# Properties

 Label 400.4.c.m.49.2 Level $400$ Weight $4$ Character 400.49 Analytic conductor $23.601$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,Mod(49,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 400.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: $$23.6007640023$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 200) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 49.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 400.49 Dual form 400.4.c.m.49.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{3} +6.00000i q^{7} +26.0000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} +6.00000i q^{7} +26.0000 q^{9} +19.0000 q^{11} -12.0000i q^{13} -75.0000i q^{17} -91.0000 q^{19} -6.00000 q^{21} +174.000i q^{23} +53.0000i q^{27} +272.000 q^{29} +230.000 q^{31} +19.0000i q^{33} -182.000i q^{37} +12.0000 q^{39} +117.000 q^{41} +372.000i q^{43} +52.0000i q^{47} +307.000 q^{49} +75.0000 q^{51} +402.000i q^{53} -91.0000i q^{57} +312.000 q^{59} +170.000 q^{61} +156.000i q^{63} -763.000i q^{67} -174.000 q^{69} +52.0000 q^{71} +981.000i q^{73} +114.000i q^{77} +1054.00 q^{79} +649.000 q^{81} +351.000i q^{83} +272.000i q^{87} -799.000 q^{89} +72.0000 q^{91} +230.000i q^{93} +962.000i q^{97} +494.000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 52 q^{9}+O(q^{10})$$ 2 * q + 52 * q^9 $$2 q + 52 q^{9} + 38 q^{11} - 182 q^{19} - 12 q^{21} + 544 q^{29} + 460 q^{31} + 24 q^{39} + 234 q^{41} + 614 q^{49} + 150 q^{51} + 624 q^{59} + 340 q^{61} - 348 q^{69} + 104 q^{71} + 2108 q^{79} + 1298 q^{81} - 1598 q^{89} + 144 q^{91} + 988 q^{99}+O(q^{100})$$ 2 * q + 52 * q^9 + 38 * q^11 - 182 * q^19 - 12 * q^21 + 544 * q^29 + 460 * q^31 + 24 * q^39 + 234 * q^41 + 614 * q^49 + 150 * q^51 + 624 * q^59 + 340 * q^61 - 348 * q^69 + 104 * q^71 + 2108 * q^79 + 1298 * q^81 - 1598 * q^89 + 144 * q^91 + 988 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000i 0.192450i 0.995360 + 0.0962250i $$0.0306768\pi$$
−0.995360 + 0.0962250i $$0.969323\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 6.00000i 0.323970i 0.986793 + 0.161985i $$0.0517895\pi$$
−0.986793 + 0.161985i $$0.948210\pi$$
$$8$$ 0 0
$$9$$ 26.0000 0.962963
$$10$$ 0 0
$$11$$ 19.0000 0.520792 0.260396 0.965502i $$-0.416147\pi$$
0.260396 + 0.965502i $$0.416147\pi$$
$$12$$ 0 0
$$13$$ − 12.0000i − 0.256015i −0.991773 0.128008i $$-0.959142\pi$$
0.991773 0.128008i $$-0.0408582\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 75.0000i − 1.07001i −0.844849 0.535005i $$-0.820310\pi$$
0.844849 0.535005i $$-0.179690\pi$$
$$18$$ 0 0
$$19$$ −91.0000 −1.09878 −0.549390 0.835566i $$-0.685140\pi$$
−0.549390 + 0.835566i $$0.685140\pi$$
$$20$$ 0 0
$$21$$ −6.00000 −0.0623480
$$22$$ 0 0
$$23$$ 174.000i 1.57746i 0.614742 + 0.788728i $$0.289260\pi$$
−0.614742 + 0.788728i $$0.710740\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 53.0000i 0.377772i
$$28$$ 0 0
$$29$$ 272.000 1.74169 0.870847 0.491554i $$-0.163571\pi$$
0.870847 + 0.491554i $$0.163571\pi$$
$$30$$ 0 0
$$31$$ 230.000 1.33256 0.666278 0.745704i $$-0.267887\pi$$
0.666278 + 0.745704i $$0.267887\pi$$
$$32$$ 0 0
$$33$$ 19.0000i 0.100227i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 182.000i − 0.808665i −0.914612 0.404333i $$-0.867504\pi$$
0.914612 0.404333i $$-0.132496\pi$$
$$38$$ 0 0
$$39$$ 12.0000 0.0492702
$$40$$ 0 0
$$41$$ 117.000 0.445667 0.222833 0.974857i $$-0.428469\pi$$
0.222833 + 0.974857i $$0.428469\pi$$
$$42$$ 0 0
$$43$$ 372.000i 1.31929i 0.751577 + 0.659645i $$0.229293\pi$$
−0.751577 + 0.659645i $$0.770707\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 52.0000i 0.161383i 0.996739 + 0.0806913i $$0.0257128\pi$$
−0.996739 + 0.0806913i $$0.974287\pi$$
$$48$$ 0 0
$$49$$ 307.000 0.895044
$$50$$ 0 0
$$51$$ 75.0000 0.205924
$$52$$ 0 0
$$53$$ 402.000i 1.04187i 0.853597 + 0.520933i $$0.174416\pi$$
−0.853597 + 0.520933i $$0.825584\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 91.0000i − 0.211460i
$$58$$ 0 0
$$59$$ 312.000 0.688457 0.344228 0.938886i $$-0.388141\pi$$
0.344228 + 0.938886i $$0.388141\pi$$
$$60$$ 0 0
$$61$$ 170.000 0.356824 0.178412 0.983956i $$-0.442904\pi$$
0.178412 + 0.983956i $$0.442904\pi$$
$$62$$ 0 0
$$63$$ 156.000i 0.311971i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 763.000i − 1.39127i −0.718394 0.695636i $$-0.755122\pi$$
