268\pi$
$660$	0	0
$661$	−3.03738e6	−0.270393	−0.135197	−	0.990819i	$-0.543167\pi$
$661$	−3.03738e6	−0.270393	−0.135197	+	0.990819i	$0.543167\pi$
$662$	− 126746.i	− 0.0112406i
$663$	− 158890.i	− 0.0140382i
$664$	−3.84086e6	−0.338071
$665$	0	0
$666$	−2.59184e6	−0.226424
$667$	− 1.76929e6i	− 0.153987i
$668$	4.96171e6i	0.430220i
$669$	9.97373e6	0.861574
$670$	0	0
$671$	430015.	0.0368704
$672$	− 691244.i	− 0.0590484i
$673$	1.29019e7i	1.09804i	0.835810	+	0.549018i	$0.184998\pi$
$673$	1.29019e7i	1.09804i	−0.835810	+	0.549018i	$0.815002\pi$
$674$	294034.	0.0249315
$675$	0	0
$676$	−5.83993e6	−0.491520
$677$	2.03709e7i	1.70820i	0.520111	+	0.854099i	$0.325891\pi$
$677$	2.03709e7i	1.70820i	−0.520111	+	0.854099i	$0.674109\pi$
$678$	− 5.65062e6i	− 0.472087i
$679$	5.59126e6	0.465409
$680$	0	0
$681$	−8.21446e6	−0.678752
$682$	− 311778.i	− 0.0256676i
$683$	3.47433e6i	0.284983i	0.989796	+	0.142492i	$0.0455114\pi$
$683$	3.47433e6i	0.284983i	−0.989796	+	0.142492i	$0.954489\pi$
$684$	−846206.	−0.0691569
$685$	0	0
$686$	−470596.	−0.0381802
$687$	1.26517e6i	0.102272i
$688$	− 68167.5i	− 0.00549043i
$689$	368707.	0.0295892
$690$	0	0
$691$	2.15415e7	1.71625	0.858124	−	0.513443i	$-0.171630\pi$
$691$	2.15415e7	1.71625	0.858124	+	0.513443i	$0.171630\pi$
$692$	8.21301e6i	0.651984i
$693$	131225.i	0.0103797i
$694$	−7.18754e6	−0.566476
$695$	0	0
$696$	3.06941e6	0.240177
$697$	− 250636.i	− 0.0195417i
$698$	− 1.07386e6i	− 0.0834277i
$699$	−3.93596e6	−0.304690
$700$	0	0
$701$	−8.76380e6	−0.673593	−0.336796	−	0.941577i	$-0.609343\pi$
$701$	−8.76380e6	−0.673593	−0.336796	+	0.941577i	$0.609343\pi$
$702$	− 1.29529e6i	− 0.0992027i
$703$	− 1.21032e7i	− 0.923658i
$704$	−206148.	−0.0156765
$705$	0	0
$706$	1.16760e6	0.0881624
$707$	2.06219e6i	0.155160i
$708$	− 4.40946e6i	− 0.330599i
$709$	1.07244e7	0.801234	0.400617	−	0.916246i	$-0.368796\pi$
$709$	1.07244e7	0.801234	0.400617	+	0.916246i	$0.368796\pi$
$710$	0	0
$711$	−237111.	−0.0175905
$712$	− 3.47287e6i	− 0.256737i
$713$	− 787092.i	− 0.0579832i
$714$	−392448.	−0.0288095
$715$	0	0
$716$	4.20952e6	0.306867
$717$	− 1.52914e6i	− 0.111083i
$718$	2.29351e6i	0.166031i
$719$	−1.77748e7	−1.28228	−0.641140	−	0.767424i	$-0.721538\pi$
$719$	−1.77748e7	−1.28228	−0.641140	+	0.767424i	$0.721538\pi$
$720$	0	0
$721$	−5.32925e6	−0.381793
$722$	5.95285e6i	0.424993i
$723$	− 1.99569e7i	− 1.41987i
$724$	422573.	0.0299610
$725$	0	0
$726$	8.73522e6	0.615081
$727$	2.71742e7i	1.90687i	0.301605	+	0.953433i	$0.402477\pi$
$727$	2.71742e7i	1.90687i	−0.301605	+	0.953433i	$0.597523\pi$
$728$	− 248855.i	− 0.0174027i
$729$	−1.59642e7	−1.11258
$730$	0	0
$731$	−38701.5	−0.00267876
$732$	1.88330e6i	0.129910i
$733$	2.34777e7i	1.61397i	0.590571	+	0.806986i	$0.298903\pi$
$733$	2.34777e7i	1.61397i	−0.590571	+	0.806986i	$0.701097\pi$