0.718394 0.695636i $$-0.244878\pi$$
$$68$$ 0 0
$$69$$ −174.000 −0.303582
$$70$$ 0 0
$$71$$ 52.0000 0.0869192 0.0434596 0.999055i $$-0.486162\pi$$
0.0434596 + 0.999055i $$0.486162\pi$$
$$72$$ 0 0
$$73$$ 981.000i 1.57284i 0.617692 + 0.786420i $$0.288068\pi$$
−0.617692 + 0.786420i $$0.711932\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 114.000i 0.168721i
$$78$$ 0 0
$$79$$ 1054.00 1.50107 0.750533 0.660833i $$-0.229797\pi$$
0.750533 + 0.660833i $$0.229797\pi$$
$$80$$ 0 0
$$81$$ 649.000 0.890261
$$82$$ 0 0
$$83$$ 351.000i 0.464184i 0.972694 + 0.232092i $$0.0745570\pi$$
−0.972694 + 0.232092i $$0.925443\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 272.000i 0.335189i
$$88$$ 0 0
$$89$$ −799.000 −0.951616 −0.475808 0.879549i $$-0.657844\pi$$
−0.475808 + 0.879549i $$0.657844\pi$$
$$90$$ 0 0
$$91$$ 72.0000 0.0829412
$$92$$ 0 0
$$93$$ 230.000i 0.256450i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 962.000i 1.00697i 0.864003 + 0.503486i $$0.167949\pi$$
−0.864003 + 0.503486i $$0.832051\pi$$
$$98$$ 0 0
$$99$$ 494.000 0.501504
$$100$$ 0 0
$$101$$ 486.000 0.478800 0.239400 0.970921i $$-0.423049\pi$$
0.239400 + 0.970921i $$0.423049\pi$$
$$102$$ 0 0
$$103$$ − 1188.00i − 1.13648i −0.822864 0.568238i $$-0.807625\pi$$
0.822864 0.568238i $$-0.192375\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 1325.00i − 1.19713i −0.801075 0.598563i $$-0.795738\pi$$
0.801075 0.598563i $$-0.204262\pi$$
$$108$$ 0 0
$$109$$ −126.000 −0.110721 −0.0553606 0.998466i $$-0.517631\pi$$
−0.0553606 + 0.998466i $$0.517631\pi$$
$$110$$ 0 0
$$111$$ 182.000 0.155628
$$112$$ 0 0
$$113$$ − 183.000i − 0.152347i −0.997095 0.0761734i $$-0.975730\pi$$
0.997095 0.0761734i $$-0.0242703\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 312.000i − 0.246533i
$$118$$ 0 0
$$119$$ 450.000 0.346651
$$120$$ 0 0
$$121$$ −970.000 −0.728775
$$122$$ 0 0
$$123$$ 117.000i 0.0857686i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 902.000i 0.630233i 0.949053 + 0.315116i $$0.102044\pi$$
−0.949053 + 0.315116i $$0.897956\pi$$
$$128$$ 0 0
$$129$$ −372.000 −0.253897
$$130$$ 0 0
$$131$$ −2068.00 −1.37925 −0.689626 0.724166i $$-0.742225\pi$$
−0.689626 + 0.724166i $$0.742225\pi$$
$$132$$ 0 0
$$133$$ − 546.000i − 0.355971i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 1339.00i − 0.835025i −0.908671 0.417513i $$-0.862902\pi$$
0.908671 0.417513i $$-0.137098\pi$$
$$138$$ 0 0
$$139$$ −2939.00 −1.79340 −0.896700 0.442638i $$-0.854043\pi$$
−0.896700 + 0.442638i $$0.854043\pi$$
$$140$$ 0 0
$$141$$ −52.0000 −0.0310581
$$142$$ 0 0
$$143$$ − 228.000i − 0.133331i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 307.000i 0.172251i
$$148$$ 0 0
$$149$$ 208.000 0.114363 0.0571813 0.998364i $$-0.481789\pi$$
0.0571813 + 0.998364i $$0.481789\pi$$
$$150$$ 0 0
$$151$$ 2678.00 1.44326 0.721631 0.692278i $$-0.243393\pi$$
0.721631 + 0.692278i $$0.243393\pi$$
$$152$$ 0 0
$$153$$ − 1950.00i − 1.03038i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 1482.00i 0.753353i 0.926345 + 0.376677i $$0.122933\pi$$
−0.926345 + 0.376677i $$0.877067\pi$$
$$158$$ 0 0
$$159$$ −402.000 −0.200507
$$160$$ 0 0
$$161$$ −1044.00 −0.511048
$$162$$ 0 0
$$163$$ − 1469.00i − 0.705895i −0.935643 0.352948i $$-0.885179\pi$$
0.935643 0.352948i $$-0.114821\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 4004.00i 1.85532i 0.373423 + 0.927661i $$0.378184\pi$$
−0.373423 + 0.927661i $$0.621816\pi$$
$$168$$ 0 0
$$169$$ 2053.00 0.934456
$$170$$ 0 0
$$171$$ −2366.00 −1.05809
$$172$$ 0 0
$$173$$ − 3224.00i − 1.41686i −0.705783 0.708428i $$-0.749405\pi$$
0.705783 0.708428i $$-0.250595\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 312.000i 0.132494i
$$178$$ 0 0
$$179$$ −4191.00 −1.75000 −0.875000 0.484123i $$-0.839139\pi$$
−0.875000 + 0.484123i $$0.839139\pi$$
$$180$$ 0 0
$$181$$ −3718.00 −1.52683 −0.763416 0.645907i $$-0.776480\pi$$
−0.763416 + 0.645907i $$0.776480\pi$$
$$182$$ 0 0
$$183$$ 170.000i 0.0686708i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 1425.00i − 0.557253i
$$188$$ 0 0
$$189$$ −318.000 −0.122387
$$190$$ 0 0
$$191$$ −870.000 −0.329586 −0.164793 0.986328i $$-0.552696\pi$$
−0.164793 + 0.986328i $$0.552696\pi$$
$$192$$ 0 0
$$193$$ − 2197.00i − 0.819396i −0.912221 0.409698i $$-0.865634\pi$$
0.912221 0.409698i $$-0.134366\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 2314.00i − 0.836882i −0.908244 0.418441i $$-0.862577\pi$$
0.908244 0.418441i $$-0.137423\pi$$
$$198$$ 0 0
$$199$$ −252.000 −0.0897679 −0.0448839 0.998992i $$-0.514292\pi$$
−0.0448839 + 0.998992i $$0.514292\pi$$
$$200$$ 0 0
$$201$$ 763.000 0.267751
$$202$$ 0 0
$$203$$ 1632.00i 0.564256i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 4524.00i 1.51903i
$$208$$ 0 0
$$209$$ −1729.00 −0.572237
$$210$$ 0 0
$$211$$ 741.000 0.241766 0.120883 0.992667i $$-0.461427\pi$$
0.120883 + 0.992667i $$0.461427\pi$$
$$212$$ 0 0
$$213$$ 52.0000i 0.0167276i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 1380.00i 0.431707i
$$218$$ 0 0
$$219$$ −981.000 −0.302693
$$220$$ 0 0
$$221$$ −900.000 −0.273939
$$222$$ 0 0
$$223$$ − 5092.00i − 1.52908i −0.644574 0.764542i $$-0.722965\pi$$
0.644574 0.764542i $$-0.277035\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 5876.00i − 1.71808i −0.511910 0.859039i $$-0.671062\pi$$
0.511910 0.859039i $$-0.328938\pi$$
$$228$$ 0 0
$$229$$ 604.000 0.174295 0.0871473 0.996195i $$-0.472225\pi$$
0.0871473 + 0.996195i $$0.472225\pi$$
$$230$$ 0 0
$$231$$ −114.000 −0.0324703
$$232$$ 0 0
$$233$$ − 278.000i − 0.0781647i −0.999236 0.0390824i $$-0.987557\pi$$
0.999236 0.0390824i $$-0.0124435\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 1054.00i 0.288880i
$$238$$ 0 0
$$239$$ 2496.00 0.675535 0.337767 0.941230i $$-0.390328\pi$$
0.337767 + 0.941230i $$0.390328\pi$$
$$240$$ 0 0
$$241$$ −2567.00 −0.686120 −0.343060 0.939313i $$-0.611463\pi$$
−0.343060 + 0.939313i $$0.611463\pi$$
$$242$$ 0 0
$$243$$ 2080.00i 0.549103i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1092.00i 0.281305i
$$248$$ 0 0
$$249$$ −351.000 −0.0893322
$$250$$ 0 0
$$251$$ −5395.00 −1.35669 −0.678345 0.734743i $$-0.737303\pi$$
−0.678345 + 0.734743i $$0.737303\pi$$
$$252$$ 0 0
$$253$$ 3306.00i 0.821527i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 1490.00i − 0.361648i −0.983515 0.180824i $$-0.942123\pi$$
0.983515 0.180824i $$-0.0578765\pi$$
$$258$$ 0 0
$$259$$ 1092.00 0.261983
$$260$$ 0 0
$$261$$ 7072.00 1.67719
$$262$$ 0 0
$$263$$ − 3330.00i − 0.780748i −0.920656 0.390374i $$-0.872346\pi$$
0.920656 0.390374i $$-0.127654\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 799.000i − 0.183139i
$$268$$ 0 0
$$269$$ −6096.00 −1.38171 −0.690854 0.722994i $$-0.742765\pi$$
−0.690854 + 0.722994i $$0.742765\pi$$
$$270$$ 0 0
$$271$$ 6006.00 1.34627 0.673134 0.739521i $$-0.264948\pi$$
0.673134 + 0.739521i $$0.264948\pi$$
$$272$$ 0 0
$$273$$ 72.0000i 0.0159620i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 6976.00i 1.51317i 0.653897 + 0.756583i $$0.273133\pi$$
−0.653897 + 0.756583i $$0.726867\pi$$
$$278$$ 0 0
$$279$$ 5980.00 1.28320
$$280$$ 0 0
$$281$$ 3998.00 0.848757 0.424378 0.905485i $$-0.360493\pi$$
0.424378 + 0.905485i $$0.360493\pi$$
$$282$$ 0 0
$$283$$ − 13.0000i − 0.00273064i −0.999999 0.00136532i $$-0.999565\pi$$
0.999999 0.00136532i $$-0.000434594\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 702.000i 0.144382i
$$288$$ 0 0
$$289$$ −712.000 −0.144922
$$290$$ 0 0
$$291$$ −962.000 −0.193792
$$292$$ 0 0
$$293$$ 4466.00i 0.890466i 0.895415 + 0.445233i $$0.146879\pi$$
−0.895415 + 0.445233i $$0.853121\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 1007.00i 0.196741i
$$298$$ 0 0
$$299$$ 2088.00 0.403853
$$300$$ 0 0
$$301$$ −2232.00 −0.427410
$$302$$ 0 0
$$303$$ 486.000i 0.0921451i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 9041.00i − 1.68077i −0.541988 0.840386i $$-0.682328\pi$$
0.541988 0.840386i $$-0.317672\pi$$
$$308$$ 0 0
$$309$$ 1188.00 0.218715
$$310$$ 0 0
$$311$$ 346.000 0.0630864 0.0315432 0.999502i $$-0.489958\pi$$
0.0315432 + 0.999502i $$0.489958\pi$$
$$312$$ 0 0
$$313$$ 10646.0i 1.92252i 0.275650 + 0.961258i $$0.411107\pi$$
−0.275650 + 0.961258i $$0.588893\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 8116.00i − 1.43798i −0.695020 0.718990i $$-0.744604\pi$$
0.695020 0.718990i $$-0.255396\pi$$
$$318$$ 0 0
$$319$$ 5168.00 0.907061
$$320$$ 0 0
$$321$$ 1325.00 0.230387
$$322$$ 0 0
$$323$$ 6825.00i 1.17571i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 126.000i − 0.0213083i
$$328$$ 0 0
$$329$$ −312.000 −0.0522830
$$330$$ 0 0