$734$	−6.94844e6	−0.476044
$735$	0	0
$736$	−520427.	−0.0354132
$737$	− 717883.i	− 0.0486838i
$738$	− 367042.i	− 0.0248071i
$739$	2.43496e7	1.64014	0.820068	−	0.572266i	$-0.193935\pi$
$739$	2.43496e7	1.64014	0.820068	+	0.572266i	$0.193935\pi$
$740$	0	0
$741$	1.08657e6	0.0726965
$742$	− 910685.i	− 0.0607237i
$743$	60731.2i	0.00403590i	0.999998	+	0.00201795i	$0.000642333\pi$
$743$	60731.2i	0.00403590i	−0.999998	+	0.00201795i	$0.999358\pi$
$744$	1.36547e6	0.0904376
$745$	0	0
$746$	−2.23802e6	−0.147237
$747$	3.19338e6i	0.209387i
$748$	117039.i	0.00764849i
$749$	7.56459e6	0.492698
$750$	0	0
$751$	9.56195e6	0.618652	0.309326	−	0.950956i	$-0.399897\pi$
$751$	9.56195e6	0.618652	0.309326	+	0.950956i	$0.399897\pi$
$752$	− 4.87657e6i	− 0.314463i
$753$	− 4.79268e6i	− 0.308029i
$754$	1.10502e6	0.0707849
$755$	0	0
$756$	−3.19928e6	−0.203586
$757$	− 8.42306e6i	− 0.534232i	−0.963664	−	0.267116i	$-0.913929\pi$
$757$	− 8.42306e6i	− 0.534232i	0.963664	−	0.267116i	$-0.0860707\pi$
$758$	7.26982e6i	0.459569i
$759$	−352383.	−0.0222029
$760$	0	0
$761$	−1.50895e6	−0.0944527	−0.0472263	−	0.998884i	$-0.515038\pi$
$761$	−1.50895e6	−0.0944527	−0.0472263	+	0.998884i	$0.515038\pi$
$762$	1.14748e7i	0.715907i
$763$	− 3.97295e6i	− 0.247060i
$764$	−5.28707e6	−0.327704
$765$	0	0
$766$	1.06953e7	0.658601
$767$	− 1.58745e6i	− 0.0974341i
$768$	− 902849.i	− 0.0552347i
$769$	1.17597e7	0.717100	0.358550	−	0.933511i	$-0.383271\pi$
$769$	1.17597e7	0.717100	0.358550	+	0.933511i	$0.383271\pi$
$770$	0	0
$771$	2.02658e7	1.22780
$772$	1.05805e7i	0.638946i
$773$	− 2.83129e7i	− 1.70426i	−0.523328	−	0.852131i	$-0.675310\pi$
$773$	− 2.83129e7i	− 1.70426i	0.523328	−	0.852131i	$-0.324690\pi$
$774$	−56676.0	−0.00340053
$775$	0	0
$776$	7.30287e6	0.435351
$777$	− 8.22011e6i	− 0.488455i
$778$	8.45261e6i	0.500659i
$779$	1.71399e6	0.101196
$780$	0	0
$781$	738545.	0.0433261
$782$	295467.i	0.0172780i
$783$	− 1.42061e7i	− 0.828079i
$784$	−614656.	−0.0357143
$785$	0	0
$786$	5.32609e6	0.307505
$787$	− 1.41914e7i	− 0.816749i	−0.912815	−	0.408374i	$-0.866096\pi$
$787$	− 1.41914e7i	− 0.816749i	0.912815	−	0.408374i	$-0.133904\pi$
$788$	− 7.19911e6i	− 0.413012i
$789$	−2.38606e7	−1.36455
$790$	0	0
$791$	−5.02455e6	−0.285533
$792$	171397.i	0.00970933i
$793$	678006.i	0.0382869i
$794$	1.45144e7	0.817050
$795$	0	0
$796$	−631299.	−0.0353145
$797$	2.49185e7i	1.38956i	0.719223	+	0.694779i	$0.244498\pi$
$797$	2.49185e7i	1.38956i	−0.719223	+	0.694779i	$0.755502\pi$
$798$	− 2.68377e6i	− 0.149189i
$799$	−2.76863e6	−0.153426
$800$	0	0
$801$	−2.88743e6	−0.159012
$802$	5.87115e6i	0.322320i
$803$	835255.i	0.0457120i
$804$	3.14405e6	0.171534
$805$	0	0
$806$	491582.	0.0266537
$807$	− 2.06895e7i	− 1.11832i
$808$	2.69347e6i	0.145139i