$$331$$ 3007.00 0.499334 0.249667 0.968332i $$-0.419679\pi$$
0.249667 + 0.968332i $$0.419679\pi$$
$$332$$ 0 0
$$333$$ − 4732.00i − 0.778715i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 83.0000i 0.0134163i 0.999978 + 0.00670816i $$0.00213529\pi$$
−0.999978 + 0.00670816i $$0.997865\pi$$
$$338$$ 0 0
$$339$$ 183.000 0.0293192
$$340$$ 0 0
$$341$$ 4370.00 0.693985
$$342$$ 0 0
$$343$$ 3900.00i 0.613936i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 6189.00i − 0.957472i −0.877959 0.478736i $$-0.841095\pi$$
0.877959 0.478736i $$-0.158905\pi$$
$$348$$ 0 0
$$349$$ 5362.00 0.822411 0.411205 0.911543i $$-0.365108\pi$$
0.411205 + 0.911543i $$0.365108\pi$$
$$350$$ 0 0
$$351$$ 636.000 0.0967156
$$352$$ 0 0
$$353$$ 1690.00i 0.254815i 0.991850 + 0.127407i $$0.0406656\pi$$
−0.991850 + 0.127407i $$0.959334\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 450.000i 0.0667130i
$$358$$ 0 0
$$359$$ 1638.00 0.240809 0.120404 0.992725i $$-0.461581\pi$$
0.120404 + 0.992725i $$0.461581\pi$$
$$360$$ 0 0
$$361$$ 1422.00 0.207319
$$362$$ 0 0
$$363$$ − 970.000i − 0.140253i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 7580.00i 1.07813i 0.842265 + 0.539064i $$0.181222\pi$$
−0.842265 + 0.539064i $$0.818778\pi$$
$$368$$ 0 0
$$369$$ 3042.00 0.429160
$$370$$ 0 0
$$371$$ −2412.00 −0.337533
$$372$$ 0 0
$$373$$ 5630.00i 0.781529i 0.920491 + 0.390765i $$0.127789\pi$$
−0.920491 + 0.390765i $$0.872211\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 3264.00i − 0.445901i
$$378$$ 0 0
$$379$$ −4385.00 −0.594307 −0.297153 0.954830i $$-0.596037\pi$$
−0.297153 + 0.954830i $$0.596037\pi$$
$$380$$ 0 0
$$381$$ −902.000 −0.121288
$$382$$ 0 0
$$383$$ 12558.0i 1.67541i 0.546119 + 0.837707i $$0.316104\pi$$
−0.546119 + 0.837707i $$0.683896\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 9672.00i 1.27043i
$$388$$ 0 0
$$389$$ 6570.00 0.856330 0.428165 0.903701i $$-0.359160\pi$$
0.428165 + 0.903701i $$0.359160\pi$$
$$390$$ 0 0
$$391$$ 13050.0 1.68789
$$392$$ 0 0
$$393$$ − 2068.00i − 0.265437i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 1268.00i 0.160300i 0.996783 + 0.0801500i $$0.0255399\pi$$
−0.996783 + 0.0801500i $$0.974460\pi$$
$$398$$ 0 0
$$399$$ 546.000 0.0685067
$$400$$ 0 0
$$401$$ −6299.00 −0.784432 −0.392216 0.919873i $$-0.628291\pi$$
−0.392216 + 0.919873i $$0.628291\pi$$
$$402$$ 0 0
$$403$$ − 2760.00i − 0.341155i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 3458.00i − 0.421147i
$$408$$ 0 0
$$409$$ 13459.0 1.62715 0.813575 0.581459i $$-0.197518\pi$$
0.813575 + 0.581459i $$0.197518\pi$$
$$410$$ 0 0
$$411$$ 1339.00 0.160701
$$412$$ 0 0
$$413$$ 1872.00i 0.223039i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 2939.00i − 0.345140i
$$418$$ 0 0
$$419$$ −9567.00 −1.11546 −0.557731 0.830022i $$-0.688328\pi$$
−0.557731 + 0.830022i $$0.688328\pi$$
$$420$$ 0 0
$$421$$ 2708.00 0.313491 0.156746 0.987639i $$-0.449900\pi$$
0.156746 + 0.987639i $$0.449900\pi$$
$$422$$ 0 0
$$423$$ 1352.00i 0.155405i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 1020.00i 0.115600i
$$428$$ 0 0
$$429$$ 228.000 0.0256595
$$430$$ 0 0
$$431$$ −5126.00 −0.572879 −0.286439 0.958098i $$-0.592472\pi$$
−0.286439 + 0.958098i $$0.592472\pi$$
$$432$$ 0 0
$$433$$ − 11445.0i − 1.27023i −0.772416 0.635117i $$-0.780952\pi$$
0.772416 0.635117i $$-0.219048\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 15834.0i − 1.73328i
$$438$$ 0 0
$$439$$ −5096.00 −0.554029 −0.277015 0.960866i $$-0.589345\pi$$
−0.277015 + 0.960866i $$0.589345\pi$$
$$440$$ 0 0
$$441$$ 7982.00 0.861894
$$442$$ 0 0
$$443$$ 13247.0i 1.42073i 0.703833 + 0.710366i $$0.251470\pi$$
−0.703833 + 0.710366i $$0.748530\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 208.000i 0.0220091i
$$448$$ 0 0
$$449$$ −7449.00 −0.782940 −0.391470 0.920191i $$-0.628033\pi$$
−0.391470 + 0.920191i $$0.628033\pi$$
$$450$$ 0 0
$$451$$ 2223.00 0.232100
$$452$$ 0 0
$$453$$ 2678.00i 0.277756i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 3497.00i 0.357949i 0.983854 + 0.178975i $$0.0572780\pi$$
−0.983854 + 0.178975i $$0.942722\pi$$
$$458$$ 0 0
$$459$$ 3975.00 0.404220
$$460$$ 0 0
$$461$$ −13108.0 −1.32430 −0.662148 0.749373i $$-0.730355\pi$$
−0.662148 + 0.749373i $$0.730355\pi$$
$$462$$ 0 0
$$463$$ − 5428.00i − 0.544839i −0.962179 0.272420i $$-0.912176\pi$$
0.962179 0.272420i $$-0.0878239\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 1516.00i 0.150219i 0.997175 + 0.0751093i $$0.0239306\pi$$
−0.997175 + 0.0751093i $$0.976069\pi$$
$$468$$ 0 0
$$469$$ 4578.00 0.450730
$$470$$ 0 0
$$471$$ −1482.00 −0.144983
$$472$$ 0 0
$$473$$ 7068.00i 0.687076i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 10452.0i 1.00328i
$$478$$ 0 0
$$479$$ −14762.0 −1.40813 −0.704064 0.710137i $$-0.748633\pi$$
−0.704064 + 0.710137i $$0.748633\pi$$
$$480$$ 0 0
$$481$$ −2184.00 −0.207031
$$482$$ 0 0
$$483$$ − 1044.00i − 0.0983512i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 3926.00i 0.365306i 0.983177 + 0.182653i $$0.0584685\pi$$
−0.983177 + 0.182653i $$0.941532\pi$$
$$488$$ 0 0
$$489$$ 1469.00 0.135850
$$490$$ 0 0
$$491$$ −996.000 −0.0915455 −0.0457728 0.998952i $$-0.514575\pi$$
−0.0457728 + 0.998952i $$0.514575\pi$$
$$492$$ 0 0
$$493$$ − 20400.0i − 1.86363i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 312.000i 0.0281592i
$$498$$ 0 0
$$499$$ 7804.00 0.700110 0.350055 0.936729i $$-0.386163\pi$$
0.350055 + 0.936729i $$0.386163\pi$$
$$500$$ 0 0
$$501$$ −4004.00 −0.357057
$$502$$ 0 0
$$503$$ − 16732.0i − 1.48319i −0.670850 0.741593i $$-0.734070\pi$$
0.670850 0.741593i $$-0.265930\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 2053.00i 0.179836i
$$508$$ 0 0
$$509$$ −10426.0 −0.907906 −0.453953 0.891026i $$-0.649987\pi$$
−0.453953 + 0.891026i $$0.649987\pi$$
$$510$$ 0 0
$$511$$ −5886.00 −0.509552
$$512$$ 0 0
$$513$$ − 4823.00i − 0.415089i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 988.000i 0.0840468i
$$518$$ 0 0
$$519$$ 3224.00 0.272674
$$520$$ 0 0
$$521$$ −2235.00 −0.187941 −0.0939704 0.995575i $$-0.529956\pi$$
−0.0939704 + 0.995575i $$0.529956\pi$$
$$522$$ 0 0
$$523$$ − 10855.0i − 0.907564i −0.891113 0.453782i $$-0.850074\pi$$
0.891113 0.453782i $$-0.149926\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 17250.0i − 1.42585i
$$528$$ 0 0
$$529$$ −18109.0 −1.48837
$$530$$ 0 0
$$531$$ 8112.00 0.662958
$$532$$ 0 0
$$533$$ − 1404.00i − 0.114098i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 4191.00i − 0.336788i
$$538$$ 0 0
$$539$$ 5833.00 0.466132
$$540$$ 0 0
$$541$$ 10608.0 0.843019 0.421510 0.906824i $$-0.361500\pi$$
0.421510 + 0.906824i $$0.361500\pi$$
$$542$$ 0 0
$$543$$ − 3718.00i − 0.293839i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 11583.0i 0.905399i 0.891663 + 0.452700i $$0.149539\pi$$
−0.891663 + 0.452700i $$0.850461\pi$$
$$548$$ 0 0
$$549$$ 4420.00 0.343608
$$550$$ 0 0
$$551$$ −24752.0 −1.91374
$$552$$ 0 0
$$553$$ 6324.00i 0.486300i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 17244.0i 1.31176i 0.754864 + 0.655881i $$0.227703\pi$$
−0.754864 + 0.655881i $$0.772297\pi$$
$$558$$ 0 0
$$559$$ 4464.00 0.337759
$$560$$ 0 0
$$561$$ 1425.00 0.107243
$$562$$ 0 0
$$563$$ − 18416.0i − 1.37858i −0.724484 0.689291i $$-0.757922\pi$$
0.724484 0.689291i $$-0.242078\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 3894.00i 0.288417i
$$568$$ 0 0
$$569$$ −11913.0 −0.877713 −0.438857 0.898557i $$-0.644616\pi$$
−0.438857 + 0.898557i $$0.644616\pi$$
$$570$$ 0 0
$$571$$ −3900.00 −0.285832 −0.142916 0.989735i $$-0.545648\pi$$
−0.142916 + 0.989735i $$0.545648\pi$$
$$572$$ 0 0
$$573$$ − 870.000i − 0.0634289i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 24899.0i 1.79646i 0.439523 + 0.898231i $$0.355147\pi$$
−0.439523 + 0.898231i $$0.644853\pi$$
$$578$$ 0 0
$$579$$ 2197.00 0.157693
$$580$$ 0 0
$$581$$ −2106.00 −0.150381
$$582$$ 0 0
$$583$$ 7638.00i 0.542596i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 1751.00i − 0.123120i −0.998103 0.0615601i $$-0.980392\pi$$
0.998103 0.0615601i $$-0.0196076\pi$$
$$588$$ 0 0
$$589$$ −20930.0 −1.46419
$$590$$ 0 0
$$591$$ 2314.00 0.161058
$$592$$ 0 0
$$593$$ − 10887.0i − 0.753922i −0.926229 0.376961i $$-0.876969\pi$$
0.926229 0.376961i $$-0.123031\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 252.000i − 0.0172758i
$$598$$ 0 0
$$599$$ −14650.0 −0.999303 −0.499652 0.866226i $$-0.666539\pi$$
−0.499652 + 0.866226i $$0.666539\pi$$
$$600$$ 0 0
$$601$$ −4237.00 −0.287572 −0.143786 0.989609i $$-0.545928\pi$$
−0.143786 + 0.989609i $$0.545928\pi$$
$$602$$ 0 0
$$603$$ − 19838.0i − 1.33974i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 11440.0i − 0.764968i −0.923962 0.382484i $$-0.875069\pi$$