$809$	2.57217e7	1.38175	0.690875	−	0.722975i	$-0.257226\pi$
$809$	2.57217e7	1.38175	0.690875	+	0.722975i	$0.257226\pi$
$810$	0	0
$811$	2.55278e7	1.36289	0.681445	−	0.731869i	$-0.261352\pi$
$811$	2.55278e7	1.36289	0.681445	+	0.731869i	$0.261352\pi$
$812$	− 2.72933e6i	− 0.145266i
$813$	− 3.12078e6i	− 0.165591i
$814$	−2.45146e6	−0.129678
$815$	0	0
$816$	−512585.	−0.0269488
$817$	− 264662.i	− 0.0138719i
$818$	1.62044e6i	0.0846738i
$819$	−206904.	−0.0107785
$820$	0	0
$821$	−2.02651e7	−1.04928	−0.524639	−	0.851325i	$-0.675800\pi$
$821$	−2.02651e7	−1.04928	−0.524639	+	0.851325i	$0.675800\pi$
$822$	− 1.02063e6i	− 0.0526852i
$823$	− 3.04345e7i	− 1.56627i	−0.621850	−	0.783136i	$-0.713619\pi$
$823$	− 3.04345e7i	− 1.56627i	0.621850	−	0.783136i	$-0.286381\pi$
$824$	−6.96066e6	−0.357135
$825$	0	0
$826$	−3.92090e6	−0.199957
$827$	3.93136e6i	0.199884i	0.994993	+	0.0999422i	$0.0318658\pi$
$827$	3.93136e6i	0.199884i	−0.994993	+	0.0999422i	$0.968134\pi$
$828$	432695.i	0.0219334i
$829$	1.86948e7	0.944788	0.472394	−	0.881387i	$-0.343390\pi$
$829$	1.86948e7	0.944788	0.472394	+	0.881387i	$0.343390\pi$
$830$	0	0
$831$	−52380.6	−0.00263129
$832$	− 325035.i	− 0.0162788i
$833$	348965.i	0.0174249i
$834$	−1.04067e7	−0.518080
$835$	0	0
$836$	−800375.	−0.0396075
$837$	− 6.31979e6i	− 0.311809i
$838$	2.23610e7i	1.09997i
$839$	4.23817e6	0.207861	0.103931	−	0.994585i	$-0.466858\pi$
$839$	4.23817e6	0.207861	0.103931	+	0.994585i	$0.466858\pi$
$840$	0	0
$841$	−8.39182e6	−0.409134
$842$	− 1.73194e7i	− 0.841883i
$843$	− 2.99451e7i	− 1.45130i
$844$	−8.95422e6	−0.432685
$845$	0	0
$846$	−4.05450e6	−0.194765
$847$	− 7.76738e6i	− 0.372020i
$848$	− 1.18947e6i	− 0.0568018i
$849$	−1.86300e6	−0.0887039
$850$	0	0
$851$	−6.18879e6	−0.292942
$852$	3.23454e6i	0.152656i
$853$	2.80984e7i	1.32224i	0.750282	+	0.661118i	$0.229918\pi$
$853$	2.80984e7i	1.32224i	−0.750282	+	0.661118i	$0.770082\pi$
$854$	1.67463e6	0.0785734
$855$	0	0
$856$	9.88028e6	0.460877
$857$	− 2.13309e7i	− 0.992106i	−0.868292	−	0.496053i	$-0.834782\pi$
$857$	− 2.13309e7i	− 0.992106i	0.868292	−	0.496053i	$-0.165218\pi$
$858$	− 220082.i	− 0.0102063i
$859$	−3.49914e6	−0.161800	−0.0809001	−	0.996722i	$-0.525779\pi$
$859$	−3.49914e6	−0.161800	−0.0809001	+	0.996722i	$0.525779\pi$
$860$	0	0
$861$	1.16409e6	0.0535152
$862$	− 9.06742e6i	− 0.415638i
$863$	− 3.86911e7i	− 1.76841i	−0.467096	−	0.884206i	$-0.654700\pi$
$863$	− 3.86911e7i	− 1.76841i	0.467096	−	0.884206i	$-0.345300\pi$
$864$	−4.17866e6	−0.190437
$865$	0	0
$866$	1.51071e7	0.684522
$867$	− 1.92695e7i	− 0.870608i
$868$	− 1.21418e6i	− 0.0546994i
$869$	−224269.	−0.0100744
$870$	0	0
$871$	1.13189e6	0.0505543
$872$	− 5.18916e6i	− 0.231103i
$873$	− 6.07178e6i	− 0.269637i
$874$	−2.02057e6	−0.0894735
$875$	0	0
$876$	−3.65809e6	−0.161062