0.923962 0.382484i $$-0.124931\pi$$
$$608$$ 0 0
$$609$$ −1632.00 −0.108591
$$610$$ 0 0
$$611$$ 624.000 0.0413164
$$612$$ 0 0
$$613$$ 19370.0i 1.27626i 0.769929 + 0.638130i $$0.220292\pi$$
−0.769929 + 0.638130i $$0.779708\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 21346.0i − 1.39280i −0.717654 0.696400i $$-0.754784\pi$$
0.717654 0.696400i $$-0.245216\pi$$
$$618$$ 0 0
$$619$$ 7436.00 0.482840 0.241420 0.970421i $$-0.422387\pi$$
0.241420 + 0.970421i $$0.422387\pi$$
$$620$$ 0 0
$$621$$ −9222.00 −0.595920
$$622$$ 0 0
$$623$$ − 4794.00i − 0.308295i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 1729.00i − 0.110127i
$$628$$ 0 0
$$629$$ −13650.0 −0.865280
$$630$$ 0 0
$$631$$ −22490.0 −1.41888 −0.709440 0.704766i $$-0.751052\pi$$
−0.709440 + 0.704766i $$0.751052\pi$$
$$632$$ 0 0
$$633$$ 741.000i 0.0465278i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 3684.00i − 0.229145i
$$638$$ 0 0
$$639$$ 1352.00 0.0837000
$$640$$ 0 0
$$641$$ −16086.0 −0.991199 −0.495600 0.868551i $$-0.665052\pi$$
−0.495600 + 0.868551i $$0.665052\pi$$
$$642$$ 0 0
$$643$$ − 2396.00i − 0.146950i −0.997297 0.0734751i $$-0.976591\pi$$
0.997297 0.0734751i $$-0.0234090\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 23244.0i − 1.41239i −0.708018 0.706195i $$-0.750410\pi$$
0.708018 0.706195i $$-0.249590\pi$$
$$648$$ 0 0
$$649$$ 5928.00 0.358543
$$650$$ 0 0
$$651$$ −1380.00 −0.0830821
$$652$$ 0 0
$$653$$ − 13598.0i − 0.814902i −0.913227 0.407451i $$-0.866418\pi$$
0.913227 0.407451i $$-0.133582\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 25506.0i 1.51459i
$$658$$ 0 0
$$659$$ 9751.00 0.576396 0.288198 0.957571i $$-0.406944\pi$$
0.288198 + 0.957571i $$0.406944\pi$$
$$660$$ 0 0
$$661$$ 19104.0 1.12414 0.562072 0.827088i $$-0.310004\pi$$
0.562072 + 0.827088i $$0.310004\pi$$
$$662$$ 0 0
$$663$$ − 900.000i − 0.0527196i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 47328.0i 2.74745i
$$668$$ 0 0
$$669$$ 5092.00 0.294272
$$670$$ 0 0
$$671$$ 3230.00 0.185831
$$672$$ 0 0
$$673$$ − 25402.0i − 1.45494i −0.686139 0.727470i $$-0.740696\pi$$
0.686139 0.727470i $$-0.259304\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 2224.00i − 0.126256i −0.998005 0.0631280i $$-0.979892\pi$$
0.998005 0.0631280i $$-0.0201076\pi$$
$$678$$ 0 0
$$679$$ −5772.00 −0.326228
$$680$$ 0 0
$$681$$ 5876.00 0.330644
$$682$$ 0 0
$$683$$ 8281.00i 0.463929i 0.972724 + 0.231965i $$0.0745154\pi$$
−0.972724 + 0.231965i $$0.925485\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 604.000i 0.0335430i
$$688$$ 0 0
$$689$$ 4824.00 0.266734
$$690$$ 0 0
$$691$$ −481.000 −0.0264806 −0.0132403 0.999912i $$-0.504215\pi$$
−0.0132403 + 0.999912i $$0.504215\pi$$
$$692$$ 0 0
$$693$$ 2964.00i 0.162472i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 8775.00i − 0.476868i
$$698$$ 0 0
$$699$$ 278.000 0.0150428
$$700$$ 0 0
$$701$$ −16788.0 −0.904528 −0.452264 0.891884i $$-0.649383\pi$$
−0.452264 + 0.891884i $$0.649383\pi$$
$$702$$ 0 0
$$703$$ 16562.0i 0.888546i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 2916.00i 0.155117i
$$708$$ 0 0
$$709$$ 23452.0 1.24225 0.621127 0.783710i $$-0.286675\pi$$
0.621127 + 0.783710i $$0.286675\pi$$
$$710$$ 0 0
$$711$$ 27404.0 1.44547
$$712$$ 0 0
$$713$$ 40020.0i 2.10205i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 2496.00i 0.130007i
$$718$$ 0 0
$$719$$ −20886.0 −1.08333 −0.541666 0.840594i $$-0.682206\pi$$
−0.541666 + 0.840594i $$0.682206\pi$$
$$720$$ 0 0
$$721$$ 7128.00 0.368184
$$722$$ 0 0
$$723$$ − 2567.00i − 0.132044i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 22576.0i − 1.15172i −0.817550 0.575858i $$-0.804668\pi$$
0.817550 0.575858i $$-0.195332\pi$$
$$728$$ 0 0
$$729$$ 15443.0 0.784586
$$730$$ 0 0
$$731$$ 27900.0 1.41165
$$732$$ 0 0
$$733$$ − 35308.0i − 1.77917i −0.456771 0.889584i $$-0.650994\pi$$
0.456771 0.889584i $$-0.349006\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 14497.0i − 0.724564i
$$738$$ 0 0
$$739$$ 188.000 0.00935818 0.00467909 0.999989i $$-0.498511\pi$$
0.00467909 + 0.999989i $$0.498511\pi$$
$$740$$ 0 0
$$741$$ −1092.00 −0.0541371
$$742$$ 0 0
$$743$$ 29870.0i 1.47486i 0.675421 + 0.737432i $$0.263962\pi$$
−0.675421 + 0.737432i $$0.736038\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 9126.00i 0.446992i
$$748$$ 0 0
$$749$$ 7950.00 0.387833
$$750$$ 0 0
$$751$$ −22784.0 −1.10706 −0.553529 0.832830i $$-0.686719\pi$$
−0.553529 + 0.832830i $$0.686719\pi$$
$$752$$ 0 0
$$753$$ − 5395.00i − 0.261095i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 28496.0i − 1.36817i −0.729402 0.684085i $$-0.760202\pi$$
0.729402 0.684085i $$-0.239798\pi$$
$$758$$ 0 0
$$759$$ −3306.00 −0.158103
$$760$$ 0 0
$$761$$ 7397.00 0.352354 0.176177 0.984359i $$-0.443627\pi$$
0.176177 + 0.984359i $$0.443627\pi$$
$$762$$ 0 0
$$763$$ − 756.000i − 0.0358703i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 3744.00i − 0.176256i
$$768$$ 0 0
$$769$$ 8883.00 0.416553 0.208276 0.978070i $$-0.433215\pi$$
0.208276 + 0.978070i $$0.433215\pi$$
$$770$$ 0 0
$$771$$ 1490.00 0.0695993
$$772$$ 0 0
$$773$$ 33960.0i 1.58015i 0.613010 + 0.790075i $$0.289959\pi$$
−0.613010 + 0.790075i $$0.710041\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 1092.00i 0.0504186i
$$778$$ 0 0
$$779$$ −10647.0 −0.489690
$$780$$ 0 0
$$781$$ 988.000 0.0452669
$$782$$ 0 0
$$783$$ 14416.0i 0.657964i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 32084.0i 1.45320i 0.687059 + 0.726602i $$0.258901\pi$$
−0.687059 + 0.726602i $$0.741099\pi$$
$$788$$ 0 0
$$789$$ 3330.00 0.150255
$$790$$ 0 0
$$791$$ 1098.00 0.0493557
$$792$$ 0 0
$$793$$ − 2040.00i − 0.0913525i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 4766.00i − 0.211820i −0.994376 0.105910i $$-0.966224\pi$$
0.994376 0.105910i $$-0.0337755\pi$$
$$798$$ 0 0
$$799$$ 3900.00 0.172681
$$800$$ 0 0
$$801$$ −20774.0 −0.916371
$$802$$ 0 0
$$803$$ 18639.0i 0.819123i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 6096.00i − 0.265910i
$$808$$ 0 0
$$809$$ 31278.0 1.35930 0.679651 0.733535i $$-0.262131\pi$$
0.679651 + 0.733535i $$0.262131\pi$$
$$810$$ 0 0
$$811$$ 29956.0 1.29704 0.648519 0.761199i $$-0.275389\pi$$
0.648519 + 0.761199i $$0.275389\pi$$
$$812$$ 0 0
$$813$$ 6006.00i 0.259089i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 33852.0i − 1.44961i
$$818$$ 0 0
$$819$$ 1872.00 0.0798693
$$820$$ 0 0
$$821$$ 14642.0 0.622423 0.311212 0.950341i $$-0.399265\pi$$
0.311212 + 0.950341i $$0.399265\pi$$
$$822$$ 0 0
$$823$$ 20844.0i 0.882839i 0.897301 + 0.441419i $$0.145525\pi$$
−0.897301 + 0.441419i $$0.854475\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 23751.0i − 0.998674i −0.866408 0.499337i $$-0.833577\pi$$
0.866408 0.499337i $$-0.166423\pi$$
$$828$$ 0 0
$$829$$ −11380.0 −0.476772 −0.238386 0.971171i $$-0.576618\pi$$
−0.238386 + 0.971171i $$0.576618\pi$$
$$830$$ 0 0
$$831$$ −6976.00 −0.291209
$$832$$ 0 0
$$833$$ − 23025.0i − 0.957706i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 12190.0i 0.503403i
$$838$$ 0 0
$$839$$ −29744.0 −1.22393 −0.611965 0.790885i $$-0.709621\pi$$
−0.611965 + 0.790885i $$0.709621\pi$$
$$840$$ 0 0
$$841$$ 49595.0 2.03350
$$842$$ 0 0
$$843$$ 3998.00i 0.163343i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 5820.00i − 0.236101i
$$848$$ 0 0
$$849$$ 13.0000 0.000525511 0
$$850$$ 0 0
$$851$$ 31668.0 1.27563
$$852$$ 0 0
$$853$$ 37726.0i 1.51432i 0.653230 + 0.757159i $$0.273413\pi$$
−0.653230 + 0.757159i $$0.726587\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 5429.00i 0.216396i 0.994129 + 0.108198i $$0.0345080\pi$$
−0.994129 + 0.108198i $$0.965492\pi$$
$$858$$ 0 0
$$859$$ 32149.0 1.27696 0.638481 0.769638i $$-0.279563\pi$$
0.638481 + 0.769638i $$0.279563\pi$$
$$860$$ 0 0
$$861$$ −702.000 −0.0277864
$$862$$ 0 0
$$863$$ − 29176.0i − 1.15083i −0.817863 0.575413i $$-0.804841\pi$$
0.817863 0.575413i $$-0.195159\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 712.000i − 0.0278902i
$$868$$ 0 0
$$869$$ 20026.0 0.781744
$$870$$ 0 0
$$871$$ −9156.00 −0.356187
$$872$$ 0 0
$$873$$ 25012.0i 0.969677i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 20068.0i 0.772689i 0.922355 + 0.386344i $$0.126262\pi$$
−0.922355 + 0.386344i $$0.873738\pi$$
$$878$$ 0 0
$$879$$ −4466.00 −0.171370
$$880$$ 0 0
$$881$$ −36850.0 −1.40920 −0.704602 0.709603i $$-0.748874\pi$$
−0.704602 + 0.709603i $$0.748874\pi$$
$$882$$ 0 0
$$883$$ 30025.0i 1.14431i 0.820147 + 0.572153i $$0.193892\pi$$
−0.820147 + 0.572153i $$0.806108\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 156.000i − 0.00590526i −0.999996 0.00295263i $$-0.999060\pi$$