$877$	− 1.56417e7i	− 0.686730i	−0.939202	−	0.343365i	$-0.888433\pi$
$877$	− 1.56417e7i	− 0.686730i	0.939202	−	0.343365i	$-0.111567\pi$
$878$	2.26650e7i	0.992246i
$879$	2.10802e7	0.920245
$880$	0	0
$881$	−1.29672e7	−0.562868	−0.281434	−	0.959581i	$-0.590810\pi$
$881$	−1.29672e7	−0.562868	−0.281434	+	0.959581i	$0.590810\pi$
$882$	511040.i	0.0221199i
$883$	4.38413e7i	1.89226i	0.323785	+	0.946131i	$0.395045\pi$
$883$	4.38413e7i	1.89226i	−0.323785	+	0.946131i	$0.604955\pi$
$884$	−184535.	−0.00794235
$885$	0	0
$886$	−1.77238e6	−0.0758531
$887$	− 1.01177e7i	− 0.431789i	−0.976417	−	0.215895i	$-0.930733\pi$
$887$	− 1.01177e7i	− 0.431789i	0.976417	−	0.215895i	$-0.0692668\pi$
$888$	− 1.07365e7i	− 0.456908i
$889$	1.02034e7	0.433003
$890$	0	0
$891$	−2.17861e6	−0.0919360
$892$	− 1.15836e7i	− 0.487450i
$893$	− 1.89334e7i	− 0.794510i
$894$	−1.62366e7	−0.679439
$895$	0	0
$896$	−802816.	−0.0334077
$897$	− 555603.i	− 0.0230560i
$898$	2.40137e7i	0.993731i
$899$	5.39145e6	0.222488
$900$	0	0
$901$	−675308.	−0.0277134
$902$	− 347163.i	− 0.0142075i
$903$	− 179750.i	− 0.00733583i
$904$	−6.56267e6	−0.267091
$905$	0	0
$906$	2.69973e7	1.09270
$907$	− 3.60373e7i	− 1.45457i	−0.686337	−	0.727284i	$-0.740782\pi$
$907$	− 3.60373e7i	− 1.45457i	0.686337	−	0.727284i	$-0.259218\pi$
$908$	9.54034e6i	0.384016i
$909$	2.23941e6	0.0898927
$910$	0	0
$911$	−1.75996e7	−0.702598	−0.351299	−	0.936263i	$-0.614260\pi$
$911$	−1.75996e7	−0.702598	−0.351299	+	0.936263i	$0.614260\pi$
$912$	− 3.50533e6i	− 0.139554i
$913$	3.02042e6i	0.119920i
$914$	−7.49789e6	−0.296875
$915$	0	0
$916$	1.46937e6	0.0578620
$917$	− 4.73597e6i	− 0.185988i
$918$	2.37239e6i	0.0929138i
$919$	−1.48834e7	−0.581319	−0.290659	−	0.956827i	$-0.593875\pi$
$919$	−1.48834e7	−0.581319	−0.290659	+	0.956827i	$0.593875\pi$
$920$	0	0
$921$	−3.06787e7	−1.19176
$922$	1.41137e7i	0.546783i
$923$	1.16447e6i	0.0449907i
$924$	−543590.	−0.0209455
$925$	0	0
$926$	−1.29585e7	−0.496624
$927$	5.78725e6i	0.221194i
$928$	− 3.56483e6i	− 0.135884i
$929$	−2.68205e7	−1.01959	−0.509797	−	0.860295i	$-0.670279\pi$
$929$	−2.68205e7	−1.01959	−0.509797	+	0.860295i	$0.670279\pi$
$930$	0	0
$931$	−2.38641e6	−0.0902343
$932$	4.57126e6i	0.172384i
$933$	6.85228e6i	0.257710i
$934$	1.87959e7	0.705011
$935$	0	0
$936$	−270241.	−0.0100824
$937$	− 1.02969e6i	− 0.0383138i	−0.999816	−	0.0191569i	$-0.993902\pi$
$937$	− 1.02969e6i	− 0.0383138i	0.999816	−	0.0191569i	$-0.00609821\pi$
$938$	− 2.79570e6i	− 0.103749i
$939$	−8.95083e6	−0.331283
$940$	0	0
$941$	−3.68245e7	−1.35570	−0.677849	−	0.735201i	$-0.737088\pi$
$941$	−3.68245e7	−1.35570	−0.677849	+	0.735201i	$0.737088\pi$
$942$	1.78223e7i	0.654389i
$943$	− 876422.i	− 0.0320948i
$944$	−5.12118e6	−0.187042
$945$	0	0
$946$	−53606.4	−0.00194755