0.999996 0.00295263i $$-0.000939853\pi$$
$$888$$ 0 0
$$889$$ −5412.00 −0.204176
$$890$$ 0 0
$$891$$ 12331.0 0.463641
$$892$$ 0 0
$$893$$ − 4732.00i − 0.177324i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 2088.00i 0.0777216i
$$898$$ 0 0
$$899$$ 62560.0 2.32090
$$900$$ 0 0
$$901$$ 30150.0 1.11481
$$902$$ 0 0
$$903$$ − 2232.00i − 0.0822550i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 356.000i 0.0130328i 0.999979 + 0.00651642i $$0.00207426\pi$$
−0.999979 + 0.00651642i $$0.997926\pi$$
$$908$$ 0 0
$$909$$ 12636.0 0.461067
$$910$$ 0 0
$$911$$ −8748.00 −0.318149 −0.159075 0.987267i $$-0.550851\pi$$
−0.159075 + 0.987267i $$0.550851\pi$$
$$912$$ 0 0
$$913$$ 6669.00i 0.241743i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 12408.0i − 0.446836i
$$918$$ 0 0
$$919$$ 36974.0 1.32716 0.663580 0.748105i $$-0.269036\pi$$
0.663580 + 0.748105i $$0.269036\pi$$
$$920$$ 0 0
$$921$$ 9041.00 0.323465
$$922$$ 0 0
$$923$$ − 624.000i − 0.0222527i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 30888.0i − 1.09439i
$$928$$ 0 0
$$929$$ −44382.0 −1.56741 −0.783706 0.621132i $$-0.786673\pi$$
−0.783706 + 0.621132i $$0.786673\pi$$
$$930$$ 0 0
$$931$$ −27937.0 −0.983457
$$932$$ 0 0
$$933$$ 346.000i 0.0121410i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 2445.00i − 0.0852451i −0.999091 0.0426226i $$-0.986429\pi$$
0.999091 0.0426226i $$-0.0135713\pi$$
$$938$$ 0 0
$$939$$ −10646.0 −0.369988
$$940$$ 0 0
$$941$$ −7076.00 −0.245134 −0.122567 0.992460i $$-0.539113\pi$$
−0.122567 + 0.992460i $$0.539113\pi$$
$$942$$ 0 0
$$943$$ 20358.0i 0.703020i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 1560.00i − 0.0535303i −0.999642 0.0267651i $$-0.991479\pi$$
0.999642 0.0267651i $$-0.00852063\pi$$
$$948$$ 0 0
$$949$$ 11772.0 0.402672
$$950$$ 0 0
$$951$$ 8116.00 0.276740
$$952$$ 0 0
$$953$$ 35087.0i 1.19263i 0.802749 + 0.596317i $$0.203370\pi$$
−0.802749 + 0.596317i $$0.796630\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 5168.00i 0.174564i
$$958$$ 0 0
$$959$$ 8034.00 0.270523
$$960$$ 0 0
$$961$$ 23109.0 0.775704
$$962$$ 0 0
$$963$$ − 34450.0i − 1.15279i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 9360.00i 0.311269i 0.987815 + 0.155635i $$0.0497422\pi$$
−0.987815 + 0.155635i $$0.950258\pi$$
$$968$$ 0 0
$$969$$ −6825.00 −0.226265
$$970$$ 0 0
$$971$$ −22269.0 −0.735990 −0.367995 0.929828i $$-0.619956\pi$$
−0.367995 + 0.929828i $$0.619956\pi$$
$$972$$ 0 0
$$973$$ − 17634.0i − 0.581007i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 37249.0i 1.21976i 0.792496 + 0.609878i $$0.208781\pi$$
−0.792496 + 0.609878i $$0.791219\pi$$
$$978$$ 0 0
$$979$$ −15181.0 −0.495594
$$980$$ 0 0
$$981$$ −3276.00 −0.106620
$$982$$ 0 0
$$983$$ − 17602.0i − 0.571126i −0.958360 0.285563i $$-0.907819\pi$$
0.958360 0.285563i $$-0.0921805\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 312.000i − 0.0100619i
$$988$$ 0 0
$$989$$ −64728.0 −2.08112
$$990$$ 0 0
$$991$$ 26402.0 0.846304 0.423152 0.906059i $$-0.360924\pi$$
0.423152 + 0.906059i $$0.360924\pi$$
$$992$$ 0 0
$$993$$ 3007.00i 0.0960969i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 16836.0i 0.534806i 0.963585 + 0.267403i $$0.0861655\pi$$
−0.963585 + 0.267403i $$0.913835\pi$$
$$998$$ 0 0
$$999$$ 9646.00 0.305491
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 400.4.c.m.49.2 2
4.3 odd 2 200.4.c.g.49.1 2
5.2 odd 4 400.4.a.k.1.1 1
5.3 odd 4 400.4.a.j.1.1 1
5.4 even 2 inner 400.4.c.m.49.1 2
12.11 even 2 1800.4.f.p.649.1 2
20.3 even 4 200.4.a.f.1.1 yes 1
20.7 even 4 200.4.a.e.1.1 1
20.19 odd 2 200.4.c.g.49.2 2
40.3 even 4 1600.4.a.w.1.1 1
40.13 odd 4 1600.4.a.be.1.1 1
40.27 even 4 1600.4.a.bf.1.1 1
40.37 odd 4 1600.4.a.v.1.1 1
60.23 odd 4 1800.4.a.l.1.1 1
60.47 odd 4 1800.4.a.w.1.1 1
60.59 even 2 1800.4.f.p.649.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
200.4.a.e.1.1 1 20.7 even 4
200.4.a.f.1.1 yes 1 20.3 even 4
200.4.c.g.49.1 2 4.3 odd 2
200.4.c.g.49.2 2 20.19 odd 2
400.4.a.j.1.1 1 5.3 odd 4
400.4.a.k.1.1 1 5.2 odd 4
400.4.c.m.49.1 2 5.4 even 2 inner
400.4.c.m.49.2 2 1.1 even 1 trivial
1600.4.a.v.1.1 1 40.37 odd 4
1600.4.a.w.1.1 1 40.3 even 4
1600.4.a.be.1.1 1 40.13 odd 4
1600.4.a.bf.1.1 1 40.27 even 4
1800.4.a.l.1.1 1 60.23 odd 4
1800.4.a.w.1.1 1 60.47 odd 4
1800.4.f.p.649.1 2 12.11 even 2
1800.4.f.p.649.2 2 60.59 even 2