$947$	− 2.74041e7i	− 0.992981i	−0.868042	−	0.496491i	$-0.834622\pi$
$947$	− 2.74041e7i	− 0.992981i	0.868042	−	0.496491i	$-0.165378\pi$
$948$	− 982210.i	− 0.0354963i
$949$	−1.31695e6	−0.0474683
$950$	0	0
$951$	3.60841e7	1.29379
$952$	455791.i	0.0162995i
$953$	3.52488e7i	1.25722i	0.777719	+	0.628612i	$0.216376\pi$
$953$	3.52488e7i	1.25722i	−0.777719	+	0.628612i	$0.783624\pi$
$954$	−988950.	−0.0351806
$955$	0	0
$956$	−1.77595e6	−0.0628472
$957$	− 2.41376e6i	− 0.0851951i
$958$	3.74019e7i	1.31668i
$959$	−907546.	−0.0318656
$960$	0	0
$961$	−2.62307e7	−0.916223
$962$	− 3.86523e6i	− 0.134660i
$963$	− 8.21470e6i	− 0.285447i
$964$	−2.31781e7	−0.803315
$965$	0	0
$966$	−1.37231e6	−0.0473160
$967$	− 2.51987e7i	− 0.866587i	−0.901253	−	0.433293i	$-0.857351\pi$
$967$	− 2.51987e7i	− 0.866587i	0.901253	−	0.433293i	$-0.142649\pi$
$968$	− 1.01452e7i	− 0.347993i
$969$	−1.99012e6	−0.0680879
$970$	0	0
$971$	−2.59136e6	−0.0882022	−0.0441011	−	0.999027i	$-0.514042\pi$
$971$	−2.59136e6	−0.0882022	−0.0441011	+	0.999027i	$0.514042\pi$
$972$	6.32436e6i	0.214709i
$973$	9.25364e6i	0.313350i
$974$	−3.72413e7	−1.25784
$975$	0	0
$976$	2.18728e6	0.0734987
$977$	2.71811e7i	0.911027i	0.890229	+	0.455514i	$0.150544\pi$
$977$	2.71811e7i	0.911027i	−0.890229	+	0.455514i	$0.849456\pi$
$978$	2.72077e7i	0.909588i
$979$	−2.73105e6	−0.0910693
$980$	0	0
$981$	−4.31439e6	−0.143135
$982$	− 3.29287e7i	− 1.08967i
$983$	− 8.13437e6i	− 0.268498i	−0.990948	−	0.134249i	$-0.957138\pi$
$983$	− 8.13437e6i	− 0.268498i	0.990948	−	0.134249i	$-0.0428621\pi$
$984$	1.52044e6	0.0500589
$985$	0	0
$986$	−2.02390e6	−0.0662975
$987$	− 1.28590e7i	− 0.420158i
$988$	− 1.26195e6i	− 0.0411293i
$989$	−135331.	−0.00439953
$990$	0	0
$991$	−702676.	−0.0227285	−0.0113643	−	0.999935i	$-0.503617\pi$
$991$	−702676.	−0.0227285	−0.0113643	+	0.999935i	$0.503617\pi$
$992$	− 1.58586e6i	− 0.0511666i
$993$	− 436526.i	− 0.0140487i
$994$	2.87616e6	0.0923309
$995$	0	0
$996$	−1.32283e7	−0.422528
$997$	1.35555e7i	0.431896i	0.976405	+	0.215948i	$0.0692841\pi$
$997$	1.35555e7i	0.431896i	−0.976405	+	0.215948i	$0.930716\pi$
$998$	− 4.20176e7i	− 1.33538i
$999$	−4.96916e7	−1.57532

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.6.c.l.99.1		4
5.2	odd	4		350.6.a.t.1.1	yes	2
5.3	odd	4		350.6.a.o.1.2	✓	2
5.4	even	2	inner	350.6.c.l.99.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.6.a.o.1.2	✓	2	5.3	odd	4
350.6.a.t.1.1	yes	2	5.2	odd	4
350.6.c.l.99.1		4	1.1	even	1	trivial
350.6.c.l.99.4		4	5.4	even	2	inner

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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

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	350.6.c.l.99.1

Level	$350$
Weight	$6$
Character	350.99
Analytic conductor	$56.134$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$